LMFDB - Embedded newform 8008.2.a.u.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 16x^{8} + 26x^{7} + 93x^{6} - 113x^{5} - 230x^{4} + 197x^{3} + 201x^{2} - 115x + 8 $ x^10 - 2x^9 - 16x^8 + 26x^7 + 93x^6 - 113x^5 - 230x^4 + 197x^3 + 201x^2 - 115*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.0822465$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0822465 q^{3} -4.16696 q^{5} -1.00000 q^{7} -2.99324 q^{9} +O(q^{10})$	$q+0.0822465 q^{3} -4.16696 q^{5} -1.00000 q^{7} -2.99324 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.342718 q^{15} +3.55835 q^{17} +5.09522 q^{19} -0.0822465 q^{21} -0.867395 q^{23} +12.3635 q^{25} -0.492922 q^{27} -8.23108 q^{29} +2.28590 q^{31} -0.0822465 q^{33} +4.16696 q^{35} +6.74793 q^{37} -0.0822465 q^{39} +2.15690 q^{41} +6.41607 q^{43} +12.4727 q^{45} -13.3368 q^{47} +1.00000 q^{49} +0.292662 q^{51} -2.36261 q^{53} +4.16696 q^{55} +0.419064 q^{57} +2.85455 q^{59} +3.08242 q^{61} +2.99324 q^{63} +4.16696 q^{65} +7.23118 q^{67} -0.0713401 q^{69} +1.56955 q^{71} +4.34423 q^{73} +1.01686 q^{75} +1.00000 q^{77} +0.0822696 q^{79} +8.93917 q^{81} -5.58917 q^{83} -14.8275 q^{85} -0.676977 q^{87} +5.43343 q^{89} +1.00000 q^{91} +0.188007 q^{93} -21.2316 q^{95} -11.8362 q^{97} +2.99324 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * 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
$3$	0.0822465	0.0474850	0.0237425	−	0.999718i	$-0.492442\pi$
$3$	0.0822465	0.0474850	0.0237425	+	0.999718i	$0.492442\pi$
$4$	0	0
$5$	−4.16696	−1.86352	−0.931760	−	0.363074i	$-0.881727\pi$
$5$	−4.16696	−1.86352	−0.931760	+	0.363074i	$0.881727\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.99324	−0.997745
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.342718	−0.0884893
$16$	0	0
$17$	3.55835	0.863027	0.431514	−	0.902106i	$-0.357980\pi$
$17$	3.55835	0.863027	0.431514	+	0.902106i	$0.357980\pi$
$18$	0	0
$19$	5.09522	1.16892	0.584462	−	0.811421i	$-0.301306\pi$
$19$	5.09522	1.16892	0.584462	+	0.811421i	$0.301306\pi$
$20$	0	0
$21$	−0.0822465	−0.0179476
$22$	0	0
$23$	−0.867395	−0.180864	−0.0904321	−	0.995903i	$-0.528825\pi$
$23$	−0.867395	−0.180864	−0.0904321	+	0.995903i	$0.528825\pi$
$24$	0	0
$25$	12.3635	2.47271
$26$	0	0
$27$	−0.492922	−0.0948629
$28$	0	0
$29$	−8.23108	−1.52847	−0.764236	−	0.644936i	$-0.776884\pi$
$29$	−8.23108	−1.52847	−0.764236	+	0.644936i	$0.776884\pi$
$30$	0	0
$31$	2.28590	0.410559	0.205280	−	0.978703i	$-0.434190\pi$
$31$	2.28590	0.410559	0.205280	+	0.978703i	$0.434190\pi$
$32$	0	0
$33$	−0.0822465	−0.0143173
$34$	0	0
$35$	4.16696	0.704345
$36$	0	0
$37$	6.74793	1.10935	0.554676	−	0.832066i	$-0.312842\pi$
$37$	6.74793	1.10935	0.554676	+	0.832066i	$0.312842\pi$
$38$	0	0
$39$	−0.0822465	−0.0131700
$40$	0	0
$41$	2.15690	0.336851	0.168425	−	0.985714i	$-0.446132\pi$
$41$	2.15690	0.336851	0.168425	+	0.985714i	$0.446132\pi$
$42$	0	0
$43$	6.41607	0.978441	0.489221	−	0.872160i	$-0.337281\pi$
$43$	6.41607	0.978441	0.489221	+	0.872160i	$0.337281\pi$
$44$	0	0
$45$	12.4727	1.85932
$46$	0	0
$47$	−13.3368	−1.94537	−0.972686	−	0.232126i	$-0.925432\pi$
$47$	−13.3368	−1.94537	−0.972686	+	0.232126i	$0.925432\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.292662	0.0409808
$52$	0	0
$53$	−2.36261	−0.324530	−0.162265	−	0.986747i	$-0.551880\pi$
$53$	−2.36261	−0.324530	−0.162265	+	0.986747i	$0.551880\pi$
$54$	0	0
$55$	4.16696	0.561873
$56$	0	0
$57$	0.419064	0.0555063
$58$	0	0
$59$	2.85455	0.371631	0.185815	−	0.982585i	$-0.440507\pi$
$59$	2.85455	0.371631	0.185815	+	0.982585i	$0.440507\pi$
$60$	0	0
$61$	3.08242	0.394663	0.197332	−	0.980337i	$-0.436772\pi$
$61$	3.08242	0.394663	0.197332	+	0.980337i	$0.436772\pi$
$62$	0	0
$63$	2.99324	0.377112
$64$	0	0
$65$	4.16696	0.516848
$66$	0	0
$67$	7.23118	0.883430	0.441715	−	0.897156i	$-0.354370\pi$
$67$	7.23118	0.883430	0.441715	+	0.897156i	$0.354370\pi$
$68$	0	0
$69$	−0.0713401	−0.00858834
$70$	0	0
$71$	1.56955	0.186271	0.0931354	−	0.995653i	$-0.470311\pi$
$71$	1.56955	0.186271	0.0931354	+	0.995653i	$0.470311\pi$
$72$	0	0
$73$	4.34423	0.508453	0.254227	−	0.967145i	$-0.418179\pi$
$73$	4.34423	0.508453	0.254227	+	0.967145i	$0.418179\pi$
$74$	0	0
$75$	1.01686	0.117417
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	0.0822696	0.00925605	0.00462802	−	0.999989i	$-0.498527\pi$
$79$	0.0822696	0.00925605	0.00462802	+	0.999989i	$0.498527\pi$
$80$	0	0
$81$	8.93917	0.993241
$82$	0	0
$83$	−5.58917	−0.613491	−0.306746	−	0.951792i	$-0.599240\pi$
$83$	−5.58917	−0.613491	−0.306746	+	0.951792i	$0.599240\pi$
$84$	0	0
$85$	−14.8275	−1.60827
$86$	0	0
$87$	−0.676977	−0.0725796
$88$	0	0
$89$	5.43343	0.575942	0.287971	−	0.957639i	$-0.407019\pi$
$89$	5.43343	0.575942	0.287971	+	0.957639i	$0.407019\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0.188007	0.0194954
$94$	0	0
$95$	−21.2316	−2.17831
$96$	0	0
$97$	−11.8362	−1.20179	−0.600894	−	0.799328i	$-0.705189\pi$
$97$	−11.8362	−1.20179	−0.600894	+	0.799328i	$0.705189\pi$
$98$	0	0
$99$	2.99324	0.300831
$100$	0	0
$101$	13.1574	1.30921	0.654605	−	0.755971i	$-0.272835\pi$
$101$	13.1574	1.30921	0.654605	+	0.755971i	$0.272835\pi$
$102$	0	0
$103$	−2.45911	−0.242303	−0.121152	−	0.992634i	$-0.538659\pi$
$103$	−2.45911	−0.242303	−0.121152	+	0.992634i	$0.538659\pi$
$104$	0	0
$105$	0.342718	0.0334458
$106$	0	0
$107$	3.15058	0.304578	0.152289	−	0.988336i	$-0.451336\pi$
$107$	3.15058	0.304578	0.152289	+	0.988336i	$0.451336\pi$
$108$	0	0
$109$	19.2113	1.84011	0.920055	−	0.391789i	$-0.128144\pi$
$109$	19.2113	1.84011	0.920055	+	0.391789i	$0.128144\pi$
$110$	0	0
$111$	0.554993	0.0526776
$112$	0	0
$113$	9.78708	0.920691	0.460345	−	0.887740i	$-0.347726\pi$
$113$	9.78708	0.920691	0.460345	+	0.887740i	$0.347726\pi$
$114$	0	0
$115$	3.61440	0.337044
$116$	0	0
$117$	2.99324	0.276725
$118$	0	0
$119$	−3.55835	−0.326194
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.177397	0.0159954
$124$	0	0
$125$	−30.6836	−2.74442
$126$	0	0
$127$	−9.25558	−0.821300	−0.410650	−	0.911793i	$-0.634698\pi$
$127$	−9.25558	−0.821300	−0.410650	+	0.911793i	$0.634698\pi$
$128$	0	0
$129$	0.527699	0.0464613
$130$	0	0
$131$	−4.72266	−0.412620	−0.206310	−	0.978487i	$-0.566146\pi$
$131$	−4.72266	−0.412620	−0.206310	+	0.978487i	$0.566146\pi$
$132$	0	0
$133$	−5.09522	−0.441812
$134$	0	0
$135$	2.05399	0.176779
$136$	0	0
$137$	−14.8620	−1.26975	−0.634875	−	0.772615i	$-0.718948\pi$
$137$	−14.8620	−1.26975	−0.634875	+	0.772615i	$0.718948\pi$
$138$	0	0
$139$	12.0767	1.02433	0.512164	−	0.858888i	$-0.328844\pi$
$139$	12.0767	1.02433	0.512164	+	0.858888i	$0.328844\pi$
$140$	0	0
$141$	−1.09690	−0.0923760
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	34.2986	2.84834
$146$	0	0
$147$	0.0822465	0.00678357
$148$	0	0
$149$	−16.0424	−1.31424	−0.657120	−	0.753786i	$-0.728226\pi$
$149$	−16.0424	−1.31424	−0.657120	+	0.753786i	$0.728226\pi$
$150$	0	0
$151$	−2.36830	−0.192729	−0.0963647	−	0.995346i	$-0.530722\pi$
$151$	−2.36830	−0.192729	−0.0963647	+	0.995346i	$0.530722\pi$
$152$	0	0
$153$	−10.6510	−0.861081
$154$	0	0
$155$	−9.52524	−0.765086
$156$	0	0
$157$	0.920932	0.0734984	0.0367492	−	0.999325i	$-0.488300\pi$
$157$	0.920932	0.0734984	0.0367492	+	0.999325i	$0.488300\pi$
$158$	0	0
$159$	−0.194316	−0.0154103
$160$	0	0
$161$	0.867395	0.0683603
$162$	0	0
$163$	−16.8658	−1.32103	−0.660516	−	0.750812i	$-0.729663\pi$
$163$	−16.8658	−1.32103	−0.660516	+	0.750812i	$0.729663\pi$
$164$	0	0
$165$	0.342718	0.0266805
$166$	0	0
$167$	−0.272664	−0.0210994	−0.0105497	−	0.999944i	$-0.503358\pi$
$167$	−0.272664	−0.0210994	−0.0105497	+	0.999944i	$0.503358\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−15.2512	−1.16629
$172$	0	0
$173$	−7.79353	−0.592531	−0.296266	−	0.955106i	$-0.595741\pi$
$173$	−7.79353	−0.592531	−0.296266	+	0.955106i	$0.595741\pi$
$174$	0	0
$175$	−12.3635	−0.934596
$176$	0	0
$177$	0.234776	0.0176469
$178$	0	0
$179$	−2.03030	−0.151752	−0.0758759	−	0.997117i	$-0.524175\pi$
$179$	−2.03030	−0.151752	−0.0758759	+	0.997117i	$0.524175\pi$
$180$	0	0
$181$	3.60659	0.268075	0.134038	−	0.990976i	$-0.457206\pi$
$181$	3.60659	0.268075	0.134038	+	0.990976i	$0.457206\pi$
$182$	0	0
$183$	0.253518	0.0187406
$184$	0	0
$185$	−28.1183	−2.06730
$186$	0	0
$187$	−3.55835	−0.260212
$188$	0	0
$189$	0.492922	0.0358548
$190$	0	0
$191$	7.49107	0.542035	0.271017	−	0.962574i	$-0.412640\pi$
$191$	7.49107	0.542035	0.271017	+	0.962574i	$0.412640\pi$
$192$	0	0
$193$	−8.29214	−0.596882	−0.298441	−	0.954428i	$-0.596467\pi$
$193$	−8.29214	−0.596882	−0.298441	+	0.954428i	$0.596467\pi$
$194$	0	0
$195$	0.342718	0.0245425
$196$	0	0
$197$	19.0417	1.35667	0.678333	−	0.734754i	$-0.262703\pi$
$197$	19.0417	1.35667	0.678333	+	0.734754i	$0.262703\pi$
$198$	0	0
$199$	−15.1220	−1.07197	−0.535986	−	0.844227i	$-0.680060\pi$
$199$	−15.1220	−1.07197	−0.535986	+	0.844227i	$0.680060\pi$
$200$	0	0
$201$	0.594739	0.0419497
$202$	0	0
$203$	8.23108	0.577708
$204$	0	0
$205$	−8.98770	−0.627728
$206$	0	0
$207$	2.59632	0.180456
$208$	0	0
$209$	−5.09522	−0.352444
$210$	0	0
$211$	−23.7605	−1.63574	−0.817872	−	0.575401i	$-0.804846\pi$
$211$	−23.7605	−1.63574	−0.817872	+	0.575401i	$0.804846\pi$
$212$	0	0
$213$	0.129090	0.00884507
$214$	0	0
$215$	−26.7355	−1.82335
$216$	0	0
$217$	−2.28590	−0.155177
$218$	0	0
$219$	0.357297	0.0241439
$220$	0	0
$221$	−3.55835	−0.239361
$222$	0	0
$223$	−5.23733	−0.350718	−0.175359	−	0.984505i	$-0.556109\pi$
$223$	−5.23733	−0.350718	−0.175359	+	0.984505i	$0.556109\pi$
$224$	0	0
$225$	−37.0070	−2.46713
$226$	0	0
$227$	0.629276	0.0417665	0.0208833	−	0.999782i	$-0.493352\pi$
$227$	0.629276	0.0417665	0.0208833	+	0.999782i	$0.493352\pi$
$228$	0	0
$229$	−15.1243	−0.999439	−0.499720	−	0.866187i	$-0.666564\pi$
$229$	−15.1243	−0.999439	−0.499720	+	0.866187i	$0.666564\pi$
$230$	0	0
$231$	0.0822465	0.00541142
$232$	0	0
$233$	−6.49987	−0.425820	−0.212910	−	0.977072i	$-0.568294\pi$
$233$	−6.49987	−0.425820	−0.212910	+	0.977072i	$0.568294\pi$
$234$	0	0
$235$	55.5739	3.62524
$236$	0	0
$237$	0.00676638	0.000439524 0
$238$	0	0
$239$	−24.6722	−1.59591	−0.797955	−	0.602717i	$-0.794085\pi$
$239$	−24.6722	−1.59591	−0.797955	+	0.602717i	$0.794085\pi$
$240$	0	0
$241$	−18.5382	−1.19415	−0.597074	−	0.802186i	$-0.703670\pi$
$241$	−18.5382	−1.19415	−0.597074	+	0.802186i	$0.703670\pi$
$242$	0	0
$243$	2.21398	0.142027
$244$	0	0
$245$	−4.16696	−0.266217
$246$	0	0
$247$	−5.09522	−0.324201
$248$	0	0
$249$	−0.459690	−0.0291316
$250$	0	0
$251$	25.0435	1.58073	0.790365	−	0.612636i	$-0.209891\pi$
$251$	25.0435	1.58073	0.790365	+	0.612636i	$0.209891\pi$
$252$	0	0
$253$	0.867395	0.0545326
$254$	0	0
$255$	−1.21951	−0.0763686
$256$	0	0
$257$	−1.87197	−0.116770	−0.0583851	−	0.998294i	$-0.518595\pi$
$257$	−1.87197	−0.116770	−0.0583851	+	0.998294i	$0.518595\pi$
$258$	0	0
$259$	−6.74793	−0.419296
$260$	0	0
$261$	24.6376	1.52503
$262$	0	0
$263$	−18.4354	−1.13678	−0.568389	−	0.822760i	$-0.692433\pi$
$263$	−18.4354	−1.13678	−0.568389	+	0.822760i	$0.692433\pi$
$264$	0	0
$265$	9.84490	0.604768
$266$	0	0
$267$	0.446880	0.0273486
$268$	0	0
$269$	−9.63833	−0.587659	−0.293830	−	0.955858i	$-0.594930\pi$
$269$	−9.63833	−0.587659	−0.293830	+	0.955858i	$0.594930\pi$
$270$	0	0
$271$	−2.11167	−0.128275	−0.0641374	−	0.997941i	$-0.520430\pi$
$271$	−2.11167	−0.128275	−0.0641374	+	0.997941i	$0.520430\pi$
$272$	0	0
$273$	0.0822465	0.00497778
$274$	0	0
$275$	−12.3635	−0.745550
$276$	0	0
$277$	−13.5147	−0.812019	−0.406010	−	0.913869i	$-0.633080\pi$
$277$	−13.5147	−0.812019	−0.406010	+	0.913869i	$0.633080\pi$
$278$	0	0
$279$	−6.84223	−0.409634
$280$	0	0
$281$	−12.4079	−0.740193	−0.370097	−	0.928993i	$-0.620675\pi$
$281$	−12.4079	−0.740193	−0.370097	+	0.928993i	$0.620675\pi$
$282$	0	0
$283$	20.5515	1.22166	0.610830	−	0.791762i	$-0.290836\pi$
$283$	20.5515	1.22166	0.610830	+	0.791762i	$0.290836\pi$
$284$	0	0
$285$	−1.74622	−0.103437
$286$	0	0
$287$	−2.15690	−0.127318
$288$	0	0
$289$	−4.33813	−0.255184
$290$	0	0
$291$	−0.973489	−0.0570669
$292$	0	0
$293$	−9.74122	−0.569088	−0.284544	−	0.958663i	$-0.591842\pi$
$293$	−9.74122	−0.569088	−0.284544	+	0.958663i	$0.591842\pi$
$294$	0	0
$295$	−11.8948	−0.692541
$296$	0	0
$297$	0.492922	0.0286023
$298$	0	0
$299$	0.867395	0.0501627
$300$	0	0
$301$	−6.41607	−0.369816
$302$	0	0
$303$	1.08215	0.0621679
$304$	0	0
$305$	−12.8443	−0.735463
$306$	0	0
$307$	8.98907	0.513034	0.256517	−	0.966540i	$-0.417425\pi$
$307$	8.98907	0.513034	0.256517	+	0.966540i	$0.417425\pi$
$308$	0	0
$309$	−0.202253	−0.0115058
$310$	0	0
$311$	−6.04639	−0.342860	−0.171430	−	0.985196i	$-0.554839\pi$
$311$	−6.04639	−0.342860	−0.171430	+	0.985196i	$0.554839\pi$
$312$	0	0
$313$	3.86492	0.218458	0.109229	−	0.994017i	$-0.465162\pi$
$313$	3.86492	0.218458	0.109229	+	0.994017i	$0.465162\pi$
$314$	0	0
$315$	−12.4727	−0.702756
$316$	0	0
$317$	20.2825	1.13918	0.569588	−	0.821930i	$-0.307103\pi$
$317$	20.2825	1.13918	0.569588	+	0.821930i	$0.307103\pi$
$318$	0	0
$319$	8.23108	0.460852
$320$	0	0
$321$	0.259124	0.0144629
$322$	0	0
$323$	18.1306	1.00881
$324$	0	0
$325$	−12.3635	−0.685806
$326$	0	0
$327$	1.58006	0.0873776
$328$	0	0
$329$	13.3368	0.735281
$330$	0	0
$331$	−13.3781	−0.735329	−0.367665	−	0.929958i	$-0.619843\pi$
$331$	−13.3781	−0.735329	−0.367665	+	0.929958i	$0.619843\pi$
$332$	0	0
$333$	−20.1981	−1.10685
$334$	0	0
$335$	−30.1320	−1.64629
$336$	0	0
$337$	−12.1252	−0.660503	−0.330251	−	0.943893i	$-0.607133\pi$
$337$	−12.1252	−0.660503	−0.330251	+	0.943893i	$0.607133\pi$
$338$	0	0
$339$	0.804952	0.0437190
$340$	0	0
$341$	−2.28590	−0.123788
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0.297271	0.0160046
$346$	0	0
$347$	−34.5746	−1.85606	−0.928032	−	0.372501i	$-0.878500\pi$
$347$	−34.5746	−1.85606	−0.928032	+	0.372501i	$0.878500\pi$
$348$	0	0
$349$	−26.7103	−1.42977	−0.714884	−	0.699243i	$-0.753521\pi$
$349$	−26.7103	−1.42977	−0.714884	+	0.699243i	$0.753521\pi$
$350$	0	0
$351$	0.492922	0.0263102
$352$	0	0
$353$	21.7511	1.15769	0.578847	−	0.815436i	$-0.303503\pi$
$353$	21.7511	1.15769	0.578847	+	0.815436i	$0.303503\pi$
$354$	0	0
$355$	−6.54023	−0.347119
$356$	0	0
$357$	−0.292662	−0.0154893
$358$	0	0
$359$	−22.6345	−1.19460	−0.597301	−	0.802017i	$-0.703760\pi$
$359$	−22.6345	−1.19460	−0.597301	+	0.802017i	$0.703760\pi$
$360$	0	0
$361$	6.96126	0.366382
$362$	0	0
$363$	0.0822465	0.00431682
$364$	0	0
$365$	−18.1022	−0.947513
$366$	0	0
$367$	28.2933	1.47690	0.738449	−	0.674309i	$-0.235558\pi$
$367$	28.2933	1.47690	0.738449	+	0.674309i	$0.235558\pi$
$368$	0	0
$369$	−6.45610	−0.336091
$370$	0	0
$371$	2.36261	0.122661
$372$	0	0
$373$	25.8464	1.33827	0.669137	−	0.743139i	$-0.266664\pi$
$373$	25.8464	1.33827	0.669137	+	0.743139i	$0.266664\pi$
$374$	0	0
$375$	−2.52361	−0.130319
$376$	0	0
$377$	8.23108	0.423922
$378$	0	0
$379$	2.00688	0.103086	0.0515431	−	0.998671i	$-0.483586\pi$
$379$	2.00688	0.103086	0.0515431	+	0.998671i	$0.483586\pi$
$380$	0	0
$381$	−0.761239	−0.0389994
$382$	0	0
$383$	25.7837	1.31749	0.658744	−	0.752367i	$-0.271088\pi$
$383$	25.7837	1.31749	0.658744	+	0.752367i	$0.271088\pi$
$384$	0	0
$385$	−4.16696	−0.212368
$386$	0	0
$387$	−19.2048	−0.976235
$388$	0	0
$389$	5.03712	0.255392	0.127696	−	0.991813i	$-0.459242\pi$
$389$	5.03712	0.255392	0.127696	+	0.991813i	$0.459242\pi$
$390$	0	0
$391$	−3.08649	−0.156091
$392$	0	0
$393$	−0.388422	−0.0195933
$394$	0	0
$395$	−0.342814	−0.0172488
$396$	0	0
$397$	−9.26949	−0.465222	−0.232611	−	0.972570i	$-0.574727\pi$
$397$	−9.26949	−0.465222	−0.232611	+	0.972570i	$0.574727\pi$
$398$	0	0
$399$	−0.419064	−0.0209794
$400$	0	0
$401$	25.3355	1.26519	0.632596	−	0.774482i	$-0.281989\pi$
$401$	25.3355	1.26519	0.632596	+	0.774482i	$0.281989\pi$
$402$	0	0
$403$	−2.28590	−0.113869
$404$	0	0
$405$	−37.2491	−1.85092
$406$	0	0
$407$	−6.74793	−0.334482
$408$	0	0
$409$	−37.9974	−1.87885	−0.939426	−	0.342751i	$-0.888641\pi$
$409$	−37.9974	−1.87885	−0.939426	+	0.342751i	$0.888641\pi$
$410$	0	0
$411$	−1.22235	−0.0602941
$412$	0	0
$413$	−2.85455	−0.140463
$414$	0	0
$415$	23.2899	1.14325
$416$	0	0
$417$	0.993262	0.0486402
$418$	0	0
$419$	16.9918	0.830105	0.415053	−	0.909797i	$-0.363763\pi$
$419$	16.9918	0.830105	0.415053	+	0.909797i	$0.363763\pi$
$420$	0	0
$421$	−1.52547	−0.0743471	−0.0371735	−	0.999309i	$-0.511835\pi$
$421$	−1.52547	−0.0743471	−0.0371735	+	0.999309i	$0.511835\pi$
$422$	0	0
$423$	39.9202	1.94099
$424$	0	0
$425$	43.9938	2.13401
$426$	0	0
$427$	−3.08242	−0.149169
$428$	0	0
$429$	0.0822465	0.00397090
$430$	0	0
$431$	−4.34282	−0.209186	−0.104593	−	0.994515i	$-0.533354\pi$
$431$	−4.34282	−0.209186	−0.104593	+	0.994515i	$0.533354\pi$
$432$	0	0
$433$	−7.42087	−0.356624	−0.178312	−	0.983974i	$-0.557064\pi$
$433$	−7.42087	−0.356624	−0.178312	+	0.983974i	$0.557064\pi$
$434$	0	0
$435$	2.82093	0.135253
$436$	0	0
$437$	−4.41957	−0.211417
$438$	0	0
$439$	34.8332	1.66250	0.831249	−	0.555901i	$-0.187627\pi$
$439$	34.8332	1.66250	0.831249	+	0.555901i	$0.187627\pi$
$440$	0	0
$441$	−2.99324	−0.142535
$442$	0	0
$443$	21.2656	1.01036	0.505180	−	0.863014i	$-0.331426\pi$
$443$	21.2656	1.01036	0.505180	+	0.863014i	$0.331426\pi$
$444$	0	0
$445$	−22.6409	−1.07328
$446$	0	0
$447$	−1.31943	−0.0624067
$448$	0	0
$449$	−40.0150	−1.88842	−0.944212	−	0.329339i	$-0.893174\pi$
$449$	−40.0150	−1.88842	−0.944212	+	0.329339i	$0.893174\pi$
$450$	0	0
$451$	−2.15690	−0.101564
$452$	0	0
$453$	−0.194784	−0.00915176
$454$	0	0
$455$	−4.16696	−0.195350
$456$	0	0
$457$	−3.96588	−0.185516	−0.0927580	−	0.995689i	$-0.529568\pi$
$457$	−3.96588	−0.185516	−0.0927580	+	0.995689i	$0.529568\pi$
$458$	0	0
$459$	−1.75399	−0.0818693
$460$	0	0
$461$	11.0325	0.513837	0.256918	−	0.966433i	$-0.417293\pi$
$461$	11.0325	0.513837	0.256918	+	0.966433i	$0.417293\pi$
$462$	0	0
$463$	−0.180837	−0.00840419	−0.00420210	−	0.999991i	$-0.501338\pi$
$463$	−0.180837	−0.00840419	−0.00420210	+	0.999991i	$0.501338\pi$
$464$	0	0
$465$	−0.783417	−0.0363301
$466$	0	0
$467$	26.2015	1.21246	0.606229	−	0.795290i	$-0.292681\pi$
$467$	26.2015	1.21246	0.606229	+	0.795290i	$0.292681\pi$
$468$	0	0
$469$	−7.23118	−0.333905
$470$	0	0
$471$	0.0757434	0.00349007
$472$	0	0
$473$	−6.41607	−0.295011
$474$	0	0
$475$	62.9950	2.89041
$476$	0	0
$477$	7.07185	0.323798
$478$	0	0
$479$	11.7491	0.536828	0.268414	−	0.963304i	$-0.413501\pi$
$479$	11.7491	0.536828	0.268414	+	0.963304i	$0.413501\pi$
$480$	0	0
$481$	−6.74793	−0.307679
$482$	0	0
$483$	0.0713401	0.00324609
$484$	0	0
$485$	49.3211	2.23956
$486$	0	0
$487$	−8.18584	−0.370936	−0.185468	−	0.982650i	$-0.559380\pi$
$487$	−8.18584	−0.370936	−0.185468	+	0.982650i	$0.559380\pi$
$488$	0	0
$489$	−1.38715	−0.0627292
$490$	0	0
$491$	−23.5403	−1.06236	−0.531179	−	0.847260i	$-0.678251\pi$
$491$	−23.5403	−1.06236	−0.531179	+	0.847260i	$0.678251\pi$
$492$	0	0
$493$	−29.2891	−1.31911
$494$	0	0
$495$	−12.4727	−0.560606
$496$	0	0
$497$	−1.56955	−0.0704037
$498$	0	0
$499$	17.2794	0.773532	0.386766	−	0.922178i	$-0.373592\pi$
$499$	17.2794	0.773532	0.386766	+	0.922178i	$0.373592\pi$
$500$	0	0
$501$	−0.0224256	−0.00100190
$502$	0	0
$503$	33.6467	1.50023	0.750117	−	0.661306i	$-0.229997\pi$
$503$	33.6467	1.50023	0.750117	+	0.661306i	$0.229997\pi$
$504$	0	0
$505$	−54.8264	−2.43974
$506$	0	0
$507$	0.0822465	0.00365269
$508$	0	0
$509$	18.4079	0.815918	0.407959	−	0.913000i	$-0.366241\pi$
$509$	18.4079	0.815918	0.407959	+	0.913000i	$0.366241\pi$
$510$	0	0
$511$	−4.34423	−0.192177
$512$	0	0
$513$	−2.51155	−0.110888
$514$	0	0
$515$	10.2470	0.451537
$516$	0	0
$517$	13.3368	0.586552
$518$	0	0
$519$	−0.640991	−0.0281364
$520$	0	0
$521$	6.80001	0.297914	0.148957	−	0.988844i	$-0.452408\pi$
$521$	6.80001	0.297914	0.148957	+	0.988844i	$0.452408\pi$
$522$	0	0
$523$	−7.68243	−0.335929	−0.167965	−	0.985793i	$-0.553719\pi$
$523$	−7.68243	−0.335929	−0.167965	+	0.985793i	$0.553719\pi$
$524$	0	0
$525$	−1.01686	−0.0443793
$526$	0	0
$527$	8.13403	0.354324
$528$	0	0
$529$	−22.2476	−0.967288
$530$	0	0
$531$	−8.54433	−0.370793
$532$	0	0
$533$	−2.15690	−0.0934256
$534$	0	0
$535$	−13.1283	−0.567587
$536$	0	0
$537$	−0.166985	−0.00720593
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	24.1763	1.03942	0.519710	−	0.854343i	$-0.326040\pi$
$541$	24.1763	1.03942	0.519710	+	0.854343i	$0.326040\pi$
$542$	0	0
$543$	0.296629	0.0127296
$544$	0	0
$545$	−80.0527	−3.42908
$546$	0	0
$547$	2.80666	0.120004	0.0600020	−	0.998198i	$-0.480889\pi$
$547$	2.80666	0.120004	0.0600020	+	0.998198i	$0.480889\pi$
$548$	0	0
$549$	−9.22640	−0.393773
$550$	0	0
$551$	−41.9392	−1.78667
$552$	0	0
$553$	−0.0822696	−0.00349846
$554$	0	0
$555$	−2.31263	−0.0981658
$556$	0	0
$557$	−39.9857	−1.69425	−0.847123	−	0.531396i	$-0.821667\pi$
$557$	−39.9857	−1.69425	−0.847123	+	0.531396i	$0.821667\pi$
$558$	0	0
$559$	−6.41607	−0.271371
$560$	0	0
$561$	−0.292662	−0.0123562
$562$	0	0
$563$	11.8498	0.499411	0.249706	−	0.968322i	$-0.419666\pi$
$563$	11.8498	0.499411	0.249706	+	0.968322i	$0.419666\pi$
$564$	0	0
$565$	−40.7823	−1.71573
$566$	0	0
$567$	−8.93917	−0.375410
$568$	0	0
$569$	23.0663	0.966990	0.483495	−	0.875347i	$-0.339367\pi$
$569$	23.0663	0.966990	0.483495	+	0.875347i	$0.339367\pi$
$570$	0	0
$571$	−44.7513	−1.87279	−0.936393	−	0.350954i	$-0.885857\pi$
$571$	−44.7513	−1.87279	−0.936393	+	0.350954i	$0.885857\pi$
$572$	0	0
$573$	0.616114	0.0257385
$574$	0	0
$575$	−10.7241	−0.447225
$576$	0	0
$577$	−29.6341	−1.23368	−0.616842	−	0.787087i	$-0.711588\pi$
$577$	−29.6341	−1.23368	−0.616842	+	0.787087i	$0.711588\pi$
$578$	0	0
$579$	−0.681999	−0.0283429
$580$	0	0
$581$	5.58917	0.231878
$582$	0	0
$583$	2.36261	0.0978494
$584$	0	0
$585$	−12.4727	−0.515682
$586$	0	0
$587$	−27.4384	−1.13250	−0.566251	−	0.824233i	$-0.691607\pi$
$587$	−27.4384	−1.13250	−0.566251	+	0.824233i	$0.691607\pi$
$588$	0	0
$589$	11.6471	0.479912
$590$	0	0
$591$	1.56611	0.0644213
$592$	0	0
$593$	22.4208	0.920712	0.460356	−	0.887734i	$-0.347722\pi$
$593$	22.4208	0.920712	0.460356	+	0.887734i	$0.347722\pi$
$594$	0	0
$595$	14.8275	0.607868
$596$	0	0
$597$	−1.24373	−0.0509026
$598$	0	0
$599$	−5.58800	−0.228319	−0.114160	−	0.993462i	$-0.536418\pi$
$599$	−5.58800	−0.228319	−0.114160	+	0.993462i	$0.536418\pi$
$600$	0	0
$601$	−42.2270	−1.72247	−0.861237	−	0.508204i	$-0.830310\pi$
$601$	−42.2270	−1.72247	−0.861237	+	0.508204i	$0.830310\pi$
$602$	0	0
$603$	−21.6446	−0.881438
$604$	0	0
$605$	−4.16696	−0.169411
$606$	0	0
$607$	46.2906	1.87888	0.939438	−	0.342720i	$-0.111348\pi$
$607$	46.2906	1.87888	0.939438	+	0.342720i	$0.111348\pi$
$608$	0	0
$609$	0.676977	0.0274325
$610$	0	0
$611$	13.3368	0.539549
$612$	0	0
$613$	−21.4942	−0.868143	−0.434071	−	0.900878i	$-0.642923\pi$
$613$	−21.4942	−0.868143	−0.434071	+	0.900878i	$0.642923\pi$
$614$	0	0
$615$	−0.739206	−0.0298077
$616$	0	0
$617$	−36.2196	−1.45815	−0.729073	−	0.684436i	$-0.760048\pi$
$617$	−36.2196	−1.45815	−0.729073	+	0.684436i	$0.760048\pi$
$618$	0	0
$619$	4.84542	0.194754	0.0973770	−	0.995248i	$-0.468955\pi$
$619$	4.84542	0.194754	0.0973770	+	0.995248i	$0.468955\pi$
$620$	0	0
$621$	0.427558	0.0171573
$622$	0	0
$623$	−5.43343	−0.217686
$624$	0	0
$625$	66.0394	2.64158
$626$	0	0
$627$	−0.419064	−0.0167358
$628$	0	0
$629$	24.0115	0.957401
$630$	0	0
$631$	−15.7269	−0.626079	−0.313040	−	0.949740i	$-0.601347\pi$
$631$	−15.7269	−0.626079	−0.313040	+	0.949740i	$0.601347\pi$
$632$	0	0
$633$	−1.95422	−0.0776733
$634$	0	0
$635$	38.5676	1.53051
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−4.69802	−0.185851
$640$	0	0
$641$	30.5557	1.20688	0.603439	−	0.797409i	$-0.293797\pi$
$641$	30.5557	1.20688	0.603439	+	0.797409i	$0.293797\pi$
$642$	0	0
$643$	30.5681	1.20549	0.602744	−	0.797934i	$-0.294074\pi$
$643$	30.5681	1.20549	0.602744	+	0.797934i	$0.294074\pi$
$644$	0	0
$645$	−2.19890	−0.0865816
$646$	0	0
$647$	41.3237	1.62460	0.812302	−	0.583237i	$-0.198214\pi$
$647$	41.3237	1.62460	0.812302	+	0.583237i	$0.198214\pi$
$648$	0	0
$649$	−2.85455	−0.112051
$650$	0	0
$651$	−0.188007	−0.00736857
$652$	0	0
$653$	−32.2029	−1.26020	−0.630099	−	0.776515i	$-0.716986\pi$
$653$	−32.2029	−1.26020	−0.630099	+	0.776515i	$0.716986\pi$
$654$	0	0
$655$	19.6791	0.768926
$656$	0	0
$657$	−13.0033	−0.507307
$658$	0	0
$659$	14.6458	0.570519	0.285260	−	0.958450i	$-0.407920\pi$
$659$	14.6458	0.570519	0.285260	+	0.958450i	$0.407920\pi$
$660$	0	0
$661$	16.9684	0.659996	0.329998	−	0.943982i	$-0.392952\pi$
$661$	16.9684	0.659996	0.329998	+	0.943982i	$0.392952\pi$
$662$	0	0
$663$	−0.292662	−0.0113660
$664$	0	0
$665$	21.2316	0.823325
$666$	0	0
$667$	7.13959	0.276446
$668$	0	0
$669$	−0.430752	−0.0166538
$670$	0	0
$671$	−3.08242	−0.118995
$672$	0	0
$673$	15.1240	0.582988	0.291494	−	0.956573i	$-0.405848\pi$
$673$	15.1240	0.582988	0.291494	+	0.956573i	$0.405848\pi$
$674$	0	0
$675$	−6.09427	−0.234568
$676$	0	0
$677$	1.25538	0.0482482	0.0241241	−	0.999709i	$-0.492320\pi$
$677$	1.25538	0.0482482	0.0241241	+	0.999709i	$0.492320\pi$
$678$	0	0
$679$	11.8362	0.454233
$680$	0	0
$681$	0.0517557	0.00198328
$682$	0	0
$683$	12.4488	0.476338	0.238169	−	0.971224i	$-0.423453\pi$
$683$	12.4488	0.476338	0.238169	+	0.971224i	$0.423453\pi$
$684$	0	0
$685$	61.9295	2.36620
$686$	0	0
$687$	−1.24392	−0.0474584
$688$	0	0
$689$	2.36261	0.0900083
$690$	0	0
$691$	−44.2277	−1.68250	−0.841251	−	0.540645i	$-0.818180\pi$
$691$	−44.2277	−1.68250	−0.841251	+	0.540645i	$0.818180\pi$
$692$	0	0
$693$	−2.99324	−0.113704
$694$	0	0
$695$	−50.3229	−1.90886
$696$	0	0
$697$	7.67500	0.290711
$698$	0	0
$699$	−0.534591	−0.0202201
$700$	0	0
$701$	3.19359	0.120620	0.0603101	−	0.998180i	$-0.480791\pi$
$701$	3.19359	0.120620	0.0603101	+	0.998180i	$0.480791\pi$
$702$	0	0
$703$	34.3822	1.29675
$704$	0	0
$705$	4.57075	0.172145
$706$	0	0
$707$	−13.1574	−0.494835
$708$	0	0
$709$	3.46491	0.130128	0.0650638	−	0.997881i	$-0.479275\pi$
$709$	3.46491	0.130128	0.0650638	+	0.997881i	$0.479275\pi$
$710$	0	0
$711$	−0.246252	−0.00923518
$712$	0	0
$713$	−1.98277	−0.0742555
$714$	0	0
$715$	−4.16696	−0.155835
$716$	0	0
$717$	−2.02920	−0.0757818
$718$	0	0
$719$	−37.9362	−1.41478	−0.707390	−	0.706823i	$-0.750128\pi$
$719$	−37.9362	−1.41478	−0.707390	+	0.706823i	$0.750128\pi$
$720$	0	0
$721$	2.45911	0.0915820
$722$	0	0
$723$	−1.52470	−0.0567042
$724$	0	0
$725$	−101.765	−3.77947
$726$	0	0
$727$	−12.4357	−0.461216	−0.230608	−	0.973047i	$-0.574071\pi$
$727$	−12.4357	−0.461216	−0.230608	+	0.973047i	$0.574071\pi$
$728$	0	0
$729$	−26.6354	−0.986496
$730$	0	0
$731$	22.8306	0.844421
$732$	0	0
$733$	−19.0836	−0.704868	−0.352434	−	0.935837i	$-0.614646\pi$
$733$	−19.0836	−0.704868	−0.352434	+	0.935837i	$0.614646\pi$
$734$	0	0
$735$	−0.342718	−0.0126413
$736$	0	0
$737$	−7.23118	−0.266364
$738$	0	0
$739$	31.7132	1.16659	0.583295	−	0.812260i	$-0.301763\pi$
$739$	31.7132	1.16659	0.583295	+	0.812260i	$0.301763\pi$
$740$	0	0
$741$	−0.419064	−0.0153947
$742$	0	0
$743$	−19.9621	−0.732339	−0.366169	−	0.930548i	$-0.619331\pi$
$743$	−19.9621	−0.732339	−0.366169	+	0.930548i	$0.619331\pi$
$744$	0	0
$745$	66.8478	2.44911
$746$	0	0
$747$	16.7297	0.612108
$748$	0	0
$749$	−3.15058	−0.115120
$750$	0	0
$751$	47.7508	1.74245	0.871225	−	0.490884i	$-0.163326\pi$
$751$	47.7508	1.74245	0.871225	+	0.490884i	$0.163326\pi$
$752$	0	0
$753$	2.05974	0.0750610
$754$	0	0
$755$	9.86860	0.359155
$756$	0	0
$757$	−42.7254	−1.55288	−0.776441	−	0.630190i	$-0.782977\pi$
$757$	−42.7254	−1.55288	−0.776441	+	0.630190i	$0.782977\pi$
$758$	0	0
$759$	0.0713401	0.00258948
$760$	0	0
$761$	27.1507	0.984214	0.492107	−	0.870535i	$-0.336227\pi$
$761$	27.1507	0.984214	0.492107	+	0.870535i	$0.336227\pi$
$762$	0	0
$763$	−19.2113	−0.695496
$764$	0	0
$765$	44.3822	1.60464
$766$	0	0
$767$	−2.85455	−0.103072
$768$	0	0
$769$	−6.51555	−0.234957	−0.117478	−	0.993075i	$-0.537481\pi$
$769$	−6.51555	−0.234957	−0.117478	+	0.993075i	$0.537481\pi$
$770$	0	0
$771$	−0.153963	−0.00554483
$772$	0	0
$773$	15.6440	0.562675	0.281338	−	0.959609i	$-0.409222\pi$
$773$	15.6440	0.562675	0.281338	+	0.959609i	$0.409222\pi$
$774$	0	0
$775$	28.2618	1.01519
$776$	0	0
$777$	−0.554993	−0.0199103
$778$	0	0
$779$	10.9899	0.393753
$780$	0	0
$781$	−1.56955	−0.0561628
$782$	0	0
$783$	4.05728	0.144995
$784$	0	0
$785$	−3.83749	−0.136966
$786$	0	0
$787$	40.6488	1.44897	0.724486	−	0.689290i	$-0.242077\pi$
$787$	40.6488	1.44897	0.724486	+	0.689290i	$0.242077\pi$
$788$	0	0
$789$	−1.51625	−0.0539799
$790$	0	0
$791$	−9.78708	−0.347988
$792$	0	0
$793$	−3.08242	−0.109460
$794$	0	0
$795$	0.809708	0.0287174
$796$	0	0
$797$	4.44704	0.157522	0.0787611	−	0.996894i	$-0.474904\pi$
$797$	4.44704	0.157522	0.0787611	+	0.996894i	$0.474904\pi$
$798$	0	0
$799$	−47.4570	−1.67891
$800$	0	0
$801$	−16.2635	−0.574644
$802$	0	0
$803$	−4.34423	−0.153304
$804$	0	0
$805$	−3.61440	−0.127391
$806$	0	0
$807$	−0.792718	−0.0279050
$808$	0	0
$809$	−45.2670	−1.59150	−0.795752	−	0.605623i	$-0.792924\pi$
$809$	−45.2670	−1.59150	−0.795752	+	0.605623i	$0.792924\pi$
$810$	0	0
$811$	9.64362	0.338633	0.169317	−	0.985562i	$-0.445844\pi$
$811$	9.64362	0.338633	0.169317	+	0.985562i	$0.445844\pi$
$812$	0	0
$813$	−0.173677	−0.00609113
$814$	0	0
$815$	70.2791	2.46177
$816$	0	0
$817$	32.6913	1.14372
$818$	0	0
$819$	−2.99324	−0.104592
$820$	0	0
$821$	−2.70516	−0.0944107	−0.0472054	−	0.998885i	$-0.515032\pi$
$821$	−2.70516	−0.0944107	−0.0472054	+	0.998885i	$0.515032\pi$
$822$	0	0
$823$	10.5217	0.366765	0.183383	−	0.983042i	$-0.441295\pi$
$823$	10.5217	0.366765	0.183383	+	0.983042i	$0.441295\pi$
$824$	0	0
$825$	−1.01686	−0.0354024
$826$	0	0
$827$	−28.2336	−0.981780	−0.490890	−	0.871222i	$-0.663328\pi$
$827$	−28.2336	−0.981780	−0.490890	+	0.871222i	$0.663328\pi$
$828$	0	0
$829$	37.9146	1.31683	0.658415	−	0.752655i	$-0.271227\pi$
$829$	37.9146	1.31683	0.658415	+	0.752655i	$0.271227\pi$
$830$	0	0
$831$	−1.11154	−0.0385587
$832$	0	0
$833$	3.55835	0.123290
$834$	0	0
$835$	1.13618	0.0393191
$836$	0	0
$837$	−1.12677	−0.0389469
$838$	0	0
$839$	7.26159	0.250698	0.125349	−	0.992113i	$-0.459995\pi$
$839$	7.26159	0.250698	0.125349	+	0.992113i	$0.459995\pi$
$840$	0	0
$841$	38.7507	1.33623
$842$	0	0
$843$	−1.02051	−0.0351481
$844$	0	0
$845$	−4.16696	−0.143348
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	1.69029	0.0580105
$850$	0	0
$851$	−5.85312	−0.200642
$852$	0	0
$853$	−1.87833	−0.0643127	−0.0321563	−	0.999483i	$-0.510237\pi$
$853$	−1.87833	−0.0643127	−0.0321563	+	0.999483i	$0.510237\pi$
$854$	0	0
$855$	63.5511	2.17340
$856$	0	0
$857$	−46.7916	−1.59837	−0.799185	−	0.601085i	$-0.794735\pi$
$857$	−46.7916	−1.59837	−0.799185	+	0.601085i	$0.794735\pi$
$858$	0	0
$859$	37.8374	1.29099	0.645497	−	0.763762i	$-0.276650\pi$
$859$	37.8374	1.29099	0.645497	+	0.763762i	$0.276650\pi$
$860$	0	0
$861$	−0.177397	−0.00604568
$862$	0	0
$863$	−14.9426	−0.508651	−0.254325	−	0.967119i	$-0.581853\pi$
$863$	−14.9426	−0.508651	−0.254325	+	0.967119i	$0.581853\pi$
$864$	0	0
$865$	32.4753	1.10419
$866$	0	0
$867$	−0.356796	−0.0121174
$868$	0	0
$869$	−0.0822696	−0.00279080
$870$	0	0
$871$	−7.23118	−0.245019
$872$	0	0
$873$	35.4287	1.19908
$874$	0	0
$875$	30.6836	1.03729
$876$	0	0
$877$	28.2606	0.954292	0.477146	−	0.878824i	$-0.341671\pi$
$877$	28.2606	0.954292	0.477146	+	0.878824i	$0.341671\pi$
$878$	0	0
$879$	−0.801181	−0.0270232
$880$	0	0
$881$	33.7099	1.13572	0.567858	−	0.823126i	$-0.307772\pi$
$881$	33.7099	1.13572	0.567858	+	0.823126i	$0.307772\pi$
$882$	0	0
$883$	−24.1577	−0.812971	−0.406485	−	0.913657i	$-0.633246\pi$
$883$	−24.1577	−0.812971	−0.406485	+	0.913657i	$0.633246\pi$
$884$	0	0
$885$	−0.978304	−0.0328853
$886$	0	0
$887$	17.5637	0.589731	0.294865	−	0.955539i	$-0.404725\pi$
$887$	17.5637	0.589731	0.294865	+	0.955539i	$0.404725\pi$
$888$	0	0
$889$	9.25558	0.310422
$890$	0	0
$891$	−8.93917	−0.299473
$892$	0	0
$893$	−67.9539	−2.27399
$894$	0	0
$895$	8.46018	0.282793
$896$	0	0
$897$	0.0713401	0.00238198
$898$	0	0
$899$	−18.8154	−0.627529
$900$	0	0
$901$	−8.40700	−0.280078
$902$	0	0
$903$	−0.527699	−0.0175607
$904$	0	0
$905$	−15.0285	−0.499564
$906$	0	0
$907$	42.3794	1.40718	0.703592	−	0.710604i	$-0.251578\pi$
$907$	42.3794	1.40718	0.703592	+	0.710604i	$0.251578\pi$
$908$	0	0
$909$	−39.3832	−1.30626
$910$	0	0
$911$	−5.13819	−0.170236	−0.0851180	−	0.996371i	$-0.527127\pi$
$911$	−5.13819	−0.170236	−0.0851180	+	0.996371i	$0.527127\pi$
$912$	0	0
$913$	5.58917	0.184975
$914$	0	0
$915$	−1.05640	−0.0349235
$916$	0	0
$917$	4.72266	0.155956
$918$	0	0
$919$	15.2237	0.502182	0.251091	−	0.967963i	$-0.419211\pi$
$919$	15.2237	0.502182	0.251091	+	0.967963i	$0.419211\pi$
$920$	0	0
$921$	0.739319	0.0243614
$922$	0	0
$923$	−1.56955	−0.0516622
$924$	0	0
$925$	83.4283	2.74311
$926$	0	0
$927$	7.36069	0.241757
$928$	0	0
$929$	32.5560	1.06813	0.534064	−	0.845444i	$-0.320664\pi$
$929$	32.5560	1.06813	0.534064	+	0.845444i	$0.320664\pi$
$930$	0	0
$931$	5.09522	0.166989
$932$	0	0
$933$	−0.497294	−0.0162807
$934$	0	0
$935$	14.8275	0.484911
$936$	0	0
$937$	31.3946	1.02562	0.512808	−	0.858503i	$-0.328605\pi$
$937$	31.3946	1.02562	0.512808	+	0.858503i	$0.328605\pi$
$938$	0	0
$939$	0.317876	0.0103735
$940$	0	0
$941$	−16.7638	−0.546484	−0.273242	−	0.961945i	$-0.588096\pi$
$941$	−16.7638	−0.546484	−0.273242	+	0.961945i	$0.588096\pi$
$942$	0	0
$943$	−1.87088	−0.0609243
$944$	0	0
$945$	−2.05399	−0.0668162
$946$	0	0
$947$	−28.5661	−0.928274	−0.464137	−	0.885763i	$-0.653635\pi$
$947$	−28.5661	−0.928274	−0.464137	+	0.885763i	$0.653635\pi$
$948$	0	0
$949$	−4.34423	−0.141020
$950$	0	0
$951$	1.66816	0.0540938
$952$	0	0
$953$	−19.7225	−0.638875	−0.319437	−	0.947607i	$-0.603494\pi$
$953$	−19.7225	−0.638875	−0.319437	+	0.947607i	$0.603494\pi$
$954$	0	0
$955$	−31.2150	−1.01009
$956$	0	0
$957$	0.676977	0.0218836
$958$	0	0
$959$	14.8620	0.479920
$960$	0	0
$961$	−25.7747	−0.831441
$962$	0	0
$963$	−9.43042	−0.303891
$964$	0	0
$965$	34.5530	1.11230
$966$	0	0
$967$	−46.9802	−1.51078	−0.755391	−	0.655274i	$-0.772553\pi$
$967$	−46.9802	−1.51078	−0.755391	+	0.655274i	$0.772553\pi$
$968$	0	0
$969$	1.49118	0.0479035
$970$	0	0
$971$	2.03048	0.0651613	0.0325807	−	0.999469i	$-0.489627\pi$
$971$	2.03048	0.0651613	0.0325807	+	0.999469i	$0.489627\pi$
$972$	0	0
$973$	−12.0767	−0.387160
$974$	0	0
$975$	−1.01686	−0.0325655
$976$	0	0
$977$	−10.7899	−0.345199	−0.172599	−	0.984992i	$-0.555217\pi$
$977$	−10.7899	−0.345199	−0.172599	+	0.984992i	$0.555217\pi$
$978$	0	0
$979$	−5.43343	−0.173653
$980$	0	0
$981$	−57.5040	−1.83596
$982$	0	0
$983$	−38.1755	−1.21761	−0.608804	−	0.793321i	$-0.708350\pi$
$983$	−38.1755	−1.21761	−0.608804	+	0.793321i	$0.708350\pi$
$984$	0	0
$985$	−79.3461	−2.52818
$986$	0	0
$987$	1.09690	0.0349148
$988$	0	0
$989$	−5.56526	−0.176965
$990$	0	0
$991$	3.98288	0.126520	0.0632602	−	0.997997i	$-0.479850\pi$
$991$	3.98288	0.126520	0.0632602	+	0.997997i	$0.479850\pi$
$992$	0	0
$993$	−1.10030	−0.0349171
$994$	0	0
$995$	63.0129	1.99764
$996$	0	0
$997$	−48.5499	−1.53759	−0.768795	−	0.639496i	$-0.779143\pi$
$997$	−48.5499	−1.53759	−0.768795	+	0.639496i	$0.779143\pi$
$998$	0	0
$999$	−3.32620	−0.105236

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.u.1.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.u.1.5	✓	10	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.0822465 q^{3} -4.16696 q^{5} -1.00000 q^{7} -2.99324 q^{9} +O(q^{10})\)	\(q+0.0822465 q^{3} -4.16696 q^{5} -1.00000 q^{7} -2.99324 q^{9} -1.00000 q^{11} -1.00000 q^{13} -0.342718 q^{15} +3.55835 q^{17} +5.09522 q^{19} -0.0822465 q^{21} -0.867395 q^{23} +12.3635 q^{25} -0.492922 q^{27} -8.23108 q^{29} +2.28590 q^{31} -0.0822465 q^{33} +4.16696 q^{35} +6.74793 q^{37} -0.0822465 q^{39} +2.15690 q^{41} +6.41607 q^{43} +12.4727 q^{45} -13.3368 q^{47} +1.00000 q^{49} +0.292662 q^{51} -2.36261 q^{53} +4.16696 q^{55} +0.419064 q^{57} +2.85455 q^{59} +3.08242 q^{61} +2.99324 q^{63} +4.16696 q^{65} +7.23118 q^{67} -0.0713401 q^{69} +1.56955 q^{71} +4.34423 q^{73} +1.01686 q^{75} +1.00000 q^{77} +0.0822696 q^{79} +8.93917 q^{81} -5.58917 q^{83} -14.8275 q^{85} -0.676977 q^{87} +5.43343 q^{89} +1.00000 q^{91} +0.188007 q^{93} -21.2316 q^{95} -11.8362 q^{97} +2.99324 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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