LMFDB - Embedded newform 8008.2.a.p.1.9

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 15x^{7} + 15x^{6} + 66x^{5} - 59x^{4} - 77x^{3} + 34x^{2} + 11x - 2 $ x^9 - x^8 - 15x^7 + 15x^6 + 66x^5 - 59x^4 - 77x^3 + 34x^2 + 11*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.67186$ 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)

$f(q)$	$=$	$q+2.67186 q^{3} -4.01631 q^{5} -1.00000 q^{7} +4.13885 q^{9} +O(q^{10})$	$q+2.67186 q^{3} -4.01631 q^{5} -1.00000 q^{7} +4.13885 q^{9} +1.00000 q^{11} +1.00000 q^{13} -10.7310 q^{15} -6.26331 q^{17} +7.26832 q^{19} -2.67186 q^{21} -3.24359 q^{23} +11.1307 q^{25} +3.04285 q^{27} +0.715129 q^{29} +2.26456 q^{31} +2.67186 q^{33} +4.01631 q^{35} -5.56287 q^{37} +2.67186 q^{39} +6.82467 q^{41} -0.466688 q^{43} -16.6229 q^{45} -2.75113 q^{47} +1.00000 q^{49} -16.7347 q^{51} -11.5024 q^{53} -4.01631 q^{55} +19.4200 q^{57} -1.46522 q^{59} +3.04010 q^{61} -4.13885 q^{63} -4.01631 q^{65} +8.56250 q^{67} -8.66643 q^{69} -3.49753 q^{71} -0.308210 q^{73} +29.7397 q^{75} -1.00000 q^{77} +9.30138 q^{79} -4.28648 q^{81} -7.19951 q^{83} +25.1553 q^{85} +1.91073 q^{87} -12.1579 q^{89} -1.00000 q^{91} +6.05060 q^{93} -29.1918 q^{95} -8.50768 q^{97} +4.13885 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9}+O(q^{10}) $ 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9	$ 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9} + 9 q^{11} + 9 q^{13} - 9 q^{15} - 11 q^{17} + 10 q^{19} - q^{21} - 14 q^{23} - q^{25} - 5 q^{27} - 10 q^{29} + 5 q^{31} + q^{33} + 4 q^{35} - 16 q^{37} + q^{39} + 2 q^{41} + 4 q^{43} - 30 q^{45} + 9 q^{49} + 3 q^{51} - 23 q^{53} - 4 q^{55} + 14 q^{57} + 9 q^{59} - 14 q^{61} - 4 q^{63} - 4 q^{65} + 8 q^{67} - 26 q^{69} - 20 q^{71} - 23 q^{73} + 32 q^{75} - 9 q^{77} + 2 q^{79} - 11 q^{81} - 9 q^{83} - 3 q^{85} - 7 q^{87} - 6 q^{89} - 9 q^{91} - 19 q^{93} - 4 q^{95} - 3 q^{97} + 4 q^{99}+O(q^{100}) $ 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9 + 9 * q^11 + 9 * q^13 - 9 * q^15 - 11 * q^17 + 10 * q^19 - q^21 - 14 * q^23 - q^25 - 5 * q^27 - 10 * q^29 + 5 * q^31 + q^33 + 4 * q^35 - 16 * q^37 + q^39 + 2 * q^41 + 4 * q^43 - 30 * q^45 + 9 * q^49 + 3 * q^51 - 23 * q^53 - 4 * q^55 + 14 * q^57 + 9 * q^59 - 14 * q^61 - 4 * q^63 - 4 * q^65 + 8 * q^67 - 26 * q^69 - 20 * q^71 - 23 * q^73 + 32 * q^75 - 9 * q^77 + 2 * q^79 - 11 * q^81 - 9 * q^83 - 3 * q^85 - 7 * q^87 - 6 * q^89 - 9 * q^91 - 19 * q^93 - 4 * q^95 - 3 * q^97 + 4 * 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
$3$	2.67186	1.54260	0.771300	−	0.636471i	$-0.219607\pi$
$3$	2.67186	1.54260	0.771300	+	0.636471i	$0.219607\pi$
$4$	0	0
$5$	−4.01631	−1.79615	−0.898073	−	0.439846i	$-0.855033\pi$
$5$	−4.01631	−1.79615	−0.898073	+	0.439846i	$0.855033\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.13885	1.37962
$10$	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
$15$	−10.7310	−2.77074
$16$	0	0
$17$	−6.26331	−1.51907	−0.759537	−	0.650464i	$-0.774575\pi$
$17$	−6.26331	−1.51907	−0.759537	+	0.650464i	$0.774575\pi$
$18$	0	0
$19$	7.26832	1.66747	0.833734	−	0.552167i	$-0.186199\pi$
$19$	7.26832	1.66747	0.833734	+	0.552167i	$0.186199\pi$
$20$	0	0
$21$	−2.67186	−0.583048
$22$	0	0
$23$	−3.24359	−0.676335	−0.338168	−	0.941086i	$-0.609807\pi$
$23$	−3.24359	−0.676335	−0.338168	+	0.941086i	$0.609807\pi$
$24$	0	0
$25$	11.1307	2.22614
$26$	0	0
$27$	3.04285	0.585596
$28$	0	0
$29$	0.715129	0.132796	0.0663981	−	0.997793i	$-0.478849\pi$
$29$	0.715129	0.132796	0.0663981	+	0.997793i	$0.478849\pi$
$30$	0	0
$31$	2.26456	0.406728	0.203364	−	0.979103i	$-0.434813\pi$
$31$	2.26456	0.406728	0.203364	+	0.979103i	$0.434813\pi$
$32$	0	0
$33$	2.67186	0.465112
$34$	0	0
$35$	4.01631	0.678879
$36$	0	0
$37$	−5.56287	−0.914531	−0.457265	−	0.889330i	$-0.651171\pi$
$37$	−5.56287	−0.914531	−0.457265	+	0.889330i	$0.651171\pi$
$38$	0	0
$39$	2.67186	0.427840
$40$	0	0
$41$	6.82467	1.06583	0.532917	−	0.846167i	$-0.321096\pi$
$41$	6.82467	1.06583	0.532917	+	0.846167i	$0.321096\pi$
$42$	0	0
$43$	−0.466688	−0.0711692	−0.0355846	−	0.999367i	$-0.511329\pi$
$43$	−0.466688	−0.0711692	−0.0355846	+	0.999367i	$0.511329\pi$
$44$	0	0
$45$	−16.6229	−2.47799
$46$	0	0
$47$	−2.75113	−0.401294	−0.200647	−	0.979664i	$-0.564304\pi$
$47$	−2.75113	−0.401294	−0.200647	+	0.979664i	$0.564304\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−16.7347	−2.34333
$52$	0	0
$53$	−11.5024	−1.57998	−0.789991	−	0.613118i	$-0.789915\pi$
$53$	−11.5024	−1.57998	−0.789991	+	0.613118i	$0.789915\pi$
$54$	0	0
$55$	−4.01631	−0.541558
$56$	0	0
$57$	19.4200	2.57224
$58$	0	0
$59$	−1.46522	−0.190755	−0.0953776	−	0.995441i	$-0.530406\pi$
$59$	−1.46522	−0.190755	−0.0953776	+	0.995441i	$0.530406\pi$
$60$	0	0
$61$	3.04010	0.389245	0.194622	−	0.980878i	$-0.437652\pi$
$61$	3.04010	0.389245	0.194622	+	0.980878i	$0.437652\pi$
$62$	0	0
$63$	−4.13885	−0.521446
$64$	0	0
$65$	−4.01631	−0.498161
$66$	0	0
$67$	8.56250	1.04608	0.523038	−	0.852309i	$-0.324799\pi$
$67$	8.56250	1.04608	0.523038	+	0.852309i	$0.324799\pi$
$68$	0	0
$69$	−8.66643	−1.04332
$70$	0	0
$71$	−3.49753	−0.415081	−0.207540	−	0.978226i	$-0.566546\pi$
$71$	−3.49753	−0.415081	−0.207540	+	0.978226i	$0.566546\pi$
$72$	0	0
$73$	−0.308210	−0.0360733	−0.0180367	−	0.999837i	$-0.505742\pi$
$73$	−0.308210	−0.0360733	−0.0180367	+	0.999837i	$0.505742\pi$
$74$	0	0
$75$	29.7397	3.43405
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	9.30138	1.04649	0.523243	−	0.852183i	$-0.324722\pi$
$79$	9.30138	1.04649	0.523243	+	0.852183i	$0.324722\pi$
$80$	0	0
$81$	−4.28648	−0.476275
$82$	0	0
$83$	−7.19951	−0.790249	−0.395125	−	0.918627i	$-0.629299\pi$
$83$	−7.19951	−0.790249	−0.395125	+	0.918627i	$0.629299\pi$
$84$	0	0
$85$	25.1553	2.72848
$86$	0	0
$87$	1.91073	0.204851
$88$	0	0
$89$	−12.1579	−1.28874	−0.644369	−	0.764715i	$-0.722880\pi$
$89$	−12.1579	−1.28874	−0.644369	+	0.764715i	$0.722880\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	6.05060	0.627418
$94$	0	0
$95$	−29.1918	−2.99501
$96$	0	0
$97$	−8.50768	−0.863824	−0.431912	−	0.901916i	$-0.642161\pi$
$97$	−8.50768	−0.863824	−0.431912	+	0.901916i	$0.642161\pi$
$98$	0	0
$99$	4.13885	0.415970
$100$	0	0
$101$	1.80021	0.179128	0.0895638	−	0.995981i	$-0.471453\pi$
$101$	1.80021	0.179128	0.0895638	+	0.995981i	$0.471453\pi$
$102$	0	0
$103$	−2.77955	−0.273878	−0.136939	−	0.990580i	$-0.543726\pi$
$103$	−2.77955	−0.273878	−0.136939	+	0.990580i	$0.543726\pi$
$104$	0	0
$105$	10.7310	1.04724
$106$	0	0
$107$	−20.0240	−1.93579	−0.967895	−	0.251357i	$-0.919123\pi$
$107$	−20.0240	−1.93579	−0.967895	+	0.251357i	$0.919123\pi$
$108$	0	0
$109$	−8.24798	−0.790014	−0.395007	−	0.918678i	$-0.629258\pi$
$109$	−8.24798	−0.790014	−0.395007	+	0.918678i	$0.629258\pi$
$110$	0	0
$111$	−14.8632	−1.41076
$112$	0	0
$113$	−3.54931	−0.333891	−0.166945	−	0.985966i	$-0.553390\pi$
$113$	−3.54931	−0.333891	−0.166945	+	0.985966i	$0.553390\pi$
$114$	0	0
$115$	13.0272	1.21480
$116$	0	0
$117$	4.13885	0.382637
$118$	0	0
$119$	6.26331	0.574156
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	18.2346	1.64416
$124$	0	0
$125$	−24.6228	−2.20233
$126$	0	0
$127$	−19.0043	−1.68636	−0.843181	−	0.537629i	$-0.819320\pi$
$127$	−19.0043	−1.68636	−0.843181	+	0.537629i	$0.819320\pi$
$128$	0	0
$129$	−1.24693	−0.109786
$130$	0	0
$131$	−12.5199	−1.09387	−0.546933	−	0.837176i	$-0.684205\pi$
$131$	−12.5199	−1.09387	−0.546933	+	0.837176i	$0.684205\pi$
$132$	0	0
$133$	−7.26832	−0.630243
$134$	0	0
$135$	−12.2210	−1.05182
$136$	0	0
$137$	12.1671	1.03951	0.519754	−	0.854316i	$-0.326024\pi$
$137$	12.1671	1.03951	0.519754	+	0.854316i	$0.326024\pi$
$138$	0	0
$139$	−4.65061	−0.394460	−0.197230	−	0.980357i	$-0.563195\pi$
$139$	−4.65061	−0.394460	−0.197230	+	0.980357i	$0.563195\pi$
$140$	0	0
$141$	−7.35064	−0.619036
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−2.87218	−0.238521
$146$	0	0
$147$	2.67186	0.220371
$148$	0	0
$149$	18.4474	1.51127	0.755634	−	0.654994i	$-0.227329\pi$
$149$	18.4474	1.51127	0.755634	+	0.654994i	$0.227329\pi$
$150$	0	0
$151$	−18.3430	−1.49273	−0.746367	−	0.665535i	$-0.768204\pi$
$151$	−18.3430	−1.49273	−0.746367	+	0.665535i	$0.768204\pi$
$152$	0	0
$153$	−25.9229	−2.09574
$154$	0	0
$155$	−9.09518	−0.730542
$156$	0	0
$157$	−11.4946	−0.917370	−0.458685	−	0.888599i	$-0.651679\pi$
$157$	−11.4946	−0.917370	−0.458685	+	0.888599i	$0.651679\pi$
$158$	0	0
$159$	−30.7330	−2.43728
$160$	0	0
$161$	3.24359	0.255631
$162$	0	0
$163$	−12.6304	−0.989290	−0.494645	−	0.869095i	$-0.664702\pi$
$163$	−12.6304	−0.989290	−0.494645	+	0.869095i	$0.664702\pi$
$164$	0	0
$165$	−10.7310	−0.835408
$166$	0	0
$167$	−13.4204	−1.03850	−0.519252	−	0.854621i	$-0.673790\pi$
$167$	−13.4204	−1.03850	−0.519252	+	0.854621i	$0.673790\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	30.0825	2.30046
$172$	0	0
$173$	−14.7173	−1.11893	−0.559466	−	0.828853i	$-0.688994\pi$
$173$	−14.7173	−1.11893	−0.559466	+	0.828853i	$0.688994\pi$
$174$	0	0
$175$	−11.1307	−0.841402
$176$	0	0
$177$	−3.91486	−0.294259
$178$	0	0
$179$	15.5835	1.16476	0.582381	−	0.812916i	$-0.302121\pi$
$179$	15.5835	1.16476	0.582381	+	0.812916i	$0.302121\pi$
$180$	0	0
$181$	−5.93317	−0.441009	−0.220505	−	0.975386i	$-0.570770\pi$
$181$	−5.93317	−0.441009	−0.220505	+	0.975386i	$0.570770\pi$
$182$	0	0
$183$	8.12273	0.600449
$184$	0	0
$185$	22.3422	1.64263
$186$	0	0
$187$	−6.26331	−0.458018
$188$	0	0
$189$	−3.04285	−0.221335
$190$	0	0
$191$	−23.6050	−1.70800	−0.853998	−	0.520277i	$-0.825829\pi$
$191$	−23.6050	−1.70800	−0.853998	+	0.520277i	$0.825829\pi$
$192$	0	0
$193$	7.47196	0.537843	0.268922	−	0.963162i	$-0.413333\pi$
$193$	7.47196	0.537843	0.268922	+	0.963162i	$0.413333\pi$
$194$	0	0
$195$	−10.7310	−0.768464
$196$	0	0
$197$	6.97053	0.496630	0.248315	−	0.968679i	$-0.420123\pi$
$197$	6.97053	0.496630	0.248315	+	0.968679i	$0.420123\pi$
$198$	0	0
$199$	19.8868	1.40974	0.704870	−	0.709337i	$-0.251005\pi$
$199$	19.8868	1.40974	0.704870	+	0.709337i	$0.251005\pi$
$200$	0	0
$201$	22.8778	1.61368
$202$	0	0
$203$	−0.715129	−0.0501922
$204$	0	0
$205$	−27.4100	−1.91439
$206$	0	0
$207$	−13.4247	−0.933083
$208$	0	0
$209$	7.26832	0.502760
$210$	0	0
$211$	−6.14141	−0.422792	−0.211396	−	0.977401i	$-0.567801\pi$
$211$	−6.14141	−0.422792	−0.211396	+	0.977401i	$0.567801\pi$
$212$	0	0
$213$	−9.34493	−0.640304
$214$	0	0
$215$	1.87436	0.127830
$216$	0	0
$217$	−2.26456	−0.153729
$218$	0	0
$219$	−0.823496	−0.0556467
$220$	0	0
$221$	−6.26331	−0.421315
$222$	0	0
$223$	27.6417	1.85102	0.925511	−	0.378720i	$-0.123636\pi$
$223$	27.6417	1.85102	0.925511	+	0.378720i	$0.123636\pi$
$224$	0	0
$225$	46.0683	3.07122
$226$	0	0
$227$	9.92405	0.658682	0.329341	−	0.944211i	$-0.393173\pi$
$227$	9.92405	0.658682	0.329341	+	0.944211i	$0.393173\pi$
$228$	0	0
$229$	−16.5781	−1.09551	−0.547757	−	0.836638i	$-0.684518\pi$
$229$	−16.5781	−1.09551	−0.547757	+	0.836638i	$0.684518\pi$
$230$	0	0
$231$	−2.67186	−0.175796
$232$	0	0
$233$	0.631312	0.0413587	0.0206793	−	0.999786i	$-0.493417\pi$
$233$	0.631312	0.0413587	0.0206793	+	0.999786i	$0.493417\pi$
$234$	0	0
$235$	11.0494	0.720782
$236$	0	0
$237$	24.8520	1.61431
$238$	0	0
$239$	1.56651	0.101329	0.0506646	−	0.998716i	$-0.483866\pi$
$239$	1.56651	0.101329	0.0506646	+	0.998716i	$0.483866\pi$
$240$	0	0
$241$	16.8799	1.08733	0.543665	−	0.839302i	$-0.317036\pi$
$241$	16.8799	1.08733	0.543665	+	0.839302i	$0.317036\pi$
$242$	0	0
$243$	−20.5814	−1.32030
$244$	0	0
$245$	−4.01631	−0.256592
$246$	0	0
$247$	7.26832	0.462472
$248$	0	0
$249$	−19.2361	−1.21904
$250$	0	0
$251$	−11.2986	−0.713159	−0.356580	−	0.934265i	$-0.616057\pi$
$251$	−11.2986	−0.713159	−0.356580	+	0.934265i	$0.616057\pi$
$252$	0	0
$253$	−3.24359	−0.203923
$254$	0	0
$255$	67.2116	4.20895
$256$	0	0
$257$	14.3554	0.895464	0.447732	−	0.894168i	$-0.352232\pi$
$257$	14.3554	0.895464	0.447732	+	0.894168i	$0.352232\pi$
$258$	0	0
$259$	5.56287	0.345660
$260$	0	0
$261$	2.95981	0.183208
$262$	0	0
$263$	−18.5994	−1.14689	−0.573445	−	0.819244i	$-0.694393\pi$
$263$	−18.5994	−1.14689	−0.573445	+	0.819244i	$0.694393\pi$
$264$	0	0
$265$	46.1973	2.83788
$266$	0	0
$267$	−32.4843	−1.98801
$268$	0	0
$269$	1.11668	0.0680850	0.0340425	−	0.999420i	$-0.489162\pi$
$269$	1.11668	0.0680850	0.0340425	+	0.999420i	$0.489162\pi$
$270$	0	0
$271$	−18.9549	−1.15143	−0.575714	−	0.817651i	$-0.695276\pi$
$271$	−18.9549	−1.15143	−0.575714	+	0.817651i	$0.695276\pi$
$272$	0	0
$273$	−2.67186	−0.161708
$274$	0	0
$275$	11.1307	0.671207
$276$	0	0
$277$	22.8561	1.37329	0.686645	−	0.726993i	$-0.259083\pi$
$277$	22.8561	1.37329	0.686645	+	0.726993i	$0.259083\pi$
$278$	0	0
$279$	9.37269	0.561128
$280$	0	0
$281$	19.0790	1.13816	0.569078	−	0.822283i	$-0.307300\pi$
$281$	19.0790	1.13816	0.569078	+	0.822283i	$0.307300\pi$
$282$	0	0
$283$	−26.9292	−1.60078	−0.800388	−	0.599483i	$-0.795373\pi$
$283$	−26.9292	−1.60078	−0.800388	+	0.599483i	$0.795373\pi$
$284$	0	0
$285$	−77.9965	−4.62011
$286$	0	0
$287$	−6.82467	−0.402847
$288$	0	0
$289$	22.2290	1.30759
$290$	0	0
$291$	−22.7314	−1.33254
$292$	0	0
$293$	31.6732	1.85037	0.925183	−	0.379520i	$-0.123911\pi$
$293$	31.6732	1.85037	0.925183	+	0.379520i	$0.123911\pi$
$294$	0	0
$295$	5.88477	0.342624
$296$	0	0
$297$	3.04285	0.176564
$298$	0	0
$299$	−3.24359	−0.187582
$300$	0	0
$301$	0.466688	0.0268994
$302$	0	0
$303$	4.80991	0.276322
$304$	0	0
$305$	−12.2100	−0.699141
$306$	0	0
$307$	−3.27175	−0.186729	−0.0933644	−	0.995632i	$-0.529762\pi$
$307$	−3.27175	−0.186729	−0.0933644	+	0.995632i	$0.529762\pi$
$308$	0	0
$309$	−7.42659	−0.422484
$310$	0	0
$311$	3.26658	0.185231	0.0926155	−	0.995702i	$-0.470477\pi$
$311$	3.26658	0.185231	0.0926155	+	0.995702i	$0.470477\pi$
$312$	0	0
$313$	−16.5394	−0.934861	−0.467431	−	0.884030i	$-0.654820\pi$
$313$	−16.5394	−0.934861	−0.467431	+	0.884030i	$0.654820\pi$
$314$	0	0
$315$	16.6229	0.936593
$316$	0	0
$317$	−12.0198	−0.675100	−0.337550	−	0.941308i	$-0.609598\pi$
$317$	−12.0198	−0.675100	−0.337550	+	0.941308i	$0.609598\pi$
$318$	0	0
$319$	0.715129	0.0400395
$320$	0	0
$321$	−53.5013	−2.98615
$322$	0	0
$323$	−45.5237	−2.53301
$324$	0	0
$325$	11.1307	0.617421
$326$	0	0
$327$	−22.0375	−1.21868
$328$	0	0
$329$	2.75113	0.151675
$330$	0	0
$331$	23.1228	1.27095	0.635473	−	0.772123i	$-0.280805\pi$
$331$	23.1228	1.27095	0.635473	+	0.772123i	$0.280805\pi$
$332$	0	0
$333$	−23.0239	−1.26170
$334$	0	0
$335$	−34.3896	−1.87891
$336$	0	0
$337$	−33.1456	−1.80556	−0.902779	−	0.430104i	$-0.858477\pi$
$337$	−33.1456	−1.80556	−0.902779	+	0.430104i	$0.858477\pi$
$338$	0	0
$339$	−9.48327	−0.515060
$340$	0	0
$341$	2.26456	0.122633
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	34.8070	1.87395
$346$	0	0
$347$	−30.3053	−1.62688	−0.813438	−	0.581652i	$-0.802407\pi$
$347$	−30.3053	−1.62688	−0.813438	+	0.581652i	$0.802407\pi$
$348$	0	0
$349$	−22.8831	−1.22491	−0.612453	−	0.790507i	$-0.709817\pi$
$349$	−22.8831	−1.22491	−0.612453	+	0.790507i	$0.709817\pi$
$350$	0	0
$351$	3.04285	0.162415
$352$	0	0
$353$	−3.53227	−0.188004	−0.0940018	−	0.995572i	$-0.529966\pi$
$353$	−3.53227	−0.188004	−0.0940018	+	0.995572i	$0.529966\pi$
$354$	0	0
$355$	14.0472	0.745546
$356$	0	0
$357$	16.7347	0.885694
$358$	0	0
$359$	−6.50637	−0.343393	−0.171697	−	0.985150i	$-0.554925\pi$
$359$	−6.50637	−0.343393	−0.171697	+	0.985150i	$0.554925\pi$
$360$	0	0
$361$	33.8285	1.78045
$362$	0	0
$363$	2.67186	0.140236
$364$	0	0
$365$	1.23787	0.0647929
$366$	0	0
$367$	7.08430	0.369798	0.184899	−	0.982758i	$-0.440804\pi$
$367$	7.08430	0.369798	0.184899	+	0.982758i	$0.440804\pi$
$368$	0	0
$369$	28.2463	1.47044
$370$	0	0
$371$	11.5024	0.597177
$372$	0	0
$373$	28.6160	1.48168	0.740840	−	0.671682i	$-0.234428\pi$
$373$	28.6160	1.48168	0.740840	+	0.671682i	$0.234428\pi$
$374$	0	0
$375$	−65.7887	−3.39731
$376$	0	0
$377$	0.715129	0.0368310
$378$	0	0
$379$	−22.0331	−1.13176	−0.565881	−	0.824487i	$-0.691464\pi$
$379$	−22.0331	−1.13176	−0.565881	+	0.824487i	$0.691464\pi$
$380$	0	0
$381$	−50.7770	−2.60138
$382$	0	0
$383$	16.0233	0.818755	0.409377	−	0.912365i	$-0.365746\pi$
$383$	16.0233	0.818755	0.409377	+	0.912365i	$0.365746\pi$
$384$	0	0
$385$	4.01631	0.204690
$386$	0	0
$387$	−1.93155	−0.0981863
$388$	0	0
$389$	−13.3706	−0.677916	−0.338958	−	0.940801i	$-0.610074\pi$
$389$	−13.3706	−0.677916	−0.338958	+	0.940801i	$0.610074\pi$
$390$	0	0
$391$	20.3156	1.02740
$392$	0	0
$393$	−33.4514	−1.68740
$394$	0	0
$395$	−37.3572	−1.87964
$396$	0	0
$397$	7.94634	0.398815	0.199408	−	0.979917i	$-0.436098\pi$
$397$	7.94634	0.398815	0.199408	+	0.979917i	$0.436098\pi$
$398$	0	0
$399$	−19.4200	−0.972214
$400$	0	0
$401$	−23.4453	−1.17080	−0.585400	−	0.810745i	$-0.699063\pi$
$401$	−23.4453	−1.17080	−0.585400	+	0.810745i	$0.699063\pi$
$402$	0	0
$403$	2.26456	0.112806
$404$	0	0
$405$	17.2158	0.855460
$406$	0	0
$407$	−5.56287	−0.275741
$408$	0	0
$409$	22.7946	1.12712	0.563561	−	0.826075i	$-0.309431\pi$
$409$	22.7946	1.12712	0.563561	+	0.826075i	$0.309431\pi$
$410$	0	0
$411$	32.5089	1.60354
$412$	0	0
$413$	1.46522	0.0720987
$414$	0	0
$415$	28.9154	1.41940
$416$	0	0
$417$	−12.4258	−0.608494
$418$	0	0
$419$	17.7350	0.866412	0.433206	−	0.901295i	$-0.357382\pi$
$419$	17.7350	0.866412	0.433206	+	0.901295i	$0.357382\pi$
$420$	0	0
$421$	−12.3004	−0.599486	−0.299743	−	0.954020i	$-0.596901\pi$
$421$	−12.3004	−0.599486	−0.299743	+	0.954020i	$0.596901\pi$
$422$	0	0
$423$	−11.3865	−0.553631
$424$	0	0
$425$	−69.7150	−3.38167
$426$	0	0
$427$	−3.04010	−0.147121
$428$	0	0
$429$	2.67186	0.128999
$430$	0	0
$431$	24.8732	1.19810	0.599051	−	0.800711i	$-0.295545\pi$
$431$	24.8732	1.19810	0.599051	+	0.800711i	$0.295545\pi$
$432$	0	0
$433$	−21.6722	−1.04150	−0.520750	−	0.853709i	$-0.674348\pi$
$433$	−21.6722	−1.04150	−0.520750	+	0.853709i	$0.674348\pi$
$434$	0	0
$435$	−7.67406	−0.367943
$436$	0	0
$437$	−23.5755	−1.12777
$438$	0	0
$439$	−4.07828	−0.194646	−0.0973228	−	0.995253i	$-0.531028\pi$
$439$	−4.07828	−0.194646	−0.0973228	+	0.995253i	$0.531028\pi$
$440$	0	0
$441$	4.13885	0.197088
$442$	0	0
$443$	−5.24459	−0.249178	−0.124589	−	0.992208i	$-0.539761\pi$
$443$	−5.24459	−0.249178	−0.124589	+	0.992208i	$0.539761\pi$
$444$	0	0
$445$	48.8300	2.31476
$446$	0	0
$447$	49.2889	2.33128
$448$	0	0
$449$	24.4449	1.15363	0.576814	−	0.816875i	$-0.304296\pi$
$449$	24.4449	1.15363	0.576814	+	0.816875i	$0.304296\pi$
$450$	0	0
$451$	6.82467	0.321361
$452$	0	0
$453$	−49.0100	−2.30269
$454$	0	0
$455$	4.01631	0.188287
$456$	0	0
$457$	−19.1223	−0.894505	−0.447252	−	0.894408i	$-0.647597\pi$
$457$	−19.1223	−0.894505	−0.447252	+	0.894408i	$0.647597\pi$
$458$	0	0
$459$	−19.0583	−0.889564
$460$	0	0
$461$	−20.9457	−0.975537	−0.487768	−	0.872973i	$-0.662189\pi$
$461$	−20.9457	−0.975537	−0.487768	+	0.872973i	$0.662189\pi$
$462$	0	0
$463$	−37.9695	−1.76459	−0.882296	−	0.470694i	$-0.844004\pi$
$463$	−37.9695	−1.76459	−0.882296	+	0.470694i	$0.844004\pi$
$464$	0	0
$465$	−24.3011	−1.12693
$466$	0	0
$467$	−18.6235	−0.861794	−0.430897	−	0.902401i	$-0.641803\pi$
$467$	−18.6235	−0.861794	−0.430897	+	0.902401i	$0.641803\pi$
$468$	0	0
$469$	−8.56250	−0.395380
$470$	0	0
$471$	−30.7120	−1.41514
$472$	0	0
$473$	−0.466688	−0.0214583
$474$	0	0
$475$	80.9015	3.71202
$476$	0	0
$477$	−47.6069	−2.17977
$478$	0	0
$479$	1.76193	0.0805048	0.0402524	−	0.999190i	$-0.487184\pi$
$479$	1.76193	0.0805048	0.0402524	+	0.999190i	$0.487184\pi$
$480$	0	0
$481$	−5.56287	−0.253645
$482$	0	0
$483$	8.66643	0.394336
$484$	0	0
$485$	34.1694	1.55155
$486$	0	0
$487$	8.00877	0.362912	0.181456	−	0.983399i	$-0.441919\pi$
$487$	8.00877	0.362912	0.181456	+	0.983399i	$0.441919\pi$
$488$	0	0
$489$	−33.7467	−1.52608
$490$	0	0
$491$	−16.6387	−0.750896	−0.375448	−	0.926843i	$-0.622511\pi$
$491$	−16.6387	−0.750896	−0.375448	+	0.926843i	$0.622511\pi$
$492$	0	0
$493$	−4.47907	−0.201727
$494$	0	0
$495$	−16.6229	−0.747143
$496$	0	0
$497$	3.49753	0.156886
$498$	0	0
$499$	15.2986	0.684861	0.342430	−	0.939543i	$-0.388750\pi$
$499$	15.2986	0.684861	0.342430	+	0.939543i	$0.388750\pi$
$500$	0	0
$501$	−35.8576	−1.60200
$502$	0	0
$503$	−13.8933	−0.619471	−0.309736	−	0.950823i	$-0.600241\pi$
$503$	−13.8933	−0.619471	−0.309736	+	0.950823i	$0.600241\pi$
$504$	0	0
$505$	−7.23019	−0.321739
$506$	0	0
$507$	2.67186	0.118662
$508$	0	0
$509$	33.2135	1.47216	0.736081	−	0.676893i	$-0.236674\pi$
$509$	33.2135	1.47216	0.736081	+	0.676893i	$0.236674\pi$
$510$	0	0
$511$	0.308210	0.0136344
$512$	0	0
$513$	22.1164	0.976462
$514$	0	0
$515$	11.1635	0.491924
$516$	0	0
$517$	−2.75113	−0.120995
$518$	0	0
$519$	−39.3225	−1.72607
$520$	0	0
$521$	34.8137	1.52521	0.762607	−	0.646862i	$-0.223919\pi$
$521$	34.8137	1.52521	0.762607	+	0.646862i	$0.223919\pi$
$522$	0	0
$523$	−24.2251	−1.05929	−0.529645	−	0.848219i	$-0.677675\pi$
$523$	−24.2251	−1.05929	−0.529645	+	0.848219i	$0.677675\pi$
$524$	0	0
$525$	−29.7397	−1.29795
$526$	0	0
$527$	−14.1837	−0.617850
$528$	0	0
$529$	−12.4791	−0.542570
$530$	0	0
$531$	−6.06432	−0.263169
$532$	0	0
$533$	6.82467	0.295609
$534$	0	0
$535$	80.4223	3.47696
$536$	0	0
$537$	41.6369	1.79676
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−3.22631	−0.138710	−0.0693551	−	0.997592i	$-0.522094\pi$
$541$	−3.22631	−0.138710	−0.0693551	+	0.997592i	$0.522094\pi$
$542$	0	0
$543$	−15.8526	−0.680301
$544$	0	0
$545$	33.1264	1.41898
$546$	0	0
$547$	−8.38590	−0.358555	−0.179278	−	0.983799i	$-0.557376\pi$
$547$	−8.38590	−0.358555	−0.179278	+	0.983799i	$0.557376\pi$
$548$	0	0
$549$	12.5825	0.537008
$550$	0	0
$551$	5.19779	0.221433
$552$	0	0
$553$	−9.30138	−0.395535
$554$	0	0
$555$	59.6953	2.53392
$556$	0	0
$557$	−38.1225	−1.61530	−0.807652	−	0.589660i	$-0.799262\pi$
$557$	−38.1225	−1.61530	−0.807652	+	0.589660i	$0.799262\pi$
$558$	0	0
$559$	−0.466688	−0.0197388
$560$	0	0
$561$	−16.7347	−0.706539
$562$	0	0
$563$	12.7531	0.537480	0.268740	−	0.963213i	$-0.413393\pi$
$563$	12.7531	0.537480	0.268740	+	0.963213i	$0.413393\pi$
$564$	0	0
$565$	14.2551	0.599717
$566$	0	0
$567$	4.28648	0.180015
$568$	0	0
$569$	−6.57158	−0.275495	−0.137747	−	0.990467i	$-0.543986\pi$
$569$	−6.57158	−0.275495	−0.137747	+	0.990467i	$0.543986\pi$
$570$	0	0
$571$	33.2885	1.39308	0.696540	−	0.717518i	$-0.254722\pi$
$571$	33.2885	1.39308	0.696540	+	0.717518i	$0.254722\pi$
$572$	0	0
$573$	−63.0692	−2.63476
$574$	0	0
$575$	−36.1035	−1.50562
$576$	0	0
$577$	24.2386	1.00906	0.504532	−	0.863393i	$-0.331665\pi$
$577$	24.2386	1.00906	0.504532	+	0.863393i	$0.331665\pi$
$578$	0	0
$579$	19.9640	0.829677
$580$	0	0
$581$	7.19951	0.298686
$582$	0	0
$583$	−11.5024	−0.476383
$584$	0	0
$585$	−16.6229	−0.687271
$586$	0	0
$587$	27.6928	1.14300	0.571501	−	0.820601i	$-0.306361\pi$
$587$	27.6928	1.14300	0.571501	+	0.820601i	$0.306361\pi$
$588$	0	0
$589$	16.4596	0.678205
$590$	0	0
$591$	18.6243	0.766102
$592$	0	0
$593$	−12.1586	−0.499295	−0.249648	−	0.968337i	$-0.580315\pi$
$593$	−12.1586	−0.499295	−0.249648	+	0.968337i	$0.580315\pi$
$594$	0	0
$595$	−25.1553	−1.03127
$596$	0	0
$597$	53.1349	2.17466
$598$	0	0
$599$	−27.2244	−1.11236	−0.556179	−	0.831062i	$-0.687733\pi$
$599$	−27.2244	−1.11236	−0.556179	+	0.831062i	$0.687733\pi$
$600$	0	0
$601$	−1.06230	−0.0433322	−0.0216661	−	0.999765i	$-0.506897\pi$
$601$	−1.06230	−0.0433322	−0.0216661	+	0.999765i	$0.506897\pi$
$602$	0	0
$603$	35.4389	1.44318
$604$	0	0
$605$	−4.01631	−0.163286
$606$	0	0
$607$	−12.8642	−0.522143	−0.261072	−	0.965319i	$-0.584076\pi$
$607$	−12.8642	−0.522143	−0.261072	+	0.965319i	$0.584076\pi$
$608$	0	0
$609$	−1.91073	−0.0774266
$610$	0	0
$611$	−2.75113	−0.111299
$612$	0	0
$613$	−35.4725	−1.43272	−0.716360	−	0.697731i	$-0.754193\pi$
$613$	−35.4725	−1.43272	−0.716360	+	0.697731i	$0.754193\pi$
$614$	0	0
$615$	−73.2356	−2.95315
$616$	0	0
$617$	−8.60699	−0.346505	−0.173252	−	0.984877i	$-0.555428\pi$
$617$	−8.60699	−0.346505	−0.173252	+	0.984877i	$0.555428\pi$
$618$	0	0
$619$	23.6484	0.950508	0.475254	−	0.879849i	$-0.342356\pi$
$619$	23.6484	0.950508	0.475254	+	0.879849i	$0.342356\pi$
$620$	0	0
$621$	−9.86975	−0.396059
$622$	0	0
$623$	12.1579	0.487097
$624$	0	0
$625$	43.2391	1.72956
$626$	0	0
$627$	19.4200	0.775558
$628$	0	0
$629$	34.8420	1.38924
$630$	0	0
$631$	−38.6976	−1.54053	−0.770263	−	0.637726i	$-0.779875\pi$
$631$	−38.6976	−1.54053	−0.770263	+	0.637726i	$0.779875\pi$
$632$	0	0
$633$	−16.4090	−0.652199
$634$	0	0
$635$	76.3272	3.02895
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−14.4758	−0.572652
$640$	0	0
$641$	−13.0769	−0.516505	−0.258252	−	0.966077i	$-0.583147\pi$
$641$	−13.0769	−0.516505	−0.258252	+	0.966077i	$0.583147\pi$
$642$	0	0
$643$	−32.5189	−1.28242	−0.641210	−	0.767365i	$-0.721567\pi$
$643$	−32.5189	−1.28242	−0.641210	+	0.767365i	$0.721567\pi$
$644$	0	0
$645$	5.00804	0.197191
$646$	0	0
$647$	47.2402	1.85720	0.928602	−	0.371078i	$-0.121012\pi$
$647$	47.2402	1.85720	0.928602	+	0.371078i	$0.121012\pi$
$648$	0	0
$649$	−1.46522	−0.0575149
$650$	0	0
$651$	−6.05060	−0.237142
$652$	0	0
$653$	30.9958	1.21296	0.606480	−	0.795099i	$-0.292581\pi$
$653$	30.9958	1.21296	0.606480	+	0.795099i	$0.292581\pi$
$654$	0	0
$655$	50.2836	1.96474
$656$	0	0
$657$	−1.27564	−0.0497673
$658$	0	0
$659$	−22.4655	−0.875130	−0.437565	−	0.899187i	$-0.644159\pi$
$659$	−22.4655	−0.875130	−0.437565	+	0.899187i	$0.644159\pi$
$660$	0	0
$661$	35.2161	1.36975	0.684874	−	0.728661i	$-0.259857\pi$
$661$	35.2161	1.36975	0.684874	+	0.728661i	$0.259857\pi$
$662$	0	0
$663$	−16.7347	−0.649921
$664$	0	0
$665$	29.1918	1.13201
$666$	0	0
$667$	−2.31959	−0.0898147
$668$	0	0
$669$	73.8547	2.85539
$670$	0	0
$671$	3.04010	0.117362
$672$	0	0
$673$	−11.7900	−0.454471	−0.227235	−	0.973840i	$-0.572969\pi$
$673$	−11.7900	−0.454471	−0.227235	+	0.973840i	$0.572969\pi$
$674$	0	0
$675$	33.8690	1.30362
$676$	0	0
$677$	44.0367	1.69247	0.846235	−	0.532810i	$-0.178864\pi$
$677$	44.0367	1.69247	0.846235	+	0.532810i	$0.178864\pi$
$678$	0	0
$679$	8.50768	0.326495
$680$	0	0
$681$	26.5157	1.01608
$682$	0	0
$683$	12.2850	0.470072	0.235036	−	0.971987i	$-0.424479\pi$
$683$	12.2850	0.470072	0.235036	+	0.971987i	$0.424479\pi$
$684$	0	0
$685$	−48.8669	−1.86711
$686$	0	0
$687$	−44.2945	−1.68994
$688$	0	0
$689$	−11.5024	−0.438208
$690$	0	0
$691$	46.5518	1.77091	0.885457	−	0.464721i	$-0.153845\pi$
$691$	46.5518	1.77091	0.885457	+	0.464721i	$0.153845\pi$
$692$	0	0
$693$	−4.13885	−0.157222
$694$	0	0
$695$	18.6783	0.708507
$696$	0	0
$697$	−42.7450	−1.61908
$698$	0	0
$699$	1.68678	0.0637999
$700$	0	0
$701$	29.9052	1.12950	0.564752	−	0.825261i	$-0.308972\pi$
$701$	29.9052	1.12950	0.564752	+	0.825261i	$0.308972\pi$
$702$	0	0
$703$	−40.4327	−1.52495
$704$	0	0
$705$	29.5224	1.11188
$706$	0	0
$707$	−1.80021	−0.0677038
$708$	0	0
$709$	2.77574	0.104245	0.0521225	−	0.998641i	$-0.483401\pi$
$709$	2.77574	0.104245	0.0521225	+	0.998641i	$0.483401\pi$
$710$	0	0
$711$	38.4970	1.44375
$712$	0	0
$713$	−7.34532	−0.275084
$714$	0	0
$715$	−4.01631	−0.150201
$716$	0	0
$717$	4.18551	0.156311
$718$	0	0
$719$	3.04149	0.113428	0.0567141	−	0.998390i	$-0.481938\pi$
$719$	3.04149	0.113428	0.0567141	+	0.998390i	$0.481938\pi$
$720$	0	0
$721$	2.77955	0.103516
$722$	0	0
$723$	45.1008	1.67732
$724$	0	0
$725$	7.95989	0.295623
$726$	0	0
$727$	48.0533	1.78220	0.891099	−	0.453809i	$-0.149935\pi$
$727$	48.0533	1.78220	0.891099	+	0.453809i	$0.149935\pi$
$728$	0	0
$729$	−42.1313	−1.56042
$730$	0	0
$731$	2.92301	0.108111
$732$	0	0
$733$	−24.5812	−0.907927	−0.453964	−	0.891020i	$-0.649990\pi$
$733$	−24.5812	−0.907927	−0.453964	+	0.891020i	$0.649990\pi$
$734$	0	0
$735$	−10.7310	−0.395819
$736$	0	0
$737$	8.56250	0.315404
$738$	0	0
$739$	−6.71310	−0.246945	−0.123473	−	0.992348i	$-0.539403\pi$
$739$	−6.71310	−0.246945	−0.123473	+	0.992348i	$0.539403\pi$
$740$	0	0
$741$	19.4200	0.713410
$742$	0	0
$743$	0.756369	0.0277485	0.0138742	−	0.999904i	$-0.495584\pi$
$743$	0.756369	0.0277485	0.0138742	+	0.999904i	$0.495584\pi$
$744$	0	0
$745$	−74.0903	−2.71446
$746$	0	0
$747$	−29.7977	−1.09024
$748$	0	0
$749$	20.0240	0.731660
$750$	0	0
$751$	−5.17216	−0.188735	−0.0943675	−	0.995537i	$-0.530083\pi$
$751$	−5.17216	−0.188735	−0.0943675	+	0.995537i	$0.530083\pi$
$752$	0	0
$753$	−30.1882	−1.10012
$754$	0	0
$755$	73.6712	2.68117
$756$	0	0
$757$	−30.2045	−1.09780	−0.548901	−	0.835888i	$-0.684953\pi$
$757$	−30.2045	−1.09780	−0.548901	+	0.835888i	$0.684953\pi$
$758$	0	0
$759$	−8.66643	−0.314571
$760$	0	0
$761$	−23.6956	−0.858964	−0.429482	−	0.903075i	$-0.641304\pi$
$761$	−23.6956	−0.858964	−0.429482	+	0.903075i	$0.641304\pi$
$762$	0	0
$763$	8.24798	0.298597
$764$	0	0
$765$	104.114	3.76426
$766$	0	0
$767$	−1.46522	−0.0529060
$768$	0	0
$769$	−12.5749	−0.453463	−0.226732	−	0.973957i	$-0.572804\pi$
$769$	−12.5749	−0.453463	−0.226732	+	0.973957i	$0.572804\pi$
$770$	0	0
$771$	38.3556	1.38134
$772$	0	0
$773$	−24.2181	−0.871065	−0.435533	−	0.900173i	$-0.643440\pi$
$773$	−24.2181	−0.871065	−0.435533	+	0.900173i	$0.643440\pi$
$774$	0	0
$775$	25.2062	0.905433
$776$	0	0
$777$	14.8632	0.533215
$778$	0	0
$779$	49.6039	1.77724
$780$	0	0
$781$	−3.49753	−0.125152
$782$	0	0
$783$	2.17603	0.0777649
$784$	0	0
$785$	46.1659	1.64773
$786$	0	0
$787$	36.2407	1.29184	0.645921	−	0.763404i	$-0.276474\pi$
$787$	36.2407	1.29184	0.645921	+	0.763404i	$0.276474\pi$
$788$	0	0
$789$	−49.6951	−1.76919
$790$	0	0
$791$	3.54931	0.126199
$792$	0	0
$793$	3.04010	0.107957
$794$	0	0
$795$	123.433	4.37771
$796$	0	0
$797$	−27.9219	−0.989046	−0.494523	−	0.869165i	$-0.664657\pi$
$797$	−27.9219	−0.989046	−0.494523	+	0.869165i	$0.664657\pi$
$798$	0	0
$799$	17.2312	0.609595
$800$	0	0
$801$	−50.3198	−1.77796
$802$	0	0
$803$	−0.308210	−0.0108765
$804$	0	0
$805$	−13.0272	−0.459150
$806$	0	0
$807$	2.98361	0.105028
$808$	0	0
$809$	−47.6242	−1.67438	−0.837189	−	0.546913i	$-0.815803\pi$
$809$	−47.6242	−1.67438	−0.837189	+	0.546913i	$0.815803\pi$
$810$	0	0
$811$	45.4415	1.59567	0.797834	−	0.602877i	$-0.205979\pi$
$811$	45.4415	1.59567	0.797834	+	0.602877i	$0.205979\pi$
$812$	0	0
$813$	−50.6449	−1.77619
$814$	0	0
$815$	50.7276	1.77691
$816$	0	0
$817$	−3.39204	−0.118672
$818$	0	0
$819$	−4.13885	−0.144623
$820$	0	0
$821$	26.2371	0.915680	0.457840	−	0.889035i	$-0.348623\pi$
$821$	26.2371	0.915680	0.457840	+	0.889035i	$0.348623\pi$
$822$	0	0
$823$	4.85077	0.169087	0.0845436	−	0.996420i	$-0.473057\pi$
$823$	4.85077	0.169087	0.0845436	+	0.996420i	$0.473057\pi$
$824$	0	0
$825$	29.7397	1.03540
$826$	0	0
$827$	36.4492	1.26746	0.633731	−	0.773553i	$-0.281523\pi$
$827$	36.4492	1.26746	0.633731	+	0.773553i	$0.281523\pi$
$828$	0	0
$829$	18.5408	0.643950	0.321975	−	0.946748i	$-0.395653\pi$
$829$	18.5408	0.643950	0.321975	+	0.946748i	$0.395653\pi$
$830$	0	0
$831$	61.0684	2.11844
$832$	0	0
$833$	−6.26331	−0.217011
$834$	0	0
$835$	53.9006	1.86531
$836$	0	0
$837$	6.89072	0.238178
$838$	0	0
$839$	19.5846	0.676136	0.338068	−	0.941122i	$-0.390227\pi$
$839$	19.5846	0.676136	0.338068	+	0.941122i	$0.390227\pi$
$840$	0	0
$841$	−28.4886	−0.982365
$842$	0	0
$843$	50.9764	1.75572
$844$	0	0
$845$	−4.01631	−0.138165
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−71.9511	−2.46936
$850$	0	0
$851$	18.0437	0.618529
$852$	0	0
$853$	−31.0957	−1.06470	−0.532349	−	0.846525i	$-0.678690\pi$
$853$	−31.0957	−1.06470	−0.532349	+	0.846525i	$0.678690\pi$
$854$	0	0
$855$	−120.820	−4.13197
$856$	0	0
$857$	38.8309	1.32644	0.663218	−	0.748426i	$-0.269190\pi$
$857$	38.8309	1.32644	0.663218	+	0.748426i	$0.269190\pi$
$858$	0	0
$859$	50.2843	1.71568	0.857839	−	0.513918i	$-0.171806\pi$
$859$	50.2843	1.71568	0.857839	+	0.513918i	$0.171806\pi$
$860$	0	0
$861$	−18.2346	−0.621433
$862$	0	0
$863$	−5.79695	−0.197330	−0.0986652	−	0.995121i	$-0.531457\pi$
$863$	−5.79695	−0.197330	−0.0986652	+	0.995121i	$0.531457\pi$
$864$	0	0
$865$	59.1090	2.00977
$866$	0	0
$867$	59.3928	2.01709
$868$	0	0
$869$	9.30138	0.315528
$870$	0	0
$871$	8.56250	0.290129
$872$	0	0
$873$	−35.2120	−1.19175
$874$	0	0
$875$	24.6228	0.832402
$876$	0	0
$877$	−1.98595	−0.0670607	−0.0335304	−	0.999438i	$-0.510675\pi$
$877$	−1.98595	−0.0670607	−0.0335304	+	0.999438i	$0.510675\pi$
$878$	0	0
$879$	84.6264	2.85438
$880$	0	0
$881$	−14.0439	−0.473150	−0.236575	−	0.971613i	$-0.576025\pi$
$881$	−14.0439	−0.473150	−0.236575	+	0.971613i	$0.576025\pi$
$882$	0	0
$883$	−28.9867	−0.975480	−0.487740	−	0.872989i	$-0.662179\pi$
$883$	−28.9867	−0.975480	−0.487740	+	0.872989i	$0.662179\pi$
$884$	0	0
$885$	15.7233	0.528532
$886$	0	0
$887$	24.0391	0.807154	0.403577	−	0.914946i	$-0.367767\pi$
$887$	24.0391	0.807154	0.403577	+	0.914946i	$0.367767\pi$
$888$	0	0
$889$	19.0043	0.637385
$890$	0	0
$891$	−4.28648	−0.143602
$892$	0	0
$893$	−19.9961	−0.669144
$894$	0	0
$895$	−62.5880	−2.09208
$896$	0	0
$897$	−8.66643	−0.289364
$898$	0	0
$899$	1.61946	0.0540119
$900$	0	0
$901$	72.0433	2.40011
$902$	0	0
$903$	1.24693	0.0414951
$904$	0	0
$905$	23.8294	0.792117
$906$	0	0
$907$	26.4792	0.879229	0.439615	−	0.898187i	$-0.355115\pi$
$907$	26.4792	0.879229	0.439615	+	0.898187i	$0.355115\pi$
$908$	0	0
$909$	7.45079	0.247127
$910$	0	0
$911$	30.6888	1.01677	0.508383	−	0.861131i	$-0.330243\pi$
$911$	30.6888	1.01677	0.508383	+	0.861131i	$0.330243\pi$
$912$	0	0
$913$	−7.19951	−0.238269
$914$	0	0
$915$	−32.6233	−1.07849
$916$	0	0
$917$	12.5199	0.413443
$918$	0	0
$919$	35.3413	1.16580	0.582900	−	0.812544i	$-0.301918\pi$
$919$	35.3413	1.16580	0.582900	+	0.812544i	$0.301918\pi$
$920$	0	0
$921$	−8.74168	−0.288048
$922$	0	0
$923$	−3.49753	−0.115123
$924$	0	0
$925$	−61.9187	−2.03587
$926$	0	0
$927$	−11.5042	−0.377846
$928$	0	0
$929$	−1.25103	−0.0410450	−0.0205225	−	0.999789i	$-0.506533\pi$
$929$	−1.25103	−0.0410450	−0.0205225	+	0.999789i	$0.506533\pi$
$930$	0	0
$931$	7.26832	0.238210
$932$	0	0
$933$	8.72787	0.285737
$934$	0	0
$935$	25.1553	0.822668
$936$	0	0
$937$	34.4687	1.12604	0.563022	−	0.826442i	$-0.309639\pi$
$937$	34.4687	1.12604	0.563022	+	0.826442i	$0.309639\pi$
$938$	0	0
$939$	−44.1910	−1.44212
$940$	0	0
$941$	30.9394	1.00860	0.504299	−	0.863529i	$-0.331751\pi$
$941$	30.9394	1.00860	0.504299	+	0.863529i	$0.331751\pi$
$942$	0	0
$943$	−22.1364	−0.720861
$944$	0	0
$945$	12.2210	0.397549
$946$	0	0
$947$	44.4862	1.44561	0.722804	−	0.691053i	$-0.242853\pi$
$947$	44.4862	1.44561	0.722804	+	0.691053i	$0.242853\pi$
$948$	0	0
$949$	−0.308210	−0.0100049
$950$	0	0
$951$	−32.1153	−1.04141
$952$	0	0
$953$	−17.3416	−0.561748	−0.280874	−	0.959745i	$-0.590624\pi$
$953$	−17.3416	−0.561748	−0.280874	+	0.959745i	$0.590624\pi$
$954$	0	0
$955$	94.8048	3.06781
$956$	0	0
$957$	1.91073	0.0617650
$958$	0	0
$959$	−12.1671	−0.392897
$960$	0	0
$961$	−25.8718	−0.834573
$962$	0	0
$963$	−82.8761	−2.67065
$964$	0	0
$965$	−30.0097	−0.966045
$966$	0	0
$967$	15.4292	0.496170	0.248085	−	0.968738i	$-0.420199\pi$
$967$	15.4292	0.496170	0.248085	+	0.968738i	$0.420199\pi$
$968$	0	0
$969$	−121.633	−3.90742
$970$	0	0
$971$	−24.4337	−0.784115	−0.392058	−	0.919941i	$-0.628237\pi$
$971$	−24.4337	−0.784115	−0.392058	+	0.919941i	$0.628237\pi$
$972$	0	0
$973$	4.65061	0.149092
$974$	0	0
$975$	29.7397	0.952433
$976$	0	0
$977$	−54.7742	−1.75238	−0.876191	−	0.481964i	$-0.839924\pi$
$977$	−54.7742	−1.75238	−0.876191	+	0.481964i	$0.839924\pi$
$978$	0	0
$979$	−12.1579	−0.388569
$980$	0	0
$981$	−34.1372	−1.08992
$982$	0	0
$983$	−30.5983	−0.975935	−0.487967	−	0.872862i	$-0.662262\pi$
$983$	−30.5983	−0.975935	−0.487967	+	0.872862i	$0.662262\pi$
$984$	0	0
$985$	−27.9958	−0.892020
$986$	0	0
$987$	7.35064	0.233973
$988$	0	0
$989$	1.51374	0.0481343
$990$	0	0
$991$	10.9390	0.347489	0.173745	−	0.984791i	$-0.444413\pi$
$991$	10.9390	0.347489	0.173745	+	0.984791i	$0.444413\pi$
$992$	0	0
$993$	61.7810	1.96056
$994$	0	0
$995$	−79.8715	−2.53210
$996$	0	0
$997$	55.2991	1.75134	0.875670	−	0.482911i	$-0.160420\pi$
$997$	55.2991	1.75134	0.875670	+	0.482911i	$0.160420\pi$
$998$	0	0
$999$	−16.9270	−0.535546

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.p.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.p.1.9	✓	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+2.67186 q^{3} -4.01631 q^{5} -1.00000 q^{7} +4.13885 q^{9} +O(q^{10})\)	\(q+2.67186 q^{3} -4.01631 q^{5} -1.00000 q^{7} +4.13885 q^{9} +1.00000 q^{11} +1.00000 q^{13} -10.7310 q^{15} -6.26331 q^{17} +7.26832 q^{19} -2.67186 q^{21} -3.24359 q^{23} +11.1307 q^{25} +3.04285 q^{27} +0.715129 q^{29} +2.26456 q^{31} +2.67186 q^{33} +4.01631 q^{35} -5.56287 q^{37} +2.67186 q^{39} +6.82467 q^{41} -0.466688 q^{43} -16.6229 q^{45} -2.75113 q^{47} +1.00000 q^{49} -16.7347 q^{51} -11.5024 q^{53} -4.01631 q^{55} +19.4200 q^{57} -1.46522 q^{59} +3.04010 q^{61} -4.13885 q^{63} -4.01631 q^{65} +8.56250 q^{67} -8.66643 q^{69} -3.49753 q^{71} -0.308210 q^{73} +29.7397 q^{75} -1.00000 q^{77} +9.30138 q^{79} -4.28648 q^{81} -7.19951 q^{83} +25.1553 q^{85} +1.91073 q^{87} -12.1579 q^{89} -1.00000 q^{91} +6.05060 q^{93} -29.1918 q^{95} -8.50768 q^{97} +4.13885 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9}+O(q^{10}) \) 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9	\( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9} + 9 q^{11} + 9 q^{13} - 9 q^{15} - 11 q^{17} + 10 q^{19} - q^{21} - 14 q^{23} - q^{25} - 5 q^{27} - 10 q^{29} + 5 q^{31} + q^{33} + 4 q^{35} - 16 q^{37} + q^{39} + 2 q^{41} + 4 q^{43} - 30 q^{45} + 9 q^{49} + 3 q^{51} - 23 q^{53} - 4 q^{55} + 14 q^{57} + 9 q^{59} - 14 q^{61} - 4 q^{63} - 4 q^{65} + 8 q^{67} - 26 q^{69} - 20 q^{71} - 23 q^{73} + 32 q^{75} - 9 q^{77} + 2 q^{79} - 11 q^{81} - 9 q^{83} - 3 q^{85} - 7 q^{87} - 6 q^{89} - 9 q^{91} - 19 q^{93} - 4 q^{95} - 3 q^{97} + 4 q^{99}+O(q^{100}) \) 9 * q + q^3 - 4 * q^5 - 9 * q^7 + 4 * q^9 + 9 * q^11 + 9 * q^13 - 9 * q^15 - 11 * q^17 + 10 * q^19 - q^21 - 14 * q^23 - q^25 - 5 * q^27 - 10 * q^29 + 5 * q^31 + q^33 + 4 * q^35 - 16 * q^37 + q^39 + 2 * q^41 + 4 * q^43 - 30 * q^45 + 9 * q^49 + 3 * q^51 - 23 * q^53 - 4 * q^55 + 14 * q^57 + 9 * q^59 - 14 * q^61 - 4 * q^63 - 4 * q^65 + 8 * q^67 - 26 * q^69 - 20 * q^71 - 23 * q^73 + 32 * q^75 - 9 * q^77 + 2 * q^79 - 11 * q^81 - 9 * q^83 - 3 * q^85 - 7 * q^87 - 6 * q^89 - 9 * q^91 - 19 * q^93 - 4 * q^95 - 3 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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