# Properties

 Label 112.6.a.e.1.1 Level $112$ Weight $6$ Character 112.1 Self dual yes Analytic conductor $17.963$ 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] = [112,6,Mod(1,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("112.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$112 = 2^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 112.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: $$17.9629878191$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 112.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000 q^{3} -96.0000 q^{5} -49.0000 q^{7} -239.000 q^{9} +O(q^{10})$$ $$q+2.00000 q^{3} -96.0000 q^{5} -49.0000 q^{7} -239.000 q^{9} +720.000 q^{11} +572.000 q^{13} -192.000 q^{15} +1254.00 q^{17} +94.0000 q^{19} -98.0000 q^{21} -96.0000 q^{23} +6091.00 q^{25} -964.000 q^{27} -4374.00 q^{29} +6244.00 q^{31} +1440.00 q^{33} +4704.00 q^{35} -10798.0 q^{37} +1144.00 q^{39} +12006.0 q^{41} +9160.00 q^{43} +22944.0 q^{45} +25836.0 q^{47} +2401.00 q^{49} +2508.00 q^{51} +1014.00 q^{53} -69120.0 q^{55} +188.000 q^{57} -1242.00 q^{59} +7592.00 q^{61} +11711.0 q^{63} -54912.0 q^{65} -41132.0 q^{67} -192.000 q^{69} +37632.0 q^{71} -13438.0 q^{73} +12182.0 q^{75} -35280.0 q^{77} -6248.00 q^{79} +56149.0 q^{81} +25254.0 q^{83} -120384. q^{85} -8748.00 q^{87} -45126.0 q^{89} -28028.0 q^{91} +12488.0 q^{93} -9024.00 q^{95} +107222. q^{97} -172080. q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 2.00000 0.128300 0.0641500 0.997940i $$-0.479566\pi$$
0.0641500 + 0.997940i $$0.479566\pi$$
$$4$$ 0 0
$$5$$ −96.0000 −1.71730 −0.858650 0.512562i $$-0.828696\pi$$
−0.858650 + 0.512562i $$0.828696\pi$$
$$6$$ 0 0
$$7$$ −49.0000 −0.377964
$$8$$ 0 0
$$9$$ −239.000 −0.983539
$$10$$ 0 0
$$11$$ 720.000 1.79412 0.897059 0.441912i $$-0.145700\pi$$
0.897059 + 0.441912i $$0.145700\pi$$
$$12$$ 0 0
$$13$$ 572.000 0.938723 0.469362 0.883006i $$-0.344484\pi$$
0.469362 + 0.883006i $$0.344484\pi$$
$$14$$ 0 0
$$15$$ −192.000 −0.220330
$$16$$ 0 0
$$17$$ 1254.00 1.05239 0.526193 0.850365i $$-0.323619\pi$$
0.526193 + 0.850365i $$0.323619\pi$$
$$18$$ 0 0
$$19$$ 94.0000 0.0597371 0.0298685 0.999554i $$-0.490491\pi$$
0.0298685 + 0.999554i $$0.490491\pi$$
$$20$$ 0 0
$$21$$ −98.0000 −0.0484929
$$22$$ 0 0
$$23$$ −96.0000 −0.0378400 −0.0189200 0.999821i $$-0.506023\pi$$
−0.0189200 + 0.999821i $$0.506023\pi$$
$$24$$ 0 0
$$25$$ 6091.00 1.94912
$$26$$ 0 0
$$27$$ −964.000 −0.254488
$$28$$ 0 0
$$29$$ −4374.00 −0.965792 −0.482896 0.875678i $$-0.660415\pi$$
−0.482896 + 0.875678i $$0.660415\pi$$
$$30$$ 0 0
$$31$$ 6244.00 1.16697 0.583484 0.812125i $$-0.301689\pi$$
0.583484 + 0.812125i $$0.301689\pi$$
$$32$$ 0 0
$$33$$ 1440.00 0.230185
$$34$$ 0 0
$$35$$ 4704.00 0.649078
$$36$$ 0 0
$$37$$ −10798.0 −1.29670 −0.648349 0.761343i $$-0.724540\pi$$
−0.648349 + 0.761343i $$0.724540\pi$$
$$38$$ 0 0
$$39$$ 1144.00 0.120438
$$40$$ 0 0
$$41$$ 12006.0 1.11542 0.557710 0.830036i $$-0.311680\pi$$
0.557710 + 0.830036i $$0.311680\pi$$
$$42$$ 0 0
$$43$$ 9160.00 0.755482 0.377741 0.925911i $$-0.376701\pi$$
0.377741 + 0.925911i $$0.376701\pi$$
$$44$$ 0 0
$$45$$ 22944.0 1.68903
$$46$$ 0 0
$$47$$ 25836.0 1.70601 0.853003 0.521906i $$-0.174779\pi$$
0.853003 + 0.521906i $$0.174779\pi$$
$$48$$ 0 0
$$49$$ 2401.00 0.142857
$$50$$ 0 0
$$51$$ 2508.00 0.135021
$$52$$ 0 0
$$53$$ 1014.00 0.0495848 0.0247924 0.999693i $$-0.492108\pi$$
0.0247924 + 0.999693i $$0.492108\pi$$
$$54$$ 0 0
$$55$$ −69120.0 −3.08104
$$56$$ 0 0
$$57$$ 188.000 0.00766427
$$58$$ 0 0
$$59$$ −1242.00 −0.0464506 −0.0232253 0.999730i $$-0.507394\pi$$
−0.0232253 + 0.999730i $$0.507394\pi$$
$$60$$ 0 0
$$61$$ 7592.00 0.261235 0.130618 0.991433i $$-0.458304\pi$$
0.130618 + 0.991433i $$0.458304\pi$$
$$62$$ 0 0
$$63$$ 11711.0 0.371743
$$64$$ 0 0
$$65$$ −54912.0 −1.61207
$$66$$ 0 0
$$67$$ −41132.0 −1.11942 −0.559710 0.828689i $$-0.689087\pi$$
−0.559710 + 0.828689i $$0.689087\pi$$
$$68$$ 0 0
$$69$$ −192.000 −0.00485488
$$70$$ 0 0
$$71$$ 37632.0 0.885955 0.442977 0.896533i $$-0.353922\pi$$
0.442977 + 0.896533i $$0.353922\pi$$
$$72$$ 0 0
$$73$$ −13438.0 −0.295140 −0.147570 0.989052i $$-0.547145\pi$$
−0.147570 + 0.989052i $$0.547145\pi$$
$$74$$ 0 0
$$75$$ 12182.0 0.250072
$$76$$ 0 0
$$77$$ −35280.0 −0.678113
$$78$$ 0 0
$$79$$ −6248.00 −0.112635 −0.0563175 0.998413i $$-0.517936\pi$$
−0.0563175 + 0.998413i $$0.517936\pi$$
$$80$$ 0 0
$$81$$ 56149.0 0.950888
$$82$$ 0 0
$$83$$ 25254.0 0.402379 0.201189 0.979552i $$-0.435519\pi$$
0.201189 + 0.979552i $$0.435519\pi$$
$$84$$ 0 0
$$85$$ −120384. −1.80726
$$86$$ 0 0
$$87$$ −8748.00 −0.123911
$$88$$ 0 0
$$89$$ −45126.0 −0.603882 −0.301941 0.953327i $$-0.597635\pi$$
−0.301941 + 0.953327i $$0.597635\pi$$
$$90$$ 0 0
$$91$$ −28028.0 −0.354804
$$92$$ 0 0
$$93$$ 12488.0 0.149722
$$94$$ 0 0
$$95$$ −9024.00 −0.102586
$$96$$ 0 0
$$97$$ 107222. 1.15706 0.578528 0.815662i $$-0.303627\pi$$
0.578528 + 0.815662i $$0.303627\pi$$
$$98$$ 0 0
$$99$$ −172080. −1.76458
$$100$$ 0 0
$$101$$ 47136.0 0.459779 0.229890 0.973217i $$-0.426163\pi$$
0.229890 + 0.973217i $$0.426163\pi$$
$$102$$ 0 0
$$103$$ −122204. −1.13499 −0.567495 0.823377i $$-0.692088\pi$$
−0.567495 + 0.823377i $$0.692088\pi$$
$$104$$ 0 0
$$105$$ 9408.00 0.0832768
$$106$$ 0 0
$$107$$ 129636. 1.09463 0.547314 0.836928i $$-0.315651\pi$$
0.547314 + 0.836928i $$0.315651\pi$$
$$108$$ 0 0
$$109$$ −220558. −1.77810 −0.889051 0.457809i $$-0.848635\pi$$
−0.889051 + 0.457809i $$0.848635\pi$$
$$110$$ 0 0
$$111$$ −21596.0 −0.166366
$$112$$ 0 0
$$113$$ 170694. 1.25754 0.628770 0.777591i $$-0.283559\pi$$
0.628770 + 0.777591i $$0.283559\pi$$
$$114$$ 0 0
$$115$$ 9216.00 0.0649827
$$116$$ 0 0
$$117$$ −136708. −0.923271
$$118$$ 0 0
$$119$$ −61446.0 −0.397765
$$120$$ 0 0
$$121$$ 357349. 2.21886
$$122$$ 0 0
$$123$$ 24012.0 0.143109
$$124$$ 0 0
$$125$$ −284736. −1.62992
$$126$$ 0 0
$$127$$ 249808. 1.37435 0.687175 0.726492i $$-0.258851\pi$$
0.687175 + 0.726492i $$0.258851\pi$$
$$128$$ 0 0
$$129$$ 18320.0 0.0969284
$$130$$ 0 0
$$131$$ −12210.0 −0.0621638 −0.0310819 0.999517i $$-0.509895\pi$$
−0.0310819 + 0.999517i $$0.509895\pi$$
$$132$$ 0 0
$$133$$ −4606.00 −0.0225785
$$134$$ 0 0
$$135$$ 92544.0 0.437033
$$136$$ 0 0
$$137$$ −13902.0 −0.0632814 −0.0316407 0.999499i $$-0.510073\pi$$
−0.0316407 + 0.999499i $$0.510073\pi$$
$$138$$ 0 0
$$139$$ 431794. 1.89557 0.947785 0.318911i $$-0.103317\pi$$
0.947785 + 0.318911i $$0.103317\pi$$
$$140$$ 0 0
$$141$$ 51672.0 0.218881
$$142$$ 0 0
$$143$$ 411840. 1.68418
$$144$$ 0 0
$$145$$ 419904. 1.65856
$$146$$ 0 0
$$147$$ 4802.00 0.0183286
$$148$$ 0 0
$$149$$ 326814. 1.20597 0.602983 0.797754i $$-0.293979\pi$$
0.602983 + 0.797754i $$0.293979\pi$$
$$150$$ 0 0
$$151$$ −173480. −0.619166 −0.309583 0.950872i $$-0.600189\pi$$
−0.309583 + 0.950872i $$0.600189\pi$$
$$152$$ 0 0
$$153$$ −299706. −1.03506
$$154$$ 0 0
$$155$$ −599424. −2.00403
$$156$$ 0 0
$$157$$ −54532.0 −0.176564 −0.0882820 0.996096i $$-0.528138\pi$$
−0.0882820 + 0.996096i $$0.528138\pi$$
$$158$$ 0 0
$$159$$ 2028.00 0.00636173
$$160$$ 0 0
$$161$$ 4704.00 0.0143022
$$162$$ 0 0
$$163$$ −104960. −0.309425 −0.154712 0.987960i $$-0.549445\pi$$
−0.154712 + 0.987960i $$0.549445\pi$$
$$164$$ 0 0
$$165$$ −138240. −0.395297
$$166$$ 0 0
$$167$$ −160788. −0.446131 −0.223066 0.974803i $$-0.571606\pi$$
−0.223066 + 0.974803i $$0.571606\pi$$
$$168$$ 0 0
$$169$$ −44109.0 −0.118798
$$170$$ 0 0
$$171$$ −22466.0 −0.0587537
$$172$$ 0 0
$$173$$ 360564. 0.915940 0.457970 0.888968i $$-0.348577\pi$$
0.457970 + 0.888968i $$0.348577\pi$$
$$174$$ 0 0
$$175$$ −298459. −0.736698
$$176$$ 0 0
$$177$$ −2484.00 −0.00595962
$$178$$ 0 0
$$179$$ 312732. 0.729524 0.364762 0.931101i $$-0.381150\pi$$
0.364762 + 0.931101i $$0.381150\pi$$
$$180$$ 0 0
$$181$$ −123820. −0.280928 −0.140464 0.990086i $$-0.544859\pi$$
−0.140464 + 0.990086i $$0.544859\pi$$
$$182$$ 0 0
$$183$$ 15184.0 0.0335165
$$184$$ 0 0
$$185$$ 1.03661e6 2.22682
$$186$$ 0 0
$$187$$ 902880. 1.88810
$$188$$ 0 0
$$189$$ 47236.0 0.0961875
$$190$$ 0 0
$$191$$ −323448. −0.641536 −0.320768 0.947158i $$-0.603941\pi$$
−0.320768 + 0.947158i $$0.603941\pi$$
$$192$$ 0 0
$$193$$ −619954. −1.19803 −0.599013 0.800739i $$-0.704440\pi$$
−0.599013 + 0.800739i $$0.704440\pi$$
$$194$$ 0 0
$$195$$ −109824. −0.206829
$$196$$ 0 0
$$197$$ −499362. −0.916748 −0.458374 0.888759i $$-0.651568\pi$$
−0.458374 + 0.888759i $$0.651568\pi$$
$$198$$ 0 0
$$199$$ 785932. 1.40686 0.703432 0.710762i $$-0.251650\pi$$
0.703432 + 0.710762i $$0.251650\pi$$
$$200$$ 0 0
$$201$$ −82264.0 −0.143622
$$202$$ 0 0
$$203$$ 214326. 0.365035
$$204$$ 0 0
$$205$$ −1.15258e6 −1.91551
$$206$$ 0 0
$$207$$ 22944.0 0.0372172
$$208$$ 0 0
$$209$$ 67680.0 0.107175
$$210$$ 0 0
$$211$$ −1.06276e6 −1.64335 −0.821676 0.569955i $$-0.806961\pi$$
−0.821676 + 0.569955i $$0.806961\pi$$
$$212$$ 0 0
$$213$$ 75264.0 0.113668
$$214$$ 0 0
$$215$$ −879360. −1.29739
$$216$$ 0 0
$$217$$ −305956. −0.441072
$$218$$ 0 0
$$219$$ −26876.0 −0.0378664
$$220$$ 0 0
$$221$$ 717288. 0.987900
$$222$$ 0 0
$$223$$ −707720. −0.953014 −0.476507 0.879171i $$-0.658097\pi$$
−0.476507 + 0.879171i $$0.658097\pi$$
$$224$$ 0 0
$$225$$ −1.45575e6 −1.91704
$$226$$ 0 0
$$227$$ 1.04437e6 1.34520 0.672602 0.740005i $$-0.265177\pi$$
0.672602 + 0.740005i $$0.265177\pi$$
$$228$$ 0 0
$$229$$ −539716. −0.680106 −0.340053 0.940406i $$-0.610445\pi$$
−0.340053 + 0.940406i $$0.610445\pi$$
$$230$$ 0 0
$$231$$ −70560.0 −0.0870019
$$232$$ 0 0
$$233$$ 177114. 0.213729 0.106864 0.994274i $$-0.465919\pi$$
0.106864 + 0.994274i $$0.465919\pi$$
$$234$$ 0 0
$$235$$ −2.48026e6 −2.92972
$$236$$ 0 0
$$237$$ −12496.0 −0.0144511
$$238$$ 0 0
$$239$$ 655464. 0.742257 0.371128 0.928582i $$-0.378971\pi$$
0.371128 + 0.928582i $$0.378971\pi$$
$$240$$ 0 0
$$241$$ 1.38709e6 1.53838 0.769189 0.639021i $$-0.220660\pi$$
0.769189 + 0.639021i $$0.220660\pi$$
$$242$$ 0 0
$$243$$ 346550. 0.376487
$$244$$ 0 0
$$245$$ −230496. −0.245329
$$246$$ 0 0
$$247$$ 53768.0 0.0560766
$$248$$ 0 0
$$249$$ 50508.0 0.0516252
$$250$$ 0 0
$$251$$ −1.88811e6 −1.89166 −0.945830 0.324663i $$-0.894749\pi$$
−0.945830 + 0.324663i $$0.894749\pi$$
$$252$$ 0 0
$$253$$ −69120.0 −0.0678895
$$254$$ 0 0
$$255$$ −240768. −0.231872
$$256$$ 0 0
$$257$$ 346194. 0.326954 0.163477 0.986547i $$-0.447729\pi$$
0.163477 + 0.986547i $$0.447729\pi$$
$$258$$ 0 0
$$259$$ 529102. 0.490106
$$260$$ 0 0
$$261$$ 1.04539e6 0.949895
$$262$$ 0 0
$$263$$ 929088. 0.828262 0.414131 0.910217i $$-0.364086\pi$$
0.414131 + 0.910217i $$0.364086\pi$$
$$264$$ 0 0
$$265$$ −97344.0 −0.0851519
$$266$$ 0 0
$$267$$ −90252.0 −0.0774780
$$268$$ 0 0
$$269$$ 382068. 0.321929 0.160964 0.986960i $$-0.448540\pi$$
0.160964 + 0.986960i $$0.448540\pi$$
$$270$$ 0 0
$$271$$ 1.58056e6 1.30734 0.653669 0.756781i $$-0.273229\pi$$
0.653669 + 0.756781i $$0.273229\pi$$
$$272$$ 0 0
$$273$$ −56056.0 −0.0455214
$$274$$ 0 0
$$275$$ 4.38552e6 3.49695
$$276$$ 0 0
$$277$$ −1.36911e6 −1.07211 −0.536056 0.844182i $$-0.680086\pi$$
−0.536056 + 0.844182i $$0.680086\pi$$
$$278$$ 0 0
$$279$$ −1.49232e6 −1.14776
$$280$$ 0 0
$$281$$ −394854. −0.298312 −0.149156 0.988814i $$-0.547656\pi$$
−0.149156 + 0.988814i $$0.547656\pi$$
$$282$$ 0 0
$$283$$ −673034. −0.499541 −0.249770 0.968305i $$-0.580355\pi$$
−0.249770 + 0.968305i $$0.580355\pi$$
$$284$$ 0 0
$$285$$ −18048.0 −0.0131618
$$286$$ 0 0
$$287$$ −588294. −0.421589
$$288$$ 0 0
$$289$$ 152659. 0.107517
$$290$$ 0 0
$$291$$ 214444. 0.148450
$$292$$ 0 0
$$293$$ 1.83468e6 1.24851 0.624254 0.781222i $$-0.285403\pi$$
0.624254 + 0.781222i $$0.285403\pi$$
$$294$$ 0 0
$$295$$ 119232. 0.0797697
$$296$$ 0 0
$$297$$ −694080. −0.456582
$$298$$ 0 0
$$299$$ −54912.0 −0.0355213
$$300$$ 0 0
$$301$$ −448840. −0.285545
$$302$$ 0 0
$$303$$ 94272.0 0.0589897
$$304$$ 0 0
$$305$$ −728832. −0.448619
$$306$$ 0 0
$$307$$ 1.51056e6 0.914727 0.457363 0.889280i $$-0.348794\pi$$
0.457363 + 0.889280i $$0.348794\pi$$
$$308$$ 0 0
$$309$$ −244408. −0.145619
$$310$$ 0 0
$$311$$ 1.87529e6 1.09943 0.549714 0.835353i $$-0.314737\pi$$
0.549714 + 0.835353i $$0.314737\pi$$
$$312$$ 0 0
$$313$$ −1.51076e6 −0.871636 −0.435818 0.900035i $$-0.643541\pi$$
−0.435818 + 0.900035i $$0.643541\pi$$
$$314$$ 0 0
$$315$$ −1.12426e6 −0.638394
$$316$$ 0 0
$$317$$ 2.02709e6 1.13299 0.566495 0.824065i $$-0.308299\pi$$
0.566495 + 0.824065i $$0.308299\pi$$
$$318$$ 0 0
$$319$$ −3.14928e6 −1.73274
$$320$$ 0 0
$$321$$ 259272. 0.140441
$$322$$ 0 0
$$323$$ 117876. 0.0628665
$$324$$ 0 0
$$325$$ 3.48405e6 1.82968
$$326$$ 0 0
$$327$$ −441116. −0.228131
$$328$$ 0 0
$$329$$ −1.26596e6 −0.644810
$$330$$ 0 0
$$331$$ −1.54009e6 −0.772637 −0.386319 0.922365i $$-0.626253\pi$$
−0.386319 + 0.922365i $$0.626253\pi$$
$$332$$ 0 0
$$333$$ 2.58072e6 1.27535
$$334$$ 0 0
$$335$$ 3.94867e6 1.92238
$$336$$ 0 0
$$337$$ 1.01166e6 0.485245 0.242622 0.970121i $$-0.421992\pi$$
0.242622 + 0.970121i $$0.421992\pi$$
$$338$$ 0 0
$$339$$ 341388. 0.161343
$$340$$ 0 0
$$341$$ 4.49568e6 2.09368
$$342$$ 0 0
$$343$$ −117649. −0.0539949
$$344$$ 0 0
$$345$$ 18432.0 0.00833729
$$346$$ 0 0
$$347$$ 2.15748e6 0.961885 0.480942 0.876752i $$-0.340295\pi$$
0.480942 + 0.876752i $$0.340295\pi$$
$$348$$ 0 0
$$349$$ −1.15798e6 −0.508906 −0.254453 0.967085i $$-0.581895\pi$$
−0.254453 + 0.967085i $$0.581895\pi$$
$$350$$ 0 0
$$351$$ −551408. −0.238894
$$352$$ 0 0
$$353$$ −3.17566e6 −1.35643 −0.678215 0.734863i $$-0.737246\pi$$
−0.678215 + 0.734863i $$0.737246\pi$$
$$354$$ 0 0
$$355$$ −3.61267e6 −1.52145
$$356$$ 0 0
$$357$$ −122892. −0.0510332
$$358$$ 0 0
$$359$$ 74616.0 0.0305560 0.0152780 0.999883i $$-0.495137\pi$$
0.0152780 + 0.999883i $$0.495137\pi$$
$$360$$ 0 0
$$361$$ −2.46726e6 −0.996431
$$362$$ 0 0
$$363$$ 714698. 0.284679
$$364$$ 0 0
$$365$$ 1.29005e6 0.506843
$$366$$ 0 0
$$367$$ 1.79807e6 0.696854 0.348427 0.937336i $$-0.386716\pi$$
0.348427 + 0.937336i $$0.386716\pi$$
$$368$$ 0 0
$$369$$ −2.86943e6 −1.09706
$$370$$ 0 0
$$371$$ −49686.0 −0.0187413
$$372$$ 0 0
$$373$$ 2.20461e6 0.820463 0.410231 0.911981i $$-0.365448\pi$$
0.410231 + 0.911981i $$0.365448\pi$$
$$374$$ 0 0
$$375$$ −569472. −0.209119
$$376$$ 0 0
$$377$$ −2.50193e6 −0.906612
$$378$$ 0 0
$$379$$ 177568. 0.0634990 0.0317495 0.999496i $$-0.489892\pi$$
0.0317495 + 0.999496i $$0.489892\pi$$
$$380$$ 0 0
$$381$$ 499616. 0.176329
$$382$$ 0 0
$$383$$ 2.87468e6 1.00137 0.500683 0.865630i $$-0.333082\pi$$
0.500683 + 0.865630i $$0.333082\pi$$
$$384$$ 0 0
$$385$$ 3.38688e6 1.16452
$$386$$ 0 0
$$387$$ −2.18924e6 −0.743046
$$388$$ 0 0
$$389$$ −4.79965e6 −1.60818 −0.804091 0.594506i $$-0.797348\pi$$
−0.804091 + 0.594506i $$0.797348\pi$$
$$390$$ 0 0
$$391$$ −120384. −0.0398223
$$392$$ 0 0
$$393$$ −24420.0 −0.00797562
$$394$$ 0 0
$$395$$ 599808. 0.193428
$$396$$ 0 0
$$397$$ −2.81643e6 −0.896855 −0.448428 0.893819i $$-0.648016\pi$$
−0.448428 + 0.893819i $$0.648016\pi$$
$$398$$ 0 0
$$399$$ −9212.00 −0.00289682
$$400$$ 0 0
$$401$$ −2.83797e6 −0.881347 −0.440673 0.897667i $$-0.645260\pi$$
−0.440673 + 0.897667i $$0.645260\pi$$
$$402$$ 0 0
$$403$$ 3.57157e6 1.09546
$$404$$ 0 0
$$405$$ −5.39030e6 −1.63296
$$406$$ 0 0
$$407$$ −7.77456e6 −2.32643
$$408$$ 0 0
$$409$$ 154286. 0.0456056 0.0228028 0.999740i $$-0.492741\pi$$
0.0228028 + 0.999740i $$0.492741\pi$$
$$410$$ 0 0
$$411$$ −27804.0 −0.00811900
$$412$$ 0 0
$$413$$ 60858.0 0.0175567
$$414$$ 0 0
$$415$$ −2.42438e6 −0.691005
$$416$$ 0 0
$$417$$ 863588. 0.243202
$$418$$ 0 0
$$419$$ −3.72865e6 −1.03757 −0.518783 0.854906i $$-0.673615\pi$$
−0.518783 + 0.854906i $$0.673615\pi$$
$$420$$ 0 0
$$421$$ −2.32623e6 −0.639658 −0.319829 0.947475i $$-0.603626\pi$$
−0.319829 + 0.947475i $$0.603626\pi$$
$$422$$ 0 0
$$423$$ −6.17480e6 −1.67792
$$424$$ 0 0
$$425$$ 7.63811e6 2.05123
$$426$$ 0 0
$$427$$ −372008. −0.0987376
$$428$$ 0 0
$$429$$ 823680. 0.216080
$$430$$ 0 0
$$431$$ 2.61482e6 0.678031 0.339015 0.940781i $$-0.389906\pi$$
0.339015 + 0.940781i $$0.389906\pi$$
$$432$$ 0 0
$$433$$ −1.19226e6 −0.305598 −0.152799 0.988257i $$-0.548829\pi$$
−0.152799 + 0.988257i $$0.548829\pi$$
$$434$$ 0 0
$$435$$ 839808. 0.212793
$$436$$ 0 0
$$437$$ −9024.00 −0.00226045
$$438$$ 0 0
$$439$$ −1.05793e6 −0.261996 −0.130998 0.991383i $$-0.541818\pi$$
−0.130998 + 0.991383i $$0.541818\pi$$
$$440$$ 0 0
$$441$$ −573839. −0.140506
$$442$$ 0 0
$$443$$ −4.12756e6 −0.999272 −0.499636 0.866235i $$-0.666533\pi$$
−0.499636 + 0.866235i $$0.666533\pi$$
$$444$$ 0 0
$$445$$ 4.33210e6 1.03705
$$446$$ 0 0
$$447$$ 653628. 0.154725
$$448$$ 0 0
$$449$$ 3.75823e6 0.879766 0.439883 0.898055i $$-0.355020\pi$$
0.439883 + 0.898055i $$0.355020\pi$$
$$450$$ 0 0
$$451$$ 8.64432e6 2.00120
$$452$$ 0 0
$$453$$ −346960. −0.0794390
$$454$$ 0 0
$$455$$ 2.69069e6 0.609305
$$456$$ 0 0
$$457$$ −451114. −0.101041 −0.0505203 0.998723i $$-0.516088\pi$$
−0.0505203 + 0.998723i $$0.516088\pi$$
$$458$$ 0 0
$$459$$ −1.20886e6 −0.267820
$$460$$ 0 0
$$461$$ −1.95186e6 −0.427756 −0.213878 0.976860i $$-0.568610\pi$$
−0.213878 + 0.976860i $$0.568610\pi$$
$$462$$ 0 0
$$463$$ 7.20218e6 1.56139 0.780695 0.624913i $$-0.214865\pi$$
0.780695 + 0.624913i $$0.214865\pi$$
$$464$$ 0 0
$$465$$ −1.19885e6 −0.257118
$$466$$ 0 0
$$467$$ −7.17801e6 −1.52304 −0.761521 0.648140i $$-0.775547\pi$$
−0.761521 + 0.648140i $$0.775547\pi$$
$$468$$ 0 0
$$469$$ 2.01547e6 0.423101
$$470$$ 0 0
$$471$$ −109064. −0.0226532
$$472$$ 0 0
$$473$$ 6.59520e6 1.35542
$$474$$ 0 0
$$475$$ 572554. 0.116435
$$476$$ 0 0
$$477$$ −242346. −0.0487686
$$478$$ 0 0
$$479$$ −6.17632e6 −1.22996 −0.614980 0.788543i $$-0.710836\pi$$
−0.614980 + 0.788543i $$0.710836\pi$$
$$480$$ 0 0
$$481$$ −6.17646e6 −1.21724
$$482$$ 0 0
$$483$$ 9408.00 0.00183497
$$484$$ 0 0
$$485$$ −1.02933e7 −1.98701
$$486$$ 0 0
$$487$$ −7.59330e6 −1.45080 −0.725401 0.688327i $$-0.758345\pi$$
−0.725401 + 0.688327i $$0.758345\pi$$
$$488$$ 0 0
$$489$$ −209920. −0.0396992
$$490$$ 0 0
$$491$$ 1.51878e6 0.284309 0.142155 0.989844i $$-0.454597\pi$$
0.142155 + 0.989844i $$0.454597\pi$$
$$492$$ 0 0
$$493$$ −5.48500e6 −1.01639
$$494$$ 0 0
$$495$$ 1.65197e7 3.03032
$$496$$ 0 0
$$497$$ −1.84397e6 −0.334859
$$498$$ 0 0
$$499$$ −1.47576e6 −0.265316 −0.132658 0.991162i $$-0.542351\pi$$
−0.132658 + 0.991162i $$0.542351\pi$$
$$500$$ 0 0
$$501$$ −321576. −0.0572386
$$502$$ 0 0
$$503$$ 1.31309e6 0.231406 0.115703 0.993284i $$-0.463088\pi$$
0.115703 + 0.993284i $$0.463088\pi$$
$$504$$ 0 0
$$505$$ −4.52506e6 −0.789579
$$506$$ 0 0
$$507$$ −88218.0 −0.0152418
$$508$$ 0 0
$$509$$ 4.40932e6 0.754357 0.377178 0.926141i $$-0.376894\pi$$
0.377178 + 0.926141i $$0.376894\pi$$
$$510$$ 0 0
$$511$$ 658462. 0.111552
$$512$$ 0 0
$$513$$ −90616.0 −0.0152024
$$514$$ 0 0
$$515$$ 1.17316e7 1.94912
$$516$$ 0 0
$$517$$ 1.86019e7 3.06078
$$518$$ 0 0
$$519$$ 721128. 0.117515
$$520$$ 0 0
$$521$$ 2.97629e6 0.480376 0.240188 0.970726i $$-0.422791\pi$$
0.240188 + 0.970726i $$0.422791\pi$$
$$522$$ 0 0
$$523$$ −6.34627e6 −1.01453 −0.507265 0.861790i $$-0.669343\pi$$
−0.507265 + 0.861790i $$0.669343\pi$$
$$524$$ 0 0
$$525$$ −596918. −0.0945184
$$526$$ 0 0
$$527$$ 7.82998e6 1.22810
$$528$$ 0 0
$$529$$ −6.42713e6 −0.998568
$$530$$ 0 0
$$531$$ 296838. 0.0456860
$$532$$ 0 0
$$533$$ 6.86743e6 1.04707
$$534$$ 0 0
$$535$$ −1.24451e7 −1.87980
$$536$$ 0 0
$$537$$ 625464. 0.0935980
$$538$$ 0 0
$$539$$ 1.72872e6 0.256302
$$540$$ 0 0
$$541$$ −1.36667e6 −0.200756 −0.100378 0.994949i $$-0.532005\pi$$
−0.100378 + 0.994949i $$0.532005\pi$$
$$542$$ 0 0
$$543$$ −247640. −0.0360430
$$544$$ 0 0
$$545$$ 2.11736e7 3.05353
$$546$$ 0 0
$$547$$ 9.55818e6 1.36586 0.682931 0.730483i $$-0.260705\pi$$
0.682931 + 0.730483i $$0.260705\pi$$
$$548$$ 0 0
$$549$$ −1.81449e6 −0.256935
$$550$$ 0 0
$$551$$ −411156. −0.0576936
$$552$$ 0 0
$$553$$ 306152. 0.0425720
$$554$$ 0 0
$$555$$ 2.07322e6 0.285701
$$556$$ 0 0
$$557$$ 6.94287e6 0.948202 0.474101 0.880470i $$-0.342773\pi$$
0.474101 + 0.880470i $$0.342773\pi$$
$$558$$ 0 0
$$559$$ 5.23952e6 0.709189
$$560$$ 0 0
$$561$$ 1.80576e6 0.242244
$$562$$ 0 0
$$563$$ −5.24662e6 −0.697604 −0.348802 0.937196i $$-0.613411\pi$$
−0.348802 + 0.937196i $$0.613411\pi$$
$$564$$ 0 0
$$565$$ −1.63866e7 −2.15958
$$566$$ 0 0
$$567$$ −2.75130e6 −0.359402
$$568$$ 0 0
$$569$$ 3.46551e6 0.448731 0.224366 0.974505i $$-0.427969\pi$$
0.224366 + 0.974505i $$0.427969\pi$$
$$570$$ 0 0
$$571$$ −4.90069e6 −0.629023 −0.314512 0.949254i $$-0.601841\pi$$
−0.314512 + 0.949254i $$0.601841\pi$$
$$572$$ 0 0
$$573$$ −646896. −0.0823091
$$574$$ 0 0
$$575$$ −584736. −0.0737548
$$576$$ 0 0
$$577$$ 2.28346e6 0.285531 0.142766 0.989757i $$-0.454401\pi$$
0.142766 + 0.989757i $$0.454401\pi$$
$$578$$ 0 0
$$579$$ −1.23991e6 −0.153707
$$580$$ 0 0
$$581$$ −1.23745e6 −0.152085
$$582$$ 0 0
$$583$$ 730080. 0.0889609
$$584$$ 0 0
$$585$$ 1.31240e7 1.58553
$$586$$ 0 0
$$587$$ −1.03157e7 −1.23568 −0.617838 0.786305i $$-0.711991\pi$$
−0.617838 + 0.786305i $$0.711991\pi$$
$$588$$ 0 0
$$589$$ 586936. 0.0697112
$$590$$ 0 0
$$591$$ −998724. −0.117619
$$592$$ 0 0
$$593$$ 3.52838e6 0.412039 0.206020 0.978548i $$-0.433949\pi$$
0.206020 + 0.978548i $$0.433949\pi$$
$$594$$ 0 0
$$595$$ 5.89882e6 0.683081
$$596$$ 0 0
$$597$$ 1.57186e6 0.180501
$$598$$ 0 0
$$599$$ −2.92260e6 −0.332815 −0.166407 0.986057i $$-0.553217\pi$$
−0.166407 + 0.986057i $$0.553217\pi$$
$$600$$ 0 0
$$601$$ −1.17567e7 −1.32770 −0.663849 0.747866i $$-0.731078\pi$$
−0.663849 + 0.747866i $$0.731078\pi$$
$$602$$ 0 0
$$603$$ 9.83055e6 1.10099
$$604$$ 0 0
$$605$$ −3.43055e7 −3.81044
$$606$$ 0 0
$$607$$ 4.71491e6 0.519400 0.259700 0.965689i $$-0.416376\pi$$
0.259700 + 0.965689i $$0.416376\pi$$
$$608$$ 0 0
$$609$$ 428652. 0.0468340
$$610$$ 0 0
$$611$$ 1.47782e7 1.60147
$$612$$ 0 0
$$613$$ 213842. 0.0229849 0.0114924 0.999934i $$-0.496342\pi$$
0.0114924 + 0.999934i $$0.496342\pi$$
$$614$$ 0 0
$$615$$ −2.30515e6 −0.245760
$$616$$ 0 0
$$617$$ −336666. −0.0356030 −0.0178015 0.999842i $$-0.505667\pi$$
−0.0178015 + 0.999842i $$0.505667\pi$$
$$618$$ 0 0
$$619$$ −1.42655e7 −1.49645 −0.748223 0.663447i $$-0.769093\pi$$
−0.748223 + 0.663447i $$0.769093\pi$$
$$620$$ 0 0
$$621$$ 92544.0 0.00962984
$$622$$ 0 0
$$623$$ 2.21117e6 0.228246
$$624$$ 0 0
$$625$$ 8.30028e6 0.849949
$$626$$ 0 0
$$627$$ 135360. 0.0137506
$$628$$ 0 0
$$629$$ −1.35407e7 −1.36463
$$630$$ 0 0
$$631$$ 6.59637e6 0.659525 0.329763 0.944064i $$-0.393031\pi$$
0.329763 + 0.944064i $$0.393031\pi$$
$$632$$ 0 0
$$633$$ −2.12553e6 −0.210842
$$634$$ 0 0
$$635$$ −2.39816e7 −2.36017
$$636$$ 0 0
$$637$$ 1.37337e6 0.134103
$$638$$ 0 0
$$639$$ −8.99405e6 −0.871371
$$640$$ 0 0
$$641$$ −1.02490e7 −0.985225 −0.492613 0.870249i $$-0.663958\pi$$
−0.492613 + 0.870249i $$0.663958\pi$$
$$642$$ 0 0
$$643$$ 4.16543e6 0.397312 0.198656 0.980069i $$-0.436342\pi$$
0.198656 + 0.980069i $$0.436342\pi$$
$$644$$ 0 0
$$645$$ −1.75872e6 −0.166455
$$646$$ 0 0
$$647$$ 3.35051e6 0.314666 0.157333 0.987546i $$-0.449710\pi$$
0.157333 + 0.987546i $$0.449710\pi$$
$$648$$ 0 0
$$649$$ −894240. −0.0833379
$$650$$ 0 0
$$651$$ −611912. −0.0565896
$$652$$ 0 0
$$653$$ −9.05408e6 −0.830924 −0.415462 0.909611i $$-0.636380\pi$$
−0.415462 + 0.909611i $$0.636380\pi$$
$$654$$ 0 0
$$655$$ 1.17216e6 0.106754
$$656$$ 0 0
$$657$$ 3.21168e6 0.290281
$$658$$ 0 0
$$659$$ −6.45382e6 −0.578899 −0.289450 0.957193i $$-0.593472\pi$$
−0.289450 + 0.957193i $$0.593472\pi$$
$$660$$ 0 0
$$661$$ 1.43167e7 1.27450 0.637250 0.770657i $$-0.280072\pi$$
0.637250 + 0.770657i $$0.280072\pi$$
$$662$$ 0 0
$$663$$ 1.43458e6 0.126748
$$664$$ 0 0
$$665$$ 442176. 0.0387740
$$666$$ 0 0
$$667$$ 419904. 0.0365456
$$668$$ 0 0
$$669$$ −1.41544e6 −0.122272
$$670$$ 0 0
$$671$$ 5.46624e6 0.468686
$$672$$ 0 0
$$673$$ 2.27250e7 1.93404 0.967020 0.254701i $$-0.0819771\pi$$
0.967020 + 0.254701i $$0.0819771\pi$$
$$674$$ 0 0
$$675$$ −5.87172e6 −0.496028
$$676$$ 0 0
$$677$$ −8.53249e6 −0.715491 −0.357746 0.933819i $$-0.616454\pi$$
−0.357746 + 0.933819i $$0.616454\pi$$
$$678$$ 0 0
$$679$$ −5.25388e6 −0.437326
$$680$$ 0 0
$$681$$ 2.08873e6 0.172590
$$682$$ 0 0
$$683$$ 2.24921e7 1.84492 0.922461 0.386090i $$-0.126175\pi$$
0.922461 + 0.386090i $$0.126175\pi$$
$$684$$ 0 0
$$685$$ 1.33459e6 0.108673
$$686$$ 0 0
$$687$$ −1.07943e6 −0.0872576
$$688$$ 0 0
$$689$$ 580008. 0.0465464
$$690$$ 0 0
$$691$$ 1.26894e7 1.01099 0.505495 0.862830i $$-0.331310\pi$$
0.505495 + 0.862830i $$0.331310\pi$$
$$692$$ 0 0
$$693$$ 8.43192e6 0.666950
$$694$$ 0 0
$$695$$ −4.14522e7 −3.25526
$$696$$ 0 0
$$697$$ 1.50555e7 1.17385
$$698$$ 0 0
$$699$$ 354228. 0.0274214
$$700$$ 0 0
$$701$$ −5.13939e6 −0.395018 −0.197509 0.980301i $$-0.563285\pi$$
−0.197509 + 0.980301i $$0.563285\pi$$
$$702$$ 0 0
$$703$$ −1.01501e6 −0.0774610
$$704$$ 0 0
$$705$$ −4.96051e6 −0.375884
$$706$$ 0 0
$$707$$ −2.30966e6 −0.173780
$$708$$ 0 0
$$709$$ −1.16065e7 −0.867132 −0.433566 0.901122i $$-0.642745\pi$$
−0.433566 + 0.901122i $$0.642745\pi$$
$$710$$ 0 0
$$711$$ 1.49327e6 0.110781
$$712$$ 0 0
$$713$$ −599424. −0.0441581
$$714$$ 0 0
$$715$$ −3.95366e7 −2.89224
$$716$$ 0 0
$$717$$ 1.31093e6 0.0952316
$$718$$ 0 0
$$719$$ −1.50998e7 −1.08930 −0.544650 0.838663i $$-0.683338\pi$$
−0.544650 + 0.838663i $$0.683338\pi$$
$$720$$ 0 0
$$721$$ 5.98800e6 0.428986
$$722$$ 0 0
$$723$$ 2.77419e6 0.197374
$$724$$ 0 0
$$725$$ −2.66420e7 −1.88245
$$726$$ 0 0
$$727$$ 2.32536e7 1.63175 0.815874 0.578229i $$-0.196256\pi$$
0.815874 + 0.578229i $$0.196256\pi$$
$$728$$ 0 0
$$729$$ −1.29511e7 −0.902585
$$730$$ 0 0
$$731$$ 1.14866e7 0.795059
$$732$$ 0 0
$$733$$ 2.37814e7 1.63485 0.817423 0.576038i $$-0.195402\pi$$
0.817423 + 0.576038i $$0.195402\pi$$
$$734$$ 0 0
$$735$$ −460992. −0.0314757
$$736$$ 0 0
$$737$$ −2.96150e7 −2.00837
$$738$$ 0 0
$$739$$ 2.51392e6 0.169333 0.0846663 0.996409i $$-0.473018\pi$$
0.0846663 + 0.996409i $$0.473018\pi$$
$$740$$ 0 0
$$741$$ 107536. 0.00719463
$$742$$ 0 0
$$743$$ 2.22646e7 1.47959 0.739797 0.672830i $$-0.234922\pi$$
0.739797 + 0.672830i $$0.234922\pi$$
$$744$$ 0 0
$$745$$ −3.13741e7 −2.07101
$$746$$ 0 0
$$747$$ −6.03571e6 −0.395755
$$748$$ 0 0
$$749$$ −6.35216e6 −0.413730
$$750$$ 0 0
$$751$$ −2.30108e7 −1.48878 −0.744392 0.667743i $$-0.767260\pi$$
−0.744392 + 0.667743i $$0.767260\pi$$
$$752$$ 0 0
$$753$$ −3.77622e6 −0.242700
$$754$$ 0 0
$$755$$ 1.66541e7 1.06329
$$756$$ 0 0
$$757$$ 1.59335e7 1.01058 0.505290 0.862950i $$-0.331386\pi$$
0.505290 + 0.862950i $$0.331386\pi$$
$$758$$ 0 0
$$759$$ −138240. −0.00871022
$$760$$ 0 0
$$761$$ 1.68629e7 1.05553 0.527764 0.849391i $$-0.323031\pi$$
0.527764 + 0.849391i $$0.323031\pi$$
$$762$$ 0 0
$$763$$ 1.08073e7 0.672059
$$764$$ 0 0
$$765$$ 2.87718e7 1.77751
$$766$$ 0 0
$$767$$ −710424. −0.0436043
$$768$$ 0 0
$$769$$ −2.75402e7 −1.67939 −0.839694 0.543060i $$-0.817265\pi$$
−0.839694 + 0.543060i $$0.817265\pi$$
$$770$$ 0 0
$$771$$ 692388. 0.0419482
$$772$$ 0 0
$$773$$ 2.26820e7 1.36532 0.682658 0.730738i $$-0.260824\pi$$
0.682658 + 0.730738i $$0.260824\pi$$
$$774$$ 0 0
$$775$$ 3.80322e7 2.27456
$$776$$ 0 0
$$777$$ 1.05820e6 0.0628806
$$778$$ 0 0
$$779$$ 1.12856e6 0.0666320
$$780$$ 0 0
$$781$$ 2.70950e7 1.58951
$$782$$ 0 0
$$783$$ 4.21654e6 0.245783
$$784$$ 0 0
$$785$$ 5.23507e6 0.303213
$$786$$ 0 0
$$787$$ 2.34266e7 1.34826 0.674129 0.738614i $$-0.264519\pi$$
0.674129 + 0.738614i $$0.264519\pi$$
$$788$$ 0 0
$$789$$ 1.85818e6 0.106266
$$790$$ 0 0
$$791$$ −8.36401e6 −0.475306
$$792$$ 0 0
$$793$$ 4.34262e6 0.245228
$$794$$ 0 0
$$795$$ −194688. −0.0109250
$$796$$ 0 0
$$797$$ −1.82051e7 −1.01519 −0.507594 0.861596i $$-0.669465\pi$$
−0.507594 + 0.861596i $$0.669465\pi$$
$$798$$ 0 0
$$799$$ 3.23983e7 1.79538
$$800$$ 0 0
$$801$$ 1.07851e7 0.593941
$$802$$ 0 0
$$803$$ −9.67536e6 −0.529515
$$804$$ 0 0
$$805$$ −451584. −0.0245612
$$806$$ 0 0
$$807$$ 764136. 0.0413035
$$808$$ 0 0
$$809$$ 1.47411e7 0.791878 0.395939 0.918277i $$-0.370419\pi$$
0.395939 + 0.918277i $$0.370419\pi$$
$$810$$ 0 0
$$811$$ −1.69629e7 −0.905625 −0.452812 0.891606i $$-0.649579\pi$$
−0.452812 + 0.891606i $$0.649579\pi$$
$$812$$ 0 0
$$813$$ 3.16112e6 0.167731
$$814$$ 0 0
$$815$$ 1.00762e7 0.531375
$$816$$ 0 0
$$817$$ 861040. 0.0451303
$$818$$ 0 0
$$819$$ 6.69869e6 0.348964
$$820$$ 0 0
$$821$$ 8.03929e6 0.416255 0.208128 0.978102i $$-0.433263\pi$$
0.208128 + 0.978102i $$0.433263\pi$$
$$822$$ 0 0
$$823$$ −386648. −0.0198983 −0.00994915 0.999951i $$-0.503167\pi$$
−0.00994915 + 0.999951i $$0.503167\pi$$
$$824$$ 0 0
$$825$$ 8.77104e6 0.448659
$$826$$ 0 0
$$827$$ −3.55021e7 −1.80505 −0.902526 0.430635i $$-0.858290\pi$$
−0.902526 + 0.430635i $$0.858290\pi$$
$$828$$ 0 0
$$829$$ 2.48814e7 1.25745 0.628723 0.777630i $$-0.283578\pi$$
0.628723 + 0.777630i $$0.283578\pi$$
$$830$$ 0 0
$$831$$ −2.73823e6 −0.137552
$$832$$ 0 0
$$833$$ 3.01085e6 0.150341
$$834$$ 0 0
$$835$$ 1.54356e7 0.766141
$$836$$ 0 0
$$837$$ −6.01922e6 −0.296979
$$838$$ 0 0
$$839$$ −3.41458e7 −1.67468 −0.837340 0.546682i $$-0.815891\pi$$
−0.837340 + 0.546682i $$0.815891\pi$$
$$840$$ 0 0
$$841$$ −1.37927e6 −0.0672450
$$842$$ 0 0
$$843$$ −789708. −0.0382734
$$844$$ 0 0
$$845$$ 4.23446e6 0.204012
$$846$$ 0 0
$$847$$ −1.75101e7 −0.838649
$$848$$ 0 0
$$849$$ −1.34607e6 −0.0640911
$$850$$ 0 0
$$851$$ 1.03661e6 0.0490671
$$852$$ 0 0
$$853$$ 2.50701e7 1.17973 0.589865 0.807502i $$-0.299181\pi$$
0.589865 + 0.807502i $$0.299181\pi$$
$$854$$ 0 0
$$855$$ 2.15674e6 0.100898
$$856$$ 0 0
$$857$$ −1.81938e7 −0.846199 −0.423099 0.906083i $$-0.639058\pi$$
−0.423099 + 0.906083i $$0.639058\pi$$
$$858$$ 0 0
$$859$$ −6.91797e6 −0.319886 −0.159943 0.987126i $$-0.551131\pi$$
−0.159943 + 0.987126i $$0.551131\pi$$
$$860$$ 0 0
$$861$$ −1.17659e6 −0.0540899
$$862$$ 0 0
$$863$$ 2.78069e7 1.27094 0.635471 0.772125i $$-0.280806\pi$$
0.635471 + 0.772125i $$0.280806\pi$$
$$864$$ 0 0
$$865$$ −3.46141e7 −1.57294
$$866$$ 0 0
$$867$$ 305318. 0.0137945
$$868$$ 0 0
$$869$$ −4.49856e6 −0.202080
$$870$$ 0 0
$$871$$ −2.35275e7 −1.05083
$$872$$ 0 0
$$873$$ −2.56261e7 −1.13801
$$874$$ 0 0
$$875$$ 1.39521e7 0.616053
$$876$$ 0 0
$$877$$ 3.79587e6 0.166653 0.0833263 0.996522i $$-0.473446\pi$$
0.0833263 + 0.996522i $$0.473446\pi$$
$$878$$ 0 0
$$879$$ 3.66936e6 0.160184
$$880$$ 0 0
$$881$$ 2.48904e7 1.08042 0.540210 0.841530i $$-0.318345\pi$$
0.540210 + 0.841530i $$0.318345\pi$$
$$882$$ 0 0
$$883$$ 3.13568e6 0.135341 0.0676705 0.997708i $$-0.478443\pi$$
0.0676705 + 0.997708i $$0.478443\pi$$
$$884$$ 0 0
$$885$$ 238464. 0.0102345
$$886$$ 0 0
$$887$$ 2.02437e7 0.863933 0.431966 0.901890i $$-0.357820\pi$$
0.431966 + 0.901890i $$0.357820\pi$$
$$888$$ 0 0
$$889$$ −1.22406e7 −0.519455
$$890$$ 0 0
$$891$$ 4.04273e7 1.70600
$$892$$ 0 0
$$893$$ 2.42858e6 0.101912
$$894$$ 0 0
$$895$$ −3.00223e7 −1.25281
$$896$$ 0 0
$$897$$ −109824. −0.00455739
$$898$$ 0 0
$$899$$ −2.73113e7 −1.12705
$$900$$ 0 0
$$901$$ 1.27156e6 0.0521823
$$902$$ 0 0
$$903$$ −897680. −0.0366355
$$904$$ 0 0
$$905$$ 1.18867e7 0.482437
$$906$$ 0 0
$$907$$ 6.86324e6 0.277020 0.138510 0.990361i $$-0.455769\pi$$
0.138510 + 0.990361i $$0.455769\pi$$
$$908$$ 0 0
$$909$$ −1.12655e7 −0.452211
$$910$$ 0 0
$$911$$ 9.40661e6 0.375523 0.187762 0.982215i $$-0.439877\pi$$
0.187762 + 0.982215i $$0.439877\pi$$
$$912$$ 0 0
$$913$$ 1.81829e7 0.721914
$$914$$ 0 0
$$915$$ −1.45766e6 −0.0575579
$$916$$ 0 0
$$917$$ 598290. 0.0234957
$$918$$ 0 0
$$919$$ 2.10280e7 0.821313 0.410656 0.911790i $$-0.365300\pi$$
0.410656 + 0.911790i $$0.365300\pi$$
$$920$$ 0 0
$$921$$ 3.02112e6 0.117360
$$922$$ 0 0
$$923$$ 2.15255e7 0.831666
$$924$$ 0 0
$$925$$ −6.57706e7 −2.52742
$$926$$ 0 0
$$927$$ 2.92068e7 1.11631
$$928$$ 0 0
$$929$$ −4.30928e7 −1.63819 −0.819096 0.573656i $$-0.805525\pi$$
−0.819096 + 0.573656i $$0.805525\pi$$
$$930$$ 0 0
$$931$$ 225694. 0.00853387
$$932$$ 0 0
$$933$$ 3.75058e6 0.141057
$$934$$ 0 0
$$935$$ −8.66765e7 −3.24244
$$936$$ 0 0
$$937$$ −3.85862e6 −0.143576 −0.0717882 0.997420i $$-0.522871\pi$$
−0.0717882 + 0.997420i $$0.522871\pi$$
$$938$$ 0 0
$$939$$ −3.02152e6 −0.111831
$$940$$ 0 0
$$941$$ 1.59601e7 0.587572 0.293786 0.955871i $$-0.405085\pi$$
0.293786 + 0.955871i $$0.405085\pi$$
$$942$$ 0 0
$$943$$ −1.15258e6 −0.0422076
$$944$$ 0 0
$$945$$ −4.53466e6 −0.165183
$$946$$ 0 0
$$947$$ 4.66831e6 0.169155 0.0845775 0.996417i $$-0.473046\pi$$
0.0845775 + 0.996417i $$0.473046\pi$$
$$948$$ 0 0
$$949$$ −7.68654e6 −0.277054
$$950$$ 0 0
$$951$$ 4.05419e6 0.145363
$$952$$ 0 0
$$953$$ −3.43457e7 −1.22501 −0.612505 0.790466i $$-0.709838\pi$$
−0.612505 + 0.790466i $$0.709838\pi$$
$$954$$ 0 0
$$955$$ 3.10510e7 1.10171
$$956$$ 0 0
$$957$$ −6.29856e6 −0.222311
$$958$$ 0 0
$$959$$ 681198. 0.0239181
$$960$$ 0 0
$$961$$ 1.03584e7 0.361813
$$962$$ 0 0
$$963$$ −3.09830e7 −1.07661
$$964$$ 0 0
$$965$$ 5.95156e7 2.05737
$$966$$ 0 0
$$967$$ 2.28181e7 0.784718 0.392359 0.919812i $$-0.371659\pi$$
0.392359 + 0.919812i $$0.371659\pi$$
$$968$$ 0 0
$$969$$ 235752. 0.00806577
$$970$$ 0 0
$$971$$ −4.94042e7 −1.68157 −0.840786 0.541367i $$-0.817907\pi$$
−0.840786 + 0.541367i $$0.817907\pi$$
$$972$$ 0 0
$$973$$ −2.11579e7 −0.716458
$$974$$ 0 0
$$975$$ 6.96810e6 0.234749
$$976$$ 0 0
$$977$$ 7.17542e6 0.240498 0.120249 0.992744i $$-0.461631\pi$$
0.120249 + 0.992744i $$0.461631\pi$$
$$978$$ 0 0
$$979$$ −3.24907e7 −1.08343
$$980$$ 0 0
$$981$$ 5.27134e7 1.74883
$$982$$ 0 0
$$983$$ 4.22279e6 0.139385 0.0696924 0.997569i $$-0.477798\pi$$
0.0696924 + 0.997569i $$0.477798\pi$$
$$984$$ 0 0
$$985$$ 4.79388e7 1.57433
$$986$$ 0 0
$$987$$ −2.53193e6 −0.0827291
$$988$$ 0 0
$$989$$ −879360. −0.0285875
$$990$$ 0 0
$$991$$ −1.65645e7 −0.535789 −0.267895 0.963448i $$-0.586328\pi$$
−0.267895 + 0.963448i $$0.586328\pi$$
$$992$$ 0 0
$$993$$ −3.08018e6 −0.0991294
$$994$$ 0 0
$$995$$ −7.54495e7 −2.41601
$$996$$ 0 0
$$997$$ −4.40973e7 −1.40499 −0.702496 0.711687i $$-0.747931\pi$$
−0.702496 + 0.711687i $$0.747931\pi$$
$$998$$ 0 0
$$999$$ 1.04093e7 0.329994
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 112.6.a.e.1.1 1
3.2 odd 2 1008.6.a.bb.1.1 1
4.3 odd 2 28.6.a.a.1.1 1
7.6 odd 2 784.6.a.f.1.1 1
8.3 odd 2 448.6.a.i.1.1 1
8.5 even 2 448.6.a.h.1.1 1
12.11 even 2 252.6.a.d.1.1 1
20.3 even 4 700.6.e.d.449.1 2
20.7 even 4 700.6.e.d.449.2 2
20.19 odd 2 700.6.a.d.1.1 1
28.3 even 6 196.6.e.e.177.1 2
28.11 odd 6 196.6.e.f.177.1 2
28.19 even 6 196.6.e.e.165.1 2
28.23 odd 6 196.6.e.f.165.1 2
28.27 even 2 196.6.a.d.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
28.6.a.a.1.1 1 4.3 odd 2
112.6.a.e.1.1 1 1.1 even 1 trivial
196.6.a.d.1.1 1 28.27 even 2
196.6.e.e.165.1 2 28.19 even 6
196.6.e.e.177.1 2 28.3 even 6
196.6.e.f.165.1 2 28.23 odd 6
196.6.e.f.177.1 2 28.11 odd 6
252.6.a.d.1.1 1 12.11 even 2
448.6.a.h.1.1 1 8.5 even 2
448.6.a.i.1.1 1 8.3 odd 2
700.6.a.d.1.1 1 20.19 odd 2
700.6.e.d.449.1 2 20.3 even 4
700.6.e.d.449.2 2 20.7 even 4
784.6.a.f.1.1 1 7.6 odd 2
1008.6.a.bb.1.1 1 3.2 odd 2