# Properties

 Label 225.4.b.f.199.2 Level $225$ Weight $4$ Character 225.199 Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(199,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("225.199");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (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: $$13.2754297513$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 225.199 Dual form 225.4.b.f.199.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{2} +7.00000 q^{4} -6.00000i q^{7} +15.0000i q^{8} +O(q^{10})$$ $$q+1.00000i q^{2} +7.00000 q^{4} -6.00000i q^{7} +15.0000i q^{8} +43.0000 q^{11} -28.0000i q^{13} +6.00000 q^{14} +41.0000 q^{16} +91.0000i q^{17} +35.0000 q^{19} +43.0000i q^{22} -162.000i q^{23} +28.0000 q^{26} -42.0000i q^{28} +160.000 q^{29} +42.0000 q^{31} +161.000i q^{32} -91.0000 q^{34} +314.000i q^{37} +35.0000i q^{38} +203.000 q^{41} +92.0000i q^{43} +301.000 q^{44} +162.000 q^{46} +196.000i q^{47} +307.000 q^{49} -196.000i q^{52} -82.0000i q^{53} +90.0000 q^{56} +160.000i q^{58} -280.000 q^{59} -518.000 q^{61} +42.0000i q^{62} +167.000 q^{64} -141.000i q^{67} +637.000i q^{68} -412.000 q^{71} -763.000i q^{73} -314.000 q^{74} +245.000 q^{76} -258.000i q^{77} -510.000 q^{79} +203.000i q^{82} -777.000i q^{83} -92.0000 q^{86} +645.000i q^{88} -945.000 q^{89} -168.000 q^{91} -1134.00i q^{92} -196.000 q^{94} -1246.00i q^{97} +307.000i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 14 q^{4}+O(q^{10})$$ 2 * q + 14 * q^4 $$2 q + 14 q^{4} + 86 q^{11} + 12 q^{14} + 82 q^{16} + 70 q^{19} + 56 q^{26} + 320 q^{29} + 84 q^{31} - 182 q^{34} + 406 q^{41} + 602 q^{44} + 324 q^{46} + 614 q^{49} + 180 q^{56} - 560 q^{59} - 1036 q^{61} + 334 q^{64} - 824 q^{71} - 628 q^{74} + 490 q^{76} - 1020 q^{79} - 184 q^{86} - 1890 q^{89} - 336 q^{91} - 392 q^{94}+O(q^{100})$$ 2 * q + 14 * q^4 + 86 * q^11 + 12 * q^14 + 82 * q^16 + 70 * q^19 + 56 * q^26 + 320 * q^29 + 84 * q^31 - 182 * q^34 + 406 * q^41 + 602 * q^44 + 324 * q^46 + 614 * q^49 + 180 * q^56 - 560 * q^59 - 1036 * q^61 + 334 * q^64 - 824 * q^71 - 628 * q^74 + 490 * q^76 - 1020 * q^79 - 184 * q^86 - 1890 * q^89 - 336 * q^91 - 392 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 1.00000i 0.353553i 0.984251 + 0.176777i $$0.0565670\pi$$
−0.984251 + 0.176777i $$0.943433\pi$$
$$3$$ 0 0
$$4$$ 7.00000 0.875000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 6.00000i − 0.323970i −0.986793 0.161985i $$-0.948210\pi$$
0.986793 0.161985i $$-0.0517895\pi$$
$$8$$ 15.0000i 0.662913i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 43.0000 1.17864 0.589318 0.807901i $$-0.299397\pi$$
0.589318 + 0.807901i $$0.299397\pi$$
$$12$$ 0 0
$$13$$ − 28.0000i − 0.597369i −0.954352 0.298685i $$-0.903452\pi$$
0.954352 0.298685i $$-0.0965479\pi$$
$$14$$ 6.00000 0.114541
$$15$$ 0 0
$$16$$ 41.0000 0.640625
$$17$$ 91.0000i 1.29828i 0.760669 + 0.649139i $$0.224871\pi$$
−0.760669 + 0.649139i $$0.775129\pi$$
$$18$$ 0 0
$$19$$ 35.0000 0.422608 0.211304 0.977420i $$-0.432229\pi$$
0.211304 + 0.977420i $$0.432229\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 43.0000i 0.416710i
$$23$$ − 162.000i − 1.46867i −0.678789 0.734333i $$-0.737495\pi$$
0.678789 0.734333i $$-0.262505\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 28.0000 0.211202
$$27$$ 0 0
$$28$$ − 42.0000i − 0.283473i
$$29$$ 160.000 1.02453 0.512263 0.858829i $$-0.328807\pi$$
0.512263 + 0.858829i $$0.328807\pi$$
$$30$$ 0 0
$$31$$ 42.0000 0.243336 0.121668 0.992571i $$-0.461176\pi$$
0.121668 + 0.992571i $$0.461176\pi$$
$$32$$ 161.000i 0.889408i
$$33$$ 0 0
$$34$$ −91.0000 −0.459011
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 314.000i 1.39517i 0.716502 + 0.697585i $$0.245742\pi$$
−0.716502 + 0.697585i $$0.754258\pi$$
$$38$$ 35.0000i 0.149414i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 203.000 0.773251 0.386625 0.922237i $$-0.373641\pi$$
0.386625 + 0.922237i $$0.373641\pi$$
$$42$$ 0 0
$$43$$ 92.0000i 0.326276i 0.986603 + 0.163138i $$0.0521616\pi$$
−0.986603 + 0.163138i $$0.947838\pi$$
$$44$$ 301.000 1.03131
$$45$$ 0 0
$$46$$ 162.000 0.519252
$$47$$ 196.000i 0.608288i 0.952626 + 0.304144i $$0.0983704\pi$$
−0.952626 + 0.304144i $$0.901630\pi$$
$$48$$ 0 0
$$49$$ 307.000 0.895044
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 196.000i − 0.522698i
$$53$$ − 82.0000i − 0.212520i −0.994338 0.106260i $$-0.966112\pi$$
0.994338 0.106260i $$-0.0338876\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 90.0000 0.214763
$$57$$ 0 0
$$58$$ 160.000i 0.362225i
$$59$$ −280.000 −0.617846 −0.308923 0.951087i $$-0.599968\pi$$
−0.308923 + 0.951087i $$0.599968\pi$$
$$60$$ 0 0
$$61$$ −518.000 −1.08726 −0.543632 0.839324i $$-0.682951\pi$$
−0.543632 + 0.839324i $$0.682951\pi$$
$$62$$ 42.0000i 0.0860323i
$$63$$ 0 0
$$64$$ 167.000 0.326172
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 141.000i − 0.257103i −0.991703 0.128551i $$-0.958967\pi$$
0.991703 0.128551i $$-0.0410327\pi$$
$$68$$ 637.000i 1.13599i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −412.000 −0.688668 −0.344334 0.938847i $$-0.611895\pi$$
−0.344334 + 0.938847i $$0.611895\pi$$
$$72$$ 0 0
$$73$$ − 763.000i − 1.22332i −0.791121 0.611660i $$-0.790502\pi$$
0.791121 0.611660i $$-0.209498\pi$$
$$74$$ −314.000 −0.493267
$$75$$ 0 0
$$76$$ 245.000 0.369782
$$77$$ − 258.000i − 0.381842i
$$78$$ 0 0
$$79$$ −510.000 −0.726323 −0.363161 0.931726i $$-0.618303\pi$$
−0.363161 + 0.931726i $$0.618303\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 203.000i 0.273385i
$$83$$ − 777.000i − 1.02755i −0.857924 0.513776i $$-0.828246\pi$$
0.857924 0.513776i $$-0.171754\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −92.0000 −0.115356
$$87$$ 0 0
$$88$$ 645.000i 0.781332i
$$89$$ −945.000 −1.12550 −0.562752 0.826626i $$-0.690257\pi$$
−0.562752 + 0.826626i $$0.690257\pi$$
$$90$$ 0 0
$$91$$ −168.000 −0.193530
$$92$$ − 1134.00i − 1.28508i
$$93$$ 0 0
$$94$$ −196.000 −0.215062
$$95$$ 0 0
$$96$$ 0 0
$$97$$ − 1246.00i − 1.30425i −0.758112 0.652124i $$-0.773878\pi$$
0.758112 0.652124i $$-0.226122\pi$$
$$98$$ 307.000i 0.316446i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1302.00 −1.28271 −0.641356 0.767244i $$-0.721628\pi$$
−0.641356 + 0.767244i $$0.721628\pi$$
$$102$$ 0 0
$$103$$ 532.000i 0.508927i 0.967082 + 0.254464i $$0.0818989\pi$$
−0.967082 + 0.254464i $$0.918101\pi$$
$$104$$ 420.000 0.396004
$$105$$ 0 0
$$106$$ 82.0000 0.0751372
$$107$$ − 1269.00i − 1.14653i −0.819370 0.573266i $$-0.805676\pi$$
0.819370 0.573266i $$-0.194324\pi$$
$$108$$ 0 0
$$109$$ −1070.00 −0.940251 −0.470126 0.882599i $$-0.655791\pi$$
−0.470126 + 0.882599i $$0.655791\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ − 246.000i − 0.207543i
$$113$$ 503.000i 0.418746i 0.977836 + 0.209373i $$0.0671422\pi$$
−0.977836 + 0.209373i $$0.932858\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 1120.00 0.896460
$$117$$ 0 0
$$118$$ − 280.000i − 0.218441i
$$119$$ 546.000 0.420603
$$120$$ 0 0
$$121$$ 518.000 0.389181
$$122$$ − 518.000i − 0.384406i
$$123$$ 0 0
$$124$$ 294.000 0.212919
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 874.000i 0.610669i 0.952245 + 0.305334i $$0.0987683\pi$$
−0.952245 + 0.305334i $$0.901232\pi$$
$$128$$ 1455.00i 1.00473i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1092.00 −0.728309 −0.364155 0.931339i $$-0.618642\pi$$
−0.364155 + 0.931339i $$0.618642\pi$$
$$132$$ 0 0
$$133$$ − 210.000i − 0.136912i
$$134$$ 141.000 0.0908996
$$135$$ 0 0
$$136$$ −1365.00 −0.860645
$$137$$ 411.000i 0.256307i 0.991754 + 0.128154i $$0.0409051\pi$$
−0.991754 + 0.128154i $$0.959095\pi$$
$$138$$ 0 0
$$139$$ 595.000 0.363074 0.181537 0.983384i $$-0.441893\pi$$
0.181537 + 0.983384i $$0.441893\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ − 412.000i − 0.243481i
$$143$$ − 1204.00i − 0.704081i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 763.000 0.432509
$$147$$ 0 0
$$148$$ 2198.00i 1.22077i
$$149$$ −3200.00 −1.75942 −0.879712 0.475507i $$-0.842265\pi$$
−0.879712 + 0.475507i $$0.842265\pi$$
$$150$$ 0 0
$$151$$ 202.000 0.108864 0.0544322 0.998517i $$-0.482665\pi$$
0.0544322 + 0.998517i $$0.482665\pi$$
$$152$$ 525.000i 0.280152i
$$153$$ 0 0
$$154$$ 258.000 0.135002
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 406.000i − 0.206384i −0.994661 0.103192i $$-0.967094\pi$$
0.994661 0.103192i $$-0.0329057\pi$$
$$158$$ − 510.000i − 0.256794i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −972.000 −0.475803
$$162$$ 0 0
$$163$$ − 3803.00i − 1.82745i −0.406336 0.913724i $$-0.633194\pi$$
0.406336 0.913724i $$-0.366806\pi$$
$$164$$ 1421.00 0.676594
$$165$$ 0 0
$$166$$ 777.000 0.363295
$$167$$ 4116.00i 1.90722i 0.301046 + 0.953610i $$0.402664\pi$$
−0.301046 + 0.953610i $$0.597336\pi$$
$$168$$ 0 0
$$169$$ 1413.00 0.643150
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 644.000i 0.285492i
$$173$$ − 1512.00i − 0.664481i −0.943195 0.332241i $$-0.892195\pi$$
0.943195 0.332241i $$-0.107805\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 1763.00 0.755063
$$177$$ 0 0
$$178$$ − 945.000i − 0.397926i
$$179$$ 2585.00 1.07940 0.539698 0.841859i $$-0.318538\pi$$
0.539698 + 0.841859i $$0.318538\pi$$
$$180$$ 0 0
$$181$$ −2758.00 −1.13260 −0.566300 0.824199i $$-0.691626\pi$$
−0.566300 + 0.824199i $$0.691626\pi$$
$$182$$ − 168.000i − 0.0684230i
$$183$$ 0 0
$$184$$ 2430.00 0.973598
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3913.00i 1.53020i
$$188$$ 1372.00i 0.532252i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 2378.00 0.900869 0.450435 0.892809i $$-0.351269\pi$$
0.450435 + 0.892809i $$0.351269\pi$$
$$192$$ 0 0
$$193$$ 3067.00i 1.14387i 0.820298 + 0.571937i $$0.193808\pi$$
−0.820298 + 0.571937i $$0.806192\pi$$
$$194$$ 1246.00 0.461122
$$195$$ 0 0
$$196$$ 2149.00 0.783163
$$197$$ 2346.00i 0.848455i 0.905556 + 0.424227i $$0.139454\pi$$
−0.905556 + 0.424227i $$0.860546\pi$$
$$198$$ 0 0
$$199$$ −4900.00 −1.74549 −0.872743 0.488180i $$-0.837661\pi$$
−0.872743 + 0.488180i $$0.837661\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 1302.00i − 0.453507i
$$203$$ − 960.000i − 0.331915i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −532.000 −0.179933
$$207$$ 0 0
$$208$$ − 1148.00i − 0.382690i
$$209$$ 1505.00 0.498101
$$210$$ 0 0
$$211$$ 4307.00 1.40524 0.702621 0.711564i $$-0.252013\pi$$
0.702621 + 0.711564i $$0.252013\pi$$
$$212$$ − 574.000i − 0.185955i
$$213$$ 0 0
$$214$$ 1269.00 0.405360
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 252.000i − 0.0788335i
$$218$$ − 1070.00i − 0.332429i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 2548.00 0.775552
$$222$$ 0 0
$$223$$ 2212.00i 0.664244i 0.943236 + 0.332122i $$0.107765\pi$$
−0.943236 + 0.332122i $$0.892235\pi$$
$$224$$ 966.000 0.288141
$$225$$ 0 0
$$226$$ −503.000 −0.148049
$$227$$ 476.000i 0.139177i 0.997576 + 0.0695886i $$0.0221687\pi$$
−0.997576 + 0.0695886i $$0.977831\pi$$
$$228$$ 0 0
$$229$$ 2940.00 0.848387 0.424194 0.905572i $$-0.360558\pi$$
0.424194 + 0.905572i $$0.360558\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2400.00i 0.679171i
$$233$$ − 1002.00i − 0.281730i −0.990029 0.140865i $$-0.955012\pi$$
0.990029 0.140865i $$-0.0449884\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −1960.00 −0.540615
$$237$$ 0 0
$$238$$ 546.000i 0.148706i
$$239$$ 2480.00 0.671204 0.335602 0.942004i $$-0.391060\pi$$
0.335602 + 0.942004i $$0.391060\pi$$
$$240$$ 0 0
$$241$$ 1897.00 0.507039 0.253520 0.967330i $$-0.418412\pi$$
0.253520 + 0.967330i $$0.418412\pi$$
$$242$$ 518.000i 0.137596i
$$243$$ 0 0
$$244$$ −3626.00 −0.951356
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 980.000i − 0.252453i
$$248$$ 630.000i 0.161311i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2373.00 0.596743 0.298371 0.954450i $$-0.403557\pi$$
0.298371 + 0.954450i $$0.403557\pi$$
$$252$$ 0 0
$$253$$ − 6966.00i − 1.73102i
$$254$$ −874.000 −0.215904
$$255$$ 0 0
$$256$$ −119.000 −0.0290527
$$257$$ − 4494.00i − 1.09077i −0.838185 0.545385i $$-0.816383\pi$$
0.838185 0.545385i $$-0.183617\pi$$
$$258$$ 0 0
$$259$$ 1884.00 0.451993
$$260$$ 0 0
$$261$$ 0 0
$$262$$ − 1092.00i − 0.257496i
$$263$$ − 722.000i − 0.169279i −0.996412 0.0846396i $$-0.973026\pi$$
0.996412 0.0846396i $$-0.0269739\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 210.000 0.0484057
$$267$$ 0 0
$$268$$ − 987.000i − 0.224965i
$$269$$ −6160.00 −1.39621 −0.698107 0.715993i $$-0.745974\pi$$
−0.698107 + 0.715993i $$0.745974\pi$$
$$270$$ 0 0
$$271$$ −7238.00 −1.62243 −0.811213 0.584751i $$-0.801192\pi$$
−0.811213 + 0.584751i $$0.801192\pi$$
$$272$$ 3731.00i 0.831710i
$$273$$ 0 0
$$274$$ −411.000 −0.0906183
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 1776.00i − 0.385233i −0.981274 0.192616i $$-0.938303\pi$$
0.981274 0.192616i $$-0.0616973\pi$$
$$278$$ 595.000i 0.128366i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −4542.00 −0.964246 −0.482123 0.876104i $$-0.660134\pi$$
−0.482123 + 0.876104i $$0.660134\pi$$
$$282$$ 0 0
$$283$$ 7077.00i 1.48652i 0.669005 + 0.743258i $$0.266720\pi$$
−0.669005 + 0.743258i $$0.733280\pi$$
$$284$$ −2884.00 −0.602584
$$285$$ 0 0
$$286$$ 1204.00 0.248930
$$287$$ − 1218.00i − 0.250510i
$$288$$ 0 0
$$289$$ −3368.00 −0.685528
$$290$$ 0 0
$$291$$ 0 0
$$292$$ − 5341.00i − 1.07041i
$$293$$ 4158.00i 0.829054i 0.910037 + 0.414527i $$0.136053\pi$$
−0.910037 + 0.414527i $$0.863947\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −4710.00 −0.924876
$$297$$ 0 0
$$298$$ − 3200.00i − 0.622050i
$$299$$ −4536.00 −0.877337
$$300$$ 0 0
$$301$$ 552.000 0.105703
$$302$$ 202.000i 0.0384894i
$$303$$ 0 0
$$304$$ 1435.00 0.270733
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2569.00i 0.477591i 0.971070 + 0.238796i $$0.0767526\pi$$
−0.971070 + 0.238796i $$0.923247\pi$$
$$308$$ − 1806.00i − 0.334112i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −2982.00 −0.543710 −0.271855 0.962338i $$-0.587637\pi$$
−0.271855 + 0.962338i $$0.587637\pi$$
$$312$$ 0 0
$$313$$ 2422.00i 0.437379i 0.975795 + 0.218689i $$0.0701781\pi$$
−0.975795 + 0.218689i $$0.929822\pi$$
$$314$$ 406.000 0.0729679
$$315$$ 0 0
$$316$$ −3570.00 −0.635532
$$317$$ − 9484.00i − 1.68036i −0.542307 0.840181i $$-0.682449\pi$$
0.542307 0.840181i $$-0.317551\pi$$
$$318$$ 0 0
$$319$$ 6880.00 1.20754
$$320$$ 0 0
$$321$$ 0 0
$$322$$ − 972.000i − 0.168222i
$$323$$ 3185.00i 0.548663i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 3803.00 0.646100
$$327$$ 0 0
$$328$$ 3045.00i 0.512598i
$$329$$ 1176.00 0.197067
$$330$$ 0 0
$$331$$ −183.000 −0.0303885 −0.0151942 0.999885i $$-0.504837\pi$$
−0.0151942 + 0.999885i $$0.504837\pi$$
$$332$$ − 5439.00i − 0.899108i
$$333$$ 0 0
$$334$$ −4116.00 −0.674304
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 2861.00i − 0.462459i −0.972899 0.231229i $$-0.925725\pi$$
0.972899 0.231229i $$-0.0742748\pi$$
$$338$$ 1413.00i 0.227388i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 1806.00 0.286805
$$342$$ 0 0
$$343$$ − 3900.00i − 0.613936i
$$344$$ −1380.00 −0.216292
$$345$$ 0 0
$$346$$ 1512.00 0.234930
$$347$$ − 629.000i − 0.0973098i −0.998816 0.0486549i $$-0.984507\pi$$
0.998816 0.0486549i $$-0.0154934\pi$$
$$348$$ 0 0
$$349$$ −5950.00 −0.912597 −0.456298 0.889827i $$-0.650825\pi$$
−0.456298 + 0.889827i $$0.650825\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 6923.00i 1.04829i
$$353$$ 11718.0i 1.76682i 0.468604 + 0.883408i $$0.344757\pi$$
−0.468604 + 0.883408i $$0.655243\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −6615.00 −0.984815
$$357$$ 0 0
$$358$$ 2585.00i 0.381624i
$$359$$ 8070.00 1.18640 0.593201 0.805054i $$-0.297864\pi$$
0.593201 + 0.805054i $$0.297864\pi$$
$$360$$ 0 0
$$361$$ −5634.00 −0.821403
$$362$$ − 2758.00i − 0.400434i
$$363$$ 0 0
$$364$$ −1176.00 −0.169338
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8316.00i − 1.18281i −0.806374 0.591406i $$-0.798573\pi$$
0.806374 0.591406i $$-0.201427\pi$$
$$368$$ − 6642.00i − 0.940865i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −492.000 −0.0688500
$$372$$ 0 0
$$373$$ 12062.0i 1.67439i 0.546906 + 0.837194i $$0.315805\pi$$
−0.546906 + 0.837194i $$0.684195\pi$$
$$374$$ −3913.00 −0.541006
$$375$$ 0 0
$$376$$ −2940.00 −0.403242
$$377$$ − 4480.00i − 0.612021i
$$378$$ 0 0
$$379$$ −1735.00 −0.235148 −0.117574 0.993064i $$-0.537512\pi$$
−0.117574 + 0.993064i $$0.537512\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 2378.00i 0.318505i
$$383$$ − 7602.00i − 1.01421i −0.861883 0.507107i $$-0.830715\pi$$
0.861883 0.507107i $$-0.169285\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3067.00 −0.404420
$$387$$ 0 0
$$388$$ − 8722.00i − 1.14122i
$$389$$ 3030.00 0.394928 0.197464 0.980310i $$-0.436729\pi$$
0.197464 + 0.980310i $$0.436729\pi$$
$$390$$ 0 0
$$391$$ 14742.0 1.90674
$$392$$ 4605.00i 0.593336i
$$393$$ 0 0
$$394$$ −2346.00 −0.299974
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 1204.00i 0.152209i 0.997100 + 0.0761046i $$0.0242483\pi$$
−0.997100 + 0.0761046i $$0.975752\pi$$
$$398$$ − 4900.00i − 0.617123i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −1077.00 −0.134122 −0.0670609 0.997749i $$-0.521362\pi$$
−0.0670609 + 0.997749i $$0.521362\pi$$
$$402$$ 0 0
$$403$$ − 1176.00i − 0.145362i
$$404$$ −9114.00 −1.12237
$$405$$ 0 0
$$406$$ 960.000 0.117350
$$407$$ 13502.0i 1.64440i
$$408$$ 0 0
$$409$$ 3955.00 0.478147 0.239074 0.971001i $$-0.423156\pi$$
0.239074 + 0.971001i $$0.423156\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 3724.00i 0.445311i
$$413$$ 1680.00i 0.200163i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 4508.00 0.531305
$$417$$ 0 0
$$418$$ 1505.00i 0.176105i
$$419$$ 6265.00 0.730466 0.365233 0.930916i $$-0.380989\pi$$
0.365233 + 0.930916i $$0.380989\pi$$
$$420$$ 0 0
$$421$$ −3788.00 −0.438517 −0.219259 0.975667i $$-0.570364\pi$$
−0.219259 + 0.975667i $$0.570364\pi$$
$$422$$ 4307.00i 0.496828i
$$423$$ 0 0
$$424$$ 1230.00 0.140882
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3108.00i 0.352240i
$$428$$ − 8883.00i − 1.00321i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 15258.0 1.70523 0.852613 0.522544i $$-0.175017\pi$$
0.852613 + 0.522544i $$0.175017\pi$$
$$432$$ 0 0
$$433$$ − 13573.0i − 1.50641i −0.657784 0.753206i $$-0.728506\pi$$
0.657784 0.753206i $$-0.271494\pi$$
$$434$$ 252.000 0.0278719
$$435$$ 0 0
$$436$$ −7490.00 −0.822720
$$437$$ − 5670.00i − 0.620670i
$$438$$ 0 0
$$439$$ 8120.00 0.882794 0.441397 0.897312i $$-0.354483\pi$$
0.441397 + 0.897312i $$0.354483\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 2548.00i 0.274199i
$$443$$ 6183.00i 0.663122i 0.943434 + 0.331561i $$0.107575\pi$$
−0.943434 + 0.331561i $$0.892425\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −2212.00 −0.234846
$$447$$ 0 0
$$448$$ − 1002.00i − 0.105670i
$$449$$ −1975.00 −0.207586 −0.103793 0.994599i $$-0.533098\pi$$
−0.103793 + 0.994599i $$0.533098\pi$$
$$450$$ 0 0
$$451$$ 8729.00 0.911380
$$452$$ 3521.00i 0.366402i
$$453$$ 0 0
$$454$$ −476.000 −0.0492066
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 11831.0i − 1.21101i −0.795842 0.605504i $$-0.792971\pi$$
0.795842 0.605504i $$-0.207029\pi$$
$$458$$ 2940.00i 0.299950i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −1932.00 −0.195189 −0.0975946 0.995226i $$-0.531115\pi$$
−0.0975946 + 0.995226i $$0.531115\pi$$
$$462$$ 0 0
$$463$$ − 9228.00i − 0.926267i −0.886289 0.463133i $$-0.846725\pi$$
0.886289 0.463133i $$-0.153275\pi$$
$$464$$ 6560.00 0.656337
$$465$$ 0 0
$$466$$ 1002.00 0.0996068
$$467$$ 13916.0i 1.37892i 0.724324 + 0.689460i $$0.242152\pi$$
−0.724324 + 0.689460i $$0.757848\pi$$
$$468$$ 0 0
$$469$$ −846.000 −0.0832935
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 4200.00i − 0.409578i
$$473$$ 3956.00i 0.384560i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 3822.00 0.368027
$$477$$ 0 0
$$478$$ 2480.00i 0.237307i
$$479$$ 2310.00 0.220348 0.110174 0.993912i $$-0.464859\pi$$
0.110174 + 0.993912i $$0.464859\pi$$
$$480$$ 0 0
$$481$$ 8792.00 0.833432
$$482$$ 1897.00i 0.179266i
$$483$$ 0 0
$$484$$ 3626.00 0.340533
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 17114.0i 1.59242i 0.605019 + 0.796211i $$0.293165\pi$$
−0.605019 + 0.796211i $$0.706835\pi$$
$$488$$ − 7770.00i − 0.720761i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 17228.0 1.58348 0.791740 0.610858i $$-0.209175\pi$$
0.791740 + 0.610858i $$0.209175\pi$$
$$492$$ 0 0
$$493$$ 14560.0i 1.33012i
$$494$$ 980.000 0.0892556
$$495$$ 0 0
$$496$$ 1722.00 0.155887
$$497$$ 2472.00i 0.223107i
$$498$$ 0 0
$$499$$ 12500.0 1.12140 0.560698 0.828020i $$-0.310533\pi$$
0.560698 + 0.828020i $$0.310533\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 2373.00i 0.210980i
$$503$$ 868.000i 0.0769428i 0.999260 + 0.0384714i $$0.0122488\pi$$
−0.999260 + 0.0384714i $$0.987751\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 6966.00 0.612009
$$507$$ 0 0
$$508$$ 6118.00i 0.534335i
$$509$$ 13370.0 1.16427 0.582136 0.813091i $$-0.302217\pi$$
0.582136 + 0.813091i $$0.302217\pi$$
$$510$$ 0 0
$$511$$ −4578.00 −0.396319
$$512$$ 11521.0i 0.994455i
$$513$$ 0 0
$$514$$ 4494.00 0.385646
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 8428.00i 0.716950i
$$518$$ 1884.00i 0.159803i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −21637.0 −1.81945 −0.909726 0.415210i $$-0.863708\pi$$
−0.909726 + 0.415210i $$0.863708\pi$$
$$522$$ 0 0
$$523$$ 287.000i 0.0239955i 0.999928 + 0.0119977i $$0.00381909\pi$$
−0.999928 + 0.0119977i $$0.996181\pi$$
$$524$$ −7644.00 −0.637270
$$525$$ 0 0
$$526$$ 722.000 0.0598492
$$527$$ 3822.00i 0.315918i
$$528$$ 0 0
$$529$$ −14077.0 −1.15698
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 1470.00i − 0.119798i
$$533$$ − 5684.00i − 0.461916i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 2115.00 0.170437
$$537$$ 0 0
$$538$$ − 6160.00i − 0.493637i
$$539$$ 13201.0 1.05493
$$540$$ 0 0
$$541$$ −5328.00 −0.423417 −0.211709 0.977333i $$-0.567903\pi$$
−0.211709 + 0.977333i $$0.567903\pi$$
$$542$$ − 7238.00i − 0.573614i
$$543$$ 0 0
$$544$$ −14651.0 −1.15470
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 71.0000i − 0.00554980i −0.999996 0.00277490i $$-0.999117\pi$$
0.999996 0.00277490i $$-0.000883279\pi$$
$$548$$ 2877.00i 0.224269i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 5600.00 0.432973
$$552$$ 0 0
$$553$$ 3060.00i 0.235306i
$$554$$ 1776.00 0.136200
$$555$$ 0 0
$$556$$ 4165.00 0.317689
$$557$$ − 18444.0i − 1.40305i −0.712646 0.701524i $$-0.752503\pi$$
0.712646 0.701524i $$-0.247497\pi$$
$$558$$ 0 0
$$559$$ 2576.00 0.194907
$$560$$ 0 0
$$561$$ 0 0
$$562$$ − 4542.00i − 0.340912i
$$563$$ − 672.000i − 0.0503045i −0.999684 0.0251522i $$-0.991993\pi$$
0.999684 0.0251522i $$-0.00800705\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −7077.00 −0.525563
$$567$$ 0 0
$$568$$ − 6180.00i − 0.456526i
$$569$$ −10935.0 −0.805657 −0.402829 0.915275i $$-0.631973\pi$$
−0.402829 + 0.915275i $$0.631973\pi$$
$$570$$ 0 0
$$571$$ −13588.0 −0.995867 −0.497934 0.867215i $$-0.665908\pi$$
−0.497934 + 0.867215i $$0.665908\pi$$
$$572$$ − 8428.00i − 0.616071i
$$573$$ 0 0
$$574$$ 1218.00 0.0885685
$$575$$ 0 0
$$576$$ 0 0
$$577$$ − 8701.00i − 0.627777i −0.949460 0.313889i $$-0.898368\pi$$
0.949460 0.313889i $$-0.101632\pi$$
$$578$$ − 3368.00i − 0.242371i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −4662.00 −0.332896
$$582$$ 0 0
$$583$$ − 3526.00i − 0.250484i
$$584$$ 11445.0 0.810955
$$585$$ 0 0
$$586$$ −4158.00 −0.293115
$$587$$ 11361.0i 0.798839i 0.916768 + 0.399420i $$0.130788\pi$$
−0.916768 + 0.399420i $$0.869212\pi$$
$$588$$ 0 0
$$589$$ 1470.00 0.102836
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 12874.0i 0.893781i
$$593$$ − 11417.0i − 0.790624i −0.918547 0.395312i $$-0.870636\pi$$
0.918547 0.395312i $$-0.129364\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −22400.0 −1.53950
$$597$$ 0 0
$$598$$ − 4536.00i − 0.310185i
$$599$$ −21050.0 −1.43586 −0.717930 0.696116i $$-0.754910\pi$$
−0.717930 + 0.696116i $$0.754910\pi$$
$$600$$ 0 0
$$601$$ 7427.00 0.504083 0.252041 0.967716i $$-0.418898\pi$$
0.252041 + 0.967716i $$0.418898\pi$$
$$602$$ 552.000i 0.0373718i
$$603$$ 0 0
$$604$$ 1414.00 0.0952564
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 4144.00i 0.277100i 0.990355 + 0.138550i $$0.0442442\pi$$
−0.990355 + 0.138550i $$0.955756\pi$$
$$608$$ 5635.00i 0.375871i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 5488.00 0.363373
$$612$$ 0 0
$$613$$ 30122.0i 1.98469i 0.123489 + 0.992346i $$0.460592\pi$$
−0.123489 + 0.992346i $$0.539408\pi$$
$$614$$ −2569.00 −0.168854
$$615$$ 0 0
$$616$$ 3870.00 0.253128
$$617$$ − 11934.0i − 0.778679i −0.921094 0.389339i $$-0.872703\pi$$
0.921094 0.389339i $$-0.127297\pi$$
$$618$$ 0 0
$$619$$ −8540.00 −0.554526 −0.277263 0.960794i $$-0.589427\pi$$
−0.277263 + 0.960794i $$0.589427\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ − 2982.00i − 0.192230i
$$623$$ 5670.00i 0.364629i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −2422.00 −0.154637
$$627$$ 0 0
$$628$$ − 2842.00i − 0.180586i
$$629$$ −28574.0 −1.81132
$$630$$ 0 0
$$631$$ −3158.00 −0.199236 −0.0996181 0.995026i $$-0.531762\pi$$
−0.0996181 + 0.995026i $$0.531762\pi$$
$$632$$ − 7650.00i − 0.481488i
$$633$$ 0 0
$$634$$ 9484.00 0.594097
$$635$$ 0 0
$$636$$ 0 0
$$637$$ − 8596.00i − 0.534672i
$$638$$ 6880.00i 0.426931i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 4278.00 0.263605 0.131803 0.991276i $$-0.457924\pi$$
0.131803 + 0.991276i $$0.457924\pi$$
$$642$$ 0 0
$$643$$ − 11508.0i − 0.705803i −0.935661 0.352901i $$-0.885195\pi$$
0.935661 0.352901i $$-0.114805\pi$$
$$644$$ −6804.00 −0.416328
$$645$$ 0 0
$$646$$ −3185.00 −0.193982
$$647$$ − 8204.00i − 0.498505i −0.968439 0.249252i $$-0.919815\pi$$
0.968439 0.249252i $$-0.0801849\pi$$
$$648$$ 0 0
$$649$$ −12040.0 −0.728215
$$650$$ 0 0
$$651$$ 0 0
$$652$$ − 26621.0i − 1.59902i
$$653$$ 5518.00i 0.330683i 0.986236 + 0.165342i $$0.0528726\pi$$
−0.986236 + 0.165342i $$0.947127\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 8323.00 0.495364
$$657$$ 0 0
$$658$$ 1176.00i 0.0696736i
$$659$$ 13295.0 0.785887 0.392944 0.919563i $$-0.371457\pi$$
0.392944 + 0.919563i $$0.371457\pi$$
$$660$$ 0 0
$$661$$ −9968.00 −0.586551 −0.293276 0.956028i $$-0.594745\pi$$
−0.293276 + 0.956028i $$0.594745\pi$$
$$662$$ − 183.000i − 0.0107440i
$$663$$ 0 0
$$664$$ 11655.0 0.681177
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 25920.0i − 1.50469i
$$668$$ 28812.0i 1.66882i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −22274.0 −1.28149
$$672$$ 0 0
$$673$$ − 15738.0i − 0.901419i −0.892671 0.450710i $$-0.851171\pi$$
0.892671 0.450710i $$-0.148829\pi$$
$$674$$ 2861.00 0.163504
$$675$$ 0 0
$$676$$ 9891.00 0.562756
$$677$$ − 19824.0i − 1.12540i −0.826660 0.562702i $$-0.809762\pi$$
0.826660 0.562702i $$-0.190238\pi$$
$$678$$ 0 0
$$679$$ −7476.00 −0.422537
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 1806.00i 0.101401i
$$683$$ 11073.0i 0.620346i 0.950680 + 0.310173i $$0.100387\pi$$
−0.950680 + 0.310173i $$0.899613\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 3900.00 0.217059
$$687$$ 0 0
$$688$$ 3772.00i 0.209021i
$$689$$ −2296.00 −0.126953
$$690$$ 0 0
$$691$$ −6503.00 −0.358011 −0.179006 0.983848i $$-0.557288\pi$$
−0.179006 + 0.983848i $$0.557288\pi$$
$$692$$ − 10584.0i − 0.581421i
$$693$$ 0 0
$$694$$ 629.000 0.0344042
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 18473.0i 1.00389i
$$698$$ − 5950.00i − 0.322652i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 10148.0 0.546768 0.273384 0.961905i $$-0.411857\pi$$
0.273384 + 0.961905i $$0.411857\pi$$
$$702$$ 0 0
$$703$$ 10990.0i 0.589610i
$$704$$ 7181.00 0.384438
$$705$$ 0 0
$$706$$ −11718.0 −0.624664
$$707$$ 7812.00i 0.415559i
$$708$$ 0 0
$$709$$ 9980.00 0.528641 0.264321 0.964435i $$-0.414852\pi$$
0.264321 + 0.964435i $$0.414852\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ − 14175.0i − 0.746110i
$$713$$ − 6804.00i − 0.357380i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 18095.0 0.944472
$$717$$ 0 0
$$718$$ 8070.00i 0.419456i
$$719$$ −27510.0 −1.42691 −0.713456 0.700700i $$-0.752871\pi$$
−0.713456 + 0.700700i $$0.752871\pi$$
$$720$$ 0 0
$$721$$ 3192.00 0.164877
$$722$$ − 5634.00i − 0.290410i
$$723$$ 0 0
$$724$$ −19306.0 −0.991025
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 17024.0i 0.868480i 0.900797 + 0.434240i $$0.142983\pi$$
−0.900797 + 0.434240i $$0.857017\pi$$
$$728$$ − 2520.00i − 0.128293i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −8372.00 −0.423597
$$732$$ 0 0
$$733$$ − 34748.0i − 1.75095i −0.483263 0.875475i $$-0.660549\pi$$
0.483263 0.875475i $$-0.339451\pi$$
$$734$$ 8316.00 0.418187
$$735$$ 0 0
$$736$$ 26082.0 1.30624
$$737$$ − 6063.00i − 0.303030i
$$738$$ 0 0
$$739$$ 12020.0 0.598326 0.299163 0.954202i $$-0.403293\pi$$
0.299163 + 0.954202i $$0.403293\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 492.000i − 0.0243422i
$$743$$ − 28642.0i − 1.41423i −0.707098 0.707115i $$-0.749996\pi$$
0.707098 0.707115i $$-0.250004\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −12062.0 −0.591986
$$747$$ 0 0
$$748$$ 27391.0i 1.33892i
$$749$$ −7614.00 −0.371441
$$750$$ 0 0
$$751$$ 8752.00 0.425253 0.212627 0.977134i $$-0.431798\pi$$
0.212627 + 0.977134i $$0.431798\pi$$
$$752$$ 8036.00i 0.389685i
$$753$$ 0 0
$$754$$ 4480.00 0.216382
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 10256.0i − 0.492418i −0.969217 0.246209i $$-0.920815\pi$$
0.969217 0.246209i $$-0.0791850\pi$$
$$758$$ − 1735.00i − 0.0831373i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −33957.0 −1.61753 −0.808765 0.588132i $$-0.799864\pi$$
−0.808765 + 0.588132i $$0.799864\pi$$
$$762$$ 0 0
$$763$$ 6420.00i 0.304613i
$$764$$ 16646.0 0.788261
$$765$$ 0 0
$$766$$ 7602.00 0.358579
$$767$$ 7840.00i 0.369082i
$$768$$ 0 0
$$769$$ −27965.0 −1.31137 −0.655685 0.755034i $$-0.727620\pi$$
−0.655685 + 0.755034i $$0.727620\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 21469.0i 1.00089i
$$773$$ − 9912.00i − 0.461203i −0.973048 0.230601i $$-0.925931\pi$$
0.973048 0.230601i $$-0.0740694\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 18690.0 0.864603
$$777$$ 0 0
$$778$$ 3030.00i 0.139628i
$$779$$ 7105.00 0.326782
$$780$$ 0 0
$$781$$ −17716.0 −0.811688
$$782$$ 14742.0i 0.674134i
$$783$$ 0 0
$$784$$ 12587.0 0.573387
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 25564.0i 1.15789i 0.815367 + 0.578944i $$0.196535\pi$$
−0.815367 + 0.578944i $$0.803465\pi$$
$$788$$ 16422.0i 0.742398i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 3018.00 0.135661
$$792$$ 0 0
$$793$$ 14504.0i 0.649498i
$$794$$ −1204.00 −0.0538141
$$795$$ 0 0
$$796$$ −34300.0 −1.52730
$$797$$ 12446.0i 0.553149i 0.960992 + 0.276575i $$0.0891993\pi$$
−0.960992 + 0.276575i $$0.910801\pi$$
$$798$$ 0 0
$$799$$ −17836.0 −0.789728
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 1077.00i − 0.0474192i
$$803$$ − 32809.0i − 1.44185i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 1176.00 0.0513931
$$807$$ 0 0
$$808$$ − 19530.0i − 0.850325i
$$809$$ 33970.0 1.47629 0.738147 0.674640i $$-0.235701\pi$$
0.738147 + 0.674640i $$0.235701\pi$$
$$810$$ 0 0
$$811$$ 18732.0 0.811060 0.405530 0.914082i $$-0.367087\pi$$
0.405530 + 0.914082i $$0.367087\pi$$
$$812$$ − 6720.00i − 0.290426i
$$813$$ 0 0
$$814$$ −13502.0 −0.581382
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 3220.00i 0.137887i
$$818$$ 3955.00i 0.169051i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −6162.00 −0.261943 −0.130972 0.991386i $$-0.541810\pi$$
−0.130972 + 0.991386i $$0.541810\pi$$
$$822$$ 0 0
$$823$$ − 25388.0i − 1.07530i −0.843169 0.537649i $$-0.819313\pi$$
0.843169 0.537649i $$-0.180687\pi$$
$$824$$ −7980.00 −0.337374
$$825$$ 0 0
$$826$$ −1680.00 −0.0707684
$$827$$ 25201.0i 1.05964i 0.848109 + 0.529821i $$0.177741\pi$$
−0.848109 + 0.529821i $$0.822259\pi$$
$$828$$ 0 0
$$829$$ 19740.0 0.827019 0.413509 0.910500i $$-0.364303\pi$$
0.413509 + 0.910500i $$0.364303\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 4676.00i − 0.194845i
$$833$$ 27937.0i 1.16202i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 10535.0 0.435838
$$837$$ 0 0
$$838$$ 6265.00i 0.258259i
$$839$$ 29680.0 1.22130 0.610648 0.791902i $$-0.290909\pi$$
0.610648 + 0.791902i $$0.290909\pi$$
$$840$$ 0 0
$$841$$ 1211.00 0.0496535
$$842$$ − 3788.00i − 0.155039i
$$843$$ 0 0
$$844$$ 30149.0 1.22959
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 3108.00i − 0.126083i
$$848$$ − 3362.00i − 0.136146i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 50868.0 2.04904
$$852$$ 0 0
$$853$$ − 1218.00i − 0.0488904i −0.999701 0.0244452i $$-0.992218\pi$$
0.999701 0.0244452i $$-0.00778193\pi$$
$$854$$ −3108.00 −0.124536
$$855$$ 0 0
$$856$$ 19035.0 0.760050
$$857$$ 38731.0i 1.54379i 0.635752 + 0.771894i $$0.280690\pi$$
−0.635752 + 0.771894i $$0.719310\pi$$
$$858$$ 0 0
$$859$$ 23555.0 0.935607 0.467803 0.883833i $$-0.345046\pi$$
0.467803 + 0.883833i $$0.345046\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 15258.0i 0.602888i
$$863$$ − 24872.0i − 0.981058i −0.871425 0.490529i $$-0.836804\pi$$
0.871425 0.490529i $$-0.163196\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 13573.0 0.532597
$$867$$ 0 0
$$868$$ − 1764.00i − 0.0689793i
$$869$$ −21930.0 −0.856069
$$870$$ 0 0
$$871$$ −3948.00 −0.153585
$$872$$ − 16050.0i − 0.623305i
$$873$$ 0 0
$$874$$ 5670.00 0.219440
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 17124.0i 0.659335i 0.944097 + 0.329667i $$0.106937\pi$$
−0.944097 + 0.329667i $$0.893063\pi$$
$$878$$ 8120.00i 0.312115i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 658.000 0.0251630 0.0125815 0.999921i $$-0.495995\pi$$
0.0125815 + 0.999921i $$0.495995\pi$$
$$882$$ 0 0
$$883$$ 33727.0i 1.28540i 0.766120 + 0.642698i $$0.222185\pi$$
−0.766120 + 0.642698i $$0.777815\pi$$
$$884$$ 17836.0 0.678608
$$885$$ 0 0
$$886$$ −6183.00 −0.234449
$$887$$ 36036.0i 1.36412i 0.731298 + 0.682058i $$0.238915\pi$$
−0.731298 + 0.682058i $$0.761085\pi$$
$$888$$ 0 0
$$889$$ 5244.00 0.197838
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 15484.0i 0.581214i
$$893$$ 6860.00i 0.257067i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 8730.00 0.325501
$$897$$ 0 0
$$898$$ − 1975.00i − 0.0733927i
$$899$$ 6720.00 0.249304
$$900$$ 0 0
$$901$$ 7462.00 0.275910
$$902$$ 8729.00i 0.322222i
$$903$$ 0 0
$$904$$ −7545.00 −0.277592
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 39156.0i − 1.43347i −0.697348 0.716733i $$-0.745637\pi$$
0.697348 0.716733i $$-0.254363\pi$$
$$908$$ 3332.00i 0.121780i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −43532.0 −1.58318 −0.791591 0.611051i $$-0.790747\pi$$
−0.791591 + 0.611051i $$0.790747\pi$$
$$912$$ 0 0
$$913$$ − 33411.0i − 1.21111i
$$914$$ 11831.0 0.428156
$$915$$ 0 0
$$916$$ 20580.0 0.742339
$$917$$ 6552.00i 0.235950i
$$918$$ 0 0
$$919$$ 28610.0 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 1932.00i − 0.0690098i
$$923$$ 11536.0i 0.411389i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 9228.00 0.327485
$$927$$ 0 0
$$928$$ 25760.0i 0.911221i
$$929$$ −24290.0 −0.857835 −0.428918 0.903344i $$-0.641105\pi$$
−0.428918 + 0.903344i $$0.641105\pi$$
$$930$$ 0 0
$$931$$ 10745.0 0.378253
$$932$$ − 7014.00i − 0.246514i
$$933$$ 0 0
$$934$$ −13916.0 −0.487522
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 34461.0i − 1.20149i −0.799442 0.600743i $$-0.794872\pi$$
0.799442 0.600743i $$-0.205128\pi$$
$$938$$ − 846.000i − 0.0294487i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 40628.0 1.40748 0.703738 0.710460i $$-0.251513\pi$$
0.703738 + 0.710460i $$0.251513\pi$$
$$942$$ 0 0
$$943$$ − 32886.0i − 1.13565i
$$944$$ −11480.0 −0.395807
$$945$$ 0 0
$$946$$ −3956.00 −0.135963
$$947$$ − 20904.0i − 0.717306i −0.933471 0.358653i $$-0.883236\pi$$
0.933471 0.358653i $$-0.116764\pi$$
$$948$$ 0 0
$$949$$ −21364.0 −0.730774
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 8190.00i 0.278823i
$$953$$ − 1807.00i − 0.0614213i −0.999528 0.0307106i $$-0.990223\pi$$
0.999528 0.0307106i $$-0.00977704\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 17360.0 0.587304
$$957$$ 0 0
$$958$$ 2310.00i 0.0779047i
$$959$$ 2466.00 0.0830358
$$960$$ 0 0
$$961$$ −28027.0 −0.940787
$$962$$ 8792.00i 0.294663i
$$963$$ 0 0
$$964$$ 13279.0 0.443660
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 57584.0i 1.91497i 0.288482 + 0.957485i $$0.406849\pi$$
−0.288482 + 0.957485i $$0.593151\pi$$
$$968$$ 7770.00i 0.257993i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −27237.0 −0.900182 −0.450091 0.892983i $$-0.648608\pi$$
−0.450091 + 0.892983i $$0.648608\pi$$
$$972$$ 0 0
$$973$$ − 3570.00i − 0.117625i
$$974$$ −17114.0 −0.563006
$$975$$ 0 0
$$976$$ −21238.0 −0.696528
$$977$$ − 13649.0i − 0.446950i −0.974710 0.223475i $$-0.928260\pi$$
0.974710 0.223475i $$-0.0717401\pi$$
$$978$$ 0 0
$$979$$ −40635.0 −1.32656
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 17228.0i 0.559845i
$$983$$ − 16002.0i − 0.519211i −0.965715 0.259606i $$-0.916407\pi$$
0.965715 0.259606i $$-0.0835925\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −14560.0 −0.470269
$$987$$ 0 0
$$988$$ − 6860.00i − 0.220896i
$$989$$ 14904.0 0.479191
$$990$$ 0 0
$$991$$ 37022.0 1.18672 0.593362 0.804936i $$-0.297800\pi$$
0.593362 + 0.804936i $$0.297800\pi$$
$$992$$ 6762.00i 0.216425i
$$993$$ 0 0
$$994$$ −2472.00 −0.0788804
$$995$$ 0 0
$$996$$ 0 0
$$997$$ − 18396.0i − 0.584360i −0.956363 0.292180i $$-0.905619\pi$$
0.956363 0.292180i $$-0.0943807\pi$$
$$998$$ 12500.0i 0.396474i
$$999$$ 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 225.4.b.f.199.2 2
3.2 odd 2 25.4.b.b.24.1 2
5.2 odd 4 225.4.a.c.1.1 1
5.3 odd 4 225.4.a.e.1.1 1
5.4 even 2 inner 225.4.b.f.199.1 2
12.11 even 2 400.4.c.e.49.1 2
15.2 even 4 25.4.a.b.1.1 yes 1
15.8 even 4 25.4.a.a.1.1 1
15.14 odd 2 25.4.b.b.24.2 2
60.23 odd 4 400.4.a.s.1.1 1
60.47 odd 4 400.4.a.c.1.1 1
60.59 even 2 400.4.c.e.49.2 2
105.62 odd 4 1225.4.a.i.1.1 1
105.83 odd 4 1225.4.a.h.1.1 1
120.53 even 4 1600.4.a.bt.1.1 1
120.77 even 4 1600.4.a.i.1.1 1
120.83 odd 4 1600.4.a.h.1.1 1
120.107 odd 4 1600.4.a.bs.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
25.4.a.a.1.1 1 15.8 even 4
25.4.a.b.1.1 yes 1 15.2 even 4
25.4.b.b.24.1 2 3.2 odd 2
25.4.b.b.24.2 2 15.14 odd 2
225.4.a.c.1.1 1 5.2 odd 4
225.4.a.e.1.1 1 5.3 odd 4
225.4.b.f.199.1 2 5.4 even 2 inner
225.4.b.f.199.2 2 1.1 even 1 trivial
400.4.a.c.1.1 1 60.47 odd 4
400.4.a.s.1.1 1 60.23 odd 4
400.4.c.e.49.1 2 12.11 even 2
400.4.c.e.49.2 2 60.59 even 2
1225.4.a.h.1.1 1 105.83 odd 4
1225.4.a.i.1.1 1 105.62 odd 4
1600.4.a.h.1.1 1 120.83 odd 4
1600.4.a.i.1.1 1 120.77 even 4
1600.4.a.bs.1.1 1 120.107 odd 4
1600.4.a.bt.1.1 1 120.53 even 4