# Properties

 Label 112.6.a.b.1.1 Level $112$ Weight $6$ Character 112.1 Self dual yes Analytic conductor $17.963$ Analytic rank $1$ 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] = [112,6,Mod(1,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("112.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 112.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: $$17.9629878191$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 112.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-26.0000 q^{3} +16.0000 q^{5} +49.0000 q^{7} +433.000 q^{9} +O(q^{10})$$ $$q-26.0000 q^{3} +16.0000 q^{5} +49.0000 q^{7} +433.000 q^{9} -8.00000 q^{11} +684.000 q^{13} -416.000 q^{15} -2218.00 q^{17} +2698.00 q^{19} -1274.00 q^{21} -3344.00 q^{23} -2869.00 q^{25} -4940.00 q^{27} -3254.00 q^{29} -4788.00 q^{31} +208.000 q^{33} +784.000 q^{35} -11470.0 q^{37} -17784.0 q^{39} +13350.0 q^{41} +928.000 q^{43} +6928.00 q^{45} -1212.00 q^{47} +2401.00 q^{49} +57668.0 q^{51} +13110.0 q^{53} -128.000 q^{55} -70148.0 q^{57} -34702.0 q^{59} -1032.00 q^{61} +21217.0 q^{63} +10944.0 q^{65} -10108.0 q^{67} +86944.0 q^{69} -62720.0 q^{71} -18926.0 q^{73} +74594.0 q^{75} -392.000 q^{77} -11400.0 q^{79} +23221.0 q^{81} -88958.0 q^{83} -35488.0 q^{85} +84604.0 q^{87} +19722.0 q^{89} +33516.0 q^{91} +124488. q^{93} +43168.0 q^{95} +17062.0 q^{97} -3464.00 q^{99} +O(q^{100})$$

## 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
$$3$$ −26.0000 −1.66790 −0.833950 0.551839i $$-0.813926\pi$$
−0.833950 + 0.551839i $$0.813926\pi$$
$$4$$ 0 0
$$5$$ 16.0000 0.286217 0.143108 0.989707i $$-0.454290\pi$$
0.143108 + 0.989707i $$0.454290\pi$$
$$6$$ 0 0
$$7$$ 49.0000 0.377964
$$8$$ 0 0
$$9$$ 433.000 1.78189
$$10$$ 0 0
$$11$$ −8.00000 −0.0199346 −0.00996732 0.999950i $$-0.503173\pi$$
−0.00996732 + 0.999950i $$0.503173\pi$$
$$12$$ 0 0
$$13$$ 684.000 1.12253 0.561265 0.827636i $$-0.310315\pi$$
0.561265 + 0.827636i $$0.310315\pi$$
$$14$$ 0 0
$$15$$ −416.000 −0.477381
$$16$$ 0 0
$$17$$ −2218.00 −1.86140 −0.930699 0.365786i $$-0.880800\pi$$
−0.930699 + 0.365786i $$0.880800\pi$$
$$18$$ 0 0
$$19$$ 2698.00 1.71458 0.857290 0.514833i $$-0.172146\pi$$
0.857290 + 0.514833i $$0.172146\pi$$
$$20$$ 0 0
$$21$$ −1274.00 −0.630407
$$22$$ 0 0
$$23$$ −3344.00 −1.31809 −0.659047 0.752101i $$-0.729040\pi$$
−0.659047 + 0.752101i $$0.729040\pi$$
$$24$$ 0 0
$$25$$ −2869.00 −0.918080
$$26$$ 0 0
$$27$$ −4940.00 −1.30412
$$28$$ 0 0
$$29$$ −3254.00 −0.718493 −0.359247 0.933243i $$-0.616966\pi$$
−0.359247 + 0.933243i $$0.616966\pi$$
$$30$$ 0 0
$$31$$ −4788.00 −0.894849 −0.447425 0.894322i $$-0.647659\pi$$
−0.447425 + 0.894322i $$0.647659\pi$$
$$32$$ 0 0
$$33$$ 208.000 0.0332490
$$34$$ 0 0
$$35$$ 784.000 0.108180
$$36$$ 0 0
$$37$$ −11470.0 −1.37740 −0.688698 0.725048i $$-0.741818\pi$$
−0.688698 + 0.725048i $$0.741818\pi$$
$$38$$ 0 0
$$39$$ −17784.0 −1.87227
$$40$$ 0 0
$$41$$ 13350.0 1.24029 0.620143 0.784489i $$-0.287075\pi$$
0.620143 + 0.784489i $$0.287075\pi$$
$$42$$ 0 0
$$43$$ 928.000 0.0765380 0.0382690 0.999267i $$-0.487816\pi$$
0.0382690 + 0.999267i $$0.487816\pi$$
$$44$$ 0 0
$$45$$ 6928.00 0.510008
$$46$$ 0 0
$$47$$ −1212.00 −0.0800310 −0.0400155 0.999199i $$-0.512741\pi$$
−0.0400155 + 0.999199i $$0.512741\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ 57668.0 3.10463
$$52$$ 0 0
$$53$$ 13110.0 0.641081 0.320541 0.947235i $$-0.396135\pi$$
0.320541 + 0.947235i $$0.396135\pi$$
$$54$$ 0 0
$$55$$ −128.000 −0.00570563
$$56$$ 0 0
$$57$$ −70148.0 −2.85975
$$58$$ 0 0
$$59$$ −34702.0 −1.29785 −0.648925 0.760852i $$-0.724781\pi$$
−0.648925 + 0.760852i $$0.724781\pi$$
$$60$$ 0 0
$$61$$ −1032.00 −0.0355104 −0.0177552 0.999842i $$-0.505652\pi$$
−0.0177552 + 0.999842i $$0.505652\pi$$
$$62$$ 0 0
$$63$$ 21217.0 0.673492
$$64$$ 0 0
$$65$$ 10944.0 0.321287
$$66$$ 0 0
$$67$$ −10108.0 −0.275092 −0.137546 0.990495i $$-0.543922\pi$$
−0.137546 + 0.990495i $$0.543922\pi$$
$$68$$ 0 0
$$69$$ 86944.0 2.19845
$$70$$ 0 0
$$71$$ −62720.0 −1.47659 −0.738295 0.674477i $$-0.764369\pi$$
−0.738295 + 0.674477i $$0.764369\pi$$
$$72$$ 0 0
$$73$$ −18926.0 −0.415673 −0.207836 0.978164i $$-0.566642\pi$$
−0.207836 + 0.978164i $$0.566642\pi$$
$$74$$ 0 0
$$75$$ 74594.0 1.53127
$$76$$ 0 0
$$77$$ −392.000 −0.00753458
$$78$$ 0 0
$$79$$ −11400.0 −0.205512 −0.102756 0.994707i $$-0.532766\pi$$
−0.102756 + 0.994707i $$0.532766\pi$$
$$80$$ 0 0
$$81$$ 23221.0 0.393250
$$82$$ 0 0
$$83$$ −88958.0 −1.41739 −0.708696 0.705514i $$-0.750716\pi$$
−0.708696 + 0.705514i $$0.750716\pi$$
$$84$$ 0 0
$$85$$ −35488.0 −0.532763
$$86$$ 0 0
$$87$$ 84604.0 1.19838
$$88$$ 0 0
$$89$$ 19722.0 0.263922 0.131961 0.991255i $$-0.457873\pi$$
0.131961 + 0.991255i $$0.457873\pi$$
$$90$$ 0 0
$$91$$ 33516.0 0.424276
$$92$$ 0 0
$$93$$ 124488. 1.49252
$$94$$ 0 0
$$95$$ 43168.0 0.490742
$$96$$ 0 0
$$97$$ 17062.0 0.184120 0.0920599 0.995753i $$-0.470655\pi$$
0.0920599 + 0.995753i $$0.470655\pi$$
$$98$$ 0 0
$$99$$ −3464.00 −0.0355214
$$100$$ 0 0
$$101$$ 45904.0 0.447762 0.223881 0.974617i $$-0.428127\pi$$
0.223881 + 0.974617i $$0.428127\pi$$
$$102$$ 0 0
$$103$$ 136012. 1.26324 0.631618 0.775280i $$-0.282391\pi$$
0.631618 + 0.775280i $$0.282391\pi$$
$$104$$ 0 0
$$105$$ −20384.0 −0.180433
$$106$$ 0 0
$$107$$ 69156.0 0.583943 0.291971 0.956427i $$-0.405689\pi$$
0.291971 + 0.956427i $$0.405689\pi$$
$$108$$ 0 0
$$109$$ −146414. −1.18037 −0.590183 0.807270i $$-0.700944\pi$$
−0.590183 + 0.807270i $$0.700944\pi$$
$$110$$ 0 0
$$111$$ 298220. 2.29736
$$112$$ 0 0
$$113$$ −80186.0 −0.590748 −0.295374 0.955382i $$-0.595444\pi$$
−0.295374 + 0.955382i $$0.595444\pi$$
$$114$$ 0 0
$$115$$ −53504.0 −0.377261
$$116$$ 0 0
$$117$$ 296172. 2.00023
$$118$$ 0 0
$$119$$ −108682. −0.703542
$$120$$ 0 0
$$121$$ −160987. −0.999603
$$122$$ 0 0
$$123$$ −347100. −2.06867
$$124$$ 0 0
$$125$$ −95904.0 −0.548987
$$126$$ 0 0
$$127$$ −274800. −1.51185 −0.755923 0.654661i $$-0.772811\pi$$
−0.755923 + 0.654661i $$0.772811\pi$$
$$128$$ 0 0
$$129$$ −24128.0 −0.127658
$$130$$ 0 0
$$131$$ −180742. −0.920197 −0.460099 0.887868i $$-0.652186\pi$$
−0.460099 + 0.887868i $$0.652186\pi$$
$$132$$ 0 0
$$133$$ 132202. 0.648051
$$134$$ 0 0
$$135$$ −79040.0 −0.373261
$$136$$ 0 0
$$137$$ −209678. −0.954446 −0.477223 0.878782i $$-0.658357\pi$$
−0.477223 + 0.878782i $$0.658357\pi$$
$$138$$ 0 0
$$139$$ −17242.0 −0.0756921 −0.0378461 0.999284i $$-0.512050\pi$$
−0.0378461 + 0.999284i $$0.512050\pi$$
$$140$$ 0 0
$$141$$ 31512.0 0.133484
$$142$$ 0 0
$$143$$ −5472.00 −0.0223772
$$144$$ 0 0
$$145$$ −52064.0 −0.205645
$$146$$ 0 0
$$147$$ −62426.0 −0.238272
$$148$$ 0 0
$$149$$ 59358.0 0.219035 0.109518 0.993985i $$-0.465069\pi$$
0.109518 + 0.993985i $$0.465069\pi$$
$$150$$ 0 0
$$151$$ 336344. 1.20044 0.600221 0.799834i $$-0.295079\pi$$
0.600221 + 0.799834i $$0.295079\pi$$
$$152$$ 0 0
$$153$$ −960394. −3.31681
$$154$$ 0 0
$$155$$ −76608.0 −0.256121
$$156$$ 0 0
$$157$$ 464588. 1.50425 0.752123 0.659023i $$-0.229030\pi$$
0.752123 + 0.659023i $$0.229030\pi$$
$$158$$ 0 0
$$159$$ −340860. −1.06926
$$160$$ 0 0
$$161$$ −163856. −0.498193
$$162$$ 0 0
$$163$$ −314792. −0.928014 −0.464007 0.885831i $$-0.653589\pi$$
−0.464007 + 0.885831i $$0.653589\pi$$
$$164$$ 0 0
$$165$$ 3328.00 0.00951642
$$166$$ 0 0
$$167$$ −285724. −0.792785 −0.396393 0.918081i $$-0.629738\pi$$
−0.396393 + 0.918081i $$0.629738\pi$$
$$168$$ 0 0
$$169$$ 96563.0 0.260072
$$170$$ 0 0
$$171$$ 1.16823e6 3.05520
$$172$$ 0 0
$$173$$ −709148. −1.80145 −0.900724 0.434392i $$-0.856963\pi$$
−0.900724 + 0.434392i $$0.856963\pi$$
$$174$$ 0 0
$$175$$ −140581. −0.347002
$$176$$ 0 0
$$177$$ 902252. 2.16468
$$178$$ 0 0
$$179$$ 617148. 1.43965 0.719825 0.694156i $$-0.244222\pi$$
0.719825 + 0.694156i $$0.244222\pi$$
$$180$$ 0 0
$$181$$ 237828. 0.539593 0.269797 0.962917i $$-0.413044\pi$$
0.269797 + 0.962917i $$0.413044\pi$$
$$182$$ 0 0
$$183$$ 26832.0 0.0592278
$$184$$ 0 0
$$185$$ −183520. −0.394234
$$186$$ 0 0
$$187$$ 17744.0 0.0371063
$$188$$ 0 0
$$189$$ −242060. −0.492911
$$190$$ 0 0
$$191$$ 133512. 0.264812 0.132406 0.991196i $$-0.457730\pi$$
0.132406 + 0.991196i $$0.457730\pi$$
$$192$$ 0 0
$$193$$ 270446. 0.522622 0.261311 0.965255i $$-0.415845\pi$$
0.261311 + 0.965255i $$0.415845\pi$$
$$194$$ 0 0
$$195$$ −284544. −0.535874
$$196$$ 0 0
$$197$$ 875102. 1.60655 0.803273 0.595611i $$-0.203090\pi$$
0.803273 + 0.595611i $$0.203090\pi$$
$$198$$ 0 0
$$199$$ 347620. 0.622260 0.311130 0.950367i $$-0.399292\pi$$
0.311130 + 0.950367i $$0.399292\pi$$
$$200$$ 0 0
$$201$$ 262808. 0.458826
$$202$$ 0 0
$$203$$ −159446. −0.271565
$$204$$ 0 0
$$205$$ 213600. 0.354990
$$206$$ 0 0
$$207$$ −1.44795e6 −2.34870
$$208$$ 0 0
$$209$$ −21584.0 −0.0341795
$$210$$ 0 0
$$211$$ 425380. 0.657765 0.328883 0.944371i $$-0.393328\pi$$
0.328883 + 0.944371i $$0.393328\pi$$
$$212$$ 0 0
$$213$$ 1.63072e6 2.46281
$$214$$ 0 0
$$215$$ 14848.0 0.0219064
$$216$$ 0 0
$$217$$ −234612. −0.338221
$$218$$ 0 0
$$219$$ 492076. 0.693301
$$220$$ 0 0
$$221$$ −1.51711e6 −2.08947
$$222$$ 0 0
$$223$$ −481592. −0.648511 −0.324255 0.945970i $$-0.605114\pi$$
−0.324255 + 0.945970i $$0.605114\pi$$
$$224$$ 0 0
$$225$$ −1.24228e6 −1.63592
$$226$$ 0 0
$$227$$ 6042.00 0.00778245 0.00389122 0.999992i $$-0.498761\pi$$
0.00389122 + 0.999992i $$0.498761\pi$$
$$228$$ 0 0
$$229$$ 1804.00 0.00227325 0.00113663 0.999999i $$-0.499638\pi$$
0.00113663 + 0.999999i $$0.499638\pi$$
$$230$$ 0 0
$$231$$ 10192.0 0.0125669
$$232$$ 0 0
$$233$$ −1.61153e6 −1.94468 −0.972339 0.233576i $$-0.924957\pi$$
−0.972339 + 0.233576i $$0.924957\pi$$
$$234$$ 0 0
$$235$$ −19392.0 −0.0229062
$$236$$ 0 0
$$237$$ 296400. 0.342774
$$238$$ 0 0
$$239$$ 987096. 1.11780 0.558901 0.829235i $$-0.311223\pi$$
0.558901 + 0.829235i $$0.311223\pi$$
$$240$$ 0 0
$$241$$ 893510. 0.990962 0.495481 0.868619i $$-0.334992\pi$$
0.495481 + 0.868619i $$0.334992\pi$$
$$242$$ 0 0
$$243$$ 596674. 0.648219
$$244$$ 0 0
$$245$$ 38416.0 0.0408881
$$246$$ 0 0
$$247$$ 1.84543e6 1.92467
$$248$$ 0 0
$$249$$ 2.31291e6 2.36407
$$250$$ 0 0
$$251$$ −365946. −0.366634 −0.183317 0.983054i $$-0.558683\pi$$
−0.183317 + 0.983054i $$0.558683\pi$$
$$252$$ 0 0
$$253$$ 26752.0 0.0262757
$$254$$ 0 0
$$255$$ 922688. 0.888596
$$256$$ 0 0
$$257$$ 1.40459e6 1.32653 0.663266 0.748383i $$-0.269170\pi$$
0.663266 + 0.748383i $$0.269170\pi$$
$$258$$ 0 0
$$259$$ −562030. −0.520607
$$260$$ 0 0
$$261$$ −1.40898e6 −1.28028
$$262$$ 0 0
$$263$$ −1.09968e6 −0.980341 −0.490170 0.871627i $$-0.663065\pi$$
−0.490170 + 0.871627i $$0.663065\pi$$
$$264$$ 0 0
$$265$$ 209760. 0.183488
$$266$$ 0 0
$$267$$ −512772. −0.440196
$$268$$ 0 0
$$269$$ 814948. 0.686672 0.343336 0.939213i $$-0.388443\pi$$
0.343336 + 0.939213i $$0.388443\pi$$
$$270$$ 0 0
$$271$$ 1.69906e6 1.40535 0.702675 0.711511i $$-0.251989\pi$$
0.702675 + 0.711511i $$0.251989\pi$$
$$272$$ 0 0
$$273$$ −871416. −0.707651
$$274$$ 0 0
$$275$$ 22952.0 0.0183016
$$276$$ 0 0
$$277$$ −1.36508e6 −1.06895 −0.534477 0.845183i $$-0.679492\pi$$
−0.534477 + 0.845183i $$0.679492\pi$$
$$278$$ 0 0
$$279$$ −2.07320e6 −1.59453
$$280$$ 0 0
$$281$$ −715846. −0.540821 −0.270411 0.962745i $$-0.587159\pi$$
−0.270411 + 0.962745i $$0.587159\pi$$
$$282$$ 0 0
$$283$$ −217726. −0.161601 −0.0808005 0.996730i $$-0.525748\pi$$
−0.0808005 + 0.996730i $$0.525748\pi$$
$$284$$ 0 0
$$285$$ −1.12237e6 −0.818508
$$286$$ 0 0
$$287$$ 654150. 0.468784
$$288$$ 0 0
$$289$$ 3.49967e6 2.46480
$$290$$ 0 0
$$291$$ −443612. −0.307094
$$292$$ 0 0
$$293$$ 1.50708e6 1.02557 0.512787 0.858516i $$-0.328613\pi$$
0.512787 + 0.858516i $$0.328613\pi$$
$$294$$ 0 0
$$295$$ −555232. −0.371466
$$296$$ 0 0
$$297$$ 39520.0 0.0259972
$$298$$ 0 0
$$299$$ −2.28730e6 −1.47960
$$300$$ 0 0
$$301$$ 45472.0 0.0289286
$$302$$ 0 0
$$303$$ −1.19350e6 −0.746822
$$304$$ 0 0
$$305$$ −16512.0 −0.0101637
$$306$$ 0 0
$$307$$ −12502.0 −0.00757066 −0.00378533 0.999993i $$-0.501205\pi$$
−0.00378533 + 0.999993i $$0.501205\pi$$
$$308$$ 0 0
$$309$$ −3.53631e6 −2.10695
$$310$$ 0 0
$$311$$ 647432. 0.379571 0.189786 0.981826i $$-0.439221\pi$$
0.189786 + 0.981826i $$0.439221\pi$$
$$312$$ 0 0
$$313$$ −935978. −0.540014 −0.270007 0.962858i $$-0.587026\pi$$
−0.270007 + 0.962858i $$0.587026\pi$$
$$314$$ 0 0
$$315$$ 339472. 0.192765
$$316$$ 0 0
$$317$$ 705942. 0.394567 0.197284 0.980346i $$-0.436788\pi$$
0.197284 + 0.980346i $$0.436788\pi$$
$$318$$ 0 0
$$319$$ 26032.0 0.0143229
$$320$$ 0 0
$$321$$ −1.79806e6 −0.973959
$$322$$ 0 0
$$323$$ −5.98416e6 −3.19152
$$324$$ 0 0
$$325$$ −1.96240e6 −1.03057
$$326$$ 0 0
$$327$$ 3.80676e6 1.96873
$$328$$ 0 0
$$329$$ −59388.0 −0.0302489
$$330$$ 0 0
$$331$$ 1.14304e6 0.573445 0.286722 0.958014i $$-0.407434\pi$$
0.286722 + 0.958014i $$0.407434\pi$$
$$332$$ 0 0
$$333$$ −4.96651e6 −2.45437
$$334$$ 0 0
$$335$$ −161728. −0.0787360
$$336$$ 0 0
$$337$$ −2.36402e6 −1.13390 −0.566952 0.823751i $$-0.691877\pi$$
−0.566952 + 0.823751i $$0.691877\pi$$
$$338$$ 0 0
$$339$$ 2.08484e6 0.985309
$$340$$ 0 0
$$341$$ 38304.0 0.0178385
$$342$$ 0 0
$$343$$ 117649. 0.0539949
$$344$$ 0 0
$$345$$ 1.39110e6 0.629234
$$346$$ 0 0
$$347$$ −726240. −0.323785 −0.161892 0.986808i $$-0.551760\pi$$
−0.161892 + 0.986808i $$0.551760\pi$$
$$348$$ 0 0
$$349$$ 136180. 0.0598480 0.0299240 0.999552i $$-0.490473\pi$$
0.0299240 + 0.999552i $$0.490473\pi$$
$$350$$ 0 0
$$351$$ −3.37896e6 −1.46391
$$352$$ 0 0
$$353$$ −1.16907e6 −0.499349 −0.249674 0.968330i $$-0.580324\pi$$
−0.249674 + 0.968330i $$0.580324\pi$$
$$354$$ 0 0
$$355$$ −1.00352e6 −0.422625
$$356$$ 0 0
$$357$$ 2.82573e6 1.17344
$$358$$ 0 0
$$359$$ 4280.00 0.00175270 0.000876350 1.00000i $$-0.499721\pi$$
0.000876350 1.00000i $$0.499721\pi$$
$$360$$ 0 0
$$361$$ 4.80310e6 1.93979
$$362$$ 0 0
$$363$$ 4.18566e6 1.66724
$$364$$ 0 0
$$365$$ −302816. −0.118973
$$366$$ 0 0
$$367$$ −2.44796e6 −0.948722 −0.474361 0.880330i $$-0.657321\pi$$
−0.474361 + 0.880330i $$0.657321\pi$$
$$368$$ 0 0
$$369$$ 5.78055e6 2.21006
$$370$$ 0 0
$$371$$ 642390. 0.242306
$$372$$ 0 0
$$373$$ −904514. −0.336623 −0.168311 0.985734i $$-0.553831\pi$$
−0.168311 + 0.985734i $$0.553831\pi$$
$$374$$ 0 0
$$375$$ 2.49350e6 0.915655
$$376$$ 0 0
$$377$$ −2.22574e6 −0.806530
$$378$$ 0 0
$$379$$ 4.23034e6 1.51279 0.756393 0.654117i $$-0.226960\pi$$
0.756393 + 0.654117i $$0.226960\pi$$
$$380$$ 0 0
$$381$$ 7.14480e6 2.52161
$$382$$ 0 0
$$383$$ −4.55400e6 −1.58634 −0.793169 0.609002i $$-0.791570\pi$$
−0.793169 + 0.609002i $$0.791570\pi$$
$$384$$ 0 0
$$385$$ −6272.00 −0.00215652
$$386$$ 0 0
$$387$$ 401824. 0.136382
$$388$$ 0 0
$$389$$ −3.98541e6 −1.33536 −0.667680 0.744448i $$-0.732713\pi$$
−0.667680 + 0.744448i $$0.732713\pi$$
$$390$$ 0 0
$$391$$ 7.41699e6 2.45350
$$392$$ 0 0
$$393$$ 4.69929e6 1.53480
$$394$$ 0 0
$$395$$ −182400. −0.0588210
$$396$$ 0 0
$$397$$ 552420. 0.175911 0.0879555 0.996124i $$-0.471967\pi$$
0.0879555 + 0.996124i $$0.471967\pi$$
$$398$$ 0 0
$$399$$ −3.43725e6 −1.08088
$$400$$ 0 0
$$401$$ 38190.0 0.0118601 0.00593006 0.999982i $$-0.498112\pi$$
0.00593006 + 0.999982i $$0.498112\pi$$
$$402$$ 0 0
$$403$$ −3.27499e6 −1.00449
$$404$$ 0 0
$$405$$ 371536. 0.112555
$$406$$ 0 0
$$407$$ 91760.0 0.0274579
$$408$$ 0 0
$$409$$ −3.92475e6 −1.16012 −0.580062 0.814573i $$-0.696972\pi$$
−0.580062 + 0.814573i $$0.696972\pi$$
$$410$$ 0 0
$$411$$ 5.45163e6 1.59192
$$412$$ 0 0
$$413$$ −1.70040e6 −0.490541
$$414$$ 0 0
$$415$$ −1.42333e6 −0.405681
$$416$$ 0 0
$$417$$ 448292. 0.126247
$$418$$ 0 0
$$419$$ −598386. −0.166512 −0.0832562 0.996528i $$-0.526532\pi$$
−0.0832562 + 0.996528i $$0.526532\pi$$
$$420$$ 0 0
$$421$$ 4.61597e6 1.26928 0.634641 0.772807i $$-0.281148\pi$$
0.634641 + 0.772807i $$0.281148\pi$$
$$422$$ 0 0
$$423$$ −524796. −0.142607
$$424$$ 0 0
$$425$$ 6.36344e6 1.70891
$$426$$ 0 0
$$427$$ −50568.0 −0.0134217
$$428$$ 0 0
$$429$$ 142272. 0.0373230
$$430$$ 0 0
$$431$$ 61560.0 0.0159627 0.00798133 0.999968i $$-0.497459\pi$$
0.00798133 + 0.999968i $$0.497459\pi$$
$$432$$ 0 0
$$433$$ −3.79727e6 −0.973310 −0.486655 0.873594i $$-0.661783\pi$$
−0.486655 + 0.873594i $$0.661783\pi$$
$$434$$ 0 0
$$435$$ 1.35366e6 0.342995
$$436$$ 0 0
$$437$$ −9.02211e6 −2.25998
$$438$$ 0 0
$$439$$ 2.28852e6 0.566752 0.283376 0.959009i $$-0.408545\pi$$
0.283376 + 0.959009i $$0.408545\pi$$
$$440$$ 0 0
$$441$$ 1.03963e6 0.254556
$$442$$ 0 0
$$443$$ 4.75976e6 1.15233 0.576163 0.817335i $$-0.304549\pi$$
0.576163 + 0.817335i $$0.304549\pi$$
$$444$$ 0 0
$$445$$ 315552. 0.0755389
$$446$$ 0 0
$$447$$ −1.54331e6 −0.365329
$$448$$ 0 0
$$449$$ −4.36715e6 −1.02231 −0.511155 0.859489i $$-0.670782\pi$$
−0.511155 + 0.859489i $$0.670782\pi$$
$$450$$ 0 0
$$451$$ −106800. −0.0247246
$$452$$ 0 0
$$453$$ −8.74494e6 −2.00222
$$454$$ 0 0
$$455$$ 536256. 0.121435
$$456$$ 0 0
$$457$$ 5.44994e6 1.22068 0.610339 0.792140i $$-0.291033\pi$$
0.610339 + 0.792140i $$0.291033\pi$$
$$458$$ 0 0
$$459$$ 1.09569e7 2.42749
$$460$$ 0 0
$$461$$ 1.66966e6 0.365911 0.182956 0.983121i $$-0.441434\pi$$
0.182956 + 0.983121i $$0.441434\pi$$
$$462$$ 0 0
$$463$$ −70768.0 −0.0153421 −0.00767104 0.999971i $$-0.502442\pi$$
−0.00767104 + 0.999971i $$0.502442\pi$$
$$464$$ 0 0
$$465$$ 1.99181e6 0.427184
$$466$$ 0 0
$$467$$ −5.66083e6 −1.20112 −0.600562 0.799578i $$-0.705056\pi$$
−0.600562 + 0.799578i $$0.705056\pi$$
$$468$$ 0 0
$$469$$ −495292. −0.103975
$$470$$ 0 0
$$471$$ −1.20793e7 −2.50893
$$472$$ 0 0
$$473$$ −7424.00 −0.00152576
$$474$$ 0 0
$$475$$ −7.74056e6 −1.57412
$$476$$ 0 0
$$477$$ 5.67663e6 1.14234
$$478$$ 0 0
$$479$$ 1.44948e6 0.288652 0.144326 0.989530i $$-0.453899\pi$$
0.144326 + 0.989530i $$0.453899\pi$$
$$480$$ 0 0
$$481$$ −7.84548e6 −1.54617
$$482$$ 0 0
$$483$$ 4.26026e6 0.830937
$$484$$ 0 0
$$485$$ 272992. 0.0526982
$$486$$ 0 0
$$487$$ −4.07504e6 −0.778591 −0.389296 0.921113i $$-0.627282\pi$$
−0.389296 + 0.921113i $$0.627282\pi$$
$$488$$ 0 0
$$489$$ 8.18459e6 1.54784
$$490$$ 0 0
$$491$$ −986100. −0.184594 −0.0922969 0.995732i $$-0.529421\pi$$
−0.0922969 + 0.995732i $$0.529421\pi$$
$$492$$ 0 0
$$493$$ 7.21737e6 1.33740
$$494$$ 0 0
$$495$$ −55424.0 −0.0101668
$$496$$ 0 0
$$497$$ −3.07328e6 −0.558099
$$498$$ 0 0
$$499$$ −5.98342e6 −1.07572 −0.537859 0.843035i $$-0.680767\pi$$
−0.537859 + 0.843035i $$0.680767\pi$$
$$500$$ 0 0
$$501$$ 7.42882e6 1.32229
$$502$$ 0 0
$$503$$ 3.49373e6 0.615700 0.307850 0.951435i $$-0.400391\pi$$
0.307850 + 0.951435i $$0.400391\pi$$
$$504$$ 0 0
$$505$$ 734464. 0.128157
$$506$$ 0 0
$$507$$ −2.51064e6 −0.433775
$$508$$ 0 0
$$509$$ 2.15711e6 0.369043 0.184522 0.982828i $$-0.440926\pi$$
0.184522 + 0.982828i $$0.440926\pi$$
$$510$$ 0 0
$$511$$ −927374. −0.157110
$$512$$ 0 0
$$513$$ −1.33281e7 −2.23602
$$514$$ 0 0
$$515$$ 2.17619e6 0.361559
$$516$$ 0 0
$$517$$ 9696.00 0.00159539
$$518$$ 0 0
$$519$$ 1.84378e7 3.00464
$$520$$ 0 0
$$521$$ −6.65817e6 −1.07463 −0.537317 0.843380i $$-0.680562\pi$$
−0.537317 + 0.843380i $$0.680562\pi$$
$$522$$ 0 0
$$523$$ −5.95223e6 −0.951537 −0.475768 0.879571i $$-0.657830\pi$$
−0.475768 + 0.879571i $$0.657830\pi$$
$$524$$ 0 0
$$525$$ 3.65511e6 0.578764
$$526$$ 0 0
$$527$$ 1.06198e7 1.66567
$$528$$ 0 0
$$529$$ 4.74599e6 0.737374
$$530$$ 0 0
$$531$$ −1.50260e7 −2.31263
$$532$$ 0 0
$$533$$ 9.13140e6 1.39226
$$534$$ 0 0
$$535$$ 1.10650e6 0.167134
$$536$$ 0 0
$$537$$ −1.60458e7 −2.40119
$$538$$ 0 0
$$539$$ −19208.0 −0.00284780
$$540$$ 0 0
$$541$$ −6.39681e6 −0.939659 −0.469830 0.882757i $$-0.655685\pi$$
−0.469830 + 0.882757i $$0.655685\pi$$
$$542$$ 0 0
$$543$$ −6.18353e6 −0.899988
$$544$$ 0 0
$$545$$ −2.34262e6 −0.337840
$$546$$ 0 0
$$547$$ 5.51851e6 0.788594 0.394297 0.918983i $$-0.370988\pi$$
0.394297 + 0.918983i $$0.370988\pi$$
$$548$$ 0 0
$$549$$ −446856. −0.0632757
$$550$$ 0 0
$$551$$ −8.77929e6 −1.23191
$$552$$ 0 0
$$553$$ −558600. −0.0776762
$$554$$ 0 0
$$555$$ 4.77152e6 0.657543
$$556$$ 0 0
$$557$$ 2.02159e6 0.276093 0.138046 0.990426i $$-0.455918\pi$$
0.138046 + 0.990426i $$0.455918\pi$$
$$558$$ 0 0
$$559$$ 634752. 0.0859161
$$560$$ 0 0
$$561$$ −461344. −0.0618896
$$562$$ 0 0
$$563$$ −8.14678e6 −1.08322 −0.541608 0.840631i $$-0.682184\pi$$
−0.541608 + 0.840631i $$0.682184\pi$$
$$564$$ 0 0
$$565$$ −1.28298e6 −0.169082
$$566$$ 0 0
$$567$$ 1.13783e6 0.148634
$$568$$ 0 0
$$569$$ −1.19824e7 −1.55154 −0.775772 0.631013i $$-0.782639\pi$$
−0.775772 + 0.631013i $$0.782639\pi$$
$$570$$ 0 0
$$571$$ −1.39582e6 −0.179159 −0.0895793 0.995980i $$-0.528552\pi$$
−0.0895793 + 0.995980i $$0.528552\pi$$
$$572$$ 0 0
$$573$$ −3.47131e6 −0.441679
$$574$$ 0 0
$$575$$ 9.59394e6 1.21012
$$576$$ 0 0
$$577$$ 1.96784e6 0.246065 0.123033 0.992403i $$-0.460738\pi$$
0.123033 + 0.992403i $$0.460738\pi$$
$$578$$ 0 0
$$579$$ −7.03160e6 −0.871681
$$580$$ 0 0
$$581$$ −4.35894e6 −0.535724
$$582$$ 0 0
$$583$$ −104880. −0.0127797
$$584$$ 0 0
$$585$$ 4.73875e6 0.572498
$$586$$ 0 0
$$587$$ 3.18897e6 0.381993 0.190997 0.981591i $$-0.438828\pi$$
0.190997 + 0.981591i $$0.438828\pi$$
$$588$$ 0 0
$$589$$ −1.29180e7 −1.53429
$$590$$ 0 0
$$591$$ −2.27527e7 −2.67956
$$592$$ 0 0
$$593$$ 1.67500e6 0.195604 0.0978022 0.995206i $$-0.468819\pi$$
0.0978022 + 0.995206i $$0.468819\pi$$
$$594$$ 0 0
$$595$$ −1.73891e6 −0.201366
$$596$$ 0 0
$$597$$ −9.03812e6 −1.03787
$$598$$ 0 0
$$599$$ 1.00635e7 1.14599 0.572994 0.819559i $$-0.305782\pi$$
0.572994 + 0.819559i $$0.305782\pi$$
$$600$$ 0 0
$$601$$ 1.72798e6 0.195143 0.0975713 0.995229i $$-0.468893\pi$$
0.0975713 + 0.995229i $$0.468893\pi$$
$$602$$ 0 0
$$603$$ −4.37676e6 −0.490185
$$604$$ 0 0
$$605$$ −2.57579e6 −0.286103
$$606$$ 0 0
$$607$$ 1.69523e7 1.86748 0.933740 0.357953i $$-0.116525\pi$$
0.933740 + 0.357953i $$0.116525\pi$$
$$608$$ 0 0
$$609$$ 4.14560e6 0.452943
$$610$$ 0 0
$$611$$ −829008. −0.0898371
$$612$$ 0 0
$$613$$ 1.01942e7 1.09572 0.547861 0.836569i $$-0.315442\pi$$
0.547861 + 0.836569i $$0.315442\pi$$
$$614$$ 0 0
$$615$$ −5.55360e6 −0.592089
$$616$$ 0 0
$$617$$ 1.57452e7 1.66508 0.832540 0.553965i $$-0.186886\pi$$
0.832540 + 0.553965i $$0.186886\pi$$
$$618$$ 0 0
$$619$$ 332690. 0.0348990 0.0174495 0.999848i $$-0.494445\pi$$
0.0174495 + 0.999848i $$0.494445\pi$$
$$620$$ 0 0
$$621$$ 1.65194e7 1.71895
$$622$$ 0 0
$$623$$ 966378. 0.0997532
$$624$$ 0 0
$$625$$ 7.43116e6 0.760951
$$626$$ 0 0
$$627$$ 561184. 0.0570081
$$628$$ 0 0
$$629$$ 2.54405e7 2.56388
$$630$$ 0 0
$$631$$ −3.59720e6 −0.359659 −0.179830 0.983698i $$-0.557555\pi$$
−0.179830 + 0.983698i $$0.557555\pi$$
$$632$$ 0 0
$$633$$ −1.10599e7 −1.09709
$$634$$ 0 0
$$635$$ −4.39680e6 −0.432715
$$636$$ 0 0
$$637$$ 1.64228e6 0.160361
$$638$$ 0 0
$$639$$ −2.71578e7 −2.63113
$$640$$ 0 0
$$641$$ −1.46389e7 −1.40723 −0.703614 0.710583i $$-0.748431\pi$$
−0.703614 + 0.710583i $$0.748431\pi$$
$$642$$ 0 0
$$643$$ 1.38386e7 1.31997 0.659987 0.751277i $$-0.270562\pi$$
0.659987 + 0.751277i $$0.270562\pi$$
$$644$$ 0 0
$$645$$ −386048. −0.0365378
$$646$$ 0 0
$$647$$ 1.40358e7 1.31819 0.659093 0.752061i $$-0.270940\pi$$
0.659093 + 0.752061i $$0.270940\pi$$
$$648$$ 0 0
$$649$$ 277616. 0.0258722
$$650$$ 0 0
$$651$$ 6.09991e6 0.564119
$$652$$ 0 0
$$653$$ 1.61063e7 1.47813 0.739064 0.673635i $$-0.235268\pi$$
0.739064 + 0.673635i $$0.235268\pi$$
$$654$$ 0 0
$$655$$ −2.89187e6 −0.263376
$$656$$ 0 0
$$657$$ −8.19496e6 −0.740685
$$658$$ 0 0
$$659$$ −4.80075e6 −0.430622 −0.215311 0.976546i $$-0.569076\pi$$
−0.215311 + 0.976546i $$0.569076\pi$$
$$660$$ 0 0
$$661$$ −1.76565e7 −1.57181 −0.785905 0.618347i $$-0.787803\pi$$
−0.785905 + 0.618347i $$0.787803\pi$$
$$662$$ 0 0
$$663$$ 3.94449e7 3.48504
$$664$$ 0 0
$$665$$ 2.11523e6 0.185483
$$666$$ 0 0
$$667$$ 1.08814e7 0.947042
$$668$$ 0 0
$$669$$ 1.25214e7 1.08165
$$670$$ 0 0
$$671$$ 8256.00 0.000707886 0
$$672$$ 0 0
$$673$$ 6.59225e6 0.561043 0.280521 0.959848i $$-0.409493\pi$$
0.280521 + 0.959848i $$0.409493\pi$$
$$674$$ 0 0
$$675$$ 1.41729e7 1.19729
$$676$$ 0 0
$$677$$ 9.77178e6 0.819411 0.409706 0.912218i $$-0.365631\pi$$
0.409706 + 0.912218i $$0.365631\pi$$
$$678$$ 0 0
$$679$$ 836038. 0.0695908
$$680$$ 0 0
$$681$$ −157092. −0.0129803
$$682$$ 0 0
$$683$$ 1.88663e7 1.54752 0.773758 0.633481i $$-0.218374\pi$$
0.773758 + 0.633481i $$0.218374\pi$$
$$684$$ 0 0
$$685$$ −3.35485e6 −0.273178
$$686$$ 0 0
$$687$$ −46904.0 −0.00379156
$$688$$ 0 0
$$689$$ 8.96724e6 0.719632
$$690$$ 0 0
$$691$$ −8.67018e6 −0.690769 −0.345385 0.938461i $$-0.612252\pi$$
−0.345385 + 0.938461i $$0.612252\pi$$
$$692$$ 0 0
$$693$$ −169736. −0.0134258
$$694$$ 0 0
$$695$$ −275872. −0.0216643
$$696$$ 0 0
$$697$$ −2.96103e7 −2.30866
$$698$$ 0 0
$$699$$ 4.18997e7 3.24353
$$700$$ 0 0
$$701$$ 7.93482e6 0.609877 0.304938 0.952372i $$-0.401364\pi$$
0.304938 + 0.952372i $$0.401364\pi$$
$$702$$ 0 0
$$703$$ −3.09461e7 −2.36166
$$704$$ 0 0
$$705$$ 504192. 0.0382053
$$706$$ 0 0
$$707$$ 2.24930e6 0.169238
$$708$$ 0 0
$$709$$ 2.62600e7 1.96191 0.980956 0.194228i $$-0.0622202\pi$$
0.980956 + 0.194228i $$0.0622202\pi$$
$$710$$ 0 0
$$711$$ −4.93620e6 −0.366200
$$712$$ 0 0
$$713$$ 1.60111e7 1.17950
$$714$$ 0 0
$$715$$ −87552.0 −0.00640473
$$716$$ 0 0
$$717$$ −2.56645e7 −1.86438
$$718$$ 0 0
$$719$$ −2.20763e7 −1.59259 −0.796295 0.604909i $$-0.793210\pi$$
−0.796295 + 0.604909i $$0.793210\pi$$
$$720$$ 0 0
$$721$$ 6.66459e6 0.477458
$$722$$ 0 0
$$723$$ −2.32313e7 −1.65283
$$724$$ 0 0
$$725$$ 9.33573e6 0.659634
$$726$$ 0 0
$$727$$ −8.49245e6 −0.595933 −0.297966 0.954576i $$-0.596308\pi$$
−0.297966 + 0.954576i $$0.596308\pi$$
$$728$$ 0 0
$$729$$ −2.11562e7 −1.47441
$$730$$ 0 0
$$731$$ −2.05830e6 −0.142468
$$732$$ 0 0
$$733$$ −1.90713e7 −1.31105 −0.655526 0.755172i $$-0.727553\pi$$
−0.655526 + 0.755172i $$0.727553\pi$$
$$734$$ 0 0
$$735$$ −998816. −0.0681973
$$736$$ 0 0
$$737$$ 80864.0 0.00548386
$$738$$ 0 0
$$739$$ 1.46832e7 0.989032 0.494516 0.869169i $$-0.335345\pi$$
0.494516 + 0.869169i $$0.335345\pi$$
$$740$$ 0 0
$$741$$ −4.79812e7 −3.21015
$$742$$ 0 0
$$743$$ −1.64265e7 −1.09162 −0.545812 0.837908i $$-0.683779\pi$$
−0.545812 + 0.837908i $$0.683779\pi$$
$$744$$ 0 0
$$745$$ 949728. 0.0626915
$$746$$ 0 0
$$747$$ −3.85188e7 −2.52564
$$748$$ 0 0
$$749$$ 3.38864e6 0.220710
$$750$$ 0 0
$$751$$ 2.44357e7 1.58097 0.790486 0.612479i $$-0.209828\pi$$
0.790486 + 0.612479i $$0.209828\pi$$
$$752$$ 0 0
$$753$$ 9.51460e6 0.611509
$$754$$ 0 0
$$755$$ 5.38150e6 0.343587
$$756$$ 0 0
$$757$$ −295566. −0.0187463 −0.00937313 0.999956i $$-0.502984\pi$$
−0.00937313 + 0.999956i $$0.502984\pi$$
$$758$$ 0 0
$$759$$ −695552. −0.0438253
$$760$$ 0 0
$$761$$ −473842. −0.0296601 −0.0148300 0.999890i $$-0.504721\pi$$
−0.0148300 + 0.999890i $$0.504721\pi$$
$$762$$ 0 0
$$763$$ −7.17429e6 −0.446136
$$764$$ 0 0
$$765$$ −1.53663e7 −0.949327
$$766$$ 0 0
$$767$$ −2.37362e7 −1.45687
$$768$$ 0 0
$$769$$ 2.33241e7 1.42229 0.711145 0.703045i $$-0.248177\pi$$
0.711145 + 0.703045i $$0.248177\pi$$
$$770$$ 0 0
$$771$$ −3.65194e7 −2.21253
$$772$$ 0 0
$$773$$ 1.55583e7 0.936511 0.468255 0.883593i $$-0.344883\pi$$
0.468255 + 0.883593i $$0.344883\pi$$
$$774$$ 0 0
$$775$$ 1.37368e7 0.821543
$$776$$ 0 0
$$777$$ 1.46128e7 0.868321
$$778$$ 0 0
$$779$$ 3.60183e7 2.12657
$$780$$ 0 0
$$781$$ 501760. 0.0294353
$$782$$ 0 0
$$783$$ 1.60748e7 0.937001
$$784$$ 0 0
$$785$$ 7.43341e6 0.430540
$$786$$ 0 0
$$787$$ −6.66843e6 −0.383784 −0.191892 0.981416i $$-0.561462\pi$$
−0.191892 + 0.981416i $$0.561462\pi$$
$$788$$ 0 0
$$789$$ 2.85917e7 1.63511
$$790$$ 0 0
$$791$$ −3.92911e6 −0.223282
$$792$$ 0 0
$$793$$ −705888. −0.0398614
$$794$$ 0 0
$$795$$ −5.45376e6 −0.306040
$$796$$ 0 0
$$797$$ −1.22461e7 −0.682892 −0.341446 0.939901i $$-0.610917\pi$$
−0.341446 + 0.939901i $$0.610917\pi$$
$$798$$ 0 0
$$799$$ 2.68822e6 0.148969
$$800$$ 0 0
$$801$$ 8.53963e6 0.470281
$$802$$ 0 0
$$803$$ 151408. 0.00828629
$$804$$ 0 0
$$805$$ −2.62170e6 −0.142591
$$806$$ 0 0
$$807$$ −2.11886e7 −1.14530
$$808$$ 0 0
$$809$$ −2.91495e7 −1.56588 −0.782941 0.622095i $$-0.786282\pi$$
−0.782941 + 0.622095i $$0.786282\pi$$
$$810$$ 0 0
$$811$$ −7.58849e6 −0.405138 −0.202569 0.979268i $$-0.564929\pi$$
−0.202569 + 0.979268i $$0.564929\pi$$
$$812$$ 0 0
$$813$$ −4.41755e7 −2.34398
$$814$$ 0 0
$$815$$ −5.03667e6 −0.265613
$$816$$ 0 0
$$817$$ 2.50374e6 0.131230
$$818$$ 0 0
$$819$$ 1.45124e7 0.756015
$$820$$ 0 0
$$821$$ −5.98849e6 −0.310070 −0.155035 0.987909i $$-0.549549\pi$$
−0.155035 + 0.987909i $$0.549549\pi$$
$$822$$ 0 0
$$823$$ −817960. −0.0420952 −0.0210476 0.999778i $$-0.506700\pi$$
−0.0210476 + 0.999778i $$0.506700\pi$$
$$824$$ 0 0
$$825$$ −596752. −0.0305252
$$826$$ 0 0
$$827$$ −2.51963e6 −0.128107 −0.0640535 0.997946i $$-0.520403\pi$$
−0.0640535 + 0.997946i $$0.520403\pi$$
$$828$$ 0 0
$$829$$ −1.61006e7 −0.813684 −0.406842 0.913499i $$-0.633370\pi$$
−0.406842 + 0.913499i $$0.633370\pi$$
$$830$$ 0 0
$$831$$ 3.54921e7 1.78291
$$832$$ 0 0
$$833$$ −5.32542e6 −0.265914
$$834$$ 0 0
$$835$$ −4.57158e6 −0.226908
$$836$$ 0 0
$$837$$ 2.36527e7 1.16699
$$838$$ 0 0
$$839$$ 2.58167e7 1.26618 0.633091 0.774077i $$-0.281786\pi$$
0.633091 + 0.774077i $$0.281786\pi$$
$$840$$ 0 0
$$841$$ −9.92263e6 −0.483768
$$842$$ 0 0
$$843$$ 1.86120e7 0.902036
$$844$$ 0 0
$$845$$ 1.54501e6 0.0744370
$$846$$ 0 0
$$847$$ −7.88836e6 −0.377814
$$848$$ 0 0
$$849$$ 5.66088e6 0.269535
$$850$$ 0 0
$$851$$ 3.83557e7 1.81554
$$852$$ 0 0
$$853$$ −1.54270e7 −0.725954 −0.362977 0.931798i $$-0.618240\pi$$
−0.362977 + 0.931798i $$0.618240\pi$$
$$854$$ 0 0
$$855$$ 1.86917e7 0.874449
$$856$$ 0 0
$$857$$ 3.60517e6 0.167677 0.0838384 0.996479i $$-0.473282\pi$$
0.0838384 + 0.996479i $$0.473282\pi$$
$$858$$ 0 0
$$859$$ −4.06995e6 −0.188194 −0.0940970 0.995563i $$-0.529996\pi$$
−0.0940970 + 0.995563i $$0.529996\pi$$
$$860$$ 0 0
$$861$$ −1.70079e7 −0.781885
$$862$$ 0 0
$$863$$ −7.25111e6 −0.331419 −0.165710 0.986175i $$-0.552991\pi$$
−0.165710 + 0.986175i $$0.552991\pi$$
$$864$$ 0 0
$$865$$ −1.13464e7 −0.515604
$$866$$ 0 0
$$867$$ −9.09913e7 −4.11105
$$868$$ 0 0
$$869$$ 91200.0 0.00409681
$$870$$ 0 0
$$871$$ −6.91387e6 −0.308799
$$872$$ 0 0
$$873$$ 7.38785e6 0.328082
$$874$$ 0 0
$$875$$ −4.69930e6 −0.207497
$$876$$ 0 0
$$877$$ 2.37414e6 0.104233 0.0521167 0.998641i $$-0.483403\pi$$
0.0521167 + 0.998641i $$0.483403\pi$$
$$878$$ 0 0
$$879$$ −3.91841e7 −1.71056
$$880$$ 0 0
$$881$$ 3.03558e7 1.31766 0.658828 0.752293i $$-0.271052\pi$$
0.658828 + 0.752293i $$0.271052\pi$$
$$882$$ 0 0
$$883$$ −1.53338e6 −0.0661832 −0.0330916 0.999452i $$-0.510535\pi$$
−0.0330916 + 0.999452i $$0.510535\pi$$
$$884$$ 0 0
$$885$$ 1.44360e7 0.619569
$$886$$ 0 0
$$887$$ 2.92379e7 1.24778 0.623888 0.781514i $$-0.285552\pi$$
0.623888 + 0.781514i $$0.285552\pi$$
$$888$$ 0 0
$$889$$ −1.34652e7 −0.571424
$$890$$ 0 0
$$891$$ −185768. −0.00783929
$$892$$ 0 0
$$893$$ −3.26998e6 −0.137220
$$894$$ 0 0
$$895$$ 9.87437e6 0.412052
$$896$$ 0 0
$$897$$ 5.94697e7 2.46783
$$898$$ 0 0
$$899$$ 1.55802e7 0.642943
$$900$$ 0 0
$$901$$ −2.90780e7 −1.19331
$$902$$ 0 0
$$903$$ −1.18227e6 −0.0482501
$$904$$ 0 0
$$905$$ 3.80525e6 0.154441
$$906$$ 0 0
$$907$$ −4.48227e7 −1.80917 −0.904587 0.426289i $$-0.859821\pi$$
−0.904587 + 0.426289i $$0.859821\pi$$
$$908$$ 0 0
$$909$$ 1.98764e7 0.797864
$$910$$ 0 0
$$911$$ 3.62906e7 1.44877 0.724384 0.689397i $$-0.242124\pi$$
0.724384 + 0.689397i $$0.242124\pi$$
$$912$$ 0 0
$$913$$ 711664. 0.0282552
$$914$$ 0 0
$$915$$ 429312. 0.0169520
$$916$$ 0 0
$$917$$ −8.85636e6 −0.347802
$$918$$ 0 0
$$919$$ −3.25350e7 −1.27076 −0.635378 0.772201i $$-0.719156\pi$$
−0.635378 + 0.772201i $$0.719156\pi$$
$$920$$ 0 0
$$921$$ 325052. 0.0126271
$$922$$ 0 0
$$923$$ −4.29005e7 −1.65752
$$924$$ 0 0
$$925$$ 3.29074e7 1.26456
$$926$$ 0 0
$$927$$ 5.88932e7 2.25095
$$928$$ 0 0
$$929$$ −4.46676e7 −1.69806 −0.849030 0.528344i $$-0.822813\pi$$
−0.849030 + 0.528344i $$0.822813\pi$$
$$930$$ 0 0
$$931$$ 6.47790e6 0.244940
$$932$$ 0 0
$$933$$ −1.68332e7 −0.633087
$$934$$ 0 0
$$935$$ 283904. 0.0106204
$$936$$ 0 0
$$937$$ 1.56680e7 0.582995 0.291498 0.956572i $$-0.405846\pi$$
0.291498 + 0.956572i $$0.405846\pi$$
$$938$$ 0 0
$$939$$ 2.43354e7 0.900689
$$940$$ 0 0
$$941$$ −2.01175e7 −0.740627 −0.370313 0.928907i $$-0.620750\pi$$
−0.370313 + 0.928907i $$0.620750\pi$$
$$942$$ 0 0
$$943$$ −4.46424e7 −1.63481
$$944$$ 0 0
$$945$$ −3.87296e6 −0.141079
$$946$$ 0 0
$$947$$ 8.84518e6 0.320503 0.160251 0.987076i $$-0.448769\pi$$
0.160251 + 0.987076i $$0.448769\pi$$
$$948$$ 0 0
$$949$$ −1.29454e7 −0.466605
$$950$$ 0 0
$$951$$ −1.83545e7 −0.658099
$$952$$ 0 0
$$953$$ −3.14364e7 −1.12124 −0.560622 0.828072i $$-0.689438\pi$$
−0.560622 + 0.828072i $$0.689438\pi$$
$$954$$ 0 0
$$955$$ 2.13619e6 0.0757935
$$956$$ 0 0
$$957$$ −676832. −0.0238892
$$958$$ 0 0
$$959$$ −1.02742e7 −0.360747
$$960$$ 0 0
$$961$$ −5.70421e6 −0.199245
$$962$$ 0 0
$$963$$ 2.99445e7 1.04052
$$964$$ 0 0
$$965$$ 4.32714e6 0.149583
$$966$$ 0 0
$$967$$ 9.52158e6 0.327449 0.163724 0.986506i $$-0.447649\pi$$
0.163724 + 0.986506i $$0.447649\pi$$
$$968$$ 0 0
$$969$$ 1.55588e8 5.32313
$$970$$ 0 0
$$971$$ −1.06520e7 −0.362564 −0.181282 0.983431i $$-0.558025\pi$$
−0.181282 + 0.983431i $$0.558025\pi$$
$$972$$ 0 0
$$973$$ −844858. −0.0286089
$$974$$ 0 0
$$975$$ 5.10223e7 1.71889
$$976$$ 0 0
$$977$$ 2.72931e7 0.914779 0.457389 0.889266i $$-0.348785\pi$$
0.457389 + 0.889266i $$0.348785\pi$$
$$978$$ 0 0
$$979$$ −157776. −0.00526119
$$980$$ 0 0
$$981$$ −6.33973e7 −2.10328
$$982$$ 0 0
$$983$$ −1.04764e7 −0.345802 −0.172901 0.984939i $$-0.555314\pi$$
−0.172901 + 0.984939i $$0.555314\pi$$
$$984$$ 0 0
$$985$$ 1.40016e7 0.459820
$$986$$ 0 0
$$987$$ 1.54409e6 0.0504521
$$988$$ 0 0
$$989$$ −3.10323e6 −0.100884
$$990$$ 0 0
$$991$$ −1.88230e6 −0.0608843 −0.0304422 0.999537i $$-0.509692\pi$$
−0.0304422 + 0.999537i $$0.509692\pi$$
$$992$$ 0 0
$$993$$ −2.97190e7 −0.956449
$$994$$ 0 0
$$995$$ 5.56192e6 0.178101
$$996$$ 0 0
$$997$$ 2.71518e7 0.865090 0.432545 0.901612i $$-0.357616\pi$$
0.432545 + 0.901612i $$0.357616\pi$$
$$998$$ 0 0
$$999$$ 5.66618e7 1.79629
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 112.6.a.b.1.1 1
3.2 odd 2 1008.6.a.l.1.1 1
4.3 odd 2 28.6.a.b.1.1 1
7.6 odd 2 784.6.a.m.1.1 1
8.3 odd 2 448.6.a.b.1.1 1
8.5 even 2 448.6.a.o.1.1 1
12.11 even 2 252.6.a.a.1.1 1
20.3 even 4 700.6.e.b.449.2 2
20.7 even 4 700.6.e.b.449.1 2
20.19 odd 2 700.6.a.b.1.1 1
28.3 even 6 196.6.e.i.177.1 2
28.11 odd 6 196.6.e.a.177.1 2
28.19 even 6 196.6.e.i.165.1 2
28.23 odd 6 196.6.e.a.165.1 2
28.27 even 2 196.6.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
28.6.a.b.1.1 1 4.3 odd 2
112.6.a.b.1.1 1 1.1 even 1 trivial
196.6.a.a.1.1 1 28.27 even 2
196.6.e.a.165.1 2 28.23 odd 6
196.6.e.a.177.1 2 28.11 odd 6
196.6.e.i.165.1 2 28.19 even 6
196.6.e.i.177.1 2 28.3 even 6
252.6.a.a.1.1 1 12.11 even 2
448.6.a.b.1.1 1 8.3 odd 2
448.6.a.o.1.1 1 8.5 even 2
700.6.a.b.1.1 1 20.19 odd 2
700.6.e.b.449.1 2 20.7 even 4
700.6.e.b.449.2 2 20.3 even 4
784.6.a.m.1.1 1 7.6 odd 2
1008.6.a.l.1.1 1 3.2 odd 2