Properties

 Label 7600.2.a.bw.1.3 Level $7600$ Weight $2$ Character 7600.1 Self dual yes Analytic conductor $60.686$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7600,2,Mod(1,7600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7600, 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("7600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7600 = 2^{4} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7600.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: $$60.6863055362$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 2x + 1$$ x^3 - x^2 - 2*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 475) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$-1.24698$$ of defining polynomial Character $$\chi$$ $$=$$ 7600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.24698 q^{3} +1.35690 q^{7} +2.04892 q^{9} +O(q^{10})$$ $$q+2.24698 q^{3} +1.35690 q^{7} +2.04892 q^{9} -4.85086 q^{11} -0.198062 q^{13} +1.13706 q^{17} +1.00000 q^{19} +3.04892 q^{21} +2.55496 q^{23} -2.13706 q^{27} -10.2349 q^{29} -2.51573 q^{31} -10.8998 q^{33} +0.137063 q^{37} -0.445042 q^{39} -11.7506 q^{41} -7.59179 q^{43} +2.69202 q^{47} -5.15883 q^{49} +2.55496 q^{51} -12.8780 q^{53} +2.24698 q^{57} -5.82371 q^{59} -7.58211 q^{61} +2.78017 q^{63} +8.01507 q^{67} +5.74094 q^{69} +8.82371 q^{71} +11.9705 q^{73} -6.58211 q^{77} -10.7409 q^{79} -10.9487 q^{81} +3.77479 q^{83} -22.9976 q^{87} +9.36658 q^{89} -0.268750 q^{91} -5.65279 q^{93} -0.198062 q^{97} -9.93900 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 3 q^{9}+O(q^{10})$$ 3 * q + 2 * q^3 - 3 * q^9 $$3 q + 2 q^{3} - 3 q^{9} - q^{11} - 5 q^{13} - 2 q^{17} + 3 q^{19} + 8 q^{23} - q^{27} - 7 q^{29} + 5 q^{31} - 10 q^{33} - 5 q^{37} - q^{39} + q^{41} + 5 q^{43} + 3 q^{47} - 7 q^{49} + 8 q^{51} - 19 q^{53} + 2 q^{57} - 10 q^{59} - 17 q^{61} + 7 q^{63} - q^{67} + 3 q^{69} + 19 q^{71} + q^{73} - 14 q^{77} - 18 q^{79} - q^{81} + 13 q^{83} - 28 q^{87} + 2 q^{89} + 7 q^{91} + q^{93} - 5 q^{97} - 20 q^{99}+O(q^{100})$$ 3 * q + 2 * q^3 - 3 * q^9 - q^11 - 5 * q^13 - 2 * q^17 + 3 * q^19 + 8 * q^23 - q^27 - 7 * q^29 + 5 * q^31 - 10 * q^33 - 5 * q^37 - q^39 + q^41 + 5 * q^43 + 3 * q^47 - 7 * q^49 + 8 * q^51 - 19 * q^53 + 2 * q^57 - 10 * q^59 - 17 * q^61 + 7 * q^63 - q^67 + 3 * q^69 + 19 * q^71 + q^73 - 14 * q^77 - 18 * q^79 - q^81 + 13 * q^83 - 28 * q^87 + 2 * q^89 + 7 * q^91 + q^93 - 5 * q^97 - 20 * 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
$$3$$ 2.24698 1.29729 0.648647 0.761089i $$-0.275335\pi$$
0.648647 + 0.761089i $$0.275335\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.35690 0.512858 0.256429 0.966563i $$-0.417454\pi$$
0.256429 + 0.966563i $$0.417454\pi$$
$$8$$ 0 0
$$9$$ 2.04892 0.682972
$$10$$ 0 0
$$11$$ −4.85086 −1.46259 −0.731294 0.682062i $$-0.761083\pi$$
−0.731294 + 0.682062i $$0.761083\pi$$
$$12$$ 0 0
$$13$$ −0.198062 −0.0549326 −0.0274663 0.999623i $$-0.508744\pi$$
−0.0274663 + 0.999623i $$0.508744\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 1.13706 0.275778 0.137889 0.990448i $$-0.455968\pi$$
0.137889 + 0.990448i $$0.455968\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 3.04892 0.665328
$$22$$ 0 0
$$23$$ 2.55496 0.532746 0.266373 0.963870i $$-0.414175\pi$$
0.266373 + 0.963870i $$0.414175\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −2.13706 −0.411278
$$28$$ 0 0
$$29$$ −10.2349 −1.90057 −0.950286 0.311377i $$-0.899210\pi$$
−0.950286 + 0.311377i $$0.899210\pi$$
$$30$$ 0 0
$$31$$ −2.51573 −0.451838 −0.225919 0.974146i $$-0.572539\pi$$
−0.225919 + 0.974146i $$0.572539\pi$$
$$32$$ 0 0
$$33$$ −10.8998 −1.89741
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0.137063 0.0225331 0.0112665 0.999937i $$-0.496414\pi$$
0.0112665 + 0.999937i $$0.496414\pi$$
$$38$$ 0 0
$$39$$ −0.445042 −0.0712637
$$40$$ 0 0
$$41$$ −11.7506 −1.83514 −0.917570 0.397575i $$-0.869852\pi$$
−0.917570 + 0.397575i $$0.869852\pi$$
$$42$$ 0 0
$$43$$ −7.59179 −1.15774 −0.578869 0.815421i $$-0.696506\pi$$
−0.578869 + 0.815421i $$0.696506\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 2.69202 0.392672 0.196336 0.980537i $$-0.437096\pi$$
0.196336 + 0.980537i $$0.437096\pi$$
$$48$$ 0 0
$$49$$ −5.15883 −0.736976
$$50$$ 0 0
$$51$$ 2.55496 0.357766
$$52$$ 0 0
$$53$$ −12.8780 −1.76893 −0.884465 0.466607i $$-0.845476\pi$$
−0.884465 + 0.466607i $$0.845476\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 2.24698 0.297620
$$58$$ 0 0
$$59$$ −5.82371 −0.758182 −0.379091 0.925359i $$-0.623763\pi$$
−0.379091 + 0.925359i $$0.623763\pi$$
$$60$$ 0 0
$$61$$ −7.58211 −0.970789 −0.485395 0.874295i $$-0.661324\pi$$
−0.485395 + 0.874295i $$0.661324\pi$$
$$62$$ 0 0
$$63$$ 2.78017 0.350268
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.01507 0.979196 0.489598 0.871948i $$-0.337144\pi$$
0.489598 + 0.871948i $$0.337144\pi$$
$$68$$ 0 0
$$69$$ 5.74094 0.691128
$$70$$ 0 0
$$71$$ 8.82371 1.04718 0.523591 0.851970i $$-0.324592\pi$$
0.523591 + 0.851970i $$0.324592\pi$$
$$72$$ 0 0
$$73$$ 11.9705 1.40104 0.700518 0.713635i $$-0.252952\pi$$
0.700518 + 0.713635i $$0.252952\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ −6.58211 −0.750101
$$78$$ 0 0
$$79$$ −10.7409 −1.20845 −0.604225 0.796814i $$-0.706517\pi$$
−0.604225 + 0.796814i $$0.706517\pi$$
$$80$$ 0 0
$$81$$ −10.9487 −1.21652
$$82$$ 0 0
$$83$$ 3.77479 0.414337 0.207169 0.978305i $$-0.433575\pi$$
0.207169 + 0.978305i $$0.433575\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −22.9976 −2.46560
$$88$$ 0 0
$$89$$ 9.36658 0.992856 0.496428 0.868078i $$-0.334645\pi$$
0.496428 + 0.868078i $$0.334645\pi$$
$$90$$ 0 0
$$91$$ −0.268750 −0.0281726
$$92$$ 0 0
$$93$$ −5.65279 −0.586167
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −0.198062 −0.0201102 −0.0100551 0.999949i $$-0.503201\pi$$
−0.0100551 + 0.999949i $$0.503201\pi$$
$$98$$ 0 0
$$99$$ −9.93900 −0.998907
$$100$$ 0 0
$$101$$ 11.5090 1.14519 0.572595 0.819838i $$-0.305937\pi$$
0.572595 + 0.819838i $$0.305937\pi$$
$$102$$ 0 0
$$103$$ 15.1564 1.49341 0.746704 0.665156i $$-0.231635\pi$$
0.746704 + 0.665156i $$0.231635\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 2.65279 0.256455 0.128228 0.991745i $$-0.459071\pi$$
0.128228 + 0.991745i $$0.459071\pi$$
$$108$$ 0 0
$$109$$ −2.49934 −0.239393 −0.119696 0.992811i $$-0.538192\pi$$
−0.119696 + 0.992811i $$0.538192\pi$$
$$110$$ 0 0
$$111$$ 0.307979 0.0292320
$$112$$ 0 0
$$113$$ −8.52781 −0.802229 −0.401114 0.916028i $$-0.631377\pi$$
−0.401114 + 0.916028i $$0.631377\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ −0.405813 −0.0375174
$$118$$ 0 0
$$119$$ 1.54288 0.141435
$$120$$ 0 0
$$121$$ 12.5308 1.13916
$$122$$ 0 0
$$123$$ −26.4034 −2.38072
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 20.4088 1.81099 0.905494 0.424359i $$-0.139501\pi$$
0.905494 + 0.424359i $$0.139501\pi$$
$$128$$ 0 0
$$129$$ −17.0586 −1.50193
$$130$$ 0 0
$$131$$ 13.2131 1.15444 0.577218 0.816590i $$-0.304138\pi$$
0.577218 + 0.816590i $$0.304138\pi$$
$$132$$ 0 0
$$133$$ 1.35690 0.117658
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 6.86054 0.586136 0.293068 0.956092i $$-0.405324\pi$$
0.293068 + 0.956092i $$0.405324\pi$$
$$138$$ 0 0
$$139$$ −4.28621 −0.363551 −0.181776 0.983340i $$-0.558184\pi$$
−0.181776 + 0.983340i $$0.558184\pi$$
$$140$$ 0 0
$$141$$ 6.04892 0.509411
$$142$$ 0 0
$$143$$ 0.960771 0.0803437
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −11.5918 −0.956075
$$148$$ 0 0
$$149$$ −15.3545 −1.25789 −0.628945 0.777450i $$-0.716513\pi$$
−0.628945 + 0.777450i $$0.716513\pi$$
$$150$$ 0 0
$$151$$ 10.2295 0.832467 0.416233 0.909258i $$-0.363350\pi$$
0.416233 + 0.909258i $$0.363350\pi$$
$$152$$ 0 0
$$153$$ 2.32975 0.188349
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −3.17092 −0.253067 −0.126533 0.991962i $$-0.540385\pi$$
−0.126533 + 0.991962i $$0.540385\pi$$
$$158$$ 0 0
$$159$$ −28.9366 −2.29482
$$160$$ 0 0
$$161$$ 3.46681 0.273223
$$162$$ 0 0
$$163$$ −4.63773 −0.363255 −0.181627 0.983367i $$-0.558136\pi$$
−0.181627 + 0.983367i $$0.558136\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −19.6286 −1.51891 −0.759454 0.650560i $$-0.774534\pi$$
−0.759454 + 0.650560i $$0.774534\pi$$
$$168$$ 0 0
$$169$$ −12.9608 −0.996982
$$170$$ 0 0
$$171$$ 2.04892 0.156685
$$172$$ 0 0
$$173$$ −20.5646 −1.56350 −0.781751 0.623591i $$-0.785673\pi$$
−0.781751 + 0.623591i $$0.785673\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −13.0858 −0.983585
$$178$$ 0 0
$$179$$ −6.92154 −0.517340 −0.258670 0.965966i $$-0.583284\pi$$
−0.258670 + 0.965966i $$0.583284\pi$$
$$180$$ 0 0
$$181$$ −17.6461 −1.31162 −0.655812 0.754925i $$-0.727673\pi$$
−0.655812 + 0.754925i $$0.727673\pi$$
$$182$$ 0 0
$$183$$ −17.0368 −1.25940
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −5.51573 −0.403350
$$188$$ 0 0
$$189$$ −2.89977 −0.210927
$$190$$ 0 0
$$191$$ −6.92931 −0.501387 −0.250694 0.968066i $$-0.580659\pi$$
−0.250694 + 0.968066i $$0.580659\pi$$
$$192$$ 0 0
$$193$$ 21.0368 1.51426 0.757132 0.653262i $$-0.226600\pi$$
0.757132 + 0.653262i $$0.226600\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −21.5646 −1.53642 −0.768209 0.640199i $$-0.778852\pi$$
−0.768209 + 0.640199i $$0.778852\pi$$
$$198$$ 0 0
$$199$$ −21.9909 −1.55889 −0.779447 0.626468i $$-0.784500\pi$$
−0.779447 + 0.626468i $$0.784500\pi$$
$$200$$ 0 0
$$201$$ 18.0097 1.27031
$$202$$ 0 0
$$203$$ −13.8877 −0.974725
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 5.23490 0.363851
$$208$$ 0 0
$$209$$ −4.85086 −0.335541
$$210$$ 0 0
$$211$$ 20.6233 1.41976 0.709882 0.704321i $$-0.248748\pi$$
0.709882 + 0.704321i $$0.248748\pi$$
$$212$$ 0 0
$$213$$ 19.8267 1.35850
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −3.41358 −0.231729
$$218$$ 0 0
$$219$$ 26.8974 1.81756
$$220$$ 0 0
$$221$$ −0.225209 −0.0151492
$$222$$ 0 0
$$223$$ 7.97716 0.534190 0.267095 0.963670i $$-0.413936\pi$$
0.267095 + 0.963670i $$0.413936\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 19.7942 1.31379 0.656893 0.753984i $$-0.271871\pi$$
0.656893 + 0.753984i $$0.271871\pi$$
$$228$$ 0 0
$$229$$ −4.03385 −0.266564 −0.133282 0.991078i $$-0.542552\pi$$
−0.133282 + 0.991078i $$0.542552\pi$$
$$230$$ 0 0
$$231$$ −14.7899 −0.973101
$$232$$ 0 0
$$233$$ −26.8159 −1.75677 −0.878385 0.477953i $$-0.841379\pi$$
−0.878385 + 0.477953i $$0.841379\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ −24.1347 −1.56772
$$238$$ 0 0
$$239$$ −3.36227 −0.217487 −0.108744 0.994070i $$-0.534683\pi$$
−0.108744 + 0.994070i $$0.534683\pi$$
$$240$$ 0 0
$$241$$ 27.7506 1.78758 0.893788 0.448491i $$-0.148038\pi$$
0.893788 + 0.448491i $$0.148038\pi$$
$$242$$ 0 0
$$243$$ −18.1903 −1.16691
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −0.198062 −0.0126024
$$248$$ 0 0
$$249$$ 8.48188 0.537517
$$250$$ 0 0
$$251$$ 5.59419 0.353102 0.176551 0.984291i $$-0.443506\pi$$
0.176551 + 0.984291i $$0.443506\pi$$
$$252$$ 0 0
$$253$$ −12.3937 −0.779187
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −10.4668 −0.652902 −0.326451 0.945214i $$-0.605853\pi$$
−0.326451 + 0.945214i $$0.605853\pi$$
$$258$$ 0 0
$$259$$ 0.185981 0.0115563
$$260$$ 0 0
$$261$$ −20.9705 −1.29804
$$262$$ 0 0
$$263$$ 15.4795 0.954506 0.477253 0.878766i $$-0.341633\pi$$
0.477253 + 0.878766i $$0.341633\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 21.0465 1.28803
$$268$$ 0 0
$$269$$ 9.13036 0.556688 0.278344 0.960481i $$-0.410215\pi$$
0.278344 + 0.960481i $$0.410215\pi$$
$$270$$ 0 0
$$271$$ −7.44265 −0.452109 −0.226054 0.974115i $$-0.572583\pi$$
−0.226054 + 0.974115i $$0.572583\pi$$
$$272$$ 0 0
$$273$$ −0.603875 −0.0365482
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 11.4155 0.685891 0.342946 0.939355i $$-0.388575\pi$$
0.342946 + 0.939355i $$0.388575\pi$$
$$278$$ 0 0
$$279$$ −5.15452 −0.308593
$$280$$ 0 0
$$281$$ 21.5060 1.28294 0.641471 0.767147i $$-0.278324\pi$$
0.641471 + 0.767147i $$0.278324\pi$$
$$282$$ 0 0
$$283$$ −5.45712 −0.324392 −0.162196 0.986759i $$-0.551858\pi$$
−0.162196 + 0.986759i $$0.551858\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −15.9444 −0.941167
$$288$$ 0 0
$$289$$ −15.7071 −0.923946
$$290$$ 0 0
$$291$$ −0.445042 −0.0260888
$$292$$ 0 0
$$293$$ −7.39075 −0.431772 −0.215886 0.976419i $$-0.569264\pi$$
−0.215886 + 0.976419i $$0.569264\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 10.3666 0.601530
$$298$$ 0 0
$$299$$ −0.506041 −0.0292651
$$300$$ 0 0
$$301$$ −10.3013 −0.593756
$$302$$ 0 0
$$303$$ 25.8605 1.48565
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −32.2131 −1.83850 −0.919250 0.393674i $$-0.871204\pi$$
−0.919250 + 0.393674i $$0.871204\pi$$
$$308$$ 0 0
$$309$$ 34.0562 1.93739
$$310$$ 0 0
$$311$$ −14.8442 −0.841735 −0.420867 0.907122i $$-0.638274\pi$$
−0.420867 + 0.907122i $$0.638274\pi$$
$$312$$ 0 0
$$313$$ 13.1491 0.743234 0.371617 0.928386i $$-0.378804\pi$$
0.371617 + 0.928386i $$0.378804\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 5.13467 0.288392 0.144196 0.989549i $$-0.453940\pi$$
0.144196 + 0.989549i $$0.453940\pi$$
$$318$$ 0 0
$$319$$ 49.6480 2.77975
$$320$$ 0 0
$$321$$ 5.96077 0.332698
$$322$$ 0 0
$$323$$ 1.13706 0.0632679
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −5.61596 −0.310563
$$328$$ 0 0
$$329$$ 3.65279 0.201385
$$330$$ 0 0
$$331$$ −2.00969 −0.110462 −0.0552312 0.998474i $$-0.517590\pi$$
−0.0552312 + 0.998474i $$0.517590\pi$$
$$332$$ 0 0
$$333$$ 0.280831 0.0153895
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.31873 0.0718359 0.0359180 0.999355i $$-0.488564\pi$$
0.0359180 + 0.999355i $$0.488564\pi$$
$$338$$ 0 0
$$339$$ −19.1618 −1.04073
$$340$$ 0 0
$$341$$ 12.2034 0.660853
$$342$$ 0 0
$$343$$ −16.4983 −0.890823
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 12.0151 0.645003 0.322501 0.946569i $$-0.395476\pi$$
0.322501 + 0.946569i $$0.395476\pi$$
$$348$$ 0 0
$$349$$ −10.5579 −0.565154 −0.282577 0.959245i $$-0.591189\pi$$
−0.282577 + 0.959245i $$0.591189\pi$$
$$350$$ 0 0
$$351$$ 0.423272 0.0225926
$$352$$ 0 0
$$353$$ 1.01102 0.0538110 0.0269055 0.999638i $$-0.491435\pi$$
0.0269055 + 0.999638i $$0.491435\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 3.46681 0.183483
$$358$$ 0 0
$$359$$ 5.14244 0.271408 0.135704 0.990749i $$-0.456670\pi$$
0.135704 + 0.990749i $$0.456670\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 28.1564 1.47783
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −29.5308 −1.54149 −0.770747 0.637141i $$-0.780117\pi$$
−0.770747 + 0.637141i $$0.780117\pi$$
$$368$$ 0 0
$$369$$ −24.0761 −1.25335
$$370$$ 0 0
$$371$$ −17.4741 −0.907210
$$372$$ 0 0
$$373$$ −8.78986 −0.455121 −0.227561 0.973764i $$-0.573075\pi$$
−0.227561 + 0.973764i $$0.573075\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 2.02715 0.104403
$$378$$ 0 0
$$379$$ −19.1511 −0.983724 −0.491862 0.870673i $$-0.663684\pi$$
−0.491862 + 0.870673i $$0.663684\pi$$
$$380$$ 0 0
$$381$$ 45.8582 2.34938
$$382$$ 0 0
$$383$$ −6.99894 −0.357629 −0.178814 0.983883i $$-0.557226\pi$$
−0.178814 + 0.983883i $$0.557226\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −15.5550 −0.790703
$$388$$ 0 0
$$389$$ 8.08575 0.409964 0.204982 0.978766i $$-0.434286\pi$$
0.204982 + 0.978766i $$0.434286\pi$$
$$390$$ 0 0
$$391$$ 2.90515 0.146920
$$392$$ 0 0
$$393$$ 29.6896 1.49764
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 17.7006 0.888370 0.444185 0.895935i $$-0.353493\pi$$
0.444185 + 0.895935i $$0.353493\pi$$
$$398$$ 0 0
$$399$$ 3.04892 0.152637
$$400$$ 0 0
$$401$$ −15.5418 −0.776121 −0.388061 0.921634i $$-0.626855\pi$$
−0.388061 + 0.921634i $$0.626855\pi$$
$$402$$ 0 0
$$403$$ 0.498271 0.0248207
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −0.664874 −0.0329566
$$408$$ 0 0
$$409$$ 13.1661 0.651023 0.325512 0.945538i $$-0.394463\pi$$
0.325512 + 0.945538i $$0.394463\pi$$
$$410$$ 0 0
$$411$$ 15.4155 0.760391
$$412$$ 0 0
$$413$$ −7.90217 −0.388840
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −9.63102 −0.471633
$$418$$ 0 0
$$419$$ 25.1739 1.22983 0.614913 0.788595i $$-0.289191\pi$$
0.614913 + 0.788595i $$0.289191\pi$$
$$420$$ 0 0
$$421$$ 26.9420 1.31307 0.656536 0.754295i $$-0.272021\pi$$
0.656536 + 0.754295i $$0.272021\pi$$
$$422$$ 0 0
$$423$$ 5.51573 0.268184
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −10.2881 −0.497877
$$428$$ 0 0
$$429$$ 2.15883 0.104229
$$430$$ 0 0
$$431$$ −18.9487 −0.912726 −0.456363 0.889794i $$-0.650848\pi$$
−0.456363 + 0.889794i $$0.650848\pi$$
$$432$$ 0 0
$$433$$ 3.68904 0.177284 0.0886419 0.996064i $$-0.471747\pi$$
0.0886419 + 0.996064i $$0.471747\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 2.55496 0.122220
$$438$$ 0 0
$$439$$ 25.6926 1.22624 0.613121 0.789989i $$-0.289914\pi$$
0.613121 + 0.789989i $$0.289914\pi$$
$$440$$ 0 0
$$441$$ −10.5700 −0.503334
$$442$$ 0 0
$$443$$ −27.3653 −1.30016 −0.650081 0.759865i $$-0.725265\pi$$
−0.650081 + 0.759865i $$0.725265\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −34.5013 −1.63185
$$448$$ 0 0
$$449$$ 7.55794 0.356681 0.178341 0.983969i $$-0.442927\pi$$
0.178341 + 0.983969i $$0.442927\pi$$
$$450$$ 0 0
$$451$$ 57.0006 2.68405
$$452$$ 0 0
$$453$$ 22.9855 1.07995
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 7.85623 0.367499 0.183750 0.982973i $$-0.441176\pi$$
0.183750 + 0.982973i $$0.441176\pi$$
$$458$$ 0 0
$$459$$ −2.42998 −0.113422
$$460$$ 0 0
$$461$$ 3.87907 0.180666 0.0903331 0.995912i $$-0.471207\pi$$
0.0903331 + 0.995912i $$0.471207\pi$$
$$462$$ 0 0
$$463$$ −13.0954 −0.608597 −0.304298 0.952577i $$-0.598422\pi$$
−0.304298 + 0.952577i $$0.598422\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 6.44026 0.298020 0.149010 0.988836i $$-0.452391\pi$$
0.149010 + 0.988836i $$0.452391\pi$$
$$468$$ 0 0
$$469$$ 10.8756 0.502189
$$470$$ 0 0
$$471$$ −7.12498 −0.328302
$$472$$ 0 0
$$473$$ 36.8267 1.69329
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −26.3860 −1.20813
$$478$$ 0 0
$$479$$ −5.69096 −0.260026 −0.130013 0.991512i $$-0.541502\pi$$
−0.130013 + 0.991512i $$0.541502\pi$$
$$480$$ 0 0
$$481$$ −0.0271471 −0.00123780
$$482$$ 0 0
$$483$$ 7.78986 0.354451
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −12.9632 −0.587417 −0.293709 0.955895i $$-0.594890\pi$$
−0.293709 + 0.955895i $$0.594890\pi$$
$$488$$ 0 0
$$489$$ −10.4209 −0.471248
$$490$$ 0 0
$$491$$ 19.6045 0.884737 0.442369 0.896833i $$-0.354138\pi$$
0.442369 + 0.896833i $$0.354138\pi$$
$$492$$ 0 0
$$493$$ −11.6377 −0.524137
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 11.9729 0.537056
$$498$$ 0 0
$$499$$ −5.49827 −0.246136 −0.123068 0.992398i $$-0.539273\pi$$
−0.123068 + 0.992398i $$0.539273\pi$$
$$500$$ 0 0
$$501$$ −44.1051 −1.97047
$$502$$ 0 0
$$503$$ −35.6939 −1.59151 −0.795757 0.605616i $$-0.792927\pi$$
−0.795757 + 0.605616i $$0.792927\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −29.1226 −1.29338
$$508$$ 0 0
$$509$$ −16.4450 −0.728914 −0.364457 0.931220i $$-0.618745\pi$$
−0.364457 + 0.931220i $$0.618745\pi$$
$$510$$ 0 0
$$511$$ 16.2427 0.718533
$$512$$ 0 0
$$513$$ −2.13706 −0.0943537
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −13.0586 −0.574317
$$518$$ 0 0
$$519$$ −46.2083 −2.02832
$$520$$ 0 0
$$521$$ −26.5435 −1.16289 −0.581445 0.813586i $$-0.697513\pi$$
−0.581445 + 0.813586i $$0.697513\pi$$
$$522$$ 0 0
$$523$$ −24.1685 −1.05682 −0.528408 0.848991i $$-0.677211\pi$$
−0.528408 + 0.848991i $$0.677211\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −2.86054 −0.124607
$$528$$ 0 0
$$529$$ −16.4722 −0.716182
$$530$$ 0 0
$$531$$ −11.9323 −0.517818
$$532$$ 0 0
$$533$$ 2.32736 0.100809
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −15.5526 −0.671143
$$538$$ 0 0
$$539$$ 25.0248 1.07789
$$540$$ 0 0
$$541$$ 9.80386 0.421501 0.210750 0.977540i $$-0.432409\pi$$
0.210750 + 0.977540i $$0.432409\pi$$
$$542$$ 0 0
$$543$$ −39.6504 −1.70156
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 21.1739 0.905331 0.452665 0.891681i $$-0.350473\pi$$
0.452665 + 0.891681i $$0.350473\pi$$
$$548$$ 0 0
$$549$$ −15.5351 −0.663022
$$550$$ 0 0
$$551$$ −10.2349 −0.436021
$$552$$ 0 0
$$553$$ −14.5743 −0.619764
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −24.4077 −1.03419 −0.517094 0.855928i $$-0.672986\pi$$
−0.517094 + 0.855928i $$0.672986\pi$$
$$558$$ 0 0
$$559$$ 1.50365 0.0635975
$$560$$ 0 0
$$561$$ −12.3937 −0.523264
$$562$$ 0 0
$$563$$ 14.4849 0.610464 0.305232 0.952278i $$-0.401266\pi$$
0.305232 + 0.952278i $$0.401266\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ −14.8562 −0.623903
$$568$$ 0 0
$$569$$ 0.811626 0.0340251 0.0170126 0.999855i $$-0.494584\pi$$
0.0170126 + 0.999855i $$0.494584\pi$$
$$570$$ 0 0
$$571$$ 37.6588 1.57597 0.787985 0.615694i $$-0.211124\pi$$
0.787985 + 0.615694i $$0.211124\pi$$
$$572$$ 0 0
$$573$$ −15.5700 −0.650447
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −8.89307 −0.370223 −0.185112 0.982718i $$-0.559265\pi$$
−0.185112 + 0.982718i $$0.559265\pi$$
$$578$$ 0 0
$$579$$ 47.2693 1.96445
$$580$$ 0 0
$$581$$ 5.12200 0.212496
$$582$$ 0 0
$$583$$ 62.4693 2.58721
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 20.3327 0.839222 0.419611 0.907704i $$-0.362167\pi$$
0.419611 + 0.907704i $$0.362167\pi$$
$$588$$ 0 0
$$589$$ −2.51573 −0.103659
$$590$$ 0 0
$$591$$ −48.4553 −1.99319
$$592$$ 0 0
$$593$$ −17.0049 −0.698308 −0.349154 0.937065i $$-0.613531\pi$$
−0.349154 + 0.937065i $$0.613531\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ −49.4131 −2.02234
$$598$$ 0 0
$$599$$ 12.4004 0.506668 0.253334 0.967379i $$-0.418473\pi$$
0.253334 + 0.967379i $$0.418473\pi$$
$$600$$ 0 0
$$601$$ −9.73125 −0.396946 −0.198473 0.980106i $$-0.563598\pi$$
−0.198473 + 0.980106i $$0.563598\pi$$
$$602$$ 0 0
$$603$$ 16.4222 0.668764
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −23.4282 −0.950920 −0.475460 0.879737i $$-0.657718\pi$$
−0.475460 + 0.879737i $$0.657718\pi$$
$$608$$ 0 0
$$609$$ −31.2054 −1.26450
$$610$$ 0 0
$$611$$ −0.533188 −0.0215705
$$612$$ 0 0
$$613$$ −28.1424 −1.13666 −0.568331 0.822800i $$-0.692411\pi$$
−0.568331 + 0.822800i $$0.692411\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −36.4805 −1.46865 −0.734326 0.678797i $$-0.762502\pi$$
−0.734326 + 0.678797i $$0.762502\pi$$
$$618$$ 0 0
$$619$$ 38.2097 1.53578 0.767888 0.640584i $$-0.221308\pi$$
0.767888 + 0.640584i $$0.221308\pi$$
$$620$$ 0 0
$$621$$ −5.46011 −0.219107
$$622$$ 0 0
$$623$$ 12.7095 0.509195
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −10.8998 −0.435295
$$628$$ 0 0
$$629$$ 0.155850 0.00621413
$$630$$ 0 0
$$631$$ 12.5235 0.498553 0.249276 0.968432i $$-0.419807\pi$$
0.249276 + 0.968432i $$0.419807\pi$$
$$632$$ 0 0
$$633$$ 46.3400 1.84185
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 1.02177 0.0404840
$$638$$ 0 0
$$639$$ 18.0790 0.715196
$$640$$ 0 0
$$641$$ 37.3564 1.47549 0.737745 0.675080i $$-0.235891\pi$$
0.737745 + 0.675080i $$0.235891\pi$$
$$642$$ 0 0
$$643$$ −14.5483 −0.573727 −0.286864 0.957971i $$-0.592613\pi$$
−0.286864 + 0.957971i $$0.592613\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −23.4403 −0.921532 −0.460766 0.887522i $$-0.652425\pi$$
−0.460766 + 0.887522i $$0.652425\pi$$
$$648$$ 0 0
$$649$$ 28.2500 1.10891
$$650$$ 0 0
$$651$$ −7.67025 −0.300621
$$652$$ 0 0
$$653$$ −27.1903 −1.06404 −0.532019 0.846732i $$-0.678567\pi$$
−0.532019 + 0.846732i $$0.678567\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 24.5265 0.956869
$$658$$ 0 0
$$659$$ −2.71486 −0.105756 −0.0528779 0.998601i $$-0.516839\pi$$
−0.0528779 + 0.998601i $$0.516839\pi$$
$$660$$ 0 0
$$661$$ −9.41311 −0.366128 −0.183064 0.983101i $$-0.558601\pi$$
−0.183064 + 0.983101i $$0.558601\pi$$
$$662$$ 0 0
$$663$$ −0.506041 −0.0196530
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −26.1497 −1.01252
$$668$$ 0 0
$$669$$ 17.9245 0.693002
$$670$$ 0 0
$$671$$ 36.7797 1.41986
$$672$$ 0 0
$$673$$ 44.8001 1.72692 0.863459 0.504419i $$-0.168293\pi$$
0.863459 + 0.504419i $$0.168293\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 32.9197 1.26521 0.632604 0.774475i $$-0.281986\pi$$
0.632604 + 0.774475i $$0.281986\pi$$
$$678$$ 0 0
$$679$$ −0.268750 −0.0103137
$$680$$ 0 0
$$681$$ 44.4771 1.70437
$$682$$ 0 0
$$683$$ 20.3448 0.778473 0.389236 0.921138i $$-0.372739\pi$$
0.389236 + 0.921138i $$0.372739\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −9.06398 −0.345813
$$688$$ 0 0
$$689$$ 2.55065 0.0971719
$$690$$ 0 0
$$691$$ 46.1473 1.75553 0.877764 0.479094i $$-0.159035\pi$$
0.877764 + 0.479094i $$0.159035\pi$$
$$692$$ 0 0
$$693$$ −13.4862 −0.512298
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −13.3612 −0.506092
$$698$$ 0 0
$$699$$ −60.2549 −2.27905
$$700$$ 0 0
$$701$$ 13.1933 0.498303 0.249152 0.968464i $$-0.419848\pi$$
0.249152 + 0.968464i $$0.419848\pi$$
$$702$$ 0 0
$$703$$ 0.137063 0.00516944
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 15.6165 0.587321
$$708$$ 0 0
$$709$$ 41.1860 1.54677 0.773386 0.633935i $$-0.218562\pi$$
0.773386 + 0.633935i $$0.218562\pi$$
$$710$$ 0 0
$$711$$ −22.0073 −0.825338
$$712$$ 0 0
$$713$$ −6.42758 −0.240715
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −7.55496 −0.282145
$$718$$ 0 0
$$719$$ 35.6256 1.32861 0.664306 0.747461i $$-0.268727\pi$$
0.664306 + 0.747461i $$0.268727\pi$$
$$720$$ 0 0
$$721$$ 20.5657 0.765907
$$722$$ 0 0
$$723$$ 62.3551 2.31901
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 20.0116 0.742189 0.371095 0.928595i $$-0.378983\pi$$
0.371095 + 0.928595i $$0.378983\pi$$
$$728$$ 0 0
$$729$$ −8.02715 −0.297302
$$730$$ 0 0
$$731$$ −8.63235 −0.319279
$$732$$ 0 0
$$733$$ 18.9952 0.701604 0.350802 0.936450i $$-0.385909\pi$$
0.350802 + 0.936450i $$0.385909\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −38.8799 −1.43216
$$738$$ 0 0
$$739$$ −48.1704 −1.77198 −0.885989 0.463706i $$-0.846519\pi$$
−0.885989 + 0.463706i $$0.846519\pi$$
$$740$$ 0 0
$$741$$ −0.445042 −0.0163490
$$742$$ 0 0
$$743$$ −13.2446 −0.485897 −0.242948 0.970039i $$-0.578115\pi$$
−0.242948 + 0.970039i $$0.578115\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 7.73423 0.282981
$$748$$ 0 0
$$749$$ 3.59956 0.131525
$$750$$ 0 0
$$751$$ −12.0562 −0.439937 −0.219969 0.975507i $$-0.570596\pi$$
−0.219969 + 0.975507i $$0.570596\pi$$
$$752$$ 0 0
$$753$$ 12.5700 0.458077
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −15.0054 −0.545380 −0.272690 0.962102i $$-0.587913\pi$$
−0.272690 + 0.962102i $$0.587913\pi$$
$$758$$ 0 0
$$759$$ −27.8485 −1.01084
$$760$$ 0 0
$$761$$ 44.3967 1.60938 0.804690 0.593695i $$-0.202332\pi$$
0.804690 + 0.593695i $$0.202332\pi$$
$$762$$ 0 0
$$763$$ −3.39134 −0.122775
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 1.15346 0.0416489
$$768$$ 0 0
$$769$$ −39.7211 −1.43238 −0.716190 0.697906i $$-0.754115\pi$$
−0.716190 + 0.697906i $$0.754115\pi$$
$$770$$ 0 0
$$771$$ −23.5187 −0.847006
$$772$$ 0 0
$$773$$ 1.72779 0.0621444 0.0310722 0.999517i $$-0.490108\pi$$
0.0310722 + 0.999517i $$0.490108\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0.417895 0.0149919
$$778$$ 0 0
$$779$$ −11.7506 −0.421010
$$780$$ 0 0
$$781$$ −42.8025 −1.53159
$$782$$ 0 0
$$783$$ 21.8726 0.781664
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 42.6329 1.51970 0.759850 0.650098i $$-0.225272\pi$$
0.759850 + 0.650098i $$0.225272\pi$$
$$788$$ 0 0
$$789$$ 34.7821 1.23828
$$790$$ 0 0
$$791$$ −11.5714 −0.411430
$$792$$ 0 0
$$793$$ 1.50173 0.0533280
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −38.5864 −1.36680 −0.683401 0.730044i $$-0.739500\pi$$
−0.683401 + 0.730044i $$0.739500\pi$$
$$798$$ 0 0
$$799$$ 3.06100 0.108290
$$800$$ 0 0
$$801$$ 19.1914 0.678093
$$802$$ 0 0
$$803$$ −58.0670 −2.04914
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 20.5157 0.722188
$$808$$ 0 0
$$809$$ −8.38298 −0.294730 −0.147365 0.989082i $$-0.547079\pi$$
−0.147365 + 0.989082i $$0.547079\pi$$
$$810$$ 0 0
$$811$$ 0.340765 0.0119659 0.00598295 0.999982i $$-0.498096\pi$$
0.00598295 + 0.999982i $$0.498096\pi$$
$$812$$ 0 0
$$813$$ −16.7235 −0.586518
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −7.59179 −0.265603
$$818$$ 0 0
$$819$$ −0.550646 −0.0192411
$$820$$ 0 0
$$821$$ −33.7506 −1.17791 −0.588953 0.808168i $$-0.700460\pi$$
−0.588953 + 0.808168i $$0.700460\pi$$
$$822$$ 0 0
$$823$$ 37.2669 1.29904 0.649522 0.760343i $$-0.274969\pi$$
0.649522 + 0.760343i $$0.274969\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −21.1691 −0.736122 −0.368061 0.929802i $$-0.619978\pi$$
−0.368061 + 0.929802i $$0.619978\pi$$
$$828$$ 0 0
$$829$$ −31.0374 −1.07797 −0.538987 0.842314i $$-0.681193\pi$$
−0.538987 + 0.842314i $$0.681193\pi$$
$$830$$ 0 0
$$831$$ 25.6504 0.889803
$$832$$ 0 0
$$833$$ −5.86592 −0.203242
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 5.37627 0.185831
$$838$$ 0 0
$$839$$ −33.2403 −1.14758 −0.573791 0.819002i $$-0.694528\pi$$
−0.573791 + 0.819002i $$0.694528\pi$$
$$840$$ 0 0
$$841$$ 75.7531 2.61218
$$842$$ 0 0
$$843$$ 48.3236 1.66435
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 17.0030 0.584229
$$848$$ 0 0
$$849$$ −12.2620 −0.420832
$$850$$ 0 0
$$851$$ 0.350191 0.0120044
$$852$$ 0 0
$$853$$ 24.3086 0.832310 0.416155 0.909294i $$-0.363377\pi$$
0.416155 + 0.909294i $$0.363377\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 57.3889 1.96037 0.980185 0.198087i $$-0.0634727\pi$$
0.980185 + 0.198087i $$0.0634727\pi$$
$$858$$ 0 0
$$859$$ 13.8135 0.471312 0.235656 0.971837i $$-0.424276\pi$$
0.235656 + 0.971837i $$0.424276\pi$$
$$860$$ 0 0
$$861$$ −35.8267 −1.22097
$$862$$ 0 0
$$863$$ −9.01938 −0.307023 −0.153512 0.988147i $$-0.549058\pi$$
−0.153512 + 0.988147i $$0.549058\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ −35.2935 −1.19863
$$868$$ 0 0
$$869$$ 52.1027 1.76746
$$870$$ 0 0
$$871$$ −1.58748 −0.0537898
$$872$$ 0 0
$$873$$ −0.405813 −0.0137347
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 2.37675 0.0802570 0.0401285 0.999195i $$-0.487223\pi$$
0.0401285 + 0.999195i $$0.487223\pi$$
$$878$$ 0 0
$$879$$ −16.6069 −0.560135
$$880$$ 0 0
$$881$$ −7.99330 −0.269301 −0.134650 0.990893i $$-0.542991\pi$$
−0.134650 + 0.990893i $$0.542991\pi$$
$$882$$ 0 0
$$883$$ −1.76377 −0.0593557 −0.0296779 0.999560i $$-0.509448\pi$$
−0.0296779 + 0.999560i $$0.509448\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 45.9197 1.54183 0.770917 0.636936i $$-0.219798\pi$$
0.770917 + 0.636936i $$0.219798\pi$$
$$888$$ 0 0
$$889$$ 27.6926 0.928780
$$890$$ 0 0
$$891$$ 53.1105 1.77927
$$892$$ 0 0
$$893$$ 2.69202 0.0900851
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −1.13706 −0.0379654
$$898$$ 0 0
$$899$$ 25.7482 0.858752
$$900$$ 0 0
$$901$$ −14.6431 −0.487833
$$902$$ 0 0
$$903$$ −23.1468 −0.770276
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 55.0549 1.82807 0.914034 0.405638i $$-0.132951\pi$$
0.914034 + 0.405638i $$0.132951\pi$$
$$908$$ 0 0
$$909$$ 23.5810 0.782134
$$910$$ 0 0
$$911$$ −51.8998 −1.71952 −0.859758 0.510702i $$-0.829386\pi$$
−0.859758 + 0.510702i $$0.829386\pi$$
$$912$$ 0 0
$$913$$ −18.3110 −0.606004
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 17.9288 0.592062
$$918$$ 0 0
$$919$$ −11.4614 −0.378078 −0.189039 0.981970i $$-0.560537\pi$$
−0.189039 + 0.981970i $$0.560537\pi$$
$$920$$ 0 0
$$921$$ −72.3822 −2.38508
$$922$$ 0 0
$$923$$ −1.74764 −0.0575244
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 31.0543 1.01996
$$928$$ 0 0
$$929$$ 36.5295 1.19849 0.599246 0.800565i $$-0.295467\pi$$
0.599246 + 0.800565i $$0.295467\pi$$
$$930$$ 0 0
$$931$$ −5.15883 −0.169074
$$932$$ 0 0
$$933$$ −33.3545 −1.09198
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 48.7845 1.59372 0.796860 0.604164i $$-0.206493\pi$$
0.796860 + 0.604164i $$0.206493\pi$$
$$938$$ 0 0
$$939$$ 29.5459 0.964193
$$940$$ 0 0
$$941$$ −18.1817 −0.592705 −0.296353 0.955079i $$-0.595770\pi$$
−0.296353 + 0.955079i $$0.595770\pi$$
$$942$$ 0 0
$$943$$ −30.0224 −0.977663
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ −7.55150 −0.245391 −0.122695 0.992444i $$-0.539154\pi$$
−0.122695 + 0.992444i $$0.539154\pi$$
$$948$$ 0 0
$$949$$ −2.37090 −0.0769626
$$950$$ 0 0
$$951$$ 11.5375 0.374129
$$952$$ 0 0
$$953$$ −13.8592 −0.448944 −0.224472 0.974481i $$-0.572066\pi$$
−0.224472 + 0.974481i $$0.572066\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 111.558 3.60616
$$958$$ 0 0
$$959$$ 9.30904 0.300605
$$960$$ 0 0
$$961$$ −24.6711 −0.795842
$$962$$ 0 0
$$963$$ 5.43535 0.175152
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 4.89977 0.157566 0.0787830 0.996892i $$-0.474897\pi$$
0.0787830 + 0.996892i $$0.474897\pi$$
$$968$$ 0 0
$$969$$ 2.55496 0.0820771
$$970$$ 0 0
$$971$$ −14.5133 −0.465755 −0.232878 0.972506i $$-0.574814\pi$$
−0.232878 + 0.972506i $$0.574814\pi$$
$$972$$ 0 0
$$973$$ −5.81594 −0.186450
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 19.6644 0.629120 0.314560 0.949238i $$-0.398143\pi$$
0.314560 + 0.949238i $$0.398143\pi$$
$$978$$ 0 0
$$979$$ −45.4359 −1.45214
$$980$$ 0 0
$$981$$ −5.12093 −0.163499
$$982$$ 0 0
$$983$$ −15.4397 −0.492449 −0.246224 0.969213i $$-0.579190\pi$$
−0.246224 + 0.969213i $$0.579190\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 8.20775 0.261256
$$988$$ 0 0
$$989$$ −19.3967 −0.616780
$$990$$ 0 0
$$991$$ −11.9377 −0.379213 −0.189606 0.981860i $$-0.560721\pi$$
−0.189606 + 0.981860i $$0.560721\pi$$
$$992$$ 0 0
$$993$$ −4.51573 −0.143302
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −17.9390 −0.568134 −0.284067 0.958804i $$-0.591684\pi$$
−0.284067 + 0.958804i $$0.591684\pi$$
$$998$$ 0 0
$$999$$ −0.292913 −0.00926736
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 7600.2.a.bw.1.3 3
4.3 odd 2 475.2.a.d.1.2 3
5.4 even 2 7600.2.a.bn.1.1 3
12.11 even 2 4275.2.a.bn.1.2 3
20.3 even 4 475.2.b.c.324.5 6
20.7 even 4 475.2.b.c.324.2 6
20.19 odd 2 475.2.a.h.1.2 yes 3
60.59 even 2 4275.2.a.z.1.2 3
76.75 even 2 9025.2.a.be.1.2 3
380.379 even 2 9025.2.a.w.1.2 3

By twisted newform
Twist Min Dim Char Parity Ord Type
475.2.a.d.1.2 3 4.3 odd 2
475.2.a.h.1.2 yes 3 20.19 odd 2
475.2.b.c.324.2 6 20.7 even 4
475.2.b.c.324.5 6 20.3 even 4
4275.2.a.z.1.2 3 60.59 even 2
4275.2.a.bn.1.2 3 12.11 even 2
7600.2.a.bn.1.1 3 5.4 even 2
7600.2.a.bw.1.3 3 1.1 even 1 trivial
9025.2.a.w.1.2 3 380.379 even 2
9025.2.a.be.1.2 3 76.75 even 2