# Properties

 Label 704.4.a.n.1.1 Level $704$ Weight $4$ Character 704.1 Self dual yes Analytic conductor $41.537$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,4,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, 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("704.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$704 = 2^{6} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 704.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: $$41.5373446440$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{12})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 3$$ x^2 - 3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 11) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.73205$$ of defining polynomial Character $$\chi$$ $$=$$ 704.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-7.92820 q^{3} -14.8564 q^{5} -3.07180 q^{7} +35.8564 q^{9} +O(q^{10})$$ $$q-7.92820 q^{3} -14.8564 q^{5} -3.07180 q^{7} +35.8564 q^{9} -11.0000 q^{11} -5.35898 q^{13} +117.785 q^{15} -41.2154 q^{17} +139.923 q^{19} +24.3538 q^{21} +111.354 q^{23} +95.7128 q^{25} -70.2154 q^{27} +24.9948 q^{29} -31.4974 q^{31} +87.2102 q^{33} +45.6359 q^{35} -13.1436 q^{37} +42.4871 q^{39} +261.072 q^{41} -57.7128 q^{43} -532.697 q^{45} +343.846 q^{47} -333.564 q^{49} +326.764 q^{51} +342.995 q^{53} +163.420 q^{55} -1109.34 q^{57} +88.3693 q^{59} -738.697 q^{61} -110.144 q^{63} +79.6152 q^{65} +342.359 q^{67} -882.836 q^{69} +207.364 q^{71} -1010.60 q^{73} -758.831 q^{75} +33.7898 q^{77} -1294.23 q^{79} -411.441 q^{81} +441.846 q^{83} +612.313 q^{85} -198.164 q^{87} -1489.11 q^{89} +16.4617 q^{91} +249.718 q^{93} -2078.75 q^{95} +1346.42 q^{97} -394.420 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 2 q^{5} - 20 q^{7} + 44 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 2 * q^5 - 20 * q^7 + 44 * q^9 $$2 q - 2 q^{3} - 2 q^{5} - 20 q^{7} + 44 q^{9} - 22 q^{11} - 80 q^{13} + 194 q^{15} - 124 q^{17} + 72 q^{19} - 76 q^{21} + 98 q^{23} + 136 q^{25} - 182 q^{27} - 144 q^{29} + 34 q^{31} + 22 q^{33} - 172 q^{35} - 54 q^{37} - 400 q^{39} + 536 q^{41} - 60 q^{43} - 428 q^{45} + 272 q^{47} - 390 q^{49} - 164 q^{51} + 492 q^{53} + 22 q^{55} - 1512 q^{57} + 634 q^{59} - 840 q^{61} - 248 q^{63} - 880 q^{65} + 754 q^{67} - 962 q^{69} + 678 q^{71} - 400 q^{73} - 520 q^{75} + 220 q^{77} - 316 q^{79} - 1294 q^{81} + 468 q^{83} - 452 q^{85} - 1200 q^{87} - 1842 q^{89} + 1280 q^{91} + 638 q^{93} - 2952 q^{95} + 2194 q^{97} - 484 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 2 * q^5 - 20 * q^7 + 44 * q^9 - 22 * q^11 - 80 * q^13 + 194 * q^15 - 124 * q^17 + 72 * q^19 - 76 * q^21 + 98 * q^23 + 136 * q^25 - 182 * q^27 - 144 * q^29 + 34 * q^31 + 22 * q^33 - 172 * q^35 - 54 * q^37 - 400 * q^39 + 536 * q^41 - 60 * q^43 - 428 * q^45 + 272 * q^47 - 390 * q^49 - 164 * q^51 + 492 * q^53 + 22 * q^55 - 1512 * q^57 + 634 * q^59 - 840 * q^61 - 248 * q^63 - 880 * q^65 + 754 * q^67 - 962 * q^69 + 678 * q^71 - 400 * q^73 - 520 * q^75 + 220 * q^77 - 316 * q^79 - 1294 * q^81 + 468 * q^83 - 452 * q^85 - 1200 * q^87 - 1842 * q^89 + 1280 * q^91 + 638 * q^93 - 2952 * q^95 + 2194 * q^97 - 484 * 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
$$3$$ −7.92820 −1.52578 −0.762892 0.646526i $$-0.776221\pi$$
−0.762892 + 0.646526i $$0.776221\pi$$
$$4$$ 0 0
$$5$$ −14.8564 −1.32880 −0.664399 0.747378i $$-0.731312\pi$$
−0.664399 + 0.747378i $$0.731312\pi$$
$$6$$ 0 0
$$7$$ −3.07180 −0.165861 −0.0829307 0.996555i $$-0.526428\pi$$
−0.0829307 + 0.996555i $$0.526428\pi$$
$$8$$ 0 0
$$9$$ 35.8564 1.32802
$$10$$ 0 0
$$11$$ −11.0000 −0.301511
$$12$$ 0 0
$$13$$ −5.35898 −0.114332 −0.0571659 0.998365i $$-0.518206\pi$$
−0.0571659 + 0.998365i $$0.518206\pi$$
$$14$$ 0 0
$$15$$ 117.785 2.02746
$$16$$ 0 0
$$17$$ −41.2154 −0.588012 −0.294006 0.955804i $$-0.594989\pi$$
−0.294006 + 0.955804i $$0.594989\pi$$
$$18$$ 0 0
$$19$$ 139.923 1.68950 0.844751 0.535159i $$-0.179748\pi$$
0.844751 + 0.535159i $$0.179748\pi$$
$$20$$ 0 0
$$21$$ 24.3538 0.253069
$$22$$ 0 0
$$23$$ 111.354 1.00952 0.504758 0.863261i $$-0.331582\pi$$
0.504758 + 0.863261i $$0.331582\pi$$
$$24$$ 0 0
$$25$$ 95.7128 0.765703
$$26$$ 0 0
$$27$$ −70.2154 −0.500480
$$28$$ 0 0
$$29$$ 24.9948 0.160049 0.0800246 0.996793i $$-0.474500\pi$$
0.0800246 + 0.996793i $$0.474500\pi$$
$$30$$ 0 0
$$31$$ −31.4974 −0.182487 −0.0912436 0.995829i $$-0.529084\pi$$
−0.0912436 + 0.995829i $$0.529084\pi$$
$$32$$ 0 0
$$33$$ 87.2102 0.460041
$$34$$ 0 0
$$35$$ 45.6359 0.220396
$$36$$ 0 0
$$37$$ −13.1436 −0.0583998 −0.0291999 0.999574i $$-0.509296\pi$$
−0.0291999 + 0.999574i $$0.509296\pi$$
$$38$$ 0 0
$$39$$ 42.4871 0.174446
$$40$$ 0 0
$$41$$ 261.072 0.994453 0.497226 0.867621i $$-0.334352\pi$$
0.497226 + 0.867621i $$0.334352\pi$$
$$42$$ 0 0
$$43$$ −57.7128 −0.204677 −0.102339 0.994750i $$-0.532633\pi$$
−0.102339 + 0.994750i $$0.532633\pi$$
$$44$$ 0 0
$$45$$ −532.697 −1.76466
$$46$$ 0 0
$$47$$ 343.846 1.06713 0.533565 0.845759i $$-0.320852\pi$$
0.533565 + 0.845759i $$0.320852\pi$$
$$48$$ 0 0
$$49$$ −333.564 −0.972490
$$50$$ 0 0
$$51$$ 326.764 0.897179
$$52$$ 0 0
$$53$$ 342.995 0.888943 0.444471 0.895793i $$-0.353392\pi$$
0.444471 + 0.895793i $$0.353392\pi$$
$$54$$ 0 0
$$55$$ 163.420 0.400647
$$56$$ 0 0
$$57$$ −1109.34 −2.57782
$$58$$ 0 0
$$59$$ 88.3693 0.194995 0.0974975 0.995236i $$-0.468916\pi$$
0.0974975 + 0.995236i $$0.468916\pi$$
$$60$$ 0 0
$$61$$ −738.697 −1.55050 −0.775250 0.631654i $$-0.782376\pi$$
−0.775250 + 0.631654i $$0.782376\pi$$
$$62$$ 0 0
$$63$$ −110.144 −0.220266
$$64$$ 0 0
$$65$$ 79.6152 0.151924
$$66$$ 0 0
$$67$$ 342.359 0.624266 0.312133 0.950038i $$-0.398957\pi$$
0.312133 + 0.950038i $$0.398957\pi$$
$$68$$ 0 0
$$69$$ −882.836 −1.54030
$$70$$ 0 0
$$71$$ 207.364 0.346614 0.173307 0.984868i $$-0.444555\pi$$
0.173307 + 0.984868i $$0.444555\pi$$
$$72$$ 0 0
$$73$$ −1010.60 −1.62030 −0.810149 0.586224i $$-0.800614\pi$$
−0.810149 + 0.586224i $$0.800614\pi$$
$$74$$ 0 0
$$75$$ −758.831 −1.16830
$$76$$ 0 0
$$77$$ 33.7898 0.0500091
$$78$$ 0 0
$$79$$ −1294.23 −1.84319 −0.921593 0.388157i $$-0.873112\pi$$
−0.921593 + 0.388157i $$0.873112\pi$$
$$80$$ 0 0
$$81$$ −411.441 −0.564391
$$82$$ 0 0
$$83$$ 441.846 0.584324 0.292162 0.956369i $$-0.405625\pi$$
0.292162 + 0.956369i $$0.405625\pi$$
$$84$$ 0 0
$$85$$ 612.313 0.781349
$$86$$ 0 0
$$87$$ −198.164 −0.244200
$$88$$ 0 0
$$89$$ −1489.11 −1.77355 −0.886773 0.462205i $$-0.847058\pi$$
−0.886773 + 0.462205i $$0.847058\pi$$
$$90$$ 0 0
$$91$$ 16.4617 0.0189633
$$92$$ 0 0
$$93$$ 249.718 0.278436
$$94$$ 0 0
$$95$$ −2078.75 −2.24501
$$96$$ 0 0
$$97$$ 1346.42 1.40936 0.704679 0.709526i $$-0.251091\pi$$
0.704679 + 0.709526i $$0.251091\pi$$
$$98$$ 0 0
$$99$$ −394.420 −0.400412
$$100$$ 0 0
$$101$$ 161.461 0.159069 0.0795347 0.996832i $$-0.474657\pi$$
0.0795347 + 0.996832i $$0.474657\pi$$
$$102$$ 0 0
$$103$$ 34.7592 0.0332517 0.0166259 0.999862i $$-0.494708\pi$$
0.0166259 + 0.999862i $$0.494708\pi$$
$$104$$ 0 0
$$105$$ −361.810 −0.336277
$$106$$ 0 0
$$107$$ 832.179 0.751867 0.375934 0.926647i $$-0.377322\pi$$
0.375934 + 0.926647i $$0.377322\pi$$
$$108$$ 0 0
$$109$$ −1044.26 −0.917629 −0.458815 0.888532i $$-0.651726\pi$$
−0.458815 + 0.888532i $$0.651726\pi$$
$$110$$ 0 0
$$111$$ 104.205 0.0891055
$$112$$ 0 0
$$113$$ 295.082 0.245654 0.122827 0.992428i $$-0.460804\pi$$
0.122827 + 0.992428i $$0.460804\pi$$
$$114$$ 0 0
$$115$$ −1654.32 −1.34144
$$116$$ 0 0
$$117$$ −192.154 −0.151834
$$118$$ 0 0
$$119$$ 126.605 0.0975285
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ −2069.83 −1.51732
$$124$$ 0 0
$$125$$ 435.102 0.311334
$$126$$ 0 0
$$127$$ 1317.60 0.920618 0.460309 0.887759i $$-0.347739\pi$$
0.460309 + 0.887759i $$0.347739\pi$$
$$128$$ 0 0
$$129$$ 457.559 0.312293
$$130$$ 0 0
$$131$$ −1600.71 −1.06759 −0.533797 0.845612i $$-0.679235\pi$$
−0.533797 + 0.845612i $$0.679235\pi$$
$$132$$ 0 0
$$133$$ −429.815 −0.280223
$$134$$ 0 0
$$135$$ 1043.15 0.665036
$$136$$ 0 0
$$137$$ 1611.68 1.00507 0.502536 0.864556i $$-0.332400\pi$$
0.502536 + 0.864556i $$0.332400\pi$$
$$138$$ 0 0
$$139$$ −31.8619 −0.0194424 −0.00972120 0.999953i $$-0.503094\pi$$
−0.00972120 + 0.999953i $$0.503094\pi$$
$$140$$ 0 0
$$141$$ −2726.08 −1.62821
$$142$$ 0 0
$$143$$ 58.9488 0.0344724
$$144$$ 0 0
$$145$$ −371.334 −0.212673
$$146$$ 0 0
$$147$$ 2644.56 1.48381
$$148$$ 0 0
$$149$$ 2428.34 1.33515 0.667576 0.744542i $$-0.267332\pi$$
0.667576 + 0.744542i $$0.267332\pi$$
$$150$$ 0 0
$$151$$ 2576.68 1.38866 0.694328 0.719659i $$-0.255702\pi$$
0.694328 + 0.719659i $$0.255702\pi$$
$$152$$ 0 0
$$153$$ −1477.84 −0.780889
$$154$$ 0 0
$$155$$ 467.939 0.242489
$$156$$ 0 0
$$157$$ −2475.94 −1.25861 −0.629305 0.777158i $$-0.716660\pi$$
−0.629305 + 0.777158i $$0.716660\pi$$
$$158$$ 0 0
$$159$$ −2719.33 −1.35633
$$160$$ 0 0
$$161$$ −342.056 −0.167440
$$162$$ 0 0
$$163$$ −2725.11 −1.30949 −0.654745 0.755850i $$-0.727224\pi$$
−0.654745 + 0.755850i $$0.727224\pi$$
$$164$$ 0 0
$$165$$ −1295.63 −0.611301
$$166$$ 0 0
$$167$$ −2737.30 −1.26837 −0.634187 0.773180i $$-0.718665\pi$$
−0.634187 + 0.773180i $$0.718665\pi$$
$$168$$ 0 0
$$169$$ −2168.28 −0.986928
$$170$$ 0 0
$$171$$ 5017.14 2.24368
$$172$$ 0 0
$$173$$ −2307.42 −1.01404 −0.507022 0.861933i $$-0.669254\pi$$
−0.507022 + 0.861933i $$0.669254\pi$$
$$174$$ 0 0
$$175$$ −294.010 −0.127001
$$176$$ 0 0
$$177$$ −700.610 −0.297520
$$178$$ 0 0
$$179$$ −1312.15 −0.547905 −0.273953 0.961743i $$-0.588331\pi$$
−0.273953 + 0.961743i $$0.588331\pi$$
$$180$$ 0 0
$$181$$ 803.174 0.329831 0.164916 0.986308i $$-0.447265\pi$$
0.164916 + 0.986308i $$0.447265\pi$$
$$182$$ 0 0
$$183$$ 5856.54 2.36573
$$184$$ 0 0
$$185$$ 195.267 0.0776015
$$186$$ 0 0
$$187$$ 453.369 0.177292
$$188$$ 0 0
$$189$$ 215.687 0.0830103
$$190$$ 0 0
$$191$$ −1718.25 −0.650932 −0.325466 0.945554i $$-0.605521\pi$$
−0.325466 + 0.945554i $$0.605521\pi$$
$$192$$ 0 0
$$193$$ 1340.18 0.499837 0.249919 0.968267i $$-0.419596\pi$$
0.249919 + 0.968267i $$0.419596\pi$$
$$194$$ 0 0
$$195$$ −631.206 −0.231803
$$196$$ 0 0
$$197$$ 3518.33 1.27244 0.636220 0.771508i $$-0.280497\pi$$
0.636220 + 0.771508i $$0.280497\pi$$
$$198$$ 0 0
$$199$$ −823.692 −0.293417 −0.146709 0.989180i $$-0.546868\pi$$
−0.146709 + 0.989180i $$0.546868\pi$$
$$200$$ 0 0
$$201$$ −2714.29 −0.952494
$$202$$ 0 0
$$203$$ −76.7791 −0.0265460
$$204$$ 0 0
$$205$$ −3878.59 −1.32143
$$206$$ 0 0
$$207$$ 3992.75 1.34065
$$208$$ 0 0
$$209$$ −1539.15 −0.509404
$$210$$ 0 0
$$211$$ −107.343 −0.0350228 −0.0175114 0.999847i $$-0.505574\pi$$
−0.0175114 + 0.999847i $$0.505574\pi$$
$$212$$ 0 0
$$213$$ −1644.03 −0.528858
$$214$$ 0 0
$$215$$ 857.405 0.271975
$$216$$ 0 0
$$217$$ 96.7537 0.0302676
$$218$$ 0 0
$$219$$ 8012.24 2.47222
$$220$$ 0 0
$$221$$ 220.873 0.0672285
$$222$$ 0 0
$$223$$ 3933.68 1.18125 0.590625 0.806946i $$-0.298881\pi$$
0.590625 + 0.806946i $$0.298881\pi$$
$$224$$ 0 0
$$225$$ 3431.92 1.01686
$$226$$ 0 0
$$227$$ −1771.90 −0.518085 −0.259042 0.965866i $$-0.583407\pi$$
−0.259042 + 0.965866i $$0.583407\pi$$
$$228$$ 0 0
$$229$$ −1915.37 −0.552713 −0.276356 0.961055i $$-0.589127\pi$$
−0.276356 + 0.961055i $$0.589127\pi$$
$$230$$ 0 0
$$231$$ −267.892 −0.0763031
$$232$$ 0 0
$$233$$ 4396.32 1.23610 0.618052 0.786137i $$-0.287922\pi$$
0.618052 + 0.786137i $$0.287922\pi$$
$$234$$ 0 0
$$235$$ −5108.32 −1.41800
$$236$$ 0 0
$$237$$ 10260.9 2.81230
$$238$$ 0 0
$$239$$ 4084.49 1.10546 0.552728 0.833362i $$-0.313587\pi$$
0.552728 + 0.833362i $$0.313587\pi$$
$$240$$ 0 0
$$241$$ 3908.58 1.04471 0.522353 0.852730i $$-0.325054\pi$$
0.522353 + 0.852730i $$0.325054\pi$$
$$242$$ 0 0
$$243$$ 5157.80 1.36162
$$244$$ 0 0
$$245$$ 4955.56 1.29224
$$246$$ 0 0
$$247$$ −749.845 −0.193164
$$248$$ 0 0
$$249$$ −3503.05 −0.891552
$$250$$ 0 0
$$251$$ 1094.89 0.275335 0.137667 0.990479i $$-0.456040\pi$$
0.137667 + 0.990479i $$0.456040\pi$$
$$252$$ 0 0
$$253$$ −1224.89 −0.304381
$$254$$ 0 0
$$255$$ −4854.54 −1.19217
$$256$$ 0 0
$$257$$ 783.179 0.190091 0.0950454 0.995473i $$-0.469700\pi$$
0.0950454 + 0.995473i $$0.469700\pi$$
$$258$$ 0 0
$$259$$ 40.3744 0.00968628
$$260$$ 0 0
$$261$$ 896.225 0.212548
$$262$$ 0 0
$$263$$ −6180.06 −1.44897 −0.724484 0.689292i $$-0.757922\pi$$
−0.724484 + 0.689292i $$0.757922\pi$$
$$264$$ 0 0
$$265$$ −5095.67 −1.18122
$$266$$ 0 0
$$267$$ 11806.0 2.70605
$$268$$ 0 0
$$269$$ −986.965 −0.223704 −0.111852 0.993725i $$-0.535678\pi$$
−0.111852 + 0.993725i $$0.535678\pi$$
$$270$$ 0 0
$$271$$ −4576.99 −1.02595 −0.512975 0.858404i $$-0.671457\pi$$
−0.512975 + 0.858404i $$0.671457\pi$$
$$272$$ 0 0
$$273$$ −130.512 −0.0289338
$$274$$ 0 0
$$275$$ −1052.84 −0.230868
$$276$$ 0 0
$$277$$ −567.836 −0.123169 −0.0615847 0.998102i $$-0.519615\pi$$
−0.0615847 + 0.998102i $$0.519615\pi$$
$$278$$ 0 0
$$279$$ −1129.38 −0.242346
$$280$$ 0 0
$$281$$ 5311.01 1.12750 0.563752 0.825944i $$-0.309357\pi$$
0.563752 + 0.825944i $$0.309357\pi$$
$$282$$ 0 0
$$283$$ −4728.44 −0.993204 −0.496602 0.867978i $$-0.665419\pi$$
−0.496602 + 0.867978i $$0.665419\pi$$
$$284$$ 0 0
$$285$$ 16480.8 3.42539
$$286$$ 0 0
$$287$$ −801.960 −0.164941
$$288$$ 0 0
$$289$$ −3214.29 −0.654242
$$290$$ 0 0
$$291$$ −10674.7 −2.15038
$$292$$ 0 0
$$293$$ −2328.92 −0.464358 −0.232179 0.972673i $$-0.574585\pi$$
−0.232179 + 0.972673i $$0.574585\pi$$
$$294$$ 0 0
$$295$$ −1312.85 −0.259109
$$296$$ 0 0
$$297$$ 772.369 0.150900
$$298$$ 0 0
$$299$$ −596.743 −0.115420
$$300$$ 0 0
$$301$$ 177.282 0.0339481
$$302$$ 0 0
$$303$$ −1280.10 −0.242705
$$304$$ 0 0
$$305$$ 10974.4 2.06030
$$306$$ 0 0
$$307$$ −1678.07 −0.311962 −0.155981 0.987760i $$-0.549854\pi$$
−0.155981 + 0.987760i $$0.549854\pi$$
$$308$$ 0 0
$$309$$ −275.578 −0.0507349
$$310$$ 0 0
$$311$$ −3572.71 −0.651413 −0.325707 0.945471i $$-0.605602\pi$$
−0.325707 + 0.945471i $$0.605602\pi$$
$$312$$ 0 0
$$313$$ 7184.36 1.29739 0.648697 0.761047i $$-0.275314\pi$$
0.648697 + 0.761047i $$0.275314\pi$$
$$314$$ 0 0
$$315$$ 1636.34 0.292690
$$316$$ 0 0
$$317$$ 15.7077 0.00278306 0.00139153 0.999999i $$-0.499557\pi$$
0.00139153 + 0.999999i $$0.499557\pi$$
$$318$$ 0 0
$$319$$ −274.943 −0.0482566
$$320$$ 0 0
$$321$$ −6597.69 −1.14719
$$322$$ 0 0
$$323$$ −5766.98 −0.993447
$$324$$ 0 0
$$325$$ −512.923 −0.0875442
$$326$$ 0 0
$$327$$ 8279.08 1.40010
$$328$$ 0 0
$$329$$ −1056.23 −0.176996
$$330$$ 0 0
$$331$$ −1318.95 −0.219022 −0.109511 0.993986i $$-0.534928\pi$$
−0.109511 + 0.993986i $$0.534928\pi$$
$$332$$ 0 0
$$333$$ −471.282 −0.0775558
$$334$$ 0 0
$$335$$ −5086.22 −0.829523
$$336$$ 0 0
$$337$$ −239.183 −0.0386621 −0.0193310 0.999813i $$-0.506154\pi$$
−0.0193310 + 0.999813i $$0.506154\pi$$
$$338$$ 0 0
$$339$$ −2339.47 −0.374816
$$340$$ 0 0
$$341$$ 346.472 0.0550220
$$342$$ 0 0
$$343$$ 2078.27 0.327160
$$344$$ 0 0
$$345$$ 13115.8 2.04675
$$346$$ 0 0
$$347$$ −5862.79 −0.907006 −0.453503 0.891255i $$-0.649826\pi$$
−0.453503 + 0.891255i $$0.649826\pi$$
$$348$$ 0 0
$$349$$ −3491.73 −0.535553 −0.267776 0.963481i $$-0.586289\pi$$
−0.267776 + 0.963481i $$0.586289\pi$$
$$350$$ 0 0
$$351$$ 376.283 0.0572208
$$352$$ 0 0
$$353$$ −10916.7 −1.64600 −0.822999 0.568043i $$-0.807701\pi$$
−0.822999 + 0.568043i $$0.807701\pi$$
$$354$$ 0 0
$$355$$ −3080.69 −0.460580
$$356$$ 0 0
$$357$$ −1003.75 −0.148807
$$358$$ 0 0
$$359$$ 11500.7 1.69077 0.845384 0.534160i $$-0.179372\pi$$
0.845384 + 0.534160i $$0.179372\pi$$
$$360$$ 0 0
$$361$$ 12719.5 1.85442
$$362$$ 0 0
$$363$$ −959.313 −0.138708
$$364$$ 0 0
$$365$$ 15013.9 2.15305
$$366$$ 0 0
$$367$$ −6767.01 −0.962493 −0.481246 0.876585i $$-0.659816\pi$$
−0.481246 + 0.876585i $$0.659816\pi$$
$$368$$ 0 0
$$369$$ 9361.10 1.32065
$$370$$ 0 0
$$371$$ −1053.61 −0.147441
$$372$$ 0 0
$$373$$ 5310.22 0.737139 0.368569 0.929600i $$-0.379848\pi$$
0.368569 + 0.929600i $$0.379848\pi$$
$$374$$ 0 0
$$375$$ −3449.58 −0.475028
$$376$$ 0 0
$$377$$ −133.947 −0.0182987
$$378$$ 0 0
$$379$$ −838.267 −0.113612 −0.0568059 0.998385i $$-0.518092\pi$$
−0.0568059 + 0.998385i $$0.518092\pi$$
$$380$$ 0 0
$$381$$ −10446.2 −1.40466
$$382$$ 0 0
$$383$$ 2832.16 0.377851 0.188925 0.981991i $$-0.439500\pi$$
0.188925 + 0.981991i $$0.439500\pi$$
$$384$$ 0 0
$$385$$ −501.994 −0.0664520
$$386$$ 0 0
$$387$$ −2069.37 −0.271814
$$388$$ 0 0
$$389$$ −3111.25 −0.405519 −0.202759 0.979229i $$-0.564991\pi$$
−0.202759 + 0.979229i $$0.564991\pi$$
$$390$$ 0 0
$$391$$ −4589.49 −0.593608
$$392$$ 0 0
$$393$$ 12690.8 1.62892
$$394$$ 0 0
$$395$$ 19227.5 2.44922
$$396$$ 0 0
$$397$$ −14208.7 −1.79626 −0.898131 0.439728i $$-0.855075\pi$$
−0.898131 + 0.439728i $$0.855075\pi$$
$$398$$ 0 0
$$399$$ 3407.66 0.427560
$$400$$ 0 0
$$401$$ −6261.68 −0.779784 −0.389892 0.920861i $$-0.627488\pi$$
−0.389892 + 0.920861i $$0.627488\pi$$
$$402$$ 0 0
$$403$$ 168.794 0.0208641
$$404$$ 0 0
$$405$$ 6112.54 0.749961
$$406$$ 0 0
$$407$$ 144.580 0.0176082
$$408$$ 0 0
$$409$$ −4192.50 −0.506860 −0.253430 0.967354i $$-0.581559\pi$$
−0.253430 + 0.967354i $$0.581559\pi$$
$$410$$ 0 0
$$411$$ −12777.7 −1.53352
$$412$$ 0 0
$$413$$ −271.453 −0.0323421
$$414$$ 0 0
$$415$$ −6564.25 −0.776448
$$416$$ 0 0
$$417$$ 252.608 0.0296649
$$418$$ 0 0
$$419$$ −9287.15 −1.08283 −0.541416 0.840755i $$-0.682112\pi$$
−0.541416 + 0.840755i $$0.682112\pi$$
$$420$$ 0 0
$$421$$ −13146.0 −1.52185 −0.760923 0.648842i $$-0.775254\pi$$
−0.760923 + 0.648842i $$0.775254\pi$$
$$422$$ 0 0
$$423$$ 12329.1 1.41716
$$424$$ 0 0
$$425$$ −3944.84 −0.450242
$$426$$ 0 0
$$427$$ 2269.13 0.257168
$$428$$ 0 0
$$429$$ −467.358 −0.0525974
$$430$$ 0 0
$$431$$ −4909.67 −0.548701 −0.274351 0.961630i $$-0.588463\pi$$
−0.274351 + 0.961630i $$0.588463\pi$$
$$432$$ 0 0
$$433$$ −11743.3 −1.30334 −0.651671 0.758502i $$-0.725932\pi$$
−0.651671 + 0.758502i $$0.725932\pi$$
$$434$$ 0 0
$$435$$ 2944.01 0.324493
$$436$$ 0 0
$$437$$ 15581.0 1.70558
$$438$$ 0 0
$$439$$ 11824.2 1.28551 0.642754 0.766073i $$-0.277792\pi$$
0.642754 + 0.766073i $$0.277792\pi$$
$$440$$ 0 0
$$441$$ −11960.4 −1.29148
$$442$$ 0 0
$$443$$ 10102.1 1.08344 0.541722 0.840558i $$-0.317772\pi$$
0.541722 + 0.840558i $$0.317772\pi$$
$$444$$ 0 0
$$445$$ 22122.9 2.35668
$$446$$ 0 0
$$447$$ −19252.4 −2.03715
$$448$$ 0 0
$$449$$ −345.254 −0.0362885 −0.0181443 0.999835i $$-0.505776\pi$$
−0.0181443 + 0.999835i $$0.505776\pi$$
$$450$$ 0 0
$$451$$ −2871.79 −0.299839
$$452$$ 0 0
$$453$$ −20428.4 −2.11879
$$454$$ 0 0
$$455$$ −244.562 −0.0251983
$$456$$ 0 0
$$457$$ −10567.1 −1.08164 −0.540821 0.841138i $$-0.681886\pi$$
−0.540821 + 0.841138i $$0.681886\pi$$
$$458$$ 0 0
$$459$$ 2893.95 0.294288
$$460$$ 0 0
$$461$$ −4733.96 −0.478270 −0.239135 0.970986i $$-0.576864\pi$$
−0.239135 + 0.970986i $$0.576864\pi$$
$$462$$ 0 0
$$463$$ −3431.20 −0.344409 −0.172204 0.985061i $$-0.555089\pi$$
−0.172204 + 0.985061i $$0.555089\pi$$
$$464$$ 0 0
$$465$$ −3709.91 −0.369985
$$466$$ 0 0
$$467$$ 5116.96 0.507034 0.253517 0.967331i $$-0.418413\pi$$
0.253517 + 0.967331i $$0.418413\pi$$
$$468$$ 0 0
$$469$$ −1051.66 −0.103542
$$470$$ 0 0
$$471$$ 19629.8 1.92037
$$472$$ 0 0
$$473$$ 634.841 0.0617125
$$474$$ 0 0
$$475$$ 13392.4 1.29366
$$476$$ 0 0
$$477$$ 12298.6 1.18053
$$478$$ 0 0
$$479$$ −11566.9 −1.10335 −0.551675 0.834059i $$-0.686011\pi$$
−0.551675 + 0.834059i $$0.686011\pi$$
$$480$$ 0 0
$$481$$ 70.4363 0.00667696
$$482$$ 0 0
$$483$$ 2711.89 0.255477
$$484$$ 0 0
$$485$$ −20002.9 −1.87275
$$486$$ 0 0
$$487$$ 18326.5 1.70525 0.852623 0.522527i $$-0.175010\pi$$
0.852623 + 0.522527i $$0.175010\pi$$
$$488$$ 0 0
$$489$$ 21605.2 1.99800
$$490$$ 0 0
$$491$$ −7617.58 −0.700156 −0.350078 0.936721i $$-0.613845\pi$$
−0.350078 + 0.936721i $$0.613845\pi$$
$$492$$ 0 0
$$493$$ −1030.17 −0.0941108
$$494$$ 0 0
$$495$$ 5859.67 0.532066
$$496$$ 0 0
$$497$$ −636.980 −0.0574899
$$498$$ 0 0
$$499$$ 12909.1 1.15810 0.579050 0.815292i $$-0.303424\pi$$
0.579050 + 0.815292i $$0.303424\pi$$
$$500$$ 0 0
$$501$$ 21701.8 1.93526
$$502$$ 0 0
$$503$$ −10165.7 −0.901121 −0.450561 0.892746i $$-0.648776\pi$$
−0.450561 + 0.892746i $$0.648776\pi$$
$$504$$ 0 0
$$505$$ −2398.74 −0.211371
$$506$$ 0 0
$$507$$ 17190.6 1.50584
$$508$$ 0 0
$$509$$ −6449.93 −0.561666 −0.280833 0.959757i $$-0.590611\pi$$
−0.280833 + 0.959757i $$0.590611\pi$$
$$510$$ 0 0
$$511$$ 3104.36 0.268745
$$512$$ 0 0
$$513$$ −9824.75 −0.845562
$$514$$ 0 0
$$515$$ −516.397 −0.0441848
$$516$$ 0 0
$$517$$ −3782.31 −0.321752
$$518$$ 0 0
$$519$$ 18293.7 1.54721
$$520$$ 0 0
$$521$$ −19327.4 −1.62524 −0.812620 0.582794i $$-0.801959\pi$$
−0.812620 + 0.582794i $$0.801959\pi$$
$$522$$ 0 0
$$523$$ 6259.09 0.523310 0.261655 0.965161i $$-0.415732\pi$$
0.261655 + 0.965161i $$0.415732\pi$$
$$524$$ 0 0
$$525$$ 2330.97 0.193775
$$526$$ 0 0
$$527$$ 1298.18 0.107305
$$528$$ 0 0
$$529$$ 232.675 0.0191235
$$530$$ 0 0
$$531$$ 3168.61 0.258956
$$532$$ 0 0
$$533$$ −1399.08 −0.113698
$$534$$ 0 0
$$535$$ −12363.2 −0.999079
$$536$$ 0 0
$$537$$ 10403.0 0.835985
$$538$$ 0 0
$$539$$ 3669.20 0.293217
$$540$$ 0 0
$$541$$ 14008.2 1.11323 0.556616 0.830770i $$-0.312100\pi$$
0.556616 + 0.830770i $$0.312100\pi$$
$$542$$ 0 0
$$543$$ −6367.72 −0.503251
$$544$$ 0 0
$$545$$ 15513.9 1.21934
$$546$$ 0 0
$$547$$ −4949.45 −0.386879 −0.193440 0.981112i $$-0.561964\pi$$
−0.193440 + 0.981112i $$0.561964\pi$$
$$548$$ 0 0
$$549$$ −26487.0 −2.05909
$$550$$ 0 0
$$551$$ 3497.35 0.270404
$$552$$ 0 0
$$553$$ 3975.60 0.305714
$$554$$ 0 0
$$555$$ −1548.11 −0.118403
$$556$$ 0 0
$$557$$ 3801.58 0.289188 0.144594 0.989491i $$-0.453812\pi$$
0.144594 + 0.989491i $$0.453812\pi$$
$$558$$ 0 0
$$559$$ 309.282 0.0234011
$$560$$ 0 0
$$561$$ −3594.40 −0.270510
$$562$$ 0 0
$$563$$ −9900.11 −0.741101 −0.370551 0.928812i $$-0.620831\pi$$
−0.370551 + 0.928812i $$0.620831\pi$$
$$564$$ 0 0
$$565$$ −4383.85 −0.326425
$$566$$ 0 0
$$567$$ 1263.86 0.0936107
$$568$$ 0 0
$$569$$ 5329.16 0.392636 0.196318 0.980540i $$-0.437102\pi$$
0.196318 + 0.980540i $$0.437102\pi$$
$$570$$ 0 0
$$571$$ −16962.6 −1.24319 −0.621597 0.783337i $$-0.713516\pi$$
−0.621597 + 0.783337i $$0.713516\pi$$
$$572$$ 0 0
$$573$$ 13622.6 0.993181
$$574$$ 0 0
$$575$$ 10658.0 0.772989
$$576$$ 0 0
$$577$$ −15487.0 −1.11738 −0.558692 0.829375i $$-0.688697\pi$$
−0.558692 + 0.829375i $$0.688697\pi$$
$$578$$ 0 0
$$579$$ −10625.3 −0.762643
$$580$$ 0 0
$$581$$ −1357.26 −0.0969169
$$582$$ 0 0
$$583$$ −3772.94 −0.268026
$$584$$ 0 0
$$585$$ 2854.72 0.201757
$$586$$ 0 0
$$587$$ 11084.2 0.779373 0.389686 0.920948i $$-0.372583\pi$$
0.389686 + 0.920948i $$0.372583\pi$$
$$588$$ 0 0
$$589$$ −4407.22 −0.308313
$$590$$ 0 0
$$591$$ −27894.0 −1.94147
$$592$$ 0 0
$$593$$ 4349.68 0.301214 0.150607 0.988594i $$-0.451877\pi$$
0.150607 + 0.988594i $$0.451877\pi$$
$$594$$ 0 0
$$595$$ −1880.90 −0.129596
$$596$$ 0 0
$$597$$ 6530.40 0.447691
$$598$$ 0 0
$$599$$ −13183.9 −0.899299 −0.449650 0.893205i $$-0.648451\pi$$
−0.449650 + 0.893205i $$0.648451\pi$$
$$600$$ 0 0
$$601$$ −18765.0 −1.27361 −0.636806 0.771024i $$-0.719745\pi$$
−0.636806 + 0.771024i $$0.719745\pi$$
$$602$$ 0 0
$$603$$ 12275.8 0.829034
$$604$$ 0 0
$$605$$ −1797.63 −0.120800
$$606$$ 0 0
$$607$$ −21871.4 −1.46249 −0.731244 0.682116i $$-0.761060\pi$$
−0.731244 + 0.682116i $$0.761060\pi$$
$$608$$ 0 0
$$609$$ 608.720 0.0405034
$$610$$ 0 0
$$611$$ −1842.67 −0.122007
$$612$$ 0 0
$$613$$ 3527.85 0.232445 0.116222 0.993223i $$-0.462921\pi$$
0.116222 + 0.993223i $$0.462921\pi$$
$$614$$ 0 0
$$615$$ 30750.2 2.01621
$$616$$ 0 0
$$617$$ −22728.1 −1.48298 −0.741490 0.670963i $$-0.765881\pi$$
−0.741490 + 0.670963i $$0.765881\pi$$
$$618$$ 0 0
$$619$$ −21443.3 −1.39237 −0.696187 0.717861i $$-0.745121\pi$$
−0.696187 + 0.717861i $$0.745121\pi$$
$$620$$ 0 0
$$621$$ −7818.75 −0.505243
$$622$$ 0 0
$$623$$ 4574.25 0.294163
$$624$$ 0 0
$$625$$ −18428.2 −1.17940
$$626$$ 0 0
$$627$$ 12202.7 0.777240
$$628$$ 0 0
$$629$$ 541.718 0.0343398
$$630$$ 0 0
$$631$$ −21532.0 −1.35844 −0.679219 0.733936i $$-0.737681\pi$$
−0.679219 + 0.733936i $$0.737681\pi$$
$$632$$ 0 0
$$633$$ 851.038 0.0534372
$$634$$ 0 0
$$635$$ −19574.9 −1.22332
$$636$$ 0 0
$$637$$ 1787.56 0.111187
$$638$$ 0 0
$$639$$ 7435.33 0.460309
$$640$$ 0 0
$$641$$ 20148.3 1.24151 0.620756 0.784004i $$-0.286826\pi$$
0.620756 + 0.784004i $$0.286826\pi$$
$$642$$ 0 0
$$643$$ 28869.7 1.77062 0.885310 0.465000i $$-0.153946\pi$$
0.885310 + 0.465000i $$0.153946\pi$$
$$644$$ 0 0
$$645$$ −6797.68 −0.414974
$$646$$ 0 0
$$647$$ 1590.02 0.0966155 0.0483077 0.998833i $$-0.484617\pi$$
0.0483077 + 0.998833i $$0.484617\pi$$
$$648$$ 0 0
$$649$$ −972.062 −0.0587932
$$650$$ 0 0
$$651$$ −767.083 −0.0461818
$$652$$ 0 0
$$653$$ −20028.1 −1.20024 −0.600122 0.799909i $$-0.704881\pi$$
−0.600122 + 0.799909i $$0.704881\pi$$
$$654$$ 0 0
$$655$$ 23780.8 1.41862
$$656$$ 0 0
$$657$$ −36236.5 −2.15178
$$658$$ 0 0
$$659$$ −10520.7 −0.621897 −0.310948 0.950427i $$-0.600647\pi$$
−0.310948 + 0.950427i $$0.600647\pi$$
$$660$$ 0 0
$$661$$ −3295.83 −0.193938 −0.0969690 0.995287i $$-0.530915\pi$$
−0.0969690 + 0.995287i $$0.530915\pi$$
$$662$$ 0 0
$$663$$ −1751.12 −0.102576
$$664$$ 0 0
$$665$$ 6385.51 0.372360
$$666$$ 0 0
$$667$$ 2783.27 0.161572
$$668$$ 0 0
$$669$$ −31187.0 −1.80233
$$670$$ 0 0
$$671$$ 8125.67 0.467493
$$672$$ 0 0
$$673$$ −1187.64 −0.0680239 −0.0340119 0.999421i $$-0.510828\pi$$
−0.0340119 + 0.999421i $$0.510828\pi$$
$$674$$ 0 0
$$675$$ −6720.51 −0.383219
$$676$$ 0 0
$$677$$ −13221.4 −0.750574 −0.375287 0.926909i $$-0.622456\pi$$
−0.375287 + 0.926909i $$0.622456\pi$$
$$678$$ 0 0
$$679$$ −4135.91 −0.233758
$$680$$ 0 0
$$681$$ 14048.0 0.790485
$$682$$ 0 0
$$683$$ −13831.4 −0.774882 −0.387441 0.921894i $$-0.626641\pi$$
−0.387441 + 0.921894i $$0.626641\pi$$
$$684$$ 0 0
$$685$$ −23943.7 −1.33554
$$686$$ 0 0
$$687$$ 15185.4 0.843320
$$688$$ 0 0
$$689$$ −1838.10 −0.101635
$$690$$ 0 0
$$691$$ −9817.07 −0.540462 −0.270231 0.962796i $$-0.587100\pi$$
−0.270231 + 0.962796i $$0.587100\pi$$
$$692$$ 0 0
$$693$$ 1211.58 0.0664128
$$694$$ 0 0
$$695$$ 473.354 0.0258350
$$696$$ 0 0
$$697$$ −10760.2 −0.584750
$$698$$ 0 0
$$699$$ −34854.9 −1.88603
$$700$$ 0 0
$$701$$ −29949.8 −1.61368 −0.806838 0.590773i $$-0.798823\pi$$
−0.806838 + 0.590773i $$0.798823\pi$$
$$702$$ 0 0
$$703$$ −1839.09 −0.0986667
$$704$$ 0 0
$$705$$ 40499.8 2.16356
$$706$$ 0 0
$$707$$ −495.976 −0.0263835
$$708$$ 0 0
$$709$$ −11307.5 −0.598959 −0.299479 0.954103i $$-0.596813\pi$$
−0.299479 + 0.954103i $$0.596813\pi$$
$$710$$ 0 0
$$711$$ −46406.3 −2.44778
$$712$$ 0 0
$$713$$ −3507.36 −0.184224
$$714$$ 0 0
$$715$$ −875.768 −0.0458068
$$716$$ 0 0
$$717$$ −32382.7 −1.68669
$$718$$ 0 0
$$719$$ 32623.4 1.69214 0.846070 0.533071i $$-0.178962\pi$$
0.846070 + 0.533071i $$0.178962\pi$$
$$720$$ 0 0
$$721$$ −106.773 −0.00551518
$$722$$ 0 0
$$723$$ −30988.0 −1.59399
$$724$$ 0 0
$$725$$ 2392.33 0.122550
$$726$$ 0 0
$$727$$ 502.545 0.0256373 0.0128187 0.999918i $$-0.495920\pi$$
0.0128187 + 0.999918i $$0.495920\pi$$
$$728$$ 0 0
$$729$$ −29783.2 −1.51314
$$730$$ 0 0
$$731$$ 2378.66 0.120353
$$732$$ 0 0
$$733$$ −8631.37 −0.434935 −0.217467 0.976068i $$-0.569780\pi$$
−0.217467 + 0.976068i $$0.569780\pi$$
$$734$$ 0 0
$$735$$ −39288.7 −1.97168
$$736$$ 0 0
$$737$$ −3765.95 −0.188223
$$738$$ 0 0
$$739$$ −18357.5 −0.913792 −0.456896 0.889520i $$-0.651039\pi$$
−0.456896 + 0.889520i $$0.651039\pi$$
$$740$$ 0 0
$$741$$ 5944.93 0.294726
$$742$$ 0 0
$$743$$ −11182.6 −0.552155 −0.276078 0.961135i $$-0.589035\pi$$
−0.276078 + 0.961135i $$0.589035\pi$$
$$744$$ 0 0
$$745$$ −36076.4 −1.77415
$$746$$ 0 0
$$747$$ 15843.0 0.775991
$$748$$ 0 0
$$749$$ −2556.29 −0.124706
$$750$$ 0 0
$$751$$ −16733.4 −0.813063 −0.406531 0.913637i $$-0.633262\pi$$
−0.406531 + 0.913637i $$0.633262\pi$$
$$752$$ 0 0
$$753$$ −8680.53 −0.420101
$$754$$ 0 0
$$755$$ −38280.2 −1.84524
$$756$$ 0 0
$$757$$ 24402.4 1.17163 0.585813 0.810446i $$-0.300775\pi$$
0.585813 + 0.810446i $$0.300775\pi$$
$$758$$ 0 0
$$759$$ 9711.19 0.464419
$$760$$ 0 0
$$761$$ 8469.33 0.403434 0.201717 0.979444i $$-0.435348\pi$$
0.201717 + 0.979444i $$0.435348\pi$$
$$762$$ 0 0
$$763$$ 3207.74 0.152199
$$764$$ 0 0
$$765$$ 21955.3 1.03764
$$766$$ 0 0
$$767$$ −473.570 −0.0222941
$$768$$ 0 0
$$769$$ 32834.7 1.53973 0.769864 0.638208i $$-0.220324\pi$$
0.769864 + 0.638208i $$0.220324\pi$$
$$770$$ 0 0
$$771$$ −6209.20 −0.290038
$$772$$ 0 0
$$773$$ 35571.4 1.65513 0.827564 0.561371i $$-0.189726\pi$$
0.827564 + 0.561371i $$0.189726\pi$$
$$774$$ 0 0
$$775$$ −3014.71 −0.139731
$$776$$ 0 0
$$777$$ −320.097 −0.0147792
$$778$$ 0 0
$$779$$ 36530.0 1.68013
$$780$$ 0 0
$$781$$ −2281.01 −0.104508
$$782$$ 0 0
$$783$$ −1755.02 −0.0801014
$$784$$ 0 0
$$785$$ 36783.6 1.67244
$$786$$ 0 0
$$787$$ 15729.6 0.712452 0.356226 0.934400i $$-0.384063\pi$$
0.356226 + 0.934400i $$0.384063\pi$$
$$788$$ 0 0
$$789$$ 48996.7 2.21081
$$790$$ 0 0
$$791$$ −906.431 −0.0407446
$$792$$ 0 0
$$793$$ 3958.67 0.177272
$$794$$ 0 0
$$795$$ 40399.5 1.80229
$$796$$ 0 0
$$797$$ −7888.07 −0.350577 −0.175288 0.984517i $$-0.556086\pi$$
−0.175288 + 0.984517i $$0.556086\pi$$
$$798$$ 0 0
$$799$$ −14171.8 −0.627485
$$800$$ 0 0
$$801$$ −53394.2 −2.35530
$$802$$ 0 0
$$803$$ 11116.6 0.488538
$$804$$ 0 0
$$805$$ 5081.73 0.222494
$$806$$ 0 0
$$807$$ 7824.86 0.341323
$$808$$ 0 0
$$809$$ 5896.97 0.256275 0.128138 0.991756i $$-0.459100\pi$$
0.128138 + 0.991756i $$0.459100\pi$$
$$810$$ 0 0
$$811$$ 14197.9 0.614744 0.307372 0.951589i $$-0.400550\pi$$
0.307372 + 0.951589i $$0.400550\pi$$
$$812$$ 0 0
$$813$$ 36287.3 1.56538
$$814$$ 0 0
$$815$$ 40485.3 1.74005
$$816$$ 0 0
$$817$$ −8075.35 −0.345803
$$818$$ 0 0
$$819$$ 590.258 0.0251835
$$820$$ 0 0
$$821$$ 19841.7 0.843459 0.421729 0.906722i $$-0.361423\pi$$
0.421729 + 0.906722i $$0.361423\pi$$
$$822$$ 0 0
$$823$$ 28202.2 1.19449 0.597246 0.802058i $$-0.296262\pi$$
0.597246 + 0.802058i $$0.296262\pi$$
$$824$$ 0 0
$$825$$ 8347.14 0.352255
$$826$$ 0 0
$$827$$ 34031.0 1.43092 0.715462 0.698651i $$-0.246216\pi$$
0.715462 + 0.698651i $$0.246216\pi$$
$$828$$ 0 0
$$829$$ −4931.55 −0.206610 −0.103305 0.994650i $$-0.532942\pi$$
−0.103305 + 0.994650i $$0.532942\pi$$
$$830$$ 0 0
$$831$$ 4501.92 0.187930
$$832$$ 0 0
$$833$$ 13748.0 0.571836
$$834$$ 0 0
$$835$$ 40666.4 1.68541
$$836$$ 0 0
$$837$$ 2211.60 0.0913312
$$838$$ 0 0
$$839$$ 38189.8 1.57146 0.785731 0.618568i $$-0.212287\pi$$
0.785731 + 0.618568i $$0.212287\pi$$
$$840$$ 0 0
$$841$$ −23764.3 −0.974384
$$842$$ 0 0
$$843$$ −42106.8 −1.72033
$$844$$ 0 0
$$845$$ 32212.9 1.31143
$$846$$ 0 0
$$847$$ −371.687 −0.0150783
$$848$$ 0 0
$$849$$ 37488.0 1.51541
$$850$$ 0 0
$$851$$ −1463.59 −0.0589556
$$852$$ 0 0
$$853$$ −42966.8 −1.72469 −0.862343 0.506325i $$-0.831003\pi$$
−0.862343 + 0.506325i $$0.831003\pi$$
$$854$$ 0 0
$$855$$ −74536.6 −2.98140
$$856$$ 0 0
$$857$$ −17281.5 −0.688828 −0.344414 0.938818i $$-0.611922\pi$$
−0.344414 + 0.938818i $$0.611922\pi$$
$$858$$ 0 0
$$859$$ 9316.75 0.370062 0.185031 0.982733i $$-0.440761\pi$$
0.185031 + 0.982733i $$0.440761\pi$$
$$860$$ 0 0
$$861$$ 6358.10 0.251665
$$862$$ 0 0
$$863$$ 9647.65 0.380544 0.190272 0.981731i $$-0.439063\pi$$
0.190272 + 0.981731i $$0.439063\pi$$
$$864$$ 0 0
$$865$$ 34279.9 1.34746
$$866$$ 0 0
$$867$$ 25483.6 0.998232
$$868$$ 0 0
$$869$$ 14236.5 0.555742
$$870$$ 0 0
$$871$$ −1834.70 −0.0713735
$$872$$ 0 0
$$873$$ 48277.6 1.87165
$$874$$ 0 0
$$875$$ −1336.55 −0.0516383
$$876$$ 0 0
$$877$$ −19728.7 −0.759624 −0.379812 0.925064i $$-0.624011\pi$$
−0.379812 + 0.925064i $$0.624011\pi$$
$$878$$ 0 0
$$879$$ 18464.1 0.708509
$$880$$ 0 0
$$881$$ 19473.9 0.744712 0.372356 0.928090i $$-0.378550\pi$$
0.372356 + 0.928090i $$0.378550\pi$$
$$882$$ 0 0
$$883$$ 49092.4 1.87100 0.935499 0.353329i $$-0.114950\pi$$
0.935499 + 0.353329i $$0.114950\pi$$
$$884$$ 0 0
$$885$$ 10408.5 0.395344
$$886$$ 0 0
$$887$$ −9292.86 −0.351774 −0.175887 0.984410i $$-0.556279\pi$$
−0.175887 + 0.984410i $$0.556279\pi$$
$$888$$ 0 0
$$889$$ −4047.41 −0.152695
$$890$$ 0 0
$$891$$ 4525.85 0.170170
$$892$$ 0 0
$$893$$ 48112.0 1.80292
$$894$$ 0 0
$$895$$ 19493.9 0.728055
$$896$$ 0 0
$$897$$ 4731.10 0.176106
$$898$$ 0 0
$$899$$ −787.273 −0.0292069
$$900$$ 0 0
$$901$$ −14136.7 −0.522709
$$902$$ 0 0
$$903$$ −1405.53 −0.0517974
$$904$$ 0 0
$$905$$ −11932.3 −0.438279
$$906$$ 0 0
$$907$$ 37688.7 1.37975 0.689875 0.723928i $$-0.257665\pi$$
0.689875 + 0.723928i $$0.257665\pi$$
$$908$$ 0 0
$$909$$ 5789.42 0.211246
$$910$$ 0 0
$$911$$ −33049.6 −1.20196 −0.600979 0.799265i $$-0.705222\pi$$
−0.600979 + 0.799265i $$0.705222\pi$$
$$912$$ 0 0
$$913$$ −4860.31 −0.176180
$$914$$ 0 0
$$915$$ −87007.2 −3.14357
$$916$$ 0 0
$$917$$ 4917.06 0.177073
$$918$$ 0 0
$$919$$ 23148.0 0.830883 0.415442 0.909620i $$-0.363627\pi$$
0.415442 + 0.909620i $$0.363627\pi$$
$$920$$ 0 0
$$921$$ 13304.1 0.475986
$$922$$ 0 0
$$923$$ −1111.26 −0.0396290
$$924$$ 0 0
$$925$$ −1258.01 −0.0447169
$$926$$ 0 0
$$927$$ 1246.34 0.0441588
$$928$$ 0 0
$$929$$ −23177.9 −0.818561 −0.409280 0.912409i $$-0.634220\pi$$
−0.409280 + 0.912409i $$0.634220\pi$$
$$930$$ 0 0
$$931$$ −46673.3 −1.64302
$$932$$ 0 0
$$933$$ 28325.1 0.993916
$$934$$ 0 0
$$935$$ −6735.44 −0.235585
$$936$$ 0 0
$$937$$ −34574.7 −1.20545 −0.602724 0.797950i $$-0.705918\pi$$
−0.602724 + 0.797950i $$0.705918\pi$$
$$938$$ 0 0
$$939$$ −56959.1 −1.97954
$$940$$ 0 0
$$941$$ −41831.2 −1.44916 −0.724578 0.689192i $$-0.757966\pi$$
−0.724578 + 0.689192i $$0.757966\pi$$
$$942$$ 0 0
$$943$$ 29071.3 1.00392
$$944$$ 0 0
$$945$$ −3204.34 −0.110304
$$946$$ 0 0
$$947$$ 27231.2 0.934419 0.467209 0.884147i $$-0.345259\pi$$
0.467209 + 0.884147i $$0.345259\pi$$
$$948$$ 0 0
$$949$$ 5415.79 0.185252
$$950$$ 0 0
$$951$$ −124.534 −0.00424635
$$952$$ 0 0
$$953$$ 40939.4 1.39156 0.695781 0.718254i $$-0.255058\pi$$
0.695781 + 0.718254i $$0.255058\pi$$
$$954$$ 0 0
$$955$$ 25527.0 0.864956
$$956$$ 0 0
$$957$$ 2179.81 0.0736292
$$958$$ 0 0
$$959$$ −4950.74 −0.166703
$$960$$ 0 0
$$961$$ −28798.9 −0.966698
$$962$$ 0 0
$$963$$ 29839.0 0.998491
$$964$$ 0 0
$$965$$ −19910.3 −0.664182
$$966$$ 0 0
$$967$$ 46173.1 1.53550 0.767750 0.640750i $$-0.221376\pi$$
0.767750 + 0.640750i $$0.221376\pi$$
$$968$$ 0 0
$$969$$ 45721.8 1.51579
$$970$$ 0 0
$$971$$ −5153.91 −0.170337 −0.0851683 0.996367i $$-0.527143\pi$$
−0.0851683 + 0.996367i $$0.527143\pi$$
$$972$$ 0 0
$$973$$ 97.8734 0.00322474
$$974$$ 0 0
$$975$$ 4066.56 0.133574
$$976$$ 0 0
$$977$$ 9692.13 0.317378 0.158689 0.987329i $$-0.449273\pi$$
0.158689 + 0.987329i $$0.449273\pi$$
$$978$$ 0 0
$$979$$ 16380.2 0.534744
$$980$$ 0 0
$$981$$ −37443.3 −1.21863
$$982$$ 0 0
$$983$$ −32915.7 −1.06800 −0.534002 0.845483i $$-0.679313\pi$$
−0.534002 + 0.845483i $$0.679313\pi$$
$$984$$ 0 0
$$985$$ −52269.7 −1.69081
$$986$$ 0 0
$$987$$ 8373.97 0.270057
$$988$$ 0 0
$$989$$ −6426.54 −0.206625
$$990$$ 0 0
$$991$$ −29477.9 −0.944901 −0.472451 0.881357i $$-0.656630\pi$$
−0.472451 + 0.881357i $$0.656630\pi$$
$$992$$ 0 0
$$993$$ 10456.9 0.334180
$$994$$ 0 0
$$995$$ 12237.1 0.389892
$$996$$ 0 0
$$997$$ 31944.4 1.01473 0.507366 0.861731i $$-0.330619\pi$$
0.507366 + 0.861731i $$0.330619\pi$$
$$998$$ 0 0
$$999$$ 922.883 0.0292279
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 704.4.a.n.1.1 2
4.3 odd 2 704.4.a.p.1.2 2
8.3 odd 2 11.4.a.a.1.2 2
8.5 even 2 176.4.a.i.1.2 2
24.5 odd 2 1584.4.a.bc.1.1 2
24.11 even 2 99.4.a.c.1.1 2
40.3 even 4 275.4.b.c.199.1 4
40.19 odd 2 275.4.a.b.1.1 2
40.27 even 4 275.4.b.c.199.4 4
56.27 even 2 539.4.a.e.1.2 2
88.3 odd 10 121.4.c.c.9.2 8
88.19 even 10 121.4.c.f.9.1 8
88.21 odd 2 1936.4.a.w.1.2 2
88.27 odd 10 121.4.c.c.3.1 8
88.35 even 10 121.4.c.f.81.2 8
88.43 even 2 121.4.a.c.1.1 2
88.51 even 10 121.4.c.f.27.1 8
88.59 odd 10 121.4.c.c.27.2 8
88.75 odd 10 121.4.c.c.81.1 8
88.83 even 10 121.4.c.f.3.2 8
104.51 odd 2 1859.4.a.a.1.1 2
120.59 even 2 2475.4.a.q.1.2 2
264.131 odd 2 1089.4.a.v.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.2 2 8.3 odd 2
99.4.a.c.1.1 2 24.11 even 2
121.4.a.c.1.1 2 88.43 even 2
121.4.c.c.3.1 8 88.27 odd 10
121.4.c.c.9.2 8 88.3 odd 10
121.4.c.c.27.2 8 88.59 odd 10
121.4.c.c.81.1 8 88.75 odd 10
121.4.c.f.3.2 8 88.83 even 10
121.4.c.f.9.1 8 88.19 even 10
121.4.c.f.27.1 8 88.51 even 10
121.4.c.f.81.2 8 88.35 even 10
176.4.a.i.1.2 2 8.5 even 2
275.4.a.b.1.1 2 40.19 odd 2
275.4.b.c.199.1 4 40.3 even 4
275.4.b.c.199.4 4 40.27 even 4
539.4.a.e.1.2 2 56.27 even 2
704.4.a.n.1.1 2 1.1 even 1 trivial
704.4.a.p.1.2 2 4.3 odd 2
1089.4.a.v.1.2 2 264.131 odd 2
1584.4.a.bc.1.1 2 24.5 odd 2
1859.4.a.a.1.1 2 104.51 odd 2
1936.4.a.w.1.2 2 88.21 odd 2
2475.4.a.q.1.2 2 120.59 even 2