LMFDB - Embedded newform 384.8.d.f.193.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,8,Mod(193,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("384.193");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	384.d (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:	$119.955849786$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} - 1020 x^{14} + 7280 x^{13} + 388150 x^{12} - 2423904 x^{11} - 70542796 x^{10} + \cdots + 694045717832241 $ x^16 - 8x^15 - 1020x^14 + 7280x^13 + 388150x^12 - 2423904x^11 - 70542796x^10 + 375107560x^9 + 6894477923x^8 - 29855519152x^7 - 372292484316x^6 + 1222974198808x^5 + 10528455140834x^4 - 23130795757272x^3 - 133992602927280x^2 + 145766920335192*x + 694045717832241
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{108}\cdot 3^{28} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			193.13
Root			$3.90187 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	384.193
Dual form			384.8.d.f.193.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.0000i q^{3} +124.096i q^{5} +1165.97 q^{7} -729.000 q^{9} +O(q^{10})$	$q+27.0000i q^{3} +124.096i q^{5} +1165.97 q^{7} -729.000 q^{9} +2265.49i q^{11} -3028.99i q^{13} -3350.59 q^{15} -18113.4 q^{17} -33226.6i q^{19} +31481.2i q^{21} +85828.5 q^{23} +62725.2 q^{25} -19683.0i q^{27} -158052. i q^{29} +199774. q^{31} -61168.3 q^{33} +144692. i q^{35} -154407. i q^{37} +81782.8 q^{39} -218071. q^{41} -248492. i q^{43} -90465.9i q^{45} -703977. q^{47} +535941. q^{49} -489061. i q^{51} +1.18948e6i q^{53} -281138. q^{55} +897118. q^{57} -1.97710e6i q^{59} -3.14364e6i q^{61} -849992. q^{63} +375885. q^{65} -1.03112e6i q^{67} +2.31737e6i q^{69} -1.24902e6 q^{71} -2.36170e6 q^{73} +1.69358e6i q^{75} +2.64150e6i q^{77} -1.65799e6 q^{79} +531441. q^{81} -2.63121e6i q^{83} -2.24779e6i q^{85} +4.26740e6 q^{87} +2.71938e6 q^{89} -3.53171e6i q^{91} +5.39390e6i q^{93} +4.12328e6 q^{95} +1.27672e7 q^{97} -1.65154e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 11664 q^{9}+O(q^{10}) $ 16 * q - 11664 * q^9	$ 16 q - 11664 q^{9} + 77280 q^{17} - 236464 q^{25} + 544320 q^{33} - 930912 q^{41} + 1163024 q^{49} - 3032640 q^{57} + 6283008 q^{65} - 2727200 q^{73} + 8503056 q^{81} - 43093152 q^{89} + 35537120 q^{97}+O(q^{100}) $ 16 * q - 11664 * q^9 + 77280 * q^17 - 236464 * q^25 + 544320 * q^33 - 930912 * q^41 + 1163024 * q^49 - 3032640 * q^57 + 6283008 * q^65 - 2727200 * q^73 + 8503056 * q^81 - 43093152 * q^89 + 35537120 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	27.0000i	0.577350i
$3$	27.0000i	0.577350i
$4$	0	0
$5$	124.096i	0.443979i	0.975049	+	0.221989i	$0.0712550\pi$
$5$	124.096i	0.443979i	−0.975049	+	0.221989i	$0.928745\pi$
$6$	0	0
$7$	1165.97	1.28482	0.642412	−	0.766359i	$-0.277934\pi$
$7$	1165.97	1.28482	0.642412	+	0.766359i	$0.277934\pi$
$8$	0	0
$9$	−729.000	−0.333333
$10$	0	0
$11$	2265.49i	0.513202i	0.966517	+	0.256601i	$0.0826026\pi$
$11$	2265.49i	0.513202i	−0.966517	+	0.256601i	$0.917397\pi$
$12$	0	0
$13$	− 3028.99i	− 0.382381i	−0.981553	−	0.191191i	$-0.938765\pi$
$13$	− 3028.99i	− 0.382381i	0.981553	−	0.191191i	$-0.0612349\pi$
$14$	0	0
$15$	−3350.59	−0.256331
$16$	0	0
$17$	−18113.4	−0.894186	−0.447093	−	0.894487i	$-0.647541\pi$
$17$	−18113.4	−0.894186	−0.447093	+	0.894487i	$0.647541\pi$
$18$	0	0
$19$	− 33226.6i	− 1.11134i	−0.831402	−	0.555672i	$-0.812461\pi$
$19$	− 33226.6i	− 1.11134i	0.831402	−	0.555672i	$-0.187539\pi$
$20$	0	0
$21$	31481.2i	0.741794i
$22$	0	0
$23$	85828.5	1.47090	0.735451	−	0.677578i	$-0.236970\pi$
$23$	85828.5	1.47090	0.735451	+	0.677578i	$0.236970\pi$
$24$	0	0
$25$	62725.2	0.802883
$26$	0	0
$27$	− 19683.0i	− 0.192450i
$28$	0	0
$29$	− 158052.i	− 1.20339i	−0.798726	−	0.601695i	$-0.794492\pi$
$29$	− 158052.i	− 1.20339i	0.798726	−	0.601695i	$-0.205508\pi$
$30$	0	0
$31$	199774.	1.20441	0.602204	−	0.798342i	$-0.294289\pi$
$31$	199774.	1.20441	0.602204	+	0.798342i	$0.294289\pi$
$32$	0	0
$33$	−61168.3	−0.296297
$34$	0	0
$35$	144692.i	0.570435i
$36$	0	0
$37$	− 154407.i	− 0.501142i	−0.968098	−	0.250571i	$-0.919382\pi$
$37$	− 154407.i	− 0.501142i	0.968098	−	0.250571i	$-0.0806183\pi$
$38$	0	0
$39$	81782.8	0.220768
$40$	0	0
$41$	−218071.	−0.494144	−0.247072	−	0.968997i	$-0.579468\pi$
$41$	−218071.	−0.494144	−0.247072	+	0.968997i	$0.579468\pi$
$42$	0	0
$43$	− 248492.i	− 0.476621i	−0.971189	−	0.238310i	$-0.923406\pi$
$43$	− 248492.i	− 0.476621i	0.971189	−	0.238310i	$-0.0765936\pi$
$44$	0	0
$45$	− 90465.9i	− 0.147993i
$46$	0	0
$47$	−703977.	−0.989045	−0.494522	−	0.869165i	$-0.664657\pi$
$47$	−703977.	−0.989045	−0.494522	+	0.869165i	$0.664657\pi$
$48$	0	0
$49$	535941.	0.650775
$50$	0	0
$51$	− 489061.i	− 0.516259i
$52$	0	0
$53$	1.18948e6i	1.09747i	0.835996	+	0.548736i	$0.184891\pi$
$53$	1.18948e6i	1.09747i	−0.835996	+	0.548736i	$0.815109\pi$
$54$	0	0
$55$	−281138.	−0.227851
$56$	0	0
$57$	897118.	0.641634
$58$	0	0
$59$	− 1.97710e6i	− 1.25328i	−0.779311	−	0.626638i	$-0.784431\pi$
$59$	− 1.97710e6i	− 1.25328i	0.779311	−	0.626638i	$-0.215569\pi$
$60$	0	0
$61$	− 3.14364e6i	− 1.77328i	−0.462456	−	0.886642i	$-0.653032\pi$
$61$	− 3.14364e6i	− 1.77328i	0.462456	−	0.886642i	$-0.346968\pi$
$62$	0	0
$63$	−849992.	−0.428275
$64$	0	0
$65$	375885.	0.169769
$66$	0	0
$67$	− 1.03112e6i	− 0.418838i	−0.977826	−	0.209419i	$-0.932843\pi$
$67$	− 1.03112e6i	− 0.418838i	0.977826	−	0.209419i	$-0.0671573\pi$
$68$	0	0
$69$	2.31737e6i	0.849226i
$70$	0	0
$71$	−1.24902e6	−0.414156	−0.207078	−	0.978324i	$-0.566395\pi$
$71$	−1.24902e6	−0.414156	−0.207078	+	0.978324i	$0.566395\pi$
$72$	0	0
$73$	−2.36170e6	−0.710551	−0.355276	−	0.934762i	$-0.615613\pi$
$73$	−2.36170e6	−0.710551	−0.355276	+	0.934762i	$0.615613\pi$
$74$	0	0
$75$	1.69358e6i	0.463545i
$76$	0	0
$77$	2.64150e6i	0.659375i
$78$	0	0
$79$	−1.65799e6	−0.378344	−0.189172	−	0.981944i	$-0.560580\pi$
$79$	−1.65799e6	−0.378344	−0.189172	+	0.981944i	$0.560580\pi$
$80$	0	0
$81$	531441.	0.111111
$82$	0	0
$83$	− 2.63121e6i	− 0.505106i	−0.967583	−	0.252553i	$-0.918730\pi$
$83$	− 2.63121e6i	− 0.505106i	0.967583	−	0.252553i	$-0.0812702\pi$
$84$	0	0
$85$	− 2.24779e6i	− 0.397000i
$86$	0	0
$87$	4.26740e6	0.694777
$88$	0	0
$89$	2.71938e6	0.408889	0.204444	−	0.978878i	$-0.434461\pi$
$89$	2.71938e6	0.408889	0.204444	+	0.978878i	$0.434461\pi$
$90$	0	0
$91$	− 3.53171e6i	− 0.491293i
$92$	0	0
$93$	5.39390e6i	0.695365i
$94$	0	0
$95$	4.12328e6	0.493413
$96$	0	0
$97$	1.27672e7	1.42034	0.710171	−	0.704029i	$-0.248618\pi$
$97$	1.27672e7	1.42034	0.710171	+	0.704029i	$0.248618\pi$
$98$	0	0
$99$	− 1.65154e6i	− 0.171067i
$100$	0	0
$101$	− 827746.i	− 0.0799415i	−0.999201	−	0.0399707i	$-0.987274\pi$
$101$	− 827746.i	− 0.0799415i	0.999201	−	0.0399707i	$-0.0127265\pi$
$102$	0	0
$103$	1.58993e7	1.43367	0.716834	−	0.697244i	$-0.245591\pi$
$103$	1.58993e7	1.43367	0.716834	+	0.697244i	$0.245591\pi$
$104$	0	0
$105$	−3.90668e6	−0.329341
$106$	0	0
$107$	7.04821e6i	0.556206i	0.960551	+	0.278103i	$0.0897056\pi$
$107$	7.04821e6i	0.556206i	−0.960551	+	0.278103i	$0.910294\pi$
$108$	0	0
$109$	− 4.25266e6i	− 0.314534i	−0.987556	−	0.157267i	$-0.949732\pi$
$109$	− 4.25266e6i	− 0.314534i	0.987556	−	0.157267i	$-0.0502684\pi$
$110$	0	0
$111$	4.16898e6	0.289334
$112$	0	0
$113$	1.77712e7	1.15862	0.579310	−	0.815107i	$-0.303322\pi$
$113$	1.77712e7	1.15862	0.579310	+	0.815107i	$0.303322\pi$
$114$	0	0
$115$	1.06510e7i	0.653050i
$116$	0	0
$117$	2.20814e6i	0.127460i
$118$	0	0
$119$	−2.11196e7	−1.14887
$120$	0	0
$121$	1.43547e7	0.736624
$122$	0	0
$123$	− 5.88791e6i	− 0.285294i
$124$	0	0
$125$	1.74789e7i	0.800442i
$126$	0	0
$127$	−1.83070e7	−0.793057	−0.396528	−	0.918022i	$-0.629785\pi$
$127$	−1.83070e7	−0.793057	−0.396528	+	0.918022i	$0.629785\pi$
$128$	0	0
$129$	6.70929e6	0.275177
$130$	0	0
$131$	1.11878e7i	0.434807i	0.976082	+	0.217404i	$0.0697588\pi$
$131$	1.11878e7i	0.434807i	−0.976082	+	0.217404i	$0.930241\pi$
$132$	0	0
$133$	− 3.87412e7i	− 1.42788i
$134$	0	0
$135$	2.44258e6	0.0854438
$136$	0	0
$137$	1.99340e7	0.662327	0.331163	−	0.943573i	$-0.392559\pi$
$137$	1.99340e7	0.662327	0.331163	+	0.943573i	$0.392559\pi$
$138$	0	0
$139$	− 8.74869e6i	− 0.276307i	−0.990411	−	0.138153i	$-0.955883\pi$
$139$	− 8.74869e6i	− 0.276307i	0.990411	−	0.138153i	$-0.0441167\pi$
$140$	0	0
$141$	− 1.90074e7i	− 0.571025i
$142$	0	0
$143$	6.86216e6	0.196239
$144$	0	0
$145$	1.96136e7	0.534279
$146$	0	0
$147$	1.44704e7i	0.375725i
$148$	0	0
$149$	− 1.47438e7i	− 0.365139i	−0.983193	−	0.182569i	$-0.941559\pi$
$149$	− 1.47438e7i	− 0.365139i	0.983193	−	0.182569i	$-0.0584414\pi$
$150$	0	0
$151$	−1.27080e7	−0.300372	−0.150186	−	0.988658i	$-0.547987\pi$
$151$	−1.27080e7	−0.300372	−0.150186	+	0.988658i	$0.547987\pi$
$152$	0	0
$153$	1.32046e7	0.298062
$154$	0	0
$155$	2.47912e7i	0.534732i
$156$	0	0
$157$	7.16246e7i	1.47711i	0.674192	+	0.738556i	$0.264492\pi$
$157$	7.16246e7i	1.47711i	−0.674192	+	0.738556i	$0.735508\pi$
$158$	0	0
$159$	−3.21161e7	−0.633626
$160$	0	0
$161$	1.00073e8	1.88985
$162$	0	0
$163$	3.94304e7i	0.713139i	0.934269	+	0.356569i	$0.116054\pi$
$163$	3.94304e7i	0.713139i	−0.934269	+	0.356569i	$0.883946\pi$
$164$	0	0
$165$	− 7.59073e6i	− 0.131550i
$166$	0	0
$167$	−8.32357e7	−1.38294	−0.691468	−	0.722407i	$-0.743036\pi$
$167$	−8.32357e7	−1.38294	−0.691468	+	0.722407i	$0.743036\pi$
$168$	0	0
$169$	5.35737e7	0.853785
$170$	0	0
$171$	2.42222e7i	0.370448i
$172$	0	0
$173$	− 1.15024e8i	− 1.68900i	−0.535559	−	0.844498i	$-0.679899\pi$
$173$	− 1.15024e8i	− 1.68900i	0.535559	−	0.844498i	$-0.320101\pi$
$174$	0	0
$175$	7.31357e7	1.03156
$176$	0	0
$177$	5.33817e7	0.723579
$178$	0	0
$179$	4.13354e6i	0.0538686i	0.999637	+	0.0269343i	$0.00857450\pi$
$179$	4.13354e6i	0.0538686i	−0.999637	+	0.0269343i	$0.991426\pi$
$180$	0	0
$181$	1.52220e8i	1.90808i	0.299674	+	0.954042i	$0.403122\pi$
$181$	1.52220e8i	1.90808i	−0.299674	+	0.954042i	$0.596878\pi$
$182$	0	0
$183$	8.48783e7	1.02381
$184$	0	0
$185$	1.91612e7	0.222496
$186$	0	0
$187$	− 4.10357e7i	− 0.458898i
$188$	0	0
$189$	− 2.29498e7i	− 0.247265i
$190$	0	0
$191$	1.37108e8	1.42379	0.711893	−	0.702288i	$-0.247838\pi$
$191$	1.37108e8	1.42379	0.711893	+	0.702288i	$0.247838\pi$
$192$	0	0
$193$	1.78651e8	1.78877	0.894386	−	0.447297i	$-0.147613\pi$
$193$	1.78651e8	1.78877	0.894386	+	0.447297i	$0.147613\pi$
$194$	0	0
$195$	1.01489e7i	0.0980163i
$196$	0	0
$197$	6.62643e7i	0.617516i	0.951141	+	0.308758i	$0.0999133\pi$
$197$	6.62643e7i	0.617516i	−0.951141	+	0.308758i	$0.900087\pi$
$198$	0	0
$199$	6.83278e7	0.614627	0.307314	−	0.951608i	$-0.400570\pi$
$199$	6.83278e7	0.614627	0.307314	+	0.951608i	$0.400570\pi$
$200$	0	0
$201$	2.78402e7	0.241817
$202$	0	0
$203$	− 1.84283e8i	− 1.54614i
$204$	0	0
$205$	− 2.70617e7i	− 0.219390i
$206$	0	0
$207$	−6.25689e7	−0.490301
$208$	0	0
$209$	7.52746e7	0.570343
$210$	0	0
$211$	− 8.64106e7i	− 0.633255i	−0.948550	−	0.316627i	$-0.897450\pi$
$211$	− 8.64106e7i	− 0.633255i	0.948550	−	0.316627i	$-0.102550\pi$
$212$	0	0
$213$	− 3.37235e7i	− 0.239113i
$214$	0	0
$215$	3.08368e7	0.211610
$216$	0	0
$217$	2.32931e8	1.54745
$218$	0	0
$219$	− 6.37660e7i	− 0.410237i
$220$	0	0
$221$	5.48653e7i	0.341920i
$222$	0	0
$223$	−2.41586e8	−1.45883	−0.729415	−	0.684072i	$-0.760207\pi$
$223$	−2.41586e8	−1.45883	−0.729415	+	0.684072i	$0.760207\pi$
$224$	0	0
$225$	−4.57267e7	−0.267628
$226$	0	0
$227$	2.39185e8i	1.35720i	0.734509	+	0.678599i	$0.237413\pi$
$227$	2.39185e8i	1.35720i	−0.734509	+	0.678599i	$0.762587\pi$
$228$	0	0
$229$	− 1.16047e8i	− 0.638573i	−0.947658	−	0.319286i	$-0.896557\pi$
$229$	− 1.16047e8i	− 0.638573i	0.947658	−	0.319286i	$-0.103443\pi$
$230$	0	0
$231$	−7.13204e7	−0.380690
$232$	0	0
$233$	3.65805e8	1.89454	0.947271	−	0.320435i	$-0.103829\pi$
$233$	3.65805e8	1.89454	0.947271	+	0.320435i	$0.103829\pi$
$234$	0	0
$235$	− 8.73606e7i	− 0.439115i
$236$	0	0
$237$	− 4.47657e7i	− 0.218437i
$238$	0	0
$239$	5.53604e7	0.262305	0.131153	−	0.991362i	$-0.458132\pi$
$239$	5.53604e7	0.262305	0.131153	+	0.991362i	$0.458132\pi$
$240$	0	0
$241$	1.27289e8	0.585777	0.292888	−	0.956147i	$-0.405384\pi$
$241$	1.27289e8	0.585777	0.292888	+	0.956147i	$0.405384\pi$
$242$	0	0
$243$	1.43489e7i	0.0641500i
$244$	0	0
$245$	6.65081e7i	0.288930i
$246$	0	0
$247$	−1.00643e8	−0.424957
$248$	0	0
$249$	7.10427e7	0.291623
$250$	0	0
$251$	− 3.90413e7i	− 0.155836i	−0.996960	−	0.0779178i	$-0.975173\pi$
$251$	− 3.90413e7i	− 0.155836i	0.996960	−	0.0779178i	$-0.0248272\pi$
$252$	0	0
$253$	1.94444e8i	0.754870i
$254$	0	0
$255$	6.06904e7	0.229208
$256$	0	0
$257$	−7.88184e7	−0.289642	−0.144821	−	0.989458i	$-0.546261\pi$
$257$	−7.88184e7	−0.289642	−0.144821	+	0.989458i	$0.546261\pi$
$258$	0	0
$259$	− 1.80034e8i	− 0.643879i
$260$	0	0
$261$	1.15220e8i	0.401130i
$262$	0	0
$263$	−7.87387e7	−0.266897	−0.133448	−	0.991056i	$-0.542605\pi$
$263$	−7.87387e7	−0.266897	−0.133448	+	0.991056i	$0.542605\pi$
$264$	0	0
$265$	−1.47610e8	−0.487254
$266$	0	0
$267$	7.34233e7i	0.236072i
$268$	0	0
$269$	− 1.18437e8i	− 0.370985i	−0.982646	−	0.185492i	$-0.940612\pi$
$269$	− 1.18437e8i	− 0.370985i	0.982646	−	0.185492i	$-0.0593880\pi$
$270$	0	0
$271$	−2.99740e7	−0.0914856	−0.0457428	−	0.998953i	$-0.514565\pi$
$271$	−2.99740e7	−0.0914856	−0.0457428	+	0.998953i	$0.514565\pi$
$272$	0	0
$273$	9.53563e7	0.283648
$274$	0	0
$275$	1.42104e8i	0.412041i
$276$	0	0
$277$	− 5.84366e8i	− 1.65198i	−0.563682	−	0.825992i	$-0.690615\pi$
$277$	− 5.84366e8i	− 1.65198i	0.563682	−	0.825992i	$-0.309385\pi$
$278$	0	0
$279$	−1.45635e8	−0.401469
$280$	0	0
$281$	3.91457e8	1.05248	0.526238	−	0.850337i	$-0.323602\pi$
$281$	3.91457e8	1.05248	0.526238	+	0.850337i	$0.323602\pi$
$282$	0	0
$283$	− 6.42194e8i	− 1.68428i	−0.539260	−	0.842139i	$-0.681296\pi$
$283$	− 6.42194e8i	− 1.68428i	0.539260	−	0.842139i	$-0.318704\pi$
$284$	0	0
$285$	1.11329e8i	0.284872i
$286$	0	0
$287$	−2.54264e8	−0.634889
$288$	0	0
$289$	−8.22445e7	−0.200431
$290$	0	0
$291$	3.44713e8i	0.820035i
$292$	0	0
$293$	− 2.73216e8i	− 0.634555i	−0.948333	−	0.317277i	$-0.897231\pi$
$293$	− 2.73216e8i	− 0.634555i	0.948333	−	0.317277i	$-0.102769\pi$
$294$	0	0
$295$	2.45350e8	0.556428
$296$	0	0
$297$	4.45917e7	0.0987658
$298$	0	0
$299$	− 2.59974e8i	− 0.562446i
$300$	0	0
$301$	− 2.89734e8i	− 0.612374i
$302$	0	0
$303$	2.23491e7	0.0461542
$304$	0	0
$305$	3.90113e8	0.787301
$306$	0	0
$307$	9.98737e8i	1.97000i	0.172548	+	0.985001i	$0.444800\pi$
$307$	9.98737e8i	1.97000i	−0.172548	+	0.985001i	$0.555200\pi$
$308$	0	0
$309$	4.29282e8i	0.827729i
$310$	0	0
$311$	8.75416e8	1.65026	0.825132	−	0.564940i	$-0.191101\pi$
$311$	8.75416e8	1.65026	0.825132	+	0.564940i	$0.191101\pi$
$312$	0	0
$313$	4.71335e8	0.868809	0.434405	−	0.900718i	$-0.356959\pi$
$313$	4.71335e8	0.868809	0.434405	+	0.900718i	$0.356959\pi$
$314$	0	0
$315$	− 1.05480e8i	− 0.190145i
$316$	0	0
$317$	7.26367e8i	1.28070i	0.768081	+	0.640352i	$0.221212\pi$
$317$	7.26367e8i	1.28070i	−0.768081	+	0.640352i	$0.778788\pi$
$318$	0	0
$319$	3.58065e8	0.617582
$320$	0	0
$321$	−1.90302e8	−0.321126
$322$	0	0
$323$	6.01845e8i	0.993748i
$324$	0	0
$325$	− 1.89994e8i	− 0.307007i
$326$	0	0
$327$	1.14822e8	0.181596
$328$	0	0
$329$	−8.20816e8	−1.27075
$330$	0	0
$331$	− 6.68322e8i	− 1.01295i	−0.862255	−	0.506475i	$-0.830948\pi$
$331$	− 6.68322e8i	− 1.01295i	0.862255	−	0.506475i	$-0.169052\pi$
$332$	0	0
$333$	1.12563e8i	0.167047i
$334$	0	0
$335$	1.27957e8	0.185955
$336$	0	0
$337$	−6.32204e8	−0.899814	−0.449907	−	0.893075i	$-0.648543\pi$
$337$	−6.32204e8	−0.899814	−0.449907	+	0.893075i	$0.648543\pi$
$338$	0	0
$339$	4.79821e8i	0.668930i
$340$	0	0
$341$	4.52587e8i	0.618104i
$342$	0	0
$343$	−3.35335e8	−0.448693
$344$	0	0
$345$	−2.87576e8	−0.377038
$346$	0	0
$347$	1.86121e8i	0.239135i	0.992826	+	0.119567i	$0.0381508\pi$
$347$	1.86121e8i	0.239135i	−0.992826	+	0.119567i	$0.961849\pi$
$348$	0	0
$349$	− 1.39636e8i	− 0.175837i	−0.996128	−	0.0879184i	$-0.971979\pi$
$349$	− 1.39636e8i	− 0.175837i	0.996128	−	0.0879184i	$-0.0280215\pi$
$350$	0	0
$351$	−5.96197e7	−0.0735893
$352$	0	0
$353$	−1.24893e9	−1.51122	−0.755609	−	0.655023i	$-0.772659\pi$
$353$	−1.24893e9	−1.51122	−0.755609	+	0.655023i	$0.772659\pi$
$354$	0	0
$355$	− 1.54998e8i	− 0.183877i
$356$	0	0
$357$	− 5.70230e8i	− 0.663302i
$358$	0	0
$359$	−4.12767e8	−0.470841	−0.235421	−	0.971894i	$-0.575647\pi$
$359$	−4.12767e8	−0.470841	−0.235421	+	0.971894i	$0.575647\pi$
$360$	0	0
$361$	−2.10134e8	−0.235083
$362$	0	0
$363$	3.87577e8i	0.425290i
$364$	0	0
$365$	− 2.93078e8i	− 0.315470i
$366$	0	0
$367$	5.29612e8	0.559276	0.279638	−	0.960105i	$-0.409785\pi$
$367$	5.29612e8	0.559276	0.279638	+	0.960105i	$0.409785\pi$
$368$	0	0
$369$	1.58974e8	0.164715
$370$	0	0
$371$	1.38690e9i	1.41006i
$372$	0	0
$373$	1.13726e9i	1.13469i	0.823480	+	0.567346i	$0.192030\pi$
$373$	1.13726e9i	1.13469i	−0.823480	+	0.567346i	$0.807970\pi$
$374$	0	0
$375$	−4.71931e8	−0.462135
$376$	0	0
$377$	−4.78737e8	−0.460153
$378$	0	0
$379$	− 1.69749e9i	− 1.60166i	−0.598894	−	0.800828i	$-0.704393\pi$
$379$	− 1.69749e9i	− 1.60166i	0.598894	−	0.800828i	$-0.295607\pi$
$380$	0	0
$381$	− 4.94289e8i	− 0.457872i
$382$	0	0
$383$	−1.86776e9	−1.69873	−0.849366	−	0.527804i	$-0.823016\pi$
$383$	−1.86776e9	−1.69873	−0.849366	+	0.527804i	$0.823016\pi$
$384$	0	0
$385$	−3.27799e8	−0.292748
$386$	0	0
$387$	1.81151e8i	0.158874i
$388$	0	0
$389$	6.73416e8i	0.580042i	0.957020	+	0.290021i	$0.0936623\pi$
$389$	6.73416e8i	0.580042i	−0.957020	+	0.290021i	$0.906338\pi$
$390$	0	0
$391$	−1.55464e9	−1.31526
$392$	0	0
$393$	−3.02072e8	−0.251036
$394$	0	0
$395$	− 2.05750e8i	− 0.167977i
$396$	0	0
$397$	1.71491e8i	0.137554i	0.997632	+	0.0687770i	$0.0219097\pi$
$397$	1.71491e8i	0.137554i	−0.997632	+	0.0687770i	$0.978090\pi$
$398$	0	0
$399$	1.04601e9	0.824388
$400$	0	0
$401$	−9.97105e8	−0.772210	−0.386105	−	0.922455i	$-0.626180\pi$
$401$	−9.97105e8	−0.772210	−0.386105	+	0.922455i	$0.626180\pi$
$402$	0	0
$403$	− 6.05115e8i	− 0.460543i
$404$	0	0
$405$	6.59496e7i	0.0493310i
$406$	0	0
$407$	3.49808e8	0.257187
$408$	0	0
$409$	−2.16978e9	−1.56813	−0.784067	−	0.620676i	$-0.786858\pi$
$409$	−2.16978e9	−1.56813	−0.784067	+	0.620676i	$0.786858\pi$
$410$	0	0
$411$	5.38218e8i	0.382394i
$412$	0	0
$413$	− 2.30524e9i	− 1.61024i
$414$	0	0
$415$	3.26522e8	0.224256
$416$	0	0
$417$	2.36215e8	0.159526
$418$	0	0
$419$	2.03805e9i	1.35353i	0.736201	+	0.676763i	$0.236618\pi$
$419$	2.03805e9i	1.35353i	−0.736201	+	0.676763i	$0.763382\pi$
$420$	0	0
$421$	1.52504e9i	0.996078i	0.867154	+	0.498039i	$0.165946\pi$
$421$	1.52504e9i	0.996078i	−0.867154	+	0.498039i	$0.834054\pi$
$422$	0	0
$423$	5.13199e8	0.329682
$424$	0	0
$425$	−1.13617e9	−0.717927
$426$	0	0
$427$	− 3.66539e9i	− 2.27836i
$428$	0	0
$429$	1.85278e8i	0.113299i
$430$	0	0
$431$	−3.38318e8	−0.203542	−0.101771	−	0.994808i	$-0.532451\pi$
$431$	−3.38318e8	−0.203542	−0.101771	+	0.994808i	$0.532451\pi$
$432$	0	0
$433$	−9.36647e8	−0.554457	−0.277229	−	0.960804i	$-0.589416\pi$
$433$	−9.36647e8	−0.554457	−0.277229	+	0.960804i	$0.589416\pi$
$434$	0	0
$435$	5.29566e8i	0.308466i
$436$	0	0
$437$	− 2.85179e9i	− 1.63468i
$438$	0	0
$439$	1.01080e9	0.570218	0.285109	−	0.958495i	$-0.407970\pi$
$439$	1.01080e9	0.570218	0.285109	+	0.958495i	$0.407970\pi$
$440$	0	0
$441$	−3.90701e8	−0.216925
$442$	0	0
$443$	− 2.81634e9i	− 1.53912i	−0.638575	−	0.769560i	$-0.720476\pi$
$443$	− 2.81634e9i	− 1.53912i	0.638575	−	0.769560i	$-0.279524\pi$
$444$	0	0
$445$	3.37464e8i	0.181538i
$446$	0	0
$447$	3.98083e8	0.210813
$448$	0	0
$449$	−8.09782e8	−0.422188	−0.211094	−	0.977466i	$-0.567703\pi$
$449$	−8.09782e8	−0.422188	−0.211094	+	0.977466i	$0.567703\pi$
$450$	0	0
$451$	− 4.94038e8i	− 0.253596i
$452$	0	0
$453$	− 3.43117e8i	− 0.173420i
$454$	0	0
$455$	4.38271e8	0.218124
$456$	0	0
$457$	−3.36937e9	−1.65136	−0.825680	−	0.564139i	$-0.809208\pi$
$457$	−3.36937e9	−1.65136	−0.825680	+	0.564139i	$0.809208\pi$
$458$	0	0
$459$	3.56525e8i	0.172086i
$460$	0	0
$461$	2.28975e9i	1.08851i	0.838918	+	0.544257i	$0.183188\pi$
$461$	2.28975e9i	1.08851i	−0.838918	+	0.544257i	$0.816812\pi$
$462$	0	0
$463$	4.02745e8	0.188581	0.0942903	−	0.995545i	$-0.469942\pi$
$463$	4.02745e8	0.188581	0.0942903	+	0.995545i	$0.469942\pi$
$464$	0	0
$465$	−6.69361e8	−0.308727
$466$	0	0
$467$	1.44824e9i	0.658010i	0.944328	+	0.329005i	$0.106713\pi$
$467$	1.44824e9i	0.658010i	−0.944328	+	0.329005i	$0.893287\pi$
$468$	0	0
$469$	− 1.20225e9i	− 0.538134i
$470$	0	0
$471$	−1.93386e9	−0.852811
$472$	0	0
$473$	5.62957e8	0.244603
$474$	0	0
$475$	− 2.08414e9i	− 0.892278i
$476$	0	0
$477$	− 8.67134e8i	− 0.365824i
$478$	0	0
$479$	3.31472e9	1.37808	0.689038	−	0.724726i	$-0.258033\pi$
$479$	3.31472e9	1.37808	0.689038	+	0.724726i	$0.258033\pi$
$480$	0	0
$481$	−4.67697e8	−0.191627
$482$	0	0
$483$	2.70198e9i	1.09111i
$484$	0	0
$485$	1.58435e9i	0.630602i
$486$	0	0
$487$	2.92864e9	1.14898	0.574492	−	0.818510i	$-0.305200\pi$
$487$	2.92864e9	1.14898	0.574492	+	0.818510i	$0.305200\pi$
$488$	0	0
$489$	−1.06462e9	−0.411731
$490$	0	0
$491$	4.22952e8i	0.161252i	0.996744	+	0.0806261i	$0.0256920\pi$
$491$	4.22952e8i	0.161252i	−0.996744	+	0.0806261i	$0.974308\pi$
$492$	0	0
$493$	2.86285e9i	1.07605i
$494$	0	0
$495$	2.04950e8	0.0759503
$496$	0	0
$497$	−1.45632e9	−0.532118
$498$	0	0
$499$	− 6.59479e8i	− 0.237601i	−0.992918	−	0.118801i	$-0.962095\pi$
$499$	− 6.59479e8i	− 0.237601i	0.992918	−	0.118801i	$-0.0379050\pi$
$500$	0	0
$501$	− 2.24736e9i	− 0.798438i
$502$	0	0
$503$	−9.85138e8	−0.345151	−0.172575	−	0.984996i	$-0.555209\pi$
$503$	−9.85138e8	−0.345151	−0.172575	+	0.984996i	$0.555209\pi$
$504$	0	0
$505$	1.02720e8	0.0354923
$506$	0	0
$507$	1.44649e9i	0.492933i
$508$	0	0
$509$	− 8.36698e8i	− 0.281227i	−0.990065	−	0.140613i	$-0.955093\pi$
$509$	− 8.36698e8i	− 0.281227i	0.990065	−	0.140613i	$-0.0449075\pi$
$510$	0	0
$511$	−2.75367e9	−0.912934
$512$	0	0
$513$	−6.53999e8	−0.213878
$514$	0	0
$515$	1.97304e9i	0.636518i
$516$	0	0
$517$	− 1.59486e9i	− 0.507580i
$518$	0	0
$519$	3.10566e9	0.975142
$520$	0	0
$521$	−4.30754e9	−1.33443	−0.667217	−	0.744863i	$-0.732515\pi$
$521$	−4.30754e9	−1.33443	−0.667217	+	0.744863i	$0.732515\pi$
$522$	0	0
$523$	− 6.11922e9i	− 1.87042i	−0.354087	−	0.935212i	$-0.615208\pi$
$523$	− 6.11922e9i	− 1.87042i	0.354087	−	0.935212i	$-0.384792\pi$
$524$	0	0
$525$	1.97466e9i	0.595574i
$526$	0	0
$527$	−3.61858e9	−1.07696
$528$	0	0
$529$	3.96170e9	1.16355
$530$	0	0
$531$	1.44131e9i	0.417758i
$532$	0	0
$533$	6.60535e8i	0.188952i
$534$	0	0
$535$	−8.74654e8	−0.246944
$536$	0	0
$537$	−1.11605e8	−0.0311011
$538$	0	0
$539$	1.21417e9i	0.333979i
$540$	0	0
$541$	− 4.35204e9i	− 1.18169i	−0.806786	−	0.590843i	$-0.798795\pi$
$541$	− 4.35204e9i	− 1.18169i	0.806786	−	0.590843i	$-0.201205\pi$
$542$	0	0
$543$	−4.10995e9	−1.10163
$544$	0	0
$545$	5.27738e8	0.139647
$546$	0	0
$547$	− 5.56738e9i	− 1.45444i	−0.686406	−	0.727219i	$-0.740813\pi$
$547$	− 5.56738e9i	− 1.45444i	0.686406	−	0.727219i	$-0.259187\pi$
$548$	0	0
$549$	2.29171e9i	0.591095i
$550$	0	0
$551$	−5.25152e9	−1.33738
$552$	0	0
$553$	−1.93317e9	−0.486106
$554$	0	0
$555$	5.17354e8i	0.128458i
$556$	0	0
$557$	4.79292e9i	1.17519i	0.809156	+	0.587594i	$0.199925\pi$
$557$	4.79292e9i	1.17519i	−0.809156	+	0.587594i	$0.800075\pi$
$558$	0	0
$559$	−7.52681e8	−0.182251
$560$	0	0
$561$	1.10796e9	0.264945
$562$	0	0
$563$	5.95275e9i	1.40585i	0.711265	+	0.702924i	$0.248123\pi$
$563$	5.95275e9i	1.40585i	−0.711265	+	0.702924i	$0.751877\pi$
$564$	0	0
$565$	2.20533e9i	0.514403i
$566$	0	0
$567$	6.19644e8	0.142758
$568$	0	0
$569$	6.50474e9	1.48026	0.740129	−	0.672465i	$-0.234765\pi$
$569$	6.50474e9	1.48026	0.740129	+	0.672465i	$0.234765\pi$
$570$	0	0
$571$	− 3.13151e9i	− 0.703926i	−0.936014	−	0.351963i	$-0.885514\pi$
$571$	− 3.13151e9i	− 0.703926i	0.936014	−	0.351963i	$-0.114486\pi$
$572$	0	0
$573$	3.70191e9i	0.822024i
$574$	0	0
$575$	5.38361e9	1.18096
$576$	0	0
$577$	−3.11667e9	−0.675423	−0.337712	−	0.941250i	$-0.609653\pi$
$577$	−3.11667e9	−0.675423	−0.337712	+	0.941250i	$0.609653\pi$
$578$	0	0
$579$	4.82358e9i	1.03275i
$580$	0	0
$581$	− 3.06791e9i	− 0.648972i
$582$	0	0
$583$	−2.69477e9	−0.563225
$584$	0	0
$585$	−2.74021e8	−0.0565897
$586$	0	0
$587$	9.06256e9i	1.84934i	0.380764	+	0.924672i	$0.375661\pi$
$587$	9.06256e9i	1.84934i	−0.380764	+	0.924672i	$0.624339\pi$
$588$	0	0
$589$	− 6.63782e9i	− 1.33851i
$590$	0	0
$591$	−1.78914e9	−0.356523
$592$	0	0
$593$	−1.27273e9	−0.250638	−0.125319	−	0.992117i	$-0.539995\pi$
$593$	−1.27273e9	−0.250638	−0.125319	+	0.992117i	$0.539995\pi$
$594$	0	0
$595$	− 2.62086e9i	− 0.510075i
$596$	0	0
$597$	1.84485e9i	0.354855i
$598$	0	0
$599$	8.35942e9	1.58921	0.794607	−	0.607125i	$-0.207677\pi$
$599$	8.35942e9	1.58921	0.794607	+	0.607125i	$0.207677\pi$
$600$	0	0
$601$	7.27337e9	1.36671	0.683353	−	0.730088i	$-0.260521\pi$
$601$	7.27337e9	1.36671	0.683353	+	0.730088i	$0.260521\pi$
$602$	0	0
$603$	7.51685e8i	0.139613i
$604$	0	0
$605$	1.78136e9i	0.327045i
$606$	0	0
$607$	1.24790e9	0.226475	0.113238	−	0.993568i	$-0.463878\pi$
$607$	1.24790e9	0.226475	0.113238	+	0.993568i	$0.463878\pi$
$608$	0	0
$609$	4.97565e9	0.892667
$610$	0	0
$611$	2.13234e9i	0.378192i
$612$	0	0
$613$	− 2.23707e8i	− 0.0392254i	−0.999808	−	0.0196127i	$-0.993757\pi$
$613$	− 2.23707e8i	− 0.0392254i	0.999808	−	0.0196127i	$-0.00624332\pi$
$614$	0	0
$615$	7.30665e8	0.126665
$616$	0	0
$617$	7.52135e9	1.28913	0.644566	−	0.764549i	$-0.277038\pi$
$617$	7.52135e9	1.28913	0.644566	+	0.764549i	$0.277038\pi$
$618$	0	0
$619$	9.47080e8i	0.160498i	0.996775	+	0.0802489i	$0.0255715\pi$
$619$	9.47080e8i	0.160498i	−0.996775	+	0.0802489i	$0.974428\pi$
$620$	0	0
$621$	− 1.68936e9i	− 0.283075i
$622$	0	0
$623$	3.17072e9	0.525351
$624$	0	0
$625$	2.73135e9	0.447504
$626$	0	0
$627$	2.03241e9i	0.329288i
$628$	0	0
$629$	2.79683e9i	0.448114i
$630$	0	0
$631$	−1.56944e9	−0.248681	−0.124340	−	0.992240i	$-0.539681\pi$
$631$	−1.56944e9	−0.248681	−0.124340	+	0.992240i	$0.539681\pi$
$632$	0	0
$633$	2.33309e9	0.365610
$634$	0	0
$635$	− 2.27182e9i	− 0.352100i
$636$	0	0
$637$	− 1.62336e9i	− 0.248844i
$638$	0	0
$639$	9.10533e8	0.138052
$640$	0	0
$641$	−2.82330e9	−0.423403	−0.211701	−	0.977334i	$-0.567900\pi$
$641$	−2.82330e9	−0.423403	−0.211701	+	0.977334i	$0.567900\pi$
$642$	0	0
$643$	− 3.51863e9i	− 0.521958i	−0.965345	−	0.260979i	$-0.915955\pi$
$643$	− 3.51863e9i	− 0.521958i	0.965345	−	0.260979i	$-0.0840453\pi$
$644$	0	0
$645$	8.32595e8i	0.122173i
$646$	0	0
$647$	−6.01687e9	−0.873384	−0.436692	−	0.899611i	$-0.643850\pi$
$647$	−6.01687e9	−0.873384	−0.436692	+	0.899611i	$0.643850\pi$
$648$	0	0
$649$	4.47910e9	0.643183
$650$	0	0
$651$	6.28913e9i	0.893423i
$652$	0	0
$653$	− 1.28919e10i	− 1.81184i	−0.423449	−	0.905920i	$-0.639181\pi$
$653$	− 1.28919e10i	− 1.81184i	0.423449	−	0.905920i	$-0.360819\pi$
$654$	0	0
$655$	−1.38837e9	−0.193045
$656$	0	0
$657$	1.72168e9	0.236850
$658$	0	0
$659$	− 9.95725e8i	− 0.135532i	−0.997701	−	0.0677658i	$-0.978413\pi$
$659$	− 9.95725e8i	− 0.135532i	0.997701	−	0.0677658i	$-0.0215871\pi$
$660$	0	0
$661$	1.93879e8i	0.0261112i	0.999915	+	0.0130556i	$0.00415584\pi$
$661$	1.93879e8i	0.0261112i	−0.999915	+	0.0130556i	$0.995844\pi$
$662$	0	0
$663$	−1.48136e9	−0.197408
$664$	0	0
$665$	4.80762e9	0.633949
$666$	0	0
$667$	− 1.35653e10i	− 1.77007i
$668$	0	0
$669$	− 6.52282e9i	− 0.842255i
$670$	0	0
$671$	7.12190e9	0.910053
$672$	0	0
$673$	4.01613e9	0.507873	0.253937	−	0.967221i	$-0.418275\pi$
$673$	4.01613e9	0.507873	0.253937	+	0.967221i	$0.418275\pi$
$674$	0	0
$675$	− 1.23462e9i	− 0.154515i
$676$	0	0
$677$	8.56492e9i	1.06087i	0.847725	+	0.530436i	$0.177972\pi$
$677$	8.56492e9i	1.06087i	−0.847725	+	0.530436i	$0.822028\pi$
$678$	0	0
$679$	1.48861e10	1.82489
$680$	0	0
$681$	−6.45799e9	−0.783579
$682$	0	0
$683$	4.38276e9i	0.526351i	0.964748	+	0.263176i	$0.0847699\pi$
$683$	4.38276e9i	0.526351i	−0.964748	+	0.263176i	$0.915230\pi$
$684$	0	0
$685$	2.47372e9i	0.294059i
$686$	0	0
$687$	3.13327e9	0.368680
$688$	0	0
$689$	3.60294e9	0.419653
$690$	0	0
$691$	− 1.47546e10i	− 1.70119i	−0.525819	−	0.850597i	$-0.676241\pi$
$691$	− 1.47546e10i	− 1.70119i	0.525819	−	0.850597i	$-0.323759\pi$
$692$	0	0
$693$	− 1.92565e9i	− 0.219792i
$694$	0	0
$695$	1.08568e9	0.122674
$696$	0	0
$697$	3.95000e9	0.441857
$698$	0	0
$699$	9.87674e9i	1.09381i
$700$	0	0
$701$	7.45363e9i	0.817250i	0.912702	+	0.408625i	$0.133992\pi$
$701$	7.45363e9i	0.817250i	−0.912702	+	0.408625i	$0.866008\pi$
$702$	0	0
$703$	−5.13041e9	−0.556940
$704$	0	0
$705$	2.35874e9	0.253523
$706$	0	0
$707$	− 9.65126e8i	− 0.102711i
$708$	0	0
$709$	− 5.77508e9i	− 0.608549i	−0.952584	−	0.304275i	$-0.901586\pi$
$709$	− 5.77508e9i	− 0.608549i	0.952584	−	0.304275i	$-0.0984141\pi$
$710$	0	0
$711$	1.20867e9	0.126115
$712$	0	0
$713$	1.71463e10	1.77157
$714$	0	0
$715$	8.51566e8i	0.0871259i
$716$	0	0
$717$	1.49473e9i	0.151442i
$718$	0	0
$719$	−6.28504e9	−0.630605	−0.315302	−	0.948991i	$-0.602106\pi$
$719$	−6.28504e9	−0.630605	−0.315302	+	0.948991i	$0.602106\pi$
$720$	0	0
$721$	1.85381e10	1.84201
$722$	0	0
$723$	3.43681e9i	0.338198i
$724$	0	0
$725$	− 9.91383e9i	− 0.966181i
$726$	0	0
$727$	−1.68427e10	−1.62570	−0.812850	−	0.582474i	$-0.802085\pi$
$727$	−1.68427e10	−1.62570	−0.812850	+	0.582474i	$0.802085\pi$
$728$	0	0
$729$	−3.87420e8	−0.0370370
$730$	0	0
$731$	4.50103e9i	0.426188i
$732$	0	0
$733$	1.92702e10i	1.80726i	0.428310	+	0.903632i	$0.359109\pi$
$733$	1.92702e10i	1.80726i	−0.428310	+	0.903632i	$0.640891\pi$
$734$	0	0
$735$	−1.79572e9	−0.166814
$736$	0	0
$737$	2.33599e9	0.214949
$738$	0	0
$739$	− 1.67042e10i	− 1.52255i	−0.648432	−	0.761273i	$-0.724575\pi$
$739$	− 1.67042e10i	− 1.52255i	0.648432	−	0.761273i	$-0.275425\pi$
$740$	0	0
$741$	− 2.71736e9i	− 0.245349i
$742$	0	0
$743$	−2.03663e10	−1.82159	−0.910796	−	0.412856i	$-0.864531\pi$
$743$	−2.03663e10	−1.82159	−0.910796	+	0.412856i	$0.864531\pi$
$744$	0	0
$745$	1.82965e9	0.162114
$746$	0	0
$747$	1.91815e9i	0.168369i
$748$	0	0
$749$	8.21800e9i	0.714627i
$750$	0	0
$751$	2.61913e8	0.0225641	0.0112820	−	0.999936i	$-0.496409\pi$
$751$	2.61913e8	0.0225641	0.0112820	+	0.999936i	$0.496409\pi$
$752$	0	0
$753$	1.05412e9	0.0899717
$754$	0	0
$755$	− 1.57701e9i	− 0.133359i
$756$	0	0
$757$	9.79252e9i	0.820463i	0.911981	+	0.410231i	$0.134552\pi$
$757$	9.79252e9i	0.820463i	−0.911981	+	0.410231i	$0.865448\pi$
$758$	0	0
$759$	−5.24998e9	−0.435824
$760$	0	0
$761$	1.36886e10	1.12593	0.562966	−	0.826480i	$-0.309660\pi$
$761$	1.36886e10	1.12593	0.562966	+	0.826480i	$0.309660\pi$
$762$	0	0
$763$	− 4.95847e9i	− 0.404121i
$764$	0	0
$765$	1.63864e9i	0.132333i
$766$	0	0
$767$	−5.98862e9	−0.479229
$768$	0	0
$769$	4.86151e9	0.385504	0.192752	−	0.981247i	$-0.438259\pi$
$769$	4.86151e9	0.385504	0.192752	+	0.981247i	$0.438259\pi$
$770$	0	0
$771$	− 2.12810e9i	− 0.167225i
$772$	0	0
$773$	− 1.94304e10i	− 1.51305i	−0.653966	−	0.756524i	$-0.726896\pi$
$773$	− 1.94304e10i	− 1.51305i	0.653966	−	0.756524i	$-0.273104\pi$
$774$	0	0
$775$	1.25309e10	0.966998
$776$	0	0
$777$	4.86091e9	0.371744
$778$	0	0
$779$	7.24575e9i	0.549164i
$780$	0	0
$781$	− 2.82964e9i	− 0.212546i
$782$	0	0
$783$	−3.11093e9	−0.231592
$784$	0	0
$785$	−8.88831e9	−0.655806
$786$	0	0
$787$	− 8.58176e9i	− 0.627574i	−0.949493	−	0.313787i	$-0.898402\pi$
$787$	− 8.58176e9i	− 0.627574i	0.949493	−	0.313787i	$-0.101598\pi$
$788$	0	0
$789$	− 2.12594e9i	− 0.154093i
$790$	0	0
$791$	2.07206e10	1.48862
$792$	0	0
$793$	−9.52207e9	−0.678071
$794$	0	0
$795$	− 3.98547e9i	− 0.281316i
$796$	0	0
$797$	− 2.34513e10i	− 1.64082i	−0.571772	−	0.820412i	$-0.693744\pi$
$797$	− 2.34513e10i	− 1.64082i	0.571772	−	0.820412i	$-0.306256\pi$
$798$	0	0
$799$	1.27514e10	0.884390
$800$	0	0
$801$	−1.98243e9	−0.136296
$802$	0	0
$803$	− 5.35042e9i	− 0.364656i
$804$	0	0
$805$	1.24187e10i	0.839054i
$806$	0	0
$807$	3.19781e9	0.214188
$808$	0	0
$809$	6.34725e9	0.421469	0.210734	−	0.977543i	$-0.432414\pi$
$809$	6.34725e9	0.421469	0.210734	+	0.977543i	$0.432414\pi$
$810$	0	0
$811$	2.67610e9i	0.176169i	0.996113	+	0.0880845i	$0.0280745\pi$
$811$	2.67610e9i	0.176169i	−0.996113	+	0.0880845i	$0.971925\pi$
$812$	0	0
$813$	− 8.09299e8i	− 0.0528192i
$814$	0	0
$815$	−4.89314e9	−0.316619
$816$	0	0
$817$	−8.25655e9	−0.529689
$818$	0	0
$819$	2.57462e9i	0.163764i
$820$	0	0
$821$	2.52971e10i	1.59540i	0.603055	+	0.797700i	$0.293950\pi$
$821$	2.52971e10i	1.59540i	−0.603055	+	0.797700i	$0.706050\pi$
$822$	0	0
$823$	−1.55792e10	−0.974195	−0.487097	−	0.873348i	$-0.661944\pi$
$823$	−1.55792e10	−0.974195	−0.487097	+	0.873348i	$0.661944\pi$
$824$	0	0
$825$	−3.83680e9	−0.237892
$826$	0	0
$827$	2.86863e10i	1.76362i	0.471605	+	0.881810i	$0.343675\pi$
$827$	2.86863e10i	1.76362i	−0.471605	+	0.881810i	$0.656325\pi$
$828$	0	0
$829$	2.67957e10i	1.63352i	0.576976	+	0.816761i	$0.304232\pi$
$829$	2.67957e10i	1.63352i	−0.576976	+	0.816761i	$0.695768\pi$
$830$	0	0
$831$	1.57779e10	0.953774
$832$	0	0
$833$	−9.70770e9	−0.581914
$834$	0	0
$835$	− 1.03292e10i	− 0.613994i
$836$	0	0
$837$	− 3.93216e9i	− 0.231788i
$838$	0	0
$839$	1.37703e10	0.804965	0.402482	−	0.915428i	$-0.368147\pi$
$839$	1.37703e10	0.804965	0.402482	+	0.915428i	$0.368147\pi$
$840$	0	0
$841$	−7.73046e9	−0.448146
$842$	0	0
$843$	1.05694e10i	0.607648i
$844$	0	0
$845$	6.64828e9i	0.379062i
$846$	0	0
$847$	1.67372e10	0.946433
$848$	0	0
$849$	1.73392e10	0.972419
$850$	0	0
$851$	− 1.32525e10i	− 0.737130i
$852$	0	0
$853$	− 2.57650e10i	− 1.42137i	−0.703509	−	0.710686i	$-0.748385\pi$
$853$	− 2.57650e10i	− 1.42137i	0.703509	−	0.710686i	$-0.251615\pi$
$854$	0	0
$855$	−3.00587e9	−0.164471
$856$	0	0
$857$	3.18920e9	0.173081	0.0865404	−	0.996248i	$-0.472419\pi$
$857$	3.18920e9	0.173081	0.0865404	+	0.996248i	$0.472419\pi$
$858$	0	0
$859$	1.18163e9i	0.0636072i	0.999494	+	0.0318036i	$0.0101251\pi$
$859$	1.18163e9i	0.0636072i	−0.999494	+	0.0318036i	$0.989875\pi$
$860$	0	0
$861$	− 6.86512e9i	− 0.366553i
$862$	0	0
$863$	−1.72234e10	−0.912183	−0.456092	−	0.889933i	$-0.650751\pi$
$863$	−1.72234e10	−0.912183	−0.456092	+	0.889933i	$0.650751\pi$
$864$	0	0
$865$	1.42740e10	0.749878
$866$	0	0
$867$	− 2.22060e9i	− 0.115719i
$868$	0	0
$869$	− 3.75616e9i	− 0.194167i
$870$	0	0
$871$	−3.12325e9	−0.160156
$872$	0	0
$873$	−9.30725e9	−0.473447
$874$	0	0
$875$	2.03799e10i	1.02843i
$876$	0	0
$877$	− 3.58497e10i	− 1.79468i	−0.441343	−	0.897339i	$-0.645498\pi$
$877$	− 3.58497e10i	− 1.79468i	0.441343	−	0.897339i	$-0.354502\pi$
$878$	0	0
$879$	7.37682e9	0.366360
$880$	0	0
$881$	−9.57198e9	−0.471614	−0.235807	−	0.971800i	$-0.575773\pi$
$881$	−9.57198e9	−0.471614	−0.235807	+	0.971800i	$0.575773\pi$
$882$	0	0
$883$	− 4.48065e9i	− 0.219017i	−0.993986	−	0.109509i	$-0.965072\pi$
$883$	− 4.48065e9i	− 0.219017i	0.993986	−	0.109509i	$-0.0349278\pi$
$884$	0	0
$885$	6.62445e9i	0.321254i
$886$	0	0
$887$	1.51174e10	0.727350	0.363675	−	0.931526i	$-0.381522\pi$
$887$	1.51174e10	0.727350	0.363675	+	0.931526i	$0.381522\pi$
$888$	0	0
$889$	−2.13454e10	−1.01894
$890$	0	0
$891$	1.20398e9i	0.0570224i
$892$	0	0
$893$	2.33908e10i	1.09917i
$894$	0	0
$895$	−5.12955e8	−0.0239165
$896$	0	0
$897$	7.01929e9	0.324728
$898$	0	0
$899$	− 3.15747e10i	− 1.44937i
$900$	0	0
$901$	− 2.15456e10i	− 0.981344i
$902$	0	0
$903$	7.82282e9	0.353555
$904$	0	0
$905$	−1.88899e10	−0.847149
$906$	0	0
$907$	3.61601e10i	1.60918i	0.593831	+	0.804590i	$0.297615\pi$
$907$	3.61601e10i	1.60918i	−0.593831	+	0.804590i	$0.702385\pi$
$908$	0	0
$909$	6.03427e8i	0.0266472i
$910$	0	0
$911$	−3.67974e10	−1.61251	−0.806256	−	0.591566i	$-0.798510\pi$
$911$	−3.67974e10	−1.61251	−0.806256	+	0.591566i	$0.798510\pi$
$912$	0	0
$913$	5.96099e9	0.259221
$914$	0	0
$915$	1.05330e10i	0.454548i
$916$	0	0
$917$	1.30447e10i	0.558651i
$918$	0	0
$919$	1.95896e10	0.832573	0.416286	−	0.909234i	$-0.363331\pi$
$919$	1.95896e10	0.832573	0.416286	+	0.909234i	$0.363331\pi$
$920$	0	0
$921$	−2.69659e10	−1.13738
$922$	0	0
$923$	3.78326e9i	0.158366i
$924$	0	0
$925$	− 9.68520e9i	− 0.402358i
$926$	0	0
$927$	−1.15906e10	−0.477889
$928$	0	0
$929$	−4.12591e10	−1.68836	−0.844180	−	0.536059i	$-0.819912\pi$
$929$	−4.12591e10	−1.68836	−0.844180	+	0.536059i	$0.819912\pi$
$930$	0	0
$931$	− 1.78075e10i	− 0.723235i
$932$	0	0
$933$	2.36362e10i	0.952780i
$934$	0	0
$935$	5.09236e9	0.203741
$936$	0	0
$937$	−2.38781e10	−0.948224	−0.474112	−	0.880464i	$-0.657231\pi$
$937$	−2.38781e10	−0.948224	−0.474112	+	0.880464i	$0.657231\pi$
$938$	0	0
$939$	1.27260e10i	0.501607i
$940$	0	0
$941$	− 3.39485e10i	− 1.32818i	−0.747653	−	0.664090i	$-0.768819\pi$
$941$	− 3.39485e10i	− 1.32818i	0.747653	−	0.664090i	$-0.231181\pi$
$942$	0	0
$943$	−1.87167e10	−0.726838
$944$	0	0
$945$	2.84797e9	0.109780
$946$	0	0
$947$	2.08467e10i	0.797648i	0.917027	+	0.398824i	$0.130582\pi$
$947$	2.08467e10i	0.797648i	−0.917027	+	0.398824i	$0.869418\pi$
$948$	0	0
$949$	7.15359e9i	0.271702i
$950$	0	0
$951$	−1.96119e10	−0.739415
$952$	0	0
$953$	1.02660e10	0.384215	0.192108	−	0.981374i	$-0.438468\pi$
$953$	1.02660e10	0.384215	0.192108	+	0.981374i	$0.438468\pi$
$954$	0	0
$955$	1.70145e10i	0.632131i
$956$	0	0
$957$	9.66775e9i	0.356561i
$958$	0	0
$959$	2.32424e10	0.850974
$960$	0	0
$961$	1.23971e10	0.450598
$962$	0	0
$963$	− 5.13814e9i	− 0.185402i
$964$	0	0
$965$	2.21698e10i	0.794177i
$966$	0	0
$967$	−4.42090e10	−1.57224	−0.786119	−	0.618076i	$-0.787913\pi$
$967$	−4.42090e10	−1.57224	−0.786119	+	0.618076i	$0.787913\pi$
$968$	0	0
$969$	−1.62498e10	−0.573741
$970$	0	0
$971$	1.99330e9i	0.0698724i	0.999390	+	0.0349362i	$0.0111228\pi$
$971$	1.99330e9i	0.0698724i	−0.999390	+	0.0349362i	$0.988877\pi$
$972$	0	0
$973$	− 1.02007e10i	− 0.355006i
$974$	0	0
$975$	5.12984e9	0.177251
$976$	0	0
$977$	−4.43760e10	−1.52236	−0.761180	−	0.648541i	$-0.775380\pi$
$977$	−4.43760e10	−1.52236	−0.761180	+	0.648541i	$0.775380\pi$
$978$	0	0
$979$	6.16074e9i	0.209843i
$980$	0	0
$981$	3.10019e9i	0.104845i
$982$	0	0
$983$	−1.36463e10	−0.458225	−0.229112	−	0.973400i	$-0.573582\pi$
$983$	−1.36463e10	−0.458225	−0.229112	+	0.973400i	$0.573582\pi$
$984$	0	0
$985$	−8.22313e9	−0.274164
$986$	0	0
$987$	− 2.21620e10i	− 0.733668i
$988$	0	0
$989$	− 2.13277e10i	− 0.701063i
$990$	0	0
$991$	−2.86452e10	−0.934963	−0.467481	−	0.884003i	$-0.654839\pi$
$991$	−2.86452e10	−0.934963	−0.467481	+	0.884003i	$0.654839\pi$
$992$	0	0
$993$	1.80447e10	0.584827
$994$	0	0
$995$	8.47920e9i	0.272881i
$996$	0	0
$997$	9.14138e9i	0.292132i	0.989275	+	0.146066i	$0.0466611\pi$
$997$	9.14138e9i	0.292132i	−0.989275	+	0.146066i	$0.953339\pi$
$998$	0	0
$999$	−3.03919e9	−0.0964447

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.8.d.f.193.13	yes	16
4.3	odd	2	inner	384.8.d.f.193.5	yes	16
8.3	odd	2	inner	384.8.d.f.193.12	yes	16
8.5	even	2	inner	384.8.d.f.193.4	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.8.d.f.193.4	✓	16	8.5	even	2	inner
384.8.d.f.193.5	yes	16	4.3	odd	2	inner
384.8.d.f.193.12	yes	16	8.3	odd	2	inner
384.8.d.f.193.13	yes	16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	384.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+27.0000i q^{3} +124.096i q^{5} +1165.97 q^{7} -729.000 q^{9} +O(q^{10})\)	\(q+27.0000i q^{3} +124.096i q^{5} +1165.97 q^{7} -729.000 q^{9} +2265.49i q^{11} -3028.99i q^{13} -3350.59 q^{15} -18113.4 q^{17} -33226.6i q^{19} +31481.2i q^{21} +85828.5 q^{23} +62725.2 q^{25} -19683.0i q^{27} -158052. i q^{29} +199774. q^{31} -61168.3 q^{33} +144692. i q^{35} -154407. i q^{37} +81782.8 q^{39} -218071. q^{41} -248492. i q^{43} -90465.9i q^{45} -703977. q^{47} +535941. q^{49} -489061. i q^{51} +1.18948e6i q^{53} -281138. q^{55} +897118. q^{57} -1.97710e6i q^{59} -3.14364e6i q^{61} -849992. q^{63} +375885. q^{65} -1.03112e6i q^{67} +2.31737e6i q^{69} -1.24902e6 q^{71} -2.36170e6 q^{73} +1.69358e6i q^{75} +2.64150e6i q^{77} -1.65799e6 q^{79} +531441. q^{81} -2.63121e6i q^{83} -2.24779e6i q^{85} +4.26740e6 q^{87} +2.71938e6 q^{89} -3.53171e6i q^{91} +5.39390e6i q^{93} +4.12328e6 q^{95} +1.27672e7 q^{97} -1.65154e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 11664 q^{9}+O(q^{10}) \) 16 * q - 11664 * q^9	\( 16 q - 11664 q^{9} + 77280 q^{17} - 236464 q^{25} + 544320 q^{33} - 930912 q^{41} + 1163024 q^{49} - 3032640 q^{57} + 6283008 q^{65} - 2727200 q^{73} + 8503056 q^{81} - 43093152 q^{89} + 35537120 q^{97}+O(q^{100}) \) 16 * q - 11664 * q^9 + 77280 * q^17 - 236464 * q^25 + 544320 * q^33 - 930912 * q^41 + 1163024 * q^49 - 3032640 * q^57 + 6283008 * q^65 - 2727200 * q^73 + 8503056 * q^81 - 43093152 * q^89 + 35537120 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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