LMFDB - Embedded newform 1296.3.e.g.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(161,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1296.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.e (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:	$35.3134422611$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1296.161
Dual form			1296.3.e.g.161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615i q^{5} -6.34847 q^{7} +O(q^{10})$	$q+5.19615i q^{5} -6.34847 q^{7} -9.43879i q^{11} +19.6969 q^{13} +1.90702i q^{17} -4.69694 q^{19} +9.43879i q^{23} -2.00000 q^{25} -3.28913i q^{29} +41.0454 q^{31} -32.9876i q^{35} +17.3031 q^{37} +61.8289i q^{41} -0.954592 q^{43} +14.1100i q^{47} -8.69694 q^{49} +9.53512i q^{53} +49.0454 q^{55} +91.5274i q^{59} -75.0908 q^{61} +102.348i q^{65} -30.9546 q^{67} +85.9026i q^{71} -96.0908 q^{73} +59.9219i q^{77} -29.7423 q^{79} +87.9060i q^{83} -9.90918 q^{85} +41.3766i q^{89} -125.045 q^{91} -24.4060i q^{95} +95.8786 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} + 20 q^{13} + 40 q^{19} - 8 q^{25} + 76 q^{31} + 128 q^{37} - 92 q^{43} + 24 q^{49} + 108 q^{55} - 124 q^{61} - 212 q^{67} - 208 q^{73} + 28 q^{79} - 216 q^{85} - 412 q^{91} - 28 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 + 20 * q^13 + 40 * q^19 - 8 * q^25 + 76 * q^31 + 128 * q^37 - 92 * q^43 + 24 * q^49 + 108 * q^55 - 124 * q^61 - 212 * q^67 - 208 * q^73 + 28 * q^79 - 216 * q^85 - 412 * q^91 - 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$-1$

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 (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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	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	$-0.649811\pi$
$7$	−6.34847	−0.906924	−0.453462	+	0.891276i	$0.649811\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 9.43879i	− 0.858072i	−0.903287	−	0.429036i	$-0.858853\pi$
$11$	− 9.43879i	− 0.858072i	0.903287	−	0.429036i	$-0.141147\pi$
$12$	0	0
$13$	19.6969	1.51515	0.757575	−	0.652749i	$-0.226384\pi$
$13$	19.6969	1.51515	0.757575	+	0.652749i	$0.226384\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	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	$-0.539445\pi$
$19$	−4.69694	−0.247207	−0.123604	+	0.992332i	$0.539445\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	9.43879i	0.410382i	0.978722	+	0.205191i	$0.0657816\pi$
$23$	9.43879i	0.410382i	−0.978722	+	0.205191i	$0.934218\pi$
$24$	0	0
$25$	−2.00000	−0.0800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 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	$-0.269698\pi$
$31$	41.0454	1.32405	0.662023	+	0.749484i	$0.269698\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 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	$0.424876\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	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	$-0.503533\pi$
$43$	−0.954592	−0.0221998	−0.0110999	+	0.999938i	$0.503533\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	14.1100i	0.300213i	0.988670	+	0.150107i	$0.0479617\pi$
$47$	14.1100i	0.300213i	−0.988670	+	0.150107i	$0.952038\pi$
$48$	0	0
$49$	−8.69694	−0.177489
$50$	0	0
$51$	0	0
$52$	0	0
$53$	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
$56$	0	0
$57$	0	0
$58$	0	0
$59$	91.5274i	1.55131i	0.631156	+	0.775656i	$0.282581\pi$
$59$	91.5274i	1.55131i	−0.631156	+	0.775656i	$0.717419\pi$
$60$	0	0
$61$	−75.0908	−1.23100	−0.615498	−	0.788138i	$-0.711045\pi$
$61$	−75.0908	−1.23100	−0.615498	+	0.788138i	$0.711045\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	102.348i	1.57459i
$66$	0	0
$67$	−30.9546	−0.462009	−0.231004	−	0.972953i	$-0.574201\pi$
$67$	−30.9546	−0.462009	−0.231004	+	0.972953i	$0.574201\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	85.9026i	1.20990i	0.796265	+	0.604948i	$0.206806\pi$
$71$	85.9026i	1.20990i	−0.796265	+	0.604948i	$0.793194\pi$
$72$	0	0
$73$	−96.0908	−1.31631	−0.658156	−	0.752881i	$-0.728663\pi$
$73$	−96.0908	−1.31631	−0.658156	+	0.752881i	$0.728663\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	59.9219i	0.778206i
$78$	0	0
$79$	−29.7423	−0.376485	−0.188243	−	0.982123i	$-0.560279\pi$
$79$	−29.7423	−0.376485	−0.188243	+	0.982123i	$0.560279\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	87.9060i	1.05911i	0.848276	+	0.529554i	$0.177641\pi$
$83$	87.9060i	1.05911i	−0.848276	+	0.529554i	$0.822359\pi$
$84$	0	0
$85$	−9.90918	−0.116579
$86$	0	0
$87$	0	0
$88$	0	0
$89$	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
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 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	$0.335454\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 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	$-0.545173\pi$
$103$	−29.1362	−0.282876	−0.141438	+	0.989947i	$0.545173\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	171.805i	1.60566i	0.596210	+	0.802829i	$0.296673\pi$
$107$	171.805i	1.60566i	−0.596210	+	0.802829i	$0.703327\pi$
$108$	0	0
$109$	116.272	1.06672	0.533360	−	0.845888i	$-0.320929\pi$
$109$	116.272	1.06672	0.533360	+	0.845888i	$0.320929\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 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
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 12.1067i	− 0.101737i
$120$	0	0
$121$	31.9092	0.263712
$122$	0	0
$123$	0	0
$124$	0	0
$125$	119.512i	0.956092i
$126$	0	0
$127$	−10.0908	−0.0794552	−0.0397276	−	0.999211i	$-0.512649\pi$
$127$	−10.0908	−0.0794552	−0.0397276	+	0.999211i	$0.512649\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	4.96021i	0.0378642i	0.999821	+	0.0189321i	$0.00602663\pi$
$131$	4.96021i	0.0378642i	−0.999821	+	0.0189321i	$0.993973\pi$
$132$	0	0
$133$	29.8184	0.224198
$134$	0	0
$135$	0	0
$136$	0	0
$137$	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	$-0.625179\pi$
$139$	−106.530	−0.766404	−0.383202	+	0.923665i	$0.625179\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 185.915i	− 1.30011i
$144$	0	0
$145$	17.0908	0.117868
$146$	0	0
$147$	0	0
$148$	0	0
$149$	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	$-0.106585\pi$
$151$	285.227	1.88892	0.944460	+	0.328625i	$0.106585\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	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	$0.716113\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 59.9219i	− 0.372186i
$162$	0	0
$163$	249.060	1.52798	0.763988	−	0.645230i	$-0.223238\pi$
$163$	249.060	1.52798	0.763988	+	0.645230i	$0.223238\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 48.4365i	− 0.290039i	−0.989429	−	0.145019i	$-0.953676\pi$
$167$	− 48.4365i	− 0.290039i	0.989429	−	0.145019i	$-0.0463244\pi$
$168$	0	0
$169$	218.969	1.29568
$170$	0	0
$171$	0	0
$172$	0	0
$173$	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
$176$	0	0
$177$	0	0
$178$	0	0
$179$	285.071i	1.59257i	0.604919	+	0.796287i	$0.293206\pi$
$179$	285.071i	1.59257i	−0.604919	+	0.796287i	$0.706794\pi$
$180$	0	0
$181$	37.1214	0.205091	0.102545	−	0.994728i	$-0.467301\pi$
$181$	37.1214	0.205091	0.102545	+	0.994728i	$0.467301\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	89.9093i	0.485996i
$186$	0	0
$187$	18.0000	0.0962567
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.9241i	0.0938433i	0.998899	+	0.0469217i	$0.0149411\pi$
$191$	17.9241i	0.0938433i	−0.998899	+	0.0469217i	$0.985059\pi$
$192$	0	0
$193$	−95.4541	−0.494581	−0.247290	−	0.968941i	$-0.579540\pi$
$193$	−95.4541	−0.494581	−0.247290	+	0.968941i	$0.579540\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	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	$-0.505211\pi$
$199$	−6.51531	−0.0327402	−0.0163701	+	0.999866i	$0.505211\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	20.8809i	0.102862i
$204$	0	0
$205$	−321.272	−1.56718
$206$	0	0
$207$	0	0
$208$	0	0
$209$	44.3334i	0.212122i
$210$	0	0
$211$	154.439	0.731940	0.365970	−	0.930627i	$-0.380737\pi$
$211$	154.439	0.731940	0.365970	+	0.930627i	$0.380737\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 4.96021i	− 0.0230707i
$216$	0	0
$217$	−260.576	−1.20081
$218$	0	0
$219$	0	0
$220$	0	0
$221$	37.5625i	0.169966i
$222$	0	0
$223$	−92.7730	−0.416022	−0.208011	−	0.978126i	$-0.566699\pi$
$223$	−92.7730	−0.416022	−0.208011	+	0.978126i	$0.566699\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 169.802i	− 0.748026i	−0.927423	−	0.374013i	$-0.877981\pi$
$227$	− 169.802i	− 0.748026i	0.927423	−	0.374013i	$-0.122019\pi$
$228$	0	0
$229$	407.545	1.77967	0.889836	−	0.456280i	$-0.150819\pi$
$229$	407.545	1.77967	0.889836	+	0.456280i	$0.150819\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	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
$236$	0	0
$237$	0	0
$238$	0	0
$239$	56.5364i	0.236554i	0.992981	+	0.118277i	$0.0377371\pi$
$239$	56.5364i	0.236554i	−0.992981	+	0.118277i	$0.962263\pi$
$240$	0	0
$241$	84.2122	0.349428	0.174714	−	0.984619i	$-0.444100\pi$
$241$	84.2122	0.349428	0.174714	+	0.984619i	$0.444100\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 45.1906i	− 0.184452i
$246$	0	0
$247$	−92.5153	−0.374556
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 218.903i	− 0.872123i	−0.899917	−	0.436062i	$-0.856373\pi$
$251$	− 218.903i	− 0.872123i	0.899917	−	0.436062i	$-0.143627\pi$
$252$	0	0
$253$	89.0908	0.352138
$254$	0	0
$255$	0	0
$256$	0	0
$257$	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
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 336.464i	− 1.27933i	−0.768653	−	0.639666i	$-0.779073\pi$
$263$	− 336.464i	− 1.27933i	0.768653	−	0.639666i	$-0.220927\pi$
$264$	0	0
$265$	−49.5459	−0.186966
$266$	0	0
$267$	0	0
$268$	0	0
$269$	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	$-0.669138\pi$
$271$	−274.636	−1.01342	−0.506708	+	0.862118i	$0.669138\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	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	$0.528191\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 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	$-0.292481\pi$
$283$	343.409	1.21346	0.606729	+	0.794909i	$0.292481\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 392.519i	− 1.36766i
$288$	0	0
$289$	285.363	0.987416
$290$	0	0
$291$	0	0
$292$	0	0
$293$	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
$296$	0	0
$297$	0	0
$298$	0	0
$299$	185.915i	0.621790i
$300$	0	0
$301$	6.06020	0.0201336
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 390.183i	− 1.27929i
$306$	0	0
$307$	−154.091	−0.501924	−0.250962	−	0.967997i	$-0.580747\pi$
$307$	−154.091	−0.501924	−0.250962	+	0.967997i	$0.580747\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 71.9853i	− 0.231464i	−0.993280	−	0.115732i	$-0.963079\pi$
$311$	− 71.9853i	− 0.231464i	0.993280	−	0.115732i	$-0.0369214\pi$
$312$	0	0
$313$	−367.606	−1.17446	−0.587230	−	0.809420i	$-0.699782\pi$
$313$	−367.606	−1.17446	−0.587230	+	0.809420i	$0.699782\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 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
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 8.95717i	− 0.0277312i
$324$	0	0
$325$	−39.3939	−0.121212
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 89.5771i	− 0.272271i
$330$	0	0
$331$	−17.1975	−0.0519561	−0.0259780	−	0.999663i	$-0.508270\pi$
$331$	−17.1975	−0.0519561	−0.0259780	+	0.999663i	$0.508270\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 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	$0.318181\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 387.419i	− 1.13613i
$342$	0	0
$343$	366.287	1.06789
$344$	0	0
$345$	0	0
$346$	0	0
$347$	583.394i	1.68125i	0.541616	+	0.840626i	$0.317813\pi$
$347$	583.394i	1.68125i	−0.541616	+	0.840626i	$0.682187\pi$
$348$	0	0
$349$	312.757	0.896152	0.448076	−	0.893995i	$-0.352109\pi$
$349$	312.757	0.896152	0.448076	+	0.893995i	$0.352109\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 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
$356$	0	0
$357$	0	0
$358$	0	0
$359$	294.028i	0.819019i	0.912306	+	0.409510i	$0.134300\pi$
$359$	294.028i	0.819019i	−0.912306	+	0.409510i	$0.865700\pi$
$360$	0	0
$361$	−338.939	−0.938889
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 499.303i	− 1.36795i
$366$	0	0
$367$	33.2270	0.0905369	0.0452684	−	0.998975i	$-0.485586\pi$
$367$	33.2270	0.0905369	0.0452684	+	0.998975i	$0.485586\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 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	$0.597537\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 64.7858i	− 0.171846i
$378$	0	0
$379$	166.334	0.438875	0.219438	−	0.975627i	$-0.429578\pi$
$379$	166.334	0.438875	0.219438	+	0.975627i	$0.429578\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 736.987i	− 1.92425i	−0.272613	−	0.962124i	$-0.587888\pi$
$383$	− 736.987i	− 1.92425i	0.272613	−	0.962124i	$-0.412112\pi$
$384$	0	0
$385$	−311.363	−0.808736
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 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
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 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	$0.604611\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 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
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 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	$0.682533\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 581.059i	− 1.40692i
$414$	0	0
$415$	−456.773	−1.10066
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.7679i	0.0256990i	0.999917	+	0.0128495i	$0.00409023\pi$
$419$	10.7679i	0.0256990i	−0.999917	+	0.0128495i	$0.995910\pi$
$420$	0	0
$421$	254.303	0.604045	0.302023	−	0.953301i	$-0.402338\pi$
$421$	254.303	0.604045	0.302023	+	0.953301i	$0.402338\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3.81405i	− 0.00897423i
$426$	0	0
$427$	476.712	1.11642
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 698.663i	− 1.62103i	−0.585719	−	0.810514i	$-0.699188\pi$
$431$	− 698.663i	− 1.62103i	0.585719	−	0.810514i	$-0.300812\pi$
$432$	0	0
$433$	211.728	0.488978	0.244489	−	0.969652i	$-0.421380\pi$
$433$	211.728	0.488978	0.244489	+	0.969652i	$0.421380\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 44.3334i	− 0.101449i
$438$	0	0
$439$	−279.863	−0.637501	−0.318750	−	0.947839i	$-0.603263\pi$
$439$	−279.863	−0.637501	−0.318750	+	0.947839i	$0.603263\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	551.254i	1.24437i	0.782872	+	0.622183i	$0.213754\pi$
$443$	551.254i	1.24437i	−0.782872	+	0.622183i	$0.786246\pi$
$444$	0	0
$445$	−214.999	−0.483144
$446$	0	0
$447$	0	0
$448$	0	0
$449$	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
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 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	$0.467778\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 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	$-0.314009\pi$
$463$	510.803	1.10325	0.551623	+	0.834094i	$0.314009\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	833.657i	1.78513i	0.450915	+	0.892567i	$0.351098\pi$
$467$	833.657i	1.78513i	−0.450915	+	0.892567i	$0.648902\pi$
$468$	0	0
$469$	196.514	0.419007
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9.01020i	0.0190490i
$474$	0	0
$475$	9.39388	0.0197766
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 657.190i	− 1.37201i	−0.727599	−	0.686003i	$-0.759364\pi$
$479$	− 657.190i	− 1.37201i	0.727599	−	0.686003i	$-0.240636\pi$
$480$	0	0
$481$	340.817	0.708560
$482$	0	0
$483$	0	0
$484$	0	0
$485$	498.200i	1.02722i
$486$	0	0
$487$	351.666	0.722107	0.361054	−	0.932545i	$-0.382417\pi$
$487$	351.666	0.722107	0.361054	+	0.932545i	$0.382417\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	245.418i	0.499834i	0.968267	+	0.249917i	$0.0804033\pi$
$491$	245.418i	0.499834i	−0.968267	+	0.249917i	$0.919597\pi$
$492$	0	0
$493$	6.27245	0.0127230
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 545.350i	− 1.09728i
$498$	0	0
$499$	−630.226	−1.26298	−0.631489	−	0.775385i	$-0.717556\pi$
$499$	−630.226	−1.26298	−0.631489	+	0.775385i	$0.717556\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 286.891i	− 0.570360i	−0.958474	−	0.285180i	$-0.907947\pi$
$503$	− 286.891i	− 0.570360i	0.958474	−	0.285180i	$-0.0920534\pi$
$504$	0	0
$505$	820.635	1.62502
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 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
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 151.396i	− 0.293973i
$516$	0	0
$517$	133.182	0.257605
$518$	0	0
$519$	0	0
$520$	0	0
$521$	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	$-0.820939\pi$
$523$	−884.817	−1.69181	−0.845906	+	0.533333i	$0.820939\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	78.2746i	0.148529i
$528$	0	0
$529$	439.909	0.831586
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1217.84i	2.28488i
$534$	0	0
$535$	−892.727	−1.66865
$536$	0	0
$537$	0	0
$538$	0	0
$539$	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	$0.655950\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	604.169i	1.10857i
$546$	0	0
$547$	−548.044	−1.00191	−0.500955	−	0.865474i	$-0.667018\pi$
$547$	−548.044	−1.00191	−0.500955	+	0.865474i	$0.667018\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	15.4488i	0.0280378i
$552$	0	0
$553$	188.818	0.341444
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 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
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 606.471i	− 1.07721i	−0.842557	−	0.538607i	$-0.818951\pi$
$563$	− 606.471i	− 1.07721i	0.842557	−	0.538607i	$-0.181049\pi$
$564$	0	0
$565$	1051.00	1.86017
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 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	$-0.475494\pi$
$571$	87.8332	0.153823	0.0769117	+	0.997038i	$0.475494\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 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	$0.536515\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 558.069i	− 0.960531i
$582$	0	0
$583$	90.0000	0.154374
$584$	0	0
$585$	0	0
$586$	0	0
$587$	567.666i	0.967064i	0.875327	+	0.483532i	$0.160646\pi$
$587$	567.666i	0.967064i	−0.875327	+	0.483532i	$0.839354\pi$
$588$	0	0
$589$	−192.788	−0.327314
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 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
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 883.250i	− 1.47454i	−0.675598	−	0.737270i	$-0.736115\pi$
$599$	− 883.250i	− 1.47454i	0.675598	−	0.737270i	$-0.263885\pi$
$600$	0	0
$601$	−795.091	−1.32295	−0.661473	−	0.749969i	$-0.730068\pi$
$601$	−795.091	−1.32295	−0.661473	+	0.749969i	$0.730068\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	165.805i	0.274058i
$606$	0	0
$607$	296.743	0.488869	0.244434	−	0.969666i	$-0.421398\pi$
$607$	296.743	0.488869	0.244434	+	0.969666i	$0.421398\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	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	$0.638617\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	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	$-0.449120\pi$
$619$	197.045	0.318329	0.159164	+	0.987252i	$0.449120\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 262.678i	− 0.421634i
$624$	0	0
$625$	−671.000	−1.07360
$626$	0	0
$627$	0	0
$628$	0	0
$629$	32.9973i	0.0524600i
$630$	0	0
$631$	160.879	0.254958	0.127479	−	0.991841i	$-0.459311\pi$
$631$	160.879	0.254958	0.127479	+	0.991841i	$0.459311\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 52.4334i	− 0.0825723i
$636$	0	0
$637$	−171.303	−0.268922
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 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	$-0.599270\pi$
$643$	−394.591	−0.613672	−0.306836	+	0.951762i	$0.599270\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 418.736i	− 0.647196i	−0.946195	−	0.323598i	$-0.895108\pi$
$647$	− 418.736i	− 0.647196i	0.946195	−	0.323598i	$-0.104892\pi$
$648$	0	0
$649$	863.908	1.33114
$650$	0	0
$651$	0	0
$652$	0	0
$653$	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
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 358.193i	− 0.543539i	−0.962362	−	0.271770i	$-0.912391\pi$
$659$	− 358.193i	− 0.543539i	0.962362	−	0.271770i	$-0.0876089\pi$
$660$	0	0
$661$	−222.271	−0.336265	−0.168133	−	0.985764i	$-0.553774\pi$
$661$	−222.271	−0.336265	−0.168133	+	0.985764i	$0.553774\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	154.941i	0.232994i
$666$	0	0
$667$	31.0454	0.0465448
$668$	0	0
$669$	0	0
$670$	0	0
$671$	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	$0.568932\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	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
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 1126.36i	− 1.64913i	−0.565767	−	0.824565i	$-0.691420\pi$
$683$	− 1126.36i	− 1.64913i	0.565767	−	0.824565i	$-0.308580\pi$
$684$	0	0
$685$	−1219.45	−1.78022
$686$	0	0
$687$	0	0
$688$	0	0
$689$	187.813i	0.272587i
$690$	0	0
$691$	−1037.68	−1.50171	−0.750855	−	0.660467i	$-0.770358\pi$
$691$	−1037.68	−1.50171	−0.750855	+	0.660467i	$0.770358\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 553.547i	− 0.796470i
$696$	0	0
$697$	−117.909	−0.169167
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 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
$704$	0	0
$705$	0	0
$706$	0	0
$707$	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	$0.809692\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	387.419i	0.543365i
$714$	0	0
$715$	966.044	1.35111
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 515.416i	− 0.716851i	−0.933558	−	0.358426i	$-0.883314\pi$
$719$	− 515.416i	− 0.716851i	0.933558	−	0.358426i	$-0.116686\pi$
$720$	0	0
$721$	184.970	0.256547
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.57826i	0.00907346i
$726$	0	0
$727$	841.409	1.15737	0.578685	−	0.815551i	$-0.303566\pi$
$727$	841.409	1.15737	0.578685	+	0.815551i	$0.303566\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 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	$0.364265\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	292.174i	0.396437i
$738$	0	0
$739$	389.362	0.526877	0.263439	−	0.964676i	$-0.415143\pi$
$739$	389.362	0.526877	0.263439	+	0.964676i	$0.415143\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1044.75i	1.40612i	0.711129	+	0.703061i	$0.248184\pi$
$743$	1044.75i	1.40612i	−0.711129	+	0.703061i	$0.751816\pi$
$744$	0	0
$745$	−546.181	−0.733128
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1090.70i	− 1.45621i
$750$	0	0
$751$	1291.83	1.72015	0.860074	−	0.510169i	$-0.170417\pi$
$751$	1291.83	1.72015	0.860074	+	0.510169i	$0.170417\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	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	$0.258277\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	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
$764$	0	0
$765$	0	0
$766$	0	0
$767$	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	$0.428474\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	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
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 290.407i	− 0.372794i
$780$	0	0
$781$	810.817	1.03818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1024.59i	− 1.30520i
$786$	0	0
$787$	103.954	0.132088	0.0660442	−	0.997817i	$-0.478962\pi$
$787$	103.954	0.132088	0.0660442	+	0.997817i	$0.478962\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1284.07i	1.62335i
$792$	0	0
$793$	−1479.06	−1.86514
$794$	0	0
$795$	0	0
$796$	0	0
$797$	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
$800$	0	0
$801$	0	0
$802$	0	0
$803$	906.981i	1.12949i
$804$	0	0
$805$	311.363	0.386787
$806$	0	0
$807$	0	0
$808$	0	0
$809$	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	$-0.649778\pi$
$811$	−735.362	−0.906735	−0.453368	+	0.891324i	$0.649778\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1294.15i	1.58792i
$816$	0	0
$817$	4.48366	0.00548796
$818$	0	0
$819$	0	0
$820$	0	0
$821$	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	$-0.113615\pi$
$823$	1542.26	1.87395	0.936973	+	0.349402i	$0.113615\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	955.707i	1.15563i	0.816167	+	0.577815i	$0.196095\pi$
$827$	955.707i	1.15563i	−0.816167	+	0.577815i	$0.803905\pi$
$828$	0	0
$829$	1082.88	1.30625	0.653123	−	0.757252i	$-0.273458\pi$
$829$	1082.88	1.30625	0.653123	+	0.757252i	$0.273458\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 16.5853i	− 0.0199103i
$834$	0	0
$835$	251.683	0.301417
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1043.59i	− 1.24385i	−0.783075	−	0.621927i	$-0.786350\pi$
$839$	− 1043.59i	− 1.24385i	0.783075	−	0.621927i	$-0.213650\pi$
$840$	0	0
$841$	830.182	0.987136
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1137.80i	1.34651i
$846$	0	0
$847$	−202.574	−0.239167
$848$	0	0
$849$	0	0
$850$	0	0
$851$	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	$0.410416\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	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	$-0.688243\pi$
$859$	−957.802	−1.11502	−0.557510	+	0.830170i	$0.688243\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 524.200i	− 0.607416i	−0.952765	−	0.303708i	$-0.901775\pi$
$863$	− 524.200i	− 0.607416i	0.952765	−	0.303708i	$-0.0982247\pi$
$864$	0	0
$865$	−521.908	−0.603362
$866$	0	0
$867$	0	0
$868$	0	0
$869$	280.732i	0.323052i
$870$	0	0
$871$	−609.711	−0.700012
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 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	$0.305178\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 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	$-0.446722\pi$
$883$	294.213	0.333197	0.166599	+	0.986025i	$0.446722\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 574.803i	− 0.648031i	−0.946052	−	0.324015i	$-0.894967\pi$
$887$	− 574.803i	− 0.648031i	0.946052	−	0.324015i	$-0.105033\pi$
$888$	0	0
$889$	64.0612	0.0720599
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 66.2739i	− 0.0742149i
$894$	0	0
$895$	−1481.27	−1.65505
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 135.004i	− 0.150171i
$900$	0	0
$901$	−18.1837	−0.0201817
$902$	0	0
$903$	0	0
$904$	0	0
$905$	192.889i	0.213137i
$906$	0	0
$907$	510.074	0.562375	0.281187	−	0.959653i	$-0.409272\pi$
$907$	510.074	0.562375	0.281187	+	0.959653i	$0.409272\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	927.371i	1.01797i	0.860775	+	0.508985i	$0.169979\pi$
$911$	927.371i	1.01797i	−0.860775	+	0.508985i	$0.830021\pi$
$912$	0	0
$913$	829.727	0.908791
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 31.4897i	− 0.0343399i
$918$	0	0
$919$	1240.63	1.34998	0.674991	−	0.737826i	$-0.264147\pi$
$919$	1240.63	1.34998	0.674991	+	0.737826i	$0.264147\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1692.02i	1.83317i
$924$	0	0
$925$	−34.6061	−0.0374120
$926$	0	0
$927$	0	0
$928$	0	0
$929$	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
$932$	0	0
$933$	0	0
$934$	0	0
$935$	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	$0.250698\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	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
$944$	0	0
$945$	0	0
$946$	0	0
$947$	775.792i	0.819210i	0.912263	+	0.409605i	$0.134333\pi$
$947$	775.792i	0.819210i	−0.912263	+	0.409605i	$0.865667\pi$
$948$	0	0
$949$	−1892.69	−1.99441
$950$	0	0
$951$	0	0
$952$	0	0
$953$	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
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1489.88i	− 1.55358i
$960$	0	0
$961$	723.725	0.753096
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 495.994i	− 0.513983i
$966$	0	0
$967$	−1224.23	−1.26600	−0.633002	−	0.774150i	$-0.718178\pi$
$967$	−1224.23	−1.26600	−0.633002	+	0.774150i	$0.718178\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 658.702i	− 0.678375i	−0.940719	−	0.339188i	$-0.889848\pi$
$971$	− 658.702i	− 0.678375i	0.940719	−	0.339188i	$-0.110152\pi$
$972$	0	0
$973$	676.303	0.695070
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 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
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 827.840i	− 0.842157i	−0.907024	−	0.421078i	$-0.861652\pi$
$983$	− 827.840i	− 0.842157i	0.907024	−	0.421078i	$-0.138348\pi$
$984$	0	0
$985$	−833.271	−0.845961
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 9.01020i	− 0.00911041i
$990$	0	0
$991$	−429.546	−0.433447	−0.216723	−	0.976233i	$-0.569537\pi$
$991$	−429.546	−0.433447	−0.216723	+	0.976233i	$0.569537\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 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	$0.613324\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.e.g.161.3		4
3.2	odd	2	inner	1296.3.e.g.161.1		4
4.3	odd	2		162.3.b.a.161.2		4
9.2	odd	6		432.3.q.d.17.2		4
9.4	even	3		432.3.q.d.305.2		4
9.5	odd	6		144.3.q.c.65.1		4
9.7	even	3		144.3.q.c.113.1		4
12.11	even	2		162.3.b.a.161.3		4
36.7	odd	6		18.3.d.a.5.1	✓	4
36.11	even	6		54.3.d.a.17.2		4
36.23	even	6		18.3.d.a.11.1	yes	4
36.31	odd	6		54.3.d.a.35.2		4
72.5	odd	6		576.3.q.e.65.2		4
72.11	even	6		1728.3.q.d.449.1		4
72.13	even	6		1728.3.q.c.1601.2		4
72.29	odd	6		1728.3.q.c.449.2		4
72.43	odd	6		576.3.q.f.257.1		4
72.59	even	6		576.3.q.f.65.1		4
72.61	even	6		576.3.q.e.257.2		4
72.67	odd	6		1728.3.q.d.1601.1		4
180.7	even	12		450.3.k.a.149.2		8
180.23	odd	12		450.3.k.a.299.2		8
180.43	even	12		450.3.k.a.149.3		8
180.47	odd	12		1350.3.k.a.449.3		8
180.59	even	6		450.3.i.b.101.2		4
180.67	even	12		1350.3.k.a.899.2		8
180.79	odd	6		450.3.i.b.401.2		4
180.83	odd	12		1350.3.k.a.449.2		8
180.103	even	12		1350.3.k.a.899.3		8
180.119	even	6		1350.3.i.b.1151.1		4
180.139	odd	6		1350.3.i.b.251.1		4
180.167	odd	12		450.3.k.a.299.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.3.d.a.5.1	✓	4	36.7	odd	6
18.3.d.a.11.1	yes	4	36.23	even	6
54.3.d.a.17.2		4	36.11	even	6
54.3.d.a.35.2		4	36.31	odd	6
144.3.q.c.65.1		4	9.5	odd	6
144.3.q.c.113.1		4	9.7	even	3
162.3.b.a.161.2		4	4.3	odd	2
162.3.b.a.161.3		4	12.11	even	2
432.3.q.d.17.2		4	9.2	odd	6
432.3.q.d.305.2		4	9.4	even	3
450.3.i.b.101.2		4	180.59	even	6
450.3.i.b.401.2		4	180.79	odd	6
450.3.k.a.149.2		8	180.7	even	12
450.3.k.a.149.3		8	180.43	even	12
450.3.k.a.299.2		8	180.23	odd	12
450.3.k.a.299.3		8	180.167	odd	12
576.3.q.e.65.2		4	72.5	odd	6
576.3.q.e.257.2		4	72.61	even	6
576.3.q.f.65.1		4	72.59	even	6
576.3.q.f.257.1		4	72.43	odd	6
1296.3.e.g.161.1		4	3.2	odd	2	inner
1296.3.e.g.161.3		4	1.1	even	1	trivial
1350.3.i.b.251.1		4	180.139	odd	6
1350.3.i.b.1151.1		4	180.119	even	6
1350.3.k.a.449.2		8	180.83	odd	12
1350.3.k.a.449.3		8	180.47	odd	12
1350.3.k.a.899.2		8	180.67	even	12
1350.3.k.a.899.3		8	180.103	even	12
1728.3.q.c.449.2		4	72.29	odd	6
1728.3.q.c.1601.2		4	72.13	even	6
1728.3.q.d.449.1		4	72.11	even	6
1728.3.q.d.1601.1		4	72.67	odd	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615i q^{5} -6.34847 q^{7} +O(q^{10})\)	\(q+5.19615i q^{5} -6.34847 q^{7} -9.43879i q^{11} +19.6969 q^{13} +1.90702i q^{17} -4.69694 q^{19} +9.43879i q^{23} -2.00000 q^{25} -3.28913i q^{29} +41.0454 q^{31} -32.9876i q^{35} +17.3031 q^{37} +61.8289i q^{41} -0.954592 q^{43} +14.1100i q^{47} -8.69694 q^{49} +9.53512i q^{53} +49.0454 q^{55} +91.5274i q^{59} -75.0908 q^{61} +102.348i q^{65} -30.9546 q^{67} +85.9026i q^{71} -96.0908 q^{73} +59.9219i q^{77} -29.7423 q^{79} +87.9060i q^{83} -9.90918 q^{85} +41.3766i q^{89} -125.045 q^{91} -24.4060i q^{95} +95.8786 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} + 20 q^{13} + 40 q^{19} - 8 q^{25} + 76 q^{31} + 128 q^{37} - 92 q^{43} + 24 q^{49} + 108 q^{55} - 124 q^{61} - 212 q^{67} - 208 q^{73} + 28 q^{79} - 216 q^{85} - 412 q^{91} - 28 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 + 20 * q^13 + 40 * q^19 - 8 * q^25 + 76 * q^31 + 128 * q^37 - 92 * q^43 + 24 * q^49 + 108 * q^55 - 124 * q^61 - 212 * q^67 - 208 * q^73 + 28 * q^79 - 216 * q^85 - 412 * q^91 - 28 * q^97

Introduction

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