# Properties

 Label 2304.4.a.bb.1.1 Level $2304$ Weight $4$ Character 2304.1 Self dual yes Analytic conductor $135.940$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2304,4,Mod(1,2304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2304, 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("2304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2304 = 2^{8} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2304.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: $$135.940400653$$ 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 $$\chi$$ $$=$$ 2304.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q-2.82843 q^{5} +14.1421 q^{7} +O(q^{10})$$ $$q-2.82843 q^{5} +14.1421 q^{7} -20.0000 q^{11} +39.5980 q^{13} +34.0000 q^{17} +52.0000 q^{19} -62.2254 q^{23} -117.000 q^{25} -200.818 q^{29} -110.309 q^{31} -40.0000 q^{35} -271.529 q^{37} +26.0000 q^{41} +252.000 q^{43} -345.068 q^{47} -143.000 q^{49} +681.651 q^{53} +56.5685 q^{55} -364.000 q^{59} +735.391 q^{61} -112.000 q^{65} +628.000 q^{67} +333.754 q^{71} +338.000 q^{73} -282.843 q^{77} -789.131 q^{79} -1036.00 q^{83} -96.1665 q^{85} -234.000 q^{89} +560.000 q^{91} -147.078 q^{95} -178.000 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q+O(q^{10})$$ 2 * q $$2 q - 40 q^{11} + 68 q^{17} + 104 q^{19} - 234 q^{25} - 80 q^{35} + 52 q^{41} + 504 q^{43} - 286 q^{49} - 728 q^{59} - 224 q^{65} + 1256 q^{67} + 676 q^{73} - 2072 q^{83} - 468 q^{89} + 1120 q^{91} - 356 q^{97}+O(q^{100})$$ 2 * q - 40 * q^11 + 68 * q^17 + 104 * q^19 - 234 * q^25 - 80 * q^35 + 52 * q^41 + 504 * q^43 - 286 * q^49 - 728 * q^59 - 224 * q^65 + 1256 * q^67 + 676 * q^73 - 2072 * q^83 - 468 * q^89 + 1120 * q^91 - 356 * q^97

## 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
$$3$$ 0 0
$$4$$ 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
$$9$$ 0 0
$$10$$ 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 $$-0.361186\pi$$
0.422404 + 0.906408i $$0.361186\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 34.0000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 0 0
$$19$$ 52.0000 0.627875 0.313937 0.949444i $$-0.398352\pi$$
0.313937 + 0.949444i $$0.398352\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 0 0
$$23$$ −62.2254 −0.564126 −0.282063 0.959396i $$-0.591019\pi$$
−0.282063 + 0.959396i $$0.591019\pi$$
$$24$$ 0 0
$$25$$ −117.000 −0.936000
$$26$$ 0 0
$$27$$ 0 0
$$28$$ 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
$$33$$ 0 0
$$34$$ 0 0
$$35$$ −40.0000 −0.193178
$$36$$ 0 0
$$37$$ −271.529 −1.20646 −0.603231 0.797567i $$-0.706120\pi$$
−0.603231 + 0.797567i $$0.706120\pi$$
$$38$$ 0 0
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 26.0000 0.0990370 0.0495185 0.998773i $$-0.484231\pi$$
0.0495185 + 0.998773i $$0.484231\pi$$
$$42$$ 0 0
$$43$$ 252.000 0.893713 0.446856 0.894606i $$-0.352544\pi$$
0.446856 + 0.894606i $$0.352544\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 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
$$51$$ 0 0
$$52$$ 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
$$57$$ 0 0
$$58$$ 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 $$-0.219367\pi$$
0.771780 + 0.635889i $$0.219367\pi$$
$$62$$ 0 0
$$63$$ 0 0
$$64$$ 0 0
$$65$$ −112.000 −0.213721
$$66$$ 0 0
$$67$$ 628.000 1.14511 0.572555 0.819866i $$-0.305952\pi$$
0.572555 + 0.819866i $$0.305952\pi$$
$$68$$ 0 0
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 333.754 0.557878 0.278939 0.960309i $$-0.410017\pi$$
0.278939 + 0.960309i $$0.410017\pi$$
$$72$$ 0 0
$$73$$ 338.000 0.541917 0.270958 0.962591i $$-0.412659\pi$$
0.270958 + 0.962591i $$0.412659\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 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
$$81$$ 0 0
$$82$$ 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
$$87$$ 0 0
$$88$$ 0 0
$$89$$ −234.000 −0.278696 −0.139348 0.990243i $$-0.544501\pi$$
−0.139348 + 0.990243i $$0.544501\pi$$
$$90$$ 0 0
$$91$$ 560.000 0.645098
$$92$$ 0 0
$$93$$ 0 0
$$94$$ 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
$$99$$ 0 0
$$100$$ 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
$$105$$ 0 0
$$106$$ 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 $$-0.505538\pi$$
−0.0173982 + 0.999849i $$0.505538\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 0 0
$$113$$ −1378.00 −1.14718 −0.573590 0.819143i $$-0.694450\pi$$
−0.573590 + 0.819143i $$0.694450\pi$$
$$114$$ 0 0
$$115$$ 176.000 0.142714
$$116$$ 0 0
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 480.833 0.370402
$$120$$ 0 0
$$121$$ −931.000 −0.699474
$$122$$ 0 0
$$123$$ 0 0
$$124$$ 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
$$129$$ 0 0
$$130$$ 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
$$135$$ 0 0
$$136$$ 0 0
$$137$$ −2854.00 −1.77981 −0.889904 0.456148i $$-0.849229\pi$$
−0.889904 + 0.456148i $$0.849229\pi$$
$$138$$ 0 0
$$139$$ 1964.00 1.19845 0.599224 0.800581i $$-0.295476\pi$$
0.599224 + 0.800581i $$0.295476\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −791.960 −0.463126
$$144$$ 0 0
$$145$$ 568.000 0.325309
$$146$$ 0 0
$$147$$ 0 0
$$148$$ 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
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 312.000 0.161680
$$156$$ 0 0
$$157$$ −3529.88 −1.79436 −0.897181 0.441663i $$-0.854389\pi$$
−0.897181 + 0.441663i $$0.854389\pi$$
$$158$$ 0 0
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −880.000 −0.430768
$$162$$ 0 0
$$163$$ 2932.00 1.40891 0.704454 0.709750i $$-0.251192\pi$$
0.704454 + 0.709750i $$0.251192\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3676.96 1.70378 0.851890 0.523720i $$-0.175456\pi$$
0.851890 + 0.523720i $$0.175456\pi$$
$$168$$ 0 0
$$169$$ −629.000 −0.286299
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 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
$$177$$ 0 0
$$178$$ 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 $$-0.365523\pi$$
0.410017 + 0.912078i $$0.365523\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 0 0
$$185$$ 768.000 0.305213
$$186$$ 0 0
$$187$$ −680.000 −0.265917
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 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
$$195$$ 0 0
$$196$$ 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
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −2840.00 −0.981916
$$204$$ 0 0
$$205$$ −73.5391 −0.0250546
$$206$$ 0 0
$$207$$ 0 0
$$208$$ 0 0
$$209$$ −1040.00 −0.344202
$$210$$ 0 0
$$211$$ −924.000 −0.301473 −0.150736 0.988574i $$-0.548164\pi$$
−0.150736 + 0.988574i $$0.548164\pi$$
$$212$$ 0 0
$$213$$ 0 0
$$214$$ 0 0
$$215$$ −712.764 −0.226093
$$216$$ 0 0
$$217$$ −1560.00 −0.488017
$$218$$ 0 0
$$219$$ 0 0
$$220$$ 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
$$225$$ 0 0
$$226$$ 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 $$-0.529399\pi$$
−0.0922296 + 0.995738i $$0.529399\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 0 0
$$233$$ −2826.00 −0.794581 −0.397291 0.917693i $$-0.630049\pi$$
−0.397291 + 0.917693i $$0.630049\pi$$
$$234$$ 0 0
$$235$$ 976.000 0.270924
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 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
$$243$$ 0 0
$$244$$ 0 0
$$245$$ 404.465 0.105471
$$246$$ 0 0
$$247$$ 2059.09 0.530433
$$248$$ 0 0
$$249$$ 0 0
$$250$$ 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
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −6882.00 −1.67038 −0.835189 0.549962i $$-0.814642\pi$$
−0.835189 + 0.549962i $$0.814642\pi$$
$$258$$ 0 0
$$259$$ −3840.00 −0.921259
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 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
$$267$$ 0 0
$$268$$ 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
$$273$$ 0 0
$$274$$ 0 0
$$275$$ 2340.00 0.513117
$$276$$ 0 0
$$277$$ −1103.09 −0.239271 −0.119635 0.992818i $$-0.538173\pi$$
−0.119635 + 0.992818i $$0.538173\pi$$
$$278$$ 0 0
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −6266.00 −1.33024 −0.665121 0.746735i $$-0.731620\pi$$
−0.665121 + 0.746735i $$0.731620\pi$$
$$282$$ 0 0
$$283$$ −8596.00 −1.80558 −0.902790 0.430082i $$-0.858485\pi$$
−0.902790 + 0.430082i $$0.858485\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 367.696 0.0756250
$$288$$ 0 0
$$289$$ −3757.00 −0.764706
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 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
$$297$$ 0 0
$$298$$ 0 0
$$299$$ −2464.00 −0.476578
$$300$$ 0 0
$$301$$ 3563.82 0.682442
$$302$$ 0 0
$$303$$ 0 0
$$304$$ 0 0
$$305$$ −2080.00 −0.390493
$$306$$ 0 0
$$307$$ −4940.00 −0.918374 −0.459187 0.888340i $$-0.651859\pi$$
−0.459187 + 0.888340i $$0.651859\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −3382.80 −0.616788 −0.308394 0.951259i $$-0.599791\pi$$
−0.308394 + 0.951259i $$0.599791\pi$$
$$312$$ 0 0
$$313$$ −3106.00 −0.560899 −0.280450 0.959869i $$-0.590484\pi$$
−0.280450 + 0.959869i $$0.590484\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 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
$$321$$ 0 0
$$322$$ 0 0
$$323$$ 1768.00 0.304564
$$324$$ 0 0
$$325$$ −4632.96 −0.790740
$$326$$ 0 0
$$327$$ 0 0
$$328$$ 0 0
$$329$$ −4880.00 −0.817760
$$330$$ 0 0
$$331$$ 2908.00 0.482895 0.241447 0.970414i $$-0.422378\pi$$
0.241447 + 0.970414i $$0.422378\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 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
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 2206.17 0.350355
$$342$$ 0 0
$$343$$ −6873.08 −1.08196
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 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 $$-0.400917\pi$$
0.306275 + 0.951943i $$0.400917\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 0 0
$$353$$ −6738.00 −1.01594 −0.507971 0.861374i $$-0.669604\pi$$
−0.507971 + 0.861374i $$0.669604\pi$$
$$354$$ 0 0
$$355$$ −944.000 −0.141133
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −2132.63 −0.313527 −0.156763 0.987636i $$-0.550106\pi$$
−0.156763 + 0.987636i $$0.550106\pi$$
$$360$$ 0 0
$$361$$ −4155.00 −0.605773
$$362$$ 0 0
$$363$$ 0 0
$$364$$ 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
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 9640.00 1.34901
$$372$$ 0 0
$$373$$ −8383.46 −1.16375 −0.581875 0.813278i $$-0.697681\pi$$
−0.581875 + 0.813278i $$0.697681\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −7952.00 −1.08634
$$378$$ 0 0
$$379$$ −12788.0 −1.73318 −0.866590 0.499020i $$-0.833693\pi$$
−0.866590 + 0.499020i $$0.833693\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 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
$$387$$ 0 0
$$388$$ 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
$$393$$ 0 0
$$394$$ 0 0
$$395$$ 2232.00 0.284314
$$396$$ 0 0
$$397$$ 2206.17 0.278903 0.139452 0.990229i $$-0.455466\pi$$
0.139452 + 0.990229i $$0.455466\pi$$
$$398$$ 0 0
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −3582.00 −0.446076 −0.223038 0.974810i $$-0.571597\pi$$
−0.223038 + 0.974810i $$0.571597\pi$$
$$402$$ 0 0
$$403$$ −4368.00 −0.539915
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 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
$$411$$ 0 0
$$412$$ 0 0
$$413$$ −5147.74 −0.613326
$$414$$ 0 0
$$415$$ 2930.25 0.346603
$$416$$ 0 0
$$417$$ 0 0
$$418$$ 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 $$-0.360847\pi$$
0.423370 + 0.905957i $$0.360847\pi$$
$$422$$ 0 0
$$423$$ 0 0
$$424$$ 0 0
$$425$$ −3978.00 −0.454027
$$426$$ 0 0
$$427$$ 10400.0 1.17867
$$428$$ 0 0
$$429$$ 0 0
$$430$$ 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
$$435$$ 0 0
$$436$$ 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
$$441$$ 0 0
$$442$$ 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
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −14190.0 −1.49146 −0.745732 0.666246i $$-0.767900\pi$$
−0.745732 + 0.666246i $$0.767900\pi$$
$$450$$ 0 0
$$451$$ −520.000 −0.0542923
$$452$$ 0 0
$$453$$ 0 0
$$454$$ 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
$$459$$ 0 0
$$460$$ 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
$$465$$ 0 0
$$466$$ 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
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −5040.00 −0.489935
$$474$$ 0 0
$$475$$ −6084.00 −0.587691
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 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
$$483$$ 0 0
$$484$$ 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
$$489$$ 0 0
$$490$$ 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
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 4720.00 0.425998
$$498$$ 0 0
$$499$$ 52.0000 0.00466501 0.00233250 0.999997i $$-0.499258\pi$$
0.00233250 + 0.999997i $$0.499258\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 0 0
$$503$$ −12428.1 −1.10167 −0.550837 0.834613i $$-0.685691\pi$$
−0.550837 + 0.834613i $$0.685691\pi$$
$$504$$ 0 0
$$505$$ 728.000 0.0641497
$$506$$ 0 0
$$507$$ 0 0
$$508$$ 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
$$513$$ 0 0
$$514$$ 0 0
$$515$$ −5336.00 −0.456567
$$516$$ 0 0
$$517$$ 6901.36 0.587082
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 714.000 0.0600401 0.0300201 0.999549i $$-0.490443\pi$$
0.0300201 + 0.999549i $$0.490443\pi$$
$$522$$ 0 0
$$523$$ 5980.00 0.499975 0.249988 0.968249i $$-0.419573\pi$$
0.249988 + 0.968249i $$0.419573\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −3750.49 −0.310008
$$528$$ 0 0
$$529$$ −8295.00 −0.681762
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 1029.55 0.0836673
$$534$$ 0 0
$$535$$ 3971.11 0.320909
$$536$$ 0 0
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 2860.00 0.228551
$$540$$ 0 0
$$541$$ −13729.2 −1.09106 −0.545530 0.838091i $$-0.683672\pi$$
−0.545530 + 0.838091i $$0.683672\pi$$
$$542$$ 0 0
$$543$$ 0 0
$$544$$ 0 0
$$545$$ 112.000 0.00880285
$$546$$ 0 0
$$547$$ 18500.0 1.44607 0.723037 0.690809i $$-0.242745\pi$$
0.723037 + 0.690809i $$0.242745\pi$$
$$548$$ 0 0
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −10442.6 −0.807382
$$552$$ 0 0
$$553$$ −11160.0 −0.858176
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 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
$$561$$ 0 0
$$562$$ 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
$$567$$ 0 0
$$568$$ 0 0
$$569$$ 13866.0 1.02160 0.510802 0.859698i $$-0.329348\pi$$
0.510802 + 0.859698i $$0.329348\pi$$
$$570$$ 0 0
$$571$$ −5140.00 −0.376712 −0.188356 0.982101i $$-0.560316\pi$$
−0.188356 + 0.982101i $$0.560316\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ 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
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −14651.3 −1.04619
$$582$$ 0 0
$$583$$ −13633.0 −0.968477
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 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
$$591$$ 0 0
$$592$$ 0 0
$$593$$ 9406.00 0.651363 0.325681 0.945480i $$-0.394406\pi$$
0.325681 + 0.945480i $$0.394406\pi$$
$$594$$ 0 0
$$595$$ −1360.00 −0.0937051
$$596$$ 0 0
$$597$$ 0 0
$$598$$ 0 0
$$599$$ 23459.0 1.60018 0.800090 0.599880i $$-0.204785\pi$$
0.800090 + 0.599880i $$0.204785\pi$$
$$600$$ 0 0
$$601$$ −1262.00 −0.0856540 −0.0428270 0.999083i $$-0.513636\pi$$
−0.0428270 + 0.999083i $$0.513636\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 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
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −13664.0 −0.904724
$$612$$ 0 0
$$613$$ −7138.95 −0.470374 −0.235187 0.971950i $$-0.575570\pi$$
−0.235187 + 0.971950i $$0.575570\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −16874.0 −1.10101 −0.550504 0.834833i $$-0.685564\pi$$
−0.550504 + 0.834833i $$0.685564\pi$$
$$618$$ 0 0
$$619$$ 20748.0 1.34723 0.673613 0.739085i $$-0.264742\pi$$
0.673613 + 0.739085i $$0.264742\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 0 0
$$623$$ −3309.26 −0.212813
$$624$$ 0 0
$$625$$ 12689.0 0.812096
$$626$$ 0 0
$$627$$ 0 0
$$628$$ 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
$$633$$ 0 0
$$634$$ 0 0
$$635$$ 5064.00 0.316470
$$636$$ 0 0
$$637$$ −5662.51 −0.352209
$$638$$ 0 0
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −17758.0 −1.09423 −0.547113 0.837059i $$-0.684273\pi$$
−0.547113 + 0.837059i $$0.684273\pi$$
$$642$$ 0 0
$$643$$ −1148.00 −0.0704086 −0.0352043 0.999380i $$-0.511208\pi$$
−0.0352043 + 0.999380i $$0.511208\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −26988.9 −1.63994 −0.819970 0.572406i $$-0.806010\pi$$
−0.819970 + 0.572406i $$0.806010\pi$$
$$648$$ 0 0
$$649$$ 7280.00 0.440316
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 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
$$657$$ 0 0
$$658$$ 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 $$-0.351885\pi$$
0.448707 + 0.893679i $$0.351885\pi$$
$$662$$ 0 0
$$663$$ 0 0
$$664$$ 0 0
$$665$$ −2080.00 −0.121292
$$666$$ 0 0
$$667$$ 12496.0 0.725408
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 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
$$675$$ 0 0
$$676$$ 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
$$681$$ 0 0
$$682$$ 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
$$687$$ 0 0
$$688$$ 0 0
$$689$$ 26992.0 1.49247
$$690$$ 0 0
$$691$$ −364.000 −0.0200394 −0.0100197 0.999950i $$-0.503189\pi$$
−0.0100197 + 0.999950i $$0.503189\pi$$
$$692$$ 0 0
$$693$$ 0 0
$$694$$ 0 0
$$695$$ −5555.03 −0.303186
$$696$$ 0 0
$$697$$ 884.000 0.0480400
$$698$$ 0 0
$$699$$ 0 0
$$700$$ 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
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −3640.00 −0.193630
$$708$$ 0 0
$$709$$ −23606.1 −1.25041 −0.625207 0.780459i $$-0.714986\pi$$
−0.625207 + 0.780459i $$0.714986\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 6864.00 0.360531
$$714$$ 0 0
$$715$$ 2240.00 0.117163
$$716$$ 0 0
$$717$$ 0 0
$$718$$ 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
$$723$$ 0 0
$$724$$ 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
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 8568.00 0.433514
$$732$$ 0 0
$$733$$ 26219.5 1.32120 0.660600 0.750738i $$-0.270302\pi$$
0.660600 + 0.750738i $$0.270302\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −12560.0 −0.627752
$$738$$ 0 0
$$739$$ 27924.0 1.38999 0.694994 0.719016i $$-0.255407\pi$$
0.694994 + 0.719016i $$0.255407\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 8937.83 0.441315 0.220658 0.975351i $$-0.429180\pi$$
0.220658 + 0.975351i $$0.429180\pi$$
$$744$$ 0 0
$$745$$ 4264.00 0.209692
$$746$$ 0 0
$$747$$ 0 0
$$748$$ 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
$$753$$ 0 0
$$754$$ 0 0
$$755$$ −6408.00 −0.308889
$$756$$ 0 0
$$757$$ −14871.9 −0.714039 −0.357019 0.934097i $$-0.616207\pi$$
−0.357019 + 0.934097i $$0.616207\pi$$
$$758$$ 0 0
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 15834.0 0.754247 0.377124 0.926163i $$-0.376913\pi$$
0.377124 + 0.926163i $$0.376913\pi$$
$$762$$ 0 0
$$763$$ −560.000 −0.0265706
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 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
$$771$$ 0 0
$$772$$ 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
$$777$$ 0 0
$$778$$ 0 0
$$779$$ 1352.00 0.0621828
$$780$$ 0 0
$$781$$ −6675.09 −0.305830
$$782$$ 0 0
$$783$$ 0 0
$$784$$ 0 0
$$785$$ 9984.00 0.453942
$$786$$ 0 0
$$787$$ 20228.0 0.916201 0.458101 0.888900i $$-0.348530\pi$$
0.458101 + 0.888900i $$0.348530\pi$$
$$788$$ 0 0
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −19487.9 −0.875991
$$792$$ 0 0
$$793$$ 29120.0 1.30401
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 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
$$801$$ 0 0
$$802$$ 0 0
$$803$$ −6760.00 −0.297080
$$804$$ 0 0
$$805$$ 2489.02 0.108977
$$806$$ 0 0
$$807$$ 0 0
$$808$$ 0 0
$$809$$ 9242.00 0.401646 0.200823 0.979628i $$-0.435638\pi$$
0.200823 + 0.979628i $$0.435638\pi$$
$$810$$ 0 0
$$811$$ 10972.0 0.475067 0.237533 0.971379i $$-0.423661\pi$$
0.237533 + 0.971379i $$0.423661\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −8292.95 −0.356429
$$816$$ 0 0
$$817$$ 13104.0 0.561139
$$818$$ 0 0
$$819$$ 0 0
$$820$$ 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
$$825$$ 0 0
$$826$$ 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 $$-0.453325\pi$$
0.146109 + 0.989269i $$0.453325\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 0 0
$$833$$ −4862.00 −0.202231
$$834$$ 0 0
$$835$$ −10400.0 −0.431026
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 0 0
$$839$$ 30077.5 1.23765 0.618826 0.785528i $$-0.287608\pi$$
0.618826 + 0.785528i $$0.287608\pi$$
$$840$$ 0 0
$$841$$ 15939.0 0.653532
$$842$$ 0 0
$$843$$ 0 0
$$844$$ 0 0
$$845$$ 1779.08 0.0724287
$$846$$ 0 0
$$847$$ −13166.3 −0.534121
$$848$$ 0 0
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 16896.0 0.680596
$$852$$ 0 0
$$853$$ −41159.3 −1.65213 −0.826065 0.563575i $$-0.809426\pi$$
−0.826065 + 0.563575i $$0.809426\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 25194.0 1.00421 0.502107 0.864806i $$-0.332559\pi$$
0.502107 + 0.864806i $$0.332559\pi$$
$$858$$ 0 0
$$859$$ 9308.00 0.369715 0.184857 0.982765i $$-0.440818\pi$$
0.184857 + 0.982765i $$0.440818\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 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
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 15782.6 0.616098
$$870$$ 0 0
$$871$$ 24867.5 0.967399
$$872$$ 0 0
$$873$$ 0 0
$$874$$ 0 0
$$875$$ 9680.00 0.373993
$$876$$ 0 0
$$877$$ 1436.84 0.0553235 0.0276617 0.999617i $$-0.491194\pi$$
0.0276617 + 0.999617i $$0.491194\pi$$
$$878$$ 0 0
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 42830.0 1.63789 0.818944 0.573873i $$-0.194560\pi$$
0.818944 + 0.573873i $$0.194560\pi$$
$$882$$ 0 0
$$883$$ −23964.0 −0.913310 −0.456655 0.889644i $$-0.650953\pi$$
−0.456655 + 0.889644i $$0.650953\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 28239.0 1.06897 0.534483 0.845179i $$-0.320506\pi$$
0.534483 + 0.845179i $$0.320506\pi$$
$$888$$ 0 0
$$889$$ −25320.0 −0.955237
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 0 0
$$893$$ −17943.5 −0.672405
$$894$$ 0 0
$$895$$ −3699.58 −0.138171
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 0 0
$$899$$ 22152.0 0.821814
$$900$$ 0 0
$$901$$ 23176.1 0.856947
$$902$$ 0 0
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −5648.00 −0.207454
$$906$$ 0 0
$$907$$ −31972.0 −1.17047 −0.585233 0.810865i $$-0.698997\pi$$
−0.585233 + 0.810865i $$0.698997\pi$$
$$908$$ 0 0
$$909$$ 0 0
$$910$$ 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
$$915$$ 0 0
$$916$$ 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
$$921$$ 0 0
$$922$$ 0 0
$$923$$ 13216.0 0.471300
$$924$$ 0 0
$$925$$ 31768.9 1.12925
$$926$$ 0 0
$$927$$ 0 0
$$928$$ 0 0
$$929$$ 13650.0 0.482069 0.241034 0.970517i $$-0.422513\pi$$
0.241034 + 0.970517i $$0.422513\pi$$
$$930$$ 0 0
$$931$$ −7436.00 −0.261767
$$932$$ 0 0
$$933$$ 0 0
$$934$$ 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
$$939$$ 0 0
$$940$$ 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
$$945$$ 0 0
$$946$$ 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
$$951$$ 0 0
$$952$$ 0 0
$$953$$ 46938.0 1.59546 0.797729 0.603016i $$-0.206035\pi$$
0.797729 + 0.603016i $$0.206035\pi$$
$$954$$ 0 0
$$955$$ 2656.00 0.0899960
$$956$$ 0 0
$$957$$ 0 0
$$958$$ 0 0
$$959$$ −40361.7 −1.35907
$$960$$ 0 0
$$961$$ −17623.0 −0.591554
$$962$$ 0 0
$$963$$ 0 0
$$964$$ 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
$$969$$ 0 0
$$970$$ 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
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 41890.0 1.37173 0.685865 0.727729i $$-0.259424\pi$$
0.685865 + 0.727729i $$0.259424\pi$$
$$978$$ 0 0
$$979$$ 4680.00 0.152782
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 0 0
$$983$$ 10861.2 0.352408 0.176204 0.984354i $$-0.443618\pi$$
0.176204 + 0.984354i $$0.443618\pi$$
$$984$$ 0 0
$$985$$ 7704.00 0.249208
$$986$$ 0 0
$$987$$ 0 0
$$988$$ 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
$$993$$ 0 0
$$994$$ 0 0
$$995$$ −6104.00 −0.194482
$$996$$ 0 0
$$997$$ 39948.7 1.26900 0.634498 0.772925i $$-0.281207\pi$$
0.634498 + 0.772925i $$0.281207\pi$$
$$998$$ 0 0
$$999$$ 0 0
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 2304.4.a.bb.1.1 2
3.2 odd 2 768.4.a.h.1.2 2
4.3 odd 2 2304.4.a.bh.1.1 2
8.3 odd 2 inner 2304.4.a.bb.1.2 2
8.5 even 2 2304.4.a.bh.1.2 2
12.11 even 2 768.4.a.m.1.2 2
16.3 odd 4 1152.4.d.n.577.4 4
16.5 even 4 1152.4.d.n.577.1 4
16.11 odd 4 1152.4.d.n.577.2 4
16.13 even 4 1152.4.d.n.577.3 4
24.5 odd 2 768.4.a.m.1.1 2
24.11 even 2 768.4.a.h.1.1 2
48.5 odd 4 384.4.d.d.193.4 yes 4
48.11 even 4 384.4.d.d.193.2 yes 4
48.29 odd 4 384.4.d.d.193.1 4
48.35 even 4 384.4.d.d.193.3 yes 4

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