# Properties

 Label 9200.2.a.bv.1.2 Level $9200$ Weight $2$ Character 9200.1 Self dual yes Analytic conductor $73.462$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9200,2,Mod(1,9200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9200, 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("9200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$9200 = 2^{4} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 9200.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: $$73.4623698596$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 4$$ x^2 - x - 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 460) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$2.56155$$ of defining polynomial Character $$\chi$$ $$=$$ 9200.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.56155 q^{3} -1.56155 q^{7} +3.56155 q^{9} +O(q^{10})$$ $$q+2.56155 q^{3} -1.56155 q^{7} +3.56155 q^{9} -2.00000 q^{11} -0.561553 q^{13} +1.56155 q^{17} -6.00000 q^{19} -4.00000 q^{21} +1.00000 q^{23} +1.43845 q^{27} -2.12311 q^{29} +9.24621 q^{31} -5.12311 q^{33} +0.438447 q^{37} -1.43845 q^{39} -4.12311 q^{41} -7.68466 q^{47} -4.56155 q^{49} +4.00000 q^{51} +0.438447 q^{53} -15.3693 q^{57} -8.68466 q^{59} +1.12311 q^{61} -5.56155 q^{63} -4.43845 q^{67} +2.56155 q^{69} -1.87689 q^{71} +8.56155 q^{73} +3.12311 q^{77} -13.1231 q^{79} -7.00000 q^{81} +14.9309 q^{83} -5.43845 q^{87} -2.24621 q^{89} +0.876894 q^{91} +23.6847 q^{93} +4.87689 q^{97} -7.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q + q^3 + q^7 + 3 * q^9 $$2 q + q^{3} + q^{7} + 3 q^{9} - 4 q^{11} + 3 q^{13} - q^{17} - 12 q^{19} - 8 q^{21} + 2 q^{23} + 7 q^{27} + 4 q^{29} + 2 q^{31} - 2 q^{33} + 5 q^{37} - 7 q^{39} - 3 q^{47} - 5 q^{49} + 8 q^{51} + 5 q^{53} - 6 q^{57} - 5 q^{59} - 6 q^{61} - 7 q^{63} - 13 q^{67} + q^{69} - 12 q^{71} + 13 q^{73} - 2 q^{77} - 18 q^{79} - 14 q^{81} + q^{83} - 15 q^{87} + 12 q^{89} + 10 q^{91} + 35 q^{93} + 18 q^{97} - 6 q^{99}+O(q^{100})$$ 2 * q + q^3 + q^7 + 3 * q^9 - 4 * q^11 + 3 * q^13 - q^17 - 12 * q^19 - 8 * q^21 + 2 * q^23 + 7 * q^27 + 4 * q^29 + 2 * q^31 - 2 * q^33 + 5 * q^37 - 7 * q^39 - 3 * q^47 - 5 * q^49 + 8 * q^51 + 5 * q^53 - 6 * q^57 - 5 * q^59 - 6 * q^61 - 7 * q^63 - 13 * q^67 + q^69 - 12 * q^71 + 13 * q^73 - 2 * q^77 - 18 * q^79 - 14 * q^81 + q^83 - 15 * q^87 + 12 * q^89 + 10 * q^91 + 35 * q^93 + 18 * q^97 - 6 * 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
$$3$$ 2.56155 1.47891 0.739457 0.673204i $$-0.235083\pi$$
0.739457 + 0.673204i $$0.235083\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −1.56155 −0.590211 −0.295106 0.955465i $$-0.595355\pi$$
−0.295106 + 0.955465i $$0.595355\pi$$
$$8$$ 0 0
$$9$$ 3.56155 1.18718
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −0.561553 −0.155747 −0.0778734 0.996963i $$-0.524813\pi$$
−0.0778734 + 0.996963i $$0.524813\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.56155 0.378732 0.189366 0.981907i $$-0.439357\pi$$
0.189366 + 0.981907i $$0.439357\pi$$
$$18$$ 0 0
$$19$$ −6.00000 −1.37649 −0.688247 0.725476i $$-0.741620\pi$$
−0.688247 + 0.725476i $$0.741620\pi$$
$$20$$ 0 0
$$21$$ −4.00000 −0.872872
$$22$$ 0 0
$$23$$ 1.00000 0.208514
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 1.43845 0.276829
$$28$$ 0 0
$$29$$ −2.12311 −0.394251 −0.197125 0.980378i $$-0.563161\pi$$
−0.197125 + 0.980378i $$0.563161\pi$$
$$30$$ 0 0
$$31$$ 9.24621 1.66067 0.830334 0.557266i $$-0.188149\pi$$
0.830334 + 0.557266i $$0.188149\pi$$
$$32$$ 0 0
$$33$$ −5.12311 −0.891818
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0.438447 0.0720803 0.0360401 0.999350i $$-0.488526\pi$$
0.0360401 + 0.999350i $$0.488526\pi$$
$$38$$ 0 0
$$39$$ −1.43845 −0.230336
$$40$$ 0 0
$$41$$ −4.12311 −0.643921 −0.321960 0.946753i $$-0.604342\pi$$
−0.321960 + 0.946753i $$0.604342\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ −7.68466 −1.12092 −0.560461 0.828181i $$-0.689376\pi$$
−0.560461 + 0.828181i $$0.689376\pi$$
$$48$$ 0 0
$$49$$ −4.56155 −0.651650
$$50$$ 0 0
$$51$$ 4.00000 0.560112
$$52$$ 0 0
$$53$$ 0.438447 0.0602254 0.0301127 0.999547i $$-0.490413\pi$$
0.0301127 + 0.999547i $$0.490413\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −15.3693 −2.03572
$$58$$ 0 0
$$59$$ −8.68466 −1.13065 −0.565323 0.824870i $$-0.691249\pi$$
−0.565323 + 0.824870i $$0.691249\pi$$
$$60$$ 0 0
$$61$$ 1.12311 0.143799 0.0718995 0.997412i $$-0.477094\pi$$
0.0718995 + 0.997412i $$0.477094\pi$$
$$62$$ 0 0
$$63$$ −5.56155 −0.700690
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.43845 −0.542243 −0.271121 0.962545i $$-0.587394\pi$$
−0.271121 + 0.962545i $$0.587394\pi$$
$$68$$ 0 0
$$69$$ 2.56155 0.308375
$$70$$ 0 0
$$71$$ −1.87689 −0.222746 −0.111373 0.993779i $$-0.535525\pi$$
−0.111373 + 0.993779i $$0.535525\pi$$
$$72$$ 0 0
$$73$$ 8.56155 1.00205 0.501027 0.865432i $$-0.332956\pi$$
0.501027 + 0.865432i $$0.332956\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.12311 0.355911
$$78$$ 0 0
$$79$$ −13.1231 −1.47646 −0.738232 0.674546i $$-0.764339\pi$$
−0.738232 + 0.674546i $$0.764339\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ 14.9309 1.63888 0.819438 0.573168i $$-0.194286\pi$$
0.819438 + 0.573168i $$0.194286\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −5.43845 −0.583063
$$88$$ 0 0
$$89$$ −2.24621 −0.238098 −0.119049 0.992888i $$-0.537985\pi$$
−0.119049 + 0.992888i $$0.537985\pi$$
$$90$$ 0 0
$$91$$ 0.876894 0.0919235
$$92$$ 0 0
$$93$$ 23.6847 2.45598
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 4.87689 0.495174 0.247587 0.968866i $$-0.420362\pi$$
0.247587 + 0.968866i $$0.420362\pi$$
$$98$$ 0 0
$$99$$ −7.12311 −0.715899
$$100$$ 0 0
$$101$$ −5.31534 −0.528896 −0.264448 0.964400i $$-0.585190\pi$$
−0.264448 + 0.964400i $$0.585190\pi$$
$$102$$ 0 0
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 7.56155 0.731003 0.365501 0.930811i $$-0.380898\pi$$
0.365501 + 0.930811i $$0.380898\pi$$
$$108$$ 0 0
$$109$$ −9.36932 −0.897418 −0.448709 0.893678i $$-0.648116\pi$$
−0.448709 + 0.893678i $$0.648116\pi$$
$$110$$ 0 0
$$111$$ 1.12311 0.106600
$$112$$ 0 0
$$113$$ 7.80776 0.734493 0.367246 0.930124i $$-0.380301\pi$$
0.367246 + 0.930124i $$0.380301\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −2.00000 −0.184900
$$118$$ 0 0
$$119$$ −2.43845 −0.223532
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ −10.5616 −0.952303
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −12.8078 −1.13651 −0.568253 0.822854i $$-0.692380\pi$$
−0.568253 + 0.822854i $$0.692380\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −20.8078 −1.81798 −0.908991 0.416815i $$-0.863146\pi$$
−0.908991 + 0.416815i $$0.863146\pi$$
$$132$$ 0 0
$$133$$ 9.36932 0.812423
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −15.1231 −1.29205 −0.646027 0.763315i $$-0.723571\pi$$
−0.646027 + 0.763315i $$0.723571\pi$$
$$138$$ 0 0
$$139$$ −5.24621 −0.444978 −0.222489 0.974935i $$-0.571418\pi$$
−0.222489 + 0.974935i $$0.571418\pi$$
$$140$$ 0 0
$$141$$ −19.6847 −1.65775
$$142$$ 0 0
$$143$$ 1.12311 0.0939188
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −11.6847 −0.963734
$$148$$ 0 0
$$149$$ 15.3693 1.25910 0.629552 0.776959i $$-0.283239\pi$$
0.629552 + 0.776959i $$0.283239\pi$$
$$150$$ 0 0
$$151$$ −23.0540 −1.87611 −0.938053 0.346492i $$-0.887373\pi$$
−0.938053 + 0.346492i $$0.887373\pi$$
$$152$$ 0 0
$$153$$ 5.56155 0.449625
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 16.6847 1.33158 0.665790 0.746139i $$-0.268095\pi$$
0.665790 + 0.746139i $$0.268095\pi$$
$$158$$ 0 0
$$159$$ 1.12311 0.0890681
$$160$$ 0 0
$$161$$ −1.56155 −0.123068
$$162$$ 0 0
$$163$$ −11.9309 −0.934498 −0.467249 0.884126i $$-0.654755\pi$$
−0.467249 + 0.884126i $$0.654755\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 0 0
$$169$$ −12.6847 −0.975743
$$170$$ 0 0
$$171$$ −21.3693 −1.63415
$$172$$ 0 0
$$173$$ 19.3693 1.47262 0.736311 0.676643i $$-0.236566\pi$$
0.736311 + 0.676643i $$0.236566\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −22.2462 −1.67213
$$178$$ 0 0
$$179$$ −21.9309 −1.63919 −0.819595 0.572943i $$-0.805802\pi$$
−0.819595 + 0.572943i $$0.805802\pi$$
$$180$$ 0 0
$$181$$ 19.3693 1.43971 0.719855 0.694124i $$-0.244208\pi$$
0.719855 + 0.694124i $$0.244208\pi$$
$$182$$ 0 0
$$183$$ 2.87689 0.212666
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −3.12311 −0.228384
$$188$$ 0 0
$$189$$ −2.24621 −0.163388
$$190$$ 0 0
$$191$$ 7.36932 0.533225 0.266613 0.963804i $$-0.414096\pi$$
0.266613 + 0.963804i $$0.414096\pi$$
$$192$$ 0 0
$$193$$ −6.56155 −0.472311 −0.236155 0.971715i $$-0.575887\pi$$
−0.236155 + 0.971715i $$0.575887\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 6.31534 0.449949 0.224975 0.974365i $$-0.427770\pi$$
0.224975 + 0.974365i $$0.427770\pi$$
$$198$$ 0 0
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ −11.3693 −0.801930
$$202$$ 0 0
$$203$$ 3.31534 0.232691
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 3.56155 0.247545
$$208$$ 0 0
$$209$$ 12.0000 0.830057
$$210$$ 0 0
$$211$$ −12.6847 −0.873248 −0.436624 0.899644i $$-0.643826\pi$$
−0.436624 + 0.899644i $$0.643826\pi$$
$$212$$ 0 0
$$213$$ −4.80776 −0.329423
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −14.4384 −0.980146
$$218$$ 0 0
$$219$$ 21.9309 1.48195
$$220$$ 0 0
$$221$$ −0.876894 −0.0589863
$$222$$ 0 0
$$223$$ 23.6155 1.58141 0.790706 0.612196i $$-0.209713\pi$$
0.790706 + 0.612196i $$0.209713\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 9.75379 0.647382 0.323691 0.946163i $$-0.395076\pi$$
0.323691 + 0.946163i $$0.395076\pi$$
$$228$$ 0 0
$$229$$ 22.7386 1.50261 0.751306 0.659954i $$-0.229424\pi$$
0.751306 + 0.659954i $$0.229424\pi$$
$$230$$ 0 0
$$231$$ 8.00000 0.526361
$$232$$ 0 0
$$233$$ −11.6847 −0.765487 −0.382744 0.923855i $$-0.625021\pi$$
−0.382744 + 0.923855i $$0.625021\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −33.6155 −2.18356
$$238$$ 0 0
$$239$$ −19.2462 −1.24493 −0.622467 0.782646i $$-0.713869\pi$$
−0.622467 + 0.782646i $$0.713869\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 0 0
$$243$$ −22.2462 −1.42710
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.36932 0.214384
$$248$$ 0 0
$$249$$ 38.2462 2.42376
$$250$$ 0 0
$$251$$ −17.3693 −1.09634 −0.548171 0.836366i $$-0.684676\pi$$
−0.548171 + 0.836366i $$0.684676\pi$$
$$252$$ 0 0
$$253$$ −2.00000 −0.125739
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −11.1922 −0.698152 −0.349076 0.937094i $$-0.613505\pi$$
−0.349076 + 0.937094i $$0.613505\pi$$
$$258$$ 0 0
$$259$$ −0.684658 −0.0425426
$$260$$ 0 0
$$261$$ −7.56155 −0.468048
$$262$$ 0 0
$$263$$ 28.9309 1.78395 0.891977 0.452081i $$-0.149318\pi$$
0.891977 + 0.452081i $$0.149318\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −5.75379 −0.352126
$$268$$ 0 0
$$269$$ 6.75379 0.411786 0.205893 0.978575i $$-0.433990\pi$$
0.205893 + 0.978575i $$0.433990\pi$$
$$270$$ 0 0
$$271$$ 6.93087 0.421020 0.210510 0.977592i $$-0.432487\pi$$
0.210510 + 0.977592i $$0.432487\pi$$
$$272$$ 0 0
$$273$$ 2.24621 0.135947
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −5.68466 −0.341558 −0.170779 0.985309i $$-0.554628\pi$$
−0.170779 + 0.985309i $$0.554628\pi$$
$$278$$ 0 0
$$279$$ 32.9309 1.97152
$$280$$ 0 0
$$281$$ −15.1231 −0.902169 −0.451084 0.892481i $$-0.648963\pi$$
−0.451084 + 0.892481i $$0.648963\pi$$
$$282$$ 0 0
$$283$$ −5.80776 −0.345236 −0.172618 0.984989i $$-0.555223\pi$$
−0.172618 + 0.984989i $$0.555223\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 6.43845 0.380050
$$288$$ 0 0
$$289$$ −14.5616 −0.856562
$$290$$ 0 0
$$291$$ 12.4924 0.732319
$$292$$ 0 0
$$293$$ −3.80776 −0.222452 −0.111226 0.993795i $$-0.535478\pi$$
−0.111226 + 0.993795i $$0.535478\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −2.87689 −0.166934
$$298$$ 0 0
$$299$$ −0.561553 −0.0324754
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ −13.6155 −0.782192
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −12.8769 −0.734923 −0.367462 0.930039i $$-0.619773\pi$$
−0.367462 + 0.930039i $$0.619773\pi$$
$$308$$ 0 0
$$309$$ −40.9848 −2.33155
$$310$$ 0 0
$$311$$ 3.68466 0.208938 0.104469 0.994528i $$-0.466686\pi$$
0.104469 + 0.994528i $$0.466686\pi$$
$$312$$ 0 0
$$313$$ −19.8078 −1.11960 −0.559801 0.828627i $$-0.689122\pi$$
−0.559801 + 0.828627i $$0.689122\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −6.87689 −0.386245 −0.193122 0.981175i $$-0.561861\pi$$
−0.193122 + 0.981175i $$0.561861\pi$$
$$318$$ 0 0
$$319$$ 4.24621 0.237742
$$320$$ 0 0
$$321$$ 19.3693 1.08109
$$322$$ 0 0
$$323$$ −9.36932 −0.521323
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −24.0000 −1.32720
$$328$$ 0 0
$$329$$ 12.0000 0.661581
$$330$$ 0 0
$$331$$ 12.6155 0.693412 0.346706 0.937974i $$-0.387300\pi$$
0.346706 + 0.937974i $$0.387300\pi$$
$$332$$ 0 0
$$333$$ 1.56155 0.0855726
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 27.6155 1.50431 0.752157 0.658984i $$-0.229014\pi$$
0.752157 + 0.658984i $$0.229014\pi$$
$$338$$ 0 0
$$339$$ 20.0000 1.08625
$$340$$ 0 0
$$341$$ −18.4924 −1.00142
$$342$$ 0 0
$$343$$ 18.0540 0.974823
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −16.4924 −0.885360 −0.442680 0.896680i $$-0.645972\pi$$
−0.442680 + 0.896680i $$0.645972\pi$$
$$348$$ 0 0
$$349$$ −1.63068 −0.0872885 −0.0436442 0.999047i $$-0.513897\pi$$
−0.0436442 + 0.999047i $$0.513897\pi$$
$$350$$ 0 0
$$351$$ −0.807764 −0.0431153
$$352$$ 0 0
$$353$$ −35.0540 −1.86573 −0.932867 0.360220i $$-0.882702\pi$$
−0.932867 + 0.360220i $$0.882702\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ −6.24621 −0.330585
$$358$$ 0 0
$$359$$ 25.6155 1.35194 0.675968 0.736931i $$-0.263726\pi$$
0.675968 + 0.736931i $$0.263726\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ −17.9309 −0.941127
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −18.4384 −0.962479 −0.481240 0.876589i $$-0.659813\pi$$
−0.481240 + 0.876589i $$0.659813\pi$$
$$368$$ 0 0
$$369$$ −14.6847 −0.764453
$$370$$ 0 0
$$371$$ −0.684658 −0.0355457
$$372$$ 0 0
$$373$$ 3.75379 0.194364 0.0971819 0.995267i $$-0.469017\pi$$
0.0971819 + 0.995267i $$0.469017\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 1.19224 0.0614033
$$378$$ 0 0
$$379$$ 3.50758 0.180172 0.0900861 0.995934i $$-0.471286\pi$$
0.0900861 + 0.995934i $$0.471286\pi$$
$$380$$ 0 0
$$381$$ −32.8078 −1.68079
$$382$$ 0 0
$$383$$ 23.8078 1.21652 0.608260 0.793738i $$-0.291868\pi$$
0.608260 + 0.793738i $$0.291868\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ −19.1231 −0.969580 −0.484790 0.874631i $$-0.661104\pi$$
−0.484790 + 0.874631i $$0.661104\pi$$
$$390$$ 0 0
$$391$$ 1.56155 0.0789711
$$392$$ 0 0
$$393$$ −53.3002 −2.68864
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 35.5464 1.78402 0.892011 0.452013i $$-0.149294\pi$$
0.892011 + 0.452013i $$0.149294\pi$$
$$398$$ 0 0
$$399$$ 24.0000 1.20150
$$400$$ 0 0
$$401$$ 2.00000 0.0998752 0.0499376 0.998752i $$-0.484098\pi$$
0.0499376 + 0.998752i $$0.484098\pi$$
$$402$$ 0 0
$$403$$ −5.19224 −0.258644
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −0.876894 −0.0434660
$$408$$ 0 0
$$409$$ 29.9848 1.48266 0.741328 0.671143i $$-0.234197\pi$$
0.741328 + 0.671143i $$0.234197\pi$$
$$410$$ 0 0
$$411$$ −38.7386 −1.91084
$$412$$ 0 0
$$413$$ 13.5616 0.667320
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −13.4384 −0.658084
$$418$$ 0 0
$$419$$ 27.1231 1.32505 0.662525 0.749040i $$-0.269485\pi$$
0.662525 + 0.749040i $$0.269485\pi$$
$$420$$ 0 0
$$421$$ 16.8769 0.822530 0.411265 0.911516i $$-0.365087\pi$$
0.411265 + 0.911516i $$0.365087\pi$$
$$422$$ 0 0
$$423$$ −27.3693 −1.33074
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1.75379 −0.0848718
$$428$$ 0 0
$$429$$ 2.87689 0.138898
$$430$$ 0 0
$$431$$ −6.24621 −0.300869 −0.150435 0.988620i $$-0.548067\pi$$
−0.150435 + 0.988620i $$0.548067\pi$$
$$432$$ 0 0
$$433$$ −11.0691 −0.531948 −0.265974 0.963980i $$-0.585694\pi$$
−0.265974 + 0.963980i $$0.585694\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −6.00000 −0.287019
$$438$$ 0 0
$$439$$ 0.807764 0.0385525 0.0192762 0.999814i $$-0.493864\pi$$
0.0192762 + 0.999814i $$0.493864\pi$$
$$440$$ 0 0
$$441$$ −16.2462 −0.773629
$$442$$ 0 0
$$443$$ 8.80776 0.418469 0.209235 0.977865i $$-0.432903\pi$$
0.209235 + 0.977865i $$0.432903\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 39.3693 1.86210
$$448$$ 0 0
$$449$$ −9.31534 −0.439618 −0.219809 0.975543i $$-0.570543\pi$$
−0.219809 + 0.975543i $$0.570543\pi$$
$$450$$ 0 0
$$451$$ 8.24621 0.388299
$$452$$ 0 0
$$453$$ −59.0540 −2.77460
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −13.5616 −0.634383 −0.317191 0.948362i $$-0.602740\pi$$
−0.317191 + 0.948362i $$0.602740\pi$$
$$458$$ 0 0
$$459$$ 2.24621 0.104844
$$460$$ 0 0
$$461$$ −12.0691 −0.562115 −0.281058 0.959691i $$-0.590685\pi$$
−0.281058 + 0.959691i $$0.590685\pi$$
$$462$$ 0 0
$$463$$ −6.63068 −0.308154 −0.154077 0.988059i $$-0.549240\pi$$
−0.154077 + 0.988059i $$0.549240\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ −34.6847 −1.60501 −0.802507 0.596642i $$-0.796501\pi$$
−0.802507 + 0.596642i $$0.796501\pi$$
$$468$$ 0 0
$$469$$ 6.93087 0.320038
$$470$$ 0 0
$$471$$ 42.7386 1.96929
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1.56155 0.0714986
$$478$$ 0 0
$$479$$ −39.2311 −1.79251 −0.896256 0.443536i $$-0.853724\pi$$
−0.896256 + 0.443536i $$0.853724\pi$$
$$480$$ 0 0
$$481$$ −0.246211 −0.0112263
$$482$$ 0 0
$$483$$ −4.00000 −0.182006
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −12.8078 −0.580375 −0.290188 0.956970i $$-0.593718\pi$$
−0.290188 + 0.956970i $$0.593718\pi$$
$$488$$ 0 0
$$489$$ −30.5616 −1.38204
$$490$$ 0 0
$$491$$ 21.4924 0.969939 0.484970 0.874531i $$-0.338831\pi$$
0.484970 + 0.874531i $$0.338831\pi$$
$$492$$ 0 0
$$493$$ −3.31534 −0.149315
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2.93087 0.131467
$$498$$ 0 0
$$499$$ 18.6155 0.833345 0.416673 0.909057i $$-0.363196\pi$$
0.416673 + 0.909057i $$0.363196\pi$$
$$500$$ 0 0
$$501$$ −20.4924 −0.915534
$$502$$ 0 0
$$503$$ 24.9309 1.11161 0.555806 0.831312i $$-0.312410\pi$$
0.555806 + 0.831312i $$0.312410\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −32.4924 −1.44304
$$508$$ 0 0
$$509$$ 18.1771 0.805685 0.402842 0.915269i $$-0.368022\pi$$
0.402842 + 0.915269i $$0.368022\pi$$
$$510$$ 0 0
$$511$$ −13.3693 −0.591424
$$512$$ 0 0
$$513$$ −8.63068 −0.381054
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 15.3693 0.675942
$$518$$ 0 0
$$519$$ 49.6155 2.17788
$$520$$ 0 0
$$521$$ 27.6155 1.20986 0.604929 0.796279i $$-0.293201\pi$$
0.604929 + 0.796279i $$0.293201\pi$$
$$522$$ 0 0
$$523$$ 34.7386 1.51901 0.759507 0.650499i $$-0.225440\pi$$
0.759507 + 0.650499i $$0.225440\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 14.4384 0.628949
$$528$$ 0 0
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ −30.9309 −1.34229
$$532$$ 0 0
$$533$$ 2.31534 0.100289
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −56.1771 −2.42422
$$538$$ 0 0
$$539$$ 9.12311 0.392960
$$540$$ 0 0
$$541$$ 9.68466 0.416376 0.208188 0.978089i $$-0.433243\pi$$
0.208188 + 0.978089i $$0.433243\pi$$
$$542$$ 0 0
$$543$$ 49.6155 2.12921
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 24.1771 1.03374 0.516869 0.856065i $$-0.327098\pi$$
0.516869 + 0.856065i $$0.327098\pi$$
$$548$$ 0 0
$$549$$ 4.00000 0.170716
$$550$$ 0 0
$$551$$ 12.7386 0.542684
$$552$$ 0 0
$$553$$ 20.4924 0.871426
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 3.31534 0.140476 0.0702378 0.997530i $$-0.477624\pi$$
0.0702378 + 0.997530i $$0.477624\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 0 0
$$563$$ −45.6695 −1.92474 −0.962370 0.271742i $$-0.912400\pi$$
−0.962370 + 0.271742i $$0.912400\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 10.9309 0.459053
$$568$$ 0 0
$$569$$ −32.7386 −1.37247 −0.686237 0.727378i $$-0.740739\pi$$
−0.686237 + 0.727378i $$0.740739\pi$$
$$570$$ 0 0
$$571$$ −36.0000 −1.50655 −0.753277 0.657704i $$-0.771528\pi$$
−0.753277 + 0.657704i $$0.771528\pi$$
$$572$$ 0 0
$$573$$ 18.8769 0.788594
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 25.4384 1.05902 0.529508 0.848305i $$-0.322376\pi$$
0.529508 + 0.848305i $$0.322376\pi$$
$$578$$ 0 0
$$579$$ −16.8078 −0.698507
$$580$$ 0 0
$$581$$ −23.3153 −0.967283
$$582$$ 0 0
$$583$$ −0.876894 −0.0363173
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 35.9309 1.48303 0.741513 0.670939i $$-0.234109\pi$$
0.741513 + 0.670939i $$0.234109\pi$$
$$588$$ 0 0
$$589$$ −55.4773 −2.28590
$$590$$ 0 0
$$591$$ 16.1771 0.665436
$$592$$ 0 0
$$593$$ 25.6155 1.05190 0.525952 0.850514i $$-0.323709\pi$$
0.525952 + 0.850514i $$0.323709\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 25.6155 1.04837
$$598$$ 0 0
$$599$$ −32.9848 −1.34772 −0.673862 0.738857i $$-0.735366\pi$$
−0.673862 + 0.738857i $$0.735366\pi$$
$$600$$ 0 0
$$601$$ 11.6307 0.474425 0.237213 0.971458i $$-0.423766\pi$$
0.237213 + 0.971458i $$0.423766\pi$$
$$602$$ 0 0
$$603$$ −15.8078 −0.643742
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −24.4924 −0.994117 −0.497058 0.867717i $$-0.665587\pi$$
−0.497058 + 0.867717i $$0.665587\pi$$
$$608$$ 0 0
$$609$$ 8.49242 0.344130
$$610$$ 0 0
$$611$$ 4.31534 0.174580
$$612$$ 0 0
$$613$$ −35.6155 −1.43850 −0.719249 0.694753i $$-0.755514\pi$$
−0.719249 + 0.694753i $$0.755514\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 3.56155 0.143383 0.0716914 0.997427i $$-0.477160\pi$$
0.0716914 + 0.997427i $$0.477160\pi$$
$$618$$ 0 0
$$619$$ 20.4924 0.823660 0.411830 0.911261i $$-0.364890\pi$$
0.411830 + 0.911261i $$0.364890\pi$$
$$620$$ 0 0
$$621$$ 1.43845 0.0577229
$$622$$ 0 0
$$623$$ 3.50758 0.140528
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 30.7386 1.22758
$$628$$ 0 0
$$629$$ 0.684658 0.0272991
$$630$$ 0 0
$$631$$ −13.7538 −0.547530 −0.273765 0.961797i $$-0.588269\pi$$
−0.273765 + 0.961797i $$0.588269\pi$$
$$632$$ 0 0
$$633$$ −32.4924 −1.29146
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 2.56155 0.101492
$$638$$ 0 0
$$639$$ −6.68466 −0.264441
$$640$$ 0 0
$$641$$ 15.1231 0.597327 0.298663 0.954359i $$-0.403459\pi$$
0.298663 + 0.954359i $$0.403459\pi$$
$$642$$ 0 0
$$643$$ −41.4233 −1.63358 −0.816788 0.576939i $$-0.804247\pi$$
−0.816788 + 0.576939i $$0.804247\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17.6847 0.695256 0.347628 0.937633i $$-0.386987\pi$$
0.347628 + 0.937633i $$0.386987\pi$$
$$648$$ 0 0
$$649$$ 17.3693 0.681805
$$650$$ 0 0
$$651$$ −36.9848 −1.44955
$$652$$ 0 0
$$653$$ 38.6695 1.51325 0.756627 0.653846i $$-0.226846\pi$$
0.756627 + 0.653846i $$0.226846\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 30.4924 1.18962
$$658$$ 0 0
$$659$$ 9.36932 0.364977 0.182488 0.983208i $$-0.441585\pi$$
0.182488 + 0.983208i $$0.441585\pi$$
$$660$$ 0 0
$$661$$ 26.4924 1.03044 0.515218 0.857059i $$-0.327711\pi$$
0.515218 + 0.857059i $$0.327711\pi$$
$$662$$ 0 0
$$663$$ −2.24621 −0.0872356
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2.12311 −0.0822070
$$668$$ 0 0
$$669$$ 60.4924 2.33877
$$670$$ 0 0
$$671$$ −2.24621 −0.0867140
$$672$$ 0 0
$$673$$ 35.5464 1.37021 0.685106 0.728443i $$-0.259756\pi$$
0.685106 + 0.728443i $$0.259756\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −6.19224 −0.237987 −0.118993 0.992895i $$-0.537967\pi$$
−0.118993 + 0.992895i $$0.537967\pi$$
$$678$$ 0 0
$$679$$ −7.61553 −0.292257
$$680$$ 0 0
$$681$$ 24.9848 0.957421
$$682$$ 0 0
$$683$$ 4.94602 0.189254 0.0946272 0.995513i $$-0.469834\pi$$
0.0946272 + 0.995513i $$0.469834\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 58.2462 2.22223
$$688$$ 0 0
$$689$$ −0.246211 −0.00937990
$$690$$ 0 0
$$691$$ 16.4924 0.627401 0.313701 0.949522i $$-0.398431\pi$$
0.313701 + 0.949522i $$0.398431\pi$$
$$692$$ 0 0
$$693$$ 11.1231 0.422532
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −6.43845 −0.243874
$$698$$ 0 0
$$699$$ −29.9309 −1.13209
$$700$$ 0 0
$$701$$ −18.2462 −0.689150 −0.344575 0.938759i $$-0.611977\pi$$
−0.344575 + 0.938759i $$0.611977\pi$$
$$702$$ 0 0
$$703$$ −2.63068 −0.0992181
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 8.30019 0.312161
$$708$$ 0 0
$$709$$ 46.2462 1.73681 0.868406 0.495853i $$-0.165145\pi$$
0.868406 + 0.495853i $$0.165145\pi$$
$$710$$ 0 0
$$711$$ −46.7386 −1.75284
$$712$$ 0 0
$$713$$ 9.24621 0.346273
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −49.3002 −1.84115
$$718$$ 0 0
$$719$$ 18.0540 0.673300 0.336650 0.941630i $$-0.390706\pi$$
0.336650 + 0.941630i $$0.390706\pi$$
$$720$$ 0 0
$$721$$ 24.9848 0.930484
$$722$$ 0 0
$$723$$ 15.3693 0.571591
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −41.8078 −1.55056 −0.775282 0.631615i $$-0.782392\pi$$
−0.775282 + 0.631615i $$0.782392\pi$$
$$728$$ 0 0
$$729$$ −35.9848 −1.33277
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ −12.6847 −0.468519 −0.234259 0.972174i $$-0.575266\pi$$
−0.234259 + 0.972174i $$0.575266\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 8.87689 0.326985
$$738$$ 0 0
$$739$$ 40.6155 1.49407 0.747033 0.664787i $$-0.231478\pi$$
0.747033 + 0.664787i $$0.231478\pi$$
$$740$$ 0 0
$$741$$ 8.63068 0.317056
$$742$$ 0 0
$$743$$ 36.4924 1.33878 0.669389 0.742912i $$-0.266556\pi$$
0.669389 + 0.742912i $$0.266556\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 53.1771 1.94565
$$748$$ 0 0
$$749$$ −11.8078 −0.431446
$$750$$ 0 0
$$751$$ 4.87689 0.177960 0.0889802 0.996033i $$-0.471639\pi$$
0.0889802 + 0.996033i $$0.471639\pi$$
$$752$$ 0 0
$$753$$ −44.4924 −1.62139
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −8.43845 −0.306701 −0.153350 0.988172i $$-0.549006\pi$$
−0.153350 + 0.988172i $$0.549006\pi$$
$$758$$ 0 0
$$759$$ −5.12311 −0.185957
$$760$$ 0 0
$$761$$ 21.9848 0.796950 0.398475 0.917179i $$-0.369540\pi$$
0.398475 + 0.917179i $$0.369540\pi$$
$$762$$ 0 0
$$763$$ 14.6307 0.529666
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 4.87689 0.176094
$$768$$ 0 0
$$769$$ −35.3693 −1.27545 −0.637725 0.770264i $$-0.720124\pi$$
−0.637725 + 0.770264i $$0.720124\pi$$
$$770$$ 0 0
$$771$$ −28.6695 −1.03251
$$772$$ 0 0
$$773$$ 32.8769 1.18250 0.591250 0.806488i $$-0.298635\pi$$
0.591250 + 0.806488i $$0.298635\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −1.75379 −0.0629168
$$778$$ 0 0
$$779$$ 24.7386 0.886354
$$780$$ 0 0
$$781$$ 3.75379 0.134321
$$782$$ 0 0
$$783$$ −3.05398 −0.109140
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −27.3153 −0.973687 −0.486843 0.873489i $$-0.661852\pi$$
−0.486843 + 0.873489i $$0.661852\pi$$
$$788$$ 0 0
$$789$$ 74.1080 2.63831
$$790$$ 0 0
$$791$$ −12.1922 −0.433506
$$792$$ 0 0
$$793$$ −0.630683 −0.0223962
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −9.42329 −0.333790 −0.166895 0.985975i $$-0.553374\pi$$
−0.166895 + 0.985975i $$0.553374\pi$$
$$798$$ 0 0
$$799$$ −12.0000 −0.424529
$$800$$ 0 0
$$801$$ −8.00000 −0.282666
$$802$$ 0 0
$$803$$ −17.1231 −0.604261
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 17.3002 0.608995
$$808$$ 0 0
$$809$$ −8.82292 −0.310197 −0.155099 0.987899i $$-0.549570\pi$$
−0.155099 + 0.987899i $$0.549570\pi$$
$$810$$ 0 0
$$811$$ −30.7538 −1.07991 −0.539956 0.841693i $$-0.681559\pi$$
−0.539956 + 0.841693i $$0.681559\pi$$
$$812$$ 0 0
$$813$$ 17.7538 0.622653
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ 3.12311 0.109130
$$820$$ 0 0
$$821$$ −1.50758 −0.0526148 −0.0263074 0.999654i $$-0.508375\pi$$
−0.0263074 + 0.999654i $$0.508375\pi$$
$$822$$ 0 0
$$823$$ −0.946025 −0.0329763 −0.0164882 0.999864i $$-0.505249\pi$$
−0.0164882 + 0.999864i $$0.505249\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 7.31534 0.254379 0.127190 0.991878i $$-0.459404\pi$$
0.127190 + 0.991878i $$0.459404\pi$$
$$828$$ 0 0
$$829$$ 40.5464 1.40823 0.704117 0.710084i $$-0.251343\pi$$
0.704117 + 0.710084i $$0.251343\pi$$
$$830$$ 0 0
$$831$$ −14.5616 −0.505135
$$832$$ 0 0
$$833$$ −7.12311 −0.246801
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 13.3002 0.459722
$$838$$ 0 0
$$839$$ 51.1231 1.76497 0.882483 0.470345i $$-0.155870\pi$$
0.882483 + 0.470345i $$0.155870\pi$$
$$840$$ 0 0
$$841$$ −24.4924 −0.844566
$$842$$ 0 0
$$843$$ −38.7386 −1.33423
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.9309 0.375589
$$848$$ 0 0
$$849$$ −14.8769 −0.510574
$$850$$ 0 0
$$851$$ 0.438447 0.0150298
$$852$$ 0 0
$$853$$ 3.75379 0.128527 0.0642636 0.997933i $$-0.479530\pi$$
0.0642636 + 0.997933i $$0.479530\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 45.6847 1.56056 0.780279 0.625431i $$-0.215077\pi$$
0.780279 + 0.625431i $$0.215077\pi$$
$$858$$ 0 0
$$859$$ 47.4924 1.62042 0.810210 0.586139i $$-0.199353\pi$$
0.810210 + 0.586139i $$0.199353\pi$$
$$860$$ 0 0
$$861$$ 16.4924 0.562060
$$862$$ 0 0
$$863$$ −3.43845 −0.117046 −0.0585231 0.998286i $$-0.518639\pi$$
−0.0585231 + 0.998286i $$0.518639\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −37.3002 −1.26678
$$868$$ 0 0
$$869$$ 26.2462 0.890342
$$870$$ 0 0
$$871$$ 2.49242 0.0844525
$$872$$ 0 0
$$873$$ 17.3693 0.587862
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −46.9848 −1.58657 −0.793283 0.608853i $$-0.791630\pi$$
−0.793283 + 0.608853i $$0.791630\pi$$
$$878$$ 0 0
$$879$$ −9.75379 −0.328987
$$880$$ 0 0
$$881$$ −57.4773 −1.93646 −0.968229 0.250065i $$-0.919548\pi$$
−0.968229 + 0.250065i $$0.919548\pi$$
$$882$$ 0 0
$$883$$ −38.7386 −1.30366 −0.651829 0.758366i $$-0.725998\pi$$
−0.651829 + 0.758366i $$0.725998\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ −43.7926 −1.47041 −0.735206 0.677844i $$-0.762915\pi$$
−0.735206 + 0.677844i $$0.762915\pi$$
$$888$$ 0 0
$$889$$ 20.0000 0.670778
$$890$$ 0 0
$$891$$ 14.0000 0.469018
$$892$$ 0 0
$$893$$ 46.1080 1.54294
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −1.43845 −0.0480284
$$898$$ 0 0
$$899$$ −19.6307 −0.654720
$$900$$ 0 0
$$901$$ 0.684658 0.0228093
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 9.31534 0.309311 0.154655 0.987968i $$-0.450573\pi$$
0.154655 + 0.987968i $$0.450573\pi$$
$$908$$ 0 0
$$909$$ −18.9309 −0.627897
$$910$$ 0 0
$$911$$ −5.12311 −0.169736 −0.0848680 0.996392i $$-0.527047\pi$$
−0.0848680 + 0.996392i $$0.527047\pi$$
$$912$$ 0 0
$$913$$ −29.8617 −0.988279
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 32.4924 1.07299
$$918$$ 0 0
$$919$$ 2.73863 0.0903392 0.0451696 0.998979i $$-0.485617\pi$$
0.0451696 + 0.998979i $$0.485617\pi$$
$$920$$ 0 0
$$921$$ −32.9848 −1.08689
$$922$$ 0 0
$$923$$ 1.05398 0.0346920
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −56.9848 −1.87163
$$928$$ 0 0
$$929$$ 58.7235 1.92665 0.963327 0.268329i $$-0.0864713\pi$$
0.963327 + 0.268329i $$0.0864713\pi$$
$$930$$ 0 0
$$931$$ 27.3693 0.896993
$$932$$ 0 0
$$933$$ 9.43845 0.309001
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −0.246211 −0.00804337 −0.00402169 0.999992i $$-0.501280\pi$$
−0.00402169 + 0.999992i $$0.501280\pi$$
$$938$$ 0 0
$$939$$ −50.7386 −1.65579
$$940$$ 0 0
$$941$$ −54.2462 −1.76838 −0.884188 0.467131i $$-0.845288\pi$$
−0.884188 + 0.467131i $$0.845288\pi$$
$$942$$ 0 0
$$943$$ −4.12311 −0.134267
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 44.3153 1.44006 0.720028 0.693945i $$-0.244129\pi$$
0.720028 + 0.693945i $$0.244129\pi$$
$$948$$ 0 0
$$949$$ −4.80776 −0.156067
$$950$$ 0 0
$$951$$ −17.6155 −0.571223
$$952$$ 0 0
$$953$$ 5.50758 0.178408 0.0892040 0.996013i $$-0.471568\pi$$
0.0892040 + 0.996013i $$0.471568\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 10.8769 0.351600
$$958$$ 0 0
$$959$$ 23.6155 0.762585
$$960$$ 0 0
$$961$$ 54.4924 1.75782
$$962$$ 0 0
$$963$$ 26.9309 0.867835
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −44.3153 −1.42509 −0.712543 0.701629i $$-0.752457\pi$$
−0.712543 + 0.701629i $$0.752457\pi$$
$$968$$ 0 0
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ 47.3693 1.52015 0.760077 0.649833i $$-0.225161\pi$$
0.760077 + 0.649833i $$0.225161\pi$$
$$972$$ 0 0
$$973$$ 8.19224 0.262631
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 26.7926 0.857172 0.428586 0.903501i $$-0.359012\pi$$
0.428586 + 0.903501i $$0.359012\pi$$
$$978$$ 0 0
$$979$$ 4.49242 0.143578
$$980$$ 0 0
$$981$$ −33.3693 −1.06540
$$982$$ 0 0
$$983$$ 0.0539753 0.00172155 0.000860773 1.00000i $$-0.499726\pi$$
0.000860773 1.00000i $$0.499726\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 30.7386 0.978421
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 0.300187 0.00953574 0.00476787 0.999989i $$-0.498482\pi$$
0.00476787 + 0.999989i $$0.498482\pi$$
$$992$$ 0 0
$$993$$ 32.3153 1.02550
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −14.3845 −0.455561 −0.227780 0.973713i $$-0.573147\pi$$
−0.227780 + 0.973713i $$0.573147\pi$$
$$998$$ 0 0
$$999$$ 0.630683 0.0199539
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 9200.2.a.bv.1.2 2
4.3 odd 2 2300.2.a.i.1.1 2
5.4 even 2 1840.2.a.m.1.1 2
20.3 even 4 2300.2.c.h.1749.1 4
20.7 even 4 2300.2.c.h.1749.4 4
20.19 odd 2 460.2.a.e.1.2 2
40.19 odd 2 7360.2.a.bi.1.1 2
40.29 even 2 7360.2.a.bo.1.2 2
60.59 even 2 4140.2.a.m.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.a.e.1.2 2 20.19 odd 2
1840.2.a.m.1.1 2 5.4 even 2
2300.2.a.i.1.1 2 4.3 odd 2
2300.2.c.h.1749.1 4 20.3 even 4
2300.2.c.h.1749.4 4 20.7 even 4
4140.2.a.m.1.1 2 60.59 even 2
7360.2.a.bi.1.1 2 40.19 odd 2
7360.2.a.bo.1.2 2 40.29 even 2
9200.2.a.bv.1.2 2 1.1 even 1 trivial