# Properties

 Label 768.4.a.m.1.1 Level $768$ Weight $4$ Character 768.1 Self dual yes Analytic conductor $45.313$ Analytic rank $1$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(1,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("768.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 768.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: $$45.3134668844$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{8})^+$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 384) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-1.41421$$ of defining polynomial Character $$\chi$$ $$=$$ 768.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000 q^{3} -2.82843 q^{5} +14.1421 q^{7} +9.00000 q^{9} +O(q^{10})$$ $$q+3.00000 q^{3} -2.82843 q^{5} +14.1421 q^{7} +9.00000 q^{9} -20.0000 q^{11} -39.5980 q^{13} -8.48528 q^{15} -34.0000 q^{17} -52.0000 q^{19} +42.4264 q^{21} +62.2254 q^{23} -117.000 q^{25} +27.0000 q^{27} -200.818 q^{29} -110.309 q^{31} -60.0000 q^{33} -40.0000 q^{35} +271.529 q^{37} -118.794 q^{39} -26.0000 q^{41} -252.000 q^{43} -25.4558 q^{45} +345.068 q^{47} -143.000 q^{49} -102.000 q^{51} +681.651 q^{53} +56.5685 q^{55} -156.000 q^{57} -364.000 q^{59} -735.391 q^{61} +127.279 q^{63} +112.000 q^{65} -628.000 q^{67} +186.676 q^{69} -333.754 q^{71} +338.000 q^{73} -351.000 q^{75} -282.843 q^{77} -789.131 q^{79} +81.0000 q^{81} -1036.00 q^{83} +96.1665 q^{85} -602.455 q^{87} +234.000 q^{89} -560.000 q^{91} -330.926 q^{93} +147.078 q^{95} -178.000 q^{97} -180.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 18 * q^9 $$2 q + 6 q^{3} + 18 q^{9} - 40 q^{11} - 68 q^{17} - 104 q^{19} - 234 q^{25} + 54 q^{27} - 120 q^{33} - 80 q^{35} - 52 q^{41} - 504 q^{43} - 286 q^{49} - 204 q^{51} - 312 q^{57} - 728 q^{59} + 224 q^{65} - 1256 q^{67} + 676 q^{73} - 702 q^{75} + 162 q^{81} - 2072 q^{83} + 468 q^{89} - 1120 q^{91} - 356 q^{97} - 360 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 18 * q^9 - 40 * q^11 - 68 * q^17 - 104 * q^19 - 234 * q^25 + 54 * q^27 - 120 * q^33 - 80 * q^35 - 52 * q^41 - 504 * q^43 - 286 * q^49 - 204 * q^51 - 312 * q^57 - 728 * q^59 + 224 * q^65 - 1256 * q^67 + 676 * q^73 - 702 * q^75 + 162 * q^81 - 2072 * q^83 + 468 * q^89 - 1120 * q^91 - 356 * q^97 - 360 * q^99

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 0 0
$$3$$ 3.00000 0.577350
$$4$$ 0 0
$$5$$ −2.82843 −0.252982 −0.126491 0.991968i $$-0.540372\pi$$
−0.126491 + 0.991968i $$0.540372\pi$$
$$6$$ 0 0
$$7$$ 14.1421 0.763604 0.381802 0.924244i $$-0.375304\pi$$
0.381802 + 0.924244i $$0.375304\pi$$
$$8$$ 0 0
$$9$$ 9.00000 0.333333
$$10$$ 0 0
$$11$$ −20.0000 −0.548202 −0.274101 0.961701i $$-0.588380\pi$$
−0.274101 + 0.961701i $$0.588380\pi$$
$$12$$ 0 0
$$13$$ −39.5980 −0.844808 −0.422404 0.906408i $$-0.638814\pi$$
−0.422404 + 0.906408i $$0.638814\pi$$
$$14$$ 0 0
$$15$$ −8.48528 −0.146059
$$16$$ 0 0
$$17$$ −34.0000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ 0 0
$$19$$ −52.0000 −0.627875 −0.313937 0.949444i $$-0.601648\pi$$
−0.313937 + 0.949444i $$0.601648\pi$$
$$20$$ 0 0
$$21$$ 42.4264 0.440867
$$22$$ 0 0
$$23$$ 62.2254 0.564126 0.282063 0.959396i $$-0.408981\pi$$
0.282063 + 0.959396i $$0.408981\pi$$
$$24$$ 0 0
$$25$$ −117.000 −0.936000
$$26$$ 0 0
$$27$$ 27.0000 0.192450
$$28$$ 0 0
$$29$$ −200.818 −1.28590 −0.642949 0.765909i $$-0.722289\pi$$
−0.642949 + 0.765909i $$0.722289\pi$$
$$30$$ 0 0
$$31$$ −110.309 −0.639097 −0.319549 0.947570i $$-0.603531\pi$$
−0.319549 + 0.947570i $$0.603531\pi$$
$$32$$ 0 0
$$33$$ −60.0000 −0.316505
$$34$$ 0 0
$$35$$ −40.0000 −0.193178
$$36$$ 0 0
$$37$$ 271.529 1.20646 0.603231 0.797567i $$-0.293880\pi$$
0.603231 + 0.797567i $$0.293880\pi$$
$$38$$ 0 0
$$39$$ −118.794 −0.487750
$$40$$ 0 0
$$41$$ −26.0000 −0.0990370 −0.0495185 0.998773i $$-0.515769\pi$$
−0.0495185 + 0.998773i $$0.515769\pi$$
$$42$$ 0 0
$$43$$ −252.000 −0.893713 −0.446856 0.894606i $$-0.647456\pi$$
−0.446856 + 0.894606i $$0.647456\pi$$
$$44$$ 0 0
$$45$$ −25.4558 −0.0843274
$$46$$ 0 0
$$47$$ 345.068 1.07092 0.535461 0.844560i $$-0.320138\pi$$
0.535461 + 0.844560i $$0.320138\pi$$
$$48$$ 0 0
$$49$$ −143.000 −0.416910
$$50$$ 0 0
$$51$$ −102.000 −0.280056
$$52$$ 0 0
$$53$$ 681.651 1.76664 0.883320 0.468770i $$-0.155303\pi$$
0.883320 + 0.468770i $$0.155303\pi$$
$$54$$ 0 0
$$55$$ 56.5685 0.138685
$$56$$ 0 0
$$57$$ −156.000 −0.362504
$$58$$ 0 0
$$59$$ −364.000 −0.803199 −0.401600 0.915815i $$-0.631546\pi$$
−0.401600 + 0.915815i $$0.631546\pi$$
$$60$$ 0 0
$$61$$ −735.391 −1.54356 −0.771780 0.635889i $$-0.780633\pi$$
−0.771780 + 0.635889i $$0.780633\pi$$
$$62$$ 0 0
$$63$$ 127.279 0.254535
$$64$$ 0 0
$$65$$ 112.000 0.213721
$$66$$ 0 0
$$67$$ −628.000 −1.14511 −0.572555 0.819866i $$-0.694048\pi$$
−0.572555 + 0.819866i $$0.694048\pi$$
$$68$$ 0 0
$$69$$ 186.676 0.325698
$$70$$ 0 0
$$71$$ −333.754 −0.557878 −0.278939 0.960309i $$-0.589983\pi$$
−0.278939 + 0.960309i $$0.589983\pi$$
$$72$$ 0 0
$$73$$ 338.000 0.541917 0.270958 0.962591i $$-0.412659\pi$$
0.270958 + 0.962591i $$0.412659\pi$$
$$74$$ 0 0
$$75$$ −351.000 −0.540400
$$76$$ 0 0
$$77$$ −282.843 −0.418609
$$78$$ 0 0
$$79$$ −789.131 −1.12385 −0.561925 0.827188i $$-0.689939\pi$$
−0.561925 + 0.827188i $$0.689939\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ −1036.00 −1.37007 −0.685035 0.728510i $$-0.740213\pi$$
−0.685035 + 0.728510i $$0.740213\pi$$
$$84$$ 0 0
$$85$$ 96.1665 0.122714
$$86$$ 0 0
$$87$$ −602.455 −0.742413
$$88$$ 0 0
$$89$$ 234.000 0.278696 0.139348 0.990243i $$-0.455499\pi$$
0.139348 + 0.990243i $$0.455499\pi$$
$$90$$ 0 0
$$91$$ −560.000 −0.645098
$$92$$ 0 0
$$93$$ −330.926 −0.368983
$$94$$ 0 0
$$95$$ 147.078 0.158841
$$96$$ 0 0
$$97$$ −178.000 −0.186321 −0.0931606 0.995651i $$-0.529697\pi$$
−0.0931606 + 0.995651i $$0.529697\pi$$
$$98$$ 0 0
$$99$$ −180.000 −0.182734
$$100$$ 0 0
$$101$$ −257.387 −0.253574 −0.126787 0.991930i $$-0.540466\pi$$
−0.126787 + 0.991930i $$0.540466\pi$$
$$102$$ 0 0
$$103$$ 1886.56 1.80474 0.902371 0.430961i $$-0.141825\pi$$
0.902371 + 0.430961i $$0.141825\pi$$
$$104$$ 0 0
$$105$$ −120.000 −0.111531
$$106$$ 0 0
$$107$$ −1404.00 −1.26850 −0.634251 0.773127i $$-0.718692\pi$$
−0.634251 + 0.773127i $$0.718692\pi$$
$$108$$ 0 0
$$109$$ 39.5980 0.0347963 0.0173982 0.999849i $$-0.494462\pi$$
0.0173982 + 0.999849i $$0.494462\pi$$
$$110$$ 0 0
$$111$$ 814.587 0.696551
$$112$$ 0 0
$$113$$ 1378.00 1.14718 0.573590 0.819143i $$-0.305550\pi$$
0.573590 + 0.819143i $$0.305550\pi$$
$$114$$ 0 0
$$115$$ −176.000 −0.142714
$$116$$ 0 0
$$117$$ −356.382 −0.281603
$$118$$ 0 0
$$119$$ −480.833 −0.370402
$$120$$ 0 0
$$121$$ −931.000 −0.699474
$$122$$ 0 0
$$123$$ −78.0000 −0.0571791
$$124$$ 0 0
$$125$$ 684.479 0.489774
$$126$$ 0 0
$$127$$ −1790.39 −1.25096 −0.625480 0.780241i $$-0.715097\pi$$
−0.625480 + 0.780241i $$0.715097\pi$$
$$128$$ 0 0
$$129$$ −756.000 −0.515985
$$130$$ 0 0
$$131$$ −1572.00 −1.04844 −0.524222 0.851581i $$-0.675644\pi$$
−0.524222 + 0.851581i $$0.675644\pi$$
$$132$$ 0 0
$$133$$ −735.391 −0.479447
$$134$$ 0 0
$$135$$ −76.3675 −0.0486864
$$136$$ 0 0
$$137$$ 2854.00 1.77981 0.889904 0.456148i $$-0.150771\pi$$
0.889904 + 0.456148i $$0.150771\pi$$
$$138$$ 0 0
$$139$$ −1964.00 −1.19845 −0.599224 0.800581i $$-0.704524\pi$$
−0.599224 + 0.800581i $$0.704524\pi$$
$$140$$ 0 0
$$141$$ 1035.20 0.618297
$$142$$ 0 0
$$143$$ 791.960 0.463126
$$144$$ 0 0
$$145$$ 568.000 0.325309
$$146$$ 0 0
$$147$$ −429.000 −0.240703
$$148$$ 0 0
$$149$$ −1507.55 −0.828882 −0.414441 0.910076i $$-0.636023\pi$$
−0.414441 + 0.910076i $$0.636023\pi$$
$$150$$ 0 0
$$151$$ 2265.57 1.22099 0.610495 0.792020i $$-0.290971\pi$$
0.610495 + 0.792020i $$0.290971\pi$$
$$152$$ 0 0
$$153$$ −306.000 −0.161690
$$154$$ 0 0
$$155$$ 312.000 0.161680
$$156$$ 0 0
$$157$$ 3529.88 1.79436 0.897181 0.441663i $$-0.145611\pi$$
0.897181 + 0.441663i $$0.145611\pi$$
$$158$$ 0 0
$$159$$ 2044.95 1.01997
$$160$$ 0 0
$$161$$ 880.000 0.430768
$$162$$ 0 0
$$163$$ −2932.00 −1.40891 −0.704454 0.709750i $$-0.748808\pi$$
−0.704454 + 0.709750i $$0.748808\pi$$
$$164$$ 0 0
$$165$$ 169.706 0.0800701
$$166$$ 0 0
$$167$$ −3676.96 −1.70378 −0.851890 0.523720i $$-0.824544\pi$$
−0.851890 + 0.523720i $$0.824544\pi$$
$$168$$ 0 0
$$169$$ −629.000 −0.286299
$$170$$ 0 0
$$171$$ −468.000 −0.209292
$$172$$ 0 0
$$173$$ 1445.33 0.635180 0.317590 0.948228i $$-0.397126\pi$$
0.317590 + 0.948228i $$0.397126\pi$$
$$174$$ 0 0
$$175$$ −1654.63 −0.714733
$$176$$ 0 0
$$177$$ −1092.00 −0.463727
$$178$$ 0 0
$$179$$ 1308.00 0.546170 0.273085 0.961990i $$-0.411956\pi$$
0.273085 + 0.961990i $$0.411956\pi$$
$$180$$ 0 0
$$181$$ −1996.87 −0.820034 −0.410017 0.912078i $$-0.634477\pi$$
−0.410017 + 0.912078i $$0.634477\pi$$
$$182$$ 0 0
$$183$$ −2206.17 −0.891175
$$184$$ 0 0
$$185$$ −768.000 −0.305213
$$186$$ 0 0
$$187$$ 680.000 0.265917
$$188$$ 0 0
$$189$$ 381.838 0.146956
$$190$$ 0 0
$$191$$ 939.038 0.355740 0.177870 0.984054i $$-0.443079\pi$$
0.177870 + 0.984054i $$0.443079\pi$$
$$192$$ 0 0
$$193$$ −2490.00 −0.928674 −0.464337 0.885659i $$-0.653707\pi$$
−0.464337 + 0.885659i $$0.653707\pi$$
$$194$$ 0 0
$$195$$ 336.000 0.123392
$$196$$ 0 0
$$197$$ −2723.78 −0.985081 −0.492540 0.870290i $$-0.663932\pi$$
−0.492540 + 0.870290i $$0.663932\pi$$
$$198$$ 0 0
$$199$$ 2158.09 0.768758 0.384379 0.923175i $$-0.374416\pi$$
0.384379 + 0.923175i $$0.374416\pi$$
$$200$$ 0 0
$$201$$ −1884.00 −0.661130
$$202$$ 0 0
$$203$$ −2840.00 −0.981916
$$204$$ 0 0
$$205$$ 73.5391 0.0250546
$$206$$ 0 0
$$207$$ 560.029 0.188042
$$208$$ 0 0
$$209$$ 1040.00 0.344202
$$210$$ 0 0
$$211$$ 924.000 0.301473 0.150736 0.988574i $$-0.451836\pi$$
0.150736 + 0.988574i $$0.451836\pi$$
$$212$$ 0 0
$$213$$ −1001.26 −0.322091
$$214$$ 0 0
$$215$$ 712.764 0.226093
$$216$$ 0 0
$$217$$ −1560.00 −0.488017
$$218$$ 0 0
$$219$$ 1014.00 0.312876
$$220$$ 0 0
$$221$$ 1346.33 0.409792
$$222$$ 0 0
$$223$$ −2276.88 −0.683728 −0.341864 0.939749i $$-0.611058\pi$$
−0.341864 + 0.939749i $$0.611058\pi$$
$$224$$ 0 0
$$225$$ −1053.00 −0.312000
$$226$$ 0 0
$$227$$ −156.000 −0.0456127 −0.0228064 0.999740i $$-0.507260\pi$$
−0.0228064 + 0.999740i $$0.507260\pi$$
$$228$$ 0 0
$$229$$ 639.225 0.184459 0.0922296 0.995738i $$-0.470601\pi$$
0.0922296 + 0.995738i $$0.470601\pi$$
$$230$$ 0 0
$$231$$ −848.528 −0.241684
$$232$$ 0 0
$$233$$ 2826.00 0.794581 0.397291 0.917693i $$-0.369951\pi$$
0.397291 + 0.917693i $$0.369951\pi$$
$$234$$ 0 0
$$235$$ −976.000 −0.270924
$$236$$ 0 0
$$237$$ −2367.39 −0.648855
$$238$$ 0 0
$$239$$ 2466.39 0.667521 0.333760 0.942658i $$-0.391682\pi$$
0.333760 + 0.942658i $$0.391682\pi$$
$$240$$ 0 0
$$241$$ −3354.00 −0.896474 −0.448237 0.893915i $$-0.647948\pi$$
−0.448237 + 0.893915i $$0.647948\pi$$
$$242$$ 0 0
$$243$$ 243.000 0.0641500
$$244$$ 0 0
$$245$$ 404.465 0.105471
$$246$$ 0 0
$$247$$ 2059.09 0.530433
$$248$$ 0 0
$$249$$ −3108.00 −0.791010
$$250$$ 0 0
$$251$$ 6396.00 1.60841 0.804207 0.594349i $$-0.202590\pi$$
0.804207 + 0.594349i $$0.202590\pi$$
$$252$$ 0 0
$$253$$ −1244.51 −0.309255
$$254$$ 0 0
$$255$$ 288.500 0.0708492
$$256$$ 0 0
$$257$$ 6882.00 1.67038 0.835189 0.549962i $$-0.185358\pi$$
0.835189 + 0.549962i $$0.185358\pi$$
$$258$$ 0 0
$$259$$ 3840.00 0.921259
$$260$$ 0 0
$$261$$ −1807.36 −0.428632
$$262$$ 0 0
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −1928.00 −0.446929
$$266$$ 0 0
$$267$$ 702.000 0.160905
$$268$$ 0 0
$$269$$ −1434.01 −0.325031 −0.162515 0.986706i $$-0.551961\pi$$
−0.162515 + 0.986706i $$0.551961\pi$$
$$270$$ 0 0
$$271$$ −5942.53 −1.33204 −0.666020 0.745934i $$-0.732003\pi$$
−0.666020 + 0.745934i $$0.732003\pi$$
$$272$$ 0 0
$$273$$ −1680.00 −0.372448
$$274$$ 0 0
$$275$$ 2340.00 0.513117
$$276$$ 0 0
$$277$$ 1103.09 0.239271 0.119635 0.992818i $$-0.461827\pi$$
0.119635 + 0.992818i $$0.461827\pi$$
$$278$$ 0 0
$$279$$ −992.778 −0.213032
$$280$$ 0 0
$$281$$ 6266.00 1.33024 0.665121 0.746735i $$-0.268380\pi$$
0.665121 + 0.746735i $$0.268380\pi$$
$$282$$ 0 0
$$283$$ 8596.00 1.80558 0.902790 0.430082i $$-0.141515\pi$$
0.902790 + 0.430082i $$0.141515\pi$$
$$284$$ 0 0
$$285$$ 441.235 0.0917070
$$286$$ 0 0
$$287$$ −367.696 −0.0756250
$$288$$ 0 0
$$289$$ −3757.00 −0.764706
$$290$$ 0 0
$$291$$ −534.000 −0.107573
$$292$$ 0 0
$$293$$ 8397.60 1.67438 0.837189 0.546913i $$-0.184197\pi$$
0.837189 + 0.546913i $$0.184197\pi$$
$$294$$ 0 0
$$295$$ 1029.55 0.203195
$$296$$ 0 0
$$297$$ −540.000 −0.105502
$$298$$ 0 0
$$299$$ −2464.00 −0.476578
$$300$$ 0 0
$$301$$ −3563.82 −0.682442
$$302$$ 0 0
$$303$$ −772.161 −0.146401
$$304$$ 0 0
$$305$$ 2080.00 0.390493
$$306$$ 0 0
$$307$$ 4940.00 0.918374 0.459187 0.888340i $$-0.348141\pi$$
0.459187 + 0.888340i $$0.348141\pi$$
$$308$$ 0 0
$$309$$ 5659.68 1.04197
$$310$$ 0 0
$$311$$ 3382.80 0.616788 0.308394 0.951259i $$-0.400209\pi$$
0.308394 + 0.951259i $$0.400209\pi$$
$$312$$ 0 0
$$313$$ −3106.00 −0.560899 −0.280450 0.959869i $$-0.590484\pi$$
−0.280450 + 0.959869i $$0.590484\pi$$
$$314$$ 0 0
$$315$$ −360.000 −0.0643927
$$316$$ 0 0
$$317$$ −6728.83 −1.19220 −0.596102 0.802909i $$-0.703285\pi$$
−0.596102 + 0.802909i $$0.703285\pi$$
$$318$$ 0 0
$$319$$ 4016.37 0.704932
$$320$$ 0 0
$$321$$ −4212.00 −0.732370
$$322$$ 0 0
$$323$$ 1768.00 0.304564
$$324$$ 0 0
$$325$$ 4632.96 0.790740
$$326$$ 0 0
$$327$$ 118.794 0.0200897
$$328$$ 0 0
$$329$$ 4880.00 0.817760
$$330$$ 0 0
$$331$$ −2908.00 −0.482895 −0.241447 0.970414i $$-0.577622\pi$$
−0.241447 + 0.970414i $$0.577622\pi$$
$$332$$ 0 0
$$333$$ 2443.76 0.402154
$$334$$ 0 0
$$335$$ 1776.25 0.289693
$$336$$ 0 0
$$337$$ 4298.00 0.694739 0.347369 0.937728i $$-0.387075\pi$$
0.347369 + 0.937728i $$0.387075\pi$$
$$338$$ 0 0
$$339$$ 4134.00 0.662325
$$340$$ 0 0
$$341$$ 2206.17 0.350355
$$342$$ 0 0
$$343$$ −6873.08 −1.08196
$$344$$ 0 0
$$345$$ −528.000 −0.0823958
$$346$$ 0 0
$$347$$ 9996.00 1.54644 0.773218 0.634140i $$-0.218646\pi$$
0.773218 + 0.634140i $$0.218646\pi$$
$$348$$ 0 0
$$349$$ −3993.74 −0.612550 −0.306275 0.951943i $$-0.599083\pi$$
−0.306275 + 0.951943i $$0.599083\pi$$
$$350$$ 0 0
$$351$$ −1069.15 −0.162583
$$352$$ 0 0
$$353$$ 6738.00 1.01594 0.507971 0.861374i $$-0.330396\pi$$
0.507971 + 0.861374i $$0.330396\pi$$
$$354$$ 0 0
$$355$$ 944.000 0.141133
$$356$$ 0 0
$$357$$ −1442.50 −0.213852
$$358$$ 0 0
$$359$$ 2132.63 0.313527 0.156763 0.987636i $$-0.449894\pi$$
0.156763 + 0.987636i $$0.449894\pi$$
$$360$$ 0 0
$$361$$ −4155.00 −0.605773
$$362$$ 0 0
$$363$$ −2793.00 −0.403842
$$364$$ 0 0
$$365$$ −956.008 −0.137095
$$366$$ 0 0
$$367$$ 7628.27 1.08499 0.542496 0.840058i $$-0.317479\pi$$
0.542496 + 0.840058i $$0.317479\pi$$
$$368$$ 0 0
$$369$$ −234.000 −0.0330123
$$370$$ 0 0
$$371$$ 9640.00 1.34901
$$372$$ 0 0
$$373$$ 8383.46 1.16375 0.581875 0.813278i $$-0.302319\pi$$
0.581875 + 0.813278i $$0.302319\pi$$
$$374$$ 0 0
$$375$$ 2053.44 0.282771
$$376$$ 0 0
$$377$$ 7952.00 1.08634
$$378$$ 0 0
$$379$$ 12788.0 1.73318 0.866590 0.499020i $$-0.166307\pi$$
0.866590 + 0.499020i $$0.166307\pi$$
$$380$$ 0 0
$$381$$ −5371.18 −0.722242
$$382$$ 0 0
$$383$$ 2319.31 0.309429 0.154714 0.987959i $$-0.450554\pi$$
0.154714 + 0.987959i $$0.450554\pi$$
$$384$$ 0 0
$$385$$ 800.000 0.105901
$$386$$ 0 0
$$387$$ −2268.00 −0.297904
$$388$$ 0 0
$$389$$ −2684.18 −0.349854 −0.174927 0.984581i $$-0.555969\pi$$
−0.174927 + 0.984581i $$0.555969\pi$$
$$390$$ 0 0
$$391$$ −2115.66 −0.273641
$$392$$ 0 0
$$393$$ −4716.00 −0.605320
$$394$$ 0 0
$$395$$ 2232.00 0.284314
$$396$$ 0 0
$$397$$ −2206.17 −0.278903 −0.139452 0.990229i $$-0.544534\pi$$
−0.139452 + 0.990229i $$0.544534\pi$$
$$398$$ 0 0
$$399$$ −2206.17 −0.276809
$$400$$ 0 0
$$401$$ 3582.00 0.446076 0.223038 0.974810i $$-0.428403\pi$$
0.223038 + 0.974810i $$0.428403\pi$$
$$402$$ 0 0
$$403$$ 4368.00 0.539915
$$404$$ 0 0
$$405$$ −229.103 −0.0281091
$$406$$ 0 0
$$407$$ −5430.58 −0.661385
$$408$$ 0 0
$$409$$ 5126.00 0.619717 0.309859 0.950783i $$-0.399718\pi$$
0.309859 + 0.950783i $$0.399718\pi$$
$$410$$ 0 0
$$411$$ 8562.00 1.02757
$$412$$ 0 0
$$413$$ −5147.74 −0.613326
$$414$$ 0 0
$$415$$ 2930.25 0.346603
$$416$$ 0 0
$$417$$ −5892.00 −0.691924
$$418$$ 0 0
$$419$$ −2924.00 −0.340923 −0.170462 0.985364i $$-0.554526\pi$$
−0.170462 + 0.985364i $$0.554526\pi$$
$$420$$ 0 0
$$421$$ −7314.31 −0.846741 −0.423370 0.905957i $$-0.639153\pi$$
−0.423370 + 0.905957i $$0.639153\pi$$
$$422$$ 0 0
$$423$$ 3105.61 0.356974
$$424$$ 0 0
$$425$$ 3978.00 0.454027
$$426$$ 0 0
$$427$$ −10400.0 −1.17867
$$428$$ 0 0
$$429$$ 2375.88 0.267386
$$430$$ 0 0
$$431$$ −15844.8 −1.77081 −0.885405 0.464819i $$-0.846119\pi$$
−0.885405 + 0.464819i $$0.846119\pi$$
$$432$$ 0 0
$$433$$ −6274.00 −0.696326 −0.348163 0.937434i $$-0.613194\pi$$
−0.348163 + 0.937434i $$0.613194\pi$$
$$434$$ 0 0
$$435$$ 1704.00 0.187817
$$436$$ 0 0
$$437$$ −3235.72 −0.354200
$$438$$ 0 0
$$439$$ −4596.19 −0.499691 −0.249846 0.968286i $$-0.580380\pi$$
−0.249846 + 0.968286i $$0.580380\pi$$
$$440$$ 0 0
$$441$$ −1287.00 −0.138970
$$442$$ 0 0
$$443$$ 5084.00 0.545255 0.272628 0.962120i $$-0.412107\pi$$
0.272628 + 0.962120i $$0.412107\pi$$
$$444$$ 0 0
$$445$$ −661.852 −0.0705051
$$446$$ 0 0
$$447$$ −4522.65 −0.478555
$$448$$ 0 0
$$449$$ 14190.0 1.49146 0.745732 0.666246i $$-0.232100\pi$$
0.745732 + 0.666246i $$0.232100\pi$$
$$450$$ 0 0
$$451$$ 520.000 0.0542923
$$452$$ 0 0
$$453$$ 6796.71 0.704939
$$454$$ 0 0
$$455$$ 1583.92 0.163198
$$456$$ 0 0
$$457$$ −6474.00 −0.662672 −0.331336 0.943513i $$-0.607499\pi$$
−0.331336 + 0.943513i $$0.607499\pi$$
$$458$$ 0 0
$$459$$ −918.000 −0.0933520
$$460$$ 0 0
$$461$$ 6321.53 0.638662 0.319331 0.947643i $$-0.396542\pi$$
0.319331 + 0.947643i $$0.396542\pi$$
$$462$$ 0 0
$$463$$ 11435.3 1.14783 0.573915 0.818915i $$-0.305424\pi$$
0.573915 + 0.818915i $$0.305424\pi$$
$$464$$ 0 0
$$465$$ 936.000 0.0933462
$$466$$ 0 0
$$467$$ 3796.00 0.376141 0.188071 0.982156i $$-0.439777\pi$$
0.188071 + 0.982156i $$0.439777\pi$$
$$468$$ 0 0
$$469$$ −8881.26 −0.874411
$$470$$ 0 0
$$471$$ 10589.6 1.03598
$$472$$ 0 0
$$473$$ 5040.00 0.489935
$$474$$ 0 0
$$475$$ 6084.00 0.587691
$$476$$ 0 0
$$477$$ 6134.86 0.588880
$$478$$ 0 0
$$479$$ −10493.5 −1.00096 −0.500479 0.865749i $$-0.666843\pi$$
−0.500479 + 0.865749i $$0.666843\pi$$
$$480$$ 0 0
$$481$$ −10752.0 −1.01923
$$482$$ 0 0
$$483$$ 2640.00 0.248704
$$484$$ 0 0
$$485$$ 503.460 0.0471360
$$486$$ 0 0
$$487$$ −15406.4 −1.43354 −0.716769 0.697311i $$-0.754380\pi$$
−0.716769 + 0.697311i $$0.754380\pi$$
$$488$$ 0 0
$$489$$ −8796.00 −0.813433
$$490$$ 0 0
$$491$$ −15452.0 −1.42024 −0.710121 0.704079i $$-0.751360\pi$$
−0.710121 + 0.704079i $$0.751360\pi$$
$$492$$ 0 0
$$493$$ 6827.82 0.623752
$$494$$ 0 0
$$495$$ 509.117 0.0462285
$$496$$ 0 0
$$497$$ −4720.00 −0.425998
$$498$$ 0 0
$$499$$ −52.0000 −0.00466501 −0.00233250 0.999997i $$-0.500742\pi$$
−0.00233250 + 0.999997i $$0.500742\pi$$
$$500$$ 0 0
$$501$$ −11030.9 −0.983678
$$502$$ 0 0
$$503$$ 12428.1 1.10167 0.550837 0.834613i $$-0.314309\pi$$
0.550837 + 0.834613i $$0.314309\pi$$
$$504$$ 0 0
$$505$$ 728.000 0.0641497
$$506$$ 0 0
$$507$$ −1887.00 −0.165295
$$508$$ 0 0
$$509$$ 16362.5 1.42486 0.712429 0.701744i $$-0.247595\pi$$
0.712429 + 0.701744i $$0.247595\pi$$
$$510$$ 0 0
$$511$$ 4780.04 0.413809
$$512$$ 0 0
$$513$$ −1404.00 −0.120835
$$514$$ 0 0
$$515$$ −5336.00 −0.456567
$$516$$ 0 0
$$517$$ −6901.36 −0.587082
$$518$$ 0 0
$$519$$ 4335.98 0.366721
$$520$$ 0 0
$$521$$ −714.000 −0.0600401 −0.0300201 0.999549i $$-0.509557\pi$$
−0.0300201 + 0.999549i $$0.509557\pi$$
$$522$$ 0 0
$$523$$ −5980.00 −0.499975 −0.249988 0.968249i $$-0.580427\pi$$
−0.249988 + 0.968249i $$0.580427\pi$$
$$524$$ 0 0
$$525$$ −4963.89 −0.412651
$$526$$ 0 0
$$527$$ 3750.49 0.310008
$$528$$ 0 0
$$529$$ −8295.00 −0.681762
$$530$$ 0 0
$$531$$ −3276.00 −0.267733
$$532$$ 0 0
$$533$$ 1029.55 0.0836673
$$534$$ 0 0
$$535$$ 3971.11 0.320909
$$536$$ 0 0
$$537$$ 3924.00 0.315332
$$538$$ 0 0
$$539$$ 2860.00 0.228551
$$540$$ 0 0
$$541$$ 13729.2 1.09106 0.545530 0.838091i $$-0.316328\pi$$
0.545530 + 0.838091i $$0.316328\pi$$
$$542$$ 0 0
$$543$$ −5990.61 −0.473447
$$544$$ 0 0
$$545$$ −112.000 −0.00880285
$$546$$ 0 0
$$547$$ −18500.0 −1.44607 −0.723037 0.690809i $$-0.757255\pi$$
−0.723037 + 0.690809i $$0.757255\pi$$
$$548$$ 0 0
$$549$$ −6618.52 −0.514520
$$550$$ 0 0
$$551$$ 10442.6 0.807382
$$552$$ 0 0
$$553$$ −11160.0 −0.858176
$$554$$ 0 0
$$555$$ −2304.00 −0.176215
$$556$$ 0 0
$$557$$ 8765.30 0.666782 0.333391 0.942789i $$-0.391807\pi$$
0.333391 + 0.942789i $$0.391807\pi$$
$$558$$ 0 0
$$559$$ 9978.69 0.755015
$$560$$ 0 0
$$561$$ 2040.00 0.153527
$$562$$ 0 0
$$563$$ −268.000 −0.0200619 −0.0100310 0.999950i $$-0.503193\pi$$
−0.0100310 + 0.999950i $$0.503193\pi$$
$$564$$ 0 0
$$565$$ −3897.57 −0.290216
$$566$$ 0 0
$$567$$ 1145.51 0.0848448
$$568$$ 0 0
$$569$$ −13866.0 −1.02160 −0.510802 0.859698i $$-0.670652\pi$$
−0.510802 + 0.859698i $$0.670652\pi$$
$$570$$ 0 0
$$571$$ 5140.00 0.376712 0.188356 0.982101i $$-0.439684\pi$$
0.188356 + 0.982101i $$0.439684\pi$$
$$572$$ 0 0
$$573$$ 2817.11 0.205387
$$574$$ 0 0
$$575$$ −7280.37 −0.528022
$$576$$ 0 0
$$577$$ 9386.00 0.677200 0.338600 0.940930i $$-0.390047\pi$$
0.338600 + 0.940930i $$0.390047\pi$$
$$578$$ 0 0
$$579$$ −7470.00 −0.536170
$$580$$ 0 0
$$581$$ −14651.3 −1.04619
$$582$$ 0 0
$$583$$ −13633.0 −0.968477
$$584$$ 0 0
$$585$$ 1008.00 0.0712405
$$586$$ 0 0
$$587$$ −8844.00 −0.621859 −0.310929 0.950433i $$-0.600640\pi$$
−0.310929 + 0.950433i $$0.600640\pi$$
$$588$$ 0 0
$$589$$ 5736.05 0.401273
$$590$$ 0 0
$$591$$ −8171.33 −0.568737
$$592$$ 0 0
$$593$$ −9406.00 −0.651363 −0.325681 0.945480i $$-0.605594\pi$$
−0.325681 + 0.945480i $$0.605594\pi$$
$$594$$ 0 0
$$595$$ 1360.00 0.0937051
$$596$$ 0 0
$$597$$ 6474.27 0.443843
$$598$$ 0 0
$$599$$ −23459.0 −1.60018 −0.800090 0.599880i $$-0.795215\pi$$
−0.800090 + 0.599880i $$0.795215\pi$$
$$600$$ 0 0
$$601$$ −1262.00 −0.0856540 −0.0428270 0.999083i $$-0.513636\pi$$
−0.0428270 + 0.999083i $$0.513636\pi$$
$$602$$ 0 0
$$603$$ −5652.00 −0.381704
$$604$$ 0 0
$$605$$ 2633.27 0.176955
$$606$$ 0 0
$$607$$ 16288.9 1.08920 0.544602 0.838695i $$-0.316681\pi$$
0.544602 + 0.838695i $$0.316681\pi$$
$$608$$ 0 0
$$609$$ −8520.00 −0.566909
$$610$$ 0 0
$$611$$ −13664.0 −0.904724
$$612$$ 0 0
$$613$$ 7138.95 0.470374 0.235187 0.971950i $$-0.424430\pi$$
0.235187 + 0.971950i $$0.424430\pi$$
$$614$$ 0 0
$$615$$ 220.617 0.0144653
$$616$$ 0 0
$$617$$ 16874.0 1.10101 0.550504 0.834833i $$-0.314436\pi$$
0.550504 + 0.834833i $$0.314436\pi$$
$$618$$ 0 0
$$619$$ −20748.0 −1.34723 −0.673613 0.739085i $$-0.735258\pi$$
−0.673613 + 0.739085i $$0.735258\pi$$
$$620$$ 0 0
$$621$$ 1680.09 0.108566
$$622$$ 0 0
$$623$$ 3309.26 0.212813
$$624$$ 0 0
$$625$$ 12689.0 0.812096
$$626$$ 0 0
$$627$$ 3120.00 0.198725
$$628$$ 0 0
$$629$$ −9231.99 −0.585220
$$630$$ 0 0
$$631$$ −14840.8 −0.936294 −0.468147 0.883651i $$-0.655078\pi$$
−0.468147 + 0.883651i $$0.655078\pi$$
$$632$$ 0 0
$$633$$ 2772.00 0.174055
$$634$$ 0 0
$$635$$ 5064.00 0.316470
$$636$$ 0 0
$$637$$ 5662.51 0.352209
$$638$$ 0 0
$$639$$ −3003.79 −0.185959
$$640$$ 0 0
$$641$$ 17758.0 1.09423 0.547113 0.837059i $$-0.315727\pi$$
0.547113 + 0.837059i $$0.315727\pi$$
$$642$$ 0 0
$$643$$ 1148.00 0.0704086 0.0352043 0.999380i $$-0.488792\pi$$
0.0352043 + 0.999380i $$0.488792\pi$$
$$644$$ 0 0
$$645$$ 2138.29 0.130535
$$646$$ 0 0
$$647$$ 26988.9 1.63994 0.819970 0.572406i $$-0.193990\pi$$
0.819970 + 0.572406i $$0.193990\pi$$
$$648$$ 0 0
$$649$$ 7280.00 0.440316
$$650$$ 0 0
$$651$$ −4680.00 −0.281757
$$652$$ 0 0
$$653$$ −21069.0 −1.26262 −0.631311 0.775530i $$-0.717483\pi$$
−0.631311 + 0.775530i $$0.717483\pi$$
$$654$$ 0 0
$$655$$ 4446.29 0.265238
$$656$$ 0 0
$$657$$ 3042.00 0.180639
$$658$$ 0 0
$$659$$ −18356.0 −1.08505 −0.542525 0.840040i $$-0.682532\pi$$
−0.542525 + 0.840040i $$0.682532\pi$$
$$660$$ 0 0
$$661$$ −15250.9 −0.897414 −0.448707 0.893679i $$-0.648115\pi$$
−0.448707 + 0.893679i $$0.648115\pi$$
$$662$$ 0 0
$$663$$ 4038.99 0.236594
$$664$$ 0 0
$$665$$ 2080.00 0.121292
$$666$$ 0 0
$$667$$ −12496.0 −0.725408
$$668$$ 0 0
$$669$$ −6830.65 −0.394751
$$670$$ 0 0
$$671$$ 14707.8 0.846184
$$672$$ 0 0
$$673$$ −12082.0 −0.692016 −0.346008 0.938232i $$-0.612463\pi$$
−0.346008 + 0.938232i $$0.612463\pi$$
$$674$$ 0 0
$$675$$ −3159.00 −0.180133
$$676$$ 0 0
$$677$$ −12742.1 −0.723364 −0.361682 0.932302i $$-0.617797\pi$$
−0.361682 + 0.932302i $$0.617797\pi$$
$$678$$ 0 0
$$679$$ −2517.30 −0.142276
$$680$$ 0 0
$$681$$ −468.000 −0.0263345
$$682$$ 0 0
$$683$$ −33508.0 −1.87723 −0.938615 0.344967i $$-0.887890\pi$$
−0.938615 + 0.344967i $$0.887890\pi$$
$$684$$ 0 0
$$685$$ −8072.33 −0.450260
$$686$$ 0 0
$$687$$ 1917.67 0.106498
$$688$$ 0 0
$$689$$ −26992.0 −1.49247
$$690$$ 0 0
$$691$$ 364.000 0.0200394 0.0100197 0.999950i $$-0.496811\pi$$
0.0100197 + 0.999950i $$0.496811\pi$$
$$692$$ 0 0
$$693$$ −2545.58 −0.139536
$$694$$ 0 0
$$695$$ 5555.03 0.303186
$$696$$ 0 0
$$697$$ 884.000 0.0480400
$$698$$ 0 0
$$699$$ 8478.00 0.458752
$$700$$ 0 0
$$701$$ 3849.49 0.207408 0.103704 0.994608i $$-0.466930\pi$$
0.103704 + 0.994608i $$0.466930\pi$$
$$702$$ 0 0
$$703$$ −14119.5 −0.757507
$$704$$ 0 0
$$705$$ −2928.00 −0.156418
$$706$$ 0 0
$$707$$ −3640.00 −0.193630
$$708$$ 0 0
$$709$$ 23606.1 1.25041 0.625207 0.780459i $$-0.285014\pi$$
0.625207 + 0.780459i $$0.285014\pi$$
$$710$$ 0 0
$$711$$ −7102.18 −0.374617
$$712$$ 0 0
$$713$$ −6864.00 −0.360531
$$714$$ 0 0
$$715$$ −2240.00 −0.117163
$$716$$ 0 0
$$717$$ 7399.17 0.385393
$$718$$ 0 0
$$719$$ −15799.6 −0.819507 −0.409753 0.912196i $$-0.634385\pi$$
−0.409753 + 0.912196i $$0.634385\pi$$
$$720$$ 0 0
$$721$$ 26680.0 1.37811
$$722$$ 0 0
$$723$$ −10062.0 −0.517579
$$724$$ 0 0
$$725$$ 23495.7 1.20360
$$726$$ 0 0
$$727$$ 4607.51 0.235052 0.117526 0.993070i $$-0.462504\pi$$
0.117526 + 0.993070i $$0.462504\pi$$
$$728$$ 0 0
$$729$$ 729.000 0.0370370
$$730$$ 0 0
$$731$$ 8568.00 0.433514
$$732$$ 0 0
$$733$$ −26219.5 −1.32120 −0.660600 0.750738i $$-0.729698\pi$$
−0.660600 + 0.750738i $$0.729698\pi$$
$$734$$ 0 0
$$735$$ 1213.40 0.0608935
$$736$$ 0 0
$$737$$ 12560.0 0.627752
$$738$$ 0 0
$$739$$ −27924.0 −1.38999 −0.694994 0.719016i $$-0.744593\pi$$
−0.694994 + 0.719016i $$0.744593\pi$$
$$740$$ 0 0
$$741$$ 6177.28 0.306246
$$742$$ 0 0
$$743$$ −8937.83 −0.441315 −0.220658 0.975351i $$-0.570820\pi$$
−0.220658 + 0.975351i $$0.570820\pi$$
$$744$$ 0 0
$$745$$ 4264.00 0.209692
$$746$$ 0 0
$$747$$ −9324.00 −0.456690
$$748$$ 0 0
$$749$$ −19855.6 −0.968633
$$750$$ 0 0
$$751$$ −14082.7 −0.684270 −0.342135 0.939651i $$-0.611150\pi$$
−0.342135 + 0.939651i $$0.611150\pi$$
$$752$$ 0 0
$$753$$ 19188.0 0.928618
$$754$$ 0 0
$$755$$ −6408.00 −0.308889
$$756$$ 0 0
$$757$$ 14871.9 0.714039 0.357019 0.934097i $$-0.383793\pi$$
0.357019 + 0.934097i $$0.383793\pi$$
$$758$$ 0 0
$$759$$ −3733.52 −0.178549
$$760$$ 0 0
$$761$$ −15834.0 −0.754247 −0.377124 0.926163i $$-0.623087\pi$$
−0.377124 + 0.926163i $$0.623087\pi$$
$$762$$ 0 0
$$763$$ 560.000 0.0265706
$$764$$ 0 0
$$765$$ 865.499 0.0409048
$$766$$ 0 0
$$767$$ 14413.7 0.678549
$$768$$ 0 0
$$769$$ −16666.0 −0.781523 −0.390762 0.920492i $$-0.627788\pi$$
−0.390762 + 0.920492i $$0.627788\pi$$
$$770$$ 0 0
$$771$$ 20646.0 0.964394
$$772$$ 0 0
$$773$$ 30957.1 1.44043 0.720214 0.693752i $$-0.244044\pi$$
0.720214 + 0.693752i $$0.244044\pi$$
$$774$$ 0 0
$$775$$ 12906.1 0.598195
$$776$$ 0 0
$$777$$ 11520.0 0.531889
$$778$$ 0 0
$$779$$ 1352.00 0.0621828
$$780$$ 0 0
$$781$$ 6675.09 0.305830
$$782$$ 0 0
$$783$$ −5422.09 −0.247471
$$784$$ 0 0
$$785$$ −9984.00 −0.453942
$$786$$ 0 0
$$787$$ −20228.0 −0.916201 −0.458101 0.888900i $$-0.651470\pi$$
−0.458101 + 0.888900i $$0.651470\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 19487.9 0.875991
$$792$$ 0 0
$$793$$ 29120.0 1.30401
$$794$$ 0 0
$$795$$ −5784.00 −0.258034
$$796$$ 0 0
$$797$$ −9008.54 −0.400375 −0.200187 0.979758i $$-0.564155\pi$$
−0.200187 + 0.979758i $$0.564155\pi$$
$$798$$ 0 0
$$799$$ −11732.3 −0.519474
$$800$$ 0 0
$$801$$ 2106.00 0.0928987
$$802$$ 0 0
$$803$$ −6760.00 −0.297080
$$804$$ 0 0
$$805$$ −2489.02 −0.108977
$$806$$ 0 0
$$807$$ −4302.04 −0.187657
$$808$$ 0 0
$$809$$ −9242.00 −0.401646 −0.200823 0.979628i $$-0.564362\pi$$
−0.200823 + 0.979628i $$0.564362\pi$$
$$810$$ 0 0
$$811$$ −10972.0 −0.475067 −0.237533 0.971379i $$-0.576339\pi$$
−0.237533 + 0.971379i $$0.576339\pi$$
$$812$$ 0 0
$$813$$ −17827.6 −0.769053
$$814$$ 0 0
$$815$$ 8292.95 0.356429
$$816$$ 0 0
$$817$$ 13104.0 0.561139
$$818$$ 0 0
$$819$$ −5040.00 −0.215033
$$820$$ 0 0
$$821$$ 9336.64 0.396895 0.198448 0.980112i $$-0.436410\pi$$
0.198448 + 0.980112i $$0.436410\pi$$
$$822$$ 0 0
$$823$$ 3566.65 0.151064 0.0755319 0.997143i $$-0.475935\pi$$
0.0755319 + 0.997143i $$0.475935\pi$$
$$824$$ 0 0
$$825$$ 7020.00 0.296249
$$826$$ 0 0
$$827$$ −18876.0 −0.793691 −0.396846 0.917885i $$-0.629895\pi$$
−0.396846 + 0.917885i $$0.629895\pi$$
$$828$$ 0 0
$$829$$ −6974.90 −0.292218 −0.146109 0.989269i $$-0.546675\pi$$
−0.146109 + 0.989269i $$0.546675\pi$$
$$830$$ 0 0
$$831$$ 3309.26 0.138143
$$832$$ 0 0
$$833$$ 4862.00 0.202231
$$834$$ 0 0
$$835$$ 10400.0 0.431026
$$836$$ 0 0
$$837$$ −2978.33 −0.122994
$$838$$ 0 0
$$839$$ −30077.5 −1.23765 −0.618826 0.785528i $$-0.712392\pi$$
−0.618826 + 0.785528i $$0.712392\pi$$
$$840$$ 0 0
$$841$$ 15939.0 0.653532
$$842$$ 0 0
$$843$$ 18798.0 0.768016
$$844$$ 0 0
$$845$$ 1779.08 0.0724287
$$846$$ 0 0
$$847$$ −13166.3 −0.534121
$$848$$ 0 0
$$849$$ 25788.0 1.04245
$$850$$ 0 0
$$851$$ 16896.0 0.680596
$$852$$ 0 0
$$853$$ 41159.3 1.65213 0.826065 0.563575i $$-0.190574\pi$$
0.826065 + 0.563575i $$0.190574\pi$$
$$854$$ 0 0
$$855$$ 1323.70 0.0529470
$$856$$ 0 0
$$857$$ −25194.0 −1.00421 −0.502107 0.864806i $$-0.667441\pi$$
−0.502107 + 0.864806i $$0.667441\pi$$
$$858$$ 0 0
$$859$$ −9308.00 −0.369715 −0.184857 0.982765i $$-0.559182\pi$$
−0.184857 + 0.982765i $$0.559182\pi$$
$$860$$ 0 0
$$861$$ −1103.09 −0.0436621
$$862$$ 0 0
$$863$$ −26802.2 −1.05719 −0.528596 0.848874i $$-0.677281\pi$$
−0.528596 + 0.848874i $$0.677281\pi$$
$$864$$ 0 0
$$865$$ −4088.00 −0.160689
$$866$$ 0 0
$$867$$ −11271.0 −0.441503
$$868$$ 0 0
$$869$$ 15782.6 0.616098
$$870$$ 0 0
$$871$$ 24867.5 0.967399
$$872$$ 0 0
$$873$$ −1602.00 −0.0621071
$$874$$ 0 0
$$875$$ 9680.00 0.373993
$$876$$ 0 0
$$877$$ −1436.84 −0.0553235 −0.0276617 0.999617i $$-0.508806\pi$$
−0.0276617 + 0.999617i $$0.508806\pi$$
$$878$$ 0 0
$$879$$ 25192.8 0.966703
$$880$$ 0 0
$$881$$ −42830.0 −1.63789 −0.818944 0.573873i $$-0.805440\pi$$
−0.818944 + 0.573873i $$0.805440\pi$$
$$882$$ 0 0
$$883$$ 23964.0 0.913310 0.456655 0.889644i $$-0.349047\pi$$
0.456655 + 0.889644i $$0.349047\pi$$
$$884$$ 0 0
$$885$$ 3088.64 0.117315
$$886$$ 0 0
$$887$$ −28239.0 −1.06897 −0.534483 0.845179i $$-0.679494\pi$$
−0.534483 + 0.845179i $$0.679494\pi$$
$$888$$ 0 0
$$889$$ −25320.0 −0.955237
$$890$$ 0 0
$$891$$ −1620.00 −0.0609114
$$892$$ 0 0
$$893$$ −17943.5 −0.672405
$$894$$ 0 0
$$895$$ −3699.58 −0.138171
$$896$$ 0 0
$$897$$ −7392.00 −0.275152
$$898$$ 0 0
$$899$$ 22152.0 0.821814
$$900$$ 0 0
$$901$$ −23176.1 −0.856947
$$902$$ 0 0
$$903$$ −10691.5 −0.394008
$$904$$ 0 0
$$905$$ 5648.00 0.207454
$$906$$ 0 0
$$907$$ 31972.0 1.17047 0.585233 0.810865i $$-0.301003\pi$$
0.585233 + 0.810865i $$0.301003\pi$$
$$908$$ 0 0
$$909$$ −2316.48 −0.0845246
$$910$$ 0 0
$$911$$ 26858.7 0.976806 0.488403 0.872618i $$-0.337580\pi$$
0.488403 + 0.872618i $$0.337580\pi$$
$$912$$ 0 0
$$913$$ 20720.0 0.751075
$$914$$ 0 0
$$915$$ 6240.00 0.225451
$$916$$ 0 0
$$917$$ −22231.4 −0.800596
$$918$$ 0 0
$$919$$ 40336.2 1.44784 0.723922 0.689882i $$-0.242338\pi$$
0.723922 + 0.689882i $$0.242338\pi$$
$$920$$ 0 0
$$921$$ 14820.0 0.530223
$$922$$ 0 0
$$923$$ 13216.0 0.471300
$$924$$ 0 0
$$925$$ −31768.9 −1.12925
$$926$$ 0 0
$$927$$ 16979.0 0.601580
$$928$$ 0 0
$$929$$ −13650.0 −0.482069 −0.241034 0.970517i $$-0.577487\pi$$
−0.241034 + 0.970517i $$0.577487\pi$$
$$930$$ 0 0
$$931$$ 7436.00 0.261767
$$932$$ 0 0
$$933$$ 10148.4 0.356102
$$934$$ 0 0
$$935$$ −1923.33 −0.0672723
$$936$$ 0 0
$$937$$ 7098.00 0.247472 0.123736 0.992315i $$-0.460512\pi$$
0.123736 + 0.992315i $$0.460512\pi$$
$$938$$ 0 0
$$939$$ −9318.00 −0.323835
$$940$$ 0 0
$$941$$ 41326.1 1.43166 0.715831 0.698274i $$-0.246048\pi$$
0.715831 + 0.698274i $$0.246048\pi$$
$$942$$ 0 0
$$943$$ −1617.86 −0.0558693
$$944$$ 0 0
$$945$$ −1080.00 −0.0371771
$$946$$ 0 0
$$947$$ 9900.00 0.339711 0.169856 0.985469i $$-0.445670\pi$$
0.169856 + 0.985469i $$0.445670\pi$$
$$948$$ 0 0
$$949$$ −13384.1 −0.457815
$$950$$ 0 0
$$951$$ −20186.5 −0.688319
$$952$$ 0 0
$$953$$ −46938.0 −1.59546 −0.797729 0.603016i $$-0.793965\pi$$
−0.797729 + 0.603016i $$0.793965\pi$$
$$954$$ 0 0
$$955$$ −2656.00 −0.0899960
$$956$$ 0 0
$$957$$ 12049.1 0.406993
$$958$$ 0 0
$$959$$ 40361.7 1.35907
$$960$$ 0 0
$$961$$ −17623.0 −0.591554
$$962$$ 0 0
$$963$$ −12636.0 −0.422834
$$964$$ 0 0
$$965$$ 7042.78 0.234938
$$966$$ 0 0
$$967$$ 6989.04 0.232422 0.116211 0.993225i $$-0.462925\pi$$
0.116211 + 0.993225i $$0.462925\pi$$
$$968$$ 0 0
$$969$$ 5304.00 0.175840
$$970$$ 0 0
$$971$$ 53052.0 1.75337 0.876684 0.481067i $$-0.159751\pi$$
0.876684 + 0.481067i $$0.159751\pi$$
$$972$$ 0 0
$$973$$ −27775.2 −0.915139
$$974$$ 0 0
$$975$$ 13898.9 0.456534
$$976$$ 0 0
$$977$$ −41890.0 −1.37173 −0.685865 0.727729i $$-0.740576\pi$$
−0.685865 + 0.727729i $$0.740576\pi$$
$$978$$ 0 0
$$979$$ −4680.00 −0.152782
$$980$$ 0 0
$$981$$ 356.382 0.0115988
$$982$$ 0 0
$$983$$ −10861.2 −0.352408 −0.176204 0.984354i $$-0.556382\pi$$
−0.176204 + 0.984354i $$0.556382\pi$$
$$984$$ 0 0
$$985$$ 7704.00 0.249208
$$986$$ 0 0
$$987$$ 14640.0 0.472134
$$988$$ 0 0
$$989$$ −15680.8 −0.504166
$$990$$ 0 0
$$991$$ 330.926 0.0106077 0.00530384 0.999986i $$-0.498312\pi$$
0.00530384 + 0.999986i $$0.498312\pi$$
$$992$$ 0 0
$$993$$ −8724.00 −0.278799
$$994$$ 0 0
$$995$$ −6104.00 −0.194482
$$996$$ 0 0
$$997$$ −39948.7 −1.26900 −0.634498 0.772925i $$-0.718793\pi$$
−0.634498 + 0.772925i $$0.718793\pi$$
$$998$$ 0 0
$$999$$ 7331.28 0.232184
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 768.4.a.m.1.1 2
3.2 odd 2 2304.4.a.bh.1.2 2
4.3 odd 2 768.4.a.h.1.1 2
8.3 odd 2 inner 768.4.a.m.1.2 2
8.5 even 2 768.4.a.h.1.2 2
12.11 even 2 2304.4.a.bb.1.2 2
16.3 odd 4 384.4.d.d.193.2 yes 4
16.5 even 4 384.4.d.d.193.1 4
16.11 odd 4 384.4.d.d.193.3 yes 4
16.13 even 4 384.4.d.d.193.4 yes 4
24.5 odd 2 2304.4.a.bb.1.1 2
24.11 even 2 2304.4.a.bh.1.1 2
48.5 odd 4 1152.4.d.n.577.3 4
48.11 even 4 1152.4.d.n.577.4 4
48.29 odd 4 1152.4.d.n.577.1 4
48.35 even 4 1152.4.d.n.577.2 4

By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.d.d.193.1 4 16.5 even 4
384.4.d.d.193.2 yes 4 16.3 odd 4
384.4.d.d.193.3 yes 4 16.11 odd 4
384.4.d.d.193.4 yes 4 16.13 even 4
768.4.a.h.1.1 2 4.3 odd 2
768.4.a.h.1.2 2 8.5 even 2
768.4.a.m.1.1 2 1.1 even 1 trivial
768.4.a.m.1.2 2 8.3 odd 2 inner
1152.4.d.n.577.1 4 48.29 odd 4
1152.4.d.n.577.2 4 48.35 even 4
1152.4.d.n.577.3 4 48.5 odd 4
1152.4.d.n.577.4 4 48.11 even 4
2304.4.a.bb.1.1 2 24.5 odd 2
2304.4.a.bb.1.2 2 12.11 even 2
2304.4.a.bh.1.1 2 24.11 even 2
2304.4.a.bh.1.2 2 3.2 odd 2