# Properties

 Label 2475.4.a.q.1.1 Level $2475$ Weight $4$ Character 2475.1 Self dual yes Analytic conductor $146.030$ Analytic rank $0$ 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] = [2475,4,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2475.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: $$146.029727264$$ Analytic rank: $$0$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\chi$$ $$=$$ 2475.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-0.732051 q^{2} -7.46410 q^{4} -16.9282 q^{7} +11.3205 q^{8} +O(q^{10})$$ $$q-0.732051 q^{2} -7.46410 q^{4} -16.9282 q^{7} +11.3205 q^{8} +11.0000 q^{11} -74.6410 q^{13} +12.3923 q^{14} +51.4256 q^{16} -82.7846 q^{17} -67.9230 q^{19} -8.05256 q^{22} +13.3538 q^{23} +54.6410 q^{26} +126.354 q^{28} -168.995 q^{29} -65.4974 q^{31} -128.210 q^{32} +60.6025 q^{34} -40.8564 q^{37} +49.7231 q^{38} -274.928 q^{41} +2.28719 q^{43} -82.1051 q^{44} -9.77568 q^{46} +71.8461 q^{47} -56.4359 q^{49} +557.128 q^{52} -149.005 q^{53} -191.636 q^{56} +123.713 q^{58} -545.631 q^{59} +101.303 q^{61} +47.9474 q^{62} -317.549 q^{64} -411.641 q^{67} +617.913 q^{68} +470.636 q^{71} -610.600 q^{73} +29.9090 q^{74} +506.985 q^{76} -186.210 q^{77} -978.225 q^{79} +201.261 q^{82} +26.1539 q^{83} -1.67434 q^{86} +124.526 q^{88} +352.887 q^{89} +1263.54 q^{91} -99.6743 q^{92} -52.5950 q^{94} -847.585 q^{97} +41.3140 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 8 q^{4} - 20 q^{7} - 12 q^{8}+O(q^{10})$$ 2 * q + 2 * q^2 - 8 * q^4 - 20 * q^7 - 12 * q^8 $$2 q + 2 q^{2} - 8 q^{4} - 20 q^{7} - 12 q^{8} + 22 q^{11} - 80 q^{13} + 4 q^{14} - 8 q^{16} - 124 q^{17} + 72 q^{19} + 22 q^{22} - 98 q^{23} + 40 q^{26} + 128 q^{28} - 144 q^{29} - 34 q^{31} - 104 q^{32} - 52 q^{34} - 54 q^{37} + 432 q^{38} - 536 q^{41} + 60 q^{43} - 88 q^{44} - 314 q^{46} - 272 q^{47} - 390 q^{49} + 560 q^{52} - 492 q^{53} - 120 q^{56} + 192 q^{58} - 634 q^{59} + 840 q^{61} + 134 q^{62} + 224 q^{64} - 754 q^{67} + 640 q^{68} + 678 q^{71} + 400 q^{73} - 6 q^{74} + 432 q^{76} - 220 q^{77} + 316 q^{79} - 512 q^{82} + 468 q^{83} + 156 q^{86} - 132 q^{88} + 1842 q^{89} + 1280 q^{91} - 40 q^{92} - 992 q^{94} - 2194 q^{97} - 870 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 8 * q^4 - 20 * q^7 - 12 * q^8 + 22 * q^11 - 80 * q^13 + 4 * q^14 - 8 * q^16 - 124 * q^17 + 72 * q^19 + 22 * q^22 - 98 * q^23 + 40 * q^26 + 128 * q^28 - 144 * q^29 - 34 * q^31 - 104 * q^32 - 52 * q^34 - 54 * q^37 + 432 * q^38 - 536 * q^41 + 60 * q^43 - 88 * q^44 - 314 * q^46 - 272 * q^47 - 390 * q^49 + 560 * q^52 - 492 * q^53 - 120 * q^56 + 192 * q^58 - 634 * q^59 + 840 * q^61 + 134 * q^62 + 224 * q^64 - 754 * q^67 + 640 * q^68 + 678 * q^71 + 400 * q^73 - 6 * q^74 + 432 * q^76 - 220 * q^77 + 316 * q^79 - 512 * q^82 + 468 * q^83 + 156 * q^86 - 132 * q^88 + 1842 * q^89 + 1280 * q^91 - 40 * q^92 - 992 * q^94 - 2194 * q^97 - 870 * q^98

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −0.732051 −0.258819 −0.129410 0.991591i $$-0.541308\pi$$
−0.129410 + 0.991591i $$0.541308\pi$$
$$3$$ 0 0
$$4$$ −7.46410 −0.933013
$$5$$ 0 0
$$6$$ 0 0
$$7$$ −16.9282 −0.914037 −0.457019 0.889457i $$-0.651083\pi$$
−0.457019 + 0.889457i $$0.651083\pi$$
$$8$$ 11.3205 0.500301
$$9$$ 0 0
$$10$$ 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 $$0.793170\pi$$
$$14$$ 12.3923 0.236570
$$15$$ 0 0
$$16$$ 51.4256 0.803525
$$17$$ −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 $$-0.634495\pi$$
−0.410069 + 0.912055i $$0.634495\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −8.05256 −0.0780369
$$23$$ 13.3538 0.121064 0.0605319 0.998166i $$-0.480720\pi$$
0.0605319 + 0.998166i $$0.480720\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 54.6410 0.412153
$$27$$ 0 0
$$28$$ 126.354 0.852808
$$29$$ −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 $$0.560763\pi$$
$$32$$ −128.210 −0.708268
$$33$$ 0 0
$$34$$ 60.6025 0.305684
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −40.8564 −0.181534 −0.0907669 0.995872i $$-0.528932\pi$$
−0.0907669 + 0.995872i $$0.528932\pi$$
$$38$$ 49.7231 0.212267
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −274.928 −1.04723 −0.523617 0.851954i $$-0.675418\pi$$
−0.523617 + 0.851954i $$0.675418\pi$$
$$42$$ 0 0
$$43$$ 2.28719 0.00811146 0.00405573 0.999992i $$-0.498709\pi$$
0.00405573 + 0.999992i $$0.498709\pi$$
$$44$$ −82.1051 −0.281314
$$45$$ 0 0
$$46$$ −9.77568 −0.0313336
$$47$$ 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
$$51$$ 0 0
$$52$$ 557.128 1.48576
$$53$$ −149.005 −0.386178 −0.193089 0.981181i $$-0.561851\pi$$
−0.193089 + 0.981181i $$0.561851\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −191.636 −0.457293
$$57$$ 0 0
$$58$$ 123.713 0.280074
$$59$$ −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 $$-0.466095\pi$$
0.106315 + 0.994332i $$0.466095\pi$$
$$62$$ 47.9474 0.0982150
$$63$$ 0 0
$$64$$ −317.549 −0.620212
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −411.641 −0.750596 −0.375298 0.926904i $$-0.622460\pi$$
−0.375298 + 0.926904i $$0.622460\pi$$
$$68$$ 617.913 1.10195
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 470.636 0.786679 0.393339 0.919393i $$-0.371320\pi$$
0.393339 + 0.919393i $$0.371320\pi$$
$$72$$ 0 0
$$73$$ −610.600 −0.978977 −0.489488 0.872010i $$-0.662816\pi$$
−0.489488 + 0.872010i $$0.662816\pi$$
$$74$$ 29.9090 0.0469844
$$75$$ 0 0
$$76$$ 506.985 0.765199
$$77$$ −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
$$81$$ 0 0
$$82$$ 201.261 0.271044
$$83$$ 26.1539 0.0345875 0.0172938 0.999850i $$-0.494495\pi$$
0.0172938 + 0.999850i $$0.494495\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1.67434 −0.00209940
$$87$$ 0 0
$$88$$ 124.526 0.150846
$$89$$ 352.887 0.420292 0.210146 0.977670i $$-0.432606\pi$$
0.210146 + 0.977670i $$0.432606\pi$$
$$90$$ 0 0
$$91$$ 1263.54 1.45555
$$92$$ −99.6743 −0.112954
$$93$$ 0 0
$$94$$ −52.5950 −0.0577102
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −847.585 −0.887208 −0.443604 0.896223i $$-0.646300\pi$$
−0.443604 + 0.896223i $$0.646300\pi$$
$$98$$ 41.3140 0.0425851
$$99$$ 0 0
$$100$$ 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 $$-0.191057\pi$$
0.825209 + 0.564828i $$0.191057\pi$$
$$104$$ −844.974 −0.796697
$$105$$ 0 0
$$106$$ 109.079 0.0999502
$$107$$ −484.179 −0.437452 −0.218726 0.975786i $$-0.570190\pi$$
−0.218726 + 0.975786i $$0.570190\pi$$
$$108$$ 0 0
$$109$$ −64.2563 −0.0564645 −0.0282323 0.999601i $$-0.508988\pi$$
−0.0282323 + 0.999601i $$0.508988\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −870.543 −0.734452
$$113$$ −2005.08 −1.66922 −0.834612 0.550839i $$-0.814308\pi$$
−0.834612 + 0.550839i $$0.814308\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1261.39 1.00963
$$117$$ 0 0
$$118$$ 399.429 0.311614
$$119$$ 1401.39 1.07954
$$120$$ 0 0
$$121$$ 121.000 0.0909091
$$122$$ −74.1587 −0.0550329
$$123$$ 0 0
$$124$$ 488.879 0.354054
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −109.605 −0.0765816 −0.0382908 0.999267i $$-0.512191\pi$$
−0.0382908 + 0.999267i $$0.512191\pi$$
$$128$$ 1258.14 0.868791
$$129$$ 0 0
$$130$$ 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
$$134$$ 301.342 0.194269
$$135$$ 0 0
$$136$$ −937.164 −0.590891
$$137$$ 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 $$-0.845735\pi$$
−0.884844 + 0.465888i $$0.845735\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −344.529 −0.203607
$$143$$ −821.051 −0.480138
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 446.990 0.253378
$$147$$ 0 0
$$148$$ 304.956 0.169373
$$149$$ −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 $$0.601494\pi$$
$$152$$ −768.923 −0.410315
$$153$$ 0 0
$$154$$ 136.315 0.0713286
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −342.057 −0.173880 −0.0869398 0.996214i $$-0.527709\pi$$
−0.0869398 + 0.996214i $$0.527709\pi$$
$$158$$ 716.111 0.360574
$$159$$ 0 0
$$160$$ 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 $$0.391216\pi$$
$$164$$ 2052.09 0.977082
$$165$$ 0 0
$$166$$ −19.1460 −0.00895191
$$167$$ 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
$$171$$ 0 0
$$172$$ −17.0718 −0.00756809
$$173$$ 1808.58 0.794822 0.397411 0.917641i $$-0.369909\pi$$
0.397411 + 0.917641i $$0.369909\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 565.682 0.242272
$$177$$ 0 0
$$178$$ −258.331 −0.108780
$$179$$ 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 $$-0.253181\pi$$
0.700005 + 0.714138i $$0.253181\pi$$
$$182$$ −924.974 −0.376723
$$183$$ 0 0
$$184$$ 151.172 0.0605682
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −910.631 −0.356106
$$188$$ −536.267 −0.208039
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2923.75 −1.10762 −0.553810 0.832643i $$-0.686827\pi$$
−0.553810 + 0.832643i $$0.686827\pi$$
$$192$$ 0 0
$$193$$ 2484.18 0.926505 0.463253 0.886226i $$-0.346682\pi$$
0.463253 + 0.886226i $$0.346682\pi$$
$$194$$ 620.475 0.229626
$$195$$ 0 0
$$196$$ 421.244 0.153514
$$197$$ −5125.67 −1.85375 −0.926876 0.375369i $$-0.877516\pi$$
−0.926876 + 0.375369i $$0.877516\pi$$
$$198$$ 0 0
$$199$$ −7.69219 −0.00274013 −0.00137006 0.999999i $$-0.500436\pi$$
−0.00137006 + 0.999999i $$0.500436\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 946.879 0.329813
$$203$$ 2860.78 0.989100
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −1262.96 −0.427160
$$207$$ 0 0
$$208$$ −3838.46 −1.27956
$$209$$ −747.154 −0.247281
$$210$$ 0 0
$$211$$ 3107.34 1.01383 0.506915 0.861996i $$-0.330786\pi$$
0.506915 + 0.861996i $$0.330786\pi$$
$$212$$ 1112.19 0.360309
$$213$$ 0 0
$$214$$ 354.444 0.113221
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1108.75 0.346853
$$218$$ 47.0388 0.0146141
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 6179.13 1.88078
$$222$$ 0 0
$$223$$ 12.3185 0.00369913 0.00184957 0.999998i $$-0.499411\pi$$
0.00184957 + 0.999998i $$0.499411\pi$$
$$224$$ 2170.37 0.647383
$$225$$ 0 0
$$226$$ 1467.82 0.432027
$$227$$ 4615.90 1.34964 0.674820 0.737983i $$-0.264221\pi$$
0.674820 + 0.737983i $$0.264221\pi$$
$$228$$ 0 0
$$229$$ 5074.63 1.46437 0.732186 0.681105i $$-0.238500\pi$$
0.732186 + 0.681105i $$0.238500\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −1913.11 −0.541386
$$233$$ 211.683 0.0595184 0.0297592 0.999557i $$-0.490526\pi$$
0.0297592 + 0.999557i $$0.490526\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 4072.64 1.12333
$$237$$ 0 0
$$238$$ −1025.89 −0.279406
$$239$$ −4312.49 −1.16716 −0.583581 0.812055i $$-0.698349\pi$$
−0.583581 + 0.812055i $$0.698349\pi$$
$$240$$ 0 0
$$241$$ −996.584 −0.266372 −0.133186 0.991091i $$-0.542521\pi$$
−0.133186 + 0.991091i $$0.542521\pi$$
$$242$$ −88.5781 −0.0235290
$$243$$ 0 0
$$244$$ −756.133 −0.198387
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 5069.85 1.30602
$$248$$ −741.464 −0.189851
$$249$$ 0 0
$$250$$ 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
$$254$$ 80.2364 0.0198208
$$255$$ 0 0
$$256$$ 1619.36 0.395352
$$257$$ −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
$$261$$ 0 0
$$262$$ 846.772 0.199671
$$263$$ 207.944 0.0487544 0.0243772 0.999703i $$-0.492240\pi$$
0.0243772 + 0.999703i $$0.492240\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −841.723 −0.194020
$$267$$ 0 0
$$268$$ 3072.53 0.700316
$$269$$ −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 $$0.446702\pi$$
$$272$$ −4257.25 −0.949021
$$273$$ 0 0
$$274$$ −145.183 −0.0320102
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 235.836 0.0511552 0.0255776 0.999673i $$-0.491858\pi$$
0.0255776 + 0.999673i $$0.491858\pi$$
$$278$$ 2123.05 0.458029
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 4915.01 1.04343 0.521717 0.853118i $$-0.325292\pi$$
0.521717 + 0.853118i $$0.325292\pi$$
$$282$$ 0 0
$$283$$ 5199.56 1.09216 0.546081 0.837733i $$-0.316119\pi$$
0.546081 + 0.837733i $$0.316119\pi$$
$$284$$ −3512.87 −0.733981
$$285$$ 0 0
$$286$$ 601.051 0.124269
$$287$$ 4654.04 0.957210
$$288$$ 0 0
$$289$$ 1940.29 0.394930
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 4557.58 0.913398
$$293$$ −8880.92 −1.77075 −0.885373 0.464881i $$-0.846097\pi$$
−0.885373 + 0.464881i $$0.846097\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −462.515 −0.0908215
$$297$$ 0 0
$$298$$ 2553.64 0.496405
$$299$$ −996.743 −0.192786
$$300$$ 0 0
$$301$$ −38.7180 −0.00741417
$$302$$ 851.612 0.162267
$$303$$ 0 0
$$304$$ −3492.99 −0.659001
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 1497.93 0.278474 0.139237 0.990259i $$-0.455535\pi$$
0.139237 + 0.990259i $$0.455535\pi$$
$$308$$ 1389.89 0.257131
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 7484.71 1.36469 0.682345 0.731030i $$-0.260960\pi$$
0.682345 + 0.731030i $$0.260960\pi$$
$$312$$ 0 0
$$313$$ 658.363 0.118891 0.0594455 0.998232i $$-0.481067\pi$$
0.0594455 + 0.998232i $$0.481067\pi$$
$$314$$ 250.403 0.0450034
$$315$$ 0 0
$$316$$ 7301.57 1.29983
$$317$$ 233.708 0.0414080 0.0207040 0.999786i $$-0.493409\pi$$
0.0207040 + 0.999786i $$0.493409\pi$$
$$318$$ 0 0
$$319$$ −1858.94 −0.326272
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 165.485 0.0286401
$$323$$ 5622.98 0.968641
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −1021.13 −0.173482
$$327$$ 0 0
$$328$$ −3112.33 −0.523931
$$329$$ −1216.23 −0.203808
$$330$$ 0 0
$$331$$ 8532.95 1.41696 0.708480 0.705731i $$-0.249381\pi$$
0.708480 + 0.705731i $$0.249381\pi$$
$$332$$ −195.215 −0.0322706
$$333$$ 0 0
$$334$$ −350.435 −0.0574100
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −11691.2 −1.88979 −0.944895 0.327373i $$-0.893837\pi$$
−0.944895 + 0.327373i $$0.893837\pi$$
$$338$$ −2470.15 −0.397509
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −720.472 −0.114416
$$342$$ 0 0
$$343$$ 6761.73 1.06443
$$344$$ 25.8921 0.00405817
$$345$$ 0 0
$$346$$ −1323.98 −0.205715
$$347$$ 4598.79 0.711459 0.355729 0.934589i $$-0.384232\pi$$
0.355729 + 0.934589i $$0.384232\pi$$
$$348$$ 0 0
$$349$$ 6720.27 1.03074 0.515369 0.856968i $$-0.327655\pi$$
0.515369 + 0.856968i $$0.327655\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −1410.31 −0.213551
$$353$$ 5738.70 0.865270 0.432635 0.901569i $$-0.357584\pi$$
0.432635 + 0.901569i $$0.357584\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −2633.99 −0.392138
$$357$$ 0 0
$$358$$ −3242.87 −0.478746
$$359$$ 4115.27 0.605001 0.302501 0.953149i $$-0.402179\pi$$
0.302501 + 0.953149i $$0.402179\pi$$
$$360$$ 0 0
$$361$$ −2245.46 −0.327374
$$362$$ −2495.69 −0.362349
$$363$$ 0 0
$$364$$ −9431.18 −1.35804
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −9662.99 −1.37440 −0.687199 0.726469i $$-0.741160\pi$$
−0.687199 + 0.726469i $$0.741160\pi$$
$$368$$ 686.729 0.0972778
$$369$$ 0 0
$$370$$ 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 $$0.496868\pi$$
$$374$$ 666.628 0.0921671
$$375$$ 0 0
$$376$$ 813.334 0.111555
$$377$$ 12613.9 1.72321
$$378$$ 0 0
$$379$$ −2819.73 −0.382163 −0.191082 0.981574i $$-0.561200\pi$$
−0.191082 + 0.981574i $$0.561200\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 2140.34 0.286673
$$383$$ −6337.84 −0.845557 −0.422778 0.906233i $$-0.638945\pi$$
−0.422778 + 0.906233i $$0.638945\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −1818.55 −0.239797
$$387$$ 0 0
$$388$$ 6326.46 0.827776
$$389$$ 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
$$392$$ −638.883 −0.0823176
$$393$$ 0 0
$$394$$ 3752.25 0.479786
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −4315.26 −0.545534 −0.272767 0.962080i $$-0.587939\pi$$
−0.272767 + 0.962080i $$0.587939\pi$$
$$398$$ 5.63108 0.000709197 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −361.681 −0.0450411 −0.0225206 0.999746i $$-0.507169\pi$$
−0.0225206 + 0.999746i $$0.507169\pi$$
$$402$$ 0 0
$$403$$ 4888.79 0.604288
$$404$$ 9654.53 1.18894
$$405$$ 0 0
$$406$$ −2094.24 −0.255998
$$407$$ −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
$$411$$ 0 0
$$412$$ −12877.4 −1.53986
$$413$$ 9236.55 1.10049
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 9569.74 1.12787
$$417$$ 0 0
$$418$$ 546.954 0.0640010
$$419$$ 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 $$-0.785088\pi$$
−0.780603 + 0.625027i $$0.785088\pi$$
$$422$$ −2274.73 −0.262399
$$423$$ 0 0
$$424$$ −1686.81 −0.193205
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −1714.87 −0.194352
$$428$$ 3613.96 0.408148
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −406.334 −0.0454116 −0.0227058 0.999742i $$-0.507228\pi$$
−0.0227058 + 0.999742i $$0.507228\pi$$
$$432$$ 0 0
$$433$$ 1766.69 0.196078 0.0980391 0.995183i $$-0.468743\pi$$
0.0980391 + 0.995183i $$0.468743\pi$$
$$434$$ −811.664 −0.0897722
$$435$$ 0 0
$$436$$ 479.615 0.0526821
$$437$$ −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
$$441$$ 0 0
$$442$$ −4523.44 −0.486783
$$443$$ 11667.9 1.25137 0.625686 0.780075i $$-0.284819\pi$$
0.625686 + 0.780075i $$0.284819\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −9.01776 −0.000957406 0
$$447$$ 0 0
$$448$$ 5375.53 0.566897
$$449$$ −16975.3 −1.78421 −0.892107 0.451825i $$-0.850773\pi$$
−0.892107 + 0.451825i $$0.850773\pi$$
$$450$$ 0 0
$$451$$ −3024.21 −0.315753
$$452$$ 14966.1 1.55741
$$453$$ 0 0
$$454$$ −3379.07 −0.349312
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 16192.9 1.65748 0.828741 0.559632i $$-0.189057\pi$$
0.828741 + 0.559632i $$0.189057\pi$$
$$458$$ −3714.89 −0.379007
$$459$$ 0 0
$$460$$ 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 $$-0.369931\pi$$
0.397347 + 0.917668i $$0.369931\pi$$
$$464$$ −8690.67 −0.869513
$$465$$ 0 0
$$466$$ −154.962 −0.0154045
$$467$$ −15155.0 −1.50169 −0.750844 0.660480i $$-0.770353\pi$$
−0.750844 + 0.660480i $$0.770353\pi$$
$$468$$ 0 0
$$469$$ 6968.34 0.686073
$$470$$ 0 0
$$471$$ 0 0
$$472$$ −6176.82 −0.602354
$$473$$ 25.1591 0.00244570
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −10460.2 −1.00723
$$477$$ 0 0
$$478$$ 3156.96 0.302084
$$479$$ −10001.1 −0.953993 −0.476996 0.878905i $$-0.658275\pi$$
−0.476996 + 0.878905i $$0.658275\pi$$
$$480$$ 0 0
$$481$$ 3049.56 0.289081
$$482$$ 729.550 0.0689421
$$483$$ 0 0
$$484$$ −903.156 −0.0848193
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −7044.54 −0.655480 −0.327740 0.944768i $$-0.606287\pi$$
−0.327740 + 0.944768i $$0.606287\pi$$
$$488$$ 1146.80 0.106379
$$489$$ 0 0
$$490$$ 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
$$494$$ −3711.38 −0.338022
$$495$$ 0 0
$$496$$ −3368.25 −0.304917
$$497$$ −7967.02 −0.719054
$$498$$ 0 0
$$499$$ −20069.1 −1.80044 −0.900218 0.435440i $$-0.856593\pi$$
−0.900218 + 0.435440i $$0.856593\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −202.699 −0.0180217
$$503$$ 7782.35 0.689856 0.344928 0.938629i $$-0.387903\pi$$
0.344928 + 0.938629i $$0.387903\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −107.532 −0.00944744
$$507$$ 0 0
$$508$$ 818.102 0.0714516
$$509$$ 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
$$512$$ −11250.6 −0.971116
$$513$$ 0 0
$$514$$ 2368.32 0.203233
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 790.307 0.0672295
$$518$$ −506.305 −0.0429455
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −7609.43 −0.639875 −0.319938 0.947439i $$-0.603662\pi$$
−0.319938 + 0.947439i $$0.603662\pi$$
$$522$$ 0 0
$$523$$ −12452.9 −1.04116 −0.520581 0.853812i $$-0.674285\pi$$
−0.520581 + 0.853812i $$0.674285\pi$$
$$524$$ 8633.82 0.719790
$$525$$ 0 0
$$526$$ −152.226 −0.0126186
$$527$$ 5422.18 0.448186
$$528$$ 0 0
$$529$$ −11988.7 −0.985344
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −8582.34 −0.699420
$$533$$ 20520.9 1.66765
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −4659.99 −0.375524
$$537$$ 0 0
$$538$$ 3684.44 0.295255
$$539$$ −620.795 −0.0496095
$$540$$ 0 0
$$541$$ 9312.17 0.740039 0.370020 0.929024i $$-0.379351\pi$$
0.370020 + 0.929024i $$0.379351\pi$$
$$542$$ −1088.57 −0.0862693
$$543$$ 0 0
$$544$$ 10613.8 0.836515
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 11018.6 0.861278 0.430639 0.902524i $$-0.358288\pi$$
0.430639 + 0.902524i $$0.358288\pi$$
$$548$$ −1480.31 −0.115393
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 11478.6 0.887490
$$552$$ 0 0
$$553$$ 16559.6 1.27339
$$554$$ −172.644 −0.0132399
$$555$$ 0 0
$$556$$ 21646.9 1.65114
$$557$$ −12018.4 −0.914250 −0.457125 0.889403i $$-0.651121\pi$$
−0.457125 + 0.889403i $$0.651121\pi$$
$$558$$ 0 0
$$559$$ −170.718 −0.0129170
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −3598.04 −0.270061
$$563$$ −8763.89 −0.656046 −0.328023 0.944670i $$-0.606382\pi$$
−0.328023 + 0.944670i $$0.606382\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −3806.34 −0.282672
$$567$$ 0 0
$$568$$ 5327.84 0.393576
$$569$$ 10273.2 0.756895 0.378447 0.925623i $$-0.376458\pi$$
0.378447 + 0.925623i $$0.376458\pi$$
$$570$$ 0 0
$$571$$ 2602.62 0.190747 0.0953734 0.995442i $$-0.469596\pi$$
0.0953734 + 0.995442i $$0.469596\pi$$
$$572$$ 6128.41 0.447975
$$573$$ 0 0
$$574$$ −3406.99 −0.247744
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 19727.0 1.42331 0.711653 0.702532i $$-0.247947\pi$$
0.711653 + 0.702532i $$0.247947\pi$$
$$578$$ −1420.39 −0.102215
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −442.739 −0.0316143
$$582$$ 0 0
$$583$$ −1639.06 −0.116437
$$584$$ −6912.30 −0.489783
$$585$$ 0 0
$$586$$ 6501.28 0.458303
$$587$$ −10116.2 −0.711309 −0.355654 0.934618i $$-0.615742\pi$$
−0.355654 + 0.934618i $$0.615742\pi$$
$$588$$ 0 0
$$589$$ 4448.78 0.311221
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −2101.07 −0.145867
$$593$$ 3130.32 0.216774 0.108387 0.994109i $$-0.465431\pi$$
0.108387 + 0.994109i $$0.465431\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 26037.3 1.78948
$$597$$ 0 0
$$598$$ 729.667 0.0498968
$$599$$ −10080.1 −0.687581 −0.343790 0.939046i $$-0.611711\pi$$
−0.343790 + 0.939046i $$0.611711\pi$$
$$600$$ 0 0
$$601$$ 4777.02 0.324224 0.162112 0.986772i $$-0.448169\pi$$
0.162112 + 0.986772i $$0.448169\pi$$
$$602$$ 28.3435 0.00191893
$$603$$ 0 0
$$604$$ 8683.16 0.584955
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 2571.35 0.171941 0.0859703 0.996298i $$-0.472601\pi$$
0.0859703 + 0.996298i $$0.472601\pi$$
$$608$$ 8708.43 0.580877
$$609$$ 0 0
$$610$$ 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 $$0.637543\pi$$
$$614$$ −1096.56 −0.0720744
$$615$$ 0 0
$$616$$ −2107.99 −0.137879
$$617$$ 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 $$-0.365202\pi$$
0.410937 + 0.911664i $$0.365202\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −5479.19 −0.353208
$$623$$ −5973.75 −0.384162
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −481.955 −0.0307713
$$627$$ 0 0
$$628$$ 2553.15 0.162232
$$629$$ 3382.28 0.214404
$$630$$ 0 0
$$631$$ −3949.97 −0.249201 −0.124600 0.992207i $$-0.539765\pi$$
−0.124600 + 0.992207i $$0.539765\pi$$
$$632$$ −11074.0 −0.696994
$$633$$ 0 0
$$634$$ −171.086 −0.0107172
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 4212.44 0.262014
$$638$$ 1360.84 0.0844455
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 7398.27 0.455872 0.227936 0.973676i $$-0.426802\pi$$
0.227936 + 0.973676i $$0.426802\pi$$
$$642$$ 0 0
$$643$$ 12491.7 0.766134 0.383067 0.923721i $$-0.374868\pi$$
0.383067 + 0.923721i $$0.374868\pi$$
$$644$$ 1687.31 0.103244
$$645$$ 0 0
$$646$$ −4116.31 −0.250703
$$647$$ −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
$$651$$ 0 0
$$652$$ −10411.6 −0.625384
$$653$$ 6337.94 0.379820 0.189910 0.981801i $$-0.439180\pi$$
0.189910 + 0.981801i $$0.439180\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −14138.4 −0.841479
$$657$$ 0 0
$$658$$ 890.339 0.0527493
$$659$$ −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 $$-0.478461\pi$$
0.0676161 + 0.997711i $$0.478461\pi$$
$$662$$ −6246.55 −0.366736
$$663$$ 0 0
$$664$$ 296.075 0.0173042
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −2256.73 −0.131006
$$668$$ −3573.09 −0.206956
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 1114.33 0.0641106
$$672$$ 0 0
$$673$$ −23199.6 −1.32880 −0.664398 0.747379i $$-0.731312\pi$$
−0.664398 + 0.747379i $$0.731312\pi$$
$$674$$ 8558.54 0.489114
$$675$$ 0 0
$$676$$ −25186.0 −1.43298
$$677$$ −2145.38 −0.121793 −0.0608963 0.998144i $$-0.519396\pi$$
−0.0608963 + 0.998144i $$0.519396\pi$$
$$678$$ 0 0
$$679$$ 14348.1 0.810941
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 527.422 0.0296129
$$683$$ −29544.6 −1.65519 −0.827593 0.561329i $$-0.810290\pi$$
−0.827593 + 0.561329i $$0.810290\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −4949.93 −0.275495
$$687$$ 0 0
$$688$$ 117.620 0.00651776
$$689$$ 11121.9 0.614964
$$690$$ 0 0
$$691$$ 27803.1 1.53065 0.765325 0.643644i $$-0.222578\pi$$
0.765325 + 0.643644i $$0.222578\pi$$
$$692$$ −13499.5 −0.741579
$$693$$ 0 0
$$694$$ −3366.55 −0.184139
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 22759.8 1.23686
$$698$$ −4919.58 −0.266775
$$699$$ 0 0
$$700$$ 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
$$704$$ −3493.03 −0.187001
$$705$$ 0 0
$$706$$ −4201.02 −0.223948
$$707$$ 21896.0 1.16476
$$708$$ 0 0
$$709$$ 19122.5 1.01292 0.506460 0.862263i $$-0.330954\pi$$
0.506460 + 0.862263i $$0.330954\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 3994.86 0.210272
$$713$$ −874.641 −0.0459405
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −33064.8 −1.72582
$$717$$ 0 0
$$718$$ −3012.59 −0.156586
$$719$$ −1837.44 −0.0953060 −0.0476530 0.998864i $$-0.515174\pi$$
−0.0476530 + 0.998864i $$0.515174\pi$$
$$720$$ 0 0
$$721$$ −29205.2 −1.50854
$$722$$ 1643.79 0.0847307
$$723$$ 0 0
$$724$$ −25446.4 −1.30623
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 7555.46 0.385442 0.192721 0.981254i $$-0.438269\pi$$
0.192721 + 0.981254i $$0.438269\pi$$
$$728$$ 14303.9 0.728211
$$729$$ 0 0
$$730$$ 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 $$0.597638\pi$$
$$734$$ 7073.80 0.355720
$$735$$ 0 0
$$736$$ −1712.10 −0.0857456
$$737$$ −4528.05 −0.226313
$$738$$ 0 0
$$739$$ −27142.5 −1.35109 −0.675543 0.737321i $$-0.736091\pi$$
−0.675543 + 0.737321i $$0.736091\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ −1846.52 −0.0913582
$$743$$ −29222.6 −1.44290 −0.721450 0.692467i $$-0.756524\pi$$
−0.721450 + 0.692467i $$0.756524\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −103.790 −0.00509388
$$747$$ 0 0
$$748$$ 6797.04 0.332252
$$749$$ 8196.29 0.399847
$$750$$ 0 0
$$751$$ −8859.39 −0.430471 −0.215236 0.976562i $$-0.569052\pi$$
−0.215236 + 0.976562i $$0.569052\pi$$
$$752$$ 3694.73 0.179166
$$753$$ 0 0
$$754$$ −9234.05 −0.446000
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −35734.4 −1.71571 −0.857853 0.513896i $$-0.828202\pi$$
−0.857853 + 0.513896i $$0.828202\pi$$
$$758$$ 2064.19 0.0989112
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −34394.7 −1.63838 −0.819189 0.573524i $$-0.805576\pi$$
−0.819189 + 0.573524i $$0.805576\pi$$
$$762$$ 0 0
$$763$$ 1087.74 0.0516107
$$764$$ 21823.2 1.03342
$$765$$ 0 0
$$766$$ 4639.62 0.218846
$$767$$ 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
$$771$$ 0 0
$$772$$ −18542.2 −0.864441
$$773$$ −12680.6 −0.590026 −0.295013 0.955493i $$-0.595324\pi$$
−0.295013 + 0.955493i $$0.595324\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −9595.09 −0.443871
$$777$$ 0 0
$$778$$ −6445.89 −0.297039
$$779$$ 18674.0 0.858876
$$780$$ 0 0
$$781$$ 5176.99 0.237193
$$782$$ 809.276 0.0370072
$$783$$ 0 0
$$784$$ −2902.25 −0.132209
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 4417.61 0.200090 0.100045 0.994983i $$-0.468101\pi$$
0.100045 + 0.994983i $$0.468101\pi$$
$$788$$ 38258.5 1.72957
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 33942.4 1.52573
$$792$$ 0 0
$$793$$ −7561.33 −0.338601
$$794$$ 3158.99 0.141194
$$795$$ 0 0
$$796$$ 57.4153 0.00255657
$$797$$ −27030.1 −1.20132 −0.600661 0.799504i $$-0.705096\pi$$
−0.600661 + 0.799504i $$0.705096\pi$$
$$798$$ 0 0
$$799$$ −5947.75 −0.263350
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 264.769 0.0116575
$$803$$ −6716.60 −0.295173
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −3578.85 −0.156401
$$807$$ 0 0
$$808$$ −14642.6 −0.637533
$$809$$ −23647.0 −1.02767 −0.513835 0.857889i $$-0.671776\pi$$
−0.513835 + 0.857889i $$0.671776\pi$$
$$810$$ 0 0
$$811$$ 33486.1 1.44988 0.724941 0.688811i $$-0.241867\pi$$
0.724941 + 0.688811i $$0.241867\pi$$
$$812$$ −21353.1 −0.922843
$$813$$ 0 0
$$814$$ 328.999 0.0141663
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −155.353 −0.00665251
$$818$$ −6749.88 −0.288513
$$819$$ 0 0
$$820$$ 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 $$-0.736793\pi$$
−0.677169 + 0.735828i $$0.736793\pi$$
$$824$$ 19530.6 0.825705
$$825$$ 0 0
$$826$$ −6761.62 −0.284827
$$827$$ −37759.0 −1.58768 −0.793839 0.608128i $$-0.791921\pi$$
−0.793839 + 0.608128i $$0.791921\pi$$
$$828$$ 0 0
$$829$$ −1137.55 −0.0476584 −0.0238292 0.999716i $$-0.507586\pi$$
−0.0238292 + 0.999716i $$0.507586\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 23702.2 0.987649
$$833$$ 4672.03 0.194329
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 5576.83 0.230716
$$837$$ 0 0
$$838$$ −10917.0 −0.450024
$$839$$ 37372.2 1.53782 0.768911 0.639356i $$-0.220799\pi$$
0.768911 + 0.639356i $$0.220799\pi$$
$$840$$ 0 0
$$841$$ 4170.26 0.170989
$$842$$ 9872.44 0.404070
$$843$$ 0 0
$$844$$ −23193.5 −0.945917
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −2048.31 −0.0830943
$$848$$ −7662.68 −0.310304
$$849$$ 0 0
$$850$$ 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 $$0.350928\pi$$
$$854$$ 1255.37 0.0503021
$$855$$ 0 0
$$856$$ −5481.16 −0.218858
$$857$$ 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 $$-0.300493\pi$$
0.586532 + 0.809926i $$0.300493\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 297.457 0.0117534
$$863$$ 14351.6 0.566090 0.283045 0.959107i $$-0.408655\pi$$
0.283045 + 0.959107i $$0.408655\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −1293.31 −0.0507488
$$867$$ 0 0
$$868$$ −8275.85 −0.323618
$$869$$ −10760.5 −0.420051
$$870$$ 0 0
$$871$$ 30725.3 1.19528
$$872$$ −727.413 −0.0282492
$$873$$ 0 0
$$874$$ 663.994 0.0256979
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 43248.7 1.66523 0.832614 0.553854i $$-0.186844\pi$$
0.832614 + 0.553854i $$0.186844\pi$$
$$878$$ −5727.71 −0.220160
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −3816.13 −0.145935 −0.0729675 0.997334i $$-0.523247\pi$$
−0.0729675 + 0.997334i $$0.523247\pi$$
$$882$$ 0 0
$$883$$ −48787.6 −1.85938 −0.929690 0.368343i $$-0.879925\pi$$
−0.929690 + 0.368343i $$0.879925\pi$$
$$884$$ −46121.6 −1.75479
$$885$$ 0 0
$$886$$ −8541.49 −0.323879
$$887$$ 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
$$891$$ 0 0
$$892$$ −91.9464 −0.00345134
$$893$$ −4880.01 −0.182870
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −21298.1 −0.794107
$$897$$ 0 0
$$898$$ 12426.7 0.461788
$$899$$ 11068.7 0.410637
$$900$$ 0 0
$$901$$ 12335.3 0.456104
$$902$$ 2213.88 0.0817228
$$903$$ 0 0
$$904$$ −22698.5 −0.835113
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −21615.3 −0.791316 −0.395658 0.918398i $$-0.629483\pi$$
−0.395658 + 0.918398i $$0.629483\pi$$
$$908$$ −34453.6 −1.25923
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −3646.35 −0.132611 −0.0663057 0.997799i $$-0.521121\pi$$
−0.0663057 + 0.997799i $$0.521121\pi$$
$$912$$ 0 0
$$913$$ 287.693 0.0104285
$$914$$ −11854.0 −0.428988
$$915$$ 0 0
$$916$$ −37877.6 −1.36628
$$917$$ 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
$$921$$ 0 0
$$922$$ 6285.42 0.224511
$$923$$ −35128.7 −1.25274
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −5795.79 −0.205682
$$927$$ 0 0
$$928$$ 21666.9 0.766433
$$929$$ −6557.92 −0.231602 −0.115801 0.993272i $$-0.536944\pi$$
−0.115801 + 0.993272i $$0.536944\pi$$
$$930$$ 0 0
$$931$$ 3833.30 0.134942
$$932$$ −1580.02 −0.0555314
$$933$$ 0 0
$$934$$ 11094.2 0.388665
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 24473.3 0.853265 0.426632 0.904425i $$-0.359700\pi$$
0.426632 + 0.904425i $$0.359700\pi$$
$$938$$ −5101.18 −0.177569
$$939$$ 0 0
$$940$$ 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
$$944$$ −28059.4 −0.967432
$$945$$ 0 0
$$946$$ −18.4177 −0.000632993 0
$$947$$ −33141.2 −1.13722 −0.568608 0.822609i $$-0.692518\pi$$
−0.568608 + 0.822609i $$0.692518\pi$$
$$948$$ 0 0
$$949$$ 45575.8 1.55896
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 15864.5 0.540096
$$953$$ −20735.4 −0.704813 −0.352406 0.935847i $$-0.614637\pi$$
−0.352406 + 0.935847i $$0.614637\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 32188.9 1.08898
$$957$$ 0 0
$$958$$ 7321.32 0.246911
$$959$$ −3357.26 −0.113046
$$960$$ 0 0
$$961$$ −25501.1 −0.856000
$$962$$ −2232.44 −0.0748198
$$963$$ 0 0
$$964$$ 7438.60 0.248528
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 8178.87 0.271990 0.135995 0.990710i $$-0.456577\pi$$
0.135995 + 0.990710i $$0.456577\pi$$
$$968$$ 1369.78 0.0454819
$$969$$ 0 0
$$970$$ 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
$$974$$ 5156.96 0.169651
$$975$$ 0 0
$$976$$ 5209.55 0.170854
$$977$$ 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
$$981$$ 0 0
$$982$$ −9755.62 −0.317021
$$983$$ −29285.7 −0.950223 −0.475111 0.879926i $$-0.657592\pi$$
−0.475111 + 0.879926i $$0.657592\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −10241.5 −0.330787
$$987$$ 0 0
$$988$$ −37841.8 −1.21853
$$989$$ 30.5427 0.000982004 0
$$990$$ 0 0
$$991$$ −38085.9 −1.22083 −0.610413 0.792083i $$-0.708997\pi$$
−0.610413 + 0.792083i $$0.708997\pi$$
$$992$$ 8397.44 0.268769
$$993$$ 0 0
$$994$$ 5832.26 0.186105
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 26803.6 0.851434 0.425717 0.904856i $$-0.360022\pi$$
0.425717 + 0.904856i $$0.360022\pi$$
$$998$$ 14691.6 0.465987
$$999$$ 0 0
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 2475.4.a.q.1.1 2
3.2 odd 2 275.4.a.b.1.2 2
5.4 even 2 99.4.a.c.1.2 2
15.2 even 4 275.4.b.c.199.3 4
15.8 even 4 275.4.b.c.199.2 4
15.14 odd 2 11.4.a.a.1.1 2
20.19 odd 2 1584.4.a.bc.1.2 2
55.54 odd 2 1089.4.a.v.1.1 2
60.59 even 2 176.4.a.i.1.1 2
105.104 even 2 539.4.a.e.1.1 2
120.29 odd 2 704.4.a.p.1.1 2
120.59 even 2 704.4.a.n.1.2 2
165.14 odd 10 121.4.c.c.9.1 8
165.29 even 10 121.4.c.f.27.2 8
165.59 odd 10 121.4.c.c.27.1 8
165.74 even 10 121.4.c.f.9.2 8
165.104 odd 10 121.4.c.c.3.2 8
165.119 odd 10 121.4.c.c.81.2 8
165.134 even 10 121.4.c.f.81.1 8
165.149 even 10 121.4.c.f.3.1 8
165.164 even 2 121.4.a.c.1.2 2
195.194 odd 2 1859.4.a.a.1.2 2
660.659 odd 2 1936.4.a.w.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
11.4.a.a.1.1 2 15.14 odd 2
99.4.a.c.1.2 2 5.4 even 2
121.4.a.c.1.2 2 165.164 even 2
121.4.c.c.3.2 8 165.104 odd 10
121.4.c.c.9.1 8 165.14 odd 10
121.4.c.c.27.1 8 165.59 odd 10
121.4.c.c.81.2 8 165.119 odd 10
121.4.c.f.3.1 8 165.149 even 10
121.4.c.f.9.2 8 165.74 even 10
121.4.c.f.27.2 8 165.29 even 10
121.4.c.f.81.1 8 165.134 even 10
176.4.a.i.1.1 2 60.59 even 2
275.4.a.b.1.2 2 3.2 odd 2
275.4.b.c.199.2 4 15.8 even 4
275.4.b.c.199.3 4 15.2 even 4
539.4.a.e.1.1 2 105.104 even 2
704.4.a.n.1.2 2 120.59 even 2
704.4.a.p.1.1 2 120.29 odd 2
1089.4.a.v.1.1 2 55.54 odd 2
1584.4.a.bc.1.2 2 20.19 odd 2
1859.4.a.a.1.2 2 195.194 odd 2
1936.4.a.w.1.1 2 660.659 odd 2
2475.4.a.q.1.1 2 1.1 even 1 trivial