LMFDB - Embedded newform 4004.2.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(1,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, 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("4004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.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:	$31.9721009693$
Analytic rank:	$0$
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} - 3x^{8} - 19x^{7} + 51x^{6} + 116x^{5} - 247x^{4} - 249x^{3} + 288x^{2} + 189x - 14 $ x^9 - 3x^8 - 19x^7 + 51x^6 + 116x^5 - 247x^4 - 249x^3 + 288x^2 + 189x - 14
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.56319$ of defining polynomial
Character	$\chi$	$=$	4004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56319 q^{3} -1.31026 q^{5} +1.00000 q^{7} +3.56996 q^{9} +O(q^{10})$	$q-2.56319 q^{3} -1.31026 q^{5} +1.00000 q^{7} +3.56996 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.35845 q^{15} -1.78120 q^{17} +4.55568 q^{19} -2.56319 q^{21} -5.54866 q^{23} -3.28322 q^{25} -1.46091 q^{27} +2.85706 q^{29} +9.25463 q^{31} -2.56319 q^{33} -1.31026 q^{35} -3.67962 q^{37} +2.56319 q^{39} +4.34204 q^{41} +1.19868 q^{43} -4.67758 q^{45} -11.6383 q^{47} +1.00000 q^{49} +4.56557 q^{51} -6.45384 q^{53} -1.31026 q^{55} -11.6771 q^{57} -0.985627 q^{59} -0.869309 q^{61} +3.56996 q^{63} +1.31026 q^{65} -3.06256 q^{67} +14.2223 q^{69} -12.6102 q^{71} +14.6810 q^{73} +8.41552 q^{75} +1.00000 q^{77} -5.88790 q^{79} -6.96527 q^{81} -5.74429 q^{83} +2.33384 q^{85} -7.32319 q^{87} +12.5572 q^{89} -1.00000 q^{91} -23.7214 q^{93} -5.96913 q^{95} -0.505117 q^{97} +3.56996 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9	$ 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + 5 q^{17} + 10 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{25} + 27 q^{27} + 14 q^{29} + 11 q^{31} + 3 q^{33} - 3 q^{39} + 14 q^{41} + 8 q^{43} + 4 q^{45} + 10 q^{47} + 9 q^{49} - 15 q^{51} + 21 q^{53} - 8 q^{57} + 23 q^{59} + 34 q^{61} + 20 q^{63} + 10 q^{67} - 16 q^{69} + 4 q^{71} + 9 q^{73} + 30 q^{75} + 9 q^{77} - 34 q^{79} + 69 q^{81} + 15 q^{83} + 5 q^{85} + 39 q^{87} - 9 q^{91} + 3 q^{93} - 64 q^{95} + 15 q^{97} + 20 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + 5 * q^17 + 10 * q^19 + 3 * q^21 + 8 * q^23 + 3 * q^25 + 27 * q^27 + 14 * q^29 + 11 * q^31 + 3 * q^33 - 3 * q^39 + 14 * q^41 + 8 * q^43 + 4 * q^45 + 10 * q^47 + 9 * q^49 - 15 * q^51 + 21 * q^53 - 8 * q^57 + 23 * q^59 + 34 * q^61 + 20 * q^63 + 10 * q^67 - 16 * q^69 + 4 * q^71 + 9 * q^73 + 30 * q^75 + 9 * q^77 - 34 * q^79 + 69 * q^81 + 15 * q^83 + 5 * q^85 + 39 * q^87 - 9 * q^91 + 3 * q^93 - 64 * q^95 + 15 * q^97 + 20 * 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
$3$	−2.56319	−1.47986	−0.739930	−	0.672684i	$-0.765141\pi$
$3$	−2.56319	−1.47986	−0.739930	+	0.672684i	$0.765141\pi$
$4$	0	0
$5$	−1.31026	−0.585966	−0.292983	−	0.956118i	$-0.594648\pi$
$5$	−1.31026	−0.585966	−0.292983	+	0.956118i	$0.594648\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.56996	1.18999
$10$	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
$15$	3.35845	0.867148
$16$	0	0
$17$	−1.78120	−0.432006	−0.216003	−	0.976393i	$-0.569302\pi$
$17$	−1.78120	−0.432006	−0.216003	+	0.976393i	$0.569302\pi$
$18$	0	0
$19$	4.55568	1.04514	0.522572	−	0.852595i	$-0.324972\pi$
$19$	4.55568	1.04514	0.522572	+	0.852595i	$0.324972\pi$
$20$	0	0
$21$	−2.56319	−0.559335
$22$	0	0
$23$	−5.54866	−1.15697	−0.578487	−	0.815691i	$-0.696357\pi$
$23$	−5.54866	−1.15697	−0.578487	+	0.815691i	$0.696357\pi$
$24$	0	0
$25$	−3.28322	−0.656643
$26$	0	0
$27$	−1.46091	−0.281153
$28$	0	0
$29$	2.85706	0.530543	0.265271	−	0.964174i	$-0.414538\pi$
$29$	2.85706	0.530543	0.265271	+	0.964174i	$0.414538\pi$
$30$	0	0
$31$	9.25463	1.66218	0.831090	−	0.556138i	$-0.187717\pi$
$31$	9.25463	1.66218	0.831090	+	0.556138i	$0.187717\pi$
$32$	0	0
$33$	−2.56319	−0.446195
$34$	0	0
$35$	−1.31026	−0.221475
$36$	0	0
$37$	−3.67962	−0.604926	−0.302463	−	0.953161i	$-0.597809\pi$
$37$	−3.67962	−0.604926	−0.302463	+	0.953161i	$0.597809\pi$
$38$	0	0
$39$	2.56319	0.410439
$40$	0	0
$41$	4.34204	0.678113	0.339057	−	0.940766i	$-0.389892\pi$
$41$	4.34204	0.678113	0.339057	+	0.940766i	$0.389892\pi$
$42$	0	0
$43$	1.19868	0.182797	0.0913983	−	0.995814i	$-0.470866\pi$
$43$	1.19868	0.182797	0.0913983	+	0.995814i	$0.470866\pi$
$44$	0	0
$45$	−4.67758	−0.697292
$46$	0	0
$47$	−11.6383	−1.69762	−0.848810	−	0.528697i	$-0.822681\pi$
$47$	−11.6383	−1.69762	−0.848810	+	0.528697i	$0.822681\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.56557	0.639308
$52$	0	0
$53$	−6.45384	−0.886503	−0.443251	−	0.896397i	$-0.646175\pi$
$53$	−6.45384	−0.886503	−0.443251	+	0.896397i	$0.646175\pi$
$54$	0	0
$55$	−1.31026	−0.176676
$56$	0	0
$57$	−11.6771	−1.54667
$58$	0	0
$59$	−0.985627	−0.128318	−0.0641589	−	0.997940i	$-0.520436\pi$
$59$	−0.985627	−0.128318	−0.0641589	+	0.997940i	$0.520436\pi$
$60$	0	0
$61$	−0.869309	−0.111304	−0.0556518	−	0.998450i	$-0.517724\pi$
$61$	−0.869309	−0.111304	−0.0556518	+	0.998450i	$0.517724\pi$
$62$	0	0
$63$	3.56996	0.449772
$64$	0	0
$65$	1.31026	0.162518
$66$	0	0
$67$	−3.06256	−0.374152	−0.187076	−	0.982345i	$-0.559901\pi$
$67$	−3.06256	−0.374152	−0.187076	+	0.982345i	$0.559901\pi$
$68$	0	0
$69$	14.2223	1.71216
$70$	0	0
$71$	−12.6102	−1.49656	−0.748278	−	0.663386i	$-0.769119\pi$
$71$	−12.6102	−1.49656	−0.748278	+	0.663386i	$0.769119\pi$
$72$	0	0
$73$	14.6810	1.71828	0.859138	−	0.511743i	$-0.171000\pi$
$73$	14.6810	1.71828	0.859138	+	0.511743i	$0.171000\pi$
$74$	0	0
$75$	8.41552	0.971740
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−5.88790	−0.662440	−0.331220	−	0.943554i	$-0.607460\pi$
$79$	−5.88790	−0.662440	−0.331220	+	0.943554i	$0.607460\pi$
$80$	0	0
$81$	−6.96527	−0.773919
$82$	0	0
$83$	−5.74429	−0.630518	−0.315259	−	0.949006i	$-0.602091\pi$
$83$	−5.74429	−0.630518	−0.315259	+	0.949006i	$0.602091\pi$
$84$	0	0
$85$	2.33384	0.253141
$86$	0	0
$87$	−7.32319	−0.785129
$88$	0	0
$89$	12.5572	1.33106	0.665532	−	0.746370i	$-0.268205\pi$
$89$	12.5572	1.33106	0.665532	+	0.746370i	$0.268205\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−23.7214	−2.45979
$94$	0	0
$95$	−5.96913	−0.612420
$96$	0	0
$97$	−0.505117	−0.0512869	−0.0256434	−	0.999671i	$-0.508163\pi$
$97$	−0.505117	−0.0512869	−0.0256434	+	0.999671i	$0.508163\pi$
$98$	0	0
$99$	3.56996	0.358794
$100$	0	0
$101$	15.0920	1.50171	0.750855	−	0.660467i	$-0.229642\pi$
$101$	15.0920	1.50171	0.750855	+	0.660467i	$0.229642\pi$
$102$	0	0
$103$	−15.3268	−1.51020	−0.755099	−	0.655611i	$-0.772411\pi$
$103$	−15.3268	−1.51020	−0.755099	+	0.655611i	$0.772411\pi$
$104$	0	0
$105$	3.35845	0.327751
$106$	0	0
$107$	−3.90090	−0.377115	−0.188557	−	0.982062i	$-0.560381\pi$
$107$	−3.90090	−0.377115	−0.188557	+	0.982062i	$0.560381\pi$
$108$	0	0
$109$	5.33805	0.511293	0.255646	−	0.966770i	$-0.417712\pi$
$109$	5.33805	0.511293	0.255646	+	0.966770i	$0.417712\pi$
$110$	0	0
$111$	9.43158	0.895206
$112$	0	0
$113$	0.947829	0.0891642	0.0445821	−	0.999006i	$-0.485804\pi$
$113$	0.947829	0.0891642	0.0445821	+	0.999006i	$0.485804\pi$
$114$	0	0
$115$	7.27019	0.677948
$116$	0	0
$117$	−3.56996	−0.330043
$118$	0	0
$119$	−1.78120	−0.163283
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−11.1295	−1.00351
$124$	0	0
$125$	10.8532	0.970737
$126$	0	0
$127$	−18.7001	−1.65936	−0.829682	−	0.558237i	$-0.811478\pi$
$127$	−18.7001	−1.65936	−0.829682	+	0.558237i	$0.811478\pi$
$128$	0	0
$129$	−3.07244	−0.270513
$130$	0	0
$131$	−13.7408	−1.20054	−0.600270	−	0.799797i	$-0.704940\pi$
$131$	−13.7408	−1.20054	−0.600270	+	0.799797i	$0.704940\pi$
$132$	0	0
$133$	4.55568	0.395028
$134$	0	0
$135$	1.91418	0.164746
$136$	0	0
$137$	−2.49101	−0.212821	−0.106411	−	0.994322i	$-0.533936\pi$
$137$	−2.49101	−0.212821	−0.106411	+	0.994322i	$0.533936\pi$
$138$	0	0
$139$	17.3501	1.47162	0.735808	−	0.677190i	$-0.236803\pi$
$139$	17.3501	1.47162	0.735808	+	0.677190i	$0.236803\pi$
$140$	0	0
$141$	29.8312	2.51224
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−3.74349	−0.310880
$146$	0	0
$147$	−2.56319	−0.211409
$148$	0	0
$149$	3.55721	0.291418	0.145709	−	0.989328i	$-0.453454\pi$
$149$	3.55721	0.291418	0.145709	+	0.989328i	$0.453454\pi$
$150$	0	0
$151$	22.0951	1.79807	0.899035	−	0.437876i	$-0.144269\pi$
$151$	22.0951	1.79807	0.899035	+	0.437876i	$0.144269\pi$
$152$	0	0
$153$	−6.35883	−0.514081
$154$	0	0
$155$	−12.1260	−0.973982
$156$	0	0
$157$	9.00759	0.718884	0.359442	−	0.933167i	$-0.382967\pi$
$157$	9.00759	0.718884	0.359442	+	0.933167i	$0.382967\pi$
$158$	0	0
$159$	16.5424	1.31190
$160$	0	0
$161$	−5.54866	−0.437295
$162$	0	0
$163$	19.4943	1.52691	0.763454	−	0.645862i	$-0.223502\pi$
$163$	19.4943	1.52691	0.763454	+	0.645862i	$0.223502\pi$
$164$	0	0
$165$	3.35845	0.261455
$166$	0	0
$167$	9.38652	0.726351	0.363175	−	0.931721i	$-0.381693\pi$
$167$	9.38652	0.726351	0.363175	+	0.931721i	$0.381693\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	16.2636	1.24371
$172$	0	0
$173$	14.4594	1.09933	0.549664	−	0.835386i	$-0.314756\pi$
$173$	14.4594	1.09933	0.549664	+	0.835386i	$0.314756\pi$
$174$	0	0
$175$	−3.28322	−0.248188
$176$	0	0
$177$	2.52635	0.189892
$178$	0	0
$179$	18.3448	1.37115	0.685576	−	0.728001i	$-0.259550\pi$
$179$	18.3448	1.37115	0.685576	+	0.728001i	$0.259550\pi$
$180$	0	0
$181$	8.34152	0.620020	0.310010	−	0.950733i	$-0.399668\pi$
$181$	8.34152	0.620020	0.310010	+	0.950733i	$0.399668\pi$
$182$	0	0
$183$	2.22821	0.164714
$184$	0	0
$185$	4.82127	0.354466
$186$	0	0
$187$	−1.78120	−0.130255
$188$	0	0
$189$	−1.46091	−0.106266
$190$	0	0
$191$	4.42461	0.320154	0.160077	−	0.987105i	$-0.448826\pi$
$191$	4.42461	0.320154	0.160077	+	0.987105i	$0.448826\pi$
$192$	0	0
$193$	27.2387	1.96068	0.980341	−	0.197308i	$-0.0632200\pi$
$193$	27.2387	1.96068	0.980341	+	0.197308i	$0.0632200\pi$
$194$	0	0
$195$	−3.35845	−0.240504
$196$	0	0
$197$	19.3664	1.37980	0.689899	−	0.723906i	$-0.257655\pi$
$197$	19.3664	1.37980	0.689899	+	0.723906i	$0.257655\pi$
$198$	0	0
$199$	−17.2746	−1.22457	−0.612283	−	0.790639i	$-0.709748\pi$
$199$	−17.2746	−1.22457	−0.612283	+	0.790639i	$0.709748\pi$
$200$	0	0
$201$	7.84995	0.553692
$202$	0	0
$203$	2.85706	0.200526
$204$	0	0
$205$	−5.68921	−0.397351
$206$	0	0
$207$	−19.8085	−1.37678
$208$	0	0
$209$	4.55568	0.315123
$210$	0	0
$211$	−10.8093	−0.744141	−0.372071	−	0.928204i	$-0.621352\pi$
$211$	−10.8093	−0.744141	−0.372071	+	0.928204i	$0.621352\pi$
$212$	0	0
$213$	32.3224	2.21469
$214$	0	0
$215$	−1.57058	−0.107113
$216$	0	0
$217$	9.25463	0.628245
$218$	0	0
$219$	−37.6301	−2.54281
$220$	0	0
$221$	1.78120	0.119817
$222$	0	0
$223$	17.3461	1.16158	0.580791	−	0.814052i	$-0.302743\pi$
$223$	17.3461	1.16158	0.580791	+	0.814052i	$0.302743\pi$
$224$	0	0
$225$	−11.7209	−0.781396
$226$	0	0
$227$	20.5753	1.36563	0.682816	−	0.730591i	$-0.260755\pi$
$227$	20.5753	1.36563	0.682816	+	0.730591i	$0.260755\pi$
$228$	0	0
$229$	−22.1895	−1.46633	−0.733163	−	0.680053i	$-0.761957\pi$
$229$	−22.1895	−1.46633	−0.733163	+	0.680053i	$0.761957\pi$
$230$	0	0
$231$	−2.56319	−0.168646
$232$	0	0
$233$	−14.9095	−0.976752	−0.488376	−	0.872633i	$-0.662411\pi$
$233$	−14.9095	−0.976752	−0.488376	+	0.872633i	$0.662411\pi$
$234$	0	0
$235$	15.2492	0.994749
$236$	0	0
$237$	15.0918	0.980319
$238$	0	0
$239$	0.691072	0.0447017	0.0223509	−	0.999750i	$-0.492885\pi$
$239$	0.691072	0.0447017	0.0223509	+	0.999750i	$0.492885\pi$
$240$	0	0
$241$	−13.3788	−0.861803	−0.430901	−	0.902399i	$-0.641804\pi$
$241$	−13.3788	−0.861803	−0.430901	+	0.902399i	$0.641804\pi$
$242$	0	0
$243$	22.2361	1.42645
$244$	0	0
$245$	−1.31026	−0.0837095
$246$	0	0
$247$	−4.55568	−0.289871
$248$	0	0
$249$	14.7237	0.933079
$250$	0	0
$251$	−12.6773	−0.800181	−0.400091	−	0.916476i	$-0.631021\pi$
$251$	−12.6773	−0.800181	−0.400091	+	0.916476i	$0.631021\pi$
$252$	0	0
$253$	−5.54866	−0.348841
$254$	0	0
$255$	−5.98209	−0.374613
$256$	0	0
$257$	29.5981	1.84628	0.923139	−	0.384466i	$-0.125615\pi$
$257$	29.5981	1.84628	0.923139	+	0.384466i	$0.125615\pi$
$258$	0	0
$259$	−3.67962	−0.228641
$260$	0	0
$261$	10.1996	0.631338
$262$	0	0
$263$	−8.62856	−0.532060	−0.266030	−	0.963965i	$-0.585712\pi$
$263$	−8.62856	−0.532060	−0.266030	+	0.963965i	$0.585712\pi$
$264$	0	0
$265$	8.45621	0.519461
$266$	0	0
$267$	−32.1866	−1.96979
$268$	0	0
$269$	3.81761	0.232764	0.116382	−	0.993205i	$-0.462870\pi$
$269$	3.81761	0.232764	0.116382	+	0.993205i	$0.462870\pi$
$270$	0	0
$271$	24.0096	1.45848	0.729239	−	0.684259i	$-0.239874\pi$
$271$	24.0096	1.45848	0.729239	+	0.684259i	$0.239874\pi$
$272$	0	0
$273$	2.56319	0.155131
$274$	0	0
$275$	−3.28322	−0.197985
$276$	0	0
$277$	15.8449	0.952025	0.476013	−	0.879438i	$-0.342082\pi$
$277$	15.8449	0.952025	0.476013	+	0.879438i	$0.342082\pi$
$278$	0	0
$279$	33.0386	1.97797
$280$	0	0
$281$	22.4112	1.33694	0.668469	−	0.743740i	$-0.266950\pi$
$281$	22.4112	1.33694	0.668469	+	0.743740i	$0.266950\pi$
$282$	0	0
$283$	12.2106	0.725844	0.362922	−	0.931820i	$-0.381779\pi$
$283$	12.2106	0.725844	0.362922	+	0.931820i	$0.381779\pi$
$284$	0	0
$285$	15.3000	0.906296
$286$	0	0
$287$	4.34204	0.256303
$288$	0	0
$289$	−13.8273	−0.813371
$290$	0	0
$291$	1.29471	0.0758974
$292$	0	0
$293$	−21.0366	−1.22897	−0.614485	−	0.788929i	$-0.710636\pi$
$293$	−21.0366	−1.22897	−0.614485	+	0.788929i	$0.710636\pi$
$294$	0	0
$295$	1.29143	0.0751899
$296$	0	0
$297$	−1.46091	−0.0847708
$298$	0	0
$299$	5.54866	0.320887
$300$	0	0
$301$	1.19868	0.0690906
$302$	0	0
$303$	−38.6837	−2.22232
$304$	0	0
$305$	1.13902	0.0652202
$306$	0	0
$307$	34.3712	1.96167	0.980834	−	0.194846i	$-0.0624207\pi$
$307$	34.3712	1.96167	0.980834	+	0.194846i	$0.0624207\pi$
$308$	0	0
$309$	39.2856	2.23488
$310$	0	0
$311$	−1.07674	−0.0610561	−0.0305281	−	0.999534i	$-0.509719\pi$
$311$	−1.07674	−0.0610561	−0.0305281	+	0.999534i	$0.509719\pi$
$312$	0	0
$313$	−3.94415	−0.222937	−0.111468	−	0.993768i	$-0.535555\pi$
$313$	−3.94415	−0.222937	−0.111468	+	0.993768i	$0.535555\pi$
$314$	0	0
$315$	−4.67758	−0.263552
$316$	0	0
$317$	−13.9574	−0.783925	−0.391963	−	0.919981i	$-0.628204\pi$
$317$	−13.9574	−0.783925	−0.391963	+	0.919981i	$0.628204\pi$
$318$	0	0
$319$	2.85706	0.159965
$320$	0	0
$321$	9.99877	0.558077
$322$	0	0
$323$	−8.11460	−0.451508
$324$	0	0
$325$	3.28322	0.182120
$326$	0	0
$327$	−13.6825	−0.756642
$328$	0	0
$329$	−11.6383	−0.641640
$330$	0	0
$331$	2.54436	0.139851	0.0699253	−	0.997552i	$-0.477724\pi$
$331$	2.54436	0.139851	0.0699253	+	0.997552i	$0.477724\pi$
$332$	0	0
$333$	−13.1361	−0.719854
$334$	0	0
$335$	4.01276	0.219240
$336$	0	0
$337$	5.41277	0.294853	0.147426	−	0.989073i	$-0.452901\pi$
$337$	5.41277	0.294853	0.147426	+	0.989073i	$0.452901\pi$
$338$	0	0
$339$	−2.42947	−0.131951
$340$	0	0
$341$	9.25463	0.501166
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−18.6349	−1.00327
$346$	0	0
$347$	7.22519	0.387868	0.193934	−	0.981015i	$-0.437875\pi$
$347$	7.22519	0.387868	0.193934	+	0.981015i	$0.437875\pi$
$348$	0	0
$349$	25.7011	1.37575	0.687873	−	0.725831i	$-0.258544\pi$
$349$	25.7011	1.37575	0.687873	+	0.725831i	$0.258544\pi$
$350$	0	0
$351$	1.46091	0.0779778
$352$	0	0
$353$	−8.21680	−0.437336	−0.218668	−	0.975799i	$-0.570171\pi$
$353$	−8.21680	−0.437336	−0.218668	+	0.975799i	$0.570171\pi$
$354$	0	0
$355$	16.5227	0.876931
$356$	0	0
$357$	4.56557	0.241636
$358$	0	0
$359$	−30.6194	−1.61603	−0.808015	−	0.589162i	$-0.799458\pi$
$359$	−30.6194	−1.61603	−0.808015	+	0.589162i	$0.799458\pi$
$360$	0	0
$361$	1.75422	0.0923275
$362$	0	0
$363$	−2.56319	−0.134533
$364$	0	0
$365$	−19.2359	−1.00685
$366$	0	0
$367$	−24.8533	−1.29733	−0.648666	−	0.761073i	$-0.724673\pi$
$367$	−24.8533	−1.29733	−0.648666	+	0.761073i	$0.724673\pi$
$368$	0	0
$369$	15.5009	0.806945
$370$	0	0
$371$	−6.45384	−0.335067
$372$	0	0
$373$	34.6264	1.79289	0.896443	−	0.443159i	$-0.146142\pi$
$373$	34.6264	1.79289	0.896443	+	0.443159i	$0.146142\pi$
$374$	0	0
$375$	−27.8188	−1.43656
$376$	0	0
$377$	−2.85706	−0.147146
$378$	0	0
$379$	19.5528	1.00436	0.502180	−	0.864763i	$-0.332531\pi$
$379$	19.5528	1.00436	0.502180	+	0.864763i	$0.332531\pi$
$380$	0	0
$381$	47.9319	2.45563
$382$	0	0
$383$	16.0689	0.821083	0.410541	−	0.911842i	$-0.365340\pi$
$383$	16.0689	0.821083	0.410541	+	0.911842i	$0.365340\pi$
$384$	0	0
$385$	−1.31026	−0.0667771
$386$	0	0
$387$	4.27923	0.217525
$388$	0	0
$389$	−6.40934	−0.324967	−0.162483	−	0.986711i	$-0.551950\pi$
$389$	−6.40934	−0.324967	−0.162483	+	0.986711i	$0.551950\pi$
$390$	0	0
$391$	9.88329	0.499820
$392$	0	0
$393$	35.2204	1.77663
$394$	0	0
$395$	7.71468	0.388168
$396$	0	0
$397$	−30.5510	−1.53331	−0.766655	−	0.642059i	$-0.778080\pi$
$397$	−30.5510	−1.53331	−0.766655	+	0.642059i	$0.778080\pi$
$398$	0	0
$399$	−11.6771	−0.584586
$400$	0	0
$401$	30.8673	1.54144	0.770718	−	0.637176i	$-0.219898\pi$
$401$	30.8673	1.54144	0.770718	+	0.637176i	$0.219898\pi$
$402$	0	0
$403$	−9.25463	−0.461006
$404$	0	0
$405$	9.12632	0.453491
$406$	0	0
$407$	−3.67962	−0.182392
$408$	0	0
$409$	−20.3423	−1.00586	−0.502931	−	0.864327i	$-0.667745\pi$
$409$	−20.3423	−1.00586	−0.502931	+	0.864327i	$0.667745\pi$
$410$	0	0
$411$	6.38494	0.314946
$412$	0	0
$413$	−0.985627	−0.0484996
$414$	0	0
$415$	7.52652	0.369462
$416$	0	0
$417$	−44.4716	−2.17779
$418$	0	0
$419$	20.4488	0.998987	0.499494	−	0.866318i	$-0.333519\pi$
$419$	20.4488	0.998987	0.499494	+	0.866318i	$0.333519\pi$
$420$	0	0
$421$	13.6834	0.666889	0.333444	−	0.942770i	$-0.391789\pi$
$421$	13.6834	0.666889	0.333444	+	0.942770i	$0.391789\pi$
$422$	0	0
$423$	−41.5483	−2.02015
$424$	0	0
$425$	5.84808	0.283674
$426$	0	0
$427$	−0.869309	−0.0420688
$428$	0	0
$429$	2.56319	0.123752
$430$	0	0
$431$	−21.8211	−1.05109	−0.525543	−	0.850767i	$-0.676138\pi$
$431$	−21.8211	−1.05109	−0.525543	+	0.850767i	$0.676138\pi$
$432$	0	0
$433$	13.1888	0.633813	0.316907	−	0.948457i	$-0.397356\pi$
$433$	13.1888	0.633813	0.316907	+	0.948457i	$0.397356\pi$
$434$	0	0
$435$	9.59530	0.460059
$436$	0	0
$437$	−25.2779	−1.20921
$438$	0	0
$439$	−18.8562	−0.899959	−0.449980	−	0.893039i	$-0.648569\pi$
$439$	−18.8562	−0.899959	−0.449980	+	0.893039i	$0.648569\pi$
$440$	0	0
$441$	3.56996	0.169998
$442$	0	0
$443$	18.7795	0.892240	0.446120	−	0.894973i	$-0.352805\pi$
$443$	18.7795	0.892240	0.446120	+	0.894973i	$0.352805\pi$
$444$	0	0
$445$	−16.4532	−0.779959
$446$	0	0
$447$	−9.11781	−0.431258
$448$	0	0
$449$	−28.6859	−1.35377	−0.676886	−	0.736088i	$-0.736671\pi$
$449$	−28.6859	−1.35377	−0.676886	+	0.736088i	$0.736671\pi$
$450$	0	0
$451$	4.34204	0.204459
$452$	0	0
$453$	−56.6339	−2.66089
$454$	0	0
$455$	1.31026	0.0614260
$456$	0	0
$457$	6.25967	0.292815	0.146408	−	0.989224i	$-0.453229\pi$
$457$	6.25967	0.292815	0.146408	+	0.989224i	$0.453229\pi$
$458$	0	0
$459$	2.60219	0.121460
$460$	0	0
$461$	29.2302	1.36139	0.680694	−	0.732568i	$-0.261678\pi$
$461$	29.2302	1.36139	0.680694	+	0.732568i	$0.261678\pi$
$462$	0	0
$463$	13.2108	0.613958	0.306979	−	0.951716i	$-0.400682\pi$
$463$	13.2108	0.613958	0.306979	+	0.951716i	$0.400682\pi$
$464$	0	0
$465$	31.0812	1.44136
$466$	0	0
$467$	−31.1731	−1.44252	−0.721260	−	0.692664i	$-0.756437\pi$
$467$	−31.1731	−1.44252	−0.721260	+	0.692664i	$0.756437\pi$
$468$	0	0
$469$	−3.06256	−0.141416
$470$	0	0
$471$	−23.0882	−1.06385
$472$	0	0
$473$	1.19868	0.0551152
$474$	0	0
$475$	−14.9573	−0.686287
$476$	0	0
$477$	−23.0399	−1.05493
$478$	0	0
$479$	22.4088	1.02388	0.511941	−	0.859020i	$-0.328926\pi$
$479$	22.4088	1.02388	0.511941	+	0.859020i	$0.328926\pi$
$480$	0	0
$481$	3.67962	0.167776
$482$	0	0
$483$	14.2223	0.647136
$484$	0	0
$485$	0.661835	0.0300524
$486$	0	0
$487$	−17.1399	−0.776684	−0.388342	−	0.921515i	$-0.626952\pi$
$487$	−17.1399	−0.776684	−0.388342	+	0.921515i	$0.626952\pi$
$488$	0	0
$489$	−49.9675	−2.25961
$490$	0	0
$491$	−2.74433	−0.123850	−0.0619250	−	0.998081i	$-0.519724\pi$
$491$	−2.74433	−0.123850	−0.0619250	+	0.998081i	$0.519724\pi$
$492$	0	0
$493$	−5.08901	−0.229197
$494$	0	0
$495$	−4.67758	−0.210241
$496$	0	0
$497$	−12.6102	−0.565645
$498$	0	0
$499$	7.28958	0.326326	0.163163	−	0.986599i	$-0.447830\pi$
$499$	7.28958	0.326326	0.163163	+	0.986599i	$0.447830\pi$
$500$	0	0
$501$	−24.0595	−1.07490
$502$	0	0
$503$	38.7200	1.72644	0.863220	−	0.504829i	$-0.168444\pi$
$503$	38.7200	1.72644	0.863220	+	0.504829i	$0.168444\pi$
$504$	0	0
$505$	−19.7744	−0.879951
$506$	0	0
$507$	−2.56319	−0.113835
$508$	0	0
$509$	−29.5527	−1.30990	−0.654950	−	0.755672i	$-0.727310\pi$
$509$	−29.5527	−1.30990	−0.654950	+	0.755672i	$0.727310\pi$
$510$	0	0
$511$	14.6810	0.649448
$512$	0	0
$513$	−6.65546	−0.293846
$514$	0	0
$515$	20.0822	0.884926
$516$	0	0
$517$	−11.6383	−0.511852
$518$	0	0
$519$	−37.0622	−1.62685
$520$	0	0
$521$	28.5726	1.25179	0.625895	−	0.779908i	$-0.284734\pi$
$521$	28.5726	1.25179	0.625895	+	0.779908i	$0.284734\pi$
$522$	0	0
$523$	−5.20207	−0.227471	−0.113735	−	0.993511i	$-0.536282\pi$
$523$	−5.20207	−0.227471	−0.113735	+	0.993511i	$0.536282\pi$
$524$	0	0
$525$	8.41552	0.367283
$526$	0	0
$527$	−16.4844	−0.718071
$528$	0	0
$529$	7.78759	0.338591
$530$	0	0
$531$	−3.51865	−0.152696
$532$	0	0
$533$	−4.34204	−0.188075
$534$	0	0
$535$	5.11120	0.220977
$536$	0	0
$537$	−47.0212	−2.02911
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	12.5568	0.539858	0.269929	−	0.962880i	$-0.413000\pi$
$541$	12.5568	0.539858	0.269929	+	0.962880i	$0.413000\pi$
$542$	0	0
$543$	−21.3809	−0.917543
$544$	0	0
$545$	−6.99424	−0.299600
$546$	0	0
$547$	−19.6383	−0.839673	−0.419837	−	0.907600i	$-0.637913\pi$
$547$	−19.6383	−0.839673	−0.419837	+	0.907600i	$0.637913\pi$
$548$	0	0
$549$	−3.10340	−0.132450
$550$	0	0
$551$	13.0158	0.554494
$552$	0	0
$553$	−5.88790	−0.250379
$554$	0	0
$555$	−12.3578	−0.524561
$556$	0	0
$557$	19.6785	0.833806	0.416903	−	0.908951i	$-0.363115\pi$
$557$	19.6785	0.833806	0.416903	+	0.908951i	$0.363115\pi$
$558$	0	0
$559$	−1.19868	−0.0506987
$560$	0	0
$561$	4.56557	0.192759
$562$	0	0
$563$	−15.0392	−0.633825	−0.316912	−	0.948455i	$-0.602646\pi$
$563$	−15.0392	−0.633825	−0.316912	+	0.948455i	$0.602646\pi$
$564$	0	0
$565$	−1.24190	−0.0522472
$566$	0	0
$567$	−6.96527	−0.292514
$568$	0	0
$569$	30.8104	1.29164	0.645819	−	0.763491i	$-0.276516\pi$
$569$	30.8104	1.29164	0.645819	+	0.763491i	$0.276516\pi$
$570$	0	0
$571$	8.26472	0.345868	0.172934	−	0.984933i	$-0.444675\pi$
$571$	8.26472	0.345868	0.172934	+	0.984933i	$0.444675\pi$
$572$	0	0
$573$	−11.3411	−0.473782
$574$	0	0
$575$	18.2174	0.759720
$576$	0	0
$577$	−18.9355	−0.788296	−0.394148	−	0.919047i	$-0.628960\pi$
$577$	−18.9355	−0.788296	−0.394148	+	0.919047i	$0.628960\pi$
$578$	0	0
$579$	−69.8180	−2.90154
$580$	0	0
$581$	−5.74429	−0.238313
$582$	0	0
$583$	−6.45384	−0.267291
$584$	0	0
$585$	4.67758	0.193394
$586$	0	0
$587$	−21.8825	−0.903187	−0.451593	−	0.892224i	$-0.649144\pi$
$587$	−21.8825	−0.903187	−0.451593	+	0.892224i	$0.649144\pi$
$588$	0	0
$589$	42.1611	1.73722
$590$	0	0
$591$	−49.6398	−2.04191
$592$	0	0
$593$	−6.74145	−0.276838	−0.138419	−	0.990374i	$-0.544202\pi$
$593$	−6.74145	−0.276838	−0.138419	+	0.990374i	$0.544202\pi$
$594$	0	0
$595$	2.33384	0.0956782
$596$	0	0
$597$	44.2782	1.81219
$598$	0	0
$599$	−19.2426	−0.786232	−0.393116	−	0.919489i	$-0.628603\pi$
$599$	−19.2426	−0.786232	−0.393116	+	0.919489i	$0.628603\pi$
$600$	0	0
$601$	−0.875564	−0.0357150	−0.0178575	−	0.999841i	$-0.505685\pi$
$601$	−0.875564	−0.0357150	−0.0178575	+	0.999841i	$0.505685\pi$
$602$	0	0
$603$	−10.9332	−0.445236
$604$	0	0
$605$	−1.31026	−0.0532697
$606$	0	0
$607$	−8.57093	−0.347883	−0.173942	−	0.984756i	$-0.555650\pi$
$607$	−8.57093	−0.347883	−0.173942	+	0.984756i	$0.555650\pi$
$608$	0	0
$609$	−7.32319	−0.296751
$610$	0	0
$611$	11.6383	0.470835
$612$	0	0
$613$	44.2024	1.78532	0.892659	−	0.450732i	$-0.148837\pi$
$613$	44.2024	1.78532	0.892659	+	0.450732i	$0.148837\pi$
$614$	0	0
$615$	14.5825	0.588025
$616$	0	0
$617$	−6.16468	−0.248181	−0.124090	−	0.992271i	$-0.539601\pi$
$617$	−6.16468	−0.248181	−0.124090	+	0.992271i	$0.539601\pi$
$618$	0	0
$619$	15.2516	0.613014	0.306507	−	0.951868i	$-0.400840\pi$
$619$	15.2516	0.613014	0.306507	+	0.951868i	$0.400840\pi$
$620$	0	0
$621$	8.10611	0.325287
$622$	0	0
$623$	12.5572	0.503095
$624$	0	0
$625$	2.19559	0.0878237
$626$	0	0
$627$	−11.6771	−0.466338
$628$	0	0
$629$	6.55416	0.261332
$630$	0	0
$631$	20.4365	0.813564	0.406782	−	0.913525i	$-0.366651\pi$
$631$	20.4365	0.813564	0.406782	+	0.913525i	$0.366651\pi$
$632$	0	0
$633$	27.7063	1.10123
$634$	0	0
$635$	24.5020	0.972331
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−45.0179	−1.78088
$640$	0	0
$641$	−24.1968	−0.955715	−0.477857	−	0.878437i	$-0.658586\pi$
$641$	−24.1968	−0.955715	−0.477857	+	0.878437i	$0.658586\pi$
$642$	0	0
$643$	14.5221	0.572694	0.286347	−	0.958126i	$-0.407559\pi$
$643$	14.5221	0.572694	0.286347	+	0.958126i	$0.407559\pi$
$644$	0	0
$645$	4.02570	0.158512
$646$	0	0
$647$	−31.9009	−1.25415	−0.627077	−	0.778958i	$-0.715749\pi$
$647$	−31.9009	−1.25415	−0.627077	+	0.778958i	$0.715749\pi$
$648$	0	0
$649$	−0.985627	−0.0386893
$650$	0	0
$651$	−23.7214	−0.929715
$652$	0	0
$653$	11.3097	0.442584	0.221292	−	0.975208i	$-0.428973\pi$
$653$	11.3097	0.442584	0.221292	+	0.975208i	$0.428973\pi$
$654$	0	0
$655$	18.0040	0.703476
$656$	0	0
$657$	52.4104	2.04473
$658$	0	0
$659$	8.30187	0.323395	0.161697	−	0.986840i	$-0.448303\pi$
$659$	8.30187	0.323395	0.161697	+	0.986840i	$0.448303\pi$
$660$	0	0
$661$	24.3382	0.946646	0.473323	−	0.880889i	$-0.343054\pi$
$661$	24.3382	0.946646	0.473323	+	0.880889i	$0.343054\pi$
$662$	0	0
$663$	−4.56557	−0.177312
$664$	0	0
$665$	−5.96913	−0.231473
$666$	0	0
$667$	−15.8528	−0.613824
$668$	0	0
$669$	−44.4615	−1.71898
$670$	0	0
$671$	−0.869309	−0.0335593
$672$	0	0
$673$	15.6886	0.604753	0.302376	−	0.953189i	$-0.402220\pi$
$673$	15.6886	0.604753	0.302376	+	0.953189i	$0.402220\pi$
$674$	0	0
$675$	4.79650	0.184617
$676$	0	0
$677$	34.1034	1.31070	0.655350	−	0.755325i	$-0.272521\pi$
$677$	34.1034	1.31070	0.655350	+	0.755325i	$0.272521\pi$
$678$	0	0
$679$	−0.505117	−0.0193846
$680$	0	0
$681$	−52.7385	−2.02094
$682$	0	0
$683$	12.2714	0.469550	0.234775	−	0.972050i	$-0.424565\pi$
$683$	12.2714	0.469550	0.234775	+	0.972050i	$0.424565\pi$
$684$	0	0
$685$	3.26387	0.124706
$686$	0	0
$687$	56.8761	2.16996
$688$	0	0
$689$	6.45384	0.245872
$690$	0	0
$691$	−30.5281	−1.16134	−0.580671	−	0.814138i	$-0.697210\pi$
$691$	−30.5281	−1.16134	−0.580671	+	0.814138i	$0.697210\pi$
$692$	0	0
$693$	3.56996	0.135612
$694$	0	0
$695$	−22.7331	−0.862317
$696$	0	0
$697$	−7.73407	−0.292949
$698$	0	0
$699$	38.2159	1.44546
$700$	0	0
$701$	−34.0064	−1.28440	−0.642202	−	0.766535i	$-0.721979\pi$
$701$	−34.0064	−1.28440	−0.642202	+	0.766535i	$0.721979\pi$
$702$	0	0
$703$	−16.7632	−0.632235
$704$	0	0
$705$	−39.0867	−1.47209
$706$	0	0
$707$	15.0920	0.567593
$708$	0	0
$709$	−18.1256	−0.680723	−0.340361	−	0.940295i	$-0.610549\pi$
$709$	−18.1256	−0.680723	−0.340361	+	0.940295i	$0.610549\pi$
$710$	0	0
$711$	−21.0195	−0.788294
$712$	0	0
$713$	−51.3507	−1.92310
$714$	0	0
$715$	1.31026	0.0490010
$716$	0	0
$717$	−1.77135	−0.0661523
$718$	0	0
$719$	−42.6444	−1.59037	−0.795184	−	0.606368i	$-0.792626\pi$
$719$	−42.6444	−1.59037	−0.795184	+	0.606368i	$0.792626\pi$
$720$	0	0
$721$	−15.3268	−0.570801
$722$	0	0
$723$	34.2924	1.27535
$724$	0	0
$725$	−9.38034	−0.348377
$726$	0	0
$727$	49.2366	1.82608	0.913041	−	0.407867i	$-0.133727\pi$
$727$	49.2366	1.82608	0.913041	+	0.407867i	$0.133727\pi$
$728$	0	0
$729$	−36.0995	−1.33702
$730$	0	0
$731$	−2.13509	−0.0789692
$732$	0	0
$733$	29.5000	1.08961	0.544804	−	0.838563i	$-0.316604\pi$
$733$	29.5000	1.08961	0.544804	+	0.838563i	$0.316604\pi$
$734$	0	0
$735$	3.35845	0.123878
$736$	0	0
$737$	−3.06256	−0.112811
$738$	0	0
$739$	33.5493	1.23413	0.617066	−	0.786912i	$-0.288321\pi$
$739$	33.5493	1.23413	0.617066	+	0.786912i	$0.288321\pi$
$740$	0	0
$741$	11.6771	0.428969
$742$	0	0
$743$	0.847165	0.0310795	0.0155397	−	0.999879i	$-0.495053\pi$
$743$	0.847165	0.0310795	0.0155397	+	0.999879i	$0.495053\pi$
$744$	0	0
$745$	−4.66087	−0.170761
$746$	0	0
$747$	−20.5069	−0.750308
$748$	0	0
$749$	−3.90090	−0.142536
$750$	0	0
$751$	−7.01203	−0.255873	−0.127936	−	0.991782i	$-0.540835\pi$
$751$	−7.01203	−0.255873	−0.127936	+	0.991782i	$0.540835\pi$
$752$	0	0
$753$	32.4943	1.18416
$754$	0	0
$755$	−28.9503	−1.05361
$756$	0	0
$757$	43.6194	1.58537	0.792686	−	0.609630i	$-0.208682\pi$
$757$	43.6194	1.58537	0.792686	+	0.609630i	$0.208682\pi$
$758$	0	0
$759$	14.2223	0.516236
$760$	0	0
$761$	−21.4859	−0.778863	−0.389432	−	0.921055i	$-0.627329\pi$
$761$	−21.4859	−0.778863	−0.389432	+	0.921055i	$0.627329\pi$
$762$	0	0
$763$	5.33805	0.193251
$764$	0	0
$765$	8.33172	0.301234
$766$	0	0
$767$	0.985627	0.0355889
$768$	0	0
$769$	−16.0566	−0.579014	−0.289507	−	0.957176i	$-0.593491\pi$
$769$	−16.0566	−0.579014	−0.289507	+	0.957176i	$0.593491\pi$
$770$	0	0
$771$	−75.8656	−2.73223
$772$	0	0
$773$	24.3606	0.876189	0.438094	−	0.898929i	$-0.355654\pi$
$773$	24.3606	0.876189	0.438094	+	0.898929i	$0.355654\pi$
$774$	0	0
$775$	−30.3849	−1.09146
$776$	0	0
$777$	9.43158	0.338356
$778$	0	0
$779$	19.7810	0.708726
$780$	0	0
$781$	−12.6102	−0.451228
$782$	0	0
$783$	−4.17392	−0.149164
$784$	0	0
$785$	−11.8023	−0.421242
$786$	0	0
$787$	13.9081	0.495769	0.247884	−	0.968790i	$-0.420265\pi$
$787$	13.9081	0.495769	0.247884	+	0.968790i	$0.420265\pi$
$788$	0	0
$789$	22.1167	0.787374
$790$	0	0
$791$	0.947829	0.0337009
$792$	0	0
$793$	0.869309	0.0308701
$794$	0	0
$795$	−21.6749	−0.768729
$796$	0	0
$797$	−42.9063	−1.51982	−0.759910	−	0.650029i	$-0.774757\pi$
$797$	−42.9063	−1.51982	−0.759910	+	0.650029i	$0.774757\pi$
$798$	0	0
$799$	20.7302	0.733382
$800$	0	0
$801$	44.8288	1.58395
$802$	0	0
$803$	14.6810	0.518080
$804$	0	0
$805$	7.27019	0.256240
$806$	0	0
$807$	−9.78527	−0.344458
$808$	0	0
$809$	−20.6998	−0.727766	−0.363883	−	0.931445i	$-0.618549\pi$
$809$	−20.6998	−0.727766	−0.363883	+	0.931445i	$0.618549\pi$
$810$	0	0
$811$	49.1031	1.72424	0.862122	−	0.506700i	$-0.169135\pi$
$811$	49.1031	1.72424	0.862122	+	0.506700i	$0.169135\pi$
$812$	0	0
$813$	−61.5412	−2.15834
$814$	0	0
$815$	−25.5426	−0.894717
$816$	0	0
$817$	5.46079	0.191049
$818$	0	0
$819$	−3.56996	−0.124744
$820$	0	0
$821$	−3.33958	−0.116552	−0.0582760	−	0.998301i	$-0.518560\pi$
$821$	−3.33958	−0.116552	−0.0582760	+	0.998301i	$0.518560\pi$
$822$	0	0
$823$	−33.2624	−1.15945	−0.579727	−	0.814811i	$-0.696841\pi$
$823$	−33.2624	−1.15945	−0.579727	+	0.814811i	$0.696841\pi$
$824$	0	0
$825$	8.41552	0.292991
$826$	0	0
$827$	−27.9417	−0.971629	−0.485815	−	0.874062i	$-0.661477\pi$
$827$	−27.9417	−0.971629	−0.485815	+	0.874062i	$0.661477\pi$
$828$	0	0
$829$	37.9194	1.31699	0.658497	−	0.752583i	$-0.271192\pi$
$829$	37.9194	1.31699	0.658497	+	0.752583i	$0.271192\pi$
$830$	0	0
$831$	−40.6134	−1.40886
$832$	0	0
$833$	−1.78120	−0.0617151
$834$	0	0
$835$	−12.2988	−0.425617
$836$	0	0
$837$	−13.5202	−0.467327
$838$	0	0
$839$	−18.8122	−0.649469	−0.324734	−	0.945805i	$-0.605275\pi$
$839$	−18.8122	−0.649469	−0.324734	+	0.945805i	$0.605275\pi$
$840$	0	0
$841$	−20.8372	−0.718525
$842$	0	0
$843$	−57.4442	−1.97848
$844$	0	0
$845$	−1.31026	−0.0450743
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−31.2981	−1.07415
$850$	0	0
$851$	20.4170	0.699884
$852$	0	0
$853$	11.6216	0.397918	0.198959	−	0.980008i	$-0.436244\pi$
$853$	11.6216	0.397918	0.198959	+	0.980008i	$0.436244\pi$
$854$	0	0
$855$	−21.3095	−0.728771
$856$	0	0
$857$	−5.52571	−0.188755	−0.0943773	−	0.995537i	$-0.530086\pi$
$857$	−5.52571	−0.188755	−0.0943773	+	0.995537i	$0.530086\pi$
$858$	0	0
$859$	−35.6294	−1.21566	−0.607829	−	0.794068i	$-0.707959\pi$
$859$	−35.6294	−1.21566	−0.607829	+	0.794068i	$0.707959\pi$
$860$	0	0
$861$	−11.1295	−0.379292
$862$	0	0
$863$	−35.3791	−1.20432	−0.602159	−	0.798377i	$-0.705692\pi$
$863$	−35.3791	−1.20432	−0.602159	+	0.798377i	$0.705692\pi$
$864$	0	0
$865$	−18.9456	−0.644169
$866$	0	0
$867$	35.4421	1.20368
$868$	0	0
$869$	−5.88790	−0.199733
$870$	0	0
$871$	3.06256	0.103771
$872$	0	0
$873$	−1.80325	−0.0610307
$874$	0	0
$875$	10.8532	0.366904
$876$	0	0
$877$	16.1409	0.545039	0.272519	−	0.962150i	$-0.412143\pi$
$877$	16.1409	0.545039	0.272519	+	0.962150i	$0.412143\pi$
$878$	0	0
$879$	53.9208	1.81870
$880$	0	0
$881$	44.0910	1.48546	0.742732	−	0.669589i	$-0.233530\pi$
$881$	44.0910	1.48546	0.742732	+	0.669589i	$0.233530\pi$
$882$	0	0
$883$	−5.23103	−0.176038	−0.0880191	−	0.996119i	$-0.528054\pi$
$883$	−5.23103	−0.176038	−0.0880191	+	0.996119i	$0.528054\pi$
$884$	0	0
$885$	−3.31018	−0.111271
$886$	0	0
$887$	−32.1171	−1.07839	−0.539193	−	0.842182i	$-0.681271\pi$
$887$	−32.1171	−1.07839	−0.539193	+	0.842182i	$0.681271\pi$
$888$	0	0
$889$	−18.7001	−0.627180
$890$	0	0
$891$	−6.96527	−0.233345
$892$	0	0
$893$	−53.0204	−1.77426
$894$	0	0
$895$	−24.0364	−0.803449
$896$	0	0
$897$	−14.2223	−0.474868
$898$	0	0
$899$	26.4410	0.881857
$900$	0	0
$901$	11.4956	0.382974
$902$	0	0
$903$	−3.07244	−0.102244
$904$	0	0
$905$	−10.9296	−0.363311
$906$	0	0
$907$	−56.4967	−1.87594	−0.937970	−	0.346716i	$-0.887297\pi$
$907$	−56.4967	−1.87594	−0.937970	+	0.346716i	$0.887297\pi$
$908$	0	0
$909$	53.8778	1.78701
$910$	0	0
$911$	37.7887	1.25200	0.625998	−	0.779825i	$-0.284692\pi$
$911$	37.7887	1.25200	0.625998	+	0.779825i	$0.284692\pi$
$912$	0	0
$913$	−5.74429	−0.190108
$914$	0	0
$915$	−2.91953	−0.0965168
$916$	0	0
$917$	−13.7408	−0.453762
$918$	0	0
$919$	−18.9068	−0.623679	−0.311839	−	0.950135i	$-0.600945\pi$
$919$	−18.9068	−0.623679	−0.311839	+	0.950135i	$0.600945\pi$
$920$	0	0
$921$	−88.1000	−2.90299
$922$	0	0
$923$	12.6102	0.415070
$924$	0	0
$925$	12.0810	0.397221
$926$	0	0
$927$	−54.7162	−1.79712
$928$	0	0
$929$	−39.1699	−1.28512	−0.642561	−	0.766234i	$-0.722128\pi$
$929$	−39.1699	−1.28512	−0.642561	+	0.766234i	$0.722128\pi$
$930$	0	0
$931$	4.55568	0.149306
$932$	0	0
$933$	2.75988	0.0903545
$934$	0	0
$935$	2.33384	0.0763248
$936$	0	0
$937$	36.9573	1.20734	0.603672	−	0.797233i	$-0.293704\pi$
$937$	36.9573	1.20734	0.603672	+	0.797233i	$0.293704\pi$
$938$	0	0
$939$	10.1096	0.329915
$940$	0	0
$941$	45.1026	1.47030	0.735152	−	0.677902i	$-0.237111\pi$
$941$	45.1026	1.47030	0.735152	+	0.677902i	$0.237111\pi$
$942$	0	0
$943$	−24.0925	−0.784560
$944$	0	0
$945$	1.91418	0.0622682
$946$	0	0
$947$	19.8420	0.644777	0.322389	−	0.946607i	$-0.395514\pi$
$947$	19.8420	0.644777	0.322389	+	0.946607i	$0.395514\pi$
$948$	0	0
$949$	−14.6810	−0.476564
$950$	0	0
$951$	35.7755	1.16010
$952$	0	0
$953$	31.0626	1.00622	0.503108	−	0.864223i	$-0.332190\pi$
$953$	31.0626	1.00622	0.503108	+	0.864223i	$0.332190\pi$
$954$	0	0
$955$	−5.79739	−0.187599
$956$	0	0
$957$	−7.32319	−0.236725
$958$	0	0
$959$	−2.49101	−0.0804389
$960$	0	0
$961$	54.6481	1.76284
$962$	0	0
$963$	−13.9261	−0.448761
$964$	0	0
$965$	−35.6898	−1.14889
$966$	0	0
$967$	−14.5885	−0.469136	−0.234568	−	0.972100i	$-0.575368\pi$
$967$	−14.5885	−0.469136	−0.234568	+	0.972100i	$0.575368\pi$
$968$	0	0
$969$	20.7993	0.668169
$970$	0	0
$971$	8.03851	0.257968	0.128984	−	0.991647i	$-0.458828\pi$
$971$	8.03851	0.257968	0.128984	+	0.991647i	$0.458828\pi$
$972$	0	0
$973$	17.3501	0.556218
$974$	0	0
$975$	−8.41552	−0.269512
$976$	0	0
$977$	21.4864	0.687410	0.343705	−	0.939078i	$-0.388318\pi$
$977$	21.4864	0.687410	0.343705	+	0.939078i	$0.388318\pi$
$978$	0	0
$979$	12.5572	0.401331
$980$	0	0
$981$	19.0566	0.608431
$982$	0	0
$983$	52.4025	1.67138	0.835691	−	0.549201i	$-0.185068\pi$
$983$	52.4025	1.67138	0.835691	+	0.549201i	$0.185068\pi$
$984$	0	0
$985$	−25.3750	−0.808515
$986$	0	0
$987$	29.8312	0.949538
$988$	0	0
$989$	−6.65105	−0.211491
$990$	0	0
$991$	33.2504	1.05623	0.528117	−	0.849172i	$-0.322898\pi$
$991$	33.2504	1.05623	0.528117	+	0.849172i	$0.322898\pi$
$992$	0	0
$993$	−6.52168	−0.206959
$994$	0	0
$995$	22.6343	0.717554
$996$	0	0
$997$	13.0215	0.412394	0.206197	−	0.978510i	$-0.433891\pi$
$997$	13.0215	0.412394	0.206197	+	0.978510i	$0.433891\pi$
$998$	0	0
$999$	5.37561	0.170077

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.h.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.h.1.2	✓	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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56319 q^{3} -1.31026 q^{5} +1.00000 q^{7} +3.56996 q^{9} +O(q^{10})\)	\(q-2.56319 q^{3} -1.31026 q^{5} +1.00000 q^{7} +3.56996 q^{9} +1.00000 q^{11} -1.00000 q^{13} +3.35845 q^{15} -1.78120 q^{17} +4.55568 q^{19} -2.56319 q^{21} -5.54866 q^{23} -3.28322 q^{25} -1.46091 q^{27} +2.85706 q^{29} +9.25463 q^{31} -2.56319 q^{33} -1.31026 q^{35} -3.67962 q^{37} +2.56319 q^{39} +4.34204 q^{41} +1.19868 q^{43} -4.67758 q^{45} -11.6383 q^{47} +1.00000 q^{49} +4.56557 q^{51} -6.45384 q^{53} -1.31026 q^{55} -11.6771 q^{57} -0.985627 q^{59} -0.869309 q^{61} +3.56996 q^{63} +1.31026 q^{65} -3.06256 q^{67} +14.2223 q^{69} -12.6102 q^{71} +14.6810 q^{73} +8.41552 q^{75} +1.00000 q^{77} -5.88790 q^{79} -6.96527 q^{81} -5.74429 q^{83} +2.33384 q^{85} -7.32319 q^{87} +12.5572 q^{89} -1.00000 q^{91} -23.7214 q^{93} -5.96913 q^{95} -0.505117 q^{97} +3.56996 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9	\( 9 q + 3 q^{3} + 9 q^{7} + 20 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + 5 q^{17} + 10 q^{19} + 3 q^{21} + 8 q^{23} + 3 q^{25} + 27 q^{27} + 14 q^{29} + 11 q^{31} + 3 q^{33} - 3 q^{39} + 14 q^{41} + 8 q^{43} + 4 q^{45} + 10 q^{47} + 9 q^{49} - 15 q^{51} + 21 q^{53} - 8 q^{57} + 23 q^{59} + 34 q^{61} + 20 q^{63} + 10 q^{67} - 16 q^{69} + 4 q^{71} + 9 q^{73} + 30 q^{75} + 9 q^{77} - 34 q^{79} + 69 q^{81} + 15 q^{83} + 5 q^{85} + 39 q^{87} - 9 q^{91} + 3 q^{93} - 64 q^{95} + 15 q^{97} + 20 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 + 9 * q^7 + 20 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + 5 * q^17 + 10 * q^19 + 3 * q^21 + 8 * q^23 + 3 * q^25 + 27 * q^27 + 14 * q^29 + 11 * q^31 + 3 * q^33 - 3 * q^39 + 14 * q^41 + 8 * q^43 + 4 * q^45 + 10 * q^47 + 9 * q^49 - 15 * q^51 + 21 * q^53 - 8 * q^57 + 23 * q^59 + 34 * q^61 + 20 * q^63 + 10 * q^67 - 16 * q^69 + 4 * q^71 + 9 * q^73 + 30 * q^75 + 9 * q^77 - 34 * q^79 + 69 * q^81 + 15 * q^83 + 5 * q^85 + 39 * q^87 - 9 * q^91 + 3 * q^93 - 64 * q^95 + 15 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more