# Properties

 Label 50.4.b.a.49.1 Level $50$ Weight $4$ Character 50.49 Analytic conductor $2.950$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [50,4,Mod(49,50)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(50, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("50.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 50.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: $$2.95009550029$$ 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 10) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.1 Root $$-1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 50.49 Dual form 50.4.b.a.49.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000i q^{2} -8.00000i q^{3} -4.00000 q^{4} -16.0000 q^{6} +4.00000i q^{7} +8.00000i q^{8} -37.0000 q^{9} +O(q^{10})$$ $$q-2.00000i q^{2} -8.00000i q^{3} -4.00000 q^{4} -16.0000 q^{6} +4.00000i q^{7} +8.00000i q^{8} -37.0000 q^{9} +12.0000 q^{11} +32.0000i q^{12} -58.0000i q^{13} +8.00000 q^{14} +16.0000 q^{16} -66.0000i q^{17} +74.0000i q^{18} +100.000 q^{19} +32.0000 q^{21} -24.0000i q^{22} +132.000i q^{23} +64.0000 q^{24} -116.000 q^{26} +80.0000i q^{27} -16.0000i q^{28} +90.0000 q^{29} +152.000 q^{31} -32.0000i q^{32} -96.0000i q^{33} -132.000 q^{34} +148.000 q^{36} +34.0000i q^{37} -200.000i q^{38} -464.000 q^{39} -438.000 q^{41} -64.0000i q^{42} +32.0000i q^{43} -48.0000 q^{44} +264.000 q^{46} +204.000i q^{47} -128.000i q^{48} +327.000 q^{49} -528.000 q^{51} +232.000i q^{52} +222.000i q^{53} +160.000 q^{54} -32.0000 q^{56} -800.000i q^{57} -180.000i q^{58} -420.000 q^{59} +902.000 q^{61} -304.000i q^{62} -148.000i q^{63} -64.0000 q^{64} -192.000 q^{66} +1024.00i q^{67} +264.000i q^{68} +1056.00 q^{69} +432.000 q^{71} -296.000i q^{72} +362.000i q^{73} +68.0000 q^{74} -400.000 q^{76} +48.0000i q^{77} +928.000i q^{78} +160.000 q^{79} -359.000 q^{81} +876.000i q^{82} +72.0000i q^{83} -128.000 q^{84} +64.0000 q^{86} -720.000i q^{87} +96.0000i q^{88} -810.000 q^{89} +232.000 q^{91} -528.000i q^{92} -1216.00i q^{93} +408.000 q^{94} -256.000 q^{96} -1106.00i q^{97} -654.000i q^{98} -444.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4} - 32 q^{6} - 74 q^{9}+O(q^{10})$$ 2 * q - 8 * q^4 - 32 * q^6 - 74 * q^9 $$2 q - 8 q^{4} - 32 q^{6} - 74 q^{9} + 24 q^{11} + 16 q^{14} + 32 q^{16} + 200 q^{19} + 64 q^{21} + 128 q^{24} - 232 q^{26} + 180 q^{29} + 304 q^{31} - 264 q^{34} + 296 q^{36} - 928 q^{39} - 876 q^{41} - 96 q^{44} + 528 q^{46} + 654 q^{49} - 1056 q^{51} + 320 q^{54} - 64 q^{56} - 840 q^{59} + 1804 q^{61} - 128 q^{64} - 384 q^{66} + 2112 q^{69} + 864 q^{71} + 136 q^{74} - 800 q^{76} + 320 q^{79} - 718 q^{81} - 256 q^{84} + 128 q^{86} - 1620 q^{89} + 464 q^{91} + 816 q^{94} - 512 q^{96} - 888 q^{99}+O(q^{100})$$ 2 * q - 8 * q^4 - 32 * q^6 - 74 * q^9 + 24 * q^11 + 16 * q^14 + 32 * q^16 + 200 * q^19 + 64 * q^21 + 128 * q^24 - 232 * q^26 + 180 * q^29 + 304 * q^31 - 264 * q^34 + 296 * q^36 - 928 * q^39 - 876 * q^41 - 96 * q^44 + 528 * q^46 + 654 * q^49 - 1056 * q^51 + 320 * q^54 - 64 * q^56 - 840 * q^59 + 1804 * q^61 - 128 * q^64 - 384 * q^66 + 2112 * q^69 + 864 * q^71 + 136 * q^74 - 800 * q^76 + 320 * q^79 - 718 * q^81 - 256 * q^84 + 128 * q^86 - 1620 * q^89 + 464 * q^91 + 816 * q^94 - 512 * q^96 - 888 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/50\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$\chi(n)$$ $$-1$$

## 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
$$3$$ − 8.00000i − 1.53960i −0.638285 0.769800i $$-0.720356\pi$$
0.638285 0.769800i $$-0.279644\pi$$
$$4$$ −4.00000 −0.500000
$$5$$ 0 0
$$6$$ −16.0000 −1.08866
$$7$$ 4.00000i 0.215980i 0.994152 + 0.107990i $$0.0344414\pi$$
−0.994152 + 0.107990i $$0.965559\pi$$
$$8$$ 8.00000i 0.353553i
$$9$$ −37.0000 −1.37037
$$10$$ 0 0
$$11$$ 12.0000 0.328921 0.164461 0.986384i $$-0.447412\pi$$
0.164461 + 0.986384i $$0.447412\pi$$
$$12$$ 32.0000i 0.769800i
$$13$$ − 58.0000i − 1.23741i −0.785624 0.618704i $$-0.787658\pi$$
0.785624 0.618704i $$-0.212342\pi$$
$$14$$ 8.00000 0.152721
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ − 66.0000i − 0.941609i −0.882238 0.470804i $$-0.843964\pi$$
0.882238 0.470804i $$-0.156036\pi$$
$$18$$ 74.0000i 0.968998i
$$19$$ 100.000 1.20745 0.603726 0.797192i $$-0.293682\pi$$
0.603726 + 0.797192i $$0.293682\pi$$
$$20$$ 0 0
$$21$$ 32.0000 0.332522
$$22$$ − 24.0000i − 0.232583i
$$23$$ 132.000i 1.19669i 0.801238 + 0.598346i $$0.204175\pi$$
−0.801238 + 0.598346i $$0.795825\pi$$
$$24$$ 64.0000 0.544331
$$25$$ 0 0
$$26$$ −116.000 −0.874980
$$27$$ 80.0000i 0.570222i
$$28$$ − 16.0000i − 0.107990i
$$29$$ 90.0000 0.576296 0.288148 0.957586i $$-0.406961\pi$$
0.288148 + 0.957586i $$0.406961\pi$$
$$30$$ 0 0
$$31$$ 152.000 0.880645 0.440323 0.897840i $$-0.354864\pi$$
0.440323 + 0.897840i $$0.354864\pi$$
$$32$$ − 32.0000i − 0.176777i
$$33$$ − 96.0000i − 0.506408i
$$34$$ −132.000 −0.665818
$$35$$ 0 0
$$36$$ 148.000 0.685185
$$37$$ 34.0000i 0.151069i 0.997143 + 0.0755347i $$0.0240664\pi$$
−0.997143 + 0.0755347i $$0.975934\pi$$
$$38$$ − 200.000i − 0.853797i
$$39$$ −464.000 −1.90511
$$40$$ 0 0
$$41$$ −438.000 −1.66839 −0.834196 0.551467i $$-0.814068\pi$$
−0.834196 + 0.551467i $$0.814068\pi$$
$$42$$ − 64.0000i − 0.235129i
$$43$$ 32.0000i 0.113487i 0.998389 + 0.0567437i $$0.0180718\pi$$
−0.998389 + 0.0567437i $$0.981928\pi$$
$$44$$ −48.0000 −0.164461
$$45$$ 0 0
$$46$$ 264.000 0.846189
$$47$$ 204.000i 0.633116i 0.948573 + 0.316558i $$0.102527\pi$$
−0.948573 + 0.316558i $$0.897473\pi$$
$$48$$ − 128.000i − 0.384900i
$$49$$ 327.000 0.953353
$$50$$ 0 0
$$51$$ −528.000 −1.44970
$$52$$ 232.000i 0.618704i
$$53$$ 222.000i 0.575359i 0.957727 + 0.287680i $$0.0928838\pi$$
−0.957727 + 0.287680i $$0.907116\pi$$
$$54$$ 160.000 0.403208
$$55$$ 0 0
$$56$$ −32.0000 −0.0763604
$$57$$ − 800.000i − 1.85899i
$$58$$ − 180.000i − 0.407503i
$$59$$ −420.000 −0.926769 −0.463384 0.886157i $$-0.653365\pi$$
−0.463384 + 0.886157i $$0.653365\pi$$
$$60$$ 0 0
$$61$$ 902.000 1.89327 0.946633 0.322312i $$-0.104460\pi$$
0.946633 + 0.322312i $$0.104460\pi$$
$$62$$ − 304.000i − 0.622710i
$$63$$ − 148.000i − 0.295972i
$$64$$ −64.0000 −0.125000
$$65$$ 0 0
$$66$$ −192.000 −0.358084
$$67$$ 1024.00i 1.86719i 0.358334 + 0.933593i $$0.383345\pi$$
−0.358334 + 0.933593i $$0.616655\pi$$
$$68$$ 264.000i 0.470804i
$$69$$ 1056.00 1.84243
$$70$$ 0 0
$$71$$ 432.000 0.722098 0.361049 0.932547i $$-0.382419\pi$$
0.361049 + 0.932547i $$0.382419\pi$$
$$72$$ − 296.000i − 0.484499i
$$73$$ 362.000i 0.580396i 0.956967 + 0.290198i $$0.0937211\pi$$
−0.956967 + 0.290198i $$0.906279\pi$$
$$74$$ 68.0000 0.106822
$$75$$ 0 0
$$76$$ −400.000 −0.603726
$$77$$ 48.0000i 0.0710404i
$$78$$ 928.000i 1.34712i
$$79$$ 160.000 0.227866 0.113933 0.993488i $$-0.463655\pi$$
0.113933 + 0.993488i $$0.463655\pi$$
$$80$$ 0 0
$$81$$ −359.000 −0.492455
$$82$$ 876.000i 1.17973i
$$83$$ 72.0000i 0.0952172i 0.998866 + 0.0476086i $$0.0151600\pi$$
−0.998866 + 0.0476086i $$0.984840\pi$$
$$84$$ −128.000 −0.166261
$$85$$ 0 0
$$86$$ 64.0000 0.0802476
$$87$$ − 720.000i − 0.887266i
$$88$$ 96.0000i 0.116291i
$$89$$ −810.000 −0.964717 −0.482359 0.875974i $$-0.660220\pi$$
−0.482359 + 0.875974i $$0.660220\pi$$
$$90$$ 0 0
$$91$$ 232.000 0.267255
$$92$$ − 528.000i − 0.598346i
$$93$$ − 1216.00i − 1.35584i
$$94$$ 408.000 0.447681
$$95$$ 0 0
$$96$$ −256.000 −0.272166
$$97$$ − 1106.00i − 1.15770i −0.815433 0.578852i $$-0.803501\pi$$
0.815433 0.578852i $$-0.196499\pi$$
$$98$$ − 654.000i − 0.674122i
$$99$$ −444.000 −0.450744
$$100$$ 0 0
$$101$$ −258.000 −0.254178 −0.127089 0.991891i $$-0.540563\pi$$
−0.127089 + 0.991891i $$0.540563\pi$$
$$102$$ 1056.00i 1.02509i
$$103$$ − 988.000i − 0.945151i −0.881290 0.472575i $$-0.843324\pi$$
0.881290 0.472575i $$-0.156676\pi$$
$$104$$ 464.000 0.437490
$$105$$ 0 0
$$106$$ 444.000 0.406840
$$107$$ 24.0000i 0.0216838i 0.999941 + 0.0108419i $$0.00345115\pi$$
−0.999941 + 0.0108419i $$0.996549\pi$$
$$108$$ − 320.000i − 0.285111i
$$109$$ −950.000 −0.834803 −0.417401 0.908722i $$-0.637059\pi$$
−0.417401 + 0.908722i $$0.637059\pi$$
$$110$$ 0 0
$$111$$ 272.000 0.232586
$$112$$ 64.0000i 0.0539949i
$$113$$ − 1038.00i − 0.864131i −0.901842 0.432066i $$-0.857785\pi$$
0.901842 0.432066i $$-0.142215\pi$$
$$114$$ −1600.00 −1.31451
$$115$$ 0 0
$$116$$ −360.000 −0.288148
$$117$$ 2146.00i 1.69571i
$$118$$ 840.000i 0.655324i
$$119$$ 264.000 0.203368
$$120$$ 0 0
$$121$$ −1187.00 −0.891811
$$122$$ − 1804.00i − 1.33874i
$$123$$ 3504.00i 2.56866i
$$124$$ −608.000 −0.440323
$$125$$ 0 0
$$126$$ −296.000 −0.209284
$$127$$ 124.000i 0.0866395i 0.999061 + 0.0433198i $$0.0137934\pi$$
−0.999061 + 0.0433198i $$0.986207\pi$$
$$128$$ 128.000i 0.0883883i
$$129$$ 256.000 0.174725
$$130$$ 0 0
$$131$$ 132.000 0.0880374 0.0440187 0.999031i $$-0.485984\pi$$
0.0440187 + 0.999031i $$0.485984\pi$$
$$132$$ 384.000i 0.253204i
$$133$$ 400.000i 0.260785i
$$134$$ 2048.00 1.32030
$$135$$ 0 0
$$136$$ 528.000 0.332909
$$137$$ 1254.00i 0.782018i 0.920387 + 0.391009i $$0.127874\pi$$
−0.920387 + 0.391009i $$0.872126\pi$$
$$138$$ − 2112.00i − 1.30279i
$$139$$ 2860.00 1.74519 0.872597 0.488440i $$-0.162434\pi$$
0.872597 + 0.488440i $$0.162434\pi$$
$$140$$ 0 0
$$141$$ 1632.00 0.974746
$$142$$ − 864.000i − 0.510600i
$$143$$ − 696.000i − 0.407010i
$$144$$ −592.000 −0.342593
$$145$$ 0 0
$$146$$ 724.000 0.410402
$$147$$ − 2616.00i − 1.46778i
$$148$$ − 136.000i − 0.0755347i
$$149$$ −750.000 −0.412365 −0.206183 0.978514i $$-0.566104\pi$$
−0.206183 + 0.978514i $$0.566104\pi$$
$$150$$ 0 0
$$151$$ −448.000 −0.241442 −0.120721 0.992686i $$-0.538521\pi$$
−0.120721 + 0.992686i $$0.538521\pi$$
$$152$$ 800.000i 0.426898i
$$153$$ 2442.00i 1.29035i
$$154$$ 96.0000 0.0502331
$$155$$ 0 0
$$156$$ 1856.00 0.952557
$$157$$ − 2246.00i − 1.14172i −0.821047 0.570861i $$-0.806610\pi$$
0.821047 0.570861i $$-0.193390\pi$$
$$158$$ − 320.000i − 0.161126i
$$159$$ 1776.00 0.885824
$$160$$ 0 0
$$161$$ −528.000 −0.258461
$$162$$ 718.000i 0.348219i
$$163$$ − 568.000i − 0.272940i −0.990644 0.136470i $$-0.956424\pi$$
0.990644 0.136470i $$-0.0435757\pi$$
$$164$$ 1752.00 0.834196
$$165$$ 0 0
$$166$$ 144.000 0.0673287
$$167$$ 1524.00i 0.706172i 0.935591 + 0.353086i $$0.114868\pi$$
−0.935591 + 0.353086i $$0.885132\pi$$
$$168$$ 256.000i 0.117564i
$$169$$ −1167.00 −0.531179
$$170$$ 0 0
$$171$$ −3700.00 −1.65466
$$172$$ − 128.000i − 0.0567437i
$$173$$ 3702.00i 1.62692i 0.581618 + 0.813462i $$0.302420\pi$$
−0.581618 + 0.813462i $$0.697580\pi$$
$$174$$ −1440.00 −0.627391
$$175$$ 0 0
$$176$$ 192.000 0.0822304
$$177$$ 3360.00i 1.42685i
$$178$$ 1620.00i 0.682158i
$$179$$ −3180.00 −1.32785 −0.663923 0.747801i $$-0.731110\pi$$
−0.663923 + 0.747801i $$0.731110\pi$$
$$180$$ 0 0
$$181$$ −2098.00 −0.861564 −0.430782 0.902456i $$-0.641762\pi$$
−0.430782 + 0.902456i $$0.641762\pi$$
$$182$$ − 464.000i − 0.188978i
$$183$$ − 7216.00i − 2.91487i
$$184$$ −1056.00 −0.423094
$$185$$ 0 0
$$186$$ −2432.00 −0.958725
$$187$$ − 792.000i − 0.309715i
$$188$$ − 816.000i − 0.316558i
$$189$$ −320.000 −0.123156
$$190$$ 0 0
$$191$$ 4392.00 1.66384 0.831921 0.554894i $$-0.187241\pi$$
0.831921 + 0.554894i $$0.187241\pi$$
$$192$$ 512.000i 0.192450i
$$193$$ − 2158.00i − 0.804851i −0.915453 0.402425i $$-0.868167\pi$$
0.915453 0.402425i $$-0.131833\pi$$
$$194$$ −2212.00 −0.818620
$$195$$ 0 0
$$196$$ −1308.00 −0.476676
$$197$$ 1074.00i 0.388423i 0.980960 + 0.194212i $$0.0622148\pi$$
−0.980960 + 0.194212i $$0.937785\pi$$
$$198$$ 888.000i 0.318724i
$$199$$ −2840.00 −1.01167 −0.505835 0.862630i $$-0.668815\pi$$
−0.505835 + 0.862630i $$0.668815\pi$$
$$200$$ 0 0
$$201$$ 8192.00 2.87472
$$202$$ 516.000i 0.179731i
$$203$$ 360.000i 0.124468i
$$204$$ 2112.00 0.724851
$$205$$ 0 0
$$206$$ −1976.00 −0.668323
$$207$$ − 4884.00i − 1.63991i
$$208$$ − 928.000i − 0.309352i
$$209$$ 1200.00 0.397157
$$210$$ 0 0
$$211$$ −2668.00 −0.870487 −0.435243 0.900313i $$-0.643338\pi$$
−0.435243 + 0.900313i $$0.643338\pi$$
$$212$$ − 888.000i − 0.287680i
$$213$$ − 3456.00i − 1.11174i
$$214$$ 48.0000 0.0153328
$$215$$ 0 0
$$216$$ −640.000 −0.201604
$$217$$ 608.000i 0.190202i
$$218$$ 1900.00i 0.590295i
$$219$$ 2896.00 0.893578
$$220$$ 0 0
$$221$$ −3828.00 −1.16515
$$222$$ − 544.000i − 0.164463i
$$223$$ 1772.00i 0.532116i 0.963957 + 0.266058i $$0.0857213\pi$$
−0.963957 + 0.266058i $$0.914279\pi$$
$$224$$ 128.000 0.0381802
$$225$$ 0 0
$$226$$ −2076.00 −0.611033
$$227$$ 2784.00i 0.814011i 0.913426 + 0.407006i $$0.133427\pi$$
−0.913426 + 0.407006i $$0.866573\pi$$
$$228$$ 3200.00i 0.929496i
$$229$$ −350.000 −0.100998 −0.0504992 0.998724i $$-0.516081\pi$$
−0.0504992 + 0.998724i $$0.516081\pi$$
$$230$$ 0 0
$$231$$ 384.000 0.109374
$$232$$ 720.000i 0.203751i
$$233$$ 1962.00i 0.551652i 0.961208 + 0.275826i $$0.0889513\pi$$
−0.961208 + 0.275826i $$0.911049\pi$$
$$234$$ 4292.00 1.19905
$$235$$ 0 0
$$236$$ 1680.00 0.463384
$$237$$ − 1280.00i − 0.350823i
$$238$$ − 528.000i − 0.143803i
$$239$$ 4320.00 1.16919 0.584597 0.811324i $$-0.301252\pi$$
0.584597 + 0.811324i $$0.301252\pi$$
$$240$$ 0 0
$$241$$ −478.000 −0.127762 −0.0638811 0.997958i $$-0.520348\pi$$
−0.0638811 + 0.997958i $$0.520348\pi$$
$$242$$ 2374.00i 0.630605i
$$243$$ 5032.00i 1.32841i
$$244$$ −3608.00 −0.946633
$$245$$ 0 0
$$246$$ 7008.00 1.81632
$$247$$ − 5800.00i − 1.49411i
$$248$$ 1216.00i 0.311355i
$$249$$ 576.000 0.146596
$$250$$ 0 0
$$251$$ 2652.00 0.666903 0.333452 0.942767i $$-0.391787\pi$$
0.333452 + 0.942767i $$0.391787\pi$$
$$252$$ 592.000i 0.147986i
$$253$$ 1584.00i 0.393617i
$$254$$ 248.000 0.0612634
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ 2334.00i 0.566502i 0.959046 + 0.283251i $$0.0914129\pi$$
−0.959046 + 0.283251i $$0.908587\pi$$
$$258$$ − 512.000i − 0.123549i
$$259$$ −136.000 −0.0326279
$$260$$ 0 0
$$261$$ −3330.00 −0.789739
$$262$$ − 264.000i − 0.0622518i
$$263$$ − 3948.00i − 0.925643i −0.886451 0.462822i $$-0.846837\pi$$
0.886451 0.462822i $$-0.153163\pi$$
$$264$$ 768.000 0.179042
$$265$$ 0 0
$$266$$ 800.000 0.184403
$$267$$ 6480.00i 1.48528i
$$268$$ − 4096.00i − 0.933593i
$$269$$ −1590.00 −0.360387 −0.180193 0.983631i $$-0.557672\pi$$
−0.180193 + 0.983631i $$0.557672\pi$$
$$270$$ 0 0
$$271$$ 4952.00 1.11001 0.555005 0.831847i $$-0.312716\pi$$
0.555005 + 0.831847i $$0.312716\pi$$
$$272$$ − 1056.00i − 0.235402i
$$273$$ − 1856.00i − 0.411466i
$$274$$ 2508.00 0.552970
$$275$$ 0 0
$$276$$ −4224.00 −0.921213
$$277$$ − 1646.00i − 0.357034i −0.983937 0.178517i $$-0.942870\pi$$
0.983937 0.178517i $$-0.0571300\pi$$
$$278$$ − 5720.00i − 1.23404i
$$279$$ −5624.00 −1.20681
$$280$$ 0 0
$$281$$ −1158.00 −0.245838 −0.122919 0.992417i $$-0.539226\pi$$
−0.122919 + 0.992417i $$0.539226\pi$$
$$282$$ − 3264.00i − 0.689250i
$$283$$ 6992.00i 1.46866i 0.678792 + 0.734331i $$0.262504\pi$$
−0.678792 + 0.734331i $$0.737496\pi$$
$$284$$ −1728.00 −0.361049
$$285$$ 0 0
$$286$$ −1392.00 −0.287800
$$287$$ − 1752.00i − 0.360339i
$$288$$ 1184.00i 0.242250i
$$289$$ 557.000 0.113373
$$290$$ 0 0
$$291$$ −8848.00 −1.78240
$$292$$ − 1448.00i − 0.290198i
$$293$$ − 258.000i − 0.0514421i −0.999669 0.0257210i $$-0.991812\pi$$
0.999669 0.0257210i $$-0.00818816\pi$$
$$294$$ −5232.00 −1.03788
$$295$$ 0 0
$$296$$ −272.000 −0.0534111
$$297$$ 960.000i 0.187558i
$$298$$ 1500.00i 0.291586i
$$299$$ 7656.00 1.48080
$$300$$ 0 0
$$301$$ −128.000 −0.0245110
$$302$$ 896.000i 0.170725i
$$303$$ 2064.00i 0.391332i
$$304$$ 1600.00 0.301863
$$305$$ 0 0
$$306$$ 4884.00 0.912417
$$307$$ 8944.00i 1.66274i 0.555720 + 0.831370i $$0.312443\pi$$
−0.555720 + 0.831370i $$0.687557\pi$$
$$308$$ − 192.000i − 0.0355202i
$$309$$ −7904.00 −1.45515
$$310$$ 0 0
$$311$$ 1392.00 0.253804 0.126902 0.991915i $$-0.459497\pi$$
0.126902 + 0.991915i $$0.459497\pi$$
$$312$$ − 3712.00i − 0.673560i
$$313$$ − 5878.00i − 1.06148i −0.847534 0.530742i $$-0.821913\pi$$
0.847534 0.530742i $$-0.178087\pi$$
$$314$$ −4492.00 −0.807319
$$315$$ 0 0
$$316$$ −640.000 −0.113933
$$317$$ − 10326.0i − 1.82955i −0.403969 0.914773i $$-0.632370\pi$$
0.403969 0.914773i $$-0.367630\pi$$
$$318$$ − 3552.00i − 0.626372i
$$319$$ 1080.00 0.189556
$$320$$ 0 0
$$321$$ 192.000 0.0333844
$$322$$ 1056.00i 0.182760i
$$323$$ − 6600.00i − 1.13695i
$$324$$ 1436.00 0.246228
$$325$$ 0 0
$$326$$ −1136.00 −0.192998
$$327$$ 7600.00i 1.28526i
$$328$$ − 3504.00i − 0.589866i
$$329$$ −816.000 −0.136740
$$330$$ 0 0
$$331$$ −4228.00 −0.702090 −0.351045 0.936359i $$-0.614174\pi$$
−0.351045 + 0.936359i $$0.614174\pi$$
$$332$$ − 288.000i − 0.0476086i
$$333$$ − 1258.00i − 0.207021i
$$334$$ 3048.00 0.499339
$$335$$ 0 0
$$336$$ 512.000 0.0831306
$$337$$ − 1106.00i − 0.178776i −0.995997 0.0893882i $$-0.971509\pi$$
0.995997 0.0893882i $$-0.0284912\pi$$
$$338$$ 2334.00i 0.375600i
$$339$$ −8304.00 −1.33042
$$340$$ 0 0
$$341$$ 1824.00 0.289663
$$342$$ 7400.00i 1.17002i
$$343$$ 2680.00i 0.421885i
$$344$$ −256.000 −0.0401238
$$345$$ 0 0
$$346$$ 7404.00 1.15041
$$347$$ − 9336.00i − 1.44433i −0.691720 0.722165i $$-0.743147\pi$$
0.691720 0.722165i $$-0.256853\pi$$
$$348$$ 2880.00i 0.443633i
$$349$$ 11770.0 1.80525 0.902627 0.430424i $$-0.141636\pi$$
0.902627 + 0.430424i $$0.141636\pi$$
$$350$$ 0 0
$$351$$ 4640.00 0.705598
$$352$$ − 384.000i − 0.0581456i
$$353$$ 8322.00i 1.25477i 0.778707 + 0.627387i $$0.215876\pi$$
−0.778707 + 0.627387i $$0.784124\pi$$
$$354$$ 6720.00 1.00894
$$355$$ 0 0
$$356$$ 3240.00 0.482359
$$357$$ − 2112.00i − 0.313106i
$$358$$ 6360.00i 0.938929i
$$359$$ −10680.0 −1.57011 −0.785054 0.619427i $$-0.787365\pi$$
−0.785054 + 0.619427i $$0.787365\pi$$
$$360$$ 0 0
$$361$$ 3141.00 0.457938
$$362$$ 4196.00i 0.609218i
$$363$$ 9496.00i 1.37303i
$$364$$ −928.000 −0.133628
$$365$$ 0 0
$$366$$ −14432.0 −2.06113
$$367$$ 5884.00i 0.836900i 0.908240 + 0.418450i $$0.137426\pi$$
−0.908240 + 0.418450i $$0.862574\pi$$
$$368$$ 2112.00i 0.299173i
$$369$$ 16206.0 2.28632
$$370$$ 0 0
$$371$$ −888.000 −0.124266
$$372$$ 4864.00i 0.677921i
$$373$$ − 2098.00i − 0.291234i −0.989341 0.145617i $$-0.953483\pi$$
0.989341 0.145617i $$-0.0465167\pi$$
$$374$$ −1584.00 −0.219002
$$375$$ 0 0
$$376$$ −1632.00 −0.223840
$$377$$ − 5220.00i − 0.713113i
$$378$$ 640.000i 0.0870848i
$$379$$ −3860.00 −0.523153 −0.261576 0.965183i $$-0.584242\pi$$
−0.261576 + 0.965183i $$0.584242\pi$$
$$380$$ 0 0
$$381$$ 992.000 0.133390
$$382$$ − 8784.00i − 1.17651i
$$383$$ − 9588.00i − 1.27917i −0.768718 0.639587i $$-0.779105\pi$$
0.768718 0.639587i $$-0.220895\pi$$
$$384$$ 1024.00 0.136083
$$385$$ 0 0
$$386$$ −4316.00 −0.569116
$$387$$ − 1184.00i − 0.155520i
$$388$$ 4424.00i 0.578852i
$$389$$ 13410.0 1.74785 0.873925 0.486060i $$-0.161566\pi$$
0.873925 + 0.486060i $$0.161566\pi$$
$$390$$ 0 0
$$391$$ 8712.00 1.12682
$$392$$ 2616.00i 0.337061i
$$393$$ − 1056.00i − 0.135542i
$$394$$ 2148.00 0.274657
$$395$$ 0 0
$$396$$ 1776.00 0.225372
$$397$$ 13114.0i 1.65787i 0.559348 + 0.828933i $$0.311052\pi$$
−0.559348 + 0.828933i $$0.688948\pi$$
$$398$$ 5680.00i 0.715358i
$$399$$ 3200.00 0.401505
$$400$$ 0 0
$$401$$ −5838.00 −0.727022 −0.363511 0.931590i $$-0.618422\pi$$
−0.363511 + 0.931590i $$0.618422\pi$$
$$402$$ − 16384.0i − 2.03274i
$$403$$ − 8816.00i − 1.08972i
$$404$$ 1032.00 0.127089
$$405$$ 0 0
$$406$$ 720.000 0.0880123
$$407$$ 408.000i 0.0496899i
$$408$$ − 4224.00i − 0.512547i
$$409$$ −9530.00 −1.15215 −0.576074 0.817398i $$-0.695416\pi$$
−0.576074 + 0.817398i $$0.695416\pi$$
$$410$$ 0 0
$$411$$ 10032.0 1.20400
$$412$$ 3952.00i 0.472575i
$$413$$ − 1680.00i − 0.200163i
$$414$$ −9768.00 −1.15959
$$415$$ 0 0
$$416$$ −1856.00 −0.218745
$$417$$ − 22880.0i − 2.68690i
$$418$$ − 2400.00i − 0.280832i
$$419$$ −7260.00 −0.846478 −0.423239 0.906018i $$-0.639107\pi$$
−0.423239 + 0.906018i $$0.639107\pi$$
$$420$$ 0 0
$$421$$ 12062.0 1.39636 0.698178 0.715924i $$-0.253994\pi$$
0.698178 + 0.715924i $$0.253994\pi$$
$$422$$ 5336.00i 0.615527i
$$423$$ − 7548.00i − 0.867604i
$$424$$ −1776.00 −0.203420
$$425$$ 0 0
$$426$$ −6912.00 −0.786121
$$427$$ 3608.00i 0.408907i
$$428$$ − 96.0000i − 0.0108419i
$$429$$ −5568.00 −0.626633
$$430$$ 0 0
$$431$$ −13608.0 −1.52082 −0.760411 0.649442i $$-0.775002\pi$$
−0.760411 + 0.649442i $$0.775002\pi$$
$$432$$ 1280.00i 0.142556i
$$433$$ − 3838.00i − 0.425964i −0.977056 0.212982i $$-0.931682\pi$$
0.977056 0.212982i $$-0.0683176\pi$$
$$434$$ 1216.00 0.134493
$$435$$ 0 0
$$436$$ 3800.00 0.417401
$$437$$ 13200.0i 1.44495i
$$438$$ − 5792.00i − 0.631855i
$$439$$ −7400.00 −0.804516 −0.402258 0.915526i $$-0.631775\pi$$
−0.402258 + 0.915526i $$0.631775\pi$$
$$440$$ 0 0
$$441$$ −12099.0 −1.30645
$$442$$ 7656.00i 0.823889i
$$443$$ 8352.00i 0.895746i 0.894097 + 0.447873i $$0.147818\pi$$
−0.894097 + 0.447873i $$0.852182\pi$$
$$444$$ −1088.00 −0.116293
$$445$$ 0 0
$$446$$ 3544.00 0.376263
$$447$$ 6000.00i 0.634878i
$$448$$ − 256.000i − 0.0269975i
$$449$$ −10770.0 −1.13200 −0.566000 0.824405i $$-0.691510\pi$$
−0.566000 + 0.824405i $$0.691510\pi$$
$$450$$ 0 0
$$451$$ −5256.00 −0.548770
$$452$$ 4152.00i 0.432066i
$$453$$ 3584.00i 0.371724i
$$454$$ 5568.00 0.575593
$$455$$ 0 0
$$456$$ 6400.00 0.657253
$$457$$ 6694.00i 0.685191i 0.939483 + 0.342595i $$0.111306\pi$$
−0.939483 + 0.342595i $$0.888694\pi$$
$$458$$ 700.000i 0.0714167i
$$459$$ 5280.00 0.536927
$$460$$ 0 0
$$461$$ −3018.00 −0.304907 −0.152454 0.988311i $$-0.548717\pi$$
−0.152454 + 0.988311i $$0.548717\pi$$
$$462$$ − 768.000i − 0.0773389i
$$463$$ 14492.0i 1.45464i 0.686296 + 0.727322i $$0.259235\pi$$
−0.686296 + 0.727322i $$0.740765\pi$$
$$464$$ 1440.00 0.144074
$$465$$ 0 0
$$466$$ 3924.00 0.390077
$$467$$ − 7776.00i − 0.770515i −0.922809 0.385257i $$-0.874113\pi$$
0.922809 0.385257i $$-0.125887\pi$$
$$468$$ − 8584.00i − 0.847854i
$$469$$ −4096.00 −0.403274
$$470$$ 0 0
$$471$$ −17968.0 −1.75780
$$472$$ − 3360.00i − 0.327662i
$$473$$ 384.000i 0.0373284i
$$474$$ −2560.00 −0.248069
$$475$$ 0 0
$$476$$ −1056.00 −0.101684
$$477$$ − 8214.00i − 0.788455i
$$478$$ − 8640.00i − 0.826746i
$$479$$ 13680.0 1.30492 0.652458 0.757825i $$-0.273738\pi$$
0.652458 + 0.757825i $$0.273738\pi$$
$$480$$ 0 0
$$481$$ 1972.00 0.186934
$$482$$ 956.000i 0.0903415i
$$483$$ 4224.00i 0.397927i
$$484$$ 4748.00 0.445905
$$485$$ 0 0
$$486$$ 10064.0 0.939326
$$487$$ − 7916.00i − 0.736567i −0.929714 0.368284i $$-0.879946\pi$$
0.929714 0.368284i $$-0.120054\pi$$
$$488$$ 7216.00i 0.669371i
$$489$$ −4544.00 −0.420218
$$490$$ 0 0
$$491$$ 13932.0 1.28053 0.640267 0.768152i $$-0.278824\pi$$
0.640267 + 0.768152i $$0.278824\pi$$
$$492$$ − 14016.0i − 1.28433i
$$493$$ − 5940.00i − 0.542645i
$$494$$ −11600.0 −1.05650
$$495$$ 0 0
$$496$$ 2432.00 0.220161
$$497$$ 1728.00i 0.155959i
$$498$$ − 1152.00i − 0.103659i
$$499$$ 8260.00 0.741019 0.370509 0.928829i $$-0.379183\pi$$
0.370509 + 0.928829i $$0.379183\pi$$
$$500$$ 0 0
$$501$$ 12192.0 1.08722
$$502$$ − 5304.00i − 0.471572i
$$503$$ − 11148.0i − 0.988200i −0.869405 0.494100i $$-0.835498\pi$$
0.869405 0.494100i $$-0.164502\pi$$
$$504$$ 1184.00 0.104642
$$505$$ 0 0
$$506$$ 3168.00 0.278330
$$507$$ 9336.00i 0.817803i
$$508$$ − 496.000i − 0.0433198i
$$509$$ 9690.00 0.843815 0.421907 0.906639i $$-0.361361\pi$$
0.421907 + 0.906639i $$0.361361\pi$$
$$510$$ 0 0
$$511$$ −1448.00 −0.125354
$$512$$ − 512.000i − 0.0441942i
$$513$$ 8000.00i 0.688516i
$$514$$ 4668.00 0.400577
$$515$$ 0 0
$$516$$ −1024.00 −0.0873626
$$517$$ 2448.00i 0.208245i
$$518$$ 272.000i 0.0230714i
$$519$$ 29616.0 2.50481
$$520$$ 0 0
$$521$$ −16038.0 −1.34863 −0.674316 0.738443i $$-0.735562\pi$$
−0.674316 + 0.738443i $$0.735562\pi$$
$$522$$ 6660.00i 0.558430i
$$523$$ 992.000i 0.0829391i 0.999140 + 0.0414695i $$0.0132039\pi$$
−0.999140 + 0.0414695i $$0.986796\pi$$
$$524$$ −528.000 −0.0440187
$$525$$ 0 0
$$526$$ −7896.00 −0.654528
$$527$$ − 10032.0i − 0.829223i
$$528$$ − 1536.00i − 0.126602i
$$529$$ −5257.00 −0.432070
$$530$$ 0 0
$$531$$ 15540.0 1.27002
$$532$$ − 1600.00i − 0.130392i
$$533$$ 25404.0i 2.06448i
$$534$$ 12960.0 1.05025
$$535$$ 0 0
$$536$$ −8192.00 −0.660150
$$537$$ 25440.0i 2.04435i
$$538$$ 3180.00i 0.254832i
$$539$$ 3924.00 0.313578
$$540$$ 0 0
$$541$$ 7142.00 0.567576 0.283788 0.958887i $$-0.408409\pi$$
0.283788 + 0.958887i $$0.408409\pi$$
$$542$$ − 9904.00i − 0.784895i
$$543$$ 16784.0i 1.32646i
$$544$$ −2112.00 −0.166455
$$545$$ 0 0
$$546$$ −3712.00 −0.290950
$$547$$ − 7616.00i − 0.595314i −0.954673 0.297657i $$-0.903795\pi$$
0.954673 0.297657i $$-0.0962051\pi$$
$$548$$ − 5016.00i − 0.391009i
$$549$$ −33374.0 −2.59448
$$550$$ 0 0
$$551$$ 9000.00 0.695849
$$552$$ 8448.00i 0.651396i
$$553$$ 640.000i 0.0492144i
$$554$$ −3292.00 −0.252462
$$555$$ 0 0
$$556$$ −11440.0 −0.872597
$$557$$ 10314.0i 0.784593i 0.919839 + 0.392296i $$0.128319\pi$$
−0.919839 + 0.392296i $$0.871681\pi$$
$$558$$ 11248.0i 0.853344i
$$559$$ 1856.00 0.140430
$$560$$ 0 0
$$561$$ −6336.00 −0.476838
$$562$$ 2316.00i 0.173834i
$$563$$ − 7128.00i − 0.533587i −0.963754 0.266793i $$-0.914036\pi$$
0.963754 0.266793i $$-0.0859641\pi$$
$$564$$ −6528.00 −0.487373
$$565$$ 0 0
$$566$$ 13984.0 1.03850
$$567$$ − 1436.00i − 0.106360i
$$568$$ 3456.00i 0.255300i
$$569$$ −2010.00 −0.148091 −0.0740453 0.997255i $$-0.523591\pi$$
−0.0740453 + 0.997255i $$0.523591\pi$$
$$570$$ 0 0
$$571$$ −23188.0 −1.69945 −0.849726 0.527224i $$-0.823233\pi$$
−0.849726 + 0.527224i $$0.823233\pi$$
$$572$$ 2784.00i 0.203505i
$$573$$ − 35136.0i − 2.56165i
$$574$$ −3504.00 −0.254798
$$575$$ 0 0
$$576$$ 2368.00 0.171296
$$577$$ − 22466.0i − 1.62092i −0.585793 0.810461i $$-0.699217\pi$$
0.585793 0.810461i $$-0.300783\pi$$
$$578$$ − 1114.00i − 0.0801666i
$$579$$ −17264.0 −1.23915
$$580$$ 0 0
$$581$$ −288.000 −0.0205650
$$582$$ 17696.0i 1.26035i
$$583$$ 2664.00i 0.189248i
$$584$$ −2896.00 −0.205201
$$585$$ 0 0
$$586$$ −516.000 −0.0363750
$$587$$ − 22776.0i − 1.60148i −0.599015 0.800738i $$-0.704441\pi$$
0.599015 0.800738i $$-0.295559\pi$$
$$588$$ 10464.0i 0.733891i
$$589$$ 15200.0 1.06334
$$590$$ 0 0
$$591$$ 8592.00 0.598016
$$592$$ 544.000i 0.0377673i
$$593$$ − 21198.0i − 1.46796i −0.679174 0.733978i $$-0.737662\pi$$
0.679174 0.733978i $$-0.262338\pi$$
$$594$$ 1920.00 0.132624
$$595$$ 0 0
$$596$$ 3000.00 0.206183
$$597$$ 22720.0i 1.55757i
$$598$$ − 15312.0i − 1.04708i
$$599$$ −15960.0 −1.08866 −0.544330 0.838871i $$-0.683216\pi$$
−0.544330 + 0.838871i $$0.683216\pi$$
$$600$$ 0 0
$$601$$ 5882.00 0.399221 0.199610 0.979875i $$-0.436032\pi$$
0.199610 + 0.979875i $$0.436032\pi$$
$$602$$ 256.000i 0.0173319i
$$603$$ − 37888.0i − 2.55874i
$$604$$ 1792.00 0.120721
$$605$$ 0 0
$$606$$ 4128.00 0.276714
$$607$$ − 8516.00i − 0.569446i −0.958610 0.284723i $$-0.908098\pi$$
0.958610 0.284723i $$-0.0919016\pi$$
$$608$$ − 3200.00i − 0.213449i
$$609$$ 2880.00 0.191631
$$610$$ 0 0
$$611$$ 11832.0 0.783423
$$612$$ − 9768.00i − 0.645176i
$$613$$ 8462.00i 0.557548i 0.960357 + 0.278774i $$0.0899280\pi$$
−0.960357 + 0.278774i $$0.910072\pi$$
$$614$$ 17888.0 1.17573
$$615$$ 0 0
$$616$$ −384.000 −0.0251166
$$617$$ 11094.0i 0.723870i 0.932203 + 0.361935i $$0.117884\pi$$
−0.932203 + 0.361935i $$0.882116\pi$$
$$618$$ 15808.0i 1.02895i
$$619$$ −2180.00 −0.141553 −0.0707767 0.997492i $$-0.522548\pi$$
−0.0707767 + 0.997492i $$0.522548\pi$$
$$620$$ 0 0
$$621$$ −10560.0 −0.682380
$$622$$ − 2784.00i − 0.179467i
$$623$$ − 3240.00i − 0.208359i
$$624$$ −7424.00 −0.476279
$$625$$ 0 0
$$626$$ −11756.0 −0.750582
$$627$$ − 9600.00i − 0.611463i
$$628$$ 8984.00i 0.570861i
$$629$$ 2244.00 0.142248
$$630$$ 0 0
$$631$$ −26848.0 −1.69382 −0.846911 0.531734i $$-0.821541\pi$$
−0.846911 + 0.531734i $$0.821541\pi$$
$$632$$ 1280.00i 0.0805628i
$$633$$ 21344.0i 1.34020i
$$634$$ −20652.0 −1.29368
$$635$$ 0 0
$$636$$ −7104.00 −0.442912
$$637$$ − 18966.0i − 1.17969i
$$638$$ − 2160.00i − 0.134036i
$$639$$ −15984.0 −0.989542
$$640$$ 0 0
$$641$$ 26322.0 1.62193 0.810965 0.585095i $$-0.198943\pi$$
0.810965 + 0.585095i $$0.198943\pi$$
$$642$$ − 384.000i − 0.0236063i
$$643$$ − 10168.0i − 0.623619i −0.950145 0.311809i $$-0.899065\pi$$
0.950145 0.311809i $$-0.100935\pi$$
$$644$$ 2112.00 0.129231
$$645$$ 0 0
$$646$$ −13200.0 −0.803943
$$647$$ 23604.0i 1.43426i 0.696937 + 0.717132i $$0.254546\pi$$
−0.696937 + 0.717132i $$0.745454\pi$$
$$648$$ − 2872.00i − 0.174109i
$$649$$ −5040.00 −0.304834
$$650$$ 0 0
$$651$$ 4864.00 0.292834
$$652$$ 2272.00i 0.136470i
$$653$$ 16422.0i 0.984139i 0.870556 + 0.492069i $$0.163759\pi$$
−0.870556 + 0.492069i $$0.836241\pi$$
$$654$$ 15200.0 0.908818
$$655$$ 0 0
$$656$$ −7008.00 −0.417098
$$657$$ − 13394.0i − 0.795357i
$$658$$ 1632.00i 0.0966899i
$$659$$ 26100.0 1.54281 0.771405 0.636345i $$-0.219554\pi$$
0.771405 + 0.636345i $$0.219554\pi$$
$$660$$ 0 0
$$661$$ −3058.00 −0.179943 −0.0899716 0.995944i $$-0.528678\pi$$
−0.0899716 + 0.995944i $$0.528678\pi$$
$$662$$ 8456.00i 0.496453i
$$663$$ 30624.0i 1.79387i
$$664$$ −576.000 −0.0336644
$$665$$ 0 0
$$666$$ −2516.00 −0.146386
$$667$$ 11880.0i 0.689648i
$$668$$ − 6096.00i − 0.353086i
$$669$$ 14176.0 0.819246
$$670$$ 0 0
$$671$$ 10824.0 0.622736
$$672$$ − 1024.00i − 0.0587822i
$$673$$ 10802.0i 0.618702i 0.950948 + 0.309351i $$0.100112\pi$$
−0.950948 + 0.309351i $$0.899888\pi$$
$$674$$ −2212.00 −0.126414
$$675$$ 0 0
$$676$$ 4668.00 0.265589
$$677$$ 10674.0i 0.605960i 0.952997 + 0.302980i $$0.0979816\pi$$
−0.952997 + 0.302980i $$0.902018\pi$$
$$678$$ 16608.0i 0.940747i
$$679$$ 4424.00 0.250041
$$680$$ 0 0
$$681$$ 22272.0 1.25325
$$682$$ − 3648.00i − 0.204823i
$$683$$ − 28608.0i − 1.60272i −0.598185 0.801358i $$-0.704111\pi$$
0.598185 0.801358i $$-0.295889\pi$$
$$684$$ 14800.0 0.827328
$$685$$ 0 0
$$686$$ 5360.00 0.298317
$$687$$ 2800.00i 0.155497i
$$688$$ 512.000i 0.0283718i
$$689$$ 12876.0 0.711954
$$690$$ 0 0
$$691$$ −2428.00 −0.133669 −0.0668346 0.997764i $$-0.521290\pi$$
−0.0668346 + 0.997764i $$0.521290\pi$$
$$692$$ − 14808.0i − 0.813462i
$$693$$ − 1776.00i − 0.0973516i
$$694$$ −18672.0 −1.02130
$$695$$ 0 0
$$696$$ 5760.00 0.313696
$$697$$ 28908.0i 1.57097i
$$698$$ − 23540.0i − 1.27651i
$$699$$ 15696.0 0.849324
$$700$$ 0 0
$$701$$ −6618.00 −0.356574 −0.178287 0.983979i $$-0.557056\pi$$
−0.178287 + 0.983979i $$0.557056\pi$$
$$702$$ − 9280.00i − 0.498933i
$$703$$ 3400.00i 0.182409i
$$704$$ −768.000 −0.0411152
$$705$$ 0 0
$$706$$ 16644.0 0.887259
$$707$$ − 1032.00i − 0.0548972i
$$708$$ − 13440.0i − 0.713427i
$$709$$ −20510.0 −1.08642 −0.543208 0.839598i $$-0.682791\pi$$
−0.543208 + 0.839598i $$0.682791\pi$$
$$710$$ 0 0
$$711$$ −5920.00 −0.312261
$$712$$ − 6480.00i − 0.341079i
$$713$$ 20064.0i 1.05386i
$$714$$ −4224.00 −0.221399
$$715$$ 0 0
$$716$$ 12720.0 0.663923
$$717$$ − 34560.0i − 1.80009i
$$718$$ 21360.0i 1.11023i
$$719$$ −31680.0 −1.64321 −0.821603 0.570061i $$-0.806920\pi$$
−0.821603 + 0.570061i $$0.806920\pi$$
$$720$$ 0 0
$$721$$ 3952.00 0.204133
$$722$$ − 6282.00i − 0.323811i
$$723$$ 3824.00i 0.196703i
$$724$$ 8392.00 0.430782
$$725$$ 0 0
$$726$$ 18992.0 0.970880
$$727$$ − 13196.0i − 0.673195i −0.941649 0.336597i $$-0.890724\pi$$
0.941649 0.336597i $$-0.109276\pi$$
$$728$$ 1856.00i 0.0944889i
$$729$$ 30563.0 1.55276
$$730$$ 0 0
$$731$$ 2112.00 0.106861
$$732$$ 28864.0i 1.45744i
$$733$$ 8102.00i 0.408259i 0.978944 + 0.204130i $$0.0654364\pi$$
−0.978944 + 0.204130i $$0.934564\pi$$
$$734$$ 11768.0 0.591778
$$735$$ 0 0
$$736$$ 4224.00 0.211547
$$737$$ 12288.0i 0.614158i
$$738$$ − 32412.0i − 1.61667i
$$739$$ 12580.0 0.626201 0.313101 0.949720i $$-0.398632\pi$$
0.313101 + 0.949720i $$0.398632\pi$$
$$740$$ 0 0
$$741$$ −46400.0 −2.30033
$$742$$ 1776.00i 0.0878693i
$$743$$ 29892.0i 1.47595i 0.674828 + 0.737975i $$0.264218\pi$$
−0.674828 + 0.737975i $$0.735782\pi$$
$$744$$ 9728.00 0.479363
$$745$$ 0 0
$$746$$ −4196.00 −0.205934
$$747$$ − 2664.00i − 0.130483i
$$748$$ 3168.00i 0.154858i
$$749$$ −96.0000 −0.00468326
$$750$$ 0 0
$$751$$ −40408.0 −1.96339 −0.981697 0.190450i $$-0.939005\pi$$
−0.981697 + 0.190450i $$0.939005\pi$$
$$752$$ 3264.00i 0.158279i
$$753$$ − 21216.0i − 1.02676i
$$754$$ −10440.0 −0.504247
$$755$$ 0 0
$$756$$ 1280.00 0.0615782
$$757$$ − 32366.0i − 1.55398i −0.629513 0.776990i $$-0.716746\pi$$
0.629513 0.776990i $$-0.283254\pi$$
$$758$$ 7720.00i 0.369925i
$$759$$ 12672.0 0.606014
$$760$$ 0 0
$$761$$ −17238.0 −0.821126 −0.410563 0.911832i $$-0.634668\pi$$
−0.410563 + 0.911832i $$0.634668\pi$$
$$762$$ − 1984.00i − 0.0943212i
$$763$$ − 3800.00i − 0.180300i
$$764$$ −17568.0 −0.831921
$$765$$ 0 0
$$766$$ −19176.0 −0.904513
$$767$$ 24360.0i 1.14679i
$$768$$ − 2048.00i − 0.0962250i
$$769$$ −10850.0 −0.508792 −0.254396 0.967100i $$-0.581877\pi$$
−0.254396 + 0.967100i $$0.581877\pi$$
$$770$$ 0 0
$$771$$ 18672.0 0.872186
$$772$$ 8632.00i 0.402425i
$$773$$ 9102.00i 0.423514i 0.977322 + 0.211757i $$0.0679185\pi$$
−0.977322 + 0.211757i $$0.932081\pi$$
$$774$$ −2368.00 −0.109969
$$775$$ 0 0
$$776$$ 8848.00 0.409310
$$777$$ 1088.00i 0.0502340i
$$778$$ − 26820.0i − 1.23592i
$$779$$ −43800.0 −2.01450
$$780$$ 0 0
$$781$$ 5184.00 0.237514
$$782$$ − 17424.0i − 0.796779i
$$783$$ 7200.00i 0.328617i
$$784$$ 5232.00 0.238338
$$785$$ 0 0
$$786$$ −2112.00 −0.0958429
$$787$$ 25504.0i 1.15517i 0.816330 + 0.577585i $$0.196005\pi$$
−0.816330 + 0.577585i $$0.803995\pi$$
$$788$$ − 4296.00i − 0.194212i
$$789$$ −31584.0 −1.42512
$$790$$ 0 0
$$791$$ 4152.00 0.186635
$$792$$ − 3552.00i − 0.159362i
$$793$$ − 52316.0i − 2.34274i
$$794$$ 26228.0 1.17229
$$795$$ 0 0
$$796$$ 11360.0 0.505835
$$797$$ − 14166.0i − 0.629593i −0.949159 0.314796i $$-0.898064\pi$$
0.949159 0.314796i $$-0.101936\pi$$
$$798$$ − 6400.00i − 0.283907i
$$799$$ 13464.0 0.596148
$$800$$ 0 0
$$801$$ 29970.0 1.32202
$$802$$ 11676.0i 0.514082i
$$803$$ 4344.00i 0.190905i
$$804$$ −32768.0 −1.43736
$$805$$ 0 0
$$806$$ −17632.0 −0.770547
$$807$$ 12720.0i 0.554852i
$$808$$ − 2064.00i − 0.0898654i
$$809$$ −33210.0 −1.44327 −0.721633 0.692276i $$-0.756608\pi$$
−0.721633 + 0.692276i $$0.756608\pi$$
$$810$$ 0 0
$$811$$ 39212.0 1.69780 0.848902 0.528550i $$-0.177264\pi$$
0.848902 + 0.528550i $$0.177264\pi$$
$$812$$ − 1440.00i − 0.0622341i
$$813$$ − 39616.0i − 1.70897i
$$814$$ 816.000 0.0351361
$$815$$ 0 0
$$816$$ −8448.00 −0.362425
$$817$$ 3200.00i 0.137030i
$$818$$ 19060.0i 0.814691i
$$819$$ −8584.00 −0.366238
$$820$$ 0 0
$$821$$ 6222.00 0.264494 0.132247 0.991217i $$-0.457781\pi$$
0.132247 + 0.991217i $$0.457781\pi$$
$$822$$ − 20064.0i − 0.851353i
$$823$$ 31172.0i 1.32028i 0.751144 + 0.660138i $$0.229502\pi$$
−0.751144 + 0.660138i $$0.770498\pi$$
$$824$$ 7904.00 0.334161
$$825$$ 0 0
$$826$$ −3360.00 −0.141537
$$827$$ 264.000i 0.0111006i 0.999985 + 0.00555029i $$0.00176672\pi$$
−0.999985 + 0.00555029i $$0.998233\pi$$
$$828$$ 19536.0i 0.819955i
$$829$$ 29050.0 1.21707 0.608533 0.793528i $$-0.291758\pi$$
0.608533 + 0.793528i $$0.291758\pi$$
$$830$$ 0 0
$$831$$ −13168.0 −0.549691
$$832$$ 3712.00i 0.154676i
$$833$$ − 21582.0i − 0.897685i
$$834$$ −45760.0 −1.89993
$$835$$ 0 0
$$836$$ −4800.00 −0.198578
$$837$$ 12160.0i 0.502164i
$$838$$ 14520.0i 0.598550i
$$839$$ 21720.0 0.893752 0.446876 0.894596i $$-0.352537\pi$$
0.446876 + 0.894596i $$0.352537\pi$$
$$840$$ 0 0
$$841$$ −16289.0 −0.667883
$$842$$ − 24124.0i − 0.987373i
$$843$$ 9264.00i 0.378492i
$$844$$ 10672.0 0.435243
$$845$$ 0 0
$$846$$ −15096.0 −0.613488
$$847$$ − 4748.00i − 0.192613i
$$848$$ 3552.00i 0.143840i
$$849$$ 55936.0 2.26115
$$850$$ 0 0
$$851$$ −4488.00 −0.180783
$$852$$ 13824.0i 0.555871i
$$853$$ − 6658.00i − 0.267252i −0.991032 0.133626i $$-0.957338\pi$$
0.991032 0.133626i $$-0.0426620\pi$$
$$854$$ 7216.00 0.289141
$$855$$ 0 0
$$856$$ −192.000 −0.00766638
$$857$$ 13974.0i 0.556993i 0.960437 + 0.278496i $$0.0898360\pi$$
−0.960437 + 0.278496i $$0.910164\pi$$
$$858$$ 11136.0i 0.443096i
$$859$$ −23780.0 −0.944544 −0.472272 0.881453i $$-0.656566\pi$$
−0.472272 + 0.881453i $$0.656566\pi$$
$$860$$ 0 0
$$861$$ −14016.0 −0.554778
$$862$$ 27216.0i 1.07538i
$$863$$ − 12228.0i − 0.482324i −0.970485 0.241162i $$-0.922471\pi$$
0.970485 0.241162i $$-0.0775286\pi$$
$$864$$ 2560.00 0.100802
$$865$$ 0 0
$$866$$ −7676.00 −0.301202
$$867$$ − 4456.00i − 0.174549i
$$868$$ − 2432.00i − 0.0951008i
$$869$$ 1920.00 0.0749500
$$870$$ 0 0
$$871$$ 59392.0 2.31047
$$872$$ − 7600.00i − 0.295147i
$$873$$ 40922.0i 1.58648i
$$874$$ 26400.0 1.02173
$$875$$ 0 0
$$876$$ −11584.0 −0.446789
$$877$$ − 11606.0i − 0.446872i −0.974719 0.223436i $$-0.928273\pi$$
0.974719 0.223436i $$-0.0717274\pi$$
$$878$$ 14800.0i 0.568879i
$$879$$ −2064.00 −0.0792002
$$880$$ 0 0
$$881$$ −32958.0 −1.26037 −0.630183 0.776446i $$-0.717020\pi$$
−0.630183 + 0.776446i $$0.717020\pi$$
$$882$$ 24198.0i 0.923797i
$$883$$ 8072.00i 0.307638i 0.988099 + 0.153819i $$0.0491573\pi$$
−0.988099 + 0.153819i $$0.950843\pi$$
$$884$$ 15312.0 0.582577
$$885$$ 0 0
$$886$$ 16704.0 0.633388
$$887$$ − 15756.0i − 0.596431i −0.954498 0.298216i $$-0.903609\pi$$
0.954498 0.298216i $$-0.0963915\pi$$
$$888$$ 2176.00i 0.0822317i
$$889$$ −496.000 −0.0187124
$$890$$ 0 0
$$891$$ −4308.00 −0.161979
$$892$$ − 7088.00i − 0.266058i
$$893$$ 20400.0i 0.764457i
$$894$$ 12000.0 0.448926
$$895$$ 0 0
$$896$$ −512.000 −0.0190901
$$897$$ − 61248.0i − 2.27983i
$$898$$ 21540.0i 0.800444i
$$899$$ 13680.0 0.507512
$$900$$ 0 0
$$901$$ 14652.0 0.541763
$$902$$ 10512.0i 0.388039i
$$903$$ 1024.00i 0.0377371i
$$904$$ 8304.00 0.305517
$$905$$ 0 0
$$906$$ 7168.00 0.262849
$$907$$ − 18776.0i − 0.687372i −0.939085 0.343686i $$-0.888324\pi$$
0.939085 0.343686i $$-0.111676\pi$$
$$908$$ − 11136.0i − 0.407006i
$$909$$ 9546.00 0.348318
$$910$$ 0 0
$$911$$ −20568.0 −0.748022 −0.374011 0.927424i $$-0.622018\pi$$
−0.374011 + 0.927424i $$0.622018\pi$$
$$912$$ − 12800.0i − 0.464748i
$$913$$ 864.000i 0.0313190i
$$914$$ 13388.0 0.484503
$$915$$ 0 0
$$916$$ 1400.00 0.0504992
$$917$$ 528.000i 0.0190143i
$$918$$ − 10560.0i − 0.379664i
$$919$$ 6280.00 0.225417 0.112708 0.993628i $$-0.464047\pi$$
0.112708 + 0.993628i $$0.464047\pi$$
$$920$$ 0 0
$$921$$ 71552.0 2.55996
$$922$$ 6036.00i 0.215602i
$$923$$ − 25056.0i − 0.893530i
$$924$$ −1536.00 −0.0546869
$$925$$ 0 0
$$926$$ 28984.0 1.02859
$$927$$ 36556.0i 1.29521i
$$928$$ − 2880.00i − 0.101876i
$$929$$ 20430.0 0.721514 0.360757 0.932660i $$-0.382518\pi$$
0.360757 + 0.932660i $$0.382518\pi$$
$$930$$ 0 0
$$931$$ 32700.0 1.15113
$$932$$ − 7848.00i − 0.275826i
$$933$$ − 11136.0i − 0.390757i
$$934$$ −15552.0 −0.544836
$$935$$ 0 0
$$936$$ −17168.0 −0.599523
$$937$$ − 8906.00i − 0.310508i −0.987875 0.155254i $$-0.950380\pi$$
0.987875 0.155254i $$-0.0496197\pi$$
$$938$$ 8192.00i 0.285158i
$$939$$ −47024.0 −1.63426
$$940$$ 0 0
$$941$$ −17418.0 −0.603412 −0.301706 0.953401i $$-0.597556\pi$$
−0.301706 + 0.953401i $$0.597556\pi$$
$$942$$ 35936.0i 1.24295i
$$943$$ − 57816.0i − 1.99655i
$$944$$ −6720.00 −0.231692
$$945$$ 0 0
$$946$$ 768.000 0.0263952
$$947$$ 2544.00i 0.0872956i 0.999047 + 0.0436478i $$0.0138979\pi$$
−0.999047 + 0.0436478i $$0.986102\pi$$
$$948$$ 5120.00i 0.175411i
$$949$$ 20996.0 0.718187
$$950$$ 0 0
$$951$$ −82608.0 −2.81677
$$952$$ 2112.00i 0.0719016i
$$953$$ 15402.0i 0.523525i 0.965132 + 0.261763i $$0.0843038\pi$$
−0.965132 + 0.261763i $$0.915696\pi$$
$$954$$ −16428.0 −0.557522
$$955$$ 0 0
$$956$$ −17280.0 −0.584597
$$957$$ − 8640.00i − 0.291841i
$$958$$ − 27360.0i − 0.922716i
$$959$$ −5016.00 −0.168900
$$960$$ 0 0
$$961$$ −6687.00 −0.224464
$$962$$ − 3944.00i − 0.132183i
$$963$$ − 888.000i − 0.0297148i
$$964$$ 1912.00 0.0638811
$$965$$ 0 0
$$966$$ 8448.00 0.281377
$$967$$ 49444.0i 1.64427i 0.569291 + 0.822136i $$0.307218\pi$$
−0.569291 + 0.822136i $$0.692782\pi$$
$$968$$ − 9496.00i − 0.315303i
$$969$$ −52800.0 −1.75044
$$970$$ 0 0
$$971$$ −25188.0 −0.832463 −0.416231 0.909259i $$-0.636649\pi$$
−0.416231 + 0.909259i $$0.636649\pi$$
$$972$$ − 20128.0i − 0.664204i
$$973$$ 11440.0i 0.376927i
$$974$$ −15832.0 −0.520832
$$975$$ 0 0
$$976$$ 14432.0 0.473317
$$977$$ − 2946.00i − 0.0964697i −0.998836 0.0482348i $$-0.984640\pi$$
0.998836 0.0482348i $$-0.0153596\pi$$
$$978$$ 9088.00i 0.297139i
$$979$$ −9720.00 −0.317316
$$980$$ 0 0
$$981$$ 35150.0 1.14399
$$982$$ − 27864.0i − 0.905475i
$$983$$ 15012.0i 0.487089i 0.969890 + 0.243544i $$0.0783102\pi$$
−0.969890 + 0.243544i $$0.921690\pi$$
$$984$$ −28032.0 −0.908158
$$985$$ 0 0
$$986$$ −11880.0 −0.383708
$$987$$ 6528.00i 0.210525i
$$988$$ 23200.0i 0.747055i
$$989$$ −4224.00 −0.135809
$$990$$ 0 0
$$991$$ −5128.00 −0.164376 −0.0821878 0.996617i $$-0.526191\pi$$
−0.0821878 + 0.996617i $$0.526191\pi$$
$$992$$ − 4864.00i − 0.155678i
$$993$$ 33824.0i 1.08094i
$$994$$ 3456.00 0.110279
$$995$$ 0 0
$$996$$ −2304.00 −0.0732982
$$997$$ 49714.0i 1.57920i 0.613625 + 0.789598i $$0.289711\pi$$
−0.613625 + 0.789598i $$0.710289\pi$$
$$998$$ − 16520.0i − 0.523979i
$$999$$ −2720.00 −0.0861431
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 50.4.b.a.49.1 2
3.2 odd 2 450.4.c.d.199.2 2
4.3 odd 2 400.4.c.c.49.2 2
5.2 odd 4 10.4.a.a.1.1 1
5.3 odd 4 50.4.a.c.1.1 1
5.4 even 2 inner 50.4.b.a.49.2 2
15.2 even 4 90.4.a.a.1.1 1
15.8 even 4 450.4.a.q.1.1 1
15.14 odd 2 450.4.c.d.199.1 2
20.3 even 4 400.4.a.b.1.1 1
20.7 even 4 80.4.a.f.1.1 1
20.19 odd 2 400.4.c.c.49.1 2
35.2 odd 12 490.4.e.i.361.1 2
35.12 even 12 490.4.e.a.361.1 2
35.13 even 4 2450.4.a.b.1.1 1
35.17 even 12 490.4.e.a.471.1 2
35.27 even 4 490.4.a.o.1.1 1
35.32 odd 12 490.4.e.i.471.1 2
40.3 even 4 1600.4.a.bx.1.1 1
40.13 odd 4 1600.4.a.d.1.1 1
40.27 even 4 320.4.a.b.1.1 1
40.37 odd 4 320.4.a.m.1.1 1
45.2 even 12 810.4.e.w.271.1 2
45.7 odd 12 810.4.e.c.271.1 2
45.22 odd 12 810.4.e.c.541.1 2
45.32 even 12 810.4.e.w.541.1 2
55.32 even 4 1210.4.a.b.1.1 1
60.47 odd 4 720.4.a.j.1.1 1
65.12 odd 4 1690.4.a.a.1.1 1
80.27 even 4 1280.4.d.g.641.1 2
80.37 odd 4 1280.4.d.j.641.2 2
80.67 even 4 1280.4.d.g.641.2 2
80.77 odd 4 1280.4.d.j.641.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
10.4.a.a.1.1 1 5.2 odd 4
50.4.a.c.1.1 1 5.3 odd 4
50.4.b.a.49.1 2 1.1 even 1 trivial
50.4.b.a.49.2 2 5.4 even 2 inner
80.4.a.f.1.1 1 20.7 even 4
90.4.a.a.1.1 1 15.2 even 4
320.4.a.b.1.1 1 40.27 even 4
320.4.a.m.1.1 1 40.37 odd 4
400.4.a.b.1.1 1 20.3 even 4
400.4.c.c.49.1 2 20.19 odd 2
400.4.c.c.49.2 2 4.3 odd 2
450.4.a.q.1.1 1 15.8 even 4
450.4.c.d.199.1 2 15.14 odd 2
450.4.c.d.199.2 2 3.2 odd 2
490.4.a.o.1.1 1 35.27 even 4
490.4.e.a.361.1 2 35.12 even 12
490.4.e.a.471.1 2 35.17 even 12
490.4.e.i.361.1 2 35.2 odd 12
490.4.e.i.471.1 2 35.32 odd 12
720.4.a.j.1.1 1 60.47 odd 4
810.4.e.c.271.1 2 45.7 odd 12
810.4.e.c.541.1 2 45.22 odd 12
810.4.e.w.271.1 2 45.2 even 12
810.4.e.w.541.1 2 45.32 even 12
1210.4.a.b.1.1 1 55.32 even 4
1280.4.d.g.641.1 2 80.27 even 4
1280.4.d.g.641.2 2 80.67 even 4
1280.4.d.j.641.1 2 80.77 odd 4
1280.4.d.j.641.2 2 80.37 odd 4
1600.4.a.d.1.1 1 40.13 odd 4
1600.4.a.bx.1.1 1 40.3 even 4
1690.4.a.a.1.1 1 65.12 odd 4
2450.4.a.b.1.1 1 35.13 even 4