# Properties

 Label 1425.2.c.a.799.2 Level $1425$ Weight $2$ Character 1425.799 Analytic conductor $11.379$ 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] = [1425,2,Mod(799,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1425 = 3 \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1425.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: $$11.3786822880$$ 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 57) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.2 Root $$1.00000i$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.799 Dual form 1425.2.c.a.799.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} +O(q^{10})$$ $$q+2.00000i q^{2} +1.00000i q^{3} -2.00000 q^{4} -2.00000 q^{6} -3.00000i q^{7} -1.00000 q^{9} -3.00000 q^{11} -2.00000i q^{12} -6.00000i q^{13} +6.00000 q^{14} -4.00000 q^{16} -3.00000i q^{17} -2.00000i q^{18} +1.00000 q^{19} +3.00000 q^{21} -6.00000i q^{22} +4.00000i q^{23} +12.0000 q^{26} -1.00000i q^{27} +6.00000i q^{28} +10.0000 q^{29} +2.00000 q^{31} -8.00000i q^{32} -3.00000i q^{33} +6.00000 q^{34} +2.00000 q^{36} -8.00000i q^{37} +2.00000i q^{38} +6.00000 q^{39} -8.00000 q^{41} +6.00000i q^{42} -1.00000i q^{43} +6.00000 q^{44} -8.00000 q^{46} -3.00000i q^{47} -4.00000i q^{48} -2.00000 q^{49} +3.00000 q^{51} +12.0000i q^{52} -6.00000i q^{53} +2.00000 q^{54} +1.00000i q^{57} +20.0000i q^{58} +7.00000 q^{61} +4.00000i q^{62} +3.00000i q^{63} +8.00000 q^{64} +6.00000 q^{66} -8.00000i q^{67} +6.00000i q^{68} -4.00000 q^{69} +12.0000 q^{71} -11.0000i q^{73} +16.0000 q^{74} -2.00000 q^{76} +9.00000i q^{77} +12.0000i q^{78} +1.00000 q^{81} -16.0000i q^{82} +4.00000i q^{83} -6.00000 q^{84} +2.00000 q^{86} +10.0000i q^{87} -10.0000 q^{89} -18.0000 q^{91} -8.00000i q^{92} +2.00000i q^{93} +6.00000 q^{94} +8.00000 q^{96} +2.00000i q^{97} -4.00000i q^{98} +3.00000 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9 $$2 q - 4 q^{4} - 4 q^{6} - 2 q^{9} - 6 q^{11} + 12 q^{14} - 8 q^{16} + 2 q^{19} + 6 q^{21} + 24 q^{26} + 20 q^{29} + 4 q^{31} + 12 q^{34} + 4 q^{36} + 12 q^{39} - 16 q^{41} + 12 q^{44} - 16 q^{46} - 4 q^{49} + 6 q^{51} + 4 q^{54} + 14 q^{61} + 16 q^{64} + 12 q^{66} - 8 q^{69} + 24 q^{71} + 32 q^{74} - 4 q^{76} + 2 q^{81} - 12 q^{84} + 4 q^{86} - 20 q^{89} - 36 q^{91} + 12 q^{94} + 16 q^{96} + 6 q^{99}+O(q^{100})$$ 2 * q - 4 * q^4 - 4 * q^6 - 2 * q^9 - 6 * q^11 + 12 * q^14 - 8 * q^16 + 2 * q^19 + 6 * q^21 + 24 * q^26 + 20 * q^29 + 4 * q^31 + 12 * q^34 + 4 * q^36 + 12 * q^39 - 16 * q^41 + 12 * q^44 - 16 * q^46 - 4 * q^49 + 6 * q^51 + 4 * q^54 + 14 * q^61 + 16 * q^64 + 12 * q^66 - 8 * q^69 + 24 * q^71 + 32 * q^74 - 4 * q^76 + 2 * q^81 - 12 * q^84 + 4 * q^86 - 20 * q^89 - 36 * q^91 + 12 * q^94 + 16 * q^96 + 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$$.

 $$n$$ $$476$$ $$1027$$ $$1351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.00000i 1.41421i 0.707107 + 0.707107i $$0.250000\pi$$
−0.707107 + 0.707107i $$0.750000\pi$$
$$3$$ 1.00000i 0.577350i
$$4$$ −2.00000 −1.00000
$$5$$ 0 0
$$6$$ −2.00000 −0.816497
$$7$$ − 3.00000i − 1.13389i −0.823754 0.566947i $$-0.808125\pi$$
0.823754 0.566947i $$-0.191875\pi$$
$$8$$ 0 0
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ − 2.00000i − 0.577350i
$$13$$ − 6.00000i − 1.66410i −0.554700 0.832050i $$-0.687167\pi$$
0.554700 0.832050i $$-0.312833\pi$$
$$14$$ 6.00000 1.60357
$$15$$ 0 0
$$16$$ −4.00000 −1.00000
$$17$$ − 3.00000i − 0.727607i −0.931476 0.363803i $$-0.881478\pi$$
0.931476 0.363803i $$-0.118522\pi$$
$$18$$ − 2.00000i − 0.471405i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ 3.00000 0.654654
$$22$$ − 6.00000i − 1.27920i
$$23$$ 4.00000i 0.834058i 0.908893 + 0.417029i $$0.136929\pi$$
−0.908893 + 0.417029i $$0.863071\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 12.0000 2.35339
$$27$$ − 1.00000i − 0.192450i
$$28$$ 6.00000i 1.13389i
$$29$$ 10.0000 1.85695 0.928477 0.371391i $$-0.121119\pi$$
0.928477 + 0.371391i $$0.121119\pi$$
$$30$$ 0 0
$$31$$ 2.00000 0.359211 0.179605 0.983739i $$-0.442518\pi$$
0.179605 + 0.983739i $$0.442518\pi$$
$$32$$ − 8.00000i − 1.41421i
$$33$$ − 3.00000i − 0.522233i
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 2.00000 0.333333
$$37$$ − 8.00000i − 1.31519i −0.753371 0.657596i $$-0.771573\pi$$
0.753371 0.657596i $$-0.228427\pi$$
$$38$$ 2.00000i 0.324443i
$$39$$ 6.00000 0.960769
$$40$$ 0 0
$$41$$ −8.00000 −1.24939 −0.624695 0.780869i $$-0.714777\pi$$
−0.624695 + 0.780869i $$0.714777\pi$$
$$42$$ 6.00000i 0.925820i
$$43$$ − 1.00000i − 0.152499i −0.997089 0.0762493i $$-0.975706\pi$$
0.997089 0.0762493i $$-0.0242945\pi$$
$$44$$ 6.00000 0.904534
$$45$$ 0 0
$$46$$ −8.00000 −1.17954
$$47$$ − 3.00000i − 0.437595i −0.975770 0.218797i $$-0.929787\pi$$
0.975770 0.218797i $$-0.0702134\pi$$
$$48$$ − 4.00000i − 0.577350i
$$49$$ −2.00000 −0.285714
$$50$$ 0 0
$$51$$ 3.00000 0.420084
$$52$$ 12.0000i 1.66410i
$$53$$ − 6.00000i − 0.824163i −0.911147 0.412082i $$-0.864802\pi$$
0.911147 0.412082i $$-0.135198\pi$$
$$54$$ 2.00000 0.272166
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 1.00000i 0.132453i
$$58$$ 20.0000i 2.62613i
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 7.00000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 4.00000i 0.508001i
$$63$$ 3.00000i 0.377964i
$$64$$ 8.00000 1.00000
$$65$$ 0 0
$$66$$ 6.00000 0.738549
$$67$$ − 8.00000i − 0.977356i −0.872464 0.488678i $$-0.837479\pi$$
0.872464 0.488678i $$-0.162521\pi$$
$$68$$ 6.00000i 0.727607i
$$69$$ −4.00000 −0.481543
$$70$$ 0 0
$$71$$ 12.0000 1.42414 0.712069 0.702109i $$-0.247758\pi$$
0.712069 + 0.702109i $$0.247758\pi$$
$$72$$ 0 0
$$73$$ − 11.0000i − 1.28745i −0.765256 0.643726i $$-0.777388\pi$$
0.765256 0.643726i $$-0.222612\pi$$
$$74$$ 16.0000 1.85996
$$75$$ 0 0
$$76$$ −2.00000 −0.229416
$$77$$ 9.00000i 1.02565i
$$78$$ 12.0000i 1.35873i
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 16.0000i − 1.76690i
$$83$$ 4.00000i 0.439057i 0.975606 + 0.219529i $$0.0704519\pi$$
−0.975606 + 0.219529i $$0.929548\pi$$
$$84$$ −6.00000 −0.654654
$$85$$ 0 0
$$86$$ 2.00000 0.215666
$$87$$ 10.0000i 1.07211i
$$88$$ 0 0
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −18.0000 −1.88691
$$92$$ − 8.00000i − 0.834058i
$$93$$ 2.00000i 0.207390i
$$94$$ 6.00000 0.618853
$$95$$ 0 0
$$96$$ 8.00000 0.816497
$$97$$ 2.00000i 0.203069i 0.994832 + 0.101535i $$0.0323753\pi$$
−0.994832 + 0.101535i $$0.967625\pi$$
$$98$$ − 4.00000i − 0.404061i
$$99$$ 3.00000 0.301511
$$100$$ 0 0
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ 6.00000i 0.594089i
$$103$$ 14.0000i 1.37946i 0.724066 + 0.689730i $$0.242271\pi$$
−0.724066 + 0.689730i $$0.757729\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 12.0000 1.16554
$$107$$ 2.00000i 0.193347i 0.995316 + 0.0966736i $$0.0308203\pi$$
−0.995316 + 0.0966736i $$0.969180\pi$$
$$108$$ 2.00000i 0.192450i
$$109$$ −20.0000 −1.91565 −0.957826 0.287348i $$-0.907226\pi$$
−0.957826 + 0.287348i $$0.907226\pi$$
$$110$$ 0 0
$$111$$ 8.00000 0.759326
$$112$$ 12.0000i 1.13389i
$$113$$ − 6.00000i − 0.564433i −0.959351 0.282216i $$-0.908930\pi$$
0.959351 0.282216i $$-0.0910696\pi$$
$$114$$ −2.00000 −0.187317
$$115$$ 0 0
$$116$$ −20.0000 −1.85695
$$117$$ 6.00000i 0.554700i
$$118$$ 0 0
$$119$$ −9.00000 −0.825029
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ 14.0000i 1.26750i
$$123$$ − 8.00000i − 0.721336i
$$124$$ −4.00000 −0.359211
$$125$$ 0 0
$$126$$ −6.00000 −0.534522
$$127$$ 2.00000i 0.177471i 0.996055 + 0.0887357i $$0.0282826\pi$$
−0.996055 + 0.0887357i $$0.971717\pi$$
$$128$$ 0 0
$$129$$ 1.00000 0.0880451
$$130$$ 0 0
$$131$$ −13.0000 −1.13582 −0.567908 0.823092i $$-0.692247\pi$$
−0.567908 + 0.823092i $$0.692247\pi$$
$$132$$ 6.00000i 0.522233i
$$133$$ − 3.00000i − 0.260133i
$$134$$ 16.0000 1.38219
$$135$$ 0 0
$$136$$ 0 0
$$137$$ − 3.00000i − 0.256307i −0.991754 0.128154i $$-0.959095\pi$$
0.991754 0.128154i $$-0.0409051\pi$$
$$138$$ − 8.00000i − 0.681005i
$$139$$ 5.00000 0.424094 0.212047 0.977259i $$-0.431987\pi$$
0.212047 + 0.977259i $$0.431987\pi$$
$$140$$ 0 0
$$141$$ 3.00000 0.252646
$$142$$ 24.0000i 2.01404i
$$143$$ 18.0000i 1.50524i
$$144$$ 4.00000 0.333333
$$145$$ 0 0
$$146$$ 22.0000 1.82073
$$147$$ − 2.00000i − 0.164957i
$$148$$ 16.0000i 1.31519i
$$149$$ −15.0000 −1.22885 −0.614424 0.788976i $$-0.710612\pi$$
−0.614424 + 0.788976i $$0.710612\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 0 0
$$153$$ 3.00000i 0.242536i
$$154$$ −18.0000 −1.45048
$$155$$ 0 0
$$156$$ −12.0000 −0.960769
$$157$$ 2.00000i 0.159617i 0.996810 + 0.0798087i $$0.0254309\pi$$
−0.996810 + 0.0798087i $$0.974569\pi$$
$$158$$ 0 0
$$159$$ 6.00000 0.475831
$$160$$ 0 0
$$161$$ 12.0000 0.945732
$$162$$ 2.00000i 0.157135i
$$163$$ − 16.0000i − 1.25322i −0.779334 0.626608i $$-0.784443\pi$$
0.779334 0.626608i $$-0.215557\pi$$
$$164$$ 16.0000 1.24939
$$165$$ 0 0
$$166$$ −8.00000 −0.620920
$$167$$ − 18.0000i − 1.39288i −0.717614 0.696441i $$-0.754766\pi$$
0.717614 0.696441i $$-0.245234\pi$$
$$168$$ 0 0
$$169$$ −23.0000 −1.76923
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 2.00000i 0.152499i
$$173$$ 14.0000i 1.06440i 0.846619 + 0.532200i $$0.178635\pi$$
−0.846619 + 0.532200i $$0.821365\pi$$
$$174$$ −20.0000 −1.51620
$$175$$ 0 0
$$176$$ 12.0000 0.904534
$$177$$ 0 0
$$178$$ − 20.0000i − 1.49906i
$$179$$ 10.0000 0.747435 0.373718 0.927543i $$-0.378083\pi$$
0.373718 + 0.927543i $$0.378083\pi$$
$$180$$ 0 0
$$181$$ 2.00000 0.148659 0.0743294 0.997234i $$-0.476318\pi$$
0.0743294 + 0.997234i $$0.476318\pi$$
$$182$$ − 36.0000i − 2.66850i
$$183$$ 7.00000i 0.517455i
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −4.00000 −0.293294
$$187$$ 9.00000i 0.658145i
$$188$$ 6.00000i 0.437595i
$$189$$ −3.00000 −0.218218
$$190$$ 0 0
$$191$$ −3.00000 −0.217072 −0.108536 0.994092i $$-0.534616\pi$$
−0.108536 + 0.994092i $$0.534616\pi$$
$$192$$ 8.00000i 0.577350i
$$193$$ 4.00000i 0.287926i 0.989583 + 0.143963i $$0.0459847\pi$$
−0.989583 + 0.143963i $$0.954015\pi$$
$$194$$ −4.00000 −0.287183
$$195$$ 0 0
$$196$$ 4.00000 0.285714
$$197$$ 2.00000i 0.142494i 0.997459 + 0.0712470i $$0.0226979\pi$$
−0.997459 + 0.0712470i $$0.977302\pi$$
$$198$$ 6.00000i 0.426401i
$$199$$ 5.00000 0.354441 0.177220 0.984171i $$-0.443289\pi$$
0.177220 + 0.984171i $$0.443289\pi$$
$$200$$ 0 0
$$201$$ 8.00000 0.564276
$$202$$ 4.00000i 0.281439i
$$203$$ − 30.0000i − 2.10559i
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ −28.0000 −1.95085
$$207$$ − 4.00000i − 0.278019i
$$208$$ 24.0000i 1.66410i
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ −28.0000 −1.92760 −0.963800 0.266627i $$-0.914091\pi$$
−0.963800 + 0.266627i $$0.914091\pi$$
$$212$$ 12.0000i 0.824163i
$$213$$ 12.0000i 0.822226i
$$214$$ −4.00000 −0.273434
$$215$$ 0 0
$$216$$ 0 0
$$217$$ − 6.00000i − 0.407307i
$$218$$ − 40.0000i − 2.70914i
$$219$$ 11.0000 0.743311
$$220$$ 0 0
$$221$$ −18.0000 −1.21081
$$222$$ 16.0000i 1.07385i
$$223$$ 4.00000i 0.267860i 0.990991 + 0.133930i $$0.0427597\pi$$
−0.990991 + 0.133930i $$0.957240\pi$$
$$224$$ −24.0000 −1.60357
$$225$$ 0 0
$$226$$ 12.0000 0.798228
$$227$$ − 18.0000i − 1.19470i −0.801980 0.597351i $$-0.796220\pi$$
0.801980 0.597351i $$-0.203780\pi$$
$$228$$ − 2.00000i − 0.132453i
$$229$$ 15.0000 0.991228 0.495614 0.868543i $$-0.334943\pi$$
0.495614 + 0.868543i $$0.334943\pi$$
$$230$$ 0 0
$$231$$ −9.00000 −0.592157
$$232$$ 0 0
$$233$$ − 11.0000i − 0.720634i −0.932830 0.360317i $$-0.882669\pi$$
0.932830 0.360317i $$-0.117331\pi$$
$$234$$ −12.0000 −0.784465
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 0 0
$$238$$ − 18.0000i − 1.16677i
$$239$$ 15.0000 0.970269 0.485135 0.874439i $$-0.338771\pi$$
0.485135 + 0.874439i $$0.338771\pi$$
$$240$$ 0 0
$$241$$ 12.0000 0.772988 0.386494 0.922292i $$-0.373686\pi$$
0.386494 + 0.922292i $$0.373686\pi$$
$$242$$ − 4.00000i − 0.257130i
$$243$$ 1.00000i 0.0641500i
$$244$$ −14.0000 −0.896258
$$245$$ 0 0
$$246$$ 16.0000 1.02012
$$247$$ − 6.00000i − 0.381771i
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ 0 0
$$251$$ 27.0000 1.70422 0.852112 0.523359i $$-0.175321\pi$$
0.852112 + 0.523359i $$0.175321\pi$$
$$252$$ − 6.00000i − 0.377964i
$$253$$ − 12.0000i − 0.754434i
$$254$$ −4.00000 −0.250982
$$255$$ 0 0
$$256$$ 16.0000 1.00000
$$257$$ − 8.00000i − 0.499026i −0.968371 0.249513i $$-0.919729\pi$$
0.968371 0.249513i $$-0.0802706\pi$$
$$258$$ 2.00000i 0.124515i
$$259$$ −24.0000 −1.49129
$$260$$ 0 0
$$261$$ −10.0000 −0.618984
$$262$$ − 26.0000i − 1.60629i
$$263$$ − 21.0000i − 1.29492i −0.762101 0.647458i $$-0.775832\pi$$
0.762101 0.647458i $$-0.224168\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 6.00000 0.367884
$$267$$ − 10.0000i − 0.611990i
$$268$$ 16.0000i 0.977356i
$$269$$ 30.0000 1.82913 0.914566 0.404436i $$-0.132532\pi$$
0.914566 + 0.404436i $$0.132532\pi$$
$$270$$ 0 0
$$271$$ 12.0000 0.728948 0.364474 0.931214i $$-0.381249\pi$$
0.364474 + 0.931214i $$0.381249\pi$$
$$272$$ 12.0000i 0.727607i
$$273$$ − 18.0000i − 1.08941i
$$274$$ 6.00000 0.362473
$$275$$ 0 0
$$276$$ 8.00000 0.481543
$$277$$ − 13.0000i − 0.781094i −0.920583 0.390547i $$-0.872286\pi$$
0.920583 0.390547i $$-0.127714\pi$$
$$278$$ 10.0000i 0.599760i
$$279$$ −2.00000 −0.119737
$$280$$ 0 0
$$281$$ 2.00000 0.119310 0.0596550 0.998219i $$-0.481000\pi$$
0.0596550 + 0.998219i $$0.481000\pi$$
$$282$$ 6.00000i 0.357295i
$$283$$ 19.0000i 1.12943i 0.825285 + 0.564716i $$0.191014\pi$$
−0.825285 + 0.564716i $$0.808986\pi$$
$$284$$ −24.0000 −1.42414
$$285$$ 0 0
$$286$$ −36.0000 −2.12872
$$287$$ 24.0000i 1.41668i
$$288$$ 8.00000i 0.471405i
$$289$$ 8.00000 0.470588
$$290$$ 0 0
$$291$$ −2.00000 −0.117242
$$292$$ 22.0000i 1.28745i
$$293$$ 4.00000i 0.233682i 0.993151 + 0.116841i $$0.0372769\pi$$
−0.993151 + 0.116841i $$0.962723\pi$$
$$294$$ 4.00000 0.233285
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.00000i 0.174078i
$$298$$ − 30.0000i − 1.73785i
$$299$$ 24.0000 1.38796
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ − 16.0000i − 0.920697i
$$303$$ 2.00000i 0.114897i
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ −6.00000 −0.342997
$$307$$ 12.0000i 0.684876i 0.939540 + 0.342438i $$0.111253\pi$$
−0.939540 + 0.342438i $$0.888747\pi$$
$$308$$ − 18.0000i − 1.02565i
$$309$$ −14.0000 −0.796432
$$310$$ 0 0
$$311$$ 7.00000 0.396934 0.198467 0.980108i $$-0.436404\pi$$
0.198467 + 0.980108i $$0.436404\pi$$
$$312$$ 0 0
$$313$$ 14.0000i 0.791327i 0.918396 + 0.395663i $$0.129485\pi$$
−0.918396 + 0.395663i $$0.870515\pi$$
$$314$$ −4.00000 −0.225733
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 12.0000i 0.673987i 0.941507 + 0.336994i $$0.109410\pi$$
−0.941507 + 0.336994i $$0.890590\pi$$
$$318$$ 12.0000i 0.672927i
$$319$$ −30.0000 −1.67968
$$320$$ 0 0
$$321$$ −2.00000 −0.111629
$$322$$ 24.0000i 1.33747i
$$323$$ − 3.00000i − 0.166924i
$$324$$ −2.00000 −0.111111
$$325$$ 0 0
$$326$$ 32.0000 1.77232
$$327$$ − 20.0000i − 1.10600i
$$328$$ 0 0
$$329$$ −9.00000 −0.496186
$$330$$ 0 0
$$331$$ 12.0000 0.659580 0.329790 0.944054i $$-0.393022\pi$$
0.329790 + 0.944054i $$0.393022\pi$$
$$332$$ − 8.00000i − 0.439057i
$$333$$ 8.00000i 0.438397i
$$334$$ 36.0000 1.96983
$$335$$ 0 0
$$336$$ −12.0000 −0.654654
$$337$$ 22.0000i 1.19842i 0.800593 + 0.599208i $$0.204518\pi$$
−0.800593 + 0.599208i $$0.795482\pi$$
$$338$$ − 46.0000i − 2.50207i
$$339$$ 6.00000 0.325875
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ − 2.00000i − 0.108148i
$$343$$ − 15.0000i − 0.809924i
$$344$$ 0 0
$$345$$ 0 0
$$346$$ −28.0000 −1.50529
$$347$$ − 3.00000i − 0.161048i −0.996753 0.0805242i $$-0.974341\pi$$
0.996753 0.0805242i $$-0.0256594\pi$$
$$348$$ − 20.0000i − 1.07211i
$$349$$ −25.0000 −1.33822 −0.669110 0.743164i $$-0.733324\pi$$
−0.669110 + 0.743164i $$0.733324\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ 24.0000i 1.27920i
$$353$$ 14.0000i 0.745145i 0.928003 + 0.372572i $$0.121524\pi$$
−0.928003 + 0.372572i $$0.878476\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 20.0000 1.06000
$$357$$ − 9.00000i − 0.476331i
$$358$$ 20.0000i 1.05703i
$$359$$ −25.0000 −1.31945 −0.659725 0.751507i $$-0.729327\pi$$
−0.659725 + 0.751507i $$0.729327\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ 4.00000i 0.210235i
$$363$$ − 2.00000i − 0.104973i
$$364$$ 36.0000 1.88691
$$365$$ 0 0
$$366$$ −14.0000 −0.731792
$$367$$ − 8.00000i − 0.417597i −0.977959 0.208798i $$-0.933045\pi$$
0.977959 0.208798i $$-0.0669552\pi$$
$$368$$ − 16.0000i − 0.834058i
$$369$$ 8.00000 0.416463
$$370$$ 0 0
$$371$$ −18.0000 −0.934513
$$372$$ − 4.00000i − 0.207390i
$$373$$ − 16.0000i − 0.828449i −0.910175 0.414224i $$-0.864053\pi$$
0.910175 0.414224i $$-0.135947\pi$$
$$374$$ −18.0000 −0.930758
$$375$$ 0 0
$$376$$ 0 0
$$377$$ − 60.0000i − 3.09016i
$$378$$ − 6.00000i − 0.308607i
$$379$$ 30.0000 1.54100 0.770498 0.637442i $$-0.220007\pi$$
0.770498 + 0.637442i $$0.220007\pi$$
$$380$$ 0 0
$$381$$ −2.00000 −0.102463
$$382$$ − 6.00000i − 0.306987i
$$383$$ 14.0000i 0.715367i 0.933843 + 0.357683i $$0.116433\pi$$
−0.933843 + 0.357683i $$0.883567\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −8.00000 −0.407189
$$387$$ 1.00000i 0.0508329i
$$388$$ − 4.00000i − 0.203069i
$$389$$ 15.0000 0.760530 0.380265 0.924878i $$-0.375833\pi$$
0.380265 + 0.924878i $$0.375833\pi$$
$$390$$ 0 0
$$391$$ 12.0000 0.606866
$$392$$ 0 0
$$393$$ − 13.0000i − 0.655763i
$$394$$ −4.00000 −0.201517
$$395$$ 0 0
$$396$$ −6.00000 −0.301511
$$397$$ 7.00000i 0.351320i 0.984451 + 0.175660i $$0.0562059\pi$$
−0.984451 + 0.175660i $$0.943794\pi$$
$$398$$ 10.0000i 0.501255i
$$399$$ 3.00000 0.150188
$$400$$ 0 0
$$401$$ −28.0000 −1.39825 −0.699127 0.714998i $$-0.746428\pi$$
−0.699127 + 0.714998i $$0.746428\pi$$
$$402$$ 16.0000i 0.798007i
$$403$$ − 12.0000i − 0.597763i
$$404$$ −4.00000 −0.199007
$$405$$ 0 0
$$406$$ 60.0000 2.97775
$$407$$ 24.0000i 1.18964i
$$408$$ 0 0
$$409$$ −10.0000 −0.494468 −0.247234 0.968956i $$-0.579522\pi$$
−0.247234 + 0.968956i $$0.579522\pi$$
$$410$$ 0 0
$$411$$ 3.00000 0.147979
$$412$$ − 28.0000i − 1.37946i
$$413$$ 0 0
$$414$$ 8.00000 0.393179
$$415$$ 0 0
$$416$$ −48.0000 −2.35339
$$417$$ 5.00000i 0.244851i
$$418$$ − 6.00000i − 0.293470i
$$419$$ −20.0000 −0.977064 −0.488532 0.872546i $$-0.662467\pi$$
−0.488532 + 0.872546i $$0.662467\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ − 56.0000i − 2.72604i
$$423$$ 3.00000i 0.145865i
$$424$$ 0 0
$$425$$ 0 0
$$426$$ −24.0000 −1.16280
$$427$$ − 21.0000i − 1.01626i
$$428$$ − 4.00000i − 0.193347i
$$429$$ −18.0000 −0.869048
$$430$$ 0 0
$$431$$ −18.0000 −0.867029 −0.433515 0.901146i $$-0.642727\pi$$
−0.433515 + 0.901146i $$0.642727\pi$$
$$432$$ 4.00000i 0.192450i
$$433$$ − 26.0000i − 1.24948i −0.780833 0.624740i $$-0.785205\pi$$
0.780833 0.624740i $$-0.214795\pi$$
$$434$$ 12.0000 0.576018
$$435$$ 0 0
$$436$$ 40.0000 1.91565
$$437$$ 4.00000i 0.191346i
$$438$$ 22.0000i 1.05120i
$$439$$ −10.0000 −0.477274 −0.238637 0.971109i $$-0.576701\pi$$
−0.238637 + 0.971109i $$0.576701\pi$$
$$440$$ 0 0
$$441$$ 2.00000 0.0952381
$$442$$ − 36.0000i − 1.71235i
$$443$$ 39.0000i 1.85295i 0.376361 + 0.926473i $$0.377175\pi$$
−0.376361 + 0.926473i $$0.622825\pi$$
$$444$$ −16.0000 −0.759326
$$445$$ 0 0
$$446$$ −8.00000 −0.378811
$$447$$ − 15.0000i − 0.709476i
$$448$$ − 24.0000i − 1.13389i
$$449$$ 20.0000 0.943858 0.471929 0.881636i $$-0.343558\pi$$
0.471929 + 0.881636i $$0.343558\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 12.0000i 0.564433i
$$453$$ − 8.00000i − 0.375873i
$$454$$ 36.0000 1.68956
$$455$$ 0 0
$$456$$ 0 0
$$457$$ − 3.00000i − 0.140334i −0.997535 0.0701670i $$-0.977647\pi$$
0.997535 0.0701670i $$-0.0223532\pi$$
$$458$$ 30.0000i 1.40181i
$$459$$ −3.00000 −0.140028
$$460$$ 0 0
$$461$$ −33.0000 −1.53696 −0.768482 0.639872i $$-0.778987\pi$$
−0.768482 + 0.639872i $$0.778987\pi$$
$$462$$ − 18.0000i − 0.837436i
$$463$$ − 31.0000i − 1.44069i −0.693615 0.720346i $$-0.743983\pi$$
0.693615 0.720346i $$-0.256017\pi$$
$$464$$ −40.0000 −1.85695
$$465$$ 0 0
$$466$$ 22.0000 1.01913
$$467$$ 17.0000i 0.786666i 0.919396 + 0.393333i $$0.128678\pi$$
−0.919396 + 0.393333i $$0.871322\pi$$
$$468$$ − 12.0000i − 0.554700i
$$469$$ −24.0000 −1.10822
$$470$$ 0 0
$$471$$ −2.00000 −0.0921551
$$472$$ 0 0
$$473$$ 3.00000i 0.137940i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 18.0000 0.825029
$$477$$ 6.00000i 0.274721i
$$478$$ 30.0000i 1.37217i
$$479$$ 40.0000 1.82765 0.913823 0.406112i $$-0.133116\pi$$
0.913823 + 0.406112i $$0.133116\pi$$
$$480$$ 0 0
$$481$$ −48.0000 −2.18861
$$482$$ 24.0000i 1.09317i
$$483$$ 12.0000i 0.546019i
$$484$$ 4.00000 0.181818
$$485$$ 0 0
$$486$$ −2.00000 −0.0907218
$$487$$ − 8.00000i − 0.362515i −0.983436 0.181257i $$-0.941983\pi$$
0.983436 0.181257i $$-0.0580167\pi$$
$$488$$ 0 0
$$489$$ 16.0000 0.723545
$$490$$ 0 0
$$491$$ −8.00000 −0.361035 −0.180517 0.983572i $$-0.557777\pi$$
−0.180517 + 0.983572i $$0.557777\pi$$
$$492$$ 16.0000i 0.721336i
$$493$$ − 30.0000i − 1.35113i
$$494$$ 12.0000 0.539906
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ − 36.0000i − 1.61482i
$$498$$ − 8.00000i − 0.358489i
$$499$$ 35.0000 1.56682 0.783408 0.621508i $$-0.213480\pi$$
0.783408 + 0.621508i $$0.213480\pi$$
$$500$$ 0 0
$$501$$ 18.0000 0.804181
$$502$$ 54.0000i 2.41014i
$$503$$ 24.0000i 1.07011i 0.844818 + 0.535054i $$0.179709\pi$$
−0.844818 + 0.535054i $$0.820291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 24.0000 1.06693
$$507$$ − 23.0000i − 1.02147i
$$508$$ − 4.00000i − 0.177471i
$$509$$ 10.0000 0.443242 0.221621 0.975133i $$-0.428865\pi$$
0.221621 + 0.975133i $$0.428865\pi$$
$$510$$ 0 0
$$511$$ −33.0000 −1.45983
$$512$$ 32.0000i 1.41421i
$$513$$ − 1.00000i − 0.0441511i
$$514$$ 16.0000 0.705730
$$515$$ 0 0
$$516$$ −2.00000 −0.0880451
$$517$$ 9.00000i 0.395820i
$$518$$ − 48.0000i − 2.10900i
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ −28.0000 −1.22670 −0.613351 0.789810i $$-0.710179\pi$$
−0.613351 + 0.789810i $$0.710179\pi$$
$$522$$ − 20.0000i − 0.875376i
$$523$$ 14.0000i 0.612177i 0.952003 + 0.306089i $$0.0990204\pi$$
−0.952003 + 0.306089i $$0.900980\pi$$
$$524$$ 26.0000 1.13582
$$525$$ 0 0
$$526$$ 42.0000 1.83129
$$527$$ − 6.00000i − 0.261364i
$$528$$ 12.0000i 0.522233i
$$529$$ 7.00000 0.304348
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 6.00000i 0.260133i
$$533$$ 48.0000i 2.07911i
$$534$$ 20.0000 0.865485
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 10.0000i 0.431532i
$$538$$ 60.0000i 2.58678i
$$539$$ 6.00000 0.258438
$$540$$ 0 0
$$541$$ −13.0000 −0.558914 −0.279457 0.960158i $$-0.590154\pi$$
−0.279457 + 0.960158i $$0.590154\pi$$
$$542$$ 24.0000i 1.03089i
$$543$$ 2.00000i 0.0858282i
$$544$$ −24.0000 −1.02899
$$545$$ 0 0
$$546$$ 36.0000 1.54066
$$547$$ 2.00000i 0.0855138i 0.999086 + 0.0427569i $$0.0136141\pi$$
−0.999086 + 0.0427569i $$0.986386\pi$$
$$548$$ 6.00000i 0.256307i
$$549$$ −7.00000 −0.298753
$$550$$ 0 0
$$551$$ 10.0000 0.426014
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 26.0000 1.10463
$$555$$ 0 0
$$556$$ −10.0000 −0.424094
$$557$$ − 3.00000i − 0.127114i −0.997978 0.0635570i $$-0.979756\pi$$
0.997978 0.0635570i $$-0.0202445\pi$$
$$558$$ − 4.00000i − 0.169334i
$$559$$ −6.00000 −0.253773
$$560$$ 0 0
$$561$$ −9.00000 −0.379980
$$562$$ 4.00000i 0.168730i
$$563$$ 44.0000i 1.85438i 0.374593 + 0.927189i $$0.377783\pi$$
−0.374593 + 0.927189i $$0.622217\pi$$
$$564$$ −6.00000 −0.252646
$$565$$ 0 0
$$566$$ −38.0000 −1.59726
$$567$$ − 3.00000i − 0.125988i
$$568$$ 0 0
$$569$$ −30.0000 −1.25767 −0.628833 0.777541i $$-0.716467\pi$$
−0.628833 + 0.777541i $$0.716467\pi$$
$$570$$ 0 0
$$571$$ 32.0000 1.33916 0.669579 0.742741i $$-0.266474\pi$$
0.669579 + 0.742741i $$0.266474\pi$$
$$572$$ − 36.0000i − 1.50524i
$$573$$ − 3.00000i − 0.125327i
$$574$$ −48.0000 −2.00348
$$575$$ 0 0
$$576$$ −8.00000 −0.333333
$$577$$ − 3.00000i − 0.124892i −0.998048 0.0624458i $$-0.980110\pi$$
0.998048 0.0624458i $$-0.0198901\pi$$
$$578$$ 16.0000i 0.665512i
$$579$$ −4.00000 −0.166234
$$580$$ 0 0
$$581$$ 12.0000 0.497844
$$582$$ − 4.00000i − 0.165805i
$$583$$ 18.0000i 0.745484i
$$584$$ 0 0
$$585$$ 0 0
$$586$$ −8.00000 −0.330477
$$587$$ 37.0000i 1.52715i 0.645717 + 0.763577i $$0.276559\pi$$
−0.645717 + 0.763577i $$0.723441\pi$$
$$588$$ 4.00000i 0.164957i
$$589$$ 2.00000 0.0824086
$$590$$ 0 0
$$591$$ −2.00000 −0.0822690
$$592$$ 32.0000i 1.31519i
$$593$$ − 6.00000i − 0.246390i −0.992382 0.123195i $$-0.960686\pi$$
0.992382 0.123195i $$-0.0393141\pi$$
$$594$$ −6.00000 −0.246183
$$595$$ 0 0
$$596$$ 30.0000 1.22885
$$597$$ 5.00000i 0.204636i
$$598$$ 48.0000i 1.96287i
$$599$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$600$$ 0 0
$$601$$ −8.00000 −0.326327 −0.163163 0.986599i $$-0.552170\pi$$
−0.163163 + 0.986599i $$0.552170\pi$$
$$602$$ − 6.00000i − 0.244542i
$$603$$ 8.00000i 0.325785i
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ −4.00000 −0.162489
$$607$$ − 18.0000i − 0.730597i −0.930890 0.365299i $$-0.880967\pi$$
0.930890 0.365299i $$-0.119033\pi$$
$$608$$ − 8.00000i − 0.324443i
$$609$$ 30.0000 1.21566
$$610$$ 0 0
$$611$$ −18.0000 −0.728202
$$612$$ − 6.00000i − 0.242536i
$$613$$ 9.00000i 0.363507i 0.983344 + 0.181753i $$0.0581772\pi$$
−0.983344 + 0.181753i $$0.941823\pi$$
$$614$$ −24.0000 −0.968561
$$615$$ 0 0
$$616$$ 0 0
$$617$$ − 23.0000i − 0.925945i −0.886373 0.462973i $$-0.846783\pi$$
0.886373 0.462973i $$-0.153217\pi$$
$$618$$ − 28.0000i − 1.12633i
$$619$$ −20.0000 −0.803868 −0.401934 0.915669i $$-0.631662\pi$$
−0.401934 + 0.915669i $$0.631662\pi$$
$$620$$ 0 0
$$621$$ 4.00000 0.160514
$$622$$ 14.0000i 0.561349i
$$623$$ 30.0000i 1.20192i
$$624$$ −24.0000 −0.960769
$$625$$ 0 0
$$626$$ −28.0000 −1.11911
$$627$$ − 3.00000i − 0.119808i
$$628$$ − 4.00000i − 0.159617i
$$629$$ −24.0000 −0.956943
$$630$$ 0 0
$$631$$ 7.00000 0.278666 0.139333 0.990246i $$-0.455504\pi$$
0.139333 + 0.990246i $$0.455504\pi$$
$$632$$ 0 0
$$633$$ − 28.0000i − 1.11290i
$$634$$ −24.0000 −0.953162
$$635$$ 0 0
$$636$$ −12.0000 −0.475831
$$637$$ 12.0000i 0.475457i
$$638$$ − 60.0000i − 2.37542i
$$639$$ −12.0000 −0.474713
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ − 4.00000i − 0.157867i
$$643$$ − 1.00000i − 0.0394362i −0.999806 0.0197181i $$-0.993723\pi$$
0.999806 0.0197181i $$-0.00627687\pi$$
$$644$$ −24.0000 −0.945732
$$645$$ 0 0
$$646$$ 6.00000 0.236067
$$647$$ 27.0000i 1.06148i 0.847535 + 0.530740i $$0.178086\pi$$
−0.847535 + 0.530740i $$0.821914\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 6.00000 0.235159
$$652$$ 32.0000i 1.25322i
$$653$$ − 1.00000i − 0.0391330i −0.999809 0.0195665i $$-0.993771\pi$$
0.999809 0.0195665i $$-0.00622861\pi$$
$$654$$ 40.0000 1.56412
$$655$$ 0 0
$$656$$ 32.0000 1.24939
$$657$$ 11.0000i 0.429151i
$$658$$ − 18.0000i − 0.701713i
$$659$$ −10.0000 −0.389545 −0.194772 0.980848i $$-0.562397\pi$$
−0.194772 + 0.980848i $$0.562397\pi$$
$$660$$ 0 0
$$661$$ 12.0000 0.466746 0.233373 0.972387i $$-0.425024\pi$$
0.233373 + 0.972387i $$0.425024\pi$$
$$662$$ 24.0000i 0.932786i
$$663$$ − 18.0000i − 0.699062i
$$664$$ 0 0
$$665$$ 0 0
$$666$$ −16.0000 −0.619987
$$667$$ 40.0000i 1.54881i
$$668$$ 36.0000i 1.39288i
$$669$$ −4.00000 −0.154649
$$670$$ 0 0
$$671$$ −21.0000 −0.810696
$$672$$ − 24.0000i − 0.925820i
$$673$$ − 16.0000i − 0.616755i −0.951264 0.308377i $$-0.900214\pi$$
0.951264 0.308377i $$-0.0997859\pi$$
$$674$$ −44.0000 −1.69482
$$675$$ 0 0
$$676$$ 46.0000 1.76923
$$677$$ 22.0000i 0.845529i 0.906240 + 0.422764i $$0.138940\pi$$
−0.906240 + 0.422764i $$0.861060\pi$$
$$678$$ 12.0000i 0.460857i
$$679$$ 6.00000 0.230259
$$680$$ 0 0
$$681$$ 18.0000 0.689761
$$682$$ − 12.0000i − 0.459504i
$$683$$ − 6.00000i − 0.229584i −0.993390 0.114792i $$-0.963380\pi$$
0.993390 0.114792i $$-0.0366201\pi$$
$$684$$ 2.00000 0.0764719
$$685$$ 0 0
$$686$$ 30.0000 1.14541
$$687$$ 15.0000i 0.572286i
$$688$$ 4.00000i 0.152499i
$$689$$ −36.0000 −1.37149
$$690$$ 0 0
$$691$$ 17.0000 0.646710 0.323355 0.946278i $$-0.395189\pi$$
0.323355 + 0.946278i $$0.395189\pi$$
$$692$$ − 28.0000i − 1.06440i
$$693$$ − 9.00000i − 0.341882i
$$694$$ 6.00000 0.227757
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 24.0000i 0.909065i
$$698$$ − 50.0000i − 1.89253i
$$699$$ 11.0000 0.416058
$$700$$ 0 0
$$701$$ 42.0000 1.58632 0.793159 0.609015i $$-0.208435\pi$$
0.793159 + 0.609015i $$0.208435\pi$$
$$702$$ − 12.0000i − 0.452911i
$$703$$ − 8.00000i − 0.301726i
$$704$$ −24.0000 −0.904534
$$705$$ 0 0
$$706$$ −28.0000 −1.05379
$$707$$ − 6.00000i − 0.225653i
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 8.00000i 0.299602i
$$714$$ 18.0000 0.673633
$$715$$ 0 0
$$716$$ −20.0000 −0.747435
$$717$$ 15.0000i 0.560185i
$$718$$ − 50.0000i − 1.86598i
$$719$$ 35.0000 1.30528 0.652640 0.757668i $$-0.273661\pi$$
0.652640 + 0.757668i $$0.273661\pi$$
$$720$$ 0 0
$$721$$ 42.0000 1.56416
$$722$$ 2.00000i 0.0744323i
$$723$$ 12.0000i 0.446285i
$$724$$ −4.00000 −0.148659
$$725$$ 0 0
$$726$$ 4.00000 0.148454
$$727$$ 7.00000i 0.259616i 0.991539 + 0.129808i $$0.0414360\pi$$
−0.991539 + 0.129808i $$0.958564\pi$$
$$728$$ 0 0
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ −3.00000 −0.110959
$$732$$ − 14.0000i − 0.517455i
$$733$$ 34.0000i 1.25582i 0.778287 + 0.627909i $$0.216089\pi$$
−0.778287 + 0.627909i $$0.783911\pi$$
$$734$$ 16.0000 0.590571
$$735$$ 0 0
$$736$$ 32.0000 1.17954
$$737$$ 24.0000i 0.884051i
$$738$$ 16.0000i 0.588968i
$$739$$ 45.0000 1.65535 0.827676 0.561206i $$-0.189663\pi$$
0.827676 + 0.561206i $$0.189663\pi$$
$$740$$ 0 0
$$741$$ 6.00000 0.220416
$$742$$ − 36.0000i − 1.32160i
$$743$$ 24.0000i 0.880475i 0.897881 + 0.440237i $$0.145106\pi$$
−0.897881 + 0.440237i $$0.854894\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 32.0000 1.17160
$$747$$ − 4.00000i − 0.146352i
$$748$$ − 18.0000i − 0.658145i
$$749$$ 6.00000 0.219235
$$750$$ 0 0
$$751$$ −8.00000 −0.291924 −0.145962 0.989290i $$-0.546628\pi$$
−0.145962 + 0.989290i $$0.546628\pi$$
$$752$$ 12.0000i 0.437595i
$$753$$ 27.0000i 0.983935i
$$754$$ 120.000 4.37014
$$755$$ 0 0
$$756$$ 6.00000 0.218218
$$757$$ − 23.0000i − 0.835949i −0.908459 0.417975i $$-0.862740\pi$$
0.908459 0.417975i $$-0.137260\pi$$
$$758$$ 60.0000i 2.17930i
$$759$$ 12.0000 0.435572
$$760$$ 0 0
$$761$$ −13.0000 −0.471250 −0.235625 0.971844i $$-0.575714\pi$$
−0.235625 + 0.971844i $$0.575714\pi$$
$$762$$ − 4.00000i − 0.144905i
$$763$$ 60.0000i 2.17215i
$$764$$ 6.00000 0.217072
$$765$$ 0 0
$$766$$ −28.0000 −1.01168
$$767$$ 0 0
$$768$$ 16.0000i 0.577350i
$$769$$ 45.0000 1.62274 0.811371 0.584532i $$-0.198722\pi$$
0.811371 + 0.584532i $$0.198722\pi$$
$$770$$ 0 0
$$771$$ 8.00000 0.288113
$$772$$ − 8.00000i − 0.287926i
$$773$$ − 36.0000i − 1.29483i −0.762138 0.647415i $$-0.775850\pi$$
0.762138 0.647415i $$-0.224150\pi$$
$$774$$ −2.00000 −0.0718885
$$775$$ 0 0
$$776$$ 0 0
$$777$$ − 24.0000i − 0.860995i
$$778$$ 30.0000i 1.07555i
$$779$$ −8.00000 −0.286630
$$780$$ 0 0
$$781$$ −36.0000 −1.28818
$$782$$ 24.0000i 0.858238i
$$783$$ − 10.0000i − 0.357371i
$$784$$ 8.00000 0.285714
$$785$$ 0 0
$$786$$ 26.0000 0.927389
$$787$$ − 8.00000i − 0.285169i −0.989783 0.142585i $$-0.954459\pi$$
0.989783 0.142585i $$-0.0455413\pi$$
$$788$$ − 4.00000i − 0.142494i
$$789$$ 21.0000 0.747620
$$790$$ 0 0
$$791$$ −18.0000 −0.640006
$$792$$ 0 0
$$793$$ − 42.0000i − 1.49146i
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ −10.0000 −0.354441
$$797$$ 12.0000i 0.425062i 0.977154 + 0.212531i $$0.0681706\pi$$
−0.977154 + 0.212531i $$0.931829\pi$$
$$798$$ 6.00000i 0.212398i
$$799$$ −9.00000 −0.318397
$$800$$ 0 0
$$801$$ 10.0000 0.353333
$$802$$ − 56.0000i − 1.97743i
$$803$$ 33.0000i 1.16454i
$$804$$ −16.0000 −0.564276
$$805$$ 0 0
$$806$$ 24.0000 0.845364
$$807$$ 30.0000i 1.05605i
$$808$$ 0 0
$$809$$ −5.00000 −0.175791 −0.0878953 0.996130i $$-0.528014\pi$$
−0.0878953 + 0.996130i $$0.528014\pi$$
$$810$$ 0 0
$$811$$ 2.00000 0.0702295 0.0351147 0.999383i $$-0.488820\pi$$
0.0351147 + 0.999383i $$0.488820\pi$$
$$812$$ 60.0000i 2.10559i
$$813$$ 12.0000i 0.420858i
$$814$$ −48.0000 −1.68240
$$815$$ 0 0
$$816$$ −12.0000 −0.420084
$$817$$ − 1.00000i − 0.0349856i
$$818$$ − 20.0000i − 0.699284i
$$819$$ 18.0000 0.628971
$$820$$ 0 0
$$821$$ −33.0000 −1.15171 −0.575854 0.817553i $$-0.695330\pi$$
−0.575854 + 0.817553i $$0.695330\pi$$
$$822$$ 6.00000i 0.209274i
$$823$$ 19.0000i 0.662298i 0.943578 + 0.331149i $$0.107436\pi$$
−0.943578 + 0.331149i $$0.892564\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 52.0000i 1.80822i 0.427303 + 0.904109i $$0.359464\pi$$
−0.427303 + 0.904109i $$0.640536\pi$$
$$828$$ 8.00000i 0.278019i
$$829$$ −20.0000 −0.694629 −0.347314 0.937749i $$-0.612906\pi$$
−0.347314 + 0.937749i $$0.612906\pi$$
$$830$$ 0 0
$$831$$ 13.0000 0.450965
$$832$$ − 48.0000i − 1.66410i
$$833$$ 6.00000i 0.207888i
$$834$$ −10.0000 −0.346272
$$835$$ 0 0
$$836$$ 6.00000 0.207514
$$837$$ − 2.00000i − 0.0691301i
$$838$$ − 40.0000i − 1.38178i
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ 71.0000 2.44828
$$842$$ 4.00000i 0.137849i
$$843$$ 2.00000i 0.0688837i
$$844$$ 56.0000 1.92760
$$845$$ 0 0
$$846$$ −6.00000 −0.206284
$$847$$ 6.00000i 0.206162i
$$848$$ 24.0000i 0.824163i
$$849$$ −19.0000 −0.652078
$$850$$ 0 0
$$851$$ 32.0000 1.09695
$$852$$ − 24.0000i − 0.822226i
$$853$$ 34.0000i 1.16414i 0.813139 + 0.582069i $$0.197757\pi$$
−0.813139 + 0.582069i $$0.802243\pi$$
$$854$$ 42.0000 1.43721
$$855$$ 0 0
$$856$$ 0 0
$$857$$ − 48.0000i − 1.63965i −0.572615 0.819824i $$-0.694071\pi$$
0.572615 0.819824i $$-0.305929\pi$$
$$858$$ − 36.0000i − 1.22902i
$$859$$ −35.0000 −1.19418 −0.597092 0.802173i $$-0.703677\pi$$
−0.597092 + 0.802173i $$0.703677\pi$$
$$860$$ 0 0
$$861$$ −24.0000 −0.817918
$$862$$ − 36.0000i − 1.22616i
$$863$$ 44.0000i 1.49778i 0.662696 + 0.748889i $$0.269412\pi$$
−0.662696 + 0.748889i $$0.730588\pi$$
$$864$$ −8.00000 −0.272166
$$865$$ 0 0
$$866$$ 52.0000 1.76703
$$867$$ 8.00000i 0.271694i
$$868$$ 12.0000i 0.407307i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ −48.0000 −1.62642
$$872$$ 0 0
$$873$$ − 2.00000i − 0.0676897i
$$874$$ −8.00000 −0.270604
$$875$$ 0 0
$$876$$ −22.0000 −0.743311
$$877$$ 22.0000i 0.742887i 0.928456 + 0.371444i $$0.121137\pi$$
−0.928456 + 0.371444i $$0.878863\pi$$
$$878$$ − 20.0000i − 0.674967i
$$879$$ −4.00000 −0.134917
$$880$$ 0 0
$$881$$ 7.00000 0.235836 0.117918 0.993023i $$-0.462378\pi$$
0.117918 + 0.993023i $$0.462378\pi$$
$$882$$ 4.00000i 0.134687i
$$883$$ − 21.0000i − 0.706706i −0.935490 0.353353i $$-0.885041\pi$$
0.935490 0.353353i $$-0.114959\pi$$
$$884$$ 36.0000 1.21081
$$885$$ 0 0
$$886$$ −78.0000 −2.62046
$$887$$ 52.0000i 1.74599i 0.487730 + 0.872995i $$0.337825\pi$$
−0.487730 + 0.872995i $$0.662175\pi$$
$$888$$ 0 0
$$889$$ 6.00000 0.201234
$$890$$ 0 0
$$891$$ −3.00000 −0.100504
$$892$$ − 8.00000i − 0.267860i
$$893$$ − 3.00000i − 0.100391i
$$894$$ 30.0000 1.00335
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 24.0000i 0.801337i
$$898$$ 40.0000i 1.33482i
$$899$$ 20.0000 0.667037
$$900$$ 0 0
$$901$$ −18.0000 −0.599667
$$902$$ 48.0000i 1.59823i
$$903$$ − 3.00000i − 0.0998337i
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 16.0000 0.531564
$$907$$ 2.00000i 0.0664089i 0.999449 + 0.0332045i $$0.0105712\pi$$
−0.999449 + 0.0332045i $$0.989429\pi$$
$$908$$ 36.0000i 1.19470i
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ 2.00000 0.0662630 0.0331315 0.999451i $$-0.489452\pi$$
0.0331315 + 0.999451i $$0.489452\pi$$
$$912$$ − 4.00000i − 0.132453i
$$913$$ − 12.0000i − 0.397142i
$$914$$ 6.00000 0.198462
$$915$$ 0 0
$$916$$ −30.0000 −0.991228
$$917$$ 39.0000i 1.28789i
$$918$$ − 6.00000i − 0.198030i
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ −12.0000 −0.395413
$$922$$ − 66.0000i − 2.17359i
$$923$$ − 72.0000i − 2.36991i
$$924$$ 18.0000 0.592157
$$925$$ 0 0
$$926$$ 62.0000 2.03745
$$927$$ − 14.0000i − 0.459820i
$$928$$ − 80.0000i − 2.62613i
$$929$$ 50.0000 1.64045 0.820223 0.572043i $$-0.193849\pi$$
0.820223 + 0.572043i $$0.193849\pi$$
$$930$$ 0 0
$$931$$ −2.00000 −0.0655474
$$932$$ 22.0000i 0.720634i
$$933$$ 7.00000i 0.229170i
$$934$$ −34.0000 −1.11251
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 53.0000i − 1.73143i −0.500533 0.865717i $$-0.666863\pi$$
0.500533 0.865717i $$-0.333137\pi$$
$$938$$ − 48.0000i − 1.56726i
$$939$$ −14.0000 −0.456873
$$940$$ 0 0
$$941$$ 2.00000 0.0651981 0.0325991 0.999469i $$-0.489622\pi$$
0.0325991 + 0.999469i $$0.489622\pi$$
$$942$$ − 4.00000i − 0.130327i
$$943$$ − 32.0000i − 1.04206i
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −6.00000 −0.195077
$$947$$ − 28.0000i − 0.909878i −0.890523 0.454939i $$-0.849661\pi$$
0.890523 0.454939i $$-0.150339\pi$$
$$948$$ 0 0
$$949$$ −66.0000 −2.14245
$$950$$ 0 0
$$951$$ −12.0000 −0.389127
$$952$$ 0 0
$$953$$ − 16.0000i − 0.518291i −0.965838 0.259145i $$-0.916559\pi$$
0.965838 0.259145i $$-0.0834409\pi$$
$$954$$ −12.0000 −0.388514
$$955$$ 0 0
$$956$$ −30.0000 −0.970269
$$957$$ − 30.0000i − 0.969762i
$$958$$ 80.0000i 2.58468i
$$959$$ −9.00000 −0.290625
$$960$$ 0 0
$$961$$ −27.0000 −0.870968
$$962$$ − 96.0000i − 3.09516i
$$963$$ − 2.00000i − 0.0644491i
$$964$$ −24.0000 −0.772988
$$965$$ 0 0
$$966$$ −24.0000 −0.772187
$$967$$ 32.0000i 1.02905i 0.857475 + 0.514525i $$0.172032\pi$$
−0.857475 + 0.514525i $$0.827968\pi$$
$$968$$ 0 0
$$969$$ 3.00000 0.0963739
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ − 2.00000i − 0.0641500i
$$973$$ − 15.0000i − 0.480878i
$$974$$ 16.0000 0.512673
$$975$$ 0 0
$$976$$ −28.0000 −0.896258
$$977$$ 42.0000i 1.34370i 0.740688 + 0.671850i $$0.234500\pi$$
−0.740688 + 0.671850i $$0.765500\pi$$
$$978$$ 32.0000i 1.02325i
$$979$$ 30.0000 0.958804
$$980$$ 0 0
$$981$$ 20.0000 0.638551
$$982$$ − 16.0000i − 0.510581i
$$983$$ 4.00000i 0.127580i 0.997963 + 0.0637901i $$0.0203188\pi$$
−0.997963 + 0.0637901i $$0.979681\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 60.0000 1.91079
$$987$$ − 9.00000i − 0.286473i
$$988$$ 12.0000i 0.381771i
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ −8.00000 −0.254128 −0.127064 0.991894i $$-0.540555\pi$$
−0.127064 + 0.991894i $$0.540555\pi$$
$$992$$ − 16.0000i − 0.508001i
$$993$$ 12.0000i 0.380808i
$$994$$ 72.0000 2.28370
$$995$$ 0 0
$$996$$ 8.00000 0.253490
$$997$$ 7.00000i 0.221692i 0.993838 + 0.110846i $$0.0353561\pi$$
−0.993838 + 0.110846i $$0.964644\pi$$
$$998$$ 70.0000i 2.21581i
$$999$$ −8.00000 −0.253109
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 1425.2.c.a.799.2 2
5.2 odd 4 57.2.a.b.1.1 1
5.3 odd 4 1425.2.a.i.1.1 1
5.4 even 2 inner 1425.2.c.a.799.1 2
15.2 even 4 171.2.a.c.1.1 1
15.8 even 4 4275.2.a.a.1.1 1
20.7 even 4 912.2.a.d.1.1 1
35.27 even 4 2793.2.a.a.1.1 1
40.27 even 4 3648.2.a.y.1.1 1
40.37 odd 4 3648.2.a.h.1.1 1
55.32 even 4 6897.2.a.g.1.1 1
60.47 odd 4 2736.2.a.h.1.1 1
65.12 odd 4 9633.2.a.p.1.1 1
95.37 even 4 1083.2.a.d.1.1 1
105.62 odd 4 8379.2.a.q.1.1 1
285.227 odd 4 3249.2.a.a.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
57.2.a.b.1.1 1 5.2 odd 4
171.2.a.c.1.1 1 15.2 even 4
912.2.a.d.1.1 1 20.7 even 4
1083.2.a.d.1.1 1 95.37 even 4
1425.2.a.i.1.1 1 5.3 odd 4
1425.2.c.a.799.1 2 5.4 even 2 inner
1425.2.c.a.799.2 2 1.1 even 1 trivial
2736.2.a.h.1.1 1 60.47 odd 4
2793.2.a.a.1.1 1 35.27 even 4
3249.2.a.a.1.1 1 285.227 odd 4
3648.2.a.h.1.1 1 40.37 odd 4
3648.2.a.y.1.1 1 40.27 even 4
4275.2.a.a.1.1 1 15.8 even 4
6897.2.a.g.1.1 1 55.32 even 4
8379.2.a.q.1.1 1 105.62 odd 4
9633.2.a.p.1.1 1 65.12 odd 4