LMFDB - Embedded newform 4004.2.a.f.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:	$1$
Dimension:	$5$
Coefficient field:	5.5.463341.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x + 1 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.53838$ 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)

$f(q)$	$=$	$q-0.538379 q^{3} +3.82180 q^{5} -1.00000 q^{7} -2.71015 q^{9} +O(q^{10})$	$q-0.538379 q^{3} +3.82180 q^{5} -1.00000 q^{7} -2.71015 q^{9} -1.00000 q^{11} +1.00000 q^{13} -2.05758 q^{15} -6.24599 q^{17} -3.44915 q^{19} +0.538379 q^{21} +4.19445 q^{23} +9.60618 q^{25} +3.07423 q^{27} -5.50927 q^{29} +7.37683 q^{31} +0.538379 q^{33} -3.82180 q^{35} -5.39508 q^{37} -0.538379 q^{39} -5.48430 q^{41} -7.33522 q^{43} -10.3576 q^{45} +1.10562 q^{47} +1.00000 q^{49} +3.36272 q^{51} +0.513417 q^{53} -3.82180 q^{55} +1.85695 q^{57} +5.87964 q^{59} -10.0042 q^{61} +2.71015 q^{63} +3.82180 q^{65} -11.2911 q^{67} -2.25821 q^{69} -3.93722 q^{71} -13.5870 q^{73} -5.17177 q^{75} +1.00000 q^{77} -1.20114 q^{79} +6.47534 q^{81} +13.8487 q^{83} -23.8710 q^{85} +2.96608 q^{87} -13.0549 q^{89} -1.00000 q^{91} -3.97153 q^{93} -13.1820 q^{95} -3.29239 q^{97} +2.71015 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9}+O(q^{10}) $ 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9	$ 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9} - 5 q^{11} + 5 q^{13} - 3 q^{15} - 9 q^{17} - 4 q^{19} - 3 q^{21} - 4 q^{23} + q^{25} + 3 q^{27} - 8 q^{29} + 7 q^{31} - 3 q^{33} - 10 q^{37} + 3 q^{39} - 18 q^{41} - 22 q^{43} - 26 q^{45} + 12 q^{47} + 5 q^{49} - 7 q^{51} + 7 q^{53} - 6 q^{57} - q^{59} - 26 q^{61} - 2 q^{63} - 8 q^{67} - 12 q^{69} + 18 q^{71} - 25 q^{73} - 16 q^{75} + 5 q^{77} - 6 q^{79} - 7 q^{81} + 11 q^{83} - 7 q^{85} - 5 q^{87} - 14 q^{89} - 5 q^{91} - 41 q^{93} - 22 q^{95} - 33 q^{97} - 2 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9 - 5 * q^11 + 5 * q^13 - 3 * q^15 - 9 * q^17 - 4 * q^19 - 3 * q^21 - 4 * q^23 + q^25 + 3 * q^27 - 8 * q^29 + 7 * q^31 - 3 * q^33 - 10 * q^37 + 3 * q^39 - 18 * q^41 - 22 * q^43 - 26 * q^45 + 12 * q^47 + 5 * q^49 - 7 * q^51 + 7 * q^53 - 6 * q^57 - q^59 - 26 * q^61 - 2 * q^63 - 8 * q^67 - 12 * q^69 + 18 * q^71 - 25 * q^73 - 16 * q^75 + 5 * q^77 - 6 * q^79 - 7 * q^81 + 11 * q^83 - 7 * q^85 - 5 * q^87 - 14 * q^89 - 5 * q^91 - 41 * q^93 - 22 * q^95 - 33 * q^97 - 2 * 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
$3$	−0.538379	−0.310834	−0.155417	−	0.987849i	$-0.549672\pi$
$3$	−0.538379	−0.310834	−0.155417	+	0.987849i	$0.549672\pi$
$4$	0	0
$5$	3.82180	1.70916	0.854581	−	0.519318i	$-0.173814\pi$
$5$	3.82180	1.70916	0.854581	+	0.519318i	$0.173814\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.71015	−0.903383
$10$	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
$15$	−2.05758	−0.531265
$16$	0	0
$17$	−6.24599	−1.51488	−0.757438	−	0.652907i	$-0.773549\pi$
$17$	−6.24599	−1.51488	−0.757438	+	0.652907i	$0.773549\pi$
$18$	0	0
$19$	−3.44915	−0.791290	−0.395645	−	0.918403i	$-0.629479\pi$
$19$	−3.44915	−0.791290	−0.395645	+	0.918403i	$0.629479\pi$
$20$	0	0
$21$	0.538379	0.117484
$22$	0	0
$23$	4.19445	0.874604	0.437302	−	0.899315i	$-0.355934\pi$
$23$	4.19445	0.874604	0.437302	+	0.899315i	$0.355934\pi$
$24$	0	0
$25$	9.60618	1.92124
$26$	0	0
$27$	3.07423	0.591635
$28$	0	0
$29$	−5.50927	−1.02305	−0.511523	−	0.859270i	$-0.670918\pi$
$29$	−5.50927	−1.02305	−0.511523	+	0.859270i	$0.670918\pi$
$30$	0	0
$31$	7.37683	1.32492	0.662459	−	0.749098i	$-0.269513\pi$
$31$	7.37683	1.32492	0.662459	+	0.749098i	$0.269513\pi$
$32$	0	0
$33$	0.538379	0.0937198
$34$	0	0
$35$	−3.82180	−0.646003
$36$	0	0
$37$	−5.39508	−0.886945	−0.443473	−	0.896288i	$-0.646254\pi$
$37$	−5.39508	−0.886945	−0.443473	+	0.896288i	$0.646254\pi$
$38$	0	0
$39$	−0.538379	−0.0862097
$40$	0	0
$41$	−5.48430	−0.856504	−0.428252	−	0.903659i	$-0.640871\pi$
$41$	−5.48430	−0.856504	−0.428252	+	0.903659i	$0.640871\pi$
$42$	0	0
$43$	−7.33522	−1.11861	−0.559305	−	0.828962i	$-0.688932\pi$
$43$	−7.33522	−1.11861	−0.559305	+	0.828962i	$0.688932\pi$
$44$	0	0
$45$	−10.3576	−1.54403
$46$	0	0
$47$	1.10562	0.161271	0.0806354	−	0.996744i	$-0.474305\pi$
$47$	1.10562	0.161271	0.0806354	+	0.996744i	$0.474305\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.36272	0.470874
$52$	0	0
$53$	0.513417	0.0705233	0.0352616	−	0.999378i	$-0.488774\pi$
$53$	0.513417	0.0705233	0.0352616	+	0.999378i	$0.488774\pi$
$54$	0	0
$55$	−3.82180	−0.515332
$56$	0	0
$57$	1.85695	0.245959
$58$	0	0
$59$	5.87964	0.765464	0.382732	−	0.923860i	$-0.374983\pi$
$59$	5.87964	0.765464	0.382732	+	0.923860i	$0.374983\pi$
$60$	0	0
$61$	−10.0042	−1.28090	−0.640452	−	0.767998i	$-0.721253\pi$
$61$	−10.0042	−1.28090	−0.640452	+	0.767998i	$0.721253\pi$
$62$	0	0
$63$	2.71015	0.341446
$64$	0	0
$65$	3.82180	0.474036
$66$	0	0
$67$	−11.2911	−1.37943	−0.689714	−	0.724082i	$-0.742264\pi$
$67$	−11.2911	−1.37943	−0.689714	+	0.724082i	$0.742264\pi$
$68$	0	0
$69$	−2.25821	−0.271856
$70$	0	0
$71$	−3.93722	−0.467262	−0.233631	−	0.972325i	$-0.575061\pi$
$71$	−3.93722	−0.467262	−0.233631	+	0.972325i	$0.575061\pi$
$72$	0	0
$73$	−13.5870	−1.59024	−0.795119	−	0.606454i	$-0.792592\pi$
$73$	−13.5870	−1.59024	−0.795119	+	0.606454i	$0.792592\pi$
$74$	0	0
$75$	−5.17177	−0.597184
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−1.20114	−0.135138	−0.0675691	−	0.997715i	$-0.521524\pi$
$79$	−1.20114	−0.135138	−0.0675691	+	0.997715i	$0.521524\pi$
$80$	0	0
$81$	6.47534	0.719482
$82$	0	0
$83$	13.8487	1.52009	0.760044	−	0.649871i	$-0.225177\pi$
$83$	13.8487	1.52009	0.760044	+	0.649871i	$0.225177\pi$
$84$	0	0
$85$	−23.8710	−2.58917
$86$	0	0
$87$	2.96608	0.317997
$88$	0	0
$89$	−13.0549	−1.38382	−0.691909	−	0.721985i	$-0.743230\pi$
$89$	−13.0549	−1.38382	−0.691909	+	0.721985i	$0.743230\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−3.97153	−0.411829
$94$	0	0
$95$	−13.1820	−1.35244
$96$	0	0
$97$	−3.29239	−0.334291	−0.167146	−	0.985932i	$-0.553455\pi$
$97$	−3.29239	−0.334291	−0.167146	+	0.985932i	$0.553455\pi$
$98$	0	0
$99$	2.71015	0.272380
$100$	0	0
$101$	−8.12440	−0.808409	−0.404204	−	0.914669i	$-0.632452\pi$
$101$	−8.12440	−0.808409	−0.404204	+	0.914669i	$0.632452\pi$
$102$	0	0
$103$	−0.776944	−0.0765546	−0.0382773	−	0.999267i	$-0.512187\pi$
$103$	−0.776944	−0.0765546	−0.0382773	+	0.999267i	$0.512187\pi$
$104$	0	0
$105$	2.05758	0.200799
$106$	0	0
$107$	−11.0198	−1.06532	−0.532660	−	0.846329i	$-0.678808\pi$
$107$	−11.0198	−1.06532	−0.532660	+	0.846329i	$0.678808\pi$
$108$	0	0
$109$	1.74279	0.166929	0.0834647	−	0.996511i	$-0.473401\pi$
$109$	1.74279	0.166929	0.0834647	+	0.996511i	$0.473401\pi$
$110$	0	0
$111$	2.90460	0.275692
$112$	0	0
$113$	1.36311	0.128230	0.0641151	−	0.997943i	$-0.479578\pi$
$113$	1.36311	0.128230	0.0641151	+	0.997943i	$0.479578\pi$
$114$	0	0
$115$	16.0304	1.49484
$116$	0	0
$117$	−2.71015	−0.250553
$118$	0	0
$119$	6.24599	0.572569
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.95264	0.266230
$124$	0	0
$125$	17.6039	1.57454
$126$	0	0
$127$	13.2209	1.17316	0.586582	−	0.809890i	$-0.300473\pi$
$127$	13.2209	1.17316	0.586582	+	0.809890i	$0.300473\pi$
$128$	0	0
$129$	3.94913	0.347702
$130$	0	0
$131$	−3.64238	−0.318236	−0.159118	−	0.987260i	$-0.550865\pi$
$131$	−3.64238	−0.318236	−0.159118	+	0.987260i	$0.550865\pi$
$132$	0	0
$133$	3.44915	0.299080
$134$	0	0
$135$	11.7491	1.01120
$136$	0	0
$137$	4.56431	0.389956	0.194978	−	0.980808i	$-0.437537\pi$
$137$	4.56431	0.389956	0.194978	+	0.980808i	$0.437537\pi$
$138$	0	0
$139$	−8.95547	−0.759593	−0.379796	−	0.925070i	$-0.624006\pi$
$139$	−8.95547	−0.759593	−0.379796	+	0.925070i	$0.624006\pi$
$140$	0	0
$141$	−0.595242	−0.0501284
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−21.0553	−1.74855
$146$	0	0
$147$	−0.538379	−0.0444048
$148$	0	0
$149$	10.6761	0.874619	0.437310	−	0.899311i	$-0.355931\pi$
$149$	10.6761	0.874619	0.437310	+	0.899311i	$0.355931\pi$
$150$	0	0
$151$	9.44421	0.768559	0.384279	−	0.923217i	$-0.374450\pi$
$151$	9.44421	0.768559	0.384279	+	0.923217i	$0.374450\pi$
$152$	0	0
$153$	16.9276	1.36851
$154$	0	0
$155$	28.1928	2.26450
$156$	0	0
$157$	−7.33801	−0.585637	−0.292818	−	0.956168i	$-0.594593\pi$
$157$	−7.33801	−0.585637	−0.292818	+	0.956168i	$0.594593\pi$
$158$	0	0
$159$	−0.276413	−0.0219210
$160$	0	0
$161$	−4.19445	−0.330569
$162$	0	0
$163$	13.2165	1.03520	0.517598	−	0.855624i	$-0.326826\pi$
$163$	13.2165	1.03520	0.517598	+	0.855624i	$0.326826\pi$
$164$	0	0
$165$	2.05758	0.160182
$166$	0	0
$167$	−17.4123	−1.34740	−0.673702	−	0.739003i	$-0.735297\pi$
$167$	−17.4123	−1.34740	−0.673702	+	0.739003i	$0.735297\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	9.34772	0.714838
$172$	0	0
$173$	−9.39436	−0.714240	−0.357120	−	0.934059i	$-0.616241\pi$
$173$	−9.39436	−0.714240	−0.357120	+	0.934059i	$0.616241\pi$
$174$	0	0
$175$	−9.60618	−0.726159
$176$	0	0
$177$	−3.16548	−0.237932
$178$	0	0
$179$	8.87688	0.663489	0.331745	−	0.943369i	$-0.392363\pi$
$179$	8.87688	0.663489	0.331745	+	0.943369i	$0.392363\pi$
$180$	0	0
$181$	−7.33083	−0.544896	−0.272448	−	0.962170i	$-0.587833\pi$
$181$	−7.33083	−0.544896	−0.272448	+	0.962170i	$0.587833\pi$
$182$	0	0
$183$	5.38604	0.398148
$184$	0	0
$185$	−20.6189	−1.51593
$186$	0	0
$187$	6.24599	0.456752
$188$	0	0
$189$	−3.07423	−0.223617
$190$	0	0
$191$	−15.3168	−1.10829	−0.554144	−	0.832421i	$-0.686954\pi$
$191$	−15.3168	−1.10829	−0.554144	+	0.832421i	$0.686954\pi$
$192$	0	0
$193$	9.27188	0.667405	0.333702	−	0.942678i	$-0.391702\pi$
$193$	9.27188	0.667405	0.333702	+	0.942678i	$0.391702\pi$
$194$	0	0
$195$	−2.05758	−0.147346
$196$	0	0
$197$	−1.78340	−0.127062	−0.0635310	−	0.997980i	$-0.520236\pi$
$197$	−1.78340	−0.127062	−0.0635310	+	0.997980i	$0.520236\pi$
$198$	0	0
$199$	13.6782	0.969622	0.484811	−	0.874619i	$-0.338888\pi$
$199$	13.6782	0.969622	0.484811	+	0.874619i	$0.338888\pi$
$200$	0	0
$201$	6.07890	0.428773
$202$	0	0
$203$	5.50927	0.386675
$204$	0	0
$205$	−20.9599	−1.46390
$206$	0	0
$207$	−11.3676	−0.790102
$208$	0	0
$209$	3.44915	0.238583
$210$	0	0
$211$	−4.10752	−0.282773	−0.141387	−	0.989954i	$-0.545156\pi$
$211$	−4.10752	−0.282773	−0.141387	+	0.989954i	$0.545156\pi$
$212$	0	0
$213$	2.11972	0.145241
$214$	0	0
$215$	−28.0338	−1.91189
$216$	0	0
$217$	−7.37683	−0.500772
$218$	0	0
$219$	7.31496	0.494299
$220$	0	0
$221$	−6.24599	−0.420151
$222$	0	0
$223$	15.4033	1.03148	0.515740	−	0.856745i	$-0.327517\pi$
$223$	15.4033	1.03148	0.515740	+	0.856745i	$0.327517\pi$
$224$	0	0
$225$	−26.0342	−1.73561
$226$	0	0
$227$	0.717412	0.0476163	0.0238082	−	0.999717i	$-0.492421\pi$
$227$	0.717412	0.0476163	0.0238082	+	0.999717i	$0.492421\pi$
$228$	0	0
$229$	−26.6403	−1.76044	−0.880220	−	0.474565i	$-0.842605\pi$
$229$	−26.6403	−1.76044	−0.880220	+	0.474565i	$0.842605\pi$
$230$	0	0
$231$	−0.538379	−0.0354228
$232$	0	0
$233$	6.46727	0.423685	0.211842	−	0.977304i	$-0.432054\pi$
$233$	6.46727	0.423685	0.211842	+	0.977304i	$0.432054\pi$
$234$	0	0
$235$	4.22545	0.275638
$236$	0	0
$237$	0.646667	0.0420055
$238$	0	0
$239$	−26.8873	−1.73920	−0.869598	−	0.493760i	$-0.835622\pi$
$239$	−26.8873	−1.73920	−0.869598	+	0.493760i	$0.835622\pi$
$240$	0	0
$241$	24.4228	1.57321	0.786606	−	0.617456i	$-0.211837\pi$
$241$	24.4228	1.57321	0.786606	+	0.617456i	$0.211837\pi$
$242$	0	0
$243$	−12.7089	−0.815274
$244$	0	0
$245$	3.82180	0.244166
$246$	0	0
$247$	−3.44915	−0.219464
$248$	0	0
$249$	−7.45584	−0.472494
$250$	0	0
$251$	−10.3635	−0.654136	−0.327068	−	0.945001i	$-0.606061\pi$
$251$	−10.3635	−0.654136	−0.327068	+	0.945001i	$0.606061\pi$
$252$	0	0
$253$	−4.19445	−0.263703
$254$	0	0
$255$	12.8516	0.804801
$256$	0	0
$257$	−30.2492	−1.88689	−0.943446	−	0.331528i	$-0.892436\pi$
$257$	−30.2492	−1.88689	−0.943446	+	0.331528i	$0.892436\pi$
$258$	0	0
$259$	5.39508	0.335234
$260$	0	0
$261$	14.9309	0.924201
$262$	0	0
$263$	−26.7082	−1.64690	−0.823450	−	0.567388i	$-0.807954\pi$
$263$	−26.7082	−1.64690	−0.823450	+	0.567388i	$0.807954\pi$
$264$	0	0
$265$	1.96218	0.120536
$266$	0	0
$267$	7.02850	0.430137
$268$	0	0
$269$	9.52361	0.580664	0.290332	−	0.956926i	$-0.406234\pi$
$269$	9.52361	0.580664	0.290332	+	0.956926i	$0.406234\pi$
$270$	0	0
$271$	−1.31988	−0.0801770	−0.0400885	−	0.999196i	$-0.512764\pi$
$271$	−1.31988	−0.0801770	−0.0400885	+	0.999196i	$0.512764\pi$
$272$	0	0
$273$	0.538379	0.0325842
$274$	0	0
$275$	−9.60618	−0.579274
$276$	0	0
$277$	−0.824968	−0.0495675	−0.0247837	−	0.999693i	$-0.507890\pi$
$277$	−0.824968	−0.0495675	−0.0247837	+	0.999693i	$0.507890\pi$
$278$	0	0
$279$	−19.9923	−1.19691
$280$	0	0
$281$	15.0105	0.895449	0.447724	−	0.894172i	$-0.352235\pi$
$281$	15.0105	0.895449	0.447724	+	0.894172i	$0.352235\pi$
$282$	0	0
$283$	28.3775	1.68686	0.843432	−	0.537236i	$-0.180531\pi$
$283$	28.3775	1.68686	0.843432	+	0.537236i	$0.180531\pi$
$284$	0	0
$285$	7.09691	0.420385
$286$	0	0
$287$	5.48430	0.323728
$288$	0	0
$289$	22.0124	1.29485
$290$	0	0
$291$	1.77255	0.103909
$292$	0	0
$293$	21.8804	1.27827	0.639133	−	0.769096i	$-0.279293\pi$
$293$	21.8804	1.27827	0.639133	+	0.769096i	$0.279293\pi$
$294$	0	0
$295$	22.4708	1.30830
$296$	0	0
$297$	−3.07423	−0.178385
$298$	0	0
$299$	4.19445	0.242571
$300$	0	0
$301$	7.33522	0.422795
$302$	0	0
$303$	4.37401	0.251280
$304$	0	0
$305$	−38.2340	−2.18927
$306$	0	0
$307$	20.1820	1.15185	0.575924	−	0.817503i	$-0.304642\pi$
$307$	20.1820	1.15185	0.575924	+	0.817503i	$0.304642\pi$
$308$	0	0
$309$	0.418291	0.0237957
$310$	0	0
$311$	5.50412	0.312110	0.156055	−	0.987748i	$-0.450122\pi$
$311$	5.50412	0.312110	0.156055	+	0.987748i	$0.450122\pi$
$312$	0	0
$313$	−13.8652	−0.783706	−0.391853	−	0.920028i	$-0.628166\pi$
$313$	−13.8652	−0.783706	−0.391853	+	0.920028i	$0.628166\pi$
$314$	0	0
$315$	10.3576	0.583587
$316$	0	0
$317$	−32.0187	−1.79835	−0.899174	−	0.437590i	$-0.855832\pi$
$317$	−32.0187	−1.79835	−0.899174	+	0.437590i	$0.855832\pi$
$318$	0	0
$319$	5.50927	0.308460
$320$	0	0
$321$	5.93281	0.331137
$322$	0	0
$323$	21.5434	1.19871
$324$	0	0
$325$	9.60618	0.532855
$326$	0	0
$327$	−0.938284	−0.0518872
$328$	0	0
$329$	−1.10562	−0.0609547
$330$	0	0
$331$	10.3578	0.569317	0.284658	−	0.958629i	$-0.408120\pi$
$331$	10.3578	0.569317	0.284658	+	0.958629i	$0.408120\pi$
$332$	0	0
$333$	14.6215	0.801251
$334$	0	0
$335$	−43.1524	−2.35767
$336$	0	0
$337$	−2.58062	−0.140575	−0.0702877	−	0.997527i	$-0.522392\pi$
$337$	−2.58062	−0.140575	−0.0702877	+	0.997527i	$0.522392\pi$
$338$	0	0
$339$	−0.733868	−0.0398583
$340$	0	0
$341$	−7.37683	−0.399478
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−8.63042	−0.464646
$346$	0	0
$347$	12.3184	0.661284	0.330642	−	0.943756i	$-0.392735\pi$
$347$	12.3184	0.661284	0.330642	+	0.943756i	$0.392735\pi$
$348$	0	0
$349$	−21.0574	−1.12717	−0.563587	−	0.826057i	$-0.690579\pi$
$349$	−21.0574	−1.12717	−0.563587	+	0.826057i	$0.690579\pi$
$350$	0	0
$351$	3.07423	0.164090
$352$	0	0
$353$	−0.214952	−0.0114407	−0.00572037	−	0.999984i	$-0.501821\pi$
$353$	−0.214952	−0.0114407	−0.00572037	+	0.999984i	$0.501821\pi$
$354$	0	0
$355$	−15.0473	−0.798626
$356$	0	0
$357$	−3.36272	−0.177974
$358$	0	0
$359$	14.6977	0.775714	0.387857	−	0.921720i	$-0.373215\pi$
$359$	14.6977	0.775714	0.387857	+	0.921720i	$0.373215\pi$
$360$	0	0
$361$	−7.10334	−0.373860
$362$	0	0
$363$	−0.538379	−0.0282576
$364$	0	0
$365$	−51.9268	−2.71797
$366$	0	0
$367$	−15.5361	−0.810977	−0.405488	−	0.914100i	$-0.632899\pi$
$367$	−15.5361	−0.810977	−0.405488	+	0.914100i	$0.632899\pi$
$368$	0	0
$369$	14.8633	0.773751
$370$	0	0
$371$	−0.513417	−0.0266553
$372$	0	0
$373$	10.9446	0.566688	0.283344	−	0.959018i	$-0.408556\pi$
$373$	10.9446	0.566688	0.283344	+	0.959018i	$0.408556\pi$
$374$	0	0
$375$	−9.47758	−0.489420
$376$	0	0
$377$	−5.50927	−0.283742
$378$	0	0
$379$	−13.5310	−0.695042	−0.347521	−	0.937672i	$-0.612976\pi$
$379$	−13.5310	−0.695042	−0.347521	+	0.937672i	$0.612976\pi$
$380$	0	0
$381$	−7.11786	−0.364659
$382$	0	0
$383$	−30.9612	−1.58205	−0.791023	−	0.611787i	$-0.790451\pi$
$383$	−30.9612	−1.58205	−0.791023	+	0.611787i	$0.790451\pi$
$384$	0	0
$385$	3.82180	0.194777
$386$	0	0
$387$	19.8795	1.01053
$388$	0	0
$389$	−0.364631	−0.0184875	−0.00924376	−	0.999957i	$-0.502942\pi$
$389$	−0.364631	−0.0184875	−0.00924376	+	0.999957i	$0.502942\pi$
$390$	0	0
$391$	−26.1985	−1.32492
$392$	0	0
$393$	1.96098	0.0989185
$394$	0	0
$395$	−4.59050	−0.230973
$396$	0	0
$397$	28.7118	1.44100	0.720502	−	0.693452i	$-0.243911\pi$
$397$	28.7118	1.44100	0.720502	+	0.693452i	$0.243911\pi$
$398$	0	0
$399$	−1.85695	−0.0929640
$400$	0	0
$401$	11.2330	0.560950	0.280475	−	0.959861i	$-0.409508\pi$
$401$	11.2330	0.560950	0.280475	+	0.959861i	$0.409508\pi$
$402$	0	0
$403$	7.37683	0.367466
$404$	0	0
$405$	24.7475	1.22971
$406$	0	0
$407$	5.39508	0.267424
$408$	0	0
$409$	−23.9493	−1.18422	−0.592109	−	0.805858i	$-0.701704\pi$
$409$	−23.9493	−1.18422	−0.592109	+	0.805858i	$0.701704\pi$
$410$	0	0
$411$	−2.45733	−0.121211
$412$	0	0
$413$	−5.87964	−0.289318
$414$	0	0
$415$	52.9269	2.59808
$416$	0	0
$417$	4.82144	0.236107
$418$	0	0
$419$	23.2783	1.13722	0.568611	−	0.822607i	$-0.307481\pi$
$419$	23.2783	1.13722	0.568611	+	0.822607i	$0.307481\pi$
$420$	0	0
$421$	−22.6348	−1.10315	−0.551577	−	0.834124i	$-0.685974\pi$
$421$	−22.6348	−1.10315	−0.551577	+	0.834124i	$0.685974\pi$
$422$	0	0
$423$	−2.99639	−0.145689
$424$	0	0
$425$	−60.0001	−2.91043
$426$	0	0
$427$	10.0042	0.484136
$428$	0	0
$429$	0.538379	0.0259932
$430$	0	0
$431$	6.43752	0.310085	0.155042	−	0.987908i	$-0.450449\pi$
$431$	6.43752	0.310085	0.155042	+	0.987908i	$0.450449\pi$
$432$	0	0
$433$	−15.8059	−0.759585	−0.379793	−	0.925072i	$-0.624005\pi$
$433$	−15.8059	−0.759585	−0.379793	+	0.925072i	$0.624005\pi$
$434$	0	0
$435$	11.3358	0.543508
$436$	0	0
$437$	−14.4673	−0.692065
$438$	0	0
$439$	−12.2523	−0.584770	−0.292385	−	0.956301i	$-0.594449\pi$
$439$	−12.2523	−0.584770	−0.292385	+	0.956301i	$0.594449\pi$
$440$	0	0
$441$	−2.71015	−0.129055
$442$	0	0
$443$	4.38888	0.208522	0.104261	−	0.994550i	$-0.466752\pi$
$443$	4.38888	0.208522	0.104261	+	0.994550i	$0.466752\pi$
$444$	0	0
$445$	−49.8933	−2.36517
$446$	0	0
$447$	−5.74779	−0.271861
$448$	0	0
$449$	30.1846	1.42450	0.712249	−	0.701926i	$-0.247676\pi$
$449$	30.1846	1.42450	0.712249	+	0.701926i	$0.247676\pi$
$450$	0	0
$451$	5.48430	0.258246
$452$	0	0
$453$	−5.08457	−0.238894
$454$	0	0
$455$	−3.82180	−0.179169
$456$	0	0
$457$	−25.8796	−1.21060	−0.605298	−	0.795999i	$-0.706946\pi$
$457$	−25.8796	−1.21060	−0.605298	+	0.795999i	$0.706946\pi$
$458$	0	0
$459$	−19.2016	−0.896254
$460$	0	0
$461$	32.9555	1.53489	0.767446	−	0.641114i	$-0.221527\pi$
$461$	32.9555	1.53489	0.767446	+	0.641114i	$0.221527\pi$
$462$	0	0
$463$	−28.8221	−1.33947	−0.669737	−	0.742598i	$-0.733593\pi$
$463$	−28.8221	−1.33947	−0.669737	+	0.742598i	$0.733593\pi$
$464$	0	0
$465$	−15.1784	−0.703882
$466$	0	0
$467$	−30.3491	−1.40439	−0.702195	−	0.711985i	$-0.747797\pi$
$467$	−30.3491	−1.40439	−0.702195	+	0.711985i	$0.747797\pi$
$468$	0	0
$469$	11.2911	0.521375
$470$	0	0
$471$	3.95063	0.182036
$472$	0	0
$473$	7.33522	0.337274
$474$	0	0
$475$	−33.1332	−1.52025
$476$	0	0
$477$	−1.39144	−0.0637095
$478$	0	0
$479$	18.1980	0.831487	0.415744	−	0.909482i	$-0.363521\pi$
$479$	18.1980	0.831487	0.415744	+	0.909482i	$0.363521\pi$
$480$	0	0
$481$	−5.39508	−0.245994
$482$	0	0
$483$	2.25821	0.102752
$484$	0	0
$485$	−12.5828	−0.571358
$486$	0	0
$487$	−6.16504	−0.279365	−0.139682	−	0.990196i	$-0.544608\pi$
$487$	−6.16504	−0.279365	−0.139682	+	0.990196i	$0.544608\pi$
$488$	0	0
$489$	−7.11549	−0.321773
$490$	0	0
$491$	4.89425	0.220874	0.110437	−	0.993883i	$-0.464775\pi$
$491$	4.89425	0.220874	0.110437	+	0.993883i	$0.464775\pi$
$492$	0	0
$493$	34.4108	1.54979
$494$	0	0
$495$	10.3576	0.465542
$496$	0	0
$497$	3.93722	0.176608
$498$	0	0
$499$	−35.0822	−1.57050	−0.785248	−	0.619181i	$-0.787465\pi$
$499$	−35.0822	−1.57050	−0.785248	+	0.619181i	$0.787465\pi$
$500$	0	0
$501$	9.37443	0.418819
$502$	0	0
$503$	19.5840	0.873207	0.436604	−	0.899654i	$-0.356181\pi$
$503$	19.5840	0.873207	0.436604	+	0.899654i	$0.356181\pi$
$504$	0	0
$505$	−31.0499	−1.38170
$506$	0	0
$507$	−0.538379	−0.0239103
$508$	0	0
$509$	42.2137	1.87109	0.935545	−	0.353207i	$-0.114909\pi$
$509$	42.2137	1.87109	0.935545	+	0.353207i	$0.114909\pi$
$510$	0	0
$511$	13.5870	0.601053
$512$	0	0
$513$	−10.6035	−0.468155
$514$	0	0
$515$	−2.96933	−0.130844
$516$	0	0
$517$	−1.10562	−0.0486250
$518$	0	0
$519$	5.05773	0.222010
$520$	0	0
$521$	−7.53081	−0.329931	−0.164965	−	0.986299i	$-0.552751\pi$
$521$	−7.53081	−0.329931	−0.164965	+	0.986299i	$0.552751\pi$
$522$	0	0
$523$	39.5287	1.72847	0.864234	−	0.503090i	$-0.167804\pi$
$523$	39.5287	1.72847	0.864234	+	0.503090i	$0.167804\pi$
$524$	0	0
$525$	5.17177	0.225714
$526$	0	0
$527$	−46.0756	−2.00709
$528$	0	0
$529$	−5.40657	−0.235068
$530$	0	0
$531$	−15.9347	−0.691506
$532$	0	0
$533$	−5.48430	−0.237552
$534$	0	0
$535$	−42.1153	−1.82081
$536$	0	0
$537$	−4.77913	−0.206235
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−10.6733	−0.458879	−0.229440	−	0.973323i	$-0.573689\pi$
$541$	−10.6733	−0.458879	−0.229440	+	0.973323i	$0.573689\pi$
$542$	0	0
$543$	3.94677	0.169372
$544$	0	0
$545$	6.66061	0.285309
$546$	0	0
$547$	−39.9192	−1.70682	−0.853411	−	0.521239i	$-0.825470\pi$
$547$	−39.9192	−1.70682	−0.853411	+	0.521239i	$0.825470\pi$
$548$	0	0
$549$	27.1128	1.15715
$550$	0	0
$551$	19.0023	0.809525
$552$	0	0
$553$	1.20114	0.0510775
$554$	0	0
$555$	11.1008	0.471203
$556$	0	0
$557$	16.4691	0.697817	0.348909	−	0.937157i	$-0.386552\pi$
$557$	16.4691	0.697817	0.348909	+	0.937157i	$0.386552\pi$
$558$	0	0
$559$	−7.33522	−0.310247
$560$	0	0
$561$	−3.36272	−0.141974
$562$	0	0
$563$	−32.3530	−1.36352	−0.681759	−	0.731577i	$-0.738785\pi$
$563$	−32.3530	−1.36352	−0.681759	+	0.731577i	$0.738785\pi$
$564$	0	0
$565$	5.20952	0.219166
$566$	0	0
$567$	−6.47534	−0.271939
$568$	0	0
$569$	2.56347	0.107466	0.0537332	−	0.998555i	$-0.482888\pi$
$569$	2.56347	0.107466	0.0537332	+	0.998555i	$0.482888\pi$
$570$	0	0
$571$	18.2800	0.764994	0.382497	−	0.923957i	$-0.375064\pi$
$571$	18.2800	0.764994	0.382497	+	0.923957i	$0.375064\pi$
$572$	0	0
$573$	8.24627	0.344493
$574$	0	0
$575$	40.2926	1.68032
$576$	0	0
$577$	38.2481	1.59229	0.796145	−	0.605105i	$-0.206869\pi$
$577$	38.2481	1.59229	0.796145	+	0.605105i	$0.206869\pi$
$578$	0	0
$579$	−4.99179	−0.207452
$580$	0	0
$581$	−13.8487	−0.574539
$582$	0	0
$583$	−0.513417	−0.0212636
$584$	0	0
$585$	−10.3576	−0.428236
$586$	0	0
$587$	45.4961	1.87783	0.938913	−	0.344153i	$-0.111834\pi$
$587$	45.4961	1.87783	0.938913	+	0.344153i	$0.111834\pi$
$588$	0	0
$589$	−25.4438	−1.04839
$590$	0	0
$591$	0.960147	0.0394952
$592$	0	0
$593$	42.4833	1.74458	0.872289	−	0.488990i	$-0.162635\pi$
$593$	42.4833	1.74458	0.872289	+	0.488990i	$0.162635\pi$
$594$	0	0
$595$	23.8710	0.978614
$596$	0	0
$597$	−7.36406	−0.301391
$598$	0	0
$599$	−31.8998	−1.30339	−0.651695	−	0.758481i	$-0.725942\pi$
$599$	−31.8998	−1.30339	−0.651695	+	0.758481i	$0.725942\pi$
$600$	0	0
$601$	−45.4902	−1.85558	−0.927792	−	0.373097i	$-0.878296\pi$
$601$	−45.4902	−1.85558	−0.927792	+	0.373097i	$0.878296\pi$
$602$	0	0
$603$	30.6006	1.24615
$604$	0	0
$605$	3.82180	0.155378
$606$	0	0
$607$	21.5639	0.875250	0.437625	−	0.899158i	$-0.355820\pi$
$607$	21.5639	0.875250	0.437625	+	0.899158i	$0.355820\pi$
$608$	0	0
$609$	−2.96608	−0.120191
$610$	0	0
$611$	1.10562	0.0447285
$612$	0	0
$613$	11.6661	0.471188	0.235594	−	0.971852i	$-0.424296\pi$
$613$	11.6661	0.471188	0.235594	+	0.971852i	$0.424296\pi$
$614$	0	0
$615$	11.2844	0.455031
$616$	0	0
$617$	41.9108	1.68726	0.843632	−	0.536922i	$-0.180413\pi$
$617$	41.9108	1.68726	0.843632	+	0.536922i	$0.180413\pi$
$618$	0	0
$619$	22.6124	0.908868	0.454434	−	0.890780i	$-0.349842\pi$
$619$	22.6124	0.908868	0.454434	+	0.890780i	$0.349842\pi$
$620$	0	0
$621$	12.8947	0.517446
$622$	0	0
$623$	13.0549	0.523034
$624$	0	0
$625$	19.2477	0.769910
$626$	0	0
$627$	−1.85695	−0.0741596
$628$	0	0
$629$	33.6976	1.34361
$630$	0	0
$631$	0.830482	0.0330610	0.0165305	−	0.999863i	$-0.494738\pi$
$631$	0.830482	0.0330610	0.0165305	+	0.999863i	$0.494738\pi$
$632$	0	0
$633$	2.21140	0.0878954
$634$	0	0
$635$	50.5277	2.00513
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	10.6704	0.422116
$640$	0	0
$641$	5.84671	0.230931	0.115466	−	0.993311i	$-0.463164\pi$
$641$	5.84671	0.230931	0.115466	+	0.993311i	$0.463164\pi$
$642$	0	0
$643$	32.4499	1.27970	0.639849	−	0.768501i	$-0.278997\pi$
$643$	32.4499	1.27970	0.639849	+	0.768501i	$0.278997\pi$
$644$	0	0
$645$	15.0928	0.594279
$646$	0	0
$647$	−12.3955	−0.487319	−0.243660	−	0.969861i	$-0.578348\pi$
$647$	−12.3955	−0.487319	−0.243660	+	0.969861i	$0.578348\pi$
$648$	0	0
$649$	−5.87964	−0.230796
$650$	0	0
$651$	3.97153	0.155657
$652$	0	0
$653$	−37.1592	−1.45415	−0.727075	−	0.686558i	$-0.759121\pi$
$653$	−37.1592	−1.45415	−0.727075	+	0.686558i	$0.759121\pi$
$654$	0	0
$655$	−13.9205	−0.543917
$656$	0	0
$657$	36.8228	1.43659
$658$	0	0
$659$	1.08274	0.0421776	0.0210888	−	0.999778i	$-0.493287\pi$
$659$	1.08274	0.0421776	0.0210888	+	0.999778i	$0.493287\pi$
$660$	0	0
$661$	15.3258	0.596105	0.298053	−	0.954549i	$-0.403663\pi$
$661$	15.3258	0.596105	0.298053	+	0.954549i	$0.403663\pi$
$662$	0	0
$663$	3.36272	0.130597
$664$	0	0
$665$	13.1820	0.511175
$666$	0	0
$667$	−23.1084	−0.894759
$668$	0	0
$669$	−8.29280	−0.320618
$670$	0	0
$671$	10.0042	0.386207
$672$	0	0
$673$	24.4256	0.941539	0.470770	−	0.882256i	$-0.343976\pi$
$673$	24.4256	0.941539	0.470770	+	0.882256i	$0.343976\pi$
$674$	0	0
$675$	29.5316	1.13667
$676$	0	0
$677$	12.2267	0.469910	0.234955	−	0.972006i	$-0.424506\pi$
$677$	12.2267	0.469910	0.234955	+	0.972006i	$0.424506\pi$
$678$	0	0
$679$	3.29239	0.126350
$680$	0	0
$681$	−0.386240	−0.0148007
$682$	0	0
$683$	−3.64058	−0.139303	−0.0696515	−	0.997571i	$-0.522189\pi$
$683$	−3.64058	−0.139303	−0.0696515	+	0.997571i	$0.522189\pi$
$684$	0	0
$685$	17.4439	0.666497
$686$	0	0
$687$	14.3426	0.547204
$688$	0	0
$689$	0.513417	0.0195596
$690$	0	0
$691$	−13.8933	−0.528527	−0.264264	−	0.964450i	$-0.585129\pi$
$691$	−13.8933	−0.528527	−0.264264	+	0.964450i	$0.585129\pi$
$692$	0	0
$693$	−2.71015	−0.102950
$694$	0	0
$695$	−34.2260	−1.29827
$696$	0	0
$697$	34.2549	1.29750
$698$	0	0
$699$	−3.48184	−0.131695
$700$	0	0
$701$	30.3282	1.14548	0.572741	−	0.819736i	$-0.305880\pi$
$701$	30.3282	1.14548	0.572741	+	0.819736i	$0.305880\pi$
$702$	0	0
$703$	18.6085	0.701831
$704$	0	0
$705$	−2.27490	−0.0856776
$706$	0	0
$707$	8.12440	0.305550
$708$	0	0
$709$	18.5981	0.698467	0.349234	−	0.937036i	$-0.386442\pi$
$709$	18.5981	0.698467	0.349234	+	0.937036i	$0.386442\pi$
$710$	0	0
$711$	3.25525	0.122082
$712$	0	0
$713$	30.9418	1.15878
$714$	0	0
$715$	−3.82180	−0.142927
$716$	0	0
$717$	14.4756	0.540600
$718$	0	0
$719$	31.0328	1.15733	0.578663	−	0.815566i	$-0.303575\pi$
$719$	31.0328	1.15733	0.578663	+	0.815566i	$0.303575\pi$
$720$	0	0
$721$	0.776944	0.0289349
$722$	0	0
$723$	−13.1487	−0.489007
$724$	0	0
$725$	−52.9230	−1.96551
$726$	0	0
$727$	11.5345	0.427791	0.213896	−	0.976857i	$-0.431385\pi$
$727$	11.5345	0.427791	0.213896	+	0.976857i	$0.431385\pi$
$728$	0	0
$729$	−12.5838	−0.466068
$730$	0	0
$731$	45.8157	1.69456
$732$	0	0
$733$	−34.1440	−1.26114	−0.630569	−	0.776133i	$-0.717179\pi$
$733$	−34.1440	−1.26114	−0.630569	+	0.776133i	$0.717179\pi$
$734$	0	0
$735$	−2.05758	−0.0758950
$736$	0	0
$737$	11.2911	0.415913
$738$	0	0
$739$	1.66418	0.0612177	0.0306089	−	0.999531i	$-0.490255\pi$
$739$	1.66418	0.0612177	0.0306089	+	0.999531i	$0.490255\pi$
$740$	0	0
$741$	1.85695	0.0682169
$742$	0	0
$743$	0.171399	0.00628804	0.00314402	−	0.999995i	$-0.498999\pi$
$743$	0.171399	0.00628804	0.00314402	+	0.999995i	$0.498999\pi$
$744$	0	0
$745$	40.8019	1.49487
$746$	0	0
$747$	−37.5319	−1.37322
$748$	0	0
$749$	11.0198	0.402653
$750$	0	0
$751$	9.39770	0.342927	0.171463	−	0.985190i	$-0.445150\pi$
$751$	9.39770	0.342927	0.171463	+	0.985190i	$0.445150\pi$
$752$	0	0
$753$	5.57947	0.203327
$754$	0	0
$755$	36.0939	1.31359
$756$	0	0
$757$	6.52118	0.237016	0.118508	−	0.992953i	$-0.462189\pi$
$757$	6.52118	0.237016	0.118508	+	0.992953i	$0.462189\pi$
$758$	0	0
$759$	2.25821	0.0819677
$760$	0	0
$761$	−45.0746	−1.63395	−0.816977	−	0.576670i	$-0.804352\pi$
$761$	−45.0746	−1.63395	−0.816977	+	0.576670i	$0.804352\pi$
$762$	0	0
$763$	−1.74279	−0.0630934
$764$	0	0
$765$	64.6938	2.33901
$766$	0	0
$767$	5.87964	0.212301
$768$	0	0
$769$	53.9933	1.94705	0.973525	−	0.228581i	$-0.0734086\pi$
$769$	53.9933	1.94705	0.973525	+	0.228581i	$0.0734086\pi$
$770$	0	0
$771$	16.2855	0.586509
$772$	0	0
$773$	−28.1956	−1.01413	−0.507063	−	0.861909i	$-0.669269\pi$
$773$	−28.1956	−1.01413	−0.507063	+	0.861909i	$0.669269\pi$
$774$	0	0
$775$	70.8631	2.54548
$776$	0	0
$777$	−2.90460	−0.104202
$778$	0	0
$779$	18.9162	0.677743
$780$	0	0
$781$	3.93722	0.140885
$782$	0	0
$783$	−16.9367	−0.605269
$784$	0	0
$785$	−28.0444	−1.00095
$786$	0	0
$787$	17.6374	0.628707	0.314354	−	0.949306i	$-0.398212\pi$
$787$	17.6374	0.628707	0.314354	+	0.949306i	$0.398212\pi$
$788$	0	0
$789$	14.3792	0.511912
$790$	0	0
$791$	−1.36311	−0.0484665
$792$	0	0
$793$	−10.0042	−0.355259
$794$	0	0
$795$	−1.05640	−0.0374665
$796$	0	0
$797$	−18.8228	−0.666737	−0.333369	−	0.942797i	$-0.608185\pi$
$797$	−18.8228	−0.666737	−0.333369	+	0.942797i	$0.608185\pi$
$798$	0	0
$799$	−6.90568	−0.244305
$800$	0	0
$801$	35.3807	1.25012
$802$	0	0
$803$	13.5870	0.479475
$804$	0	0
$805$	−16.0304	−0.564996
$806$	0	0
$807$	−5.12731	−0.180490
$808$	0	0
$809$	2.44058	0.0858063	0.0429032	−	0.999079i	$-0.486339\pi$
$809$	2.44058	0.0858063	0.0429032	+	0.999079i	$0.486339\pi$
$810$	0	0
$811$	−15.5240	−0.545120	−0.272560	−	0.962139i	$-0.587870\pi$
$811$	−15.5240	−0.545120	−0.272560	+	0.962139i	$0.587870\pi$
$812$	0	0
$813$	0.710596	0.0249217
$814$	0	0
$815$	50.5108	1.76932
$816$	0	0
$817$	25.3003	0.885145
$818$	0	0
$819$	2.71015	0.0947002
$820$	0	0
$821$	−50.9286	−1.77742	−0.888711	−	0.458468i	$-0.848398\pi$
$821$	−50.9286	−1.77742	−0.888711	+	0.458468i	$0.848398\pi$
$822$	0	0
$823$	−47.0887	−1.64141	−0.820704	−	0.571353i	$-0.806419\pi$
$823$	−47.0887	−1.64141	−0.820704	+	0.571353i	$0.806419\pi$
$824$	0	0
$825$	5.17177	0.180058
$826$	0	0
$827$	−27.1153	−0.942892	−0.471446	−	0.881895i	$-0.656268\pi$
$827$	−27.1153	−0.942892	−0.471446	+	0.881895i	$0.656268\pi$
$828$	0	0
$829$	5.52894	0.192028	0.0960140	−	0.995380i	$-0.469391\pi$
$829$	5.52894	0.192028	0.0960140	+	0.995380i	$0.469391\pi$
$830$	0	0
$831$	0.444146	0.0154072
$832$	0	0
$833$	−6.24599	−0.216411
$834$	0	0
$835$	−66.5464	−2.30293
$836$	0	0
$837$	22.6780	0.783868
$838$	0	0
$839$	−11.3774	−0.392791	−0.196396	−	0.980525i	$-0.562924\pi$
$839$	−11.3774	−0.392791	−0.196396	+	0.980525i	$0.562924\pi$
$840$	0	0
$841$	1.35202	0.0466213
$842$	0	0
$843$	−8.08132	−0.278336
$844$	0	0
$845$	3.82180	0.131474
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−15.2778	−0.524334
$850$	0	0
$851$	−22.6294	−0.775726
$852$	0	0
$853$	18.2971	0.626479	0.313240	−	0.949674i	$-0.398586\pi$
$853$	18.2971	0.626479	0.313240	+	0.949674i	$0.398586\pi$
$854$	0	0
$855$	35.7251	1.22177
$856$	0	0
$857$	19.8166	0.676922	0.338461	−	0.940980i	$-0.390094\pi$
$857$	19.8166	0.676922	0.338461	+	0.940980i	$0.390094\pi$
$858$	0	0
$859$	−33.4986	−1.14296	−0.571479	−	0.820617i	$-0.693630\pi$
$859$	−33.4986	−1.14296	−0.571479	+	0.820617i	$0.693630\pi$
$860$	0	0
$861$	−2.95264	−0.100626
$862$	0	0
$863$	1.11196	0.0378516	0.0189258	−	0.999821i	$-0.493975\pi$
$863$	1.11196	0.0378516	0.0189258	+	0.999821i	$0.493975\pi$
$864$	0	0
$865$	−35.9034	−1.22075
$866$	0	0
$867$	−11.8510	−0.402483
$868$	0	0
$869$	1.20114	0.0407457
$870$	0	0
$871$	−11.2911	−0.382585
$872$	0	0
$873$	8.92285	0.301993
$874$	0	0
$875$	−17.6039	−0.595120
$876$	0	0
$877$	−21.4233	−0.723415	−0.361707	−	0.932292i	$-0.617806\pi$
$877$	−21.4233	−0.723415	−0.361707	+	0.932292i	$0.617806\pi$
$878$	0	0
$879$	−11.7799	−0.397328
$880$	0	0
$881$	40.0563	1.34953	0.674765	−	0.738033i	$-0.264245\pi$
$881$	40.0563	1.34953	0.674765	+	0.738033i	$0.264245\pi$
$882$	0	0
$883$	16.3232	0.549319	0.274660	−	0.961541i	$-0.411435\pi$
$883$	16.3232	0.549319	0.274660	+	0.961541i	$0.411435\pi$
$884$	0	0
$885$	−12.0978	−0.406664
$886$	0	0
$887$	44.7860	1.50377	0.751884	−	0.659295i	$-0.229145\pi$
$887$	44.7860	1.50377	0.751884	+	0.659295i	$0.229145\pi$
$888$	0	0
$889$	−13.2209	−0.443415
$890$	0	0
$891$	−6.47534	−0.216932
$892$	0	0
$893$	−3.81344	−0.127612
$894$	0	0
$895$	33.9257	1.13401
$896$	0	0
$897$	−2.25821	−0.0753993
$898$	0	0
$899$	−40.6409	−1.35545
$900$	0	0
$901$	−3.20680	−0.106834
$902$	0	0
$903$	−3.94913	−0.131419
$904$	0	0
$905$	−28.0170	−0.931316
$906$	0	0
$907$	10.6318	0.353023	0.176512	−	0.984299i	$-0.443519\pi$
$907$	10.6318	0.353023	0.176512	+	0.984299i	$0.443519\pi$
$908$	0	0
$909$	22.0183	0.730302
$910$	0	0
$911$	22.7741	0.754539	0.377269	−	0.926104i	$-0.376863\pi$
$911$	22.7741	0.754539	0.377269	+	0.926104i	$0.376863\pi$
$912$	0	0
$913$	−13.8487	−0.458324
$914$	0	0
$915$	20.5844	0.680499
$916$	0	0
$917$	3.64238	0.120282
$918$	0	0
$919$	−26.2637	−0.866361	−0.433180	−	0.901307i	$-0.642609\pi$
$919$	−26.2637	−0.866361	−0.433180	+	0.901307i	$0.642609\pi$
$920$	0	0
$921$	−10.8656	−0.358033
$922$	0	0
$923$	−3.93722	−0.129595
$924$	0	0
$925$	−51.8261	−1.70403
$926$	0	0
$927$	2.10563	0.0691581
$928$	0	0
$929$	18.0149	0.591051	0.295525	−	0.955335i	$-0.404505\pi$
$929$	18.0149	0.591051	0.295525	+	0.955335i	$0.404505\pi$
$930$	0	0
$931$	−3.44915	−0.113041
$932$	0	0
$933$	−2.96330	−0.0970142
$934$	0	0
$935$	23.8710	0.780664
$936$	0	0
$937$	31.8333	1.03995	0.519974	−	0.854182i	$-0.325942\pi$
$937$	31.8333	1.03995	0.519974	+	0.854182i	$0.325942\pi$
$938$	0	0
$939$	7.46473	0.243602
$940$	0	0
$941$	38.5684	1.25730	0.628648	−	0.777690i	$-0.283609\pi$
$941$	38.5684	1.25730	0.628648	+	0.777690i	$0.283609\pi$
$942$	0	0
$943$	−23.0037	−0.749102
$944$	0	0
$945$	−11.7491	−0.382198
$946$	0	0
$947$	15.5473	0.505219	0.252610	−	0.967568i	$-0.418711\pi$
$947$	15.5473	0.505219	0.252610	+	0.967568i	$0.418711\pi$
$948$	0	0
$949$	−13.5870	−0.441053
$950$	0	0
$951$	17.2382	0.558987
$952$	0	0
$953$	39.3745	1.27546	0.637732	−	0.770258i	$-0.279873\pi$
$953$	39.3745	1.27546	0.637732	+	0.770258i	$0.279873\pi$
$954$	0	0
$955$	−58.5379	−1.89424
$956$	0	0
$957$	−2.96608	−0.0958796
$958$	0	0
$959$	−4.56431	−0.147389
$960$	0	0
$961$	23.4176	0.755406
$962$	0	0
$963$	29.8652	0.962392
$964$	0	0
$965$	35.4353	1.14070
$966$	0	0
$967$	−16.6674	−0.535988	−0.267994	−	0.963421i	$-0.586361\pi$
$967$	−16.6674	−0.535988	−0.267994	+	0.963421i	$0.586361\pi$
$968$	0	0
$969$	−11.5985	−0.372598
$970$	0	0
$971$	−27.4767	−0.881769	−0.440884	−	0.897564i	$-0.645335\pi$
$971$	−27.4767	−0.881769	−0.440884	+	0.897564i	$0.645335\pi$
$972$	0	0
$973$	8.95547	0.287099
$974$	0	0
$975$	−5.17177	−0.165629
$976$	0	0
$977$	30.2812	0.968783	0.484391	−	0.874851i	$-0.339041\pi$
$977$	30.2812	0.968783	0.484391	+	0.874851i	$0.339041\pi$
$978$	0	0
$979$	13.0549	0.417237
$980$	0	0
$981$	−4.72323	−0.150801
$982$	0	0
$983$	43.0010	1.37152	0.685759	−	0.727829i	$-0.259470\pi$
$983$	43.0010	1.37152	0.685759	+	0.727829i	$0.259470\pi$
$984$	0	0
$985$	−6.81581	−0.217170
$986$	0	0
$987$	0.595242	0.0189468
$988$	0	0
$989$	−30.7672	−0.978341
$990$	0	0
$991$	28.1185	0.893212	0.446606	−	0.894731i	$-0.352633\pi$
$991$	28.1185	0.893212	0.446606	+	0.894731i	$0.352633\pi$
$992$	0	0
$993$	−5.57643	−0.176963
$994$	0	0
$995$	52.2754	1.65724
$996$	0	0
$997$	40.4594	1.28136	0.640681	−	0.767807i	$-0.278652\pi$
$997$	40.4594	1.28136	0.640681	+	0.767807i	$0.278652\pi$
$998$	0	0
$999$	−16.5857	−0.524748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.a.f.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.a.f.1.2	✓	5	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)

\(f(q)\)	\(=\)	\(q-0.538379 q^{3} +3.82180 q^{5} -1.00000 q^{7} -2.71015 q^{9} +O(q^{10})\)	\(q-0.538379 q^{3} +3.82180 q^{5} -1.00000 q^{7} -2.71015 q^{9} -1.00000 q^{11} +1.00000 q^{13} -2.05758 q^{15} -6.24599 q^{17} -3.44915 q^{19} +0.538379 q^{21} +4.19445 q^{23} +9.60618 q^{25} +3.07423 q^{27} -5.50927 q^{29} +7.37683 q^{31} +0.538379 q^{33} -3.82180 q^{35} -5.39508 q^{37} -0.538379 q^{39} -5.48430 q^{41} -7.33522 q^{43} -10.3576 q^{45} +1.10562 q^{47} +1.00000 q^{49} +3.36272 q^{51} +0.513417 q^{53} -3.82180 q^{55} +1.85695 q^{57} +5.87964 q^{59} -10.0042 q^{61} +2.71015 q^{63} +3.82180 q^{65} -11.2911 q^{67} -2.25821 q^{69} -3.93722 q^{71} -13.5870 q^{73} -5.17177 q^{75} +1.00000 q^{77} -1.20114 q^{79} +6.47534 q^{81} +13.8487 q^{83} -23.8710 q^{85} +2.96608 q^{87} -13.0549 q^{89} -1.00000 q^{91} -3.97153 q^{93} -13.1820 q^{95} -3.29239 q^{97} +2.71015 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9}+O(q^{10}) \) 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9	\( 5 q + 3 q^{3} - 5 q^{7} + 2 q^{9} - 5 q^{11} + 5 q^{13} - 3 q^{15} - 9 q^{17} - 4 q^{19} - 3 q^{21} - 4 q^{23} + q^{25} + 3 q^{27} - 8 q^{29} + 7 q^{31} - 3 q^{33} - 10 q^{37} + 3 q^{39} - 18 q^{41} - 22 q^{43} - 26 q^{45} + 12 q^{47} + 5 q^{49} - 7 q^{51} + 7 q^{53} - 6 q^{57} - q^{59} - 26 q^{61} - 2 q^{63} - 8 q^{67} - 12 q^{69} + 18 q^{71} - 25 q^{73} - 16 q^{75} + 5 q^{77} - 6 q^{79} - 7 q^{81} + 11 q^{83} - 7 q^{85} - 5 q^{87} - 14 q^{89} - 5 q^{91} - 41 q^{93} - 22 q^{95} - 33 q^{97} - 2 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 - 5 * q^7 + 2 * q^9 - 5 * q^11 + 5 * q^13 - 3 * q^15 - 9 * q^17 - 4 * q^19 - 3 * q^21 - 4 * q^23 + q^25 + 3 * q^27 - 8 * q^29 + 7 * q^31 - 3 * q^33 - 10 * q^37 + 3 * q^39 - 18 * q^41 - 22 * q^43 - 26 * q^45 + 12 * q^47 + 5 * q^49 - 7 * q^51 + 7 * q^53 - 6 * q^57 - q^59 - 26 * q^61 - 2 * q^63 - 8 * q^67 - 12 * q^69 + 18 * q^71 - 25 * q^73 - 16 * q^75 + 5 * q^77 - 6 * q^79 - 7 * q^81 + 11 * q^83 - 7 * q^85 - 5 * q^87 - 14 * q^89 - 5 * q^91 - 41 * q^93 - 22 * q^95 - 33 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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