# Properties

 Label 704.4.a.p.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 $1$

# Related objects

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)

 $$f(q)$$ $$=$$ $$q-5.92820 q^{3} +12.8564 q^{5} +16.9282 q^{7} +8.14359 q^{9} +O(q^{10})$$ $$q-5.92820 q^{3} +12.8564 q^{5} +16.9282 q^{7} +8.14359 q^{9} +11.0000 q^{11} -74.6410 q^{13} -76.2154 q^{15} -82.7846 q^{17} +67.9230 q^{19} -100.354 q^{21} +13.3538 q^{23} +40.2872 q^{25} +111.785 q^{27} -168.995 q^{29} -65.4974 q^{31} -65.2102 q^{33} +217.636 q^{35} -40.8564 q^{37} +442.487 q^{39} +274.928 q^{41} +2.28719 q^{43} +104.697 q^{45} +71.8461 q^{47} -56.4359 q^{49} +490.764 q^{51} +149.005 q^{53} +141.420 q^{55} -402.662 q^{57} -545.631 q^{59} -101.303 q^{61} +137.856 q^{63} -959.615 q^{65} -411.641 q^{67} -79.1642 q^{69} -470.636 q^{71} +610.600 q^{73} -238.831 q^{75} +186.210 q^{77} -978.225 q^{79} -882.559 q^{81} -26.1539 q^{83} -1064.31 q^{85} +1001.84 q^{87} -352.887 q^{89} -1263.54 q^{91} +388.282 q^{93} +873.246 q^{95} +847.585 q^{97} +89.5795 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{5} + 20 q^{7} + 44 q^{9}+O(q^{10})$$ 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
$$3$$ −5.92820 −1.14088 −0.570442 0.821338i $$-0.693228\pi$$
−0.570442 + 0.821338i $$0.693228\pi$$
$$4$$ 0 0
$$5$$ 12.8564 1.14991 0.574956 0.818184i $$-0.305019\pi$$
0.574956 + 0.818184i $$0.305019\pi$$
$$6$$ 0 0
$$7$$ 16.9282 0.914037 0.457019 0.889457i $$-0.348917\pi$$
0.457019 + 0.889457i $$0.348917\pi$$
$$8$$ 0 0
$$9$$ 8.14359 0.301615
$$10$$ 0 0
$$11$$ 11.0000 0.301511
$$12$$ 0 0
$$13$$ −74.6410 −1.59244 −0.796219 0.605009i $$-0.793170\pi$$
−0.796219 + 0.605009i $$0.793170\pi$$
$$14$$ 0 0
$$15$$ −76.2154 −1.31192
$$16$$ 0 0
$$17$$ −82.7846 −1.18107 −0.590536 0.807011i $$-0.701084\pi$$
−0.590536 + 0.807011i $$0.701084\pi$$
$$18$$ 0 0
$$19$$ 67.9230 0.820138 0.410069 0.912055i $$-0.365505\pi$$
0.410069 + 0.912055i $$0.365505\pi$$
$$20$$ 0 0
$$21$$ −100.354 −1.04281
$$22$$ 0 0
$$23$$ 13.3538 0.121064 0.0605319 0.998166i $$-0.480720\pi$$
0.0605319 + 0.998166i $$0.480720\pi$$
$$24$$ 0 0
$$25$$ 40.2872 0.322297
$$26$$ 0 0
$$27$$ 111.785 0.796776
$$28$$ 0 0
$$29$$ −168.995 −1.08212 −0.541061 0.840983i $$-0.681977\pi$$
−0.541061 + 0.840983i $$0.681977\pi$$
$$30$$ 0 0
$$31$$ −65.4974 −0.379474 −0.189737 0.981835i $$-0.560763\pi$$
−0.189737 + 0.981835i $$0.560763\pi$$
$$32$$ 0 0
$$33$$ −65.2102 −0.343989
$$34$$ 0 0
$$35$$ 217.636 1.05106
$$36$$ 0 0
$$37$$ −40.8564 −0.181534 −0.0907669 0.995872i $$-0.528932\pi$$
−0.0907669 + 0.995872i $$0.528932\pi$$
$$38$$ 0 0
$$39$$ 442.487 1.81679
$$40$$ 0 0
$$41$$ 274.928 1.04723 0.523617 0.851954i $$-0.324582\pi$$
0.523617 + 0.851954i $$0.324582\pi$$
$$42$$ 0 0
$$43$$ 2.28719 0.00811146 0.00405573 0.999992i $$-0.498709\pi$$
0.00405573 + 0.999992i $$0.498709\pi$$
$$44$$ 0 0
$$45$$ 104.697 0.346830
$$46$$ 0 0
$$47$$ 71.8461 0.222975 0.111488 0.993766i $$-0.464438\pi$$
0.111488 + 0.993766i $$0.464438\pi$$
$$48$$ 0 0
$$49$$ −56.4359 −0.164536
$$50$$ 0 0
$$51$$ 490.764 1.34746
$$52$$ 0 0
$$53$$ 149.005 0.386178 0.193089 0.981181i $$-0.438149\pi$$
0.193089 + 0.981181i $$0.438149\pi$$
$$54$$ 0 0
$$55$$ 141.420 0.346711
$$56$$ 0 0
$$57$$ −402.662 −0.935681
$$58$$ 0 0
$$59$$ −545.631 −1.20398 −0.601992 0.798502i $$-0.705626\pi$$
−0.601992 + 0.798502i $$0.705626\pi$$
$$60$$ 0 0
$$61$$ −101.303 −0.212631 −0.106315 0.994332i $$-0.533905\pi$$
−0.106315 + 0.994332i $$0.533905\pi$$
$$62$$ 0 0
$$63$$ 137.856 0.275687
$$64$$ 0 0
$$65$$ −959.615 −1.83116
$$66$$ 0 0
$$67$$ −411.641 −0.750596 −0.375298 0.926904i $$-0.622460\pi$$
−0.375298 + 0.926904i $$0.622460\pi$$
$$68$$ 0 0
$$69$$ −79.1642 −0.138120
$$70$$ 0 0
$$71$$ −470.636 −0.786679 −0.393339 0.919393i $$-0.628680\pi$$
−0.393339 + 0.919393i $$0.628680\pi$$
$$72$$ 0 0
$$73$$ 610.600 0.978977 0.489488 0.872010i $$-0.337184\pi$$
0.489488 + 0.872010i $$0.337184\pi$$
$$74$$ 0 0
$$75$$ −238.831 −0.367704
$$76$$ 0 0
$$77$$ 186.210 0.275593
$$78$$ 0 0
$$79$$ −978.225 −1.39315 −0.696576 0.717483i $$-0.745294\pi$$
−0.696576 + 0.717483i $$0.745294\pi$$
$$80$$ 0 0
$$81$$ −882.559 −1.21064
$$82$$ 0 0
$$83$$ −26.1539 −0.0345875 −0.0172938 0.999850i $$-0.505505\pi$$
−0.0172938 + 0.999850i $$0.505505\pi$$
$$84$$ 0 0
$$85$$ −1064.31 −1.35813
$$86$$ 0 0
$$87$$ 1001.84 1.23458
$$88$$ 0 0
$$89$$ −352.887 −0.420292 −0.210146 0.977670i $$-0.567394\pi$$
−0.210146 + 0.977670i $$0.567394\pi$$
$$90$$ 0 0
$$91$$ −1263.54 −1.45555
$$92$$ 0 0
$$93$$ 388.282 0.432935
$$94$$ 0 0
$$95$$ 873.246 0.943086
$$96$$ 0 0
$$97$$ 847.585 0.887208 0.443604 0.896223i $$-0.353700\pi$$
0.443604 + 0.896223i $$0.353700\pi$$
$$98$$ 0 0
$$99$$ 89.5795 0.0909402
$$100$$ 0 0
$$101$$ −1293.46 −1.27430 −0.637150 0.770740i $$-0.719887\pi$$
−0.637150 + 0.770740i $$0.719887\pi$$
$$102$$ 0 0
$$103$$ −1725.24 −1.65042 −0.825209 0.564828i $$-0.808943\pi$$
−0.825209 + 0.564828i $$0.808943\pi$$
$$104$$ 0 0
$$105$$ −1290.19 −1.19914
$$106$$ 0 0
$$107$$ 484.179 0.437452 0.218726 0.975786i $$-0.429810\pi$$
0.218726 + 0.975786i $$0.429810\pi$$
$$108$$ 0 0
$$109$$ 64.2563 0.0564645 0.0282323 0.999601i $$-0.491012\pi$$
0.0282323 + 0.999601i $$0.491012\pi$$
$$110$$ 0 0
$$111$$ 242.205 0.207109
$$112$$ 0 0
$$113$$ −2005.08 −1.66922 −0.834612 0.550839i $$-0.814308\pi$$
−0.834612 + 0.550839i $$0.814308\pi$$
$$114$$ 0 0
$$115$$ 171.682 0.139213
$$116$$ 0 0
$$117$$ −607.846 −0.480302
$$118$$ 0 0
$$119$$ −1401.39 −1.07954
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ 0 0
$$123$$ −1629.83 −1.19477
$$124$$ 0 0
$$125$$ −1089.10 −0.779298
$$126$$ 0 0
$$127$$ 109.605 0.0765816 0.0382908 0.999267i $$-0.487809\pi$$
0.0382908 + 0.999267i $$0.487809\pi$$
$$128$$ 0 0
$$129$$ −13.5589 −0.00925423
$$130$$ 0 0
$$131$$ −1156.71 −0.771469 −0.385734 0.922610i $$-0.626052\pi$$
−0.385734 + 0.922610i $$0.626052\pi$$
$$132$$ 0 0
$$133$$ 1149.82 0.749636
$$134$$ 0 0
$$135$$ 1437.15 0.916223
$$136$$ 0 0
$$137$$ 198.323 0.123678 0.0618391 0.998086i $$-0.480303\pi$$
0.0618391 + 0.998086i $$0.480303\pi$$
$$138$$ 0 0
$$139$$ 2900.14 1.76969 0.884844 0.465888i $$-0.154265\pi$$
0.884844 + 0.465888i $$0.154265\pi$$
$$140$$ 0 0
$$141$$ −425.918 −0.254389
$$142$$ 0 0
$$143$$ −821.051 −0.480138
$$144$$ 0 0
$$145$$ −2172.67 −1.24435
$$146$$ 0 0
$$147$$ 334.564 0.187717
$$148$$ 0 0
$$149$$ −3488.34 −1.91796 −0.958980 0.283472i $$-0.908514\pi$$
−0.958980 + 0.283472i $$0.908514\pi$$
$$150$$ 0 0
$$151$$ −1163.32 −0.626953 −0.313477 0.949596i $$-0.601494\pi$$
−0.313477 + 0.949596i $$0.601494\pi$$
$$152$$ 0 0
$$153$$ −674.164 −0.356228
$$154$$ 0 0
$$155$$ −842.061 −0.436361
$$156$$ 0 0
$$157$$ −342.057 −0.173880 −0.0869398 0.996214i $$-0.527709\pi$$
−0.0869398 + 0.996214i $$0.527709\pi$$
$$158$$ 0 0
$$159$$ −883.333 −0.440584
$$160$$ 0 0
$$161$$ 226.056 0.110657
$$162$$ 0 0
$$163$$ 1394.89 0.670285 0.335142 0.942167i $$-0.391216\pi$$
0.335142 + 0.942167i $$0.391216\pi$$
$$164$$ 0 0
$$165$$ −838.369 −0.395557
$$166$$ 0 0
$$167$$ 478.703 0.221815 0.110908 0.993831i $$-0.464624\pi$$
0.110908 + 0.993831i $$0.464624\pi$$
$$168$$ 0 0
$$169$$ 3374.28 1.53586
$$170$$ 0 0
$$171$$ 553.138 0.247365
$$172$$ 0 0
$$173$$ −1808.58 −0.794822 −0.397411 0.917641i $$-0.630091\pi$$
−0.397411 + 0.917641i $$0.630091\pi$$
$$174$$ 0 0
$$175$$ 681.990 0.294592
$$176$$ 0 0
$$177$$ 3234.61 1.37361
$$178$$ 0 0
$$179$$ 4429.85 1.84973 0.924867 0.380292i $$-0.124176\pi$$
0.924867 + 0.380292i $$0.124176\pi$$
$$180$$ 0 0
$$181$$ −3409.17 −1.40001 −0.700005 0.714138i $$-0.746819\pi$$
−0.700005 + 0.714138i $$0.746819\pi$$
$$182$$ 0 0
$$183$$ 600.543 0.242587
$$184$$ 0 0
$$185$$ −525.267 −0.208748
$$186$$ 0 0
$$187$$ −910.631 −0.356106
$$188$$ 0 0
$$189$$ 1892.31 0.728283
$$190$$ 0 0
$$191$$ 2923.75 1.10762 0.553810 0.832643i $$-0.313173\pi$$
0.553810 + 0.832643i $$0.313173\pi$$
$$192$$ 0 0
$$193$$ −2484.18 −0.926505 −0.463253 0.886226i $$-0.653318\pi$$
−0.463253 + 0.886226i $$0.653318\pi$$
$$194$$ 0 0
$$195$$ 5688.79 2.08914
$$196$$ 0 0
$$197$$ 5125.67 1.85375 0.926876 0.375369i $$-0.122484\pi$$
0.926876 + 0.375369i $$0.122484\pi$$
$$198$$ 0 0
$$199$$ −7.69219 −0.00274013 −0.00137006 0.999999i $$-0.500436\pi$$
−0.00137006 + 0.999999i $$0.500436\pi$$
$$200$$ 0 0
$$201$$ 2440.29 0.856343
$$202$$ 0 0
$$203$$ −2860.78 −0.989100
$$204$$ 0 0
$$205$$ 3534.59 1.20423
$$206$$ 0 0
$$207$$ 108.748 0.0365146
$$208$$ 0 0
$$209$$ 747.154 0.247281
$$210$$ 0 0
$$211$$ −3107.34 −1.01383 −0.506915 0.861996i $$-0.669214\pi$$
−0.506915 + 0.861996i $$0.669214\pi$$
$$212$$ 0 0
$$213$$ 2790.03 0.897509
$$214$$ 0 0
$$215$$ 29.4050 0.00932746
$$216$$ 0 0
$$217$$ −1108.75 −0.346853
$$218$$ 0 0
$$219$$ −3619.76 −1.11690
$$220$$ 0 0
$$221$$ 6179.13 1.88078
$$222$$ 0 0
$$223$$ −12.3185 −0.00369913 −0.00184957 0.999998i $$-0.500589\pi$$
−0.00184957 + 0.999998i $$0.500589\pi$$
$$224$$ 0 0
$$225$$ 328.082 0.0972096
$$226$$ 0 0
$$227$$ −4615.90 −1.34964 −0.674820 0.737983i $$-0.735779\pi$$
−0.674820 + 0.737983i $$0.735779\pi$$
$$228$$ 0 0
$$229$$ −5074.63 −1.46437 −0.732186 0.681105i $$-0.761500\pi$$
−0.732186 + 0.681105i $$0.761500\pi$$
$$230$$ 0 0
$$231$$ −1103.89 −0.314419
$$232$$ 0 0
$$233$$ 211.683 0.0595184 0.0297592 0.999557i $$-0.490526\pi$$
0.0297592 + 0.999557i $$0.490526\pi$$
$$234$$ 0 0
$$235$$ 923.683 0.256402
$$236$$ 0 0
$$237$$ 5799.12 1.58942
$$238$$ 0 0
$$239$$ 4312.49 1.16716 0.583581 0.812055i $$-0.301651\pi$$
0.583581 + 0.812055i $$0.301651\pi$$
$$240$$ 0 0
$$241$$ −996.584 −0.266372 −0.133186 0.991091i $$-0.542521\pi$$
−0.133186 + 0.991091i $$0.542521\pi$$
$$242$$ 0 0
$$243$$ 2213.80 0.584426
$$244$$ 0 0
$$245$$ −725.563 −0.189202
$$246$$ 0 0
$$247$$ −5069.85 −1.30602
$$248$$ 0 0
$$249$$ 155.046 0.0394603
$$250$$ 0 0
$$251$$ 276.892 0.0696306 0.0348153 0.999394i $$-0.488916\pi$$
0.0348153 + 0.999394i $$0.488916\pi$$
$$252$$ 0 0
$$253$$ 146.892 0.0365021
$$254$$ 0 0
$$255$$ 6309.46 1.54947
$$256$$ 0 0
$$257$$ −3235.18 −0.785233 −0.392617 0.919702i $$-0.628430\pi$$
−0.392617 + 0.919702i $$0.628430\pi$$
$$258$$ 0 0
$$259$$ −691.626 −0.165929
$$260$$ 0 0
$$261$$ −1376.23 −0.326384
$$262$$ 0 0
$$263$$ 207.944 0.0487544 0.0243772 0.999703i $$-0.492240\pi$$
0.0243772 + 0.999703i $$0.492240\pi$$
$$264$$ 0 0
$$265$$ 1915.67 0.444071
$$266$$ 0 0
$$267$$ 2091.99 0.479504
$$268$$ 0 0
$$269$$ −5033.04 −1.14078 −0.570390 0.821374i $$-0.693208\pi$$
−0.570390 + 0.821374i $$0.693208\pi$$
$$270$$ 0 0
$$271$$ 1487.01 0.333319 0.166660 0.986015i $$-0.446702\pi$$
0.166660 + 0.986015i $$0.446702\pi$$
$$272$$ 0 0
$$273$$ 7490.51 1.66061
$$274$$ 0 0
$$275$$ 443.159 0.0971764
$$276$$ 0 0
$$277$$ 235.836 0.0511552 0.0255776 0.999673i $$-0.491858\pi$$
0.0255776 + 0.999673i $$0.491858\pi$$
$$278$$ 0 0
$$279$$ −533.384 −0.114455
$$280$$ 0 0
$$281$$ −4915.01 −1.04343 −0.521717 0.853118i $$-0.674708\pi$$
−0.521717 + 0.853118i $$0.674708\pi$$
$$282$$ 0 0
$$283$$ 5199.56 1.09216 0.546081 0.837733i $$-0.316119\pi$$
0.546081 + 0.837733i $$0.316119\pi$$
$$284$$ 0 0
$$285$$ −5176.78 −1.07595
$$286$$ 0 0
$$287$$ 4654.04 0.957210
$$288$$ 0 0
$$289$$ 1940.29 0.394930
$$290$$ 0 0
$$291$$ −5024.65 −1.01220
$$292$$ 0 0
$$293$$ 8880.92 1.77075 0.885373 0.464881i $$-0.153903\pi$$
0.885373 + 0.464881i $$0.153903\pi$$
$$294$$ 0 0
$$295$$ −7014.85 −1.38448
$$296$$ 0 0
$$297$$ 1229.63 0.240237
$$298$$ 0 0
$$299$$ −996.743 −0.192786
$$300$$ 0 0
$$301$$ 38.7180 0.00741417
$$302$$ 0 0
$$303$$ 7667.90 1.45383
$$304$$ 0 0
$$305$$ −1302.39 −0.244507
$$306$$ 0 0
$$307$$ 1497.93 0.278474 0.139237 0.990259i $$-0.455535\pi$$
0.139237 + 0.990259i $$0.455535\pi$$
$$308$$ 0 0
$$309$$ 10227.6 1.88293
$$310$$ 0 0
$$311$$ −7484.71 −1.36469 −0.682345 0.731030i $$-0.739040\pi$$
−0.682345 + 0.731030i $$0.739040\pi$$
$$312$$ 0 0
$$313$$ −658.363 −0.118891 −0.0594455 0.998232i $$-0.518933\pi$$
−0.0594455 + 0.998232i $$0.518933\pi$$
$$314$$ 0 0
$$315$$ 1772.34 0.317016
$$316$$ 0 0
$$317$$ −233.708 −0.0414080 −0.0207040 0.999786i $$-0.506591\pi$$
−0.0207040 + 0.999786i $$0.506591\pi$$
$$318$$ 0 0
$$319$$ −1858.94 −0.326272
$$320$$ 0 0
$$321$$ −2870.31 −0.499082
$$322$$ 0 0
$$323$$ −5622.98 −0.968641
$$324$$ 0 0
$$325$$ −3007.08 −0.513239
$$326$$ 0 0
$$327$$ −380.924 −0.0644194
$$328$$ 0 0
$$329$$ 1216.23 0.203808
$$330$$ 0 0
$$331$$ −8532.95 −1.41696 −0.708480 0.705731i $$-0.750619\pi$$
−0.708480 + 0.705731i $$0.750619\pi$$
$$332$$ 0 0
$$333$$ −332.718 −0.0547533
$$334$$ 0 0
$$335$$ −5292.22 −0.863120
$$336$$ 0 0
$$337$$ 11691.2 1.88979 0.944895 0.327373i $$-0.106163\pi$$
0.944895 + 0.327373i $$0.106163\pi$$
$$338$$ 0 0
$$339$$ 11886.5 1.90439
$$340$$ 0 0
$$341$$ −720.472 −0.114416
$$342$$ 0 0
$$343$$ −6761.73 −1.06443
$$344$$ 0 0
$$345$$ −1017.77 −0.158825
$$346$$ 0 0
$$347$$ −4598.79 −0.711459 −0.355729 0.934589i $$-0.615768\pi$$
−0.355729 + 0.934589i $$0.615768\pi$$
$$348$$ 0 0
$$349$$ −6720.27 −1.03074 −0.515369 0.856968i $$-0.672345\pi$$
−0.515369 + 0.856968i $$0.672345\pi$$
$$350$$ 0 0
$$351$$ −8343.72 −1.26882
$$352$$ 0 0
$$353$$ 5738.70 0.865270 0.432635 0.901569i $$-0.357584\pi$$
0.432635 + 0.901569i $$0.357584\pi$$
$$354$$ 0 0
$$355$$ −6050.69 −0.904611
$$356$$ 0 0
$$357$$ 8307.75 1.23163
$$358$$ 0 0
$$359$$ −4115.27 −0.605001 −0.302501 0.953149i $$-0.597821\pi$$
−0.302501 + 0.953149i $$0.597821\pi$$
$$360$$ 0 0
$$361$$ −2245.46 −0.327374
$$362$$ 0 0
$$363$$ −717.313 −0.103717
$$364$$ 0 0
$$365$$ 7850.12 1.12574
$$366$$ 0 0
$$367$$ 9662.99 1.37440 0.687199 0.726469i $$-0.258840\pi$$
0.687199 + 0.726469i $$0.258840\pi$$
$$368$$ 0 0
$$369$$ 2238.90 0.315861
$$370$$ 0 0
$$371$$ 2522.39 0.352981
$$372$$ 0 0
$$373$$ 141.780 0.0196812 0.00984062 0.999952i $$-0.496868\pi$$
0.00984062 + 0.999952i $$0.496868\pi$$
$$374$$ 0 0
$$375$$ 6456.42 0.889088
$$376$$ 0 0
$$377$$ 12613.9 1.72321
$$378$$ 0 0
$$379$$ 2819.73 0.382163 0.191082 0.981574i $$-0.438800\pi$$
0.191082 + 0.981574i $$0.438800\pi$$
$$380$$ 0 0
$$381$$ −649.760 −0.0873707
$$382$$ 0 0
$$383$$ −6337.84 −0.845557 −0.422778 0.906233i $$-0.638945\pi$$
−0.422778 + 0.906233i $$0.638945\pi$$
$$384$$ 0 0
$$385$$ 2393.99 0.316907
$$386$$ 0 0
$$387$$ 18.6259 0.00244653
$$388$$ 0 0
$$389$$ 8805.25 1.14767 0.573836 0.818970i $$-0.305455\pi$$
0.573836 + 0.818970i $$0.305455\pi$$
$$390$$ 0 0
$$391$$ −1105.49 −0.142985
$$392$$ 0 0
$$393$$ 6857.23 0.880156
$$394$$ 0 0
$$395$$ −12576.5 −1.60200
$$396$$ 0 0
$$397$$ −4315.26 −0.545534 −0.272767 0.962080i $$-0.587939\pi$$
−0.272767 + 0.962080i $$0.587939\pi$$
$$398$$ 0 0
$$399$$ −6816.34 −0.855247
$$400$$ 0 0
$$401$$ 361.681 0.0450411 0.0225206 0.999746i $$-0.492831\pi$$
0.0225206 + 0.999746i $$0.492831\pi$$
$$402$$ 0 0
$$403$$ 4888.79 0.604288
$$404$$ 0 0
$$405$$ −11346.5 −1.39213
$$406$$ 0 0
$$407$$ −449.420 −0.0547345
$$408$$ 0 0
$$409$$ 9220.50 1.11473 0.557365 0.830268i $$-0.311812\pi$$
0.557365 + 0.830268i $$0.311812\pi$$
$$410$$ 0 0
$$411$$ −1175.70 −0.141102
$$412$$ 0 0
$$413$$ −9236.55 −1.10049
$$414$$ 0 0
$$415$$ −336.245 −0.0397726
$$416$$ 0 0
$$417$$ −17192.6 −2.01901
$$418$$ 0 0
$$419$$ 14912.9 1.73876 0.869380 0.494144i $$-0.164519\pi$$
0.869380 + 0.494144i $$0.164519\pi$$
$$420$$ 0 0
$$421$$ 13486.0 1.56121 0.780603 0.625027i $$-0.214912\pi$$
0.780603 + 0.625027i $$0.214912\pi$$
$$422$$ 0 0
$$423$$ 585.085 0.0672525
$$424$$ 0 0
$$425$$ −3335.16 −0.380656
$$426$$ 0 0
$$427$$ −1714.87 −0.194352
$$428$$ 0 0
$$429$$ 4867.36 0.547782
$$430$$ 0 0
$$431$$ 406.334 0.0454116 0.0227058 0.999742i $$-0.492772\pi$$
0.0227058 + 0.999742i $$0.492772\pi$$
$$432$$ 0 0
$$433$$ −1766.69 −0.196078 −0.0980391 0.995183i $$-0.531257\pi$$
−0.0980391 + 0.995183i $$0.531257\pi$$
$$434$$ 0 0
$$435$$ 12880.0 1.41965
$$436$$ 0 0
$$437$$ 907.033 0.0992889
$$438$$ 0 0
$$439$$ 7824.19 0.850634 0.425317 0.905044i $$-0.360163\pi$$
0.425317 + 0.905044i $$0.360163\pi$$
$$440$$ 0 0
$$441$$ −459.591 −0.0496265
$$442$$ 0 0
$$443$$ −11667.9 −1.25137 −0.625686 0.780075i $$-0.715181\pi$$
−0.625686 + 0.780075i $$0.715181\pi$$
$$444$$ 0 0
$$445$$ −4536.86 −0.483299
$$446$$ 0 0
$$447$$ 20679.6 2.18817
$$448$$ 0 0
$$449$$ 16975.3 1.78421 0.892107 0.451825i $$-0.149227\pi$$
0.892107 + 0.451825i $$0.149227\pi$$
$$450$$ 0 0
$$451$$ 3024.21 0.315753
$$452$$ 0 0
$$453$$ 6896.42 0.715280
$$454$$ 0 0
$$455$$ −16244.6 −1.67375
$$456$$ 0 0
$$457$$ −16192.9 −1.65748 −0.828741 0.559632i $$-0.810943\pi$$
−0.828741 + 0.559632i $$0.810943\pi$$
$$458$$ 0 0
$$459$$ −9254.05 −0.941050
$$460$$ 0 0
$$461$$ −8586.04 −0.867444 −0.433722 0.901047i $$-0.642800\pi$$
−0.433722 + 0.901047i $$0.642800\pi$$
$$462$$ 0 0
$$463$$ −7917.20 −0.794694 −0.397347 0.917668i $$-0.630069\pi$$
−0.397347 + 0.917668i $$0.630069\pi$$
$$464$$ 0 0
$$465$$ 4991.91 0.497837
$$466$$ 0 0
$$467$$ 15155.0 1.50169 0.750844 0.660480i $$-0.229647\pi$$
0.750844 + 0.660480i $$0.229647\pi$$
$$468$$ 0 0
$$469$$ −6968.34 −0.686073
$$470$$ 0 0
$$471$$ 2027.78 0.198376
$$472$$ 0 0
$$473$$ 25.1591 0.00244570
$$474$$ 0 0
$$475$$ 2736.43 0.264328
$$476$$ 0 0
$$477$$ 1213.44 0.116477
$$478$$ 0 0
$$479$$ 10001.1 0.953993 0.476996 0.878905i $$-0.341725\pi$$
0.476996 + 0.878905i $$0.341725\pi$$
$$480$$ 0 0
$$481$$ 3049.56 0.289081
$$482$$ 0 0
$$483$$ −1340.11 −0.126246
$$484$$ 0 0
$$485$$ 10896.9 1.02021
$$486$$ 0 0
$$487$$ 7044.54 0.655480 0.327740 0.944768i $$-0.393713\pi$$
0.327740 + 0.944768i $$0.393713\pi$$
$$488$$ 0 0
$$489$$ −8269.21 −0.764717
$$490$$ 0 0
$$491$$ 13326.4 1.22487 0.612437 0.790520i $$-0.290189\pi$$
0.612437 + 0.790520i $$0.290189\pi$$
$$492$$ 0 0
$$493$$ 13990.2 1.27806
$$494$$ 0 0
$$495$$ 1151.67 0.104573
$$496$$ 0 0
$$497$$ −7967.02 −0.719054
$$498$$ 0 0
$$499$$ 20069.1 1.80044 0.900218 0.435440i $$-0.143407\pi$$
0.900218 + 0.435440i $$0.143407\pi$$
$$500$$ 0 0
$$501$$ −2837.85 −0.253065
$$502$$ 0 0
$$503$$ 7782.35 0.689856 0.344928 0.938629i $$-0.387903\pi$$
0.344928 + 0.938629i $$0.387903\pi$$
$$504$$ 0 0
$$505$$ −16629.3 −1.46533
$$506$$ 0 0
$$507$$ −20003.4 −1.75224
$$508$$ 0 0
$$509$$ 1475.93 0.128526 0.0642628 0.997933i $$-0.479530\pi$$
0.0642628 + 0.997933i $$0.479530\pi$$
$$510$$ 0 0
$$511$$ 10336.4 0.894821
$$512$$ 0 0
$$513$$ 7592.75 0.653466
$$514$$ 0 0
$$515$$ −22180.4 −1.89784
$$516$$ 0 0
$$517$$ 790.307 0.0672295
$$518$$ 0 0
$$519$$ 10721.7 0.906799
$$520$$ 0 0
$$521$$ 7609.43 0.639875 0.319938 0.947439i $$-0.396338\pi$$
0.319938 + 0.947439i $$0.396338\pi$$
$$522$$ 0 0
$$523$$ −12452.9 −1.04116 −0.520581 0.853812i $$-0.674285\pi$$
−0.520581 + 0.853812i $$0.674285\pi$$
$$524$$ 0 0
$$525$$ −4042.97 −0.336095
$$526$$ 0 0
$$527$$ 5422.18 0.448186
$$528$$ 0 0
$$529$$ −11988.7 −0.985344
$$530$$ 0 0
$$531$$ −4443.39 −0.363139
$$532$$ 0 0
$$533$$ −20520.9 −1.66765
$$534$$ 0 0
$$535$$ 6224.81 0.503031
$$536$$ 0 0
$$537$$ −26261.0 −2.11033
$$538$$ 0 0
$$539$$ −620.795 −0.0496095
$$540$$ 0 0
$$541$$ −9312.17 −0.740039 −0.370020 0.929024i $$-0.620649\pi$$
−0.370020 + 0.929024i $$0.620649\pi$$
$$542$$ 0 0
$$543$$ 20210.3 1.59725
$$544$$ 0 0
$$545$$ 826.105 0.0649292
$$546$$ 0 0
$$547$$ 11018.6 0.861278 0.430639 0.902524i $$-0.358288\pi$$
0.430639 + 0.902524i $$0.358288\pi$$
$$548$$ 0 0
$$549$$ −824.968 −0.0641325
$$550$$ 0 0
$$551$$ −11478.6 −0.887490
$$552$$ 0 0
$$553$$ −16559.6 −1.27339
$$554$$ 0 0
$$555$$ 3113.89 0.238157
$$556$$ 0 0
$$557$$ 12018.4 0.914250 0.457125 0.889403i $$-0.348879\pi$$
0.457125 + 0.889403i $$0.348879\pi$$
$$558$$ 0 0
$$559$$ −170.718 −0.0129170
$$560$$ 0 0
$$561$$ 5398.40 0.406276
$$562$$ 0 0
$$563$$ 8763.89 0.656046 0.328023 0.944670i $$-0.393618\pi$$
0.328023 + 0.944670i $$0.393618\pi$$
$$564$$ 0 0
$$565$$ −25778.1 −1.91946
$$566$$ 0 0
$$567$$ −14940.1 −1.10657
$$568$$ 0 0
$$569$$ −10273.2 −0.756895 −0.378447 0.925623i $$-0.623542\pi$$
−0.378447 + 0.925623i $$0.623542\pi$$
$$570$$ 0 0
$$571$$ −2602.62 −0.190747 −0.0953734 0.995442i $$-0.530404\pi$$
−0.0953734 + 0.995442i $$0.530404\pi$$
$$572$$ 0 0
$$573$$ −17332.6 −1.26366
$$574$$ 0 0
$$575$$ 537.988 0.0390185
$$576$$ 0 0
$$577$$ −19727.0 −1.42331 −0.711653 0.702532i $$-0.752053\pi$$
−0.711653 + 0.702532i $$0.752053\pi$$
$$578$$ 0 0
$$579$$ 14726.7 1.05703
$$580$$ 0 0
$$581$$ −442.739 −0.0316143
$$582$$ 0 0
$$583$$ 1639.06 0.116437
$$584$$ 0 0
$$585$$ −7814.72 −0.552306
$$586$$ 0 0
$$587$$ 10116.2 0.711309 0.355654 0.934618i $$-0.384258\pi$$
0.355654 + 0.934618i $$0.384258\pi$$
$$588$$ 0 0
$$589$$ −4448.78 −0.311221
$$590$$ 0 0
$$591$$ −30386.0 −2.11491
$$592$$ 0 0
$$593$$ 3130.32 0.216774 0.108387 0.994109i $$-0.465431\pi$$
0.108387 + 0.994109i $$0.465431\pi$$
$$594$$ 0 0
$$595$$ −18016.9 −1.24138
$$596$$ 0 0
$$597$$ 45.6009 0.00312616
$$598$$ 0 0
$$599$$ 10080.1 0.687581 0.343790 0.939046i $$-0.388289\pi$$
0.343790 + 0.939046i $$0.388289\pi$$
$$600$$ 0 0
$$601$$ 4777.02 0.324224 0.162112 0.986772i $$-0.448169\pi$$
0.162112 + 0.986772i $$0.448169\pi$$
$$602$$ 0 0
$$603$$ −3352.24 −0.226391
$$604$$ 0 0
$$605$$ 1555.63 0.104537
$$606$$ 0 0
$$607$$ −2571.35 −0.171941 −0.0859703 0.996298i $$-0.527399\pi$$
−0.0859703 + 0.996298i $$0.527399\pi$$
$$608$$ 0 0
$$609$$ 16959.3 1.12845
$$610$$ 0 0
$$611$$ −5362.67 −0.355074
$$612$$ 0 0
$$613$$ −12711.9 −0.837564 −0.418782 0.908087i $$-0.637543\pi$$
−0.418782 + 0.908087i $$0.637543\pi$$
$$614$$ 0 0
$$615$$ −20953.8 −1.37388
$$616$$ 0 0
$$617$$ 16236.1 1.05939 0.529693 0.848189i $$-0.322307\pi$$
0.529693 + 0.848189i $$0.322307\pi$$
$$618$$ 0 0
$$619$$ −12657.3 −0.821874 −0.410937 0.911664i $$-0.634798\pi$$
−0.410937 + 0.911664i $$0.634798\pi$$
$$620$$ 0 0
$$621$$ 1492.75 0.0964607
$$622$$ 0 0
$$623$$ −5973.75 −0.384162
$$624$$ 0 0
$$625$$ −19037.8 −1.21842
$$626$$ 0 0
$$627$$ −4429.28 −0.282119
$$628$$ 0 0
$$629$$ 3382.28 0.214404
$$630$$ 0 0
$$631$$ −3949.97 −0.249201 −0.124600 0.992207i $$-0.539765\pi$$
−0.124600 + 0.992207i $$0.539765\pi$$
$$632$$ 0 0
$$633$$ 18421.0 1.15666
$$634$$ 0 0
$$635$$ 1409.13 0.0880621
$$636$$ 0 0
$$637$$ 4212.44 0.262014
$$638$$ 0 0
$$639$$ −3832.67 −0.237274
$$640$$ 0 0
$$641$$ −7398.27 −0.455872 −0.227936 0.973676i $$-0.573198\pi$$
−0.227936 + 0.973676i $$0.573198\pi$$
$$642$$ 0 0
$$643$$ 12491.7 0.766134 0.383067 0.923721i $$-0.374868\pi$$
0.383067 + 0.923721i $$0.374868\pi$$
$$644$$ 0 0
$$645$$ −174.319 −0.0106415
$$646$$ 0 0
$$647$$ −10472.0 −0.636315 −0.318158 0.948038i $$-0.603064\pi$$
−0.318158 + 0.948038i $$0.603064\pi$$
$$648$$ 0 0
$$649$$ −6001.94 −0.363015
$$650$$ 0 0
$$651$$ 6572.92 0.395719
$$652$$ 0 0
$$653$$ −6337.94 −0.379820 −0.189910 0.981801i $$-0.560820\pi$$
−0.189910 + 0.981801i $$0.560820\pi$$
$$654$$ 0 0
$$655$$ −14871.2 −0.887121
$$656$$ 0 0
$$657$$ 4972.48 0.295274
$$658$$ 0 0
$$659$$ −15196.7 −0.898302 −0.449151 0.893456i $$-0.648274\pi$$
−0.449151 + 0.893456i $$0.648274\pi$$
$$660$$ 0 0
$$661$$ −2298.17 −0.135232 −0.0676161 0.997711i $$-0.521539\pi$$
−0.0676161 + 0.997711i $$0.521539\pi$$
$$662$$ 0 0
$$663$$ −36631.1 −2.14575
$$664$$ 0 0
$$665$$ 14782.5 0.862016
$$666$$ 0 0
$$667$$ −2256.73 −0.131006
$$668$$ 0 0
$$669$$ 73.0265 0.00422028
$$670$$ 0 0
$$671$$ −1114.33 −0.0641106
$$672$$ 0 0
$$673$$ 23199.6 1.32880 0.664398 0.747379i $$-0.268688\pi$$
0.664398 + 0.747379i $$0.268688\pi$$
$$674$$ 0 0
$$675$$ 4503.49 0.256799
$$676$$ 0 0
$$677$$ 2145.38 0.121793 0.0608963 0.998144i $$-0.480604\pi$$
0.0608963 + 0.998144i $$0.480604\pi$$
$$678$$ 0 0
$$679$$ 14348.1 0.810941
$$680$$ 0 0
$$681$$ 27364.0 1.53978
$$682$$ 0 0
$$683$$ 29544.6 1.65519 0.827593 0.561329i $$-0.189710\pi$$
0.827593 + 0.561329i $$0.189710\pi$$
$$684$$ 0 0
$$685$$ 2549.72 0.142219
$$686$$ 0 0
$$687$$ 30083.4 1.67068
$$688$$ 0 0
$$689$$ −11121.9 −0.614964
$$690$$ 0 0
$$691$$ −27803.1 −1.53065 −0.765325 0.643644i $$-0.777422\pi$$
−0.765325 + 0.643644i $$0.777422\pi$$
$$692$$ 0 0
$$693$$ 1516.42 0.0831227
$$694$$ 0 0
$$695$$ 37285.4 2.03498
$$696$$ 0 0
$$697$$ −22759.8 −1.23686
$$698$$ 0 0
$$699$$ −1254.90 −0.0679036
$$700$$ 0 0
$$701$$ 19697.8 1.06130 0.530652 0.847590i $$-0.321947\pi$$
0.530652 + 0.847590i $$0.321947\pi$$
$$702$$ 0 0
$$703$$ −2775.09 −0.148883
$$704$$ 0 0
$$705$$ −5475.78 −0.292524
$$706$$ 0 0
$$707$$ −21896.0 −1.16476
$$708$$ 0 0
$$709$$ −19122.5 −1.01292 −0.506460 0.862263i $$-0.669046\pi$$
−0.506460 + 0.862263i $$0.669046\pi$$
$$710$$ 0 0
$$711$$ −7966.27 −0.420195
$$712$$ 0 0
$$713$$ −874.641 −0.0459405
$$714$$ 0 0
$$715$$ −10555.8 −0.552117
$$716$$ 0 0
$$717$$ −25565.3 −1.33160
$$718$$ 0 0
$$719$$ 1837.44 0.0953060 0.0476530 0.998864i $$-0.484826\pi$$
0.0476530 + 0.998864i $$0.484826\pi$$
$$720$$ 0 0
$$721$$ −29205.2 −1.50854
$$722$$ 0 0
$$723$$ 5907.95 0.303899
$$724$$ 0 0
$$725$$ −6808.33 −0.348765
$$726$$ 0 0
$$727$$ −7555.46 −0.385442 −0.192721 0.981254i $$-0.561731\pi$$
−0.192721 + 0.981254i $$0.561731\pi$$
$$728$$ 0 0
$$729$$ 10705.2 0.543881
$$730$$ 0 0
$$731$$ −189.344 −0.00958021
$$732$$ 0 0
$$733$$ −11984.6 −0.603905 −0.301952 0.953323i $$-0.597638\pi$$
−0.301952 + 0.953323i $$0.597638\pi$$
$$734$$ 0 0
$$735$$ 4301.29 0.215858
$$736$$ 0 0
$$737$$ −4528.05 −0.226313
$$738$$ 0 0
$$739$$ 27142.5 1.35109 0.675543 0.737321i $$-0.263909\pi$$
0.675543 + 0.737321i $$0.263909\pi$$
$$740$$ 0 0
$$741$$ 30055.1 1.49001
$$742$$ 0 0
$$743$$ −29222.6 −1.44290 −0.721450 0.692467i $$-0.756524\pi$$
−0.721450 + 0.692467i $$0.756524\pi$$
$$744$$ 0 0
$$745$$ −44847.6 −2.20549
$$746$$ 0 0
$$747$$ −212.987 −0.0104321
$$748$$ 0 0
$$749$$ 8196.29 0.399847
$$750$$ 0 0
$$751$$ −8859.39 −0.430471 −0.215236 0.976562i $$-0.569052\pi$$
−0.215236 + 0.976562i $$0.569052\pi$$
$$752$$ 0 0
$$753$$ −1641.47 −0.0794404
$$754$$ 0 0
$$755$$ −14956.2 −0.720941
$$756$$ 0 0
$$757$$ −35734.4 −1.71571 −0.857853 0.513896i $$-0.828202\pi$$
−0.857853 + 0.513896i $$0.828202\pi$$
$$758$$ 0 0
$$759$$ −870.806 −0.0416446
$$760$$ 0 0
$$761$$ 34394.7 1.63838 0.819189 0.573524i $$-0.194424\pi$$
0.819189 + 0.573524i $$0.194424\pi$$
$$762$$ 0 0
$$763$$ 1087.74 0.0516107
$$764$$ 0 0
$$765$$ −8667.33 −0.409631
$$766$$ 0 0
$$767$$ 40726.4 1.91727
$$768$$ 0 0
$$769$$ −11602.7 −0.544091 −0.272045 0.962284i $$-0.587700\pi$$
−0.272045 + 0.962284i $$0.587700\pi$$
$$770$$ 0 0
$$771$$ 19178.8 0.895859
$$772$$ 0 0
$$773$$ 12680.6 0.590026 0.295013 0.955493i $$-0.404676\pi$$
0.295013 + 0.955493i $$0.404676\pi$$
$$774$$ 0 0
$$775$$ −2638.71 −0.122303
$$776$$ 0 0
$$777$$ 4100.10 0.189305
$$778$$ 0 0
$$779$$ 18674.0 0.858876
$$780$$ 0 0
$$781$$ −5176.99 −0.237193
$$782$$ 0 0
$$783$$ −18891.0 −0.862210
$$784$$ 0 0
$$785$$ −4397.62 −0.199946
$$786$$ 0 0
$$787$$ 4417.61 0.200090 0.100045 0.994983i $$-0.468101\pi$$
0.100045 + 0.994983i $$0.468101\pi$$
$$788$$ 0 0
$$789$$ −1232.74 −0.0556231
$$790$$ 0 0
$$791$$ −33942.4 −1.52573
$$792$$ 0 0
$$793$$ 7561.33 0.338601
$$794$$ 0 0
$$795$$ −11356.5 −0.506633
$$796$$ 0 0
$$797$$ 27030.1 1.20132 0.600661 0.799504i $$-0.294904\pi$$
0.600661 + 0.799504i $$0.294904\pi$$
$$798$$ 0 0
$$799$$ −5947.75 −0.263350
$$800$$ 0 0
$$801$$ −2873.77 −0.126766
$$802$$ 0 0
$$803$$ 6716.60 0.295173
$$804$$ 0 0
$$805$$ 2906.27 0.127246
$$806$$ 0 0
$$807$$ 29836.9 1.30150
$$808$$ 0 0
$$809$$ 23647.0 1.02767 0.513835 0.857889i $$-0.328224\pi$$
0.513835 + 0.857889i $$0.328224\pi$$
$$810$$ 0 0
$$811$$ −33486.1 −1.44988 −0.724941 0.688811i $$-0.758133\pi$$
−0.724941 + 0.688811i $$0.758133\pi$$
$$812$$ 0 0
$$813$$ −8815.30 −0.380278
$$814$$ 0 0
$$815$$ 17933.3 0.770768
$$816$$ 0 0
$$817$$ 155.353 0.00665251
$$818$$ 0 0
$$819$$ −10289.7 −0.439014
$$820$$ 0 0
$$821$$ −2605.69 −0.110766 −0.0553832 0.998465i $$-0.517638\pi$$
−0.0553832 + 0.998465i $$0.517638\pi$$
$$822$$ 0 0
$$823$$ 31976.2 1.35434 0.677169 0.735828i $$-0.263207\pi$$
0.677169 + 0.735828i $$0.263207\pi$$
$$824$$ 0 0
$$825$$ −2627.14 −0.110867
$$826$$ 0 0
$$827$$ 37759.0 1.58768 0.793839 0.608128i $$-0.208079\pi$$
0.793839 + 0.608128i $$0.208079\pi$$
$$828$$ 0 0
$$829$$ 1137.55 0.0476584 0.0238292 0.999716i $$-0.492414\pi$$
0.0238292 + 0.999716i $$0.492414\pi$$
$$830$$ 0 0
$$831$$ −1398.08 −0.0583621
$$832$$ 0 0
$$833$$ 4672.03 0.194329
$$834$$ 0 0
$$835$$ 6154.40 0.255068
$$836$$ 0 0
$$837$$ −7321.60 −0.302356
$$838$$ 0 0
$$839$$ −37372.2 −1.53782 −0.768911 0.639356i $$-0.779201\pi$$
−0.768911 + 0.639356i $$0.779201\pi$$
$$840$$ 0 0
$$841$$ 4170.26 0.170989
$$842$$ 0 0
$$843$$ 29137.2 1.19044
$$844$$ 0 0
$$845$$ 43381.1 1.76610
$$846$$ 0 0
$$847$$ 2048.31 0.0830943
$$848$$ 0 0
$$849$$ −30824.0 −1.24603
$$850$$ 0 0
$$851$$ −545.589 −0.0219772
$$852$$ 0 0
$$853$$ 22490.8 0.902780 0.451390 0.892327i $$-0.350928\pi$$
0.451390 + 0.892327i $$0.350928\pi$$
$$854$$ 0 0
$$855$$ 7111.36 0.284449
$$856$$ 0 0
$$857$$ 43409.5 1.73027 0.865135 0.501539i $$-0.167233\pi$$
0.865135 + 0.501539i $$0.167233\pi$$
$$858$$ 0 0
$$859$$ −29533.2 −1.17306 −0.586532 0.809926i $$-0.699507\pi$$
−0.586532 + 0.809926i $$0.699507\pi$$
$$860$$ 0 0
$$861$$ −27590.1 −1.09207
$$862$$ 0 0
$$863$$ 14351.6 0.566090 0.283045 0.959107i $$-0.408655\pi$$
0.283045 + 0.959107i $$0.408655\pi$$
$$864$$ 0 0
$$865$$ −23251.9 −0.913975
$$866$$ 0 0
$$867$$ −11502.4 −0.450569
$$868$$ 0 0
$$869$$ −10760.5 −0.420051
$$870$$ 0 0
$$871$$ 30725.3 1.19528
$$872$$ 0 0
$$873$$ 6902.39 0.267595
$$874$$ 0 0
$$875$$ −18436.5 −0.712307
$$876$$ 0 0
$$877$$ 43248.7 1.66523 0.832614 0.553854i $$-0.186844\pi$$
0.832614 + 0.553854i $$0.186844\pi$$
$$878$$ 0 0
$$879$$ −52647.9 −2.02021
$$880$$ 0 0
$$881$$ 3816.13 0.145935 0.0729675 0.997334i $$-0.476753\pi$$
0.0729675 + 0.997334i $$0.476753\pi$$
$$882$$ 0 0
$$883$$ −48787.6 −1.85938 −0.929690 0.368343i $$-0.879925\pi$$
−0.929690 + 0.368343i $$0.879925\pi$$
$$884$$ 0 0
$$885$$ 41585.5 1.57953
$$886$$ 0 0
$$887$$ 41495.1 1.57077 0.785384 0.619009i $$-0.212466\pi$$
0.785384 + 0.619009i $$0.212466\pi$$
$$888$$ 0 0
$$889$$ 1855.41 0.0699984
$$890$$ 0 0
$$891$$ −9708.15 −0.365023
$$892$$ 0 0
$$893$$ 4880.01 0.182870
$$894$$ 0 0
$$895$$ 56951.9 2.12703
$$896$$ 0 0
$$897$$ 5908.90 0.219947
$$898$$ 0 0
$$899$$ 11068.7 0.410637
$$900$$ 0 0
$$901$$ −12335.3 −0.456104
$$902$$ 0 0
$$903$$ −229.528 −0.00845871
$$904$$ 0 0
$$905$$ −43829.7 −1.60989
$$906$$ 0 0
$$907$$ −21615.3 −0.791316 −0.395658 0.918398i $$-0.629483\pi$$
−0.395658 + 0.918398i $$0.629483\pi$$
$$908$$ 0 0
$$909$$ −10533.4 −0.384347
$$910$$ 0 0
$$911$$ 3646.35 0.132611 0.0663057 0.997799i $$-0.478879\pi$$
0.0663057 + 0.997799i $$0.478879\pi$$
$$912$$ 0 0
$$913$$ −287.693 −0.0104285
$$914$$ 0 0
$$915$$ 7720.82 0.278954
$$916$$ 0 0
$$917$$ −19581.1 −0.705151
$$918$$ 0 0
$$919$$ 31280.0 1.12278 0.561388 0.827553i $$-0.310267\pi$$
0.561388 + 0.827553i $$0.310267\pi$$
$$920$$ 0 0
$$921$$ −8880.05 −0.317707
$$922$$ 0 0
$$923$$ 35128.7 1.25274
$$924$$ 0 0
$$925$$ −1645.99 −0.0585079
$$926$$ 0 0
$$927$$ −14049.7 −0.497790
$$928$$ 0 0
$$929$$ 6557.92 0.231602 0.115801 0.993272i $$-0.463056\pi$$
0.115801 + 0.993272i $$0.463056\pi$$
$$930$$ 0 0
$$931$$ −3833.30 −0.134942
$$932$$ 0 0
$$933$$ 44370.9 1.55695
$$934$$ 0 0
$$935$$ −11707.4 −0.409491
$$936$$ 0 0
$$937$$ −24473.3 −0.853265 −0.426632 0.904425i $$-0.640300\pi$$
−0.426632 + 0.904425i $$0.640300\pi$$
$$938$$ 0 0
$$939$$ 3902.91 0.135641
$$940$$ 0 0
$$941$$ −15420.8 −0.534224 −0.267112 0.963665i $$-0.586069\pi$$
−0.267112 + 0.963665i $$0.586069\pi$$
$$942$$ 0 0
$$943$$ 3671.34 0.126782
$$944$$ 0 0
$$945$$ 24328.3 0.837461
$$946$$ 0 0
$$947$$ 33141.2 1.13722 0.568608 0.822609i $$-0.307482\pi$$
0.568608 + 0.822609i $$0.307482\pi$$
$$948$$ 0 0
$$949$$ −45575.8 −1.55896
$$950$$ 0 0
$$951$$ 1385.47 0.0472417
$$952$$ 0 0
$$953$$ −20735.4 −0.704813 −0.352406 0.935847i $$-0.614637\pi$$
−0.352406 + 0.935847i $$0.614637\pi$$
$$954$$ 0 0
$$955$$ 37589.0 1.27367
$$956$$ 0 0
$$957$$ 11020.2 0.372239
$$958$$ 0 0
$$959$$ 3357.26 0.113046
$$960$$ 0 0
$$961$$ −25501.1 −0.856000
$$962$$ 0 0
$$963$$ 3942.96 0.131942
$$964$$ 0 0
$$965$$ −31937.7 −1.06540
$$966$$ 0 0
$$967$$ −8178.87 −0.271990 −0.135995 0.990710i $$-0.543423\pi$$
−0.135995 + 0.990710i $$0.543423\pi$$
$$968$$ 0 0
$$969$$ 33334.2 1.10511
$$970$$ 0 0
$$971$$ 20576.1 0.680039 0.340020 0.940418i $$-0.389566\pi$$
0.340020 + 0.940418i $$0.389566\pi$$
$$972$$ 0 0
$$973$$ 49094.1 1.61756
$$974$$ 0 0
$$975$$ 17826.6 0.585546
$$976$$ 0 0
$$977$$ 14541.9 0.476188 0.238094 0.971242i $$-0.423477\pi$$
0.238094 + 0.971242i $$0.423477\pi$$
$$978$$ 0 0
$$979$$ −3881.76 −0.126723
$$980$$ 0 0
$$981$$ 523.277 0.0170305
$$982$$ 0 0
$$983$$ −29285.7 −0.950223 −0.475111 0.879926i $$-0.657592\pi$$
−0.475111 + 0.879926i $$0.657592\pi$$
$$984$$ 0 0
$$985$$ 65897.7 2.13165
$$986$$ 0 0
$$987$$ −7210.03 −0.232521
$$988$$ 0 0
$$989$$ 30.5427 0.000982004 0
$$990$$ 0 0
$$991$$ −38085.9 −1.22083 −0.610413 0.792083i $$-0.708997\pi$$
−0.610413 + 0.792083i $$0.708997\pi$$
$$992$$ 0 0
$$993$$ 50585.1 1.61658
$$994$$ 0 0
$$995$$ −98.8940 −0.00315090
$$996$$ 0 0
$$997$$ 26803.6 0.851434 0.425717 0.904856i $$-0.360022\pi$$
0.425717 + 0.904856i $$0.360022\pi$$
$$998$$ 0 0
$$999$$ −4567.12 −0.144642
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.p.1.1 2
4.3 odd 2 704.4.a.n.1.2 2
8.3 odd 2 176.4.a.i.1.1 2
8.5 even 2 11.4.a.a.1.1 2
24.5 odd 2 99.4.a.c.1.2 2
24.11 even 2 1584.4.a.bc.1.2 2
40.13 odd 4 275.4.b.c.199.3 4
40.29 even 2 275.4.a.b.1.2 2
40.37 odd 4 275.4.b.c.199.2 4
56.13 odd 2 539.4.a.e.1.1 2
88.5 even 10 121.4.c.c.3.2 8
88.13 odd 10 121.4.c.f.81.1 8
88.21 odd 2 121.4.a.c.1.2 2
88.29 odd 10 121.4.c.f.27.2 8
88.37 even 10 121.4.c.c.27.1 8
88.43 even 2 1936.4.a.w.1.1 2
88.53 even 10 121.4.c.c.81.2 8
88.61 odd 10 121.4.c.f.3.1 8
88.69 even 10 121.4.c.c.9.1 8
88.85 odd 10 121.4.c.f.9.2 8
104.77 even 2 1859.4.a.a.1.2 2
120.29 odd 2 2475.4.a.q.1.1 2
264.197 even 2 1089.4.a.v.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.1 2 8.5 even 2
99.4.a.c.1.2 2 24.5 odd 2
121.4.a.c.1.2 2 88.21 odd 2
121.4.c.c.3.2 8 88.5 even 10
121.4.c.c.9.1 8 88.69 even 10
121.4.c.c.27.1 8 88.37 even 10
121.4.c.c.81.2 8 88.53 even 10
121.4.c.f.3.1 8 88.61 odd 10
121.4.c.f.9.2 8 88.85 odd 10
121.4.c.f.27.2 8 88.29 odd 10
121.4.c.f.81.1 8 88.13 odd 10
176.4.a.i.1.1 2 8.3 odd 2
275.4.a.b.1.2 2 40.29 even 2
275.4.b.c.199.2 4 40.37 odd 4
275.4.b.c.199.3 4 40.13 odd 4
539.4.a.e.1.1 2 56.13 odd 2
704.4.a.n.1.2 2 4.3 odd 2
704.4.a.p.1.1 2 1.1 even 1 trivial
1089.4.a.v.1.1 2 264.197 even 2
1584.4.a.bc.1.2 2 24.11 even 2
1859.4.a.a.1.2 2 104.77 even 2
1936.4.a.w.1.1 2 88.43 even 2
2475.4.a.q.1.1 2 120.29 odd 2