# Properties

 Label 400.4.c.l.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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 100) 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.l.49.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+1.00000i q^{3} -26.0000i q^{7} +26.0000 q^{9} +O(q^{10})$$ $$q+1.00000i q^{3} -26.0000i q^{7} +26.0000 q^{9} -45.0000 q^{11} -44.0000i q^{13} +117.000i q^{17} -91.0000 q^{19} +26.0000 q^{21} -18.0000i q^{23} +53.0000i q^{27} -144.000 q^{29} -26.0000 q^{31} -45.0000i q^{33} -214.000i q^{37} +44.0000 q^{39} -459.000 q^{41} -460.000i q^{43} +468.000i q^{47} -333.000 q^{49} -117.000 q^{51} -558.000i q^{53} -91.0000i q^{57} -72.0000 q^{59} -118.000 q^{61} -676.000i q^{63} -251.000i q^{67} +18.0000 q^{69} -108.000 q^{71} -299.000i q^{73} +1170.00i q^{77} -898.000 q^{79} +649.000 q^{81} +927.000i q^{83} -144.000i q^{87} -351.000 q^{89} -1144.00 q^{91} -26.0000i q^{93} +386.000i q^{97} -1170.00 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 52 q^{9}+O(q^{10})$$ 2 * q + 52 * q^9 $$2 q + 52 q^{9} - 90 q^{11} - 182 q^{19} + 52 q^{21} - 288 q^{29} - 52 q^{31} + 88 q^{39} - 918 q^{41} - 666 q^{49} - 234 q^{51} - 144 q^{59} - 236 q^{61} + 36 q^{69} - 216 q^{71} - 1796 q^{79} + 1298 q^{81} - 702 q^{89} - 2288 q^{91} - 2340 q^{99}+O(q^{100})$$ 2 * q + 52 * q^9 - 90 * q^11 - 182 * q^19 + 52 * q^21 - 288 * q^29 - 52 * q^31 + 88 * q^39 - 918 * q^41 - 666 * q^49 - 234 * q^51 - 144 * q^59 - 236 * q^61 + 36 * q^69 - 216 * q^71 - 1796 * q^79 + 1298 * q^81 - 702 * q^89 - 2288 * q^91 - 2340 * 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
$$3$$ 1.00000i 0.192450i 0.995360 + 0.0962250i $$0.0306768\pi$$
−0.995360 + 0.0962250i $$0.969323\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 26.0000i − 1.40387i −0.712242 0.701934i $$-0.752320\pi$$
0.712242 0.701934i $$-0.247680\pi$$
$$8$$ 0 0
$$9$$ 26.0000 0.962963
$$10$$ 0 0
$$11$$ −45.0000 −1.23346 −0.616728 0.787177i $$-0.711542\pi$$
−0.616728 + 0.787177i $$0.711542\pi$$
$$12$$ 0 0
$$13$$ − 44.0000i − 0.938723i −0.883006 0.469362i $$-0.844484\pi$$
0.883006 0.469362i $$-0.155516\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 117.000i 1.66922i 0.550845 + 0.834608i $$0.314306\pi$$
−0.550845 + 0.834608i $$0.685694\pi$$
$$18$$ 0 0
$$19$$ −91.0000 −1.09878 −0.549390 0.835566i $$-0.685140\pi$$
−0.549390 + 0.835566i $$0.685140\pi$$
$$20$$ 0 0
$$21$$ 26.0000 0.270175
$$22$$ 0 0
$$23$$ − 18.0000i − 0.163185i −0.996666 0.0815926i $$-0.973999\pi$$
0.996666 0.0815926i $$-0.0260006\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 53.0000i 0.377772i
$$28$$ 0 0
$$29$$ −144.000 −0.922073 −0.461037 0.887381i $$-0.652522\pi$$
−0.461037 + 0.887381i $$0.652522\pi$$
$$30$$ 0 0
$$31$$ −26.0000 −0.150637 −0.0753184 0.997160i $$-0.523997\pi$$
−0.0753184 + 0.997160i $$0.523997\pi$$
$$32$$ 0 0
$$33$$ − 45.0000i − 0.237379i
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 214.000i − 0.950848i −0.879757 0.475424i $$-0.842295\pi$$
0.879757 0.475424i $$-0.157705\pi$$
$$38$$ 0 0
$$39$$ 44.0000 0.180657
$$40$$ 0 0
$$41$$ −459.000 −1.74838 −0.874192 0.485580i $$-0.838608\pi$$
−0.874192 + 0.485580i $$0.838608\pi$$
$$42$$ 0 0
$$43$$ − 460.000i − 1.63138i −0.578489 0.815690i $$-0.696358\pi$$
0.578489 0.815690i $$-0.303642\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 468.000i 1.45244i 0.687461 + 0.726221i $$0.258725\pi$$
−0.687461 + 0.726221i $$0.741275\pi$$
$$48$$ 0 0
$$49$$ −333.000 −0.970845
$$50$$ 0 0
$$51$$ −117.000 −0.321241
$$52$$ 0 0
$$53$$ − 558.000i − 1.44617i −0.690757 0.723087i $$-0.742723\pi$$
0.690757 0.723087i $$-0.257277\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ − 91.0000i − 0.211460i
$$58$$ 0 0
$$59$$ −72.0000 −0.158875 −0.0794373 0.996840i $$-0.525312\pi$$
−0.0794373 + 0.996840i $$0.525312\pi$$
$$60$$ 0 0
$$61$$ −118.000 −0.247678 −0.123839 0.992302i $$-0.539521\pi$$
−0.123839 + 0.992302i $$0.539521\pi$$
$$62$$ 0 0
$$63$$ − 676.000i − 1.35187i
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 251.000i − 0.457680i −0.973464 0.228840i $$-0.926507\pi$$
0.973464 0.228840i $$-0.0734932\pi$$
$$68$$ 0 0
$$69$$ 18.0000 0.0314050
$$70$$ 0 0
$$71$$ −108.000 −0.180525 −0.0902623 0.995918i $$-0.528771\pi$$
−0.0902623 + 0.995918i $$0.528771\pi$$
$$72$$ 0 0
$$73$$ − 299.000i − 0.479388i −0.970849 0.239694i $$-0.922953\pi$$
0.970849 0.239694i $$-0.0770471\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1170.00i 1.73161i
$$78$$ 0 0
$$79$$ −898.000 −1.27890 −0.639449 0.768834i $$-0.720837\pi$$
−0.639449 + 0.768834i $$0.720837\pi$$
$$80$$ 0 0
$$81$$ 649.000 0.890261
$$82$$ 0 0
$$83$$ 927.000i 1.22592i 0.790113 + 0.612961i $$0.210022\pi$$
−0.790113 + 0.612961i $$0.789978\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ − 144.000i − 0.177453i
$$88$$ 0 0
$$89$$ −351.000 −0.418044 −0.209022 0.977911i $$-0.567028\pi$$
−0.209022 + 0.977911i $$0.567028\pi$$
$$90$$ 0 0
$$91$$ −1144.00 −1.31784
$$92$$ 0 0
$$93$$ − 26.0000i − 0.0289900i
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 386.000i 0.404045i 0.979381 + 0.202022i $$0.0647514\pi$$
−0.979381 + 0.202022i $$0.935249\pi$$
$$98$$ 0 0
$$99$$ −1170.00 −1.18777
$$100$$ 0 0
$$101$$ −954.000 −0.939867 −0.469933 0.882702i $$-0.655722\pi$$
−0.469933 + 0.882702i $$0.655722\pi$$
$$102$$ 0 0
$$103$$ − 772.000i − 0.738519i −0.929326 0.369259i $$-0.879611\pi$$
0.929326 0.369259i $$-0.120389\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ − 1197.00i − 1.08148i −0.841190 0.540740i $$-0.818144\pi$$
0.841190 0.540740i $$-0.181856\pi$$
$$108$$ 0 0
$$109$$ 802.000 0.704749 0.352375 0.935859i $$-0.385374\pi$$
0.352375 + 0.935859i $$0.385374\pi$$
$$110$$ 0 0
$$111$$ 214.000 0.182991
$$112$$ 0 0
$$113$$ − 1143.00i − 0.951543i −0.879569 0.475772i $$-0.842169\pi$$
0.879569 0.475772i $$-0.157831\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ − 1144.00i − 0.903956i
$$118$$ 0 0
$$119$$ 3042.00 2.34336
$$120$$ 0 0
$$121$$ 694.000 0.521412
$$122$$ 0 0
$$123$$ − 459.000i − 0.336477i
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 2374.00i 1.65873i 0.558709 + 0.829364i $$0.311297\pi$$
−0.558709 + 0.829364i $$0.688703\pi$$
$$128$$ 0 0
$$129$$ 460.000 0.313959
$$130$$ 0 0
$$131$$ 1260.00 0.840357 0.420178 0.907442i $$-0.361968\pi$$
0.420178 + 0.907442i $$0.361968\pi$$
$$132$$ 0 0
$$133$$ 2366.00i 1.54254i
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 891.000i − 0.555644i −0.960633 0.277822i $$-0.910387\pi$$
0.960633 0.277822i $$-0.0896126\pi$$
$$138$$ 0 0
$$139$$ 389.000 0.237371 0.118685 0.992932i $$-0.462132\pi$$
0.118685 + 0.992932i $$0.462132\pi$$
$$140$$ 0 0
$$141$$ −468.000 −0.279523
$$142$$ 0 0
$$143$$ 1980.00i 1.15787i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ − 333.000i − 0.186839i
$$148$$ 0 0
$$149$$ −1296.00 −0.712567 −0.356283 0.934378i $$-0.615956\pi$$
−0.356283 + 0.934378i $$0.615956\pi$$
$$150$$ 0 0
$$151$$ 2710.00 1.46051 0.730254 0.683176i $$-0.239402\pi$$
0.730254 + 0.683176i $$0.239402\pi$$
$$152$$ 0 0
$$153$$ 3042.00i 1.60739i
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 1846.00i − 0.938388i −0.883095 0.469194i $$-0.844545\pi$$
0.883095 0.469194i $$-0.155455\pi$$
$$158$$ 0 0
$$159$$ 558.000 0.278316
$$160$$ 0 0
$$161$$ −468.000 −0.229090
$$162$$ 0 0
$$163$$ 1475.00i 0.708779i 0.935098 + 0.354389i $$0.115311\pi$$
−0.935098 + 0.354389i $$0.884689\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 1476.00i 0.683930i 0.939713 + 0.341965i $$0.111092\pi$$
−0.939713 + 0.341965i $$0.888908\pi$$
$$168$$ 0 0
$$169$$ 261.000 0.118798
$$170$$ 0 0
$$171$$ −2366.00 −1.05809
$$172$$ 0 0
$$173$$ − 1368.00i − 0.601197i −0.953751 0.300599i $$-0.902814\pi$$
0.953751 0.300599i $$-0.0971864\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ − 72.0000i − 0.0305754i
$$178$$ 0 0
$$179$$ −1503.00 −0.627595 −0.313797 0.949490i $$-0.601601\pi$$
−0.313797 + 0.949490i $$0.601601\pi$$
$$180$$ 0 0
$$181$$ 3770.00 1.54819 0.774094 0.633071i $$-0.218206\pi$$
0.774094 + 0.633071i $$0.218206\pi$$
$$182$$ 0 0
$$183$$ − 118.000i − 0.0476656i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 5265.00i − 2.05890i
$$188$$ 0 0
$$189$$ 1378.00 0.530343
$$190$$ 0 0
$$191$$ 4122.00 1.56156 0.780779 0.624808i $$-0.214823\pi$$
0.780779 + 0.624808i $$0.214823\pi$$
$$192$$ 0 0
$$193$$ 1963.00i 0.732123i 0.930591 + 0.366062i $$0.119294\pi$$
−0.930591 + 0.366062i $$0.880706\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 2934.00i 1.06111i 0.847650 + 0.530555i $$0.178017\pi$$
−0.847650 + 0.530555i $$0.821983\pi$$
$$198$$ 0 0
$$199$$ 1412.00 0.502985 0.251493 0.967859i $$-0.419079\pi$$
0.251493 + 0.967859i $$0.419079\pi$$
$$200$$ 0 0
$$201$$ 251.000 0.0880805
$$202$$ 0 0
$$203$$ 3744.00i 1.29447i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ − 468.000i − 0.157141i
$$208$$ 0 0
$$209$$ 4095.00 1.35530
$$210$$ 0 0
$$211$$ −3419.00 −1.11552 −0.557758 0.830004i $$-0.688338\pi$$
−0.557758 + 0.830004i $$0.688338\pi$$
$$212$$ 0 0
$$213$$ − 108.000i − 0.0347420i
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 676.000i 0.211474i
$$218$$ 0 0
$$219$$ 299.000 0.0922582
$$220$$ 0 0
$$221$$ 5148.00 1.56693
$$222$$ 0 0
$$223$$ − 100.000i − 0.0300291i −0.999887 0.0150146i $$-0.995221\pi$$
0.999887 0.0150146i $$-0.00477946\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ − 4212.00i − 1.23154i −0.787925 0.615771i $$-0.788844\pi$$
0.787925 0.615771i $$-0.211156\pi$$
$$228$$ 0 0
$$229$$ 3484.00 1.00537 0.502684 0.864470i $$-0.332346\pi$$
0.502684 + 0.864470i $$0.332346\pi$$
$$230$$ 0 0
$$231$$ −1170.00 −0.333248
$$232$$ 0 0
$$233$$ − 918.000i − 0.258112i −0.991637 0.129056i $$-0.958805\pi$$
0.991637 0.129056i $$-0.0411948\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ − 898.000i − 0.246124i
$$238$$ 0 0
$$239$$ 3744.00 1.01330 0.506651 0.862151i $$-0.330883\pi$$
0.506651 + 0.862151i $$0.330883\pi$$
$$240$$ 0 0
$$241$$ −4231.00 −1.13088 −0.565441 0.824789i $$-0.691294\pi$$
−0.565441 + 0.824789i $$0.691294\pi$$
$$242$$ 0 0
$$243$$ 2080.00i 0.549103i
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 4004.00i 1.03145i
$$248$$ 0 0
$$249$$ −927.000 −0.235929
$$250$$ 0 0
$$251$$ 2925.00 0.735555 0.367778 0.929914i $$-0.380119\pi$$
0.367778 + 0.929914i $$0.380119\pi$$
$$252$$ 0 0
$$253$$ 810.000i 0.201282i
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ − 18.0000i − 0.00436891i −0.999998 0.00218445i $$-0.999305\pi$$
0.999998 0.00218445i $$-0.000695334\pi$$
$$258$$ 0 0
$$259$$ −5564.00 −1.33487
$$260$$ 0 0
$$261$$ −3744.00 −0.887923
$$262$$ 0 0
$$263$$ − 6786.00i − 1.59104i −0.605929 0.795518i $$-0.707199\pi$$
0.605929 0.795518i $$-0.292801\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ − 351.000i − 0.0804526i
$$268$$ 0 0
$$269$$ −7632.00 −1.72986 −0.864928 0.501896i $$-0.832636\pi$$
−0.864928 + 0.501896i $$0.832636\pi$$
$$270$$ 0 0
$$271$$ −650.000 −0.145700 −0.0728500 0.997343i $$-0.523209\pi$$
−0.0728500 + 0.997343i $$0.523209\pi$$
$$272$$ 0 0
$$273$$ − 1144.00i − 0.253619i
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ − 3232.00i − 0.701054i −0.936553 0.350527i $$-0.886002\pi$$
0.936553 0.350527i $$-0.113998\pi$$
$$278$$ 0 0
$$279$$ −676.000 −0.145058
$$280$$ 0 0
$$281$$ 4446.00 0.943865 0.471933 0.881635i $$-0.343557\pi$$
0.471933 + 0.881635i $$0.343557\pi$$
$$282$$ 0 0
$$283$$ 2483.00i 0.521551i 0.965399 + 0.260776i $$0.0839783\pi$$
−0.965399 + 0.260776i $$0.916022\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ 11934.0i 2.45450i
$$288$$ 0 0
$$289$$ −8776.00 −1.78628
$$290$$ 0 0
$$291$$ −386.000 −0.0777585
$$292$$ 0 0
$$293$$ 4050.00i 0.807521i 0.914865 + 0.403760i $$0.132297\pi$$
−0.914865 + 0.403760i $$0.867703\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ − 2385.00i − 0.465965i
$$298$$ 0 0
$$299$$ −792.000 −0.153186
$$300$$ 0 0
$$301$$ −11960.0 −2.29024
$$302$$ 0 0
$$303$$ − 954.000i − 0.180877i
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ − 2321.00i − 0.431487i −0.976450 0.215743i $$-0.930783\pi$$
0.976450 0.215743i $$-0.0692175\pi$$
$$308$$ 0 0
$$309$$ 772.000 0.142128
$$310$$ 0 0
$$311$$ 3258.00 0.594033 0.297016 0.954872i $$-0.404008\pi$$
0.297016 + 0.954872i $$0.404008\pi$$
$$312$$ 0 0
$$313$$ − 3626.00i − 0.654804i −0.944885 0.327402i $$-0.893827\pi$$
0.944885 0.327402i $$-0.106173\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 3852.00i 0.682492i 0.939974 + 0.341246i $$0.110849\pi$$
−0.939974 + 0.341246i $$0.889151\pi$$
$$318$$ 0 0
$$319$$ 6480.00 1.13734
$$320$$ 0 0
$$321$$ 1197.00 0.208131
$$322$$ 0 0
$$323$$ − 10647.0i − 1.83410i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 802.000i 0.135629i
$$328$$ 0 0
$$329$$ 12168.0 2.03904
$$330$$ 0 0
$$331$$ −7553.00 −1.25423 −0.627115 0.778926i $$-0.715765\pi$$
−0.627115 + 0.778926i $$0.715765\pi$$
$$332$$ 0 0
$$333$$ − 5564.00i − 0.915632i
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ − 109.000i − 0.0176190i −0.999961 0.00880951i $$-0.997196\pi$$
0.999961 0.00880951i $$-0.00280419\pi$$
$$338$$ 0 0
$$339$$ 1143.00 0.183125
$$340$$ 0 0
$$341$$ 1170.00 0.185804
$$342$$ 0 0
$$343$$ − 260.000i − 0.0409291i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 2835.00i 0.438590i 0.975659 + 0.219295i $$0.0703757\pi$$
−0.975659 + 0.219295i $$0.929624\pi$$
$$348$$ 0 0
$$349$$ −2990.00 −0.458599 −0.229299 0.973356i $$-0.573644\pi$$
−0.229299 + 0.973356i $$0.573644\pi$$
$$350$$ 0 0
$$351$$ 2332.00 0.354624
$$352$$ 0 0
$$353$$ − 9126.00i − 1.37600i −0.725711 0.688000i $$-0.758489\pi$$
0.725711 0.688000i $$-0.241511\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 3042.00i 0.450980i
$$358$$ 0 0
$$359$$ −9594.00 −1.41045 −0.705226 0.708983i $$-0.749154\pi$$
−0.705226 + 0.708983i $$0.749154\pi$$
$$360$$ 0 0
$$361$$ 1422.00 0.207319
$$362$$ 0 0
$$363$$ 694.000i 0.100346i
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 9764.00i − 1.38876i −0.719606 0.694382i $$-0.755678\pi$$
0.719606 0.694382i $$-0.244322\pi$$
$$368$$ 0 0
$$369$$ −11934.0 −1.68363
$$370$$ 0 0
$$371$$ −14508.0 −2.03024
$$372$$ 0 0
$$373$$ − 6722.00i − 0.933115i −0.884491 0.466558i $$-0.845494\pi$$
0.884491 0.466558i $$-0.154506\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6336.00i 0.865572i
$$378$$ 0 0
$$379$$ −13537.0 −1.83469 −0.917347 0.398089i $$-0.869674\pi$$
−0.917347 + 0.398089i $$0.869674\pi$$
$$380$$ 0 0
$$381$$ −2374.00 −0.319222
$$382$$ 0 0
$$383$$ − 8658.00i − 1.15510i −0.816355 0.577550i $$-0.804009\pi$$
0.816355 0.577550i $$-0.195991\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ − 11960.0i − 1.57096i
$$388$$ 0 0
$$389$$ 8874.00 1.15663 0.578316 0.815813i $$-0.303710\pi$$
0.578316 + 0.815813i $$0.303710\pi$$
$$390$$ 0 0
$$391$$ 2106.00 0.272391
$$392$$ 0 0
$$393$$ 1260.00i 0.161727i
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 5876.00i 0.742841i 0.928465 + 0.371421i $$0.121129\pi$$
−0.928465 + 0.371421i $$0.878871\pi$$
$$398$$ 0 0
$$399$$ −2366.00 −0.296863
$$400$$ 0 0
$$401$$ −1755.00 −0.218555 −0.109277 0.994011i $$-0.534854\pi$$
−0.109277 + 0.994011i $$0.534854\pi$$
$$402$$ 0 0
$$403$$ 1144.00i 0.141406i
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 9630.00i 1.17283i
$$408$$ 0 0
$$409$$ −4589.00 −0.554796 −0.277398 0.960755i $$-0.589472\pi$$
−0.277398 + 0.960755i $$0.589472\pi$$
$$410$$ 0 0
$$411$$ 891.000 0.106934
$$412$$ 0 0
$$413$$ 1872.00i 0.223039i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 389.000i 0.0456820i
$$418$$ 0 0
$$419$$ 5409.00 0.630661 0.315330 0.948982i $$-0.397885\pi$$
0.315330 + 0.948982i $$0.397885\pi$$
$$420$$ 0 0
$$421$$ 12116.0 1.40261 0.701304 0.712863i $$-0.252602\pi$$
0.701304 + 0.712863i $$0.252602\pi$$
$$422$$ 0 0
$$423$$ 12168.0i 1.39865i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 3068.00i 0.347707i
$$428$$ 0 0
$$429$$ −1980.00 −0.222833
$$430$$ 0 0
$$431$$ −9126.00 −1.01992 −0.509958 0.860199i $$-0.670339\pi$$
−0.509958 + 0.860199i $$0.670339\pi$$
$$432$$ 0 0
$$433$$ − 629.000i − 0.0698102i −0.999391 0.0349051i $$-0.988887\pi$$
0.999391 0.0349051i $$-0.0111129\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ 1638.00i 0.179305i
$$438$$ 0 0
$$439$$ 4472.00 0.486189 0.243094 0.970003i $$-0.421838\pi$$
0.243094 + 0.970003i $$0.421838\pi$$
$$440$$ 0 0
$$441$$ −8658.00 −0.934888
$$442$$ 0 0
$$443$$ − 3393.00i − 0.363897i −0.983308 0.181948i $$-0.941760\pi$$
0.983308 0.181948i $$-0.0582404\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ − 1296.00i − 0.137134i
$$448$$ 0 0
$$449$$ 5031.00 0.528792 0.264396 0.964414i $$-0.414827\pi$$
0.264396 + 0.964414i $$0.414827\pi$$
$$450$$ 0 0
$$451$$ 20655.0 2.15655
$$452$$ 0 0
$$453$$ 2710.00i 0.281075i
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 6487.00i − 0.664002i −0.943279 0.332001i $$-0.892276\pi$$
0.943279 0.332001i $$-0.107724\pi$$
$$458$$ 0 0
$$459$$ −6201.00 −0.630584
$$460$$ 0 0
$$461$$ 2700.00 0.272780 0.136390 0.990655i $$-0.456450\pi$$
0.136390 + 0.990655i $$0.456450\pi$$
$$462$$ 0 0
$$463$$ − 2932.00i − 0.294302i −0.989114 0.147151i $$-0.952990\pi$$
0.989114 0.147151i $$-0.0470102\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 15660.0i 1.55173i 0.630898 + 0.775866i $$0.282686\pi$$
−0.630898 + 0.775866i $$0.717314\pi$$
$$468$$ 0 0
$$469$$ −6526.00 −0.642522
$$470$$ 0 0
$$471$$ 1846.00 0.180593
$$472$$ 0 0
$$473$$ 20700.0i 2.01223i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ − 14508.0i − 1.39261i
$$478$$ 0 0
$$479$$ 10134.0 0.966669 0.483334 0.875436i $$-0.339426\pi$$
0.483334 + 0.875436i $$0.339426\pi$$
$$480$$ 0 0
$$481$$ −9416.00 −0.892583
$$482$$ 0 0
$$483$$ − 468.000i − 0.0440885i
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 1898.00i − 0.176605i −0.996094 0.0883025i $$-0.971856\pi$$
0.996094 0.0883025i $$-0.0281442\pi$$
$$488$$ 0 0
$$489$$ −1475.00 −0.136405
$$490$$ 0 0
$$491$$ 6300.00 0.579053 0.289526 0.957170i $$-0.406502\pi$$
0.289526 + 0.957170i $$0.406502\pi$$
$$492$$ 0 0
$$493$$ − 16848.0i − 1.53914i
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 2808.00i 0.253433i
$$498$$ 0 0
$$499$$ 18044.0 1.61876 0.809379 0.587286i $$-0.199804\pi$$
0.809379 + 0.587286i $$0.199804\pi$$
$$500$$ 0 0
$$501$$ −1476.00 −0.131622
$$502$$ 0 0
$$503$$ − 6876.00i − 0.609514i −0.952430 0.304757i $$-0.901425\pi$$
0.952430 0.304757i $$-0.0985753\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ 261.000i 0.0228628i
$$508$$ 0 0
$$509$$ 4806.00 0.418511 0.209256 0.977861i $$-0.432896\pi$$
0.209256 + 0.977861i $$0.432896\pi$$
$$510$$ 0 0
$$511$$ −7774.00 −0.672997
$$512$$ 0 0
$$513$$ − 4823.00i − 0.415089i
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ − 21060.0i − 1.79152i
$$518$$ 0 0
$$519$$ 1368.00 0.115700
$$520$$ 0 0
$$521$$ 7749.00 0.651612 0.325806 0.945437i $$-0.394364\pi$$
0.325806 + 0.945437i $$0.394364\pi$$
$$522$$ 0 0
$$523$$ 8153.00i 0.681655i 0.940126 + 0.340828i $$0.110707\pi$$
−0.940126 + 0.340828i $$0.889293\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ − 3042.00i − 0.251445i
$$528$$ 0 0
$$529$$ 11843.0 0.973371
$$530$$ 0 0
$$531$$ −1872.00 −0.152990
$$532$$ 0 0
$$533$$ 20196.0i 1.64125i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ − 1503.00i − 0.120781i
$$538$$ 0 0
$$539$$ 14985.0 1.19749
$$540$$ 0 0
$$541$$ −10576.0 −0.840476 −0.420238 0.907414i $$-0.638053\pi$$
−0.420238 + 0.907414i $$0.638053\pi$$
$$542$$ 0 0
$$543$$ 3770.00i 0.297949i
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 7553.00i − 0.590389i −0.955437 0.295195i $$-0.904615\pi$$
0.955437 0.295195i $$-0.0953845\pi$$
$$548$$ 0 0
$$549$$ −3068.00 −0.238505
$$550$$ 0 0
$$551$$ 13104.0 1.01316
$$552$$ 0 0
$$553$$ 23348.0i 1.79540i
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 13500.0i 1.02695i 0.858103 + 0.513477i $$0.171643\pi$$
−0.858103 + 0.513477i $$0.828357\pi$$
$$558$$ 0 0
$$559$$ −20240.0 −1.53141
$$560$$ 0 0
$$561$$ 5265.00 0.396236
$$562$$ 0 0
$$563$$ 23184.0i 1.73550i 0.496997 + 0.867752i $$0.334436\pi$$
−0.496997 + 0.867752i $$0.665564\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ − 16874.0i − 1.24981i
$$568$$ 0 0
$$569$$ 8055.00 0.593468 0.296734 0.954960i $$-0.404103\pi$$
0.296734 + 0.954960i $$0.404103\pi$$
$$570$$ 0 0
$$571$$ −3068.00 −0.224854 −0.112427 0.993660i $$-0.535862\pi$$
−0.112427 + 0.993660i $$0.535862\pi$$
$$572$$ 0 0
$$573$$ 4122.00i 0.300522i
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 12419.0i 0.896031i 0.894026 + 0.448015i $$0.147869\pi$$
−0.894026 + 0.448015i $$0.852131\pi$$
$$578$$ 0 0
$$579$$ −1963.00 −0.140897
$$580$$ 0 0
$$581$$ 24102.0 1.72103
$$582$$ 0 0
$$583$$ 25110.0i 1.78379i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 12393.0i 0.871403i 0.900091 + 0.435702i $$0.143500\pi$$
−0.900091 + 0.435702i $$0.856500\pi$$
$$588$$ 0 0
$$589$$ 2366.00 0.165517
$$590$$ 0 0
$$591$$ −2934.00 −0.204211
$$592$$ 0 0
$$593$$ − 23751.0i − 1.64475i −0.568946 0.822375i $$-0.692649\pi$$
0.568946 0.822375i $$-0.307351\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 1412.00i 0.0967995i
$$598$$ 0 0
$$599$$ −11610.0 −0.791939 −0.395970 0.918264i $$-0.629591\pi$$
−0.395970 + 0.918264i $$0.629591\pi$$
$$600$$ 0 0
$$601$$ 26675.0 1.81048 0.905238 0.424905i $$-0.139693\pi$$
0.905238 + 0.424905i $$0.139693\pi$$
$$602$$ 0 0
$$603$$ − 6526.00i − 0.440728i
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 17264.0i − 1.15441i −0.816601 0.577203i $$-0.804144\pi$$
0.816601 0.577203i $$-0.195856\pi$$
$$608$$ 0 0
$$609$$ −3744.00 −0.249121
$$610$$ 0 0
$$611$$ 20592.0 1.36344
$$612$$ 0 0
$$613$$ 26026.0i 1.71481i 0.514640 + 0.857406i $$0.327926\pi$$
−0.514640 + 0.857406i $$0.672074\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 5022.00i 0.327679i 0.986487 + 0.163840i $$0.0523880\pi$$
−0.986487 + 0.163840i $$0.947612\pi$$
$$618$$ 0 0
$$619$$ 7820.00 0.507774 0.253887 0.967234i $$-0.418291\pi$$
0.253887 + 0.967234i $$0.418291\pi$$
$$620$$ 0 0
$$621$$ 954.000 0.0616469
$$622$$ 0 0
$$623$$ 9126.00i 0.586879i
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ 4095.00i 0.260827i
$$628$$ 0 0
$$629$$ 25038.0 1.58717
$$630$$ 0 0
$$631$$ −15002.0 −0.946466 −0.473233 0.880937i $$-0.656913\pi$$
−0.473233 + 0.880937i $$0.656913\pi$$
$$632$$ 0 0
$$633$$ − 3419.00i − 0.214681i
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 14652.0i 0.911355i
$$638$$ 0 0
$$639$$ −2808.00 −0.173838
$$640$$ 0 0
$$641$$ −918.000 −0.0565660 −0.0282830 0.999600i $$-0.509004\pi$$
−0.0282830 + 0.999600i $$0.509004\pi$$
$$642$$ 0 0
$$643$$ − 23452.0i − 1.43835i −0.694831 0.719173i $$-0.744521\pi$$
0.694831 0.719173i $$-0.255479\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ − 20556.0i − 1.24906i −0.781002 0.624528i $$-0.785291\pi$$
0.781002 0.624528i $$-0.214709\pi$$
$$648$$ 0 0
$$649$$ 3240.00 0.195965
$$650$$ 0 0
$$651$$ −676.000 −0.0406982
$$652$$ 0 0
$$653$$ − 30654.0i − 1.83703i −0.395381 0.918517i $$-0.629387\pi$$
0.395381 0.918517i $$-0.370613\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ − 7774.00i − 0.461633i
$$658$$ 0 0
$$659$$ 8919.00 0.527215 0.263608 0.964630i $$-0.415088\pi$$
0.263608 + 0.964630i $$0.415088\pi$$
$$660$$ 0 0
$$661$$ −22912.0 −1.34822 −0.674110 0.738631i $$-0.735473\pi$$
−0.674110 + 0.738631i $$0.735473\pi$$
$$662$$ 0 0
$$663$$ 5148.00i 0.301556i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 2592.00i 0.150469i
$$668$$ 0 0
$$669$$ 100.000 0.00577911
$$670$$ 0 0
$$671$$ 5310.00 0.305500
$$672$$ 0 0
$$673$$ 1222.00i 0.0699920i 0.999387 + 0.0349960i $$0.0111419\pi$$
−0.999387 + 0.0349960i $$0.988858\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 144.000i 0.00817484i 0.999992 + 0.00408742i $$0.00130107\pi$$
−0.999992 + 0.00408742i $$0.998699\pi$$
$$678$$ 0 0
$$679$$ 10036.0 0.567226
$$680$$ 0 0
$$681$$ 4212.00 0.237011
$$682$$ 0 0
$$683$$ − 12519.0i − 0.701356i −0.936496 0.350678i $$-0.885951\pi$$
0.936496 0.350678i $$-0.114049\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 3484.00i 0.193483i
$$688$$ 0 0
$$689$$ −24552.0 −1.35756
$$690$$ 0 0
$$691$$ −11873.0 −0.653647 −0.326824 0.945085i $$-0.605978\pi$$
−0.326824 + 0.945085i $$0.605978\pi$$
$$692$$ 0 0
$$693$$ 30420.0i 1.66748i
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ − 53703.0i − 2.91843i
$$698$$ 0 0
$$699$$ 918.000 0.0496737
$$700$$ 0 0
$$701$$ −3060.00 −0.164871 −0.0824355 0.996596i $$-0.526270\pi$$
−0.0824355 + 0.996596i $$0.526270\pi$$
$$702$$ 0 0
$$703$$ 19474.0i 1.04477i
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 24804.0i 1.31945i
$$708$$ 0 0
$$709$$ −4004.00 −0.212092 −0.106046 0.994361i $$-0.533819\pi$$
−0.106046 + 0.994361i $$0.533819\pi$$
$$710$$ 0 0
$$711$$ −23348.0 −1.23153
$$712$$ 0 0
$$713$$ 468.000i 0.0245817i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 3744.00i 0.195010i
$$718$$ 0 0
$$719$$ 10314.0 0.534975 0.267488 0.963561i $$-0.413807\pi$$
0.267488 + 0.963561i $$0.413807\pi$$
$$720$$ 0 0
$$721$$ −20072.0 −1.03678
$$722$$ 0 0
$$723$$ − 4231.00i − 0.217638i
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 9872.00i − 0.503621i −0.967777 0.251810i $$-0.918974\pi$$
0.967777 0.251810i $$-0.0810259\pi$$
$$728$$ 0 0
$$729$$ 15443.0 0.784586
$$730$$ 0 0
$$731$$ 53820.0 2.72313
$$732$$ 0 0
$$733$$ − 7436.00i − 0.374700i −0.982293 0.187350i $$-0.940010\pi$$
0.982293 0.187350i $$-0.0599898\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 11295.0i 0.564527i
$$738$$ 0 0
$$739$$ −16900.0 −0.841240 −0.420620 0.907237i $$-0.638187\pi$$
−0.420620 + 0.907237i $$0.638187\pi$$
$$740$$ 0 0
$$741$$ −4004.00 −0.198503
$$742$$ 0 0
$$743$$ − 23058.0i − 1.13851i −0.822160 0.569257i $$-0.807231\pi$$
0.822160 0.569257i $$-0.192769\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 24102.0i 1.18052i
$$748$$ 0 0
$$749$$ −31122.0 −1.51826
$$750$$ 0 0
$$751$$ 8224.00 0.399598 0.199799 0.979837i $$-0.435971\pi$$
0.199799 + 0.979837i $$0.435971\pi$$
$$752$$ 0 0
$$753$$ 2925.00i 0.141558i
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ − 7696.00i − 0.369506i −0.982785 0.184753i $$-0.940852\pi$$
0.982785 0.184753i $$-0.0591485\pi$$
$$758$$ 0 0
$$759$$ −810.000 −0.0387367
$$760$$ 0 0
$$761$$ −6363.00 −0.303099 −0.151550 0.988450i $$-0.548426\pi$$
−0.151550 + 0.988450i $$0.548426\pi$$
$$762$$ 0 0
$$763$$ − 20852.0i − 0.989375i
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ 3168.00i 0.149139i
$$768$$ 0 0
$$769$$ −8333.00 −0.390762 −0.195381 0.980727i $$-0.562594\pi$$
−0.195381 + 0.980727i $$0.562594\pi$$
$$770$$ 0 0
$$771$$ 18.0000 0.000840797 0
$$772$$ 0 0
$$773$$ − 32760.0i − 1.52431i −0.647392 0.762157i $$-0.724140\pi$$
0.647392 0.762157i $$-0.275860\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 5564.00i − 0.256895i
$$778$$ 0 0
$$779$$ 41769.0 1.92109
$$780$$ 0 0
$$781$$ 4860.00 0.222669
$$782$$ 0 0
$$783$$ − 7632.00i − 0.348334i
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 43732.0i 1.98078i 0.138286 + 0.990392i $$0.455841\pi$$
−0.138286 + 0.990392i $$0.544159\pi$$
$$788$$ 0 0
$$789$$ 6786.00 0.306195
$$790$$ 0 0
$$791$$ −29718.0 −1.33584
$$792$$ 0 0
$$793$$ 5192.00i 0.232501i
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 16866.0i 0.749591i 0.927107 + 0.374796i $$0.122287\pi$$
−0.927107 + 0.374796i $$0.877713\pi$$
$$798$$ 0 0
$$799$$ −54756.0 −2.42444
$$800$$ 0 0
$$801$$ −9126.00 −0.402561
$$802$$ 0 0
$$803$$ 13455.0i 0.591303i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ − 7632.00i − 0.332911i
$$808$$ 0 0
$$809$$ −16146.0 −0.701685 −0.350842 0.936434i $$-0.614105\pi$$
−0.350842 + 0.936434i $$0.614105\pi$$
$$810$$ 0 0
$$811$$ −32444.0 −1.40476 −0.702382 0.711801i $$-0.747880\pi$$
−0.702382 + 0.711801i $$0.747880\pi$$
$$812$$ 0 0
$$813$$ − 650.000i − 0.0280400i
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 41860.0i 1.79253i
$$818$$ 0 0
$$819$$ −29744.0 −1.26903
$$820$$ 0 0
$$821$$ −2574.00 −0.109419 −0.0547096 0.998502i $$-0.517423\pi$$
−0.0547096 + 0.998502i $$0.517423\pi$$
$$822$$ 0 0
$$823$$ − 27604.0i − 1.16916i −0.811338 0.584578i $$-0.801260\pi$$
0.811338 0.584578i $$-0.198740\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ − 11655.0i − 0.490065i −0.969515 0.245033i $$-0.921201\pi$$
0.969515 0.245033i $$-0.0787987\pi$$
$$828$$ 0 0
$$829$$ −33428.0 −1.40049 −0.700243 0.713905i $$-0.746925\pi$$
−0.700243 + 0.713905i $$0.746925\pi$$
$$830$$ 0 0
$$831$$ 3232.00 0.134918
$$832$$ 0 0
$$833$$ − 38961.0i − 1.62055i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ − 1378.00i − 0.0569064i
$$838$$ 0 0
$$839$$ −17712.0 −0.728827 −0.364414 0.931237i $$-0.618731\pi$$
−0.364414 + 0.931237i $$0.618731\pi$$
$$840$$ 0 0
$$841$$ −3653.00 −0.149781
$$842$$ 0 0
$$843$$ 4446.00i 0.181647i
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 18044.0i − 0.731994i
$$848$$ 0 0
$$849$$ −2483.00 −0.100373
$$850$$ 0 0
$$851$$ −3852.00 −0.155164
$$852$$ 0 0
$$853$$ 10270.0i 0.412237i 0.978527 + 0.206118i $$0.0660832\pi$$
−0.978527 + 0.206118i $$0.933917\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ 38709.0i 1.54291i 0.636284 + 0.771455i $$0.280471\pi$$
−0.636284 + 0.771455i $$0.719529\pi$$
$$858$$ 0 0
$$859$$ 15509.0 0.616019 0.308009 0.951383i $$-0.400337\pi$$
0.308009 + 0.951383i $$0.400337\pi$$
$$860$$ 0 0
$$861$$ −11934.0 −0.472369
$$862$$ 0 0
$$863$$ 15912.0i 0.627637i 0.949483 + 0.313819i $$0.101608\pi$$
−0.949483 + 0.313819i $$0.898392\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ − 8776.00i − 0.343770i
$$868$$ 0 0
$$869$$ 40410.0 1.57746
$$870$$ 0 0
$$871$$ −11044.0 −0.429635
$$872$$ 0 0
$$873$$ 10036.0i 0.389080i
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ − 10972.0i − 0.422461i −0.977436 0.211230i $$-0.932253\pi$$
0.977436 0.211230i $$-0.0677470\pi$$
$$878$$ 0 0
$$879$$ −4050.00 −0.155407
$$880$$ 0 0
$$881$$ −18738.0 −0.716571 −0.358286 0.933612i $$-0.616639\pi$$
−0.358286 + 0.933612i $$0.616639\pi$$
$$882$$ 0 0
$$883$$ − 21367.0i − 0.814334i −0.913354 0.407167i $$-0.866517\pi$$
0.913354 0.407167i $$-0.133483\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ − 20124.0i − 0.761779i −0.924621 0.380889i $$-0.875618\pi$$
0.924621 0.380889i $$-0.124382\pi$$
$$888$$ 0 0
$$889$$ 61724.0 2.32864
$$890$$ 0 0
$$891$$ −29205.0 −1.09810
$$892$$ 0 0
$$893$$ − 42588.0i − 1.59592i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ − 792.000i − 0.0294806i
$$898$$ 0 0
$$899$$ 3744.00 0.138898
$$900$$ 0 0
$$901$$ 65286.0 2.41398
$$902$$ 0 0
$$903$$ − 11960.0i − 0.440757i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 23132.0i − 0.846842i −0.905933 0.423421i $$-0.860829\pi$$
0.905933 0.423421i $$-0.139171\pi$$
$$908$$ 0 0
$$909$$ −24804.0 −0.905057
$$910$$ 0 0
$$911$$ −31212.0 −1.13513 −0.567563 0.823330i $$-0.692114\pi$$
−0.567563 + 0.823330i $$0.692114\pi$$
$$912$$ 0 0
$$913$$ − 41715.0i − 1.51212i
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ − 32760.0i − 1.17975i
$$918$$ 0 0
$$919$$ −6994.00 −0.251045 −0.125523 0.992091i $$-0.540061\pi$$
−0.125523 + 0.992091i $$0.540061\pi$$
$$920$$ 0 0
$$921$$ 2321.00 0.0830397
$$922$$ 0 0
$$923$$ 4752.00i 0.169463i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ − 20072.0i − 0.711166i
$$928$$ 0 0
$$929$$ −19422.0 −0.685915 −0.342958 0.939351i $$-0.611429\pi$$
−0.342958 + 0.939351i $$0.611429\pi$$
$$930$$ 0 0
$$931$$ 30303.0 1.06675
$$932$$ 0 0
$$933$$ 3258.00i 0.114322i
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 11699.0i 0.407887i 0.978983 + 0.203943i $$0.0653758\pi$$
−0.978983 + 0.203943i $$0.934624\pi$$
$$938$$ 0 0
$$939$$ 3626.00 0.126017
$$940$$ 0 0
$$941$$ −42948.0 −1.48785 −0.743924 0.668264i $$-0.767038\pi$$
−0.743924 + 0.668264i $$0.767038\pi$$
$$942$$ 0 0
$$943$$ 8262.00i 0.285310i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 3816.00i 0.130943i 0.997854 + 0.0654717i $$0.0208552\pi$$
−0.997854 + 0.0654717i $$0.979145\pi$$
$$948$$ 0 0
$$949$$ −13156.0 −0.450012
$$950$$ 0 0
$$951$$ −3852.00 −0.131346
$$952$$ 0 0
$$953$$ 43407.0i 1.47544i 0.675109 + 0.737718i $$0.264097\pi$$
−0.675109 + 0.737718i $$0.735903\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 6480.00i 0.218881i
$$958$$ 0 0
$$959$$ −23166.0 −0.780051
$$960$$ 0 0
$$961$$ −29115.0 −0.977309
$$962$$ 0 0
$$963$$ − 31122.0i − 1.04143i
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 43216.0i 1.43716i 0.695445 + 0.718580i $$0.255207\pi$$
−0.695445 + 0.718580i $$0.744793\pi$$
$$968$$ 0 0
$$969$$ 10647.0 0.352973
$$970$$ 0 0
$$971$$ 47619.0 1.57381 0.786903 0.617076i $$-0.211683\pi$$
0.786903 + 0.617076i $$0.211683\pi$$
$$972$$ 0 0
$$973$$ − 10114.0i − 0.333237i
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ − 4671.00i − 0.152957i −0.997071 0.0764783i $$-0.975632\pi$$
0.997071 0.0764783i $$-0.0243676\pi$$
$$978$$ 0 0
$$979$$ 15795.0 0.515639
$$980$$ 0 0
$$981$$ 20852.0 0.678647
$$982$$ 0 0
$$983$$ 9054.00i 0.293772i 0.989153 + 0.146886i $$0.0469250\pi$$
−0.989153 + 0.146886i $$0.953075\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 12168.0i 0.392413i
$$988$$ 0 0
$$989$$ −8280.00 −0.266217
$$990$$ 0 0
$$991$$ −8126.00 −0.260475 −0.130238 0.991483i $$-0.541574\pi$$
−0.130238 + 0.991483i $$0.541574\pi$$
$$992$$ 0 0
$$993$$ − 7553.00i − 0.241377i
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 38468.0i 1.22196i 0.791646 + 0.610980i $$0.209224\pi$$
−0.791646 + 0.610980i $$0.790776\pi$$
$$998$$ 0 0
$$999$$ 11342.0 0.359204
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.l.49.2 2
4.3 odd 2 100.4.c.b.49.1 2
5.2 odd 4 400.4.a.l.1.1 1
5.3 odd 4 400.4.a.i.1.1 1
5.4 even 2 inner 400.4.c.l.49.1 2
12.11 even 2 900.4.d.a.649.2 2
20.3 even 4 100.4.a.c.1.1 yes 1
20.7 even 4 100.4.a.b.1.1 1
20.19 odd 2 100.4.c.b.49.2 2
40.3 even 4 1600.4.a.x.1.1 1
40.13 odd 4 1600.4.a.bd.1.1 1
40.27 even 4 1600.4.a.bc.1.1 1
40.37 odd 4 1600.4.a.y.1.1 1
60.23 odd 4 900.4.a.p.1.1 1
60.47 odd 4 900.4.a.c.1.1 1
60.59 even 2 900.4.d.a.649.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
100.4.a.b.1.1 1 20.7 even 4
100.4.a.c.1.1 yes 1 20.3 even 4
100.4.c.b.49.1 2 4.3 odd 2
100.4.c.b.49.2 2 20.19 odd 2
400.4.a.i.1.1 1 5.3 odd 4
400.4.a.l.1.1 1 5.2 odd 4
400.4.c.l.49.1 2 5.4 even 2 inner
400.4.c.l.49.2 2 1.1 even 1 trivial
900.4.a.c.1.1 1 60.47 odd 4
900.4.a.p.1.1 1 60.23 odd 4
900.4.d.a.649.1 2 60.59 even 2
900.4.d.a.649.2 2 12.11 even 2
1600.4.a.x.1.1 1 40.3 even 4
1600.4.a.y.1.1 1 40.37 odd 4
1600.4.a.bc.1.1 1 40.27 even 4
1600.4.a.bd.1.1 1 40.13 odd 4