# Properties

 Label 1452.4.a.q.1.1 Level $1452$ Weight $4$ Character 1452.1 Self dual yes Analytic conductor $85.671$ Analytic rank $0$ Dimension $4$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1452,4,Mod(1,1452)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1452, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1452.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1452 = 2^{2} \cdot 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1452.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: $$85.6707733283$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.20959101.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 90x^{2} + 91x + 2026$$ x^4 - 2*x^3 - 90*x^2 + 91*x + 2026 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{5}\cdot 3$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-5.75283$$ of defining polynomial Character $$\chi$$ $$=$$ 1452.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -10.3041 q^{5} -18.4086 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -10.3041 q^{5} -18.4086 q^{7} +9.00000 q^{9} -93.4426 q^{13} -30.9124 q^{15} +75.0340 q^{17} -38.2168 q^{19} -55.2258 q^{21} -107.345 q^{23} -18.8248 q^{25} +27.0000 q^{27} -185.486 q^{29} +285.474 q^{31} +189.685 q^{35} +35.8248 q^{37} -280.328 q^{39} +446.005 q^{41} -295.937 q^{43} -92.7372 q^{45} -66.3868 q^{47} -4.12404 q^{49} +225.102 q^{51} +430.562 q^{53} -114.651 q^{57} -382.691 q^{59} -277.528 q^{61} -165.677 q^{63} +962.845 q^{65} -363.299 q^{67} -322.036 q^{69} +991.954 q^{71} +223.702 q^{73} -56.4744 q^{75} +310.147 q^{79} +81.0000 q^{81} +1073.30 q^{83} -773.161 q^{85} -556.457 q^{87} -519.917 q^{89} +1720.15 q^{91} +856.423 q^{93} +393.791 q^{95} +1091.12 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{3} + 12 q^{5} + 36 q^{9}+O(q^{10})$$ 4 * q + 12 * q^3 + 12 * q^5 + 36 * q^9 $$4 q + 12 q^{3} + 12 q^{5} + 36 q^{9} + 36 q^{15} + 156 q^{23} + 244 q^{25} + 108 q^{27} + 184 q^{31} - 176 q^{37} + 108 q^{45} + 852 q^{47} + 1580 q^{49} + 924 q^{53} - 360 q^{59} - 176 q^{67} + 468 q^{69} + 3276 q^{71} + 732 q^{75} + 324 q^{81} - 3144 q^{89} - 144 q^{91} + 552 q^{93} + 2768 q^{97}+O(q^{100})$$ 4 * q + 12 * q^3 + 12 * q^5 + 36 * q^9 + 36 * q^15 + 156 * q^23 + 244 * q^25 + 108 * q^27 + 184 * q^31 - 176 * q^37 + 108 * q^45 + 852 * q^47 + 1580 * q^49 + 924 * q^53 - 360 * q^59 - 176 * q^67 + 468 * q^69 + 3276 * q^71 + 732 * q^75 + 324 * q^81 - 3144 * q^89 - 144 * q^91 + 552 * q^93 + 2768 * q^97

## 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
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −10.3041 −0.921630 −0.460815 0.887496i $$-0.652443\pi$$
−0.460815 + 0.887496i $$0.652443\pi$$
$$6$$ 0 0
$$7$$ −18.4086 −0.993970 −0.496985 0.867759i $$-0.665560\pi$$
−0.496985 + 0.867759i $$0.665560\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ 0 0
$$12$$ 0 0
$$13$$ −93.4426 −1.99356 −0.996781 0.0801697i $$-0.974454\pi$$
−0.996781 + 0.0801697i $$0.974454\pi$$
$$14$$ 0 0
$$15$$ −30.9124 −0.532103
$$16$$ 0 0
$$17$$ 75.0340 1.07050 0.535248 0.844695i $$-0.320218\pi$$
0.535248 + 0.844695i $$0.320218\pi$$
$$18$$ 0 0
$$19$$ −38.2168 −0.461450 −0.230725 0.973019i $$-0.574110\pi$$
−0.230725 + 0.973019i $$0.574110\pi$$
$$20$$ 0 0
$$21$$ −55.2258 −0.573869
$$22$$ 0 0
$$23$$ −107.345 −0.973177 −0.486589 0.873631i $$-0.661759\pi$$
−0.486589 + 0.873631i $$0.661759\pi$$
$$24$$ 0 0
$$25$$ −18.8248 −0.150598
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −185.486 −1.18772 −0.593859 0.804569i $$-0.702396\pi$$
−0.593859 + 0.804569i $$0.702396\pi$$
$$30$$ 0 0
$$31$$ 285.474 1.65396 0.826979 0.562232i $$-0.190057\pi$$
0.826979 + 0.562232i $$0.190057\pi$$
$$32$$ 0 0
$$33$$ 0 0
$$34$$ 0 0
$$35$$ 189.685 0.916072
$$36$$ 0 0
$$37$$ 35.8248 0.159177 0.0795887 0.996828i $$-0.474639\pi$$
0.0795887 + 0.996828i $$0.474639\pi$$
$$38$$ 0 0
$$39$$ −280.328 −1.15098
$$40$$ 0 0
$$41$$ 446.005 1.69888 0.849442 0.527681i $$-0.176938\pi$$
0.849442 + 0.527681i $$0.176938\pi$$
$$42$$ 0 0
$$43$$ −295.937 −1.04953 −0.524767 0.851246i $$-0.675848\pi$$
−0.524767 + 0.851246i $$0.675848\pi$$
$$44$$ 0 0
$$45$$ −92.7372 −0.307210
$$46$$ 0 0
$$47$$ −66.3868 −0.206032 −0.103016 0.994680i $$-0.532849\pi$$
−0.103016 + 0.994680i $$0.532849\pi$$
$$48$$ 0 0
$$49$$ −4.12404 −0.0120234
$$50$$ 0 0
$$51$$ 225.102 0.618051
$$52$$ 0 0
$$53$$ 430.562 1.11589 0.557946 0.829877i $$-0.311590\pi$$
0.557946 + 0.829877i $$0.311590\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −114.651 −0.266418
$$58$$ 0 0
$$59$$ −382.691 −0.844443 −0.422221 0.906493i $$-0.638750\pi$$
−0.422221 + 0.906493i $$0.638750\pi$$
$$60$$ 0 0
$$61$$ −277.528 −0.582523 −0.291261 0.956644i $$-0.594075\pi$$
−0.291261 + 0.956644i $$0.594075\pi$$
$$62$$ 0 0
$$63$$ −165.677 −0.331323
$$64$$ 0 0
$$65$$ 962.845 1.83733
$$66$$ 0 0
$$67$$ −363.299 −0.662449 −0.331224 0.943552i $$-0.607462\pi$$
−0.331224 + 0.943552i $$0.607462\pi$$
$$68$$ 0 0
$$69$$ −322.036 −0.561864
$$70$$ 0 0
$$71$$ 991.954 1.65807 0.829037 0.559194i $$-0.188889\pi$$
0.829037 + 0.559194i $$0.188889\pi$$
$$72$$ 0 0
$$73$$ 223.702 0.358663 0.179331 0.983789i $$-0.442607\pi$$
0.179331 + 0.983789i $$0.442607\pi$$
$$74$$ 0 0
$$75$$ −56.4744 −0.0869481
$$76$$ 0 0
$$77$$ 0 0
$$78$$ 0 0
$$79$$ 310.147 0.441699 0.220849 0.975308i $$-0.429117\pi$$
0.220849 + 0.975308i $$0.429117\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ 1073.30 1.41939 0.709696 0.704508i $$-0.248832\pi$$
0.709696 + 0.704508i $$0.248832\pi$$
$$84$$ 0 0
$$85$$ −773.161 −0.986600
$$86$$ 0 0
$$87$$ −556.457 −0.685729
$$88$$ 0 0
$$89$$ −519.917 −0.619226 −0.309613 0.950863i $$-0.600200\pi$$
−0.309613 + 0.950863i $$0.600200\pi$$
$$90$$ 0 0
$$91$$ 1720.15 1.98154
$$92$$ 0 0
$$93$$ 856.423 0.954914
$$94$$ 0 0
$$95$$ 393.791 0.425286
$$96$$ 0 0
$$97$$ 1091.12 1.14213 0.571066 0.820904i $$-0.306530\pi$$
0.571066 + 0.820904i $$0.306530\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1187.95 −1.17035 −0.585174 0.810908i $$-0.698974\pi$$
−0.585174 + 0.810908i $$0.698974\pi$$
$$102$$ 0 0
$$103$$ 833.124 0.796992 0.398496 0.917170i $$-0.369532\pi$$
0.398496 + 0.917170i $$0.369532\pi$$
$$104$$ 0 0
$$105$$ 569.054 0.528895
$$106$$ 0 0
$$107$$ −829.573 −0.749513 −0.374756 0.927123i $$-0.622274\pi$$
−0.374756 + 0.927123i $$0.622274\pi$$
$$108$$ 0 0
$$109$$ 201.095 0.176710 0.0883550 0.996089i $$-0.471839\pi$$
0.0883550 + 0.996089i $$0.471839\pi$$
$$110$$ 0 0
$$111$$ 107.474 0.0919011
$$112$$ 0 0
$$113$$ 889.835 0.740784 0.370392 0.928876i $$-0.379223\pi$$
0.370392 + 0.928876i $$0.379223\pi$$
$$114$$ 0 0
$$115$$ 1106.10 0.896909
$$116$$ 0 0
$$117$$ −840.983 −0.664521
$$118$$ 0 0
$$119$$ −1381.27 −1.06404
$$120$$ 0 0
$$121$$ 0 0
$$122$$ 0 0
$$123$$ 1338.02 0.980852
$$124$$ 0 0
$$125$$ 1481.99 1.06043
$$126$$ 0 0
$$127$$ 15.6092 0.0109063 0.00545313 0.999985i $$-0.498264\pi$$
0.00545313 + 0.999985i $$0.498264\pi$$
$$128$$ 0 0
$$129$$ −887.811 −0.605949
$$130$$ 0 0
$$131$$ −418.986 −0.279442 −0.139721 0.990191i $$-0.544621\pi$$
−0.139721 + 0.990191i $$0.544621\pi$$
$$132$$ 0 0
$$133$$ 703.518 0.458667
$$134$$ 0 0
$$135$$ −278.212 −0.177368
$$136$$ 0 0
$$137$$ 2197.54 1.37043 0.685213 0.728343i $$-0.259709\pi$$
0.685213 + 0.728343i $$0.259709\pi$$
$$138$$ 0 0
$$139$$ −32.6182 −0.0199039 −0.00995193 0.999950i $$-0.503168\pi$$
−0.00995193 + 0.999950i $$0.503168\pi$$
$$140$$ 0 0
$$141$$ −199.160 −0.118953
$$142$$ 0 0
$$143$$ 0 0
$$144$$ 0 0
$$145$$ 1911.27 1.09464
$$146$$ 0 0
$$147$$ −12.3721 −0.00694174
$$148$$ 0 0
$$149$$ −3359.73 −1.84725 −0.923625 0.383298i $$-0.874788\pi$$
−0.923625 + 0.383298i $$0.874788\pi$$
$$150$$ 0 0
$$151$$ 3073.81 1.65658 0.828288 0.560303i $$-0.189315\pi$$
0.828288 + 0.560303i $$0.189315\pi$$
$$152$$ 0 0
$$153$$ 675.306 0.356832
$$154$$ 0 0
$$155$$ −2941.57 −1.52434
$$156$$ 0 0
$$157$$ 2295.20 1.16673 0.583365 0.812210i $$-0.301736\pi$$
0.583365 + 0.812210i $$0.301736\pi$$
$$158$$ 0 0
$$159$$ 1291.69 0.644260
$$160$$ 0 0
$$161$$ 1976.08 0.967309
$$162$$ 0 0
$$163$$ 1173.55 0.563922 0.281961 0.959426i $$-0.409015\pi$$
0.281961 + 0.959426i $$0.409015\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ −2874.11 −1.33177 −0.665885 0.746055i $$-0.731946\pi$$
−0.665885 + 0.746055i $$0.731946\pi$$
$$168$$ 0 0
$$169$$ 6534.52 2.97429
$$170$$ 0 0
$$171$$ −343.952 −0.153817
$$172$$ 0 0
$$173$$ 2640.61 1.16047 0.580237 0.814448i $$-0.302960\pi$$
0.580237 + 0.814448i $$0.302960\pi$$
$$174$$ 0 0
$$175$$ 346.538 0.149690
$$176$$ 0 0
$$177$$ −1148.07 −0.487539
$$178$$ 0 0
$$179$$ 3375.89 1.40964 0.704820 0.709386i $$-0.251028\pi$$
0.704820 + 0.709386i $$0.251028\pi$$
$$180$$ 0 0
$$181$$ 785.927 0.322749 0.161374 0.986893i $$-0.448407\pi$$
0.161374 + 0.986893i $$0.448407\pi$$
$$182$$ 0 0
$$183$$ −832.585 −0.336320
$$184$$ 0 0
$$185$$ −369.144 −0.146703
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −497.032 −0.191290
$$190$$ 0 0
$$191$$ −1039.91 −0.393956 −0.196978 0.980408i $$-0.563113\pi$$
−0.196978 + 0.980408i $$0.563113\pi$$
$$192$$ 0 0
$$193$$ 1245.76 0.464620 0.232310 0.972642i $$-0.425372\pi$$
0.232310 + 0.972642i $$0.425372\pi$$
$$194$$ 0 0
$$195$$ 2888.54 1.06078
$$196$$ 0 0
$$197$$ −2892.73 −1.04619 −0.523093 0.852275i $$-0.675222\pi$$
−0.523093 + 0.852275i $$0.675222\pi$$
$$198$$ 0 0
$$199$$ −5130.39 −1.82756 −0.913779 0.406212i $$-0.866849\pi$$
−0.913779 + 0.406212i $$0.866849\pi$$
$$200$$ 0 0
$$201$$ −1089.90 −0.382465
$$202$$ 0 0
$$203$$ 3414.53 1.18056
$$204$$ 0 0
$$205$$ −4595.70 −1.56574
$$206$$ 0 0
$$207$$ −966.109 −0.324392
$$208$$ 0 0
$$209$$ 0 0
$$210$$ 0 0
$$211$$ −2977.57 −0.971489 −0.485744 0.874101i $$-0.661451\pi$$
−0.485744 + 0.874101i $$0.661451\pi$$
$$212$$ 0 0
$$213$$ 2975.86 0.957289
$$214$$ 0 0
$$215$$ 3049.37 0.967282
$$216$$ 0 0
$$217$$ −5255.18 −1.64399
$$218$$ 0 0
$$219$$ 671.107 0.207074
$$220$$ 0 0
$$221$$ −7011.37 −2.13410
$$222$$ 0 0
$$223$$ −1865.55 −0.560208 −0.280104 0.959970i $$-0.590369\pi$$
−0.280104 + 0.959970i $$0.590369\pi$$
$$224$$ 0 0
$$225$$ −169.423 −0.0501995
$$226$$ 0 0
$$227$$ −1950.88 −0.570417 −0.285209 0.958465i $$-0.592063\pi$$
−0.285209 + 0.958465i $$0.592063\pi$$
$$228$$ 0 0
$$229$$ 443.547 0.127993 0.0639966 0.997950i $$-0.479615\pi$$
0.0639966 + 0.997950i $$0.479615\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ 5814.86 1.63495 0.817477 0.575961i $$-0.195372\pi$$
0.817477 + 0.575961i $$0.195372\pi$$
$$234$$ 0 0
$$235$$ 684.059 0.189885
$$236$$ 0 0
$$237$$ 930.440 0.255015
$$238$$ 0 0
$$239$$ −5208.02 −1.40953 −0.704767 0.709439i $$-0.748948\pi$$
−0.704767 + 0.709439i $$0.748948\pi$$
$$240$$ 0 0
$$241$$ −1871.23 −0.500150 −0.250075 0.968226i $$-0.580455\pi$$
−0.250075 + 0.968226i $$0.580455\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 42.4947 0.0110812
$$246$$ 0 0
$$247$$ 3571.08 0.919929
$$248$$ 0 0
$$249$$ 3219.89 0.819487
$$250$$ 0 0
$$251$$ 501.085 0.126009 0.0630044 0.998013i $$-0.479932\pi$$
0.0630044 + 0.998013i $$0.479932\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −2319.48 −0.569614
$$256$$ 0 0
$$257$$ 3578.85 0.868647 0.434324 0.900757i $$-0.356987\pi$$
0.434324 + 0.900757i $$0.356987\pi$$
$$258$$ 0 0
$$259$$ −659.484 −0.158218
$$260$$ 0 0
$$261$$ −1669.37 −0.395906
$$262$$ 0 0
$$263$$ 1769.60 0.414898 0.207449 0.978246i $$-0.433484\pi$$
0.207449 + 0.978246i $$0.433484\pi$$
$$264$$ 0 0
$$265$$ −4436.57 −1.02844
$$266$$ 0 0
$$267$$ −1559.75 −0.357510
$$268$$ 0 0
$$269$$ −3356.68 −0.760819 −0.380409 0.924818i $$-0.624217\pi$$
−0.380409 + 0.924818i $$0.624217\pi$$
$$270$$ 0 0
$$271$$ 7222.10 1.61886 0.809430 0.587216i $$-0.199776\pi$$
0.809430 + 0.587216i $$0.199776\pi$$
$$272$$ 0 0
$$273$$ 5160.44 1.14404
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −5161.74 −1.11963 −0.559817 0.828616i $$-0.689129\pi$$
−0.559817 + 0.828616i $$0.689129\pi$$
$$278$$ 0 0
$$279$$ 2569.27 0.551320
$$280$$ 0 0
$$281$$ −958.646 −0.203516 −0.101758 0.994809i $$-0.532447\pi$$
−0.101758 + 0.994809i $$0.532447\pi$$
$$282$$ 0 0
$$283$$ 1283.98 0.269698 0.134849 0.990866i $$-0.456945\pi$$
0.134849 + 0.990866i $$0.456945\pi$$
$$284$$ 0 0
$$285$$ 1181.37 0.245539
$$286$$ 0 0
$$287$$ −8210.32 −1.68864
$$288$$ 0 0
$$289$$ 717.102 0.145960
$$290$$ 0 0
$$291$$ 3273.37 0.659411
$$292$$ 0 0
$$293$$ −6400.71 −1.27622 −0.638112 0.769944i $$-0.720284\pi$$
−0.638112 + 0.769944i $$0.720284\pi$$
$$294$$ 0 0
$$295$$ 3943.30 0.778264
$$296$$ 0 0
$$297$$ 0 0
$$298$$ 0 0
$$299$$ 10030.6 1.94009
$$300$$ 0 0
$$301$$ 5447.78 1.04321
$$302$$ 0 0
$$303$$ −3563.84 −0.675701
$$304$$ 0 0
$$305$$ 2859.69 0.536870
$$306$$ 0 0
$$307$$ 5637.23 1.04799 0.523996 0.851721i $$-0.324441\pi$$
0.523996 + 0.851721i $$0.324441\pi$$
$$308$$ 0 0
$$309$$ 2499.37 0.460143
$$310$$ 0 0
$$311$$ 8785.63 1.60189 0.800944 0.598739i $$-0.204331\pi$$
0.800944 + 0.598739i $$0.204331\pi$$
$$312$$ 0 0
$$313$$ 5970.23 1.07814 0.539069 0.842261i $$-0.318776\pi$$
0.539069 + 0.842261i $$0.318776\pi$$
$$314$$ 0 0
$$315$$ 1707.16 0.305357
$$316$$ 0 0
$$317$$ −2490.40 −0.441245 −0.220622 0.975359i $$-0.570809\pi$$
−0.220622 + 0.975359i $$0.570809\pi$$
$$318$$ 0 0
$$319$$ 0 0
$$320$$ 0 0
$$321$$ −2488.72 −0.432731
$$322$$ 0 0
$$323$$ −2867.56 −0.493980
$$324$$ 0 0
$$325$$ 1759.04 0.300227
$$326$$ 0 0
$$327$$ 603.284 0.102024
$$328$$ 0 0
$$329$$ 1222.09 0.204790
$$330$$ 0 0
$$331$$ 297.606 0.0494197 0.0247098 0.999695i $$-0.492134\pi$$
0.0247098 + 0.999695i $$0.492134\pi$$
$$332$$ 0 0
$$333$$ 322.423 0.0530591
$$334$$ 0 0
$$335$$ 3743.48 0.610533
$$336$$ 0 0
$$337$$ 11726.6 1.89552 0.947758 0.318991i $$-0.103344\pi$$
0.947758 + 0.318991i $$0.103344\pi$$
$$338$$ 0 0
$$339$$ 2669.50 0.427692
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 0 0
$$343$$ 6390.06 1.00592
$$344$$ 0 0
$$345$$ 3318.31 0.517831
$$346$$ 0 0
$$347$$ −1627.93 −0.251849 −0.125925 0.992040i $$-0.540190\pi$$
−0.125925 + 0.992040i $$0.540190\pi$$
$$348$$ 0 0
$$349$$ 11665.9 1.78929 0.894644 0.446780i $$-0.147429\pi$$
0.894644 + 0.446780i $$0.147429\pi$$
$$350$$ 0 0
$$351$$ −2522.95 −0.383661
$$352$$ 0 0
$$353$$ −2269.34 −0.342166 −0.171083 0.985257i $$-0.554727\pi$$
−0.171083 + 0.985257i $$0.554727\pi$$
$$354$$ 0 0
$$355$$ −10221.2 −1.52813
$$356$$ 0 0
$$357$$ −4143.81 −0.614324
$$358$$ 0 0
$$359$$ 2809.30 0.413006 0.206503 0.978446i $$-0.433792\pi$$
0.206503 + 0.978446i $$0.433792\pi$$
$$360$$ 0 0
$$361$$ −5398.47 −0.787064
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 0 0
$$365$$ −2305.06 −0.330554
$$366$$ 0 0
$$367$$ −3366.60 −0.478842 −0.239421 0.970916i $$-0.576958\pi$$
−0.239421 + 0.970916i $$0.576958\pi$$
$$368$$ 0 0
$$369$$ 4014.05 0.566295
$$370$$ 0 0
$$371$$ −7926.04 −1.10916
$$372$$ 0 0
$$373$$ 2071.89 0.287610 0.143805 0.989606i $$-0.454066\pi$$
0.143805 + 0.989606i $$0.454066\pi$$
$$374$$ 0 0
$$375$$ 4445.97 0.612237
$$376$$ 0 0
$$377$$ 17332.2 2.36779
$$378$$ 0 0
$$379$$ 9563.59 1.29617 0.648085 0.761568i $$-0.275570\pi$$
0.648085 + 0.761568i $$0.275570\pi$$
$$380$$ 0 0
$$381$$ 46.8277 0.00629674
$$382$$ 0 0
$$383$$ 956.509 0.127612 0.0638059 0.997962i $$-0.479676\pi$$
0.0638059 + 0.997962i $$0.479676\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ −2663.43 −0.349845
$$388$$ 0 0
$$389$$ −14004.3 −1.82531 −0.912655 0.408731i $$-0.865972\pi$$
−0.912655 + 0.408731i $$0.865972\pi$$
$$390$$ 0 0
$$391$$ −8054.56 −1.04178
$$392$$ 0 0
$$393$$ −1256.96 −0.161336
$$394$$ 0 0
$$395$$ −3195.79 −0.407083
$$396$$ 0 0
$$397$$ 11361.4 1.43630 0.718152 0.695887i $$-0.244988\pi$$
0.718152 + 0.695887i $$0.244988\pi$$
$$398$$ 0 0
$$399$$ 2110.55 0.264812
$$400$$ 0 0
$$401$$ −6752.88 −0.840954 −0.420477 0.907303i $$-0.638137\pi$$
−0.420477 + 0.907303i $$0.638137\pi$$
$$402$$ 0 0
$$403$$ −26675.5 −3.29727
$$404$$ 0 0
$$405$$ −834.635 −0.102403
$$406$$ 0 0
$$407$$ 0 0
$$408$$ 0 0
$$409$$ −1362.24 −0.164690 −0.0823450 0.996604i $$-0.526241\pi$$
−0.0823450 + 0.996604i $$0.526241\pi$$
$$410$$ 0 0
$$411$$ 6592.61 0.791216
$$412$$ 0 0
$$413$$ 7044.80 0.839351
$$414$$ 0 0
$$415$$ −11059.4 −1.30815
$$416$$ 0 0
$$417$$ −97.8545 −0.0114915
$$418$$ 0 0
$$419$$ 6496.80 0.757493 0.378746 0.925500i $$-0.376355\pi$$
0.378746 + 0.925500i $$0.376355\pi$$
$$420$$ 0 0
$$421$$ −2898.48 −0.335542 −0.167771 0.985826i $$-0.553657\pi$$
−0.167771 + 0.985826i $$0.553657\pi$$
$$422$$ 0 0
$$423$$ −597.481 −0.0686774
$$424$$ 0 0
$$425$$ −1412.50 −0.161215
$$426$$ 0 0
$$427$$ 5108.91 0.579010
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −16041.8 −1.79282 −0.896409 0.443228i $$-0.853833\pi$$
−0.896409 + 0.443228i $$0.853833\pi$$
$$432$$ 0 0
$$433$$ 8413.94 0.933830 0.466915 0.884302i $$-0.345365\pi$$
0.466915 + 0.884302i $$0.345365\pi$$
$$434$$ 0 0
$$435$$ 5733.80 0.631988
$$436$$ 0 0
$$437$$ 4102.41 0.449072
$$438$$ 0 0
$$439$$ −15686.4 −1.70540 −0.852700 0.522400i $$-0.825037\pi$$
−0.852700 + 0.522400i $$0.825037\pi$$
$$440$$ 0 0
$$441$$ −37.1164 −0.00400781
$$442$$ 0 0
$$443$$ 8628.26 0.925374 0.462687 0.886522i $$-0.346885\pi$$
0.462687 + 0.886522i $$0.346885\pi$$
$$444$$ 0 0
$$445$$ 5357.30 0.570697
$$446$$ 0 0
$$447$$ −10079.2 −1.06651
$$448$$ 0 0
$$449$$ −5832.05 −0.612988 −0.306494 0.951873i $$-0.599156\pi$$
−0.306494 + 0.951873i $$0.599156\pi$$
$$450$$ 0 0
$$451$$ 0 0
$$452$$ 0 0
$$453$$ 9221.42 0.956424
$$454$$ 0 0
$$455$$ −17724.6 −1.82625
$$456$$ 0 0
$$457$$ 4795.06 0.490817 0.245408 0.969420i $$-0.421078\pi$$
0.245408 + 0.969420i $$0.421078\pi$$
$$458$$ 0 0
$$459$$ 2025.92 0.206017
$$460$$ 0 0
$$461$$ −11452.6 −1.15706 −0.578528 0.815663i $$-0.696372\pi$$
−0.578528 + 0.815663i $$0.696372\pi$$
$$462$$ 0 0
$$463$$ 13765.5 1.38172 0.690859 0.722989i $$-0.257232\pi$$
0.690859 + 0.722989i $$0.257232\pi$$
$$464$$ 0 0
$$465$$ −8824.70 −0.880077
$$466$$ 0 0
$$467$$ 8670.67 0.859166 0.429583 0.903027i $$-0.358661\pi$$
0.429583 + 0.903027i $$0.358661\pi$$
$$468$$ 0 0
$$469$$ 6687.82 0.658454
$$470$$ 0 0
$$471$$ 6885.59 0.673612
$$472$$ 0 0
$$473$$ 0 0
$$474$$ 0 0
$$475$$ 719.425 0.0694936
$$476$$ 0 0
$$477$$ 3875.06 0.371964
$$478$$ 0 0
$$479$$ −12507.3 −1.19306 −0.596528 0.802592i $$-0.703453\pi$$
−0.596528 + 0.802592i $$0.703453\pi$$
$$480$$ 0 0
$$481$$ −3347.56 −0.317330
$$482$$ 0 0
$$483$$ 5928.23 0.558476
$$484$$ 0 0
$$485$$ −11243.1 −1.05262
$$486$$ 0 0
$$487$$ −17148.5 −1.59563 −0.797817 0.602900i $$-0.794012\pi$$
−0.797817 + 0.602900i $$0.794012\pi$$
$$488$$ 0 0
$$489$$ 3520.64 0.325581
$$490$$ 0 0
$$491$$ 15140.1 1.39157 0.695786 0.718249i $$-0.255056\pi$$
0.695786 + 0.718249i $$0.255056\pi$$
$$492$$ 0 0
$$493$$ −13917.7 −1.27145
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ −18260.5 −1.64808
$$498$$ 0 0
$$499$$ −11101.7 −0.995952 −0.497976 0.867191i $$-0.665923\pi$$
−0.497976 + 0.867191i $$0.665923\pi$$
$$500$$ 0 0
$$501$$ −8622.34 −0.768898
$$502$$ 0 0
$$503$$ 18622.9 1.65080 0.825400 0.564549i $$-0.190950\pi$$
0.825400 + 0.564549i $$0.190950\pi$$
$$504$$ 0 0
$$505$$ 12240.8 1.07863
$$506$$ 0 0
$$507$$ 19603.6 1.71721
$$508$$ 0 0
$$509$$ 2403.28 0.209280 0.104640 0.994510i $$-0.466631\pi$$
0.104640 + 0.994510i $$0.466631\pi$$
$$510$$ 0 0
$$511$$ −4118.04 −0.356500
$$512$$ 0 0
$$513$$ −1031.85 −0.0888060
$$514$$ 0 0
$$515$$ −8584.62 −0.734531
$$516$$ 0 0
$$517$$ 0 0
$$518$$ 0 0
$$519$$ 7921.84 0.670000
$$520$$ 0 0
$$521$$ −7624.25 −0.641122 −0.320561 0.947228i $$-0.603871\pi$$
−0.320561 + 0.947228i $$0.603871\pi$$
$$522$$ 0 0
$$523$$ 12095.3 1.01126 0.505632 0.862749i $$-0.331259\pi$$
0.505632 + 0.862749i $$0.331259\pi$$
$$524$$ 0 0
$$525$$ 1039.61 0.0864238
$$526$$ 0 0
$$527$$ 21420.3 1.77056
$$528$$ 0 0
$$529$$ −643.948 −0.0529257
$$530$$ 0 0
$$531$$ −3444.22 −0.281481
$$532$$ 0 0
$$533$$ −41675.9 −3.38683
$$534$$ 0 0
$$535$$ 8548.03 0.690773
$$536$$ 0 0
$$537$$ 10127.7 0.813857
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 16614.3 1.32034 0.660169 0.751117i $$-0.270485\pi$$
0.660169 + 0.751117i $$0.270485\pi$$
$$542$$ 0 0
$$543$$ 2357.78 0.186339
$$544$$ 0 0
$$545$$ −2072.11 −0.162861
$$546$$ 0 0
$$547$$ 12871.3 1.00610 0.503050 0.864258i $$-0.332211\pi$$
0.503050 + 0.864258i $$0.332211\pi$$
$$548$$ 0 0
$$549$$ −2497.76 −0.194174
$$550$$ 0 0
$$551$$ 7088.67 0.548072
$$552$$ 0 0
$$553$$ −5709.36 −0.439036
$$554$$ 0 0
$$555$$ −1107.43 −0.0846988
$$556$$ 0 0
$$557$$ 5691.26 0.432938 0.216469 0.976289i $$-0.430546\pi$$
0.216469 + 0.976289i $$0.430546\pi$$
$$558$$ 0 0
$$559$$ 27653.1 2.09231
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 17890.6 1.33925 0.669626 0.742699i $$-0.266455\pi$$
0.669626 + 0.742699i $$0.266455\pi$$
$$564$$ 0 0
$$565$$ −9168.98 −0.682729
$$566$$ 0 0
$$567$$ −1491.10 −0.110441
$$568$$ 0 0
$$569$$ −9099.56 −0.670428 −0.335214 0.942142i $$-0.608809\pi$$
−0.335214 + 0.942142i $$0.608809\pi$$
$$570$$ 0 0
$$571$$ −3320.97 −0.243394 −0.121697 0.992567i $$-0.538834\pi$$
−0.121697 + 0.992567i $$0.538834\pi$$
$$572$$ 0 0
$$573$$ −3119.74 −0.227451
$$574$$ 0 0
$$575$$ 2020.76 0.146559
$$576$$ 0 0
$$577$$ −24181.1 −1.74467 −0.872333 0.488912i $$-0.837394\pi$$
−0.872333 + 0.488912i $$0.837394\pi$$
$$578$$ 0 0
$$579$$ 3737.28 0.268249
$$580$$ 0 0
$$581$$ −19757.9 −1.41083
$$582$$ 0 0
$$583$$ 0 0
$$584$$ 0 0
$$585$$ 8665.61 0.612442
$$586$$ 0 0
$$587$$ −26367.9 −1.85403 −0.927017 0.375018i $$-0.877636\pi$$
−0.927017 + 0.375018i $$0.877636\pi$$
$$588$$ 0 0
$$589$$ −10909.9 −0.763219
$$590$$ 0 0
$$591$$ −8678.20 −0.604016
$$592$$ 0 0
$$593$$ 3262.61 0.225935 0.112967 0.993599i $$-0.463964\pi$$
0.112967 + 0.993599i $$0.463964\pi$$
$$594$$ 0 0
$$595$$ 14232.8 0.980651
$$596$$ 0 0
$$597$$ −15391.2 −1.05514
$$598$$ 0 0
$$599$$ −10205.7 −0.696148 −0.348074 0.937467i $$-0.613164\pi$$
−0.348074 + 0.937467i $$0.613164\pi$$
$$600$$ 0 0
$$601$$ −19629.0 −1.33225 −0.666125 0.745840i $$-0.732048\pi$$
−0.666125 + 0.745840i $$0.732048\pi$$
$$602$$ 0 0
$$603$$ −3269.69 −0.220816
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −18047.2 −1.20678 −0.603388 0.797448i $$-0.706183\pi$$
−0.603388 + 0.797448i $$0.706183\pi$$
$$608$$ 0 0
$$609$$ 10243.6 0.681594
$$610$$ 0 0
$$611$$ 6203.36 0.410738
$$612$$ 0 0
$$613$$ −21199.2 −1.39678 −0.698390 0.715717i $$-0.746100\pi$$
−0.698390 + 0.715717i $$0.746100\pi$$
$$614$$ 0 0
$$615$$ −13787.1 −0.903982
$$616$$ 0 0
$$617$$ −24424.5 −1.59367 −0.796835 0.604196i $$-0.793494\pi$$
−0.796835 + 0.604196i $$0.793494\pi$$
$$618$$ 0 0
$$619$$ −6954.04 −0.451545 −0.225773 0.974180i $$-0.572491\pi$$
−0.225773 + 0.974180i $$0.572491\pi$$
$$620$$ 0 0
$$621$$ −2898.33 −0.187288
$$622$$ 0 0
$$623$$ 9570.94 0.615492
$$624$$ 0 0
$$625$$ −12917.5 −0.826722
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 0 0
$$629$$ 2688.08 0.170399
$$630$$ 0 0
$$631$$ 10767.5 0.679314 0.339657 0.940549i $$-0.389689\pi$$
0.339657 + 0.940549i $$0.389689\pi$$
$$632$$ 0 0
$$633$$ −8932.70 −0.560889
$$634$$ 0 0
$$635$$ −160.840 −0.0100515
$$636$$ 0 0
$$637$$ 385.361 0.0239695
$$638$$ 0 0
$$639$$ 8927.58 0.552691
$$640$$ 0 0
$$641$$ −4627.73 −0.285155 −0.142577 0.989784i $$-0.545539\pi$$
−0.142577 + 0.989784i $$0.545539\pi$$
$$642$$ 0 0
$$643$$ −21050.2 −1.29104 −0.645521 0.763743i $$-0.723360\pi$$
−0.645521 + 0.763743i $$0.723360\pi$$
$$644$$ 0 0
$$645$$ 9148.12 0.558461
$$646$$ 0 0
$$647$$ 4869.46 0.295886 0.147943 0.988996i $$-0.452735\pi$$
0.147943 + 0.988996i $$0.452735\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ −15765.5 −0.949155
$$652$$ 0 0
$$653$$ −30026.5 −1.79943 −0.899715 0.436477i $$-0.856226\pi$$
−0.899715 + 0.436477i $$0.856226\pi$$
$$654$$ 0 0
$$655$$ 4317.28 0.257542
$$656$$ 0 0
$$657$$ 2013.32 0.119554
$$658$$ 0 0
$$659$$ 10255.0 0.606189 0.303095 0.952960i $$-0.401980\pi$$
0.303095 + 0.952960i $$0.401980\pi$$
$$660$$ 0 0
$$661$$ 15754.2 0.927031 0.463515 0.886089i $$-0.346588\pi$$
0.463515 + 0.886089i $$0.346588\pi$$
$$662$$ 0 0
$$663$$ −21034.1 −1.23212
$$664$$ 0 0
$$665$$ −7249.14 −0.422721
$$666$$ 0 0
$$667$$ 19911.0 1.15586
$$668$$ 0 0
$$669$$ −5596.64 −0.323436
$$670$$ 0 0
$$671$$ 0 0
$$672$$ 0 0
$$673$$ 6016.90 0.344628 0.172314 0.985042i $$-0.444876\pi$$
0.172314 + 0.985042i $$0.444876\pi$$
$$674$$ 0 0
$$675$$ −508.270 −0.0289827
$$676$$ 0 0
$$677$$ 25373.0 1.44042 0.720209 0.693757i $$-0.244046\pi$$
0.720209 + 0.693757i $$0.244046\pi$$
$$678$$ 0 0
$$679$$ −20086.0 −1.13525
$$680$$ 0 0
$$681$$ −5852.65 −0.329331
$$682$$ 0 0
$$683$$ −3803.49 −0.213084 −0.106542 0.994308i $$-0.533978\pi$$
−0.106542 + 0.994308i $$0.533978\pi$$
$$684$$ 0 0
$$685$$ −22643.7 −1.26303
$$686$$ 0 0
$$687$$ 1330.64 0.0738969
$$688$$ 0 0
$$689$$ −40232.8 −2.22460
$$690$$ 0 0
$$691$$ 15703.7 0.864540 0.432270 0.901744i $$-0.357713\pi$$
0.432270 + 0.901744i $$0.357713\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 336.102 0.0183440
$$696$$ 0 0
$$697$$ 33465.5 1.81865
$$698$$ 0 0
$$699$$ 17444.6 0.943941
$$700$$ 0 0
$$701$$ −16367.6 −0.881879 −0.440939 0.897537i $$-0.645355\pi$$
−0.440939 + 0.897537i $$0.645355\pi$$
$$702$$ 0 0
$$703$$ −1369.11 −0.0734523
$$704$$ 0 0
$$705$$ 2052.18 0.109630
$$706$$ 0 0
$$707$$ 21868.4 1.16329
$$708$$ 0 0
$$709$$ −31262.6 −1.65598 −0.827992 0.560739i $$-0.810517\pi$$
−0.827992 + 0.560739i $$0.810517\pi$$
$$710$$ 0 0
$$711$$ 2791.32 0.147233
$$712$$ 0 0
$$713$$ −30644.4 −1.60960
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −15624.1 −0.813795
$$718$$ 0 0
$$719$$ −29448.9 −1.52748 −0.763741 0.645523i $$-0.776640\pi$$
−0.763741 + 0.645523i $$0.776640\pi$$
$$720$$ 0 0
$$721$$ −15336.6 −0.792186
$$722$$ 0 0
$$723$$ −5613.68 −0.288762
$$724$$ 0 0
$$725$$ 3491.73 0.178868
$$726$$ 0 0
$$727$$ 23185.3 1.18280 0.591399 0.806379i $$-0.298576\pi$$
0.591399 + 0.806379i $$0.298576\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ −22205.3 −1.12352
$$732$$ 0 0
$$733$$ 19688.4 0.992097 0.496049 0.868295i $$-0.334784\pi$$
0.496049 + 0.868295i $$0.334784\pi$$
$$734$$ 0 0
$$735$$ 127.484 0.00639771
$$736$$ 0 0
$$737$$ 0 0
$$738$$ 0 0
$$739$$ −3261.15 −0.162332 −0.0811660 0.996701i $$-0.525864\pi$$
−0.0811660 + 0.996701i $$0.525864\pi$$
$$740$$ 0 0
$$741$$ 10713.2 0.531121
$$742$$ 0 0
$$743$$ 20001.0 0.987573 0.493787 0.869583i $$-0.335612\pi$$
0.493787 + 0.869583i $$0.335612\pi$$
$$744$$ 0 0
$$745$$ 34619.2 1.70248
$$746$$ 0 0
$$747$$ 9659.67 0.473131
$$748$$ 0 0
$$749$$ 15271.3 0.744993
$$750$$ 0 0
$$751$$ −10624.2 −0.516224 −0.258112 0.966115i $$-0.583100\pi$$
−0.258112 + 0.966115i $$0.583100\pi$$
$$752$$ 0 0
$$753$$ 1503.25 0.0727512
$$754$$ 0 0
$$755$$ −31672.9 −1.52675
$$756$$ 0 0
$$757$$ −14839.5 −0.712486 −0.356243 0.934393i $$-0.615942\pi$$
−0.356243 + 0.934393i $$0.615942\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 4401.81 0.209679 0.104839 0.994489i $$-0.466567\pi$$
0.104839 + 0.994489i $$0.466567\pi$$
$$762$$ 0 0
$$763$$ −3701.87 −0.175644
$$764$$ 0 0
$$765$$ −6958.44 −0.328867
$$766$$ 0 0
$$767$$ 35759.6 1.68345
$$768$$ 0 0
$$769$$ −34100.9 −1.59910 −0.799552 0.600597i $$-0.794929\pi$$
−0.799552 + 0.600597i $$0.794929\pi$$
$$770$$ 0 0
$$771$$ 10736.5 0.501514
$$772$$ 0 0
$$773$$ −15004.5 −0.698157 −0.349079 0.937093i $$-0.613505\pi$$
−0.349079 + 0.937093i $$0.613505\pi$$
$$774$$ 0 0
$$775$$ −5374.00 −0.249084
$$776$$ 0 0
$$777$$ −1978.45 −0.0913469
$$778$$ 0 0
$$779$$ −17044.9 −0.783950
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −5008.11 −0.228576
$$784$$ 0 0
$$785$$ −23650.0 −1.07529
$$786$$ 0 0
$$787$$ −15766.9 −0.714143 −0.357072 0.934077i $$-0.616225\pi$$
−0.357072 + 0.934077i $$0.616225\pi$$
$$788$$ 0 0
$$789$$ 5308.79 0.239541
$$790$$ 0 0
$$791$$ −16380.6 −0.736317
$$792$$ 0 0
$$793$$ 25933.0 1.16130
$$794$$ 0 0
$$795$$ −13309.7 −0.593769
$$796$$ 0 0
$$797$$ −8490.24 −0.377340 −0.188670 0.982041i $$-0.560418\pi$$
−0.188670 + 0.982041i $$0.560418\pi$$
$$798$$ 0 0
$$799$$ −4981.27 −0.220557
$$800$$ 0 0
$$801$$ −4679.26 −0.206409
$$802$$ 0 0
$$803$$ 0 0
$$804$$ 0 0
$$805$$ −20361.8 −0.891501
$$806$$ 0 0
$$807$$ −10070.0 −0.439259
$$808$$ 0 0
$$809$$ −41661.0 −1.81054 −0.905268 0.424841i $$-0.860330\pi$$
−0.905268 + 0.424841i $$0.860330\pi$$
$$810$$ 0 0
$$811$$ 38286.1 1.65772 0.828858 0.559458i $$-0.188991\pi$$
0.828858 + 0.559458i $$0.188991\pi$$
$$812$$ 0 0
$$813$$ 21666.3 0.934650
$$814$$ 0 0
$$815$$ −12092.4 −0.519728
$$816$$ 0 0
$$817$$ 11309.8 0.484307
$$818$$ 0 0
$$819$$ 15481.3 0.660514
$$820$$ 0 0
$$821$$ 18942.7 0.805244 0.402622 0.915366i $$-0.368099\pi$$
0.402622 + 0.915366i $$0.368099\pi$$
$$822$$ 0 0
$$823$$ 22445.5 0.950668 0.475334 0.879805i $$-0.342327\pi$$
0.475334 + 0.879805i $$0.342327\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 31648.8 1.33076 0.665380 0.746505i $$-0.268270\pi$$
0.665380 + 0.746505i $$0.268270\pi$$
$$828$$ 0 0
$$829$$ 13695.0 0.573760 0.286880 0.957967i $$-0.407382\pi$$
0.286880 + 0.957967i $$0.407382\pi$$
$$830$$ 0 0
$$831$$ −15485.2 −0.646421
$$832$$ 0 0
$$833$$ −309.443 −0.0128710
$$834$$ 0 0
$$835$$ 29615.2 1.22740
$$836$$ 0 0
$$837$$ 7707.81 0.318305
$$838$$ 0 0
$$839$$ 34249.7 1.40934 0.704668 0.709537i $$-0.251096\pi$$
0.704668 + 0.709537i $$0.251096\pi$$
$$840$$ 0 0
$$841$$ 10015.9 0.410672
$$842$$ 0 0
$$843$$ −2875.94 −0.117500
$$844$$ 0 0
$$845$$ −67332.6 −2.74120
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 0 0
$$849$$ 3851.93 0.155710
$$850$$ 0 0
$$851$$ −3845.63 −0.154908
$$852$$ 0 0
$$853$$ 2046.03 0.0821275 0.0410637 0.999157i $$-0.486925\pi$$
0.0410637 + 0.999157i $$0.486925\pi$$
$$854$$ 0 0
$$855$$ 3544.12 0.141762
$$856$$ 0 0
$$857$$ 281.695 0.0112281 0.00561407 0.999984i $$-0.498213\pi$$
0.00561407 + 0.999984i $$0.498213\pi$$
$$858$$ 0 0
$$859$$ 19534.2 0.775901 0.387951 0.921680i $$-0.373183\pi$$
0.387951 + 0.921680i $$0.373183\pi$$
$$860$$ 0 0
$$861$$ −24631.0 −0.974937
$$862$$ 0 0
$$863$$ 25836.9 1.01912 0.509559 0.860435i $$-0.329808\pi$$
0.509559 + 0.860435i $$0.329808\pi$$
$$864$$ 0 0
$$865$$ −27209.2 −1.06953
$$866$$ 0 0
$$867$$ 2151.31 0.0842701
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 33947.6 1.32063
$$872$$ 0 0
$$873$$ 9820.12 0.380711
$$874$$ 0 0
$$875$$ −27281.3 −1.05403
$$876$$ 0 0
$$877$$ −7433.88 −0.286230 −0.143115 0.989706i $$-0.545712\pi$$
−0.143115 + 0.989706i $$0.545712\pi$$
$$878$$ 0 0
$$879$$ −19202.1 −0.736828
$$880$$ 0 0
$$881$$ −26580.5 −1.01648 −0.508240 0.861215i $$-0.669704\pi$$
−0.508240 + 0.861215i $$0.669704\pi$$
$$882$$ 0 0
$$883$$ 47009.3 1.79161 0.895804 0.444449i $$-0.146600\pi$$
0.895804 + 0.444449i $$0.146600\pi$$
$$884$$ 0 0
$$885$$ 11829.9 0.449331
$$886$$ 0 0
$$887$$ 23948.4 0.906550 0.453275 0.891371i $$-0.350256\pi$$
0.453275 + 0.891371i $$0.350256\pi$$
$$888$$ 0 0
$$889$$ −287.344 −0.0108405
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ 2537.09 0.0950735
$$894$$ 0 0
$$895$$ −34785.6 −1.29917
$$896$$ 0 0
$$897$$ 30091.9 1.12011
$$898$$ 0 0
$$899$$ −52951.4 −1.96443
$$900$$ 0 0
$$901$$ 32306.8 1.19456
$$902$$ 0 0
$$903$$ 16343.3 0.602295
$$904$$ 0 0
$$905$$ −8098.30 −0.297455
$$906$$ 0 0
$$907$$ 8876.24 0.324951 0.162475 0.986713i $$-0.448052\pi$$
0.162475 + 0.986713i $$0.448052\pi$$
$$908$$ 0 0
$$909$$ −10691.5 −0.390116
$$910$$ 0 0
$$911$$ 29594.4 1.07630 0.538148 0.842851i $$-0.319124\pi$$
0.538148 + 0.842851i $$0.319124\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 0 0
$$915$$ 8579.07 0.309962
$$916$$ 0 0
$$917$$ 7712.93 0.277757
$$918$$ 0 0
$$919$$ 37066.9 1.33049 0.665247 0.746623i $$-0.268326\pi$$
0.665247 + 0.746623i $$0.268326\pi$$
$$920$$ 0 0
$$921$$ 16911.7 0.605058
$$922$$ 0 0
$$923$$ −92690.7 −3.30547
$$924$$ 0 0
$$925$$ −674.395 −0.0239719
$$926$$ 0 0
$$927$$ 7498.12 0.265664
$$928$$ 0 0
$$929$$ −39465.4 −1.39378 −0.696888 0.717180i $$-0.745432\pi$$
−0.696888 + 0.717180i $$0.745432\pi$$
$$930$$ 0 0
$$931$$ 157.608 0.00554821
$$932$$ 0 0
$$933$$ 26356.9 0.924851
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −29964.1 −1.04470 −0.522350 0.852731i $$-0.674944\pi$$
−0.522350 + 0.852731i $$0.674944\pi$$
$$938$$ 0 0
$$939$$ 17910.7 0.622464
$$940$$ 0 0
$$941$$ 13533.3 0.468835 0.234417 0.972136i $$-0.424682\pi$$
0.234417 + 0.972136i $$0.424682\pi$$
$$942$$ 0 0
$$943$$ −47876.6 −1.65332
$$944$$ 0 0
$$945$$ 5121.48 0.176298
$$946$$ 0 0
$$947$$ 23764.6 0.815466 0.407733 0.913101i $$-0.366319\pi$$
0.407733 + 0.913101i $$0.366319\pi$$
$$948$$ 0 0
$$949$$ −20903.3 −0.715017
$$950$$ 0 0
$$951$$ −7471.19 −0.254753
$$952$$ 0 0
$$953$$ 44164.0 1.50117 0.750583 0.660776i $$-0.229773\pi$$
0.750583 + 0.660776i $$0.229773\pi$$
$$954$$ 0 0
$$955$$ 10715.4 0.363081
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −40453.6 −1.36216
$$960$$ 0 0
$$961$$ 51704.6 1.73558
$$962$$ 0 0
$$963$$ −7466.16 −0.249838
$$964$$ 0 0
$$965$$ −12836.5 −0.428208
$$966$$ 0 0
$$967$$ 12183.8 0.405174 0.202587 0.979264i $$-0.435065\pi$$
0.202587 + 0.979264i $$0.435065\pi$$
$$968$$ 0 0
$$969$$ −8602.69 −0.285199
$$970$$ 0 0
$$971$$ 53903.0 1.78149 0.890747 0.454500i $$-0.150182\pi$$
0.890747 + 0.454500i $$0.150182\pi$$
$$972$$ 0 0
$$973$$ 600.454 0.0197838
$$974$$ 0 0
$$975$$ 5277.12 0.173336
$$976$$ 0 0
$$977$$ 23888.9 0.782264 0.391132 0.920334i $$-0.372084\pi$$
0.391132 + 0.920334i $$0.372084\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 1809.85 0.0589033
$$982$$ 0 0
$$983$$ −54001.2 −1.75216 −0.876079 0.482167i $$-0.839850\pi$$
−0.876079 + 0.482167i $$0.839850\pi$$
$$984$$ 0 0
$$985$$ 29807.1 0.964197
$$986$$ 0 0
$$987$$ 3666.26 0.118235
$$988$$ 0 0
$$989$$ 31767.5 1.02138
$$990$$ 0 0
$$991$$ −8358.61 −0.267931 −0.133966 0.990986i $$-0.542771\pi$$
−0.133966 + 0.990986i $$0.542771\pi$$
$$992$$ 0 0
$$993$$ 892.818 0.0285325
$$994$$ 0 0
$$995$$ 52864.3 1.68433
$$996$$ 0 0
$$997$$ −11623.2 −0.369219 −0.184610 0.982812i $$-0.559102\pi$$
−0.184610 + 0.982812i $$0.559102\pi$$
$$998$$ 0 0
$$999$$ 967.270 0.0306337
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 1452.4.a.q.1.1 4
11.10 odd 2 inner 1452.4.a.q.1.2 yes 4

By twisted newform
Twist Min Dim Char Parity Ord Type
1452.4.a.q.1.1 4 1.1 even 1 trivial
1452.4.a.q.1.2 yes 4 11.10 odd 2 inner