# Properties

 Label 338.4.a.c.1.1 Level $338$ Weight $4$ Character 338.1 Self dual yes Analytic conductor $19.943$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 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)

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +4.00000 q^{3} +4.00000 q^{4} +18.0000 q^{5} -8.00000 q^{6} -20.0000 q^{7} -8.00000 q^{8} -11.0000 q^{9} +O(q^{10})$$ $$q-2.00000 q^{2} +4.00000 q^{3} +4.00000 q^{4} +18.0000 q^{5} -8.00000 q^{6} -20.0000 q^{7} -8.00000 q^{8} -11.0000 q^{9} -36.0000 q^{10} +48.0000 q^{11} +16.0000 q^{12} +40.0000 q^{14} +72.0000 q^{15} +16.0000 q^{16} +66.0000 q^{17} +22.0000 q^{18} +16.0000 q^{19} +72.0000 q^{20} -80.0000 q^{21} -96.0000 q^{22} +168.000 q^{23} -32.0000 q^{24} +199.000 q^{25} -152.000 q^{27} -80.0000 q^{28} +6.00000 q^{29} -144.000 q^{30} -20.0000 q^{31} -32.0000 q^{32} +192.000 q^{33} -132.000 q^{34} -360.000 q^{35} -44.0000 q^{36} -254.000 q^{37} -32.0000 q^{38} -144.000 q^{40} +390.000 q^{41} +160.000 q^{42} -124.000 q^{43} +192.000 q^{44} -198.000 q^{45} -336.000 q^{46} +468.000 q^{47} +64.0000 q^{48} +57.0000 q^{49} -398.000 q^{50} +264.000 q^{51} +558.000 q^{53} +304.000 q^{54} +864.000 q^{55} +160.000 q^{56} +64.0000 q^{57} -12.0000 q^{58} +96.0000 q^{59} +288.000 q^{60} -826.000 q^{61} +40.0000 q^{62} +220.000 q^{63} +64.0000 q^{64} -384.000 q^{66} +160.000 q^{67} +264.000 q^{68} +672.000 q^{69} +720.000 q^{70} +420.000 q^{71} +88.0000 q^{72} -362.000 q^{73} +508.000 q^{74} +796.000 q^{75} +64.0000 q^{76} -960.000 q^{77} +776.000 q^{79} +288.000 q^{80} -311.000 q^{81} -780.000 q^{82} -320.000 q^{84} +1188.00 q^{85} +248.000 q^{86} +24.0000 q^{87} -384.000 q^{88} -1626.00 q^{89} +396.000 q^{90} +672.000 q^{92} -80.0000 q^{93} -936.000 q^{94} +288.000 q^{95} -128.000 q^{96} +1294.00 q^{97} -114.000 q^{98} -528.000 q^{99} +O(q^{100})$$

## 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
$$3$$ 4.00000 0.769800 0.384900 0.922958i $$-0.374236\pi$$
0.384900 + 0.922958i $$0.374236\pi$$
$$4$$ 4.00000 0.500000
$$5$$ 18.0000 1.60997 0.804984 0.593296i $$-0.202174\pi$$
0.804984 + 0.593296i $$0.202174\pi$$
$$6$$ −8.00000 −0.544331
$$7$$ −20.0000 −1.07990 −0.539949 0.841698i $$-0.681557\pi$$
−0.539949 + 0.841698i $$0.681557\pi$$
$$8$$ −8.00000 −0.353553
$$9$$ −11.0000 −0.407407
$$10$$ −36.0000 −1.13842
$$11$$ 48.0000 1.31569 0.657843 0.753155i $$-0.271469\pi$$
0.657843 + 0.753155i $$0.271469\pi$$
$$12$$ 16.0000 0.384900
$$13$$ 0 0
$$14$$ 40.0000 0.763604
$$15$$ 72.0000 1.23935
$$16$$ 16.0000 0.250000
$$17$$ 66.0000 0.941609 0.470804 0.882238i $$-0.343964\pi$$
0.470804 + 0.882238i $$0.343964\pi$$
$$18$$ 22.0000 0.288081
$$19$$ 16.0000 0.193192 0.0965961 0.995324i $$-0.469204\pi$$
0.0965961 + 0.995324i $$0.469204\pi$$
$$20$$ 72.0000 0.804984
$$21$$ −80.0000 −0.831306
$$22$$ −96.0000 −0.930330
$$23$$ 168.000 1.52306 0.761531 0.648129i $$-0.224448\pi$$
0.761531 + 0.648129i $$0.224448\pi$$
$$24$$ −32.0000 −0.272166
$$25$$ 199.000 1.59200
$$26$$ 0 0
$$27$$ −152.000 −1.08342
$$28$$ −80.0000 −0.539949
$$29$$ 6.00000 0.0384197 0.0192099 0.999815i $$-0.493885\pi$$
0.0192099 + 0.999815i $$0.493885\pi$$
$$30$$ −144.000 −0.876356
$$31$$ −20.0000 −0.115874 −0.0579372 0.998320i $$-0.518452\pi$$
−0.0579372 + 0.998320i $$0.518452\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 192.000 1.01282
$$34$$ −132.000 −0.665818
$$35$$ −360.000 −1.73860
$$36$$ −44.0000 −0.203704
$$37$$ −254.000 −1.12858 −0.564288 0.825578i $$-0.690849\pi$$
−0.564288 + 0.825578i $$0.690849\pi$$
$$38$$ −32.0000 −0.136608
$$39$$ 0 0
$$40$$ −144.000 −0.569210
$$41$$ 390.000 1.48556 0.742778 0.669538i $$-0.233508\pi$$
0.742778 + 0.669538i $$0.233508\pi$$
$$42$$ 160.000 0.587822
$$43$$ −124.000 −0.439763 −0.219882 0.975527i $$-0.570567\pi$$
−0.219882 + 0.975527i $$0.570567\pi$$
$$44$$ 192.000 0.657843
$$45$$ −198.000 −0.655913
$$46$$ −336.000 −1.07697
$$47$$ 468.000 1.45244 0.726221 0.687461i $$-0.241275\pi$$
0.726221 + 0.687461i $$0.241275\pi$$
$$48$$ 64.0000 0.192450
$$49$$ 57.0000 0.166181
$$50$$ −398.000 −1.12571
$$51$$ 264.000 0.724851
$$52$$ 0 0
$$53$$ 558.000 1.44617 0.723087 0.690757i $$-0.242723\pi$$
0.723087 + 0.690757i $$0.242723\pi$$
$$54$$ 304.000 0.766096
$$55$$ 864.000 2.11821
$$56$$ 160.000 0.381802
$$57$$ 64.0000 0.148719
$$58$$ −12.0000 −0.0271668
$$59$$ 96.0000 0.211833 0.105916 0.994375i $$-0.466222\pi$$
0.105916 + 0.994375i $$0.466222\pi$$
$$60$$ 288.000 0.619677
$$61$$ −826.000 −1.73375 −0.866873 0.498530i $$-0.833873\pi$$
−0.866873 + 0.498530i $$0.833873\pi$$
$$62$$ 40.0000 0.0819356
$$63$$ 220.000 0.439959
$$64$$ 64.0000 0.125000
$$65$$ 0 0
$$66$$ −384.000 −0.716169
$$67$$ 160.000 0.291748 0.145874 0.989303i $$-0.453401\pi$$
0.145874 + 0.989303i $$0.453401\pi$$
$$68$$ 264.000 0.470804
$$69$$ 672.000 1.17245
$$70$$ 720.000 1.22938
$$71$$ 420.000 0.702040 0.351020 0.936368i $$-0.385835\pi$$
0.351020 + 0.936368i $$0.385835\pi$$
$$72$$ 88.0000 0.144040
$$73$$ −362.000 −0.580396 −0.290198 0.956967i $$-0.593721\pi$$
−0.290198 + 0.956967i $$0.593721\pi$$
$$74$$ 508.000 0.798024
$$75$$ 796.000 1.22552
$$76$$ 64.0000 0.0965961
$$77$$ −960.000 −1.42081
$$78$$ 0 0
$$79$$ 776.000 1.10515 0.552575 0.833463i $$-0.313645\pi$$
0.552575 + 0.833463i $$0.313645\pi$$
$$80$$ 288.000 0.402492
$$81$$ −311.000 −0.426612
$$82$$ −780.000 −1.05045
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ −320.000 −0.415653
$$85$$ 1188.00 1.51596
$$86$$ 248.000 0.310960
$$87$$ 24.0000 0.0295755
$$88$$ −384.000 −0.465165
$$89$$ −1626.00 −1.93658 −0.968290 0.249828i $$-0.919626\pi$$
−0.968290 + 0.249828i $$0.919626\pi$$
$$90$$ 396.000 0.463801
$$91$$ 0 0
$$92$$ 672.000 0.761531
$$93$$ −80.0000 −0.0892001
$$94$$ −936.000 −1.02703
$$95$$ 288.000 0.311033
$$96$$ −128.000 −0.136083
$$97$$ 1294.00 1.35449 0.677246 0.735756i $$-0.263173\pi$$
0.677246 + 0.735756i $$0.263173\pi$$
$$98$$ −114.000 −0.117508
$$99$$ −528.000 −0.536020
$$100$$ 796.000 0.796000
$$101$$ 222.000 0.218711 0.109356 0.994003i $$-0.465121\pi$$
0.109356 + 0.994003i $$0.465121\pi$$
$$102$$ −528.000 −0.512547
$$103$$ 632.000 0.604590 0.302295 0.953214i $$-0.402247\pi$$
0.302295 + 0.953214i $$0.402247\pi$$
$$104$$ 0 0
$$105$$ −1440.00 −1.33838
$$106$$ −1116.00 −1.02260
$$107$$ −948.000 −0.856510 −0.428255 0.903658i $$-0.640872\pi$$
−0.428255 + 0.903658i $$0.640872\pi$$
$$108$$ −608.000 −0.541711
$$109$$ −758.000 −0.666085 −0.333042 0.942912i $$-0.608075\pi$$
−0.333042 + 0.942912i $$0.608075\pi$$
$$110$$ −1728.00 −1.49780
$$111$$ −1016.00 −0.868779
$$112$$ −320.000 −0.269975
$$113$$ 642.000 0.534463 0.267231 0.963632i $$-0.413891\pi$$
0.267231 + 0.963632i $$0.413891\pi$$
$$114$$ −128.000 −0.105161
$$115$$ 3024.00 2.45208
$$116$$ 24.0000 0.0192099
$$117$$ 0 0
$$118$$ −192.000 −0.149788
$$119$$ −1320.00 −1.01684
$$120$$ −576.000 −0.438178
$$121$$ 973.000 0.731029
$$122$$ 1652.00 1.22594
$$123$$ 1560.00 1.14358
$$124$$ −80.0000 −0.0579372
$$125$$ 1332.00 0.953102
$$126$$ −440.000 −0.311098
$$127$$ −880.000 −0.614861 −0.307431 0.951571i $$-0.599469\pi$$
−0.307431 + 0.951571i $$0.599469\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ −496.000 −0.338530
$$130$$ 0 0
$$131$$ 324.000 0.216092 0.108046 0.994146i $$-0.465541\pi$$
0.108046 + 0.994146i $$0.465541\pi$$
$$132$$ 768.000 0.506408
$$133$$ −320.000 −0.208628
$$134$$ −320.000 −0.206297
$$135$$ −2736.00 −1.74428
$$136$$ −528.000 −0.332909
$$137$$ −1722.00 −1.07387 −0.536936 0.843623i $$-0.680418\pi$$
−0.536936 + 0.843623i $$0.680418\pi$$
$$138$$ −1344.00 −0.829050
$$139$$ −340.000 −0.207471 −0.103735 0.994605i $$-0.533079\pi$$
−0.103735 + 0.994605i $$0.533079\pi$$
$$140$$ −1440.00 −0.869302
$$141$$ 1872.00 1.11809
$$142$$ −840.000 −0.496417
$$143$$ 0 0
$$144$$ −176.000 −0.101852
$$145$$ 108.000 0.0618546
$$146$$ 724.000 0.410402
$$147$$ 228.000 0.127926
$$148$$ −1016.00 −0.564288
$$149$$ −750.000 −0.412365 −0.206183 0.978514i $$-0.566104\pi$$
−0.206183 + 0.978514i $$0.566104\pi$$
$$150$$ −1592.00 −0.866575
$$151$$ −1748.00 −0.942054 −0.471027 0.882119i $$-0.656117\pi$$
−0.471027 + 0.882119i $$0.656117\pi$$
$$152$$ −128.000 −0.0683038
$$153$$ −726.000 −0.383618
$$154$$ 1920.00 1.00466
$$155$$ −360.000 −0.186554
$$156$$ 0 0
$$157$$ 614.000 0.312118 0.156059 0.987748i $$-0.450121\pi$$
0.156059 + 0.987748i $$0.450121\pi$$
$$158$$ −1552.00 −0.781459
$$159$$ 2232.00 1.11326
$$160$$ −576.000 −0.284605
$$161$$ −3360.00 −1.64475
$$162$$ 622.000 0.301660
$$163$$ 808.000 0.388267 0.194133 0.980975i $$-0.437811\pi$$
0.194133 + 0.980975i $$0.437811\pi$$
$$164$$ 1560.00 0.742778
$$165$$ 3456.00 1.63060
$$166$$ 0 0
$$167$$ 2028.00 0.939709 0.469854 0.882744i $$-0.344306\pi$$
0.469854 + 0.882744i $$0.344306\pi$$
$$168$$ 640.000 0.293911
$$169$$ 0 0
$$170$$ −2376.00 −1.07195
$$171$$ −176.000 −0.0787079
$$172$$ −496.000 −0.219882
$$173$$ −1194.00 −0.524729 −0.262365 0.964969i $$-0.584502\pi$$
−0.262365 + 0.964969i $$0.584502\pi$$
$$174$$ −48.0000 −0.0209130
$$175$$ −3980.00 −1.71920
$$176$$ 768.000 0.328921
$$177$$ 384.000 0.163069
$$178$$ 3252.00 1.36937
$$179$$ −2820.00 −1.17752 −0.588762 0.808307i $$-0.700384\pi$$
−0.588762 + 0.808307i $$0.700384\pi$$
$$180$$ −792.000 −0.327957
$$181$$ −754.000 −0.309637 −0.154819 0.987943i $$-0.549479\pi$$
−0.154819 + 0.987943i $$0.549479\pi$$
$$182$$ 0 0
$$183$$ −3304.00 −1.33464
$$184$$ −1344.00 −0.538484
$$185$$ −4572.00 −1.81697
$$186$$ 160.000 0.0630740
$$187$$ 3168.00 1.23886
$$188$$ 1872.00 0.726221
$$189$$ 3040.00 1.16999
$$190$$ −576.000 −0.219934
$$191$$ −2328.00 −0.881928 −0.440964 0.897525i $$-0.645363\pi$$
−0.440964 + 0.897525i $$0.645363\pi$$
$$192$$ 256.000 0.0962250
$$193$$ −2450.00 −0.913756 −0.456878 0.889529i $$-0.651032\pi$$
−0.456878 + 0.889529i $$0.651032\pi$$
$$194$$ −2588.00 −0.957771
$$195$$ 0 0
$$196$$ 228.000 0.0830904
$$197$$ −4542.00 −1.64266 −0.821330 0.570453i $$-0.806768\pi$$
−0.821330 + 0.570453i $$0.806768\pi$$
$$198$$ 1056.00 0.379023
$$199$$ −664.000 −0.236531 −0.118266 0.992982i $$-0.537733\pi$$
−0.118266 + 0.992982i $$0.537733\pi$$
$$200$$ −1592.00 −0.562857
$$201$$ 640.000 0.224588
$$202$$ −444.000 −0.154652
$$203$$ −120.000 −0.0414894
$$204$$ 1056.00 0.362425
$$205$$ 7020.00 2.39170
$$206$$ −1264.00 −0.427510
$$207$$ −1848.00 −0.620507
$$208$$ 0 0
$$209$$ 768.000 0.254180
$$210$$ 2880.00 0.946376
$$211$$ −4156.00 −1.35598 −0.677988 0.735073i $$-0.737148\pi$$
−0.677988 + 0.735073i $$0.737148\pi$$
$$212$$ 2232.00 0.723087
$$213$$ 1680.00 0.540431
$$214$$ 1896.00 0.605644
$$215$$ −2232.00 −0.708005
$$216$$ 1216.00 0.383048
$$217$$ 400.000 0.125133
$$218$$ 1516.00 0.470993
$$219$$ −1448.00 −0.446789
$$220$$ 3456.00 1.05911
$$221$$ 0 0
$$222$$ 2032.00 0.614319
$$223$$ 3292.00 0.988559 0.494279 0.869303i $$-0.335432\pi$$
0.494279 + 0.869303i $$0.335432\pi$$
$$224$$ 640.000 0.190901
$$225$$ −2189.00 −0.648593
$$226$$ −1284.00 −0.377922
$$227$$ −2352.00 −0.687699 −0.343850 0.939025i $$-0.611731\pi$$
−0.343850 + 0.939025i $$0.611731\pi$$
$$228$$ 256.000 0.0743597
$$229$$ −686.000 −0.197957 −0.0989785 0.995090i $$-0.531558\pi$$
−0.0989785 + 0.995090i $$0.531558\pi$$
$$230$$ −6048.00 −1.73388
$$231$$ −3840.00 −1.09374
$$232$$ −48.0000 −0.0135834
$$233$$ 1818.00 0.511164 0.255582 0.966787i $$-0.417733\pi$$
0.255582 + 0.966787i $$0.417733\pi$$
$$234$$ 0 0
$$235$$ 8424.00 2.33839
$$236$$ 384.000 0.105916
$$237$$ 3104.00 0.850745
$$238$$ 2640.00 0.719016
$$239$$ 540.000 0.146149 0.0730747 0.997326i $$-0.476719\pi$$
0.0730747 + 0.997326i $$0.476719\pi$$
$$240$$ 1152.00 0.309839
$$241$$ 862.000 0.230400 0.115200 0.993342i $$-0.463249\pi$$
0.115200 + 0.993342i $$0.463249\pi$$
$$242$$ −1946.00 −0.516916
$$243$$ 2860.00 0.755017
$$244$$ −3304.00 −0.866873
$$245$$ 1026.00 0.267546
$$246$$ −3120.00 −0.808634
$$247$$ 0 0
$$248$$ 160.000 0.0409678
$$249$$ 0 0
$$250$$ −2664.00 −0.673945
$$251$$ 4836.00 1.21612 0.608059 0.793892i $$-0.291948\pi$$
0.608059 + 0.793892i $$0.291948\pi$$
$$252$$ 880.000 0.219979
$$253$$ 8064.00 2.00387
$$254$$ 1760.00 0.434773
$$255$$ 4752.00 1.16699
$$256$$ 256.000 0.0625000
$$257$$ 1410.00 0.342231 0.171116 0.985251i $$-0.445263\pi$$
0.171116 + 0.985251i $$0.445263\pi$$
$$258$$ 992.000 0.239377
$$259$$ 5080.00 1.21875
$$260$$ 0 0
$$261$$ −66.0000 −0.0156525
$$262$$ −648.000 −0.152800
$$263$$ 8304.00 1.94695 0.973473 0.228804i $$-0.0734814\pi$$
0.973473 + 0.228804i $$0.0734814\pi$$
$$264$$ −1536.00 −0.358084
$$265$$ 10044.0 2.32829
$$266$$ 640.000 0.147522
$$267$$ −6504.00 −1.49078
$$268$$ 640.000 0.145874
$$269$$ −2634.00 −0.597018 −0.298509 0.954407i $$-0.596489\pi$$
−0.298509 + 0.954407i $$0.596489\pi$$
$$270$$ 5472.00 1.23339
$$271$$ −7436.00 −1.66681 −0.833404 0.552665i $$-0.813611\pi$$
−0.833404 + 0.552665i $$0.813611\pi$$
$$272$$ 1056.00 0.235402
$$273$$ 0 0
$$274$$ 3444.00 0.759342
$$275$$ 9552.00 2.09457
$$276$$ 2688.00 0.586227
$$277$$ −5074.00 −1.10060 −0.550302 0.834966i $$-0.685487\pi$$
−0.550302 + 0.834966i $$0.685487\pi$$
$$278$$ 680.000 0.146704
$$279$$ 220.000 0.0472081
$$280$$ 2880.00 0.614689
$$281$$ 1638.00 0.347740 0.173870 0.984769i $$-0.444373\pi$$
0.173870 + 0.984769i $$0.444373\pi$$
$$282$$ −3744.00 −0.790610
$$283$$ −4588.00 −0.963704 −0.481852 0.876253i $$-0.660036\pi$$
−0.481852 + 0.876253i $$0.660036\pi$$
$$284$$ 1680.00 0.351020
$$285$$ 1152.00 0.239434
$$286$$ 0 0
$$287$$ −7800.00 −1.60425
$$288$$ 352.000 0.0720201
$$289$$ −557.000 −0.113373
$$290$$ −216.000 −0.0437378
$$291$$ 5176.00 1.04269
$$292$$ −1448.00 −0.290198
$$293$$ 2850.00 0.568255 0.284128 0.958786i $$-0.408296\pi$$
0.284128 + 0.958786i $$0.408296\pi$$
$$294$$ −456.000 −0.0904573
$$295$$ 1728.00 0.341044
$$296$$ 2032.00 0.399012
$$297$$ −7296.00 −1.42544
$$298$$ 1500.00 0.291586
$$299$$ 0 0
$$300$$ 3184.00 0.612761
$$301$$ 2480.00 0.474900
$$302$$ 3496.00 0.666133
$$303$$ 888.000 0.168364
$$304$$ 256.000 0.0482980
$$305$$ −14868.0 −2.79128
$$306$$ 1452.00 0.271259
$$307$$ −8120.00 −1.50955 −0.754777 0.655982i $$-0.772255\pi$$
−0.754777 + 0.655982i $$0.772255\pi$$
$$308$$ −3840.00 −0.710404
$$309$$ 2528.00 0.465414
$$310$$ 720.000 0.131914
$$311$$ −3528.00 −0.643262 −0.321631 0.946865i $$-0.604231\pi$$
−0.321631 + 0.946865i $$0.604231\pi$$
$$312$$ 0 0
$$313$$ −6982.00 −1.26085 −0.630425 0.776250i $$-0.717119\pi$$
−0.630425 + 0.776250i $$0.717119\pi$$
$$314$$ −1228.00 −0.220701
$$315$$ 3960.00 0.708320
$$316$$ 3104.00 0.552575
$$317$$ −9270.00 −1.64245 −0.821223 0.570608i $$-0.806708\pi$$
−0.821223 + 0.570608i $$0.806708\pi$$
$$318$$ −4464.00 −0.787197
$$319$$ 288.000 0.0505483
$$320$$ 1152.00 0.201246
$$321$$ −3792.00 −0.659342
$$322$$ 6720.00 1.16302
$$323$$ 1056.00 0.181911
$$324$$ −1244.00 −0.213306
$$325$$ 0 0
$$326$$ −1616.00 −0.274546
$$327$$ −3032.00 −0.512752
$$328$$ −3120.00 −0.525223
$$329$$ −9360.00 −1.56849
$$330$$ −6912.00 −1.15301
$$331$$ 7720.00 1.28196 0.640981 0.767557i $$-0.278528\pi$$
0.640981 + 0.767557i $$0.278528\pi$$
$$332$$ 0 0
$$333$$ 2794.00 0.459791
$$334$$ −4056.00 −0.664474
$$335$$ 2880.00 0.469705
$$336$$ −1280.00 −0.207827
$$337$$ −1726.00 −0.278995 −0.139497 0.990222i $$-0.544549\pi$$
−0.139497 + 0.990222i $$0.544549\pi$$
$$338$$ 0 0
$$339$$ 2568.00 0.411430
$$340$$ 4752.00 0.757981
$$341$$ −960.000 −0.152454
$$342$$ 352.000 0.0556549
$$343$$ 5720.00 0.900440
$$344$$ 992.000 0.155480
$$345$$ 12096.0 1.88761
$$346$$ 2388.00 0.371040
$$347$$ −4020.00 −0.621916 −0.310958 0.950424i $$-0.600650\pi$$
−0.310958 + 0.950424i $$0.600650\pi$$
$$348$$ 96.0000 0.0147878
$$349$$ −1910.00 −0.292951 −0.146476 0.989214i $$-0.546793\pi$$
−0.146476 + 0.989214i $$0.546793\pi$$
$$350$$ 7960.00 1.21566
$$351$$ 0 0
$$352$$ −1536.00 −0.232583
$$353$$ −5442.00 −0.820534 −0.410267 0.911965i $$-0.634564\pi$$
−0.410267 + 0.911965i $$0.634564\pi$$
$$354$$ −768.000 −0.115307
$$355$$ 7560.00 1.13026
$$356$$ −6504.00 −0.968290
$$357$$ −5280.00 −0.782765
$$358$$ 5640.00 0.832635
$$359$$ 9324.00 1.37076 0.685379 0.728187i $$-0.259637\pi$$
0.685379 + 0.728187i $$0.259637\pi$$
$$360$$ 1584.00 0.231900
$$361$$ −6603.00 −0.962677
$$362$$ 1508.00 0.218947
$$363$$ 3892.00 0.562747
$$364$$ 0 0
$$365$$ −6516.00 −0.934419
$$366$$ 6608.00 0.943731
$$367$$ 4520.00 0.642894 0.321447 0.946928i $$-0.395831\pi$$
0.321447 + 0.946928i $$0.395831\pi$$
$$368$$ 2688.00 0.380765
$$369$$ −4290.00 −0.605226
$$370$$ 9144.00 1.28479
$$371$$ −11160.0 −1.56172
$$372$$ −320.000 −0.0446001
$$373$$ −5938.00 −0.824284 −0.412142 0.911120i $$-0.635219\pi$$
−0.412142 + 0.911120i $$0.635219\pi$$
$$374$$ −6336.00 −0.876007
$$375$$ 5328.00 0.733698
$$376$$ −3744.00 −0.513516
$$377$$ 0 0
$$378$$ −6080.00 −0.827305
$$379$$ −2216.00 −0.300338 −0.150169 0.988660i $$-0.547982\pi$$
−0.150169 + 0.988660i $$0.547982\pi$$
$$380$$ 1152.00 0.155517
$$381$$ −3520.00 −0.473320
$$382$$ 4656.00 0.623617
$$383$$ −3828.00 −0.510709 −0.255355 0.966847i $$-0.582192\pi$$
−0.255355 + 0.966847i $$0.582192\pi$$
$$384$$ −512.000 −0.0680414
$$385$$ −17280.0 −2.28746
$$386$$ 4900.00 0.646123
$$387$$ 1364.00 0.179163
$$388$$ 5176.00 0.677246
$$389$$ 5022.00 0.654564 0.327282 0.944927i $$-0.393867\pi$$
0.327282 + 0.944927i $$0.393867\pi$$
$$390$$ 0 0
$$391$$ 11088.0 1.43413
$$392$$ −456.000 −0.0587538
$$393$$ 1296.00 0.166347
$$394$$ 9084.00 1.16154
$$395$$ 13968.0 1.77926
$$396$$ −2112.00 −0.268010
$$397$$ −6086.00 −0.769389 −0.384695 0.923044i $$-0.625693\pi$$
−0.384695 + 0.923044i $$0.625693\pi$$
$$398$$ 1328.00 0.167253
$$399$$ −1280.00 −0.160602
$$400$$ 3184.00 0.398000
$$401$$ −1122.00 −0.139726 −0.0698629 0.997557i $$-0.522256\pi$$
−0.0698629 + 0.997557i $$0.522256\pi$$
$$402$$ −1280.00 −0.158807
$$403$$ 0 0
$$404$$ 888.000 0.109356
$$405$$ −5598.00 −0.686832
$$406$$ 240.000 0.0293374
$$407$$ −12192.0 −1.48485
$$408$$ −2112.00 −0.256273
$$409$$ −362.000 −0.0437647 −0.0218823 0.999761i $$-0.506966\pi$$
−0.0218823 + 0.999761i $$0.506966\pi$$
$$410$$ −14040.0 −1.69119
$$411$$ −6888.00 −0.826667
$$412$$ 2528.00 0.302295
$$413$$ −1920.00 −0.228758
$$414$$ 3696.00 0.438764
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −1360.00 −0.159711
$$418$$ −1536.00 −0.179733
$$419$$ −2316.00 −0.270033 −0.135017 0.990843i $$-0.543109\pi$$
−0.135017 + 0.990843i $$0.543109\pi$$
$$420$$ −5760.00 −0.669189
$$421$$ −5006.00 −0.579519 −0.289760 0.957099i $$-0.593575\pi$$
−0.289760 + 0.957099i $$0.593575\pi$$
$$422$$ 8312.00 0.958820
$$423$$ −5148.00 −0.591736
$$424$$ −4464.00 −0.511300
$$425$$ 13134.0 1.49904
$$426$$ −3360.00 −0.382142
$$427$$ 16520.0 1.87227
$$428$$ −3792.00 −0.428255
$$429$$ 0 0
$$430$$ 4464.00 0.500635
$$431$$ 11244.0 1.25662 0.628311 0.777962i $$-0.283746\pi$$
0.628311 + 0.777962i $$0.283746\pi$$
$$432$$ −2432.00 −0.270856
$$433$$ 13106.0 1.45458 0.727291 0.686329i $$-0.240779\pi$$
0.727291 + 0.686329i $$0.240779\pi$$
$$434$$ −800.000 −0.0884821
$$435$$ 432.000 0.0476157
$$436$$ −3032.00 −0.333042
$$437$$ 2688.00 0.294244
$$438$$ 2896.00 0.315927
$$439$$ −13480.0 −1.46552 −0.732762 0.680485i $$-0.761769\pi$$
−0.732762 + 0.680485i $$0.761769\pi$$
$$440$$ −6912.00 −0.748902
$$441$$ −627.000 −0.0677033
$$442$$ 0 0
$$443$$ 14508.0 1.55597 0.777986 0.628281i $$-0.216241\pi$$
0.777986 + 0.628281i $$0.216241\pi$$
$$444$$ −4064.00 −0.434389
$$445$$ −29268.0 −3.11783
$$446$$ −6584.00 −0.699017
$$447$$ −3000.00 −0.317439
$$448$$ −1280.00 −0.134987
$$449$$ 7566.00 0.795237 0.397619 0.917551i $$-0.369837\pi$$
0.397619 + 0.917551i $$0.369837\pi$$
$$450$$ 4378.00 0.458624
$$451$$ 18720.0 1.95452
$$452$$ 2568.00 0.267231
$$453$$ −6992.00 −0.725194
$$454$$ 4704.00 0.486277
$$455$$ 0 0
$$456$$ −512.000 −0.0525803
$$457$$ −5402.00 −0.552943 −0.276471 0.961022i $$-0.589165\pi$$
−0.276471 + 0.961022i $$0.589165\pi$$
$$458$$ 1372.00 0.139977
$$459$$ −10032.0 −1.02016
$$460$$ 12096.0 1.22604
$$461$$ 4650.00 0.469788 0.234894 0.972021i $$-0.424526\pi$$
0.234894 + 0.972021i $$0.424526\pi$$
$$462$$ 7680.00 0.773389
$$463$$ 17188.0 1.72526 0.862629 0.505838i $$-0.168817\pi$$
0.862629 + 0.505838i $$0.168817\pi$$
$$464$$ 96.0000 0.00960493
$$465$$ −1440.00 −0.143609
$$466$$ −3636.00 −0.361447
$$467$$ −14580.0 −1.44472 −0.722358 0.691520i $$-0.756941\pi$$
−0.722358 + 0.691520i $$0.756941\pi$$
$$468$$ 0 0
$$469$$ −3200.00 −0.315058
$$470$$ −16848.0 −1.65349
$$471$$ 2456.00 0.240269
$$472$$ −768.000 −0.0748942
$$473$$ −5952.00 −0.578590
$$474$$ −6208.00 −0.601567
$$475$$ 3184.00 0.307562
$$476$$ −5280.00 −0.508421
$$477$$ −6138.00 −0.589182
$$478$$ −1080.00 −0.103343
$$479$$ 2100.00 0.200316 0.100158 0.994972i $$-0.468065\pi$$
0.100158 + 0.994972i $$0.468065\pi$$
$$480$$ −2304.00 −0.219089
$$481$$ 0 0
$$482$$ −1724.00 −0.162917
$$483$$ −13440.0 −1.26613
$$484$$ 3892.00 0.365515
$$485$$ 23292.0 2.18069
$$486$$ −5720.00 −0.533878
$$487$$ 12004.0 1.11695 0.558473 0.829522i $$-0.311387\pi$$
0.558473 + 0.829522i $$0.311387\pi$$
$$488$$ 6608.00 0.612972
$$489$$ 3232.00 0.298888
$$490$$ −2052.00 −0.189183
$$491$$ 13236.0 1.21656 0.608281 0.793721i $$-0.291859\pi$$
0.608281 + 0.793721i $$0.291859\pi$$
$$492$$ 6240.00 0.571791
$$493$$ 396.000 0.0361764
$$494$$ 0 0
$$495$$ −9504.00 −0.862976
$$496$$ −320.000 −0.0289686
$$497$$ −8400.00 −0.758132
$$498$$ 0 0
$$499$$ −18560.0 −1.66505 −0.832525 0.553988i $$-0.813105\pi$$
−0.832525 + 0.553988i $$0.813105\pi$$
$$500$$ 5328.00 0.476551
$$501$$ 8112.00 0.723388
$$502$$ −9672.00 −0.859925
$$503$$ 12432.0 1.10202 0.551009 0.834499i $$-0.314243\pi$$
0.551009 + 0.834499i $$0.314243\pi$$
$$504$$ −1760.00 −0.155549
$$505$$ 3996.00 0.352118
$$506$$ −16128.0 −1.41695
$$507$$ 0 0
$$508$$ −3520.00 −0.307431
$$509$$ 7914.00 0.689159 0.344579 0.938757i $$-0.388022\pi$$
0.344579 + 0.938757i $$0.388022\pi$$
$$510$$ −9504.00 −0.825185
$$511$$ 7240.00 0.626769
$$512$$ −512.000 −0.0441942
$$513$$ −2432.00 −0.209309
$$514$$ −2820.00 −0.241994
$$515$$ 11376.0 0.973372
$$516$$ −1984.00 −0.169265
$$517$$ 22464.0 1.91096
$$518$$ −10160.0 −0.861785
$$519$$ −4776.00 −0.403937
$$520$$ 0 0
$$521$$ −14742.0 −1.23965 −0.619826 0.784739i $$-0.712797\pi$$
−0.619826 + 0.784739i $$0.712797\pi$$
$$522$$ 132.000 0.0110680
$$523$$ −2500.00 −0.209020 −0.104510 0.994524i $$-0.533327\pi$$
−0.104510 + 0.994524i $$0.533327\pi$$
$$524$$ 1296.00 0.108046
$$525$$ −15920.0 −1.32344
$$526$$ −16608.0 −1.37670
$$527$$ −1320.00 −0.109108
$$528$$ 3072.00 0.253204
$$529$$ 16057.0 1.31972
$$530$$ −20088.0 −1.64635
$$531$$ −1056.00 −0.0863023
$$532$$ −1280.00 −0.104314
$$533$$ 0 0
$$534$$ 13008.0 1.05414
$$535$$ −17064.0 −1.37896
$$536$$ −1280.00 −0.103148
$$537$$ −11280.0 −0.906458
$$538$$ 5268.00 0.422155
$$539$$ 2736.00 0.218642
$$540$$ −10944.0 −0.872138
$$541$$ −17894.0 −1.42204 −0.711020 0.703172i $$-0.751766\pi$$
−0.711020 + 0.703172i $$0.751766\pi$$
$$542$$ 14872.0 1.17861
$$543$$ −3016.00 −0.238359
$$544$$ −2112.00 −0.166455
$$545$$ −13644.0 −1.07238
$$546$$ 0 0
$$547$$ 17444.0 1.36353 0.681766 0.731571i $$-0.261212\pi$$
0.681766 + 0.731571i $$0.261212\pi$$
$$548$$ −6888.00 −0.536936
$$549$$ 9086.00 0.706341
$$550$$ −19104.0 −1.48109
$$551$$ 96.0000 0.00742239
$$552$$ −5376.00 −0.414525
$$553$$ −15520.0 −1.19345
$$554$$ 10148.0 0.778244
$$555$$ −18288.0 −1.39871
$$556$$ −1360.00 −0.103735
$$557$$ 1002.00 0.0762228 0.0381114 0.999273i $$-0.487866\pi$$
0.0381114 + 0.999273i $$0.487866\pi$$
$$558$$ −440.000 −0.0333812
$$559$$ 0 0
$$560$$ −5760.00 −0.434651
$$561$$ 12672.0 0.953676
$$562$$ −3276.00 −0.245889
$$563$$ −4740.00 −0.354826 −0.177413 0.984136i $$-0.556773\pi$$
−0.177413 + 0.984136i $$0.556773\pi$$
$$564$$ 7488.00 0.559046
$$565$$ 11556.0 0.860468
$$566$$ 9176.00 0.681442
$$567$$ 6220.00 0.460697
$$568$$ −3360.00 −0.248209
$$569$$ 8682.00 0.639663 0.319832 0.947474i $$-0.396374\pi$$
0.319832 + 0.947474i $$0.396374\pi$$
$$570$$ −2304.00 −0.169305
$$571$$ 5492.00 0.402510 0.201255 0.979539i $$-0.435498\pi$$
0.201255 + 0.979539i $$0.435498\pi$$
$$572$$ 0 0
$$573$$ −9312.00 −0.678908
$$574$$ 15600.0 1.13438
$$575$$ 33432.0 2.42471
$$576$$ −704.000 −0.0509259
$$577$$ 17278.0 1.24661 0.623304 0.781980i $$-0.285790\pi$$
0.623304 + 0.781980i $$0.285790\pi$$
$$578$$ 1114.00 0.0801666
$$579$$ −9800.00 −0.703410
$$580$$ 432.000 0.0309273
$$581$$ 0 0
$$582$$ −10352.0 −0.737292
$$583$$ 26784.0 1.90271
$$584$$ 2896.00 0.205201
$$585$$ 0 0
$$586$$ −5700.00 −0.401817
$$587$$ 15240.0 1.07159 0.535794 0.844349i $$-0.320012\pi$$
0.535794 + 0.844349i $$0.320012\pi$$
$$588$$ 912.000 0.0639630
$$589$$ −320.000 −0.0223860
$$590$$ −3456.00 −0.241155
$$591$$ −18168.0 −1.26452
$$592$$ −4064.00 −0.282144
$$593$$ 9198.00 0.636959 0.318479 0.947930i $$-0.396828\pi$$
0.318479 + 0.947930i $$0.396828\pi$$
$$594$$ 14592.0 1.00794
$$595$$ −23760.0 −1.63708
$$596$$ −3000.00 −0.206183
$$597$$ −2656.00 −0.182082
$$598$$ 0 0
$$599$$ 7200.00 0.491125 0.245563 0.969381i $$-0.421027\pi$$
0.245563 + 0.969381i $$0.421027\pi$$
$$600$$ −6368.00 −0.433288
$$601$$ −14470.0 −0.982103 −0.491051 0.871131i $$-0.663387\pi$$
−0.491051 + 0.871131i $$0.663387\pi$$
$$602$$ −4960.00 −0.335805
$$603$$ −1760.00 −0.118860
$$604$$ −6992.00 −0.471027
$$605$$ 17514.0 1.17693
$$606$$ −1776.00 −0.119051
$$607$$ −20824.0 −1.39245 −0.696227 0.717821i $$-0.745139\pi$$
−0.696227 + 0.717821i $$0.745139\pi$$
$$608$$ −512.000 −0.0341519
$$609$$ −480.000 −0.0319386
$$610$$ 29736.0 1.97373
$$611$$ 0 0
$$612$$ −2904.00 −0.191809
$$613$$ −8606.00 −0.567036 −0.283518 0.958967i $$-0.591502\pi$$
−0.283518 + 0.958967i $$0.591502\pi$$
$$614$$ 16240.0 1.06742
$$615$$ 28080.0 1.84113
$$616$$ 7680.00 0.502331
$$617$$ 9654.00 0.629912 0.314956 0.949106i $$-0.398010\pi$$
0.314956 + 0.949106i $$0.398010\pi$$
$$618$$ −5056.00 −0.329097
$$619$$ −14384.0 −0.933993 −0.466997 0.884259i $$-0.654664\pi$$
−0.466997 + 0.884259i $$0.654664\pi$$
$$620$$ −1440.00 −0.0932771
$$621$$ −25536.0 −1.65012
$$622$$ 7056.00 0.454855
$$623$$ 32520.0 2.09131
$$624$$ 0 0
$$625$$ −899.000 −0.0575360
$$626$$ 13964.0 0.891555
$$627$$ 3072.00 0.195668
$$628$$ 2456.00 0.156059
$$629$$ −16764.0 −1.06268
$$630$$ −7920.00 −0.500858
$$631$$ −1460.00 −0.0921104 −0.0460552 0.998939i $$-0.514665\pi$$
−0.0460552 + 0.998939i $$0.514665\pi$$
$$632$$ −6208.00 −0.390729
$$633$$ −16624.0 −1.04383
$$634$$ 18540.0 1.16138
$$635$$ −15840.0 −0.989907
$$636$$ 8928.00 0.556632
$$637$$ 0 0
$$638$$ −576.000 −0.0357430
$$639$$ −4620.00 −0.286016
$$640$$ −2304.00 −0.142302
$$641$$ −12462.0 −0.767893 −0.383946 0.923355i $$-0.625435\pi$$
−0.383946 + 0.923355i $$0.625435\pi$$
$$642$$ 7584.00 0.466225
$$643$$ 9952.00 0.610371 0.305186 0.952293i $$-0.401282\pi$$
0.305186 + 0.952293i $$0.401282\pi$$
$$644$$ −13440.0 −0.822376
$$645$$ −8928.00 −0.545023
$$646$$ −2112.00 −0.128631
$$647$$ 26088.0 1.58520 0.792601 0.609741i $$-0.208727\pi$$
0.792601 + 0.609741i $$0.208727\pi$$
$$648$$ 2488.00 0.150830
$$649$$ 4608.00 0.278705
$$650$$ 0 0
$$651$$ 1600.00 0.0963271
$$652$$ 3232.00 0.194133
$$653$$ 3894.00 0.233360 0.116680 0.993170i $$-0.462775\pi$$
0.116680 + 0.993170i $$0.462775\pi$$
$$654$$ 6064.00 0.362571
$$655$$ 5832.00 0.347901
$$656$$ 6240.00 0.371389
$$657$$ 3982.00 0.236458
$$658$$ 18720.0 1.10909
$$659$$ −23820.0 −1.40804 −0.704018 0.710182i $$-0.748612\pi$$
−0.704018 + 0.710182i $$0.748612\pi$$
$$660$$ 13824.0 0.815301
$$661$$ −7742.00 −0.455566 −0.227783 0.973712i $$-0.573148\pi$$
−0.227783 + 0.973712i $$0.573148\pi$$
$$662$$ −15440.0 −0.906484
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −5760.00 −0.335885
$$666$$ −5588.00 −0.325121
$$667$$ 1008.00 0.0585156
$$668$$ 8112.00 0.469854
$$669$$ 13168.0 0.760993
$$670$$ −5760.00 −0.332132
$$671$$ −39648.0 −2.28106
$$672$$ 2560.00 0.146956
$$673$$ 21170.0 1.21255 0.606273 0.795257i $$-0.292664\pi$$
0.606273 + 0.795257i $$0.292664\pi$$
$$674$$ 3452.00 0.197279
$$675$$ −30248.0 −1.72481
$$676$$ 0 0
$$677$$ 17982.0 1.02083 0.510417 0.859927i $$-0.329491\pi$$
0.510417 + 0.859927i $$0.329491\pi$$
$$678$$ −5136.00 −0.290925
$$679$$ −25880.0 −1.46271
$$680$$ −9504.00 −0.535973
$$681$$ −9408.00 −0.529391
$$682$$ 1920.00 0.107801
$$683$$ 17520.0 0.981529 0.490764 0.871292i $$-0.336718\pi$$
0.490764 + 0.871292i $$0.336718\pi$$
$$684$$ −704.000 −0.0393540
$$685$$ −30996.0 −1.72890
$$686$$ −11440.0 −0.636707
$$687$$ −2744.00 −0.152387
$$688$$ −1984.00 −0.109941
$$689$$ 0 0
$$690$$ −24192.0 −1.33474
$$691$$ 28096.0 1.54678 0.773388 0.633933i $$-0.218560\pi$$
0.773388 + 0.633933i $$0.218560\pi$$
$$692$$ −4776.00 −0.262365
$$693$$ 10560.0 0.578847
$$694$$ 8040.00 0.439761
$$695$$ −6120.00 −0.334021
$$696$$ −192.000 −0.0104565
$$697$$ 25740.0 1.39881
$$698$$ 3820.00 0.207148
$$699$$ 7272.00 0.393494
$$700$$ −15920.0 −0.859599
$$701$$ 18342.0 0.988256 0.494128 0.869389i $$-0.335487\pi$$
0.494128 + 0.869389i $$0.335487\pi$$
$$702$$ 0 0
$$703$$ −4064.00 −0.218032
$$704$$ 3072.00 0.164461
$$705$$ 33696.0 1.80009
$$706$$ 10884.0 0.580205
$$707$$ −4440.00 −0.236186
$$708$$ 1536.00 0.0815345
$$709$$ 37330.0 1.97737 0.988687 0.149996i $$-0.0479261\pi$$
0.988687 + 0.149996i $$0.0479261\pi$$
$$710$$ −15120.0 −0.799216
$$711$$ −8536.00 −0.450246
$$712$$ 13008.0 0.684685
$$713$$ −3360.00 −0.176484
$$714$$ 10560.0 0.553499
$$715$$ 0 0
$$716$$ −11280.0 −0.588762
$$717$$ 2160.00 0.112506
$$718$$ −18648.0 −0.969272
$$719$$ 4800.00 0.248971 0.124485 0.992221i $$-0.460272\pi$$
0.124485 + 0.992221i $$0.460272\pi$$
$$720$$ −3168.00 −0.163978
$$721$$ −12640.0 −0.652896
$$722$$ 13206.0 0.680715
$$723$$ 3448.00 0.177362
$$724$$ −3016.00 −0.154819
$$725$$ 1194.00 0.0611642
$$726$$ −7784.00 −0.397922
$$727$$ 23960.0 1.22232 0.611160 0.791507i $$-0.290703\pi$$
0.611160 + 0.791507i $$0.290703\pi$$
$$728$$ 0 0
$$729$$ 19837.0 1.00782
$$730$$ 13032.0 0.660734
$$731$$ −8184.00 −0.414085
$$732$$ −13216.0 −0.667319
$$733$$ 21418.0 1.07925 0.539626 0.841905i $$-0.318566\pi$$
0.539626 + 0.841905i $$0.318566\pi$$
$$734$$ −9040.00 −0.454595
$$735$$ 4104.00 0.205957
$$736$$ −5376.00 −0.269242
$$737$$ 7680.00 0.383849
$$738$$ 8580.00 0.427960
$$739$$ −5384.00 −0.268002 −0.134001 0.990981i $$-0.542783\pi$$
−0.134001 + 0.990981i $$0.542783\pi$$
$$740$$ −18288.0 −0.908487
$$741$$ 0 0
$$742$$ 22320.0 1.10430
$$743$$ 1524.00 0.0752492 0.0376246 0.999292i $$-0.488021\pi$$
0.0376246 + 0.999292i $$0.488021\pi$$
$$744$$ 640.000 0.0315370
$$745$$ −13500.0 −0.663895
$$746$$ 11876.0 0.582857
$$747$$ 0 0
$$748$$ 12672.0 0.619431
$$749$$ 18960.0 0.924944
$$750$$ −10656.0 −0.518803
$$751$$ −19312.0 −0.938355 −0.469178 0.883104i $$-0.655450\pi$$
−0.469178 + 0.883104i $$0.655450\pi$$
$$752$$ 7488.00 0.363111
$$753$$ 19344.0 0.936168
$$754$$ 0 0
$$755$$ −31464.0 −1.51668
$$756$$ 12160.0 0.584993
$$757$$ 35246.0 1.69226 0.846128 0.532980i $$-0.178928\pi$$
0.846128 + 0.532980i $$0.178928\pi$$
$$758$$ 4432.00 0.212371
$$759$$ 32256.0 1.54258
$$760$$ −2304.00 −0.109967
$$761$$ −12522.0 −0.596481 −0.298241 0.954491i $$-0.596400\pi$$
−0.298241 + 0.954491i $$0.596400\pi$$
$$762$$ 7040.00 0.334688
$$763$$ 15160.0 0.719304
$$764$$ −9312.00 −0.440964
$$765$$ −13068.0 −0.617614
$$766$$ 7656.00 0.361126
$$767$$ 0 0
$$768$$ 1024.00 0.0481125
$$769$$ −24050.0 −1.12778 −0.563892 0.825849i $$-0.690696\pi$$
−0.563892 + 0.825849i $$0.690696\pi$$
$$770$$ 34560.0 1.61748
$$771$$ 5640.00 0.263450
$$772$$ −9800.00 −0.456878
$$773$$ −25806.0 −1.20075 −0.600373 0.799720i $$-0.704981\pi$$
−0.600373 + 0.799720i $$0.704981\pi$$
$$774$$ −2728.00 −0.126687
$$775$$ −3980.00 −0.184472
$$776$$ −10352.0 −0.478885
$$777$$ 20320.0 0.938193
$$778$$ −10044.0 −0.462847
$$779$$ 6240.00 0.286998
$$780$$ 0 0
$$781$$ 20160.0 0.923664
$$782$$ −22176.0 −1.01408
$$783$$ −912.000 −0.0416248
$$784$$ 912.000 0.0415452
$$785$$ 11052.0 0.502500
$$786$$ −2592.00 −0.117625
$$787$$ −18632.0 −0.843912 −0.421956 0.906616i $$-0.638656\pi$$
−0.421956 + 0.906616i $$0.638656\pi$$
$$788$$ −18168.0 −0.821330
$$789$$ 33216.0 1.49876
$$790$$ −27936.0 −1.25812
$$791$$ −12840.0 −0.577165
$$792$$ 4224.00 0.189512
$$793$$ 0 0
$$794$$ 12172.0 0.544040
$$795$$ 40176.0 1.79232
$$796$$ −2656.00 −0.118266
$$797$$ −16314.0 −0.725058 −0.362529 0.931972i $$-0.618087\pi$$
−0.362529 + 0.931972i $$0.618087\pi$$
$$798$$ 2560.00 0.113563
$$799$$ 30888.0 1.36763
$$800$$ −6368.00 −0.281428
$$801$$ 17886.0 0.788977
$$802$$ 2244.00 0.0988010
$$803$$ −17376.0 −0.763619
$$804$$ 2560.00 0.112294
$$805$$ −60480.0 −2.64800
$$806$$ 0 0
$$807$$ −10536.0 −0.459585
$$808$$ −1776.00 −0.0773261
$$809$$ −4278.00 −0.185917 −0.0929583 0.995670i $$-0.529632\pi$$
−0.0929583 + 0.995670i $$0.529632\pi$$
$$810$$ 11196.0 0.485663
$$811$$ −18632.0 −0.806730 −0.403365 0.915039i $$-0.632159\pi$$
−0.403365 + 0.915039i $$0.632159\pi$$
$$812$$ −480.000 −0.0207447
$$813$$ −29744.0 −1.28311
$$814$$ 24384.0 1.04995
$$815$$ 14544.0 0.625097
$$816$$ 4224.00 0.181213
$$817$$ −1984.00 −0.0849588
$$818$$ 724.000 0.0309463
$$819$$ 0 0
$$820$$ 28080.0 1.19585
$$821$$ 46434.0 1.97388 0.986941 0.161080i $$-0.0514976\pi$$
0.986941 + 0.161080i $$0.0514976\pi$$
$$822$$ 13776.0 0.584542
$$823$$ 24968.0 1.05751 0.528754 0.848775i $$-0.322659\pi$$
0.528754 + 0.848775i $$0.322659\pi$$
$$824$$ −5056.00 −0.213755
$$825$$ 38208.0 1.61240
$$826$$ 3840.00 0.161756
$$827$$ 14112.0 0.593376 0.296688 0.954974i $$-0.404118\pi$$
0.296688 + 0.954974i $$0.404118\pi$$
$$828$$ −7392.00 −0.310253
$$829$$ 37190.0 1.55810 0.779048 0.626964i $$-0.215703\pi$$
0.779048 + 0.626964i $$0.215703\pi$$
$$830$$ 0 0
$$831$$ −20296.0 −0.847245
$$832$$ 0 0
$$833$$ 3762.00 0.156477
$$834$$ 2720.00 0.112933
$$835$$ 36504.0 1.51290
$$836$$ 3072.00 0.127090
$$837$$ 3040.00 0.125541
$$838$$ 4632.00 0.190942
$$839$$ −1380.00 −0.0567853 −0.0283927 0.999597i $$-0.509039\pi$$
−0.0283927 + 0.999597i $$0.509039\pi$$
$$840$$ 11520.0 0.473188
$$841$$ −24353.0 −0.998524
$$842$$ 10012.0 0.409782
$$843$$ 6552.00 0.267690
$$844$$ −16624.0 −0.677988
$$845$$ 0 0
$$846$$ 10296.0 0.418421
$$847$$ −19460.0 −0.789437
$$848$$ 8928.00 0.361543
$$849$$ −18352.0 −0.741860
$$850$$ −26268.0 −1.05998
$$851$$ −42672.0 −1.71889
$$852$$ 6720.00 0.270215
$$853$$ −5150.00 −0.206721 −0.103360 0.994644i $$-0.532959\pi$$
−0.103360 + 0.994644i $$0.532959\pi$$
$$854$$ −33040.0 −1.32389
$$855$$ −3168.00 −0.126717
$$856$$ 7584.00 0.302822
$$857$$ 23562.0 0.939163 0.469581 0.882889i $$-0.344405\pi$$
0.469581 + 0.882889i $$0.344405\pi$$
$$858$$ 0 0
$$859$$ −34612.0 −1.37479 −0.687396 0.726283i $$-0.741246\pi$$
−0.687396 + 0.726283i $$0.741246\pi$$
$$860$$ −8928.00 −0.354003
$$861$$ −31200.0 −1.23495
$$862$$ −22488.0 −0.888566
$$863$$ 14940.0 0.589297 0.294649 0.955606i $$-0.404797\pi$$
0.294649 + 0.955606i $$0.404797\pi$$
$$864$$ 4864.00 0.191524
$$865$$ −21492.0 −0.844798
$$866$$ −26212.0 −1.02855
$$867$$ −2228.00 −0.0872743
$$868$$ 1600.00 0.0625663
$$869$$ 37248.0 1.45403
$$870$$ −864.000 −0.0336694
$$871$$ 0 0
$$872$$ 6064.00 0.235497
$$873$$ −14234.0 −0.551830
$$874$$ −5376.00 −0.208062
$$875$$ −26640.0 −1.02925
$$876$$ −5792.00 −0.223394
$$877$$ −17030.0 −0.655715 −0.327858 0.944727i $$-0.606327\pi$$
−0.327858 + 0.944727i $$0.606327\pi$$
$$878$$ 26960.0 1.03628
$$879$$ 11400.0 0.437443
$$880$$ 13824.0 0.529553
$$881$$ −27246.0 −1.04193 −0.520965 0.853578i $$-0.674428\pi$$
−0.520965 + 0.853578i $$0.674428\pi$$
$$882$$ 1254.00 0.0478734
$$883$$ −8260.00 −0.314803 −0.157402 0.987535i $$-0.550312\pi$$
−0.157402 + 0.987535i $$0.550312\pi$$
$$884$$ 0 0
$$885$$ 6912.00 0.262536
$$886$$ −29016.0 −1.10024
$$887$$ −43392.0 −1.64257 −0.821286 0.570517i $$-0.806743\pi$$
−0.821286 + 0.570517i $$0.806743\pi$$
$$888$$ 8128.00 0.307160
$$889$$ 17600.0 0.663988
$$890$$ 58536.0 2.20464
$$891$$ −14928.0 −0.561287
$$892$$ 13168.0 0.494279
$$893$$ 7488.00 0.280601
$$894$$ 6000.00 0.224463
$$895$$ −50760.0 −1.89578
$$896$$ 2560.00 0.0954504
$$897$$ 0 0
$$898$$ −15132.0 −0.562318
$$899$$ −120.000 −0.00445186
$$900$$ −8756.00 −0.324296
$$901$$ 36828.0 1.36173
$$902$$ −37440.0 −1.38206
$$903$$ 9920.00 0.365578
$$904$$ −5136.00 −0.188961
$$905$$ −13572.0 −0.498507
$$906$$ 13984.0 0.512789
$$907$$ 1028.00 0.0376342 0.0188171 0.999823i $$-0.494010\pi$$
0.0188171 + 0.999823i $$0.494010\pi$$
$$908$$ −9408.00 −0.343850
$$909$$ −2442.00 −0.0891045
$$910$$ 0 0
$$911$$ −21816.0 −0.793410 −0.396705 0.917946i $$-0.629846\pi$$
−0.396705 + 0.917946i $$0.629846\pi$$
$$912$$ 1024.00 0.0371799
$$913$$ 0 0
$$914$$ 10804.0 0.390990
$$915$$ −59472.0 −2.14873
$$916$$ −2744.00 −0.0989785
$$917$$ −6480.00 −0.233357
$$918$$ 20064.0 0.721362
$$919$$ 42752.0 1.53456 0.767279 0.641314i $$-0.221610\pi$$
0.767279 + 0.641314i $$0.221610\pi$$
$$920$$ −24192.0 −0.866942
$$921$$ −32480.0 −1.16205
$$922$$ −9300.00 −0.332190
$$923$$ 0 0
$$924$$ −15360.0 −0.546869
$$925$$ −50546.0 −1.79669
$$926$$ −34376.0 −1.21994
$$927$$ −6952.00 −0.246315
$$928$$ −192.000 −0.00679171
$$929$$ −24978.0 −0.882133 −0.441067 0.897474i $$-0.645400\pi$$
−0.441067 + 0.897474i $$0.645400\pi$$
$$930$$ 2880.00 0.101547
$$931$$ 912.000 0.0321048
$$932$$ 7272.00 0.255582
$$933$$ −14112.0 −0.495183
$$934$$ 29160.0 1.02157
$$935$$ 57024.0 1.99453
$$936$$ 0 0
$$937$$ 33914.0 1.18241 0.591207 0.806520i $$-0.298652\pi$$
0.591207 + 0.806520i $$0.298652\pi$$
$$938$$ 6400.00 0.222780
$$939$$ −27928.0 −0.970603
$$940$$ 33696.0 1.16919
$$941$$ 8442.00 0.292456 0.146228 0.989251i $$-0.453287\pi$$
0.146228 + 0.989251i $$0.453287\pi$$
$$942$$ −4912.00 −0.169896
$$943$$ 65520.0 2.26259
$$944$$ 1536.00 0.0529582
$$945$$ 54720.0 1.88364
$$946$$ 11904.0 0.409125
$$947$$ 43176.0 1.48155 0.740777 0.671751i $$-0.234458\pi$$
0.740777 + 0.671751i $$0.234458\pi$$
$$948$$ 12416.0 0.425372
$$949$$ 0 0
$$950$$ −6368.00 −0.217479
$$951$$ −37080.0 −1.26435
$$952$$ 10560.0 0.359508
$$953$$ −43926.0 −1.49308 −0.746539 0.665342i $$-0.768286\pi$$
−0.746539 + 0.665342i $$0.768286\pi$$
$$954$$ 12276.0 0.416614
$$955$$ −41904.0 −1.41988
$$956$$ 2160.00 0.0730747
$$957$$ 1152.00 0.0389121
$$958$$ −4200.00 −0.141645
$$959$$ 34440.0 1.15967
$$960$$ 4608.00 0.154919
$$961$$ −29391.0 −0.986573
$$962$$ 0 0
$$963$$ 10428.0 0.348949
$$964$$ 3448.00 0.115200
$$965$$ −44100.0 −1.47112
$$966$$ 26880.0 0.895290
$$967$$ 11572.0 0.384830 0.192415 0.981314i $$-0.438368\pi$$
0.192415 + 0.981314i $$0.438368\pi$$
$$968$$ −7784.00 −0.258458
$$969$$ 4224.00 0.140036
$$970$$ −46584.0 −1.54198
$$971$$ 14412.0 0.476316 0.238158 0.971226i $$-0.423456\pi$$
0.238158 + 0.971226i $$0.423456\pi$$
$$972$$ 11440.0 0.377508
$$973$$ 6800.00 0.224047
$$974$$ −24008.0 −0.789801
$$975$$ 0 0
$$976$$ −13216.0 −0.433436
$$977$$ −25602.0 −0.838363 −0.419181 0.907902i $$-0.637683\pi$$
−0.419181 + 0.907902i $$0.637683\pi$$
$$978$$ −6464.00 −0.211346
$$979$$ −78048.0 −2.54793
$$980$$ 4104.00 0.133773
$$981$$ 8338.00 0.271368
$$982$$ −26472.0 −0.860240
$$983$$ −32148.0 −1.04309 −0.521547 0.853222i $$-0.674645\pi$$
−0.521547 + 0.853222i $$0.674645\pi$$
$$984$$ −12480.0 −0.404317
$$985$$ −81756.0 −2.64463
$$986$$ −792.000 −0.0255805
$$987$$ −37440.0 −1.20742
$$988$$ 0 0
$$989$$ −20832.0 −0.669787
$$990$$ 19008.0 0.610216
$$991$$ −9736.00 −0.312083 −0.156041 0.987751i $$-0.549873\pi$$
−0.156041 + 0.987751i $$0.549873\pi$$
$$992$$ 640.000 0.0204839
$$993$$ 30880.0 0.986855
$$994$$ 16800.0 0.536080
$$995$$ −11952.0 −0.380808
$$996$$ 0 0
$$997$$ 6878.00 0.218484 0.109242 0.994015i $$-0.465158\pi$$
0.109242 + 0.994015i $$0.465158\pi$$
$$998$$ 37120.0 1.17737
$$999$$ 38608.0 1.22273
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.c.1.1 1
13.2 odd 12 338.4.e.a.147.1 4
13.3 even 3 338.4.c.e.191.1 2
13.4 even 6 338.4.c.a.315.1 2
13.5 odd 4 338.4.b.d.337.2 2
13.6 odd 12 338.4.e.a.23.2 4
13.7 odd 12 338.4.e.a.23.1 4
13.8 odd 4 338.4.b.d.337.1 2
13.9 even 3 338.4.c.e.315.1 2
13.10 even 6 338.4.c.a.191.1 2
13.11 odd 12 338.4.e.a.147.2 4
13.12 even 2 26.4.a.c.1.1 1
39.38 odd 2 234.4.a.e.1.1 1
52.51 odd 2 208.4.a.b.1.1 1
65.12 odd 4 650.4.b.f.599.2 2
65.38 odd 4 650.4.b.f.599.1 2
65.64 even 2 650.4.a.b.1.1 1
91.90 odd 2 1274.4.a.d.1.1 1
104.51 odd 2 832.4.a.o.1.1 1
104.77 even 2 832.4.a.d.1.1 1
156.155 even 2 1872.4.a.q.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
26.4.a.c.1.1 1 13.12 even 2
208.4.a.b.1.1 1 52.51 odd 2
234.4.a.e.1.1 1 39.38 odd 2
338.4.a.c.1.1 1 1.1 even 1 trivial
338.4.b.d.337.1 2 13.8 odd 4
338.4.b.d.337.2 2 13.5 odd 4
338.4.c.a.191.1 2 13.10 even 6
338.4.c.a.315.1 2 13.4 even 6
338.4.c.e.191.1 2 13.3 even 3
338.4.c.e.315.1 2 13.9 even 3
338.4.e.a.23.1 4 13.7 odd 12
338.4.e.a.23.2 4 13.6 odd 12
338.4.e.a.147.1 4 13.2 odd 12
338.4.e.a.147.2 4 13.11 odd 12
650.4.a.b.1.1 1 65.64 even 2
650.4.b.f.599.1 2 65.38 odd 4
650.4.b.f.599.2 2 65.12 odd 4
832.4.a.d.1.1 1 104.77 even 2
832.4.a.o.1.1 1 104.51 odd 2
1274.4.a.d.1.1 1 91.90 odd 2
1872.4.a.q.1.1 1 156.155 even 2