# Properties

 Label 112.6.a.d.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 14) 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)

 $$f(q)$$ $$=$$ $$q-8.00000 q^{3} +10.0000 q^{5} +49.0000 q^{7} -179.000 q^{9} +O(q^{10})$$ $$q-8.00000 q^{3} +10.0000 q^{5} +49.0000 q^{7} -179.000 q^{9} +340.000 q^{11} -294.000 q^{13} -80.0000 q^{15} +1226.00 q^{17} -2432.00 q^{19} -392.000 q^{21} -2000.00 q^{23} -3025.00 q^{25} +3376.00 q^{27} -6746.00 q^{29} -8856.00 q^{31} -2720.00 q^{33} +490.000 q^{35} +9182.00 q^{37} +2352.00 q^{39} -14574.0 q^{41} -8108.00 q^{43} -1790.00 q^{45} +312.000 q^{47} +2401.00 q^{49} -9808.00 q^{51} -14634.0 q^{53} +3400.00 q^{55} +19456.0 q^{57} +27656.0 q^{59} +34338.0 q^{61} -8771.00 q^{63} -2940.00 q^{65} -12316.0 q^{67} +16000.0 q^{69} -36920.0 q^{71} -61718.0 q^{73} +24200.0 q^{75} +16660.0 q^{77} +64752.0 q^{79} +16489.0 q^{81} +77056.0 q^{83} +12260.0 q^{85} +53968.0 q^{87} -8166.00 q^{89} -14406.0 q^{91} +70848.0 q^{93} -24320.0 q^{95} +20650.0 q^{97} -60860.0 q^{99} +O(q^{100})$$

## 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
$$3$$ −8.00000 −0.513200 −0.256600 0.966518i $$-0.582602\pi$$
−0.256600 + 0.966518i $$0.582602\pi$$
$$4$$ 0 0
$$5$$ 10.0000 0.178885 0.0894427 0.995992i $$-0.471491\pi$$
0.0894427 + 0.995992i $$0.471491\pi$$
$$6$$ 0 0
$$7$$ 49.0000 0.377964
$$8$$ 0 0
$$9$$ −179.000 −0.736626
$$10$$ 0 0
$$11$$ 340.000 0.847222 0.423611 0.905844i $$-0.360762\pi$$
0.423611 + 0.905844i $$0.360762\pi$$
$$12$$ 0 0
$$13$$ −294.000 −0.482491 −0.241245 0.970464i $$-0.577556\pi$$
−0.241245 + 0.970464i $$0.577556\pi$$
$$14$$ 0 0
$$15$$ −80.0000 −0.0918040
$$16$$ 0 0
$$17$$ 1226.00 1.02889 0.514444 0.857524i $$-0.327998\pi$$
0.514444 + 0.857524i $$0.327998\pi$$
$$18$$ 0 0
$$19$$ −2432.00 −1.54554 −0.772769 0.634688i $$-0.781129\pi$$
−0.772769 + 0.634688i $$0.781129\pi$$
$$20$$ 0 0
$$21$$ −392.000 −0.193971
$$22$$ 0 0
$$23$$ −2000.00 −0.788334 −0.394167 0.919039i $$-0.628967\pi$$
−0.394167 + 0.919039i $$0.628967\pi$$
$$24$$ 0 0
$$25$$ −3025.00 −0.968000
$$26$$ 0 0
$$27$$ 3376.00 0.891237
$$28$$ 0 0
$$29$$ −6746.00 −1.48954 −0.744769 0.667323i $$-0.767440\pi$$
−0.744769 + 0.667323i $$0.767440\pi$$
$$30$$ 0 0
$$31$$ −8856.00 −1.65513 −0.827567 0.561366i $$-0.810276\pi$$
−0.827567 + 0.561366i $$0.810276\pi$$
$$32$$ 0 0
$$33$$ −2720.00 −0.434795
$$34$$ 0 0
$$35$$ 490.000 0.0676123
$$36$$ 0 0
$$37$$ 9182.00 1.10264 0.551319 0.834295i $$-0.314125\pi$$
0.551319 + 0.834295i $$0.314125\pi$$
$$38$$ 0 0
$$39$$ 2352.00 0.247614
$$40$$ 0 0
$$41$$ −14574.0 −1.35400 −0.677001 0.735982i $$-0.736721\pi$$
−0.677001 + 0.735982i $$0.736721\pi$$
$$42$$ 0 0
$$43$$ −8108.00 −0.668717 −0.334359 0.942446i $$-0.608520\pi$$
−0.334359 + 0.942446i $$0.608520\pi$$
$$44$$ 0 0
$$45$$ −1790.00 −0.131772
$$46$$ 0 0
$$47$$ 312.000 0.0206020 0.0103010 0.999947i $$-0.496721\pi$$
0.0103010 + 0.999947i $$0.496721\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ −9808.00 −0.528026
$$52$$ 0 0
$$53$$ −14634.0 −0.715605 −0.357803 0.933797i $$-0.616474\pi$$
−0.357803 + 0.933797i $$0.616474\pi$$
$$54$$ 0 0
$$55$$ 3400.00 0.151556
$$56$$ 0 0
$$57$$ 19456.0 0.793170
$$58$$ 0 0
$$59$$ 27656.0 1.03433 0.517165 0.855886i $$-0.326987\pi$$
0.517165 + 0.855886i $$0.326987\pi$$
$$60$$ 0 0
$$61$$ 34338.0 1.18155 0.590773 0.806838i $$-0.298823\pi$$
0.590773 + 0.806838i $$0.298823\pi$$
$$62$$ 0 0
$$63$$ −8771.00 −0.278418
$$64$$ 0 0
$$65$$ −2940.00 −0.0863106
$$66$$ 0 0
$$67$$ −12316.0 −0.335184 −0.167592 0.985856i $$-0.553599\pi$$
−0.167592 + 0.985856i $$0.553599\pi$$
$$68$$ 0 0
$$69$$ 16000.0 0.404573
$$70$$ 0 0
$$71$$ −36920.0 −0.869192 −0.434596 0.900625i $$-0.643109\pi$$
−0.434596 + 0.900625i $$0.643109\pi$$
$$72$$ 0 0
$$73$$ −61718.0 −1.35552 −0.677758 0.735285i $$-0.737048\pi$$
−0.677758 + 0.735285i $$0.737048\pi$$
$$74$$ 0 0
$$75$$ 24200.0 0.496778
$$76$$ 0 0
$$77$$ 16660.0 0.320220
$$78$$ 0 0
$$79$$ 64752.0 1.16731 0.583654 0.812002i $$-0.301622\pi$$
0.583654 + 0.812002i $$0.301622\pi$$
$$80$$ 0 0
$$81$$ 16489.0 0.279243
$$82$$ 0 0
$$83$$ 77056.0 1.22775 0.613877 0.789402i $$-0.289609\pi$$
0.613877 + 0.789402i $$0.289609\pi$$
$$84$$ 0 0
$$85$$ 12260.0 0.184053
$$86$$ 0 0
$$87$$ 53968.0 0.764431
$$88$$ 0 0
$$89$$ −8166.00 −0.109278 −0.0546392 0.998506i $$-0.517401\pi$$
−0.0546392 + 0.998506i $$0.517401\pi$$
$$90$$ 0 0
$$91$$ −14406.0 −0.182364
$$92$$ 0 0
$$93$$ 70848.0 0.849416
$$94$$ 0 0
$$95$$ −24320.0 −0.276474
$$96$$ 0 0
$$97$$ 20650.0 0.222839 0.111419 0.993773i $$-0.464460\pi$$
0.111419 + 0.993773i $$0.464460\pi$$
$$98$$ 0 0
$$99$$ −60860.0 −0.624085
$$100$$ 0 0
$$101$$ 186250. 1.81674 0.908370 0.418167i $$-0.137327\pi$$
0.908370 + 0.418167i $$0.137327\pi$$
$$102$$ 0 0
$$103$$ 60064.0 0.557855 0.278927 0.960312i $$-0.410021\pi$$
0.278927 + 0.960312i $$0.410021\pi$$
$$104$$ 0 0
$$105$$ −3920.00 −0.0346987
$$106$$ 0 0
$$107$$ −47892.0 −0.404393 −0.202196 0.979345i $$-0.564808\pi$$
−0.202196 + 0.979345i $$0.564808\pi$$
$$108$$ 0 0
$$109$$ 22102.0 0.178183 0.0890913 0.996023i $$-0.471604\pi$$
0.0890913 + 0.996023i $$0.471604\pi$$
$$110$$ 0 0
$$111$$ −73456.0 −0.565874
$$112$$ 0 0
$$113$$ −245054. −1.80537 −0.902684 0.430304i $$-0.858406\pi$$
−0.902684 + 0.430304i $$0.858406\pi$$
$$114$$ 0 0
$$115$$ −20000.0 −0.141022
$$116$$ 0 0
$$117$$ 52626.0 0.355415
$$118$$ 0 0
$$119$$ 60074.0 0.388883
$$120$$ 0 0
$$121$$ −45451.0 −0.282215
$$122$$ 0 0
$$123$$ 116592. 0.694874
$$124$$ 0 0
$$125$$ −61500.0 −0.352047
$$126$$ 0 0
$$127$$ 96696.0 0.531985 0.265992 0.963975i $$-0.414300\pi$$
0.265992 + 0.963975i $$0.414300\pi$$
$$128$$ 0 0
$$129$$ 64864.0 0.343186
$$130$$ 0 0
$$131$$ −134368. −0.684097 −0.342048 0.939682i $$-0.611121\pi$$
−0.342048 + 0.939682i $$0.611121\pi$$
$$132$$ 0 0
$$133$$ −119168. −0.584158
$$134$$ 0 0
$$135$$ 33760.0 0.159429
$$136$$ 0 0
$$137$$ −294662. −1.34129 −0.670645 0.741778i $$-0.733983\pi$$
−0.670645 + 0.741778i $$0.733983\pi$$
$$138$$ 0 0
$$139$$ −314944. −1.38260 −0.691300 0.722568i $$-0.742962\pi$$
−0.691300 + 0.722568i $$0.742962\pi$$
$$140$$ 0 0
$$141$$ −2496.00 −0.0105730
$$142$$ 0 0
$$143$$ −99960.0 −0.408777
$$144$$ 0 0
$$145$$ −67460.0 −0.266457
$$146$$ 0 0
$$147$$ −19208.0 −0.0733143
$$148$$ 0 0
$$149$$ 113622. 0.419273 0.209636 0.977779i $$-0.432772\pi$$
0.209636 + 0.977779i $$0.432772\pi$$
$$150$$ 0 0
$$151$$ −408208. −1.45693 −0.728466 0.685082i $$-0.759766\pi$$
−0.728466 + 0.685082i $$0.759766\pi$$
$$152$$ 0 0
$$153$$ −219454. −0.757905
$$154$$ 0 0
$$155$$ −88560.0 −0.296080
$$156$$ 0 0
$$157$$ 293546. 0.950445 0.475223 0.879866i $$-0.342368\pi$$
0.475223 + 0.879866i $$0.342368\pi$$
$$158$$ 0 0
$$159$$ 117072. 0.367249
$$160$$ 0 0
$$161$$ −98000.0 −0.297962
$$162$$ 0 0
$$163$$ 317116. 0.934866 0.467433 0.884029i $$-0.345179\pi$$
0.467433 + 0.884029i $$0.345179\pi$$
$$164$$ 0 0
$$165$$ −27200.0 −0.0777784
$$166$$ 0 0
$$167$$ −141568. −0.392802 −0.196401 0.980524i $$-0.562925\pi$$
−0.196401 + 0.980524i $$0.562925\pi$$
$$168$$ 0 0
$$169$$ −284857. −0.767203
$$170$$ 0 0
$$171$$ 435328. 1.13848
$$172$$ 0 0
$$173$$ −71222.0 −0.180925 −0.0904626 0.995900i $$-0.528835\pi$$
−0.0904626 + 0.995900i $$0.528835\pi$$
$$174$$ 0 0
$$175$$ −148225. −0.365870
$$176$$ 0 0
$$177$$ −221248. −0.530819
$$178$$ 0 0
$$179$$ −485628. −1.13285 −0.566423 0.824114i $$-0.691673\pi$$
−0.566423 + 0.824114i $$0.691673\pi$$
$$180$$ 0 0
$$181$$ 657090. 1.49083 0.745416 0.666600i $$-0.232251\pi$$
0.745416 + 0.666600i $$0.232251\pi$$
$$182$$ 0 0
$$183$$ −274704. −0.606369
$$184$$ 0 0
$$185$$ 91820.0 0.197246
$$186$$ 0 0
$$187$$ 416840. 0.871697
$$188$$ 0 0
$$189$$ 165424. 0.336856
$$190$$ 0 0
$$191$$ −68304.0 −0.135476 −0.0677381 0.997703i $$-0.521578\pi$$
−0.0677381 + 0.997703i $$0.521578\pi$$
$$192$$ 0 0
$$193$$ 352754. 0.681677 0.340839 0.940122i $$-0.389289\pi$$
0.340839 + 0.940122i $$0.389289\pi$$
$$194$$ 0 0
$$195$$ 23520.0 0.0442946
$$196$$ 0 0
$$197$$ 196982. 0.361627 0.180814 0.983517i $$-0.442127\pi$$
0.180814 + 0.983517i $$0.442127\pi$$
$$198$$ 0 0
$$199$$ 1.10392e6 1.97608 0.988041 0.154192i $$-0.0492775\pi$$
0.988041 + 0.154192i $$0.0492775\pi$$
$$200$$ 0 0
$$201$$ 98528.0 0.172016
$$202$$ 0 0
$$203$$ −330554. −0.562992
$$204$$ 0 0
$$205$$ −145740. −0.242211
$$206$$ 0 0
$$207$$ 358000. 0.580707
$$208$$ 0 0
$$209$$ −826880. −1.30941
$$210$$ 0 0
$$211$$ 103444. 0.159955 0.0799777 0.996797i $$-0.474515\pi$$
0.0799777 + 0.996797i $$0.474515\pi$$
$$212$$ 0 0
$$213$$ 295360. 0.446070
$$214$$ 0 0
$$215$$ −81080.0 −0.119624
$$216$$ 0 0
$$217$$ −433944. −0.625582
$$218$$ 0 0
$$219$$ 493744. 0.695651
$$220$$ 0 0
$$221$$ −360444. −0.496429
$$222$$ 0 0
$$223$$ −307328. −0.413847 −0.206924 0.978357i $$-0.566345\pi$$
−0.206924 + 0.978357i $$0.566345\pi$$
$$224$$ 0 0
$$225$$ 541475. 0.713053
$$226$$ 0 0
$$227$$ 891792. 1.14868 0.574340 0.818617i $$-0.305259\pi$$
0.574340 + 0.818617i $$0.305259\pi$$
$$228$$ 0 0
$$229$$ 276706. 0.348682 0.174341 0.984685i $$-0.444220\pi$$
0.174341 + 0.984685i $$0.444220\pi$$
$$230$$ 0 0
$$231$$ −133280. −0.164337
$$232$$ 0 0
$$233$$ 1.47943e6 1.78528 0.892639 0.450772i $$-0.148851\pi$$
0.892639 + 0.450772i $$0.148851\pi$$
$$234$$ 0 0
$$235$$ 3120.00 0.00368540
$$236$$ 0 0
$$237$$ −518016. −0.599063
$$238$$ 0 0
$$239$$ −1.00034e6 −1.13280 −0.566402 0.824129i $$-0.691665\pi$$
−0.566402 + 0.824129i $$0.691665\pi$$
$$240$$ 0 0
$$241$$ 1.35833e6 1.50648 0.753239 0.657747i $$-0.228490\pi$$
0.753239 + 0.657747i $$0.228490\pi$$
$$242$$ 0 0
$$243$$ −952280. −1.03454
$$244$$ 0 0
$$245$$ 24010.0 0.0255551
$$246$$ 0 0
$$247$$ 715008. 0.745708
$$248$$ 0 0
$$249$$ −616448. −0.630083
$$250$$ 0 0
$$251$$ 177408. 0.177742 0.0888708 0.996043i $$-0.471674\pi$$
0.0888708 + 0.996043i $$0.471674\pi$$
$$252$$ 0 0
$$253$$ −680000. −0.667894
$$254$$ 0 0
$$255$$ −98080.0 −0.0944561
$$256$$ 0 0
$$257$$ 326658. 0.308504 0.154252 0.988032i $$-0.450703\pi$$
0.154252 + 0.988032i $$0.450703\pi$$
$$258$$ 0 0
$$259$$ 449918. 0.416758
$$260$$ 0 0
$$261$$ 1.20753e6 1.09723
$$262$$ 0 0
$$263$$ 34920.0 0.0311304 0.0155652 0.999879i $$-0.495045\pi$$
0.0155652 + 0.999879i $$0.495045\pi$$
$$264$$ 0 0
$$265$$ −146340. −0.128011
$$266$$ 0 0
$$267$$ 65328.0 0.0560817
$$268$$ 0 0
$$269$$ 716458. 0.603685 0.301842 0.953358i $$-0.402398\pi$$
0.301842 + 0.953358i $$0.402398\pi$$
$$270$$ 0 0
$$271$$ 953376. 0.788571 0.394286 0.918988i $$-0.370992\pi$$
0.394286 + 0.918988i $$0.370992\pi$$
$$272$$ 0 0
$$273$$ 115248. 0.0935894
$$274$$ 0 0
$$275$$ −1.02850e6 −0.820111
$$276$$ 0 0
$$277$$ −1.84729e6 −1.44656 −0.723279 0.690556i $$-0.757366\pi$$
−0.723279 + 0.690556i $$0.757366\pi$$
$$278$$ 0 0
$$279$$ 1.58522e6 1.21921
$$280$$ 0 0
$$281$$ −1.99601e6 −1.50798 −0.753991 0.656885i $$-0.771874\pi$$
−0.753991 + 0.656885i $$0.771874\pi$$
$$282$$ 0 0
$$283$$ −234088. −0.173745 −0.0868726 0.996219i $$-0.527687\pi$$
−0.0868726 + 0.996219i $$0.527687\pi$$
$$284$$ 0 0
$$285$$ 194560. 0.141887
$$286$$ 0 0
$$287$$ −714126. −0.511764
$$288$$ 0 0
$$289$$ 83219.0 0.0586108
$$290$$ 0 0
$$291$$ −165200. −0.114361
$$292$$ 0 0
$$293$$ −2.50081e6 −1.70181 −0.850905 0.525320i $$-0.823946\pi$$
−0.850905 + 0.525320i $$0.823946\pi$$
$$294$$ 0 0
$$295$$ 276560. 0.185027
$$296$$ 0 0
$$297$$ 1.14784e6 0.755075
$$298$$ 0 0
$$299$$ 588000. 0.380364
$$300$$ 0 0
$$301$$ −397292. −0.252751
$$302$$ 0 0
$$303$$ −1.49000e6 −0.932352
$$304$$ 0 0
$$305$$ 343380. 0.211361
$$306$$ 0 0
$$307$$ −2.34203e6 −1.41823 −0.709115 0.705092i $$-0.750905\pi$$
−0.709115 + 0.705092i $$0.750905\pi$$
$$308$$ 0 0
$$309$$ −480512. −0.286291
$$310$$ 0 0
$$311$$ 163064. 0.0955998 0.0477999 0.998857i $$-0.484779\pi$$
0.0477999 + 0.998857i $$0.484779\pi$$
$$312$$ 0 0
$$313$$ 1.73965e6 1.00369 0.501847 0.864957i $$-0.332654\pi$$
0.501847 + 0.864957i $$0.332654\pi$$
$$314$$ 0 0
$$315$$ −87710.0 −0.0498050
$$316$$ 0 0
$$317$$ −1.79771e6 −1.00478 −0.502392 0.864640i $$-0.667546\pi$$
−0.502392 + 0.864640i $$0.667546\pi$$
$$318$$ 0 0
$$319$$ −2.29364e6 −1.26197
$$320$$ 0 0
$$321$$ 383136. 0.207535
$$322$$ 0 0
$$323$$ −2.98163e6 −1.59019
$$324$$ 0 0
$$325$$ 889350. 0.467051
$$326$$ 0 0
$$327$$ −176816. −0.0914434
$$328$$ 0 0
$$329$$ 15288.0 0.00778683
$$330$$ 0 0
$$331$$ 2.47541e6 1.24187 0.620937 0.783861i $$-0.286752\pi$$
0.620937 + 0.783861i $$0.286752\pi$$
$$332$$ 0 0
$$333$$ −1.64358e6 −0.812231
$$334$$ 0 0
$$335$$ −123160. −0.0599595
$$336$$ 0 0
$$337$$ 89154.0 0.0427628 0.0213814 0.999771i $$-0.493194\pi$$
0.0213814 + 0.999771i $$0.493194\pi$$
$$338$$ 0 0
$$339$$ 1.96043e6 0.926515
$$340$$ 0 0
$$341$$ −3.01104e6 −1.40227
$$342$$ 0 0
$$343$$ 117649. 0.0539949
$$344$$ 0 0
$$345$$ 160000. 0.0723723
$$346$$ 0 0
$$347$$ −938556. −0.418443 −0.209222 0.977868i $$-0.567093\pi$$
−0.209222 + 0.977868i $$0.567093\pi$$
$$348$$ 0 0
$$349$$ 3.34268e6 1.46903 0.734516 0.678591i $$-0.237409\pi$$
0.734516 + 0.678591i $$0.237409\pi$$
$$350$$ 0 0
$$351$$ −992544. −0.430013
$$352$$ 0 0
$$353$$ −3.76606e6 −1.60861 −0.804305 0.594217i $$-0.797462\pi$$
−0.804305 + 0.594217i $$0.797462\pi$$
$$354$$ 0 0
$$355$$ −369200. −0.155486
$$356$$ 0 0
$$357$$ −480592. −0.199575
$$358$$ 0 0
$$359$$ 1.53934e6 0.630376 0.315188 0.949029i $$-0.397932\pi$$
0.315188 + 0.949029i $$0.397932\pi$$
$$360$$ 0 0
$$361$$ 3.43852e6 1.38869
$$362$$ 0 0
$$363$$ 363608. 0.144833
$$364$$ 0 0
$$365$$ −617180. −0.242482
$$366$$ 0 0
$$367$$ 859312. 0.333032 0.166516 0.986039i $$-0.446748\pi$$
0.166516 + 0.986039i $$0.446748\pi$$
$$368$$ 0 0
$$369$$ 2.60875e6 0.997392
$$370$$ 0 0
$$371$$ −717066. −0.270473
$$372$$ 0 0
$$373$$ −976586. −0.363445 −0.181722 0.983350i $$-0.558167\pi$$
−0.181722 + 0.983350i $$0.558167\pi$$
$$374$$ 0 0
$$375$$ 492000. 0.180670
$$376$$ 0 0
$$377$$ 1.98332e6 0.718688
$$378$$ 0 0
$$379$$ −106444. −0.0380648 −0.0190324 0.999819i $$-0.506059\pi$$
−0.0190324 + 0.999819i $$0.506059\pi$$
$$380$$ 0 0
$$381$$ −773568. −0.273015
$$382$$ 0 0
$$383$$ 2.00634e6 0.698889 0.349445 0.936957i $$-0.386370\pi$$
0.349445 + 0.936957i $$0.386370\pi$$
$$384$$ 0 0
$$385$$ 166600. 0.0572827
$$386$$ 0 0
$$387$$ 1.45133e6 0.492594
$$388$$ 0 0
$$389$$ −684002. −0.229184 −0.114592 0.993413i $$-0.536556\pi$$
−0.114592 + 0.993413i $$0.536556\pi$$
$$390$$ 0 0
$$391$$ −2.45200e6 −0.811108
$$392$$ 0 0
$$393$$ 1.07494e6 0.351079
$$394$$ 0 0
$$395$$ 647520. 0.208814
$$396$$ 0 0
$$397$$ −222870. −0.0709701 −0.0354850 0.999370i $$-0.511298\pi$$
−0.0354850 + 0.999370i $$0.511298\pi$$
$$398$$ 0 0
$$399$$ 953344. 0.299790
$$400$$ 0 0
$$401$$ 1.90072e6 0.590279 0.295140 0.955454i $$-0.404634\pi$$
0.295140 + 0.955454i $$0.404634\pi$$
$$402$$ 0 0
$$403$$ 2.60366e6 0.798587
$$404$$ 0 0
$$405$$ 164890. 0.0499524
$$406$$ 0 0
$$407$$ 3.12188e6 0.934179
$$408$$ 0 0
$$409$$ 1.77715e6 0.525311 0.262656 0.964890i $$-0.415402\pi$$
0.262656 + 0.964890i $$0.415402\pi$$
$$410$$ 0 0
$$411$$ 2.35730e6 0.688350
$$412$$ 0 0
$$413$$ 1.35514e6 0.390940
$$414$$ 0 0
$$415$$ 770560. 0.219627
$$416$$ 0 0
$$417$$ 2.51955e6 0.709550
$$418$$ 0 0
$$419$$ −28056.0 −0.00780712 −0.00390356 0.999992i $$-0.501243\pi$$
−0.00390356 + 0.999992i $$0.501243\pi$$
$$420$$ 0 0
$$421$$ −2.70897e6 −0.744902 −0.372451 0.928052i $$-0.621482\pi$$
−0.372451 + 0.928052i $$0.621482\pi$$
$$422$$ 0 0
$$423$$ −55848.0 −0.0151760
$$424$$ 0 0
$$425$$ −3.70865e6 −0.995964
$$426$$ 0 0
$$427$$ 1.68256e6 0.446582
$$428$$ 0 0
$$429$$ 799680. 0.209784
$$430$$ 0 0
$$431$$ −5.53898e6 −1.43627 −0.718136 0.695902i $$-0.755005\pi$$
−0.718136 + 0.695902i $$0.755005\pi$$
$$432$$ 0 0
$$433$$ −868294. −0.222560 −0.111280 0.993789i $$-0.535495\pi$$
−0.111280 + 0.993789i $$0.535495\pi$$
$$434$$ 0 0
$$435$$ 539680. 0.136746
$$436$$ 0 0
$$437$$ 4.86400e6 1.21840
$$438$$ 0 0
$$439$$ 1.13767e6 0.281745 0.140872 0.990028i $$-0.455009\pi$$
0.140872 + 0.990028i $$0.455009\pi$$
$$440$$ 0 0
$$441$$ −429779. −0.105232
$$442$$ 0 0
$$443$$ −1.75399e6 −0.424636 −0.212318 0.977201i $$-0.568101\pi$$
−0.212318 + 0.977201i $$0.568101\pi$$
$$444$$ 0 0
$$445$$ −81660.0 −0.0195483
$$446$$ 0 0
$$447$$ −908976. −0.215171
$$448$$ 0 0
$$449$$ 2.41674e6 0.565736 0.282868 0.959159i $$-0.408714\pi$$
0.282868 + 0.959159i $$0.408714\pi$$
$$450$$ 0 0
$$451$$ −4.95516e6 −1.14714
$$452$$ 0 0
$$453$$ 3.26566e6 0.747698
$$454$$ 0 0
$$455$$ −144060. −0.0326223
$$456$$ 0 0
$$457$$ −127430. −0.0285418 −0.0142709 0.999898i $$-0.504543\pi$$
−0.0142709 + 0.999898i $$0.504543\pi$$
$$458$$ 0 0
$$459$$ 4.13898e6 0.916983
$$460$$ 0 0
$$461$$ −128198. −0.0280950 −0.0140475 0.999901i $$-0.504472\pi$$
−0.0140475 + 0.999901i $$0.504472\pi$$
$$462$$ 0 0
$$463$$ 4.01653e6 0.870760 0.435380 0.900247i $$-0.356614\pi$$
0.435380 + 0.900247i $$0.356614\pi$$
$$464$$ 0 0
$$465$$ 708480. 0.151948
$$466$$ 0 0
$$467$$ −8.67246e6 −1.84014 −0.920069 0.391757i $$-0.871867\pi$$
−0.920069 + 0.391757i $$0.871867\pi$$
$$468$$ 0 0
$$469$$ −603484. −0.126687
$$470$$ 0 0
$$471$$ −2.34837e6 −0.487769
$$472$$ 0 0
$$473$$ −2.75672e6 −0.566552
$$474$$ 0 0
$$475$$ 7.35680e6 1.49608
$$476$$ 0 0
$$477$$ 2.61949e6 0.527133
$$478$$ 0 0
$$479$$ −8.28946e6 −1.65077 −0.825387 0.564567i $$-0.809043\pi$$
−0.825387 + 0.564567i $$0.809043\pi$$
$$480$$ 0 0
$$481$$ −2.69951e6 −0.532013
$$482$$ 0 0
$$483$$ 784000. 0.152914
$$484$$ 0 0
$$485$$ 206500. 0.0398626
$$486$$ 0 0
$$487$$ 8.91770e6 1.70385 0.851923 0.523667i $$-0.175437\pi$$
0.851923 + 0.523667i $$0.175437\pi$$
$$488$$ 0 0
$$489$$ −2.53693e6 −0.479773
$$490$$ 0 0
$$491$$ 5.71537e6 1.06989 0.534947 0.844886i $$-0.320332\pi$$
0.534947 + 0.844886i $$0.320332\pi$$
$$492$$ 0 0
$$493$$ −8.27060e6 −1.53257
$$494$$ 0 0
$$495$$ −608600. −0.111640
$$496$$ 0 0
$$497$$ −1.80908e6 −0.328524
$$498$$ 0 0
$$499$$ −125116. −0.0224937 −0.0112469 0.999937i $$-0.503580\pi$$
−0.0112469 + 0.999937i $$0.503580\pi$$
$$500$$ 0 0
$$501$$ 1.13254e6 0.201586
$$502$$ 0 0
$$503$$ 2.77116e6 0.488362 0.244181 0.969730i $$-0.421481\pi$$
0.244181 + 0.969730i $$0.421481\pi$$
$$504$$ 0 0
$$505$$ 1.86250e6 0.324988
$$506$$ 0 0
$$507$$ 2.27886e6 0.393729
$$508$$ 0 0
$$509$$ −138534. −0.0237007 −0.0118504 0.999930i $$-0.503772\pi$$
−0.0118504 + 0.999930i $$0.503772\pi$$
$$510$$ 0 0
$$511$$ −3.02418e6 −0.512337
$$512$$ 0 0
$$513$$ −8.21043e6 −1.37744
$$514$$ 0 0
$$515$$ 600640. 0.0997921
$$516$$ 0 0
$$517$$ 106080. 0.0174545
$$518$$ 0 0
$$519$$ 569776. 0.0928508
$$520$$ 0 0
$$521$$ −1.80281e6 −0.290976 −0.145488 0.989360i $$-0.546475\pi$$
−0.145488 + 0.989360i $$0.546475\pi$$
$$522$$ 0 0
$$523$$ −9.77247e6 −1.56225 −0.781124 0.624375i $$-0.785354\pi$$
−0.781124 + 0.624375i $$0.785354\pi$$
$$524$$ 0 0
$$525$$ 1.18580e6 0.187764
$$526$$ 0 0
$$527$$ −1.08575e7 −1.70295
$$528$$ 0 0
$$529$$ −2.43634e6 −0.378529
$$530$$ 0 0
$$531$$ −4.95042e6 −0.761914
$$532$$ 0 0
$$533$$ 4.28476e6 0.653293
$$534$$ 0 0
$$535$$ −478920. −0.0723400
$$536$$ 0 0
$$537$$ 3.88502e6 0.581377
$$538$$ 0 0
$$539$$ 816340. 0.121032
$$540$$ 0 0
$$541$$ 2.45504e6 0.360633 0.180316 0.983609i $$-0.442288\pi$$
0.180316 + 0.983609i $$0.442288\pi$$
$$542$$ 0 0
$$543$$ −5.25672e6 −0.765095
$$544$$ 0 0
$$545$$ 221020. 0.0318743
$$546$$ 0 0
$$547$$ −1.32081e7 −1.88744 −0.943721 0.330743i $$-0.892701\pi$$
−0.943721 + 0.330743i $$0.892701\pi$$
$$548$$ 0 0
$$549$$ −6.14650e6 −0.870356
$$550$$ 0 0
$$551$$ 1.64063e7 2.30214
$$552$$ 0 0
$$553$$ 3.17285e6 0.441201
$$554$$ 0 0
$$555$$ −734560. −0.101227
$$556$$ 0 0
$$557$$ 7.83293e6 1.06976 0.534880 0.844928i $$-0.320357\pi$$
0.534880 + 0.844928i $$0.320357\pi$$
$$558$$ 0 0
$$559$$ 2.38375e6 0.322650
$$560$$ 0 0
$$561$$ −3.33472e6 −0.447355
$$562$$ 0 0
$$563$$ −3.57908e6 −0.475883 −0.237942 0.971279i $$-0.576473\pi$$
−0.237942 + 0.971279i $$0.576473\pi$$
$$564$$ 0 0
$$565$$ −2.45054e6 −0.322954
$$566$$ 0 0
$$567$$ 807961. 0.105544
$$568$$ 0 0
$$569$$ −3.39581e6 −0.439707 −0.219853 0.975533i $$-0.570558\pi$$
−0.219853 + 0.975533i $$0.570558\pi$$
$$570$$ 0 0
$$571$$ 1.47695e6 0.189572 0.0947862 0.995498i $$-0.469783\pi$$
0.0947862 + 0.995498i $$0.469783\pi$$
$$572$$ 0 0
$$573$$ 546432. 0.0695264
$$574$$ 0 0
$$575$$ 6.05000e6 0.763108
$$576$$ 0 0
$$577$$ −1.49961e7 −1.87516 −0.937580 0.347771i $$-0.886939\pi$$
−0.937580 + 0.347771i $$0.886939\pi$$
$$578$$ 0 0
$$579$$ −2.82203e6 −0.349837
$$580$$ 0 0
$$581$$ 3.77574e6 0.464047
$$582$$ 0 0
$$583$$ −4.97556e6 −0.606276
$$584$$ 0 0
$$585$$ 526260. 0.0635786
$$586$$ 0 0
$$587$$ 3.29291e6 0.394444 0.197222 0.980359i $$-0.436808\pi$$
0.197222 + 0.980359i $$0.436808\pi$$
$$588$$ 0 0
$$589$$ 2.15378e7 2.55807
$$590$$ 0 0
$$591$$ −1.57586e6 −0.185587
$$592$$ 0 0
$$593$$ −1.17908e7 −1.37692 −0.688459 0.725275i $$-0.741713\pi$$
−0.688459 + 0.725275i $$0.741713\pi$$
$$594$$ 0 0
$$595$$ 600740. 0.0695655
$$596$$ 0 0
$$597$$ −8.83136e6 −1.01413
$$598$$ 0 0
$$599$$ 1.52642e6 0.173823 0.0869117 0.996216i $$-0.472300\pi$$
0.0869117 + 0.996216i $$0.472300\pi$$
$$600$$ 0 0
$$601$$ −1.00142e7 −1.13092 −0.565458 0.824777i $$-0.691301\pi$$
−0.565458 + 0.824777i $$0.691301\pi$$
$$602$$ 0 0
$$603$$ 2.20456e6 0.246905
$$604$$ 0 0
$$605$$ −454510. −0.0504841
$$606$$ 0 0
$$607$$ −1.20660e7 −1.32920 −0.664599 0.747200i $$-0.731398\pi$$
−0.664599 + 0.747200i $$0.731398\pi$$
$$608$$ 0 0
$$609$$ 2.64443e6 0.288928
$$610$$ 0 0
$$611$$ −91728.0 −0.00994029
$$612$$ 0 0
$$613$$ 5.81950e6 0.625511 0.312755 0.949834i $$-0.398748\pi$$
0.312755 + 0.949834i $$0.398748\pi$$
$$614$$ 0 0
$$615$$ 1.16592e6 0.124303
$$616$$ 0 0
$$617$$ −4.16589e6 −0.440550 −0.220275 0.975438i $$-0.570695\pi$$
−0.220275 + 0.975438i $$0.570695\pi$$
$$618$$ 0 0
$$619$$ 8.08090e6 0.847683 0.423841 0.905736i $$-0.360681\pi$$
0.423841 + 0.905736i $$0.360681\pi$$
$$620$$ 0 0
$$621$$ −6.75200e6 −0.702592
$$622$$ 0 0
$$623$$ −400134. −0.0413034
$$624$$ 0 0
$$625$$ 8.83812e6 0.905024
$$626$$ 0 0
$$627$$ 6.61504e6 0.671991
$$628$$ 0 0
$$629$$ 1.12571e7 1.13449
$$630$$ 0 0
$$631$$ 8.40878e6 0.840735 0.420368 0.907354i $$-0.361901\pi$$
0.420368 + 0.907354i $$0.361901\pi$$
$$632$$ 0 0
$$633$$ −827552. −0.0820892
$$634$$ 0 0
$$635$$ 966960. 0.0951643
$$636$$ 0 0
$$637$$ −705894. −0.0689272
$$638$$ 0 0
$$639$$ 6.60868e6 0.640269
$$640$$ 0 0
$$641$$ 6.29760e6 0.605383 0.302691 0.953089i $$-0.402115\pi$$
0.302691 + 0.953089i $$0.402115\pi$$
$$642$$ 0 0
$$643$$ −4.39762e6 −0.419460 −0.209730 0.977759i $$-0.567259\pi$$
−0.209730 + 0.977759i $$0.567259\pi$$
$$644$$ 0 0
$$645$$ 648640. 0.0613910
$$646$$ 0 0
$$647$$ −6.55397e6 −0.615522 −0.307761 0.951464i $$-0.599580\pi$$
−0.307761 + 0.951464i $$0.599580\pi$$
$$648$$ 0 0
$$649$$ 9.40304e6 0.876308
$$650$$ 0 0
$$651$$ 3.47155e6 0.321049
$$652$$ 0 0
$$653$$ 3.79652e6 0.348420 0.174210 0.984709i $$-0.444263\pi$$
0.174210 + 0.984709i $$0.444263\pi$$
$$654$$ 0 0
$$655$$ −1.34368e6 −0.122375
$$656$$ 0 0
$$657$$ 1.10475e7 0.998508
$$658$$ 0 0
$$659$$ 8.82684e6 0.791757 0.395879 0.918303i $$-0.370440\pi$$
0.395879 + 0.918303i $$0.370440\pi$$
$$660$$ 0 0
$$661$$ −341270. −0.0303805 −0.0151902 0.999885i $$-0.504835\pi$$
−0.0151902 + 0.999885i $$0.504835\pi$$
$$662$$ 0 0
$$663$$ 2.88355e6 0.254767
$$664$$ 0 0
$$665$$ −1.19168e6 −0.104497
$$666$$ 0 0
$$667$$ 1.34920e7 1.17425
$$668$$ 0 0
$$669$$ 2.45862e6 0.212386
$$670$$ 0 0
$$671$$ 1.16749e7 1.00103
$$672$$ 0 0
$$673$$ 4.41807e6 0.376006 0.188003 0.982168i $$-0.439799\pi$$
0.188003 + 0.982168i $$0.439799\pi$$
$$674$$ 0 0
$$675$$ −1.02124e7 −0.862717
$$676$$ 0 0
$$677$$ 1.63858e7 1.37403 0.687014 0.726644i $$-0.258921\pi$$
0.687014 + 0.726644i $$0.258921\pi$$
$$678$$ 0 0
$$679$$ 1.01185e6 0.0842251
$$680$$ 0 0
$$681$$ −7.13434e6 −0.589503
$$682$$ 0 0
$$683$$ 1.75399e7 1.43872 0.719360 0.694638i $$-0.244435\pi$$
0.719360 + 0.694638i $$0.244435\pi$$
$$684$$ 0 0
$$685$$ −2.94662e6 −0.239937
$$686$$ 0 0
$$687$$ −2.21365e6 −0.178944
$$688$$ 0 0
$$689$$ 4.30240e6 0.345273
$$690$$ 0 0
$$691$$ −3.14638e6 −0.250678 −0.125339 0.992114i $$-0.540002\pi$$
−0.125339 + 0.992114i $$0.540002\pi$$
$$692$$ 0 0
$$693$$ −2.98214e6 −0.235882
$$694$$ 0 0
$$695$$ −3.14944e6 −0.247327
$$696$$ 0 0
$$697$$ −1.78677e7 −1.39312
$$698$$ 0 0
$$699$$ −1.18355e7 −0.916205
$$700$$ 0 0
$$701$$ −1.90919e7 −1.46742 −0.733709 0.679464i $$-0.762212\pi$$
−0.733709 + 0.679464i $$0.762212\pi$$
$$702$$ 0 0
$$703$$ −2.23306e7 −1.70417
$$704$$ 0 0
$$705$$ −24960.0 −0.00189135
$$706$$ 0 0
$$707$$ 9.12625e6 0.686663
$$708$$ 0 0
$$709$$ 990974. 0.0740366 0.0370183 0.999315i $$-0.488214\pi$$
0.0370183 + 0.999315i $$0.488214\pi$$
$$710$$ 0 0
$$711$$ −1.15906e7 −0.859869
$$712$$ 0 0
$$713$$ 1.77120e7 1.30480
$$714$$ 0 0
$$715$$ −999600. −0.0731242
$$716$$ 0 0
$$717$$ 8.00275e6 0.581355
$$718$$ 0 0
$$719$$ 1.69014e7 1.21928 0.609638 0.792680i $$-0.291315\pi$$
0.609638 + 0.792680i $$0.291315\pi$$
$$720$$ 0 0
$$721$$ 2.94314e6 0.210849
$$722$$ 0 0
$$723$$ −1.08666e7 −0.773125
$$724$$ 0 0
$$725$$ 2.04066e7 1.44187
$$726$$ 0 0
$$727$$ 2.34302e7 1.64414 0.822071 0.569384i $$-0.192818\pi$$
0.822071 + 0.569384i $$0.192818\pi$$
$$728$$ 0 0
$$729$$ 3.61141e6 0.251686
$$730$$ 0 0
$$731$$ −9.94041e6 −0.688035
$$732$$ 0 0
$$733$$ 975810. 0.0670819 0.0335409 0.999437i $$-0.489322\pi$$
0.0335409 + 0.999437i $$0.489322\pi$$
$$734$$ 0 0
$$735$$ −192080. −0.0131149
$$736$$ 0 0
$$737$$ −4.18744e6 −0.283975
$$738$$ 0 0
$$739$$ 6.30208e6 0.424495 0.212247 0.977216i $$-0.431922\pi$$
0.212247 + 0.977216i $$0.431922\pi$$
$$740$$ 0 0
$$741$$ −5.72006e6 −0.382697
$$742$$ 0 0
$$743$$ 6.95698e6 0.462326 0.231163 0.972915i $$-0.425747\pi$$
0.231163 + 0.972915i $$0.425747\pi$$
$$744$$ 0 0
$$745$$ 1.13622e6 0.0750018
$$746$$ 0 0
$$747$$ −1.37930e7 −0.904395
$$748$$ 0 0
$$749$$ −2.34671e6 −0.152846
$$750$$ 0 0
$$751$$ −2.74535e7 −1.77622 −0.888112 0.459628i $$-0.847983\pi$$
−0.888112 + 0.459628i $$0.847983\pi$$
$$752$$ 0 0
$$753$$ −1.41926e6 −0.0912170
$$754$$ 0 0
$$755$$ −4.08208e6 −0.260624
$$756$$ 0 0
$$757$$ −1.96889e7 −1.24877 −0.624384 0.781118i $$-0.714650\pi$$
−0.624384 + 0.781118i $$0.714650\pi$$
$$758$$ 0 0
$$759$$ 5.44000e6 0.342763
$$760$$ 0 0
$$761$$ −2.82079e7 −1.76567 −0.882835 0.469684i $$-0.844368\pi$$
−0.882835 + 0.469684i $$0.844368\pi$$
$$762$$ 0 0
$$763$$ 1.08300e6 0.0673467
$$764$$ 0 0
$$765$$ −2.19454e6 −0.135578
$$766$$ 0 0
$$767$$ −8.13086e6 −0.499055
$$768$$ 0 0
$$769$$ −1.38081e6 −0.0842009 −0.0421005 0.999113i $$-0.513405\pi$$
−0.0421005 + 0.999113i $$0.513405\pi$$
$$770$$ 0 0
$$771$$ −2.61326e6 −0.158324
$$772$$ 0 0
$$773$$ −1.54347e7 −0.929074 −0.464537 0.885554i $$-0.653779\pi$$
−0.464537 + 0.885554i $$0.653779\pi$$
$$774$$ 0 0
$$775$$ 2.67894e7 1.60217
$$776$$ 0 0
$$777$$ −3.59934e6 −0.213880
$$778$$ 0 0
$$779$$ 3.54440e7 2.09266
$$780$$ 0 0
$$781$$ −1.25528e7 −0.736399
$$782$$ 0 0
$$783$$ −2.27745e7 −1.32753
$$784$$ 0 0
$$785$$ 2.93546e6 0.170021
$$786$$ 0 0
$$787$$ 7.10107e6 0.408683 0.204342 0.978900i $$-0.434495\pi$$
0.204342 + 0.978900i $$0.434495\pi$$
$$788$$ 0 0
$$789$$ −279360. −0.0159761
$$790$$ 0 0
$$791$$ −1.20076e7 −0.682365
$$792$$ 0 0
$$793$$ −1.00954e7 −0.570085
$$794$$ 0 0
$$795$$ 1.17072e6 0.0656954
$$796$$ 0 0
$$797$$ 6.48182e6 0.361452 0.180726 0.983533i $$-0.442155\pi$$
0.180726 + 0.983533i $$0.442155\pi$$
$$798$$ 0 0
$$799$$ 382512. 0.0211972
$$800$$ 0 0
$$801$$ 1.46171e6 0.0804973
$$802$$ 0 0
$$803$$ −2.09841e7 −1.14842
$$804$$ 0 0
$$805$$ −980000. −0.0533011
$$806$$ 0 0
$$807$$ −5.73166e6 −0.309811
$$808$$ 0 0
$$809$$ 1.60578e7 0.862610 0.431305 0.902206i $$-0.358053\pi$$
0.431305 + 0.902206i $$0.358053\pi$$
$$810$$ 0 0
$$811$$ −4.84775e6 −0.258814 −0.129407 0.991592i $$-0.541307\pi$$
−0.129407 + 0.991592i $$0.541307\pi$$
$$812$$ 0 0
$$813$$ −7.62701e6 −0.404695
$$814$$ 0 0
$$815$$ 3.17116e6 0.167234
$$816$$ 0 0
$$817$$ 1.97187e7 1.03353
$$818$$ 0 0
$$819$$ 2.57867e6 0.134334
$$820$$ 0 0
$$821$$ 2.17976e7 1.12863 0.564314 0.825560i $$-0.309141\pi$$
0.564314 + 0.825560i $$0.309141\pi$$
$$822$$ 0 0
$$823$$ −3.20206e7 −1.64790 −0.823948 0.566665i $$-0.808233\pi$$
−0.823948 + 0.566665i $$0.808233\pi$$
$$824$$ 0 0
$$825$$ 8.22800e6 0.420881
$$826$$ 0 0
$$827$$ −2.19008e7 −1.11352 −0.556758 0.830675i $$-0.687955\pi$$
−0.556758 + 0.830675i $$0.687955\pi$$
$$828$$ 0 0
$$829$$ −1.45999e7 −0.737844 −0.368922 0.929460i $$-0.620273\pi$$
−0.368922 + 0.929460i $$0.620273\pi$$
$$830$$ 0 0
$$831$$ 1.47783e7 0.742374
$$832$$ 0 0
$$833$$ 2.94363e6 0.146984
$$834$$ 0 0
$$835$$ −1.41568e6 −0.0702666
$$836$$ 0 0
$$837$$ −2.98979e7 −1.47512
$$838$$ 0 0
$$839$$ −4.60947e6 −0.226072 −0.113036 0.993591i $$-0.536058\pi$$
−0.113036 + 0.993591i $$0.536058\pi$$
$$840$$ 0 0
$$841$$ 2.49974e7 1.21872
$$842$$ 0 0
$$843$$ 1.59680e7 0.773897
$$844$$ 0 0
$$845$$ −2.84857e6 −0.137241
$$846$$ 0 0
$$847$$ −2.22710e6 −0.106667
$$848$$ 0 0
$$849$$ 1.87270e6 0.0891661
$$850$$ 0 0
$$851$$ −1.83640e7 −0.869247
$$852$$ 0 0
$$853$$ −1.98437e7 −0.933793 −0.466897 0.884312i $$-0.654628\pi$$
−0.466897 + 0.884312i $$0.654628\pi$$
$$854$$ 0 0
$$855$$ 4.35328e6 0.203658
$$856$$ 0 0
$$857$$ −1.22960e6 −0.0571888 −0.0285944 0.999591i $$-0.509103\pi$$
−0.0285944 + 0.999591i $$0.509103\pi$$
$$858$$ 0 0
$$859$$ −3.33041e7 −1.53998 −0.769989 0.638058i $$-0.779738\pi$$
−0.769989 + 0.638058i $$0.779738\pi$$
$$860$$ 0 0
$$861$$ 5.71301e6 0.262638
$$862$$ 0 0
$$863$$ 2.36616e7 1.08148 0.540738 0.841191i $$-0.318145\pi$$
0.540738 + 0.841191i $$0.318145\pi$$
$$864$$ 0 0
$$865$$ −712220. −0.0323649
$$866$$ 0 0
$$867$$ −665752. −0.0300791
$$868$$ 0 0
$$869$$ 2.20157e7 0.988969
$$870$$ 0 0
$$871$$ 3.62090e6 0.161723
$$872$$ 0 0
$$873$$ −3.69635e6 −0.164149
$$874$$ 0 0
$$875$$ −3.01350e6 −0.133061
$$876$$ 0 0
$$877$$ −2.37812e7 −1.04408 −0.522042 0.852920i $$-0.674830\pi$$
−0.522042 + 0.852920i $$0.674830\pi$$
$$878$$ 0 0
$$879$$ 2.00064e7 0.873369
$$880$$ 0 0
$$881$$ −1.41871e7 −0.615818 −0.307909 0.951416i $$-0.599629\pi$$
−0.307909 + 0.951416i $$0.599629\pi$$
$$882$$ 0 0
$$883$$ −2.09281e7 −0.903293 −0.451647 0.892197i $$-0.649163\pi$$
−0.451647 + 0.892197i $$0.649163\pi$$
$$884$$ 0 0
$$885$$ −2.21248e6 −0.0949557
$$886$$ 0 0
$$887$$ 7.98586e6 0.340810 0.170405 0.985374i $$-0.445492\pi$$
0.170405 + 0.985374i $$0.445492\pi$$
$$888$$ 0 0
$$889$$ 4.73810e6 0.201071
$$890$$ 0 0
$$891$$ 5.60626e6 0.236581
$$892$$ 0 0
$$893$$ −758784. −0.0318412
$$894$$ 0 0
$$895$$ −4.85628e6 −0.202650
$$896$$ 0 0
$$897$$ −4.70400e6 −0.195203
$$898$$ 0 0
$$899$$ 5.97426e7 2.46538
$$900$$ 0 0
$$901$$ −1.79413e7 −0.736278
$$902$$ 0 0
$$903$$ 3.17834e6 0.129712
$$904$$ 0 0
$$905$$ 6.57090e6 0.266688
$$906$$ 0 0
$$907$$ 2.31861e7 0.935856 0.467928 0.883767i $$-0.345001\pi$$
0.467928 + 0.883767i $$0.345001\pi$$
$$908$$ 0 0
$$909$$ −3.33388e7 −1.33826
$$910$$ 0 0
$$911$$ −1.65299e7 −0.659895 −0.329948 0.943999i $$-0.607031\pi$$
−0.329948 + 0.943999i $$0.607031\pi$$
$$912$$ 0 0
$$913$$ 2.61990e7 1.04018
$$914$$ 0 0
$$915$$ −2.74704e6 −0.108471
$$916$$ 0 0
$$917$$ −6.58403e6 −0.258564
$$918$$ 0 0
$$919$$ −1.28087e7 −0.500283 −0.250142 0.968209i $$-0.580477\pi$$
−0.250142 + 0.968209i $$0.580477\pi$$
$$920$$ 0 0
$$921$$ 1.87363e7 0.727836
$$922$$ 0 0
$$923$$ 1.08545e7 0.419377
$$924$$ 0 0
$$925$$ −2.77756e7 −1.06735
$$926$$ 0 0
$$927$$ −1.07515e7 −0.410930
$$928$$ 0 0
$$929$$ 2.97319e7 1.13027 0.565136 0.824998i $$-0.308824\pi$$
0.565136 + 0.824998i $$0.308824\pi$$
$$930$$ 0 0
$$931$$ −5.83923e6 −0.220791
$$932$$ 0 0
$$933$$ −1.30451e6 −0.0490619
$$934$$ 0 0
$$935$$ 4.16840e6 0.155934
$$936$$ 0 0
$$937$$ 1.10970e7 0.412911 0.206456 0.978456i $$-0.433807\pi$$
0.206456 + 0.978456i $$0.433807\pi$$
$$938$$ 0 0
$$939$$ −1.39172e7 −0.515096
$$940$$ 0 0
$$941$$ 3.74313e7 1.37804 0.689019 0.724743i $$-0.258042\pi$$
0.689019 + 0.724743i $$0.258042\pi$$
$$942$$ 0 0
$$943$$ 2.91480e7 1.06741
$$944$$ 0 0
$$945$$ 1.65424e6 0.0602586
$$946$$ 0 0
$$947$$ −1.50907e7 −0.546808 −0.273404 0.961899i $$-0.588150\pi$$
−0.273404 + 0.961899i $$0.588150\pi$$
$$948$$ 0 0
$$949$$ 1.81451e7 0.654024
$$950$$ 0 0
$$951$$ 1.43817e7 0.515655
$$952$$ 0 0
$$953$$ −2.15741e7 −0.769484 −0.384742 0.923024i $$-0.625710\pi$$
−0.384742 + 0.923024i $$0.625710\pi$$
$$954$$ 0 0
$$955$$ −683040. −0.0242347
$$956$$ 0 0
$$957$$ 1.83491e7 0.647643
$$958$$ 0 0
$$959$$ −1.44384e7 −0.506960
$$960$$ 0 0
$$961$$ 4.97996e7 1.73947
$$962$$ 0 0
$$963$$ 8.57267e6 0.297886
$$964$$ 0 0
$$965$$ 3.52754e6 0.121942
$$966$$ 0 0
$$967$$ 3.29467e7 1.13304 0.566520 0.824048i $$-0.308289\pi$$
0.566520 + 0.824048i $$0.308289\pi$$
$$968$$ 0 0
$$969$$ 2.38531e7 0.816083
$$970$$ 0 0
$$971$$ −2.24599e7 −0.764470 −0.382235 0.924065i $$-0.624846\pi$$
−0.382235 + 0.924065i $$0.624846\pi$$
$$972$$ 0 0
$$973$$ −1.54323e7 −0.522573
$$974$$ 0 0
$$975$$ −7.11480e6 −0.239691
$$976$$ 0 0
$$977$$ −5.16236e7 −1.73026 −0.865132 0.501545i $$-0.832765\pi$$
−0.865132 + 0.501545i $$0.832765\pi$$
$$978$$ 0 0
$$979$$ −2.77644e6 −0.0925831
$$980$$ 0 0
$$981$$ −3.95626e6 −0.131254
$$982$$ 0 0
$$983$$ 1.10202e7 0.363751 0.181876 0.983322i $$-0.441783\pi$$
0.181876 + 0.983322i $$0.441783\pi$$
$$984$$ 0 0
$$985$$ 1.96982e6 0.0646898
$$986$$ 0 0
$$987$$ −122304. −0.00399621
$$988$$ 0 0
$$989$$ 1.62160e7 0.527173
$$990$$ 0 0
$$991$$ −3.21029e7 −1.03839 −0.519194 0.854656i $$-0.673768\pi$$
−0.519194 + 0.854656i $$0.673768\pi$$
$$992$$ 0 0
$$993$$ −1.98033e7 −0.637330
$$994$$ 0 0
$$995$$ 1.10392e7 0.353492
$$996$$ 0 0
$$997$$ 2.81772e7 0.897759 0.448879 0.893592i $$-0.351823\pi$$
0.448879 + 0.893592i $$0.351823\pi$$
$$998$$ 0 0
$$999$$ 3.09984e7 0.982711
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.d.1.1 1
3.2 odd 2 1008.6.a.n.1.1 1
4.3 odd 2 14.6.a.b.1.1 1
7.6 odd 2 784.6.a.h.1.1 1
8.3 odd 2 448.6.a.f.1.1 1
8.5 even 2 448.6.a.k.1.1 1
12.11 even 2 126.6.a.c.1.1 1
20.3 even 4 350.6.c.f.99.1 2
20.7 even 4 350.6.c.f.99.2 2
20.19 odd 2 350.6.a.b.1.1 1
28.3 even 6 98.6.c.b.79.1 2
28.11 odd 6 98.6.c.a.79.1 2
28.19 even 6 98.6.c.b.67.1 2
28.23 odd 6 98.6.c.a.67.1 2
28.27 even 2 98.6.a.b.1.1 1
84.83 odd 2 882.6.a.g.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
14.6.a.b.1.1 1 4.3 odd 2
98.6.a.b.1.1 1 28.27 even 2
98.6.c.a.67.1 2 28.23 odd 6
98.6.c.a.79.1 2 28.11 odd 6
98.6.c.b.67.1 2 28.19 even 6
98.6.c.b.79.1 2 28.3 even 6
112.6.a.d.1.1 1 1.1 even 1 trivial
126.6.a.c.1.1 1 12.11 even 2
350.6.a.b.1.1 1 20.19 odd 2
350.6.c.f.99.1 2 20.3 even 4
350.6.c.f.99.2 2 20.7 even 4
448.6.a.f.1.1 1 8.3 odd 2
448.6.a.k.1.1 1 8.5 even 2
784.6.a.h.1.1 1 7.6 odd 2
882.6.a.g.1.1 1 84.83 odd 2
1008.6.a.n.1.1 1 3.2 odd 2