# Properties

 Label 1840.2.a.q.1.3 Level $1840$ Weight $2$ Character 1840.1 Self dual yes Analytic conductor $14.692$ Analytic rank $1$ 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] = [1840,2,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, 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("1840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1840 = 2^{4} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1840.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: $$14.6924739719$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 4x - 1$$ x^3 - 4*x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.3 Root $$-0.254102$$ of defining polynomial Character $$\chi$$ $$=$$ 1840.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.93543 q^{3} +1.00000 q^{5} -4.93543 q^{7} +0.745898 q^{9} +O(q^{10})$$ $$q+1.93543 q^{3} +1.00000 q^{5} -4.93543 q^{7} +0.745898 q^{9} -0.745898 q^{11} +1.74590 q^{13} +1.93543 q^{15} -6.10856 q^{17} -5.44364 q^{19} -9.55220 q^{21} +1.00000 q^{23} +1.00000 q^{25} -4.36266 q^{27} -1.66492 q^{29} +1.61676 q^{31} -1.44364 q^{33} -4.93543 q^{35} +4.34625 q^{37} +3.37907 q^{39} -6.95184 q^{41} -5.01641 q^{43} +0.745898 q^{45} -2.68133 q^{47} +17.3585 q^{49} -11.8227 q^{51} +13.7417 q^{53} -0.745898 q^{55} -10.5358 q^{57} -12.2171 q^{59} -13.9794 q^{61} -3.68133 q^{63} +1.74590 q^{65} -13.1044 q^{67} +1.93543 q^{69} +9.67716 q^{71} +5.69774 q^{73} +1.93543 q^{75} +3.68133 q^{77} -10.3791 q^{79} -10.6813 q^{81} -0.637339 q^{83} -6.10856 q^{85} -3.22235 q^{87} +2.72532 q^{89} -8.61676 q^{91} +3.12914 q^{93} -5.44364 q^{95} +7.12497 q^{97} -0.556364 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9}+O(q^{10})$$ 3 * q - 2 * q^3 + 3 * q^5 - 7 * q^7 + 3 * q^9 $$3 q - 2 q^{3} + 3 q^{5} - 7 q^{7} + 3 q^{9} - 3 q^{11} + 6 q^{13} - 2 q^{15} - 5 q^{17} - 7 q^{19} - 6 q^{21} + 3 q^{23} + 3 q^{25} + q^{27} - q^{29} - 10 q^{31} + 5 q^{33} - 7 q^{35} + 2 q^{37} - 7 q^{39} - 10 q^{41} - 12 q^{43} + 3 q^{45} - q^{47} + 6 q^{49} - 9 q^{51} + 10 q^{53} - 3 q^{55} - 12 q^{57} - 10 q^{59} - 13 q^{61} - 4 q^{63} + 6 q^{65} + 6 q^{67} - 2 q^{69} - 10 q^{71} + 7 q^{73} - 2 q^{75} + 4 q^{77} - 14 q^{79} - 25 q^{81} - 16 q^{83} - 5 q^{85} + 5 q^{87} - 20 q^{89} - 11 q^{91} + 25 q^{93} - 7 q^{95} + 5 q^{97} - 11 q^{99}+O(q^{100})$$ 3 * q - 2 * q^3 + 3 * q^5 - 7 * q^7 + 3 * q^9 - 3 * q^11 + 6 * q^13 - 2 * q^15 - 5 * q^17 - 7 * q^19 - 6 * q^21 + 3 * q^23 + 3 * q^25 + q^27 - q^29 - 10 * q^31 + 5 * q^33 - 7 * q^35 + 2 * q^37 - 7 * q^39 - 10 * q^41 - 12 * q^43 + 3 * q^45 - q^47 + 6 * q^49 - 9 * q^51 + 10 * q^53 - 3 * q^55 - 12 * q^57 - 10 * q^59 - 13 * q^61 - 4 * q^63 + 6 * q^65 + 6 * q^67 - 2 * q^69 - 10 * q^71 + 7 * q^73 - 2 * q^75 + 4 * q^77 - 14 * q^79 - 25 * q^81 - 16 * q^83 - 5 * q^85 + 5 * q^87 - 20 * q^89 - 11 * q^91 + 25 * q^93 - 7 * q^95 + 5 * q^97 - 11 * 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
$$3$$ 1.93543 1.11742 0.558711 0.829362i $$-0.311296\pi$$
0.558711 + 0.829362i $$0.311296\pi$$
$$4$$ 0 0
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ −4.93543 −1.86542 −0.932709 0.360630i $$-0.882562\pi$$
−0.932709 + 0.360630i $$0.882562\pi$$
$$8$$ 0 0
$$9$$ 0.745898 0.248633
$$10$$ 0 0
$$11$$ −0.745898 −0.224897 −0.112448 0.993658i $$-0.535869\pi$$
−0.112448 + 0.993658i $$0.535869\pi$$
$$12$$ 0 0
$$13$$ 1.74590 0.484225 0.242113 0.970248i $$-0.422160\pi$$
0.242113 + 0.970248i $$0.422160\pi$$
$$14$$ 0 0
$$15$$ 1.93543 0.499726
$$16$$ 0 0
$$17$$ −6.10856 −1.48154 −0.740772 0.671757i $$-0.765540\pi$$
−0.740772 + 0.671757i $$0.765540\pi$$
$$18$$ 0 0
$$19$$ −5.44364 −1.24886 −0.624428 0.781083i $$-0.714668\pi$$
−0.624428 + 0.781083i $$0.714668\pi$$
$$20$$ 0 0
$$21$$ −9.55220 −2.08446
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 0 0
$$27$$ −4.36266 −0.839595
$$28$$ 0 0
$$29$$ −1.66492 −0.309169 −0.154584 0.987980i $$-0.549404\pi$$
−0.154584 + 0.987980i $$0.549404\pi$$
$$30$$ 0 0
$$31$$ 1.61676 0.290379 0.145190 0.989404i $$-0.453621\pi$$
0.145190 + 0.989404i $$0.453621\pi$$
$$32$$ 0 0
$$33$$ −1.44364 −0.251305
$$34$$ 0 0
$$35$$ −4.93543 −0.834240
$$36$$ 0 0
$$37$$ 4.34625 0.714520 0.357260 0.934005i $$-0.383711\pi$$
0.357260 + 0.934005i $$0.383711\pi$$
$$38$$ 0 0
$$39$$ 3.37907 0.541084
$$40$$ 0 0
$$41$$ −6.95184 −1.08569 −0.542847 0.839831i $$-0.682654\pi$$
−0.542847 + 0.839831i $$0.682654\pi$$
$$42$$ 0 0
$$43$$ −5.01641 −0.764995 −0.382497 0.923957i $$-0.624936\pi$$
−0.382497 + 0.923957i $$0.624936\pi$$
$$44$$ 0 0
$$45$$ 0.745898 0.111192
$$46$$ 0 0
$$47$$ −2.68133 −0.391112 −0.195556 0.980693i $$-0.562651\pi$$
−0.195556 + 0.980693i $$0.562651\pi$$
$$48$$ 0 0
$$49$$ 17.3585 2.47978
$$50$$ 0 0
$$51$$ −11.8227 −1.65551
$$52$$ 0 0
$$53$$ 13.7417 1.88757 0.943786 0.330558i $$-0.107237\pi$$
0.943786 + 0.330558i $$0.107237\pi$$
$$54$$ 0 0
$$55$$ −0.745898 −0.100577
$$56$$ 0 0
$$57$$ −10.5358 −1.39550
$$58$$ 0 0
$$59$$ −12.2171 −1.59053 −0.795267 0.606260i $$-0.792669\pi$$
−0.795267 + 0.606260i $$0.792669\pi$$
$$60$$ 0 0
$$61$$ −13.9794 −1.78988 −0.894941 0.446185i $$-0.852782\pi$$
−0.894941 + 0.446185i $$0.852782\pi$$
$$62$$ 0 0
$$63$$ −3.68133 −0.463804
$$64$$ 0 0
$$65$$ 1.74590 0.216552
$$66$$ 0 0
$$67$$ −13.1044 −1.60096 −0.800478 0.599362i $$-0.795421\pi$$
−0.800478 + 0.599362i $$0.795421\pi$$
$$68$$ 0 0
$$69$$ 1.93543 0.232999
$$70$$ 0 0
$$71$$ 9.67716 1.14847 0.574234 0.818691i $$-0.305300\pi$$
0.574234 + 0.818691i $$0.305300\pi$$
$$72$$ 0 0
$$73$$ 5.69774 0.666870 0.333435 0.942773i $$-0.391792\pi$$
0.333435 + 0.942773i $$0.391792\pi$$
$$74$$ 0 0
$$75$$ 1.93543 0.223484
$$76$$ 0 0
$$77$$ 3.68133 0.419527
$$78$$ 0 0
$$79$$ −10.3791 −1.16774 −0.583868 0.811848i $$-0.698462\pi$$
−0.583868 + 0.811848i $$0.698462\pi$$
$$80$$ 0 0
$$81$$ −10.6813 −1.18681
$$82$$ 0 0
$$83$$ −0.637339 −0.0699570 −0.0349785 0.999388i $$-0.511136\pi$$
−0.0349785 + 0.999388i $$0.511136\pi$$
$$84$$ 0 0
$$85$$ −6.10856 −0.662566
$$86$$ 0 0
$$87$$ −3.22235 −0.345472
$$88$$ 0 0
$$89$$ 2.72532 0.288884 0.144442 0.989513i $$-0.453861\pi$$
0.144442 + 0.989513i $$0.453861\pi$$
$$90$$ 0 0
$$91$$ −8.61676 −0.903282
$$92$$ 0 0
$$93$$ 3.12914 0.324476
$$94$$ 0 0
$$95$$ −5.44364 −0.558505
$$96$$ 0 0
$$97$$ 7.12497 0.723431 0.361715 0.932289i $$-0.382191\pi$$
0.361715 + 0.932289i $$0.382191\pi$$
$$98$$ 0 0
$$99$$ −0.556364 −0.0559167
$$100$$ 0 0
$$101$$ 14.9753 1.49009 0.745047 0.667012i $$-0.232427\pi$$
0.745047 + 0.667012i $$0.232427\pi$$
$$102$$ 0 0
$$103$$ 10.8667 1.07073 0.535364 0.844622i $$-0.320175\pi$$
0.535364 + 0.844622i $$0.320175\pi$$
$$104$$ 0 0
$$105$$ −9.55220 −0.932199
$$106$$ 0 0
$$107$$ 7.49180 0.724259 0.362130 0.932128i $$-0.382050\pi$$
0.362130 + 0.932128i $$0.382050\pi$$
$$108$$ 0 0
$$109$$ −18.3309 −1.75578 −0.877891 0.478860i $$-0.841050\pi$$
−0.877891 + 0.478860i $$0.841050\pi$$
$$110$$ 0 0
$$111$$ 8.41188 0.798420
$$112$$ 0 0
$$113$$ 0.379068 0.0356597 0.0178299 0.999841i $$-0.494324\pi$$
0.0178299 + 0.999841i $$0.494324\pi$$
$$114$$ 0 0
$$115$$ 1.00000 0.0932505
$$116$$ 0 0
$$117$$ 1.30226 0.120394
$$118$$ 0 0
$$119$$ 30.1484 2.76370
$$120$$ 0 0
$$121$$ −10.4436 −0.949421
$$122$$ 0 0
$$123$$ −13.4548 −1.21318
$$124$$ 0 0
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −2.64852 −0.235018 −0.117509 0.993072i $$-0.537491\pi$$
−0.117509 + 0.993072i $$0.537491\pi$$
$$128$$ 0 0
$$129$$ −9.70892 −0.854822
$$130$$ 0 0
$$131$$ 5.53579 0.483664 0.241832 0.970318i $$-0.422252\pi$$
0.241832 + 0.970318i $$0.422252\pi$$
$$132$$ 0 0
$$133$$ 26.8667 2.32964
$$134$$ 0 0
$$135$$ −4.36266 −0.375478
$$136$$ 0 0
$$137$$ 6.16896 0.527050 0.263525 0.964653i $$-0.415115\pi$$
0.263525 + 0.964653i $$0.415115\pi$$
$$138$$ 0 0
$$139$$ 20.7693 1.76163 0.880815 0.473460i $$-0.156995\pi$$
0.880815 + 0.473460i $$0.156995\pi$$
$$140$$ 0 0
$$141$$ −5.18953 −0.437038
$$142$$ 0 0
$$143$$ −1.30226 −0.108901
$$144$$ 0 0
$$145$$ −1.66492 −0.138264
$$146$$ 0 0
$$147$$ 33.5962 2.77097
$$148$$ 0 0
$$149$$ −1.06457 −0.0872128 −0.0436064 0.999049i $$-0.513885\pi$$
−0.0436064 + 0.999049i $$0.513885\pi$$
$$150$$ 0 0
$$151$$ 12.6608 1.03032 0.515159 0.857095i $$-0.327733\pi$$
0.515159 + 0.857095i $$0.327733\pi$$
$$152$$ 0 0
$$153$$ −4.55636 −0.368360
$$154$$ 0 0
$$155$$ 1.61676 0.129862
$$156$$ 0 0
$$157$$ 14.8873 1.18813 0.594067 0.804416i $$-0.297521\pi$$
0.594067 + 0.804416i $$0.297521\pi$$
$$158$$ 0 0
$$159$$ 26.5962 2.10921
$$160$$ 0 0
$$161$$ −4.93543 −0.388967
$$162$$ 0 0
$$163$$ −6.72949 −0.527094 −0.263547 0.964646i $$-0.584892\pi$$
−0.263547 + 0.964646i $$0.584892\pi$$
$$164$$ 0 0
$$165$$ −1.44364 −0.112387
$$166$$ 0 0
$$167$$ 2.98359 0.230877 0.115439 0.993315i $$-0.463173\pi$$
0.115439 + 0.993315i $$0.463173\pi$$
$$168$$ 0 0
$$169$$ −9.95184 −0.765526
$$170$$ 0 0
$$171$$ −4.06040 −0.310506
$$172$$ 0 0
$$173$$ −15.8503 −1.20508 −0.602538 0.798091i $$-0.705844\pi$$
−0.602538 + 0.798091i $$0.705844\pi$$
$$174$$ 0 0
$$175$$ −4.93543 −0.373084
$$176$$ 0 0
$$177$$ −23.6454 −1.77730
$$178$$ 0 0
$$179$$ −3.02759 −0.226292 −0.113146 0.993578i $$-0.536093\pi$$
−0.113146 + 0.993578i $$0.536093\pi$$
$$180$$ 0 0
$$181$$ 1.82270 0.135481 0.0677403 0.997703i $$-0.478421\pi$$
0.0677403 + 0.997703i $$0.478421\pi$$
$$182$$ 0 0
$$183$$ −27.0562 −2.00005
$$184$$ 0 0
$$185$$ 4.34625 0.319543
$$186$$ 0 0
$$187$$ 4.55636 0.333194
$$188$$ 0 0
$$189$$ 21.5316 1.56619
$$190$$ 0 0
$$191$$ −19.6126 −1.41912 −0.709559 0.704646i $$-0.751106\pi$$
−0.709559 + 0.704646i $$0.751106\pi$$
$$192$$ 0 0
$$193$$ −0.335076 −0.0241193 −0.0120597 0.999927i $$-0.503839\pi$$
−0.0120597 + 0.999927i $$0.503839\pi$$
$$194$$ 0 0
$$195$$ 3.37907 0.241980
$$196$$ 0 0
$$197$$ −20.5316 −1.46282 −0.731409 0.681939i $$-0.761137\pi$$
−0.731409 + 0.681939i $$0.761137\pi$$
$$198$$ 0 0
$$199$$ −16.7253 −1.18563 −0.592813 0.805340i $$-0.701983\pi$$
−0.592813 + 0.805340i $$0.701983\pi$$
$$200$$ 0 0
$$201$$ −25.3627 −1.78894
$$202$$ 0 0
$$203$$ 8.21712 0.576729
$$204$$ 0 0
$$205$$ −6.95184 −0.485538
$$206$$ 0 0
$$207$$ 0.745898 0.0518435
$$208$$ 0 0
$$209$$ 4.06040 0.280864
$$210$$ 0 0
$$211$$ 13.3955 0.922183 0.461091 0.887353i $$-0.347458\pi$$
0.461091 + 0.887353i $$0.347458\pi$$
$$212$$ 0 0
$$213$$ 18.7295 1.28332
$$214$$ 0 0
$$215$$ −5.01641 −0.342116
$$216$$ 0 0
$$217$$ −7.97942 −0.541679
$$218$$ 0 0
$$219$$ 11.0276 0.745175
$$220$$ 0 0
$$221$$ −10.6649 −0.717400
$$222$$ 0 0
$$223$$ −1.27468 −0.0853587 −0.0426794 0.999089i $$-0.513589\pi$$
−0.0426794 + 0.999089i $$0.513589\pi$$
$$224$$ 0 0
$$225$$ 0.745898 0.0497266
$$226$$ 0 0
$$227$$ −3.57978 −0.237598 −0.118799 0.992918i $$-0.537904\pi$$
−0.118799 + 0.992918i $$0.537904\pi$$
$$228$$ 0 0
$$229$$ 10.4119 0.688036 0.344018 0.938963i $$-0.388212\pi$$
0.344018 + 0.938963i $$0.388212\pi$$
$$230$$ 0 0
$$231$$ 7.12497 0.468788
$$232$$ 0 0
$$233$$ 3.06040 0.200493 0.100247 0.994963i $$-0.468037\pi$$
0.100247 + 0.994963i $$0.468037\pi$$
$$234$$ 0 0
$$235$$ −2.68133 −0.174911
$$236$$ 0 0
$$237$$ −20.0880 −1.30485
$$238$$ 0 0
$$239$$ 19.1895 1.24127 0.620634 0.784100i $$-0.286875\pi$$
0.620634 + 0.784100i $$0.286875\pi$$
$$240$$ 0 0
$$241$$ 0.346255 0.0223042 0.0111521 0.999938i $$-0.496450\pi$$
0.0111521 + 0.999938i $$0.496450\pi$$
$$242$$ 0 0
$$243$$ −7.58501 −0.486579
$$244$$ 0 0
$$245$$ 17.3585 1.10899
$$246$$ 0 0
$$247$$ −9.50403 −0.604727
$$248$$ 0 0
$$249$$ −1.23353 −0.0781715
$$250$$ 0 0
$$251$$ −14.3861 −0.908041 −0.454021 0.890991i $$-0.650011\pi$$
−0.454021 + 0.890991i $$0.650011\pi$$
$$252$$ 0 0
$$253$$ −0.745898 −0.0468942
$$254$$ 0 0
$$255$$ −11.8227 −0.740366
$$256$$ 0 0
$$257$$ 1.28586 0.0802095 0.0401047 0.999195i $$-0.487231\pi$$
0.0401047 + 0.999195i $$0.487231\pi$$
$$258$$ 0 0
$$259$$ −21.4506 −1.33288
$$260$$ 0 0
$$261$$ −1.24186 −0.0768694
$$262$$ 0 0
$$263$$ −28.2294 −1.74070 −0.870348 0.492437i $$-0.836106\pi$$
−0.870348 + 0.492437i $$0.836106\pi$$
$$264$$ 0 0
$$265$$ 13.7417 0.844148
$$266$$ 0 0
$$267$$ 5.27468 0.322805
$$268$$ 0 0
$$269$$ −19.4395 −1.18525 −0.592623 0.805480i $$-0.701907\pi$$
−0.592623 + 0.805480i $$0.701907\pi$$
$$270$$ 0 0
$$271$$ 6.42723 0.390426 0.195213 0.980761i $$-0.437460\pi$$
0.195213 + 0.980761i $$0.437460\pi$$
$$272$$ 0 0
$$273$$ −16.6772 −1.00935
$$274$$ 0 0
$$275$$ −0.745898 −0.0449794
$$276$$ 0 0
$$277$$ 4.08514 0.245452 0.122726 0.992441i $$-0.460836\pi$$
0.122726 + 0.992441i $$0.460836\pi$$
$$278$$ 0 0
$$279$$ 1.20594 0.0721978
$$280$$ 0 0
$$281$$ −6.38741 −0.381041 −0.190520 0.981683i $$-0.561018\pi$$
−0.190520 + 0.981683i $$0.561018\pi$$
$$282$$ 0 0
$$283$$ 21.4835 1.27706 0.638530 0.769597i $$-0.279543\pi$$
0.638530 + 0.769597i $$0.279543\pi$$
$$284$$ 0 0
$$285$$ −10.5358 −0.624086
$$286$$ 0 0
$$287$$ 34.3103 2.02527
$$288$$ 0 0
$$289$$ 20.3145 1.19497
$$290$$ 0 0
$$291$$ 13.7899 0.808378
$$292$$ 0 0
$$293$$ −16.1208 −0.941787 −0.470894 0.882190i $$-0.656068\pi$$
−0.470894 + 0.882190i $$0.656068\pi$$
$$294$$ 0 0
$$295$$ −12.2171 −0.711308
$$296$$ 0 0
$$297$$ 3.25410 0.188822
$$298$$ 0 0
$$299$$ 1.74590 0.100968
$$300$$ 0 0
$$301$$ 24.7581 1.42704
$$302$$ 0 0
$$303$$ 28.9836 1.66506
$$304$$ 0 0
$$305$$ −13.9794 −0.800460
$$306$$ 0 0
$$307$$ 6.96302 0.397400 0.198700 0.980060i $$-0.436328\pi$$
0.198700 + 0.980060i $$0.436328\pi$$
$$308$$ 0 0
$$309$$ 21.0318 1.19645
$$310$$ 0 0
$$311$$ 4.14031 0.234776 0.117388 0.993086i $$-0.462548\pi$$
0.117388 + 0.993086i $$0.462548\pi$$
$$312$$ 0 0
$$313$$ 6.26528 0.354135 0.177067 0.984199i $$-0.443339\pi$$
0.177067 + 0.984199i $$0.443339\pi$$
$$314$$ 0 0
$$315$$ −3.68133 −0.207419
$$316$$ 0 0
$$317$$ 9.53996 0.535817 0.267909 0.963444i $$-0.413667\pi$$
0.267909 + 0.963444i $$0.413667\pi$$
$$318$$ 0 0
$$319$$ 1.24186 0.0695310
$$320$$ 0 0
$$321$$ 14.4999 0.809304
$$322$$ 0 0
$$323$$ 33.2528 1.85023
$$324$$ 0 0
$$325$$ 1.74590 0.0968450
$$326$$ 0 0
$$327$$ −35.4782 −1.96195
$$328$$ 0 0
$$329$$ 13.2335 0.729588
$$330$$ 0 0
$$331$$ 14.3023 0.786123 0.393062 0.919512i $$-0.371416\pi$$
0.393062 + 0.919512i $$0.371416\pi$$
$$332$$ 0 0
$$333$$ 3.24186 0.177653
$$334$$ 0 0
$$335$$ −13.1044 −0.715969
$$336$$ 0 0
$$337$$ −3.34731 −0.182340 −0.0911699 0.995835i $$-0.529061\pi$$
−0.0911699 + 0.995835i $$0.529061\pi$$
$$338$$ 0 0
$$339$$ 0.733661 0.0398470
$$340$$ 0 0
$$341$$ −1.20594 −0.0653054
$$342$$ 0 0
$$343$$ −51.1236 −2.76042
$$344$$ 0 0
$$345$$ 1.93543 0.104200
$$346$$ 0 0
$$347$$ 23.9383 1.28507 0.642537 0.766255i $$-0.277882\pi$$
0.642537 + 0.766255i $$0.277882\pi$$
$$348$$ 0 0
$$349$$ −11.9149 −0.637788 −0.318894 0.947790i $$-0.603311\pi$$
−0.318894 + 0.947790i $$0.603311\pi$$
$$350$$ 0 0
$$351$$ −7.61676 −0.406553
$$352$$ 0 0
$$353$$ −25.2447 −1.34364 −0.671820 0.740714i $$-0.734487\pi$$
−0.671820 + 0.740714i $$0.734487\pi$$
$$354$$ 0 0
$$355$$ 9.67716 0.513610
$$356$$ 0 0
$$357$$ 58.3502 3.08822
$$358$$ 0 0
$$359$$ −20.4671 −1.08021 −0.540105 0.841598i $$-0.681615\pi$$
−0.540105 + 0.841598i $$0.681615\pi$$
$$360$$ 0 0
$$361$$ 10.6332 0.559641
$$362$$ 0 0
$$363$$ −20.2130 −1.06090
$$364$$ 0 0
$$365$$ 5.69774 0.298233
$$366$$ 0 0
$$367$$ 36.4999 1.90528 0.952639 0.304104i $$-0.0983571\pi$$
0.952639 + 0.304104i $$0.0983571\pi$$
$$368$$ 0 0
$$369$$ −5.18537 −0.269939
$$370$$ 0 0
$$371$$ −67.8214 −3.52111
$$372$$ 0 0
$$373$$ −5.57978 −0.288910 −0.144455 0.989511i $$-0.546143\pi$$
−0.144455 + 0.989511i $$0.546143\pi$$
$$374$$ 0 0
$$375$$ 1.93543 0.0999453
$$376$$ 0 0
$$377$$ −2.90679 −0.149707
$$378$$ 0 0
$$379$$ −36.6168 −1.88088 −0.940438 0.339964i $$-0.889585\pi$$
−0.940438 + 0.339964i $$0.889585\pi$$
$$380$$ 0 0
$$381$$ −5.12603 −0.262614
$$382$$ 0 0
$$383$$ 28.7581 1.46947 0.734736 0.678353i $$-0.237306\pi$$
0.734736 + 0.678353i $$0.237306\pi$$
$$384$$ 0 0
$$385$$ 3.68133 0.187618
$$386$$ 0 0
$$387$$ −3.74173 −0.190203
$$388$$ 0 0
$$389$$ 26.3913 1.33809 0.669046 0.743221i $$-0.266703\pi$$
0.669046 + 0.743221i $$0.266703\pi$$
$$390$$ 0 0
$$391$$ −6.10856 −0.308923
$$392$$ 0 0
$$393$$ 10.7141 0.540457
$$394$$ 0 0
$$395$$ −10.3791 −0.522228
$$396$$ 0 0
$$397$$ 1.08931 0.0546710 0.0273355 0.999626i $$-0.491298\pi$$
0.0273355 + 0.999626i $$0.491298\pi$$
$$398$$ 0 0
$$399$$ 51.9987 2.60319
$$400$$ 0 0
$$401$$ −22.5962 −1.12840 −0.564200 0.825638i $$-0.690815\pi$$
−0.564200 + 0.825638i $$0.690815\pi$$
$$402$$ 0 0
$$403$$ 2.82270 0.140609
$$404$$ 0 0
$$405$$ −10.6813 −0.530760
$$406$$ 0 0
$$407$$ −3.24186 −0.160693
$$408$$ 0 0
$$409$$ −16.0510 −0.793671 −0.396835 0.917890i $$-0.629892\pi$$
−0.396835 + 0.917890i $$0.629892\pi$$
$$410$$ 0 0
$$411$$ 11.9396 0.588937
$$412$$ 0 0
$$413$$ 60.2968 2.96701
$$414$$ 0 0
$$415$$ −0.637339 −0.0312857
$$416$$ 0 0
$$417$$ 40.1976 1.96849
$$418$$ 0 0
$$419$$ 24.4342 1.19369 0.596845 0.802356i $$-0.296421\pi$$
0.596845 + 0.802356i $$0.296421\pi$$
$$420$$ 0 0
$$421$$ −11.3473 −0.553034 −0.276517 0.961009i $$-0.589180\pi$$
−0.276517 + 0.961009i $$0.589180\pi$$
$$422$$ 0 0
$$423$$ −2.00000 −0.0972433
$$424$$ 0 0
$$425$$ −6.10856 −0.296309
$$426$$ 0 0
$$427$$ 68.9945 3.33888
$$428$$ 0 0
$$429$$ −2.52044 −0.121688
$$430$$ 0 0
$$431$$ −7.83805 −0.377546 −0.188773 0.982021i $$-0.560451\pi$$
−0.188773 + 0.982021i $$0.560451\pi$$
$$432$$ 0 0
$$433$$ −37.9023 −1.82147 −0.910735 0.412990i $$-0.864484\pi$$
−0.910735 + 0.412990i $$0.864484\pi$$
$$434$$ 0 0
$$435$$ −3.22235 −0.154500
$$436$$ 0 0
$$437$$ −5.44364 −0.260404
$$438$$ 0 0
$$439$$ 17.7704 0.848134 0.424067 0.905631i $$-0.360602\pi$$
0.424067 + 0.905631i $$0.360602\pi$$
$$440$$ 0 0
$$441$$ 12.9477 0.616556
$$442$$ 0 0
$$443$$ −16.9466 −0.805158 −0.402579 0.915385i $$-0.631886\pi$$
−0.402579 + 0.915385i $$0.631886\pi$$
$$444$$ 0 0
$$445$$ 2.72532 0.129193
$$446$$ 0 0
$$447$$ −2.06040 −0.0974535
$$448$$ 0 0
$$449$$ −21.0890 −0.995253 −0.497627 0.867391i $$-0.665795\pi$$
−0.497627 + 0.867391i $$0.665795\pi$$
$$450$$ 0 0
$$451$$ 5.18537 0.244169
$$452$$ 0 0
$$453$$ 24.5040 1.15130
$$454$$ 0 0
$$455$$ −8.61676 −0.403960
$$456$$ 0 0
$$457$$ 13.1372 0.614532 0.307266 0.951624i $$-0.400586\pi$$
0.307266 + 0.951624i $$0.400586\pi$$
$$458$$ 0 0
$$459$$ 26.6496 1.24390
$$460$$ 0 0
$$461$$ −14.5439 −0.677375 −0.338687 0.940899i $$-0.609983\pi$$
−0.338687 + 0.940899i $$0.609983\pi$$
$$462$$ 0 0
$$463$$ −19.4283 −0.902909 −0.451455 0.892294i $$-0.649095\pi$$
−0.451455 + 0.892294i $$0.649095\pi$$
$$464$$ 0 0
$$465$$ 3.12914 0.145110
$$466$$ 0 0
$$467$$ 3.61259 0.167171 0.0835855 0.996501i $$-0.473363\pi$$
0.0835855 + 0.996501i $$0.473363\pi$$
$$468$$ 0 0
$$469$$ 64.6758 2.98645
$$470$$ 0 0
$$471$$ 28.8133 1.32765
$$472$$ 0 0
$$473$$ 3.74173 0.172045
$$474$$ 0 0
$$475$$ −5.44364 −0.249771
$$476$$ 0 0
$$477$$ 10.2499 0.469312
$$478$$ 0 0
$$479$$ −5.13720 −0.234725 −0.117362 0.993089i $$-0.537444\pi$$
−0.117362 + 0.993089i $$0.537444\pi$$
$$480$$ 0 0
$$481$$ 7.58812 0.345988
$$482$$ 0 0
$$483$$ −9.55220 −0.434640
$$484$$ 0 0
$$485$$ 7.12497 0.323528
$$486$$ 0 0
$$487$$ 24.7693 1.12240 0.561202 0.827679i $$-0.310339\pi$$
0.561202 + 0.827679i $$0.310339\pi$$
$$488$$ 0 0
$$489$$ −13.0245 −0.588987
$$490$$ 0 0
$$491$$ −3.54413 −0.159944 −0.0799721 0.996797i $$-0.525483\pi$$
−0.0799721 + 0.996797i $$0.525483\pi$$
$$492$$ 0 0
$$493$$ 10.1703 0.458047
$$494$$ 0 0
$$495$$ −0.556364 −0.0250067
$$496$$ 0 0
$$497$$ −47.7610 −2.14237
$$498$$ 0 0
$$499$$ −31.5358 −1.41174 −0.705868 0.708344i $$-0.749443\pi$$
−0.705868 + 0.708344i $$0.749443\pi$$
$$500$$ 0 0
$$501$$ 5.77454 0.257988
$$502$$ 0 0
$$503$$ 32.2346 1.43727 0.718635 0.695388i $$-0.244767\pi$$
0.718635 + 0.695388i $$0.244767\pi$$
$$504$$ 0 0
$$505$$ 14.9753 0.666390
$$506$$ 0 0
$$507$$ −19.2611 −0.855416
$$508$$ 0 0
$$509$$ 0.302263 0.0133976 0.00669878 0.999978i $$-0.497868\pi$$
0.00669878 + 0.999978i $$0.497868\pi$$
$$510$$ 0 0
$$511$$ −28.1208 −1.24399
$$512$$ 0 0
$$513$$ 23.7487 1.04853
$$514$$ 0 0
$$515$$ 10.8667 0.478844
$$516$$ 0 0
$$517$$ 2.00000 0.0879599
$$518$$ 0 0
$$519$$ −30.6772 −1.34658
$$520$$ 0 0
$$521$$ −30.3463 −1.32949 −0.664747 0.747069i $$-0.731461\pi$$
−0.664747 + 0.747069i $$0.731461\pi$$
$$522$$ 0 0
$$523$$ 42.5878 1.86224 0.931118 0.364717i $$-0.118834\pi$$
0.931118 + 0.364717i $$0.118834\pi$$
$$524$$ 0 0
$$525$$ −9.55220 −0.416892
$$526$$ 0 0
$$527$$ −9.87609 −0.430209
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −9.11273 −0.395459
$$532$$ 0 0
$$533$$ −12.1372 −0.525721
$$534$$ 0 0
$$535$$ 7.49180 0.323899
$$536$$ 0 0
$$537$$ −5.85969 −0.252864
$$538$$ 0 0
$$539$$ −12.9477 −0.557696
$$540$$ 0 0
$$541$$ 38.9313 1.67379 0.836893 0.547367i $$-0.184370\pi$$
0.836893 + 0.547367i $$0.184370\pi$$
$$542$$ 0 0
$$543$$ 3.52772 0.151389
$$544$$ 0 0
$$545$$ −18.3309 −0.785210
$$546$$ 0 0
$$547$$ −16.6004 −0.709780 −0.354890 0.934908i $$-0.615482\pi$$
−0.354890 + 0.934908i $$0.615482\pi$$
$$548$$ 0 0
$$549$$ −10.4272 −0.445023
$$550$$ 0 0
$$551$$ 9.06324 0.386107
$$552$$ 0 0
$$553$$ 51.2252 2.17832
$$554$$ 0 0
$$555$$ 8.41188 0.357064
$$556$$ 0 0
$$557$$ 17.4506 0.739408 0.369704 0.929150i $$-0.379459\pi$$
0.369704 + 0.929150i $$0.379459\pi$$
$$558$$ 0 0
$$559$$ −8.75814 −0.370430
$$560$$ 0 0
$$561$$ 8.81853 0.372319
$$562$$ 0 0
$$563$$ −8.25827 −0.348045 −0.174022 0.984742i $$-0.555676\pi$$
−0.174022 + 0.984742i $$0.555676\pi$$
$$564$$ 0 0
$$565$$ 0.379068 0.0159475
$$566$$ 0 0
$$567$$ 52.7170 2.21391
$$568$$ 0 0
$$569$$ −41.4178 −1.73633 −0.868163 0.496279i $$-0.834699\pi$$
−0.868163 + 0.496279i $$0.834699\pi$$
$$570$$ 0 0
$$571$$ −7.19059 −0.300917 −0.150458 0.988616i $$-0.548075\pi$$
−0.150458 + 0.988616i $$0.548075\pi$$
$$572$$ 0 0
$$573$$ −37.9588 −1.58575
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 0 0
$$577$$ 39.3103 1.63651 0.818255 0.574855i $$-0.194942\pi$$
0.818255 + 0.574855i $$0.194942\pi$$
$$578$$ 0 0
$$579$$ −0.648517 −0.0269515
$$580$$ 0 0
$$581$$ 3.14554 0.130499
$$582$$ 0 0
$$583$$ −10.2499 −0.424509
$$584$$ 0 0
$$585$$ 1.30226 0.0538419
$$586$$ 0 0
$$587$$ −28.7899 −1.18829 −0.594143 0.804359i $$-0.702509\pi$$
−0.594143 + 0.804359i $$0.702509\pi$$
$$588$$ 0 0
$$589$$ −8.80107 −0.362642
$$590$$ 0 0
$$591$$ −39.7376 −1.63458
$$592$$ 0 0
$$593$$ −40.8873 −1.67904 −0.839519 0.543330i $$-0.817163\pi$$
−0.839519 + 0.543330i $$0.817163\pi$$
$$594$$ 0 0
$$595$$ 30.1484 1.23596
$$596$$ 0 0
$$597$$ −32.3707 −1.32485
$$598$$ 0 0
$$599$$ −19.1801 −0.783679 −0.391840 0.920034i $$-0.628161\pi$$
−0.391840 + 0.920034i $$0.628161\pi$$
$$600$$ 0 0
$$601$$ 3.36683 0.137336 0.0686679 0.997640i $$-0.478125\pi$$
0.0686679 + 0.997640i $$0.478125\pi$$
$$602$$ 0 0
$$603$$ −9.77454 −0.398050
$$604$$ 0 0
$$605$$ −10.4436 −0.424594
$$606$$ 0 0
$$607$$ −13.1924 −0.535462 −0.267731 0.963494i $$-0.586274\pi$$
−0.267731 + 0.963494i $$0.586274\pi$$
$$608$$ 0 0
$$609$$ 15.9037 0.644450
$$610$$ 0 0
$$611$$ −4.68133 −0.189386
$$612$$ 0 0
$$613$$ 32.6207 1.31754 0.658768 0.752346i $$-0.271078\pi$$
0.658768 + 0.752346i $$0.271078\pi$$
$$614$$ 0 0
$$615$$ −13.4548 −0.542550
$$616$$ 0 0
$$617$$ 41.7212 1.67963 0.839815 0.542872i $$-0.182663\pi$$
0.839815 + 0.542872i $$0.182663\pi$$
$$618$$ 0 0
$$619$$ 0.339245 0.0136354 0.00681771 0.999977i $$-0.497830\pi$$
0.00681771 + 0.999977i $$0.497830\pi$$
$$620$$ 0 0
$$621$$ −4.36266 −0.175068
$$622$$ 0 0
$$623$$ −13.4506 −0.538889
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 0 0
$$627$$ 7.85863 0.313843
$$628$$ 0 0
$$629$$ −26.5494 −1.05859
$$630$$ 0 0
$$631$$ −3.71725 −0.147982 −0.0739908 0.997259i $$-0.523574\pi$$
−0.0739908 + 0.997259i $$0.523574\pi$$
$$632$$ 0 0
$$633$$ 25.9260 1.03047
$$634$$ 0 0
$$635$$ −2.64852 −0.105103
$$636$$ 0 0
$$637$$ 30.3062 1.20077
$$638$$ 0 0
$$639$$ 7.21818 0.285547
$$640$$ 0 0
$$641$$ −22.0328 −0.870244 −0.435122 0.900372i $$-0.643295\pi$$
−0.435122 + 0.900372i $$0.643295\pi$$
$$642$$ 0 0
$$643$$ 24.6842 0.973449 0.486724 0.873556i $$-0.338192\pi$$
0.486724 + 0.873556i $$0.338192\pi$$
$$644$$ 0 0
$$645$$ −9.70892 −0.382288
$$646$$ 0 0
$$647$$ −32.1731 −1.26486 −0.632428 0.774619i $$-0.717942\pi$$
−0.632428 + 0.774619i $$0.717942\pi$$
$$648$$ 0 0
$$649$$ 9.11273 0.357706
$$650$$ 0 0
$$651$$ −15.4436 −0.605284
$$652$$ 0 0
$$653$$ −26.9466 −1.05450 −0.527251 0.849709i $$-0.676777\pi$$
−0.527251 + 0.849709i $$0.676777\pi$$
$$654$$ 0 0
$$655$$ 5.53579 0.216301
$$656$$ 0 0
$$657$$ 4.24993 0.165806
$$658$$ 0 0
$$659$$ 19.7417 0.769029 0.384514 0.923119i $$-0.374369\pi$$
0.384514 + 0.923119i $$0.374369\pi$$
$$660$$ 0 0
$$661$$ −1.28692 −0.0500552 −0.0250276 0.999687i $$-0.507967\pi$$
−0.0250276 + 0.999687i $$0.507967\pi$$
$$662$$ 0 0
$$663$$ −20.6412 −0.801639
$$664$$ 0 0
$$665$$ 26.8667 1.04185
$$666$$ 0 0
$$667$$ −1.66492 −0.0644661
$$668$$ 0 0
$$669$$ −2.46705 −0.0953817
$$670$$ 0 0
$$671$$ 10.4272 0.402539
$$672$$ 0 0
$$673$$ 12.4970 0.481725 0.240862 0.970559i $$-0.422570\pi$$
0.240862 + 0.970559i $$0.422570\pi$$
$$674$$ 0 0
$$675$$ −4.36266 −0.167919
$$676$$ 0 0
$$677$$ −2.69251 −0.103482 −0.0517408 0.998661i $$-0.516477\pi$$
−0.0517408 + 0.998661i $$0.516477\pi$$
$$678$$ 0 0
$$679$$ −35.1648 −1.34950
$$680$$ 0 0
$$681$$ −6.92842 −0.265498
$$682$$ 0 0
$$683$$ 2.72115 0.104122 0.0520610 0.998644i $$-0.483421\pi$$
0.0520610 + 0.998644i $$0.483421\pi$$
$$684$$ 0 0
$$685$$ 6.16896 0.235704
$$686$$ 0 0
$$687$$ 20.1515 0.768827
$$688$$ 0 0
$$689$$ 23.9917 0.914010
$$690$$ 0 0
$$691$$ −38.3051 −1.45719 −0.728597 0.684942i $$-0.759827\pi$$
−0.728597 + 0.684942i $$0.759827\pi$$
$$692$$ 0 0
$$693$$ 2.74590 0.104308
$$694$$ 0 0
$$695$$ 20.7693 0.787825
$$696$$ 0 0
$$697$$ 42.4657 1.60850
$$698$$ 0 0
$$699$$ 5.92319 0.224036
$$700$$ 0 0
$$701$$ 39.8472 1.50501 0.752504 0.658588i $$-0.228846\pi$$
0.752504 + 0.658588i $$0.228846\pi$$
$$702$$ 0 0
$$703$$ −23.6594 −0.892332
$$704$$ 0 0
$$705$$ −5.18953 −0.195449
$$706$$ 0 0
$$707$$ −73.9094 −2.77965
$$708$$ 0 0
$$709$$ −44.8995 −1.68624 −0.843118 0.537728i $$-0.819283\pi$$
−0.843118 + 0.537728i $$0.819283\pi$$
$$710$$ 0 0
$$711$$ −7.74173 −0.290338
$$712$$ 0 0
$$713$$ 1.61676 0.0605482
$$714$$ 0 0
$$715$$ −1.30226 −0.0487019
$$716$$ 0 0
$$717$$ 37.1400 1.38702
$$718$$ 0 0
$$719$$ 31.5400 1.17624 0.588121 0.808773i $$-0.299868\pi$$
0.588121 + 0.808773i $$0.299868\pi$$
$$720$$ 0 0
$$721$$ −53.6318 −1.99735
$$722$$ 0 0
$$723$$ 0.670152 0.0249232
$$724$$ 0 0
$$725$$ −1.66492 −0.0618337
$$726$$ 0 0
$$727$$ 48.9219 1.81441 0.907206 0.420687i $$-0.138211\pi$$
0.907206 + 0.420687i $$0.138211\pi$$
$$728$$ 0 0
$$729$$ 17.3637 0.643101
$$730$$ 0 0
$$731$$ 30.6430 1.13337
$$732$$ 0 0
$$733$$ −14.1536 −0.522776 −0.261388 0.965234i $$-0.584180\pi$$
−0.261388 + 0.965234i $$0.584180\pi$$
$$734$$ 0 0
$$735$$ 33.5962 1.23921
$$736$$ 0 0
$$737$$ 9.77454 0.360050
$$738$$ 0 0
$$739$$ −4.46421 −0.164219 −0.0821093 0.996623i $$-0.526166\pi$$
−0.0821093 + 0.996623i $$0.526166\pi$$
$$740$$ 0 0
$$741$$ −18.3944 −0.675736
$$742$$ 0 0
$$743$$ 34.2898 1.25797 0.628985 0.777418i $$-0.283471\pi$$
0.628985 + 0.777418i $$0.283471\pi$$
$$744$$ 0 0
$$745$$ −1.06457 −0.0390027
$$746$$ 0 0
$$747$$ −0.475390 −0.0173936
$$748$$ 0 0
$$749$$ −36.9753 −1.35105
$$750$$ 0 0
$$751$$ 8.28275 0.302242 0.151121 0.988515i $$-0.451712\pi$$
0.151121 + 0.988515i $$0.451712\pi$$
$$752$$ 0 0
$$753$$ −27.8433 −1.01467
$$754$$ 0 0
$$755$$ 12.6608 0.460772
$$756$$ 0 0
$$757$$ 3.32985 0.121025 0.0605127 0.998167i $$-0.480726\pi$$
0.0605127 + 0.998167i $$0.480726\pi$$
$$758$$ 0 0
$$759$$ −1.44364 −0.0524007
$$760$$ 0 0
$$761$$ −18.4077 −0.667279 −0.333640 0.942701i $$-0.608277\pi$$
−0.333640 + 0.942701i $$0.608277\pi$$
$$762$$ 0 0
$$763$$ 90.4710 3.27527
$$764$$ 0 0
$$765$$ −4.55636 −0.164736
$$766$$ 0 0
$$767$$ −21.3298 −0.770176
$$768$$ 0 0
$$769$$ 51.4506 1.85536 0.927679 0.373379i $$-0.121801\pi$$
0.927679 + 0.373379i $$0.121801\pi$$
$$770$$ 0 0
$$771$$ 2.48869 0.0896279
$$772$$ 0 0
$$773$$ 17.4506 0.627656 0.313828 0.949480i $$-0.398388\pi$$
0.313828 + 0.949480i $$0.398388\pi$$
$$774$$ 0 0
$$775$$ 1.61676 0.0580758
$$776$$ 0 0
$$777$$ −41.5163 −1.48939
$$778$$ 0 0
$$779$$ 37.8433 1.35588
$$780$$ 0 0
$$781$$ −7.21818 −0.258287
$$782$$ 0 0
$$783$$ 7.26350 0.259576
$$784$$ 0 0
$$785$$ 14.8873 0.531350
$$786$$ 0 0
$$787$$ 21.2335 0.756893 0.378447 0.925623i $$-0.376458\pi$$
0.378447 + 0.925623i $$0.376458\pi$$
$$788$$ 0 0
$$789$$ −54.6360 −1.94509
$$790$$ 0 0
$$791$$ −1.87086 −0.0665203
$$792$$ 0 0
$$793$$ −24.4067 −0.866706
$$794$$ 0 0
$$795$$ 26.5962 0.943270
$$796$$ 0 0
$$797$$ −25.3215 −0.896934 −0.448467 0.893799i $$-0.648030\pi$$
−0.448467 + 0.893799i $$0.648030\pi$$
$$798$$ 0 0
$$799$$ 16.3791 0.579450
$$800$$ 0 0
$$801$$ 2.03281 0.0718259
$$802$$ 0 0
$$803$$ −4.24993 −0.149977
$$804$$ 0 0
$$805$$ −4.93543 −0.173951
$$806$$ 0 0
$$807$$ −37.6238 −1.32442
$$808$$ 0 0
$$809$$ 6.14137 0.215919 0.107960 0.994155i $$-0.465568\pi$$
0.107960 + 0.994155i $$0.465568\pi$$
$$810$$ 0 0
$$811$$ −15.5051 −0.544457 −0.272229 0.962233i $$-0.587761\pi$$
−0.272229 + 0.962233i $$0.587761\pi$$
$$812$$ 0 0
$$813$$ 12.4395 0.436271
$$814$$ 0 0
$$815$$ −6.72949 −0.235724
$$816$$ 0 0
$$817$$ 27.3075 0.955368
$$818$$ 0 0
$$819$$ −6.42723 −0.224586
$$820$$ 0 0
$$821$$ −21.4835 −0.749778 −0.374889 0.927070i $$-0.622319\pi$$
−0.374889 + 0.927070i $$0.622319\pi$$
$$822$$ 0 0
$$823$$ −40.1565 −1.39977 −0.699883 0.714258i $$-0.746765\pi$$
−0.699883 + 0.714258i $$0.746765\pi$$
$$824$$ 0 0
$$825$$ −1.44364 −0.0502609
$$826$$ 0 0
$$827$$ −38.0573 −1.32338 −0.661691 0.749777i $$-0.730161\pi$$
−0.661691 + 0.749777i $$0.730161\pi$$
$$828$$ 0 0
$$829$$ −51.2580 −1.78026 −0.890132 0.455703i $$-0.849388\pi$$
−0.890132 + 0.455703i $$0.849388\pi$$
$$830$$ 0 0
$$831$$ 7.90652 0.274274
$$832$$ 0 0
$$833$$ −106.035 −3.67391
$$834$$ 0 0
$$835$$ 2.98359 0.103252
$$836$$ 0 0
$$837$$ −7.05339 −0.243801
$$838$$ 0 0
$$839$$ −45.4200 −1.56807 −0.784035 0.620716i $$-0.786842\pi$$
−0.784035 + 0.620716i $$0.786842\pi$$
$$840$$ 0 0
$$841$$ −26.2280 −0.904415
$$842$$ 0 0
$$843$$ −12.3624 −0.425783
$$844$$ 0 0
$$845$$ −9.95184 −0.342354
$$846$$ 0 0
$$847$$ 51.5439 1.77107
$$848$$ 0 0
$$849$$ 41.5798 1.42701
$$850$$ 0 0
$$851$$ 4.34625 0.148988
$$852$$ 0 0
$$853$$ 40.0867 1.37254 0.686270 0.727346i $$-0.259247\pi$$
0.686270 + 0.727346i $$0.259247\pi$$
$$854$$ 0 0
$$855$$ −4.06040 −0.138863
$$856$$ 0 0
$$857$$ −3.63423 −0.124143 −0.0620715 0.998072i $$-0.519771\pi$$
−0.0620715 + 0.998072i $$0.519771\pi$$
$$858$$ 0 0
$$859$$ 37.7529 1.28811 0.644056 0.764978i $$-0.277250\pi$$
0.644056 + 0.764978i $$0.277250\pi$$
$$860$$ 0 0
$$861$$ 66.4053 2.26309
$$862$$ 0 0
$$863$$ 3.94555 0.134308 0.0671541 0.997743i $$-0.478608\pi$$
0.0671541 + 0.997743i $$0.478608\pi$$
$$864$$ 0 0
$$865$$ −15.8503 −0.538926
$$866$$ 0 0
$$867$$ 39.3173 1.33529
$$868$$ 0 0
$$869$$ 7.74173 0.262620
$$870$$ 0 0
$$871$$ −22.8789 −0.775223
$$872$$ 0 0
$$873$$ 5.31450 0.179869
$$874$$ 0 0
$$875$$ −4.93543 −0.166848
$$876$$ 0 0
$$877$$ −8.64958 −0.292075 −0.146038 0.989279i $$-0.546652\pi$$
−0.146038 + 0.989279i $$0.546652\pi$$
$$878$$ 0 0
$$879$$ −31.2007 −1.05237
$$880$$ 0 0
$$881$$ −56.1760 −1.89262 −0.946308 0.323266i $$-0.895219\pi$$
−0.946308 + 0.323266i $$0.895219\pi$$
$$882$$ 0 0
$$883$$ 12.1054 0.407381 0.203690 0.979035i $$-0.434706\pi$$
0.203690 + 0.979035i $$0.434706\pi$$
$$884$$ 0 0
$$885$$ −23.6454 −0.794832
$$886$$ 0 0
$$887$$ 24.7058 0.829540 0.414770 0.909926i $$-0.363862\pi$$
0.414770 + 0.909926i $$0.363862\pi$$
$$888$$ 0 0
$$889$$ 13.0716 0.438407
$$890$$ 0 0
$$891$$ 7.96719 0.266911
$$892$$ 0 0
$$893$$ 14.5962 0.488443
$$894$$ 0 0
$$895$$ −3.02759 −0.101201
$$896$$ 0 0
$$897$$ 3.37907 0.112824
$$898$$ 0 0
$$899$$ −2.69179 −0.0897761
$$900$$ 0 0
$$901$$ −83.9422 −2.79652
$$902$$ 0 0
$$903$$ 47.9177 1.59460
$$904$$ 0 0
$$905$$ 1.82270 0.0605887
$$906$$ 0 0
$$907$$ −22.7170 −0.754305 −0.377153 0.926151i $$-0.623097\pi$$
−0.377153 + 0.926151i $$0.623097\pi$$
$$908$$ 0 0
$$909$$ 11.1700 0.370486
$$910$$ 0 0
$$911$$ 12.3156 0.408033 0.204016 0.978967i $$-0.434600\pi$$
0.204016 + 0.978967i $$0.434600\pi$$
$$912$$ 0 0
$$913$$ 0.475390 0.0157331
$$914$$ 0 0
$$915$$ −27.0562 −0.894451
$$916$$ 0 0
$$917$$ −27.3215 −0.902236
$$918$$ 0 0
$$919$$ −46.2088 −1.52429 −0.762144 0.647408i $$-0.775853\pi$$
−0.762144 + 0.647408i $$0.775853\pi$$
$$920$$ 0 0
$$921$$ 13.4764 0.444064
$$922$$ 0 0
$$923$$ 16.8953 0.556117
$$924$$ 0 0
$$925$$ 4.34625 0.142904
$$926$$ 0 0
$$927$$ 8.10545 0.266218
$$928$$ 0 0
$$929$$ 9.12391 0.299346 0.149673 0.988736i $$-0.452178\pi$$
0.149673 + 0.988736i $$0.452178\pi$$
$$930$$ 0 0
$$931$$ −94.4933 −3.09689
$$932$$ 0 0
$$933$$ 8.01330 0.262344
$$934$$ 0 0
$$935$$ 4.55636 0.149009
$$936$$ 0 0
$$937$$ −33.9190 −1.10809 −0.554043 0.832488i $$-0.686916\pi$$
−0.554043 + 0.832488i $$0.686916\pi$$
$$938$$ 0 0
$$939$$ 12.1260 0.395718
$$940$$ 0 0
$$941$$ −37.2926 −1.21570 −0.607852 0.794050i $$-0.707969\pi$$
−0.607852 + 0.794050i $$0.707969\pi$$
$$942$$ 0 0
$$943$$ −6.95184 −0.226383
$$944$$ 0 0
$$945$$ 21.5316 0.700424
$$946$$ 0 0
$$947$$ −33.7047 −1.09526 −0.547629 0.836722i $$-0.684469\pi$$
−0.547629 + 0.836722i $$0.684469\pi$$
$$948$$ 0 0
$$949$$ 9.94767 0.322915
$$950$$ 0 0
$$951$$ 18.4639 0.598734
$$952$$ 0 0
$$953$$ −48.4465 −1.56934 −0.784668 0.619917i $$-0.787166\pi$$
−0.784668 + 0.619917i $$0.787166\pi$$
$$954$$ 0 0
$$955$$ −19.6126 −0.634649
$$956$$ 0 0
$$957$$ 2.40354 0.0776955
$$958$$ 0 0
$$959$$ −30.4465 −0.983168
$$960$$ 0 0
$$961$$ −28.3861 −0.915680
$$962$$ 0 0
$$963$$ 5.58812 0.180075
$$964$$ 0 0
$$965$$ −0.335076 −0.0107865
$$966$$ 0 0
$$967$$ 28.2528 0.908548 0.454274 0.890862i $$-0.349899\pi$$
0.454274 + 0.890862i $$0.349899\pi$$
$$968$$ 0 0
$$969$$ 64.3585 2.06749
$$970$$ 0 0
$$971$$ −1.65553 −0.0531284 −0.0265642 0.999647i $$-0.508457\pi$$
−0.0265642 + 0.999647i $$0.508457\pi$$
$$972$$ 0 0
$$973$$ −102.506 −3.28618
$$974$$ 0 0
$$975$$ 3.37907 0.108217
$$976$$ 0 0
$$977$$ −11.1630 −0.357136 −0.178568 0.983928i $$-0.557146\pi$$
−0.178568 + 0.983928i $$0.557146\pi$$
$$978$$ 0 0
$$979$$ −2.03281 −0.0649690
$$980$$ 0 0
$$981$$ −13.6730 −0.436545
$$982$$ 0 0
$$983$$ −15.0615 −0.480386 −0.240193 0.970725i $$-0.577211\pi$$
−0.240193 + 0.970725i $$0.577211\pi$$
$$984$$ 0 0
$$985$$ −20.5316 −0.654192
$$986$$ 0 0
$$987$$ 25.6126 0.815258
$$988$$ 0 0
$$989$$ −5.01641 −0.159512
$$990$$ 0 0
$$991$$ 43.8971 1.39444 0.697219 0.716858i $$-0.254421\pi$$
0.697219 + 0.716858i $$0.254421\pi$$
$$992$$ 0 0
$$993$$ 27.6811 0.878432
$$994$$ 0 0
$$995$$ −16.7253 −0.530228
$$996$$ 0 0
$$997$$ 38.5655 1.22138 0.610691 0.791869i $$-0.290892\pi$$
0.610691 + 0.791869i $$0.290892\pi$$
$$998$$ 0 0
$$999$$ −18.9612 −0.599907
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 1840.2.a.q.1.3 3
4.3 odd 2 920.2.a.i.1.1 3
5.4 even 2 9200.2.a.ci.1.1 3
8.3 odd 2 7360.2.a.bw.1.3 3
8.5 even 2 7360.2.a.cf.1.1 3
12.11 even 2 8280.2.a.bl.1.3 3
20.3 even 4 4600.2.e.q.4049.2 6
20.7 even 4 4600.2.e.q.4049.5 6
20.19 odd 2 4600.2.a.v.1.3 3

By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.i.1.1 3 4.3 odd 2
1840.2.a.q.1.3 3 1.1 even 1 trivial
4600.2.a.v.1.3 3 20.19 odd 2
4600.2.e.q.4049.2 6 20.3 even 4
4600.2.e.q.4049.5 6 20.7 even 4
7360.2.a.bw.1.3 3 8.3 odd 2
7360.2.a.cf.1.1 3 8.5 even 2
8280.2.a.bl.1.3 3 12.11 even 2
9200.2.a.ci.1.1 3 5.4 even 2