# Properties

 Label 6080.2.a.y.1.1 Level $6080$ Weight $2$ Character 6080.1 Self dual yes Analytic conductor $48.549$ Analytic rank $0$ 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] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6080 = 2^{6} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6080.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: $$48.5490444289$$ Analytic rank: $$0$$ 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: no (minimal twist has level 380) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 6080.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-3.41421 q^{3} -1.00000 q^{5} -0.828427 q^{7} +8.65685 q^{9} +O(q^{10})$$ $$q-3.41421 q^{3} -1.00000 q^{5} -0.828427 q^{7} +8.65685 q^{9} -2.00000 q^{11} +6.24264 q^{13} +3.41421 q^{15} +0.828427 q^{17} -1.00000 q^{19} +2.82843 q^{21} +6.00000 q^{23} +1.00000 q^{25} -19.3137 q^{27} +6.48528 q^{29} +6.82843 q^{31} +6.82843 q^{33} +0.828427 q^{35} +1.75736 q^{37} -21.3137 q^{39} +3.65685 q^{41} +4.82843 q^{43} -8.65685 q^{45} +4.82843 q^{47} -6.31371 q^{49} -2.82843 q^{51} -9.07107 q^{53} +2.00000 q^{55} +3.41421 q^{57} +13.6569 q^{59} +13.6569 q^{61} -7.17157 q^{63} -6.24264 q^{65} -3.41421 q^{67} -20.4853 q^{69} -5.17157 q^{71} -2.48528 q^{73} -3.41421 q^{75} +1.65685 q^{77} -1.65685 q^{79} +39.9706 q^{81} -13.3137 q^{83} -0.828427 q^{85} -22.1421 q^{87} -6.48528 q^{89} -5.17157 q^{91} -23.3137 q^{93} +1.00000 q^{95} -10.2426 q^{97} -17.3137 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10})$$ 2 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 6 * q^9 $$2 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 6 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 2 q^{19} + 12 q^{23} + 2 q^{25} - 16 q^{27} - 4 q^{29} + 8 q^{31} + 8 q^{33} - 4 q^{35} + 12 q^{37} - 20 q^{39} - 4 q^{41} + 4 q^{43} - 6 q^{45} + 4 q^{47} + 10 q^{49} - 4 q^{53} + 4 q^{55} + 4 q^{57} + 16 q^{59} + 16 q^{61} - 20 q^{63} - 4 q^{65} - 4 q^{67} - 24 q^{69} - 16 q^{71} + 12 q^{73} - 4 q^{75} - 8 q^{77} + 8 q^{79} + 46 q^{81} - 4 q^{83} + 4 q^{85} - 16 q^{87} + 4 q^{89} - 16 q^{91} - 24 q^{93} + 2 q^{95} - 12 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 6 * q^9 - 4 * q^11 + 4 * q^13 + 4 * q^15 - 4 * q^17 - 2 * q^19 + 12 * q^23 + 2 * q^25 - 16 * q^27 - 4 * q^29 + 8 * q^31 + 8 * q^33 - 4 * q^35 + 12 * q^37 - 20 * q^39 - 4 * q^41 + 4 * q^43 - 6 * q^45 + 4 * q^47 + 10 * q^49 - 4 * q^53 + 4 * q^55 + 4 * q^57 + 16 * q^59 + 16 * q^61 - 20 * q^63 - 4 * q^65 - 4 * q^67 - 24 * q^69 - 16 * q^71 + 12 * q^73 - 4 * q^75 - 8 * q^77 + 8 * q^79 + 46 * q^81 - 4 * q^83 + 4 * q^85 - 16 * q^87 + 4 * q^89 - 16 * q^91 - 24 * q^93 + 2 * q^95 - 12 * q^97 - 12 * 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
$$3$$ −3.41421 −1.97120 −0.985599 0.169102i $$-0.945913\pi$$
−0.985599 + 0.169102i $$0.945913\pi$$
$$4$$ 0 0
$$5$$ −1.00000 −0.447214
$$6$$ 0 0
$$7$$ −0.828427 −0.313116 −0.156558 0.987669i $$-0.550040\pi$$
−0.156558 + 0.987669i $$0.550040\pi$$
$$8$$ 0 0
$$9$$ 8.65685 2.88562
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ 6.24264 1.73140 0.865699 0.500566i $$-0.166875\pi$$
0.865699 + 0.500566i $$0.166875\pi$$
$$14$$ 0 0
$$15$$ 3.41421 0.881546
$$16$$ 0 0
$$17$$ 0.828427 0.200923 0.100462 0.994941i $$-0.467968\pi$$
0.100462 + 0.994941i $$0.467968\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ 2.82843 0.617213
$$22$$ 0 0
$$23$$ 6.00000 1.25109 0.625543 0.780189i $$-0.284877\pi$$
0.625543 + 0.780189i $$0.284877\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −19.3137 −3.71692
$$28$$ 0 0
$$29$$ 6.48528 1.20429 0.602143 0.798388i $$-0.294314\pi$$
0.602143 + 0.798388i $$0.294314\pi$$
$$30$$ 0 0
$$31$$ 6.82843 1.22642 0.613211 0.789919i $$-0.289878\pi$$
0.613211 + 0.789919i $$0.289878\pi$$
$$32$$ 0 0
$$33$$ 6.82843 1.18868
$$34$$ 0 0
$$35$$ 0.828427 0.140030
$$36$$ 0 0
$$37$$ 1.75736 0.288908 0.144454 0.989512i $$-0.453857\pi$$
0.144454 + 0.989512i $$0.453857\pi$$
$$38$$ 0 0
$$39$$ −21.3137 −3.41292
$$40$$ 0 0
$$41$$ 3.65685 0.571105 0.285552 0.958363i $$-0.407823\pi$$
0.285552 + 0.958363i $$0.407823\pi$$
$$42$$ 0 0
$$43$$ 4.82843 0.736328 0.368164 0.929761i $$-0.379986\pi$$
0.368164 + 0.929761i $$0.379986\pi$$
$$44$$ 0 0
$$45$$ −8.65685 −1.29049
$$46$$ 0 0
$$47$$ 4.82843 0.704298 0.352149 0.935944i $$-0.385451\pi$$
0.352149 + 0.935944i $$0.385451\pi$$
$$48$$ 0 0
$$49$$ −6.31371 −0.901958
$$50$$ 0 0
$$51$$ −2.82843 −0.396059
$$52$$ 0 0
$$53$$ −9.07107 −1.24601 −0.623003 0.782219i $$-0.714088\pi$$
−0.623003 + 0.782219i $$0.714088\pi$$
$$54$$ 0 0
$$55$$ 2.00000 0.269680
$$56$$ 0 0
$$57$$ 3.41421 0.452224
$$58$$ 0 0
$$59$$ 13.6569 1.77797 0.888985 0.457935i $$-0.151411\pi$$
0.888985 + 0.457935i $$0.151411\pi$$
$$60$$ 0 0
$$61$$ 13.6569 1.74858 0.874291 0.485403i $$-0.161327\pi$$
0.874291 + 0.485403i $$0.161327\pi$$
$$62$$ 0 0
$$63$$ −7.17157 −0.903533
$$64$$ 0 0
$$65$$ −6.24264 −0.774304
$$66$$ 0 0
$$67$$ −3.41421 −0.417113 −0.208556 0.978010i $$-0.566876\pi$$
−0.208556 + 0.978010i $$0.566876\pi$$
$$68$$ 0 0
$$69$$ −20.4853 −2.46614
$$70$$ 0 0
$$71$$ −5.17157 −0.613753 −0.306876 0.951749i $$-0.599284\pi$$
−0.306876 + 0.951749i $$0.599284\pi$$
$$72$$ 0 0
$$73$$ −2.48528 −0.290880 −0.145440 0.989367i $$-0.546460\pi$$
−0.145440 + 0.989367i $$0.546460\pi$$
$$74$$ 0 0
$$75$$ −3.41421 −0.394239
$$76$$ 0 0
$$77$$ 1.65685 0.188816
$$78$$ 0 0
$$79$$ −1.65685 −0.186411 −0.0932053 0.995647i $$-0.529711\pi$$
−0.0932053 + 0.995647i $$0.529711\pi$$
$$80$$ 0 0
$$81$$ 39.9706 4.44117
$$82$$ 0 0
$$83$$ −13.3137 −1.46137 −0.730685 0.682715i $$-0.760799\pi$$
−0.730685 + 0.682715i $$0.760799\pi$$
$$84$$ 0 0
$$85$$ −0.828427 −0.0898555
$$86$$ 0 0
$$87$$ −22.1421 −2.37389
$$88$$ 0 0
$$89$$ −6.48528 −0.687438 −0.343719 0.939072i $$-0.611687\pi$$
−0.343719 + 0.939072i $$0.611687\pi$$
$$90$$ 0 0
$$91$$ −5.17157 −0.542128
$$92$$ 0 0
$$93$$ −23.3137 −2.41752
$$94$$ 0 0
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ −10.2426 −1.03998 −0.519991 0.854172i $$-0.674065\pi$$
−0.519991 + 0.854172i $$0.674065\pi$$
$$98$$ 0 0
$$99$$ −17.3137 −1.74009
$$100$$ 0 0
$$101$$ −4.00000 −0.398015 −0.199007 0.979998i $$-0.563772\pi$$
−0.199007 + 0.979998i $$0.563772\pi$$
$$102$$ 0 0
$$103$$ −3.89949 −0.384229 −0.192114 0.981373i $$-0.561534\pi$$
−0.192114 + 0.981373i $$0.561534\pi$$
$$104$$ 0 0
$$105$$ −2.82843 −0.276026
$$106$$ 0 0
$$107$$ 7.41421 0.716759 0.358380 0.933576i $$-0.383329\pi$$
0.358380 + 0.933576i $$0.383329\pi$$
$$108$$ 0 0
$$109$$ 3.17157 0.303782 0.151891 0.988397i $$-0.451464\pi$$
0.151891 + 0.988397i $$0.451464\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ −9.07107 −0.853334 −0.426667 0.904409i $$-0.640312\pi$$
−0.426667 + 0.904409i $$0.640312\pi$$
$$114$$ 0 0
$$115$$ −6.00000 −0.559503
$$116$$ 0 0
$$117$$ 54.0416 4.99615
$$118$$ 0 0
$$119$$ −0.686292 −0.0629122
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ −12.4853 −1.12576
$$124$$ 0 0
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ −9.55635 −0.847989 −0.423994 0.905665i $$-0.639372\pi$$
−0.423994 + 0.905665i $$0.639372\pi$$
$$128$$ 0 0
$$129$$ −16.4853 −1.45145
$$130$$ 0 0
$$131$$ 5.65685 0.494242 0.247121 0.968985i $$-0.420516\pi$$
0.247121 + 0.968985i $$0.420516\pi$$
$$132$$ 0 0
$$133$$ 0.828427 0.0718337
$$134$$ 0 0
$$135$$ 19.3137 1.66226
$$136$$ 0 0
$$137$$ 8.14214 0.695630 0.347815 0.937563i $$-0.386924\pi$$
0.347815 + 0.937563i $$0.386924\pi$$
$$138$$ 0 0
$$139$$ 5.31371 0.450703 0.225351 0.974278i $$-0.427647\pi$$
0.225351 + 0.974278i $$0.427647\pi$$
$$140$$ 0 0
$$141$$ −16.4853 −1.38831
$$142$$ 0 0
$$143$$ −12.4853 −1.04407
$$144$$ 0 0
$$145$$ −6.48528 −0.538573
$$146$$ 0 0
$$147$$ 21.5563 1.77794
$$148$$ 0 0
$$149$$ 8.00000 0.655386 0.327693 0.944784i $$-0.393729\pi$$
0.327693 + 0.944784i $$0.393729\pi$$
$$150$$ 0 0
$$151$$ 10.1421 0.825355 0.412678 0.910877i $$-0.364594\pi$$
0.412678 + 0.910877i $$0.364594\pi$$
$$152$$ 0 0
$$153$$ 7.17157 0.579787
$$154$$ 0 0
$$155$$ −6.82843 −0.548472
$$156$$ 0 0
$$157$$ −8.82843 −0.704585 −0.352293 0.935890i $$-0.614598\pi$$
−0.352293 + 0.935890i $$0.614598\pi$$
$$158$$ 0 0
$$159$$ 30.9706 2.45613
$$160$$ 0 0
$$161$$ −4.97056 −0.391735
$$162$$ 0 0
$$163$$ 14.4853 1.13457 0.567287 0.823520i $$-0.307993\pi$$
0.567287 + 0.823520i $$0.307993\pi$$
$$164$$ 0 0
$$165$$ −6.82843 −0.531592
$$166$$ 0 0
$$167$$ −14.7279 −1.13968 −0.569840 0.821755i $$-0.692995\pi$$
−0.569840 + 0.821755i $$0.692995\pi$$
$$168$$ 0 0
$$169$$ 25.9706 1.99774
$$170$$ 0 0
$$171$$ −8.65685 −0.662006
$$172$$ 0 0
$$173$$ −15.8995 −1.20882 −0.604408 0.796675i $$-0.706590\pi$$
−0.604408 + 0.796675i $$0.706590\pi$$
$$174$$ 0 0
$$175$$ −0.828427 −0.0626232
$$176$$ 0 0
$$177$$ −46.6274 −3.50473
$$178$$ 0 0
$$179$$ 10.3431 0.773083 0.386542 0.922272i $$-0.373670\pi$$
0.386542 + 0.922272i $$0.373670\pi$$
$$180$$ 0 0
$$181$$ 18.0000 1.33793 0.668965 0.743294i $$-0.266738\pi$$
0.668965 + 0.743294i $$0.266738\pi$$
$$182$$ 0 0
$$183$$ −46.6274 −3.44680
$$184$$ 0 0
$$185$$ −1.75736 −0.129204
$$186$$ 0 0
$$187$$ −1.65685 −0.121161
$$188$$ 0 0
$$189$$ 16.0000 1.16383
$$190$$ 0 0
$$191$$ −4.00000 −0.289430 −0.144715 0.989473i $$-0.546227\pi$$
−0.144715 + 0.989473i $$0.546227\pi$$
$$192$$ 0 0
$$193$$ 3.41421 0.245760 0.122880 0.992422i $$-0.460787\pi$$
0.122880 + 0.992422i $$0.460787\pi$$
$$194$$ 0 0
$$195$$ 21.3137 1.52631
$$196$$ 0 0
$$197$$ −17.3137 −1.23355 −0.616775 0.787139i $$-0.711561\pi$$
−0.616775 + 0.787139i $$0.711561\pi$$
$$198$$ 0 0
$$199$$ 21.6569 1.53521 0.767607 0.640921i $$-0.221447\pi$$
0.767607 + 0.640921i $$0.221447\pi$$
$$200$$ 0 0
$$201$$ 11.6569 0.822211
$$202$$ 0 0
$$203$$ −5.37258 −0.377081
$$204$$ 0 0
$$205$$ −3.65685 −0.255406
$$206$$ 0 0
$$207$$ 51.9411 3.61016
$$208$$ 0 0
$$209$$ 2.00000 0.138343
$$210$$ 0 0
$$211$$ −28.4853 −1.96101 −0.980504 0.196500i $$-0.937042\pi$$
−0.980504 + 0.196500i $$0.937042\pi$$
$$212$$ 0 0
$$213$$ 17.6569 1.20983
$$214$$ 0 0
$$215$$ −4.82843 −0.329296
$$216$$ 0 0
$$217$$ −5.65685 −0.384012
$$218$$ 0 0
$$219$$ 8.48528 0.573382
$$220$$ 0 0
$$221$$ 5.17157 0.347878
$$222$$ 0 0
$$223$$ 23.2132 1.55447 0.777236 0.629210i $$-0.216621\pi$$
0.777236 + 0.629210i $$0.216621\pi$$
$$224$$ 0 0
$$225$$ 8.65685 0.577124
$$226$$ 0 0
$$227$$ 23.4142 1.55406 0.777028 0.629466i $$-0.216726\pi$$
0.777028 + 0.629466i $$0.216726\pi$$
$$228$$ 0 0
$$229$$ 10.3431 0.683494 0.341747 0.939792i $$-0.388981\pi$$
0.341747 + 0.939792i $$0.388981\pi$$
$$230$$ 0 0
$$231$$ −5.65685 −0.372194
$$232$$ 0 0
$$233$$ 10.0000 0.655122 0.327561 0.944830i $$-0.393773\pi$$
0.327561 + 0.944830i $$0.393773\pi$$
$$234$$ 0 0
$$235$$ −4.82843 −0.314972
$$236$$ 0 0
$$237$$ 5.65685 0.367452
$$238$$ 0 0
$$239$$ −21.6569 −1.40087 −0.700433 0.713718i $$-0.747010\pi$$
−0.700433 + 0.713718i $$0.747010\pi$$
$$240$$ 0 0
$$241$$ 26.2843 1.69312 0.846559 0.532294i $$-0.178670\pi$$
0.846559 + 0.532294i $$0.178670\pi$$
$$242$$ 0 0
$$243$$ −78.5269 −5.03750
$$244$$ 0 0
$$245$$ 6.31371 0.403368
$$246$$ 0 0
$$247$$ −6.24264 −0.397210
$$248$$ 0 0
$$249$$ 45.4558 2.88065
$$250$$ 0 0
$$251$$ −2.34315 −0.147898 −0.0739490 0.997262i $$-0.523560\pi$$
−0.0739490 + 0.997262i $$0.523560\pi$$
$$252$$ 0 0
$$253$$ −12.0000 −0.754434
$$254$$ 0 0
$$255$$ 2.82843 0.177123
$$256$$ 0 0
$$257$$ 6.24264 0.389405 0.194703 0.980862i $$-0.437626\pi$$
0.194703 + 0.980862i $$0.437626\pi$$
$$258$$ 0 0
$$259$$ −1.45584 −0.0904618
$$260$$ 0 0
$$261$$ 56.1421 3.47511
$$262$$ 0 0
$$263$$ 7.65685 0.472142 0.236071 0.971736i $$-0.424140\pi$$
0.236071 + 0.971736i $$0.424140\pi$$
$$264$$ 0 0
$$265$$ 9.07107 0.557231
$$266$$ 0 0
$$267$$ 22.1421 1.35508
$$268$$ 0 0
$$269$$ 5.51472 0.336238 0.168119 0.985767i $$-0.446231\pi$$
0.168119 + 0.985767i $$0.446231\pi$$
$$270$$ 0 0
$$271$$ −12.3431 −0.749793 −0.374896 0.927067i $$-0.622322\pi$$
−0.374896 + 0.927067i $$0.622322\pi$$
$$272$$ 0 0
$$273$$ 17.6569 1.06864
$$274$$ 0 0
$$275$$ −2.00000 −0.120605
$$276$$ 0 0
$$277$$ −11.1716 −0.671235 −0.335617 0.941998i $$-0.608945\pi$$
−0.335617 + 0.941998i $$0.608945\pi$$
$$278$$ 0 0
$$279$$ 59.1127 3.53898
$$280$$ 0 0
$$281$$ −14.9706 −0.893069 −0.446534 0.894766i $$-0.647342\pi$$
−0.446534 + 0.894766i $$0.647342\pi$$
$$282$$ 0 0
$$283$$ −25.3137 −1.50474 −0.752372 0.658739i $$-0.771090\pi$$
−0.752372 + 0.658739i $$0.771090\pi$$
$$284$$ 0 0
$$285$$ −3.41421 −0.202241
$$286$$ 0 0
$$287$$ −3.02944 −0.178822
$$288$$ 0 0
$$289$$ −16.3137 −0.959630
$$290$$ 0 0
$$291$$ 34.9706 2.05001
$$292$$ 0 0
$$293$$ 20.5858 1.20263 0.601317 0.799010i $$-0.294643\pi$$
0.601317 + 0.799010i $$0.294643\pi$$
$$294$$ 0 0
$$295$$ −13.6569 −0.795133
$$296$$ 0 0
$$297$$ 38.6274 2.24139
$$298$$ 0 0
$$299$$ 37.4558 2.16613
$$300$$ 0 0
$$301$$ −4.00000 −0.230556
$$302$$ 0 0
$$303$$ 13.6569 0.784566
$$304$$ 0 0
$$305$$ −13.6569 −0.781989
$$306$$ 0 0
$$307$$ −11.8995 −0.679140 −0.339570 0.940581i $$-0.610282\pi$$
−0.339570 + 0.940581i $$0.610282\pi$$
$$308$$ 0 0
$$309$$ 13.3137 0.757390
$$310$$ 0 0
$$311$$ 14.0000 0.793867 0.396934 0.917847i $$-0.370074\pi$$
0.396934 + 0.917847i $$0.370074\pi$$
$$312$$ 0 0
$$313$$ 18.0000 1.01742 0.508710 0.860938i $$-0.330123\pi$$
0.508710 + 0.860938i $$0.330123\pi$$
$$314$$ 0 0
$$315$$ 7.17157 0.404072
$$316$$ 0 0
$$317$$ 14.2426 0.799946 0.399973 0.916527i $$-0.369019\pi$$
0.399973 + 0.916527i $$0.369019\pi$$
$$318$$ 0 0
$$319$$ −12.9706 −0.726212
$$320$$ 0 0
$$321$$ −25.3137 −1.41287
$$322$$ 0 0
$$323$$ −0.828427 −0.0460949
$$324$$ 0 0
$$325$$ 6.24264 0.346279
$$326$$ 0 0
$$327$$ −10.8284 −0.598813
$$328$$ 0 0
$$329$$ −4.00000 −0.220527
$$330$$ 0 0
$$331$$ 6.82843 0.375324 0.187662 0.982234i $$-0.439909\pi$$
0.187662 + 0.982234i $$0.439909\pi$$
$$332$$ 0 0
$$333$$ 15.2132 0.833678
$$334$$ 0 0
$$335$$ 3.41421 0.186538
$$336$$ 0 0
$$337$$ 26.2426 1.42953 0.714764 0.699366i $$-0.246534\pi$$
0.714764 + 0.699366i $$0.246534\pi$$
$$338$$ 0 0
$$339$$ 30.9706 1.68209
$$340$$ 0 0
$$341$$ −13.6569 −0.739560
$$342$$ 0 0
$$343$$ 11.0294 0.595534
$$344$$ 0 0
$$345$$ 20.4853 1.10289
$$346$$ 0 0
$$347$$ −10.9706 −0.588931 −0.294465 0.955662i $$-0.595142\pi$$
−0.294465 + 0.955662i $$0.595142\pi$$
$$348$$ 0 0
$$349$$ 11.6569 0.623977 0.311989 0.950086i $$-0.399005\pi$$
0.311989 + 0.950086i $$0.399005\pi$$
$$350$$ 0 0
$$351$$ −120.569 −6.43547
$$352$$ 0 0
$$353$$ −32.1421 −1.71075 −0.855377 0.518007i $$-0.826674\pi$$
−0.855377 + 0.518007i $$0.826674\pi$$
$$354$$ 0 0
$$355$$ 5.17157 0.274479
$$356$$ 0 0
$$357$$ 2.34315 0.124012
$$358$$ 0 0
$$359$$ −1.31371 −0.0693349 −0.0346674 0.999399i $$-0.511037\pi$$
−0.0346674 + 0.999399i $$0.511037\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 23.8995 1.25440
$$364$$ 0 0
$$365$$ 2.48528 0.130086
$$366$$ 0 0
$$367$$ −0.142136 −0.00741942 −0.00370971 0.999993i $$-0.501181\pi$$
−0.00370971 + 0.999993i $$0.501181\pi$$
$$368$$ 0 0
$$369$$ 31.6569 1.64799
$$370$$ 0 0
$$371$$ 7.51472 0.390145
$$372$$ 0 0
$$373$$ 21.5563 1.11615 0.558073 0.829792i $$-0.311541\pi$$
0.558073 + 0.829792i $$0.311541\pi$$
$$374$$ 0 0
$$375$$ 3.41421 0.176309
$$376$$ 0 0
$$377$$ 40.4853 2.08510
$$378$$ 0 0
$$379$$ 33.6569 1.72884 0.864418 0.502773i $$-0.167687\pi$$
0.864418 + 0.502773i $$0.167687\pi$$
$$380$$ 0 0
$$381$$ 32.6274 1.67155
$$382$$ 0 0
$$383$$ −17.0711 −0.872291 −0.436145 0.899876i $$-0.643657\pi$$
−0.436145 + 0.899876i $$0.643657\pi$$
$$384$$ 0 0
$$385$$ −1.65685 −0.0844411
$$386$$ 0 0
$$387$$ 41.7990 2.12476
$$388$$ 0 0
$$389$$ −16.6274 −0.843044 −0.421522 0.906818i $$-0.638504\pi$$
−0.421522 + 0.906818i $$0.638504\pi$$
$$390$$ 0 0
$$391$$ 4.97056 0.251372
$$392$$ 0 0
$$393$$ −19.3137 −0.974248
$$394$$ 0 0
$$395$$ 1.65685 0.0833654
$$396$$ 0 0
$$397$$ 3.85786 0.193621 0.0968103 0.995303i $$-0.469136\pi$$
0.0968103 + 0.995303i $$0.469136\pi$$
$$398$$ 0 0
$$399$$ −2.82843 −0.141598
$$400$$ 0 0
$$401$$ −29.3137 −1.46386 −0.731928 0.681382i $$-0.761379\pi$$
−0.731928 + 0.681382i $$0.761379\pi$$
$$402$$ 0 0
$$403$$ 42.6274 2.12342
$$404$$ 0 0
$$405$$ −39.9706 −1.98615
$$406$$ 0 0
$$407$$ −3.51472 −0.174218
$$408$$ 0 0
$$409$$ 26.4853 1.30961 0.654806 0.755797i $$-0.272750\pi$$
0.654806 + 0.755797i $$0.272750\pi$$
$$410$$ 0 0
$$411$$ −27.7990 −1.37122
$$412$$ 0 0
$$413$$ −11.3137 −0.556711
$$414$$ 0 0
$$415$$ 13.3137 0.653544
$$416$$ 0 0
$$417$$ −18.1421 −0.888424
$$418$$ 0 0
$$419$$ −1.65685 −0.0809426 −0.0404713 0.999181i $$-0.512886\pi$$
−0.0404713 + 0.999181i $$0.512886\pi$$
$$420$$ 0 0
$$421$$ 19.6569 0.958016 0.479008 0.877810i $$-0.340996\pi$$
0.479008 + 0.877810i $$0.340996\pi$$
$$422$$ 0 0
$$423$$ 41.7990 2.03234
$$424$$ 0 0
$$425$$ 0.828427 0.0401846
$$426$$ 0 0
$$427$$ −11.3137 −0.547509
$$428$$ 0 0
$$429$$ 42.6274 2.05807
$$430$$ 0 0
$$431$$ 17.4558 0.840818 0.420409 0.907335i $$-0.361887\pi$$
0.420409 + 0.907335i $$0.361887\pi$$
$$432$$ 0 0
$$433$$ 27.2132 1.30778 0.653892 0.756588i $$-0.273135\pi$$
0.653892 + 0.756588i $$0.273135\pi$$
$$434$$ 0 0
$$435$$ 22.1421 1.06163
$$436$$ 0 0
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ 18.3431 0.875471 0.437735 0.899104i $$-0.355781\pi$$
0.437735 + 0.899104i $$0.355781\pi$$
$$440$$ 0 0
$$441$$ −54.6569 −2.60271
$$442$$ 0 0
$$443$$ −22.2843 −1.05876 −0.529379 0.848386i $$-0.677575\pi$$
−0.529379 + 0.848386i $$0.677575\pi$$
$$444$$ 0 0
$$445$$ 6.48528 0.307432
$$446$$ 0 0
$$447$$ −27.3137 −1.29189
$$448$$ 0 0
$$449$$ −30.4853 −1.43869 −0.719345 0.694653i $$-0.755558\pi$$
−0.719345 + 0.694653i $$0.755558\pi$$
$$450$$ 0 0
$$451$$ −7.31371 −0.344389
$$452$$ 0 0
$$453$$ −34.6274 −1.62694
$$454$$ 0 0
$$455$$ 5.17157 0.242447
$$456$$ 0 0
$$457$$ −14.9706 −0.700293 −0.350147 0.936695i $$-0.613868\pi$$
−0.350147 + 0.936695i $$0.613868\pi$$
$$458$$ 0 0
$$459$$ −16.0000 −0.746816
$$460$$ 0 0
$$461$$ −17.3137 −0.806380 −0.403190 0.915116i $$-0.632099\pi$$
−0.403190 + 0.915116i $$0.632099\pi$$
$$462$$ 0 0
$$463$$ −13.3137 −0.618741 −0.309370 0.950942i $$-0.600118\pi$$
−0.309370 + 0.950942i $$0.600118\pi$$
$$464$$ 0 0
$$465$$ 23.3137 1.08115
$$466$$ 0 0
$$467$$ −13.3137 −0.616085 −0.308042 0.951373i $$-0.599674\pi$$
−0.308042 + 0.951373i $$0.599674\pi$$
$$468$$ 0 0
$$469$$ 2.82843 0.130605
$$470$$ 0 0
$$471$$ 30.1421 1.38888
$$472$$ 0 0
$$473$$ −9.65685 −0.444023
$$474$$ 0 0
$$475$$ −1.00000 −0.0458831
$$476$$ 0 0
$$477$$ −78.5269 −3.59550
$$478$$ 0 0
$$479$$ 10.0000 0.456912 0.228456 0.973554i $$-0.426632\pi$$
0.228456 + 0.973554i $$0.426632\pi$$
$$480$$ 0 0
$$481$$ 10.9706 0.500215
$$482$$ 0 0
$$483$$ 16.9706 0.772187
$$484$$ 0 0
$$485$$ 10.2426 0.465094
$$486$$ 0 0
$$487$$ 12.3848 0.561208 0.280604 0.959824i $$-0.409465\pi$$
0.280604 + 0.959824i $$0.409465\pi$$
$$488$$ 0 0
$$489$$ −49.4558 −2.23647
$$490$$ 0 0
$$491$$ 12.6863 0.572524 0.286262 0.958151i $$-0.407587\pi$$
0.286262 + 0.958151i $$0.407587\pi$$
$$492$$ 0 0
$$493$$ 5.37258 0.241969
$$494$$ 0 0
$$495$$ 17.3137 0.778193
$$496$$ 0 0
$$497$$ 4.28427 0.192176
$$498$$ 0 0
$$499$$ 6.97056 0.312045 0.156023 0.987753i $$-0.450133\pi$$
0.156023 + 0.987753i $$0.450133\pi$$
$$500$$ 0 0
$$501$$ 50.2843 2.24654
$$502$$ 0 0
$$503$$ 30.9706 1.38091 0.690455 0.723376i $$-0.257411\pi$$
0.690455 + 0.723376i $$0.257411\pi$$
$$504$$ 0 0
$$505$$ 4.00000 0.177998
$$506$$ 0 0
$$507$$ −88.6690 −3.93793
$$508$$ 0 0
$$509$$ −5.79899 −0.257036 −0.128518 0.991707i $$-0.541022\pi$$
−0.128518 + 0.991707i $$0.541022\pi$$
$$510$$ 0 0
$$511$$ 2.05887 0.0910792
$$512$$ 0 0
$$513$$ 19.3137 0.852721
$$514$$ 0 0
$$515$$ 3.89949 0.171832
$$516$$ 0 0
$$517$$ −9.65685 −0.424708
$$518$$ 0 0
$$519$$ 54.2843 2.38282
$$520$$ 0 0
$$521$$ 39.6569 1.73740 0.868699 0.495340i $$-0.164957\pi$$
0.868699 + 0.495340i $$0.164957\pi$$
$$522$$ 0 0
$$523$$ 18.7279 0.818915 0.409457 0.912329i $$-0.365718\pi$$
0.409457 + 0.912329i $$0.365718\pi$$
$$524$$ 0 0
$$525$$ 2.82843 0.123443
$$526$$ 0 0
$$527$$ 5.65685 0.246416
$$528$$ 0 0
$$529$$ 13.0000 0.565217
$$530$$ 0 0
$$531$$ 118.225 5.13055
$$532$$ 0 0
$$533$$ 22.8284 0.988809
$$534$$ 0 0
$$535$$ −7.41421 −0.320544
$$536$$ 0 0
$$537$$ −35.3137 −1.52390
$$538$$ 0 0
$$539$$ 12.6274 0.543901
$$540$$ 0 0
$$541$$ 11.3137 0.486414 0.243207 0.969974i $$-0.421801\pi$$
0.243207 + 0.969974i $$0.421801\pi$$
$$542$$ 0 0
$$543$$ −61.4558 −2.63732
$$544$$ 0 0
$$545$$ −3.17157 −0.135855
$$546$$ 0 0
$$547$$ −0.384776 −0.0164518 −0.00822592 0.999966i $$-0.502618\pi$$
−0.00822592 + 0.999966i $$0.502618\pi$$
$$548$$ 0 0
$$549$$ 118.225 5.04574
$$550$$ 0 0
$$551$$ −6.48528 −0.276282
$$552$$ 0 0
$$553$$ 1.37258 0.0583682
$$554$$ 0 0
$$555$$ 6.00000 0.254686
$$556$$ 0 0
$$557$$ −2.20101 −0.0932598 −0.0466299 0.998912i $$-0.514848\pi$$
−0.0466299 + 0.998912i $$0.514848\pi$$
$$558$$ 0 0
$$559$$ 30.1421 1.27488
$$560$$ 0 0
$$561$$ 5.65685 0.238833
$$562$$ 0 0
$$563$$ −8.58579 −0.361848 −0.180924 0.983497i $$-0.557909\pi$$
−0.180924 + 0.983497i $$0.557909\pi$$
$$564$$ 0 0
$$565$$ 9.07107 0.381623
$$566$$ 0 0
$$567$$ −33.1127 −1.39060
$$568$$ 0 0
$$569$$ −4.82843 −0.202418 −0.101209 0.994865i $$-0.532271\pi$$
−0.101209 + 0.994865i $$0.532271\pi$$
$$570$$ 0 0
$$571$$ −37.3137 −1.56153 −0.780765 0.624825i $$-0.785170\pi$$
−0.780765 + 0.624825i $$0.785170\pi$$
$$572$$ 0 0
$$573$$ 13.6569 0.570523
$$574$$ 0 0
$$575$$ 6.00000 0.250217
$$576$$ 0 0
$$577$$ −6.00000 −0.249783 −0.124892 0.992170i $$-0.539858\pi$$
−0.124892 + 0.992170i $$0.539858\pi$$
$$578$$ 0 0
$$579$$ −11.6569 −0.484442
$$580$$ 0 0
$$581$$ 11.0294 0.457578
$$582$$ 0 0
$$583$$ 18.1421 0.751370
$$584$$ 0 0
$$585$$ −54.0416 −2.23435
$$586$$ 0 0
$$587$$ 18.9706 0.782999 0.391499 0.920178i $$-0.371956\pi$$
0.391499 + 0.920178i $$0.371956\pi$$
$$588$$ 0 0
$$589$$ −6.82843 −0.281360
$$590$$ 0 0
$$591$$ 59.1127 2.43157
$$592$$ 0 0
$$593$$ −36.6274 −1.50411 −0.752054 0.659102i $$-0.770937\pi$$
−0.752054 + 0.659102i $$0.770937\pi$$
$$594$$ 0 0
$$595$$ 0.686292 0.0281352
$$596$$ 0 0
$$597$$ −73.9411 −3.02621
$$598$$ 0 0
$$599$$ 27.3137 1.11601 0.558004 0.829838i $$-0.311567\pi$$
0.558004 + 0.829838i $$0.311567\pi$$
$$600$$ 0 0
$$601$$ −18.0000 −0.734235 −0.367118 0.930175i $$-0.619655\pi$$
−0.367118 + 0.930175i $$0.619655\pi$$
$$602$$ 0 0
$$603$$ −29.5563 −1.20363
$$604$$ 0 0
$$605$$ 7.00000 0.284590
$$606$$ 0 0
$$607$$ 9.07107 0.368183 0.184092 0.982909i $$-0.441066\pi$$
0.184092 + 0.982909i $$0.441066\pi$$
$$608$$ 0 0
$$609$$ 18.3431 0.743302
$$610$$ 0 0
$$611$$ 30.1421 1.21942
$$612$$ 0 0
$$613$$ −34.4853 −1.39285 −0.696424 0.717631i $$-0.745227\pi$$
−0.696424 + 0.717631i $$0.745227\pi$$
$$614$$ 0 0
$$615$$ 12.4853 0.503455
$$616$$ 0 0
$$617$$ −45.7990 −1.84380 −0.921899 0.387430i $$-0.873363\pi$$
−0.921899 + 0.387430i $$0.873363\pi$$
$$618$$ 0 0
$$619$$ 18.0000 0.723481 0.361741 0.932279i $$-0.382183\pi$$
0.361741 + 0.932279i $$0.382183\pi$$
$$620$$ 0 0
$$621$$ −115.882 −4.65019
$$622$$ 0 0
$$623$$ 5.37258 0.215248
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ −6.82843 −0.272701
$$628$$ 0 0
$$629$$ 1.45584 0.0580483
$$630$$ 0 0
$$631$$ −7.65685 −0.304815 −0.152407 0.988318i $$-0.548703\pi$$
−0.152407 + 0.988318i $$0.548703\pi$$
$$632$$ 0 0
$$633$$ 97.2548 3.86553
$$634$$ 0 0
$$635$$ 9.55635 0.379232
$$636$$ 0 0
$$637$$ −39.4142 −1.56165
$$638$$ 0 0
$$639$$ −44.7696 −1.77106
$$640$$ 0 0
$$641$$ 1.31371 0.0518884 0.0259442 0.999663i $$-0.491741\pi$$
0.0259442 + 0.999663i $$0.491741\pi$$
$$642$$ 0 0
$$643$$ −20.6274 −0.813466 −0.406733 0.913547i $$-0.633332\pi$$
−0.406733 + 0.913547i $$0.633332\pi$$
$$644$$ 0 0
$$645$$ 16.4853 0.649107
$$646$$ 0 0
$$647$$ −29.3137 −1.15244 −0.576220 0.817294i $$-0.695473\pi$$
−0.576220 + 0.817294i $$0.695473\pi$$
$$648$$ 0 0
$$649$$ −27.3137 −1.07216
$$650$$ 0 0
$$651$$ 19.3137 0.756964
$$652$$ 0 0
$$653$$ 25.7990 1.00959 0.504796 0.863239i $$-0.331568\pi$$
0.504796 + 0.863239i $$0.331568\pi$$
$$654$$ 0 0
$$655$$ −5.65685 −0.221032
$$656$$ 0 0
$$657$$ −21.5147 −0.839369
$$658$$ 0 0
$$659$$ 10.6274 0.413985 0.206993 0.978342i $$-0.433632\pi$$
0.206993 + 0.978342i $$0.433632\pi$$
$$660$$ 0 0
$$661$$ −32.6274 −1.26906 −0.634530 0.772898i $$-0.718806\pi$$
−0.634530 + 0.772898i $$0.718806\pi$$
$$662$$ 0 0
$$663$$ −17.6569 −0.685735
$$664$$ 0 0
$$665$$ −0.828427 −0.0321250
$$666$$ 0 0
$$667$$ 38.9117 1.50667
$$668$$ 0 0
$$669$$ −79.2548 −3.06417
$$670$$ 0 0
$$671$$ −27.3137 −1.05443
$$672$$ 0 0
$$673$$ 1.75736 0.0677412 0.0338706 0.999426i $$-0.489217\pi$$
0.0338706 + 0.999426i $$0.489217\pi$$
$$674$$ 0 0
$$675$$ −19.3137 −0.743385
$$676$$ 0 0
$$677$$ 5.55635 0.213548 0.106774 0.994283i $$-0.465948\pi$$
0.106774 + 0.994283i $$0.465948\pi$$
$$678$$ 0 0
$$679$$ 8.48528 0.325635
$$680$$ 0 0
$$681$$ −79.9411 −3.06335
$$682$$ 0 0
$$683$$ −13.2721 −0.507842 −0.253921 0.967225i $$-0.581720\pi$$
−0.253921 + 0.967225i $$0.581720\pi$$
$$684$$ 0 0
$$685$$ −8.14214 −0.311095
$$686$$ 0 0
$$687$$ −35.3137 −1.34730
$$688$$ 0 0
$$689$$ −56.6274 −2.15733
$$690$$ 0 0
$$691$$ −4.34315 −0.165221 −0.0826105 0.996582i $$-0.526326\pi$$
−0.0826105 + 0.996582i $$0.526326\pi$$
$$692$$ 0 0
$$693$$ 14.3431 0.544851
$$694$$ 0 0
$$695$$ −5.31371 −0.201560
$$696$$ 0 0
$$697$$ 3.02944 0.114748
$$698$$ 0 0
$$699$$ −34.1421 −1.29137
$$700$$ 0 0
$$701$$ −38.3431 −1.44820 −0.724100 0.689695i $$-0.757745\pi$$
−0.724100 + 0.689695i $$0.757745\pi$$
$$702$$ 0 0
$$703$$ −1.75736 −0.0662801
$$704$$ 0 0
$$705$$ 16.4853 0.620872
$$706$$ 0 0
$$707$$ 3.31371 0.124625
$$708$$ 0 0
$$709$$ 14.9706 0.562231 0.281116 0.959674i $$-0.409296\pi$$
0.281116 + 0.959674i $$0.409296\pi$$
$$710$$ 0 0
$$711$$ −14.3431 −0.537910
$$712$$ 0 0
$$713$$ 40.9706 1.53436
$$714$$ 0 0
$$715$$ 12.4853 0.466923
$$716$$ 0 0
$$717$$ 73.9411 2.76138
$$718$$ 0 0
$$719$$ 18.2843 0.681888 0.340944 0.940084i $$-0.389253\pi$$
0.340944 + 0.940084i $$0.389253\pi$$
$$720$$ 0 0
$$721$$ 3.23045 0.120308
$$722$$ 0 0
$$723$$ −89.7401 −3.33747
$$724$$ 0 0
$$725$$ 6.48528 0.240857
$$726$$ 0 0
$$727$$ 46.4853 1.72404 0.862022 0.506871i $$-0.169198\pi$$
0.862022 + 0.506871i $$0.169198\pi$$
$$728$$ 0 0
$$729$$ 148.196 5.48874
$$730$$ 0 0
$$731$$ 4.00000 0.147945
$$732$$ 0 0
$$733$$ −28.3431 −1.04688 −0.523439 0.852063i $$-0.675351\pi$$
−0.523439 + 0.852063i $$0.675351\pi$$
$$734$$ 0 0
$$735$$ −21.5563 −0.795118
$$736$$ 0 0
$$737$$ 6.82843 0.251528
$$738$$ 0 0
$$739$$ −10.6274 −0.390936 −0.195468 0.980710i $$-0.562623\pi$$
−0.195468 + 0.980710i $$0.562623\pi$$
$$740$$ 0 0
$$741$$ 21.3137 0.782979
$$742$$ 0 0
$$743$$ 34.0416 1.24887 0.624433 0.781078i $$-0.285330\pi$$
0.624433 + 0.781078i $$0.285330\pi$$
$$744$$ 0 0
$$745$$ −8.00000 −0.293097
$$746$$ 0 0
$$747$$ −115.255 −4.21695
$$748$$ 0 0
$$749$$ −6.14214 −0.224429
$$750$$ 0 0
$$751$$ 14.1421 0.516054 0.258027 0.966138i $$-0.416928\pi$$
0.258027 + 0.966138i $$0.416928\pi$$
$$752$$ 0 0
$$753$$ 8.00000 0.291536
$$754$$ 0 0
$$755$$ −10.1421 −0.369110
$$756$$ 0 0
$$757$$ −8.34315 −0.303237 −0.151618 0.988439i $$-0.548448\pi$$
−0.151618 + 0.988439i $$0.548448\pi$$
$$758$$ 0 0
$$759$$ 40.9706 1.48714
$$760$$ 0 0
$$761$$ −14.6274 −0.530243 −0.265122 0.964215i $$-0.585412\pi$$
−0.265122 + 0.964215i $$0.585412\pi$$
$$762$$ 0 0
$$763$$ −2.62742 −0.0951189
$$764$$ 0 0
$$765$$ −7.17157 −0.259289
$$766$$ 0 0
$$767$$ 85.2548 3.07837
$$768$$ 0 0
$$769$$ −6.62742 −0.238991 −0.119495 0.992835i $$-0.538128\pi$$
−0.119495 + 0.992835i $$0.538128\pi$$
$$770$$ 0 0
$$771$$ −21.3137 −0.767594
$$772$$ 0 0
$$773$$ 19.8995 0.715735 0.357868 0.933772i $$-0.383504\pi$$
0.357868 + 0.933772i $$0.383504\pi$$
$$774$$ 0 0
$$775$$ 6.82843 0.245284
$$776$$ 0 0
$$777$$ 4.97056 0.178318
$$778$$ 0 0
$$779$$ −3.65685 −0.131020
$$780$$ 0 0
$$781$$ 10.3431 0.370107
$$782$$ 0 0
$$783$$ −125.255 −4.47624
$$784$$ 0 0
$$785$$ 8.82843 0.315100
$$786$$ 0 0
$$787$$ 45.8406 1.63404 0.817021 0.576608i $$-0.195624\pi$$
0.817021 + 0.576608i $$0.195624\pi$$
$$788$$ 0 0
$$789$$ −26.1421 −0.930685
$$790$$ 0 0
$$791$$ 7.51472 0.267193
$$792$$ 0 0
$$793$$ 85.2548 3.02749
$$794$$ 0 0
$$795$$ −30.9706 −1.09841
$$796$$ 0 0
$$797$$ 7.89949 0.279814 0.139907 0.990165i $$-0.455320\pi$$
0.139907 + 0.990165i $$0.455320\pi$$
$$798$$ 0 0
$$799$$ 4.00000 0.141510
$$800$$ 0 0
$$801$$ −56.1421 −1.98368
$$802$$ 0 0
$$803$$ 4.97056 0.175407
$$804$$ 0 0
$$805$$ 4.97056 0.175189
$$806$$ 0 0
$$807$$ −18.8284 −0.662792
$$808$$ 0 0
$$809$$ 51.2548 1.80202 0.901012 0.433794i $$-0.142825\pi$$
0.901012 + 0.433794i $$0.142825\pi$$
$$810$$ 0 0
$$811$$ −19.7990 −0.695237 −0.347618 0.937636i $$-0.613009\pi$$
−0.347618 + 0.937636i $$0.613009\pi$$
$$812$$ 0 0
$$813$$ 42.1421 1.47799
$$814$$ 0 0
$$815$$ −14.4853 −0.507397
$$816$$ 0 0
$$817$$ −4.82843 −0.168925
$$818$$ 0 0
$$819$$ −44.7696 −1.56437
$$820$$ 0 0
$$821$$ −10.0000 −0.349002 −0.174501 0.984657i $$-0.555831\pi$$
−0.174501 + 0.984657i $$0.555831\pi$$
$$822$$ 0 0
$$823$$ 20.8284 0.726033 0.363017 0.931783i $$-0.381747\pi$$
0.363017 + 0.931783i $$0.381747\pi$$
$$824$$ 0 0
$$825$$ 6.82843 0.237735
$$826$$ 0 0
$$827$$ −14.9289 −0.519130 −0.259565 0.965726i $$-0.583579\pi$$
−0.259565 + 0.965726i $$0.583579\pi$$
$$828$$ 0 0
$$829$$ −44.4264 −1.54299 −0.771496 0.636234i $$-0.780491\pi$$
−0.771496 + 0.636234i $$0.780491\pi$$
$$830$$ 0 0
$$831$$ 38.1421 1.32314
$$832$$ 0 0
$$833$$ −5.23045 −0.181224
$$834$$ 0 0
$$835$$ 14.7279 0.509681
$$836$$ 0 0
$$837$$ −131.882 −4.55852
$$838$$ 0 0
$$839$$ 35.5980 1.22898 0.614489 0.788925i $$-0.289362\pi$$
0.614489 + 0.788925i $$0.289362\pi$$
$$840$$ 0 0
$$841$$ 13.0589 0.450306
$$842$$ 0 0
$$843$$ 51.1127 1.76041
$$844$$ 0 0
$$845$$ −25.9706 −0.893415
$$846$$ 0 0
$$847$$ 5.79899 0.199256
$$848$$ 0 0
$$849$$ 86.4264 2.96615
$$850$$ 0 0
$$851$$ 10.5442 0.361449
$$852$$ 0 0
$$853$$ −42.0000 −1.43805 −0.719026 0.694983i $$-0.755412\pi$$
−0.719026 + 0.694983i $$0.755412\pi$$
$$854$$ 0 0
$$855$$ 8.65685 0.296058
$$856$$ 0 0
$$857$$ 15.2132 0.519673 0.259837 0.965653i $$-0.416331\pi$$
0.259837 + 0.965653i $$0.416331\pi$$
$$858$$ 0 0
$$859$$ 32.2843 1.10153 0.550763 0.834662i $$-0.314337\pi$$
0.550763 + 0.834662i $$0.314337\pi$$
$$860$$ 0 0
$$861$$ 10.3431 0.352493
$$862$$ 0 0
$$863$$ −15.8995 −0.541225 −0.270613 0.962688i $$-0.587226\pi$$
−0.270613 + 0.962688i $$0.587226\pi$$
$$864$$ 0 0
$$865$$ 15.8995 0.540599
$$866$$ 0 0
$$867$$ 55.6985 1.89162
$$868$$ 0 0
$$869$$ 3.31371 0.112410
$$870$$ 0 0
$$871$$ −21.3137 −0.722187
$$872$$ 0 0
$$873$$ −88.6690 −3.00099
$$874$$ 0 0
$$875$$ 0.828427 0.0280059
$$876$$ 0 0
$$877$$ 22.9289 0.774255 0.387128 0.922026i $$-0.373467\pi$$
0.387128 + 0.922026i $$0.373467\pi$$
$$878$$ 0 0
$$879$$ −70.2843 −2.37063
$$880$$ 0 0
$$881$$ −12.2843 −0.413868 −0.206934 0.978355i $$-0.566348\pi$$
−0.206934 + 0.978355i $$0.566348\pi$$
$$882$$ 0 0
$$883$$ −44.8284 −1.50860 −0.754298 0.656532i $$-0.772023\pi$$
−0.754298 + 0.656532i $$0.772023\pi$$
$$884$$ 0 0
$$885$$ 46.6274 1.56736
$$886$$ 0 0
$$887$$ 34.9289 1.17280 0.586399 0.810022i $$-0.300545\pi$$
0.586399 + 0.810022i $$0.300545\pi$$
$$888$$ 0 0
$$889$$ 7.91674 0.265519
$$890$$ 0 0
$$891$$ −79.9411 −2.67813
$$892$$ 0 0
$$893$$ −4.82843 −0.161577
$$894$$ 0 0
$$895$$ −10.3431 −0.345733
$$896$$ 0 0
$$897$$ −127.882 −4.26986
$$898$$ 0 0
$$899$$ 44.2843 1.47696
$$900$$ 0 0
$$901$$ −7.51472 −0.250352
$$902$$ 0 0
$$903$$ 13.6569 0.454472
$$904$$ 0 0
$$905$$ −18.0000 −0.598340
$$906$$ 0 0
$$907$$ 16.3848 0.544048 0.272024 0.962291i $$-0.412307\pi$$
0.272024 + 0.962291i $$0.412307\pi$$
$$908$$ 0 0
$$909$$ −34.6274 −1.14852
$$910$$ 0 0
$$911$$ −43.7990 −1.45113 −0.725563 0.688156i $$-0.758420\pi$$
−0.725563 + 0.688156i $$0.758420\pi$$
$$912$$ 0 0
$$913$$ 26.6274 0.881239
$$914$$ 0 0
$$915$$ 46.6274 1.54145
$$916$$ 0 0
$$917$$ −4.68629 −0.154755
$$918$$ 0 0
$$919$$ −12.6863 −0.418482 −0.209241 0.977864i $$-0.567099\pi$$
−0.209241 + 0.977864i $$0.567099\pi$$
$$920$$ 0 0
$$921$$ 40.6274 1.33872
$$922$$ 0 0
$$923$$ −32.2843 −1.06265
$$924$$ 0 0
$$925$$ 1.75736 0.0577816
$$926$$ 0 0
$$927$$ −33.7574 −1.10874
$$928$$ 0 0
$$929$$ 29.3137 0.961752 0.480876 0.876789i $$-0.340319\pi$$
0.480876 + 0.876789i $$0.340319\pi$$
$$930$$ 0 0
$$931$$ 6.31371 0.206923
$$932$$ 0 0
$$933$$ −47.7990 −1.56487
$$934$$ 0 0
$$935$$ 1.65685 0.0541849
$$936$$ 0 0
$$937$$ −11.8579 −0.387380 −0.193690 0.981063i $$-0.562046\pi$$
−0.193690 + 0.981063i $$0.562046\pi$$
$$938$$ 0 0
$$939$$ −61.4558 −2.00554
$$940$$ 0 0
$$941$$ −26.2843 −0.856843 −0.428421 0.903579i $$-0.640930\pi$$
−0.428421 + 0.903579i $$0.640930\pi$$
$$942$$ 0 0
$$943$$ 21.9411 0.714501
$$944$$ 0 0
$$945$$ −16.0000 −0.520480
$$946$$ 0 0
$$947$$ −19.6569 −0.638762 −0.319381 0.947626i $$-0.603475\pi$$
−0.319381 + 0.947626i $$0.603475\pi$$
$$948$$ 0 0
$$949$$ −15.5147 −0.503629
$$950$$ 0 0
$$951$$ −48.6274 −1.57685
$$952$$ 0 0
$$953$$ 14.2426 0.461364 0.230682 0.973029i $$-0.425904\pi$$
0.230682 + 0.973029i $$0.425904\pi$$
$$954$$ 0 0
$$955$$ 4.00000 0.129437
$$956$$ 0 0
$$957$$ 44.2843 1.43151
$$958$$ 0 0
$$959$$ −6.74517 −0.217813
$$960$$ 0 0
$$961$$ 15.6274 0.504110
$$962$$ 0 0
$$963$$ 64.1838 2.06829
$$964$$ 0 0
$$965$$ −3.41421 −0.109907
$$966$$ 0 0
$$967$$ 24.6274 0.791964 0.395982 0.918258i $$-0.370404\pi$$
0.395982 + 0.918258i $$0.370404\pi$$
$$968$$ 0 0
$$969$$ 2.82843 0.0908622
$$970$$ 0 0
$$971$$ 27.1127 0.870088 0.435044 0.900409i $$-0.356733\pi$$
0.435044 + 0.900409i $$0.356733\pi$$
$$972$$ 0 0
$$973$$ −4.40202 −0.141122
$$974$$ 0 0
$$975$$ −21.3137 −0.682585
$$976$$ 0 0
$$977$$ 34.7279 1.11104 0.555522 0.831502i $$-0.312518\pi$$
0.555522 + 0.831502i $$0.312518\pi$$
$$978$$ 0 0
$$979$$ 12.9706 0.414541
$$980$$ 0 0
$$981$$ 27.4558 0.876598
$$982$$ 0 0
$$983$$ 20.5858 0.656585 0.328292 0.944576i $$-0.393527\pi$$
0.328292 + 0.944576i $$0.393527\pi$$
$$984$$ 0 0
$$985$$ 17.3137 0.551661
$$986$$ 0 0
$$987$$ 13.6569 0.434702
$$988$$ 0 0
$$989$$ 28.9706 0.921210
$$990$$ 0 0
$$991$$ −11.1127 −0.353006 −0.176503 0.984300i $$-0.556479\pi$$
−0.176503 + 0.984300i $$0.556479\pi$$
$$992$$ 0 0
$$993$$ −23.3137 −0.739838
$$994$$ 0 0
$$995$$ −21.6569 −0.686568
$$996$$ 0 0
$$997$$ 30.4853 0.965479 0.482739 0.875764i $$-0.339642\pi$$
0.482739 + 0.875764i $$0.339642\pi$$
$$998$$ 0 0
$$999$$ −33.9411 −1.07385
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 6080.2.a.y.1.1 2
4.3 odd 2 6080.2.a.bl.1.2 2
8.3 odd 2 380.2.a.c.1.1 2
8.5 even 2 1520.2.a.o.1.2 2
24.11 even 2 3420.2.a.g.1.2 2
40.3 even 4 1900.2.c.d.1749.1 4
40.19 odd 2 1900.2.a.e.1.2 2
40.27 even 4 1900.2.c.d.1749.4 4
40.29 even 2 7600.2.a.u.1.1 2
152.75 even 2 7220.2.a.m.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.a.c.1.1 2 8.3 odd 2
1520.2.a.o.1.2 2 8.5 even 2
1900.2.a.e.1.2 2 40.19 odd 2
1900.2.c.d.1749.1 4 40.3 even 4
1900.2.c.d.1749.4 4 40.27 even 4
3420.2.a.g.1.2 2 24.11 even 2
6080.2.a.y.1.1 2 1.1 even 1 trivial
6080.2.a.bl.1.2 2 4.3 odd 2
7220.2.a.m.1.2 2 152.75 even 2
7600.2.a.u.1.1 2 40.29 even 2