LMFDB - Embedded newform 384.10.a.f.1.3

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} - 2070x^{2} - 13768x + 561570 $ x^4 - 2070x^2 - 13768x + 561570
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			1.3
Root			$-25.0977$ 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)

$f(q)$	$=$	$q+81.0000 q^{3} -126.806 q^{5} -5939.66 q^{7} +6561.00 q^{9} +O(q^{10})$	$q+81.0000 q^{3} -126.806 q^{5} -5939.66 q^{7} +6561.00 q^{9} +4237.38 q^{11} +136583. q^{13} -10271.3 q^{15} -230372. q^{17} +450474. q^{19} -481112. q^{21} -1.33693e6 q^{23} -1.93705e6 q^{25} +531441. q^{27} -4.08853e6 q^{29} +4.71901e6 q^{31} +343228. q^{33} +753184. q^{35} +1.93239e7 q^{37} +1.10632e7 q^{39} -1.33619e7 q^{41} +4.48837e6 q^{43} -831974. q^{45} +1.27952e7 q^{47} -5.07408e6 q^{49} -1.86602e7 q^{51} +5.11895e7 q^{53} -537325. q^{55} +3.64884e7 q^{57} -4.93950e7 q^{59} +1.72852e7 q^{61} -3.89701e7 q^{63} -1.73196e7 q^{65} +2.72355e8 q^{67} -1.08291e8 q^{69} -2.24504e8 q^{71} -2.69433e8 q^{73} -1.56901e8 q^{75} -2.51686e7 q^{77} +1.14996e8 q^{79} +4.30467e7 q^{81} -6.29829e8 q^{83} +2.92126e7 q^{85} -3.31171e8 q^{87} +6.35980e8 q^{89} -8.11257e8 q^{91} +3.82240e8 q^{93} -5.71228e7 q^{95} -6.48896e8 q^{97} +2.78014e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 324 q^{3} - 240 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) $ 4 * q + 324 * q^3 - 240 * q^5 + 4840 * q^7 + 26244 * q^9	$ 4 q + 324 q^{3} - 240 q^{5} + 4840 q^{7} + 26244 q^{9} - 99664 q^{11} - 60840 q^{13} - 19440 q^{15} - 434952 q^{17} + 631776 q^{19} + 392040 q^{21} + 749392 q^{23} + 5991532 q^{25} + 2125764 q^{27} - 7908544 q^{29} + 11351240 q^{31} - 8072784 q^{33} - 25567008 q^{35} - 13592920 q^{37} - 4928040 q^{39} - 18838888 q^{41} + 14177920 q^{43} - 1574640 q^{45} - 37779120 q^{47} + 9409332 q^{49} - 35231112 q^{51} + 115336512 q^{53} - 184580544 q^{55} + 51173856 q^{57} - 115028080 q^{59} + 173228648 q^{61} + 31755240 q^{63} - 328077984 q^{65} - 231785104 q^{67} + 60700752 q^{69} + 197476208 q^{71} + 44629400 q^{73} + 485314092 q^{75} - 308117920 q^{77} - 355774584 q^{79} + 172186884 q^{81} - 607613328 q^{83} + 1087351392 q^{85} - 640592064 q^{87} - 1157146424 q^{89} - 847629840 q^{91} + 919450440 q^{93} - 329699328 q^{95} - 1599536472 q^{97} - 653895504 q^{99}+O(q^{100}) $ 4 * q + 324 * q^3 - 240 * q^5 + 4840 * q^7 + 26244 * q^9 - 99664 * q^11 - 60840 * q^13 - 19440 * q^15 - 434952 * q^17 + 631776 * q^19 + 392040 * q^21 + 749392 * q^23 + 5991532 * q^25 + 2125764 * q^27 - 7908544 * q^29 + 11351240 * q^31 - 8072784 * q^33 - 25567008 * q^35 - 13592920 * q^37 - 4928040 * q^39 - 18838888 * q^41 + 14177920 * q^43 - 1574640 * q^45 - 37779120 * q^47 + 9409332 * q^49 - 35231112 * q^51 + 115336512 * q^53 - 184580544 * q^55 + 51173856 * q^57 - 115028080 * q^59 + 173228648 * q^61 + 31755240 * q^63 - 328077984 * q^65 - 231785104 * q^67 + 60700752 * q^69 + 197476208 * q^71 + 44629400 * q^73 + 485314092 * q^75 - 308117920 * q^77 - 355774584 * q^79 + 172186884 * q^81 - 607613328 * q^83 + 1087351392 * q^85 - 640592064 * q^87 - 1157146424 * q^89 - 847629840 * q^91 + 919450440 * q^93 - 329699328 * q^95 - 1599536472 * q^97 - 653895504 * 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
$5$	−126.806	−0.0907350	−0.0453675	−	0.998970i	$-0.514446\pi$
$5$	−126.806	−0.0907350	−0.0453675	+	0.998970i	$0.514446\pi$
$6$	0	0
$7$	−5939.66	−0.935018	−0.467509	−	0.883988i	$-0.654849\pi$
$7$	−5939.66	−0.935018	−0.467509	+	0.883988i	$0.654849\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	4237.38	0.0872630	0.0436315	−	0.999048i	$-0.486107\pi$
$11$	4237.38	0.0872630	0.0436315	+	0.999048i	$0.486107\pi$
$12$	0	0
$13$	136583.	1.32633	0.663165	−	0.748473i	$-0.269213\pi$
$13$	136583.	1.32633	0.663165	+	0.748473i	$0.269213\pi$
$14$	0	0
$15$	−10271.3	−0.0523859
$16$	0	0
$17$	−230372.	−0.668976	−0.334488	−	0.942400i	$-0.608563\pi$
$17$	−230372.	−0.668976	−0.334488	+	0.942400i	$0.608563\pi$
$18$	0	0
$19$	450474.	0.793010	0.396505	−	0.918033i	$-0.370223\pi$
$19$	450474.	0.793010	0.396505	+	0.918033i	$0.370223\pi$
$20$	0	0
$21$	−481112.	−0.539833
$22$	0	0
$23$	−1.33693e6	−0.996167	−0.498084	−	0.867129i	$-0.665963\pi$
$23$	−1.33693e6	−0.996167	−0.498084	+	0.867129i	$0.665963\pi$
$24$	0	0
$25$	−1.93705e6	−0.991767
$26$	0	0
$27$	531441.	0.192450
$28$	0	0
$29$	−4.08853e6	−1.07344	−0.536718	−	0.843761i	$-0.680336\pi$
$29$	−4.08853e6	−1.07344	−0.536718	+	0.843761i	$0.680336\pi$
$30$	0	0
$31$	4.71901e6	0.917748	0.458874	−	0.888501i	$-0.348253\pi$
$31$	4.71901e6	0.917748	0.458874	+	0.888501i	$0.348253\pi$
$32$	0	0
$33$	343228.	0.0503813
$34$	0	0
$35$	753184.	0.0848389
$36$	0	0
$37$	1.93239e7	1.69507	0.847535	−	0.530739i	$-0.178085\pi$
$37$	1.93239e7	1.69507	0.847535	+	0.530739i	$0.178085\pi$
$38$	0	0
$39$	1.10632e7	0.765757
$40$	0	0
$41$	−1.33619e7	−0.738485	−0.369242	−	0.929333i	$-0.620383\pi$
$41$	−1.33619e7	−0.738485	−0.369242	+	0.929333i	$0.620383\pi$
$42$	0	0
$43$	4.48837e6	0.200208	0.100104	−	0.994977i	$-0.468082\pi$
$43$	4.48837e6	0.200208	0.100104	+	0.994977i	$0.468082\pi$
$44$	0	0
$45$	−831974.	−0.0302450
$46$	0	0
$47$	1.27952e7	0.382477	0.191238	−	0.981544i	$-0.438750\pi$
$47$	1.27952e7	0.382477	0.191238	+	0.981544i	$0.438750\pi$
$48$	0	0
$49$	−5.07408e6	−0.125741
$50$	0	0
$51$	−1.86602e7	−0.386233
$52$	0	0
$53$	5.11895e7	0.891127	0.445563	−	0.895250i	$-0.353003\pi$
$53$	5.11895e7	0.891127	0.445563	+	0.895250i	$0.353003\pi$
$54$	0	0
$55$	−537325.	−0.00791781
$56$	0	0
$57$	3.64884e7	0.457845
$58$	0	0
$59$	−4.93950e7	−0.530700	−0.265350	−	0.964152i	$-0.585487\pi$
$59$	−4.93950e7	−0.530700	−0.265350	+	0.964152i	$0.585487\pi$
$60$	0	0
$61$	1.72852e7	0.159842	0.0799210	−	0.996801i	$-0.474533\pi$
$61$	1.72852e7	0.159842	0.0799210	+	0.996801i	$0.474533\pi$
$62$	0	0
$63$	−3.89701e7	−0.311673
$64$	0	0
$65$	−1.73196e7	−0.120345
$66$	0	0
$67$	2.72355e8	1.65120	0.825599	−	0.564257i	$-0.190837\pi$
$67$	2.72355e8	1.65120	0.825599	+	0.564257i	$0.190837\pi$
$68$	0	0
$69$	−1.08291e8	−0.575137
$70$	0	0
$71$	−2.24504e8	−1.04848	−0.524242	−	0.851569i	$-0.675651\pi$
$71$	−2.24504e8	−1.04848	−0.524242	+	0.851569i	$0.675651\pi$
$72$	0	0
$73$	−2.69433e8	−1.11045	−0.555223	−	0.831702i	$-0.687367\pi$
$73$	−2.69433e8	−1.11045	−0.555223	+	0.831702i	$0.687367\pi$
$74$	0	0
$75$	−1.56901e8	−0.572597
$76$	0	0
$77$	−2.51686e7	−0.0815925
$78$	0	0
$79$	1.14996e8	0.332172	0.166086	−	0.986111i	$-0.446887\pi$
$79$	1.14996e8	0.332172	0.166086	+	0.986111i	$0.446887\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	−6.29829e8	−1.45670	−0.728351	−	0.685204i	$-0.759713\pi$
$83$	−6.29829e8	−1.45670	−0.728351	+	0.685204i	$0.759713\pi$
$84$	0	0
$85$	2.92126e7	0.0606995
$86$	0	0
$87$	−3.31171e8	−0.619749
$88$	0	0
$89$	6.35980e8	1.07446	0.537228	−	0.843437i	$-0.319471\pi$
$89$	6.35980e8	1.07446	0.537228	+	0.843437i	$0.319471\pi$
$90$	0	0
$91$	−8.11257e8	−1.24014
$92$	0	0
$93$	3.82240e8	0.529862
$94$	0	0
$95$	−5.71228e7	−0.0719538
$96$	0	0
$97$	−6.48896e8	−0.744221	−0.372111	−	0.928188i	$-0.621366\pi$
$97$	−6.48896e8	−0.744221	−0.372111	+	0.928188i	$0.621366\pi$
$98$	0	0
$99$	2.78014e7	0.0290877
$100$	0	0
$101$	2.08357e8	0.199233	0.0996164	−	0.995026i	$-0.468238\pi$
$101$	2.08357e8	0.199233	0.0996164	+	0.995026i	$0.468238\pi$
$102$	0	0
$103$	−1.16057e9	−1.01603	−0.508013	−	0.861349i	$-0.669620\pi$
$103$	−1.16057e9	−1.01603	−0.508013	+	0.861349i	$0.669620\pi$
$104$	0	0
$105$	6.10079e7	0.0489818
$106$	0	0
$107$	6.55331e8	0.483319	0.241659	−	0.970361i	$-0.422308\pi$
$107$	6.55331e8	0.483319	0.241659	+	0.970361i	$0.422308\pi$
$108$	0	0
$109$	−2.98216e8	−0.202354	−0.101177	−	0.994868i	$-0.532261\pi$
$109$	−2.98216e8	−0.202354	−0.101177	+	0.994868i	$0.532261\pi$
$110$	0	0
$111$	1.56524e9	0.978650
$112$	0	0
$113$	−9.44019e8	−0.544663	−0.272331	−	0.962204i	$-0.587795\pi$
$113$	−9.44019e8	−0.544663	−0.272331	+	0.962204i	$0.587795\pi$
$114$	0	0
$115$	1.69530e8	0.0903872
$116$	0	0
$117$	8.96122e8	0.442110
$118$	0	0
$119$	1.36833e9	0.625505
$120$	0	0
$121$	−2.33999e9	−0.992385
$122$	0	0
$123$	−1.08232e9	−0.426364
$124$	0	0
$125$	4.93297e8	0.180723
$126$	0	0
$127$	−2.03041e9	−0.692574	−0.346287	−	0.938129i	$-0.612558\pi$
$127$	−2.03041e9	−0.692574	−0.346287	+	0.938129i	$0.612558\pi$
$128$	0	0
$129$	3.63558e8	0.115590
$130$	0	0
$131$	−1.21820e9	−0.361407	−0.180704	−	0.983538i	$-0.557837\pi$
$131$	−1.21820e9	−0.361407	−0.180704	+	0.983538i	$0.557837\pi$
$132$	0	0
$133$	−2.67566e9	−0.741479
$134$	0	0
$135$	−6.73899e7	−0.0174620
$136$	0	0
$137$	−4.85728e9	−1.17802	−0.589008	−	0.808128i	$-0.700481\pi$
$137$	−4.85728e9	−1.17802	−0.589008	+	0.808128i	$0.700481\pi$
$138$	0	0
$139$	3.29557e9	0.748798	0.374399	−	0.927268i	$-0.377849\pi$
$139$	3.29557e9	0.748798	0.374399	+	0.927268i	$0.377849\pi$
$140$	0	0
$141$	1.03641e9	0.220823
$142$	0	0
$143$	5.78754e8	0.115740
$144$	0	0
$145$	5.18451e8	0.0973983
$146$	0	0
$147$	−4.11001e8	−0.0725963
$148$	0	0
$149$	5.03213e9	0.836400	0.418200	−	0.908355i	$-0.362661\pi$
$149$	5.03213e9	0.836400	0.418200	+	0.908355i	$0.362661\pi$
$150$	0	0
$151$	−2.76425e9	−0.432694	−0.216347	−	0.976317i	$-0.569414\pi$
$151$	−2.76425e9	−0.432694	−0.216347	+	0.976317i	$0.569414\pi$
$152$	0	0
$153$	−1.51147e9	−0.222992
$154$	0	0
$155$	−5.98399e8	−0.0832718
$156$	0	0
$157$	−3.19336e9	−0.419469	−0.209734	−	0.977758i	$-0.567260\pi$
$157$	−3.19336e9	−0.419469	−0.209734	+	0.977758i	$0.567260\pi$
$158$	0	0
$159$	4.14635e9	0.514492
$160$	0	0
$161$	7.94088e9	0.931435
$162$	0	0
$163$	−3.86760e9	−0.429139	−0.214569	−	0.976709i	$-0.568835\pi$
$163$	−3.86760e9	−0.429139	−0.214569	+	0.976709i	$0.568835\pi$
$164$	0	0
$165$	−4.35233e7	−0.00457135
$166$	0	0
$167$	8.96510e9	0.891931	0.445965	−	0.895050i	$-0.352860\pi$
$167$	8.96510e9	0.891931	0.445965	+	0.895050i	$0.352860\pi$
$168$	0	0
$169$	8.05044e9	0.759153
$170$	0	0
$171$	2.95556e9	0.264337
$172$	0	0
$173$	−9.70609e9	−0.823829	−0.411914	−	0.911223i	$-0.635140\pi$
$173$	−9.70609e9	−0.823829	−0.411914	+	0.911223i	$0.635140\pi$
$174$	0	0
$175$	1.15054e10	0.927321
$176$	0	0
$177$	−4.00100e9	−0.306400
$178$	0	0
$179$	−2.67643e10	−1.94857	−0.974287	−	0.225312i	$-0.927660\pi$
$179$	−2.67643e10	−1.94857	−0.974287	+	0.225312i	$0.927660\pi$
$180$	0	0
$181$	−1.73116e10	−1.19890	−0.599450	−	0.800412i	$-0.704614\pi$
$181$	−1.73116e10	−1.19890	−0.599450	+	0.800412i	$0.704614\pi$
$182$	0	0
$183$	1.40010e9	0.0922848
$184$	0	0
$185$	−2.45039e9	−0.153802
$186$	0	0
$187$	−9.76176e8	−0.0583769
$188$	0	0
$189$	−3.15658e9	−0.179944
$190$	0	0
$191$	−3.60854e9	−0.196192	−0.0980959	−	0.995177i	$-0.531275\pi$
$191$	−3.60854e9	−0.196192	−0.0980959	+	0.995177i	$0.531275\pi$
$192$	0	0
$193$	−2.86520e10	−1.48644	−0.743221	−	0.669046i	$-0.766703\pi$
$193$	−2.86520e10	−1.48644	−0.743221	+	0.669046i	$0.766703\pi$
$194$	0	0
$195$	−1.40288e9	−0.0694810
$196$	0	0
$197$	−2.52639e10	−1.19510	−0.597549	−	0.801833i	$-0.703858\pi$
$197$	−2.52639e10	−1.19510	−0.597549	+	0.801833i	$0.703858\pi$
$198$	0	0
$199$	7.39547e9	0.334293	0.167146	−	0.985932i	$-0.446545\pi$
$199$	7.39547e9	0.334293	0.167146	+	0.985932i	$0.446545\pi$
$200$	0	0
$201$	2.20608e10	0.953320
$202$	0	0
$203$	2.42845e10	1.00368
$204$	0	0
$205$	1.69437e9	0.0670064
$206$	0	0
$207$	−8.77157e9	−0.332056
$208$	0	0
$209$	1.90883e9	0.0692005
$210$	0	0
$211$	−8.16676e9	−0.283647	−0.141824	−	0.989892i	$-0.545297\pi$
$211$	−8.16676e9	−0.283647	−0.141824	+	0.989892i	$0.545297\pi$
$212$	0	0
$213$	−1.81848e10	−0.605342
$214$	0	0
$215$	−5.69153e8	−0.0181659
$216$	0	0
$217$	−2.80293e10	−0.858111
$218$	0	0
$219$	−2.18240e10	−0.641116
$220$	0	0
$221$	−3.14650e10	−0.887283
$222$	0	0
$223$	6.20049e10	1.67901	0.839507	−	0.543349i	$-0.182844\pi$
$223$	6.20049e10	1.67901	0.839507	+	0.543349i	$0.182844\pi$
$224$	0	0
$225$	−1.27090e10	−0.330589
$226$	0	0
$227$	2.12021e10	0.529984	0.264992	−	0.964251i	$-0.414631\pi$
$227$	2.12021e10	0.529984	0.264992	+	0.964251i	$0.414631\pi$
$228$	0	0
$229$	2.34925e10	0.564508	0.282254	−	0.959340i	$-0.408918\pi$
$229$	2.34925e10	0.564508	0.282254	+	0.959340i	$0.408918\pi$
$230$	0	0
$231$	−2.03866e9	−0.0471075
$232$	0	0
$233$	−3.64418e10	−0.810025	−0.405012	−	0.914311i	$-0.632733\pi$
$233$	−3.64418e10	−0.810025	−0.405012	+	0.914311i	$0.632733\pi$
$234$	0	0
$235$	−1.62250e9	−0.0347040
$236$	0	0
$237$	9.31471e9	0.191779
$238$	0	0
$239$	9.03924e10	1.79201	0.896007	−	0.444039i	$-0.146455\pi$
$239$	9.03924e10	1.79201	0.896007	+	0.444039i	$0.146455\pi$
$240$	0	0
$241$	3.80041e10	0.725695	0.362848	−	0.931848i	$-0.381805\pi$
$241$	3.80041e10	0.725695	0.362848	+	0.931848i	$0.381805\pi$
$242$	0	0
$243$	3.48678e9	0.0641500
$244$	0	0
$245$	6.43424e8	0.0114091
$246$	0	0
$247$	6.15272e10	1.05179
$248$	0	0
$249$	−5.10161e10	−0.841028
$250$	0	0
$251$	−9.45362e10	−1.50337	−0.751686	−	0.659522i	$-0.770759\pi$
$251$	−9.45362e10	−1.50337	−0.751686	+	0.659522i	$0.770759\pi$
$252$	0	0
$253$	−5.66506e9	−0.0869286
$254$	0	0
$255$	2.36622e9	0.0350449
$256$	0	0
$257$	−3.68326e10	−0.526664	−0.263332	−	0.964705i	$-0.584821\pi$
$257$	−3.68326e10	−0.526664	−0.263332	+	0.964705i	$0.584821\pi$
$258$	0	0
$259$	−1.14778e11	−1.58492
$260$	0	0
$261$	−2.68249e10	−0.357812
$262$	0	0
$263$	−6.98154e10	−0.899810	−0.449905	−	0.893077i	$-0.648542\pi$
$263$	−6.98154e10	−0.899810	−0.449905	+	0.893077i	$0.648542\pi$
$264$	0	0
$265$	−6.49114e9	−0.0808564
$266$	0	0
$267$	5.15144e10	0.620337
$268$	0	0
$269$	−6.08188e10	−0.708194	−0.354097	−	0.935209i	$-0.615212\pi$
$269$	−6.08188e10	−0.708194	−0.354097	+	0.935209i	$0.615212\pi$
$270$	0	0
$271$	−5.33650e9	−0.0601028	−0.0300514	−	0.999548i	$-0.509567\pi$
$271$	−5.33650e9	−0.0601028	−0.0300514	+	0.999548i	$0.509567\pi$
$272$	0	0
$273$	−6.57118e10	−0.715997
$274$	0	0
$275$	−8.20800e9	−0.0865446
$276$	0	0
$277$	−1.34420e11	−1.37184	−0.685922	−	0.727675i	$-0.740601\pi$
$277$	−1.34420e11	−1.37184	−0.685922	+	0.727675i	$0.740601\pi$
$278$	0	0
$279$	3.09614e10	0.305916
$280$	0	0
$281$	−1.92090e11	−1.83791	−0.918957	−	0.394357i	$-0.870967\pi$
$281$	−1.92090e11	−1.83791	−0.918957	+	0.394357i	$0.870967\pi$
$282$	0	0
$283$	−4.72340e10	−0.437739	−0.218870	−	0.975754i	$-0.570237\pi$
$283$	−4.72340e10	−0.437739	−0.218870	+	0.975754i	$0.570237\pi$
$284$	0	0
$285$	−4.62695e9	−0.0415425
$286$	0	0
$287$	7.93652e10	0.690497
$288$	0	0
$289$	−6.55164e10	−0.552471
$290$	0	0
$291$	−5.25605e10	−0.429676
$292$	0	0
$293$	−1.72252e10	−0.136540	−0.0682702	−	0.997667i	$-0.521748\pi$
$293$	−1.72252e10	−0.136540	−0.0682702	+	0.997667i	$0.521748\pi$
$294$	0	0
$295$	6.26358e9	0.0481530
$296$	0	0
$297$	2.25192e9	0.0167938
$298$	0	0
$299$	−1.82602e11	−1.32125
$300$	0	0
$301$	−2.66594e10	−0.187198
$302$	0	0
$303$	1.68769e10	0.115027
$304$	0	0
$305$	−2.19187e9	−0.0145033
$306$	0	0
$307$	−9.42172e10	−0.605351	−0.302676	−	0.953094i	$-0.597880\pi$
$307$	−9.42172e10	−0.605351	−0.302676	+	0.953094i	$0.597880\pi$
$308$	0	0
$309$	−9.40064e10	−0.586603
$310$	0	0
$311$	7.95356e10	0.482103	0.241051	−	0.970512i	$-0.422508\pi$
$311$	7.95356e10	0.482103	0.241051	+	0.970512i	$0.422508\pi$
$312$	0	0
$313$	−6.27784e10	−0.369709	−0.184855	−	0.982766i	$-0.559181\pi$
$313$	−6.27784e10	−0.369709	−0.184855	+	0.982766i	$0.559181\pi$
$314$	0	0
$315$	4.94164e9	0.0282796
$316$	0	0
$317$	−9.48671e10	−0.527653	−0.263827	−	0.964570i	$-0.584985\pi$
$317$	−9.48671e10	−0.527653	−0.263827	+	0.964570i	$0.584985\pi$
$318$	0	0
$319$	−1.73247e10	−0.0936714
$320$	0	0
$321$	5.30818e10	0.279044
$322$	0	0
$323$	−1.03777e11	−0.530505
$324$	0	0
$325$	−2.64568e11	−1.31541
$326$	0	0
$327$	−2.41555e10	−0.116829
$328$	0	0
$329$	−7.59988e10	−0.357623
$330$	0	0
$331$	−4.21508e11	−1.93010	−0.965051	−	0.262063i	$-0.915597\pi$
$331$	−4.21508e11	−1.93010	−0.965051	+	0.262063i	$0.915597\pi$
$332$	0	0
$333$	1.26784e11	0.565024
$334$	0	0
$335$	−3.45363e10	−0.149821
$336$	0	0
$337$	−1.95896e11	−0.827355	−0.413678	−	0.910423i	$-0.635756\pi$
$337$	−1.95896e11	−0.827355	−0.413678	+	0.910423i	$0.635756\pi$
$338$	0	0
$339$	−7.64655e10	−0.314461
$340$	0	0
$341$	1.99962e10	0.0800854
$342$	0	0
$343$	2.69825e11	1.05259
$344$	0	0
$345$	1.37320e10	0.0521851
$346$	0	0
$347$	1.00666e11	0.372736	0.186368	−	0.982480i	$-0.440328\pi$
$347$	1.00666e11	0.372736	0.186368	+	0.982480i	$0.440328\pi$
$348$	0	0
$349$	1.21109e11	0.436980	0.218490	−	0.975839i	$-0.429887\pi$
$349$	1.21109e11	0.436980	0.218490	+	0.975839i	$0.429887\pi$
$350$	0	0
$351$	7.25858e10	0.255252
$352$	0	0
$353$	−1.50575e11	−0.516139	−0.258069	−	0.966126i	$-0.583086\pi$
$353$	−1.50575e11	−0.516139	−0.258069	+	0.966126i	$0.583086\pi$
$354$	0	0
$355$	2.84685e10	0.0951342
$356$	0	0
$357$	1.10835e11	0.361135
$358$	0	0
$359$	4.85699e11	1.54327	0.771636	−	0.636065i	$-0.219439\pi$
$359$	4.85699e11	1.54327	0.771636	+	0.636065i	$0.219439\pi$
$360$	0	0
$361$	−1.19761e11	−0.371135
$362$	0	0
$363$	−1.89539e11	−0.572954
$364$	0	0
$365$	3.41657e10	0.100756
$366$	0	0
$367$	1.61310e11	0.464155	0.232078	−	0.972697i	$-0.425448\pi$
$367$	1.61310e11	0.464155	0.232078	+	0.972697i	$0.425448\pi$
$368$	0	0
$369$	−8.76675e10	−0.246162
$370$	0	0
$371$	−3.04048e11	−0.833220
$372$	0	0
$373$	−1.73064e11	−0.462932	−0.231466	−	0.972843i	$-0.574352\pi$
$373$	−1.73064e11	−0.462932	−0.231466	+	0.972843i	$0.574352\pi$
$374$	0	0
$375$	3.99571e10	0.104340
$376$	0	0
$377$	−5.58424e11	−1.42373
$378$	0	0
$379$	6.65005e11	1.65557	0.827787	−	0.561043i	$-0.189600\pi$
$379$	6.65005e11	1.65557	0.827787	+	0.561043i	$0.189600\pi$
$380$	0	0
$381$	−1.64463e11	−0.399858
$382$	0	0
$383$	−1.90834e11	−0.453171	−0.226585	−	0.973991i	$-0.572756\pi$
$383$	−1.90834e11	−0.453171	−0.226585	+	0.973991i	$0.572756\pi$
$384$	0	0
$385$	3.19153e9	0.00740330
$386$	0	0
$387$	2.94482e10	0.0667359
$388$	0	0
$389$	−5.83430e11	−1.29186	−0.645930	−	0.763397i	$-0.723530\pi$
$389$	−5.83430e11	−1.29186	−0.645930	+	0.763397i	$0.723530\pi$
$390$	0	0
$391$	3.07991e11	0.666412
$392$	0	0
$393$	−9.86740e10	−0.208659
$394$	0	0
$395$	−1.45822e10	−0.0301396
$396$	0	0
$397$	−5.15367e11	−1.04126	−0.520630	−	0.853782i	$-0.674303\pi$
$397$	−5.15367e11	−1.04126	−0.520630	+	0.853782i	$0.674303\pi$
$398$	0	0
$399$	−2.16729e11	−0.428093
$400$	0	0
$401$	9.54840e11	1.84409	0.922043	−	0.387088i	$-0.126519\pi$
$401$	9.54840e11	1.84409	0.922043	+	0.387088i	$0.126519\pi$
$402$	0	0
$403$	6.44537e11	1.21724
$404$	0	0
$405$	−5.45858e9	−0.0100817
$406$	0	0
$407$	8.18829e10	0.147917
$408$	0	0
$409$	−1.02451e12	−1.81034	−0.905168	−	0.425054i	$-0.860255\pi$
$409$	−1.02451e12	−1.81034	−0.905168	+	0.425054i	$0.860255\pi$
$410$	0	0
$411$	−3.93440e11	−0.680127
$412$	0	0
$413$	2.93389e11	0.496214
$414$	0	0
$415$	7.98661e10	0.132174
$416$	0	0
$417$	2.66941e11	0.432318
$418$	0	0
$419$	−3.45721e11	−0.547978	−0.273989	−	0.961733i	$-0.588343\pi$
$419$	−3.45721e11	−0.547978	−0.273989	+	0.961733i	$0.588343\pi$
$420$	0	0
$421$	1.62812e10	0.0252590	0.0126295	−	0.999920i	$-0.495980\pi$
$421$	1.62812e10	0.0252590	0.0126295	+	0.999920i	$0.495980\pi$
$422$	0	0
$423$	8.39490e10	0.127492
$424$	0	0
$425$	4.46242e11	0.663468
$426$	0	0
$427$	−1.02668e11	−0.149455
$428$	0	0
$429$	4.68791e10	0.0668223
$430$	0	0
$431$	7.54079e11	1.05261	0.526307	−	0.850295i	$-0.323576\pi$
$431$	7.54079e11	1.05261	0.526307	+	0.850295i	$0.323576\pi$
$432$	0	0
$433$	5.87173e11	0.802732	0.401366	−	0.915918i	$-0.368535\pi$
$433$	5.87173e11	0.802732	0.401366	+	0.915918i	$0.368535\pi$
$434$	0	0
$435$	4.19945e10	0.0562329
$436$	0	0
$437$	−6.02251e11	−0.789971
$438$	0	0
$439$	−2.07502e11	−0.266643	−0.133322	−	0.991073i	$-0.542564\pi$
$439$	−2.07502e11	−0.266643	−0.133322	+	0.991073i	$0.542564\pi$
$440$	0	0
$441$	−3.32911e10	−0.0419135
$442$	0	0
$443$	−7.76397e11	−0.957783	−0.478891	−	0.877874i	$-0.658961\pi$
$443$	−7.76397e11	−0.957783	−0.478891	+	0.877874i	$0.658961\pi$
$444$	0	0
$445$	−8.06461e10	−0.0974908
$446$	0	0
$447$	4.07603e11	0.482896
$448$	0	0
$449$	−1.20244e12	−1.39623	−0.698113	−	0.715988i	$-0.745977\pi$
$449$	−1.20244e12	−1.39623	−0.698113	+	0.715988i	$0.745977\pi$
$450$	0	0
$451$	−5.66195e10	−0.0644424
$452$	0	0
$453$	−2.23904e11	−0.249816
$454$	0	0
$455$	1.02872e11	0.112524
$456$	0	0
$457$	7.35408e11	0.788688	0.394344	−	0.918963i	$-0.370972\pi$
$457$	7.35408e11	0.788688	0.394344	+	0.918963i	$0.370972\pi$
$458$	0	0
$459$	−1.22429e11	−0.128744
$460$	0	0
$461$	1.28508e12	1.32518	0.662590	−	0.748982i	$-0.269457\pi$
$461$	1.28508e12	1.32518	0.662590	+	0.748982i	$0.269457\pi$
$462$	0	0
$463$	−4.47746e11	−0.452811	−0.226406	−	0.974033i	$-0.572698\pi$
$463$	−4.47746e11	−0.452811	−0.226406	+	0.974033i	$0.572698\pi$
$464$	0	0
$465$	−4.84703e10	−0.0480770
$466$	0	0
$467$	1.21634e12	1.18340	0.591698	−	0.806160i	$-0.298458\pi$
$467$	1.21634e12	1.18340	0.591698	+	0.806160i	$0.298458\pi$
$468$	0	0
$469$	−1.61770e12	−1.54390
$470$	0	0
$471$	−2.58662e11	−0.242181
$472$	0	0
$473$	1.90189e10	0.0174707
$474$	0	0
$475$	−8.72589e11	−0.786481
$476$	0	0
$477$	3.35854e11	0.297042
$478$	0	0
$479$	−1.70420e12	−1.47915	−0.739574	−	0.673075i	$-0.764973\pi$
$479$	−1.70420e12	−1.47915	−0.739574	+	0.673075i	$0.764973\pi$
$480$	0	0
$481$	2.63932e12	2.24822
$482$	0	0
$483$	6.43212e11	0.537764
$484$	0	0
$485$	8.22839e10	0.0675269
$486$	0	0
$487$	−2.60755e11	−0.210064	−0.105032	−	0.994469i	$-0.533495\pi$
$487$	−2.60755e11	−0.210064	−0.105032	+	0.994469i	$0.533495\pi$
$488$	0	0
$489$	−3.13276e11	−0.247763
$490$	0	0
$491$	1.16626e12	0.905586	0.452793	−	0.891616i	$-0.350428\pi$
$491$	1.16626e12	0.905586	0.452793	+	0.891616i	$0.350428\pi$
$492$	0	0
$493$	9.41885e11	0.718103
$494$	0	0
$495$	−3.52539e9	−0.00263927
$496$	0	0
$497$	1.33348e12	0.980352
$498$	0	0
$499$	−2.38683e11	−0.172333	−0.0861664	−	0.996281i	$-0.527462\pi$
$499$	−2.38683e11	−0.172333	−0.0861664	+	0.996281i	$0.527462\pi$
$500$	0	0
$501$	7.26173e11	0.514957
$502$	0	0
$503$	−1.01016e12	−0.703614	−0.351807	−	0.936073i	$-0.614433\pi$
$503$	−1.01016e12	−0.703614	−0.351807	+	0.936073i	$0.614433\pi$
$504$	0	0
$505$	−2.64209e10	−0.0180774
$506$	0	0
$507$	6.52086e11	0.438297
$508$	0	0
$509$	1.36854e12	0.903710	0.451855	−	0.892091i	$-0.350762\pi$
$509$	1.36854e12	0.903710	0.451855	+	0.892091i	$0.350762\pi$
$510$	0	0
$511$	1.60034e12	1.03829
$512$	0	0
$513$	2.39400e11	0.152615
$514$	0	0
$515$	1.47168e11	0.0921891
$516$	0	0
$517$	5.42179e10	0.0333761
$518$	0	0
$519$	−7.86193e11	−0.475638
$520$	0	0
$521$	−2.43050e12	−1.44519	−0.722596	−	0.691271i	$-0.757051\pi$
$521$	−2.43050e12	−1.44519	−0.722596	+	0.691271i	$0.757051\pi$
$522$	0	0
$523$	1.95220e12	1.14095	0.570475	−	0.821315i	$-0.306759\pi$
$523$	1.95220e12	1.14095	0.570475	+	0.821315i	$0.306759\pi$
$524$	0	0
$525$	9.31936e11	0.535389
$526$	0	0
$527$	−1.08713e12	−0.613951
$528$	0	0
$529$	−1.37806e10	−0.00765098
$530$	0	0
$531$	−3.24081e11	−0.176900
$532$	0	0
$533$	−1.82501e12	−0.979475
$534$	0	0
$535$	−8.30999e10	−0.0438539
$536$	0	0
$537$	−2.16791e12	−1.12501
$538$	0	0
$539$	−2.15008e10	−0.0109725
$540$	0	0
$541$	2.91282e12	1.46193	0.730964	−	0.682416i	$-0.239071\pi$
$541$	2.91282e12	1.46193	0.730964	+	0.682416i	$0.239071\pi$
$542$	0	0
$543$	−1.40224e12	−0.692186
$544$	0	0
$545$	3.78156e10	0.0183606
$546$	0	0
$547$	1.98365e12	0.947377	0.473689	−	0.880692i	$-0.342922\pi$
$547$	1.98365e12	0.947377	0.473689	+	0.880692i	$0.342922\pi$
$548$	0	0
$549$	1.13408e11	0.0532807
$550$	0	0
$551$	−1.84178e12	−0.851246
$552$	0	0
$553$	−6.83039e11	−0.310587
$554$	0	0
$555$	−1.98482e11	−0.0887978
$556$	0	0
$557$	3.14140e12	1.38285	0.691425	−	0.722448i	$-0.256983\pi$
$557$	3.14140e12	1.38285	0.691425	+	0.722448i	$0.256983\pi$
$558$	0	0
$559$	6.13036e11	0.265542
$560$	0	0
$561$	−7.90702e10	−0.0337039
$562$	0	0
$563$	−1.06824e12	−0.448107	−0.224054	−	0.974577i	$-0.571929\pi$
$563$	−1.06824e12	−0.448107	−0.224054	+	0.974577i	$0.571929\pi$
$564$	0	0
$565$	1.19707e11	0.0494200
$566$	0	0
$567$	−2.55683e11	−0.103891
$568$	0	0
$569$	3.48022e12	1.39188	0.695940	−	0.718100i	$-0.254988\pi$
$569$	3.48022e12	1.39188	0.695940	+	0.718100i	$0.254988\pi$
$570$	0	0
$571$	6.19984e11	0.244072	0.122036	−	0.992526i	$-0.461058\pi$
$571$	6.19984e11	0.244072	0.122036	+	0.992526i	$0.461058\pi$
$572$	0	0
$573$	−2.92291e11	−0.113271
$574$	0	0
$575$	2.58969e12	0.987966
$576$	0	0
$577$	5.08907e12	1.91138	0.955691	−	0.294372i	$-0.0951105\pi$
$577$	5.08907e12	1.91138	0.955691	+	0.294372i	$0.0951105\pi$
$578$	0	0
$579$	−2.32082e12	−0.858197
$580$	0	0
$581$	3.74097e12	1.36204
$582$	0	0
$583$	2.16909e11	0.0777624
$584$	0	0
$585$	−1.13634e11	−0.0401149
$586$	0	0
$587$	5.10341e11	0.177414	0.0887071	−	0.996058i	$-0.471726\pi$
$587$	5.10341e11	0.177414	0.0887071	+	0.996058i	$0.471726\pi$
$588$	0	0
$589$	2.12579e12	0.727783
$590$	0	0
$591$	−2.04638e12	−0.689990
$592$	0	0
$593$	2.15153e12	0.714498	0.357249	−	0.934009i	$-0.383715\pi$
$593$	2.15153e12	0.714498	0.357249	+	0.934009i	$0.383715\pi$
$594$	0	0
$595$	−1.73513e11	−0.0567552
$596$	0	0
$597$	5.99033e11	0.193004
$598$	0	0
$599$	−3.47411e12	−1.10261	−0.551307	−	0.834303i	$-0.685871\pi$
$599$	−3.47411e12	−1.10261	−0.551307	+	0.834303i	$0.685871\pi$
$600$	0	0
$601$	8.52520e11	0.266544	0.133272	−	0.991079i	$-0.457452\pi$
$601$	8.52520e11	0.266544	0.133272	+	0.991079i	$0.457452\pi$
$602$	0	0
$603$	1.78692e12	0.550399
$604$	0	0
$605$	2.96725e11	0.0900441
$606$	0	0
$607$	−4.73779e12	−1.41653	−0.708267	−	0.705945i	$-0.750523\pi$
$607$	−4.73779e12	−1.41653	−0.708267	+	0.705945i	$0.750523\pi$
$608$	0	0
$609$	1.96704e12	0.579477
$610$	0	0
$611$	1.74760e12	0.507291
$612$	0	0
$613$	7.29889e11	0.208778	0.104389	−	0.994537i	$-0.466711\pi$
$613$	7.29889e11	0.208778	0.104389	+	0.994537i	$0.466711\pi$
$614$	0	0
$615$	1.37244e11	0.0386862
$616$	0	0
$617$	−5.10485e12	−1.41808	−0.709038	−	0.705170i	$-0.750871\pi$
$617$	−5.10485e12	−1.41808	−0.709038	+	0.705170i	$0.750871\pi$
$618$	0	0
$619$	5.41159e12	1.48155	0.740776	−	0.671752i	$-0.234458\pi$
$619$	5.41159e12	1.48155	0.740776	+	0.671752i	$0.234458\pi$
$620$	0	0
$621$	−7.10497e11	−0.191712
$622$	0	0
$623$	−3.77750e12	−1.00464
$624$	0	0
$625$	3.72074e12	0.975369
$626$	0	0
$627$	1.54615e11	0.0399529
$628$	0	0
$629$	−4.45170e12	−1.13396
$630$	0	0
$631$	2.54599e12	0.639329	0.319665	−	0.947531i	$-0.396430\pi$
$631$	2.54599e12	0.639329	0.319665	+	0.947531i	$0.396430\pi$
$632$	0	0
$633$	−6.61508e11	−0.163764
$634$	0	0
$635$	2.57468e11	0.0628407
$636$	0	0
$637$	−6.93034e11	−0.166774
$638$	0	0
$639$	−1.47297e12	−0.349495
$640$	0	0
$641$	2.55558e12	0.597899	0.298950	−	0.954269i	$-0.403364\pi$
$641$	2.55558e12	0.597899	0.298950	+	0.954269i	$0.403364\pi$
$642$	0	0
$643$	1.94766e12	0.449328	0.224664	−	0.974436i	$-0.427872\pi$
$643$	1.94766e12	0.449328	0.224664	+	0.974436i	$0.427872\pi$
$644$	0	0
$645$	−4.61014e10	−0.0104881
$646$	0	0
$647$	−6.82986e12	−1.53229	−0.766147	−	0.642665i	$-0.777829\pi$
$647$	−6.82986e12	−1.53229	−0.766147	+	0.642665i	$0.777829\pi$
$648$	0	0
$649$	−2.09305e11	−0.0463105
$650$	0	0
$651$	−2.27037e12	−0.495431
$652$	0	0
$653$	7.20283e12	1.55022	0.775111	−	0.631825i	$-0.217694\pi$
$653$	7.20283e12	1.55022	0.775111	+	0.631825i	$0.217694\pi$
$654$	0	0
$655$	1.54475e11	0.0327923
$656$	0	0
$657$	−1.76775e12	−0.370149
$658$	0	0
$659$	−3.37156e12	−0.696381	−0.348191	−	0.937424i	$-0.613204\pi$
$659$	−3.37156e12	−0.696381	−0.348191	+	0.937424i	$0.613204\pi$
$660$	0	0
$661$	−5.04657e12	−1.02823	−0.514115	−	0.857721i	$-0.671879\pi$
$661$	−5.04657e12	−1.02823	−0.514115	+	0.857721i	$0.671879\pi$
$662$	0	0
$663$	−2.54866e12	−0.512273
$664$	0	0
$665$	3.39290e11	0.0672781
$666$	0	0
$667$	5.46607e12	1.06932
$668$	0	0
$669$	5.02240e12	0.969379
$670$	0	0
$671$	7.32441e10	0.0139483
$672$	0	0
$673$	4.03136e12	0.757503	0.378751	−	0.925498i	$-0.376354\pi$
$673$	4.03136e12	0.757503	0.378751	+	0.925498i	$0.376354\pi$
$674$	0	0
$675$	−1.02943e12	−0.190866
$676$	0	0
$677$	−4.67581e12	−0.855476	−0.427738	−	0.903903i	$-0.640689\pi$
$677$	−4.67581e12	−0.855476	−0.427738	+	0.903903i	$0.640689\pi$
$678$	0	0
$679$	3.85422e12	0.695861
$680$	0	0
$681$	1.71737e12	0.305986
$682$	0	0
$683$	4.04924e12	0.712001	0.356001	−	0.934486i	$-0.384140\pi$
$683$	4.04924e12	0.712001	0.356001	+	0.934486i	$0.384140\pi$
$684$	0	0
$685$	6.15933e11	0.106887
$686$	0	0
$687$	1.90289e12	0.325919
$688$	0	0
$689$	6.99162e12	1.18193
$690$	0	0
$691$	7.85336e12	1.31040	0.655200	−	0.755455i	$-0.272584\pi$
$691$	7.85336e12	1.31040	0.655200	+	0.755455i	$0.272584\pi$
$692$	0	0
$693$	−1.65131e11	−0.0271975
$694$	0	0
$695$	−4.17899e11	−0.0679421
$696$	0	0
$697$	3.07822e12	0.494028
$698$	0	0
$699$	−2.95179e12	−0.467668
$700$	0	0
$701$	9.77407e12	1.52878	0.764389	−	0.644756i	$-0.223041\pi$
$701$	9.77407e12	1.52878	0.764389	+	0.644756i	$0.223041\pi$
$702$	0	0
$703$	8.70494e12	1.34421
$704$	0	0
$705$	−1.31423e11	−0.0200364
$706$	0	0
$707$	−1.23757e12	−0.186286
$708$	0	0
$709$	−1.08047e13	−1.60585	−0.802924	−	0.596082i	$-0.796723\pi$
$709$	−1.08047e13	−1.60585	−0.802924	+	0.596082i	$0.796723\pi$
$710$	0	0
$711$	7.54492e11	0.110724
$712$	0	0
$713$	−6.30897e12	−0.914230
$714$	0	0
$715$	−7.33895e10	−0.0105016
$716$	0	0
$717$	7.32179e12	1.03462
$718$	0	0
$719$	1.39028e12	0.194009	0.0970047	−	0.995284i	$-0.469074\pi$
$719$	1.39028e12	0.194009	0.0970047	+	0.995284i	$0.469074\pi$
$720$	0	0
$721$	6.89340e12	0.950003
$722$	0	0
$723$	3.07834e12	0.418980
$724$	0	0
$725$	7.91967e12	1.06460
$726$	0	0
$727$	−6.60854e12	−0.877406	−0.438703	−	0.898632i	$-0.644562\pi$
$727$	−6.60854e12	−0.877406	−0.438703	+	0.898632i	$0.644562\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−1.03400e12	−0.133934
$732$	0	0
$733$	−7.06680e12	−0.904180	−0.452090	−	0.891972i	$-0.649321\pi$
$733$	−7.06680e12	−0.904180	−0.452090	+	0.891972i	$0.649321\pi$
$734$	0	0
$735$	5.21174e10	0.00658703
$736$	0	0
$737$	1.15407e12	0.144089
$738$	0	0
$739$	4.31518e12	0.532230	0.266115	−	0.963941i	$-0.414260\pi$
$739$	4.31518e12	0.532230	0.266115	+	0.963941i	$0.414260\pi$
$740$	0	0
$741$	4.98370e12	0.607253
$742$	0	0
$743$	1.46782e13	1.76694	0.883472	−	0.468484i	$-0.155200\pi$
$743$	1.46782e13	1.76694	0.883472	+	0.468484i	$0.155200\pi$
$744$	0	0
$745$	−6.38105e11	−0.0758907
$746$	0	0
$747$	−4.13231e12	−0.485568
$748$	0	0
$749$	−3.89244e12	−0.451912
$750$	0	0
$751$	8.75994e12	1.00490	0.502448	−	0.864607i	$-0.332433\pi$
$751$	8.75994e12	1.00490	0.502448	+	0.864607i	$0.332433\pi$
$752$	0	0
$753$	−7.65743e12	−0.867972
$754$	0	0
$755$	3.50523e11	0.0392605
$756$	0	0
$757$	−1.42430e13	−1.57642	−0.788209	−	0.615408i	$-0.788991\pi$
$757$	−1.42430e13	−1.57642	−0.788209	+	0.615408i	$0.788991\pi$
$758$	0	0
$759$	−4.58870e11	−0.0501882
$760$	0	0
$761$	−1.15114e13	−1.24422	−0.622109	−	0.782931i	$-0.713724\pi$
$761$	−1.15114e13	−1.24422	−0.622109	+	0.782931i	$0.713724\pi$
$762$	0	0
$763$	1.77130e12	0.189205
$764$	0	0
$765$	1.91664e11	0.0202332
$766$	0	0
$767$	−6.74652e12	−0.703884
$768$	0	0
$769$	2.69503e12	0.277905	0.138952	−	0.990299i	$-0.455627\pi$
$769$	2.69503e12	0.277905	0.138952	+	0.990299i	$0.455627\pi$
$770$	0	0
$771$	−2.98344e12	−0.304070
$772$	0	0
$773$	1.36336e13	1.37342	0.686710	−	0.726931i	$-0.259054\pi$
$773$	1.36336e13	1.37342	0.686710	+	0.726931i	$0.259054\pi$
$774$	0	0
$775$	−9.14094e12	−0.910192
$776$	0	0
$777$	−9.29698e12	−0.915055
$778$	0	0
$779$	−6.01920e12	−0.585626
$780$	0	0
$781$	−9.51309e11	−0.0914939
$782$	0	0
$783$	−2.17281e12	−0.206583
$784$	0	0
$785$	4.04938e11	0.0380605
$786$	0	0
$787$	9.76393e12	0.907274	0.453637	−	0.891187i	$-0.350126\pi$
$787$	9.76393e12	0.907274	0.453637	+	0.891187i	$0.350126\pi$
$788$	0	0
$789$	−5.65505e12	−0.519505
$790$	0	0
$791$	5.60715e12	0.509270
$792$	0	0
$793$	2.36087e12	0.212003
$794$	0	0
$795$	−5.25782e11	−0.0466825
$796$	0	0
$797$	−9.96680e12	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$797$	−9.96680e12	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$798$	0	0
$799$	−2.94765e12	−0.255868
$800$	0	0
$801$	4.17267e12	0.358152
$802$	0	0
$803$	−1.14169e12	−0.0969009
$804$	0	0
$805$	−1.00695e12	−0.0845137
$806$	0	0
$807$	−4.92632e12	−0.408876
$808$	0	0
$809$	−4.77919e12	−0.392271	−0.196135	−	0.980577i	$-0.562839\pi$
$809$	−4.77919e12	−0.392271	−0.196135	+	0.980577i	$0.562839\pi$
$810$	0	0
$811$	−1.25488e13	−1.01861	−0.509307	−	0.860585i	$-0.670098\pi$
$811$	−1.25488e13	−1.01861	−0.509307	+	0.860585i	$0.670098\pi$
$812$	0	0
$813$	−4.32257e11	−0.0347004
$814$	0	0
$815$	4.90436e11	0.0389379
$816$	0	0
$817$	2.02190e12	0.158767
$818$	0	0
$819$	−5.32265e12	−0.413381
$820$	0	0
$821$	−1.77488e13	−1.36340	−0.681701	−	0.731631i	$-0.738760\pi$
$821$	−1.77488e13	−1.36340	−0.681701	+	0.731631i	$0.738760\pi$
$822$	0	0
$823$	2.59964e13	1.97521	0.987607	−	0.156947i	$-0.0501653\pi$
$823$	2.59964e13	1.97521	0.987607	+	0.156947i	$0.0501653\pi$
$824$	0	0
$825$	−6.64848e11	−0.0499666
$826$	0	0
$827$	−7.89884e12	−0.587203	−0.293602	−	0.955928i	$-0.594854\pi$
$827$	−7.89884e12	−0.587203	−0.293602	+	0.955928i	$0.594854\pi$
$828$	0	0
$829$	−1.75453e12	−0.129023	−0.0645113	−	0.997917i	$-0.520549\pi$
$829$	−1.75453e12	−0.129023	−0.0645113	+	0.997917i	$0.520549\pi$
$830$	0	0
$831$	−1.08880e13	−0.792035
$832$	0	0
$833$	1.16893e12	0.0841174
$834$	0	0
$835$	−1.13683e12	−0.0809294
$836$	0	0
$837$	2.50788e12	0.176621
$838$	0	0
$839$	−2.22719e13	−1.55177	−0.775886	−	0.630873i	$-0.782697\pi$
$839$	−2.22719e13	−1.55177	−0.775886	+	0.630873i	$0.782697\pi$
$840$	0	0
$841$	2.20896e12	0.152267
$842$	0	0
$843$	−1.55593e13	−1.06112
$844$	0	0
$845$	−1.02084e12	−0.0688818
$846$	0	0
$847$	1.38988e13	0.927898
$848$	0	0
$849$	−3.82595e12	−0.252729
$850$	0	0
$851$	−2.58347e13	−1.68857
$852$	0	0
$853$	−1.05383e13	−0.681555	−0.340777	−	0.940144i	$-0.610690\pi$
$853$	−1.05383e13	−0.681555	−0.340777	+	0.940144i	$0.610690\pi$
$854$	0	0
$855$	−3.74783e11	−0.0239846
$856$	0	0
$857$	−1.26793e13	−0.802936	−0.401468	−	0.915873i	$-0.631500\pi$
$857$	−1.26793e13	−0.802936	−0.401468	+	0.915873i	$0.631500\pi$
$858$	0	0
$859$	1.77348e13	1.11136	0.555682	−	0.831395i	$-0.312457\pi$
$859$	1.77348e13	1.11136	0.555682	+	0.831395i	$0.312457\pi$
$860$	0	0
$861$	6.42858e12	0.398658
$862$	0	0
$863$	−2.02098e12	−0.124026	−0.0620131	−	0.998075i	$-0.519752\pi$
$863$	−2.02098e12	−0.124026	−0.0620131	+	0.998075i	$0.519752\pi$
$864$	0	0
$865$	1.23079e12	0.0747501
$866$	0	0
$867$	−5.30683e12	−0.318969
$868$	0	0
$869$	4.87284e11	0.0289863
$870$	0	0
$871$	3.71991e13	2.19004
$872$	0	0
$873$	−4.25740e12	−0.248074
$874$	0	0
$875$	−2.93001e12	−0.168979
$876$	0	0
$877$	8.27496e12	0.472354	0.236177	−	0.971710i	$-0.424105\pi$
$877$	8.27496e12	0.472354	0.236177	+	0.971710i	$0.424105\pi$
$878$	0	0
$879$	−1.39524e12	−0.0788316
$880$	0	0
$881$	−2.74131e13	−1.53309	−0.766543	−	0.642193i	$-0.778025\pi$
$881$	−2.74131e13	−1.53309	−0.766543	+	0.642193i	$0.778025\pi$
$882$	0	0
$883$	5.35680e12	0.296539	0.148270	−	0.988947i	$-0.452630\pi$
$883$	5.35680e12	0.296539	0.148270	+	0.988947i	$0.452630\pi$
$884$	0	0
$885$	5.07350e11	0.0278012
$886$	0	0
$887$	−5.91189e12	−0.320679	−0.160339	−	0.987062i	$-0.551259\pi$
$887$	−5.91189e12	−0.320679	−0.160339	+	0.987062i	$0.551259\pi$
$888$	0	0
$889$	1.20599e13	0.647569
$890$	0	0
$891$	1.82405e11	0.00969589
$892$	0	0
$893$	5.76389e12	0.303308
$894$	0	0
$895$	3.39387e12	0.176804
$896$	0	0
$897$	−1.47907e13	−0.762822
$898$	0	0
$899$	−1.92938e13	−0.985144
$900$	0	0
$901$	−1.17927e13	−0.596142
$902$	0	0
$903$	−2.15941e12	−0.108079
$904$	0	0
$905$	2.19521e12	0.108782
$906$	0	0
$907$	−1.45625e13	−0.714502	−0.357251	−	0.934008i	$-0.616286\pi$
$907$	−1.45625e13	−0.714502	−0.357251	+	0.934008i	$0.616286\pi$
$908$	0	0
$909$	1.36703e12	0.0664109
$910$	0	0
$911$	−9.50127e12	−0.457035	−0.228517	−	0.973540i	$-0.573388\pi$
$911$	−9.50127e12	−0.457035	−0.228517	+	0.973540i	$0.573388\pi$
$912$	0	0
$913$	−2.66882e12	−0.127116
$914$	0	0
$915$	−1.77542e11	−0.00837346
$916$	0	0
$917$	7.23568e12	0.337923
$918$	0	0
$919$	2.17748e13	1.00701	0.503506	−	0.863992i	$-0.332043\pi$
$919$	2.17748e13	1.00701	0.503506	+	0.863992i	$0.332043\pi$
$920$	0	0
$921$	−7.63159e12	−0.349500
$922$	0	0
$923$	−3.06635e13	−1.39064
$924$	0	0
$925$	−3.74313e13	−1.68112
$926$	0	0
$927$	−7.61452e12	−0.338675
$928$	0	0
$929$	−3.48742e13	−1.53615	−0.768075	−	0.640360i	$-0.778785\pi$
$929$	−3.48742e13	−1.53615	−0.768075	+	0.640360i	$0.778785\pi$
$930$	0	0
$931$	−2.28574e12	−0.0997135
$932$	0	0
$933$	6.44238e12	0.278342
$934$	0	0
$935$	1.23785e11	0.00529682
$936$	0	0
$937$	−5.31211e12	−0.225133	−0.112566	−	0.993644i	$-0.535907\pi$
$937$	−5.31211e12	−0.225133	−0.112566	+	0.993644i	$0.535907\pi$
$938$	0	0
$939$	−5.08505e12	−0.213452
$940$	0	0
$941$	−6.60648e11	−0.0274673	−0.0137337	−	0.999906i	$-0.504372\pi$
$941$	−6.60648e11	−0.0274673	−0.0137337	+	0.999906i	$0.504372\pi$
$942$	0	0
$943$	1.78639e13	0.735654
$944$	0	0
$945$	4.00273e11	0.0163273
$946$	0	0
$947$	−7.16022e11	−0.0289302	−0.0144651	−	0.999895i	$-0.504605\pi$
$947$	−7.16022e11	−0.0289302	−0.0144651	+	0.999895i	$0.504605\pi$
$948$	0	0
$949$	−3.67999e13	−1.47282
$950$	0	0
$951$	−7.68423e12	−0.304641
$952$	0	0
$953$	−3.31789e13	−1.30300	−0.651500	−	0.758649i	$-0.725860\pi$
$953$	−3.31789e13	−1.30300	−0.651500	+	0.758649i	$0.725860\pi$
$954$	0	0
$955$	4.57584e11	0.0178015
$956$	0	0
$957$	−1.40330e12	−0.0540812
$958$	0	0
$959$	2.88506e13	1.10147
$960$	0	0
$961$	−4.17057e12	−0.157739
$962$	0	0
$963$	4.29963e12	0.161106
$964$	0	0
$965$	3.63325e12	0.134872
$966$	0	0
$967$	3.96440e13	1.45800	0.729002	−	0.684511i	$-0.239984\pi$
$967$	3.96440e13	1.45800	0.729002	+	0.684511i	$0.239984\pi$
$968$	0	0
$969$	−8.40593e12	−0.306287
$970$	0	0
$971$	5.13178e13	1.85260	0.926300	−	0.376786i	$-0.122971\pi$
$971$	5.13178e13	1.85260	0.926300	+	0.376786i	$0.122971\pi$
$972$	0	0
$973$	−1.95746e13	−0.700140
$974$	0	0
$975$	−2.14300e13	−0.759453
$976$	0	0
$977$	2.68486e12	0.0942750	0.0471375	−	0.998888i	$-0.484990\pi$
$977$	2.68486e12	0.0942750	0.0471375	+	0.998888i	$0.484990\pi$
$978$	0	0
$979$	2.69489e12	0.0937603
$980$	0	0
$981$	−1.95660e12	−0.0674514
$982$	0	0
$983$	3.79462e12	0.129622	0.0648109	−	0.997898i	$-0.479356\pi$
$983$	3.79462e12	0.129622	0.0648109	+	0.997898i	$0.479356\pi$
$984$	0	0
$985$	3.20362e12	0.108437
$986$	0	0
$987$	−6.15591e12	−0.206474
$988$	0	0
$989$	−6.00063e12	−0.199440
$990$	0	0
$991$	7.48099e12	0.246393	0.123196	−	0.992382i	$-0.460686\pi$
$991$	7.48099e12	0.246393	0.123196	+	0.992382i	$0.460686\pi$
$992$	0	0
$993$	−3.41422e13	−1.11434
$994$	0	0
$995$	−9.37790e11	−0.0303320
$996$	0	0
$997$	4.91760e13	1.57625	0.788125	−	0.615515i	$-0.211052\pi$
$997$	4.91760e13	1.57625	0.788125	+	0.615515i	$0.211052\pi$
$998$	0	0
$999$	1.02695e13	0.326217

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.a.f.1.3	yes	4
4.3	odd	2		384.10.a.b.1.3	✓	4
8.3	odd	2		384.10.a.g.1.2	yes	4
8.5	even	2		384.10.a.c.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.a.b.1.3	✓	4	4.3	odd	2
384.10.a.c.1.2	yes	4	8.5	even	2
384.10.a.f.1.3	yes	4	1.1	even	1	trivial
384.10.a.g.1.2	yes	4	8.3	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)

\(f(q)\)	\(=\)	\(q+81.0000 q^{3} -126.806 q^{5} -5939.66 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q+81.0000 q^{3} -126.806 q^{5} -5939.66 q^{7} +6561.00 q^{9} +4237.38 q^{11} +136583. q^{13} -10271.3 q^{15} -230372. q^{17} +450474. q^{19} -481112. q^{21} -1.33693e6 q^{23} -1.93705e6 q^{25} +531441. q^{27} -4.08853e6 q^{29} +4.71901e6 q^{31} +343228. q^{33} +753184. q^{35} +1.93239e7 q^{37} +1.10632e7 q^{39} -1.33619e7 q^{41} +4.48837e6 q^{43} -831974. q^{45} +1.27952e7 q^{47} -5.07408e6 q^{49} -1.86602e7 q^{51} +5.11895e7 q^{53} -537325. q^{55} +3.64884e7 q^{57} -4.93950e7 q^{59} +1.72852e7 q^{61} -3.89701e7 q^{63} -1.73196e7 q^{65} +2.72355e8 q^{67} -1.08291e8 q^{69} -2.24504e8 q^{71} -2.69433e8 q^{73} -1.56901e8 q^{75} -2.51686e7 q^{77} +1.14996e8 q^{79} +4.30467e7 q^{81} -6.29829e8 q^{83} +2.92126e7 q^{85} -3.31171e8 q^{87} +6.35980e8 q^{89} -8.11257e8 q^{91} +3.82240e8 q^{93} -5.71228e7 q^{95} -6.48896e8 q^{97} +2.78014e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 324 q^{3} - 240 q^{5} + 4840 q^{7} + 26244 q^{9}+O(q^{10}) \) 4 * q + 324 * q^3 - 240 * q^5 + 4840 * q^7 + 26244 * q^9	\( 4 q + 324 q^{3} - 240 q^{5} + 4840 q^{7} + 26244 q^{9} - 99664 q^{11} - 60840 q^{13} - 19440 q^{15} - 434952 q^{17} + 631776 q^{19} + 392040 q^{21} + 749392 q^{23} + 5991532 q^{25} + 2125764 q^{27} - 7908544 q^{29} + 11351240 q^{31} - 8072784 q^{33} - 25567008 q^{35} - 13592920 q^{37} - 4928040 q^{39} - 18838888 q^{41} + 14177920 q^{43} - 1574640 q^{45} - 37779120 q^{47} + 9409332 q^{49} - 35231112 q^{51} + 115336512 q^{53} - 184580544 q^{55} + 51173856 q^{57} - 115028080 q^{59} + 173228648 q^{61} + 31755240 q^{63} - 328077984 q^{65} - 231785104 q^{67} + 60700752 q^{69} + 197476208 q^{71} + 44629400 q^{73} + 485314092 q^{75} - 308117920 q^{77} - 355774584 q^{79} + 172186884 q^{81} - 607613328 q^{83} + 1087351392 q^{85} - 640592064 q^{87} - 1157146424 q^{89} - 847629840 q^{91} + 919450440 q^{93} - 329699328 q^{95} - 1599536472 q^{97} - 653895504 q^{99}+O(q^{100}) \) 4 * q + 324 * q^3 - 240 * q^5 + 4840 * q^7 + 26244 * q^9 - 99664 * q^11 - 60840 * q^13 - 19440 * q^15 - 434952 * q^17 + 631776 * q^19 + 392040 * q^21 + 749392 * q^23 + 5991532 * q^25 + 2125764 * q^27 - 7908544 * q^29 + 11351240 * q^31 - 8072784 * q^33 - 25567008 * q^35 - 13592920 * q^37 - 4928040 * q^39 - 18838888 * q^41 + 14177920 * q^43 - 1574640 * q^45 - 37779120 * q^47 + 9409332 * q^49 - 35231112 * q^51 + 115336512 * q^53 - 184580544 * q^55 + 51173856 * q^57 - 115028080 * q^59 + 173228648 * q^61 + 31755240 * q^63 - 328077984 * q^65 - 231785104 * q^67 + 60700752 * q^69 + 197476208 * q^71 + 44629400 * q^73 + 485314092 * q^75 - 308117920 * q^77 - 355774584 * q^79 + 172186884 * q^81 - 607613328 * q^83 + 1087351392 * q^85 - 640592064 * q^87 - 1157146424 * q^89 - 847629840 * q^91 + 919450440 * q^93 - 329699328 * q^95 - 1599536472 * q^97 - 653895504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more