LMFDB - Embedded newform 162.3.b.a.161.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,3,Mod(161,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("162.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	162.b (of order $2$, degree $1$, 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:	$4.41418028264$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	162.161
Dual form			162.3.b.a.161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -2.00000 q^{4} +5.19615i q^{5} +6.34847 q^{7} +2.82843i q^{8} +O(q^{10})$	\(q-1.41421i q^{2} -2.00000 q^{4} +5.19615i q^{5} +6.34847 q^{7} +2.82843i q^{8} +7.34847 q^{10} +9.43879i q^{11} +19.6969 q^{13} -8.97809i q^{14} +4.00000 q^{16} +1.90702i q^{17} +4.69694 q^{19} -10.3923i q^{20} +13.3485 q^{22} -9.43879i q^{23} -2.00000 q^{25} -27.8557i q^{26} -12.6969 q^{28} -3.28913i q^{29} -41.0454 q^{31} -5.65685i q^{32} +2.69694 q^{34} +32.9876i q^{35} +17.3031 q^{37} -6.64247i q^{38} -14.6969 q^{40} +61.8289i q^{41} +0.954592 q^{43} -18.8776i q^{44} -13.3485 q^{46} -14.1100i q^{47} -8.69694 q^{49} +2.82843i q^{50} -39.3939 q^{52} +9.53512i q^{53} -49.0454 q^{55} +17.9562i q^{56} -4.65153 q^{58} -91.5274i q^{59} -75.0908 q^{61} +58.0470i q^{62} -8.00000 q^{64} +102.348i q^{65} +30.9546 q^{67} -3.81405i q^{68} +46.6515 q^{70} -85.9026i q^{71} -96.0908 q^{73} -24.4702i q^{74} -9.39388 q^{76} +59.9219i q^{77} +29.7423 q^{79} +20.7846i q^{80} +87.4393 q^{82} -87.9060i q^{83} -9.90918 q^{85} -1.35000i q^{86} -26.6969 q^{88} +41.3766i q^{89} +125.045 q^{91} +18.8776i q^{92} -19.9546 q^{94} +24.4060i q^{95} +95.8786 q^{97} +12.2993i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 4 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 - 4 * q^7	$ 4 q - 8 q^{4} - 4 q^{7} + 20 q^{13} + 16 q^{16} - 40 q^{19} + 24 q^{22} - 8 q^{25} + 8 q^{28} - 76 q^{31} - 48 q^{34} + 128 q^{37} + 92 q^{43} - 24 q^{46} + 24 q^{49} - 40 q^{52} - 108 q^{55} - 48 q^{58} - 124 q^{61} - 32 q^{64} + 212 q^{67} + 216 q^{70} - 208 q^{73} + 80 q^{76} - 28 q^{79} + 144 q^{82} - 216 q^{85} - 48 q^{88} + 412 q^{91} - 168 q^{94} - 28 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 - 4 * q^7 + 20 * q^13 + 16 * q^16 - 40 * q^19 + 24 * q^22 - 8 * q^25 + 8 * q^28 - 76 * q^31 - 48 * q^34 + 128 * q^37 + 92 * q^43 - 24 * q^46 + 24 * q^49 - 40 * q^52 - 108 * q^55 - 48 * q^58 - 124 * q^61 - 32 * q^64 + 212 * q^67 + 216 * q^70 - 208 * q^73 + 80 * q^76 - 28 * q^79 + 144 * q^82 - 216 * q^85 - 48 * q^88 + 412 * q^91 - 168 * q^94 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$-1$

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 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	5.19615i	1.03923i	0.854400	+	0.519615i	$0.173925\pi$
$5$	5.19615i	1.03923i	−0.854400	+	0.519615i	$0.826075\pi$
$6$	0	0
$7$	6.34847	0.906924	0.453462	−	0.891276i	$-0.350189\pi$
$7$	6.34847	0.906924	0.453462	+	0.891276i	$0.350189\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	7.34847	0.734847
$11$	9.43879i	0.858072i	0.903287	+	0.429036i	$0.141147\pi$
$11$	9.43879i	0.858072i	−0.903287	+	0.429036i	$0.858853\pi$
$12$	0	0
$13$	19.6969	1.51515	0.757575	−	0.652749i	$-0.226384\pi$
$13$	19.6969	1.51515	0.757575	+	0.652749i	$0.226384\pi$
$14$	− 8.97809i	− 0.641292i
$15$	0	0
$16$	4.00000	0.250000
$17$	1.90702i	0.112178i	0.998426	+	0.0560889i	$0.0178630\pi$
$17$	1.90702i	0.112178i	−0.998426	+	0.0560889i	$0.982137\pi$
$18$	0	0
$19$	4.69694	0.247207	0.123604	−	0.992332i	$-0.460555\pi$
$19$	4.69694	0.247207	0.123604	+	0.992332i	$0.460555\pi$
$20$	− 10.3923i	− 0.519615i
$21$	0	0
$22$	13.3485	0.606749
$23$	− 9.43879i	− 0.410382i	−0.978722	−	0.205191i	$-0.934218\pi$
$23$	− 9.43879i	− 0.410382i	0.978722	−	0.205191i	$-0.0657816\pi$
$24$	0	0
$25$	−2.00000	−0.0800000
$26$	− 27.8557i	− 1.07137i
$27$	0	0
$28$	−12.6969	−0.453462
$29$	− 3.28913i	− 0.113418i	−0.998391	−	0.0567091i	$-0.981939\pi$
$29$	− 3.28913i	− 0.113418i	0.998391	−	0.0567091i	$-0.0180608\pi$
$30$	0	0
$31$	−41.0454	−1.32405	−0.662023	−	0.749484i	$-0.730302\pi$
$31$	−41.0454	−1.32405	−0.662023	+	0.749484i	$0.730302\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	2.69694	0.0793217
$35$	32.9876i	0.942503i
$36$	0	0
$37$	17.3031	0.467650	0.233825	−	0.972279i	$-0.424876\pi$
$37$	17.3031	0.467650	0.233825	+	0.972279i	$0.424876\pi$
$38$	− 6.64247i	− 0.174802i
$39$	0	0
$40$	−14.6969	−0.367423
$41$	61.8289i	1.50802i	0.656862	+	0.754011i	$0.271884\pi$
$41$	61.8289i	1.50802i	−0.656862	+	0.754011i	$0.728116\pi$
$42$	0	0
$43$	0.954592	0.0221998	0.0110999	−	0.999938i	$-0.496467\pi$
$43$	0.954592	0.0221998	0.0110999	+	0.999938i	$0.496467\pi$
$44$	− 18.8776i	− 0.429036i
$45$	0	0
$46$	−13.3485	−0.290184
$47$	− 14.1100i	− 0.300213i	−0.988670	−	0.150107i	$-0.952038\pi$
$47$	− 14.1100i	− 0.300213i	0.988670	−	0.150107i	$-0.0479617\pi$
$48$	0	0
$49$	−8.69694	−0.177489
$50$	2.82843i	0.0565685i
$51$	0	0
$52$	−39.3939	−0.757575
$53$	9.53512i	0.179908i	0.995946	+	0.0899539i	$0.0286720\pi$
$53$	9.53512i	0.179908i	−0.995946	+	0.0899539i	$0.971328\pi$
$54$	0	0
$55$	−49.0454	−0.891735
$56$	17.9562i	0.320646i
$57$	0	0
$58$	−4.65153	−0.0801988
$59$	− 91.5274i	− 1.55131i	−0.631156	−	0.775656i	$-0.717419\pi$
$59$	− 91.5274i	− 1.55131i	0.631156	−	0.775656i	$-0.282581\pi$
$60$	0	0
$61$	−75.0908	−1.23100	−0.615498	−	0.788138i	$-0.711045\pi$
$61$	−75.0908	−1.23100	−0.615498	+	0.788138i	$0.711045\pi$
$62$	58.0470i	0.936241i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	102.348i	1.57459i
$66$	0	0
$67$	30.9546	0.462009	0.231004	−	0.972953i	$-0.425799\pi$
$67$	30.9546	0.462009	0.231004	+	0.972953i	$0.425799\pi$
$68$	− 3.81405i	− 0.0560889i
$69$	0	0
$70$	46.6515	0.666450
$71$	− 85.9026i	− 1.20990i	−0.796265	−	0.604948i	$-0.793194\pi$
$71$	− 85.9026i	− 1.20990i	0.796265	−	0.604948i	$-0.206806\pi$
$72$	0	0
$73$	−96.0908	−1.31631	−0.658156	−	0.752881i	$-0.728663\pi$
$73$	−96.0908	−1.31631	−0.658156	+	0.752881i	$0.728663\pi$
$74$	− 24.4702i	− 0.330679i
$75$	0	0
$76$	−9.39388	−0.123604
$77$	59.9219i	0.778206i
$78$	0	0
$79$	29.7423	0.376485	0.188243	−	0.982123i	$-0.439721\pi$
$79$	29.7423	0.376485	0.188243	+	0.982123i	$0.439721\pi$
$80$	20.7846i	0.259808i
$81$	0	0
$82$	87.4393	1.06633
$83$	− 87.9060i	− 1.05911i	−0.848276	−	0.529554i	$-0.822359\pi$
$83$	− 87.9060i	− 1.05911i	0.848276	−	0.529554i	$-0.177641\pi$
$84$	0	0
$85$	−9.90918	−0.116579
$86$	− 1.35000i	− 0.0156976i
$87$	0	0
$88$	−26.6969	−0.303374
$89$	41.3766i	0.464905i	0.972608	+	0.232453i	$0.0746751\pi$
$89$	41.3766i	0.464905i	−0.972608	+	0.232453i	$0.925325\pi$
$90$	0	0
$91$	125.045	1.37413
$92$	18.8776i	0.205191i
$93$	0	0
$94$	−19.9546	−0.212283
$95$	24.4060i	0.256905i
$96$	0	0
$97$	95.8786	0.988439	0.494219	−	0.869337i	$-0.335454\pi$
$97$	95.8786	0.988439	0.494219	+	0.869337i	$0.335454\pi$
$98$	12.2993i	0.125503i
$99$	0	0
$100$	4.00000	0.0400000
$101$	− 157.931i	− 1.56368i	−0.623482	−	0.781838i	$-0.714283\pi$
$101$	− 157.931i	− 1.56368i	0.623482	−	0.781838i	$-0.285717\pi$
$102$	0	0
$103$	29.1362	0.282876	0.141438	−	0.989947i	$-0.454827\pi$
$103$	29.1362	0.282876	0.141438	+	0.989947i	$0.454827\pi$
$104$	55.7114i	0.535686i
$105$	0	0
$106$	13.4847	0.127214
$107$	− 171.805i	− 1.60566i	−0.596210	−	0.802829i	$-0.703327\pi$
$107$	− 171.805i	− 1.60566i	0.596210	−	0.802829i	$-0.296673\pi$
$108$	0	0
$109$	116.272	1.06672	0.533360	−	0.845888i	$-0.320929\pi$
$109$	116.272	1.06672	0.533360	+	0.845888i	$0.320929\pi$
$110$	69.3607i	0.630552i
$111$	0	0
$112$	25.3939	0.226731
$113$	− 202.265i	− 1.78995i	−0.446114	−	0.894976i	$-0.647192\pi$
$113$	− 202.265i	− 1.78995i	0.446114	−	0.894976i	$-0.352808\pi$
$114$	0	0
$115$	49.0454	0.426482
$116$	6.57826i	0.0567091i
$117$	0	0
$118$	−129.439	−1.09694
$119$	12.1067i	0.101737i
$120$	0	0
$121$	31.9092	0.263712
$122$	106.194i	0.870446i
$123$	0	0
$124$	82.0908	0.662023
$125$	119.512i	0.956092i
$126$	0	0
$127$	10.0908	0.0794552	0.0397276	−	0.999211i	$-0.487351\pi$
$127$	10.0908	0.0794552	0.0397276	+	0.999211i	$0.487351\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	144.742	1.11340
$131$	− 4.96021i	− 0.0378642i	−0.999821	−	0.0189321i	$-0.993973\pi$
$131$	− 4.96021i	− 0.0378642i	0.999821	−	0.0189321i	$-0.00602663\pi$
$132$	0	0
$133$	29.8184	0.224198
$134$	− 43.7764i	− 0.326690i
$135$	0	0
$136$	−5.39388	−0.0396609
$137$	234.684i	1.71302i	0.516129	+	0.856511i	$0.327373\pi$
$137$	234.684i	1.71302i	−0.516129	+	0.856511i	$0.672627\pi$
$138$	0	0
$139$	106.530	0.766404	0.383202	−	0.923665i	$-0.374821\pi$
$139$	106.530	0.766404	0.383202	+	0.923665i	$0.374821\pi$
$140$	− 65.9752i	− 0.471252i
$141$	0	0
$142$	−121.485	−0.855526
$143$	185.915i	1.30011i
$144$	0	0
$145$	17.0908	0.117868
$146$	135.893i	0.930774i
$147$	0	0
$148$	−34.6061	−0.233825
$149$	105.113i	0.705453i	0.935726	+	0.352727i	$0.114745\pi$
$149$	105.113i	0.705453i	−0.935726	+	0.352727i	$0.885255\pi$
$150$	0	0
$151$	−285.227	−1.88892	−0.944460	−	0.328625i	$-0.893415\pi$
$151$	−285.227	−1.88892	−0.944460	+	0.328625i	$0.893415\pi$
$152$	13.2849i	0.0874010i
$153$	0	0
$154$	84.7423	0.550275
$155$	− 213.278i	− 1.37599i
$156$	0	0
$157$	−197.182	−1.25593	−0.627967	−	0.778240i	$-0.716113\pi$
$157$	−197.182	−1.25593	−0.627967	+	0.778240i	$0.716113\pi$
$158$	− 42.0620i	− 0.266215i
$159$	0	0
$160$	29.3939	0.183712
$161$	− 59.9219i	− 0.372186i
$162$	0	0
$163$	−249.060	−1.52798	−0.763988	−	0.645230i	$-0.776762\pi$
$163$	−249.060	−1.52798	−0.763988	+	0.645230i	$0.776762\pi$
$164$	− 123.658i	− 0.754011i
$165$	0	0
$166$	−124.318	−0.748903
$167$	48.4365i	0.290039i	0.989429	+	0.145019i	$0.0463244\pi$
$167$	48.4365i	0.290039i	−0.989429	+	0.145019i	$0.953676\pi$
$168$	0	0
$169$	218.969	1.29568
$170$	14.0137i	0.0824335i
$171$	0	0
$172$	−1.90918	−0.0110999
$173$	100.441i	0.580585i	0.956938	+	0.290293i	$0.0937527\pi$
$173$	100.441i	0.580585i	−0.956938	+	0.290293i	$0.906247\pi$
$174$	0	0
$175$	−12.6969	−0.0725539
$176$	37.7552i	0.214518i
$177$	0	0
$178$	58.5153	0.328738
$179$	− 285.071i	− 1.59257i	−0.604919	−	0.796287i	$-0.706794\pi$
$179$	− 285.071i	− 1.59257i	0.604919	−	0.796287i	$-0.293206\pi$
$180$	0	0
$181$	37.1214	0.205091	0.102545	−	0.994728i	$-0.467301\pi$
$181$	37.1214	0.205091	0.102545	+	0.994728i	$0.467301\pi$
$182$	− 176.841i	− 0.971653i
$183$	0	0
$184$	26.6969	0.145092
$185$	89.9093i	0.485996i
$186$	0	0
$187$	−18.0000	−0.0962567
$188$	28.2201i	0.150107i
$189$	0	0
$190$	34.5153	0.181660
$191$	− 17.9241i	− 0.0938433i	−0.998899	−	0.0469217i	$-0.985059\pi$
$191$	− 17.9241i	− 0.0938433i	0.998899	−	0.0469217i	$-0.0149411\pi$
$192$	0	0
$193$	−95.4541	−0.494581	−0.247290	−	0.968941i	$-0.579540\pi$
$193$	−95.4541	−0.494581	−0.247290	+	0.968941i	$0.579540\pi$
$194$	− 135.593i	− 0.698932i
$195$	0	0
$196$	17.3939	0.0887443
$197$	160.363i	0.814026i	0.913422	+	0.407013i	$0.133430\pi$
$197$	160.363i	0.814026i	−0.913422	+	0.407013i	$0.866570\pi$
$198$	0	0
$199$	6.51531	0.0327402	0.0163701	−	0.999866i	$-0.494789\pi$
$199$	6.51531	0.0327402	0.0163701	+	0.999866i	$0.494789\pi$
$200$	− 5.65685i	− 0.0282843i
$201$	0	0
$202$	−223.348	−1.10569
$203$	− 20.8809i	− 0.102862i
$204$	0	0
$205$	−321.272	−1.56718
$206$	− 41.2048i	− 0.200024i
$207$	0	0
$208$	78.7878	0.378787
$209$	44.3334i	0.212122i
$210$	0	0
$211$	−154.439	−0.731940	−0.365970	−	0.930627i	$-0.619263\pi$
$211$	−154.439	−0.731940	−0.365970	+	0.930627i	$0.619263\pi$
$212$	− 19.0702i	− 0.0899539i
$213$	0	0
$214$	−242.969	−1.13537
$215$	4.96021i	0.0230707i
$216$	0	0
$217$	−260.576	−1.20081
$218$	− 164.434i	− 0.754285i
$219$	0	0
$220$	98.0908	0.445867
$221$	37.5625i	0.169966i
$222$	0	0
$223$	92.7730	0.416022	0.208011	−	0.978126i	$-0.433301\pi$
$223$	92.7730	0.416022	0.208011	+	0.978126i	$0.433301\pi$
$224$	− 35.9124i	− 0.160323i
$225$	0	0
$226$	−286.045	−1.26569
$227$	169.802i	0.748026i	0.927423	+	0.374013i	$0.122019\pi$
$227$	169.802i	0.748026i	−0.927423	+	0.374013i	$0.877981\pi$
$228$	0	0
$229$	407.545	1.77967	0.889836	−	0.456280i	$-0.150819\pi$
$229$	407.545	1.77967	0.889836	+	0.456280i	$0.150819\pi$
$230$	− 69.3607i	− 0.301568i
$231$	0	0
$232$	9.30306	0.0400994
$233$	15.2562i	0.0654772i	0.999464	+	0.0327386i	$0.0104229\pi$
$233$	15.2562i	0.0654772i	−0.999464	+	0.0327386i	$0.989577\pi$
$234$	0	0
$235$	73.3179	0.311991
$236$	183.055i	0.775656i
$237$	0	0
$238$	17.1214	0.0719388
$239$	− 56.5364i	− 0.236554i	−0.992981	−	0.118277i	$-0.962263\pi$
$239$	− 56.5364i	− 0.236554i	0.992981	−	0.118277i	$-0.0377371\pi$
$240$	0	0
$241$	84.2122	0.349428	0.174714	−	0.984619i	$-0.444100\pi$
$241$	84.2122	0.349428	0.174714	+	0.984619i	$0.444100\pi$
$242$	− 45.1264i	− 0.186473i
$243$	0	0
$244$	150.182	0.615498
$245$	− 45.1906i	− 0.184452i
$246$	0	0
$247$	92.5153	0.374556
$248$	− 116.094i	− 0.468121i
$249$	0	0
$250$	169.015	0.676059
$251$	218.903i	0.872123i	0.899917	+	0.436062i	$0.143627\pi$
$251$	218.903i	0.872123i	−0.899917	+	0.436062i	$0.856373\pi$
$252$	0	0
$253$	89.0908	0.352138
$254$	− 14.2706i	− 0.0561833i
$255$	0	0
$256$	16.0000	0.0625000
$257$	12.8242i	0.0498998i	0.999689	+	0.0249499i	$0.00794262\pi$
$257$	12.8242i	0.0498998i	−0.999689	+	0.0249499i	$0.992057\pi$
$258$	0	0
$259$	109.848	0.424123
$260$	− 204.697i	− 0.787295i
$261$	0	0
$262$	−7.01479	−0.0267740
$263$	336.464i	1.27933i	0.768653	+	0.639666i	$0.220927\pi$
$263$	336.464i	1.27933i	−0.768653	+	0.639666i	$0.779073\pi$
$264$	0	0
$265$	−49.5459	−0.186966
$266$	− 42.1695i	− 0.158532i
$267$	0	0
$268$	−61.9092	−0.231004
$269$	60.4468i	0.224709i	0.993668	+	0.112355i	$0.0358393\pi$
$269$	60.4468i	0.224709i	−0.993668	+	0.112355i	$0.964161\pi$
$270$	0	0
$271$	274.636	1.01342	0.506708	−	0.862118i	$-0.330862\pi$
$271$	274.636	1.01342	0.506708	+	0.862118i	$0.330862\pi$
$272$	7.62809i	0.0280445i
$273$	0	0
$274$	331.893	1.21129
$275$	− 18.8776i	− 0.0686458i
$276$	0	0
$277$	−49.0000	−0.176895	−0.0884477	−	0.996081i	$-0.528191\pi$
$277$	−49.0000	−0.176895	−0.0884477	+	0.996081i	$0.528191\pi$
$278$	− 150.656i	− 0.541929i
$279$	0	0
$280$	−93.3031	−0.333225
$281$	− 343.086i	− 1.22095i	−0.792037	−	0.610473i	$-0.790979\pi$
$281$	− 343.086i	− 1.22095i	0.792037	−	0.610473i	$-0.209021\pi$
$282$	0	0
$283$	−343.409	−1.21346	−0.606729	−	0.794909i	$-0.707519\pi$
$283$	−343.409	−1.21346	−0.606729	+	0.794909i	$0.707519\pi$
$284$	171.805i	0.604948i
$285$	0	0
$286$	262.924	0.919315
$287$	392.519i	1.36766i
$288$	0	0
$289$	285.363	0.987416
$290$	− 24.1701i	− 0.0833450i
$291$	0	0
$292$	192.182	0.658156
$293$	286.453i	0.977655i	0.872380	+	0.488828i	$0.162575\pi$
$293$	286.453i	0.977655i	−0.872380	+	0.488828i	$0.837425\pi$
$294$	0	0
$295$	475.590	1.61217
$296$	48.9404i	0.165339i
$297$	0	0
$298$	148.652	0.498831
$299$	− 185.915i	− 0.621790i
$300$	0	0
$301$	6.06020	0.0201336
$302$	403.372i	1.33567i
$303$	0	0
$304$	18.7878	0.0618018
$305$	− 390.183i	− 1.27929i
$306$	0	0
$307$	154.091	0.501924	0.250962	−	0.967997i	$-0.419253\pi$
$307$	154.091	0.501924	0.250962	+	0.967997i	$0.419253\pi$
$308$	− 119.844i	− 0.389103i
$309$	0	0
$310$	−301.621	−0.972971
$311$	71.9853i	0.231464i	0.993280	+	0.115732i	$0.0369214\pi$
$311$	71.9853i	0.231464i	−0.993280	+	0.115732i	$0.963079\pi$
$312$	0	0
$313$	−367.606	−1.17446	−0.587230	−	0.809420i	$-0.699782\pi$
$313$	−367.606	−1.17446	−0.587230	+	0.809420i	$0.699782\pi$
$314$	278.857i	0.888079i
$315$	0	0
$316$	−59.4847	−0.188243
$317$	− 107.597i	− 0.339424i	−0.985494	−	0.169712i	$-0.945716\pi$
$317$	− 107.597i	− 0.339424i	0.985494	−	0.169712i	$-0.0542838\pi$
$318$	0	0
$319$	31.0454	0.0973210
$320$	− 41.5692i	− 0.129904i
$321$	0	0
$322$	−84.7423	−0.263175
$323$	8.95717i	0.0277312i
$324$	0	0
$325$	−39.3939	−0.121212
$326$	352.224i	1.08044i
$327$	0	0
$328$	−174.879	−0.533166
$329$	− 89.5771i	− 0.272271i
$330$	0	0
$331$	17.1975	0.0519561	0.0259780	−	0.999663i	$-0.491730\pi$
$331$	17.1975	0.0519561	0.0259780	+	0.999663i	$0.491730\pi$
$332$	175.812i	0.529554i
$333$	0	0
$334$	68.4995	0.205088
$335$	160.845i	0.480134i
$336$	0	0
$337$	364.394	1.08129	0.540644	−	0.841252i	$-0.318181\pi$
$337$	364.394	1.08129	0.540644	+	0.841252i	$0.318181\pi$
$338$	− 309.669i	− 0.916182i
$339$	0	0
$340$	19.8184	0.0582893
$341$	− 387.419i	− 1.13613i
$342$	0	0
$343$	−366.287	−1.06789
$344$	2.69999i	0.00784882i
$345$	0	0
$346$	142.045	0.410536
$347$	− 583.394i	− 1.68125i	−0.541616	−	0.840626i	$-0.682187\pi$
$347$	− 583.394i	− 1.68125i	0.541616	−	0.840626i	$-0.317813\pi$
$348$	0	0
$349$	312.757	0.896152	0.448076	−	0.893995i	$-0.352109\pi$
$349$	312.757	0.896152	0.448076	+	0.893995i	$0.352109\pi$
$350$	17.9562i	0.0513034i
$351$	0	0
$352$	53.3939	0.151687
$353$	− 37.6156i	− 0.106560i	−0.998580	−	0.0532798i	$-0.983032\pi$
$353$	− 37.6156i	− 0.106560i	0.998580	−	0.0532798i	$-0.0169675\pi$
$354$	0	0
$355$	446.363	1.25736
$356$	− 82.7531i	− 0.232453i
$357$	0	0
$358$	−403.151	−1.12612
$359$	− 294.028i	− 0.819019i	−0.912306	−	0.409510i	$-0.865700\pi$
$359$	− 294.028i	− 0.819019i	0.912306	−	0.409510i	$-0.134300\pi$
$360$	0	0
$361$	−338.939	−0.938889
$362$	− 52.4976i	− 0.145021i
$363$	0	0
$364$	−250.091	−0.687063
$365$	− 499.303i	− 1.36795i
$366$	0	0
$367$	−33.2270	−0.0905369	−0.0452684	−	0.998975i	$-0.514414\pi$
$367$	−33.2270	−0.0905369	−0.0452684	+	0.998975i	$0.514414\pi$
$368$	− 37.7552i	− 0.102596i
$369$	0	0
$370$	127.151	0.343651
$371$	60.5334i	0.163163i
$372$	0	0
$373$	−225.030	−0.603296	−0.301648	−	0.953419i	$-0.597537\pi$
$373$	−225.030	−0.603296	−0.301648	+	0.953419i	$0.597537\pi$
$374$	25.4558i	0.0680638i
$375$	0	0
$376$	39.9092	0.106141
$377$	− 64.7858i	− 0.171846i
$378$	0	0
$379$	−166.334	−0.438875	−0.219438	−	0.975627i	$-0.570422\pi$
$379$	−166.334	−0.438875	−0.219438	+	0.975627i	$0.570422\pi$
$380$	− 48.8120i	− 0.128453i
$381$	0	0
$382$	−25.3485	−0.0663572
$383$	736.987i	1.92425i	0.272613	+	0.962124i	$0.412112\pi$
$383$	736.987i	1.92425i	−0.272613	+	0.962124i	$0.587888\pi$
$384$	0	0
$385$	−311.363	−0.808736
$386$	134.992i	0.349721i
$387$	0	0
$388$	−191.757	−0.494219
$389$	− 169.373i	− 0.435407i	−0.976015	−	0.217704i	$-0.930143\pi$
$389$	− 169.373i	− 0.435407i	0.976015	−	0.217704i	$-0.0698566\pi$
$390$	0	0
$391$	18.0000	0.0460358
$392$	− 24.5987i	− 0.0627517i
$393$	0	0
$394$	226.788	0.575603
$395$	154.546i	0.391255i
$396$	0	0
$397$	−256.272	−0.645523	−0.322761	−	0.946480i	$-0.604611\pi$
$397$	−256.272	−0.645523	−0.322761	+	0.946480i	$0.604611\pi$
$398$	− 9.21404i	− 0.0231508i
$399$	0	0
$400$	−8.00000	−0.0200000
$401$	− 261.382i	− 0.651826i	−0.945400	−	0.325913i	$-0.894328\pi$
$401$	− 261.382i	− 0.651826i	0.945400	−	0.325913i	$-0.105672\pi$
$402$	0	0
$403$	−808.469	−2.00613
$404$	315.862i	0.781838i
$405$	0	0
$406$	−29.5301	−0.0727342
$407$	163.320i	0.401278i
$408$	0	0
$409$	−443.788	−1.08506	−0.542528	−	0.840038i	$-0.682533\pi$
$409$	−443.788	−1.08506	−0.542528	+	0.840038i	$0.682533\pi$
$410$	454.348i	1.10817i
$411$	0	0
$412$	−58.2724	−0.141438
$413$	− 581.059i	− 1.40692i
$414$	0	0
$415$	456.773	1.10066
$416$	− 111.423i	− 0.267843i
$417$	0	0
$418$	62.6969	0.149993
$419$	− 10.7679i	− 0.0256990i	−0.999917	−	0.0128495i	$-0.995910\pi$
$419$	− 10.7679i	− 0.0256990i	0.999917	−	0.0128495i	$-0.00409023\pi$
$420$	0	0
$421$	254.303	0.604045	0.302023	−	0.953301i	$-0.402338\pi$
$421$	254.303	0.604045	0.302023	+	0.953301i	$0.402338\pi$
$422$	218.410i	0.517560i
$423$	0	0
$424$	−26.9694	−0.0636070
$425$	− 3.81405i	− 0.00897423i
$426$	0	0
$427$	−476.712	−1.11642
$428$	343.611i	0.802829i
$429$	0	0
$430$	7.01479	0.0163135
$431$	698.663i	1.62103i	0.585719	+	0.810514i	$0.300812\pi$
$431$	698.663i	1.62103i	−0.585719	+	0.810514i	$0.699188\pi$
$432$	0	0
$433$	211.728	0.488978	0.244489	−	0.969652i	$-0.421380\pi$
$433$	211.728	0.488978	0.244489	+	0.969652i	$0.421380\pi$
$434$	368.509i	0.849100i
$435$	0	0
$436$	−232.545	−0.533360
$437$	− 44.3334i	− 0.101449i
$438$	0	0
$439$	279.863	0.637501	0.318750	−	0.947839i	$-0.396737\pi$
$439$	279.863	0.637501	0.318750	+	0.947839i	$0.396737\pi$
$440$	− 138.721i	− 0.315276i
$441$	0	0
$442$	53.1214	0.120184
$443$	− 551.254i	− 1.24437i	−0.782872	−	0.622183i	$-0.786246\pi$
$443$	− 551.254i	− 1.24437i	0.782872	−	0.622183i	$-0.213754\pi$
$444$	0	0
$445$	−214.999	−0.483144
$446$	− 131.201i	− 0.294172i
$447$	0	0
$448$	−50.7878	−0.113366
$449$	542.865i	1.20905i	0.796585	+	0.604527i	$0.206638\pi$
$449$	542.865i	1.20905i	−0.796585	+	0.604527i	$0.793362\pi$
$450$	0	0
$451$	−583.590	−1.29399
$452$	404.529i	0.894976i
$453$	0	0
$454$	240.136	0.528934
$455$	649.755i	1.42803i
$456$	0	0
$457$	92.3643	0.202110	0.101055	−	0.994881i	$-0.467778\pi$
$457$	92.3643	0.202110	0.101055	+	0.994881i	$0.467778\pi$
$458$	− 576.356i	− 1.25842i
$459$	0	0
$460$	−98.0908	−0.213241
$461$	− 229.820i	− 0.498525i	−0.968436	−	0.249263i	$-0.919812\pi$
$461$	− 229.820i	− 0.498525i	0.968436	−	0.249263i	$-0.0801882\pi$
$462$	0	0
$463$	−510.803	−1.10325	−0.551623	−	0.834094i	$-0.685991\pi$
$463$	−510.803	−1.10325	−0.551623	+	0.834094i	$0.685991\pi$
$464$	− 13.1565i	− 0.0283546i
$465$	0	0
$466$	21.5755	0.0462994
$467$	− 833.657i	− 1.78513i	−0.450915	−	0.892567i	$-0.648902\pi$
$467$	− 833.657i	− 1.78513i	0.450915	−	0.892567i	$-0.351098\pi$
$468$	0	0
$469$	196.514	0.419007
$470$	− 103.687i	− 0.220611i
$471$	0	0
$472$	258.879	0.548472
$473$	9.01020i	0.0190490i
$474$	0	0
$475$	−9.39388	−0.0197766
$476$	− 24.2134i	− 0.0508684i
$477$	0	0
$478$	−79.9546	−0.167269
$479$	657.190i	1.37201i	0.727599	+	0.686003i	$0.240636\pi$
$479$	657.190i	1.37201i	−0.727599	+	0.686003i	$0.759364\pi$
$480$	0	0
$481$	340.817	0.708560
$482$	− 119.094i	− 0.247083i
$483$	0	0
$484$	−63.8184	−0.131856
$485$	498.200i	1.02722i
$486$	0	0
$487$	−351.666	−0.722107	−0.361054	−	0.932545i	$-0.617583\pi$
$487$	−351.666	−0.722107	−0.361054	+	0.932545i	$0.617583\pi$
$488$	− 212.389i	− 0.435223i
$489$	0	0
$490$	−63.9092	−0.130427
$491$	− 245.418i	− 0.499834i	−0.968267	−	0.249917i	$-0.919597\pi$
$491$	− 245.418i	− 0.499834i	0.968267	−	0.249917i	$-0.0804033\pi$
$492$	0	0
$493$	6.27245	0.0127230
$494$	− 130.836i	− 0.264851i
$495$	0	0
$496$	−164.182	−0.331011
$497$	− 545.350i	− 1.09728i
$498$	0	0
$499$	630.226	1.26298	0.631489	−	0.775385i	$-0.282444\pi$
$499$	630.226	1.26298	0.631489	+	0.775385i	$0.282444\pi$
$500$	− 239.023i	− 0.478046i
$501$	0	0
$502$	309.576	0.616684
$503$	286.891i	0.570360i	0.958474	+	0.285180i	$0.0920534\pi$
$503$	286.891i	0.570360i	−0.958474	+	0.285180i	$0.907947\pi$
$504$	0	0
$505$	820.635	1.62502
$506$	− 125.993i	− 0.248999i
$507$	0	0
$508$	−20.1816	−0.0397276
$509$	− 872.323i	− 1.71380i	−0.515485	−	0.856898i	$-0.672388\pi$
$509$	− 872.323i	− 1.71380i	0.515485	−	0.856898i	$-0.327612\pi$
$510$	0	0
$511$	−610.030	−1.19380
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	18.1362	0.0352845
$515$	151.396i	0.293973i
$516$	0	0
$517$	133.182	0.257605
$518$	− 155.348i	− 0.299901i
$519$	0	0
$520$	−289.485	−0.556701
$521$	206.132i	0.395646i	0.980238	+	0.197823i	$0.0633872\pi$
$521$	206.132i	0.395646i	−0.980238	+	0.197823i	$0.936613\pi$
$522$	0	0
$523$	884.817	1.69181	0.845906	−	0.533333i	$-0.179061\pi$
$523$	884.817	1.69181	0.845906	+	0.533333i	$0.179061\pi$
$524$	9.92041i	0.0189321i
$525$	0	0
$526$	475.832	0.904624
$527$	− 78.2746i	− 0.148529i
$528$	0	0
$529$	439.909	0.831586
$530$	70.0685i	0.132205i
$531$	0	0
$532$	−59.6367	−0.112099
$533$	1217.84i	2.28488i
$534$	0	0
$535$	892.727	1.66865
$536$	87.5528i	0.163345i
$537$	0	0
$538$	85.4847	0.158893
$539$	− 82.0886i	− 0.152298i
$540$	0	0
$541$	−509.151	−0.941129	−0.470565	−	0.882365i	$-0.655950\pi$
$541$	−509.151	−0.941129	−0.470565	+	0.882365i	$0.655950\pi$
$542$	− 388.394i	− 0.716593i
$543$	0	0
$544$	10.7878	0.0198304
$545$	604.169i	1.10857i
$546$	0	0
$547$	548.044	1.00191	0.500955	−	0.865474i	$-0.332982\pi$
$547$	548.044	1.00191	0.500955	+	0.865474i	$0.332982\pi$
$548$	− 469.368i	− 0.856511i
$549$	0	0
$550$	−26.6969	−0.0485399
$551$	− 15.4488i	− 0.0280378i
$552$	0	0
$553$	188.818	0.341444
$554$	69.2965i	0.125084i
$555$	0	0
$556$	−213.060	−0.383202
$557$	− 406.542i	− 0.729879i	−0.931031	−	0.364939i	$-0.881090\pi$
$557$	− 406.542i	− 0.729879i	0.931031	−	0.364939i	$-0.118910\pi$
$558$	0	0
$559$	18.8025	0.0336360
$560$	131.950i	0.235626i
$561$	0	0
$562$	−485.196	−0.863339
$563$	606.471i	1.07721i	0.842557	+	0.538607i	$0.181049\pi$
$563$	606.471i	1.07721i	−0.842557	+	0.538607i	$0.818951\pi$
$564$	0	0
$565$	1051.00	1.86017
$566$	485.653i	0.858045i
$567$	0	0
$568$	242.969	0.427763
$569$	− 259.755i	− 0.456511i	−0.973601	−	0.228255i	$-0.926698\pi$
$569$	− 259.755i	− 0.456511i	0.973601	−	0.228255i	$-0.0733021\pi$
$570$	0	0
$571$	−87.8332	−0.153823	−0.0769117	−	0.997038i	$-0.524506\pi$
$571$	−87.8332	−0.153823	−0.0769117	+	0.997038i	$0.524506\pi$
$572$	− 371.831i	− 0.650054i
$573$	0	0
$574$	555.106	0.967083
$575$	18.8776i	0.0328306i
$576$	0	0
$577$	−132.091	−0.228927	−0.114463	−	0.993427i	$-0.536515\pi$
$577$	−132.091	−0.228927	−0.114463	+	0.993427i	$0.536515\pi$
$578$	− 403.565i	− 0.698209i
$579$	0	0
$580$	−34.1816	−0.0589338
$581$	− 558.069i	− 0.960531i
$582$	0	0
$583$	−90.0000	−0.154374
$584$	− 271.786i	− 0.465387i
$585$	0	0
$586$	405.106	0.691306
$587$	− 567.666i	− 0.967064i	−0.875327	−	0.483532i	$-0.839354\pi$
$587$	− 567.666i	− 0.967064i	0.875327	−	0.483532i	$-0.160646\pi$
$588$	0	0
$589$	−192.788	−0.327314
$590$	− 672.586i	− 1.13998i
$591$	0	0
$592$	69.2122	0.116913
$593$	− 77.0321i	− 0.129902i	−0.997888	−	0.0649512i	$-0.979311\pi$
$593$	− 77.0321i	− 0.129902i	0.997888	−	0.0649512i	$-0.0206892\pi$
$594$	0	0
$595$	−62.9082	−0.105728
$596$	− 210.225i	− 0.352727i
$597$	0	0
$598$	−262.924	−0.439672
$599$	883.250i	1.47454i	0.675598	+	0.737270i	$0.263885\pi$
$599$	883.250i	1.47454i	−0.675598	+	0.737270i	$0.736115\pi$
$600$	0	0
$601$	−795.091	−1.32295	−0.661473	−	0.749969i	$-0.730068\pi$
$601$	−795.091	−1.32295	−0.661473	+	0.749969i	$0.730068\pi$
$602$	− 8.57042i	− 0.0142366i
$603$	0	0
$604$	570.454	0.944460
$605$	165.805i	0.274058i
$606$	0	0
$607$	−296.743	−0.488869	−0.244434	−	0.969666i	$-0.578602\pi$
$607$	−296.743	−0.488869	−0.244434	+	0.969666i	$0.578602\pi$
$608$	− 26.5699i	− 0.0437005i
$609$	0	0
$610$	−551.803	−0.904594
$611$	− 277.924i	− 0.454868i
$612$	0	0
$613$	−517.181	−0.843688	−0.421844	−	0.906668i	$-0.638617\pi$
$613$	−517.181	−0.843688	−0.421844	+	0.906668i	$0.638617\pi$
$614$	− 217.917i	− 0.354914i
$615$	0	0
$616$	−169.485	−0.275137
$617$	265.476i	0.430268i	0.976584	+	0.215134i	$0.0690189\pi$
$617$	265.476i	0.430268i	−0.976584	+	0.215134i	$0.930981\pi$
$618$	0	0
$619$	−197.045	−0.318329	−0.159164	−	0.987252i	$-0.550880\pi$
$619$	−197.045	−0.318329	−0.159164	+	0.987252i	$0.550880\pi$
$620$	426.556i	0.687994i
$621$	0	0
$622$	101.803	0.163670
$623$	262.678i	0.421634i
$624$	0	0
$625$	−671.000	−1.07360
$626$	519.874i	0.830469i
$627$	0	0
$628$	394.363	0.627967
$629$	32.9973i	0.0524600i
$630$	0	0
$631$	−160.879	−0.254958	−0.127479	−	0.991841i	$-0.540689\pi$
$631$	−160.879	−0.254958	−0.127479	+	0.991841i	$0.540689\pi$
$632$	84.1241i	0.133108i
$633$	0	0
$634$	−152.166	−0.240009
$635$	52.4334i	0.0825723i
$636$	0	0
$637$	−171.303	−0.268922
$638$	− 43.9048i	− 0.0688164i
$639$	0	0
$640$	−58.7878	−0.0918559
$641$	− 309.337i	− 0.482585i	−0.970452	−	0.241293i	$-0.922429\pi$
$641$	− 309.337i	− 0.482585i	0.970452	−	0.241293i	$-0.0775714\pi$
$642$	0	0
$643$	394.591	0.613672	0.306836	−	0.951762i	$-0.400730\pi$
$643$	394.591	0.613672	0.306836	+	0.951762i	$0.400730\pi$
$644$	119.844i	0.186093i
$645$	0	0
$646$	12.6674	0.0196089
$647$	418.736i	0.647196i	0.946195	+	0.323598i	$0.104892\pi$
$647$	418.736i	0.647196i	−0.946195	+	0.323598i	$0.895108\pi$
$648$	0	0
$649$	863.908	1.33114
$650$	55.7114i	0.0857098i
$651$	0	0
$652$	498.120	0.763988
$653$	530.725i	0.812749i	0.913707	+	0.406375i	$0.133207\pi$
$653$	530.725i	0.812749i	−0.913707	+	0.406375i	$0.866793\pi$
$654$	0	0
$655$	25.7740	0.0393496
$656$	247.316i	0.377006i
$657$	0	0
$658$	−126.681	−0.192524
$659$	358.193i	0.543539i	0.962362	+	0.271770i	$0.0876089\pi$
$659$	358.193i	0.543539i	−0.962362	+	0.271770i	$0.912391\pi$
$660$	0	0
$661$	−222.271	−0.336265	−0.168133	−	0.985764i	$-0.553774\pi$
$661$	−222.271	−0.336265	−0.168133	+	0.985764i	$0.553774\pi$
$662$	− 24.3209i	− 0.0367385i
$663$	0	0
$664$	248.636	0.374451
$665$	154.941i	0.232994i
$666$	0	0
$667$	−31.0454	−0.0465448
$668$	− 96.8729i	− 0.145019i
$669$	0	0
$670$	227.469	0.339506
$671$	− 708.767i	− 1.05628i
$672$	0	0
$673$	−289.211	−0.429734	−0.214867	−	0.976643i	$-0.568932\pi$
$673$	−289.211	−0.429734	−0.214867	+	0.976643i	$0.568932\pi$
$674$	− 515.331i	− 0.764586i
$675$	0	0
$676$	−437.939	−0.647838
$677$	464.451i	0.686043i	0.939328	+	0.343022i	$0.111450\pi$
$677$	464.451i	0.686043i	−0.939328	+	0.343022i	$0.888550\pi$
$678$	0	0
$679$	608.682	0.896439
$680$	− 28.0274i	− 0.0412168i
$681$	0	0
$682$	−547.893	−0.803363
$683$	1126.36i	1.64913i	0.565767	+	0.824565i	$0.308580\pi$
$683$	1126.36i	1.64913i	−0.565767	+	0.824565i	$0.691420\pi$
$684$	0	0
$685$	−1219.45	−1.78022
$686$	518.008i	0.755114i
$687$	0	0
$688$	3.81837	0.00554996
$689$	187.813i	0.272587i
$690$	0	0
$691$	1037.68	1.50171	0.750855	−	0.660467i	$-0.229642\pi$
$691$	1037.68	1.50171	0.750855	+	0.660467i	$0.229642\pi$
$692$	− 200.883i	− 0.290293i
$693$	0	0
$694$	−825.044	−1.18882
$695$	553.547i	0.796470i
$696$	0	0
$697$	−117.909	−0.169167
$698$	− 442.305i	− 0.633675i
$699$	0	0
$700$	25.3939	0.0362770
$701$	− 778.180i	− 1.11010i	−0.831817	−	0.555050i	$-0.812699\pi$
$701$	− 778.180i	− 1.11010i	0.831817	−	0.555050i	$-0.187301\pi$
$702$	0	0
$703$	81.2714	0.115607
$704$	− 75.5103i	− 0.107259i
$705$	0	0
$706$	−53.1964	−0.0753490
$707$	− 1002.62i	− 1.41814i
$708$	0	0
$709$	−1172.03	−1.65307	−0.826536	−	0.562883i	$-0.809692\pi$
$709$	−1172.03	−1.65307	−0.826536	+	0.562883i	$0.809692\pi$
$710$	− 631.253i	− 0.889089i
$711$	0	0
$712$	−117.031	−0.164369
$713$	387.419i	0.543365i
$714$	0	0
$715$	−966.044	−1.35111
$716$	570.142i	0.796287i
$717$	0	0
$718$	−415.818	−0.579134
$719$	515.416i	0.716851i	0.933558	+	0.358426i	$0.116686\pi$
$719$	515.416i	0.716851i	−0.933558	+	0.358426i	$0.883314\pi$
$720$	0	0
$721$	184.970	0.256547
$722$	479.332i	0.663894i
$723$	0	0
$724$	−74.2429	−0.102545
$725$	6.57826i	0.00907346i
$726$	0	0
$727$	−841.409	−1.15737	−0.578685	−	0.815551i	$-0.696434\pi$
$727$	−841.409	−1.15737	−0.578685	+	0.815551i	$0.696434\pi$
$728$	353.682i	0.485827i
$729$	0	0
$730$	−706.120	−0.967288
$731$	1.82043i	0.00249033i
$732$	0	0
$733$	606.362	0.827234	0.413617	−	0.910451i	$-0.364265\pi$
$733$	606.362	0.827234	0.413617	+	0.910451i	$0.364265\pi$
$734$	46.9901i	0.0640192i
$735$	0	0
$736$	−53.3939	−0.0725460
$737$	292.174i	0.396437i
$738$	0	0
$739$	−389.362	−0.526877	−0.263439	−	0.964676i	$-0.584857\pi$
$739$	−389.362	−0.526877	−0.263439	+	0.964676i	$0.584857\pi$
$740$	− 179.819i	− 0.242998i
$741$	0	0
$742$	85.6072	0.115374
$743$	− 1044.75i	− 1.40612i	−0.711129	−	0.703061i	$-0.751816\pi$
$743$	− 1044.75i	− 1.40612i	0.711129	−	0.703061i	$-0.248184\pi$
$744$	0	0
$745$	−546.181	−0.733128
$746$	318.240i	0.426595i
$747$	0	0
$748$	36.0000	0.0481283
$749$	− 1090.70i	− 1.45621i
$750$	0	0
$751$	−1291.83	−1.72015	−0.860074	−	0.510169i	$-0.829583\pi$
$751$	−1291.83	−1.72015	−0.860074	+	0.510169i	$0.829583\pi$
$752$	− 56.4401i	− 0.0750533i
$753$	0	0
$754$	−91.6209	−0.121513
$755$	− 1482.08i	− 1.96302i
$756$	0	0
$757$	1042.36	1.37697	0.688483	−	0.725252i	$-0.258277\pi$
$757$	1042.36	1.37697	0.688483	+	0.725252i	$0.258277\pi$
$758$	235.231i	0.310332i
$759$	0	0
$760$	−69.0306	−0.0908298
$761$	325.171i	0.427295i	0.976911	+	0.213647i	$0.0685344\pi$
$761$	325.171i	0.427295i	−0.976911	+	0.213647i	$0.931466\pi$
$762$	0	0
$763$	738.152	0.967434
$764$	35.8481i	0.0469217i
$765$	0	0
$766$	1042.26	1.36065
$767$	− 1802.81i	− 2.35047i
$768$	0	0
$769$	342.696	0.445638	0.222819	−	0.974860i	$-0.428474\pi$
$769$	342.696	0.445638	0.222819	+	0.974860i	$0.428474\pi$
$770$	440.334i	0.571863i
$771$	0	0
$772$	190.908	0.247290
$773$	532.579i	0.688977i	0.938791	+	0.344488i	$0.111948\pi$
$773$	532.579i	0.688977i	−0.938791	+	0.344488i	$0.888052\pi$
$774$	0	0
$775$	82.0908	0.105924
$776$	271.186i	0.349466i
$777$	0	0
$778$	−239.530	−0.307879
$779$	290.407i	0.372794i
$780$	0	0
$781$	810.817	1.03818
$782$	− 25.4558i	− 0.0325522i
$783$	0	0
$784$	−34.7878	−0.0443721
$785$	− 1024.59i	− 1.30520i
$786$	0	0
$787$	−103.954	−0.132088	−0.0660442	−	0.997817i	$-0.521038\pi$
$787$	−103.954	−0.132088	−0.0660442	+	0.997817i	$0.521038\pi$
$788$	− 320.726i	− 0.407013i
$789$	0	0
$790$	218.561	0.276659
$791$	− 1284.07i	− 1.62335i
$792$	0	0
$793$	−1479.06	−1.86514
$794$	362.424i	0.456453i
$795$	0	0
$796$	−13.0306	−0.0163701
$797$	1104.28i	1.38554i	0.721158	+	0.692770i	$0.243610\pi$
$797$	1104.28i	1.38554i	−0.721158	+	0.692770i	$0.756390\pi$
$798$	0	0
$799$	26.9082	0.0336773
$800$	11.3137i	0.0141421i
$801$	0	0
$802$	−369.650	−0.460911
$803$	− 906.981i	− 1.12949i
$804$	0	0
$805$	311.363	0.386787
$806$	1143.35i	1.41855i
$807$	0	0
$808$	446.697	0.552843
$809$	256.465i	0.317015i	0.987358	+	0.158508i	$0.0506683\pi$
$809$	256.465i	0.317015i	−0.987358	+	0.158508i	$0.949332\pi$
$810$	0	0
$811$	735.362	0.906735	0.453368	−	0.891324i	$-0.350222\pi$
$811$	735.362	0.906735	0.453368	+	0.891324i	$0.350222\pi$
$812$	41.7619i	0.0514309i
$813$	0	0
$814$	230.969	0.283746
$815$	− 1294.15i	− 1.58792i
$816$	0	0
$817$	4.48366	0.00548796
$818$	627.611i	0.767250i
$819$	0	0
$820$	642.545	0.783591
$821$	1245.29i	1.51680i	0.651792	+	0.758398i	$0.274018\pi$
$821$	1245.29i	1.51680i	−0.651792	+	0.758398i	$0.725982\pi$
$822$	0	0
$823$	−1542.26	−1.87395	−0.936973	−	0.349402i	$-0.886385\pi$
$823$	−1542.26	−1.87395	−0.936973	+	0.349402i	$0.886385\pi$
$824$	82.4097i	0.100012i
$825$	0	0
$826$	−821.741	−0.994844
$827$	− 955.707i	− 1.15563i	−0.816167	−	0.577815i	$-0.803905\pi$
$827$	− 955.707i	− 1.15563i	0.816167	−	0.577815i	$-0.196095\pi$
$828$	0	0
$829$	1082.88	1.30625	0.653123	−	0.757252i	$-0.273458\pi$
$829$	1082.88	1.30625	0.653123	+	0.757252i	$0.273458\pi$
$830$	− 645.975i	− 0.778283i
$831$	0	0
$832$	−157.576	−0.189394
$833$	− 16.5853i	− 0.0199103i
$834$	0	0
$835$	−251.683	−0.301417
$836$	− 88.6669i	− 0.106061i
$837$	0	0
$838$	−15.2281	−0.0181719
$839$	1043.59i	1.24385i	0.783075	+	0.621927i	$0.213650\pi$
$839$	1043.59i	1.24385i	−0.783075	+	0.621927i	$0.786350\pi$
$840$	0	0
$841$	830.182	0.987136
$842$	− 359.639i	− 0.427125i
$843$	0	0
$844$	308.879	0.365970
$845$	1137.80i	1.34651i
$846$	0	0
$847$	202.574	0.239167
$848$	38.1405i	0.0449770i
$849$	0	0
$850$	−5.39388	−0.00634574
$851$	− 163.320i	− 0.191915i
$852$	0	0
$853$	473.817	0.555472	0.277736	−	0.960657i	$-0.410416\pi$
$853$	473.817	0.555472	0.277736	+	0.960657i	$0.410416\pi$
$854$	674.172i	0.789429i
$855$	0	0
$856$	485.939	0.567685
$857$	916.762i	1.06973i	0.844936	+	0.534867i	$0.179638\pi$
$857$	916.762i	1.06973i	−0.844936	+	0.534867i	$0.820362\pi$
$858$	0	0
$859$	957.802	1.11502	0.557510	−	0.830170i	$-0.311757\pi$
$859$	957.802	1.11502	0.557510	+	0.830170i	$0.311757\pi$
$860$	− 9.92041i	− 0.0115354i
$861$	0	0
$862$	988.059	1.14624
$863$	524.200i	0.607416i	0.952765	+	0.303708i	$0.0982247\pi$
$863$	524.200i	0.607416i	−0.952765	+	0.303708i	$0.901775\pi$
$864$	0	0
$865$	−521.908	−0.603362
$866$	− 299.428i	− 0.345760i
$867$	0	0
$868$	521.151	0.600404
$869$	280.732i	0.323052i
$870$	0	0
$871$	609.711	0.700012
$872$	328.868i	0.377142i
$873$	0	0
$874$	−62.6969	−0.0717356
$875$	758.715i	0.867103i
$876$	0	0
$877$	1007.76	1.14909	0.574547	−	0.818471i	$-0.305178\pi$
$877$	1007.76	1.14909	0.574547	+	0.818471i	$0.305178\pi$
$878$	− 395.786i	− 0.450781i
$879$	0	0
$880$	−196.182	−0.222934
$881$	− 1536.71i	− 1.74428i	−0.489254	−	0.872141i	$-0.662731\pi$
$881$	− 1536.71i	− 1.74428i	0.489254	−	0.872141i	$-0.337269\pi$
$882$	0	0
$883$	−294.213	−0.333197	−0.166599	−	0.986025i	$-0.553278\pi$
$883$	−294.213	−0.333197	−0.166599	+	0.986025i	$0.553278\pi$
$884$	− 75.1250i	− 0.0849831i
$885$	0	0
$886$	−779.591	−0.879900
$887$	574.803i	0.648031i	0.946052	+	0.324015i	$0.105033\pi$
$887$	574.803i	0.648031i	−0.946052	+	0.324015i	$0.894967\pi$
$888$	0	0
$889$	64.0612	0.0720599
$890$	304.054i	0.341634i
$891$	0	0
$892$	−185.546	−0.208011
$893$	− 66.2739i	− 0.0742149i
$894$	0	0
$895$	1481.27	1.65505
$896$	71.8247i	0.0801615i
$897$	0	0
$898$	767.728	0.854930
$899$	135.004i	0.150171i
$900$	0	0
$901$	−18.1837	−0.0201817
$902$	825.321i	0.914990i
$903$	0	0
$904$	572.091	0.632844
$905$	192.889i	0.213137i
$906$	0	0
$907$	−510.074	−0.562375	−0.281187	−	0.959653i	$-0.590728\pi$
$907$	−510.074	−0.562375	−0.281187	+	0.959653i	$0.590728\pi$
$908$	− 339.604i	− 0.374013i
$909$	0	0
$910$	918.892	1.00977
$911$	− 927.371i	− 1.01797i	−0.860775	−	0.508985i	$-0.830021\pi$
$911$	− 927.371i	− 1.01797i	0.860775	−	0.508985i	$-0.169979\pi$
$912$	0	0
$913$	829.727	0.908791
$914$	− 130.623i	− 0.142913i
$915$	0	0
$916$	−815.090	−0.889836
$917$	− 31.4897i	− 0.0343399i
$918$	0	0
$919$	−1240.63	−1.34998	−0.674991	−	0.737826i	$-0.735853\pi$
$919$	−1240.63	−1.34998	−0.674991	+	0.737826i	$0.735853\pi$
$920$	138.721i	0.150784i
$921$	0	0
$922$	−325.015	−0.352511
$923$	− 1692.02i	− 1.83317i
$924$	0	0
$925$	−34.6061	−0.0374120
$926$	722.384i	0.780112i
$927$	0	0
$928$	−18.6061	−0.0200497
$929$	338.992i	0.364900i	0.983215	+	0.182450i	$0.0584028\pi$
$929$	338.992i	0.364900i	−0.983215	+	0.182450i	$0.941597\pi$
$930$	0	0
$931$	−40.8490	−0.0438765
$932$	− 30.5124i	− 0.0327386i
$933$	0	0
$934$	−1178.97	−1.26228
$935$	− 93.5307i	− 0.100033i
$936$	0	0
$937$	1322.21	1.41111	0.705556	−	0.708655i	$-0.250698\pi$
$937$	1322.21	1.41111	0.705556	+	0.708655i	$0.250698\pi$
$938$	− 277.913i	− 0.296283i
$939$	0	0
$940$	−146.636	−0.155995
$941$	359.093i	0.381608i	0.981628	+	0.190804i	$0.0611095\pi$
$941$	359.093i	0.381608i	−0.981628	+	0.190804i	$0.938891\pi$
$942$	0	0
$943$	583.590	0.618866
$944$	− 366.110i	− 0.387828i
$945$	0	0
$946$	12.7423	0.0134697
$947$	− 775.792i	− 0.819210i	−0.912263	−	0.409605i	$-0.865667\pi$
$947$	− 775.792i	− 0.819210i	0.912263	−	0.409605i	$-0.134333\pi$
$948$	0	0
$949$	−1892.69	−1.99441
$950$	13.2849i	0.0139842i
$951$	0	0
$952$	−34.2429	−0.0359694
$953$	465.082i	0.488019i	0.969773	+	0.244010i	$0.0784628\pi$
$953$	465.082i	0.488019i	−0.969773	+	0.244010i	$0.921537\pi$
$954$	0	0
$955$	93.1362	0.0975248
$956$	113.073i	0.118277i
$957$	0	0
$958$	929.408	0.970154
$959$	1489.88i	1.55358i
$960$	0	0
$961$	723.725	0.753096
$962$	− 481.989i	− 0.501028i
$963$	0	0
$964$	−168.424	−0.174714
$965$	− 495.994i	− 0.513983i
$966$	0	0
$967$	1224.23	1.26600	0.633002	−	0.774150i	$-0.281822\pi$
$967$	1224.23	1.26600	0.633002	+	0.774150i	$0.281822\pi$
$968$	90.2528i	0.0932364i
$969$	0	0
$970$	704.561	0.726351
$971$	658.702i	0.678375i	0.940719	+	0.339188i	$0.110152\pi$
$971$	658.702i	0.678375i	−0.940719	+	0.339188i	$0.889848\pi$
$972$	0	0
$973$	676.303	0.695070
$974$	497.331i	0.510607i
$975$	0	0
$976$	−300.363	−0.307749
$977$	− 1518.34i	− 1.55408i	−0.629449	−	0.777042i	$-0.716719\pi$
$977$	− 1518.34i	− 1.55408i	0.629449	−	0.777042i	$-0.283281\pi$
$978$	0	0
$979$	−390.545	−0.398922
$980$	90.3812i	0.0922258i
$981$	0	0
$982$	−347.074	−0.353436
$983$	827.840i	0.842157i	0.907024	+	0.421078i	$0.138348\pi$
$983$	827.840i	0.842157i	−0.907024	+	0.421078i	$0.861652\pi$
$984$	0	0
$985$	−833.271	−0.845961
$986$	− 8.87058i	− 0.00899653i
$987$	0	0
$988$	−185.031	−0.187278
$989$	− 9.01020i	− 0.00911041i
$990$	0	0
$991$	429.546	0.433447	0.216723	−	0.976233i	$-0.430463\pi$
$991$	429.546	0.433447	0.216723	+	0.976233i	$0.430463\pi$
$992$	232.188i	0.234060i
$993$	0	0
$994$	−771.242	−0.775897
$995$	33.8545i	0.0340247i
$996$	0	0
$997$	−694.998	−0.697089	−0.348545	−	0.937292i	$-0.613324\pi$
$997$	−694.998	−0.697089	−0.348545	+	0.937292i	$0.613324\pi$
$998$	− 891.274i	− 0.893060i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.3.b.a.161.2		4
3.2	odd	2	inner	162.3.b.a.161.3		4
4.3	odd	2		1296.3.e.g.161.3		4
9.2	odd	6		54.3.d.a.17.2		4
9.4	even	3		54.3.d.a.35.2		4
9.5	odd	6		18.3.d.a.11.1	yes	4
9.7	even	3		18.3.d.a.5.1	✓	4
12.11	even	2		1296.3.e.g.161.1		4
36.7	odd	6		144.3.q.c.113.1		4
36.11	even	6		432.3.q.d.17.2		4
36.23	even	6		144.3.q.c.65.1		4
36.31	odd	6		432.3.q.d.305.2		4
45.2	even	12		1350.3.k.a.449.3		8
45.4	even	6		1350.3.i.b.251.1		4
45.7	odd	12		450.3.k.a.149.2		8
45.13	odd	12		1350.3.k.a.899.3		8
45.14	odd	6		450.3.i.b.101.2		4
45.22	odd	12		1350.3.k.a.899.2		8
45.23	even	12		450.3.k.a.299.2		8
45.29	odd	6		1350.3.i.b.1151.1		4
45.32	even	12		450.3.k.a.299.3		8
45.34	even	6		450.3.i.b.401.2		4
45.38	even	12		1350.3.k.a.449.2		8
45.43	odd	12		450.3.k.a.149.3		8
72.5	odd	6		576.3.q.f.65.1		4
72.11	even	6		1728.3.q.c.449.2		4
72.13	even	6		1728.3.q.d.1601.1		4
72.29	odd	6		1728.3.q.d.449.1		4
72.43	odd	6		576.3.q.e.257.2		4
72.59	even	6		576.3.q.e.65.2		4
72.61	even	6		576.3.q.f.257.1		4
72.67	odd	6		1728.3.q.c.1601.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.d.a.5.1	✓	4	9.7	even	3
18.3.d.a.11.1	yes	4	9.5	odd	6
54.3.d.a.17.2		4	9.2	odd	6
54.3.d.a.35.2		4	9.4	even	3
144.3.q.c.65.1		4	36.23	even	6
144.3.q.c.113.1		4	36.7	odd	6
162.3.b.a.161.2		4	1.1	even	1	trivial
162.3.b.a.161.3		4	3.2	odd	2	inner
432.3.q.d.17.2		4	36.11	even	6
432.3.q.d.305.2		4	36.31	odd	6
450.3.i.b.101.2		4	45.14	odd	6
450.3.i.b.401.2		4	45.34	even	6
450.3.k.a.149.2		8	45.7	odd	12
450.3.k.a.149.3		8	45.43	odd	12
450.3.k.a.299.2		8	45.23	even	12
450.3.k.a.299.3		8	45.32	even	12
576.3.q.e.65.2		4	72.59	even	6
576.3.q.e.257.2		4	72.43	odd	6
576.3.q.f.65.1		4	72.5	odd	6
576.3.q.f.257.1		4	72.61	even	6
1296.3.e.g.161.1		4	12.11	even	2
1296.3.e.g.161.3		4	4.3	odd	2
1350.3.i.b.251.1		4	45.4	even	6
1350.3.i.b.1151.1		4	45.29	odd	6
1350.3.k.a.449.2		8	45.38	even	12
1350.3.k.a.449.3		8	45.2	even	12
1350.3.k.a.899.2		8	45.22	odd	12
1350.3.k.a.899.3		8	45.13	odd	12
1728.3.q.c.449.2		4	72.11	even	6
1728.3.q.c.1601.2		4	72.67	odd	6
1728.3.q.d.449.1		4	72.29	odd	6
1728.3.q.d.1601.1		4	72.13	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	162.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +5.19615i q^{5} +6.34847 q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q-1.41421i q^{2} -2.00000 q^{4} +5.19615i q^{5} +6.34847 q^{7} +2.82843i q^{8} +7.34847 q^{10} +9.43879i q^{11} +19.6969 q^{13} -8.97809i q^{14} +4.00000 q^{16} +1.90702i q^{17} +4.69694 q^{19} -10.3923i q^{20} +13.3485 q^{22} -9.43879i q^{23} -2.00000 q^{25} -27.8557i q^{26} -12.6969 q^{28} -3.28913i q^{29} -41.0454 q^{31} -5.65685i q^{32} +2.69694 q^{34} +32.9876i q^{35} +17.3031 q^{37} -6.64247i q^{38} -14.6969 q^{40} +61.8289i q^{41} +0.954592 q^{43} -18.8776i q^{44} -13.3485 q^{46} -14.1100i q^{47} -8.69694 q^{49} +2.82843i q^{50} -39.3939 q^{52} +9.53512i q^{53} -49.0454 q^{55} +17.9562i q^{56} -4.65153 q^{58} -91.5274i q^{59} -75.0908 q^{61} +58.0470i q^{62} -8.00000 q^{64} +102.348i q^{65} +30.9546 q^{67} -3.81405i q^{68} +46.6515 q^{70} -85.9026i q^{71} -96.0908 q^{73} -24.4702i q^{74} -9.39388 q^{76} +59.9219i q^{77} +29.7423 q^{79} +20.7846i q^{80} +87.4393 q^{82} -87.9060i q^{83} -9.90918 q^{85} -1.35000i q^{86} -26.6969 q^{88} +41.3766i q^{89} +125.045 q^{91} +18.8776i q^{92} -19.9546 q^{94} +24.4060i q^{95} +95.8786 q^{97} +12.2993i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 4 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 - 4 * q^7	\( 4 q - 8 q^{4} - 4 q^{7} + 20 q^{13} + 16 q^{16} - 40 q^{19} + 24 q^{22} - 8 q^{25} + 8 q^{28} - 76 q^{31} - 48 q^{34} + 128 q^{37} + 92 q^{43} - 24 q^{46} + 24 q^{49} - 40 q^{52} - 108 q^{55} - 48 q^{58} - 124 q^{61} - 32 q^{64} + 212 q^{67} + 216 q^{70} - 208 q^{73} + 80 q^{76} - 28 q^{79} + 144 q^{82} - 216 q^{85} - 48 q^{88} + 412 q^{91} - 168 q^{94} - 28 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 - 4 * q^7 + 20 * q^13 + 16 * q^16 - 40 * q^19 + 24 * q^22 - 8 * q^25 + 8 * q^28 - 76 * q^31 - 48 * q^34 + 128 * q^37 + 92 * q^43 - 24 * q^46 + 24 * q^49 - 40 * q^52 - 108 * q^55 - 48 * q^58 - 124 * q^61 - 32 * q^64 + 212 * q^67 + 216 * q^70 - 208 * q^73 + 80 * q^76 - 28 * q^79 + 144 * q^82 - 216 * q^85 - 48 * q^88 + 412 * q^91 - 168 * q^94 - 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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