# Properties

 Label 338.6.a.c.1.1 Level $338$ Weight $6$ Character 338.1 Self dual yes Analytic conductor $54.210$ Analytic rank $1$ 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] = [338,6,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, 6, names="a")

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

chi := DirichletCharacter("338.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$338 = 2 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$54.2097310968$$ Analytic rank: $$1$$ 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)

 $$f(q)$$ $$=$$ $$q+4.00000 q^{2} -13.0000 q^{3} +16.0000 q^{4} -51.0000 q^{5} -52.0000 q^{6} +105.000 q^{7} +64.0000 q^{8} -74.0000 q^{9} +O(q^{10})$$ $$q+4.00000 q^{2} -13.0000 q^{3} +16.0000 q^{4} -51.0000 q^{5} -52.0000 q^{6} +105.000 q^{7} +64.0000 q^{8} -74.0000 q^{9} -204.000 q^{10} +120.000 q^{11} -208.000 q^{12} +420.000 q^{14} +663.000 q^{15} +256.000 q^{16} +1101.00 q^{17} -296.000 q^{18} +1170.00 q^{19} -816.000 q^{20} -1365.00 q^{21} +480.000 q^{22} -1050.00 q^{23} -832.000 q^{24} -524.000 q^{25} +4121.00 q^{27} +1680.00 q^{28} -4104.00 q^{29} +2652.00 q^{30} -9624.00 q^{31} +1024.00 q^{32} -1560.00 q^{33} +4404.00 q^{34} -5355.00 q^{35} -1184.00 q^{36} +8709.00 q^{37} +4680.00 q^{38} -3264.00 q^{40} +9480.00 q^{41} -5460.00 q^{42} -9995.00 q^{43} +1920.00 q^{44} +3774.00 q^{45} -4200.00 q^{46} -2943.00 q^{47} -3328.00 q^{48} -5782.00 q^{49} -2096.00 q^{50} -14313.0 q^{51} -750.000 q^{53} +16484.0 q^{54} -6120.00 q^{55} +6720.00 q^{56} -15210.0 q^{57} -16416.0 q^{58} -40938.0 q^{59} +10608.0 q^{60} -57920.0 q^{61} -38496.0 q^{62} -7770.00 q^{63} +4096.00 q^{64} -6240.00 q^{66} -22812.0 q^{67} +17616.0 q^{68} +13650.0 q^{69} -21420.0 q^{70} -63741.0 q^{71} -4736.00 q^{72} +58866.0 q^{73} +34836.0 q^{74} +6812.00 q^{75} +18720.0 q^{76} +12600.0 q^{77} +63202.0 q^{79} -13056.0 q^{80} -35591.0 q^{81} +37920.0 q^{82} -55458.0 q^{83} -21840.0 q^{84} -56151.0 q^{85} -39980.0 q^{86} +53352.0 q^{87} +7680.00 q^{88} -104778. q^{89} +15096.0 q^{90} -16800.0 q^{92} +125112. q^{93} -11772.0 q^{94} -59670.0 q^{95} -13312.0 q^{96} -160452. q^{97} -23128.0 q^{98} -8880.00 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 4.00000 0.707107
$$3$$ −13.0000 −0.833950 −0.416975 0.908918i $$-0.636910\pi$$
−0.416975 + 0.908918i $$0.636910\pi$$
$$4$$ 16.0000 0.500000
$$5$$ −51.0000 −0.912316 −0.456158 0.889899i $$-0.650775\pi$$
−0.456158 + 0.889899i $$0.650775\pi$$
$$6$$ −52.0000 −0.589692
$$7$$ 105.000 0.809924 0.404962 0.914334i $$-0.367285\pi$$
0.404962 + 0.914334i $$0.367285\pi$$
$$8$$ 64.0000 0.353553
$$9$$ −74.0000 −0.304527
$$10$$ −204.000 −0.645105
$$11$$ 120.000 0.299020 0.149510 0.988760i $$-0.452230\pi$$
0.149510 + 0.988760i $$0.452230\pi$$
$$12$$ −208.000 −0.416975
$$13$$ 0 0
$$14$$ 420.000 0.572703
$$15$$ 663.000 0.760826
$$16$$ 256.000 0.250000
$$17$$ 1101.00 0.923985 0.461993 0.886884i $$-0.347135\pi$$
0.461993 + 0.886884i $$0.347135\pi$$
$$18$$ −296.000 −0.215333
$$19$$ 1170.00 0.743536 0.371768 0.928326i $$-0.378752\pi$$
0.371768 + 0.928326i $$0.378752\pi$$
$$20$$ −816.000 −0.456158
$$21$$ −1365.00 −0.675436
$$22$$ 480.000 0.211439
$$23$$ −1050.00 −0.413875 −0.206938 0.978354i $$-0.566350\pi$$
−0.206938 + 0.978354i $$0.566350\pi$$
$$24$$ −832.000 −0.294846
$$25$$ −524.000 −0.167680
$$26$$ 0 0
$$27$$ 4121.00 1.08791
$$28$$ 1680.00 0.404962
$$29$$ −4104.00 −0.906176 −0.453088 0.891466i $$-0.649678\pi$$
−0.453088 + 0.891466i $$0.649678\pi$$
$$30$$ 2652.00 0.537985
$$31$$ −9624.00 −1.79867 −0.899335 0.437261i $$-0.855949\pi$$
−0.899335 + 0.437261i $$0.855949\pi$$
$$32$$ 1024.00 0.176777
$$33$$ −1560.00 −0.249367
$$34$$ 4404.00 0.653356
$$35$$ −5355.00 −0.738906
$$36$$ −1184.00 −0.152263
$$37$$ 8709.00 1.04584 0.522918 0.852383i $$-0.324843\pi$$
0.522918 + 0.852383i $$0.324843\pi$$
$$38$$ 4680.00 0.525759
$$39$$ 0 0
$$40$$ −3264.00 −0.322552
$$41$$ 9480.00 0.880742 0.440371 0.897816i $$-0.354847\pi$$
0.440371 + 0.897816i $$0.354847\pi$$
$$42$$ −5460.00 −0.477606
$$43$$ −9995.00 −0.824350 −0.412175 0.911105i $$-0.635231\pi$$
−0.412175 + 0.911105i $$0.635231\pi$$
$$44$$ 1920.00 0.149510
$$45$$ 3774.00 0.277825
$$46$$ −4200.00 −0.292654
$$47$$ −2943.00 −0.194333 −0.0971663 0.995268i $$-0.530978\pi$$
−0.0971663 + 0.995268i $$0.530978\pi$$
$$48$$ −3328.00 −0.208488
$$49$$ −5782.00 −0.344023
$$50$$ −2096.00 −0.118568
$$51$$ −14313.0 −0.770558
$$52$$ 0 0
$$53$$ −750.000 −0.0366751 −0.0183376 0.999832i $$-0.505837\pi$$
−0.0183376 + 0.999832i $$0.505837\pi$$
$$54$$ 16484.0 0.769269
$$55$$ −6120.00 −0.272800
$$56$$ 6720.00 0.286351
$$57$$ −15210.0 −0.620072
$$58$$ −16416.0 −0.640763
$$59$$ −40938.0 −1.53108 −0.765538 0.643391i $$-0.777527\pi$$
−0.765538 + 0.643391i $$0.777527\pi$$
$$60$$ 10608.0 0.380413
$$61$$ −57920.0 −1.99298 −0.996492 0.0836839i $$-0.973331\pi$$
−0.996492 + 0.0836839i $$0.973331\pi$$
$$62$$ −38496.0 −1.27185
$$63$$ −7770.00 −0.246643
$$64$$ 4096.00 0.125000
$$65$$ 0 0
$$66$$ −6240.00 −0.176329
$$67$$ −22812.0 −0.620835 −0.310418 0.950600i $$-0.600469\pi$$
−0.310418 + 0.950600i $$0.600469\pi$$
$$68$$ 17616.0 0.461993
$$69$$ 13650.0 0.345152
$$70$$ −21420.0 −0.522486
$$71$$ −63741.0 −1.50063 −0.750314 0.661082i $$-0.770098\pi$$
−0.750314 + 0.661082i $$0.770098\pi$$
$$72$$ −4736.00 −0.107666
$$73$$ 58866.0 1.29288 0.646439 0.762966i $$-0.276258\pi$$
0.646439 + 0.762966i $$0.276258\pi$$
$$74$$ 34836.0 0.739518
$$75$$ 6812.00 0.139837
$$76$$ 18720.0 0.371768
$$77$$ 12600.0 0.242183
$$78$$ 0 0
$$79$$ 63202.0 1.13937 0.569683 0.821865i $$-0.307066\pi$$
0.569683 + 0.821865i $$0.307066\pi$$
$$80$$ −13056.0 −0.228079
$$81$$ −35591.0 −0.602737
$$82$$ 37920.0 0.622779
$$83$$ −55458.0 −0.883627 −0.441813 0.897107i $$-0.645665\pi$$
−0.441813 + 0.897107i $$0.645665\pi$$
$$84$$ −21840.0 −0.337718
$$85$$ −56151.0 −0.842966
$$86$$ −39980.0 −0.582903
$$87$$ 53352.0 0.755705
$$88$$ 7680.00 0.105719
$$89$$ −104778. −1.40215 −0.701076 0.713087i $$-0.747297\pi$$
−0.701076 + 0.713087i $$0.747297\pi$$
$$90$$ 15096.0 0.196452
$$91$$ 0 0
$$92$$ −16800.0 −0.206938
$$93$$ 125112. 1.50000
$$94$$ −11772.0 −0.137414
$$95$$ −59670.0 −0.678339
$$96$$ −13312.0 −0.147423
$$97$$ −160452. −1.73147 −0.865737 0.500500i $$-0.833150\pi$$
−0.865737 + 0.500500i $$0.833150\pi$$
$$98$$ −23128.0 −0.243261
$$99$$ −8880.00 −0.0910594
$$100$$ −8384.00 −0.0838400
$$101$$ 113124. 1.10345 0.551723 0.834027i $$-0.313970\pi$$
0.551723 + 0.834027i $$0.313970\pi$$
$$102$$ −57252.0 −0.544867
$$103$$ −25046.0 −0.232619 −0.116310 0.993213i $$-0.537106\pi$$
−0.116310 + 0.993213i $$0.537106\pi$$
$$104$$ 0 0
$$105$$ 69615.0 0.616211
$$106$$ −3000.00 −0.0259332
$$107$$ 24924.0 0.210455 0.105227 0.994448i $$-0.466443\pi$$
0.105227 + 0.994448i $$0.466443\pi$$
$$108$$ 65936.0 0.543955
$$109$$ −144831. −1.16760 −0.583802 0.811896i $$-0.698435\pi$$
−0.583802 + 0.811896i $$0.698435\pi$$
$$110$$ −24480.0 −0.192899
$$111$$ −113217. −0.872176
$$112$$ 26880.0 0.202481
$$113$$ 100266. 0.738682 0.369341 0.929294i $$-0.379583\pi$$
0.369341 + 0.929294i $$0.379583\pi$$
$$114$$ −60840.0 −0.438457
$$115$$ 53550.0 0.377585
$$116$$ −65664.0 −0.453088
$$117$$ 0 0
$$118$$ −163752. −1.08263
$$119$$ 115605. 0.748358
$$120$$ 42432.0 0.268993
$$121$$ −146651. −0.910587
$$122$$ −231680. −1.40925
$$123$$ −123240. −0.734495
$$124$$ −153984. −0.899335
$$125$$ 186099. 1.06529
$$126$$ −31080.0 −0.174403
$$127$$ 202754. 1.11548 0.557738 0.830017i $$-0.311669\pi$$
0.557738 + 0.830017i $$0.311669\pi$$
$$128$$ 16384.0 0.0883883
$$129$$ 129935. 0.687467
$$130$$ 0 0
$$131$$ −303855. −1.54699 −0.773496 0.633801i $$-0.781494\pi$$
−0.773496 + 0.633801i $$0.781494\pi$$
$$132$$ −24960.0 −0.124684
$$133$$ 122850. 0.602207
$$134$$ −91248.0 −0.438997
$$135$$ −210171. −0.992518
$$136$$ 70464.0 0.326678
$$137$$ −63738.0 −0.290133 −0.145066 0.989422i $$-0.546340\pi$$
−0.145066 + 0.989422i $$0.546340\pi$$
$$138$$ 54600.0 0.244059
$$139$$ 13841.0 0.0607618 0.0303809 0.999538i $$-0.490328\pi$$
0.0303809 + 0.999538i $$0.490328\pi$$
$$140$$ −85680.0 −0.369453
$$141$$ 38259.0 0.162064
$$142$$ −254964. −1.06110
$$143$$ 0 0
$$144$$ −18944.0 −0.0761317
$$145$$ 209304. 0.826718
$$146$$ 235464. 0.914202
$$147$$ 75166.0 0.286898
$$148$$ 139344. 0.522918
$$149$$ 276426. 1.02003 0.510015 0.860165i $$-0.329640\pi$$
0.510015 + 0.860165i $$0.329640\pi$$
$$150$$ 27248.0 0.0988796
$$151$$ −321333. −1.14687 −0.573433 0.819252i $$-0.694389\pi$$
−0.573433 + 0.819252i $$0.694389\pi$$
$$152$$ 74880.0 0.262880
$$153$$ −81474.0 −0.281378
$$154$$ 50400.0 0.171249
$$155$$ 490824. 1.64095
$$156$$ 0 0
$$157$$ 339506. 1.09925 0.549627 0.835410i $$-0.314770\pi$$
0.549627 + 0.835410i $$0.314770\pi$$
$$158$$ 252808. 0.805653
$$159$$ 9750.00 0.0305852
$$160$$ −52224.0 −0.161276
$$161$$ −110250. −0.335208
$$162$$ −142364. −0.426199
$$163$$ −395718. −1.16659 −0.583293 0.812262i $$-0.698236\pi$$
−0.583293 + 0.812262i $$0.698236\pi$$
$$164$$ 151680. 0.440371
$$165$$ 79560.0 0.227502
$$166$$ −221832. −0.624819
$$167$$ 426708. 1.18397 0.591984 0.805950i $$-0.298345\pi$$
0.591984 + 0.805950i $$0.298345\pi$$
$$168$$ −87360.0 −0.238803
$$169$$ 0 0
$$170$$ −224604. −0.596067
$$171$$ −86580.0 −0.226427
$$172$$ −159920. −0.412175
$$173$$ −16026.0 −0.0407108 −0.0203554 0.999793i $$-0.506480\pi$$
−0.0203554 + 0.999793i $$0.506480\pi$$
$$174$$ 213408. 0.534364
$$175$$ −55020.0 −0.135808
$$176$$ 30720.0 0.0747549
$$177$$ 532194. 1.27684
$$178$$ −419112. −0.991471
$$179$$ 690045. 1.60970 0.804850 0.593479i $$-0.202246\pi$$
0.804850 + 0.593479i $$0.202246\pi$$
$$180$$ 60384.0 0.138912
$$181$$ 96478.0 0.218893 0.109446 0.993993i $$-0.465092\pi$$
0.109446 + 0.993993i $$0.465092\pi$$
$$182$$ 0 0
$$183$$ 752960. 1.66205
$$184$$ −67200.0 −0.146327
$$185$$ −444159. −0.954134
$$186$$ 500448. 1.06066
$$187$$ 132120. 0.276290
$$188$$ −47088.0 −0.0971663
$$189$$ 432705. 0.881125
$$190$$ −238680. −0.479658
$$191$$ 708180. 1.40462 0.702312 0.711869i $$-0.252151\pi$$
0.702312 + 0.711869i $$0.252151\pi$$
$$192$$ −53248.0 −0.104244
$$193$$ −347862. −0.672224 −0.336112 0.941822i $$-0.609112\pi$$
−0.336112 + 0.941822i $$0.609112\pi$$
$$194$$ −641808. −1.22434
$$195$$ 0 0
$$196$$ −92512.0 −0.172012
$$197$$ −899589. −1.65150 −0.825750 0.564036i $$-0.809248\pi$$
−0.825750 + 0.564036i $$0.809248\pi$$
$$198$$ −35520.0 −0.0643887
$$199$$ −143116. −0.256186 −0.128093 0.991762i $$-0.540886\pi$$
−0.128093 + 0.991762i $$0.540886\pi$$
$$200$$ −33536.0 −0.0592838
$$201$$ 296556. 0.517746
$$202$$ 452496. 0.780255
$$203$$ −430920. −0.733933
$$204$$ −229008. −0.385279
$$205$$ −483480. −0.803515
$$206$$ −100184. −0.164487
$$207$$ 77700.0 0.126036
$$208$$ 0 0
$$209$$ 140400. 0.222332
$$210$$ 278460. 0.435727
$$211$$ −339731. −0.525326 −0.262663 0.964888i $$-0.584601\pi$$
−0.262663 + 0.964888i $$0.584601\pi$$
$$212$$ −12000.0 −0.0183376
$$213$$ 828633. 1.25145
$$214$$ 99696.0 0.148814
$$215$$ 509745. 0.752068
$$216$$ 263744. 0.384634
$$217$$ −1.01052e6 −1.45679
$$218$$ −579324. −0.825620
$$219$$ −765258. −1.07820
$$220$$ −97920.0 −0.136400
$$221$$ 0 0
$$222$$ −452868. −0.616722
$$223$$ −623757. −0.839950 −0.419975 0.907536i $$-0.637961\pi$$
−0.419975 + 0.907536i $$0.637961\pi$$
$$224$$ 107520. 0.143176
$$225$$ 38776.0 0.0510630
$$226$$ 401064. 0.522327
$$227$$ 177612. 0.228775 0.114387 0.993436i $$-0.463510\pi$$
0.114387 + 0.993436i $$0.463510\pi$$
$$228$$ −243360. −0.310036
$$229$$ −1.18705e6 −1.49582 −0.747911 0.663799i $$-0.768943\pi$$
−0.747911 + 0.663799i $$0.768943\pi$$
$$230$$ 214200. 0.266993
$$231$$ −163800. −0.201969
$$232$$ −262656. −0.320381
$$233$$ 112317. 0.135536 0.0677682 0.997701i $$-0.478412\pi$$
0.0677682 + 0.997701i $$0.478412\pi$$
$$234$$ 0 0
$$235$$ 150093. 0.177293
$$236$$ −655008. −0.765538
$$237$$ −821626. −0.950174
$$238$$ 462420. 0.529169
$$239$$ 1.19805e6 1.35669 0.678346 0.734743i $$-0.262697\pi$$
0.678346 + 0.734743i $$0.262697\pi$$
$$240$$ 169728. 0.190207
$$241$$ 1.16629e6 1.29349 0.646744 0.762707i $$-0.276130\pi$$
0.646744 + 0.762707i $$0.276130\pi$$
$$242$$ −586604. −0.643882
$$243$$ −538720. −0.585258
$$244$$ −926720. −0.996492
$$245$$ 294882. 0.313858
$$246$$ −492960. −0.519366
$$247$$ 0 0
$$248$$ −615936. −0.635926
$$249$$ 720954. 0.736901
$$250$$ 744396. 0.753276
$$251$$ −648996. −0.650216 −0.325108 0.945677i $$-0.605401\pi$$
−0.325108 + 0.945677i $$0.605401\pi$$
$$252$$ −124320. −0.123322
$$253$$ −126000. −0.123757
$$254$$ 811016. 0.788760
$$255$$ 729963. 0.702992
$$256$$ 65536.0 0.0625000
$$257$$ 945885. 0.893317 0.446658 0.894705i $$-0.352614\pi$$
0.446658 + 0.894705i $$0.352614\pi$$
$$258$$ 519740. 0.486113
$$259$$ 914445. 0.847048
$$260$$ 0 0
$$261$$ 303696. 0.275955
$$262$$ −1.21542e6 −1.09389
$$263$$ 1.01222e6 0.902375 0.451188 0.892429i $$-0.351000\pi$$
0.451188 + 0.892429i $$0.351000\pi$$
$$264$$ −99840.0 −0.0881647
$$265$$ 38250.0 0.0334593
$$266$$ 491400. 0.425825
$$267$$ 1.36211e6 1.16933
$$268$$ −364992. −0.310418
$$269$$ −1.01772e6 −0.857527 −0.428763 0.903417i $$-0.641051\pi$$
−0.428763 + 0.903417i $$0.641051\pi$$
$$270$$ −840684. −0.701816
$$271$$ 463461. 0.383345 0.191673 0.981459i $$-0.438609\pi$$
0.191673 + 0.981459i $$0.438609\pi$$
$$272$$ 281856. 0.230996
$$273$$ 0 0
$$274$$ −254952. −0.205155
$$275$$ −62880.0 −0.0501396
$$276$$ 218400. 0.172576
$$277$$ −332528. −0.260393 −0.130196 0.991488i $$-0.541561\pi$$
−0.130196 + 0.991488i $$0.541561\pi$$
$$278$$ 55364.0 0.0429651
$$279$$ 712176. 0.547743
$$280$$ −342720. −0.261243
$$281$$ −49122.0 −0.0371116 −0.0185558 0.999828i $$-0.505907\pi$$
−0.0185558 + 0.999828i $$0.505907\pi$$
$$282$$ 153036. 0.114596
$$283$$ 1.55848e6 1.15674 0.578371 0.815774i $$-0.303689\pi$$
0.578371 + 0.815774i $$0.303689\pi$$
$$284$$ −1.01986e6 −0.750314
$$285$$ 775710. 0.565701
$$286$$ 0 0
$$287$$ 995400. 0.713334
$$288$$ −75776.0 −0.0538332
$$289$$ −207656. −0.146251
$$290$$ 837216. 0.584578
$$291$$ 2.08588e6 1.44396
$$292$$ 941856. 0.646439
$$293$$ −218463. −0.148665 −0.0743325 0.997234i $$-0.523683\pi$$
−0.0743325 + 0.997234i $$0.523683\pi$$
$$294$$ 300664. 0.202868
$$295$$ 2.08784e6 1.39682
$$296$$ 557376. 0.369759
$$297$$ 494520. 0.325306
$$298$$ 1.10570e6 0.721271
$$299$$ 0 0
$$300$$ 108992. 0.0699184
$$301$$ −1.04948e6 −0.667661
$$302$$ −1.28533e6 −0.810957
$$303$$ −1.47061e6 −0.920220
$$304$$ 299520. 0.185884
$$305$$ 2.95392e6 1.81823
$$306$$ −325896. −0.198964
$$307$$ 321102. 0.194445 0.0972226 0.995263i $$-0.469004\pi$$
0.0972226 + 0.995263i $$0.469004\pi$$
$$308$$ 201600. 0.121092
$$309$$ 325598. 0.193993
$$310$$ 1.96330e6 1.16033
$$311$$ −3.33725e6 −1.95654 −0.978269 0.207340i $$-0.933519\pi$$
−0.978269 + 0.207340i $$0.933519\pi$$
$$312$$ 0 0
$$313$$ 1.16568e6 0.672538 0.336269 0.941766i $$-0.390835\pi$$
0.336269 + 0.941766i $$0.390835\pi$$
$$314$$ 1.35802e6 0.777290
$$315$$ 396270. 0.225017
$$316$$ 1.01123e6 0.569683
$$317$$ −73518.0 −0.0410909 −0.0205454 0.999789i $$-0.506540\pi$$
−0.0205454 + 0.999789i $$0.506540\pi$$
$$318$$ 39000.0 0.0216270
$$319$$ −492480. −0.270964
$$320$$ −208896. −0.114039
$$321$$ −324012. −0.175509
$$322$$ −441000. −0.237028
$$323$$ 1.28817e6 0.687016
$$324$$ −569456. −0.301368
$$325$$ 0 0
$$326$$ −1.58287e6 −0.824901
$$327$$ 1.88280e6 0.973723
$$328$$ 606720. 0.311389
$$329$$ −309015. −0.157395
$$330$$ 318240. 0.160868
$$331$$ −632682. −0.317406 −0.158703 0.987326i $$-0.550731\pi$$
−0.158703 + 0.987326i $$0.550731\pi$$
$$332$$ −887328. −0.441813
$$333$$ −644466. −0.318485
$$334$$ 1.70683e6 0.837191
$$335$$ 1.16341e6 0.566398
$$336$$ −349440. −0.168859
$$337$$ 326843. 0.156771 0.0783853 0.996923i $$-0.475024\pi$$
0.0783853 + 0.996923i $$0.475024\pi$$
$$338$$ 0 0
$$339$$ −1.30346e6 −0.616024
$$340$$ −898416. −0.421483
$$341$$ −1.15488e6 −0.537837
$$342$$ −346320. −0.160108
$$343$$ −2.37184e6 −1.08856
$$344$$ −639680. −0.291452
$$345$$ −696150. −0.314887
$$346$$ −64104.0 −0.0287869
$$347$$ 2.96275e6 1.32090 0.660452 0.750868i $$-0.270365\pi$$
0.660452 + 0.750868i $$0.270365\pi$$
$$348$$ 853632. 0.377853
$$349$$ 866325. 0.380730 0.190365 0.981713i $$-0.439033\pi$$
0.190365 + 0.981713i $$0.439033\pi$$
$$350$$ −220080. −0.0960308
$$351$$ 0 0
$$352$$ 122880. 0.0528597
$$353$$ −1.66291e6 −0.710282 −0.355141 0.934813i $$-0.615567\pi$$
−0.355141 + 0.934813i $$0.615567\pi$$
$$354$$ 2.12878e6 0.902863
$$355$$ 3.25079e6 1.36905
$$356$$ −1.67645e6 −0.701076
$$357$$ −1.50286e6 −0.624093
$$358$$ 2.76018e6 1.13823
$$359$$ 625536. 0.256163 0.128081 0.991764i $$-0.459118\pi$$
0.128081 + 0.991764i $$0.459118\pi$$
$$360$$ 241536. 0.0982258
$$361$$ −1.10720e6 −0.447155
$$362$$ 385912. 0.154781
$$363$$ 1.90646e6 0.759385
$$364$$ 0 0
$$365$$ −3.00217e6 −1.17951
$$366$$ 3.01184e6 1.17525
$$367$$ 1.08327e6 0.419829 0.209914 0.977720i $$-0.432681\pi$$
0.209914 + 0.977720i $$0.432681\pi$$
$$368$$ −268800. −0.103469
$$369$$ −701520. −0.268209
$$370$$ −1.77664e6 −0.674674
$$371$$ −78750.0 −0.0297041
$$372$$ 2.00179e6 0.750001
$$373$$ −1.78896e6 −0.665775 −0.332888 0.942967i $$-0.608023\pi$$
−0.332888 + 0.942967i $$0.608023\pi$$
$$374$$ 528480. 0.195366
$$375$$ −2.41929e6 −0.888401
$$376$$ −188352. −0.0687069
$$377$$ 0 0
$$378$$ 1.73082e6 0.623049
$$379$$ −868614. −0.310620 −0.155310 0.987866i $$-0.549638\pi$$
−0.155310 + 0.987866i $$0.549638\pi$$
$$380$$ −954720. −0.339170
$$381$$ −2.63580e6 −0.930251
$$382$$ 2.83272e6 0.993220
$$383$$ 1.07972e6 0.376108 0.188054 0.982159i $$-0.439782\pi$$
0.188054 + 0.982159i $$0.439782\pi$$
$$384$$ −212992. −0.0737115
$$385$$ −642600. −0.220947
$$386$$ −1.39145e6 −0.475334
$$387$$ 739630. 0.251037
$$388$$ −2.56723e6 −0.865737
$$389$$ −1.28822e6 −0.431634 −0.215817 0.976434i $$-0.569241\pi$$
−0.215817 + 0.976434i $$0.569241\pi$$
$$390$$ 0 0
$$391$$ −1.15605e6 −0.382415
$$392$$ −370048. −0.121631
$$393$$ 3.95012e6 1.29011
$$394$$ −3.59836e6 −1.16779
$$395$$ −3.22330e6 −1.03946
$$396$$ −142080. −0.0455297
$$397$$ −5.46909e6 −1.74156 −0.870781 0.491672i $$-0.836386\pi$$
−0.870781 + 0.491672i $$0.836386\pi$$
$$398$$ −572464. −0.181151
$$399$$ −1.59705e6 −0.502211
$$400$$ −134144. −0.0419200
$$401$$ −1.58612e6 −0.492577 −0.246289 0.969196i $$-0.579211\pi$$
−0.246289 + 0.969196i $$0.579211\pi$$
$$402$$ 1.18622e6 0.366102
$$403$$ 0 0
$$404$$ 1.80998e6 0.551723
$$405$$ 1.81514e6 0.549886
$$406$$ −1.72368e6 −0.518969
$$407$$ 1.04508e6 0.312726
$$408$$ −916032. −0.272433
$$409$$ −6.44192e6 −1.90418 −0.952088 0.305825i $$-0.901068\pi$$
−0.952088 + 0.305825i $$0.901068\pi$$
$$410$$ −1.93392e6 −0.568171
$$411$$ 828594. 0.241956
$$412$$ −400736. −0.116310
$$413$$ −4.29849e6 −1.24005
$$414$$ 310800. 0.0891210
$$415$$ 2.82836e6 0.806147
$$416$$ 0 0
$$417$$ −179933. −0.0506723
$$418$$ 561600. 0.157212
$$419$$ −4.30545e6 −1.19807 −0.599037 0.800721i $$-0.704450\pi$$
−0.599037 + 0.800721i $$0.704450\pi$$
$$420$$ 1.11384e6 0.308106
$$421$$ −1.51346e6 −0.416164 −0.208082 0.978111i $$-0.566722\pi$$
−0.208082 + 0.978111i $$0.566722\pi$$
$$422$$ −1.35892e6 −0.371462
$$423$$ 217782. 0.0591795
$$424$$ −48000.0 −0.0129666
$$425$$ −576924. −0.154934
$$426$$ 3.31453e6 0.884908
$$427$$ −6.08160e6 −1.61417
$$428$$ 398784. 0.105227
$$429$$ 0 0
$$430$$ 2.03898e6 0.531792
$$431$$ 1.43116e6 0.371105 0.185552 0.982634i $$-0.440593\pi$$
0.185552 + 0.982634i $$0.440593\pi$$
$$432$$ 1.05498e6 0.271978
$$433$$ −429613. −0.110118 −0.0550589 0.998483i $$-0.517535\pi$$
−0.0550589 + 0.998483i $$0.517535\pi$$
$$434$$ −4.04208e6 −1.03010
$$435$$ −2.72095e6 −0.689442
$$436$$ −2.31730e6 −0.583802
$$437$$ −1.22850e6 −0.307731
$$438$$ −3.06103e6 −0.762399
$$439$$ −552038. −0.136712 −0.0683562 0.997661i $$-0.521775\pi$$
−0.0683562 + 0.997661i $$0.521775\pi$$
$$440$$ −391680. −0.0964494
$$441$$ 427868. 0.104764
$$442$$ 0 0
$$443$$ 2.15255e6 0.521128 0.260564 0.965457i $$-0.416092\pi$$
0.260564 + 0.965457i $$0.416092\pi$$
$$444$$ −1.81147e6 −0.436088
$$445$$ 5.34368e6 1.27921
$$446$$ −2.49503e6 −0.593934
$$447$$ −3.59354e6 −0.850655
$$448$$ 430080. 0.101240
$$449$$ 1.40429e6 0.328731 0.164365 0.986400i $$-0.447442\pi$$
0.164365 + 0.986400i $$0.447442\pi$$
$$450$$ 155104. 0.0361070
$$451$$ 1.13760e6 0.263359
$$452$$ 1.60426e6 0.369341
$$453$$ 4.17733e6 0.956430
$$454$$ 710448. 0.161768
$$455$$ 0 0
$$456$$ −973440. −0.219229
$$457$$ 1.32818e6 0.297485 0.148743 0.988876i $$-0.452477\pi$$
0.148743 + 0.988876i $$0.452477\pi$$
$$458$$ −4.74820e6 −1.05771
$$459$$ 4.53722e6 1.00521
$$460$$ 856800. 0.188793
$$461$$ −5.89070e6 −1.29096 −0.645482 0.763775i $$-0.723344\pi$$
−0.645482 + 0.763775i $$0.723344\pi$$
$$462$$ −655200. −0.142813
$$463$$ 2.37139e6 0.514104 0.257052 0.966398i $$-0.417249\pi$$
0.257052 + 0.966398i $$0.417249\pi$$
$$464$$ −1.05062e6 −0.226544
$$465$$ −6.38071e6 −1.36847
$$466$$ 449268. 0.0958387
$$467$$ −7.17827e6 −1.52310 −0.761548 0.648108i $$-0.775560\pi$$
−0.761548 + 0.648108i $$0.775560\pi$$
$$468$$ 0 0
$$469$$ −2.39526e6 −0.502829
$$470$$ 600372. 0.125365
$$471$$ −4.41358e6 −0.916724
$$472$$ −2.62003e6 −0.541317
$$473$$ −1.19940e6 −0.246497
$$474$$ −3.28650e6 −0.671875
$$475$$ −613080. −0.124676
$$476$$ 1.84968e6 0.374179
$$477$$ 55500.0 0.0111686
$$478$$ 4.79221e6 0.959326
$$479$$ 7.25193e6 1.44416 0.722079 0.691810i $$-0.243186\pi$$
0.722079 + 0.691810i $$0.243186\pi$$
$$480$$ 678912. 0.134496
$$481$$ 0 0
$$482$$ 4.66514e6 0.914634
$$483$$ 1.43325e6 0.279547
$$484$$ −2.34642e6 −0.455294
$$485$$ 8.18305e6 1.57965
$$486$$ −2.15488e6 −0.413840
$$487$$ 2.53364e6 0.484087 0.242043 0.970265i $$-0.422182\pi$$
0.242043 + 0.970265i $$0.422182\pi$$
$$488$$ −3.70688e6 −0.704626
$$489$$ 5.14433e6 0.972875
$$490$$ 1.17953e6 0.221931
$$491$$ −8.46186e6 −1.58403 −0.792013 0.610504i $$-0.790967\pi$$
−0.792013 + 0.610504i $$0.790967\pi$$
$$492$$ −1.97184e6 −0.367248
$$493$$ −4.51850e6 −0.837293
$$494$$ 0 0
$$495$$ 452880. 0.0830750
$$496$$ −2.46374e6 −0.449667
$$497$$ −6.69280e6 −1.21539
$$498$$ 2.88382e6 0.521068
$$499$$ 1.95383e6 0.351265 0.175633 0.984456i $$-0.443803\pi$$
0.175633 + 0.984456i $$0.443803\pi$$
$$500$$ 2.97758e6 0.532646
$$501$$ −5.54720e6 −0.987370
$$502$$ −2.59598e6 −0.459772
$$503$$ 119778. 0.0211085 0.0105542 0.999944i $$-0.496640\pi$$
0.0105542 + 0.999944i $$0.496640\pi$$
$$504$$ −497280. −0.0872016
$$505$$ −5.76932e6 −1.00669
$$506$$ −504000. −0.0875093
$$507$$ 0 0
$$508$$ 3.24406e6 0.557738
$$509$$ 1.03653e7 1.77332 0.886661 0.462420i $$-0.153019\pi$$
0.886661 + 0.462420i $$0.153019\pi$$
$$510$$ 2.91985e6 0.497090
$$511$$ 6.18093e6 1.04713
$$512$$ 262144. 0.0441942
$$513$$ 4.82157e6 0.808900
$$514$$ 3.78354e6 0.631670
$$515$$ 1.27735e6 0.212222
$$516$$ 2.07896e6 0.343734
$$517$$ −353160. −0.0581092
$$518$$ 3.65778e6 0.598954
$$519$$ 208338. 0.0339508
$$520$$ 0 0
$$521$$ −1.04899e7 −1.69307 −0.846537 0.532330i $$-0.821316\pi$$
−0.846537 + 0.532330i $$0.821316\pi$$
$$522$$ 1.21478e6 0.195129
$$523$$ 4.42662e6 0.707649 0.353824 0.935312i $$-0.384881\pi$$
0.353824 + 0.935312i $$0.384881\pi$$
$$524$$ −4.86168e6 −0.773496
$$525$$ 715260. 0.113257
$$526$$ 4.04890e6 0.638076
$$527$$ −1.05960e7 −1.66194
$$528$$ −399360. −0.0623419
$$529$$ −5.33384e6 −0.828707
$$530$$ 153000. 0.0236593
$$531$$ 3.02941e6 0.466253
$$532$$ 1.96560e6 0.301104
$$533$$ 0 0
$$534$$ 5.44846e6 0.826838
$$535$$ −1.27112e6 −0.192001
$$536$$ −1.45997e6 −0.219498
$$537$$ −8.97058e6 −1.34241
$$538$$ −4.07088e6 −0.606363
$$539$$ −693840. −0.102870
$$540$$ −3.36274e6 −0.496259
$$541$$ 2.26377e6 0.332536 0.166268 0.986081i $$-0.446828\pi$$
0.166268 + 0.986081i $$0.446828\pi$$
$$542$$ 1.85384e6 0.271066
$$543$$ −1.25421e6 −0.182546
$$544$$ 1.12742e6 0.163339
$$545$$ 7.38638e6 1.06522
$$546$$ 0 0
$$547$$ 7.21090e6 1.03044 0.515218 0.857059i $$-0.327711\pi$$
0.515218 + 0.857059i $$0.327711\pi$$
$$548$$ −1.01981e6 −0.145066
$$549$$ 4.28608e6 0.606917
$$550$$ −251520. −0.0354540
$$551$$ −4.80168e6 −0.673774
$$552$$ 873600. 0.122030
$$553$$ 6.63621e6 0.922799
$$554$$ −1.33011e6 −0.184125
$$555$$ 5.77407e6 0.795700
$$556$$ 221456. 0.0303809
$$557$$ 273507. 0.0373534 0.0186767 0.999826i $$-0.494055\pi$$
0.0186767 + 0.999826i $$0.494055\pi$$
$$558$$ 2.84870e6 0.387313
$$559$$ 0 0
$$560$$ −1.37088e6 −0.184727
$$561$$ −1.71756e6 −0.230412
$$562$$ −196488. −0.0262419
$$563$$ 959349. 0.127557 0.0637787 0.997964i $$-0.479685\pi$$
0.0637787 + 0.997964i $$0.479685\pi$$
$$564$$ 612144. 0.0810319
$$565$$ −5.11357e6 −0.673911
$$566$$ 6.23394e6 0.817940
$$567$$ −3.73706e6 −0.488171
$$568$$ −4.07942e6 −0.530552
$$569$$ 1.19403e7 1.54609 0.773044 0.634352i $$-0.218733\pi$$
0.773044 + 0.634352i $$0.218733\pi$$
$$570$$ 3.10284e6 0.400011
$$571$$ −7.20205e6 −0.924413 −0.462206 0.886772i $$-0.652942\pi$$
−0.462206 + 0.886772i $$0.652942\pi$$
$$572$$ 0 0
$$573$$ −9.20634e6 −1.17139
$$574$$ 3.98160e6 0.504403
$$575$$ 550200. 0.0693986
$$576$$ −303104. −0.0380658
$$577$$ 1.66990e6 0.208810 0.104405 0.994535i $$-0.466706\pi$$
0.104405 + 0.994535i $$0.466706\pi$$
$$578$$ −830624. −0.103415
$$579$$ 4.52221e6 0.560601
$$580$$ 3.34886e6 0.413359
$$581$$ −5.82309e6 −0.715671
$$582$$ 8.34350e6 1.02104
$$583$$ −90000.0 −0.0109666
$$584$$ 3.76742e6 0.457101
$$585$$ 0 0
$$586$$ −873852. −0.105122
$$587$$ −8.29913e6 −0.994117 −0.497059 0.867717i $$-0.665587\pi$$
−0.497059 + 0.867717i $$0.665587\pi$$
$$588$$ 1.20266e6 0.143449
$$589$$ −1.12601e7 −1.33738
$$590$$ 8.35135e6 0.987704
$$591$$ 1.16947e7 1.37727
$$592$$ 2.22950e6 0.261459
$$593$$ −4.48969e6 −0.524300 −0.262150 0.965027i $$-0.584432\pi$$
−0.262150 + 0.965027i $$0.584432\pi$$
$$594$$ 1.97808e6 0.230026
$$595$$ −5.89586e6 −0.682738
$$596$$ 4.42282e6 0.510015
$$597$$ 1.86051e6 0.213646
$$598$$ 0 0
$$599$$ 1.38261e6 0.157446 0.0787232 0.996897i $$-0.474916\pi$$
0.0787232 + 0.996897i $$0.474916\pi$$
$$600$$ 435968. 0.0494398
$$601$$ 1.04021e7 1.17472 0.587359 0.809327i $$-0.300168\pi$$
0.587359 + 0.809327i $$0.300168\pi$$
$$602$$ −4.19790e6 −0.472107
$$603$$ 1.68809e6 0.189061
$$604$$ −5.14133e6 −0.573433
$$605$$ 7.47920e6 0.830743
$$606$$ −5.88245e6 −0.650694
$$607$$ −4.78668e6 −0.527306 −0.263653 0.964618i $$-0.584927\pi$$
−0.263653 + 0.964618i $$0.584927\pi$$
$$608$$ 1.19808e6 0.131440
$$609$$ 5.60196e6 0.612064
$$610$$ 1.18157e7 1.28568
$$611$$ 0 0
$$612$$ −1.30358e6 −0.140689
$$613$$ −1.04783e7 −1.12627 −0.563134 0.826366i $$-0.690404\pi$$
−0.563134 + 0.826366i $$0.690404\pi$$
$$614$$ 1.28441e6 0.137493
$$615$$ 6.28524e6 0.670091
$$616$$ 806400. 0.0856246
$$617$$ 1.79106e7 1.89407 0.947036 0.321128i $$-0.104062\pi$$
0.947036 + 0.321128i $$0.104062\pi$$
$$618$$ 1.30239e6 0.137174
$$619$$ −4.43222e6 −0.464938 −0.232469 0.972604i $$-0.574680\pi$$
−0.232469 + 0.972604i $$0.574680\pi$$
$$620$$ 7.85318e6 0.820477
$$621$$ −4.32705e6 −0.450260
$$622$$ −1.33490e7 −1.38348
$$623$$ −1.10017e7 −1.13564
$$624$$ 0 0
$$625$$ −7.85355e6 −0.804203
$$626$$ 4.66270e6 0.475556
$$627$$ −1.82520e6 −0.185414
$$628$$ 5.43210e6 0.549627
$$629$$ 9.58861e6 0.966338
$$630$$ 1.58508e6 0.159111
$$631$$ 1.43291e7 1.43267 0.716335 0.697756i $$-0.245818\pi$$
0.716335 + 0.697756i $$0.245818\pi$$
$$632$$ 4.04493e6 0.402827
$$633$$ 4.41650e6 0.438096
$$634$$ −294072. −0.0290556
$$635$$ −1.03405e7 −1.01767
$$636$$ 156000. 0.0152926
$$637$$ 0 0
$$638$$ −1.96992e6 −0.191601
$$639$$ 4.71683e6 0.456981
$$640$$ −835584. −0.0806381
$$641$$ −6.65869e6 −0.640094 −0.320047 0.947402i $$-0.603699\pi$$
−0.320047 + 0.947402i $$0.603699\pi$$
$$642$$ −1.29605e6 −0.124103
$$643$$ −1.55224e7 −1.48058 −0.740291 0.672286i $$-0.765312\pi$$
−0.740291 + 0.672286i $$0.765312\pi$$
$$644$$ −1.76400e6 −0.167604
$$645$$ −6.62668e6 −0.627187
$$646$$ 5.15268e6 0.485794
$$647$$ 2.44454e6 0.229581 0.114791 0.993390i $$-0.463380\pi$$
0.114791 + 0.993390i $$0.463380\pi$$
$$648$$ −2.27782e6 −0.213100
$$649$$ −4.91256e6 −0.457821
$$650$$ 0 0
$$651$$ 1.31368e7 1.21489
$$652$$ −6.33149e6 −0.583293
$$653$$ 1.16500e7 1.06916 0.534580 0.845118i $$-0.320470\pi$$
0.534580 + 0.845118i $$0.320470\pi$$
$$654$$ 7.53121e6 0.688526
$$655$$ 1.54966e7 1.41135
$$656$$ 2.42688e6 0.220185
$$657$$ −4.35608e6 −0.393716
$$658$$ −1.23606e6 −0.111295
$$659$$ 1.33185e7 1.19465 0.597326 0.801999i $$-0.296230\pi$$
0.597326 + 0.801999i $$0.296230\pi$$
$$660$$ 1.27296e6 0.113751
$$661$$ −1.35722e7 −1.20822 −0.604112 0.796900i $$-0.706472\pi$$
−0.604112 + 0.796900i $$0.706472\pi$$
$$662$$ −2.53073e6 −0.224440
$$663$$ 0 0
$$664$$ −3.54931e6 −0.312409
$$665$$ −6.26535e6 −0.549403
$$666$$ −2.57786e6 −0.225203
$$667$$ 4.30920e6 0.375044
$$668$$ 6.82733e6 0.591984
$$669$$ 8.10884e6 0.700476
$$670$$ 4.65365e6 0.400504
$$671$$ −6.95040e6 −0.595941
$$672$$ −1.39776e6 −0.119401
$$673$$ 1.58674e7 1.35042 0.675209 0.737626i $$-0.264053\pi$$
0.675209 + 0.737626i $$0.264053\pi$$
$$674$$ 1.30737e6 0.110854
$$675$$ −2.15940e6 −0.182421
$$676$$ 0 0
$$677$$ −2.24264e7 −1.88056 −0.940281 0.340398i $$-0.889438\pi$$
−0.940281 + 0.340398i $$0.889438\pi$$
$$678$$ −5.21383e6 −0.435595
$$679$$ −1.68475e7 −1.40236
$$680$$ −3.59366e6 −0.298034
$$681$$ −2.30896e6 −0.190787
$$682$$ −4.61952e6 −0.380308
$$683$$ 8.11034e6 0.665254 0.332627 0.943059i $$-0.392065\pi$$
0.332627 + 0.943059i $$0.392065\pi$$
$$684$$ −1.38528e6 −0.113213
$$685$$ 3.25064e6 0.264693
$$686$$ −9.48738e6 −0.769726
$$687$$ 1.54316e7 1.24744
$$688$$ −2.55872e6 −0.206088
$$689$$ 0 0
$$690$$ −2.78460e6 −0.222659
$$691$$ −2.00020e7 −1.59359 −0.796797 0.604246i $$-0.793474\pi$$
−0.796797 + 0.604246i $$0.793474\pi$$
$$692$$ −256416. −0.0203554
$$693$$ −932400. −0.0737512
$$694$$ 1.18510e7 0.934020
$$695$$ −705891. −0.0554339
$$696$$ 3.41453e6 0.267182
$$697$$ 1.04375e7 0.813793
$$698$$ 3.46530e6 0.269217
$$699$$ −1.46012e6 −0.113031
$$700$$ −880320. −0.0679040
$$701$$ 2.22272e6 0.170840 0.0854200 0.996345i $$-0.472777\pi$$
0.0854200 + 0.996345i $$0.472777\pi$$
$$702$$ 0 0
$$703$$ 1.01895e7 0.777617
$$704$$ 491520. 0.0373774
$$705$$ −1.95121e6 −0.147853
$$706$$ −6.65162e6 −0.502245
$$707$$ 1.18780e7 0.893708
$$708$$ 8.51510e6 0.638420
$$709$$ 2.03634e7 1.52137 0.760684 0.649122i $$-0.224864\pi$$
0.760684 + 0.649122i $$0.224864\pi$$
$$710$$ 1.30032e7 0.968062
$$711$$ −4.67695e6 −0.346967
$$712$$ −6.70579e6 −0.495736
$$713$$ 1.01052e7 0.744425
$$714$$ −6.01146e6 −0.441301
$$715$$ 0 0
$$716$$ 1.10407e7 0.804850
$$717$$ −1.55747e7 −1.13141
$$718$$ 2.50214e6 0.181135
$$719$$ 1.98255e7 1.43022 0.715108 0.699014i $$-0.246377\pi$$
0.715108 + 0.699014i $$0.246377\pi$$
$$720$$ 966144. 0.0694561
$$721$$ −2.62983e6 −0.188404
$$722$$ −4.42880e6 −0.316186
$$723$$ −1.51617e7 −1.07870
$$724$$ 1.54365e6 0.109446
$$725$$ 2.15050e6 0.151948
$$726$$ 7.62585e6 0.536966
$$727$$ 9.24667e6 0.648857 0.324429 0.945910i $$-0.394828\pi$$
0.324429 + 0.945910i $$0.394828\pi$$
$$728$$ 0 0
$$729$$ 1.56520e7 1.09081
$$730$$ −1.20087e7 −0.834041
$$731$$ −1.10045e7 −0.761687
$$732$$ 1.20474e7 0.831025
$$733$$ 1.48114e7 1.01821 0.509105 0.860704i $$-0.329976\pi$$
0.509105 + 0.860704i $$0.329976\pi$$
$$734$$ 4.33309e6 0.296864
$$735$$ −3.83347e6 −0.261742
$$736$$ −1.07520e6 −0.0731635
$$737$$ −2.73744e6 −0.185642
$$738$$ −2.80608e6 −0.189653
$$739$$ −5.67210e6 −0.382061 −0.191031 0.981584i $$-0.561183\pi$$
−0.191031 + 0.981584i $$0.561183\pi$$
$$740$$ −7.10654e6 −0.477067
$$741$$ 0 0
$$742$$ −315000. −0.0210039
$$743$$ 2.75704e7 1.83219 0.916095 0.400960i $$-0.131323\pi$$
0.916095 + 0.400960i $$0.131323\pi$$
$$744$$ 8.00717e6 0.530330
$$745$$ −1.40977e7 −0.930590
$$746$$ −7.15582e6 −0.470774
$$747$$ 4.10389e6 0.269088
$$748$$ 2.11392e6 0.138145
$$749$$ 2.61702e6 0.170452
$$750$$ −9.67715e6 −0.628195
$$751$$ 4.09636e6 0.265032 0.132516 0.991181i $$-0.457694\pi$$
0.132516 + 0.991181i $$0.457694\pi$$
$$752$$ −753408. −0.0485831
$$753$$ 8.43695e6 0.542248
$$754$$ 0 0
$$755$$ 1.63880e7 1.04630
$$756$$ 6.92328e6 0.440562
$$757$$ 1.09396e7 0.693844 0.346922 0.937894i $$-0.387227\pi$$
0.346922 + 0.937894i $$0.387227\pi$$
$$758$$ −3.47446e6 −0.219641
$$759$$ 1.63800e6 0.103207
$$760$$ −3.81888e6 −0.239829
$$761$$ 1.36940e6 0.0857172 0.0428586 0.999081i $$-0.486354\pi$$
0.0428586 + 0.999081i $$0.486354\pi$$
$$762$$ −1.05432e7 −0.657787
$$763$$ −1.52073e7 −0.945670
$$764$$ 1.13309e7 0.702312
$$765$$ 4.15517e6 0.256706
$$766$$ 4.31886e6 0.265948
$$767$$ 0 0
$$768$$ −851968. −0.0521219
$$769$$ 1.08375e7 0.660867 0.330433 0.943829i $$-0.392805\pi$$
0.330433 + 0.943829i $$0.392805\pi$$
$$770$$ −2.57040e6 −0.156233
$$771$$ −1.22965e7 −0.744982
$$772$$ −5.56579e6 −0.336112
$$773$$ 2.05445e7 1.23665 0.618325 0.785922i $$-0.287812\pi$$
0.618325 + 0.785922i $$0.287812\pi$$
$$774$$ 2.95852e6 0.177510
$$775$$ 5.04298e6 0.301601
$$776$$ −1.02689e7 −0.612168
$$777$$ −1.18878e7 −0.706396
$$778$$ −5.15287e6 −0.305211
$$779$$ 1.10916e7 0.654863
$$780$$ 0 0
$$781$$ −7.64892e6 −0.448717
$$782$$ −4.62420e6 −0.270408
$$783$$ −1.69126e7 −0.985838
$$784$$ −1.48019e6 −0.0860058
$$785$$ −1.73148e7 −1.00287
$$786$$ 1.58005e7 0.912249
$$787$$ −1.34637e7 −0.774869 −0.387435 0.921897i $$-0.626639\pi$$
−0.387435 + 0.921897i $$0.626639\pi$$
$$788$$ −1.43934e7 −0.825750
$$789$$ −1.31589e7 −0.752536
$$790$$ −1.28932e7 −0.735010
$$791$$ 1.05279e7 0.598276
$$792$$ −568320. −0.0321944
$$793$$ 0 0
$$794$$ −2.18764e7 −1.23147
$$795$$ −497250. −0.0279034
$$796$$ −2.28986e6 −0.128093
$$797$$ −2.02451e7 −1.12895 −0.564475 0.825450i $$-0.690921\pi$$
−0.564475 + 0.825450i $$0.690921\pi$$
$$798$$ −6.38820e6 −0.355117
$$799$$ −3.24024e6 −0.179560
$$800$$ −536576. −0.0296419
$$801$$ 7.75357e6 0.426993
$$802$$ −6.34447e6 −0.348305
$$803$$ 7.06392e6 0.386596
$$804$$ 4.74490e6 0.258873
$$805$$ 5.62275e6 0.305815
$$806$$ 0 0
$$807$$ 1.32304e7 0.715135
$$808$$ 7.23994e6 0.390127
$$809$$ 2.48958e7 1.33738 0.668689 0.743542i $$-0.266856\pi$$
0.668689 + 0.743542i $$0.266856\pi$$
$$810$$ 7.26056e6 0.388828
$$811$$ 2.53328e7 1.35248 0.676241 0.736681i $$-0.263608\pi$$
0.676241 + 0.736681i $$0.263608\pi$$
$$812$$ −6.89472e6 −0.366967
$$813$$ −6.02499e6 −0.319691
$$814$$ 4.18032e6 0.221130
$$815$$ 2.01816e7 1.06429
$$816$$ −3.66413e6 −0.192639
$$817$$ −1.16942e7 −0.612934
$$818$$ −2.57677e7 −1.34646
$$819$$ 0 0
$$820$$ −7.73568e6 −0.401757
$$821$$ 1.25922e7 0.651993 0.325996 0.945371i $$-0.394300\pi$$
0.325996 + 0.945371i $$0.394300\pi$$
$$822$$ 3.31438e6 0.171089
$$823$$ 3.32776e7 1.71259 0.856294 0.516490i $$-0.172761\pi$$
0.856294 + 0.516490i $$0.172761\pi$$
$$824$$ −1.60294e6 −0.0822433
$$825$$ 817440. 0.0418139
$$826$$ −1.71940e7 −0.876851
$$827$$ 2.98630e7 1.51834 0.759171 0.650891i $$-0.225604\pi$$
0.759171 + 0.650891i $$0.225604\pi$$
$$828$$ 1.24320e6 0.0630181
$$829$$ −3.04980e7 −1.54129 −0.770647 0.637262i $$-0.780067\pi$$
−0.770647 + 0.637262i $$0.780067\pi$$
$$830$$ 1.13134e7 0.570032
$$831$$ 4.32286e6 0.217155
$$832$$ 0 0
$$833$$ −6.36598e6 −0.317872
$$834$$ −719732. −0.0358307
$$835$$ −2.17621e7 −1.08015
$$836$$ 2.24640e6 0.111166
$$837$$ −3.96605e7 −1.95679
$$838$$ −1.72218e7 −0.847167
$$839$$ −1.46249e7 −0.717278 −0.358639 0.933476i $$-0.616759\pi$$
−0.358639 + 0.933476i $$0.616759\pi$$
$$840$$ 4.45536e6 0.217864
$$841$$ −3.66833e6 −0.178846
$$842$$ −6.05382e6 −0.294272
$$843$$ 638586. 0.0309493
$$844$$ −5.43570e6 −0.262663
$$845$$ 0 0
$$846$$ 871128. 0.0418462
$$847$$ −1.53984e7 −0.737506
$$848$$ −192000. −0.00916878
$$849$$ −2.02603e7 −0.964665
$$850$$ −2.30770e6 −0.109555
$$851$$ −9.14445e6 −0.432846
$$852$$ 1.32581e7 0.625725
$$853$$ −1.54032e7 −0.724832 −0.362416 0.932016i $$-0.618048\pi$$
−0.362416 + 0.932016i $$0.618048\pi$$
$$854$$ −2.43264e7 −1.14139
$$855$$ 4.41558e6 0.206572
$$856$$ 1.59514e6 0.0744069
$$857$$ 2.59910e7 1.20884 0.604422 0.796664i $$-0.293404\pi$$
0.604422 + 0.796664i $$0.293404\pi$$
$$858$$ 0 0
$$859$$ 1.26690e7 0.585815 0.292908 0.956141i $$-0.405377\pi$$
0.292908 + 0.956141i $$0.405377\pi$$
$$860$$ 8.15592e6 0.376034
$$861$$ −1.29402e7 −0.594885
$$862$$ 5.72466e6 0.262411
$$863$$ −3.93618e7 −1.79907 −0.899535 0.436849i $$-0.856094\pi$$
−0.899535 + 0.436849i $$0.856094\pi$$
$$864$$ 4.21990e6 0.192317
$$865$$ 817326. 0.0371411
$$866$$ −1.71845e6 −0.0778651
$$867$$ 2.69953e6 0.121966
$$868$$ −1.61683e7 −0.728393
$$869$$ 7.58424e6 0.340693
$$870$$ −1.08838e7 −0.487509
$$871$$ 0 0
$$872$$ −9.26918e6 −0.412810
$$873$$ 1.18734e7 0.527280
$$874$$ −4.91400e6 −0.217599
$$875$$ 1.95404e7 0.862806
$$876$$ −1.22441e7 −0.539098
$$877$$ −2.93636e7 −1.28917 −0.644585 0.764532i $$-0.722970\pi$$
−0.644585 + 0.764532i $$0.722970\pi$$
$$878$$ −2.20815e6 −0.0966702
$$879$$ 2.84002e6 0.123979
$$880$$ −1.56672e6 −0.0682001
$$881$$ 2.47421e7 1.07398 0.536990 0.843589i $$-0.319561\pi$$
0.536990 + 0.843589i $$0.319561\pi$$
$$882$$ 1.71147e6 0.0740796
$$883$$ −1.56178e7 −0.674092 −0.337046 0.941488i $$-0.609428\pi$$
−0.337046 + 0.941488i $$0.609428\pi$$
$$884$$ 0 0
$$885$$ −2.71419e7 −1.16488
$$886$$ 8.61020e6 0.368493
$$887$$ 1.41193e6 0.0602566 0.0301283 0.999546i $$-0.490408\pi$$
0.0301283 + 0.999546i $$0.490408\pi$$
$$888$$ −7.24589e6 −0.308361
$$889$$ 2.12892e7 0.903450
$$890$$ 2.13747e7 0.904535
$$891$$ −4.27092e6 −0.180230
$$892$$ −9.98011e6 −0.419975
$$893$$ −3.44331e6 −0.144493
$$894$$ −1.43742e7 −0.601504
$$895$$ −3.51923e7 −1.46855
$$896$$ 1.72032e6 0.0715878
$$897$$ 0 0
$$898$$ 5.61715e6 0.232448
$$899$$ 3.94969e7 1.62991
$$900$$ 620416. 0.0255315
$$901$$ −825750. −0.0338873
$$902$$ 4.55040e6 0.186223
$$903$$ 1.36432e7 0.556796
$$904$$ 6.41702e6 0.261164
$$905$$ −4.92038e6 −0.199700
$$906$$ 1.67093e7 0.676298
$$907$$ 1.48543e7 0.599563 0.299781 0.954008i $$-0.403086\pi$$
0.299781 + 0.954008i $$0.403086\pi$$
$$908$$ 2.84179e6 0.114387
$$909$$ −8.37118e6 −0.336029
$$910$$ 0 0
$$911$$ 4.29118e7 1.71309 0.856547 0.516069i $$-0.172605\pi$$
0.856547 + 0.516069i $$0.172605\pi$$
$$912$$ −3.89376e6 −0.155018
$$913$$ −6.65496e6 −0.264222
$$914$$ 5.31271e6 0.210354
$$915$$ −3.84010e7 −1.51631
$$916$$ −1.89928e7 −0.747911
$$917$$ −3.19048e7 −1.25295
$$918$$ 1.81489e7 0.710793
$$919$$ 4.34706e7 1.69788 0.848939 0.528491i $$-0.177242\pi$$
0.848939 + 0.528491i $$0.177242\pi$$
$$920$$ 3.42720e6 0.133497
$$921$$ −4.17433e6 −0.162158
$$922$$ −2.35628e7 −0.912850
$$923$$ 0 0
$$924$$ −2.62080e6 −0.100984
$$925$$ −4.56352e6 −0.175366
$$926$$ 9.48557e6 0.363526
$$927$$ 1.85340e6 0.0708387
$$928$$ −4.20250e6 −0.160191
$$929$$ −2.97375e7 −1.13049 −0.565243 0.824925i $$-0.691217\pi$$
−0.565243 + 0.824925i $$0.691217\pi$$
$$930$$ −2.55228e7 −0.967658
$$931$$ −6.76494e6 −0.255794
$$932$$ 1.79707e6 0.0677682
$$933$$ 4.33843e7 1.63166
$$934$$ −2.87131e7 −1.07699
$$935$$ −6.73812e6 −0.252063
$$936$$ 0 0
$$937$$ 1.46550e7 0.545303 0.272651 0.962113i $$-0.412099\pi$$
0.272651 + 0.962113i $$0.412099\pi$$
$$938$$ −9.58104e6 −0.355554
$$939$$ −1.51538e7 −0.560863
$$940$$ 2.40149e6 0.0886463
$$941$$ −8.80233e6 −0.324059 −0.162029 0.986786i $$-0.551804\pi$$
−0.162029 + 0.986786i $$0.551804\pi$$
$$942$$ −1.76543e7 −0.648222
$$943$$ −9.95400e6 −0.364518
$$944$$ −1.04801e7 −0.382769
$$945$$ −2.20680e7 −0.803864
$$946$$ −4.79760e6 −0.174300
$$947$$ 9.00847e6 0.326420 0.163210 0.986591i $$-0.447815\pi$$
0.163210 + 0.986591i $$0.447815\pi$$
$$948$$ −1.31460e7 −0.475087
$$949$$ 0 0
$$950$$ −2.45232e6 −0.0881593
$$951$$ 955734. 0.0342678
$$952$$ 7.39872e6 0.264584
$$953$$ −1.31122e7 −0.467675 −0.233837 0.972276i $$-0.575128\pi$$
−0.233837 + 0.972276i $$0.575128\pi$$
$$954$$ 222000. 0.00789736
$$955$$ −3.61172e7 −1.28146
$$956$$ 1.91688e7 0.678346
$$957$$ 6.40224e6 0.225971
$$958$$ 2.90077e7 1.02117
$$959$$ −6.69249e6 −0.234986
$$960$$ 2.71565e6 0.0951033
$$961$$ 6.39922e7 2.23521
$$962$$ 0 0
$$963$$ −1.84438e6 −0.0640890
$$964$$ 1.86606e7 0.646744
$$965$$ 1.77410e7 0.613280
$$966$$ 5.73300e6 0.197669
$$967$$ 1.53210e6 0.0526892 0.0263446 0.999653i $$-0.491613\pi$$
0.0263446 + 0.999653i $$0.491613\pi$$
$$968$$ −9.38566e6 −0.321941
$$969$$ −1.67462e7 −0.572937
$$970$$ 3.27322e7 1.11698
$$971$$ 2.37514e7 0.808426 0.404213 0.914665i $$-0.367545\pi$$
0.404213 + 0.914665i $$0.367545\pi$$
$$972$$ −8.61952e6 −0.292629
$$973$$ 1.45330e6 0.0492124
$$974$$ 1.01346e7 0.342301
$$975$$ 0 0
$$976$$ −1.48275e7 −0.498246
$$977$$ −1.97751e7 −0.662798 −0.331399 0.943491i $$-0.607521\pi$$
−0.331399 + 0.943491i $$0.607521\pi$$
$$978$$ 2.05773e7 0.687926
$$979$$ −1.25734e7 −0.419271
$$980$$ 4.71811e6 0.156929
$$981$$ 1.07175e7 0.355566
$$982$$ −3.38475e7 −1.12008
$$983$$ 1.57006e7 0.518241 0.259121 0.965845i $$-0.416567\pi$$
0.259121 + 0.965845i $$0.416567\pi$$
$$984$$ −7.88736e6 −0.259683
$$985$$ 4.58790e7 1.50669
$$986$$ −1.80740e7 −0.592055
$$987$$ 4.01720e6 0.131259
$$988$$ 0 0
$$989$$ 1.04948e7 0.341178
$$990$$ 1.81152e6 0.0587429
$$991$$ 1.65835e7 0.536406 0.268203 0.963362i $$-0.413570\pi$$
0.268203 + 0.963362i $$0.413570\pi$$
$$992$$ −9.85498e6 −0.317963
$$993$$ 8.22487e6 0.264701
$$994$$ −2.67712e7 −0.859414
$$995$$ 7.29892e6 0.233723
$$996$$ 1.15353e7 0.368451
$$997$$ −2.08432e7 −0.664091 −0.332045 0.943263i $$-0.607739\pi$$
−0.332045 + 0.943263i $$0.607739\pi$$
$$998$$ 7.81531e6 0.248382
$$999$$ 3.58898e7 1.13778
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 338.6.a.c.1.1 1
13.5 odd 4 26.6.b.a.25.1 2
13.8 odd 4 26.6.b.a.25.2 yes 2
13.12 even 2 338.6.a.a.1.1 1
39.5 even 4 234.6.b.b.181.2 2
39.8 even 4 234.6.b.b.181.1 2
52.31 even 4 208.6.f.b.129.2 2
52.47 even 4 208.6.f.b.129.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
26.6.b.a.25.1 2 13.5 odd 4
26.6.b.a.25.2 yes 2 13.8 odd 4
208.6.f.b.129.1 2 52.47 even 4
208.6.f.b.129.2 2 52.31 even 4
234.6.b.b.181.1 2 39.8 even 4
234.6.b.b.181.2 2 39.5 even 4
338.6.a.a.1.1 1 13.12 even 2
338.6.a.c.1.1 1 1.1 even 1 trivial