# Properties

 Label 980.2.a.j.1.2 Level $980$ Weight $2$ Character 980.1 Self dual yes Analytic conductor $7.825$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,2,Mod(1,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("980.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 980.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: $$7.82533939809$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 980.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} +O(q^{10})$$ $$q+0.414214 q^{3} +1.00000 q^{5} -2.82843 q^{9} -3.82843 q^{11} -3.58579 q^{13} +0.414214 q^{15} -6.41421 q^{17} -3.65685 q^{19} +0.585786 q^{23} +1.00000 q^{25} -2.41421 q^{27} +6.65685 q^{29} -4.58579 q^{31} -1.58579 q^{33} -3.41421 q^{37} -1.48528 q^{39} -0.585786 q^{41} +11.6569 q^{43} -2.82843 q^{45} +8.89949 q^{47} -2.65685 q^{51} -3.75736 q^{53} -3.82843 q^{55} -1.51472 q^{57} -3.41421 q^{59} -5.17157 q^{61} -3.58579 q^{65} -11.0711 q^{67} +0.242641 q^{69} +6.48528 q^{71} -5.17157 q^{73} +0.414214 q^{75} +13.1421 q^{79} +7.48528 q^{81} +8.00000 q^{83} -6.41421 q^{85} +2.75736 q^{87} -16.9706 q^{89} -1.89949 q^{93} -3.65685 q^{95} -15.7279 q^{97} +10.8284 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^5 $$2 q - 2 q^{3} + 2 q^{5} - 2 q^{11} - 10 q^{13} - 2 q^{15} - 10 q^{17} + 4 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{27} + 2 q^{29} - 12 q^{31} - 6 q^{33} - 4 q^{37} + 14 q^{39} - 4 q^{41} + 12 q^{43} - 2 q^{47} + 6 q^{51} - 16 q^{53} - 2 q^{55} - 20 q^{57} - 4 q^{59} - 16 q^{61} - 10 q^{65} - 8 q^{67} - 8 q^{69} - 4 q^{71} - 16 q^{73} - 2 q^{75} - 2 q^{79} - 2 q^{81} + 16 q^{83} - 10 q^{85} + 14 q^{87} + 16 q^{93} + 4 q^{95} - 6 q^{97} + 16 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^11 - 10 * q^13 - 2 * q^15 - 10 * q^17 + 4 * q^19 + 4 * q^23 + 2 * q^25 - 2 * q^27 + 2 * q^29 - 12 * q^31 - 6 * q^33 - 4 * q^37 + 14 * q^39 - 4 * q^41 + 12 * q^43 - 2 * q^47 + 6 * q^51 - 16 * q^53 - 2 * q^55 - 20 * q^57 - 4 * q^59 - 16 * q^61 - 10 * q^65 - 8 * q^67 - 8 * q^69 - 4 * q^71 - 16 * q^73 - 2 * q^75 - 2 * q^79 - 2 * q^81 + 16 * q^83 - 10 * q^85 + 14 * q^87 + 16 * q^93 + 4 * q^95 - 6 * q^97 + 16 * q^99

## 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
$$3$$ 0.414214 0.239146 0.119573 0.992825i $$-0.461847\pi$$
0.119573 + 0.992825i $$0.461847\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 0 0
$$9$$ −2.82843 −0.942809
$$10$$ 0 0
$$11$$ −3.82843 −1.15431 −0.577157 0.816633i $$-0.695838\pi$$
−0.577157 + 0.816633i $$0.695838\pi$$
$$12$$ 0 0
$$13$$ −3.58579 −0.994518 −0.497259 0.867602i $$-0.665660\pi$$
−0.497259 + 0.867602i $$0.665660\pi$$
$$14$$ 0 0
$$15$$ 0.414214 0.106949
$$16$$ 0 0
$$17$$ −6.41421 −1.55568 −0.777838 0.628465i $$-0.783683\pi$$
−0.777838 + 0.628465i $$0.783683\pi$$
$$18$$ 0 0
$$19$$ −3.65685 −0.838940 −0.419470 0.907769i $$-0.637784\pi$$
−0.419470 + 0.907769i $$0.637784\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ 0.585786 0.122145 0.0610725 0.998133i $$-0.480548\pi$$
0.0610725 + 0.998133i $$0.480548\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −2.41421 −0.464616
$$28$$ 0 0
$$29$$ 6.65685 1.23615 0.618073 0.786120i $$-0.287913\pi$$
0.618073 + 0.786120i $$0.287913\pi$$
$$30$$ 0 0
$$31$$ −4.58579 −0.823632 −0.411816 0.911267i $$-0.635105\pi$$
−0.411816 + 0.911267i $$0.635105\pi$$
$$32$$ 0 0
$$33$$ −1.58579 −0.276050
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −3.41421 −0.561293 −0.280647 0.959811i $$-0.590549\pi$$
−0.280647 + 0.959811i $$0.590549\pi$$
$$38$$ 0 0
$$39$$ −1.48528 −0.237835
$$40$$ 0 0
$$41$$ −0.585786 −0.0914845 −0.0457422 0.998953i $$-0.514565\pi$$
−0.0457422 + 0.998953i $$0.514565\pi$$
$$42$$ 0 0
$$43$$ 11.6569 1.77765 0.888827 0.458243i $$-0.151521\pi$$
0.888827 + 0.458243i $$0.151521\pi$$
$$44$$ 0 0
$$45$$ −2.82843 −0.421637
$$46$$ 0 0
$$47$$ 8.89949 1.29812 0.649062 0.760735i $$-0.275161\pi$$
0.649062 + 0.760735i $$0.275161\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 0 0
$$51$$ −2.65685 −0.372034
$$52$$ 0 0
$$53$$ −3.75736 −0.516113 −0.258056 0.966130i $$-0.583082\pi$$
−0.258056 + 0.966130i $$0.583082\pi$$
$$54$$ 0 0
$$55$$ −3.82843 −0.516225
$$56$$ 0 0
$$57$$ −1.51472 −0.200629
$$58$$ 0 0
$$59$$ −3.41421 −0.444493 −0.222246 0.974991i $$-0.571339\pi$$
−0.222246 + 0.974991i $$0.571339\pi$$
$$60$$ 0 0
$$61$$ −5.17157 −0.662152 −0.331076 0.943604i $$-0.607412\pi$$
−0.331076 + 0.943604i $$0.607412\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −3.58579 −0.444762
$$66$$ 0 0
$$67$$ −11.0711 −1.35255 −0.676273 0.736651i $$-0.736406\pi$$
−0.676273 + 0.736651i $$0.736406\pi$$
$$68$$ 0 0
$$69$$ 0.242641 0.0292105
$$70$$ 0 0
$$71$$ 6.48528 0.769661 0.384831 0.922987i $$-0.374260\pi$$
0.384831 + 0.922987i $$0.374260\pi$$
$$72$$ 0 0
$$73$$ −5.17157 −0.605287 −0.302643 0.953104i $$-0.597869\pi$$
−0.302643 + 0.953104i $$0.597869\pi$$
$$74$$ 0 0
$$75$$ 0.414214 0.0478293
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 13.1421 1.47861 0.739303 0.673373i $$-0.235155\pi$$
0.739303 + 0.673373i $$0.235155\pi$$
$$80$$ 0 0
$$81$$ 7.48528 0.831698
$$82$$ 0 0
$$83$$ 8.00000 0.878114 0.439057 0.898459i $$-0.355313\pi$$
0.439057 + 0.898459i $$0.355313\pi$$
$$84$$ 0 0
$$85$$ −6.41421 −0.695719
$$86$$ 0 0
$$87$$ 2.75736 0.295620
$$88$$ 0 0
$$89$$ −16.9706 −1.79888 −0.899438 0.437048i $$-0.856024\pi$$
−0.899438 + 0.437048i $$0.856024\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ −1.89949 −0.196968
$$94$$ 0 0
$$95$$ −3.65685 −0.375185
$$96$$ 0 0
$$97$$ −15.7279 −1.59693 −0.798464 0.602042i $$-0.794354\pi$$
−0.798464 + 0.602042i $$0.794354\pi$$
$$98$$ 0 0
$$99$$ 10.8284 1.08830
$$100$$ 0 0
$$101$$ 4.82843 0.480446 0.240223 0.970718i $$-0.422779\pi$$
0.240223 + 0.970718i $$0.422779\pi$$
$$102$$ 0 0
$$103$$ −1.58579 −0.156252 −0.0781261 0.996943i $$-0.524894\pi$$
−0.0781261 + 0.996943i $$0.524894\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −16.8284 −1.62687 −0.813433 0.581659i $$-0.802404\pi$$
−0.813433 + 0.581659i $$0.802404\pi$$
$$108$$ 0 0
$$109$$ 9.00000 0.862044 0.431022 0.902342i $$-0.358153\pi$$
0.431022 + 0.902342i $$0.358153\pi$$
$$110$$ 0 0
$$111$$ −1.41421 −0.134231
$$112$$ 0 0
$$113$$ −5.07107 −0.477046 −0.238523 0.971137i $$-0.576663\pi$$
−0.238523 + 0.971137i $$0.576663\pi$$
$$114$$ 0 0
$$115$$ 0.585786 0.0546249
$$116$$ 0 0
$$117$$ 10.1421 0.937641
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 3.65685 0.332441
$$122$$ 0 0
$$123$$ −0.242641 −0.0218782
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −21.8995 −1.94327 −0.971633 0.236494i $$-0.924002\pi$$
−0.971633 + 0.236494i $$0.924002\pi$$
$$128$$ 0 0
$$129$$ 4.82843 0.425119
$$130$$ 0 0
$$131$$ 11.7574 1.02725 0.513623 0.858016i $$-0.328303\pi$$
0.513623 + 0.858016i $$0.328303\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 0 0
$$135$$ −2.41421 −0.207782
$$136$$ 0 0
$$137$$ 12.9706 1.10815 0.554075 0.832467i $$-0.313072\pi$$
0.554075 + 0.832467i $$0.313072\pi$$
$$138$$ 0 0
$$139$$ 13.8995 1.17894 0.589470 0.807790i $$-0.299337\pi$$
0.589470 + 0.807790i $$0.299337\pi$$
$$140$$ 0 0
$$141$$ 3.68629 0.310442
$$142$$ 0 0
$$143$$ 13.7279 1.14799
$$144$$ 0 0
$$145$$ 6.65685 0.552822
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 0 0
$$149$$ 3.51472 0.287937 0.143968 0.989582i $$-0.454014\pi$$
0.143968 + 0.989582i $$0.454014\pi$$
$$150$$ 0 0
$$151$$ −11.8284 −0.962584 −0.481292 0.876560i $$-0.659832\pi$$
−0.481292 + 0.876560i $$0.659832\pi$$
$$152$$ 0 0
$$153$$ 18.1421 1.46670
$$154$$ 0 0
$$155$$ −4.58579 −0.368339
$$156$$ 0 0
$$157$$ −10.4853 −0.836817 −0.418408 0.908259i $$-0.637412\pi$$
−0.418408 + 0.908259i $$0.637412\pi$$
$$158$$ 0 0
$$159$$ −1.55635 −0.123427
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 21.5563 1.68842 0.844212 0.536010i $$-0.180069\pi$$
0.844212 + 0.536010i $$0.180069\pi$$
$$164$$ 0 0
$$165$$ −1.58579 −0.123453
$$166$$ 0 0
$$167$$ −2.41421 −0.186817 −0.0934087 0.995628i $$-0.529776\pi$$
−0.0934087 + 0.995628i $$0.529776\pi$$
$$168$$ 0 0
$$169$$ −0.142136 −0.0109335
$$170$$ 0 0
$$171$$ 10.3431 0.790960
$$172$$ 0 0
$$173$$ −18.5563 −1.41081 −0.705407 0.708803i $$-0.749236\pi$$
−0.705407 + 0.708803i $$0.749236\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −1.41421 −0.106299
$$178$$ 0 0
$$179$$ 6.48528 0.484733 0.242366 0.970185i $$-0.422076\pi$$
0.242366 + 0.970185i $$0.422076\pi$$
$$180$$ 0 0
$$181$$ −1.75736 −0.130623 −0.0653117 0.997865i $$-0.520804\pi$$
−0.0653117 + 0.997865i $$0.520804\pi$$
$$182$$ 0 0
$$183$$ −2.14214 −0.158351
$$184$$ 0 0
$$185$$ −3.41421 −0.251018
$$186$$ 0 0
$$187$$ 24.5563 1.79574
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 16.6569 1.20525 0.602624 0.798025i $$-0.294122\pi$$
0.602624 + 0.798025i $$0.294122\pi$$
$$192$$ 0 0
$$193$$ 5.65685 0.407189 0.203595 0.979055i $$-0.434738\pi$$
0.203595 + 0.979055i $$0.434738\pi$$
$$194$$ 0 0
$$195$$ −1.48528 −0.106363
$$196$$ 0 0
$$197$$ −27.5563 −1.96331 −0.981654 0.190669i $$-0.938934\pi$$
−0.981654 + 0.190669i $$0.938934\pi$$
$$198$$ 0 0
$$199$$ 26.7279 1.89469 0.947346 0.320212i $$-0.103754\pi$$
0.947346 + 0.320212i $$0.103754\pi$$
$$200$$ 0 0
$$201$$ −4.58579 −0.323456
$$202$$ 0 0
$$203$$ 0 0
$$204$$ 0 0
$$205$$ −0.585786 −0.0409131
$$206$$ 0 0
$$207$$ −1.65685 −0.115159
$$208$$ 0 0
$$209$$ 14.0000 0.968400
$$210$$ 0 0
$$211$$ −24.3137 −1.67382 −0.836912 0.547337i $$-0.815642\pi$$
−0.836912 + 0.547337i $$0.815642\pi$$
$$212$$ 0 0
$$213$$ 2.68629 0.184062
$$214$$ 0 0
$$215$$ 11.6569 0.794991
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 0 0
$$219$$ −2.14214 −0.144752
$$220$$ 0 0
$$221$$ 23.0000 1.54715
$$222$$ 0 0
$$223$$ −24.0711 −1.61192 −0.805959 0.591971i $$-0.798350\pi$$
−0.805959 + 0.591971i $$0.798350\pi$$
$$224$$ 0 0
$$225$$ −2.82843 −0.188562
$$226$$ 0 0
$$227$$ −7.72792 −0.512920 −0.256460 0.966555i $$-0.582556\pi$$
−0.256460 + 0.966555i $$0.582556\pi$$
$$228$$ 0 0
$$229$$ −23.8995 −1.57932 −0.789662 0.613543i $$-0.789744\pi$$
−0.789662 + 0.613543i $$0.789744\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −9.17157 −0.600850 −0.300425 0.953805i $$-0.597128\pi$$
−0.300425 + 0.953805i $$0.597128\pi$$
$$234$$ 0 0
$$235$$ 8.89949 0.580539
$$236$$ 0 0
$$237$$ 5.44365 0.353603
$$238$$ 0 0
$$239$$ 14.1716 0.916683 0.458341 0.888776i $$-0.348444\pi$$
0.458341 + 0.888776i $$0.348444\pi$$
$$240$$ 0 0
$$241$$ 3.55635 0.229085 0.114542 0.993418i $$-0.463460\pi$$
0.114542 + 0.993418i $$0.463460\pi$$
$$242$$ 0 0
$$243$$ 10.3431 0.663513
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 13.1127 0.834341
$$248$$ 0 0
$$249$$ 3.31371 0.209998
$$250$$ 0 0
$$251$$ 27.0711 1.70871 0.854355 0.519689i $$-0.173952\pi$$
0.854355 + 0.519689i $$0.173952\pi$$
$$252$$ 0 0
$$253$$ −2.24264 −0.140994
$$254$$ 0 0
$$255$$ −2.65685 −0.166379
$$256$$ 0 0
$$257$$ 25.7990 1.60930 0.804648 0.593752i $$-0.202354\pi$$
0.804648 + 0.593752i $$0.202354\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −18.8284 −1.16545
$$262$$ 0 0
$$263$$ 10.0000 0.616626 0.308313 0.951285i $$-0.400236\pi$$
0.308313 + 0.951285i $$0.400236\pi$$
$$264$$ 0 0
$$265$$ −3.75736 −0.230813
$$266$$ 0 0
$$267$$ −7.02944 −0.430195
$$268$$ 0 0
$$269$$ 16.7279 1.01992 0.509960 0.860198i $$-0.329660\pi$$
0.509960 + 0.860198i $$0.329660\pi$$
$$270$$ 0 0
$$271$$ −28.9706 −1.75984 −0.879918 0.475125i $$-0.842403\pi$$
−0.879918 + 0.475125i $$0.842403\pi$$
$$272$$ 0 0
$$273$$ 0 0
$$274$$ 0 0
$$275$$ −3.82843 −0.230863
$$276$$ 0 0
$$277$$ −7.41421 −0.445477 −0.222738 0.974878i $$-0.571500\pi$$
−0.222738 + 0.974878i $$0.571500\pi$$
$$278$$ 0 0
$$279$$ 12.9706 0.776527
$$280$$ 0 0
$$281$$ −1.00000 −0.0596550 −0.0298275 0.999555i $$-0.509496\pi$$
−0.0298275 + 0.999555i $$0.509496\pi$$
$$282$$ 0 0
$$283$$ 18.7574 1.11501 0.557505 0.830174i $$-0.311759\pi$$
0.557505 + 0.830174i $$0.311759\pi$$
$$284$$ 0 0
$$285$$ −1.51472 −0.0897242
$$286$$ 0 0
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 24.1421 1.42013
$$290$$ 0 0
$$291$$ −6.51472 −0.381900
$$292$$ 0 0
$$293$$ 14.4142 0.842087 0.421044 0.907040i $$-0.361664\pi$$
0.421044 + 0.907040i $$0.361664\pi$$
$$294$$ 0 0
$$295$$ −3.41421 −0.198783
$$296$$ 0 0
$$297$$ 9.24264 0.536312
$$298$$ 0 0
$$299$$ −2.10051 −0.121475
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 2.00000 0.114897
$$304$$ 0 0
$$305$$ −5.17157 −0.296123
$$306$$ 0 0
$$307$$ −3.58579 −0.204652 −0.102326 0.994751i $$-0.532628\pi$$
−0.102326 + 0.994751i $$0.532628\pi$$
$$308$$ 0 0
$$309$$ −0.656854 −0.0373671
$$310$$ 0 0
$$311$$ −6.97056 −0.395264 −0.197632 0.980276i $$-0.563325\pi$$
−0.197632 + 0.980276i $$0.563325\pi$$
$$312$$ 0 0
$$313$$ −16.5563 −0.935820 −0.467910 0.883776i $$-0.654993\pi$$
−0.467910 + 0.883776i $$0.654993\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −14.9706 −0.840831 −0.420415 0.907332i $$-0.638116\pi$$
−0.420415 + 0.907332i $$0.638116\pi$$
$$318$$ 0 0
$$319$$ −25.4853 −1.42690
$$320$$ 0 0
$$321$$ −6.97056 −0.389059
$$322$$ 0 0
$$323$$ 23.4558 1.30512
$$324$$ 0 0
$$325$$ −3.58579 −0.198904
$$326$$ 0 0
$$327$$ 3.72792 0.206155
$$328$$ 0 0
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 4.82843 0.265394 0.132697 0.991157i $$-0.457636\pi$$
0.132697 + 0.991157i $$0.457636\pi$$
$$332$$ 0 0
$$333$$ 9.65685 0.529192
$$334$$ 0 0
$$335$$ −11.0711 −0.604877
$$336$$ 0 0
$$337$$ −18.7279 −1.02017 −0.510087 0.860123i $$-0.670387\pi$$
−0.510087 + 0.860123i $$0.670387\pi$$
$$338$$ 0 0
$$339$$ −2.10051 −0.114084
$$340$$ 0 0
$$341$$ 17.5563 0.950730
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 0 0
$$345$$ 0.242641 0.0130633
$$346$$ 0 0
$$347$$ −7.41421 −0.398016 −0.199008 0.979998i $$-0.563772\pi$$
−0.199008 + 0.979998i $$0.563772\pi$$
$$348$$ 0 0
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 0 0
$$351$$ 8.65685 0.462069
$$352$$ 0 0
$$353$$ 2.07107 0.110232 0.0551159 0.998480i $$-0.482447\pi$$
0.0551159 + 0.998480i $$0.482447\pi$$
$$354$$ 0 0
$$355$$ 6.48528 0.344203
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ 11.3137 0.597115 0.298557 0.954392i $$-0.403495\pi$$
0.298557 + 0.954392i $$0.403495\pi$$
$$360$$ 0 0
$$361$$ −5.62742 −0.296180
$$362$$ 0 0
$$363$$ 1.51472 0.0795021
$$364$$ 0 0
$$365$$ −5.17157 −0.270692
$$366$$ 0 0
$$367$$ −19.7279 −1.02979 −0.514895 0.857254i $$-0.672169\pi$$
−0.514895 + 0.857254i $$0.672169\pi$$
$$368$$ 0 0
$$369$$ 1.65685 0.0862524
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 16.4853 0.853576 0.426788 0.904352i $$-0.359645\pi$$
0.426788 + 0.904352i $$0.359645\pi$$
$$374$$ 0 0
$$375$$ 0.414214 0.0213899
$$376$$ 0 0
$$377$$ −23.8701 −1.22937
$$378$$ 0 0
$$379$$ 14.0000 0.719132 0.359566 0.933120i $$-0.382925\pi$$
0.359566 + 0.933120i $$0.382925\pi$$
$$380$$ 0 0
$$381$$ −9.07107 −0.464725
$$382$$ 0 0
$$383$$ 3.51472 0.179594 0.0897969 0.995960i $$-0.471378\pi$$
0.0897969 + 0.995960i $$0.471378\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −32.9706 −1.67599
$$388$$ 0 0
$$389$$ 13.1421 0.666333 0.333166 0.942868i $$-0.391883\pi$$
0.333166 + 0.942868i $$0.391883\pi$$
$$390$$ 0 0
$$391$$ −3.75736 −0.190018
$$392$$ 0 0
$$393$$ 4.87006 0.245662
$$394$$ 0 0
$$395$$ 13.1421 0.661253
$$396$$ 0 0
$$397$$ −14.4142 −0.723429 −0.361714 0.932289i $$-0.617808\pi$$
−0.361714 + 0.932289i $$0.617808\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 0.514719 0.0257038 0.0128519 0.999917i $$-0.495909\pi$$
0.0128519 + 0.999917i $$0.495909\pi$$
$$402$$ 0 0
$$403$$ 16.4437 0.819117
$$404$$ 0 0
$$405$$ 7.48528 0.371947
$$406$$ 0 0
$$407$$ 13.0711 0.647909
$$408$$ 0 0
$$409$$ −32.8284 −1.62326 −0.811631 0.584171i $$-0.801420\pi$$
−0.811631 + 0.584171i $$0.801420\pi$$
$$410$$ 0 0
$$411$$ 5.37258 0.265010
$$412$$ 0 0
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 8.00000 0.392705
$$416$$ 0 0
$$417$$ 5.75736 0.281939
$$418$$ 0 0
$$419$$ 9.55635 0.466858 0.233429 0.972374i $$-0.425005\pi$$
0.233429 + 0.972374i $$0.425005\pi$$
$$420$$ 0 0
$$421$$ 20.3137 0.990030 0.495015 0.868885i $$-0.335163\pi$$
0.495015 + 0.868885i $$0.335163\pi$$
$$422$$ 0 0
$$423$$ −25.1716 −1.22388
$$424$$ 0 0
$$425$$ −6.41421 −0.311135
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 0 0
$$429$$ 5.68629 0.274537
$$430$$ 0 0
$$431$$ −9.82843 −0.473419 −0.236709 0.971581i $$-0.576069\pi$$
−0.236709 + 0.971581i $$0.576069\pi$$
$$432$$ 0 0
$$433$$ 22.2843 1.07091 0.535457 0.844563i $$-0.320139\pi$$
0.535457 + 0.844563i $$0.320139\pi$$
$$434$$ 0 0
$$435$$ 2.75736 0.132205
$$436$$ 0 0
$$437$$ −2.14214 −0.102472
$$438$$ 0 0
$$439$$ −24.2426 −1.15704 −0.578519 0.815669i $$-0.696369\pi$$
−0.578519 + 0.815669i $$0.696369\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 0 0
$$443$$ −35.4558 −1.68456 −0.842279 0.539042i $$-0.818786\pi$$
−0.842279 + 0.539042i $$0.818786\pi$$
$$444$$ 0 0
$$445$$ −16.9706 −0.804482
$$446$$ 0 0
$$447$$ 1.45584 0.0688591
$$448$$ 0 0
$$449$$ 23.8284 1.12453 0.562267 0.826956i $$-0.309930\pi$$
0.562267 + 0.826956i $$0.309930\pi$$
$$450$$ 0 0
$$451$$ 2.24264 0.105602
$$452$$ 0 0
$$453$$ −4.89949 −0.230198
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −15.0711 −0.704995 −0.352497 0.935813i $$-0.614667\pi$$
−0.352497 + 0.935813i $$0.614667\pi$$
$$458$$ 0 0
$$459$$ 15.4853 0.722791
$$460$$ 0 0
$$461$$ −22.3431 −1.04062 −0.520312 0.853976i $$-0.674184\pi$$
−0.520312 + 0.853976i $$0.674184\pi$$
$$462$$ 0 0
$$463$$ −13.4558 −0.625346 −0.312673 0.949861i $$-0.601224\pi$$
−0.312673 + 0.949861i $$0.601224\pi$$
$$464$$ 0 0
$$465$$ −1.89949 −0.0880870
$$466$$ 0 0
$$467$$ −0.899495 −0.0416237 −0.0208118 0.999783i $$-0.506625\pi$$
−0.0208118 + 0.999783i $$0.506625\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 0 0
$$471$$ −4.34315 −0.200122
$$472$$ 0 0
$$473$$ −44.6274 −2.05197
$$474$$ 0 0
$$475$$ −3.65685 −0.167788
$$476$$ 0 0
$$477$$ 10.6274 0.486596
$$478$$ 0 0
$$479$$ −9.41421 −0.430146 −0.215073 0.976598i $$-0.568999\pi$$
−0.215073 + 0.976598i $$0.568999\pi$$
$$480$$ 0 0
$$481$$ 12.2426 0.558216
$$482$$ 0 0
$$483$$ 0 0
$$484$$ 0 0
$$485$$ −15.7279 −0.714168
$$486$$ 0 0
$$487$$ −7.41421 −0.335970 −0.167985 0.985790i $$-0.553726\pi$$
−0.167985 + 0.985790i $$0.553726\pi$$
$$488$$ 0 0
$$489$$ 8.92893 0.403780
$$490$$ 0 0
$$491$$ 17.2843 0.780028 0.390014 0.920809i $$-0.372470\pi$$
0.390014 + 0.920809i $$0.372470\pi$$
$$492$$ 0 0
$$493$$ −42.6985 −1.92304
$$494$$ 0 0
$$495$$ 10.8284 0.486702
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −13.6274 −0.610047 −0.305023 0.952345i $$-0.598664\pi$$
−0.305023 + 0.952345i $$0.598664\pi$$
$$500$$ 0 0
$$501$$ −1.00000 −0.0446767
$$502$$ 0 0
$$503$$ −39.0416 −1.74078 −0.870390 0.492363i $$-0.836133\pi$$
−0.870390 + 0.492363i $$0.836133\pi$$
$$504$$ 0 0
$$505$$ 4.82843 0.214862
$$506$$ 0 0
$$507$$ −0.0588745 −0.00261471
$$508$$ 0 0
$$509$$ 35.0711 1.55450 0.777249 0.629193i $$-0.216615\pi$$
0.777249 + 0.629193i $$0.216615\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 0 0
$$513$$ 8.82843 0.389785
$$514$$ 0 0
$$515$$ −1.58579 −0.0698781
$$516$$ 0 0
$$517$$ −34.0711 −1.49844
$$518$$ 0 0
$$519$$ −7.68629 −0.337391
$$520$$ 0 0
$$521$$ −6.68629 −0.292932 −0.146466 0.989216i $$-0.546790\pi$$
−0.146466 + 0.989216i $$0.546790\pi$$
$$522$$ 0 0
$$523$$ −1.85786 −0.0812387 −0.0406194 0.999175i $$-0.512933\pi$$
−0.0406194 + 0.999175i $$0.512933\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 29.4142 1.28130
$$528$$ 0 0
$$529$$ −22.6569 −0.985081
$$530$$ 0 0
$$531$$ 9.65685 0.419072
$$532$$ 0 0
$$533$$ 2.10051 0.0909830
$$534$$ 0 0
$$535$$ −16.8284 −0.727556
$$536$$ 0 0
$$537$$ 2.68629 0.115922
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 15.9706 0.686628 0.343314 0.939221i $$-0.388450\pi$$
0.343314 + 0.939221i $$0.388450\pi$$
$$542$$ 0 0
$$543$$ −0.727922 −0.0312381
$$544$$ 0 0
$$545$$ 9.00000 0.385518
$$546$$ 0 0
$$547$$ 7.51472 0.321306 0.160653 0.987011i $$-0.448640\pi$$
0.160653 + 0.987011i $$0.448640\pi$$
$$548$$ 0 0
$$549$$ 14.6274 0.624283
$$550$$ 0 0
$$551$$ −24.3431 −1.03705
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 0 0
$$555$$ −1.41421 −0.0600300
$$556$$ 0 0
$$557$$ −7.79899 −0.330454 −0.165227 0.986256i $$-0.552836\pi$$
−0.165227 + 0.986256i $$0.552836\pi$$
$$558$$ 0 0
$$559$$ −41.7990 −1.76791
$$560$$ 0 0
$$561$$ 10.1716 0.429444
$$562$$ 0 0
$$563$$ 20.6274 0.869342 0.434671 0.900589i $$-0.356865\pi$$
0.434671 + 0.900589i $$0.356865\pi$$
$$564$$ 0 0
$$565$$ −5.07107 −0.213341
$$566$$ 0 0
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −31.7990 −1.33308 −0.666542 0.745468i $$-0.732226\pi$$
−0.666542 + 0.745468i $$0.732226\pi$$
$$570$$ 0 0
$$571$$ 4.82843 0.202063 0.101032 0.994883i $$-0.467786\pi$$
0.101032 + 0.994883i $$0.467786\pi$$
$$572$$ 0 0
$$573$$ 6.89949 0.288231
$$574$$ 0 0
$$575$$ 0.585786 0.0244290
$$576$$ 0 0
$$577$$ 14.0711 0.585786 0.292893 0.956145i $$-0.405382\pi$$
0.292893 + 0.956145i $$0.405382\pi$$
$$578$$ 0 0
$$579$$ 2.34315 0.0973778
$$580$$ 0 0
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 14.3848 0.595757
$$584$$ 0 0
$$585$$ 10.1421 0.419326
$$586$$ 0 0
$$587$$ 30.8284 1.27243 0.636213 0.771514i $$-0.280500\pi$$
0.636213 + 0.771514i $$0.280500\pi$$
$$588$$ 0 0
$$589$$ 16.7696 0.690977
$$590$$ 0 0
$$591$$ −11.4142 −0.469518
$$592$$ 0 0
$$593$$ −17.7279 −0.727999 −0.363999 0.931399i $$-0.618589\pi$$
−0.363999 + 0.931399i $$0.618589\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 11.0711 0.453109
$$598$$ 0 0
$$599$$ −12.7990 −0.522953 −0.261476 0.965210i $$-0.584209\pi$$
−0.261476 + 0.965210i $$0.584209\pi$$
$$600$$ 0 0
$$601$$ −39.6569 −1.61764 −0.808818 0.588058i $$-0.799892\pi$$
−0.808818 + 0.588058i $$0.799892\pi$$
$$602$$ 0 0
$$603$$ 31.3137 1.27519
$$604$$ 0 0
$$605$$ 3.65685 0.148672
$$606$$ 0 0
$$607$$ −13.9289 −0.565358 −0.282679 0.959215i $$-0.591223\pi$$
−0.282679 + 0.959215i $$0.591223\pi$$
$$608$$ 0 0
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −31.9117 −1.29101
$$612$$ 0 0
$$613$$ −41.3137 −1.66864 −0.834322 0.551277i $$-0.814141\pi$$
−0.834322 + 0.551277i $$0.814141\pi$$
$$614$$ 0 0
$$615$$ −0.242641 −0.00978422
$$616$$ 0 0
$$617$$ 40.8701 1.64537 0.822683 0.568500i $$-0.192476\pi$$
0.822683 + 0.568500i $$0.192476\pi$$
$$618$$ 0 0
$$619$$ −10.8701 −0.436905 −0.218452 0.975848i $$-0.570101\pi$$
−0.218452 + 0.975848i $$0.570101\pi$$
$$620$$ 0 0
$$621$$ −1.41421 −0.0567504
$$622$$ 0 0
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 5.79899 0.231589
$$628$$ 0 0
$$629$$ 21.8995 0.873190
$$630$$ 0 0
$$631$$ −39.4853 −1.57188 −0.785942 0.618300i $$-0.787822\pi$$
−0.785942 + 0.618300i $$0.787822\pi$$
$$632$$ 0 0
$$633$$ −10.0711 −0.400289
$$634$$ 0 0
$$635$$ −21.8995 −0.869055
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 0 0
$$639$$ −18.3431 −0.725644
$$640$$ 0 0
$$641$$ 39.3137 1.55280 0.776399 0.630242i $$-0.217044\pi$$
0.776399 + 0.630242i $$0.217044\pi$$
$$642$$ 0 0
$$643$$ −4.21320 −0.166153 −0.0830763 0.996543i $$-0.526475\pi$$
−0.0830763 + 0.996543i $$0.526475\pi$$
$$644$$ 0 0
$$645$$ 4.82843 0.190119
$$646$$ 0 0
$$647$$ −35.1127 −1.38042 −0.690211 0.723608i $$-0.742482\pi$$
−0.690211 + 0.723608i $$0.742482\pi$$
$$648$$ 0 0
$$649$$ 13.0711 0.513084
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 0 0
$$653$$ −26.6863 −1.04432 −0.522158 0.852849i $$-0.674873\pi$$
−0.522158 + 0.852849i $$0.674873\pi$$
$$654$$ 0 0
$$655$$ 11.7574 0.459398
$$656$$ 0 0
$$657$$ 14.6274 0.570670
$$658$$ 0 0
$$659$$ −20.6569 −0.804677 −0.402338 0.915491i $$-0.631802\pi$$
−0.402338 + 0.915491i $$0.631802\pi$$
$$660$$ 0 0
$$661$$ 34.1421 1.32798 0.663988 0.747744i $$-0.268863\pi$$
0.663988 + 0.747744i $$0.268863\pi$$
$$662$$ 0 0
$$663$$ 9.52691 0.369995
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 3.89949 0.150989
$$668$$ 0 0
$$669$$ −9.97056 −0.385484
$$670$$ 0 0
$$671$$ 19.7990 0.764332
$$672$$ 0 0
$$673$$ −27.5147 −1.06061 −0.530307 0.847806i $$-0.677923\pi$$
−0.530307 + 0.847806i $$0.677923\pi$$
$$674$$ 0 0
$$675$$ −2.41421 −0.0929231
$$676$$ 0 0
$$677$$ 10.2721 0.394788 0.197394 0.980324i $$-0.436752\pi$$
0.197394 + 0.980324i $$0.436752\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −3.20101 −0.122663
$$682$$ 0 0
$$683$$ 35.1127 1.34355 0.671775 0.740755i $$-0.265532\pi$$
0.671775 + 0.740755i $$0.265532\pi$$
$$684$$ 0 0
$$685$$ 12.9706 0.495580
$$686$$ 0 0
$$687$$ −9.89949 −0.377689
$$688$$ 0 0
$$689$$ 13.4731 0.513284
$$690$$ 0 0
$$691$$ −3.51472 −0.133706 −0.0668531 0.997763i $$-0.521296\pi$$
−0.0668531 + 0.997763i $$0.521296\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 13.8995 0.527238
$$696$$ 0 0
$$697$$ 3.75736 0.142320
$$698$$ 0 0
$$699$$ −3.79899 −0.143691
$$700$$ 0 0
$$701$$ 0.514719 0.0194407 0.00972033 0.999953i $$-0.496906\pi$$
0.00972033 + 0.999953i $$0.496906\pi$$
$$702$$ 0 0
$$703$$ 12.4853 0.470891
$$704$$ 0 0
$$705$$ 3.68629 0.138834
$$706$$ 0 0
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 3.62742 0.136231 0.0681153 0.997677i $$-0.478301\pi$$
0.0681153 + 0.997677i $$0.478301\pi$$
$$710$$ 0 0
$$711$$ −37.1716 −1.39404
$$712$$ 0 0
$$713$$ −2.68629 −0.100602
$$714$$ 0 0
$$715$$ 13.7279 0.513395
$$716$$ 0 0
$$717$$ 5.87006 0.219221
$$718$$ 0 0
$$719$$ −15.7574 −0.587650 −0.293825 0.955859i $$-0.594928\pi$$
−0.293825 + 0.955859i $$0.594928\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 0 0
$$723$$ 1.47309 0.0547847
$$724$$ 0 0
$$725$$ 6.65685 0.247229
$$726$$ 0 0
$$727$$ 18.0000 0.667583 0.333792 0.942647i $$-0.391672\pi$$
0.333792 + 0.942647i $$0.391672\pi$$
$$728$$ 0 0
$$729$$ −18.1716 −0.673021
$$730$$ 0 0
$$731$$ −74.7696 −2.76545
$$732$$ 0 0
$$733$$ 30.6985 1.13387 0.566937 0.823761i $$-0.308128\pi$$
0.566937 + 0.823761i $$0.308128\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 42.3848 1.56126
$$738$$ 0 0
$$739$$ 20.6569 0.759875 0.379937 0.925012i $$-0.375946\pi$$
0.379937 + 0.925012i $$0.375946\pi$$
$$740$$ 0 0
$$741$$ 5.43146 0.199530
$$742$$ 0 0
$$743$$ −8.92893 −0.327571 −0.163785 0.986496i $$-0.552370\pi$$
−0.163785 + 0.986496i $$0.552370\pi$$
$$744$$ 0 0
$$745$$ 3.51472 0.128769
$$746$$ 0 0
$$747$$ −22.6274 −0.827894
$$748$$ 0 0
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 1.14214 0.0416771 0.0208386 0.999783i $$-0.493366\pi$$
0.0208386 + 0.999783i $$0.493366\pi$$
$$752$$ 0 0
$$753$$ 11.2132 0.408632
$$754$$ 0 0
$$755$$ −11.8284 −0.430481
$$756$$ 0 0
$$757$$ 37.1716 1.35102 0.675512 0.737349i $$-0.263923\pi$$
0.675512 + 0.737349i $$0.263923\pi$$
$$758$$ 0 0
$$759$$ −0.928932 −0.0337181
$$760$$ 0 0
$$761$$ 18.8701 0.684039 0.342020 0.939693i $$-0.388889\pi$$
0.342020 + 0.939693i $$0.388889\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 18.1421 0.655930
$$766$$ 0 0
$$767$$ 12.2426 0.442056
$$768$$ 0 0
$$769$$ −28.1421 −1.01483 −0.507416 0.861701i $$-0.669399\pi$$
−0.507416 + 0.861701i $$0.669399\pi$$
$$770$$ 0 0
$$771$$ 10.6863 0.384857
$$772$$ 0 0
$$773$$ 43.7279 1.57278 0.786392 0.617728i $$-0.211947\pi$$
0.786392 + 0.617728i $$0.211947\pi$$
$$774$$ 0 0
$$775$$ −4.58579 −0.164726
$$776$$ 0 0
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 2.14214 0.0767500
$$780$$ 0 0
$$781$$ −24.8284 −0.888431
$$782$$ 0 0
$$783$$ −16.0711 −0.574333
$$784$$ 0 0
$$785$$ −10.4853 −0.374236
$$786$$ 0 0
$$787$$ 43.7279 1.55873 0.779366 0.626569i $$-0.215541\pi$$
0.779366 + 0.626569i $$0.215541\pi$$
$$788$$ 0 0
$$789$$ 4.14214 0.147464
$$790$$ 0 0
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 18.5442 0.658522
$$794$$ 0 0
$$795$$ −1.55635 −0.0551980
$$796$$ 0 0
$$797$$ 35.3848 1.25339 0.626697 0.779263i $$-0.284407\pi$$
0.626697 + 0.779263i $$0.284407\pi$$
$$798$$ 0 0
$$799$$ −57.0833 −2.01946
$$800$$ 0 0
$$801$$ 48.0000 1.69600
$$802$$ 0 0
$$803$$ 19.7990 0.698691
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 6.92893 0.243910
$$808$$ 0 0
$$809$$ 6.02944 0.211984 0.105992 0.994367i $$-0.466198\pi$$
0.105992 + 0.994367i $$0.466198\pi$$
$$810$$ 0 0
$$811$$ 19.5563 0.686716 0.343358 0.939205i $$-0.388436\pi$$
0.343358 + 0.939205i $$0.388436\pi$$
$$812$$ 0 0
$$813$$ −12.0000 −0.420858
$$814$$ 0 0
$$815$$ 21.5563 0.755086
$$816$$ 0 0
$$817$$ −42.6274 −1.49134
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 47.8284 1.66922 0.834612 0.550839i $$-0.185692\pi$$
0.834612 + 0.550839i $$0.185692\pi$$
$$822$$ 0 0
$$823$$ 36.5269 1.27325 0.636624 0.771174i $$-0.280330\pi$$
0.636624 + 0.771174i $$0.280330\pi$$
$$824$$ 0 0
$$825$$ −1.58579 −0.0552100
$$826$$ 0 0
$$827$$ −10.0416 −0.349182 −0.174591 0.984641i $$-0.555860\pi$$
−0.174591 + 0.984641i $$0.555860\pi$$
$$828$$ 0 0
$$829$$ −22.7279 −0.789373 −0.394687 0.918816i $$-0.629147\pi$$
−0.394687 + 0.918816i $$0.629147\pi$$
$$830$$ 0 0
$$831$$ −3.07107 −0.106534
$$832$$ 0 0
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −2.41421 −0.0835473
$$836$$ 0 0
$$837$$ 11.0711 0.382672
$$838$$ 0 0
$$839$$ −22.3848 −0.772808 −0.386404 0.922330i $$-0.626283\pi$$
−0.386404 + 0.922330i $$0.626283\pi$$
$$840$$ 0 0
$$841$$ 15.3137 0.528059
$$842$$ 0 0
$$843$$ −0.414214 −0.0142663
$$844$$ 0 0
$$845$$ −0.142136 −0.00488961
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 7.76955 0.266650
$$850$$ 0 0
$$851$$ −2.00000 −0.0685591
$$852$$ 0 0
$$853$$ 30.2843 1.03691 0.518457 0.855104i $$-0.326507\pi$$
0.518457 + 0.855104i $$0.326507\pi$$
$$854$$ 0 0
$$855$$ 10.3431 0.353728
$$856$$ 0 0
$$857$$ 20.8284 0.711486 0.355743 0.934584i $$-0.384228\pi$$
0.355743 + 0.934584i $$0.384228\pi$$
$$858$$ 0 0
$$859$$ 45.4558 1.55093 0.775467 0.631388i $$-0.217515\pi$$
0.775467 + 0.631388i $$0.217515\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 0 0
$$863$$ −19.7574 −0.672548 −0.336274 0.941764i $$-0.609167\pi$$
−0.336274 + 0.941764i $$0.609167\pi$$
$$864$$ 0 0
$$865$$ −18.5563 −0.630935
$$866$$ 0 0
$$867$$ 10.0000 0.339618
$$868$$ 0 0
$$869$$ −50.3137 −1.70678
$$870$$ 0 0
$$871$$ 39.6985 1.34513
$$872$$ 0 0
$$873$$ 44.4853 1.50560
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 10.6863 0.360850 0.180425 0.983589i $$-0.442253\pi$$
0.180425 + 0.983589i $$0.442253\pi$$
$$878$$ 0 0
$$879$$ 5.97056 0.201382
$$880$$ 0 0
$$881$$ 56.2843 1.89627 0.948133 0.317875i $$-0.102969\pi$$
0.948133 + 0.317875i $$0.102969\pi$$
$$882$$ 0 0
$$883$$ −7.11270 −0.239361 −0.119681 0.992812i $$-0.538187\pi$$
−0.119681 + 0.992812i $$0.538187\pi$$
$$884$$ 0 0
$$885$$ −1.41421 −0.0475383
$$886$$ 0 0
$$887$$ −57.5980 −1.93395 −0.966975 0.254870i $$-0.917967\pi$$
−0.966975 + 0.254870i $$0.917967\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −28.6569 −0.960041
$$892$$ 0 0
$$893$$ −32.5442 −1.08905
$$894$$ 0 0
$$895$$ 6.48528 0.216779
$$896$$ 0 0
$$897$$ −0.870058 −0.0290504
$$898$$ 0 0
$$899$$ −30.5269 −1.01813
$$900$$ 0 0
$$901$$ 24.1005 0.802904
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −1.75736 −0.0584166
$$906$$ 0 0
$$907$$ 22.5269 0.747994 0.373997 0.927430i $$-0.377987\pi$$
0.373997 + 0.927430i $$0.377987\pi$$
$$908$$ 0 0
$$909$$ −13.6569 −0.452969
$$910$$ 0 0
$$911$$ −55.5980 −1.84204 −0.921022 0.389511i $$-0.872644\pi$$
−0.921022 + 0.389511i $$0.872644\pi$$
$$912$$ 0 0
$$913$$ −30.6274 −1.01362
$$914$$ 0 0
$$915$$ −2.14214 −0.0708168
$$916$$ 0 0
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −18.5147 −0.610744 −0.305372 0.952233i $$-0.598781\pi$$
−0.305372 + 0.952233i $$0.598781\pi$$
$$920$$ 0 0
$$921$$ −1.48528 −0.0489417
$$922$$ 0 0
$$923$$ −23.2548 −0.765442
$$924$$ 0 0
$$925$$ −3.41421 −0.112259
$$926$$ 0 0
$$927$$ 4.48528 0.147316
$$928$$ 0 0
$$929$$ 21.2721 0.697914 0.348957 0.937139i $$-0.386536\pi$$
0.348957 + 0.937139i $$0.386536\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 0 0
$$933$$ −2.88730 −0.0945260
$$934$$ 0 0
$$935$$ 24.5563 0.803078
$$936$$ 0 0
$$937$$ 7.72792 0.252460 0.126230 0.992001i $$-0.459712\pi$$
0.126230 + 0.992001i $$0.459712\pi$$
$$938$$ 0 0
$$939$$ −6.85786 −0.223798
$$940$$ 0 0
$$941$$ 4.00000 0.130396 0.0651981 0.997872i $$-0.479232\pi$$
0.0651981 + 0.997872i $$0.479232\pi$$
$$942$$ 0 0
$$943$$ −0.343146 −0.0111744
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −32.0416 −1.04121 −0.520607 0.853797i $$-0.674294\pi$$
−0.520607 + 0.853797i $$0.674294\pi$$
$$948$$ 0 0
$$949$$ 18.5442 0.601969
$$950$$ 0 0
$$951$$ −6.20101 −0.201082
$$952$$ 0 0
$$953$$ −38.8701 −1.25912 −0.629562 0.776950i $$-0.716766\pi$$
−0.629562 + 0.776950i $$0.716766\pi$$
$$954$$ 0 0
$$955$$ 16.6569 0.539003
$$956$$ 0 0
$$957$$ −10.5563 −0.341238
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ −9.97056 −0.321631
$$962$$ 0 0
$$963$$ 47.5980 1.53382
$$964$$ 0 0
$$965$$ 5.65685 0.182101
$$966$$ 0 0
$$967$$ −7.45584 −0.239764 −0.119882 0.992788i $$-0.538252\pi$$
−0.119882 + 0.992788i $$0.538252\pi$$
$$968$$ 0 0
$$969$$ 9.71573 0.312114
$$970$$ 0 0
$$971$$ 22.5269 0.722923 0.361462 0.932387i $$-0.382278\pi$$
0.361462 + 0.932387i $$0.382278\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ 0 0
$$975$$ −1.48528 −0.0475671
$$976$$ 0 0
$$977$$ −8.14214 −0.260490 −0.130245 0.991482i $$-0.541576\pi$$
−0.130245 + 0.991482i $$0.541576\pi$$
$$978$$ 0 0
$$979$$ 64.9706 2.07647
$$980$$ 0 0
$$981$$ −25.4558 −0.812743
$$982$$ 0 0
$$983$$ 42.0122 1.33998 0.669990 0.742370i $$-0.266298\pi$$
0.669990 + 0.742370i $$0.266298\pi$$
$$984$$ 0 0
$$985$$ −27.5563 −0.878018
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 0 0
$$989$$ 6.82843 0.217131
$$990$$ 0 0
$$991$$ 25.3137 0.804116 0.402058 0.915614i $$-0.368295\pi$$
0.402058 + 0.915614i $$0.368295\pi$$
$$992$$ 0 0
$$993$$ 2.00000 0.0634681
$$994$$ 0 0
$$995$$ 26.7279 0.847332
$$996$$ 0 0
$$997$$ 50.8406 1.61014 0.805069 0.593181i $$-0.202128\pi$$
0.805069 + 0.593181i $$0.202128\pi$$
$$998$$ 0 0
$$999$$ 8.24264 0.260786
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 980.2.a.j.1.2 2
3.2 odd 2 8820.2.a.bg.1.2 2
4.3 odd 2 3920.2.a.bx.1.1 2
5.2 odd 4 4900.2.e.q.2549.2 4
5.3 odd 4 4900.2.e.q.2549.3 4
5.4 even 2 4900.2.a.z.1.1 2
7.2 even 3 980.2.i.l.361.1 4
7.3 odd 6 980.2.i.k.961.2 4
7.4 even 3 980.2.i.l.961.1 4
7.5 odd 6 980.2.i.k.361.2 4
7.6 odd 2 980.2.a.k.1.1 yes 2
21.20 even 2 8820.2.a.bl.1.2 2
28.27 even 2 3920.2.a.bo.1.2 2
35.13 even 4 4900.2.e.r.2549.2 4
35.27 even 4 4900.2.e.r.2549.3 4
35.34 odd 2 4900.2.a.x.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
980.2.a.j.1.2 2 1.1 even 1 trivial
980.2.a.k.1.1 yes 2 7.6 odd 2
980.2.i.k.361.2 4 7.5 odd 6
980.2.i.k.961.2 4 7.3 odd 6
980.2.i.l.361.1 4 7.2 even 3
980.2.i.l.961.1 4 7.4 even 3
3920.2.a.bo.1.2 2 28.27 even 2
3920.2.a.bx.1.1 2 4.3 odd 2
4900.2.a.x.1.2 2 35.34 odd 2
4900.2.a.z.1.1 2 5.4 even 2
4900.2.e.q.2549.2 4 5.2 odd 4
4900.2.e.q.2549.3 4 5.3 odd 4
4900.2.e.r.2549.2 4 35.13 even 4
4900.2.e.r.2549.3 4 35.27 even 4
8820.2.a.bg.1.2 2 3.2 odd 2
8820.2.a.bl.1.2 2 21.20 even 2