# Properties

 Label 400.4.c.e.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 25) 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.e.49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+7.00000i q^{3} -6.00000i q^{7} -22.0000 q^{9} +O(q^{10})$$ $$q+7.00000i q^{3} -6.00000i q^{7} -22.0000 q^{9} +43.0000 q^{11} +28.0000i q^{13} +91.0000i q^{17} -35.0000 q^{19} +42.0000 q^{21} +162.000i q^{23} +35.0000i q^{27} -160.000 q^{29} -42.0000 q^{31} +301.000i q^{33} -314.000i q^{37} -196.000 q^{39} -203.000 q^{41} +92.0000i q^{43} -196.000i q^{47} +307.000 q^{49} -637.000 q^{51} -82.0000i q^{53} -245.000i q^{57} -280.000 q^{59} -518.000 q^{61} +132.000i q^{63} -141.000i q^{67} -1134.00 q^{69} -412.000 q^{71} +763.000i q^{73} -258.000i q^{77} +510.000 q^{79} -839.000 q^{81} +777.000i q^{83} -1120.00i q^{87} +945.000 q^{89} +168.000 q^{91} -294.000i q^{93} +1246.00i q^{97} -946.000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 44 q^{9}+O(q^{10})$$ 2 * q - 44 * q^9 $$2 q - 44 q^{9} + 86 q^{11} - 70 q^{19} + 84 q^{21} - 320 q^{29} - 84 q^{31} - 392 q^{39} - 406 q^{41} + 614 q^{49} - 1274 q^{51} - 560 q^{59} - 1036 q^{61} - 2268 q^{69} - 824 q^{71} + 1020 q^{79} - 1678 q^{81} + 1890 q^{89} + 336 q^{91} - 1892 q^{99}+O(q^{100})$$ 2 * q - 44 * q^9 + 86 * q^11 - 70 * q^19 + 84 * q^21 - 320 * q^29 - 84 * q^31 - 392 * q^39 - 406 * q^41 + 614 * q^49 - 1274 * q^51 - 560 * q^59 - 1036 * q^61 - 2268 * q^69 - 824 * q^71 + 1020 * q^79 - 1678 * q^81 + 1890 * q^89 + 336 * q^91 - 1892 * 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
$$3$$ 7.00000i 1.34715i 0.739119 + 0.673575i $$0.235242\pi$$
−0.739119 + 0.673575i $$0.764758\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 6.00000i − 0.323970i −0.986793 0.161985i $$-0.948210\pi$$
0.986793 0.161985i $$-0.0517895\pi$$
$$8$$ 0 0
$$9$$ −22.0000 −0.814815
$$10$$ 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 $$0.0965479\pi$$
−0.954352 + 0.298685i $$0.903452\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 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 $$-0.567771\pi$$
−0.211304 + 0.977420i $$0.567771\pi$$
$$20$$ 0 0
$$21$$ 42.0000 0.436436
$$22$$ 0 0
$$23$$ 162.000i 1.46867i 0.678789 + 0.734333i $$0.262505\pi$$
−0.678789 + 0.734333i $$0.737495\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 35.0000i 0.249472i
$$28$$ 0 0
$$29$$ −160.000 −1.02453 −0.512263 0.858829i $$-0.671193\pi$$
−0.512263 + 0.858829i $$0.671193\pi$$
$$30$$ 0 0
$$31$$ −42.0000 −0.243336 −0.121668 0.992571i $$-0.538824\pi$$
−0.121668 + 0.992571i $$0.538824\pi$$
$$32$$ 0 0
$$33$$ 301.000i 1.58780i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 314.000i − 1.39517i −0.716502 0.697585i $$-0.754258\pi$$
0.716502 0.697585i $$-0.245742\pi$$
$$38$$ 0 0
$$39$$ −196.000 −0.804747
$$40$$ 0 0
$$41$$ −203.000 −0.773251 −0.386625 0.922237i $$-0.626359\pi$$
−0.386625 + 0.922237i $$0.626359\pi$$
$$42$$ 0 0
$$43$$ 92.0000i 0.326276i 0.986603 + 0.163138i $$0.0521616\pi$$
−0.986603 + 0.163138i $$0.947838\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ − 196.000i − 0.608288i −0.952626 0.304144i $$-0.901630\pi$$
0.952626 0.304144i $$-0.0983704\pi$$
$$48$$ 0 0
$$49$$ 307.000 0.895044
$$50$$ 0 0
$$51$$ −637.000 −1.74898
$$52$$ 0 0
$$53$$ − 82.0000i − 0.212520i −0.994338 0.106260i $$-0.966112\pi$$
0.994338 0.106260i $$-0.0338876\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 245.000i − 0.569317i
$$58$$ 0 0
$$59$$ −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 $$0.682951\pi$$
$$62$$ 0 0
$$63$$ 132.000i 0.263975i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 141.000i − 0.257103i −0.991703 0.128551i $$-0.958967\pi$$
0.991703 0.128551i $$-0.0410327\pi$$
$$68$$ 0 0
$$69$$ −1134.00 −1.97852
$$70$$ 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 $$0.209498\pi$$
−0.791121 + 0.611660i $$0.790502\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ − 258.000i − 0.381842i
$$78$$ 0 0
$$79$$ 510.000 0.726323 0.363161 0.931726i $$-0.381697\pi$$
0.363161 + 0.931726i $$0.381697\pi$$
$$80$$ 0 0
$$81$$ −839.000 −1.15089
$$82$$ 0 0
$$83$$ 777.000i 1.02755i 0.857924 + 0.513776i $$0.171754\pi$$
−0.857924 + 0.513776i $$0.828246\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 1120.00i − 1.38019i
$$88$$ 0 0
$$89$$ 945.000 1.12550 0.562752 0.826626i $$-0.309743\pi$$
0.562752 + 0.826626i $$0.309743\pi$$
$$90$$ 0 0
$$91$$ 168.000 0.193530
$$92$$ 0 0
$$93$$ − 294.000i − 0.327811i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1246.00i 1.30425i 0.758112 + 0.652124i $$0.226122\pi$$
−0.758112 + 0.652124i $$0.773878\pi$$
$$98$$ 0 0
$$99$$ −946.000 −0.960369
$$100$$ 0 0
$$101$$ 1302.00 1.28271 0.641356 0.767244i $$-0.278372\pi$$
0.641356 + 0.767244i $$0.278372\pi$$
$$102$$ 0 0
$$103$$ 532.000i 0.508927i 0.967082 + 0.254464i $$0.0818989\pi$$
−0.967082 + 0.254464i $$0.918101\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ 1269.00i 1.14653i 0.819370 + 0.573266i $$0.194324\pi$$
−0.819370 + 0.573266i $$0.805676\pi$$
$$108$$ 0 0
$$109$$ −1070.00 −0.940251 −0.470126 0.882599i $$-0.655791\pi$$
−0.470126 + 0.882599i $$0.655791\pi$$
$$110$$ 0 0
$$111$$ 2198.00 1.87950
$$112$$ 0 0
$$113$$ 503.000i 0.418746i 0.977836 + 0.209373i $$0.0671422\pi$$
−0.977836 + 0.209373i $$0.932858\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 616.000i − 0.486745i
$$118$$ 0 0
$$119$$ 546.000 0.420603
$$120$$ 0 0
$$121$$ 518.000 0.389181
$$122$$ 0 0
$$123$$ − 1421.00i − 1.04169i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 874.000i 0.610669i 0.952245 + 0.305334i $$0.0987683\pi$$
−0.952245 + 0.305334i $$0.901232\pi$$
$$128$$ 0 0
$$129$$ −644.000 −0.439543
$$130$$ 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
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 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 $$-0.558107\pi$$
−0.181537 + 0.983384i $$0.558107\pi$$
$$140$$ 0 0
$$141$$ 1372.00 0.819456
$$142$$ 0 0
$$143$$ 1204.00i 0.704081i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 2149.00i 1.20576i
$$148$$ 0 0
$$149$$ 3200.00 1.75942 0.879712 0.475507i $$-0.157735\pi$$
0.879712 + 0.475507i $$0.157735\pi$$
$$150$$ 0 0
$$151$$ −202.000 −0.108864 −0.0544322 0.998517i $$-0.517335\pi$$
−0.0544322 + 0.998517i $$0.517335\pi$$
$$152$$ 0 0
$$153$$ − 2002.00i − 1.05786i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 406.000i 0.206384i 0.994661 + 0.103192i $$0.0329057\pi$$
−0.994661 + 0.103192i $$0.967094\pi$$
$$158$$ 0 0
$$159$$ 574.000 0.286297
$$160$$ 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 $$-0.366806\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ − 4116.00i − 1.90722i −0.301046 0.953610i $$-0.597336\pi$$
0.301046 0.953610i $$-0.402664\pi$$
$$168$$ 0 0
$$169$$ 1413.00 0.643150
$$170$$ 0 0
$$171$$ 770.000 0.344347
$$172$$ 0 0
$$173$$ − 1512.00i − 0.664481i −0.943195 0.332241i $$-0.892195\pi$$
0.943195 0.332241i $$-0.107805\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 1960.00i − 0.832331i
$$178$$ 0 0
$$179$$ 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 $$0.691626\pi$$
$$182$$ 0 0
$$183$$ − 3626.00i − 1.46471i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3913.00i 1.53020i
$$188$$ 0 0
$$189$$ 210.000 0.0808214
$$190$$ 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 $$-0.806192\pi$$
0.820298 0.571937i $$-0.193808\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 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 $$-0.162339\pi$$
0.872743 + 0.488180i $$0.162339\pi$$
$$200$$ 0 0
$$201$$ 987.000 0.346356
$$202$$ 0 0
$$203$$ 960.000i 0.331915i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 3564.00i − 1.19669i
$$208$$ 0 0
$$209$$ −1505.00 −0.498101
$$210$$ 0 0
$$211$$ −4307.00 −1.40524 −0.702621 0.711564i $$-0.747987\pi$$
−0.702621 + 0.711564i $$0.747987\pi$$
$$212$$ 0 0
$$213$$ − 2884.00i − 0.927739i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 252.000i 0.0788335i
$$218$$ 0 0
$$219$$ −5341.00 −1.64800
$$220$$ 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 $$0.892235\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 476.000i − 0.139177i −0.997576 0.0695886i $$-0.977831\pi$$
0.997576 0.0695886i $$-0.0221687\pi$$
$$228$$ 0 0
$$229$$ 2940.00 0.848387 0.424194 0.905572i $$-0.360558\pi$$
0.424194 + 0.905572i $$0.360558\pi$$
$$230$$ 0 0
$$231$$ 1806.00 0.514399
$$232$$ 0 0
$$233$$ − 1002.00i − 0.281730i −0.990029 0.140865i $$-0.955012\pi$$
0.990029 0.140865i $$-0.0449884\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 3570.00i 0.978466i
$$238$$ 0 0
$$239$$ 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 $$0.418412\pi$$
$$242$$ 0 0
$$243$$ − 4928.00i − 1.30095i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 980.000i − 0.252453i
$$248$$ 0 0
$$249$$ −5439.00 −1.38427
$$250$$ 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
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 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
$$261$$ 3520.00 0.834799
$$262$$ 0 0
$$263$$ 722.000i 0.169279i 0.996412 + 0.0846396i $$0.0269739\pi$$
−0.996412 + 0.0846396i $$0.973026\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 6615.00i 1.51622i
$$268$$ 0 0
$$269$$ 6160.00 1.39621 0.698107 0.715993i $$-0.254026\pi$$
0.698107 + 0.715993i $$0.254026\pi$$
$$270$$ 0 0
$$271$$ 7238.00 1.62243 0.811213 0.584751i $$-0.198808\pi$$
0.811213 + 0.584751i $$0.198808\pi$$
$$272$$ 0 0
$$273$$ 1176.00i 0.260713i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 1776.00i 0.385233i 0.981274 + 0.192616i $$0.0616973\pi$$
−0.981274 + 0.192616i $$0.938303\pi$$
$$278$$ 0 0
$$279$$ 924.000 0.198274
$$280$$ 0 0
$$281$$ 4542.00 0.964246 0.482123 0.876104i $$-0.339866\pi$$
0.482123 + 0.876104i $$0.339866\pi$$
$$282$$ 0 0
$$283$$ 7077.00i 1.48652i 0.669005 + 0.743258i $$0.266720\pi$$
−0.669005 + 0.743258i $$0.733280\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 1218.00i 0.250510i
$$288$$ 0 0
$$289$$ −3368.00 −0.685528
$$290$$ 0 0
$$291$$ −8722.00 −1.75702
$$292$$ 0 0
$$293$$ 4158.00i 0.829054i 0.910037 + 0.414527i $$0.136053\pi$$
−0.910037 + 0.414527i $$0.863947\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 1505.00i 0.294037i
$$298$$ 0 0
$$299$$ −4536.00 −0.877337
$$300$$ 0 0
$$301$$ 552.000 0.105703
$$302$$ 0 0
$$303$$ 9114.00i 1.72801i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2569.00i 0.477591i 0.971070 + 0.238796i $$0.0767526\pi$$
−0.971070 + 0.238796i $$0.923247\pi$$
$$308$$ 0 0
$$309$$ −3724.00 −0.685602
$$310$$ 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 $$-0.929822\pi$$
0.975795 0.218689i $$-0.0701781\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 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
$$321$$ −8883.00 −1.54455
$$322$$ 0 0
$$323$$ − 3185.00i − 0.548663i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ − 7490.00i − 1.26666i
$$328$$ 0 0
$$329$$ −1176.00 −0.197067
$$330$$ 0 0
$$331$$ 183.000 0.0303885 0.0151942 0.999885i $$-0.495163\pi$$
0.0151942 + 0.999885i $$0.495163\pi$$
$$332$$ 0 0
$$333$$ 6908.00i 1.13681i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 2861.00i 0.462459i 0.972899 + 0.231229i $$0.0742748\pi$$
−0.972899 + 0.231229i $$0.925725\pi$$
$$338$$ 0 0
$$339$$ −3521.00 −0.564113
$$340$$ 0 0
$$341$$ −1806.00 −0.286805
$$342$$ 0 0
$$343$$ − 3900.00i − 0.613936i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 629.000i 0.0973098i 0.998816 + 0.0486549i $$0.0154934\pi$$
−0.998816 + 0.0486549i $$0.984507\pi$$
$$348$$ 0 0
$$349$$ −5950.00 −0.912597 −0.456298 0.889827i $$-0.650825\pi$$
−0.456298 + 0.889827i $$0.650825\pi$$
$$350$$ 0 0
$$351$$ −980.000 −0.149027
$$352$$ 0 0
$$353$$ 11718.0i 1.76682i 0.468604 + 0.883408i $$0.344757\pi$$
−0.468604 + 0.883408i $$0.655243\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 3822.00i 0.566615i
$$358$$ 0 0
$$359$$ 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
$$362$$ 0 0
$$363$$ 3626.00i 0.524286i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8316.00i − 1.18281i −0.806374 0.591406i $$-0.798573\pi$$
0.806374 0.591406i $$-0.201427\pi$$
$$368$$ 0 0
$$369$$ 4466.00 0.630056
$$370$$ 0 0
$$371$$ −492.000 −0.0688500
$$372$$ 0 0
$$373$$ − 12062.0i − 1.67439i −0.546906 0.837194i $$-0.684195\pi$$
0.546906 0.837194i $$-0.315805\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 4480.00i − 0.612021i
$$378$$ 0 0
$$379$$ 1735.00 0.235148 0.117574 0.993064i $$-0.462488\pi$$
0.117574 + 0.993064i $$0.462488\pi$$
$$380$$ 0 0
$$381$$ −6118.00 −0.822663
$$382$$ 0 0
$$383$$ 7602.00i 1.01421i 0.861883 + 0.507107i $$0.169285\pi$$
−0.861883 + 0.507107i $$0.830715\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 2024.00i − 0.265855i
$$388$$ 0 0
$$389$$ −3030.00 −0.394928 −0.197464 0.980310i $$-0.563271\pi$$
−0.197464 + 0.980310i $$0.563271\pi$$
$$390$$ 0 0
$$391$$ −14742.0 −1.90674
$$392$$ 0 0
$$393$$ − 7644.00i − 0.981142i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ − 1204.00i − 0.152209i −0.997100 0.0761046i $$-0.975752\pi$$
0.997100 0.0761046i $$-0.0242483\pi$$
$$398$$ 0 0
$$399$$ −1470.00 −0.184441
$$400$$ 0 0
$$401$$ 1077.00 0.134122 0.0670609 0.997749i $$-0.478638\pi$$
0.0670609 + 0.997749i $$0.478638\pi$$
$$402$$ 0 0
$$403$$ − 1176.00i − 0.145362i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ − 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
$$411$$ −2877.00 −0.345285
$$412$$ 0 0
$$413$$ 1680.00i 0.200163i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ − 4165.00i − 0.489115i
$$418$$ 0 0
$$419$$ 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 $$0.570364\pi$$
$$422$$ 0 0
$$423$$ 4312.00i 0.495642i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3108.00i 0.352240i
$$428$$ 0 0
$$429$$ −8428.00 −0.948503
$$430$$ 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 $$0.271494\pi$$
−0.657784 + 0.753206i $$0.728506\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ − 5670.00i − 0.620670i
$$438$$ 0 0
$$439$$ −8120.00 −0.882794 −0.441397 0.897312i $$-0.645517\pi$$
−0.441397 + 0.897312i $$0.645517\pi$$
$$440$$ 0 0
$$441$$ −6754.00 −0.729295
$$442$$ 0 0
$$443$$ − 6183.00i − 0.663122i −0.943434 0.331561i $$-0.892425\pi$$
0.943434 0.331561i $$-0.107575\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 22400.0i 2.37021i
$$448$$ 0 0
$$449$$ 1975.00 0.207586 0.103793 0.994599i $$-0.466902\pi$$
0.103793 + 0.994599i $$0.466902\pi$$
$$450$$ 0 0
$$451$$ −8729.00 −0.911380
$$452$$ 0 0
$$453$$ − 1414.00i − 0.146657i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 11831.0i 1.21101i 0.795842 + 0.605504i $$0.207029\pi$$
−0.795842 + 0.605504i $$0.792971\pi$$
$$458$$ 0 0
$$459$$ −3185.00 −0.323885
$$460$$ 0 0
$$461$$ 1932.00 0.195189 0.0975946 0.995226i $$-0.468885\pi$$
0.0975946 + 0.995226i $$0.468885\pi$$
$$462$$ 0 0
$$463$$ − 9228.00i − 0.926267i −0.886289 0.463133i $$-0.846725\pi$$
0.886289 0.463133i $$-0.153275\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ − 13916.0i − 1.37892i −0.724324 0.689460i $$-0.757848\pi$$
0.724324 0.689460i $$-0.242152\pi$$
$$468$$ 0 0
$$469$$ −846.000 −0.0832935
$$470$$ 0 0
$$471$$ −2842.00 −0.278031
$$472$$ 0 0
$$473$$ 3956.00i 0.384560i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 1804.00i 0.173165i
$$478$$ 0 0
$$479$$ 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
$$482$$ 0 0
$$483$$ 6804.00i 0.640979i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 17114.0i 1.59242i 0.605019 + 0.796211i $$0.293165\pi$$
−0.605019 + 0.796211i $$0.706835\pi$$
$$488$$ 0 0
$$489$$ 26621.0 2.46185
$$490$$ 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
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2472.00i 0.223107i
$$498$$ 0 0
$$499$$ −12500.0 −1.12140 −0.560698 0.828020i $$-0.689467\pi$$
−0.560698 + 0.828020i $$0.689467\pi$$
$$500$$ 0 0
$$501$$ 28812.0 2.56931
$$502$$ 0 0
$$503$$ − 868.000i − 0.0769428i −0.999260 0.0384714i $$-0.987751\pi$$
0.999260 0.0384714i $$-0.0122488\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 9891.00i 0.866420i
$$508$$ 0 0
$$509$$ −13370.0 −1.16427 −0.582136 0.813091i $$-0.697783\pi$$
−0.582136 + 0.813091i $$0.697783\pi$$
$$510$$ 0 0
$$511$$ 4578.00 0.396319
$$512$$ 0 0
$$513$$ − 1225.00i − 0.105429i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 8428.00i − 0.716950i
$$518$$ 0 0
$$519$$ 10584.0 0.895156
$$520$$ 0 0
$$521$$ 21637.0 1.81945 0.909726 0.415210i $$-0.136292\pi$$
0.909726 + 0.415210i $$0.136292\pi$$
$$522$$ 0 0
$$523$$ 287.000i 0.0239955i 0.999928 + 0.0119977i $$0.00381909\pi$$
−0.999928 + 0.0119977i $$0.996181\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 3822.00i − 0.315918i
$$528$$ 0 0
$$529$$ −14077.0 −1.15698
$$530$$ 0 0
$$531$$ 6160.00 0.503430
$$532$$ 0 0
$$533$$ − 5684.00i − 0.461916i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 18095.0i 1.45411i
$$538$$ 0 0
$$539$$ 13201.0 1.05493
$$540$$ 0 0
$$541$$ −5328.00 −0.423417 −0.211709 0.977333i $$-0.567903\pi$$
−0.211709 + 0.977333i $$0.567903\pi$$
$$542$$ 0 0
$$543$$ − 19306.0i − 1.52578i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 71.0000i − 0.00554980i −0.999996 0.00277490i $$-0.999117\pi$$
0.999996 0.00277490i $$-0.000883279\pi$$
$$548$$ 0 0
$$549$$ 11396.0 0.885919
$$550$$ 0 0
$$551$$ 5600.00 0.432973
$$552$$ 0 0
$$553$$ − 3060.00i − 0.235306i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ − 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
$$561$$ −27391.0 −2.06141
$$562$$ 0 0
$$563$$ 672.000i 0.0503045i 0.999684 + 0.0251522i $$0.00800705\pi$$
−0.999684 + 0.0251522i $$0.991993\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 5034.00i 0.372854i
$$568$$ 0 0
$$569$$ 10935.0 0.805657 0.402829 0.915275i $$-0.368027\pi$$
0.402829 + 0.915275i $$0.368027\pi$$
$$570$$ 0 0
$$571$$ 13588.0 0.995867 0.497934 0.867215i $$-0.334092\pi$$
0.497934 + 0.867215i $$0.334092\pi$$
$$572$$ 0 0
$$573$$ 16646.0i 1.21361i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 8701.00i 0.627777i 0.949460 + 0.313889i $$0.101632\pi$$
−0.949460 + 0.313889i $$0.898368\pi$$
$$578$$ 0 0
$$579$$ 21469.0 1.54097
$$580$$ 0 0
$$581$$ 4662.00 0.332896
$$582$$ 0 0
$$583$$ − 3526.00i − 0.250484i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ − 11361.0i − 0.798839i −0.916768 0.399420i $$-0.869212\pi$$
0.916768 0.399420i $$-0.130788\pi$$
$$588$$ 0 0
$$589$$ 1470.00 0.102836
$$590$$ 0 0
$$591$$ −16422.0 −1.14300
$$592$$ 0 0
$$593$$ − 11417.0i − 0.790624i −0.918547 0.395312i $$-0.870636\pi$$
0.918547 0.395312i $$-0.129364\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 34300.0i 2.35143i
$$598$$ 0 0
$$599$$ −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 $$0.418898\pi$$
$$602$$ 0 0
$$603$$ 3102.00i 0.209491i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 4144.00i 0.277100i 0.990355 + 0.138550i $$0.0442442\pi$$
−0.990355 + 0.138550i $$0.955756\pi$$
$$608$$ 0 0
$$609$$ −6720.00 −0.447140
$$610$$ 0 0
$$611$$ 5488.00 0.363373
$$612$$ 0 0
$$613$$ − 30122.0i − 1.98469i −0.123489 0.992346i $$-0.539408\pi$$
0.123489 0.992346i $$-0.460592\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 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 $$-0.410573\pi$$
0.277263 + 0.960794i $$0.410573\pi$$
$$620$$ 0 0
$$621$$ −5670.00 −0.366392
$$622$$ 0 0
$$623$$ − 5670.00i − 0.364629i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ − 10535.0i − 0.671017i
$$628$$ 0 0
$$629$$ 28574.0 1.81132
$$630$$ 0 0
$$631$$ 3158.00 0.199236 0.0996181 0.995026i $$-0.468238\pi$$
0.0996181 + 0.995026i $$0.468238\pi$$
$$632$$ 0 0
$$633$$ − 30149.0i − 1.89307i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 8596.00i 0.534672i
$$638$$ 0 0
$$639$$ 9064.00 0.561137
$$640$$ 0 0
$$641$$ −4278.00 −0.263605 −0.131803 0.991276i $$-0.542076\pi$$
−0.131803 + 0.991276i $$0.542076\pi$$
$$642$$ 0 0
$$643$$ − 11508.0i − 0.705803i −0.935661 0.352901i $$-0.885195\pi$$
0.935661 0.352901i $$-0.114805\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ 8204.00i 0.498505i 0.968439 + 0.249252i $$0.0801849\pi$$
−0.968439 + 0.249252i $$0.919815\pi$$
$$648$$ 0 0
$$649$$ −12040.0 −0.728215
$$650$$ 0 0
$$651$$ −1764.00 −0.106201
$$652$$ 0 0
$$653$$ 5518.00i 0.330683i 0.986236 + 0.165342i $$0.0528726\pi$$
−0.986236 + 0.165342i $$0.947127\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 16786.0i − 0.996780i
$$658$$ 0 0
$$659$$ 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 $$0.594745\pi$$
$$662$$ 0 0
$$663$$ − 17836.0i − 1.04479i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 25920.0i − 1.50469i
$$668$$ 0 0
$$669$$ −15484.0 −0.894837
$$670$$ 0 0
$$671$$ −22274.0 −1.28149
$$672$$ 0 0
$$673$$ 15738.0i 0.901419i 0.892671 + 0.450710i $$0.148829\pi$$
−0.892671 + 0.450710i $$0.851171\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ − 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
$$681$$ 3332.00 0.187493
$$682$$ 0 0
$$683$$ − 11073.0i − 0.620346i −0.950680 0.310173i $$-0.899613\pi$$
0.950680 0.310173i $$-0.100387\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 20580.0i 1.14291i
$$688$$ 0 0
$$689$$ 2296.00 0.126953
$$690$$ 0 0
$$691$$ 6503.00 0.358011 0.179006 0.983848i $$-0.442712\pi$$
0.179006 + 0.983848i $$0.442712\pi$$
$$692$$ 0 0
$$693$$ 5676.00i 0.311130i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 18473.0i − 1.00389i
$$698$$ 0 0
$$699$$ 7014.00 0.379533
$$700$$ 0 0
$$701$$ −10148.0 −0.546768 −0.273384 0.961905i $$-0.588143\pi$$
−0.273384 + 0.961905i $$0.588143\pi$$
$$702$$ 0 0
$$703$$ 10990.0i 0.589610i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ − 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
$$711$$ −11220.0 −0.591818
$$712$$ 0 0
$$713$$ − 6804.00i − 0.357380i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 17360.0i 0.904214i
$$718$$ 0 0
$$719$$ −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
$$722$$ 0 0
$$723$$ 13279.0i 0.683059i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 17024.0i 0.868480i 0.900797 + 0.434240i $$0.142983\pi$$
−0.900797 + 0.434240i $$0.857017\pi$$
$$728$$ 0 0
$$729$$ 11843.0 0.601687
$$730$$ 0 0
$$731$$ −8372.00 −0.423597
$$732$$ 0 0
$$733$$ 34748.0i 1.75095i 0.483263 + 0.875475i $$0.339451\pi$$
−0.483263 + 0.875475i $$0.660549\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ − 6063.00i − 0.303030i
$$738$$ 0 0
$$739$$ −12020.0 −0.598326 −0.299163 0.954202i $$-0.596707\pi$$
−0.299163 + 0.954202i $$0.596707\pi$$
$$740$$ 0 0
$$741$$ 6860.00 0.340092
$$742$$ 0 0
$$743$$ 28642.0i 1.41423i 0.707098 + 0.707115i $$0.250004\pi$$
−0.707098 + 0.707115i $$0.749996\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ − 17094.0i − 0.837265i
$$748$$ 0 0
$$749$$ 7614.00 0.371441
$$750$$ 0 0
$$751$$ −8752.00 −0.425253 −0.212627 0.977134i $$-0.568202\pi$$
−0.212627 + 0.977134i $$0.568202\pi$$
$$752$$ 0 0
$$753$$ 16611.0i 0.803902i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 10256.0i 0.492418i 0.969217 + 0.246209i $$0.0791850\pi$$
−0.969217 + 0.246209i $$0.920815\pi$$
$$758$$ 0 0
$$759$$ −48762.0 −2.33195
$$760$$ 0 0
$$761$$ 33957.0 1.61753 0.808765 0.588132i $$-0.200136\pi$$
0.808765 + 0.588132i $$0.200136\pi$$
$$762$$ 0 0
$$763$$ 6420.00i 0.304613i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ − 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
$$771$$ 31458.0 1.46943
$$772$$ 0 0
$$773$$ − 9912.00i − 0.461203i −0.973048 0.230601i $$-0.925931\pi$$
0.973048 0.230601i $$-0.0740694\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 13188.0i − 0.608902i
$$778$$ 0 0
$$779$$ 7105.00 0.326782
$$780$$ 0 0
$$781$$ −17716.0 −0.811688
$$782$$ 0 0
$$783$$ − 5600.00i − 0.255591i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 25564.0i 1.15789i 0.815367 + 0.578944i $$0.196535\pi$$
−0.815367 + 0.578944i $$0.803465\pi$$
$$788$$ 0 0
$$789$$ −5054.00 −0.228045
$$790$$ 0 0
$$791$$ 3018.00 0.135661
$$792$$ 0 0
$$793$$ − 14504.0i − 0.649498i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 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
$$801$$ −20790.0 −0.917077
$$802$$ 0 0
$$803$$ 32809.0i 1.44185i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 43120.0i 1.88091i
$$808$$ 0 0
$$809$$ −33970.0 −1.47629 −0.738147 0.674640i $$-0.764299\pi$$
−0.738147 + 0.674640i $$0.764299\pi$$
$$810$$ 0 0
$$811$$ −18732.0 −0.811060 −0.405530 0.914082i $$-0.632913\pi$$
−0.405530 + 0.914082i $$0.632913\pi$$
$$812$$ 0 0
$$813$$ 50666.0i 2.18565i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 3220.00i − 0.137887i
$$818$$ 0 0
$$819$$ −3696.00 −0.157691
$$820$$ 0 0
$$821$$ 6162.00 0.261943 0.130972 0.991386i $$-0.458190\pi$$
0.130972 + 0.991386i $$0.458190\pi$$
$$822$$ 0 0
$$823$$ − 25388.0i − 1.07530i −0.843169 0.537649i $$-0.819313\pi$$
0.843169 0.537649i $$-0.180687\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 25201.0i − 1.05964i −0.848109 0.529821i $$-0.822259\pi$$
0.848109 0.529821i $$-0.177741\pi$$
$$828$$ 0 0
$$829$$ 19740.0 0.827019 0.413509 0.910500i $$-0.364303\pi$$
0.413509 + 0.910500i $$0.364303\pi$$
$$830$$ 0 0
$$831$$ −12432.0 −0.518967
$$832$$ 0 0
$$833$$ 27937.0i 1.16202i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 1470.00i − 0.0607057i
$$838$$ 0 0
$$839$$ 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
$$842$$ 0 0
$$843$$ 31794.0i 1.29898i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 3108.00i − 0.126083i
$$848$$ 0 0
$$849$$ −49539.0 −2.00256
$$850$$ 0 0
$$851$$ 50868.0 2.04904
$$852$$ 0 0
$$853$$ 1218.00i 0.0488904i 0.999701 + 0.0244452i $$0.00778193\pi$$
−0.999701 + 0.0244452i $$0.992218\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 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 $$-0.654954\pi$$
−0.467803 + 0.883833i $$0.654954\pi$$
$$860$$ 0 0
$$861$$ −8526.00 −0.337474
$$862$$ 0 0
$$863$$ 24872.0i 0.981058i 0.871425 + 0.490529i $$0.163196\pi$$
−0.871425 + 0.490529i $$0.836804\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 23576.0i − 0.923510i
$$868$$ 0 0
$$869$$ 21930.0 0.856069
$$870$$ 0 0
$$871$$ 3948.00 0.153585
$$872$$ 0 0
$$873$$ − 27412.0i − 1.06272i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 17124.0i − 0.659335i −0.944097 0.329667i $$-0.893063\pi$$
0.944097 0.329667i $$-0.106937\pi$$
$$878$$ 0 0
$$879$$ −29106.0 −1.11686
$$880$$ 0 0
$$881$$ −658.000 −0.0251630 −0.0125815 0.999921i $$-0.504005\pi$$
−0.0125815 + 0.999921i $$0.504005\pi$$
$$882$$ 0 0
$$883$$ 33727.0i 1.28540i 0.766120 + 0.642698i $$0.222185\pi$$
−0.766120 + 0.642698i $$0.777815\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 36036.0i − 1.36412i −0.731298 0.682058i $$-0.761085\pi$$
0.731298 0.682058i $$-0.238915\pi$$
$$888$$ 0 0
$$889$$ 5244.00 0.197838
$$890$$ 0 0
$$891$$ −36077.0 −1.35648
$$892$$ 0 0
$$893$$ 6860.00i 0.257067i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 31752.0i − 1.18190i
$$898$$ 0 0
$$899$$ 6720.00 0.249304
$$900$$ 0 0
$$901$$ 7462.00 0.275910
$$902$$ 0 0
$$903$$ 3864.00i 0.142399i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 39156.0i − 1.43347i −0.697348 0.716733i $$-0.745637\pi$$
0.697348 0.716733i $$-0.254363\pi$$
$$908$$ 0 0
$$909$$ −28644.0 −1.04517
$$910$$ 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
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 6552.00i 0.235950i
$$918$$ 0 0
$$919$$ −28610.0 −1.02694 −0.513469 0.858108i $$-0.671640\pi$$
−0.513469 + 0.858108i $$0.671640\pi$$
$$920$$ 0 0
$$921$$ −17983.0 −0.643388
$$922$$ 0 0
$$923$$ − 11536.0i − 0.411389i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 11704.0i − 0.414682i
$$928$$ 0 0
$$929$$ 24290.0 0.857835 0.428918 0.903344i $$-0.358895\pi$$
0.428918 + 0.903344i $$0.358895\pi$$
$$930$$ 0 0
$$931$$ −10745.0 −0.378253
$$932$$ 0 0
$$933$$ − 20874.0i − 0.732459i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 34461.0i 1.20149i 0.799442 + 0.600743i $$0.205128\pi$$
−0.799442 + 0.600743i $$0.794872\pi$$
$$938$$ 0 0
$$939$$ 16954.0 0.589215
$$940$$ 0 0
$$941$$ −40628.0 −1.40748 −0.703738 0.710460i $$-0.748487\pi$$
−0.703738 + 0.710460i $$0.748487\pi$$
$$942$$ 0 0
$$943$$ − 32886.0i − 1.13565i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 20904.0i 0.717306i 0.933471 + 0.358653i $$0.116764\pi$$
−0.933471 + 0.358653i $$0.883236\pi$$
$$948$$ 0 0
$$949$$ −21364.0 −0.730774
$$950$$ 0 0
$$951$$ 66388.0 2.26370
$$952$$ 0 0
$$953$$ − 1807.00i − 0.0614213i −0.999528 0.0307106i $$-0.990223\pi$$
0.999528 0.0307106i $$-0.00977704\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ − 48160.0i − 1.62674i
$$958$$ 0 0
$$959$$ 2466.00 0.0830358
$$960$$ 0 0
$$961$$ −28027.0 −0.940787
$$962$$ 0 0
$$963$$ − 27918.0i − 0.934211i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 57584.0i 1.91497i 0.288482 + 0.957485i $$0.406849\pi$$
−0.288482 + 0.957485i $$0.593151\pi$$
$$968$$ 0 0
$$969$$ 22295.0 0.739132
$$970$$ 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
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 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
$$981$$ 23540.0 0.766131
$$982$$ 0 0
$$983$$ 16002.0i 0.519211i 0.965715 + 0.259606i $$0.0835925\pi$$
−0.965715 + 0.259606i $$0.916407\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ − 8232.00i − 0.265479i
$$988$$ 0 0
$$989$$ −14904.0 −0.479191
$$990$$ 0 0
$$991$$ −37022.0 −1.18672 −0.593362 0.804936i $$-0.702200\pi$$
−0.593362 + 0.804936i $$0.702200\pi$$
$$992$$ 0 0
$$993$$ 1281.00i 0.0409379i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 18396.0i 0.584360i 0.956363 + 0.292180i $$0.0943807\pi$$
−0.956363 + 0.292180i $$0.905619\pi$$
$$998$$ 0 0
$$999$$ 10990.0 0.348056
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.e.49.2 2
4.3 odd 2 25.4.b.b.24.2 2
5.2 odd 4 400.4.a.s.1.1 1
5.3 odd 4 400.4.a.c.1.1 1
5.4 even 2 inner 400.4.c.e.49.1 2
12.11 even 2 225.4.b.f.199.1 2
20.3 even 4 25.4.a.b.1.1 yes 1
20.7 even 4 25.4.a.a.1.1 1
20.19 odd 2 25.4.b.b.24.1 2
40.3 even 4 1600.4.a.i.1.1 1
40.13 odd 4 1600.4.a.bs.1.1 1
40.27 even 4 1600.4.a.bt.1.1 1
40.37 odd 4 1600.4.a.h.1.1 1
60.23 odd 4 225.4.a.c.1.1 1
60.47 odd 4 225.4.a.e.1.1 1
60.59 even 2 225.4.b.f.199.2 2
140.27 odd 4 1225.4.a.h.1.1 1
140.83 odd 4 1225.4.a.i.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
25.4.a.a.1.1 1 20.7 even 4
25.4.a.b.1.1 yes 1 20.3 even 4
25.4.b.b.24.1 2 20.19 odd 2
25.4.b.b.24.2 2 4.3 odd 2
225.4.a.c.1.1 1 60.23 odd 4
225.4.a.e.1.1 1 60.47 odd 4
225.4.b.f.199.1 2 12.11 even 2
225.4.b.f.199.2 2 60.59 even 2
400.4.a.c.1.1 1 5.3 odd 4
400.4.a.s.1.1 1 5.2 odd 4
400.4.c.e.49.1 2 5.4 even 2 inner
400.4.c.e.49.2 2 1.1 even 1 trivial
1225.4.a.h.1.1 1 140.27 odd 4
1225.4.a.i.1.1 1 140.83 odd 4
1600.4.a.h.1.1 1 40.37 odd 4
1600.4.a.i.1.1 1 40.3 even 4
1600.4.a.bs.1.1 1 40.13 odd 4
1600.4.a.bt.1.1 1 40.27 even 4