LMFDB - Embedded newform 384.10.a.p.1.1

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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 3854x^{3} + 12258x^{2} + 2877633x + 16772643 $ x^5 - x^4 - 3854x^3 + 12258x^2 + 2877633*x + 16772643
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{29}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-56.6510$ 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)

$f(q)$	$=$	$q+81.0000 q^{3} -1765.98 q^{5} +255.174 q^{7} +6561.00 q^{9} +O(q^{10})$	$q+81.0000 q^{3} -1765.98 q^{5} +255.174 q^{7} +6561.00 q^{9} -61110.8 q^{11} +50850.4 q^{13} -143044. q^{15} +389771. q^{17} -866221. q^{19} +20669.1 q^{21} -981759. q^{23} +1.16554e6 q^{25} +531441. q^{27} -2.03815e6 q^{29} +2.77154e6 q^{31} -4.94998e6 q^{33} -450630. q^{35} -3.50171e6 q^{37} +4.11888e6 q^{39} +5.84336e6 q^{41} -2.78482e7 q^{43} -1.15866e7 q^{45} +1.99825e7 q^{47} -4.02885e7 q^{49} +3.15714e7 q^{51} -4.12389e7 q^{53} +1.07920e8 q^{55} -7.01639e7 q^{57} -9.84523e7 q^{59} +1.02345e8 q^{61} +1.67419e6 q^{63} -8.98005e7 q^{65} +6.49550e7 q^{67} -7.95224e7 q^{69} +2.07696e8 q^{71} -3.01570e8 q^{73} +9.44091e7 q^{75} -1.55939e7 q^{77} -3.25591e8 q^{79} +4.30467e7 q^{81} -2.33927e8 q^{83} -6.88326e8 q^{85} -1.65090e8 q^{87} +9.45256e8 q^{89} +1.29757e7 q^{91} +2.24495e8 q^{93} +1.52973e9 q^{95} -6.41020e8 q^{97} -4.00948e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 405 q^{3} + 772 q^{5} - 38 q^{7} + 32805 q^{9}+O(q^{10}) $ 5 * q + 405 * q^3 + 772 * q^5 - 38 * q^7 + 32805 * q^9	$ 5 q + 405 q^{3} + 772 q^{5} - 38 q^{7} + 32805 q^{9} + 41100 q^{11} + 22486 q^{13} + 62532 q^{15} - 1130 q^{17} - 664712 q^{19} - 3078 q^{21} + 369972 q^{23} + 823383 q^{25} + 2657205 q^{27} + 7390736 q^{29} + 9149938 q^{31} + 3329100 q^{33} - 3167800 q^{35} + 11922058 q^{37} + 1821366 q^{39} - 8471746 q^{41} - 8948896 q^{43} + 5065092 q^{45} - 5051660 q^{47} + 39616113 q^{49} - 91530 q^{51} - 31431984 q^{53} - 67216 q^{55} - 53841672 q^{57} + 204260948 q^{59} + 190850874 q^{61} - 249318 q^{63} - 165466760 q^{65} + 274483500 q^{67} + 29967732 q^{69} + 162722908 q^{71} - 508927538 q^{73} + 66694023 q^{75} + 428895960 q^{77} + 491411266 q^{79} + 215233605 q^{81} + 766279260 q^{83} + 713985400 q^{85} + 598649616 q^{87} - 954097990 q^{89} + 503505932 q^{91} + 741144978 q^{93} + 968680288 q^{95} - 677085326 q^{97} + 269657100 q^{99}+O(q^{100}) $ 5 * q + 405 * q^3 + 772 * q^5 - 38 * q^7 + 32805 * q^9 + 41100 * q^11 + 22486 * q^13 + 62532 * q^15 - 1130 * q^17 - 664712 * q^19 - 3078 * q^21 + 369972 * q^23 + 823383 * q^25 + 2657205 * q^27 + 7390736 * q^29 + 9149938 * q^31 + 3329100 * q^33 - 3167800 * q^35 + 11922058 * q^37 + 1821366 * q^39 - 8471746 * q^41 - 8948896 * q^43 + 5065092 * q^45 - 5051660 * q^47 + 39616113 * q^49 - 91530 * q^51 - 31431984 * q^53 - 67216 * q^55 - 53841672 * q^57 + 204260948 * q^59 + 190850874 * q^61 - 249318 * q^63 - 165466760 * q^65 + 274483500 * q^67 + 29967732 * q^69 + 162722908 * q^71 - 508927538 * q^73 + 66694023 * q^75 + 428895960 * q^77 + 491411266 * q^79 + 215233605 * q^81 + 766279260 * q^83 + 713985400 * q^85 + 598649616 * q^87 - 954097990 * q^89 + 503505932 * q^91 + 741144978 * q^93 + 968680288 * q^95 - 677085326 * q^97 + 269657100 * 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
$5$	−1765.98	−1.26363	−0.631815	−	0.775120i	$-0.717690\pi$
$5$	−1765.98	−1.26363	−0.631815	+	0.775120i	$0.717690\pi$
$6$	0	0
$7$	255.174	0.0401693	0.0200847	−	0.999798i	$-0.493606\pi$
$7$	255.174	0.0401693	0.0200847	+	0.999798i	$0.493606\pi$
$8$	0	0
$9$	6561.00	0.333333
$10$	0	0
$11$	−61110.8	−1.25849	−0.629247	−	0.777205i	$-0.716637\pi$
$11$	−61110.8	−1.25849	−0.629247	+	0.777205i	$0.716637\pi$
$12$	0	0
$13$	50850.4	0.493798	0.246899	−	0.969041i	$-0.420588\pi$
$13$	50850.4	0.493798	0.246899	+	0.969041i	$0.420588\pi$
$14$	0	0
$15$	−143044.	−0.729557
$16$	0	0
$17$	389771.	1.13185	0.565926	−	0.824456i	$-0.308519\pi$
$17$	389771.	1.13185	0.565926	+	0.824456i	$0.308519\pi$
$18$	0	0
$19$	−866221.	−1.52489	−0.762444	−	0.647055i	$-0.776001\pi$
$19$	−866221.	−1.52489	−0.762444	+	0.647055i	$0.776001\pi$
$20$	0	0
$21$	20669.1	0.0231918
$22$	0	0
$23$	−981759.	−0.731525	−0.365763	−	0.930708i	$-0.619192\pi$
$23$	−981759.	−0.731525	−0.365763	+	0.930708i	$0.619192\pi$
$24$	0	0
$25$	1.16554e6	0.596759
$26$	0	0
$27$	531441.	0.192450
$28$	0	0
$29$	−2.03815e6	−0.535113	−0.267556	−	0.963542i	$-0.586216\pi$
$29$	−2.03815e6	−0.535113	−0.267556	+	0.963542i	$0.586216\pi$
$30$	0	0
$31$	2.77154e6	0.539005	0.269503	−	0.963000i	$-0.413141\pi$
$31$	2.77154e6	0.539005	0.269503	+	0.963000i	$0.413141\pi$
$32$	0	0
$33$	−4.94998e6	−0.726592
$34$	0	0
$35$	−450630.	−0.0507591
$36$	0	0
$37$	−3.50171e6	−0.307166	−0.153583	−	0.988136i	$-0.549081\pi$
$37$	−3.50171e6	−0.307166	−0.153583	+	0.988136i	$0.549081\pi$
$38$	0	0
$39$	4.11888e6	0.285094
$40$	0	0
$41$	5.84336e6	0.322950	0.161475	−	0.986877i	$-0.448375\pi$
$41$	5.84336e6	0.322950	0.161475	+	0.986877i	$0.448375\pi$
$42$	0	0
$43$	−2.78482e7	−1.24219	−0.621095	−	0.783735i	$-0.713312\pi$
$43$	−2.78482e7	−1.24219	−0.621095	+	0.783735i	$0.713312\pi$
$44$	0	0
$45$	−1.15866e7	−0.421210
$46$	0	0
$47$	1.99825e7	0.597324	0.298662	−	0.954359i	$-0.403460\pi$
$47$	1.99825e7	0.597324	0.298662	+	0.954359i	$0.403460\pi$
$48$	0	0
$49$	−4.02885e7	−0.998386
$50$	0	0
$51$	3.15714e7	0.653475
$52$	0	0
$53$	−4.12389e7	−0.717903	−0.358952	−	0.933356i	$-0.616866\pi$
$53$	−4.12389e7	−0.717903	−0.358952	+	0.933356i	$0.616866\pi$
$54$	0	0
$55$	1.07920e8	1.59027
$56$	0	0
$57$	−7.01639e7	−0.880394
$58$	0	0
$59$	−9.84523e7	−1.05777	−0.528886	−	0.848693i	$-0.677390\pi$
$59$	−9.84523e7	−1.05777	−0.528886	+	0.848693i	$0.677390\pi$
$60$	0	0
$61$	1.02345e8	0.946420	0.473210	−	0.880950i	$-0.343095\pi$
$61$	1.02345e8	0.946420	0.473210	+	0.880950i	$0.343095\pi$
$62$	0	0
$63$	1.67419e6	0.0133898
$64$	0	0
$65$	−8.98005e7	−0.623977
$66$	0	0
$67$	6.49550e7	0.393800	0.196900	−	0.980424i	$-0.436913\pi$
$67$	6.49550e7	0.393800	0.196900	+	0.980424i	$0.436913\pi$
$68$	0	0
$69$	−7.95224e7	−0.422346
$70$	0	0
$71$	2.07696e8	0.969989	0.484994	−	0.874517i	$-0.338822\pi$
$71$	2.07696e8	0.969989	0.484994	+	0.874517i	$0.338822\pi$
$72$	0	0
$73$	−3.01570e8	−1.24290	−0.621448	−	0.783455i	$-0.713455\pi$
$73$	−3.01570e8	−1.24290	−0.621448	+	0.783455i	$0.713455\pi$
$74$	0	0
$75$	9.44091e7	0.344539
$76$	0	0
$77$	−1.55939e7	−0.0505528
$78$	0	0
$79$	−3.25591e8	−0.940481	−0.470240	−	0.882538i	$-0.655833\pi$
$79$	−3.25591e8	−0.940481	−0.470240	+	0.882538i	$0.655833\pi$
$80$	0	0
$81$	4.30467e7	0.111111
$82$	0	0
$83$	−2.33927e8	−0.541040	−0.270520	−	0.962714i	$-0.587196\pi$
$83$	−2.33927e8	−0.541040	−0.270520	+	0.962714i	$0.587196\pi$
$84$	0	0
$85$	−6.88326e8	−1.43024
$86$	0	0
$87$	−1.65090e8	−0.308948
$88$	0	0
$89$	9.45256e8	1.59696	0.798480	−	0.602021i	$-0.205638\pi$
$89$	9.45256e8	1.59696	0.798480	+	0.602021i	$0.205638\pi$
$90$	0	0
$91$	1.29757e7	0.0198355
$92$	0	0
$93$	2.24495e8	0.311195
$94$	0	0
$95$	1.52973e9	1.92689
$96$	0	0
$97$	−6.41020e8	−0.735189	−0.367594	−	0.929986i	$-0.619819\pi$
$97$	−6.41020e8	−0.735189	−0.367594	+	0.929986i	$0.619819\pi$
$98$	0	0
$99$	−4.00948e8	−0.419498
$100$	0	0
$101$	1.69442e9	1.62022	0.810111	−	0.586277i	$-0.199407\pi$
$101$	1.69442e9	1.62022	0.810111	+	0.586277i	$0.199407\pi$
$102$	0	0
$103$	−2.98594e8	−0.261405	−0.130702	−	0.991422i	$-0.541723\pi$
$103$	−2.98594e8	−0.261405	−0.130702	+	0.991422i	$0.541723\pi$
$104$	0	0
$105$	−3.65010e7	−0.0293058
$106$	0	0
$107$	1.62589e9	1.19913	0.599563	−	0.800328i	$-0.295341\pi$
$107$	1.62589e9	1.19913	0.599563	+	0.800328i	$0.295341\pi$
$108$	0	0
$109$	2.56904e9	1.74322	0.871610	−	0.490200i	$-0.163076\pi$
$109$	2.56904e9	1.74322	0.871610	+	0.490200i	$0.163076\pi$
$110$	0	0
$111$	−2.83639e8	−0.177342
$112$	0	0
$113$	−1.20387e9	−0.694587	−0.347293	−	0.937757i	$-0.612899\pi$
$113$	−1.20387e9	−0.694587	−0.347293	+	0.937757i	$0.612899\pi$
$114$	0	0
$115$	1.73376e9	0.924377
$116$	0	0
$117$	3.33629e8	0.164599
$118$	0	0
$119$	9.94592e7	0.0454657
$120$	0	0
$121$	1.37659e9	0.583807
$122$	0	0
$123$	4.73312e8	0.186455
$124$	0	0
$125$	1.39085e9	0.509547
$126$	0	0
$127$	−5.11703e8	−0.174542	−0.0872712	−	0.996185i	$-0.527815\pi$
$127$	−5.11703e8	−0.174542	−0.0872712	+	0.996185i	$0.527815\pi$
$128$	0	0
$129$	−2.25570e9	−0.717179
$130$	0	0
$131$	4.99470e9	1.48180	0.740898	−	0.671618i	$-0.234400\pi$
$131$	4.99470e9	1.48180	0.740898	+	0.671618i	$0.234400\pi$
$132$	0	0
$133$	−2.21037e8	−0.0612537
$134$	0	0
$135$	−9.38512e8	−0.243186
$136$	0	0
$137$	5.84686e9	1.41801	0.709007	−	0.705202i	$-0.249144\pi$
$137$	5.84686e9	1.41801	0.709007	+	0.705202i	$0.249144\pi$
$138$	0	0
$139$	−3.52787e9	−0.801578	−0.400789	−	0.916170i	$-0.631264\pi$
$139$	−3.52787e9	−0.801578	−0.400789	+	0.916170i	$0.631264\pi$
$140$	0	0
$141$	1.61858e9	0.344865
$142$	0	0
$143$	−3.10751e9	−0.621441
$144$	0	0
$145$	3.59933e9	0.676184
$146$	0	0
$147$	−3.26337e9	−0.576419
$148$	0	0
$149$	−1.14788e10	−1.90792	−0.953958	−	0.299940i	$-0.903033\pi$
$149$	−1.14788e10	−1.90792	−0.953958	+	0.299940i	$0.903033\pi$
$150$	0	0
$151$	7.12972e9	1.11603	0.558016	−	0.829830i	$-0.311563\pi$
$151$	7.12972e9	1.11603	0.558016	+	0.829830i	$0.311563\pi$
$152$	0	0
$153$	2.55729e9	0.377284
$154$	0	0
$155$	−4.89447e9	−0.681103
$156$	0	0
$157$	9.51212e9	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$157$	9.51212e9	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$158$	0	0
$159$	−3.34035e9	−0.414482
$160$	0	0
$161$	−2.50519e8	−0.0293849
$162$	0	0
$163$	−8.55828e9	−0.949604	−0.474802	−	0.880093i	$-0.657480\pi$
$163$	−8.55828e9	−0.949604	−0.474802	+	0.880093i	$0.657480\pi$
$164$	0	0
$165$	8.74154e9	0.918143
$166$	0	0
$167$	1.07544e10	1.06995	0.534973	−	0.844869i	$-0.320322\pi$
$167$	1.07544e10	1.06995	0.534973	+	0.844869i	$0.320322\pi$
$168$	0	0
$169$	−8.01874e9	−0.756164
$170$	0	0
$171$	−5.68328e9	−0.508296
$172$	0	0
$173$	−1.71422e10	−1.45499	−0.727494	−	0.686114i	$-0.759315\pi$
$173$	−1.71422e10	−1.45499	−0.727494	+	0.686114i	$0.759315\pi$
$174$	0	0
$175$	2.97416e8	0.0239714
$176$	0	0
$177$	−7.97464e9	−0.610705
$178$	0	0
$179$	3.40407e9	0.247833	0.123917	−	0.992293i	$-0.460454\pi$
$179$	3.40407e9	0.247833	0.123917	+	0.992293i	$0.460454\pi$
$180$	0	0
$181$	2.10401e9	0.145712	0.0728558	−	0.997342i	$-0.476789\pi$
$181$	2.10401e9	0.145712	0.0728558	+	0.997342i	$0.476789\pi$
$182$	0	0
$183$	8.28997e9	0.546416
$184$	0	0
$185$	6.18394e9	0.388143
$186$	0	0
$187$	−2.38192e10	−1.42443
$188$	0	0
$189$	1.35610e8	0.00773059
$190$	0	0
$191$	1.67085e10	0.908423	0.454211	−	0.890894i	$-0.349921\pi$
$191$	1.67085e10	0.908423	0.454211	+	0.890894i	$0.349921\pi$
$192$	0	0
$193$	2.79394e9	0.144947	0.0724735	−	0.997370i	$-0.476911\pi$
$193$	2.79394e9	0.144947	0.0724735	+	0.997370i	$0.476911\pi$
$194$	0	0
$195$	−7.27384e9	−0.360253
$196$	0	0
$197$	5.15498e9	0.243854	0.121927	−	0.992539i	$-0.461093\pi$
$197$	5.15498e9	0.243854	0.121927	+	0.992539i	$0.461093\pi$
$198$	0	0
$199$	−1.37113e10	−0.619784	−0.309892	−	0.950772i	$-0.600293\pi$
$199$	−1.37113e10	−0.619784	−0.309892	+	0.950772i	$0.600293\pi$
$200$	0	0
$201$	5.26135e9	0.227361
$202$	0	0
$203$	−5.20082e8	−0.0214951
$204$	0	0
$205$	−1.03192e10	−0.408089
$206$	0	0
$207$	−6.44132e9	−0.243842
$208$	0	0
$209$	5.29355e10	1.91906
$210$	0	0
$211$	−6.96605e9	−0.241944	−0.120972	−	0.992656i	$-0.538601\pi$
$211$	−6.96605e9	−0.241944	−0.120972	+	0.992656i	$0.538601\pi$
$212$	0	0
$213$	1.68234e10	0.560023
$214$	0	0
$215$	4.91792e10	1.56967
$216$	0	0
$217$	7.07223e8	0.0216515
$218$	0	0
$219$	−2.44271e10	−0.717586
$220$	0	0
$221$	1.98200e10	0.558905
$222$	0	0
$223$	2.38221e9	0.0645073	0.0322536	−	0.999480i	$-0.489732\pi$
$223$	2.38221e9	0.0645073	0.0322536	+	0.999480i	$0.489732\pi$
$224$	0	0
$225$	7.64714e9	0.198920
$226$	0	0
$227$	−1.13762e10	−0.284368	−0.142184	−	0.989840i	$-0.545412\pi$
$227$	−1.13762e10	−0.284368	−0.142184	+	0.989840i	$0.545412\pi$
$228$	0	0
$229$	2.57913e10	0.619745	0.309872	−	0.950778i	$-0.399714\pi$
$229$	2.57913e10	0.619745	0.309872	+	0.950778i	$0.399714\pi$
$230$	0	0
$231$	−1.26310e9	−0.0291867
$232$	0	0
$233$	2.58759e10	0.575167	0.287584	−	0.957756i	$-0.407148\pi$
$233$	2.58759e10	0.575167	0.287584	+	0.957756i	$0.407148\pi$
$234$	0	0
$235$	−3.52887e10	−0.754796
$236$	0	0
$237$	−2.63728e10	−0.542987
$238$	0	0
$239$	6.20648e9	0.123042	0.0615212	−	0.998106i	$-0.480405\pi$
$239$	6.20648e9	0.123042	0.0615212	+	0.998106i	$0.480405\pi$
$240$	0	0
$241$	5.04336e10	0.963038	0.481519	−	0.876436i	$-0.340085\pi$
$241$	5.04336e10	0.963038	0.481519	+	0.876436i	$0.340085\pi$
$242$	0	0
$243$	3.48678e9	0.0641500
$244$	0	0
$245$	7.11485e10	1.26159
$246$	0	0
$247$	−4.40477e10	−0.752986
$248$	0	0
$249$	−1.89481e10	−0.312369
$250$	0	0
$251$	−1.09334e10	−0.173869	−0.0869344	−	0.996214i	$-0.527707\pi$
$251$	−1.09334e10	−0.173869	−0.0869344	+	0.996214i	$0.527707\pi$
$252$	0	0
$253$	5.99961e10	0.920620
$254$	0	0
$255$	−5.57544e10	−0.825750
$256$	0	0
$257$	7.28890e10	1.04223	0.521114	−	0.853487i	$-0.325517\pi$
$257$	7.28890e10	1.04223	0.521114	+	0.853487i	$0.325517\pi$
$258$	0	0
$259$	−8.93544e8	−0.0123386
$260$	0	0
$261$	−1.33723e10	−0.178371
$262$	0	0
$263$	4.69186e10	0.604706	0.302353	−	0.953196i	$-0.402228\pi$
$263$	4.69186e10	0.604706	0.302353	+	0.953196i	$0.402228\pi$
$264$	0	0
$265$	7.28270e10	0.907164
$266$	0	0
$267$	7.65657e10	0.922006
$268$	0	0
$269$	−2.13518e10	−0.248627	−0.124314	−	0.992243i	$-0.539673\pi$
$269$	−2.13518e10	−0.248627	−0.124314	+	0.992243i	$0.539673\pi$
$270$	0	0
$271$	1.42937e11	1.60984	0.804919	−	0.593384i	$-0.202209\pi$
$271$	1.42937e11	1.60984	0.804919	+	0.593384i	$0.202209\pi$
$272$	0	0
$273$	1.05103e9	0.0114520
$274$	0	0
$275$	−7.12274e10	−0.751018
$276$	0	0
$277$	1.48422e11	1.51475	0.757374	−	0.652981i	$-0.226482\pi$
$277$	1.48422e11	1.51475	0.757374	+	0.652981i	$0.226482\pi$
$278$	0	0
$279$	1.81841e10	0.179668
$280$	0	0
$281$	9.41397e10	0.900729	0.450365	−	0.892845i	$-0.351294\pi$
$281$	9.41397e10	0.900729	0.450365	+	0.892845i	$0.351294\pi$
$282$	0	0
$283$	−9.67561e10	−0.896684	−0.448342	−	0.893862i	$-0.647985\pi$
$283$	−9.67561e10	−0.896684	−0.448342	+	0.893862i	$0.647985\pi$
$284$	0	0
$285$	1.23908e11	1.11249
$286$	0	0
$287$	1.49107e9	0.0129727
$288$	0	0
$289$	3.33335e10	0.281087
$290$	0	0
$291$	−5.19226e10	−0.424461
$292$	0	0
$293$	2.33621e11	1.85186	0.925929	−	0.377696i	$-0.123284\pi$
$293$	2.33621e11	1.85186	0.925929	+	0.377696i	$0.123284\pi$
$294$	0	0
$295$	1.73864e11	1.33663
$296$	0	0
$297$	−3.24768e10	−0.242197
$298$	0	0
$299$	−4.99228e10	−0.361226
$300$	0	0
$301$	−7.10611e9	−0.0498980
$302$	0	0
$303$	1.37248e11	0.935435
$304$	0	0
$305$	−1.80739e11	−1.19592
$306$	0	0
$307$	2.47267e11	1.58870	0.794352	−	0.607458i	$-0.207811\pi$
$307$	2.47267e11	1.58870	0.794352	+	0.607458i	$0.207811\pi$
$308$	0	0
$309$	−2.41861e10	−0.150922
$310$	0	0
$311$	2.40670e11	1.45882	0.729408	−	0.684079i	$-0.239796\pi$
$311$	2.40670e11	1.45882	0.729408	+	0.684079i	$0.239796\pi$
$312$	0	0
$313$	1.61332e11	0.950102	0.475051	−	0.879958i	$-0.342430\pi$
$313$	1.61332e11	0.950102	0.475051	+	0.879958i	$0.342430\pi$
$314$	0	0
$315$	−2.95658e9	−0.0169197
$316$	0	0
$317$	1.37836e11	0.766649	0.383324	−	0.923614i	$-0.374779\pi$
$317$	1.37836e11	0.766649	0.383324	+	0.923614i	$0.374779\pi$
$318$	0	0
$319$	1.24553e11	0.673437
$320$	0	0
$321$	1.31697e11	0.692316
$322$	0	0
$323$	−3.37628e11	−1.72595
$324$	0	0
$325$	5.92684e10	0.294678
$326$	0	0
$327$	2.08093e11	1.00645
$328$	0	0
$329$	5.09901e9	0.0239941
$330$	0	0
$331$	−3.71092e11	−1.69925	−0.849623	−	0.527391i	$-0.823170\pi$
$331$	−3.71092e11	−1.69925	−0.849623	+	0.527391i	$0.823170\pi$
$332$	0	0
$333$	−2.29747e10	−0.102389
$334$	0	0
$335$	−1.14709e11	−0.497617
$336$	0	0
$337$	−1.49457e11	−0.631221	−0.315611	−	0.948889i	$-0.602209\pi$
$337$	−1.49457e11	−0.631221	−0.315611	+	0.948889i	$0.602209\pi$
$338$	0	0
$339$	−9.75134e10	−0.401020
$340$	0	0
$341$	−1.69371e11	−0.678335
$342$	0	0
$343$	−2.05777e10	−0.0802738
$344$	0	0
$345$	1.40435e11	0.533689
$346$	0	0
$347$	−1.58799e11	−0.587982	−0.293991	−	0.955808i	$-0.594984\pi$
$347$	−1.58799e11	−0.587982	−0.293991	+	0.955808i	$0.594984\pi$
$348$	0	0
$349$	−4.09993e11	−1.47932	−0.739660	−	0.672980i	$-0.765014\pi$
$349$	−4.09993e11	−1.47932	−0.739660	+	0.672980i	$0.765014\pi$
$350$	0	0
$351$	2.70240e10	0.0950314
$352$	0	0
$353$	−1.85118e11	−0.634544	−0.317272	−	0.948335i	$-0.602767\pi$
$353$	−1.85118e11	−0.634544	−0.317272	+	0.948335i	$0.602767\pi$
$354$	0	0
$355$	−3.66787e11	−1.22571
$356$	0	0
$357$	8.05620e9	0.0262496
$358$	0	0
$359$	−3.74919e11	−1.19127	−0.595637	−	0.803253i	$-0.703100\pi$
$359$	−3.74919e11	−1.19127	−0.595637	+	0.803253i	$0.703100\pi$
$360$	0	0
$361$	4.27652e11	1.32528
$362$	0	0
$363$	1.11504e11	0.337061
$364$	0	0
$365$	5.32565e11	1.57056
$366$	0	0
$367$	5.41320e11	1.55760	0.778801	−	0.627271i	$-0.215828\pi$
$367$	5.41320e11	1.55760	0.778801	+	0.627271i	$0.215828\pi$
$368$	0	0
$369$	3.83383e10	0.107650
$370$	0	0
$371$	−1.05231e10	−0.0288377
$372$	0	0
$373$	6.46809e10	0.173016	0.0865079	−	0.996251i	$-0.472429\pi$
$373$	6.46809e10	0.173016	0.0865079	+	0.996251i	$0.472429\pi$
$374$	0	0
$375$	1.12659e11	0.294187
$376$	0	0
$377$	−1.03641e11	−0.264237
$378$	0	0
$379$	−2.96558e10	−0.0738302	−0.0369151	−	0.999318i	$-0.511753\pi$
$379$	−2.96558e10	−0.0738302	−0.0369151	+	0.999318i	$0.511753\pi$
$380$	0	0
$381$	−4.14479e10	−0.100772
$382$	0	0
$383$	5.50984e11	1.30841	0.654206	−	0.756316i	$-0.273003\pi$
$383$	5.50984e11	1.30841	0.654206	+	0.756316i	$0.273003\pi$
$384$	0	0
$385$	2.75384e10	0.0638801
$386$	0	0
$387$	−1.82712e11	−0.414064
$388$	0	0
$389$	2.21843e11	0.491217	0.245608	−	0.969369i	$-0.421012\pi$
$389$	2.21843e11	0.491217	0.245608	+	0.969369i	$0.421012\pi$
$390$	0	0
$391$	−3.82661e11	−0.827978
$392$	0	0
$393$	4.04570e11	0.855515
$394$	0	0
$395$	5.74985e11	1.18842
$396$	0	0
$397$	8.69009e11	1.75577	0.877884	−	0.478874i	$-0.158955\pi$
$397$	8.69009e11	1.75577	0.877884	+	0.478874i	$0.158955\pi$
$398$	0	0
$399$	−1.79040e10	−0.0353648
$400$	0	0
$401$	2.05871e11	0.397600	0.198800	−	0.980040i	$-0.436296\pi$
$401$	2.05871e11	0.397600	0.198800	+	0.980040i	$0.436296\pi$
$402$	0	0
$403$	1.40934e11	0.266160
$404$	0	0
$405$	−7.60195e10	−0.140403
$406$	0	0
$407$	2.13993e11	0.386566
$408$	0	0
$409$	9.85642e11	1.74166	0.870832	−	0.491580i	$-0.163581\pi$
$409$	9.85642e11	1.74166	0.870832	+	0.491580i	$0.163581\pi$
$410$	0	0
$411$	4.73596e11	0.818691
$412$	0	0
$413$	−2.51224e10	−0.0424900
$414$	0	0
$415$	4.13110e11	0.683674
$416$	0	0
$417$	−2.85757e11	−0.462791
$418$	0	0
$419$	−1.03568e12	−1.64159	−0.820794	−	0.571224i	$-0.806469\pi$
$419$	−1.03568e12	−1.64159	−0.820794	+	0.571224i	$0.806469\pi$
$420$	0	0
$421$	2.83666e11	0.440086	0.220043	−	0.975490i	$-0.429380\pi$
$421$	2.83666e11	0.440086	0.220043	+	0.975490i	$0.429380\pi$
$422$	0	0
$423$	1.31105e11	0.199108
$424$	0	0
$425$	4.54295e11	0.675442
$426$	0	0
$427$	2.61158e10	0.0380170
$428$	0	0
$429$	−2.51708e11	−0.358789
$430$	0	0
$431$	−1.02143e12	−1.42580	−0.712902	−	0.701264i	$-0.752619\pi$
$431$	−1.02143e12	−1.42580	−0.712902	+	0.701264i	$0.752619\pi$
$432$	0	0
$433$	4.75012e10	0.0649396	0.0324698	−	0.999473i	$-0.489663\pi$
$433$	4.75012e10	0.0649396	0.0324698	+	0.999473i	$0.489663\pi$
$434$	0	0
$435$	2.91545e11	0.390395
$436$	0	0
$437$	8.50420e11	1.11549
$438$	0	0
$439$	−1.30311e12	−1.67452	−0.837258	−	0.546808i	$-0.815843\pi$
$439$	−1.30311e12	−1.67452	−0.837258	+	0.546808i	$0.815843\pi$
$440$	0	0
$441$	−2.64333e11	−0.332795
$442$	0	0
$443$	−6.46062e11	−0.796998	−0.398499	−	0.917169i	$-0.630469\pi$
$443$	−6.46062e11	−0.796998	−0.398499	+	0.917169i	$0.630469\pi$
$444$	0	0
$445$	−1.66930e12	−2.01797
$446$	0	0
$447$	−9.29785e11	−1.10154
$448$	0	0
$449$	−4.80878e11	−0.558376	−0.279188	−	0.960237i	$-0.590065\pi$
$449$	−4.80878e11	−0.558376	−0.279188	+	0.960237i	$0.590065\pi$
$450$	0	0
$451$	−3.57093e11	−0.406431
$452$	0	0
$453$	5.77508e11	0.644341
$454$	0	0
$455$	−2.29147e10	−0.0250647
$456$	0	0
$457$	3.78544e11	0.405970	0.202985	−	0.979182i	$-0.434936\pi$
$457$	3.78544e11	0.405970	0.202985	+	0.979182i	$0.434936\pi$
$458$	0	0
$459$	2.07140e11	0.217825
$460$	0	0
$461$	−1.36127e12	−1.40375	−0.701876	−	0.712299i	$-0.747654\pi$
$461$	−1.36127e12	−1.40375	−0.701876	+	0.712299i	$0.747654\pi$
$462$	0	0
$463$	−7.64822e11	−0.773475	−0.386737	−	0.922190i	$-0.626398\pi$
$463$	−7.64822e11	−0.773475	−0.386737	+	0.922190i	$0.626398\pi$
$464$	0	0
$465$	−3.96452e11	−0.393235
$466$	0	0
$467$	1.21947e12	1.18643	0.593217	−	0.805043i	$-0.297858\pi$
$467$	1.21947e12	1.18643	0.593217	+	0.805043i	$0.297858\pi$
$468$	0	0
$469$	1.65748e10	0.0158187
$470$	0	0
$471$	7.70482e11	0.721387
$472$	0	0
$473$	1.70182e12	1.56329
$474$	0	0
$475$	−1.00962e12	−0.909990
$476$	0	0
$477$	−2.70569e11	−0.239301
$478$	0	0
$479$	4.14989e11	0.360186	0.180093	−	0.983650i	$-0.442360\pi$
$479$	4.14989e11	0.360186	0.180093	+	0.983650i	$0.442360\pi$
$480$	0	0
$481$	−1.78063e11	−0.151678
$482$	0	0
$483$	−2.02920e10	−0.0169654
$484$	0	0
$485$	1.13203e12	0.929006
$486$	0	0
$487$	4.81842e11	0.388172	0.194086	−	0.980984i	$-0.437826\pi$
$487$	4.81842e11	0.388172	0.194086	+	0.980984i	$0.437826\pi$
$488$	0	0
$489$	−6.93221e11	−0.548254
$490$	0	0
$491$	−5.39338e10	−0.0418788	−0.0209394	−	0.999781i	$-0.506666\pi$
$491$	−5.39338e10	−0.0418788	−0.0209394	+	0.999781i	$0.506666\pi$
$492$	0	0
$493$	−7.94412e11	−0.605668
$494$	0	0
$495$	7.08065e11	0.530090
$496$	0	0
$497$	5.29986e10	0.0389638
$498$	0	0
$499$	1.83450e12	1.32454	0.662271	−	0.749264i	$-0.269593\pi$
$499$	1.83450e12	1.32454	0.662271	+	0.749264i	$0.269593\pi$
$500$	0	0
$501$	8.71106e11	0.617734
$502$	0	0
$503$	1.05329e12	0.733656	0.366828	−	0.930289i	$-0.380444\pi$
$503$	1.05329e12	0.733656	0.366828	+	0.930289i	$0.380444\pi$
$504$	0	0
$505$	−2.99230e12	−2.04736
$506$	0	0
$507$	−6.49518e11	−0.436571
$508$	0	0
$509$	−1.99487e12	−1.31730	−0.658649	−	0.752451i	$-0.728872\pi$
$509$	−1.99487e12	−1.31730	−0.658649	+	0.752451i	$0.728872\pi$
$510$	0	0
$511$	−7.69526e10	−0.0499263
$512$	0	0
$513$	−4.60346e11	−0.293465
$514$	0	0
$515$	5.27309e11	0.330318
$516$	0	0
$517$	−1.22115e12	−0.751729
$518$	0	0
$519$	−1.38852e12	−0.840038
$520$	0	0
$521$	−1.06379e12	−0.632538	−0.316269	−	0.948670i	$-0.602430\pi$
$521$	−1.06379e12	−0.632538	−0.316269	+	0.948670i	$0.602430\pi$
$522$	0	0
$523$	1.62100e12	0.947380	0.473690	−	0.880692i	$-0.342922\pi$
$523$	1.62100e12	0.947380	0.473690	+	0.880692i	$0.342922\pi$
$524$	0	0
$525$	2.40907e10	0.0138399
$526$	0	0
$527$	1.08026e12	0.610074
$528$	0	0
$529$	−8.37303e11	−0.464870
$530$	0	0
$531$	−6.45946e11	−0.352590
$532$	0	0
$533$	2.97137e11	0.159472
$534$	0	0
$535$	−2.87129e12	−1.51525
$536$	0	0
$537$	2.75730e11	0.143087
$538$	0	0
$539$	2.46206e12	1.25646
$540$	0	0
$541$	−1.79940e12	−0.903110	−0.451555	−	0.892243i	$-0.649131\pi$
$541$	−1.79940e12	−0.903110	−0.451555	+	0.892243i	$0.649131\pi$
$542$	0	0
$543$	1.70425e11	0.0841266
$544$	0	0
$545$	−4.53687e12	−2.20278
$546$	0	0
$547$	−2.72449e12	−1.30119	−0.650597	−	0.759423i	$-0.725481\pi$
$547$	−2.72449e12	−1.30119	−0.650597	+	0.759423i	$0.725481\pi$
$548$	0	0
$549$	6.71488e11	0.315473
$550$	0	0
$551$	1.76549e12	0.815987
$552$	0	0
$553$	−8.30821e10	−0.0377785
$554$	0	0
$555$	5.00899e11	0.224095
$556$	0	0
$557$	−3.69214e12	−1.62529	−0.812643	−	0.582762i	$-0.801972\pi$
$557$	−3.69214e12	−1.62529	−0.812643	+	0.582762i	$0.801972\pi$
$558$	0	0
$559$	−1.41609e12	−0.613391
$560$	0	0
$561$	−1.92936e12	−0.822394
$562$	0	0
$563$	4.11205e12	1.72493	0.862464	−	0.506119i	$-0.168920\pi$
$563$	4.11205e12	1.72493	0.862464	+	0.506119i	$0.168920\pi$
$564$	0	0
$565$	2.12600e12	0.877700
$566$	0	0
$567$	1.09844e10	0.00446326
$568$	0	0
$569$	−3.22207e11	−0.128863	−0.0644317	−	0.997922i	$-0.520523\pi$
$569$	−3.22207e11	−0.128863	−0.0644317	+	0.997922i	$0.520523\pi$
$570$	0	0
$571$	−2.63593e12	−1.03770	−0.518849	−	0.854866i	$-0.673639\pi$
$571$	−2.63593e12	−1.03770	−0.518849	+	0.854866i	$0.673639\pi$
$572$	0	0
$573$	1.35339e12	0.524478
$574$	0	0
$575$	−1.14428e12	−0.436544
$576$	0	0
$577$	1.31833e11	0.0495144	0.0247572	−	0.999693i	$-0.492119\pi$
$577$	1.31833e11	0.0495144	0.0247572	+	0.999693i	$0.492119\pi$
$578$	0	0
$579$	2.26309e11	0.0836852
$580$	0	0
$581$	−5.96920e10	−0.0217332
$582$	0	0
$583$	2.52015e12	0.903477
$584$	0	0
$585$	−5.89181e11	−0.207992
$586$	0	0
$587$	4.79743e12	1.66777	0.833887	−	0.551935i	$-0.186110\pi$
$587$	4.79743e12	1.66777	0.833887	+	0.551935i	$0.186110\pi$
$588$	0	0
$589$	−2.40076e12	−0.821922
$590$	0	0
$591$	4.17554e11	0.140789
$592$	0	0
$593$	−3.72680e12	−1.23763	−0.618814	−	0.785537i	$-0.712387\pi$
$593$	−3.72680e12	−1.23763	−0.618814	+	0.785537i	$0.712387\pi$
$594$	0	0
$595$	−1.75643e11	−0.0574518
$596$	0	0
$597$	−1.11062e12	−0.357833
$598$	0	0
$599$	−1.85008e12	−0.587177	−0.293589	−	0.955932i	$-0.594850\pi$
$599$	−1.85008e12	−0.587177	−0.293589	+	0.955932i	$0.594850\pi$
$600$	0	0
$601$	5.76495e12	1.80244	0.901220	−	0.433362i	$-0.142673\pi$
$601$	5.76495e12	1.80244	0.901220	+	0.433362i	$0.142673\pi$
$602$	0	0
$603$	4.26170e11	0.131267
$604$	0	0
$605$	−2.43102e12	−0.737716
$606$	0	0
$607$	−6.35819e12	−1.90101	−0.950506	−	0.310707i	$-0.899434\pi$
$607$	−6.35819e12	−1.90101	−0.950506	+	0.310707i	$0.899434\pi$
$608$	0	0
$609$	−4.21267e10	−0.0124102
$610$	0	0
$611$	1.01612e12	0.294957
$612$	0	0
$613$	2.88766e12	0.825989	0.412994	−	0.910734i	$-0.364483\pi$
$613$	2.88766e12	0.825989	0.412994	+	0.910734i	$0.364483\pi$
$614$	0	0
$615$	−8.35858e11	−0.235610
$616$	0	0
$617$	2.42599e12	0.673916	0.336958	−	0.941520i	$-0.390602\pi$
$617$	2.42599e12	0.673916	0.336958	+	0.941520i	$0.390602\pi$
$618$	0	0
$619$	−3.50743e11	−0.0960244	−0.0480122	−	0.998847i	$-0.515289\pi$
$619$	−3.50743e11	−0.0960244	−0.0480122	+	0.998847i	$0.515289\pi$
$620$	0	0
$621$	−5.21747e11	−0.140782
$622$	0	0
$623$	2.41204e11	0.0641488
$624$	0	0
$625$	−4.73266e12	−1.24064
$626$	0	0
$627$	4.28778e12	1.10797
$628$	0	0
$629$	−1.36487e12	−0.347666
$630$	0	0
$631$	−2.98370e12	−0.749244	−0.374622	−	0.927178i	$-0.622228\pi$
$631$	−2.98370e12	−0.749244	−0.374622	+	0.927178i	$0.622228\pi$
$632$	0	0
$633$	−5.64250e11	−0.139687
$634$	0	0
$635$	9.03654e11	0.220557
$636$	0	0
$637$	−2.04868e12	−0.493001
$638$	0	0
$639$	1.36270e12	0.323330
$640$	0	0
$641$	1.36067e10	0.00318340	0.00159170	−	0.999999i	$-0.499493\pi$
$641$	1.36067e10	0.00318340	0.00159170	+	0.999999i	$0.499493\pi$
$642$	0	0
$643$	3.62977e12	0.837395	0.418697	−	0.908126i	$-0.362487\pi$
$643$	3.62977e12	0.837395	0.418697	+	0.908126i	$0.362487\pi$
$644$	0	0
$645$	3.98351e12	0.906249
$646$	0	0
$647$	−5.53275e12	−1.24129	−0.620643	−	0.784093i	$-0.713128\pi$
$647$	−5.53275e12	−1.24129	−0.620643	+	0.784093i	$0.713128\pi$
$648$	0	0
$649$	6.01650e12	1.33120
$650$	0	0
$651$	5.72851e10	0.0125005
$652$	0	0
$653$	2.34566e12	0.504842	0.252421	−	0.967617i	$-0.418773\pi$
$653$	2.34566e12	0.504842	0.252421	+	0.967617i	$0.418773\pi$
$654$	0	0
$655$	−8.82051e12	−1.87244
$656$	0	0
$657$	−1.97860e12	−0.414299
$658$	0	0
$659$	6.32585e12	1.30657	0.653287	−	0.757110i	$-0.273389\pi$
$659$	6.32585e12	1.30657	0.653287	+	0.757110i	$0.273389\pi$
$660$	0	0
$661$	−9.50863e12	−1.93736	−0.968682	−	0.248303i	$-0.920127\pi$
$661$	−9.50863e12	−1.93736	−0.968682	+	0.248303i	$0.920127\pi$
$662$	0	0
$663$	1.60542e12	0.322684
$664$	0	0
$665$	3.90346e11	0.0774019
$666$	0	0
$667$	2.00097e12	0.391449
$668$	0	0
$669$	1.92959e11	0.0372433
$670$	0	0
$671$	−6.25441e12	−1.19106
$672$	0	0
$673$	−7.55373e12	−1.41936	−0.709682	−	0.704522i	$-0.751161\pi$
$673$	−7.55373e12	−1.41936	−0.709682	+	0.704522i	$0.751161\pi$
$674$	0	0
$675$	6.19418e11	0.114846
$676$	0	0
$677$	3.60386e12	0.659355	0.329677	−	0.944094i	$-0.393060\pi$
$677$	3.60386e12	0.659355	0.329677	+	0.944094i	$0.393060\pi$
$678$	0	0
$679$	−1.63571e11	−0.0295320
$680$	0	0
$681$	−9.21472e11	−0.164180
$682$	0	0
$683$	−6.27816e12	−1.10392	−0.551962	−	0.833869i	$-0.686121\pi$
$683$	−6.27816e12	−1.10392	−0.551962	+	0.833869i	$0.686121\pi$
$684$	0	0
$685$	−1.03254e13	−1.79184
$686$	0	0
$687$	2.08909e12	0.357810
$688$	0	0
$689$	−2.09701e12	−0.354499
$690$	0	0
$691$	−4.11525e12	−0.686665	−0.343332	−	0.939214i	$-0.611556\pi$
$691$	−4.11525e12	−0.686665	−0.343332	+	0.939214i	$0.611556\pi$
$692$	0	0
$693$	−1.02311e11	−0.0168509
$694$	0	0
$695$	6.23013e12	1.01290
$696$	0	0
$697$	2.27757e12	0.365531
$698$	0	0
$699$	2.09595e12	0.332073
$700$	0	0
$701$	5.90783e12	0.924052	0.462026	−	0.886866i	$-0.347123\pi$
$701$	5.90783e12	0.924052	0.462026	+	0.886866i	$0.347123\pi$
$702$	0	0
$703$	3.03326e12	0.468393
$704$	0	0
$705$	−2.85838e12	−0.435782
$706$	0	0
$707$	4.32371e11	0.0650832
$708$	0	0
$709$	8.42990e12	1.25289	0.626447	−	0.779464i	$-0.284509\pi$
$709$	8.42990e12	1.25289	0.626447	+	0.779464i	$0.284509\pi$
$710$	0	0
$711$	−2.13620e12	−0.313494
$712$	0	0
$713$	−2.72098e12	−0.394296
$714$	0	0
$715$	5.48778e12	0.785271
$716$	0	0
$717$	5.02725e11	0.0710385
$718$	0	0
$719$	9.25307e12	1.29124	0.645618	−	0.763660i	$-0.276600\pi$
$719$	9.25307e12	1.29124	0.645618	+	0.763660i	$0.276600\pi$
$720$	0	0
$721$	−7.61932e10	−0.0105004
$722$	0	0
$723$	4.08512e12	0.556010
$724$	0	0
$725$	−2.37556e12	−0.319333
$726$	0	0
$727$	−4.41342e12	−0.585963	−0.292982	−	0.956118i	$-0.594648\pi$
$727$	−4.41342e12	−0.585963	−0.292982	+	0.956118i	$0.594648\pi$
$728$	0	0
$729$	2.82430e11	0.0370370
$730$	0	0
$731$	−1.08544e13	−1.40598
$732$	0	0
$733$	−1.35813e13	−1.73770	−0.868850	−	0.495075i	$-0.835140\pi$
$733$	−1.35813e13	−1.73770	−0.868850	+	0.495075i	$0.835140\pi$
$734$	0	0
$735$	5.76303e12	0.728379
$736$	0	0
$737$	−3.96945e12	−0.495595
$738$	0	0
$739$	−1.35822e13	−1.67521	−0.837606	−	0.546275i	$-0.816046\pi$
$739$	−1.35822e13	−1.67521	−0.837606	+	0.546275i	$0.816046\pi$
$740$	0	0
$741$	−3.56786e12	−0.434736
$742$	0	0
$743$	4.02646e11	0.0484700	0.0242350	−	0.999706i	$-0.492285\pi$
$743$	4.02646e11	0.0484700	0.0242350	+	0.999706i	$0.492285\pi$
$744$	0	0
$745$	2.02713e13	2.41090
$746$	0	0
$747$	−1.53480e12	−0.180347
$748$	0	0
$749$	4.14885e11	0.0481681
$750$	0	0
$751$	−9.57602e12	−1.09851	−0.549257	−	0.835654i	$-0.685089\pi$
$751$	−9.57602e12	−1.09851	−0.549257	+	0.835654i	$0.685089\pi$
$752$	0	0
$753$	−8.85602e11	−0.100383
$754$	0	0
$755$	−1.25909e13	−1.41025
$756$	0	0
$757$	−1.37982e13	−1.52719	−0.763594	−	0.645697i	$-0.776567\pi$
$757$	−1.37982e13	−1.52719	−0.763594	+	0.645697i	$0.776567\pi$
$758$	0	0
$759$	4.85968e12	0.531520
$760$	0	0
$761$	1.41019e11	0.0152422	0.00762111	−	0.999971i	$-0.497574\pi$
$761$	1.41019e11	0.0152422	0.00762111	+	0.999971i	$0.497574\pi$
$762$	0	0
$763$	6.55552e11	0.0700240
$764$	0	0
$765$	−4.51611e12	−0.476747
$766$	0	0
$767$	−5.00634e12	−0.522325
$768$	0	0
$769$	3.67984e12	0.379455	0.189728	−	0.981837i	$-0.439240\pi$
$769$	3.67984e12	0.379455	0.189728	+	0.981837i	$0.439240\pi$
$770$	0	0
$771$	5.90401e12	0.601731
$772$	0	0
$773$	1.83821e13	1.85177	0.925887	−	0.377800i	$-0.123319\pi$
$773$	1.83821e13	1.85177	0.925887	+	0.377800i	$0.123319\pi$
$774$	0	0
$775$	3.23035e12	0.321656
$776$	0	0
$777$	−7.23771e10	−0.00712371
$778$	0	0
$779$	−5.06164e12	−0.492462
$780$	0	0
$781$	−1.26925e13	−1.22072
$782$	0	0
$783$	−1.08316e12	−0.102983
$784$	0	0
$785$	−1.67982e13	−1.57888
$786$	0	0
$787$	−1.68352e13	−1.56434	−0.782172	−	0.623063i	$-0.785888\pi$
$787$	−1.68352e13	−1.56434	−0.782172	+	0.623063i	$0.785888\pi$
$788$	0	0
$789$	3.80041e12	0.349127
$790$	0	0
$791$	−3.07196e11	−0.0279011
$792$	0	0
$793$	5.20430e12	0.467340
$794$	0	0
$795$	5.89898e12	0.523751
$796$	0	0
$797$	−7.08925e12	−0.622355	−0.311177	−	0.950352i	$-0.600723\pi$
$797$	−7.08925e12	−0.622355	−0.311177	+	0.950352i	$0.600723\pi$
$798$	0	0
$799$	7.78861e12	0.676082
$800$	0	0
$801$	6.20182e12	0.532320
$802$	0	0
$803$	1.84292e13	1.56418
$804$	0	0
$805$	4.42410e11	0.0371316
$806$	0	0
$807$	−1.72949e12	−0.143545
$808$	0	0
$809$	−5.68225e12	−0.466393	−0.233196	−	0.972430i	$-0.574918\pi$
$809$	−5.68225e12	−0.466393	−0.233196	+	0.972430i	$0.574918\pi$
$810$	0	0
$811$	−7.46976e12	−0.606335	−0.303167	−	0.952937i	$-0.598044\pi$
$811$	−7.46976e12	−0.606335	−0.303167	+	0.952937i	$0.598044\pi$
$812$	0	0
$813$	1.15779e13	0.929441
$814$	0	0
$815$	1.51137e13	1.19995
$816$	0	0
$817$	2.41227e13	1.89420
$818$	0	0
$819$	8.51333e10	0.00661184
$820$	0	0
$821$	1.64789e13	1.26585	0.632927	−	0.774212i	$-0.281853\pi$
$821$	1.64789e13	1.26585	0.632927	+	0.774212i	$0.281853\pi$
$822$	0	0
$823$	−3.57261e12	−0.271448	−0.135724	−	0.990747i	$-0.543336\pi$
$823$	−3.57261e12	−0.271448	−0.135724	+	0.990747i	$0.543336\pi$
$824$	0	0
$825$	−5.76942e12	−0.433600
$826$	0	0
$827$	1.47344e13	1.09536	0.547682	−	0.836686i	$-0.315510\pi$
$827$	1.47344e13	1.09536	0.547682	+	0.836686i	$0.315510\pi$
$828$	0	0
$829$	1.19382e13	0.877896	0.438948	−	0.898512i	$-0.355351\pi$
$829$	1.19382e13	0.877896	0.438948	+	0.898512i	$0.355351\pi$
$830$	0	0
$831$	1.20222e13	0.874540
$832$	0	0
$833$	−1.57033e13	−1.13002
$834$	0	0
$835$	−1.89920e13	−1.35201
$836$	0	0
$837$	1.47291e12	0.103732
$838$	0	0
$839$	−4.33007e12	−0.301693	−0.150847	−	0.988557i	$-0.548200\pi$
$839$	−4.33007e12	−0.301693	−0.150847	+	0.988557i	$0.548200\pi$
$840$	0	0
$841$	−1.03531e13	−0.713654
$842$	0	0
$843$	7.62532e12	0.520036
$844$	0	0
$845$	1.41609e13	0.955511
$846$	0	0
$847$	3.51269e11	0.0234511
$848$	0	0
$849$	−7.83725e12	−0.517701
$850$	0	0
$851$	3.43784e12	0.224699
$852$	0	0
$853$	2.13253e13	1.37919	0.689595	−	0.724195i	$-0.257789\pi$
$853$	2.13253e13	1.37919	0.689595	+	0.724195i	$0.257789\pi$
$854$	0	0
$855$	1.00365e13	0.642297
$856$	0	0
$857$	6.44731e12	0.408286	0.204143	−	0.978941i	$-0.434559\pi$
$857$	6.44731e12	0.408286	0.204143	+	0.978941i	$0.434559\pi$
$858$	0	0
$859$	6.97717e12	0.437230	0.218615	−	0.975811i	$-0.429846\pi$
$859$	6.97717e12	0.437230	0.218615	+	0.975811i	$0.429846\pi$
$860$	0	0
$861$	1.20777e11	0.00748978
$862$	0	0
$863$	1.14224e13	0.700985	0.350493	−	0.936565i	$-0.386014\pi$
$863$	1.14224e13	0.700985	0.350493	+	0.936565i	$0.386014\pi$
$864$	0	0
$865$	3.02727e13	1.83857
$866$	0	0
$867$	2.70001e12	0.162286
$868$	0	0
$869$	1.98971e13	1.18359
$870$	0	0
$871$	3.30298e12	0.194458
$872$	0	0
$873$	−4.20573e12	−0.245063
$874$	0	0
$875$	3.54907e11	0.0204682
$876$	0	0
$877$	1.27113e13	0.725590	0.362795	−	0.931869i	$-0.381823\pi$
$877$	1.27113e13	0.725590	0.362795	+	0.931869i	$0.381823\pi$
$878$	0	0
$879$	1.89233e13	1.06917
$880$	0	0
$881$	3.02640e13	1.69252	0.846262	−	0.532767i	$-0.178848\pi$
$881$	3.02640e13	1.69252	0.846262	+	0.532767i	$0.178848\pi$
$882$	0	0
$883$	−8.08954e12	−0.447817	−0.223909	−	0.974610i	$-0.571882\pi$
$883$	−8.08954e12	−0.447817	−0.223909	+	0.974610i	$0.571882\pi$
$884$	0	0
$885$	1.40830e13	0.771704
$886$	0	0
$887$	−1.44945e11	−0.00786225	−0.00393113	−	0.999992i	$-0.501251\pi$
$887$	−1.44945e11	−0.00786225	−0.00393113	+	0.999992i	$0.501251\pi$
$888$	0	0
$889$	−1.30573e11	−0.00701125
$890$	0	0
$891$	−2.63062e12	−0.139833
$892$	0	0
$893$	−1.73093e13	−0.910852
$894$	0	0
$895$	−6.01151e12	−0.313170
$896$	0	0
$897$	−4.04375e12	−0.208554
$898$	0	0
$899$	−5.64881e12	−0.288429
$900$	0	0
$901$	−1.60737e13	−0.812560
$902$	0	0
$903$	−5.75595e11	−0.0288086
$904$	0	0
$905$	−3.71563e12	−0.184125
$906$	0	0
$907$	−3.52261e13	−1.72835	−0.864174	−	0.503193i	$-0.832159\pi$
$907$	−3.52261e13	−1.72835	−0.864174	+	0.503193i	$0.832159\pi$
$908$	0	0
$909$	1.11171e13	0.540074
$910$	0	0
$911$	−5.75344e12	−0.276755	−0.138377	−	0.990380i	$-0.544189\pi$
$911$	−5.75344e12	−0.276755	−0.138377	+	0.990380i	$0.544189\pi$
$912$	0	0
$913$	1.42955e13	0.680895
$914$	0	0
$915$	−1.46399e13	−0.690467
$916$	0	0
$917$	1.27451e12	0.0595227
$918$	0	0
$919$	−1.91115e13	−0.883843	−0.441922	−	0.897054i	$-0.645703\pi$
$919$	−1.91115e13	−0.883843	−0.441922	+	0.897054i	$0.645703\pi$
$920$	0	0
$921$	2.00286e13	0.917239
$922$	0	0
$923$	1.05614e13	0.478978
$924$	0	0
$925$	−4.08140e12	−0.183304
$926$	0	0
$927$	−1.95907e12	−0.0871349
$928$	0	0
$929$	3.18407e13	1.40253	0.701264	−	0.712901i	$-0.252619\pi$
$929$	3.18407e13	1.40253	0.701264	+	0.712901i	$0.252619\pi$
$930$	0	0
$931$	3.48988e13	1.52243
$932$	0	0
$933$	1.94943e13	0.842247
$934$	0	0
$935$	4.20642e13	1.79995
$936$	0	0
$937$	3.47694e13	1.47356	0.736782	−	0.676130i	$-0.236344\pi$
$937$	3.47694e13	1.47356	0.736782	+	0.676130i	$0.236344\pi$
$938$	0	0
$939$	1.30679e13	0.548542
$940$	0	0
$941$	2.44021e13	1.01455	0.507276	−	0.861784i	$-0.330652\pi$
$941$	2.44021e13	1.01455	0.507276	+	0.861784i	$0.330652\pi$
$942$	0	0
$943$	−5.73677e12	−0.236246
$944$	0	0
$945$	−2.39483e11	−0.00976860
$946$	0	0
$947$	−1.03205e13	−0.416988	−0.208494	−	0.978024i	$-0.566856\pi$
$947$	−1.03205e13	−0.416988	−0.208494	+	0.978024i	$0.566856\pi$
$948$	0	0
$949$	−1.53349e13	−0.613739
$950$	0	0
$951$	1.11647e13	0.442625
$952$	0	0
$953$	8.27804e12	0.325095	0.162547	−	0.986701i	$-0.448029\pi$
$953$	8.27804e12	0.325095	0.162547	+	0.986701i	$0.448029\pi$
$954$	0	0
$955$	−2.95068e13	−1.14791
$956$	0	0
$957$	1.00888e13	0.388809
$958$	0	0
$959$	1.49196e12	0.0569606
$960$	0	0
$961$	−1.87582e13	−0.709473
$962$	0	0
$963$	1.06675e13	0.399709
$964$	0	0
$965$	−4.93403e12	−0.183159
$966$	0	0
$967$	−2.32863e13	−0.856409	−0.428204	−	0.903682i	$-0.640854\pi$
$967$	−2.32863e13	−0.856409	−0.428204	+	0.903682i	$0.640854\pi$
$968$	0	0
$969$	−2.73479e13	−0.996475
$970$	0	0
$971$	−9.82596e12	−0.354722	−0.177361	−	0.984146i	$-0.556756\pi$
$971$	−9.82596e12	−0.354722	−0.177361	+	0.984146i	$0.556756\pi$
$972$	0	0
$973$	−9.00219e11	−0.0321989
$974$	0	0
$975$	4.80074e12	0.170132
$976$	0	0
$977$	−3.11832e13	−1.09495	−0.547475	−	0.836822i	$-0.684411\pi$
$977$	−3.11832e13	−1.09495	−0.547475	+	0.836822i	$0.684411\pi$
$978$	0	0
$979$	−5.77654e13	−2.00977
$980$	0	0
$981$	1.68555e13	0.581073
$982$	0	0
$983$	5.75795e13	1.96688	0.983439	−	0.181240i	$-0.0580110\pi$
$983$	5.75795e13	1.96688	0.983439	+	0.181240i	$0.0580110\pi$
$984$	0	0
$985$	−9.10358e12	−0.308141
$986$	0	0
$987$	4.13020e11	0.0138530
$988$	0	0
$989$	2.73402e13	0.908694
$990$	0	0
$991$	−2.07393e13	−0.683068	−0.341534	−	0.939870i	$-0.610946\pi$
$991$	−2.07393e13	−0.683068	−0.341534	+	0.939870i	$0.610946\pi$
$992$	0	0
$993$	−3.00585e13	−0.981060
$994$	0	0
$995$	2.42139e13	0.783178
$996$	0	0
$997$	4.79589e10	0.00153724	0.000768619	−	1.00000i	$-0.499755\pi$
$997$	4.79589e10	0.00153724	0.000768619		1.00000i	$0.499755\pi$
$998$	0	0
$999$	−1.86095e12	−0.0591140

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.10.a.p.1.1	yes	5
4.3	odd	2		384.10.a.l.1.1	yes	5
8.3	odd	2		384.10.a.m.1.5	yes	5
8.5	even	2		384.10.a.i.1.5	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.10.a.i.1.5	✓	5	8.5	even	2
384.10.a.l.1.1	yes	5	4.3	odd	2
384.10.a.m.1.5	yes	5	8.3	odd	2
384.10.a.p.1.1	yes	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+81.0000 q^{3} -1765.98 q^{5} +255.174 q^{7} +6561.00 q^{9} +O(q^{10})\)	\(q+81.0000 q^{3} -1765.98 q^{5} +255.174 q^{7} +6561.00 q^{9} -61110.8 q^{11} +50850.4 q^{13} -143044. q^{15} +389771. q^{17} -866221. q^{19} +20669.1 q^{21} -981759. q^{23} +1.16554e6 q^{25} +531441. q^{27} -2.03815e6 q^{29} +2.77154e6 q^{31} -4.94998e6 q^{33} -450630. q^{35} -3.50171e6 q^{37} +4.11888e6 q^{39} +5.84336e6 q^{41} -2.78482e7 q^{43} -1.15866e7 q^{45} +1.99825e7 q^{47} -4.02885e7 q^{49} +3.15714e7 q^{51} -4.12389e7 q^{53} +1.07920e8 q^{55} -7.01639e7 q^{57} -9.84523e7 q^{59} +1.02345e8 q^{61} +1.67419e6 q^{63} -8.98005e7 q^{65} +6.49550e7 q^{67} -7.95224e7 q^{69} +2.07696e8 q^{71} -3.01570e8 q^{73} +9.44091e7 q^{75} -1.55939e7 q^{77} -3.25591e8 q^{79} +4.30467e7 q^{81} -2.33927e8 q^{83} -6.88326e8 q^{85} -1.65090e8 q^{87} +9.45256e8 q^{89} +1.29757e7 q^{91} +2.24495e8 q^{93} +1.52973e9 q^{95} -6.41020e8 q^{97} -4.00948e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 405 q^{3} + 772 q^{5} - 38 q^{7} + 32805 q^{9}+O(q^{10}) \) 5 * q + 405 * q^3 + 772 * q^5 - 38 * q^7 + 32805 * q^9	\( 5 q + 405 q^{3} + 772 q^{5} - 38 q^{7} + 32805 q^{9} + 41100 q^{11} + 22486 q^{13} + 62532 q^{15} - 1130 q^{17} - 664712 q^{19} - 3078 q^{21} + 369972 q^{23} + 823383 q^{25} + 2657205 q^{27} + 7390736 q^{29} + 9149938 q^{31} + 3329100 q^{33} - 3167800 q^{35} + 11922058 q^{37} + 1821366 q^{39} - 8471746 q^{41} - 8948896 q^{43} + 5065092 q^{45} - 5051660 q^{47} + 39616113 q^{49} - 91530 q^{51} - 31431984 q^{53} - 67216 q^{55} - 53841672 q^{57} + 204260948 q^{59} + 190850874 q^{61} - 249318 q^{63} - 165466760 q^{65} + 274483500 q^{67} + 29967732 q^{69} + 162722908 q^{71} - 508927538 q^{73} + 66694023 q^{75} + 428895960 q^{77} + 491411266 q^{79} + 215233605 q^{81} + 766279260 q^{83} + 713985400 q^{85} + 598649616 q^{87} - 954097990 q^{89} + 503505932 q^{91} + 741144978 q^{93} + 968680288 q^{95} - 677085326 q^{97} + 269657100 q^{99}+O(q^{100}) \) 5 * q + 405 * q^3 + 772 * q^5 - 38 * q^7 + 32805 * q^9 + 41100 * q^11 + 22486 * q^13 + 62532 * q^15 - 1130 * q^17 - 664712 * q^19 - 3078 * q^21 + 369972 * q^23 + 823383 * q^25 + 2657205 * q^27 + 7390736 * q^29 + 9149938 * q^31 + 3329100 * q^33 - 3167800 * q^35 + 11922058 * q^37 + 1821366 * q^39 - 8471746 * q^41 - 8948896 * q^43 + 5065092 * q^45 - 5051660 * q^47 + 39616113 * q^49 - 91530 * q^51 - 31431984 * q^53 - 67216 * q^55 - 53841672 * q^57 + 204260948 * q^59 + 190850874 * q^61 - 249318 * q^63 - 165466760 * q^65 + 274483500 * q^67 + 29967732 * q^69 + 162722908 * q^71 - 508927538 * q^73 + 66694023 * q^75 + 428895960 * q^77 + 491411266 * q^79 + 215233605 * q^81 + 766279260 * q^83 + 713985400 * q^85 + 598649616 * q^87 - 954097990 * q^89 + 503505932 * q^91 + 741144978 * q^93 + 968680288 * q^95 - 677085326 * q^97 + 269657100 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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