# Properties

 Label 1520.2.a.o.1.2 Level $1520$ Weight $2$ Character 1520.1 Self dual yes Analytic conductor $12.137$ 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] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1520 = 2^{4} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1520.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: $$12.1372611072$$ 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 1.2 Root $$1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 1520.1

## $q$-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 $$-0.0540867\pi$$
0.985599 + 0.169102i $$0.0540867\pi$$
$$4$$ 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 $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ −6.24264 −1.73140 −0.865699 0.500566i $$-0.833125\pi$$
−0.865699 + 0.500566i $$0.833125\pi$$
$$14$$ 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 $$-0.705686\pi$$
−0.602143 + 0.798388i $$0.705686\pi$$
$$30$$ 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 $$-0.546143\pi$$
−0.144454 + 0.989512i $$0.546143\pi$$
$$38$$ 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 $$-0.620014\pi$$
−0.368164 + 0.929761i $$0.620014\pi$$
$$44$$ 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 $$-0.285912\pi$$
0.623003 + 0.782219i $$0.285912\pi$$
$$54$$ 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 $$-0.848589\pi$$
−0.888985 + 0.457935i $$0.848589\pi$$
$$60$$ 0 0
$$61$$ −13.6569 −1.74858 −0.874291 0.485403i $$-0.838673\pi$$
−0.874291 + 0.485403i $$0.838673\pi$$
$$62$$ 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 $$-0.433124\pi$$
0.208556 + 0.978010i $$0.433124\pi$$
$$68$$ 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 $$-0.239201\pi$$
0.730685 + 0.682715i $$0.239201\pi$$
$$84$$ 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 $$-0.436228\pi$$
0.199007 + 0.979998i $$0.436228\pi$$
$$102$$ 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 $$-0.616671\pi$$
−0.358380 + 0.933576i $$0.616671\pi$$
$$108$$ 0 0
$$109$$ −3.17157 −0.303782 −0.151891 0.988397i $$-0.548536\pi$$
−0.151891 + 0.988397i $$0.548536\pi$$
$$110$$ 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 $$-0.579484\pi$$
−0.247121 + 0.968985i $$0.579484\pi$$
$$132$$ 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 $$-0.572353\pi$$
−0.225351 + 0.974278i $$0.572353\pi$$
$$140$$ 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 $$-0.606271\pi$$
−0.327693 + 0.944784i $$0.606271\pi$$
$$150$$ 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 $$-0.385402\pi$$
0.352293 + 0.935890i $$0.385402\pi$$
$$158$$ 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 $$-0.692007\pi$$
−0.567287 + 0.823520i $$0.692007\pi$$
$$164$$ 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 $$-0.293410\pi$$
0.604408 + 0.796675i $$0.293410\pi$$
$$174$$ 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 $$-0.626330\pi$$
−0.386542 + 0.922272i $$0.626330\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 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 $$-0.288439\pi$$
0.616775 + 0.787139i $$0.288439\pi$$
$$198$$ 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 $$-0.0629576\pi$$
0.980504 + 0.196500i $$0.0629576\pi$$
$$212$$ 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 $$-0.783274\pi$$
−0.777028 + 0.629466i $$0.783274\pi$$
$$228$$ 0 0
$$229$$ −10.3431 −0.683494 −0.341747 0.939792i $$-0.611019\pi$$
−0.341747 + 0.939792i $$0.611019\pi$$
$$230$$ 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 $$-0.476440\pi$$
0.0739490 + 0.997262i $$0.476440\pi$$
$$252$$ 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 $$-0.553769\pi$$
−0.168119 + 0.985767i $$0.553769\pi$$
$$270$$ 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 $$-0.391055\pi$$
0.335617 + 0.941998i $$0.391055\pi$$
$$278$$ 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 $$-0.228910\pi$$
0.752372 + 0.658739i $$0.228910\pi$$
$$284$$ 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 $$-0.705357\pi$$
−0.601317 + 0.799010i $$0.705357\pi$$
$$294$$ 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 $$-0.389718\pi$$
0.339570 + 0.940581i $$0.389718\pi$$
$$308$$ 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 $$-0.630981\pi$$
−0.399973 + 0.916527i $$0.630981\pi$$
$$318$$ 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 $$-0.560091\pi$$
−0.187662 + 0.982234i $$0.560091\pi$$
$$332$$ 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 $$-0.404858\pi$$
0.294465 + 0.955662i $$0.404858\pi$$
$$348$$ 0 0
$$349$$ −11.6569 −0.623977 −0.311989 0.950086i $$-0.600995\pi$$
−0.311989 + 0.950086i $$0.600995\pi$$
$$350$$ 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 $$-0.688459\pi$$
−0.558073 + 0.829792i $$0.688459\pi$$
$$374$$ 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 $$-0.832313\pi$$
−0.864418 + 0.502773i $$0.832313\pi$$
$$380$$ 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 $$-0.361496\pi$$
0.421522 + 0.906818i $$0.361496\pi$$
$$390$$ 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 $$-0.530864\pi$$
−0.0968103 + 0.995303i $$0.530864\pi$$
$$398$$ 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 $$-0.487114\pi$$
0.0404713 + 0.999181i $$0.487114\pi$$
$$420$$ 0 0
$$421$$ −19.6569 −0.958016 −0.479008 0.877810i $$-0.659004\pi$$
−0.479008 + 0.877810i $$0.659004\pi$$
$$422$$ 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 $$-0.322425\pi$$
0.529379 + 0.848386i $$0.322425\pi$$
$$444$$ 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 $$-0.367901\pi$$
0.403190 + 0.915116i $$0.367901\pi$$
$$462$$ 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 $$-0.400326\pi$$
0.308042 + 0.951373i $$0.400326\pi$$
$$468$$ 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 $$-0.592413\pi$$
−0.286262 + 0.958151i $$0.592413\pi$$
$$492$$ 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 $$-0.549867\pi$$
−0.156023 + 0.987753i $$0.549867\pi$$
$$500$$ 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 $$-0.458978\pi$$
0.128518 + 0.991707i $$0.458978\pi$$
$$510$$ 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 $$-0.634282\pi$$
−0.409457 + 0.912329i $$0.634282\pi$$
$$524$$ 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 $$-0.578199\pi$$
−0.243207 + 0.969974i $$0.578199\pi$$
$$542$$ 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 $$-0.497382\pi$$
0.00822592 + 0.999966i $$0.497382\pi$$
$$548$$ 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 $$-0.485152\pi$$
0.0466299 + 0.998912i $$0.485152\pi$$
$$558$$ 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 $$-0.442091\pi$$
0.180924 + 0.983497i $$0.442091\pi$$
$$564$$ 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 $$-0.214830\pi$$
0.780765 + 0.624825i $$0.214830\pi$$
$$572$$ 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 $$-0.628044\pi$$
−0.391499 + 0.920178i $$0.628044\pi$$
$$588$$ 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 $$-0.254773\pi$$
0.696424 + 0.717631i $$0.254773\pi$$
$$614$$ 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 $$-0.617817\pi$$
−0.361741 + 0.932279i $$0.617817\pi$$
$$620$$ 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 $$-0.366668\pi$$
0.406733 + 0.913547i $$0.366668\pi$$
$$644$$ 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 $$-0.668432\pi$$
−0.504796 + 0.863239i $$0.668432\pi$$
$$654$$ 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 $$-0.566368\pi$$
−0.206993 + 0.978342i $$0.566368\pi$$
$$660$$ 0 0
$$661$$ 32.6274 1.26906 0.634530 0.772898i $$-0.281194\pi$$
0.634530 + 0.772898i $$0.281194\pi$$
$$662$$ 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 $$-0.534052\pi$$
−0.106774 + 0.994283i $$0.534052\pi$$
$$678$$ 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 $$-0.418280\pi$$
0.253921 + 0.967225i $$0.418280\pi$$
$$684$$ 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 $$-0.473674\pi$$
0.0826105 + 0.996582i $$0.473674\pi$$
$$692$$ 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 $$-0.242255\pi$$
0.724100 + 0.689695i $$0.242255\pi$$
$$702$$ 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 $$-0.590704\pi$$
−0.281116 + 0.959674i $$0.590704\pi$$
$$710$$ 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 $$-0.324649\pi$$
0.523439 + 0.852063i $$0.324649\pi$$
$$734$$ 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 $$-0.437377\pi$$
0.195468 + 0.980710i $$0.437377\pi$$
$$740$$ 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 $$-0.451552\pi$$
0.151618 + 0.988439i $$0.451552\pi$$
$$758$$ 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 $$-0.616496\pi$$
−0.357868 + 0.933772i $$0.616496\pi$$
$$774$$ 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 $$-0.804376\pi$$
−0.817021 + 0.576608i $$0.804376\pi$$
$$788$$ 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 $$-0.544680\pi$$
−0.139907 + 0.990165i $$0.544680\pi$$
$$798$$ 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 $$-0.386991\pi$$
0.347618 + 0.937636i $$0.386991\pi$$
$$812$$ 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 $$-0.444169\pi$$
0.174501 + 0.984657i $$0.444169\pi$$
$$822$$ 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 $$-0.416421\pi$$
0.259565 + 0.965726i $$0.416421\pi$$
$$828$$ 0 0
$$829$$ 44.4264 1.54299 0.771496 0.636234i $$-0.219509\pi$$
0.771496 + 0.636234i $$0.219509\pi$$
$$830$$ 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 $$-0.244588\pi$$
0.719026 + 0.694983i $$0.244588\pi$$
$$854$$ 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 $$-0.685663\pi$$
−0.550763 + 0.834662i $$0.685663\pi$$
$$860$$ 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 $$-0.626533\pi$$
−0.387128 + 0.922026i $$0.626533\pi$$
$$878$$ 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 $$-0.227977\pi$$
0.754298 + 0.656532i $$0.227977\pi$$
$$884$$ 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 $$-0.587693\pi$$
−0.272024 + 0.962291i $$0.587693\pi$$
$$908$$ 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 $$-0.359070\pi$$
0.428421 + 0.903579i $$0.359070\pi$$
$$942$$ 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 $$-0.396525\pi$$
0.319381 + 0.947626i $$0.396525\pi$$
$$948$$ 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 $$-0.643267\pi$$
−0.435044 + 0.900409i $$0.643267\pi$$
$$972$$ 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 $$-0.660358\pi$$
−0.482739 + 0.875764i $$0.660358\pi$$
$$998$$ 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 1520.2.a.o.1.2 2
4.3 odd 2 380.2.a.c.1.1 2
5.4 even 2 7600.2.a.u.1.1 2
8.3 odd 2 6080.2.a.bl.1.2 2
8.5 even 2 6080.2.a.y.1.1 2
12.11 even 2 3420.2.a.g.1.2 2
20.3 even 4 1900.2.c.d.1749.1 4
20.7 even 4 1900.2.c.d.1749.4 4
20.19 odd 2 1900.2.a.e.1.2 2
76.75 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 4.3 odd 2
1520.2.a.o.1.2 2 1.1 even 1 trivial
1900.2.a.e.1.2 2 20.19 odd 2
1900.2.c.d.1749.1 4 20.3 even 4
1900.2.c.d.1749.4 4 20.7 even 4
3420.2.a.g.1.2 2 12.11 even 2
6080.2.a.y.1.1 2 8.5 even 2
6080.2.a.bl.1.2 2 8.3 odd 2
7220.2.a.m.1.2 2 76.75 even 2
7600.2.a.u.1.1 2 5.4 even 2