Properties

 Label 825.4.c.i.199.4 Level $825$ Weight $4$ Character 825.199 Analytic conductor $48.677$ Analytic rank $0$ Dimension $4$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,4,Mod(199,825)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("825.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 825.c (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: $$48.6765757547$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 17x^{2} + 64$$ x^4 + 17*x^2 + 64 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 199.4 Root $$3.37228i$$ of defining polynomial Character $$\chi$$ $$=$$ 825.199 Dual form 825.4.c.i.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.37228i q^{2} -3.00000i q^{3} -3.37228 q^{4} +10.1168 q^{6} -4.74456i q^{7} +15.6060i q^{8} -9.00000 q^{9} +O(q^{10})$$ $$q+3.37228i q^{2} -3.00000i q^{3} -3.37228 q^{4} +10.1168 q^{6} -4.74456i q^{7} +15.6060i q^{8} -9.00000 q^{9} +11.0000 q^{11} +10.1168i q^{12} +15.0217i q^{13} +16.0000 q^{14} -79.6060 q^{16} +73.1684i q^{17} -30.3505i q^{18} +78.7011 q^{19} -14.2337 q^{21} +37.0951i q^{22} -112.000i q^{23} +46.8179 q^{24} -50.6576 q^{26} +27.0000i q^{27} +16.0000i q^{28} -243.125 q^{29} +278.717 q^{31} -143.606i q^{32} -33.0000i q^{33} -246.745 q^{34} +30.3505 q^{36} +102.380i q^{37} +265.402i q^{38} +45.0652 q^{39} -241.255 q^{41} -48.0000i q^{42} +280.016i q^{43} -37.0951 q^{44} +377.696 q^{46} -169.870i q^{47} +238.818i q^{48} +320.489 q^{49} +219.505 q^{51} -50.6576i q^{52} +409.652i q^{53} -91.0516 q^{54} +74.0435 q^{56} -236.103i q^{57} -819.886i q^{58} -196.000 q^{59} -701.359 q^{61} +939.913i q^{62} +42.7011i q^{63} -152.568 q^{64} +111.285 q^{66} +900.587i q^{67} -246.745i q^{68} -336.000 q^{69} +756.500 q^{71} -140.454i q^{72} +1019.81i q^{73} -345.255 q^{74} -265.402 q^{76} -52.1902i q^{77} +151.973i q^{78} +327.549 q^{79} +81.0000 q^{81} -813.581i q^{82} +756.619i q^{83} +48.0000 q^{84} -944.293 q^{86} +729.375i q^{87} +171.666i q^{88} -508.978 q^{89} +71.2716 q^{91} +377.696i q^{92} -836.152i q^{93} +572.848 q^{94} -430.818 q^{96} +614.358i q^{97} +1080.78i q^{98} -99.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{4} + 6 q^{6} - 36 q^{9}+O(q^{10})$$ 4 * q - 2 * q^4 + 6 * q^6 - 36 * q^9 $$4 q - 2 q^{4} + 6 q^{6} - 36 q^{9} + 44 q^{11} + 64 q^{14} - 238 q^{16} + 108 q^{19} + 12 q^{21} - 54 q^{24} + 188 q^{26} - 444 q^{29} - 80 q^{31} - 964 q^{34} + 18 q^{36} + 456 q^{39} - 988 q^{41} - 22 q^{44} + 224 q^{46} + 1236 q^{49} - 156 q^{51} - 54 q^{54} + 480 q^{56} - 784 q^{59} - 2208 q^{61} - 1426 q^{64} + 66 q^{66} - 1344 q^{69} + 912 q^{71} - 1404 q^{74} - 648 q^{76} + 460 q^{79} + 324 q^{81} + 192 q^{84} - 2904 q^{86} - 1944 q^{89} - 680 q^{91} + 1648 q^{94} - 1482 q^{96} - 396 q^{99}+O(q^{100})$$ 4 * q - 2 * q^4 + 6 * q^6 - 36 * q^9 + 44 * q^11 + 64 * q^14 - 238 * q^16 + 108 * q^19 + 12 * q^21 - 54 * q^24 + 188 * q^26 - 444 * q^29 - 80 * q^31 - 964 * q^34 + 18 * q^36 + 456 * q^39 - 988 * q^41 - 22 * q^44 + 224 * q^46 + 1236 * q^49 - 156 * q^51 - 54 * q^54 + 480 * q^56 - 784 * q^59 - 2208 * q^61 - 1426 * q^64 + 66 * q^66 - 1344 * q^69 + 912 * q^71 - 1404 * q^74 - 648 * q^76 + 460 * q^79 + 324 * q^81 + 192 * q^84 - 2904 * q^86 - 1944 * q^89 - 680 * q^91 + 1648 * q^94 - 1482 * q^96 - 396 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$$.

 $$n$$ $$376$$ $$551$$ $$727$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.37228i 1.19228i 0.802880 + 0.596141i $$0.203300\pi$$
−0.802880 + 0.596141i $$0.796700\pi$$
$$3$$ − 3.00000i − 0.577350i
$$4$$ −3.37228 −0.421535
$$5$$ 0 0
$$6$$ 10.1168 0.688364
$$7$$ − 4.74456i − 0.256182i −0.991762 0.128091i $$-0.959115\pi$$
0.991762 0.128091i $$-0.0408850\pi$$
$$8$$ 15.6060i 0.689693i
$$9$$ −9.00000 −0.333333
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 10.1168i 0.243373i
$$13$$ 15.0217i 0.320483i 0.987078 + 0.160242i $$0.0512274\pi$$
−0.987078 + 0.160242i $$0.948773\pi$$
$$14$$ 16.0000 0.305441
$$15$$ 0 0
$$16$$ −79.6060 −1.24384
$$17$$ 73.1684i 1.04388i 0.852982 + 0.521940i $$0.174791\pi$$
−0.852982 + 0.521940i $$0.825209\pi$$
$$18$$ − 30.3505i − 0.397427i
$$19$$ 78.7011 0.950277 0.475138 0.879911i $$-0.342398\pi$$
0.475138 + 0.879911i $$0.342398\pi$$
$$20$$ 0 0
$$21$$ −14.2337 −0.147907
$$22$$ 37.0951i 0.359486i
$$23$$ − 112.000i − 1.01537i −0.861541 0.507687i $$-0.830501\pi$$
0.861541 0.507687i $$-0.169499\pi$$
$$24$$ 46.8179 0.398194
$$25$$ 0 0
$$26$$ −50.6576 −0.382106
$$27$$ 27.0000i 0.192450i
$$28$$ 16.0000i 0.107990i
$$29$$ −243.125 −1.55680 −0.778399 0.627769i $$-0.783968\pi$$
−0.778399 + 0.627769i $$0.783968\pi$$
$$30$$ 0 0
$$31$$ 278.717 1.61481 0.807405 0.589998i $$-0.200871\pi$$
0.807405 + 0.589998i $$0.200871\pi$$
$$32$$ − 143.606i − 0.793318i
$$33$$ − 33.0000i − 0.174078i
$$34$$ −246.745 −1.24460
$$35$$ 0 0
$$36$$ 30.3505 0.140512
$$37$$ 102.380i 0.454898i 0.973790 + 0.227449i $$0.0730385\pi$$
−0.973790 + 0.227449i $$0.926961\pi$$
$$38$$ 265.402i 1.13300i
$$39$$ 45.0652 0.185031
$$40$$ 0 0
$$41$$ −241.255 −0.918970 −0.459485 0.888186i $$-0.651966\pi$$
−0.459485 + 0.888186i $$0.651966\pi$$
$$42$$ − 48.0000i − 0.176347i
$$43$$ 280.016i 0.993071i 0.868016 + 0.496536i $$0.165395\pi$$
−0.868016 + 0.496536i $$0.834605\pi$$
$$44$$ −37.0951 −0.127098
$$45$$ 0 0
$$46$$ 377.696 1.21061
$$47$$ − 169.870i − 0.527192i −0.964633 0.263596i $$-0.915091\pi$$
0.964633 0.263596i $$-0.0849085\pi$$
$$48$$ 238.818i 0.718133i
$$49$$ 320.489 0.934371
$$50$$ 0 0
$$51$$ 219.505 0.602684
$$52$$ − 50.6576i − 0.135095i
$$53$$ 409.652i 1.06170i 0.847466 + 0.530849i $$0.178127\pi$$
−0.847466 + 0.530849i $$0.821873\pi$$
$$54$$ −91.0516 −0.229455
$$55$$ 0 0
$$56$$ 74.0435 0.176687
$$57$$ − 236.103i − 0.548643i
$$58$$ − 819.886i − 1.85614i
$$59$$ −196.000 −0.432492 −0.216246 0.976339i $$-0.569381\pi$$
−0.216246 + 0.976339i $$0.569381\pi$$
$$60$$ 0 0
$$61$$ −701.359 −1.47213 −0.736064 0.676912i $$-0.763318\pi$$
−0.736064 + 0.676912i $$0.763318\pi$$
$$62$$ 939.913i 1.92531i
$$63$$ 42.7011i 0.0853941i
$$64$$ −152.568 −0.297984
$$65$$ 0 0
$$66$$ 111.285 0.207550
$$67$$ 900.587i 1.64215i 0.570819 + 0.821076i $$0.306626\pi$$
−0.570819 + 0.821076i $$0.693374\pi$$
$$68$$ − 246.745i − 0.440032i
$$69$$ −336.000 −0.586227
$$70$$ 0 0
$$71$$ 756.500 1.26451 0.632254 0.774762i $$-0.282130\pi$$
0.632254 + 0.774762i $$0.282130\pi$$
$$72$$ − 140.454i − 0.229898i
$$73$$ 1019.81i 1.63507i 0.575877 + 0.817536i $$0.304661\pi$$
−0.575877 + 0.817536i $$0.695339\pi$$
$$74$$ −345.255 −0.542367
$$75$$ 0 0
$$76$$ −265.402 −0.400575
$$77$$ − 52.1902i − 0.0772419i
$$78$$ 151.973i 0.220609i
$$79$$ 327.549 0.466483 0.233241 0.972419i $$-0.425067\pi$$
0.233241 + 0.972419i $$0.425067\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ − 813.581i − 1.09567i
$$83$$ 756.619i 1.00060i 0.865852 + 0.500300i $$0.166777\pi$$
−0.865852 + 0.500300i $$0.833223\pi$$
$$84$$ 48.0000 0.0623480
$$85$$ 0 0
$$86$$ −944.293 −1.18402
$$87$$ 729.375i 0.898818i
$$88$$ 171.666i 0.207950i
$$89$$ −508.978 −0.606198 −0.303099 0.952959i $$-0.598021\pi$$
−0.303099 + 0.952959i $$0.598021\pi$$
$$90$$ 0 0
$$91$$ 71.2716 0.0821022
$$92$$ 377.696i 0.428016i
$$93$$ − 836.152i − 0.932311i
$$94$$ 572.848 0.628561
$$95$$ 0 0
$$96$$ −430.818 −0.458023
$$97$$ 614.358i 0.643079i 0.946896 + 0.321539i $$0.104200\pi$$
−0.946896 + 0.321539i $$0.895800\pi$$
$$98$$ 1080.78i 1.11403i
$$99$$ −99.0000 −0.100504
$$100$$ 0 0
$$101$$ −1015.92 −1.00087 −0.500434 0.865775i $$-0.666826\pi$$
−0.500434 + 0.865775i $$0.666826\pi$$
$$102$$ 740.234i 0.718569i
$$103$$ − 1102.16i − 1.05436i −0.849753 0.527181i $$-0.823249\pi$$
0.849753 0.527181i $$-0.176751\pi$$
$$104$$ −234.429 −0.221035
$$105$$ 0 0
$$106$$ −1381.46 −1.26584
$$107$$ 1377.58i 1.24463i 0.782767 + 0.622315i $$0.213808\pi$$
−0.782767 + 0.622315i $$0.786192\pi$$
$$108$$ − 91.0516i − 0.0811245i
$$109$$ −320.217 −0.281388 −0.140694 0.990053i $$-0.544933\pi$$
−0.140694 + 0.990053i $$0.544933\pi$$
$$110$$ 0 0
$$111$$ 307.141 0.262636
$$112$$ 377.696i 0.318651i
$$113$$ 1629.45i 1.35651i 0.734828 + 0.678254i $$0.237263\pi$$
−0.734828 + 0.678254i $$0.762737\pi$$
$$114$$ 796.206 0.654136
$$115$$ 0 0
$$116$$ 819.886 0.656245
$$117$$ − 135.196i − 0.106828i
$$118$$ − 660.967i − 0.515652i
$$119$$ 347.152 0.267423
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ − 2365.18i − 1.75519i
$$123$$ 723.766i 0.530568i
$$124$$ −939.913 −0.680699
$$125$$ 0 0
$$126$$ −144.000 −0.101814
$$127$$ 2291.26i 1.60091i 0.599390 + 0.800457i $$0.295410\pi$$
−0.599390 + 0.800457i $$0.704590\pi$$
$$128$$ − 1663.35i − 1.14860i
$$129$$ 840.049 0.573350
$$130$$ 0 0
$$131$$ −1147.41 −0.765267 −0.382633 0.923900i $$-0.624983\pi$$
−0.382633 + 0.923900i $$0.624983\pi$$
$$132$$ 111.285i 0.0733799i
$$133$$ − 373.402i − 0.243444i
$$134$$ −3037.03 −1.95791
$$135$$ 0 0
$$136$$ −1141.86 −0.719956
$$137$$ 1268.60i 0.791121i 0.918440 + 0.395561i $$0.129450\pi$$
−0.918440 + 0.395561i $$0.870550\pi$$
$$138$$ − 1133.09i − 0.698947i
$$139$$ 486.288 0.296737 0.148368 0.988932i $$-0.452598\pi$$
0.148368 + 0.988932i $$0.452598\pi$$
$$140$$ 0 0
$$141$$ −509.609 −0.304374
$$142$$ 2551.13i 1.50765i
$$143$$ 165.239i 0.0966294i
$$144$$ 716.454 0.414614
$$145$$ 0 0
$$146$$ −3439.10 −1.94947
$$147$$ − 961.467i − 0.539459i
$$148$$ − 345.255i − 0.191756i
$$149$$ −2354.11 −1.29434 −0.647169 0.762346i $$-0.724047\pi$$
−0.647169 + 0.762346i $$0.724047\pi$$
$$150$$ 0 0
$$151$$ −570.070 −0.307229 −0.153615 0.988131i $$-0.549091\pi$$
−0.153615 + 0.988131i $$0.549091\pi$$
$$152$$ 1228.21i 0.655399i
$$153$$ − 658.516i − 0.347960i
$$154$$ 176.000 0.0920941
$$155$$ 0 0
$$156$$ −151.973 −0.0779971
$$157$$ − 2072.67i − 1.05361i −0.849985 0.526807i $$-0.823389\pi$$
0.849985 0.526807i $$-0.176611\pi$$
$$158$$ 1104.59i 0.556179i
$$159$$ 1228.96 0.612972
$$160$$ 0 0
$$161$$ −531.391 −0.260121
$$162$$ 273.155i 0.132476i
$$163$$ − 2676.51i − 1.28614i −0.765808 0.643069i $$-0.777661\pi$$
0.765808 0.643069i $$-0.222339\pi$$
$$164$$ 813.581 0.387378
$$165$$ 0 0
$$166$$ −2551.53 −1.19300
$$167$$ − 1188.12i − 0.550536i −0.961368 0.275268i $$-0.911233\pi$$
0.961368 0.275268i $$-0.0887665\pi$$
$$168$$ − 222.130i − 0.102010i
$$169$$ 1971.35 0.897290
$$170$$ 0 0
$$171$$ −708.310 −0.316759
$$172$$ − 944.293i − 0.418615i
$$173$$ − 807.147i − 0.354718i −0.984146 0.177359i $$-0.943245\pi$$
0.984146 0.177359i $$-0.0567554\pi$$
$$174$$ −2459.66 −1.07164
$$175$$ 0 0
$$176$$ −875.666 −0.375033
$$177$$ 588.000i 0.249699i
$$178$$ − 1716.42i − 0.722758i
$$179$$ 1950.39 0.814408 0.407204 0.913337i $$-0.366504\pi$$
0.407204 + 0.913337i $$0.366504\pi$$
$$180$$ 0 0
$$181$$ 1061.61 0.435959 0.217980 0.975953i $$-0.430053\pi$$
0.217980 + 0.975953i $$0.430053\pi$$
$$182$$ 240.348i 0.0978889i
$$183$$ 2104.08i 0.849933i
$$184$$ 1747.87 0.700297
$$185$$ 0 0
$$186$$ 2819.74 1.11158
$$187$$ 804.853i 0.314742i
$$188$$ 572.848i 0.222230i
$$189$$ 128.103 0.0493023
$$190$$ 0 0
$$191$$ 2136.41 0.809348 0.404674 0.914461i $$-0.367385\pi$$
0.404674 + 0.914461i $$0.367385\pi$$
$$192$$ 457.704i 0.172041i
$$193$$ − 3947.76i − 1.47236i −0.676784 0.736181i $$-0.736627\pi$$
0.676784 0.736181i $$-0.263373\pi$$
$$194$$ −2071.79 −0.766731
$$195$$ 0 0
$$196$$ −1080.78 −0.393870
$$197$$ 923.886i 0.334133i 0.985946 + 0.167066i $$0.0534294\pi$$
−0.985946 + 0.167066i $$0.946571\pi$$
$$198$$ − 333.856i − 0.119829i
$$199$$ 476.152 0.169616 0.0848078 0.996397i $$-0.472972\pi$$
0.0848078 + 0.996397i $$0.472972\pi$$
$$200$$ 0 0
$$201$$ 2701.76 0.948097
$$202$$ − 3425.96i − 1.19332i
$$203$$ 1153.52i 0.398824i
$$204$$ −740.234 −0.254053
$$205$$ 0 0
$$206$$ 3716.80 1.25710
$$207$$ 1008.00i 0.338458i
$$208$$ − 1195.82i − 0.398631i
$$209$$ 865.712 0.286519
$$210$$ 0 0
$$211$$ −4918.24 −1.60467 −0.802336 0.596872i $$-0.796410\pi$$
−0.802336 + 0.596872i $$0.796410\pi$$
$$212$$ − 1381.46i − 0.447543i
$$213$$ − 2269.50i − 0.730064i
$$214$$ −4645.57 −1.48395
$$215$$ 0 0
$$216$$ −421.361 −0.132731
$$217$$ − 1322.39i − 0.413686i
$$218$$ − 1079.86i − 0.335494i
$$219$$ 3059.44 0.944010
$$220$$ 0 0
$$221$$ −1099.12 −0.334546
$$222$$ 1035.77i 0.313136i
$$223$$ − 2100.29i − 0.630700i −0.948975 0.315350i $$-0.897878\pi$$
0.948975 0.315350i $$-0.102122\pi$$
$$224$$ −681.348 −0.203234
$$225$$ 0 0
$$226$$ −5494.95 −1.61734
$$227$$ − 2257.16i − 0.659970i −0.943986 0.329985i $$-0.892956\pi$$
0.943986 0.329985i $$-0.107044\pi$$
$$228$$ 796.206i 0.231272i
$$229$$ 5311.07 1.53260 0.766301 0.642482i $$-0.222095\pi$$
0.766301 + 0.642482i $$0.222095\pi$$
$$230$$ 0 0
$$231$$ −156.571 −0.0445956
$$232$$ − 3794.20i − 1.07371i
$$233$$ − 2466.27i − 0.693435i −0.937970 0.346718i $$-0.887296\pi$$
0.937970 0.346718i $$-0.112704\pi$$
$$234$$ 455.918 0.127369
$$235$$ 0 0
$$236$$ 660.967 0.182311
$$237$$ − 982.646i − 0.269324i
$$238$$ 1170.70i 0.318844i
$$239$$ −1429.40 −0.386863 −0.193432 0.981114i $$-0.561962\pi$$
−0.193432 + 0.981114i $$0.561962\pi$$
$$240$$ 0 0
$$241$$ −978.989 −0.261669 −0.130835 0.991404i $$-0.541766\pi$$
−0.130835 + 0.991404i $$0.541766\pi$$
$$242$$ 408.046i 0.108389i
$$243$$ − 243.000i − 0.0641500i
$$244$$ 2365.18 0.620553
$$245$$ 0 0
$$246$$ −2440.74 −0.632586
$$247$$ 1182.23i 0.304548i
$$248$$ 4349.65i 1.11372i
$$249$$ 2269.86 0.577696
$$250$$ 0 0
$$251$$ −6530.63 −1.64227 −0.821135 0.570734i $$-0.806659\pi$$
−0.821135 + 0.570734i $$0.806659\pi$$
$$252$$ − 144.000i − 0.0359966i
$$253$$ − 1232.00i − 0.306147i
$$254$$ −7726.76 −1.90874
$$255$$ 0 0
$$256$$ 4388.74 1.07147
$$257$$ 8130.26i 1.97335i 0.162696 + 0.986676i $$0.447981\pi$$
−0.162696 + 0.986676i $$0.552019\pi$$
$$258$$ 2832.88i 0.683595i
$$259$$ 485.750 0.116537
$$260$$ 0 0
$$261$$ 2188.12 0.518933
$$262$$ − 3869.40i − 0.912414i
$$263$$ 4549.42i 1.06665i 0.845910 + 0.533326i $$0.179058\pi$$
−0.845910 + 0.533326i $$0.820942\pi$$
$$264$$ 514.997 0.120060
$$265$$ 0 0
$$266$$ 1259.22 0.290254
$$267$$ 1526.93i 0.349988i
$$268$$ − 3037.03i − 0.692225i
$$269$$ 29.1522 0.00660760 0.00330380 0.999995i $$-0.498948\pi$$
0.00330380 + 0.999995i $$0.498948\pi$$
$$270$$ 0 0
$$271$$ 7711.22 1.72850 0.864250 0.503063i $$-0.167794\pi$$
0.864250 + 0.503063i $$0.167794\pi$$
$$272$$ − 5824.64i − 1.29842i
$$273$$ − 213.815i − 0.0474017i
$$274$$ −4278.07 −0.943239
$$275$$ 0 0
$$276$$ 1133.09 0.247115
$$277$$ 1127.52i 0.244571i 0.992495 + 0.122286i $$0.0390224\pi$$
−0.992495 + 0.122286i $$0.960978\pi$$
$$278$$ 1639.90i 0.353794i
$$279$$ −2508.46 −0.538270
$$280$$ 0 0
$$281$$ −1872.47 −0.397517 −0.198758 0.980049i $$-0.563691\pi$$
−0.198758 + 0.980049i $$0.563691\pi$$
$$282$$ − 1718.54i − 0.362900i
$$283$$ − 2124.48i − 0.446245i −0.974790 0.223123i $$-0.928375\pi$$
0.974790 0.223123i $$-0.0716251\pi$$
$$284$$ −2551.13 −0.533034
$$285$$ 0 0
$$286$$ −557.233 −0.115209
$$287$$ 1144.65i 0.235424i
$$288$$ 1292.45i 0.264439i
$$289$$ −440.621 −0.0896846
$$290$$ 0 0
$$291$$ 1843.07 0.371282
$$292$$ − 3439.10i − 0.689241i
$$293$$ 3324.19i 0.662802i 0.943490 + 0.331401i $$0.107521\pi$$
−0.943490 + 0.331401i $$0.892479\pi$$
$$294$$ 3242.34 0.643187
$$295$$ 0 0
$$296$$ −1597.75 −0.313740
$$297$$ 297.000i 0.0580259i
$$298$$ − 7938.73i − 1.54322i
$$299$$ 1682.44 0.325411
$$300$$ 0 0
$$301$$ 1328.55 0.254407
$$302$$ − 1922.44i − 0.366304i
$$303$$ 3047.75i 0.577851i
$$304$$ −6265.07 −1.18200
$$305$$ 0 0
$$306$$ 2220.70 0.414866
$$307$$ − 1698.94i − 0.315843i −0.987452 0.157921i $$-0.949521\pi$$
0.987452 0.157921i $$-0.0504793\pi$$
$$308$$ 176.000i 0.0325602i
$$309$$ −3306.49 −0.608736
$$310$$ 0 0
$$311$$ 6928.83 1.26334 0.631668 0.775239i $$-0.282370\pi$$
0.631668 + 0.775239i $$0.282370\pi$$
$$312$$ 703.287i 0.127615i
$$313$$ 3560.75i 0.643020i 0.946906 + 0.321510i $$0.104190\pi$$
−0.946906 + 0.321510i $$0.895810\pi$$
$$314$$ 6989.64 1.25620
$$315$$ 0 0
$$316$$ −1104.59 −0.196639
$$317$$ 332.750i 0.0589561i 0.999565 + 0.0294780i $$0.00938451\pi$$
−0.999565 + 0.0294780i $$0.990615\pi$$
$$318$$ 4144.39i 0.730835i
$$319$$ −2674.37 −0.469393
$$320$$ 0 0
$$321$$ 4132.73 0.718587
$$322$$ − 1792.00i − 0.310137i
$$323$$ 5758.43i 0.991975i
$$324$$ −273.155 −0.0468372
$$325$$ 0 0
$$326$$ 9025.94 1.53344
$$327$$ 960.652i 0.162459i
$$328$$ − 3765.02i − 0.633807i
$$329$$ −805.957 −0.135057
$$330$$ 0 0
$$331$$ −541.445 −0.0899108 −0.0449554 0.998989i $$-0.514315\pi$$
−0.0449554 + 0.998989i $$0.514315\pi$$
$$332$$ − 2551.53i − 0.421788i
$$333$$ − 921.423i − 0.151633i
$$334$$ 4006.67 0.656393
$$335$$ 0 0
$$336$$ 1133.09 0.183973
$$337$$ 816.531i 0.131986i 0.997820 + 0.0659930i $$0.0210215\pi$$
−0.997820 + 0.0659930i $$0.978978\pi$$
$$338$$ 6647.94i 1.06982i
$$339$$ 4888.34 0.783180
$$340$$ 0 0
$$341$$ 3065.89 0.486883
$$342$$ − 2388.62i − 0.377666i
$$343$$ − 3147.97i − 0.495552i
$$344$$ −4369.92 −0.684914
$$345$$ 0 0
$$346$$ 2721.93 0.422924
$$347$$ 6260.53i 0.968539i 0.874919 + 0.484269i $$0.160914\pi$$
−0.874919 + 0.484269i $$0.839086\pi$$
$$348$$ − 2459.66i − 0.378884i
$$349$$ 12768.5 1.95840 0.979198 0.202906i $$-0.0650386\pi$$
0.979198 + 0.202906i $$0.0650386\pi$$
$$350$$ 0 0
$$351$$ −405.587 −0.0616771
$$352$$ − 1579.67i − 0.239194i
$$353$$ 2649.28i 0.399453i 0.979852 + 0.199727i $$0.0640054\pi$$
−0.979852 + 0.199727i $$0.935995\pi$$
$$354$$ −1982.90 −0.297712
$$355$$ 0 0
$$356$$ 1716.42 0.255534
$$357$$ − 1041.46i − 0.154397i
$$358$$ 6577.27i 0.971004i
$$359$$ 3203.91 0.471020 0.235510 0.971872i $$-0.424324\pi$$
0.235510 + 0.971872i $$0.424324\pi$$
$$360$$ 0 0
$$361$$ −665.143 −0.0969737
$$362$$ 3580.04i 0.519786i
$$363$$ − 363.000i − 0.0524864i
$$364$$ −240.348 −0.0346089
$$365$$ 0 0
$$366$$ −7095.54 −1.01336
$$367$$ − 8429.40i − 1.19894i −0.800397 0.599470i $$-0.795378\pi$$
0.800397 0.599470i $$-0.204622\pi$$
$$368$$ 8915.87i 1.26297i
$$369$$ 2171.30 0.306323
$$370$$ 0 0
$$371$$ 1943.62 0.271988
$$372$$ 2819.74i 0.393002i
$$373$$ 9388.53i 1.30327i 0.758533 + 0.651635i $$0.225917\pi$$
−0.758533 + 0.651635i $$0.774083\pi$$
$$374$$ −2714.19 −0.375261
$$375$$ 0 0
$$376$$ 2650.98 0.363600
$$377$$ − 3652.16i − 0.498928i
$$378$$ 432.000i 0.0587822i
$$379$$ 14264.5 1.93329 0.966647 0.256112i $$-0.0824415\pi$$
0.966647 + 0.256112i $$0.0824415\pi$$
$$380$$ 0 0
$$381$$ 6873.77 0.924288
$$382$$ 7204.58i 0.964970i
$$383$$ − 13462.2i − 1.79605i −0.439942 0.898026i $$-0.645001\pi$$
0.439942 0.898026i $$-0.354999\pi$$
$$384$$ −4990.05 −0.663144
$$385$$ 0 0
$$386$$ 13313.0 1.75547
$$387$$ − 2520.15i − 0.331024i
$$388$$ − 2071.79i − 0.271080i
$$389$$ 941.881 0.122764 0.0613821 0.998114i $$-0.480449\pi$$
0.0613821 + 0.998114i $$0.480449\pi$$
$$390$$ 0 0
$$391$$ 8194.87 1.05993
$$392$$ 5001.54i 0.644429i
$$393$$ 3442.24i 0.441827i
$$394$$ −3115.60 −0.398380
$$395$$ 0 0
$$396$$ 333.856 0.0423659
$$397$$ − 847.839i − 0.107183i −0.998563 0.0535917i $$-0.982933\pi$$
0.998563 0.0535917i $$-0.0170669\pi$$
$$398$$ 1605.72i 0.202230i
$$399$$ −1120.21 −0.140553
$$400$$ 0 0
$$401$$ 12203.6 1.51975 0.759875 0.650069i $$-0.225260\pi$$
0.759875 + 0.650069i $$0.225260\pi$$
$$402$$ 9111.10i 1.13040i
$$403$$ 4186.82i 0.517520i
$$404$$ 3425.96 0.421901
$$405$$ 0 0
$$406$$ −3890.00 −0.475511
$$407$$ 1126.18i 0.137157i
$$408$$ 3425.59i 0.415667i
$$409$$ −8759.53 −1.05900 −0.529500 0.848310i $$-0.677620\pi$$
−0.529500 + 0.848310i $$0.677620\pi$$
$$410$$ 0 0
$$411$$ 3805.79 0.456754
$$412$$ 3716.80i 0.444451i
$$413$$ 929.934i 0.110797i
$$414$$ −3399.26 −0.403537
$$415$$ 0 0
$$416$$ 2157.21 0.254245
$$417$$ − 1458.86i − 0.171321i
$$418$$ 2919.42i 0.341612i
$$419$$ 11188.4 1.30451 0.652256 0.757999i $$-0.273823\pi$$
0.652256 + 0.757999i $$0.273823\pi$$
$$420$$ 0 0
$$421$$ −14082.3 −1.63023 −0.815116 0.579298i $$-0.803327\pi$$
−0.815116 + 0.579298i $$0.803327\pi$$
$$422$$ − 16585.7i − 1.91322i
$$423$$ 1528.83i 0.175731i
$$424$$ −6393.02 −0.732246
$$425$$ 0 0
$$426$$ 7653.39 0.870441
$$427$$ 3327.64i 0.377133i
$$428$$ − 4645.57i − 0.524655i
$$429$$ 495.718 0.0557890
$$430$$ 0 0
$$431$$ −5616.05 −0.627647 −0.313823 0.949481i $$-0.601610\pi$$
−0.313823 + 0.949481i $$0.601610\pi$$
$$432$$ − 2149.36i − 0.239378i
$$433$$ − 7195.75i − 0.798627i −0.916814 0.399314i $$-0.869248\pi$$
0.916814 0.399314i $$-0.130752\pi$$
$$434$$ 4459.48 0.493230
$$435$$ 0 0
$$436$$ 1079.86 0.118615
$$437$$ − 8814.52i − 0.964887i
$$438$$ 10317.3i 1.12553i
$$439$$ −101.959 −0.0110848 −0.00554240 0.999985i $$-0.501764\pi$$
−0.00554240 + 0.999985i $$0.501764\pi$$
$$440$$ 0 0
$$441$$ −2884.40 −0.311457
$$442$$ − 3706.53i − 0.398873i
$$443$$ − 4953.74i − 0.531285i −0.964072 0.265642i $$-0.914416\pi$$
0.964072 0.265642i $$-0.0855841\pi$$
$$444$$ −1035.77 −0.110710
$$445$$ 0 0
$$446$$ 7082.78 0.751972
$$447$$ 7062.34i 0.747287i
$$448$$ 723.869i 0.0763383i
$$449$$ 11602.0 1.21945 0.609723 0.792615i $$-0.291281\pi$$
0.609723 + 0.792615i $$0.291281\pi$$
$$450$$ 0 0
$$451$$ −2653.81 −0.277080
$$452$$ − 5494.95i − 0.571816i
$$453$$ 1710.21i 0.177379i
$$454$$ 7611.79 0.786870
$$455$$ 0 0
$$456$$ 3684.62 0.378395
$$457$$ − 3530.68i − 0.361397i −0.983539 0.180698i $$-0.942164\pi$$
0.983539 0.180698i $$-0.0578358\pi$$
$$458$$ 17910.4i 1.82729i
$$459$$ −1975.55 −0.200895
$$460$$ 0 0
$$461$$ 11566.3 1.16854 0.584271 0.811559i $$-0.301381\pi$$
0.584271 + 0.811559i $$0.301381\pi$$
$$462$$ − 528.000i − 0.0531705i
$$463$$ − 10888.5i − 1.09294i −0.837479 0.546470i $$-0.815971\pi$$
0.837479 0.546470i $$-0.184029\pi$$
$$464$$ 19354.2 1.93641
$$465$$ 0 0
$$466$$ 8316.94 0.826770
$$467$$ 10688.0i 1.05906i 0.848292 + 0.529529i $$0.177631\pi$$
−0.848292 + 0.529529i $$0.822369\pi$$
$$468$$ 455.918i 0.0450317i
$$469$$ 4272.89 0.420690
$$470$$ 0 0
$$471$$ −6218.02 −0.608304
$$472$$ − 3058.77i − 0.298287i
$$473$$ 3080.18i 0.299422i
$$474$$ 3313.76 0.321110
$$475$$ 0 0
$$476$$ −1170.70 −0.112728
$$477$$ − 3686.87i − 0.353900i
$$478$$ − 4820.35i − 0.461250i
$$479$$ −2341.90 −0.223391 −0.111696 0.993742i $$-0.535628\pi$$
−0.111696 + 0.993742i $$0.535628\pi$$
$$480$$ 0 0
$$481$$ −1537.93 −0.145787
$$482$$ − 3301.43i − 0.311983i
$$483$$ 1594.17i 0.150181i
$$484$$ −408.046 −0.0383214
$$485$$ 0 0
$$486$$ 819.464 0.0764849
$$487$$ 6748.91i 0.627972i 0.949428 + 0.313986i $$0.101665\pi$$
−0.949428 + 0.313986i $$0.898335\pi$$
$$488$$ − 10945.4i − 1.01532i
$$489$$ −8029.53 −0.742552
$$490$$ 0 0
$$491$$ 7361.40 0.676609 0.338305 0.941037i $$-0.390147\pi$$
0.338305 + 0.941037i $$0.390147\pi$$
$$492$$ − 2440.74i − 0.223653i
$$493$$ − 17789.1i − 1.62511i
$$494$$ −3986.80 −0.363107
$$495$$ 0 0
$$496$$ −22187.6 −2.00857
$$497$$ − 3589.26i − 0.323944i
$$498$$ 7654.60i 0.688777i
$$499$$ −10381.7 −0.931359 −0.465680 0.884953i $$-0.654190\pi$$
−0.465680 + 0.884953i $$0.654190\pi$$
$$500$$ 0 0
$$501$$ −3564.36 −0.317852
$$502$$ − 22023.1i − 1.95805i
$$503$$ − 19149.0i − 1.69744i −0.528840 0.848721i $$-0.677373\pi$$
0.528840 0.848721i $$-0.322627\pi$$
$$504$$ −666.391 −0.0588957
$$505$$ 0 0
$$506$$ 4154.65 0.365013
$$507$$ − 5914.04i − 0.518051i
$$508$$ − 7726.76i − 0.674841i
$$509$$ −16073.2 −1.39967 −0.699836 0.714303i $$-0.746744\pi$$
−0.699836 + 0.714303i $$0.746744\pi$$
$$510$$ 0 0
$$511$$ 4838.58 0.418877
$$512$$ 1493.27i 0.128894i
$$513$$ 2124.93i 0.182881i
$$514$$ −27417.5 −2.35279
$$515$$ 0 0
$$516$$ −2832.88 −0.241687
$$517$$ − 1868.56i − 0.158954i
$$518$$ 1638.09i 0.138945i
$$519$$ −2421.44 −0.204797
$$520$$ 0 0
$$521$$ −18955.3 −1.59395 −0.796975 0.604012i $$-0.793568\pi$$
−0.796975 + 0.604012i $$0.793568\pi$$
$$522$$ 7378.97i 0.618714i
$$523$$ 4442.19i 0.371402i 0.982606 + 0.185701i $$0.0594556\pi$$
−0.982606 + 0.185701i $$0.940544\pi$$
$$524$$ 3869.40 0.322587
$$525$$ 0 0
$$526$$ −15341.9 −1.27175
$$527$$ 20393.3i 1.68567i
$$528$$ 2627.00i 0.216525i
$$529$$ −377.000 −0.0309855
$$530$$ 0 0
$$531$$ 1764.00 0.144164
$$532$$ 1259.22i 0.102620i
$$533$$ − 3624.08i − 0.294515i
$$534$$ −5149.25 −0.417285
$$535$$ 0 0
$$536$$ −14054.5 −1.13258
$$537$$ − 5851.17i − 0.470199i
$$538$$ 98.3096i 0.00787812i
$$539$$ 3525.38 0.281723
$$540$$ 0 0
$$541$$ 2180.90 0.173316 0.0866580 0.996238i $$-0.472381\pi$$
0.0866580 + 0.996238i $$0.472381\pi$$
$$542$$ 26004.4i 2.06086i
$$543$$ − 3184.82i − 0.251701i
$$544$$ 10507.4 0.828129
$$545$$ 0 0
$$546$$ 721.044 0.0565162
$$547$$ 8225.04i 0.642920i 0.946923 + 0.321460i $$0.104174\pi$$
−0.946923 + 0.321460i $$0.895826\pi$$
$$548$$ − 4278.07i − 0.333485i
$$549$$ 6312.23 0.490709
$$550$$ 0 0
$$551$$ −19134.2 −1.47939
$$552$$ − 5243.61i − 0.404316i
$$553$$ − 1554.08i − 0.119505i
$$554$$ −3802.32 −0.291598
$$555$$ 0 0
$$556$$ −1639.90 −0.125085
$$557$$ − 25181.9i − 1.91561i −0.287423 0.957804i $$-0.592799\pi$$
0.287423 0.957804i $$-0.407201\pi$$
$$558$$ − 8459.22i − 0.641769i
$$559$$ −4206.33 −0.318263
$$560$$ 0 0
$$561$$ 2414.56 0.181716
$$562$$ − 6314.50i − 0.473952i
$$563$$ 4504.50i 0.337197i 0.985685 + 0.168599i $$0.0539242\pi$$
−0.985685 + 0.168599i $$0.946076\pi$$
$$564$$ 1718.54 0.128304
$$565$$ 0 0
$$566$$ 7164.36 0.532050
$$567$$ − 384.310i − 0.0284647i
$$568$$ 11805.9i 0.872122i
$$569$$ 13447.0 0.990732 0.495366 0.868684i $$-0.335034\pi$$
0.495366 + 0.868684i $$0.335034\pi$$
$$570$$ 0 0
$$571$$ −2605.52 −0.190959 −0.0954795 0.995431i $$-0.530438\pi$$
−0.0954795 + 0.995431i $$0.530438\pi$$
$$572$$ − 557.233i − 0.0407327i
$$573$$ − 6409.24i − 0.467277i
$$574$$ −3860.09 −0.280691
$$575$$ 0 0
$$576$$ 1373.11 0.0993281
$$577$$ 6339.65i 0.457406i 0.973496 + 0.228703i $$0.0734484\pi$$
−0.973496 + 0.228703i $$0.926552\pi$$
$$578$$ − 1485.90i − 0.106929i
$$579$$ −11843.3 −0.850069
$$580$$ 0 0
$$581$$ 3589.83 0.256336
$$582$$ 6215.37i 0.442672i
$$583$$ 4506.17i 0.320114i
$$584$$ −15915.2 −1.12770
$$585$$ 0 0
$$586$$ −11210.1 −0.790247
$$587$$ − 13370.6i − 0.940140i −0.882629 0.470070i $$-0.844229\pi$$
0.882629 0.470070i $$-0.155771\pi$$
$$588$$ 3242.34i 0.227401i
$$589$$ 21935.3 1.53452
$$590$$ 0 0
$$591$$ 2771.66 0.192912
$$592$$ − 8150.09i − 0.565822i
$$593$$ − 14319.3i − 0.991608i −0.868434 0.495804i $$-0.834873\pi$$
0.868434 0.495804i $$-0.165127\pi$$
$$594$$ −1001.57 −0.0691832
$$595$$ 0 0
$$596$$ 7938.73 0.545609
$$597$$ − 1428.46i − 0.0979276i
$$598$$ 5673.65i 0.387981i
$$599$$ 5788.63 0.394853 0.197427 0.980318i $$-0.436742\pi$$
0.197427 + 0.980318i $$0.436742\pi$$
$$600$$ 0 0
$$601$$ 23968.1 1.62675 0.813375 0.581739i $$-0.197628\pi$$
0.813375 + 0.581739i $$0.197628\pi$$
$$602$$ 4480.26i 0.303325i
$$603$$ − 8105.28i − 0.547384i
$$604$$ 1922.44 0.129508
$$605$$ 0 0
$$606$$ −10277.9 −0.688961
$$607$$ − 23526.6i − 1.57317i −0.617482 0.786585i $$-0.711847\pi$$
0.617482 0.786585i $$-0.288153\pi$$
$$608$$ − 11301.9i − 0.753872i
$$609$$ 3460.56 0.230261
$$610$$ 0 0
$$611$$ 2551.74 0.168956
$$612$$ 2220.70i 0.146677i
$$613$$ − 1228.07i − 0.0809159i −0.999181 0.0404579i $$-0.987118\pi$$
0.999181 0.0404579i $$-0.0128817\pi$$
$$614$$ 5729.31 0.376573
$$615$$ 0 0
$$616$$ 814.478 0.0532732
$$617$$ − 9844.90i − 0.642368i −0.947017 0.321184i $$-0.895919\pi$$
0.947017 0.321184i $$-0.104081\pi$$
$$618$$ − 11150.4i − 0.725785i
$$619$$ 6551.68 0.425419 0.212709 0.977115i $$-0.431771\pi$$
0.212709 + 0.977115i $$0.431771\pi$$
$$620$$ 0 0
$$621$$ 3024.00 0.195409
$$622$$ 23365.9i 1.50625i
$$623$$ 2414.88i 0.155297i
$$624$$ −3587.46 −0.230150
$$625$$ 0 0
$$626$$ −12007.8 −0.766661
$$627$$ − 2597.14i − 0.165422i
$$628$$ 6989.64i 0.444135i
$$629$$ −7491.01 −0.474859
$$630$$ 0 0
$$631$$ −26440.5 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$632$$ 5111.72i 0.321730i
$$633$$ 14754.7i 0.926458i
$$634$$ −1122.13 −0.0702923
$$635$$ 0 0
$$636$$ −4144.39 −0.258389
$$637$$ 4814.31i 0.299450i
$$638$$ − 9018.74i − 0.559648i
$$639$$ −6808.50 −0.421502
$$640$$ 0 0
$$641$$ −27927.2 −1.72084 −0.860421 0.509584i $$-0.829799\pi$$
−0.860421 + 0.509584i $$0.829799\pi$$
$$642$$ 13936.7i 0.856758i
$$643$$ 16737.7i 1.02655i 0.858225 + 0.513274i $$0.171568\pi$$
−0.858225 + 0.513274i $$0.828432\pi$$
$$644$$ 1792.00 0.109650
$$645$$ 0 0
$$646$$ −19419.1 −1.18271
$$647$$ 7818.70i 0.475092i 0.971376 + 0.237546i $$0.0763431\pi$$
−0.971376 + 0.237546i $$0.923657\pi$$
$$648$$ 1264.08i 0.0766325i
$$649$$ −2156.00 −0.130401
$$650$$ 0 0
$$651$$ −3967.17 −0.238842
$$652$$ 9025.94i 0.542152i
$$653$$ − 19747.6i − 1.18344i −0.806144 0.591719i $$-0.798450\pi$$
0.806144 0.591719i $$-0.201550\pi$$
$$654$$ −3239.59 −0.193697
$$655$$ 0 0
$$656$$ 19205.4 1.14305
$$657$$ − 9178.33i − 0.545024i
$$658$$ − 2717.91i − 0.161026i
$$659$$ −7867.72 −0.465072 −0.232536 0.972588i $$-0.574702\pi$$
−0.232536 + 0.972588i $$0.574702\pi$$
$$660$$ 0 0
$$661$$ 4227.41 0.248755 0.124378 0.992235i $$-0.460307\pi$$
0.124378 + 0.992235i $$0.460307\pi$$
$$662$$ − 1825.90i − 0.107199i
$$663$$ 3297.35i 0.193150i
$$664$$ −11807.8 −0.690106
$$665$$ 0 0
$$666$$ 3107.30 0.180789
$$667$$ 27230.0i 1.58073i
$$668$$ 4006.67i 0.232070i
$$669$$ −6300.88 −0.364135
$$670$$ 0 0
$$671$$ −7714.94 −0.443863
$$672$$ 2044.04i 0.117337i
$$673$$ − 29397.6i − 1.68379i −0.539638 0.841897i $$-0.681439\pi$$
0.539638 0.841897i $$-0.318561\pi$$
$$674$$ −2753.57 −0.157364
$$675$$ 0 0
$$676$$ −6647.94 −0.378239
$$677$$ 5737.14i 0.325696i 0.986651 + 0.162848i $$0.0520680\pi$$
−0.986651 + 0.162848i $$0.947932\pi$$
$$678$$ 16484.8i 0.933771i
$$679$$ 2914.86 0.164745
$$680$$ 0 0
$$681$$ −6771.49 −0.381034
$$682$$ 10339.0i 0.580502i
$$683$$ − 32097.6i − 1.79821i −0.437729 0.899107i $$-0.644217\pi$$
0.437729 0.899107i $$-0.355783\pi$$
$$684$$ 2388.62 0.133525
$$685$$ 0 0
$$686$$ 10615.8 0.590837
$$687$$ − 15933.2i − 0.884848i
$$688$$ − 22291.0i − 1.23523i
$$689$$ −6153.69 −0.340257
$$690$$ 0 0
$$691$$ −16456.2 −0.905965 −0.452983 0.891519i $$-0.649640\pi$$
−0.452983 + 0.891519i $$0.649640\pi$$
$$692$$ 2721.93i 0.149526i
$$693$$ 469.712i 0.0257473i
$$694$$ −21112.3 −1.15477
$$695$$ 0 0
$$696$$ −11382.6 −0.619909
$$697$$ − 17652.3i − 0.959294i
$$698$$ 43058.9i 2.33496i
$$699$$ −7398.80 −0.400355
$$700$$ 0 0
$$701$$ 27238.1 1.46758 0.733788 0.679379i $$-0.237751\pi$$
0.733788 + 0.679379i $$0.237751\pi$$
$$702$$ − 1367.75i − 0.0735364i
$$703$$ 8057.44i 0.432279i
$$704$$ −1678.25 −0.0898457
$$705$$ 0 0
$$706$$ −8934.12 −0.476261
$$707$$ 4820.09i 0.256405i
$$708$$ − 1982.90i − 0.105257i
$$709$$ −28761.4 −1.52349 −0.761747 0.647875i $$-0.775658\pi$$
−0.761747 + 0.647875i $$0.775658\pi$$
$$710$$ 0 0
$$711$$ −2947.94 −0.155494
$$712$$ − 7943.10i − 0.418090i
$$713$$ − 31216.3i − 1.63964i
$$714$$ 3512.09 0.184085
$$715$$ 0 0
$$716$$ −6577.27 −0.343302
$$717$$ 4288.21i 0.223356i
$$718$$ 10804.5i 0.561588i
$$719$$ 27272.0 1.41456 0.707282 0.706931i $$-0.249921\pi$$
0.707282 + 0.706931i $$0.249921\pi$$
$$720$$ 0 0
$$721$$ −5229.28 −0.270109
$$722$$ − 2243.05i − 0.115620i
$$723$$ 2936.97i 0.151075i
$$724$$ −3580.04 −0.183772
$$725$$ 0 0
$$726$$ 1224.14 0.0625785
$$727$$ 3979.75i 0.203027i 0.994834 + 0.101514i $$0.0323685\pi$$
−0.994834 + 0.101514i $$0.967631\pi$$
$$728$$ 1112.26i 0.0566253i
$$729$$ −729.000 −0.0370370
$$730$$ 0 0
$$731$$ −20488.3 −1.03665
$$732$$ − 7095.54i − 0.358277i
$$733$$ − 9342.48i − 0.470767i −0.971903 0.235384i $$-0.924365\pi$$
0.971903 0.235384i $$-0.0756346\pi$$
$$734$$ 28426.3 1.42947
$$735$$ 0 0
$$736$$ −16083.9 −0.805515
$$737$$ 9906.45i 0.495127i
$$738$$ 7322.23i 0.365224i
$$739$$ 28928.0 1.43997 0.719983 0.693992i $$-0.244150\pi$$
0.719983 + 0.693992i $$0.244150\pi$$
$$740$$ 0 0
$$741$$ 3546.68 0.175831
$$742$$ 6554.43i 0.324287i
$$743$$ − 4857.04i − 0.239822i −0.992785 0.119911i $$-0.961739\pi$$
0.992785 0.119911i $$-0.0382609\pi$$
$$744$$ 13049.0 0.643008
$$745$$ 0 0
$$746$$ −31660.8 −1.55386
$$747$$ − 6809.57i − 0.333533i
$$748$$ − 2714.19i − 0.132675i
$$749$$ 6536.00 0.318852
$$750$$ 0 0
$$751$$ 14355.4 0.697517 0.348759 0.937213i $$-0.386603\pi$$
0.348759 + 0.937213i $$0.386603\pi$$
$$752$$ 13522.6i 0.655744i
$$753$$ 19591.9i 0.948165i
$$754$$ 12316.1 0.594863
$$755$$ 0 0
$$756$$ −432.000 −0.0207827
$$757$$ − 17714.9i − 0.850538i −0.905067 0.425269i $$-0.860179\pi$$
0.905067 0.425269i $$-0.139821\pi$$
$$758$$ 48103.9i 2.30503i
$$759$$ −3696.00 −0.176754
$$760$$ 0 0
$$761$$ −7945.82 −0.378497 −0.189248 0.981929i $$-0.560605\pi$$
−0.189248 + 0.981929i $$0.560605\pi$$
$$762$$ 23180.3i 1.10201i
$$763$$ 1519.29i 0.0720866i
$$764$$ −7204.58 −0.341168
$$765$$ 0 0
$$766$$ 45398.4 2.14140
$$767$$ − 2944.26i − 0.138606i
$$768$$ − 13166.2i − 0.618613i
$$769$$ −27308.1 −1.28057 −0.640284 0.768139i $$-0.721183\pi$$
−0.640284 + 0.768139i $$0.721183\pi$$
$$770$$ 0 0
$$771$$ 24390.8 1.13932
$$772$$ 13313.0i 0.620653i
$$773$$ 18872.6i 0.878136i 0.898454 + 0.439068i $$0.144691\pi$$
−0.898454 + 0.439068i $$0.855309\pi$$
$$774$$ 8498.64 0.394674
$$775$$ 0 0
$$776$$ −9587.65 −0.443527
$$777$$ − 1457.25i − 0.0672826i
$$778$$ 3176.29i 0.146369i
$$779$$ −18987.1 −0.873276
$$780$$ 0 0
$$781$$ 8321.50 0.381263
$$782$$ 27635.4i 1.26373i
$$783$$ − 6564.37i − 0.299606i
$$784$$ −25512.8 −1.16221
$$785$$ 0 0
$$786$$ −11608.2 −0.526782
$$787$$ − 14512.1i − 0.657307i −0.944450 0.328654i $$-0.893405\pi$$
0.944450 0.328654i $$-0.106595\pi$$
$$788$$ − 3115.60i − 0.140849i
$$789$$ 13648.3 0.615832
$$790$$ 0 0
$$791$$ 7731.01 0.347513
$$792$$ − 1544.99i − 0.0693167i
$$793$$ − 10535.6i − 0.471792i
$$794$$ 2859.15 0.127793
$$795$$ 0 0
$$796$$ −1605.72 −0.0714989
$$797$$ 29108.9i 1.29371i 0.762612 + 0.646856i $$0.223917\pi$$
−0.762612 + 0.646856i $$0.776083\pi$$
$$798$$ − 3777.65i − 0.167578i
$$799$$ 12429.1 0.550325
$$800$$ 0 0
$$801$$ 4580.80 0.202066
$$802$$ 41154.0i 1.81197i
$$803$$ 11218.0i 0.492993i
$$804$$ −9111.10 −0.399656
$$805$$ 0 0
$$806$$ −14119.1 −0.617029
$$807$$ − 87.4567i − 0.00381490i
$$808$$ − 15854.4i − 0.690291i
$$809$$ 3000.83 0.130413 0.0652063 0.997872i $$-0.479229\pi$$
0.0652063 + 0.997872i $$0.479229\pi$$
$$810$$ 0 0
$$811$$ 6239.39 0.270154 0.135077 0.990835i $$-0.456872\pi$$
0.135077 + 0.990835i $$0.456872\pi$$
$$812$$ − 3890.00i − 0.168118i
$$813$$ − 23133.7i − 0.997950i
$$814$$ −3797.81 −0.163530
$$815$$ 0 0
$$816$$ −17473.9 −0.749645
$$817$$ 22037.6i 0.943693i
$$818$$ − 29539.6i − 1.26263i
$$819$$ −641.445 −0.0273674
$$820$$ 0 0
$$821$$ 14922.4 0.634342 0.317171 0.948368i $$-0.397267\pi$$
0.317171 + 0.948368i $$0.397267\pi$$
$$822$$ 12834.2i 0.544580i
$$823$$ 25737.8i 1.09011i 0.838399 + 0.545057i $$0.183492\pi$$
−0.838399 + 0.545057i $$0.816508\pi$$
$$824$$ 17200.3 0.727186
$$825$$ 0 0
$$826$$ −3136.00 −0.132101
$$827$$ 27043.4i 1.13711i 0.822645 + 0.568555i $$0.192497\pi$$
−0.822645 + 0.568555i $$0.807503\pi$$
$$828$$ − 3399.26i − 0.142672i
$$829$$ 9795.41 0.410384 0.205192 0.978722i $$-0.434218\pi$$
0.205192 + 0.978722i $$0.434218\pi$$
$$830$$ 0 0
$$831$$ 3382.56 0.141203
$$832$$ − 2291.84i − 0.0954990i
$$833$$ 23449.7i 0.975370i
$$834$$ 4919.70 0.204263
$$835$$ 0 0
$$836$$ −2919.42 −0.120778
$$837$$ 7525.37i 0.310770i
$$838$$ 37730.5i 1.55535i
$$839$$ −28875.5 −1.18819 −0.594095 0.804395i $$-0.702490\pi$$
−0.594095 + 0.804395i $$0.702490\pi$$
$$840$$ 0 0
$$841$$ 34720.7 1.42362
$$842$$ − 47489.3i − 1.94369i
$$843$$ 5617.41i 0.229506i
$$844$$ 16585.7 0.676426
$$845$$ 0 0
$$846$$ −5155.63 −0.209520
$$847$$ − 574.092i − 0.0232893i
$$848$$ − 32610.7i − 1.32059i
$$849$$ −6373.45 −0.257640
$$850$$ 0 0
$$851$$ 11466.6 0.461892
$$852$$ 7653.39i 0.307747i
$$853$$ 47157.1i 1.89288i 0.322878 + 0.946441i $$0.395350\pi$$
−0.322878 + 0.946441i $$0.604650\pi$$
$$854$$ −11221.7 −0.449649
$$855$$ 0 0
$$856$$ −21498.4 −0.858412
$$857$$ 5021.31i 0.200145i 0.994980 + 0.100073i $$0.0319075\pi$$
−0.994980 + 0.100073i $$0.968092\pi$$
$$858$$ 1671.70i 0.0665162i
$$859$$ 22921.1 0.910428 0.455214 0.890382i $$-0.349563\pi$$
0.455214 + 0.890382i $$0.349563\pi$$
$$860$$ 0 0
$$861$$ 3433.95 0.135922
$$862$$ − 18938.9i − 0.748332i
$$863$$ 19488.1i 0.768693i 0.923189 + 0.384347i $$0.125573\pi$$
−0.923189 + 0.384347i $$0.874427\pi$$
$$864$$ 3877.36 0.152674
$$865$$ 0 0
$$866$$ 24266.1 0.952188
$$867$$ 1321.86i 0.0517794i
$$868$$ 4459.48i 0.174383i
$$869$$ 3603.04 0.140650
$$870$$ 0 0
$$871$$ −13528.4 −0.526282
$$872$$ − 4997.30i − 0.194071i
$$873$$ − 5529.22i − 0.214360i
$$874$$ 29725.0 1.15042
$$875$$ 0 0
$$876$$ −10317.3 −0.397933
$$877$$ − 8455.67i − 0.325573i −0.986661 0.162787i $$-0.947952\pi$$
0.986661 0.162787i $$-0.0520482\pi$$
$$878$$ − 343.834i − 0.0132162i
$$879$$ 9972.56 0.382669
$$880$$ 0 0
$$881$$ −11291.2 −0.431794 −0.215897 0.976416i $$-0.569268\pi$$
−0.215897 + 0.976416i $$0.569268\pi$$
$$882$$ − 9727.02i − 0.371344i
$$883$$ − 31818.1i − 1.21264i −0.795219 0.606322i $$-0.792644\pi$$
0.795219 0.606322i $$-0.207356\pi$$
$$884$$ 3706.53 0.141023
$$885$$ 0 0
$$886$$ 16705.4 0.633441
$$887$$ 17481.1i 0.661732i 0.943678 + 0.330866i $$0.107341\pi$$
−0.943678 + 0.330866i $$0.892659\pi$$
$$888$$ 4793.24i 0.181138i
$$889$$ 10871.0 0.410126
$$890$$ 0 0
$$891$$ 891.000 0.0335013
$$892$$ 7082.78i 0.265862i
$$893$$ − 13368.9i − 0.500978i
$$894$$ −23816.2 −0.890976
$$895$$ 0 0
$$896$$ −7891.87 −0.294251
$$897$$ − 5047.31i − 0.187876i
$$898$$ 39125.1i 1.45392i
$$899$$ −67763.1 −2.51393
$$900$$ 0 0
$$901$$ −29973.6 −1.10829
$$902$$ − 8949.39i − 0.330357i
$$903$$ − 3985.66i − 0.146882i
$$904$$ −25429.1 −0.935574
$$905$$ 0 0
$$906$$ −5767.31 −0.211486
$$907$$ 10607.4i 0.388326i 0.980969 + 0.194163i $$0.0621990\pi$$
−0.980969 + 0.194163i $$0.937801\pi$$
$$908$$ 7611.79i 0.278201i
$$909$$ 9143.26 0.333623
$$910$$ 0 0
$$911$$ −41249.2 −1.50016 −0.750080 0.661347i $$-0.769985\pi$$
−0.750080 + 0.661347i $$0.769985\pi$$
$$912$$ 18795.2i 0.682425i
$$913$$ 8322.81i 0.301692i
$$914$$ 11906.5 0.430887
$$915$$ 0 0
$$916$$ −17910.4 −0.646045
$$917$$ 5443.97i 0.196048i
$$918$$ − 6662.10i − 0.239523i
$$919$$ 13858.1 0.497429 0.248714 0.968577i $$-0.419992\pi$$
0.248714 + 0.968577i $$0.419992\pi$$
$$920$$ 0 0
$$921$$ −5096.83 −0.182352
$$922$$ 39004.9i 1.39323i
$$923$$ 11363.9i 0.405253i
$$924$$ 528.000 0.0187986
$$925$$ 0 0
$$926$$ 36719.0 1.30309
$$927$$ 9919.47i 0.351454i
$$928$$ 34914.2i 1.23504i
$$929$$ −20893.7 −0.737890 −0.368945 0.929451i $$-0.620281\pi$$
−0.368945 + 0.929451i $$0.620281\pi$$
$$930$$ 0 0
$$931$$ 25222.8 0.887911
$$932$$ 8316.94i 0.292307i
$$933$$ − 20786.5i − 0.729388i
$$934$$ −36042.9 −1.26270
$$935$$ 0 0
$$936$$ 2109.86 0.0736784
$$937$$ 3203.52i 0.111691i 0.998439 + 0.0558454i $$0.0177854\pi$$
−0.998439 + 0.0558454i $$0.982215\pi$$
$$938$$ 14409.4i 0.501581i
$$939$$ 10682.2 0.371248
$$940$$ 0 0
$$941$$ 19951.6 0.691182 0.345591 0.938385i $$-0.387678\pi$$
0.345591 + 0.938385i $$0.387678\pi$$
$$942$$ − 20968.9i − 0.725270i
$$943$$ 27020.6i 0.933099i
$$944$$ 15602.8 0.537952
$$945$$ 0 0
$$946$$ −10387.2 −0.356996
$$947$$ − 38216.7i − 1.31138i −0.755031 0.655689i $$-0.772378\pi$$
0.755031 0.655689i $$-0.227622\pi$$
$$948$$ 3313.76i 0.113529i
$$949$$ −15319.4 −0.524014
$$950$$ 0 0
$$951$$ 998.249 0.0340383
$$952$$ 5417.65i 0.184440i
$$953$$ 47661.4i 1.62004i 0.586399 + 0.810022i $$0.300545\pi$$
−0.586399 + 0.810022i $$0.699455\pi$$
$$954$$ 12433.2 0.421948
$$955$$ 0 0
$$956$$ 4820.35 0.163077
$$957$$ 8023.12i 0.271004i
$$958$$ − 7897.56i − 0.266345i
$$959$$ 6018.94 0.202671
$$960$$ 0 0
$$961$$ 47892.3 1.60761
$$962$$ − 5186.34i − 0.173819i
$$963$$ − 12398.2i − 0.414876i
$$964$$ 3301.43 0.110303
$$965$$ 0 0
$$966$$ −5376.00 −0.179058
$$967$$ 18933.2i 0.629628i 0.949153 + 0.314814i $$0.101942\pi$$
−0.949153 + 0.314814i $$0.898058\pi$$
$$968$$ 1888.32i 0.0626994i
$$969$$ 17275.3 0.572717
$$970$$ 0 0
$$971$$ −40660.3 −1.34382 −0.671911 0.740632i $$-0.734526\pi$$
−0.671911 + 0.740632i $$0.734526\pi$$
$$972$$ 819.464i 0.0270415i
$$973$$ − 2307.23i − 0.0760188i
$$974$$ −22759.2 −0.748720
$$975$$ 0 0
$$976$$ 55832.3 1.83110
$$977$$ − 22502.8i − 0.736876i −0.929652 0.368438i $$-0.879893\pi$$
0.929652 0.368438i $$-0.120107\pi$$
$$978$$ − 27077.8i − 0.885331i
$$979$$ −5598.76 −0.182775
$$980$$ 0 0
$$981$$ 2881.96 0.0937959
$$982$$ 24824.7i 0.806709i
$$983$$ 4435.20i 0.143907i 0.997408 + 0.0719536i $$0.0229234\pi$$
−0.997408 + 0.0719536i $$0.977077\pi$$
$$984$$ −11295.1 −0.365929
$$985$$ 0 0
$$986$$ 59989.8 1.93759
$$987$$ 2417.87i 0.0779753i
$$988$$ − 3986.80i − 0.128378i
$$989$$ 31361.8 1.00834
$$990$$ 0 0
$$991$$ 7362.76 0.236010 0.118005 0.993013i $$-0.462350\pi$$
0.118005 + 0.993013i $$0.462350\pi$$
$$992$$ − 40025.5i − 1.28106i
$$993$$ 1624.33i 0.0519101i
$$994$$ 12104.0 0.386233
$$995$$ 0 0
$$996$$ −7654.60 −0.243519
$$997$$ − 53480.1i − 1.69883i −0.527728 0.849413i $$-0.676956\pi$$
0.527728 0.849413i $$-0.323044\pi$$
$$998$$ − 35010.0i − 1.11044i
$$999$$ −2764.27 −0.0875452
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 825.4.c.i.199.4 4
5.2 odd 4 825.4.a.k.1.1 2
5.3 odd 4 33.4.a.d.1.2 2
5.4 even 2 inner 825.4.c.i.199.1 4
15.2 even 4 2475.4.a.o.1.2 2
15.8 even 4 99.4.a.e.1.1 2
20.3 even 4 528.4.a.o.1.1 2
35.13 even 4 1617.4.a.j.1.2 2
40.3 even 4 2112.4.a.bh.1.2 2
40.13 odd 4 2112.4.a.ba.1.2 2
55.43 even 4 363.4.a.j.1.1 2
60.23 odd 4 1584.4.a.x.1.2 2
165.98 odd 4 1089.4.a.t.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 5.3 odd 4
99.4.a.e.1.1 2 15.8 even 4
363.4.a.j.1.1 2 55.43 even 4
528.4.a.o.1.1 2 20.3 even 4
825.4.a.k.1.1 2 5.2 odd 4
825.4.c.i.199.1 4 5.4 even 2 inner
825.4.c.i.199.4 4 1.1 even 1 trivial
1089.4.a.t.1.2 2 165.98 odd 4
1584.4.a.x.1.2 2 60.23 odd 4
1617.4.a.j.1.2 2 35.13 even 4
2112.4.a.ba.1.2 2 40.13 odd 4
2112.4.a.bh.1.2 2 40.3 even 4
2475.4.a.o.1.2 2 15.2 even 4