LMFDB - Embedded newform 8004.2.a.d.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 19x^{5} + 19x^{4} - 35x^{3} - 10x^{2} + 18x - 4 $ x^8 - 3x^7 - 8x^6 + 19x^5 + 19x^4 - 35x^3 - 10x^2 + 18*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.58311$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.270840 q^{5} +0.409136 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.270840 q^{5} +0.409136 q^{7} +1.00000 q^{9} -3.18144 q^{11} +0.382284 q^{13} -0.270840 q^{15} -4.23453 q^{17} +5.44767 q^{19} +0.409136 q^{21} +1.00000 q^{23} -4.92665 q^{25} +1.00000 q^{27} +1.00000 q^{29} +2.08063 q^{31} -3.18144 q^{33} -0.110810 q^{35} -5.55848 q^{37} +0.382284 q^{39} +7.07476 q^{41} -5.46885 q^{43} -0.270840 q^{45} -11.6721 q^{47} -6.83261 q^{49} -4.23453 q^{51} +13.1074 q^{53} +0.861661 q^{55} +5.44767 q^{57} +4.76314 q^{59} -2.32894 q^{61} +0.409136 q^{63} -0.103538 q^{65} +0.562865 q^{67} +1.00000 q^{69} -7.04632 q^{71} -10.9653 q^{73} -4.92665 q^{75} -1.30164 q^{77} -16.6996 q^{79} +1.00000 q^{81} +9.44555 q^{83} +1.14688 q^{85} +1.00000 q^{87} +8.74304 q^{89} +0.156406 q^{91} +2.08063 q^{93} -1.47545 q^{95} -2.87535 q^{97} -3.18144 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9	$ 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9} - 5 q^{11} - 4 q^{13} - 5 q^{15} - 3 q^{17} - 5 q^{19} - 4 q^{21} + 8 q^{23} - 5 q^{25} + 8 q^{27} + 8 q^{29} - 2 q^{31} - 5 q^{33} - 15 q^{35} - 10 q^{37} - 4 q^{39} - 11 q^{41} - 7 q^{43} - 5 q^{45} - 14 q^{47} - 18 q^{49} - 3 q^{51} - 15 q^{53} - 17 q^{55} - 5 q^{57} + 4 q^{59} + q^{61} - 4 q^{63} - 5 q^{67} + 8 q^{69} - q^{71} - 21 q^{73} - 5 q^{75} - 8 q^{79} + 8 q^{81} + 3 q^{83} + 8 q^{87} - 20 q^{89} - 7 q^{91} - 2 q^{93} - 3 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9 - 5 * q^11 - 4 * q^13 - 5 * q^15 - 3 * q^17 - 5 * q^19 - 4 * q^21 + 8 * q^23 - 5 * q^25 + 8 * q^27 + 8 * q^29 - 2 * q^31 - 5 * q^33 - 15 * q^35 - 10 * q^37 - 4 * q^39 - 11 * q^41 - 7 * q^43 - 5 * q^45 - 14 * q^47 - 18 * q^49 - 3 * q^51 - 15 * q^53 - 17 * q^55 - 5 * q^57 + 4 * q^59 + q^61 - 4 * q^63 - 5 * q^67 + 8 * q^69 - q^71 - 21 * q^73 - 5 * q^75 - 8 * q^79 + 8 * q^81 + 3 * q^83 + 8 * q^87 - 20 * q^89 - 7 * q^91 - 2 * q^93 - 3 * q^95 - 7 * q^97 - 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.270840	−0.121123	−0.0605617	−	0.998164i	$-0.519289\pi$
$5$	−0.270840	−0.121123	−0.0605617	+	0.998164i	$0.519289\pi$
$6$	0	0
$7$	0.409136	0.154639	0.0773195	−	0.997006i	$-0.475364\pi$
$7$	0.409136	0.154639	0.0773195	+	0.997006i	$0.475364\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.18144	−0.959240	−0.479620	−	0.877476i	$-0.659226\pi$
$11$	−3.18144	−0.959240	−0.479620	+	0.877476i	$0.659226\pi$
$12$	0	0
$13$	0.382284	0.106027	0.0530133	−	0.998594i	$-0.483117\pi$
$13$	0.382284	0.106027	0.0530133	+	0.998594i	$0.483117\pi$
$14$	0	0
$15$	−0.270840	−0.0699306
$16$	0	0
$17$	−4.23453	−1.02702	−0.513512	−	0.858083i	$-0.671656\pi$
$17$	−4.23453	−1.02702	−0.513512	+	0.858083i	$0.671656\pi$
$18$	0	0
$19$	5.44767	1.24978	0.624891	−	0.780712i	$-0.285144\pi$
$19$	5.44767	1.24978	0.624891	+	0.780712i	$0.285144\pi$
$20$	0	0
$21$	0.409136	0.0892808
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.92665	−0.985329
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	2.08063	0.373691	0.186846	−	0.982389i	$-0.440174\pi$
$31$	2.08063	0.373691	0.186846	+	0.982389i	$0.440174\pi$
$32$	0	0
$33$	−3.18144	−0.553818
$34$	0	0
$35$	−0.110810	−0.0187304
$36$	0	0
$37$	−5.55848	−0.913808	−0.456904	−	0.889516i	$-0.651042\pi$
$37$	−5.55848	−0.913808	−0.456904	+	0.889516i	$0.651042\pi$
$38$	0	0
$39$	0.382284	0.0612145
$40$	0	0
$41$	7.07476	1.10489	0.552446	−	0.833549i	$-0.313695\pi$
$41$	7.07476	1.10489	0.552446	+	0.833549i	$0.313695\pi$
$42$	0	0
$43$	−5.46885	−0.833992	−0.416996	−	0.908908i	$-0.636917\pi$
$43$	−5.46885	−0.833992	−0.416996	+	0.908908i	$0.636917\pi$
$44$	0	0
$45$	−0.270840	−0.0403744
$46$	0	0
$47$	−11.6721	−1.70255	−0.851276	−	0.524717i	$-0.824171\pi$
$47$	−11.6721	−1.70255	−0.851276	+	0.524717i	$0.824171\pi$
$48$	0	0
$49$	−6.83261	−0.976087
$50$	0	0
$51$	−4.23453	−0.592952
$52$	0	0
$53$	13.1074	1.80043	0.900217	−	0.435441i	$-0.143408\pi$
$53$	13.1074	1.80043	0.900217	+	0.435441i	$0.143408\pi$
$54$	0	0
$55$	0.861661	0.116186
$56$	0	0
$57$	5.44767	0.721561
$58$	0	0
$59$	4.76314	0.620108	0.310054	−	0.950719i	$-0.399653\pi$
$59$	4.76314	0.620108	0.310054	+	0.950719i	$0.399653\pi$
$60$	0	0
$61$	−2.32894	−0.298190	−0.149095	−	0.988823i	$-0.547636\pi$
$61$	−2.32894	−0.298190	−0.149095	+	0.988823i	$0.547636\pi$
$62$	0	0
$63$	0.409136	0.0515463
$64$	0	0
$65$	−0.103538	−0.0128423
$66$	0	0
$67$	0.562865	0.0687649	0.0343824	−	0.999409i	$-0.489054\pi$
$67$	0.562865	0.0687649	0.0343824	+	0.999409i	$0.489054\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−7.04632	−0.836245	−0.418122	−	0.908391i	$-0.637312\pi$
$71$	−7.04632	−0.836245	−0.418122	+	0.908391i	$0.637312\pi$
$72$	0	0
$73$	−10.9653	−1.28340	−0.641698	−	0.766957i	$-0.721770\pi$
$73$	−10.9653	−1.28340	−0.641698	+	0.766957i	$0.721770\pi$
$74$	0	0
$75$	−4.92665	−0.568880
$76$	0	0
$77$	−1.30164	−0.148336
$78$	0	0
$79$	−16.6996	−1.87885	−0.939424	−	0.342758i	$-0.888639\pi$
$79$	−16.6996	−1.87885	−0.939424	+	0.342758i	$0.888639\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.44555	1.03678	0.518392	−	0.855143i	$-0.326531\pi$
$83$	9.44555	1.03678	0.518392	+	0.855143i	$0.326531\pi$
$84$	0	0
$85$	1.14688	0.124396
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	8.74304	0.926760	0.463380	−	0.886160i	$-0.346637\pi$
$89$	8.74304	0.926760	0.463380	+	0.886160i	$0.346637\pi$
$90$	0	0
$91$	0.156406	0.0163958
$92$	0	0
$93$	2.08063	0.215751
$94$	0	0
$95$	−1.47545	−0.151378
$96$	0	0
$97$	−2.87535	−0.291948	−0.145974	−	0.989288i	$-0.546632\pi$
$97$	−2.87535	−0.291948	−0.145974	+	0.989288i	$0.546632\pi$
$98$	0	0
$99$	−3.18144	−0.319747
$100$	0	0
$101$	13.7261	1.36580	0.682900	−	0.730512i	$-0.260719\pi$
$101$	13.7261	1.36580	0.682900	+	0.730512i	$0.260719\pi$
$102$	0	0
$103$	−12.0130	−1.18367	−0.591837	−	0.806058i	$-0.701597\pi$
$103$	−12.0130	−1.18367	−0.591837	+	0.806058i	$0.701597\pi$
$104$	0	0
$105$	−0.110810	−0.0108140
$106$	0	0
$107$	−0.954655	−0.0922900	−0.0461450	−	0.998935i	$-0.514694\pi$
$107$	−0.954655	−0.0922900	−0.0461450	+	0.998935i	$0.514694\pi$
$108$	0	0
$109$	−9.85059	−0.943516	−0.471758	−	0.881728i	$-0.656380\pi$
$109$	−9.85059	−0.943516	−0.471758	+	0.881728i	$0.656380\pi$
$110$	0	0
$111$	−5.55848	−0.527588
$112$	0	0
$113$	3.54743	0.333714	0.166857	−	0.985981i	$-0.446638\pi$
$113$	3.54743	0.333714	0.166857	+	0.985981i	$0.446638\pi$
$114$	0	0
$115$	−0.270840	−0.0252560
$116$	0	0
$117$	0.382284	0.0353422
$118$	0	0
$119$	−1.73250	−0.158818
$120$	0	0
$121$	−0.878442	−0.0798584
$122$	0	0
$123$	7.07476	0.637910
$124$	0	0
$125$	2.68853	0.240470
$126$	0	0
$127$	2.20435	0.195605	0.0978024	−	0.995206i	$-0.468819\pi$
$127$	2.20435	0.195605	0.0978024	+	0.995206i	$0.468819\pi$
$128$	0	0
$129$	−5.46885	−0.481505
$130$	0	0
$131$	−11.6263	−1.01580	−0.507899	−	0.861417i	$-0.669578\pi$
$131$	−11.6263	−1.01580	−0.507899	+	0.861417i	$0.669578\pi$
$132$	0	0
$133$	2.22884	0.193265
$134$	0	0
$135$	−0.270840	−0.0233102
$136$	0	0
$137$	−8.00999	−0.684340	−0.342170	−	0.939638i	$-0.611162\pi$
$137$	−8.00999	−0.684340	−0.342170	+	0.939638i	$0.611162\pi$
$138$	0	0
$139$	−16.7849	−1.42368	−0.711838	−	0.702343i	$-0.752137\pi$
$139$	−16.7849	−1.42368	−0.711838	+	0.702343i	$0.752137\pi$
$140$	0	0
$141$	−11.6721	−0.982969
$142$	0	0
$143$	−1.21621	−0.101705
$144$	0	0
$145$	−0.270840	−0.0224920
$146$	0	0
$147$	−6.83261	−0.563544
$148$	0	0
$149$	−16.5791	−1.35821	−0.679107	−	0.734039i	$-0.737633\pi$
$149$	−16.5791	−1.35821	−0.679107	+	0.734039i	$0.737633\pi$
$150$	0	0
$151$	11.8936	0.967888	0.483944	−	0.875099i	$-0.339204\pi$
$151$	11.8936	0.967888	0.483944	+	0.875099i	$0.339204\pi$
$152$	0	0
$153$	−4.23453	−0.342341
$154$	0	0
$155$	−0.563517	−0.0452628
$156$	0	0
$157$	8.71347	0.695411	0.347705	−	0.937604i	$-0.386961\pi$
$157$	8.71347	0.695411	0.347705	+	0.937604i	$0.386961\pi$
$158$	0	0
$159$	13.1074	1.03948
$160$	0	0
$161$	0.409136	0.0322444
$162$	0	0
$163$	15.7742	1.23553	0.617767	−	0.786362i	$-0.288038\pi$
$163$	15.7742	1.23553	0.617767	+	0.786362i	$0.288038\pi$
$164$	0	0
$165$	0.861661	0.0670802
$166$	0	0
$167$	−19.0619	−1.47505	−0.737525	−	0.675320i	$-0.764006\pi$
$167$	−19.0619	−1.47505	−0.737525	+	0.675320i	$0.764006\pi$
$168$	0	0
$169$	−12.8539	−0.988758
$170$	0	0
$171$	5.44767	0.416594
$172$	0	0
$173$	2.35020	0.178682	0.0893410	−	0.996001i	$-0.471524\pi$
$173$	2.35020	0.178682	0.0893410	+	0.996001i	$0.471524\pi$
$174$	0	0
$175$	−2.01567	−0.152370
$176$	0	0
$177$	4.76314	0.358019
$178$	0	0
$179$	−6.80243	−0.508437	−0.254219	−	0.967147i	$-0.581818\pi$
$179$	−6.80243	−0.508437	−0.254219	+	0.967147i	$0.581818\pi$
$180$	0	0
$181$	−8.35756	−0.621212	−0.310606	−	0.950539i	$-0.600532\pi$
$181$	−8.35756	−0.621212	−0.310606	+	0.950539i	$0.600532\pi$
$182$	0	0
$183$	−2.32894	−0.172160
$184$	0	0
$185$	1.50546	0.110684
$186$	0	0
$187$	13.4719	0.985162
$188$	0	0
$189$	0.409136	0.0297603
$190$	0	0
$191$	16.5128	1.19483	0.597413	−	0.801934i	$-0.296195\pi$
$191$	16.5128	1.19483	0.597413	+	0.801934i	$0.296195\pi$
$192$	0	0
$193$	−20.6858	−1.48900	−0.744499	−	0.667623i	$-0.767312\pi$
$193$	−20.6858	−1.48900	−0.744499	+	0.667623i	$0.767312\pi$
$194$	0	0
$195$	−0.103538	−0.00741450
$196$	0	0
$197$	6.56222	0.467539	0.233769	−	0.972292i	$-0.424894\pi$
$197$	6.56222	0.467539	0.233769	+	0.972292i	$0.424894\pi$
$198$	0	0
$199$	0.673220	0.0477233	0.0238616	−	0.999715i	$-0.492404\pi$
$199$	0.673220	0.0477233	0.0238616	+	0.999715i	$0.492404\pi$
$200$	0	0
$201$	0.562865	0.0397014
$202$	0	0
$203$	0.409136	0.0287157
$204$	0	0
$205$	−1.91613	−0.133828
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−17.3314	−1.19884
$210$	0	0
$211$	−24.8206	−1.70872	−0.854359	−	0.519684i	$-0.826050\pi$
$211$	−24.8206	−1.70872	−0.854359	+	0.519684i	$0.826050\pi$
$212$	0	0
$213$	−7.04632	−0.482806
$214$	0	0
$215$	1.48118	0.101016
$216$	0	0
$217$	0.851259	0.0577872
$218$	0	0
$219$	−10.9653	−0.740969
$220$	0	0
$221$	−1.61879	−0.108892
$222$	0	0
$223$	−27.7648	−1.85927	−0.929633	−	0.368487i	$-0.879876\pi$
$223$	−27.7648	−1.85927	−0.929633	+	0.368487i	$0.879876\pi$
$224$	0	0
$225$	−4.92665	−0.328443
$226$	0	0
$227$	−12.5430	−0.832511	−0.416255	−	0.909248i	$-0.636658\pi$
$227$	−12.5430	−0.832511	−0.416255	+	0.909248i	$0.636658\pi$
$228$	0	0
$229$	−12.0808	−0.798321	−0.399161	−	0.916881i	$-0.630698\pi$
$229$	−12.0808	−0.798321	−0.399161	+	0.916881i	$0.630698\pi$
$230$	0	0
$231$	−1.30164	−0.0856418
$232$	0	0
$233$	7.64163	0.500620	0.250310	−	0.968166i	$-0.419468\pi$
$233$	7.64163	0.500620	0.250310	+	0.968166i	$0.419468\pi$
$234$	0	0
$235$	3.16128	0.206219
$236$	0	0
$237$	−16.6996	−1.08475
$238$	0	0
$239$	10.0402	0.649445	0.324723	−	0.945809i	$-0.394729\pi$
$239$	10.0402	0.649445	0.324723	+	0.945809i	$0.394729\pi$
$240$	0	0
$241$	2.88233	0.185667	0.0928335	−	0.995682i	$-0.470408\pi$
$241$	2.88233	0.185667	0.0928335	+	0.995682i	$0.470408\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.85054	0.118227
$246$	0	0
$247$	2.08256	0.132510
$248$	0	0
$249$	9.44555	0.598587
$250$	0	0
$251$	17.4367	1.10059	0.550297	−	0.834969i	$-0.314514\pi$
$251$	17.4367	1.10059	0.550297	+	0.834969i	$0.314514\pi$
$252$	0	0
$253$	−3.18144	−0.200015
$254$	0	0
$255$	1.14688	0.0718204
$256$	0	0
$257$	−23.8712	−1.48904	−0.744522	−	0.667597i	$-0.767323\pi$
$257$	−23.8712	−1.48904	−0.744522	+	0.667597i	$0.767323\pi$
$258$	0	0
$259$	−2.27418	−0.141310
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	13.4132	0.827090	0.413545	−	0.910484i	$-0.364290\pi$
$263$	13.4132	0.827090	0.413545	+	0.910484i	$0.364290\pi$
$264$	0	0
$265$	−3.55000	−0.218075
$266$	0	0
$267$	8.74304	0.535065
$268$	0	0
$269$	−1.27424	−0.0776915	−0.0388457	−	0.999245i	$-0.512368\pi$
$269$	−1.27424	−0.0776915	−0.0388457	+	0.999245i	$0.512368\pi$
$270$	0	0
$271$	8.32920	0.505963	0.252981	−	0.967471i	$-0.418589\pi$
$271$	8.32920	0.505963	0.252981	+	0.967471i	$0.418589\pi$
$272$	0	0
$273$	0.156406	0.00946614
$274$	0	0
$275$	15.6738	0.945167
$276$	0	0
$277$	12.4631	0.748834	0.374417	−	0.927261i	$-0.377843\pi$
$277$	12.4631	0.748834	0.374417	+	0.927261i	$0.377843\pi$
$278$	0	0
$279$	2.08063	0.124564
$280$	0	0
$281$	4.79090	0.285801	0.142901	−	0.989737i	$-0.454357\pi$
$281$	4.79090	0.285801	0.142901	+	0.989737i	$0.454357\pi$
$282$	0	0
$283$	8.47563	0.503824	0.251912	−	0.967750i	$-0.418941\pi$
$283$	8.47563	0.503824	0.251912	+	0.967750i	$0.418941\pi$
$284$	0	0
$285$	−1.47545	−0.0873979
$286$	0	0
$287$	2.89454	0.170859
$288$	0	0
$289$	0.931210	0.0547770
$290$	0	0
$291$	−2.87535	−0.168556
$292$	0	0
$293$	−25.4167	−1.48486	−0.742431	−	0.669923i	$-0.766327\pi$
$293$	−25.4167	−1.48486	−0.742431	+	0.669923i	$0.766327\pi$
$294$	0	0
$295$	−1.29005	−0.0751095
$296$	0	0
$297$	−3.18144	−0.184606
$298$	0	0
$299$	0.382284	0.0221081
$300$	0	0
$301$	−2.23750	−0.128968
$302$	0	0
$303$	13.7261	0.788545
$304$	0	0
$305$	0.630769	0.0361177
$306$	0	0
$307$	30.3855	1.73419	0.867097	−	0.498140i	$-0.165983\pi$
$307$	30.3855	1.73419	0.867097	+	0.498140i	$0.165983\pi$
$308$	0	0
$309$	−12.0130	−0.683394
$310$	0	0
$311$	−27.4460	−1.55632	−0.778159	−	0.628068i	$-0.783846\pi$
$311$	−27.4460	−1.55632	−0.778159	+	0.628068i	$0.783846\pi$
$312$	0	0
$313$	13.1030	0.740625	0.370312	−	0.928907i	$-0.379251\pi$
$313$	13.1030	0.740625	0.370312	+	0.928907i	$0.379251\pi$
$314$	0	0
$315$	−0.110810	−0.00624346
$316$	0	0
$317$	28.8773	1.62191	0.810956	−	0.585107i	$-0.198948\pi$
$317$	28.8773	1.62191	0.810956	+	0.585107i	$0.198948\pi$
$318$	0	0
$319$	−3.18144	−0.178126
$320$	0	0
$321$	−0.954655	−0.0532836
$322$	0	0
$323$	−23.0683	−1.28355
$324$	0	0
$325$	−1.88338	−0.104471
$326$	0	0
$327$	−9.85059	−0.544739
$328$	0	0
$329$	−4.77548	−0.263281
$330$	0	0
$331$	9.77951	0.537531	0.268765	−	0.963206i	$-0.413384\pi$
$331$	9.77951	0.537531	0.268765	+	0.963206i	$0.413384\pi$
$332$	0	0
$333$	−5.55848	−0.304603
$334$	0	0
$335$	−0.152446	−0.00832903
$336$	0	0
$337$	32.0122	1.74382	0.871909	−	0.489668i	$-0.162882\pi$
$337$	32.0122	1.74382	0.871909	+	0.489668i	$0.162882\pi$
$338$	0	0
$339$	3.54743	0.192670
$340$	0	0
$341$	−6.61939	−0.358460
$342$	0	0
$343$	−5.65942	−0.305580
$344$	0	0
$345$	−0.270840	−0.0145815
$346$	0	0
$347$	−5.80201	−0.311468	−0.155734	−	0.987799i	$-0.549774\pi$
$347$	−5.80201	−0.311468	−0.155734	+	0.987799i	$0.549774\pi$
$348$	0	0
$349$	−22.3183	−1.19467	−0.597334	−	0.801992i	$-0.703773\pi$
$349$	−22.3183	−1.19467	−0.597334	+	0.801992i	$0.703773\pi$
$350$	0	0
$351$	0.382284	0.0204048
$352$	0	0
$353$	−7.93967	−0.422586	−0.211293	−	0.977423i	$-0.567767\pi$
$353$	−7.93967	−0.422586	−0.211293	+	0.977423i	$0.567767\pi$
$354$	0	0
$355$	1.90843	0.101289
$356$	0	0
$357$	−1.73250	−0.0916935
$358$	0	0
$359$	−4.91659	−0.259488	−0.129744	−	0.991548i	$-0.541416\pi$
$359$	−4.91659	−0.259488	−0.129744	+	0.991548i	$0.541416\pi$
$360$	0	0
$361$	10.6771	0.561953
$362$	0	0
$363$	−0.878442	−0.0461063
$364$	0	0
$365$	2.96985	0.155449
$366$	0	0
$367$	−6.68277	−0.348838	−0.174419	−	0.984672i	$-0.555805\pi$
$367$	−6.68277	−0.348838	−0.174419	+	0.984672i	$0.555805\pi$
$368$	0	0
$369$	7.07476	0.368297
$370$	0	0
$371$	5.36270	0.278417
$372$	0	0
$373$	−12.5263	−0.648587	−0.324293	−	0.945957i	$-0.605126\pi$
$373$	−12.5263	−0.648587	−0.324293	+	0.945957i	$0.605126\pi$
$374$	0	0
$375$	2.68853	0.138835
$376$	0	0
$377$	0.382284	0.0196886
$378$	0	0
$379$	14.0180	0.720057	0.360028	−	0.932941i	$-0.382767\pi$
$379$	14.0180	0.720057	0.360028	+	0.932941i	$0.382767\pi$
$380$	0	0
$381$	2.20435	0.112933
$382$	0	0
$383$	24.8668	1.27063	0.635317	−	0.772252i	$-0.280870\pi$
$383$	24.8668	1.27063	0.635317	+	0.772252i	$0.280870\pi$
$384$	0	0
$385$	0.352537	0.0179669
$386$	0	0
$387$	−5.46885	−0.277997
$388$	0	0
$389$	−35.5368	−1.80179	−0.900895	−	0.434037i	$-0.857089\pi$
$389$	−35.5368	−1.80179	−0.900895	+	0.434037i	$0.857089\pi$
$390$	0	0
$391$	−4.23453	−0.214149
$392$	0	0
$393$	−11.6263	−0.586471
$394$	0	0
$395$	4.52291	0.227572
$396$	0	0
$397$	1.45246	0.0728967	0.0364484	−	0.999336i	$-0.488396\pi$
$397$	1.45246	0.0728967	0.0364484	+	0.999336i	$0.488396\pi$
$398$	0	0
$399$	2.22884	0.111581
$400$	0	0
$401$	−14.4573	−0.721964	−0.360982	−	0.932573i	$-0.617558\pi$
$401$	−14.4573	−0.721964	−0.360982	+	0.932573i	$0.617558\pi$
$402$	0	0
$403$	0.795391	0.0396212
$404$	0	0
$405$	−0.270840	−0.0134581
$406$	0	0
$407$	17.6840	0.876562
$408$	0	0
$409$	−4.78037	−0.236374	−0.118187	−	0.992991i	$-0.537708\pi$
$409$	−4.78037	−0.236374	−0.118187	+	0.992991i	$0.537708\pi$
$410$	0	0
$411$	−8.00999	−0.395104
$412$	0	0
$413$	1.94877	0.0958928
$414$	0	0
$415$	−2.55823	−0.125579
$416$	0	0
$417$	−16.7849	−0.821960
$418$	0	0
$419$	−38.0916	−1.86090	−0.930448	−	0.366424i	$-0.880582\pi$
$419$	−38.0916	−1.86090	−0.930448	+	0.366424i	$0.880582\pi$
$420$	0	0
$421$	−24.7113	−1.20436	−0.602178	−	0.798362i	$-0.705700\pi$
$421$	−24.7113	−1.20436	−0.602178	+	0.798362i	$0.705700\pi$
$422$	0	0
$423$	−11.6721	−0.567518
$424$	0	0
$425$	20.8620	1.01196
$426$	0	0
$427$	−0.952852	−0.0461117
$428$	0	0
$429$	−1.21621	−0.0587194
$430$	0	0
$431$	−0.932560	−0.0449198	−0.0224599	−	0.999748i	$-0.507150\pi$
$431$	−0.932560	−0.0449198	−0.0224599	+	0.999748i	$0.507150\pi$
$432$	0	0
$433$	−10.5454	−0.506780	−0.253390	−	0.967364i	$-0.581546\pi$
$433$	−10.5454	−0.506780	−0.253390	+	0.967364i	$0.581546\pi$
$434$	0	0
$435$	−0.270840	−0.0129858
$436$	0	0
$437$	5.44767	0.260597
$438$	0	0
$439$	27.6249	1.31847	0.659233	−	0.751938i	$-0.270881\pi$
$439$	27.6249	1.31847	0.659233	+	0.751938i	$0.270881\pi$
$440$	0	0
$441$	−6.83261	−0.325362
$442$	0	0
$443$	29.2217	1.38837	0.694184	−	0.719798i	$-0.255766\pi$
$443$	29.2217	1.38837	0.694184	+	0.719798i	$0.255766\pi$
$444$	0	0
$445$	−2.36796	−0.112252
$446$	0	0
$447$	−16.5791	−0.784165
$448$	0	0
$449$	−16.7444	−0.790215	−0.395108	−	0.918635i	$-0.629293\pi$
$449$	−16.7444	−0.790215	−0.395108	+	0.918635i	$0.629293\pi$
$450$	0	0
$451$	−22.5079	−1.05986
$452$	0	0
$453$	11.8936	0.558811
$454$	0	0
$455$	−0.0423611	−0.00198592
$456$	0	0
$457$	4.05973	0.189906	0.0949530	−	0.995482i	$-0.469730\pi$
$457$	4.05973	0.189906	0.0949530	+	0.995482i	$0.469730\pi$
$458$	0	0
$459$	−4.23453	−0.197651
$460$	0	0
$461$	−32.7229	−1.52406	−0.762028	−	0.647544i	$-0.775796\pi$
$461$	−32.7229	−1.52406	−0.762028	+	0.647544i	$0.775796\pi$
$462$	0	0
$463$	−7.16747	−0.333101	−0.166550	−	0.986033i	$-0.553263\pi$
$463$	−7.16747	−0.333101	−0.166550	+	0.986033i	$0.553263\pi$
$464$	0	0
$465$	−0.563517	−0.0261325
$466$	0	0
$467$	15.7373	0.728234	0.364117	−	0.931353i	$-0.381371\pi$
$467$	15.7373	0.728234	0.364117	+	0.931353i	$0.381371\pi$
$468$	0	0
$469$	0.230288	0.0106337
$470$	0	0
$471$	8.71347	0.401496
$472$	0	0
$473$	17.3988	0.799998
$474$	0	0
$475$	−26.8387	−1.23145
$476$	0	0
$477$	13.1074	0.600145
$478$	0	0
$479$	−11.3495	−0.518573	−0.259286	−	0.965800i	$-0.583487\pi$
$479$	−11.3495	−0.518573	−0.259286	+	0.965800i	$0.583487\pi$
$480$	0	0
$481$	−2.12492	−0.0968880
$482$	0	0
$483$	0.409136	0.0186163
$484$	0	0
$485$	0.778760	0.0353617
$486$	0	0
$487$	2.67846	0.121373	0.0606863	−	0.998157i	$-0.480671\pi$
$487$	2.67846	0.121373	0.0606863	+	0.998157i	$0.480671\pi$
$488$	0	0
$489$	15.7742	0.713335
$490$	0	0
$491$	30.9044	1.39470	0.697349	−	0.716732i	$-0.254363\pi$
$491$	30.9044	1.39470	0.697349	+	0.716732i	$0.254363\pi$
$492$	0	0
$493$	−4.23453	−0.190713
$494$	0	0
$495$	0.861661	0.0387288
$496$	0	0
$497$	−2.88291	−0.129316
$498$	0	0
$499$	−23.6171	−1.05725	−0.528623	−	0.848857i	$-0.677291\pi$
$499$	−23.6171	−1.05725	−0.528623	+	0.848857i	$0.677291\pi$
$500$	0	0
$501$	−19.0619	−0.851621
$502$	0	0
$503$	38.8329	1.73147	0.865736	−	0.500501i	$-0.166851\pi$
$503$	38.8329	1.73147	0.865736	+	0.500501i	$0.166851\pi$
$504$	0	0
$505$	−3.71758	−0.165430
$506$	0	0
$507$	−12.8539	−0.570860
$508$	0	0
$509$	−13.6346	−0.604342	−0.302171	−	0.953254i	$-0.597711\pi$
$509$	−13.6346	−0.604342	−0.302171	+	0.953254i	$0.597711\pi$
$510$	0	0
$511$	−4.48632	−0.198463
$512$	0	0
$513$	5.44767	0.240520
$514$	0	0
$515$	3.25359	0.143370
$516$	0	0
$517$	37.1341	1.63316
$518$	0	0
$519$	2.35020	0.103162
$520$	0	0
$521$	5.43489	0.238107	0.119053	−	0.992888i	$-0.462014\pi$
$521$	5.43489	0.238107	0.119053	+	0.992888i	$0.462014\pi$
$522$	0	0
$523$	23.5508	1.02980	0.514902	−	0.857249i	$-0.327828\pi$
$523$	23.5508	1.02980	0.514902	+	0.857249i	$0.327828\pi$
$524$	0	0
$525$	−2.01567	−0.0879710
$526$	0	0
$527$	−8.81046	−0.383790
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.76314	0.206703
$532$	0	0
$533$	2.70457	0.117148
$534$	0	0
$535$	0.258559	0.0111785
$536$	0	0
$537$	−6.80243	−0.293546
$538$	0	0
$539$	21.7375	0.936302
$540$	0	0
$541$	18.7530	0.806254	0.403127	−	0.915144i	$-0.367923\pi$
$541$	18.7530	0.806254	0.403127	+	0.915144i	$0.367923\pi$
$542$	0	0
$543$	−8.35756	−0.358657
$544$	0	0
$545$	2.66793	0.114282
$546$	0	0
$547$	−16.6602	−0.712338	−0.356169	−	0.934421i	$-0.615917\pi$
$547$	−16.6602	−0.712338	−0.356169	+	0.934421i	$0.615917\pi$
$548$	0	0
$549$	−2.32894	−0.0993966
$550$	0	0
$551$	5.44767	0.232079
$552$	0	0
$553$	−6.83239	−0.290543
$554$	0	0
$555$	1.50546	0.0639032
$556$	0	0
$557$	9.73836	0.412627	0.206314	−	0.978486i	$-0.433853\pi$
$557$	9.73836	0.412627	0.206314	+	0.978486i	$0.433853\pi$
$558$	0	0
$559$	−2.09066	−0.0884253
$560$	0	0
$561$	13.4719	0.568784
$562$	0	0
$563$	−9.64200	−0.406362	−0.203181	−	0.979141i	$-0.565128\pi$
$563$	−9.64200	−0.406362	−0.203181	+	0.979141i	$0.565128\pi$
$564$	0	0
$565$	−0.960786	−0.0404206
$566$	0	0
$567$	0.409136	0.0171821
$568$	0	0
$569$	19.0987	0.800659	0.400329	−	0.916371i	$-0.368896\pi$
$569$	19.0987	0.800659	0.400329	+	0.916371i	$0.368896\pi$
$570$	0	0
$571$	−11.8736	−0.496896	−0.248448	−	0.968645i	$-0.579920\pi$
$571$	−11.8736	−0.496896	−0.248448	+	0.968645i	$0.579920\pi$
$572$	0	0
$573$	16.5128	0.689833
$574$	0	0
$575$	−4.92665	−0.205455
$576$	0	0
$577$	−11.2776	−0.469493	−0.234746	−	0.972057i	$-0.575426\pi$
$577$	−11.2776	−0.469493	−0.234746	+	0.972057i	$0.575426\pi$
$578$	0	0
$579$	−20.6858	−0.859674
$580$	0	0
$581$	3.86452	0.160327
$582$	0	0
$583$	−41.7003	−1.72705
$584$	0	0
$585$	−0.103538	−0.00428076
$586$	0	0
$587$	−0.805937	−0.0332646	−0.0166323	−	0.999862i	$-0.505294\pi$
$587$	−0.805937	−0.0332646	−0.0166323	+	0.999862i	$0.505294\pi$
$588$	0	0
$589$	11.3346	0.467032
$590$	0	0
$591$	6.56222	0.269934
$592$	0	0
$593$	−4.45928	−0.183121	−0.0915603	−	0.995800i	$-0.529185\pi$
$593$	−4.45928	−0.183121	−0.0915603	+	0.995800i	$0.529185\pi$
$594$	0	0
$595$	0.469230	0.0192365
$596$	0	0
$597$	0.673220	0.0275530
$598$	0	0
$599$	−13.7095	−0.560155	−0.280078	−	0.959977i	$-0.590360\pi$
$599$	−13.7095	−0.560155	−0.280078	+	0.959977i	$0.590360\pi$
$600$	0	0
$601$	−2.45512	−0.100146	−0.0500731	−	0.998746i	$-0.515945\pi$
$601$	−2.45512	−0.100146	−0.0500731	+	0.998746i	$0.515945\pi$
$602$	0	0
$603$	0.562865	0.0229216
$604$	0	0
$605$	0.237917	0.00967271
$606$	0	0
$607$	2.83771	0.115179	0.0575895	−	0.998340i	$-0.481659\pi$
$607$	2.83771	0.115179	0.0575895	+	0.998340i	$0.481659\pi$
$608$	0	0
$609$	0.409136	0.0165790
$610$	0	0
$611$	−4.46207	−0.180516
$612$	0	0
$613$	30.1424	1.21744	0.608720	−	0.793385i	$-0.291683\pi$
$613$	30.1424	1.21744	0.608720	+	0.793385i	$0.291683\pi$
$614$	0	0
$615$	−1.91613	−0.0772657
$616$	0	0
$617$	20.1814	0.812474	0.406237	−	0.913768i	$-0.366841\pi$
$617$	20.1814	0.812474	0.406237	+	0.913768i	$0.366841\pi$
$618$	0	0
$619$	−34.0792	−1.36976	−0.684880	−	0.728656i	$-0.740145\pi$
$619$	−34.0792	−1.36976	−0.684880	+	0.728656i	$0.740145\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	3.57709	0.143313
$624$	0	0
$625$	23.9051	0.956203
$626$	0	0
$627$	−17.3314	−0.692151
$628$	0	0
$629$	23.5375	0.938503
$630$	0	0
$631$	−0.593809	−0.0236392	−0.0118196	−	0.999930i	$-0.503762\pi$
$631$	−0.593809	−0.0236392	−0.0118196	+	0.999930i	$0.503762\pi$
$632$	0	0
$633$	−24.8206	−0.986528
$634$	0	0
$635$	−0.597028	−0.0236923
$636$	0	0
$637$	−2.61200	−0.103491
$638$	0	0
$639$	−7.04632	−0.278748
$640$	0	0
$641$	14.9940	0.592225	0.296113	−	0.955153i	$-0.404310\pi$
$641$	14.9940	0.592225	0.296113	+	0.955153i	$0.404310\pi$
$642$	0	0
$643$	−47.2413	−1.86302	−0.931508	−	0.363720i	$-0.881506\pi$
$643$	−47.2413	−1.86302	−0.931508	+	0.363720i	$0.881506\pi$
$644$	0	0
$645$	1.48118	0.0583215
$646$	0	0
$647$	−5.46767	−0.214956	−0.107478	−	0.994207i	$-0.534278\pi$
$647$	−5.46767	−0.214956	−0.107478	+	0.994207i	$0.534278\pi$
$648$	0	0
$649$	−15.1536	−0.594832
$650$	0	0
$651$	0.851259	0.0333635
$652$	0	0
$653$	−24.7535	−0.968679	−0.484339	−	0.874880i	$-0.660940\pi$
$653$	−24.7535	−0.968679	−0.484339	+	0.874880i	$0.660940\pi$
$654$	0	0
$655$	3.14888	0.123037
$656$	0	0
$657$	−10.9653	−0.427799
$658$	0	0
$659$	−18.2057	−0.709192	−0.354596	−	0.935020i	$-0.615382\pi$
$659$	−18.2057	−0.709192	−0.354596	+	0.935020i	$0.615382\pi$
$660$	0	0
$661$	46.9906	1.82772	0.913861	−	0.406028i	$-0.133087\pi$
$661$	46.9906	1.82772	0.913861	+	0.406028i	$0.133087\pi$
$662$	0	0
$663$	−1.61879	−0.0628687
$664$	0	0
$665$	−0.603659	−0.0234089
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	−27.7648	−1.07345
$670$	0	0
$671$	7.40937	0.286036
$672$	0	0
$673$	−34.6358	−1.33511	−0.667556	−	0.744559i	$-0.732660\pi$
$673$	−34.6358	−1.33511	−0.667556	+	0.744559i	$0.732660\pi$
$674$	0	0
$675$	−4.92665	−0.189627
$676$	0	0
$677$	−24.6121	−0.945920	−0.472960	−	0.881084i	$-0.656815\pi$
$677$	−24.6121	−0.945920	−0.472960	+	0.881084i	$0.656815\pi$
$678$	0	0
$679$	−1.17641	−0.0451465
$680$	0	0
$681$	−12.5430	−0.480650
$682$	0	0
$683$	−28.3828	−1.08604	−0.543019	−	0.839720i	$-0.682719\pi$
$683$	−28.3828	−1.08604	−0.543019	+	0.839720i	$0.682719\pi$
$684$	0	0
$685$	2.16943	0.0828895
$686$	0	0
$687$	−12.0808	−0.460911
$688$	0	0
$689$	5.01074	0.190894
$690$	0	0
$691$	8.08477	0.307559	0.153780	−	0.988105i	$-0.450855\pi$
$691$	8.08477	0.307559	0.153780	+	0.988105i	$0.450855\pi$
$692$	0	0
$693$	−1.30164	−0.0494453
$694$	0	0
$695$	4.54602	0.172440
$696$	0	0
$697$	−29.9583	−1.13475
$698$	0	0
$699$	7.64163	0.289033
$700$	0	0
$701$	−18.1267	−0.684637	−0.342318	−	0.939584i	$-0.611212\pi$
$701$	−18.1267	−0.684637	−0.342318	+	0.939584i	$0.611212\pi$
$702$	0	0
$703$	−30.2808	−1.14206
$704$	0	0
$705$	3.16128	0.119061
$706$	0	0
$707$	5.61585	0.211206
$708$	0	0
$709$	−14.4856	−0.544018	−0.272009	−	0.962295i	$-0.587688\pi$
$709$	−14.4856	−0.544018	−0.272009	+	0.962295i	$0.587688\pi$
$710$	0	0
$711$	−16.6996	−0.626282
$712$	0	0
$713$	2.08063	0.0779200
$714$	0	0
$715$	0.329400	0.0123188
$716$	0	0
$717$	10.0402	0.374957
$718$	0	0
$719$	19.6488	0.732778	0.366389	−	0.930462i	$-0.380594\pi$
$719$	19.6488	0.732778	0.366389	+	0.930462i	$0.380594\pi$
$720$	0	0
$721$	−4.91494	−0.183042
$722$	0	0
$723$	2.88233	0.107195
$724$	0	0
$725$	−4.92665	−0.182971
$726$	0	0
$727$	−14.6587	−0.543659	−0.271830	−	0.962345i	$-0.587629\pi$
$727$	−14.6587	−0.543659	−0.271830	+	0.962345i	$0.587629\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	23.1580	0.856529
$732$	0	0
$733$	−14.9313	−0.551499	−0.275749	−	0.961230i	$-0.588926\pi$
$733$	−14.9313	−0.551499	−0.275749	+	0.961230i	$0.588926\pi$
$734$	0	0
$735$	1.85054	0.0682583
$736$	0	0
$737$	−1.79072	−0.0659620
$738$	0	0
$739$	−17.8041	−0.654935	−0.327468	−	0.944862i	$-0.606195\pi$
$739$	−17.8041	−0.654935	−0.327468	+	0.944862i	$0.606195\pi$
$740$	0	0
$741$	2.08256	0.0765047
$742$	0	0
$743$	28.7026	1.05300	0.526498	−	0.850177i	$-0.323505\pi$
$743$	28.7026	1.05300	0.526498	+	0.850177i	$0.323505\pi$
$744$	0	0
$745$	4.49029	0.164511
$746$	0	0
$747$	9.44555	0.345595
$748$	0	0
$749$	−0.390584	−0.0142716
$750$	0	0
$751$	43.6902	1.59428	0.797139	−	0.603796i	$-0.206346\pi$
$751$	43.6902	1.59428	0.797139	+	0.603796i	$0.206346\pi$
$752$	0	0
$753$	17.4367	0.635429
$754$	0	0
$755$	−3.22127	−0.117234
$756$	0	0
$757$	4.34039	0.157754	0.0788771	−	0.996884i	$-0.474867\pi$
$757$	4.34039	0.157754	0.0788771	+	0.996884i	$0.474867\pi$
$758$	0	0
$759$	−3.18144	−0.115479
$760$	0	0
$761$	−23.0433	−0.835319	−0.417660	−	0.908604i	$-0.637150\pi$
$761$	−23.0433	−0.835319	−0.417660	+	0.908604i	$0.637150\pi$
$762$	0	0
$763$	−4.03023	−0.145904
$764$	0	0
$765$	1.14688	0.0414655
$766$	0	0
$767$	1.82087	0.0657479
$768$	0	0
$769$	17.2849	0.623310	0.311655	−	0.950195i	$-0.399117\pi$
$769$	17.2849	0.623310	0.311655	+	0.950195i	$0.399117\pi$
$770$	0	0
$771$	−23.8712	−0.859700
$772$	0	0
$773$	49.4741	1.77946	0.889729	−	0.456489i	$-0.150893\pi$
$773$	49.4741	1.77946	0.889729	+	0.456489i	$0.150893\pi$
$774$	0	0
$775$	−10.2505	−0.368209
$776$	0	0
$777$	−2.27418	−0.0815856
$778$	0	0
$779$	38.5410	1.38087
$780$	0	0
$781$	22.4174	0.802159
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	−2.35996	−0.0842305
$786$	0	0
$787$	22.9575	0.818345	0.409173	−	0.912457i	$-0.365817\pi$
$787$	22.9575	0.818345	0.409173	+	0.912457i	$0.365817\pi$
$788$	0	0
$789$	13.4132	0.477521
$790$	0	0
$791$	1.45138	0.0516052
$792$	0	0
$793$	−0.890316	−0.0316160
$794$	0	0
$795$	−3.55000	−0.125905
$796$	0	0
$797$	47.6929	1.68937	0.844684	−	0.535265i	$-0.179788\pi$
$797$	47.6929	1.68937	0.844684	+	0.535265i	$0.179788\pi$
$798$	0	0
$799$	49.4259	1.74856
$800$	0	0
$801$	8.74304	0.308920
$802$	0	0
$803$	34.8856	1.23108
$804$	0	0
$805$	−0.110810	−0.00390556
$806$	0	0
$807$	−1.27424	−0.0448552
$808$	0	0
$809$	−11.4679	−0.403191	−0.201596	−	0.979469i	$-0.564613\pi$
$809$	−11.4679	−0.403191	−0.201596	+	0.979469i	$0.564613\pi$
$810$	0	0
$811$	−19.4479	−0.682909	−0.341454	−	0.939898i	$-0.610919\pi$
$811$	−19.4479	−0.682909	−0.341454	+	0.939898i	$0.610919\pi$
$812$	0	0
$813$	8.32920	0.292118
$814$	0	0
$815$	−4.27229	−0.149652
$816$	0	0
$817$	−29.7925	−1.04231
$818$	0	0
$819$	0.156406	0.00546528
$820$	0	0
$821$	54.2605	1.89370	0.946852	−	0.321670i	$-0.104244\pi$
$821$	54.2605	1.89370	0.946852	+	0.321670i	$0.104244\pi$
$822$	0	0
$823$	12.7804	0.445496	0.222748	−	0.974876i	$-0.428497\pi$
$823$	12.7804	0.445496	0.222748	+	0.974876i	$0.428497\pi$
$824$	0	0
$825$	15.6738	0.545693
$826$	0	0
$827$	−9.84339	−0.342288	−0.171144	−	0.985246i	$-0.554746\pi$
$827$	−9.84339	−0.342288	−0.171144	+	0.985246i	$0.554746\pi$
$828$	0	0
$829$	−22.9313	−0.796437	−0.398219	−	0.917291i	$-0.630371\pi$
$829$	−22.9313	−0.796437	−0.398219	+	0.917291i	$0.630371\pi$
$830$	0	0
$831$	12.4631	0.432339
$832$	0	0
$833$	28.9329	1.00246
$834$	0	0
$835$	5.16271	0.178663
$836$	0	0
$837$	2.08063	0.0719169
$838$	0	0
$839$	−12.2808	−0.423980	−0.211990	−	0.977272i	$-0.567994\pi$
$839$	−12.2808	−0.423980	−0.211990	+	0.977272i	$0.567994\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	4.79090	0.165007
$844$	0	0
$845$	3.48134	0.119762
$846$	0	0
$847$	−0.359403	−0.0123492
$848$	0	0
$849$	8.47563	0.290883
$850$	0	0
$851$	−5.55848	−0.190542
$852$	0	0
$853$	20.4262	0.699380	0.349690	−	0.936865i	$-0.386287\pi$
$853$	20.4262	0.699380	0.349690	+	0.936865i	$0.386287\pi$
$854$	0	0
$855$	−1.47545	−0.0504592
$856$	0	0
$857$	50.6690	1.73082	0.865410	−	0.501065i	$-0.167058\pi$
$857$	50.6690	1.73082	0.865410	+	0.501065i	$0.167058\pi$
$858$	0	0
$859$	20.0628	0.684532	0.342266	−	0.939603i	$-0.388806\pi$
$859$	20.0628	0.684532	0.342266	+	0.939603i	$0.388806\pi$
$860$	0	0
$861$	2.89454	0.0986457
$862$	0	0
$863$	−18.6708	−0.635561	−0.317780	−	0.948164i	$-0.602938\pi$
$863$	−18.6708	−0.635561	−0.317780	+	0.948164i	$0.602938\pi$
$864$	0	0
$865$	−0.636527	−0.0216426
$866$	0	0
$867$	0.931210	0.0316255
$868$	0	0
$869$	53.1286	1.80227
$870$	0	0
$871$	0.215174	0.00729090
$872$	0	0
$873$	−2.87535	−0.0973159
$874$	0	0
$875$	1.09998	0.0371860
$876$	0	0
$877$	−41.1010	−1.38788	−0.693941	−	0.720032i	$-0.744127\pi$
$877$	−41.1010	−1.38788	−0.693941	+	0.720032i	$0.744127\pi$
$878$	0	0
$879$	−25.4167	−0.857285
$880$	0	0
$881$	−19.2512	−0.648590	−0.324295	−	0.945956i	$-0.605127\pi$
$881$	−19.2512	−0.648590	−0.324295	+	0.945956i	$0.605127\pi$
$882$	0	0
$883$	33.2113	1.11765	0.558825	−	0.829286i	$-0.311252\pi$
$883$	33.2113	1.11765	0.558825	+	0.829286i	$0.311252\pi$
$884$	0	0
$885$	−1.29005	−0.0433645
$886$	0	0
$887$	−21.2053	−0.712005	−0.356002	−	0.934485i	$-0.615860\pi$
$887$	−21.2053	−0.712005	−0.356002	+	0.934485i	$0.615860\pi$
$888$	0	0
$889$	0.901881	0.0302481
$890$	0	0
$891$	−3.18144	−0.106582
$892$	0	0
$893$	−63.5858	−2.12782
$894$	0	0
$895$	1.84237	0.0615836
$896$	0	0
$897$	0.382284	0.0127641
$898$	0	0
$899$	2.08063	0.0693928
$900$	0	0
$901$	−55.5035	−1.84909
$902$	0	0
$903$	−2.23750	−0.0744595
$904$	0	0
$905$	2.26356	0.0752433
$906$	0	0
$907$	−12.8168	−0.425575	−0.212788	−	0.977098i	$-0.568254\pi$
$907$	−12.8168	−0.425575	−0.212788	+	0.977098i	$0.568254\pi$
$908$	0	0
$909$	13.7261	0.455266
$910$	0	0
$911$	16.5644	0.548802	0.274401	−	0.961615i	$-0.411520\pi$
$911$	16.5644	0.548802	0.274401	+	0.961615i	$0.411520\pi$
$912$	0	0
$913$	−30.0504	−0.994525
$914$	0	0
$915$	0.630769	0.0208526
$916$	0	0
$917$	−4.75676	−0.157082
$918$	0	0
$919$	18.7651	0.619005	0.309503	−	0.950899i	$-0.399837\pi$
$919$	18.7651	0.619005	0.309503	+	0.950899i	$0.399837\pi$
$920$	0	0
$921$	30.3855	1.00124
$922$	0	0
$923$	−2.69370	−0.0886642
$924$	0	0
$925$	27.3847	0.900402
$926$	0	0
$927$	−12.0130	−0.394558
$928$	0	0
$929$	−18.0666	−0.592747	−0.296373	−	0.955072i	$-0.595777\pi$
$929$	−18.0666	−0.592747	−0.296373	+	0.955072i	$0.595777\pi$
$930$	0	0
$931$	−37.2218	−1.21989
$932$	0	0
$933$	−27.4460	−0.898540
$934$	0	0
$935$	−3.64873	−0.119326
$936$	0	0
$937$	58.5144	1.91158	0.955792	−	0.294044i	$-0.0950012\pi$
$937$	58.5144	1.91158	0.955792	+	0.294044i	$0.0950012\pi$
$938$	0	0
$939$	13.1030	0.427600
$940$	0	0
$941$	−46.9778	−1.53143	−0.765716	−	0.643178i	$-0.777615\pi$
$941$	−46.9778	−1.53143	−0.765716	+	0.643178i	$0.777615\pi$
$942$	0	0
$943$	7.07476	0.230386
$944$	0	0
$945$	−0.110810	−0.00360466
$946$	0	0
$947$	16.2887	0.529310	0.264655	−	0.964343i	$-0.414742\pi$
$947$	16.2887	0.529310	0.264655	+	0.964343i	$0.414742\pi$
$948$	0	0
$949$	−4.19188	−0.136074
$950$	0	0
$951$	28.8773	0.936411
$952$	0	0
$953$	−48.2390	−1.56261	−0.781307	−	0.624147i	$-0.785447\pi$
$953$	−48.2390	−1.56261	−0.781307	+	0.624147i	$0.785447\pi$
$954$	0	0
$955$	−4.47233	−0.144721
$956$	0	0
$957$	−3.18144	−0.102841
$958$	0	0
$959$	−3.27718	−0.105826
$960$	0	0
$961$	−26.6710	−0.860355
$962$	0	0
$963$	−0.954655	−0.0307633
$964$	0	0
$965$	5.60255	0.180352
$966$	0	0
$967$	−13.1024	−0.421345	−0.210672	−	0.977557i	$-0.567565\pi$
$967$	−13.1024	−0.421345	−0.210672	+	0.977557i	$0.567565\pi$
$968$	0	0
$969$	−23.0683	−0.741060
$970$	0	0
$971$	18.6526	0.598592	0.299296	−	0.954160i	$-0.403248\pi$
$971$	18.6526	0.598592	0.299296	+	0.954160i	$0.403248\pi$
$972$	0	0
$973$	−6.86731	−0.220156
$974$	0	0
$975$	−1.88338	−0.0603164
$976$	0	0
$977$	25.1509	0.804649	0.402325	−	0.915497i	$-0.368202\pi$
$977$	25.1509	0.804649	0.402325	+	0.915497i	$0.368202\pi$
$978$	0	0
$979$	−27.8154	−0.888985
$980$	0	0
$981$	−9.85059	−0.314505
$982$	0	0
$983$	10.4413	0.333025	0.166512	−	0.986039i	$-0.446749\pi$
$983$	10.4413	0.333025	0.166512	+	0.986039i	$0.446749\pi$
$984$	0	0
$985$	−1.77731	−0.0566299
$986$	0	0
$987$	−4.77548	−0.152005
$988$	0	0
$989$	−5.46885	−0.173899
$990$	0	0
$991$	46.4100	1.47426	0.737131	−	0.675750i	$-0.236180\pi$
$991$	46.4100	1.47426	0.737131	+	0.675750i	$0.236180\pi$
$992$	0	0
$993$	9.77951	0.310343
$994$	0	0
$995$	−0.182335	−0.00578040
$996$	0	0
$997$	−39.3398	−1.24590	−0.622951	−	0.782261i	$-0.714067\pi$
$997$	−39.3398	−1.24590	−0.622951	+	0.782261i	$0.714067\pi$
$998$	0	0
$999$	−5.55848	−0.175863

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.d.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.d.1.5	✓	8	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:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.270840 q^{5} +0.409136 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.270840 q^{5} +0.409136 q^{7} +1.00000 q^{9} -3.18144 q^{11} +0.382284 q^{13} -0.270840 q^{15} -4.23453 q^{17} +5.44767 q^{19} +0.409136 q^{21} +1.00000 q^{23} -4.92665 q^{25} +1.00000 q^{27} +1.00000 q^{29} +2.08063 q^{31} -3.18144 q^{33} -0.110810 q^{35} -5.55848 q^{37} +0.382284 q^{39} +7.07476 q^{41} -5.46885 q^{43} -0.270840 q^{45} -11.6721 q^{47} -6.83261 q^{49} -4.23453 q^{51} +13.1074 q^{53} +0.861661 q^{55} +5.44767 q^{57} +4.76314 q^{59} -2.32894 q^{61} +0.409136 q^{63} -0.103538 q^{65} +0.562865 q^{67} +1.00000 q^{69} -7.04632 q^{71} -10.9653 q^{73} -4.92665 q^{75} -1.30164 q^{77} -16.6996 q^{79} +1.00000 q^{81} +9.44555 q^{83} +1.14688 q^{85} +1.00000 q^{87} +8.74304 q^{89} +0.156406 q^{91} +2.08063 q^{93} -1.47545 q^{95} -2.87535 q^{97} -3.18144 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9	\( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9} - 5 q^{11} - 4 q^{13} - 5 q^{15} - 3 q^{17} - 5 q^{19} - 4 q^{21} + 8 q^{23} - 5 q^{25} + 8 q^{27} + 8 q^{29} - 2 q^{31} - 5 q^{33} - 15 q^{35} - 10 q^{37} - 4 q^{39} - 11 q^{41} - 7 q^{43} - 5 q^{45} - 14 q^{47} - 18 q^{49} - 3 q^{51} - 15 q^{53} - 17 q^{55} - 5 q^{57} + 4 q^{59} + q^{61} - 4 q^{63} - 5 q^{67} + 8 q^{69} - q^{71} - 21 q^{73} - 5 q^{75} - 8 q^{79} + 8 q^{81} + 3 q^{83} + 8 q^{87} - 20 q^{89} - 7 q^{91} - 2 q^{93} - 3 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 - 5 * q^5 - 4 * q^7 + 8 * q^9 - 5 * q^11 - 4 * q^13 - 5 * q^15 - 3 * q^17 - 5 * q^19 - 4 * q^21 + 8 * q^23 - 5 * q^25 + 8 * q^27 + 8 * q^29 - 2 * q^31 - 5 * q^33 - 15 * q^35 - 10 * q^37 - 4 * q^39 - 11 * q^41 - 7 * q^43 - 5 * q^45 - 14 * q^47 - 18 * q^49 - 3 * q^51 - 15 * q^53 - 17 * q^55 - 5 * q^57 + 4 * q^59 + q^61 - 4 * q^63 - 5 * q^67 + 8 * q^69 - q^71 - 21 * q^73 - 5 * q^75 - 8 * q^79 + 8 * q^81 + 3 * q^83 + 8 * q^87 - 20 * q^89 - 7 * q^91 - 2 * q^93 - 3 * q^95 - 7 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more