LMFDB - Embedded newform 4016.2.a.h.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 7x^{6} + 40x^{5} - 11x^{4} - 53x^{3} - 2x^{2} + 13x + 1 $ x^9 - x^8 - 11x^7 + 7x^6 + 40x^5 - 11x^4 - 53x^3 - 2x^2 + 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.71581$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.71581 q^{3} -0.169057 q^{5} +0.581118 q^{7} -0.0559961 q^{9} +O(q^{10})$	$q+1.71581 q^{3} -0.169057 q^{5} +0.581118 q^{7} -0.0559961 q^{9} -2.33231 q^{11} -3.50252 q^{13} -0.290070 q^{15} +3.99859 q^{17} -0.490816 q^{19} +0.997088 q^{21} -5.91408 q^{23} -4.97142 q^{25} -5.24351 q^{27} +6.85536 q^{29} +5.85812 q^{31} -4.00179 q^{33} -0.0982423 q^{35} -7.03243 q^{37} -6.00966 q^{39} -6.77752 q^{41} +3.26938 q^{43} +0.00946657 q^{45} -0.120365 q^{47} -6.66230 q^{49} +6.86082 q^{51} -5.84573 q^{53} +0.394294 q^{55} -0.842148 q^{57} -8.47096 q^{59} +6.77570 q^{61} -0.0325404 q^{63} +0.592128 q^{65} +2.24906 q^{67} -10.1474 q^{69} +3.60320 q^{71} -0.738613 q^{73} -8.53001 q^{75} -1.35534 q^{77} -5.77695 q^{79} -8.82888 q^{81} -5.12046 q^{83} -0.675992 q^{85} +11.7625 q^{87} -8.10695 q^{89} -2.03538 q^{91} +10.0514 q^{93} +0.0829762 q^{95} -13.9224 q^{97} +0.130600 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

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
$3$	1.71581	0.990623	0.495312	−	0.868715i	$-0.335054\pi$
$3$	1.71581	0.990623	0.495312	+	0.868715i	$0.335054\pi$
$4$	0	0
$5$	−0.169057	−0.0756048	−0.0378024	−	0.999285i	$-0.512036\pi$
$5$	−0.169057	−0.0756048	−0.0378024	+	0.999285i	$0.512036\pi$
$6$	0	0
$7$	0.581118	0.219642	0.109821	−	0.993951i	$-0.464972\pi$
$7$	0.581118	0.219642	0.109821	+	0.993951i	$0.464972\pi$
$8$	0	0
$9$	−0.0559961	−0.0186654
$10$	0	0
$11$	−2.33231	−0.703216	−0.351608	−	0.936147i	$-0.614365\pi$
$11$	−2.33231	−0.703216	−0.351608	+	0.936147i	$0.614365\pi$
$12$	0	0
$13$	−3.50252	−0.971425	−0.485712	−	0.874119i	$-0.661440\pi$
$13$	−3.50252	−0.971425	−0.485712	+	0.874119i	$0.661440\pi$
$14$	0	0
$15$	−0.290070	−0.0748959
$16$	0	0
$17$	3.99859	0.969801	0.484901	−	0.874569i	$-0.338856\pi$
$17$	3.99859	0.969801	0.484901	+	0.874569i	$0.338856\pi$
$18$	0	0
$19$	−0.490816	−0.112601	−0.0563005	−	0.998414i	$-0.517930\pi$
$19$	−0.490816	−0.112601	−0.0563005	+	0.998414i	$0.517930\pi$
$20$	0	0
$21$	0.997088	0.217582
$22$	0	0
$23$	−5.91408	−1.23317	−0.616585	−	0.787288i	$-0.711484\pi$
$23$	−5.91408	−1.23317	−0.616585	+	0.787288i	$0.711484\pi$
$24$	0	0
$25$	−4.97142	−0.994284
$26$	0	0
$27$	−5.24351	−1.00911
$28$	0	0
$29$	6.85536	1.27301	0.636505	−	0.771273i	$-0.280380\pi$
$29$	6.85536	1.27301	0.636505	+	0.771273i	$0.280380\pi$
$30$	0	0
$31$	5.85812	1.05215	0.526075	−	0.850438i	$-0.323663\pi$
$31$	5.85812	1.05215	0.526075	+	0.850438i	$0.323663\pi$
$32$	0	0
$33$	−4.00179	−0.696623
$34$	0	0
$35$	−0.0982423	−0.0166060
$36$	0	0
$37$	−7.03243	−1.15612	−0.578062	−	0.815993i	$-0.696191\pi$
$37$	−7.03243	−1.15612	−0.578062	+	0.815993i	$0.696191\pi$
$38$	0	0
$39$	−6.00966	−0.962316
$40$	0	0
$41$	−6.77752	−1.05847	−0.529236	−	0.848475i	$-0.677521\pi$
$41$	−6.77752	−1.05847	−0.529236	+	0.848475i	$0.677521\pi$
$42$	0	0
$43$	3.26938	0.498576	0.249288	−	0.968429i	$-0.419803\pi$
$43$	3.26938	0.498576	0.249288	+	0.968429i	$0.419803\pi$
$44$	0	0
$45$	0.00946657	0.00141119
$46$	0	0
$47$	−0.120365	−0.0175570	−0.00877851	−	0.999961i	$-0.502794\pi$
$47$	−0.120365	−0.0175570	−0.00877851	+	0.999961i	$0.502794\pi$
$48$	0	0
$49$	−6.66230	−0.951757
$50$	0	0
$51$	6.86082	0.960708
$52$	0	0
$53$	−5.84573	−0.802973	−0.401486	−	0.915865i	$-0.631506\pi$
$53$	−5.84573	−0.802973	−0.401486	+	0.915865i	$0.631506\pi$
$54$	0	0
$55$	0.394294	0.0531665
$56$	0	0
$57$	−0.842148	−0.111545
$58$	0	0
$59$	−8.47096	−1.10283	−0.551413	−	0.834233i	$-0.685911\pi$
$59$	−8.47096	−1.10283	−0.551413	+	0.834233i	$0.685911\pi$
$60$	0	0
$61$	6.77570	0.867539	0.433769	−	0.901024i	$-0.357183\pi$
$61$	6.77570	0.867539	0.433769	+	0.901024i	$0.357183\pi$
$62$	0	0
$63$	−0.0325404	−0.00409970
$64$	0	0
$65$	0.592128	0.0734444
$66$	0	0
$67$	2.24906	0.274767	0.137383	−	0.990518i	$-0.456131\pi$
$67$	2.24906	0.274767	0.137383	+	0.990518i	$0.456131\pi$
$68$	0	0
$69$	−10.1474	−1.22161
$70$	0	0
$71$	3.60320	0.427621	0.213811	−	0.976875i	$-0.431412\pi$
$71$	3.60320	0.427621	0.213811	+	0.976875i	$0.431412\pi$
$72$	0	0
$73$	−0.738613	−0.0864481	−0.0432241	−	0.999065i	$-0.513763\pi$
$73$	−0.738613	−0.0864481	−0.0432241	+	0.999065i	$0.513763\pi$
$74$	0	0
$75$	−8.53001	−0.984961
$76$	0	0
$77$	−1.35534	−0.154456
$78$	0	0
$79$	−5.77695	−0.649957	−0.324979	−	0.945721i	$-0.605357\pi$
$79$	−5.77695	−0.649957	−0.324979	+	0.945721i	$0.605357\pi$
$80$	0	0
$81$	−8.82888	−0.980986
$82$	0	0
$83$	−5.12046	−0.562044	−0.281022	−	0.959701i	$-0.590673\pi$
$83$	−5.12046	−0.562044	−0.281022	+	0.959701i	$0.590673\pi$
$84$	0	0
$85$	−0.675992	−0.0733216
$86$	0	0
$87$	11.7625	1.26107
$88$	0	0
$89$	−8.10695	−0.859335	−0.429667	−	0.902987i	$-0.641369\pi$
$89$	−8.10695	−0.859335	−0.429667	+	0.902987i	$0.641369\pi$
$90$	0	0
$91$	−2.03538	−0.213366
$92$	0	0
$93$	10.0514	1.04228
$94$	0	0
$95$	0.0829762	0.00851318
$96$	0	0
$97$	−13.9224	−1.41361	−0.706804	−	0.707409i	$-0.749864\pi$
$97$	−13.9224	−1.41361	−0.706804	+	0.707409i	$0.749864\pi$
$98$	0	0
$99$	0.130600	0.0131258
$100$	0	0
$101$	−7.04890	−0.701392	−0.350696	−	0.936489i	$-0.614055\pi$
$101$	−7.04890	−0.701392	−0.350696	+	0.936489i	$0.614055\pi$
$102$	0	0
$103$	−7.55678	−0.744592	−0.372296	−	0.928114i	$-0.621429\pi$
$103$	−7.55678	−0.744592	−0.372296	+	0.928114i	$0.621429\pi$
$104$	0	0
$105$	−0.168565	−0.0164503
$106$	0	0
$107$	18.6525	1.80321	0.901604	−	0.432563i	$-0.142391\pi$
$107$	18.6525	1.80321	0.901604	+	0.432563i	$0.142391\pi$
$108$	0	0
$109$	−5.17256	−0.495441	−0.247721	−	0.968832i	$-0.579682\pi$
$109$	−5.17256	−0.495441	−0.247721	+	0.968832i	$0.579682\pi$
$110$	0	0
$111$	−12.0663	−1.14528
$112$	0	0
$113$	−12.9381	−1.21712	−0.608558	−	0.793510i	$-0.708252\pi$
$113$	−12.9381	−1.21712	−0.608558	+	0.793510i	$0.708252\pi$
$114$	0	0
$115$	0.999819	0.0932336
$116$	0	0
$117$	0.196128	0.0181320
$118$	0	0
$119$	2.32365	0.213009
$120$	0	0
$121$	−5.56035	−0.505487
$122$	0	0
$123$	−11.6289	−1.04855
$124$	0	0
$125$	1.68574	0.150777
$126$	0	0
$127$	9.26615	0.822238	0.411119	−	0.911582i	$-0.365138\pi$
$127$	9.26615	0.822238	0.411119	+	0.911582i	$0.365138\pi$
$128$	0	0
$129$	5.60964	0.493901
$130$	0	0
$131$	17.1968	1.50249	0.751244	−	0.660025i	$-0.229454\pi$
$131$	17.1968	1.50249	0.751244	+	0.660025i	$0.229454\pi$
$132$	0	0
$133$	−0.285222	−0.0247319
$134$	0	0
$135$	0.886454	0.0762938
$136$	0	0
$137$	−14.3146	−1.22298	−0.611488	−	0.791254i	$-0.709429\pi$
$137$	−14.3146	−1.22298	−0.611488	+	0.791254i	$0.709429\pi$
$138$	0	0
$139$	−5.29763	−0.449340	−0.224670	−	0.974435i	$-0.572130\pi$
$139$	−5.29763	−0.449340	−0.224670	+	0.974435i	$0.572130\pi$
$140$	0	0
$141$	−0.206523	−0.0173924
$142$	0	0
$143$	8.16895	0.683122
$144$	0	0
$145$	−1.15895	−0.0962456
$146$	0	0
$147$	−11.4312	−0.942833
$148$	0	0
$149$	−9.87143	−0.808699	−0.404349	−	0.914605i	$-0.632502\pi$
$149$	−9.87143	−0.808699	−0.404349	+	0.914605i	$0.632502\pi$
$150$	0	0
$151$	5.22717	0.425381	0.212690	−	0.977120i	$-0.431777\pi$
$151$	5.22717	0.425381	0.212690	+	0.977120i	$0.431777\pi$
$152$	0	0
$153$	−0.223906	−0.0181017
$154$	0	0
$155$	−0.990359	−0.0795475
$156$	0	0
$157$	11.6215	0.927500	0.463750	−	0.885966i	$-0.346504\pi$
$157$	11.6215	0.927500	0.463750	+	0.885966i	$0.346504\pi$
$158$	0	0
$159$	−10.0302	−0.795443
$160$	0	0
$161$	−3.43677	−0.270856
$162$	0	0
$163$	18.1348	1.42042	0.710212	−	0.703988i	$-0.248599\pi$
$163$	18.1348	1.42042	0.710212	+	0.703988i	$0.248599\pi$
$164$	0	0
$165$	0.676533	0.0526680
$166$	0	0
$167$	19.0652	1.47531	0.737653	−	0.675180i	$-0.235934\pi$
$167$	19.0652	1.47531	0.737653	+	0.675180i	$0.235934\pi$
$168$	0	0
$169$	−0.732340	−0.0563338
$170$	0	0
$171$	0.0274838	0.00210174
$172$	0	0
$173$	−1.00543	−0.0764413	−0.0382207	−	0.999269i	$-0.512169\pi$
$173$	−1.00543	−0.0764413	−0.0382207	+	0.999269i	$0.512169\pi$
$174$	0	0
$175$	−2.88898	−0.218386
$176$	0	0
$177$	−14.5346	−1.09248
$178$	0	0
$179$	−22.3820	−1.67291	−0.836456	−	0.548034i	$-0.815377\pi$
$179$	−22.3820	−1.67291	−0.836456	+	0.548034i	$0.815377\pi$
$180$	0	0
$181$	2.93650	0.218268	0.109134	−	0.994027i	$-0.465192\pi$
$181$	2.93650	0.218268	0.109134	+	0.994027i	$0.465192\pi$
$182$	0	0
$183$	11.6258	0.859404
$184$	0	0
$185$	1.18889	0.0874086
$186$	0	0
$187$	−9.32594	−0.681980
$188$	0	0
$189$	−3.04710	−0.221644
$190$	0	0
$191$	−12.7628	−0.923481	−0.461741	−	0.887015i	$-0.652775\pi$
$191$	−12.7628	−0.923481	−0.461741	+	0.887015i	$0.652775\pi$
$192$	0	0
$193$	17.6103	1.26762	0.633809	−	0.773490i	$-0.281491\pi$
$193$	17.6103	1.26762	0.633809	+	0.773490i	$0.281491\pi$
$194$	0	0
$195$	1.01598	0.0727557
$196$	0	0
$197$	−6.35041	−0.452448	−0.226224	−	0.974075i	$-0.572638\pi$
$197$	−6.35041	−0.452448	−0.226224	+	0.974075i	$0.572638\pi$
$198$	0	0
$199$	19.4936	1.38186	0.690931	−	0.722921i	$-0.257201\pi$
$199$	19.4936	1.38186	0.690931	+	0.722921i	$0.257201\pi$
$200$	0	0
$201$	3.85897	0.272191
$202$	0	0
$203$	3.98377	0.279606
$204$	0	0
$205$	1.14579	0.0800255
$206$	0	0
$207$	0.331165	0.0230176
$208$	0	0
$209$	1.14473	0.0791829
$210$	0	0
$211$	−25.6640	−1.76679	−0.883393	−	0.468634i	$-0.844746\pi$
$211$	−25.6640	−1.76679	−0.883393	+	0.468634i	$0.844746\pi$
$212$	0	0
$213$	6.18241	0.423612
$214$	0	0
$215$	−0.552714	−0.0376948
$216$	0	0
$217$	3.40426	0.231096
$218$	0	0
$219$	−1.26732	−0.0856375
$220$	0	0
$221$	−14.0052	−0.942089
$222$	0	0
$223$	−15.9595	−1.06873	−0.534365	−	0.845254i	$-0.679449\pi$
$223$	−15.9595	−1.06873	−0.534365	+	0.845254i	$0.679449\pi$
$224$	0	0
$225$	0.278380	0.0185587
$226$	0	0
$227$	5.54266	0.367879	0.183940	−	0.982938i	$-0.441115\pi$
$227$	5.54266	0.367879	0.183940	+	0.982938i	$0.441115\pi$
$228$	0	0
$229$	2.34671	0.155075	0.0775374	−	0.996989i	$-0.475294\pi$
$229$	2.34671	0.155075	0.0775374	+	0.996989i	$0.475294\pi$
$230$	0	0
$231$	−2.32551	−0.153007
$232$	0	0
$233$	−4.09138	−0.268035	−0.134018	−	0.990979i	$-0.542788\pi$
$233$	−4.09138	−0.268035	−0.134018	+	0.990979i	$0.542788\pi$
$234$	0	0
$235$	0.0203486	0.00132740
$236$	0	0
$237$	−9.91214	−0.643863
$238$	0	0
$239$	27.5464	1.78183	0.890916	−	0.454169i	$-0.150064\pi$
$239$	27.5464	1.78183	0.890916	+	0.454169i	$0.150064\pi$
$240$	0	0
$241$	9.44175	0.608196	0.304098	−	0.952641i	$-0.401645\pi$
$241$	9.44175	0.608196	0.304098	+	0.952641i	$0.401645\pi$
$242$	0	0
$243$	0.581853	0.0373259
$244$	0	0
$245$	1.12631	0.0719574
$246$	0	0
$247$	1.71910	0.109383
$248$	0	0
$249$	−8.78574	−0.556774
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	13.7934	0.867186
$254$	0	0
$255$	−1.15987	−0.0726341
$256$	0	0
$257$	20.1781	1.25867	0.629337	−	0.777132i	$-0.283326\pi$
$257$	20.1781	1.25867	0.629337	+	0.777132i	$0.283326\pi$
$258$	0	0
$259$	−4.08667	−0.253933
$260$	0	0
$261$	−0.383874	−0.0237612
$262$	0	0
$263$	15.8544	0.977621	0.488811	−	0.872390i	$-0.337431\pi$
$263$	15.8544	0.977621	0.488811	+	0.872390i	$0.337431\pi$
$264$	0	0
$265$	0.988264	0.0607086
$266$	0	0
$267$	−13.9100	−0.851277
$268$	0	0
$269$	19.1409	1.16704	0.583521	−	0.812098i	$-0.301675\pi$
$269$	19.1409	1.16704	0.583521	+	0.812098i	$0.301675\pi$
$270$	0	0
$271$	−17.2231	−1.04623	−0.523115	−	0.852262i	$-0.675230\pi$
$271$	−17.2231	−1.04623	−0.523115	+	0.852262i	$0.675230\pi$
$272$	0	0
$273$	−3.49232	−0.211365
$274$	0	0
$275$	11.5949	0.699197
$276$	0	0
$277$	−26.6822	−1.60318	−0.801590	−	0.597875i	$-0.796012\pi$
$277$	−26.6822	−1.60318	−0.801590	+	0.597875i	$0.796012\pi$
$278$	0	0
$279$	−0.328032	−0.0196388
$280$	0	0
$281$	24.5596	1.46510	0.732552	−	0.680711i	$-0.238329\pi$
$281$	24.5596	1.46510	0.732552	+	0.680711i	$0.238329\pi$
$282$	0	0
$283$	−5.28784	−0.314330	−0.157165	−	0.987572i	$-0.550235\pi$
$283$	−5.28784	−0.314330	−0.157165	+	0.987572i	$0.550235\pi$
$284$	0	0
$285$	0.142371	0.00843335
$286$	0	0
$287$	−3.93854	−0.232485
$288$	0	0
$289$	−1.01126	−0.0594859
$290$	0	0
$291$	−23.8882	−1.40035
$292$	0	0
$293$	20.6169	1.20445	0.602227	−	0.798325i	$-0.294280\pi$
$293$	20.6169	1.20445	0.602227	+	0.798325i	$0.294280\pi$
$294$	0	0
$295$	1.43208	0.0833789
$296$	0	0
$297$	12.2295	0.709625
$298$	0	0
$299$	20.7142	1.19793
$300$	0	0
$301$	1.89990	0.109508
$302$	0	0
$303$	−12.0946	−0.694815
$304$	0	0
$305$	−1.14548	−0.0655901
$306$	0	0
$307$	15.4374	0.881058	0.440529	−	0.897738i	$-0.354791\pi$
$307$	15.4374	0.881058	0.440529	+	0.897738i	$0.354791\pi$
$308$	0	0
$309$	−12.9660	−0.737610
$310$	0	0
$311$	−23.1242	−1.31126	−0.655628	−	0.755084i	$-0.727596\pi$
$311$	−23.1242	−1.31126	−0.655628	+	0.755084i	$0.727596\pi$
$312$	0	0
$313$	−27.7536	−1.56872	−0.784362	−	0.620303i	$-0.787010\pi$
$313$	−27.7536	−1.56872	−0.784362	+	0.620303i	$0.787010\pi$
$314$	0	0
$315$	0.00550119	0.000309957 0
$316$	0	0
$317$	−0.0523253	−0.00293888	−0.00146944	−	0.999999i	$-0.500468\pi$
$317$	−0.0523253	−0.00293888	−0.00146944	+	0.999999i	$0.500468\pi$
$318$	0	0
$319$	−15.9888	−0.895201
$320$	0	0
$321$	32.0042	1.78630
$322$	0	0
$323$	−1.96257	−0.109201
$324$	0	0
$325$	17.4125	0.965872
$326$	0	0
$327$	−8.87513	−0.490796
$328$	0	0
$329$	−0.0699462	−0.00385626
$330$	0	0
$331$	−1.24522	−0.0684434	−0.0342217	−	0.999414i	$-0.510895\pi$
$331$	−1.24522	−0.0684434	−0.0342217	+	0.999414i	$0.510895\pi$
$332$	0	0
$333$	0.393789	0.0215795
$334$	0	0
$335$	−0.380221	−0.0207737
$336$	0	0
$337$	−13.1228	−0.714842	−0.357421	−	0.933943i	$-0.616344\pi$
$337$	−13.1228	−0.714842	−0.357421	+	0.933943i	$0.616344\pi$
$338$	0	0
$339$	−22.1993	−1.20570
$340$	0	0
$341$	−13.6629	−0.739889
$342$	0	0
$343$	−7.93941	−0.428688
$344$	0	0
$345$	1.71550	0.0923594
$346$	0	0
$347$	1.79689	0.0964622	0.0482311	−	0.998836i	$-0.484642\pi$
$347$	1.79689	0.0964622	0.0482311	+	0.998836i	$0.484642\pi$
$348$	0	0
$349$	−21.0387	−1.12618	−0.563089	−	0.826396i	$-0.690387\pi$
$349$	−21.0387	−1.12618	−0.563089	+	0.826396i	$0.690387\pi$
$350$	0	0
$351$	18.3655	0.980278
$352$	0	0
$353$	−25.7370	−1.36984	−0.684922	−	0.728616i	$-0.740164\pi$
$353$	−25.7370	−1.36984	−0.684922	+	0.728616i	$0.740164\pi$
$354$	0	0
$355$	−0.609148	−0.0323302
$356$	0	0
$357$	3.98695	0.211012
$358$	0	0
$359$	24.3588	1.28561	0.642803	−	0.766032i	$-0.277771\pi$
$359$	24.3588	1.28561	0.642803	+	0.766032i	$0.277771\pi$
$360$	0	0
$361$	−18.7591	−0.987321
$362$	0	0
$363$	−9.54051	−0.500747
$364$	0	0
$365$	0.124868	0.00653589
$366$	0	0
$367$	−17.4846	−0.912687	−0.456344	−	0.889804i	$-0.650841\pi$
$367$	−17.4846	−0.912687	−0.456344	+	0.889804i	$0.650841\pi$
$368$	0	0
$369$	0.379515	0.0197568
$370$	0	0
$371$	−3.39706	−0.176366
$372$	0	0
$373$	31.2745	1.61933	0.809667	−	0.586889i	$-0.199648\pi$
$373$	31.2745	1.61933	0.809667	+	0.586889i	$0.199648\pi$
$374$	0	0
$375$	2.89241	0.149364
$376$	0	0
$377$	−24.0111	−1.23663
$378$	0	0
$379$	−8.85702	−0.454955	−0.227477	−	0.973783i	$-0.573048\pi$
$379$	−8.85702	−0.454955	−0.227477	+	0.973783i	$0.573048\pi$
$380$	0	0
$381$	15.8990	0.814528
$382$	0	0
$383$	−3.13151	−0.160013	−0.0800064	−	0.996794i	$-0.525494\pi$
$383$	−3.13151	−0.160013	−0.0800064	+	0.996794i	$0.525494\pi$
$384$	0	0
$385$	0.229131	0.0116776
$386$	0	0
$387$	−0.183073	−0.00930612
$388$	0	0
$389$	24.4818	1.24128	0.620639	−	0.784096i	$-0.286873\pi$
$389$	24.4818	1.24128	0.620639	+	0.784096i	$0.286873\pi$
$390$	0	0
$391$	−23.6480	−1.19593
$392$	0	0
$393$	29.5064	1.48840
$394$	0	0
$395$	0.976636	0.0491399
$396$	0	0
$397$	14.8243	0.744009	0.372005	−	0.928231i	$-0.378671\pi$
$397$	14.8243	0.744009	0.372005	+	0.928231i	$0.378671\pi$
$398$	0	0
$399$	−0.489387	−0.0245000
$400$	0	0
$401$	35.9221	1.79386	0.896931	−	0.442170i	$-0.145791\pi$
$401$	35.9221	1.79386	0.896931	+	0.442170i	$0.145791\pi$
$402$	0	0
$403$	−20.5182	−1.02208
$404$	0	0
$405$	1.49259	0.0741673
$406$	0	0
$407$	16.4018	0.813006
$408$	0	0
$409$	−18.6949	−0.924403	−0.462202	−	0.886775i	$-0.652940\pi$
$409$	−18.6949	−0.924403	−0.462202	+	0.886775i	$0.652940\pi$
$410$	0	0
$411$	−24.5611	−1.21151
$412$	0	0
$413$	−4.92263	−0.242227
$414$	0	0
$415$	0.865652	0.0424932
$416$	0	0
$417$	−9.08973	−0.445126
$418$	0	0
$419$	−36.9208	−1.80370	−0.901849	−	0.432051i	$-0.857790\pi$
$419$	−36.9208	−1.80370	−0.901849	+	0.432051i	$0.857790\pi$
$420$	0	0
$421$	17.6190	0.858697	0.429349	−	0.903139i	$-0.358743\pi$
$421$	17.6190	0.858697	0.429349	+	0.903139i	$0.358743\pi$
$422$	0	0
$423$	0.00673997	0.000327709 0
$424$	0	0
$425$	−19.8787	−0.964258
$426$	0	0
$427$	3.93748	0.190548
$428$	0	0
$429$	14.0164	0.676717
$430$	0	0
$431$	1.98507	0.0956174	0.0478087	−	0.998857i	$-0.484776\pi$
$431$	1.98507	0.0956174	0.0478087	+	0.998857i	$0.484776\pi$
$432$	0	0
$433$	18.1129	0.870450	0.435225	−	0.900322i	$-0.356669\pi$
$433$	18.1129	0.870450	0.435225	+	0.900322i	$0.356669\pi$
$434$	0	0
$435$	−1.98854	−0.0953431
$436$	0	0
$437$	2.90273	0.138856
$438$	0	0
$439$	−16.8855	−0.805899	−0.402950	−	0.915222i	$-0.632015\pi$
$439$	−16.8855	−0.805899	−0.402950	+	0.915222i	$0.632015\pi$
$440$	0	0
$441$	0.373063	0.0177649
$442$	0	0
$443$	−9.09276	−0.432010	−0.216005	−	0.976392i	$-0.569303\pi$
$443$	−9.09276	−0.432010	−0.216005	+	0.976392i	$0.569303\pi$
$444$	0	0
$445$	1.37054	0.0649698
$446$	0	0
$447$	−16.9375	−0.801116
$448$	0	0
$449$	13.3849	0.631675	0.315837	−	0.948813i	$-0.397715\pi$
$449$	13.3849	0.631675	0.315837	+	0.948813i	$0.397715\pi$
$450$	0	0
$451$	15.8073	0.744335
$452$	0	0
$453$	8.96883	0.421392
$454$	0	0
$455$	0.344096	0.0161315
$456$	0	0
$457$	−22.7434	−1.06389	−0.531945	−	0.846779i	$-0.678539\pi$
$457$	−22.7434	−1.06389	−0.531945	+	0.846779i	$0.678539\pi$
$458$	0	0
$459$	−20.9667	−0.978640
$460$	0	0
$461$	−1.64686	−0.0767019	−0.0383510	−	0.999264i	$-0.512210\pi$
$461$	−1.64686	−0.0767019	−0.0383510	+	0.999264i	$0.512210\pi$
$462$	0	0
$463$	35.6070	1.65480	0.827398	−	0.561616i	$-0.189820\pi$
$463$	35.6070	1.65480	0.827398	+	0.561616i	$0.189820\pi$
$464$	0	0
$465$	−1.69927	−0.0788017
$466$	0	0
$467$	5.05914	0.234109	0.117054	−	0.993126i	$-0.462655\pi$
$467$	5.05914	0.234109	0.117054	+	0.993126i	$0.462655\pi$
$468$	0	0
$469$	1.30697	0.0603503
$470$	0	0
$471$	19.9404	0.918803
$472$	0	0
$473$	−7.62520	−0.350607
$474$	0	0
$475$	2.44005	0.111957
$476$	0	0
$477$	0.327338	0.0149878
$478$	0	0
$479$	−0.0540342	−0.00246889	−0.00123444	−	0.999999i	$-0.500393\pi$
$479$	−0.0540342	−0.00246889	−0.00123444	+	0.999999i	$0.500393\pi$
$480$	0	0
$481$	24.6313	1.12309
$482$	0	0
$483$	−5.89685	−0.268316
$484$	0	0
$485$	2.35369	0.106876
$486$	0	0
$487$	25.1224	1.13840	0.569201	−	0.822198i	$-0.307253\pi$
$487$	25.1224	1.13840	0.569201	+	0.822198i	$0.307253\pi$
$488$	0	0
$489$	31.1158	1.40711
$490$	0	0
$491$	28.5127	1.28676	0.643380	−	0.765547i	$-0.277531\pi$
$491$	28.5127	1.28676	0.643380	+	0.765547i	$0.277531\pi$
$492$	0	0
$493$	27.4118	1.23457
$494$	0	0
$495$	−0.0220789	−0.000992374 0
$496$	0	0
$497$	2.09388	0.0939236
$498$	0	0
$499$	10.1775	0.455606	0.227803	−	0.973707i	$-0.426846\pi$
$499$	10.1775	0.455606	0.227803	+	0.973707i	$0.426846\pi$
$500$	0	0
$501$	32.7122	1.46147
$502$	0	0
$503$	30.8738	1.37659	0.688297	−	0.725429i	$-0.258358\pi$
$503$	30.8738	1.37659	0.688297	+	0.725429i	$0.258358\pi$
$504$	0	0
$505$	1.19167	0.0530286
$506$	0	0
$507$	−1.25656	−0.0558056
$508$	0	0
$509$	20.3292	0.901077	0.450539	−	0.892757i	$-0.351232\pi$
$509$	20.3292	0.901077	0.450539	+	0.892757i	$0.351232\pi$
$510$	0	0
$511$	−0.429221	−0.0189876
$512$	0	0
$513$	2.57360	0.113627
$514$	0	0
$515$	1.27753	0.0562947
$516$	0	0
$517$	0.280728	0.0123464
$518$	0	0
$519$	−1.72512	−0.0757245
$520$	0	0
$521$	−10.6296	−0.465693	−0.232847	−	0.972513i	$-0.574804\pi$
$521$	−10.6296	−0.465693	−0.232847	+	0.972513i	$0.574804\pi$
$522$	0	0
$523$	−15.4235	−0.674424	−0.337212	−	0.941429i	$-0.609484\pi$
$523$	−15.4235	−0.674424	−0.337212	+	0.941429i	$0.609484\pi$
$524$	0	0
$525$	−4.95694	−0.216339
$526$	0	0
$527$	23.4242	1.02038
$528$	0	0
$529$	11.9763	0.520708
$530$	0	0
$531$	0.474341	0.0205847
$532$	0	0
$533$	23.7384	1.02823
$534$	0	0
$535$	−3.15335	−0.136331
$536$	0	0
$537$	−38.4033	−1.65723
$538$	0	0
$539$	15.5385	0.669292
$540$	0	0
$541$	15.6927	0.674684	0.337342	−	0.941382i	$-0.390472\pi$
$541$	15.6927	0.674684	0.337342	+	0.941382i	$0.390472\pi$
$542$	0	0
$543$	5.03847	0.216221
$544$	0	0
$545$	0.874460	0.0374577
$546$	0	0
$547$	−16.6425	−0.711581	−0.355791	−	0.934566i	$-0.615788\pi$
$547$	−16.6425	−0.711581	−0.355791	+	0.934566i	$0.615788\pi$
$548$	0	0
$549$	−0.379413	−0.0161929
$550$	0	0
$551$	−3.36472	−0.143342
$552$	0	0
$553$	−3.35709	−0.142758
$554$	0	0
$555$	2.03990	0.0865890
$556$	0	0
$557$	−11.3613	−0.481394	−0.240697	−	0.970600i	$-0.577376\pi$
$557$	−11.3613	−0.481394	−0.240697	+	0.970600i	$0.577376\pi$
$558$	0	0
$559$	−11.4511	−0.484329
$560$	0	0
$561$	−16.0015	−0.675585
$562$	0	0
$563$	0.0380992	0.00160569	0.000802845	−	1.00000i	$-0.499744\pi$
$563$	0.0380992	0.00160569	0.000802845		1.00000i	$0.499744\pi$
$564$	0	0
$565$	2.18729	0.0920198
$566$	0	0
$567$	−5.13062	−0.215466
$568$	0	0
$569$	7.39383	0.309965	0.154983	−	0.987917i	$-0.450468\pi$
$569$	7.39383	0.309965	0.154983	+	0.987917i	$0.450468\pi$
$570$	0	0
$571$	12.9675	0.542673	0.271336	−	0.962485i	$-0.412534\pi$
$571$	12.9675	0.542673	0.271336	+	0.962485i	$0.412534\pi$
$572$	0	0
$573$	−21.8985	−0.914822
$574$	0	0
$575$	29.4014	1.22612
$576$	0	0
$577$	−30.0079	−1.24924	−0.624622	−	0.780927i	$-0.714747\pi$
$577$	−30.0079	−1.24924	−0.624622	+	0.780927i	$0.714747\pi$
$578$	0	0
$579$	30.2159	1.25573
$580$	0	0
$581$	−2.97559	−0.123448
$582$	0	0
$583$	13.6340	0.564664
$584$	0	0
$585$	−0.0331569	−0.00137087
$586$	0	0
$587$	29.6482	1.22371	0.611856	−	0.790969i	$-0.290423\pi$
$587$	29.6482	1.22371	0.611856	+	0.790969i	$0.290423\pi$
$588$	0	0
$589$	−2.87526	−0.118473
$590$	0	0
$591$	−10.8961	−0.448205
$592$	0	0
$593$	5.92930	0.243487	0.121744	−	0.992562i	$-0.461151\pi$
$593$	5.92930	0.243487	0.121744	+	0.992562i	$0.461151\pi$
$594$	0	0
$595$	−0.392831	−0.0161045
$596$	0	0
$597$	33.4473	1.36891
$598$	0	0
$599$	−14.5880	−0.596051	−0.298025	−	0.954558i	$-0.596328\pi$
$599$	−14.5880	−0.596051	−0.298025	+	0.954558i	$0.596328\pi$
$600$	0	0
$601$	−42.0079	−1.71354	−0.856769	−	0.515701i	$-0.827532\pi$
$601$	−42.0079	−1.71354	−0.856769	+	0.515701i	$0.827532\pi$
$602$	0	0
$603$	−0.125939	−0.00512863
$604$	0	0
$605$	0.940019	0.0382172
$606$	0	0
$607$	−21.8340	−0.886217	−0.443108	−	0.896468i	$-0.646124\pi$
$607$	−21.8340	−0.886217	−0.443108	+	0.896468i	$0.646124\pi$
$608$	0	0
$609$	6.83540	0.276984
$610$	0	0
$611$	0.421581	0.0170553
$612$	0	0
$613$	−14.7946	−0.597550	−0.298775	−	0.954324i	$-0.596578\pi$
$613$	−14.7946	−0.597550	−0.298775	+	0.954324i	$0.596578\pi$
$614$	0	0
$615$	1.96596	0.0792751
$616$	0	0
$617$	−46.0947	−1.85570	−0.927852	−	0.372948i	$-0.878347\pi$
$617$	−46.0947	−1.85570	−0.927852	+	0.372948i	$0.878347\pi$
$618$	0	0
$619$	−14.5161	−0.583449	−0.291725	−	0.956502i	$-0.594229\pi$
$619$	−14.5161	−0.583449	−0.291725	+	0.956502i	$0.594229\pi$
$620$	0	0
$621$	31.0105	1.24441
$622$	0	0
$623$	−4.71109	−0.188746
$624$	0	0
$625$	24.5721	0.982884
$626$	0	0
$627$	1.96415	0.0784404
$628$	0	0
$629$	−28.1198	−1.12121
$630$	0	0
$631$	−30.4058	−1.21043	−0.605217	−	0.796060i	$-0.706914\pi$
$631$	−30.4058	−1.21043	−0.605217	+	0.796060i	$0.706914\pi$
$632$	0	0
$633$	−44.0346	−1.75022
$634$	0	0
$635$	−1.56651	−0.0621651
$636$	0	0
$637$	23.3349	0.924561
$638$	0	0
$639$	−0.201765	−0.00798172
$640$	0	0
$641$	−43.5345	−1.71951	−0.859754	−	0.510708i	$-0.829383\pi$
$641$	−43.5345	−1.71951	−0.859754	+	0.510708i	$0.829383\pi$
$642$	0	0
$643$	8.18036	0.322602	0.161301	−	0.986905i	$-0.448431\pi$
$643$	8.18036	0.322602	0.161301	+	0.986905i	$0.448431\pi$
$644$	0	0
$645$	−0.948352	−0.0373413
$646$	0	0
$647$	−37.3257	−1.46742	−0.733712	−	0.679460i	$-0.762214\pi$
$647$	−37.3257	−1.46742	−0.733712	+	0.679460i	$0.762214\pi$
$648$	0	0
$649$	19.7569	0.775525
$650$	0	0
$651$	5.84106	0.228929
$652$	0	0
$653$	−3.63854	−0.142387	−0.0711935	−	0.997463i	$-0.522681\pi$
$653$	−3.63854	−0.142387	−0.0711935	+	0.997463i	$0.522681\pi$
$654$	0	0
$655$	−2.90724	−0.113595
$656$	0	0
$657$	0.0413595	0.00161359
$658$	0	0
$659$	−0.523744	−0.0204022	−0.0102011	−	0.999948i	$-0.503247\pi$
$659$	−0.523744	−0.0204022	−0.0102011	+	0.999948i	$0.503247\pi$
$660$	0	0
$661$	8.91116	0.346604	0.173302	−	0.984869i	$-0.444556\pi$
$661$	8.91116	0.346604	0.173302	+	0.984869i	$0.444556\pi$
$662$	0	0
$663$	−24.0302	−0.933255
$664$	0	0
$665$	0.0482189	0.00186985
$666$	0	0
$667$	−40.5431	−1.56984
$668$	0	0
$669$	−27.3835	−1.05871
$670$	0	0
$671$	−15.8030	−0.610068
$672$	0	0
$673$	−38.2697	−1.47519	−0.737593	−	0.675245i	$-0.764038\pi$
$673$	−38.2697	−1.47519	−0.737593	+	0.675245i	$0.764038\pi$
$674$	0	0
$675$	26.0677	1.00335
$676$	0	0
$677$	−11.6913	−0.449332	−0.224666	−	0.974436i	$-0.572129\pi$
$677$	−11.6913	−0.449332	−0.224666	+	0.974436i	$0.572129\pi$
$678$	0	0
$679$	−8.09057	−0.310488
$680$	0	0
$681$	9.51015	0.364430
$682$	0	0
$683$	−35.7197	−1.36677	−0.683387	−	0.730056i	$-0.739494\pi$
$683$	−35.7197	−1.36677	−0.683387	+	0.730056i	$0.739494\pi$
$684$	0	0
$685$	2.41998	0.0924628
$686$	0	0
$687$	4.02650	0.153621
$688$	0	0
$689$	20.4748	0.780028
$690$	0	0
$691$	16.4539	0.625937	0.312968	−	0.949764i	$-0.398677\pi$
$691$	16.4539	0.625937	0.312968	+	0.949764i	$0.398677\pi$
$692$	0	0
$693$	0.0758940	0.00288298
$694$	0	0
$695$	0.895605	0.0339722
$696$	0	0
$697$	−27.1006	−1.02651
$698$	0	0
$699$	−7.02003	−0.265522
$700$	0	0
$701$	44.7635	1.69069	0.845347	−	0.534218i	$-0.179394\pi$
$701$	44.7635	1.69069	0.845347	+	0.534218i	$0.179394\pi$
$702$	0	0
$703$	3.45163	0.130181
$704$	0	0
$705$	0.0349143	0.00131495
$706$	0	0
$707$	−4.09624	−0.154055
$708$	0	0
$709$	−52.9403	−1.98821	−0.994107	−	0.108405i	$-0.965426\pi$
$709$	−52.9403	−1.98821	−0.994107	+	0.108405i	$0.965426\pi$
$710$	0	0
$711$	0.323487	0.0121317
$712$	0	0
$713$	−34.6454	−1.29748
$714$	0	0
$715$	−1.38102	−0.0516473
$716$	0	0
$717$	47.2645	1.76512
$718$	0	0
$719$	26.5904	0.991654	0.495827	−	0.868421i	$-0.334865\pi$
$719$	26.5904	0.991654	0.495827	+	0.868421i	$0.334865\pi$
$720$	0	0
$721$	−4.39138	−0.163544
$722$	0	0
$723$	16.2002	0.602493
$724$	0	0
$725$	−34.0809	−1.26573
$726$	0	0
$727$	24.2894	0.900843	0.450421	−	0.892816i	$-0.351274\pi$
$727$	24.2894	0.900843	0.450421	+	0.892816i	$0.351274\pi$
$728$	0	0
$729$	27.4850	1.01796
$730$	0	0
$731$	13.0729	0.483520
$732$	0	0
$733$	18.9342	0.699349	0.349675	−	0.936871i	$-0.386292\pi$
$733$	18.9342	0.699349	0.349675	+	0.936871i	$0.386292\pi$
$734$	0	0
$735$	1.93254	0.0712827
$736$	0	0
$737$	−5.24551	−0.193221
$738$	0	0
$739$	28.1476	1.03543	0.517713	−	0.855554i	$-0.326784\pi$
$739$	28.1476	1.03543	0.517713	+	0.855554i	$0.326784\pi$
$740$	0	0
$741$	2.94964	0.108358
$742$	0	0
$743$	15.9931	0.586731	0.293365	−	0.956000i	$-0.405225\pi$
$743$	15.9931	0.586731	0.293365	+	0.956000i	$0.405225\pi$
$744$	0	0
$745$	1.66884	0.0611415
$746$	0	0
$747$	0.286726	0.0104908
$748$	0	0
$749$	10.8393	0.396060
$750$	0	0
$751$	−14.7039	−0.536553	−0.268276	−	0.963342i	$-0.586454\pi$
$751$	−14.7039	−0.536553	−0.268276	+	0.963342i	$0.586454\pi$
$752$	0	0
$753$	−1.71581	−0.0625276
$754$	0	0
$755$	−0.883692	−0.0321608
$756$	0	0
$757$	28.9305	1.05150	0.525748	−	0.850640i	$-0.323786\pi$
$757$	28.9305	1.05150	0.525748	+	0.850640i	$0.323786\pi$
$758$	0	0
$759$	23.6669	0.859054
$760$	0	0
$761$	20.0188	0.725682	0.362841	−	0.931851i	$-0.381807\pi$
$761$	20.0188	0.725682	0.362841	+	0.931851i	$0.381807\pi$
$762$	0	0
$763$	−3.00587	−0.108820
$764$	0	0
$765$	0.0378529	0.00136858
$766$	0	0
$767$	29.6697	1.07131
$768$	0	0
$769$	17.4955	0.630903	0.315452	−	0.948942i	$-0.397844\pi$
$769$	17.4955	0.630903	0.315452	+	0.948942i	$0.397844\pi$
$770$	0	0
$771$	34.6218	1.24687
$772$	0	0
$773$	−21.0294	−0.756375	−0.378187	−	0.925729i	$-0.623452\pi$
$773$	−21.0294	−0.756375	−0.378187	+	0.925729i	$0.623452\pi$
$774$	0	0
$775$	−29.1232	−1.04614
$776$	0	0
$777$	−7.01195	−0.251552
$778$	0	0
$779$	3.32652	0.119185
$780$	0	0
$781$	−8.40377	−0.300710
$782$	0	0
$783$	−35.9462	−1.28461
$784$	0	0
$785$	−1.96471	−0.0701234
$786$	0	0
$787$	−21.7521	−0.775377	−0.387689	−	0.921790i	$-0.626726\pi$
$787$	−21.7521	−0.775377	−0.387689	+	0.921790i	$0.626726\pi$
$788$	0	0
$789$	27.2031	0.968454
$790$	0	0
$791$	−7.51857	−0.267330
$792$	0	0
$793$	−23.7320	−0.842749
$794$	0	0
$795$	1.69567	0.0601393
$796$	0	0
$797$	9.04449	0.320372	0.160186	−	0.987087i	$-0.448791\pi$
$797$	9.04449	0.320372	0.160186	+	0.987087i	$0.448791\pi$
$798$	0	0
$799$	−0.481290	−0.0170268
$800$	0	0
$801$	0.453958	0.0160398
$802$	0	0
$803$	1.72267	0.0607917
$804$	0	0
$805$	0.581012	0.0204780
$806$	0	0
$807$	32.8422	1.15610
$808$	0	0
$809$	50.9612	1.79170	0.895850	−	0.444356i	$-0.146567\pi$
$809$	50.9612	1.79170	0.895850	+	0.444356i	$0.146567\pi$
$810$	0	0
$811$	21.3282	0.748936	0.374468	−	0.927240i	$-0.377825\pi$
$811$	21.3282	0.748936	0.374468	+	0.927240i	$0.377825\pi$
$812$	0	0
$813$	−29.5516	−1.03642
$814$	0	0
$815$	−3.06582	−0.107391
$816$	0	0
$817$	−1.60467	−0.0561402
$818$	0	0
$819$	0.113973	0.00398255
$820$	0	0
$821$	41.5291	1.44937	0.724687	−	0.689078i	$-0.241984\pi$
$821$	41.5291	1.44937	0.724687	+	0.689078i	$0.241984\pi$
$822$	0	0
$823$	−23.3514	−0.813978	−0.406989	−	0.913433i	$-0.633421\pi$
$823$	−23.3514	−0.813978	−0.406989	+	0.913433i	$0.633421\pi$
$824$	0	0
$825$	19.8946	0.692641
$826$	0	0
$827$	28.2721	0.983117	0.491558	−	0.870845i	$-0.336427\pi$
$827$	28.2721	0.983117	0.491558	+	0.870845i	$0.336427\pi$
$828$	0	0
$829$	19.7146	0.684718	0.342359	−	0.939569i	$-0.388774\pi$
$829$	19.7146	0.684718	0.342359	+	0.939569i	$0.388774\pi$
$830$	0	0
$831$	−45.7816	−1.58815
$832$	0	0
$833$	−26.6398	−0.923015
$834$	0	0
$835$	−3.22311	−0.111540
$836$	0	0
$837$	−30.7171	−1.06174
$838$	0	0
$839$	−19.6500	−0.678392	−0.339196	−	0.940716i	$-0.610155\pi$
$839$	−19.6500	−0.678392	−0.339196	+	0.940716i	$0.610155\pi$
$840$	0	0
$841$	17.9960	0.620552
$842$	0	0
$843$	42.1397	1.45137
$844$	0	0
$845$	0.123808	0.00425911
$846$	0	0
$847$	−3.23122	−0.111026
$848$	0	0
$849$	−9.07293	−0.311382
$850$	0	0
$851$	41.5903	1.42570
$852$	0	0
$853$	−12.9236	−0.442496	−0.221248	−	0.975218i	$-0.571013\pi$
$853$	−12.9236	−0.442496	−0.221248	+	0.975218i	$0.571013\pi$
$854$	0	0
$855$	−0.00464635	−0.000158902 0
$856$	0	0
$857$	35.1963	1.20228	0.601141	−	0.799143i	$-0.294713\pi$
$857$	35.1963	1.20228	0.601141	+	0.799143i	$0.294713\pi$
$858$	0	0
$859$	13.3647	0.455997	0.227998	−	0.973662i	$-0.426782\pi$
$859$	13.3647	0.455997	0.227998	+	0.973662i	$0.426782\pi$
$860$	0	0
$861$	−6.75778	−0.230305
$862$	0	0
$863$	−19.8721	−0.676456	−0.338228	−	0.941064i	$-0.609827\pi$
$863$	−19.8721	−0.676456	−0.338228	+	0.941064i	$0.609827\pi$
$864$	0	0
$865$	0.169975	0.00577933
$866$	0	0
$867$	−1.73513	−0.0589281
$868$	0	0
$869$	13.4736	0.457061
$870$	0	0
$871$	−7.87740	−0.266915
$872$	0	0
$873$	0.779602	0.0263855
$874$	0	0
$875$	0.979615	0.0331170
$876$	0	0
$877$	−13.8345	−0.467158	−0.233579	−	0.972338i	$-0.575044\pi$
$877$	−13.8345	−0.467158	−0.233579	+	0.972338i	$0.575044\pi$
$878$	0	0
$879$	35.3748	1.19316
$880$	0	0
$881$	−20.5036	−0.690784	−0.345392	−	0.938459i	$-0.612254\pi$
$881$	−20.5036	−0.690784	−0.345392	+	0.938459i	$0.612254\pi$
$882$	0	0
$883$	−4.93984	−0.166239	−0.0831193	−	0.996540i	$-0.526488\pi$
$883$	−4.93984	−0.166239	−0.0831193	+	0.996540i	$0.526488\pi$
$884$	0	0
$885$	2.45718	0.0825971
$886$	0	0
$887$	19.4050	0.651557	0.325779	−	0.945446i	$-0.394374\pi$
$887$	19.4050	0.651557	0.325779	+	0.945446i	$0.394374\pi$
$888$	0	0
$889$	5.38473	0.180598
$890$	0	0
$891$	20.5916	0.689846
$892$	0	0
$893$	0.0590771	0.00197694
$894$	0	0
$895$	3.78385	0.126480
$896$	0	0
$897$	35.5416	1.18670
$898$	0	0
$899$	40.1595	1.33940
$900$	0	0
$901$	−23.3747	−0.778724
$902$	0	0
$903$	3.25986	0.108481
$904$	0	0
$905$	−0.496437	−0.0165021
$906$	0	0
$907$	37.2500	1.23686	0.618432	−	0.785838i	$-0.287768\pi$
$907$	37.2500	1.23686	0.618432	+	0.785838i	$0.287768\pi$
$908$	0	0
$909$	0.394711	0.0130918
$910$	0	0
$911$	−36.7007	−1.21595	−0.607975	−	0.793956i	$-0.708018\pi$
$911$	−36.7007	−1.21595	−0.607975	+	0.793956i	$0.708018\pi$
$912$	0	0
$913$	11.9425	0.395238
$914$	0	0
$915$	−1.96543	−0.0649751
$916$	0	0
$917$	9.99334	0.330009
$918$	0	0
$919$	3.78133	0.124735	0.0623673	−	0.998053i	$-0.480135\pi$
$919$	3.78133	0.124735	0.0623673	+	0.998053i	$0.480135\pi$
$920$	0	0
$921$	26.4876	0.872797
$922$	0	0
$923$	−12.6203	−0.415402
$924$	0	0
$925$	34.9612	1.14952
$926$	0	0
$927$	0.423151	0.0138981
$928$	0	0
$929$	−40.3540	−1.32397	−0.661986	−	0.749516i	$-0.730286\pi$
$929$	−40.3540	−1.32397	−0.661986	+	0.749516i	$0.730286\pi$
$930$	0	0
$931$	3.26997	0.107169
$932$	0	0
$933$	−39.6768	−1.29896
$934$	0	0
$935$	1.57662	0.0515610
$936$	0	0
$937$	−15.5294	−0.507323	−0.253661	−	0.967293i	$-0.581635\pi$
$937$	−15.5294	−0.507323	−0.253661	+	0.967293i	$0.581635\pi$
$938$	0	0
$939$	−47.6198	−1.55401
$940$	0	0
$941$	−44.4373	−1.44861	−0.724307	−	0.689478i	$-0.757840\pi$
$941$	−44.4373	−1.44861	−0.724307	+	0.689478i	$0.757840\pi$
$942$	0	0
$943$	40.0828	1.30528
$944$	0	0
$945$	0.515134	0.0167573
$946$	0	0
$947$	44.0218	1.43052	0.715258	−	0.698860i	$-0.246309\pi$
$947$	44.0218	1.43052	0.715258	+	0.698860i	$0.246309\pi$
$948$	0	0
$949$	2.58701	0.0839778
$950$	0	0
$951$	−0.0897803	−0.00291133
$952$	0	0
$953$	−16.9402	−0.548747	−0.274373	−	0.961623i	$-0.588470\pi$
$953$	−16.9402	−0.548747	−0.274373	+	0.961623i	$0.588470\pi$
$954$	0	0
$955$	2.15764	0.0698196
$956$	0	0
$957$	−27.4337	−0.886807
$958$	0	0
$959$	−8.31844	−0.268617
$960$	0	0
$961$	3.31757	0.107018
$962$	0	0
$963$	−1.04447	−0.0336576
$964$	0	0
$965$	−2.97715	−0.0958380
$966$	0	0
$967$	−10.4446	−0.335876	−0.167938	−	0.985798i	$-0.553711\pi$
$967$	−10.4446	−0.335876	−0.167938	+	0.985798i	$0.553711\pi$
$968$	0	0
$969$	−3.36741	−0.108177
$970$	0	0
$971$	43.6903	1.40209	0.701044	−	0.713118i	$-0.252717\pi$
$971$	43.6903	1.40209	0.701044	+	0.713118i	$0.252717\pi$
$972$	0	0
$973$	−3.07855	−0.0986938
$974$	0	0
$975$	29.8766	0.956815
$976$	0	0
$977$	40.2204	1.28677	0.643383	−	0.765545i	$-0.277530\pi$
$977$	40.2204	1.28677	0.643383	+	0.765545i	$0.277530\pi$
$978$	0	0
$979$	18.9079	0.604298
$980$	0	0
$981$	0.289643	0.00924760
$982$	0	0
$983$	−27.7769	−0.885946	−0.442973	−	0.896535i	$-0.646076\pi$
$983$	−27.7769	−0.885946	−0.442973	+	0.896535i	$0.646076\pi$
$984$	0	0
$985$	1.07358	0.0342072
$986$	0	0
$987$	−0.120014	−0.00382010
$988$	0	0
$989$	−19.3354	−0.614829
$990$	0	0
$991$	42.9248	1.36355	0.681776	−	0.731561i	$-0.261208\pi$
$991$	42.9248	1.36355	0.681776	+	0.731561i	$0.261208\pi$
$992$	0	0
$993$	−2.13656	−0.0678016
$994$	0	0
$995$	−3.29553	−0.104475
$996$	0	0
$997$	5.03519	0.159466	0.0797330	−	0.996816i	$-0.474593\pi$
$997$	5.03519	0.159466	0.0797330	+	0.996816i	$0.474593\pi$
$998$	0	0
$999$	36.8746	1.16666

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.h.1.7		9
4.3	odd	2		2008.2.a.a.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.a.1.3	✓	9	4.3	odd	2
4016.2.a.h.1.7		9	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.71581 q^{3} -0.169057 q^{5} +0.581118 q^{7} -0.0559961 q^{9} +O(q^{10})\)	\(q+1.71581 q^{3} -0.169057 q^{5} +0.581118 q^{7} -0.0559961 q^{9} -2.33231 q^{11} -3.50252 q^{13} -0.290070 q^{15} +3.99859 q^{17} -0.490816 q^{19} +0.997088 q^{21} -5.91408 q^{23} -4.97142 q^{25} -5.24351 q^{27} +6.85536 q^{29} +5.85812 q^{31} -4.00179 q^{33} -0.0982423 q^{35} -7.03243 q^{37} -6.00966 q^{39} -6.77752 q^{41} +3.26938 q^{43} +0.00946657 q^{45} -0.120365 q^{47} -6.66230 q^{49} +6.86082 q^{51} -5.84573 q^{53} +0.394294 q^{55} -0.842148 q^{57} -8.47096 q^{59} +6.77570 q^{61} -0.0325404 q^{63} +0.592128 q^{65} +2.24906 q^{67} -10.1474 q^{69} +3.60320 q^{71} -0.738613 q^{73} -8.53001 q^{75} -1.35534 q^{77} -5.77695 q^{79} -8.82888 q^{81} -5.12046 q^{83} -0.675992 q^{85} +11.7625 q^{87} -8.10695 q^{89} -2.03538 q^{91} +10.0514 q^{93} +0.0829762 q^{95} -13.9224 q^{97} +0.130600 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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