# Properties

 Label 384.4.d.d.193.3 Level $384$ Weight $4$ Character 384.193 Analytic conductor $22.657$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

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

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$384 = 2^{7} \cdot 3$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 384.d (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$22.6567334422$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 193.3 Root $$-0.707107 - 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 384.193 Dual form 384.4.d.d.193.2

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+3.00000i q^{3} -2.82843i q^{5} +14.1421 q^{7} -9.00000 q^{9} +O(q^{10})$$ $$q+3.00000i q^{3} -2.82843i q^{5} +14.1421 q^{7} -9.00000 q^{9} +20.0000i q^{11} +39.5980i q^{13} +8.48528 q^{15} -34.0000 q^{17} -52.0000i q^{19} +42.4264i q^{21} +62.2254 q^{23} +117.000 q^{25} -27.0000i q^{27} +200.818i q^{29} +110.309 q^{31} -60.0000 q^{33} -40.0000i q^{35} +271.529i q^{37} -118.794 q^{39} +26.0000 q^{41} +252.000i q^{43} +25.4558i q^{45} -345.068 q^{47} -143.000 q^{49} -102.000i q^{51} +681.651i q^{53} +56.5685 q^{55} +156.000 q^{57} +364.000i q^{59} +735.391i q^{61} -127.279 q^{63} +112.000 q^{65} -628.000i q^{67} +186.676i q^{69} -333.754 q^{71} -338.000 q^{73} +351.000i q^{75} +282.843i q^{77} +789.131 q^{79} +81.0000 q^{81} -1036.00i q^{83} +96.1665i q^{85} -602.455 q^{87} -234.000 q^{89} +560.000i q^{91} +330.926i q^{93} -147.078 q^{95} -178.000 q^{97} -180.000i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 36 q^{9}+O(q^{10})$$ 4 * q - 36 * q^9 $$4 q - 36 q^{9} - 136 q^{17} + 468 q^{25} - 240 q^{33} + 104 q^{41} - 572 q^{49} + 624 q^{57} + 448 q^{65} - 1352 q^{73} + 324 q^{81} - 936 q^{89} - 712 q^{97}+O(q^{100})$$ 4 * q - 36 * q^9 - 136 * q^17 + 468 * q^25 - 240 * q^33 + 104 * q^41 - 572 * q^49 + 624 * q^57 + 448 * q^65 - 1352 * q^73 + 324 * q^81 - 936 * q^89 - 712 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$133$$ $$257$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 3.00000i 0.577350i
$$4$$ 0 0
$$5$$ − 2.82843i − 0.252982i −0.991968 0.126491i $$-0.959628\pi$$
0.991968 0.126491i $$-0.0403715\pi$$
$$6$$ 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
$$11$$ 20.0000i 0.548202i 0.961701 + 0.274101i $$0.0883803\pi$$
−0.961701 + 0.274101i $$0.911620\pi$$
$$12$$ 0 0
$$13$$ 39.5980i 0.844808i 0.906408 + 0.422404i $$0.138814\pi$$
−0.906408 + 0.422404i $$0.861186\pi$$
$$14$$ 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
$$19$$ − 52.0000i − 0.627875i −0.949444 0.313937i $$-0.898352\pi$$
0.949444 0.313937i $$-0.101648\pi$$
$$20$$ 0 0
$$21$$ 42.4264i 0.440867i
$$22$$ 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
$$27$$ − 27.0000i − 0.192450i
$$28$$ 0 0
$$29$$ 200.818i 1.28590i 0.765909 + 0.642949i $$0.222289\pi$$
−0.765909 + 0.642949i $$0.777711\pi$$
$$30$$ 0 0
$$31$$ 110.309 0.639097 0.319549 0.947570i $$-0.396469\pi$$
0.319549 + 0.947570i $$0.396469\pi$$
$$32$$ 0 0
$$33$$ −60.0000 −0.316505
$$34$$ 0 0
$$35$$ − 40.0000i − 0.193178i
$$36$$ 0 0
$$37$$ 271.529i 1.20646i 0.797567 + 0.603231i $$0.206120\pi$$
−0.797567 + 0.603231i $$0.793880\pi$$
$$38$$ 0 0
$$39$$ −118.794 −0.487750
$$40$$ 0 0
$$41$$ 26.0000 0.0990370 0.0495185 0.998773i $$-0.484231\pi$$
0.0495185 + 0.998773i $$0.484231\pi$$
$$42$$ 0 0
$$43$$ 252.000i 0.893713i 0.894606 + 0.446856i $$0.147456\pi$$
−0.894606 + 0.446856i $$0.852544\pi$$
$$44$$ 0 0
$$45$$ 25.4558i 0.0843274i
$$46$$ 0 0
$$47$$ −345.068 −1.07092 −0.535461 0.844560i $$-0.679862\pi$$
−0.535461 + 0.844560i $$0.679862\pi$$
$$48$$ 0 0
$$49$$ −143.000 −0.416910
$$50$$ 0 0
$$51$$ − 102.000i − 0.280056i
$$52$$ 0 0
$$53$$ 681.651i 1.76664i 0.468770 + 0.883320i $$0.344697\pi$$
−0.468770 + 0.883320i $$0.655303\pi$$
$$54$$ 0 0
$$55$$ 56.5685 0.138685
$$56$$ 0 0
$$57$$ 156.000 0.362504
$$58$$ 0 0
$$59$$ 364.000i 0.803199i 0.915815 + 0.401600i $$0.131546\pi$$
−0.915815 + 0.401600i $$0.868454\pi$$
$$60$$ 0 0
$$61$$ 735.391i 1.54356i 0.635889 + 0.771780i $$0.280633\pi$$
−0.635889 + 0.771780i $$0.719367\pi$$
$$62$$ 0 0
$$63$$ −127.279 −0.254535
$$64$$ 0 0
$$65$$ 112.000 0.213721
$$66$$ 0 0
$$67$$ − 628.000i − 1.14511i −0.819866 0.572555i $$-0.805952\pi$$
0.819866 0.572555i $$-0.194048\pi$$
$$68$$ 0 0
$$69$$ 186.676i 0.325698i
$$70$$ 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 $$-0.587341\pi$$
−0.270958 + 0.962591i $$0.587341\pi$$
$$74$$ 0 0
$$75$$ 351.000i 0.540400i
$$76$$ 0 0
$$77$$ 282.843i 0.418609i
$$78$$ 0 0
$$79$$ 789.131 1.12385 0.561925 0.827188i $$-0.310061\pi$$
0.561925 + 0.827188i $$0.310061\pi$$
$$80$$ 0 0
$$81$$ 81.0000 0.111111
$$82$$ 0 0
$$83$$ − 1036.00i − 1.37007i −0.728510 0.685035i $$-0.759787\pi$$
0.728510 0.685035i $$-0.240213\pi$$
$$84$$ 0 0
$$85$$ 96.1665i 0.122714i
$$86$$ 0 0
$$87$$ −602.455 −0.742413
$$88$$ 0 0
$$89$$ −234.000 −0.278696 −0.139348 0.990243i $$-0.544501\pi$$
−0.139348 + 0.990243i $$0.544501\pi$$
$$90$$ 0 0
$$91$$ 560.000i 0.645098i
$$92$$ 0 0
$$93$$ 330.926i 0.368983i
$$94$$ 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
$$99$$ − 180.000i − 0.182734i
$$100$$ 0 0
$$101$$ − 257.387i − 0.253574i −0.991930 0.126787i $$-0.959534\pi$$
0.991930 0.126787i $$-0.0404664\pi$$
$$102$$ 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
$$107$$ 1404.00i 1.26850i 0.773127 + 0.634251i $$0.218692\pi$$
−0.773127 + 0.634251i $$0.781308\pi$$
$$108$$ 0 0
$$109$$ − 39.5980i − 0.0347963i −0.999849 0.0173982i $$-0.994462\pi$$
0.999849 0.0173982i $$-0.00553829\pi$$
$$110$$ 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
$$115$$ − 176.000i − 0.142714i
$$116$$ 0 0
$$117$$ − 356.382i − 0.281603i
$$118$$ 0 0
$$119$$ −480.833 −0.370402
$$120$$ 0 0
$$121$$ 931.000 0.699474
$$122$$ 0 0
$$123$$ 78.0000i 0.0571791i
$$124$$ 0 0
$$125$$ − 684.479i − 0.489774i
$$126$$ 0 0
$$127$$ 1790.39 1.25096 0.625480 0.780241i $$-0.284903\pi$$
0.625480 + 0.780241i $$0.284903\pi$$
$$128$$ 0 0
$$129$$ −756.000 −0.515985
$$130$$ 0 0
$$131$$ − 1572.00i − 1.04844i −0.851581 0.524222i $$-0.824356\pi$$
0.851581 0.524222i $$-0.175644\pi$$
$$132$$ 0 0
$$133$$ − 735.391i − 0.479447i
$$134$$ 0 0
$$135$$ −76.3675 −0.0486864
$$136$$ 0 0
$$137$$ −2854.00 −1.77981 −0.889904 0.456148i $$-0.849229\pi$$
−0.889904 + 0.456148i $$0.849229\pi$$
$$138$$ 0 0
$$139$$ 1964.00i 1.19845i 0.800581 + 0.599224i $$0.204524\pi$$
−0.800581 + 0.599224i $$0.795476\pi$$
$$140$$ 0 0
$$141$$ − 1035.20i − 0.618297i
$$142$$ 0 0
$$143$$ −791.960 −0.463126
$$144$$ 0 0
$$145$$ 568.000 0.325309
$$146$$ 0 0
$$147$$ − 429.000i − 0.240703i
$$148$$ 0 0
$$149$$ − 1507.55i − 0.828882i −0.910076 0.414441i $$-0.863977\pi$$
0.910076 0.414441i $$-0.136023\pi$$
$$150$$ 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
$$155$$ − 312.000i − 0.161680i
$$156$$ 0 0
$$157$$ − 3529.88i − 1.79436i −0.441663 0.897181i $$-0.645611\pi$$
0.441663 0.897181i $$-0.354389\pi$$
$$158$$ 0 0
$$159$$ −2044.95 −1.01997
$$160$$ 0 0
$$161$$ 880.000 0.430768
$$162$$ 0 0
$$163$$ − 2932.00i − 1.40891i −0.709750 0.704454i $$-0.751192\pi$$
0.709750 0.704454i $$-0.248808\pi$$
$$164$$ 0 0
$$165$$ 169.706i 0.0800701i
$$166$$ 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
$$171$$ 468.000i 0.209292i
$$172$$ 0 0
$$173$$ − 1445.33i − 0.635180i −0.948228 0.317590i $$-0.897126\pi$$
0.948228 0.317590i $$-0.102874\pi$$
$$174$$ 0 0
$$175$$ 1654.63 0.714733
$$176$$ 0 0
$$177$$ −1092.00 −0.463727
$$178$$ 0 0
$$179$$ 1308.00i 0.546170i 0.961990 + 0.273085i $$0.0880441\pi$$
−0.961990 + 0.273085i $$0.911956\pi$$
$$180$$ 0 0
$$181$$ − 1996.87i − 0.820034i −0.912078 0.410017i $$-0.865523\pi$$
0.912078 0.410017i $$-0.134477\pi$$
$$182$$ 0 0
$$183$$ −2206.17 −0.891175
$$184$$ 0 0
$$185$$ 768.000 0.305213
$$186$$ 0 0
$$187$$ − 680.000i − 0.265917i
$$188$$ 0 0
$$189$$ − 381.838i − 0.146956i
$$190$$ 0 0
$$191$$ −939.038 −0.355740 −0.177870 0.984054i $$-0.556921\pi$$
−0.177870 + 0.984054i $$0.556921\pi$$
$$192$$ 0 0
$$193$$ −2490.00 −0.928674 −0.464337 0.885659i $$-0.653707\pi$$
−0.464337 + 0.885659i $$0.653707\pi$$
$$194$$ 0 0
$$195$$ 336.000i 0.123392i
$$196$$ 0 0
$$197$$ − 2723.78i − 0.985081i −0.870290 0.492540i $$-0.836068\pi$$
0.870290 0.492540i $$-0.163932\pi$$
$$198$$ 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
$$203$$ 2840.00i 0.981916i
$$204$$ 0 0
$$205$$ − 73.5391i − 0.0250546i
$$206$$ 0 0
$$207$$ −560.029 −0.188042
$$208$$ 0 0
$$209$$ 1040.00 0.344202
$$210$$ 0 0
$$211$$ 924.000i 0.301473i 0.988574 + 0.150736i $$0.0481645\pi$$
−0.988574 + 0.150736i $$0.951836\pi$$
$$212$$ 0 0
$$213$$ − 1001.26i − 0.322091i
$$214$$ 0 0
$$215$$ 712.764 0.226093
$$216$$ 0 0
$$217$$ 1560.00 0.488017
$$218$$ 0 0
$$219$$ − 1014.00i − 0.312876i
$$220$$ 0 0
$$221$$ − 1346.33i − 0.409792i
$$222$$ 0 0
$$223$$ 2276.88 0.683728 0.341864 0.939749i $$-0.388942\pi$$
0.341864 + 0.939749i $$0.388942\pi$$
$$224$$ 0 0
$$225$$ −1053.00 −0.312000
$$226$$ 0 0
$$227$$ − 156.000i − 0.0456127i −0.999740 0.0228064i $$-0.992740\pi$$
0.999740 0.0228064i $$-0.00726012\pi$$
$$228$$ 0 0
$$229$$ 639.225i 0.184459i 0.995738 + 0.0922296i $$0.0293994\pi$$
−0.995738 + 0.0922296i $$0.970601\pi$$
$$230$$ 0 0
$$231$$ −848.528 −0.241684
$$232$$ 0 0
$$233$$ −2826.00 −0.794581 −0.397291 0.917693i $$-0.630049\pi$$
−0.397291 + 0.917693i $$0.630049\pi$$
$$234$$ 0 0
$$235$$ 976.000i 0.270924i
$$236$$ 0 0
$$237$$ 2367.39i 0.648855i
$$238$$ 0 0
$$239$$ −2466.39 −0.667521 −0.333760 0.942658i $$-0.608318\pi$$
−0.333760 + 0.942658i $$0.608318\pi$$
$$240$$ 0 0
$$241$$ −3354.00 −0.896474 −0.448237 0.893915i $$-0.647948\pi$$
−0.448237 + 0.893915i $$0.647948\pi$$
$$242$$ 0 0
$$243$$ 243.000i 0.0641500i
$$244$$ 0 0
$$245$$ 404.465i 0.105471i
$$246$$ 0 0
$$247$$ 2059.09 0.530433
$$248$$ 0 0
$$249$$ 3108.00 0.791010
$$250$$ 0 0
$$251$$ − 6396.00i − 1.60841i −0.594349 0.804207i $$-0.702590\pi$$
0.594349 0.804207i $$-0.297410\pi$$
$$252$$ 0 0
$$253$$ 1244.51i 0.309255i
$$254$$ 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
$$259$$ 3840.00i 0.921259i
$$260$$ 0 0
$$261$$ − 1807.36i − 0.428632i
$$262$$ 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
$$267$$ − 702.000i − 0.160905i
$$268$$ 0 0
$$269$$ 1434.01i 0.325031i 0.986706 + 0.162515i $$0.0519607\pi$$
−0.986706 + 0.162515i $$0.948039\pi$$
$$270$$ 0 0
$$271$$ 5942.53 1.33204 0.666020 0.745934i $$-0.267997\pi$$
0.666020 + 0.745934i $$0.267997\pi$$
$$272$$ 0 0
$$273$$ −1680.00 −0.372448
$$274$$ 0 0
$$275$$ 2340.00i 0.513117i
$$276$$ 0 0
$$277$$ 1103.09i 0.239271i 0.992818 + 0.119635i $$0.0381726\pi$$
−0.992818 + 0.119635i $$0.961827\pi$$
$$278$$ 0 0
$$279$$ −992.778 −0.213032
$$280$$ 0 0
$$281$$ −6266.00 −1.33024 −0.665121 0.746735i $$-0.731620\pi$$
−0.665121 + 0.746735i $$0.731620\pi$$
$$282$$ 0 0
$$283$$ − 8596.00i − 1.80558i −0.430082 0.902790i $$-0.641515\pi$$
0.430082 0.902790i $$-0.358485\pi$$
$$284$$ 0 0
$$285$$ − 441.235i − 0.0917070i
$$286$$ 0 0
$$287$$ 367.696 0.0756250
$$288$$ 0 0
$$289$$ −3757.00 −0.764706
$$290$$ 0 0
$$291$$ − 534.000i − 0.107573i
$$292$$ 0 0
$$293$$ 8397.60i 1.67438i 0.546913 + 0.837189i $$0.315803\pi$$
−0.546913 + 0.837189i $$0.684197\pi$$
$$294$$ 0 0
$$295$$ 1029.55 0.203195
$$296$$ 0 0
$$297$$ 540.000 0.105502
$$298$$ 0 0
$$299$$ 2464.00i 0.476578i
$$300$$ 0 0
$$301$$ 3563.82i 0.682442i
$$302$$ 0 0
$$303$$ 772.161 0.146401
$$304$$ 0 0
$$305$$ 2080.00 0.390493
$$306$$ 0 0
$$307$$ 4940.00i 0.918374i 0.888340 + 0.459187i $$0.151859\pi$$
−0.888340 + 0.459187i $$0.848141\pi$$
$$308$$ 0 0
$$309$$ 5659.68i 1.04197i
$$310$$ 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 $$-0.409516\pi$$
0.280450 + 0.959869i $$0.409516\pi$$
$$314$$ 0 0
$$315$$ 360.000i 0.0643927i
$$316$$ 0 0
$$317$$ 6728.83i 1.19220i 0.802909 + 0.596102i $$0.203285\pi$$
−0.802909 + 0.596102i $$0.796715\pi$$
$$318$$ 0 0
$$319$$ −4016.37 −0.704932
$$320$$ 0 0
$$321$$ −4212.00 −0.732370
$$322$$ 0 0
$$323$$ 1768.00i 0.304564i
$$324$$ 0 0
$$325$$ 4632.96i 0.790740i
$$326$$ 0 0
$$327$$ 118.794 0.0200897
$$328$$ 0 0
$$329$$ −4880.00 −0.817760
$$330$$ 0 0
$$331$$ 2908.00i 0.482895i 0.970414 + 0.241447i $$0.0776221\pi$$
−0.970414 + 0.241447i $$0.922378\pi$$
$$332$$ 0 0
$$333$$ − 2443.76i − 0.402154i
$$334$$ 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
$$339$$ 4134.00i 0.662325i
$$340$$ 0 0
$$341$$ 2206.17i 0.350355i
$$342$$ 0 0
$$343$$ −6873.08 −1.08196
$$344$$ 0 0
$$345$$ 528.000 0.0823958
$$346$$ 0 0
$$347$$ − 9996.00i − 1.54644i −0.634140 0.773218i $$-0.718646\pi$$
0.634140 0.773218i $$-0.281354\pi$$
$$348$$ 0 0
$$349$$ 3993.74i 0.612550i 0.951943 + 0.306275i $$0.0990827\pi$$
−0.951943 + 0.306275i $$0.900917\pi$$
$$350$$ 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
$$355$$ 944.000i 0.141133i
$$356$$ 0 0
$$357$$ − 1442.50i − 0.213852i
$$358$$ 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
$$363$$ 2793.00i 0.403842i
$$364$$ 0 0
$$365$$ 956.008i 0.137095i
$$366$$ 0 0
$$367$$ −7628.27 −1.08499 −0.542496 0.840058i $$-0.682521\pi$$
−0.542496 + 0.840058i $$0.682521\pi$$
$$368$$ 0 0
$$369$$ −234.000 −0.0330123
$$370$$ 0 0
$$371$$ 9640.00i 1.34901i
$$372$$ 0 0
$$373$$ 8383.46i 1.16375i 0.813278 + 0.581875i $$0.197681\pi$$
−0.813278 + 0.581875i $$0.802319\pi$$
$$374$$ 0 0
$$375$$ 2053.44 0.282771
$$376$$ 0 0
$$377$$ −7952.00 −1.08634
$$378$$ 0 0
$$379$$ − 12788.0i − 1.73318i −0.499020 0.866590i $$-0.666307\pi$$
0.499020 0.866590i $$-0.333693\pi$$
$$380$$ 0 0
$$381$$ 5371.18i 0.722242i
$$382$$ 0 0
$$383$$ −2319.31 −0.309429 −0.154714 0.987959i $$-0.549446\pi$$
−0.154714 + 0.987959i $$0.549446\pi$$
$$384$$ 0 0
$$385$$ 800.000 0.105901
$$386$$ 0 0
$$387$$ − 2268.00i − 0.297904i
$$388$$ 0 0
$$389$$ − 2684.18i − 0.349854i −0.984581 0.174927i $$-0.944031\pi$$
0.984581 0.174927i $$-0.0559689\pi$$
$$390$$ 0 0
$$391$$ −2115.66 −0.273641
$$392$$ 0 0
$$393$$ 4716.00 0.605320
$$394$$ 0 0
$$395$$ − 2232.00i − 0.284314i
$$396$$ 0 0
$$397$$ 2206.17i 0.278903i 0.990229 + 0.139452i $$0.0445340\pi$$
−0.990229 + 0.139452i $$0.955466\pi$$
$$398$$ 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
$$403$$ 4368.00i 0.539915i
$$404$$ 0 0
$$405$$ − 229.103i − 0.0281091i
$$406$$ 0 0
$$407$$ −5430.58 −0.661385
$$408$$ 0 0
$$409$$ −5126.00 −0.619717 −0.309859 0.950783i $$-0.600282\pi$$
−0.309859 + 0.950783i $$0.600282\pi$$
$$410$$ 0 0
$$411$$ − 8562.00i − 1.02757i
$$412$$ 0 0
$$413$$ 5147.74i 0.613326i
$$414$$ 0 0
$$415$$ −2930.25 −0.346603
$$416$$ 0 0
$$417$$ −5892.00 −0.691924
$$418$$ 0 0
$$419$$ − 2924.00i − 0.340923i −0.985364 0.170462i $$-0.945474\pi$$
0.985364 0.170462i $$-0.0545259\pi$$
$$420$$ 0 0
$$421$$ − 7314.31i − 0.846741i −0.905957 0.423370i $$-0.860847\pi$$
0.905957 0.423370i $$-0.139153\pi$$
$$422$$ 0 0
$$423$$ 3105.61 0.356974
$$424$$ 0 0
$$425$$ −3978.00 −0.454027
$$426$$ 0 0
$$427$$ 10400.0i 1.17867i
$$428$$ 0 0
$$429$$ − 2375.88i − 0.267386i
$$430$$ 0 0
$$431$$ 15844.8 1.77081 0.885405 0.464819i $$-0.153881\pi$$
0.885405 + 0.464819i $$0.153881\pi$$
$$432$$ 0 0
$$433$$ −6274.00 −0.696326 −0.348163 0.937434i $$-0.613194\pi$$
−0.348163 + 0.937434i $$0.613194\pi$$
$$434$$ 0 0
$$435$$ 1704.00i 0.187817i
$$436$$ 0 0
$$437$$ − 3235.72i − 0.354200i
$$438$$ 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
$$443$$ − 5084.00i − 0.545255i −0.962120 0.272628i $$-0.912107\pi$$
0.962120 0.272628i $$-0.0878927\pi$$
$$444$$ 0 0
$$445$$ 661.852i 0.0705051i
$$446$$ 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
$$451$$ 520.000i 0.0542923i
$$452$$ 0 0
$$453$$ 6796.71i 0.704939i
$$454$$ 0 0
$$455$$ 1583.92 0.163198
$$456$$ 0 0
$$457$$ 6474.00 0.662672 0.331336 0.943513i $$-0.392501\pi$$
0.331336 + 0.943513i $$0.392501\pi$$
$$458$$ 0 0
$$459$$ 918.000i 0.0933520i
$$460$$ 0 0
$$461$$ − 6321.53i − 0.638662i −0.947643 0.319331i $$-0.896542\pi$$
0.947643 0.319331i $$-0.103458\pi$$
$$462$$ 0 0
$$463$$ −11435.3 −1.14783 −0.573915 0.818915i $$-0.694576\pi$$
−0.573915 + 0.818915i $$0.694576\pi$$
$$464$$ 0 0
$$465$$ 936.000 0.0933462
$$466$$ 0 0
$$467$$ 3796.00i 0.376141i 0.982156 + 0.188071i $$0.0602234\pi$$
−0.982156 + 0.188071i $$0.939777\pi$$
$$468$$ 0 0
$$469$$ − 8881.26i − 0.874411i
$$470$$ 0 0
$$471$$ 10589.6 1.03598
$$472$$ 0 0
$$473$$ −5040.00 −0.489935
$$474$$ 0 0
$$475$$ − 6084.00i − 0.587691i
$$476$$ 0 0
$$477$$ − 6134.86i − 0.588880i
$$478$$ 0 0
$$479$$ 10493.5 1.00096 0.500479 0.865749i $$-0.333157\pi$$
0.500479 + 0.865749i $$0.333157\pi$$
$$480$$ 0 0
$$481$$ −10752.0 −1.01923
$$482$$ 0 0
$$483$$ 2640.00i 0.248704i
$$484$$ 0 0
$$485$$ 503.460i 0.0471360i
$$486$$ 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
$$491$$ 15452.0i 1.42024i 0.704079 + 0.710121i $$0.251360\pi$$
−0.704079 + 0.710121i $$0.748640\pi$$
$$492$$ 0 0
$$493$$ − 6827.82i − 0.623752i
$$494$$ 0 0
$$495$$ −509.117 −0.0462285
$$496$$ 0 0
$$497$$ −4720.00 −0.425998
$$498$$ 0 0
$$499$$ − 52.0000i − 0.00466501i −0.999997 0.00233250i $$-0.999258\pi$$
0.999997 0.00233250i $$-0.000742460\pi$$
$$500$$ 0 0
$$501$$ − 11030.9i − 0.983678i
$$502$$ 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
$$507$$ 1887.00i 0.165295i
$$508$$ 0 0
$$509$$ − 16362.5i − 1.42486i −0.701744 0.712429i $$-0.747595\pi$$
0.701744 0.712429i $$-0.252405\pi$$
$$510$$ 0 0
$$511$$ −4780.04 −0.413809
$$512$$ 0 0
$$513$$ −1404.00 −0.120835
$$514$$ 0 0
$$515$$ − 5336.00i − 0.456567i
$$516$$ 0 0
$$517$$ − 6901.36i − 0.587082i
$$518$$ 0 0
$$519$$ 4335.98 0.366721
$$520$$ 0 0
$$521$$ 714.000 0.0600401 0.0300201 0.999549i $$-0.490443\pi$$
0.0300201 + 0.999549i $$0.490443\pi$$
$$522$$ 0 0
$$523$$ 5980.00i 0.499975i 0.968249 + 0.249988i $$0.0804266\pi$$
−0.968249 + 0.249988i $$0.919573\pi$$
$$524$$ 0 0
$$525$$ 4963.89i 0.412651i
$$526$$ 0 0
$$527$$ −3750.49 −0.310008
$$528$$ 0 0
$$529$$ −8295.00 −0.681762
$$530$$ 0 0
$$531$$ − 3276.00i − 0.267733i
$$532$$ 0 0
$$533$$ 1029.55i 0.0836673i
$$534$$ 0 0
$$535$$ 3971.11 0.320909
$$536$$ 0 0
$$537$$ −3924.00 −0.315332
$$538$$ 0 0
$$539$$ − 2860.00i − 0.228551i
$$540$$ 0 0
$$541$$ − 13729.2i − 1.09106i −0.838091 0.545530i $$-0.816328\pi$$
0.838091 0.545530i $$-0.183672\pi$$
$$542$$ 0 0
$$543$$ 5990.61 0.473447
$$544$$ 0 0
$$545$$ −112.000 −0.00880285
$$546$$ 0 0
$$547$$ − 18500.0i − 1.44607i −0.690809 0.723037i $$-0.742745\pi$$
0.690809 0.723037i $$-0.257255\pi$$
$$548$$ 0 0
$$549$$ − 6618.52i − 0.514520i
$$550$$ 0 0
$$551$$ 10442.6 0.807382
$$552$$ 0 0
$$553$$ 11160.0 0.858176
$$554$$ 0 0
$$555$$ 2304.00i 0.176215i
$$556$$ 0 0
$$557$$ − 8765.30i − 0.666782i −0.942789 0.333391i $$-0.891807\pi$$
0.942789 0.333391i $$-0.108193\pi$$
$$558$$ 0 0
$$559$$ −9978.69 −0.755015
$$560$$ 0 0
$$561$$ 2040.00 0.153527
$$562$$ 0 0
$$563$$ − 268.000i − 0.0200619i −0.999950 0.0100310i $$-0.996807\pi$$
0.999950 0.0100310i $$-0.00319301\pi$$
$$564$$ 0 0
$$565$$ − 3897.57i − 0.290216i
$$566$$ 0 0
$$567$$ 1145.51 0.0848448
$$568$$ 0 0
$$569$$ 13866.0 1.02160 0.510802 0.859698i $$-0.329348\pi$$
0.510802 + 0.859698i $$0.329348\pi$$
$$570$$ 0 0
$$571$$ − 5140.00i − 0.376712i −0.982101 0.188356i $$-0.939684\pi$$
0.982101 0.188356i $$-0.0603158\pi$$
$$572$$ 0 0
$$573$$ − 2817.11i − 0.205387i
$$574$$ 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
$$579$$ − 7470.00i − 0.536170i
$$580$$ 0 0
$$581$$ − 14651.3i − 1.04619i
$$582$$ 0 0
$$583$$ −13633.0 −0.968477
$$584$$ 0 0
$$585$$ −1008.00 −0.0712405
$$586$$ 0 0
$$587$$ 8844.00i 0.621859i 0.950433 + 0.310929i $$0.100640\pi$$
−0.950433 + 0.310929i $$0.899360\pi$$
$$588$$ 0 0
$$589$$ − 5736.05i − 0.401273i
$$590$$ 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
$$595$$ 1360.00i 0.0937051i
$$596$$ 0 0
$$597$$ 6474.27i 0.443843i
$$598$$ 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 $$-0.486364\pi$$
0.0428270 + 0.999083i $$0.486364\pi$$
$$602$$ 0 0
$$603$$ 5652.00i 0.381704i
$$604$$ 0 0
$$605$$ − 2633.27i − 0.176955i
$$606$$ 0 0
$$607$$ −16288.9 −1.08920 −0.544602 0.838695i $$-0.683319\pi$$
−0.544602 + 0.838695i $$0.683319\pi$$
$$608$$ 0 0
$$609$$ −8520.00 −0.566909
$$610$$ 0 0
$$611$$ − 13664.0i − 0.904724i
$$612$$ 0 0
$$613$$ 7138.95i 0.470374i 0.971950 + 0.235187i $$0.0755703\pi$$
−0.971950 + 0.235187i $$0.924430\pi$$
$$614$$ 0 0
$$615$$ 220.617 0.0144653
$$616$$ 0 0
$$617$$ −16874.0 −1.10101 −0.550504 0.834833i $$-0.685564\pi$$
−0.550504 + 0.834833i $$0.685564\pi$$
$$618$$ 0 0
$$619$$ 20748.0i 1.34723i 0.739085 + 0.673613i $$0.235258\pi$$
−0.739085 + 0.673613i $$0.764742\pi$$
$$620$$ 0 0
$$621$$ − 1680.09i − 0.108566i
$$622$$ 0 0
$$623$$ −3309.26 −0.212813
$$624$$ 0 0
$$625$$ 12689.0 0.812096
$$626$$ 0 0
$$627$$ 3120.00i 0.198725i
$$628$$ 0 0
$$629$$ − 9231.99i − 0.585220i
$$630$$ 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
$$635$$ − 5064.00i − 0.316470i
$$636$$ 0 0
$$637$$ − 5662.51i − 0.352209i
$$638$$ 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
$$643$$ 1148.00i 0.0704086i 0.999380 + 0.0352043i $$0.0112082\pi$$
−0.999380 + 0.0352043i $$0.988792\pi$$
$$644$$ 0 0
$$645$$ 2138.29i 0.130535i
$$646$$ 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
$$651$$ 4680.00i 0.281757i
$$652$$ 0 0
$$653$$ 21069.0i 1.26262i 0.775530 + 0.631311i $$0.217483\pi$$
−0.775530 + 0.631311i $$0.782517\pi$$
$$654$$ 0 0
$$655$$ −4446.29 −0.265238
$$656$$ 0 0
$$657$$ 3042.00 0.180639
$$658$$ 0 0
$$659$$ − 18356.0i − 1.08505i −0.840040 0.542525i $$-0.817468\pi$$
0.840040 0.542525i $$-0.182532\pi$$
$$660$$ 0 0
$$661$$ − 15250.9i − 0.897414i −0.893679 0.448707i $$-0.851885\pi$$
0.893679 0.448707i $$-0.148115\pi$$
$$662$$ 0 0
$$663$$ 4038.99 0.236594
$$664$$ 0 0
$$665$$ −2080.00 −0.121292
$$666$$ 0 0
$$667$$ 12496.0i 0.725408i
$$668$$ 0 0
$$669$$ 6830.65i 0.394751i
$$670$$ 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
$$675$$ − 3159.00i − 0.180133i
$$676$$ 0 0
$$677$$ − 12742.1i − 0.723364i −0.932302 0.361682i $$-0.882203\pi$$
0.932302 0.361682i $$-0.117797\pi$$
$$678$$ 0 0
$$679$$ −2517.30 −0.142276
$$680$$ 0 0
$$681$$ 468.000 0.0263345
$$682$$ 0 0
$$683$$ 33508.0i 1.87723i 0.344967 + 0.938615i $$0.387890\pi$$
−0.344967 + 0.938615i $$0.612110\pi$$
$$684$$ 0 0
$$685$$ 8072.33i 0.450260i
$$686$$ 0 0
$$687$$ −1917.67 −0.106498
$$688$$ 0 0
$$689$$ −26992.0 −1.49247
$$690$$ 0 0
$$691$$ 364.000i 0.0200394i 0.999950 + 0.0100197i $$0.00318942\pi$$
−0.999950 + 0.0100197i $$0.996811\pi$$
$$692$$ 0 0
$$693$$ − 2545.58i − 0.139536i
$$694$$ 0 0
$$695$$ 5555.03 0.303186
$$696$$ 0 0
$$697$$ −884.000 −0.0480400
$$698$$ 0 0
$$699$$ − 8478.00i − 0.458752i
$$700$$ 0 0
$$701$$ − 3849.49i − 0.207408i −0.994608 0.103704i $$-0.966930\pi$$
0.994608 0.103704i $$-0.0330695\pi$$
$$702$$ 0 0
$$703$$ 14119.5 0.757507
$$704$$ 0 0
$$705$$ −2928.00 −0.156418
$$706$$ 0 0
$$707$$ − 3640.00i − 0.193630i
$$708$$ 0 0
$$709$$ 23606.1i 1.25041i 0.780459 + 0.625207i $$0.214986\pi$$
−0.780459 + 0.625207i $$0.785014\pi$$
$$710$$ 0 0
$$711$$ −7102.18 −0.374617
$$712$$ 0 0
$$713$$ 6864.00 0.360531
$$714$$ 0 0
$$715$$ 2240.00i 0.117163i
$$716$$ 0 0
$$717$$ − 7399.17i − 0.385393i
$$718$$ 0 0
$$719$$ 15799.6 0.819507 0.409753 0.912196i $$-0.365615\pi$$
0.409753 + 0.912196i $$0.365615\pi$$
$$720$$ 0 0
$$721$$ 26680.0 1.37811
$$722$$ 0 0
$$723$$ − 10062.0i − 0.517579i
$$724$$ 0 0
$$725$$ 23495.7i 1.20360i
$$726$$ 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
$$731$$ − 8568.00i − 0.433514i
$$732$$ 0 0
$$733$$ 26219.5i 1.32120i 0.750738 + 0.660600i $$0.229698\pi$$
−0.750738 + 0.660600i $$0.770302\pi$$
$$734$$ 0 0
$$735$$ −1213.40 −0.0608935
$$736$$ 0 0
$$737$$ 12560.0 0.627752
$$738$$ 0 0
$$739$$ − 27924.0i − 1.38999i −0.719016 0.694994i $$-0.755407\pi$$
0.719016 0.694994i $$-0.244593\pi$$
$$740$$ 0 0
$$741$$ 6177.28i 0.306246i
$$742$$ 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
$$747$$ 9324.00i 0.456690i
$$748$$ 0 0
$$749$$ 19855.6i 0.968633i
$$750$$ 0 0
$$751$$ 14082.7 0.684270 0.342135 0.939651i $$-0.388850\pi$$
0.342135 + 0.939651i $$0.388850\pi$$
$$752$$ 0 0
$$753$$ 19188.0 0.928618
$$754$$ 0 0
$$755$$ − 6408.00i − 0.308889i
$$756$$ 0 0
$$757$$ 14871.9i 0.714039i 0.934097 + 0.357019i $$0.116207\pi$$
−0.934097 + 0.357019i $$0.883793\pi$$
$$758$$ 0 0
$$759$$ −3733.52 −0.178549
$$760$$ 0 0
$$761$$ 15834.0 0.754247 0.377124 0.926163i $$-0.376913\pi$$
0.377124 + 0.926163i $$0.376913\pi$$
$$762$$ 0 0
$$763$$ − 560.000i − 0.0265706i
$$764$$ 0 0
$$765$$ − 865.499i − 0.0409048i
$$766$$ 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
$$771$$ 20646.0i 0.964394i
$$772$$ 0 0
$$773$$ 30957.1i 1.44043i 0.693752 + 0.720214i $$0.255956\pi$$
−0.693752 + 0.720214i $$0.744044\pi$$
$$774$$ 0 0
$$775$$ 12906.1 0.598195
$$776$$ 0 0
$$777$$ −11520.0 −0.531889
$$778$$ 0 0
$$779$$ − 1352.00i − 0.0621828i
$$780$$ 0 0
$$781$$ − 6675.09i − 0.305830i
$$782$$ 0 0
$$783$$ 5422.09 0.247471
$$784$$ 0 0
$$785$$ −9984.00 −0.453942
$$786$$ 0 0
$$787$$ − 20228.0i − 0.916201i −0.888900 0.458101i $$-0.848530\pi$$
0.888900 0.458101i $$-0.151470\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 19487.9 0.875991
$$792$$ 0 0
$$793$$ −29120.0 −1.30401
$$794$$ 0 0
$$795$$ 5784.00i 0.258034i
$$796$$ 0 0
$$797$$ 9008.54i 0.400375i 0.979758 + 0.200187i $$0.0641551\pi$$
−0.979758 + 0.200187i $$0.935845\pi$$
$$798$$ 0 0
$$799$$ 11732.3 0.519474
$$800$$ 0 0
$$801$$ 2106.00 0.0928987
$$802$$ 0 0
$$803$$ − 6760.00i − 0.297080i
$$804$$ 0 0
$$805$$ − 2489.02i − 0.108977i
$$806$$ 0 0
$$807$$ −4302.04 −0.187657
$$808$$ 0 0
$$809$$ 9242.00 0.401646 0.200823 0.979628i $$-0.435638\pi$$
0.200823 + 0.979628i $$0.435638\pi$$
$$810$$ 0 0
$$811$$ 10972.0i 0.475067i 0.971379 + 0.237533i $$0.0763389\pi$$
−0.971379 + 0.237533i $$0.923661\pi$$
$$812$$ 0 0
$$813$$ 17827.6i 0.769053i
$$814$$ 0 0
$$815$$ −8292.95 −0.356429
$$816$$ 0 0
$$817$$ 13104.0 0.561139
$$818$$ 0 0
$$819$$ − 5040.00i − 0.215033i
$$820$$ 0 0
$$821$$ 9336.64i 0.396895i 0.980112 + 0.198448i $$0.0635900\pi$$
−0.980112 + 0.198448i $$0.936410\pi$$
$$822$$ 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
$$827$$ 18876.0i 0.793691i 0.917885 + 0.396846i $$0.129895\pi$$
−0.917885 + 0.396846i $$0.870105\pi$$
$$828$$ 0 0
$$829$$ 6974.90i 0.292218i 0.989269 + 0.146109i $$0.0466749\pi$$
−0.989269 + 0.146109i $$0.953325\pi$$
$$830$$ 0 0
$$831$$ −3309.26 −0.138143
$$832$$ 0 0
$$833$$ 4862.00 0.202231
$$834$$ 0 0
$$835$$ 10400.0i 0.431026i
$$836$$ 0 0
$$837$$ − 2978.33i − 0.122994i
$$838$$ 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
$$843$$ − 18798.0i − 0.768016i
$$844$$ 0 0
$$845$$ − 1779.08i − 0.0724287i
$$846$$ 0 0
$$847$$ 13166.3 0.534121
$$848$$ 0 0
$$849$$ 25788.0 1.04245
$$850$$ 0 0
$$851$$ 16896.0i 0.680596i
$$852$$ 0 0
$$853$$ 41159.3i 1.65213i 0.563575 + 0.826065i $$0.309426\pi$$
−0.563575 + 0.826065i $$0.690574\pi$$
$$854$$ 0 0
$$855$$ 1323.70 0.0529470
$$856$$ 0 0
$$857$$ 25194.0 1.00421 0.502107 0.864806i $$-0.332559\pi$$
0.502107 + 0.864806i $$0.332559\pi$$
$$858$$ 0 0
$$859$$ 9308.00i 0.369715i 0.982765 + 0.184857i $$0.0591823\pi$$
−0.982765 + 0.184857i $$0.940818\pi$$
$$860$$ 0 0
$$861$$ 1103.09i 0.0436621i
$$862$$ 0 0
$$863$$ 26802.2 1.05719 0.528596 0.848874i $$-0.322719\pi$$
0.528596 + 0.848874i $$0.322719\pi$$
$$864$$ 0 0
$$865$$ −4088.00 −0.160689
$$866$$ 0 0
$$867$$ − 11271.0i − 0.441503i
$$868$$ 0 0
$$869$$ 15782.6i 0.616098i
$$870$$ 0 0
$$871$$ 24867.5 0.967399
$$872$$ 0 0
$$873$$ 1602.00 0.0621071
$$874$$ 0 0
$$875$$ − 9680.00i − 0.373993i
$$876$$ 0 0
$$877$$ 1436.84i 0.0553235i 0.999617 + 0.0276617i $$0.00880613\pi$$
−0.999617 + 0.0276617i $$0.991194\pi$$
$$878$$ 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
$$883$$ 23964.0i 0.913310i 0.889644 + 0.456655i $$0.150953\pi$$
−0.889644 + 0.456655i $$0.849047\pi$$
$$884$$ 0 0
$$885$$ 3088.64i 0.117315i
$$886$$ 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
$$891$$ 1620.00i 0.0609114i
$$892$$ 0 0
$$893$$ 17943.5i 0.672405i
$$894$$ 0 0
$$895$$ 3699.58 0.138171
$$896$$ 0 0
$$897$$ −7392.00 −0.275152
$$898$$ 0 0
$$899$$ 22152.0i 0.821814i
$$900$$ 0 0
$$901$$ − 23176.1i − 0.856947i
$$902$$ 0 0
$$903$$ −10691.5 −0.394008
$$904$$ 0 0
$$905$$ −5648.00 −0.207454
$$906$$ 0 0
$$907$$ − 31972.0i − 1.17047i −0.810865 0.585233i $$-0.801003\pi$$
0.810865 0.585233i $$-0.198997\pi$$
$$908$$ 0 0
$$909$$ 2316.48i 0.0845246i
$$910$$ 0 0
$$911$$ −26858.7 −0.976806 −0.488403 0.872618i $$-0.662420\pi$$
−0.488403 + 0.872618i $$0.662420\pi$$
$$912$$ 0 0
$$913$$ 20720.0 0.751075
$$914$$ 0 0
$$915$$ 6240.00i 0.225451i
$$916$$ 0 0
$$917$$ − 22231.4i − 0.800596i
$$918$$ 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
$$923$$ − 13216.0i − 0.471300i
$$924$$ 0 0
$$925$$ 31768.9i 1.12925i
$$926$$ 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
$$931$$ 7436.00i 0.261767i
$$932$$ 0 0
$$933$$ 10148.4i 0.356102i
$$934$$ 0 0
$$935$$ −1923.33 −0.0672723
$$936$$ 0 0
$$937$$ −7098.00 −0.247472 −0.123736 0.992315i $$-0.539488\pi$$
−0.123736 + 0.992315i $$0.539488\pi$$
$$938$$ 0 0
$$939$$ 9318.00i 0.323835i
$$940$$ 0 0
$$941$$ − 41326.1i − 1.43166i −0.698274 0.715831i $$-0.746048\pi$$
0.698274 0.715831i $$-0.253952\pi$$
$$942$$ 0 0
$$943$$ 1617.86 0.0558693
$$944$$ 0 0
$$945$$ −1080.00 −0.0371771
$$946$$ 0 0
$$947$$ 9900.00i 0.339711i 0.985469 + 0.169856i $$0.0543302\pi$$
−0.985469 + 0.169856i $$0.945670\pi$$
$$948$$ 0 0
$$949$$ − 13384.1i − 0.457815i
$$950$$ 0 0
$$951$$ −20186.5 −0.688319
$$952$$ 0 0
$$953$$ 46938.0 1.59546 0.797729 0.603016i $$-0.206035\pi$$
0.797729 + 0.603016i $$0.206035\pi$$
$$954$$ 0 0
$$955$$ 2656.00i 0.0899960i
$$956$$ 0 0
$$957$$ − 12049.1i − 0.406993i
$$958$$ 0 0
$$959$$ −40361.7 −1.35907
$$960$$ 0 0
$$961$$ −17623.0 −0.591554
$$962$$ 0 0
$$963$$ − 12636.0i − 0.422834i
$$964$$ 0 0
$$965$$ 7042.78i 0.234938i
$$966$$ 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
$$971$$ − 53052.0i − 1.75337i −0.481067 0.876684i $$-0.659751\pi$$
0.481067 0.876684i $$-0.340249\pi$$
$$972$$ 0 0
$$973$$ 27775.2i 0.915139i
$$974$$ 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
$$979$$ − 4680.00i − 0.152782i
$$980$$ 0 0
$$981$$ 356.382i 0.0115988i
$$982$$ 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
$$987$$ − 14640.0i − 0.472134i
$$988$$ 0 0
$$989$$ 15680.8i 0.504166i
$$990$$ 0 0
$$991$$ −330.926 −0.0106077 −0.00530384 0.999986i $$-0.501688\pi$$
−0.00530384 + 0.999986i $$0.501688\pi$$
$$992$$ 0 0
$$993$$ −8724.00 −0.278799
$$994$$ 0 0
$$995$$ − 6104.00i − 0.194482i
$$996$$ 0 0
$$997$$ − 39948.7i − 1.26900i −0.772925 0.634498i $$-0.781207\pi$$
0.772925 0.634498i $$-0.218793\pi$$
$$998$$ 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 384.4.d.d.193.3 yes 4
3.2 odd 2 1152.4.d.n.577.4 4
4.3 odd 2 inner 384.4.d.d.193.1 4
8.3 odd 2 inner 384.4.d.d.193.4 yes 4
8.5 even 2 inner 384.4.d.d.193.2 yes 4
12.11 even 2 1152.4.d.n.577.3 4
16.3 odd 4 768.4.a.m.1.1 2
16.5 even 4 768.4.a.m.1.2 2
16.11 odd 4 768.4.a.h.1.2 2
16.13 even 4 768.4.a.h.1.1 2
24.5 odd 2 1152.4.d.n.577.2 4
24.11 even 2 1152.4.d.n.577.1 4
48.5 odd 4 2304.4.a.bh.1.1 2
48.11 even 4 2304.4.a.bb.1.1 2
48.29 odd 4 2304.4.a.bb.1.2 2
48.35 even 4 2304.4.a.bh.1.2 2

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