# Properties

 Label 1323.4.a.bg.1.6 Level $1323$ Weight $4$ Character 1323.1 Self dual yes Analytic conductor $78.060$ Analytic rank $0$ 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: $$0$$ 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: $$x^{6} - 42x^{4} + 369x^{2} - 112$$ x^6 - 42*x^4 + 369*x^2 - 112 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{3}\cdot 3^{4}$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.6 Root $$5.45019$$ 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)

 $$f(q)$$ $$=$$ $$q+5.45019 q^{2} +21.7046 q^{4} +19.3432 q^{5} +74.6929 q^{8} +O(q^{10})$$ $$q+5.45019 q^{2} +21.7046 q^{4} +19.3432 q^{5} +74.6929 q^{8} +105.424 q^{10} -11.2132 q^{11} +46.3102 q^{13} +233.454 q^{16} -97.6541 q^{17} +98.9176 q^{19} +419.836 q^{20} -61.1139 q^{22} -138.173 q^{23} +249.158 q^{25} +252.400 q^{26} -180.123 q^{29} +31.9046 q^{31} +674.825 q^{32} -532.234 q^{34} -205.452 q^{37} +539.120 q^{38} +1444.80 q^{40} +234.404 q^{41} -320.568 q^{43} -243.377 q^{44} -753.067 q^{46} -312.715 q^{47} +1357.96 q^{50} +1005.15 q^{52} -53.7942 q^{53} -216.898 q^{55} -981.706 q^{58} +400.291 q^{59} +97.3536 q^{61} +173.886 q^{62} +1810.30 q^{64} +895.786 q^{65} +257.525 q^{67} -2119.55 q^{68} -253.601 q^{71} +1161.99 q^{73} -1119.75 q^{74} +2146.97 q^{76} -1070.16 q^{79} +4515.73 q^{80} +1277.55 q^{82} -889.495 q^{83} -1888.94 q^{85} -1747.16 q^{86} -837.543 q^{88} -647.303 q^{89} -2998.98 q^{92} -1704.36 q^{94} +1913.38 q^{95} -673.379 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 36 q^{4}+O(q^{10})$$ 6 * q + 36 * q^4 $$6 q + 36 q^{4} + 180 q^{10} + 108 q^{13} + 420 q^{16} + 198 q^{19} - 84 q^{22} + 420 q^{25} - 90 q^{31} - 648 q^{34} - 402 q^{37} + 2844 q^{40} - 660 q^{43} - 1332 q^{46} + 1224 q^{52} + 846 q^{55} - 1800 q^{58} + 1152 q^{61} + 2964 q^{64} + 924 q^{67} + 1260 q^{73} + 5868 q^{76} - 1500 q^{79} + 4500 q^{82} - 2232 q^{85} - 2460 q^{88} - 4968 q^{94} + 3312 q^{97}+O(q^{100})$$ 6 * q + 36 * q^4 + 180 * q^10 + 108 * q^13 + 420 * q^16 + 198 * q^19 - 84 * q^22 + 420 * q^25 - 90 * q^31 - 648 * q^34 - 402 * q^37 + 2844 * q^40 - 660 * q^43 - 1332 * q^46 + 1224 * q^52 + 846 * q^55 - 1800 * q^58 + 1152 * q^61 + 2964 * q^64 + 924 * q^67 + 1260 * q^73 + 5868 * q^76 - 1500 * q^79 + 4500 * q^82 - 2232 * q^85 - 2460 * q^88 - 4968 * q^94 + 3312 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 5.45019 1.92693 0.963467 0.267826i $$-0.0863051\pi$$
0.963467 + 0.267826i $$0.0863051\pi$$
$$3$$ 0 0
$$4$$ 21.7046 2.71308
$$5$$ 19.3432 1.73011 0.865053 0.501681i $$-0.167285\pi$$
0.865053 + 0.501681i $$0.167285\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 74.6929 3.30099
$$9$$ 0 0
$$10$$ 105.424 3.33380
$$11$$ −11.2132 −0.307354 −0.153677 0.988121i $$-0.549112\pi$$
−0.153677 + 0.988121i $$0.549112\pi$$
$$12$$ 0 0
$$13$$ 46.3102 0.988010 0.494005 0.869459i $$-0.335532\pi$$
0.494005 + 0.869459i $$0.335532\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 233.454 3.64771
$$17$$ −97.6541 −1.39321 −0.696606 0.717454i $$-0.745307\pi$$
−0.696606 + 0.717454i $$0.745307\pi$$
$$18$$ 0 0
$$19$$ 98.9176 1.19438 0.597191 0.802099i $$-0.296284\pi$$
0.597191 + 0.802099i $$0.296284\pi$$
$$20$$ 419.836 4.69391
$$21$$ 0 0
$$22$$ −61.1139 −0.592251
$$23$$ −138.173 −1.25265 −0.626325 0.779562i $$-0.715442\pi$$
−0.626325 + 0.779562i $$0.715442\pi$$
$$24$$ 0 0
$$25$$ 249.158 1.99327
$$26$$ 252.400 1.90383
$$27$$ 0 0
$$28$$ 0 0
$$29$$ −180.123 −1.15338 −0.576690 0.816963i $$-0.695656\pi$$
−0.576690 + 0.816963i $$0.695656\pi$$
$$30$$ 0 0
$$31$$ 31.9046 0.184846 0.0924230 0.995720i $$-0.470539\pi$$
0.0924230 + 0.995720i $$0.470539\pi$$
$$32$$ 674.825 3.72792
$$33$$ 0 0
$$34$$ −532.234 −2.68463
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −205.452 −0.912867 −0.456433 0.889758i $$-0.650873\pi$$
−0.456433 + 0.889758i $$0.650873\pi$$
$$38$$ 539.120 2.30149
$$39$$ 0 0
$$40$$ 1444.80 5.71106
$$41$$ 234.404 0.892874 0.446437 0.894815i $$-0.352693\pi$$
0.446437 + 0.894815i $$0.352693\pi$$
$$42$$ 0 0
$$43$$ −320.568 −1.13689 −0.568443 0.822723i $$-0.692454\pi$$
−0.568443 + 0.822723i $$0.692454\pi$$
$$44$$ −243.377 −0.833875
$$45$$ 0 0
$$46$$ −753.067 −2.41378
$$47$$ −312.715 −0.970516 −0.485258 0.874371i $$-0.661274\pi$$
−0.485258 + 0.874371i $$0.661274\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 1357.96 3.84089
$$51$$ 0 0
$$52$$ 1005.15 2.68055
$$53$$ −53.7942 −0.139419 −0.0697094 0.997567i $$-0.522207\pi$$
−0.0697094 + 0.997567i $$0.522207\pi$$
$$54$$ 0 0
$$55$$ −216.898 −0.531755
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −981.706 −2.22249
$$59$$ 400.291 0.883278 0.441639 0.897193i $$-0.354397\pi$$
0.441639 + 0.897193i $$0.354397\pi$$
$$60$$ 0 0
$$61$$ 97.3536 0.204342 0.102171 0.994767i $$-0.467421\pi$$
0.102171 + 0.994767i $$0.467421\pi$$
$$62$$ 173.886 0.356186
$$63$$ 0 0
$$64$$ 1810.30 3.53574
$$65$$ 895.786 1.70936
$$66$$ 0 0
$$67$$ 257.525 0.469577 0.234789 0.972046i $$-0.424560\pi$$
0.234789 + 0.972046i $$0.424560\pi$$
$$68$$ −2119.55 −3.77989
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −253.601 −0.423900 −0.211950 0.977280i $$-0.567981\pi$$
−0.211950 + 0.977280i $$0.567981\pi$$
$$72$$ 0 0
$$73$$ 1161.99 1.86303 0.931514 0.363706i $$-0.118489\pi$$
0.931514 + 0.363706i $$0.118489\pi$$
$$74$$ −1119.75 −1.75904
$$75$$ 0 0
$$76$$ 2146.97 3.24045
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −1070.16 −1.52408 −0.762039 0.647531i $$-0.775802\pi$$
−0.762039 + 0.647531i $$0.775802\pi$$
$$80$$ 4515.73 6.31093
$$81$$ 0 0
$$82$$ 1277.55 1.72051
$$83$$ −889.495 −1.17632 −0.588161 0.808744i $$-0.700148\pi$$
−0.588161 + 0.808744i $$0.700148\pi$$
$$84$$ 0 0
$$85$$ −1888.94 −2.41040
$$86$$ −1747.16 −2.19070
$$87$$ 0 0
$$88$$ −837.543 −1.01457
$$89$$ −647.303 −0.770943 −0.385472 0.922720i $$-0.625961\pi$$
−0.385472 + 0.922720i $$0.625961\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −2998.98 −3.39854
$$93$$ 0 0
$$94$$ −1704.36 −1.87012
$$95$$ 1913.38 2.06641
$$96$$ 0 0
$$97$$ −673.379 −0.704859 −0.352429 0.935838i $$-0.614644\pi$$
−0.352429 + 0.935838i $$0.614644\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 5407.89 5.40789
$$101$$ −243.254 −0.239650 −0.119825 0.992795i $$-0.538233\pi$$
−0.119825 + 0.992795i $$0.538233\pi$$
$$102$$ 0 0
$$103$$ −486.156 −0.465072 −0.232536 0.972588i $$-0.574702\pi$$
−0.232536 + 0.972588i $$0.574702\pi$$
$$104$$ 3459.04 3.26141
$$105$$ 0 0
$$106$$ −293.189 −0.268651
$$107$$ 384.679 0.347555 0.173777 0.984785i $$-0.444403\pi$$
0.173777 + 0.984785i $$0.444403\pi$$
$$108$$ 0 0
$$109$$ 994.781 0.874153 0.437077 0.899424i $$-0.356014\pi$$
0.437077 + 0.899424i $$0.356014\pi$$
$$110$$ −1182.14 −1.02466
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −189.932 −0.158118 −0.0790589 0.996870i $$-0.525192\pi$$
−0.0790589 + 0.996870i $$0.525192\pi$$
$$114$$ 0 0
$$115$$ −2672.70 −2.16722
$$116$$ −3909.50 −3.12921
$$117$$ 0 0
$$118$$ 2181.66 1.70202
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −1205.27 −0.905534
$$122$$ 530.596 0.393754
$$123$$ 0 0
$$124$$ 692.476 0.501502
$$125$$ 2401.62 1.71846
$$126$$ 0 0
$$127$$ 1679.82 1.17370 0.586851 0.809695i $$-0.300367\pi$$
0.586851 + 0.809695i $$0.300367\pi$$
$$128$$ 4467.88 3.08522
$$129$$ 0 0
$$130$$ 4882.21 3.29383
$$131$$ 237.457 0.158372 0.0791859 0.996860i $$-0.474768\pi$$
0.0791859 + 0.996860i $$0.474768\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 1403.56 0.904845
$$135$$ 0 0
$$136$$ −7294.07 −4.59898
$$137$$ 526.657 0.328433 0.164217 0.986424i $$-0.447490\pi$$
0.164217 + 0.986424i $$0.447490\pi$$
$$138$$ 0 0
$$139$$ −580.607 −0.354291 −0.177145 0.984185i $$-0.556686\pi$$
−0.177145 + 0.984185i $$0.556686\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1382.18 −0.816828
$$143$$ −519.283 −0.303669
$$144$$ 0 0
$$145$$ −3484.15 −1.99547
$$146$$ 6333.09 3.58993
$$147$$ 0 0
$$148$$ −4459.26 −2.47668
$$149$$ 2781.48 1.52931 0.764657 0.644438i $$-0.222909\pi$$
0.764657 + 0.644438i $$0.222909\pi$$
$$150$$ 0 0
$$151$$ −1413.35 −0.761702 −0.380851 0.924636i $$-0.624369\pi$$
−0.380851 + 0.924636i $$0.624369\pi$$
$$152$$ 7388.44 3.94264
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 617.135 0.319803
$$156$$ 0 0
$$157$$ 2356.59 1.19794 0.598970 0.800772i $$-0.295577\pi$$
0.598970 + 0.800772i $$0.295577\pi$$
$$158$$ −5832.57 −2.93680
$$159$$ 0 0
$$160$$ 13053.3 6.44969
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 1309.51 0.629255 0.314627 0.949215i $$-0.398120\pi$$
0.314627 + 0.949215i $$0.398120\pi$$
$$164$$ 5087.66 2.42244
$$165$$ 0 0
$$166$$ −4847.92 −2.26670
$$167$$ 1623.77 0.752401 0.376201 0.926538i $$-0.377230\pi$$
0.376201 + 0.926538i $$0.377230\pi$$
$$168$$ 0 0
$$169$$ −52.3668 −0.0238356
$$170$$ −10295.1 −4.64469
$$171$$ 0 0
$$172$$ −6957.80 −3.08446
$$173$$ −2219.78 −0.975529 −0.487765 0.872975i $$-0.662188\pi$$
−0.487765 + 0.872975i $$0.662188\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2617.75 −1.12114
$$177$$ 0 0
$$178$$ −3527.92 −1.48556
$$179$$ 1124.59 0.469584 0.234792 0.972046i $$-0.424559\pi$$
0.234792 + 0.972046i $$0.424559\pi$$
$$180$$ 0 0
$$181$$ 3951.04 1.62253 0.811267 0.584676i $$-0.198778\pi$$
0.811267 + 0.584676i $$0.198778\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −10320.5 −4.13499
$$185$$ −3974.09 −1.57936
$$186$$ 0 0
$$187$$ 1095.01 0.428209
$$188$$ −6787.37 −2.63308
$$189$$ 0 0
$$190$$ 10428.3 3.98183
$$191$$ 1177.40 0.446039 0.223019 0.974814i $$-0.428409\pi$$
0.223019 + 0.974814i $$0.428409\pi$$
$$192$$ 0 0
$$193$$ −1772.08 −0.660917 −0.330458 0.943821i $$-0.607203\pi$$
−0.330458 + 0.943821i $$0.607203\pi$$
$$194$$ −3670.05 −1.35822
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −794.274 −0.287257 −0.143629 0.989632i $$-0.545877\pi$$
−0.143629 + 0.989632i $$0.545877\pi$$
$$198$$ 0 0
$$199$$ 3662.14 1.30453 0.652267 0.757990i $$-0.273818\pi$$
0.652267 + 0.757990i $$0.273818\pi$$
$$200$$ 18610.3 6.57975
$$201$$ 0 0
$$202$$ −1325.78 −0.461791
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 4534.12 1.54477
$$206$$ −2649.64 −0.896162
$$207$$ 0 0
$$208$$ 10811.3 3.60398
$$209$$ −1109.18 −0.367098
$$210$$ 0 0
$$211$$ 2063.69 0.673318 0.336659 0.941627i $$-0.390703\pi$$
0.336659 + 0.941627i $$0.390703\pi$$
$$212$$ −1167.58 −0.378254
$$213$$ 0 0
$$214$$ 2096.58 0.669715
$$215$$ −6200.79 −1.96693
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 5421.75 1.68444
$$219$$ 0 0
$$220$$ −4707.69 −1.44269
$$221$$ −4522.38 −1.37651
$$222$$ 0 0
$$223$$ −3728.28 −1.11957 −0.559784 0.828638i $$-0.689116\pi$$
−0.559784 + 0.828638i $$0.689116\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −1035.17 −0.304683
$$227$$ −4558.22 −1.33277 −0.666387 0.745606i $$-0.732160\pi$$
−0.666387 + 0.745606i $$0.732160\pi$$
$$228$$ 0 0
$$229$$ 2068.22 0.596819 0.298410 0.954438i $$-0.403544\pi$$
0.298410 + 0.954438i $$0.403544\pi$$
$$230$$ −14566.7 −4.17609
$$231$$ 0 0
$$232$$ −13453.9 −3.80730
$$233$$ 1459.56 0.410383 0.205191 0.978722i $$-0.434218\pi$$
0.205191 + 0.978722i $$0.434218\pi$$
$$234$$ 0 0
$$235$$ −6048.91 −1.67909
$$236$$ 8688.16 2.39640
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −2220.74 −0.601035 −0.300518 0.953776i $$-0.597159\pi$$
−0.300518 + 0.953776i $$0.597159\pi$$
$$240$$ 0 0
$$241$$ −5064.37 −1.35363 −0.676815 0.736153i $$-0.736640\pi$$
−0.676815 + 0.736153i $$0.736640\pi$$
$$242$$ −6568.93 −1.74490
$$243$$ 0 0
$$244$$ 2113.02 0.554395
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4580.89 1.18006
$$248$$ 2383.04 0.610175
$$249$$ 0 0
$$250$$ 13089.3 3.31135
$$251$$ −1797.86 −0.452112 −0.226056 0.974114i $$-0.572583\pi$$
−0.226056 + 0.974114i $$0.572583\pi$$
$$252$$ 0 0
$$253$$ 1549.35 0.385007
$$254$$ 9155.37 2.26165
$$255$$ 0 0
$$256$$ 9868.42 2.40928
$$257$$ −2734.28 −0.663656 −0.331828 0.943340i $$-0.607665\pi$$
−0.331828 + 0.943340i $$0.607665\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 19442.7 4.63763
$$261$$ 0 0
$$262$$ 1294.19 0.305172
$$263$$ 8197.71 1.92203 0.961013 0.276504i $$-0.0891758\pi$$
0.961013 + 0.276504i $$0.0891758\pi$$
$$264$$ 0 0
$$265$$ −1040.55 −0.241209
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 5589.48 1.27400
$$269$$ 237.512 0.0538342 0.0269171 0.999638i $$-0.491431\pi$$
0.0269171 + 0.999638i $$0.491431\pi$$
$$270$$ 0 0
$$271$$ −4349.95 −0.975057 −0.487528 0.873107i $$-0.662101\pi$$
−0.487528 + 0.873107i $$0.662101\pi$$
$$272$$ −22797.7 −5.08204
$$273$$ 0 0
$$274$$ 2870.38 0.632869
$$275$$ −2793.85 −0.612638
$$276$$ 0 0
$$277$$ 6228.51 1.35103 0.675515 0.737347i $$-0.263921\pi$$
0.675515 + 0.737347i $$0.263921\pi$$
$$278$$ −3164.42 −0.682695
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −3360.95 −0.713515 −0.356757 0.934197i $$-0.616118\pi$$
−0.356757 + 0.934197i $$0.616118\pi$$
$$282$$ 0 0
$$283$$ 8009.37 1.68236 0.841180 0.540755i $$-0.181862\pi$$
0.841180 + 0.540755i $$0.181862\pi$$
$$284$$ −5504.32 −1.15007
$$285$$ 0 0
$$286$$ −2830.19 −0.585150
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 4623.33 0.941041
$$290$$ −18989.3 −3.84514
$$291$$ 0 0
$$292$$ 25220.6 5.05454
$$293$$ 9247.12 1.84376 0.921881 0.387472i $$-0.126652\pi$$
0.921881 + 0.387472i $$0.126652\pi$$
$$294$$ 0 0
$$295$$ 7742.89 1.52816
$$296$$ −15345.8 −3.01336
$$297$$ 0 0
$$298$$ 15159.6 2.94689
$$299$$ −6398.80 −1.23763
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −7703.04 −1.46775
$$303$$ 0 0
$$304$$ 23092.7 4.35676
$$305$$ 1883.13 0.353533
$$306$$ 0 0
$$307$$ 3367.11 0.625965 0.312983 0.949759i $$-0.398672\pi$$
0.312983 + 0.949759i $$0.398672\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 3363.51 0.616240
$$311$$ −6555.92 −1.19534 −0.597672 0.801740i $$-0.703908\pi$$
−0.597672 + 0.801740i $$0.703908\pi$$
$$312$$ 0 0
$$313$$ −616.028 −0.111246 −0.0556229 0.998452i $$-0.517714\pi$$
−0.0556229 + 0.998452i $$0.517714\pi$$
$$314$$ 12843.9 2.30835
$$315$$ 0 0
$$316$$ −23227.4 −4.13494
$$317$$ 3481.49 0.616845 0.308422 0.951250i $$-0.400199\pi$$
0.308422 + 0.951250i $$0.400199\pi$$
$$318$$ 0 0
$$319$$ 2019.75 0.354496
$$320$$ 35016.9 6.11720
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −9659.71 −1.66403
$$324$$ 0 0
$$325$$ 11538.6 1.96937
$$326$$ 7137.07 1.21253
$$327$$ 0 0
$$328$$ 17508.3 2.94737
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −7187.44 −1.19353 −0.596763 0.802417i $$-0.703547\pi$$
−0.596763 + 0.802417i $$0.703547\pi$$
$$332$$ −19306.2 −3.19145
$$333$$ 0 0
$$334$$ 8849.86 1.44983
$$335$$ 4981.35 0.812418
$$336$$ 0 0
$$337$$ 3575.89 0.578015 0.289007 0.957327i $$-0.406675\pi$$
0.289007 + 0.957327i $$0.406675\pi$$
$$338$$ −285.410 −0.0459297
$$339$$ 0 0
$$340$$ −40998.8 −6.53962
$$341$$ −357.751 −0.0568131
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −23944.1 −3.75285
$$345$$ 0 0
$$346$$ −12098.2 −1.87978
$$347$$ −574.458 −0.0888719 −0.0444359 0.999012i $$-0.514149\pi$$
−0.0444359 + 0.999012i $$0.514149\pi$$
$$348$$ 0 0
$$349$$ −11491.0 −1.76246 −0.881229 0.472689i $$-0.843283\pi$$
−0.881229 + 0.472689i $$0.843283\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −7566.92 −1.14579
$$353$$ −1369.32 −0.206464 −0.103232 0.994657i $$-0.532918\pi$$
−0.103232 + 0.994657i $$0.532918\pi$$
$$354$$ 0 0
$$355$$ −4905.45 −0.733392
$$356$$ −14049.5 −2.09163
$$357$$ 0 0
$$358$$ 6129.21 0.904857
$$359$$ −12915.0 −1.89868 −0.949339 0.314254i $$-0.898246\pi$$
−0.949339 + 0.314254i $$0.898246\pi$$
$$360$$ 0 0
$$361$$ 2925.68 0.426546
$$362$$ 21533.9 3.12652
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 22476.6 3.22323
$$366$$ 0 0
$$367$$ 1186.30 0.168731 0.0843653 0.996435i $$-0.473114\pi$$
0.0843653 + 0.996435i $$0.473114\pi$$
$$368$$ −32256.9 −4.56931
$$369$$ 0 0
$$370$$ −21659.6 −3.04332
$$371$$ 0 0
$$372$$ 0 0
$$373$$ −6462.68 −0.897117 −0.448559 0.893753i $$-0.648062\pi$$
−0.448559 + 0.893753i $$0.648062\pi$$
$$374$$ 5968.02 0.825131
$$375$$ 0 0
$$376$$ −23357.6 −3.20366
$$377$$ −8341.54 −1.13955
$$378$$ 0 0
$$379$$ −8555.31 −1.15952 −0.579758 0.814789i $$-0.696853\pi$$
−0.579758 + 0.814789i $$0.696853\pi$$
$$380$$ 41529.2 5.60632
$$381$$ 0 0
$$382$$ 6417.04 0.859488
$$383$$ 570.982 0.0761770 0.0380885 0.999274i $$-0.487873\pi$$
0.0380885 + 0.999274i $$0.487873\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −9658.17 −1.27354
$$387$$ 0 0
$$388$$ −14615.4 −1.91234
$$389$$ −14437.2 −1.88174 −0.940869 0.338771i $$-0.889989\pi$$
−0.940869 + 0.338771i $$0.889989\pi$$
$$390$$ 0 0
$$391$$ 13493.1 1.74521
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −4328.95 −0.553526
$$395$$ −20700.3 −2.63682
$$396$$ 0 0
$$397$$ 9384.81 1.18642 0.593212 0.805047i $$-0.297860\pi$$
0.593212 + 0.805047i $$0.297860\pi$$
$$398$$ 19959.4 2.51375
$$399$$ 0 0
$$400$$ 58166.9 7.27086
$$401$$ 8167.50 1.01712 0.508560 0.861026i $$-0.330178\pi$$
0.508560 + 0.861026i $$0.330178\pi$$
$$402$$ 0 0
$$403$$ 1477.51 0.182630
$$404$$ −5279.74 −0.650190
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 2303.76 0.280573
$$408$$ 0 0
$$409$$ −11655.7 −1.40913 −0.704567 0.709637i $$-0.748859\pi$$
−0.704567 + 0.709637i $$0.748859\pi$$
$$410$$ 24711.9 2.97666
$$411$$ 0 0
$$412$$ −10551.8 −1.26178
$$413$$ 0 0
$$414$$ 0 0
$$415$$ −17205.7 −2.03516
$$416$$ 31251.3 3.68322
$$417$$ 0 0
$$418$$ −6045.23 −0.707373
$$419$$ 10257.6 1.19598 0.597992 0.801502i $$-0.295965\pi$$
0.597992 + 0.801502i $$0.295965\pi$$
$$420$$ 0 0
$$421$$ −8160.70 −0.944723 −0.472361 0.881405i $$-0.656598\pi$$
−0.472361 + 0.881405i $$0.656598\pi$$
$$422$$ 11247.5 1.29744
$$423$$ 0 0
$$424$$ −4018.04 −0.460220
$$425$$ −24331.3 −2.77704
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 8349.32 0.942943
$$429$$ 0 0
$$430$$ −33795.5 −3.79015
$$431$$ −8722.62 −0.974835 −0.487417 0.873169i $$-0.662061\pi$$
−0.487417 + 0.873169i $$0.662061\pi$$
$$432$$ 0 0
$$433$$ 12628.4 1.40157 0.700787 0.713371i $$-0.252832\pi$$
0.700787 + 0.713371i $$0.252832\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 21591.3 2.37165
$$437$$ −13667.7 −1.49614
$$438$$ 0 0
$$439$$ −3487.47 −0.379153 −0.189576 0.981866i $$-0.560711\pi$$
−0.189576 + 0.981866i $$0.560711\pi$$
$$440$$ −16200.7 −1.75532
$$441$$ 0 0
$$442$$ −24647.9 −2.65244
$$443$$ −14138.0 −1.51629 −0.758144 0.652087i $$-0.773894\pi$$
−0.758144 + 0.652087i $$0.773894\pi$$
$$444$$ 0 0
$$445$$ −12520.9 −1.33381
$$446$$ −20319.8 −2.15734
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 14007.0 1.47223 0.736114 0.676858i $$-0.236659\pi$$
0.736114 + 0.676858i $$0.236659\pi$$
$$450$$ 0 0
$$451$$ −2628.41 −0.274428
$$452$$ −4122.40 −0.428986
$$453$$ 0 0
$$454$$ −24843.2 −2.56817
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 4837.29 0.495140 0.247570 0.968870i $$-0.420368\pi$$
0.247570 + 0.968870i $$0.420368\pi$$
$$458$$ 11272.2 1.15003
$$459$$ 0 0
$$460$$ −58009.9 −5.87983
$$461$$ −10034.5 −1.01378 −0.506890 0.862011i $$-0.669205\pi$$
−0.506890 + 0.862011i $$0.669205\pi$$
$$462$$ 0 0
$$463$$ −15501.9 −1.55602 −0.778008 0.628254i $$-0.783770\pi$$
−0.778008 + 0.628254i $$0.783770\pi$$
$$464$$ −42050.4 −4.20720
$$465$$ 0 0
$$466$$ 7954.91 0.790781
$$467$$ −9616.13 −0.952851 −0.476426 0.879215i $$-0.658068\pi$$
−0.476426 + 0.879215i $$0.658068\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −32967.7 −3.23551
$$471$$ 0 0
$$472$$ 29898.8 2.91569
$$473$$ 3594.57 0.349426
$$474$$ 0 0
$$475$$ 24646.1 2.38072
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −12103.4 −1.15816
$$479$$ 14159.7 1.35067 0.675337 0.737509i $$-0.263998\pi$$
0.675337 + 0.737509i $$0.263998\pi$$
$$480$$ 0 0
$$481$$ −9514.51 −0.901922
$$482$$ −27601.8 −2.60836
$$483$$ 0 0
$$484$$ −26159.8 −2.45678
$$485$$ −13025.3 −1.21948
$$486$$ 0 0
$$487$$ 15584.6 1.45012 0.725058 0.688688i $$-0.241813\pi$$
0.725058 + 0.688688i $$0.241813\pi$$
$$488$$ 7271.62 0.674530
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −12898.0 −1.18549 −0.592747 0.805389i $$-0.701957\pi$$
−0.592747 + 0.805389i $$0.701957\pi$$
$$492$$ 0 0
$$493$$ 17589.8 1.60690
$$494$$ 24966.7 2.27390
$$495$$ 0 0
$$496$$ 7448.23 0.674265
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 6707.11 0.601707 0.300853 0.953670i $$-0.402729\pi$$
0.300853 + 0.953670i $$0.402729\pi$$
$$500$$ 52126.1 4.66230
$$501$$ 0 0
$$502$$ −9798.70 −0.871190
$$503$$ 14186.3 1.25753 0.628763 0.777597i $$-0.283562\pi$$
0.628763 + 0.777597i $$0.283562\pi$$
$$504$$ 0 0
$$505$$ −4705.31 −0.414620
$$506$$ 8444.26 0.741884
$$507$$ 0 0
$$508$$ 36459.9 3.18435
$$509$$ −12086.8 −1.05253 −0.526263 0.850322i $$-0.676407\pi$$
−0.526263 + 0.850322i $$0.676407\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 18041.8 1.55731
$$513$$ 0 0
$$514$$ −14902.3 −1.27882
$$515$$ −9403.80 −0.804623
$$516$$ 0 0
$$517$$ 3506.53 0.298292
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 66908.8 5.64259
$$521$$ 12142.6 1.02107 0.510534 0.859857i $$-0.329448\pi$$
0.510534 + 0.859857i $$0.329448\pi$$
$$522$$ 0 0
$$523$$ −3408.38 −0.284967 −0.142484 0.989797i $$-0.545509\pi$$
−0.142484 + 0.989797i $$0.545509\pi$$
$$524$$ 5153.91 0.429675
$$525$$ 0 0
$$526$$ 44679.1 3.70362
$$527$$ −3115.61 −0.257530
$$528$$ 0 0
$$529$$ 6924.66 0.569135
$$530$$ −5671.20 −0.464795
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 10855.3 0.882168
$$534$$ 0 0
$$535$$ 7440.92 0.601306
$$536$$ 19235.3 1.55007
$$537$$ 0 0
$$538$$ 1294.49 0.103735
$$539$$ 0 0
$$540$$ 0 0
$$541$$ −16737.5 −1.33013 −0.665066 0.746785i $$-0.731597\pi$$
−0.665066 + 0.746785i $$0.731597\pi$$
$$542$$ −23708.1 −1.87887
$$543$$ 0 0
$$544$$ −65899.5 −5.19378
$$545$$ 19242.2 1.51238
$$546$$ 0 0
$$547$$ −13940.6 −1.08968 −0.544840 0.838540i $$-0.683410\pi$$
−0.544840 + 0.838540i $$0.683410\pi$$
$$548$$ 11430.9 0.891065
$$549$$ 0 0
$$550$$ −15227.0 −1.18051
$$551$$ −17817.3 −1.37758
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 33946.6 2.60335
$$555$$ 0 0
$$556$$ −12601.8 −0.961218
$$557$$ 818.376 0.0622544 0.0311272 0.999515i $$-0.490090\pi$$
0.0311272 + 0.999515i $$0.490090\pi$$
$$558$$ 0 0
$$559$$ −14845.5 −1.12325
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −18317.9 −1.37490
$$563$$ −8957.98 −0.670575 −0.335288 0.942116i $$-0.608834\pi$$
−0.335288 + 0.942116i $$0.608834\pi$$
$$564$$ 0 0
$$565$$ −3673.89 −0.273560
$$566$$ 43652.6 3.24180
$$567$$ 0 0
$$568$$ −18942.2 −1.39929
$$569$$ 19997.8 1.47338 0.736688 0.676233i $$-0.236389\pi$$
0.736688 + 0.676233i $$0.236389\pi$$
$$570$$ 0 0
$$571$$ −3391.26 −0.248546 −0.124273 0.992248i $$-0.539660\pi$$
−0.124273 + 0.992248i $$0.539660\pi$$
$$572$$ −11270.8 −0.823877
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −34426.8 −2.49687
$$576$$ 0 0
$$577$$ −3957.72 −0.285549 −0.142775 0.989755i $$-0.545602\pi$$
−0.142775 + 0.989755i $$0.545602\pi$$
$$578$$ 25198.1 1.81332
$$579$$ 0 0
$$580$$ −75622.2 −5.41387
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 603.202 0.0428509
$$584$$ 86792.6 6.14983
$$585$$ 0 0
$$586$$ 50398.6 3.55281
$$587$$ −12647.3 −0.889283 −0.444642 0.895709i $$-0.646669\pi$$
−0.444642 + 0.895709i $$0.646669\pi$$
$$588$$ 0 0
$$589$$ 3155.92 0.220777
$$590$$ 42200.3 2.94467
$$591$$ 0 0
$$592$$ −47963.5 −3.32988
$$593$$ −3910.60 −0.270808 −0.135404 0.990790i $$-0.543233\pi$$
−0.135404 + 0.990790i $$0.543233\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 60371.0 4.14915
$$597$$ 0 0
$$598$$ −34874.7 −2.38484
$$599$$ −13933.9 −0.950455 −0.475228 0.879863i $$-0.657634\pi$$
−0.475228 + 0.879863i $$0.657634\pi$$
$$600$$ 0 0
$$601$$ 16095.1 1.09240 0.546202 0.837654i $$-0.316073\pi$$
0.546202 + 0.837654i $$0.316073\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −30676.3 −2.06656
$$605$$ −23313.7 −1.56667
$$606$$ 0 0
$$607$$ −3050.91 −0.204008 −0.102004 0.994784i $$-0.532525\pi$$
−0.102004 + 0.994784i $$0.532525\pi$$
$$608$$ 66752.0 4.45255
$$609$$ 0 0
$$610$$ 10263.4 0.681235
$$611$$ −14481.9 −0.958879
$$612$$ 0 0
$$613$$ −1512.24 −0.0996394 −0.0498197 0.998758i $$-0.515865\pi$$
−0.0498197 + 0.998758i $$0.515865\pi$$
$$614$$ 18351.4 1.20619
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 23466.1 1.53114 0.765569 0.643354i $$-0.222458\pi$$
0.765569 + 0.643354i $$0.222458\pi$$
$$618$$ 0 0
$$619$$ 20736.1 1.34645 0.673226 0.739437i $$-0.264908\pi$$
0.673226 + 0.739437i $$0.264908\pi$$
$$620$$ 13394.7 0.867651
$$621$$ 0 0
$$622$$ −35731.1 −2.30335
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 15310.1 0.979844
$$626$$ −3357.47 −0.214364
$$627$$ 0 0
$$628$$ 51148.9 3.25010
$$629$$ 20063.2 1.27182
$$630$$ 0 0
$$631$$ 15623.5 0.985673 0.492837 0.870122i $$-0.335960\pi$$
0.492837 + 0.870122i $$0.335960\pi$$
$$632$$ −79933.2 −5.03097
$$633$$ 0 0
$$634$$ 18974.8 1.18862
$$635$$ 32493.1 2.03063
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 11008.0 0.683091
$$639$$ 0 0
$$640$$ 86423.0 5.33776
$$641$$ 12284.6 0.756961 0.378481 0.925609i $$-0.376447\pi$$
0.378481 + 0.925609i $$0.376447\pi$$
$$642$$ 0 0
$$643$$ −924.041 −0.0566728 −0.0283364 0.999598i $$-0.509021\pi$$
−0.0283364 + 0.999598i $$0.509021\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −52647.3 −3.20647
$$647$$ −10330.2 −0.627701 −0.313850 0.949472i $$-0.601619\pi$$
−0.313850 + 0.949472i $$0.601619\pi$$
$$648$$ 0 0
$$649$$ −4488.52 −0.271479
$$650$$ 62887.4 3.79484
$$651$$ 0 0
$$652$$ 28422.4 1.70722
$$653$$ 3064.12 0.183627 0.0918134 0.995776i $$-0.470734\pi$$
0.0918134 + 0.995776i $$0.470734\pi$$
$$654$$ 0 0
$$655$$ 4593.17 0.274000
$$656$$ 54722.6 3.25695
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −9157.78 −0.541330 −0.270665 0.962674i $$-0.587244\pi$$
−0.270665 + 0.962674i $$0.587244\pi$$
$$660$$ 0 0
$$661$$ 23367.4 1.37502 0.687508 0.726177i $$-0.258705\pi$$
0.687508 + 0.726177i $$0.258705\pi$$
$$662$$ −39172.9 −2.29985
$$663$$ 0 0
$$664$$ −66438.9 −3.88303
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 24888.1 1.44478
$$668$$ 35243.3 2.04132
$$669$$ 0 0
$$670$$ 27149.3 1.56548
$$671$$ −1091.64 −0.0628053
$$672$$ 0 0
$$673$$ 34214.8 1.95971 0.979853 0.199719i $$-0.0640029\pi$$
0.979853 + 0.199719i $$0.0640029\pi$$
$$674$$ 19489.3 1.11380
$$675$$ 0 0
$$676$$ −1136.60 −0.0646679
$$677$$ 19288.8 1.09502 0.547511 0.836799i $$-0.315576\pi$$
0.547511 + 0.836799i $$0.315576\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −141090. −7.95672
$$681$$ 0 0
$$682$$ −1949.81 −0.109475
$$683$$ 31119.7 1.74343 0.871714 0.490015i $$-0.163009\pi$$
0.871714 + 0.490015i $$0.163009\pi$$
$$684$$ 0 0
$$685$$ 10187.2 0.568224
$$686$$ 0 0
$$687$$ 0 0
$$688$$ −74837.7 −4.14703
$$689$$ −2491.22 −0.137747
$$690$$ 0 0
$$691$$ 12732.5 0.700967 0.350483 0.936569i $$-0.386017\pi$$
0.350483 + 0.936569i $$0.386017\pi$$
$$692$$ −48179.4 −2.64669
$$693$$ 0 0
$$694$$ −3130.91 −0.171250
$$695$$ −11230.8 −0.612961
$$696$$ 0 0
$$697$$ −22890.6 −1.24396
$$698$$ −62628.1 −3.39614
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −9506.11 −0.512184 −0.256092 0.966652i $$-0.582435\pi$$
−0.256092 + 0.966652i $$0.582435\pi$$
$$702$$ 0 0
$$703$$ −20322.8 −1.09031
$$704$$ −20299.2 −1.08672
$$705$$ 0 0
$$706$$ −7463.08 −0.397842
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 2500.48 0.132450 0.0662252 0.997805i $$-0.478904\pi$$
0.0662252 + 0.997805i $$0.478904\pi$$
$$710$$ −26735.7 −1.41320
$$711$$ 0 0
$$712$$ −48348.9 −2.54487
$$713$$ −4408.33 −0.231548
$$714$$ 0 0
$$715$$ −10044.6 −0.525379
$$716$$ 24408.7 1.27402
$$717$$ 0 0
$$718$$ −70389.0 −3.65863
$$719$$ −19519.3 −1.01245 −0.506223 0.862403i $$-0.668959\pi$$
−0.506223 + 0.862403i $$0.668959\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 15945.5 0.821927
$$723$$ 0 0
$$724$$ 85755.9 4.40206
$$725$$ −44879.2 −2.29899
$$726$$ 0 0
$$727$$ 35008.3 1.78595 0.892975 0.450105i $$-0.148614\pi$$
0.892975 + 0.450105i $$0.148614\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 122502. 6.21096
$$731$$ 31304.7 1.58392
$$732$$ 0 0
$$733$$ 33284.0 1.67718 0.838590 0.544763i $$-0.183381\pi$$
0.838590 + 0.544763i $$0.183381\pi$$
$$734$$ 6465.54 0.325133
$$735$$ 0 0
$$736$$ −93242.3 −4.66978
$$737$$ −2887.67 −0.144326
$$738$$ 0 0
$$739$$ −25554.2 −1.27203 −0.636013 0.771678i $$-0.719418\pi$$
−0.636013 + 0.771678i $$0.719418\pi$$
$$740$$ −86256.1 −4.28492
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 3926.32 0.193867 0.0969333 0.995291i $$-0.469097\pi$$
0.0969333 + 0.995291i $$0.469097\pi$$
$$744$$ 0 0
$$745$$ 53802.6 2.64587
$$746$$ −35222.8 −1.72869
$$747$$ 0 0
$$748$$ 23766.8 1.16177
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 2496.80 0.121318 0.0606589 0.998159i $$-0.480680\pi$$
0.0606589 + 0.998159i $$0.480680\pi$$
$$752$$ −73004.6 −3.54016
$$753$$ 0 0
$$754$$ −45463.0 −2.19584
$$755$$ −27338.7 −1.31782
$$756$$ 0 0
$$757$$ 9014.06 0.432789 0.216395 0.976306i $$-0.430570\pi$$
0.216395 + 0.976306i $$0.430570\pi$$
$$758$$ −46628.1 −2.23431
$$759$$ 0 0
$$760$$ 142916. 6.82118
$$761$$ 2393.59 0.114018 0.0570089 0.998374i $$-0.481844\pi$$
0.0570089 + 0.998374i $$0.481844\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 25554.9 1.21014
$$765$$ 0 0
$$766$$ 3111.96 0.146788
$$767$$ 18537.5 0.872688
$$768$$ 0 0
$$769$$ 9474.06 0.444270 0.222135 0.975016i $$-0.428697\pi$$
0.222135 + 0.975016i $$0.428697\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −38462.3 −1.79312
$$773$$ 12878.4 0.599231 0.299615 0.954060i $$-0.403142\pi$$
0.299615 + 0.954060i $$0.403142\pi$$
$$774$$ 0 0
$$775$$ 7949.28 0.368447
$$776$$ −50296.6 −2.32673
$$777$$ 0 0
$$778$$ −78685.6 −3.62599
$$779$$ 23186.7 1.06643
$$780$$ 0 0
$$781$$ 2843.67 0.130287
$$782$$ 73540.2 3.36290
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 45584.0 2.07256
$$786$$ 0 0
$$787$$ −16966.5 −0.768477 −0.384239 0.923234i $$-0.625536\pi$$
−0.384239 + 0.923234i $$0.625536\pi$$
$$788$$ −17239.4 −0.779351
$$789$$ 0 0
$$790$$ −112820. −5.08097
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 4508.46 0.201892
$$794$$ 51149.0 2.28616
$$795$$ 0 0
$$796$$ 79485.3 3.53930
$$797$$ 21.6935 0.000964143 0 0.000482072 1.00000i $$-0.499847\pi$$
0.000482072 1.00000i $$0.499847\pi$$
$$798$$ 0 0
$$799$$ 30538.0 1.35213
$$800$$ 168138. 7.43073
$$801$$ 0 0
$$802$$ 44514.5 1.95993
$$803$$ −13029.6 −0.572609
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 8052.69 0.351916
$$807$$ 0 0
$$808$$ −18169.3 −0.791083
$$809$$ 31766.3 1.38052 0.690262 0.723560i $$-0.257495\pi$$
0.690262 + 0.723560i $$0.257495\pi$$
$$810$$ 0 0
$$811$$ −31081.1 −1.34575 −0.672875 0.739756i $$-0.734941\pi$$
−0.672875 + 0.739756i $$0.734941\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 12556.0 0.540646
$$815$$ 25330.0 1.08868
$$816$$ 0 0
$$817$$ −31709.8 −1.35788
$$818$$ −63525.7 −2.71531
$$819$$ 0 0
$$820$$ 98411.5 4.19107
$$821$$ 5021.74 0.213472 0.106736 0.994287i $$-0.465960\pi$$
0.106736 + 0.994287i $$0.465960\pi$$
$$822$$ 0 0
$$823$$ 13443.5 0.569394 0.284697 0.958617i $$-0.408107\pi$$
0.284697 + 0.958617i $$0.408107\pi$$
$$824$$ −36312.4 −1.53520
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −37439.2 −1.57423 −0.787116 0.616805i $$-0.788427\pi$$
−0.787116 + 0.616805i $$0.788427\pi$$
$$828$$ 0 0
$$829$$ 3890.21 0.162982 0.0814912 0.996674i $$-0.474032\pi$$
0.0814912 + 0.996674i $$0.474032\pi$$
$$830$$ −93774.2 −3.92162
$$831$$ 0 0
$$832$$ 83835.3 3.49335
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 31408.9 1.30173
$$836$$ −24074.3 −0.995965
$$837$$ 0 0
$$838$$ 55906.0 2.30458
$$839$$ 15062.0 0.619783 0.309891 0.950772i $$-0.399707\pi$$
0.309891 + 0.950772i $$0.399707\pi$$
$$840$$ 0 0
$$841$$ 8055.35 0.330286
$$842$$ −44477.4 −1.82042
$$843$$ 0 0
$$844$$ 44791.5 1.82676
$$845$$ −1012.94 −0.0412381
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −12558.4 −0.508560
$$849$$ 0 0
$$850$$ −132611. −5.35118
$$851$$ 28387.8 1.14350
$$852$$ 0 0
$$853$$ 39207.9 1.57380 0.786900 0.617081i $$-0.211685\pi$$
0.786900 + 0.617081i $$0.211685\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 28732.8 1.14727
$$857$$ −37029.6 −1.47597 −0.737986 0.674816i $$-0.764223\pi$$
−0.737986 + 0.674816i $$0.764223\pi$$
$$858$$ 0 0
$$859$$ −37646.3 −1.49531 −0.747657 0.664085i $$-0.768821\pi$$
−0.747657 + 0.664085i $$0.768821\pi$$
$$860$$ −134586. −5.33644
$$861$$ 0 0
$$862$$ −47540.0 −1.87844
$$863$$ −26698.1 −1.05309 −0.526543 0.850149i $$-0.676512\pi$$
−0.526543 + 0.850149i $$0.676512\pi$$
$$864$$ 0 0
$$865$$ −42937.5 −1.68777
$$866$$ 68827.2 2.70074
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 11999.8 0.468431
$$870$$ 0 0
$$871$$ 11926.0 0.463947
$$872$$ 74303.0 2.88557
$$873$$ 0 0
$$874$$ −74491.6 −2.88297
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −22177.5 −0.853912 −0.426956 0.904272i $$-0.640414\pi$$
−0.426956 + 0.904272i $$0.640414\pi$$
$$878$$ −19007.4 −0.730602
$$879$$ 0 0
$$880$$ −50635.6 −1.93969
$$881$$ −11503.1 −0.439898 −0.219949 0.975511i $$-0.570589\pi$$
−0.219949 + 0.975511i $$0.570589\pi$$
$$882$$ 0 0
$$883$$ 41751.6 1.59123 0.795613 0.605806i $$-0.207149\pi$$
0.795613 + 0.605806i $$0.207149\pi$$
$$884$$ −98156.6 −3.73457
$$885$$ 0 0
$$886$$ −77054.7 −2.92179
$$887$$ −4670.18 −0.176786 −0.0883930 0.996086i $$-0.528173\pi$$
−0.0883930 + 0.996086i $$0.528173\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ −68241.3 −2.57017
$$891$$ 0 0
$$892$$ −80920.8 −3.03748
$$893$$ −30933.0 −1.15917
$$894$$ 0 0
$$895$$ 21753.1 0.812430
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 76340.8 2.83689
$$899$$ −5746.75 −0.213198
$$900$$ 0 0
$$901$$ 5253.22 0.194240
$$902$$ −14325.4 −0.528805
$$903$$ 0 0
$$904$$ −14186.6 −0.521945
$$905$$ 76425.7 2.80715
$$906$$ 0 0
$$907$$ 33783.8 1.23679 0.618397 0.785866i $$-0.287782\pi$$
0.618397 + 0.785866i $$0.287782\pi$$
$$908$$ −98934.5 −3.61592
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −34880.5 −1.26854 −0.634271 0.773111i $$-0.718700\pi$$
−0.634271 + 0.773111i $$0.718700\pi$$
$$912$$ 0 0
$$913$$ 9974.04 0.361547
$$914$$ 26364.2 0.954102
$$915$$ 0 0
$$916$$ 44889.9 1.61922
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −5964.51 −0.214092 −0.107046 0.994254i $$-0.534139\pi$$
−0.107046 + 0.994254i $$0.534139\pi$$
$$920$$ −199631. −7.15397
$$921$$ 0 0
$$922$$ −54689.9 −1.95349
$$923$$ −11744.3 −0.418818
$$924$$ 0 0
$$925$$ −51190.0 −1.81959
$$926$$ −84488.5 −2.99834
$$927$$ 0 0
$$928$$ −121552. −4.29971
$$929$$ 54065.1 1.90939 0.954693 0.297594i $$-0.0961841\pi$$
0.954693 + 0.297594i $$0.0961841\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 31679.3 1.11340
$$933$$ 0 0
$$934$$ −52409.8 −1.83608
$$935$$ 21181.0 0.740847
$$936$$ 0 0
$$937$$ 14678.5 0.511766 0.255883 0.966708i $$-0.417634\pi$$
0.255883 + 0.966708i $$0.417634\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ −131289. −4.55551
$$941$$ 22198.8 0.769032 0.384516 0.923118i $$-0.374368\pi$$
0.384516 + 0.923118i $$0.374368\pi$$
$$942$$ 0 0
$$943$$ −32388.3 −1.11846
$$944$$ 93449.3 3.22194
$$945$$ 0 0
$$946$$ 19591.1 0.673322
$$947$$ 3314.30 0.113728 0.0568639 0.998382i $$-0.481890\pi$$
0.0568639 + 0.998382i $$0.481890\pi$$
$$948$$ 0 0
$$949$$ 53812.1 1.84069
$$950$$ 134326. 4.58749
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −45137.3 −1.53425 −0.767125 0.641498i $$-0.778313\pi$$
−0.767125 + 0.641498i $$0.778313\pi$$
$$954$$ 0 0
$$955$$ 22774.6 0.771694
$$956$$ −48200.2 −1.63066
$$957$$ 0 0
$$958$$ 77173.1 2.60266
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −28773.1 −0.965832
$$962$$ −51856.0 −1.73794
$$963$$ 0 0
$$964$$ −109920. −3.67250
$$965$$ −34277.6 −1.14346
$$966$$ 0 0
$$967$$ 13423.9 0.446416 0.223208 0.974771i $$-0.428347\pi$$
0.223208 + 0.974771i $$0.428347\pi$$
$$968$$ −90024.7 −2.98916
$$969$$ 0 0
$$970$$ −70990.4 −2.34986
$$971$$ 50282.5 1.66183 0.830917 0.556396i $$-0.187816\pi$$
0.830917 + 0.556396i $$0.187816\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 84939.2 2.79428
$$975$$ 0 0
$$976$$ 22727.6 0.745381
$$977$$ 24918.5 0.815980 0.407990 0.912986i $$-0.366230\pi$$
0.407990 + 0.912986i $$0.366230\pi$$
$$978$$ 0 0
$$979$$ 7258.30 0.236952
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −70296.5 −2.28437
$$983$$ −5830.48 −0.189179 −0.0945897 0.995516i $$-0.530154\pi$$
−0.0945897 + 0.995516i $$0.530154\pi$$
$$984$$ 0 0
$$985$$ −15363.8 −0.496986
$$986$$ 95867.7 3.09640
$$987$$ 0 0
$$988$$ 99426.5 3.20160
$$989$$ 44293.6 1.42412
$$990$$ 0 0
$$991$$ 15669.9 0.502293 0.251146 0.967949i $$-0.419192\pi$$
0.251146 + 0.967949i $$0.419192\pi$$
$$992$$ 21530.0 0.689091
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 70837.4 2.25698
$$996$$ 0 0
$$997$$ −11367.5 −0.361095 −0.180548 0.983566i $$-0.557787\pi$$
−0.180548 + 0.983566i $$0.557787\pi$$
$$998$$ 36555.1 1.15945
$$999$$ 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.bg.1.6 yes 6
3.2 odd 2 inner 1323.4.a.bg.1.1 yes 6
7.6 odd 2 1323.4.a.bf.1.6 yes 6
21.20 even 2 1323.4.a.bf.1.1 6

By twisted newform
Twist Min Dim Char Parity Ord Type
1323.4.a.bf.1.1 6 21.20 even 2
1323.4.a.bf.1.6 yes 6 7.6 odd 2
1323.4.a.bg.1.1 yes 6 3.2 odd 2 inner
1323.4.a.bg.1.6 yes 6 1.1 even 1 trivial