LMFDB - Embedded newform 216.5.e.a.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [216,5,Mod(161,216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(216, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("216.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 216 = 2^{3} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	216.e (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:	$22.3279120261$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$-1.16372i$ of defining polynomial
Character	$\chi$	$=$	216.161
Dual form			216.5.e.a.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+16.7931i q^{5} -54.4980 q^{7} +O(q^{10})$	$q+16.7931i q^{5} -54.4980 q^{7} +27.7519i q^{11} +70.4980 q^{13} -522.715i q^{17} -157.996 q^{19} -368.583i q^{23} +342.992 q^{25} -61.1828i q^{29} +683.961 q^{31} -915.190i q^{35} -344.467 q^{37} -2251.07i q^{41} -2331.94 q^{43} -2827.47i q^{47} +569.035 q^{49} -2258.17i q^{53} -466.039 q^{55} -5556.56i q^{59} +5151.41 q^{61} +1183.88i q^{65} -7004.90 q^{67} +936.775i q^{71} +5613.87 q^{73} -1512.42i q^{77} -1495.60 q^{79} +11070.7i q^{83} +8778.01 q^{85} -8524.75i q^{89} -3842.00 q^{91} -2653.24i q^{95} -6355.67 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 36 q^{7}+O(q^{10}) $ 4 * q + 36 * q^7	$ 4 q + 36 q^{7} + 28 q^{13} - 124 q^{19} + 356 q^{25} - 2344 q^{31} + 2940 q^{37} - 2216 q^{43} + 6848 q^{49} - 6944 q^{55} + 8668 q^{61} - 14812 q^{67} + 5692 q^{73} - 18428 q^{79} + 36128 q^{85} - 15876 q^{91} + 17756 q^{97}+O(q^{100}) $ 4 * q + 36 * q^7 + 28 * q^13 - 124 * q^19 + 356 * q^25 - 2344 * q^31 + 2940 * q^37 - 2216 * q^43 + 6848 * q^49 - 6944 * q^55 + 8668 * q^61 - 14812 * q^67 + 5692 * q^73 - 18428 * q^79 + 36128 * q^85 - 15876 * q^91 + 17756 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/216\mathbb{Z}\right)^\times$.

$n$	$55$	$109$	$137$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	16.7931i	0.671724i	0.941911	+	0.335862i	$0.109028\pi$
$5$	16.7931i	0.671724i	−0.941911	+	0.335862i	$0.890972\pi$
$6$	0	0
$7$	−54.4980	−1.11220	−0.556102	−	0.831114i	$-0.687704\pi$
$7$	−54.4980	−1.11220	−0.556102	+	0.831114i	$0.687704\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	27.7519i	0.229354i	0.993403	+	0.114677i	$0.0365833\pi$
$11$	27.7519i	0.229354i	−0.993403	+	0.114677i	$0.963417\pi$
$12$	0	0
$13$	70.4980	0.417148	0.208574	−	0.978007i	$-0.433118\pi$
$13$	70.4980	0.417148	0.208574	+	0.978007i	$0.433118\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 522.715i	− 1.80870i	−0.426787	−	0.904352i	$-0.640355\pi$
$17$	− 522.715i	− 1.80870i	0.426787	−	0.904352i	$-0.359645\pi$
$18$	0	0
$19$	−157.996	−0.437662	−0.218831	−	0.975763i	$-0.570224\pi$
$19$	−157.996	−0.437662	−0.218831	+	0.975763i	$0.570224\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 368.583i	− 0.696754i	−0.937354	−	0.348377i	$-0.886733\pi$
$23$	− 368.583i	− 0.696754i	0.937354	−	0.348377i	$-0.113267\pi$
$24$	0	0
$25$	342.992	0.548787
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 61.1828i	− 0.0727501i	−0.999338	−	0.0363750i	$-0.988419\pi$
$29$	− 61.1828i	− 0.0727501i	0.999338	−	0.0363750i	$-0.0115811\pi$
$30$	0	0
$31$	683.961	0.711718	0.355859	−	0.934540i	$-0.384188\pi$
$31$	683.961	0.711718	0.355859	+	0.934540i	$0.384188\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 915.190i	− 0.747094i
$36$	0	0
$37$	−344.467	−0.251619	−0.125810	−	0.992054i	$-0.540153\pi$
$37$	−344.467	−0.251619	−0.125810	+	0.992054i	$0.540153\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2251.07i	− 1.33913i	−0.742755	−	0.669564i	$-0.766481\pi$
$41$	− 2251.07i	− 1.33913i	0.742755	−	0.669564i	$-0.233519\pi$
$42$	0	0
$43$	−2331.94	−1.26119	−0.630596	−	0.776111i	$-0.717190\pi$
$43$	−2331.94	−1.26119	−0.630596	+	0.776111i	$0.717190\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2827.47i	− 1.27998i	−0.768384	−	0.639989i	$-0.778939\pi$
$47$	− 2827.47i	− 1.27998i	0.768384	−	0.639989i	$-0.221061\pi$
$48$	0	0
$49$	569.035	0.236999
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2258.17i	− 0.803906i	−0.915660	−	0.401953i	$-0.868332\pi$
$53$	− 2258.17i	− 0.803906i	0.915660	−	0.401953i	$-0.131668\pi$
$54$	0	0
$55$	−466.039	−0.154063
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 5556.56i	− 1.59625i	−0.602489	−	0.798127i	$-0.705824\pi$
$59$	− 5556.56i	− 1.59625i	0.602489	−	0.798127i	$-0.294176\pi$
$60$	0	0
$61$	5151.41	1.38441	0.692207	−	0.721699i	$-0.256638\pi$
$61$	5151.41	1.38441	0.692207	+	0.721699i	$0.256638\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1183.88i	0.280208i
$66$	0	0
$67$	−7004.90	−1.56046	−0.780229	−	0.625494i	$-0.784898\pi$
$67$	−7004.90	−1.56046	−0.780229	+	0.625494i	$0.784898\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	936.775i	0.185831i	0.995674	+	0.0929155i	$0.0296187\pi$
$71$	936.775i	0.185831i	−0.995674	+	0.0929155i	$0.970381\pi$
$72$	0	0
$73$	5613.87	1.05346	0.526728	−	0.850034i	$-0.323419\pi$
$73$	5613.87	1.05346	0.526728	+	0.850034i	$0.323419\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1512.42i	− 0.255089i
$78$	0	0
$79$	−1495.60	−0.239641	−0.119820	−	0.992796i	$-0.538232\pi$
$79$	−1495.60	−0.239641	−0.119820	+	0.992796i	$0.538232\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11070.7i	1.60701i	0.595296	+	0.803506i	$0.297035\pi$
$83$	11070.7i	1.60701i	−0.595296	+	0.803506i	$0.702965\pi$
$84$	0	0
$85$	8778.01	1.21495
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 8524.75i	− 1.07622i	−0.842874	−	0.538110i	$-0.819138\pi$
$89$	− 8524.75i	− 1.07622i	0.842874	−	0.538110i	$-0.180862\pi$
$90$	0	0
$91$	−3842.00	−0.463954
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 2653.24i	− 0.293988i
$96$	0	0
$97$	−6355.67	−0.675488	−0.337744	−	0.941238i	$-0.609664\pi$
$97$	−6355.67	−0.675488	−0.337744	+	0.941238i	$0.609664\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18722.7i	1.83538i	0.397296	+	0.917691i	$0.369949\pi$
$101$	18722.7i	1.83538i	−0.397296	+	0.917691i	$0.630051\pi$
$102$	0	0
$103$	−12421.4	−1.17084	−0.585420	−	0.810731i	$-0.699070\pi$
$103$	−12421.4	−1.17084	−0.585420	+	0.810731i	$0.699070\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6532.11i	− 0.570540i	−0.958447	−	0.285270i	$-0.907917\pi$
$107$	− 6532.11i	− 0.570540i	0.958447	−	0.285270i	$-0.0920833\pi$
$108$	0	0
$109$	1100.20	0.0926014	0.0463007	−	0.998928i	$-0.485257\pi$
$109$	1100.20	0.0926014	0.0463007	+	0.998928i	$0.485257\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 4380.56i	− 0.343062i	−0.985179	−	0.171531i	$-0.945129\pi$
$113$	− 4380.56i	− 0.343062i	0.985179	−	0.171531i	$-0.0548713\pi$
$114$	0	0
$115$	6189.65	0.468026
$116$	0	0
$117$	0	0
$118$	0	0
$119$	28487.0i	2.01165i
$120$	0	0
$121$	13870.8	0.947397
$122$	0	0
$123$	0	0
$124$	0	0
$125$	16255.6i	1.04036i
$126$	0	0
$127$	−9545.65	−0.591832	−0.295916	−	0.955214i	$-0.595625\pi$
$127$	−9545.65	−0.591832	−0.295916	+	0.955214i	$0.595625\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13920.6i	0.811177i	0.914056	+	0.405588i	$0.132933\pi$
$131$	13920.6i	0.811177i	−0.914056	+	0.405588i	$0.867067\pi$
$132$	0	0
$133$	8610.47	0.486770
$134$	0	0
$135$	0	0
$136$	0	0
$137$	32465.5i	1.72974i	0.501994	+	0.864871i	$0.332600\pi$
$137$	32465.5i	1.72974i	−0.501994	+	0.864871i	$0.667400\pi$
$138$	0	0
$139$	−10915.3	−0.564943	−0.282472	−	0.959276i	$-0.591154\pi$
$139$	−10915.3	−0.564943	−0.282472	+	0.959276i	$0.591154\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1956.45i	0.0956747i
$144$	0	0
$145$	1027.45	0.0488680
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 16681.0i	− 0.751363i	−0.926749	−	0.375682i	$-0.877409\pi$
$149$	− 16681.0i	− 0.751363i	0.926749	−	0.375682i	$-0.122591\pi$
$150$	0	0
$151$	−14388.9	−0.631064	−0.315532	−	0.948915i	$-0.602183\pi$
$151$	−14388.9	−0.631064	−0.315532	+	0.948915i	$0.602183\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11485.8i	0.478078i
$156$	0	0
$157$	47631.3	1.93238	0.966191	−	0.257829i	$-0.0830070\pi$
$157$	47631.3	1.93238	0.966191	+	0.257829i	$0.0830070\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	20087.0i	0.774933i
$162$	0	0
$163$	20759.8	0.781353	0.390677	−	0.920528i	$-0.372241\pi$
$163$	20759.8	0.781353	0.390677	+	0.920528i	$0.372241\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 51588.7i	− 1.84979i	−0.380226	−	0.924893i	$-0.624154\pi$
$167$	− 51588.7i	− 1.84979i	0.380226	−	0.924893i	$-0.375846\pi$
$168$	0	0
$169$	−23591.0	−0.825987
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1946.18i	− 0.0650264i	−0.999471	−	0.0325132i	$-0.989649\pi$
$173$	− 1946.18i	− 0.0650264i	0.999471	−	0.0325132i	$-0.0103511\pi$
$174$	0	0
$175$	−18692.4	−0.610364
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 10628.6i	− 0.331718i	−0.986149	−	0.165859i	$-0.946960\pi$
$179$	− 10628.6i	− 0.331718i	0.986149	−	0.165859i	$-0.0530397\pi$
$180$	0	0
$181$	−50545.9	−1.54287	−0.771434	−	0.636309i	$-0.780460\pi$
$181$	−50545.9	−1.54287	−0.771434	+	0.636309i	$0.780460\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 5784.66i	− 0.169018i
$186$	0	0
$187$	14506.3	0.414834
$188$	0	0
$189$	0	0
$190$	0	0
$191$	56168.5i	1.53967i	0.638246	+	0.769833i	$0.279660\pi$
$191$	56168.5i	1.53967i	−0.638246	+	0.769833i	$0.720340\pi$
$192$	0	0
$193$	−53259.2	−1.42982	−0.714908	−	0.699219i	$-0.753531\pi$
$193$	−53259.2	−1.42982	−0.714908	+	0.699219i	$0.753531\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 58910.9i	− 1.51797i	−0.651109	−	0.758984i	$-0.725696\pi$
$197$	− 58910.9i	− 1.51797i	0.651109	−	0.758984i	$-0.274304\pi$
$198$	0	0
$199$	55975.9	1.41350	0.706748	−	0.707465i	$-0.250161\pi$
$199$	55975.9	1.41350	0.706748	+	0.707465i	$0.250161\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3334.34i	0.0809130i
$204$	0	0
$205$	37802.5	0.899523
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4384.68i	− 0.100380i
$210$	0	0
$211$	−6355.39	−0.142750	−0.0713752	−	0.997450i	$-0.522739\pi$
$211$	−6355.39	−0.142750	−0.0713752	+	0.997450i	$0.522739\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 39160.6i	− 0.847173i
$216$	0	0
$217$	−37274.5	−0.791576
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 36850.4i	− 0.754497i
$222$	0	0
$223$	46703.3	0.939156	0.469578	−	0.882891i	$-0.344406\pi$
$223$	46703.3	0.939156	0.469578	+	0.882891i	$0.344406\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 45773.4i	− 0.888304i	−0.895952	−	0.444152i	$-0.853505\pi$
$227$	− 45773.4i	− 0.888304i	0.895952	−	0.444152i	$-0.146495\pi$
$228$	0	0
$229$	28850.9	0.550159	0.275079	−	0.961422i	$-0.411296\pi$
$229$	28850.9	0.550159	0.275079	+	0.961422i	$0.411296\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 52269.2i	− 0.962796i	−0.876502	−	0.481398i	$-0.840129\pi$
$233$	− 52269.2i	− 0.962796i	0.876502	−	0.481398i	$-0.159871\pi$
$234$	0	0
$235$	47482.0	0.859792
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 28580.2i	− 0.500345i	−0.968201	−	0.250172i	$-0.919513\pi$
$239$	− 28580.2i	− 0.500345i	0.968201	−	0.250172i	$-0.0804873\pi$
$240$	0	0
$241$	−105372.	−1.81423	−0.907115	−	0.420882i	$-0.861721\pi$
$241$	−105372.	−1.81423	−0.907115	+	0.420882i	$0.861721\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9555.86i	0.159198i
$246$	0	0
$247$	−11138.4	−0.182570
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 52937.9i	− 0.840271i	−0.907461	−	0.420135i	$-0.861983\pi$
$251$	− 52937.9i	− 0.840271i	0.907461	−	0.420135i	$-0.138017\pi$
$252$	0	0
$253$	10228.9	0.159803
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 48237.2i	− 0.730324i	−0.930944	−	0.365162i	$-0.881014\pi$
$257$	− 48237.2i	− 0.730324i	0.930944	−	0.365162i	$-0.118986\pi$
$258$	0	0
$259$	18772.7	0.279852
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 84023.0i	− 1.21475i	−0.794415	−	0.607375i	$-0.792223\pi$
$263$	− 84023.0i	− 1.21475i	0.794415	−	0.607375i	$-0.207777\pi$
$264$	0	0
$265$	37921.7	0.540003
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11092.5i	0.153294i	0.997058	+	0.0766470i	$0.0244215\pi$
$269$	11092.5i	0.153294i	−0.997058	+	0.0766470i	$0.975579\pi$
$270$	0	0
$271$	72287.5	0.984293	0.492147	−	0.870512i	$-0.336212\pi$
$271$	72287.5	0.984293	0.492147	+	0.870512i	$0.336212\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9518.67i	0.125867i
$276$	0	0
$277$	−18391.1	−0.239689	−0.119845	−	0.992793i	$-0.538240\pi$
$277$	−18391.1	−0.239689	−0.119845	+	0.992793i	$0.538240\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 23661.1i	− 0.299656i	−0.988712	−	0.149828i	$-0.952128\pi$
$281$	− 23661.1i	− 0.299656i	0.988712	−	0.149828i	$-0.0478720\pi$
$282$	0	0
$283$	−110406.	−1.37854	−0.689269	−	0.724505i	$-0.742068\pi$
$283$	−110406.	−1.37854	−0.689269	+	0.724505i	$0.742068\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	122679.i	1.48938i
$288$	0	0
$289$	−189710.	−2.27141
$290$	0	0
$291$	0	0
$292$	0	0
$293$	84688.1i	0.986478i	0.869894	+	0.493239i	$0.164187\pi$
$293$	84688.1i	0.986478i	−0.869894	+	0.493239i	$0.835813\pi$
$294$	0	0
$295$	93311.8	1.07224
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 25984.4i	− 0.290650i
$300$	0	0
$301$	127086.	1.40270
$302$	0	0
$303$	0	0
$304$	0	0
$305$	86508.0i	0.929944i
$306$	0	0
$307$	−16361.1	−0.173595	−0.0867973	−	0.996226i	$-0.527663\pi$
$307$	−16361.1	−0.173595	−0.0867973	+	0.996226i	$0.527663\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14948.3i	0.154550i	0.997010	+	0.0772752i	$0.0246220\pi$
$311$	14948.3i	0.154550i	−0.997010	+	0.0772752i	$0.975378\pi$
$312$	0	0
$313$	−134060.	−1.36839	−0.684196	−	0.729298i	$-0.739847\pi$
$313$	−134060.	−1.36839	−0.684196	+	0.729298i	$0.739847\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 86061.7i	− 0.856429i	−0.903677	−	0.428214i	$-0.859143\pi$
$317$	− 86061.7i	− 0.856429i	0.903677	−	0.428214i	$-0.140857\pi$
$318$	0	0
$319$	1697.94	0.0166855
$320$	0	0
$321$	0	0
$322$	0	0
$323$	82587.0i	0.791601i
$324$	0	0
$325$	24180.3	0.228926
$326$	0	0
$327$	0	0
$328$	0	0
$329$	154092.i	1.42360i
$330$	0	0
$331$	149207.	1.36187	0.680933	−	0.732346i	$-0.261575\pi$
$331$	149207.	1.36187	0.680933	+	0.732346i	$0.261575\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 117634.i	− 1.04820i
$336$	0	0
$337$	36926.1	0.325142	0.162571	−	0.986697i	$-0.448021\pi$
$337$	36926.1	0.325142	0.162571	+	0.986697i	$0.448021\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	18981.2i	0.163235i
$342$	0	0
$343$	99838.5	0.848613
$344$	0	0
$345$	0	0
$346$	0	0
$347$	88812.4i	0.737590i	0.929511	+	0.368795i	$0.120230\pi$
$347$	88812.4i	0.737590i	−0.929511	+	0.368795i	$0.879770\pi$
$348$	0	0
$349$	−14758.3	−0.121167	−0.0605835	−	0.998163i	$-0.519296\pi$
$349$	−14758.3	−0.121167	−0.0605835	+	0.998163i	$0.519296\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 77766.3i	− 0.624083i	−0.950069	−	0.312041i	$-0.898987\pi$
$353$	− 77766.3i	− 0.624083i	0.950069	−	0.312041i	$-0.101013\pi$
$354$	0	0
$355$	−15731.3	−0.124827
$356$	0	0
$357$	0	0
$358$	0	0
$359$	114321.i	0.887025i	0.896268	+	0.443513i	$0.146268\pi$
$359$	114321.i	0.887025i	−0.896268	+	0.443513i	$0.853732\pi$
$360$	0	0
$361$	−105358.	−0.808452
$362$	0	0
$363$	0	0
$364$	0	0
$365$	94274.2i	0.707632i
$366$	0	0
$367$	−200142.	−1.48596	−0.742978	−	0.669316i	$-0.766587\pi$
$367$	−200142.	−1.48596	−0.742978	+	0.669316i	$0.766587\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	123066.i	0.894108i
$372$	0	0
$373$	173908.	1.24997	0.624987	−	0.780635i	$-0.285104\pi$
$373$	173908.	1.24997	0.624987	+	0.780635i	$0.285104\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4313.27i	− 0.0303476i
$378$	0	0
$379$	52888.6	0.368200	0.184100	−	0.982908i	$-0.441063\pi$
$379$	52888.6	0.368200	0.184100	+	0.982908i	$0.441063\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	284885.i	1.94210i	0.238872	+	0.971051i	$0.423222\pi$
$383$	284885.i	1.94210i	−0.238872	+	0.971051i	$0.576778\pi$
$384$	0	0
$385$	25398.2	0.171349
$386$	0	0
$387$	0	0
$388$	0	0
$389$	76777.6i	0.507382i	0.967285	+	0.253691i	$0.0816447\pi$
$389$	76777.6i	0.507382i	−0.967285	+	0.253691i	$0.918355\pi$
$390$	0	0
$391$	−192664.	−1.26022
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 25115.7i	− 0.160972i
$396$	0	0
$397$	207521.	1.31668	0.658342	−	0.752719i	$-0.271258\pi$
$397$	207521.	1.31668	0.658342	+	0.752719i	$0.271258\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26074.8i	0.162156i	0.996708	+	0.0810778i	$0.0258362\pi$
$401$	26074.8i	0.162156i	−0.996708	+	0.0810778i	$0.974164\pi$
$402$	0	0
$403$	48217.9	0.296892
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 9559.58i	− 0.0577099i
$408$	0	0
$409$	−103331.	−0.617708	−0.308854	−	0.951109i	$-0.599946\pi$
$409$	−103331.	−0.617708	−0.308854	+	0.951109i	$0.599946\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	302822.i	1.77536i
$414$	0	0
$415$	−185911.	−1.07947
$416$	0	0
$417$	0	0
$418$	0	0
$419$	111666.i	0.636053i	0.948082	+	0.318027i	$0.103020\pi$
$419$	111666.i	0.636053i	−0.948082	+	0.318027i	$0.896980\pi$
$420$	0	0
$421$	244363.	1.37870	0.689352	−	0.724427i	$-0.257895\pi$
$421$	244363.	1.37870	0.689352	+	0.724427i	$0.257895\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 179287.i	− 0.992594i
$426$	0	0
$427$	−280742.	−1.53975
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 225431.i	− 1.21356i	−0.794871	−	0.606778i	$-0.792462\pi$
$431$	− 225431.i	− 1.21356i	0.794871	−	0.606778i	$-0.207538\pi$
$432$	0	0
$433$	−156965.	−0.837196	−0.418598	−	0.908172i	$-0.637478\pi$
$433$	−156965.	−0.837196	−0.418598	+	0.908172i	$0.637478\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	58234.6i	0.304943i
$438$	0	0
$439$	−135624.	−0.703734	−0.351867	−	0.936050i	$-0.614453\pi$
$439$	−135624.	−0.703734	−0.351867	+	0.936050i	$0.614453\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 296902.i	− 1.51289i	−0.654060	−	0.756443i	$-0.726935\pi$
$443$	− 296902.i	− 1.51289i	0.654060	−	0.756443i	$-0.273065\pi$
$444$	0	0
$445$	143157.	0.722923
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 33504.9i	− 0.166194i	−0.996541	−	0.0830970i	$-0.973519\pi$
$449$	− 33504.9i	− 0.166194i	0.996541	−	0.0830970i	$-0.0264811\pi$
$450$	0	0
$451$	62471.4	0.307134
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 64519.1i	− 0.311649i
$456$	0	0
$457$	105859.	0.506867	0.253433	−	0.967353i	$-0.418440\pi$
$457$	105859.	0.506867	0.253433	+	0.967353i	$0.418440\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 252698.i	− 1.18905i	−0.804078	−	0.594524i	$-0.797341\pi$
$461$	− 252698.i	− 1.18905i	0.804078	−	0.594524i	$-0.202659\pi$
$462$	0	0
$463$	138659.	0.646824	0.323412	−	0.946258i	$-0.395170\pi$
$463$	138659.	0.646824	0.323412	+	0.946258i	$0.395170\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	93323.7i	0.427916i	0.976843	+	0.213958i	$0.0686355\pi$
$467$	93323.7i	0.427916i	−0.976843	+	0.213958i	$0.931364\pi$
$468$	0	0
$469$	381753.	1.73555
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 64715.8i	− 0.289260i
$474$	0	0
$475$	−54191.4	−0.240184
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 11792.8i	− 0.0513981i	−0.999670	−	0.0256991i	$-0.991819\pi$
$479$	− 11792.8i	− 0.0513981i	0.999670	−	0.0256991i	$-0.00818117\pi$
$480$	0	0
$481$	−24284.2	−0.104962
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 106731.i	− 0.453741i
$486$	0	0
$487$	82751.4	0.348913	0.174457	−	0.984665i	$-0.444183\pi$
$487$	82751.4	0.348913	0.174457	+	0.984665i	$0.444183\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 139655.i	− 0.579285i	−0.957135	−	0.289643i	$-0.906464\pi$
$491$	− 139655.i	− 0.579285i	0.957135	−	0.289643i	$-0.0935364\pi$
$492$	0	0
$493$	−31981.2	−0.131583
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 51052.4i	− 0.206682i
$498$	0	0
$499$	89055.0	0.357649	0.178825	−	0.983881i	$-0.442771\pi$
$499$	89055.0	0.357649	0.178825	+	0.983881i	$0.442771\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16199.6i	0.0640277i	0.999487	+	0.0320139i	$0.0101921\pi$
$503$	16199.6i	0.0640277i	−0.999487	+	0.0320139i	$0.989808\pi$
$504$	0	0
$505$	−314412.	−1.23287
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 45421.9i	− 0.175319i	−0.996150	−	0.0876597i	$-0.972061\pi$
$509$	− 45421.9i	− 0.175319i	0.996150	−	0.0876597i	$-0.0279388\pi$
$510$	0	0
$511$	−305945.	−1.17166
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 208594.i	− 0.786480i
$516$	0	0
$517$	78467.6	0.293568
$518$	0	0
$519$	0	0
$520$	0	0
$521$	148324.i	0.546433i	0.961953	+	0.273217i	$0.0880876\pi$
$521$	148324.i	0.546433i	−0.961953	+	0.273217i	$0.911912\pi$
$522$	0	0
$523$	−220169.	−0.804922	−0.402461	−	0.915437i	$-0.631845\pi$
$523$	−220169.	−0.804922	−0.402461	+	0.915437i	$0.631845\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 357517.i	− 1.28729i
$528$	0	0
$529$	143988.	0.514534
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 158696.i	− 0.558614i
$534$	0	0
$535$	109694.	0.383245
$536$	0	0
$537$	0	0
$538$	0	0
$539$	15791.8i	0.0543568i
$540$	0	0
$541$	104022.	0.355410	0.177705	−	0.984084i	$-0.443133\pi$
$541$	104022.	0.355410	0.177705	+	0.984084i	$0.443133\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18475.7i	0.0622025i
$546$	0	0
$547$	106892.	0.357249	0.178624	−	0.983917i	$-0.442835\pi$
$547$	106892.	0.357249	0.178624	+	0.983917i	$0.442835\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9666.65i	0.0318400i
$552$	0	0
$553$	81507.1	0.266529
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 528557.i	− 1.70366i	−0.523822	−	0.851828i	$-0.675494\pi$
$557$	− 528557.i	− 1.70366i	0.523822	−	0.851828i	$-0.324506\pi$
$558$	0	0
$559$	−164398.	−0.526104
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 81419.0i	− 0.256867i	−0.991718	−	0.128434i	$-0.959005\pi$
$563$	− 81419.0i	− 0.256867i	0.991718	−	0.128434i	$-0.0409949\pi$
$564$	0	0
$565$	73563.1	0.230443
$566$	0	0
$567$	0	0
$568$	0	0
$569$	187928.i	0.580452i	0.956958	+	0.290226i	$0.0937305\pi$
$569$	187928.i	0.580452i	−0.956958	+	0.290226i	$0.906270\pi$
$570$	0	0
$571$	−417033.	−1.27908	−0.639540	−	0.768758i	$-0.720875\pi$
$571$	−417033.	−1.27908	−0.639540	+	0.768758i	$0.720875\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 126421.i	− 0.382370i
$576$	0	0
$577$	227914.	0.684571	0.342286	−	0.939596i	$-0.388799\pi$
$577$	227914.	0.684571	0.342286	+	0.939596i	$0.388799\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 603332.i	− 1.78733i
$582$	0	0
$583$	62668.5	0.184379
$584$	0	0
$585$	0	0
$586$	0	0
$587$	166988.i	0.484628i	0.970198	+	0.242314i	$0.0779064\pi$
$587$	166988.i	0.484628i	−0.970198	+	0.242314i	$0.922094\pi$
$588$	0	0
$589$	−108063.	−0.311492
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 318532.i	− 0.905824i	−0.891555	−	0.452912i	$-0.850385\pi$
$593$	− 318532.i	− 0.905824i	0.891555	−	0.452912i	$-0.149615\pi$
$594$	0	0
$595$	−478384.	−1.35127
$596$	0	0
$597$	0	0
$598$	0	0
$599$	302637.i	0.843468i	0.906720	+	0.421734i	$0.138578\pi$
$599$	302637.i	0.843468i	−0.906720	+	0.421734i	$0.861422\pi$
$600$	0	0
$601$	−422926.	−1.17089	−0.585445	−	0.810712i	$-0.699080\pi$
$601$	−422926.	−1.17089	−0.585445	+	0.810712i	$0.699080\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	232934.i	0.636389i
$606$	0	0
$607$	520864.	1.41367	0.706833	−	0.707381i	$-0.250123\pi$
$607$	520864.	1.41367	0.706833	+	0.707381i	$0.250123\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 199331.i	− 0.533941i
$612$	0	0
$613$	178145.	0.474080	0.237040	−	0.971500i	$-0.423823\pi$
$613$	178145.	0.474080	0.237040	+	0.971500i	$0.423823\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 137889.i	− 0.362208i	−0.983464	−	0.181104i	$-0.942033\pi$
$617$	− 137889.i	− 0.362208i	0.983464	−	0.181104i	$-0.0579671\pi$
$618$	0	0
$619$	251133.	0.655425	0.327713	−	0.944777i	$-0.393722\pi$
$619$	251133.	0.655425	0.327713	+	0.944777i	$0.393722\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	464582.i	1.19698i
$624$	0	0
$625$	−58611.3	−0.150045
$626$	0	0
$627$	0	0
$628$	0	0
$629$	180058.i	0.455104i
$630$	0	0
$631$	445643.	1.11925	0.559626	−	0.828745i	$-0.310945\pi$
$631$	445643.	1.11925	0.559626	+	0.828745i	$0.310945\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 160301.i	− 0.397547i
$636$	0	0
$637$	40115.9	0.0988638
$638$	0	0
$639$	0	0
$640$	0	0
$641$	775880.i	1.88833i	0.329467	+	0.944167i	$0.393131\pi$
$641$	775880.i	1.88833i	−0.329467	+	0.944167i	$0.606869\pi$
$642$	0	0
$643$	409296.	0.989955	0.494977	−	0.868906i	$-0.335176\pi$
$643$	409296.	0.989955	0.494977	+	0.868906i	$0.335176\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 443379.i	− 1.05917i	−0.848256	−	0.529586i	$-0.822347\pi$
$647$	− 443379.i	− 1.05917i	0.848256	−	0.529586i	$-0.177653\pi$
$648$	0	0
$649$	154205.	0.366108
$650$	0	0
$651$	0	0
$652$	0	0
$653$	601542.i	1.41072i	0.708851	+	0.705359i	$0.249214\pi$
$653$	601542.i	1.41072i	−0.708851	+	0.705359i	$0.750786\pi$
$654$	0	0
$655$	−233770.	−0.544887
$656$	0	0
$657$	0	0
$658$	0	0
$659$	489818.i	1.12788i	0.825814	+	0.563942i	$0.190716\pi$
$659$	489818.i	1.12788i	−0.825814	+	0.563942i	$0.809284\pi$
$660$	0	0
$661$	−331368.	−0.758417	−0.379209	−	0.925311i	$-0.623804\pi$
$661$	−331368.	−0.758417	−0.379209	+	0.925311i	$0.623804\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	144596.i	0.326975i
$666$	0	0
$667$	−22550.9	−0.0506889
$668$	0	0
$669$	0	0
$670$	0	0
$671$	142961.i	0.317521i
$672$	0	0
$673$	−253335.	−0.559326	−0.279663	−	0.960098i	$-0.590223\pi$
$673$	−253335.	−0.559326	−0.279663	+	0.960098i	$0.590223\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	696282.i	1.51917i	0.650406	+	0.759587i	$0.274599\pi$
$677$	696282.i	1.51917i	−0.650406	+	0.759587i	$0.725401\pi$
$678$	0	0
$679$	346371.	0.751281
$680$	0	0
$681$	0	0
$682$	0	0
$683$	733575.i	1.57254i	0.617880	+	0.786272i	$0.287992\pi$
$683$	733575.i	1.57254i	−0.617880	+	0.786272i	$0.712008\pi$
$684$	0	0
$685$	−545197.	−1.16191
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 159197.i	− 0.335348i
$690$	0	0
$691$	737689.	1.54496	0.772480	−	0.635039i	$-0.219016\pi$
$691$	737689.	1.54496	0.772480	+	0.635039i	$0.219016\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 183301.i	− 0.379486i
$696$	0	0
$697$	−1.17667e6	−2.42209
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 30577.8i	− 0.0622258i	−0.999516	−	0.0311129i	$-0.990095\pi$
$701$	− 30577.8i	− 0.0622258i	0.999516	−	0.0311129i	$-0.00990514\pi$
$702$	0	0
$703$	54424.4	0.110124
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.02035e6i	− 2.04132i
$708$	0	0
$709$	147706.	0.293836	0.146918	−	0.989149i	$-0.453065\pi$
$709$	147706.	0.293836	0.146918	+	0.989149i	$0.453065\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 252096.i	− 0.495892i
$714$	0	0
$715$	−32854.9	−0.0642669
$716$	0	0
$717$	0	0
$718$	0	0
$719$	636542.i	1.23131i	0.788014	+	0.615657i	$0.211109\pi$
$719$	636542.i	1.23131i	−0.788014	+	0.615657i	$0.788891\pi$
$720$	0	0
$721$	676944.	1.30221
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 20985.2i	− 0.0399243i
$726$	0	0
$727$	−190722.	−0.360853	−0.180427	−	0.983588i	$-0.557748\pi$
$727$	−190722.	−0.360853	−0.180427	+	0.983588i	$0.557748\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.21894e6i	2.28112i
$732$	0	0
$733$	−650363.	−1.21045	−0.605226	−	0.796054i	$-0.706917\pi$
$733$	−650363.	−1.21045	−0.605226	+	0.796054i	$0.706917\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 194399.i	− 0.357898i
$738$	0	0
$739$	−183014.	−0.335117	−0.167558	−	0.985862i	$-0.553588\pi$
$739$	−183014.	−0.335117	−0.167558	+	0.985862i	$0.553588\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	20638.0i	0.0373844i	0.999825	+	0.0186922i	$0.00595025\pi$
$743$	20638.0i	0.0373844i	−0.999825	+	0.0186922i	$0.994050\pi$
$744$	0	0
$745$	280126.	0.504709
$746$	0	0
$747$	0	0
$748$	0	0
$749$	355987.i	0.634557i
$750$	0	0
$751$	593868.	1.05296	0.526478	−	0.850189i	$-0.323512\pi$
$751$	593868.	1.05296	0.526478	+	0.850189i	$0.323512\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 241634.i	− 0.423901i
$756$	0	0
$757$	−463873.	−0.809483	−0.404741	−	0.914431i	$-0.632638\pi$
$757$	−463873.	−0.809483	−0.404741	+	0.914431i	$0.632638\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	324294.i	0.559977i	0.960003	+	0.279989i	$0.0903307\pi$
$761$	324294.i	0.559977i	−0.960003	+	0.279989i	$0.909669\pi$
$762$	0	0
$763$	−59958.6	−0.102992
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 391727.i	− 0.665874i
$768$	0	0
$769$	−323637.	−0.547275	−0.273638	−	0.961833i	$-0.588227\pi$
$769$	−323637.	−0.547275	−0.273638	+	0.961833i	$0.588227\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	542127.i	0.907282i	0.891185	+	0.453641i	$0.149875\pi$
$773$	542127.i	0.907282i	−0.891185	+	0.453641i	$0.850125\pi$
$774$	0	0
$775$	234593.	0.390582
$776$	0	0
$777$	0	0
$778$	0	0
$779$	355661.i	0.586085i
$780$	0	0
$781$	−25997.2	−0.0426211
$782$	0	0
$783$	0	0
$784$	0	0
$785$	799876.i	1.29803i
$786$	0	0
$787$	1.00397e6	1.62096	0.810481	−	0.585765i	$-0.199206\pi$
$787$	1.00397e6	1.62096	0.810481	+	0.585765i	$0.199206\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	238732.i	0.381555i
$792$	0	0
$793$	363164.	0.577506
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.01967e6i	− 1.60526i	−0.596480	−	0.802628i	$-0.703435\pi$
$797$	− 1.01967e6i	− 1.60526i	0.596480	−	0.802628i	$-0.296565\pi$
$798$	0	0
$799$	−1.47796e6	−2.31510
$800$	0	0
$801$	0	0
$802$	0	0
$803$	155795.i	0.241615i
$804$	0	0
$805$	−337324.	−0.520541
$806$	0	0
$807$	0	0
$808$	0	0
$809$	323388.i	0.494114i	0.969001	+	0.247057i	$0.0794635\pi$
$809$	323388.i	0.494114i	−0.969001	+	0.247057i	$0.920536\pi$
$810$	0	0
$811$	98256.1	0.149389	0.0746944	−	0.997206i	$-0.476202\pi$
$811$	98256.1	0.149389	0.0746944	+	0.997206i	$0.476202\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	348621.i	0.524853i
$816$	0	0
$817$	368438.	0.551976
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 221130.i	− 0.328067i	−0.986455	−	0.164033i	$-0.947550\pi$
$821$	− 221130.i	− 0.328067i	0.986455	−	0.164033i	$-0.0524505\pi$
$822$	0	0
$823$	1.09099e6	1.61072	0.805359	−	0.592787i	$-0.201973\pi$
$823$	1.09099e6	1.61072	0.805359	+	0.592787i	$0.201973\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.00999e6i	− 1.47675i	−0.674392	−	0.738373i	$-0.735594\pi$
$827$	− 1.00999e6i	− 1.47675i	0.674392	−	0.738373i	$-0.264406\pi$
$828$	0	0
$829$	91383.0	0.132971	0.0664854	−	0.997787i	$-0.478821\pi$
$829$	91383.0	0.132971	0.0664854	+	0.997787i	$0.478821\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 297444.i	− 0.428662i
$834$	0	0
$835$	866334.	1.24255
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1.09255e6i	1.55209i	0.630677	+	0.776045i	$0.282777\pi$
$839$	1.09255e6i	1.55209i	−0.630677	+	0.776045i	$0.717223\pi$
$840$	0	0
$841$	703538.	0.994707
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 396166.i	− 0.554835i
$846$	0	0
$847$	−755933.	−1.05370
$848$	0	0
$849$	0	0
$850$	0	0
$851$	126964.i	0.175317i
$852$	0	0
$853$	−128093.	−0.176047	−0.0880233	−	0.996118i	$-0.528055\pi$
$853$	−128093.	−0.176047	−0.0880233	+	0.996118i	$0.528055\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1.29528e6i	− 1.76361i	−0.471610	−	0.881807i	$-0.656327\pi$
$857$	− 1.29528e6i	− 1.76361i	0.471610	−	0.881807i	$-0.343673\pi$
$858$	0	0
$859$	−530958.	−0.719571	−0.359785	−	0.933035i	$-0.617150\pi$
$859$	−530958.	−0.719571	−0.359785	+	0.933035i	$0.617150\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1.28810e6i	− 1.72953i	−0.502177	−	0.864765i	$-0.667467\pi$
$863$	− 1.28810e6i	− 1.72953i	0.502177	−	0.864765i	$-0.332533\pi$
$864$	0	0
$865$	32682.3	0.0436798
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 41505.6i	− 0.0549625i
$870$	0	0
$871$	−493831.	−0.650942
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 885897.i	− 1.15709i
$876$	0	0
$877$	−1.12511e6	−1.46284	−0.731418	−	0.681930i	$-0.761141\pi$
$877$	−1.12511e6	−1.46284	−0.731418	+	0.681930i	$0.761141\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 584300.i	− 0.752807i	−0.926456	−	0.376404i	$-0.877161\pi$
$881$	− 584300.i	− 0.752807i	0.926456	−	0.376404i	$-0.122839\pi$
$882$	0	0
$883$	−278108.	−0.356692	−0.178346	−	0.983968i	$-0.557075\pi$
$883$	−278108.	−0.356692	−0.178346	+	0.983968i	$0.557075\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 309151.i	− 0.392938i	−0.980510	−	0.196469i	$-0.937052\pi$
$887$	− 309151.i	− 0.392938i	0.980510	−	0.196469i	$-0.0629475\pi$
$888$	0	0
$889$	520219.	0.658238
$890$	0	0
$891$	0	0
$892$	0	0
$893$	446730.i	0.560198i
$894$	0	0
$895$	178487.	0.222823
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 41846.6i	− 0.0517775i
$900$	0	0
$901$	−1.18038e6	−1.45403
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 848822.i	− 1.03638i
$906$	0	0
$907$	727666.	0.884540	0.442270	−	0.896882i	$-0.354173\pi$
$907$	727666.	0.884540	0.442270	+	0.896882i	$0.354173\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 587369.i	− 0.707740i	−0.935295	−	0.353870i	$-0.884865\pi$
$911$	− 587369.i	− 0.707740i	0.935295	−	0.353870i	$-0.115135\pi$
$912$	0	0
$913$	−307233.	−0.368575
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 758646.i	− 0.902195i
$918$	0	0
$919$	324267.	0.383947	0.191974	−	0.981400i	$-0.438511\pi$
$919$	324267.	0.383947	0.191974	+	0.981400i	$0.438511\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	66040.8i	0.0775191i
$924$	0	0
$925$	−118149.	−0.138085
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 421994.i	− 0.488962i	−0.969654	−	0.244481i	$-0.921382\pi$
$929$	− 421994.i	− 0.488962i	0.969654	−	0.244481i	$-0.0786175\pi$
$930$	0	0
$931$	−89905.4	−0.103726
$932$	0	0
$933$	0	0
$934$	0	0
$935$	243606.i	0.278654i
$936$	0	0
$937$	1.56511e6	1.78265	0.891327	−	0.453362i	$-0.149775\pi$
$937$	1.56511e6	1.78265	0.891327	+	0.453362i	$0.149775\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.69092e6i	1.90961i	0.297234	+	0.954805i	$0.403936\pi$
$941$	1.69092e6i	1.90961i	−0.297234	+	0.954805i	$0.596064\pi$
$942$	0	0
$943$	−829707.	−0.933042
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 810385.i	− 0.903632i	−0.892111	−	0.451816i	$-0.850776\pi$
$947$	− 810385.i	− 0.903632i	0.892111	−	0.451816i	$-0.149224\pi$
$948$	0	0
$949$	395767.	0.439447
$950$	0	0
$951$	0	0
$952$	0	0
$953$	639808.i	0.704472i	0.935911	+	0.352236i	$0.114579\pi$
$953$	639808.i	0.704472i	−0.935911	+	0.352236i	$0.885421\pi$
$954$	0	0
$955$	−943243.	−1.03423
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.76931e6i	− 1.92383i
$960$	0	0
$961$	−455719.	−0.493458
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 894386.i	− 0.960441i
$966$	0	0
$967$	1.20128e6	1.28467	0.642337	−	0.766422i	$-0.277965\pi$
$967$	1.20128e6	1.28467	0.642337	+	0.766422i	$0.277965\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	280285.i	0.297278i	0.988892	+	0.148639i	$0.0474892\pi$
$971$	280285.i	0.297278i	−0.988892	+	0.148639i	$0.952511\pi$
$972$	0	0
$973$	594861.	0.628333
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 432819.i	− 0.453437i	−0.973960	−	0.226719i	$-0.927200\pi$
$977$	− 432819.i	− 0.453437i	0.973960	−	0.226719i	$-0.0727998\pi$
$978$	0	0
$979$	236577.	0.246836
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 261450.i	− 0.270571i	−0.990807	−	0.135285i	$-0.956805\pi$
$983$	− 261450.i	− 0.270571i	0.990807	−	0.135285i	$-0.0431951\pi$
$984$	0	0
$985$	989295.	1.01966
$986$	0	0
$987$	0	0
$988$	0	0
$989$	859515.i	0.878741i
$990$	0	0
$991$	−982664.	−1.00059	−0.500297	−	0.865854i	$-0.666776\pi$
$991$	−982664.	−1.00059	−0.500297	+	0.865854i	$0.666776\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	940008.i	0.949479i
$996$	0	0
$997$	−1.04183e6	−1.04811	−0.524053	−	0.851686i	$-0.675580\pi$
$997$	−1.04183e6	−1.04811	−0.524053	+	0.851686i	$0.675580\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	216.5.e.a.161.3	yes	4
3.2	odd	2	inner	216.5.e.a.161.2	✓	4
4.3	odd	2		432.5.e.j.161.3		4
9.2	odd	6		648.5.m.d.377.3		8
9.4	even	3		648.5.m.d.593.3		8
9.5	odd	6		648.5.m.d.593.2		8
9.7	even	3		648.5.m.d.377.2		8
12.11	even	2		432.5.e.j.161.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.5.e.a.161.2	✓	4	3.2	odd	2	inner
216.5.e.a.161.3	yes	4	1.1	even	1	trivial
432.5.e.j.161.2		4	12.11	even	2
432.5.e.j.161.3		4	4.3	odd	2
648.5.m.d.377.2		8	9.7	even	3
648.5.m.d.377.3		8	9.2	odd	6
648.5.m.d.593.2		8	9.5	odd	6
648.5.m.d.593.3		8	9.4	even	3

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:	\( N \)	\(=\)	\( 216 = 2^{3} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	216.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+16.7931i q^{5} -54.4980 q^{7} +O(q^{10})\)	\(q+16.7931i q^{5} -54.4980 q^{7} +27.7519i q^{11} +70.4980 q^{13} -522.715i q^{17} -157.996 q^{19} -368.583i q^{23} +342.992 q^{25} -61.1828i q^{29} +683.961 q^{31} -915.190i q^{35} -344.467 q^{37} -2251.07i q^{41} -2331.94 q^{43} -2827.47i q^{47} +569.035 q^{49} -2258.17i q^{53} -466.039 q^{55} -5556.56i q^{59} +5151.41 q^{61} +1183.88i q^{65} -7004.90 q^{67} +936.775i q^{71} +5613.87 q^{73} -1512.42i q^{77} -1495.60 q^{79} +11070.7i q^{83} +8778.01 q^{85} -8524.75i q^{89} -3842.00 q^{91} -2653.24i q^{95} -6355.67 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 36 q^{7}+O(q^{10}) \) 4 * q + 36 * q^7	\( 4 q + 36 q^{7} + 28 q^{13} - 124 q^{19} + 356 q^{25} - 2344 q^{31} + 2940 q^{37} - 2216 q^{43} + 6848 q^{49} - 6944 q^{55} + 8668 q^{61} - 14812 q^{67} + 5692 q^{73} - 18428 q^{79} + 36128 q^{85} - 15876 q^{91} + 17756 q^{97}+O(q^{100}) \) 4 * q + 36 * q^7 + 28 * q^13 - 124 * q^19 + 356 * q^25 - 2344 * q^31 + 2940 * q^37 - 2216 * q^43 + 6848 * q^49 - 6944 * q^55 + 8668 * q^61 - 14812 * q^67 + 5692 * q^73 - 18428 * q^79 + 36128 * q^85 - 15876 * q^91 + 17756 * q^97

Introduction

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