LMFDB - Embedded newform 8004.2.a.i.1.16

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:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} + \cdots - 208 $ x^16 - 5x^15 - 43x^14 + 234x^13 + 634x^12 - 4048x^11 - 3483x^10 + 32512x^9 - 137x^8 - 121665x^7 + 60168x^6 + 172218x^5 - 138024x^4 - 17844x^3 + 14352x^2 + 72*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Root			$4.15437$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +4.15437 q^{5} -4.14427 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.15437 q^{5} -4.14427 q^{7} +1.00000 q^{9} +0.229135 q^{11} -4.10323 q^{13} -4.15437 q^{15} +6.38557 q^{17} -2.89203 q^{19} +4.14427 q^{21} +1.00000 q^{23} +12.2588 q^{25} -1.00000 q^{27} +1.00000 q^{29} +7.43114 q^{31} -0.229135 q^{33} -17.2168 q^{35} -6.31719 q^{37} +4.10323 q^{39} +5.76434 q^{41} +0.993037 q^{43} +4.15437 q^{45} -6.88898 q^{47} +10.1750 q^{49} -6.38557 q^{51} +6.39600 q^{53} +0.951911 q^{55} +2.89203 q^{57} -3.98312 q^{59} -4.98317 q^{61} -4.14427 q^{63} -17.0464 q^{65} +2.09792 q^{67} -1.00000 q^{69} -13.7993 q^{71} -13.8931 q^{73} -12.2588 q^{75} -0.949596 q^{77} +12.1000 q^{79} +1.00000 q^{81} +9.00171 q^{83} +26.5280 q^{85} -1.00000 q^{87} +7.22544 q^{89} +17.0049 q^{91} -7.43114 q^{93} -12.0146 q^{95} +2.29533 q^{97} +0.229135 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	$ 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) $ 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * 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
$5$	4.15437	1.85789	0.928946	−	0.370216i	$-0.120716\pi$
$5$	4.15437	1.85789	0.928946	+	0.370216i	$0.120716\pi$
$6$	0	0
$7$	−4.14427	−1.56639	−0.783193	−	0.621778i	$-0.786410\pi$
$7$	−4.14427	−1.56639	−0.783193	+	0.621778i	$0.786410\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.229135	0.0690867	0.0345434	−	0.999403i	$-0.489002\pi$
$11$	0.229135	0.0690867	0.0345434	+	0.999403i	$0.489002\pi$
$12$	0	0
$13$	−4.10323	−1.13803	−0.569016	−	0.822326i	$-0.692676\pi$
$13$	−4.10323	−1.13803	−0.569016	+	0.822326i	$0.692676\pi$
$14$	0	0
$15$	−4.15437	−1.07265
$16$	0	0
$17$	6.38557	1.54873	0.774364	−	0.632741i	$-0.218070\pi$
$17$	6.38557	1.54873	0.774364	+	0.632741i	$0.218070\pi$
$18$	0	0
$19$	−2.89203	−0.663478	−0.331739	−	0.943371i	$-0.607635\pi$
$19$	−2.89203	−0.663478	−0.331739	+	0.943371i	$0.607635\pi$
$20$	0	0
$21$	4.14427	0.904354
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	12.2588	2.45176
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	7.43114	1.33467	0.667336	−	0.744757i	$-0.267435\pi$
$31$	7.43114	1.33467	0.667336	+	0.744757i	$0.267435\pi$
$32$	0	0
$33$	−0.229135	−0.0398872
$34$	0	0
$35$	−17.2168	−2.91018
$36$	0	0
$37$	−6.31719	−1.03854	−0.519270	−	0.854610i	$-0.673796\pi$
$37$	−6.31719	−1.03854	−0.519270	+	0.854610i	$0.673796\pi$
$38$	0	0
$39$	4.10323	0.657043
$40$	0	0
$41$	5.76434	0.900239	0.450119	−	0.892968i	$-0.351381\pi$
$41$	5.76434	0.900239	0.450119	+	0.892968i	$0.351381\pi$
$42$	0	0
$43$	0.993037	0.151437	0.0757183	−	0.997129i	$-0.475875\pi$
$43$	0.993037	0.151437	0.0757183	+	0.997129i	$0.475875\pi$
$44$	0	0
$45$	4.15437	0.619297
$46$	0	0
$47$	−6.88898	−1.00486	−0.502430	−	0.864618i	$-0.667561\pi$
$47$	−6.88898	−1.00486	−0.502430	+	0.864618i	$0.667561\pi$
$48$	0	0
$49$	10.1750	1.45357
$50$	0	0
$51$	−6.38557	−0.894158
$52$	0	0
$53$	6.39600	0.878558	0.439279	−	0.898351i	$-0.355234\pi$
$53$	6.39600	0.878558	0.439279	+	0.898351i	$0.355234\pi$
$54$	0	0
$55$	0.951911	0.128356
$56$	0	0
$57$	2.89203	0.383059
$58$	0	0
$59$	−3.98312	−0.518558	−0.259279	−	0.965802i	$-0.583485\pi$
$59$	−3.98312	−0.518558	−0.259279	+	0.965802i	$0.583485\pi$
$60$	0	0
$61$	−4.98317	−0.638030	−0.319015	−	0.947750i	$-0.603352\pi$
$61$	−4.98317	−0.638030	−0.319015	+	0.947750i	$0.603352\pi$
$62$	0	0
$63$	−4.14427	−0.522129
$64$	0	0
$65$	−17.0464	−2.11434
$66$	0	0
$67$	2.09792	0.256302	0.128151	−	0.991755i	$-0.459096\pi$
$67$	2.09792	0.256302	0.128151	+	0.991755i	$0.459096\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−13.7993	−1.63768	−0.818840	−	0.574021i	$-0.805383\pi$
$71$	−13.7993	−1.63768	−0.818840	+	0.574021i	$0.805383\pi$
$72$	0	0
$73$	−13.8931	−1.62606	−0.813032	−	0.582219i	$-0.802185\pi$
$73$	−13.8931	−1.62606	−0.813032	+	0.582219i	$0.802185\pi$
$74$	0	0
$75$	−12.2588	−1.41552
$76$	0	0
$77$	−0.949596	−0.108216
$78$	0	0
$79$	12.1000	1.36135	0.680677	−	0.732584i	$-0.261686\pi$
$79$	12.1000	1.36135	0.680677	+	0.732584i	$0.261686\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.00171	0.988066	0.494033	−	0.869443i	$-0.335522\pi$
$83$	9.00171	0.988066	0.494033	+	0.869443i	$0.335522\pi$
$84$	0	0
$85$	26.5280	2.87737
$86$	0	0
$87$	−1.00000	−0.107211
$88$	0	0
$89$	7.22544	0.765895	0.382948	−	0.923770i	$-0.374909\pi$
$89$	7.22544	0.765895	0.382948	+	0.923770i	$0.374909\pi$
$90$	0	0
$91$	17.0049	1.78260
$92$	0	0
$93$	−7.43114	−0.770573
$94$	0	0
$95$	−12.0146	−1.23267
$96$	0	0
$97$	2.29533	0.233055	0.116528	−	0.993187i	$-0.462824\pi$
$97$	2.29533	0.233055	0.116528	+	0.993187i	$0.462824\pi$
$98$	0	0
$99$	0.229135	0.0230289
$100$	0	0
$101$	−1.84478	−0.183562	−0.0917812	−	0.995779i	$-0.529256\pi$
$101$	−1.84478	−0.183562	−0.0917812	+	0.995779i	$0.529256\pi$
$102$	0	0
$103$	−10.0687	−0.992095	−0.496048	−	0.868295i	$-0.665216\pi$
$103$	−10.0687	−0.992095	−0.496048	+	0.868295i	$0.665216\pi$
$104$	0	0
$105$	17.2168	1.68019
$106$	0	0
$107$	−2.97463	−0.287568	−0.143784	−	0.989609i	$-0.545927\pi$
$107$	−2.97463	−0.287568	−0.143784	+	0.989609i	$0.545927\pi$
$108$	0	0
$109$	17.2466	1.65193	0.825965	−	0.563722i	$-0.190631\pi$
$109$	17.2466	1.65193	0.825965	+	0.563722i	$0.190631\pi$
$110$	0	0
$111$	6.31719	0.599602
$112$	0	0
$113$	14.2563	1.34112	0.670561	−	0.741854i	$-0.266053\pi$
$113$	14.2563	1.34112	0.670561	+	0.741854i	$0.266053\pi$
$114$	0	0
$115$	4.15437	0.387397
$116$	0	0
$117$	−4.10323	−0.379344
$118$	0	0
$119$	−26.4635	−2.42591
$120$	0	0
$121$	−10.9475	−0.995227
$122$	0	0
$123$	−5.76434	−0.519753
$124$	0	0
$125$	30.1558	2.69721
$126$	0	0
$127$	8.12633	0.721095	0.360548	−	0.932741i	$-0.382590\pi$
$127$	8.12633	0.721095	0.360548	+	0.932741i	$0.382590\pi$
$128$	0	0
$129$	−0.993037	−0.0874320
$130$	0	0
$131$	−6.66793	−0.582579	−0.291290	−	0.956635i	$-0.594084\pi$
$131$	−6.66793	−0.582579	−0.291290	+	0.956635i	$0.594084\pi$
$132$	0	0
$133$	11.9854	1.03926
$134$	0	0
$135$	−4.15437	−0.357551
$136$	0	0
$137$	8.92799	0.762770	0.381385	−	0.924416i	$-0.375447\pi$
$137$	8.92799	0.762770	0.381385	+	0.924416i	$0.375447\pi$
$138$	0	0
$139$	15.6891	1.33073	0.665365	−	0.746518i	$-0.268276\pi$
$139$	15.6891	1.33073	0.665365	+	0.746518i	$0.268276\pi$
$140$	0	0
$141$	6.88898	0.580156
$142$	0	0
$143$	−0.940193	−0.0786229
$144$	0	0
$145$	4.15437	0.345002
$146$	0	0
$147$	−10.1750	−0.839217
$148$	0	0
$149$	−21.2764	−1.74303	−0.871515	−	0.490369i	$-0.836862\pi$
$149$	−21.2764	−1.74303	−0.871515	+	0.490369i	$0.836862\pi$
$150$	0	0
$151$	16.0301	1.30451	0.652254	−	0.758000i	$-0.273823\pi$
$151$	16.0301	1.30451	0.652254	+	0.758000i	$0.273823\pi$
$152$	0	0
$153$	6.38557	0.516242
$154$	0	0
$155$	30.8717	2.47967
$156$	0	0
$157$	21.8133	1.74089	0.870447	−	0.492262i	$-0.163830\pi$
$157$	21.8133	1.74089	0.870447	+	0.492262i	$0.163830\pi$
$158$	0	0
$159$	−6.39600	−0.507236
$160$	0	0
$161$	−4.14427	−0.326614
$162$	0	0
$163$	20.1201	1.57593	0.787965	−	0.615719i	$-0.211135\pi$
$163$	20.1201	1.57593	0.787965	+	0.615719i	$0.211135\pi$
$164$	0	0
$165$	−0.951911	−0.0741061
$166$	0	0
$167$	−14.8101	−1.14604	−0.573018	−	0.819543i	$-0.694228\pi$
$167$	−14.8101	−1.14604	−0.573018	+	0.819543i	$0.694228\pi$
$168$	0	0
$169$	3.83652	0.295117
$170$	0	0
$171$	−2.89203	−0.221159
$172$	0	0
$173$	−3.05807	−0.232501	−0.116250	−	0.993220i	$-0.537087\pi$
$173$	−3.05807	−0.232501	−0.116250	+	0.993220i	$0.537087\pi$
$174$	0	0
$175$	−50.8038	−3.84040
$176$	0	0
$177$	3.98312	0.299390
$178$	0	0
$179$	−6.09443	−0.455519	−0.227759	−	0.973717i	$-0.573140\pi$
$179$	−6.09443	−0.455519	−0.227759	+	0.973717i	$0.573140\pi$
$180$	0	0
$181$	24.1648	1.79616	0.898079	−	0.439834i	$-0.144963\pi$
$181$	24.1648	1.79616	0.898079	+	0.439834i	$0.144963\pi$
$182$	0	0
$183$	4.98317	0.368367
$184$	0	0
$185$	−26.2440	−1.92950
$186$	0	0
$187$	1.46315	0.106996
$188$	0	0
$189$	4.14427	0.301451
$190$	0	0
$191$	4.15674	0.300771	0.150385	−	0.988627i	$-0.451949\pi$
$191$	4.15674	0.300771	0.150385	+	0.988627i	$0.451949\pi$
$192$	0	0
$193$	−11.5694	−0.832784	−0.416392	−	0.909185i	$-0.636706\pi$
$193$	−11.5694	−0.832784	−0.416392	+	0.909185i	$0.636706\pi$
$194$	0	0
$195$	17.0464	1.22071
$196$	0	0
$197$	26.6706	1.90020	0.950100	−	0.311947i	$-0.100981\pi$
$197$	26.6706	1.90020	0.950100	+	0.311947i	$0.100981\pi$
$198$	0	0
$199$	2.14916	0.152350	0.0761750	−	0.997094i	$-0.475729\pi$
$199$	2.14916	0.152350	0.0761750	+	0.997094i	$0.475729\pi$
$200$	0	0
$201$	−2.09792	−0.147976
$202$	0	0
$203$	−4.14427	−0.290871
$204$	0	0
$205$	23.9472	1.67255
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−0.662665	−0.0458375
$210$	0	0
$211$	25.3278	1.74364	0.871819	−	0.489828i	$-0.162940\pi$
$211$	25.3278	1.74364	0.871819	+	0.489828i	$0.162940\pi$
$212$	0	0
$213$	13.7993	0.945515
$214$	0	0
$215$	4.12544	0.281353
$216$	0	0
$217$	−30.7966	−2.09061
$218$	0	0
$219$	13.8931	0.938809
$220$	0	0
$221$	−26.2015	−1.76250
$222$	0	0
$223$	−2.20969	−0.147972	−0.0739858	−	0.997259i	$-0.523572\pi$
$223$	−2.20969	−0.147972	−0.0739858	+	0.997259i	$0.523572\pi$
$224$	0	0
$225$	12.2588	0.817253
$226$	0	0
$227$	2.88544	0.191514	0.0957568	−	0.995405i	$-0.469473\pi$
$227$	2.88544	0.191514	0.0957568	+	0.995405i	$0.469473\pi$
$228$	0	0
$229$	−21.1310	−1.39638	−0.698188	−	0.715915i	$-0.746010\pi$
$229$	−21.1310	−1.39638	−0.698188	+	0.715915i	$0.746010\pi$
$230$	0	0
$231$	0.949596	0.0624788
$232$	0	0
$233$	−14.1662	−0.928057	−0.464029	−	0.885820i	$-0.653597\pi$
$233$	−14.1662	−0.928057	−0.464029	+	0.885820i	$0.653597\pi$
$234$	0	0
$235$	−28.6194	−1.86692
$236$	0	0
$237$	−12.1000	−0.785978
$238$	0	0
$239$	19.0504	1.23227	0.616133	−	0.787642i	$-0.288698\pi$
$239$	19.0504	1.23227	0.616133	+	0.787642i	$0.288698\pi$
$240$	0	0
$241$	28.9548	1.86514	0.932570	−	0.360989i	$-0.117561\pi$
$241$	28.9548	1.86514	0.932570	+	0.360989i	$0.117561\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	42.2706	2.70057
$246$	0	0
$247$	11.8667	0.755059
$248$	0	0
$249$	−9.00171	−0.570460
$250$	0	0
$251$	13.5507	0.855313	0.427656	−	0.903941i	$-0.359339\pi$
$251$	13.5507	0.855313	0.427656	+	0.903941i	$0.359339\pi$
$252$	0	0
$253$	0.229135	0.0144056
$254$	0	0
$255$	−26.5280	−1.66125
$256$	0	0
$257$	1.47343	0.0919100	0.0459550	−	0.998944i	$-0.485367\pi$
$257$	1.47343	0.0919100	0.0459550	+	0.998944i	$0.485367\pi$
$258$	0	0
$259$	26.1802	1.62676
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	22.8721	1.41035	0.705176	−	0.709032i	$-0.250868\pi$
$263$	22.8721	1.41035	0.705176	+	0.709032i	$0.250868\pi$
$264$	0	0
$265$	26.5714	1.63227
$266$	0	0
$267$	−7.22544	−0.442190
$268$	0	0
$269$	25.0211	1.52556	0.762781	−	0.646657i	$-0.223833\pi$
$269$	25.0211	1.52556	0.762781	+	0.646657i	$0.223833\pi$
$270$	0	0
$271$	−2.80847	−0.170602	−0.0853012	−	0.996355i	$-0.527185\pi$
$271$	−2.80847	−0.170602	−0.0853012	+	0.996355i	$0.527185\pi$
$272$	0	0
$273$	−17.0049	−1.02918
$274$	0	0
$275$	2.80892	0.169384
$276$	0	0
$277$	−3.13857	−0.188578	−0.0942892	−	0.995545i	$-0.530058\pi$
$277$	−3.13857	−0.188578	−0.0942892	+	0.995545i	$0.530058\pi$
$278$	0	0
$279$	7.43114	0.444890
$280$	0	0
$281$	−7.87615	−0.469852	−0.234926	−	0.972013i	$-0.575485\pi$
$281$	−7.87615	−0.469852	−0.234926	+	0.972013i	$0.575485\pi$
$282$	0	0
$283$	−13.1634	−0.782484	−0.391242	−	0.920288i	$-0.627954\pi$
$283$	−13.1634	−0.782484	−0.391242	+	0.920288i	$0.627954\pi$
$284$	0	0
$285$	12.0146	0.711683
$286$	0	0
$287$	−23.8890	−1.41012
$288$	0	0
$289$	23.7754	1.39856
$290$	0	0
$291$	−2.29533	−0.134554
$292$	0	0
$293$	−32.9068	−1.92243	−0.961216	−	0.275795i	$-0.911059\pi$
$293$	−32.9068	−1.92243	−0.961216	+	0.275795i	$0.911059\pi$
$294$	0	0
$295$	−16.5474	−0.963425
$296$	0	0
$297$	−0.229135	−0.0132957
$298$	0	0
$299$	−4.10323	−0.237296
$300$	0	0
$301$	−4.11541	−0.237208
$302$	0	0
$303$	1.84478	0.105980
$304$	0	0
$305$	−20.7019	−1.18539
$306$	0	0
$307$	3.01583	0.172123	0.0860613	−	0.996290i	$-0.472572\pi$
$307$	3.01583	0.172123	0.0860613	+	0.996290i	$0.472572\pi$
$308$	0	0
$309$	10.0687	0.572786
$310$	0	0
$311$	−7.45318	−0.422631	−0.211315	−	0.977418i	$-0.567775\pi$
$311$	−7.45318	−0.422631	−0.211315	+	0.977418i	$0.567775\pi$
$312$	0	0
$313$	−3.43534	−0.194177	−0.0970886	−	0.995276i	$-0.530953\pi$
$313$	−3.43534	−0.194177	−0.0970886	+	0.995276i	$0.530953\pi$
$314$	0	0
$315$	−17.2168	−0.970059
$316$	0	0
$317$	0.318803	0.0179057	0.00895287	−	0.999960i	$-0.497150\pi$
$317$	0.318803	0.0179057	0.00895287	+	0.999960i	$0.497150\pi$
$318$	0	0
$319$	0.229135	0.0128291
$320$	0	0
$321$	2.97463	0.166028
$322$	0	0
$323$	−18.4673	−1.02755
$324$	0	0
$325$	−50.3007	−2.79018
$326$	0	0
$327$	−17.2466	−0.953742
$328$	0	0
$329$	28.5498	1.57400
$330$	0	0
$331$	26.0762	1.43328	0.716638	−	0.697446i	$-0.245680\pi$
$331$	26.0762	1.43328	0.716638	+	0.697446i	$0.245680\pi$
$332$	0	0
$333$	−6.31719	−0.346180
$334$	0	0
$335$	8.71556	0.476182
$336$	0	0
$337$	−15.0255	−0.818489	−0.409244	−	0.912425i	$-0.634208\pi$
$337$	−15.0255	−0.818489	−0.409244	+	0.912425i	$0.634208\pi$
$338$	0	0
$339$	−14.2563	−0.774298
$340$	0	0
$341$	1.70273	0.0922080
$342$	0	0
$343$	−13.1579	−0.710460
$344$	0	0
$345$	−4.15437	−0.223664
$346$	0	0
$347$	−10.0828	−0.541274	−0.270637	−	0.962681i	$-0.587234\pi$
$347$	−10.0828	−0.541274	−0.270637	+	0.962681i	$0.587234\pi$
$348$	0	0
$349$	−4.11851	−0.220459	−0.110229	−	0.993906i	$-0.535159\pi$
$349$	−4.11851	−0.220459	−0.110229	+	0.993906i	$0.535159\pi$
$350$	0	0
$351$	4.10323	0.219014
$352$	0	0
$353$	27.9036	1.48516	0.742579	−	0.669759i	$-0.233603\pi$
$353$	27.9036	1.48516	0.742579	+	0.669759i	$0.233603\pi$
$354$	0	0
$355$	−57.3276	−3.04263
$356$	0	0
$357$	26.4635	1.40060
$358$	0	0
$359$	−17.4428	−0.920598	−0.460299	−	0.887764i	$-0.652258\pi$
$359$	−17.4428	−0.920598	−0.460299	+	0.887764i	$0.652258\pi$
$360$	0	0
$361$	−10.6361	−0.559797
$362$	0	0
$363$	10.9475	0.574595
$364$	0	0
$365$	−57.7171	−3.02105
$366$	0	0
$367$	−33.8999	−1.76956	−0.884781	−	0.466006i	$-0.845692\pi$
$367$	−33.8999	−1.76956	−0.884781	+	0.466006i	$0.845692\pi$
$368$	0	0
$369$	5.76434	0.300080
$370$	0	0
$371$	−26.5067	−1.37616
$372$	0	0
$373$	−18.2447	−0.944676	−0.472338	−	0.881417i	$-0.656590\pi$
$373$	−18.2447	−0.944676	−0.472338	+	0.881417i	$0.656590\pi$
$374$	0	0
$375$	−30.1558	−1.55724
$376$	0	0
$377$	−4.10323	−0.211327
$378$	0	0
$379$	−12.1061	−0.621848	−0.310924	−	0.950435i	$-0.600638\pi$
$379$	−12.1061	−0.621848	−0.310924	+	0.950435i	$0.600638\pi$
$380$	0	0
$381$	−8.12633	−0.416325
$382$	0	0
$383$	22.0811	1.12829	0.564147	−	0.825675i	$-0.309205\pi$
$383$	22.0811	1.12829	0.564147	+	0.825675i	$0.309205\pi$
$384$	0	0
$385$	−3.94497	−0.201054
$386$	0	0
$387$	0.993037	0.0504789
$388$	0	0
$389$	31.1346	1.57859	0.789293	−	0.614016i	$-0.210447\pi$
$389$	31.1346	1.57859	0.789293	+	0.614016i	$0.210447\pi$
$390$	0	0
$391$	6.38557	0.322932
$392$	0	0
$393$	6.66793	0.336352
$394$	0	0
$395$	50.2678	2.52925
$396$	0	0
$397$	6.74900	0.338723	0.169361	−	0.985554i	$-0.445830\pi$
$397$	6.74900	0.338723	0.169361	+	0.985554i	$0.445830\pi$
$398$	0	0
$399$	−11.9854	−0.600019
$400$	0	0
$401$	21.2099	1.05917	0.529586	−	0.848256i	$-0.322347\pi$
$401$	21.2099	1.05917	0.529586	+	0.848256i	$0.322347\pi$
$402$	0	0
$403$	−30.4917	−1.51890
$404$	0	0
$405$	4.15437	0.206432
$406$	0	0
$407$	−1.44749	−0.0717493
$408$	0	0
$409$	15.9041	0.786406	0.393203	−	0.919452i	$-0.371367\pi$
$409$	15.9041	0.786406	0.393203	+	0.919452i	$0.371367\pi$
$410$	0	0
$411$	−8.92799	−0.440385
$412$	0	0
$413$	16.5071	0.812262
$414$	0	0
$415$	37.3964	1.83572
$416$	0	0
$417$	−15.6891	−0.768297
$418$	0	0
$419$	31.7059	1.54894	0.774468	−	0.632613i	$-0.218018\pi$
$419$	31.7059	1.54894	0.774468	+	0.632613i	$0.218018\pi$
$420$	0	0
$421$	21.4102	1.04347	0.521735	−	0.853108i	$-0.325285\pi$
$421$	21.4102	1.04347	0.521735	+	0.853108i	$0.325285\pi$
$422$	0	0
$423$	−6.88898	−0.334953
$424$	0	0
$425$	78.2794	3.79711
$426$	0	0
$427$	20.6516	0.999401
$428$	0	0
$429$	0.940193	0.0453929
$430$	0	0
$431$	−2.77251	−0.133547	−0.0667735	−	0.997768i	$-0.521270\pi$
$431$	−2.77251	−0.133547	−0.0667735	+	0.997768i	$0.521270\pi$
$432$	0	0
$433$	1.68490	0.0809711	0.0404856	−	0.999180i	$-0.487110\pi$
$433$	1.68490	0.0809711	0.0404856	+	0.999180i	$0.487110\pi$
$434$	0	0
$435$	−4.15437	−0.199187
$436$	0	0
$437$	−2.89203	−0.138345
$438$	0	0
$439$	−14.6829	−0.700777	−0.350389	−	0.936604i	$-0.613950\pi$
$439$	−14.6829	−0.700777	−0.350389	+	0.936604i	$0.613950\pi$
$440$	0	0
$441$	10.1750	0.484522
$442$	0	0
$443$	18.2376	0.866494	0.433247	−	0.901275i	$-0.357368\pi$
$443$	18.2376	0.866494	0.433247	+	0.901275i	$0.357368\pi$
$444$	0	0
$445$	30.0172	1.42295
$446$	0	0
$447$	21.2764	1.00634
$448$	0	0
$449$	6.91352	0.326269	0.163135	−	0.986604i	$-0.447840\pi$
$449$	6.91352	0.326269	0.163135	+	0.986604i	$0.447840\pi$
$450$	0	0
$451$	1.32081	0.0621945
$452$	0	0
$453$	−16.0301	−0.753158
$454$	0	0
$455$	70.6447	3.31187
$456$	0	0
$457$	1.39169	0.0651005	0.0325502	−	0.999470i	$-0.489637\pi$
$457$	1.39169	0.0651005	0.0325502	+	0.999470i	$0.489637\pi$
$458$	0	0
$459$	−6.38557	−0.298053
$460$	0	0
$461$	17.4374	0.812140	0.406070	−	0.913842i	$-0.366899\pi$
$461$	17.4374	0.812140	0.406070	+	0.913842i	$0.366899\pi$
$462$	0	0
$463$	−13.7841	−0.640599	−0.320300	−	0.947316i	$-0.603784\pi$
$463$	−13.7841	−0.640599	−0.320300	+	0.947316i	$0.603784\pi$
$464$	0	0
$465$	−30.8717	−1.43164
$466$	0	0
$467$	−10.2314	−0.473454	−0.236727	−	0.971576i	$-0.576075\pi$
$467$	−10.2314	−0.473454	−0.236727	+	0.971576i	$0.576075\pi$
$468$	0	0
$469$	−8.69436	−0.401468
$470$	0	0
$471$	−21.8133	−1.00511
$472$	0	0
$473$	0.227539	0.0104623
$474$	0	0
$475$	−35.4529	−1.62669
$476$	0	0
$477$	6.39600	0.292853
$478$	0	0
$479$	−23.0206	−1.05184	−0.525920	−	0.850534i	$-0.676279\pi$
$479$	−23.0206	−1.05184	−0.525920	+	0.850534i	$0.676279\pi$
$480$	0	0
$481$	25.9209	1.18189
$482$	0	0
$483$	4.14427	0.188571
$484$	0	0
$485$	9.53564	0.432991
$486$	0	0
$487$	−22.2168	−1.00674	−0.503370	−	0.864071i	$-0.667907\pi$
$487$	−22.2168	−1.00674	−0.503370	+	0.864071i	$0.667907\pi$
$488$	0	0
$489$	−20.1201	−0.909864
$490$	0	0
$491$	25.0400	1.13004	0.565020	−	0.825077i	$-0.308868\pi$
$491$	25.0400	1.13004	0.565020	+	0.825077i	$0.308868\pi$
$492$	0	0
$493$	6.38557	0.287591
$494$	0	0
$495$	0.951911	0.0427852
$496$	0	0
$497$	57.1882	2.56524
$498$	0	0
$499$	23.6498	1.05871	0.529356	−	0.848400i	$-0.322434\pi$
$499$	23.6498	1.05871	0.529356	+	0.848400i	$0.322434\pi$
$500$	0	0
$501$	14.8101	0.661664
$502$	0	0
$503$	−20.9246	−0.932980	−0.466490	−	0.884526i	$-0.654482\pi$
$503$	−20.9246	−0.932980	−0.466490	+	0.884526i	$0.654482\pi$
$504$	0	0
$505$	−7.66390	−0.341039
$506$	0	0
$507$	−3.83652	−0.170386
$508$	0	0
$509$	−27.1188	−1.20202	−0.601009	−	0.799242i	$-0.705235\pi$
$509$	−27.1188	−1.20202	−0.601009	+	0.799242i	$0.705235\pi$
$510$	0	0
$511$	57.5767	2.54704
$512$	0	0
$513$	2.89203	0.127686
$514$	0	0
$515$	−41.8290	−1.84320
$516$	0	0
$517$	−1.57850	−0.0694225
$518$	0	0
$519$	3.05807	0.134234
$520$	0	0
$521$	38.8200	1.70073	0.850367	−	0.526190i	$-0.176380\pi$
$521$	38.8200	1.70073	0.850367	+	0.526190i	$0.176380\pi$
$522$	0	0
$523$	39.7447	1.73791	0.868957	−	0.494887i	$-0.164791\pi$
$523$	39.7447	1.73791	0.868957	+	0.494887i	$0.164791\pi$
$524$	0	0
$525$	50.8038	2.21726
$526$	0	0
$527$	47.4520	2.06704
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.98312	−0.172853
$532$	0	0
$533$	−23.6524	−1.02450
$534$	0	0
$535$	−12.3577	−0.534270
$536$	0	0
$537$	6.09443	0.262994
$538$	0	0
$539$	2.33144	0.100422
$540$	0	0
$541$	−35.9805	−1.54692	−0.773462	−	0.633843i	$-0.781477\pi$
$541$	−35.9805	−1.54692	−0.773462	+	0.633843i	$0.781477\pi$
$542$	0	0
$543$	−24.1648	−1.03701
$544$	0	0
$545$	71.6490	3.06911
$546$	0	0
$547$	16.3011	0.696986	0.348493	−	0.937311i	$-0.386693\pi$
$547$	16.3011	0.696986	0.348493	+	0.937311i	$0.386693\pi$
$548$	0	0
$549$	−4.98317	−0.212677
$550$	0	0
$551$	−2.89203	−0.123205
$552$	0	0
$553$	−50.1456	−2.13241
$554$	0	0
$555$	26.2440	1.11399
$556$	0	0
$557$	4.14809	0.175760	0.0878800	−	0.996131i	$-0.471991\pi$
$557$	4.14809	0.175760	0.0878800	+	0.996131i	$0.471991\pi$
$558$	0	0
$559$	−4.07466	−0.172340
$560$	0	0
$561$	−1.46315	−0.0617744
$562$	0	0
$563$	−25.5993	−1.07888	−0.539440	−	0.842024i	$-0.681364\pi$
$563$	−25.5993	−1.07888	−0.539440	+	0.842024i	$0.681364\pi$
$564$	0	0
$565$	59.2261	2.49166
$566$	0	0
$567$	−4.14427	−0.174043
$568$	0	0
$569$	−19.4642	−0.815984	−0.407992	−	0.912986i	$-0.633771\pi$
$569$	−19.4642	−0.815984	−0.407992	+	0.912986i	$0.633771\pi$
$570$	0	0
$571$	37.7938	1.58162	0.790810	−	0.612062i	$-0.209660\pi$
$571$	37.7938	1.58162	0.790810	+	0.612062i	$0.209660\pi$
$572$	0	0
$573$	−4.15674	−0.173650
$574$	0	0
$575$	12.2588	0.511227
$576$	0	0
$577$	8.28292	0.344822	0.172411	−	0.985025i	$-0.444844\pi$
$577$	8.28292	0.344822	0.172411	+	0.985025i	$0.444844\pi$
$578$	0	0
$579$	11.5694	0.480808
$580$	0	0
$581$	−37.3055	−1.54769
$582$	0	0
$583$	1.46555	0.0606967
$584$	0	0
$585$	−17.0464	−0.704780
$586$	0	0
$587$	25.5240	1.05349	0.526744	−	0.850024i	$-0.323413\pi$
$587$	25.5240	1.05349	0.526744	+	0.850024i	$0.323413\pi$
$588$	0	0
$589$	−21.4911	−0.885525
$590$	0	0
$591$	−26.6706	−1.09708
$592$	0	0
$593$	35.2203	1.44633	0.723163	−	0.690677i	$-0.242688\pi$
$593$	35.2203	1.44633	0.723163	+	0.690677i	$0.242688\pi$
$594$	0	0
$595$	−109.939	−4.50707
$596$	0	0
$597$	−2.14916	−0.0879593
$598$	0	0
$599$	−26.8228	−1.09595	−0.547974	−	0.836495i	$-0.684601\pi$
$599$	−26.8228	−1.09595	−0.547974	+	0.836495i	$0.684601\pi$
$600$	0	0
$601$	−32.5755	−1.32878	−0.664390	−	0.747386i	$-0.731309\pi$
$601$	−32.5755	−1.32878	−0.664390	+	0.747386i	$0.731309\pi$
$602$	0	0
$603$	2.09792	0.0854341
$604$	0	0
$605$	−45.4800	−1.84902
$606$	0	0
$607$	15.2479	0.618892	0.309446	−	0.950917i	$-0.399856\pi$
$607$	15.2479	0.618892	0.309446	+	0.950917i	$0.399856\pi$
$608$	0	0
$609$	4.14427	0.167934
$610$	0	0
$611$	28.2671	1.14356
$612$	0	0
$613$	−10.1136	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$613$	−10.1136	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$614$	0	0
$615$	−23.9472	−0.965645
$616$	0	0
$617$	8.26929	0.332909	0.166454	−	0.986049i	$-0.446768\pi$
$617$	8.26929	0.332909	0.166454	+	0.986049i	$0.446768\pi$
$618$	0	0
$619$	0.388463	0.0156137	0.00780683	−	0.999970i	$-0.497515\pi$
$619$	0.388463	0.0156137	0.00780683	+	0.999970i	$0.497515\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−29.9442	−1.19969
$624$	0	0
$625$	63.9842	2.55937
$626$	0	0
$627$	0.662665	0.0264643
$628$	0	0
$629$	−40.3389	−1.60842
$630$	0	0
$631$	−11.3853	−0.453241	−0.226621	−	0.973983i	$-0.572768\pi$
$631$	−11.3853	−0.453241	−0.226621	+	0.973983i	$0.572768\pi$
$632$	0	0
$633$	−25.3278	−1.00669
$634$	0	0
$635$	33.7598	1.33972
$636$	0	0
$637$	−41.7502	−1.65420
$638$	0	0
$639$	−13.7993	−0.545894
$640$	0	0
$641$	−16.0469	−0.633815	−0.316907	−	0.948457i	$-0.602644\pi$
$641$	−16.0469	−0.633815	−0.316907	+	0.948457i	$0.602644\pi$
$642$	0	0
$643$	8.54739	0.337076	0.168538	−	0.985695i	$-0.446095\pi$
$643$	8.54739	0.337076	0.168538	+	0.985695i	$0.446095\pi$
$644$	0	0
$645$	−4.12544	−0.162439
$646$	0	0
$647$	20.7023	0.813891	0.406945	−	0.913453i	$-0.366594\pi$
$647$	20.7023	0.813891	0.406945	+	0.913453i	$0.366594\pi$
$648$	0	0
$649$	−0.912671	−0.0358255
$650$	0	0
$651$	30.7966	1.20701
$652$	0	0
$653$	−31.2691	−1.22366	−0.611828	−	0.790991i	$-0.709565\pi$
$653$	−31.2691	−1.22366	−0.611828	+	0.790991i	$0.709565\pi$
$654$	0	0
$655$	−27.7010	−1.08237
$656$	0	0
$657$	−13.8931	−0.542021
$658$	0	0
$659$	−2.05556	−0.0800734	−0.0400367	−	0.999198i	$-0.512747\pi$
$659$	−2.05556	−0.0800734	−0.0400367	+	0.999198i	$0.512747\pi$
$660$	0	0
$661$	23.8207	0.926517	0.463258	−	0.886223i	$-0.346680\pi$
$661$	23.8207	0.926517	0.463258	+	0.886223i	$0.346680\pi$
$662$	0	0
$663$	26.2015	1.01758
$664$	0	0
$665$	49.7917	1.93084
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	2.20969	0.0854314
$670$	0	0
$671$	−1.14182	−0.0440794
$672$	0	0
$673$	−3.62405	−0.139697	−0.0698483	−	0.997558i	$-0.522252\pi$
$673$	−3.62405	−0.139697	−0.0698483	+	0.997558i	$0.522252\pi$
$674$	0	0
$675$	−12.2588	−0.471841
$676$	0	0
$677$	8.74392	0.336056	0.168028	−	0.985782i	$-0.446260\pi$
$677$	8.74392	0.336056	0.168028	+	0.985782i	$0.446260\pi$
$678$	0	0
$679$	−9.51245	−0.365054
$680$	0	0
$681$	−2.88544	−0.110570
$682$	0	0
$683$	−10.7533	−0.411464	−0.205732	−	0.978608i	$-0.565958\pi$
$683$	−10.7533	−0.411464	−0.205732	+	0.978608i	$0.565958\pi$
$684$	0	0
$685$	37.0902	1.41714
$686$	0	0
$687$	21.1310	0.806198
$688$	0	0
$689$	−26.2443	−0.999827
$690$	0	0
$691$	−29.9770	−1.14038	−0.570190	−	0.821513i	$-0.693130\pi$
$691$	−29.9770	−1.14038	−0.570190	+	0.821513i	$0.693130\pi$
$692$	0	0
$693$	−0.949596	−0.0360722
$694$	0	0
$695$	65.1782	2.47235
$696$	0	0
$697$	36.8086	1.39422
$698$	0	0
$699$	14.1662	0.535814
$700$	0	0
$701$	−12.1979	−0.460710	−0.230355	−	0.973107i	$-0.573989\pi$
$701$	−12.1979	−0.460710	−0.230355	+	0.973107i	$0.573989\pi$
$702$	0	0
$703$	18.2695	0.689049
$704$	0	0
$705$	28.6194	1.07787
$706$	0	0
$707$	7.64526	0.287530
$708$	0	0
$709$	15.6535	0.587878	0.293939	−	0.955824i	$-0.405034\pi$
$709$	15.6535	0.587878	0.293939	+	0.955824i	$0.405034\pi$
$710$	0	0
$711$	12.1000	0.453785
$712$	0	0
$713$	7.43114	0.278298
$714$	0	0
$715$	−3.90591	−0.146073
$716$	0	0
$717$	−19.0504	−0.711449
$718$	0	0
$719$	−0.652811	−0.0243457	−0.0121729	−	0.999926i	$-0.503875\pi$
$719$	−0.652811	−0.0243457	−0.0121729	+	0.999926i	$0.503875\pi$
$720$	0	0
$721$	41.7273	1.55400
$722$	0	0
$723$	−28.9548	−1.07684
$724$	0	0
$725$	12.2588	0.455280
$726$	0	0
$727$	27.4190	1.01692	0.508458	−	0.861087i	$-0.330216\pi$
$727$	27.4190	1.01692	0.508458	+	0.861087i	$0.330216\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.34110	0.234534
$732$	0	0
$733$	40.6326	1.50080	0.750400	−	0.660984i	$-0.229861\pi$
$733$	40.6326	1.50080	0.750400	+	0.660984i	$0.229861\pi$
$734$	0	0
$735$	−42.2706	−1.55917
$736$	0	0
$737$	0.480707	0.0177071
$738$	0	0
$739$	28.6218	1.05287	0.526435	−	0.850215i	$-0.323528\pi$
$739$	28.6218	1.05287	0.526435	+	0.850215i	$0.323528\pi$
$740$	0	0
$741$	−11.8667	−0.435934
$742$	0	0
$743$	−7.39927	−0.271453	−0.135726	−	0.990746i	$-0.543337\pi$
$743$	−7.39927	−0.271453	−0.135726	+	0.990746i	$0.543337\pi$
$744$	0	0
$745$	−88.3900	−3.23836
$746$	0	0
$747$	9.00171	0.329355
$748$	0	0
$749$	12.3277	0.450443
$750$	0	0
$751$	19.6285	0.716253	0.358126	−	0.933673i	$-0.383416\pi$
$751$	19.6285	0.716253	0.358126	+	0.933673i	$0.383416\pi$
$752$	0	0
$753$	−13.5507	−0.493815
$754$	0	0
$755$	66.5949	2.42364
$756$	0	0
$757$	27.8155	1.01097	0.505486	−	0.862835i	$-0.331313\pi$
$757$	27.8155	1.01097	0.505486	+	0.862835i	$0.331313\pi$
$758$	0	0
$759$	−0.229135	−0.00831706
$760$	0	0
$761$	−23.6768	−0.858285	−0.429143	−	0.903237i	$-0.641184\pi$
$761$	−23.6768	−0.858285	−0.429143	+	0.903237i	$0.641184\pi$
$762$	0	0
$763$	−71.4747	−2.58756
$764$	0	0
$765$	26.5280	0.959122
$766$	0	0
$767$	16.3437	0.590136
$768$	0	0
$769$	−15.4026	−0.555433	−0.277716	−	0.960663i	$-0.589578\pi$
$769$	−15.4026	−0.555433	−0.277716	+	0.960663i	$0.589578\pi$
$770$	0	0
$771$	−1.47343	−0.0530643
$772$	0	0
$773$	−37.7016	−1.35603	−0.678016	−	0.735047i	$-0.737160\pi$
$773$	−37.7016	−1.35603	−0.678016	+	0.735047i	$0.737160\pi$
$774$	0	0
$775$	91.0968	3.27229
$776$	0	0
$777$	−26.1802	−0.939208
$778$	0	0
$779$	−16.6707	−0.597289
$780$	0	0
$781$	−3.16191	−0.113142
$782$	0	0
$783$	−1.00000	−0.0357371
$784$	0	0
$785$	90.6207	3.23439
$786$	0	0
$787$	−30.1332	−1.07413	−0.537067	−	0.843540i	$-0.680468\pi$
$787$	−30.1332	−1.07413	−0.537067	+	0.843540i	$0.680468\pi$
$788$	0	0
$789$	−22.8721	−0.814268
$790$	0	0
$791$	−59.0821	−2.10072
$792$	0	0
$793$	20.4471	0.726098
$794$	0	0
$795$	−26.5714	−0.942389
$796$	0	0
$797$	−33.8038	−1.19739	−0.598696	−	0.800976i	$-0.704314\pi$
$797$	−33.8038	−1.19739	−0.598696	+	0.800976i	$0.704314\pi$
$798$	0	0
$799$	−43.9900	−1.55625
$800$	0	0
$801$	7.22544	0.255298
$802$	0	0
$803$	−3.18339	−0.112339
$804$	0	0
$805$	−17.2168	−0.606814
$806$	0	0
$807$	−25.0211	−0.880784
$808$	0	0
$809$	22.2495	0.782252	0.391126	−	0.920337i	$-0.372086\pi$
$809$	22.2495	0.782252	0.391126	+	0.920337i	$0.372086\pi$
$810$	0	0
$811$	−32.0653	−1.12596	−0.562982	−	0.826469i	$-0.690346\pi$
$811$	−32.0653	−1.12596	−0.562982	+	0.826469i	$0.690346\pi$
$812$	0	0
$813$	2.80847	0.0984973
$814$	0	0
$815$	83.5865	2.92791
$816$	0	0
$817$	−2.87190	−0.100475
$818$	0	0
$819$	17.0049	0.594199
$820$	0	0
$821$	−32.1049	−1.12047	−0.560234	−	0.828334i	$-0.689289\pi$
$821$	−32.1049	−1.12047	−0.560234	+	0.828334i	$0.689289\pi$
$822$	0	0
$823$	−24.7750	−0.863603	−0.431801	−	0.901969i	$-0.642122\pi$
$823$	−24.7750	−0.863603	−0.431801	+	0.901969i	$0.642122\pi$
$824$	0	0
$825$	−2.80892	−0.0977939
$826$	0	0
$827$	−37.7080	−1.31124	−0.655618	−	0.755093i	$-0.727592\pi$
$827$	−37.7080	−1.31124	−0.655618	+	0.755093i	$0.727592\pi$
$828$	0	0
$829$	33.3634	1.15876	0.579379	−	0.815059i	$-0.303295\pi$
$829$	33.3634	1.15876	0.579379	+	0.815059i	$0.303295\pi$
$830$	0	0
$831$	3.13857	0.108876
$832$	0	0
$833$	64.9729	2.25118
$834$	0	0
$835$	−61.5265	−2.12921
$836$	0	0
$837$	−7.43114	−0.256858
$838$	0	0
$839$	15.9819	0.551756	0.275878	−	0.961193i	$-0.411031\pi$
$839$	15.9819	0.551756	0.275878	+	0.961193i	$0.411031\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	7.87615	0.271269
$844$	0	0
$845$	15.9383	0.548295
$846$	0	0
$847$	45.3694	1.55891
$848$	0	0
$849$	13.1634	0.451767
$850$	0	0
$851$	−6.31719	−0.216551
$852$	0	0
$853$	−3.22370	−0.110378	−0.0551888	−	0.998476i	$-0.517576\pi$
$853$	−3.22370	−0.110378	−0.0551888	+	0.998476i	$0.517576\pi$
$854$	0	0
$855$	−12.0146	−0.410890
$856$	0	0
$857$	35.1231	1.19978	0.599891	−	0.800082i	$-0.295211\pi$
$857$	35.1231	1.19978	0.599891	+	0.800082i	$0.295211\pi$
$858$	0	0
$859$	−51.7428	−1.76544	−0.882720	−	0.469899i	$-0.844290\pi$
$859$	−51.7428	−1.76544	−0.882720	+	0.469899i	$0.844290\pi$
$860$	0	0
$861$	23.8890	0.814134
$862$	0	0
$863$	13.7214	0.467080	0.233540	−	0.972347i	$-0.424969\pi$
$863$	13.7214	0.467080	0.233540	+	0.972347i	$0.424969\pi$
$864$	0	0
$865$	−12.7044	−0.431961
$866$	0	0
$867$	−23.7754	−0.807456
$868$	0	0
$869$	2.77252	0.0940515
$870$	0	0
$871$	−8.60827	−0.291680
$872$	0	0
$873$	2.29533	0.0776850
$874$	0	0
$875$	−124.974	−4.22488
$876$	0	0
$877$	29.5106	0.996502	0.498251	−	0.867033i	$-0.333976\pi$
$877$	29.5106	0.996502	0.498251	+	0.867033i	$0.333976\pi$
$878$	0	0
$879$	32.9068	1.10992
$880$	0	0
$881$	23.4896	0.791386	0.395693	−	0.918383i	$-0.370504\pi$
$881$	23.4896	0.791386	0.395693	+	0.918383i	$0.370504\pi$
$882$	0	0
$883$	44.0606	1.48276	0.741378	−	0.671088i	$-0.234173\pi$
$883$	44.0606	1.48276	0.741378	+	0.671088i	$0.234173\pi$
$884$	0	0
$885$	16.5474	0.556234
$886$	0	0
$887$	−59.3172	−1.99168	−0.995839	−	0.0911296i	$-0.970952\pi$
$887$	−59.3172	−1.99168	−0.995839	+	0.0911296i	$0.970952\pi$
$888$	0	0
$889$	−33.6777	−1.12951
$890$	0	0
$891$	0.229135	0.00767630
$892$	0	0
$893$	19.9232	0.666703
$894$	0	0
$895$	−25.3185	−0.846305
$896$	0	0
$897$	4.10323	0.137003
$898$	0	0
$899$	7.43114	0.247842
$900$	0	0
$901$	40.8421	1.36065
$902$	0	0
$903$	4.11541	0.136952
$904$	0	0
$905$	100.390	3.33707
$906$	0	0
$907$	32.9700	1.09475	0.547375	−	0.836887i	$-0.315627\pi$
$907$	32.9700	1.09475	0.547375	+	0.836887i	$0.315627\pi$
$908$	0	0
$909$	−1.84478	−0.0611874
$910$	0	0
$911$	49.0127	1.62386	0.811931	−	0.583753i	$-0.198416\pi$
$911$	49.0127	1.62386	0.811931	+	0.583753i	$0.198416\pi$
$912$	0	0
$913$	2.06260	0.0682622
$914$	0	0
$915$	20.7019	0.684385
$916$	0	0
$917$	27.6337	0.912544
$918$	0	0
$919$	24.1928	0.798048	0.399024	−	0.916940i	$-0.369349\pi$
$919$	24.1928	0.798048	0.399024	+	0.916940i	$0.369349\pi$
$920$	0	0
$921$	−3.01583	−0.0993751
$922$	0	0
$923$	56.6219	1.86373
$924$	0	0
$925$	−77.4412	−2.54625
$926$	0	0
$927$	−10.0687	−0.330698
$928$	0	0
$929$	−37.1169	−1.21777	−0.608883	−	0.793260i	$-0.708382\pi$
$929$	−37.1169	−1.21777	−0.608883	+	0.793260i	$0.708382\pi$
$930$	0	0
$931$	−29.4263	−0.964409
$932$	0	0
$933$	7.45318	0.244006
$934$	0	0
$935$	6.07849	0.198788
$936$	0	0
$937$	−37.2959	−1.21840	−0.609202	−	0.793015i	$-0.708510\pi$
$937$	−37.2959	−1.21840	−0.609202	+	0.793015i	$0.708510\pi$
$938$	0	0
$939$	3.43534	0.112108
$940$	0	0
$941$	48.0014	1.56480	0.782401	−	0.622775i	$-0.213995\pi$
$941$	48.0014	1.56480	0.782401	+	0.622775i	$0.213995\pi$
$942$	0	0
$943$	5.76434	0.187713
$944$	0	0
$945$	17.2168	0.560064
$946$	0	0
$947$	−32.5481	−1.05767	−0.528835	−	0.848725i	$-0.677371\pi$
$947$	−32.5481	−1.05767	−0.528835	+	0.848725i	$0.677371\pi$
$948$	0	0
$949$	57.0066	1.85051
$950$	0	0
$951$	−0.318803	−0.0103379
$952$	0	0
$953$	−54.1571	−1.75432	−0.877161	−	0.480196i	$-0.840565\pi$
$953$	−54.1571	−1.75432	−0.877161	+	0.480196i	$0.840565\pi$
$954$	0	0
$955$	17.2686	0.558800
$956$	0	0
$957$	−0.229135	−0.00740687
$958$	0	0
$959$	−37.0000	−1.19479
$960$	0	0
$961$	24.2218	0.781348
$962$	0	0
$963$	−2.97463	−0.0958560
$964$	0	0
$965$	−48.0636	−1.54722
$966$	0	0
$967$	55.4514	1.78320	0.891598	−	0.452828i	$-0.149585\pi$
$967$	55.4514	1.78320	0.891598	+	0.452828i	$0.149585\pi$
$968$	0	0
$969$	18.4673	0.593254
$970$	0	0
$971$	22.7413	0.729803	0.364902	−	0.931046i	$-0.381103\pi$
$971$	22.7413	0.729803	0.364902	+	0.931046i	$0.381103\pi$
$972$	0	0
$973$	−65.0197	−2.08444
$974$	0	0
$975$	50.3007	1.61091
$976$	0	0
$977$	−0.809033	−0.0258833	−0.0129416	−	0.999916i	$-0.504120\pi$
$977$	−0.809033	−0.0258833	−0.0129416	+	0.999916i	$0.504120\pi$
$978$	0	0
$979$	1.65560	0.0529132
$980$	0	0
$981$	17.2466	0.550643
$982$	0	0
$983$	3.46754	0.110597	0.0552986	−	0.998470i	$-0.482389\pi$
$983$	3.46754	0.110597	0.0552986	+	0.998470i	$0.482389\pi$
$984$	0	0
$985$	110.799	3.53036
$986$	0	0
$987$	−28.5498	−0.908749
$988$	0	0
$989$	0.993037	0.0315767
$990$	0	0
$991$	−34.3123	−1.08997	−0.544984	−	0.838447i	$-0.683464\pi$
$991$	−34.3123	−1.08997	−0.544984	+	0.838447i	$0.683464\pi$
$992$	0	0
$993$	−26.0762	−0.827502
$994$	0	0
$995$	8.92841	0.283050
$996$	0	0
$997$	−19.7495	−0.625473	−0.312736	−	0.949840i	$-0.601246\pi$
$997$	−19.7495	−0.625473	−0.312736	+	0.949840i	$0.601246\pi$
$998$	0	0
$999$	6.31719	0.199867

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.i.1.16	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.i.1.16	✓	16	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.15437 q^{5} -4.14427 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.15437 q^{5} -4.14427 q^{7} +1.00000 q^{9} +0.229135 q^{11} -4.10323 q^{13} -4.15437 q^{15} +6.38557 q^{17} -2.89203 q^{19} +4.14427 q^{21} +1.00000 q^{23} +12.2588 q^{25} -1.00000 q^{27} +1.00000 q^{29} +7.43114 q^{31} -0.229135 q^{33} -17.2168 q^{35} -6.31719 q^{37} +4.10323 q^{39} +5.76434 q^{41} +0.993037 q^{43} +4.15437 q^{45} -6.88898 q^{47} +10.1750 q^{49} -6.38557 q^{51} +6.39600 q^{53} +0.951911 q^{55} +2.89203 q^{57} -3.98312 q^{59} -4.98317 q^{61} -4.14427 q^{63} -17.0464 q^{65} +2.09792 q^{67} -1.00000 q^{69} -13.7993 q^{71} -13.8931 q^{73} -12.2588 q^{75} -0.949596 q^{77} +12.1000 q^{79} +1.00000 q^{81} +9.00171 q^{83} +26.5280 q^{85} -1.00000 q^{87} +7.22544 q^{89} +17.0049 q^{91} -7.43114 q^{93} -12.0146 q^{95} +2.29533 q^{97} +0.229135 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9}+O(q^{10}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9	\( 16 q - 16 q^{3} + 5 q^{5} - 4 q^{7} + 16 q^{9} + 5 q^{11} + 8 q^{13} - 5 q^{15} + 7 q^{17} + q^{19} + 4 q^{21} + 16 q^{23} + 31 q^{25} - 16 q^{27} + 16 q^{29} - 2 q^{31} - 5 q^{33} + 5 q^{35} + 14 q^{37} - 8 q^{39} - q^{41} - 13 q^{43} + 5 q^{45} - 4 q^{47} + 30 q^{49} - 7 q^{51} + 19 q^{53} - 37 q^{55} - q^{57} + 12 q^{59} + 21 q^{61} - 4 q^{63} + 26 q^{65} - 11 q^{67} - 16 q^{69} + 7 q^{71} - 13 q^{73} - 31 q^{75} + 4 q^{77} - 18 q^{79} + 16 q^{81} + 25 q^{83} + 48 q^{85} - 16 q^{87} + 12 q^{89} - 11 q^{91} + 2 q^{93} - q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) \) 16 * q - 16 * q^3 + 5 * q^5 - 4 * q^7 + 16 * q^9 + 5 * q^11 + 8 * q^13 - 5 * q^15 + 7 * q^17 + q^19 + 4 * q^21 + 16 * q^23 + 31 * q^25 - 16 * q^27 + 16 * q^29 - 2 * q^31 - 5 * q^33 + 5 * q^35 + 14 * q^37 - 8 * q^39 - q^41 - 13 * q^43 + 5 * q^45 - 4 * q^47 + 30 * q^49 - 7 * q^51 + 19 * q^53 - 37 * q^55 - q^57 + 12 * q^59 + 21 * q^61 - 4 * q^63 + 26 * q^65 - 11 * q^67 - 16 * q^69 + 7 * q^71 - 13 * q^73 - 31 * q^75 + 4 * q^77 - 18 * q^79 + 16 * q^81 + 25 * q^83 + 48 * q^85 - 16 * q^87 + 12 * q^89 - 11 * q^91 + 2 * q^93 - q^95 + 5 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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