# Properties

 Label 400.4.c.g.49.2 Level $400$ Weight $4$ Character 400.49 Analytic conductor $23.601$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (of order $$2$$, degree $$1$$, not 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: $$23.6007640023$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 49.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 400.49 Dual form 400.4.c.g.49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+5.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} +O(q^{10})$$ $$q+5.00000i q^{3} -2.00000i q^{7} +2.00000 q^{9} -39.0000 q^{11} -84.0000i q^{13} -61.0000i q^{17} +151.000 q^{19} +10.0000 q^{21} -58.0000i q^{23} +145.000i q^{27} -192.000 q^{29} +18.0000 q^{31} -195.000i q^{33} -138.000i q^{37} +420.000 q^{39} +229.000 q^{41} -164.000i q^{43} +212.000i q^{47} +339.000 q^{49} +305.000 q^{51} -578.000i q^{53} +755.000i q^{57} -336.000 q^{59} +858.000 q^{61} -4.00000i q^{63} +209.000i q^{67} +290.000 q^{69} +780.000 q^{71} +403.000i q^{73} +78.0000i q^{77} -230.000 q^{79} -671.000 q^{81} -1293.00i q^{83} -960.000i q^{87} +1369.00 q^{89} -168.000 q^{91} +90.0000i q^{93} +382.000i q^{97} -78.0000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} - 78 q^{11} + 302 q^{19} + 20 q^{21} - 384 q^{29} + 36 q^{31} + 840 q^{39} + 458 q^{41} + 678 q^{49} + 610 q^{51} - 672 q^{59} + 1716 q^{61} + 580 q^{69} + 1560 q^{71} - 460 q^{79} - 1342 q^{81} + 2738 q^{89} - 336 q^{91} - 156 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 - 78 * q^11 + 302 * q^19 + 20 * q^21 - 384 * q^29 + 36 * q^31 + 840 * q^39 + 458 * q^41 + 678 * q^49 + 610 * q^51 - 672 * q^59 + 1716 * q^61 + 580 * q^69 + 1560 * q^71 - 460 * q^79 - 1342 * q^81 + 2738 * q^89 - 336 * q^91 - 156 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$3$$ 5.00000i 0.962250i 0.876652 + 0.481125i $$0.159772\pi$$
−0.876652 + 0.481125i $$0.840228\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 2.00000i − 0.107990i −0.998541 0.0539949i $$-0.982805\pi$$
0.998541 0.0539949i $$-0.0171955\pi$$
$$8$$ 0 0
$$9$$ 2.00000 0.0740741
$$10$$ 0 0
$$11$$ −39.0000 −1.06899 −0.534497 0.845170i $$-0.679499\pi$$
−0.534497 + 0.845170i $$0.679499\pi$$
$$12$$ 0 0
$$13$$ − 84.0000i − 1.79211i −0.443945 0.896054i $$-0.646421\pi$$
0.443945 0.896054i $$-0.353579\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ − 61.0000i − 0.870275i −0.900364 0.435137i $$-0.856700\pi$$
0.900364 0.435137i $$-0.143300\pi$$
$$18$$ 0 0
$$19$$ 151.000 1.82325 0.911626 0.411021i $$-0.134828\pi$$
0.911626 + 0.411021i $$0.134828\pi$$
$$20$$ 0 0
$$21$$ 10.0000 0.103913
$$22$$ 0 0
$$23$$ − 58.0000i − 0.525819i −0.964821 0.262909i $$-0.915318\pi$$
0.964821 0.262909i $$-0.0846821\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 145.000i 1.03353i
$$28$$ 0 0
$$29$$ −192.000 −1.22943 −0.614716 0.788749i $$-0.710729\pi$$
−0.614716 + 0.788749i $$0.710729\pi$$
$$30$$ 0 0
$$31$$ 18.0000 0.104287 0.0521435 0.998640i $$-0.483395\pi$$
0.0521435 + 0.998640i $$0.483395\pi$$
$$32$$ 0 0
$$33$$ − 195.000i − 1.02864i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 138.000i − 0.613164i −0.951844 0.306582i $$-0.900815\pi$$
0.951844 0.306582i $$-0.0991853\pi$$
$$38$$ 0 0
$$39$$ 420.000 1.72446
$$40$$ 0 0
$$41$$ 229.000 0.872288 0.436144 0.899877i $$-0.356344\pi$$
0.436144 + 0.899877i $$0.356344\pi$$
$$42$$ 0 0
$$43$$ − 164.000i − 0.581622i −0.956780 0.290811i $$-0.906075\pi$$
0.956780 0.290811i $$-0.0939252\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 212.000i 0.657944i 0.944340 + 0.328972i $$0.106702\pi$$
−0.944340 + 0.328972i $$0.893298\pi$$
$$48$$ 0 0
$$49$$ 339.000 0.988338
$$50$$ 0 0
$$51$$ 305.000 0.837422
$$52$$ 0 0
$$53$$ − 578.000i − 1.49801i −0.662566 0.749004i $$-0.730532\pi$$
0.662566 0.749004i $$-0.269468\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 755.000i 1.75442i
$$58$$ 0 0
$$59$$ −336.000 −0.741415 −0.370707 0.928750i $$-0.620885\pi$$
−0.370707 + 0.928750i $$0.620885\pi$$
$$60$$ 0 0
$$61$$ 858.000 1.80091 0.900456 0.434947i $$-0.143233\pi$$
0.900456 + 0.434947i $$0.143233\pi$$
$$62$$ 0 0
$$63$$ − 4.00000i − 0.00799925i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 209.000i 0.381096i 0.981678 + 0.190548i $$0.0610264\pi$$
−0.981678 + 0.190548i $$0.938974\pi$$
$$68$$ 0 0
$$69$$ 290.000 0.505970
$$70$$ 0 0
$$71$$ 780.000 1.30379 0.651894 0.758310i $$-0.273975\pi$$
0.651894 + 0.758310i $$0.273975\pi$$
$$72$$ 0 0
$$73$$ 403.000i 0.646131i 0.946377 + 0.323066i $$0.104713\pi$$
−0.946377 + 0.323066i $$0.895287\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 78.0000i 0.115441i
$$78$$ 0 0
$$79$$ −230.000 −0.327557 −0.163779 0.986497i $$-0.552368\pi$$
−0.163779 + 0.986497i $$0.552368\pi$$
$$80$$ 0 0
$$81$$ −671.000 −0.920439
$$82$$ 0 0
$$83$$ − 1293.00i − 1.70994i −0.518676 0.854971i $$-0.673575\pi$$
0.518676 0.854971i $$-0.326425\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 960.000i − 1.18302i
$$88$$ 0 0
$$89$$ 1369.00 1.63049 0.815246 0.579115i $$-0.196602\pi$$
0.815246 + 0.579115i $$0.196602\pi$$
$$90$$ 0 0
$$91$$ −168.000 −0.193530
$$92$$ 0 0
$$93$$ 90.0000i 0.100350i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 382.000i 0.399858i 0.979810 + 0.199929i $$0.0640711\pi$$
−0.979810 + 0.199929i $$0.935929\pi$$
$$98$$ 0 0
$$99$$ −78.0000 −0.0791848
$$100$$ 0 0
$$101$$ −794.000 −0.782237 −0.391119 0.920340i $$-0.627912\pi$$
−0.391119 + 0.920340i $$0.627912\pi$$
$$102$$ 0 0
$$103$$ − 1348.00i − 1.28954i −0.764378 0.644769i $$-0.776954\pi$$
0.764378 0.644769i $$-0.223046\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 775.000i 0.700206i 0.936711 + 0.350103i $$0.113853\pi$$
−0.936711 + 0.350103i $$0.886147\pi$$
$$108$$ 0 0
$$109$$ −446.000 −0.391918 −0.195959 0.980612i $$-0.562782\pi$$
−0.195959 + 0.980612i $$0.562782\pi$$
$$110$$ 0 0
$$111$$ 690.000 0.590017
$$112$$ 0 0
$$113$$ 231.000i 0.192307i 0.995367 + 0.0961533i $$0.0306539\pi$$
−0.995367 + 0.0961533i $$0.969346\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 168.000i − 0.132749i
$$118$$ 0 0
$$119$$ −122.000 −0.0939809
$$120$$ 0 0
$$121$$ 190.000 0.142750
$$122$$ 0 0
$$123$$ 1145.00i 0.839359i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 2386.00i − 1.66711i −0.552435 0.833556i $$-0.686301\pi$$
0.552435 0.833556i $$-0.313699\pi$$
$$128$$ 0 0
$$129$$ 820.000 0.559666
$$130$$ 0 0
$$131$$ −2452.00 −1.63536 −0.817680 0.575673i $$-0.804740\pi$$
−0.817680 + 0.575673i $$0.804740\pi$$
$$132$$ 0 0
$$133$$ − 302.000i − 0.196893i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 1125.00i − 0.701571i −0.936456 0.350786i $$-0.885915\pi$$
0.936456 0.350786i $$-0.114085\pi$$
$$138$$ 0 0
$$139$$ −1377.00 −0.840256 −0.420128 0.907465i $$-0.638015\pi$$
−0.420128 + 0.907465i $$0.638015\pi$$
$$140$$ 0 0
$$141$$ −1060.00 −0.633107
$$142$$ 0 0
$$143$$ 3276.00i 1.91575i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 1695.00i 0.951029i
$$148$$ 0 0
$$149$$ −1920.00 −1.05565 −0.527827 0.849352i $$-0.676993\pi$$
−0.527827 + 0.849352i $$0.676993\pi$$
$$150$$ 0 0
$$151$$ −1854.00 −0.999181 −0.499591 0.866262i $$-0.666516\pi$$
−0.499591 + 0.866262i $$0.666516\pi$$
$$152$$ 0 0
$$153$$ − 122.000i − 0.0644648i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 634.000i − 0.322285i −0.986931 0.161142i $$-0.948482\pi$$
0.986931 0.161142i $$-0.0515178\pi$$
$$158$$ 0 0
$$159$$ 2890.00 1.44146
$$160$$ 0 0
$$161$$ −116.000 −0.0567831
$$162$$ 0 0
$$163$$ 103.000i 0.0494944i 0.999694 + 0.0247472i $$0.00787808\pi$$
−0.999694 + 0.0247472i $$0.992122\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 44.0000i − 0.0203882i −0.999948 0.0101941i $$-0.996755\pi$$
0.999948 0.0101941i $$-0.00324493\pi$$
$$168$$ 0 0
$$169$$ −4859.00 −2.21165
$$170$$ 0 0
$$171$$ 302.000 0.135056
$$172$$ 0 0
$$173$$ 1128.00i 0.495724i 0.968795 + 0.247862i $$0.0797280\pi$$
−0.968795 + 0.247862i $$0.920272\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 1680.00i − 0.713427i
$$178$$ 0 0
$$179$$ −2245.00 −0.937426 −0.468713 0.883351i $$-0.655282\pi$$
−0.468713 + 0.883351i $$0.655282\pi$$
$$180$$ 0 0
$$181$$ 3050.00 1.25251 0.626256 0.779617i $$-0.284586\pi$$
0.626256 + 0.779617i $$0.284586\pi$$
$$182$$ 0 0
$$183$$ 4290.00i 1.73293i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2379.00i 0.930319i
$$188$$ 0 0
$$189$$ 290.000 0.111611
$$190$$ 0 0
$$191$$ 4222.00 1.59944 0.799720 0.600373i $$-0.204981\pi$$
0.799720 + 0.600373i $$0.204981\pi$$
$$192$$ 0 0
$$193$$ 3357.00i 1.25203i 0.779810 + 0.626016i $$0.215316\pi$$
−0.779810 + 0.626016i $$0.784684\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ − 166.000i − 0.0600356i −0.999549 0.0300178i $$-0.990444\pi$$
0.999549 0.0300178i $$-0.00955640\pi$$
$$198$$ 0 0
$$199$$ 3372.00 1.20118 0.600590 0.799557i $$-0.294932\pi$$
0.600590 + 0.799557i $$0.294932\pi$$
$$200$$ 0 0
$$201$$ −1045.00 −0.366710
$$202$$ 0 0
$$203$$ 384.000i 0.132766i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 116.000i − 0.0389496i
$$208$$ 0 0
$$209$$ −5889.00 −1.94905
$$210$$ 0 0
$$211$$ −5601.00 −1.82743 −0.913717 0.406350i $$-0.866801\pi$$
−0.913717 + 0.406350i $$0.866801\pi$$
$$212$$ 0 0
$$213$$ 3900.00i 1.25457i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 36.0000i − 0.0112619i
$$218$$ 0 0
$$219$$ −2015.00 −0.621740
$$220$$ 0 0
$$221$$ −5124.00 −1.55963
$$222$$ 0 0
$$223$$ 828.000i 0.248641i 0.992242 + 0.124321i $$0.0396751\pi$$
−0.992242 + 0.124321i $$0.960325\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 2388.00i 0.698225i 0.937081 + 0.349113i $$0.113517\pi$$
−0.937081 + 0.349113i $$0.886483\pi$$
$$228$$ 0 0
$$229$$ 2844.00 0.820685 0.410342 0.911932i $$-0.365409\pi$$
0.410342 + 0.911932i $$0.365409\pi$$
$$230$$ 0 0
$$231$$ −390.000 −0.111083
$$232$$ 0 0
$$233$$ − 5962.00i − 1.67632i −0.545421 0.838162i $$-0.683630\pi$$
0.545421 0.838162i $$-0.316370\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 1150.00i − 0.315192i
$$238$$ 0 0
$$239$$ −4320.00 −1.16919 −0.584597 0.811324i $$-0.698748\pi$$
−0.584597 + 0.811324i $$0.698748\pi$$
$$240$$ 0 0
$$241$$ 3857.00 1.03092 0.515459 0.856914i $$-0.327621\pi$$
0.515459 + 0.856914i $$0.327621\pi$$
$$242$$ 0 0
$$243$$ 560.000i 0.147835i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 12684.0i − 3.26746i
$$248$$ 0 0
$$249$$ 6465.00 1.64539
$$250$$ 0 0
$$251$$ 287.000 0.0721724 0.0360862 0.999349i $$-0.488511\pi$$
0.0360862 + 0.999349i $$0.488511\pi$$
$$252$$ 0 0
$$253$$ 2262.00i 0.562098i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 2130.00i 0.516987i 0.966013 + 0.258494i $$0.0832261\pi$$
−0.966013 + 0.258494i $$0.916774\pi$$
$$258$$ 0 0
$$259$$ −276.000 −0.0662155
$$260$$ 0 0
$$261$$ −384.000 −0.0910690
$$262$$ 0 0
$$263$$ − 3066.00i − 0.718850i −0.933174 0.359425i $$-0.882973\pi$$
0.933174 0.359425i $$-0.117027\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6845.00i 1.56894i
$$268$$ 0 0
$$269$$ 3744.00 0.848609 0.424304 0.905520i $$-0.360519\pi$$
0.424304 + 0.905520i $$0.360519\pi$$
$$270$$ 0 0
$$271$$ 3346.00 0.750019 0.375009 0.927021i $$-0.377640\pi$$
0.375009 + 0.927021i $$0.377640\pi$$
$$272$$ 0 0
$$273$$ − 840.000i − 0.186224i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7040.00i 1.52705i 0.645779 + 0.763525i $$0.276533\pi$$
−0.645779 + 0.763525i $$0.723467\pi$$
$$278$$ 0 0
$$279$$ 36.0000 0.00772496
$$280$$ 0 0
$$281$$ −3010.00 −0.639009 −0.319505 0.947585i $$-0.603516\pi$$
−0.319505 + 0.947585i $$0.603516\pi$$
$$282$$ 0 0
$$283$$ − 6001.00i − 1.26050i −0.776391 0.630252i $$-0.782952\pi$$
0.776391 0.630252i $$-0.217048\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ − 458.000i − 0.0941982i
$$288$$ 0 0
$$289$$ 1192.00 0.242622
$$290$$ 0 0
$$291$$ −1910.00 −0.384764
$$292$$ 0 0
$$293$$ − 4802.00i − 0.957460i −0.877962 0.478730i $$-0.841097\pi$$
0.877962 0.478730i $$-0.158903\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 5655.00i − 1.10484i
$$298$$ 0 0
$$299$$ −4872.00 −0.942325
$$300$$ 0 0
$$301$$ −328.000 −0.0628093
$$302$$ 0 0
$$303$$ − 3970.00i − 0.752708i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 6149.00i − 1.14313i −0.820556 0.571567i $$-0.806336\pi$$
0.820556 0.571567i $$-0.193664\pi$$
$$308$$ 0 0
$$309$$ 6740.00 1.24086
$$310$$ 0 0
$$311$$ 878.000 0.160086 0.0800431 0.996791i $$-0.474494\pi$$
0.0800431 + 0.996791i $$0.474494\pi$$
$$312$$ 0 0
$$313$$ 4042.00i 0.729928i 0.931022 + 0.364964i $$0.118919\pi$$
−0.931022 + 0.364964i $$0.881081\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3844.00i 0.681074i 0.940231 + 0.340537i $$0.110609\pi$$
−0.940231 + 0.340537i $$0.889391\pi$$
$$318$$ 0 0
$$319$$ 7488.00 1.31426
$$320$$ 0 0
$$321$$ −3875.00 −0.673774
$$322$$ 0 0
$$323$$ − 9211.00i − 1.58673i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 2230.00i − 0.377123i
$$328$$ 0 0
$$329$$ 424.000 0.0710513
$$330$$ 0 0
$$331$$ 2717.00 0.451178 0.225589 0.974223i $$-0.427569\pi$$
0.225589 + 0.974223i $$0.427569\pi$$
$$332$$ 0 0
$$333$$ − 276.000i − 0.0454195i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 1603.00i − 0.259113i −0.991572 0.129556i $$-0.958645\pi$$
0.991572 0.129556i $$-0.0413553\pi$$
$$338$$ 0 0
$$339$$ −1155.00 −0.185047
$$340$$ 0 0
$$341$$ −702.000 −0.111482
$$342$$ 0 0
$$343$$ − 1364.00i − 0.214720i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 11607.0i 1.79567i 0.440335 + 0.897833i $$0.354860\pi$$
−0.440335 + 0.897833i $$0.645140\pi$$
$$348$$ 0 0
$$349$$ −4030.00 −0.618112 −0.309056 0.951044i $$-0.600013\pi$$
−0.309056 + 0.951044i $$0.600013\pi$$
$$350$$ 0 0
$$351$$ 12180.0 1.85219
$$352$$ 0 0
$$353$$ − 2106.00i − 0.317538i −0.987316 0.158769i $$-0.949247\pi$$
0.987316 0.158769i $$-0.0507526\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ − 610.000i − 0.0904331i
$$358$$ 0 0
$$359$$ 7394.00 1.08702 0.543510 0.839402i $$-0.317095\pi$$
0.543510 + 0.839402i $$0.317095\pi$$
$$360$$ 0 0
$$361$$ 15942.0 2.32425
$$362$$ 0 0
$$363$$ 950.000i 0.137361i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 6940.00i 0.987098i 0.869718 + 0.493549i $$0.164301\pi$$
−0.869718 + 0.493549i $$0.835699\pi$$
$$368$$ 0 0
$$369$$ 458.000 0.0646139
$$370$$ 0 0
$$371$$ −1156.00 −0.161770
$$372$$ 0 0
$$373$$ − 7486.00i − 1.03917i −0.854419 0.519585i $$-0.826087\pi$$
0.854419 0.519585i $$-0.173913\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 16128.0i 2.20327i
$$378$$ 0 0
$$379$$ 1285.00 0.174158 0.0870792 0.996201i $$-0.472247\pi$$
0.0870792 + 0.996201i $$0.472247\pi$$
$$380$$ 0 0
$$381$$ 11930.0 1.60418
$$382$$ 0 0
$$383$$ 9622.00i 1.28371i 0.766826 + 0.641855i $$0.221835\pi$$
−0.766826 + 0.641855i $$0.778165\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 328.000i − 0.0430831i
$$388$$ 0 0
$$389$$ −1974.00 −0.257290 −0.128645 0.991691i $$-0.541063\pi$$
−0.128645 + 0.991691i $$0.541063\pi$$
$$390$$ 0 0
$$391$$ −3538.00 −0.457607
$$392$$ 0 0
$$393$$ − 12260.0i − 1.57363i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 8084.00i − 1.02198i −0.859588 0.510988i $$-0.829280\pi$$
0.859588 0.510988i $$-0.170720\pi$$
$$398$$ 0 0
$$399$$ 1510.00 0.189460
$$400$$ 0 0
$$401$$ −5667.00 −0.705727 −0.352863 0.935675i $$-0.614792\pi$$
−0.352863 + 0.935675i $$0.614792\pi$$
$$402$$ 0 0
$$403$$ − 1512.00i − 0.186894i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 5382.00i 0.655469i
$$408$$ 0 0
$$409$$ 4835.00 0.584536 0.292268 0.956336i $$-0.405590\pi$$
0.292268 + 0.956336i $$0.405590\pi$$
$$410$$ 0 0
$$411$$ 5625.00 0.675087
$$412$$ 0 0
$$413$$ 672.000i 0.0800653i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 6885.00i − 0.808537i
$$418$$ 0 0
$$419$$ 4619.00 0.538551 0.269276 0.963063i $$-0.413216\pi$$
0.269276 + 0.963063i $$0.413216\pi$$
$$420$$ 0 0
$$421$$ 7476.00 0.865458 0.432729 0.901524i $$-0.357551\pi$$
0.432729 + 0.901524i $$0.357551\pi$$
$$422$$ 0 0
$$423$$ 424.000i 0.0487366i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 1716.00i − 0.194480i
$$428$$ 0 0
$$429$$ −16380.0 −1.84344
$$430$$ 0 0
$$431$$ −7810.00 −0.872841 −0.436420 0.899743i $$-0.643754\pi$$
−0.436420 + 0.899743i $$0.643754\pi$$
$$432$$ 0 0
$$433$$ 2029.00i 0.225191i 0.993641 + 0.112595i $$0.0359164\pi$$
−0.993641 + 0.112595i $$0.964084\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 8758.00i − 0.958700i
$$438$$ 0 0
$$439$$ 3208.00 0.348769 0.174384 0.984678i $$-0.444206\pi$$
0.174384 + 0.984678i $$0.444206\pi$$
$$440$$ 0 0
$$441$$ 678.000 0.0732102
$$442$$ 0 0
$$443$$ 13227.0i 1.41859i 0.704914 + 0.709293i $$0.250986\pi$$
−0.704914 + 0.709293i $$0.749014\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 9600.00i − 1.01580i
$$448$$ 0 0
$$449$$ −3617.00 −0.380171 −0.190086 0.981768i $$-0.560877\pi$$
−0.190086 + 0.981768i $$0.560877\pi$$
$$450$$ 0 0
$$451$$ −8931.00 −0.932471
$$452$$ 0 0
$$453$$ − 9270.00i − 0.961463i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6215.00i 0.636161i 0.948064 + 0.318080i $$0.103038\pi$$
−0.948064 + 0.318080i $$0.896962\pi$$
$$458$$ 0 0
$$459$$ 8845.00 0.899454
$$460$$ 0 0
$$461$$ −7108.00 −0.718118 −0.359059 0.933315i $$-0.616902\pi$$
−0.359059 + 0.933315i $$0.616902\pi$$
$$462$$ 0 0
$$463$$ − 3364.00i − 0.337664i −0.985645 0.168832i $$-0.946000\pi$$
0.985645 0.168832i $$-0.0539995\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 18964.0i 1.87912i 0.342384 + 0.939560i $$0.388766\pi$$
−0.342384 + 0.939560i $$0.611234\pi$$
$$468$$ 0 0
$$469$$ 418.000 0.0411545
$$470$$ 0 0
$$471$$ 3170.00 0.310119
$$472$$ 0 0
$$473$$ 6396.00i 0.621751i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 1156.00i − 0.110964i
$$478$$ 0 0
$$479$$ −10926.0 −1.04222 −0.521108 0.853491i $$-0.674481\pi$$
−0.521108 + 0.853491i $$0.674481\pi$$
$$480$$ 0 0
$$481$$ −11592.0 −1.09886
$$482$$ 0 0
$$483$$ − 580.000i − 0.0546396i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 4350.00i 0.404758i 0.979307 + 0.202379i $$0.0648673\pi$$
−0.979307 + 0.202379i $$0.935133\pi$$
$$488$$ 0 0
$$489$$ −515.000 −0.0476260
$$490$$ 0 0
$$491$$ 1324.00 0.121693 0.0608465 0.998147i $$-0.480620\pi$$
0.0608465 + 0.998147i $$0.480620\pi$$
$$492$$ 0 0
$$493$$ 11712.0i 1.06994i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ − 1560.00i − 0.140796i
$$498$$ 0 0
$$499$$ 9068.00 0.813506 0.406753 0.913538i $$-0.366661\pi$$
0.406753 + 0.913538i $$0.366661\pi$$
$$500$$ 0 0
$$501$$ 220.000 0.0196185
$$502$$ 0 0
$$503$$ − 19836.0i − 1.75834i −0.476511 0.879169i $$-0.658099\pi$$
0.476511 0.879169i $$-0.341901\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ − 24295.0i − 2.12816i
$$508$$ 0 0
$$509$$ −2682.00 −0.233551 −0.116776 0.993158i $$-0.537256\pi$$
−0.116776 + 0.993158i $$0.537256\pi$$
$$510$$ 0 0
$$511$$ 806.000 0.0697756
$$512$$ 0 0
$$513$$ 21895.0i 1.88438i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 8268.00i − 0.703339i
$$518$$ 0 0
$$519$$ −5640.00 −0.477011
$$520$$ 0 0
$$521$$ −3035.00 −0.255213 −0.127606 0.991825i $$-0.540729\pi$$
−0.127606 + 0.991825i $$0.540729\pi$$
$$522$$ 0 0
$$523$$ 7701.00i 0.643865i 0.946763 + 0.321932i $$0.104332\pi$$
−0.946763 + 0.321932i $$0.895668\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 1098.00i − 0.0907583i
$$528$$ 0 0
$$529$$ 8803.00 0.723514
$$530$$ 0 0
$$531$$ −672.000 −0.0549196
$$532$$ 0 0
$$533$$ − 19236.0i − 1.56323i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 11225.0i − 0.902038i
$$538$$ 0 0
$$539$$ −13221.0 −1.05653
$$540$$ 0 0
$$541$$ −18112.0 −1.43936 −0.719682 0.694304i $$-0.755712\pi$$
−0.719682 + 0.694304i $$0.755712\pi$$
$$542$$ 0 0
$$543$$ 15250.0i 1.20523i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 19541.0i − 1.52745i −0.645544 0.763723i $$-0.723369\pi$$
0.645544 0.763723i $$-0.276631\pi$$
$$548$$ 0 0
$$549$$ 1716.00 0.133401
$$550$$ 0 0
$$551$$ −28992.0 −2.24156
$$552$$ 0 0
$$553$$ 460.000i 0.0353729i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 13508.0i 1.02756i 0.857921 + 0.513781i $$0.171756\pi$$
−0.857921 + 0.513781i $$0.828244\pi$$
$$558$$ 0 0
$$559$$ −13776.0 −1.04233
$$560$$ 0 0
$$561$$ −11895.0 −0.895200
$$562$$ 0 0
$$563$$ − 8712.00i − 0.652162i −0.945342 0.326081i $$-0.894272\pi$$
0.945342 0.326081i $$-0.105728\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1342.00i 0.0993981i
$$568$$ 0 0
$$569$$ 9623.00 0.708993 0.354497 0.935057i $$-0.384652\pi$$
0.354497 + 0.935057i $$0.384652\pi$$
$$570$$ 0 0
$$571$$ −604.000 −0.0442673 −0.0221336 0.999755i $$-0.507046\pi$$
−0.0221336 + 0.999755i $$0.507046\pi$$
$$572$$ 0 0
$$573$$ 21110.0i 1.53906i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 3629.00i 0.261832i 0.991393 + 0.130916i $$0.0417919\pi$$
−0.991393 + 0.130916i $$0.958208\pi$$
$$578$$ 0 0
$$579$$ −16785.0 −1.20477
$$580$$ 0 0
$$581$$ −2586.00 −0.184656
$$582$$ 0 0
$$583$$ 22542.0i 1.60136i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 9219.00i − 0.648226i −0.946018 0.324113i $$-0.894934\pi$$
0.946018 0.324113i $$-0.105066\pi$$
$$588$$ 0 0
$$589$$ 2718.00 0.190141
$$590$$ 0 0
$$591$$ 830.000 0.0577693
$$592$$ 0 0
$$593$$ 19111.0i 1.32343i 0.749755 + 0.661716i $$0.230171\pi$$
−0.749755 + 0.661716i $$0.769829\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 16860.0i 1.15584i
$$598$$ 0 0
$$599$$ −17086.0 −1.16547 −0.582734 0.812663i $$-0.698017\pi$$
−0.582734 + 0.812663i $$0.698017\pi$$
$$600$$ 0 0
$$601$$ 9035.00 0.613220 0.306610 0.951835i $$-0.400805\pi$$
0.306610 + 0.951835i $$0.400805\pi$$
$$602$$ 0 0
$$603$$ 418.000i 0.0282293i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 14784.0i 0.988573i 0.869299 + 0.494287i $$0.164571\pi$$
−0.869299 + 0.494287i $$0.835429\pi$$
$$608$$ 0 0
$$609$$ −1920.00 −0.127754
$$610$$ 0 0
$$611$$ 17808.0 1.17911
$$612$$ 0 0
$$613$$ 17846.0i 1.17585i 0.808917 + 0.587923i $$0.200054\pi$$
−0.808917 + 0.587923i $$0.799946\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 11618.0i 0.758060i 0.925384 + 0.379030i $$0.123742\pi$$
−0.925384 + 0.379030i $$0.876258\pi$$
$$618$$ 0 0
$$619$$ −9556.00 −0.620498 −0.310249 0.950655i $$-0.600412\pi$$
−0.310249 + 0.950655i $$0.600412\pi$$
$$620$$ 0 0
$$621$$ 8410.00 0.543449
$$622$$ 0 0
$$623$$ − 2738.00i − 0.176076i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 29445.0i − 1.87547i
$$628$$ 0 0
$$629$$ −8418.00 −0.533621
$$630$$ 0 0
$$631$$ 19394.0 1.22355 0.611777 0.791030i $$-0.290455\pi$$
0.611777 + 0.791030i $$0.290455\pi$$
$$632$$ 0 0
$$633$$ − 28005.0i − 1.75845i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 28476.0i − 1.77121i
$$638$$ 0 0
$$639$$ 1560.00 0.0965769
$$640$$ 0 0
$$641$$ 12138.0 0.747929 0.373964 0.927443i $$-0.377998\pi$$
0.373964 + 0.927443i $$0.377998\pi$$
$$642$$ 0 0
$$643$$ 27036.0i 1.65816i 0.559131 + 0.829079i $$0.311135\pi$$
−0.559131 + 0.829079i $$0.688865\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 17556.0i 1.06677i 0.845874 + 0.533383i $$0.179080\pi$$
−0.845874 + 0.533383i $$0.820920\pi$$
$$648$$ 0 0
$$649$$ 13104.0 0.792569
$$650$$ 0 0
$$651$$ 180.000 0.0108368
$$652$$ 0 0
$$653$$ 17262.0i 1.03448i 0.855841 + 0.517239i $$0.173040\pi$$
−0.855841 + 0.517239i $$0.826960\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 806.000i 0.0478616i
$$658$$ 0 0
$$659$$ 10517.0 0.621675 0.310838 0.950463i $$-0.399390\pi$$
0.310838 + 0.950463i $$0.399390\pi$$
$$660$$ 0 0
$$661$$ 1408.00 0.0828515 0.0414258 0.999142i $$-0.486810\pi$$
0.0414258 + 0.999142i $$0.486810\pi$$
$$662$$ 0 0
$$663$$ − 25620.0i − 1.50075i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 11136.0i 0.646458i
$$668$$ 0 0
$$669$$ −4140.00 −0.239255
$$670$$ 0 0
$$671$$ −33462.0 −1.92517
$$672$$ 0 0
$$673$$ 9626.00i 0.551345i 0.961252 + 0.275672i $$0.0889005\pi$$
−0.961252 + 0.275672i $$0.911100\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 28464.0i 1.61589i 0.589255 + 0.807947i $$0.299421\pi$$
−0.589255 + 0.807947i $$0.700579\pi$$
$$678$$ 0 0
$$679$$ 764.000 0.0431806
$$680$$ 0 0
$$681$$ −11940.0 −0.671868
$$682$$ 0 0
$$683$$ − 3963.00i − 0.222020i −0.993819 0.111010i $$-0.964591\pi$$
0.993819 0.111010i $$-0.0354086\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 14220.0i 0.789704i
$$688$$ 0 0
$$689$$ −48552.0 −2.68459
$$690$$ 0 0
$$691$$ 31781.0 1.74965 0.874824 0.484442i $$-0.160977\pi$$
0.874824 + 0.484442i $$0.160977\pi$$
$$692$$ 0 0
$$693$$ 156.000i 0.00855115i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 13969.0i − 0.759130i
$$698$$ 0 0
$$699$$ 29810.0 1.61304
$$700$$ 0 0
$$701$$ −28004.0 −1.50884 −0.754420 0.656392i $$-0.772082\pi$$
−0.754420 + 0.656392i $$0.772082\pi$$
$$702$$ 0 0
$$703$$ − 20838.0i − 1.11795i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 1588.00i 0.0844737i
$$708$$ 0 0
$$709$$ 35228.0 1.86603 0.933015 0.359837i $$-0.117168\pi$$
0.933015 + 0.359837i $$0.117168\pi$$
$$710$$ 0 0
$$711$$ −460.000 −0.0242635
$$712$$ 0 0
$$713$$ − 1044.00i − 0.0548361i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ − 21600.0i − 1.12506i
$$718$$ 0 0
$$719$$ −8658.00 −0.449081 −0.224540 0.974465i $$-0.572088\pi$$
−0.224540 + 0.974465i $$0.572088\pi$$
$$720$$ 0 0
$$721$$ −2696.00 −0.139257
$$722$$ 0 0
$$723$$ 19285.0i 0.992001i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 5728.00i − 0.292214i −0.989269 0.146107i $$-0.953326\pi$$
0.989269 0.146107i $$-0.0466744\pi$$
$$728$$ 0 0
$$729$$ −20917.0 −1.06269
$$730$$ 0 0
$$731$$ −10004.0 −0.506171
$$732$$ 0 0
$$733$$ − 21460.0i − 1.08137i −0.841226 0.540684i $$-0.818165\pi$$
0.841226 0.540684i $$-0.181835\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 8151.00i − 0.407389i
$$738$$ 0 0
$$739$$ 29164.0 1.45171 0.725856 0.687847i $$-0.241444\pi$$
0.725856 + 0.687847i $$0.241444\pi$$
$$740$$ 0 0
$$741$$ 63420.0 3.14412
$$742$$ 0 0
$$743$$ 29478.0i 1.45551i 0.685838 + 0.727754i $$0.259436\pi$$
−0.685838 + 0.727754i $$0.740564\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 2586.00i − 0.126662i
$$748$$ 0 0
$$749$$ 1550.00 0.0756152
$$750$$ 0 0
$$751$$ −576.000 −0.0279874 −0.0139937 0.999902i $$-0.504454\pi$$
−0.0139937 + 0.999902i $$0.504454\pi$$
$$752$$ 0 0
$$753$$ 1435.00i 0.0694480i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2880.00i 0.138277i 0.997607 + 0.0691383i $$0.0220250\pi$$
−0.997607 + 0.0691383i $$0.977975\pi$$
$$758$$ 0 0
$$759$$ −11310.0 −0.540879
$$760$$ 0 0
$$761$$ 20789.0 0.990277 0.495138 0.868814i $$-0.335117\pi$$
0.495138 + 0.868814i $$0.335117\pi$$
$$762$$ 0 0
$$763$$ 892.000i 0.0423232i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 28224.0i 1.32870i
$$768$$ 0 0
$$769$$ −26421.0 −1.23897 −0.619484 0.785010i $$-0.712658\pi$$
−0.619484 + 0.785010i $$0.712658\pi$$
$$770$$ 0 0
$$771$$ −10650.0 −0.497471
$$772$$ 0 0
$$773$$ − 32504.0i − 1.51240i −0.654339 0.756202i $$-0.727053\pi$$
0.654339 0.756202i $$-0.272947\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 1380.00i − 0.0637159i
$$778$$ 0 0
$$779$$ 34579.0 1.59040
$$780$$ 0 0
$$781$$ −30420.0 −1.39374
$$782$$ 0 0
$$783$$ − 27840.0i − 1.27065i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 996.000i − 0.0451125i −0.999746 0.0225563i $$-0.992820\pi$$
0.999746 0.0225563i $$-0.00718049\pi$$
$$788$$ 0 0
$$789$$ 15330.0 0.691714
$$790$$ 0 0
$$791$$ 462.000 0.0207672
$$792$$ 0 0
$$793$$ − 72072.0i − 3.22743i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 15134.0i 0.672615i 0.941752 + 0.336307i $$0.109178\pi$$
−0.941752 + 0.336307i $$0.890822\pi$$
$$798$$ 0 0
$$799$$ 12932.0 0.572592
$$800$$ 0 0
$$801$$ 2738.00 0.120777
$$802$$ 0 0
$$803$$ − 15717.0i − 0.690711i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 18720.0i 0.816574i
$$808$$ 0 0
$$809$$ 36942.0 1.60545 0.802727 0.596347i $$-0.203382\pi$$
0.802727 + 0.596347i $$0.203382\pi$$
$$810$$ 0 0
$$811$$ 11748.0 0.508666 0.254333 0.967117i $$-0.418144\pi$$
0.254333 + 0.967117i $$0.418144\pi$$
$$812$$ 0 0
$$813$$ 16730.0i 0.721706i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 24764.0i − 1.06044i
$$818$$ 0 0
$$819$$ −336.000 −0.0143355
$$820$$ 0 0
$$821$$ −1198.00 −0.0509263 −0.0254631 0.999676i $$-0.508106\pi$$
−0.0254631 + 0.999676i $$0.508106\pi$$
$$822$$ 0 0
$$823$$ − 6788.00i − 0.287503i −0.989614 0.143751i $$-0.954083\pi$$
0.989614 0.143751i $$-0.0459166\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 33011.0i − 1.38803i −0.719958 0.694017i $$-0.755839\pi$$
0.719958 0.694017i $$-0.244161\pi$$
$$828$$ 0 0
$$829$$ −17732.0 −0.742892 −0.371446 0.928454i $$-0.621138\pi$$
−0.371446 + 0.928454i $$0.621138\pi$$
$$830$$ 0 0
$$831$$ −35200.0 −1.46940
$$832$$ 0 0
$$833$$ − 20679.0i − 0.860126i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 2610.00i 0.107784i
$$838$$ 0 0
$$839$$ 8480.00 0.348942 0.174471 0.984662i $$-0.444179\pi$$
0.174471 + 0.984662i $$0.444179\pi$$
$$840$$ 0 0
$$841$$ 12475.0 0.511501
$$842$$ 0 0
$$843$$ − 15050.0i − 0.614887i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 380.000i − 0.0154155i
$$848$$ 0 0
$$849$$ 30005.0 1.21292
$$850$$ 0 0
$$851$$ −8004.00 −0.322413
$$852$$ 0 0
$$853$$ − 30014.0i − 1.20476i −0.798210 0.602380i $$-0.794219\pi$$
0.798210 0.602380i $$-0.205781\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 21643.0i 0.862673i 0.902191 + 0.431337i $$0.141958\pi$$
−0.902191 + 0.431337i $$0.858042\pi$$
$$858$$ 0 0
$$859$$ 2799.00 0.111177 0.0555883 0.998454i $$-0.482297\pi$$
0.0555883 + 0.998454i $$0.482297\pi$$
$$860$$ 0 0
$$861$$ 2290.00 0.0906423
$$862$$ 0 0
$$863$$ − 19384.0i − 0.764588i −0.924041 0.382294i $$-0.875134\pi$$
0.924041 0.382294i $$-0.124866\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 5960.00i 0.233463i
$$868$$ 0 0
$$869$$ 8970.00 0.350157
$$870$$ 0 0
$$871$$ 17556.0 0.682965
$$872$$ 0 0
$$873$$ 764.000i 0.0296191i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 5132.00i 0.197600i 0.995107 + 0.0988001i $$0.0315004\pi$$
−0.995107 + 0.0988001i $$0.968500\pi$$
$$878$$ 0 0
$$879$$ 24010.0 0.921316
$$880$$ 0 0
$$881$$ 4430.00 0.169410 0.0847052 0.996406i $$-0.473005\pi$$
0.0847052 + 0.996406i $$0.473005\pi$$
$$882$$ 0 0
$$883$$ 24317.0i 0.926764i 0.886159 + 0.463382i $$0.153364\pi$$
−0.886159 + 0.463382i $$0.846636\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 26100.0i 0.987996i 0.869463 + 0.493998i $$0.164465\pi$$
−0.869463 + 0.493998i $$0.835535\pi$$
$$888$$ 0 0
$$889$$ −4772.00 −0.180031
$$890$$ 0 0
$$891$$ 26169.0 0.983944
$$892$$ 0 0
$$893$$ 32012.0i 1.19960i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 24360.0i − 0.906752i
$$898$$ 0 0
$$899$$ −3456.00 −0.128214
$$900$$ 0 0
$$901$$ −35258.0 −1.30368
$$902$$ 0 0
$$903$$ − 1640.00i − 0.0604383i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 24356.0i − 0.891651i −0.895120 0.445826i $$-0.852910\pi$$
0.895120 0.445826i $$-0.147090\pi$$
$$908$$ 0 0
$$909$$ −1588.00 −0.0579435
$$910$$ 0 0
$$911$$ 29900.0 1.08741 0.543705 0.839276i $$-0.317021\pi$$
0.543705 + 0.839276i $$0.317021\pi$$
$$912$$ 0 0
$$913$$ 50427.0i 1.82792i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 4904.00i 0.176602i
$$918$$ 0 0
$$919$$ −34838.0 −1.25049 −0.625245 0.780429i $$-0.715001\pi$$
−0.625245 + 0.780429i $$0.715001\pi$$
$$920$$ 0 0
$$921$$ 30745.0 1.09998
$$922$$ 0 0
$$923$$ − 65520.0i − 2.33653i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 2696.00i − 0.0955213i
$$928$$ 0 0
$$929$$ −26334.0 −0.930022 −0.465011 0.885305i $$-0.653950\pi$$
−0.465011 + 0.885305i $$0.653950\pi$$
$$930$$ 0 0
$$931$$ 51189.0 1.80199
$$932$$ 0 0
$$933$$ 4390.00i 0.154043i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 30949.0i 1.07904i 0.841973 + 0.539520i $$0.181394\pi$$
−0.841973 + 0.539520i $$0.818606\pi$$
$$938$$ 0 0
$$939$$ −20210.0 −0.702373
$$940$$ 0 0
$$941$$ 25276.0 0.875637 0.437818 0.899063i $$-0.355751\pi$$
0.437818 + 0.899063i $$0.355751\pi$$
$$942$$ 0 0
$$943$$ − 13282.0i − 0.458665i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 1216.00i 0.0417262i 0.999782 + 0.0208631i $$0.00664141\pi$$
−0.999782 + 0.0208631i $$0.993359\pi$$
$$948$$ 0 0
$$949$$ 33852.0 1.15794
$$950$$ 0 0
$$951$$ −19220.0 −0.655364
$$952$$ 0 0
$$953$$ 6033.00i 0.205066i 0.994730 + 0.102533i $$0.0326947\pi$$
−0.994730 + 0.102533i $$0.967305\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 37440.0i 1.26464i
$$958$$ 0 0
$$959$$ −2250.00 −0.0757626
$$960$$ 0 0
$$961$$ −29467.0 −0.989124
$$962$$ 0 0
$$963$$ 1550.00i 0.0518671i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 41792.0i 1.38980i 0.719105 + 0.694902i $$0.244552\pi$$
−0.719105 + 0.694902i $$0.755448\pi$$
$$968$$ 0 0
$$969$$ 46055.0 1.52683
$$970$$ 0 0
$$971$$ 2105.00 0.0695702 0.0347851 0.999395i $$-0.488925\pi$$
0.0347851 + 0.999395i $$0.488925\pi$$
$$972$$ 0 0
$$973$$ 2754.00i 0.0907391i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 30119.0i 0.986277i 0.869951 + 0.493138i $$0.164150\pi$$
−0.869951 + 0.493138i $$0.835850\pi$$
$$978$$ 0 0
$$979$$ −53391.0 −1.74299
$$980$$ 0 0
$$981$$ −892.000 −0.0290310
$$982$$ 0 0
$$983$$ 18438.0i 0.598251i 0.954214 + 0.299126i $$0.0966949\pi$$
−0.954214 + 0.299126i $$0.903305\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 2120.00i 0.0683691i
$$988$$ 0 0
$$989$$ −9512.00 −0.305828
$$990$$ 0 0
$$991$$ 2230.00 0.0714816 0.0357408 0.999361i $$-0.488621\pi$$
0.0357408 + 0.999361i $$0.488621\pi$$
$$992$$ 0 0
$$993$$ 13585.0i 0.434146i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 6804.00i − 0.216133i −0.994144 0.108067i $$-0.965534\pi$$
0.994144 0.108067i $$-0.0344660\pi$$
$$998$$ 0 0
$$999$$ 20010.0 0.633722
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 400.4.c.g.49.2 2
4.3 odd 2 200.4.c.d.49.1 2
5.2 odd 4 400.4.a.q.1.1 1
5.3 odd 4 400.4.a.f.1.1 1
5.4 even 2 inner 400.4.c.g.49.1 2
12.11 even 2 1800.4.f.c.649.2 2
20.3 even 4 200.4.a.h.1.1 yes 1
20.7 even 4 200.4.a.c.1.1 1
20.19 odd 2 200.4.c.d.49.2 2
40.3 even 4 1600.4.a.m.1.1 1
40.13 odd 4 1600.4.a.bo.1.1 1
40.27 even 4 1600.4.a.bn.1.1 1
40.37 odd 4 1600.4.a.n.1.1 1
60.23 odd 4 1800.4.a.t.1.1 1
60.47 odd 4 1800.4.a.p.1.1 1
60.59 even 2 1800.4.f.c.649.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
200.4.a.c.1.1 1 20.7 even 4
200.4.a.h.1.1 yes 1 20.3 even 4
200.4.c.d.49.1 2 4.3 odd 2
200.4.c.d.49.2 2 20.19 odd 2
400.4.a.f.1.1 1 5.3 odd 4
400.4.a.q.1.1 1 5.2 odd 4
400.4.c.g.49.1 2 5.4 even 2 inner
400.4.c.g.49.2 2 1.1 even 1 trivial
1600.4.a.m.1.1 1 40.3 even 4
1600.4.a.n.1.1 1 40.37 odd 4
1600.4.a.bn.1.1 1 40.27 even 4
1600.4.a.bo.1.1 1 40.13 odd 4
1800.4.a.p.1.1 1 60.47 odd 4
1800.4.a.t.1.1 1 60.23 odd 4
1800.4.f.c.649.1 2 60.59 even 2
1800.4.f.c.649.2 2 12.11 even 2