# Properties

 Label 3040.2.a.k.1.1 Level $3040$ Weight $2$ Character 3040.1 Self dual yes Analytic conductor $24.275$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3040,2,Mod(1,3040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3040, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3040.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3040 = 2^{5} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3040.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: $$24.2745222145$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.564.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 3$$ x^3 - x^2 - 5*x + 3 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 1.1 Root $$-2.08613$$ of defining polynomial Character $$\chi$$ $$=$$ 3040.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.08613 q^{3} -1.00000 q^{5} +0.648061 q^{7} +6.52420 q^{9} +O(q^{10})$$ $$q-3.08613 q^{3} -1.00000 q^{5} +0.648061 q^{7} +6.52420 q^{9} -1.35194 q^{11} +4.43807 q^{13} +3.08613 q^{15} +2.00000 q^{17} +1.00000 q^{19} -2.00000 q^{21} +3.35194 q^{23} +1.00000 q^{25} -10.8761 q^{27} -4.17226 q^{29} -4.17226 q^{31} +4.17226 q^{33} -0.648061 q^{35} -2.43807 q^{37} -13.6965 q^{39} +10.1723 q^{41} -6.82032 q^{43} -6.52420 q^{45} +0.648061 q^{47} -6.58002 q^{49} -6.17226 q^{51} +6.43807 q^{53} +1.35194 q^{55} -3.08613 q^{57} -10.3445 q^{59} -2.11644 q^{61} +4.22808 q^{63} -4.43807 q^{65} +12.1345 q^{67} -10.3445 q^{69} +3.82774 q^{71} -4.17226 q^{73} -3.08613 q^{75} -0.876139 q^{77} -2.00000 q^{79} +13.9926 q^{81} -9.52420 q^{83} -2.00000 q^{85} +12.8761 q^{87} +3.82774 q^{89} +2.87614 q^{91} +12.8761 q^{93} -1.00000 q^{95} +7.73419 q^{97} -8.82032 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 - 3 * q^5 + 4 * q^7 + 3 * q^9 $$3 q - 2 q^{3} - 3 q^{5} + 4 q^{7} + 3 q^{9} - 2 q^{11} + 4 q^{13} + 2 q^{15} + 6 q^{17} + 3 q^{19} - 6 q^{21} + 8 q^{23} + 3 q^{25} - 14 q^{27} + 2 q^{29} + 2 q^{31} - 2 q^{33} - 4 q^{35} + 2 q^{37} - 10 q^{39} + 16 q^{41} - 8 q^{43} - 3 q^{45} + 4 q^{47} + 3 q^{49} - 4 q^{51} + 10 q^{53} + 2 q^{55} - 2 q^{57} - 2 q^{59} + 2 q^{61} - 8 q^{63} - 4 q^{65} - 4 q^{67} - 2 q^{69} + 26 q^{71} + 2 q^{73} - 2 q^{75} + 16 q^{77} - 6 q^{79} + 15 q^{81} - 12 q^{83} - 6 q^{85} + 20 q^{87} + 26 q^{89} - 10 q^{91} + 20 q^{93} - 3 q^{95} + 18 q^{97} - 14 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 - 3 * q^5 + 4 * q^7 + 3 * q^9 - 2 * q^11 + 4 * q^13 + 2 * q^15 + 6 * q^17 + 3 * q^19 - 6 * q^21 + 8 * q^23 + 3 * q^25 - 14 * q^27 + 2 * q^29 + 2 * q^31 - 2 * q^33 - 4 * q^35 + 2 * q^37 - 10 * q^39 + 16 * q^41 - 8 * q^43 - 3 * q^45 + 4 * q^47 + 3 * q^49 - 4 * q^51 + 10 * q^53 + 2 * q^55 - 2 * q^57 - 2 * q^59 + 2 * q^61 - 8 * q^63 - 4 * q^65 - 4 * q^67 - 2 * q^69 + 26 * q^71 + 2 * q^73 - 2 * q^75 + 16 * q^77 - 6 * q^79 + 15 * q^81 - 12 * q^83 - 6 * q^85 + 20 * q^87 + 26 * q^89 - 10 * q^91 + 20 * q^93 - 3 * q^95 + 18 * q^97 - 14 * 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
$$3$$ −3.08613 −1.78178 −0.890889 0.454221i $$-0.849918\pi$$
−0.890889 + 0.454221i $$0.849918\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ 0.648061 0.244944 0.122472 0.992472i $$-0.460918\pi$$
0.122472 + 0.992472i $$0.460918\pi$$
$$8$$ 0 0
$$9$$ 6.52420 2.17473
$$10$$ 0 0
$$11$$ −1.35194 −0.407625 −0.203813 0.979010i $$-0.565333\pi$$
−0.203813 + 0.979010i $$0.565333\pi$$
$$12$$ 0 0
$$13$$ 4.43807 1.23090 0.615449 0.788176i $$-0.288975\pi$$
0.615449 + 0.788176i $$0.288975\pi$$
$$14$$ 0 0
$$15$$ 3.08613 0.796835
$$16$$ 0 0
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −2.00000 −0.436436
$$22$$ 0 0
$$23$$ 3.35194 0.698928 0.349464 0.936950i $$-0.386364\pi$$
0.349464 + 0.936950i $$0.386364\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −10.8761 −2.09311
$$28$$ 0 0
$$29$$ −4.17226 −0.774769 −0.387385 0.921918i $$-0.626621\pi$$
−0.387385 + 0.921918i $$0.626621\pi$$
$$30$$ 0 0
$$31$$ −4.17226 −0.749360 −0.374680 0.927154i $$-0.622247\pi$$
−0.374680 + 0.927154i $$0.622247\pi$$
$$32$$ 0 0
$$33$$ 4.17226 0.726297
$$34$$ 0 0
$$35$$ −0.648061 −0.109542
$$36$$ 0 0
$$37$$ −2.43807 −0.400816 −0.200408 0.979713i $$-0.564227\pi$$
−0.200408 + 0.979713i $$0.564227\pi$$
$$38$$ 0 0
$$39$$ −13.6965 −2.19319
$$40$$ 0 0
$$41$$ 10.1723 1.58864 0.794320 0.607499i $$-0.207827\pi$$
0.794320 + 0.607499i $$0.207827\pi$$
$$42$$ 0 0
$$43$$ −6.82032 −1.04009 −0.520045 0.854139i $$-0.674085\pi$$
−0.520045 + 0.854139i $$0.674085\pi$$
$$44$$ 0 0
$$45$$ −6.52420 −0.972570
$$46$$ 0 0
$$47$$ 0.648061 0.0945294 0.0472647 0.998882i $$-0.484950\pi$$
0.0472647 + 0.998882i $$0.484950\pi$$
$$48$$ 0 0
$$49$$ −6.58002 −0.940002
$$50$$ 0 0
$$51$$ −6.17226 −0.864289
$$52$$ 0 0
$$53$$ 6.43807 0.884337 0.442168 0.896932i $$-0.354209\pi$$
0.442168 + 0.896932i $$0.354209\pi$$
$$54$$ 0 0
$$55$$ 1.35194 0.182295
$$56$$ 0 0
$$57$$ −3.08613 −0.408768
$$58$$ 0 0
$$59$$ −10.3445 −1.34674 −0.673371 0.739305i $$-0.735154\pi$$
−0.673371 + 0.739305i $$0.735154\pi$$
$$60$$ 0 0
$$61$$ −2.11644 −0.270983 −0.135491 0.990779i $$-0.543261\pi$$
−0.135491 + 0.990779i $$0.543261\pi$$
$$62$$ 0 0
$$63$$ 4.22808 0.532688
$$64$$ 0 0
$$65$$ −4.43807 −0.550475
$$66$$ 0 0
$$67$$ 12.1345 1.48247 0.741234 0.671246i $$-0.234241\pi$$
0.741234 + 0.671246i $$0.234241\pi$$
$$68$$ 0 0
$$69$$ −10.3445 −1.24533
$$70$$ 0 0
$$71$$ 3.82774 0.454269 0.227135 0.973863i $$-0.427064\pi$$
0.227135 + 0.973863i $$0.427064\pi$$
$$72$$ 0 0
$$73$$ −4.17226 −0.488326 −0.244163 0.969734i $$-0.578513\pi$$
−0.244163 + 0.969734i $$0.578513\pi$$
$$74$$ 0 0
$$75$$ −3.08613 −0.356356
$$76$$ 0 0
$$77$$ −0.876139 −0.0998453
$$78$$ 0 0
$$79$$ −2.00000 −0.225018 −0.112509 0.993651i $$-0.535889\pi$$
−0.112509 + 0.993651i $$0.535889\pi$$
$$80$$ 0 0
$$81$$ 13.9926 1.55473
$$82$$ 0 0
$$83$$ −9.52420 −1.04542 −0.522708 0.852512i $$-0.675078\pi$$
−0.522708 + 0.852512i $$0.675078\pi$$
$$84$$ 0 0
$$85$$ −2.00000 −0.216930
$$86$$ 0 0
$$87$$ 12.8761 1.38047
$$88$$ 0 0
$$89$$ 3.82774 0.405740 0.202870 0.979206i $$-0.434973\pi$$
0.202870 + 0.979206i $$0.434973\pi$$
$$90$$ 0 0
$$91$$ 2.87614 0.301501
$$92$$ 0 0
$$93$$ 12.8761 1.33519
$$94$$ 0 0
$$95$$ −1.00000 −0.102598
$$96$$ 0 0
$$97$$ 7.73419 0.785288 0.392644 0.919691i $$-0.371560\pi$$
0.392644 + 0.919691i $$0.371560\pi$$
$$98$$ 0 0
$$99$$ −8.82032 −0.886476
$$100$$ 0 0
$$101$$ −1.35194 −0.134523 −0.0672615 0.997735i $$-0.521426\pi$$
−0.0672615 + 0.997735i $$0.521426\pi$$
$$102$$ 0 0
$$103$$ 3.79001 0.373441 0.186720 0.982413i $$-0.440214\pi$$
0.186720 + 0.982413i $$0.440214\pi$$
$$104$$ 0 0
$$105$$ 2.00000 0.195180
$$106$$ 0 0
$$107$$ 8.55451 0.826996 0.413498 0.910505i $$-0.364307\pi$$
0.413498 + 0.910505i $$0.364307\pi$$
$$108$$ 0 0
$$109$$ −7.04840 −0.675114 −0.337557 0.941305i $$-0.609601\pi$$
−0.337557 + 0.941305i $$0.609601\pi$$
$$110$$ 0 0
$$111$$ 7.52420 0.714165
$$112$$ 0 0
$$113$$ 18.6103 1.75071 0.875356 0.483478i $$-0.160627\pi$$
0.875356 + 0.483478i $$0.160627\pi$$
$$114$$ 0 0
$$115$$ −3.35194 −0.312570
$$116$$ 0 0
$$117$$ 28.9549 2.67688
$$118$$ 0 0
$$119$$ 1.29612 0.118815
$$120$$ 0 0
$$121$$ −9.17226 −0.833842
$$122$$ 0 0
$$123$$ −31.3929 −2.83060
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −1.79001 −0.158838 −0.0794188 0.996841i $$-0.525306\pi$$
−0.0794188 + 0.996841i $$0.525306\pi$$
$$128$$ 0 0
$$129$$ 21.0484 1.85321
$$130$$ 0 0
$$131$$ 17.2207 1.50458 0.752288 0.658834i $$-0.228950\pi$$
0.752288 + 0.658834i $$0.228950\pi$$
$$132$$ 0 0
$$133$$ 0.648061 0.0561940
$$134$$ 0 0
$$135$$ 10.8761 0.936069
$$136$$ 0 0
$$137$$ 10.1116 0.863895 0.431948 0.901899i $$-0.357827\pi$$
0.431948 + 0.901899i $$0.357827\pi$$
$$138$$ 0 0
$$139$$ 9.69646 0.822443 0.411222 0.911535i $$-0.365102\pi$$
0.411222 + 0.911535i $$0.365102\pi$$
$$140$$ 0 0
$$141$$ −2.00000 −0.168430
$$142$$ 0 0
$$143$$ −6.00000 −0.501745
$$144$$ 0 0
$$145$$ 4.17226 0.346487
$$146$$ 0 0
$$147$$ 20.3068 1.67488
$$148$$ 0 0
$$149$$ 5.88356 0.482000 0.241000 0.970525i $$-0.422525\pi$$
0.241000 + 0.970525i $$0.422525\pi$$
$$150$$ 0 0
$$151$$ −18.6284 −1.51596 −0.757980 0.652278i $$-0.773813\pi$$
−0.757980 + 0.652278i $$0.773813\pi$$
$$152$$ 0 0
$$153$$ 13.0484 1.05490
$$154$$ 0 0
$$155$$ 4.17226 0.335124
$$156$$ 0 0
$$157$$ −17.3929 −1.38811 −0.694053 0.719924i $$-0.744177\pi$$
−0.694053 + 0.719924i $$0.744177\pi$$
$$158$$ 0 0
$$159$$ −19.8687 −1.57569
$$160$$ 0 0
$$161$$ 2.17226 0.171198
$$162$$ 0 0
$$163$$ −1.17968 −0.0923996 −0.0461998 0.998932i $$-0.514711\pi$$
−0.0461998 + 0.998932i $$0.514711\pi$$
$$164$$ 0 0
$$165$$ −4.17226 −0.324810
$$166$$ 0 0
$$167$$ 24.7268 1.91342 0.956708 0.291051i $$-0.0940049\pi$$
0.956708 + 0.291051i $$0.0940049\pi$$
$$168$$ 0 0
$$169$$ 6.69646 0.515112
$$170$$ 0 0
$$171$$ 6.52420 0.498918
$$172$$ 0 0
$$173$$ 17.2026 1.30789 0.653944 0.756543i $$-0.273113\pi$$
0.653944 + 0.756543i $$0.273113\pi$$
$$174$$ 0 0
$$175$$ 0.648061 0.0489888
$$176$$ 0 0
$$177$$ 31.9245 2.39960
$$178$$ 0 0
$$179$$ −6.00000 −0.448461 −0.224231 0.974536i $$-0.571987\pi$$
−0.224231 + 0.974536i $$0.571987\pi$$
$$180$$ 0 0
$$181$$ 16.0968 1.19647 0.598233 0.801322i $$-0.295870\pi$$
0.598233 + 0.801322i $$0.295870\pi$$
$$182$$ 0 0
$$183$$ 6.53162 0.482831
$$184$$ 0 0
$$185$$ 2.43807 0.179250
$$186$$ 0 0
$$187$$ −2.70388 −0.197727
$$188$$ 0 0
$$189$$ −7.04840 −0.512696
$$190$$ 0 0
$$191$$ −0.111635 −0.00807764 −0.00403882 0.999992i $$-0.501286\pi$$
−0.00403882 + 0.999992i $$0.501286\pi$$
$$192$$ 0 0
$$193$$ −6.01809 −0.433191 −0.216596 0.976261i $$-0.569495\pi$$
−0.216596 + 0.976261i $$0.569495\pi$$
$$194$$ 0 0
$$195$$ 13.6965 0.980824
$$196$$ 0 0
$$197$$ 14.7645 1.05193 0.525964 0.850507i $$-0.323705\pi$$
0.525964 + 0.850507i $$0.323705\pi$$
$$198$$ 0 0
$$199$$ −9.75228 −0.691321 −0.345660 0.938360i $$-0.612345\pi$$
−0.345660 + 0.938360i $$0.612345\pi$$
$$200$$ 0 0
$$201$$ −37.4487 −2.64143
$$202$$ 0 0
$$203$$ −2.70388 −0.189775
$$204$$ 0 0
$$205$$ −10.1723 −0.710461
$$206$$ 0 0
$$207$$ 21.8687 1.51998
$$208$$ 0 0
$$209$$ −1.35194 −0.0935156
$$210$$ 0 0
$$211$$ −4.87614 −0.335687 −0.167844 0.985814i $$-0.553680\pi$$
−0.167844 + 0.985814i $$0.553680\pi$$
$$212$$ 0 0
$$213$$ −11.8129 −0.809407
$$214$$ 0 0
$$215$$ 6.82032 0.465142
$$216$$ 0 0
$$217$$ −2.70388 −0.183551
$$218$$ 0 0
$$219$$ 12.8761 0.870089
$$220$$ 0 0
$$221$$ 8.87614 0.597074
$$222$$ 0 0
$$223$$ 20.8384 1.39544 0.697721 0.716369i $$-0.254197\pi$$
0.697721 + 0.716369i $$0.254197\pi$$
$$224$$ 0 0
$$225$$ 6.52420 0.434947
$$226$$ 0 0
$$227$$ 25.3552 1.68288 0.841441 0.540348i $$-0.181707\pi$$
0.841441 + 0.540348i $$0.181707\pi$$
$$228$$ 0 0
$$229$$ 26.0410 1.72084 0.860418 0.509589i $$-0.170202\pi$$
0.860418 + 0.509589i $$0.170202\pi$$
$$230$$ 0 0
$$231$$ 2.70388 0.177902
$$232$$ 0 0
$$233$$ −15.1090 −0.989825 −0.494913 0.868943i $$-0.664800\pi$$
−0.494913 + 0.868943i $$0.664800\pi$$
$$234$$ 0 0
$$235$$ −0.648061 −0.0422748
$$236$$ 0 0
$$237$$ 6.17226 0.400931
$$238$$ 0 0
$$239$$ 20.9878 1.35759 0.678793 0.734330i $$-0.262503\pi$$
0.678793 + 0.734330i $$0.262503\pi$$
$$240$$ 0 0
$$241$$ −8.87614 −0.571762 −0.285881 0.958265i $$-0.592286\pi$$
−0.285881 + 0.958265i $$0.592286\pi$$
$$242$$ 0 0
$$243$$ −10.5545 −0.677072
$$244$$ 0 0
$$245$$ 6.58002 0.420382
$$246$$ 0 0
$$247$$ 4.43807 0.282388
$$248$$ 0 0
$$249$$ 29.3929 1.86270
$$250$$ 0 0
$$251$$ −4.11164 −0.259524 −0.129762 0.991545i $$-0.541421\pi$$
−0.129762 + 0.991545i $$0.541421\pi$$
$$252$$ 0 0
$$253$$ −4.53162 −0.284900
$$254$$ 0 0
$$255$$ 6.17226 0.386522
$$256$$ 0 0
$$257$$ −5.37483 −0.335273 −0.167636 0.985849i $$-0.553613\pi$$
−0.167636 + 0.985849i $$0.553613\pi$$
$$258$$ 0 0
$$259$$ −1.58002 −0.0981775
$$260$$ 0 0
$$261$$ −27.2207 −1.68492
$$262$$ 0 0
$$263$$ 7.58482 0.467700 0.233850 0.972273i $$-0.424867\pi$$
0.233850 + 0.972273i $$0.424867\pi$$
$$264$$ 0 0
$$265$$ −6.43807 −0.395487
$$266$$ 0 0
$$267$$ −11.8129 −0.722938
$$268$$ 0 0
$$269$$ −2.81551 −0.171665 −0.0858324 0.996310i $$-0.527355\pi$$
−0.0858324 + 0.996310i $$0.527355\pi$$
$$270$$ 0 0
$$271$$ 16.7449 1.01718 0.508589 0.861009i $$-0.330167\pi$$
0.508589 + 0.861009i $$0.330167\pi$$
$$272$$ 0 0
$$273$$ −8.87614 −0.537208
$$274$$ 0 0
$$275$$ −1.35194 −0.0815250
$$276$$ 0 0
$$277$$ 25.0484 1.50501 0.752506 0.658585i $$-0.228845\pi$$
0.752506 + 0.658585i $$0.228845\pi$$
$$278$$ 0 0
$$279$$ −27.2207 −1.62966
$$280$$ 0 0
$$281$$ −26.8613 −1.60241 −0.801205 0.598389i $$-0.795808\pi$$
−0.801205 + 0.598389i $$0.795808\pi$$
$$282$$ 0 0
$$283$$ 21.5242 1.27948 0.639740 0.768591i $$-0.279042\pi$$
0.639740 + 0.768591i $$0.279042\pi$$
$$284$$ 0 0
$$285$$ 3.08613 0.182807
$$286$$ 0 0
$$287$$ 6.59224 0.389128
$$288$$ 0 0
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ −23.8687 −1.39921
$$292$$ 0 0
$$293$$ 18.6710 1.09077 0.545384 0.838186i $$-0.316384\pi$$
0.545384 + 0.838186i $$0.316384\pi$$
$$294$$ 0 0
$$295$$ 10.3445 0.602281
$$296$$ 0 0
$$297$$ 14.7039 0.853206
$$298$$ 0 0
$$299$$ 14.8761 0.860309
$$300$$ 0 0
$$301$$ −4.41998 −0.254764
$$302$$ 0 0
$$303$$ 4.17226 0.239690
$$304$$ 0 0
$$305$$ 2.11644 0.121187
$$306$$ 0 0
$$307$$ 10.3068 0.588240 0.294120 0.955769i $$-0.404974\pi$$
0.294120 + 0.955769i $$0.404974\pi$$
$$308$$ 0 0
$$309$$ −11.6965 −0.665388
$$310$$ 0 0
$$311$$ 12.4003 0.703159 0.351579 0.936158i $$-0.385645\pi$$
0.351579 + 0.936158i $$0.385645\pi$$
$$312$$ 0 0
$$313$$ −27.4535 −1.55177 −0.775883 0.630877i $$-0.782695\pi$$
−0.775883 + 0.630877i $$0.782695\pi$$
$$314$$ 0 0
$$315$$ −4.22808 −0.238225
$$316$$ 0 0
$$317$$ −34.1149 −1.91608 −0.958041 0.286630i $$-0.907465\pi$$
−0.958041 + 0.286630i $$0.907465\pi$$
$$318$$ 0 0
$$319$$ 5.64064 0.315815
$$320$$ 0 0
$$321$$ −26.4003 −1.47352
$$322$$ 0 0
$$323$$ 2.00000 0.111283
$$324$$ 0 0
$$325$$ 4.43807 0.246180
$$326$$ 0 0
$$327$$ 21.7523 1.20290
$$328$$ 0 0
$$329$$ 0.419983 0.0231544
$$330$$ 0 0
$$331$$ 16.2691 0.894228 0.447114 0.894477i $$-0.352452\pi$$
0.447114 + 0.894477i $$0.352452\pi$$
$$332$$ 0 0
$$333$$ −15.9065 −0.871668
$$334$$ 0 0
$$335$$ −12.1345 −0.662980
$$336$$ 0 0
$$337$$ 23.0665 1.25651 0.628256 0.778007i $$-0.283769\pi$$
0.628256 + 0.778007i $$0.283769\pi$$
$$338$$ 0 0
$$339$$ −57.4339 −3.11938
$$340$$ 0 0
$$341$$ 5.64064 0.305458
$$342$$ 0 0
$$343$$ −8.80068 −0.475192
$$344$$ 0 0
$$345$$ 10.3445 0.556930
$$346$$ 0 0
$$347$$ −26.4003 −1.41724 −0.708622 0.705588i $$-0.750683\pi$$
−0.708622 + 0.705588i $$0.750683\pi$$
$$348$$ 0 0
$$349$$ −35.4535 −1.89778 −0.948892 0.315600i $$-0.897794\pi$$
−0.948892 + 0.315600i $$0.897794\pi$$
$$350$$ 0 0
$$351$$ −48.2691 −2.57641
$$352$$ 0 0
$$353$$ −19.0336 −1.01305 −0.506527 0.862224i $$-0.669071\pi$$
−0.506527 + 0.862224i $$0.669071\pi$$
$$354$$ 0 0
$$355$$ −3.82774 −0.203155
$$356$$ 0 0
$$357$$ −4.00000 −0.211702
$$358$$ 0 0
$$359$$ 4.16745 0.219950 0.109975 0.993934i $$-0.464923\pi$$
0.109975 + 0.993934i $$0.464923\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 28.3068 1.48572
$$364$$ 0 0
$$365$$ 4.17226 0.218386
$$366$$ 0 0
$$367$$ 34.3249 1.79174 0.895872 0.444312i $$-0.146552\pi$$
0.895872 + 0.444312i $$0.146552\pi$$
$$368$$ 0 0
$$369$$ 66.3659 3.45487
$$370$$ 0 0
$$371$$ 4.17226 0.216613
$$372$$ 0 0
$$373$$ −1.32905 −0.0688153 −0.0344077 0.999408i $$-0.510954\pi$$
−0.0344077 + 0.999408i $$0.510954\pi$$
$$374$$ 0 0
$$375$$ 3.08613 0.159367
$$376$$ 0 0
$$377$$ −18.5168 −0.953663
$$378$$ 0 0
$$379$$ −10.5922 −0.544087 −0.272043 0.962285i $$-0.587699\pi$$
−0.272043 + 0.962285i $$0.587699\pi$$
$$380$$ 0 0
$$381$$ 5.52420 0.283013
$$382$$ 0 0
$$383$$ −2.38225 −0.121727 −0.0608637 0.998146i $$-0.519385\pi$$
−0.0608637 + 0.998146i $$0.519385\pi$$
$$384$$ 0 0
$$385$$ 0.876139 0.0446522
$$386$$ 0 0
$$387$$ −44.4971 −2.26192
$$388$$ 0 0
$$389$$ 23.3323 1.18299 0.591497 0.806307i $$-0.298537\pi$$
0.591497 + 0.806307i $$0.298537\pi$$
$$390$$ 0 0
$$391$$ 6.70388 0.339030
$$392$$ 0 0
$$393$$ −53.1452 −2.68082
$$394$$ 0 0
$$395$$ 2.00000 0.100631
$$396$$ 0 0
$$397$$ 34.3807 1.72552 0.862759 0.505616i $$-0.168735\pi$$
0.862759 + 0.505616i $$0.168735\pi$$
$$398$$ 0 0
$$399$$ −2.00000 −0.100125
$$400$$ 0 0
$$401$$ 24.3297 1.21497 0.607483 0.794333i $$-0.292179\pi$$
0.607483 + 0.794333i $$0.292179\pi$$
$$402$$ 0 0
$$403$$ −18.5168 −0.922387
$$404$$ 0 0
$$405$$ −13.9926 −0.695297
$$406$$ 0 0
$$407$$ 3.29612 0.163383
$$408$$ 0 0
$$409$$ −9.75228 −0.482219 −0.241110 0.970498i $$-0.577511\pi$$
−0.241110 + 0.970498i $$0.577511\pi$$
$$410$$ 0 0
$$411$$ −31.2058 −1.53927
$$412$$ 0 0
$$413$$ −6.70388 −0.329876
$$414$$ 0 0
$$415$$ 9.52420 0.467525
$$416$$ 0 0
$$417$$ −29.9245 −1.46541
$$418$$ 0 0
$$419$$ −6.51678 −0.318366 −0.159183 0.987249i $$-0.550886\pi$$
−0.159183 + 0.987249i $$0.550886\pi$$
$$420$$ 0 0
$$421$$ 2.40515 0.117220 0.0586098 0.998281i $$-0.481333\pi$$
0.0586098 + 0.998281i $$0.481333\pi$$
$$422$$ 0 0
$$423$$ 4.22808 0.205576
$$424$$ 0 0
$$425$$ 2.00000 0.0970143
$$426$$ 0 0
$$427$$ −1.37158 −0.0663756
$$428$$ 0 0
$$429$$ 18.5168 0.893999
$$430$$ 0 0
$$431$$ −13.1090 −0.631439 −0.315720 0.948852i $$-0.602246\pi$$
−0.315720 + 0.948852i $$0.602246\pi$$
$$432$$ 0 0
$$433$$ −20.8942 −1.00411 −0.502056 0.864835i $$-0.667423\pi$$
−0.502056 + 0.864835i $$0.667423\pi$$
$$434$$ 0 0
$$435$$ −12.8761 −0.617364
$$436$$ 0 0
$$437$$ 3.35194 0.160345
$$438$$ 0 0
$$439$$ 41.3929 1.97558 0.987788 0.155803i $$-0.0497965\pi$$
0.987788 + 0.155803i $$0.0497965\pi$$
$$440$$ 0 0
$$441$$ −42.9293 −2.04425
$$442$$ 0 0
$$443$$ −10.0558 −0.477766 −0.238883 0.971048i $$-0.576781\pi$$
−0.238883 + 0.971048i $$0.576781\pi$$
$$444$$ 0 0
$$445$$ −3.82774 −0.181452
$$446$$ 0 0
$$447$$ −18.1574 −0.858817
$$448$$ 0 0
$$449$$ 12.1116 0.571583 0.285792 0.958292i $$-0.407743\pi$$
0.285792 + 0.958292i $$0.407743\pi$$
$$450$$ 0 0
$$451$$ −13.7523 −0.647569
$$452$$ 0 0
$$453$$ 57.4897 2.70110
$$454$$ 0 0
$$455$$ −2.87614 −0.134835
$$456$$ 0 0
$$457$$ −30.9123 −1.44602 −0.723008 0.690839i $$-0.757241\pi$$
−0.723008 + 0.690839i $$0.757241\pi$$
$$458$$ 0 0
$$459$$ −21.7523 −1.01531
$$460$$ 0 0
$$461$$ −6.00000 −0.279448 −0.139724 0.990190i $$-0.544622\pi$$
−0.139724 + 0.990190i $$0.544622\pi$$
$$462$$ 0 0
$$463$$ −31.0894 −1.44485 −0.722423 0.691451i $$-0.756972\pi$$
−0.722423 + 0.691451i $$0.756972\pi$$
$$464$$ 0 0
$$465$$ −12.8761 −0.597117
$$466$$ 0 0
$$467$$ 12.0410 0.557190 0.278595 0.960409i $$-0.410131\pi$$
0.278595 + 0.960409i $$0.410131\pi$$
$$468$$ 0 0
$$469$$ 7.86391 0.363122
$$470$$ 0 0
$$471$$ 53.6768 2.47330
$$472$$ 0 0
$$473$$ 9.22066 0.423966
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 42.0032 1.92320
$$478$$ 0 0
$$479$$ −10.3035 −0.470781 −0.235390 0.971901i $$-0.575637\pi$$
−0.235390 + 0.971901i $$0.575637\pi$$
$$480$$ 0 0
$$481$$ −10.8203 −0.493364
$$482$$ 0 0
$$483$$ −6.70388 −0.305037
$$484$$ 0 0
$$485$$ −7.73419 −0.351192
$$486$$ 0 0
$$487$$ 22.4939 1.01930 0.509648 0.860383i $$-0.329776\pi$$
0.509648 + 0.860383i $$0.329776\pi$$
$$488$$ 0 0
$$489$$ 3.64064 0.164636
$$490$$ 0 0
$$491$$ 35.6162 1.60734 0.803668 0.595078i $$-0.202879\pi$$
0.803668 + 0.595078i $$0.202879\pi$$
$$492$$ 0 0
$$493$$ −8.34452 −0.375818
$$494$$ 0 0
$$495$$ 8.82032 0.396444
$$496$$ 0 0
$$497$$ 2.48061 0.111270
$$498$$ 0 0
$$499$$ 0.0558176 0.00249874 0.00124937 0.999999i $$-0.499602\pi$$
0.00124937 + 0.999999i $$0.499602\pi$$
$$500$$ 0 0
$$501$$ −76.3100 −3.40928
$$502$$ 0 0
$$503$$ −23.8081 −1.06155 −0.530775 0.847513i $$-0.678099\pi$$
−0.530775 + 0.847513i $$0.678099\pi$$
$$504$$ 0 0
$$505$$ 1.35194 0.0601605
$$506$$ 0 0
$$507$$ −20.6661 −0.917816
$$508$$ 0 0
$$509$$ 10.5316 0.466806 0.233403 0.972380i $$-0.425014\pi$$
0.233403 + 0.972380i $$0.425014\pi$$
$$510$$ 0 0
$$511$$ −2.70388 −0.119613
$$512$$ 0 0
$$513$$ −10.8761 −0.480193
$$514$$ 0 0
$$515$$ −3.79001 −0.167008
$$516$$ 0 0
$$517$$ −0.876139 −0.0385325
$$518$$ 0 0
$$519$$ −53.0894 −2.33037
$$520$$ 0 0
$$521$$ −11.2961 −0.494892 −0.247446 0.968902i $$-0.579591\pi$$
−0.247446 + 0.968902i $$0.579591\pi$$
$$522$$ 0 0
$$523$$ 11.0861 0.484763 0.242381 0.970181i $$-0.422071\pi$$
0.242381 + 0.970181i $$0.422071\pi$$
$$524$$ 0 0
$$525$$ −2.00000 −0.0872872
$$526$$ 0 0
$$527$$ −8.34452 −0.363493
$$528$$ 0 0
$$529$$ −11.7645 −0.511500
$$530$$ 0 0
$$531$$ −67.4897 −2.92880
$$532$$ 0 0
$$533$$ 45.1452 1.95546
$$534$$ 0 0
$$535$$ −8.55451 −0.369844
$$536$$ 0 0
$$537$$ 18.5168 0.799058
$$538$$ 0 0
$$539$$ 8.89578 0.383169
$$540$$ 0 0
$$541$$ 16.8565 0.724717 0.362359 0.932039i $$-0.381972\pi$$
0.362359 + 0.932039i $$0.381972\pi$$
$$542$$ 0 0
$$543$$ −49.6768 −2.13184
$$544$$ 0 0
$$545$$ 7.04840 0.301920
$$546$$ 0 0
$$547$$ −16.6151 −0.710412 −0.355206 0.934788i $$-0.615589\pi$$
−0.355206 + 0.934788i $$0.615589\pi$$
$$548$$ 0 0
$$549$$ −13.8081 −0.589315
$$550$$ 0 0
$$551$$ −4.17226 −0.177744
$$552$$ 0 0
$$553$$ −1.29612 −0.0551167
$$554$$ 0 0
$$555$$ −7.52420 −0.319384
$$556$$ 0 0
$$557$$ 38.4562 1.62944 0.814720 0.579855i $$-0.196891\pi$$
0.814720 + 0.579855i $$0.196891\pi$$
$$558$$ 0 0
$$559$$ −30.2691 −1.28024
$$560$$ 0 0
$$561$$ 8.34452 0.352306
$$562$$ 0 0
$$563$$ −40.4184 −1.70343 −0.851717 0.524002i $$-0.824438\pi$$
−0.851717 + 0.524002i $$0.824438\pi$$
$$564$$ 0 0
$$565$$ −18.6103 −0.782942
$$566$$ 0 0
$$567$$ 9.06804 0.380822
$$568$$ 0 0
$$569$$ 26.9368 1.12925 0.564624 0.825348i $$-0.309021\pi$$
0.564624 + 0.825348i $$0.309021\pi$$
$$570$$ 0 0
$$571$$ 21.4636 0.898223 0.449111 0.893476i $$-0.351741\pi$$
0.449111 + 0.893476i $$0.351741\pi$$
$$572$$ 0 0
$$573$$ 0.344521 0.0143926
$$574$$ 0 0
$$575$$ 3.35194 0.139786
$$576$$ 0 0
$$577$$ −12.8613 −0.535423 −0.267712 0.963499i $$-0.586267\pi$$
−0.267712 + 0.963499i $$0.586267\pi$$
$$578$$ 0 0
$$579$$ 18.5726 0.771851
$$580$$ 0 0
$$581$$ −6.17226 −0.256069
$$582$$ 0 0
$$583$$ −8.70388 −0.360478
$$584$$ 0 0
$$585$$ −28.9549 −1.19714
$$586$$ 0 0
$$587$$ 13.8687 0.572423 0.286212 0.958166i $$-0.407604\pi$$
0.286212 + 0.958166i $$0.407604\pi$$
$$588$$ 0 0
$$589$$ −4.17226 −0.171915
$$590$$ 0 0
$$591$$ −45.5652 −1.87430
$$592$$ 0 0
$$593$$ −20.9368 −0.859770 −0.429885 0.902884i $$-0.641446\pi$$
−0.429885 + 0.902884i $$0.641446\pi$$
$$594$$ 0 0
$$595$$ −1.29612 −0.0531358
$$596$$ 0 0
$$597$$ 30.0968 1.23178
$$598$$ 0 0
$$599$$ 0.703878 0.0287597 0.0143798 0.999897i $$-0.495423\pi$$
0.0143798 + 0.999897i $$0.495423\pi$$
$$600$$ 0 0
$$601$$ 5.22066 0.212955 0.106478 0.994315i $$-0.466043\pi$$
0.106478 + 0.994315i $$0.466043\pi$$
$$602$$ 0 0
$$603$$ 79.1681 3.22397
$$604$$ 0 0
$$605$$ 9.17226 0.372905
$$606$$ 0 0
$$607$$ 16.1345 0.654880 0.327440 0.944872i $$-0.393814\pi$$
0.327440 + 0.944872i $$0.393814\pi$$
$$608$$ 0 0
$$609$$ 8.34452 0.338137
$$610$$ 0 0
$$611$$ 2.87614 0.116356
$$612$$ 0 0
$$613$$ −19.3323 −0.780824 −0.390412 0.920640i $$-0.627667\pi$$
−0.390412 + 0.920640i $$0.627667\pi$$
$$614$$ 0 0
$$615$$ 31.3929 1.26588
$$616$$ 0 0
$$617$$ 1.01223 0.0407507 0.0203753 0.999792i $$-0.493514\pi$$
0.0203753 + 0.999792i $$0.493514\pi$$
$$618$$ 0 0
$$619$$ 18.8809 0.758889 0.379445 0.925214i $$-0.376115\pi$$
0.379445 + 0.925214i $$0.376115\pi$$
$$620$$ 0 0
$$621$$ −36.4562 −1.46294
$$622$$ 0 0
$$623$$ 2.48061 0.0993835
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 4.17226 0.166624
$$628$$ 0 0
$$629$$ −4.87614 −0.194424
$$630$$ 0 0
$$631$$ 17.4636 0.695214 0.347607 0.937640i $$-0.386994\pi$$
0.347607 + 0.937640i $$0.386994\pi$$
$$632$$ 0 0
$$633$$ 15.0484 0.598120
$$634$$ 0 0
$$635$$ 1.79001 0.0710343
$$636$$ 0 0
$$637$$ −29.2026 −1.15705
$$638$$ 0 0
$$639$$ 24.9729 0.987914
$$640$$ 0 0
$$641$$ −22.4051 −0.884950 −0.442475 0.896781i $$-0.645899\pi$$
−0.442475 + 0.896781i $$0.645899\pi$$
$$642$$ 0 0
$$643$$ −22.6332 −0.892567 −0.446284 0.894892i $$-0.647253\pi$$
−0.446284 + 0.894892i $$0.647253\pi$$
$$644$$ 0 0
$$645$$ −21.0484 −0.828780
$$646$$ 0 0
$$647$$ 34.8203 1.36893 0.684464 0.729047i $$-0.260036\pi$$
0.684464 + 0.729047i $$0.260036\pi$$
$$648$$ 0 0
$$649$$ 13.9852 0.548966
$$650$$ 0 0
$$651$$ 8.34452 0.327048
$$652$$ 0 0
$$653$$ −23.1090 −0.904326 −0.452163 0.891935i $$-0.649347\pi$$
−0.452163 + 0.891935i $$0.649347\pi$$
$$654$$ 0 0
$$655$$ −17.2207 −0.672867
$$656$$ 0 0
$$657$$ −27.2207 −1.06198
$$658$$ 0 0
$$659$$ −4.24772 −0.165468 −0.0827339 0.996572i $$-0.526365\pi$$
−0.0827339 + 0.996572i $$0.526365\pi$$
$$660$$ 0 0
$$661$$ 8.15742 0.317287 0.158644 0.987336i $$-0.449288\pi$$
0.158644 + 0.987336i $$0.449288\pi$$
$$662$$ 0 0
$$663$$ −27.3929 −1.06385
$$664$$ 0 0
$$665$$ −0.648061 −0.0251307
$$666$$ 0 0
$$667$$ −13.9852 −0.541508
$$668$$ 0 0
$$669$$ −64.3100 −2.48637
$$670$$ 0 0
$$671$$ 2.86130 0.110459
$$672$$ 0 0
$$673$$ 39.2239 1.51197 0.755985 0.654589i $$-0.227158\pi$$
0.755985 + 0.654589i $$0.227158\pi$$
$$674$$ 0 0
$$675$$ −10.8761 −0.418623
$$676$$ 0 0
$$677$$ 32.4232 1.24613 0.623063 0.782172i $$-0.285888\pi$$
0.623063 + 0.782172i $$0.285888\pi$$
$$678$$ 0 0
$$679$$ 5.01223 0.192352
$$680$$ 0 0
$$681$$ −78.2494 −2.99852
$$682$$ 0 0
$$683$$ 35.6539 1.36426 0.682130 0.731231i $$-0.261054\pi$$
0.682130 + 0.731231i $$0.261054\pi$$
$$684$$ 0 0
$$685$$ −10.1116 −0.386346
$$686$$ 0 0
$$687$$ −80.3659 −3.06615
$$688$$ 0 0
$$689$$ 28.5726 1.08853
$$690$$ 0 0
$$691$$ 44.0261 1.67483 0.837417 0.546565i $$-0.184065\pi$$
0.837417 + 0.546565i $$0.184065\pi$$
$$692$$ 0 0
$$693$$ −5.71610 −0.217137
$$694$$ 0 0
$$695$$ −9.69646 −0.367808
$$696$$ 0 0
$$697$$ 20.3445 0.770604
$$698$$ 0 0
$$699$$ 46.6284 1.76365
$$700$$ 0 0
$$701$$ 28.9023 1.09162 0.545812 0.837908i $$-0.316221\pi$$
0.545812 + 0.837908i $$0.316221\pi$$
$$702$$ 0 0
$$703$$ −2.43807 −0.0919535
$$704$$ 0 0
$$705$$ 2.00000 0.0753244
$$706$$ 0 0
$$707$$ −0.876139 −0.0329506
$$708$$ 0 0
$$709$$ −10.6433 −0.399716 −0.199858 0.979825i $$-0.564048\pi$$
−0.199858 + 0.979825i $$0.564048\pi$$
$$710$$ 0 0
$$711$$ −13.0484 −0.489353
$$712$$ 0 0
$$713$$ −13.9852 −0.523748
$$714$$ 0 0
$$715$$ 6.00000 0.224387
$$716$$ 0 0
$$717$$ −64.7710 −2.41892
$$718$$ 0 0
$$719$$ 41.4339 1.54522 0.772612 0.634879i $$-0.218950\pi$$
0.772612 + 0.634879i $$0.218950\pi$$
$$720$$ 0 0
$$721$$ 2.45616 0.0914720
$$722$$ 0 0
$$723$$ 27.3929 1.01875
$$724$$ 0 0
$$725$$ −4.17226 −0.154954
$$726$$ 0 0
$$727$$ 4.72352 0.175186 0.0875929 0.996156i $$-0.472083\pi$$
0.0875929 + 0.996156i $$0.472083\pi$$
$$728$$ 0 0
$$729$$ −9.40515 −0.348339
$$730$$ 0 0
$$731$$ −13.6406 −0.504517
$$732$$ 0 0
$$733$$ 17.6917 0.653456 0.326728 0.945118i $$-0.394054\pi$$
0.326728 + 0.945118i $$0.394054\pi$$
$$734$$ 0 0
$$735$$ −20.3068 −0.749027
$$736$$ 0 0
$$737$$ −16.4051 −0.604291
$$738$$ 0 0
$$739$$ −12.2233 −0.449640 −0.224820 0.974400i $$-0.572179\pi$$
−0.224820 + 0.974400i $$0.572179\pi$$
$$740$$ 0 0
$$741$$ −13.6965 −0.503152
$$742$$ 0 0
$$743$$ −19.6635 −0.721385 −0.360693 0.932685i $$-0.617460\pi$$
−0.360693 + 0.932685i $$0.617460\pi$$
$$744$$ 0 0
$$745$$ −5.88356 −0.215557
$$746$$ 0 0
$$747$$ −62.1378 −2.27350
$$748$$ 0 0
$$749$$ 5.54384 0.202568
$$750$$ 0 0
$$751$$ 41.6768 1.52081 0.760404 0.649450i $$-0.225001\pi$$
0.760404 + 0.649450i $$0.225001\pi$$
$$752$$ 0 0
$$753$$ 12.6890 0.462414
$$754$$ 0 0
$$755$$ 18.6284 0.677957
$$756$$ 0 0
$$757$$ −42.6136 −1.54882 −0.774408 0.632686i $$-0.781952\pi$$
−0.774408 + 0.632686i $$0.781952\pi$$
$$758$$ 0 0
$$759$$ 13.9852 0.507629
$$760$$ 0 0
$$761$$ −30.8809 −1.11943 −0.559717 0.828684i $$-0.689090\pi$$
−0.559717 + 0.828684i $$0.689090\pi$$
$$762$$ 0 0
$$763$$ −4.56779 −0.165365
$$764$$ 0 0
$$765$$ −13.0484 −0.471766
$$766$$ 0 0
$$767$$ −45.9097 −1.65770
$$768$$ 0 0
$$769$$ −5.12867 −0.184945 −0.0924723 0.995715i $$-0.529477\pi$$
−0.0924723 + 0.995715i $$0.529477\pi$$
$$770$$ 0 0
$$771$$ 16.5874 0.597382
$$772$$ 0 0
$$773$$ 42.0787 1.51347 0.756733 0.653724i $$-0.226794\pi$$
0.756733 + 0.653724i $$0.226794\pi$$
$$774$$ 0 0
$$775$$ −4.17226 −0.149872
$$776$$ 0 0
$$777$$ 4.87614 0.174931
$$778$$ 0 0
$$779$$ 10.1723 0.364459
$$780$$ 0 0
$$781$$ −5.17487 −0.185171
$$782$$ 0 0
$$783$$ 45.3781 1.62168
$$784$$ 0 0
$$785$$ 17.3929 0.620780
$$786$$ 0 0
$$787$$ −43.4307 −1.54814 −0.774068 0.633103i $$-0.781781\pi$$
−0.774068 + 0.633103i $$0.781781\pi$$
$$788$$ 0 0
$$789$$ −23.4078 −0.833338
$$790$$ 0 0
$$791$$ 12.0606 0.428826
$$792$$ 0 0
$$793$$ −9.39292 −0.333552
$$794$$ 0 0
$$795$$ 19.8687 0.704671
$$796$$ 0 0
$$797$$ −40.0032 −1.41699 −0.708494 0.705717i $$-0.750625\pi$$
−0.708494 + 0.705717i $$0.750625\pi$$
$$798$$ 0 0
$$799$$ 1.29612 0.0458535
$$800$$ 0 0
$$801$$ 24.9729 0.882375
$$802$$ 0 0
$$803$$ 5.64064 0.199054
$$804$$ 0 0
$$805$$ −2.17226 −0.0765621
$$806$$ 0 0
$$807$$ 8.68904 0.305869
$$808$$ 0 0
$$809$$ −28.7401 −1.01045 −0.505223 0.862989i $$-0.668590\pi$$
−0.505223 + 0.862989i $$0.668590\pi$$
$$810$$ 0 0
$$811$$ −16.0510 −0.563627 −0.281814 0.959469i $$-0.590936\pi$$
−0.281814 + 0.959469i $$0.590936\pi$$
$$812$$ 0 0
$$813$$ −51.6768 −1.81239
$$814$$ 0 0
$$815$$ 1.17968 0.0413223
$$816$$ 0 0
$$817$$ −6.82032 −0.238613
$$818$$ 0 0
$$819$$ 18.7645 0.655685
$$820$$ 0 0
$$821$$ 7.82774 0.273190 0.136595 0.990627i $$-0.456384\pi$$
0.136595 + 0.990627i $$0.456384\pi$$
$$822$$ 0 0
$$823$$ −9.78676 −0.341145 −0.170572 0.985345i $$-0.554562\pi$$
−0.170572 + 0.985345i $$0.554562\pi$$
$$824$$ 0 0
$$825$$ 4.17226 0.145259
$$826$$ 0 0
$$827$$ −50.1707 −1.74461 −0.872303 0.488965i $$-0.837374\pi$$
−0.872303 + 0.488965i $$0.837374\pi$$
$$828$$ 0 0
$$829$$ −21.2961 −0.739645 −0.369822 0.929102i $$-0.620581\pi$$
−0.369822 + 0.929102i $$0.620581\pi$$
$$830$$ 0 0
$$831$$ −77.3026 −2.68160
$$832$$ 0 0
$$833$$ −13.1600 −0.455968
$$834$$ 0 0
$$835$$ −24.7268 −0.855705
$$836$$ 0 0
$$837$$ 45.3781 1.56850
$$838$$ 0 0
$$839$$ −30.9516 −1.06857 −0.534284 0.845305i $$-0.679419\pi$$
−0.534284 + 0.845305i $$0.679419\pi$$
$$840$$ 0 0
$$841$$ −11.5922 −0.399733
$$842$$ 0 0
$$843$$ 82.8975 2.85514
$$844$$ 0 0
$$845$$ −6.69646 −0.230365
$$846$$ 0 0
$$847$$ −5.94418 −0.204245
$$848$$ 0 0
$$849$$ −66.4265 −2.27975
$$850$$ 0 0
$$851$$ −8.17226 −0.280141
$$852$$ 0 0
$$853$$ 38.9219 1.33266 0.666331 0.745656i $$-0.267864\pi$$
0.666331 + 0.745656i $$0.267864\pi$$
$$854$$ 0 0
$$855$$ −6.52420 −0.223123
$$856$$ 0 0
$$857$$ 52.1245 1.78054 0.890270 0.455434i $$-0.150516\pi$$
0.890270 + 0.455434i $$0.150516\pi$$
$$858$$ 0 0
$$859$$ 4.41998 0.150808 0.0754039 0.997153i $$-0.475975\pi$$
0.0754039 + 0.997153i $$0.475975\pi$$
$$860$$ 0 0
$$861$$ −20.3445 −0.693339
$$862$$ 0 0
$$863$$ 12.4184 0.422728 0.211364 0.977407i $$-0.432209\pi$$
0.211364 + 0.977407i $$0.432209\pi$$
$$864$$ 0 0
$$865$$ −17.2026 −0.584905
$$866$$ 0 0
$$867$$ 40.1197 1.36254
$$868$$ 0 0
$$869$$ 2.70388 0.0917228
$$870$$ 0 0
$$871$$ 53.8539 1.82477
$$872$$ 0 0
$$873$$ 50.4594 1.70779
$$874$$ 0 0
$$875$$ −0.648061 −0.0219085
$$876$$ 0 0
$$877$$ 40.3020 1.36090 0.680451 0.732794i $$-0.261784\pi$$
0.680451 + 0.732794i $$0.261784\pi$$
$$878$$ 0 0
$$879$$ −57.6210 −1.94351
$$880$$ 0 0
$$881$$ −4.47580 −0.150794 −0.0753968 0.997154i $$-0.524022\pi$$
−0.0753968 + 0.997154i $$0.524022\pi$$
$$882$$ 0 0
$$883$$ 27.5455 0.926981 0.463491 0.886102i $$-0.346597\pi$$
0.463491 + 0.886102i $$0.346597\pi$$
$$884$$ 0 0
$$885$$ −31.9245 −1.07313
$$886$$ 0 0
$$887$$ 48.5152 1.62898 0.814491 0.580176i $$-0.197016\pi$$
0.814491 + 0.580176i $$0.197016\pi$$
$$888$$ 0 0
$$889$$ −1.16003 −0.0389063
$$890$$ 0 0
$$891$$ −18.9171 −0.633747
$$892$$ 0 0
$$893$$ 0.648061 0.0216865
$$894$$ 0 0
$$895$$ 6.00000 0.200558
$$896$$ 0 0
$$897$$ −45.9097 −1.53288
$$898$$ 0 0
$$899$$ 17.4078 0.580581
$$900$$ 0 0
$$901$$ 12.8761 0.428966
$$902$$ 0 0
$$903$$ 13.6406 0.453932
$$904$$ 0 0
$$905$$ −16.0968 −0.535076
$$906$$ 0 0
$$907$$ 4.55451 0.151230 0.0756150 0.997137i $$-0.475908\pi$$
0.0756150 + 0.997137i $$0.475908\pi$$
$$908$$ 0 0
$$909$$ −8.82032 −0.292552
$$910$$ 0 0
$$911$$ −0.298732 −0.00989745 −0.00494872 0.999988i $$-0.501575\pi$$
−0.00494872 + 0.999988i $$0.501575\pi$$
$$912$$ 0 0
$$913$$ 12.8761 0.426138
$$914$$ 0 0
$$915$$ −6.53162 −0.215929
$$916$$ 0 0
$$917$$ 11.1600 0.368537
$$918$$ 0 0
$$919$$ 0.0754620 0.00248926 0.00124463 0.999999i $$-0.499604\pi$$
0.00124463 + 0.999999i $$0.499604\pi$$
$$920$$ 0 0
$$921$$ −31.8081 −1.04811
$$922$$ 0 0
$$923$$ 16.9878 0.559159
$$924$$ 0 0
$$925$$ −2.43807 −0.0801632
$$926$$ 0 0
$$927$$ 24.7268 0.812134
$$928$$ 0 0
$$929$$ 30.0000 0.984268 0.492134 0.870519i $$-0.336217\pi$$
0.492134 + 0.870519i $$0.336217\pi$$
$$930$$ 0 0
$$931$$ −6.58002 −0.215651
$$932$$ 0 0
$$933$$ −38.2691 −1.25287
$$934$$ 0 0
$$935$$ 2.70388 0.0884263
$$936$$ 0 0
$$937$$ 42.9581 1.40338 0.701690 0.712482i $$-0.252429\pi$$
0.701690 + 0.712482i $$0.252429\pi$$
$$938$$ 0 0
$$939$$ 84.7252 2.76490
$$940$$ 0 0
$$941$$ 20.3297 0.662729 0.331364 0.943503i $$-0.392491\pi$$
0.331364 + 0.943503i $$0.392491\pi$$
$$942$$ 0 0
$$943$$ 34.0968 1.11034
$$944$$ 0 0
$$945$$ 7.04840 0.229284
$$946$$ 0 0
$$947$$ −57.0745 −1.85467 −0.927337 0.374228i $$-0.877908\pi$$
−0.927337 + 0.374228i $$0.877908\pi$$
$$948$$ 0 0
$$949$$ −18.5168 −0.601080
$$950$$ 0 0
$$951$$ 105.283 3.41403
$$952$$ 0 0
$$953$$ 44.4594 1.44018 0.720091 0.693880i $$-0.244100\pi$$
0.720091 + 0.693880i $$0.244100\pi$$
$$954$$ 0 0
$$955$$ 0.111635 0.00361243
$$956$$ 0 0
$$957$$ −17.4078 −0.562713
$$958$$ 0 0
$$959$$ 6.55295 0.211606
$$960$$ 0 0
$$961$$ −13.5922 −0.438459
$$962$$ 0 0
$$963$$ 55.8113 1.79850
$$964$$ 0 0
$$965$$ 6.01809 0.193729
$$966$$ 0 0
$$967$$ −38.1016 −1.22527 −0.612633 0.790368i $$-0.709889\pi$$
−0.612633 + 0.790368i $$0.709889\pi$$
$$968$$ 0 0
$$969$$ −6.17226 −0.198282
$$970$$ 0 0
$$971$$ 15.3716 0.493298 0.246649 0.969105i $$-0.420671\pi$$
0.246649 + 0.969105i $$0.420671\pi$$
$$972$$ 0 0
$$973$$ 6.28390 0.201452
$$974$$ 0 0
$$975$$ −13.6965 −0.438638
$$976$$ 0 0
$$977$$ 25.1271 0.803888 0.401944 0.915664i $$-0.368335\pi$$
0.401944 + 0.915664i $$0.368335\pi$$
$$978$$ 0 0
$$979$$ −5.17487 −0.165390
$$980$$ 0 0
$$981$$ −45.9852 −1.46819
$$982$$ 0 0
$$983$$ −27.3042 −0.870868 −0.435434 0.900221i $$-0.643405\pi$$
−0.435434 + 0.900221i $$0.643405\pi$$
$$984$$ 0 0
$$985$$ −14.7645 −0.470436
$$986$$ 0 0
$$987$$ −1.29612 −0.0412560
$$988$$ 0 0
$$989$$ −22.8613 −0.726947
$$990$$ 0 0
$$991$$ 57.0697 1.81288 0.906440 0.422335i $$-0.138789\pi$$
0.906440 + 0.422335i $$0.138789\pi$$
$$992$$ 0 0
$$993$$ −50.2084 −1.59332
$$994$$ 0 0
$$995$$ 9.75228 0.309168
$$996$$ 0 0
$$997$$ 18.1213 0.573906 0.286953 0.957945i $$-0.407358\pi$$
0.286953 + 0.957945i $$0.407358\pi$$
$$998$$ 0 0
$$999$$ 26.5168 0.838954
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 3040.2.a.k.1.1 3
4.3 odd 2 3040.2.a.n.1.3 yes 3
8.3 odd 2 6080.2.a.bp.1.1 3
8.5 even 2 6080.2.a.bz.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.k.1.1 3 1.1 even 1 trivial
3040.2.a.n.1.3 yes 3 4.3 odd 2
6080.2.a.bp.1.1 3 8.3 odd 2
6080.2.a.bz.1.3 3 8.5 even 2