# Properties

 Label 2300.2.c.h.1749.2 Level $2300$ Weight $2$ Character 2300.1749 Analytic conductor $18.366$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2300,2,Mod(1749,2300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2300, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2300.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2300 = 2^{2} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2300.c (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$18.3655924649$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{17})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 9x^{2} + 16$$ x^4 + 9*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 460) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 1749.2 Root $$-1.56155i$$ of defining polynomial Character $$\chi$$ $$=$$ 2300.1749 Dual form 2300.2.c.h.1749.3

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.56155i q^{3} -2.56155i q^{7} +0.561553 q^{9} +O(q^{10})$$ $$q-1.56155i q^{3} -2.56155i q^{7} +0.561553 q^{9} +2.00000 q^{11} -3.56155i q^{13} -2.56155i q^{17} -6.00000 q^{19} -4.00000 q^{21} +1.00000i q^{23} -5.56155i q^{27} -6.12311 q^{29} +7.24621 q^{31} -3.12311i q^{33} +4.56155i q^{37} -5.56155 q^{39} +4.12311 q^{41} -4.68466i q^{47} +0.438447 q^{49} -4.00000 q^{51} -4.56155i q^{53} +9.36932i q^{57} +3.68466 q^{59} -7.12311 q^{61} -1.43845i q^{63} +8.56155i q^{67} +1.56155 q^{69} +10.1231 q^{71} -4.43845i q^{73} -5.12311i q^{77} -4.87689 q^{79} -7.00000 q^{81} -13.9309i q^{83} +9.56155i q^{87} -14.2462 q^{89} -9.12311 q^{91} -11.3153i q^{93} +13.1231i q^{97} +1.12311 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{9}+O(q^{10})$$ 4 * q - 6 * q^9 $$4 q - 6 q^{9} + 8 q^{11} - 24 q^{19} - 16 q^{21} - 8 q^{29} - 4 q^{31} - 14 q^{39} + 10 q^{49} - 16 q^{51} - 10 q^{59} - 12 q^{61} - 2 q^{69} + 24 q^{71} - 36 q^{79} - 28 q^{81} - 24 q^{89} - 20 q^{91} - 12 q^{99}+O(q^{100})$$ 4 * q - 6 * q^9 + 8 * q^11 - 24 * q^19 - 16 * q^21 - 8 * q^29 - 4 * q^31 - 14 * q^39 + 10 * q^49 - 16 * q^51 - 10 * q^59 - 12 * q^61 - 2 * q^69 + 24 * q^71 - 36 * q^79 - 28 * q^81 - 24 * q^89 - 20 * q^91 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times$$.

 $$n$$ $$277$$ $$1151$$ $$1201$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ − 1.56155i − 0.901563i −0.892634 0.450781i $$-0.851145\pi$$
0.892634 0.450781i $$-0.148855\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.56155i − 0.968176i −0.875019 0.484088i $$-0.839151\pi$$
0.875019 0.484088i $$-0.160849\pi$$
$$8$$ 0 0
$$9$$ 0.561553 0.187184
$$10$$ 0 0
$$11$$ 2.00000 0.603023 0.301511 0.953463i $$-0.402509\pi$$
0.301511 + 0.953463i $$0.402509\pi$$
$$12$$ 0 0
$$13$$ − 3.56155i − 0.987797i −0.869520 0.493899i $$-0.835571\pi$$
0.869520 0.493899i $$-0.164429\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 2.56155i − 0.621268i −0.950530 0.310634i $$-0.899459\pi$$
0.950530 0.310634i $$-0.100541\pi$$
$$18$$ 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
$$23$$ 1.00000i 0.208514i
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 5.56155i − 1.07032i
$$28$$ 0 0
$$29$$ −6.12311 −1.13703 −0.568516 0.822672i $$-0.692482\pi$$
−0.568516 + 0.822672i $$0.692482\pi$$
$$30$$ 0 0
$$31$$ 7.24621 1.30146 0.650729 0.759310i $$-0.274463\pi$$
0.650729 + 0.759310i $$0.274463\pi$$
$$32$$ 0 0
$$33$$ − 3.12311i − 0.543663i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.56155i 0.749915i 0.927042 + 0.374957i $$0.122343\pi$$
−0.927042 + 0.374957i $$0.877657\pi$$
$$38$$ 0 0
$$39$$ −5.56155 −0.890561
$$40$$ 0 0
$$41$$ 4.12311 0.643921 0.321960 0.946753i $$-0.395658\pi$$
0.321960 + 0.946753i $$0.395658\pi$$
$$42$$ 0 0
$$43$$ 0 0 1.00000 $$0$$
−1.00000 $$\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 4.68466i − 0.683328i −0.939822 0.341664i $$-0.889010\pi$$
0.939822 0.341664i $$-0.110990\pi$$
$$48$$ 0 0
$$49$$ 0.438447 0.0626353
$$50$$ 0 0
$$51$$ −4.00000 −0.560112
$$52$$ 0 0
$$53$$ − 4.56155i − 0.626577i −0.949658 0.313289i $$-0.898569\pi$$
0.949658 0.313289i $$-0.101431\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 9.36932i 1.24100i
$$58$$ 0 0
$$59$$ 3.68466 0.479702 0.239851 0.970810i $$-0.422901\pi$$
0.239851 + 0.970810i $$0.422901\pi$$
$$60$$ 0 0
$$61$$ −7.12311 −0.912020 −0.456010 0.889975i $$-0.650722\pi$$
−0.456010 + 0.889975i $$0.650722\pi$$
$$62$$ 0 0
$$63$$ − 1.43845i − 0.181227i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 8.56155i 1.04596i 0.852345 + 0.522980i $$0.175180\pi$$
−0.852345 + 0.522980i $$0.824820\pi$$
$$68$$ 0 0
$$69$$ 1.56155 0.187989
$$70$$ 0 0
$$71$$ 10.1231 1.20139 0.600696 0.799478i $$-0.294890\pi$$
0.600696 + 0.799478i $$0.294890\pi$$
$$72$$ 0 0
$$73$$ − 4.43845i − 0.519481i −0.965678 0.259740i $$-0.916363\pi$$
0.965678 0.259740i $$-0.0836370\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 5.12311i − 0.583832i
$$78$$ 0 0
$$79$$ −4.87689 −0.548693 −0.274347 0.961631i $$-0.588462\pi$$
−0.274347 + 0.961631i $$0.588462\pi$$
$$80$$ 0 0
$$81$$ −7.00000 −0.777778
$$82$$ 0 0
$$83$$ − 13.9309i − 1.52911i −0.644558 0.764556i $$-0.722958\pi$$
0.644558 0.764556i $$-0.277042\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 9.56155i 1.02511i
$$88$$ 0 0
$$89$$ −14.2462 −1.51010 −0.755048 0.655670i $$-0.772386\pi$$
−0.755048 + 0.655670i $$0.772386\pi$$
$$90$$ 0 0
$$91$$ −9.12311 −0.956361
$$92$$ 0 0
$$93$$ − 11.3153i − 1.17335i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 13.1231i 1.33245i 0.745751 + 0.666225i $$0.232091\pi$$
−0.745751 + 0.666225i $$0.767909\pi$$
$$98$$ 0 0
$$99$$ 1.12311 0.112876
$$100$$ 0 0
$$101$$ −17.6847 −1.75969 −0.879845 0.475261i $$-0.842353\pi$$
−0.879845 + 0.475261i $$0.842353\pi$$
$$102$$ 0 0
$$103$$ − 16.0000i − 1.57653i −0.615338 0.788263i $$-0.710980\pi$$
0.615338 0.788263i $$-0.289020\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 3.43845i − 0.332407i −0.986091 0.166204i $$-0.946849\pi$$
0.986091 0.166204i $$-0.0531509\pi$$
$$108$$ 0 0
$$109$$ −15.3693 −1.47211 −0.736057 0.676920i $$-0.763314\pi$$
−0.736057 + 0.676920i $$0.763314\pi$$
$$110$$ 0 0
$$111$$ 7.12311 0.676095
$$112$$ 0 0
$$113$$ 12.8078i 1.20485i 0.798174 + 0.602427i $$0.205799\pi$$
−0.798174 + 0.602427i $$0.794201\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 2.00000i − 0.184900i
$$118$$ 0 0
$$119$$ −6.56155 −0.601497
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 0 0
$$123$$ − 6.43845i − 0.580535i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 7.80776i − 0.692827i −0.938082 0.346414i $$-0.887399\pi$$
0.938082 0.346414i $$-0.112601\pi$$
$$128$$ 0 0
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 0.192236 0.0167957 0.00839787 0.999965i $$-0.497327\pi$$
0.00839787 + 0.999965i $$0.497327\pi$$
$$132$$ 0 0
$$133$$ 15.3693i 1.33269i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 6.87689i − 0.587533i −0.955877 0.293766i $$-0.905091\pi$$
0.955877 0.293766i $$-0.0949088\pi$$
$$138$$ 0 0
$$139$$ 11.2462 0.953891 0.476946 0.878933i $$-0.341744\pi$$
0.476946 + 0.878933i $$0.341744\pi$$
$$140$$ 0 0
$$141$$ −7.31534 −0.616063
$$142$$ 0 0
$$143$$ − 7.12311i − 0.595664i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 0.684658i − 0.0564697i
$$148$$ 0 0
$$149$$ 9.36932 0.767564 0.383782 0.923424i $$-0.374621\pi$$
0.383782 + 0.923424i $$0.374621\pi$$
$$150$$ 0 0
$$151$$ −14.0540 −1.14370 −0.571848 0.820359i $$-0.693773\pi$$
−0.571848 + 0.820359i $$0.693773\pi$$
$$152$$ 0 0
$$153$$ − 1.43845i − 0.116292i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.31534i 0.344402i 0.985062 + 0.172201i $$0.0550878\pi$$
−0.985062 + 0.172201i $$0.944912\pi$$
$$158$$ 0 0
$$159$$ −7.12311 −0.564899
$$160$$ 0 0
$$161$$ 2.56155 0.201879
$$162$$ 0 0
$$163$$ 16.9309i 1.32613i 0.748563 + 0.663064i $$0.230744\pi$$
−0.748563 + 0.663064i $$0.769256\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 8.00000i 0.619059i 0.950890 + 0.309529i $$0.100171\pi$$
−0.950890 + 0.309529i $$0.899829\pi$$
$$168$$ 0 0
$$169$$ 0.315342 0.0242570
$$170$$ 0 0
$$171$$ −3.36932 −0.257658
$$172$$ 0 0
$$173$$ 5.36932i 0.408222i 0.978948 + 0.204111i $$0.0654303\pi$$
−0.978948 + 0.204111i $$0.934570\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 5.75379i − 0.432481i
$$178$$ 0 0
$$179$$ 6.93087 0.518038 0.259019 0.965872i $$-0.416601\pi$$
0.259019 + 0.965872i $$0.416601\pi$$
$$180$$ 0 0
$$181$$ −5.36932 −0.399098 −0.199549 0.979888i $$-0.563948\pi$$
−0.199549 + 0.979888i $$0.563948\pi$$
$$182$$ 0 0
$$183$$ 11.1231i 0.822244i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 5.12311i − 0.374639i
$$188$$ 0 0
$$189$$ −14.2462 −1.03626
$$190$$ 0 0
$$191$$ 17.3693 1.25680 0.628400 0.777891i $$-0.283710\pi$$
0.628400 + 0.777891i $$0.283710\pi$$
$$192$$ 0 0
$$193$$ 2.43845i 0.175523i 0.996142 + 0.0877616i $$0.0279714\pi$$
−0.996142 + 0.0877616i $$0.972029\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 18.6847i 1.33123i 0.746297 + 0.665613i $$0.231830\pi$$
−0.746297 + 0.665613i $$0.768170\pi$$
$$198$$ 0 0
$$199$$ 10.0000 0.708881 0.354441 0.935079i $$-0.384671\pi$$
0.354441 + 0.935079i $$0.384671\pi$$
$$200$$ 0 0
$$201$$ 13.3693 0.942999
$$202$$ 0 0
$$203$$ 15.6847i 1.10085i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 0.561553i 0.0390306i
$$208$$ 0 0
$$209$$ −12.0000 −0.830057
$$210$$ 0 0
$$211$$ 0.315342 0.0217090 0.0108545 0.999941i $$-0.496545\pi$$
0.0108545 + 0.999941i $$0.496545\pi$$
$$212$$ 0 0
$$213$$ − 15.8078i − 1.08313i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 18.5616i − 1.26004i
$$218$$ 0 0
$$219$$ −6.93087 −0.468345
$$220$$ 0 0
$$221$$ −9.12311 −0.613686
$$222$$ 0 0
$$223$$ − 17.6155i − 1.17962i −0.807541 0.589812i $$-0.799202\pi$$
0.807541 0.589812i $$-0.200798\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 26.2462i − 1.74202i −0.491263 0.871011i $$-0.663465\pi$$
0.491263 0.871011i $$-0.336535\pi$$
$$228$$ 0 0
$$229$$ 26.7386 1.76694 0.883469 0.468489i $$-0.155201\pi$$
0.883469 + 0.468489i $$0.155201\pi$$
$$230$$ 0 0
$$231$$ −8.00000 −0.526361
$$232$$ 0 0
$$233$$ − 0.684658i − 0.0448535i −0.999748 0.0224267i $$-0.992861\pi$$
0.999748 0.0224267i $$-0.00713925\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 7.61553i 0.494682i
$$238$$ 0 0
$$239$$ −2.75379 −0.178128 −0.0890639 0.996026i $$-0.528388\pi$$
−0.0890639 + 0.996026i $$0.528388\pi$$
$$240$$ 0 0
$$241$$ 6.00000 0.386494 0.193247 0.981150i $$-0.438098\pi$$
0.193247 + 0.981150i $$0.438098\pi$$
$$242$$ 0 0
$$243$$ − 5.75379i − 0.369106i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 21.3693i 1.35970i
$$248$$ 0 0
$$249$$ −21.7538 −1.37859
$$250$$ 0 0
$$251$$ −7.36932 −0.465147 −0.232574 0.972579i $$-0.574715\pi$$
−0.232574 + 0.972579i $$0.574715\pi$$
$$252$$ 0 0
$$253$$ 2.00000i 0.125739i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 31.8078i − 1.98411i −0.125790 0.992057i $$-0.540147\pi$$
0.125790 0.992057i $$-0.459853\pi$$
$$258$$ 0 0
$$259$$ 11.6847 0.726049
$$260$$ 0 0
$$261$$ −3.43845 −0.212835
$$262$$ 0 0
$$263$$ 0.0691303i 0.00426276i 0.999998 + 0.00213138i $$0.000678439\pi$$
−0.999998 + 0.00213138i $$0.999322\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 22.2462i 1.36145i
$$268$$ 0 0
$$269$$ −23.2462 −1.41735 −0.708673 0.705537i $$-0.750706\pi$$
−0.708673 + 0.705537i $$0.750706\pi$$
$$270$$ 0 0
$$271$$ 21.9309 1.33221 0.666103 0.745860i $$-0.267961\pi$$
0.666103 + 0.745860i $$0.267961\pi$$
$$272$$ 0 0
$$273$$ 14.2462i 0.862220i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 6.68466i 0.401642i 0.979628 + 0.200821i $$0.0643610\pi$$
−0.979628 + 0.200821i $$0.935639\pi$$
$$278$$ 0 0
$$279$$ 4.06913 0.243612
$$280$$ 0 0
$$281$$ −6.87689 −0.410241 −0.205121 0.978737i $$-0.565759\pi$$
−0.205121 + 0.978737i $$0.565759\pi$$
$$282$$ 0 0
$$283$$ 14.8078i 0.880230i 0.897941 + 0.440115i $$0.145062\pi$$
−0.897941 + 0.440115i $$0.854938\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 10.5616i − 0.623429i
$$288$$ 0 0
$$289$$ 10.4384 0.614026
$$290$$ 0 0
$$291$$ 20.4924 1.20129
$$292$$ 0 0
$$293$$ − 16.8078i − 0.981920i −0.871182 0.490960i $$-0.836646\pi$$
0.871182 0.490960i $$-0.163354\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 11.1231i − 0.645428i
$$298$$ 0 0
$$299$$ 3.56155 0.205970
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 0 0
$$303$$ 27.6155i 1.58647i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 21.1231i 1.20556i 0.797908 + 0.602780i $$0.205940\pi$$
−0.797908 + 0.602780i $$0.794060\pi$$
$$308$$ 0 0
$$309$$ −24.9848 −1.42134
$$310$$ 0 0
$$311$$ 8.68466 0.492462 0.246231 0.969211i $$-0.420808\pi$$
0.246231 + 0.969211i $$0.420808\pi$$
$$312$$ 0 0
$$313$$ − 0.807764i − 0.0456575i −0.999739 0.0228288i $$-0.992733\pi$$
0.999739 0.0228288i $$-0.00726725\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 15.1231i − 0.849398i −0.905335 0.424699i $$-0.860380\pi$$
0.905335 0.424699i $$-0.139620\pi$$
$$318$$ 0 0
$$319$$ −12.2462 −0.685656
$$320$$ 0 0
$$321$$ −5.36932 −0.299686
$$322$$ 0 0
$$323$$ 15.3693i 0.855172i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 24.0000i 1.32720i
$$328$$ 0 0
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ 28.6155 1.57285 0.786426 0.617685i $$-0.211929\pi$$
0.786426 + 0.617685i $$0.211929\pi$$
$$332$$ 0 0
$$333$$ 2.56155i 0.140372i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 13.6155i − 0.741685i −0.928696 0.370843i $$-0.879069\pi$$
0.928696 0.370843i $$-0.120931\pi$$
$$338$$ 0 0
$$339$$ 20.0000 1.08625
$$340$$ 0 0
$$341$$ 14.4924 0.784809
$$342$$ 0 0
$$343$$ − 19.0540i − 1.02882i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 16.4924i − 0.885360i −0.896680 0.442680i $$-0.854028\pi$$
0.896680 0.442680i $$-0.145972\pi$$
$$348$$ 0 0
$$349$$ 26.3693 1.41152 0.705759 0.708452i $$-0.250606\pi$$
0.705759 + 0.708452i $$0.250606\pi$$
$$350$$ 0 0
$$351$$ −19.8078 −1.05726
$$352$$ 0 0
$$353$$ − 2.05398i − 0.109322i −0.998505 0.0546610i $$-0.982592\pi$$
0.998505 0.0546610i $$-0.0174078\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 10.2462i 0.542287i
$$358$$ 0 0
$$359$$ −15.6155 −0.824156 −0.412078 0.911149i $$-0.635197\pi$$
−0.412078 + 0.911149i $$0.635197\pi$$
$$360$$ 0 0
$$361$$ 17.0000 0.894737
$$362$$ 0 0
$$363$$ 10.9309i 0.573722i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 22.5616i 1.17770i 0.808241 + 0.588852i $$0.200420\pi$$
−0.808241 + 0.588852i $$0.799580\pi$$
$$368$$ 0 0
$$369$$ 2.31534 0.120532
$$370$$ 0 0
$$371$$ −11.6847 −0.606637
$$372$$ 0 0
$$373$$ − 20.2462i − 1.04831i −0.851623 0.524155i $$-0.824381\pi$$
0.851623 0.524155i $$-0.175619\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 21.8078i 1.12316i
$$378$$ 0 0
$$379$$ 36.4924 1.87449 0.937245 0.348672i $$-0.113367\pi$$
0.937245 + 0.348672i $$0.113367\pi$$
$$380$$ 0 0
$$381$$ −12.1922 −0.624627
$$382$$ 0 0
$$383$$ 3.19224i 0.163116i 0.996669 + 0.0815578i $$0.0259895\pi$$
−0.996669 + 0.0815578i $$0.974010\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 0 0
$$388$$ 0 0
$$389$$ 10.8769 0.551480 0.275740 0.961232i $$-0.411077\pi$$
0.275740 + 0.961232i $$0.411077\pi$$
$$390$$ 0 0
$$391$$ 2.56155 0.129543
$$392$$ 0 0
$$393$$ − 0.300187i − 0.0151424i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 34.5464i − 1.73383i −0.498453 0.866917i $$-0.666098\pi$$
0.498453 0.866917i $$-0.333902\pi$$
$$398$$ 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
$$403$$ − 25.8078i − 1.28558i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9.12311i 0.452216i
$$408$$ 0 0
$$409$$ 35.9848 1.77934 0.889668 0.456608i $$-0.150936\pi$$
0.889668 + 0.456608i $$0.150936\pi$$
$$410$$ 0 0
$$411$$ −10.7386 −0.529698
$$412$$ 0 0
$$413$$ − 9.43845i − 0.464436i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 17.5616i − 0.859993i
$$418$$ 0 0
$$419$$ 18.8769 0.922197 0.461098 0.887349i $$-0.347456\pi$$
0.461098 + 0.887349i $$0.347456\pi$$
$$420$$ 0 0
$$421$$ 25.1231 1.22443 0.612213 0.790693i $$-0.290280\pi$$
0.612213 + 0.790693i $$0.290280\pi$$
$$422$$ 0 0
$$423$$ − 2.63068i − 0.127908i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 18.2462i 0.882996i
$$428$$ 0 0
$$429$$ −11.1231 −0.537029
$$430$$ 0 0
$$431$$ −10.2462 −0.493543 −0.246771 0.969074i $$-0.579370\pi$$
−0.246771 + 0.969074i $$0.579370\pi$$
$$432$$ 0 0
$$433$$ 39.9309i 1.91896i 0.281785 + 0.959478i $$0.409074\pi$$
−0.281785 + 0.959478i $$0.590926\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 6.00000i − 0.287019i
$$438$$ 0 0
$$439$$ −19.8078 −0.945373 −0.472686 0.881231i $$-0.656716\pi$$
−0.472686 + 0.881231i $$0.656716\pi$$
$$440$$ 0 0
$$441$$ 0.246211 0.0117243
$$442$$ 0 0
$$443$$ − 11.8078i − 0.561004i −0.959853 0.280502i $$-0.909499\pi$$
0.959853 0.280502i $$-0.0905009\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 14.6307i − 0.692008i
$$448$$ 0 0
$$449$$ 21.6847 1.02336 0.511681 0.859175i $$-0.329023\pi$$
0.511681 + 0.859175i $$0.329023\pi$$
$$450$$ 0 0
$$451$$ 8.24621 0.388299
$$452$$ 0 0
$$453$$ 21.9460i 1.03111i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 9.43845i − 0.441512i −0.975329 0.220756i $$-0.929148\pi$$
0.975329 0.220756i $$-0.0708524\pi$$
$$458$$ 0 0
$$459$$ −14.2462 −0.664956
$$460$$ 0 0
$$461$$ −40.9309 −1.90634 −0.953170 0.302434i $$-0.902201\pi$$
−0.953170 + 0.302434i $$0.902201\pi$$
$$462$$ 0 0
$$463$$ − 31.3693i − 1.45786i −0.684590 0.728928i $$-0.740019\pi$$
0.684590 0.728928i $$-0.259981\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 22.3153i 1.03263i 0.856398 + 0.516315i $$0.172697\pi$$
−0.856398 + 0.516315i $$0.827303\pi$$
$$468$$ 0 0
$$469$$ 21.9309 1.01267
$$470$$ 0 0
$$471$$ 6.73863 0.310500
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 2.56155i − 0.117285i
$$478$$ 0 0
$$479$$ 43.2311 1.97528 0.987639 0.156748i $$-0.0501009\pi$$
0.987639 + 0.156748i $$0.0501009\pi$$
$$480$$ 0 0
$$481$$ 16.2462 0.740763
$$482$$ 0 0
$$483$$ − 4.00000i − 0.182006i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 7.80776i − 0.353804i −0.984229 0.176902i $$-0.943393\pi$$
0.984229 0.176902i $$-0.0566075\pi$$
$$488$$ 0 0
$$489$$ 26.4384 1.19559
$$490$$ 0 0
$$491$$ 11.4924 0.518646 0.259323 0.965791i $$-0.416501\pi$$
0.259323 + 0.965791i $$0.416501\pi$$
$$492$$ 0 0
$$493$$ 15.6847i 0.706401i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 25.9309i − 1.16316i
$$498$$ 0 0
$$499$$ −22.6155 −1.01241 −0.506205 0.862413i $$-0.668952\pi$$
−0.506205 + 0.862413i $$0.668952\pi$$
$$500$$ 0 0
$$501$$ 12.4924 0.558120
$$502$$ 0 0
$$503$$ − 3.93087i − 0.175269i −0.996153 0.0876344i $$-0.972069\pi$$
0.996153 0.0876344i $$-0.0279307\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 0.492423i − 0.0218693i
$$508$$ 0 0
$$509$$ 27.1771 1.20460 0.602301 0.798269i $$-0.294251\pi$$
0.602301 + 0.798269i $$0.294251\pi$$
$$510$$ 0 0
$$511$$ −11.3693 −0.502949
$$512$$ 0 0
$$513$$ 33.3693i 1.47329i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 9.36932i − 0.412062i
$$518$$ 0 0
$$519$$ 8.38447 0.368037
$$520$$ 0 0
$$521$$ −13.6155 −0.596507 −0.298254 0.954487i $$-0.596404\pi$$
−0.298254 + 0.954487i $$0.596404\pi$$
$$522$$ 0 0
$$523$$ − 14.7386i − 0.644475i −0.946659 0.322238i $$-0.895565\pi$$
0.946659 0.322238i $$-0.104435\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 18.5616i − 0.808554i
$$528$$ 0 0
$$529$$ −1.00000 −0.0434783
$$530$$ 0 0
$$531$$ 2.06913 0.0897926
$$532$$ 0 0
$$533$$ − 14.6847i − 0.636063i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 10.8229i − 0.467043i
$$538$$ 0 0
$$539$$ 0.876894 0.0377705
$$540$$ 0 0
$$541$$ −2.68466 −0.115422 −0.0577112 0.998333i $$-0.518380\pi$$
−0.0577112 + 0.998333i $$0.518380\pi$$
$$542$$ 0 0
$$543$$ 8.38447i 0.359812i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 21.1771i 0.905467i 0.891646 + 0.452733i $$0.149551\pi$$
−0.891646 + 0.452733i $$0.850449\pi$$
$$548$$ 0 0
$$549$$ −4.00000 −0.170716
$$550$$ 0 0
$$551$$ 36.7386 1.56512
$$552$$ 0 0
$$553$$ 12.4924i 0.531232i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 15.6847i 0.664580i 0.943177 + 0.332290i $$0.107821\pi$$
−0.943177 + 0.332290i $$0.892179\pi$$
$$558$$ 0 0
$$559$$ 0 0
$$560$$ 0 0
$$561$$ −8.00000 −0.337760
$$562$$ 0 0
$$563$$ 32.6695i 1.37686i 0.725305 + 0.688428i $$0.241699\pi$$
−0.725305 + 0.688428i $$0.758301\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 17.9309i 0.753026i
$$568$$ 0 0
$$569$$ −16.7386 −0.701720 −0.350860 0.936428i $$-0.614111\pi$$
−0.350860 + 0.936428i $$0.614111\pi$$
$$570$$ 0 0
$$571$$ 36.0000 1.50655 0.753277 0.657704i $$-0.228472\pi$$
0.753277 + 0.657704i $$0.228472\pi$$
$$572$$ 0 0
$$573$$ − 27.1231i − 1.13308i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 29.5616i 1.23066i 0.788268 + 0.615332i $$0.210978\pi$$
−0.788268 + 0.615332i $$0.789022\pi$$
$$578$$ 0 0
$$579$$ 3.80776 0.158245
$$580$$ 0 0
$$581$$ −35.6847 −1.48045
$$582$$ 0 0
$$583$$ − 9.12311i − 0.377840i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 7.06913i − 0.291774i −0.989301 0.145887i $$-0.953396\pi$$
0.989301 0.145887i $$-0.0466036\pi$$
$$588$$ 0 0
$$589$$ −43.4773 −1.79145
$$590$$ 0 0
$$591$$ 29.1771 1.20018
$$592$$ 0 0
$$593$$ 15.6155i 0.641253i 0.947206 + 0.320626i $$0.103893\pi$$
−0.947206 + 0.320626i $$0.896107\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ − 15.6155i − 0.639101i
$$598$$ 0 0
$$599$$ 32.9848 1.34772 0.673862 0.738857i $$-0.264634\pi$$
0.673862 + 0.738857i $$0.264634\pi$$
$$600$$ 0 0
$$601$$ 36.3693 1.48354 0.741768 0.670657i $$-0.233988\pi$$
0.741768 + 0.670657i $$0.233988\pi$$
$$602$$ 0 0
$$603$$ 4.80776i 0.195787i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 8.49242i − 0.344697i −0.985036 0.172348i $$-0.944865\pi$$
0.985036 0.172348i $$-0.0551355\pi$$
$$608$$ 0 0
$$609$$ 24.4924 0.992483
$$610$$ 0 0
$$611$$ −16.6847 −0.674989
$$612$$ 0 0
$$613$$ − 5.61553i − 0.226809i −0.993549 0.113405i $$-0.963824\pi$$
0.993549 0.113405i $$-0.0361756\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 0.561553i − 0.0226073i −0.999936 0.0113036i $$-0.996402\pi$$
0.999936 0.0113036i $$-0.00359813\pi$$
$$618$$ 0 0
$$619$$ −12.4924 −0.502113 −0.251056 0.967972i $$-0.580778\pi$$
−0.251056 + 0.967972i $$0.580778\pi$$
$$620$$ 0 0
$$621$$ 5.56155 0.223177
$$622$$ 0 0
$$623$$ 36.4924i 1.46204i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 18.7386i 0.748349i
$$628$$ 0 0
$$629$$ 11.6847 0.465898
$$630$$ 0 0
$$631$$ 30.2462 1.20408 0.602041 0.798465i $$-0.294354\pi$$
0.602041 + 0.798465i $$0.294354\pi$$
$$632$$ 0 0
$$633$$ − 0.492423i − 0.0195720i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 1.56155i − 0.0618710i
$$638$$ 0 0
$$639$$ 5.68466 0.224882
$$640$$ 0 0
$$641$$ 6.87689 0.271621 0.135810 0.990735i $$-0.456636\pi$$
0.135810 + 0.990735i $$0.456636\pi$$
$$642$$ 0 0
$$643$$ 20.4233i 0.805416i 0.915328 + 0.402708i $$0.131931\pi$$
−0.915328 + 0.402708i $$0.868069\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 5.31534i − 0.208968i −0.994527 0.104484i $$-0.966681\pi$$
0.994527 0.104484i $$-0.0333190\pi$$
$$648$$ 0 0
$$649$$ 7.36932 0.289271
$$650$$ 0 0
$$651$$ −28.9848 −1.13601
$$652$$ 0 0
$$653$$ 39.6695i 1.55239i 0.630494 + 0.776194i $$0.282852\pi$$
−0.630494 + 0.776194i $$0.717148\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 2.49242i − 0.0972387i
$$658$$ 0 0
$$659$$ −15.3693 −0.598704 −0.299352 0.954143i $$-0.596770\pi$$
−0.299352 + 0.954143i $$0.596770\pi$$
$$660$$ 0 0
$$661$$ −6.49242 −0.252526 −0.126263 0.991997i $$-0.540298\pi$$
−0.126263 + 0.991997i $$0.540298\pi$$
$$662$$ 0 0
$$663$$ 14.2462i 0.553277i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 6.12311i − 0.237088i
$$668$$ 0 0
$$669$$ −27.5076 −1.06350
$$670$$ 0 0
$$671$$ −14.2462 −0.549969
$$672$$ 0 0
$$673$$ 34.5464i 1.33167i 0.746101 + 0.665833i $$0.231924\pi$$
−0.746101 + 0.665833i $$0.768076\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 26.8078i − 1.03031i −0.857098 0.515153i $$-0.827735\pi$$
0.857098 0.515153i $$-0.172265\pi$$
$$678$$ 0 0
$$679$$ 33.6155 1.29005
$$680$$ 0 0
$$681$$ −40.9848 −1.57054
$$682$$ 0 0
$$683$$ 42.0540i 1.60915i 0.593851 + 0.804575i $$0.297607\pi$$
−0.593851 + 0.804575i $$0.702393\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ − 41.7538i − 1.59301i
$$688$$ 0 0
$$689$$ −16.2462 −0.618931
$$690$$ 0 0
$$691$$ 16.4924 0.627401 0.313701 0.949522i $$-0.398431\pi$$
0.313701 + 0.949522i $$0.398431\pi$$
$$692$$ 0 0
$$693$$ − 2.87689i − 0.109284i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 10.5616i − 0.400047i
$$698$$ 0 0
$$699$$ −1.06913 −0.0404382
$$700$$ 0 0
$$701$$ −1.75379 −0.0662397 −0.0331198 0.999451i $$-0.510544\pi$$
−0.0331198 + 0.999451i $$0.510544\pi$$
$$702$$ 0 0
$$703$$ − 27.3693i − 1.03225i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 45.3002i 1.70369i
$$708$$ 0 0
$$709$$ −29.7538 −1.11743 −0.558713 0.829361i $$-0.688705\pi$$
−0.558713 + 0.829361i $$0.688705\pi$$
$$710$$ 0 0
$$711$$ −2.73863 −0.102707
$$712$$ 0 0
$$713$$ 7.24621i 0.271373i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 4.30019i 0.160593i
$$718$$ 0 0
$$719$$ −19.0540 −0.710593 −0.355297 0.934754i $$-0.615620\pi$$
−0.355297 + 0.934754i $$0.615620\pi$$
$$720$$ 0 0
$$721$$ −40.9848 −1.52636
$$722$$ 0 0
$$723$$ − 9.36932i − 0.348449i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 21.1922i 0.785977i 0.919543 + 0.392988i $$0.128559\pi$$
−0.919543 + 0.392988i $$0.871441\pi$$
$$728$$ 0 0
$$729$$ −29.9848 −1.11055
$$730$$ 0 0
$$731$$ 0 0
$$732$$ 0 0
$$733$$ 0.315342i 0.0116474i 0.999983 + 0.00582370i $$0.00185375\pi$$
−0.999983 + 0.00582370i $$0.998146\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 17.1231i 0.630738i
$$738$$ 0 0
$$739$$ −0.615528 −0.0226426 −0.0113213 0.999936i $$-0.503604\pi$$
−0.0113213 + 0.999936i $$0.503604\pi$$
$$740$$ 0 0
$$741$$ 33.3693 1.22585
$$742$$ 0 0
$$743$$ 3.50758i 0.128681i 0.997928 + 0.0643403i $$0.0204943\pi$$
−0.997928 + 0.0643403i $$0.979506\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 7.82292i − 0.286226i
$$748$$ 0 0
$$749$$ −8.80776 −0.321829
$$750$$ 0 0
$$751$$ −13.1231 −0.478869 −0.239434 0.970913i $$-0.576962\pi$$
−0.239434 + 0.970913i $$0.576962\pi$$
$$752$$ 0 0
$$753$$ 11.5076i 0.419359i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 12.5616i − 0.456557i −0.973596 0.228279i $$-0.926690\pi$$
0.973596 0.228279i $$-0.0733097\pi$$
$$758$$ 0 0
$$759$$ 3.12311 0.113362
$$760$$ 0 0
$$761$$ −43.9848 −1.59445 −0.797225 0.603683i $$-0.793699\pi$$
−0.797225 + 0.603683i $$0.793699\pi$$
$$762$$ 0 0
$$763$$ 39.3693i 1.42526i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 13.1231i − 0.473848i
$$768$$ 0 0
$$769$$ 10.6307 0.383352 0.191676 0.981458i $$-0.438608\pi$$
0.191676 + 0.981458i $$0.438608\pi$$
$$770$$ 0 0
$$771$$ −49.6695 −1.78880
$$772$$ 0 0
$$773$$ − 41.1231i − 1.47910i −0.673104 0.739548i $$-0.735039\pi$$
0.673104 0.739548i $$-0.264961\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 18.2462i − 0.654579i
$$778$$ 0 0
$$779$$ −24.7386 −0.886354
$$780$$ 0 0
$$781$$ 20.2462 0.724466
$$782$$ 0 0
$$783$$ 34.0540i 1.21699i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 39.6847i 1.41461i 0.706911 + 0.707303i $$0.250088\pi$$
−0.706911 + 0.707303i $$0.749912\pi$$
$$788$$ 0 0
$$789$$ 0.107951 0.00384314
$$790$$ 0 0
$$791$$ 32.8078 1.16651
$$792$$ 0 0
$$793$$ 25.3693i 0.900891i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 52.4233i 1.85693i 0.371422 + 0.928464i $$0.378870\pi$$
−0.371422 + 0.928464i $$0.621130\pi$$
$$798$$ 0 0
$$799$$ −12.0000 −0.424529
$$800$$ 0 0
$$801$$ −8.00000 −0.282666
$$802$$ 0 0
$$803$$ − 8.87689i − 0.313259i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 36.3002i 1.27783i
$$808$$ 0 0
$$809$$ 54.1771 1.90476 0.952382 0.304906i $$-0.0986251\pi$$
0.952382 + 0.304906i $$0.0986251\pi$$
$$810$$ 0 0
$$811$$ 47.2462 1.65904 0.829519 0.558478i $$-0.188614\pi$$
0.829519 + 0.558478i $$0.188614\pi$$
$$812$$ 0 0
$$813$$ − 34.2462i − 1.20107i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 0 0
$$818$$ 0 0
$$819$$ −5.12311 −0.179016
$$820$$ 0 0
$$821$$ −34.4924 −1.20379 −0.601897 0.798574i $$-0.705588\pi$$
−0.601897 + 0.798574i $$0.705588\pi$$
$$822$$ 0 0
$$823$$ − 38.0540i − 1.32648i −0.748408 0.663239i $$-0.769181\pi$$
0.748408 0.663239i $$-0.230819\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 19.6847i − 0.684503i −0.939608 0.342251i $$-0.888811\pi$$
0.939608 0.342251i $$-0.111189\pi$$
$$828$$ 0 0
$$829$$ 29.5464 1.02619 0.513094 0.858332i $$-0.328499\pi$$
0.513094 + 0.858332i $$0.328499\pi$$
$$830$$ 0 0
$$831$$ 10.4384 0.362106
$$832$$ 0 0
$$833$$ − 1.12311i − 0.0389133i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 40.3002i − 1.39298i
$$838$$ 0 0
$$839$$ 42.8769 1.48027 0.740137 0.672456i $$-0.234760\pi$$
0.740137 + 0.672456i $$0.234760\pi$$
$$840$$ 0 0
$$841$$ 8.49242 0.292842
$$842$$ 0 0
$$843$$ 10.7386i 0.369858i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 17.9309i 0.616112i
$$848$$ 0 0
$$849$$ 23.1231 0.793583
$$850$$ 0 0
$$851$$ −4.56155 −0.156368
$$852$$ 0 0
$$853$$ − 20.2462i − 0.693217i −0.938010 0.346609i $$-0.887333\pi$$
0.938010 0.346609i $$-0.112667\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 33.3153i 1.13803i 0.822327 + 0.569015i $$0.192675\pi$$
−0.822327 + 0.569015i $$0.807325\pi$$
$$858$$ 0 0
$$859$$ 14.5076 0.494992 0.247496 0.968889i $$-0.420392\pi$$
0.247496 + 0.968889i $$0.420392\pi$$
$$860$$ 0 0
$$861$$ −16.4924 −0.562060
$$862$$ 0 0
$$863$$ − 7.56155i − 0.257398i −0.991684 0.128699i $$-0.958920\pi$$
0.991684 0.128699i $$-0.0410801\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 16.3002i − 0.553583i
$$868$$ 0 0
$$869$$ −9.75379 −0.330875
$$870$$ 0 0
$$871$$ 30.4924 1.03320
$$872$$ 0 0
$$873$$ 7.36932i 0.249414i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 18.9848i 0.641073i 0.947236 + 0.320536i $$0.103863\pi$$
−0.947236 + 0.320536i $$0.896137\pi$$
$$878$$ 0 0
$$879$$ −26.2462 −0.885263
$$880$$ 0 0
$$881$$ 41.4773 1.39740 0.698702 0.715413i $$-0.253761\pi$$
0.698702 + 0.715413i $$0.253761\pi$$
$$882$$ 0 0
$$883$$ 10.7386i 0.361384i 0.983540 + 0.180692i $$0.0578337\pi$$
−0.983540 + 0.180692i $$0.942166\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 42.7926i − 1.43684i −0.695612 0.718418i $$-0.744867\pi$$
0.695612 0.718418i $$-0.255133\pi$$
$$888$$ 0 0
$$889$$ −20.0000 −0.670778
$$890$$ 0 0
$$891$$ −14.0000 −0.469018
$$892$$ 0 0
$$893$$ 28.1080i 0.940597i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 5.56155i − 0.185695i
$$898$$ 0 0
$$899$$ −44.3693 −1.47980
$$900$$ 0 0
$$901$$ −11.6847 −0.389272
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 21.6847i − 0.720027i −0.932947 0.360014i $$-0.882772\pi$$
0.932947 0.360014i $$-0.117228\pi$$
$$908$$ 0 0
$$909$$ −9.93087 −0.329386
$$910$$ 0 0
$$911$$ −3.12311 −0.103473 −0.0517366 0.998661i $$-0.516476\pi$$
−0.0517366 + 0.998661i $$0.516476\pi$$
$$912$$ 0 0
$$913$$ − 27.8617i − 0.922089i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 0.492423i − 0.0162612i
$$918$$ 0 0
$$919$$ −46.7386 −1.54177 −0.770883 0.636977i $$-0.780185\pi$$
−0.770883 + 0.636977i $$0.780185\pi$$
$$920$$ 0 0
$$921$$ 32.9848 1.08689
$$922$$ 0 0
$$923$$ − 36.0540i − 1.18673i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 8.98485i − 0.295101i
$$928$$ 0 0
$$929$$ 56.7235 1.86104 0.930518 0.366245i $$-0.119357\pi$$
0.930518 + 0.366245i $$0.119357\pi$$
$$930$$ 0 0
$$931$$ −2.63068 −0.0862172
$$932$$ 0 0
$$933$$ − 13.5616i − 0.443985i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 16.2462i 0.530741i 0.964147 + 0.265370i $$0.0854942\pi$$
−0.964147 + 0.265370i $$0.914506\pi$$
$$938$$ 0 0
$$939$$ −1.26137 −0.0411631
$$940$$ 0 0
$$941$$ −37.7538 −1.23074 −0.615369 0.788239i $$-0.710993\pi$$
−0.615369 + 0.788239i $$0.710993\pi$$
$$942$$ 0 0
$$943$$ 4.12311i 0.134267i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ − 56.6847i − 1.84200i −0.389558 0.921002i $$-0.627372\pi$$
0.389558 0.921002i $$-0.372628\pi$$
$$948$$ 0 0
$$949$$ −15.8078 −0.513142
$$950$$ 0 0
$$951$$ −23.6155 −0.765786
$$952$$ 0 0
$$953$$ − 38.4924i − 1.24689i −0.781866 0.623446i $$-0.785732\pi$$
0.781866 0.623446i $$-0.214268\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 19.1231i 0.618162i
$$958$$ 0 0
$$959$$ −17.6155 −0.568835
$$960$$ 0 0
$$961$$ 21.5076 0.693793
$$962$$ 0 0
$$963$$ − 1.93087i − 0.0622214i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 56.6847i 1.82286i 0.411460 + 0.911428i $$0.365019\pi$$
−0.411460 + 0.911428i $$0.634981\pi$$
$$968$$ 0 0
$$969$$ 24.0000 0.770991
$$970$$ 0 0
$$971$$ −22.6307 −0.726253 −0.363127 0.931740i $$-0.618291\pi$$
−0.363127 + 0.931740i $$0.618291\pi$$
$$972$$ 0 0
$$973$$ − 28.8078i − 0.923535i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 59.7926i − 1.91294i −0.291839 0.956468i $$-0.594267\pi$$
0.291839 0.956468i $$-0.405733\pi$$
$$978$$ 0 0
$$979$$ −28.4924 −0.910622
$$980$$ 0 0
$$981$$ −8.63068 −0.275557
$$982$$ 0 0
$$983$$ − 37.0540i − 1.18184i −0.806731 0.590919i $$-0.798765\pi$$
0.806731 0.590919i $$-0.201235\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 18.7386i 0.596457i
$$988$$ 0 0
$$989$$ 0 0
$$990$$ 0 0
$$991$$ 53.3002 1.69314 0.846568 0.532280i $$-0.178665\pi$$
0.846568 + 0.532280i $$0.178665\pi$$
$$992$$ 0 0
$$993$$ − 44.6847i − 1.41802i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 55.6155i − 1.76136i −0.473710 0.880681i $$-0.657086\pi$$
0.473710 0.880681i $$-0.342914\pi$$
$$998$$ 0 0
$$999$$ 25.3693 0.802650
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 2300.2.c.h.1749.2 4
5.2 odd 4 460.2.a.e.1.1 2
5.3 odd 4 2300.2.a.i.1.2 2
5.4 even 2 inner 2300.2.c.h.1749.3 4
15.2 even 4 4140.2.a.m.1.2 2
20.3 even 4 9200.2.a.bv.1.1 2
20.7 even 4 1840.2.a.m.1.2 2
40.27 even 4 7360.2.a.bo.1.1 2
40.37 odd 4 7360.2.a.bi.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.a.e.1.1 2 5.2 odd 4
1840.2.a.m.1.2 2 20.7 even 4
2300.2.a.i.1.2 2 5.3 odd 4
2300.2.c.h.1749.2 4 1.1 even 1 trivial
2300.2.c.h.1749.3 4 5.4 even 2 inner
4140.2.a.m.1.2 2 15.2 even 4
7360.2.a.bi.1.2 2 40.37 odd 4
7360.2.a.bo.1.1 2 40.27 even 4
9200.2.a.bv.1.1 2 20.3 even 4