Properties

 Label 3800.2.d.n.3649.2 Level $3800$ Weight $2$ Character 3800.3649 Analytic conductor $30.343$ Analytic rank $0$ Dimension $6$ Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(3649,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3800.3649");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3800.d (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: $$30.3431527681$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.399424.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$$ x^6 - 2*x^5 + 3*x^4 - 6*x^3 + 6*x^2 - 8*x + 8 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 760) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

 Embedding label 3649.2 Root $$1.40680 + 0.144584i$$ of defining polynomial Character $$\chi$$ $$=$$ 3800.3649 Dual form 3800.2.d.n.3649.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-1.81361i q^{3} -4.91638i q^{7} -0.289169 q^{9} +O(q^{10})$$ $$q-1.81361i q^{3} -4.91638i q^{7} -0.289169 q^{9} +0.578337 q^{11} +6.39194i q^{13} -0.710831i q^{17} -1.00000 q^{19} -8.91638 q^{21} +2.71083i q^{23} -4.91638i q^{27} -6.54359 q^{29} +1.42166 q^{31} -1.04888i q^{33} -9.10278i q^{37} +11.5925 q^{39} -11.0489 q^{41} -5.83276i q^{43} -1.15667i q^{47} -17.1708 q^{49} -1.28917 q^{51} -13.2736i q^{53} +1.81361i q^{57} -11.3869 q^{59} -9.04888 q^{61} +1.42166i q^{63} +2.97028i q^{67} +4.91638 q^{69} +9.38692i q^{73} -2.84333i q^{77} -4.37279 q^{79} -9.78389 q^{81} +0.372787i q^{83} +11.8675i q^{87} +16.6167 q^{89} +31.4252 q^{91} -2.57834i q^{93} +3.94610i q^{97} -0.167237 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q+O(q^{10})$$ 6 * q $$6 q - 6 q^{19} - 26 q^{21} + 14 q^{29} + 12 q^{31} - 6 q^{39} - 44 q^{41} - 24 q^{49} - 6 q^{51} - 22 q^{59} - 32 q^{61} + 2 q^{69} - 52 q^{79} - 26 q^{81} + 12 q^{89} + 58 q^{91} - 56 q^{99}+O(q^{100})$$ 6 * q - 6 * q^19 - 26 * q^21 + 14 * q^29 + 12 * q^31 - 6 * q^39 - 44 * q^41 - 24 * q^49 - 6 * q^51 - 22 * q^59 - 32 * q^61 + 2 * q^69 - 52 * q^79 - 26 * q^81 + 12 * q^89 + 58 * q^91 - 56 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

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
$$3$$ − 1.81361i − 1.04709i −0.851999 0.523543i $$-0.824610\pi$$
0.851999 0.523543i $$-0.175390\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 4.91638i − 1.85822i −0.369807 0.929109i $$-0.620576\pi$$
0.369807 0.929109i $$-0.379424\pi$$
$$8$$ 0 0
$$9$$ −0.289169 −0.0963895
$$10$$ 0 0
$$11$$ 0.578337 0.174375 0.0871876 0.996192i $$-0.472212\pi$$
0.0871876 + 0.996192i $$0.472212\pi$$
$$12$$ 0 0
$$13$$ 6.39194i 1.77281i 0.462914 + 0.886403i $$0.346804\pi$$
−0.462914 + 0.886403i $$0.653196\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 0.710831i − 0.172402i −0.996278 0.0862010i $$-0.972527\pi$$
0.996278 0.0862010i $$-0.0274727\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416
$$20$$ 0 0
$$21$$ −8.91638 −1.94571
$$22$$ 0 0
$$23$$ 2.71083i 0.565247i 0.959231 + 0.282624i $$0.0912048\pi$$
−0.959231 + 0.282624i $$0.908795\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ − 4.91638i − 0.946158i
$$28$$ 0 0
$$29$$ −6.54359 −1.21512 −0.607558 0.794276i $$-0.707851\pi$$
−0.607558 + 0.794276i $$0.707851\pi$$
$$30$$ 0 0
$$31$$ 1.42166 0.255338 0.127669 0.991817i $$-0.459250\pi$$
0.127669 + 0.991817i $$0.459250\pi$$
$$32$$ 0 0
$$33$$ − 1.04888i − 0.182586i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 9.10278i − 1.49649i −0.663424 0.748243i $$-0.730897\pi$$
0.663424 0.748243i $$-0.269103\pi$$
$$38$$ 0 0
$$39$$ 11.5925 1.85628
$$40$$ 0 0
$$41$$ −11.0489 −1.72554 −0.862772 0.505593i $$-0.831274\pi$$
−0.862772 + 0.505593i $$0.831274\pi$$
$$42$$ 0 0
$$43$$ − 5.83276i − 0.889488i −0.895658 0.444744i $$-0.853295\pi$$
0.895658 0.444744i $$-0.146705\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 1.15667i − 0.168718i −0.996435 0.0843591i $$-0.973116\pi$$
0.996435 0.0843591i $$-0.0268843\pi$$
$$48$$ 0 0
$$49$$ −17.1708 −2.45297
$$50$$ 0 0
$$51$$ −1.28917 −0.180520
$$52$$ 0 0
$$53$$ − 13.2736i − 1.82327i −0.411004 0.911633i $$-0.634822\pi$$
0.411004 0.911633i $$-0.365178\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.81361i 0.240218i
$$58$$ 0 0
$$59$$ −11.3869 −1.48245 −0.741225 0.671256i $$-0.765755\pi$$
−0.741225 + 0.671256i $$0.765755\pi$$
$$60$$ 0 0
$$61$$ −9.04888 −1.15859 −0.579295 0.815118i $$-0.696672\pi$$
−0.579295 + 0.815118i $$0.696672\pi$$
$$62$$ 0 0
$$63$$ 1.42166i 0.179113i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 2.97028i 0.362878i 0.983402 + 0.181439i $$0.0580754\pi$$
−0.983402 + 0.181439i $$0.941925\pi$$
$$68$$ 0 0
$$69$$ 4.91638 0.591863
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 9.38692i 1.09866i 0.835607 + 0.549328i $$0.185116\pi$$
−0.835607 + 0.549328i $$0.814884\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 2.84333i − 0.324027i
$$78$$ 0 0
$$79$$ −4.37279 −0.491977 −0.245988 0.969273i $$-0.579113\pi$$
−0.245988 + 0.969273i $$0.579113\pi$$
$$80$$ 0 0
$$81$$ −9.78389 −1.08710
$$82$$ 0 0
$$83$$ 0.372787i 0.0409187i 0.999791 + 0.0204593i $$0.00651287\pi$$
−0.999791 + 0.0204593i $$0.993487\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 11.8675i 1.27233i
$$88$$ 0 0
$$89$$ 16.6167 1.76136 0.880681 0.473710i $$-0.157086\pi$$
0.880681 + 0.473710i $$0.157086\pi$$
$$90$$ 0 0
$$91$$ 31.4252 3.29426
$$92$$ 0 0
$$93$$ − 2.57834i − 0.267361i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 3.94610i 0.400666i 0.979728 + 0.200333i $$0.0642024\pi$$
−0.979728 + 0.200333i $$0.935798\pi$$
$$98$$ 0 0
$$99$$ −0.167237 −0.0168079
$$100$$ 0 0
$$101$$ −6.20555 −0.617475 −0.308738 0.951147i $$-0.599907\pi$$
−0.308738 + 0.951147i $$0.599907\pi$$
$$102$$ 0 0
$$103$$ − 8.15165i − 0.803206i −0.915814 0.401603i $$-0.868453\pi$$
0.915814 0.401603i $$-0.131547\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 13.0680i − 1.26333i −0.775240 0.631667i $$-0.782371\pi$$
0.775240 0.631667i $$-0.217629\pi$$
$$108$$ 0 0
$$109$$ 11.5925 1.11036 0.555179 0.831731i $$-0.312650\pi$$
0.555179 + 0.831731i $$0.312650\pi$$
$$110$$ 0 0
$$111$$ −16.5089 −1.56695
$$112$$ 0 0
$$113$$ 14.9355i 1.40502i 0.711675 + 0.702509i $$0.247937\pi$$
−0.711675 + 0.702509i $$0.752063\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 1.84835i − 0.170880i
$$118$$ 0 0
$$119$$ −3.49472 −0.320360
$$120$$ 0 0
$$121$$ −10.6655 −0.969593
$$122$$ 0 0
$$123$$ 20.0383i 1.80679i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 11.8867i 1.05477i 0.849626 + 0.527385i $$0.176828\pi$$
−0.849626 + 0.527385i $$0.823172\pi$$
$$128$$ 0 0
$$129$$ −10.5783 −0.931371
$$130$$ 0 0
$$131$$ −9.15667 −0.800022 −0.400011 0.916510i $$-0.630994\pi$$
−0.400011 + 0.916510i $$0.630994\pi$$
$$132$$ 0 0
$$133$$ 4.91638i 0.426304i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 13.3869i 1.14372i 0.820351 + 0.571861i $$0.193778\pi$$
−0.820351 + 0.571861i $$0.806222\pi$$
$$138$$ 0 0
$$139$$ −3.42166 −0.290222 −0.145111 0.989415i $$-0.546354\pi$$
−0.145111 + 0.989415i $$0.546354\pi$$
$$140$$ 0 0
$$141$$ −2.09775 −0.176663
$$142$$ 0 0
$$143$$ 3.69670i 0.309133i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 31.1411i 2.56847i
$$148$$ 0 0
$$149$$ 3.36222 0.275444 0.137722 0.990471i $$-0.456022\pi$$
0.137722 + 0.990471i $$0.456022\pi$$
$$150$$ 0 0
$$151$$ 21.1950 1.72482 0.862412 0.506207i $$-0.168953\pi$$
0.862412 + 0.506207i $$0.168953\pi$$
$$152$$ 0 0
$$153$$ 0.205550i 0.0166177i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 11.7944i − 0.941300i −0.882320 0.470650i $$-0.844020\pi$$
0.882320 0.470650i $$-0.155980\pi$$
$$158$$ 0 0
$$159$$ −24.0731 −1.90912
$$160$$ 0 0
$$161$$ 13.3275 1.05035
$$162$$ 0 0
$$163$$ 14.2439i 1.11567i 0.829953 + 0.557833i $$0.188367\pi$$
−0.829953 + 0.557833i $$0.811633\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 7.47556i − 0.578476i −0.957257 0.289238i $$-0.906598\pi$$
0.957257 0.289238i $$-0.0934020\pi$$
$$168$$ 0 0
$$169$$ −27.8569 −2.14284
$$170$$ 0 0
$$171$$ 0.289169 0.0221133
$$172$$ 0 0
$$173$$ − 4.05390i − 0.308212i −0.988054 0.154106i $$-0.950750\pi$$
0.988054 0.154106i $$-0.0492498\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 20.6514i 1.55225i
$$178$$ 0 0
$$179$$ −2.95112 −0.220577 −0.110289 0.993900i $$-0.535178\pi$$
−0.110289 + 0.993900i $$0.535178\pi$$
$$180$$ 0 0
$$181$$ 9.66553 0.718433 0.359216 0.933254i $$-0.383044\pi$$
0.359216 + 0.933254i $$0.383044\pi$$
$$182$$ 0 0
$$183$$ 16.4111i 1.21314i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 0.411100i − 0.0300626i
$$188$$ 0 0
$$189$$ −24.1708 −1.75817
$$190$$ 0 0
$$191$$ −15.9058 −1.15090 −0.575452 0.817835i $$-0.695174\pi$$
−0.575452 + 0.817835i $$0.695174\pi$$
$$192$$ 0 0
$$193$$ − 15.6116i − 1.12375i −0.827222 0.561875i $$-0.810080\pi$$
0.827222 0.561875i $$-0.189920\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 13.6655i − 0.973628i −0.873506 0.486814i $$-0.838159\pi$$
0.873506 0.486814i $$-0.161841\pi$$
$$198$$ 0 0
$$199$$ −8.91638 −0.632066 −0.316033 0.948748i $$-0.602351\pi$$
−0.316033 + 0.948748i $$0.602351\pi$$
$$200$$ 0 0
$$201$$ 5.38692 0.379964
$$202$$ 0 0
$$203$$ 32.1708i 2.25795i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 0.783887i − 0.0544839i
$$208$$ 0 0
$$209$$ −0.578337 −0.0400044
$$210$$ 0 0
$$211$$ 19.7980 1.36295 0.681476 0.731840i $$-0.261338\pi$$
0.681476 + 0.731840i $$0.261338\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 6.98944i − 0.474474i
$$218$$ 0 0
$$219$$ 17.0242 1.15039
$$220$$ 0 0
$$221$$ 4.54359 0.305635
$$222$$ 0 0
$$223$$ 1.57331i 0.105357i 0.998612 + 0.0526784i $$0.0167758\pi$$
−0.998612 + 0.0526784i $$0.983224\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 28.0575i − 1.86224i −0.364713 0.931120i $$-0.618833\pi$$
0.364713 0.931120i $$-0.381167\pi$$
$$228$$ 0 0
$$229$$ −2.47054 −0.163258 −0.0816289 0.996663i $$-0.526012\pi$$
−0.0816289 + 0.996663i $$0.526012\pi$$
$$230$$ 0 0
$$231$$ −5.15667 −0.339284
$$232$$ 0 0
$$233$$ − 3.15667i − 0.206801i −0.994640 0.103400i $$-0.967028\pi$$
0.994640 0.103400i $$-0.0329723\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 7.93051i 0.515142i
$$238$$ 0 0
$$239$$ 2.74914 0.177827 0.0889137 0.996039i $$-0.471660\pi$$
0.0889137 + 0.996039i $$0.471660\pi$$
$$240$$ 0 0
$$241$$ 8.88164 0.572117 0.286058 0.958212i $$-0.407655\pi$$
0.286058 + 0.958212i $$0.407655\pi$$
$$242$$ 0 0
$$243$$ 2.99498i 0.192128i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 6.39194i − 0.406710i
$$248$$ 0 0
$$249$$ 0.676089 0.0428454
$$250$$ 0 0
$$251$$ 6.31335 0.398495 0.199248 0.979949i $$-0.436150\pi$$
0.199248 + 0.979949i $$0.436150\pi$$
$$252$$ 0 0
$$253$$ 1.56777i 0.0985651i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 7.83830i 0.488940i 0.969657 + 0.244470i $$0.0786140\pi$$
−0.969657 + 0.244470i $$0.921386\pi$$
$$258$$ 0 0
$$259$$ −44.7527 −2.78080
$$260$$ 0 0
$$261$$ 1.89220 0.117124
$$262$$ 0 0
$$263$$ − 13.8711i − 0.855327i −0.903938 0.427664i $$-0.859337\pi$$
0.903938 0.427664i $$-0.140663\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 30.1361i − 1.84430i
$$268$$ 0 0
$$269$$ 14.0000 0.853595 0.426798 0.904347i $$-0.359642\pi$$
0.426798 + 0.904347i $$0.359642\pi$$
$$270$$ 0 0
$$271$$ −4.07306 −0.247421 −0.123710 0.992318i $$-0.539479\pi$$
−0.123710 + 0.992318i $$0.539479\pi$$
$$272$$ 0 0
$$273$$ − 56.9930i − 3.44937i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 16.1361i 0.969522i 0.874647 + 0.484761i $$0.161093\pi$$
−0.874647 + 0.484761i $$0.838907\pi$$
$$278$$ 0 0
$$279$$ −0.411100 −0.0246119
$$280$$ 0 0
$$281$$ −11.4600 −0.683645 −0.341822 0.939765i $$-0.611044\pi$$
−0.341822 + 0.939765i $$0.611044\pi$$
$$282$$ 0 0
$$283$$ 21.7250i 1.29142i 0.763585 + 0.645708i $$0.223437\pi$$
−0.763585 + 0.645708i $$0.776563\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 54.3205i 3.20644i
$$288$$ 0 0
$$289$$ 16.4947 0.970278
$$290$$ 0 0
$$291$$ 7.15667 0.419532
$$292$$ 0 0
$$293$$ 5.91136i 0.345345i 0.984979 + 0.172673i $$0.0552403\pi$$
−0.984979 + 0.172673i $$0.944760\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 2.84333i − 0.164986i
$$298$$ 0 0
$$299$$ −17.3275 −1.00207
$$300$$ 0 0
$$301$$ −28.6761 −1.65286
$$302$$ 0 0
$$303$$ 11.2544i 0.646550i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 29.7194i − 1.69618i −0.529854 0.848089i $$-0.677753\pi$$
0.529854 0.848089i $$-0.322247\pi$$
$$308$$ 0 0
$$309$$ −14.7839 −0.841026
$$310$$ 0 0
$$311$$ −0.407530 −0.0231089 −0.0115544 0.999933i $$-0.503678\pi$$
−0.0115544 + 0.999933i $$0.503678\pi$$
$$312$$ 0 0
$$313$$ 0.338044i 0.0191074i 0.999954 + 0.00955370i $$0.00304108\pi$$
−0.999954 + 0.00955370i $$0.996959\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 8.75468i − 0.491712i −0.969306 0.245856i $$-0.920931\pi$$
0.969306 0.245856i $$-0.0790690\pi$$
$$318$$ 0 0
$$319$$ −3.78440 −0.211886
$$320$$ 0 0
$$321$$ −23.7003 −1.32282
$$322$$ 0 0
$$323$$ 0.710831i 0.0395517i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 21.0242i − 1.16264i
$$328$$ 0 0
$$329$$ −5.68665 −0.313515
$$330$$ 0 0
$$331$$ 23.8675 1.31188 0.655938 0.754814i $$-0.272273\pi$$
0.655938 + 0.754814i $$0.272273\pi$$
$$332$$ 0 0
$$333$$ 2.63224i 0.144246i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 10.0439i − 0.547124i −0.961854 0.273562i $$-0.911798\pi$$
0.961854 0.273562i $$-0.0882018\pi$$
$$338$$ 0 0
$$339$$ 27.0872 1.47117
$$340$$ 0 0
$$341$$ 0.822200 0.0445246
$$342$$ 0 0
$$343$$ 50.0036i 2.69994i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ − 15.6272i − 0.838913i −0.907775 0.419456i $$-0.862221\pi$$
0.907775 0.419456i $$-0.137779\pi$$
$$348$$ 0 0
$$349$$ −17.2544 −0.923608 −0.461804 0.886982i $$-0.652798\pi$$
−0.461804 + 0.886982i $$0.652798\pi$$
$$350$$ 0 0
$$351$$ 31.4252 1.67735
$$352$$ 0 0
$$353$$ − 27.2197i − 1.44876i −0.689402 0.724379i $$-0.742127\pi$$
0.689402 0.724379i $$-0.257873\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 6.33804i 0.335445i
$$358$$ 0 0
$$359$$ 7.18137 0.379018 0.189509 0.981879i $$-0.439310\pi$$
0.189509 + 0.981879i $$0.439310\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 0 0
$$363$$ 19.3431i 1.01525i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 9.15667i 0.477975i 0.971023 + 0.238987i $$0.0768154\pi$$
−0.971023 + 0.238987i $$0.923185\pi$$
$$368$$ 0 0
$$369$$ 3.19499 0.166324
$$370$$ 0 0
$$371$$ −65.2580 −3.38803
$$372$$ 0 0
$$373$$ 8.75468i 0.453300i 0.973976 + 0.226650i $$0.0727774\pi$$
−0.973976 + 0.226650i $$0.927223\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 41.8263i − 2.15416i
$$378$$ 0 0
$$379$$ 12.9164 0.663470 0.331735 0.943373i $$-0.392366\pi$$
0.331735 + 0.943373i $$0.392366\pi$$
$$380$$ 0 0
$$381$$ 21.5577 1.10444
$$382$$ 0 0
$$383$$ 1.64280i 0.0839431i 0.999119 + 0.0419716i $$0.0133639\pi$$
−0.999119 + 0.0419716i $$0.986636\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 1.68665i 0.0857373i
$$388$$ 0 0
$$389$$ 11.8328 0.599945 0.299972 0.953948i $$-0.403023\pi$$
0.299972 + 0.953948i $$0.403023\pi$$
$$390$$ 0 0
$$391$$ 1.92694 0.0974498
$$392$$ 0 0
$$393$$ 16.6066i 0.837692i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 24.1361i − 1.21135i −0.795710 0.605677i $$-0.792902\pi$$
0.795710 0.605677i $$-0.207098\pi$$
$$398$$ 0 0
$$399$$ 8.91638 0.446377
$$400$$ 0 0
$$401$$ 19.9789 0.997697 0.498849 0.866689i $$-0.333756\pi$$
0.498849 + 0.866689i $$0.333756\pi$$
$$402$$ 0 0
$$403$$ 9.08719i 0.452665i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 5.26447i − 0.260950i
$$408$$ 0 0
$$409$$ −5.66553 −0.280142 −0.140071 0.990141i $$-0.544733\pi$$
−0.140071 + 0.990141i $$0.544733\pi$$
$$410$$ 0 0
$$411$$ 24.2786 1.19758
$$412$$ 0 0
$$413$$ 55.9824i 2.75472i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 6.20555i 0.303887i
$$418$$ 0 0
$$419$$ −22.6550 −1.10677 −0.553384 0.832926i $$-0.686664\pi$$
−0.553384 + 0.832926i $$0.686664\pi$$
$$420$$ 0 0
$$421$$ −25.5330 −1.24440 −0.622202 0.782857i $$-0.713762\pi$$
−0.622202 + 0.782857i $$0.713762\pi$$
$$422$$ 0 0
$$423$$ 0.334474i 0.0162627i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 44.4877i 2.15291i
$$428$$ 0 0
$$429$$ 6.70436 0.323689
$$430$$ 0 0
$$431$$ −13.7944 −0.664455 −0.332228 0.943199i $$-0.607800\pi$$
−0.332228 + 0.943199i $$0.607800\pi$$
$$432$$ 0 0
$$433$$ 29.4444i 1.41501i 0.706710 + 0.707504i $$0.250179\pi$$
−0.706710 + 0.707504i $$0.749821\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 2.71083i − 0.129677i
$$438$$ 0 0
$$439$$ 34.5472 1.64885 0.824423 0.565974i $$-0.191500\pi$$
0.824423 + 0.565974i $$0.191500\pi$$
$$440$$ 0 0
$$441$$ 4.96526 0.236441
$$442$$ 0 0
$$443$$ 23.5577i 1.11926i 0.828742 + 0.559631i $$0.189057\pi$$
−0.828742 + 0.559631i $$0.810943\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 6.09775i − 0.288414i
$$448$$ 0 0
$$449$$ 7.49115 0.353529 0.176765 0.984253i $$-0.443437\pi$$
0.176765 + 0.984253i $$0.443437\pi$$
$$450$$ 0 0
$$451$$ −6.38997 −0.300892
$$452$$ 0 0
$$453$$ − 38.4394i − 1.80604i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 38.5819i 1.80479i 0.430914 + 0.902393i $$0.358191\pi$$
−0.430914 + 0.902393i $$0.641809\pi$$
$$458$$ 0 0
$$459$$ −3.49472 −0.163119
$$460$$ 0 0
$$461$$ −31.9789 −1.48940 −0.744702 0.667397i $$-0.767409\pi$$
−0.744702 + 0.667397i $$0.767409\pi$$
$$462$$ 0 0
$$463$$ − 30.8122i − 1.43196i −0.698120 0.715981i $$-0.745980\pi$$
0.698120 0.715981i $$-0.254020\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 41.4600i 1.91854i 0.282493 + 0.959269i $$0.408839\pi$$
−0.282493 + 0.959269i $$0.591161\pi$$
$$468$$ 0 0
$$469$$ 14.6030 0.674305
$$470$$ 0 0
$$471$$ −21.3905 −0.985622
$$472$$ 0 0
$$473$$ − 3.37330i − 0.155105i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 3.83830i 0.175744i
$$478$$ 0 0
$$479$$ 9.93051 0.453737 0.226868 0.973925i $$-0.427151\pi$$
0.226868 + 0.973925i $$0.427151\pi$$
$$480$$ 0 0
$$481$$ 58.1844 2.65298
$$482$$ 0 0
$$483$$ − 24.1708i − 1.09981i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 19.7789i − 0.896266i −0.893967 0.448133i $$-0.852089\pi$$
0.893967 0.448133i $$-0.147911\pi$$
$$488$$ 0 0
$$489$$ 25.8328 1.16820
$$490$$ 0 0
$$491$$ −16.1461 −0.728664 −0.364332 0.931269i $$-0.618703\pi$$
−0.364332 + 0.931269i $$0.618703\pi$$
$$492$$ 0 0
$$493$$ 4.65139i 0.209488i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 0 0
$$498$$ 0 0
$$499$$ 38.8222 1.73792 0.868960 0.494882i $$-0.164789\pi$$
0.868960 + 0.494882i $$0.164789\pi$$
$$500$$ 0 0
$$501$$ −13.5577 −0.605715
$$502$$ 0 0
$$503$$ − 12.5436i − 0.559291i −0.960103 0.279646i $$-0.909783\pi$$
0.960103 0.279646i $$-0.0902170\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 50.5215i 2.24374i
$$508$$ 0 0
$$509$$ −25.0278 −1.10934 −0.554668 0.832072i $$-0.687155\pi$$
−0.554668 + 0.832072i $$0.687155\pi$$
$$510$$ 0 0
$$511$$ 46.1497 2.04154
$$512$$ 0 0
$$513$$ 4.91638i 0.217064i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 0.668948i − 0.0294203i
$$518$$ 0 0
$$519$$ −7.35218 −0.322725
$$520$$ 0 0
$$521$$ −16.6550 −0.729667 −0.364834 0.931073i $$-0.618874\pi$$
−0.364834 + 0.931073i $$0.618874\pi$$
$$522$$ 0 0
$$523$$ − 24.3608i − 1.06522i −0.846360 0.532611i $$-0.821211\pi$$
0.846360 0.532611i $$-0.178789\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 1.01056i − 0.0440208i
$$528$$ 0 0
$$529$$ 15.6514 0.680495
$$530$$ 0 0
$$531$$ 3.29274 0.142893
$$532$$ 0 0
$$533$$ − 70.6238i − 3.05906i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 5.35218i 0.230964i
$$538$$ 0 0
$$539$$ −9.93051 −0.427738
$$540$$ 0 0
$$541$$ 15.3622 0.660474 0.330237 0.943898i $$-0.392871\pi$$
0.330237 + 0.943898i $$0.392871\pi$$
$$542$$ 0 0
$$543$$ − 17.5295i − 0.752261i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 26.5527i − 1.13531i −0.823266 0.567656i $$-0.807850\pi$$
0.823266 0.567656i $$-0.192150\pi$$
$$548$$ 0 0
$$549$$ 2.61665 0.111676
$$550$$ 0 0
$$551$$ 6.54359 0.278767
$$552$$ 0 0
$$553$$ 21.4983i 0.914200i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 28.9583i − 1.22700i −0.789694 0.613501i $$-0.789761\pi$$
0.789694 0.613501i $$-0.210239\pi$$
$$558$$ 0 0
$$559$$ 37.2827 1.57689
$$560$$ 0 0
$$561$$ −0.745574 −0.0314782
$$562$$ 0 0
$$563$$ 1.64280i 0.0692357i 0.999401 + 0.0346179i $$0.0110214\pi$$
−0.999401 + 0.0346179i $$0.988979\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 48.1013i 2.02007i
$$568$$ 0 0
$$569$$ −14.3416 −0.601232 −0.300616 0.953745i $$-0.597192\pi$$
−0.300616 + 0.953745i $$0.597192\pi$$
$$570$$ 0 0
$$571$$ −39.0177 −1.63284 −0.816420 0.577458i $$-0.804045\pi$$
−0.816420 + 0.577458i $$0.804045\pi$$
$$572$$ 0 0
$$573$$ 28.8469i 1.20510i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 25.4947i 1.06136i 0.847573 + 0.530680i $$0.178063\pi$$
−0.847573 + 0.530680i $$0.821937\pi$$
$$578$$ 0 0
$$579$$ −28.3133 −1.17666
$$580$$ 0 0
$$581$$ 1.83276 0.0760358
$$582$$ 0 0
$$583$$ − 7.67661i − 0.317933i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 9.72496i − 0.401392i −0.979654 0.200696i $$-0.935680\pi$$
0.979654 0.200696i $$-0.0643204\pi$$
$$588$$ 0 0
$$589$$ −1.42166 −0.0585786
$$590$$ 0 0
$$591$$ −24.7839 −1.01947
$$592$$ 0 0
$$593$$ 13.0388i 0.535441i 0.963497 + 0.267720i $$0.0862703\pi$$
−0.963497 + 0.267720i $$0.913730\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16.1708i 0.661827i
$$598$$ 0 0
$$599$$ 18.6861 0.763495 0.381747 0.924267i $$-0.375322\pi$$
0.381747 + 0.924267i $$0.375322\pi$$
$$600$$ 0 0
$$601$$ 12.7355 0.519493 0.259747 0.965677i $$-0.416361\pi$$
0.259747 + 0.965677i $$0.416361\pi$$
$$602$$ 0 0
$$603$$ − 0.858912i − 0.0349776i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 23.0645i 0.936158i 0.883687 + 0.468079i $$0.155054\pi$$
−0.883687 + 0.468079i $$0.844946\pi$$
$$608$$ 0 0
$$609$$ 58.3452 2.36427
$$610$$ 0 0
$$611$$ 7.39340 0.299105
$$612$$ 0 0
$$613$$ − 25.7038i − 1.03817i −0.854723 0.519084i $$-0.826273\pi$$
0.854723 0.519084i $$-0.173727\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 0.205550i − 0.00827514i −0.999991 0.00413757i $$-0.998683\pi$$
0.999991 0.00413757i $$-0.00131703\pi$$
$$618$$ 0 0
$$619$$ 11.0872 0.445632 0.222816 0.974861i $$-0.428475\pi$$
0.222816 + 0.974861i $$0.428475\pi$$
$$620$$ 0 0
$$621$$ 13.3275 0.534813
$$622$$ 0 0
$$623$$ − 81.6938i − 3.27299i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 1.04888i 0.0418881i
$$628$$ 0 0
$$629$$ −6.47054 −0.257997
$$630$$ 0 0
$$631$$ 10.4806 0.417226 0.208613 0.977998i $$-0.433105\pi$$
0.208613 + 0.977998i $$0.433105\pi$$
$$632$$ 0 0
$$633$$ − 35.9058i − 1.42713i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 109.755i − 4.34864i
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 37.3694 1.47600 0.738001 0.674800i $$-0.235770\pi$$
0.738001 + 0.674800i $$0.235770\pi$$
$$642$$ 0 0
$$643$$ − 42.0172i − 1.65700i −0.559992 0.828498i $$-0.689196\pi$$
0.559992 0.828498i $$-0.310804\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 23.4635i 0.922447i 0.887284 + 0.461224i $$0.152589\pi$$
−0.887284 + 0.461224i $$0.847411\pi$$
$$648$$ 0 0
$$649$$ −6.58548 −0.258503
$$650$$ 0 0
$$651$$ −12.6761 −0.496815
$$652$$ 0 0
$$653$$ − 33.6272i − 1.31593i −0.753047 0.657967i $$-0.771417\pi$$
0.753047 0.657967i $$-0.228583\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 2.71440i − 0.105899i
$$658$$ 0 0
$$659$$ −17.9653 −0.699827 −0.349914 0.936782i $$-0.613789\pi$$
−0.349914 + 0.936782i $$0.613789\pi$$
$$660$$ 0 0
$$661$$ 15.5542 0.604987 0.302493 0.953152i $$-0.402181\pi$$
0.302493 + 0.953152i $$0.402181\pi$$
$$662$$ 0 0
$$663$$ − 8.24029i − 0.320026i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 17.7386i − 0.686841i
$$668$$ 0 0
$$669$$ 2.85337 0.110318
$$670$$ 0 0
$$671$$ −5.23330 −0.202029
$$672$$ 0 0
$$673$$ − 12.2111i − 0.470703i −0.971910 0.235351i $$-0.924376\pi$$
0.971910 0.235351i $$-0.0756241\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 43.0680i − 1.65524i −0.561290 0.827619i $$-0.689695\pi$$
0.561290 0.827619i $$-0.310305\pi$$
$$678$$ 0 0
$$679$$ 19.4005 0.744524
$$680$$ 0 0
$$681$$ −50.8852 −1.94993
$$682$$ 0 0
$$683$$ 45.3850i 1.73661i 0.496033 + 0.868303i $$0.334789\pi$$
−0.496033 + 0.868303i $$0.665211\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 4.48059i 0.170945i
$$688$$ 0 0
$$689$$ 84.8440 3.23230
$$690$$ 0 0
$$691$$ −0.508852 −0.0193576 −0.00967882 0.999953i $$-0.503081\pi$$
−0.00967882 + 0.999953i $$0.503081\pi$$
$$692$$ 0 0
$$693$$ 0.822200i 0.0312328i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 7.85389i 0.297487i
$$698$$ 0 0
$$699$$ −5.72496 −0.216538
$$700$$ 0 0
$$701$$ −40.1844 −1.51774 −0.758872 0.651239i $$-0.774249\pi$$
−0.758872 + 0.651239i $$0.774249\pi$$
$$702$$ 0 0
$$703$$ 9.10278i 0.343318i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 30.5089i 1.14740i
$$708$$ 0 0
$$709$$ −20.0978 −0.754787 −0.377393 0.926053i $$-0.623180\pi$$
−0.377393 + 0.926053i $$0.623180\pi$$
$$710$$ 0 0
$$711$$ 1.26447 0.0474214
$$712$$ 0 0
$$713$$ 3.85389i 0.144329i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 4.98587i − 0.186201i
$$718$$ 0 0
$$719$$ −35.5925 −1.32738 −0.663688 0.748010i $$-0.731010\pi$$
−0.663688 + 0.748010i $$0.731010\pi$$
$$720$$ 0 0
$$721$$ −40.0766 −1.49253
$$722$$ 0 0
$$723$$ − 16.1078i − 0.599055i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 20.1708i − 0.748094i −0.927410 0.374047i $$-0.877970\pi$$
0.927410 0.374047i $$-0.122030\pi$$
$$728$$ 0 0
$$729$$ −23.9200 −0.885924
$$730$$ 0 0
$$731$$ −4.14611 −0.153349
$$732$$ 0 0
$$733$$ − 30.1461i − 1.11347i −0.830689 0.556736i $$-0.812053\pi$$
0.830689 0.556736i $$-0.187947\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 1.71782i 0.0632768i
$$738$$ 0 0
$$739$$ −26.0283 −0.957465 −0.478733 0.877961i $$-0.658904\pi$$
−0.478733 + 0.877961i $$0.658904\pi$$
$$740$$ 0 0
$$741$$ −11.5925 −0.425860
$$742$$ 0 0
$$743$$ − 0.221136i − 0.00811269i −0.999992 0.00405635i $$-0.998709\pi$$
0.999992 0.00405635i $$-0.00129118\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 0.107798i − 0.00394413i
$$748$$ 0 0
$$749$$ −64.2474 −2.34755
$$750$$ 0 0
$$751$$ 41.9406 1.53043 0.765216 0.643773i $$-0.222632\pi$$
0.765216 + 0.643773i $$0.222632\pi$$
$$752$$ 0 0
$$753$$ − 11.4499i − 0.417259i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 26.7456i − 0.972084i −0.873935 0.486042i $$-0.838440\pi$$
0.873935 0.486042i $$-0.161560\pi$$
$$758$$ 0 0
$$759$$ 2.84333 0.103206
$$760$$ 0 0
$$761$$ −11.6519 −0.422381 −0.211191 0.977445i $$-0.567734\pi$$
−0.211191 + 0.977445i $$0.567734\pi$$
$$762$$ 0 0
$$763$$ − 56.9930i − 2.06329i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 72.7846i − 2.62810i
$$768$$ 0 0
$$769$$ 9.28917 0.334976 0.167488 0.985874i $$-0.446434\pi$$
0.167488 + 0.985874i $$0.446434\pi$$
$$770$$ 0 0
$$771$$ 14.2156 0.511962
$$772$$ 0 0
$$773$$ 10.3225i 0.371273i 0.982618 + 0.185637i $$0.0594347\pi$$
−0.982618 + 0.185637i $$0.940565\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 81.1638i 2.91174i
$$778$$ 0 0
$$779$$ 11.0489 0.395867
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ 32.1708i 1.14969i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 49.8902i 1.77839i 0.457524 + 0.889197i $$0.348736\pi$$
−0.457524 + 0.889197i $$0.651264\pi$$
$$788$$ 0 0
$$789$$ −25.1567 −0.895601
$$790$$ 0 0
$$791$$ 73.4288 2.61083
$$792$$ 0 0
$$793$$ − 57.8399i − 2.05396i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 28.7436i 1.01815i 0.860722 + 0.509075i $$0.170013\pi$$
−0.860722 + 0.509075i $$0.829987\pi$$
$$798$$ 0 0
$$799$$ −0.822200 −0.0290874
$$800$$ 0 0
$$801$$ −4.80501 −0.169777
$$802$$ 0 0
$$803$$ 5.42880i 0.191578i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 25.3905i − 0.893788i
$$808$$ 0 0
$$809$$ −38.5124 −1.35402 −0.677012 0.735972i $$-0.736726\pi$$
−0.677012 + 0.735972i $$0.736726\pi$$
$$810$$ 0 0
$$811$$ −28.1013 −0.986771 −0.493385 0.869811i $$-0.664241\pi$$
−0.493385 + 0.869811i $$0.664241\pi$$
$$812$$ 0 0
$$813$$ 7.38692i 0.259071i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 5.83276i 0.204063i
$$818$$ 0 0
$$819$$ −9.08719 −0.317532
$$820$$ 0 0
$$821$$ 22.2650 0.777053 0.388527 0.921437i $$-0.372984\pi$$
0.388527 + 0.921437i $$0.372984\pi$$
$$822$$ 0 0
$$823$$ − 39.8363i − 1.38861i −0.719682 0.694304i $$-0.755712\pi$$
0.719682 0.694304i $$-0.244288\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 48.5874i − 1.68955i −0.535121 0.844776i $$-0.679734\pi$$
0.535121 0.844776i $$-0.320266\pi$$
$$828$$ 0 0
$$829$$ 45.9058 1.59437 0.797187 0.603732i $$-0.206320\pi$$
0.797187 + 0.603732i $$0.206320\pi$$
$$830$$ 0 0
$$831$$ 29.2645 1.01517
$$832$$ 0 0
$$833$$ 12.2056i 0.422897i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 6.98944i − 0.241590i
$$838$$ 0 0
$$839$$ 27.9688 0.965591 0.482796 0.875733i $$-0.339621\pi$$
0.482796 + 0.875733i $$0.339621\pi$$
$$840$$ 0 0
$$841$$ 13.8186 0.476504
$$842$$ 0 0
$$843$$ 20.7839i 0.715835i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 52.4358i 1.80172i
$$848$$ 0 0
$$849$$ 39.4005 1.35222
$$850$$ 0 0
$$851$$ 24.6761 0.845885
$$852$$ 0 0
$$853$$ − 7.56777i − 0.259116i −0.991572 0.129558i $$-0.958644\pi$$
0.991572 0.129558i $$-0.0413558\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 21.4756i 0.733591i 0.930302 + 0.366796i $$0.119545\pi$$
−0.930302 + 0.366796i $$0.880455\pi$$
$$858$$ 0 0
$$859$$ 33.0177 1.12655 0.563275 0.826270i $$-0.309541\pi$$
0.563275 + 0.826270i $$0.309541\pi$$
$$860$$ 0 0
$$861$$ 98.5160 3.35742
$$862$$ 0 0
$$863$$ − 18.6605i − 0.635211i −0.948223 0.317605i $$-0.897121\pi$$
0.948223 0.317605i $$-0.102879\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 29.9149i − 1.01596i
$$868$$ 0 0
$$869$$ −2.52894 −0.0857886
$$870$$ 0 0
$$871$$ −18.9859 −0.643312
$$872$$ 0 0
$$873$$ − 1.14109i − 0.0386200i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 57.7925i − 1.95151i −0.218858 0.975757i $$-0.570233\pi$$
0.218858 0.975757i $$-0.429767\pi$$
$$878$$ 0 0
$$879$$ 10.7209 0.361606
$$880$$ 0 0
$$881$$ 36.2338 1.22075 0.610374 0.792113i $$-0.291019\pi$$
0.610374 + 0.792113i $$0.291019\pi$$
$$882$$ 0 0
$$883$$ 2.98944i 0.100603i 0.998734 + 0.0503013i $$0.0160182\pi$$
−0.998734 + 0.0503013i $$0.983982\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 42.2494i − 1.41860i −0.704909 0.709298i $$-0.749012\pi$$
0.704909 0.709298i $$-0.250988\pi$$
$$888$$ 0 0
$$889$$ 58.4394 1.95999
$$890$$ 0 0
$$891$$ −5.65838 −0.189563
$$892$$ 0 0
$$893$$ 1.15667i 0.0387066i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 31.4252i 1.04926i
$$898$$ 0 0
$$899$$ −9.30279 −0.310265
$$900$$ 0 0
$$901$$ −9.43528 −0.314335
$$902$$ 0 0
$$903$$ 52.0071i 1.73069i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 1.97080i 0.0654392i 0.999465 + 0.0327196i $$0.0104168\pi$$
−0.999465 + 0.0327196i $$0.989583\pi$$
$$908$$ 0 0
$$909$$ 1.79445 0.0595181
$$910$$ 0 0
$$911$$ 9.98995 0.330982 0.165491 0.986211i $$-0.447079\pi$$
0.165491 + 0.986211i $$0.447079\pi$$
$$912$$ 0 0
$$913$$ 0.215597i 0.00713520i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 45.0177i 1.48662i
$$918$$ 0 0
$$919$$ −4.24029 −0.139874 −0.0699372 0.997551i $$-0.522280\pi$$
−0.0699372 + 0.997551i $$0.522280\pi$$
$$920$$ 0 0
$$921$$ −53.8993 −1.77604
$$922$$ 0 0
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 2.35720i 0.0774206i
$$928$$ 0 0
$$929$$ 13.1531 0.431539 0.215770 0.976444i $$-0.430774\pi$$
0.215770 + 0.976444i $$0.430774\pi$$
$$930$$ 0 0
$$931$$ 17.1708 0.562750
$$932$$ 0 0
$$933$$ 0.739098i 0.0241970i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 38.5436i − 1.25916i −0.776934 0.629582i $$-0.783226\pi$$
0.776934 0.629582i $$-0.216774\pi$$
$$938$$ 0 0
$$939$$ 0.613080 0.0200071
$$940$$ 0 0
$$941$$ −2.91638 −0.0950713 −0.0475357 0.998870i $$-0.515137\pi$$
−0.0475357 + 0.998870i $$0.515137\pi$$
$$942$$ 0 0
$$943$$ − 29.9516i − 0.975360i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 58.4011i 1.89778i 0.315608 + 0.948890i $$0.397792\pi$$
−0.315608 + 0.948890i $$0.602208\pi$$
$$948$$ 0 0
$$949$$ −60.0007 −1.94770
$$950$$ 0 0
$$951$$ −15.8776 −0.514865
$$952$$ 0 0
$$953$$ 36.8378i 1.19329i 0.802504 + 0.596646i $$0.203501\pi$$
−0.802504 + 0.596646i $$0.796499\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 6.86342i 0.221863i
$$958$$ 0 0
$$959$$ 65.8152 2.12528
$$960$$ 0 0
$$961$$ −28.9789 −0.934802
$$962$$ 0 0
$$963$$ 3.77886i 0.121772i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ − 43.0278i − 1.38368i −0.722051 0.691840i $$-0.756801\pi$$
0.722051 0.691840i $$-0.243199\pi$$
$$968$$ 0 0
$$969$$ 1.28917 0.0414141
$$970$$ 0 0
$$971$$ −49.0177 −1.57305 −0.786526 0.617557i $$-0.788123\pi$$
−0.786526 + 0.617557i $$0.788123\pi$$
$$972$$ 0 0
$$973$$ 16.8222i 0.539295i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 20.2383i 0.647481i 0.946146 + 0.323741i $$0.104941\pi$$
−0.946146 + 0.323741i $$0.895059\pi$$
$$978$$ 0 0
$$979$$ 9.61003 0.307138
$$980$$ 0 0
$$981$$ −3.35218 −0.107027
$$982$$ 0 0
$$983$$ − 25.6811i − 0.819100i −0.912288 0.409550i $$-0.865686\pi$$
0.912288 0.409550i $$-0.134314\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 10.3133i 0.328277i
$$988$$ 0 0
$$989$$ 15.8116 0.502781
$$990$$ 0 0
$$991$$ 8.56829 0.272181 0.136090 0.990696i $$-0.456546\pi$$
0.136090 + 0.990696i $$0.456546\pi$$
$$992$$ 0 0
$$993$$ − 43.2863i − 1.37365i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 29.9688i 0.949122i 0.880223 + 0.474561i $$0.157393\pi$$
−0.880223 + 0.474561i $$0.842607\pi$$
$$998$$ 0 0
$$999$$ −44.7527 −1.41591
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 3800.2.d.n.3649.2 6
5.2 odd 4 3800.2.a.w.1.1 3
5.3 odd 4 760.2.a.i.1.3 3
5.4 even 2 inner 3800.2.d.n.3649.5 6
15.8 even 4 6840.2.a.bm.1.1 3
20.3 even 4 1520.2.a.q.1.1 3
20.7 even 4 7600.2.a.bp.1.3 3
40.3 even 4 6080.2.a.br.1.3 3
40.13 odd 4 6080.2.a.bx.1.1 3

By twisted newform
Twist Min Dim Char Parity Ord Type
760.2.a.i.1.3 3 5.3 odd 4
1520.2.a.q.1.1 3 20.3 even 4
3800.2.a.w.1.1 3 5.2 odd 4
3800.2.d.n.3649.2 6 1.1 even 1 trivial
3800.2.d.n.3649.5 6 5.4 even 2 inner
6080.2.a.br.1.3 3 40.3 even 4
6080.2.a.bx.1.1 3 40.13 odd 4
6840.2.a.bm.1.1 3 15.8 even 4
7600.2.a.bp.1.3 3 20.7 even 4