LMFDB - Embedded newform 6004.2.a.e.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.34688 q^{3} +3.43943 q^{5} +0.297446 q^{7} -1.18593 q^{9} +O(q^{10})$	$q+1.34688 q^{3} +3.43943 q^{5} +0.297446 q^{7} -1.18593 q^{9} +1.35909 q^{11} +3.70816 q^{13} +4.63248 q^{15} -2.04988 q^{17} +1.00000 q^{19} +0.400623 q^{21} +3.78241 q^{23} +6.82966 q^{25} -5.63792 q^{27} +2.03684 q^{29} +1.91636 q^{31} +1.83053 q^{33} +1.02304 q^{35} +9.15978 q^{37} +4.99443 q^{39} +2.26137 q^{41} -3.78436 q^{43} -4.07890 q^{45} +5.94492 q^{47} -6.91153 q^{49} -2.76093 q^{51} -9.31015 q^{53} +4.67450 q^{55} +1.34688 q^{57} +5.93394 q^{59} -5.48794 q^{61} -0.352749 q^{63} +12.7540 q^{65} -5.15909 q^{67} +5.09444 q^{69} -9.11383 q^{71} +13.9511 q^{73} +9.19870 q^{75} +0.404257 q^{77} +1.00000 q^{79} -4.03581 q^{81} -1.39578 q^{83} -7.05040 q^{85} +2.74337 q^{87} +9.88312 q^{89} +1.10298 q^{91} +2.58109 q^{93} +3.43943 q^{95} +9.57145 q^{97} -1.61178 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	$ 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) $ 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * 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
$3$	1.34688	0.777619	0.388810	−	0.921318i	$-0.372886\pi$
$3$	1.34688	0.777619	0.388810	+	0.921318i	$0.372886\pi$
$4$	0	0
$5$	3.43943	1.53816	0.769079	−	0.639153i	$-0.220715\pi$
$5$	3.43943	1.53816	0.769079	+	0.639153i	$0.220715\pi$
$6$	0	0
$7$	0.297446	0.112424	0.0562120	−	0.998419i	$-0.482098\pi$
$7$	0.297446	0.112424	0.0562120	+	0.998419i	$0.482098\pi$
$8$	0	0
$9$	−1.18593	−0.395308
$10$	0	0
$11$	1.35909	0.409782	0.204891	−	0.978785i	$-0.434316\pi$
$11$	1.35909	0.409782	0.204891	+	0.978785i	$0.434316\pi$
$12$	0	0
$13$	3.70816	1.02846	0.514230	−	0.857653i	$-0.328078\pi$
$13$	3.70816	1.02846	0.514230	+	0.857653i	$0.328078\pi$
$14$	0	0
$15$	4.63248	1.19610
$16$	0	0
$17$	−2.04988	−0.497168	−0.248584	−	0.968610i	$-0.579965\pi$
$17$	−2.04988	−0.497168	−0.248584	+	0.968610i	$0.579965\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.400623	0.0874230
$22$	0	0
$23$	3.78241	0.788687	0.394344	−	0.918963i	$-0.370972\pi$
$23$	3.78241	0.788687	0.394344	+	0.918963i	$0.370972\pi$
$24$	0	0
$25$	6.82966	1.36593
$26$	0	0
$27$	−5.63792	−1.08502
$28$	0	0
$29$	2.03684	0.378232	0.189116	−	0.981955i	$-0.439438\pi$
$29$	2.03684	0.378232	0.189116	+	0.981955i	$0.439438\pi$
$30$	0	0
$31$	1.91636	0.344188	0.172094	−	0.985081i	$-0.444947\pi$
$31$	1.91636	0.344188	0.172094	+	0.985081i	$0.444947\pi$
$32$	0	0
$33$	1.83053	0.318654
$34$	0	0
$35$	1.02304	0.172926
$36$	0	0
$37$	9.15978	1.50586	0.752929	−	0.658101i	$-0.228640\pi$
$37$	9.15978	1.50586	0.752929	+	0.658101i	$0.228640\pi$
$38$	0	0
$39$	4.99443	0.799750
$40$	0	0
$41$	2.26137	0.353167	0.176584	−	0.984286i	$-0.443495\pi$
$41$	2.26137	0.353167	0.176584	+	0.984286i	$0.443495\pi$
$42$	0	0
$43$	−3.78436	−0.577109	−0.288555	−	0.957463i	$-0.593175\pi$
$43$	−3.78436	−0.577109	−0.288555	+	0.957463i	$0.593175\pi$
$44$	0	0
$45$	−4.07890	−0.608047
$46$	0	0
$47$	5.94492	0.867155	0.433578	−	0.901116i	$-0.357251\pi$
$47$	5.94492	0.867155	0.433578	+	0.901116i	$0.357251\pi$
$48$	0	0
$49$	−6.91153	−0.987361
$50$	0	0
$51$	−2.76093	−0.386608
$52$	0	0
$53$	−9.31015	−1.27885	−0.639423	−	0.768855i	$-0.720827\pi$
$53$	−9.31015	−1.27885	−0.639423	+	0.768855i	$0.720827\pi$
$54$	0	0
$55$	4.67450	0.630310
$56$	0	0
$57$	1.34688	0.178398
$58$	0	0
$59$	5.93394	0.772533	0.386266	−	0.922387i	$-0.373765\pi$
$59$	5.93394	0.772533	0.386266	+	0.922387i	$0.373765\pi$
$60$	0	0
$61$	−5.48794	−0.702658	−0.351329	−	0.936252i	$-0.614270\pi$
$61$	−5.48794	−0.702658	−0.351329	+	0.936252i	$0.614270\pi$
$62$	0	0
$63$	−0.352749	−0.0444421
$64$	0	0
$65$	12.7540	1.58193
$66$	0	0
$67$	−5.15909	−0.630283	−0.315141	−	0.949045i	$-0.602052\pi$
$67$	−5.15909	−0.630283	−0.315141	+	0.949045i	$0.602052\pi$
$68$	0	0
$69$	5.09444	0.613298
$70$	0	0
$71$	−9.11383	−1.08161	−0.540806	−	0.841147i	$-0.681881\pi$
$71$	−9.11383	−1.08161	−0.540806	+	0.841147i	$0.681881\pi$
$72$	0	0
$73$	13.9511	1.63285	0.816427	−	0.577449i	$-0.195952\pi$
$73$	13.9511	1.63285	0.816427	+	0.577449i	$0.195952\pi$
$74$	0	0
$75$	9.19870	1.06217
$76$	0	0
$77$	0.404257	0.0460693
$78$	0	0
$79$	1.00000	0.112509
$79$	1.00000	0.112509
$80$	0	0
$81$	−4.03581	−0.448423
$82$	0	0
$83$	−1.39578	−0.153207	−0.0766036	−	0.997062i	$-0.524408\pi$
$83$	−1.39578	−0.153207	−0.0766036	+	0.997062i	$0.524408\pi$
$84$	0	0
$85$	−7.05040	−0.764724
$86$	0	0
$87$	2.74337	0.294120
$88$	0	0
$89$	9.88312	1.04761	0.523804	−	0.851839i	$-0.324512\pi$
$89$	9.88312	1.04761	0.523804	+	0.851839i	$0.324512\pi$
$90$	0	0
$91$	1.10298	0.115623
$92$	0	0
$93$	2.58109	0.267647
$94$	0	0
$95$	3.43943	0.352878
$96$	0	0
$97$	9.57145	0.971834	0.485917	−	0.874005i	$-0.338486\pi$
$97$	9.57145	0.971834	0.485917	+	0.874005i	$0.338486\pi$
$98$	0	0
$99$	−1.61178	−0.161990
$100$	0	0
$101$	1.57001	0.156222	0.0781108	−	0.996945i	$-0.475111\pi$
$101$	1.57001	0.156222	0.0781108	+	0.996945i	$0.475111\pi$
$102$	0	0
$103$	−4.14248	−0.408171	−0.204085	−	0.978953i	$-0.565422\pi$
$103$	−4.14248	−0.408171	−0.204085	+	0.978953i	$0.565422\pi$
$104$	0	0
$105$	1.37791	0.134470
$106$	0	0
$107$	−2.52731	−0.244325	−0.122162	−	0.992510i	$-0.538983\pi$
$107$	−2.52731	−0.244325	−0.122162	+	0.992510i	$0.538983\pi$
$108$	0	0
$109$	−0.379208	−0.0363215	−0.0181608	−	0.999835i	$-0.505781\pi$
$109$	−0.379208	−0.0363215	−0.0181608	+	0.999835i	$0.505781\pi$
$110$	0	0
$111$	12.3371	1.17098
$112$	0	0
$113$	7.26933	0.683841	0.341921	−	0.939729i	$-0.388923\pi$
$113$	7.26933	0.683841	0.341921	+	0.939729i	$0.388923\pi$
$114$	0	0
$115$	13.0093	1.21313
$116$	0	0
$117$	−4.39760	−0.406559
$118$	0	0
$119$	−0.609728	−0.0558936
$120$	0	0
$121$	−9.15287	−0.832079
$122$	0	0
$123$	3.04579	0.274630
$124$	0	0
$125$	6.29297	0.562860
$126$	0	0
$127$	−2.53880	−0.225282	−0.112641	−	0.993636i	$-0.535931\pi$
$127$	−2.53880	−0.225282	−0.112641	+	0.993636i	$0.535931\pi$
$128$	0	0
$129$	−5.09706	−0.448771
$130$	0	0
$131$	14.6343	1.27860	0.639300	−	0.768957i	$-0.279224\pi$
$131$	14.6343	1.27860	0.639300	+	0.768957i	$0.279224\pi$
$132$	0	0
$133$	0.297446	0.0257918
$134$	0	0
$135$	−19.3912	−1.66893
$136$	0	0
$137$	9.40451	0.803481	0.401741	−	0.915754i	$-0.368405\pi$
$137$	9.40451	0.803481	0.401741	+	0.915754i	$0.368405\pi$
$138$	0	0
$139$	14.1967	1.20415	0.602076	−	0.798439i	$-0.294341\pi$
$139$	14.1967	1.20415	0.602076	+	0.798439i	$0.294341\pi$
$140$	0	0
$141$	8.00707	0.674317
$142$	0	0
$143$	5.03974	0.421444
$144$	0	0
$145$	7.00556	0.581780
$146$	0	0
$147$	−9.30897	−0.767791
$148$	0	0
$149$	−9.64741	−0.790346	−0.395173	−	0.918607i	$-0.629315\pi$
$149$	−9.64741	−0.790346	−0.395173	+	0.918607i	$0.629315\pi$
$150$	0	0
$151$	−8.82365	−0.718058	−0.359029	−	0.933326i	$-0.616892\pi$
$151$	−8.82365	−0.718058	−0.359029	+	0.933326i	$0.616892\pi$
$152$	0	0
$153$	2.43100	0.196535
$154$	0	0
$155$	6.59117	0.529415
$156$	0	0
$157$	−3.07954	−0.245774	−0.122887	−	0.992421i	$-0.539215\pi$
$157$	−3.07954	−0.245774	−0.122887	+	0.992421i	$0.539215\pi$
$158$	0	0
$159$	−12.5396	−0.994456
$160$	0	0
$161$	1.12506	0.0886673
$162$	0	0
$163$	5.14634	0.403092	0.201546	−	0.979479i	$-0.435403\pi$
$163$	5.14634	0.403092	0.201546	+	0.979479i	$0.435403\pi$
$164$	0	0
$165$	6.29597	0.490141
$166$	0	0
$167$	6.27292	0.485413	0.242706	−	0.970100i	$-0.421965\pi$
$167$	6.27292	0.485413	0.242706	+	0.970100i	$0.421965\pi$
$168$	0	0
$169$	0.750465	0.0577281
$170$	0	0
$171$	−1.18593	−0.0906900
$172$	0	0
$173$	−11.0082	−0.836935	−0.418467	−	0.908232i	$-0.637433\pi$
$173$	−11.0082	−0.836935	−0.418467	+	0.908232i	$0.637433\pi$
$174$	0	0
$175$	2.03145	0.153563
$176$	0	0
$177$	7.99228	0.600736
$178$	0	0
$179$	−16.6901	−1.24747	−0.623737	−	0.781634i	$-0.714386\pi$
$179$	−16.6901	−1.24747	−0.623737	+	0.781634i	$0.714386\pi$
$180$	0	0
$181$	−18.2312	−1.35511	−0.677556	−	0.735471i	$-0.736961\pi$
$181$	−18.2312	−1.35511	−0.677556	+	0.735471i	$0.736961\pi$
$182$	0	0
$183$	−7.39157	−0.546401
$184$	0	0
$185$	31.5044	2.31625
$186$	0	0
$187$	−2.78597	−0.203731
$188$	0	0
$189$	−1.67698	−0.121982
$190$	0	0
$191$	0.937535	0.0678377	0.0339188	−	0.999425i	$-0.489201\pi$
$191$	0.937535	0.0678377	0.0339188	+	0.999425i	$0.489201\pi$
$192$	0	0
$193$	−5.44554	−0.391978	−0.195989	−	0.980606i	$-0.562792\pi$
$193$	−5.44554	−0.391978	−0.195989	+	0.980606i	$0.562792\pi$
$194$	0	0
$195$	17.1780	1.23014
$196$	0	0
$197$	4.66307	0.332230	0.166115	−	0.986106i	$-0.446878\pi$
$197$	4.66307	0.332230	0.166115	+	0.986106i	$0.446878\pi$
$198$	0	0
$199$	−1.82394	−0.129296	−0.0646478	−	0.997908i	$-0.520592\pi$
$199$	−1.82394	−0.129296	−0.0646478	+	0.997908i	$0.520592\pi$
$200$	0	0
$201$	−6.94865	−0.490120
$202$	0	0
$203$	0.605849	0.0425223
$204$	0	0
$205$	7.77783	0.543227
$206$	0	0
$207$	−4.48566	−0.311775
$208$	0	0
$209$	1.35909	0.0940104
$210$	0	0
$211$	4.94954	0.340741	0.170370	−	0.985380i	$-0.445504\pi$
$211$	4.94954	0.340741	0.170370	+	0.985380i	$0.445504\pi$
$212$	0	0
$213$	−12.2752	−0.841083
$214$	0	0
$215$	−13.0160	−0.887686
$216$	0	0
$217$	0.570012	0.0386950
$218$	0	0
$219$	18.7904	1.26974
$220$	0	0
$221$	−7.60128	−0.511317
$222$	0	0
$223$	9.43173	0.631595	0.315798	−	0.948827i	$-0.397728\pi$
$223$	9.43173	0.631595	0.315798	+	0.948827i	$0.397728\pi$
$224$	0	0
$225$	−8.09946	−0.539964
$226$	0	0
$227$	−24.9762	−1.65773	−0.828865	−	0.559449i	$-0.811013\pi$
$227$	−24.9762	−1.65773	−0.828865	+	0.559449i	$0.811013\pi$
$228$	0	0
$229$	23.7516	1.56955	0.784774	−	0.619781i	$-0.212779\pi$
$229$	23.7516	1.56955	0.784774	+	0.619781i	$0.212779\pi$
$230$	0	0
$231$	0.544484	0.0358244
$232$	0	0
$233$	28.0553	1.83796	0.918981	−	0.394302i	$-0.129014\pi$
$233$	28.0553	1.83796	0.918981	+	0.394302i	$0.129014\pi$
$234$	0	0
$235$	20.4471	1.33382
$236$	0	0
$237$	1.34688	0.0874890
$238$	0	0
$239$	−10.5733	−0.683929	−0.341965	−	0.939713i	$-0.611092\pi$
$239$	−10.5733	−0.683929	−0.341965	+	0.939713i	$0.611092\pi$
$240$	0	0
$241$	15.9003	1.02423	0.512115	−	0.858917i	$-0.328862\pi$
$241$	15.9003	1.02423	0.512115	+	0.858917i	$0.328862\pi$
$242$	0	0
$243$	11.4780	0.736316
$244$	0	0
$245$	−23.7717	−1.51872
$246$	0	0
$247$	3.70816	0.235945
$248$	0	0
$249$	−1.87995	−0.119137
$250$	0	0
$251$	6.35844	0.401341	0.200671	−	0.979659i	$-0.435688\pi$
$251$	6.35844	0.401341	0.200671	+	0.979659i	$0.435688\pi$
$252$	0	0
$253$	5.14065	0.323190
$254$	0	0
$255$	−9.49602	−0.594664
$256$	0	0
$257$	5.93035	0.369925	0.184963	−	0.982746i	$-0.440784\pi$
$257$	5.93035	0.369925	0.184963	+	0.982746i	$0.440784\pi$
$258$	0	0
$259$	2.72454	0.169295
$260$	0	0
$261$	−2.41554	−0.149518
$262$	0	0
$263$	−13.7282	−0.846519	−0.423260	−	0.906008i	$-0.639114\pi$
$263$	−13.7282	−0.846519	−0.423260	+	0.906008i	$0.639114\pi$
$264$	0	0
$265$	−32.0216	−1.96707
$266$	0	0
$267$	13.3113	0.814640
$268$	0	0
$269$	31.0438	1.89277	0.946387	−	0.323034i	$-0.104703\pi$
$269$	31.0438	1.89277	0.946387	+	0.323034i	$0.104703\pi$
$270$	0	0
$271$	20.2955	1.23286	0.616432	−	0.787408i	$-0.288577\pi$
$271$	20.2955	1.23286	0.616432	+	0.787408i	$0.288577\pi$
$272$	0	0
$273$	1.48557	0.0899110
$274$	0	0
$275$	9.28214	0.559734
$276$	0	0
$277$	−4.44535	−0.267095	−0.133548	−	0.991042i	$-0.542637\pi$
$277$	−4.44535	−0.267095	−0.133548	+	0.991042i	$0.542637\pi$
$278$	0	0
$279$	−2.27266	−0.136060
$280$	0	0
$281$	−6.58004	−0.392532	−0.196266	−	0.980551i	$-0.562882\pi$
$281$	−6.58004	−0.392532	−0.196266	+	0.980551i	$0.562882\pi$
$282$	0	0
$283$	−25.9630	−1.54334	−0.771670	−	0.636024i	$-0.780578\pi$
$283$	−25.9630	−1.54334	−0.771670	+	0.636024i	$0.780578\pi$
$284$	0	0
$285$	4.63248	0.274405
$286$	0	0
$287$	0.672636	0.0397045
$288$	0	0
$289$	−12.7980	−0.752824
$290$	0	0
$291$	12.8916	0.755717
$292$	0	0
$293$	17.5594	1.02583	0.512915	−	0.858440i	$-0.328566\pi$
$293$	17.5594	1.02583	0.512915	+	0.858440i	$0.328566\pi$
$294$	0	0
$295$	20.4093	1.18828
$296$	0	0
$297$	−7.66246	−0.444621
$298$	0	0
$299$	14.0258	0.811132
$300$	0	0
$301$	−1.12564	−0.0648809
$302$	0	0
$303$	2.11461	0.121481
$304$	0	0
$305$	−18.8754	−1.08080
$306$	0	0
$307$	−13.3869	−0.764033	−0.382017	−	0.924155i	$-0.624770\pi$
$307$	−13.3869	−0.764033	−0.382017	+	0.924155i	$0.624770\pi$
$308$	0	0
$309$	−5.57941	−0.317401
$310$	0	0
$311$	−18.4633	−1.04696	−0.523478	−	0.852039i	$-0.675366\pi$
$311$	−18.4633	−1.04696	−0.523478	+	0.852039i	$0.675366\pi$
$312$	0	0
$313$	−22.1865	−1.25406	−0.627028	−	0.778997i	$-0.715729\pi$
$313$	−22.1865	−1.25406	−0.627028	+	0.778997i	$0.715729\pi$
$314$	0	0
$315$	−1.21325	−0.0683591
$316$	0	0
$317$	−3.16354	−0.177682	−0.0888411	−	0.996046i	$-0.528316\pi$
$317$	−3.16354	−0.177682	−0.0888411	+	0.996046i	$0.528316\pi$
$318$	0	0
$319$	2.76825	0.154992
$320$	0	0
$321$	−3.40398	−0.189992
$322$	0	0
$323$	−2.04988	−0.114058
$324$	0	0
$325$	25.3255	1.40480
$326$	0	0
$327$	−0.510746	−0.0282443
$328$	0	0
$329$	1.76829	0.0974890
$330$	0	0
$331$	−29.5792	−1.62582	−0.812910	−	0.582389i	$-0.802118\pi$
$331$	−29.5792	−1.62582	−0.812910	+	0.582389i	$0.802118\pi$
$332$	0	0
$333$	−10.8628	−0.595279
$334$	0	0
$335$	−17.7443	−0.969475
$336$	0	0
$337$	29.1777	1.58941	0.794704	−	0.606997i	$-0.207626\pi$
$337$	29.1777	1.58941	0.794704	+	0.606997i	$0.207626\pi$
$338$	0	0
$339$	9.79089	0.531768
$340$	0	0
$341$	2.60451	0.141042
$342$	0	0
$343$	−4.13793	−0.223427
$344$	0	0
$345$	17.5219	0.943350
$346$	0	0
$347$	8.09513	0.434569	0.217285	−	0.976108i	$-0.430280\pi$
$347$	8.09513	0.434569	0.217285	+	0.976108i	$0.430280\pi$
$348$	0	0
$349$	−0.476581	−0.0255108	−0.0127554	−	0.999919i	$-0.504060\pi$
$349$	−0.476581	−0.0255108	−0.0127554	+	0.999919i	$0.504060\pi$
$350$	0	0
$351$	−20.9063	−1.11590
$352$	0	0
$353$	−24.1300	−1.28431	−0.642156	−	0.766574i	$-0.721960\pi$
$353$	−24.1300	−1.28431	−0.642156	+	0.766574i	$0.721960\pi$
$354$	0	0
$355$	−31.3464	−1.66369
$356$	0	0
$357$	−0.821227	−0.0434640
$358$	0	0
$359$	−30.7865	−1.62485	−0.812424	−	0.583067i	$-0.801853\pi$
$359$	−30.7865	−1.62485	−0.812424	+	0.583067i	$0.801853\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−12.3278	−0.647040
$364$	0	0
$365$	47.9838	2.51159
$366$	0	0
$367$	−1.45503	−0.0759519	−0.0379759	−	0.999279i	$-0.512091\pi$
$367$	−1.45503	−0.0759519	−0.0379759	+	0.999279i	$0.512091\pi$
$368$	0	0
$369$	−2.68182	−0.139610
$370$	0	0
$371$	−2.76926	−0.143773
$372$	0	0
$373$	−19.8929	−1.03002	−0.515009	−	0.857185i	$-0.672211\pi$
$373$	−19.8929	−1.03002	−0.515009	+	0.857185i	$0.672211\pi$
$374$	0	0
$375$	8.47585	0.437691
$376$	0	0
$377$	7.55293	0.388996
$378$	0	0
$379$	−7.46500	−0.383451	−0.191726	−	0.981449i	$-0.561408\pi$
$379$	−7.46500	−0.383451	−0.191726	+	0.981449i	$0.561408\pi$
$380$	0	0
$381$	−3.41945	−0.175184
$382$	0	0
$383$	−21.5139	−1.09931	−0.549654	−	0.835392i	$-0.685241\pi$
$383$	−21.5139	−1.09931	−0.549654	+	0.835392i	$0.685241\pi$
$384$	0	0
$385$	1.39041	0.0708619
$386$	0	0
$387$	4.48797	0.228136
$388$	0	0
$389$	2.75959	0.139917	0.0699583	−	0.997550i	$-0.477713\pi$
$389$	2.75959	0.139917	0.0699583	+	0.997550i	$0.477713\pi$
$390$	0	0
$391$	−7.75348	−0.392110
$392$	0	0
$393$	19.7105	0.994264
$394$	0	0
$395$	3.43943	0.173056
$396$	0	0
$397$	−10.1903	−0.511437	−0.255719	−	0.966751i	$-0.582312\pi$
$397$	−10.1903	−0.511437	−0.255719	+	0.966751i	$0.582312\pi$
$398$	0	0
$399$	0.400623	0.0200562
$400$	0	0
$401$	−10.6052	−0.529598	−0.264799	−	0.964304i	$-0.585306\pi$
$401$	−10.6052	−0.529598	−0.264799	+	0.964304i	$0.585306\pi$
$402$	0	0
$403$	7.10616	0.353983
$404$	0	0
$405$	−13.8809	−0.689745
$406$	0	0
$407$	12.4490	0.617074
$408$	0	0
$409$	−21.4432	−1.06030	−0.530149	−	0.847904i	$-0.677864\pi$
$409$	−21.4432	−1.06030	−0.530149	+	0.847904i	$0.677864\pi$
$410$	0	0
$411$	12.6667	0.624802
$412$	0	0
$413$	1.76502	0.0868512
$414$	0	0
$415$	−4.80070	−0.235657
$416$	0	0
$417$	19.1212	0.936371
$418$	0	0
$419$	17.9009	0.874514	0.437257	−	0.899337i	$-0.355950\pi$
$419$	17.9009	0.874514	0.437257	+	0.899337i	$0.355950\pi$
$420$	0	0
$421$	−2.37614	−0.115806	−0.0579030	−	0.998322i	$-0.518441\pi$
$421$	−2.37614	−0.115806	−0.0579030	+	0.998322i	$0.518441\pi$
$422$	0	0
$423$	−7.05023	−0.342794
$424$	0	0
$425$	−14.0000	−0.679098
$426$	0	0
$427$	−1.63236	−0.0789956
$428$	0	0
$429$	6.78790	0.327723
$430$	0	0
$431$	−31.0088	−1.49364	−0.746820	−	0.665026i	$-0.768420\pi$
$431$	−31.0088	−1.49364	−0.746820	+	0.665026i	$0.768420\pi$
$432$	0	0
$433$	24.5286	1.17877	0.589384	−	0.807853i	$-0.299371\pi$
$433$	24.5286	1.17877	0.589384	+	0.807853i	$0.299371\pi$
$434$	0	0
$435$	9.43562	0.452403
$436$	0	0
$437$	3.78241	0.180937
$438$	0	0
$439$	35.4518	1.69202	0.846011	−	0.533166i	$-0.178998\pi$
$439$	35.4518	1.69202	0.846011	+	0.533166i	$0.178998\pi$
$440$	0	0
$441$	8.19655	0.390312
$442$	0	0
$443$	−27.8361	−1.32253	−0.661267	−	0.750150i	$-0.729981\pi$
$443$	−27.8361	−1.32253	−0.661267	+	0.750150i	$0.729981\pi$
$444$	0	0
$445$	33.9923	1.61139
$446$	0	0
$447$	−12.9939	−0.614588
$448$	0	0
$449$	−7.30547	−0.344766	−0.172383	−	0.985030i	$-0.555147\pi$
$449$	−7.30547	−0.344766	−0.172383	+	0.985030i	$0.555147\pi$
$450$	0	0
$451$	3.07342	0.144722
$452$	0	0
$453$	−11.8844	−0.558376
$454$	0	0
$455$	3.79361	0.177847
$456$	0	0
$457$	31.6924	1.48251	0.741255	−	0.671224i	$-0.234231\pi$
$457$	31.6924	1.48251	0.741255	+	0.671224i	$0.234231\pi$
$458$	0	0
$459$	11.5570	0.539437
$460$	0	0
$461$	5.17203	0.240886	0.120443	−	0.992720i	$-0.461569\pi$
$461$	5.17203	0.240886	0.120443	+	0.992720i	$0.461569\pi$
$462$	0	0
$463$	−28.9859	−1.34709	−0.673545	−	0.739146i	$-0.735229\pi$
$463$	−28.9859	−1.34709	−0.673545	+	0.739146i	$0.735229\pi$
$464$	0	0
$465$	8.87749	0.411684
$466$	0	0
$467$	7.94674	0.367731	0.183866	−	0.982951i	$-0.441139\pi$
$467$	7.94674	0.367731	0.183866	+	0.982951i	$0.441139\pi$
$468$	0	0
$469$	−1.53455	−0.0708589
$470$	0	0
$471$	−4.14776	−0.191119
$472$	0	0
$473$	−5.14330	−0.236489
$474$	0	0
$475$	6.82966	0.313366
$476$	0	0
$477$	11.0411	0.505539
$478$	0	0
$479$	−32.7410	−1.49597	−0.747986	−	0.663714i	$-0.768979\pi$
$479$	−32.7410	−1.49597	−0.747986	+	0.663714i	$0.768979\pi$
$480$	0	0
$481$	33.9660	1.54871
$482$	0	0
$483$	1.51532	0.0689494
$484$	0	0
$485$	32.9203	1.49483
$486$	0	0
$487$	0.998888	0.0452639	0.0226320	−	0.999744i	$-0.492795\pi$
$487$	0.998888	0.0452639	0.0226320	+	0.999744i	$0.492795\pi$
$488$	0	0
$489$	6.93148	0.313452
$490$	0	0
$491$	−11.1738	−0.504268	−0.252134	−	0.967692i	$-0.581132\pi$
$491$	−11.1738	−0.504268	−0.252134	+	0.967692i	$0.581132\pi$
$492$	0	0
$493$	−4.17527	−0.188045
$494$	0	0
$495$	−5.54361	−0.249167
$496$	0	0
$497$	−2.71087	−0.121599
$498$	0	0
$499$	2.76634	0.123838	0.0619192	−	0.998081i	$-0.480278\pi$
$499$	2.76634	0.123838	0.0619192	+	0.998081i	$0.480278\pi$
$500$	0	0
$501$	8.44884	0.377466
$502$	0	0
$503$	−10.9170	−0.486765	−0.243382	−	0.969930i	$-0.578257\pi$
$503$	−10.9170	−0.486765	−0.243382	+	0.969930i	$0.578257\pi$
$504$	0	0
$505$	5.39993	0.240294
$506$	0	0
$507$	1.01078	0.0448905
$508$	0	0
$509$	10.9746	0.486442	0.243221	−	0.969971i	$-0.421796\pi$
$509$	10.9746	0.486442	0.243221	+	0.969971i	$0.421796\pi$
$510$	0	0
$511$	4.14970	0.183572
$512$	0	0
$513$	−5.63792	−0.248920
$514$	0	0
$515$	−14.2478	−0.627831
$516$	0	0
$517$	8.07970	0.355345
$518$	0	0
$519$	−14.8266	−0.650816
$520$	0	0
$521$	−30.5095	−1.33664	−0.668322	−	0.743872i	$-0.732987\pi$
$521$	−30.5095	−1.33664	−0.668322	+	0.743872i	$0.732987\pi$
$522$	0	0
$523$	23.7750	1.03961	0.519804	−	0.854286i	$-0.326005\pi$
$523$	23.7750	1.03961	0.519804	+	0.854286i	$0.326005\pi$
$524$	0	0
$525$	2.73612	0.119414
$526$	0	0
$527$	−3.92830	−0.171119
$528$	0	0
$529$	−8.69337	−0.377973
$530$	0	0
$531$	−7.03720	−0.305389
$532$	0	0
$533$	8.38554	0.363218
$534$	0	0
$535$	−8.69251	−0.375810
$536$	0	0
$537$	−22.4794	−0.970060
$538$	0	0
$539$	−9.39341	−0.404603
$540$	0	0
$541$	−4.80643	−0.206644	−0.103322	−	0.994648i	$-0.532947\pi$
$541$	−4.80643	−0.206644	−0.103322	+	0.994648i	$0.532947\pi$
$542$	0	0
$543$	−24.5551	−1.05376
$544$	0	0
$545$	−1.30426	−0.0558682
$546$	0	0
$547$	13.7538	0.588071	0.294035	−	0.955795i	$-0.405002\pi$
$547$	13.7538	0.588071	0.294035	+	0.955795i	$0.405002\pi$
$548$	0	0
$549$	6.50828	0.277767
$550$	0	0
$551$	2.03684	0.0867723
$552$	0	0
$553$	0.297446	0.0126487
$554$	0	0
$555$	42.4325	1.80116
$556$	0	0
$557$	43.5298	1.84442	0.922208	−	0.386695i	$-0.126384\pi$
$557$	43.5298	1.84442	0.922208	+	0.386695i	$0.126384\pi$
$558$	0	0
$559$	−14.0330	−0.593533
$560$	0	0
$561$	−3.75236	−0.158425
$562$	0	0
$563$	10.3318	0.435435	0.217717	−	0.976012i	$-0.430139\pi$
$563$	10.3318	0.435435	0.217717	+	0.976012i	$0.430139\pi$
$564$	0	0
$565$	25.0023	1.05186
$566$	0	0
$567$	−1.20043	−0.0504135
$568$	0	0
$569$	−2.89474	−0.121354	−0.0606770	−	0.998157i	$-0.519326\pi$
$569$	−2.89474	−0.121354	−0.0606770	+	0.998157i	$0.519326\pi$
$570$	0	0
$571$	14.0026	0.585992	0.292996	−	0.956114i	$-0.405348\pi$
$571$	14.0026	0.585992	0.292996	+	0.956114i	$0.405348\pi$
$572$	0	0
$573$	1.26274	0.0527519
$574$	0	0
$575$	25.8326	1.07729
$576$	0	0
$577$	−10.1939	−0.424376	−0.212188	−	0.977229i	$-0.568059\pi$
$577$	−10.1939	−0.424376	−0.212188	+	0.977229i	$0.568059\pi$
$578$	0	0
$579$	−7.33446	−0.304810
$580$	0	0
$581$	−0.415170	−0.0172242
$582$	0	0
$583$	−12.6534	−0.524049
$584$	0	0
$585$	−15.1252	−0.625351
$586$	0	0
$587$	−5.86124	−0.241919	−0.120960	−	0.992657i	$-0.538597\pi$
$587$	−5.86124	−0.241919	−0.120960	+	0.992657i	$0.538597\pi$
$588$	0	0
$589$	1.91636	0.0789621
$590$	0	0
$591$	6.28057	0.258348
$592$	0	0
$593$	−33.7014	−1.38395	−0.691975	−	0.721922i	$-0.743259\pi$
$593$	−33.7014	−1.38395	−0.691975	+	0.721922i	$0.743259\pi$
$594$	0	0
$595$	−2.09711	−0.0859733
$596$	0	0
$597$	−2.45662	−0.100543
$598$	0	0
$599$	17.7665	0.725918	0.362959	−	0.931805i	$-0.381767\pi$
$599$	17.7665	0.725918	0.362959	+	0.931805i	$0.381767\pi$
$600$	0	0
$601$	−13.3128	−0.543040	−0.271520	−	0.962433i	$-0.587526\pi$
$601$	−13.3128	−0.543040	−0.271520	+	0.962433i	$0.587526\pi$
$602$	0	0
$603$	6.11829	0.249156
$604$	0	0
$605$	−31.4806	−1.27987
$606$	0	0
$607$	22.3791	0.908339	0.454170	−	0.890915i	$-0.349936\pi$
$607$	22.3791	0.908339	0.454170	+	0.890915i	$0.349936\pi$
$608$	0	0
$609$	0.816004	0.0330661
$610$	0	0
$611$	22.0447	0.891834
$612$	0	0
$613$	3.65730	0.147717	0.0738585	−	0.997269i	$-0.476469\pi$
$613$	3.65730	0.147717	0.0738585	+	0.997269i	$0.476469\pi$
$614$	0	0
$615$	10.4758	0.422424
$616$	0	0
$617$	24.9447	1.00424	0.502119	−	0.864799i	$-0.332554\pi$
$617$	24.9447	1.00424	0.502119	+	0.864799i	$0.332554\pi$
$618$	0	0
$619$	−3.25300	−0.130749	−0.0653746	−	0.997861i	$-0.520824\pi$
$619$	−3.25300	−0.130749	−0.0653746	+	0.997861i	$0.520824\pi$
$620$	0	0
$621$	−21.3249	−0.855740
$622$	0	0
$623$	2.93969	0.117776
$624$	0	0
$625$	−12.5041	−0.500163
$626$	0	0
$627$	1.83053	0.0731043
$628$	0	0
$629$	−18.7764	−0.748665
$630$	0	0
$631$	23.4364	0.932987	0.466494	−	0.884525i	$-0.345517\pi$
$631$	23.4364	0.932987	0.466494	+	0.884525i	$0.345517\pi$
$632$	0	0
$633$	6.66642	0.264966
$634$	0	0
$635$	−8.73203	−0.346520
$636$	0	0
$637$	−25.6291	−1.01546
$638$	0	0
$639$	10.8083	0.427571
$640$	0	0
$641$	0.0637270	0.00251706	0.00125853	−	0.999999i	$-0.499599\pi$
$641$	0.0637270	0.00251706	0.00125853	+	0.999999i	$0.499599\pi$
$642$	0	0
$643$	−26.9659	−1.06343	−0.531715	−	0.846923i	$-0.678452\pi$
$643$	−26.9659	−1.06343	−0.531715	+	0.846923i	$0.678452\pi$
$644$	0	0
$645$	−17.5310	−0.690281
$646$	0	0
$647$	32.6105	1.28205	0.641025	−	0.767520i	$-0.278510\pi$
$647$	32.6105	1.28205	0.641025	+	0.767520i	$0.278510\pi$
$648$	0	0
$649$	8.06477	0.316570
$650$	0	0
$651$	0.767736	0.0300899
$652$	0	0
$653$	34.9059	1.36597	0.682986	−	0.730432i	$-0.260681\pi$
$653$	34.9059	1.36597	0.682986	+	0.730432i	$0.260681\pi$
$654$	0	0
$655$	50.3334	1.96669
$656$	0	0
$657$	−16.5450	−0.645481
$658$	0	0
$659$	−2.18397	−0.0850753	−0.0425376	−	0.999095i	$-0.513544\pi$
$659$	−2.18397	−0.0850753	−0.0425376	+	0.999095i	$0.513544\pi$
$660$	0	0
$661$	−5.46077	−0.212399	−0.106200	−	0.994345i	$-0.533868\pi$
$661$	−5.46077	−0.212399	−0.106200	+	0.994345i	$0.533868\pi$
$662$	0	0
$663$	−10.2380	−0.397610
$664$	0	0
$665$	1.02304	0.0396719
$666$	0	0
$667$	7.70416	0.298306
$668$	0	0
$669$	12.7034	0.491141
$670$	0	0
$671$	−7.45862	−0.287937
$672$	0	0
$673$	28.0802	1.08241	0.541206	−	0.840890i	$-0.317968\pi$
$673$	28.0802	1.08241	0.541206	+	0.840890i	$0.317968\pi$
$674$	0	0
$675$	−38.5051	−1.48206
$676$	0	0
$677$	28.8554	1.10900	0.554502	−	0.832183i	$-0.312909\pi$
$677$	28.8554	1.10900	0.554502	+	0.832183i	$0.312909\pi$
$678$	0	0
$679$	2.84699	0.109257
$680$	0	0
$681$	−33.6399	−1.28908
$682$	0	0
$683$	−48.4346	−1.85330	−0.926649	−	0.375928i	$-0.877324\pi$
$683$	−48.4346	−1.85330	−0.926649	+	0.375928i	$0.877324\pi$
$684$	0	0
$685$	32.3461	1.23588
$686$	0	0
$687$	31.9904	1.22051
$688$	0	0
$689$	−34.5235	−1.31524
$690$	0	0
$691$	−9.30232	−0.353877	−0.176938	−	0.984222i	$-0.556619\pi$
$691$	−9.30232	−0.353877	−0.176938	+	0.984222i	$0.556619\pi$
$692$	0	0
$693$	−0.479418	−0.0182116
$694$	0	0
$695$	48.8286	1.85218
$696$	0	0
$697$	−4.63554	−0.175584
$698$	0	0
$699$	37.7870	1.42923
$700$	0	0
$701$	−7.62336	−0.287930	−0.143965	−	0.989583i	$-0.545985\pi$
$701$	−7.62336	−0.287930	−0.143965	+	0.989583i	$0.545985\pi$
$702$	0	0
$703$	9.15978	0.345468
$704$	0	0
$705$	27.5397	1.03721
$706$	0	0
$707$	0.466992	0.0175631
$708$	0	0
$709$	46.9466	1.76312	0.881558	−	0.472076i	$-0.156495\pi$
$709$	46.9466	1.76312	0.881558	+	0.472076i	$0.156495\pi$
$710$	0	0
$711$	−1.18593	−0.0444757
$712$	0	0
$713$	7.24845	0.271456
$714$	0	0
$715$	17.3338	0.648248
$716$	0	0
$717$	−14.2409	−0.531836
$718$	0	0
$719$	18.0681	0.673828	0.336914	−	0.941535i	$-0.390617\pi$
$719$	18.0681	0.673828	0.336914	+	0.941535i	$0.390617\pi$
$720$	0	0
$721$	−1.23216	−0.0458882
$722$	0	0
$723$	21.4158	0.796462
$724$	0	0
$725$	13.9109	0.516638
$726$	0	0
$727$	−0.973868	−0.0361188	−0.0180594	−	0.999837i	$-0.505749\pi$
$727$	−0.973868	−0.0361188	−0.0180594	+	0.999837i	$0.505749\pi$
$728$	0	0
$729$	27.5669	1.02100
$730$	0	0
$731$	7.75747	0.286920
$732$	0	0
$733$	−18.2180	−0.672899	−0.336449	−	0.941702i	$-0.609226\pi$
$733$	−18.2180	−0.672899	−0.336449	+	0.941702i	$0.609226\pi$
$734$	0	0
$735$	−32.0175	−1.18098
$736$	0	0
$737$	−7.01168	−0.258279
$738$	0	0
$739$	−14.5933	−0.536823	−0.268411	−	0.963304i	$-0.586499\pi$
$739$	−14.5933	−0.536823	−0.268411	+	0.963304i	$0.586499\pi$
$740$	0	0
$741$	4.99443	0.183475
$742$	0	0
$743$	8.16279	0.299464	0.149732	−	0.988727i	$-0.452159\pi$
$743$	8.16279	0.299464	0.149732	+	0.988727i	$0.452159\pi$
$744$	0	0
$745$	−33.1815	−1.21568
$746$	0	0
$747$	1.65530	0.0605641
$748$	0	0
$749$	−0.751739	−0.0274680
$750$	0	0
$751$	−19.7782	−0.721717	−0.360858	−	0.932621i	$-0.617516\pi$
$751$	−19.7782	−0.721717	−0.360858	+	0.932621i	$0.617516\pi$
$752$	0	0
$753$	8.56404	0.312091
$754$	0	0
$755$	−30.3483	−1.10449
$756$	0	0
$757$	−37.5588	−1.36510	−0.682549	−	0.730839i	$-0.739129\pi$
$757$	−37.5588	−1.36510	−0.682549	+	0.730839i	$0.739129\pi$
$758$	0	0
$759$	6.92382	0.251319
$760$	0	0
$761$	34.8610	1.26371	0.631856	−	0.775086i	$-0.282294\pi$
$761$	34.8610	1.26371	0.631856	+	0.775086i	$0.282294\pi$
$762$	0	0
$763$	−0.112794	−0.00408341
$764$	0	0
$765$	8.36125	0.302302
$766$	0	0
$767$	22.0040	0.794518
$768$	0	0
$769$	28.2390	1.01832	0.509162	−	0.860671i	$-0.329955\pi$
$769$	28.2390	1.01832	0.509162	+	0.860671i	$0.329955\pi$
$770$	0	0
$771$	7.98745	0.287661
$772$	0	0
$773$	−16.3073	−0.586532	−0.293266	−	0.956031i	$-0.594742\pi$
$773$	−16.3073	−0.586532	−0.293266	+	0.956031i	$0.594742\pi$
$774$	0	0
$775$	13.0881	0.470137
$776$	0	0
$777$	3.66962	0.131647
$778$	0	0
$779$	2.26137	0.0810221
$780$	0	0
$781$	−12.3865	−0.443225
$782$	0	0
$783$	−11.4835	−0.410388
$784$	0	0
$785$	−10.5919	−0.378040
$786$	0	0
$787$	42.0033	1.49726	0.748629	−	0.662990i	$-0.230713\pi$
$787$	42.0033	1.49726	0.748629	+	0.662990i	$0.230713\pi$
$788$	0	0
$789$	−18.4902	−0.658270
$790$	0	0
$791$	2.16223	0.0768802
$792$	0	0
$793$	−20.3502	−0.722655
$794$	0	0
$795$	−43.1291	−1.52963
$796$	0	0
$797$	17.4998	0.619875	0.309937	−	0.950757i	$-0.399692\pi$
$797$	17.4998	0.619875	0.309937	+	0.950757i	$0.399692\pi$
$798$	0	0
$799$	−12.1864	−0.431122
$800$	0	0
$801$	−11.7206	−0.414128
$802$	0	0
$803$	18.9608	0.669114
$804$	0	0
$805$	3.86957	0.136384
$806$	0	0
$807$	41.8122	1.47186
$808$	0	0
$809$	10.5173	0.369770	0.184885	−	0.982760i	$-0.440809\pi$
$809$	10.5173	0.369770	0.184885	+	0.982760i	$0.440809\pi$
$810$	0	0
$811$	47.4567	1.66643	0.833215	−	0.552949i	$-0.186498\pi$
$811$	47.4567	1.66643	0.833215	+	0.552949i	$0.186498\pi$
$812$	0	0
$813$	27.3355	0.958699
$814$	0	0
$815$	17.7005	0.620020
$816$	0	0
$817$	−3.78436	−0.132398
$818$	0	0
$819$	−1.30805	−0.0457069
$820$	0	0
$821$	−27.4214	−0.957013	−0.478506	−	0.878084i	$-0.658822\pi$
$821$	−27.4214	−0.957013	−0.478506	+	0.878084i	$0.658822\pi$
$822$	0	0
$823$	−11.8338	−0.412501	−0.206251	−	0.978499i	$-0.566126\pi$
$823$	−11.8338	−0.412501	−0.206251	+	0.978499i	$0.566126\pi$
$824$	0	0
$825$	12.5019	0.435260
$826$	0	0
$827$	8.04868	0.279880	0.139940	−	0.990160i	$-0.455309\pi$
$827$	8.04868	0.279880	0.139940	+	0.990160i	$0.455309\pi$
$828$	0	0
$829$	31.0128	1.07712	0.538559	−	0.842588i	$-0.318969\pi$
$829$	31.0128	1.07712	0.538559	+	0.842588i	$0.318969\pi$
$830$	0	0
$831$	−5.98733	−0.207698
$832$	0	0
$833$	14.1678	0.490884
$834$	0	0
$835$	21.5752	0.746642
$836$	0	0
$837$	−10.8043	−0.373450
$838$	0	0
$839$	−5.29748	−0.182889	−0.0914446	−	0.995810i	$-0.529148\pi$
$839$	−5.29748	−0.182889	−0.0914446	+	0.995810i	$0.529148\pi$
$840$	0	0
$841$	−24.8513	−0.856941
$842$	0	0
$843$	−8.86249	−0.305241
$844$	0	0
$845$	2.58117	0.0887950
$846$	0	0
$847$	−2.72248	−0.0935456
$848$	0	0
$849$	−34.9689	−1.20013
$850$	0	0
$851$	34.6461	1.18765
$852$	0	0
$853$	−50.1445	−1.71692	−0.858458	−	0.512883i	$-0.828577\pi$
$853$	−50.1445	−1.71692	−0.858458	+	0.512883i	$0.828577\pi$
$854$	0	0
$855$	−4.07890	−0.139496
$856$	0	0
$857$	−6.32360	−0.216010	−0.108005	−	0.994150i	$-0.534446\pi$
$857$	−6.32360	−0.216010	−0.108005	+	0.994150i	$0.534446\pi$
$858$	0	0
$859$	−9.25778	−0.315871	−0.157936	−	0.987449i	$-0.550484\pi$
$859$	−9.25778	−0.315871	−0.157936	+	0.987449i	$0.550484\pi$
$860$	0	0
$861$	0.905957	0.0308749
$862$	0	0
$863$	8.49298	0.289104	0.144552	−	0.989497i	$-0.453826\pi$
$863$	8.49298	0.289104	0.144552	+	0.989497i	$0.453826\pi$
$864$	0	0
$865$	−37.8617	−1.28734
$866$	0	0
$867$	−17.2373	−0.585410
$868$	0	0
$869$	1.35909	0.0461041
$870$	0	0
$871$	−19.1307	−0.648220
$872$	0	0
$873$	−11.3510	−0.384174
$874$	0	0
$875$	1.87182	0.0632790
$876$	0	0
$877$	−12.9924	−0.438721	−0.219361	−	0.975644i	$-0.570397\pi$
$877$	−12.9924	−0.438721	−0.219361	+	0.975644i	$0.570397\pi$
$878$	0	0
$879$	23.6503	0.797705
$880$	0	0
$881$	10.6775	0.359735	0.179867	−	0.983691i	$-0.442433\pi$
$881$	10.6775	0.359735	0.179867	+	0.983691i	$0.442433\pi$
$882$	0	0
$883$	−28.4961	−0.958971	−0.479486	−	0.877550i	$-0.659177\pi$
$883$	−28.4961	−0.958971	−0.479486	+	0.877550i	$0.659177\pi$
$884$	0	0
$885$	27.4888	0.924027
$886$	0	0
$887$	−44.7321	−1.50196	−0.750979	−	0.660326i	$-0.770418\pi$
$887$	−44.7321	−1.50196	−0.750979	+	0.660326i	$0.770418\pi$
$888$	0	0
$889$	−0.755157	−0.0253271
$890$	0	0
$891$	−5.48504	−0.183756
$892$	0	0
$893$	5.94492	0.198939
$894$	0	0
$895$	−57.4043	−1.91881
$896$	0	0
$897$	18.8910	0.630752
$898$	0	0
$899$	3.90331	0.130183
$900$	0	0
$901$	19.0847	0.635802
$902$	0	0
$903$	−1.51610	−0.0504527
$904$	0	0
$905$	−62.7048	−2.08438
$906$	0	0
$907$	32.2452	1.07068	0.535342	−	0.844635i	$-0.320183\pi$
$907$	32.2452	1.07068	0.535342	+	0.844635i	$0.320183\pi$
$908$	0	0
$909$	−1.86191	−0.0617557
$910$	0	0
$911$	−6.88165	−0.227999	−0.114000	−	0.993481i	$-0.536366\pi$
$911$	−6.88165	−0.227999	−0.114000	+	0.993481i	$0.536366\pi$
$912$	0	0
$913$	−1.89700	−0.0627816
$914$	0	0
$915$	−25.4228	−0.840451
$916$	0	0
$917$	4.35290	0.143745
$918$	0	0
$919$	4.10588	0.135440	0.0677202	−	0.997704i	$-0.478427\pi$
$919$	4.10588	0.135440	0.0677202	+	0.997704i	$0.478427\pi$
$920$	0	0
$921$	−18.0306	−0.594127
$922$	0	0
$923$	−33.7956	−1.11239
$924$	0	0
$925$	62.5582	2.05690
$926$	0	0
$927$	4.91267	0.161353
$928$	0	0
$929$	6.24461	0.204879	0.102439	−	0.994739i	$-0.467335\pi$
$929$	6.24461	0.204879	0.102439	+	0.994739i	$0.467335\pi$
$930$	0	0
$931$	−6.91153	−0.226516
$932$	0	0
$933$	−24.8677	−0.814133
$934$	0	0
$935$	−9.58216	−0.313370
$936$	0	0
$937$	−26.4519	−0.864146	−0.432073	−	0.901839i	$-0.642218\pi$
$937$	−26.4519	−0.864146	−0.432073	+	0.901839i	$0.642218\pi$
$938$	0	0
$939$	−29.8825	−0.975178
$940$	0	0
$941$	−20.3990	−0.664987	−0.332494	−	0.943105i	$-0.607890\pi$
$941$	−20.3990	−0.664987	−0.332494	+	0.943105i	$0.607890\pi$
$942$	0	0
$943$	8.55344	0.278538
$944$	0	0
$945$	−5.76784	−0.187628
$946$	0	0
$947$	−56.1409	−1.82433	−0.912166	−	0.409820i	$-0.865591\pi$
$947$	−56.1409	−1.82433	−0.912166	+	0.409820i	$0.865591\pi$
$948$	0	0
$949$	51.7329	1.67932
$950$	0	0
$951$	−4.26090	−0.138169
$952$	0	0
$953$	2.30565	0.0746872	0.0373436	−	0.999302i	$-0.488110\pi$
$953$	2.30565	0.0746872	0.0373436	+	0.999302i	$0.488110\pi$
$954$	0	0
$955$	3.22458	0.104345
$956$	0	0
$957$	3.72850	0.120525
$958$	0	0
$959$	2.79733	0.0903306
$960$	0	0
$961$	−27.3276	−0.881535
$962$	0	0
$963$	2.99721	0.0965836
$964$	0	0
$965$	−18.7295	−0.602925
$966$	0	0
$967$	−19.6894	−0.633169	−0.316585	−	0.948564i	$-0.602536\pi$
$967$	−19.6894	−0.633169	−0.316585	+	0.948564i	$0.602536\pi$
$968$	0	0
$969$	−2.76093	−0.0886939
$970$	0	0
$971$	−34.6432	−1.11175	−0.555877	−	0.831264i	$-0.687618\pi$
$971$	−34.6432	−1.11175	−0.555877	+	0.831264i	$0.687618\pi$
$972$	0	0
$973$	4.22276	0.135375
$974$	0	0
$975$	34.1103	1.09240
$976$	0	0
$977$	38.8406	1.24262	0.621310	−	0.783565i	$-0.286601\pi$
$977$	38.8406	1.24262	0.621310	+	0.783565i	$0.286601\pi$
$978$	0	0
$979$	13.4321	0.429291
$980$	0	0
$981$	0.449712	0.0143582
$982$	0	0
$983$	−14.1611	−0.451668	−0.225834	−	0.974166i	$-0.572511\pi$
$983$	−14.1611	−0.451668	−0.225834	+	0.974166i	$0.572511\pi$
$984$	0	0
$985$	16.0383	0.511022
$986$	0	0
$987$	2.38167	0.0758093
$988$	0	0
$989$	−14.3140	−0.455159
$990$	0	0
$991$	−49.9601	−1.58703	−0.793517	−	0.608548i	$-0.791752\pi$
$991$	−49.9601	−1.58703	−0.793517	+	0.608548i	$0.791752\pi$
$992$	0	0
$993$	−39.8395	−1.26427
$994$	0	0
$995$	−6.27331	−0.198877
$996$	0	0
$997$	−29.2597	−0.926665	−0.463332	−	0.886185i	$-0.653346\pi$
$997$	−29.2597	−0.926665	−0.463332	+	0.886185i	$0.653346\pi$
$998$	0	0
$999$	−51.6421	−1.63388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.e.1.13	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.e.1.13	✓	24	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q+1.34688 q^{3} +3.43943 q^{5} +0.297446 q^{7} -1.18593 q^{9} +O(q^{10})\)	\(q+1.34688 q^{3} +3.43943 q^{5} +0.297446 q^{7} -1.18593 q^{9} +1.35909 q^{11} +3.70816 q^{13} +4.63248 q^{15} -2.04988 q^{17} +1.00000 q^{19} +0.400623 q^{21} +3.78241 q^{23} +6.82966 q^{25} -5.63792 q^{27} +2.03684 q^{29} +1.91636 q^{31} +1.83053 q^{33} +1.02304 q^{35} +9.15978 q^{37} +4.99443 q^{39} +2.26137 q^{41} -3.78436 q^{43} -4.07890 q^{45} +5.94492 q^{47} -6.91153 q^{49} -2.76093 q^{51} -9.31015 q^{53} +4.67450 q^{55} +1.34688 q^{57} +5.93394 q^{59} -5.48794 q^{61} -0.352749 q^{63} +12.7540 q^{65} -5.15909 q^{67} +5.09444 q^{69} -9.11383 q^{71} +13.9511 q^{73} +9.19870 q^{75} +0.404257 q^{77} +1.00000 q^{79} -4.03581 q^{81} -1.39578 q^{83} -7.05040 q^{85} +2.74337 q^{87} +9.88312 q^{89} +1.10298 q^{91} +2.58109 q^{93} +3.43943 q^{95} +9.57145 q^{97} -1.61178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9}+O(q^{10}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9	\( 24 q + q^{3} + 9 q^{5} + 2 q^{7} + 75 q^{9} + 10 q^{11} + 18 q^{13} + 16 q^{15} + 18 q^{17} + 24 q^{19} + 25 q^{21} + 9 q^{23} + 25 q^{25} + 4 q^{27} + 32 q^{29} + 20 q^{31} - 4 q^{33} + 3 q^{35} + 20 q^{37} + 13 q^{39} + 41 q^{41} - 8 q^{43} + 48 q^{45} - 5 q^{47} + 12 q^{49} + 24 q^{51} + 15 q^{53} + 14 q^{55} + q^{57} + 5 q^{59} - 13 q^{61} + 9 q^{63} + 59 q^{65} - 30 q^{67} + 51 q^{69} + 20 q^{73} - 31 q^{75} + 6 q^{77} + 24 q^{79} + 32 q^{81} + 8 q^{83} + 4 q^{85} - 32 q^{87} + 47 q^{89} - 27 q^{91} + 34 q^{93} + 9 q^{95} + 69 q^{97} - 34 q^{99}+O(q^{100}) \) 24 * q + q^3 + 9 * q^5 + 2 * q^7 + 75 * q^9 + 10 * q^11 + 18 * q^13 + 16 * q^15 + 18 * q^17 + 24 * q^19 + 25 * q^21 + 9 * q^23 + 25 * q^25 + 4 * q^27 + 32 * q^29 + 20 * q^31 - 4 * q^33 + 3 * q^35 + 20 * q^37 + 13 * q^39 + 41 * q^41 - 8 * q^43 + 48 * q^45 - 5 * q^47 + 12 * q^49 + 24 * q^51 + 15 * q^53 + 14 * q^55 + q^57 + 5 * q^59 - 13 * q^61 + 9 * q^63 + 59 * q^65 - 30 * q^67 + 51 * q^69 + 20 * q^73 - 31 * q^75 + 6 * q^77 + 24 * q^79 + 32 * q^81 + 8 * q^83 + 4 * q^85 - 32 * q^87 + 47 * q^89 - 27 * q^91 + 34 * q^93 + 9 * q^95 + 69 * q^97 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more