LMFDB - Embedded newform 400.8.c.s.49.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [400,8,Mod(49,400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("400.49"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,4552] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$124.954010194$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{649})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 325x^{2} + 26244 $ x^4 + 325*x^2 + 26244
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$-13.2377i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.8.c.s.49.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.47548i q^{3} +769.970i q^{7} +2157.02 q^{9} -5356.43 q^{11} +14037.0i q^{13} -12402.4i q^{17} -5038.89 q^{19} -4215.95 q^{21} +75190.5i q^{23} +23785.6i q^{27} -195529. q^{29} +93568.2 q^{31} -29329.0i q^{33} -161554. i q^{37} -76859.3 q^{39} -28767.5 q^{41} -739076. i q^{43} +1.06799e6i q^{47} +230689. q^{49} +67908.9 q^{51} +626442. i q^{53} -27590.3i q^{57} +2.14861e6 q^{59} -2.57497e6 q^{61} +1.66084e6i q^{63} -807676. i q^{67} -411704. q^{69} +1.72722e6 q^{71} -1.74519e6i q^{73} -4.12429e6i q^{77} -2.46887e6 q^{79} +4.58716e6 q^{81} -6.90850e6i q^{83} -1.07061e6i q^{87} -2.63567e6 q^{89} -1.08081e7 q^{91} +512331. i q^{93} +1.01234e7i q^{97} -1.15539e7 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4552 q^{9} - 8688 q^{11} + 36400 q^{19} - 133032 q^{21} - 111600 q^{29} + 603552 q^{31} + 177616 q^{39} - 216972 q^{41} - 1645172 q^{49} + 638992 q^{51} + 4135200 q^{59} + 1164088 q^{61} - 9582936 q^{69}+ \cdots - 22866944 q^{99}+O(q^{100}) $ 4 * q + 4552 * q^9 - 8688 * q^11 + 36400 * q^19 - 133032 * q^21 - 111600 * q^29 + 603552 * q^31 + 177616 * q^39 - 216972 * q^41 - 1645172 * q^49 + 638992 * q^51 + 4135200 * q^59 + 1164088 * q^61 - 9582936 * q^69 + 9456432 * q^71 - 14372400 * q^79 + 157124 * q^81 + 11981700 * q^89 - 11634528 * q^91 - 22866944 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	5.47548i	0.117084i	0.998285	+	0.0585420i	$0.0186452\pi$
$3$	5.47548i	0.117084i	−0.998285	+	0.0585420i	$0.981355\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	769.970i	0.848459i	0.905555	+	0.424229i	$0.139455\pi$
$7$	769.970i	0.848459i	−0.905555	+	0.424229i	$0.860545\pi$
$8$	0	0
$9$	2157.02	0.986291
$10$	0	0
$11$	−5356.43	−1.21339	−0.606696	−	0.794934i	$-0.707506\pi$
$11$	−5356.43	−1.21339	−0.606696	+	0.794934i	$0.707506\pi$
$12$	0	0
$13$	14037.0i	1.77204i	0.463651	+	0.886018i	$0.346539\pi$
$13$	14037.0i	1.77204i	−0.463651	+	0.886018i	$0.653461\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 12402.4i	− 0.612256i	−0.951990	−	0.306128i	$-0.900966\pi$
$17$	− 12402.4i	− 0.612256i	0.951990	−	0.306128i	$-0.0990336\pi$
$18$	0	0
$19$	−5038.89	−0.168538	−0.0842689	−	0.996443i	$-0.526855\pi$
$19$	−5038.89	−0.168538	−0.0842689	+	0.996443i	$0.526855\pi$
$20$	0	0
$21$	−4215.95	−0.0993410
$22$	0	0
$23$	75190.5i	1.28859i	0.764776	+	0.644296i	$0.222849\pi$
$23$	75190.5i	1.28859i	−0.764776	+	0.644296i	$0.777151\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	23785.6i	0.232563i
$28$	0	0
$29$	−195529.	−1.48874	−0.744368	−	0.667770i	$-0.767249\pi$
$29$	−195529.	−1.48874	−0.744368	+	0.667770i	$0.767249\pi$
$30$	0	0
$31$	93568.2	0.564108	0.282054	−	0.959399i	$-0.408984\pi$
$31$	93568.2	0.564108	0.282054	+	0.959399i	$0.408984\pi$
$32$	0	0
$33$	− 29329.0i	− 0.142069i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 161554.i	− 0.524338i	−0.965022	−	0.262169i	$-0.915562\pi$
$37$	− 161554.i	− 0.524338i	0.965022	−	0.262169i	$-0.0844379\pi$
$38$	0	0
$39$	−76859.3	−0.207477
$40$	0	0
$41$	−28767.5	−0.0651867	−0.0325933	−	0.999469i	$-0.510377\pi$
$41$	−28767.5	−0.0651867	−0.0325933	+	0.999469i	$0.510377\pi$
$42$	0	0
$43$	− 739076.i	− 1.41759i	−0.705417	−	0.708793i	$-0.749240\pi$
$43$	− 739076.i	− 1.41759i	0.705417	−	0.708793i	$-0.250760\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.06799e6i	1.50046i	0.661178	+	0.750229i	$0.270057\pi$
$47$	1.06799e6i	1.50046i	−0.661178	+	0.750229i	$0.729943\pi$
$48$	0	0
$49$	230689.	0.280118
$50$	0	0
$51$	67908.9	0.0716855
$52$	0	0
$53$	626442.i	0.577983i	0.957332	+	0.288992i	$0.0933200\pi$
$53$	626442.i	0.577983i	−0.957332	+	0.288992i	$0.906680\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 27590.3i	− 0.0197331i
$58$	0	0
$59$	2.14861e6	1.36199	0.680997	−	0.732287i	$-0.261547\pi$
$59$	2.14861e6	1.36199	0.680997	+	0.732287i	$0.261547\pi$
$60$	0	0
$61$	−2.57497e6	−1.45251	−0.726253	−	0.687428i	$-0.758740\pi$
$61$	−2.57497e6	−1.45251	−0.726253	+	0.687428i	$0.758740\pi$
$62$	0	0
$63$	1.66084e6i	0.836827i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 807676.i	− 0.328077i	−0.986454	−	0.164038i	$-0.947548\pi$
$67$	− 807676.i	− 0.328077i	0.986454	−	0.164038i	$-0.0524521\pi$
$68$	0	0
$69$	−411704.	−0.150874
$70$	0	0
$71$	1.72722e6	0.572722	0.286361	−	0.958122i	$-0.407554\pi$
$71$	1.72722e6	0.572722	0.286361	+	0.958122i	$0.407554\pi$
$72$	0	0
$73$	− 1.74519e6i	− 0.525064i	−0.964923	−	0.262532i	$-0.915442\pi$
$73$	− 1.74519e6i	− 0.525064i	0.964923	−	0.262532i	$-0.0845575\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.12429e6i	− 1.02951i
$78$	0	0
$79$	−2.46887e6	−0.563382	−0.281691	−	0.959505i	$-0.590895\pi$
$79$	−2.46887e6	−0.563382	−0.281691	+	0.959505i	$0.590895\pi$
$80$	0	0
$81$	4.58716e6	0.959062
$82$	0	0
$83$	− 6.90850e6i	− 1.32620i	−0.748529	−	0.663102i	$-0.769240\pi$
$83$	− 6.90850e6i	− 1.32620i	0.748529	−	0.663102i	$-0.230760\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1.07061e6i	− 0.174307i
$88$	0	0
$89$	−2.63567e6	−0.396302	−0.198151	−	0.980171i	$-0.563494\pi$
$89$	−2.63567e6	−0.396302	−0.198151	+	0.980171i	$0.563494\pi$
$90$	0	0
$91$	−1.08081e7	−1.50350
$92$	0	0
$93$	512331.i	0.0660480i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.01234e7i	1.12622i	0.826381	+	0.563112i	$0.190396\pi$
$97$	1.01234e7i	1.12622i	−0.826381	+	0.563112i	$0.809604\pi$
$98$	0	0
$99$	−1.15539e7	−1.19676
$100$	0	0
$101$	−8.33196e6	−0.804679	−0.402339	−	0.915491i	$-0.631803\pi$
$101$	−8.33196e6	−0.804679	−0.402339	+	0.915491i	$0.631803\pi$
$102$	0	0
$103$	7.75782e6i	0.699535i	0.936837	+	0.349767i	$0.113739\pi$
$103$	7.75782e6i	0.699535i	−0.936837	+	0.349767i	$0.886261\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.86598e6i	0.699654i	0.936814	+	0.349827i	$0.113760\pi$
$107$	8.86598e6i	0.699654i	−0.936814	+	0.349827i	$0.886240\pi$
$108$	0	0
$109$	−1.55125e6	−0.114734	−0.0573668	−	0.998353i	$-0.518270\pi$
$109$	−1.55125e6	−0.114734	−0.0573668	+	0.998353i	$0.518270\pi$
$110$	0	0
$111$	884585.	0.0613917
$112$	0	0
$113$	− 1.80311e7i	− 1.17557i	−0.809019	−	0.587783i	$-0.800001\pi$
$113$	− 1.80311e7i	− 1.17557i	0.809019	−	0.587783i	$-0.199999\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.02781e7i	1.74774i
$118$	0	0
$119$	9.54945e6	0.519474
$120$	0	0
$121$	9.20422e6	0.472322
$122$	0	0
$123$	− 157516.i	− 0.00763232i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1.98814e7i	− 0.861258i	−0.902529	−	0.430629i	$-0.858292\pi$
$127$	− 1.98814e7i	− 0.861258i	0.902529	−	0.430629i	$-0.141708\pi$
$128$	0	0
$129$	4.04679e6	0.165977
$130$	0	0
$131$	5.81990e6	0.226186	0.113093	−	0.993584i	$-0.463924\pi$
$131$	5.81990e6	0.226186	0.113093	+	0.993584i	$0.463924\pi$
$132$	0	0
$133$	− 3.87979e6i	− 0.142997i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.83328e7i	− 1.27365i	−0.771010	−	0.636823i	$-0.780248\pi$
$137$	− 3.83328e7i	− 1.27365i	0.771010	−	0.636823i	$-0.219752\pi$
$138$	0	0
$139$	−3.60826e7	−1.13958	−0.569791	−	0.821789i	$-0.692976\pi$
$139$	−3.60826e7	−1.13958	−0.569791	+	0.821789i	$0.692976\pi$
$140$	0	0
$141$	−5.84775e6	−0.175680
$142$	0	0
$143$	− 7.51883e7i	− 2.15018i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.26313e6i	0.0327973i
$148$	0	0
$149$	4.91341e6	0.121683	0.0608416	−	0.998147i	$-0.480622\pi$
$149$	4.91341e6	0.121683	0.0608416	+	0.998147i	$0.480622\pi$
$150$	0	0
$151$	−4.95186e7	−1.17044	−0.585220	−	0.810874i	$-0.698992\pi$
$151$	−4.95186e7	−1.17044	−0.585220	+	0.810874i	$0.698992\pi$
$152$	0	0
$153$	− 2.67521e7i	− 0.603863i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.84517e6i	0.120545i	0.998182	+	0.0602724i	$0.0191969\pi$
$157$	5.84517e6i	0.120545i	−0.998182	+	0.0602724i	$0.980803\pi$
$158$	0	0
$159$	−3.43007e6	−0.0676726
$160$	0	0
$161$	−5.78944e7	−1.09332
$162$	0	0
$163$	− 1.79731e7i	− 0.325062i	−0.986703	−	0.162531i	$-0.948034\pi$
$163$	− 1.79731e7i	− 0.325062i	0.986703	−	0.162531i	$-0.0519658\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.72460e7i	0.951124i	0.879682	+	0.475562i	$0.157755\pi$
$167$	5.72460e7i	0.951124i	−0.879682	+	0.475562i	$0.842245\pi$
$168$	0	0
$169$	−1.34289e8	−2.14011
$170$	0	0
$171$	−1.08690e7	−0.166227
$172$	0	0
$173$	− 1.09265e8i	− 1.60442i	−0.597039	−	0.802212i	$-0.703656\pi$
$173$	− 1.09265e8i	− 1.60442i	0.597039	−	0.802212i	$-0.296344\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.17647e7i	0.159468i
$178$	0	0
$179$	1.98072e7	0.258130	0.129065	−	0.991636i	$-0.458802\pi$
$179$	1.98072e7	0.258130	0.129065	+	0.991636i	$0.458802\pi$
$180$	0	0
$181$	−1.00547e8	−1.26036	−0.630178	−	0.776451i	$-0.717018\pi$
$181$	−1.00547e8	−1.26036	−0.630178	+	0.776451i	$0.717018\pi$
$182$	0	0
$183$	− 1.40992e7i	− 0.170065i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.64325e7i	0.742908i
$188$	0	0
$189$	−1.83142e7	−0.197320
$190$	0	0
$191$	−2.28106e7	−0.236875	−0.118437	−	0.992962i	$-0.537789\pi$
$191$	−2.28106e7	−0.236875	−0.118437	+	0.992962i	$0.537789\pi$
$192$	0	0
$193$	− 1.92250e6i	− 0.0192493i	−0.999954	−	0.00962465i	$-0.996936\pi$
$193$	− 1.92250e6i	− 0.0192493i	0.999954	−	0.00962465i	$-0.00306367\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.98118e7i	0.184626i	0.995730	+	0.0923129i	$0.0294260\pi$
$197$	1.98118e7i	0.184626i	−0.995730	+	0.0923129i	$0.970574\pi$
$198$	0	0
$199$	−1.14071e8	−1.02610	−0.513048	−	0.858360i	$-0.671484\pi$
$199$	−1.14071e8	−1.02610	−0.513048	+	0.858360i	$0.671484\pi$
$200$	0	0
$201$	4.42241e6	0.0384125
$202$	0	0
$203$	− 1.50551e8i	− 1.26313i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.62187e8i	1.27093i
$208$	0	0
$209$	2.69905e7	0.204503
$210$	0	0
$211$	−9.50285e6	−0.0696410	−0.0348205	−	0.999394i	$-0.511086\pi$
$211$	−9.50285e6	−0.0696410	−0.0348205	+	0.999394i	$0.511086\pi$
$212$	0	0
$213$	9.45736e6i	0.0670566i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.20447e7i	0.478622i
$218$	0	0
$219$	9.55574e6	0.0614766
$220$	0	0
$221$	1.74092e8	1.08494
$222$	0	0
$223$	− 1.92299e8i	− 1.16121i	−0.814187	−	0.580603i	$-0.802817\pi$
$223$	− 1.92299e8i	− 1.16121i	0.814187	−	0.580603i	$-0.197183\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.60189e8i	− 0.908955i	−0.890758	−	0.454478i	$-0.849826\pi$
$227$	− 1.60189e8i	− 0.908955i	0.890758	−	0.454478i	$-0.150174\pi$
$228$	0	0
$229$	−1.97660e8	−1.08766	−0.543832	−	0.839194i	$-0.683027\pi$
$229$	−1.97660e8	−1.08766	−0.543832	+	0.839194i	$0.683027\pi$
$230$	0	0
$231$	2.25825e7	0.120540
$232$	0	0
$233$	− 1.99180e8i	− 1.03157i	−0.856717	−	0.515786i	$-0.827500\pi$
$233$	− 1.99180e8i	− 1.03157i	0.856717	−	0.515786i	$-0.172500\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 1.35182e7i	− 0.0659630i
$238$	0	0
$239$	−3.31111e8	−1.56885	−0.784424	−	0.620225i	$-0.787041\pi$
$239$	−3.31111e8	−1.56885	−0.784424	+	0.620225i	$0.787041\pi$
$240$	0	0
$241$	−3.68270e8	−1.69475	−0.847377	−	0.530991i	$-0.821820\pi$
$241$	−3.68270e8	−1.69475	−0.847377	+	0.530991i	$0.821820\pi$
$242$	0	0
$243$	7.71360e7i	0.344854i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 7.07309e7i	− 0.298655i
$248$	0	0
$249$	3.78273e7	0.155277
$250$	0	0
$251$	−3.74255e8	−1.49386	−0.746929	−	0.664903i	$-0.768473\pi$
$251$	−3.74255e8	−1.49386	−0.746929	+	0.664903i	$0.768473\pi$
$252$	0	0
$253$	− 4.02753e8i	− 1.56357i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.57062e8i	1.67961i	0.542885	+	0.839807i	$0.317332\pi$
$257$	4.57062e8i	1.67961i	−0.542885	+	0.839807i	$0.682668\pi$
$258$	0	0
$259$	1.24392e8	0.444880
$260$	0	0
$261$	−4.21759e8	−1.46833
$262$	0	0
$263$	− 6.77338e7i	− 0.229594i	−0.993389	−	0.114797i	$-0.963378\pi$
$263$	− 6.77338e7i	− 0.229594i	0.993389	−	0.114797i	$-0.0366217\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 1.44316e7i	− 0.0464007i
$268$	0	0
$269$	−2.21000e7	−0.0692244	−0.0346122	−	0.999401i	$-0.511020\pi$
$269$	−2.21000e7	−0.0692244	−0.0346122	+	0.999401i	$0.511020\pi$
$270$	0	0
$271$	−4.22269e8	−1.28883	−0.644416	−	0.764675i	$-0.722900\pi$
$271$	−4.22269e8	−1.28883	−0.644416	+	0.764675i	$0.722900\pi$
$272$	0	0
$273$	− 5.91793e7i	− 0.176036i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.98868e8i	− 0.562195i	−0.959679	−	0.281097i	$-0.909302\pi$
$277$	− 1.98868e8i	− 0.562195i	0.959679	−	0.281097i	$-0.0906984\pi$
$278$	0	0
$279$	2.01828e8	0.556375
$280$	0	0
$281$	3.88135e8	1.04354	0.521771	−	0.853085i	$-0.325271\pi$
$281$	3.88135e8	1.04354	0.521771	+	0.853085i	$0.325271\pi$
$282$	0	0
$283$	− 2.98951e8i	− 0.784056i	−0.919953	−	0.392028i	$-0.871774\pi$
$283$	− 2.98951e8i	− 0.784056i	0.919953	−	0.392028i	$-0.128226\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 2.21501e7i	− 0.0553082i
$288$	0	0
$289$	2.56520e8	0.625142
$290$	0	0
$291$	−5.54304e7	−0.131863
$292$	0	0
$293$	1.77029e8i	0.411158i	0.978641	+	0.205579i	$0.0659077\pi$
$293$	1.77029e8i	0.411158i	−0.978641	+	0.205579i	$0.934092\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.27406e8i	− 0.282190i
$298$	0	0
$299$	−1.05545e9	−2.28343
$300$	0	0
$301$	5.69066e8	1.20276
$302$	0	0
$303$	− 4.56215e7i	− 0.0942150i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.25726e8i	0.445243i	0.974905	+	0.222622i	$0.0714614\pi$
$307$	2.25726e8i	0.445243i	−0.974905	+	0.222622i	$0.928539\pi$
$308$	0	0
$309$	−4.24778e7	−0.0819044
$310$	0	0
$311$	8.14288e8	1.53503	0.767515	−	0.641031i	$-0.221493\pi$
$311$	8.14288e8	1.53503	0.767515	+	0.641031i	$0.221493\pi$
$312$	0	0
$313$	6.60615e7i	0.121771i	0.998145	+	0.0608854i	$0.0193924\pi$
$313$	6.60615e7i	0.121771i	−0.998145	+	0.0608854i	$0.980608\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.63770e7i	− 0.117033i	−0.998286	−	0.0585167i	$-0.981363\pi$
$317$	− 6.63770e7i	− 0.117033i	0.998286	−	0.0585167i	$-0.0186371\pi$
$318$	0	0
$319$	1.04734e9	1.80642
$320$	0	0
$321$	−4.85455e7	−0.0819184
$322$	0	0
$323$	6.24942e7i	0.103188i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 8.49386e6i	− 0.0134335i
$328$	0	0
$329$	−8.22319e8	−1.27308
$330$	0	0
$331$	5.59199e8	0.847557	0.423778	−	0.905766i	$-0.360704\pi$
$331$	5.59199e8	0.847557	0.423778	+	0.905766i	$0.360704\pi$
$332$	0	0
$333$	− 3.48475e8i	− 0.517150i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.77074e8i	0.679018i	0.940603	+	0.339509i	$0.110261\pi$
$337$	4.77074e8i	0.679018i	−0.940603	+	0.339509i	$0.889739\pi$
$338$	0	0
$339$	9.87287e7	0.137640
$340$	0	0
$341$	−5.01192e8	−0.684485
$342$	0	0
$343$	8.11727e8i	1.08613i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.49825e8i	0.706433i	0.935542	+	0.353217i	$0.114912\pi$
$347$	5.49825e8i	0.706433i	−0.935542	+	0.353217i	$0.885088\pi$
$348$	0	0
$349$	2.51578e8	0.316799	0.158400	−	0.987375i	$-0.449367\pi$
$349$	2.51578e8	0.316799	0.158400	+	0.987375i	$0.449367\pi$
$350$	0	0
$351$	−3.33878e8	−0.412110
$352$	0	0
$353$	8.04432e8i	0.973370i	0.873577	+	0.486685i	$0.161794\pi$
$353$	8.04432e8i	0.973370i	−0.873577	+	0.486685i	$0.838206\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	5.22878e7i	0.0608222i
$358$	0	0
$359$	1.13834e8	0.129850	0.0649250	−	0.997890i	$-0.479319\pi$
$359$	1.13834e8	0.129850	0.0649250	+	0.997890i	$0.479319\pi$
$360$	0	0
$361$	−8.68481e8	−0.971595
$362$	0	0
$363$	5.03975e7i	0.0553014i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.25920e8i	0.238574i	0.992860	+	0.119287i	$0.0380609\pi$
$367$	2.25920e8i	0.238574i	−0.992860	+	0.119287i	$0.961939\pi$
$368$	0	0
$369$	−6.20521e7	−0.0642931
$370$	0	0
$371$	−4.82342e8	−0.490395
$372$	0	0
$373$	8.19128e8i	0.817280i	0.912696	+	0.408640i	$0.133997\pi$
$373$	8.19128e8i	0.817280i	−0.912696	+	0.408640i	$0.866003\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.74464e9i	− 2.63809i
$378$	0	0
$379$	−1.86356e8	−0.175836	−0.0879178	−	0.996128i	$-0.528021\pi$
$379$	−1.86356e8	−0.175836	−0.0879178	+	0.996128i	$0.528021\pi$
$380$	0	0
$381$	1.08860e8	0.100840
$382$	0	0
$383$	− 1.43839e9i	− 1.30822i	−0.756400	−	0.654109i	$-0.773044\pi$
$383$	− 1.43839e9i	− 1.30822i	0.756400	−	0.654109i	$-0.226956\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.59420e9i	− 1.39815i
$388$	0	0
$389$	1.00373e9	0.864558	0.432279	−	0.901740i	$-0.357710\pi$
$389$	1.00373e9	0.864558	0.432279	+	0.901740i	$0.357710\pi$
$390$	0	0
$391$	9.32540e8	0.788949
$392$	0	0
$393$	3.18668e7i	0.0264828i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9.05187e8i	0.726058i	0.931778	+	0.363029i	$0.118257\pi$
$397$	9.05187e8i	0.726058i	−0.931778	+	0.363029i	$0.881743\pi$
$398$	0	0
$399$	2.12437e7	0.0167427
$400$	0	0
$401$	2.05917e9	1.59473	0.797363	−	0.603500i	$-0.206228\pi$
$401$	2.05917e9	1.59473	0.797363	+	0.603500i	$0.206228\pi$
$402$	0	0
$403$	1.31342e9i	0.999619i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.65354e8i	0.636229i
$408$	0	0
$409$	6.09853e8	0.440751	0.220376	−	0.975415i	$-0.429272\pi$
$409$	6.09853e8	0.440751	0.220376	+	0.975415i	$0.429272\pi$
$410$	0	0
$411$	2.09891e8	0.149124
$412$	0	0
$413$	1.65436e9i	1.15559i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 1.97569e8i	− 0.133427i
$418$	0	0
$419$	2.76626e9	1.83715	0.918574	−	0.395248i	$-0.129341\pi$
$419$	2.76626e9	1.83715	0.918574	+	0.395248i	$0.129341\pi$
$420$	0	0
$421$	1.11034e9	0.725219	0.362609	−	0.931941i	$-0.381886\pi$
$421$	1.11034e9	0.725219	0.362609	+	0.931941i	$0.381886\pi$
$422$	0	0
$423$	2.30367e9i	1.47989i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 1.98265e9i	− 1.23239i
$428$	0	0
$429$	4.11692e8	0.251751
$430$	0	0
$431$	1.55134e9	0.933331	0.466666	−	0.884434i	$-0.345455\pi$
$431$	1.55134e9	0.933331	0.466666	+	0.884434i	$0.345455\pi$
$432$	0	0
$433$	− 1.19534e9i	− 0.707593i	−0.935322	−	0.353797i	$-0.884891\pi$
$433$	− 1.19534e9i	− 0.707593i	0.935322	−	0.353797i	$-0.115109\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.78876e8i	− 0.217176i
$438$	0	0
$439$	−2.81223e8	−0.158644	−0.0793221	−	0.996849i	$-0.525276\pi$
$439$	−2.81223e8	−0.158644	−0.0793221	+	0.996849i	$0.525276\pi$
$440$	0	0
$441$	4.97601e8	0.276278
$442$	0	0
$443$	2.07780e9i	1.13551i	0.823197	+	0.567756i	$0.192188\pi$
$443$	2.07780e9i	1.13551i	−0.823197	+	0.567756i	$0.807812\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.69033e7i	0.0142472i
$448$	0	0
$449$	1.73277e9	0.903397	0.451698	−	0.892171i	$-0.350818\pi$
$449$	1.73277e9	0.903397	0.451698	+	0.892171i	$0.350818\pi$
$450$	0	0
$451$	1.54091e8	0.0790971
$452$	0	0
$453$	− 2.71138e8i	− 0.137040i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.78846e9i	0.876541i	0.898843	+	0.438270i	$0.144409\pi$
$457$	1.78846e9i	0.876541i	−0.898843	+	0.438270i	$0.855591\pi$
$458$	0	0
$459$	2.94998e8	0.142388
$460$	0	0
$461$	−3.91667e8	−0.186193	−0.0930965	−	0.995657i	$-0.529677\pi$
$461$	−3.91667e8	−0.186193	−0.0930965	+	0.995657i	$0.529677\pi$
$462$	0	0
$463$	1.05509e9i	0.494034i	0.969011	+	0.247017i	$0.0794503\pi$
$463$	1.05509e9i	0.494034i	−0.969011	+	0.247017i	$0.920550\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.73541e9i	− 0.788485i	−0.919006	−	0.394243i	$-0.871007\pi$
$467$	− 1.73541e9i	− 0.788485i	0.919006	−	0.394243i	$-0.128993\pi$
$468$	0	0
$469$	6.21886e8	0.278360
$470$	0	0
$471$	−3.20051e7	−0.0141139
$472$	0	0
$473$	3.95881e9i	1.72009i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.35125e9i	0.570060i
$478$	0	0
$479$	2.46088e8	0.102310	0.0511548	−	0.998691i	$-0.483710\pi$
$479$	2.46088e8	0.102310	0.0511548	+	0.998691i	$0.483710\pi$
$480$	0	0
$481$	2.26773e9	0.929146
$482$	0	0
$483$	− 3.17000e8i	− 0.128010i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.60089e9i	1.02040i	0.860055	+	0.510201i	$0.170429\pi$
$487$	2.60089e9i	1.02040i	−0.860055	+	0.510201i	$0.829571\pi$
$488$	0	0
$489$	9.84114e7	0.0380596
$490$	0	0
$491$	−3.18729e9	−1.21517	−0.607584	−	0.794256i	$-0.707861\pi$
$491$	−3.18729e9	−1.21517	−0.607584	+	0.794256i	$0.707861\pi$
$492$	0	0
$493$	2.42502e9i	0.911488i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.32991e9i	0.485931i
$498$	0	0
$499$	−3.08098e9	−1.11004	−0.555018	−	0.831838i	$-0.687289\pi$
$499$	−3.08098e9	−1.11004	−0.555018	+	0.831838i	$0.687289\pi$
$500$	0	0
$501$	−3.13449e8	−0.111361
$502$	0	0
$503$	− 2.17354e9i	− 0.761517i	−0.924674	−	0.380759i	$-0.875663\pi$
$503$	− 2.17354e9i	− 0.761517i	0.924674	−	0.380759i	$-0.124337\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 7.35295e8i	− 0.250573i
$508$	0	0
$509$	1.91522e9	0.643733	0.321866	−	0.946785i	$-0.395690\pi$
$509$	1.91522e9	0.643733	0.321866	+	0.946785i	$0.395690\pi$
$510$	0	0
$511$	1.34374e9	0.445495
$512$	0	0
$513$	− 1.19853e8i	− 0.0391957i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 5.72061e9i	− 1.82065i
$518$	0	0
$519$	5.98278e8	0.187853
$520$	0	0
$521$	−6.78513e8	−0.210197	−0.105098	−	0.994462i	$-0.533516\pi$
$521$	−6.78513e8	−0.210197	−0.105098	+	0.994462i	$0.533516\pi$
$522$	0	0
$523$	5.85186e9i	1.78870i	0.447366	+	0.894351i	$0.352362\pi$
$523$	5.85186e9i	1.78870i	−0.447366	+	0.894351i	$0.647638\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.16047e9i	− 0.345379i
$528$	0	0
$529$	−2.24878e9	−0.660468
$530$	0	0
$531$	4.63459e9	1.34332
$532$	0	0
$533$	− 4.03810e8i	− 0.115513i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	1.08454e8i	0.0302229i
$538$	0	0
$539$	−1.23567e9	−0.339893
$540$	0	0
$541$	−7.71811e8	−0.209566	−0.104783	−	0.994495i	$-0.533415\pi$
$541$	−7.71811e8	−0.209566	−0.104783	+	0.994495i	$0.533415\pi$
$542$	0	0
$543$	− 5.50542e8i	− 0.147568i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 5.26686e9i	− 1.37593i	−0.725744	−	0.687965i	$-0.758504\pi$
$547$	− 5.26686e9i	− 1.37593i	0.725744	−	0.687965i	$-0.241496\pi$
$548$	0	0
$549$	−5.55426e9	−1.43259
$550$	0	0
$551$	9.85247e8	0.250908
$552$	0	0
$553$	− 1.90095e9i	− 0.478006i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.78323e7i	0.0215358i	0.999942	+	0.0107679i	$0.00342760\pi$
$557$	8.78323e7i	0.0215358i	−0.999942	+	0.0107679i	$0.996572\pi$
$558$	0	0
$559$	1.03744e10	2.51201
$560$	0	0
$561$	−3.63750e8	−0.0869826
$562$	0	0
$563$	− 5.32226e8i	− 0.125695i	−0.998023	−	0.0628473i	$-0.979982\pi$
$563$	− 5.32226e8i	− 0.125695i	0.998023	−	0.0628473i	$-0.0200181\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.53198e9i	0.813724i
$568$	0	0
$569$	7.55216e9	1.71861	0.859306	−	0.511461i	$-0.170896\pi$
$569$	7.55216e9	1.71861	0.859306	+	0.511461i	$0.170896\pi$
$570$	0	0
$571$	2.55147e9	0.573541	0.286771	−	0.957999i	$-0.407418\pi$
$571$	2.55147e9	0.573541	0.286771	+	0.957999i	$0.407418\pi$
$572$	0	0
$573$	− 1.24899e8i	− 0.0277343i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 5.53852e9i	− 1.20027i	−0.799899	−	0.600135i	$-0.795114\pi$
$577$	− 5.53852e9i	− 1.20027i	0.799899	−	0.600135i	$-0.204886\pi$
$578$	0	0
$579$	1.05266e7	0.00225379
$580$	0	0
$581$	5.31934e9	1.12523
$582$	0	0
$583$	− 3.35550e9i	− 0.701321i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.73642e9i	0.966533i	0.875473	+	0.483266i	$0.160550\pi$
$587$	4.73642e9i	0.966533i	−0.875473	+	0.483266i	$0.839450\pi$
$588$	0	0
$589$	−4.71480e8	−0.0950735
$590$	0	0
$591$	−1.08479e8	−0.0216167
$592$	0	0
$593$	1.44860e9i	0.285272i	0.989775	+	0.142636i	$0.0455578\pi$
$593$	1.44860e9i	0.285272i	−0.989775	+	0.142636i	$0.954442\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 6.24591e8i	− 0.120139i
$598$	0	0
$599$	1.04150e9	0.198001	0.0990005	−	0.995087i	$-0.468435\pi$
$599$	1.04150e9	0.198001	0.0990005	+	0.995087i	$0.468435\pi$
$600$	0	0
$601$	−7.31995e9	−1.37546	−0.687729	−	0.725967i	$-0.741392\pi$
$601$	−7.31995e9	−1.37546	−0.687729	+	0.725967i	$0.741392\pi$
$602$	0	0
$603$	− 1.74217e9i	− 0.323579i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	6.16291e9i	1.11847i	0.829008	+	0.559237i	$0.188906\pi$
$607$	6.16291e9i	1.11847i	−0.829008	+	0.559237i	$0.811094\pi$
$608$	0	0
$609$	8.24340e8	0.147892
$610$	0	0
$611$	−1.49914e10	−2.65887
$612$	0	0
$613$	3.18382e8i	0.0558260i	0.999610	+	0.0279130i	$0.00888614\pi$
$613$	3.18382e8i	0.0558260i	−0.999610	+	0.0279130i	$0.991114\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.13555e9i	1.22301i	0.791241	+	0.611505i	$0.209435\pi$
$617$	7.13555e9i	1.22301i	−0.791241	+	0.611505i	$0.790565\pi$
$618$	0	0
$619$	−9.96303e9	−1.68840	−0.844198	−	0.536031i	$-0.819923\pi$
$619$	−9.96303e9	−1.68840	−0.844198	+	0.536031i	$0.819923\pi$
$620$	0	0
$621$	−1.78845e9	−0.299679
$622$	0	0
$623$	− 2.02939e9i	− 0.336246i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.47786e8i	0.0239440i
$628$	0	0
$629$	−2.00365e9	−0.321030
$630$	0	0
$631$	4.94300e9	0.783227	0.391614	−	0.920130i	$-0.371917\pi$
$631$	4.94300e9	0.783227	0.391614	+	0.920130i	$0.371917\pi$
$632$	0	0
$633$	− 5.20326e7i	− 0.00815385i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.23818e9i	0.496379i
$638$	0	0
$639$	3.72565e9	0.564871
$640$	0	0
$641$	−2.18024e9	−0.326965	−0.163482	−	0.986546i	$-0.552273\pi$
$641$	−2.18024e9	−0.326965	−0.163482	+	0.986546i	$0.552273\pi$
$642$	0	0
$643$	− 8.70849e9i	− 1.29183i	−0.763411	−	0.645913i	$-0.776477\pi$
$643$	− 8.70849e9i	− 1.29183i	0.763411	−	0.645913i	$-0.223523\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.57610e9i	0.664249i	0.943236	+	0.332124i	$0.107765\pi$
$647$	4.57610e9i	0.664249i	−0.943236	+	0.332124i	$0.892235\pi$
$648$	0	0
$649$	−1.15089e10	−1.65263
$650$	0	0
$651$	−3.94479e8	−0.0560390
$652$	0	0
$653$	− 9.23098e9i	− 1.29733i	−0.761072	−	0.648667i	$-0.775327\pi$
$653$	− 9.23098e9i	− 1.29733i	0.761072	−	0.648667i	$-0.224673\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3.76440e9i	− 0.517866i
$658$	0	0
$659$	−8.30221e9	−1.13004	−0.565021	−	0.825077i	$-0.691132\pi$
$659$	−8.30221e9	−1.13004	−0.565021	+	0.825077i	$0.691132\pi$
$660$	0	0
$661$	2.67892e9	0.360791	0.180395	−	0.983594i	$-0.442262\pi$
$661$	2.67892e9	0.360791	0.180395	+	0.983594i	$0.442262\pi$
$662$	0	0
$663$	9.53237e8i	0.127029i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1.47019e10i	− 1.91837i
$668$	0	0
$669$	1.05293e9	0.135959
$670$	0	0
$671$	1.37927e10	1.76246
$672$	0	0
$673$	− 8.63768e9i	− 1.09231i	−0.837685	−	0.546153i	$-0.816092\pi$
$673$	− 8.63768e9i	− 1.09231i	0.837685	−	0.546153i	$-0.183908\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.17701e9i	1.01282i	0.862291	+	0.506412i	$0.169029\pi$
$677$	8.17701e9i	1.01282i	−0.862291	+	0.506412i	$0.830971\pi$
$678$	0	0
$679$	−7.79471e9	−0.955555
$680$	0	0
$681$	8.77112e8	0.106424
$682$	0	0
$683$	3.81811e9i	0.458539i	0.973363	+	0.229269i	$0.0736337\pi$
$683$	3.81811e9i	0.458539i	−0.973363	+	0.229269i	$0.926366\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1.08228e9i	− 0.127348i
$688$	0	0
$689$	−8.79336e9	−1.02421
$690$	0	0
$691$	−1.38590e10	−1.59794	−0.798969	−	0.601372i	$-0.794621\pi$
$691$	−1.38590e10	−1.59794	−0.798969	+	0.601372i	$0.794621\pi$
$692$	0	0
$693$	− 8.89618e9i	− 1.01540i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.56785e8i	0.0399110i
$698$	0	0
$699$	1.09061e9	0.120781
$700$	0	0
$701$	−4.38274e9	−0.480543	−0.240271	−	0.970706i	$-0.577236\pi$
$701$	−4.38274e9	−0.480543	−0.240271	+	0.970706i	$0.577236\pi$
$702$	0	0
$703$	8.14053e8i	0.0883708i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.41536e9i	− 0.682737i
$708$	0	0
$709$	5.46537e9	0.575914	0.287957	−	0.957643i	$-0.407024\pi$
$709$	5.46537e9	0.575914	0.287957	+	0.957643i	$0.407024\pi$
$710$	0	0
$711$	−5.32539e9	−0.555659
$712$	0	0
$713$	7.03543e9i	0.726905i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.81299e9i	− 0.183687i
$718$	0	0
$719$	−7.24738e9	−0.727160	−0.363580	−	0.931563i	$-0.618446\pi$
$719$	−7.24738e9	−0.727160	−0.363580	+	0.931563i	$0.618446\pi$
$720$	0	0
$721$	−5.97329e9	−0.593526
$722$	0	0
$723$	− 2.01646e9i	− 0.198429i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.61701e10i	1.56078i	0.625291	+	0.780392i	$0.284980\pi$
$727$	1.61701e10i	1.56078i	−0.625291	+	0.780392i	$0.715020\pi$
$728$	0	0
$729$	9.60977e9	0.918685
$730$	0	0
$731$	−9.16629e9	−0.867926
$732$	0	0
$733$	6.59871e9i	0.618864i	0.950922	+	0.309432i	$0.100139\pi$
$733$	6.59871e9i	0.618864i	−0.950922	+	0.309432i	$0.899861\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4.32626e9i	0.398086i
$738$	0	0
$739$	−4.52765e8	−0.0412683	−0.0206342	−	0.999787i	$-0.506569\pi$
$739$	−4.52765e8	−0.0412683	−0.0206342	+	0.999787i	$0.506569\pi$
$740$	0	0
$741$	3.87285e8	0.0349677
$742$	0	0
$743$	− 2.03152e10i	− 1.81702i	−0.417864	−	0.908509i	$-0.637221\pi$
$743$	− 2.03152e10i	− 1.81702i	0.417864	−	0.908509i	$-0.362779\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.49018e10i	− 1.30802i
$748$	0	0
$749$	−6.82654e9	−0.593628
$750$	0	0
$751$	−1.40167e10	−1.20755	−0.603776	−	0.797154i	$-0.706338\pi$
$751$	−1.40167e10	−1.20755	−0.603776	+	0.797154i	$0.706338\pi$
$752$	0	0
$753$	− 2.04922e9i	− 0.174907i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.74979e10i	1.46605i	0.680200	+	0.733026i	$0.261893\pi$
$757$	1.74979e10i	1.46605i	−0.680200	+	0.733026i	$0.738107\pi$
$758$	0	0
$759$	2.20526e9	0.183069
$760$	0	0
$761$	7.69298e9	0.632774	0.316387	−	0.948630i	$-0.397530\pi$
$761$	7.69298e9	0.632774	0.316387	+	0.948630i	$0.397530\pi$
$762$	0	0
$763$	− 1.19442e9i	− 0.0973466i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.01600e10i	2.41350i
$768$	0	0
$769$	6.85942e9	0.543933	0.271967	−	0.962307i	$-0.412326\pi$
$769$	6.85942e9	0.543933	0.271967	+	0.962307i	$0.412326\pi$
$770$	0	0
$771$	−2.50263e9	−0.196656
$772$	0	0
$773$	− 8.93170e9i	− 0.695514i	−0.937585	−	0.347757i	$-0.886943\pi$
$773$	− 8.93170e9i	− 0.695514i	0.937585	−	0.347757i	$-0.113057\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	6.81104e8i	0.0520883i
$778$	0	0
$779$	1.44956e8	0.0109864
$780$	0	0
$781$	−9.25175e9	−0.694937
$782$	0	0
$783$	− 4.65076e9i	− 0.346225i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 5.20674e9i	− 0.380762i	−0.981710	−	0.190381i	$-0.939028\pi$
$787$	− 5.20674e9i	− 0.380762i	0.981710	−	0.190381i	$-0.0609724\pi$
$788$	0	0
$789$	3.70875e8	0.0268818
$790$	0	0
$791$	1.38834e10	0.997419
$792$	0	0
$793$	− 3.61448e10i	− 2.57389i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1.96111e10i	− 1.37214i	−0.727538	−	0.686068i	$-0.759335\pi$
$797$	− 1.96111e10i	− 1.37214i	0.727538	−	0.686068i	$-0.240665\pi$
$798$	0	0
$799$	1.32456e10	0.918666
$800$	0	0
$801$	−5.68520e9	−0.390870
$802$	0	0
$803$	9.34798e9i	0.637109i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.21008e8i	− 0.00810508i
$808$	0	0
$809$	−1.99512e10	−1.32479	−0.662397	−	0.749153i	$-0.730461\pi$
$809$	−1.99512e10	−1.32479	−0.662397	+	0.749153i	$0.730461\pi$
$810$	0	0
$811$	−2.18039e9	−0.143536	−0.0717679	−	0.997421i	$-0.522864\pi$
$811$	−2.18039e9	−0.143536	−0.0717679	+	0.997421i	$0.522864\pi$
$812$	0	0
$813$	− 2.31212e9i	− 0.150902i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.72412e9i	0.238917i
$818$	0	0
$819$	−2.33132e10	−1.48289
$820$	0	0
$821$	−2.52847e10	−1.59462	−0.797310	−	0.603570i	$-0.793744\pi$
$821$	−2.52847e10	−1.59462	−0.797310	+	0.603570i	$0.793744\pi$
$822$	0	0
$823$	− 1.86851e10i	− 1.16841i	−0.811606	−	0.584205i	$-0.801406\pi$
$823$	− 1.86851e10i	− 1.16841i	0.811606	−	0.584205i	$-0.198594\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 7.19901e9i	− 0.442592i	−0.975207	−	0.221296i	$-0.928971\pi$
$827$	− 7.19901e9i	− 0.442592i	0.975207	−	0.221296i	$-0.0710287\pi$
$828$	0	0
$829$	−1.83330e10	−1.11762	−0.558808	−	0.829297i	$-0.688741\pi$
$829$	−1.83330e10	−1.11762	−0.558808	+	0.829297i	$0.688741\pi$
$830$	0	0
$831$	1.08890e9	0.0658240
$832$	0	0
$833$	− 2.86109e9i	− 0.171504i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.22557e9i	0.131191i
$838$	0	0
$839$	2.42360e10	1.41675	0.708376	−	0.705835i	$-0.249428\pi$
$839$	2.42360e10	1.41675	0.708376	+	0.705835i	$0.249428\pi$
$840$	0	0
$841$	2.09816e10	1.21633
$842$	0	0
$843$	2.12522e9i	0.122182i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.08698e9i	0.400746i
$848$	0	0
$849$	1.63690e9	0.0918004
$850$	0	0
$851$	1.21473e10	0.675658
$852$	0	0
$853$	3.21411e10i	1.77312i	0.462612	+	0.886561i	$0.346912\pi$
$853$	3.21411e10i	1.77312i	−0.462612	+	0.886561i	$0.653088\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.48584e9i	− 0.514805i	−0.966304	−	0.257403i	$-0.917133\pi$
$857$	− 9.48584e9i	− 0.514805i	0.966304	−	0.257403i	$-0.0828667\pi$
$858$	0	0
$859$	−1.33721e10	−0.719819	−0.359910	−	0.932987i	$-0.617192\pi$
$859$	−1.33721e10	−0.719819	−0.359910	+	0.932987i	$0.617192\pi$
$860$	0	0
$861$	1.21283e8	0.00647571
$862$	0	0
$863$	− 1.13928e10i	− 0.603381i	−0.953406	−	0.301690i	$-0.902449\pi$
$863$	− 1.13928e10i	− 0.603381i	0.953406	−	0.301690i	$-0.0975509\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.40457e9i	0.0731941i
$868$	0	0
$869$	1.32243e10	0.683604
$870$	0	0
$871$	1.13373e10	0.581364
$872$	0	0
$873$	2.18363e10i	1.11079i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.45509e10i	− 0.728435i	−0.931314	−	0.364218i	$-0.881336\pi$
$877$	− 1.45509e10i	− 0.728435i	0.931314	−	0.364218i	$-0.118664\pi$
$878$	0	0
$879$	−9.69320e8	−0.0481400
$880$	0	0
$881$	−5.46095e9	−0.269062	−0.134531	−	0.990909i	$-0.542953\pi$
$881$	−5.46095e9	−0.269062	−0.134531	+	0.990909i	$0.542953\pi$
$882$	0	0
$883$	1.95772e10i	0.956948i	0.878102	+	0.478474i	$0.158810\pi$
$883$	1.95772e10i	0.956948i	−0.878102	+	0.478474i	$0.841190\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.35654e10i	0.652678i	0.945253	+	0.326339i	$0.105815\pi$
$887$	1.35654e10i	0.652678i	−0.945253	+	0.326339i	$0.894185\pi$
$888$	0	0
$889$	1.53081e10	0.730742
$890$	0	0
$891$	−2.45708e10	−1.16372
$892$	0	0
$893$	− 5.38148e9i	− 0.252884i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5.77908e9i	− 0.267353i
$898$	0	0
$899$	−1.82953e10	−0.839807
$900$	0	0
$901$	7.76936e9	0.353874
$902$	0	0
$903$	3.11591e9i	0.140824i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 7.63541e9i	− 0.339787i	−0.985462	−	0.169894i	$-0.945658\pi$
$907$	− 7.63541e9i	− 0.339787i	0.985462	−	0.169894i	$-0.0543424\pi$
$908$	0	0
$909$	−1.79722e10	−0.793648
$910$	0	0
$911$	2.03069e10	0.889875	0.444937	−	0.895562i	$-0.353226\pi$
$911$	2.03069e10	0.889875	0.444937	+	0.895562i	$0.353226\pi$
$912$	0	0
$913$	3.70049e10i	1.60921i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.48115e9i	0.191910i
$918$	0	0
$919$	1.93832e10	0.823797	0.411899	−	0.911230i	$-0.364866\pi$
$919$	1.93832e10	0.823797	0.411899	+	0.911230i	$0.364866\pi$
$920$	0	0
$921$	−1.23596e9	−0.0521309
$922$	0	0
$923$	2.42450e10i	1.01488i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.67338e10i	0.689945i
$928$	0	0
$929$	−1.17298e10	−0.479993	−0.239997	−	0.970774i	$-0.577146\pi$
$929$	−1.17298e10	−0.479993	−0.239997	+	0.970774i	$0.577146\pi$
$930$	0	0
$931$	−1.16242e9	−0.0472104
$932$	0	0
$933$	4.45862e9i	0.179727i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.41104e10i	0.560340i	0.959950	+	0.280170i	$0.0903908\pi$
$937$	1.41104e10i	0.560340i	−0.959950	+	0.280170i	$0.909609\pi$
$938$	0	0
$939$	−3.61718e8	−0.0142574
$940$	0	0
$941$	1.98332e10	0.775942	0.387971	−	0.921672i	$-0.373176\pi$
$941$	1.98332e10	0.775942	0.387971	+	0.921672i	$0.373176\pi$
$942$	0	0
$943$	− 2.16304e9i	− 0.0839990i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 4.96249e10i	− 1.89878i	−0.314098	−	0.949390i	$-0.601702\pi$
$947$	− 4.96249e10i	− 1.89878i	0.314098	−	0.949390i	$-0.398298\pi$
$948$	0	0
$949$	2.44972e10	0.930432
$950$	0	0
$951$	3.63446e8	0.0137028
$952$	0	0
$953$	− 3.24475e10i	− 1.21438i	−0.794555	−	0.607192i	$-0.792296\pi$
$953$	− 3.24475e10i	− 1.21438i	0.794555	−	0.607192i	$-0.207704\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	5.73467e9i	0.211503i
$958$	0	0
$959$	2.95151e10	1.08064
$960$	0	0
$961$	−1.87576e10	−0.681782
$962$	0	0
$963$	1.91241e10i	0.690063i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6.43301e9i	0.228782i	0.993436	+	0.114391i	$0.0364917\pi$
$967$	6.43301e9i	0.228782i	−0.993436	+	0.114391i	$0.963508\pi$
$968$	0	0
$969$	−3.42186e8	−0.0120817
$970$	0	0
$971$	−7.99885e9	−0.280389	−0.140194	−	0.990124i	$-0.544773\pi$
$971$	−7.99885e9	−0.280389	−0.140194	+	0.990124i	$0.544773\pi$
$972$	0	0
$973$	− 2.77825e10i	− 0.966889i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.43029e10i	0.833733i	0.908968	+	0.416866i	$0.136872\pi$
$977$	2.43029e10i	0.833733i	−0.908968	+	0.416866i	$0.863128\pi$
$978$	0	0
$979$	1.41178e10	0.480871
$980$	0	0
$981$	−3.34609e9	−0.113161
$982$	0	0
$983$	2.05054e10i	0.688544i	0.938870	+	0.344272i	$0.111874\pi$
$983$	2.05054e10i	0.688544i	−0.938870	+	0.344272i	$0.888126\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.50259e9i	− 0.149057i
$988$	0	0
$989$	5.55714e10	1.82669
$990$	0	0
$991$	3.93587e10	1.28464	0.642322	−	0.766435i	$-0.277971\pi$
$991$	3.93587e10	1.28464	0.642322	+	0.766435i	$0.277971\pi$
$992$	0	0
$993$	3.06188e9i	0.0992353i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.18052e10i	− 0.696831i	−0.937340	−	0.348416i	$-0.886720\pi$
$997$	− 2.18052e10i	− 0.696831i	0.937340	−	0.348416i	$-0.113280\pi$
$998$	0	0
$999$	3.84266e9	0.121942

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.8.c.s.49.3		4
4.3	odd	2		25.8.b.b.24.3		4
5.2	odd	4		400.8.a.v.1.2		2
5.3	odd	4		400.8.a.bd.1.1		2
5.4	even	2	inner	400.8.c.s.49.2		4
12.11	even	2		225.8.b.l.199.2		4
20.3	even	4		25.8.a.c.1.2	✓	2
20.7	even	4		25.8.a.e.1.1	yes	2
20.19	odd	2		25.8.b.b.24.2		4
60.23	odd	4		225.8.a.v.1.1		2
60.47	odd	4		225.8.a.k.1.2		2
60.59	even	2		225.8.b.l.199.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.8.a.c.1.2	✓	2	20.3	even	4
25.8.a.e.1.1	yes	2	20.7	even	4
25.8.b.b.24.2		4	20.19	odd	2
25.8.b.b.24.3		4	4.3	odd	2
225.8.a.k.1.2		2	60.47	odd	4
225.8.a.v.1.1		2	60.23	odd	4
225.8.b.l.199.2		4	12.11	even	2
225.8.b.l.199.3		4	60.59	even	2
400.8.a.v.1.2		2	5.2	odd	4
400.8.a.bd.1.1		2	5.3	odd	4
400.8.c.s.49.2		4	5.4	even	2	inner
400.8.c.s.49.3		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.47548i q^{3} +769.970i q^{7} +2157.02 q^{9} -5356.43 q^{11} +14037.0i q^{13} -12402.4i q^{17} -5038.89 q^{19} -4215.95 q^{21} +75190.5i q^{23} +23785.6i q^{27} -195529. q^{29} +93568.2 q^{31} -29329.0i q^{33} -161554. i q^{37} -76859.3 q^{39} -28767.5 q^{41} -739076. i q^{43} +1.06799e6i q^{47} +230689. q^{49} +67908.9 q^{51} +626442. i q^{53} -27590.3i q^{57} +2.14861e6 q^{59} -2.57497e6 q^{61} +1.66084e6i q^{63} -807676. i q^{67} -411704. q^{69} +1.72722e6 q^{71} -1.74519e6i q^{73} -4.12429e6i q^{77} -2.46887e6 q^{79} +4.58716e6 q^{81} -6.90850e6i q^{83} -1.07061e6i q^{87} -2.63567e6 q^{89} -1.08081e7 q^{91} +512331. i q^{93} +1.01234e7i q^{97} -1.15539e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4552 q^{9} - 8688 q^{11} + 36400 q^{19} - 133032 q^{21} - 111600 q^{29} + 603552 q^{31} + 177616 q^{39} - 216972 q^{41} - 1645172 q^{49} + 638992 q^{51} + 4135200 q^{59} + 1164088 q^{61} - 9582936 q^{69}+ \cdots - 22866944 q^{99}+O(q^{100}) \) 4 * q + 4552 * q^9 - 8688 * q^11 + 36400 * q^19 - 133032 * q^21 - 111600 * q^29 + 603552 * q^31 + 177616 * q^39 - 216972 * q^41 - 1645172 * q^49 + 638992 * q^51 + 4135200 * q^59 + 1164088 * q^61 - 9582936 * q^69 + 9456432 * q^71 - 14372400 * q^79 + 157124 * q^81 + 11981700 * q^89 - 11634528 * q^91 - 22866944 * q^99

Introduction

L-functions

Modular forms

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Fields

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