# Properties

 Label 1323.4.a.bd.1.1 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 40x^{4} + 453x^{2} - 1278$$ x^6 - 40*x^4 + 453*x^2 - 1278 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 189) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-4.70753$$ of defining polynomial Character $$\chi$$ $$=$$ 1323.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.70753 q^{2} +14.1608 q^{4} -0.830453 q^{5} -29.0024 q^{8} +O(q^{10})$$ $$q-4.70753 q^{2} +14.1608 q^{4} -0.830453 q^{5} -29.0024 q^{8} +3.90938 q^{10} +68.2503 q^{11} -35.7721 q^{13} +23.2429 q^{16} -14.6779 q^{17} +86.7693 q^{19} -11.7599 q^{20} -321.290 q^{22} -58.4087 q^{23} -124.310 q^{25} +168.398 q^{26} +67.1398 q^{29} -69.5002 q^{31} +122.603 q^{32} +69.0965 q^{34} -289.240 q^{37} -408.469 q^{38} +24.0851 q^{40} +357.750 q^{41} -235.801 q^{43} +966.482 q^{44} +274.961 q^{46} -450.067 q^{47} +585.195 q^{50} -506.563 q^{52} -172.464 q^{53} -56.6787 q^{55} -316.063 q^{58} +445.856 q^{59} +476.314 q^{61} +327.174 q^{62} -763.099 q^{64} +29.7070 q^{65} -804.320 q^{67} -207.851 q^{68} -353.412 q^{71} -778.045 q^{73} +1361.61 q^{74} +1228.73 q^{76} -74.1996 q^{79} -19.3021 q^{80} -1684.12 q^{82} +1168.49 q^{83} +12.1893 q^{85} +1110.04 q^{86} -1979.42 q^{88} -1514.72 q^{89} -827.116 q^{92} +2118.70 q^{94} -72.0579 q^{95} +1631.46 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 32 q^{4}+O(q^{10})$$ 6 * q + 32 * q^4 $$6 q + 32 q^{4} - 20 q^{10} + 52 q^{13} - 148 q^{16} - 62 q^{19} - 356 q^{22} - 46 q^{25} - 82 q^{31} + 420 q^{34} - 1132 q^{37} - 444 q^{40} - 1566 q^{43} - 888 q^{46} - 72 q^{52} + 224 q^{55} + 4 q^{58} + 886 q^{61} - 924 q^{64} - 2084 q^{67} - 2398 q^{73} + 3204 q^{76} + 984 q^{79} - 3892 q^{82} - 3600 q^{85} - 5796 q^{88} + 2772 q^{94} - 682 q^{97}+O(q^{100})$$ 6 * q + 32 * q^4 - 20 * q^10 + 52 * q^13 - 148 * q^16 - 62 * q^19 - 356 * q^22 - 46 * q^25 - 82 * q^31 + 420 * q^34 - 1132 * q^37 - 444 * q^40 - 1566 * q^43 - 888 * q^46 - 72 * q^52 + 224 * q^55 + 4 * q^58 + 886 * q^61 - 924 * q^64 - 2084 * q^67 - 2398 * q^73 + 3204 * q^76 + 984 * q^79 - 3892 * q^82 - 3600 * q^85 - 5796 * q^88 + 2772 * q^94 - 682 * q^97

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.70753 −1.66436 −0.832182 0.554503i $$-0.812908\pi$$
−0.832182 + 0.554503i $$0.812908\pi$$
$$3$$ 0 0
$$4$$ 14.1608 1.77011
$$5$$ −0.830453 −0.0742780 −0.0371390 0.999310i $$-0.511824\pi$$
−0.0371390 + 0.999310i $$0.511824\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −29.0024 −1.28174
$$9$$ 0 0
$$10$$ 3.90938 0.123626
$$11$$ 68.2503 1.87075 0.935374 0.353659i $$-0.115063\pi$$
0.935374 + 0.353659i $$0.115063\pi$$
$$12$$ 0 0
$$13$$ −35.7721 −0.763184 −0.381592 0.924331i $$-0.624624\pi$$
−0.381592 + 0.924331i $$0.624624\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 23.2429 0.363170
$$17$$ −14.6779 −0.209406 −0.104703 0.994504i $$-0.533389\pi$$
−0.104703 + 0.994504i $$0.533389\pi$$
$$18$$ 0 0
$$19$$ 86.7693 1.04770 0.523849 0.851811i $$-0.324496\pi$$
0.523849 + 0.851811i $$0.324496\pi$$
$$20$$ −11.7599 −0.131480
$$21$$ 0 0
$$22$$ −321.290 −3.11361
$$23$$ −58.4087 −0.529524 −0.264762 0.964314i $$-0.585293\pi$$
−0.264762 + 0.964314i $$0.585293\pi$$
$$24$$ 0 0
$$25$$ −124.310 −0.994483
$$26$$ 168.398 1.27022
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 67.1398 0.429915 0.214958 0.976623i $$-0.431039\pi$$
0.214958 + 0.976623i $$0.431039\pi$$
$$30$$ 0 0
$$31$$ −69.5002 −0.402665 −0.201332 0.979523i $$-0.564527\pi$$
−0.201332 + 0.979523i $$0.564527\pi$$
$$32$$ 122.603 0.677290
$$33$$ 0 0
$$34$$ 69.0965 0.348528
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −289.240 −1.28516 −0.642578 0.766220i $$-0.722135\pi$$
−0.642578 + 0.766220i $$0.722135\pi$$
$$38$$ −408.469 −1.74375
$$39$$ 0 0
$$40$$ 24.0851 0.0952048
$$41$$ 357.750 1.36271 0.681355 0.731953i $$-0.261391\pi$$
0.681355 + 0.731953i $$0.261391\pi$$
$$42$$ 0 0
$$43$$ −235.801 −0.836263 −0.418132 0.908386i $$-0.637315\pi$$
−0.418132 + 0.908386i $$0.637315\pi$$
$$44$$ 966.482 3.31142
$$45$$ 0 0
$$46$$ 274.961 0.881320
$$47$$ −450.067 −1.39679 −0.698394 0.715714i $$-0.746102\pi$$
−0.698394 + 0.715714i $$0.746102\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 585.195 1.65518
$$51$$ 0 0
$$52$$ −506.563 −1.35092
$$53$$ −172.464 −0.446977 −0.223489 0.974707i $$-0.571745\pi$$
−0.223489 + 0.974707i $$0.571745\pi$$
$$54$$ 0 0
$$55$$ −56.6787 −0.138955
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −316.063 −0.715535
$$59$$ 445.856 0.983823 0.491912 0.870645i $$-0.336298\pi$$
0.491912 + 0.870645i $$0.336298\pi$$
$$60$$ 0 0
$$61$$ 476.314 0.999767 0.499883 0.866093i $$-0.333376\pi$$
0.499883 + 0.866093i $$0.333376\pi$$
$$62$$ 327.174 0.670180
$$63$$ 0 0
$$64$$ −763.099 −1.49043
$$65$$ 29.7070 0.0566878
$$66$$ 0 0
$$67$$ −804.320 −1.46662 −0.733309 0.679896i $$-0.762025\pi$$
−0.733309 + 0.679896i $$0.762025\pi$$
$$68$$ −207.851 −0.370671
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −353.412 −0.590737 −0.295368 0.955383i $$-0.595442\pi$$
−0.295368 + 0.955383i $$0.595442\pi$$
$$72$$ 0 0
$$73$$ −778.045 −1.24744 −0.623721 0.781647i $$-0.714380\pi$$
−0.623721 + 0.781647i $$0.714380\pi$$
$$74$$ 1361.61 2.13897
$$75$$ 0 0
$$76$$ 1228.73 1.85454
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −74.1996 −0.105672 −0.0528361 0.998603i $$-0.516826\pi$$
−0.0528361 + 0.998603i $$0.516826\pi$$
$$80$$ −19.3021 −0.0269755
$$81$$ 0 0
$$82$$ −1684.12 −2.26804
$$83$$ 1168.49 1.54529 0.772643 0.634840i $$-0.218934\pi$$
0.772643 + 0.634840i $$0.218934\pi$$
$$84$$ 0 0
$$85$$ 12.1893 0.0155543
$$86$$ 1110.04 1.39185
$$87$$ 0 0
$$88$$ −1979.42 −2.39781
$$89$$ −1514.72 −1.80405 −0.902023 0.431687i $$-0.857918\pi$$
−0.902023 + 0.431687i $$0.857918\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −827.116 −0.937313
$$93$$ 0 0
$$94$$ 2118.70 2.32476
$$95$$ −72.0579 −0.0778209
$$96$$ 0 0
$$97$$ 1631.46 1.70772 0.853862 0.520499i $$-0.174254\pi$$
0.853862 + 0.520499i $$0.174254\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −1760.34 −1.76034
$$101$$ −1205.73 −1.18787 −0.593934 0.804514i $$-0.702426\pi$$
−0.593934 + 0.804514i $$0.702426\pi$$
$$102$$ 0 0
$$103$$ −485.363 −0.464313 −0.232156 0.972679i $$-0.574578\pi$$
−0.232156 + 0.972679i $$0.574578\pi$$
$$104$$ 1037.48 0.978201
$$105$$ 0 0
$$106$$ 811.881 0.743933
$$107$$ 258.101 0.233192 0.116596 0.993179i $$-0.462802\pi$$
0.116596 + 0.993179i $$0.462802\pi$$
$$108$$ 0 0
$$109$$ 581.246 0.510764 0.255382 0.966840i $$-0.417799\pi$$
0.255382 + 0.966840i $$0.417799\pi$$
$$110$$ 266.817 0.231272
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −708.244 −0.589610 −0.294805 0.955557i $$-0.595255\pi$$
−0.294805 + 0.955557i $$0.595255\pi$$
$$114$$ 0 0
$$115$$ 48.5057 0.0393320
$$116$$ 950.756 0.760996
$$117$$ 0 0
$$118$$ −2098.88 −1.63744
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 3327.10 2.49970
$$122$$ −2242.26 −1.66398
$$123$$ 0 0
$$124$$ −984.182 −0.712759
$$125$$ 207.041 0.148146
$$126$$ 0 0
$$127$$ 2348.04 1.64059 0.820296 0.571939i $$-0.193809\pi$$
0.820296 + 0.571939i $$0.193809\pi$$
$$128$$ 2611.49 1.80332
$$129$$ 0 0
$$130$$ −139.847 −0.0943491
$$131$$ 2405.82 1.60456 0.802279 0.596949i $$-0.203621\pi$$
0.802279 + 0.596949i $$0.203621\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 3786.36 2.44098
$$135$$ 0 0
$$136$$ 425.693 0.268403
$$137$$ 1187.55 0.740579 0.370289 0.928916i $$-0.379258\pi$$
0.370289 + 0.928916i $$0.379258\pi$$
$$138$$ 0 0
$$139$$ −712.439 −0.434736 −0.217368 0.976090i $$-0.569747\pi$$
−0.217368 + 0.976090i $$0.569747\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 1663.70 0.983201
$$143$$ −2441.46 −1.42773
$$144$$ 0 0
$$145$$ −55.7564 −0.0319332
$$146$$ 3662.67 2.07620
$$147$$ 0 0
$$148$$ −4095.89 −2.27486
$$149$$ 2209.24 1.21468 0.607342 0.794441i $$-0.292236\pi$$
0.607342 + 0.794441i $$0.292236\pi$$
$$150$$ 0 0
$$151$$ −546.983 −0.294787 −0.147394 0.989078i $$-0.547088\pi$$
−0.147394 + 0.989078i $$0.547088\pi$$
$$152$$ −2516.52 −1.34287
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 57.7166 0.0299091
$$156$$ 0 0
$$157$$ 1870.19 0.950686 0.475343 0.879801i $$-0.342324\pi$$
0.475343 + 0.879801i $$0.342324\pi$$
$$158$$ 349.297 0.175877
$$159$$ 0 0
$$160$$ −101.816 −0.0503078
$$161$$ 0 0
$$162$$ 0 0
$$163$$ −3037.40 −1.45955 −0.729777 0.683685i $$-0.760376\pi$$
−0.729777 + 0.683685i $$0.760376\pi$$
$$164$$ 5066.04 2.41214
$$165$$ 0 0
$$166$$ −5500.72 −2.57192
$$167$$ −2967.09 −1.37485 −0.687426 0.726255i $$-0.741259\pi$$
−0.687426 + 0.726255i $$0.741259\pi$$
$$168$$ 0 0
$$169$$ −917.358 −0.417550
$$170$$ −57.3814 −0.0258879
$$171$$ 0 0
$$172$$ −3339.14 −1.48028
$$173$$ 3866.81 1.69935 0.849677 0.527303i $$-0.176797\pi$$
0.849677 + 0.527303i $$0.176797\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1586.33 0.679399
$$177$$ 0 0
$$178$$ 7130.60 3.00259
$$179$$ −334.726 −0.139769 −0.0698844 0.997555i $$-0.522263\pi$$
−0.0698844 + 0.997555i $$0.522263\pi$$
$$180$$ 0 0
$$181$$ −1564.56 −0.642502 −0.321251 0.946994i $$-0.604103\pi$$
−0.321251 + 0.946994i $$0.604103\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 1693.99 0.678710
$$185$$ 240.200 0.0954588
$$186$$ 0 0
$$187$$ −1001.77 −0.391746
$$188$$ −6373.33 −2.47246
$$189$$ 0 0
$$190$$ 339.215 0.129522
$$191$$ −3067.14 −1.16194 −0.580971 0.813925i $$-0.697327\pi$$
−0.580971 + 0.813925i $$0.697327\pi$$
$$192$$ 0 0
$$193$$ −2223.60 −0.829316 −0.414658 0.909977i $$-0.636099\pi$$
−0.414658 + 0.909977i $$0.636099\pi$$
$$194$$ −7680.13 −2.84227
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −1856.18 −0.671306 −0.335653 0.941986i $$-0.608957\pi$$
−0.335653 + 0.941986i $$0.608957\pi$$
$$198$$ 0 0
$$199$$ −993.602 −0.353943 −0.176971 0.984216i $$-0.556630\pi$$
−0.176971 + 0.984216i $$0.556630\pi$$
$$200$$ 3605.30 1.27467
$$201$$ 0 0
$$202$$ 5676.02 1.97704
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −297.094 −0.101219
$$206$$ 2284.86 0.772785
$$207$$ 0 0
$$208$$ −831.446 −0.277165
$$209$$ 5922.03 1.95998
$$210$$ 0 0
$$211$$ −545.079 −0.177843 −0.0889213 0.996039i $$-0.528342\pi$$
−0.0889213 + 0.996039i $$0.528342\pi$$
$$212$$ −2442.24 −0.791197
$$213$$ 0 0
$$214$$ −1215.02 −0.388116
$$215$$ 195.822 0.0621160
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −2736.23 −0.850097
$$219$$ 0 0
$$220$$ −802.618 −0.245966
$$221$$ 525.058 0.159815
$$222$$ 0 0
$$223$$ −3446.72 −1.03502 −0.517510 0.855677i $$-0.673141\pi$$
−0.517510 + 0.855677i $$0.673141\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 3334.08 0.981326
$$227$$ 486.297 0.142188 0.0710940 0.997470i $$-0.477351\pi$$
0.0710940 + 0.997470i $$0.477351\pi$$
$$228$$ 0 0
$$229$$ 4214.68 1.21622 0.608109 0.793853i $$-0.291928\pi$$
0.608109 + 0.793853i $$0.291928\pi$$
$$230$$ −228.342 −0.0654627
$$231$$ 0 0
$$232$$ −1947.21 −0.551038
$$233$$ 1099.34 0.309100 0.154550 0.987985i $$-0.450607\pi$$
0.154550 + 0.987985i $$0.450607\pi$$
$$234$$ 0 0
$$235$$ 373.760 0.103751
$$236$$ 6313.71 1.74147
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −4558.99 −1.23388 −0.616939 0.787011i $$-0.711627\pi$$
−0.616939 + 0.787011i $$0.711627\pi$$
$$240$$ 0 0
$$241$$ −1089.40 −0.291181 −0.145591 0.989345i $$-0.546508\pi$$
−0.145591 + 0.989345i $$0.546508\pi$$
$$242$$ −15662.4 −4.16041
$$243$$ 0 0
$$244$$ 6745.01 1.76969
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −3103.92 −0.799586
$$248$$ 2015.67 0.516110
$$249$$ 0 0
$$250$$ −974.650 −0.246569
$$251$$ −6356.75 −1.59855 −0.799273 0.600969i $$-0.794782\pi$$
−0.799273 + 0.600969i $$0.794782\pi$$
$$252$$ 0 0
$$253$$ −3986.41 −0.990606
$$254$$ −11053.5 −2.73054
$$255$$ 0 0
$$256$$ −6188.88 −1.51096
$$257$$ 1063.43 0.258113 0.129056 0.991637i $$-0.458805\pi$$
0.129056 + 0.991637i $$0.458805\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 420.677 0.100343
$$261$$ 0 0
$$262$$ −11325.5 −2.67057
$$263$$ −3674.59 −0.861539 −0.430770 0.902462i $$-0.641758\pi$$
−0.430770 + 0.902462i $$0.641758\pi$$
$$264$$ 0 0
$$265$$ 143.224 0.0332006
$$266$$ 0 0
$$267$$ 0 0
$$268$$ −11389.9 −2.59607
$$269$$ −6959.94 −1.57753 −0.788764 0.614696i $$-0.789279\pi$$
−0.788764 + 0.614696i $$0.789279\pi$$
$$270$$ 0 0
$$271$$ −6595.78 −1.47847 −0.739235 0.673448i $$-0.764813\pi$$
−0.739235 + 0.673448i $$0.764813\pi$$
$$272$$ −341.155 −0.0760499
$$273$$ 0 0
$$274$$ −5590.43 −1.23259
$$275$$ −8484.22 −1.86043
$$276$$ 0 0
$$277$$ −3121.06 −0.676990 −0.338495 0.940968i $$-0.609918\pi$$
−0.338495 + 0.940968i $$0.609918\pi$$
$$278$$ 3353.83 0.723559
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1662.38 −0.352915 −0.176457 0.984308i $$-0.556464\pi$$
−0.176457 + 0.984308i $$0.556464\pi$$
$$282$$ 0 0
$$283$$ 453.279 0.0952107 0.0476053 0.998866i $$-0.484841\pi$$
0.0476053 + 0.998866i $$0.484841\pi$$
$$284$$ −5004.62 −1.04567
$$285$$ 0 0
$$286$$ 11493.2 2.37625
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −4697.56 −0.956149
$$290$$ 262.475 0.0531485
$$291$$ 0 0
$$292$$ −11017.8 −2.20811
$$293$$ 772.774 0.154082 0.0770408 0.997028i $$-0.475453\pi$$
0.0770408 + 0.997028i $$0.475453\pi$$
$$294$$ 0 0
$$295$$ −370.263 −0.0730764
$$296$$ 8388.65 1.64723
$$297$$ 0 0
$$298$$ −10400.1 −2.02168
$$299$$ 2089.40 0.404124
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 2574.94 0.490633
$$303$$ 0 0
$$304$$ 2016.77 0.380492
$$305$$ −395.557 −0.0742607
$$306$$ 0 0
$$307$$ 4193.64 0.779621 0.389810 0.920895i $$-0.372541\pi$$
0.389810 + 0.920895i $$0.372541\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −271.703 −0.0497796
$$311$$ 399.740 0.0728848 0.0364424 0.999336i $$-0.488397\pi$$
0.0364424 + 0.999336i $$0.488397\pi$$
$$312$$ 0 0
$$313$$ 3730.48 0.673672 0.336836 0.941563i $$-0.390643\pi$$
0.336836 + 0.941563i $$0.390643\pi$$
$$314$$ −8803.99 −1.58229
$$315$$ 0 0
$$316$$ −1050.73 −0.187051
$$317$$ −8373.47 −1.48360 −0.741800 0.670622i $$-0.766027\pi$$
−0.741800 + 0.670622i $$0.766027\pi$$
$$318$$ 0 0
$$319$$ 4582.31 0.804263
$$320$$ 633.718 0.110706
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −1273.59 −0.219394
$$324$$ 0 0
$$325$$ 4446.84 0.758973
$$326$$ 14298.6 2.42923
$$327$$ 0 0
$$328$$ −10375.6 −1.74664
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −1349.72 −0.224130 −0.112065 0.993701i $$-0.535747\pi$$
−0.112065 + 0.993701i $$0.535747\pi$$
$$332$$ 16546.9 2.73532
$$333$$ 0 0
$$334$$ 13967.7 2.28825
$$335$$ 667.950 0.108937
$$336$$ 0 0
$$337$$ −6629.72 −1.07164 −0.535822 0.844331i $$-0.679998\pi$$
−0.535822 + 0.844331i $$0.679998\pi$$
$$338$$ 4318.49 0.694955
$$339$$ 0 0
$$340$$ 172.610 0.0275327
$$341$$ −4743.41 −0.753284
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 6838.79 1.07187
$$345$$ 0 0
$$346$$ −18203.1 −2.82834
$$347$$ −4952.94 −0.766246 −0.383123 0.923697i $$-0.625152\pi$$
−0.383123 + 0.923697i $$0.625152\pi$$
$$348$$ 0 0
$$349$$ −4498.12 −0.689911 −0.344955 0.938619i $$-0.612106\pi$$
−0.344955 + 0.938619i $$0.612106\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 8367.67 1.26704
$$353$$ 1613.35 0.243258 0.121629 0.992576i $$-0.461188\pi$$
0.121629 + 0.992576i $$0.461188\pi$$
$$354$$ 0 0
$$355$$ 293.492 0.0438787
$$356$$ −21449.7 −3.19335
$$357$$ 0 0
$$358$$ 1575.73 0.232626
$$359$$ −13186.9 −1.93866 −0.969328 0.245770i $$-0.920959\pi$$
−0.969328 + 0.245770i $$0.920959\pi$$
$$360$$ 0 0
$$361$$ 669.919 0.0976700
$$362$$ 7365.22 1.06936
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 646.130 0.0926575
$$366$$ 0 0
$$367$$ 5103.39 0.725871 0.362936 0.931814i $$-0.381774\pi$$
0.362936 + 0.931814i $$0.381774\pi$$
$$368$$ −1357.58 −0.192307
$$369$$ 0 0
$$370$$ −1130.75 −0.158878
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −1459.19 −0.202557 −0.101279 0.994858i $$-0.532293\pi$$
−0.101279 + 0.994858i $$0.532293\pi$$
$$374$$ 4715.85 0.652008
$$375$$ 0 0
$$376$$ 13053.0 1.79031
$$377$$ −2401.73 −0.328104
$$378$$ 0 0
$$379$$ 2332.75 0.316162 0.158081 0.987426i $$-0.449469\pi$$
0.158081 + 0.987426i $$0.449469\pi$$
$$380$$ −1020.40 −0.137751
$$381$$ 0 0
$$382$$ 14438.7 1.93389
$$383$$ 5651.11 0.753938 0.376969 0.926226i $$-0.376966\pi$$
0.376969 + 0.926226i $$0.376966\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 10467.7 1.38028
$$387$$ 0 0
$$388$$ 23102.8 3.02285
$$389$$ 9040.82 1.17838 0.589188 0.807996i $$-0.299448\pi$$
0.589188 + 0.807996i $$0.299448\pi$$
$$390$$ 0 0
$$391$$ 857.314 0.110886
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 8738.02 1.11730
$$395$$ 61.6193 0.00784912
$$396$$ 0 0
$$397$$ 3524.35 0.445547 0.222774 0.974870i $$-0.428489\pi$$
0.222774 + 0.974870i $$0.428489\pi$$
$$398$$ 4677.41 0.589089
$$399$$ 0 0
$$400$$ −2889.33 −0.361166
$$401$$ 4469.81 0.556638 0.278319 0.960489i $$-0.410223\pi$$
0.278319 + 0.960489i $$0.410223\pi$$
$$402$$ 0 0
$$403$$ 2486.17 0.307307
$$404$$ −17074.2 −2.10265
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −19740.7 −2.40420
$$408$$ 0 0
$$409$$ −7532.36 −0.910639 −0.455320 0.890328i $$-0.650475\pi$$
−0.455320 + 0.890328i $$0.650475\pi$$
$$410$$ 1398.58 0.168466
$$411$$ 0 0
$$412$$ −6873.15 −0.821882
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −970.379 −0.114781
$$416$$ −4385.75 −0.516897
$$417$$ 0 0
$$418$$ −27878.2 −3.26212
$$419$$ 15486.4 1.80563 0.902814 0.430031i $$-0.141498\pi$$
0.902814 + 0.430031i $$0.141498\pi$$
$$420$$ 0 0
$$421$$ 4408.87 0.510393 0.255196 0.966889i $$-0.417860\pi$$
0.255196 + 0.966889i $$0.417860\pi$$
$$422$$ 2565.98 0.295995
$$423$$ 0 0
$$424$$ 5001.88 0.572907
$$425$$ 1824.61 0.208251
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 3654.93 0.412774
$$429$$ 0 0
$$430$$ −921.837 −0.103384
$$431$$ −10356.6 −1.15745 −0.578726 0.815522i $$-0.696450\pi$$
−0.578726 + 0.815522i $$0.696450\pi$$
$$432$$ 0 0
$$433$$ −1398.15 −0.155175 −0.0775875 0.996986i $$-0.524722\pi$$
−0.0775875 + 0.996986i $$0.524722\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 8230.94 0.904107
$$437$$ −5068.08 −0.554781
$$438$$ 0 0
$$439$$ −3449.32 −0.375005 −0.187502 0.982264i $$-0.560039\pi$$
−0.187502 + 0.982264i $$0.560039\pi$$
$$440$$ 1643.82 0.178104
$$441$$ 0 0
$$442$$ −2471.73 −0.265991
$$443$$ 2244.65 0.240737 0.120369 0.992729i $$-0.461592\pi$$
0.120369 + 0.992729i $$0.461592\pi$$
$$444$$ 0 0
$$445$$ 1257.91 0.134001
$$446$$ 16225.6 1.72265
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −7639.18 −0.802929 −0.401465 0.915875i $$-0.631499\pi$$
−0.401465 + 0.915875i $$0.631499\pi$$
$$450$$ 0 0
$$451$$ 24416.5 2.54929
$$452$$ −10029.3 −1.04367
$$453$$ 0 0
$$454$$ −2289.26 −0.236652
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 540.823 0.0553581 0.0276790 0.999617i $$-0.491188\pi$$
0.0276790 + 0.999617i $$0.491188\pi$$
$$458$$ −19840.8 −2.02423
$$459$$ 0 0
$$460$$ 686.881 0.0696217
$$461$$ 8627.25 0.871608 0.435804 0.900042i $$-0.356464\pi$$
0.435804 + 0.900042i $$0.356464\pi$$
$$462$$ 0 0
$$463$$ 4497.71 0.451461 0.225730 0.974190i $$-0.427523\pi$$
0.225730 + 0.974190i $$0.427523\pi$$
$$464$$ 1560.52 0.156132
$$465$$ 0 0
$$466$$ −5175.19 −0.514455
$$467$$ −2493.07 −0.247035 −0.123518 0.992342i $$-0.539418\pi$$
−0.123518 + 0.992342i $$0.539418\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −1759.48 −0.172679
$$471$$ 0 0
$$472$$ −12930.9 −1.26100
$$473$$ −16093.5 −1.56444
$$474$$ 0 0
$$475$$ −10786.3 −1.04192
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 21461.6 2.05362
$$479$$ −8339.27 −0.795472 −0.397736 0.917500i $$-0.630204\pi$$
−0.397736 + 0.917500i $$0.630204\pi$$
$$480$$ 0 0
$$481$$ 10346.7 0.980811
$$482$$ 5128.41 0.484632
$$483$$ 0 0
$$484$$ 47114.6 4.42474
$$485$$ −1354.85 −0.126846
$$486$$ 0 0
$$487$$ −3104.85 −0.288899 −0.144450 0.989512i $$-0.546141\pi$$
−0.144450 + 0.989512i $$0.546141\pi$$
$$488$$ −13814.2 −1.28144
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −496.579 −0.0456421 −0.0228211 0.999740i $$-0.507265\pi$$
−0.0228211 + 0.999740i $$0.507265\pi$$
$$492$$ 0 0
$$493$$ −985.468 −0.0900269
$$494$$ 14611.8 1.33080
$$495$$ 0 0
$$496$$ −1615.38 −0.146236
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −13253.6 −1.18900 −0.594501 0.804095i $$-0.702650\pi$$
−0.594501 + 0.804095i $$0.702650\pi$$
$$500$$ 2931.87 0.262234
$$501$$ 0 0
$$502$$ 29924.6 2.66056
$$503$$ −1625.59 −0.144099 −0.0720493 0.997401i $$-0.522954\pi$$
−0.0720493 + 0.997401i $$0.522954\pi$$
$$504$$ 0 0
$$505$$ 1001.30 0.0882325
$$506$$ 18766.1 1.64873
$$507$$ 0 0
$$508$$ 33250.3 2.90402
$$509$$ −7083.05 −0.616799 −0.308399 0.951257i $$-0.599793\pi$$
−0.308399 + 0.951257i $$0.599793\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 8242.42 0.711459
$$513$$ 0 0
$$514$$ −5006.14 −0.429594
$$515$$ 403.071 0.0344882
$$516$$ 0 0
$$517$$ −30717.2 −2.61304
$$518$$ 0 0
$$519$$ 0 0
$$520$$ −861.575 −0.0726588
$$521$$ −16565.3 −1.39298 −0.696489 0.717568i $$-0.745255\pi$$
−0.696489 + 0.717568i $$0.745255\pi$$
$$522$$ 0 0
$$523$$ −6472.69 −0.541168 −0.270584 0.962696i $$-0.587217\pi$$
−0.270584 + 0.962696i $$0.587217\pi$$
$$524$$ 34068.4 2.84024
$$525$$ 0 0
$$526$$ 17298.2 1.43391
$$527$$ 1020.11 0.0843204
$$528$$ 0 0
$$529$$ −8755.43 −0.719605
$$530$$ −674.229 −0.0552578
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −12797.5 −1.04000
$$534$$ 0 0
$$535$$ −214.341 −0.0173210
$$536$$ 23327.2 1.87982
$$537$$ 0 0
$$538$$ 32764.1 2.62558
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −3388.92 −0.269318 −0.134659 0.990892i $$-0.542994\pi$$
−0.134659 + 0.990892i $$0.542994\pi$$
$$542$$ 31049.8 2.46071
$$543$$ 0 0
$$544$$ −1799.54 −0.141829
$$545$$ −482.698 −0.0379385
$$546$$ 0 0
$$547$$ 10055.2 0.785980 0.392990 0.919543i $$-0.371441\pi$$
0.392990 + 0.919543i $$0.371441\pi$$
$$548$$ 16816.7 1.31090
$$549$$ 0 0
$$550$$ 39939.7 3.09643
$$551$$ 5825.67 0.450421
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 14692.5 1.12676
$$555$$ 0 0
$$556$$ −10088.7 −0.769529
$$557$$ 5632.51 0.428469 0.214234 0.976782i $$-0.431274\pi$$
0.214234 + 0.976782i $$0.431274\pi$$
$$558$$ 0 0
$$559$$ 8435.10 0.638223
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 7825.68 0.587378
$$563$$ −1450.41 −0.108575 −0.0542874 0.998525i $$-0.517289\pi$$
−0.0542874 + 0.998525i $$0.517289\pi$$
$$564$$ 0 0
$$565$$ 588.163 0.0437951
$$566$$ −2133.82 −0.158465
$$567$$ 0 0
$$568$$ 10249.8 0.757169
$$569$$ 16030.4 1.18107 0.590536 0.807012i $$-0.298916\pi$$
0.590536 + 0.807012i $$0.298916\pi$$
$$570$$ 0 0
$$571$$ −16540.8 −1.21228 −0.606139 0.795359i $$-0.707282\pi$$
−0.606139 + 0.795359i $$0.707282\pi$$
$$572$$ −34573.1 −2.52723
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 7260.80 0.526602
$$576$$ 0 0
$$577$$ −2039.27 −0.147133 −0.0735666 0.997290i $$-0.523438\pi$$
−0.0735666 + 0.997290i $$0.523438\pi$$
$$578$$ 22113.9 1.59138
$$579$$ 0 0
$$580$$ −789.558 −0.0565252
$$581$$ 0 0
$$582$$ 0 0
$$583$$ −11770.7 −0.836182
$$584$$ 22565.2 1.59889
$$585$$ 0 0
$$586$$ −3637.86 −0.256448
$$587$$ −20392.1 −1.43385 −0.716927 0.697148i $$-0.754452\pi$$
−0.716927 + 0.697148i $$0.754452\pi$$
$$588$$ 0 0
$$589$$ −6030.49 −0.421871
$$590$$ 1743.02 0.121626
$$591$$ 0 0
$$592$$ −6722.77 −0.466730
$$593$$ 20269.0 1.40362 0.701810 0.712364i $$-0.252375\pi$$
0.701810 + 0.712364i $$0.252375\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 31284.7 2.15012
$$597$$ 0 0
$$598$$ −9835.91 −0.672609
$$599$$ 22317.8 1.52234 0.761169 0.648554i $$-0.224626\pi$$
0.761169 + 0.648554i $$0.224626\pi$$
$$600$$ 0 0
$$601$$ 1880.78 0.127652 0.0638259 0.997961i $$-0.479670\pi$$
0.0638259 + 0.997961i $$0.479670\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −7745.74 −0.521804
$$605$$ −2763.00 −0.185673
$$606$$ 0 0
$$607$$ −16361.6 −1.09407 −0.547033 0.837111i $$-0.684243\pi$$
−0.547033 + 0.837111i $$0.684243\pi$$
$$608$$ 10638.2 0.709595
$$609$$ 0 0
$$610$$ 1862.09 0.123597
$$611$$ 16099.8 1.06601
$$612$$ 0 0
$$613$$ −15921.4 −1.04904 −0.524518 0.851399i $$-0.675755\pi$$
−0.524518 + 0.851399i $$0.675755\pi$$
$$614$$ −19741.7 −1.29757
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −7690.80 −0.501815 −0.250908 0.968011i $$-0.580729\pi$$
−0.250908 + 0.968011i $$0.580729\pi$$
$$618$$ 0 0
$$619$$ −604.012 −0.0392202 −0.0196101 0.999808i $$-0.506242\pi$$
−0.0196101 + 0.999808i $$0.506242\pi$$
$$620$$ 817.317 0.0529423
$$621$$ 0 0
$$622$$ −1881.79 −0.121307
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 15366.9 0.983479
$$626$$ −17561.3 −1.12123
$$627$$ 0 0
$$628$$ 26483.5 1.68281
$$629$$ 4245.43 0.269120
$$630$$ 0 0
$$631$$ −10806.1 −0.681752 −0.340876 0.940108i $$-0.610724\pi$$
−0.340876 + 0.940108i $$0.610724\pi$$
$$632$$ 2151.97 0.135444
$$633$$ 0 0
$$634$$ 39418.4 2.46925
$$635$$ −1949.94 −0.121860
$$636$$ 0 0
$$637$$ 0 0
$$638$$ −21571.4 −1.33859
$$639$$ 0 0
$$640$$ −2168.72 −0.133947
$$641$$ 4447.62 0.274057 0.137029 0.990567i $$-0.456245\pi$$
0.137029 + 0.990567i $$0.456245\pi$$
$$642$$ 0 0
$$643$$ 13531.3 0.829897 0.414949 0.909845i $$-0.363800\pi$$
0.414949 + 0.909845i $$0.363800\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 5995.46 0.365152
$$647$$ −18976.9 −1.15311 −0.576553 0.817059i $$-0.695603\pi$$
−0.576553 + 0.817059i $$0.695603\pi$$
$$648$$ 0 0
$$649$$ 30429.8 1.84049
$$650$$ −20933.6 −1.26321
$$651$$ 0 0
$$652$$ −43012.1 −2.58357
$$653$$ 8796.72 0.527171 0.263585 0.964636i $$-0.415095\pi$$
0.263585 + 0.964636i $$0.415095\pi$$
$$654$$ 0 0
$$655$$ −1997.92 −0.119183
$$656$$ 8315.13 0.494895
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 24279.6 1.43520 0.717601 0.696455i $$-0.245240\pi$$
0.717601 + 0.696455i $$0.245240\pi$$
$$660$$ 0 0
$$661$$ −25635.3 −1.50847 −0.754234 0.656605i $$-0.771992\pi$$
−0.754234 + 0.656605i $$0.771992\pi$$
$$662$$ 6353.83 0.373034
$$663$$ 0 0
$$664$$ −33889.1 −1.98065
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −3921.54 −0.227650
$$668$$ −42016.5 −2.43363
$$669$$ 0 0
$$670$$ −3144.40 −0.181311
$$671$$ 32508.6 1.87031
$$672$$ 0 0
$$673$$ −23355.1 −1.33770 −0.668852 0.743396i $$-0.733214\pi$$
−0.668852 + 0.743396i $$0.733214\pi$$
$$674$$ 31209.6 1.78361
$$675$$ 0 0
$$676$$ −12990.6 −0.739108
$$677$$ 17993.7 1.02150 0.510750 0.859730i $$-0.329368\pi$$
0.510750 + 0.859730i $$0.329368\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −353.518 −0.0199365
$$681$$ 0 0
$$682$$ 22329.7 1.25374
$$683$$ 17930.9 1.00455 0.502275 0.864708i $$-0.332496\pi$$
0.502275 + 0.864708i $$0.332496\pi$$
$$684$$ 0 0
$$685$$ −986.205 −0.0550087
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −5480.69 −0.303706
$$689$$ 6169.41 0.341126
$$690$$ 0 0
$$691$$ 25818.0 1.42137 0.710683 0.703512i $$-0.248386\pi$$
0.710683 + 0.703512i $$0.248386\pi$$
$$692$$ 54757.4 3.00804
$$693$$ 0 0
$$694$$ 23316.1 1.27531
$$695$$ 591.647 0.0322913
$$696$$ 0 0
$$697$$ −5251.00 −0.285360
$$698$$ 21175.0 1.14826
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 16727.2 0.901250 0.450625 0.892713i $$-0.351201\pi$$
0.450625 + 0.892713i $$0.351201\pi$$
$$702$$ 0 0
$$703$$ −25097.2 −1.34646
$$704$$ −52081.7 −2.78821
$$705$$ 0 0
$$706$$ −7594.92 −0.404870
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −6891.41 −0.365039 −0.182519 0.983202i $$-0.558425\pi$$
−0.182519 + 0.983202i $$0.558425\pi$$
$$710$$ −1381.62 −0.0730302
$$711$$ 0 0
$$712$$ 43930.5 2.31231
$$713$$ 4059.41 0.213220
$$714$$ 0 0
$$715$$ 2027.51 0.106049
$$716$$ −4740.01 −0.247406
$$717$$ 0 0
$$718$$ 62077.7 3.22663
$$719$$ 10406.4 0.539767 0.269884 0.962893i $$-0.413015\pi$$
0.269884 + 0.962893i $$0.413015\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −3153.66 −0.162558
$$723$$ 0 0
$$724$$ −22155.5 −1.13730
$$725$$ −8346.17 −0.427543
$$726$$ 0 0
$$727$$ 2203.46 0.112410 0.0562049 0.998419i $$-0.482100\pi$$
0.0562049 + 0.998419i $$0.482100\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −3041.68 −0.154216
$$731$$ 3461.05 0.175119
$$732$$ 0 0
$$733$$ −8773.70 −0.442106 −0.221053 0.975262i $$-0.570949\pi$$
−0.221053 + 0.975262i $$0.570949\pi$$
$$734$$ −24024.4 −1.20811
$$735$$ 0 0
$$736$$ −7161.06 −0.358641
$$737$$ −54895.1 −2.74367
$$738$$ 0 0
$$739$$ 27385.9 1.36320 0.681601 0.731724i $$-0.261284\pi$$
0.681601 + 0.731724i $$0.261284\pi$$
$$740$$ 3401.44 0.168972
$$741$$ 0 0
$$742$$ 0 0
$$743$$ −28890.0 −1.42648 −0.713239 0.700921i $$-0.752772\pi$$
−0.713239 + 0.700921i $$0.752772\pi$$
$$744$$ 0 0
$$745$$ −1834.67 −0.0902243
$$746$$ 6869.17 0.337129
$$747$$ 0 0
$$748$$ −14185.9 −0.693432
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −20807.7 −1.01103 −0.505515 0.862818i $$-0.668697\pi$$
−0.505515 + 0.862818i $$0.668697\pi$$
$$752$$ −10460.8 −0.507271
$$753$$ 0 0
$$754$$ 11306.2 0.546085
$$755$$ 454.244 0.0218962
$$756$$ 0 0
$$757$$ 14372.8 0.690078 0.345039 0.938588i $$-0.387866\pi$$
0.345039 + 0.938588i $$0.387866\pi$$
$$758$$ −10981.5 −0.526208
$$759$$ 0 0
$$760$$ 2089.85 0.0997459
$$761$$ −26973.6 −1.28488 −0.642439 0.766337i $$-0.722077\pi$$
−0.642439 + 0.766337i $$0.722077\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −43433.4 −2.05676
$$765$$ 0 0
$$766$$ −26602.8 −1.25483
$$767$$ −15949.2 −0.750838
$$768$$ 0 0
$$769$$ −3264.38 −0.153077 −0.0765387 0.997067i $$-0.524387\pi$$
−0.0765387 + 0.997067i $$0.524387\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −31488.0 −1.46798
$$773$$ 22135.4 1.02995 0.514977 0.857204i $$-0.327800\pi$$
0.514977 + 0.857204i $$0.327800\pi$$
$$774$$ 0 0
$$775$$ 8639.59 0.400443
$$776$$ −47316.1 −2.18885
$$777$$ 0 0
$$778$$ −42560.0 −1.96124
$$779$$ 31041.7 1.42771
$$780$$ 0 0
$$781$$ −24120.5 −1.10512
$$782$$ −4035.83 −0.184554
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −1553.11 −0.0706150
$$786$$ 0 0
$$787$$ 29347.2 1.32924 0.664622 0.747180i $$-0.268593\pi$$
0.664622 + 0.747180i $$0.268593\pi$$
$$788$$ −26285.1 −1.18828
$$789$$ 0 0
$$790$$ −290.075 −0.0130638
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −17038.7 −0.763006
$$794$$ −16591.0 −0.741553
$$795$$ 0 0
$$796$$ −14070.3 −0.626516
$$797$$ −26622.6 −1.18321 −0.591606 0.806227i $$-0.701506\pi$$
−0.591606 + 0.806227i $$0.701506\pi$$
$$798$$ 0 0
$$799$$ 6606.02 0.292496
$$800$$ −15240.8 −0.673554
$$801$$ 0 0
$$802$$ −21041.8 −0.926448
$$803$$ −53101.8 −2.33365
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −11703.7 −0.511471
$$807$$ 0 0
$$808$$ 34969.1 1.52253
$$809$$ 21385.7 0.929395 0.464697 0.885470i $$-0.346163\pi$$
0.464697 + 0.885470i $$0.346163\pi$$
$$810$$ 0 0
$$811$$ −26846.2 −1.16239 −0.581195 0.813764i $$-0.697415\pi$$
−0.581195 + 0.813764i $$0.697415\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 92930.1 4.00147
$$815$$ 2522.42 0.108413
$$816$$ 0 0
$$817$$ −20460.3 −0.876151
$$818$$ 35458.8 1.51563
$$819$$ 0 0
$$820$$ −4207.11 −0.179169
$$821$$ −23679.3 −1.00659 −0.503296 0.864114i $$-0.667880\pi$$
−0.503296 + 0.864114i $$0.667880\pi$$
$$822$$ 0 0
$$823$$ −20546.6 −0.870242 −0.435121 0.900372i $$-0.643294\pi$$
−0.435121 + 0.900372i $$0.643294\pi$$
$$824$$ 14076.7 0.595126
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 24006.6 1.00942 0.504712 0.863288i $$-0.331599\pi$$
0.504712 + 0.863288i $$0.331599\pi$$
$$828$$ 0 0
$$829$$ −20544.1 −0.860706 −0.430353 0.902661i $$-0.641611\pi$$
−0.430353 + 0.902661i $$0.641611\pi$$
$$830$$ 4568.09 0.191037
$$831$$ 0 0
$$832$$ 27297.6 1.13747
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 2464.03 0.102121
$$836$$ 83861.0 3.46937
$$837$$ 0 0
$$838$$ −72902.5 −3.00522
$$839$$ −8213.78 −0.337987 −0.168993 0.985617i $$-0.554052\pi$$
−0.168993 + 0.985617i $$0.554052\pi$$
$$840$$ 0 0
$$841$$ −19881.3 −0.815173
$$842$$ −20754.9 −0.849479
$$843$$ 0 0
$$844$$ −7718.79 −0.314800
$$845$$ 761.823 0.0310148
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −4008.56 −0.162329
$$849$$ 0 0
$$850$$ −8589.41 −0.346605
$$851$$ 16894.1 0.680521
$$852$$ 0 0
$$853$$ 9158.34 0.367615 0.183808 0.982962i $$-0.441158\pi$$
0.183808 + 0.982962i $$0.441158\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −7485.54 −0.298891
$$857$$ 25763.8 1.02692 0.513462 0.858112i $$-0.328363\pi$$
0.513462 + 0.858112i $$0.328363\pi$$
$$858$$ 0 0
$$859$$ 41844.0 1.66205 0.831024 0.556236i $$-0.187755\pi$$
0.831024 + 0.556236i $$0.187755\pi$$
$$860$$ 2773.00 0.109952
$$861$$ 0 0
$$862$$ 48754.2 1.92642
$$863$$ −43821.4 −1.72850 −0.864250 0.503062i $$-0.832207\pi$$
−0.864250 + 0.503062i $$0.832207\pi$$
$$864$$ 0 0
$$865$$ −3211.21 −0.126225
$$866$$ 6581.83 0.258268
$$867$$ 0 0
$$868$$ 0 0
$$869$$ −5064.15 −0.197686
$$870$$ 0 0
$$871$$ 28772.2 1.11930
$$872$$ −16857.5 −0.654665
$$873$$ 0 0
$$874$$ 23858.1 0.923357
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −6690.44 −0.257606 −0.128803 0.991670i $$-0.541113\pi$$
−0.128803 + 0.991670i $$0.541113\pi$$
$$878$$ 16237.8 0.624144
$$879$$ 0 0
$$880$$ −1317.37 −0.0504644
$$881$$ −11946.0 −0.456835 −0.228417 0.973563i $$-0.573355\pi$$
−0.228417 + 0.973563i $$0.573355\pi$$
$$882$$ 0 0
$$883$$ −4878.25 −0.185919 −0.0929593 0.995670i $$-0.529633\pi$$
−0.0929593 + 0.995670i $$0.529633\pi$$
$$884$$ 7435.26 0.282890
$$885$$ 0 0
$$886$$ −10566.8 −0.400674
$$887$$ 30111.7 1.13986 0.569928 0.821695i $$-0.306971\pi$$
0.569928 + 0.821695i $$0.306971\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −5921.63 −0.223026
$$891$$ 0 0
$$892$$ −48808.5 −1.83210
$$893$$ −39052.0 −1.46341
$$894$$ 0 0
$$895$$ 277.974 0.0103817
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 35961.7 1.33637
$$899$$ −4666.23 −0.173112
$$900$$ 0 0
$$901$$ 2531.41 0.0935998
$$902$$ −114942. −4.24294
$$903$$ 0 0
$$904$$ 20540.8 0.755725
$$905$$ 1299.30 0.0477238
$$906$$ 0 0
$$907$$ −1986.84 −0.0727364 −0.0363682 0.999338i $$-0.511579\pi$$
−0.0363682 + 0.999338i $$0.511579\pi$$
$$908$$ 6886.38 0.251688
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 31604.7 1.14941 0.574703 0.818362i $$-0.305118\pi$$
0.574703 + 0.818362i $$0.305118\pi$$
$$912$$ 0 0
$$913$$ 79750.0 2.89084
$$914$$ −2545.94 −0.0921360
$$915$$ 0 0
$$916$$ 59683.5 2.15284
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −26999.8 −0.969140 −0.484570 0.874752i $$-0.661024\pi$$
−0.484570 + 0.874752i $$0.661024\pi$$
$$920$$ −1406.78 −0.0504132
$$921$$ 0 0
$$922$$ −40613.1 −1.45067
$$923$$ 12642.3 0.450841
$$924$$ 0 0
$$925$$ 35955.5 1.27807
$$926$$ −21173.1 −0.751395
$$927$$ 0 0
$$928$$ 8231.51 0.291177
$$929$$ −17312.9 −0.611431 −0.305715 0.952123i $$-0.598896\pi$$
−0.305715 + 0.952123i $$0.598896\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 15567.6 0.547140
$$933$$ 0 0
$$934$$ 11736.2 0.411156
$$935$$ 831.922 0.0290981
$$936$$ 0 0
$$937$$ −31294.1 −1.09107 −0.545535 0.838088i $$-0.683673\pi$$
−0.545535 + 0.838088i $$0.683673\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 5292.75 0.183650
$$941$$ 44549.5 1.54333 0.771665 0.636029i $$-0.219424\pi$$
0.771665 + 0.636029i $$0.219424\pi$$
$$942$$ 0 0
$$943$$ −20895.7 −0.721587
$$944$$ 10363.0 0.357295
$$945$$ 0 0
$$946$$ 75760.6 2.60380
$$947$$ 54541.4 1.87155 0.935775 0.352596i $$-0.114701\pi$$
0.935775 + 0.352596i $$0.114701\pi$$
$$948$$ 0 0
$$949$$ 27832.3 0.952028
$$950$$ 50777.0 1.73413
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 19349.7 0.657710 0.328855 0.944380i $$-0.393337\pi$$
0.328855 + 0.944380i $$0.393337\pi$$
$$954$$ 0 0
$$955$$ 2547.12 0.0863066
$$956$$ −64559.2 −2.18409
$$957$$ 0 0
$$958$$ 39257.4 1.32395
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −24960.7 −0.837861
$$962$$ −48707.5 −1.63243
$$963$$ 0 0
$$964$$ −15426.9 −0.515422
$$965$$ 1846.59 0.0615999
$$966$$ 0 0
$$967$$ 52450.4 1.74425 0.872125 0.489283i $$-0.162742\pi$$
0.872125 + 0.489283i $$0.162742\pi$$
$$968$$ −96493.9 −3.20396
$$969$$ 0 0
$$970$$ 6377.99 0.211118
$$971$$ −13635.8 −0.450662 −0.225331 0.974282i $$-0.572346\pi$$
−0.225331 + 0.974282i $$0.572346\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 14616.2 0.480834
$$975$$ 0 0
$$976$$ 11070.9 0.363085
$$977$$ −30018.6 −0.982989 −0.491495 0.870881i $$-0.663549\pi$$
−0.491495 + 0.870881i $$0.663549\pi$$
$$978$$ 0 0
$$979$$ −103380. −3.37492
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 2337.66 0.0759651
$$983$$ −6470.42 −0.209943 −0.104972 0.994475i $$-0.533475\pi$$
−0.104972 + 0.994475i $$0.533475\pi$$
$$984$$ 0 0
$$985$$ 1541.47 0.0498633
$$986$$ 4639.12 0.149837
$$987$$ 0 0
$$988$$ −43954.2 −1.41535
$$989$$ 13772.8 0.442821
$$990$$ 0 0
$$991$$ −23884.4 −0.765605 −0.382802 0.923830i $$-0.625041\pi$$
−0.382802 + 0.923830i $$0.625041\pi$$
$$992$$ −8520.91 −0.272721
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 825.140 0.0262902
$$996$$ 0 0
$$997$$ 7790.90 0.247483 0.123741 0.992315i $$-0.460511\pi$$
0.123741 + 0.992315i $$0.460511\pi$$
$$998$$ 62391.6 1.97893
$$999$$ 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 1323.4.a.bd.1.1 6
3.2 odd 2 inner 1323.4.a.bd.1.6 6
7.2 even 3 189.4.e.e.109.6 yes 12
7.4 even 3 189.4.e.e.163.6 yes 12
7.6 odd 2 1323.4.a.be.1.1 6
21.2 odd 6 189.4.e.e.109.1 12
21.11 odd 6 189.4.e.e.163.1 yes 12
21.20 even 2 1323.4.a.be.1.6 6

By twisted newform
Twist Min Dim Char Parity Ord Type
189.4.e.e.109.1 12 21.2 odd 6
189.4.e.e.109.6 yes 12 7.2 even 3
189.4.e.e.163.1 yes 12 21.11 odd 6
189.4.e.e.163.6 yes 12 7.4 even 3
1323.4.a.bd.1.1 6 1.1 even 1 trivial
1323.4.a.bd.1.6 6 3.2 odd 2 inner
1323.4.a.be.1.1 6 7.6 odd 2
1323.4.a.be.1.6 6 21.20 even 2