# Properties

 Label 252.6.a.g.1.2 Level $252$ Weight $6$ Character 252.1 Self dual yes Analytic conductor $40.417$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,6,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, 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("252.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 252.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: $$40.4167225929$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{91})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 91$$ x^2 - 91 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$9.53939$$ of defining polynomial Character $$\chi$$ $$=$$ 252.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+57.2364 q^{5} +49.0000 q^{7} +O(q^{10})$$ $$q+57.2364 q^{5} +49.0000 q^{7} -515.127 q^{11} -670.000 q^{13} -973.018 q^{17} +284.000 q^{19} -1774.33 q^{23} +151.000 q^{25} +6868.36 q^{29} +1532.00 q^{31} +2804.58 q^{35} -15118.0 q^{37} +5322.98 q^{41} -10996.0 q^{43} -19345.9 q^{47} +2401.00 q^{49} +23466.9 q^{53} -29484.0 q^{55} -23695.8 q^{59} -14602.0 q^{61} -38348.4 q^{65} -36628.0 q^{67} +67939.5 q^{71} -54802.0 q^{73} -25241.2 q^{77} -31768.0 q^{79} -74178.3 q^{83} -55692.0 q^{85} -858.545 q^{89} -32830.0 q^{91} +16255.1 q^{95} +14126.0 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 98 q^{7}+O(q^{10})$$ 2 * q + 98 * q^7 $$2 q + 98 q^{7} - 1340 q^{13} + 568 q^{19} + 302 q^{25} + 3064 q^{31} - 30236 q^{37} - 21992 q^{43} + 4802 q^{49} - 58968 q^{55} - 29204 q^{61} - 73256 q^{67} - 109604 q^{73} - 63536 q^{79} - 111384 q^{85} - 65660 q^{91} + 28252 q^{97}+O(q^{100})$$ 2 * q + 98 * q^7 - 1340 * q^13 + 568 * q^19 + 302 * q^25 + 3064 * q^31 - 30236 * q^37 - 21992 * q^43 + 4802 * q^49 - 58968 * q^55 - 29204 * q^61 - 73256 * q^67 - 109604 * q^73 - 63536 * q^79 - 111384 * q^85 - 65660 * q^91 + 28252 * q^97

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 0 0
$$4$$ 0 0
$$5$$ 57.2364 1.02387 0.511937 0.859023i $$-0.328928\pi$$
0.511937 + 0.859023i $$0.328928\pi$$
$$6$$ 0 0
$$7$$ 49.0000 0.377964
$$8$$ 0 0
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −515.127 −1.28361 −0.641804 0.766868i $$-0.721814\pi$$
−0.641804 + 0.766868i $$0.721814\pi$$
$$12$$ 0 0
$$13$$ −670.000 −1.09955 −0.549777 0.835312i $$-0.685287\pi$$
−0.549777 + 0.835312i $$0.685287\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −973.018 −0.816580 −0.408290 0.912852i $$-0.633875\pi$$
−0.408290 + 0.912852i $$0.633875\pi$$
$$18$$ 0 0
$$19$$ 284.000 0.180482 0.0902411 0.995920i $$-0.471236\pi$$
0.0902411 + 0.995920i $$0.471236\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −1774.33 −0.699381 −0.349691 0.936865i $$-0.613713\pi$$
−0.349691 + 0.936865i $$0.613713\pi$$
$$24$$ 0 0
$$25$$ 151.000 0.0483200
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 6868.36 1.51656 0.758278 0.651932i $$-0.226041\pi$$
0.758278 + 0.651932i $$0.226041\pi$$
$$30$$ 0 0
$$31$$ 1532.00 0.286322 0.143161 0.989699i $$-0.454273\pi$$
0.143161 + 0.989699i $$0.454273\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 2804.58 0.386988
$$36$$ 0 0
$$37$$ −15118.0 −1.81547 −0.907737 0.419540i $$-0.862192\pi$$
−0.907737 + 0.419540i $$0.862192\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 5322.98 0.494533 0.247266 0.968948i $$-0.420468\pi$$
0.247266 + 0.968948i $$0.420468\pi$$
$$42$$ 0 0
$$43$$ −10996.0 −0.906909 −0.453454 0.891279i $$-0.649809\pi$$
−0.453454 + 0.891279i $$0.649809\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −19345.9 −1.27745 −0.638725 0.769435i $$-0.720538\pi$$
−0.638725 + 0.769435i $$0.720538\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 0 0
$$53$$ 23466.9 1.14754 0.573768 0.819018i $$-0.305481\pi$$
0.573768 + 0.819018i $$0.305481\pi$$
$$54$$ 0 0
$$55$$ −29484.0 −1.31426
$$56$$ 0 0
$$57$$ 0 0
$$58$$ 0 0
$$59$$ −23695.8 −0.886221 −0.443111 0.896467i $$-0.646125\pi$$
−0.443111 + 0.896467i $$0.646125\pi$$
$$60$$ 0 0
$$61$$ −14602.0 −0.502444 −0.251222 0.967929i $$-0.580832\pi$$
−0.251222 + 0.967929i $$0.580832\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −38348.4 −1.12581
$$66$$ 0 0
$$67$$ −36628.0 −0.996842 −0.498421 0.866935i $$-0.666087\pi$$
−0.498421 + 0.866935i $$0.666087\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 67939.5 1.59947 0.799736 0.600351i $$-0.204973\pi$$
0.799736 + 0.600351i $$0.204973\pi$$
$$72$$ 0 0
$$73$$ −54802.0 −1.20362 −0.601810 0.798639i $$-0.705553\pi$$
−0.601810 + 0.798639i $$0.705553\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −25241.2 −0.485159
$$78$$ 0 0
$$79$$ −31768.0 −0.572693 −0.286347 0.958126i $$-0.592441\pi$$
−0.286347 + 0.958126i $$0.592441\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 0 0
$$83$$ −74178.3 −1.18190 −0.590951 0.806707i $$-0.701247\pi$$
−0.590951 + 0.806707i $$0.701247\pi$$
$$84$$ 0 0
$$85$$ −55692.0 −0.836076
$$86$$ 0 0
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −858.545 −0.0114892 −0.00574458 0.999983i $$-0.501829\pi$$
−0.00574458 + 0.999983i $$0.501829\pi$$
$$90$$ 0 0
$$91$$ −32830.0 −0.415592
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 16255.1 0.184791
$$96$$ 0 0
$$97$$ 14126.0 0.152437 0.0762184 0.997091i $$-0.475715\pi$$
0.0762184 + 0.997091i $$0.475715\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 177604. 1.73241 0.866204 0.499690i $$-0.166553\pi$$
0.866204 + 0.499690i $$0.166553\pi$$
$$102$$ 0 0
$$103$$ −128356. −1.19213 −0.596064 0.802937i $$-0.703270\pi$$
−0.596064 + 0.802937i $$0.703270\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 74350.0 0.627800 0.313900 0.949456i $$-0.398364\pi$$
0.313900 + 0.949456i $$0.398364\pi$$
$$108$$ 0 0
$$109$$ −103918. −0.837769 −0.418885 0.908039i $$-0.637579\pi$$
−0.418885 + 0.908039i $$0.637579\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 96157.1 0.708411 0.354205 0.935168i $$-0.384751\pi$$
0.354205 + 0.935168i $$0.384751\pi$$
$$114$$ 0 0
$$115$$ −101556. −0.716079
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ −47677.9 −0.308638
$$120$$ 0 0
$$121$$ 104305. 0.647652
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 0 0
$$125$$ −170221. −0.974401
$$126$$ 0 0
$$127$$ 127112. 0.699322 0.349661 0.936876i $$-0.386297\pi$$
0.349661 + 0.936876i $$0.386297\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −328537. −1.67265 −0.836326 0.548232i $$-0.815301\pi$$
−0.836326 + 0.548232i $$0.815301\pi$$
$$132$$ 0 0
$$133$$ 13916.0 0.0682159
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −176631. −0.804019 −0.402010 0.915635i $$-0.631688\pi$$
−0.402010 + 0.915635i $$0.631688\pi$$
$$138$$ 0 0
$$139$$ −55792.0 −0.244926 −0.122463 0.992473i $$-0.539079\pi$$
−0.122463 + 0.992473i $$0.539079\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 345135. 1.41140
$$144$$ 0 0
$$145$$ 393120. 1.55276
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ −261570. −0.965211 −0.482606 0.875838i $$-0.660310\pi$$
−0.482606 + 0.875838i $$0.660310\pi$$
$$150$$ 0 0
$$151$$ 149288. 0.532822 0.266411 0.963859i $$-0.414162\pi$$
0.266411 + 0.963859i $$0.414162\pi$$
$$152$$ 0 0
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 87686.1 0.293158
$$156$$ 0 0
$$157$$ 299222. 0.968823 0.484411 0.874840i $$-0.339034\pi$$
0.484411 + 0.874840i $$0.339034\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −86942.0 −0.264341
$$162$$ 0 0
$$163$$ −336220. −0.991185 −0.495592 0.868555i $$-0.665049\pi$$
−0.495592 + 0.868555i $$0.665049\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 262257. 0.727672 0.363836 0.931463i $$-0.381467\pi$$
0.363836 + 0.931463i $$0.381467\pi$$
$$168$$ 0 0
$$169$$ 77607.0 0.209018
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 0 0
$$173$$ 541857. 1.37648 0.688239 0.725484i $$-0.258384\pi$$
0.688239 + 0.725484i $$0.258384\pi$$
$$174$$ 0 0
$$175$$ 7399.00 0.0182632
$$176$$ 0 0
$$177$$ 0 0
$$178$$ 0 0
$$179$$ 296542. 0.691756 0.345878 0.938279i $$-0.387581\pi$$
0.345878 + 0.938279i $$0.387581\pi$$
$$180$$ 0 0
$$181$$ 587522. 1.33299 0.666496 0.745508i $$-0.267793\pi$$
0.666496 + 0.745508i $$0.267793\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ −865299. −1.85882
$$186$$ 0 0
$$187$$ 501228. 1.04817
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 188136. 0.373154 0.186577 0.982440i $$-0.440261\pi$$
0.186577 + 0.982440i $$0.440261\pi$$
$$192$$ 0 0
$$193$$ 403022. 0.778817 0.389409 0.921065i $$-0.372679\pi$$
0.389409 + 0.921065i $$0.372679\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 254702. 0.467591 0.233796 0.972286i $$-0.424885\pi$$
0.233796 + 0.972286i $$0.424885\pi$$
$$198$$ 0 0
$$199$$ 353360. 0.632535 0.316268 0.948670i $$-0.397570\pi$$
0.316268 + 0.948670i $$0.397570\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ 336550. 0.573204
$$204$$ 0 0
$$205$$ 304668. 0.506340
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −146296. −0.231669
$$210$$ 0 0
$$211$$ 616292. 0.952973 0.476486 0.879182i $$-0.341910\pi$$
0.476486 + 0.879182i $$0.341910\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −629371. −0.928561
$$216$$ 0 0
$$217$$ 75068.0 0.108219
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 651922. 0.897873
$$222$$ 0 0
$$223$$ −328216. −0.441975 −0.220987 0.975277i $$-0.570928\pi$$
−0.220987 + 0.975277i $$0.570928\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −550270. −0.708780 −0.354390 0.935098i $$-0.615311\pi$$
−0.354390 + 0.935098i $$0.615311\pi$$
$$228$$ 0 0
$$229$$ 1.24299e6 1.56631 0.783155 0.621827i $$-0.213609\pi$$
0.783155 + 0.621827i $$0.213609\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 172968. 0.208726 0.104363 0.994539i $$-0.466720\pi$$
0.104363 + 0.994539i $$0.466720\pi$$
$$234$$ 0 0
$$235$$ −1.10729e6 −1.30795
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ −1.35278e6 −1.53191 −0.765954 0.642895i $$-0.777733\pi$$
−0.765954 + 0.642895i $$0.777733\pi$$
$$240$$ 0 0
$$241$$ −605290. −0.671307 −0.335653 0.941986i $$-0.608957\pi$$
−0.335653 + 0.941986i $$0.608957\pi$$
$$242$$ 0 0
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 137424. 0.146268
$$246$$ 0 0
$$247$$ −190280. −0.198450
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 606820. 0.607961 0.303980 0.952678i $$-0.401684\pi$$
0.303980 + 0.952678i $$0.401684\pi$$
$$252$$ 0 0
$$253$$ 914004. 0.897732
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 1.29314e6 1.22127 0.610637 0.791911i $$-0.290914\pi$$
0.610637 + 0.791911i $$0.290914\pi$$
$$258$$ 0 0
$$259$$ −740782. −0.686185
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 0 0
$$263$$ −1.53456e6 −1.36803 −0.684015 0.729468i $$-0.739768\pi$$
−0.684015 + 0.729468i $$0.739768\pi$$
$$264$$ 0 0
$$265$$ 1.34316e6 1.17493
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 0 0
$$269$$ −430131. −0.362427 −0.181213 0.983444i $$-0.558002\pi$$
−0.181213 + 0.983444i $$0.558002\pi$$
$$270$$ 0 0
$$271$$ −769804. −0.636732 −0.318366 0.947968i $$-0.603134\pi$$
−0.318366 + 0.947968i $$0.603134\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −77784.2 −0.0620240
$$276$$ 0 0
$$277$$ −391066. −0.306232 −0.153116 0.988208i $$-0.548931\pi$$
−0.153116 + 0.988208i $$0.548931\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1.18559e6 −0.895716 −0.447858 0.894105i $$-0.647813\pi$$
−0.447858 + 0.894105i $$0.647813\pi$$
$$282$$ 0 0
$$283$$ −2.02464e6 −1.50274 −0.751368 0.659884i $$-0.770606\pi$$
−0.751368 + 0.659884i $$0.770606\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 260826. 0.186916
$$288$$ 0 0
$$289$$ −473093. −0.333198
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 0 0
$$293$$ 2.16874e6 1.47584 0.737919 0.674889i $$-0.235808\pi$$
0.737919 + 0.674889i $$0.235808\pi$$
$$294$$ 0 0
$$295$$ −1.35626e6 −0.907380
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 1.18880e6 0.769007
$$300$$ 0 0
$$301$$ −538804. −0.342779
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −835765. −0.514440
$$306$$ 0 0
$$307$$ −743164. −0.450027 −0.225014 0.974356i $$-0.572243\pi$$
−0.225014 + 0.974356i $$0.572243\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 2.81546e6 1.65062 0.825311 0.564678i $$-0.191000\pi$$
0.825311 + 0.564678i $$0.191000\pi$$
$$312$$ 0 0
$$313$$ 1.86240e6 1.07452 0.537258 0.843418i $$-0.319460\pi$$
0.537258 + 0.843418i $$0.319460\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 1.64234e6 0.917941 0.458971 0.888451i $$-0.348218\pi$$
0.458971 + 0.888451i $$0.348218\pi$$
$$318$$ 0 0
$$319$$ −3.53808e6 −1.94666
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −276337. −0.147378
$$324$$ 0 0
$$325$$ −101170. −0.0531304
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −947948. −0.482831
$$330$$ 0 0
$$331$$ −3.80969e6 −1.91126 −0.955630 0.294569i $$-0.904824\pi$$
−0.955630 + 0.294569i $$0.904824\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 0 0
$$335$$ −2.09645e6 −1.02064
$$336$$ 0 0
$$337$$ 2.19707e6 1.05383 0.526913 0.849919i $$-0.323349\pi$$
0.526913 + 0.849919i $$0.323349\pi$$
$$338$$ 0 0
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −789175. −0.367525
$$342$$ 0 0
$$343$$ 117649. 0.0539949
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −1.02493e6 −0.456953 −0.228476 0.973549i $$-0.573374\pi$$
−0.228476 + 0.973549i $$0.573374\pi$$
$$348$$ 0 0
$$349$$ −2.16719e6 −0.952429 −0.476215 0.879329i $$-0.657991\pi$$
−0.476215 + 0.879329i $$0.657991\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −582494. −0.248803 −0.124401 0.992232i $$-0.539701\pi$$
−0.124401 + 0.992232i $$0.539701\pi$$
$$354$$ 0 0
$$355$$ 3.88861e6 1.63766
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 203704. 0.0834188 0.0417094 0.999130i $$-0.486720\pi$$
0.0417094 + 0.999130i $$0.486720\pi$$
$$360$$ 0 0
$$361$$ −2.39544e6 −0.967426
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −3.13667e6 −1.23236
$$366$$ 0 0
$$367$$ −4.08011e6 −1.58127 −0.790637 0.612286i $$-0.790250\pi$$
−0.790637 + 0.612286i $$0.790250\pi$$
$$368$$ 0 0
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 1.14988e6 0.433728
$$372$$ 0 0
$$373$$ −3.40643e6 −1.26773 −0.633865 0.773444i $$-0.718533\pi$$
−0.633865 + 0.773444i $$0.718533\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.60180e6 −1.66753
$$378$$ 0 0
$$379$$ −1.35617e6 −0.484972 −0.242486 0.970155i $$-0.577963\pi$$
−0.242486 + 0.970155i $$0.577963\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −1.20219e6 −0.418771 −0.209386 0.977833i $$-0.567146\pi$$
−0.209386 + 0.977833i $$0.567146\pi$$
$$384$$ 0 0
$$385$$ −1.44472e6 −0.496742
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 5.21103e6 1.74602 0.873010 0.487702i $$-0.162165\pi$$
0.873010 + 0.487702i $$0.162165\pi$$
$$390$$ 0 0
$$391$$ 1.72645e6 0.571101
$$392$$ 0 0
$$393$$ 0 0
$$394$$ 0 0
$$395$$ −1.81828e6 −0.586366
$$396$$ 0 0
$$397$$ 4.05383e6 1.29089 0.645445 0.763807i $$-0.276672\pi$$
0.645445 + 0.763807i $$0.276672\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −468079. −0.145364 −0.0726822 0.997355i $$-0.523156\pi$$
−0.0726822 + 0.997355i $$0.523156\pi$$
$$402$$ 0 0
$$403$$ −1.02644e6 −0.314826
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 7.78769e6 2.33036
$$408$$ 0 0
$$409$$ 3.46254e6 1.02350 0.511749 0.859135i $$-0.328998\pi$$
0.511749 + 0.859135i $$0.328998\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −1.16110e6 −0.334960
$$414$$ 0 0
$$415$$ −4.24570e6 −1.21012
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 0 0
$$419$$ 6.91747e6 1.92492 0.962459 0.271427i $$-0.0874954\pi$$
0.962459 + 0.271427i $$0.0874954\pi$$
$$420$$ 0 0
$$421$$ 3.57661e6 0.983483 0.491741 0.870741i $$-0.336361\pi$$
0.491741 + 0.870741i $$0.336361\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −146926. −0.0394571
$$426$$ 0 0
$$427$$ −715498. −0.189906
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −1.99921e6 −0.518400 −0.259200 0.965824i $$-0.583459\pi$$
−0.259200 + 0.965824i $$0.583459\pi$$
$$432$$ 0 0
$$433$$ 4.79321e6 1.22859 0.614295 0.789077i $$-0.289441\pi$$
0.614295 + 0.789077i $$0.289441\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −503909. −0.126226
$$438$$ 0 0
$$439$$ 6.94501e6 1.71993 0.859966 0.510351i $$-0.170484\pi$$
0.859966 + 0.510351i $$0.170484\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −7.09496e6 −1.71767 −0.858837 0.512249i $$-0.828812\pi$$
−0.858837 + 0.512249i $$0.828812\pi$$
$$444$$ 0 0
$$445$$ −49140.0 −0.0117635
$$446$$ 0 0
$$447$$ 0 0
$$448$$ 0 0
$$449$$ 6.71016e6 1.57079 0.785393 0.618997i $$-0.212461\pi$$
0.785393 + 0.618997i $$0.212461\pi$$
$$450$$ 0 0
$$451$$ −2.74201e6 −0.634787
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 0 0
$$455$$ −1.87907e6 −0.425514
$$456$$ 0 0
$$457$$ 164054. 0.0367448 0.0183724 0.999831i $$-0.494152\pi$$
0.0183724 + 0.999831i $$0.494152\pi$$
$$458$$ 0 0
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −5.64173e6 −1.23640 −0.618201 0.786020i $$-0.712138\pi$$
−0.618201 + 0.786020i $$0.712138\pi$$
$$462$$ 0 0
$$463$$ 7.50483e6 1.62700 0.813502 0.581562i $$-0.197558\pi$$
0.813502 + 0.581562i $$0.197558\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −2.83446e6 −0.601420 −0.300710 0.953716i $$-0.597224\pi$$
−0.300710 + 0.953716i $$0.597224\pi$$
$$468$$ 0 0
$$469$$ −1.79477e6 −0.376771
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0 0
$$473$$ 5.66434e6 1.16412
$$474$$ 0 0
$$475$$ 42884.0 0.00872090
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 0 0
$$479$$ 1.12012e6 0.223061 0.111531 0.993761i $$-0.464425\pi$$
0.111531 + 0.993761i $$0.464425\pi$$
$$480$$ 0 0
$$481$$ 1.01291e7 1.99621
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ 808521. 0.156076
$$486$$ 0 0
$$487$$ −9.75086e6 −1.86303 −0.931516 0.363700i $$-0.881513\pi$$
−0.931516 + 0.363700i $$0.881513\pi$$
$$488$$ 0 0
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 2.08277e6 0.389887 0.194943 0.980815i $$-0.437548\pi$$
0.194943 + 0.980815i $$0.437548\pi$$
$$492$$ 0 0
$$493$$ −6.68304e6 −1.23839
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 3.32904e6 0.604544
$$498$$ 0 0
$$499$$ 7.75963e6 1.39505 0.697525 0.716561i $$-0.254285\pi$$
0.697525 + 0.716561i $$0.254285\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ 3.56697e6 0.628607 0.314304 0.949323i $$-0.398229\pi$$
0.314304 + 0.949323i $$0.398229\pi$$
$$504$$ 0 0
$$505$$ 1.01654e7 1.77377
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 0 0
$$509$$ −8.86105e6 −1.51597 −0.757985 0.652272i $$-0.773816\pi$$
−0.757985 + 0.652272i $$0.773816\pi$$
$$510$$ 0 0
$$511$$ −2.68530e6 −0.454925
$$512$$ 0 0
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −7.34663e6 −1.22059
$$516$$ 0 0
$$517$$ 9.96559e6 1.63975
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 1.43039e6 0.230867 0.115433 0.993315i $$-0.463174\pi$$
0.115433 + 0.993315i $$0.463174\pi$$
$$522$$ 0 0
$$523$$ 1.08653e7 1.73694 0.868472 0.495737i $$-0.165102\pi$$
0.868472 + 0.495737i $$0.165102\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −1.49066e6 −0.233805
$$528$$ 0 0
$$529$$ −3.28811e6 −0.510866
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −3.56640e6 −0.543766
$$534$$ 0 0
$$535$$ 4.25552e6 0.642789
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ −1.23682e6 −0.183373
$$540$$ 0 0
$$541$$ −6.46621e6 −0.949854 −0.474927 0.880025i $$-0.657525\pi$$
−0.474927 + 0.880025i $$0.657525\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ −5.94789e6 −0.857771
$$546$$ 0 0
$$547$$ 153932. 0.0219969 0.0109984 0.999940i $$-0.496499\pi$$
0.0109984 + 0.999940i $$0.496499\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 1.95061e6 0.273711
$$552$$ 0 0
$$553$$ −1.55663e6 −0.216458
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 3.26625e6 0.446079 0.223039 0.974809i $$-0.428402\pi$$
0.223039 + 0.974809i $$0.428402\pi$$
$$558$$ 0 0
$$559$$ 7.36732e6 0.997195
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 6.21209e6 0.825975 0.412987 0.910737i $$-0.364485\pi$$
0.412987 + 0.910737i $$0.364485\pi$$
$$564$$ 0 0
$$565$$ 5.50368e6 0.725324
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 6.00375e6 0.777395 0.388698 0.921365i $$-0.372925\pi$$
0.388698 + 0.921365i $$0.372925\pi$$
$$570$$ 0 0
$$571$$ 5.08418e6 0.652575 0.326288 0.945271i $$-0.394202\pi$$
0.326288 + 0.945271i $$0.394202\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −267923. −0.0337941
$$576$$ 0 0
$$577$$ −4.96896e6 −0.621335 −0.310667 0.950519i $$-0.600553\pi$$
−0.310667 + 0.950519i $$0.600553\pi$$
$$578$$ 0 0
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3.63474e6 −0.446717
$$582$$ 0 0
$$583$$ −1.20884e7 −1.47299
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −9.18769e6 −1.10055 −0.550277 0.834982i $$-0.685478\pi$$
−0.550277 + 0.834982i $$0.685478\pi$$
$$588$$ 0 0
$$589$$ 435088. 0.0516760
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 1.35521e7 1.58260 0.791300 0.611428i $$-0.209405\pi$$
0.791300 + 0.611428i $$0.209405\pi$$
$$594$$ 0 0
$$595$$ −2.72891e6 −0.316007
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ −4.97161e6 −0.566148 −0.283074 0.959098i $$-0.591354\pi$$
−0.283074 + 0.959098i $$0.591354\pi$$
$$600$$ 0 0
$$601$$ −5.49055e6 −0.620054 −0.310027 0.950728i $$-0.600338\pi$$
−0.310027 + 0.950728i $$0.600338\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 0 0
$$605$$ 5.97004e6 0.663115
$$606$$ 0 0
$$607$$ 1.76305e7 1.94220 0.971098 0.238682i $$-0.0767152\pi$$
0.971098 + 0.238682i $$0.0767152\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 1.29617e7 1.40463
$$612$$ 0 0
$$613$$ −1.47183e7 −1.58200 −0.791002 0.611813i $$-0.790441\pi$$
−0.791002 + 0.611813i $$0.790441\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −7.00355e6 −0.740637 −0.370319 0.928905i $$-0.620752\pi$$
−0.370319 + 0.928905i $$0.620752\pi$$
$$618$$ 0 0
$$619$$ −6.85036e6 −0.718599 −0.359300 0.933222i $$-0.616984\pi$$
−0.359300 + 0.933222i $$0.616984\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −42068.7 −0.00434249
$$624$$ 0 0
$$625$$ −1.02147e7 −1.04599
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 1.47101e7 1.48248
$$630$$ 0 0
$$631$$ 1.36802e7 1.36779 0.683894 0.729581i $$-0.260285\pi$$
0.683894 + 0.729581i $$0.260285\pi$$
$$632$$ 0 0
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 7.27543e6 0.716018
$$636$$ 0 0
$$637$$ −1.60867e6 −0.157079
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 1.90471e6 0.183098 0.0915491 0.995801i $$-0.470818\pi$$
0.0915491 + 0.995801i $$0.470818\pi$$
$$642$$ 0 0
$$643$$ −5.19569e6 −0.495583 −0.247791 0.968813i $$-0.579705\pi$$
−0.247791 + 0.968813i $$0.579705\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −9.37749e6 −0.880696 −0.440348 0.897827i $$-0.645145\pi$$
−0.440348 + 0.897827i $$0.645145\pi$$
$$648$$ 0 0
$$649$$ 1.22064e7 1.13756
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −1.31149e7 −1.20360 −0.601800 0.798647i $$-0.705550\pi$$
−0.601800 + 0.798647i $$0.705550\pi$$
$$654$$ 0 0
$$655$$ −1.88042e7 −1.71259
$$656$$ 0 0
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −1.62527e7 −1.45784 −0.728922 0.684597i $$-0.759978\pi$$
−0.728922 + 0.684597i $$0.759978\pi$$
$$660$$ 0 0
$$661$$ −4.31300e6 −0.383951 −0.191976 0.981400i $$-0.561489\pi$$
−0.191976 + 0.981400i $$0.561489\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 796501. 0.0698445
$$666$$ 0 0
$$667$$ −1.21867e7 −1.06065
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 7.52189e6 0.644942
$$672$$ 0 0
$$673$$ −1.61027e7 −1.37044 −0.685220 0.728336i $$-0.740294\pi$$
−0.685220 + 0.728336i $$0.740294\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 4.02526e6 0.337538 0.168769 0.985656i $$-0.446021\pi$$
0.168769 + 0.985656i $$0.446021\pi$$
$$678$$ 0 0
$$679$$ 692174. 0.0576157
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −5.70526e6 −0.467976 −0.233988 0.972239i $$-0.575178\pi$$
−0.233988 + 0.972239i $$0.575178\pi$$
$$684$$ 0 0
$$685$$ −1.01097e7 −0.823215
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 0 0
$$689$$ −1.57228e7 −1.26178
$$690$$ 0 0
$$691$$ −610648. −0.0486515 −0.0243257 0.999704i $$-0.507744\pi$$
−0.0243257 + 0.999704i $$0.507744\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −3.19333e6 −0.250774
$$696$$ 0 0
$$697$$ −5.17936e6 −0.403826
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −1.55440e7 −1.19473 −0.597363 0.801971i $$-0.703785\pi$$
−0.597363 + 0.801971i $$0.703785\pi$$
$$702$$ 0 0
$$703$$ −4.29351e6 −0.327661
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.70262e6 0.654789
$$708$$ 0 0
$$709$$ −2.53702e7 −1.89543 −0.947717 0.319111i $$-0.896616\pi$$
−0.947717 + 0.319111i $$0.896616\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −2.71827e6 −0.200248
$$714$$ 0 0
$$715$$ 1.97543e7 1.44509
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 0 0
$$719$$ −1.71869e7 −1.23987 −0.619935 0.784653i $$-0.712841\pi$$
−0.619935 + 0.784653i $$0.712841\pi$$
$$720$$ 0 0
$$721$$ −6.28944e6 −0.450582
$$722$$ 0 0
$$723$$ 0 0
$$724$$ 0 0
$$725$$ 1.03712e6 0.0732799
$$726$$ 0 0
$$727$$ 3.03580e6 0.213028 0.106514 0.994311i $$-0.466031\pi$$
0.106514 + 0.994311i $$0.466031\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 1.06993e7 0.740563
$$732$$ 0 0
$$733$$ 3.64299e6 0.250436 0.125218 0.992129i $$-0.460037\pi$$
0.125218 + 0.992129i $$0.460037\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.88681e7 1.27955
$$738$$ 0 0
$$739$$ 8.20260e6 0.552510 0.276255 0.961084i $$-0.410907\pi$$
0.276255 + 0.961084i $$0.410907\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 1.44384e7 0.959504 0.479752 0.877404i $$-0.340727\pi$$
0.479752 + 0.877404i $$0.340727\pi$$
$$744$$ 0 0
$$745$$ −1.49713e7 −0.988256
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 0 0
$$749$$ 3.64315e6 0.237286
$$750$$ 0 0
$$751$$ 8.34049e6 0.539624 0.269812 0.962913i $$-0.413038\pi$$
0.269812 + 0.962913i $$0.413038\pi$$
$$752$$ 0 0
$$753$$ 0 0
$$754$$ 0 0
$$755$$ 8.54470e6 0.545544
$$756$$ 0 0
$$757$$ −2.78547e7 −1.76668 −0.883342 0.468729i $$-0.844712\pi$$
−0.883342 + 0.468729i $$0.844712\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −2.79193e6 −0.174760 −0.0873802 0.996175i $$-0.527850\pi$$
−0.0873802 + 0.996175i $$0.527850\pi$$
$$762$$ 0 0
$$763$$ −5.09198e6 −0.316647
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1.58762e7 0.974448
$$768$$ 0 0
$$769$$ 2.56309e7 1.56296 0.781481 0.623929i $$-0.214465\pi$$
0.781481 + 0.623929i $$0.214465\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 0 0
$$773$$ −2.07212e7 −1.24729 −0.623644 0.781709i $$-0.714348\pi$$
−0.623644 + 0.781709i $$0.714348\pi$$
$$774$$ 0 0
$$775$$ 231332. 0.0138351
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 1.51173e6 0.0892544
$$780$$ 0 0
$$781$$ −3.49975e7 −2.05310
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 1.71264e7 0.991954
$$786$$ 0 0
$$787$$ −2.42099e6 −0.139334 −0.0696669 0.997570i $$-0.522194\pi$$
−0.0696669 + 0.997570i $$0.522194\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 4.71170e6 0.267754
$$792$$ 0 0
$$793$$ 9.78334e6 0.552464
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 1.82162e7 1.01581 0.507905 0.861413i $$-0.330420\pi$$
0.507905 + 0.861413i $$0.330420\pi$$
$$798$$ 0 0
$$799$$ 1.88239e7 1.04314
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 0 0
$$803$$ 2.82300e7 1.54498
$$804$$ 0 0
$$805$$ −4.97624e6 −0.270652
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −3.25647e7 −1.74935 −0.874674 0.484711i $$-0.838925\pi$$
−0.874674 + 0.484711i $$0.838925\pi$$
$$810$$ 0 0
$$811$$ −1.52243e7 −0.812801 −0.406401 0.913695i $$-0.633216\pi$$
−0.406401 + 0.913695i $$0.633216\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −1.92440e7 −1.01485
$$816$$ 0 0
$$817$$ −3.12286e6 −0.163681
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −3.47838e7 −1.80102 −0.900511 0.434833i $$-0.856807\pi$$
−0.900511 + 0.434833i $$0.856807\pi$$
$$822$$ 0 0
$$823$$ −4.15043e6 −0.213596 −0.106798 0.994281i $$-0.534060\pi$$
−0.106798 + 0.994281i $$0.534060\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −1.23139e7 −0.626082 −0.313041 0.949740i $$-0.601348\pi$$
−0.313041 + 0.949740i $$0.601348\pi$$
$$828$$ 0 0
$$829$$ −2.50392e7 −1.26542 −0.632710 0.774389i $$-0.718058\pi$$
−0.632710 + 0.774389i $$0.718058\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −2.33622e6 −0.116654
$$834$$ 0 0
$$835$$ 1.50106e7 0.745045
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ −6.76247e6 −0.331666 −0.165833 0.986154i $$-0.553031\pi$$
−0.165833 + 0.986154i $$0.553031\pi$$
$$840$$ 0 0
$$841$$ 2.66633e7 1.29994
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 4.44194e6 0.214009
$$846$$ 0 0
$$847$$ 5.11094e6 0.244789
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 2.68243e7 1.26971
$$852$$ 0 0
$$853$$ −601162. −0.0282891 −0.0141445 0.999900i $$-0.504502\pi$$
−0.0141445 + 0.999900i $$0.504502\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 1.81694e7 0.845062 0.422531 0.906349i $$-0.361142\pi$$
0.422531 + 0.906349i $$0.361142\pi$$
$$858$$ 0 0
$$859$$ 4.04414e7 1.87001 0.935004 0.354636i $$-0.115395\pi$$
0.935004 + 0.354636i $$0.115395\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ 8.66358e6 0.395977 0.197989 0.980204i $$-0.436559\pi$$
0.197989 + 0.980204i $$0.436559\pi$$
$$864$$ 0 0
$$865$$ 3.10139e7 1.40934
$$866$$ 0 0
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 1.63646e7 0.735114
$$870$$ 0 0
$$871$$ 2.45408e7 1.09608
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ −8.34082e6 −0.368289
$$876$$ 0 0
$$877$$ 1.30182e6 0.0571548 0.0285774 0.999592i $$-0.490902\pi$$
0.0285774 + 0.999592i $$0.490902\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −2.03826e7 −0.884749 −0.442374 0.896830i $$-0.645864\pi$$
−0.442374 + 0.896830i $$0.645864\pi$$
$$882$$ 0 0
$$883$$ −3.24432e6 −0.140030 −0.0700152 0.997546i $$-0.522305\pi$$
−0.0700152 + 0.997546i $$0.522305\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −2.57482e7 −1.09885 −0.549425 0.835543i $$-0.685153\pi$$
−0.549425 + 0.835543i $$0.685153\pi$$
$$888$$ 0 0
$$889$$ 6.22849e6 0.264319
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −5.49423e6 −0.230557
$$894$$ 0 0
$$895$$ 1.69730e7 0.708272
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 1.05223e7 0.434223
$$900$$ 0 0
$$901$$ −2.28337e7 −0.937054
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 3.36276e7 1.36482
$$906$$ 0 0
$$907$$ −2.75038e6 −0.111013 −0.0555066 0.998458i $$-0.517677\pi$$
−0.0555066 + 0.998458i $$0.517677\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −3.07918e7 −1.22925 −0.614624 0.788820i $$-0.710692\pi$$
−0.614624 + 0.788820i $$0.710692\pi$$
$$912$$ 0 0
$$913$$ 3.82113e7 1.51710
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −1.60983e7 −0.632203
$$918$$ 0 0
$$919$$ 1.55324e7 0.606666 0.303333 0.952885i $$-0.401901\pi$$
0.303333 + 0.952885i $$0.401901\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 0 0
$$923$$ −4.55195e7 −1.75871
$$924$$ 0 0
$$925$$ −2.28282e6 −0.0877237
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ −4.23841e6 −0.161125 −0.0805626 0.996750i $$-0.525672\pi$$
−0.0805626 + 0.996750i $$0.525672\pi$$
$$930$$ 0 0
$$931$$ 681884. 0.0257832
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 0 0
$$935$$ 2.86885e7 1.07319
$$936$$ 0 0
$$937$$ 2.16194e7 0.804443 0.402222 0.915542i $$-0.368238\pi$$
0.402222 + 0.915542i $$0.368238\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 4.75669e7 1.75118 0.875590 0.483055i $$-0.160473\pi$$
0.875590 + 0.483055i $$0.160473\pi$$
$$942$$ 0 0
$$943$$ −9.44471e6 −0.345867
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −4.54109e7 −1.64545 −0.822726 0.568438i $$-0.807548\pi$$
−0.822726 + 0.568438i $$0.807548\pi$$
$$948$$ 0 0
$$949$$ 3.67173e7 1.32344
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 2.93139e7 1.04554 0.522772 0.852473i $$-0.324898\pi$$
0.522772 + 0.852473i $$0.324898\pi$$
$$954$$ 0 0
$$955$$ 1.07682e7 0.382063
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −8.65494e6 −0.303891
$$960$$ 0 0
$$961$$ −2.62821e7 −0.918020
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 0 0
$$965$$ 2.30675e7 0.797411
$$966$$ 0 0
$$967$$ −1.45552e7 −0.500557 −0.250278 0.968174i $$-0.580522\pi$$
−0.250278 + 0.968174i $$0.580522\pi$$
$$968$$ 0 0
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −4.70222e7 −1.60050 −0.800248 0.599669i $$-0.795299\pi$$
−0.800248 + 0.599669i $$0.795299\pi$$
$$972$$ 0 0
$$973$$ −2.73381e6 −0.0925733
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −3.46666e7 −1.16192 −0.580958 0.813934i $$-0.697322\pi$$
−0.580958 + 0.813934i $$0.697322\pi$$
$$978$$ 0 0
$$979$$ 442260. 0.0147476
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 2.35848e7 0.778482 0.389241 0.921136i $$-0.372737\pi$$
0.389241 + 0.921136i $$0.372737\pi$$
$$984$$ 0 0
$$985$$ 1.45782e7 0.478755
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 1.95105e7 0.634275
$$990$$ 0 0
$$991$$ 2.63456e7 0.852166 0.426083 0.904684i $$-0.359893\pi$$
0.426083 + 0.904684i $$0.359893\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 2.02250e7 0.647637
$$996$$ 0 0
$$997$$ 1.40009e7 0.446086 0.223043 0.974809i $$-0.428401\pi$$
0.223043 + 0.974809i $$0.428401\pi$$
$$998$$ 0 0
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 252.6.a.g.1.2 yes 2
3.2 odd 2 inner 252.6.a.g.1.1 2
4.3 odd 2 1008.6.a.bk.1.2 2
12.11 even 2 1008.6.a.bk.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
252.6.a.g.1.1 2 3.2 odd 2 inner
252.6.a.g.1.2 yes 2 1.1 even 1 trivial
1008.6.a.bk.1.1 2 12.11 even 2
1008.6.a.bk.1.2 2 4.3 odd 2