# Properties

 Label 784.4.a.y.1.2 Level $784$ Weight $4$ Character 784.1 Self dual yes Analytic conductor $46.257$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 784.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: $$46.2574974445$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 98) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 784.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+7.07107 q^{3} -19.7990 q^{5} +23.0000 q^{9} +O(q^{10})$$ $$q+7.07107 q^{3} -19.7990 q^{5} +23.0000 q^{9} +14.0000 q^{11} +50.9117 q^{13} -140.000 q^{15} +1.41421 q^{17} +1.41421 q^{19} -140.000 q^{23} +267.000 q^{25} -28.2843 q^{27} -286.000 q^{29} +93.3381 q^{31} +98.9949 q^{33} -38.0000 q^{37} +360.000 q^{39} -125.865 q^{41} +34.0000 q^{43} -455.377 q^{45} -523.259 q^{47} +10.0000 q^{51} -74.0000 q^{53} -277.186 q^{55} +10.0000 q^{57} -434.164 q^{59} +14.1421 q^{61} -1008.00 q^{65} -684.000 q^{67} -989.949 q^{69} -588.000 q^{71} -270.115 q^{73} +1887.98 q^{75} -1220.00 q^{79} -821.000 q^{81} -422.850 q^{83} -28.0000 q^{85} -2022.33 q^{87} +618.011 q^{89} +660.000 q^{93} -28.0000 q^{95} +1483.51 q^{97} +322.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 46 q^{9}+O(q^{10})$$ 2 * q + 46 * q^9 $$2 q + 46 q^{9} + 28 q^{11} - 280 q^{15} - 280 q^{23} + 534 q^{25} - 572 q^{29} - 76 q^{37} + 720 q^{39} + 68 q^{43} + 20 q^{51} - 148 q^{53} + 20 q^{57} - 2016 q^{65} - 1368 q^{67} - 1176 q^{71} - 2440 q^{79} - 1642 q^{81} - 56 q^{85} + 1320 q^{93} - 56 q^{95} + 644 q^{99}+O(q^{100})$$ 2 * q + 46 * q^9 + 28 * q^11 - 280 * q^15 - 280 * q^23 + 534 * q^25 - 572 * q^29 - 76 * q^37 + 720 * q^39 + 68 * q^43 + 20 * q^51 - 148 * q^53 + 20 * q^57 - 2016 * q^65 - 1368 * q^67 - 1176 * q^71 - 2440 * q^79 - 1642 * q^81 - 56 * q^85 + 1320 * q^93 - 56 * q^95 + 644 * q^99

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 7.07107 1.36083 0.680414 0.732828i $$-0.261800\pi$$
0.680414 + 0.732828i $$0.261800\pi$$
$$4$$ 0 0
$$5$$ −19.7990 −1.77088 −0.885438 0.464758i $$-0.846141\pi$$
−0.885438 + 0.464758i $$0.846141\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ 23.0000 0.851852
$$10$$ 0 0
$$11$$ 14.0000 0.383742 0.191871 0.981420i $$-0.438545\pi$$
0.191871 + 0.981420i $$0.438545\pi$$
$$12$$ 0 0
$$13$$ 50.9117 1.08618 0.543091 0.839674i $$-0.317254\pi$$
0.543091 + 0.839674i $$0.317254\pi$$
$$14$$ 0 0
$$15$$ −140.000 −2.40986
$$16$$ 0 0
$$17$$ 1.41421 0.0201763 0.0100882 0.999949i $$-0.496789\pi$$
0.0100882 + 0.999949i $$0.496789\pi$$
$$18$$ 0 0
$$19$$ 1.41421 0.0170759 0.00853797 0.999964i $$-0.497282\pi$$
0.00853797 + 0.999964i $$0.497282\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −140.000 −1.26922 −0.634609 0.772833i $$-0.718839\pi$$
−0.634609 + 0.772833i $$0.718839\pi$$
$$24$$ 0 0
$$25$$ 267.000 2.13600
$$26$$ 0 0
$$27$$ −28.2843 −0.201604
$$28$$ 0 0
$$29$$ −286.000 −1.83134 −0.915670 0.401931i $$-0.868339\pi$$
−0.915670 + 0.401931i $$0.868339\pi$$
$$30$$ 0 0
$$31$$ 93.3381 0.540775 0.270387 0.962752i $$-0.412848\pi$$
0.270387 + 0.962752i $$0.412848\pi$$
$$32$$ 0 0
$$33$$ 98.9949 0.522206
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −38.0000 −0.168842 −0.0844211 0.996430i $$-0.526904\pi$$
−0.0844211 + 0.996430i $$0.526904\pi$$
$$38$$ 0 0
$$39$$ 360.000 1.47811
$$40$$ 0 0
$$41$$ −125.865 −0.479434 −0.239717 0.970843i $$-0.577055\pi$$
−0.239717 + 0.970843i $$0.577055\pi$$
$$42$$ 0 0
$$43$$ 34.0000 0.120580 0.0602901 0.998181i $$-0.480797\pi$$
0.0602901 + 0.998181i $$0.480797\pi$$
$$44$$ 0 0
$$45$$ −455.377 −1.50852
$$46$$ 0 0
$$47$$ −523.259 −1.62394 −0.811970 0.583699i $$-0.801605\pi$$
−0.811970 + 0.583699i $$0.801605\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ 10.0000 0.0274565
$$52$$ 0 0
$$53$$ −74.0000 −0.191786 −0.0958932 0.995392i $$-0.530571\pi$$
−0.0958932 + 0.995392i $$0.530571\pi$$
$$54$$ 0 0
$$55$$ −277.186 −0.679559
$$56$$ 0 0
$$57$$ 10.0000 0.0232374
$$58$$ 0 0
$$59$$ −434.164 −0.958022 −0.479011 0.877809i $$-0.659005\pi$$
−0.479011 + 0.877809i $$0.659005\pi$$
$$60$$ 0 0
$$61$$ 14.1421 0.0296839 0.0148419 0.999890i $$-0.495275\pi$$
0.0148419 + 0.999890i $$0.495275\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −1008.00 −1.92349
$$66$$ 0 0
$$67$$ −684.000 −1.24722 −0.623611 0.781735i $$-0.714335\pi$$
−0.623611 + 0.781735i $$0.714335\pi$$
$$68$$ 0 0
$$69$$ −989.949 −1.72719
$$70$$ 0 0
$$71$$ −588.000 −0.982856 −0.491428 0.870918i $$-0.663525\pi$$
−0.491428 + 0.870918i $$0.663525\pi$$
$$72$$ 0 0
$$73$$ −270.115 −0.433076 −0.216538 0.976274i $$-0.569477\pi$$
−0.216538 + 0.976274i $$0.569477\pi$$
$$74$$ 0 0
$$75$$ 1887.98 2.90673
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1220.00 −1.73748 −0.868739 0.495271i $$-0.835069\pi$$
−0.868739 + 0.495271i $$0.835069\pi$$
$$80$$ 0 0
$$81$$ −821.000 −1.12620
$$82$$ 0 0
$$83$$ −422.850 −0.559202 −0.279601 0.960116i $$-0.590202\pi$$
−0.279601 + 0.960116i $$0.590202\pi$$
$$84$$ 0 0
$$85$$ −28.0000 −0.0357297
$$86$$ 0 0
$$87$$ −2022.33 −2.49214
$$88$$ 0 0
$$89$$ 618.011 0.736057 0.368028 0.929815i $$-0.380033\pi$$
0.368028 + 0.929815i $$0.380033\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 660.000 0.735901
$$94$$ 0 0
$$95$$ −28.0000 −0.0302394
$$96$$ 0 0
$$97$$ 1483.51 1.55286 0.776431 0.630202i $$-0.217028\pi$$
0.776431 + 0.630202i $$0.217028\pi$$
$$98$$ 0 0
$$99$$ 322.000 0.326891
$$100$$ 0 0
$$101$$ 1128.54 1.11182 0.555912 0.831241i $$-0.312369\pi$$
0.555912 + 0.831241i $$0.312369\pi$$
$$102$$ 0 0
$$103$$ 868.327 0.830668 0.415334 0.909669i $$-0.363665\pi$$
0.415334 + 0.909669i $$0.363665\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1684.00 1.52148 0.760740 0.649056i $$-0.224836\pi$$
0.760740 + 0.649056i $$0.224836\pi$$
$$108$$ 0 0
$$109$$ −818.000 −0.718809 −0.359405 0.933182i $$-0.617020\pi$$
−0.359405 + 0.933182i $$0.617020\pi$$
$$110$$ 0 0
$$111$$ −268.701 −0.229765
$$112$$ 0 0
$$113$$ −540.000 −0.449548 −0.224774 0.974411i $$-0.572164\pi$$
−0.224774 + 0.974411i $$0.572164\pi$$
$$114$$ 0 0
$$115$$ 2771.86 2.24763
$$116$$ 0 0
$$117$$ 1170.97 0.925266
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1135.00 −0.852742
$$122$$ 0 0
$$123$$ −890.000 −0.652428
$$124$$ 0 0
$$125$$ −2811.46 −2.01171
$$126$$ 0 0
$$127$$ −1720.00 −1.20177 −0.600887 0.799334i $$-0.705186\pi$$
−0.600887 + 0.799334i $$0.705186\pi$$
$$128$$ 0 0
$$129$$ 240.416 0.164089
$$130$$ 0 0
$$131$$ 1735.24 1.15732 0.578659 0.815570i $$-0.303576\pi$$
0.578659 + 0.815570i $$0.303576\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ 560.000 0.357016
$$136$$ 0 0
$$137$$ 828.000 0.516356 0.258178 0.966097i $$-0.416878\pi$$
0.258178 + 0.966097i $$0.416878\pi$$
$$138$$ 0 0
$$139$$ 425.678 0.259752 0.129876 0.991530i $$-0.458542\pi$$
0.129876 + 0.991530i $$0.458542\pi$$
$$140$$ 0 0
$$141$$ −3700.00 −2.20990
$$142$$ 0 0
$$143$$ 712.764 0.416813
$$144$$ 0 0
$$145$$ 5662.51 3.24308
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 2050.00 1.12713 0.563566 0.826071i $$-0.309429\pi$$
0.563566 + 0.826071i $$0.309429\pi$$
$$150$$ 0 0
$$151$$ 472.000 0.254376 0.127188 0.991879i $$-0.459405\pi$$
0.127188 + 0.991879i $$0.459405\pi$$
$$152$$ 0 0
$$153$$ 32.5269 0.0171872
$$154$$ 0 0
$$155$$ −1848.00 −0.957645
$$156$$ 0 0
$$157$$ −2211.83 −1.12435 −0.562176 0.827018i $$-0.690036\pi$$
−0.562176 + 0.827018i $$0.690036\pi$$
$$158$$ 0 0
$$159$$ −523.259 −0.260988
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3286.00 −1.57901 −0.789507 0.613741i $$-0.789664\pi$$
−0.789507 + 0.613741i $$0.789664\pi$$
$$164$$ 0 0
$$165$$ −1960.00 −0.924762
$$166$$ 0 0
$$167$$ 1490.58 0.690686 0.345343 0.938476i $$-0.387763\pi$$
0.345343 + 0.938476i $$0.387763\pi$$
$$168$$ 0 0
$$169$$ 395.000 0.179791
$$170$$ 0 0
$$171$$ 32.5269 0.0145462
$$172$$ 0 0
$$173$$ −2070.41 −0.909886 −0.454943 0.890521i $$-0.650340\pi$$
−0.454943 + 0.890521i $$0.650340\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −3070.00 −1.30370
$$178$$ 0 0
$$179$$ −540.000 −0.225483 −0.112742 0.993624i $$-0.535963\pi$$
−0.112742 + 0.993624i $$0.535963\pi$$
$$180$$ 0 0
$$181$$ 3784.44 1.55412 0.777058 0.629429i $$-0.216711\pi$$
0.777058 + 0.629429i $$0.216711\pi$$
$$182$$ 0 0
$$183$$ 100.000 0.0403946
$$184$$ 0 0
$$185$$ 752.362 0.298999
$$186$$ 0 0
$$187$$ 19.7990 0.00774249
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −1028.00 −0.389442 −0.194721 0.980859i $$-0.562380\pi$$
−0.194721 + 0.980859i $$0.562380\pi$$
$$192$$ 0 0
$$193$$ 4592.00 1.71264 0.856320 0.516446i $$-0.172745\pi$$
0.856320 + 0.516446i $$0.172745\pi$$
$$194$$ 0 0
$$195$$ −7127.64 −2.61754
$$196$$ 0 0
$$197$$ 794.000 0.287158 0.143579 0.989639i $$-0.454139\pi$$
0.143579 + 0.989639i $$0.454139\pi$$
$$198$$ 0 0
$$199$$ −2486.19 −0.885634 −0.442817 0.896612i $$-0.646021\pi$$
−0.442817 + 0.896612i $$0.646021\pi$$
$$200$$ 0 0
$$201$$ −4836.61 −1.69725
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 2492.00 0.849019
$$206$$ 0 0
$$207$$ −3220.00 −1.08119
$$208$$ 0 0
$$209$$ 19.7990 0.00655275
$$210$$ 0 0
$$211$$ 2748.00 0.896588 0.448294 0.893886i $$-0.352032\pi$$
0.448294 + 0.893886i $$0.352032\pi$$
$$212$$ 0 0
$$213$$ −4157.79 −1.33750
$$214$$ 0 0
$$215$$ −673.166 −0.213533
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −1910.00 −0.589342
$$220$$ 0 0
$$221$$ 72.0000 0.0219151
$$222$$ 0 0
$$223$$ 3428.05 1.02941 0.514707 0.857366i $$-0.327901\pi$$
0.514707 + 0.857366i $$0.327901\pi$$
$$224$$ 0 0
$$225$$ 6141.00 1.81956
$$226$$ 0 0
$$227$$ −5290.57 −1.54691 −0.773453 0.633854i $$-0.781472\pi$$
−0.773453 + 0.633854i $$0.781472\pi$$
$$228$$ 0 0
$$229$$ −2749.23 −0.793338 −0.396669 0.917962i $$-0.629834\pi$$
−0.396669 + 0.917962i $$0.629834\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 72.0000 0.0202441 0.0101221 0.999949i $$-0.496778\pi$$
0.0101221 + 0.999949i $$0.496778\pi$$
$$234$$ 0 0
$$235$$ 10360.0 2.87580
$$236$$ 0 0
$$237$$ −8626.70 −2.36441
$$238$$ 0 0
$$239$$ −4308.00 −1.16595 −0.582974 0.812491i $$-0.698111\pi$$
−0.582974 + 0.812491i $$0.698111\pi$$
$$240$$ 0 0
$$241$$ −1540.08 −0.411640 −0.205820 0.978590i $$-0.565986\pi$$
−0.205820 + 0.978590i $$0.565986\pi$$
$$242$$ 0 0
$$243$$ −5041.67 −1.33096
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 72.0000 0.0185476
$$248$$ 0 0
$$249$$ −2990.00 −0.760978
$$250$$ 0 0
$$251$$ −931.967 −0.234363 −0.117182 0.993110i $$-0.537386\pi$$
−0.117182 + 0.993110i $$0.537386\pi$$
$$252$$ 0 0
$$253$$ −1960.00 −0.487052
$$254$$ 0 0
$$255$$ −197.990 −0.0486220
$$256$$ 0 0
$$257$$ 937.624 0.227577 0.113789 0.993505i $$-0.463701\pi$$
0.113789 + 0.993505i $$0.463701\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6578.00 −1.56003
$$262$$ 0 0
$$263$$ −7140.00 −1.67404 −0.837018 0.547176i $$-0.815703\pi$$
−0.837018 + 0.547176i $$0.815703\pi$$
$$264$$ 0 0
$$265$$ 1465.13 0.339630
$$266$$ 0 0
$$267$$ 4370.00 1.00165
$$268$$ 0 0
$$269$$ 4610.34 1.04497 0.522485 0.852648i $$-0.325005\pi$$
0.522485 + 0.852648i $$0.325005\pi$$
$$270$$ 0 0
$$271$$ 2364.57 0.530026 0.265013 0.964245i $$-0.414624\pi$$
0.265013 + 0.964245i $$0.414624\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 3738.00 0.819672
$$276$$ 0 0
$$277$$ 4006.00 0.868943 0.434472 0.900686i $$-0.356935\pi$$
0.434472 + 0.900686i $$0.356935\pi$$
$$278$$ 0 0
$$279$$ 2146.78 0.460660
$$280$$ 0 0
$$281$$ −5984.00 −1.27038 −0.635188 0.772358i $$-0.719077\pi$$
−0.635188 + 0.772358i $$0.719077\pi$$
$$282$$ 0 0
$$283$$ 4928.53 1.03523 0.517617 0.855613i $$-0.326819\pi$$
0.517617 + 0.855613i $$0.326819\pi$$
$$284$$ 0 0
$$285$$ −197.990 −0.0411506
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4911.00 −0.999593
$$290$$ 0 0
$$291$$ 10490.0 2.11318
$$292$$ 0 0
$$293$$ 1971.41 0.393076 0.196538 0.980496i $$-0.437030\pi$$
0.196538 + 0.980496i $$0.437030\pi$$
$$294$$ 0 0
$$295$$ 8596.00 1.69654
$$296$$ 0 0
$$297$$ −395.980 −0.0773639
$$298$$ 0 0
$$299$$ −7127.64 −1.37860
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 7980.00 1.51300
$$304$$ 0 0
$$305$$ −280.000 −0.0525664
$$306$$ 0 0
$$307$$ 4767.31 0.886270 0.443135 0.896455i $$-0.353866\pi$$
0.443135 + 0.896455i $$0.353866\pi$$
$$308$$ 0 0
$$309$$ 6140.00 1.13040
$$310$$ 0 0
$$311$$ 6776.91 1.23564 0.617819 0.786320i $$-0.288016\pi$$
0.617819 + 0.786320i $$0.288016\pi$$
$$312$$ 0 0
$$313$$ −6190.01 −1.11783 −0.558914 0.829226i $$-0.688782\pi$$
−0.558914 + 0.829226i $$0.688782\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 9826.00 1.74096 0.870478 0.492207i $$-0.163810\pi$$
0.870478 + 0.492207i $$0.163810\pi$$
$$318$$ 0 0
$$319$$ −4004.00 −0.702762
$$320$$ 0 0
$$321$$ 11907.7 2.07047
$$322$$ 0 0
$$323$$ 2.00000 0.000344529 0
$$324$$ 0 0
$$325$$ 13593.4 2.32008
$$326$$ 0 0
$$327$$ −5784.13 −0.978175
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 5738.00 0.952837 0.476418 0.879219i $$-0.341935\pi$$
0.476418 + 0.879219i $$0.341935\pi$$
$$332$$ 0 0
$$333$$ −874.000 −0.143829
$$334$$ 0 0
$$335$$ 13542.5 2.20868
$$336$$ 0 0
$$337$$ −2254.00 −0.364342 −0.182171 0.983267i $$-0.558312\pi$$
−0.182171 + 0.983267i $$0.558312\pi$$
$$338$$ 0 0
$$339$$ −3818.38 −0.611757
$$340$$ 0 0
$$341$$ 1306.73 0.207518
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 19600.0 3.05863
$$346$$ 0 0
$$347$$ 1986.00 0.307245 0.153623 0.988130i $$-0.450906\pi$$
0.153623 + 0.988130i $$0.450906\pi$$
$$348$$ 0 0
$$349$$ 6771.25 1.03856 0.519279 0.854605i $$-0.326200\pi$$
0.519279 + 0.854605i $$0.326200\pi$$
$$350$$ 0 0
$$351$$ −1440.00 −0.218979
$$352$$ 0 0
$$353$$ 6993.29 1.05443 0.527217 0.849731i $$-0.323236\pi$$
0.527217 + 0.849731i $$0.323236\pi$$
$$354$$ 0 0
$$355$$ 11641.8 1.74052
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −5944.00 −0.873850 −0.436925 0.899498i $$-0.643933\pi$$
−0.436925 + 0.899498i $$0.643933\pi$$
$$360$$ 0 0
$$361$$ −6857.00 −0.999708
$$362$$ 0 0
$$363$$ −8025.66 −1.16044
$$364$$ 0 0
$$365$$ 5348.00 0.766924
$$366$$ 0 0
$$367$$ 842.871 0.119884 0.0599421 0.998202i $$-0.480908\pi$$
0.0599421 + 0.998202i $$0.480908\pi$$
$$368$$ 0 0
$$369$$ −2894.90 −0.408407
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −5726.00 −0.794855 −0.397428 0.917634i $$-0.630097\pi$$
−0.397428 + 0.917634i $$0.630097\pi$$
$$374$$ 0 0
$$375$$ −19880.0 −2.73760
$$376$$ 0 0
$$377$$ −14560.7 −1.98917
$$378$$ 0 0
$$379$$ −10330.0 −1.40004 −0.700022 0.714122i $$-0.746826\pi$$
−0.700022 + 0.714122i $$0.746826\pi$$
$$380$$ 0 0
$$381$$ −12162.2 −1.63541
$$382$$ 0 0
$$383$$ −1004.09 −0.133960 −0.0669800 0.997754i $$-0.521336\pi$$
−0.0669800 + 0.997754i $$0.521336\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 782.000 0.102717
$$388$$ 0 0
$$389$$ 5210.00 0.679068 0.339534 0.940594i $$-0.389731\pi$$
0.339534 + 0.940594i $$0.389731\pi$$
$$390$$ 0 0
$$391$$ −197.990 −0.0256081
$$392$$ 0 0
$$393$$ 12270.0 1.57491
$$394$$ 0 0
$$395$$ 24154.8 3.07686
$$396$$ 0 0
$$397$$ −73.5391 −0.00929678 −0.00464839 0.999989i $$-0.501480\pi$$
−0.00464839 + 0.999989i $$0.501480\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −498.000 −0.0620173 −0.0310086 0.999519i $$-0.509872\pi$$
−0.0310086 + 0.999519i $$0.509872\pi$$
$$402$$ 0 0
$$403$$ 4752.00 0.587380
$$404$$ 0 0
$$405$$ 16255.0 1.99436
$$406$$ 0 0
$$407$$ −532.000 −0.0647918
$$408$$ 0 0
$$409$$ 3355.93 0.405721 0.202861 0.979208i $$-0.434976\pi$$
0.202861 + 0.979208i $$0.434976\pi$$
$$410$$ 0 0
$$411$$ 5854.84 0.702672
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 8372.00 0.990278
$$416$$ 0 0
$$417$$ 3010.00 0.353478
$$418$$ 0 0
$$419$$ −14545.2 −1.69589 −0.847946 0.530082i $$-0.822161\pi$$
−0.847946 + 0.530082i $$0.822161\pi$$
$$420$$ 0 0
$$421$$ 10854.0 1.25651 0.628256 0.778007i $$-0.283769\pi$$
0.628256 + 0.778007i $$0.283769\pi$$
$$422$$ 0 0
$$423$$ −12035.0 −1.38336
$$424$$ 0 0
$$425$$ 377.595 0.0430966
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 5040.00 0.567211
$$430$$ 0 0
$$431$$ 5364.00 0.599477 0.299739 0.954021i $$-0.403100\pi$$
0.299739 + 0.954021i $$0.403100\pi$$
$$432$$ 0 0
$$433$$ −6487.00 −0.719966 −0.359983 0.932959i $$-0.617217\pi$$
−0.359983 + 0.932959i $$0.617217\pi$$
$$434$$ 0 0
$$435$$ 40040.0 4.41327
$$436$$ 0 0
$$437$$ −197.990 −0.0216731
$$438$$ 0 0
$$439$$ −13932.8 −1.51476 −0.757378 0.652977i $$-0.773520\pi$$
−0.757378 + 0.652977i $$0.773520\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −5996.00 −0.643067 −0.321533 0.946898i $$-0.604198\pi$$
−0.321533 + 0.946898i $$0.604198\pi$$
$$444$$ 0 0
$$445$$ −12236.0 −1.30347
$$446$$ 0 0
$$447$$ 14495.7 1.53383
$$448$$ 0 0
$$449$$ 2622.00 0.275590 0.137795 0.990461i $$-0.455999\pi$$
0.137795 + 0.990461i $$0.455999\pi$$
$$450$$ 0 0
$$451$$ −1762.11 −0.183979
$$452$$ 0 0
$$453$$ 3337.54 0.346162
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 11208.0 1.14724 0.573619 0.819122i $$-0.305539\pi$$
0.573619 + 0.819122i $$0.305539\pi$$
$$458$$ 0 0
$$459$$ −40.0000 −0.00406763
$$460$$ 0 0
$$461$$ 9786.36 0.988712 0.494356 0.869260i $$-0.335404\pi$$
0.494356 + 0.869260i $$0.335404\pi$$
$$462$$ 0 0
$$463$$ −3952.00 −0.396685 −0.198342 0.980133i $$-0.563556\pi$$
−0.198342 + 0.980133i $$0.563556\pi$$
$$464$$ 0 0
$$465$$ −13067.3 −1.30319
$$466$$ 0 0
$$467$$ −17506.5 −1.73470 −0.867352 0.497696i $$-0.834180\pi$$
−0.867352 + 0.497696i $$0.834180\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −15640.0 −1.53005
$$472$$ 0 0
$$473$$ 476.000 0.0462717
$$474$$ 0 0
$$475$$ 377.595 0.0364742
$$476$$ 0 0
$$477$$ −1702.00 −0.163374
$$478$$ 0 0
$$479$$ −2288.20 −0.218268 −0.109134 0.994027i $$-0.534808\pi$$
−0.109134 + 0.994027i $$0.534808\pi$$
$$480$$ 0 0
$$481$$ −1934.64 −0.183393
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −29372.0 −2.74993
$$486$$ 0 0
$$487$$ −972.000 −0.0904426 −0.0452213 0.998977i $$-0.514399\pi$$
−0.0452213 + 0.998977i $$0.514399\pi$$
$$488$$ 0 0
$$489$$ −23235.5 −2.14877
$$490$$ 0 0
$$491$$ 7404.00 0.680525 0.340263 0.940330i $$-0.389484\pi$$
0.340263 + 0.940330i $$0.389484\pi$$
$$492$$ 0 0
$$493$$ −404.465 −0.0369497
$$494$$ 0 0
$$495$$ −6375.27 −0.578883
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 12244.0 1.09843 0.549215 0.835681i $$-0.314927\pi$$
0.549215 + 0.835681i $$0.314927\pi$$
$$500$$ 0 0
$$501$$ 10540.0 0.939905
$$502$$ 0 0
$$503$$ 2415.48 0.214117 0.107058 0.994253i $$-0.465857\pi$$
0.107058 + 0.994253i $$0.465857\pi$$
$$504$$ 0 0
$$505$$ −22344.0 −1.96890
$$506$$ 0 0
$$507$$ 2793.07 0.244664
$$508$$ 0 0
$$509$$ −5707.77 −0.497038 −0.248519 0.968627i $$-0.579944\pi$$
−0.248519 + 0.968627i $$0.579944\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ −40.0000 −0.00344258
$$514$$ 0 0
$$515$$ −17192.0 −1.47101
$$516$$ 0 0
$$517$$ −7325.63 −0.623173
$$518$$ 0 0
$$519$$ −14640.0 −1.23820
$$520$$ 0 0
$$521$$ 1.41421 0.000118921 0 5.94605e−5 1.00000i $$-0.499981\pi$$
5.94605e−5 1.00000i $$0.499981\pi$$
$$522$$ 0 0
$$523$$ 12257.0 1.02478 0.512391 0.858752i $$-0.328760\pi$$
0.512391 + 0.858752i $$0.328760\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 132.000 0.0109108
$$528$$ 0 0
$$529$$ 7433.00 0.610915
$$530$$ 0 0
$$531$$ −9985.76 −0.816093
$$532$$ 0 0
$$533$$ −6408.00 −0.520753
$$534$$ 0 0
$$535$$ −33341.5 −2.69435
$$536$$ 0 0
$$537$$ −3818.38 −0.306844
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 2050.00 0.162914 0.0814569 0.996677i $$-0.474043\pi$$
0.0814569 + 0.996677i $$0.474043\pi$$
$$542$$ 0 0
$$543$$ 26760.0 2.11488
$$544$$ 0 0
$$545$$ 16195.6 1.27292
$$546$$ 0 0
$$547$$ −14554.0 −1.13763 −0.568815 0.822465i $$-0.692598\pi$$
−0.568815 + 0.822465i $$0.692598\pi$$
$$548$$ 0 0
$$549$$ 325.269 0.0252862
$$550$$ 0 0
$$551$$ −404.465 −0.0312719
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ 5320.00 0.406885
$$556$$ 0 0
$$557$$ 6954.00 0.528995 0.264498 0.964386i $$-0.414794\pi$$
0.264498 + 0.964386i $$0.414794\pi$$
$$558$$ 0 0
$$559$$ 1731.00 0.130972
$$560$$ 0 0
$$561$$ 140.000 0.0105362
$$562$$ 0 0
$$563$$ 1636.25 0.122486 0.0612429 0.998123i $$-0.480494\pi$$
0.0612429 + 0.998123i $$0.480494\pi$$
$$564$$ 0 0
$$565$$ 10691.5 0.796094
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −7142.00 −0.526201 −0.263100 0.964768i $$-0.584745\pi$$
−0.263100 + 0.964768i $$0.584745\pi$$
$$570$$ 0 0
$$571$$ 20606.0 1.51022 0.755109 0.655599i $$-0.227584\pi$$
0.755109 + 0.655599i $$0.227584\pi$$
$$572$$ 0 0
$$573$$ −7269.06 −0.529964
$$574$$ 0 0
$$575$$ −37380.0 −2.71105
$$576$$ 0 0
$$577$$ −8803.48 −0.635171 −0.317585 0.948230i $$-0.602872\pi$$
−0.317585 + 0.948230i $$0.602872\pi$$
$$578$$ 0 0
$$579$$ 32470.3 2.33061
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −1036.00 −0.0735965
$$584$$ 0 0
$$585$$ −23184.0 −1.63853
$$586$$ 0 0
$$587$$ −6503.97 −0.457321 −0.228661 0.973506i $$-0.573435\pi$$
−0.228661 + 0.973506i $$0.573435\pi$$
$$588$$ 0 0
$$589$$ 132.000 0.00923424
$$590$$ 0 0
$$591$$ 5614.43 0.390773
$$592$$ 0 0
$$593$$ −23140.8 −1.60249 −0.801246 0.598335i $$-0.795829\pi$$
−0.801246 + 0.598335i $$0.795829\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −17580.0 −1.20520
$$598$$ 0 0
$$599$$ 11296.0 0.770521 0.385260 0.922808i $$-0.374112\pi$$
0.385260 + 0.922808i $$0.374112\pi$$
$$600$$ 0 0
$$601$$ 8727.11 0.592323 0.296162 0.955138i $$-0.404293\pi$$
0.296162 + 0.955138i $$0.404293\pi$$
$$602$$ 0 0
$$603$$ −15732.0 −1.06245
$$604$$ 0 0
$$605$$ 22471.9 1.51010
$$606$$ 0 0
$$607$$ −19736.8 −1.31975 −0.659877 0.751374i $$-0.729392\pi$$
−0.659877 + 0.751374i $$0.729392\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −26640.0 −1.76389
$$612$$ 0 0
$$613$$ 16962.0 1.11760 0.558800 0.829302i $$-0.311262\pi$$
0.558800 + 0.829302i $$0.311262\pi$$
$$614$$ 0 0
$$615$$ 17621.1 1.15537
$$616$$ 0 0
$$617$$ −19034.0 −1.24194 −0.620972 0.783832i $$-0.713262\pi$$
−0.620972 + 0.783832i $$0.713262\pi$$
$$618$$ 0 0
$$619$$ 18677.5 1.21278 0.606392 0.795166i $$-0.292616\pi$$
0.606392 + 0.795166i $$0.292616\pi$$
$$620$$ 0 0
$$621$$ 3959.80 0.255880
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 22289.0 1.42650
$$626$$ 0 0
$$627$$ 140.000 0.00891716
$$628$$ 0 0
$$629$$ −53.7401 −0.00340661
$$630$$ 0 0
$$631$$ 14716.0 0.928423 0.464211 0.885724i $$-0.346338\pi$$
0.464211 + 0.885724i $$0.346338\pi$$
$$632$$ 0 0
$$633$$ 19431.3 1.22010
$$634$$ 0 0
$$635$$ 34054.3 2.12819
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −13524.0 −0.837248
$$640$$ 0 0
$$641$$ 4730.00 0.291457 0.145728 0.989325i $$-0.453447\pi$$
0.145728 + 0.989325i $$0.453447\pi$$
$$642$$ 0 0
$$643$$ −19056.5 −1.16877 −0.584383 0.811478i $$-0.698663\pi$$
−0.584383 + 0.811478i $$0.698663\pi$$
$$644$$ 0 0
$$645$$ −4760.00 −0.290581
$$646$$ 0 0
$$647$$ −9342.29 −0.567672 −0.283836 0.958873i $$-0.591607\pi$$
−0.283836 + 0.958873i $$0.591607\pi$$
$$648$$ 0 0
$$649$$ −6078.29 −0.367633
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ 3774.00 0.226169 0.113084 0.993585i $$-0.463927\pi$$
0.113084 + 0.993585i $$0.463927\pi$$
$$654$$ 0 0
$$655$$ −34356.0 −2.04947
$$656$$ 0 0
$$657$$ −6212.64 −0.368917
$$658$$ 0 0
$$659$$ 21150.0 1.25021 0.625104 0.780541i $$-0.285057\pi$$
0.625104 + 0.780541i $$0.285057\pi$$
$$660$$ 0 0
$$661$$ −10377.5 −0.610647 −0.305324 0.952249i $$-0.598765\pi$$
−0.305324 + 0.952249i $$0.598765\pi$$
$$662$$ 0 0
$$663$$ 509.117 0.0298227
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 40040.0 2.32437
$$668$$ 0 0
$$669$$ 24240.0 1.40086
$$670$$ 0 0
$$671$$ 197.990 0.0113909
$$672$$ 0 0
$$673$$ −1164.00 −0.0666700 −0.0333350 0.999444i $$-0.510613\pi$$
−0.0333350 + 0.999444i $$0.510613\pi$$
$$674$$ 0 0
$$675$$ −7551.90 −0.430626
$$676$$ 0 0
$$677$$ 27152.9 1.54146 0.770732 0.637160i $$-0.219891\pi$$
0.770732 + 0.637160i $$0.219891\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −37410.0 −2.10507
$$682$$ 0 0
$$683$$ 16596.0 0.929763 0.464882 0.885373i $$-0.346097\pi$$
0.464882 + 0.885373i $$0.346097\pi$$
$$684$$ 0 0
$$685$$ −16393.6 −0.914403
$$686$$ 0 0
$$687$$ −19440.0 −1.07960
$$688$$ 0 0
$$689$$ −3767.46 −0.208315
$$690$$ 0 0
$$691$$ −11298.2 −0.622000 −0.311000 0.950410i $$-0.600664\pi$$
−0.311000 + 0.950410i $$0.600664\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −8428.00 −0.459989
$$696$$ 0 0
$$697$$ −178.000 −0.00967321
$$698$$ 0 0
$$699$$ 509.117 0.0275487
$$700$$ 0 0
$$701$$ −2754.00 −0.148384 −0.0741920 0.997244i $$-0.523638\pi$$
−0.0741920 + 0.997244i $$0.523638\pi$$
$$702$$ 0 0
$$703$$ −53.7401 −0.00288314
$$704$$ 0 0
$$705$$ 73256.3 3.91346
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 29434.0 1.55912 0.779561 0.626327i $$-0.215442\pi$$
0.779561 + 0.626327i $$0.215442\pi$$
$$710$$ 0 0
$$711$$ −28060.0 −1.48007
$$712$$ 0 0
$$713$$ −13067.3 −0.686361
$$714$$ 0 0
$$715$$ −14112.0 −0.738124
$$716$$ 0 0
$$717$$ −30462.2 −1.58665
$$718$$ 0 0
$$719$$ 17669.2 0.916480 0.458240 0.888828i $$-0.348480\pi$$
0.458240 + 0.888828i $$0.348480\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ −10890.0 −0.560171
$$724$$ 0 0
$$725$$ −76362.0 −3.91174
$$726$$ 0 0
$$727$$ −28445.5 −1.45115 −0.725574 0.688144i $$-0.758426\pi$$
−0.725574 + 0.688144i $$0.758426\pi$$
$$728$$ 0 0
$$729$$ −13483.0 −0.685007
$$730$$ 0 0
$$731$$ 48.0833 0.00243286
$$732$$ 0 0
$$733$$ 22341.7 1.12580 0.562900 0.826525i $$-0.309686\pi$$
0.562900 + 0.826525i $$0.309686\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −9576.00 −0.478611
$$738$$ 0 0
$$739$$ −20670.0 −1.02890 −0.514451 0.857520i $$-0.672004\pi$$
−0.514451 + 0.857520i $$0.672004\pi$$
$$740$$ 0 0
$$741$$ 509.117 0.0252400
$$742$$ 0 0
$$743$$ 25400.0 1.25415 0.627076 0.778958i $$-0.284251\pi$$
0.627076 + 0.778958i $$0.284251\pi$$
$$744$$ 0 0
$$745$$ −40587.9 −1.99601
$$746$$ 0 0
$$747$$ −9725.55 −0.476358
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −29180.0 −1.41783 −0.708917 0.705292i $$-0.750816\pi$$
−0.708917 + 0.705292i $$0.750816\pi$$
$$752$$ 0 0
$$753$$ −6590.00 −0.318928
$$754$$ 0 0
$$755$$ −9345.12 −0.450469
$$756$$ 0 0
$$757$$ −26206.0 −1.25822 −0.629110 0.777316i $$-0.716581\pi$$
−0.629110 + 0.777316i $$0.716581\pi$$
$$758$$ 0 0
$$759$$ −13859.3 −0.662794
$$760$$ 0 0
$$761$$ 6863.18 0.326925 0.163463 0.986550i $$-0.447734\pi$$
0.163463 + 0.986550i $$0.447734\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ −644.000 −0.0304364
$$766$$ 0 0
$$767$$ −22104.0 −1.04059
$$768$$ 0 0
$$769$$ −9058.04 −0.424761 −0.212380 0.977187i $$-0.568122\pi$$
−0.212380 + 0.977187i $$0.568122\pi$$
$$770$$ 0 0
$$771$$ 6630.00 0.309693
$$772$$ 0 0
$$773$$ −132.936 −0.00618548 −0.00309274 0.999995i $$-0.500984\pi$$
−0.00309274 + 0.999995i $$0.500984\pi$$
$$774$$ 0 0
$$775$$ 24921.3 1.15509
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ −178.000 −0.00818679
$$780$$ 0 0
$$781$$ −8232.00 −0.377163
$$782$$ 0 0
$$783$$ 8089.30 0.369206
$$784$$ 0 0
$$785$$ 43792.0 1.99109
$$786$$ 0 0
$$787$$ 8729.94 0.395411 0.197706 0.980261i $$-0.436651\pi$$
0.197706 + 0.980261i $$0.436651\pi$$
$$788$$ 0 0
$$789$$ −50487.4 −2.27807
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 720.000 0.0322421
$$794$$ 0 0
$$795$$ 10360.0 0.462178
$$796$$ 0 0
$$797$$ 7517.96 0.334128 0.167064 0.985946i $$-0.446571\pi$$
0.167064 + 0.985946i $$0.446571\pi$$
$$798$$ 0 0
$$799$$ −740.000 −0.0327651
$$800$$ 0 0
$$801$$ 14214.3 0.627011
$$802$$ 0 0
$$803$$ −3781.61 −0.166189
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 32600.0 1.42203
$$808$$ 0 0
$$809$$ 3776.00 0.164100 0.0820501 0.996628i $$-0.473853\pi$$
0.0820501 + 0.996628i $$0.473853\pi$$
$$810$$ 0 0
$$811$$ 36227.9 1.56860 0.784300 0.620382i $$-0.213023\pi$$
0.784300 + 0.620382i $$0.213023\pi$$
$$812$$ 0 0
$$813$$ 16720.0 0.721274
$$814$$ 0 0
$$815$$ 65059.5 2.79624
$$816$$ 0 0
$$817$$ 48.0833 0.00205902
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −16410.0 −0.697580 −0.348790 0.937201i $$-0.613407\pi$$
−0.348790 + 0.937201i $$0.613407\pi$$
$$822$$ 0 0
$$823$$ −22072.0 −0.934850 −0.467425 0.884033i $$-0.654818\pi$$
−0.467425 + 0.884033i $$0.654818\pi$$
$$824$$ 0 0
$$825$$ 26431.7 1.11543
$$826$$ 0 0
$$827$$ 11628.0 0.488930 0.244465 0.969658i $$-0.421388\pi$$
0.244465 + 0.969658i $$0.421388\pi$$
$$828$$ 0 0
$$829$$ −30906.2 −1.29483 −0.647417 0.762136i $$-0.724151\pi$$
−0.647417 + 0.762136i $$0.724151\pi$$
$$830$$ 0 0
$$831$$ 28326.7 1.18248
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −29512.0 −1.22312
$$836$$ 0 0
$$837$$ −2640.00 −0.109022
$$838$$ 0 0
$$839$$ −17884.1 −0.735911 −0.367955 0.929843i $$-0.619942\pi$$
−0.367955 + 0.929843i $$0.619942\pi$$
$$840$$ 0 0
$$841$$ 57407.0 2.35381
$$842$$ 0 0
$$843$$ −42313.3 −1.72876
$$844$$ 0 0
$$845$$ −7820.60 −0.318387
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 34850.0 1.40877
$$850$$ 0 0
$$851$$ 5320.00 0.214298
$$852$$ 0 0
$$853$$ −20755.0 −0.833104 −0.416552 0.909112i $$-0.636762\pi$$
−0.416552 + 0.909112i $$0.636762\pi$$
$$854$$ 0 0
$$855$$ −644.000 −0.0257595
$$856$$ 0 0
$$857$$ 44919.7 1.79046 0.895231 0.445602i $$-0.147010\pi$$
0.895231 + 0.445602i $$0.147010\pi$$
$$858$$ 0 0
$$859$$ −69.2965 −0.00275246 −0.00137623 0.999999i $$-0.500438\pi$$
−0.00137623 + 0.999999i $$0.500438\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 5452.00 0.215050 0.107525 0.994202i $$-0.465707\pi$$
0.107525 + 0.994202i $$0.465707\pi$$
$$864$$ 0 0
$$865$$ 40992.0 1.61129
$$866$$ 0 0
$$867$$ −34726.0 −1.36027
$$868$$ 0 0
$$869$$ −17080.0 −0.666743
$$870$$ 0 0
$$871$$ −34823.6 −1.35471
$$872$$ 0 0
$$873$$ 34120.7 1.32281
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 31106.0 1.19769 0.598845 0.800865i $$-0.295626\pi$$
0.598845 + 0.800865i $$0.295626\pi$$
$$878$$ 0 0
$$879$$ 13940.0 0.534908
$$880$$ 0 0
$$881$$ 5943.94 0.227306 0.113653 0.993521i $$-0.463745\pi$$
0.113653 + 0.993521i $$0.463745\pi$$
$$882$$ 0 0
$$883$$ −34796.0 −1.32614 −0.663068 0.748559i $$-0.730746\pi$$
−0.663068 + 0.748559i $$0.730746\pi$$
$$884$$ 0 0
$$885$$ 60782.9 2.30869
$$886$$ 0 0
$$887$$ −9964.55 −0.377200 −0.188600 0.982054i $$-0.560395\pi$$
−0.188600 + 0.982054i $$0.560395\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −11494.0 −0.432170
$$892$$ 0 0
$$893$$ −740.000 −0.0277303
$$894$$ 0 0
$$895$$ 10691.5 0.399303
$$896$$ 0 0
$$897$$ −50400.0 −1.87604
$$898$$ 0 0
$$899$$ −26694.7 −0.990343
$$900$$ 0 0
$$901$$ −104.652 −0.00386954
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −74928.0 −2.75214
$$906$$ 0 0
$$907$$ 29756.0 1.08934 0.544670 0.838650i $$-0.316655\pi$$
0.544670 + 0.838650i $$0.316655\pi$$
$$908$$ 0 0
$$909$$ 25956.5 0.947109
$$910$$ 0 0
$$911$$ −21440.0 −0.779735 −0.389868 0.920871i $$-0.627479\pi$$
−0.389868 + 0.920871i $$0.627479\pi$$
$$912$$ 0 0
$$913$$ −5919.90 −0.214589
$$914$$ 0 0
$$915$$ −1979.90 −0.0715338
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ 8288.00 0.297493 0.148746 0.988875i $$-0.452476\pi$$
0.148746 + 0.988875i $$0.452476\pi$$
$$920$$ 0 0
$$921$$ 33710.0 1.20606
$$922$$ 0 0
$$923$$ −29936.1 −1.06756
$$924$$ 0 0
$$925$$ −10146.0 −0.360647
$$926$$ 0 0
$$927$$ 19971.5 0.707606
$$928$$ 0 0
$$929$$ −45581.5 −1.60978 −0.804888 0.593427i $$-0.797774\pi$$
−0.804888 + 0.593427i $$0.797774\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ 47920.0 1.68149
$$934$$ 0 0
$$935$$ −392.000 −0.0137110
$$936$$ 0 0
$$937$$ 11665.8 0.406731 0.203365 0.979103i $$-0.434812\pi$$
0.203365 + 0.979103i $$0.434812\pi$$
$$938$$ 0 0
$$939$$ −43770.0 −1.52117
$$940$$ 0 0
$$941$$ 14.1421 0.000489926 0 0.000244963 1.00000i $$-0.499922\pi$$
0.000244963 1.00000i $$0.499922\pi$$
$$942$$ 0 0
$$943$$ 17621.1 0.608507
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −14034.0 −0.481567 −0.240783 0.970579i $$-0.577404\pi$$
−0.240783 + 0.970579i $$0.577404\pi$$
$$948$$ 0 0
$$949$$ −13752.0 −0.470399
$$950$$ 0 0
$$951$$ 69480.3 2.36914
$$952$$ 0 0
$$953$$ −42698.0 −1.45134 −0.725668 0.688045i $$-0.758469\pi$$
−0.725668 + 0.688045i $$0.758469\pi$$
$$954$$ 0 0
$$955$$ 20353.4 0.689654
$$956$$ 0 0
$$957$$ −28312.6 −0.956337
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −21079.0 −0.707563
$$962$$ 0 0
$$963$$ 38732.0 1.29608
$$964$$ 0 0
$$965$$ −90917.0 −3.03287
$$966$$ 0 0
$$967$$ 48492.0 1.61261 0.806307 0.591497i $$-0.201463\pi$$
0.806307 + 0.591497i $$0.201463\pi$$
$$968$$ 0 0
$$969$$ 14.1421 0.000468845 0
$$970$$ 0 0
$$971$$ −52669.6 −1.74073 −0.870364 0.492409i $$-0.836116\pi$$
−0.870364 + 0.492409i $$0.836116\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ 96120.0 3.15723
$$976$$ 0 0
$$977$$ −55380.0 −1.81347 −0.906737 0.421698i $$-0.861434\pi$$
−0.906737 + 0.421698i $$0.861434\pi$$
$$978$$ 0 0
$$979$$ 8652.16 0.282456
$$980$$ 0 0
$$981$$ −18814.0 −0.612319
$$982$$ 0 0
$$983$$ 50535.5 1.63971 0.819854 0.572573i $$-0.194055\pi$$
0.819854 + 0.572573i $$0.194055\pi$$
$$984$$ 0 0
$$985$$ −15720.4 −0.508521
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ −4760.00 −0.153043
$$990$$ 0 0
$$991$$ 39712.0 1.27295 0.636475 0.771297i $$-0.280392\pi$$
0.636475 + 0.771297i $$0.280392\pi$$
$$992$$ 0 0
$$993$$ 40573.8 1.29665
$$994$$ 0 0
$$995$$ 49224.0 1.56835
$$996$$ 0 0
$$997$$ 2186.37 0.0694515 0.0347258 0.999397i $$-0.488944\pi$$
0.0347258 + 0.999397i $$0.488944\pi$$
$$998$$ 0 0
$$999$$ 1074.80 0.0340393
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 784.4.a.y.1.2 2
4.3 odd 2 98.4.a.g.1.1 2
7.6 odd 2 inner 784.4.a.y.1.1 2
12.11 even 2 882.4.a.bg.1.2 2
20.19 odd 2 2450.4.a.bx.1.2 2
28.3 even 6 98.4.c.h.79.1 4
28.11 odd 6 98.4.c.h.79.2 4
28.19 even 6 98.4.c.h.67.1 4
28.23 odd 6 98.4.c.h.67.2 4
28.27 even 2 98.4.a.g.1.2 yes 2
84.11 even 6 882.4.g.ba.667.1 4
84.23 even 6 882.4.g.ba.361.1 4
84.47 odd 6 882.4.g.ba.361.2 4
84.59 odd 6 882.4.g.ba.667.2 4
84.83 odd 2 882.4.a.bg.1.1 2
140.139 even 2 2450.4.a.bx.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.g.1.1 2 4.3 odd 2
98.4.a.g.1.2 yes 2 28.27 even 2
98.4.c.h.67.1 4 28.19 even 6
98.4.c.h.67.2 4 28.23 odd 6
98.4.c.h.79.1 4 28.3 even 6
98.4.c.h.79.2 4 28.11 odd 6
784.4.a.y.1.1 2 7.6 odd 2 inner
784.4.a.y.1.2 2 1.1 even 1 trivial
882.4.a.bg.1.1 2 84.83 odd 2
882.4.a.bg.1.2 2 12.11 even 2
882.4.g.ba.361.1 4 84.23 even 6
882.4.g.ba.361.2 4 84.47 odd 6
882.4.g.ba.667.1 4 84.11 even 6
882.4.g.ba.667.2 4 84.59 odd 6
2450.4.a.bx.1.1 2 140.139 even 2
2450.4.a.bx.1.2 2 20.19 odd 2