# Properties

 Label 1216.4.a.bg.1.1 Level $1216$ Weight $4$ Character 1216.1 Self dual yes Analytic conductor $71.746$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,4,Mod(1,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, 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("1216.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1216.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: $$71.7463225670$$ Analytic rank: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664$$ x^7 - 3*x^6 - 121*x^5 + 402*x^4 + 4234*x^3 - 14542*x^2 - 40996*x + 141664 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 608) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$3.39349$$ of defining polynomial Character $$\chi$$ $$=$$ 1216.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-9.45545 q^{3} +11.5945 q^{5} -15.9032 q^{7} +62.4055 q^{9} +O(q^{10})$$ $$q-9.45545 q^{3} +11.5945 q^{5} -15.9032 q^{7} +62.4055 q^{9} -19.4189 q^{11} +69.4979 q^{13} -109.631 q^{15} -112.461 q^{17} -19.0000 q^{19} +150.372 q^{21} -57.3974 q^{23} +9.43209 q^{25} -334.775 q^{27} +59.8778 q^{29} +223.000 q^{31} +183.614 q^{33} -184.390 q^{35} +258.002 q^{37} -657.133 q^{39} -305.175 q^{41} +322.813 q^{43} +723.559 q^{45} +213.936 q^{47} -90.0873 q^{49} +1063.37 q^{51} -348.651 q^{53} -225.152 q^{55} +179.654 q^{57} -56.3816 q^{59} -87.2763 q^{61} -992.449 q^{63} +805.792 q^{65} +1074.10 q^{67} +542.718 q^{69} -725.349 q^{71} +889.389 q^{73} -89.1846 q^{75} +308.823 q^{77} +1258.90 q^{79} +1480.50 q^{81} +227.650 q^{83} -1303.93 q^{85} -566.171 q^{87} -927.529 q^{89} -1105.24 q^{91} -2108.56 q^{93} -220.295 q^{95} +603.489 q^{97} -1211.84 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + 3 q^{3} + 17 q^{5} - 42 q^{7} + 86 q^{9}+O(q^{10})$$ 7 * q + 3 * q^3 + 17 * q^5 - 42 * q^7 + 86 * q^9 $$7 q + 3 q^{3} + 17 q^{5} - 42 q^{7} + 86 q^{9} - 33 q^{11} + 35 q^{13} - 120 q^{15} + 66 q^{17} - 133 q^{19} - 33 q^{21} - 389 q^{23} + 44 q^{25} - 39 q^{27} - 233 q^{29} - 158 q^{31} - 206 q^{33} + 123 q^{35} + 436 q^{37} - 807 q^{39} - 94 q^{41} + 645 q^{43} - 103 q^{45} - 1451 q^{47} + 93 q^{49} + 1741 q^{51} - 3 q^{53} - 1971 q^{55} - 57 q^{57} + 297 q^{59} - 93 q^{61} - 2999 q^{63} - 788 q^{65} + 1641 q^{67} - 945 q^{69} - 2392 q^{71} + 324 q^{73} + 1909 q^{75} + 711 q^{77} - 2492 q^{79} + 143 q^{81} + 310 q^{83} - 2353 q^{85} - 4795 q^{87} - 440 q^{89} - 107 q^{91} - 900 q^{93} - 323 q^{95} - 532 q^{97} - 1591 q^{99}+O(q^{100})$$ 7 * q + 3 * q^3 + 17 * q^5 - 42 * q^7 + 86 * q^9 - 33 * q^11 + 35 * q^13 - 120 * q^15 + 66 * q^17 - 133 * q^19 - 33 * q^21 - 389 * q^23 + 44 * q^25 - 39 * q^27 - 233 * q^29 - 158 * q^31 - 206 * q^33 + 123 * q^35 + 436 * q^37 - 807 * q^39 - 94 * q^41 + 645 * q^43 - 103 * q^45 - 1451 * q^47 + 93 * q^49 + 1741 * q^51 - 3 * q^53 - 1971 * q^55 - 57 * q^57 + 297 * q^59 - 93 * q^61 - 2999 * q^63 - 788 * q^65 + 1641 * q^67 - 945 * q^69 - 2392 * q^71 + 324 * q^73 + 1909 * q^75 + 711 * q^77 - 2492 * q^79 + 143 * q^81 + 310 * q^83 - 2353 * q^85 - 4795 * q^87 - 440 * q^89 - 107 * q^91 - 900 * q^93 - 323 * q^95 - 532 * q^97 - 1591 * q^99

## Coefficient data

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

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ −9.45545 −1.81970 −0.909851 0.414935i $$-0.863804\pi$$
−0.909851 + 0.414935i $$0.863804\pi$$
$$4$$ 0 0
$$5$$ 11.5945 1.03704 0.518521 0.855065i $$-0.326483\pi$$
0.518521 + 0.855065i $$0.326483\pi$$
$$6$$ 0 0
$$7$$ −15.9032 −0.858694 −0.429347 0.903140i $$-0.641256\pi$$
−0.429347 + 0.903140i $$0.641256\pi$$
$$8$$ 0 0
$$9$$ 62.4055 2.31131
$$10$$ 0 0
$$11$$ −19.4189 −0.532274 −0.266137 0.963935i $$-0.585747\pi$$
−0.266137 + 0.963935i $$0.585747\pi$$
$$12$$ 0 0
$$13$$ 69.4979 1.48271 0.741355 0.671113i $$-0.234183\pi$$
0.741355 + 0.671113i $$0.234183\pi$$
$$14$$ 0 0
$$15$$ −109.631 −1.88711
$$16$$ 0 0
$$17$$ −112.461 −1.60446 −0.802230 0.597015i $$-0.796353\pi$$
−0.802230 + 0.597015i $$0.796353\pi$$
$$18$$ 0 0
$$19$$ −19.0000 −0.229416
$$20$$ 0 0
$$21$$ 150.372 1.56257
$$22$$ 0 0
$$23$$ −57.3974 −0.520356 −0.260178 0.965561i $$-0.583781\pi$$
−0.260178 + 0.965561i $$0.583781\pi$$
$$24$$ 0 0
$$25$$ 9.43209 0.0754567
$$26$$ 0 0
$$27$$ −334.775 −2.38620
$$28$$ 0 0
$$29$$ 59.8778 0.383415 0.191707 0.981452i $$-0.438598\pi$$
0.191707 + 0.981452i $$0.438598\pi$$
$$30$$ 0 0
$$31$$ 223.000 1.29200 0.645998 0.763339i $$-0.276441\pi$$
0.645998 + 0.763339i $$0.276441\pi$$
$$32$$ 0 0
$$33$$ 183.614 0.968580
$$34$$ 0 0
$$35$$ −184.390 −0.890502
$$36$$ 0 0
$$37$$ 258.002 1.14636 0.573180 0.819429i $$-0.305709\pi$$
0.573180 + 0.819429i $$0.305709\pi$$
$$38$$ 0 0
$$39$$ −657.133 −2.69809
$$40$$ 0 0
$$41$$ −305.175 −1.16245 −0.581224 0.813743i $$-0.697426\pi$$
−0.581224 + 0.813743i $$0.697426\pi$$
$$42$$ 0 0
$$43$$ 322.813 1.14485 0.572424 0.819958i $$-0.306003\pi$$
0.572424 + 0.819958i $$0.306003\pi$$
$$44$$ 0 0
$$45$$ 723.559 2.39693
$$46$$ 0 0
$$47$$ 213.936 0.663952 0.331976 0.943288i $$-0.392285\pi$$
0.331976 + 0.943288i $$0.392285\pi$$
$$48$$ 0 0
$$49$$ −90.0873 −0.262645
$$50$$ 0 0
$$51$$ 1063.37 2.91964
$$52$$ 0 0
$$53$$ −348.651 −0.903603 −0.451802 0.892118i $$-0.649218\pi$$
−0.451802 + 0.892118i $$0.649218\pi$$
$$54$$ 0 0
$$55$$ −225.152 −0.551990
$$56$$ 0 0
$$57$$ 179.654 0.417468
$$58$$ 0 0
$$59$$ −56.3816 −0.124411 −0.0622056 0.998063i $$-0.519813\pi$$
−0.0622056 + 0.998063i $$0.519813\pi$$
$$60$$ 0 0
$$61$$ −87.2763 −0.183190 −0.0915950 0.995796i $$-0.529197\pi$$
−0.0915950 + 0.995796i $$0.529197\pi$$
$$62$$ 0 0
$$63$$ −992.449 −1.98471
$$64$$ 0 0
$$65$$ 805.792 1.53763
$$66$$ 0 0
$$67$$ 1074.10 1.95854 0.979272 0.202549i $$-0.0649224\pi$$
0.979272 + 0.202549i $$0.0649224\pi$$
$$68$$ 0 0
$$69$$ 542.718 0.946892
$$70$$ 0 0
$$71$$ −725.349 −1.21244 −0.606219 0.795298i $$-0.707315\pi$$
−0.606219 + 0.795298i $$0.707315\pi$$
$$72$$ 0 0
$$73$$ 889.389 1.42596 0.712980 0.701184i $$-0.247345\pi$$
0.712980 + 0.701184i $$0.247345\pi$$
$$74$$ 0 0
$$75$$ −89.1846 −0.137309
$$76$$ 0 0
$$77$$ 308.823 0.457060
$$78$$ 0 0
$$79$$ 1258.90 1.79288 0.896439 0.443168i $$-0.146145\pi$$
0.896439 + 0.443168i $$0.146145\pi$$
$$80$$ 0 0
$$81$$ 1480.50 2.03086
$$82$$ 0 0
$$83$$ 227.650 0.301058 0.150529 0.988606i $$-0.451902\pi$$
0.150529 + 0.988606i $$0.451902\pi$$
$$84$$ 0 0
$$85$$ −1303.93 −1.66389
$$86$$ 0 0
$$87$$ −566.171 −0.697700
$$88$$ 0 0
$$89$$ −927.529 −1.10470 −0.552348 0.833614i $$-0.686268\pi$$
−0.552348 + 0.833614i $$0.686268\pi$$
$$90$$ 0 0
$$91$$ −1105.24 −1.27319
$$92$$ 0 0
$$93$$ −2108.56 −2.35105
$$94$$ 0 0
$$95$$ −220.295 −0.237914
$$96$$ 0 0
$$97$$ 603.489 0.631701 0.315851 0.948809i $$-0.397710\pi$$
0.315851 + 0.948809i $$0.397710\pi$$
$$98$$ 0 0
$$99$$ −1211.84 −1.23025
$$100$$ 0 0
$$101$$ 1035.55 1.02021 0.510106 0.860112i $$-0.329606\pi$$
0.510106 + 0.860112i $$0.329606\pi$$
$$102$$ 0 0
$$103$$ −666.961 −0.638035 −0.319018 0.947749i $$-0.603353\pi$$
−0.319018 + 0.947749i $$0.603353\pi$$
$$104$$ 0 0
$$105$$ 1743.49 1.62045
$$106$$ 0 0
$$107$$ −1369.29 −1.23714 −0.618572 0.785728i $$-0.712289\pi$$
−0.618572 + 0.785728i $$0.712289\pi$$
$$108$$ 0 0
$$109$$ −1326.45 −1.16560 −0.582800 0.812615i $$-0.698043\pi$$
−0.582800 + 0.812615i $$0.698043\pi$$
$$110$$ 0 0
$$111$$ −2439.53 −2.08603
$$112$$ 0 0
$$113$$ −1204.31 −1.00258 −0.501291 0.865279i $$-0.667141\pi$$
−0.501291 + 0.865279i $$0.667141\pi$$
$$114$$ 0 0
$$115$$ −665.493 −0.539631
$$116$$ 0 0
$$117$$ 4337.05 3.42701
$$118$$ 0 0
$$119$$ 1788.49 1.37774
$$120$$ 0 0
$$121$$ −953.907 −0.716685
$$122$$ 0 0
$$123$$ 2885.57 2.11531
$$124$$ 0 0
$$125$$ −1339.95 −0.958790
$$126$$ 0 0
$$127$$ −375.658 −0.262475 −0.131237 0.991351i $$-0.541895\pi$$
−0.131237 + 0.991351i $$0.541895\pi$$
$$128$$ 0 0
$$129$$ −3052.34 −2.08328
$$130$$ 0 0
$$131$$ −2063.01 −1.37592 −0.687962 0.725747i $$-0.741495\pi$$
−0.687962 + 0.725747i $$0.741495\pi$$
$$132$$ 0 0
$$133$$ 302.161 0.196998
$$134$$ 0 0
$$135$$ −3881.54 −2.47459
$$136$$ 0 0
$$137$$ 1905.46 1.18828 0.594141 0.804361i $$-0.297492\pi$$
0.594141 + 0.804361i $$0.297492\pi$$
$$138$$ 0 0
$$139$$ 1869.18 1.14059 0.570293 0.821441i $$-0.306830\pi$$
0.570293 + 0.821441i $$0.306830\pi$$
$$140$$ 0 0
$$141$$ −2022.86 −1.20819
$$142$$ 0 0
$$143$$ −1349.57 −0.789208
$$144$$ 0 0
$$145$$ 694.252 0.397617
$$146$$ 0 0
$$147$$ 851.816 0.477936
$$148$$ 0 0
$$149$$ −2041.70 −1.12257 −0.561284 0.827623i $$-0.689692\pi$$
−0.561284 + 0.827623i $$0.689692\pi$$
$$150$$ 0 0
$$151$$ −1970.09 −1.06175 −0.530874 0.847451i $$-0.678136\pi$$
−0.530874 + 0.847451i $$0.678136\pi$$
$$152$$ 0 0
$$153$$ −7018.19 −3.70841
$$154$$ 0 0
$$155$$ 2585.56 1.33986
$$156$$ 0 0
$$157$$ −945.030 −0.480392 −0.240196 0.970724i $$-0.577212\pi$$
−0.240196 + 0.970724i $$0.577212\pi$$
$$158$$ 0 0
$$159$$ 3296.66 1.64429
$$160$$ 0 0
$$161$$ 912.803 0.446826
$$162$$ 0 0
$$163$$ −345.511 −0.166028 −0.0830139 0.996548i $$-0.526455\pi$$
−0.0830139 + 0.996548i $$0.526455\pi$$
$$164$$ 0 0
$$165$$ 2128.91 1.00446
$$166$$ 0 0
$$167$$ −1554.01 −0.720079 −0.360040 0.932937i $$-0.617237\pi$$
−0.360040 + 0.932937i $$0.617237\pi$$
$$168$$ 0 0
$$169$$ 2632.95 1.19843
$$170$$ 0 0
$$171$$ −1185.70 −0.530252
$$172$$ 0 0
$$173$$ 2065.34 0.907658 0.453829 0.891089i $$-0.350058\pi$$
0.453829 + 0.891089i $$0.350058\pi$$
$$174$$ 0 0
$$175$$ −150.001 −0.0647942
$$176$$ 0 0
$$177$$ 533.113 0.226391
$$178$$ 0 0
$$179$$ −3565.91 −1.48899 −0.744493 0.667631i $$-0.767309\pi$$
−0.744493 + 0.667631i $$0.767309\pi$$
$$180$$ 0 0
$$181$$ −1620.30 −0.665393 −0.332697 0.943034i $$-0.607958\pi$$
−0.332697 + 0.943034i $$0.607958\pi$$
$$182$$ 0 0
$$183$$ 825.237 0.333351
$$184$$ 0 0
$$185$$ 2991.41 1.18882
$$186$$ 0 0
$$187$$ 2183.87 0.854012
$$188$$ 0 0
$$189$$ 5324.00 2.04902
$$190$$ 0 0
$$191$$ 2138.73 0.810226 0.405113 0.914267i $$-0.367232\pi$$
0.405113 + 0.914267i $$0.367232\pi$$
$$192$$ 0 0
$$193$$ −3704.25 −1.38154 −0.690772 0.723073i $$-0.742729\pi$$
−0.690772 + 0.723073i $$0.742729\pi$$
$$194$$ 0 0
$$195$$ −7619.12 −2.79803
$$196$$ 0 0
$$197$$ 520.563 0.188267 0.0941334 0.995560i $$-0.469992\pi$$
0.0941334 + 0.995560i $$0.469992\pi$$
$$198$$ 0 0
$$199$$ 67.7611 0.0241380 0.0120690 0.999927i $$-0.496158\pi$$
0.0120690 + 0.999927i $$0.496158\pi$$
$$200$$ 0 0
$$201$$ −10156.1 −3.56397
$$202$$ 0 0
$$203$$ −952.250 −0.329236
$$204$$ 0 0
$$205$$ −3538.35 −1.20551
$$206$$ 0 0
$$207$$ −3581.91 −1.20271
$$208$$ 0 0
$$209$$ 368.959 0.122112
$$210$$ 0 0
$$211$$ −860.566 −0.280776 −0.140388 0.990097i $$-0.544835\pi$$
−0.140388 + 0.990097i $$0.544835\pi$$
$$212$$ 0 0
$$213$$ 6858.50 2.20628
$$214$$ 0 0
$$215$$ 3742.85 1.18726
$$216$$ 0 0
$$217$$ −3546.41 −1.10943
$$218$$ 0 0
$$219$$ −8409.57 −2.59482
$$220$$ 0 0
$$221$$ −7815.80 −2.37895
$$222$$ 0 0
$$223$$ −6297.10 −1.89096 −0.945482 0.325674i $$-0.894409\pi$$
−0.945482 + 0.325674i $$0.894409\pi$$
$$224$$ 0 0
$$225$$ 588.614 0.174404
$$226$$ 0 0
$$227$$ 10.8392 0.00316928 0.00158464 0.999999i $$-0.499496\pi$$
0.00158464 + 0.999999i $$0.499496\pi$$
$$228$$ 0 0
$$229$$ −4424.48 −1.27676 −0.638379 0.769722i $$-0.720395\pi$$
−0.638379 + 0.769722i $$0.720395\pi$$
$$230$$ 0 0
$$231$$ −2920.06 −0.831713
$$232$$ 0 0
$$233$$ −2410.23 −0.677681 −0.338840 0.940844i $$-0.610035\pi$$
−0.338840 + 0.940844i $$0.610035\pi$$
$$234$$ 0 0
$$235$$ 2480.47 0.688546
$$236$$ 0 0
$$237$$ −11903.5 −3.26250
$$238$$ 0 0
$$239$$ −2833.55 −0.766893 −0.383446 0.923563i $$-0.625263\pi$$
−0.383446 + 0.923563i $$0.625263\pi$$
$$240$$ 0 0
$$241$$ 3255.23 0.870074 0.435037 0.900413i $$-0.356735\pi$$
0.435037 + 0.900413i $$0.356735\pi$$
$$242$$ 0 0
$$243$$ −4959.84 −1.30936
$$244$$ 0 0
$$245$$ −1044.52 −0.272374
$$246$$ 0 0
$$247$$ −1320.46 −0.340157
$$248$$ 0 0
$$249$$ −2152.53 −0.547836
$$250$$ 0 0
$$251$$ −4125.96 −1.03756 −0.518782 0.854907i $$-0.673614\pi$$
−0.518782 + 0.854907i $$0.673614\pi$$
$$252$$ 0 0
$$253$$ 1114.59 0.276972
$$254$$ 0 0
$$255$$ 12329.2 3.02779
$$256$$ 0 0
$$257$$ 6627.29 1.60856 0.804278 0.594253i $$-0.202552\pi$$
0.804278 + 0.594253i $$0.202552\pi$$
$$258$$ 0 0
$$259$$ −4103.07 −0.984372
$$260$$ 0 0
$$261$$ 3736.70 0.886192
$$262$$ 0 0
$$263$$ 823.636 0.193109 0.0965543 0.995328i $$-0.469218\pi$$
0.0965543 + 0.995328i $$0.469218\pi$$
$$264$$ 0 0
$$265$$ −4042.43 −0.937075
$$266$$ 0 0
$$267$$ 8770.20 2.01022
$$268$$ 0 0
$$269$$ 5760.24 1.30561 0.652803 0.757527i $$-0.273593\pi$$
0.652803 + 0.757527i $$0.273593\pi$$
$$270$$ 0 0
$$271$$ −8413.28 −1.88587 −0.942934 0.332980i $$-0.891946\pi$$
−0.942934 + 0.332980i $$0.891946\pi$$
$$272$$ 0 0
$$273$$ 10450.5 2.31683
$$274$$ 0 0
$$275$$ −183.161 −0.0401636
$$276$$ 0 0
$$277$$ 4635.95 1.00558 0.502792 0.864407i $$-0.332306\pi$$
0.502792 + 0.864407i $$0.332306\pi$$
$$278$$ 0 0
$$279$$ 13916.4 2.98621
$$280$$ 0 0
$$281$$ −4108.20 −0.872153 −0.436076 0.899910i $$-0.643632\pi$$
−0.436076 + 0.899910i $$0.643632\pi$$
$$282$$ 0 0
$$283$$ 5079.96 1.06704 0.533519 0.845788i $$-0.320869\pi$$
0.533519 + 0.845788i $$0.320869\pi$$
$$284$$ 0 0
$$285$$ 2082.99 0.432932
$$286$$ 0 0
$$287$$ 4853.27 0.998187
$$288$$ 0 0
$$289$$ 7734.49 1.57429
$$290$$ 0 0
$$291$$ −5706.26 −1.14951
$$292$$ 0 0
$$293$$ −7988.86 −1.59288 −0.796441 0.604716i $$-0.793286\pi$$
−0.796441 + 0.604716i $$0.793286\pi$$
$$294$$ 0 0
$$295$$ −653.716 −0.129020
$$296$$ 0 0
$$297$$ 6500.95 1.27011
$$298$$ 0 0
$$299$$ −3988.99 −0.771537
$$300$$ 0 0
$$301$$ −5133.77 −0.983074
$$302$$ 0 0
$$303$$ −9791.62 −1.85648
$$304$$ 0 0
$$305$$ −1011.92 −0.189976
$$306$$ 0 0
$$307$$ −7003.80 −1.30205 −0.651023 0.759058i $$-0.725660\pi$$
−0.651023 + 0.759058i $$0.725660\pi$$
$$308$$ 0 0
$$309$$ 6306.42 1.16103
$$310$$ 0 0
$$311$$ −2835.42 −0.516983 −0.258492 0.966013i $$-0.583225\pi$$
−0.258492 + 0.966013i $$0.583225\pi$$
$$312$$ 0 0
$$313$$ 669.394 0.120883 0.0604415 0.998172i $$-0.480749\pi$$
0.0604415 + 0.998172i $$0.480749\pi$$
$$314$$ 0 0
$$315$$ −11506.9 −2.05823
$$316$$ 0 0
$$317$$ −5609.44 −0.993872 −0.496936 0.867787i $$-0.665542\pi$$
−0.496936 + 0.867787i $$0.665542\pi$$
$$318$$ 0 0
$$319$$ −1162.76 −0.204082
$$320$$ 0 0
$$321$$ 12947.3 2.25123
$$322$$ 0 0
$$323$$ 2136.76 0.368088
$$324$$ 0 0
$$325$$ 655.510 0.111880
$$326$$ 0 0
$$327$$ 12542.1 2.12105
$$328$$ 0 0
$$329$$ −3402.27 −0.570131
$$330$$ 0 0
$$331$$ −10139.4 −1.68373 −0.841864 0.539689i $$-0.818542\pi$$
−0.841864 + 0.539689i $$0.818542\pi$$
$$332$$ 0 0
$$333$$ 16100.8 2.64960
$$334$$ 0 0
$$335$$ 12453.7 2.03109
$$336$$ 0 0
$$337$$ 946.788 0.153041 0.0765205 0.997068i $$-0.475619\pi$$
0.0765205 + 0.997068i $$0.475619\pi$$
$$338$$ 0 0
$$339$$ 11387.3 1.82440
$$340$$ 0 0
$$341$$ −4330.40 −0.687696
$$342$$ 0 0
$$343$$ 6887.49 1.08423
$$344$$ 0 0
$$345$$ 6292.53 0.981967
$$346$$ 0 0
$$347$$ 6022.07 0.931648 0.465824 0.884878i $$-0.345758\pi$$
0.465824 + 0.884878i $$0.345758\pi$$
$$348$$ 0 0
$$349$$ −2271.70 −0.348427 −0.174214 0.984708i $$-0.555738\pi$$
−0.174214 + 0.984708i $$0.555738\pi$$
$$350$$ 0 0
$$351$$ −23266.1 −3.53805
$$352$$ 0 0
$$353$$ 3519.71 0.530695 0.265348 0.964153i $$-0.414513\pi$$
0.265348 + 0.964153i $$0.414513\pi$$
$$354$$ 0 0
$$355$$ −8410.05 −1.25735
$$356$$ 0 0
$$357$$ −16911.0 −2.50707
$$358$$ 0 0
$$359$$ −3485.84 −0.512467 −0.256234 0.966615i $$-0.582482\pi$$
−0.256234 + 0.966615i $$0.582482\pi$$
$$360$$ 0 0
$$361$$ 361.000 0.0526316
$$362$$ 0 0
$$363$$ 9019.62 1.30415
$$364$$ 0 0
$$365$$ 10312.0 1.47878
$$366$$ 0 0
$$367$$ −13100.4 −1.86331 −0.931654 0.363348i $$-0.881634\pi$$
−0.931654 + 0.363348i $$0.881634\pi$$
$$368$$ 0 0
$$369$$ −19044.6 −2.68678
$$370$$ 0 0
$$371$$ 5544.68 0.775918
$$372$$ 0 0
$$373$$ 13692.7 1.90075 0.950377 0.311101i $$-0.100698\pi$$
0.950377 + 0.311101i $$0.100698\pi$$
$$374$$ 0 0
$$375$$ 12669.8 1.74471
$$376$$ 0 0
$$377$$ 4161.38 0.568493
$$378$$ 0 0
$$379$$ −3198.56 −0.433506 −0.216753 0.976226i $$-0.569547\pi$$
−0.216753 + 0.976226i $$0.569547\pi$$
$$380$$ 0 0
$$381$$ 3552.02 0.477625
$$382$$ 0 0
$$383$$ 10724.5 1.43080 0.715400 0.698716i $$-0.246245\pi$$
0.715400 + 0.698716i $$0.246245\pi$$
$$384$$ 0 0
$$385$$ 3580.64 0.473991
$$386$$ 0 0
$$387$$ 20145.3 2.64610
$$388$$ 0 0
$$389$$ 6806.29 0.887128 0.443564 0.896243i $$-0.353714\pi$$
0.443564 + 0.896243i $$0.353714\pi$$
$$390$$ 0 0
$$391$$ 6454.97 0.834889
$$392$$ 0 0
$$393$$ 19506.7 2.50377
$$394$$ 0 0
$$395$$ 14596.3 1.85929
$$396$$ 0 0
$$397$$ 6878.98 0.869637 0.434819 0.900518i $$-0.356812\pi$$
0.434819 + 0.900518i $$0.356812\pi$$
$$398$$ 0 0
$$399$$ −2857.07 −0.358477
$$400$$ 0 0
$$401$$ 10972.1 1.36639 0.683194 0.730237i $$-0.260590\pi$$
0.683194 + 0.730237i $$0.260590\pi$$
$$402$$ 0 0
$$403$$ 15498.0 1.91566
$$404$$ 0 0
$$405$$ 17165.6 2.10609
$$406$$ 0 0
$$407$$ −5010.12 −0.610178
$$408$$ 0 0
$$409$$ −13952.8 −1.68685 −0.843424 0.537249i $$-0.819464\pi$$
−0.843424 + 0.537249i $$0.819464\pi$$
$$410$$ 0 0
$$411$$ −18017.0 −2.16232
$$412$$ 0 0
$$413$$ 896.650 0.106831
$$414$$ 0 0
$$415$$ 2639.48 0.312210
$$416$$ 0 0
$$417$$ −17673.9 −2.07553
$$418$$ 0 0
$$419$$ 3901.77 0.454925 0.227463 0.973787i $$-0.426957\pi$$
0.227463 + 0.973787i $$0.426957\pi$$
$$420$$ 0 0
$$421$$ −5739.67 −0.664452 −0.332226 0.943200i $$-0.607800\pi$$
−0.332226 + 0.943200i $$0.607800\pi$$
$$422$$ 0 0
$$423$$ 13350.8 1.53460
$$424$$ 0 0
$$425$$ −1060.74 −0.121067
$$426$$ 0 0
$$427$$ 1387.98 0.157304
$$428$$ 0 0
$$429$$ 12760.8 1.43612
$$430$$ 0 0
$$431$$ 4016.82 0.448917 0.224459 0.974484i $$-0.427939\pi$$
0.224459 + 0.974484i $$0.427939\pi$$
$$432$$ 0 0
$$433$$ −10619.1 −1.17857 −0.589286 0.807925i $$-0.700591\pi$$
−0.589286 + 0.807925i $$0.700591\pi$$
$$434$$ 0 0
$$435$$ −6564.46 −0.723545
$$436$$ 0 0
$$437$$ 1090.55 0.119378
$$438$$ 0 0
$$439$$ −1594.11 −0.173309 −0.0866545 0.996238i $$-0.527618\pi$$
−0.0866545 + 0.996238i $$0.527618\pi$$
$$440$$ 0 0
$$441$$ −5621.94 −0.607056
$$442$$ 0 0
$$443$$ 3512.03 0.376662 0.188331 0.982106i $$-0.439692\pi$$
0.188331 + 0.982106i $$0.439692\pi$$
$$444$$ 0 0
$$445$$ −10754.2 −1.14562
$$446$$ 0 0
$$447$$ 19305.2 2.04274
$$448$$ 0 0
$$449$$ 3239.71 0.340515 0.170258 0.985400i $$-0.445540\pi$$
0.170258 + 0.985400i $$0.445540\pi$$
$$450$$ 0 0
$$451$$ 5926.16 0.618741
$$452$$ 0 0
$$453$$ 18628.1 1.93206
$$454$$ 0 0
$$455$$ −12814.7 −1.32036
$$456$$ 0 0
$$457$$ −7149.11 −0.731775 −0.365887 0.930659i $$-0.619234\pi$$
−0.365887 + 0.930659i $$0.619234\pi$$
$$458$$ 0 0
$$459$$ 37649.1 3.82856
$$460$$ 0 0
$$461$$ 1452.36 0.146731 0.0733657 0.997305i $$-0.476626\pi$$
0.0733657 + 0.997305i $$0.476626\pi$$
$$462$$ 0 0
$$463$$ −4887.26 −0.490562 −0.245281 0.969452i $$-0.578880\pi$$
−0.245281 + 0.969452i $$0.578880\pi$$
$$464$$ 0 0
$$465$$ −24447.7 −2.43814
$$466$$ 0 0
$$467$$ −15576.4 −1.54344 −0.771722 0.635960i $$-0.780604\pi$$
−0.771722 + 0.635960i $$0.780604\pi$$
$$468$$ 0 0
$$469$$ −17081.7 −1.68179
$$470$$ 0 0
$$471$$ 8935.68 0.874171
$$472$$ 0 0
$$473$$ −6268.66 −0.609373
$$474$$ 0 0
$$475$$ −179.210 −0.0173110
$$476$$ 0 0
$$477$$ −21757.8 −2.08851
$$478$$ 0 0
$$479$$ 9155.87 0.873366 0.436683 0.899616i $$-0.356153\pi$$
0.436683 + 0.899616i $$0.356153\pi$$
$$480$$ 0 0
$$481$$ 17930.6 1.69972
$$482$$ 0 0
$$483$$ −8630.97 −0.813090
$$484$$ 0 0
$$485$$ 6997.14 0.655101
$$486$$ 0 0
$$487$$ 4855.86 0.451827 0.225914 0.974147i $$-0.427463\pi$$
0.225914 + 0.974147i $$0.427463\pi$$
$$488$$ 0 0
$$489$$ 3266.96 0.302121
$$490$$ 0 0
$$491$$ 13319.0 1.22419 0.612094 0.790785i $$-0.290327\pi$$
0.612094 + 0.790785i $$0.290327\pi$$
$$492$$ 0 0
$$493$$ −6733.92 −0.615173
$$494$$ 0 0
$$495$$ −14050.7 −1.27582
$$496$$ 0 0
$$497$$ 11535.4 1.04111
$$498$$ 0 0
$$499$$ −9859.86 −0.884545 −0.442273 0.896881i $$-0.645828\pi$$
−0.442273 + 0.896881i $$0.645828\pi$$
$$500$$ 0 0
$$501$$ 14693.9 1.31033
$$502$$ 0 0
$$503$$ −9994.26 −0.885929 −0.442964 0.896539i $$-0.646073\pi$$
−0.442964 + 0.896539i $$0.646073\pi$$
$$504$$ 0 0
$$505$$ 12006.7 1.05800
$$506$$ 0 0
$$507$$ −24895.8 −2.18079
$$508$$ 0 0
$$509$$ −9489.87 −0.826387 −0.413194 0.910643i $$-0.635587\pi$$
−0.413194 + 0.910643i $$0.635587\pi$$
$$510$$ 0 0
$$511$$ −14144.2 −1.22446
$$512$$ 0 0
$$513$$ 6360.72 0.547432
$$514$$ 0 0
$$515$$ −7733.07 −0.661670
$$516$$ 0 0
$$517$$ −4154.39 −0.353404
$$518$$ 0 0
$$519$$ −19528.7 −1.65167
$$520$$ 0 0
$$521$$ −14621.5 −1.22952 −0.614759 0.788715i $$-0.710747\pi$$
−0.614759 + 0.788715i $$0.710747\pi$$
$$522$$ 0 0
$$523$$ 7500.48 0.627099 0.313550 0.949572i $$-0.398482\pi$$
0.313550 + 0.949572i $$0.398482\pi$$
$$524$$ 0 0
$$525$$ 1418.32 0.117906
$$526$$ 0 0
$$527$$ −25078.8 −2.07296
$$528$$ 0 0
$$529$$ −8872.54 −0.729230
$$530$$ 0 0
$$531$$ −3518.52 −0.287553
$$532$$ 0 0
$$533$$ −21209.0 −1.72358
$$534$$ 0 0
$$535$$ −15876.2 −1.28297
$$536$$ 0 0
$$537$$ 33717.2 2.70951
$$538$$ 0 0
$$539$$ 1749.39 0.139799
$$540$$ 0 0
$$541$$ 2413.87 0.191831 0.0959153 0.995390i $$-0.469422\pi$$
0.0959153 + 0.995390i $$0.469422\pi$$
$$542$$ 0 0
$$543$$ 15320.7 1.21082
$$544$$ 0 0
$$545$$ −15379.5 −1.20878
$$546$$ 0 0
$$547$$ 21120.6 1.65092 0.825458 0.564464i $$-0.190917\pi$$
0.825458 + 0.564464i $$0.190917\pi$$
$$548$$ 0 0
$$549$$ −5446.52 −0.423410
$$550$$ 0 0
$$551$$ −1137.68 −0.0879614
$$552$$ 0 0
$$553$$ −20020.6 −1.53953
$$554$$ 0 0
$$555$$ −28285.1 −2.16331
$$556$$ 0 0
$$557$$ −16059.4 −1.22165 −0.610823 0.791767i $$-0.709161\pi$$
−0.610823 + 0.791767i $$0.709161\pi$$
$$558$$ 0 0
$$559$$ 22434.8 1.69748
$$560$$ 0 0
$$561$$ −20649.4 −1.55405
$$562$$ 0 0
$$563$$ −19306.8 −1.44527 −0.722634 0.691231i $$-0.757069\pi$$
−0.722634 + 0.691231i $$0.757069\pi$$
$$564$$ 0 0
$$565$$ −13963.3 −1.03972
$$566$$ 0 0
$$567$$ −23544.7 −1.74389
$$568$$ 0 0
$$569$$ −11379.0 −0.838373 −0.419187 0.907900i $$-0.637685\pi$$
−0.419187 + 0.907900i $$0.637685\pi$$
$$570$$ 0 0
$$571$$ 16788.6 1.23044 0.615219 0.788356i $$-0.289068\pi$$
0.615219 + 0.788356i $$0.289068\pi$$
$$572$$ 0 0
$$573$$ −20222.7 −1.47437
$$574$$ 0 0
$$575$$ −541.377 −0.0392643
$$576$$ 0 0
$$577$$ 4084.31 0.294683 0.147342 0.989086i $$-0.452928\pi$$
0.147342 + 0.989086i $$0.452928\pi$$
$$578$$ 0 0
$$579$$ 35025.4 2.51400
$$580$$ 0 0
$$581$$ −3620.37 −0.258516
$$582$$ 0 0
$$583$$ 6770.42 0.480964
$$584$$ 0 0
$$585$$ 50285.8 3.55396
$$586$$ 0 0
$$587$$ −10687.2 −0.751458 −0.375729 0.926730i $$-0.622608\pi$$
−0.375729 + 0.926730i $$0.622608\pi$$
$$588$$ 0 0
$$589$$ −4236.99 −0.296404
$$590$$ 0 0
$$591$$ −4922.15 −0.342590
$$592$$ 0 0
$$593$$ −18697.2 −1.29477 −0.647387 0.762162i $$-0.724138\pi$$
−0.647387 + 0.762162i $$0.724138\pi$$
$$594$$ 0 0
$$595$$ 20736.7 1.42877
$$596$$ 0 0
$$597$$ −640.712 −0.0439239
$$598$$ 0 0
$$599$$ 3455.17 0.235683 0.117842 0.993032i $$-0.462402\pi$$
0.117842 + 0.993032i $$0.462402\pi$$
$$600$$ 0 0
$$601$$ −15030.6 −1.02015 −0.510075 0.860130i $$-0.670382\pi$$
−0.510075 + 0.860130i $$0.670382\pi$$
$$602$$ 0 0
$$603$$ 67029.9 4.52681
$$604$$ 0 0
$$605$$ −11060.1 −0.743232
$$606$$ 0 0
$$607$$ −2229.47 −0.149080 −0.0745398 0.997218i $$-0.523749\pi$$
−0.0745398 + 0.997218i $$0.523749\pi$$
$$608$$ 0 0
$$609$$ 9003.95 0.599111
$$610$$ 0 0
$$611$$ 14868.1 0.984448
$$612$$ 0 0
$$613$$ −14821.0 −0.976534 −0.488267 0.872694i $$-0.662371\pi$$
−0.488267 + 0.872694i $$0.662371\pi$$
$$614$$ 0 0
$$615$$ 33456.7 2.19367
$$616$$ 0 0
$$617$$ −19301.7 −1.25941 −0.629706 0.776833i $$-0.716825\pi$$
−0.629706 + 0.776833i $$0.716825\pi$$
$$618$$ 0 0
$$619$$ 20550.4 1.33440 0.667198 0.744880i $$-0.267493\pi$$
0.667198 + 0.744880i $$0.267493\pi$$
$$620$$ 0 0
$$621$$ 19215.2 1.24167
$$622$$ 0 0
$$623$$ 14750.7 0.948595
$$624$$ 0 0
$$625$$ −16715.0 −1.06976
$$626$$ 0 0
$$627$$ −3488.67 −0.222207
$$628$$ 0 0
$$629$$ −29015.2 −1.83929
$$630$$ 0 0
$$631$$ 6019.78 0.379784 0.189892 0.981805i $$-0.439186\pi$$
0.189892 + 0.981805i $$0.439186\pi$$
$$632$$ 0 0
$$633$$ 8137.04 0.510929
$$634$$ 0 0
$$635$$ −4355.56 −0.272197
$$636$$ 0 0
$$637$$ −6260.88 −0.389427
$$638$$ 0 0
$$639$$ −45265.8 −2.80233
$$640$$ 0 0
$$641$$ 554.836 0.0341883 0.0170941 0.999854i $$-0.494559\pi$$
0.0170941 + 0.999854i $$0.494559\pi$$
$$642$$ 0 0
$$643$$ 25135.5 1.54160 0.770800 0.637077i $$-0.219857\pi$$
0.770800 + 0.637077i $$0.219857\pi$$
$$644$$ 0 0
$$645$$ −35390.3 −2.16045
$$646$$ 0 0
$$647$$ −17081.4 −1.03792 −0.518962 0.854797i $$-0.673682\pi$$
−0.518962 + 0.854797i $$0.673682\pi$$
$$648$$ 0 0
$$649$$ 1094.87 0.0662208
$$650$$ 0 0
$$651$$ 33532.9 2.01883
$$652$$ 0 0
$$653$$ 17375.8 1.04130 0.520649 0.853771i $$-0.325690\pi$$
0.520649 + 0.853771i $$0.325690\pi$$
$$654$$ 0 0
$$655$$ −23919.5 −1.42689
$$656$$ 0 0
$$657$$ 55502.8 3.29584
$$658$$ 0 0
$$659$$ −14944.2 −0.883377 −0.441688 0.897169i $$-0.645620\pi$$
−0.441688 + 0.897169i $$0.645620\pi$$
$$660$$ 0 0
$$661$$ 2829.77 0.166513 0.0832565 0.996528i $$-0.473468\pi$$
0.0832565 + 0.996528i $$0.473468\pi$$
$$662$$ 0 0
$$663$$ 73901.9 4.32898
$$664$$ 0 0
$$665$$ 3503.41 0.204295
$$666$$ 0 0
$$667$$ −3436.83 −0.199512
$$668$$ 0 0
$$669$$ 59541.9 3.44099
$$670$$ 0 0
$$671$$ 1694.81 0.0975072
$$672$$ 0 0
$$673$$ −18325.9 −1.04965 −0.524823 0.851211i $$-0.675869\pi$$
−0.524823 + 0.851211i $$0.675869\pi$$
$$674$$ 0 0
$$675$$ −3157.62 −0.180055
$$676$$ 0 0
$$677$$ −12887.9 −0.731645 −0.365823 0.930685i $$-0.619212\pi$$
−0.365823 + 0.930685i $$0.619212\pi$$
$$678$$ 0 0
$$679$$ −9597.42 −0.542438
$$680$$ 0 0
$$681$$ −102.490 −0.00576714
$$682$$ 0 0
$$683$$ −21490.8 −1.20399 −0.601993 0.798501i $$-0.705626\pi$$
−0.601993 + 0.798501i $$0.705626\pi$$
$$684$$ 0 0
$$685$$ 22092.8 1.23230
$$686$$ 0 0
$$687$$ 41835.4 2.32332
$$688$$ 0 0
$$689$$ −24230.5 −1.33978
$$690$$ 0 0
$$691$$ −7795.43 −0.429164 −0.214582 0.976706i $$-0.568839\pi$$
−0.214582 + 0.976706i $$0.568839\pi$$
$$692$$ 0 0
$$693$$ 19272.2 1.05641
$$694$$ 0 0
$$695$$ 21672.1 1.18284
$$696$$ 0 0
$$697$$ 34320.3 1.86510
$$698$$ 0 0
$$699$$ 22789.8 1.23318
$$700$$ 0 0
$$701$$ 4462.67 0.240446 0.120223 0.992747i $$-0.461639\pi$$
0.120223 + 0.992747i $$0.461639\pi$$
$$702$$ 0 0
$$703$$ −4902.05 −0.262993
$$704$$ 0 0
$$705$$ −23454.0 −1.25295
$$706$$ 0 0
$$707$$ −16468.6 −0.876050
$$708$$ 0 0
$$709$$ 27093.8 1.43516 0.717581 0.696475i $$-0.245249\pi$$
0.717581 + 0.696475i $$0.245249\pi$$
$$710$$ 0 0
$$711$$ 78562.3 4.14390
$$712$$ 0 0
$$713$$ −12799.6 −0.672298
$$714$$ 0 0
$$715$$ −15647.6 −0.818442
$$716$$ 0 0
$$717$$ 26792.5 1.39552
$$718$$ 0 0
$$719$$ 5743.87 0.297928 0.148964 0.988843i $$-0.452406\pi$$
0.148964 + 0.988843i $$0.452406\pi$$
$$720$$ 0 0
$$721$$ 10606.8 0.547877
$$722$$ 0 0
$$723$$ −30779.6 −1.58327
$$724$$ 0 0
$$725$$ 564.772 0.0289312
$$726$$ 0 0
$$727$$ 33173.9 1.69237 0.846185 0.532889i $$-0.178894\pi$$
0.846185 + 0.532889i $$0.178894\pi$$
$$728$$ 0 0
$$729$$ 6924.11 0.351781
$$730$$ 0 0
$$731$$ −36303.9 −1.83686
$$732$$ 0 0
$$733$$ −9406.08 −0.473972 −0.236986 0.971513i $$-0.576160\pi$$
−0.236986 + 0.971513i $$0.576160\pi$$
$$734$$ 0 0
$$735$$ 9876.37 0.495640
$$736$$ 0 0
$$737$$ −20857.9 −1.04248
$$738$$ 0 0
$$739$$ 11407.9 0.567858 0.283929 0.958845i $$-0.408362\pi$$
0.283929 + 0.958845i $$0.408362\pi$$
$$740$$ 0 0
$$741$$ 12485.5 0.618985
$$742$$ 0 0
$$743$$ −25469.4 −1.25758 −0.628790 0.777575i $$-0.716449\pi$$
−0.628790 + 0.777575i $$0.716449\pi$$
$$744$$ 0 0
$$745$$ −23672.5 −1.16415
$$746$$ 0 0
$$747$$ 14206.6 0.695839
$$748$$ 0 0
$$749$$ 21776.2 1.06233
$$750$$ 0 0
$$751$$ −23819.5 −1.15737 −0.578686 0.815550i $$-0.696434\pi$$
−0.578686 + 0.815550i $$0.696434\pi$$
$$752$$ 0 0
$$753$$ 39012.8 1.88806
$$754$$ 0 0
$$755$$ −22842.2 −1.10108
$$756$$ 0 0
$$757$$ −6611.66 −0.317444 −0.158722 0.987323i $$-0.550737\pi$$
−0.158722 + 0.987323i $$0.550737\pi$$
$$758$$ 0 0
$$759$$ −10539.0 −0.504006
$$760$$ 0 0
$$761$$ 29643.4 1.41205 0.706026 0.708185i $$-0.250486\pi$$
0.706026 + 0.708185i $$0.250486\pi$$
$$762$$ 0 0
$$763$$ 21094.8 1.00089
$$764$$ 0 0
$$765$$ −81372.3 −3.84578
$$766$$ 0 0
$$767$$ −3918.40 −0.184466
$$768$$ 0 0
$$769$$ −6329.72 −0.296821 −0.148411 0.988926i $$-0.547416\pi$$
−0.148411 + 0.988926i $$0.547416\pi$$
$$770$$ 0 0
$$771$$ −62664.0 −2.92709
$$772$$ 0 0
$$773$$ 24504.8 1.14020 0.570101 0.821575i $$-0.306904\pi$$
0.570101 + 0.821575i $$0.306904\pi$$
$$774$$ 0 0
$$775$$ 2103.35 0.0974898
$$776$$ 0 0
$$777$$ 38796.4 1.79126
$$778$$ 0 0
$$779$$ 5798.33 0.266684
$$780$$ 0 0
$$781$$ 14085.5 0.645349
$$782$$ 0 0
$$783$$ −20045.6 −0.914904
$$784$$ 0 0
$$785$$ −10957.1 −0.498187
$$786$$ 0 0
$$787$$ 33851.3 1.53325 0.766627 0.642093i $$-0.221934\pi$$
0.766627 + 0.642093i $$0.221934\pi$$
$$788$$ 0 0
$$789$$ −7787.84 −0.351400
$$790$$ 0 0
$$791$$ 19152.4 0.860910
$$792$$ 0 0
$$793$$ −6065.52 −0.271618
$$794$$ 0 0
$$795$$ 38223.0 1.70520
$$796$$ 0 0
$$797$$ 18282.7 0.812556 0.406278 0.913750i $$-0.366826\pi$$
0.406278 + 0.913750i $$0.366826\pi$$
$$798$$ 0 0
$$799$$ −24059.4 −1.06528
$$800$$ 0 0
$$801$$ −57882.9 −2.55330
$$802$$ 0 0
$$803$$ −17270.9 −0.759002
$$804$$ 0 0
$$805$$ 10583.5 0.463377
$$806$$ 0 0
$$807$$ −54465.7 −2.37582
$$808$$ 0 0
$$809$$ −12733.6 −0.553386 −0.276693 0.960958i $$-0.589238\pi$$
−0.276693 + 0.960958i $$0.589238\pi$$
$$810$$ 0 0
$$811$$ 10126.7 0.438469 0.219234 0.975672i $$-0.429644\pi$$
0.219234 + 0.975672i $$0.429644\pi$$
$$812$$ 0 0
$$813$$ 79551.3 3.43172
$$814$$ 0 0
$$815$$ −4006.03 −0.172178
$$816$$ 0 0
$$817$$ −6133.44 −0.262646
$$818$$ 0 0
$$819$$ −68973.1 −2.94275
$$820$$ 0 0
$$821$$ 1944.98 0.0826802 0.0413401 0.999145i $$-0.486837\pi$$
0.0413401 + 0.999145i $$0.486837\pi$$
$$822$$ 0 0
$$823$$ −31712.7 −1.34318 −0.671589 0.740924i $$-0.734388\pi$$
−0.671589 + 0.740924i $$0.734388\pi$$
$$824$$ 0 0
$$825$$ 1731.86 0.0730858
$$826$$ 0 0
$$827$$ −3728.06 −0.156756 −0.0783781 0.996924i $$-0.524974\pi$$
−0.0783781 + 0.996924i $$0.524974\pi$$
$$828$$ 0 0
$$829$$ −13200.4 −0.553037 −0.276518 0.961009i $$-0.589181\pi$$
−0.276518 + 0.961009i $$0.589181\pi$$
$$830$$ 0 0
$$831$$ −43834.9 −1.82986
$$832$$ 0 0
$$833$$ 10131.3 0.421404
$$834$$ 0 0
$$835$$ −18018.0 −0.746753
$$836$$ 0 0
$$837$$ −74654.6 −3.08296
$$838$$ 0 0
$$839$$ 11445.0 0.470949 0.235475 0.971881i $$-0.424336\pi$$
0.235475 + 0.971881i $$0.424336\pi$$
$$840$$ 0 0
$$841$$ −20803.7 −0.852993
$$842$$ 0 0
$$843$$ 38844.9 1.58706
$$844$$ 0 0
$$845$$ 30527.7 1.24282
$$846$$ 0 0
$$847$$ 15170.2 0.615413
$$848$$ 0 0
$$849$$ −48033.2 −1.94169
$$850$$ 0 0
$$851$$ −14808.7 −0.596515
$$852$$ 0 0
$$853$$ 8701.48 0.349276 0.174638 0.984633i $$-0.444124\pi$$
0.174638 + 0.984633i $$0.444124\pi$$
$$854$$ 0 0
$$855$$ −13747.6 −0.549894
$$856$$ 0 0
$$857$$ −720.765 −0.0287291 −0.0143646 0.999897i $$-0.504573\pi$$
−0.0143646 + 0.999897i $$0.504573\pi$$
$$858$$ 0 0
$$859$$ 4678.63 0.185836 0.0929179 0.995674i $$-0.470381\pi$$
0.0929179 + 0.995674i $$0.470381\pi$$
$$860$$ 0 0
$$861$$ −45889.9 −1.81640
$$862$$ 0 0
$$863$$ −7009.92 −0.276501 −0.138250 0.990397i $$-0.544148\pi$$
−0.138250 + 0.990397i $$0.544148\pi$$
$$864$$ 0 0
$$865$$ 23946.5 0.941280
$$866$$ 0 0
$$867$$ −73133.0 −2.86474
$$868$$ 0 0
$$869$$ −24446.4 −0.954302
$$870$$ 0 0
$$871$$ 74647.8 2.90396
$$872$$ 0 0
$$873$$ 37661.0 1.46006
$$874$$ 0 0
$$875$$ 21309.5 0.823307
$$876$$ 0 0
$$877$$ 22737.0 0.875453 0.437727 0.899108i $$-0.355784\pi$$
0.437727 + 0.899108i $$0.355784\pi$$
$$878$$ 0 0
$$879$$ 75538.3 2.89857
$$880$$ 0 0
$$881$$ 7697.02 0.294346 0.147173 0.989111i $$-0.452983\pi$$
0.147173 + 0.989111i $$0.452983\pi$$
$$882$$ 0 0
$$883$$ −7842.91 −0.298907 −0.149454 0.988769i $$-0.547751\pi$$
−0.149454 + 0.988769i $$0.547751\pi$$
$$884$$ 0 0
$$885$$ 6181.18 0.234777
$$886$$ 0 0
$$887$$ −41079.7 −1.55504 −0.777521 0.628857i $$-0.783523\pi$$
−0.777521 + 0.628857i $$0.783523\pi$$
$$888$$ 0 0
$$889$$ 5974.18 0.225385
$$890$$ 0 0
$$891$$ −28749.6 −1.08097
$$892$$ 0 0
$$893$$ −4064.78 −0.152321
$$894$$ 0 0
$$895$$ −41344.9 −1.54414
$$896$$ 0 0
$$897$$ 37717.7 1.40397
$$898$$ 0 0
$$899$$ 13352.7 0.495370
$$900$$ 0 0
$$901$$ 39209.7 1.44979
$$902$$ 0 0
$$903$$ 48542.0 1.78890
$$904$$ 0 0
$$905$$ −18786.6 −0.690041
$$906$$ 0 0
$$907$$ −38639.4 −1.41455 −0.707277 0.706937i $$-0.750076\pi$$
−0.707277 + 0.706937i $$0.750076\pi$$
$$908$$ 0 0
$$909$$ 64624.2 2.35803
$$910$$ 0 0
$$911$$ −18398.0 −0.669103 −0.334551 0.942377i $$-0.608585\pi$$
−0.334551 + 0.942377i $$0.608585\pi$$
$$912$$ 0 0
$$913$$ −4420.70 −0.160245
$$914$$ 0 0
$$915$$ 9568.19 0.345699
$$916$$ 0 0
$$917$$ 32808.5 1.18150
$$918$$ 0 0
$$919$$ −24502.3 −0.879495 −0.439748 0.898121i $$-0.644932\pi$$
−0.439748 + 0.898121i $$0.644932\pi$$
$$920$$ 0 0
$$921$$ 66224.1 2.36934
$$922$$ 0 0
$$923$$ −50410.2 −1.79770
$$924$$ 0 0
$$925$$ 2433.50 0.0865006
$$926$$ 0 0
$$927$$ −41622.0 −1.47470
$$928$$ 0 0
$$929$$ −47616.6 −1.68165 −0.840823 0.541310i $$-0.817928\pi$$
−0.840823 + 0.541310i $$0.817928\pi$$
$$930$$ 0 0
$$931$$ 1711.66 0.0602549
$$932$$ 0 0
$$933$$ 26810.1 0.940755
$$934$$ 0 0
$$935$$ 25320.8 0.885646
$$936$$ 0 0
$$937$$ 21360.6 0.744739 0.372369 0.928085i $$-0.378545\pi$$
0.372369 + 0.928085i $$0.378545\pi$$
$$938$$ 0 0
$$939$$ −6329.42 −0.219971
$$940$$ 0 0
$$941$$ −23923.3 −0.828776 −0.414388 0.910100i $$-0.636004\pi$$
−0.414388 + 0.910100i $$0.636004\pi$$
$$942$$ 0 0
$$943$$ 17516.3 0.604887
$$944$$ 0 0
$$945$$ 61729.0 2.12492
$$946$$ 0 0
$$947$$ −23330.5 −0.800569 −0.400285 0.916391i $$-0.631089\pi$$
−0.400285 + 0.916391i $$0.631089\pi$$
$$948$$ 0 0
$$949$$ 61810.7 2.11429
$$950$$ 0 0
$$951$$ 53039.8 1.80855
$$952$$ 0 0
$$953$$ −25756.2 −0.875472 −0.437736 0.899104i $$-0.644220\pi$$
−0.437736 + 0.899104i $$0.644220\pi$$
$$954$$ 0 0
$$955$$ 24797.5 0.840238
$$956$$ 0 0
$$957$$ 10994.4 0.371368
$$958$$ 0 0
$$959$$ −30303.0 −1.02037
$$960$$ 0 0
$$961$$ 19937.8 0.669255
$$962$$ 0 0
$$963$$ −85451.3 −2.85943
$$964$$ 0 0
$$965$$ −42948.9 −1.43272
$$966$$ 0 0
$$967$$ −2488.74 −0.0827635 −0.0413818 0.999143i $$-0.513176\pi$$
−0.0413818 + 0.999143i $$0.513176\pi$$
$$968$$ 0 0
$$969$$ −20204.0 −0.669811
$$970$$ 0 0
$$971$$ −4909.90 −0.162272 −0.0811360 0.996703i $$-0.525855\pi$$
−0.0811360 + 0.996703i $$0.525855\pi$$
$$972$$ 0 0
$$973$$ −29726.0 −0.979415
$$974$$ 0 0
$$975$$ −6198.14 −0.203589
$$976$$ 0 0
$$977$$ 47452.4 1.55388 0.776938 0.629578i $$-0.216772\pi$$
0.776938 + 0.629578i $$0.216772\pi$$
$$978$$ 0 0
$$979$$ 18011.6 0.588000
$$980$$ 0 0
$$981$$ −82777.5 −2.69407
$$982$$ 0 0
$$983$$ 27802.2 0.902088 0.451044 0.892502i $$-0.351052\pi$$
0.451044 + 0.892502i $$0.351052\pi$$
$$984$$ 0 0
$$985$$ 6035.66 0.195241
$$986$$ 0 0
$$987$$ 32170.0 1.03747
$$988$$ 0 0
$$989$$ −18528.6 −0.595728
$$990$$ 0 0
$$991$$ 26200.5 0.839845 0.419922 0.907560i $$-0.362057\pi$$
0.419922 + 0.907560i $$0.362057\pi$$
$$992$$ 0 0
$$993$$ 95873.0 3.06388
$$994$$ 0 0
$$995$$ 785.655 0.0250321
$$996$$ 0 0
$$997$$ −25254.6 −0.802227 −0.401113 0.916028i $$-0.631377\pi$$
−0.401113 + 0.916028i $$0.631377\pi$$
$$998$$ 0 0
$$999$$ −86372.7 −2.73545
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 1216.4.a.bg.1.1 7
4.3 odd 2 1216.4.a.bf.1.7 7
8.3 odd 2 608.4.a.k.1.1 yes 7
8.5 even 2 608.4.a.j.1.7 7

By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.j.1.7 7 8.5 even 2
608.4.a.k.1.1 yes 7 8.3 odd 2
1216.4.a.bf.1.7 7 4.3 odd 2
1216.4.a.bg.1.1 7 1.1 even 1 trivial