# Properties

 Label 6525.2.a.bf.1.2 Level $6525$ Weight $2$ Character 6525.1 Self dual yes Analytic conductor $52.102$ 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] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$6525 = 3^{2} \cdot 5^{2} \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6525.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: $$52.1023873189$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.469.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 5x + 4$$ x^3 - x^2 - 5*x + 4 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$0.772866$$ of defining polynomial Character $$\chi$$ $$=$$ 6525.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+0.772866 q^{2} -1.40268 q^{4} -2.17554 q^{7} -2.62981 q^{8} +O(q^{10})$$ $$q+0.772866 q^{2} -1.40268 q^{4} -2.17554 q^{7} -2.62981 q^{8} -3.00000 q^{11} +0.629813 q^{13} -1.68140 q^{14} +0.772866 q^{16} +4.17554 q^{17} +4.80536 q^{19} -2.31860 q^{22} +2.08408 q^{23} +0.486761 q^{26} +3.05159 q^{28} +1.00000 q^{29} +5.85695 q^{32} +3.22713 q^{34} -6.08408 q^{37} +3.71390 q^{38} -0.824456 q^{41} +8.72128 q^{43} +4.20804 q^{44} +1.61072 q^{46} -8.98090 q^{47} -2.26701 q^{49} -0.883426 q^{52} +6.88944 q^{53} +5.72128 q^{56} +0.772866 q^{58} -6.45427 q^{59} -2.80536 q^{61} +2.98090 q^{64} +11.0841 q^{67} -5.85695 q^{68} -2.63719 q^{71} +14.7863 q^{73} -4.70218 q^{74} -6.74037 q^{76} +6.52663 q^{77} -12.0650 q^{79} -0.637193 q^{82} -7.62981 q^{83} +6.74037 q^{86} +7.88944 q^{88} -17.8894 q^{89} -1.37019 q^{91} -2.92330 q^{92} -6.94103 q^{94} -0.538351 q^{97} -1.75209 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + q^{2} + 5 q^{4} + 4 q^{7}+O(q^{10})$$ 3 * q + q^2 + 5 * q^4 + 4 * q^7 $$3 q + q^{2} + 5 q^{4} + 4 q^{7} - 9 q^{11} - 6 q^{13} - 9 q^{14} + q^{16} + 2 q^{17} - 4 q^{19} - 3 q^{22} + q^{23} - 13 q^{26} + 21 q^{28} + 3 q^{29} + 11 q^{32} + 11 q^{34} - 13 q^{37} - 2 q^{38} - 13 q^{41} + 13 q^{43} - 15 q^{44} - 32 q^{46} + 2 q^{47} + 9 q^{49} - 25 q^{52} - 3 q^{53} + 4 q^{56} + q^{58} - 22 q^{59} + 10 q^{61} - 20 q^{64} + 28 q^{67} - 11 q^{68} - 3 q^{73} + 28 q^{74} - 36 q^{76} - 12 q^{77} - 2 q^{79} + 6 q^{82} - 15 q^{83} + 36 q^{86} - 30 q^{89} - 12 q^{91} - 14 q^{92} - 9 q^{94} + q^{97} - 50 q^{98}+O(q^{100})$$ 3 * q + q^2 + 5 * q^4 + 4 * q^7 - 9 * q^11 - 6 * q^13 - 9 * q^14 + q^16 + 2 * q^17 - 4 * q^19 - 3 * q^22 + q^23 - 13 * q^26 + 21 * q^28 + 3 * q^29 + 11 * q^32 + 11 * q^34 - 13 * q^37 - 2 * q^38 - 13 * q^41 + 13 * q^43 - 15 * q^44 - 32 * q^46 + 2 * q^47 + 9 * q^49 - 25 * q^52 - 3 * q^53 + 4 * q^56 + q^58 - 22 * q^59 + 10 * q^61 - 20 * q^64 + 28 * q^67 - 11 * q^68 - 3 * q^73 + 28 * q^74 - 36 * q^76 - 12 * q^77 - 2 * q^79 + 6 * q^82 - 15 * q^83 + 36 * q^86 - 30 * q^89 - 12 * q^91 - 14 * q^92 - 9 * q^94 + q^97 - 50 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0.772866 0.546498 0.273249 0.961943i $$-0.411902\pi$$
0.273249 + 0.961943i $$0.411902\pi$$
$$3$$ 0 0
$$4$$ −1.40268 −0.701339
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −2.17554 −0.822278 −0.411139 0.911573i $$-0.634869\pi$$
−0.411139 + 0.911573i $$0.634869\pi$$
$$8$$ −2.62981 −0.929779
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 0.629813 0.174679 0.0873394 0.996179i $$-0.472164\pi$$
0.0873394 + 0.996179i $$0.472164\pi$$
$$14$$ −1.68140 −0.449374
$$15$$ 0 0
$$16$$ 0.772866 0.193216
$$17$$ 4.17554 1.01272 0.506359 0.862323i $$-0.330991\pi$$
0.506359 + 0.862323i $$0.330991\pi$$
$$18$$ 0 0
$$19$$ 4.80536 1.10242 0.551212 0.834365i $$-0.314165\pi$$
0.551212 + 0.834365i $$0.314165\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −2.31860 −0.494326
$$23$$ 2.08408 0.434561 0.217281 0.976109i $$-0.430281\pi$$
0.217281 + 0.976109i $$0.430281\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0.486761 0.0954617
$$27$$ 0 0
$$28$$ 3.05159 0.576696
$$29$$ 1.00000 0.185695
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 5.85695 1.03537
$$33$$ 0 0
$$34$$ 3.22713 0.553449
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −6.08408 −1.00022 −0.500108 0.865963i $$-0.666707\pi$$
−0.500108 + 0.865963i $$0.666707\pi$$
$$38$$ 3.71390 0.602473
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −0.824456 −0.128758 −0.0643792 0.997926i $$-0.520507\pi$$
−0.0643792 + 0.997926i $$0.520507\pi$$
$$42$$ 0 0
$$43$$ 8.72128 1.32998 0.664991 0.746851i $$-0.268435\pi$$
0.664991 + 0.746851i $$0.268435\pi$$
$$44$$ 4.20804 0.634385
$$45$$ 0 0
$$46$$ 1.61072 0.237487
$$47$$ −8.98090 −1.31000 −0.655000 0.755629i $$-0.727331\pi$$
−0.655000 + 0.755629i $$0.727331\pi$$
$$48$$ 0 0
$$49$$ −2.26701 −0.323858
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −0.883426 −0.122509
$$53$$ 6.88944 0.946337 0.473169 0.880972i $$-0.343110\pi$$
0.473169 + 0.880972i $$0.343110\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 5.72128 0.764538
$$57$$ 0 0
$$58$$ 0.772866 0.101482
$$59$$ −6.45427 −0.840274 −0.420137 0.907461i $$-0.638018\pi$$
−0.420137 + 0.907461i $$0.638018\pi$$
$$60$$ 0 0
$$61$$ −2.80536 −0.359189 −0.179595 0.983741i $$-0.557479\pi$$
−0.179595 + 0.983741i $$0.557479\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 2.98090 0.372613
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 11.0841 1.35414 0.677068 0.735920i $$-0.263250\pi$$
0.677068 + 0.735920i $$0.263250\pi$$
$$68$$ −5.85695 −0.710259
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −2.63719 −0.312977 −0.156489 0.987680i $$-0.550017\pi$$
−0.156489 + 0.987680i $$0.550017\pi$$
$$72$$ 0 0
$$73$$ 14.7863 1.73060 0.865300 0.501254i $$-0.167128\pi$$
0.865300 + 0.501254i $$0.167128\pi$$
$$74$$ −4.70218 −0.546617
$$75$$ 0 0
$$76$$ −6.74037 −0.773174
$$77$$ 6.52663 0.743779
$$78$$ 0 0
$$79$$ −12.0650 −1.35742 −0.678708 0.734408i $$-0.737460\pi$$
−0.678708 + 0.734408i $$0.737460\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −0.637193 −0.0703662
$$83$$ −7.62981 −0.837481 −0.418740 0.908106i $$-0.637528\pi$$
−0.418740 + 0.908106i $$0.637528\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 6.74037 0.726833
$$87$$ 0 0
$$88$$ 7.88944 0.841017
$$89$$ −17.8894 −1.89628 −0.948138 0.317858i $$-0.897037\pi$$
−0.948138 + 0.317858i $$0.897037\pi$$
$$90$$ 0 0
$$91$$ −1.37019 −0.143635
$$92$$ −2.92330 −0.304775
$$93$$ 0 0
$$94$$ −6.94103 −0.715913
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −0.538351 −0.0546613 −0.0273306 0.999626i $$-0.508701\pi$$
−0.0273306 + 0.999626i $$0.508701\pi$$
$$98$$ −1.75209 −0.176988
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −13.8054 −1.37368 −0.686842 0.726807i $$-0.741004\pi$$
−0.686842 + 0.726807i $$0.741004\pi$$
$$102$$ 0 0
$$103$$ 13.4426 1.32453 0.662267 0.749268i $$-0.269594\pi$$
0.662267 + 0.749268i $$0.269594\pi$$
$$104$$ −1.65629 −0.162413
$$105$$ 0 0
$$106$$ 5.32461 0.517172
$$107$$ −9.25963 −0.895162 −0.447581 0.894243i $$-0.647714\pi$$
−0.447581 + 0.894243i $$0.647714\pi$$
$$108$$ 0 0
$$109$$ 8.61072 0.824757 0.412378 0.911013i $$-0.364698\pi$$
0.412378 + 0.911013i $$0.364698\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −1.68140 −0.158878
$$113$$ 11.2670 1.05991 0.529955 0.848026i $$-0.322209\pi$$
0.529955 + 0.848026i $$0.322209\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.40268 −0.130235
$$117$$ 0 0
$$118$$ −4.98828 −0.459209
$$119$$ −9.08408 −0.832736
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −2.16816 −0.196296
$$123$$ 0 0
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −16.7863 −1.48954 −0.744770 0.667321i $$-0.767441\pi$$
−0.744770 + 0.667321i $$0.767441\pi$$
$$128$$ −9.41006 −0.831740
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −11.2670 −0.984403 −0.492201 0.870481i $$-0.663808\pi$$
−0.492201 + 0.870481i $$0.663808\pi$$
$$132$$ 0 0
$$133$$ −10.4543 −0.906500
$$134$$ 8.56651 0.740033
$$135$$ 0 0
$$136$$ −10.9809 −0.941605
$$137$$ −18.8703 −1.61220 −0.806101 0.591778i $$-0.798426\pi$$
−0.806101 + 0.591778i $$0.798426\pi$$
$$138$$ 0 0
$$139$$ 3.80536 0.322766 0.161383 0.986892i $$-0.448405\pi$$
0.161383 + 0.986892i $$0.448405\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −2.03820 −0.171042
$$143$$ −1.88944 −0.158003
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 11.4278 0.945771
$$147$$ 0 0
$$148$$ 8.53401 0.701492
$$149$$ −4.45427 −0.364908 −0.182454 0.983214i $$-0.558404\pi$$
−0.182454 + 0.983214i $$0.558404\pi$$
$$150$$ 0 0
$$151$$ 1.17554 0.0956644 0.0478322 0.998855i $$-0.484769\pi$$
0.0478322 + 0.998855i $$0.484769\pi$$
$$152$$ −12.6372 −1.02501
$$153$$ 0 0
$$154$$ 5.04421 0.406474
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 0.805358 0.0642745 0.0321373 0.999483i $$-0.489769\pi$$
0.0321373 + 0.999483i $$0.489769\pi$$
$$158$$ −9.32461 −0.741826
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −4.53401 −0.357330
$$162$$ 0 0
$$163$$ 1.11056 0.0869858 0.0434929 0.999054i $$-0.486151\pi$$
0.0434929 + 0.999054i $$0.486151\pi$$
$$164$$ 1.15645 0.0903033
$$165$$ 0 0
$$166$$ −5.89682 −0.457682
$$167$$ −8.70218 −0.673395 −0.336697 0.941613i $$-0.609310\pi$$
−0.336697 + 0.941613i $$0.609310\pi$$
$$168$$ 0 0
$$169$$ −12.6033 −0.969487
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −12.2331 −0.932769
$$173$$ −7.69480 −0.585025 −0.292512 0.956262i $$-0.594491\pi$$
−0.292512 + 0.956262i $$0.594491\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.31860 −0.174771
$$177$$ 0 0
$$178$$ −13.8261 −1.03631
$$179$$ 6.06498 0.453318 0.226659 0.973974i $$-0.427220\pi$$
0.226659 + 0.973974i $$0.427220\pi$$
$$180$$ 0 0
$$181$$ 4.09146 0.304116 0.152058 0.988372i $$-0.451410\pi$$
0.152058 + 0.988372i $$0.451410\pi$$
$$182$$ −1.05897 −0.0784961
$$183$$ 0 0
$$184$$ −5.48075 −0.404046
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −12.5266 −0.916038
$$188$$ 12.5973 0.918754
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0.267007 0.0193199 0.00965996 0.999953i $$-0.496925\pi$$
0.00965996 + 0.999953i $$0.496925\pi$$
$$192$$ 0 0
$$193$$ −10.7022 −0.770360 −0.385180 0.922842i $$-0.625861\pi$$
−0.385180 + 0.922842i $$0.625861\pi$$
$$194$$ −0.416073 −0.0298723
$$195$$ 0 0
$$196$$ 3.17988 0.227134
$$197$$ 8.08408 0.575967 0.287984 0.957635i $$-0.407015\pi$$
0.287984 + 0.957635i $$0.407015\pi$$
$$198$$ 0 0
$$199$$ −15.8054 −1.12041 −0.560206 0.828353i $$-0.689278\pi$$
−0.560206 + 0.828353i $$0.689278\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ −10.6697 −0.750716
$$203$$ −2.17554 −0.152693
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 10.3893 0.723856
$$207$$ 0 0
$$208$$ 0.486761 0.0337508
$$209$$ −14.4161 −0.997181
$$210$$ 0 0
$$211$$ −25.2214 −1.73631 −0.868157 0.496289i $$-0.834696\pi$$
−0.868157 + 0.496289i $$0.834696\pi$$
$$212$$ −9.66367 −0.663704
$$213$$ 0 0
$$214$$ −7.15645 −0.489205
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 0 0
$$218$$ 6.65493 0.450728
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2.62981 0.176900
$$222$$ 0 0
$$223$$ 12.5266 0.838845 0.419423 0.907791i $$-0.362233\pi$$
0.419423 + 0.907791i $$0.362233\pi$$
$$224$$ −12.7420 −0.851364
$$225$$ 0 0
$$226$$ 8.70788 0.579240
$$227$$ −7.46165 −0.495247 −0.247624 0.968856i $$-0.579650\pi$$
−0.247624 + 0.968856i $$0.579650\pi$$
$$228$$ 0 0
$$229$$ 21.7936 1.44016 0.720082 0.693889i $$-0.244104\pi$$
0.720082 + 0.693889i $$0.244104\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −2.62981 −0.172656
$$233$$ −9.69480 −0.635127 −0.317564 0.948237i $$-0.602865\pi$$
−0.317564 + 0.948237i $$0.602865\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 9.05327 0.589317
$$237$$ 0 0
$$238$$ −7.02077 −0.455089
$$239$$ 10.7022 0.692266 0.346133 0.938185i $$-0.387495\pi$$
0.346133 + 0.938185i $$0.387495\pi$$
$$240$$ 0 0
$$241$$ 11.1682 0.719405 0.359702 0.933067i $$-0.382878\pi$$
0.359702 + 0.933067i $$0.382878\pi$$
$$242$$ −1.54573 −0.0993634
$$243$$ 0 0
$$244$$ 3.93502 0.251914
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3.02648 0.192570
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −14.3585 −0.906299 −0.453149 0.891435i $$-0.649700\pi$$
−0.453149 + 0.891435i $$0.649700\pi$$
$$252$$ 0 0
$$253$$ −6.25225 −0.393075
$$254$$ −12.9735 −0.814031
$$255$$ 0 0
$$256$$ −13.2345 −0.827157
$$257$$ −7.11056 −0.443545 −0.221772 0.975098i $$-0.571184\pi$$
−0.221772 + 0.975098i $$0.571184\pi$$
$$258$$ 0 0
$$259$$ 13.2362 0.822457
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −8.70788 −0.537975
$$263$$ 19.2214 1.18524 0.592622 0.805481i $$-0.298093\pi$$
0.592622 + 0.805481i $$0.298093\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −8.07974 −0.495401
$$267$$ 0 0
$$268$$ −15.5474 −0.949709
$$269$$ 9.88944 0.602970 0.301485 0.953471i $$-0.402518\pi$$
0.301485 + 0.953471i $$0.402518\pi$$
$$270$$ 0 0
$$271$$ −13.7936 −0.837904 −0.418952 0.908008i $$-0.637602\pi$$
−0.418952 + 0.908008i $$0.637602\pi$$
$$272$$ 3.22713 0.195674
$$273$$ 0 0
$$274$$ −14.5842 −0.881066
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 14.9661 0.899228 0.449614 0.893223i $$-0.351561\pi$$
0.449614 + 0.893223i $$0.351561\pi$$
$$278$$ 2.94103 0.176391
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −11.7287 −0.699673 −0.349836 0.936811i $$-0.613763\pi$$
−0.349836 + 0.936811i $$0.613763\pi$$
$$282$$ 0 0
$$283$$ 3.96180 0.235505 0.117752 0.993043i $$-0.462431\pi$$
0.117752 + 0.993043i $$0.462431\pi$$
$$284$$ 3.69914 0.219503
$$285$$ 0 0
$$286$$ −1.46028 −0.0863483
$$287$$ 1.79364 0.105875
$$288$$ 0 0
$$289$$ 0.435171 0.0255983
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −20.7404 −1.21374
$$293$$ −3.26701 −0.190861 −0.0954303 0.995436i $$-0.530423\pi$$
−0.0954303 + 0.995436i $$0.530423\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 16.0000 0.929981
$$297$$ 0 0
$$298$$ −3.44255 −0.199422
$$299$$ 1.31258 0.0759086
$$300$$ 0 0
$$301$$ −18.9735 −1.09362
$$302$$ 0.908538 0.0522805
$$303$$ 0 0
$$304$$ 3.71390 0.213007
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −20.4352 −1.16630 −0.583148 0.812366i $$-0.698179\pi$$
−0.583148 + 0.812366i $$0.698179\pi$$
$$308$$ −9.15477 −0.521641
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −11.0000 −0.623753 −0.311876 0.950123i $$-0.600957\pi$$
−0.311876 + 0.950123i $$0.600957\pi$$
$$312$$ 0 0
$$313$$ −27.6683 −1.56391 −0.781953 0.623337i $$-0.785776\pi$$
−0.781953 + 0.623337i $$0.785776\pi$$
$$314$$ 0.622433 0.0351259
$$315$$ 0 0
$$316$$ 16.9233 0.952010
$$317$$ −12.8777 −0.723285 −0.361642 0.932317i $$-0.617784\pi$$
−0.361642 + 0.932317i $$0.617784\pi$$
$$318$$ 0 0
$$319$$ −3.00000 −0.167968
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −3.50418 −0.195280
$$323$$ 20.0650 1.11645
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0.858314 0.0475376
$$327$$ 0 0
$$328$$ 2.16816 0.119717
$$329$$ 19.5384 1.07718
$$330$$ 0 0
$$331$$ 24.6874 1.35694 0.678472 0.734627i $$-0.262643\pi$$
0.678472 + 0.734627i $$0.262643\pi$$
$$332$$ 10.7022 0.587358
$$333$$ 0 0
$$334$$ −6.72561 −0.368009
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 21.0915 1.14893 0.574463 0.818531i $$-0.305211\pi$$
0.574463 + 0.818531i $$0.305211\pi$$
$$338$$ −9.74068 −0.529823
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 20.1608 1.08858
$$344$$ −22.9353 −1.23659
$$345$$ 0 0
$$346$$ −5.94704 −0.319715
$$347$$ −18.8245 −1.01055 −0.505275 0.862958i $$-0.668609\pi$$
−0.505275 + 0.862958i $$0.668609\pi$$
$$348$$ 0 0
$$349$$ 17.4884 0.936135 0.468067 0.883693i $$-0.344950\pi$$
0.468067 + 0.883693i $$0.344950\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −17.5708 −0.936529
$$353$$ −8.38928 −0.446517 −0.223258 0.974759i $$-0.571669\pi$$
−0.223258 + 0.974759i $$0.571669\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 25.0931 1.32993
$$357$$ 0 0
$$358$$ 4.68742 0.247738
$$359$$ 27.6948 1.46168 0.730838 0.682551i $$-0.239130\pi$$
0.730838 + 0.682551i $$0.239130\pi$$
$$360$$ 0 0
$$361$$ 4.09146 0.215340
$$362$$ 3.16215 0.166199
$$363$$ 0 0
$$364$$ 1.92193 0.100737
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −9.00738 −0.470181 −0.235091 0.971973i $$-0.575539\pi$$
−0.235091 + 0.971973i $$0.575539\pi$$
$$368$$ 1.61072 0.0839643
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −14.9883 −0.778153
$$372$$ 0 0
$$373$$ 25.0533 1.29721 0.648604 0.761126i $$-0.275353\pi$$
0.648604 + 0.761126i $$0.275353\pi$$
$$374$$ −9.68140 −0.500613
$$375$$ 0 0
$$376$$ 23.6181 1.21801
$$377$$ 0.629813 0.0324370
$$378$$ 0 0
$$379$$ −27.5725 −1.41631 −0.708153 0.706059i $$-0.750471\pi$$
−0.708153 + 0.706059i $$0.750471\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0.206360 0.0105583
$$383$$ −13.0724 −0.667967 −0.333983 0.942579i $$-0.608393\pi$$
−0.333983 + 0.942579i $$0.608393\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −8.27134 −0.421000
$$387$$ 0 0
$$388$$ 0.755134 0.0383361
$$389$$ 23.0797 1.17019 0.585095 0.810965i $$-0.301057\pi$$
0.585095 + 0.810965i $$0.301057\pi$$
$$390$$ 0 0
$$391$$ 8.70218 0.440088
$$392$$ 5.96180 0.301117
$$393$$ 0 0
$$394$$ 6.24791 0.314765
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 12.3129 0.617966 0.308983 0.951067i $$-0.400011\pi$$
0.308983 + 0.951067i $$0.400011\pi$$
$$398$$ −12.2154 −0.612304
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 36.4308 1.81927 0.909634 0.415410i $$-0.136362\pi$$
0.909634 + 0.415410i $$0.136362\pi$$
$$402$$ 0 0
$$403$$ 0 0
$$404$$ 19.3645 0.963419
$$405$$ 0 0
$$406$$ −1.68140 −0.0834466
$$407$$ 18.2522 0.904730
$$408$$ 0 0
$$409$$ −22.5842 −1.11672 −0.558359 0.829599i $$-0.688569\pi$$
−0.558359 + 0.829599i $$0.688569\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −18.8556 −0.928948
$$413$$ 14.0415 0.690939
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 3.68878 0.180857
$$417$$ 0 0
$$418$$ −11.1417 −0.544958
$$419$$ 2.90854 0.142091 0.0710457 0.997473i $$-0.477366\pi$$
0.0710457 + 0.997473i $$0.477366\pi$$
$$420$$ 0 0
$$421$$ 27.1183 1.32166 0.660831 0.750534i $$-0.270204\pi$$
0.660831 + 0.750534i $$0.270204\pi$$
$$422$$ −19.4928 −0.948893
$$423$$ 0 0
$$424$$ −18.1179 −0.879885
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 6.10318 0.295354
$$428$$ 12.9883 0.627812
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −19.5960 −0.943904 −0.471952 0.881624i $$-0.656450\pi$$
−0.471952 + 0.881624i $$0.656450\pi$$
$$432$$ 0 0
$$433$$ −24.1638 −1.16124 −0.580620 0.814175i $$-0.697190\pi$$
−0.580620 + 0.814175i $$0.697190\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −12.0781 −0.578435
$$437$$ 10.0148 0.479071
$$438$$ 0 0
$$439$$ −30.4235 −1.45203 −0.726016 0.687678i $$-0.758630\pi$$
−0.726016 + 0.687678i $$0.758630\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 2.03249 0.0966758
$$443$$ −32.3853 −1.53867 −0.769335 0.638846i $$-0.779412\pi$$
−0.769335 + 0.638846i $$0.779412\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 9.68140 0.458428
$$447$$ 0 0
$$448$$ −6.48508 −0.306391
$$449$$ 5.62243 0.265339 0.132670 0.991160i $$-0.457645\pi$$
0.132670 + 0.991160i $$0.457645\pi$$
$$450$$ 0 0
$$451$$ 2.47337 0.116466
$$452$$ −15.8040 −0.743357
$$453$$ 0 0
$$454$$ −5.76685 −0.270652
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −3.33199 −0.155864 −0.0779320 0.996959i $$-0.524832\pi$$
−0.0779320 + 0.996959i $$0.524832\pi$$
$$458$$ 16.8436 0.787048
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −23.3437 −1.08722 −0.543612 0.839336i $$-0.682944\pi$$
−0.543612 + 0.839336i $$0.682944\pi$$
$$462$$ 0 0
$$463$$ −4.13735 −0.192279 −0.0961394 0.995368i $$-0.530649\pi$$
−0.0961394 + 0.995368i $$0.530649\pi$$
$$464$$ 0.772866 0.0358794
$$465$$ 0 0
$$466$$ −7.49278 −0.347096
$$467$$ 35.7554 1.65456 0.827282 0.561786i $$-0.189886\pi$$
0.827282 + 0.561786i $$0.189886\pi$$
$$468$$ 0 0
$$469$$ −24.1139 −1.11348
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 16.9735 0.781270
$$473$$ −26.1638 −1.20301
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 12.7420 0.584031
$$477$$ 0 0
$$478$$ 8.27134 0.378322
$$479$$ 29.6107 1.35295 0.676474 0.736466i $$-0.263507\pi$$
0.676474 + 0.736466i $$0.263507\pi$$
$$480$$ 0 0
$$481$$ −3.83184 −0.174717
$$482$$ 8.63149 0.393154
$$483$$ 0 0
$$484$$ 2.80536 0.127516
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4.33633 0.196498 0.0982489 0.995162i $$-0.468676\pi$$
0.0982489 + 0.995162i $$0.468676\pi$$
$$488$$ 7.37757 0.333967
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −22.5193 −1.01628 −0.508140 0.861275i $$-0.669667\pi$$
−0.508140 + 0.861275i $$0.669667\pi$$
$$492$$ 0 0
$$493$$ 4.17554 0.188057
$$494$$ 2.33906 0.105239
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 5.73733 0.257354
$$498$$ 0 0
$$499$$ −41.1638 −1.84275 −0.921373 0.388680i $$-0.872931\pi$$
−0.921373 + 0.388680i $$0.872931\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −11.0972 −0.495291
$$503$$ 31.5149 1.40518 0.702590 0.711595i $$-0.252027\pi$$
0.702590 + 0.711595i $$0.252027\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −4.83215 −0.214815
$$507$$ 0 0
$$508$$ 23.5457 1.04467
$$509$$ −22.6224 −1.00272 −0.501361 0.865238i $$-0.667167\pi$$
−0.501361 + 0.865238i $$0.667167\pi$$
$$510$$ 0 0
$$511$$ −32.1682 −1.42304
$$512$$ 8.59162 0.379699
$$513$$ 0 0
$$514$$ −5.49551 −0.242396
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 26.9427 1.18494
$$518$$ 10.2298 0.449471
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 2.06498 0.0904686 0.0452343 0.998976i $$-0.485597\pi$$
0.0452343 + 0.998976i $$0.485597\pi$$
$$522$$ 0 0
$$523$$ −24.5266 −1.07247 −0.536237 0.844067i $$-0.680155\pi$$
−0.536237 + 0.844067i $$0.680155\pi$$
$$524$$ 15.8040 0.690401
$$525$$ 0 0
$$526$$ 14.8556 0.647734
$$527$$ 0 0
$$528$$ 0 0
$$529$$ −18.6566 −0.811157
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 14.6640 0.635764
$$533$$ −0.519253 −0.0224913
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −29.1491 −1.25905
$$537$$ 0 0
$$538$$ 7.64321 0.329522
$$539$$ 6.80102 0.292941
$$540$$ 0 0
$$541$$ 0.389285 0.0167367 0.00836833 0.999965i $$-0.497336\pi$$
0.00836833 + 0.999965i $$0.497336\pi$$
$$542$$ −10.6606 −0.457913
$$543$$ 0 0
$$544$$ 24.4559 1.04854
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −21.4352 −0.916502 −0.458251 0.888823i $$-0.651524\pi$$
−0.458251 + 0.888823i $$0.651524\pi$$
$$548$$ 26.4690 1.13070
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 4.80536 0.204715
$$552$$ 0 0
$$553$$ 26.2479 1.11617
$$554$$ 11.5668 0.491427
$$555$$ 0 0
$$556$$ −5.33769 −0.226369
$$557$$ −19.7598 −0.837249 −0.418624 0.908159i $$-0.637488\pi$$
−0.418624 + 0.908159i $$0.637488\pi$$
$$558$$ 0 0
$$559$$ 5.49278 0.232320
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −9.06467 −0.382370
$$563$$ 6.46165 0.272326 0.136163 0.990686i $$-0.456523\pi$$
0.136163 + 0.990686i $$0.456523\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 3.06194 0.128703
$$567$$ 0 0
$$568$$ 6.93533 0.291000
$$569$$ 14.9427 0.626431 0.313215 0.949682i $$-0.398594\pi$$
0.313215 + 0.949682i $$0.398594\pi$$
$$570$$ 0 0
$$571$$ 35.1521 1.47107 0.735535 0.677487i $$-0.236931\pi$$
0.735535 + 0.677487i $$0.236931\pi$$
$$572$$ 2.65028 0.110814
$$573$$ 0 0
$$574$$ 1.38624 0.0578606
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 0.740373 0.0308222 0.0154111 0.999881i $$-0.495094\pi$$
0.0154111 + 0.999881i $$0.495094\pi$$
$$578$$ 0.336329 0.0139894
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 16.5990 0.688642
$$582$$ 0 0
$$583$$ −20.6683 −0.855994
$$584$$ −38.8851 −1.60908
$$585$$ 0 0
$$586$$ −2.52496 −0.104305
$$587$$ −18.5842 −0.767054 −0.383527 0.923530i $$-0.625291\pi$$
−0.383527 + 0.923530i $$0.625291\pi$$
$$588$$ 0 0
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −4.70218 −0.193258
$$593$$ 24.0268 0.986662 0.493331 0.869842i $$-0.335779\pi$$
0.493331 + 0.869842i $$0.335779\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.24791 0.255924
$$597$$ 0 0
$$598$$ 1.01445 0.0414839
$$599$$ −22.3585 −0.913542 −0.456771 0.889584i $$-0.650994\pi$$
−0.456771 + 0.889584i $$0.650994\pi$$
$$600$$ 0 0
$$601$$ −20.3511 −0.830138 −0.415069 0.909790i $$-0.636243\pi$$
−0.415069 + 0.909790i $$0.636243\pi$$
$$602$$ −14.6640 −0.597659
$$603$$ 0 0
$$604$$ −1.64891 −0.0670932
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −19.4928 −0.791187 −0.395594 0.918426i $$-0.629461\pi$$
−0.395594 + 0.918426i $$0.629461\pi$$
$$608$$ 28.1447 1.14142
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −5.65629 −0.228829
$$612$$ 0 0
$$613$$ −39.1638 −1.58181 −0.790906 0.611938i $$-0.790390\pi$$
−0.790906 + 0.611938i $$0.790390\pi$$
$$614$$ −15.7936 −0.637379
$$615$$ 0 0
$$616$$ −17.1638 −0.691550
$$617$$ 26.8321 1.08022 0.540111 0.841594i $$-0.318382\pi$$
0.540111 + 0.841594i $$0.318382\pi$$
$$618$$ 0 0
$$619$$ −33.4278 −1.34358 −0.671788 0.740743i $$-0.734473\pi$$
−0.671788 + 0.740743i $$0.734473\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −8.50152 −0.340880
$$623$$ 38.9193 1.55927
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −21.3839 −0.854672
$$627$$ 0 0
$$628$$ −1.12966 −0.0450783
$$629$$ −25.4044 −1.01294
$$630$$ 0 0
$$631$$ −32.3705 −1.28865 −0.644325 0.764752i $$-0.722861\pi$$
−0.644325 + 0.764752i $$0.722861\pi$$
$$632$$ 31.7287 1.26210
$$633$$ 0 0
$$634$$ −9.95275 −0.395274
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −1.42779 −0.0565711
$$638$$ −2.31860 −0.0917941
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −33.0415 −1.30506 −0.652531 0.757762i $$-0.726293\pi$$
−0.652531 + 0.757762i $$0.726293\pi$$
$$642$$ 0 0
$$643$$ −27.4734 −1.08344 −0.541722 0.840558i $$-0.682227\pi$$
−0.541722 + 0.840558i $$0.682227\pi$$
$$644$$ 6.35976 0.250610
$$645$$ 0 0
$$646$$ 15.5075 0.610136
$$647$$ −23.2023 −0.912178 −0.456089 0.889934i $$-0.650750\pi$$
−0.456089 + 0.889934i $$0.650750\pi$$
$$648$$ 0 0
$$649$$ 19.3628 0.760057
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −1.55776 −0.0610066
$$653$$ −28.9159 −1.13157 −0.565784 0.824554i $$-0.691426\pi$$
−0.565784 + 0.824554i $$0.691426\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −0.637193 −0.0248782
$$657$$ 0 0
$$658$$ 15.1005 0.588680
$$659$$ −36.1832 −1.40950 −0.704749 0.709456i $$-0.748941\pi$$
−0.704749 + 0.709456i $$0.748941\pi$$
$$660$$ 0 0
$$661$$ 45.0918 1.75387 0.876933 0.480612i $$-0.159585\pi$$
0.876933 + 0.480612i $$0.159585\pi$$
$$662$$ 19.0801 0.741567
$$663$$ 0 0
$$664$$ 20.0650 0.778672
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2.08408 0.0806960
$$668$$ 12.2064 0.472278
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 8.41607 0.324899
$$672$$ 0 0
$$673$$ 23.1491 0.892331 0.446165 0.894950i $$-0.352789\pi$$
0.446165 + 0.894950i $$0.352789\pi$$
$$674$$ 16.3009 0.627886
$$675$$ 0 0
$$676$$ 17.6784 0.679940
$$677$$ 38.1521 1.46630 0.733152 0.680064i $$-0.238048\pi$$
0.733152 + 0.680064i $$0.238048\pi$$
$$678$$ 0 0
$$679$$ 1.17121 0.0449468
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 0 0
$$683$$ −0.228811 −0.00875520 −0.00437760 0.999990i $$-0.501393\pi$$
−0.00437760 + 0.999990i $$0.501393\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 15.5816 0.594907
$$687$$ 0 0
$$688$$ 6.74037 0.256974
$$689$$ 4.33906 0.165305
$$690$$ 0 0
$$691$$ 16.2258 0.617257 0.308629 0.951183i $$-0.400130\pi$$
0.308629 + 0.951183i $$0.400130\pi$$
$$692$$ 10.7933 0.410301
$$693$$ 0 0
$$694$$ −14.5488 −0.552264
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −3.44255 −0.130396
$$698$$ 13.5162 0.511596
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −9.69914 −0.366331 −0.183166 0.983082i $$-0.558634\pi$$
−0.183166 + 0.983082i $$0.558634\pi$$
$$702$$ 0 0
$$703$$ −29.2362 −1.10266
$$704$$ −8.94271 −0.337041
$$705$$ 0 0
$$706$$ −6.48379 −0.244021
$$707$$ 30.0342 1.12955
$$708$$ 0 0
$$709$$ 49.4884 1.85858 0.929289 0.369354i $$-0.120421\pi$$
0.929289 + 0.369354i $$0.120421\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 47.0459 1.76312
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −8.50722 −0.317930
$$717$$ 0 0
$$718$$ 21.4044 0.798803
$$719$$ −15.7139 −0.586029 −0.293015 0.956108i $$-0.594658\pi$$
−0.293015 + 0.956108i $$0.594658\pi$$
$$720$$ 0 0
$$721$$ −29.2449 −1.08914
$$722$$ 3.16215 0.117683
$$723$$ 0 0
$$724$$ −5.73901 −0.213289
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −6.51925 −0.241786 −0.120893 0.992666i $$-0.538576\pi$$
−0.120893 + 0.992666i $$0.538576\pi$$
$$728$$ 3.60334 0.133548
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 36.4161 1.34690
$$732$$ 0 0
$$733$$ 6.76716 0.249951 0.124975 0.992160i $$-0.460115\pi$$
0.124975 + 0.992160i $$0.460115\pi$$
$$734$$ −6.96149 −0.256953
$$735$$ 0 0
$$736$$ 12.2064 0.449932
$$737$$ −33.2522 −1.22486
$$738$$ 0 0
$$739$$ −41.1183 −1.51256 −0.756280 0.654248i $$-0.772985\pi$$
−0.756280 + 0.654248i $$0.772985\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −11.5839 −0.425259
$$743$$ 36.5534 1.34101 0.670507 0.741903i $$-0.266076\pi$$
0.670507 + 0.741903i $$0.266076\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 19.3628 0.708923
$$747$$ 0 0
$$748$$ 17.5708 0.642454
$$749$$ 20.1447 0.736072
$$750$$ 0 0
$$751$$ −45.1980 −1.64930 −0.824649 0.565645i $$-0.808627\pi$$
−0.824649 + 0.565645i $$0.808627\pi$$
$$752$$ −6.94103 −0.253113
$$753$$ 0 0
$$754$$ 0.486761 0.0177268
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −53.4737 −1.94353 −0.971767 0.235943i $$-0.924182\pi$$
−0.971767 + 0.235943i $$0.924182\pi$$
$$758$$ −21.3099 −0.774009
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 49.4693 1.79326 0.896631 0.442778i $$-0.146007\pi$$
0.896631 + 0.442778i $$0.146007\pi$$
$$762$$ 0 0
$$763$$ −18.7330 −0.678180
$$764$$ −0.374525 −0.0135498
$$765$$ 0 0
$$766$$ −10.1032 −0.365043
$$767$$ −4.06498 −0.146778
$$768$$ 0 0
$$769$$ −14.2331 −0.513260 −0.256630 0.966510i $$-0.582612\pi$$
−0.256630 + 0.966510i $$0.582612\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 15.0117 0.540284
$$773$$ 21.7936 0.783863 0.391931 0.919994i $$-0.371807\pi$$
0.391931 + 0.919994i $$0.371807\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 1.41576 0.0508229
$$777$$ 0 0
$$778$$ 17.8375 0.639507
$$779$$ −3.96180 −0.141946
$$780$$ 0 0
$$781$$ 7.91158 0.283099
$$782$$ 6.72561 0.240507
$$783$$ 0 0
$$784$$ −1.75209 −0.0625747
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 14.1829 0.505567 0.252783 0.967523i $$-0.418654\pi$$
0.252783 + 0.967523i $$0.418654\pi$$
$$788$$ −11.3394 −0.403948
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −24.5119 −0.871542
$$792$$ 0 0
$$793$$ −1.76685 −0.0627427
$$794$$ 9.51621 0.337718
$$795$$ 0 0
$$796$$ 22.1698 0.785789
$$797$$ −2.00000 −0.0708436 −0.0354218 0.999372i $$-0.511277\pi$$
−0.0354218 + 0.999372i $$0.511277\pi$$
$$798$$ 0 0
$$799$$ −37.5002 −1.32666
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 28.1561 0.994228
$$803$$ −44.3588 −1.56539
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 36.3055 1.27722
$$809$$ −30.6757 −1.07850 −0.539250 0.842146i $$-0.681292\pi$$
−0.539250 + 0.842146i $$0.681292\pi$$
$$810$$ 0 0
$$811$$ 36.6609 1.28734 0.643670 0.765303i $$-0.277411\pi$$
0.643670 + 0.765303i $$0.277411\pi$$
$$812$$ 3.05159 0.107090
$$813$$ 0 0
$$814$$ 14.1065 0.494434
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 41.9088 1.46620
$$818$$ −17.4546 −0.610285
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 36.1447 1.26146 0.630730 0.776002i $$-0.282756\pi$$
0.630730 + 0.776002i $$0.282756\pi$$
$$822$$ 0 0
$$823$$ 10.4395 0.363898 0.181949 0.983308i $$-0.441759\pi$$
0.181949 + 0.983308i $$0.441759\pi$$
$$824$$ −35.3514 −1.23152
$$825$$ 0 0
$$826$$ 10.8522 0.377597
$$827$$ −34.1447 −1.18733 −0.593664 0.804713i $$-0.702319\pi$$
−0.593664 + 0.804713i $$0.702319\pi$$
$$828$$ 0 0
$$829$$ 42.6874 1.48260 0.741298 0.671176i $$-0.234211\pi$$
0.741298 + 0.671176i $$0.234211\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.87741 0.0650875
$$833$$ −9.46599 −0.327977
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 20.2211 0.699362
$$837$$ 0 0
$$838$$ 2.24791 0.0776527
$$839$$ −13.9926 −0.483079 −0.241539 0.970391i $$-0.577652\pi$$
−0.241539 + 0.970391i $$0.577652\pi$$
$$840$$ 0 0
$$841$$ 1.00000 0.0344828
$$842$$ 20.9588 0.722287
$$843$$ 0 0
$$844$$ 35.3776 1.21775
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4.35109 0.149505
$$848$$ 5.32461 0.182848
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −12.6797 −0.434655
$$852$$ 0 0
$$853$$ −34.6978 −1.18803 −0.594016 0.804453i $$-0.702458\pi$$
−0.594016 + 0.804453i $$0.702458\pi$$
$$854$$ 4.71694 0.161410
$$855$$ 0 0
$$856$$ 24.3511 0.832303
$$857$$ 48.0077 1.63991 0.819956 0.572427i $$-0.193998\pi$$
0.819956 + 0.572427i $$0.193998\pi$$
$$858$$ 0 0
$$859$$ 16.2627 0.554875 0.277438 0.960744i $$-0.410515\pi$$
0.277438 + 0.960744i $$0.410515\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −15.1450 −0.515842
$$863$$ 13.5605 0.461604 0.230802 0.973001i $$-0.425865\pi$$
0.230802 + 0.973001i $$0.425865\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −18.6754 −0.634616
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 36.1950 1.22783
$$870$$ 0 0
$$871$$ 6.98090 0.236539
$$872$$ −22.6446 −0.766842
$$873$$ 0 0
$$874$$ 7.74006 0.261812
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 8.01476 0.270639 0.135320 0.990802i $$-0.456794\pi$$
0.135320 + 0.990802i $$0.456794\pi$$
$$878$$ −23.5132 −0.793533
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 44.1564 1.48767 0.743834 0.668364i $$-0.233005\pi$$
0.743834 + 0.668364i $$0.233005\pi$$
$$882$$ 0 0
$$883$$ 20.7022 0.696684 0.348342 0.937368i $$-0.386745\pi$$
0.348342 + 0.937368i $$0.386745\pi$$
$$884$$ −3.68878 −0.124067
$$885$$ 0 0
$$886$$ −25.0294 −0.840881
$$887$$ 48.5387 1.62977 0.814884 0.579624i $$-0.196800\pi$$
0.814884 + 0.579624i $$0.196800\pi$$
$$888$$ 0 0
$$889$$ 36.5193 1.22482
$$890$$ 0 0
$$891$$ 0 0
$$892$$ −17.5708 −0.588315
$$893$$ −43.1564 −1.44418
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 20.4720 0.683922
$$897$$ 0 0
$$898$$ 4.34538 0.145007
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 28.7672 0.958373
$$902$$ 1.91158 0.0636487
$$903$$ 0 0
$$904$$ −29.6301 −0.985483
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 26.5149 0.880413 0.440207 0.897896i $$-0.354905\pi$$
0.440207 + 0.897896i $$0.354905\pi$$
$$908$$ 10.4663 0.347336
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −53.8469 −1.78403 −0.892014 0.452008i $$-0.850708\pi$$
−0.892014 + 0.452008i $$0.850708\pi$$
$$912$$ 0 0
$$913$$ 22.8894 0.757530
$$914$$ −2.57518 −0.0851794
$$915$$ 0 0
$$916$$ −30.5695 −1.01004
$$917$$ 24.5119 0.809453
$$918$$ 0 0
$$919$$ −11.4084 −0.376328 −0.188164 0.982138i $$-0.560254\pi$$
−0.188164 + 0.982138i $$0.560254\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −18.0415 −0.594167
$$923$$ −1.66094 −0.0546705
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −3.19761 −0.105080
$$927$$ 0 0
$$928$$ 5.85695 0.192264
$$929$$ −13.7554 −0.451301 −0.225651 0.974208i $$-0.572451\pi$$
−0.225651 + 0.974208i $$0.572451\pi$$
$$930$$ 0 0
$$931$$ −10.8938 −0.357029
$$932$$ 13.5987 0.445440
$$933$$ 0 0
$$934$$ 27.6342 0.904217
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −1.29379 −0.0422664 −0.0211332 0.999777i $$-0.506727\pi$$
−0.0211332 + 0.999777i $$0.506727\pi$$
$$938$$ −18.6368 −0.608514
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 49.8586 1.62534 0.812672 0.582721i $$-0.198012\pi$$
0.812672 + 0.582721i $$0.198012\pi$$
$$942$$ 0 0
$$943$$ −1.71823 −0.0559534
$$944$$ −4.98828 −0.162355
$$945$$ 0 0
$$946$$ −20.2211 −0.657445
$$947$$ −21.1109 −0.686011 −0.343006 0.939333i $$-0.611445\pi$$
−0.343006 + 0.939333i $$0.611445\pi$$
$$948$$ 0 0
$$949$$ 9.31258 0.302299
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 23.8894 0.774261
$$953$$ −42.7672 −1.38536 −0.692682 0.721243i $$-0.743571\pi$$
−0.692682 + 0.721243i $$0.743571\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −15.0117 −0.485514
$$957$$ 0 0
$$958$$ 22.8851 0.739384
$$959$$ 41.0533 1.32568
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ −2.96149 −0.0954824
$$963$$ 0 0
$$964$$ −15.6653 −0.504547
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 33.0342 1.06231 0.531154 0.847276i $$-0.321759\pi$$
0.531154 + 0.847276i $$0.321759\pi$$
$$968$$ 5.25963 0.169051
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 26.0841 0.837078 0.418539 0.908199i $$-0.362542\pi$$
0.418539 + 0.908199i $$0.362542\pi$$
$$972$$ 0 0
$$973$$ −8.27872 −0.265404
$$974$$ 3.35140 0.107386
$$975$$ 0 0
$$976$$ −2.16816 −0.0694012
$$977$$ −43.6948 −1.39792 −0.698960 0.715161i $$-0.746354\pi$$
−0.698960 + 0.715161i $$0.746354\pi$$
$$978$$ 0 0
$$979$$ 53.6683 1.71525
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −17.4044 −0.555395
$$983$$ 2.55745 0.0815700 0.0407850 0.999168i $$-0.487014\pi$$
0.0407850 + 0.999168i $$0.487014\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 3.22713 0.102773
$$987$$ 0 0
$$988$$ −4.24518 −0.135057
$$989$$ 18.1759 0.577959
$$990$$ 0 0
$$991$$ 14.7287 0.467871 0.233936 0.972252i $$-0.424840\pi$$
0.233936 + 0.972252i $$0.424840\pi$$
$$992$$ 0 0
$$993$$ 0 0
$$994$$ 4.43419 0.140644
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −33.0992 −1.04826 −0.524130 0.851638i $$-0.675610\pi$$
−0.524130 + 0.851638i $$0.675610\pi$$
$$998$$ −31.8141 −1.00706
$$999$$ 0 0
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 6525.2.a.bf.1.2 3
3.2 odd 2 2175.2.a.u.1.2 3
5.4 even 2 1305.2.a.q.1.2 3
15.2 even 4 2175.2.c.m.349.3 6
15.8 even 4 2175.2.c.m.349.4 6
15.14 odd 2 435.2.a.i.1.2 3
60.59 even 2 6960.2.a.cl.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.i.1.2 3 15.14 odd 2
1305.2.a.q.1.2 3 5.4 even 2
2175.2.a.u.1.2 3 3.2 odd 2
2175.2.c.m.349.3 6 15.2 even 4
2175.2.c.m.349.4 6 15.8 even 4
6525.2.a.bf.1.2 3 1.1 even 1 trivial
6960.2.a.cl.1.1 3 60.59 even 2