LMFDB - Embedded newform 108.9.c.b.53.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [108,9,Mod(53,108)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("108.53"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(108, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	108.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$43.9968898866$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-30}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 30 $ x^2 + 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{2}\cdot 7 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			53.1
Root			$-5.47723i$ of defining polynomial
Character	$\chi$	$=$	108.53
Dual form			108.9.c.b.53.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-690.130i q^{5} -7.00000 q^{7} -13112.5i q^{11} -769.000 q^{13} -69703.2i q^{17} -11863.0 q^{19} +46238.7i q^{23} -85655.0 q^{25} +1.02001e6i q^{29} -356266. q^{31} +4830.91i q^{35} -1.05113e6 q^{37} -5.13319e6i q^{41} -2.75679e6 q^{43} +4.66873e6i q^{47} -5.76475e6 q^{49} -6.14354e6i q^{53} -9.04932e6 q^{55} +1.05390e7i q^{59} -1.76932e7 q^{61} +530710. i q^{65} -2.40562e7 q^{67} +1.89841e7i q^{71} -2.55876e7 q^{73} +91787.3i q^{77} -4.16597e7 q^{79} -7.71386e7i q^{83} -4.81043e7 q^{85} -8.64112e6i q^{89} +5383.00 q^{91} +8.18702e6i q^{95} -5.69294e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 14 q^{7} - 1538 q^{13} - 23726 q^{19} - 171310 q^{25} - 712532 q^{31} - 2102258 q^{37} - 5513588 q^{43} - 11529504 q^{49} - 18098640 q^{55} - 35386466 q^{61} - 48112478 q^{67} - 51175298 q^{73} - 83319422 q^{79}+ \cdots - 113858834 q^{97}+O(q^{100}) $ 2 * q - 14 * q^7 - 1538 * q^13 - 23726 * q^19 - 171310 * q^25 - 712532 * q^31 - 2102258 * q^37 - 5513588 * q^43 - 11529504 * q^49 - 18098640 * q^55 - 35386466 * q^61 - 48112478 * q^67 - 51175298 * q^73 - 83319422 * q^79 - 96208560 * q^85 + 10766 * q^91 - 113858834 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$-1$	$1$

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
$5$	− 690.130i	− 1.10421i	−0.833775	−	0.552104i	$-0.813825\pi$
$5$	− 690.130i	− 1.10421i	0.833775	−	0.552104i	$-0.186175\pi$
$6$	0	0
$7$	−7.00000	−0.00291545	−0.00145773	−	0.999999i	$-0.500464\pi$
$7$	−7.00000	−0.00291545	−0.00145773	+	0.999999i	$0.500464\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 13112.5i	− 0.895600i	−0.894134	−	0.447800i	$-0.852208\pi$
$11$	− 13112.5i	− 0.895600i	0.894134	−	0.447800i	$-0.147792\pi$
$12$	0	0
$13$	−769.000	−0.0269248	−0.0134624	−	0.999909i	$-0.504285\pi$
$13$	−769.000	−0.0269248	−0.0134624	+	0.999909i	$0.504285\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 69703.2i	− 0.834559i	−0.908778	−	0.417279i	$-0.862984\pi$
$17$	− 69703.2i	− 0.834559i	0.908778	−	0.417279i	$-0.137016\pi$
$18$	0	0
$19$	−11863.0	−0.0910291	−0.0455145	−	0.998964i	$-0.514493\pi$
$19$	−11863.0	−0.0910291	−0.0455145	+	0.998964i	$0.514493\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	46238.7i	0.165232i	0.996581	+	0.0826161i	$0.0263275\pi$
$23$	46238.7i	0.165232i	−0.996581	+	0.0826161i	$0.973672\pi$
$24$	0	0
$25$	−85655.0	−0.219277
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.02001e6i	1.44216i	0.692852	+	0.721080i	$0.256354\pi$
$29$	1.02001e6i	1.44216i	−0.692852	+	0.721080i	$0.743646\pi$
$30$	0	0
$31$	−356266.	−0.385769	−0.192885	−	0.981221i	$-0.561784\pi$
$31$	−356266.	−0.385769	−0.192885	+	0.981221i	$0.561784\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4830.91i	0.00321927i
$36$	0	0
$37$	−1.05113e6	−0.560853	−0.280427	−	0.959875i	$-0.590476\pi$
$37$	−1.05113e6	−0.560853	−0.280427	+	0.959875i	$0.590476\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.13319e6i	− 1.81657i	−0.418353	−	0.908285i	$-0.637392\pi$
$41$	− 5.13319e6i	− 1.81657i	0.418353	−	0.908285i	$-0.362608\pi$
$42$	0	0
$43$	−2.75679e6	−0.806363	−0.403181	−	0.915120i	$-0.632096\pi$
$43$	−2.75679e6	−0.806363	−0.403181	+	0.915120i	$0.632096\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.66873e6i	0.956770i	0.878150	+	0.478385i	$0.158778\pi$
$47$	4.66873e6i	0.956770i	−0.878150	+	0.478385i	$0.841222\pi$
$48$	0	0
$49$	−5.76475e6	−0.999992
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.14354e6i	− 0.778602i	−0.921111	−	0.389301i	$-0.872717\pi$
$53$	− 6.14354e6i	− 0.778602i	0.921111	−	0.389301i	$-0.127283\pi$
$54$	0	0
$55$	−9.04932e6	−0.988929
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.05390e7i	0.869742i	0.900493	+	0.434871i	$0.143206\pi$
$59$	1.05390e7i	0.869742i	−0.900493	+	0.434871i	$0.856794\pi$
$60$	0	0
$61$	−1.76932e7	−1.27787	−0.638937	−	0.769259i	$-0.720625\pi$
$61$	−1.76932e7	−1.27787	−0.638937	+	0.769259i	$0.720625\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	530710.i	0.0297306i
$66$	0	0
$67$	−2.40562e7	−1.19379	−0.596896	−	0.802319i	$-0.703599\pi$
$67$	−2.40562e7	−1.19379	−0.596896	+	0.802319i	$0.703599\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.89841e7i	0.747062i	0.927618	+	0.373531i	$0.121853\pi$
$71$	1.89841e7i	0.747062i	−0.927618	+	0.373531i	$0.878147\pi$
$72$	0	0
$73$	−2.55876e7	−0.901029	−0.450515	−	0.892769i	$-0.648760\pi$
$73$	−2.55876e7	−0.901029	−0.450515	+	0.892769i	$0.648760\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	91787.3i	0.00261108i
$78$	0	0
$79$	−4.16597e7	−1.06957	−0.534783	−	0.844989i	$-0.679607\pi$
$79$	−4.16597e7	−1.06957	−0.534783	+	0.844989i	$0.679607\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 7.71386e7i	− 1.62540i	−0.582684	−	0.812699i	$-0.697997\pi$
$83$	− 7.71386e7i	− 1.62540i	0.582684	−	0.812699i	$-0.302003\pi$
$84$	0	0
$85$	−4.81043e7	−0.921527
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 8.64112e6i	− 0.137724i	−0.997626	−	0.0688621i	$-0.978063\pi$
$89$	− 8.64112e6i	− 0.137724i	0.997626	−	0.0688621i	$-0.0219368\pi$
$90$	0	0
$91$	5383.00	7.84980e−5 0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.18702e6i	0.100515i
$96$	0	0
$97$	−5.69294e7	−0.643057	−0.321529	−	0.946900i	$-0.604197\pi$
$97$	−5.69294e7	−0.643057	−0.321529	+	0.946900i	$0.604197\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.25327e7i	0.504829i	0.967619	+	0.252415i	$0.0812247\pi$
$101$	5.25327e7i	0.504829i	−0.967619	+	0.252415i	$0.918775\pi$
$102$	0	0
$103$	−4.52804e6	−0.0402310	−0.0201155	−	0.999798i	$-0.506403\pi$
$103$	−4.52804e6	−0.0402310	−0.0201155	+	0.999798i	$0.506403\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.60696e7i	0.504042i	0.967722	+	0.252021i	$0.0810952\pi$
$107$	6.60696e7i	0.504042i	−0.967722	+	0.252021i	$0.918905\pi$
$108$	0	0
$109$	1.10676e8	0.784054	0.392027	−	0.919954i	$-0.371774\pi$
$109$	1.10676e8	0.784054	0.392027	+	0.919954i	$0.371774\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.45708e8i	0.893654i	0.894620	+	0.446827i	$0.147446\pi$
$113$	1.45708e8i	0.893654i	−0.894620	+	0.446827i	$0.852554\pi$
$114$	0	0
$115$	3.19108e7	0.182451
$116$	0	0
$117$	0	0
$118$	0	0
$119$	487922.i	0.00243312i
$120$	0	0
$121$	4.24218e7	0.197901
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 2.10469e8i	− 0.862081i
$126$	0	0
$127$	4.79927e8	1.84485	0.922424	−	0.386179i	$-0.126205\pi$
$127$	4.79927e8	1.84485	0.922424	+	0.386179i	$0.126205\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 4.42458e8i	− 1.50240i	−0.660073	−	0.751202i	$-0.729474\pi$
$131$	− 4.42458e8i	− 1.50240i	0.660073	−	0.751202i	$-0.270526\pi$
$132$	0	0
$133$	83041.0	0.000265391 0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.41320e8i	0.968901i	0.874819	+	0.484450i	$0.160980\pi$
$137$	3.41320e8i	0.968901i	−0.874819	+	0.484450i	$0.839020\pi$
$138$	0	0
$139$	2.80647e8	0.751798	0.375899	−	0.926661i	$-0.377334\pi$
$139$	2.80647e8	0.751798	0.375899	+	0.926661i	$0.377334\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.00835e7i	0.0241139i
$144$	0	0
$145$	7.03942e8	1.59245
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 2.05115e8i	− 0.416152i	−0.978113	−	0.208076i	$-0.933280\pi$
$149$	− 2.05115e8i	− 0.416152i	0.978113	−	0.208076i	$-0.0667202\pi$
$150$	0	0
$151$	1.28831e8	0.247806	0.123903	−	0.992294i	$-0.460459\pi$
$151$	1.28831e8	0.247806	0.123903	+	0.992294i	$0.460459\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.45870e8i	0.425970i
$156$	0	0
$157$	−1.19282e8	−0.196326	−0.0981630	−	0.995170i	$-0.531297\pi$
$157$	−1.19282e8	−0.196326	−0.0981630	+	0.995170i	$0.531297\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 323671.i	0 0.000481726i
$162$	0	0
$163$	1.39482e9	1.97592	0.987959	−	0.154714i	$-0.0494457\pi$
$163$	1.39482e9	1.97592	0.987959	+	0.154714i	$0.0494457\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.33558e8i	0.814556i	0.913304	+	0.407278i	$0.133522\pi$
$167$	6.33558e8i	0.814556i	−0.913304	+	0.407278i	$0.866478\pi$
$168$	0	0
$169$	−8.15139e8	−0.999275
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 5.78389e8i	− 0.645707i	−0.946449	−	0.322853i	$-0.895358\pi$
$173$	− 5.78389e8i	− 0.645707i	0.946449	−	0.322853i	$-0.104642\pi$
$174$	0	0
$175$	599585.	0.000639291 0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.23738e9i	1.20529i	0.798010	+	0.602644i	$0.205886\pi$
$179$	1.23738e9i	1.20529i	−0.798010	+	0.602644i	$0.794114\pi$
$180$	0	0
$181$	8.20752e8	0.764712	0.382356	−	0.924015i	$-0.375113\pi$
$181$	8.20752e8	0.764712	0.382356	+	0.924015i	$0.375113\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.25416e8i	0.619299i
$186$	0	0
$187$	−9.13981e8	−0.747431
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.86929e9i	− 1.40457i	−0.711898	−	0.702283i	$-0.752164\pi$
$191$	− 1.86929e9i	− 1.40457i	0.711898	−	0.702283i	$-0.247836\pi$
$192$	0	0
$193$	−1.99991e9	−1.44139	−0.720696	−	0.693252i	$-0.756177\pi$
$193$	−1.99991e9	−1.44139	−0.720696	+	0.693252i	$0.756177\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.58090e9i	− 1.71358i	−0.515662	−	0.856792i	$-0.672454\pi$
$197$	− 2.58090e9i	− 1.71358i	0.515662	−	0.856792i	$-0.327546\pi$
$198$	0	0
$199$	1.91233e9	1.21941	0.609707	−	0.792627i	$-0.291287\pi$
$199$	1.91233e9	1.21941	0.609707	+	0.792627i	$0.291287\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 7.14009e6i	− 0.00420455i
$204$	0	0
$205$	−3.54257e9	−2.00587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.55553e8i	0.0815256i
$210$	0	0
$211$	−1.17629e9	−0.593453	−0.296726	−	0.954963i	$-0.595895\pi$
$211$	−1.17629e9	−0.593453	−0.296726	+	0.954963i	$0.595895\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.90255e9i	0.890393i
$216$	0	0
$217$	2.49386e6	0.00112469
$218$	0	0
$219$	0	0
$220$	0	0
$221$	5.36017e7i	0.0224703i
$222$	0	0
$223$	−2.35462e9	−0.952141	−0.476071	−	0.879407i	$-0.657939\pi$
$223$	−2.35462e9	−0.952141	−0.476071	+	0.879407i	$0.657939\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.62436e9i	0.611758i	0.952070	+	0.305879i	$0.0989503\pi$
$227$	1.62436e9i	0.611758i	−0.952070	+	0.305879i	$0.901050\pi$
$228$	0	0
$229$	−4.53972e9	−1.65077	−0.825387	−	0.564567i	$-0.809043\pi$
$229$	−4.53972e9	−1.65077	−0.825387	+	0.564567i	$0.809043\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.08114e9i	1.04541i	0.852513	+	0.522706i	$0.175077\pi$
$233$	3.08114e9i	1.04541i	−0.852513	+	0.522706i	$0.824923\pi$
$234$	0	0
$235$	3.22203e9	1.05647
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 3.01841e9i	− 0.925095i	−0.886595	−	0.462547i	$-0.846936\pi$
$239$	− 3.01841e9i	− 0.925095i	0.886595	−	0.462547i	$-0.153064\pi$
$240$	0	0
$241$	−5.77217e9	−1.71108	−0.855542	−	0.517734i	$-0.826776\pi$
$241$	−5.77217e9	−1.71108	−0.855542	+	0.517734i	$0.826776\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.97843e9i	1.10420i
$246$	0	0
$247$	9.12265e6	0.00245094
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 4.69528e9i	− 1.18295i	−0.806323	−	0.591476i	$-0.798546\pi$
$251$	− 4.69528e9i	− 1.18295i	0.806323	−	0.591476i	$-0.201454\pi$
$252$	0	0
$253$	6.06304e8	0.147982
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.43060e8i	0.0786389i	0.999227	+	0.0393194i	$0.0125190\pi$
$257$	3.43060e8i	0.0786389i	−0.999227	+	0.0393194i	$0.987481\pi$
$258$	0	0
$259$	7.35790e6	0.00163514
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 2.71809e9i	− 0.568121i	−0.958806	−	0.284060i	$-0.908318\pi$
$263$	− 2.71809e9i	− 0.568121i	0.958806	−	0.284060i	$-0.0916816\pi$
$264$	0	0
$265$	−4.23984e9	−0.859739
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4.53999e9i	− 0.867053i	−0.901141	−	0.433526i	$-0.857269\pi$
$269$	− 4.53999e9i	− 0.867053i	0.901141	−	0.433526i	$-0.142731\pi$
$270$	0	0
$271$	7.14392e9	1.32452	0.662261	−	0.749273i	$-0.269597\pi$
$271$	7.14392e9	1.32452	0.662261	+	0.749273i	$0.269597\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.12315e9i	0.196384i
$276$	0	0
$277$	−2.88521e9	−0.490070	−0.245035	−	0.969514i	$-0.578799\pi$
$277$	−2.88521e9	−0.490070	−0.245035	+	0.969514i	$0.578799\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.90927e9i	0.627005i	0.949587	+	0.313502i	$0.101502\pi$
$281$	3.90927e9i	0.627005i	−0.949587	+	0.313502i	$0.898498\pi$
$282$	0	0
$283$	6.17503e9	0.962705	0.481353	−	0.876527i	$-0.340146\pi$
$283$	6.17503e9	0.962705	0.481353	+	0.876527i	$0.340146\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.59323e7i	0.00529612i
$288$	0	0
$289$	2.11723e9	0.303512
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 5.58398e9i	− 0.757658i	−0.925467	−	0.378829i	$-0.876327\pi$
$293$	− 5.58398e9i	− 0.757658i	0.925467	−	0.378829i	$-0.123673\pi$
$294$	0	0
$295$	7.27327e9	0.960377
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3.55576e7i	− 0.00444885i
$300$	0	0
$301$	1.92976e7	0.00235091
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.22106e10i	1.41104i
$306$	0	0
$307$	9.50467e9	1.07000	0.535000	−	0.844852i	$-0.320312\pi$
$307$	9.50467e9	1.07000	0.535000	+	0.844852i	$0.320312\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.25626e10i	1.34288i	0.741057	+	0.671442i	$0.234325\pi$
$311$	1.25626e10i	1.34288i	−0.741057	+	0.671442i	$0.765675\pi$
$312$	0	0
$313$	2.28958e9	0.238549	0.119275	−	0.992861i	$-0.461943\pi$
$313$	2.28958e9	0.238549	0.119275	+	0.992861i	$0.461943\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 1.76089e10i	− 1.74379i	−0.489691	−	0.871896i	$-0.662891\pi$
$317$	− 1.76089e10i	− 1.74379i	0.489691	−	0.871896i	$-0.337109\pi$
$318$	0	0
$319$	1.33749e10	1.29160
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.26889e8i	0.0759691i
$324$	0	0
$325$	6.58687e7	0.00590399
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 3.26811e7i	− 0.00278942i
$330$	0	0
$331$	1.27344e10	1.06088	0.530441	−	0.847722i	$-0.322026\pi$
$331$	1.27344e10	1.06088	0.530441	+	0.847722i	$0.322026\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.66019e10i	1.31820i
$336$	0	0
$337$	−5.69592e8	−0.0441616	−0.0220808	−	0.999756i	$-0.507029\pi$
$337$	−5.69592e8	−0.0441616	−0.0220808	+	0.999756i	$0.507029\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.67153e9i	0.345495i
$342$	0	0
$343$	8.07069e7	0.00583088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.06446e10i	− 0.734196i	−0.930182	−	0.367098i	$-0.880351\pi$
$347$	− 1.06446e10i	− 0.734196i	0.930182	−	0.367098i	$-0.119649\pi$
$348$	0	0
$349$	6.40044e9	0.431428	0.215714	−	0.976457i	$-0.430792\pi$
$349$	6.40044e9	0.431428	0.215714	+	0.976457i	$0.430792\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2.81785e10i	− 1.81476i	−0.420309	−	0.907381i	$-0.638078\pi$
$353$	− 2.81785e10i	− 1.81476i	0.420309	−	0.907381i	$-0.361922\pi$
$354$	0	0
$355$	1.31015e10	0.824913
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 2.53861e10i	− 1.52833i	−0.645020	−	0.764166i	$-0.723151\pi$
$359$	− 2.53861e10i	− 1.52833i	0.645020	−	0.764166i	$-0.276849\pi$
$360$	0	0
$361$	−1.68428e10	−0.991714
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.76588e10i	0.994924i
$366$	0	0
$367$	1.03958e10	0.573053	0.286527	−	0.958072i	$-0.407499\pi$
$367$	1.03958e10	0.573053	0.286527	+	0.958072i	$0.407499\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.30048e7i	0.00226998i
$372$	0	0
$373$	−5.19259e9	−0.268256	−0.134128	−	0.990964i	$-0.542823\pi$
$373$	−5.19259e9	−0.268256	−0.134128	+	0.990964i	$0.542823\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 7.84390e8i	− 0.0388299i
$378$	0	0
$379$	−2.38155e10	−1.15426	−0.577129	−	0.816653i	$-0.695827\pi$
$379$	−2.38155e10	−1.15426	−0.577129	+	0.816653i	$0.695827\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.32272e9i	0.293838i	0.989148	+	0.146919i	$0.0469357\pi$
$383$	6.32272e9i	0.293838i	−0.989148	+	0.146919i	$0.953064\pi$
$384$	0	0
$385$	6.33452e7	0.00288318
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.08685e10i	1.78480i	0.451242	+	0.892402i	$0.350981\pi$
$389$	4.08685e10i	1.78480i	−0.451242	+	0.892402i	$0.649019\pi$
$390$	0	0
$391$	3.22299e9	0.137896
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.87506e10i	1.18102i
$396$	0	0
$397$	−3.14239e10	−1.26502	−0.632510	−	0.774552i	$-0.717975\pi$
$397$	−3.14239e10	−1.26502	−0.632510	+	0.774552i	$0.717975\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 3.17958e10i	− 1.22968i	−0.788652	−	0.614840i	$-0.789220\pi$
$401$	− 3.17958e10i	− 1.22968i	0.788652	−	0.614840i	$-0.210780\pi$
$402$	0	0
$403$	2.73969e8	0.0103868
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.37829e10i	0.502300i
$408$	0	0
$409$	−1.70740e10	−0.610158	−0.305079	−	0.952327i	$-0.598683\pi$
$409$	−1.70740e10	−0.610158	−0.305079	+	0.952327i	$0.598683\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 7.37729e7i	− 0.00253569i
$414$	0	0
$415$	−5.32357e10	−1.79478
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.95501e10i	− 0.634299i	−0.948376	−	0.317149i	$-0.897274\pi$
$419$	− 1.95501e10i	− 0.634299i	0.948376	−	0.317149i	$-0.102726\pi$
$420$	0	0
$421$	9.28362e9	0.295521	0.147761	−	0.989023i	$-0.452793\pi$
$421$	9.28362e9	0.295521	0.147761	+	0.989023i	$0.452793\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.97043e9i	0.182999i
$426$	0	0
$427$	1.23853e8	0.00372558
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 3.40413e10i	− 0.986500i	−0.869888	−	0.493250i	$-0.835809\pi$
$431$	− 3.40413e10i	− 0.986500i	0.869888	−	0.493250i	$-0.164191\pi$
$432$	0	0
$433$	−4.15503e10	−1.18201	−0.591006	−	0.806667i	$-0.701269\pi$
$433$	−4.15503e10	−1.18201	−0.591006	+	0.806667i	$0.701269\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.48530e8i	− 0.0150409i
$438$	0	0
$439$	−6.36157e9	−0.171280	−0.0856400	−	0.996326i	$-0.527293\pi$
$439$	−6.36157e9	−0.171280	−0.0856400	+	0.996326i	$0.527293\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 9.86245e9i	− 0.256076i	−0.991769	−	0.128038i	$-0.959132\pi$
$443$	− 9.86245e9i	− 0.256076i	0.991769	−	0.128038i	$-0.0408680\pi$
$444$	0	0
$445$	−5.96350e9	−0.152076
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 2.27951e9i	− 0.0560861i	−0.999607	−	0.0280431i	$-0.991072\pi$
$449$	− 2.27951e9i	− 0.0560861i	0.999607	−	0.0280431i	$-0.00892756\pi$
$450$	0	0
$451$	−6.73088e10	−1.62692
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 3.71497e6i	0 8.66782e-5i
$456$	0	0
$457$	2.31852e10	0.531553	0.265777	−	0.964035i	$-0.414372\pi$
$457$	2.31852e10	0.531553	0.265777	+	0.964035i	$0.414372\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.08781e9i	0.0905080i	0.998976	+	0.0452540i	$0.0144097\pi$
$461$	4.08781e9i	0.0905080i	−0.998976	+	0.0452540i	$0.985590\pi$
$462$	0	0
$463$	1.17258e10	0.255164	0.127582	−	0.991828i	$-0.459278\pi$
$463$	1.17258e10	0.255164	0.127582	+	0.991828i	$0.459278\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7.99859e9i	− 0.168169i	−0.996459	−	0.0840845i	$-0.973203\pi$
$467$	− 7.99859e9i	− 0.168169i	0.996459	−	0.0840845i	$-0.0267966\pi$
$468$	0	0
$469$	1.68394e8	0.00348044
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.61484e10i	0.722178i
$474$	0	0
$475$	1.01613e9	0.0199606
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 8.52117e10i	− 1.61867i	−0.587350	−	0.809333i	$-0.699829\pi$
$479$	− 8.52117e10i	− 1.61867i	0.587350	−	0.809333i	$-0.300171\pi$
$480$	0	0
$481$	8.08318e8	0.0151009
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.92887e10i	0.710070i
$486$	0	0
$487$	−4.17235e10	−0.741763	−0.370882	−	0.928680i	$-0.620944\pi$
$487$	−4.17235e10	−0.741763	−0.370882	+	0.928680i	$0.620944\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.99080e10i	0.858705i	0.903137	+	0.429352i	$0.141258\pi$
$491$	4.99080e10i	0.858705i	−0.903137	+	0.429352i	$0.858742\pi$
$492$	0	0
$493$	7.10981e10	1.20357
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.32889e8i	− 0.00217802i
$498$	0	0
$499$	3.20557e10	0.517014	0.258507	−	0.966009i	$-0.416769\pi$
$499$	3.20557e10	0.517014	0.258507	+	0.966009i	$0.416769\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8.35856e10i	1.30575i	0.757467	+	0.652874i	$0.226437\pi$
$503$	8.35856e10i	1.30575i	−0.757467	+	0.652874i	$0.773563\pi$
$504$	0	0
$505$	3.62544e10	0.557437
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.06110e10i	0.307063i	0.988144	+	0.153531i	$0.0490646\pi$
$509$	2.06110e10i	0.307063i	−0.988144	+	0.153531i	$0.950935\pi$
$510$	0	0
$511$	1.79114e8	0.00262691
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.12494e9i	0.0444235i
$516$	0	0
$517$	6.12186e10	0.856883
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.22695e11i	− 1.66524i	−0.553844	−	0.832621i	$-0.686840\pi$
$521$	− 1.22695e11i	− 1.66524i	0.553844	−	0.832621i	$-0.313160\pi$
$522$	0	0
$523$	−1.71228e10	−0.228859	−0.114429	−	0.993431i	$-0.536504\pi$
$523$	−1.71228e10	−0.228859	−0.114429	+	0.993431i	$0.536504\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.48329e10i	0.321947i
$528$	0	0
$529$	7.61730e10	0.972698
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.94742e9i	0.0489108i
$534$	0	0
$535$	4.55967e10	0.556568
$536$	0	0
$537$	0	0
$538$	0	0
$539$	7.55902e10i	0.895592i
$540$	0	0
$541$	1.42085e11	1.65866	0.829331	−	0.558757i	$-0.188722\pi$
$541$	1.42085e11	1.65866	0.829331	+	0.558757i	$0.188722\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 7.63806e10i	− 0.865759i
$546$	0	0
$547$	−1.91860e10	−0.214307	−0.107153	−	0.994243i	$-0.534174\pi$
$547$	−1.91860e10	−0.214307	−0.107153	+	0.994243i	$0.534174\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1.21004e10i	− 0.131279i
$552$	0	0
$553$	2.91618e8	0.00311827
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 5.88630e10i	− 0.611535i	−0.952106	−	0.305767i	$-0.901087\pi$
$557$	− 5.88630e10i	− 0.611535i	0.952106	−	0.305767i	$-0.0989130\pi$
$558$	0	0
$559$	2.11997e9	0.0217112
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 1.68404e11i	− 1.67617i	−0.545539	−	0.838085i	$-0.683675\pi$
$563$	− 1.68404e11i	− 1.67617i	0.545539	−	0.838085i	$-0.316325\pi$
$564$	0	0
$565$	1.00557e11	0.986780
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.37920e10i	0.226977i	0.993539	+	0.113488i	$0.0362025\pi$
$569$	2.37920e10i	0.226977i	−0.993539	+	0.113488i	$0.963798\pi$
$570$	0	0
$571$	2.57853e10	0.242565	0.121283	−	0.992618i	$-0.461299\pi$
$571$	2.57853e10	0.242565	0.121283	+	0.992618i	$0.461299\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 3.96058e9i	− 0.0362316i
$576$	0	0
$577$	1.43649e11	1.29598	0.647991	−	0.761648i	$-0.275609\pi$
$577$	1.43649e11	1.29598	0.647991	+	0.761648i	$0.275609\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	5.39970e8i	0.00473877i
$582$	0	0
$583$	−8.05570e10	−0.697315
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.74988e11i	1.47386i	0.675971	+	0.736929i	$0.263725\pi$
$587$	1.74988e11i	1.47386i	−0.675971	+	0.736929i	$0.736275\pi$
$588$	0	0
$589$	4.22638e9	0.0351162
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 7.19394e9i	− 0.0581766i	−0.999577	−	0.0290883i	$-0.990740\pi$
$593$	− 7.19394e9i	− 0.0581766i	0.999577	−	0.0290883i	$-0.00926039\pi$
$594$	0	0
$595$	3.36730e8	0.00268667
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.52262e11i	1.18272i	0.806406	+	0.591362i	$0.201410\pi$
$599$	1.52262e11i	1.18272i	−0.806406	+	0.591362i	$0.798590\pi$
$600$	0	0
$601$	−6.12297e10	−0.469315	−0.234657	−	0.972078i	$-0.575397\pi$
$601$	−6.12297e10	−0.469315	−0.234657	+	0.972078i	$0.575397\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2.92766e10i	− 0.218524i
$606$	0	0
$607$	−1.50544e10	−0.110894	−0.0554472	−	0.998462i	$-0.517658\pi$
$607$	−1.50544e10	−0.110894	−0.0554472	+	0.998462i	$0.517658\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 3.59026e9i	− 0.0257609i
$612$	0	0
$613$	2.11739e11	1.49954	0.749770	−	0.661698i	$-0.230164\pi$
$613$	2.11739e11	1.49954	0.749770	+	0.661698i	$0.230164\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.93848e11i	1.33758i	0.743449	+	0.668792i	$0.233188\pi$
$617$	1.93848e11i	1.33758i	−0.743449	+	0.668792i	$0.766812\pi$
$618$	0	0
$619$	−1.73290e11	−1.18035	−0.590176	−	0.807274i	$-0.700942\pi$
$619$	−1.73290e11	−1.18035	−0.590176	+	0.807274i	$0.700942\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.04879e7i	0 0.000401528i
$624$	0	0
$625$	−1.78710e11	−1.17119
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.32670e10i	0.468065i
$630$	0	0
$631$	−6.63235e10	−0.418360	−0.209180	−	0.977877i	$-0.567079\pi$
$631$	−6.63235e10	−0.418360	−0.209180	+	0.977877i	$0.567079\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 3.31212e11i	− 2.03710i
$636$	0	0
$637$	4.43309e9	0.0269246
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.45297e11i	0.860644i	0.902676	+	0.430322i	$0.141600\pi$
$641$	1.45297e11i	0.860644i	−0.902676	+	0.430322i	$0.858400\pi$
$642$	0	0
$643$	−6.21621e10	−0.363648	−0.181824	−	0.983331i	$-0.558200\pi$
$643$	−6.21621e10	−0.363648	−0.181824	+	0.983331i	$0.558200\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.22471e11i	− 1.26957i	−0.772688	−	0.634786i	$-0.781088\pi$
$647$	− 2.22471e11i	− 1.26957i	0.772688	−	0.634786i	$-0.218912\pi$
$648$	0	0
$649$	1.38192e11	0.778941
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 7.49669e10i	− 0.412303i	−0.978520	−	0.206152i	$-0.933906\pi$
$653$	− 7.49669e10i	− 0.412303i	0.978520	−	0.206152i	$-0.0660940\pi$
$654$	0	0
$655$	−3.05354e11	−1.65897
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1.65866e11i	− 0.879458i	−0.898130	−	0.439729i	$-0.855074\pi$
$659$	− 1.65866e11i	− 0.879458i	0.898130	−	0.439729i	$-0.144926\pi$
$660$	0	0
$661$	−2.26604e11	−1.18703	−0.593516	−	0.804822i	$-0.702261\pi$
$661$	−2.26604e11	−1.18703	−0.593516	+	0.804822i	$0.702261\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 5.73091e7i	0 0.000293047i
$666$	0	0
$667$	−4.71641e10	−0.238291
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.32002e11i	1.14446i
$672$	0	0
$673$	−2.06445e11	−1.00634	−0.503170	−	0.864188i	$-0.667833\pi$
$673$	−2.06445e11	−1.00634	−0.503170	+	0.864188i	$0.667833\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 2.08742e11i	− 0.993699i	−0.867837	−	0.496850i	$-0.834490\pi$
$677$	− 2.08742e11i	− 0.993699i	0.867837	−	0.496850i	$-0.165510\pi$
$678$	0	0
$679$	3.98506e8	0.00187480
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.51957e11i	1.15783i	0.815390	+	0.578913i	$0.196523\pi$
$683$	2.51957e11i	1.15783i	−0.815390	+	0.578913i	$0.803477\pi$
$684$	0	0
$685$	2.35555e11	1.06987
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.72438e9i	0.0209637i
$690$	0	0
$691$	−2.80957e11	−1.23233	−0.616165	−	0.787617i	$-0.711315\pi$
$691$	−2.80957e11	−1.23233	−0.616165	+	0.787617i	$0.711315\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.93683e11i	− 0.830142i
$696$	0	0
$697$	−3.57800e11	−1.51603
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.29537e11i	0.950564i	0.879833	+	0.475282i	$0.157654\pi$
$701$	2.29537e11i	0.950564i	−0.879833	+	0.475282i	$0.842346\pi$
$702$	0	0
$703$	1.24695e10	0.0510539
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.67729e8i	− 0.00147181i
$708$	0	0
$709$	−3.69232e10	−0.146121	−0.0730607	−	0.997327i	$-0.523277\pi$
$709$	−3.69232e10	−0.146121	−0.0730607	+	0.997327i	$0.523277\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1.64733e10i	− 0.0637415i
$714$	0	0
$715$	6.95893e9	0.0266267
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 2.85751e11i	− 1.06923i	−0.845095	−	0.534617i	$-0.820456\pi$
$719$	− 2.85751e11i	− 1.06923i	0.845095	−	0.534617i	$-0.179544\pi$
$720$	0	0
$721$	3.16963e7	0.000117292 0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 8.73692e10i	− 0.316232i
$726$	0	0
$727$	−9.18263e10	−0.328723	−0.164361	−	0.986400i	$-0.552556\pi$
$727$	−9.18263e10	−0.328723	−0.164361	+	0.986400i	$0.552556\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.92157e11i	0.672957i
$732$	0	0
$733$	−9.28369e10	−0.321592	−0.160796	−	0.986988i	$-0.551406\pi$
$733$	−9.28369e10	−0.321592	−0.160796	+	0.986988i	$0.551406\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.15437e11i	1.06916i
$738$	0	0
$739$	4.57349e10	0.153345	0.0766725	−	0.997056i	$-0.475570\pi$
$739$	4.57349e10	0.153345	0.0766725	+	0.997056i	$0.475570\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5.69333e11i	1.86815i	0.357080	+	0.934074i	$0.383772\pi$
$743$	5.69333e11i	1.86815i	−0.357080	+	0.934074i	$0.616228\pi$
$744$	0	0
$745$	−1.41556e11	−0.459519
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 4.62487e8i	− 0.00146951i
$750$	0	0
$751$	2.18221e11	0.686021	0.343010	−	0.939332i	$-0.388553\pi$
$751$	2.18221e11	0.686021	0.343010	+	0.939332i	$0.388553\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 8.89101e10i	− 0.273630i
$756$	0	0
$757$	5.02194e11	1.52928	0.764642	−	0.644455i	$-0.222916\pi$
$757$	5.02194e11	1.52928	0.764642	+	0.644455i	$0.222916\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 1.51690e11i	− 0.452293i	−0.974093	−	0.226146i	$-0.927387\pi$
$761$	− 1.51690e11i	− 0.452293i	0.974093	−	0.226146i	$-0.0726127\pi$
$762$	0	0
$763$	−7.74729e8	−0.00228587
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 8.10448e9i	− 0.0234177i
$768$	0	0
$769$	−3.99244e11	−1.14165	−0.570825	−	0.821072i	$-0.693377\pi$
$769$	−3.99244e11	−1.14165	−0.570825	+	0.821072i	$0.693377\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 6.32593e11i	− 1.77177i	−0.463909	−	0.885883i	$-0.653554\pi$
$773$	− 6.32593e11i	− 1.77177i	0.463909	−	0.885883i	$-0.346446\pi$
$774$	0	0
$775$	3.05160e10	0.0845902
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.08950e10i	0.165361i
$780$	0	0
$781$	2.48929e11	0.669069
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.23204e10i	0.216785i
$786$	0	0
$787$	6.79091e11	1.77023	0.885114	−	0.465374i	$-0.154080\pi$
$787$	6.79091e11	1.77023	0.885114	+	0.465374i	$0.154080\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1.01996e9i	− 0.00260541i
$792$	0	0
$793$	1.36061e10	0.0344065
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 4.84277e11i	− 1.20022i	−0.799918	−	0.600110i	$-0.795124\pi$
$797$	− 4.84277e11i	− 1.20022i	0.799918	−	0.600110i	$-0.204876\pi$
$798$	0	0
$799$	3.25425e11	0.798481
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.35517e11i	0.806962i
$804$	0	0
$805$	−2.23375e8	−0.000531927 0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.06645e11i	0.248970i	0.992222	+	0.124485i	$0.0397278\pi$
$809$	1.06645e11i	0.248970i	−0.992222	+	0.124485i	$0.960272\pi$
$810$	0	0
$811$	4.87943e10	0.112794	0.0563970	−	0.998408i	$-0.482039\pi$
$811$	4.87943e10	0.112794	0.0563970	+	0.998408i	$0.482039\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 9.62611e11i	− 2.18183i
$816$	0	0
$817$	3.27038e10	0.0734025
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.32476e11i	− 0.511688i	−0.966718	−	0.255844i	$-0.917647\pi$
$821$	− 2.32476e11i	− 0.511688i	0.966718	−	0.255844i	$-0.0823534\pi$
$822$	0	0
$823$	6.54441e11	1.42650	0.713249	−	0.700911i	$-0.247223\pi$
$823$	6.54441e11	1.42650	0.713249	+	0.700911i	$0.247223\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.24792e11i	0.480573i	0.970702	+	0.240286i	$0.0772414\pi$
$827$	2.24792e11i	0.480573i	−0.970702	+	0.240286i	$0.922759\pi$
$828$	0	0
$829$	2.89004e11	0.611908	0.305954	−	0.952046i	$-0.401025\pi$
$829$	2.89004e11	0.611908	0.305954	+	0.952046i	$0.401025\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.01822e11i	0.834552i
$834$	0	0
$835$	4.37238e11	0.899439
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.66131e11i	1.34435i	0.740393	+	0.672175i	$0.234640\pi$
$839$	6.66131e11i	1.34435i	−0.740393	+	0.672175i	$0.765360\pi$
$840$	0	0
$841$	−5.40180e11	−1.07983
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.62552e11i	1.10341i
$846$	0	0
$847$	−2.96953e8	−0.000576970 0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 4.86029e10i	− 0.0926710i
$852$	0	0
$853$	6.81076e11	1.28647	0.643235	−	0.765669i	$-0.277592\pi$
$853$	6.81076e11	1.28647	0.643235	+	0.765669i	$0.277592\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.46083e10i	0.0270817i	0.999908	+	0.0135409i	$0.00431032\pi$
$857$	1.46083e10i	0.0270817i	−0.999908	+	0.0135409i	$0.995690\pi$
$858$	0	0
$859$	8.44458e11	1.55098	0.775488	−	0.631362i	$-0.217504\pi$
$859$	8.44458e11	1.55098	0.775488	+	0.631362i	$0.217504\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 4.33171e11i	− 0.780937i	−0.920616	−	0.390469i	$-0.872313\pi$
$863$	− 4.33171e11i	− 0.780937i	0.920616	−	0.390469i	$-0.127687\pi$
$864$	0	0
$865$	−3.99164e11	−0.712995
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.46262e11i	0.957904i
$870$	0	0
$871$	1.84992e10	0.0321426
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.47328e9i	0.00251336i
$876$	0	0
$877$	−2.79400e10	−0.0472311	−0.0236155	−	0.999721i	$-0.507518\pi$
$877$	−2.79400e10	−0.0472311	−0.0236155	+	0.999721i	$0.507518\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1.09316e11i	− 0.181460i	−0.995876	−	0.0907298i	$-0.971080\pi$
$881$	− 1.09316e11i	− 0.181460i	0.995876	−	0.0907298i	$-0.0289200\pi$
$882$	0	0
$883$	−5.39725e11	−0.887829	−0.443915	−	0.896069i	$-0.646411\pi$
$883$	−5.39725e11	−0.887829	−0.443915	+	0.896069i	$0.646411\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.82837e11i	1.26467i	0.774695	+	0.632335i	$0.217903\pi$
$887$	7.82837e11i	1.26467i	−0.774695	+	0.632335i	$0.782097\pi$
$888$	0	0
$889$	−3.35949e9	−0.00537856
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5.53852e10i	− 0.0870939i
$894$	0	0
$895$	8.53953e11	1.33089
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 3.63396e11i	− 0.556341i
$900$	0	0
$901$	−4.28224e11	−0.649789
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 5.66426e11i	− 0.844402i
$906$	0	0
$907$	6.27214e11	0.926801	0.463400	−	0.886149i	$-0.346629\pi$
$907$	6.27214e11	0.926801	0.463400	+	0.886149i	$0.346629\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 9.11957e11i	− 1.32404i	−0.749487	−	0.662019i	$-0.769700\pi$
$911$	− 9.11957e11i	− 1.32404i	0.749487	−	0.662019i	$-0.230300\pi$
$912$	0	0
$913$	−1.01148e12	−1.45571
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.09720e9i	0.00438019i
$918$	0	0
$919$	−4.03048e11	−0.565060	−0.282530	−	0.959258i	$-0.591174\pi$
$919$	−4.03048e11	−0.565060	−0.282530	+	0.959258i	$0.591174\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.45988e10i	− 0.0201145i
$924$	0	0
$925$	9.00345e10	0.122982
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 8.24710e11i	− 1.10723i	−0.832772	−	0.553616i	$-0.813248\pi$
$929$	− 8.24710e11i	− 1.10723i	0.832772	−	0.553616i	$-0.186752\pi$
$930$	0	0
$931$	6.83873e10	0.0910283
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.30766e11i	0.825319i
$936$	0	0
$937$	−1.17037e12	−1.51832	−0.759161	−	0.650903i	$-0.774391\pi$
$937$	−1.17037e12	−1.51832	−0.759161	+	0.650903i	$0.774391\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.67926e11i	0.724324i	0.932115	+	0.362162i	$0.117961\pi$
$941$	5.67926e11i	0.724324i	−0.932115	+	0.362162i	$0.882039\pi$
$942$	0	0
$943$	2.37352e11	0.300156
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.05909e12i	− 1.31684i	−0.752653	−	0.658418i	$-0.771226\pi$
$947$	− 1.05909e12i	− 1.31684i	0.752653	−	0.658418i	$-0.228774\pi$
$948$	0	0
$949$	1.96769e10	0.0242601
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 2.69349e11i	− 0.326546i	−0.986581	−	0.163273i	$-0.947795\pi$
$953$	− 2.69349e11i	− 0.326546i	0.986581	−	0.163273i	$-0.0522051\pi$
$954$	0	0
$955$	−1.29005e12	−1.55093
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2.38924e9i	− 0.00282478i
$960$	0	0
$961$	−7.25966e11	−0.851182
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.38020e12i	1.59160i
$966$	0	0
$967$	6.87113e11	0.785819	0.392909	−	0.919577i	$-0.371469\pi$
$967$	6.87113e11	0.785819	0.392909	+	0.919577i	$0.371469\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.61624e11i	0.631785i	0.948795	+	0.315892i	$0.102304\pi$
$971$	5.61624e11i	0.631785i	−0.948795	+	0.315892i	$0.897696\pi$
$972$	0	0
$973$	−1.96453e9	−0.00219183
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.86949e11i	0.424693i	0.977194	+	0.212347i	$0.0681106\pi$
$977$	3.86949e11i	0.424693i	−0.977194	+	0.212347i	$0.931889\pi$
$978$	0	0
$979$	−1.13307e11	−0.123346
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.58823e10i	− 0.0170098i	−0.999964	−	0.00850491i	$-0.997293\pi$
$983$	− 1.58823e10i	− 0.0170098i	0.999964	−	0.00850491i	$-0.00270723\pi$
$984$	0	0
$985$	−1.78115e12	−1.89215
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.27471e11i	− 0.133237i
$990$	0	0
$991$	−1.78722e12	−1.85303	−0.926516	−	0.376256i	$-0.877211\pi$
$991$	−1.78722e12	−1.85303	−0.926516	+	0.376256i	$0.877211\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.31976e12i	− 1.34649i
$996$	0	0
$997$	1.01042e12	1.02263	0.511316	−	0.859393i	$-0.329158\pi$
$997$	1.01042e12	1.02263	0.511316	+	0.859393i	$0.329158\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.9.c.b.53.1	✓	2
3.2	odd	2	inner	108.9.c.b.53.2	yes	2
4.3	odd	2		432.9.e.e.161.1		2
9.2	odd	6		324.9.g.e.53.1		4
9.4	even	3		324.9.g.e.269.1		4
9.5	odd	6		324.9.g.e.269.2		4
9.7	even	3		324.9.g.e.53.2		4
12.11	even	2		432.9.e.e.161.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.9.c.b.53.1	✓	2	1.1	even	1	trivial
108.9.c.b.53.2	yes	2	3.2	odd	2	inner
324.9.g.e.53.1		4	9.2	odd	6
324.9.g.e.53.2		4	9.7	even	3
324.9.g.e.269.1		4	9.4	even	3
324.9.g.e.269.2		4	9.5	odd	6
432.9.e.e.161.1		2	4.3	odd	2
432.9.e.e.161.2		2	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	108.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-690.130i q^{5} -7.00000 q^{7} -13112.5i q^{11} -769.000 q^{13} -69703.2i q^{17} -11863.0 q^{19} +46238.7i q^{23} -85655.0 q^{25} +1.02001e6i q^{29} -356266. q^{31} +4830.91i q^{35} -1.05113e6 q^{37} -5.13319e6i q^{41} -2.75679e6 q^{43} +4.66873e6i q^{47} -5.76475e6 q^{49} -6.14354e6i q^{53} -9.04932e6 q^{55} +1.05390e7i q^{59} -1.76932e7 q^{61} +530710. i q^{65} -2.40562e7 q^{67} +1.89841e7i q^{71} -2.55876e7 q^{73} +91787.3i q^{77} -4.16597e7 q^{79} -7.71386e7i q^{83} -4.81043e7 q^{85} -8.64112e6i q^{89} +5383.00 q^{91} +8.18702e6i q^{95} -5.69294e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 14 q^{7} - 1538 q^{13} - 23726 q^{19} - 171310 q^{25} - 712532 q^{31} - 2102258 q^{37} - 5513588 q^{43} - 11529504 q^{49} - 18098640 q^{55} - 35386466 q^{61} - 48112478 q^{67} - 51175298 q^{73} - 83319422 q^{79}+ \cdots - 113858834 q^{97}+O(q^{100}) \) 2 * q - 14 * q^7 - 1538 * q^13 - 23726 * q^19 - 171310 * q^25 - 712532 * q^31 - 2102258 * q^37 - 5513588 * q^43 - 11529504 * q^49 - 18098640 * q^55 - 35386466 * q^61 - 48112478 * q^67 - 51175298 * q^73 - 83319422 * q^79 - 96208560 * q^85 + 10766 * q^91 - 113858834 * q^97

Introduction

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