# Properties

 Label 80.4.a.a.1.1 Level $80$ Weight $4$ Character 80.1 Self dual yes Analytic conductor $4.720$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,4,Mod(1,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("80.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 80.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: $$4.72015280046$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 80.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-10.0000 q^{3} -5.00000 q^{5} +18.0000 q^{7} +73.0000 q^{9} +O(q^{10})$$ $$q-10.0000 q^{3} -5.00000 q^{5} +18.0000 q^{7} +73.0000 q^{9} +16.0000 q^{11} -6.00000 q^{13} +50.0000 q^{15} -6.00000 q^{17} +124.000 q^{19} -180.000 q^{21} -42.0000 q^{23} +25.0000 q^{25} -460.000 q^{27} +142.000 q^{29} +188.000 q^{31} -160.000 q^{33} -90.0000 q^{35} +202.000 q^{37} +60.0000 q^{39} +54.0000 q^{41} -66.0000 q^{43} -365.000 q^{45} -38.0000 q^{47} -19.0000 q^{49} +60.0000 q^{51} +738.000 q^{53} -80.0000 q^{55} -1240.00 q^{57} -564.000 q^{59} -262.000 q^{61} +1314.00 q^{63} +30.0000 q^{65} +554.000 q^{67} +420.000 q^{69} -140.000 q^{71} +882.000 q^{73} -250.000 q^{75} +288.000 q^{77} +1160.00 q^{79} +2629.00 q^{81} -642.000 q^{83} +30.0000 q^{85} -1420.00 q^{87} -854.000 q^{89} -108.000 q^{91} -1880.00 q^{93} -620.000 q^{95} -478.000 q^{97} +1168.00 q^{99} +O(q^{100})$$

## 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
$$3$$ −10.0000 −1.92450 −0.962250 0.272166i $$-0.912260\pi$$
−0.962250 + 0.272166i $$0.912260\pi$$
$$4$$ 0 0
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ 18.0000 0.971909 0.485954 0.873984i $$-0.338472\pi$$
0.485954 + 0.873984i $$0.338472\pi$$
$$8$$ 0 0
$$9$$ 73.0000 2.70370
$$10$$ 0 0
$$11$$ 16.0000 0.438562 0.219281 0.975662i $$-0.429629\pi$$
0.219281 + 0.975662i $$0.429629\pi$$
$$12$$ 0 0
$$13$$ −6.00000 −0.128008 −0.0640039 0.997950i $$-0.520387\pi$$
−0.0640039 + 0.997950i $$0.520387\pi$$
$$14$$ 0 0
$$15$$ 50.0000 0.860663
$$16$$ 0 0
$$17$$ −6.00000 −0.0856008 −0.0428004 0.999084i $$-0.513628\pi$$
−0.0428004 + 0.999084i $$0.513628\pi$$
$$18$$ 0 0
$$19$$ 124.000 1.49724 0.748620 0.663000i $$-0.230717\pi$$
0.748620 + 0.663000i $$0.230717\pi$$
$$20$$ 0 0
$$21$$ −180.000 −1.87044
$$22$$ 0 0
$$23$$ −42.0000 −0.380765 −0.190383 0.981710i $$-0.560973\pi$$
−0.190383 + 0.981710i $$0.560973\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 0 0
$$27$$ −460.000 −3.27878
$$28$$ 0 0
$$29$$ 142.000 0.909267 0.454633 0.890679i $$-0.349770\pi$$
0.454633 + 0.890679i $$0.349770\pi$$
$$30$$ 0 0
$$31$$ 188.000 1.08922 0.544610 0.838690i $$-0.316678\pi$$
0.544610 + 0.838690i $$0.316678\pi$$
$$32$$ 0 0
$$33$$ −160.000 −0.844013
$$34$$ 0 0
$$35$$ −90.0000 −0.434651
$$36$$ 0 0
$$37$$ 202.000 0.897530 0.448765 0.893650i $$-0.351864\pi$$
0.448765 + 0.893650i $$0.351864\pi$$
$$38$$ 0 0
$$39$$ 60.0000 0.246351
$$40$$ 0 0
$$41$$ 54.0000 0.205692 0.102846 0.994697i $$-0.467205\pi$$
0.102846 + 0.994697i $$0.467205\pi$$
$$42$$ 0 0
$$43$$ −66.0000 −0.234068 −0.117034 0.993128i $$-0.537339\pi$$
−0.117034 + 0.993128i $$0.537339\pi$$
$$44$$ 0 0
$$45$$ −365.000 −1.20913
$$46$$ 0 0
$$47$$ −38.0000 −0.117933 −0.0589667 0.998260i $$-0.518781\pi$$
−0.0589667 + 0.998260i $$0.518781\pi$$
$$48$$ 0 0
$$49$$ −19.0000 −0.0553936
$$50$$ 0 0
$$51$$ 60.0000 0.164739
$$52$$ 0 0
$$53$$ 738.000 1.91268 0.956341 0.292255i $$-0.0944055\pi$$
0.956341 + 0.292255i $$0.0944055\pi$$
$$54$$ 0 0
$$55$$ −80.0000 −0.196131
$$56$$ 0 0
$$57$$ −1240.00 −2.88144
$$58$$ 0 0
$$59$$ −564.000 −1.24452 −0.622259 0.782812i $$-0.713785\pi$$
−0.622259 + 0.782812i $$0.713785\pi$$
$$60$$ 0 0
$$61$$ −262.000 −0.549929 −0.274964 0.961454i $$-0.588666\pi$$
−0.274964 + 0.961454i $$0.588666\pi$$
$$62$$ 0 0
$$63$$ 1314.00 2.62775
$$64$$ 0 0
$$65$$ 30.0000 0.0572468
$$66$$ 0 0
$$67$$ 554.000 1.01018 0.505089 0.863067i $$-0.331460\pi$$
0.505089 + 0.863067i $$0.331460\pi$$
$$68$$ 0 0
$$69$$ 420.000 0.732783
$$70$$ 0 0
$$71$$ −140.000 −0.234013 −0.117007 0.993131i $$-0.537330\pi$$
−0.117007 + 0.993131i $$0.537330\pi$$
$$72$$ 0 0
$$73$$ 882.000 1.41411 0.707057 0.707157i $$-0.250023\pi$$
0.707057 + 0.707157i $$0.250023\pi$$
$$74$$ 0 0
$$75$$ −250.000 −0.384900
$$76$$ 0 0
$$77$$ 288.000 0.426242
$$78$$ 0 0
$$79$$ 1160.00 1.65203 0.826014 0.563650i $$-0.190603\pi$$
0.826014 + 0.563650i $$0.190603\pi$$
$$80$$ 0 0
$$81$$ 2629.00 3.60631
$$82$$ 0 0
$$83$$ −642.000 −0.849020 −0.424510 0.905423i $$-0.639554\pi$$
−0.424510 + 0.905423i $$0.639554\pi$$
$$84$$ 0 0
$$85$$ 30.0000 0.0382818
$$86$$ 0 0
$$87$$ −1420.00 −1.74988
$$88$$ 0 0
$$89$$ −854.000 −1.01712 −0.508561 0.861026i $$-0.669822\pi$$
−0.508561 + 0.861026i $$0.669822\pi$$
$$90$$ 0 0
$$91$$ −108.000 −0.124412
$$92$$ 0 0
$$93$$ −1880.00 −2.09620
$$94$$ 0 0
$$95$$ −620.000 −0.669586
$$96$$ 0 0
$$97$$ −478.000 −0.500346 −0.250173 0.968201i $$-0.580487\pi$$
−0.250173 + 0.968201i $$0.580487\pi$$
$$98$$ 0 0
$$99$$ 1168.00 1.18574
$$100$$ 0 0
$$101$$ −1794.00 −1.76742 −0.883711 0.468033i $$-0.844963\pi$$
−0.883711 + 0.468033i $$0.844963\pi$$
$$102$$ 0 0
$$103$$ −642.000 −0.614157 −0.307078 0.951684i $$-0.599351\pi$$
−0.307078 + 0.951684i $$0.599351\pi$$
$$104$$ 0 0
$$105$$ 900.000 0.836486
$$106$$ 0 0
$$107$$ 850.000 0.767968 0.383984 0.923340i $$-0.374552\pi$$
0.383984 + 0.923340i $$0.374552\pi$$
$$108$$ 0 0
$$109$$ 666.000 0.585241 0.292620 0.956229i $$-0.405473\pi$$
0.292620 + 0.956229i $$0.405473\pi$$
$$110$$ 0 0
$$111$$ −2020.00 −1.72730
$$112$$ 0 0
$$113$$ −1446.00 −1.20379 −0.601895 0.798575i $$-0.705587\pi$$
−0.601895 + 0.798575i $$0.705587\pi$$
$$114$$ 0 0
$$115$$ 210.000 0.170283
$$116$$ 0 0
$$117$$ −438.000 −0.346095
$$118$$ 0 0
$$119$$ −108.000 −0.0831962
$$120$$ 0 0
$$121$$ −1075.00 −0.807663
$$122$$ 0 0
$$123$$ −540.000 −0.395855
$$124$$ 0 0
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ 1154.00 0.806307 0.403153 0.915132i $$-0.367914\pi$$
0.403153 + 0.915132i $$0.367914\pi$$
$$128$$ 0 0
$$129$$ 660.000 0.450463
$$130$$ 0 0
$$131$$ 368.000 0.245437 0.122719 0.992441i $$-0.460839\pi$$
0.122719 + 0.992441i $$0.460839\pi$$
$$132$$ 0 0
$$133$$ 2232.00 1.45518
$$134$$ 0 0
$$135$$ 2300.00 1.46631
$$136$$ 0 0
$$137$$ −670.000 −0.417825 −0.208912 0.977934i $$-0.566992\pi$$
−0.208912 + 0.977934i $$0.566992\pi$$
$$138$$ 0 0
$$139$$ 572.000 0.349039 0.174519 0.984654i $$-0.444163\pi$$
0.174519 + 0.984654i $$0.444163\pi$$
$$140$$ 0 0
$$141$$ 380.000 0.226963
$$142$$ 0 0
$$143$$ −96.0000 −0.0561393
$$144$$ 0 0
$$145$$ −710.000 −0.406636
$$146$$ 0 0
$$147$$ 190.000 0.106605
$$148$$ 0 0
$$149$$ 1730.00 0.951189 0.475594 0.879665i $$-0.342233\pi$$
0.475594 + 0.879665i $$0.342233\pi$$
$$150$$ 0 0
$$151$$ −1324.00 −0.713547 −0.356773 0.934191i $$-0.616123\pi$$
−0.356773 + 0.934191i $$0.616123\pi$$
$$152$$ 0 0
$$153$$ −438.000 −0.231439
$$154$$ 0 0
$$155$$ −940.000 −0.487114
$$156$$ 0 0
$$157$$ 2946.00 1.49756 0.748778 0.662820i $$-0.230641\pi$$
0.748778 + 0.662820i $$0.230641\pi$$
$$158$$ 0 0
$$159$$ −7380.00 −3.68096
$$160$$ 0 0
$$161$$ −756.000 −0.370069
$$162$$ 0 0
$$163$$ −2098.00 −1.00815 −0.504074 0.863661i $$-0.668166\pi$$
−0.504074 + 0.863661i $$0.668166\pi$$
$$164$$ 0 0
$$165$$ 800.000 0.377454
$$166$$ 0 0
$$167$$ 866.000 0.401276 0.200638 0.979665i $$-0.435699\pi$$
0.200638 + 0.979665i $$0.435699\pi$$
$$168$$ 0 0
$$169$$ −2161.00 −0.983614
$$170$$ 0 0
$$171$$ 9052.00 4.04809
$$172$$ 0 0
$$173$$ −1678.00 −0.737433 −0.368717 0.929542i $$-0.620203\pi$$
−0.368717 + 0.929542i $$0.620203\pi$$
$$174$$ 0 0
$$175$$ 450.000 0.194382
$$176$$ 0 0
$$177$$ 5640.00 2.39508
$$178$$ 0 0
$$179$$ 1620.00 0.676450 0.338225 0.941065i $$-0.390174\pi$$
0.338225 + 0.941065i $$0.390174\pi$$
$$180$$ 0 0
$$181$$ 2510.00 1.03076 0.515378 0.856963i $$-0.327652\pi$$
0.515378 + 0.856963i $$0.327652\pi$$
$$182$$ 0 0
$$183$$ 2620.00 1.05834
$$184$$ 0 0
$$185$$ −1010.00 −0.401387
$$186$$ 0 0
$$187$$ −96.0000 −0.0375413
$$188$$ 0 0
$$189$$ −8280.00 −3.18667
$$190$$ 0 0
$$191$$ 372.000 0.140927 0.0704633 0.997514i $$-0.477552\pi$$
0.0704633 + 0.997514i $$0.477552\pi$$
$$192$$ 0 0
$$193$$ 2938.00 1.09576 0.547880 0.836557i $$-0.315435\pi$$
0.547880 + 0.836557i $$0.315435\pi$$
$$194$$ 0 0
$$195$$ −300.000 −0.110172
$$196$$ 0 0
$$197$$ 2234.00 0.807949 0.403974 0.914770i $$-0.367628\pi$$
0.403974 + 0.914770i $$0.367628\pi$$
$$198$$ 0 0
$$199$$ 3048.00 1.08576 0.542882 0.839809i $$-0.317333\pi$$
0.542882 + 0.839809i $$0.317333\pi$$
$$200$$ 0 0
$$201$$ −5540.00 −1.94409
$$202$$ 0 0
$$203$$ 2556.00 0.883724
$$204$$ 0 0
$$205$$ −270.000 −0.0919884
$$206$$ 0 0
$$207$$ −3066.00 −1.02948
$$208$$ 0 0
$$209$$ 1984.00 0.656632
$$210$$ 0 0
$$211$$ −4896.00 −1.59741 −0.798707 0.601720i $$-0.794482\pi$$
−0.798707 + 0.601720i $$0.794482\pi$$
$$212$$ 0 0
$$213$$ 1400.00 0.450359
$$214$$ 0 0
$$215$$ 330.000 0.104678
$$216$$ 0 0
$$217$$ 3384.00 1.05862
$$218$$ 0 0
$$219$$ −8820.00 −2.72146
$$220$$ 0 0
$$221$$ 36.0000 0.0109576
$$222$$ 0 0
$$223$$ 5302.00 1.59214 0.796072 0.605202i $$-0.206908\pi$$
0.796072 + 0.605202i $$0.206908\pi$$
$$224$$ 0 0
$$225$$ 1825.00 0.540741
$$226$$ 0 0
$$227$$ 3778.00 1.10465 0.552323 0.833630i $$-0.313741\pi$$
0.552323 + 0.833630i $$0.313741\pi$$
$$228$$ 0 0
$$229$$ −3034.00 −0.875513 −0.437756 0.899094i $$-0.644227\pi$$
−0.437756 + 0.899094i $$0.644227\pi$$
$$230$$ 0 0
$$231$$ −2880.00 −0.820303
$$232$$ 0 0
$$233$$ −3478.00 −0.977903 −0.488951 0.872311i $$-0.662620\pi$$
−0.488951 + 0.872311i $$0.662620\pi$$
$$234$$ 0 0
$$235$$ 190.000 0.0527414
$$236$$ 0 0
$$237$$ −11600.0 −3.17933
$$238$$ 0 0
$$239$$ −1560.00 −0.422209 −0.211105 0.977463i $$-0.567706\pi$$
−0.211105 + 0.977463i $$0.567706\pi$$
$$240$$ 0 0
$$241$$ −3218.00 −0.860123 −0.430061 0.902800i $$-0.641508\pi$$
−0.430061 + 0.902800i $$0.641508\pi$$
$$242$$ 0 0
$$243$$ −13870.0 −3.66157
$$244$$ 0 0
$$245$$ 95.0000 0.0247728
$$246$$ 0 0
$$247$$ −744.000 −0.191658
$$248$$ 0 0
$$249$$ 6420.00 1.63394
$$250$$ 0 0
$$251$$ −688.000 −0.173013 −0.0865063 0.996251i $$-0.527570\pi$$
−0.0865063 + 0.996251i $$0.527570\pi$$
$$252$$ 0 0
$$253$$ −672.000 −0.166989
$$254$$ 0 0
$$255$$ −300.000 −0.0736734
$$256$$ 0 0
$$257$$ 2170.00 0.526696 0.263348 0.964701i $$-0.415173\pi$$
0.263348 + 0.964701i $$0.415173\pi$$
$$258$$ 0 0
$$259$$ 3636.00 0.872317
$$260$$ 0 0
$$261$$ 10366.0 2.45839
$$262$$ 0 0
$$263$$ −2274.00 −0.533159 −0.266580 0.963813i $$-0.585894\pi$$
−0.266580 + 0.963813i $$0.585894\pi$$
$$264$$ 0 0
$$265$$ −3690.00 −0.855377
$$266$$ 0 0
$$267$$ 8540.00 1.95745
$$268$$ 0 0
$$269$$ 7146.00 1.61970 0.809850 0.586637i $$-0.199548\pi$$
0.809850 + 0.586637i $$0.199548\pi$$
$$270$$ 0 0
$$271$$ −2604.00 −0.583696 −0.291848 0.956465i $$-0.594270\pi$$
−0.291848 + 0.956465i $$0.594270\pi$$
$$272$$ 0 0
$$273$$ 1080.00 0.239431
$$274$$ 0 0
$$275$$ 400.000 0.0877124
$$276$$ 0 0
$$277$$ −5150.00 −1.11709 −0.558544 0.829475i $$-0.688640\pi$$
−0.558544 + 0.829475i $$0.688640\pi$$
$$278$$ 0 0
$$279$$ 13724.0 2.94493
$$280$$ 0 0
$$281$$ 5270.00 1.11880 0.559398 0.828899i $$-0.311032\pi$$
0.559398 + 0.828899i $$0.311032\pi$$
$$282$$ 0 0
$$283$$ −3434.00 −0.721308 −0.360654 0.932700i $$-0.617446\pi$$
−0.360654 + 0.932700i $$0.617446\pi$$
$$284$$ 0 0
$$285$$ 6200.00 1.28862
$$286$$ 0 0
$$287$$ 972.000 0.199914
$$288$$ 0 0
$$289$$ −4877.00 −0.992673
$$290$$ 0 0
$$291$$ 4780.00 0.962916
$$292$$ 0 0
$$293$$ −9878.00 −1.96955 −0.984776 0.173826i $$-0.944387\pi$$
−0.984776 + 0.173826i $$0.944387\pi$$
$$294$$ 0 0
$$295$$ 2820.00 0.556565
$$296$$ 0 0
$$297$$ −7360.00 −1.43795
$$298$$ 0 0
$$299$$ 252.000 0.0487409
$$300$$ 0 0
$$301$$ −1188.00 −0.227492
$$302$$ 0 0
$$303$$ 17940.0 3.40141
$$304$$ 0 0
$$305$$ 1310.00 0.245936
$$306$$ 0 0
$$307$$ −8054.00 −1.49728 −0.748642 0.662975i $$-0.769294\pi$$
−0.748642 + 0.662975i $$0.769294\pi$$
$$308$$ 0 0
$$309$$ 6420.00 1.18195
$$310$$ 0 0
$$311$$ −5492.00 −1.00136 −0.500680 0.865633i $$-0.666917\pi$$
−0.500680 + 0.865633i $$0.666917\pi$$
$$312$$ 0 0
$$313$$ −422.000 −0.0762072 −0.0381036 0.999274i $$-0.512132\pi$$
−0.0381036 + 0.999274i $$0.512132\pi$$
$$314$$ 0 0
$$315$$ −6570.00 −1.17517
$$316$$ 0 0
$$317$$ 6194.00 1.09744 0.548722 0.836005i $$-0.315115\pi$$
0.548722 + 0.836005i $$0.315115\pi$$
$$318$$ 0 0
$$319$$ 2272.00 0.398770
$$320$$ 0 0
$$321$$ −8500.00 −1.47796
$$322$$ 0 0
$$323$$ −744.000 −0.128165
$$324$$ 0 0
$$325$$ −150.000 −0.0256015
$$326$$ 0 0
$$327$$ −6660.00 −1.12630
$$328$$ 0 0
$$329$$ −684.000 −0.114620
$$330$$ 0 0
$$331$$ −7688.00 −1.27665 −0.638324 0.769768i $$-0.720372\pi$$
−0.638324 + 0.769768i $$0.720372\pi$$
$$332$$ 0 0
$$333$$ 14746.0 2.42665
$$334$$ 0 0
$$335$$ −2770.00 −0.451765
$$336$$ 0 0
$$337$$ −1438.00 −0.232442 −0.116221 0.993223i $$-0.537078\pi$$
−0.116221 + 0.993223i $$0.537078\pi$$
$$338$$ 0 0
$$339$$ 14460.0 2.31669
$$340$$ 0 0
$$341$$ 3008.00 0.477690
$$342$$ 0 0
$$343$$ −6516.00 −1.02575
$$344$$ 0 0
$$345$$ −2100.00 −0.327711
$$346$$ 0 0
$$347$$ −8838.00 −1.36729 −0.683644 0.729816i $$-0.739606\pi$$
−0.683644 + 0.729816i $$0.739606\pi$$
$$348$$ 0 0
$$349$$ −7810.00 −1.19788 −0.598939 0.800794i $$-0.704411\pi$$
−0.598939 + 0.800794i $$0.704411\pi$$
$$350$$ 0 0
$$351$$ 2760.00 0.419709
$$352$$ 0 0
$$353$$ 5906.00 0.890495 0.445247 0.895408i $$-0.353116\pi$$
0.445247 + 0.895408i $$0.353116\pi$$
$$354$$ 0 0
$$355$$ 700.000 0.104654
$$356$$ 0 0
$$357$$ 1080.00 0.160111
$$358$$ 0 0
$$359$$ −8904.00 −1.30901 −0.654506 0.756057i $$-0.727123\pi$$
−0.654506 + 0.756057i $$0.727123\pi$$
$$360$$ 0 0
$$361$$ 8517.00 1.24173
$$362$$ 0 0
$$363$$ 10750.0 1.55435
$$364$$ 0 0
$$365$$ −4410.00 −0.632411
$$366$$ 0 0
$$367$$ 7370.00 1.04826 0.524129 0.851639i $$-0.324391\pi$$
0.524129 + 0.851639i $$0.324391\pi$$
$$368$$ 0 0
$$369$$ 3942.00 0.556131
$$370$$ 0 0
$$371$$ 13284.0 1.85895
$$372$$ 0 0
$$373$$ −734.000 −0.101890 −0.0509451 0.998701i $$-0.516223\pi$$
−0.0509451 + 0.998701i $$0.516223\pi$$
$$374$$ 0 0
$$375$$ 1250.00 0.172133
$$376$$ 0 0
$$377$$ −852.000 −0.116393
$$378$$ 0 0
$$379$$ −10300.0 −1.39598 −0.697989 0.716109i $$-0.745921\pi$$
−0.697989 + 0.716109i $$0.745921\pi$$
$$380$$ 0 0
$$381$$ −11540.0 −1.55174
$$382$$ 0 0
$$383$$ −2682.00 −0.357817 −0.178908 0.983866i $$-0.557257\pi$$
−0.178908 + 0.983866i $$0.557257\pi$$
$$384$$ 0 0
$$385$$ −1440.00 −0.190621
$$386$$ 0 0
$$387$$ −4818.00 −0.632849
$$388$$ 0 0
$$389$$ 6114.00 0.796895 0.398447 0.917191i $$-0.369549\pi$$
0.398447 + 0.917191i $$0.369549\pi$$
$$390$$ 0 0
$$391$$ 252.000 0.0325938
$$392$$ 0 0
$$393$$ −3680.00 −0.472345
$$394$$ 0 0
$$395$$ −5800.00 −0.738809
$$396$$ 0 0
$$397$$ −7174.00 −0.906934 −0.453467 0.891273i $$-0.649813\pi$$
−0.453467 + 0.891273i $$0.649813\pi$$
$$398$$ 0 0
$$399$$ −22320.0 −2.80050
$$400$$ 0 0
$$401$$ 10498.0 1.30734 0.653672 0.756778i $$-0.273228\pi$$
0.653672 + 0.756778i $$0.273228\pi$$
$$402$$ 0 0
$$403$$ −1128.00 −0.139428
$$404$$ 0 0
$$405$$ −13145.0 −1.61279
$$406$$ 0 0
$$407$$ 3232.00 0.393622
$$408$$ 0 0
$$409$$ −1810.00 −0.218823 −0.109412 0.993997i $$-0.534897\pi$$
−0.109412 + 0.993997i $$0.534897\pi$$
$$410$$ 0 0
$$411$$ 6700.00 0.804104
$$412$$ 0 0
$$413$$ −10152.0 −1.20956
$$414$$ 0 0
$$415$$ 3210.00 0.379693
$$416$$ 0 0
$$417$$ −5720.00 −0.671726
$$418$$ 0 0
$$419$$ 3396.00 0.395956 0.197978 0.980206i $$-0.436563\pi$$
0.197978 + 0.980206i $$0.436563\pi$$
$$420$$ 0 0
$$421$$ −14974.0 −1.73346 −0.866732 0.498775i $$-0.833784\pi$$
−0.866732 + 0.498775i $$0.833784\pi$$
$$422$$ 0 0
$$423$$ −2774.00 −0.318857
$$424$$ 0 0
$$425$$ −150.000 −0.0171202
$$426$$ 0 0
$$427$$ −4716.00 −0.534481
$$428$$ 0 0
$$429$$ 960.000 0.108040
$$430$$ 0 0
$$431$$ 13540.0 1.51322 0.756611 0.653865i $$-0.226854\pi$$
0.756611 + 0.653865i $$0.226854\pi$$
$$432$$ 0 0
$$433$$ 15426.0 1.71207 0.856035 0.516918i $$-0.172921\pi$$
0.856035 + 0.516918i $$0.172921\pi$$
$$434$$ 0 0
$$435$$ 7100.00 0.782572
$$436$$ 0 0
$$437$$ −5208.00 −0.570097
$$438$$ 0 0
$$439$$ 10472.0 1.13850 0.569250 0.822165i $$-0.307234\pi$$
0.569250 + 0.822165i $$0.307234\pi$$
$$440$$ 0 0
$$441$$ −1387.00 −0.149768
$$442$$ 0 0
$$443$$ −722.000 −0.0774340 −0.0387170 0.999250i $$-0.512327\pi$$
−0.0387170 + 0.999250i $$0.512327\pi$$
$$444$$ 0 0
$$445$$ 4270.00 0.454871
$$446$$ 0 0
$$447$$ −17300.0 −1.83056
$$448$$ 0 0
$$449$$ −11898.0 −1.25056 −0.625280 0.780401i $$-0.715015\pi$$
−0.625280 + 0.780401i $$0.715015\pi$$
$$450$$ 0 0
$$451$$ 864.000 0.0902088
$$452$$ 0 0
$$453$$ 13240.0 1.37322
$$454$$ 0 0
$$455$$ 540.000 0.0556387
$$456$$ 0 0
$$457$$ −790.000 −0.0808635 −0.0404318 0.999182i $$-0.512873\pi$$
−0.0404318 + 0.999182i $$0.512873\pi$$
$$458$$ 0 0
$$459$$ 2760.00 0.280666
$$460$$ 0 0
$$461$$ −3418.00 −0.345319 −0.172660 0.984982i $$-0.555236\pi$$
−0.172660 + 0.984982i $$0.555236\pi$$
$$462$$ 0 0
$$463$$ 7534.00 0.756230 0.378115 0.925759i $$-0.376572\pi$$
0.378115 + 0.925759i $$0.376572\pi$$
$$464$$ 0 0
$$465$$ 9400.00 0.937451
$$466$$ 0 0
$$467$$ 14314.0 1.41836 0.709179 0.705029i $$-0.249066\pi$$
0.709179 + 0.705029i $$0.249066\pi$$
$$468$$ 0 0
$$469$$ 9972.00 0.981800
$$470$$ 0 0
$$471$$ −29460.0 −2.88205
$$472$$ 0 0
$$473$$ −1056.00 −0.102653
$$474$$ 0 0
$$475$$ 3100.00 0.299448
$$476$$ 0 0
$$477$$ 53874.0 5.17132
$$478$$ 0 0
$$479$$ 7016.00 0.669247 0.334623 0.942352i $$-0.391391\pi$$
0.334623 + 0.942352i $$0.391391\pi$$
$$480$$ 0 0
$$481$$ −1212.00 −0.114891
$$482$$ 0 0
$$483$$ 7560.00 0.712199
$$484$$ 0 0
$$485$$ 2390.00 0.223761
$$486$$ 0 0
$$487$$ −15190.0 −1.41340 −0.706699 0.707515i $$-0.749816\pi$$
−0.706699 + 0.707515i $$0.749816\pi$$
$$488$$ 0 0
$$489$$ 20980.0 1.94018
$$490$$ 0 0
$$491$$ 12624.0 1.16031 0.580156 0.814505i $$-0.302992\pi$$
0.580156 + 0.814505i $$0.302992\pi$$
$$492$$ 0 0
$$493$$ −852.000 −0.0778340
$$494$$ 0 0
$$495$$ −5840.00 −0.530280
$$496$$ 0 0
$$497$$ −2520.00 −0.227440
$$498$$ 0 0
$$499$$ 2492.00 0.223562 0.111781 0.993733i $$-0.464345\pi$$
0.111781 + 0.993733i $$0.464345\pi$$
$$500$$ 0 0
$$501$$ −8660.00 −0.772256
$$502$$ 0 0
$$503$$ −11714.0 −1.03837 −0.519186 0.854661i $$-0.673765\pi$$
−0.519186 + 0.854661i $$0.673765\pi$$
$$504$$ 0 0
$$505$$ 8970.00 0.790415
$$506$$ 0 0
$$507$$ 21610.0 1.89297
$$508$$ 0 0
$$509$$ −5618.00 −0.489221 −0.244610 0.969621i $$-0.578660\pi$$
−0.244610 + 0.969621i $$0.578660\pi$$
$$510$$ 0 0
$$511$$ 15876.0 1.37439
$$512$$ 0 0
$$513$$ −57040.0 −4.90912
$$514$$ 0 0
$$515$$ 3210.00 0.274659
$$516$$ 0 0
$$517$$ −608.000 −0.0517211
$$518$$ 0 0
$$519$$ 16780.0 1.41919
$$520$$ 0 0
$$521$$ 13770.0 1.15792 0.578958 0.815357i $$-0.303459\pi$$
0.578958 + 0.815357i $$0.303459\pi$$
$$522$$ 0 0
$$523$$ −6986.00 −0.584085 −0.292042 0.956405i $$-0.594335\pi$$
−0.292042 + 0.956405i $$0.594335\pi$$
$$524$$ 0 0
$$525$$ −4500.00 −0.374088
$$526$$ 0 0
$$527$$ −1128.00 −0.0932380
$$528$$ 0 0
$$529$$ −10403.0 −0.855018
$$530$$ 0 0
$$531$$ −41172.0 −3.36481
$$532$$ 0 0
$$533$$ −324.000 −0.0263302
$$534$$ 0 0
$$535$$ −4250.00 −0.343446
$$536$$ 0 0
$$537$$ −16200.0 −1.30183
$$538$$ 0 0
$$539$$ −304.000 −0.0242935
$$540$$ 0 0
$$541$$ 11958.0 0.950304 0.475152 0.879904i $$-0.342393\pi$$
0.475152 + 0.879904i $$0.342393\pi$$
$$542$$ 0 0
$$543$$ −25100.0 −1.98369
$$544$$ 0 0
$$545$$ −3330.00 −0.261728
$$546$$ 0 0
$$547$$ 4194.00 0.327829 0.163915 0.986475i $$-0.447588\pi$$
0.163915 + 0.986475i $$0.447588\pi$$
$$548$$ 0 0
$$549$$ −19126.0 −1.48684
$$550$$ 0 0
$$551$$ 17608.0 1.36139
$$552$$ 0 0
$$553$$ 20880.0 1.60562
$$554$$ 0 0
$$555$$ 10100.0 0.772470
$$556$$ 0 0
$$557$$ −5382.00 −0.409412 −0.204706 0.978823i $$-0.565624\pi$$
−0.204706 + 0.978823i $$0.565624\pi$$
$$558$$ 0 0
$$559$$ 396.000 0.0299625
$$560$$ 0 0
$$561$$ 960.000 0.0722482
$$562$$ 0 0
$$563$$ −15418.0 −1.15416 −0.577079 0.816688i $$-0.695808\pi$$
−0.577079 + 0.816688i $$0.695808\pi$$
$$564$$ 0 0
$$565$$ 7230.00 0.538351
$$566$$ 0 0
$$567$$ 47322.0 3.50500
$$568$$ 0 0
$$569$$ −5778.00 −0.425705 −0.212853 0.977084i $$-0.568275\pi$$
−0.212853 + 0.977084i $$0.568275\pi$$
$$570$$ 0 0
$$571$$ −6024.00 −0.441500 −0.220750 0.975330i $$-0.570851\pi$$
−0.220750 + 0.975330i $$0.570851\pi$$
$$572$$ 0 0
$$573$$ −3720.00 −0.271213
$$574$$ 0 0
$$575$$ −1050.00 −0.0761531
$$576$$ 0 0
$$577$$ 554.000 0.0399711 0.0199855 0.999800i $$-0.493638\pi$$
0.0199855 + 0.999800i $$0.493638\pi$$
$$578$$ 0 0
$$579$$ −29380.0 −2.10879
$$580$$ 0 0
$$581$$ −11556.0 −0.825170
$$582$$ 0 0
$$583$$ 11808.0 0.838829
$$584$$ 0 0
$$585$$ 2190.00 0.154778
$$586$$ 0 0
$$587$$ 2386.00 0.167770 0.0838848 0.996475i $$-0.473267\pi$$
0.0838848 + 0.996475i $$0.473267\pi$$
$$588$$ 0 0
$$589$$ 23312.0 1.63082
$$590$$ 0 0
$$591$$ −22340.0 −1.55490
$$592$$ 0 0
$$593$$ −846.000 −0.0585853 −0.0292926 0.999571i $$-0.509325\pi$$
−0.0292926 + 0.999571i $$0.509325\pi$$
$$594$$ 0 0
$$595$$ 540.000 0.0372065
$$596$$ 0 0
$$597$$ −30480.0 −2.08955
$$598$$ 0 0
$$599$$ −22304.0 −1.52140 −0.760698 0.649105i $$-0.775143\pi$$
−0.760698 + 0.649105i $$0.775143\pi$$
$$600$$ 0 0
$$601$$ 5510.00 0.373973 0.186986 0.982363i $$-0.440128\pi$$
0.186986 + 0.982363i $$0.440128\pi$$
$$602$$ 0 0
$$603$$ 40442.0 2.73122
$$604$$ 0 0
$$605$$ 5375.00 0.361198
$$606$$ 0 0
$$607$$ 8234.00 0.550589 0.275295 0.961360i $$-0.411225\pi$$
0.275295 + 0.961360i $$0.411225\pi$$
$$608$$ 0 0
$$609$$ −25560.0 −1.70073
$$610$$ 0 0
$$611$$ 228.000 0.0150964
$$612$$ 0 0
$$613$$ −1046.00 −0.0689193 −0.0344597 0.999406i $$-0.510971\pi$$
−0.0344597 + 0.999406i $$0.510971\pi$$
$$614$$ 0 0
$$615$$ 2700.00 0.177032
$$616$$ 0 0
$$617$$ −3862.00 −0.251991 −0.125995 0.992031i $$-0.540212\pi$$
−0.125995 + 0.992031i $$0.540212\pi$$
$$618$$ 0 0
$$619$$ −13964.0 −0.906721 −0.453361 0.891327i $$-0.649775\pi$$
−0.453361 + 0.891327i $$0.649775\pi$$
$$620$$ 0 0
$$621$$ 19320.0 1.24845
$$622$$ 0 0
$$623$$ −15372.0 −0.988549
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ 0 0
$$627$$ −19840.0 −1.26369
$$628$$ 0 0
$$629$$ −1212.00 −0.0768293
$$630$$ 0 0
$$631$$ 14884.0 0.939022 0.469511 0.882927i $$-0.344430\pi$$
0.469511 + 0.882927i $$0.344430\pi$$
$$632$$ 0 0
$$633$$ 48960.0 3.07423
$$634$$ 0 0
$$635$$ −5770.00 −0.360591
$$636$$ 0 0
$$637$$ 114.000 0.00709081
$$638$$ 0 0
$$639$$ −10220.0 −0.632703
$$640$$ 0 0
$$641$$ 17838.0 1.09916 0.549578 0.835443i $$-0.314789\pi$$
0.549578 + 0.835443i $$0.314789\pi$$
$$642$$ 0 0
$$643$$ 7814.00 0.479244 0.239622 0.970866i $$-0.422976\pi$$
0.239622 + 0.970866i $$0.422976\pi$$
$$644$$ 0 0
$$645$$ −3300.00 −0.201453
$$646$$ 0 0
$$647$$ −774.000 −0.0470310 −0.0235155 0.999723i $$-0.507486\pi$$
−0.0235155 + 0.999723i $$0.507486\pi$$
$$648$$ 0 0
$$649$$ −9024.00 −0.545798
$$650$$ 0 0
$$651$$ −33840.0 −2.03732
$$652$$ 0 0
$$653$$ −23422.0 −1.40364 −0.701818 0.712357i $$-0.747628\pi$$
−0.701818 + 0.712357i $$0.747628\pi$$
$$654$$ 0 0
$$655$$ −1840.00 −0.109763
$$656$$ 0 0
$$657$$ 64386.0 3.82334
$$658$$ 0 0
$$659$$ 13508.0 0.798478 0.399239 0.916847i $$-0.369274\pi$$
0.399239 + 0.916847i $$0.369274\pi$$
$$660$$ 0 0
$$661$$ −6222.00 −0.366124 −0.183062 0.983101i $$-0.558601\pi$$
−0.183062 + 0.983101i $$0.558601\pi$$
$$662$$ 0 0
$$663$$ −360.000 −0.0210878
$$664$$ 0 0
$$665$$ −11160.0 −0.650776
$$666$$ 0 0
$$667$$ −5964.00 −0.346217
$$668$$ 0 0
$$669$$ −53020.0 −3.06408
$$670$$ 0 0
$$671$$ −4192.00 −0.241178
$$672$$ 0 0
$$673$$ −15566.0 −0.891568 −0.445784 0.895141i $$-0.647075\pi$$
−0.445784 + 0.895141i $$0.647075\pi$$
$$674$$ 0 0
$$675$$ −11500.0 −0.655756
$$676$$ 0 0
$$677$$ 2234.00 0.126824 0.0634118 0.997987i $$-0.479802\pi$$
0.0634118 + 0.997987i $$0.479802\pi$$
$$678$$ 0 0
$$679$$ −8604.00 −0.486290
$$680$$ 0 0
$$681$$ −37780.0 −2.12589
$$682$$ 0 0
$$683$$ −13282.0 −0.744102 −0.372051 0.928212i $$-0.621345\pi$$
−0.372051 + 0.928212i $$0.621345\pi$$
$$684$$ 0 0
$$685$$ 3350.00 0.186857
$$686$$ 0 0
$$687$$ 30340.0 1.68492
$$688$$ 0 0
$$689$$ −4428.00 −0.244838
$$690$$ 0 0
$$691$$ 27416.0 1.50934 0.754670 0.656105i $$-0.227797\pi$$
0.754670 + 0.656105i $$0.227797\pi$$
$$692$$ 0 0
$$693$$ 21024.0 1.15243
$$694$$ 0 0
$$695$$ −2860.00 −0.156095
$$696$$ 0 0
$$697$$ −324.000 −0.0176074
$$698$$ 0 0
$$699$$ 34780.0 1.88197
$$700$$ 0 0
$$701$$ 25626.0 1.38071 0.690357 0.723469i $$-0.257453\pi$$
0.690357 + 0.723469i $$0.257453\pi$$
$$702$$ 0 0
$$703$$ 25048.0 1.34382
$$704$$ 0 0
$$705$$ −1900.00 −0.101501
$$706$$ 0 0
$$707$$ −32292.0 −1.71777
$$708$$ 0 0
$$709$$ 11702.0 0.619856 0.309928 0.950760i $$-0.399695\pi$$
0.309928 + 0.950760i $$0.399695\pi$$
$$710$$ 0 0
$$711$$ 84680.0 4.46659
$$712$$ 0 0
$$713$$ −7896.00 −0.414737
$$714$$ 0 0
$$715$$ 480.000 0.0251063
$$716$$ 0 0
$$717$$ 15600.0 0.812542
$$718$$ 0 0
$$719$$ 28008.0 1.45274 0.726371 0.687302i $$-0.241205\pi$$
0.726371 + 0.687302i $$0.241205\pi$$
$$720$$ 0 0
$$721$$ −11556.0 −0.596904
$$722$$ 0 0
$$723$$ 32180.0 1.65531
$$724$$ 0 0
$$725$$ 3550.00 0.181853
$$726$$ 0 0
$$727$$ 7682.00 0.391898 0.195949 0.980614i $$-0.437221\pi$$
0.195949 + 0.980614i $$0.437221\pi$$
$$728$$ 0 0
$$729$$ 67717.0 3.44038
$$730$$ 0 0
$$731$$ 396.000 0.0200364
$$732$$ 0 0
$$733$$ −14270.0 −0.719065 −0.359532 0.933133i $$-0.617064\pi$$
−0.359532 + 0.933133i $$0.617064\pi$$
$$734$$ 0 0
$$735$$ −950.000 −0.0476752
$$736$$ 0 0
$$737$$ 8864.00 0.443025
$$738$$ 0 0
$$739$$ −29324.0 −1.45968 −0.729838 0.683620i $$-0.760405\pi$$
−0.729838 + 0.683620i $$0.760405\pi$$
$$740$$ 0 0
$$741$$ 7440.00 0.368846
$$742$$ 0 0
$$743$$ −29258.0 −1.44465 −0.722323 0.691556i $$-0.756926\pi$$
−0.722323 + 0.691556i $$0.756926\pi$$
$$744$$ 0 0
$$745$$ −8650.00 −0.425385
$$746$$ 0 0
$$747$$ −46866.0 −2.29550
$$748$$ 0 0
$$749$$ 15300.0 0.746395
$$750$$ 0 0
$$751$$ −19076.0 −0.926888 −0.463444 0.886126i $$-0.653387\pi$$
−0.463444 + 0.886126i $$0.653387\pi$$
$$752$$ 0 0
$$753$$ 6880.00 0.332963
$$754$$ 0 0
$$755$$ 6620.00 0.319108
$$756$$ 0 0
$$757$$ −22670.0 −1.08845 −0.544224 0.838940i $$-0.683176\pi$$
−0.544224 + 0.838940i $$0.683176\pi$$
$$758$$ 0 0
$$759$$ 6720.00 0.321371
$$760$$ 0 0
$$761$$ −23206.0 −1.10541 −0.552705 0.833377i $$-0.686404\pi$$
−0.552705 + 0.833377i $$0.686404\pi$$
$$762$$ 0 0
$$763$$ 11988.0 0.568800
$$764$$ 0 0
$$765$$ 2190.00 0.103503
$$766$$ 0 0
$$767$$ 3384.00 0.159308
$$768$$ 0 0
$$769$$ −1854.00 −0.0869401 −0.0434701 0.999055i $$-0.513841\pi$$
−0.0434701 + 0.999055i $$0.513841\pi$$
$$770$$ 0 0
$$771$$ −21700.0 −1.01363
$$772$$ 0 0
$$773$$ 6474.00 0.301234 0.150617 0.988592i $$-0.451874\pi$$
0.150617 + 0.988592i $$0.451874\pi$$
$$774$$ 0 0
$$775$$ 4700.00 0.217844
$$776$$ 0 0
$$777$$ −36360.0 −1.67877
$$778$$ 0 0
$$779$$ 6696.00 0.307971
$$780$$ 0 0
$$781$$ −2240.00 −0.102629
$$782$$ 0 0
$$783$$ −65320.0 −2.98129
$$784$$ 0 0
$$785$$ −14730.0 −0.669728
$$786$$ 0 0
$$787$$ 20354.0 0.921908 0.460954 0.887424i $$-0.347507\pi$$
0.460954 + 0.887424i $$0.347507\pi$$
$$788$$ 0 0
$$789$$ 22740.0 1.02607
$$790$$ 0 0
$$791$$ −26028.0 −1.16997
$$792$$ 0 0
$$793$$ 1572.00 0.0703952
$$794$$ 0 0
$$795$$ 36900.0 1.64617
$$796$$ 0 0
$$797$$ −1886.00 −0.0838213 −0.0419106 0.999121i $$-0.513344\pi$$
−0.0419106 + 0.999121i $$0.513344\pi$$
$$798$$ 0 0
$$799$$ 228.000 0.0100952
$$800$$ 0 0
$$801$$ −62342.0 −2.75000
$$802$$ 0 0
$$803$$ 14112.0 0.620176
$$804$$ 0 0
$$805$$ 3780.00 0.165500
$$806$$ 0 0
$$807$$ −71460.0 −3.11711
$$808$$ 0 0
$$809$$ −9462.00 −0.411207 −0.205603 0.978635i $$-0.565916\pi$$
−0.205603 + 0.978635i $$0.565916\pi$$
$$810$$ 0 0
$$811$$ −24512.0 −1.06132 −0.530661 0.847584i $$-0.678056\pi$$
−0.530661 + 0.847584i $$0.678056\pi$$
$$812$$ 0 0
$$813$$ 26040.0 1.12332
$$814$$ 0 0
$$815$$ 10490.0 0.450857
$$816$$ 0 0
$$817$$ −8184.00 −0.350455
$$818$$ 0 0
$$819$$ −7884.00 −0.336373
$$820$$ 0 0
$$821$$ 36242.0 1.54063 0.770313 0.637666i $$-0.220100\pi$$
0.770313 + 0.637666i $$0.220100\pi$$
$$822$$ 0 0
$$823$$ 17718.0 0.750438 0.375219 0.926936i $$-0.377567\pi$$
0.375219 + 0.926936i $$0.377567\pi$$
$$824$$ 0 0
$$825$$ −4000.00 −0.168803
$$826$$ 0 0
$$827$$ −6726.00 −0.282812 −0.141406 0.989952i $$-0.545162\pi$$
−0.141406 + 0.989952i $$0.545162\pi$$
$$828$$ 0 0
$$829$$ 41722.0 1.74797 0.873984 0.485955i $$-0.161528\pi$$
0.873984 + 0.485955i $$0.161528\pi$$
$$830$$ 0 0
$$831$$ 51500.0 2.14984
$$832$$ 0 0
$$833$$ 114.000 0.00474174
$$834$$ 0 0
$$835$$ −4330.00 −0.179456
$$836$$ 0 0
$$837$$ −86480.0 −3.57131
$$838$$ 0 0
$$839$$ −16720.0 −0.688008 −0.344004 0.938968i $$-0.611783\pi$$
−0.344004 + 0.938968i $$0.611783\pi$$
$$840$$ 0 0
$$841$$ −4225.00 −0.173234
$$842$$ 0 0
$$843$$ −52700.0 −2.15313
$$844$$ 0 0
$$845$$ 10805.0 0.439886
$$846$$ 0 0
$$847$$ −19350.0 −0.784975
$$848$$ 0 0
$$849$$ 34340.0 1.38816
$$850$$ 0 0
$$851$$ −8484.00 −0.341748
$$852$$ 0 0
$$853$$ −33286.0 −1.33610 −0.668049 0.744118i $$-0.732870\pi$$
−0.668049 + 0.744118i $$0.732870\pi$$
$$854$$ 0 0
$$855$$ −45260.0 −1.81036
$$856$$ 0 0
$$857$$ 38978.0 1.55363 0.776816 0.629727i $$-0.216833\pi$$
0.776816 + 0.629727i $$0.216833\pi$$
$$858$$ 0 0
$$859$$ 1916.00 0.0761037 0.0380518 0.999276i $$-0.487885\pi$$
0.0380518 + 0.999276i $$0.487885\pi$$
$$860$$ 0 0
$$861$$ −9720.00 −0.384735
$$862$$ 0 0
$$863$$ 2374.00 0.0936407 0.0468203 0.998903i $$-0.485091\pi$$
0.0468203 + 0.998903i $$0.485091\pi$$
$$864$$ 0 0
$$865$$ 8390.00 0.329790
$$866$$ 0 0
$$867$$ 48770.0 1.91040
$$868$$ 0 0
$$869$$ 18560.0 0.724517
$$870$$ 0 0
$$871$$ −3324.00 −0.129310
$$872$$ 0 0
$$873$$ −34894.0 −1.35279
$$874$$ 0 0
$$875$$ −2250.00 −0.0869302
$$876$$ 0 0
$$877$$ 32722.0 1.25991 0.629956 0.776631i $$-0.283073\pi$$
0.629956 + 0.776631i $$0.283073\pi$$
$$878$$ 0 0
$$879$$ 98780.0 3.79041
$$880$$ 0 0
$$881$$ 5390.00 0.206122 0.103061 0.994675i $$-0.467136\pi$$
0.103061 + 0.994675i $$0.467136\pi$$
$$882$$ 0 0
$$883$$ 43238.0 1.64788 0.823938 0.566680i $$-0.191772\pi$$
0.823938 + 0.566680i $$0.191772\pi$$
$$884$$ 0 0
$$885$$ −28200.0 −1.07111
$$886$$ 0 0
$$887$$ 11010.0 0.416775 0.208388 0.978046i $$-0.433178\pi$$
0.208388 + 0.978046i $$0.433178\pi$$
$$888$$ 0 0
$$889$$ 20772.0 0.783656
$$890$$ 0 0
$$891$$ 42064.0 1.58159
$$892$$ 0 0
$$893$$ −4712.00 −0.176575
$$894$$ 0 0
$$895$$ −8100.00 −0.302517
$$896$$ 0 0
$$897$$ −2520.00 −0.0938020
$$898$$ 0 0
$$899$$ 26696.0 0.990391
$$900$$ 0 0
$$901$$ −4428.00 −0.163727
$$902$$ 0 0
$$903$$ 11880.0 0.437809
$$904$$ 0 0
$$905$$ −12550.0 −0.460968
$$906$$ 0 0
$$907$$ 74.0000 0.00270907 0.00135454 0.999999i $$-0.499569\pi$$
0.00135454 + 0.999999i $$0.499569\pi$$
$$908$$ 0 0
$$909$$ −130962. −4.77859
$$910$$ 0 0
$$911$$ −17460.0 −0.634990 −0.317495 0.948260i $$-0.602842\pi$$
−0.317495 + 0.948260i $$0.602842\pi$$
$$912$$ 0 0
$$913$$ −10272.0 −0.372348
$$914$$ 0 0
$$915$$ −13100.0 −0.473303
$$916$$ 0 0
$$917$$ 6624.00 0.238543
$$918$$ 0 0
$$919$$ −17072.0 −0.612789 −0.306395 0.951905i $$-0.599123\pi$$
−0.306395 + 0.951905i $$0.599123\pi$$
$$920$$ 0 0
$$921$$ 80540.0 2.88152
$$922$$ 0 0
$$923$$ 840.000 0.0299555
$$924$$ 0 0
$$925$$ 5050.00 0.179506
$$926$$ 0 0
$$927$$ −46866.0 −1.66050
$$928$$ 0 0
$$929$$ −14826.0 −0.523601 −0.261800 0.965122i $$-0.584316\pi$$
−0.261800 + 0.965122i $$0.584316\pi$$
$$930$$ 0 0
$$931$$ −2356.00 −0.0829375
$$932$$ 0 0
$$933$$ 54920.0 1.92712
$$934$$ 0 0
$$935$$ 480.000 0.0167890
$$936$$ 0 0
$$937$$ 3354.00 0.116937 0.0584687 0.998289i $$-0.481378\pi$$
0.0584687 + 0.998289i $$0.481378\pi$$
$$938$$ 0 0
$$939$$ 4220.00 0.146661
$$940$$ 0 0
$$941$$ −15434.0 −0.534680 −0.267340 0.963602i $$-0.586145\pi$$
−0.267340 + 0.963602i $$0.586145\pi$$
$$942$$ 0 0
$$943$$ −2268.00 −0.0783205
$$944$$ 0 0
$$945$$ 41400.0 1.42512
$$946$$ 0 0
$$947$$ 9306.00 0.319329 0.159664 0.987171i $$-0.448959\pi$$
0.159664 + 0.987171i $$0.448959\pi$$
$$948$$ 0 0
$$949$$ −5292.00 −0.181017
$$950$$ 0 0
$$951$$ −61940.0 −2.11203
$$952$$ 0 0
$$953$$ 12202.0 0.414755 0.207378 0.978261i $$-0.433507\pi$$
0.207378 + 0.978261i $$0.433507\pi$$
$$954$$ 0 0
$$955$$ −1860.00 −0.0630243
$$956$$ 0 0
$$957$$ −22720.0 −0.767433
$$958$$ 0 0
$$959$$ −12060.0 −0.406087
$$960$$ 0 0
$$961$$ 5553.00 0.186399
$$962$$ 0 0
$$963$$ 62050.0 2.07636
$$964$$ 0 0
$$965$$ −14690.0 −0.490039
$$966$$ 0 0
$$967$$ −17478.0 −0.581235 −0.290618 0.956839i $$-0.593861\pi$$
−0.290618 + 0.956839i $$0.593861\pi$$
$$968$$ 0 0
$$969$$ 7440.00 0.246653
$$970$$ 0 0
$$971$$ −10920.0 −0.360906 −0.180453 0.983584i $$-0.557756\pi$$
−0.180453 + 0.983584i $$0.557756\pi$$
$$972$$ 0 0
$$973$$ 10296.0 0.339234
$$974$$ 0 0
$$975$$ 1500.00 0.0492702
$$976$$ 0 0
$$977$$ 10834.0 0.354770 0.177385 0.984142i $$-0.443236\pi$$
0.177385 + 0.984142i $$0.443236\pi$$
$$978$$ 0 0
$$979$$ −13664.0 −0.446071
$$980$$ 0 0
$$981$$ 48618.0 1.58232
$$982$$ 0 0
$$983$$ 36862.0 1.19605 0.598024 0.801478i $$-0.295953\pi$$
0.598024 + 0.801478i $$0.295953\pi$$
$$984$$ 0 0
$$985$$ −11170.0 −0.361326
$$986$$ 0 0
$$987$$ 6840.00 0.220587
$$988$$ 0 0
$$989$$ 2772.00 0.0891248
$$990$$ 0 0
$$991$$ −5380.00 −0.172453 −0.0862267 0.996276i $$-0.527481\pi$$
−0.0862267 + 0.996276i $$0.527481\pi$$
$$992$$ 0 0
$$993$$ 76880.0 2.45691
$$994$$ 0 0
$$995$$ −15240.0 −0.485568
$$996$$ 0 0
$$997$$ 31266.0 0.993184 0.496592 0.867984i $$-0.334585\pi$$
0.496592 + 0.867984i $$0.334585\pi$$
$$998$$ 0 0
$$999$$ −92920.0 −2.94280
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 80.4.a.a.1.1 1
3.2 odd 2 720.4.a.ba.1.1 1
4.3 odd 2 40.4.a.c.1.1 1
5.2 odd 4 400.4.c.a.49.2 2
5.3 odd 4 400.4.c.a.49.1 2
5.4 even 2 400.4.a.u.1.1 1
8.3 odd 2 320.4.a.a.1.1 1
8.5 even 2 320.4.a.n.1.1 1
12.11 even 2 360.4.a.i.1.1 1
16.3 odd 4 1280.4.d.o.641.2 2
16.5 even 4 1280.4.d.b.641.2 2
16.11 odd 4 1280.4.d.o.641.1 2
16.13 even 4 1280.4.d.b.641.1 2
20.3 even 4 200.4.c.a.49.2 2
20.7 even 4 200.4.c.a.49.1 2
20.19 odd 2 200.4.a.a.1.1 1
28.27 even 2 1960.4.a.a.1.1 1
40.19 odd 2 1600.4.a.ca.1.1 1
40.29 even 2 1600.4.a.a.1.1 1
60.23 odd 4 1800.4.f.n.649.2 2
60.47 odd 4 1800.4.f.n.649.1 2
60.59 even 2 1800.4.a.bd.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
40.4.a.c.1.1 1 4.3 odd 2
80.4.a.a.1.1 1 1.1 even 1 trivial
200.4.a.a.1.1 1 20.19 odd 2
200.4.c.a.49.1 2 20.7 even 4
200.4.c.a.49.2 2 20.3 even 4
320.4.a.a.1.1 1 8.3 odd 2
320.4.a.n.1.1 1 8.5 even 2
360.4.a.i.1.1 1 12.11 even 2
400.4.a.u.1.1 1 5.4 even 2
400.4.c.a.49.1 2 5.3 odd 4
400.4.c.a.49.2 2 5.2 odd 4
720.4.a.ba.1.1 1 3.2 odd 2
1280.4.d.b.641.1 2 16.13 even 4
1280.4.d.b.641.2 2 16.5 even 4
1280.4.d.o.641.1 2 16.11 odd 4
1280.4.d.o.641.2 2 16.3 odd 4
1600.4.a.a.1.1 1 40.29 even 2
1600.4.a.ca.1.1 1 40.19 odd 2
1800.4.a.bd.1.1 1 60.59 even 2
1800.4.f.n.649.1 2 60.47 odd 4
1800.4.f.n.649.2 2 60.23 odd 4
1960.4.a.a.1.1 1 28.27 even 2