LMFDB - Embedded newform 8008.2.a.u.1.8

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			1.8
Root			$1.76763$ 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)

$f(q)$	$=$	$q+1.76763 q^{3} +0.187208 q^{5} -1.00000 q^{7} +0.124526 q^{9} +O(q^{10})$	$q+1.76763 q^{3} +0.187208 q^{5} -1.00000 q^{7} +0.124526 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.330915 q^{15} +4.32648 q^{17} -5.79897 q^{19} -1.76763 q^{21} +5.19778 q^{23} -4.96495 q^{25} -5.08278 q^{27} +1.25596 q^{29} +6.52310 q^{31} -1.76763 q^{33} -0.187208 q^{35} -5.96866 q^{37} -1.76763 q^{39} +5.23412 q^{41} -3.99762 q^{43} +0.0233124 q^{45} -5.21957 q^{47} +1.00000 q^{49} +7.64763 q^{51} -1.33361 q^{53} -0.187208 q^{55} -10.2504 q^{57} +7.28024 q^{59} -3.40345 q^{61} -0.124526 q^{63} -0.187208 q^{65} -2.43127 q^{67} +9.18776 q^{69} -2.65660 q^{71} -12.4341 q^{73} -8.77621 q^{75} +1.00000 q^{77} +3.58710 q^{79} -9.35807 q^{81} -9.04444 q^{83} +0.809952 q^{85} +2.22008 q^{87} -12.7021 q^{89} +1.00000 q^{91} +11.5304 q^{93} -1.08561 q^{95} -4.31505 q^{97} -0.124526 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.76763	1.02054	0.510272	−	0.860013i	$-0.329545\pi$
$3$	1.76763	1.02054	0.510272	+	0.860013i	$0.329545\pi$
$4$	0	0
$5$	0.187208	0.0837220	0.0418610	−	0.999123i	$-0.486671\pi$
$5$	0.187208	0.0837220	0.0418610	+	0.999123i	$0.486671\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.124526	0.0415088
$10$	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
$15$	0.330915	0.0854420
$16$	0	0
$17$	4.32648	1.04933	0.524663	−	0.851310i	$-0.324191\pi$
$17$	4.32648	1.04933	0.524663	+	0.851310i	$0.324191\pi$
$18$	0	0
$19$	−5.79897	−1.33037	−0.665187	−	0.746677i	$-0.731648\pi$
$19$	−5.79897	−1.33037	−0.665187	+	0.746677i	$0.731648\pi$
$20$	0	0
$21$	−1.76763	−0.385729
$22$	0	0
$23$	5.19778	1.08381	0.541906	−	0.840439i	$-0.317703\pi$
$23$	5.19778	1.08381	0.541906	+	0.840439i	$0.317703\pi$
$24$	0	0
$25$	−4.96495	−0.992991
$26$	0	0
$27$	−5.08278	−0.978182
$28$	0	0
$29$	1.25596	0.233226	0.116613	−	0.993177i	$-0.462796\pi$
$29$	1.25596	0.233226	0.116613	+	0.993177i	$0.462796\pi$
$30$	0	0
$31$	6.52310	1.17158	0.585792	−	0.810462i	$-0.300784\pi$
$31$	6.52310	1.17158	0.585792	+	0.810462i	$0.300784\pi$
$32$	0	0
$33$	−1.76763	−0.307705
$34$	0	0
$35$	−0.187208	−0.0316440
$36$	0	0
$37$	−5.96866	−0.981241	−0.490621	−	0.871373i	$-0.663230\pi$
$37$	−5.96866	−0.981241	−0.490621	+	0.871373i	$0.663230\pi$
$38$	0	0
$39$	−1.76763	−0.283048
$40$	0	0
$41$	5.23412	0.817432	0.408716	−	0.912662i	$-0.365977\pi$
$41$	5.23412	0.817432	0.408716	+	0.912662i	$0.365977\pi$
$42$	0	0
$43$	−3.99762	−0.609631	−0.304815	−	0.952411i	$-0.598595\pi$
$43$	−3.99762	−0.609631	−0.304815	+	0.952411i	$0.598595\pi$
$44$	0	0
$45$	0.0233124	0.00347520
$46$	0	0
$47$	−5.21957	−0.761352	−0.380676	−	0.924708i	$-0.624309\pi$
$47$	−5.21957	−0.761352	−0.380676	+	0.924708i	$0.624309\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.64763	1.07088
$52$	0	0
$53$	−1.33361	−0.183186	−0.0915930	−	0.995797i	$-0.529196\pi$
$53$	−1.33361	−0.183186	−0.0915930	+	0.995797i	$0.529196\pi$
$54$	0	0
$55$	−0.187208	−0.0252431
$56$	0	0
$57$	−10.2504	−1.35770
$58$	0	0
$59$	7.28024	0.947806	0.473903	−	0.880577i	$-0.342845\pi$
$59$	7.28024	0.947806	0.473903	+	0.880577i	$0.342845\pi$
$60$	0	0
$61$	−3.40345	−0.435767	−0.217884	−	0.975975i	$-0.569915\pi$
$61$	−3.40345	−0.435767	−0.217884	+	0.975975i	$0.569915\pi$
$62$	0	0
$63$	−0.124526	−0.0156889
$64$	0	0
$65$	−0.187208	−0.0232203
$66$	0	0
$67$	−2.43127	−0.297027	−0.148514	−	0.988910i	$-0.547449\pi$
$67$	−2.43127	−0.297027	−0.148514	+	0.988910i	$0.547449\pi$
$68$	0	0
$69$	9.18776	1.10608
$70$	0	0
$71$	−2.65660	−0.315281	−0.157640	−	0.987497i	$-0.550389\pi$
$71$	−2.65660	−0.315281	−0.157640	+	0.987497i	$0.550389\pi$
$72$	0	0
$73$	−12.4341	−1.45530	−0.727649	−	0.685950i	$-0.759387\pi$
$73$	−12.4341	−1.45530	−0.727649	+	0.685950i	$0.759387\pi$
$74$	0	0
$75$	−8.77621	−1.01339
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	3.58710	0.403580	0.201790	−	0.979429i	$-0.435324\pi$
$79$	3.58710	0.403580	0.201790	+	0.979429i	$0.435324\pi$
$80$	0	0
$81$	−9.35807	−1.03979
$82$	0	0
$83$	−9.04444	−0.992756	−0.496378	−	0.868107i	$-0.665337\pi$
$83$	−9.04444	−0.992756	−0.496378	+	0.868107i	$0.665337\pi$
$84$	0	0
$85$	0.809952	0.0878516
$86$	0	0
$87$	2.22008	0.238018
$88$	0	0
$89$	−12.7021	−1.34642	−0.673210	−	0.739451i	$-0.735085\pi$
$89$	−12.7021	−1.34642	−0.673210	+	0.739451i	$0.735085\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	11.5304	1.19565
$94$	0	0
$95$	−1.08561	−0.111382
$96$	0	0
$97$	−4.31505	−0.438127	−0.219064	−	0.975711i	$-0.570300\pi$
$97$	−4.31505	−0.438127	−0.219064	+	0.975711i	$0.570300\pi$
$98$	0	0
$99$	−0.124526	−0.0125154
$100$	0	0
$101$	2.48362	0.247129	0.123564	−	0.992337i	$-0.460567\pi$
$101$	2.48362	0.247129	0.123564	+	0.992337i	$0.460567\pi$
$102$	0	0
$103$	−14.0267	−1.38209	−0.691047	−	0.722810i	$-0.742850\pi$
$103$	−14.0267	−1.38209	−0.691047	+	0.722810i	$0.742850\pi$
$104$	0	0
$105$	−0.330915	−0.0322940
$106$	0	0
$107$	12.9847	1.25528	0.627638	−	0.778505i	$-0.284022\pi$
$107$	12.9847	1.25528	0.627638	+	0.778505i	$0.284022\pi$
$108$	0	0
$109$	3.60242	0.345049	0.172524	−	0.985005i	$-0.444808\pi$
$109$	3.60242	0.345049	0.172524	+	0.985005i	$0.444808\pi$
$110$	0	0
$111$	−10.5504	−1.00140
$112$	0	0
$113$	−16.7312	−1.57394	−0.786970	−	0.616991i	$-0.788351\pi$
$113$	−16.7312	−1.57394	−0.786970	+	0.616991i	$0.788351\pi$
$114$	0	0
$115$	0.973066	0.0907389
$116$	0	0
$117$	−0.124526	−0.0115125
$118$	0	0
$119$	−4.32648	−0.396608
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	9.25200	0.834224
$124$	0	0
$125$	−1.86552	−0.166857
$126$	0	0
$127$	−13.1367	−1.16570	−0.582848	−	0.812582i	$-0.698062\pi$
$127$	−13.1367	−1.16570	−0.582848	+	0.812582i	$0.698062\pi$
$128$	0	0
$129$	−7.06632	−0.622155
$130$	0	0
$131$	6.10959	0.533798	0.266899	−	0.963725i	$-0.414001\pi$
$131$	6.10959	0.533798	0.266899	+	0.963725i	$0.414001\pi$
$132$	0	0
$133$	5.79897	0.502834
$134$	0	0
$135$	−0.951538	−0.0818954
$136$	0	0
$137$	13.6249	1.16405	0.582025	−	0.813171i	$-0.302260\pi$
$137$	13.6249	1.16405	0.582025	+	0.813171i	$0.302260\pi$
$138$	0	0
$139$	1.19652	0.101488	0.0507439	−	0.998712i	$-0.483841\pi$
$139$	1.19652	0.101488	0.0507439	+	0.998712i	$0.483841\pi$
$140$	0	0
$141$	−9.22628	−0.776993
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0.235126	0.0195262
$146$	0	0
$147$	1.76763	0.145792
$148$	0	0
$149$	−8.15546	−0.668121	−0.334061	−	0.942552i	$-0.608419\pi$
$149$	−8.15546	−0.668121	−0.334061	+	0.942552i	$0.608419\pi$
$150$	0	0
$151$	−12.2259	−0.994934	−0.497467	−	0.867483i	$-0.665736\pi$
$151$	−12.2259	−0.994934	−0.497467	+	0.867483i	$0.665736\pi$
$152$	0	0
$153$	0.538761	0.0435562
$154$	0	0
$155$	1.22118	0.0980873
$156$	0	0
$157$	−20.2040	−1.61246	−0.806228	−	0.591605i	$-0.798494\pi$
$157$	−20.2040	−1.61246	−0.806228	+	0.591605i	$0.798494\pi$
$158$	0	0
$159$	−2.35734	−0.186949
$160$	0	0
$161$	−5.19778	−0.409642
$162$	0	0
$163$	−16.7786	−1.31420	−0.657099	−	0.753804i	$-0.728217\pi$
$163$	−16.7786	−1.31420	−0.657099	+	0.753804i	$0.728217\pi$
$164$	0	0
$165$	−0.330915	−0.0257617
$166$	0	0
$167$	1.21006	0.0936376	0.0468188	−	0.998903i	$-0.485092\pi$
$167$	1.21006	0.0936376	0.0468188	+	0.998903i	$0.485092\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.722124	−0.0552222
$172$	0	0
$173$	13.2712	1.00899	0.504494	−	0.863415i	$-0.331679\pi$
$173$	13.2712	1.00899	0.504494	+	0.863415i	$0.331679\pi$
$174$	0	0
$175$	4.96495	0.375315
$176$	0	0
$177$	12.8688	0.967277
$178$	0	0
$179$	18.5205	1.38429	0.692144	−	0.721759i	$-0.256666\pi$
$179$	18.5205	1.38429	0.692144	+	0.721759i	$0.256666\pi$
$180$	0	0
$181$	14.9939	1.11449	0.557244	−	0.830348i	$-0.311859\pi$
$181$	14.9939	1.11449	0.557244	+	0.830348i	$0.311859\pi$
$182$	0	0
$183$	−6.01605	−0.444719
$184$	0	0
$185$	−1.11738	−0.0821515
$186$	0	0
$187$	−4.32648	−0.316383
$188$	0	0
$189$	5.08278	0.369718
$190$	0	0
$191$	−8.49034	−0.614339	−0.307170	−	0.951655i	$-0.599382\pi$
$191$	−8.49034	−0.614339	−0.307170	+	0.951655i	$0.599382\pi$
$192$	0	0
$193$	−10.9141	−0.785611	−0.392805	−	0.919622i	$-0.628495\pi$
$193$	−10.9141	−0.785611	−0.392805	+	0.919622i	$0.628495\pi$
$194$	0	0
$195$	−0.330915	−0.0236973
$196$	0	0
$197$	−19.8771	−1.41619	−0.708094	−	0.706119i	$-0.750445\pi$
$197$	−19.8771	−1.41619	−0.708094	+	0.706119i	$0.750445\pi$
$198$	0	0
$199$	−26.7949	−1.89944	−0.949720	−	0.313100i	$-0.898633\pi$
$199$	−26.7949	−1.89944	−0.949720	+	0.313100i	$0.898633\pi$
$200$	0	0
$201$	−4.29760	−0.303129
$202$	0	0
$203$	−1.25596	−0.0881513
$204$	0	0
$205$	0.979869	0.0684370
$206$	0	0
$207$	0.647260	0.0449877
$208$	0	0
$209$	5.79897	0.401123
$210$	0	0
$211$	11.3739	0.783015	0.391507	−	0.920175i	$-0.371954\pi$
$211$	11.3739	0.783015	0.391507	+	0.920175i	$0.371954\pi$
$212$	0	0
$213$	−4.69590	−0.321758
$214$	0	0
$215$	−0.748387	−0.0510395
$216$	0	0
$217$	−6.52310	−0.442817
$218$	0	0
$219$	−21.9789	−1.48519
$220$	0	0
$221$	−4.32648	−0.291030
$222$	0	0
$223$	−23.3245	−1.56193	−0.780963	−	0.624578i	$-0.785271\pi$
$223$	−23.3245	−1.56193	−0.780963	+	0.624578i	$0.785271\pi$
$224$	0	0
$225$	−0.618268	−0.0412178
$226$	0	0
$227$	8.19009	0.543596	0.271798	−	0.962354i	$-0.412382\pi$
$227$	8.19009	0.543596	0.271798	+	0.962354i	$0.412382\pi$
$228$	0	0
$229$	−16.2055	−1.07089	−0.535446	−	0.844569i	$-0.679857\pi$
$229$	−16.2055	−1.07089	−0.535446	+	0.844569i	$0.679857\pi$
$230$	0	0
$231$	1.76763	0.116302
$232$	0	0
$233$	−18.0871	−1.18492	−0.592462	−	0.805598i	$-0.701844\pi$
$233$	−18.0871	−1.18492	−0.592462	+	0.805598i	$0.701844\pi$
$234$	0	0
$235$	−0.977146	−0.0637419
$236$	0	0
$237$	6.34068	0.411871
$238$	0	0
$239$	11.8444	0.766152	0.383076	−	0.923717i	$-0.374865\pi$
$239$	11.8444	0.766152	0.383076	+	0.923717i	$0.374865\pi$
$240$	0	0
$241$	20.7427	1.33615	0.668077	−	0.744092i	$-0.267118\pi$
$241$	20.7427	1.33615	0.668077	+	0.744092i	$0.267118\pi$
$242$	0	0
$243$	−1.29329	−0.0829647
$244$	0	0
$245$	0.187208	0.0119603
$246$	0	0
$247$	5.79897	0.368979
$248$	0	0
$249$	−15.9872	−1.01315
$250$	0	0
$251$	9.39562	0.593046	0.296523	−	0.955026i	$-0.404173\pi$
$251$	9.39562	0.593046	0.296523	+	0.955026i	$0.404173\pi$
$252$	0	0
$253$	−5.19778	−0.326781
$254$	0	0
$255$	1.43170	0.0896564
$256$	0	0
$257$	−7.86272	−0.490463	−0.245232	−	0.969465i	$-0.578864\pi$
$257$	−7.86272	−0.490463	−0.245232	+	0.969465i	$0.578864\pi$
$258$	0	0
$259$	5.96866	0.370874
$260$	0	0
$261$	0.156400	0.00968094
$262$	0	0
$263$	−0.230621	−0.0142207	−0.00711036	−	0.999975i	$-0.502263\pi$
$263$	−0.230621	−0.0142207	−0.00711036	+	0.999975i	$0.502263\pi$
$264$	0	0
$265$	−0.249663	−0.0153367
$266$	0	0
$267$	−22.4526	−1.37408
$268$	0	0
$269$	17.4203	1.06214	0.531069	−	0.847329i	$-0.321791\pi$
$269$	17.4203	1.06214	0.531069	+	0.847329i	$0.321791\pi$
$270$	0	0
$271$	−16.7770	−1.01913	−0.509564	−	0.860433i	$-0.670193\pi$
$271$	−16.7770	−1.01913	−0.509564	+	0.860433i	$0.670193\pi$
$272$	0	0
$273$	1.76763	0.106982
$274$	0	0
$275$	4.96495	0.299398
$276$	0	0
$277$	3.16058	0.189901	0.0949504	−	0.995482i	$-0.469731\pi$
$277$	3.16058	0.189901	0.0949504	+	0.995482i	$0.469731\pi$
$278$	0	0
$279$	0.812298	0.0486310
$280$	0	0
$281$	11.2627	0.671877	0.335938	−	0.941884i	$-0.390947\pi$
$281$	11.2627	0.671877	0.335938	+	0.941884i	$0.390947\pi$
$282$	0	0
$283$	16.4640	0.978682	0.489341	−	0.872092i	$-0.337237\pi$
$283$	16.4640	0.978682	0.489341	+	0.872092i	$0.337237\pi$
$284$	0	0
$285$	−1.91897	−0.113670
$286$	0	0
$287$	−5.23412	−0.308960
$288$	0	0
$289$	1.71842	0.101083
$290$	0	0
$291$	−7.62743	−0.447128
$292$	0	0
$293$	−10.3742	−0.606068	−0.303034	−	0.952980i	$-0.598000\pi$
$293$	−10.3742	−0.606068	−0.303034	+	0.952980i	$0.598000\pi$
$294$	0	0
$295$	1.36292	0.0793523
$296$	0	0
$297$	5.08278	0.294933
$298$	0	0
$299$	−5.19778	−0.300595
$300$	0	0
$301$	3.99762	0.230419
$302$	0	0
$303$	4.39012	0.252206
$304$	0	0
$305$	−0.637154	−0.0364833
$306$	0	0
$307$	8.28375	0.472778	0.236389	−	0.971658i	$-0.424036\pi$
$307$	8.28375	0.472778	0.236389	+	0.971658i	$0.424036\pi$
$308$	0	0
$309$	−24.7941	−1.41049
$310$	0	0
$311$	−5.90180	−0.334660	−0.167330	−	0.985901i	$-0.553515\pi$
$311$	−5.90180	−0.334660	−0.167330	+	0.985901i	$0.553515\pi$
$312$	0	0
$313$	−9.92094	−0.560765	−0.280382	−	0.959888i	$-0.590461\pi$
$313$	−9.92094	−0.560765	−0.280382	+	0.959888i	$0.590461\pi$
$314$	0	0
$315$	−0.0233124	−0.00131350
$316$	0	0
$317$	−6.69036	−0.375768	−0.187884	−	0.982191i	$-0.560163\pi$
$317$	−6.69036	−0.375768	−0.187884	+	0.982191i	$0.560163\pi$
$318$	0	0
$319$	−1.25596	−0.0703204
$320$	0	0
$321$	22.9521	1.28106
$322$	0	0
$323$	−25.0891	−1.39599
$324$	0	0
$325$	4.96495	0.275406
$326$	0	0
$327$	6.36775	0.352137
$328$	0	0
$329$	5.21957	0.287764
$330$	0	0
$331$	18.8852	1.03803	0.519013	−	0.854766i	$-0.326299\pi$
$331$	18.8852	1.03803	0.519013	+	0.854766i	$0.326299\pi$
$332$	0	0
$333$	−0.743256	−0.0407302
$334$	0	0
$335$	−0.455154	−0.0248677
$336$	0	0
$337$	−11.3932	−0.620627	−0.310313	−	0.950634i	$-0.600434\pi$
$337$	−11.3932	−0.620627	−0.310313	+	0.950634i	$0.600434\pi$
$338$	0	0
$339$	−29.5746	−1.60627
$340$	0	0
$341$	−6.52310	−0.353246
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.72002	0.0926030
$346$	0	0
$347$	−15.6924	−0.842411	−0.421205	−	0.906965i	$-0.638393\pi$
$347$	−15.6924	−0.842411	−0.421205	+	0.906965i	$0.638393\pi$
$348$	0	0
$349$	−14.1132	−0.755465	−0.377732	−	0.925915i	$-0.623296\pi$
$349$	−14.1132	−0.755465	−0.377732	+	0.925915i	$0.623296\pi$
$350$	0	0
$351$	5.08278	0.271299
$352$	0	0
$353$	18.8716	1.00443	0.502217	−	0.864741i	$-0.332518\pi$
$353$	18.8716	1.00443	0.502217	+	0.864741i	$0.332518\pi$
$354$	0	0
$355$	−0.497338	−0.0263959
$356$	0	0
$357$	−7.64763	−0.404755
$358$	0	0
$359$	4.09500	0.216126	0.108063	−	0.994144i	$-0.465535\pi$
$359$	4.09500	0.216126	0.108063	+	0.994144i	$0.465535\pi$
$360$	0	0
$361$	14.6280	0.769895
$362$	0	0
$363$	1.76763	0.0927767
$364$	0	0
$365$	−2.32776	−0.121840
$366$	0	0
$367$	10.8091	0.564230	0.282115	−	0.959381i	$-0.408964\pi$
$367$	10.8091	0.564230	0.282115	+	0.959381i	$0.408964\pi$
$368$	0	0
$369$	0.651786	0.0339306
$370$	0	0
$371$	1.33361	0.0692378
$372$	0	0
$373$	20.5928	1.06625	0.533127	−	0.846035i	$-0.321017\pi$
$373$	20.5928	1.06625	0.533127	+	0.846035i	$0.321017\pi$
$374$	0	0
$375$	−3.29756	−0.170285
$376$	0	0
$377$	−1.25596	−0.0646853
$378$	0	0
$379$	−31.8324	−1.63512	−0.817560	−	0.575843i	$-0.804674\pi$
$379$	−31.8324	−1.63512	−0.817560	+	0.575843i	$0.804674\pi$
$380$	0	0
$381$	−23.2209	−1.18964
$382$	0	0
$383$	1.47019	0.0751232	0.0375616	−	0.999294i	$-0.488041\pi$
$383$	1.47019	0.0751232	0.0375616	+	0.999294i	$0.488041\pi$
$384$	0	0
$385$	0.187208	0.00954101
$386$	0	0
$387$	−0.497809	−0.0253050
$388$	0	0
$389$	−1.97538	−0.100156	−0.0500780	−	0.998745i	$-0.515947\pi$
$389$	−1.97538	−0.100156	−0.0500780	+	0.998745i	$0.515947\pi$
$390$	0	0
$391$	22.4881	1.13727
$392$	0	0
$393$	10.7995	0.544764
$394$	0	0
$395$	0.671535	0.0337886
$396$	0	0
$397$	27.8123	1.39586	0.697929	−	0.716166i	$-0.254105\pi$
$397$	27.8123	1.39586	0.697929	+	0.716166i	$0.254105\pi$
$398$	0	0
$399$	10.2504	0.513164
$400$	0	0
$401$	7.71223	0.385130	0.192565	−	0.981284i	$-0.438319\pi$
$401$	7.71223	0.385130	0.192565	+	0.981284i	$0.438319\pi$
$402$	0	0
$403$	−6.52310	−0.324939
$404$	0	0
$405$	−1.75191	−0.0870530
$406$	0	0
$407$	5.96866	0.295855
$408$	0	0
$409$	−30.7736	−1.52166	−0.760828	−	0.648953i	$-0.775207\pi$
$409$	−30.7736	−1.52166	−0.760828	+	0.648953i	$0.775207\pi$
$410$	0	0
$411$	24.0838	1.18796
$412$	0	0
$413$	−7.28024	−0.358237
$414$	0	0
$415$	−1.69319	−0.0831155
$416$	0	0
$417$	2.11501	0.103573
$418$	0	0
$419$	−22.0196	−1.07573	−0.537865	−	0.843031i	$-0.680769\pi$
$419$	−22.0196	−1.07573	−0.537865	+	0.843031i	$0.680769\pi$
$420$	0	0
$421$	−8.39136	−0.408970	−0.204485	−	0.978870i	$-0.565552\pi$
$421$	−8.39136	−0.408970	−0.204485	+	0.978870i	$0.565552\pi$
$422$	0	0
$423$	−0.649974	−0.0316028
$424$	0	0
$425$	−21.4808	−1.04197
$426$	0	0
$427$	3.40345	0.164704
$428$	0	0
$429$	1.76763	0.0853421
$430$	0	0
$431$	2.60175	0.125322	0.0626610	−	0.998035i	$-0.480041\pi$
$431$	2.60175	0.125322	0.0626610	+	0.998035i	$0.480041\pi$
$432$	0	0
$433$	6.06670	0.291547	0.145773	−	0.989318i	$-0.453433\pi$
$433$	6.06670	0.291547	0.145773	+	0.989318i	$0.453433\pi$
$434$	0	0
$435$	0.415617	0.0199273
$436$	0	0
$437$	−30.1417	−1.44187
$438$	0	0
$439$	31.5583	1.50619	0.753097	−	0.657910i	$-0.228559\pi$
$439$	31.5583	1.50619	0.753097	+	0.657910i	$0.228559\pi$
$440$	0	0
$441$	0.124526	0.00592983
$442$	0	0
$443$	−7.90800	−0.375720	−0.187860	−	0.982196i	$-0.560155\pi$
$443$	−7.90800	−0.375720	−0.187860	+	0.982196i	$0.560155\pi$
$444$	0	0
$445$	−2.37794	−0.112725
$446$	0	0
$447$	−14.4159	−0.681847
$448$	0	0
$449$	35.2243	1.66234	0.831170	−	0.556019i	$-0.187672\pi$
$449$	35.2243	1.66234	0.831170	+	0.556019i	$0.187672\pi$
$450$	0	0
$451$	−5.23412	−0.246465
$452$	0	0
$453$	−21.6110	−1.01537
$454$	0	0
$455$	0.187208	0.00877645
$456$	0	0
$457$	−15.2350	−0.712662	−0.356331	−	0.934360i	$-0.615972\pi$
$457$	−15.2350	−0.712662	−0.356331	+	0.934360i	$0.615972\pi$
$458$	0	0
$459$	−21.9905	−1.02643
$460$	0	0
$461$	−10.0347	−0.467362	−0.233681	−	0.972313i	$-0.575077\pi$
$461$	−10.0347	−0.467362	−0.233681	+	0.972313i	$0.575077\pi$
$462$	0	0
$463$	12.1534	0.564819	0.282409	−	0.959294i	$-0.408866\pi$
$463$	12.1534	0.564819	0.282409	+	0.959294i	$0.408866\pi$
$464$	0	0
$465$	2.15859	0.100102
$466$	0	0
$467$	14.4572	0.669001	0.334501	−	0.942396i	$-0.391432\pi$
$467$	14.4572	0.669001	0.334501	+	0.942396i	$0.391432\pi$
$468$	0	0
$469$	2.43127	0.112266
$470$	0	0
$471$	−35.7133	−1.64558
$472$	0	0
$473$	3.99762	0.183811
$474$	0	0
$475$	28.7916	1.32105
$476$	0	0
$477$	−0.166070	−0.00760383
$478$	0	0
$479$	11.5569	0.528049	0.264024	−	0.964516i	$-0.414950\pi$
$479$	11.5569	0.528049	0.264024	+	0.964516i	$0.414950\pi$
$480$	0	0
$481$	5.96866	0.272147
$482$	0	0
$483$	−9.18776	−0.418058
$484$	0	0
$485$	−0.807813	−0.0366809
$486$	0	0
$487$	15.0710	0.682932	0.341466	−	0.939894i	$-0.389077\pi$
$487$	15.0710	0.682932	0.341466	+	0.939894i	$0.389077\pi$
$488$	0	0
$489$	−29.6583	−1.34120
$490$	0	0
$491$	42.9728	1.93933	0.969667	−	0.244430i	$-0.0786009\pi$
$491$	42.9728	1.93933	0.969667	+	0.244430i	$0.0786009\pi$
$492$	0	0
$493$	5.43389	0.244730
$494$	0	0
$495$	−0.0233124	−0.00104781
$496$	0	0
$497$	2.65660	0.119165
$498$	0	0
$499$	5.91705	0.264884	0.132442	−	0.991191i	$-0.457718\pi$
$499$	5.91705	0.264884	0.132442	+	0.991191i	$0.457718\pi$
$500$	0	0
$501$	2.13895	0.0955612
$502$	0	0
$503$	−43.6034	−1.94418	−0.972090	−	0.234607i	$-0.924620\pi$
$503$	−43.6034	−1.94418	−0.972090	+	0.234607i	$0.924620\pi$
$504$	0	0
$505$	0.464953	0.0206901
$506$	0	0
$507$	1.76763	0.0785033
$508$	0	0
$509$	−12.5945	−0.558242	−0.279121	−	0.960256i	$-0.590043\pi$
$509$	−12.5945	−0.558242	−0.279121	+	0.960256i	$0.590043\pi$
$510$	0	0
$511$	12.4341	0.550051
$512$	0	0
$513$	29.4749	1.30135
$514$	0	0
$515$	−2.62592	−0.115712
$516$	0	0
$517$	5.21957	0.229556
$518$	0	0
$519$	23.4585	1.02972
$520$	0	0
$521$	4.89362	0.214393	0.107197	−	0.994238i	$-0.465813\pi$
$521$	4.89362	0.214393	0.107197	+	0.994238i	$0.465813\pi$
$522$	0	0
$523$	10.0064	0.437551	0.218775	−	0.975775i	$-0.429794\pi$
$523$	10.0064	0.437551	0.218775	+	0.975775i	$0.429794\pi$
$524$	0	0
$525$	8.77621	0.383025
$526$	0	0
$527$	28.2221	1.22937
$528$	0	0
$529$	4.01688	0.174647
$530$	0	0
$531$	0.906582	0.0393423
$532$	0	0
$533$	−5.23412	−0.226715
$534$	0	0
$535$	2.43084	0.105094
$536$	0	0
$537$	32.7375	1.41273
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−5.34680	−0.229877	−0.114938	−	0.993373i	$-0.536667\pi$
$541$	−5.34680	−0.229877	−0.114938	+	0.993373i	$0.536667\pi$
$542$	0	0
$543$	26.5037	1.13738
$544$	0	0
$545$	0.674402	0.0288882
$546$	0	0
$547$	30.1478	1.28903	0.644513	−	0.764593i	$-0.277060\pi$
$547$	30.1478	1.28903	0.644513	+	0.764593i	$0.277060\pi$
$548$	0	0
$549$	−0.423819	−0.0180882
$550$	0	0
$551$	−7.28328	−0.310278
$552$	0	0
$553$	−3.58710	−0.152539
$554$	0	0
$555$	−1.97512	−0.0838392
$556$	0	0
$557$	11.8352	0.501474	0.250737	−	0.968055i	$-0.419327\pi$
$557$	11.8352	0.501474	0.250737	+	0.968055i	$0.419327\pi$
$558$	0	0
$559$	3.99762	0.169081
$560$	0	0
$561$	−7.64763	−0.322883
$562$	0	0
$563$	−6.17425	−0.260214	−0.130107	−	0.991500i	$-0.541532\pi$
$563$	−6.17425	−0.260214	−0.130107	+	0.991500i	$0.541532\pi$
$564$	0	0
$565$	−3.13222	−0.131773
$566$	0	0
$567$	9.35807	0.393002
$568$	0	0
$569$	17.9554	0.752730	0.376365	−	0.926472i	$-0.377174\pi$
$569$	17.9554	0.752730	0.376365	+	0.926472i	$0.377174\pi$
$570$	0	0
$571$	25.5500	1.06923	0.534617	−	0.845094i	$-0.320456\pi$
$571$	25.5500	1.06923	0.534617	+	0.845094i	$0.320456\pi$
$572$	0	0
$573$	−15.0078	−0.626960
$574$	0	0
$575$	−25.8067	−1.07621
$576$	0	0
$577$	−36.6591	−1.52614	−0.763069	−	0.646317i	$-0.776308\pi$
$577$	−36.6591	−1.52614	−0.763069	+	0.646317i	$0.776308\pi$
$578$	0	0
$579$	−19.2920	−0.801750
$580$	0	0
$581$	9.04444	0.375226
$582$	0	0
$583$	1.33361	0.0552326
$584$	0	0
$585$	−0.0233124	−0.000963847 0
$586$	0	0
$587$	42.7022	1.76251	0.881254	−	0.472643i	$-0.156700\pi$
$587$	42.7022	1.76251	0.881254	+	0.472643i	$0.156700\pi$
$588$	0	0
$589$	−37.8272	−1.55864
$590$	0	0
$591$	−35.1355	−1.44528
$592$	0	0
$593$	−4.02047	−0.165101	−0.0825504	−	0.996587i	$-0.526307\pi$
$593$	−4.02047	−0.165101	−0.0825504	+	0.996587i	$0.526307\pi$
$594$	0	0
$595$	−0.809952	−0.0332048
$596$	0	0
$597$	−47.3636	−1.93846
$598$	0	0
$599$	−8.16880	−0.333768	−0.166884	−	0.985977i	$-0.553371\pi$
$599$	−8.16880	−0.333768	−0.166884	+	0.985977i	$0.553371\pi$
$600$	0	0
$601$	−13.7887	−0.562452	−0.281226	−	0.959642i	$-0.590741\pi$
$601$	−13.7887	−0.562452	−0.281226	+	0.959642i	$0.590741\pi$
$602$	0	0
$603$	−0.302757	−0.0123292
$604$	0	0
$605$	0.187208	0.00761109
$606$	0	0
$607$	38.7177	1.57150	0.785752	−	0.618541i	$-0.212276\pi$
$607$	38.7177	1.57150	0.785752	+	0.618541i	$0.212276\pi$
$608$	0	0
$609$	−2.22008	−0.0899622
$610$	0	0
$611$	5.21957	0.211161
$612$	0	0
$613$	29.7798	1.20280	0.601398	−	0.798949i	$-0.294610\pi$
$613$	29.7798	1.20280	0.601398	+	0.798949i	$0.294610\pi$
$614$	0	0
$615$	1.73205	0.0698430
$616$	0	0
$617$	−32.1833	−1.29565	−0.647826	−	0.761788i	$-0.724322\pi$
$617$	−32.1833	−1.29565	−0.647826	+	0.761788i	$0.724322\pi$
$618$	0	0
$619$	33.9357	1.36399	0.681996	−	0.731356i	$-0.261112\pi$
$619$	33.9357	1.36399	0.681996	+	0.731356i	$0.261112\pi$
$620$	0	0
$621$	−26.4192	−1.06016
$622$	0	0
$623$	12.7021	0.508899
$624$	0	0
$625$	24.4755	0.979021
$626$	0	0
$627$	10.2504	0.409363
$628$	0	0
$629$	−25.8233	−1.02964
$630$	0	0
$631$	−37.7604	−1.50322	−0.751610	−	0.659608i	$-0.770722\pi$
$631$	−37.7604	−1.50322	−0.751610	+	0.659608i	$0.770722\pi$
$632$	0	0
$633$	20.1050	0.799101
$634$	0	0
$635$	−2.45930	−0.0975944
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−0.330817	−0.0130869
$640$	0	0
$641$	17.9804	0.710184	0.355092	−	0.934831i	$-0.384450\pi$
$641$	17.9804	0.710184	0.355092	+	0.934831i	$0.384450\pi$
$642$	0	0
$643$	−26.5417	−1.04670	−0.523351	−	0.852117i	$-0.675318\pi$
$643$	−26.5417	−1.04670	−0.523351	+	0.852117i	$0.675318\pi$
$644$	0	0
$645$	−1.32287	−0.0520881
$646$	0	0
$647$	−43.3705	−1.70507	−0.852534	−	0.522671i	$-0.824936\pi$
$647$	−43.3705	−1.70507	−0.852534	+	0.522671i	$0.824936\pi$
$648$	0	0
$649$	−7.28024	−0.285774
$650$	0	0
$651$	−11.5304	−0.451914
$652$	0	0
$653$	−14.5353	−0.568812	−0.284406	−	0.958704i	$-0.591796\pi$
$653$	−14.5353	−0.568812	−0.284406	+	0.958704i	$0.591796\pi$
$654$	0	0
$655$	1.14377	0.0446906
$656$	0	0
$657$	−1.54837	−0.0604076
$658$	0	0
$659$	44.7224	1.74214	0.871069	−	0.491161i	$-0.163427\pi$
$659$	44.7224	1.74214	0.871069	+	0.491161i	$0.163427\pi$
$660$	0	0
$661$	−18.4491	−0.717589	−0.358794	−	0.933417i	$-0.616812\pi$
$661$	−18.4491	−0.717589	−0.358794	+	0.933417i	$0.616812\pi$
$662$	0	0
$663$	−7.64763	−0.297009
$664$	0	0
$665$	1.08561	0.0420983
$666$	0	0
$667$	6.52821	0.252773
$668$	0	0
$669$	−41.2292	−1.59401
$670$	0	0
$671$	3.40345	0.131389
$672$	0	0
$673$	43.8438	1.69005	0.845026	−	0.534725i	$-0.179585\pi$
$673$	43.8438	1.69005	0.845026	+	0.534725i	$0.179585\pi$
$674$	0	0
$675$	25.2358	0.971325
$676$	0	0
$677$	−22.1603	−0.851690	−0.425845	−	0.904796i	$-0.640023\pi$
$677$	−22.1603	−0.851690	−0.425845	+	0.904796i	$0.640023\pi$
$678$	0	0
$679$	4.31505	0.165596
$680$	0	0
$681$	14.4771	0.554763
$682$	0	0
$683$	32.2063	1.23234	0.616170	−	0.787613i	$-0.288683\pi$
$683$	32.2063	1.23234	0.616170	+	0.787613i	$0.288683\pi$
$684$	0	0
$685$	2.55069	0.0974567
$686$	0	0
$687$	−28.6455	−1.09289
$688$	0	0
$689$	1.33361	0.0508066
$690$	0	0
$691$	−1.70081	−0.0647020	−0.0323510	−	0.999477i	$-0.510299\pi$
$691$	−1.70081	−0.0647020	−0.0323510	+	0.999477i	$0.510299\pi$
$692$	0	0
$693$	0.124526	0.00473037
$694$	0	0
$695$	0.223999	0.00849676
$696$	0	0
$697$	22.6453	0.857751
$698$	0	0
$699$	−31.9713	−1.20927
$700$	0	0
$701$	8.11443	0.306478	0.153239	−	0.988189i	$-0.451030\pi$
$701$	8.11443	0.306478	0.153239	+	0.988189i	$0.451030\pi$
$702$	0	0
$703$	34.6120	1.30542
$704$	0	0
$705$	−1.72723	−0.0650514
$706$	0	0
$707$	−2.48362	−0.0934060
$708$	0	0
$709$	−41.6051	−1.56251	−0.781256	−	0.624211i	$-0.785421\pi$
$709$	−41.6051	−1.56251	−0.781256	+	0.624211i	$0.785421\pi$
$710$	0	0
$711$	0.446689	0.0167521
$712$	0	0
$713$	33.9056	1.26978
$714$	0	0
$715$	0.187208	0.00700119
$716$	0	0
$717$	20.9366	0.781891
$718$	0	0
$719$	18.5432	0.691545	0.345773	−	0.938318i	$-0.387617\pi$
$719$	18.5432	0.691545	0.345773	+	0.938318i	$0.387617\pi$
$720$	0	0
$721$	14.0267	0.522383
$722$	0	0
$723$	36.6655	1.36360
$724$	0	0
$725$	−6.23579	−0.231592
$726$	0	0
$727$	32.6335	1.21031	0.605155	−	0.796107i	$-0.293111\pi$
$727$	32.6335	1.21031	0.605155	+	0.796107i	$0.293111\pi$
$728$	0	0
$729$	25.7882	0.955117
$730$	0	0
$731$	−17.2956	−0.639701
$732$	0	0
$733$	4.50958	0.166565	0.0832826	−	0.996526i	$-0.473460\pi$
$733$	4.50958	0.166565	0.0832826	+	0.996526i	$0.473460\pi$
$734$	0	0
$735$	0.330915	0.0122060
$736$	0	0
$737$	2.43127	0.0895570
$738$	0	0
$739$	18.1237	0.666690	0.333345	−	0.942805i	$-0.391823\pi$
$739$	18.1237	0.666690	0.333345	+	0.942805i	$0.391823\pi$
$740$	0	0
$741$	10.2504	0.376559
$742$	0	0
$743$	−44.2109	−1.62194	−0.810971	−	0.585087i	$-0.801061\pi$
$743$	−44.2109	−1.62194	−0.810971	+	0.585087i	$0.801061\pi$
$744$	0	0
$745$	−1.52677	−0.0559365
$746$	0	0
$747$	−1.12627	−0.0412081
$748$	0	0
$749$	−12.9847	−0.474450
$750$	0	0
$751$	−21.1389	−0.771369	−0.385684	−	0.922631i	$-0.626035\pi$
$751$	−21.1389	−0.771369	−0.385684	+	0.922631i	$0.626035\pi$
$752$	0	0
$753$	16.6080	0.605229
$754$	0	0
$755$	−2.28880	−0.0832979
$756$	0	0
$757$	25.2934	0.919305	0.459653	−	0.888099i	$-0.347974\pi$
$757$	25.2934	0.919305	0.459653	+	0.888099i	$0.347974\pi$
$758$	0	0
$759$	−9.18776	−0.333495
$760$	0	0
$761$	−18.7837	−0.680910	−0.340455	−	0.940261i	$-0.610581\pi$
$761$	−18.7837	−0.680910	−0.340455	+	0.940261i	$0.610581\pi$
$762$	0	0
$763$	−3.60242	−0.130416
$764$	0	0
$765$	0.100860	0.00364662
$766$	0	0
$767$	−7.28024	−0.262874
$768$	0	0
$769$	−44.0937	−1.59006	−0.795030	−	0.606570i	$-0.792545\pi$
$769$	−44.0937	−1.59006	−0.795030	+	0.606570i	$0.792545\pi$
$770$	0	0
$771$	−13.8984	−0.500539
$772$	0	0
$773$	−28.7377	−1.03362	−0.516811	−	0.856100i	$-0.672881\pi$
$773$	−28.7377	−1.03362	−0.516811	+	0.856100i	$0.672881\pi$
$774$	0	0
$775$	−32.3869	−1.16337
$776$	0	0
$777$	10.5504	0.378493
$778$	0	0
$779$	−30.3525	−1.08749
$780$	0	0
$781$	2.65660	0.0950607
$782$	0	0
$783$	−6.38378	−0.228138
$784$	0	0
$785$	−3.78236	−0.134998
$786$	0	0
$787$	20.0356	0.714193	0.357097	−	0.934067i	$-0.383767\pi$
$787$	20.0356	0.714193	0.357097	+	0.934067i	$0.383767\pi$
$788$	0	0
$789$	−0.407654	−0.0145129
$790$	0	0
$791$	16.7312	0.594893
$792$	0	0
$793$	3.40345	0.120860
$794$	0	0
$795$	−0.441313	−0.0156518
$796$	0	0
$797$	12.4961	0.442633	0.221317	−	0.975202i	$-0.428965\pi$
$797$	12.4961	0.442633	0.221317	+	0.975202i	$0.428965\pi$
$798$	0	0
$799$	−22.5823	−0.798906
$800$	0	0
$801$	−1.58175	−0.0558883
$802$	0	0
$803$	12.4341	0.438789
$804$	0	0
$805$	−0.973066	−0.0342961
$806$	0	0
$807$	30.7928	1.08396
$808$	0	0
$809$	−14.5390	−0.511162	−0.255581	−	0.966788i	$-0.582267\pi$
$809$	−14.5390	−0.511162	−0.255581	+	0.966788i	$0.582267\pi$
$810$	0	0
$811$	33.8243	1.18773	0.593866	−	0.804564i	$-0.297601\pi$
$811$	33.8243	1.18773	0.593866	+	0.804564i	$0.297601\pi$
$812$	0	0
$813$	−29.6555	−1.04006
$814$	0	0
$815$	−3.14108	−0.110027
$816$	0	0
$817$	23.1820	0.811037
$818$	0	0
$819$	0.124526	0.00435130
$820$	0	0
$821$	−13.6665	−0.476966	−0.238483	−	0.971147i	$-0.576650\pi$
$821$	−13.6665	−0.476966	−0.238483	+	0.971147i	$0.576650\pi$
$822$	0	0
$823$	−3.95988	−0.138033	−0.0690163	−	0.997616i	$-0.521986\pi$
$823$	−3.95988	−0.138033	−0.0690163	+	0.997616i	$0.521986\pi$
$824$	0	0
$825$	8.77621	0.305549
$826$	0	0
$827$	−45.6515	−1.58746	−0.793728	−	0.608272i	$-0.791863\pi$
$827$	−45.6515	−1.58746	−0.793728	+	0.608272i	$0.791863\pi$
$828$	0	0
$829$	−43.4499	−1.50908	−0.754538	−	0.656257i	$-0.772139\pi$
$829$	−43.4499	−1.50908	−0.754538	+	0.656257i	$0.772139\pi$
$830$	0	0
$831$	5.58674	0.193802
$832$	0	0
$833$	4.32648	0.149904
$834$	0	0
$835$	0.226534	0.00783953
$836$	0	0
$837$	−33.1555	−1.14602
$838$	0	0
$839$	17.2572	0.595785	0.297893	−	0.954599i	$-0.403716\pi$
$839$	17.2572	0.595785	0.297893	+	0.954599i	$0.403716\pi$
$840$	0	0
$841$	−27.4226	−0.945605
$842$	0	0
$843$	19.9083	0.685680
$844$	0	0
$845$	0.187208	0.00644016
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	29.1023	0.998788
$850$	0	0
$851$	−31.0238	−1.06348
$852$	0	0
$853$	2.52750	0.0865401	0.0432700	−	0.999063i	$-0.486222\pi$
$853$	2.52750	0.0865401	0.0432700	+	0.999063i	$0.486222\pi$
$854$	0	0
$855$	−0.135188	−0.00462332
$856$	0	0
$857$	38.7291	1.32296	0.661480	−	0.749963i	$-0.269929\pi$
$857$	38.7291	1.32296	0.661480	+	0.749963i	$0.269929\pi$
$858$	0	0
$859$	−5.04338	−0.172078	−0.0860390	−	0.996292i	$-0.527421\pi$
$859$	−5.04338	−0.172078	−0.0860390	+	0.996292i	$0.527421\pi$
$860$	0	0
$861$	−9.25200	−0.315307
$862$	0	0
$863$	−27.5510	−0.937846	−0.468923	−	0.883239i	$-0.655358\pi$
$863$	−27.5510	−0.937846	−0.468923	+	0.883239i	$0.655358\pi$
$864$	0	0
$865$	2.48447	0.0844745
$866$	0	0
$867$	3.03753	0.103160
$868$	0	0
$869$	−3.58710	−0.121684
$870$	0	0
$871$	2.43127	0.0823805
$872$	0	0
$873$	−0.537338	−0.0181861
$874$	0	0
$875$	1.86552	0.0630661
$876$	0	0
$877$	24.3947	0.823751	0.411876	−	0.911240i	$-0.364874\pi$
$877$	24.3947	0.823751	0.411876	+	0.911240i	$0.364874\pi$
$878$	0	0
$879$	−18.3378	−0.618518
$880$	0	0
$881$	19.5933	0.660117	0.330058	−	0.943961i	$-0.392932\pi$
$881$	19.5933	0.660117	0.330058	+	0.943961i	$0.392932\pi$
$882$	0	0
$883$	−52.4809	−1.76612	−0.883061	−	0.469258i	$-0.844521\pi$
$883$	−52.4809	−1.76612	−0.883061	+	0.469258i	$0.844521\pi$
$884$	0	0
$885$	2.40914	0.0809824
$886$	0	0
$887$	−7.77851	−0.261177	−0.130588	−	0.991437i	$-0.541687\pi$
$887$	−7.77851	−0.261177	−0.130588	+	0.991437i	$0.541687\pi$
$888$	0	0
$889$	13.1367	0.440591
$890$	0	0
$891$	9.35807	0.313507
$892$	0	0
$893$	30.2681	1.01288
$894$	0	0
$895$	3.46719	0.115895
$896$	0	0
$897$	−9.18776	−0.306770
$898$	0	0
$899$	8.19277	0.273244
$900$	0	0
$901$	−5.76985	−0.192222
$902$	0	0
$903$	7.06632	0.235152
$904$	0	0
$905$	2.80698	0.0933073
$906$	0	0
$907$	24.8514	0.825178	0.412589	−	0.910917i	$-0.364625\pi$
$907$	24.8514	0.825178	0.412589	+	0.910917i	$0.364625\pi$
$908$	0	0
$909$	0.309276	0.0102580
$910$	0	0
$911$	14.1481	0.468748	0.234374	−	0.972147i	$-0.424696\pi$
$911$	14.1481	0.468748	0.234374	+	0.972147i	$0.424696\pi$
$912$	0	0
$913$	9.04444	0.299327
$914$	0	0
$915$	−1.12625	−0.0372328
$916$	0	0
$917$	−6.10959	−0.201756
$918$	0	0
$919$	34.8296	1.14892	0.574462	−	0.818532i	$-0.305211\pi$
$919$	34.8296	1.14892	0.574462	+	0.818532i	$0.305211\pi$
$920$	0	0
$921$	14.6426	0.482491
$922$	0	0
$923$	2.65660	0.0874431
$924$	0	0
$925$	29.6341	0.974364
$926$	0	0
$927$	−1.74670	−0.0573691
$928$	0	0
$929$	−39.7577	−1.30441	−0.652204	−	0.758043i	$-0.726156\pi$
$929$	−39.7577	−1.30441	−0.652204	+	0.758043i	$0.726156\pi$
$930$	0	0
$931$	−5.79897	−0.190053
$932$	0	0
$933$	−10.4322	−0.341535
$934$	0	0
$935$	−0.809952	−0.0264883
$936$	0	0
$937$	7.61195	0.248672	0.124336	−	0.992240i	$-0.460320\pi$
$937$	7.61195	0.248672	0.124336	+	0.992240i	$0.460320\pi$
$938$	0	0
$939$	−17.5366	−0.572285
$940$	0	0
$941$	−20.6401	−0.672849	−0.336424	−	0.941710i	$-0.609218\pi$
$941$	−20.6401	−0.672849	−0.336424	+	0.941710i	$0.609218\pi$
$942$	0	0
$943$	27.2058	0.885942
$944$	0	0
$945$	0.951538	0.0309535
$946$	0	0
$947$	41.8689	1.36056	0.680278	−	0.732954i	$-0.261859\pi$
$947$	41.8689	1.36056	0.680278	+	0.732954i	$0.261859\pi$
$948$	0	0
$949$	12.4341	0.403627
$950$	0	0
$951$	−11.8261	−0.383488
$952$	0	0
$953$	8.13721	0.263590	0.131795	−	0.991277i	$-0.457926\pi$
$953$	8.13721	0.263590	0.131795	+	0.991277i	$0.457926\pi$
$954$	0	0
$955$	−1.58946	−0.0514337
$956$	0	0
$957$	−2.22008	−0.0717650
$958$	0	0
$959$	−13.6249	−0.439970
$960$	0	0
$961$	11.5508	0.372607
$962$	0	0
$963$	1.61693	0.0521050
$964$	0	0
$965$	−2.04320	−0.0657729
$966$	0	0
$967$	48.5726	1.56199	0.780995	−	0.624538i	$-0.214713\pi$
$967$	48.5726	1.56199	0.780995	+	0.624538i	$0.214713\pi$
$968$	0	0
$969$	−44.3483	−1.42467
$970$	0	0
$971$	29.5249	0.947500	0.473750	−	0.880659i	$-0.342900\pi$
$971$	29.5249	0.947500	0.473750	+	0.880659i	$0.342900\pi$
$972$	0	0
$973$	−1.19652	−0.0383588
$974$	0	0
$975$	8.77621	0.281064
$976$	0	0
$977$	−8.68546	−0.277873	−0.138936	−	0.990301i	$-0.544368\pi$
$977$	−8.68546	−0.277873	−0.138936	+	0.990301i	$0.544368\pi$
$978$	0	0
$979$	12.7021	0.405961
$980$	0	0
$981$	0.448596	0.0143226
$982$	0	0
$983$	−16.3899	−0.522756	−0.261378	−	0.965237i	$-0.584177\pi$
$983$	−16.3899	−0.522756	−0.261378	+	0.965237i	$0.584177\pi$
$984$	0	0
$985$	−3.72116	−0.118566
$986$	0	0
$987$	9.22628	0.293676
$988$	0	0
$989$	−20.7787	−0.660725
$990$	0	0
$991$	−37.3457	−1.18633	−0.593163	−	0.805083i	$-0.702121\pi$
$991$	−37.3457	−1.18633	−0.593163	+	0.805083i	$0.702121\pi$
$992$	0	0
$993$	33.3822	1.05935
$994$	0	0
$995$	−5.01622	−0.159025
$996$	0	0
$997$	12.5699	0.398092	0.199046	−	0.979990i	$-0.436216\pi$
$997$	12.5699	0.398092	0.199046	+	0.979990i	$0.436216\pi$
$998$	0	0
$999$	30.3374	0.959833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.u.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.u.1.8	✓	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)

\(f(q)\)	\(=\)	\(q+1.76763 q^{3} +0.187208 q^{5} -1.00000 q^{7} +0.124526 q^{9} +O(q^{10})\)	\(q+1.76763 q^{3} +0.187208 q^{5} -1.00000 q^{7} +0.124526 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.330915 q^{15} +4.32648 q^{17} -5.79897 q^{19} -1.76763 q^{21} +5.19778 q^{23} -4.96495 q^{25} -5.08278 q^{27} +1.25596 q^{29} +6.52310 q^{31} -1.76763 q^{33} -0.187208 q^{35} -5.96866 q^{37} -1.76763 q^{39} +5.23412 q^{41} -3.99762 q^{43} +0.0233124 q^{45} -5.21957 q^{47} +1.00000 q^{49} +7.64763 q^{51} -1.33361 q^{53} -0.187208 q^{55} -10.2504 q^{57} +7.28024 q^{59} -3.40345 q^{61} -0.124526 q^{63} -0.187208 q^{65} -2.43127 q^{67} +9.18776 q^{69} -2.65660 q^{71} -12.4341 q^{73} -8.77621 q^{75} +1.00000 q^{77} +3.58710 q^{79} -9.35807 q^{81} -9.04444 q^{83} +0.809952 q^{85} +2.22008 q^{87} -12.7021 q^{89} +1.00000 q^{91} +11.5304 q^{93} -1.08561 q^{95} -4.31505 q^{97} -0.124526 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) \) 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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