LMFDB - Embedded newform 384.10.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,10,Mod(1,384)]

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, 0, 0]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("384.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	384.a (trivial)

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:	yes
Analytic conductor:	$197.773761087$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 14124x^{2} - 170336x + 18391464 $ x^4 - 14124x^2 - 170336x + 18391464
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{15}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$119.418$ of defining polynomial
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-81.0000 q^{3} +1818.41 q^{5} +3817.40 q^{7} +6561.00 q^{9} +O(q^{10})$	$q-81.0000 q^{3} +1818.41 q^{5} +3817.40 q^{7} +6561.00 q^{9} -79487.2 q^{11} +154447. q^{13} -147291. q^{15} +343554. q^{17} -25500.6 q^{19} -309209. q^{21} -2.10072e6 q^{23} +1.35350e6 q^{25} -531441. q^{27} +1.13554e6 q^{29} -5.73826e6 q^{31} +6.43847e6 q^{33} +6.94160e6 q^{35} -9.67864e6 q^{37} -1.25102e7 q^{39} -1.08921e7 q^{41} +108899. q^{43} +1.19306e7 q^{45} +4.24010e7 q^{47} -2.57811e7 q^{49} -2.78279e7 q^{51} -7.58656e7 q^{53} -1.44541e8 q^{55} +2.06555e6 q^{57} -9.39466e7 q^{59} -1.38466e8 q^{61} +2.50459e7 q^{63} +2.80848e8 q^{65} -6.57461e7 q^{67} +1.70159e8 q^{69} -3.75829e7 q^{71} +2.83407e8 q^{73} -1.09633e8 q^{75} -3.03434e8 q^{77} -4.68780e8 q^{79} +4.30467e7 q^{81} +4.19210e8 q^{83} +6.24723e8 q^{85} -9.19785e7 q^{87} +8.32398e8 q^{89} +5.89586e8 q^{91} +4.64799e8 q^{93} -4.63705e7 q^{95} -1.37090e9 q^{97} -5.21516e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 324 q^{3} + 1728 q^{5} - 4840 q^{7} + 26244 q^{9}+O(q^{10}) $ 4 * q - 324 * q^3 + 1728 * q^5 - 4840 * q^7 + 26244 * q^9	$ 4 q - 324 q^{3} + 1728 q^{5} - 4840 q^{7} + 26244 q^{9} - 15824 q^{11} + 82440 q^{13} - 139968 q^{15} + 165912 q^{17} - 539904 q^{19} + 392040 q^{21} - 729680 q^{23} + 224812 q^{25} - 2125764 q^{27} + 1850864 q^{29} - 3197960 q^{31} + 1281744 q^{33} - 8574912 q^{35} + 4187992 q^{37} - 6677640 q^{39} + 227704 q^{41} - 1600352 q^{43} + 11337408 q^{45} - 18053904 q^{47} + 57728820 q^{49} - 13438872 q^{51} + 29418288 q^{53} - 45906816 q^{55} + 43732224 q^{57} - 38300048 q^{59} - 99764648 q^{61} - 31755240 q^{63} + 314120832 q^{65} + 183717008 q^{67} + 59104080 q^{69} - 181868080 q^{71} + 254539160 q^{73} - 18209772 q^{75} - 230564704 q^{77} - 831578184 q^{79} + 172186884 q^{81} + 687923952 q^{83} + 1041391104 q^{85} - 149919984 q^{87} + 627627272 q^{89} + 1018147632 q^{91} + 259034760 q^{93} - 2167118208 q^{95} - 889385880 q^{97} - 103821264 q^{99}+O(q^{100}) $ 4 * q - 324 * q^3 + 1728 * q^5 - 4840 * q^7 + 26244 * q^9 - 15824 * q^11 + 82440 * q^13 - 139968 * q^15 + 165912 * q^17 - 539904 * q^19 + 392040 * q^21 - 729680 * q^23 + 224812 * q^25 - 2125764 * q^27 + 1850864 * q^29 - 3197960 * q^31 + 1281744 * q^33 - 8574912 * q^35 + 4187992 * q^37 - 6677640 * q^39 + 227704 * q^41 - 1600352 * q^43 + 11337408 * q^45 - 18053904 * q^47 + 57728820 * q^49 - 13438872 * q^51 + 29418288 * q^53 - 45906816 * q^55 + 43732224 * q^57 - 38300048 * q^59 - 99764648 * q^61 - 31755240 * q^63 + 314120832 * q^65 + 183717008 * q^67 + 59104080 * q^69 - 181868080 * q^71 + 254539160 * q^73 - 18209772 * q^75 - 230564704 * q^77 - 831578184 * q^79 + 172186884 * q^81 + 687923952 * q^83 + 1041391104 * q^85 - 149919984 * q^87 + 627627272 * q^89 + 1018147632 * q^91 + 259034760 * q^93 - 2167118208 * q^95 - 889385880 * q^97 - 103821264 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−81.0000	−0.577350
$3$	−81.0000	−0.577350
$4$	0	0
$5$	1818.41	1.30115	0.650575	−	0.759442i	$-0.274528\pi$
$5$	1818.41	1.30115	0.650575	+	0.759442i	$0.274528\pi$
$6$	0	0
$7$	3817.40	0.600933	0.300467	−	0.953792i	$-0.402858\pi$
$7$	3817.40	0.600933	0.300467	+	0.953792i	$0.402858\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	−79487.2	−1.63693	−0.818465	−	0.574556i	$-0.805175\pi$
$11$	−79487.2	−1.63693	−0.818465	+	0.574556i	$0.805175\pi$
$12$	0	0
$13$	154447.	1.49980	0.749902	−	0.661549i	$-0.230101\pi$
$13$	154447.	1.49980	0.749902	+	0.661549i	$0.230101\pi$
$14$	0	0
$15$	−147291.	−0.751219
$16$	0	0
$17$	343554.	0.997643	0.498822	−	0.866705i	$-0.333766\pi$
$17$	343554.	0.997643	0.498822	+	0.866705i	$0.333766\pi$
$18$	0	0
$19$	−25500.6	−0.0448909	−0.0224455	−	0.999748i	$-0.507145\pi$
$19$	−25500.6	−0.0448909	−0.0224455	+	0.999748i	$0.507145\pi$
$20$	0	0
$21$	−309209.	−0.346949
$22$	0	0
$23$	−2.10072e6	−1.56529	−0.782643	−	0.622470i	$-0.786129\pi$
$23$	−2.10072e6	−1.56529	−0.782643	+	0.622470i	$0.786129\pi$
$24$	0	0
$25$	1.35350e6	0.692990
$26$	0	0
$27$	−531441.	−0.192450
$28$	0	0
$29$	1.13554e6	0.298133	0.149067	−	0.988827i	$-0.452373\pi$
$29$	1.13554e6	0.298133	0.149067	+	0.988827i	$0.452373\pi$
$30$	0	0
$31$	−5.73826e6	−1.11597	−0.557985	−	0.829851i	$-0.688426\pi$
$31$	−5.73826e6	−1.11597	−0.557985	+	0.829851i	$0.688426\pi$
$32$	0	0
$33$	6.43847e6	0.945082
$34$	0	0
$35$	6.94160e6	0.781904
$36$	0	0
$37$	−9.67864e6	−0.848998	−0.424499	−	0.905429i	$-0.639550\pi$
$37$	−9.67864e6	−0.848998	−0.424499	+	0.905429i	$0.639550\pi$
$38$	0	0
$39$	−1.25102e7	−0.865912
$40$	0	0
$41$	−1.08921e7	−0.601982	−0.300991	−	0.953627i	$-0.597317\pi$
$41$	−1.08921e7	−0.601982	−0.300991	+	0.953627i	$0.597317\pi$
$42$	0	0
$43$	108899.	0.00485754	0.00242877	−	0.999997i	$-0.499227\pi$
$43$	108899.	0.00485754	0.00242877	+	0.999997i	$0.499227\pi$
$44$	0	0
$45$	1.19306e7	0.433717
$46$	0	0
$47$	4.24010e7	1.26747	0.633733	−	0.773552i	$-0.281522\pi$
$47$	4.24010e7	1.26747	0.633733	+	0.773552i	$0.281522\pi$
$48$	0	0
$49$	−2.57811e7	−0.638879
$50$	0	0
$51$	−2.78279e7	−0.575990
$52$	0	0
$53$	−7.58656e7	−1.32070	−0.660349	−	0.750959i	$-0.729592\pi$
$53$	−7.58656e7	−1.32070	−0.660349	+	0.750959i	$0.729592\pi$
$54$	0	0
$55$	−1.44541e8	−2.12989
$56$	0	0
$57$	2.06555e6	0.0259178
$58$	0	0
$59$	−9.39466e7	−1.00936	−0.504681	−	0.863306i	$-0.668390\pi$
$59$	−9.39466e7	−1.00936	−0.504681	+	0.863306i	$0.668390\pi$
$60$	0	0
$61$	−1.38466e8	−1.28044	−0.640221	−	0.768190i	$-0.721157\pi$
$61$	−1.38466e8	−1.28044	−0.640221	+	0.768190i	$0.721157\pi$
$62$	0	0
$63$	2.50459e7	0.200311
$64$	0	0
$65$	2.80848e8	1.95147
$66$	0	0
$67$	−6.57461e7	−0.398597	−0.199298	−	0.979939i	$-0.563866\pi$
$67$	−6.57461e7	−0.398597	−0.199298	+	0.979939i	$0.563866\pi$
$68$	0	0
$69$	1.70159e8	0.903719
$70$	0	0
$71$	−3.75829e7	−0.175520	−0.0877602	−	0.996142i	$-0.527971\pi$
$71$	−3.75829e7	−0.175520	−0.0877602	+	0.996142i	$0.527971\pi$
$72$	0	0
$73$	2.83407e8	1.16804	0.584019	−	0.811740i	$-0.301479\pi$
$73$	2.83407e8	1.16804	0.584019	+	0.811740i	$0.301479\pi$
$74$	0	0
$75$	−1.09633e8	−0.400098
$76$	0	0
$77$	−3.03434e8	−0.983686
$78$	0	0
$79$	−4.68780e8	−1.35409	−0.677045	−	0.735942i	$-0.736740\pi$
$79$	−4.68780e8	−1.35409	−0.677045	+	0.735942i	$0.736740\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	4.19210e8	0.969573	0.484786	−	0.874633i	$-0.338897\pi$
$83$	4.19210e8	0.969573	0.484786	+	0.874633i	$0.338897\pi$
$84$	0	0
$85$	6.24723e8	1.29808
$86$	0	0
$87$	−9.19785e7	−0.172127
$88$	0	0
$89$	8.32398e8	1.40629	0.703147	−	0.711045i	$-0.251778\pi$
$89$	8.32398e8	1.40629	0.703147	+	0.711045i	$0.251778\pi$
$90$	0	0
$91$	5.89586e8	0.901282
$92$	0	0
$93$	4.64799e8	0.644306
$94$	0	0
$95$	−4.63705e7	−0.0584098
$96$	0	0
$97$	−1.37090e9	−1.57229	−0.786143	−	0.618044i	$-0.787925\pi$
$97$	−1.37090e9	−1.57229	−0.786143	+	0.618044i	$0.787925\pi$
$98$	0	0
$99$	−5.21516e8	−0.545643
$100$	0	0
$101$	1.29635e9	1.23958	0.619791	−	0.784767i	$-0.287218\pi$
$101$	1.29635e9	1.23958	0.619791	+	0.784767i	$0.287218\pi$
$102$	0	0
$103$	2.02804e8	0.177545	0.0887727	−	0.996052i	$-0.471706\pi$
$103$	2.02804e8	0.177545	0.0887727	+	0.996052i	$0.471706\pi$
$104$	0	0
$105$	−5.62270e8	−0.451433
$106$	0	0
$107$	2.25915e9	1.66616	0.833082	−	0.553150i	$-0.186574\pi$
$107$	2.25915e9	1.66616	0.833082	+	0.553150i	$0.186574\pi$
$108$	0	0
$109$	2.66475e8	0.180816	0.0904082	−	0.995905i	$-0.471183\pi$
$109$	2.66475e8	0.180816	0.0904082	+	0.995905i	$0.471183\pi$
$110$	0	0
$111$	7.83970e8	0.490169
$112$	0	0
$113$	−2.42429e9	−1.39872	−0.699360	−	0.714770i	$-0.746532\pi$
$113$	−2.42429e9	−1.39872	−0.699360	+	0.714770i	$0.746532\pi$
$114$	0	0
$115$	−3.81998e9	−2.03667
$116$	0	0
$117$	1.01333e9	0.499935
$118$	0	0
$119$	1.31148e9	0.599517
$120$	0	0
$121$	3.96027e9	1.67954
$122$	0	0
$123$	8.82259e8	0.347555
$124$	0	0
$125$	−1.09037e9	−0.399465
$126$	0	0
$127$	9.12647e8	0.311305	0.155652	−	0.987812i	$-0.450252\pi$
$127$	9.12647e8	0.311305	0.155652	+	0.987812i	$0.450252\pi$
$128$	0	0
$129$	−8.82084e6	−0.00280450
$130$	0	0
$131$	5.96692e9	1.77023	0.885114	−	0.465375i	$-0.154080\pi$
$131$	5.96692e9	1.77023	0.885114	+	0.465375i	$0.154080\pi$
$132$	0	0
$133$	−9.73458e7	−0.0269764
$134$	0	0
$135$	−9.66379e8	−0.250406
$136$	0	0
$137$	−7.06316e9	−1.71300	−0.856498	−	0.516150i	$-0.827365\pi$
$137$	−7.06316e9	−1.71300	−0.856498	+	0.516150i	$0.827365\pi$
$138$	0	0
$139$	2.13879e9	0.485961	0.242980	−	0.970031i	$-0.421875\pi$
$139$	2.13879e9	0.485961	0.242980	+	0.970031i	$0.421875\pi$
$140$	0	0
$141$	−3.43448e9	−0.731772
$142$	0	0
$143$	−1.22766e10	−2.45508
$144$	0	0
$145$	2.06487e9	0.387916
$146$	0	0
$147$	2.08827e9	0.368857
$148$	0	0
$149$	1.97318e9	0.327965	0.163983	−	0.986463i	$-0.447566\pi$
$149$	1.97318e9	0.327965	0.163983	+	0.986463i	$0.447566\pi$
$150$	0	0
$151$	2.03152e8	0.0317998	0.0158999	−	0.999874i	$-0.494939\pi$
$151$	2.03152e8	0.0317998	0.0158999	+	0.999874i	$0.494939\pi$
$152$	0	0
$153$	2.25406e9	0.332548
$154$	0	0
$155$	−1.04345e10	−1.45205
$156$	0	0
$157$	1.37720e10	1.80904	0.904520	−	0.426432i	$-0.140230\pi$
$157$	1.37720e10	1.80904	0.904520	+	0.426432i	$0.140230\pi$
$158$	0	0
$159$	6.14511e9	0.762505
$160$	0	0
$161$	−8.01930e9	−0.940633
$162$	0	0
$163$	−3.01690e9	−0.334747	−0.167374	−	0.985894i	$-0.553529\pi$
$163$	−3.01690e9	−0.334747	−0.167374	+	0.985894i	$0.553529\pi$
$164$	0	0
$165$	1.17078e10	1.22969
$166$	0	0
$167$	−4.92894e9	−0.490376	−0.245188	−	0.969476i	$-0.578850\pi$
$167$	−4.92894e9	−0.490376	−0.245188	+	0.969476i	$0.578850\pi$
$168$	0	0
$169$	1.32494e10	1.24941
$170$	0	0
$171$	−1.67309e8	−0.0149636
$172$	0	0
$173$	−1.34775e10	−1.14393	−0.571967	−	0.820277i	$-0.693819\pi$
$173$	−1.34775e10	−1.14393	−0.571967	+	0.820277i	$0.693819\pi$
$174$	0	0
$175$	5.16684e9	0.416441
$176$	0	0
$177$	7.60968e9	0.582756
$178$	0	0
$179$	−1.28594e10	−0.936231	−0.468116	−	0.883667i	$-0.655067\pi$
$179$	−1.28594e10	−0.936231	−0.468116	+	0.883667i	$0.655067\pi$
$180$	0	0
$181$	−1.46209e10	−1.01256	−0.506281	−	0.862368i	$-0.668980\pi$
$181$	−1.46209e10	−1.01256	−0.506281	+	0.862368i	$0.668980\pi$
$182$	0	0
$183$	1.12158e10	0.739264
$184$	0	0
$185$	−1.75997e10	−1.10467
$186$	0	0
$187$	−2.73082e10	−1.63307
$188$	0	0
$189$	−2.02872e9	−0.115650
$190$	0	0
$191$	−2.49838e10	−1.35834	−0.679170	−	0.733981i	$-0.737660\pi$
$191$	−2.49838e10	−1.35834	−0.679170	+	0.733981i	$0.737660\pi$
$192$	0	0
$193$	−2.45447e10	−1.27336	−0.636678	−	0.771129i	$-0.719692\pi$
$193$	−2.45447e10	−1.27336	−0.636678	+	0.771129i	$0.719692\pi$
$194$	0	0
$195$	−2.27487e10	−1.12668
$196$	0	0
$197$	−1.14198e10	−0.540209	−0.270104	−	0.962831i	$-0.587058\pi$
$197$	−1.14198e10	−0.540209	−0.270104	+	0.962831i	$0.587058\pi$
$198$	0	0
$199$	−1.08567e10	−0.490747	−0.245373	−	0.969429i	$-0.578911\pi$
$199$	−1.08567e10	−0.490747	−0.245373	+	0.969429i	$0.578911\pi$
$200$	0	0
$201$	5.32544e9	0.230130
$202$	0	0
$203$	4.33480e9	0.179158
$204$	0	0
$205$	−1.98063e10	−0.783269
$206$	0	0
$207$	−1.37829e10	−0.521762
$208$	0	0
$209$	2.02697e9	0.0734833
$210$	0	0
$211$	−4.64028e10	−1.61166	−0.805830	−	0.592147i	$-0.798281\pi$
$211$	−4.64028e10	−1.61166	−0.805830	+	0.592147i	$0.798281\pi$
$212$	0	0
$213$	3.04422e9	0.101337
$214$	0	0
$215$	1.98024e8	0.00632039
$216$	0	0
$217$	−2.19052e10	−0.670624
$218$	0	0
$219$	−2.29559e10	−0.674367
$220$	0	0
$221$	5.30610e10	1.49627
$222$	0	0
$223$	−2.71348e10	−0.734775	−0.367388	−	0.930068i	$-0.619748\pi$
$223$	−2.71348e10	−0.734775	−0.367388	+	0.930068i	$0.619748\pi$
$224$	0	0
$225$	8.88029e9	0.230997
$226$	0	0
$227$	7.40226e9	0.185032	0.0925162	−	0.995711i	$-0.470509\pi$
$227$	7.40226e9	0.185032	0.0925162	+	0.995711i	$0.470509\pi$
$228$	0	0
$229$	4.21764e10	1.01347	0.506733	−	0.862103i	$-0.330853\pi$
$229$	4.21764e10	1.01347	0.506733	+	0.862103i	$0.330853\pi$
$230$	0	0
$231$	2.45782e10	0.567931
$232$	0	0
$233$	−7.28793e10	−1.61995	−0.809977	−	0.586462i	$-0.800520\pi$
$233$	−7.28793e10	−1.61995	−0.809977	+	0.586462i	$0.800520\pi$
$234$	0	0
$235$	7.71026e10	1.64916
$236$	0	0
$237$	3.79712e10	0.781784
$238$	0	0
$239$	−1.38185e10	−0.273950	−0.136975	−	0.990574i	$-0.543738\pi$
$239$	−1.38185e10	−0.273950	−0.136975	+	0.990574i	$0.543738\pi$
$240$	0	0
$241$	−4.86065e10	−0.928148	−0.464074	−	0.885796i	$-0.653613\pi$
$241$	−4.86065e10	−0.928148	−0.464074	+	0.885796i	$0.653613\pi$
$242$	0	0
$243$	−3.48678e9	−0.0641500
$244$	0	0
$245$	−4.68806e10	−0.831277
$246$	0	0
$247$	−3.93849e9	−0.0673276
$248$	0	0
$249$	−3.39560e10	−0.559783
$250$	0	0
$251$	9.95300e10	1.58279	0.791393	−	0.611308i	$-0.209356\pi$
$251$	9.95300e10	1.58279	0.791393	+	0.611308i	$0.209356\pi$
$252$	0	0
$253$	1.66981e11	2.56227
$254$	0	0
$255$	−5.06026e10	−0.749449
$256$	0	0
$257$	4.82008e10	0.689216	0.344608	−	0.938747i	$-0.388012\pi$
$257$	4.82008e10	0.689216	0.344608	+	0.938747i	$0.388012\pi$
$258$	0	0
$259$	−3.69472e10	−0.510191
$260$	0	0
$261$	7.45026e9	0.0993777
$262$	0	0
$263$	−3.11820e10	−0.401886	−0.200943	−	0.979603i	$-0.564401\pi$
$263$	−3.11820e10	−0.401886	−0.200943	+	0.979603i	$0.564401\pi$
$264$	0	0
$265$	−1.37955e11	−1.71843
$266$	0	0
$267$	−6.74243e10	−0.811924
$268$	0	0
$269$	4.79418e10	0.558251	0.279125	−	0.960255i	$-0.409955\pi$
$269$	4.79418e10	0.558251	0.279125	+	0.960255i	$0.409955\pi$
$270$	0	0
$271$	6.20036e10	0.698321	0.349160	−	0.937063i	$-0.386467\pi$
$271$	6.20036e10	0.698321	0.349160	+	0.937063i	$0.386467\pi$
$272$	0	0
$273$	−4.77565e10	−0.520356
$274$	0	0
$275$	−1.07586e11	−1.13438
$276$	0	0
$277$	7.51491e10	0.766946	0.383473	−	0.923552i	$-0.374728\pi$
$277$	7.51491e10	0.766946	0.383473	+	0.923552i	$0.374728\pi$
$278$	0	0
$279$	−3.76487e10	−0.371990
$280$	0	0
$281$	7.27701e9	0.0696265	0.0348133	−	0.999394i	$-0.488916\pi$
$281$	7.27701e9	0.0696265	0.0348133	+	0.999394i	$0.488916\pi$
$282$	0	0
$283$	−1.07526e11	−0.996492	−0.498246	−	0.867036i	$-0.666022\pi$
$283$	−1.07526e11	−0.996492	−0.498246	+	0.867036i	$0.666022\pi$
$284$	0	0
$285$	3.75601e9	0.0337229
$286$	0	0
$287$	−4.15794e10	−0.361751
$288$	0	0
$289$	−5.58306e8	−0.00470795
$290$	0	0
$291$	1.11043e11	0.907760
$292$	0	0
$293$	−2.31989e10	−0.183892	−0.0919462	−	0.995764i	$-0.529309\pi$
$293$	−2.31989e10	−0.183892	−0.0919462	+	0.995764i	$0.529309\pi$
$294$	0	0
$295$	−1.70834e11	−1.31333
$296$	0	0
$297$	4.22428e10	0.315027
$298$	0	0
$299$	−3.24451e11	−2.34762
$300$	0	0
$301$	4.15712e8	0.00291906
$302$	0	0
$303$	−1.05004e11	−0.715672
$304$	0	0
$305$	−2.51789e11	−1.66605
$306$	0	0
$307$	−1.84503e11	−1.18544	−0.592721	−	0.805408i	$-0.701946\pi$
$307$	−1.84503e11	−1.18544	−0.592721	+	0.805408i	$0.701946\pi$
$308$	0	0
$309$	−1.64271e10	−0.102506
$310$	0	0
$311$	−1.83522e11	−1.11241	−0.556205	−	0.831045i	$-0.687743\pi$
$311$	−1.83522e11	−1.11241	−0.556205	+	0.831045i	$0.687743\pi$
$312$	0	0
$313$	1.67363e11	0.985620	0.492810	−	0.870137i	$-0.335970\pi$
$313$	1.67363e11	0.985620	0.492810	+	0.870137i	$0.335970\pi$
$314$	0	0
$315$	4.55439e10	0.260635
$316$	0	0
$317$	3.07406e11	1.70980	0.854899	−	0.518794i	$-0.173619\pi$
$317$	3.07406e11	1.70980	0.854899	+	0.518794i	$0.173619\pi$
$318$	0	0
$319$	−9.02607e10	−0.488023
$320$	0	0
$321$	−1.82991e11	−0.961960
$322$	0	0
$323$	−8.76083e9	−0.0447851
$324$	0	0
$325$	2.09044e11	1.03935
$326$	0	0
$327$	−2.15845e10	−0.104394
$328$	0	0
$329$	1.61862e11	0.761662
$330$	0	0
$331$	2.22502e10	0.101884	0.0509421	−	0.998702i	$-0.483778\pi$
$331$	2.22502e10	0.101884	0.0509421	+	0.998702i	$0.483778\pi$
$332$	0	0
$333$	−6.35015e10	−0.282999
$334$	0	0
$335$	−1.19554e11	−0.518634
$336$	0	0
$337$	−6.04892e10	−0.255472	−0.127736	−	0.991808i	$-0.540771\pi$
$337$	−6.04892e10	−0.255472	−0.127736	+	0.991808i	$0.540771\pi$
$338$	0	0
$339$	1.96367e11	0.807551
$340$	0	0
$341$	4.56119e11	1.82677
$342$	0	0
$343$	−2.52462e11	−0.984857
$344$	0	0
$345$	3.09419e11	1.17587
$346$	0	0
$347$	−1.97594e11	−0.731628	−0.365814	−	0.930688i	$-0.619209\pi$
$347$	−1.97594e11	−0.731628	−0.365814	+	0.930688i	$0.619209\pi$
$348$	0	0
$349$	−4.32441e11	−1.56032	−0.780159	−	0.625582i	$-0.784862\pi$
$349$	−4.32441e11	−1.56032	−0.780159	+	0.625582i	$0.784862\pi$
$350$	0	0
$351$	−8.20795e10	−0.288637
$352$	0	0
$353$	3.49455e10	0.119786	0.0598928	−	0.998205i	$-0.480924\pi$
$353$	3.49455e10	0.119786	0.0598928	+	0.998205i	$0.480924\pi$
$354$	0	0
$355$	−6.83412e10	−0.228378
$356$	0	0
$357$	−1.06230e11	−0.346131
$358$	0	0
$359$	2.44648e10	0.0777351	0.0388676	−	0.999244i	$-0.487625\pi$
$359$	2.44648e10	0.0777351	0.0388676	+	0.999244i	$0.487625\pi$
$360$	0	0
$361$	−3.22037e11	−0.997985
$362$	0	0
$363$	−3.20782e11	−0.969684
$364$	0	0
$365$	5.15350e11	1.51979
$366$	0	0
$367$	−1.54134e11	−0.443508	−0.221754	−	0.975103i	$-0.571178\pi$
$367$	−1.54134e11	−0.443508	−0.221754	+	0.975103i	$0.571178\pi$
$368$	0	0
$369$	−7.14630e10	−0.200661
$370$	0	0
$371$	−2.89609e11	−0.793651
$372$	0	0
$373$	−2.83594e11	−0.758590	−0.379295	−	0.925276i	$-0.623833\pi$
$373$	−2.83594e11	−0.758590	−0.379295	+	0.925276i	$0.623833\pi$
$374$	0	0
$375$	8.83200e10	0.230631
$376$	0	0
$377$	1.75380e11	0.447141
$378$	0	0
$379$	4.98473e11	1.24098	0.620490	−	0.784214i	$-0.286934\pi$
$379$	4.98473e11	1.24098	0.620490	+	0.784214i	$0.286934\pi$
$380$	0	0
$381$	−7.39244e10	−0.179732
$382$	0	0
$383$	−2.80570e11	−0.666264	−0.333132	−	0.942880i	$-0.608105\pi$
$383$	−2.80570e11	−0.666264	−0.333132	+	0.942880i	$0.608105\pi$
$384$	0	0
$385$	−5.51769e11	−1.27992
$386$	0	0
$387$	7.14488e8	0.00161918
$388$	0	0
$389$	3.48132e10	0.0770853	0.0385426	−	0.999257i	$-0.487728\pi$
$389$	3.48132e10	0.0770853	0.0385426	+	0.999257i	$0.487728\pi$
$390$	0	0
$391$	−7.21713e11	−1.56160
$392$	0	0
$393$	−4.83320e11	−1.02204
$394$	0	0
$395$	−8.52436e11	−1.76187
$396$	0	0
$397$	−2.60260e10	−0.0525835	−0.0262917	−	0.999654i	$-0.508370\pi$
$397$	−2.60260e10	−0.0525835	−0.0262917	+	0.999654i	$0.508370\pi$
$398$	0	0
$399$	7.88501e9	0.0155749
$400$	0	0
$401$	−7.66429e11	−1.48021	−0.740103	−	0.672494i	$-0.765223\pi$
$401$	−7.66429e11	−1.48021	−0.740103	+	0.672494i	$0.765223\pi$
$402$	0	0
$403$	−8.86258e11	−1.67374
$404$	0	0
$405$	7.82767e10	0.144572
$406$	0	0
$407$	7.69328e11	1.38975
$408$	0	0
$409$	−3.76343e11	−0.665011	−0.332506	−	0.943101i	$-0.607894\pi$
$409$	−3.76343e11	−0.665011	−0.332506	+	0.943101i	$0.607894\pi$
$410$	0	0
$411$	5.72116e11	0.988999
$412$	0	0
$413$	−3.58632e11	−0.606559
$414$	0	0
$415$	7.62297e11	1.26156
$416$	0	0
$417$	−1.73242e11	−0.280569
$418$	0	0
$419$	9.24941e11	1.46606	0.733028	−	0.680198i	$-0.238106\pi$
$419$	9.24941e11	1.46606	0.733028	+	0.680198i	$0.238106\pi$
$420$	0	0
$421$	−5.98004e11	−0.927758	−0.463879	−	0.885899i	$-0.653543\pi$
$421$	−5.98004e11	−0.927758	−0.463879	+	0.885899i	$0.653543\pi$
$422$	0	0
$423$	2.78193e11	0.422489
$424$	0	0
$425$	4.65000e11	0.691357
$426$	0	0
$427$	−5.28581e11	−0.769461
$428$	0	0
$429$	9.94402e11	1.41744
$430$	0	0
$431$	−3.60269e10	−0.0502897	−0.0251448	−	0.999684i	$-0.508005\pi$
$431$	−3.60269e10	−0.0502897	−0.0251448	+	0.999684i	$0.508005\pi$
$432$	0	0
$433$	−8.52251e11	−1.16512	−0.582562	−	0.812786i	$-0.697950\pi$
$433$	−8.52251e11	−1.16512	−0.582562	+	0.812786i	$0.697950\pi$
$434$	0	0
$435$	−1.67255e11	−0.223963
$436$	0	0
$437$	5.35697e10	0.0702672
$438$	0	0
$439$	1.94785e11	0.250302	0.125151	−	0.992138i	$-0.460058\pi$
$439$	1.94785e11	0.250302	0.125151	+	0.992138i	$0.460058\pi$
$440$	0	0
$441$	−1.69150e11	−0.212960
$442$	0	0
$443$	−2.03960e11	−0.251611	−0.125805	−	0.992055i	$-0.540151\pi$
$443$	−2.03960e11	−0.251611	−0.125805	+	0.992055i	$0.540151\pi$
$444$	0	0
$445$	1.51364e12	1.82980
$446$	0	0
$447$	−1.59827e11	−0.189351
$448$	0	0
$449$	3.72352e11	0.432360	0.216180	−	0.976354i	$-0.430640\pi$
$449$	3.72352e11	0.432360	0.216180	+	0.976354i	$0.430640\pi$
$450$	0	0
$451$	8.65782e11	0.985403
$452$	0	0
$453$	−1.64553e10	−0.0183596
$454$	0	0
$455$	1.07211e12	1.17270
$456$	0	0
$457$	1.43400e12	1.53789	0.768947	−	0.639313i	$-0.220781\pi$
$457$	1.43400e12	1.53789	0.768947	+	0.639313i	$0.220781\pi$
$458$	0	0
$459$	−1.82579e11	−0.191997
$460$	0	0
$461$	1.21208e12	1.24991	0.624953	−	0.780662i	$-0.285118\pi$
$461$	1.21208e12	1.24991	0.624953	+	0.780662i	$0.285118\pi$
$462$	0	0
$463$	1.68275e12	1.70179	0.850895	−	0.525336i	$-0.176061\pi$
$463$	1.68275e12	1.70179	0.850895	+	0.525336i	$0.176061\pi$
$464$	0	0
$465$	8.45197e11	0.838339
$466$	0	0
$467$	9.42499e11	0.916969	0.458485	−	0.888702i	$-0.348392\pi$
$467$	9.42499e11	0.916969	0.458485	+	0.888702i	$0.348392\pi$
$468$	0	0
$469$	−2.50979e11	−0.239530
$470$	0	0
$471$	−1.11553e12	−1.04445
$472$	0	0
$473$	−8.65610e9	−0.00795146
$474$	0	0
$475$	−3.45149e10	−0.0311090
$476$	0	0
$477$	−4.97754e11	−0.440233
$478$	0	0
$479$	9.15579e11	0.794669	0.397334	−	0.917674i	$-0.369935\pi$
$479$	9.15579e11	0.794669	0.397334	+	0.917674i	$0.369935\pi$
$480$	0	0
$481$	−1.49484e12	−1.27333
$482$	0	0
$483$	6.49564e11	0.543075
$484$	0	0
$485$	−2.49285e12	−2.04578
$486$	0	0
$487$	−1.70542e12	−1.37389	−0.686943	−	0.726712i	$-0.741048\pi$
$487$	−1.70542e12	−1.37389	−0.686943	+	0.726712i	$0.741048\pi$
$488$	0	0
$489$	2.44369e11	0.193266
$490$	0	0
$491$	1.95142e12	1.51525	0.757625	−	0.652690i	$-0.226360\pi$
$491$	1.95142e12	1.51525	0.757625	+	0.652690i	$0.226360\pi$
$492$	0	0
$493$	3.90119e11	0.297431
$494$	0	0
$495$	−9.48330e11	−0.709964
$496$	0	0
$497$	−1.43469e11	−0.105476
$498$	0	0
$499$	−2.72335e12	−1.96630	−0.983152	−	0.182790i	$-0.941487\pi$
$499$	−2.72335e12	−1.96630	−0.983152	+	0.182790i	$0.941487\pi$
$500$	0	0
$501$	3.99244e11	0.283119
$502$	0	0
$503$	−2.70028e12	−1.88084	−0.940422	−	0.340011i	$-0.889569\pi$
$503$	−2.70028e12	−1.88084	−0.940422	+	0.340011i	$0.889569\pi$
$504$	0	0
$505$	2.35729e12	1.61288
$506$	0	0
$507$	−1.07320e12	−0.721349
$508$	0	0
$509$	−2.15879e12	−1.42554	−0.712770	−	0.701398i	$-0.752560\pi$
$509$	−2.15879e12	−1.42554	−0.712770	+	0.701398i	$0.752560\pi$
$510$	0	0
$511$	1.08188e12	0.701913
$512$	0	0
$513$	1.35520e10	0.00863926
$514$	0	0
$515$	3.68782e11	0.231013
$516$	0	0
$517$	−3.37034e12	−2.07475
$518$	0	0
$519$	1.09167e12	0.660450
$520$	0	0
$521$	6.59041e11	0.391871	0.195935	−	0.980617i	$-0.437226\pi$
$521$	6.59041e11	0.391871	0.195935	+	0.980617i	$0.437226\pi$
$522$	0	0
$523$	−7.33748e11	−0.428834	−0.214417	−	0.976742i	$-0.568785\pi$
$523$	−7.33748e11	−0.428834	−0.214417	+	0.976742i	$0.568785\pi$
$524$	0	0
$525$	−4.18514e11	−0.240432
$526$	0	0
$527$	−1.97141e12	−1.11334
$528$	0	0
$529$	2.61189e12	1.45012
$530$	0	0
$531$	−6.16384e11	−0.336454
$532$	0	0
$533$	−1.68225e12	−0.902856
$534$	0	0
$535$	4.10806e12	2.16793
$536$	0	0
$537$	1.04161e12	0.540533
$538$	0	0
$539$	2.04927e12	1.04580
$540$	0	0
$541$	2.56238e12	1.28604	0.643022	−	0.765848i	$-0.277680\pi$
$541$	2.56238e12	1.28604	0.643022	+	0.765848i	$0.277680\pi$
$542$	0	0
$543$	1.18430e12	0.584603
$544$	0	0
$545$	4.84562e11	0.235269
$546$	0	0
$547$	2.42235e12	1.15689	0.578447	−	0.815720i	$-0.303659\pi$
$547$	2.42235e12	1.15689	0.578447	+	0.815720i	$0.303659\pi$
$548$	0	0
$549$	−9.08478e11	−0.426814
$550$	0	0
$551$	−2.89568e10	−0.0133835
$552$	0	0
$553$	−1.78952e12	−0.813718
$554$	0	0
$555$	1.42558e12	0.637783
$556$	0	0
$557$	−2.99530e11	−0.131854	−0.0659269	−	0.997824i	$-0.521000\pi$
$557$	−2.99530e11	−0.131854	−0.0659269	+	0.997824i	$0.521000\pi$
$558$	0	0
$559$	1.68192e10	0.00728537
$560$	0	0
$561$	2.21196e12	0.942855
$562$	0	0
$563$	−2.58923e12	−1.08613	−0.543066	−	0.839690i	$-0.682737\pi$
$563$	−2.58923e12	−1.08613	−0.543066	+	0.839690i	$0.682737\pi$
$564$	0	0
$565$	−4.40835e12	−1.81994
$566$	0	0
$567$	1.64326e11	0.0667704
$568$	0	0
$569$	3.09519e12	1.23789	0.618945	−	0.785434i	$-0.287560\pi$
$569$	3.09519e12	1.23789	0.618945	+	0.785434i	$0.287560\pi$
$570$	0	0
$571$	1.97701e12	0.778298	0.389149	−	0.921175i	$-0.372769\pi$
$571$	1.97701e12	0.778298	0.389149	+	0.921175i	$0.372769\pi$
$572$	0	0
$573$	2.02369e12	0.784238
$574$	0	0
$575$	−2.84332e12	−1.08473
$576$	0	0
$577$	−4.50291e11	−0.169123	−0.0845614	−	0.996418i	$-0.526949\pi$
$577$	−4.50291e11	−0.169123	−0.0845614	+	0.996418i	$0.526949\pi$
$578$	0	0
$579$	1.98812e12	0.735173
$580$	0	0
$581$	1.60029e12	0.582648
$582$	0	0
$583$	6.03035e12	2.16189
$584$	0	0
$585$	1.84265e12	0.650490
$586$	0	0
$587$	3.02724e12	1.05239	0.526193	−	0.850365i	$-0.323619\pi$
$587$	3.02724e12	1.05239	0.526193	+	0.850365i	$0.323619\pi$
$588$	0	0
$589$	1.46329e11	0.0500970
$590$	0	0
$591$	9.25007e11	0.311890
$592$	0	0
$593$	−4.55472e12	−1.51257	−0.756284	−	0.654243i	$-0.772987\pi$
$593$	−4.55472e12	−1.51257	−0.756284	+	0.654243i	$0.772987\pi$
$594$	0	0
$595$	2.38482e12	0.780061
$596$	0	0
$597$	8.79390e11	0.283333
$598$	0	0
$599$	−2.02438e12	−0.642497	−0.321249	−	0.946995i	$-0.604103\pi$
$599$	−2.02438e12	−0.642497	−0.321249	+	0.946995i	$0.604103\pi$
$600$	0	0
$601$	5.89526e11	0.184318	0.0921591	−	0.995744i	$-0.470623\pi$
$601$	5.89526e11	0.184318	0.0921591	+	0.995744i	$0.470623\pi$
$602$	0	0
$603$	−4.31360e11	−0.132866
$604$	0	0
$605$	7.20140e12	2.18533
$606$	0	0
$607$	−5.99382e12	−1.79207	−0.896035	−	0.443984i	$-0.853565\pi$
$607$	−5.99382e12	−1.79207	−0.896035	+	0.443984i	$0.853565\pi$
$608$	0	0
$609$	−3.51119e11	−0.103437
$610$	0	0
$611$	6.54872e12	1.90095
$612$	0	0
$613$	−3.98091e12	−1.13870	−0.569351	−	0.822094i	$-0.692806\pi$
$613$	−3.98091e12	−1.13870	−0.569351	+	0.822094i	$0.692806\pi$
$614$	0	0
$615$	1.60431e12	0.452221
$616$	0	0
$617$	6.04076e12	1.67806	0.839032	−	0.544082i	$-0.183122\pi$
$617$	6.04076e12	1.67806	0.839032	+	0.544082i	$0.183122\pi$
$618$	0	0
$619$	1.19083e12	0.326019	0.163010	−	0.986624i	$-0.447880\pi$
$619$	1.19083e12	0.326019	0.163010	+	0.986624i	$0.447880\pi$
$620$	0	0
$621$	1.11641e12	0.301240
$622$	0	0
$623$	3.17760e12	0.845089
$624$	0	0
$625$	−4.62629e12	−1.21275
$626$	0	0
$627$	−1.64184e11	−0.0424256
$628$	0	0
$629$	−3.32514e12	−0.846997
$630$	0	0
$631$	7.09497e12	1.78163	0.890816	−	0.454363i	$-0.150133\pi$
$631$	7.09497e12	1.78163	0.890816	+	0.454363i	$0.150133\pi$
$632$	0	0
$633$	3.75863e12	0.930493
$634$	0	0
$635$	1.65957e12	0.405054
$636$	0	0
$637$	−3.98181e12	−0.958194
$638$	0	0
$639$	−2.46581e11	−0.0585068
$640$	0	0
$641$	4.21353e12	0.985792	0.492896	−	0.870088i	$-0.335938\pi$
$641$	4.21353e12	0.985792	0.492896	+	0.870088i	$0.335938\pi$
$642$	0	0
$643$	−2.10169e12	−0.484864	−0.242432	−	0.970168i	$-0.577945\pi$
$643$	−2.10169e12	−0.484864	−0.242432	+	0.970168i	$0.577945\pi$
$644$	0	0
$645$	−1.60399e10	−0.00364908
$646$	0	0
$647$	3.45447e12	0.775018	0.387509	−	0.921866i	$-0.373336\pi$
$647$	3.45447e12	0.775018	0.387509	+	0.921866i	$0.373336\pi$
$648$	0	0
$649$	7.46755e12	1.65226
$650$	0	0
$651$	1.77432e12	0.387185
$652$	0	0
$653$	−5.38398e12	−1.15876	−0.579380	−	0.815057i	$-0.696705\pi$
$653$	−5.38398e12	−1.15876	−0.579380	+	0.815057i	$0.696705\pi$
$654$	0	0
$655$	1.08503e13	2.30333
$656$	0	0
$657$	1.85943e12	0.389346
$658$	0	0
$659$	−6.08428e12	−1.25668	−0.628340	−	0.777939i	$-0.716265\pi$
$659$	−6.08428e12	−1.25668	−0.628340	+	0.777939i	$0.716265\pi$
$660$	0	0
$661$	4.81235e12	0.980506	0.490253	−	0.871580i	$-0.336904\pi$
$661$	4.81235e12	0.980506	0.490253	+	0.871580i	$0.336904\pi$
$662$	0	0
$663$	−4.29794e12	−0.863872
$664$	0	0
$665$	−1.77015e11	−0.0351004
$666$	0	0
$667$	−2.38545e12	−0.466664
$668$	0	0
$669$	2.19792e12	0.424223
$670$	0	0
$671$	1.10063e13	2.09600
$672$	0	0
$673$	−8.82624e12	−1.65847	−0.829235	−	0.558899i	$-0.811224\pi$
$673$	−8.82624e12	−1.65847	−0.829235	+	0.558899i	$0.811224\pi$
$674$	0	0
$675$	−7.19304e11	−0.133366
$676$	0	0
$677$	−3.95703e12	−0.723969	−0.361984	−	0.932184i	$-0.617901\pi$
$677$	−3.95703e12	−0.723969	−0.361984	+	0.932184i	$0.617901\pi$
$678$	0	0
$679$	−5.23326e12	−0.944839
$680$	0	0
$681$	−5.99583e11	−0.106829
$682$	0	0
$683$	5.76613e12	1.01389	0.506946	−	0.861978i	$-0.330774\pi$
$683$	5.76613e12	1.01389	0.506946	+	0.861978i	$0.330774\pi$
$684$	0	0
$685$	−1.28437e13	−2.22887
$686$	0	0
$687$	−3.41629e12	−0.585125
$688$	0	0
$689$	−1.17172e13	−1.98079
$690$	0	0
$691$	4.26637e12	0.711881	0.355941	−	0.934509i	$-0.384161\pi$
$691$	4.26637e12	0.711881	0.355941	+	0.934509i	$0.384161\pi$
$692$	0	0
$693$	−1.99083e12	−0.327895
$694$	0	0
$695$	3.88920e12	0.632307
$696$	0	0
$697$	−3.74202e12	−0.600564
$698$	0	0
$699$	5.90322e12	0.935281
$700$	0	0
$701$	5.62492e12	0.879802	0.439901	−	0.898046i	$-0.355013\pi$
$701$	5.62492e12	0.879802	0.439901	+	0.898046i	$0.355013\pi$
$702$	0	0
$703$	2.46811e11	0.0381123
$704$	0	0
$705$	−6.24531e12	−0.952144
$706$	0	0
$707$	4.94867e12	0.744906
$708$	0	0
$709$	−2.74160e12	−0.407470	−0.203735	−	0.979026i	$-0.565308\pi$
$709$	−2.74160e12	−0.407470	−0.203735	+	0.979026i	$0.565308\pi$
$710$	0	0
$711$	−3.07567e12	−0.451363
$712$	0	0
$713$	1.20545e13	1.74681
$714$	0	0
$715$	−2.23239e13	−3.19442
$716$	0	0
$717$	1.11930e12	0.158165
$718$	0	0
$719$	−3.15437e12	−0.440182	−0.220091	−	0.975479i	$-0.570635\pi$
$719$	−3.15437e12	−0.440182	−0.220091	+	0.975479i	$0.570635\pi$
$720$	0	0
$721$	7.74184e11	0.106693
$722$	0	0
$723$	3.93712e12	0.535866
$724$	0	0
$725$	1.53695e12	0.206603
$726$	0	0
$727$	8.74710e12	1.16134	0.580670	−	0.814139i	$-0.302791\pi$
$727$	8.74710e12	1.16134	0.580670	+	0.814139i	$0.302791\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	3.74128e10	0.00484610
$732$	0	0
$733$	−9.44871e10	−0.0120894	−0.00604470	−	0.999982i	$-0.501924\pi$
$733$	−9.44871e10	−0.0120894	−0.00604470	+	0.999982i	$0.501924\pi$
$734$	0	0
$735$	3.79733e12	0.479938
$736$	0	0
$737$	5.22598e12	0.652475
$738$	0	0
$739$	−7.12664e12	−0.878992	−0.439496	−	0.898244i	$-0.644843\pi$
$739$	−7.12664e12	−0.878992	−0.439496	+	0.898244i	$0.644843\pi$
$740$	0	0
$741$	3.19017e11	0.0388716
$742$	0	0
$743$	−7.11604e11	−0.0856621	−0.0428310	−	0.999082i	$-0.513638\pi$
$743$	−7.11604e11	−0.0856621	−0.0428310	+	0.999082i	$0.513638\pi$
$744$	0	0
$745$	3.58805e12	0.426732
$746$	0	0
$747$	2.75044e12	0.323191
$748$	0	0
$749$	8.62407e12	1.00125
$750$	0	0
$751$	−1.09650e13	−1.25785	−0.628927	−	0.777464i	$-0.716506\pi$
$751$	−1.09650e13	−1.25785	−0.628927	+	0.777464i	$0.716506\pi$
$752$	0	0
$753$	−8.06193e12	−0.913822
$754$	0	0
$755$	3.69414e11	0.0413763
$756$	0	0
$757$	−1.22871e13	−1.35993	−0.679966	−	0.733243i	$-0.738006\pi$
$757$	−1.22871e13	−1.35993	−0.679966	+	0.733243i	$0.738006\pi$
$758$	0	0
$759$	−1.35254e13	−1.47932
$760$	0	0
$761$	1.02506e13	1.10794	0.553971	−	0.832536i	$-0.313112\pi$
$761$	1.02506e13	1.10794	0.553971	+	0.832536i	$0.313112\pi$
$762$	0	0
$763$	1.01724e12	0.108659
$764$	0	0
$765$	4.09881e12	0.432694
$766$	0	0
$767$	−1.45098e13	−1.51385
$768$	0	0
$769$	−9.06562e12	−0.934822	−0.467411	−	0.884040i	$-0.654813\pi$
$769$	−9.06562e12	−0.934822	−0.467411	+	0.884040i	$0.654813\pi$
$770$	0	0
$771$	−3.90427e12	−0.397919
$772$	0	0
$773$	−1.84509e13	−1.85870	−0.929350	−	0.369201i	$-0.879632\pi$
$773$	−1.84509e13	−1.85870	−0.929350	+	0.369201i	$0.879632\pi$
$774$	0	0
$775$	−7.76672e12	−0.773357
$776$	0	0
$777$	2.99272e12	0.294559
$778$	0	0
$779$	2.77754e11	0.0270235
$780$	0	0
$781$	2.98736e12	0.287315
$782$	0	0
$783$	−6.03471e11	−0.0573758
$784$	0	0
$785$	2.50431e13	2.35383
$786$	0	0
$787$	−3.03454e12	−0.281972	−0.140986	−	0.990012i	$-0.545027\pi$
$787$	−3.03454e12	−0.281972	−0.140986	+	0.990012i	$0.545027\pi$
$788$	0	0
$789$	2.52574e12	0.232029
$790$	0	0
$791$	−9.25446e12	−0.840537
$792$	0	0
$793$	−2.13857e13	−1.92041
$794$	0	0
$795$	1.11743e13	0.992134
$796$	0	0
$797$	1.71637e13	1.50677	0.753386	−	0.657579i	$-0.228419\pi$
$797$	1.71637e13	1.50677	0.753386	+	0.657579i	$0.228419\pi$
$798$	0	0
$799$	1.45671e13	1.26448
$800$	0	0
$801$	5.46136e12	0.468765
$802$	0	0
$803$	−2.25272e13	−1.91200
$804$	0	0
$805$	−1.45824e13	−1.22390
$806$	0	0
$807$	−3.88329e12	−0.322306
$808$	0	0
$809$	−3.84376e12	−0.315491	−0.157746	−	0.987480i	$-0.550423\pi$
$809$	−3.84376e12	−0.315491	−0.157746	+	0.987480i	$0.550423\pi$
$810$	0	0
$811$	−5.81504e12	−0.472019	−0.236009	−	0.971751i	$-0.575840\pi$
$811$	−5.81504e12	−0.472019	−0.236009	+	0.971751i	$0.575840\pi$
$812$	0	0
$813$	−5.02229e12	−0.403176
$814$	0	0
$815$	−5.48597e12	−0.435556
$816$	0	0
$817$	−2.77699e9	−0.000218060 0
$818$	0	0
$819$	3.86827e12	0.300427
$820$	0	0
$821$	1.98531e13	1.52505	0.762526	−	0.646957i	$-0.223959\pi$
$821$	1.98531e13	1.52505	0.762526	+	0.646957i	$0.223959\pi$
$822$	0	0
$823$	1.43301e13	1.08881	0.544403	−	0.838824i	$-0.316756\pi$
$823$	1.43301e13	1.08881	0.544403	+	0.838824i	$0.316756\pi$
$824$	0	0
$825$	8.71444e12	0.654933
$826$	0	0
$827$	−8.25152e12	−0.613422	−0.306711	−	0.951803i	$-0.599228\pi$
$827$	−8.25152e12	−0.613422	−0.306711	+	0.951803i	$0.599228\pi$
$828$	0	0
$829$	−3.31194e12	−0.243549	−0.121775	−	0.992558i	$-0.538859\pi$
$829$	−3.31194e12	−0.243549	−0.121775	+	0.992558i	$0.538859\pi$
$830$	0	0
$831$	−6.08708e12	−0.442797
$832$	0	0
$833$	−8.85720e12	−0.637373
$834$	0	0
$835$	−8.96284e12	−0.638053
$836$	0	0
$837$	3.04955e12	0.214769
$838$	0	0
$839$	1.20859e13	0.842072	0.421036	−	0.907044i	$-0.361667\pi$
$839$	1.20859e13	0.842072	0.421036	+	0.907044i	$0.361667\pi$
$840$	0	0
$841$	−1.32177e13	−0.911117
$842$	0	0
$843$	−5.89438e11	−0.0401989
$844$	0	0
$845$	2.40929e13	1.62567
$846$	0	0
$847$	1.51179e13	1.00929
$848$	0	0
$849$	8.70959e12	0.575325
$850$	0	0
$851$	2.03322e13	1.32892
$852$	0	0
$853$	−3.65968e12	−0.236686	−0.118343	−	0.992973i	$-0.537758\pi$
$853$	−3.65968e12	−0.236686	−0.118343	+	0.992973i	$0.537758\pi$
$854$	0	0
$855$	−3.04237e11	−0.0194699
$856$	0	0
$857$	−1.91677e13	−1.21383	−0.606913	−	0.794768i	$-0.707592\pi$
$857$	−1.91677e13	−1.21383	−0.606913	+	0.794768i	$0.707592\pi$
$858$	0	0
$859$	−2.06228e13	−1.29234	−0.646172	−	0.763192i	$-0.723631\pi$
$859$	−2.06228e13	−1.29234	−0.646172	+	0.763192i	$0.723631\pi$
$860$	0	0
$861$	3.36793e12	0.208857
$862$	0	0
$863$	3.08452e12	0.189295	0.0946475	−	0.995511i	$-0.469828\pi$
$863$	3.08452e12	0.189295	0.0946475	+	0.995511i	$0.469828\pi$
$864$	0	0
$865$	−2.45076e13	−1.48843
$866$	0	0
$867$	4.52228e10	0.00271814
$868$	0	0
$869$	3.72621e13	2.21655
$870$	0	0
$871$	−1.01543e13	−0.597817
$872$	0	0
$873$	−8.99445e12	−0.524096
$874$	0	0
$875$	−4.16238e12	−0.240052
$876$	0	0
$877$	2.90264e13	1.65689	0.828446	−	0.560069i	$-0.189225\pi$
$877$	2.90264e13	1.65689	0.828446	+	0.560069i	$0.189225\pi$
$878$	0	0
$879$	1.87911e12	0.106170
$880$	0	0
$881$	−1.69337e13	−0.947021	−0.473511	−	0.880788i	$-0.657013\pi$
$881$	−1.69337e13	−0.947021	−0.473511	+	0.880788i	$0.657013\pi$
$882$	0	0
$883$	1.73642e13	0.961242	0.480621	−	0.876928i	$-0.340411\pi$
$883$	1.73642e13	0.961242	0.480621	+	0.876928i	$0.340411\pi$
$884$	0	0
$885$	1.38375e13	0.758252
$886$	0	0
$887$	−1.69498e13	−0.919406	−0.459703	−	0.888073i	$-0.652044\pi$
$887$	−1.69498e13	−0.919406	−0.459703	+	0.888073i	$0.652044\pi$
$888$	0	0
$889$	3.48394e12	0.187073
$890$	0	0
$891$	−3.42166e12	−0.181881
$892$	0	0
$893$	−1.08125e12	−0.0568977
$894$	0	0
$895$	−2.33837e13	−1.21818
$896$	0	0
$897$	2.62805e13	1.35540
$898$	0	0
$899$	−6.51601e12	−0.332708
$900$	0	0
$901$	−2.60640e13	−1.31759
$902$	0	0
$903$	−3.36726e10	−0.00168532
$904$	0	0
$905$	−2.65869e13	−1.31750
$906$	0	0
$907$	1.95992e13	0.961622	0.480811	−	0.876824i	$-0.340342\pi$
$907$	1.95992e13	0.961622	0.480811	+	0.876824i	$0.340342\pi$
$908$	0	0
$909$	8.50533e12	0.413194
$910$	0	0
$911$	−1.77979e13	−0.856124	−0.428062	−	0.903749i	$-0.640803\pi$
$911$	−1.77979e13	−0.856124	−0.428062	+	0.903749i	$0.640803\pi$
$912$	0	0
$913$	−3.33218e13	−1.58712
$914$	0	0
$915$	2.03949e13	0.961893
$916$	0	0
$917$	2.27781e13	1.06379
$918$	0	0
$919$	−6.61520e11	−0.0305931	−0.0152965	−	0.999883i	$-0.504869\pi$
$919$	−6.61520e11	−0.0305931	−0.0152965	+	0.999883i	$0.504869\pi$
$920$	0	0
$921$	1.49447e13	0.684415
$922$	0	0
$923$	−5.80457e12	−0.263246
$924$	0	0
$925$	−1.31000e13	−0.588347
$926$	0	0
$927$	1.33060e12	0.0591818
$928$	0	0
$929$	1.85586e12	0.0817475	0.0408737	−	0.999164i	$-0.486986\pi$
$929$	1.85586e12	0.0817475	0.0408737	+	0.999164i	$0.486986\pi$
$930$	0	0
$931$	6.57432e11	0.0286799
$932$	0	0
$933$	1.48652e13	0.642251
$934$	0	0
$935$	−4.96575e13	−2.12487
$936$	0	0
$937$	−1.97151e13	−0.835548	−0.417774	−	0.908551i	$-0.637190\pi$
$937$	−1.97151e13	−0.835548	−0.417774	+	0.908551i	$0.637190\pi$
$938$	0	0
$939$	−1.35564e13	−0.569048
$940$	0	0
$941$	−1.18755e13	−0.493739	−0.246869	−	0.969049i	$-0.579402\pi$
$941$	−1.18755e13	−0.493739	−0.246869	+	0.969049i	$0.579402\pi$
$942$	0	0
$943$	2.28813e13	0.942275
$944$	0	0
$945$	−3.68905e12	−0.150478
$946$	0	0
$947$	1.15724e13	0.467572	0.233786	−	0.972288i	$-0.424888\pi$
$947$	1.15724e13	0.467572	0.233786	+	0.972288i	$0.424888\pi$
$948$	0	0
$949$	4.37713e13	1.75183
$950$	0	0
$951$	−2.48998e13	−0.987153
$952$	0	0
$953$	4.16575e13	1.63597	0.817985	−	0.575239i	$-0.195091\pi$
$953$	4.16575e13	1.63597	0.817985	+	0.575239i	$0.195091\pi$
$954$	0	0
$955$	−4.54309e13	−1.76740
$956$	0	0
$957$	7.31112e12	0.281760
$958$	0	0
$959$	−2.69629e13	−1.02940
$960$	0	0
$961$	6.48804e12	0.245391
$962$	0	0
$963$	1.48223e13	0.555388
$964$	0	0
$965$	−4.46324e13	−1.65683
$966$	0	0
$967$	2.31308e13	0.850692	0.425346	−	0.905031i	$-0.360152\pi$
$967$	2.31308e13	0.850692	0.425346	+	0.905031i	$0.360152\pi$
$968$	0	0
$969$	7.09627e11	0.0258567
$970$	0	0
$971$	2.35529e13	0.850272	0.425136	−	0.905129i	$-0.360226\pi$
$971$	2.35529e13	0.850272	0.425136	+	0.905129i	$0.360226\pi$
$972$	0	0
$973$	8.16460e12	0.292030
$974$	0	0
$975$	−1.69325e13	−0.600069
$976$	0	0
$977$	3.82311e13	1.34243	0.671215	−	0.741263i	$-0.265773\pi$
$977$	3.82311e13	1.34243	0.671215	+	0.741263i	$0.265773\pi$
$978$	0	0
$979$	−6.61650e13	−2.30201
$980$	0	0
$981$	1.74834e12	0.0602721
$982$	0	0
$983$	−2.76306e13	−0.943843	−0.471922	−	0.881641i	$-0.656439\pi$
$983$	−2.76306e13	−0.943843	−0.471922	+	0.881641i	$0.656439\pi$
$984$	0	0
$985$	−2.07660e13	−0.702893
$986$	0	0
$987$	−1.31108e13	−0.439746
$988$	0	0
$989$	−2.28767e11	−0.00760345
$990$	0	0
$991$	2.13754e13	0.704017	0.352008	−	0.935997i	$-0.385499\pi$
$991$	2.13754e13	0.704017	0.352008	+	0.935997i	$0.385499\pi$
$992$	0	0
$993$	−1.80226e12	−0.0588229
$994$	0	0
$995$	−1.97419e13	−0.638535
$996$	0	0
$997$	−5.63793e12	−0.180714	−0.0903569	−	0.995909i	$-0.528801\pi$
$997$	−5.63793e12	−0.180714	−0.0903569	+	0.995909i	$0.528801\pi$
$998$	0	0
$999$	5.14362e12	0.163390

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.a.d.1.4	yes	4
4.3	odd	2		384.10.a.h.1.4	yes	4
8.3	odd	2		384.10.a.a.1.1	✓	4
8.5	even	2		384.10.a.e.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.a.a.1.1	✓	4	8.3	odd	2
384.10.a.d.1.4	yes	4	1.1	even	1	trivial
384.10.a.e.1.1	yes	4	8.5	even	2
384.10.a.h.1.4	yes	4	4.3	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

\(f(q)\)	\(=\)	\(q-81.0000 q^{3} +1818.41 q^{5} +3817.40 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q-81.0000 q^{3} +1818.41 q^{5} +3817.40 q^{7} +6561.00 q^{9} -79487.2 q^{11} +154447. q^{13} -147291. q^{15} +343554. q^{17} -25500.6 q^{19} -309209. q^{21} -2.10072e6 q^{23} +1.35350e6 q^{25} -531441. q^{27} +1.13554e6 q^{29} -5.73826e6 q^{31} +6.43847e6 q^{33} +6.94160e6 q^{35} -9.67864e6 q^{37} -1.25102e7 q^{39} -1.08921e7 q^{41} +108899. q^{43} +1.19306e7 q^{45} +4.24010e7 q^{47} -2.57811e7 q^{49} -2.78279e7 q^{51} -7.58656e7 q^{53} -1.44541e8 q^{55} +2.06555e6 q^{57} -9.39466e7 q^{59} -1.38466e8 q^{61} +2.50459e7 q^{63} +2.80848e8 q^{65} -6.57461e7 q^{67} +1.70159e8 q^{69} -3.75829e7 q^{71} +2.83407e8 q^{73} -1.09633e8 q^{75} -3.03434e8 q^{77} -4.68780e8 q^{79} +4.30467e7 q^{81} +4.19210e8 q^{83} +6.24723e8 q^{85} -9.19785e7 q^{87} +8.32398e8 q^{89} +5.89586e8 q^{91} +4.64799e8 q^{93} -4.63705e7 q^{95} -1.37090e9 q^{97} -5.21516e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 324 q^{3} + 1728 q^{5} - 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) 4 * q - 324 * q^3 + 1728 * q^5 - 4840 * q^7 + 26244 * q^9	\( 4 q - 324 q^{3} + 1728 q^{5} - 4840 q^{7} + 26244 q^{9} - 15824 q^{11} + 82440 q^{13} - 139968 q^{15} + 165912 q^{17} - 539904 q^{19} + 392040 q^{21} - 729680 q^{23} + 224812 q^{25} - 2125764 q^{27} + 1850864 q^{29} - 3197960 q^{31} + 1281744 q^{33} - 8574912 q^{35} + 4187992 q^{37} - 6677640 q^{39} + 227704 q^{41} - 1600352 q^{43} + 11337408 q^{45} - 18053904 q^{47} + 57728820 q^{49} - 13438872 q^{51} + 29418288 q^{53} - 45906816 q^{55} + 43732224 q^{57} - 38300048 q^{59} - 99764648 q^{61} - 31755240 q^{63} + 314120832 q^{65} + 183717008 q^{67} + 59104080 q^{69} - 181868080 q^{71} + 254539160 q^{73} - 18209772 q^{75} - 230564704 q^{77} - 831578184 q^{79} + 172186884 q^{81} + 687923952 q^{83} + 1041391104 q^{85} - 149919984 q^{87} + 627627272 q^{89} + 1018147632 q^{91} + 259034760 q^{93} - 2167118208 q^{95} - 889385880 q^{97} - 103821264 q^{99}+O(q^{100}) \) 4 * q - 324 * q^3 + 1728 * q^5 - 4840 * q^7 + 26244 * q^9 - 15824 * q^11 + 82440 * q^13 - 139968 * q^15 + 165912 * q^17 - 539904 * q^19 + 392040 * q^21 - 729680 * q^23 + 224812 * q^25 - 2125764 * q^27 + 1850864 * q^29 - 3197960 * q^31 + 1281744 * q^33 - 8574912 * q^35 + 4187992 * q^37 - 6677640 * q^39 + 227704 * q^41 - 1600352 * q^43 + 11337408 * q^45 - 18053904 * q^47 + 57728820 * q^49 - 13438872 * q^51 + 29418288 * q^53 - 45906816 * q^55 + 43732224 * q^57 - 38300048 * q^59 - 99764648 * q^61 - 31755240 * q^63 + 314120832 * q^65 + 183717008 * q^67 + 59104080 * q^69 - 181868080 * q^71 + 254539160 * q^73 - 18209772 * q^75 - 230564704 * q^77 - 831578184 * q^79 + 172186884 * q^81 + 687923952 * q^83 + 1041391104 * q^85 - 149919984 * q^87 + 627627272 * q^89 + 1018147632 * q^91 + 259034760 * q^93 - 2167118208 * q^95 - 889385880 * q^97 - 103821264 * q^99

Introduction

L-functions

Modular forms

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Database

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