# Properties

 Label 336.6.a.l.1.1 Level $336$ Weight $6$ Character 336.1 Self dual yes Analytic conductor $53.889$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 336.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: $$53.8889634572$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 21) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 336.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+9.00000 q^{3} -34.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$$ $$q+9.00000 q^{3} -34.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +340.000 q^{11} +454.000 q^{13} -306.000 q^{15} -798.000 q^{17} -892.000 q^{19} +441.000 q^{21} +3192.00 q^{23} -1969.00 q^{25} +729.000 q^{27} -8242.00 q^{29} +2496.00 q^{31} +3060.00 q^{33} -1666.00 q^{35} +9798.00 q^{37} +4086.00 q^{39} +19834.0 q^{41} +17236.0 q^{43} -2754.00 q^{45} -8928.00 q^{47} +2401.00 q^{49} -7182.00 q^{51} +150.000 q^{53} -11560.0 q^{55} -8028.00 q^{57} +42396.0 q^{59} +14758.0 q^{61} +3969.00 q^{63} -15436.0 q^{65} +1676.00 q^{67} +28728.0 q^{69} -14568.0 q^{71} +78378.0 q^{73} -17721.0 q^{75} +16660.0 q^{77} +2272.00 q^{79} +6561.00 q^{81} +37764.0 q^{83} +27132.0 q^{85} -74178.0 q^{87} -117286. q^{89} +22246.0 q^{91} +22464.0 q^{93} +30328.0 q^{95} +10002.0 q^{97} +27540.0 q^{99} +O(q^{100})$$

## 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
$$3$$ 9.00000 0.577350
$$4$$ 0 0
$$5$$ −34.0000 −0.608210 −0.304105 0.952638i $$-0.598357\pi$$
−0.304105 + 0.952638i $$0.598357\pi$$
$$6$$ 0 0
$$7$$ 49.0000 0.377964
$$8$$ 0 0
$$9$$ 81.0000 0.333333
$$10$$ 0 0
$$11$$ 340.000 0.847222 0.423611 0.905844i $$-0.360762\pi$$
0.423611 + 0.905844i $$0.360762\pi$$
$$12$$ 0 0
$$13$$ 454.000 0.745071 0.372535 0.928018i $$-0.378489\pi$$
0.372535 + 0.928018i $$0.378489\pi$$
$$14$$ 0 0
$$15$$ −306.000 −0.351150
$$16$$ 0 0
$$17$$ −798.000 −0.669700 −0.334850 0.942271i $$-0.608686\pi$$
−0.334850 + 0.942271i $$0.608686\pi$$
$$18$$ 0 0
$$19$$ −892.000 −0.566867 −0.283433 0.958992i $$-0.591473\pi$$
−0.283433 + 0.958992i $$0.591473\pi$$
$$20$$ 0 0
$$21$$ 441.000 0.218218
$$22$$ 0 0
$$23$$ 3192.00 1.25818 0.629091 0.777332i $$-0.283427\pi$$
0.629091 + 0.777332i $$0.283427\pi$$
$$24$$ 0 0
$$25$$ −1969.00 −0.630080
$$26$$ 0 0
$$27$$ 729.000 0.192450
$$28$$ 0 0
$$29$$ −8242.00 −1.81986 −0.909929 0.414764i $$-0.863864\pi$$
−0.909929 + 0.414764i $$0.863864\pi$$
$$30$$ 0 0
$$31$$ 2496.00 0.466488 0.233244 0.972418i $$-0.425066\pi$$
0.233244 + 0.972418i $$0.425066\pi$$
$$32$$ 0 0
$$33$$ 3060.00 0.489144
$$34$$ 0 0
$$35$$ −1666.00 −0.229882
$$36$$ 0 0
$$37$$ 9798.00 1.17661 0.588306 0.808639i $$-0.299795\pi$$
0.588306 + 0.808639i $$0.299795\pi$$
$$38$$ 0 0
$$39$$ 4086.00 0.430167
$$40$$ 0 0
$$41$$ 19834.0 1.84268 0.921342 0.388754i $$-0.127094\pi$$
0.921342 + 0.388754i $$0.127094\pi$$
$$42$$ 0 0
$$43$$ 17236.0 1.42156 0.710780 0.703414i $$-0.248342\pi$$
0.710780 + 0.703414i $$0.248342\pi$$
$$44$$ 0 0
$$45$$ −2754.00 −0.202737
$$46$$ 0 0
$$47$$ −8928.00 −0.589535 −0.294767 0.955569i $$-0.595242\pi$$
−0.294767 + 0.955569i $$0.595242\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ −7182.00 −0.386652
$$52$$ 0 0
$$53$$ 150.000 0.00733502 0.00366751 0.999993i $$-0.498833\pi$$
0.00366751 + 0.999993i $$0.498833\pi$$
$$54$$ 0 0
$$55$$ −11560.0 −0.515289
$$56$$ 0 0
$$57$$ −8028.00 −0.327281
$$58$$ 0 0
$$59$$ 42396.0 1.58560 0.792802 0.609479i $$-0.208621\pi$$
0.792802 + 0.609479i $$0.208621\pi$$
$$60$$ 0 0
$$61$$ 14758.0 0.507812 0.253906 0.967229i $$-0.418285\pi$$
0.253906 + 0.967229i $$0.418285\pi$$
$$62$$ 0 0
$$63$$ 3969.00 0.125988
$$64$$ 0 0
$$65$$ −15436.0 −0.453160
$$66$$ 0 0
$$67$$ 1676.00 0.0456128 0.0228064 0.999740i $$-0.492740\pi$$
0.0228064 + 0.999740i $$0.492740\pi$$
$$68$$ 0 0
$$69$$ 28728.0 0.726411
$$70$$ 0 0
$$71$$ −14568.0 −0.342968 −0.171484 0.985187i $$-0.554856\pi$$
−0.171484 + 0.985187i $$0.554856\pi$$
$$72$$ 0 0
$$73$$ 78378.0 1.72142 0.860710 0.509095i $$-0.170020\pi$$
0.860710 + 0.509095i $$0.170020\pi$$
$$74$$ 0 0
$$75$$ −17721.0 −0.363777
$$76$$ 0 0
$$77$$ 16660.0 0.320220
$$78$$ 0 0
$$79$$ 2272.00 0.0409582 0.0204791 0.999790i $$-0.493481\pi$$
0.0204791 + 0.999790i $$0.493481\pi$$
$$80$$ 0 0
$$81$$ 6561.00 0.111111
$$82$$ 0 0
$$83$$ 37764.0 0.601704 0.300852 0.953671i $$-0.402729\pi$$
0.300852 + 0.953671i $$0.402729\pi$$
$$84$$ 0 0
$$85$$ 27132.0 0.407319
$$86$$ 0 0
$$87$$ −74178.0 −1.05070
$$88$$ 0 0
$$89$$ −117286. −1.56954 −0.784768 0.619790i $$-0.787218\pi$$
−0.784768 + 0.619790i $$0.787218\pi$$
$$90$$ 0 0
$$91$$ 22246.0 0.281610
$$92$$ 0 0
$$93$$ 22464.0 0.269327
$$94$$ 0 0
$$95$$ 30328.0 0.344774
$$96$$ 0 0
$$97$$ 10002.0 0.107934 0.0539669 0.998543i $$-0.482813\pi$$
0.0539669 + 0.998543i $$0.482813\pi$$
$$98$$ 0 0
$$99$$ 27540.0 0.282407
$$100$$ 0 0
$$101$$ −108770. −1.06098 −0.530488 0.847692i $$-0.677991\pi$$
−0.530488 + 0.847692i $$0.677991\pi$$
$$102$$ 0 0
$$103$$ 199192. 1.85003 0.925015 0.379930i $$-0.124052\pi$$
0.925015 + 0.379930i $$0.124052\pi$$
$$104$$ 0 0
$$105$$ −14994.0 −0.132722
$$106$$ 0 0
$$107$$ 79972.0 0.675272 0.337636 0.941277i $$-0.390373\pi$$
0.337636 + 0.941277i $$0.390373\pi$$
$$108$$ 0 0
$$109$$ −46098.0 −0.371634 −0.185817 0.982584i $$-0.559493\pi$$
−0.185817 + 0.982584i $$0.559493\pi$$
$$110$$ 0 0
$$111$$ 88182.0 0.679317
$$112$$ 0 0
$$113$$ 262706. 1.93541 0.967707 0.252078i $$-0.0811138\pi$$
0.967707 + 0.252078i $$0.0811138\pi$$
$$114$$ 0 0
$$115$$ −108528. −0.765239
$$116$$ 0 0
$$117$$ 36774.0 0.248357
$$118$$ 0 0
$$119$$ −39102.0 −0.253123
$$120$$ 0 0
$$121$$ −45451.0 −0.282215
$$122$$ 0 0
$$123$$ 178506. 1.06387
$$124$$ 0 0
$$125$$ 173196. 0.991432
$$126$$ 0 0
$$127$$ −196608. −1.08166 −0.540831 0.841131i $$-0.681890\pi$$
−0.540831 + 0.841131i $$0.681890\pi$$
$$128$$ 0 0
$$129$$ 155124. 0.820738
$$130$$ 0 0
$$131$$ 77140.0 0.392737 0.196368 0.980530i $$-0.437085\pi$$
0.196368 + 0.980530i $$0.437085\pi$$
$$132$$ 0 0
$$133$$ −43708.0 −0.214255
$$134$$ 0 0
$$135$$ −24786.0 −0.117050
$$136$$ 0 0
$$137$$ 208170. 0.947582 0.473791 0.880637i $$-0.342885\pi$$
0.473791 + 0.880637i $$0.342885\pi$$
$$138$$ 0 0
$$139$$ 275580. 1.20979 0.604896 0.796304i $$-0.293215\pi$$
0.604896 + 0.796304i $$0.293215\pi$$
$$140$$ 0 0
$$141$$ −80352.0 −0.340368
$$142$$ 0 0
$$143$$ 154360. 0.631240
$$144$$ 0 0
$$145$$ 280228. 1.10686
$$146$$ 0 0
$$147$$ 21609.0 0.0824786
$$148$$ 0 0
$$149$$ −296106. −1.09265 −0.546326 0.837573i $$-0.683974\pi$$
−0.546326 + 0.837573i $$0.683974\pi$$
$$150$$ 0 0
$$151$$ 426472. 1.52212 0.761059 0.648683i $$-0.224680\pi$$
0.761059 + 0.648683i $$0.224680\pi$$
$$152$$ 0 0
$$153$$ −64638.0 −0.223233
$$154$$ 0 0
$$155$$ −84864.0 −0.283723
$$156$$ 0 0
$$157$$ 178486. 0.577903 0.288952 0.957344i $$-0.406693\pi$$
0.288952 + 0.957344i $$0.406693\pi$$
$$158$$ 0 0
$$159$$ 1350.00 0.00423488
$$160$$ 0 0
$$161$$ 156408. 0.475548
$$162$$ 0 0
$$163$$ −252772. −0.745178 −0.372589 0.927996i $$-0.621530\pi$$
−0.372589 + 0.927996i $$0.621530\pi$$
$$164$$ 0 0
$$165$$ −104040. −0.297502
$$166$$ 0 0
$$167$$ −508088. −1.40977 −0.704884 0.709322i $$-0.749001\pi$$
−0.704884 + 0.709322i $$0.749001\pi$$
$$168$$ 0 0
$$169$$ −165177. −0.444870
$$170$$ 0 0
$$171$$ −72252.0 −0.188956
$$172$$ 0 0
$$173$$ −221834. −0.563525 −0.281762 0.959484i $$-0.590919\pi$$
−0.281762 + 0.959484i $$0.590919\pi$$
$$174$$ 0 0
$$175$$ −96481.0 −0.238148
$$176$$ 0 0
$$177$$ 381564. 0.915449
$$178$$ 0 0
$$179$$ 113564. 0.264916 0.132458 0.991189i $$-0.457713\pi$$
0.132458 + 0.991189i $$0.457713\pi$$
$$180$$ 0 0
$$181$$ 663118. 1.50451 0.752254 0.658873i $$-0.228967\pi$$
0.752254 + 0.658873i $$0.228967\pi$$
$$182$$ 0 0
$$183$$ 132822. 0.293185
$$184$$ 0 0
$$185$$ −333132. −0.715628
$$186$$ 0 0
$$187$$ −271320. −0.567385
$$188$$ 0 0
$$189$$ 35721.0 0.0727393
$$190$$ 0 0
$$191$$ −505664. −1.00295 −0.501474 0.865173i $$-0.667209\pi$$
−0.501474 + 0.865173i $$0.667209\pi$$
$$192$$ 0 0
$$193$$ −432382. −0.835554 −0.417777 0.908550i $$-0.637191\pi$$
−0.417777 + 0.908550i $$0.637191\pi$$
$$194$$ 0 0
$$195$$ −138924. −0.261632
$$196$$ 0 0
$$197$$ −131962. −0.242261 −0.121130 0.992637i $$-0.538652\pi$$
−0.121130 + 0.992637i $$0.538652\pi$$
$$198$$ 0 0
$$199$$ −298536. −0.534397 −0.267199 0.963642i $$-0.586098\pi$$
−0.267199 + 0.963642i $$0.586098\pi$$
$$200$$ 0 0
$$201$$ 15084.0 0.0263346
$$202$$ 0 0
$$203$$ −403858. −0.687842
$$204$$ 0 0
$$205$$ −674356. −1.12074
$$206$$ 0 0
$$207$$ 258552. 0.419394
$$208$$ 0 0
$$209$$ −303280. −0.480262
$$210$$ 0 0
$$211$$ 1.17062e6 1.81013 0.905065 0.425273i $$-0.139822\pi$$
0.905065 + 0.425273i $$0.139822\pi$$
$$212$$ 0 0
$$213$$ −131112. −0.198013
$$214$$ 0 0
$$215$$ −586024. −0.864608
$$216$$ 0 0
$$217$$ 122304. 0.176316
$$218$$ 0 0
$$219$$ 705402. 0.993863
$$220$$ 0 0
$$221$$ −362292. −0.498974
$$222$$ 0 0
$$223$$ −399376. −0.537799 −0.268899 0.963168i $$-0.586660\pi$$
−0.268899 + 0.963168i $$0.586660\pi$$
$$224$$ 0 0
$$225$$ −159489. −0.210027
$$226$$ 0 0
$$227$$ −707916. −0.911837 −0.455918 0.890022i $$-0.650689\pi$$
−0.455918 + 0.890022i $$0.650689\pi$$
$$228$$ 0 0
$$229$$ −735778. −0.927167 −0.463584 0.886053i $$-0.653437\pi$$
−0.463584 + 0.886053i $$0.653437\pi$$
$$230$$ 0 0
$$231$$ 149940. 0.184879
$$232$$ 0 0
$$233$$ −208758. −0.251915 −0.125957 0.992036i $$-0.540200\pi$$
−0.125957 + 0.992036i $$0.540200\pi$$
$$234$$ 0 0
$$235$$ 303552. 0.358561
$$236$$ 0 0
$$237$$ 20448.0 0.0236472
$$238$$ 0 0
$$239$$ −713376. −0.807837 −0.403919 0.914795i $$-0.632352\pi$$
−0.403919 + 0.914795i $$0.632352\pi$$
$$240$$ 0 0
$$241$$ −505246. −0.560351 −0.280176 0.959949i $$-0.590393\pi$$
−0.280176 + 0.959949i $$0.590393\pi$$
$$242$$ 0 0
$$243$$ 59049.0 0.0641500
$$244$$ 0 0
$$245$$ −81634.0 −0.0868872
$$246$$ 0 0
$$247$$ −404968. −0.422356
$$248$$ 0 0
$$249$$ 339876. 0.347394
$$250$$ 0 0
$$251$$ −317108. −0.317704 −0.158852 0.987302i $$-0.550779\pi$$
−0.158852 + 0.987302i $$0.550779\pi$$
$$252$$ 0 0
$$253$$ 1.08528e6 1.06596
$$254$$ 0 0
$$255$$ 244188. 0.235166
$$256$$ 0 0
$$257$$ −1.44285e6 −1.36266 −0.681329 0.731977i $$-0.738598\pi$$
−0.681329 + 0.731977i $$0.738598\pi$$
$$258$$ 0 0
$$259$$ 480102. 0.444717
$$260$$ 0 0
$$261$$ −667602. −0.606619
$$262$$ 0 0
$$263$$ −271496. −0.242033 −0.121016 0.992651i $$-0.538615\pi$$
−0.121016 + 0.992651i $$0.538615\pi$$
$$264$$ 0 0
$$265$$ −5100.00 −0.00446124
$$266$$ 0 0
$$267$$ −1.05557e6 −0.906172
$$268$$ 0 0
$$269$$ 850614. 0.716724 0.358362 0.933583i $$-0.383335\pi$$
0.358362 + 0.933583i $$0.383335\pi$$
$$270$$ 0 0
$$271$$ 540128. 0.446759 0.223380 0.974732i $$-0.428291\pi$$
0.223380 + 0.974732i $$0.428291\pi$$
$$272$$ 0 0
$$273$$ 200214. 0.162588
$$274$$ 0 0
$$275$$ −669460. −0.533818
$$276$$ 0 0
$$277$$ 513574. 0.402164 0.201082 0.979574i $$-0.435554\pi$$
0.201082 + 0.979574i $$0.435554\pi$$
$$278$$ 0 0
$$279$$ 202176. 0.155496
$$280$$ 0 0
$$281$$ −1.35642e6 −1.02478 −0.512388 0.858754i $$-0.671239\pi$$
−0.512388 + 0.858754i $$0.671239\pi$$
$$282$$ 0 0
$$283$$ −286756. −0.212837 −0.106418 0.994321i $$-0.533938\pi$$
−0.106418 + 0.994321i $$0.533938\pi$$
$$284$$ 0 0
$$285$$ 272952. 0.199055
$$286$$ 0 0
$$287$$ 971866. 0.696469
$$288$$ 0 0
$$289$$ −783053. −0.551501
$$290$$ 0 0
$$291$$ 90018.0 0.0623156
$$292$$ 0 0
$$293$$ −1.70727e6 −1.16180 −0.580901 0.813974i $$-0.697300\pi$$
−0.580901 + 0.813974i $$0.697300\pi$$
$$294$$ 0 0
$$295$$ −1.44146e6 −0.964381
$$296$$ 0 0
$$297$$ 247860. 0.163048
$$298$$ 0 0
$$299$$ 1.44917e6 0.937434
$$300$$ 0 0
$$301$$ 844564. 0.537299
$$302$$ 0 0
$$303$$ −978930. −0.612555
$$304$$ 0 0
$$305$$ −501772. −0.308857
$$306$$ 0 0
$$307$$ 546788. 0.331111 0.165555 0.986201i $$-0.447058\pi$$
0.165555 + 0.986201i $$0.447058\pi$$
$$308$$ 0 0
$$309$$ 1.79273e6 1.06812
$$310$$ 0 0
$$311$$ −3.23426e6 −1.89616 −0.948079 0.318035i $$-0.896977\pi$$
−0.948079 + 0.318035i $$0.896977\pi$$
$$312$$ 0 0
$$313$$ 1.81313e6 1.04609 0.523044 0.852306i $$-0.324796\pi$$
0.523044 + 0.852306i $$0.324796\pi$$
$$314$$ 0 0
$$315$$ −134946. −0.0766273
$$316$$ 0 0
$$317$$ −1.27658e6 −0.713509 −0.356754 0.934198i $$-0.616117\pi$$
−0.356754 + 0.934198i $$0.616117\pi$$
$$318$$ 0 0
$$319$$ −2.80228e6 −1.54182
$$320$$ 0 0
$$321$$ 719748. 0.389868
$$322$$ 0 0
$$323$$ 711816. 0.379631
$$324$$ 0 0
$$325$$ −893926. −0.469454
$$326$$ 0 0
$$327$$ −414882. −0.214563
$$328$$ 0 0
$$329$$ −437472. −0.222823
$$330$$ 0 0
$$331$$ 1.73621e6 0.871029 0.435515 0.900182i $$-0.356566\pi$$
0.435515 + 0.900182i $$0.356566\pi$$
$$332$$ 0 0
$$333$$ 793638. 0.392204
$$334$$ 0 0
$$335$$ −56984.0 −0.0277422
$$336$$ 0 0
$$337$$ 2.07215e6 0.993907 0.496953 0.867777i $$-0.334452\pi$$
0.496953 + 0.867777i $$0.334452\pi$$
$$338$$ 0 0
$$339$$ 2.36435e6 1.11741
$$340$$ 0 0
$$341$$ 848640. 0.395219
$$342$$ 0 0
$$343$$ 117649. 0.0539949
$$344$$ 0 0
$$345$$ −976752. −0.441811
$$346$$ 0 0
$$347$$ 1.65146e6 0.736282 0.368141 0.929770i $$-0.379994\pi$$
0.368141 + 0.929770i $$0.379994\pi$$
$$348$$ 0 0
$$349$$ 1.26645e6 0.556578 0.278289 0.960497i $$-0.410233\pi$$
0.278289 + 0.960497i $$0.410233\pi$$
$$350$$ 0 0
$$351$$ 330966. 0.143389
$$352$$ 0 0
$$353$$ 573218. 0.244840 0.122420 0.992478i $$-0.460934\pi$$
0.122420 + 0.992478i $$0.460934\pi$$
$$354$$ 0 0
$$355$$ 495312. 0.208597
$$356$$ 0 0
$$357$$ −351918. −0.146141
$$358$$ 0 0
$$359$$ −4.46322e6 −1.82773 −0.913866 0.406016i $$-0.866918\pi$$
−0.913866 + 0.406016i $$0.866918\pi$$
$$360$$ 0 0
$$361$$ −1.68044e6 −0.678662
$$362$$ 0 0
$$363$$ −409059. −0.162937
$$364$$ 0 0
$$365$$ −2.66485e6 −1.04699
$$366$$ 0 0
$$367$$ 4.50797e6 1.74709 0.873546 0.486742i $$-0.161815\pi$$
0.873546 + 0.486742i $$0.161815\pi$$
$$368$$ 0 0
$$369$$ 1.60655e6 0.614228
$$370$$ 0 0
$$371$$ 7350.00 0.00277238
$$372$$ 0 0
$$373$$ 1.66535e6 0.619774 0.309887 0.950773i $$-0.399709\pi$$
0.309887 + 0.950773i $$0.399709\pi$$
$$374$$ 0 0
$$375$$ 1.55876e6 0.572403
$$376$$ 0 0
$$377$$ −3.74187e6 −1.35592
$$378$$ 0 0
$$379$$ 2.53232e6 0.905568 0.452784 0.891620i $$-0.350431\pi$$
0.452784 + 0.891620i $$0.350431\pi$$
$$380$$ 0 0
$$381$$ −1.76947e6 −0.624498
$$382$$ 0 0
$$383$$ −796368. −0.277407 −0.138703 0.990334i $$-0.544293\pi$$
−0.138703 + 0.990334i $$0.544293\pi$$
$$384$$ 0 0
$$385$$ −566440. −0.194761
$$386$$ 0 0
$$387$$ 1.39612e6 0.473853
$$388$$ 0 0
$$389$$ 1.94799e6 0.652699 0.326349 0.945249i $$-0.394181\pi$$
0.326349 + 0.945249i $$0.394181\pi$$
$$390$$ 0 0
$$391$$ −2.54722e6 −0.842605
$$392$$ 0 0
$$393$$ 694260. 0.226747
$$394$$ 0 0
$$395$$ −77248.0 −0.0249112
$$396$$ 0 0
$$397$$ 1.08116e6 0.344281 0.172140 0.985072i $$-0.444932\pi$$
0.172140 + 0.985072i $$0.444932\pi$$
$$398$$ 0 0
$$399$$ −393372. −0.123700
$$400$$ 0 0
$$401$$ 2.76770e6 0.859524 0.429762 0.902942i $$-0.358598\pi$$
0.429762 + 0.902942i $$0.358598\pi$$
$$402$$ 0 0
$$403$$ 1.13318e6 0.347566
$$404$$ 0 0
$$405$$ −223074. −0.0675789
$$406$$ 0 0
$$407$$ 3.33132e6 0.996851
$$408$$ 0 0
$$409$$ 2.36350e6 0.698630 0.349315 0.937005i $$-0.386414\pi$$
0.349315 + 0.937005i $$0.386414\pi$$
$$410$$ 0 0
$$411$$ 1.87353e6 0.547087
$$412$$ 0 0
$$413$$ 2.07740e6 0.599302
$$414$$ 0 0
$$415$$ −1.28398e6 −0.365963
$$416$$ 0 0
$$417$$ 2.48022e6 0.698474
$$418$$ 0 0
$$419$$ 2.98669e6 0.831104 0.415552 0.909569i $$-0.363588\pi$$
0.415552 + 0.909569i $$0.363588\pi$$
$$420$$ 0 0
$$421$$ −3.46331e6 −0.952326 −0.476163 0.879357i $$-0.657973\pi$$
−0.476163 + 0.879357i $$0.657973\pi$$
$$422$$ 0 0
$$423$$ −723168. −0.196512
$$424$$ 0 0
$$425$$ 1.57126e6 0.421965
$$426$$ 0 0
$$427$$ 723142. 0.191935
$$428$$ 0 0
$$429$$ 1.38924e6 0.364447
$$430$$ 0 0
$$431$$ −2.33693e6 −0.605971 −0.302986 0.952995i $$-0.597983\pi$$
−0.302986 + 0.952995i $$0.597983\pi$$
$$432$$ 0 0
$$433$$ −3.50838e6 −0.899264 −0.449632 0.893214i $$-0.648445\pi$$
−0.449632 + 0.893214i $$0.648445\pi$$
$$434$$ 0 0
$$435$$ 2.52205e6 0.639044
$$436$$ 0 0
$$437$$ −2.84726e6 −0.713221
$$438$$ 0 0
$$439$$ −3.54833e6 −0.878744 −0.439372 0.898305i $$-0.644799\pi$$
−0.439372 + 0.898305i $$0.644799\pi$$
$$440$$ 0 0
$$441$$ 194481. 0.0476190
$$442$$ 0 0
$$443$$ −1.76833e6 −0.428109 −0.214055 0.976822i $$-0.568667\pi$$
−0.214055 + 0.976822i $$0.568667\pi$$
$$444$$ 0 0
$$445$$ 3.98772e6 0.954608
$$446$$ 0 0
$$447$$ −2.66495e6 −0.630842
$$448$$ 0 0
$$449$$ −5.52579e6 −1.29354 −0.646768 0.762687i $$-0.723880\pi$$
−0.646768 + 0.762687i $$0.723880\pi$$
$$450$$ 0 0
$$451$$ 6.74356e6 1.56116
$$452$$ 0 0
$$453$$ 3.83825e6 0.878795
$$454$$ 0 0
$$455$$ −756364. −0.171278
$$456$$ 0 0
$$457$$ −2.96226e6 −0.663488 −0.331744 0.943369i $$-0.607637\pi$$
−0.331744 + 0.943369i $$0.607637\pi$$
$$458$$ 0 0
$$459$$ −581742. −0.128884
$$460$$ 0 0
$$461$$ 2.11884e6 0.464350 0.232175 0.972674i $$-0.425416\pi$$
0.232175 + 0.972674i $$0.425416\pi$$
$$462$$ 0 0
$$463$$ −3.19226e6 −0.692062 −0.346031 0.938223i $$-0.612471\pi$$
−0.346031 + 0.938223i $$0.612471\pi$$
$$464$$ 0 0
$$465$$ −763776. −0.163807
$$466$$ 0 0
$$467$$ 7.42621e6 1.57571 0.787853 0.615863i $$-0.211193\pi$$
0.787853 + 0.615863i $$0.211193\pi$$
$$468$$ 0 0
$$469$$ 82124.0 0.0172400
$$470$$ 0 0
$$471$$ 1.60637e6 0.333653
$$472$$ 0 0
$$473$$ 5.86024e6 1.20438
$$474$$ 0 0
$$475$$ 1.75635e6 0.357171
$$476$$ 0 0
$$477$$ 12150.0 0.00244501
$$478$$ 0 0
$$479$$ 3.39685e6 0.676453 0.338226 0.941065i $$-0.390173\pi$$
0.338226 + 0.941065i $$0.390173\pi$$
$$480$$ 0 0
$$481$$ 4.44829e6 0.876659
$$482$$ 0 0
$$483$$ 1.40767e6 0.274558
$$484$$ 0 0
$$485$$ −340068. −0.0656465
$$486$$ 0 0
$$487$$ 3.71382e6 0.709574 0.354787 0.934947i $$-0.384553\pi$$
0.354787 + 0.934947i $$0.384553\pi$$
$$488$$ 0 0
$$489$$ −2.27495e6 −0.430229
$$490$$ 0 0
$$491$$ −5.57494e6 −1.04361 −0.521803 0.853066i $$-0.674740\pi$$
−0.521803 + 0.853066i $$0.674740\pi$$
$$492$$ 0 0
$$493$$ 6.57712e6 1.21876
$$494$$ 0 0
$$495$$ −936360. −0.171763
$$496$$ 0 0
$$497$$ −713832. −0.129630
$$498$$ 0 0
$$499$$ −3.92698e6 −0.706004 −0.353002 0.935623i $$-0.614839\pi$$
−0.353002 + 0.935623i $$0.614839\pi$$
$$500$$ 0 0
$$501$$ −4.57279e6 −0.813930
$$502$$ 0 0
$$503$$ −6.42079e6 −1.13154 −0.565768 0.824564i $$-0.691420\pi$$
−0.565768 + 0.824564i $$0.691420\pi$$
$$504$$ 0 0
$$505$$ 3.69818e6 0.645297
$$506$$ 0 0
$$507$$ −1.48659e6 −0.256846
$$508$$ 0 0
$$509$$ 146278. 0.0250256 0.0125128 0.999922i $$-0.496017\pi$$
0.0125128 + 0.999922i $$0.496017\pi$$
$$510$$ 0 0
$$511$$ 3.84052e6 0.650636
$$512$$ 0 0
$$513$$ −650268. −0.109094
$$514$$ 0 0
$$515$$ −6.77253e6 −1.12521
$$516$$ 0 0
$$517$$ −3.03552e6 −0.499467
$$518$$ 0 0
$$519$$ −1.99651e6 −0.325351
$$520$$ 0 0
$$521$$ 7.70937e6 1.24430 0.622149 0.782899i $$-0.286260\pi$$
0.622149 + 0.782899i $$0.286260\pi$$
$$522$$ 0 0
$$523$$ 569420. 0.0910287 0.0455144 0.998964i $$-0.485507\pi$$
0.0455144 + 0.998964i $$0.485507\pi$$
$$524$$ 0 0
$$525$$ −868329. −0.137495
$$526$$ 0 0
$$527$$ −1.99181e6 −0.312407
$$528$$ 0 0
$$529$$ 3.75252e6 0.583021
$$530$$ 0 0
$$531$$ 3.43408e6 0.528535
$$532$$ 0 0
$$533$$ 9.00464e6 1.37293
$$534$$ 0 0
$$535$$ −2.71905e6 −0.410707
$$536$$ 0 0
$$537$$ 1.02208e6 0.152949
$$538$$ 0 0
$$539$$ 816340. 0.121032
$$540$$ 0 0
$$541$$ −9.44802e6 −1.38787 −0.693933 0.720040i $$-0.744124\pi$$
−0.693933 + 0.720040i $$0.744124\pi$$
$$542$$ 0 0
$$543$$ 5.96806e6 0.868628
$$544$$ 0 0
$$545$$ 1.56733e6 0.226032
$$546$$ 0 0
$$547$$ 1.35321e6 0.193374 0.0966869 0.995315i $$-0.469175\pi$$
0.0966869 + 0.995315i $$0.469175\pi$$
$$548$$ 0 0
$$549$$ 1.19540e6 0.169271
$$550$$ 0 0
$$551$$ 7.35186e6 1.03162
$$552$$ 0 0
$$553$$ 111328. 0.0154807
$$554$$ 0 0
$$555$$ −2.99819e6 −0.413168
$$556$$ 0 0
$$557$$ 8.19390e6 1.11906 0.559529 0.828811i $$-0.310982\pi$$
0.559529 + 0.828811i $$0.310982\pi$$
$$558$$ 0 0
$$559$$ 7.82514e6 1.05916
$$560$$ 0 0
$$561$$ −2.44188e6 −0.327580
$$562$$ 0 0
$$563$$ 1.05796e7 1.40669 0.703347 0.710847i $$-0.251688\pi$$
0.703347 + 0.710847i $$0.251688\pi$$
$$564$$ 0 0
$$565$$ −8.93200e6 −1.17714
$$566$$ 0 0
$$567$$ 321489. 0.0419961
$$568$$ 0 0
$$569$$ −1.20205e7 −1.55648 −0.778238 0.627969i $$-0.783886\pi$$
−0.778238 + 0.627969i $$0.783886\pi$$
$$570$$ 0 0
$$571$$ 2.48948e6 0.319534 0.159767 0.987155i $$-0.448926\pi$$
0.159767 + 0.987155i $$0.448926\pi$$
$$572$$ 0 0
$$573$$ −4.55098e6 −0.579053
$$574$$ 0 0
$$575$$ −6.28505e6 −0.792755
$$576$$ 0 0
$$577$$ 8.21322e6 1.02701 0.513504 0.858087i $$-0.328347\pi$$
0.513504 + 0.858087i $$0.328347\pi$$
$$578$$ 0 0
$$579$$ −3.89144e6 −0.482407
$$580$$ 0 0
$$581$$ 1.85044e6 0.227423
$$582$$ 0 0
$$583$$ 51000.0 0.00621439
$$584$$ 0 0
$$585$$ −1.25032e6 −0.151053
$$586$$ 0 0
$$587$$ 1.21827e6 0.145931 0.0729655 0.997334i $$-0.476754\pi$$
0.0729655 + 0.997334i $$0.476754\pi$$
$$588$$ 0 0
$$589$$ −2.22643e6 −0.264436
$$590$$ 0 0
$$591$$ −1.18766e6 −0.139869
$$592$$ 0 0
$$593$$ −8.42379e6 −0.983718 −0.491859 0.870675i $$-0.663683\pi$$
−0.491859 + 0.870675i $$0.663683\pi$$
$$594$$ 0 0
$$595$$ 1.32947e6 0.153952
$$596$$ 0 0
$$597$$ −2.68682e6 −0.308534
$$598$$ 0 0
$$599$$ −8.21254e6 −0.935212 −0.467606 0.883937i $$-0.654883\pi$$
−0.467606 + 0.883937i $$0.654883\pi$$
$$600$$ 0 0
$$601$$ 3.25478e6 0.367566 0.183783 0.982967i $$-0.441166\pi$$
0.183783 + 0.982967i $$0.441166\pi$$
$$602$$ 0 0
$$603$$ 135756. 0.0152043
$$604$$ 0 0
$$605$$ 1.54533e6 0.171646
$$606$$ 0 0
$$607$$ −7.82101e6 −0.861571 −0.430785 0.902454i $$-0.641763\pi$$
−0.430785 + 0.902454i $$0.641763\pi$$
$$608$$ 0 0
$$609$$ −3.63472e6 −0.397126
$$610$$ 0 0
$$611$$ −4.05331e6 −0.439245
$$612$$ 0 0
$$613$$ −9.51670e6 −1.02290 −0.511452 0.859312i $$-0.670892\pi$$
−0.511452 + 0.859312i $$0.670892\pi$$
$$614$$ 0 0
$$615$$ −6.06920e6 −0.647059
$$616$$ 0 0
$$617$$ −7.04895e6 −0.745438 −0.372719 0.927944i $$-0.621574\pi$$
−0.372719 + 0.927944i $$0.621574\pi$$
$$618$$ 0 0
$$619$$ 6.32174e6 0.663147 0.331574 0.943429i $$-0.392420\pi$$
0.331574 + 0.943429i $$0.392420\pi$$
$$620$$ 0 0
$$621$$ 2.32697e6 0.242137
$$622$$ 0 0
$$623$$ −5.74701e6 −0.593229
$$624$$ 0 0
$$625$$ 264461. 0.0270808
$$626$$ 0 0
$$627$$ −2.72952e6 −0.277279
$$628$$ 0 0
$$629$$ −7.81880e6 −0.787977
$$630$$ 0 0
$$631$$ −8.61236e6 −0.861090 −0.430545 0.902569i $$-0.641679\pi$$
−0.430545 + 0.902569i $$0.641679\pi$$
$$632$$ 0 0
$$633$$ 1.05356e7 1.04508
$$634$$ 0 0
$$635$$ 6.68467e6 0.657879
$$636$$ 0 0
$$637$$ 1.09005e6 0.106439
$$638$$ 0 0
$$639$$ −1.18001e6 −0.114323
$$640$$ 0 0
$$641$$ −5.22829e6 −0.502590 −0.251295 0.967910i $$-0.580857\pi$$
−0.251295 + 0.967910i $$0.580857\pi$$
$$642$$ 0 0
$$643$$ −1.61373e7 −1.53923 −0.769615 0.638508i $$-0.779552\pi$$
−0.769615 + 0.638508i $$0.779552\pi$$
$$644$$ 0 0
$$645$$ −5.27422e6 −0.499182
$$646$$ 0 0
$$647$$ 1.58749e7 1.49090 0.745451 0.666560i $$-0.232234\pi$$
0.745451 + 0.666560i $$0.232234\pi$$
$$648$$ 0 0
$$649$$ 1.44146e7 1.34336
$$650$$ 0 0
$$651$$ 1.10074e6 0.101796
$$652$$ 0 0
$$653$$ −5.94112e6 −0.545237 −0.272619 0.962122i $$-0.587890\pi$$
−0.272619 + 0.962122i $$0.587890\pi$$
$$654$$ 0 0
$$655$$ −2.62276e6 −0.238867
$$656$$ 0 0
$$657$$ 6.34862e6 0.573807
$$658$$ 0 0
$$659$$ 7.64430e6 0.685684 0.342842 0.939393i $$-0.388610\pi$$
0.342842 + 0.939393i $$0.388610\pi$$
$$660$$ 0 0
$$661$$ −7.58688e6 −0.675398 −0.337699 0.941254i $$-0.609649\pi$$
−0.337699 + 0.941254i $$0.609649\pi$$
$$662$$ 0 0
$$663$$ −3.26063e6 −0.288083
$$664$$ 0 0
$$665$$ 1.48607e6 0.130312
$$666$$ 0 0
$$667$$ −2.63085e7 −2.28971
$$668$$ 0 0
$$669$$ −3.59438e6 −0.310498
$$670$$ 0 0
$$671$$ 5.01772e6 0.430229
$$672$$ 0 0
$$673$$ −2.06681e7 −1.75899 −0.879494 0.475910i $$-0.842119\pi$$
−0.879494 + 0.475910i $$0.842119\pi$$
$$674$$ 0 0
$$675$$ −1.43540e6 −0.121259
$$676$$ 0 0
$$677$$ 7.89541e6 0.662068 0.331034 0.943619i $$-0.392602\pi$$
0.331034 + 0.943619i $$0.392602\pi$$
$$678$$ 0 0
$$679$$ 490098. 0.0407951
$$680$$ 0 0
$$681$$ −6.37124e6 −0.526449
$$682$$ 0 0
$$683$$ 1.96015e7 1.60782 0.803911 0.594750i $$-0.202749\pi$$
0.803911 + 0.594750i $$0.202749\pi$$
$$684$$ 0 0
$$685$$ −7.07778e6 −0.576329
$$686$$ 0 0
$$687$$ −6.62200e6 −0.535300
$$688$$ 0 0
$$689$$ 68100.0 0.00546511
$$690$$ 0 0
$$691$$ 1.72710e7 1.37601 0.688005 0.725706i $$-0.258487\pi$$
0.688005 + 0.725706i $$0.258487\pi$$
$$692$$ 0 0
$$693$$ 1.34946e6 0.106740
$$694$$ 0 0
$$695$$ −9.36972e6 −0.735808
$$696$$ 0 0
$$697$$ −1.58275e7 −1.23405
$$698$$ 0 0
$$699$$ −1.87882e6 −0.145443
$$700$$ 0 0
$$701$$ −5.36344e6 −0.412238 −0.206119 0.978527i $$-0.566083\pi$$
−0.206119 + 0.978527i $$0.566083\pi$$
$$702$$ 0 0
$$703$$ −8.73982e6 −0.666982
$$704$$ 0 0
$$705$$ 2.73197e6 0.207015
$$706$$ 0 0
$$707$$ −5.32973e6 −0.401011
$$708$$ 0 0
$$709$$ −1.73733e7 −1.29798 −0.648988 0.760798i $$-0.724808\pi$$
−0.648988 + 0.760798i $$0.724808\pi$$
$$710$$ 0 0
$$711$$ 184032. 0.0136527
$$712$$ 0 0
$$713$$ 7.96723e6 0.586926
$$714$$ 0 0
$$715$$ −5.24824e6 −0.383927
$$716$$ 0 0
$$717$$ −6.42038e6 −0.466405
$$718$$ 0 0
$$719$$ −424608. −0.0306313 −0.0153157 0.999883i $$-0.504875\pi$$
−0.0153157 + 0.999883i $$0.504875\pi$$
$$720$$ 0 0
$$721$$ 9.76041e6 0.699246
$$722$$ 0 0
$$723$$ −4.54721e6 −0.323519
$$724$$ 0 0
$$725$$ 1.62285e7 1.14666
$$726$$ 0 0
$$727$$ −2.18290e7 −1.53179 −0.765893 0.642968i $$-0.777703\pi$$
−0.765893 + 0.642968i $$0.777703\pi$$
$$728$$ 0 0
$$729$$ 531441. 0.0370370
$$730$$ 0 0
$$731$$ −1.37543e7 −0.952020
$$732$$ 0 0
$$733$$ 2.17675e7 1.49640 0.748202 0.663470i $$-0.230917\pi$$
0.748202 + 0.663470i $$0.230917\pi$$
$$734$$ 0 0
$$735$$ −734706. −0.0501644
$$736$$ 0 0
$$737$$ 569840. 0.0386442
$$738$$ 0 0
$$739$$ −6.21786e6 −0.418822 −0.209411 0.977828i $$-0.567155\pi$$
−0.209411 + 0.977828i $$0.567155\pi$$
$$740$$ 0 0
$$741$$ −3.64471e6 −0.243847
$$742$$ 0 0
$$743$$ −3.77647e6 −0.250966 −0.125483 0.992096i $$-0.540048\pi$$
−0.125483 + 0.992096i $$0.540048\pi$$
$$744$$ 0 0
$$745$$ 1.00676e7 0.664562
$$746$$ 0 0
$$747$$ 3.05888e6 0.200568
$$748$$ 0 0
$$749$$ 3.91863e6 0.255229
$$750$$ 0 0
$$751$$ 2.88795e6 0.186849 0.0934244 0.995626i $$-0.470219\pi$$
0.0934244 + 0.995626i $$0.470219\pi$$
$$752$$ 0 0
$$753$$ −2.85397e6 −0.183427
$$754$$ 0 0
$$755$$ −1.45000e7 −0.925768
$$756$$ 0 0
$$757$$ 1.25519e6 0.0796104 0.0398052 0.999207i $$-0.487326\pi$$
0.0398052 + 0.999207i $$0.487326\pi$$
$$758$$ 0 0
$$759$$ 9.76752e6 0.615432
$$760$$ 0 0
$$761$$ −1.42623e7 −0.892746 −0.446373 0.894847i $$-0.647284\pi$$
−0.446373 + 0.894847i $$0.647284\pi$$
$$762$$ 0 0
$$763$$ −2.25880e6 −0.140465
$$764$$ 0 0
$$765$$ 2.19769e6 0.135773
$$766$$ 0 0
$$767$$ 1.92478e7 1.18139
$$768$$ 0 0
$$769$$ −2.02261e7 −1.23338 −0.616689 0.787207i $$-0.711526\pi$$
−0.616689 + 0.787207i $$0.711526\pi$$
$$770$$ 0 0
$$771$$ −1.29856e7 −0.786732
$$772$$ 0 0
$$773$$ 2.62288e7 1.57881 0.789406 0.613872i $$-0.210389\pi$$
0.789406 + 0.613872i $$0.210389\pi$$
$$774$$ 0 0
$$775$$ −4.91462e6 −0.293925
$$776$$ 0 0
$$777$$ 4.32092e6 0.256758
$$778$$ 0 0
$$779$$ −1.76919e7 −1.04456
$$780$$ 0 0
$$781$$ −4.95312e6 −0.290570
$$782$$ 0 0
$$783$$ −6.00842e6 −0.350232
$$784$$ 0 0
$$785$$ −6.06852e6 −0.351487
$$786$$ 0 0
$$787$$ 9.92829e6 0.571397 0.285698 0.958320i $$-0.407774\pi$$
0.285698 + 0.958320i $$0.407774\pi$$
$$788$$ 0 0
$$789$$ −2.44346e6 −0.139738
$$790$$ 0 0
$$791$$ 1.28726e7 0.731518
$$792$$ 0 0
$$793$$ 6.70013e6 0.378356
$$794$$ 0 0
$$795$$ −45900.0 −0.00257570
$$796$$ 0 0
$$797$$ 1.09033e7 0.608014 0.304007 0.952670i $$-0.401675\pi$$
0.304007 + 0.952670i $$0.401675\pi$$
$$798$$ 0 0
$$799$$ 7.12454e6 0.394812
$$800$$ 0 0
$$801$$ −9.50017e6 −0.523179
$$802$$ 0 0
$$803$$ 2.66485e7 1.45843
$$804$$ 0 0
$$805$$ −5.31787e6 −0.289233
$$806$$ 0 0
$$807$$ 7.65553e6 0.413801
$$808$$ 0 0
$$809$$ 6.06398e6 0.325751 0.162876 0.986647i $$-0.447923\pi$$
0.162876 + 0.986647i $$0.447923\pi$$
$$810$$ 0 0
$$811$$ 8.59438e6 0.458841 0.229421 0.973327i $$-0.426317\pi$$
0.229421 + 0.973327i $$0.426317\pi$$
$$812$$ 0 0
$$813$$ 4.86115e6 0.257937
$$814$$ 0 0
$$815$$ 8.59425e6 0.453225
$$816$$ 0 0
$$817$$ −1.53745e7 −0.805835
$$818$$ 0 0
$$819$$ 1.80193e6 0.0938701
$$820$$ 0 0
$$821$$ −2.01396e6 −0.104278 −0.0521391 0.998640i $$-0.516604\pi$$
−0.0521391 + 0.998640i $$0.516604\pi$$
$$822$$ 0 0
$$823$$ 2.64679e7 1.36213 0.681067 0.732221i $$-0.261516\pi$$
0.681067 + 0.732221i $$0.261516\pi$$
$$824$$ 0 0
$$825$$ −6.02514e6 −0.308200
$$826$$ 0 0
$$827$$ 3.90229e6 0.198407 0.0992033 0.995067i $$-0.468371\pi$$
0.0992033 + 0.995067i $$0.468371\pi$$
$$828$$ 0 0
$$829$$ −1.95595e7 −0.988487 −0.494244 0.869323i $$-0.664555\pi$$
−0.494244 + 0.869323i $$0.664555\pi$$
$$830$$ 0 0
$$831$$ 4.62217e6 0.232190
$$832$$ 0 0
$$833$$ −1.91600e6 −0.0956715
$$834$$ 0 0
$$835$$ 1.72750e7 0.857436
$$836$$ 0 0
$$837$$ 1.81958e6 0.0897756
$$838$$ 0 0
$$839$$ 2.45448e7 1.20380 0.601901 0.798570i $$-0.294410\pi$$
0.601901 + 0.798570i $$0.294410\pi$$
$$840$$ 0 0
$$841$$ 4.74194e7 2.31188
$$842$$ 0 0
$$843$$ −1.22078e7 −0.591655
$$844$$ 0 0
$$845$$ 5.61602e6 0.270574
$$846$$ 0 0
$$847$$ −2.22710e6 −0.106667
$$848$$ 0 0
$$849$$ −2.58080e6 −0.122881
$$850$$ 0 0
$$851$$ 3.12752e7 1.48039
$$852$$ 0 0
$$853$$ 3.38305e7 1.59197 0.795987 0.605314i $$-0.206952\pi$$
0.795987 + 0.605314i $$0.206952\pi$$
$$854$$ 0 0
$$855$$ 2.45657e6 0.114925
$$856$$ 0 0
$$857$$ 3.18009e7 1.47907 0.739534 0.673120i $$-0.235046\pi$$
0.739534 + 0.673120i $$0.235046\pi$$
$$858$$ 0 0
$$859$$ −638420. −0.0295205 −0.0147602 0.999891i $$-0.504699\pi$$
−0.0147602 + 0.999891i $$0.504699\pi$$
$$860$$ 0 0
$$861$$ 8.74679e6 0.402106
$$862$$ 0 0
$$863$$ 4.22256e6 0.192996 0.0964981 0.995333i $$-0.469236\pi$$
0.0964981 + 0.995333i $$0.469236\pi$$
$$864$$ 0 0
$$865$$ 7.54236e6 0.342742
$$866$$ 0 0
$$867$$ −7.04748e6 −0.318409
$$868$$ 0 0
$$869$$ 772480. 0.0347007
$$870$$ 0 0
$$871$$ 760904. 0.0339848
$$872$$ 0 0
$$873$$ 810162. 0.0359779
$$874$$ 0 0
$$875$$ 8.48660e6 0.374726
$$876$$ 0 0
$$877$$ −2.45043e7 −1.07583 −0.537915 0.842999i $$-0.680788\pi$$
−0.537915 + 0.842999i $$0.680788\pi$$
$$878$$ 0 0
$$879$$ −1.53654e7 −0.670767
$$880$$ 0 0
$$881$$ −2.77630e7 −1.20511 −0.602555 0.798078i $$-0.705850\pi$$
−0.602555 + 0.798078i $$0.705850\pi$$
$$882$$ 0 0
$$883$$ −3.30170e7 −1.42507 −0.712534 0.701638i $$-0.752452\pi$$
−0.712534 + 0.701638i $$0.752452\pi$$
$$884$$ 0 0
$$885$$ −1.29732e7 −0.556786
$$886$$ 0 0
$$887$$ −4.34462e6 −0.185414 −0.0927070 0.995693i $$-0.529552\pi$$
−0.0927070 + 0.995693i $$0.529552\pi$$
$$888$$ 0 0
$$889$$ −9.63379e6 −0.408830
$$890$$ 0 0
$$891$$ 2.23074e6 0.0941358
$$892$$ 0 0
$$893$$ 7.96378e6 0.334188
$$894$$ 0 0
$$895$$ −3.86118e6 −0.161125
$$896$$ 0 0
$$897$$ 1.30425e7 0.541228
$$898$$ 0 0
$$899$$ −2.05720e7 −0.848942
$$900$$ 0 0
$$901$$ −119700. −0.00491227
$$902$$ 0 0
$$903$$ 7.60108e6 0.310210
$$904$$ 0 0
$$905$$ −2.25460e7 −0.915057
$$906$$ 0 0
$$907$$ −1.96499e7 −0.793128 −0.396564 0.918007i $$-0.629797\pi$$
−0.396564 + 0.918007i $$0.629797\pi$$
$$908$$ 0 0
$$909$$ −8.81037e6 −0.353659
$$910$$ 0 0
$$911$$ 7.26518e6 0.290035 0.145018 0.989429i $$-0.453676\pi$$
0.145018 + 0.989429i $$0.453676\pi$$
$$912$$ 0 0
$$913$$ 1.28398e7 0.509777
$$914$$ 0 0
$$915$$ −4.51595e6 −0.178318
$$916$$ 0 0
$$917$$ 3.77986e6 0.148440
$$918$$ 0 0
$$919$$ −9.82532e6 −0.383758 −0.191879 0.981419i $$-0.561458\pi$$
−0.191879 + 0.981419i $$0.561458\pi$$
$$920$$ 0 0
$$921$$ 4.92109e6 0.191167
$$922$$ 0 0
$$923$$ −6.61387e6 −0.255536
$$924$$ 0 0
$$925$$ −1.92923e7 −0.741359
$$926$$ 0 0
$$927$$ 1.61346e7 0.616677
$$928$$ 0 0
$$929$$ 2.71152e7 1.03080 0.515399 0.856951i $$-0.327644\pi$$
0.515399 + 0.856951i $$0.327644\pi$$
$$930$$ 0 0
$$931$$ −2.14169e6 −0.0809809
$$932$$ 0 0
$$933$$ −2.91084e7 −1.09475
$$934$$ 0 0
$$935$$ 9.22488e6 0.345089
$$936$$ 0 0
$$937$$ −4.53522e7 −1.68752 −0.843761 0.536720i $$-0.819663\pi$$
−0.843761 + 0.536720i $$0.819663\pi$$
$$938$$ 0 0
$$939$$ 1.63182e7 0.603959
$$940$$ 0 0
$$941$$ 4.65780e7 1.71477 0.857387 0.514672i $$-0.172086\pi$$
0.857387 + 0.514672i $$0.172086\pi$$
$$942$$ 0 0
$$943$$ 6.33101e7 2.31843
$$944$$ 0 0
$$945$$ −1.21451e6 −0.0442408
$$946$$ 0 0
$$947$$ −2.53799e7 −0.919632 −0.459816 0.888014i $$-0.652085\pi$$
−0.459816 + 0.888014i $$0.652085\pi$$
$$948$$ 0 0
$$949$$ 3.55836e7 1.28258
$$950$$ 0 0
$$951$$ −1.14892e7 −0.411944
$$952$$ 0 0
$$953$$ 1.52948e7 0.545520 0.272760 0.962082i $$-0.412063\pi$$
0.272760 + 0.962082i $$0.412063\pi$$
$$954$$ 0 0
$$955$$ 1.71926e7 0.610004
$$956$$ 0 0
$$957$$ −2.52205e7 −0.890173
$$958$$ 0 0
$$959$$ 1.02003e7 0.358152
$$960$$ 0 0
$$961$$ −2.23991e7 −0.782389
$$962$$ 0 0
$$963$$ 6.47773e6 0.225091
$$964$$ 0 0
$$965$$ 1.47010e7 0.508192
$$966$$ 0 0
$$967$$ 5.71465e6 0.196527 0.0982637 0.995160i $$-0.468671\pi$$
0.0982637 + 0.995160i $$0.468671\pi$$
$$968$$ 0 0
$$969$$ 6.40634e6 0.219180
$$970$$ 0 0
$$971$$ −1.30250e7 −0.443332 −0.221666 0.975123i $$-0.571149\pi$$
−0.221666 + 0.975123i $$0.571149\pi$$
$$972$$ 0 0
$$973$$ 1.35034e7 0.457258
$$974$$ 0 0
$$975$$ −8.04533e6 −0.271039
$$976$$ 0 0
$$977$$ −1.70360e7 −0.570992 −0.285496 0.958380i $$-0.592158\pi$$
−0.285496 + 0.958380i $$0.592158\pi$$
$$978$$ 0 0
$$979$$ −3.98772e7 −1.32974
$$980$$ 0 0
$$981$$ −3.73394e6 −0.123878
$$982$$ 0 0
$$983$$ 1.36985e7 0.452156 0.226078 0.974109i $$-0.427410\pi$$
0.226078 + 0.974109i $$0.427410\pi$$
$$984$$ 0 0
$$985$$ 4.48671e6 0.147346
$$986$$ 0 0
$$987$$ −3.93725e6 −0.128647
$$988$$ 0 0
$$989$$ 5.50173e7 1.78858
$$990$$ 0 0
$$991$$ 3.49088e7 1.12915 0.564574 0.825383i $$-0.309041\pi$$
0.564574 + 0.825383i $$0.309041\pi$$
$$992$$ 0 0
$$993$$ 1.56259e7 0.502889
$$994$$ 0 0
$$995$$ 1.01502e7 0.325026
$$996$$ 0 0
$$997$$ 875662. 0.0278996 0.0139498 0.999903i $$-0.495559\pi$$
0.0139498 + 0.999903i $$0.495559\pi$$
$$998$$ 0 0
$$999$$ 7.14274e6 0.226439
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 336.6.a.l.1.1 1
3.2 odd 2 1008.6.a.t.1.1 1
4.3 odd 2 21.6.a.b.1.1 1
12.11 even 2 63.6.a.c.1.1 1
20.3 even 4 525.6.d.d.274.1 2
20.7 even 4 525.6.d.d.274.2 2
20.19 odd 2 525.6.a.c.1.1 1
28.3 even 6 147.6.e.e.79.1 2
28.11 odd 6 147.6.e.f.79.1 2
28.19 even 6 147.6.e.e.67.1 2
28.23 odd 6 147.6.e.f.67.1 2
28.27 even 2 147.6.a.e.1.1 1
84.83 odd 2 441.6.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.b.1.1 1 4.3 odd 2
63.6.a.c.1.1 1 12.11 even 2
147.6.a.e.1.1 1 28.27 even 2
147.6.e.e.67.1 2 28.19 even 6
147.6.e.e.79.1 2 28.3 even 6
147.6.e.f.67.1 2 28.23 odd 6
147.6.e.f.79.1 2 28.11 odd 6
336.6.a.l.1.1 1 1.1 even 1 trivial
441.6.a.d.1.1 1 84.83 odd 2
525.6.a.c.1.1 1 20.19 odd 2
525.6.d.d.274.1 2 20.3 even 4
525.6.d.d.274.2 2 20.7 even 4
1008.6.a.t.1.1 1 3.2 odd 2