# Properties

 Label 315.4.a.i.1.1 Level $315$ Weight $4$ Character 315.1 Self dual yes Analytic conductor $18.586$ 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] = [315,4,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, 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("315.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$315 = 3^{2} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 315.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: $$18.5856016518$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{65})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 16$$ x^2 - x - 16 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$4.53113$$ of defining polynomial Character $$\chi$$ $$=$$ 315.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-4.53113 q^{2} +12.5311 q^{4} -5.00000 q^{5} -7.00000 q^{7} -20.5311 q^{8} +O(q^{10})$$ $$q-4.53113 q^{2} +12.5311 q^{4} -5.00000 q^{5} -7.00000 q^{7} -20.5311 q^{8} +22.6556 q^{10} +19.0623 q^{11} -2.93774 q^{13} +31.7179 q^{14} -7.21984 q^{16} +6.49806 q^{17} -5.43580 q^{19} -62.6556 q^{20} -86.3735 q^{22} -49.3774 q^{23} +25.0000 q^{25} +13.3113 q^{26} -87.7179 q^{28} +291.494 q^{29} +244.307 q^{31} +196.963 q^{32} -29.4436 q^{34} +35.0000 q^{35} -193.121 q^{37} +24.6303 q^{38} +102.656 q^{40} -315.113 q^{41} -300.996 q^{43} +238.872 q^{44} +223.735 q^{46} -86.5058 q^{47} +49.0000 q^{49} -113.278 q^{50} -36.8132 q^{52} -509.677 q^{53} -95.3113 q^{55} +143.718 q^{56} -1320.80 q^{58} +83.3852 q^{59} -5.25291 q^{61} -1106.99 q^{62} -834.706 q^{64} +14.6887 q^{65} +205.992 q^{67} +81.4281 q^{68} -158.590 q^{70} -1004.31 q^{71} -1007.29 q^{73} +875.055 q^{74} -68.1168 q^{76} -133.436 q^{77} -863.237 q^{79} +36.0992 q^{80} +1427.82 q^{82} -1334.72 q^{83} -32.4903 q^{85} +1363.85 q^{86} -391.370 q^{88} -326.249 q^{89} +20.5642 q^{91} -618.755 q^{92} +391.969 q^{94} +27.1790 q^{95} +1526.77 q^{97} -222.025 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 17 q^{4} - 10 q^{5} - 14 q^{7} - 33 q^{8}+O(q^{10})$$ 2 * q - q^2 + 17 * q^4 - 10 * q^5 - 14 * q^7 - 33 * q^8 $$2 q - q^{2} + 17 q^{4} - 10 q^{5} - 14 q^{7} - 33 q^{8} + 5 q^{10} + 22 q^{11} - 22 q^{13} + 7 q^{14} - 87 q^{16} - 116 q^{17} + 102 q^{19} - 85 q^{20} - 76 q^{22} - 260 q^{23} + 50 q^{25} - 54 q^{26} - 119 q^{28} + 196 q^{29} + 150 q^{31} + 15 q^{32} - 462 q^{34} + 70 q^{35} - 96 q^{37} + 404 q^{38} + 165 q^{40} + 176 q^{41} - 344 q^{43} + 252 q^{44} - 520 q^{46} - 560 q^{47} + 98 q^{49} - 25 q^{50} - 122 q^{52} - 326 q^{53} - 110 q^{55} + 231 q^{56} - 1658 q^{58} + 844 q^{59} - 204 q^{61} - 1440 q^{62} - 839 q^{64} + 110 q^{65} - 104 q^{67} - 466 q^{68} - 35 q^{70} - 1670 q^{71} - 386 q^{73} + 1218 q^{74} + 412 q^{76} - 154 q^{77} - 888 q^{79} + 435 q^{80} + 3162 q^{82} - 928 q^{83} + 580 q^{85} + 1212 q^{86} - 428 q^{88} - 588 q^{89} + 154 q^{91} - 1560 q^{92} - 1280 q^{94} - 510 q^{95} + 522 q^{97} - 49 q^{98}+O(q^{100})$$ 2 * q - q^2 + 17 * q^4 - 10 * q^5 - 14 * q^7 - 33 * q^8 + 5 * q^10 + 22 * q^11 - 22 * q^13 + 7 * q^14 - 87 * q^16 - 116 * q^17 + 102 * q^19 - 85 * q^20 - 76 * q^22 - 260 * q^23 + 50 * q^25 - 54 * q^26 - 119 * q^28 + 196 * q^29 + 150 * q^31 + 15 * q^32 - 462 * q^34 + 70 * q^35 - 96 * q^37 + 404 * q^38 + 165 * q^40 + 176 * q^41 - 344 * q^43 + 252 * q^44 - 520 * q^46 - 560 * q^47 + 98 * q^49 - 25 * q^50 - 122 * q^52 - 326 * q^53 - 110 * q^55 + 231 * q^56 - 1658 * q^58 + 844 * q^59 - 204 * q^61 - 1440 * q^62 - 839 * q^64 + 110 * q^65 - 104 * q^67 - 466 * q^68 - 35 * q^70 - 1670 * q^71 - 386 * q^73 + 1218 * q^74 + 412 * q^76 - 154 * q^77 - 888 * q^79 + 435 * q^80 + 3162 * q^82 - 928 * q^83 + 580 * q^85 + 1212 * q^86 - 428 * q^88 - 588 * q^89 + 154 * q^91 - 1560 * q^92 - 1280 * q^94 - 510 * q^95 + 522 * q^97 - 49 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −4.53113 −1.60200 −0.800998 0.598667i $$-0.795697\pi$$
−0.800998 + 0.598667i $$0.795697\pi$$
$$3$$ 0 0
$$4$$ 12.5311 1.56639
$$5$$ −5.00000 −0.447214
$$6$$ 0 0
$$7$$ −7.00000 −0.377964
$$8$$ −20.5311 −0.907356
$$9$$ 0 0
$$10$$ 22.6556 0.716434
$$11$$ 19.0623 0.522499 0.261249 0.965271i $$-0.415865\pi$$
0.261249 + 0.965271i $$0.415865\pi$$
$$12$$ 0 0
$$13$$ −2.93774 −0.0626756 −0.0313378 0.999509i $$-0.509977\pi$$
−0.0313378 + 0.999509i $$0.509977\pi$$
$$14$$ 31.7179 0.605498
$$15$$ 0 0
$$16$$ −7.21984 −0.112810
$$17$$ 6.49806 0.0927066 0.0463533 0.998925i $$-0.485240\pi$$
0.0463533 + 0.998925i $$0.485240\pi$$
$$18$$ 0 0
$$19$$ −5.43580 −0.0656347 −0.0328173 0.999461i $$-0.510448\pi$$
−0.0328173 + 0.999461i $$0.510448\pi$$
$$20$$ −62.6556 −0.700511
$$21$$ 0 0
$$22$$ −86.3735 −0.837041
$$23$$ −49.3774 −0.447648 −0.223824 0.974630i $$-0.571854\pi$$
−0.223824 + 0.974630i $$0.571854\pi$$
$$24$$ 0 0
$$25$$ 25.0000 0.200000
$$26$$ 13.3113 0.100406
$$27$$ 0 0
$$28$$ −87.7179 −0.592040
$$29$$ 291.494 1.86652 0.933261 0.359200i $$-0.116950\pi$$
0.933261 + 0.359200i $$0.116950\pi$$
$$30$$ 0 0
$$31$$ 244.307 1.41545 0.707724 0.706489i $$-0.249722\pi$$
0.707724 + 0.706489i $$0.249722\pi$$
$$32$$ 196.963 1.08808
$$33$$ 0 0
$$34$$ −29.4436 −0.148516
$$35$$ 35.0000 0.169031
$$36$$ 0 0
$$37$$ −193.121 −0.858077 −0.429038 0.903286i $$-0.641147\pi$$
−0.429038 + 0.903286i $$0.641147\pi$$
$$38$$ 24.6303 0.105147
$$39$$ 0 0
$$40$$ 102.656 0.405782
$$41$$ −315.113 −1.20030 −0.600151 0.799887i $$-0.704893\pi$$
−0.600151 + 0.799887i $$0.704893\pi$$
$$42$$ 0 0
$$43$$ −300.996 −1.06748 −0.533738 0.845650i $$-0.679213\pi$$
−0.533738 + 0.845650i $$0.679213\pi$$
$$44$$ 238.872 0.818437
$$45$$ 0 0
$$46$$ 223.735 0.717130
$$47$$ −86.5058 −0.268472 −0.134236 0.990949i $$-0.542858\pi$$
−0.134236 + 0.990949i $$0.542858\pi$$
$$48$$ 0 0
$$49$$ 49.0000 0.142857
$$50$$ −113.278 −0.320399
$$51$$ 0 0
$$52$$ −36.8132 −0.0981745
$$53$$ −509.677 −1.32093 −0.660467 0.750855i $$-0.729642\pi$$
−0.660467 + 0.750855i $$0.729642\pi$$
$$54$$ 0 0
$$55$$ −95.3113 −0.233669
$$56$$ 143.718 0.342948
$$57$$ 0 0
$$58$$ −1320.80 −2.99016
$$59$$ 83.3852 0.183997 0.0919985 0.995759i $$-0.470674\pi$$
0.0919985 + 0.995759i $$0.470674\pi$$
$$60$$ 0 0
$$61$$ −5.25291 −0.0110257 −0.00551283 0.999985i $$-0.501755\pi$$
−0.00551283 + 0.999985i $$0.501755\pi$$
$$62$$ −1106.99 −2.26754
$$63$$ 0 0
$$64$$ −834.706 −1.63029
$$65$$ 14.6887 0.0280294
$$66$$ 0 0
$$67$$ 205.992 0.375611 0.187806 0.982206i $$-0.439862\pi$$
0.187806 + 0.982206i $$0.439862\pi$$
$$68$$ 81.4281 0.145215
$$69$$ 0 0
$$70$$ −158.590 −0.270787
$$71$$ −1004.31 −1.67872 −0.839362 0.543573i $$-0.817071\pi$$
−0.839362 + 0.543573i $$0.817071\pi$$
$$72$$ 0 0
$$73$$ −1007.29 −1.61499 −0.807494 0.589876i $$-0.799177\pi$$
−0.807494 + 0.589876i $$0.799177\pi$$
$$74$$ 875.055 1.37464
$$75$$ 0 0
$$76$$ −68.1168 −0.102810
$$77$$ −133.436 −0.197486
$$78$$ 0 0
$$79$$ −863.237 −1.22939 −0.614695 0.788765i $$-0.710721\pi$$
−0.614695 + 0.788765i $$0.710721\pi$$
$$80$$ 36.0992 0.0504502
$$81$$ 0 0
$$82$$ 1427.82 1.92288
$$83$$ −1334.72 −1.76512 −0.882560 0.470200i $$-0.844182\pi$$
−0.882560 + 0.470200i $$0.844182\pi$$
$$84$$ 0 0
$$85$$ −32.4903 −0.0414596
$$86$$ 1363.85 1.71009
$$87$$ 0 0
$$88$$ −391.370 −0.474093
$$89$$ −326.249 −0.388565 −0.194283 0.980946i $$-0.562238\pi$$
−0.194283 + 0.980946i $$0.562238\pi$$
$$90$$ 0 0
$$91$$ 20.5642 0.0236892
$$92$$ −618.755 −0.701192
$$93$$ 0 0
$$94$$ 391.969 0.430091
$$95$$ 27.1790 0.0293527
$$96$$ 0 0
$$97$$ 1526.77 1.59815 0.799075 0.601232i $$-0.205323\pi$$
0.799075 + 0.601232i $$0.205323\pi$$
$$98$$ −222.025 −0.228857
$$99$$ 0 0
$$100$$ 313.278 0.313278
$$101$$ −96.8716 −0.0954365 −0.0477182 0.998861i $$-0.515195\pi$$
−0.0477182 + 0.998861i $$0.515195\pi$$
$$102$$ 0 0
$$103$$ 1321.99 1.26466 0.632329 0.774700i $$-0.282099\pi$$
0.632329 + 0.774700i $$0.282099\pi$$
$$104$$ 60.3152 0.0568691
$$105$$ 0 0
$$106$$ 2309.41 2.11613
$$107$$ −1745.71 −1.57724 −0.788619 0.614883i $$-0.789203\pi$$
−0.788619 + 0.614883i $$0.789203\pi$$
$$108$$ 0 0
$$109$$ 476.856 0.419032 0.209516 0.977805i $$-0.432811\pi$$
0.209516 + 0.977805i $$0.432811\pi$$
$$110$$ 431.868 0.374336
$$111$$ 0 0
$$112$$ 50.5389 0.0426382
$$113$$ −1641.65 −1.36666 −0.683332 0.730108i $$-0.739470\pi$$
−0.683332 + 0.730108i $$0.739470\pi$$
$$114$$ 0 0
$$115$$ 246.887 0.200194
$$116$$ 3652.75 2.92370
$$117$$ 0 0
$$118$$ −377.829 −0.294763
$$119$$ −45.4864 −0.0350398
$$120$$ 0 0
$$121$$ −967.630 −0.726995
$$122$$ 23.8016 0.0176631
$$123$$ 0 0
$$124$$ 3061.45 2.21715
$$125$$ −125.000 −0.0894427
$$126$$ 0 0
$$127$$ −844.016 −0.589719 −0.294859 0.955541i $$-0.595273\pi$$
−0.294859 + 0.955541i $$0.595273\pi$$
$$128$$ 2206.46 1.52363
$$129$$ 0 0
$$130$$ −66.5564 −0.0449030
$$131$$ 2796.20 1.86493 0.932463 0.361265i $$-0.117655\pi$$
0.932463 + 0.361265i $$0.117655\pi$$
$$132$$ 0 0
$$133$$ 38.0506 0.0248076
$$134$$ −933.377 −0.601728
$$135$$ 0 0
$$136$$ −133.413 −0.0841179
$$137$$ 2057.13 1.28287 0.641433 0.767179i $$-0.278340\pi$$
0.641433 + 0.767179i $$0.278340\pi$$
$$138$$ 0 0
$$139$$ −1745.12 −1.06489 −0.532444 0.846465i $$-0.678726\pi$$
−0.532444 + 0.846465i $$0.678726\pi$$
$$140$$ 438.590 0.264768
$$141$$ 0 0
$$142$$ 4550.65 2.68931
$$143$$ −56.0000 −0.0327479
$$144$$ 0 0
$$145$$ −1457.47 −0.834734
$$146$$ 4564.15 2.58720
$$147$$ 0 0
$$148$$ −2420.02 −1.34408
$$149$$ 1173.57 0.645254 0.322627 0.946526i $$-0.395434\pi$$
0.322627 + 0.946526i $$0.395434\pi$$
$$150$$ 0 0
$$151$$ 1540.07 0.829994 0.414997 0.909823i $$-0.363783\pi$$
0.414997 + 0.909823i $$0.363783\pi$$
$$152$$ 111.603 0.0595540
$$153$$ 0 0
$$154$$ 604.615 0.316372
$$155$$ −1221.54 −0.633008
$$156$$ 0 0
$$157$$ −2544.53 −1.29348 −0.646738 0.762712i $$-0.723867\pi$$
−0.646738 + 0.762712i $$0.723867\pi$$
$$158$$ 3911.44 1.96948
$$159$$ 0 0
$$160$$ −984.815 −0.486603
$$161$$ 345.642 0.169195
$$162$$ 0 0
$$163$$ −594.708 −0.285774 −0.142887 0.989739i $$-0.545639\pi$$
−0.142887 + 0.989739i $$0.545639\pi$$
$$164$$ −3948.72 −1.88014
$$165$$ 0 0
$$166$$ 6047.81 2.82772
$$167$$ −928.498 −0.430236 −0.215118 0.976588i $$-0.569014\pi$$
−0.215118 + 0.976588i $$0.569014\pi$$
$$168$$ 0 0
$$169$$ −2188.37 −0.996072
$$170$$ 147.218 0.0664182
$$171$$ 0 0
$$172$$ −3771.82 −1.67209
$$173$$ 315.642 0.138716 0.0693578 0.997592i $$-0.477905\pi$$
0.0693578 + 0.997592i $$0.477905\pi$$
$$174$$ 0 0
$$175$$ −175.000 −0.0755929
$$176$$ −137.626 −0.0589431
$$177$$ 0 0
$$178$$ 1478.28 0.622480
$$179$$ 1445.49 0.603581 0.301791 0.953374i $$-0.402416\pi$$
0.301791 + 0.953374i $$0.402416\pi$$
$$180$$ 0 0
$$181$$ −1843.81 −0.757180 −0.378590 0.925564i $$-0.623591\pi$$
−0.378590 + 0.925564i $$0.623591\pi$$
$$182$$ −93.1790 −0.0379499
$$183$$ 0 0
$$184$$ 1013.77 0.406176
$$185$$ 965.603 0.383744
$$186$$ 0 0
$$187$$ 123.868 0.0484391
$$188$$ −1084.02 −0.420532
$$189$$ 0 0
$$190$$ −123.152 −0.0470229
$$191$$ 244.074 0.0924637 0.0462318 0.998931i $$-0.485279\pi$$
0.0462318 + 0.998931i $$0.485279\pi$$
$$192$$ 0 0
$$193$$ −1733.03 −0.646355 −0.323178 0.946338i $$-0.604751\pi$$
−0.323178 + 0.946338i $$0.604751\pi$$
$$194$$ −6918.01 −2.56023
$$195$$ 0 0
$$196$$ 614.025 0.223770
$$197$$ −358.230 −0.129557 −0.0647787 0.997900i $$-0.520634\pi$$
−0.0647787 + 0.997900i $$0.520634\pi$$
$$198$$ 0 0
$$199$$ −3203.63 −1.14120 −0.570601 0.821227i $$-0.693290\pi$$
−0.570601 + 0.821227i $$0.693290\pi$$
$$200$$ −513.278 −0.181471
$$201$$ 0 0
$$202$$ 438.938 0.152889
$$203$$ −2040.46 −0.705479
$$204$$ 0 0
$$205$$ 1575.56 0.536791
$$206$$ −5990.12 −2.02598
$$207$$ 0 0
$$208$$ 21.2100 0.00707044
$$209$$ −103.619 −0.0342940
$$210$$ 0 0
$$211$$ 4943.16 1.61280 0.806401 0.591369i $$-0.201412\pi$$
0.806401 + 0.591369i $$0.201412\pi$$
$$212$$ −6386.83 −2.06910
$$213$$ 0 0
$$214$$ 7910.05 2.52673
$$215$$ 1504.98 0.477390
$$216$$ 0 0
$$217$$ −1710.15 −0.534989
$$218$$ −2160.70 −0.671288
$$219$$ 0 0
$$220$$ −1194.36 −0.366016
$$221$$ −19.0896 −0.00581044
$$222$$ 0 0
$$223$$ 3160.15 0.948965 0.474482 0.880265i $$-0.342635\pi$$
0.474482 + 0.880265i $$0.342635\pi$$
$$224$$ −1378.74 −0.411255
$$225$$ 0 0
$$226$$ 7438.51 2.18939
$$227$$ 3651.11 1.06755 0.533773 0.845628i $$-0.320774\pi$$
0.533773 + 0.845628i $$0.320774\pi$$
$$228$$ 0 0
$$229$$ −4083.70 −1.17842 −0.589210 0.807980i $$-0.700561\pi$$
−0.589210 + 0.807980i $$0.700561\pi$$
$$230$$ −1118.68 −0.320710
$$231$$ 0 0
$$232$$ −5984.70 −1.69360
$$233$$ 3682.51 1.03540 0.517702 0.855561i $$-0.326788\pi$$
0.517702 + 0.855561i $$0.326788\pi$$
$$234$$ 0 0
$$235$$ 432.529 0.120064
$$236$$ 1044.91 0.288211
$$237$$ 0 0
$$238$$ 206.105 0.0561336
$$239$$ −2658.78 −0.719591 −0.359796 0.933031i $$-0.617154\pi$$
−0.359796 + 0.933031i $$0.617154\pi$$
$$240$$ 0 0
$$241$$ −4820.39 −1.28842 −0.644209 0.764850i $$-0.722813\pi$$
−0.644209 + 0.764850i $$0.722813\pi$$
$$242$$ 4384.46 1.16464
$$243$$ 0 0
$$244$$ −65.8249 −0.0172705
$$245$$ −245.000 −0.0638877
$$246$$ 0 0
$$247$$ 15.9690 0.00411369
$$248$$ −5015.91 −1.28432
$$249$$ 0 0
$$250$$ 566.391 0.143287
$$251$$ −1672.27 −0.420530 −0.210265 0.977644i $$-0.567433\pi$$
−0.210265 + 0.977644i $$0.567433\pi$$
$$252$$ 0 0
$$253$$ −941.245 −0.233896
$$254$$ 3824.34 0.944727
$$255$$ 0 0
$$256$$ −3320.09 −0.810569
$$257$$ 3697.74 0.897506 0.448753 0.893656i $$-0.351868\pi$$
0.448753 + 0.893656i $$0.351868\pi$$
$$258$$ 0 0
$$259$$ 1351.84 0.324323
$$260$$ 184.066 0.0439050
$$261$$ 0 0
$$262$$ −12670.0 −2.98760
$$263$$ −7319.00 −1.71600 −0.858002 0.513646i $$-0.828294\pi$$
−0.858002 + 0.513646i $$0.828294\pi$$
$$264$$ 0 0
$$265$$ 2548.39 0.590740
$$266$$ −172.412 −0.0397416
$$267$$ 0 0
$$268$$ 2581.32 0.588354
$$269$$ 815.097 0.184749 0.0923743 0.995724i $$-0.470554\pi$$
0.0923743 + 0.995724i $$0.470554\pi$$
$$270$$ 0 0
$$271$$ −5106.02 −1.14453 −0.572267 0.820068i $$-0.693936\pi$$
−0.572267 + 0.820068i $$0.693936\pi$$
$$272$$ −46.9150 −0.0104582
$$273$$ 0 0
$$274$$ −9321.13 −2.05515
$$275$$ 476.556 0.104500
$$276$$ 0 0
$$277$$ 1398.72 0.303398 0.151699 0.988427i $$-0.451526\pi$$
0.151699 + 0.988427i $$0.451526\pi$$
$$278$$ 7907.39 1.70595
$$279$$ 0 0
$$280$$ −718.590 −0.153371
$$281$$ 7102.38 1.50780 0.753901 0.656988i $$-0.228170\pi$$
0.753901 + 0.656988i $$0.228170\pi$$
$$282$$ 0 0
$$283$$ 4465.18 0.937907 0.468953 0.883223i $$-0.344631\pi$$
0.468953 + 0.883223i $$0.344631\pi$$
$$284$$ −12585.1 −2.62954
$$285$$ 0 0
$$286$$ 253.743 0.0524621
$$287$$ 2205.79 0.453671
$$288$$ 0 0
$$289$$ −4870.78 −0.991405
$$290$$ 6603.99 1.33724
$$291$$ 0 0
$$292$$ −12622.5 −2.52970
$$293$$ −7590.61 −1.51348 −0.756738 0.653718i $$-0.773208\pi$$
−0.756738 + 0.653718i $$0.773208\pi$$
$$294$$ 0 0
$$295$$ −416.926 −0.0822860
$$296$$ 3964.98 0.778581
$$297$$ 0 0
$$298$$ −5317.61 −1.03369
$$299$$ 145.058 0.0280566
$$300$$ 0 0
$$301$$ 2106.97 0.403468
$$302$$ −6978.26 −1.32965
$$303$$ 0 0
$$304$$ 39.2456 0.00740425
$$305$$ 26.2645 0.00493083
$$306$$ 0 0
$$307$$ 9480.12 1.76241 0.881203 0.472737i $$-0.156734\pi$$
0.881203 + 0.472737i $$0.156734\pi$$
$$308$$ −1672.10 −0.309340
$$309$$ 0 0
$$310$$ 5534.94 1.01408
$$311$$ −7078.01 −1.29054 −0.645268 0.763956i $$-0.723254\pi$$
−0.645268 + 0.763956i $$0.723254\pi$$
$$312$$ 0 0
$$313$$ 5593.84 1.01017 0.505084 0.863070i $$-0.331461\pi$$
0.505084 + 0.863070i $$0.331461\pi$$
$$314$$ 11529.6 2.07214
$$315$$ 0 0
$$316$$ −10817.3 −1.92571
$$317$$ −3567.81 −0.632139 −0.316070 0.948736i $$-0.602363\pi$$
−0.316070 + 0.948736i $$0.602363\pi$$
$$318$$ 0 0
$$319$$ 5556.54 0.975255
$$320$$ 4173.53 0.729086
$$321$$ 0 0
$$322$$ −1566.15 −0.271050
$$323$$ −35.3222 −0.00608477
$$324$$ 0 0
$$325$$ −73.4436 −0.0125351
$$326$$ 2694.70 0.457809
$$327$$ 0 0
$$328$$ 6469.62 1.08910
$$329$$ 605.541 0.101473
$$330$$ 0 0
$$331$$ −4389.67 −0.728936 −0.364468 0.931216i $$-0.618749\pi$$
−0.364468 + 0.931216i $$0.618749\pi$$
$$332$$ −16725.6 −2.76487
$$333$$ 0 0
$$334$$ 4207.14 0.689236
$$335$$ −1029.96 −0.167978
$$336$$ 0 0
$$337$$ 2348.83 0.379671 0.189835 0.981816i $$-0.439205\pi$$
0.189835 + 0.981816i $$0.439205\pi$$
$$338$$ 9915.79 1.59570
$$339$$ 0 0
$$340$$ −407.140 −0.0649420
$$341$$ 4657.05 0.739570
$$342$$ 0 0
$$343$$ −343.000 −0.0539949
$$344$$ 6179.79 0.968581
$$345$$ 0 0
$$346$$ −1430.21 −0.222222
$$347$$ −558.436 −0.0863931 −0.0431965 0.999067i $$-0.513754\pi$$
−0.0431965 + 0.999067i $$0.513754\pi$$
$$348$$ 0 0
$$349$$ 3233.89 0.496006 0.248003 0.968759i $$-0.420226\pi$$
0.248003 + 0.968759i $$0.420226\pi$$
$$350$$ 792.948 0.121100
$$351$$ 0 0
$$352$$ 3754.56 0.568519
$$353$$ 7516.35 1.13330 0.566650 0.823959i $$-0.308239\pi$$
0.566650 + 0.823959i $$0.308239\pi$$
$$354$$ 0 0
$$355$$ 5021.54 0.750748
$$356$$ −4088.27 −0.608646
$$357$$ 0 0
$$358$$ −6549.70 −0.966935
$$359$$ 6577.76 0.967021 0.483511 0.875338i $$-0.339361\pi$$
0.483511 + 0.875338i $$0.339361\pi$$
$$360$$ 0 0
$$361$$ −6829.45 −0.995692
$$362$$ 8354.56 1.21300
$$363$$ 0 0
$$364$$ 257.693 0.0371065
$$365$$ 5036.44 0.722245
$$366$$ 0 0
$$367$$ 8307.17 1.18155 0.590777 0.806835i $$-0.298821\pi$$
0.590777 + 0.806835i $$0.298821\pi$$
$$368$$ 356.497 0.0504992
$$369$$ 0 0
$$370$$ −4375.27 −0.614756
$$371$$ 3567.74 0.499266
$$372$$ 0 0
$$373$$ −4551.09 −0.631760 −0.315880 0.948799i $$-0.602300\pi$$
−0.315880 + 0.948799i $$0.602300\pi$$
$$374$$ −561.261 −0.0775992
$$375$$ 0 0
$$376$$ 1776.06 0.243599
$$377$$ −856.335 −0.116985
$$378$$ 0 0
$$379$$ −1788.29 −0.242370 −0.121185 0.992630i $$-0.538669\pi$$
−0.121185 + 0.992630i $$0.538669\pi$$
$$380$$ 340.584 0.0459778
$$381$$ 0 0
$$382$$ −1105.93 −0.148126
$$383$$ 1358.47 0.181240 0.0906199 0.995886i $$-0.471115\pi$$
0.0906199 + 0.995886i $$0.471115\pi$$
$$384$$ 0 0
$$385$$ 667.179 0.0883184
$$386$$ 7852.60 1.03546
$$387$$ 0 0
$$388$$ 19132.2 2.50333
$$389$$ −9722.54 −1.26723 −0.633615 0.773649i $$-0.718430\pi$$
−0.633615 + 0.773649i $$0.718430\pi$$
$$390$$ 0 0
$$391$$ −320.858 −0.0414999
$$392$$ −1006.03 −0.129622
$$393$$ 0 0
$$394$$ 1623.19 0.207551
$$395$$ 4316.19 0.549800
$$396$$ 0 0
$$397$$ −4788.04 −0.605302 −0.302651 0.953101i $$-0.597872\pi$$
−0.302651 + 0.953101i $$0.597872\pi$$
$$398$$ 14516.1 1.82820
$$399$$ 0 0
$$400$$ −180.496 −0.0225620
$$401$$ −9681.41 −1.20565 −0.602826 0.797873i $$-0.705959\pi$$
−0.602826 + 0.797873i $$0.705959\pi$$
$$402$$ 0 0
$$403$$ −717.712 −0.0887141
$$404$$ −1213.91 −0.149491
$$405$$ 0 0
$$406$$ 9245.58 1.13017
$$407$$ −3681.32 −0.448344
$$408$$ 0 0
$$409$$ 11113.1 1.34353 0.671767 0.740763i $$-0.265536\pi$$
0.671767 + 0.740763i $$0.265536\pi$$
$$410$$ −7139.09 −0.859937
$$411$$ 0 0
$$412$$ 16566.1 1.98095
$$413$$ −583.696 −0.0695443
$$414$$ 0 0
$$415$$ 6673.62 0.789386
$$416$$ −578.627 −0.0681959
$$417$$ 0 0
$$418$$ 469.510 0.0549389
$$419$$ 1230.09 0.143421 0.0717107 0.997425i $$-0.477154\pi$$
0.0717107 + 0.997425i $$0.477154\pi$$
$$420$$ 0 0
$$421$$ −12356.5 −1.43044 −0.715222 0.698897i $$-0.753674\pi$$
−0.715222 + 0.698897i $$0.753674\pi$$
$$422$$ −22398.1 −2.58370
$$423$$ 0 0
$$424$$ 10464.2 1.19856
$$425$$ 162.452 0.0185413
$$426$$ 0 0
$$427$$ 36.7703 0.00416731
$$428$$ −21875.7 −2.47057
$$429$$ 0 0
$$430$$ −6819.26 −0.764777
$$431$$ 7375.27 0.824256 0.412128 0.911126i $$-0.364786\pi$$
0.412128 + 0.911126i $$0.364786\pi$$
$$432$$ 0 0
$$433$$ −690.067 −0.0765877 −0.0382939 0.999267i $$-0.512192\pi$$
−0.0382939 + 0.999267i $$0.512192\pi$$
$$434$$ 7748.92 0.857051
$$435$$ 0 0
$$436$$ 5975.55 0.656369
$$437$$ 268.406 0.0293812
$$438$$ 0 0
$$439$$ 8408.79 0.914191 0.457095 0.889418i $$-0.348890\pi$$
0.457095 + 0.889418i $$0.348890\pi$$
$$440$$ 1956.85 0.212021
$$441$$ 0 0
$$442$$ 86.4976 0.00930830
$$443$$ −6568.55 −0.704473 −0.352236 0.935911i $$-0.614579\pi$$
−0.352236 + 0.935911i $$0.614579\pi$$
$$444$$ 0 0
$$445$$ 1631.25 0.173772
$$446$$ −14319.0 −1.52024
$$447$$ 0 0
$$448$$ 5842.94 0.616190
$$449$$ −2954.55 −0.310543 −0.155271 0.987872i $$-0.549625\pi$$
−0.155271 + 0.987872i $$0.549625\pi$$
$$450$$ 0 0
$$451$$ −6006.76 −0.627156
$$452$$ −20571.7 −2.14073
$$453$$ 0 0
$$454$$ −16543.7 −1.71020
$$455$$ −102.821 −0.0105941
$$456$$ 0 0
$$457$$ −8144.84 −0.833697 −0.416849 0.908976i $$-0.636865\pi$$
−0.416849 + 0.908976i $$0.636865\pi$$
$$458$$ 18503.8 1.88782
$$459$$ 0 0
$$460$$ 3093.77 0.313583
$$461$$ −2495.26 −0.252095 −0.126048 0.992024i $$-0.540229\pi$$
−0.126048 + 0.992024i $$0.540229\pi$$
$$462$$ 0 0
$$463$$ −5755.66 −0.577728 −0.288864 0.957370i $$-0.593278\pi$$
−0.288864 + 0.957370i $$0.593278\pi$$
$$464$$ −2104.54 −0.210562
$$465$$ 0 0
$$466$$ −16685.9 −1.65871
$$467$$ 4143.73 0.410598 0.205299 0.978699i $$-0.434183\pi$$
0.205299 + 0.978699i $$0.434183\pi$$
$$468$$ 0 0
$$469$$ −1441.95 −0.141968
$$470$$ −1959.84 −0.192342
$$471$$ 0 0
$$472$$ −1711.99 −0.166951
$$473$$ −5737.67 −0.557755
$$474$$ 0 0
$$475$$ −135.895 −0.0131269
$$476$$ −569.996 −0.0548860
$$477$$ 0 0
$$478$$ 12047.3 1.15278
$$479$$ 6765.96 0.645396 0.322698 0.946502i $$-0.395410\pi$$
0.322698 + 0.946502i $$0.395410\pi$$
$$480$$ 0 0
$$481$$ 567.339 0.0537805
$$482$$ 21841.8 2.06404
$$483$$ 0 0
$$484$$ −12125.5 −1.13876
$$485$$ −7633.87 −0.714714
$$486$$ 0 0
$$487$$ 6360.42 0.591824 0.295912 0.955215i $$-0.404377\pi$$
0.295912 + 0.955215i $$0.404377\pi$$
$$488$$ 107.848 0.0100042
$$489$$ 0 0
$$490$$ 1110.13 0.102348
$$491$$ 7072.54 0.650060 0.325030 0.945704i $$-0.394626\pi$$
0.325030 + 0.945704i $$0.394626\pi$$
$$492$$ 0 0
$$493$$ 1894.15 0.173039
$$494$$ −72.3576 −0.00659012
$$495$$ 0 0
$$496$$ −1763.86 −0.159677
$$497$$ 7030.15 0.634498
$$498$$ 0 0
$$499$$ −18473.9 −1.65732 −0.828661 0.559751i $$-0.810897\pi$$
−0.828661 + 0.559751i $$0.810897\pi$$
$$500$$ −1566.39 −0.140102
$$501$$ 0 0
$$502$$ 7577.28 0.673687
$$503$$ −11379.2 −1.00869 −0.504347 0.863501i $$-0.668267\pi$$
−0.504347 + 0.863501i $$0.668267\pi$$
$$504$$ 0 0
$$505$$ 484.358 0.0426805
$$506$$ 4264.90 0.374700
$$507$$ 0 0
$$508$$ −10576.5 −0.923730
$$509$$ −6064.48 −0.528101 −0.264051 0.964509i $$-0.585059\pi$$
−0.264051 + 0.964509i $$0.585059\pi$$
$$510$$ 0 0
$$511$$ 7051.02 0.610408
$$512$$ −2607.89 −0.225105
$$513$$ 0 0
$$514$$ −16755.0 −1.43780
$$515$$ −6609.96 −0.565572
$$516$$ 0 0
$$517$$ −1649.00 −0.140276
$$518$$ −6125.38 −0.519563
$$519$$ 0 0
$$520$$ −301.576 −0.0254326
$$521$$ −2682.88 −0.225603 −0.112801 0.993618i $$-0.535982\pi$$
−0.112801 + 0.993618i $$0.535982\pi$$
$$522$$ 0 0
$$523$$ −4309.02 −0.360268 −0.180134 0.983642i $$-0.557653\pi$$
−0.180134 + 0.983642i $$0.557653\pi$$
$$524$$ 35039.6 2.92120
$$525$$ 0 0
$$526$$ 33163.3 2.74903
$$527$$ 1587.52 0.131221
$$528$$ 0 0
$$529$$ −9728.87 −0.799611
$$530$$ −11547.1 −0.946363
$$531$$ 0 0
$$532$$ 476.817 0.0388584
$$533$$ 925.720 0.0752296
$$534$$ 0 0
$$535$$ 8728.56 0.705362
$$536$$ −4229.25 −0.340813
$$537$$ 0 0
$$538$$ −3693.31 −0.295966
$$539$$ 934.051 0.0746427
$$540$$ 0 0
$$541$$ 4081.47 0.324355 0.162178 0.986762i $$-0.448148\pi$$
0.162178 + 0.986762i $$0.448148\pi$$
$$542$$ 23136.0 1.83354
$$543$$ 0 0
$$544$$ 1279.88 0.100872
$$545$$ −2384.28 −0.187397
$$546$$ 0 0
$$547$$ 8844.82 0.691366 0.345683 0.938351i $$-0.387647\pi$$
0.345683 + 0.938351i $$0.387647\pi$$
$$548$$ 25778.2 2.00947
$$549$$ 0 0
$$550$$ −2159.34 −0.167408
$$551$$ −1584.51 −0.122509
$$552$$ 0 0
$$553$$ 6042.66 0.464666
$$554$$ −6337.80 −0.486042
$$555$$ 0 0
$$556$$ −21868.4 −1.66803
$$557$$ 11144.7 0.847787 0.423894 0.905712i $$-0.360663\pi$$
0.423894 + 0.905712i $$0.360663\pi$$
$$558$$ 0 0
$$559$$ 884.249 0.0669047
$$560$$ −252.694 −0.0190684
$$561$$ 0 0
$$562$$ −32181.8 −2.41549
$$563$$ 21857.5 1.63621 0.818104 0.575071i $$-0.195025\pi$$
0.818104 + 0.575071i $$0.195025\pi$$
$$564$$ 0 0
$$565$$ 8208.23 0.611191
$$566$$ −20232.3 −1.50252
$$567$$ 0 0
$$568$$ 20619.6 1.52320
$$569$$ −23496.4 −1.73115 −0.865573 0.500783i $$-0.833046\pi$$
−0.865573 + 0.500783i $$0.833046\pi$$
$$570$$ 0 0
$$571$$ 11067.8 0.811164 0.405582 0.914059i $$-0.367069\pi$$
0.405582 + 0.914059i $$0.367069\pi$$
$$572$$ −701.743 −0.0512961
$$573$$ 0 0
$$574$$ −9994.72 −0.726780
$$575$$ −1234.44 −0.0895296
$$576$$ 0 0
$$577$$ 20482.9 1.47784 0.738922 0.673791i $$-0.235335\pi$$
0.738922 + 0.673791i $$0.235335\pi$$
$$578$$ 22070.1 1.58823
$$579$$ 0 0
$$580$$ −18263.8 −1.30752
$$581$$ 9343.07 0.667153
$$582$$ 0 0
$$583$$ −9715.60 −0.690187
$$584$$ 20680.8 1.46537
$$585$$ 0 0
$$586$$ 34394.1 2.42458
$$587$$ 23444.3 1.64847 0.824235 0.566248i $$-0.191606\pi$$
0.824235 + 0.566248i $$0.191606\pi$$
$$588$$ 0 0
$$589$$ −1328.01 −0.0929025
$$590$$ 1889.14 0.131822
$$591$$ 0 0
$$592$$ 1394.30 0.0967996
$$593$$ −4404.69 −0.305024 −0.152512 0.988302i $$-0.548736\pi$$
−0.152512 + 0.988302i $$0.548736\pi$$
$$594$$ 0 0
$$595$$ 227.432 0.0156703
$$596$$ 14706.2 1.01072
$$597$$ 0 0
$$598$$ −657.277 −0.0449466
$$599$$ −3327.05 −0.226945 −0.113472 0.993541i $$-0.536197\pi$$
−0.113472 + 0.993541i $$0.536197\pi$$
$$600$$ 0 0
$$601$$ −14244.8 −0.966818 −0.483409 0.875395i $$-0.660602\pi$$
−0.483409 + 0.875395i $$0.660602\pi$$
$$602$$ −9546.97 −0.646354
$$603$$ 0 0
$$604$$ 19298.8 1.30010
$$605$$ 4838.15 0.325122
$$606$$ 0 0
$$607$$ 11446.5 0.765402 0.382701 0.923872i $$-0.374994\pi$$
0.382701 + 0.923872i $$0.374994\pi$$
$$608$$ −1070.65 −0.0714156
$$609$$ 0 0
$$610$$ −119.008 −0.00789917
$$611$$ 254.132 0.0168266
$$612$$ 0 0
$$613$$ −19436.4 −1.28063 −0.640316 0.768111i $$-0.721197\pi$$
−0.640316 + 0.768111i $$0.721197\pi$$
$$614$$ −42955.6 −2.82337
$$615$$ 0 0
$$616$$ 2739.59 0.179190
$$617$$ 20530.1 1.33956 0.669781 0.742558i $$-0.266388\pi$$
0.669781 + 0.742558i $$0.266388\pi$$
$$618$$ 0 0
$$619$$ 5833.35 0.378776 0.189388 0.981902i $$-0.439350\pi$$
0.189388 + 0.981902i $$0.439350\pi$$
$$620$$ −15307.2 −0.991538
$$621$$ 0 0
$$622$$ 32071.4 2.06744
$$623$$ 2283.74 0.146864
$$624$$ 0 0
$$625$$ 625.000 0.0400000
$$626$$ −25346.4 −1.61829
$$627$$ 0 0
$$628$$ −31885.9 −2.02609
$$629$$ −1254.91 −0.0795493
$$630$$ 0 0
$$631$$ 24776.6 1.56314 0.781568 0.623820i $$-0.214420\pi$$
0.781568 + 0.623820i $$0.214420\pi$$
$$632$$ 17723.2 1.11549
$$633$$ 0 0
$$634$$ 16166.2 1.01268
$$635$$ 4220.08 0.263730
$$636$$ 0 0
$$637$$ −143.949 −0.00895366
$$638$$ −25177.4 −1.56235
$$639$$ 0 0
$$640$$ −11032.3 −0.681390
$$641$$ −27219.4 −1.67723 −0.838613 0.544728i $$-0.816633\pi$$
−0.838613 + 0.544728i $$0.816633\pi$$
$$642$$ 0 0
$$643$$ 7091.79 0.434950 0.217475 0.976066i $$-0.430218\pi$$
0.217475 + 0.976066i $$0.430218\pi$$
$$644$$ 4331.28 0.265026
$$645$$ 0 0
$$646$$ 160.049 0.00974777
$$647$$ −27773.0 −1.68758 −0.843792 0.536670i $$-0.819682\pi$$
−0.843792 + 0.536670i $$0.819682\pi$$
$$648$$ 0 0
$$649$$ 1589.51 0.0961382
$$650$$ 332.782 0.0200812
$$651$$ 0 0
$$652$$ −7452.37 −0.447634
$$653$$ −21380.4 −1.28129 −0.640643 0.767839i $$-0.721332\pi$$
−0.640643 + 0.767839i $$0.721332\pi$$
$$654$$ 0 0
$$655$$ −13981.0 −0.834020
$$656$$ 2275.06 0.135406
$$657$$ 0 0
$$658$$ −2743.78 −0.162559
$$659$$ 17232.3 1.01863 0.509315 0.860580i $$-0.329899\pi$$
0.509315 + 0.860580i $$0.329899\pi$$
$$660$$ 0 0
$$661$$ 26577.7 1.56392 0.781962 0.623326i $$-0.214219\pi$$
0.781962 + 0.623326i $$0.214219\pi$$
$$662$$ 19890.1 1.16775
$$663$$ 0 0
$$664$$ 27403.4 1.60159
$$665$$ −190.253 −0.0110943
$$666$$ 0 0
$$667$$ −14393.2 −0.835544
$$668$$ −11635.1 −0.673917
$$669$$ 0 0
$$670$$ 4666.89 0.269101
$$671$$ −100.132 −0.00576090
$$672$$ 0 0
$$673$$ −31695.2 −1.81540 −0.907698 0.419624i $$-0.862162\pi$$
−0.907698 + 0.419624i $$0.862162\pi$$
$$674$$ −10642.9 −0.608231
$$675$$ 0 0
$$676$$ −27422.7 −1.56024
$$677$$ −20440.3 −1.16039 −0.580195 0.814477i $$-0.697024\pi$$
−0.580195 + 0.814477i $$0.697024\pi$$
$$678$$ 0 0
$$679$$ −10687.4 −0.604044
$$680$$ 667.063 0.0376187
$$681$$ 0 0
$$682$$ −21101.7 −1.18479
$$683$$ 22896.9 1.28276 0.641381 0.767223i $$-0.278362\pi$$
0.641381 + 0.767223i $$0.278362\pi$$
$$684$$ 0 0
$$685$$ −10285.7 −0.573715
$$686$$ 1554.18 0.0864997
$$687$$ 0 0
$$688$$ 2173.14 0.120422
$$689$$ 1497.30 0.0827904
$$690$$ 0 0
$$691$$ −23764.0 −1.30829 −0.654143 0.756371i $$-0.726971\pi$$
−0.654143 + 0.756371i $$0.726971\pi$$
$$692$$ 3955.35 0.217283
$$693$$ 0 0
$$694$$ 2530.34 0.138401
$$695$$ 8725.62 0.476233
$$696$$ 0 0
$$697$$ −2047.62 −0.111276
$$698$$ −14653.2 −0.794600
$$699$$ 0 0
$$700$$ −2192.95 −0.118408
$$701$$ −26259.5 −1.41485 −0.707423 0.706791i $$-0.750142\pi$$
−0.707423 + 0.706791i $$0.750142\pi$$
$$702$$ 0 0
$$703$$ 1049.77 0.0563196
$$704$$ −15911.4 −0.851822
$$705$$ 0 0
$$706$$ −34057.6 −1.81554
$$707$$ 678.101 0.0360716
$$708$$ 0 0
$$709$$ 12783.0 0.677116 0.338558 0.940945i $$-0.390061\pi$$
0.338558 + 0.940945i $$0.390061\pi$$
$$710$$ −22753.2 −1.20270
$$711$$ 0 0
$$712$$ 6698.26 0.352567
$$713$$ −12063.3 −0.633623
$$714$$ 0 0
$$715$$ 280.000 0.0146453
$$716$$ 18113.6 0.945444
$$717$$ 0 0
$$718$$ −29804.7 −1.54916
$$719$$ 27609.0 1.43205 0.716025 0.698075i $$-0.245960\pi$$
0.716025 + 0.698075i $$0.245960\pi$$
$$720$$ 0 0
$$721$$ −9253.95 −0.477996
$$722$$ 30945.1 1.59509
$$723$$ 0 0
$$724$$ −23105.1 −1.18604
$$725$$ 7287.35 0.373304
$$726$$ 0 0
$$727$$ 31306.2 1.59709 0.798544 0.601937i $$-0.205604\pi$$
0.798544 + 0.601937i $$0.205604\pi$$
$$728$$ −422.206 −0.0214945
$$729$$ 0 0
$$730$$ −22820.8 −1.15703
$$731$$ −1955.89 −0.0989621
$$732$$ 0 0
$$733$$ −15765.8 −0.794441 −0.397220 0.917723i $$-0.630025\pi$$
−0.397220 + 0.917723i $$0.630025\pi$$
$$734$$ −37640.8 −1.89285
$$735$$ 0 0
$$736$$ −9725.53 −0.487076
$$737$$ 3926.68 0.196256
$$738$$ 0 0
$$739$$ 3966.51 0.197443 0.0987216 0.995115i $$-0.468525\pi$$
0.0987216 + 0.995115i $$0.468525\pi$$
$$740$$ 12100.1 0.601093
$$741$$ 0 0
$$742$$ −16165.9 −0.799823
$$743$$ −8224.50 −0.406094 −0.203047 0.979169i $$-0.565084\pi$$
−0.203047 + 0.979169i $$0.565084\pi$$
$$744$$ 0 0
$$745$$ −5867.86 −0.288566
$$746$$ 20621.6 1.01208
$$747$$ 0 0
$$748$$ 1552.20 0.0758745
$$749$$ 12220.0 0.596140
$$750$$ 0 0
$$751$$ 18929.2 0.919754 0.459877 0.887983i $$-0.347894\pi$$
0.459877 + 0.887983i $$0.347894\pi$$
$$752$$ 624.558 0.0302863
$$753$$ 0 0
$$754$$ 3880.16 0.187410
$$755$$ −7700.35 −0.371185
$$756$$ 0 0
$$757$$ −34906.8 −1.67597 −0.837984 0.545695i $$-0.816266\pi$$
−0.837984 + 0.545695i $$0.816266\pi$$
$$758$$ 8102.96 0.388276
$$759$$ 0 0
$$760$$ −558.016 −0.0266334
$$761$$ 13683.4 0.651803 0.325902 0.945404i $$-0.394332\pi$$
0.325902 + 0.945404i $$0.394332\pi$$
$$762$$ 0 0
$$763$$ −3337.99 −0.158379
$$764$$ 3058.52 0.144834
$$765$$ 0 0
$$766$$ −6155.42 −0.290345
$$767$$ −244.964 −0.0115321
$$768$$ 0 0
$$769$$ 41837.3 1.96189 0.980943 0.194294i $$-0.0622416\pi$$
0.980943 + 0.194294i $$0.0622416\pi$$
$$770$$ −3023.07 −0.141486
$$771$$ 0 0
$$772$$ −21716.9 −1.01245
$$773$$ −19640.0 −0.913843 −0.456921 0.889507i $$-0.651048\pi$$
−0.456921 + 0.889507i $$0.651048\pi$$
$$774$$ 0 0
$$775$$ 6107.69 0.283090
$$776$$ −31346.4 −1.45009
$$777$$ 0 0
$$778$$ 44054.1 2.03010
$$779$$ 1712.89 0.0787814
$$780$$ 0 0
$$781$$ −19144.4 −0.877131
$$782$$ 1453.85 0.0664827
$$783$$ 0 0
$$784$$ −353.772 −0.0161157
$$785$$ 12722.7 0.578460
$$786$$ 0 0
$$787$$ 24935.3 1.12941 0.564705 0.825293i $$-0.308990\pi$$
0.564705 + 0.825293i $$0.308990\pi$$
$$788$$ −4489.02 −0.202938
$$789$$ 0 0
$$790$$ −19557.2 −0.880777
$$791$$ 11491.5 0.516551
$$792$$ 0 0
$$793$$ 15.4317 0.000691041 0
$$794$$ 21695.2 0.969692
$$795$$ 0 0
$$796$$ −40145.1 −1.78757
$$797$$ 1168.33 0.0519251 0.0259625 0.999663i $$-0.491735\pi$$
0.0259625 + 0.999663i $$0.491735\pi$$
$$798$$ 0 0
$$799$$ −562.120 −0.0248891
$$800$$ 4924.08 0.217615
$$801$$ 0 0
$$802$$ 43867.7 1.93145
$$803$$ −19201.2 −0.843829
$$804$$ 0 0
$$805$$ −1728.21 −0.0756663
$$806$$ 3252.05 0.142120
$$807$$ 0 0
$$808$$ 1988.88 0.0865949
$$809$$ 35175.7 1.52869 0.764345 0.644807i $$-0.223062\pi$$
0.764345 + 0.644807i $$0.223062\pi$$
$$810$$ 0 0
$$811$$ −15256.5 −0.660577 −0.330288 0.943880i $$-0.607146\pi$$
−0.330288 + 0.943880i $$0.607146\pi$$
$$812$$ −25569.3 −1.10506
$$813$$ 0 0
$$814$$ 16680.5 0.718245
$$815$$ 2973.54 0.127802
$$816$$ 0 0
$$817$$ 1636.16 0.0700635
$$818$$ −50354.7 −2.15233
$$819$$ 0 0
$$820$$ 19743.6 0.840825
$$821$$ 15971.9 0.678956 0.339478 0.940614i $$-0.389750\pi$$
0.339478 + 0.940614i $$0.389750\pi$$
$$822$$ 0 0
$$823$$ 2312.41 0.0979409 0.0489705 0.998800i $$-0.484406\pi$$
0.0489705 + 0.998800i $$0.484406\pi$$
$$824$$ −27142.0 −1.14750
$$825$$ 0 0
$$826$$ 2644.80 0.111410
$$827$$ −10422.4 −0.438238 −0.219119 0.975698i $$-0.570318\pi$$
−0.219119 + 0.975698i $$0.570318\pi$$
$$828$$ 0 0
$$829$$ −13213.4 −0.553584 −0.276792 0.960930i $$-0.589271\pi$$
−0.276792 + 0.960930i $$0.589271\pi$$
$$830$$ −30239.0 −1.26459
$$831$$ 0 0
$$832$$ 2452.15 0.102179
$$833$$ 318.405 0.0132438
$$834$$ 0 0
$$835$$ 4642.49 0.192407
$$836$$ −1298.46 −0.0537179
$$837$$ 0 0
$$838$$ −5573.67 −0.229761
$$839$$ 10119.6 0.416409 0.208205 0.978085i $$-0.433238\pi$$
0.208205 + 0.978085i $$0.433238\pi$$
$$840$$ 0 0
$$841$$ 60579.9 2.48390
$$842$$ 55988.7 2.29157
$$843$$ 0 0
$$844$$ 61943.4 2.52628
$$845$$ 10941.8 0.445457
$$846$$ 0 0
$$847$$ 6773.41 0.274778
$$848$$ 3679.79 0.149015
$$849$$ 0 0
$$850$$ −736.089 −0.0297031
$$851$$ 9535.80 0.384116
$$852$$ 0 0
$$853$$ 35378.1 1.42007 0.710037 0.704165i $$-0.248678\pi$$
0.710037 + 0.704165i $$0.248678\pi$$
$$854$$ −166.611 −0.00667602
$$855$$ 0 0
$$856$$ 35841.4 1.43112
$$857$$ −6697.57 −0.266960 −0.133480 0.991052i $$-0.542615\pi$$
−0.133480 + 0.991052i $$0.542615\pi$$
$$858$$ 0 0
$$859$$ −24298.4 −0.965135 −0.482568 0.875859i $$-0.660296\pi$$
−0.482568 + 0.875859i $$0.660296\pi$$
$$860$$ 18859.1 0.747779
$$861$$ 0 0
$$862$$ −33418.3 −1.32045
$$863$$ −24942.9 −0.983853 −0.491926 0.870637i $$-0.663707\pi$$
−0.491926 + 0.870637i $$0.663707\pi$$
$$864$$ 0 0
$$865$$ −1578.21 −0.0620355
$$866$$ 3126.78 0.122693
$$867$$ 0 0
$$868$$ −21430.1 −0.838002
$$869$$ −16455.3 −0.642355
$$870$$ 0 0
$$871$$ −605.152 −0.0235417
$$872$$ −9790.39 −0.380212
$$873$$ 0 0
$$874$$ −1216.18 −0.0470686
$$875$$ 875.000 0.0338062
$$876$$ 0 0
$$877$$ −16276.6 −0.626705 −0.313353 0.949637i $$-0.601452\pi$$
−0.313353 + 0.949637i $$0.601452\pi$$
$$878$$ −38101.3 −1.46453
$$879$$ 0 0
$$880$$ 688.132 0.0263602
$$881$$ −26636.5 −1.01862 −0.509311 0.860582i $$-0.670100\pi$$
−0.509311 + 0.860582i $$0.670100\pi$$
$$882$$ 0 0
$$883$$ −21788.3 −0.830392 −0.415196 0.909732i $$-0.636287\pi$$
−0.415196 + 0.909732i $$0.636287\pi$$
$$884$$ −239.215 −0.00910142
$$885$$ 0 0
$$886$$ 29763.0 1.12856
$$887$$ 26813.2 1.01499 0.507496 0.861654i $$-0.330571\pi$$
0.507496 + 0.861654i $$0.330571\pi$$
$$888$$ 0 0
$$889$$ 5908.11 0.222893
$$890$$ −7391.38 −0.278382
$$891$$ 0 0
$$892$$ 39600.2 1.48645
$$893$$ 470.229 0.0176211
$$894$$ 0 0
$$895$$ −7227.45 −0.269930
$$896$$ −15445.2 −0.575879
$$897$$ 0 0
$$898$$ 13387.4 0.497488
$$899$$ 71214.2 2.64196
$$900$$ 0 0
$$901$$ −3311.91 −0.122459
$$902$$ 27217.4 1.00470
$$903$$ 0 0
$$904$$ 33704.8 1.24005
$$905$$ 9219.07 0.338621
$$906$$ 0 0
$$907$$ 15543.0 0.569014 0.284507 0.958674i $$-0.408170\pi$$
0.284507 + 0.958674i $$0.408170\pi$$
$$908$$ 45752.6 1.67219
$$909$$ 0 0
$$910$$ 465.895 0.0169717
$$911$$ −48711.1 −1.77154 −0.885768 0.464128i $$-0.846368\pi$$
−0.885768 + 0.464128i $$0.846368\pi$$
$$912$$ 0 0
$$913$$ −25442.8 −0.922273
$$914$$ 36905.3 1.33558
$$915$$ 0 0
$$916$$ −51173.3 −1.84587
$$917$$ −19573.4 −0.704876
$$918$$ 0 0
$$919$$ 1030.47 0.0369883 0.0184941 0.999829i $$-0.494113\pi$$
0.0184941 + 0.999829i $$0.494113\pi$$
$$920$$ −5068.87 −0.181648
$$921$$ 0 0
$$922$$ 11306.3 0.403855
$$923$$ 2950.40 0.105215
$$924$$ 0 0
$$925$$ −4828.02 −0.171615
$$926$$ 26079.6 0.925518
$$927$$ 0 0
$$928$$ 57413.6 2.03092
$$929$$ −879.756 −0.0310698 −0.0155349 0.999879i $$-0.504945\pi$$
−0.0155349 + 0.999879i $$0.504945\pi$$
$$930$$ 0 0
$$931$$ −266.354 −0.00937638
$$932$$ 46146.0 1.62185
$$933$$ 0 0
$$934$$ −18775.8 −0.657776
$$935$$ −619.339 −0.0216626
$$936$$ 0 0
$$937$$ −18668.1 −0.650864 −0.325432 0.945565i $$-0.605510\pi$$
−0.325432 + 0.945565i $$0.605510\pi$$
$$938$$ 6533.64 0.227432
$$939$$ 0 0
$$940$$ 5420.08 0.188067
$$941$$ −29613.4 −1.02590 −0.512948 0.858420i $$-0.671447\pi$$
−0.512948 + 0.858420i $$0.671447\pi$$
$$942$$ 0 0
$$943$$ 15559.5 0.537313
$$944$$ −602.028 −0.0207567
$$945$$ 0 0
$$946$$ 25998.1 0.893521
$$947$$ −20738.9 −0.711640 −0.355820 0.934554i $$-0.615798\pi$$
−0.355820 + 0.934554i $$0.615798\pi$$
$$948$$ 0 0
$$949$$ 2959.15 0.101220
$$950$$ 615.758 0.0210293
$$951$$ 0 0
$$952$$ 933.888 0.0317936
$$953$$ 45776.5 1.55598 0.777988 0.628279i $$-0.216240\pi$$
0.777988 + 0.628279i $$0.216240\pi$$
$$954$$ 0 0
$$955$$ −1220.37 −0.0413510
$$956$$ −33317.5 −1.12716
$$957$$ 0 0
$$958$$ −30657.4 −1.03392
$$959$$ −14399.9 −0.484878
$$960$$ 0 0
$$961$$ 29895.1 1.00349
$$962$$ −2570.68 −0.0861561
$$963$$ 0 0
$$964$$ −60404.9 −2.01817
$$965$$ 8665.17 0.289059
$$966$$ 0 0
$$967$$ 34461.0 1.14601 0.573005 0.819552i $$-0.305778\pi$$
0.573005 + 0.819552i $$0.305778\pi$$
$$968$$ 19866.5 0.659643
$$969$$ 0 0
$$970$$ 34590.1 1.14497
$$971$$ 22762.8 0.752309 0.376154 0.926557i $$-0.377246\pi$$
0.376154 + 0.926557i $$0.377246\pi$$
$$972$$ 0 0
$$973$$ 12215.9 0.402490
$$974$$ −28819.9 −0.948099
$$975$$ 0 0
$$976$$ 37.9251 0.00124381
$$977$$ −4809.57 −0.157494 −0.0787470 0.996895i $$-0.525092\pi$$
−0.0787470 + 0.996895i $$0.525092\pi$$
$$978$$ 0 0
$$979$$ −6219.04 −0.203025
$$980$$ −3070.13 −0.100073
$$981$$ 0 0
$$982$$ −32046.6 −1.04139
$$983$$ 27591.6 0.895256 0.447628 0.894220i $$-0.352269\pi$$
0.447628 + 0.894220i $$0.352269\pi$$
$$984$$ 0 0
$$985$$ 1791.15 0.0579399
$$986$$ −8582.63 −0.277207
$$987$$ 0 0
$$988$$ 200.109 0.00644365
$$989$$ 14862.4 0.477854
$$990$$ 0 0
$$991$$ −22263.4 −0.713643 −0.356822 0.934173i $$-0.616140\pi$$
−0.356822 + 0.934173i $$0.616140\pi$$
$$992$$ 48119.5 1.54012
$$993$$ 0 0
$$994$$ −31854.5 −1.01646
$$995$$ 16018.2 0.510361
$$996$$ 0 0
$$997$$ 30378.2 0.964983 0.482491 0.875901i $$-0.339732\pi$$
0.482491 + 0.875901i $$0.339732\pi$$
$$998$$ 83707.4 2.65502
$$999$$ 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 315.4.a.i.1.1 2
3.2 odd 2 105.4.a.f.1.2 2
5.4 even 2 1575.4.a.w.1.2 2
7.6 odd 2 2205.4.a.z.1.1 2
12.11 even 2 1680.4.a.bg.1.2 2
15.2 even 4 525.4.d.h.274.4 4
15.8 even 4 525.4.d.h.274.1 4
15.14 odd 2 525.4.a.k.1.1 2
21.20 even 2 735.4.a.p.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.f.1.2 2 3.2 odd 2
315.4.a.i.1.1 2 1.1 even 1 trivial
525.4.a.k.1.1 2 15.14 odd 2
525.4.d.h.274.1 4 15.8 even 4
525.4.d.h.274.4 4 15.2 even 4
735.4.a.p.1.2 2 21.20 even 2
1575.4.a.w.1.2 2 5.4 even 2
1680.4.a.bg.1.2 2 12.11 even 2
2205.4.a.z.1.1 2 7.6 odd 2