Properties

 Label 338.4.a.n.1.3 Level $338$ Weight $4$ Character 338.1 Self dual yes Analytic conductor $19.943$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [338,4,Mod(1,338)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(338, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 338.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: $$19.9426455819$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.6681389953.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 107x^{4} + 85x^{3} + 3703x^{2} - 1659x - 41951$$ x^6 - x^5 - 107*x^4 + 85*x^3 + 3703*x^2 - 1659*x - 41951 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$13^{2}$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.3 Root $$-5.40469$$ of defining polynomial Character $$\chi$$ $$=$$ 338.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} -1.24585 q^{3} +4.00000 q^{4} -20.5002 q^{5} +2.49171 q^{6} -21.2545 q^{7} -8.00000 q^{8} -25.4478 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} -1.24585 q^{3} +4.00000 q^{4} -20.5002 q^{5} +2.49171 q^{6} -21.2545 q^{7} -8.00000 q^{8} -25.4478 q^{9} +41.0004 q^{10} -53.0579 q^{11} -4.98341 q^{12} +42.5090 q^{14} +25.5403 q^{15} +16.0000 q^{16} -69.2150 q^{17} +50.8957 q^{18} +46.0603 q^{19} -82.0009 q^{20} +26.4800 q^{21} +106.116 q^{22} -87.2424 q^{23} +9.96683 q^{24} +295.259 q^{25} +65.3423 q^{27} -85.0181 q^{28} -162.062 q^{29} -51.0805 q^{30} -28.3466 q^{31} -32.0000 q^{32} +66.1024 q^{33} +138.430 q^{34} +435.722 q^{35} -101.791 q^{36} -111.972 q^{37} -92.1206 q^{38} +164.002 q^{40} -84.2751 q^{41} -52.9600 q^{42} -328.359 q^{43} -212.232 q^{44} +521.686 q^{45} +174.485 q^{46} -63.2021 q^{47} -19.9337 q^{48} +108.754 q^{49} -590.518 q^{50} +86.2318 q^{51} +721.631 q^{53} -130.685 q^{54} +1087.70 q^{55} +170.036 q^{56} -57.3844 q^{57} +324.123 q^{58} -819.081 q^{59} +102.161 q^{60} -397.828 q^{61} +56.6933 q^{62} +540.882 q^{63} +64.0000 q^{64} -132.205 q^{66} +77.8816 q^{67} -276.860 q^{68} +108.691 q^{69} -871.444 q^{70} +721.069 q^{71} +203.583 q^{72} -57.1470 q^{73} +223.945 q^{74} -367.849 q^{75} +184.241 q^{76} +1127.72 q^{77} -419.705 q^{79} -328.003 q^{80} +605.685 q^{81} +168.550 q^{82} +917.600 q^{83} +105.920 q^{84} +1418.92 q^{85} +656.717 q^{86} +201.905 q^{87} +424.463 q^{88} -378.804 q^{89} -1043.37 q^{90} -348.970 q^{92} +35.3158 q^{93} +126.404 q^{94} -944.246 q^{95} +39.8673 q^{96} -346.444 q^{97} -217.509 q^{98} +1350.21 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9}+O(q^{10})$$ 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 $$6 q - 12 q^{2} + 9 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 25 q^{7} - 48 q^{8} + 113 q^{9} + 36 q^{10} - 37 q^{11} + 36 q^{12} + 50 q^{14} - 118 q^{15} + 96 q^{16} + 99 q^{17} - 226 q^{18} + 81 q^{19} - 72 q^{20} + 26 q^{21} + 74 q^{22} + 267 q^{23} - 72 q^{24} + 368 q^{25} + 669 q^{27} - 100 q^{28} - 119 q^{29} + 236 q^{30} + 625 q^{31} - 192 q^{32} - 762 q^{33} - 198 q^{34} + 614 q^{35} + 452 q^{36} - 274 q^{37} - 162 q^{38} + 144 q^{40} - 1140 q^{41} - 52 q^{42} + 428 q^{43} - 148 q^{44} + 1215 q^{45} - 534 q^{46} + 986 q^{47} + 144 q^{48} + 899 q^{49} - 736 q^{50} + 289 q^{51} + 89 q^{53} - 1338 q^{54} + 1126 q^{55} + 200 q^{56} + 2553 q^{57} + 238 q^{58} - 1088 q^{59} - 472 q^{60} + 1704 q^{61} - 1250 q^{62} + 3222 q^{63} + 384 q^{64} + 1524 q^{66} + 1692 q^{67} + 396 q^{68} + 1168 q^{69} - 1228 q^{70} + 1221 q^{71} - 904 q^{72} + 1554 q^{73} + 548 q^{74} + 1798 q^{75} + 324 q^{76} + 2790 q^{77} - 875 q^{79} - 288 q^{80} + 3338 q^{81} + 2280 q^{82} + 126 q^{83} + 104 q^{84} + 3721 q^{85} - 856 q^{86} + 1602 q^{87} + 296 q^{88} + 374 q^{89} - 2430 q^{90} + 1068 q^{92} + 1868 q^{93} - 1972 q^{94} - 4093 q^{95} - 288 q^{96} + 330 q^{97} - 1798 q^{98} + 1344 q^{99}+O(q^{100})$$ 6 * q - 12 * q^2 + 9 * q^3 + 24 * q^4 - 18 * q^5 - 18 * q^6 - 25 * q^7 - 48 * q^8 + 113 * q^9 + 36 * q^10 - 37 * q^11 + 36 * q^12 + 50 * q^14 - 118 * q^15 + 96 * q^16 + 99 * q^17 - 226 * q^18 + 81 * q^19 - 72 * q^20 + 26 * q^21 + 74 * q^22 + 267 * q^23 - 72 * q^24 + 368 * q^25 + 669 * q^27 - 100 * q^28 - 119 * q^29 + 236 * q^30 + 625 * q^31 - 192 * q^32 - 762 * q^33 - 198 * q^34 + 614 * q^35 + 452 * q^36 - 274 * q^37 - 162 * q^38 + 144 * q^40 - 1140 * q^41 - 52 * q^42 + 428 * q^43 - 148 * q^44 + 1215 * q^45 - 534 * q^46 + 986 * q^47 + 144 * q^48 + 899 * q^49 - 736 * q^50 + 289 * q^51 + 89 * q^53 - 1338 * q^54 + 1126 * q^55 + 200 * q^56 + 2553 * q^57 + 238 * q^58 - 1088 * q^59 - 472 * q^60 + 1704 * q^61 - 1250 * q^62 + 3222 * q^63 + 384 * q^64 + 1524 * q^66 + 1692 * q^67 + 396 * q^68 + 1168 * q^69 - 1228 * q^70 + 1221 * q^71 - 904 * q^72 + 1554 * q^73 + 548 * q^74 + 1798 * q^75 + 324 * q^76 + 2790 * q^77 - 875 * q^79 - 288 * q^80 + 3338 * q^81 + 2280 * q^82 + 126 * q^83 + 104 * q^84 + 3721 * q^85 - 856 * q^86 + 1602 * q^87 + 296 * q^88 + 374 * q^89 - 2430 * q^90 + 1068 * q^92 + 1868 * q^93 - 1972 * q^94 - 4093 * q^95 - 288 * q^96 + 330 * q^97 - 1798 * q^98 + 1344 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.707107
$$3$$ −1.24585 −0.239765 −0.119882 0.992788i $$-0.538252\pi$$
−0.119882 + 0.992788i $$0.538252\pi$$
$$4$$ 4.00000 0.500000
$$5$$ −20.5002 −1.83359 −0.916797 0.399352i $$-0.869235\pi$$
−0.916797 + 0.399352i $$0.869235\pi$$
$$6$$ 2.49171 0.169539
$$7$$ −21.2545 −1.14764 −0.573818 0.818983i $$-0.694538\pi$$
−0.573818 + 0.818983i $$0.694538\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ −25.4478 −0.942513
$$10$$ 41.0004 1.29655
$$11$$ −53.0579 −1.45432 −0.727162 0.686466i $$-0.759161\pi$$
−0.727162 + 0.686466i $$0.759161\pi$$
$$12$$ −4.98341 −0.119882
$$13$$ 0 0
$$14$$ 42.5090 0.811501
$$15$$ 25.5403 0.439631
$$16$$ 16.0000 0.250000
$$17$$ −69.2150 −0.987477 −0.493738 0.869611i $$-0.664370\pi$$
−0.493738 + 0.869611i $$0.664370\pi$$
$$18$$ 50.8957 0.666457
$$19$$ 46.0603 0.556156 0.278078 0.960559i $$-0.410303\pi$$
0.278078 + 0.960559i $$0.410303\pi$$
$$20$$ −82.0009 −0.916797
$$21$$ 26.4800 0.275163
$$22$$ 106.116 1.02836
$$23$$ −87.2424 −0.790926 −0.395463 0.918482i $$-0.629416\pi$$
−0.395463 + 0.918482i $$0.629416\pi$$
$$24$$ 9.96683 0.0847696
$$25$$ 295.259 2.36207
$$26$$ 0 0
$$27$$ 65.3423 0.465746
$$28$$ −85.0181 −0.573818
$$29$$ −162.062 −1.03773 −0.518864 0.854857i $$-0.673645\pi$$
−0.518864 + 0.854857i $$0.673645\pi$$
$$30$$ −51.0805 −0.310866
$$31$$ −28.3466 −0.164232 −0.0821162 0.996623i $$-0.526168\pi$$
−0.0821162 + 0.996623i $$0.526168\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 66.1024 0.348695
$$34$$ 138.430 0.698251
$$35$$ 435.722 2.10430
$$36$$ −101.791 −0.471256
$$37$$ −111.972 −0.497517 −0.248759 0.968566i $$-0.580023\pi$$
−0.248759 + 0.968566i $$0.580023\pi$$
$$38$$ −92.1206 −0.393261
$$39$$ 0 0
$$40$$ 164.002 0.648274
$$41$$ −84.2751 −0.321014 −0.160507 0.987035i $$-0.551313\pi$$
−0.160507 + 0.987035i $$0.551313\pi$$
$$42$$ −52.9600 −0.194569
$$43$$ −328.359 −1.16452 −0.582258 0.813004i $$-0.697831\pi$$
−0.582258 + 0.813004i $$0.697831\pi$$
$$44$$ −212.232 −0.727162
$$45$$ 521.686 1.72819
$$46$$ 174.485 0.559269
$$47$$ −63.2021 −0.196148 −0.0980742 0.995179i $$-0.531268\pi$$
−0.0980742 + 0.995179i $$0.531268\pi$$
$$48$$ −19.9337 −0.0599412
$$49$$ 108.754 0.317068
$$50$$ −590.518 −1.67024
$$51$$ 86.2318 0.236762
$$52$$ 0 0
$$53$$ 721.631 1.87026 0.935129 0.354308i $$-0.115284\pi$$
0.935129 + 0.354308i $$0.115284\pi$$
$$54$$ −130.685 −0.329332
$$55$$ 1087.70 2.66664
$$56$$ 170.036 0.405751
$$57$$ −57.3844 −0.133346
$$58$$ 324.123 0.733784
$$59$$ −819.081 −1.80738 −0.903689 0.428190i $$-0.859151\pi$$
−0.903689 + 0.428190i $$0.859151\pi$$
$$60$$ 102.161 0.219816
$$61$$ −397.828 −0.835027 −0.417514 0.908671i $$-0.637098\pi$$
−0.417514 + 0.908671i $$0.637098\pi$$
$$62$$ 56.6933 0.116130
$$63$$ 540.882 1.08166
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ −132.205 −0.246565
$$67$$ 77.8816 0.142011 0.0710056 0.997476i $$-0.477379\pi$$
0.0710056 + 0.997476i $$0.477379\pi$$
$$68$$ −276.860 −0.493738
$$69$$ 108.691 0.189636
$$70$$ −871.444 −1.48796
$$71$$ 721.069 1.20528 0.602642 0.798011i $$-0.294115\pi$$
0.602642 + 0.798011i $$0.294115\pi$$
$$72$$ 203.583 0.333229
$$73$$ −57.1470 −0.0916240 −0.0458120 0.998950i $$-0.514588\pi$$
−0.0458120 + 0.998950i $$0.514588\pi$$
$$74$$ 223.945 0.351798
$$75$$ −367.849 −0.566341
$$76$$ 184.241 0.278078
$$77$$ 1127.72 1.66903
$$78$$ 0 0
$$79$$ −419.705 −0.597728 −0.298864 0.954296i $$-0.596608\pi$$
−0.298864 + 0.954296i $$0.596608\pi$$
$$80$$ −328.003 −0.458399
$$81$$ 605.685 0.830843
$$82$$ 168.550 0.226991
$$83$$ 917.600 1.21349 0.606745 0.794896i $$-0.292475\pi$$
0.606745 + 0.794896i $$0.292475\pi$$
$$84$$ 105.920 0.137581
$$85$$ 1418.92 1.81063
$$86$$ 656.717 0.823438
$$87$$ 201.905 0.248810
$$88$$ 424.463 0.514181
$$89$$ −378.804 −0.451159 −0.225580 0.974225i $$-0.572428\pi$$
−0.225580 + 0.974225i $$0.572428\pi$$
$$90$$ −1043.37 −1.22201
$$91$$ 0 0
$$92$$ −348.970 −0.395463
$$93$$ 35.3158 0.0393771
$$94$$ 126.404 0.138698
$$95$$ −944.246 −1.01976
$$96$$ 39.8673 0.0423848
$$97$$ −346.444 −0.362640 −0.181320 0.983424i $$-0.558037\pi$$
−0.181320 + 0.983424i $$0.558037\pi$$
$$98$$ −217.509 −0.224201
$$99$$ 1350.21 1.37072
$$100$$ 1181.04 1.18104
$$101$$ −519.293 −0.511600 −0.255800 0.966730i $$-0.582339\pi$$
−0.255800 + 0.966730i $$0.582339\pi$$
$$102$$ −172.464 −0.167416
$$103$$ −386.911 −0.370130 −0.185065 0.982726i $$-0.559250\pi$$
−0.185065 + 0.982726i $$0.559250\pi$$
$$104$$ 0 0
$$105$$ −542.846 −0.504537
$$106$$ −1443.26 −1.32247
$$107$$ −1542.78 −1.39389 −0.696944 0.717125i $$-0.745457\pi$$
−0.696944 + 0.717125i $$0.745457\pi$$
$$108$$ 261.369 0.232873
$$109$$ 535.630 0.470679 0.235340 0.971913i $$-0.424380\pi$$
0.235340 + 0.971913i $$0.424380\pi$$
$$110$$ −2175.40 −1.88560
$$111$$ 139.501 0.119287
$$112$$ −340.072 −0.286909
$$113$$ −997.890 −0.830739 −0.415370 0.909653i $$-0.636348\pi$$
−0.415370 + 0.909653i $$0.636348\pi$$
$$114$$ 114.769 0.0942902
$$115$$ 1788.49 1.45024
$$116$$ −648.247 −0.518864
$$117$$ 0 0
$$118$$ 1638.16 1.27801
$$119$$ 1471.13 1.13326
$$120$$ −204.322 −0.155433
$$121$$ 1484.14 1.11506
$$122$$ 795.656 0.590454
$$123$$ 104.994 0.0769678
$$124$$ −113.387 −0.0821162
$$125$$ −3490.34 −2.49749
$$126$$ −1081.76 −0.764850
$$127$$ −612.391 −0.427881 −0.213941 0.976847i $$-0.568630\pi$$
−0.213941 + 0.976847i $$0.568630\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 409.087 0.279210
$$130$$ 0 0
$$131$$ −2115.00 −1.41060 −0.705298 0.708911i $$-0.749187\pi$$
−0.705298 + 0.708911i $$0.749187\pi$$
$$132$$ 264.409 0.174348
$$133$$ −978.990 −0.638264
$$134$$ −155.763 −0.100417
$$135$$ −1339.53 −0.853989
$$136$$ 553.720 0.349126
$$137$$ −2925.55 −1.82443 −0.912214 0.409713i $$-0.865629\pi$$
−0.912214 + 0.409713i $$0.865629\pi$$
$$138$$ −217.382 −0.134093
$$139$$ −2162.95 −1.31985 −0.659923 0.751333i $$-0.729411\pi$$
−0.659923 + 0.751333i $$0.729411\pi$$
$$140$$ 1742.89 1.05215
$$141$$ 78.7406 0.0470294
$$142$$ −1442.14 −0.852265
$$143$$ 0 0
$$144$$ −407.166 −0.235628
$$145$$ 3322.30 1.90277
$$146$$ 114.294 0.0647879
$$147$$ −135.492 −0.0760218
$$148$$ −447.889 −0.248759
$$149$$ −1185.39 −0.651751 −0.325876 0.945413i $$-0.605659\pi$$
−0.325876 + 0.945413i $$0.605659\pi$$
$$150$$ 735.699 0.400464
$$151$$ −1757.32 −0.947077 −0.473538 0.880773i $$-0.657024\pi$$
−0.473538 + 0.880773i $$0.657024\pi$$
$$152$$ −368.482 −0.196631
$$153$$ 1761.37 0.930710
$$154$$ −2255.44 −1.18019
$$155$$ 581.112 0.301136
$$156$$ 0 0
$$157$$ 509.980 0.259241 0.129620 0.991564i $$-0.458624\pi$$
0.129620 + 0.991564i $$0.458624\pi$$
$$158$$ 839.411 0.422658
$$159$$ −899.047 −0.448422
$$160$$ 656.007 0.324137
$$161$$ 1854.29 0.907695
$$162$$ −1211.37 −0.587495
$$163$$ 1763.88 0.847591 0.423796 0.905758i $$-0.360697\pi$$
0.423796 + 0.905758i $$0.360697\pi$$
$$164$$ −337.101 −0.160507
$$165$$ −1355.11 −0.639366
$$166$$ −1835.20 −0.858067
$$167$$ −397.025 −0.183968 −0.0919842 0.995760i $$-0.529321\pi$$
−0.0919842 + 0.995760i $$0.529321\pi$$
$$168$$ −211.840 −0.0972847
$$169$$ 0 0
$$170$$ −2837.84 −1.28031
$$171$$ −1172.14 −0.524184
$$172$$ −1313.43 −0.582258
$$173$$ −12.3237 −0.00541593 −0.00270796 0.999996i $$-0.500862\pi$$
−0.00270796 + 0.999996i $$0.500862\pi$$
$$174$$ −403.810 −0.175936
$$175$$ −6275.58 −2.71080
$$176$$ −848.926 −0.363581
$$177$$ 1020.46 0.433345
$$178$$ 757.609 0.319018
$$179$$ −278.459 −0.116274 −0.0581368 0.998309i $$-0.518516\pi$$
−0.0581368 + 0.998309i $$0.518516\pi$$
$$180$$ 2086.75 0.864093
$$181$$ −3482.28 −1.43003 −0.715017 0.699107i $$-0.753581\pi$$
−0.715017 + 0.699107i $$0.753581\pi$$
$$182$$ 0 0
$$183$$ 495.636 0.200210
$$184$$ 697.939 0.279635
$$185$$ 2295.46 0.912245
$$186$$ −70.6315 −0.0278438
$$187$$ 3672.40 1.43611
$$188$$ −252.808 −0.0980742
$$189$$ −1388.82 −0.534507
$$190$$ 1888.49 0.721082
$$191$$ 3784.38 1.43366 0.716828 0.697250i $$-0.245593\pi$$
0.716828 + 0.697250i $$0.245593\pi$$
$$192$$ −79.7346 −0.0299706
$$193$$ 1990.44 0.742356 0.371178 0.928562i $$-0.378954\pi$$
0.371178 + 0.928562i $$0.378954\pi$$
$$194$$ 692.888 0.256425
$$195$$ 0 0
$$196$$ 435.018 0.158534
$$197$$ −4088.79 −1.47875 −0.739377 0.673292i $$-0.764880\pi$$
−0.739377 + 0.673292i $$0.764880\pi$$
$$198$$ −2700.42 −0.969244
$$199$$ 3461.38 1.23302 0.616510 0.787347i $$-0.288546\pi$$
0.616510 + 0.787347i $$0.288546\pi$$
$$200$$ −2362.07 −0.835118
$$201$$ −97.0291 −0.0340493
$$202$$ 1038.59 0.361756
$$203$$ 3444.54 1.19093
$$204$$ 344.927 0.118381
$$205$$ 1727.66 0.588609
$$206$$ 773.821 0.261722
$$207$$ 2220.13 0.745458
$$208$$ 0 0
$$209$$ −2443.86 −0.808830
$$210$$ 1085.69 0.356761
$$211$$ −197.608 −0.0644736 −0.0322368 0.999480i $$-0.510263\pi$$
−0.0322368 + 0.999480i $$0.510263\pi$$
$$212$$ 2886.52 0.935129
$$213$$ −898.347 −0.288985
$$214$$ 3085.56 0.985628
$$215$$ 6731.42 2.13525
$$216$$ −522.739 −0.164666
$$217$$ 602.494 0.188479
$$218$$ −1071.26 −0.332821
$$219$$ 71.1968 0.0219682
$$220$$ 4350.79 1.33332
$$221$$ 0 0
$$222$$ −279.002 −0.0843487
$$223$$ 3622.66 1.08785 0.543927 0.839132i $$-0.316937\pi$$
0.543927 + 0.839132i $$0.316937\pi$$
$$224$$ 680.145 0.202875
$$225$$ −7513.70 −2.22628
$$226$$ 1995.78 0.587421
$$227$$ 5788.77 1.69257 0.846287 0.532727i $$-0.178833\pi$$
0.846287 + 0.532727i $$0.178833\pi$$
$$228$$ −229.538 −0.0666732
$$229$$ −988.340 −0.285203 −0.142601 0.989780i $$-0.545547\pi$$
−0.142601 + 0.989780i $$0.545547\pi$$
$$230$$ −3576.98 −1.02547
$$231$$ −1404.97 −0.400175
$$232$$ 1296.49 0.366892
$$233$$ −6121.48 −1.72116 −0.860582 0.509312i $$-0.829900\pi$$
−0.860582 + 0.509312i $$0.829900\pi$$
$$234$$ 0 0
$$235$$ 1295.66 0.359657
$$236$$ −3276.32 −0.903689
$$237$$ 522.891 0.143314
$$238$$ −2942.26 −0.801338
$$239$$ −224.263 −0.0606961 −0.0303480 0.999539i $$-0.509662\pi$$
−0.0303480 + 0.999539i $$0.509662\pi$$
$$240$$ 408.644 0.109908
$$241$$ −5636.03 −1.50642 −0.753212 0.657777i $$-0.771497\pi$$
−0.753212 + 0.657777i $$0.771497\pi$$
$$242$$ −2968.28 −0.788464
$$243$$ −2518.84 −0.664953
$$244$$ −1591.31 −0.417514
$$245$$ −2229.49 −0.581375
$$246$$ −209.989 −0.0544244
$$247$$ 0 0
$$248$$ 226.773 0.0580649
$$249$$ −1143.20 −0.290952
$$250$$ 6980.68 1.76599
$$251$$ 2416.44 0.607666 0.303833 0.952725i $$-0.401734\pi$$
0.303833 + 0.952725i $$0.401734\pi$$
$$252$$ 2163.53 0.540831
$$253$$ 4628.90 1.15026
$$254$$ 1224.78 0.302558
$$255$$ −1767.77 −0.434126
$$256$$ 256.000 0.0625000
$$257$$ −907.064 −0.220160 −0.110080 0.993923i $$-0.535111\pi$$
−0.110080 + 0.993923i $$0.535111\pi$$
$$258$$ −818.174 −0.197431
$$259$$ 2379.92 0.570969
$$260$$ 0 0
$$261$$ 4124.12 0.978072
$$262$$ 4229.99 0.997442
$$263$$ 3034.33 0.711425 0.355712 0.934596i $$-0.384238\pi$$
0.355712 + 0.934596i $$0.384238\pi$$
$$264$$ −528.819 −0.123282
$$265$$ −14793.6 −3.42929
$$266$$ 1957.98 0.451321
$$267$$ 471.935 0.108172
$$268$$ 311.526 0.0710056
$$269$$ −4047.61 −0.917424 −0.458712 0.888585i $$-0.651689\pi$$
−0.458712 + 0.888585i $$0.651689\pi$$
$$270$$ 2679.06 0.603862
$$271$$ −3613.26 −0.809926 −0.404963 0.914333i $$-0.632716\pi$$
−0.404963 + 0.914333i $$0.632716\pi$$
$$272$$ −1107.44 −0.246869
$$273$$ 0 0
$$274$$ 5851.10 1.29007
$$275$$ −15665.8 −3.43521
$$276$$ 434.765 0.0948180
$$277$$ 4858.36 1.05383 0.526914 0.849918i $$-0.323349\pi$$
0.526914 + 0.849918i $$0.323349\pi$$
$$278$$ 4325.89 0.933272
$$279$$ 721.361 0.154791
$$280$$ −3485.78 −0.743982
$$281$$ 4570.79 0.970358 0.485179 0.874415i $$-0.338754\pi$$
0.485179 + 0.874415i $$0.338754\pi$$
$$282$$ −157.481 −0.0332548
$$283$$ 867.036 0.182120 0.0910600 0.995845i $$-0.470974\pi$$
0.0910600 + 0.995845i $$0.470974\pi$$
$$284$$ 2884.28 0.602642
$$285$$ 1176.39 0.244503
$$286$$ 0 0
$$287$$ 1791.23 0.368407
$$288$$ 814.331 0.166614
$$289$$ −122.284 −0.0248898
$$290$$ −6644.60 −1.34546
$$291$$ 431.619 0.0869482
$$292$$ −228.588 −0.0458120
$$293$$ −1149.40 −0.229176 −0.114588 0.993413i $$-0.536555\pi$$
−0.114588 + 0.993413i $$0.536555\pi$$
$$294$$ 270.984 0.0537555
$$295$$ 16791.3 3.31400
$$296$$ 895.779 0.175899
$$297$$ −3466.93 −0.677345
$$298$$ 2370.78 0.460858
$$299$$ 0 0
$$300$$ −1471.40 −0.283171
$$301$$ 6979.10 1.33644
$$302$$ 3514.64 0.669685
$$303$$ 646.963 0.122664
$$304$$ 736.965 0.139039
$$305$$ 8155.56 1.53110
$$306$$ −3522.75 −0.658111
$$307$$ 6457.99 1.20058 0.600288 0.799784i $$-0.295053\pi$$
0.600288 + 0.799784i $$0.295053\pi$$
$$308$$ 4510.88 0.834517
$$309$$ 482.034 0.0887442
$$310$$ −1162.22 −0.212935
$$311$$ 4856.58 0.885504 0.442752 0.896644i $$-0.354002\pi$$
0.442752 + 0.896644i $$0.354002\pi$$
$$312$$ 0 0
$$313$$ 2594.40 0.468511 0.234255 0.972175i $$-0.424735\pi$$
0.234255 + 0.972175i $$0.424735\pi$$
$$314$$ −1019.96 −0.183311
$$315$$ −11088.2 −1.98333
$$316$$ −1678.82 −0.298864
$$317$$ −2690.19 −0.476643 −0.238322 0.971186i $$-0.576597\pi$$
−0.238322 + 0.971186i $$0.576597\pi$$
$$318$$ 1798.09 0.317082
$$319$$ 8598.65 1.50919
$$320$$ −1312.01 −0.229199
$$321$$ 1922.08 0.334205
$$322$$ −3708.59 −0.641837
$$323$$ −3188.06 −0.549191
$$324$$ 2422.74 0.415422
$$325$$ 0 0
$$326$$ −3527.75 −0.599338
$$327$$ −667.317 −0.112852
$$328$$ 674.201 0.113496
$$329$$ 1343.33 0.225107
$$330$$ 2710.23 0.452100
$$331$$ −10641.6 −1.76711 −0.883556 0.468326i $$-0.844857\pi$$
−0.883556 + 0.468326i $$0.844857\pi$$
$$332$$ 3670.40 0.606745
$$333$$ 2849.46 0.468916
$$334$$ 794.050 0.130085
$$335$$ −1596.59 −0.260391
$$336$$ 423.680 0.0687906
$$337$$ 670.259 0.108342 0.0541711 0.998532i $$-0.482748\pi$$
0.0541711 + 0.998532i $$0.482748\pi$$
$$338$$ 0 0
$$339$$ 1243.22 0.199182
$$340$$ 5675.69 0.905316
$$341$$ 1504.01 0.238847
$$342$$ 2344.27 0.370654
$$343$$ 4978.78 0.783757
$$344$$ 2626.87 0.411719
$$345$$ −2228.19 −0.347716
$$346$$ 24.6474 0.00382964
$$347$$ 1047.94 0.162122 0.0810609 0.996709i $$-0.474169\pi$$
0.0810609 + 0.996709i $$0.474169\pi$$
$$348$$ 807.621 0.124405
$$349$$ −8329.53 −1.27756 −0.638782 0.769388i $$-0.720561\pi$$
−0.638782 + 0.769388i $$0.720561\pi$$
$$350$$ 12551.2 1.91682
$$351$$ 0 0
$$352$$ 1697.85 0.257090
$$353$$ 2824.64 0.425893 0.212947 0.977064i $$-0.431694\pi$$
0.212947 + 0.977064i $$0.431694\pi$$
$$354$$ −2040.91 −0.306421
$$355$$ −14782.1 −2.21000
$$356$$ −1515.22 −0.225580
$$357$$ −1832.81 −0.271717
$$358$$ 556.918 0.0822179
$$359$$ 8753.05 1.28682 0.643410 0.765522i $$-0.277519\pi$$
0.643410 + 0.765522i $$0.277519\pi$$
$$360$$ −4173.49 −0.611006
$$361$$ −4737.45 −0.690691
$$362$$ 6964.57 1.01119
$$363$$ −1849.02 −0.267351
$$364$$ 0 0
$$365$$ 1171.53 0.168001
$$366$$ −991.271 −0.141570
$$367$$ 7378.20 1.04942 0.524712 0.851280i $$-0.324173\pi$$
0.524712 + 0.851280i $$0.324173\pi$$
$$368$$ −1395.88 −0.197731
$$369$$ 2144.62 0.302560
$$370$$ −4590.91 −0.645055
$$371$$ −15337.9 −2.14637
$$372$$ 141.263 0.0196886
$$373$$ 2626.42 0.364587 0.182293 0.983244i $$-0.441648\pi$$
0.182293 + 0.983244i $$0.441648\pi$$
$$374$$ −7344.80 −1.01548
$$375$$ 4348.46 0.598809
$$376$$ 505.617 0.0693489
$$377$$ 0 0
$$378$$ 2777.64 0.377953
$$379$$ −503.491 −0.0682391 −0.0341195 0.999418i $$-0.510863\pi$$
−0.0341195 + 0.999418i $$0.510863\pi$$
$$380$$ −3776.98 −0.509882
$$381$$ 762.949 0.102591
$$382$$ −7568.76 −1.01375
$$383$$ −264.479 −0.0352852 −0.0176426 0.999844i $$-0.505616\pi$$
−0.0176426 + 0.999844i $$0.505616\pi$$
$$384$$ 159.469 0.0211924
$$385$$ −23118.5 −3.06033
$$386$$ −3980.87 −0.524925
$$387$$ 8356.02 1.09757
$$388$$ −1385.78 −0.181320
$$389$$ −5074.42 −0.661397 −0.330698 0.943736i $$-0.607284\pi$$
−0.330698 + 0.943736i $$0.607284\pi$$
$$390$$ 0 0
$$391$$ 6038.48 0.781021
$$392$$ −870.036 −0.112101
$$393$$ 2634.98 0.338211
$$394$$ 8177.59 1.04564
$$395$$ 8604.05 1.09599
$$396$$ 5400.84 0.685359
$$397$$ 1633.19 0.206467 0.103234 0.994657i $$-0.467081\pi$$
0.103234 + 0.994657i $$0.467081\pi$$
$$398$$ −6922.77 −0.871877
$$399$$ 1219.68 0.153033
$$400$$ 4724.14 0.590518
$$401$$ −3512.59 −0.437432 −0.218716 0.975789i $$-0.570187\pi$$
−0.218716 + 0.975789i $$0.570187\pi$$
$$402$$ 194.058 0.0240765
$$403$$ 0 0
$$404$$ −2077.17 −0.255800
$$405$$ −12416.7 −1.52343
$$406$$ −6889.09 −0.842117
$$407$$ 5941.02 0.723551
$$408$$ −689.854 −0.0837080
$$409$$ 1502.01 0.181589 0.0907944 0.995870i $$-0.471059\pi$$
0.0907944 + 0.995870i $$0.471059\pi$$
$$410$$ −3455.32 −0.416210
$$411$$ 3644.81 0.437434
$$412$$ −1547.64 −0.185065
$$413$$ 17409.2 2.07421
$$414$$ −4440.26 −0.527118
$$415$$ −18811.0 −2.22505
$$416$$ 0 0
$$417$$ 2694.71 0.316452
$$418$$ 4887.73 0.571929
$$419$$ −13040.3 −1.52043 −0.760214 0.649673i $$-0.774906\pi$$
−0.760214 + 0.649673i $$0.774906\pi$$
$$420$$ −2171.38 −0.252268
$$421$$ −13790.6 −1.59647 −0.798234 0.602348i $$-0.794232\pi$$
−0.798234 + 0.602348i $$0.794232\pi$$
$$422$$ 395.217 0.0455897
$$423$$ 1608.36 0.184872
$$424$$ −5773.05 −0.661236
$$425$$ −20436.3 −2.33249
$$426$$ 1796.69 0.204343
$$427$$ 8455.64 0.958308
$$428$$ −6171.12 −0.696944
$$429$$ 0 0
$$430$$ −13462.8 −1.50985
$$431$$ 13366.5 1.49383 0.746915 0.664920i $$-0.231534\pi$$
0.746915 + 0.664920i $$0.231534\pi$$
$$432$$ 1045.48 0.116436
$$433$$ 12375.5 1.37351 0.686753 0.726891i $$-0.259036\pi$$
0.686753 + 0.726891i $$0.259036\pi$$
$$434$$ −1204.99 −0.133275
$$435$$ −4139.10 −0.456218
$$436$$ 2142.52 0.235340
$$437$$ −4018.41 −0.439878
$$438$$ −142.394 −0.0155339
$$439$$ −12339.8 −1.34156 −0.670782 0.741655i $$-0.734041\pi$$
−0.670782 + 0.741655i $$0.734041\pi$$
$$440$$ −8701.59 −0.942800
$$441$$ −2767.57 −0.298841
$$442$$ 0 0
$$443$$ −15065.3 −1.61575 −0.807873 0.589356i $$-0.799381\pi$$
−0.807873 + 0.589356i $$0.799381\pi$$
$$444$$ 558.005 0.0596435
$$445$$ 7765.57 0.827244
$$446$$ −7245.33 −0.769229
$$447$$ 1476.82 0.156267
$$448$$ −1360.29 −0.143454
$$449$$ 4223.62 0.443931 0.221966 0.975055i $$-0.428753\pi$$
0.221966 + 0.975055i $$0.428753\pi$$
$$450$$ 15027.4 1.57422
$$451$$ 4471.46 0.466858
$$452$$ −3991.56 −0.415370
$$453$$ 2189.36 0.227076
$$454$$ −11577.5 −1.19683
$$455$$ 0 0
$$456$$ 459.075 0.0471451
$$457$$ −5141.66 −0.526295 −0.263148 0.964756i $$-0.584761\pi$$
−0.263148 + 0.964756i $$0.584761\pi$$
$$458$$ 1976.68 0.201669
$$459$$ −4522.67 −0.459913
$$460$$ 7153.95 0.725119
$$461$$ −11630.3 −1.17500 −0.587501 0.809223i $$-0.699888\pi$$
−0.587501 + 0.809223i $$0.699888\pi$$
$$462$$ 2809.95 0.282967
$$463$$ 1028.85 0.103272 0.0516359 0.998666i $$-0.483556\pi$$
0.0516359 + 0.998666i $$0.483556\pi$$
$$464$$ −2592.99 −0.259432
$$465$$ −723.981 −0.0722017
$$466$$ 12243.0 1.21705
$$467$$ 14089.0 1.39606 0.698030 0.716068i $$-0.254060\pi$$
0.698030 + 0.716068i $$0.254060\pi$$
$$468$$ 0 0
$$469$$ −1655.34 −0.162977
$$470$$ −2591.31 −0.254316
$$471$$ −635.360 −0.0621568
$$472$$ 6552.65 0.639004
$$473$$ 17422.0 1.69358
$$474$$ −1045.78 −0.101338
$$475$$ 13599.7 1.31368
$$476$$ 5884.53 0.566632
$$477$$ −18364.0 −1.76274
$$478$$ 448.526 0.0429186
$$479$$ 14176.7 1.35230 0.676149 0.736765i $$-0.263648\pi$$
0.676149 + 0.736765i $$0.263648\pi$$
$$480$$ −817.289 −0.0777166
$$481$$ 0 0
$$482$$ 11272.1 1.06520
$$483$$ −2310.18 −0.217633
$$484$$ 5936.56 0.557528
$$485$$ 7102.18 0.664934
$$486$$ 5037.68 0.470193
$$487$$ −5713.18 −0.531600 −0.265800 0.964028i $$-0.585636\pi$$
−0.265800 + 0.964028i $$0.585636\pi$$
$$488$$ 3182.63 0.295227
$$489$$ −2197.53 −0.203222
$$490$$ 4458.98 0.411094
$$491$$ −14868.4 −1.36660 −0.683301 0.730137i $$-0.739456\pi$$
−0.683301 + 0.730137i $$0.739456\pi$$
$$492$$ 419.978 0.0384839
$$493$$ 11217.1 1.02473
$$494$$ 0 0
$$495$$ −27679.6 −2.51334
$$496$$ −453.546 −0.0410581
$$497$$ −15326.0 −1.38323
$$498$$ 2286.39 0.205734
$$499$$ −6429.79 −0.576828 −0.288414 0.957506i $$-0.593128\pi$$
−0.288414 + 0.957506i $$0.593128\pi$$
$$500$$ −13961.4 −1.24874
$$501$$ 494.635 0.0441091
$$502$$ −4832.87 −0.429685
$$503$$ 11799.0 1.04590 0.522952 0.852362i $$-0.324831\pi$$
0.522952 + 0.852362i $$0.324831\pi$$
$$504$$ −4327.05 −0.382425
$$505$$ 10645.6 0.938067
$$506$$ −9257.79 −0.813358
$$507$$ 0 0
$$508$$ −2449.56 −0.213941
$$509$$ 9183.02 0.799667 0.399833 0.916588i $$-0.369068\pi$$
0.399833 + 0.916588i $$0.369068\pi$$
$$510$$ 3535.54 0.306973
$$511$$ 1214.63 0.105151
$$512$$ −512.000 −0.0441942
$$513$$ 3009.69 0.259027
$$514$$ 1814.13 0.155677
$$515$$ 7931.75 0.678669
$$516$$ 1636.35 0.139605
$$517$$ 3353.37 0.285263
$$518$$ −4759.84 −0.403736
$$519$$ 15.3536 0.00129855
$$520$$ 0 0
$$521$$ −2688.15 −0.226046 −0.113023 0.993592i $$-0.536053\pi$$
−0.113023 + 0.993592i $$0.536053\pi$$
$$522$$ −8248.24 −0.691601
$$523$$ 15920.5 1.33108 0.665538 0.746364i $$-0.268202\pi$$
0.665538 + 0.746364i $$0.268202\pi$$
$$524$$ −8459.98 −0.705298
$$525$$ 7818.46 0.649953
$$526$$ −6068.66 −0.503053
$$527$$ 1962.01 0.162176
$$528$$ 1057.64 0.0871738
$$529$$ −4555.77 −0.374436
$$530$$ 29587.2 2.42488
$$531$$ 20843.9 1.70348
$$532$$ −3915.96 −0.319132
$$533$$ 0 0
$$534$$ −943.870 −0.0764892
$$535$$ 31627.3 2.55583
$$536$$ −623.053 −0.0502086
$$537$$ 346.919 0.0278783
$$538$$ 8095.22 0.648717
$$539$$ −5770.28 −0.461120
$$540$$ −5358.13 −0.426995
$$541$$ 1967.63 0.156368 0.0781838 0.996939i $$-0.475088\pi$$
0.0781838 + 0.996939i $$0.475088\pi$$
$$542$$ 7226.52 0.572704
$$543$$ 4338.42 0.342871
$$544$$ 2214.88 0.174563
$$545$$ −10980.5 −0.863035
$$546$$ 0 0
$$547$$ −6075.05 −0.474863 −0.237432 0.971404i $$-0.576306\pi$$
−0.237432 + 0.971404i $$0.576306\pi$$
$$548$$ −11702.2 −0.912214
$$549$$ 10123.9 0.787024
$$550$$ 31331.6 2.42906
$$551$$ −7464.61 −0.577138
$$552$$ −869.530 −0.0670465
$$553$$ 8920.63 0.685974
$$554$$ −9716.72 −0.745169
$$555$$ −2859.80 −0.218724
$$556$$ −8651.78 −0.659923
$$557$$ −9690.09 −0.737132 −0.368566 0.929602i $$-0.620151\pi$$
−0.368566 + 0.929602i $$0.620151\pi$$
$$558$$ −1442.72 −0.109454
$$559$$ 0 0
$$560$$ 6971.55 0.526075
$$561$$ −4575.28 −0.344329
$$562$$ −9141.58 −0.686147
$$563$$ −6132.79 −0.459088 −0.229544 0.973298i $$-0.573723\pi$$
−0.229544 + 0.973298i $$0.573723\pi$$
$$564$$ 314.962 0.0235147
$$565$$ 20457.0 1.52324
$$566$$ −1734.07 −0.128778
$$567$$ −12873.5 −0.953506
$$568$$ −5768.56 −0.426132
$$569$$ 12081.1 0.890098 0.445049 0.895506i $$-0.353186\pi$$
0.445049 + 0.895506i $$0.353186\pi$$
$$570$$ −2352.79 −0.172890
$$571$$ 712.570 0.0522244 0.0261122 0.999659i $$-0.491687\pi$$
0.0261122 + 0.999659i $$0.491687\pi$$
$$572$$ 0 0
$$573$$ −4714.79 −0.343740
$$574$$ −3582.45 −0.260503
$$575$$ −25759.1 −1.86822
$$576$$ −1628.66 −0.117814
$$577$$ −1413.94 −0.102016 −0.0510078 0.998698i $$-0.516243\pi$$
−0.0510078 + 0.998698i $$0.516243\pi$$
$$578$$ 244.568 0.0175998
$$579$$ −2479.79 −0.177991
$$580$$ 13289.2 0.951386
$$581$$ −19503.1 −1.39265
$$582$$ −863.237 −0.0614817
$$583$$ −38288.2 −2.71996
$$584$$ 457.176 0.0323940
$$585$$ 0 0
$$586$$ 2298.79 0.162052
$$587$$ −7352.50 −0.516985 −0.258492 0.966013i $$-0.583226\pi$$
−0.258492 + 0.966013i $$0.583226\pi$$
$$588$$ −541.968 −0.0380109
$$589$$ −1305.65 −0.0913388
$$590$$ −33582.7 −2.34335
$$591$$ 5094.04 0.354553
$$592$$ −1791.56 −0.124379
$$593$$ −15534.1 −1.07573 −0.537866 0.843031i $$-0.680769\pi$$
−0.537866 + 0.843031i $$0.680769\pi$$
$$594$$ 6933.85 0.478955
$$595$$ −30158.5 −2.07795
$$596$$ −4741.56 −0.325876
$$597$$ −4312.38 −0.295635
$$598$$ 0 0
$$599$$ 20257.5 1.38180 0.690901 0.722949i $$-0.257214\pi$$
0.690901 + 0.722949i $$0.257214\pi$$
$$600$$ 2942.79 0.200232
$$601$$ 25021.6 1.69826 0.849128 0.528187i $$-0.177128\pi$$
0.849128 + 0.528187i $$0.177128\pi$$
$$602$$ −13958.2 −0.945007
$$603$$ −1981.92 −0.133847
$$604$$ −7029.28 −0.473538
$$605$$ −30425.2 −2.04456
$$606$$ −1293.93 −0.0867363
$$607$$ −4754.85 −0.317946 −0.158973 0.987283i $$-0.550818\pi$$
−0.158973 + 0.987283i $$0.550818\pi$$
$$608$$ −1473.93 −0.0983154
$$609$$ −4291.40 −0.285544
$$610$$ −16311.1 −1.08265
$$611$$ 0 0
$$612$$ 7045.49 0.465355
$$613$$ 11141.9 0.734125 0.367063 0.930196i $$-0.380363\pi$$
0.367063 + 0.930196i $$0.380363\pi$$
$$614$$ −12916.0 −0.848936
$$615$$ −2152.41 −0.141128
$$616$$ −9021.76 −0.590093
$$617$$ −8032.61 −0.524118 −0.262059 0.965052i $$-0.584401\pi$$
−0.262059 + 0.965052i $$0.584401\pi$$
$$618$$ −964.068 −0.0627516
$$619$$ −5142.70 −0.333930 −0.166965 0.985963i $$-0.553397\pi$$
−0.166965 + 0.985963i $$0.553397\pi$$
$$620$$ 2324.45 0.150568
$$621$$ −5700.62 −0.368371
$$622$$ −9713.17 −0.626146
$$623$$ 8051.30 0.517767
$$624$$ 0 0
$$625$$ 34645.4 2.21731
$$626$$ −5188.79 −0.331287
$$627$$ 3044.70 0.193929
$$628$$ 2039.92 0.129620
$$629$$ 7750.17 0.491287
$$630$$ 22176.4 1.40243
$$631$$ −9537.79 −0.601733 −0.300866 0.953666i $$-0.597276\pi$$
−0.300866 + 0.953666i $$0.597276\pi$$
$$632$$ 3357.64 0.211329
$$633$$ 246.191 0.0154585
$$634$$ 5380.37 0.337038
$$635$$ 12554.1 0.784561
$$636$$ −3596.19 −0.224211
$$637$$ 0 0
$$638$$ −17197.3 −1.06716
$$639$$ −18349.7 −1.13600
$$640$$ 2624.03 0.162068
$$641$$ −25332.4 −1.56095 −0.780476 0.625186i $$-0.785023\pi$$
−0.780476 + 0.625186i $$0.785023\pi$$
$$642$$ −3844.15 −0.236319
$$643$$ −813.455 −0.0498904 −0.0249452 0.999689i $$-0.507941\pi$$
−0.0249452 + 0.999689i $$0.507941\pi$$
$$644$$ 7417.18 0.453847
$$645$$ −8386.37 −0.511958
$$646$$ 6376.13 0.388337
$$647$$ 18110.9 1.10048 0.550241 0.835006i $$-0.314536\pi$$
0.550241 + 0.835006i $$0.314536\pi$$
$$648$$ −4845.48 −0.293748
$$649$$ 43458.7 2.62851
$$650$$ 0 0
$$651$$ −750.619 −0.0451906
$$652$$ 7055.50 0.423796
$$653$$ −11793.7 −0.706773 −0.353387 0.935477i $$-0.614970\pi$$
−0.353387 + 0.935477i $$0.614970\pi$$
$$654$$ 1334.63 0.0797986
$$655$$ 43357.9 2.58646
$$656$$ −1348.40 −0.0802534
$$657$$ 1454.27 0.0863568
$$658$$ −2686.66 −0.159175
$$659$$ 8729.73 0.516027 0.258014 0.966141i $$-0.416932\pi$$
0.258014 + 0.966141i $$0.416932\pi$$
$$660$$ −5420.45 −0.319683
$$661$$ 28673.9 1.68727 0.843636 0.536916i $$-0.180411\pi$$
0.843636 + 0.536916i $$0.180411\pi$$
$$662$$ 21283.2 1.24954
$$663$$ 0 0
$$664$$ −7340.80 −0.429034
$$665$$ 20069.5 1.17032
$$666$$ −5698.91 −0.331574
$$667$$ 14138.7 0.820766
$$668$$ −1588.10 −0.0919842
$$669$$ −4513.31 −0.260829
$$670$$ 3193.18 0.184124
$$671$$ 21107.9 1.21440
$$672$$ −847.361 −0.0486423
$$673$$ 13868.6 0.794346 0.397173 0.917744i $$-0.369991\pi$$
0.397173 + 0.917744i $$0.369991\pi$$
$$674$$ −1340.52 −0.0766095
$$675$$ 19292.9 1.10012
$$676$$ 0 0
$$677$$ 15320.7 0.869753 0.434876 0.900490i $$-0.356792\pi$$
0.434876 + 0.900490i $$0.356792\pi$$
$$678$$ −2486.45 −0.140843
$$679$$ 7363.50 0.416178
$$680$$ −11351.4 −0.640155
$$681$$ −7211.96 −0.405819
$$682$$ −3008.03 −0.168890
$$683$$ 7588.13 0.425112 0.212556 0.977149i $$-0.431821\pi$$
0.212556 + 0.977149i $$0.431821\pi$$
$$684$$ −4688.54 −0.262092
$$685$$ 59974.4 3.34526
$$686$$ −9957.55 −0.554200
$$687$$ 1231.33 0.0683815
$$688$$ −5253.74 −0.291129
$$689$$ 0 0
$$690$$ 4456.39 0.245872
$$691$$ −18737.6 −1.03156 −0.515782 0.856720i $$-0.672498\pi$$
−0.515782 + 0.856720i $$0.672498\pi$$
$$692$$ −49.2949 −0.00270796
$$693$$ −28698.0 −1.57309
$$694$$ −2095.88 −0.114637
$$695$$ 44340.8 2.42006
$$696$$ −1615.24 −0.0879678
$$697$$ 5833.10 0.316994
$$698$$ 16659.1 0.903374
$$699$$ 7626.47 0.412674
$$700$$ −25102.3 −1.35540
$$701$$ −20758.4 −1.11845 −0.559226 0.829015i $$-0.688902\pi$$
−0.559226 + 0.829015i $$0.688902\pi$$
$$702$$ 0 0
$$703$$ −5157.48 −0.276697
$$704$$ −3395.71 −0.181790
$$705$$ −1614.20 −0.0862330
$$706$$ −5649.27 −0.301152
$$707$$ 11037.3 0.587131
$$708$$ 4081.82 0.216673
$$709$$ −11658.7 −0.617564 −0.308782 0.951133i $$-0.599921\pi$$
−0.308782 + 0.951133i $$0.599921\pi$$
$$710$$ 29564.2 1.56271
$$711$$ 10680.6 0.563367
$$712$$ 3030.43 0.159509
$$713$$ 2473.03 0.129896
$$714$$ 3665.63 0.192133
$$715$$ 0 0
$$716$$ −1113.84 −0.0581368
$$717$$ 279.399 0.0145528
$$718$$ −17506.1 −0.909919
$$719$$ −11213.2 −0.581614 −0.290807 0.956782i $$-0.593924\pi$$
−0.290807 + 0.956782i $$0.593924\pi$$
$$720$$ 8346.98 0.432047
$$721$$ 8223.60 0.424775
$$722$$ 9474.90 0.488392
$$723$$ 7021.66 0.361187
$$724$$ −13929.1 −0.715017
$$725$$ −47850.1 −2.45119
$$726$$ 3698.04 0.189046
$$727$$ 31125.2 1.58785 0.793927 0.608014i $$-0.208033\pi$$
0.793927 + 0.608014i $$0.208033\pi$$
$$728$$ 0 0
$$729$$ −13215.4 −0.671411
$$730$$ −2343.05 −0.118795
$$731$$ 22727.3 1.14993
$$732$$ 1982.54 0.100105
$$733$$ −38636.1 −1.94687 −0.973437 0.228955i $$-0.926469\pi$$
−0.973437 + 0.228955i $$0.926469\pi$$
$$734$$ −14756.4 −0.742055
$$735$$ 2777.62 0.139393
$$736$$ 2791.76 0.139817
$$737$$ −4132.23 −0.206530
$$738$$ −4289.24 −0.213942
$$739$$ −23806.6 −1.18503 −0.592517 0.805558i $$-0.701866\pi$$
−0.592517 + 0.805558i $$0.701866\pi$$
$$740$$ 9181.83 0.456123
$$741$$ 0 0
$$742$$ 30675.8 1.51772
$$743$$ −12575.0 −0.620903 −0.310451 0.950589i $$-0.600480\pi$$
−0.310451 + 0.950589i $$0.600480\pi$$
$$744$$ −282.526 −0.0139219
$$745$$ 24300.8 1.19505
$$746$$ −5252.84 −0.257802
$$747$$ −23351.0 −1.14373
$$748$$ 14689.6 0.718055
$$749$$ 32791.0 1.59968
$$750$$ −8696.91 −0.423422
$$751$$ 17709.8 0.860506 0.430253 0.902708i $$-0.358424\pi$$
0.430253 + 0.902708i $$0.358424\pi$$
$$752$$ −1011.23 −0.0490371
$$753$$ −3010.53 −0.145697
$$754$$ 0 0
$$755$$ 36025.4 1.73656
$$756$$ −5555.28 −0.267253
$$757$$ −5976.41 −0.286943 −0.143472 0.989654i $$-0.545827\pi$$
−0.143472 + 0.989654i $$0.545827\pi$$
$$758$$ 1006.98 0.0482523
$$759$$ −5766.93 −0.275792
$$760$$ 7553.97 0.360541
$$761$$ −14868.4 −0.708249 −0.354125 0.935198i $$-0.615221\pi$$
−0.354125 + 0.935198i $$0.615221\pi$$
$$762$$ −1525.90 −0.0725426
$$763$$ −11384.6 −0.540168
$$764$$ 15137.5 0.716828
$$765$$ −36108.5 −1.70654
$$766$$ 528.957 0.0249504
$$767$$ 0 0
$$768$$ −318.939 −0.0149853
$$769$$ −6592.53 −0.309145 −0.154573 0.987981i $$-0.549400\pi$$
−0.154573 + 0.987981i $$0.549400\pi$$
$$770$$ 46237.0 2.16398
$$771$$ 1130.07 0.0527866
$$772$$ 7961.74 0.371178
$$773$$ 33862.6 1.57562 0.787810 0.615918i $$-0.211215\pi$$
0.787810 + 0.615918i $$0.211215\pi$$
$$774$$ −16712.0 −0.776101
$$775$$ −8369.59 −0.387929
$$776$$ 2771.55 0.128213
$$777$$ −2965.03 −0.136898
$$778$$ 10148.8 0.467678
$$779$$ −3881.74 −0.178534
$$780$$ 0 0
$$781$$ −38258.4 −1.75287
$$782$$ −12077.0 −0.552265
$$783$$ −10589.5 −0.483317
$$784$$ 1740.07 0.0792671
$$785$$ −10454.7 −0.475343
$$786$$ −5269.95 −0.239151
$$787$$ 15674.1 0.709939 0.354970 0.934878i $$-0.384491\pi$$
0.354970 + 0.934878i $$0.384491\pi$$
$$788$$ −16355.2 −0.739377
$$789$$ −3780.33 −0.170574
$$790$$ −17208.1 −0.774983
$$791$$ 21209.7 0.953386
$$792$$ −10801.7 −0.484622
$$793$$ 0 0
$$794$$ −3266.38 −0.145994
$$795$$ 18430.7 0.822224
$$796$$ 13845.5 0.616510
$$797$$ 18387.2 0.817200 0.408600 0.912714i $$-0.366017\pi$$
0.408600 + 0.912714i $$0.366017\pi$$
$$798$$ −2439.36 −0.108211
$$799$$ 4374.53 0.193692
$$800$$ −9448.28 −0.417559
$$801$$ 9639.76 0.425224
$$802$$ 7025.18 0.309311
$$803$$ 3032.10 0.133251
$$804$$ −388.116 −0.0170246
$$805$$ −38013.4 −1.66434
$$806$$ 0 0
$$807$$ 5042.73 0.219966
$$808$$ 4154.35 0.180878
$$809$$ 17954.5 0.780280 0.390140 0.920756i $$-0.372427\pi$$
0.390140 + 0.920756i $$0.372427\pi$$
$$810$$ 24833.3 1.07723
$$811$$ −42308.9 −1.83189 −0.915947 0.401300i $$-0.868559\pi$$
−0.915947 + 0.401300i $$0.868559\pi$$
$$812$$ 13778.2 0.595467
$$813$$ 4501.59 0.194192
$$814$$ −11882.0 −0.511628
$$815$$ −36159.8 −1.55414
$$816$$ 1379.71 0.0591905
$$817$$ −15124.3 −0.647653
$$818$$ −3004.03 −0.128403
$$819$$ 0 0
$$820$$ 6910.63 0.294305
$$821$$ 6656.89 0.282981 0.141490 0.989940i $$-0.454811\pi$$
0.141490 + 0.989940i $$0.454811\pi$$
$$822$$ −7289.62 −0.309312
$$823$$ −31429.9 −1.33120 −0.665600 0.746308i $$-0.731824\pi$$
−0.665600 + 0.746308i $$0.731824\pi$$
$$824$$ 3095.28 0.130861
$$825$$ 19517.3 0.823643
$$826$$ −34818.3 −1.46669
$$827$$ −31475.2 −1.32346 −0.661730 0.749742i $$-0.730178\pi$$
−0.661730 + 0.749742i $$0.730178\pi$$
$$828$$ 8880.52 0.372729
$$829$$ 17919.1 0.750732 0.375366 0.926877i $$-0.377517\pi$$
0.375366 + 0.926877i $$0.377517\pi$$
$$830$$ 37622.0 1.57335
$$831$$ −6052.81 −0.252671
$$832$$ 0 0
$$833$$ −7527.44 −0.313098
$$834$$ −5389.43 −0.223766
$$835$$ 8139.10 0.337324
$$836$$ −9775.45 −0.404415
$$837$$ −1852.24 −0.0764906
$$838$$ 26080.6 1.07510
$$839$$ 48376.2 1.99062 0.995310 0.0967360i $$-0.0308403\pi$$
0.995310 + 0.0967360i $$0.0308403\pi$$
$$840$$ 4342.77 0.178381
$$841$$ 1875.00 0.0768788
$$842$$ 27581.2 1.12887
$$843$$ −5694.54 −0.232657
$$844$$ −790.434 −0.0322368
$$845$$ 0 0
$$846$$ −3216.71 −0.130725
$$847$$ −31544.7 −1.27968
$$848$$ 11546.1 0.467564
$$849$$ −1080.20 −0.0436659
$$850$$ 40872.7 1.64932
$$851$$ 9768.73 0.393499
$$852$$ −3593.39 −0.144492
$$853$$ −36926.5 −1.48222 −0.741112 0.671381i $$-0.765701\pi$$
−0.741112 + 0.671381i $$0.765701\pi$$
$$854$$ −16911.3 −0.677626
$$855$$ 24029.0 0.961141
$$856$$ 12342.2 0.492814
$$857$$ 17315.0 0.690161 0.345080 0.938573i $$-0.387852\pi$$
0.345080 + 0.938573i $$0.387852\pi$$
$$858$$ 0 0
$$859$$ 37342.8 1.48326 0.741630 0.670809i $$-0.234053\pi$$
0.741630 + 0.670809i $$0.234053\pi$$
$$860$$ 26925.7 1.06763
$$861$$ −2231.61 −0.0883310
$$862$$ −26733.0 −1.05630
$$863$$ 13285.9 0.524053 0.262026 0.965061i $$-0.415609\pi$$
0.262026 + 0.965061i $$0.415609\pi$$
$$864$$ −2090.96 −0.0823330
$$865$$ 252.639 0.00993062
$$866$$ −24751.0 −0.971215
$$867$$ 152.348 0.00596770
$$868$$ 2409.98 0.0942395
$$869$$ 22268.7 0.869290
$$870$$ 8278.20 0.322595
$$871$$ 0 0
$$872$$ −4285.04 −0.166410
$$873$$ 8816.25 0.341793
$$874$$ 8036.82 0.311041
$$875$$ 74185.5 2.86620
$$876$$ 284.787 0.0109841
$$877$$ −44228.8 −1.70296 −0.851482 0.524384i $$-0.824296\pi$$
−0.851482 + 0.524384i $$0.824296\pi$$
$$878$$ 24679.6 0.948628
$$879$$ 1431.98 0.0549482
$$880$$ 17403.2 0.666660
$$881$$ −49187.9 −1.88102 −0.940511 0.339762i $$-0.889653\pi$$
−0.940511 + 0.339762i $$0.889653\pi$$
$$882$$ 5535.13 0.211312
$$883$$ −19458.9 −0.741612 −0.370806 0.928710i $$-0.620918\pi$$
−0.370806 + 0.928710i $$0.620918\pi$$
$$884$$ 0 0
$$885$$ −20919.6 −0.794580
$$886$$ 30130.7 1.14251
$$887$$ 13529.3 0.512141 0.256071 0.966658i $$-0.417572\pi$$
0.256071 + 0.966658i $$0.417572\pi$$
$$888$$ −1116.01 −0.0421743
$$889$$ 13016.1 0.491052
$$890$$ −15531.1 −0.584950
$$891$$ −32136.4 −1.20832
$$892$$ 14490.7 0.543927
$$893$$ −2911.11 −0.109089
$$894$$ −2953.65 −0.110497
$$895$$ 5708.47 0.213199
$$896$$ 2720.58 0.101438
$$897$$ 0 0
$$898$$ −8447.24 −0.313907
$$899$$ 4593.90 0.170429
$$900$$ −30054.8 −1.11314
$$901$$ −49947.7 −1.84684
$$902$$ −8942.92 −0.330118
$$903$$ −8694.94 −0.320431
$$904$$ 7983.12 0.293711
$$905$$ 71387.6 2.62210
$$906$$ −4378.73 −0.160567
$$907$$ −49365.3 −1.80722 −0.903609 0.428357i $$-0.859092\pi$$
−0.903609 + 0.428357i $$0.859092\pi$$
$$908$$ 23155.1 0.846287
$$909$$ 13214.9 0.482190
$$910$$ 0 0
$$911$$ −38465.7 −1.39893 −0.699465 0.714667i $$-0.746578\pi$$
−0.699465 + 0.714667i $$0.746578\pi$$
$$912$$ −918.150 −0.0333366
$$913$$ −48685.9 −1.76481
$$914$$ 10283.3 0.372147
$$915$$ −10160.6 −0.367104
$$916$$ −3953.36 −0.142601
$$917$$ 44953.2 1.61885
$$918$$ 9045.34 0.325208
$$919$$ −14358.4 −0.515386 −0.257693 0.966227i $$-0.582962\pi$$
−0.257693 + 0.966227i $$0.582962\pi$$
$$920$$ −14307.9 −0.512736
$$921$$ −8045.71 −0.287856
$$922$$ 23260.6 0.830852
$$923$$ 0 0
$$924$$ −5619.90 −0.200088
$$925$$ −33060.8 −1.17517
$$926$$ −2057.71 −0.0730243
$$927$$ 9846.04 0.348853
$$928$$ 5185.97 0.183446
$$929$$ −48022.5 −1.69598 −0.847992 0.530009i $$-0.822188\pi$$
−0.847992 + 0.530009i $$0.822188\pi$$
$$930$$ 1447.96 0.0510543
$$931$$ 5009.26 0.176339
$$932$$ −24485.9 −0.860582
$$933$$ −6050.59 −0.212312
$$934$$ −28178.0 −0.987164
$$935$$ −75285.0 −2.63324
$$936$$ 0 0
$$937$$ −38908.9 −1.35656 −0.678281 0.734803i $$-0.737275\pi$$
−0.678281 + 0.734803i $$0.737275\pi$$
$$938$$ 3310.67 0.115242
$$939$$ −3232.24 −0.112332
$$940$$ 5182.63 0.179828
$$941$$ 19540.6 0.676946 0.338473 0.940976i $$-0.390090\pi$$
0.338473 + 0.940976i $$0.390090\pi$$
$$942$$ 1270.72 0.0439515
$$943$$ 7352.36 0.253898
$$944$$ −13105.3 −0.451844
$$945$$ 28471.1 0.980069
$$946$$ −34844.0 −1.19754
$$947$$ −8232.59 −0.282496 −0.141248 0.989974i $$-0.545111\pi$$
−0.141248 + 0.989974i $$0.545111\pi$$
$$948$$ 2091.57 0.0716571
$$949$$ 0 0
$$950$$ −27199.4 −0.928911
$$951$$ 3351.58 0.114282
$$952$$ −11769.1 −0.400669
$$953$$ 16636.2 0.565478 0.282739 0.959197i $$-0.408757\pi$$
0.282739 + 0.959197i $$0.408757\pi$$
$$954$$ 36727.9 1.24645
$$955$$ −77580.7 −2.62874
$$956$$ −897.052 −0.0303480
$$957$$ −10712.7 −0.361851
$$958$$ −28353.4 −0.956219
$$959$$ 62181.2 2.09378
$$960$$ 1634.58 0.0549539
$$961$$ −28987.5 −0.973028
$$962$$ 0 0
$$963$$ 39260.4 1.31376
$$964$$ −22544.1 −0.753212
$$965$$ −40804.4 −1.36118
$$966$$ 4620.36 0.153890
$$967$$ 9036.53 0.300512 0.150256 0.988647i $$-0.451990\pi$$
0.150256 + 0.988647i $$0.451990\pi$$
$$968$$ −11873.1 −0.394232
$$969$$ 3971.86 0.131677
$$970$$ −14204.4 −0.470180
$$971$$ 42397.3 1.40123 0.700614 0.713540i $$-0.252909\pi$$
0.700614 + 0.713540i $$0.252909\pi$$
$$972$$ −10075.4 −0.332476
$$973$$ 45972.3 1.51470
$$974$$ 11426.4 0.375898
$$975$$ 0 0
$$976$$ −6365.25 −0.208757
$$977$$ −33074.7 −1.08307 −0.541533 0.840680i $$-0.682156\pi$$
−0.541533 + 0.840680i $$0.682156\pi$$
$$978$$ 4395.06 0.143700
$$979$$ 20098.6 0.656132
$$980$$ −8917.96 −0.290687
$$981$$ −13630.6 −0.443621
$$982$$ 29736.8 0.966334
$$983$$ 17120.9 0.555517 0.277759 0.960651i $$-0.410408\pi$$
0.277759 + 0.960651i $$0.410408\pi$$
$$984$$ −839.956 −0.0272122
$$985$$ 83821.2 2.71144
$$986$$ −22434.2 −0.724595
$$987$$ −1673.59 −0.0539727
$$988$$ 0 0
$$989$$ 28646.8 0.921046
$$990$$ 55359.2 1.77720
$$991$$ −43698.5 −1.40074 −0.700368 0.713782i $$-0.746981\pi$$
−0.700368 + 0.713782i $$0.746981\pi$$
$$992$$ 907.092 0.0290325
$$993$$ 13257.8 0.423691
$$994$$ 30652.0 0.978090
$$995$$ −70959.1 −2.26086
$$996$$ −4572.78 −0.145476
$$997$$ −2611.53 −0.0829570 −0.0414785 0.999139i $$-0.513207\pi$$
−0.0414785 + 0.999139i $$0.513207\pi$$
$$998$$ 12859.6 0.407879
$$999$$ −7316.54 −0.231717
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 338.4.a.n.1.3 6
13.2 odd 12 338.4.e.i.147.2 24
13.3 even 3 338.4.c.p.191.4 12
13.4 even 6 338.4.c.o.315.4 12
13.5 odd 4 338.4.b.h.337.9 12
13.6 odd 12 338.4.e.i.23.8 24
13.7 odd 12 338.4.e.i.23.2 24
13.8 odd 4 338.4.b.h.337.3 12
13.9 even 3 338.4.c.p.315.4 12
13.10 even 6 338.4.c.o.191.4 12
13.11 odd 12 338.4.e.i.147.8 24
13.12 even 2 338.4.a.o.1.3 yes 6

By twisted newform
Twist Min Dim Char Parity Ord Type
338.4.a.n.1.3 6 1.1 even 1 trivial
338.4.a.o.1.3 yes 6 13.12 even 2
338.4.b.h.337.3 12 13.8 odd 4
338.4.b.h.337.9 12 13.5 odd 4
338.4.c.o.191.4 12 13.10 even 6
338.4.c.o.315.4 12 13.4 even 6
338.4.c.p.191.4 12 13.3 even 3
338.4.c.p.315.4 12 13.9 even 3
338.4.e.i.23.2 24 13.7 odd 12
338.4.e.i.23.8 24 13.6 odd 12
338.4.e.i.147.2 24 13.2 odd 12
338.4.e.i.147.8 24 13.11 odd 12