# Properties

 Label 2475.4.a.o.1.2 Level $2475$ Weight $4$ Character 2475.1 Self dual yes Analytic conductor $146.030$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2475.a (trivial)

## Newform invariants

comment: select newform

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

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

 Self dual: yes Analytic conductor: $$146.029727264$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 33) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$3.37228$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.37228 q^{2} +3.37228 q^{4} +4.74456 q^{7} -15.6060 q^{8} +O(q^{10})$$ $$q+3.37228 q^{2} +3.37228 q^{4} +4.74456 q^{7} -15.6060 q^{8} -11.0000 q^{11} +15.0217 q^{13} +16.0000 q^{14} -79.6060 q^{16} +73.1684 q^{17} -78.7011 q^{19} -37.0951 q^{22} +112.000 q^{23} +50.6576 q^{26} +16.0000 q^{28} -243.125 q^{29} +278.717 q^{31} -143.606 q^{32} +246.745 q^{34} -102.380 q^{37} -265.402 q^{38} +241.255 q^{41} +280.016 q^{43} -37.0951 q^{44} +377.696 q^{46} -169.870 q^{47} -320.489 q^{49} +50.6576 q^{52} -409.652 q^{53} -74.0435 q^{56} -819.886 q^{58} -196.000 q^{59} -701.359 q^{61} +939.913 q^{62} +152.568 q^{64} -900.587 q^{67} +246.745 q^{68} -756.500 q^{71} +1019.81 q^{73} -345.255 q^{74} -265.402 q^{76} -52.1902 q^{77} -327.549 q^{79} +813.581 q^{82} -756.619 q^{83} +944.293 q^{86} +171.666 q^{88} -508.978 q^{89} +71.2716 q^{91} +377.696 q^{92} -572.848 q^{94} -614.358 q^{97} -1080.78 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} - 2 q^{7} + 9 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 - 2 * q^7 + 9 * q^8 $$2 q + q^{2} + q^{4} - 2 q^{7} + 9 q^{8} - 22 q^{11} + 76 q^{13} + 32 q^{14} - 119 q^{16} - 26 q^{17} - 54 q^{19} - 11 q^{22} + 224 q^{23} - 94 q^{26} + 32 q^{28} - 222 q^{29} - 40 q^{31} - 247 q^{32} + 482 q^{34} + 48 q^{37} - 324 q^{38} + 494 q^{41} + 66 q^{43} - 11 q^{44} + 112 q^{46} - 64 q^{47} - 618 q^{49} - 94 q^{52} - 84 q^{53} - 240 q^{56} - 870 q^{58} - 392 q^{59} - 1104 q^{61} + 1696 q^{62} + 713 q^{64} - 928 q^{67} + 482 q^{68} - 456 q^{71} + 592 q^{73} - 702 q^{74} - 324 q^{76} + 22 q^{77} - 230 q^{79} + 214 q^{82} + 348 q^{83} + 1452 q^{86} - 99 q^{88} - 972 q^{89} - 340 q^{91} + 112 q^{92} - 824 q^{94} + 1184 q^{97} - 375 q^{98}+O(q^{100})$$ 2 * q + q^2 + q^4 - 2 * q^7 + 9 * q^8 - 22 * q^11 + 76 * q^13 + 32 * q^14 - 119 * q^16 - 26 * q^17 - 54 * q^19 - 11 * q^22 + 224 * q^23 - 94 * q^26 + 32 * q^28 - 222 * q^29 - 40 * q^31 - 247 * q^32 + 482 * q^34 + 48 * q^37 - 324 * q^38 + 494 * q^41 + 66 * q^43 - 11 * q^44 + 112 * q^46 - 64 * q^47 - 618 * q^49 - 94 * q^52 - 84 * q^53 - 240 * q^56 - 870 * q^58 - 392 * q^59 - 1104 * q^61 + 1696 * q^62 + 713 * q^64 - 928 * q^67 + 482 * q^68 - 456 * q^71 + 592 * q^73 - 702 * q^74 - 324 * q^76 + 22 * q^77 - 230 * q^79 + 214 * q^82 + 348 * q^83 + 1452 * q^86 - 99 * q^88 - 972 * q^89 - 340 * q^91 + 112 * q^92 - 824 * q^94 + 1184 * q^97 - 375 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 3.37228 1.19228 0.596141 0.802880i $$-0.296700\pi$$
0.596141 + 0.802880i $$0.296700\pi$$
$$3$$ 0 0
$$4$$ 3.37228 0.421535
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.74456 0.256182 0.128091 0.991762i $$-0.459115\pi$$
0.128091 + 0.991762i $$0.459115\pi$$
$$8$$ −15.6060 −0.689693
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ 15.0217 0.320483 0.160242 0.987078i $$-0.448773\pi$$
0.160242 + 0.987078i $$0.448773\pi$$
$$14$$ 16.0000 0.305441
$$15$$ 0 0
$$16$$ −79.6060 −1.24384
$$17$$ 73.1684 1.04388 0.521940 0.852982i $$-0.325209\pi$$
0.521940 + 0.852982i $$0.325209\pi$$
$$18$$ 0 0
$$19$$ −78.7011 −0.950277 −0.475138 0.879911i $$-0.657602\pi$$
−0.475138 + 0.879911i $$0.657602\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −37.0951 −0.359486
$$23$$ 112.000 1.01537 0.507687 0.861541i $$-0.330501\pi$$
0.507687 + 0.861541i $$0.330501\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 50.6576 0.382106
$$27$$ 0 0
$$28$$ 16.0000 0.107990
$$29$$ −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 $$0.200871\pi$$
$$32$$ −143.606 −0.793318
$$33$$ 0 0
$$34$$ 246.745 1.24460
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −102.380 −0.454898 −0.227449 0.973790i $$-0.573039\pi$$
−0.227449 + 0.973790i $$0.573039\pi$$
$$38$$ −265.402 −1.13300
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 241.255 0.918970 0.459485 0.888186i $$-0.348034\pi$$
0.459485 + 0.888186i $$0.348034\pi$$
$$42$$ 0 0
$$43$$ 280.016 0.993071 0.496536 0.868016i $$-0.334605\pi$$
0.496536 + 0.868016i $$0.334605\pi$$
$$44$$ −37.0951 −0.127098
$$45$$ 0 0
$$46$$ 377.696 1.21061
$$47$$ −169.870 −0.527192 −0.263596 0.964633i $$-0.584909\pi$$
−0.263596 + 0.964633i $$0.584909\pi$$
$$48$$ 0 0
$$49$$ −320.489 −0.934371
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 50.6576 0.135095
$$53$$ −409.652 −1.06170 −0.530849 0.847466i $$-0.678127\pi$$
−0.530849 + 0.847466i $$0.678127\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −74.0435 −0.176687
$$57$$ 0 0
$$58$$ −819.886 −1.85614
$$59$$ −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 $$0.763318\pi$$
$$62$$ 939.913 1.92531
$$63$$ 0 0
$$64$$ 152.568 0.297984
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −900.587 −1.64215 −0.821076 0.570819i $$-0.806626\pi$$
−0.821076 + 0.570819i $$0.806626\pi$$
$$68$$ 246.745 0.440032
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −756.500 −1.26451 −0.632254 0.774762i $$-0.717870\pi$$
−0.632254 + 0.774762i $$0.717870\pi$$
$$72$$ 0 0
$$73$$ 1019.81 1.63507 0.817536 0.575877i $$-0.195339\pi$$
0.817536 + 0.575877i $$0.195339\pi$$
$$74$$ −345.255 −0.542367
$$75$$ 0 0
$$76$$ −265.402 −0.400575
$$77$$ −52.1902 −0.0772419
$$78$$ 0 0
$$79$$ −327.549 −0.466483 −0.233241 0.972419i $$-0.574933\pi$$
−0.233241 + 0.972419i $$0.574933\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 813.581 1.09567
$$83$$ −756.619 −1.00060 −0.500300 0.865852i $$-0.666777\pi$$
−0.500300 + 0.865852i $$0.666777\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 944.293 1.18402
$$87$$ 0 0
$$88$$ 171.666 0.207950
$$89$$ −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
$$92$$ 377.696 0.428016
$$93$$ 0 0
$$94$$ −572.848 −0.628561
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −614.358 −0.643079 −0.321539 0.946896i $$-0.604200\pi$$
−0.321539 + 0.946896i $$0.604200\pi$$
$$98$$ −1080.78 −1.11403
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1015.92 1.00087 0.500434 0.865775i $$-0.333174\pi$$
0.500434 + 0.865775i $$0.333174\pi$$
$$102$$ 0 0
$$103$$ −1102.16 −1.05436 −0.527181 0.849753i $$-0.676751\pi$$
−0.527181 + 0.849753i $$0.676751\pi$$
$$104$$ −234.429 −0.221035
$$105$$ 0 0
$$106$$ −1381.46 −1.26584
$$107$$ 1377.58 1.24463 0.622315 0.782767i $$-0.286192\pi$$
0.622315 + 0.782767i $$0.286192\pi$$
$$108$$ 0 0
$$109$$ 320.217 0.281388 0.140694 0.990053i $$-0.455067\pi$$
0.140694 + 0.990053i $$0.455067\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −377.696 −0.318651
$$113$$ −1629.45 −1.35651 −0.678254 0.734828i $$-0.737263\pi$$
−0.678254 + 0.734828i $$0.737263\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −819.886 −0.656245
$$117$$ 0 0
$$118$$ −660.967 −0.515652
$$119$$ 347.152 0.267423
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ −2365.18 −1.75519
$$123$$ 0 0
$$124$$ 939.913 0.680699
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −2291.26 −1.60091 −0.800457 0.599390i $$-0.795410\pi$$
−0.800457 + 0.599390i $$0.795410\pi$$
$$128$$ 1663.35 1.14860
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1147.41 0.765267 0.382633 0.923900i $$-0.375017\pi$$
0.382633 + 0.923900i $$0.375017\pi$$
$$132$$ 0 0
$$133$$ −373.402 −0.243444
$$134$$ −3037.03 −1.95791
$$135$$ 0 0
$$136$$ −1141.86 −0.719956
$$137$$ 1268.60 0.791121 0.395561 0.918440i $$-0.370550\pi$$
0.395561 + 0.918440i $$0.370550\pi$$
$$138$$ 0 0
$$139$$ −486.288 −0.296737 −0.148368 0.988932i $$-0.547402\pi$$
−0.148368 + 0.988932i $$0.547402\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −2551.13 −1.50765
$$143$$ −165.239 −0.0966294
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 3439.10 1.94947
$$147$$ 0 0
$$148$$ −345.255 −0.191756
$$149$$ −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 $$0.549091\pi$$
$$152$$ 1228.21 0.655399
$$153$$ 0 0
$$154$$ −176.000 −0.0920941
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 2072.67 1.05361 0.526807 0.849985i $$-0.323389\pi$$
0.526807 + 0.849985i $$0.323389\pi$$
$$158$$ −1104.59 −0.556179
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 531.391 0.260121
$$162$$ 0 0
$$163$$ −2676.51 −1.28614 −0.643069 0.765808i $$-0.722339\pi$$
−0.643069 + 0.765808i $$0.722339\pi$$
$$164$$ 813.581 0.387378
$$165$$ 0 0
$$166$$ −2551.53 −1.19300
$$167$$ −1188.12 −0.550536 −0.275268 0.961368i $$-0.588767\pi$$
−0.275268 + 0.961368i $$0.588767\pi$$
$$168$$ 0 0
$$169$$ −1971.35 −0.897290
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 944.293 0.418615
$$173$$ 807.147 0.354718 0.177359 0.984146i $$-0.443245\pi$$
0.177359 + 0.984146i $$0.443245\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 875.666 0.375033
$$177$$ 0 0
$$178$$ −1716.42 −0.722758
$$179$$ 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 $$0.430053\pi$$
$$182$$ 240.348 0.0978889
$$183$$ 0 0
$$184$$ −1747.87 −0.700297
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −804.853 −0.314742
$$188$$ −572.848 −0.222230
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2136.41 −0.809348 −0.404674 0.914461i $$-0.632615\pi$$
−0.404674 + 0.914461i $$0.632615\pi$$
$$192$$ 0 0
$$193$$ −3947.76 −1.47236 −0.736181 0.676784i $$-0.763373\pi$$
−0.736181 + 0.676784i $$0.763373\pi$$
$$194$$ −2071.79 −0.766731
$$195$$ 0 0
$$196$$ −1080.78 −0.393870
$$197$$ 923.886 0.334133 0.167066 0.985946i $$-0.446571\pi$$
0.167066 + 0.985946i $$0.446571\pi$$
$$198$$ 0 0
$$199$$ −476.152 −0.169616 −0.0848078 0.996397i $$-0.527028\pi$$
−0.0848078 + 0.996397i $$0.527028\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 3425.96 1.19332
$$203$$ −1153.52 −0.398824
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −3716.80 −1.25710
$$207$$ 0 0
$$208$$ −1195.82 −0.398631
$$209$$ 865.712 0.286519
$$210$$ 0 0
$$211$$ −4918.24 −1.60467 −0.802336 0.596872i $$-0.796410\pi$$
−0.802336 + 0.596872i $$0.796410\pi$$
$$212$$ −1381.46 −0.447543
$$213$$ 0 0
$$214$$ 4645.57 1.48395
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1322.39 0.413686
$$218$$ 1079.86 0.335494
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1099.12 0.334546
$$222$$ 0 0
$$223$$ −2100.29 −0.630700 −0.315350 0.948975i $$-0.602122\pi$$
−0.315350 + 0.948975i $$0.602122\pi$$
$$224$$ −681.348 −0.203234
$$225$$ 0 0
$$226$$ −5494.95 −1.61734
$$227$$ −2257.16 −0.659970 −0.329985 0.943986i $$-0.607044\pi$$
−0.329985 + 0.943986i $$0.607044\pi$$
$$228$$ 0 0
$$229$$ −5311.07 −1.53260 −0.766301 0.642482i $$-0.777905\pi$$
−0.766301 + 0.642482i $$0.777905\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 3794.20 1.07371
$$233$$ 2466.27 0.693435 0.346718 0.937970i $$-0.387296\pi$$
0.346718 + 0.937970i $$0.387296\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −660.967 −0.182311
$$237$$ 0 0
$$238$$ 1170.70 0.318844
$$239$$ −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 $$0.541766\pi$$
$$242$$ 408.046 0.108389
$$243$$ 0 0
$$244$$ −2365.18 −0.620553
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −1182.23 −0.304548
$$248$$ −4349.65 −1.11372
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 6530.63 1.64227 0.821135 0.570734i $$-0.193341\pi$$
0.821135 + 0.570734i $$0.193341\pi$$
$$252$$ 0 0
$$253$$ −1232.00 −0.306147
$$254$$ −7726.76 −1.90874
$$255$$ 0 0
$$256$$ 4388.74 1.07147
$$257$$ 8130.26 1.97335 0.986676 0.162696i $$-0.0520188\pi$$
0.986676 + 0.162696i $$0.0520188\pi$$
$$258$$ 0 0
$$259$$ −485.750 −0.116537
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 3869.40 0.912414
$$263$$ −4549.42 −1.06665 −0.533326 0.845910i $$-0.679058\pi$$
−0.533326 + 0.845910i $$0.679058\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −1259.22 −0.290254
$$267$$ 0 0
$$268$$ −3037.03 −0.692225
$$269$$ 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 $$0.167794\pi$$
$$272$$ −5824.64 −1.29842
$$273$$ 0 0
$$274$$ 4278.07 0.943239
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −1127.52 −0.244571 −0.122286 0.992495i $$-0.539022\pi$$
−0.122286 + 0.992495i $$0.539022\pi$$
$$278$$ −1639.90 −0.353794
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 1872.47 0.397517 0.198758 0.980049i $$-0.436309\pi$$
0.198758 + 0.980049i $$0.436309\pi$$
$$282$$ 0 0
$$283$$ −2124.48 −0.446245 −0.223123 0.974790i $$-0.571625\pi$$
−0.223123 + 0.974790i $$0.571625\pi$$
$$284$$ −2551.13 −0.533034
$$285$$ 0 0
$$286$$ −557.233 −0.115209
$$287$$ 1144.65 0.235424
$$288$$ 0 0
$$289$$ 440.621 0.0896846
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 3439.10 0.689241
$$293$$ −3324.19 −0.662802 −0.331401 0.943490i $$-0.607521\pi$$
−0.331401 + 0.943490i $$0.607521\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 1597.75 0.313740
$$297$$ 0 0
$$298$$ −7938.73 −1.54322
$$299$$ 1682.44 0.325411
$$300$$ 0 0
$$301$$ 1328.55 0.254407
$$302$$ −1922.44 −0.366304
$$303$$ 0 0
$$304$$ 6265.07 1.18200
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1698.94 0.315843 0.157921 0.987452i $$-0.449521\pi$$
0.157921 + 0.987452i $$0.449521\pi$$
$$308$$ −176.000 −0.0325602
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −6928.83 −1.26334 −0.631668 0.775239i $$-0.717630\pi$$
−0.631668 + 0.775239i $$0.717630\pi$$
$$312$$ 0 0
$$313$$ 3560.75 0.643020 0.321510 0.946906i $$-0.395810\pi$$
0.321510 + 0.946906i $$0.395810\pi$$
$$314$$ 6989.64 1.25620
$$315$$ 0 0
$$316$$ −1104.59 −0.196639
$$317$$ 332.750 0.0589561 0.0294780 0.999565i $$-0.490615\pi$$
0.0294780 + 0.999565i $$0.490615\pi$$
$$318$$ 0 0
$$319$$ 2674.37 0.469393
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 1792.00 0.310137
$$323$$ −5758.43 −0.991975
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −9025.94 −1.53344
$$327$$ 0 0
$$328$$ −3765.02 −0.633807
$$329$$ −805.957 −0.135057
$$330$$ 0 0
$$331$$ −541.445 −0.0899108 −0.0449554 0.998989i $$-0.514315\pi$$
−0.0449554 + 0.998989i $$0.514315\pi$$
$$332$$ −2551.53 −0.421788
$$333$$ 0 0
$$334$$ −4006.67 −0.656393
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −816.531 −0.131986 −0.0659930 0.997820i $$-0.521022\pi$$
−0.0659930 + 0.997820i $$0.521022\pi$$
$$338$$ −6647.94 −1.06982
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −3065.89 −0.486883
$$342$$ 0 0
$$343$$ −3147.97 −0.495552
$$344$$ −4369.92 −0.684914
$$345$$ 0 0
$$346$$ 2721.93 0.422924
$$347$$ 6260.53 0.968539 0.484269 0.874919i $$-0.339086\pi$$
0.484269 + 0.874919i $$0.339086\pi$$
$$348$$ 0 0
$$349$$ −12768.5 −1.95840 −0.979198 0.202906i $$-0.934961\pi$$
−0.979198 + 0.202906i $$0.934961\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 1579.67 0.239194
$$353$$ −2649.28 −0.399453 −0.199727 0.979852i $$-0.564005\pi$$
−0.199727 + 0.979852i $$0.564005\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −1716.42 −0.255534
$$357$$ 0 0
$$358$$ 6577.27 0.971004
$$359$$ 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
$$362$$ 3580.04 0.519786
$$363$$ 0 0
$$364$$ 240.348 0.0346089
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8429.40 1.19894 0.599470 0.800397i $$-0.295378\pi$$
0.599470 + 0.800397i $$0.295378\pi$$
$$368$$ −8915.87 −1.26297
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1943.62 −0.271988
$$372$$ 0 0
$$373$$ 9388.53 1.30327 0.651635 0.758533i $$-0.274083\pi$$
0.651635 + 0.758533i $$0.274083\pi$$
$$374$$ −2714.19 −0.375261
$$375$$ 0 0
$$376$$ 2650.98 0.363600
$$377$$ −3652.16 −0.498928
$$378$$ 0 0
$$379$$ −14264.5 −1.93329 −0.966647 0.256112i $$-0.917558\pi$$
−0.966647 + 0.256112i $$0.917558\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ −7204.58 −0.964970
$$383$$ 13462.2 1.79605 0.898026 0.439942i $$-0.145001\pi$$
0.898026 + 0.439942i $$0.145001\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −13313.0 −1.75547
$$387$$ 0 0
$$388$$ −2071.79 −0.271080
$$389$$ 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
$$392$$ 5001.54 0.644429
$$393$$ 0 0
$$394$$ 3115.60 0.398380
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 847.839 0.107183 0.0535917 0.998563i $$-0.482933\pi$$
0.0535917 + 0.998563i $$0.482933\pi$$
$$398$$ −1605.72 −0.202230
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −12203.6 −1.51975 −0.759875 0.650069i $$-0.774740\pi$$
−0.759875 + 0.650069i $$0.774740\pi$$
$$402$$ 0 0
$$403$$ 4186.82 0.517520
$$404$$ 3425.96 0.421901
$$405$$ 0 0
$$406$$ −3890.00 −0.475511
$$407$$ 1126.18 0.137157
$$408$$ 0 0
$$409$$ 8759.53 1.05900 0.529500 0.848310i $$-0.322380\pi$$
0.529500 + 0.848310i $$0.322380\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −3716.80 −0.444451
$$413$$ −929.934 −0.110797
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2157.21 −0.254245
$$417$$ 0 0
$$418$$ 2919.42 0.341612
$$419$$ 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 $$0.803327\pi$$
$$422$$ −16585.7 −1.91322
$$423$$ 0 0
$$424$$ 6393.02 0.732246
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −3327.64 −0.377133
$$428$$ 4645.57 0.524655
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 5616.05 0.627647 0.313823 0.949481i $$-0.398390\pi$$
0.313823 + 0.949481i $$0.398390\pi$$
$$432$$ 0 0
$$433$$ −7195.75 −0.798627 −0.399314 0.916814i $$-0.630752\pi$$
−0.399314 + 0.916814i $$0.630752\pi$$
$$434$$ 4459.48 0.493230
$$435$$ 0 0
$$436$$ 1079.86 0.118615
$$437$$ −8814.52 −0.964887
$$438$$ 0 0
$$439$$ 101.959 0.0110848 0.00554240 0.999985i $$-0.498236\pi$$
0.00554240 + 0.999985i $$0.498236\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 3706.53 0.398873
$$443$$ 4953.74 0.531285 0.265642 0.964072i $$-0.414416\pi$$
0.265642 + 0.964072i $$0.414416\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −7082.78 −0.751972
$$447$$ 0 0
$$448$$ 723.869 0.0763383
$$449$$ 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
$$452$$ −5494.95 −0.571816
$$453$$ 0 0
$$454$$ −7611.79 −0.786870
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 3530.68 0.361397 0.180698 0.983539i $$-0.442164\pi$$
0.180698 + 0.983539i $$0.442164\pi$$
$$458$$ −17910.4 −1.82729
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −11566.3 −1.16854 −0.584271 0.811559i $$-0.698619\pi$$
−0.584271 + 0.811559i $$0.698619\pi$$
$$462$$ 0 0
$$463$$ −10888.5 −1.09294 −0.546470 0.837479i $$-0.684029\pi$$
−0.546470 + 0.837479i $$0.684029\pi$$
$$464$$ 19354.2 1.93641
$$465$$ 0 0
$$466$$ 8316.94 0.826770
$$467$$ 10688.0 1.05906 0.529529 0.848292i $$-0.322369\pi$$
0.529529 + 0.848292i $$0.322369\pi$$
$$468$$ 0 0
$$469$$ −4272.89 −0.420690
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 3058.77 0.298287
$$473$$ −3080.18 −0.299422
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 1170.70 0.112728
$$477$$ 0 0
$$478$$ −4820.35 −0.461250
$$479$$ −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
$$482$$ −3301.43 −0.311983
$$483$$ 0 0
$$484$$ 408.046 0.0383214
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −6748.91 −0.627972 −0.313986 0.949428i $$-0.601665\pi$$
−0.313986 + 0.949428i $$0.601665\pi$$
$$488$$ 10945.4 1.01532
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −7361.40 −0.676609 −0.338305 0.941037i $$-0.609853\pi$$
−0.338305 + 0.941037i $$0.609853\pi$$
$$492$$ 0 0
$$493$$ −17789.1 −1.62511
$$494$$ −3986.80 −0.363107
$$495$$ 0 0
$$496$$ −22187.6 −2.00857
$$497$$ −3589.26 −0.323944
$$498$$ 0 0
$$499$$ 10381.7 0.931359 0.465680 0.884953i $$-0.345810\pi$$
0.465680 + 0.884953i $$0.345810\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 22023.1 1.95805
$$503$$ 19149.0 1.69744 0.848721 0.528840i $$-0.177373\pi$$
0.848721 + 0.528840i $$0.177373\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −4154.65 −0.365013
$$507$$ 0 0
$$508$$ −7726.76 −0.674841
$$509$$ −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
$$512$$ 1493.27 0.128894
$$513$$ 0 0
$$514$$ 27417.5 2.35279
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1868.56 0.158954
$$518$$ −1638.09 −0.138945
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 18955.3 1.59395 0.796975 0.604012i $$-0.206432\pi$$
0.796975 + 0.604012i $$0.206432\pi$$
$$522$$ 0 0
$$523$$ 4442.19 0.371402 0.185701 0.982606i $$-0.440544\pi$$
0.185701 + 0.982606i $$0.440544\pi$$
$$524$$ 3869.40 0.322587
$$525$$ 0 0
$$526$$ −15341.9 −1.27175
$$527$$ 20393.3 1.68567
$$528$$ 0 0
$$529$$ 377.000 0.0309855
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −1259.22 −0.102620
$$533$$ 3624.08 0.294515
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 14054.5 1.13258
$$537$$ 0 0
$$538$$ 98.3096 0.00787812
$$539$$ 3525.38 0.281723
$$540$$ 0 0
$$541$$ 2180.90 0.173316 0.0866580 0.996238i $$-0.472381\pi$$
0.0866580 + 0.996238i $$0.472381\pi$$
$$542$$ 26004.4 2.06086
$$543$$ 0 0
$$544$$ −10507.4 −0.828129
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −8225.04 −0.642920 −0.321460 0.946923i $$-0.604174\pi$$
−0.321460 + 0.946923i $$0.604174\pi$$
$$548$$ 4278.07 0.333485
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 19134.2 1.47939
$$552$$ 0 0
$$553$$ −1554.08 −0.119505
$$554$$ −3802.32 −0.291598
$$555$$ 0 0
$$556$$ −1639.90 −0.125085
$$557$$ −25181.9 −1.91561 −0.957804 0.287423i $$-0.907201\pi$$
−0.957804 + 0.287423i $$0.907201\pi$$
$$558$$ 0 0
$$559$$ 4206.33 0.318263
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 6314.50 0.473952
$$563$$ −4504.50 −0.337197 −0.168599 0.985685i $$-0.553924\pi$$
−0.168599 + 0.985685i $$0.553924\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −7164.36 −0.532050
$$567$$ 0 0
$$568$$ 11805.9 0.872122
$$569$$ 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 $$0.530438\pi$$
$$572$$ −557.233 −0.0407327
$$573$$ 0 0
$$574$$ 3860.09 0.280691
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −6339.65 −0.457406 −0.228703 0.973496i $$-0.573448\pi$$
−0.228703 + 0.973496i $$0.573448\pi$$
$$578$$ 1485.90 0.106929
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3589.83 −0.256336
$$582$$ 0 0
$$583$$ 4506.17 0.320114
$$584$$ −15915.2 −1.12770
$$585$$ 0 0
$$586$$ −11210.1 −0.790247
$$587$$ −13370.6 −0.940140 −0.470070 0.882629i $$-0.655771\pi$$
−0.470070 + 0.882629i $$0.655771\pi$$
$$588$$ 0 0
$$589$$ −21935.3 −1.53452
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 8150.09 0.565822
$$593$$ 14319.3 0.991608 0.495804 0.868434i $$-0.334873\pi$$
0.495804 + 0.868434i $$0.334873\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −7938.73 −0.545609
$$597$$ 0 0
$$598$$ 5673.65 0.387981
$$599$$ 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 $$0.197628\pi$$
$$602$$ 4480.26 0.303325
$$603$$ 0 0
$$604$$ −1922.44 −0.129508
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 23526.6 1.57317 0.786585 0.617482i $$-0.211847\pi$$
0.786585 + 0.617482i $$0.211847\pi$$
$$608$$ 11301.9 0.753872
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −2551.74 −0.168956
$$612$$ 0 0
$$613$$ −1228.07 −0.0809159 −0.0404579 0.999181i $$-0.512882\pi$$
−0.0404579 + 0.999181i $$0.512882\pi$$
$$614$$ 5729.31 0.376573
$$615$$ 0 0
$$616$$ 814.478 0.0532732
$$617$$ −9844.90 −0.642368 −0.321184 0.947017i $$-0.604081\pi$$
−0.321184 + 0.947017i $$0.604081\pi$$
$$618$$ 0 0
$$619$$ −6551.68 −0.425419 −0.212709 0.977115i $$-0.568229\pi$$
−0.212709 + 0.977115i $$0.568229\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −23365.9 −1.50625
$$623$$ −2414.88 −0.155297
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 12007.8 0.766661
$$627$$ 0 0
$$628$$ 6989.64 0.444135
$$629$$ −7491.01 −0.474859
$$630$$ 0 0
$$631$$ −26440.5 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$632$$ 5111.72 0.321730
$$633$$ 0 0
$$634$$ 1122.13 0.0702923
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −4814.31 −0.299450
$$638$$ 9018.74 0.559648
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 27927.2 1.72084 0.860421 0.509584i $$-0.170201\pi$$
0.860421 + 0.509584i $$0.170201\pi$$
$$642$$ 0 0
$$643$$ 16737.7 1.02655 0.513274 0.858225i $$-0.328432\pi$$
0.513274 + 0.858225i $$0.328432\pi$$
$$644$$ 1792.00 0.109650
$$645$$ 0 0
$$646$$ −19419.1 −1.18271
$$647$$ 7818.70 0.475092 0.237546 0.971376i $$-0.423657\pi$$
0.237546 + 0.971376i $$0.423657\pi$$
$$648$$ 0 0
$$649$$ 2156.00 0.130401
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −9025.94 −0.542152
$$653$$ 19747.6 1.18344 0.591719 0.806144i $$-0.298450\pi$$
0.591719 + 0.806144i $$0.298450\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −19205.4 −1.14305
$$657$$ 0 0
$$658$$ −2717.91 −0.161026
$$659$$ −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 $$0.460307\pi$$
$$662$$ −1825.90 −0.107199
$$663$$ 0 0
$$664$$ 11807.8 0.690106
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −27230.0 −1.58073
$$668$$ −4006.67 −0.232070
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 7714.94 0.443863
$$672$$ 0 0
$$673$$ −29397.6 −1.68379 −0.841897 0.539638i $$-0.818561\pi$$
−0.841897 + 0.539638i $$0.818561\pi$$
$$674$$ −2753.57 −0.157364
$$675$$ 0 0
$$676$$ −6647.94 −0.378239
$$677$$ 5737.14 0.325696 0.162848 0.986651i $$-0.447932\pi$$
0.162848 + 0.986651i $$0.447932\pi$$
$$678$$ 0 0
$$679$$ −2914.86 −0.164745
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −10339.0 −0.580502
$$683$$ 32097.6 1.79821 0.899107 0.437729i $$-0.144217\pi$$
0.899107 + 0.437729i $$0.144217\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −10615.8 −0.590837
$$687$$ 0 0
$$688$$ −22291.0 −1.23523
$$689$$ −6153.69 −0.340257
$$690$$ 0 0
$$691$$ −16456.2 −0.905965 −0.452983 0.891519i $$-0.649640\pi$$
−0.452983 + 0.891519i $$0.649640\pi$$
$$692$$ 2721.93 0.149526
$$693$$ 0 0
$$694$$ 21112.3 1.15477
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 17652.3 0.959294
$$698$$ −43058.9 −2.33496
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −27238.1 −1.46758 −0.733788 0.679379i $$-0.762249\pi$$
−0.733788 + 0.679379i $$0.762249\pi$$
$$702$$ 0 0
$$703$$ 8057.44 0.432279
$$704$$ −1678.25 −0.0898457
$$705$$ 0 0
$$706$$ −8934.12 −0.476261
$$707$$ 4820.09 0.256405
$$708$$ 0 0
$$709$$ 28761.4 1.52349 0.761747 0.647875i $$-0.224342\pi$$
0.761747 + 0.647875i $$0.224342\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 7943.10 0.418090
$$713$$ 31216.3 1.63964
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 6577.27 0.343302
$$717$$ 0 0
$$718$$ 10804.5 0.561588
$$719$$ 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
$$722$$ −2243.05 −0.115620
$$723$$ 0 0
$$724$$ 3580.04 0.183772
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −3979.75 −0.203027 −0.101514 0.994834i $$-0.532369\pi$$
−0.101514 + 0.994834i $$0.532369\pi$$
$$728$$ −1112.26 −0.0566253
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 20488.3 1.03665
$$732$$ 0 0
$$733$$ −9342.48 −0.470767 −0.235384 0.971903i $$-0.575635\pi$$
−0.235384 + 0.971903i $$0.575635\pi$$
$$734$$ 28426.3 1.42947
$$735$$ 0 0
$$736$$ −16083.9 −0.805515
$$737$$ 9906.45 0.495127
$$738$$ 0 0
$$739$$ −28928.0 −1.43997 −0.719983 0.693992i $$-0.755850\pi$$
−0.719983 + 0.693992i $$0.755850\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −6554.43 −0.324287
$$743$$ 4857.04 0.239822 0.119911 0.992785i $$-0.461739\pi$$
0.119911 + 0.992785i $$0.461739\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 31660.8 1.55386
$$747$$ 0 0
$$748$$ −2714.19 −0.132675
$$749$$ 6536.00 0.318852
$$750$$ 0 0
$$751$$ 14355.4 0.697517 0.348759 0.937213i $$-0.386603\pi$$
0.348759 + 0.937213i $$0.386603\pi$$
$$752$$ 13522.6 0.655744
$$753$$ 0 0
$$754$$ −12316.1 −0.594863
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 17714.9 0.850538 0.425269 0.905067i $$-0.360179\pi$$
0.425269 + 0.905067i $$0.360179\pi$$
$$758$$ −48103.9 −2.30503
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 7945.82 0.378497 0.189248 0.981929i $$-0.439395\pi$$
0.189248 + 0.981929i $$0.439395\pi$$
$$762$$ 0 0
$$763$$ 1519.29 0.0720866
$$764$$ −7204.58 −0.341168
$$765$$ 0 0
$$766$$ 45398.4 2.14140
$$767$$ −2944.26 −0.138606
$$768$$ 0 0
$$769$$ 27308.1 1.28057 0.640284 0.768139i $$-0.278817\pi$$
0.640284 + 0.768139i $$0.278817\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −13313.0 −0.620653
$$773$$ −18872.6 −0.878136 −0.439068 0.898454i $$-0.644691\pi$$
−0.439068 + 0.898454i $$0.644691\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 9587.65 0.443527
$$777$$ 0 0
$$778$$ 3176.29 0.146369
$$779$$ −18987.1 −0.873276
$$780$$ 0 0
$$781$$ 8321.50 0.381263
$$782$$ 27635.4 1.26373
$$783$$ 0 0
$$784$$ 25512.8 1.16221
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 14512.1 0.657307 0.328654 0.944450i $$-0.393405\pi$$
0.328654 + 0.944450i $$0.393405\pi$$
$$788$$ 3115.60 0.140849
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −7731.01 −0.347513
$$792$$ 0 0
$$793$$ −10535.6 −0.471792
$$794$$ 2859.15 0.127793
$$795$$ 0 0
$$796$$ −1605.72 −0.0714989
$$797$$ 29108.9 1.29371 0.646856 0.762612i $$-0.276083\pi$$
0.646856 + 0.762612i $$0.276083\pi$$
$$798$$ 0 0
$$799$$ −12429.1 −0.550325
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −41154.0 −1.81197
$$803$$ −11218.0 −0.492993
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 14119.1 0.617029
$$807$$ 0 0
$$808$$ −15854.4 −0.690291
$$809$$ 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 $$0.456872\pi$$
$$812$$ −3890.00 −0.168118
$$813$$ 0 0
$$814$$ 3797.81 0.163530
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −22037.6 −0.943693
$$818$$ 29539.6 1.26263
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −14922.4 −0.634342 −0.317171 0.948368i $$-0.602733\pi$$
−0.317171 + 0.948368i $$0.602733\pi$$
$$822$$ 0 0
$$823$$ 25737.8 1.09011 0.545057 0.838399i $$-0.316508\pi$$
0.545057 + 0.838399i $$0.316508\pi$$
$$824$$ 17200.3 0.727186
$$825$$ 0 0
$$826$$ −3136.00 −0.132101
$$827$$ 27043.4 1.13711 0.568555 0.822645i $$-0.307503\pi$$
0.568555 + 0.822645i $$0.307503\pi$$
$$828$$ 0 0
$$829$$ −9795.41 −0.410384 −0.205192 0.978722i $$-0.565782\pi$$
−0.205192 + 0.978722i $$0.565782\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 2291.84 0.0954990
$$833$$ −23449.7 −0.975370
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 2919.42 0.120778
$$837$$ 0 0
$$838$$ 37730.5 1.55535
$$839$$ −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
$$842$$ −47489.3 −1.94369
$$843$$ 0 0
$$844$$ −16585.7 −0.676426
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 574.092 0.0232893
$$848$$ 32610.7 1.32059
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −11466.6 −0.461892
$$852$$ 0 0
$$853$$ 47157.1 1.89288 0.946441 0.322878i $$-0.104650\pi$$
0.946441 + 0.322878i $$0.104650\pi$$
$$854$$ −11221.7 −0.449649
$$855$$ 0 0
$$856$$ −21498.4 −0.858412
$$857$$ 5021.31 0.200145 0.100073 0.994980i $$-0.468092\pi$$
0.100073 + 0.994980i $$0.468092\pi$$
$$858$$ 0 0
$$859$$ −22921.1 −0.910428 −0.455214 0.890382i $$-0.650437\pi$$
−0.455214 + 0.890382i $$0.650437\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 18938.9 0.748332
$$863$$ −19488.1 −0.768693 −0.384347 0.923189i $$-0.625573\pi$$
−0.384347 + 0.923189i $$0.625573\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −24266.1 −0.952188
$$867$$ 0 0
$$868$$ 4459.48 0.174383
$$869$$ 3603.04 0.140650
$$870$$ 0 0
$$871$$ −13528.4 −0.526282
$$872$$ −4997.30 −0.194071
$$873$$ 0 0
$$874$$ −29725.0 −1.15042
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8455.67 0.325573 0.162787 0.986661i $$-0.447952\pi$$
0.162787 + 0.986661i $$0.447952\pi$$
$$878$$ 343.834 0.0132162
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 11291.2 0.431794 0.215897 0.976416i $$-0.430732\pi$$
0.215897 + 0.976416i $$0.430732\pi$$
$$882$$ 0 0
$$883$$ −31818.1 −1.21264 −0.606322 0.795219i $$-0.707356\pi$$
−0.606322 + 0.795219i $$0.707356\pi$$
$$884$$ 3706.53 0.141023
$$885$$ 0 0
$$886$$ 16705.4 0.633441
$$887$$ 17481.1 0.661732 0.330866 0.943678i $$-0.392659\pi$$
0.330866 + 0.943678i $$0.392659\pi$$
$$888$$ 0 0
$$889$$ −10871.0 −0.410126
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −7082.78 −0.265862
$$893$$ 13368.9 0.500978
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 7891.87 0.294251
$$897$$ 0 0
$$898$$ 39125.1 1.45392
$$899$$ −67763.1 −2.51393
$$900$$ 0 0
$$901$$ −29973.6 −1.10829
$$902$$ −8949.39 −0.330357
$$903$$ 0 0
$$904$$ 25429.1 0.935574
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −10607.4 −0.388326 −0.194163 0.980969i $$-0.562199\pi$$
−0.194163 + 0.980969i $$0.562199\pi$$
$$908$$ −7611.79 −0.278201
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 41249.2 1.50016 0.750080 0.661347i $$-0.230015\pi$$
0.750080 + 0.661347i $$0.230015\pi$$
$$912$$ 0 0
$$913$$ 8322.81 0.301692
$$914$$ 11906.5 0.430887
$$915$$ 0 0
$$916$$ −17910.4 −0.646045
$$917$$ 5443.97 0.196048
$$918$$ 0 0
$$919$$ −13858.1 −0.497429 −0.248714 0.968577i $$-0.580008\pi$$
−0.248714 + 0.968577i $$0.580008\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −39004.9 −1.39323
$$923$$ −11363.9 −0.405253
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −36719.0 −1.30309
$$927$$ 0 0
$$928$$ 34914.2 1.23504
$$929$$ −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
$$932$$ 8316.94 0.292307
$$933$$ 0 0
$$934$$ 36042.9 1.26270
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −3203.52 −0.111691 −0.0558454 0.998439i $$-0.517785\pi$$
−0.0558454 + 0.998439i $$0.517785\pi$$
$$938$$ −14409.4 −0.501581
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −19951.6 −0.691182 −0.345591 0.938385i $$-0.612322\pi$$
−0.345591 + 0.938385i $$0.612322\pi$$
$$942$$ 0 0
$$943$$ 27020.6 0.933099
$$944$$ 15602.8 0.537952
$$945$$ 0 0
$$946$$ −10387.2 −0.356996
$$947$$ −38216.7 −1.31138 −0.655689 0.755031i $$-0.727622\pi$$
−0.655689 + 0.755031i $$0.727622\pi$$
$$948$$ 0 0
$$949$$ 15319.4 0.524014
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −5417.65 −0.184440
$$953$$ −47661.4 −1.62004 −0.810022 0.586399i $$-0.800545\pi$$
−0.810022 + 0.586399i $$0.800545\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −4820.35 −0.163077
$$957$$ 0 0
$$958$$ −7897.56 −0.266345
$$959$$ 6018.94 0.202671
$$960$$ 0 0
$$961$$ 47892.3 1.60761
$$962$$ −5186.34 −0.173819
$$963$$ 0 0
$$964$$ −3301.43 −0.110303
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −18933.2 −0.629628 −0.314814 0.949153i $$-0.601942\pi$$
−0.314814 + 0.949153i $$0.601942\pi$$
$$968$$ −1888.32 −0.0626994
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 40660.3 1.34382 0.671911 0.740632i $$-0.265474\pi$$
0.671911 + 0.740632i $$0.265474\pi$$
$$972$$ 0 0
$$973$$ −2307.23 −0.0760188
$$974$$ −22759.2 −0.748720
$$975$$ 0 0
$$976$$ 55832.3 1.83110
$$977$$ −22502.8 −0.736876 −0.368438 0.929652i $$-0.620107\pi$$
−0.368438 + 0.929652i $$0.620107\pi$$
$$978$$ 0 0
$$979$$ 5598.76 0.182775
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −24824.7 −0.806709
$$983$$ −4435.20 −0.143907 −0.0719536 0.997408i $$-0.522923\pi$$
−0.0719536 + 0.997408i $$0.522923\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −59989.8 −1.93759
$$987$$ 0 0
$$988$$ −3986.80 −0.128378
$$989$$ 31361.8 1.00834
$$990$$ 0 0
$$991$$ 7362.76 0.236010 0.118005 0.993013i $$-0.462350\pi$$
0.118005 + 0.993013i $$0.462350\pi$$
$$992$$ −40025.5 −1.28106
$$993$$ 0 0
$$994$$ −12104.0 −0.386233
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 53480.1 1.69883 0.849413 0.527728i $$-0.176956\pi$$
0.849413 + 0.527728i $$0.176956\pi$$
$$998$$ 35010.0 1.11044
$$999$$ 0 0
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 2475.4.a.o.1.2 2
3.2 odd 2 825.4.a.k.1.1 2
5.4 even 2 99.4.a.e.1.1 2
15.2 even 4 825.4.c.i.199.1 4
15.8 even 4 825.4.c.i.199.4 4
15.14 odd 2 33.4.a.d.1.2 2
20.19 odd 2 1584.4.a.x.1.2 2
55.54 odd 2 1089.4.a.t.1.2 2
60.59 even 2 528.4.a.o.1.1 2
105.104 even 2 1617.4.a.j.1.2 2
120.29 odd 2 2112.4.a.ba.1.2 2
120.59 even 2 2112.4.a.bh.1.2 2
165.164 even 2 363.4.a.j.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.d.1.2 2 15.14 odd 2
99.4.a.e.1.1 2 5.4 even 2
363.4.a.j.1.1 2 165.164 even 2
528.4.a.o.1.1 2 60.59 even 2
825.4.a.k.1.1 2 3.2 odd 2
825.4.c.i.199.1 4 15.2 even 4
825.4.c.i.199.4 4 15.8 even 4
1089.4.a.t.1.2 2 55.54 odd 2
1584.4.a.x.1.2 2 20.19 odd 2
1617.4.a.j.1.2 2 105.104 even 2
2112.4.a.ba.1.2 2 120.29 odd 2
2112.4.a.bh.1.2 2 120.59 even 2
2475.4.a.o.1.2 2 1.1 even 1 trivial