# Properties

 Label 1425.2.c.l.799.4 Level $1425$ Weight $2$ Character 1425.799 Analytic conductor $11.379$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 285) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 799.4 Root $$0.707107 + 0.707107i$$ of defining polynomial Character $$\chi$$ $$=$$ 1425.799 Dual form 1425.2.c.l.799.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q+2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} +2.41421 q^{6} -1.41421i q^{7} -4.41421i q^{8} -1.00000 q^{9} +O(q^{10})$$ $$q+2.41421i q^{2} -1.00000i q^{3} -3.82843 q^{4} +2.41421 q^{6} -1.41421i q^{7} -4.41421i q^{8} -1.00000 q^{9} -2.24264 q^{11} +3.82843i q^{12} +3.41421i q^{13} +3.41421 q^{14} +3.00000 q^{16} +1.17157i q^{17} -2.41421i q^{18} +1.00000 q^{19} -1.41421 q^{21} -5.41421i q^{22} -7.65685i q^{23} -4.41421 q^{24} -8.24264 q^{26} +1.00000i q^{27} +5.41421i q^{28} -1.41421 q^{29} -3.17157 q^{31} -1.58579i q^{32} +2.24264i q^{33} -2.82843 q^{34} +3.82843 q^{36} -3.41421i q^{37} +2.41421i q^{38} +3.41421 q^{39} -0.242641 q^{41} -3.41421i q^{42} -12.2426i q^{43} +8.58579 q^{44} +18.4853 q^{46} -7.65685i q^{47} -3.00000i q^{48} +5.00000 q^{49} +1.17157 q^{51} -13.0711i q^{52} -8.00000i q^{53} -2.41421 q^{54} -6.24264 q^{56} -1.00000i q^{57} -3.41421i q^{58} -12.4853 q^{59} +7.31371 q^{61} -7.65685i q^{62} +1.41421i q^{63} +9.82843 q^{64} -5.41421 q^{66} -9.65685i q^{67} -4.48528i q^{68} -7.65685 q^{69} -10.8284 q^{71} +4.41421i q^{72} +7.65685i q^{73} +8.24264 q^{74} -3.82843 q^{76} +3.17157i q^{77} +8.24264i q^{78} +1.00000 q^{81} -0.585786i q^{82} -12.8284i q^{83} +5.41421 q^{84} +29.5563 q^{86} +1.41421i q^{87} +9.89949i q^{88} -5.89949 q^{89} +4.82843 q^{91} +29.3137i q^{92} +3.17157i q^{93} +18.4853 q^{94} -1.58579 q^{96} -9.75736i q^{97} +12.0711i q^{98} +2.24264 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 $$4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + 8 q^{11} + 8 q^{14} + 12 q^{16} + 4 q^{19} - 12 q^{24} - 16 q^{26} - 24 q^{31} + 4 q^{36} + 8 q^{39} + 16 q^{41} + 40 q^{44} + 40 q^{46} + 20 q^{49} + 16 q^{51} - 4 q^{54} - 8 q^{56} - 16 q^{59} - 16 q^{61} + 28 q^{64} - 16 q^{66} - 8 q^{69} - 32 q^{71} + 16 q^{74} - 4 q^{76} + 4 q^{81} + 16 q^{84} + 56 q^{86} + 16 q^{89} + 8 q^{91} + 40 q^{94} - 12 q^{96} - 8 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 + 4 * q^6 - 4 * q^9 + 8 * q^11 + 8 * q^14 + 12 * q^16 + 4 * q^19 - 12 * q^24 - 16 * q^26 - 24 * q^31 + 4 * q^36 + 8 * q^39 + 16 * q^41 + 40 * q^44 + 40 * q^46 + 20 * q^49 + 16 * q^51 - 4 * q^54 - 8 * q^56 - 16 * q^59 - 16 * q^61 + 28 * q^64 - 16 * q^66 - 8 * q^69 - 32 * q^71 + 16 * q^74 - 4 * q^76 + 4 * q^81 + 16 * q^84 + 56 * q^86 + 16 * q^89 + 8 * q^91 + 40 * q^94 - 12 * q^96 - 8 * 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ 2.41421i 1.70711i 0.521005 + 0.853553i $$0.325557\pi$$
−0.521005 + 0.853553i $$0.674443\pi$$
$$3$$ − 1.00000i − 0.577350i
$$4$$ −3.82843 −1.91421
$$5$$ 0 0
$$6$$ 2.41421 0.985599
$$7$$ − 1.41421i − 0.534522i −0.963624 0.267261i $$-0.913881\pi$$
0.963624 0.267261i $$-0.0861187\pi$$
$$8$$ − 4.41421i − 1.56066i
$$9$$ −1.00000 −0.333333
$$10$$ 0 0
$$11$$ −2.24264 −0.676182 −0.338091 0.941113i $$-0.609781\pi$$
−0.338091 + 0.941113i $$0.609781\pi$$
$$12$$ 3.82843i 1.10517i
$$13$$ 3.41421i 0.946932i 0.880812 + 0.473466i $$0.156997\pi$$
−0.880812 + 0.473466i $$0.843003\pi$$
$$14$$ 3.41421 0.912487
$$15$$ 0 0
$$16$$ 3.00000 0.750000
$$17$$ 1.17157i 0.284148i 0.989856 + 0.142074i $$0.0453771\pi$$
−0.989856 + 0.142074i $$0.954623\pi$$
$$18$$ − 2.41421i − 0.569036i
$$19$$ 1.00000 0.229416
$$20$$ 0 0
$$21$$ −1.41421 −0.308607
$$22$$ − 5.41421i − 1.15431i
$$23$$ − 7.65685i − 1.59656i −0.602284 0.798282i $$-0.705742\pi$$
0.602284 0.798282i $$-0.294258\pi$$
$$24$$ −4.41421 −0.901048
$$25$$ 0 0
$$26$$ −8.24264 −1.61651
$$27$$ 1.00000i 0.192450i
$$28$$ 5.41421i 1.02319i
$$29$$ −1.41421 −0.262613 −0.131306 0.991342i $$-0.541917\pi$$
−0.131306 + 0.991342i $$0.541917\pi$$
$$30$$ 0 0
$$31$$ −3.17157 −0.569631 −0.284816 0.958582i $$-0.591932\pi$$
−0.284816 + 0.958582i $$0.591932\pi$$
$$32$$ − 1.58579i − 0.280330i
$$33$$ 2.24264i 0.390394i
$$34$$ −2.82843 −0.485071
$$35$$ 0 0
$$36$$ 3.82843 0.638071
$$37$$ − 3.41421i − 0.561293i −0.959811 0.280647i $$-0.909451\pi$$
0.959811 0.280647i $$-0.0905489\pi$$
$$38$$ 2.41421i 0.391637i
$$39$$ 3.41421 0.546712
$$40$$ 0 0
$$41$$ −0.242641 −0.0378941 −0.0189471 0.999820i $$-0.506031\pi$$
−0.0189471 + 0.999820i $$0.506031\pi$$
$$42$$ − 3.41421i − 0.526825i
$$43$$ − 12.2426i − 1.86699i −0.358597 0.933493i $$-0.616745\pi$$
0.358597 0.933493i $$-0.383255\pi$$
$$44$$ 8.58579 1.29436
$$45$$ 0 0
$$46$$ 18.4853 2.72551
$$47$$ − 7.65685i − 1.11687i −0.829549 0.558433i $$-0.811403\pi$$
0.829549 0.558433i $$-0.188597\pi$$
$$48$$ − 3.00000i − 0.433013i
$$49$$ 5.00000 0.714286
$$50$$ 0 0
$$51$$ 1.17157 0.164053
$$52$$ − 13.0711i − 1.81263i
$$53$$ − 8.00000i − 1.09888i −0.835532 0.549442i $$-0.814840\pi$$
0.835532 0.549442i $$-0.185160\pi$$
$$54$$ −2.41421 −0.328533
$$55$$ 0 0
$$56$$ −6.24264 −0.834208
$$57$$ − 1.00000i − 0.132453i
$$58$$ − 3.41421i − 0.448308i
$$59$$ −12.4853 −1.62545 −0.812723 0.582651i $$-0.802016\pi$$
−0.812723 + 0.582651i $$0.802016\pi$$
$$60$$ 0 0
$$61$$ 7.31371 0.936424 0.468212 0.883616i $$-0.344898\pi$$
0.468212 + 0.883616i $$0.344898\pi$$
$$62$$ − 7.65685i − 0.972421i
$$63$$ 1.41421i 0.178174i
$$64$$ 9.82843 1.22855
$$65$$ 0 0
$$66$$ −5.41421 −0.666444
$$67$$ − 9.65685i − 1.17977i −0.807486 0.589886i $$-0.799173\pi$$
0.807486 0.589886i $$-0.200827\pi$$
$$68$$ − 4.48528i − 0.543920i
$$69$$ −7.65685 −0.921777
$$70$$ 0 0
$$71$$ −10.8284 −1.28510 −0.642549 0.766245i $$-0.722123\pi$$
−0.642549 + 0.766245i $$0.722123\pi$$
$$72$$ 4.41421i 0.520220i
$$73$$ 7.65685i 0.896167i 0.893992 + 0.448084i $$0.147893\pi$$
−0.893992 + 0.448084i $$0.852107\pi$$
$$74$$ 8.24264 0.958188
$$75$$ 0 0
$$76$$ −3.82843 −0.439151
$$77$$ 3.17157i 0.361434i
$$78$$ 8.24264i 0.933295i
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ − 0.585786i − 0.0646893i
$$83$$ − 12.8284i − 1.40810i −0.710149 0.704051i $$-0.751372\pi$$
0.710149 0.704051i $$-0.248628\pi$$
$$84$$ 5.41421 0.590739
$$85$$ 0 0
$$86$$ 29.5563 3.18714
$$87$$ 1.41421i 0.151620i
$$88$$ 9.89949i 1.05529i
$$89$$ −5.89949 −0.625345 −0.312673 0.949861i $$-0.601224\pi$$
−0.312673 + 0.949861i $$0.601224\pi$$
$$90$$ 0 0
$$91$$ 4.82843 0.506157
$$92$$ 29.3137i 3.05617i
$$93$$ 3.17157i 0.328877i
$$94$$ 18.4853 1.90661
$$95$$ 0 0
$$96$$ −1.58579 −0.161849
$$97$$ − 9.75736i − 0.990710i −0.868691 0.495355i $$-0.835038\pi$$
0.868691 0.495355i $$-0.164962\pi$$
$$98$$ 12.0711i 1.21936i
$$99$$ 2.24264 0.225394
$$100$$ 0 0
$$101$$ −3.17157 −0.315583 −0.157792 0.987472i $$-0.550437\pi$$
−0.157792 + 0.987472i $$0.550437\pi$$
$$102$$ 2.82843i 0.280056i
$$103$$ 7.31371i 0.720641i 0.932829 + 0.360321i $$0.117333\pi$$
−0.932829 + 0.360321i $$0.882667\pi$$
$$104$$ 15.0711 1.47784
$$105$$ 0 0
$$106$$ 19.3137 1.87591
$$107$$ 19.3137i 1.86713i 0.358412 + 0.933563i $$0.383318\pi$$
−0.358412 + 0.933563i $$0.616682\pi$$
$$108$$ − 3.82843i − 0.368391i
$$109$$ 6.48528 0.621177 0.310589 0.950544i $$-0.399474\pi$$
0.310589 + 0.950544i $$0.399474\pi$$
$$110$$ 0 0
$$111$$ −3.41421 −0.324063
$$112$$ − 4.24264i − 0.400892i
$$113$$ − 10.1421i − 0.954092i −0.878878 0.477046i $$-0.841708\pi$$
0.878878 0.477046i $$-0.158292\pi$$
$$114$$ 2.41421 0.226112
$$115$$ 0 0
$$116$$ 5.41421 0.502697
$$117$$ − 3.41421i − 0.315644i
$$118$$ − 30.1421i − 2.77481i
$$119$$ 1.65685 0.151884
$$120$$ 0 0
$$121$$ −5.97056 −0.542778
$$122$$ 17.6569i 1.59858i
$$123$$ 0.242641i 0.0218782i
$$124$$ 12.1421 1.09040
$$125$$ 0 0
$$126$$ −3.41421 −0.304162
$$127$$ 19.3137i 1.71381i 0.515471 + 0.856907i $$0.327617\pi$$
−0.515471 + 0.856907i $$0.672383\pi$$
$$128$$ 20.5563i 1.81694i
$$129$$ −12.2426 −1.07790
$$130$$ 0 0
$$131$$ −8.58579 −0.750144 −0.375072 0.926996i $$-0.622382\pi$$
−0.375072 + 0.926996i $$0.622382\pi$$
$$132$$ − 8.58579i − 0.747297i
$$133$$ − 1.41421i − 0.122628i
$$134$$ 23.3137 2.01400
$$135$$ 0 0
$$136$$ 5.17157 0.443459
$$137$$ 10.0000i 0.854358i 0.904167 + 0.427179i $$0.140493\pi$$
−0.904167 + 0.427179i $$0.859507\pi$$
$$138$$ − 18.4853i − 1.57357i
$$139$$ 14.1421 1.19952 0.599760 0.800180i $$-0.295263\pi$$
0.599760 + 0.800180i $$0.295263\pi$$
$$140$$ 0 0
$$141$$ −7.65685 −0.644823
$$142$$ − 26.1421i − 2.19380i
$$143$$ − 7.65685i − 0.640298i
$$144$$ −3.00000 −0.250000
$$145$$ 0 0
$$146$$ −18.4853 −1.52985
$$147$$ − 5.00000i − 0.412393i
$$148$$ 13.0711i 1.07444i
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −6.48528 −0.527765 −0.263882 0.964555i $$-0.585003\pi$$
−0.263882 + 0.964555i $$0.585003\pi$$
$$152$$ − 4.41421i − 0.358040i
$$153$$ − 1.17157i − 0.0947161i
$$154$$ −7.65685 −0.617007
$$155$$ 0 0
$$156$$ −13.0711 −1.04652
$$157$$ 10.4853i 0.836817i 0.908259 + 0.418408i $$0.137412\pi$$
−0.908259 + 0.418408i $$0.862588\pi$$
$$158$$ 0 0
$$159$$ −8.00000 −0.634441
$$160$$ 0 0
$$161$$ −10.8284 −0.853400
$$162$$ 2.41421i 0.189679i
$$163$$ − 21.8995i − 1.71530i −0.514233 0.857650i $$-0.671923\pi$$
0.514233 0.857650i $$-0.328077\pi$$
$$164$$ 0.928932 0.0725374
$$165$$ 0 0
$$166$$ 30.9706 2.40378
$$167$$ − 17.3137i − 1.33977i −0.742463 0.669887i $$-0.766342\pi$$
0.742463 0.669887i $$-0.233658\pi$$
$$168$$ 6.24264i 0.481630i
$$169$$ 1.34315 0.103319
$$170$$ 0 0
$$171$$ −1.00000 −0.0764719
$$172$$ 46.8701i 3.57381i
$$173$$ 19.7990i 1.50529i 0.658427 + 0.752645i $$0.271222\pi$$
−0.658427 + 0.752645i $$0.728778\pi$$
$$174$$ −3.41421 −0.258831
$$175$$ 0 0
$$176$$ −6.72792 −0.507136
$$177$$ 12.4853i 0.938451i
$$178$$ − 14.2426i − 1.06753i
$$179$$ −16.4853 −1.23217 −0.616084 0.787681i $$-0.711282\pi$$
−0.616084 + 0.787681i $$0.711282\pi$$
$$180$$ 0 0
$$181$$ −20.8284 −1.54816 −0.774082 0.633085i $$-0.781788\pi$$
−0.774082 + 0.633085i $$0.781788\pi$$
$$182$$ 11.6569i 0.864064i
$$183$$ − 7.31371i − 0.540645i
$$184$$ −33.7990 −2.49169
$$185$$ 0 0
$$186$$ −7.65685 −0.561428
$$187$$ − 2.62742i − 0.192136i
$$188$$ 29.3137i 2.13792i
$$189$$ 1.41421 0.102869
$$190$$ 0 0
$$191$$ 10.2426 0.741131 0.370566 0.928806i $$-0.379164\pi$$
0.370566 + 0.928806i $$0.379164\pi$$
$$192$$ − 9.82843i − 0.709306i
$$193$$ − 5.07107i − 0.365023i −0.983204 0.182512i $$-0.941577\pi$$
0.983204 0.182512i $$-0.0584227\pi$$
$$194$$ 23.5563 1.69125
$$195$$ 0 0
$$196$$ −19.1421 −1.36730
$$197$$ 6.82843i 0.486505i 0.969963 + 0.243253i $$0.0782144\pi$$
−0.969963 + 0.243253i $$0.921786\pi$$
$$198$$ 5.41421i 0.384771i
$$199$$ −18.1421 −1.28606 −0.643031 0.765840i $$-0.722323\pi$$
−0.643031 + 0.765840i $$0.722323\pi$$
$$200$$ 0 0
$$201$$ −9.65685 −0.681142
$$202$$ − 7.65685i − 0.538734i
$$203$$ 2.00000i 0.140372i
$$204$$ −4.48528 −0.314033
$$205$$ 0 0
$$206$$ −17.6569 −1.23021
$$207$$ 7.65685i 0.532188i
$$208$$ 10.2426i 0.710199i
$$209$$ −2.24264 −0.155127
$$210$$ 0 0
$$211$$ 7.31371 0.503496 0.251748 0.967793i $$-0.418995\pi$$
0.251748 + 0.967793i $$0.418995\pi$$
$$212$$ 30.6274i 2.10350i
$$213$$ 10.8284i 0.741952i
$$214$$ −46.6274 −3.18738
$$215$$ 0 0
$$216$$ 4.41421 0.300349
$$217$$ 4.48528i 0.304481i
$$218$$ 15.6569i 1.06042i
$$219$$ 7.65685 0.517402
$$220$$ 0 0
$$221$$ −4.00000 −0.269069
$$222$$ − 8.24264i − 0.553210i
$$223$$ − 18.6274i − 1.24738i −0.781670 0.623692i $$-0.785632\pi$$
0.781670 0.623692i $$-0.214368\pi$$
$$224$$ −2.24264 −0.149843
$$225$$ 0 0
$$226$$ 24.4853 1.62874
$$227$$ 14.9706i 0.993631i 0.867856 + 0.496816i $$0.165497\pi$$
−0.867856 + 0.496816i $$0.834503\pi$$
$$228$$ 3.82843i 0.253544i
$$229$$ 22.6274 1.49526 0.747631 0.664114i $$-0.231191\pi$$
0.747631 + 0.664114i $$0.231191\pi$$
$$230$$ 0 0
$$231$$ 3.17157 0.208674
$$232$$ 6.24264i 0.409849i
$$233$$ 0.343146i 0.0224802i 0.999937 + 0.0112401i $$0.00357791\pi$$
−0.999937 + 0.0112401i $$0.996422\pi$$
$$234$$ 8.24264 0.538838
$$235$$ 0 0
$$236$$ 47.7990 3.11145
$$237$$ 0 0
$$238$$ 4.00000i 0.259281i
$$239$$ 26.7279 1.72889 0.864443 0.502731i $$-0.167671\pi$$
0.864443 + 0.502731i $$0.167671\pi$$
$$240$$ 0 0
$$241$$ −19.6569 −1.26621 −0.633105 0.774066i $$-0.718220\pi$$
−0.633105 + 0.774066i $$0.718220\pi$$
$$242$$ − 14.4142i − 0.926581i
$$243$$ − 1.00000i − 0.0641500i
$$244$$ −28.0000 −1.79252
$$245$$ 0 0
$$246$$ −0.585786 −0.0373484
$$247$$ 3.41421i 0.217241i
$$248$$ 14.0000i 0.889001i
$$249$$ −12.8284 −0.812969
$$250$$ 0 0
$$251$$ −1.75736 −0.110924 −0.0554618 0.998461i $$-0.517663\pi$$
−0.0554618 + 0.998461i $$0.517663\pi$$
$$252$$ − 5.41421i − 0.341063i
$$253$$ 17.1716i 1.07957i
$$254$$ −46.6274 −2.92566
$$255$$ 0 0
$$256$$ −29.9706 −1.87316
$$257$$ − 4.48528i − 0.279784i −0.990167 0.139892i $$-0.955324\pi$$
0.990167 0.139892i $$-0.0446756\pi$$
$$258$$ − 29.5563i − 1.84010i
$$259$$ −4.82843 −0.300024
$$260$$ 0 0
$$261$$ 1.41421 0.0875376
$$262$$ − 20.7279i − 1.28058i
$$263$$ 23.4558i 1.44635i 0.690665 + 0.723175i $$0.257318\pi$$
−0.690665 + 0.723175i $$0.742682\pi$$
$$264$$ 9.89949 0.609272
$$265$$ 0 0
$$266$$ 3.41421 0.209339
$$267$$ 5.89949i 0.361043i
$$268$$ 36.9706i 2.25834i
$$269$$ 3.07107 0.187246 0.0936232 0.995608i $$-0.470155\pi$$
0.0936232 + 0.995608i $$0.470155\pi$$
$$270$$ 0 0
$$271$$ −10.8284 −0.657780 −0.328890 0.944368i $$-0.606675\pi$$
−0.328890 + 0.944368i $$0.606675\pi$$
$$272$$ 3.51472i 0.213111i
$$273$$ − 4.82843i − 0.292230i
$$274$$ −24.1421 −1.45848
$$275$$ 0 0
$$276$$ 29.3137 1.76448
$$277$$ − 22.0000i − 1.32185i −0.750451 0.660926i $$-0.770164\pi$$
0.750451 0.660926i $$-0.229836\pi$$
$$278$$ 34.1421i 2.04771i
$$279$$ 3.17157 0.189877
$$280$$ 0 0
$$281$$ −14.5858 −0.870115 −0.435058 0.900403i $$-0.643272\pi$$
−0.435058 + 0.900403i $$0.643272\pi$$
$$282$$ − 18.4853i − 1.10078i
$$283$$ 10.3848i 0.617311i 0.951174 + 0.308655i $$0.0998790\pi$$
−0.951174 + 0.308655i $$0.900121\pi$$
$$284$$ 41.4558 2.45995
$$285$$ 0 0
$$286$$ 18.4853 1.09306
$$287$$ 0.343146i 0.0202553i
$$288$$ 1.58579i 0.0934434i
$$289$$ 15.6274 0.919260
$$290$$ 0 0
$$291$$ −9.75736 −0.571987
$$292$$ − 29.3137i − 1.71546i
$$293$$ 11.5147i 0.672697i 0.941738 + 0.336349i $$0.109192\pi$$
−0.941738 + 0.336349i $$0.890808\pi$$
$$294$$ 12.0711 0.703999
$$295$$ 0 0
$$296$$ −15.0711 −0.875988
$$297$$ − 2.24264i − 0.130131i
$$298$$ 14.4853i 0.839110i
$$299$$ 26.1421 1.51184
$$300$$ 0 0
$$301$$ −17.3137 −0.997946
$$302$$ − 15.6569i − 0.900951i
$$303$$ 3.17157i 0.182202i
$$304$$ 3.00000 0.172062
$$305$$ 0 0
$$306$$ 2.82843 0.161690
$$307$$ 21.1716i 1.20833i 0.796861 + 0.604163i $$0.206492\pi$$
−0.796861 + 0.604163i $$0.793508\pi$$
$$308$$ − 12.1421i − 0.691862i
$$309$$ 7.31371 0.416062
$$310$$ 0 0
$$311$$ −6.24264 −0.353988 −0.176994 0.984212i $$-0.556637\pi$$
−0.176994 + 0.984212i $$0.556637\pi$$
$$312$$ − 15.0711i − 0.853231i
$$313$$ − 5.79899i − 0.327778i −0.986479 0.163889i $$-0.947596\pi$$
0.986479 0.163889i $$-0.0524039\pi$$
$$314$$ −25.3137 −1.42854
$$315$$ 0 0
$$316$$ 0 0
$$317$$ − 26.6274i − 1.49554i −0.663955 0.747772i $$-0.731123\pi$$
0.663955 0.747772i $$-0.268877\pi$$
$$318$$ − 19.3137i − 1.08306i
$$319$$ 3.17157 0.177574
$$320$$ 0 0
$$321$$ 19.3137 1.07799
$$322$$ − 26.1421i − 1.45684i
$$323$$ 1.17157i 0.0651881i
$$324$$ −3.82843 −0.212690
$$325$$ 0 0
$$326$$ 52.8701 2.92820
$$327$$ − 6.48528i − 0.358637i
$$328$$ 1.07107i 0.0591398i
$$329$$ −10.8284 −0.596991
$$330$$ 0 0
$$331$$ 28.1421 1.54683 0.773416 0.633899i $$-0.218547\pi$$
0.773416 + 0.633899i $$0.218547\pi$$
$$332$$ 49.1127i 2.69541i
$$333$$ 3.41421i 0.187098i
$$334$$ 41.7990 2.28714
$$335$$ 0 0
$$336$$ −4.24264 −0.231455
$$337$$ − 24.5858i − 1.33927i −0.742689 0.669637i $$-0.766450\pi$$
0.742689 0.669637i $$-0.233550\pi$$
$$338$$ 3.24264i 0.176376i
$$339$$ −10.1421 −0.550845
$$340$$ 0 0
$$341$$ 7.11270 0.385174
$$342$$ − 2.41421i − 0.130546i
$$343$$ − 16.9706i − 0.916324i
$$344$$ −54.0416 −2.91373
$$345$$ 0 0
$$346$$ −47.7990 −2.56969
$$347$$ − 27.4558i − 1.47391i −0.675943 0.736953i $$-0.736264\pi$$
0.675943 0.736953i $$-0.263736\pi$$
$$348$$ − 5.41421i − 0.290232i
$$349$$ −18.0000 −0.963518 −0.481759 0.876304i $$-0.660002\pi$$
−0.481759 + 0.876304i $$0.660002\pi$$
$$350$$ 0 0
$$351$$ −3.41421 −0.182237
$$352$$ 3.55635i 0.189554i
$$353$$ − 3.65685i − 0.194635i −0.995253 0.0973174i $$-0.968974\pi$$
0.995253 0.0973174i $$-0.0310262\pi$$
$$354$$ −30.1421 −1.60204
$$355$$ 0 0
$$356$$ 22.5858 1.19704
$$357$$ − 1.65685i − 0.0876900i
$$358$$ − 39.7990i − 2.10344i
$$359$$ −24.8701 −1.31259 −0.656296 0.754504i $$-0.727878\pi$$
−0.656296 + 0.754504i $$0.727878\pi$$
$$360$$ 0 0
$$361$$ 1.00000 0.0526316
$$362$$ − 50.2843i − 2.64288i
$$363$$ 5.97056i 0.313373i
$$364$$ −18.4853 −0.968892
$$365$$ 0 0
$$366$$ 17.6569 0.922939
$$367$$ − 14.3848i − 0.750879i −0.926847 0.375440i $$-0.877492\pi$$
0.926847 0.375440i $$-0.122508\pi$$
$$368$$ − 22.9706i − 1.19742i
$$369$$ 0.242641 0.0126314
$$370$$ 0 0
$$371$$ −11.3137 −0.587378
$$372$$ − 12.1421i − 0.629540i
$$373$$ − 0.585786i − 0.0303309i −0.999885 0.0151654i $$-0.995173\pi$$
0.999885 0.0151654i $$-0.00482749\pi$$
$$374$$ 6.34315 0.327996
$$375$$ 0 0
$$376$$ −33.7990 −1.74305
$$377$$ − 4.82843i − 0.248677i
$$378$$ 3.41421i 0.175608i
$$379$$ −3.17157 −0.162913 −0.0814564 0.996677i $$-0.525957\pi$$
−0.0814564 + 0.996677i $$0.525957\pi$$
$$380$$ 0 0
$$381$$ 19.3137 0.989471
$$382$$ 24.7279i 1.26519i
$$383$$ − 28.0000i − 1.43073i −0.698749 0.715367i $$-0.746260\pi$$
0.698749 0.715367i $$-0.253740\pi$$
$$384$$ 20.5563 1.04901
$$385$$ 0 0
$$386$$ 12.2426 0.623134
$$387$$ 12.2426i 0.622328i
$$388$$ 37.3553i 1.89643i
$$389$$ −30.9706 −1.57027 −0.785135 0.619325i $$-0.787406\pi$$
−0.785135 + 0.619325i $$0.787406\pi$$
$$390$$ 0 0
$$391$$ 8.97056 0.453661
$$392$$ − 22.0711i − 1.11476i
$$393$$ 8.58579i 0.433096i
$$394$$ −16.4853 −0.830516
$$395$$ 0 0
$$396$$ −8.58579 −0.431452
$$397$$ 28.6274i 1.43677i 0.695646 + 0.718384i $$0.255118\pi$$
−0.695646 + 0.718384i $$0.744882\pi$$
$$398$$ − 43.7990i − 2.19544i
$$399$$ −1.41421 −0.0707992
$$400$$ 0 0
$$401$$ −7.75736 −0.387384 −0.193692 0.981062i $$-0.562046\pi$$
−0.193692 + 0.981062i $$0.562046\pi$$
$$402$$ − 23.3137i − 1.16278i
$$403$$ − 10.8284i − 0.539402i
$$404$$ 12.1421 0.604094
$$405$$ 0 0
$$406$$ −4.82843 −0.239631
$$407$$ 7.65685i 0.379536i
$$408$$ − 5.17157i − 0.256031i
$$409$$ 12.8284 0.634325 0.317162 0.948371i $$-0.397270\pi$$
0.317162 + 0.948371i $$0.397270\pi$$
$$410$$ 0 0
$$411$$ 10.0000 0.493264
$$412$$ − 28.0000i − 1.37946i
$$413$$ 17.6569i 0.868837i
$$414$$ −18.4853 −0.908502
$$415$$ 0 0
$$416$$ 5.41421 0.265454
$$417$$ − 14.1421i − 0.692543i
$$418$$ − 5.41421i − 0.264818i
$$419$$ −3.41421 −0.166795 −0.0833976 0.996516i $$-0.526577\pi$$
−0.0833976 + 0.996516i $$0.526577\pi$$
$$420$$ 0 0
$$421$$ 9.31371 0.453922 0.226961 0.973904i $$-0.427121\pi$$
0.226961 + 0.973904i $$0.427121\pi$$
$$422$$ 17.6569i 0.859522i
$$423$$ 7.65685i 0.372289i
$$424$$ −35.3137 −1.71499
$$425$$ 0 0
$$426$$ −26.1421 −1.26659
$$427$$ − 10.3431i − 0.500540i
$$428$$ − 73.9411i − 3.57408i
$$429$$ −7.65685 −0.369676
$$430$$ 0 0
$$431$$ −31.1127 −1.49865 −0.749323 0.662205i $$-0.769621\pi$$
−0.749323 + 0.662205i $$0.769621\pi$$
$$432$$ 3.00000i 0.144338i
$$433$$ 10.9289i 0.525211i 0.964903 + 0.262605i $$0.0845818\pi$$
−0.964903 + 0.262605i $$0.915418\pi$$
$$434$$ −10.8284 −0.519781
$$435$$ 0 0
$$436$$ −24.8284 −1.18907
$$437$$ − 7.65685i − 0.366277i
$$438$$ 18.4853i 0.883261i
$$439$$ 21.6569 1.03363 0.516813 0.856099i $$-0.327118\pi$$
0.516813 + 0.856099i $$0.327118\pi$$
$$440$$ 0 0
$$441$$ −5.00000 −0.238095
$$442$$ − 9.65685i − 0.459330i
$$443$$ − 18.0000i − 0.855206i −0.903967 0.427603i $$-0.859358\pi$$
0.903967 0.427603i $$-0.140642\pi$$
$$444$$ 13.0711 0.620325
$$445$$ 0 0
$$446$$ 44.9706 2.12942
$$447$$ − 6.00000i − 0.283790i
$$448$$ − 13.8995i − 0.656689i
$$449$$ 26.8701 1.26808 0.634038 0.773302i $$-0.281396\pi$$
0.634038 + 0.773302i $$0.281396\pi$$
$$450$$ 0 0
$$451$$ 0.544156 0.0256233
$$452$$ 38.8284i 1.82634i
$$453$$ 6.48528i 0.304705i
$$454$$ −36.1421 −1.69623
$$455$$ 0 0
$$456$$ −4.41421 −0.206714
$$457$$ 32.8284i 1.53565i 0.640660 + 0.767825i $$0.278661\pi$$
−0.640660 + 0.767825i $$0.721339\pi$$
$$458$$ 54.6274i 2.55257i
$$459$$ −1.17157 −0.0546843
$$460$$ 0 0
$$461$$ 19.6569 0.915511 0.457755 0.889078i $$-0.348654\pi$$
0.457755 + 0.889078i $$0.348654\pi$$
$$462$$ 7.65685i 0.356229i
$$463$$ − 24.2426i − 1.12665i −0.826235 0.563326i $$-0.809522\pi$$
0.826235 0.563326i $$-0.190478\pi$$
$$464$$ −4.24264 −0.196960
$$465$$ 0 0
$$466$$ −0.828427 −0.0383761
$$467$$ 35.6569i 1.65000i 0.565131 + 0.825001i $$0.308826\pi$$
−0.565131 + 0.825001i $$0.691174\pi$$
$$468$$ 13.0711i 0.604210i
$$469$$ −13.6569 −0.630615
$$470$$ 0 0
$$471$$ 10.4853 0.483136
$$472$$ 55.1127i 2.53677i
$$473$$ 27.4558i 1.26242i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −6.34315 −0.290738
$$477$$ 8.00000i 0.366295i
$$478$$ 64.5269i 2.95139i
$$479$$ −17.0711 −0.779997 −0.389998 0.920815i $$-0.627524\pi$$
−0.389998 + 0.920815i $$0.627524\pi$$
$$480$$ 0 0
$$481$$ 11.6569 0.531507
$$482$$ − 47.4558i − 2.16155i
$$483$$ 10.8284i 0.492710i
$$484$$ 22.8579 1.03899
$$485$$ 0 0
$$486$$ 2.41421 0.109511
$$487$$ 20.4853i 0.928277i 0.885763 + 0.464138i $$0.153636\pi$$
−0.885763 + 0.464138i $$0.846364\pi$$
$$488$$ − 32.2843i − 1.46144i
$$489$$ −21.8995 −0.990329
$$490$$ 0 0
$$491$$ −1.75736 −0.0793085 −0.0396543 0.999213i $$-0.512626\pi$$
−0.0396543 + 0.999213i $$0.512626\pi$$
$$492$$ − 0.928932i − 0.0418795i
$$493$$ − 1.65685i − 0.0746210i
$$494$$ −8.24264 −0.370854
$$495$$ 0 0
$$496$$ −9.51472 −0.427223
$$497$$ 15.3137i 0.686914i
$$498$$ − 30.9706i − 1.38782i
$$499$$ 5.17157 0.231511 0.115756 0.993278i $$-0.463071\pi$$
0.115756 + 0.993278i $$0.463071\pi$$
$$500$$ 0 0
$$501$$ −17.3137 −0.773519
$$502$$ − 4.24264i − 0.189358i
$$503$$ − 4.82843i − 0.215289i −0.994189 0.107644i $$-0.965669\pi$$
0.994189 0.107644i $$-0.0343308\pi$$
$$504$$ 6.24264 0.278069
$$505$$ 0 0
$$506$$ −41.4558 −1.84294
$$507$$ − 1.34315i − 0.0596512i
$$508$$ − 73.9411i − 3.28061i
$$509$$ 0.727922 0.0322646 0.0161323 0.999870i $$-0.494865\pi$$
0.0161323 + 0.999870i $$0.494865\pi$$
$$510$$ 0 0
$$511$$ 10.8284 0.479021
$$512$$ − 31.2426i − 1.38074i
$$513$$ 1.00000i 0.0441511i
$$514$$ 10.8284 0.477621
$$515$$ 0 0
$$516$$ 46.8701 2.06334
$$517$$ 17.1716i 0.755205i
$$518$$ − 11.6569i − 0.512173i
$$519$$ 19.7990 0.869079
$$520$$ 0 0
$$521$$ 7.75736 0.339856 0.169928 0.985456i $$-0.445646\pi$$
0.169928 + 0.985456i $$0.445646\pi$$
$$522$$ 3.41421i 0.149436i
$$523$$ 15.5147i 0.678411i 0.940712 + 0.339206i $$0.110158\pi$$
−0.940712 + 0.339206i $$0.889842\pi$$
$$524$$ 32.8701 1.43594
$$525$$ 0 0
$$526$$ −56.6274 −2.46907
$$527$$ − 3.71573i − 0.161860i
$$528$$ 6.72792i 0.292795i
$$529$$ −35.6274 −1.54902
$$530$$ 0 0
$$531$$ 12.4853 0.541815
$$532$$ 5.41421i 0.234736i
$$533$$ − 0.828427i − 0.0358832i
$$534$$ −14.2426 −0.616339
$$535$$ 0 0
$$536$$ −42.6274 −1.84122
$$537$$ 16.4853i 0.711392i
$$538$$ 7.41421i 0.319649i
$$539$$ −11.2132 −0.482987
$$540$$ 0 0
$$541$$ 22.0000 0.945854 0.472927 0.881102i $$-0.343197\pi$$
0.472927 + 0.881102i $$0.343197\pi$$
$$542$$ − 26.1421i − 1.12290i
$$543$$ 20.8284i 0.893833i
$$544$$ 1.85786 0.0796553
$$545$$ 0 0
$$546$$ 11.6569 0.498867
$$547$$ 24.4853i 1.04692i 0.852052 + 0.523458i $$0.175358\pi$$
−0.852052 + 0.523458i $$0.824642\pi$$
$$548$$ − 38.2843i − 1.63542i
$$549$$ −7.31371 −0.312141
$$550$$ 0 0
$$551$$ −1.41421 −0.0602475
$$552$$ 33.7990i 1.43858i
$$553$$ 0 0
$$554$$ 53.1127 2.25654
$$555$$ 0 0
$$556$$ −54.1421 −2.29614
$$557$$ − 25.3137i − 1.07258i −0.844035 0.536288i $$-0.819826\pi$$
0.844035 0.536288i $$-0.180174\pi$$
$$558$$ 7.65685i 0.324140i
$$559$$ 41.7990 1.76791
$$560$$ 0 0
$$561$$ −2.62742 −0.110930
$$562$$ − 35.2132i − 1.48538i
$$563$$ 30.2843i 1.27633i 0.769900 + 0.638165i $$0.220306\pi$$
−0.769900 + 0.638165i $$0.779694\pi$$
$$564$$ 29.3137 1.23433
$$565$$ 0 0
$$566$$ −25.0711 −1.05382
$$567$$ − 1.41421i − 0.0593914i
$$568$$ 47.7990i 2.00560i
$$569$$ −31.0711 −1.30257 −0.651283 0.758835i $$-0.725769\pi$$
−0.651283 + 0.758835i $$0.725769\pi$$
$$570$$ 0 0
$$571$$ 9.17157 0.383818 0.191909 0.981413i $$-0.438532\pi$$
0.191909 + 0.981413i $$0.438532\pi$$
$$572$$ 29.3137i 1.22567i
$$573$$ − 10.2426i − 0.427892i
$$574$$ −0.828427 −0.0345779
$$575$$ 0 0
$$576$$ −9.82843 −0.409518
$$577$$ 19.4558i 0.809957i 0.914326 + 0.404979i $$0.132721\pi$$
−0.914326 + 0.404979i $$0.867279\pi$$
$$578$$ 37.7279i 1.56927i
$$579$$ −5.07107 −0.210746
$$580$$ 0 0
$$581$$ −18.1421 −0.752663
$$582$$ − 23.5563i − 0.976442i
$$583$$ 17.9411i 0.743045i
$$584$$ 33.7990 1.39861
$$585$$ 0 0
$$586$$ −27.7990 −1.14837
$$587$$ − 12.3431i − 0.509456i −0.967013 0.254728i $$-0.918014\pi$$
0.967013 0.254728i $$-0.0819860\pi$$
$$588$$ 19.1421i 0.789408i
$$589$$ −3.17157 −0.130682
$$590$$ 0 0
$$591$$ 6.82843 0.280884
$$592$$ − 10.2426i − 0.420970i
$$593$$ − 36.6274i − 1.50411i −0.659102 0.752054i $$-0.729063\pi$$
0.659102 0.752054i $$-0.270937\pi$$
$$594$$ 5.41421 0.222148
$$595$$ 0 0
$$596$$ −22.9706 −0.940911
$$597$$ 18.1421i 0.742508i
$$598$$ 63.1127i 2.58087i
$$599$$ 3.02944 0.123779 0.0618897 0.998083i $$-0.480287\pi$$
0.0618897 + 0.998083i $$0.480287\pi$$
$$600$$ 0 0
$$601$$ 8.14214 0.332125 0.166062 0.986115i $$-0.446895\pi$$
0.166062 + 0.986115i $$0.446895\pi$$
$$602$$ − 41.7990i − 1.70360i
$$603$$ 9.65685i 0.393258i
$$604$$ 24.8284 1.01025
$$605$$ 0 0
$$606$$ −7.65685 −0.311038
$$607$$ 16.4853i 0.669117i 0.942375 + 0.334558i $$0.108587\pi$$
−0.942375 + 0.334558i $$0.891413\pi$$
$$608$$ − 1.58579i − 0.0643121i
$$609$$ 2.00000 0.0810441
$$610$$ 0 0
$$611$$ 26.1421 1.05760
$$612$$ 4.48528i 0.181307i
$$613$$ 10.4853i 0.423497i 0.977324 + 0.211748i $$0.0679157\pi$$
−0.977324 + 0.211748i $$0.932084\pi$$
$$614$$ −51.1127 −2.06274
$$615$$ 0 0
$$616$$ 14.0000 0.564076
$$617$$ 28.4853i 1.14677i 0.819285 + 0.573387i $$0.194371\pi$$
−0.819285 + 0.573387i $$0.805629\pi$$
$$618$$ 17.6569i 0.710263i
$$619$$ 20.4853 0.823373 0.411686 0.911326i $$-0.364940\pi$$
0.411686 + 0.911326i $$0.364940\pi$$
$$620$$ 0 0
$$621$$ 7.65685 0.307259
$$622$$ − 15.0711i − 0.604295i
$$623$$ 8.34315i 0.334261i
$$624$$ 10.2426 0.410034
$$625$$ 0 0
$$626$$ 14.0000 0.559553
$$627$$ 2.24264i 0.0895624i
$$628$$ − 40.1421i − 1.60185i
$$629$$ 4.00000 0.159490
$$630$$ 0 0
$$631$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$632$$ 0 0
$$633$$ − 7.31371i − 0.290694i
$$634$$ 64.2843 2.55305
$$635$$ 0 0
$$636$$ 30.6274 1.21446
$$637$$ 17.0711i 0.676380i
$$638$$ 7.65685i 0.303138i
$$639$$ 10.8284 0.428366
$$640$$ 0 0
$$641$$ −31.3553 −1.23846 −0.619231 0.785209i $$-0.712555\pi$$
−0.619231 + 0.785209i $$0.712555\pi$$
$$642$$ 46.6274i 1.84024i
$$643$$ − 42.3848i − 1.67149i −0.549116 0.835746i $$-0.685035\pi$$
0.549116 0.835746i $$-0.314965\pi$$
$$644$$ 41.4558 1.63359
$$645$$ 0 0
$$646$$ −2.82843 −0.111283
$$647$$ − 36.1421i − 1.42089i −0.703751 0.710447i $$-0.748493\pi$$
0.703751 0.710447i $$-0.251507\pi$$
$$648$$ − 4.41421i − 0.173407i
$$649$$ 28.0000 1.09910
$$650$$ 0 0
$$651$$ 4.48528 0.175792
$$652$$ 83.8406i 3.28345i
$$653$$ − 42.8284i − 1.67601i −0.545666 0.838003i $$-0.683723\pi$$
0.545666 0.838003i $$-0.316277\pi$$
$$654$$ 15.6569 0.612231
$$655$$ 0 0
$$656$$ −0.727922 −0.0284206
$$657$$ − 7.65685i − 0.298722i
$$658$$ − 26.1421i − 1.01913i
$$659$$ 18.6274 0.725621 0.362811 0.931863i $$-0.381817\pi$$
0.362811 + 0.931863i $$0.381817\pi$$
$$660$$ 0 0
$$661$$ 7.45584 0.289999 0.144999 0.989432i $$-0.453682\pi$$
0.144999 + 0.989432i $$0.453682\pi$$
$$662$$ 67.9411i 2.64061i
$$663$$ 4.00000i 0.155347i
$$664$$ −56.6274 −2.19757
$$665$$ 0 0
$$666$$ −8.24264 −0.319396
$$667$$ 10.8284i 0.419278i
$$668$$ 66.2843i 2.56462i
$$669$$ −18.6274 −0.720178
$$670$$ 0 0
$$671$$ −16.4020 −0.633193
$$672$$ 2.24264i 0.0865117i
$$673$$ 44.1838i 1.70316i 0.524225 + 0.851580i $$0.324355\pi$$
−0.524225 + 0.851580i $$0.675645\pi$$
$$674$$ 59.3553 2.28628
$$675$$ 0 0
$$676$$ −5.14214 −0.197774
$$677$$ 16.9706i 0.652232i 0.945330 + 0.326116i $$0.105740\pi$$
−0.945330 + 0.326116i $$0.894260\pi$$
$$678$$ − 24.4853i − 0.940352i
$$679$$ −13.7990 −0.529557
$$680$$ 0 0
$$681$$ 14.9706 0.573673
$$682$$ 17.1716i 0.657534i
$$683$$ − 18.3431i − 0.701881i −0.936398 0.350940i $$-0.885862\pi$$
0.936398 0.350940i $$-0.114138\pi$$
$$684$$ 3.82843 0.146384
$$685$$ 0 0
$$686$$ 40.9706 1.56426
$$687$$ − 22.6274i − 0.863290i
$$688$$ − 36.7279i − 1.40024i
$$689$$ 27.3137 1.04057
$$690$$ 0 0
$$691$$ 46.8284 1.78144 0.890719 0.454555i $$-0.150202\pi$$
0.890719 + 0.454555i $$0.150202\pi$$
$$692$$ − 75.7990i − 2.88145i
$$693$$ − 3.17157i − 0.120478i
$$694$$ 66.2843 2.51612
$$695$$ 0 0
$$696$$ 6.24264 0.236627
$$697$$ − 0.284271i − 0.0107675i
$$698$$ − 43.4558i − 1.64483i
$$699$$ 0.343146 0.0129790
$$700$$ 0 0
$$701$$ 6.68629 0.252538 0.126269 0.991996i $$-0.459700\pi$$
0.126269 + 0.991996i $$0.459700\pi$$
$$702$$ − 8.24264i − 0.311098i
$$703$$ − 3.41421i − 0.128770i
$$704$$ −22.0416 −0.830725
$$705$$ 0 0
$$706$$ 8.82843 0.332262
$$707$$ 4.48528i 0.168686i
$$708$$ − 47.7990i − 1.79640i
$$709$$ −28.9706 −1.08801 −0.544006 0.839081i $$-0.683093\pi$$
−0.544006 + 0.839081i $$0.683093\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 26.0416i 0.975951i
$$713$$ 24.2843i 0.909453i
$$714$$ 4.00000 0.149696
$$715$$ 0 0
$$716$$ 63.1127 2.35863
$$717$$ − 26.7279i − 0.998173i
$$718$$ − 60.0416i − 2.24073i
$$719$$ 1.07107 0.0399441 0.0199720 0.999801i $$-0.493642\pi$$
0.0199720 + 0.999801i $$0.493642\pi$$
$$720$$ 0 0
$$721$$ 10.3431 0.385199
$$722$$ 2.41421i 0.0898477i
$$723$$ 19.6569i 0.731046i
$$724$$ 79.7401 2.96352
$$725$$ 0 0
$$726$$ −14.4142 −0.534962
$$727$$ 47.3553i 1.75631i 0.478374 + 0.878156i $$0.341226\pi$$
−0.478374 + 0.878156i $$0.658774\pi$$
$$728$$ − 21.3137i − 0.789939i
$$729$$ −1.00000 −0.0370370
$$730$$ 0 0
$$731$$ 14.3431 0.530500
$$732$$ 28.0000i 1.03491i
$$733$$ 22.0000i 0.812589i 0.913742 + 0.406294i $$0.133179\pi$$
−0.913742 + 0.406294i $$0.866821\pi$$
$$734$$ 34.7279 1.28183
$$735$$ 0 0
$$736$$ −12.1421 −0.447565
$$737$$ 21.6569i 0.797740i
$$738$$ 0.585786i 0.0215631i
$$739$$ −14.3431 −0.527621 −0.263811 0.964575i $$-0.584979\pi$$
−0.263811 + 0.964575i $$0.584979\pi$$
$$740$$ 0 0
$$741$$ 3.41421 0.125424
$$742$$ − 27.3137i − 1.00272i
$$743$$ 4.00000i 0.146746i 0.997305 + 0.0733729i $$0.0233763\pi$$
−0.997305 + 0.0733729i $$0.976624\pi$$
$$744$$ 14.0000 0.513265
$$745$$ 0 0
$$746$$ 1.41421 0.0517780
$$747$$ 12.8284i 0.469368i
$$748$$ 10.0589i 0.367789i
$$749$$ 27.3137 0.998021
$$750$$ 0 0
$$751$$ −24.1421 −0.880959 −0.440480 0.897763i $$-0.645192\pi$$
−0.440480 + 0.897763i $$0.645192\pi$$
$$752$$ − 22.9706i − 0.837650i
$$753$$ 1.75736i 0.0640417i
$$754$$ 11.6569 0.424518
$$755$$ 0 0
$$756$$ −5.41421 −0.196913
$$757$$ − 32.4264i − 1.17856i −0.807930 0.589279i $$-0.799412\pi$$
0.807930 0.589279i $$-0.200588\pi$$
$$758$$ − 7.65685i − 0.278109i
$$759$$ 17.1716 0.623289
$$760$$ 0 0
$$761$$ 43.9411 1.59286 0.796432 0.604728i $$-0.206718\pi$$
0.796432 + 0.604728i $$0.206718\pi$$
$$762$$ 46.6274i 1.68913i
$$763$$ − 9.17157i − 0.332033i
$$764$$ −39.2132 −1.41868
$$765$$ 0 0
$$766$$ 67.5980 2.44241
$$767$$ − 42.6274i − 1.53919i
$$768$$ 29.9706i 1.08147i
$$769$$ −42.9706 −1.54956 −0.774779 0.632232i $$-0.782139\pi$$
−0.774779 + 0.632232i $$0.782139\pi$$
$$770$$ 0 0
$$771$$ −4.48528 −0.161533
$$772$$ 19.4142i 0.698733i
$$773$$ 25.6569i 0.922813i 0.887189 + 0.461406i $$0.152655\pi$$
−0.887189 + 0.461406i $$0.847345\pi$$
$$774$$ −29.5563 −1.06238
$$775$$ 0 0
$$776$$ −43.0711 −1.54616
$$777$$ 4.82843i 0.173219i
$$778$$ − 74.7696i − 2.68062i
$$779$$ −0.242641 −0.00869350
$$780$$ 0 0
$$781$$ 24.2843 0.868960
$$782$$ 21.6569i 0.774448i
$$783$$ − 1.41421i − 0.0505399i
$$784$$ 15.0000 0.535714
$$785$$ 0 0
$$786$$ −20.7279 −0.739340
$$787$$ − 42.8284i − 1.52667i −0.646004 0.763334i $$-0.723561\pi$$
0.646004 0.763334i $$-0.276439\pi$$
$$788$$ − 26.1421i − 0.931275i
$$789$$ 23.4558 0.835050
$$790$$ 0 0
$$791$$ −14.3431 −0.509984
$$792$$ − 9.89949i − 0.351763i
$$793$$ 24.9706i 0.886731i
$$794$$ −69.1127 −2.45272
$$795$$ 0 0
$$796$$ 69.4558 2.46180
$$797$$ − 14.1421i − 0.500940i −0.968124 0.250470i $$-0.919415\pi$$
0.968124 0.250470i $$-0.0805852\pi$$
$$798$$ − 3.41421i − 0.120862i
$$799$$ 8.97056 0.317356
$$800$$ 0 0
$$801$$ 5.89949 0.208448
$$802$$ − 18.7279i − 0.661306i
$$803$$ − 17.1716i − 0.605972i
$$804$$ 36.9706 1.30385
$$805$$ 0 0
$$806$$ 26.1421 0.920817
$$807$$ − 3.07107i − 0.108107i
$$808$$ 14.0000i 0.492518i
$$809$$ 2.00000 0.0703163 0.0351581 0.999382i $$-0.488807\pi$$
0.0351581 + 0.999382i $$0.488807\pi$$
$$810$$ 0 0
$$811$$ 8.68629 0.305017 0.152508 0.988302i $$-0.451265\pi$$
0.152508 + 0.988302i $$0.451265\pi$$
$$812$$ − 7.65685i − 0.268703i
$$813$$ 10.8284i 0.379770i
$$814$$ −18.4853 −0.647909
$$815$$ 0 0
$$816$$ 3.51472 0.123040
$$817$$ − 12.2426i − 0.428316i
$$818$$ 30.9706i 1.08286i
$$819$$ −4.82843 −0.168719
$$820$$ 0 0
$$821$$ 14.4853 0.505540 0.252770 0.967526i $$-0.418658\pi$$
0.252770 + 0.967526i $$0.418658\pi$$
$$822$$ 24.1421i 0.842054i
$$823$$ 25.0122i 0.871870i 0.899978 + 0.435935i $$0.143582\pi$$
−0.899978 + 0.435935i $$0.856418\pi$$
$$824$$ 32.2843 1.12468
$$825$$ 0 0
$$826$$ −42.6274 −1.48320
$$827$$ − 2.68629i − 0.0934115i −0.998909 0.0467058i $$-0.985128\pi$$
0.998909 0.0467058i $$-0.0148723\pi$$
$$828$$ − 29.3137i − 1.01872i
$$829$$ −46.4853 −1.61450 −0.807250 0.590209i $$-0.799045\pi$$
−0.807250 + 0.590209i $$0.799045\pi$$
$$830$$ 0 0
$$831$$ −22.0000 −0.763172
$$832$$ 33.5563i 1.16336i
$$833$$ 5.85786i 0.202963i
$$834$$ 34.1421 1.18225
$$835$$ 0 0
$$836$$ 8.58579 0.296946
$$837$$ − 3.17157i − 0.109626i
$$838$$ − 8.24264i − 0.284737i
$$839$$ 9.85786 0.340331 0.170166 0.985415i $$-0.445570\pi$$
0.170166 + 0.985415i $$0.445570\pi$$
$$840$$ 0 0
$$841$$ −27.0000 −0.931034
$$842$$ 22.4853i 0.774894i
$$843$$ 14.5858i 0.502361i
$$844$$ −28.0000 −0.963800
$$845$$ 0 0
$$846$$ −18.4853 −0.635537
$$847$$ 8.44365i 0.290127i
$$848$$ − 24.0000i − 0.824163i
$$849$$ 10.3848 0.356405
$$850$$ 0 0
$$851$$ −26.1421 −0.896141
$$852$$ − 41.4558i − 1.42025i
$$853$$ 16.8284i 0.576194i 0.957601 + 0.288097i $$0.0930226\pi$$
−0.957601 + 0.288097i $$0.906977\pi$$
$$854$$ 24.9706 0.854475
$$855$$ 0 0
$$856$$ 85.2548 2.91395
$$857$$ 12.6863i 0.433355i 0.976243 + 0.216678i $$0.0695221\pi$$
−0.976243 + 0.216678i $$0.930478\pi$$
$$858$$ − 18.4853i − 0.631077i
$$859$$ 49.9411 1.70397 0.851985 0.523567i $$-0.175399\pi$$
0.851985 + 0.523567i $$0.175399\pi$$
$$860$$ 0 0
$$861$$ 0.343146 0.0116944
$$862$$ − 75.1127i − 2.55835i
$$863$$ − 8.68629i − 0.295685i −0.989011 0.147842i $$-0.952767\pi$$
0.989011 0.147842i $$-0.0472328\pi$$
$$864$$ 1.58579 0.0539496
$$865$$ 0 0
$$866$$ −26.3848 −0.896591
$$867$$ − 15.6274i − 0.530735i
$$868$$ − 17.1716i − 0.582841i
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 32.9706 1.11716
$$872$$ − 28.6274i − 0.969447i
$$873$$ 9.75736i 0.330237i
$$874$$ 18.4853 0.625274
$$875$$ 0 0
$$876$$ −29.3137 −0.990418
$$877$$ − 34.9289i − 1.17947i −0.807598 0.589733i $$-0.799233\pi$$
0.807598 0.589733i $$-0.200767\pi$$
$$878$$ 52.2843i 1.76451i
$$879$$ 11.5147 0.388382
$$880$$ 0 0
$$881$$ −5.79899 −0.195373 −0.0976865 0.995217i $$-0.531144\pi$$
−0.0976865 + 0.995217i $$0.531144\pi$$
$$882$$ − 12.0711i − 0.406454i
$$883$$ 12.0416i 0.405233i 0.979258 + 0.202617i $$0.0649445\pi$$
−0.979258 + 0.202617i $$0.935055\pi$$
$$884$$ 15.3137 0.515056
$$885$$ 0 0
$$886$$ 43.4558 1.45993
$$887$$ 25.9411i 0.871018i 0.900184 + 0.435509i $$0.143432\pi$$
−0.900184 + 0.435509i $$0.856568\pi$$
$$888$$ 15.0711i 0.505752i
$$889$$ 27.3137 0.916072
$$890$$ 0 0
$$891$$ −2.24264 −0.0751313
$$892$$ 71.3137i 2.38776i
$$893$$ − 7.65685i − 0.256227i
$$894$$ 14.4853 0.484460
$$895$$ 0 0
$$896$$ 29.0711 0.971196
$$897$$ − 26.1421i − 0.872861i
$$898$$ 64.8701i 2.16474i
$$899$$ 4.48528 0.149593
$$900$$ 0 0
$$901$$ 9.37258 0.312246
$$902$$ 1.31371i 0.0437417i
$$903$$ 17.3137i 0.576164i
$$904$$ −44.7696 −1.48901
$$905$$ 0 0
$$906$$ −15.6569 −0.520164
$$907$$ − 18.1421i − 0.602400i −0.953561 0.301200i $$-0.902613\pi$$
0.953561 0.301200i $$-0.0973871\pi$$
$$908$$ − 57.3137i − 1.90202i
$$909$$ 3.17157 0.105194
$$910$$ 0 0
$$911$$ 8.68629 0.287790 0.143895 0.989593i $$-0.454037\pi$$
0.143895 + 0.989593i $$0.454037\pi$$
$$912$$ − 3.00000i − 0.0993399i
$$913$$ 28.7696i 0.952133i
$$914$$ −79.2548 −2.62152
$$915$$ 0 0
$$916$$ −86.6274 −2.86225
$$917$$ 12.1421i 0.400969i
$$918$$ − 2.82843i − 0.0933520i
$$919$$ 12.0000 0.395843 0.197922 0.980218i $$-0.436581\pi$$
0.197922 + 0.980218i $$0.436581\pi$$
$$920$$ 0 0
$$921$$ 21.1716 0.697627
$$922$$ 47.4558i 1.56287i
$$923$$ − 36.9706i − 1.21690i
$$924$$ −12.1421 −0.399447
$$925$$ 0 0
$$926$$ 58.5269 1.92331
$$927$$ − 7.31371i − 0.240214i
$$928$$ 2.24264i 0.0736183i
$$929$$ −33.1127 −1.08639 −0.543196 0.839606i $$-0.682786\pi$$
−0.543196 + 0.839606i $$0.682786\pi$$
$$930$$ 0 0
$$931$$ 5.00000 0.163868
$$932$$ − 1.31371i − 0.0430320i
$$933$$ 6.24264i 0.204375i
$$934$$ −86.0833 −2.81673
$$935$$ 0 0
$$936$$ −15.0711 −0.492613
$$937$$ − 29.7990i − 0.973491i −0.873544 0.486745i $$-0.838184\pi$$
0.873544 0.486745i $$-0.161816\pi$$
$$938$$ − 32.9706i − 1.07653i
$$939$$ −5.79899 −0.189243
$$940$$ 0 0
$$941$$ 22.8701 0.745543 0.372771 0.927923i $$-0.378408\pi$$
0.372771 + 0.927923i $$0.378408\pi$$
$$942$$ 25.3137i 0.824765i
$$943$$ 1.85786i 0.0605004i
$$944$$ −37.4558 −1.21908
$$945$$ 0 0
$$946$$ −66.2843 −2.15509
$$947$$ − 47.1716i − 1.53287i −0.642322 0.766435i $$-0.722029\pi$$
0.642322 0.766435i $$-0.277971\pi$$
$$948$$ 0 0
$$949$$ −26.1421 −0.848610
$$950$$ 0 0
$$951$$ −26.6274 −0.863453
$$952$$ − 7.31371i − 0.237039i
$$953$$ 53.4558i 1.73160i 0.500386 + 0.865802i $$0.333191\pi$$
−0.500386 + 0.865802i $$0.666809\pi$$
$$954$$ −19.3137 −0.625304
$$955$$ 0 0
$$956$$ −102.326 −3.30946
$$957$$ − 3.17157i − 0.102522i
$$958$$ − 41.2132i − 1.33154i
$$959$$ 14.1421 0.456673
$$960$$ 0 0
$$961$$ −20.9411 −0.675520
$$962$$ 28.1421i 0.907339i
$$963$$ − 19.3137i − 0.622376i
$$964$$ 75.2548 2.42379
$$965$$ 0 0
$$966$$ −26.1421 −0.841109
$$967$$ 8.04163i 0.258601i 0.991605 + 0.129301i $$0.0412732\pi$$
−0.991605 + 0.129301i $$0.958727\pi$$
$$968$$ 26.3553i 0.847093i
$$969$$ 1.17157 0.0376363
$$970$$ 0 0
$$971$$ 22.3431 0.717026 0.358513 0.933525i $$-0.383284\pi$$
0.358513 + 0.933525i $$0.383284\pi$$
$$972$$ 3.82843i 0.122797i
$$973$$ − 20.0000i − 0.641171i
$$974$$ −49.4558 −1.58467
$$975$$ 0 0
$$976$$ 21.9411 0.702318
$$977$$ 32.2843i 1.03287i 0.856328 + 0.516433i $$0.172740\pi$$
−0.856328 + 0.516433i $$0.827260\pi$$
$$978$$ − 52.8701i − 1.69060i
$$979$$ 13.2304 0.422847
$$980$$ 0 0
$$981$$ −6.48528 −0.207059
$$982$$ − 4.24264i − 0.135388i
$$983$$ 24.6274i 0.785493i 0.919647 + 0.392746i $$0.128475\pi$$
−0.919647 + 0.392746i $$0.871525\pi$$
$$984$$ 1.07107 0.0341444
$$985$$ 0 0
$$986$$ 4.00000 0.127386
$$987$$ 10.8284i 0.344673i
$$988$$ − 13.0711i − 0.415846i
$$989$$ −93.7401 −2.98076
$$990$$ 0 0
$$991$$ 2.34315 0.0744325 0.0372162 0.999307i $$-0.488151\pi$$
0.0372162 + 0.999307i $$0.488151\pi$$
$$992$$ 5.02944i 0.159685i
$$993$$ − 28.1421i − 0.893064i
$$994$$ −36.9706 −1.17264
$$995$$ 0 0
$$996$$ 49.1127 1.55620
$$997$$ − 38.0833i − 1.20611i −0.797700 0.603054i $$-0.793950\pi$$
0.797700 0.603054i $$-0.206050\pi$$
$$998$$ 12.4853i 0.395215i
$$999$$ 3.41421 0.108021
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.l.799.4 4
5.2 odd 4 1425.2.a.k.1.1 2
5.3 odd 4 285.2.a.g.1.2 2
5.4 even 2 inner 1425.2.c.l.799.1 4
15.2 even 4 4275.2.a.y.1.2 2
15.8 even 4 855.2.a.d.1.1 2
20.3 even 4 4560.2.a.bf.1.2 2
95.18 even 4 5415.2.a.n.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
285.2.a.g.1.2 2 5.3 odd 4
855.2.a.d.1.1 2 15.8 even 4
1425.2.a.k.1.1 2 5.2 odd 4
1425.2.c.l.799.1 4 5.4 even 2 inner
1425.2.c.l.799.4 4 1.1 even 1 trivial
4275.2.a.y.1.2 2 15.2 even 4
4560.2.a.bf.1.2 2 20.3 even 4
5415.2.a.n.1.1 2 95.18 even 4