LMFDB - Embedded newform 6004.2.a.g.1.7

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:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
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)

$f(q)$	$=$	$q-1.94547 q^{3} -1.68966 q^{5} -1.54452 q^{7} +0.784840 q^{9} +O(q^{10})$	$q-1.94547 q^{3} -1.68966 q^{5} -1.54452 q^{7} +0.784840 q^{9} -2.23286 q^{11} -1.94704 q^{13} +3.28718 q^{15} -2.28556 q^{17} +1.00000 q^{19} +3.00481 q^{21} +0.628320 q^{23} -2.14505 q^{25} +4.30952 q^{27} +9.85349 q^{29} +8.81551 q^{31} +4.34395 q^{33} +2.60971 q^{35} -2.52622 q^{37} +3.78791 q^{39} +8.82145 q^{41} -2.39479 q^{43} -1.32611 q^{45} -1.22440 q^{47} -4.61447 q^{49} +4.44649 q^{51} +5.98586 q^{53} +3.77277 q^{55} -1.94547 q^{57} -8.90536 q^{59} -0.227978 q^{61} -1.21220 q^{63} +3.28984 q^{65} +10.6449 q^{67} -1.22238 q^{69} -11.5083 q^{71} +10.5499 q^{73} +4.17311 q^{75} +3.44869 q^{77} -1.00000 q^{79} -10.7385 q^{81} +11.1289 q^{83} +3.86183 q^{85} -19.1696 q^{87} +0.0896524 q^{89} +3.00724 q^{91} -17.1503 q^{93} -1.68966 q^{95} -6.86913 q^{97} -1.75244 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	$ 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) $ 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * 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
$3$	−1.94547	−1.12322	−0.561608	−	0.827404i	$-0.689817\pi$
$3$	−1.94547	−1.12322	−0.561608	+	0.827404i	$0.689817\pi$
$4$	0	0
$5$	−1.68966	−0.755639	−0.377820	−	0.925879i	$-0.623326\pi$
$5$	−1.68966	−0.755639	−0.377820	+	0.925879i	$0.623326\pi$
$6$	0	0
$7$	−1.54452	−0.583773	−0.291886	−	0.956453i	$-0.594283\pi$
$7$	−1.54452	−0.583773	−0.291886	+	0.956453i	$0.594283\pi$
$8$	0	0
$9$	0.784840	0.261613
$10$	0	0
$11$	−2.23286	−0.673232	−0.336616	−	0.941642i	$-0.609282\pi$
$11$	−2.23286	−0.673232	−0.336616	+	0.941642i	$0.609282\pi$
$12$	0	0
$13$	−1.94704	−0.540012	−0.270006	−	0.962859i	$-0.587026\pi$
$13$	−1.94704	−0.540012	−0.270006	+	0.962859i	$0.587026\pi$
$14$	0	0
$15$	3.28718	0.848746
$16$	0	0
$17$	−2.28556	−0.554331	−0.277165	−	0.960822i	$-0.589395\pi$
$17$	−2.28556	−0.554331	−0.277165	+	0.960822i	$0.589395\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	3.00481	0.655702
$22$	0	0
$23$	0.628320	0.131014	0.0655069	−	0.997852i	$-0.479134\pi$
$23$	0.628320	0.131014	0.0655069	+	0.997852i	$0.479134\pi$
$24$	0	0
$25$	−2.14505	−0.429009
$26$	0	0
$27$	4.30952	0.829367
$28$	0	0
$29$	9.85349	1.82975	0.914874	−	0.403740i	$-0.132290\pi$
$29$	9.85349	1.82975	0.914874	+	0.403740i	$0.132290\pi$
$30$	0	0
$31$	8.81551	1.58331	0.791656	−	0.610967i	$-0.209219\pi$
$31$	8.81551	1.58331	0.791656	+	0.610967i	$0.209219\pi$
$32$	0	0
$33$	4.34395	0.756185
$34$	0	0
$35$	2.60971	0.441122
$36$	0	0
$37$	−2.52622	−0.415309	−0.207654	−	0.978202i	$-0.566583\pi$
$37$	−2.52622	−0.415309	−0.207654	+	0.978202i	$0.566583\pi$
$38$	0	0
$39$	3.78791	0.606550
$40$	0	0
$41$	8.82145	1.37768	0.688840	−	0.724914i	$-0.258120\pi$
$41$	8.82145	1.37768	0.688840	+	0.724914i	$0.258120\pi$
$42$	0	0
$43$	−2.39479	−0.365203	−0.182601	−	0.983187i	$-0.558452\pi$
$43$	−2.39479	−0.365203	−0.182601	+	0.983187i	$0.558452\pi$
$44$	0	0
$45$	−1.32611	−0.197685
$46$	0	0
$47$	−1.22440	−0.178597	−0.0892986	−	0.996005i	$-0.528463\pi$
$47$	−1.22440	−0.178597	−0.0892986	+	0.996005i	$0.528463\pi$
$48$	0	0
$49$	−4.61447	−0.659210
$50$	0	0
$51$	4.44649	0.622633
$52$	0	0
$53$	5.98586	0.822221	0.411111	−	0.911585i	$-0.365141\pi$
$53$	5.98586	0.822221	0.411111	+	0.911585i	$0.365141\pi$
$54$	0	0
$55$	3.77277	0.508721
$56$	0	0
$57$	−1.94547	−0.257683
$58$	0	0
$59$	−8.90536	−1.15938	−0.579690	−	0.814837i	$-0.696826\pi$
$59$	−8.90536	−1.15938	−0.579690	+	0.814837i	$0.696826\pi$
$60$	0	0
$61$	−0.227978	−0.0291896	−0.0145948	−	0.999893i	$-0.504646\pi$
$61$	−0.227978	−0.0291896	−0.0145948	+	0.999893i	$0.504646\pi$
$62$	0	0
$63$	−1.21220	−0.152723
$64$	0	0
$65$	3.28984	0.408055
$66$	0	0
$67$	10.6449	1.30048	0.650239	−	0.759729i	$-0.274669\pi$
$67$	10.6449	1.30048	0.650239	+	0.759729i	$0.274669\pi$
$68$	0	0
$69$	−1.22238	−0.147157
$70$	0	0
$71$	−11.5083	−1.36579	−0.682894	−	0.730517i	$-0.739279\pi$
$71$	−11.5083	−1.36579	−0.682894	+	0.730517i	$0.739279\pi$
$72$	0	0
$73$	10.5499	1.23477	0.617384	−	0.786662i	$-0.288192\pi$
$73$	10.5499	1.23477	0.617384	+	0.786662i	$0.288192\pi$
$74$	0	0
$75$	4.17311	0.481870
$76$	0	0
$77$	3.44869	0.393014
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−10.7385	−1.19317
$82$	0	0
$83$	11.1289	1.22155	0.610777	−	0.791802i	$-0.290857\pi$
$83$	11.1289	1.22155	0.610777	+	0.791802i	$0.290857\pi$
$84$	0	0
$85$	3.86183	0.418874
$86$	0	0
$87$	−19.1696	−2.05520
$88$	0	0
$89$	0.0896524	0.00950313	0.00475157	−	0.999989i	$-0.498488\pi$
$89$	0.0896524	0.00950313	0.00475157	+	0.999989i	$0.498488\pi$
$90$	0	0
$91$	3.00724	0.315244
$92$	0	0
$93$	−17.1503	−1.77840
$94$	0	0
$95$	−1.68966	−0.173356
$96$	0	0
$97$	−6.86913	−0.697455	−0.348727	−	0.937224i	$-0.613386\pi$
$97$	−6.86913	−0.697455	−0.348727	+	0.937224i	$0.613386\pi$
$98$	0	0
$99$	−1.75244	−0.176127
$100$	0	0
$101$	−19.5738	−1.94767	−0.973834	−	0.227261i	$-0.927023\pi$
$101$	−19.5738	−1.94767	−0.973834	+	0.227261i	$0.927023\pi$
$102$	0	0
$103$	−2.97112	−0.292753	−0.146377	−	0.989229i	$-0.546761\pi$
$103$	−2.97112	−0.292753	−0.146377	+	0.989229i	$0.546761\pi$
$104$	0	0
$105$	−5.07710	−0.495475
$106$	0	0
$107$	4.77671	0.461782	0.230891	−	0.972980i	$-0.425836\pi$
$107$	4.77671	0.461782	0.230891	+	0.972980i	$0.425836\pi$
$108$	0	0
$109$	9.04370	0.866230	0.433115	−	0.901339i	$-0.357414\pi$
$109$	9.04370	0.866230	0.433115	+	0.901339i	$0.357414\pi$
$110$	0	0
$111$	4.91469	0.466481
$112$	0	0
$113$	9.42838	0.886947	0.443474	−	0.896287i	$-0.353746\pi$
$113$	9.42838	0.886947	0.443474	+	0.896287i	$0.353746\pi$
$114$	0	0
$115$	−1.06165	−0.0989992
$116$	0	0
$117$	−1.52812	−0.141274
$118$	0	0
$119$	3.53009	0.323603
$120$	0	0
$121$	−6.01434	−0.546759
$122$	0	0
$123$	−17.1618	−1.54743
$124$	0	0
$125$	12.0727	1.07982
$126$	0	0
$127$	10.2289	0.907667	0.453834	−	0.891086i	$-0.350056\pi$
$127$	10.2289	0.907667	0.453834	+	0.891086i	$0.350056\pi$
$128$	0	0
$129$	4.65899	0.410201
$130$	0	0
$131$	−1.42246	−0.124281	−0.0621406	−	0.998067i	$-0.519793\pi$
$131$	−1.42246	−0.124281	−0.0621406	+	0.998067i	$0.519793\pi$
$132$	0	0
$133$	−1.54452	−0.133927
$134$	0	0
$135$	−7.28163	−0.626703
$136$	0	0
$137$	−6.43714	−0.549962	−0.274981	−	0.961450i	$-0.588672\pi$
$137$	−6.43714	−0.549962	−0.274981	+	0.961450i	$0.588672\pi$
$138$	0	0
$139$	−18.9811	−1.60996	−0.804978	−	0.593304i	$-0.797823\pi$
$139$	−18.9811	−1.60996	−0.804978	+	0.593304i	$0.797823\pi$
$140$	0	0
$141$	2.38203	0.200603
$142$	0	0
$143$	4.34747	0.363554
$144$	0	0
$145$	−16.6491	−1.38263
$146$	0	0
$147$	8.97729	0.740435
$148$	0	0
$149$	−18.7653	−1.53731	−0.768657	−	0.639661i	$-0.779075\pi$
$149$	−18.7653	−1.53731	−0.768657	+	0.639661i	$0.779075\pi$
$150$	0	0
$151$	1.69858	0.138228	0.0691141	−	0.997609i	$-0.477983\pi$
$151$	1.69858	0.138228	0.0691141	+	0.997609i	$0.477983\pi$
$152$	0	0
$153$	−1.79380	−0.145020
$154$	0	0
$155$	−14.8952	−1.19641
$156$	0	0
$157$	−1.35957	−0.108506	−0.0542529	−	0.998527i	$-0.517278\pi$
$157$	−1.35957	−0.108506	−0.0542529	+	0.998527i	$0.517278\pi$
$158$	0	0
$159$	−11.6453	−0.923532
$160$	0	0
$161$	−0.970451	−0.0764823
$162$	0	0
$163$	10.2339	0.801583	0.400791	−	0.916169i	$-0.368735\pi$
$163$	10.2339	0.801583	0.400791	+	0.916169i	$0.368735\pi$
$164$	0	0
$165$	−7.33981	−0.571403
$166$	0	0
$167$	23.7932	1.84117	0.920587	−	0.390538i	$-0.127711\pi$
$167$	23.7932	1.84117	0.920587	+	0.390538i	$0.127711\pi$
$168$	0	0
$169$	−9.20903	−0.708387
$170$	0	0
$171$	0.784840	0.0600182
$172$	0	0
$173$	4.59167	0.349098	0.174549	−	0.984648i	$-0.444153\pi$
$173$	4.59167	0.349098	0.174549	+	0.984648i	$0.444153\pi$
$174$	0	0
$175$	3.31306	0.250444
$176$	0	0
$177$	17.3251	1.30223
$178$	0	0
$179$	18.5661	1.38770	0.693849	−	0.720120i	$-0.255913\pi$
$179$	18.5661	1.38770	0.693849	+	0.720120i	$0.255913\pi$
$180$	0	0
$181$	−5.19440	−0.386096	−0.193048	−	0.981189i	$-0.561837\pi$
$181$	−5.19440	−0.386096	−0.193048	+	0.981189i	$0.561837\pi$
$182$	0	0
$183$	0.443524	0.0327863
$184$	0	0
$185$	4.26846	0.313824
$186$	0	0
$187$	5.10334	0.373193
$188$	0	0
$189$	−6.65613	−0.484162
$190$	0	0
$191$	16.9035	1.22309	0.611546	−	0.791209i	$-0.290548\pi$
$191$	16.9035	1.22309	0.611546	+	0.791209i	$0.290548\pi$
$192$	0	0
$193$	9.36408	0.674041	0.337021	−	0.941497i	$-0.390581\pi$
$193$	9.36408	0.674041	0.337021	+	0.941497i	$0.390581\pi$
$194$	0	0
$195$	−6.40028	−0.458333
$196$	0	0
$197$	5.78091	0.411873	0.205936	−	0.978565i	$-0.433976\pi$
$197$	5.78091	0.411873	0.205936	+	0.978565i	$0.433976\pi$
$198$	0	0
$199$	8.99799	0.637851	0.318925	−	0.947780i	$-0.396678\pi$
$199$	8.99799	0.637851	0.318925	+	0.947780i	$0.396678\pi$
$200$	0	0
$201$	−20.7093	−1.46072
$202$	0	0
$203$	−15.2189	−1.06816
$204$	0	0
$205$	−14.9053	−1.04103
$206$	0	0
$207$	0.493131	0.0342750
$208$	0	0
$209$	−2.23286	−0.154450
$210$	0	0
$211$	12.7868	0.880281	0.440141	−	0.897929i	$-0.354929\pi$
$211$	12.7868	0.880281	0.440141	+	0.897929i	$0.354929\pi$
$212$	0	0
$213$	22.3891	1.53408
$214$	0	0
$215$	4.04639	0.275961
$216$	0	0
$217$	−13.6157	−0.924294
$218$	0	0
$219$	−20.5244	−1.38691
$220$	0	0
$221$	4.45009	0.299345
$222$	0	0
$223$	16.2912	1.09094	0.545469	−	0.838131i	$-0.316352\pi$
$223$	16.2912	1.09094	0.545469	+	0.838131i	$0.316352\pi$
$224$	0	0
$225$	−1.68352	−0.112235
$226$	0	0
$227$	−15.2508	−1.01223	−0.506116	−	0.862465i	$-0.668919\pi$
$227$	−15.2508	−1.01223	−0.506116	+	0.862465i	$0.668919\pi$
$228$	0	0
$229$	6.04728	0.399616	0.199808	−	0.979835i	$-0.435968\pi$
$229$	6.04728	0.399616	0.199808	+	0.979835i	$0.435968\pi$
$230$	0	0
$231$	−6.70931	−0.441440
$232$	0	0
$233$	−25.5034	−1.67078	−0.835390	−	0.549657i	$-0.814758\pi$
$233$	−25.5034	−1.67078	−0.835390	+	0.549657i	$0.814758\pi$
$234$	0	0
$235$	2.06882	0.134955
$236$	0	0
$237$	1.94547	0.126372
$238$	0	0
$239$	−27.0987	−1.75287	−0.876435	−	0.481520i	$-0.840085\pi$
$239$	−27.0987	−1.75287	−0.876435	+	0.481520i	$0.840085\pi$
$240$	0	0
$241$	−20.2881	−1.30687	−0.653437	−	0.756981i	$-0.726673\pi$
$241$	−20.2881	−1.30687	−0.653437	+	0.756981i	$0.726673\pi$
$242$	0	0
$243$	7.96293	0.510822
$244$	0	0
$245$	7.79689	0.498125
$246$	0	0
$247$	−1.94704	−0.123887
$248$	0	0
$249$	−21.6509	−1.37207
$250$	0	0
$251$	−16.6565	−1.05135	−0.525674	−	0.850686i	$-0.676187\pi$
$251$	−16.6565	−1.05135	−0.525674	+	0.850686i	$0.676187\pi$
$252$	0	0
$253$	−1.40295	−0.0882027
$254$	0	0
$255$	−7.51306	−0.470486
$256$	0	0
$257$	−10.5528	−0.658264	−0.329132	−	0.944284i	$-0.606756\pi$
$257$	−10.5528	−0.658264	−0.329132	+	0.944284i	$0.606756\pi$
$258$	0	0
$259$	3.90180	0.242446
$260$	0	0
$261$	7.73342	0.478687
$262$	0	0
$263$	−15.7219	−0.969453	−0.484727	−	0.874666i	$-0.661081\pi$
$263$	−15.7219	−0.969453	−0.484727	+	0.874666i	$0.661081\pi$
$264$	0	0
$265$	−10.1141	−0.621303
$266$	0	0
$267$	−0.174416	−0.0106741
$268$	0	0
$269$	−15.8377	−0.965644	−0.482822	−	0.875719i	$-0.660388\pi$
$269$	−15.8377	−0.965644	−0.482822	+	0.875719i	$0.660388\pi$
$270$	0	0
$271$	−19.3871	−1.17768	−0.588841	−	0.808249i	$-0.700416\pi$
$271$	−19.3871	−1.17768	−0.588841	+	0.808249i	$0.700416\pi$
$272$	0	0
$273$	−5.85048	−0.354087
$274$	0	0
$275$	4.78958	0.288823
$276$	0	0
$277$	−14.1338	−0.849219	−0.424610	−	0.905377i	$-0.639589\pi$
$277$	−14.1338	−0.849219	−0.424610	+	0.905377i	$0.639589\pi$
$278$	0	0
$279$	6.91877	0.414216
$280$	0	0
$281$	−31.7341	−1.89310	−0.946549	−	0.322559i	$-0.895457\pi$
$281$	−31.7341	−1.89310	−0.946549	+	0.322559i	$0.895457\pi$
$282$	0	0
$283$	21.2319	1.26210	0.631052	−	0.775740i	$-0.282623\pi$
$283$	21.2319	1.26210	0.631052	+	0.775740i	$0.282623\pi$
$284$	0	0
$285$	3.28718	0.194716
$286$	0	0
$287$	−13.6249	−0.804252
$288$	0	0
$289$	−11.7762	−0.692718
$290$	0	0
$291$	13.3637	0.783392
$292$	0	0
$293$	11.2583	0.657715	0.328857	−	0.944380i	$-0.393336\pi$
$293$	11.2583	0.657715	0.328857	+	0.944380i	$0.393336\pi$
$294$	0	0
$295$	15.0470	0.876073
$296$	0	0
$297$	−9.62255	−0.558357
$298$	0	0
$299$	−1.22337	−0.0707491
$300$	0	0
$301$	3.69880	0.213195
$302$	0	0
$303$	38.0802	2.18765
$304$	0	0
$305$	0.385206	0.0220568
$306$	0	0
$307$	14.3753	0.820442	0.410221	−	0.911986i	$-0.365452\pi$
$307$	14.3753	0.820442	0.410221	+	0.911986i	$0.365452\pi$
$308$	0	0
$309$	5.78021	0.328825
$310$	0	0
$311$	19.5755	1.11002	0.555011	−	0.831843i	$-0.312714\pi$
$311$	19.5755	1.11002	0.555011	+	0.831843i	$0.312714\pi$
$312$	0	0
$313$	−17.0882	−0.965883	−0.482942	−	0.875652i	$-0.660432\pi$
$313$	−17.0882	−0.965883	−0.482942	+	0.875652i	$0.660432\pi$
$314$	0	0
$315$	2.04821	0.115403
$316$	0	0
$317$	−5.37892	−0.302110	−0.151055	−	0.988525i	$-0.548267\pi$
$317$	−5.37892	−0.302110	−0.151055	+	0.988525i	$0.548267\pi$
$318$	0	0
$319$	−22.0015	−1.23184
$320$	0	0
$321$	−9.29293	−0.518681
$322$	0	0
$323$	−2.28556	−0.127172
$324$	0	0
$325$	4.17649	0.231670
$326$	0	0
$327$	−17.5942	−0.972963
$328$	0	0
$329$	1.89111	0.104260
$330$	0	0
$331$	−6.89404	−0.378931	−0.189465	−	0.981887i	$-0.560675\pi$
$331$	−6.89404	−0.378931	−0.189465	+	0.981887i	$0.560675\pi$
$332$	0	0
$333$	−1.98268	−0.108650
$334$	0	0
$335$	−17.9862	−0.982693
$336$	0	0
$337$	16.8477	0.917751	0.458876	−	0.888500i	$-0.348252\pi$
$337$	16.8477	0.917751	0.458876	+	0.888500i	$0.348252\pi$
$338$	0	0
$339$	−18.3426	−0.996233
$340$	0	0
$341$	−19.6838	−1.06594
$342$	0	0
$343$	17.9387	0.968601
$344$	0	0
$345$	2.06540	0.111197
$346$	0	0
$347$	−0.878020	−0.0471346	−0.0235673	−	0.999722i	$-0.507502\pi$
$347$	−0.878020	−0.0471346	−0.0235673	+	0.999722i	$0.507502\pi$
$348$	0	0
$349$	16.5273	0.884685	0.442342	−	0.896846i	$-0.354148\pi$
$349$	16.5273	0.884685	0.442342	+	0.896846i	$0.354148\pi$
$350$	0	0
$351$	−8.39082	−0.447869
$352$	0	0
$353$	−16.6844	−0.888020	−0.444010	−	0.896022i	$-0.646444\pi$
$353$	−16.6844	−0.888020	−0.444010	+	0.896022i	$0.646444\pi$
$354$	0	0
$355$	19.4452	1.03204
$356$	0	0
$357$	−6.86768	−0.363476
$358$	0	0
$359$	−22.3069	−1.17732	−0.588658	−	0.808382i	$-0.700343\pi$
$359$	−22.3069	−1.17732	−0.588658	+	0.808382i	$0.700343\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	11.7007	0.614128
$364$	0	0
$365$	−17.8257	−0.933040
$366$	0	0
$367$	−2.87588	−0.150120	−0.0750598	−	0.997179i	$-0.523915\pi$
$367$	−2.87588	−0.150120	−0.0750598	+	0.997179i	$0.523915\pi$
$368$	0	0
$369$	6.92343	0.360420
$370$	0	0
$371$	−9.24526	−0.479990
$372$	0	0
$373$	−7.25843	−0.375827	−0.187914	−	0.982186i	$-0.560172\pi$
$373$	−7.25843	−0.375827	−0.187914	+	0.982186i	$0.560172\pi$
$374$	0	0
$375$	−23.4870	−1.21287
$376$	0	0
$377$	−19.1852	−0.988086
$378$	0	0
$379$	12.9269	0.664011	0.332005	−	0.943278i	$-0.392275\pi$
$379$	12.9269	0.664011	0.332005	+	0.943278i	$0.392275\pi$
$380$	0	0
$381$	−19.9000	−1.01951
$382$	0	0
$383$	−28.3525	−1.44874	−0.724372	−	0.689409i	$-0.757870\pi$
$383$	−28.3525	−1.44874	−0.724372	+	0.689409i	$0.757870\pi$
$384$	0	0
$385$	−5.82711	−0.296977
$386$	0	0
$387$	−1.87953	−0.0955419
$388$	0	0
$389$	24.0792	1.22086	0.610431	−	0.792069i	$-0.290996\pi$
$389$	24.0792	1.22086	0.610431	+	0.792069i	$0.290996\pi$
$390$	0	0
$391$	−1.43607	−0.0726250
$392$	0	0
$393$	2.76735	0.139595
$394$	0	0
$395$	1.68966	0.0850161
$396$	0	0
$397$	−8.21626	−0.412362	−0.206181	−	0.978514i	$-0.566104\pi$
$397$	−8.21626	−0.412362	−0.206181	+	0.978514i	$0.566104\pi$
$398$	0	0
$399$	3.00481	0.150428
$400$	0	0
$401$	−13.3379	−0.666065	−0.333032	−	0.942915i	$-0.608072\pi$
$401$	−13.3379	−0.666065	−0.333032	+	0.942915i	$0.608072\pi$
$402$	0	0
$403$	−17.1642	−0.855008
$404$	0	0
$405$	18.1445	0.901608
$406$	0	0
$407$	5.64070	0.279599
$408$	0	0
$409$	22.1978	1.09761	0.548805	−	0.835950i	$-0.315083\pi$
$409$	22.1978	1.09761	0.548805	+	0.835950i	$0.315083\pi$
$410$	0	0
$411$	12.5232	0.617726
$412$	0	0
$413$	13.7545	0.676814
$414$	0	0
$415$	−18.8041	−0.923055
$416$	0	0
$417$	36.9271	1.80833
$418$	0	0
$419$	−3.89237	−0.190155	−0.0950774	−	0.995470i	$-0.530310\pi$
$419$	−3.89237	−0.190155	−0.0950774	+	0.995470i	$0.530310\pi$
$420$	0	0
$421$	1.31264	0.0639741	0.0319870	−	0.999488i	$-0.489816\pi$
$421$	1.31264	0.0639741	0.0319870	+	0.999488i	$0.489816\pi$
$422$	0	0
$423$	−0.960959	−0.0467234
$424$	0	0
$425$	4.90264	0.237813
$426$	0	0
$427$	0.352116	0.0170401
$428$	0	0
$429$	−8.45786	−0.408349
$430$	0	0
$431$	−0.429092	−0.0206686	−0.0103343	−	0.999947i	$-0.503290\pi$
$431$	−0.429092	−0.0206686	−0.0103343	+	0.999947i	$0.503290\pi$
$432$	0	0
$433$	−21.4872	−1.03261	−0.516303	−	0.856406i	$-0.672692\pi$
$433$	−21.4872	−1.03261	−0.516303	+	0.856406i	$0.672692\pi$
$434$	0	0
$435$	32.3902	1.55299
$436$	0	0
$437$	0.628320	0.0300566
$438$	0	0
$439$	26.2766	1.25411	0.627057	−	0.778973i	$-0.284259\pi$
$439$	26.2766	1.25411	0.627057	+	0.778973i	$0.284259\pi$
$440$	0	0
$441$	−3.62162	−0.172458
$442$	0	0
$443$	4.58447	0.217815	0.108907	−	0.994052i	$-0.465265\pi$
$443$	4.58447	0.217815	0.108907	+	0.994052i	$0.465265\pi$
$444$	0	0
$445$	−0.151482	−0.00718094
$446$	0	0
$447$	36.5073	1.72673
$448$	0	0
$449$	−39.8035	−1.87844	−0.939222	−	0.343310i	$-0.888452\pi$
$449$	−39.8035	−1.87844	−0.939222	+	0.343310i	$0.888452\pi$
$450$	0	0
$451$	−19.6971	−0.927498
$452$	0	0
$453$	−3.30452	−0.155260
$454$	0	0
$455$	−5.08122	−0.238211
$456$	0	0
$457$	−16.2471	−0.760006	−0.380003	−	0.924985i	$-0.624077\pi$
$457$	−16.2471	−0.760006	−0.380003	+	0.924985i	$0.624077\pi$
$458$	0	0
$459$	−9.84968	−0.459744
$460$	0	0
$461$	5.66095	0.263657	0.131828	−	0.991273i	$-0.457915\pi$
$461$	5.66095	0.263657	0.131828	+	0.991273i	$0.457915\pi$
$462$	0	0
$463$	−15.9993	−0.743550	−0.371775	−	0.928323i	$-0.621251\pi$
$463$	−15.9993	−0.743550	−0.371775	+	0.928323i	$0.621251\pi$
$464$	0	0
$465$	28.9782	1.34383
$466$	0	0
$467$	−38.9373	−1.80180	−0.900902	−	0.434022i	$-0.857094\pi$
$467$	−38.9373	−1.80180	−0.900902	+	0.434022i	$0.857094\pi$
$468$	0	0
$469$	−16.4412	−0.759184
$470$	0	0
$471$	2.64501	0.121875
$472$	0	0
$473$	5.34724	0.245866
$474$	0	0
$475$	−2.14505	−0.0984214
$476$	0	0
$477$	4.69795	0.215104
$478$	0	0
$479$	21.3865	0.977175	0.488588	−	0.872515i	$-0.337512\pi$
$479$	21.3865	0.977175	0.488588	+	0.872515i	$0.337512\pi$
$480$	0	0
$481$	4.91867	0.224272
$482$	0	0
$483$	1.88798	0.0859061
$484$	0	0
$485$	11.6065	0.527024
$486$	0	0
$487$	15.0325	0.681185	0.340593	−	0.940211i	$-0.389372\pi$
$487$	15.0325	0.681185	0.340593	+	0.940211i	$0.389372\pi$
$488$	0	0
$489$	−19.9098	−0.900351
$490$	0	0
$491$	−4.71053	−0.212583	−0.106292	−	0.994335i	$-0.533898\pi$
$491$	−4.71053	−0.212583	−0.106292	+	0.994335i	$0.533898\pi$
$492$	0	0
$493$	−22.5208	−1.01429
$494$	0	0
$495$	2.96102	0.133088
$496$	0	0
$497$	17.7748	0.797310
$498$	0	0
$499$	39.0601	1.74857	0.874286	−	0.485411i	$-0.161330\pi$
$499$	39.0601	1.74857	0.874286	+	0.485411i	$0.161330\pi$
$500$	0	0
$501$	−46.2889	−2.06804
$502$	0	0
$503$	−17.2588	−0.769533	−0.384766	−	0.923014i	$-0.625718\pi$
$503$	−17.2588	−0.769533	−0.384766	+	0.923014i	$0.625718\pi$
$504$	0	0
$505$	33.0731	1.47173
$506$	0	0
$507$	17.9159	0.795671
$508$	0	0
$509$	−2.59053	−0.114823	−0.0574116	−	0.998351i	$-0.518285\pi$
$509$	−2.59053	−0.114823	−0.0574116	+	0.998351i	$0.518285\pi$
$510$	0	0
$511$	−16.2945	−0.720824
$512$	0	0
$513$	4.30952	0.190270
$514$	0	0
$515$	5.02018	0.221216
$516$	0	0
$517$	2.73391	0.120237
$518$	0	0
$519$	−8.93293	−0.392112
$520$	0	0
$521$	4.33931	0.190109	0.0950543	−	0.995472i	$-0.469698\pi$
$521$	4.33931	0.190109	0.0950543	+	0.995472i	$0.469698\pi$
$522$	0	0
$523$	44.4972	1.94573	0.972863	−	0.231382i	$-0.0743247\pi$
$523$	44.4972	1.94573	0.972863	+	0.231382i	$0.0743247\pi$
$524$	0	0
$525$	−6.44545	−0.281302
$526$	0	0
$527$	−20.1484	−0.877679
$528$	0	0
$529$	−22.6052	−0.982835
$530$	0	0
$531$	−6.98929	−0.303309
$532$	0	0
$533$	−17.1757	−0.743964
$534$	0	0
$535$	−8.07102	−0.348941
$536$	0	0
$537$	−36.1198	−1.55868
$538$	0	0
$539$	10.3035	0.443801
$540$	0	0
$541$	−36.8530	−1.58443	−0.792217	−	0.610239i	$-0.791073\pi$
$541$	−36.8530	−1.58443	−0.792217	+	0.610239i	$0.791073\pi$
$542$	0	0
$543$	10.1055	0.433670
$544$	0	0
$545$	−15.2808	−0.654557
$546$	0	0
$547$	19.0232	0.813371	0.406686	−	0.913568i	$-0.366684\pi$
$547$	19.0232	0.813371	0.406686	+	0.913568i	$0.366684\pi$
$548$	0	0
$549$	−0.178927	−0.00763640
$550$	0	0
$551$	9.85349	0.419773
$552$	0	0
$553$	1.54452	0.0656795
$554$	0	0
$555$	−8.30415	−0.352492
$556$	0	0
$557$	5.76386	0.244223	0.122111	−	0.992516i	$-0.461033\pi$
$557$	5.76386	0.244223	0.122111	+	0.992516i	$0.461033\pi$
$558$	0	0
$559$	4.66276	0.197214
$560$	0	0
$561$	−9.92838	−0.419176
$562$	0	0
$563$	11.1413	0.469551	0.234776	−	0.972050i	$-0.424564\pi$
$563$	11.1413	0.469551	0.234776	+	0.972050i	$0.424564\pi$
$564$	0	0
$565$	−15.9308	−0.670212
$566$	0	0
$567$	16.5859	0.696541
$568$	0	0
$569$	−9.34362	−0.391705	−0.195852	−	0.980633i	$-0.562747\pi$
$569$	−9.34362	−0.391705	−0.195852	+	0.980633i	$0.562747\pi$
$570$	0	0
$571$	−24.4434	−1.02292	−0.511462	−	0.859306i	$-0.670896\pi$
$571$	−24.4434	−1.02292	−0.511462	+	0.859306i	$0.670896\pi$
$572$	0	0
$573$	−32.8851	−1.37380
$574$	0	0
$575$	−1.34778	−0.0562061
$576$	0	0
$577$	31.9054	1.32824	0.664119	−	0.747627i	$-0.268807\pi$
$577$	31.9054	1.32824	0.664119	+	0.747627i	$0.268807\pi$
$578$	0	0
$579$	−18.2175	−0.757094
$580$	0	0
$581$	−17.1888	−0.713110
$582$	0	0
$583$	−13.3656	−0.553546
$584$	0	0
$585$	2.58200	0.106753
$586$	0	0
$587$	16.8863	0.696974	0.348487	−	0.937314i	$-0.386696\pi$
$587$	16.8863	0.696974	0.348487	+	0.937314i	$0.386696\pi$
$588$	0	0
$589$	8.81551	0.363237
$590$	0	0
$591$	−11.2466	−0.462622
$592$	0	0
$593$	27.6755	1.13650	0.568248	−	0.822857i	$-0.307621\pi$
$593$	27.6755	1.13650	0.568248	+	0.822857i	$0.307621\pi$
$594$	0	0
$595$	−5.96466	−0.244527
$596$	0	0
$597$	−17.5053	−0.716444
$598$	0	0
$599$	−4.44481	−0.181610	−0.0908050	−	0.995869i	$-0.528944\pi$
$599$	−4.44481	−0.181610	−0.0908050	+	0.995869i	$0.528944\pi$
$600$	0	0
$601$	−22.4815	−0.917039	−0.458519	−	0.888684i	$-0.651620\pi$
$601$	−22.4815	−0.917039	−0.458519	+	0.888684i	$0.651620\pi$
$602$	0	0
$603$	8.35453	0.340223
$604$	0	0
$605$	10.1622	0.413152
$606$	0	0
$607$	−36.9989	−1.50174	−0.750869	−	0.660451i	$-0.770365\pi$
$607$	−36.9989	−1.50174	−0.750869	+	0.660451i	$0.770365\pi$
$608$	0	0
$609$	29.6078	1.19977
$610$	0	0
$611$	2.38396	0.0964447
$612$	0	0
$613$	41.5150	1.67677	0.838387	−	0.545075i	$-0.183499\pi$
$613$	41.5150	1.67677	0.838387	+	0.545075i	$0.183499\pi$
$614$	0	0
$615$	28.9977	1.16930
$616$	0	0
$617$	38.1286	1.53500	0.767499	−	0.641050i	$-0.221501\pi$
$617$	38.1286	1.53500	0.767499	+	0.641050i	$0.221501\pi$
$618$	0	0
$619$	−43.5820	−1.75171	−0.875854	−	0.482575i	$-0.839701\pi$
$619$	−43.5820	−1.75171	−0.875854	+	0.482575i	$0.839701\pi$
$620$	0	0
$621$	2.70776	0.108659
$622$	0	0
$623$	−0.138470	−0.00554767
$624$	0	0
$625$	−9.67355	−0.386942
$626$	0	0
$627$	4.34395	0.173481
$628$	0	0
$629$	5.77385	0.230218
$630$	0	0
$631$	16.4660	0.655503	0.327751	−	0.944764i	$-0.393709\pi$
$631$	16.4660	0.655503	0.327751	+	0.944764i	$0.393709\pi$
$632$	0	0
$633$	−24.8763	−0.988746
$634$	0	0
$635$	−17.2834	−0.685869
$636$	0	0
$637$	8.98456	0.355981
$638$	0	0
$639$	−9.03221	−0.357309
$640$	0	0
$641$	−26.2420	−1.03650	−0.518249	−	0.855230i	$-0.673416\pi$
$641$	−26.2420	−1.03650	−0.518249	+	0.855230i	$0.673416\pi$
$642$	0	0
$643$	−23.0795	−0.910169	−0.455084	−	0.890448i	$-0.650391\pi$
$643$	−23.0795	−0.910169	−0.455084	+	0.890448i	$0.650391\pi$
$644$	0	0
$645$	−7.87212	−0.309964
$646$	0	0
$647$	38.9237	1.53025	0.765124	−	0.643883i	$-0.222677\pi$
$647$	38.9237	1.53025	0.765124	+	0.643883i	$0.222677\pi$
$648$	0	0
$649$	19.8844	0.780531
$650$	0	0
$651$	26.4889	1.03818
$652$	0	0
$653$	−27.3366	−1.06976	−0.534882	−	0.844927i	$-0.679644\pi$
$653$	−27.3366	−1.06976	−0.534882	+	0.844927i	$0.679644\pi$
$654$	0	0
$655$	2.40348	0.0939118
$656$	0	0
$657$	8.27996	0.323032
$658$	0	0
$659$	−14.0685	−0.548031	−0.274015	−	0.961725i	$-0.588352\pi$
$659$	−14.0685	−0.548031	−0.274015	+	0.961725i	$0.588352\pi$
$660$	0	0
$661$	0.0153979	0.000598911 0	0.000299455	−	1.00000i	$-0.499905\pi$
$661$	0.0153979	0.000598911 0	0.000299455		1.00000i	$0.499905\pi$
$662$	0	0
$663$	−8.65750	−0.336229
$664$	0	0
$665$	2.60971	0.101200
$666$	0	0
$667$	6.19115	0.239722
$668$	0	0
$669$	−31.6939	−1.22536
$670$	0	0
$671$	0.509043	0.0196514
$672$	0	0
$673$	12.6003	0.485705	0.242853	−	0.970063i	$-0.421917\pi$
$673$	12.6003	0.485705	0.242853	+	0.970063i	$0.421917\pi$
$674$	0	0
$675$	−9.24412	−0.355806
$676$	0	0
$677$	46.5166	1.78778	0.893890	−	0.448287i	$-0.147966\pi$
$677$	46.5166	1.78778	0.893890	+	0.448287i	$0.147966\pi$
$678$	0	0
$679$	10.6095	0.407155
$680$	0	0
$681$	29.6699	1.13695
$682$	0	0
$683$	2.26070	0.0865034	0.0432517	−	0.999064i	$-0.486228\pi$
$683$	2.26070	0.0865034	0.0432517	+	0.999064i	$0.486228\pi$
$684$	0	0
$685$	10.8766	0.415573
$686$	0	0
$687$	−11.7648	−0.448855
$688$	0	0
$689$	−11.6547	−0.444010
$690$	0	0
$691$	47.6231	1.81167	0.905834	−	0.423634i	$-0.139246\pi$
$691$	47.6231	1.81167	0.905834	+	0.423634i	$0.139246\pi$
$692$	0	0
$693$	2.70667	0.102818
$694$	0	0
$695$	32.0716	1.21655
$696$	0	0
$697$	−20.1620	−0.763690
$698$	0	0
$699$	49.6159	1.87665
$700$	0	0
$701$	−3.00884	−0.113642	−0.0568212	−	0.998384i	$-0.518096\pi$
$701$	−3.00884	−0.113642	−0.0568212	+	0.998384i	$0.518096\pi$
$702$	0	0
$703$	−2.52622	−0.0952784
$704$	0	0
$705$	−4.02483	−0.151584
$706$	0	0
$707$	30.2321	1.13700
$708$	0	0
$709$	44.3215	1.66453	0.832264	−	0.554380i	$-0.187044\pi$
$709$	44.3215	1.66453	0.832264	+	0.554380i	$0.187044\pi$
$710$	0	0
$711$	−0.784840	−0.0294338
$712$	0	0
$713$	5.53896	0.207436
$714$	0	0
$715$	−7.34575	−0.274715
$716$	0	0
$717$	52.7197	1.96885
$718$	0	0
$719$	−34.7352	−1.29541	−0.647703	−	0.761893i	$-0.724270\pi$
$719$	−34.7352	−1.29541	−0.647703	+	0.761893i	$0.724270\pi$
$720$	0	0
$721$	4.58894	0.170901
$722$	0	0
$723$	39.4699	1.46790
$724$	0	0
$725$	−21.1362	−0.784978
$726$	0	0
$727$	−43.8591	−1.62665	−0.813323	−	0.581813i	$-0.802344\pi$
$727$	−43.8591	−1.62665	−0.813323	+	0.581813i	$0.802344\pi$
$728$	0	0
$729$	16.7240	0.619409
$730$	0	0
$731$	5.47345	0.202443
$732$	0	0
$733$	−20.9794	−0.774890	−0.387445	−	0.921893i	$-0.626642\pi$
$733$	−20.9794	−0.774890	−0.387445	+	0.921893i	$0.626642\pi$
$734$	0	0
$735$	−15.1686	−0.559502
$736$	0	0
$737$	−23.7685	−0.875524
$738$	0	0
$739$	−11.0946	−0.408121	−0.204060	−	0.978958i	$-0.565414\pi$
$739$	−11.0946	−0.408121	−0.204060	+	0.978958i	$0.565414\pi$
$740$	0	0
$741$	3.78791	0.139152
$742$	0	0
$743$	−48.3365	−1.77329	−0.886646	−	0.462448i	$-0.846971\pi$
$743$	−48.3365	−1.77329	−0.886646	+	0.462448i	$0.846971\pi$
$744$	0	0
$745$	31.7070	1.16165
$746$	0	0
$747$	8.73440	0.319575
$748$	0	0
$749$	−7.37771	−0.269576
$750$	0	0
$751$	5.57879	0.203573	0.101786	−	0.994806i	$-0.467544\pi$
$751$	5.57879	0.203573	0.101786	+	0.994806i	$0.467544\pi$
$752$	0	0
$753$	32.4047	1.18089
$754$	0	0
$755$	−2.87002	−0.104451
$756$	0	0
$757$	−29.6358	−1.07713	−0.538565	−	0.842584i	$-0.681034\pi$
$757$	−29.6358	−1.07713	−0.538565	+	0.842584i	$0.681034\pi$
$758$	0	0
$759$	2.72939	0.0990707
$760$	0	0
$761$	24.3976	0.884412	0.442206	−	0.896913i	$-0.354196\pi$
$761$	24.3976	0.884412	0.442206	+	0.896913i	$0.354196\pi$
$762$	0	0
$763$	−13.9682	−0.505681
$764$	0	0
$765$	3.03092	0.109583
$766$	0	0
$767$	17.3391	0.626079
$768$	0	0
$769$	−29.1113	−1.04978	−0.524891	−	0.851170i	$-0.675894\pi$
$769$	−29.1113	−1.04978	−0.524891	+	0.851170i	$0.675894\pi$
$770$	0	0
$771$	20.5301	0.739372
$772$	0	0
$773$	−24.6129	−0.885266	−0.442633	−	0.896703i	$-0.645955\pi$
$773$	−24.6129	−0.885266	−0.442633	+	0.896703i	$0.645955\pi$
$774$	0	0
$775$	−18.9097	−0.679256
$776$	0	0
$777$	−7.59082	−0.272319
$778$	0	0
$779$	8.82145	0.316061
$780$	0	0
$781$	25.6965	0.919493
$782$	0	0
$783$	42.4638	1.51753
$784$	0	0
$785$	2.29722	0.0819913
$786$	0	0
$787$	23.2216	0.827760	0.413880	−	0.910331i	$-0.364173\pi$
$787$	23.2216	0.827760	0.413880	+	0.910331i	$0.364173\pi$
$788$	0	0
$789$	30.5864	1.08891
$790$	0	0
$791$	−14.5623	−0.517775
$792$	0	0
$793$	0.443883	0.0157628
$794$	0	0
$795$	19.6766	0.697857
$796$	0	0
$797$	−28.1051	−0.995535	−0.497768	−	0.867310i	$-0.665847\pi$
$797$	−28.1051	−0.995535	−0.497768	+	0.867310i	$0.665847\pi$
$798$	0	0
$799$	2.79845	0.0990019
$800$	0	0
$801$	0.0703628	0.00248615
$802$	0	0
$803$	−23.5564	−0.831286
$804$	0	0
$805$	1.63973	0.0577930
$806$	0	0
$807$	30.8118	1.08463
$808$	0	0
$809$	−2.22258	−0.0781419	−0.0390709	−	0.999236i	$-0.512440\pi$
$809$	−2.22258	−0.0781419	−0.0390709	+	0.999236i	$0.512440\pi$
$810$	0	0
$811$	−5.34496	−0.187687	−0.0938435	−	0.995587i	$-0.529915\pi$
$811$	−5.34496	−0.187687	−0.0938435	+	0.995587i	$0.529915\pi$
$812$	0	0
$813$	37.7169	1.32279
$814$	0	0
$815$	−17.2919	−0.605708
$816$	0	0
$817$	−2.39479	−0.0837832
$818$	0	0
$819$	2.36020	0.0824722
$820$	0	0
$821$	−8.49136	−0.296351	−0.148175	−	0.988961i	$-0.547340\pi$
$821$	−8.49136	−0.296351	−0.148175	+	0.988961i	$0.547340\pi$
$822$	0	0
$823$	−49.2158	−1.71555	−0.857777	−	0.514022i	$-0.828155\pi$
$823$	−49.2158	−1.71555	−0.857777	+	0.514022i	$0.828155\pi$
$824$	0	0
$825$	−9.31797	−0.324410
$826$	0	0
$827$	32.5404	1.13154	0.565770	−	0.824563i	$-0.308579\pi$
$827$	32.5404	1.13154	0.565770	+	0.824563i	$0.308579\pi$
$828$	0	0
$829$	−11.2391	−0.390349	−0.195175	−	0.980769i	$-0.562527\pi$
$829$	−11.2391	−0.390349	−0.195175	+	0.980769i	$0.562527\pi$
$830$	0	0
$831$	27.4969	0.953856
$832$	0	0
$833$	10.5467	0.365420
$834$	0	0
$835$	−40.2025	−1.39126
$836$	0	0
$837$	37.9906	1.31315
$838$	0	0
$839$	6.38218	0.220337	0.110169	−	0.993913i	$-0.464861\pi$
$839$	6.38218	0.220337	0.110169	+	0.993913i	$0.464861\pi$
$840$	0	0
$841$	68.0913	2.34798
$842$	0	0
$843$	61.7377	2.12636
$844$	0	0
$845$	15.5601	0.535285
$846$	0	0
$847$	9.28926	0.319183
$848$	0	0
$849$	−41.3059	−1.41762
$850$	0	0
$851$	−1.58728	−0.0544112
$852$	0	0
$853$	43.2401	1.48051	0.740257	−	0.672324i	$-0.234704\pi$
$853$	43.2401	1.48051	0.740257	+	0.672324i	$0.234704\pi$
$854$	0	0
$855$	−1.32611	−0.0453521
$856$	0	0
$857$	24.2471	0.828264	0.414132	−	0.910217i	$-0.364085\pi$
$857$	24.2471	0.828264	0.414132	+	0.910217i	$0.364085\pi$
$858$	0	0
$859$	36.6247	1.24962	0.624809	−	0.780778i	$-0.285177\pi$
$859$	36.6247	1.24962	0.624809	+	0.780778i	$0.285177\pi$
$860$	0	0
$861$	26.5068	0.903348
$862$	0	0
$863$	−2.93754	−0.0999951	−0.0499976	−	0.998749i	$-0.515921\pi$
$863$	−2.93754	−0.0999951	−0.0499976	+	0.998749i	$0.515921\pi$
$864$	0	0
$865$	−7.75836	−0.263792
$866$	0	0
$867$	22.9102	0.778071
$868$	0	0
$869$	2.23286	0.0757445
$870$	0	0
$871$	−20.7260	−0.702274
$872$	0	0
$873$	−5.39117	−0.182464
$874$	0	0
$875$	−18.6465	−0.630367
$876$	0	0
$877$	−53.4490	−1.80485	−0.902423	−	0.430852i	$-0.858213\pi$
$877$	−53.4490	−1.80485	−0.902423	+	0.430852i	$0.858213\pi$
$878$	0	0
$879$	−21.9026	−0.738755
$880$	0	0
$881$	−20.6903	−0.697074	−0.348537	−	0.937295i	$-0.613321\pi$
$881$	−20.6903	−0.697074	−0.348537	+	0.937295i	$0.613321\pi$
$882$	0	0
$883$	−9.44494	−0.317848	−0.158924	−	0.987291i	$-0.550802\pi$
$883$	−9.44494	−0.317848	−0.158924	+	0.987291i	$0.550802\pi$
$884$	0	0
$885$	−29.2735	−0.984018
$886$	0	0
$887$	−17.7502	−0.595993	−0.297997	−	0.954567i	$-0.596318\pi$
$887$	−17.7502	−0.595993	−0.297997	+	0.954567i	$0.596318\pi$
$888$	0	0
$889$	−15.7987	−0.529871
$890$	0	0
$891$	23.9777	0.803282
$892$	0	0
$893$	−1.22440	−0.0409730
$894$	0	0
$895$	−31.3705	−1.04860
$896$	0	0
$897$	2.38002	0.0794665
$898$	0	0
$899$	86.8636	2.89706
$900$	0	0
$901$	−13.6811	−0.455782
$902$	0	0
$903$	−7.19589	−0.239464
$904$	0	0
$905$	8.77677	0.291750
$906$	0	0
$907$	−37.7606	−1.25382	−0.626911	−	0.779091i	$-0.715681\pi$
$907$	−37.7606	−1.25382	−0.626911	+	0.779091i	$0.715681\pi$
$908$	0	0
$909$	−15.3623	−0.509536
$910$	0	0
$911$	41.5142	1.37543	0.687714	−	0.725982i	$-0.258614\pi$
$911$	41.5142	1.37543	0.687714	+	0.725982i	$0.258614\pi$
$912$	0	0
$913$	−24.8492	−0.822390
$914$	0	0
$915$	−0.749406	−0.0247746
$916$	0	0
$917$	2.19702	0.0725519
$918$	0	0
$919$	14.3875	0.474598	0.237299	−	0.971437i	$-0.423738\pi$
$919$	14.3875	0.474598	0.237299	+	0.971437i	$0.423738\pi$
$920$	0	0
$921$	−27.9667	−0.921533
$922$	0	0
$923$	22.4072	0.737543
$924$	0	0
$925$	5.41887	0.178171
$926$	0	0
$927$	−2.33185	−0.0765881
$928$	0	0
$929$	−56.6503	−1.85864	−0.929319	−	0.369279i	$-0.879605\pi$
$929$	−56.6503	−1.85864	−0.929319	+	0.369279i	$0.879605\pi$
$930$	0	0
$931$	−4.61447	−0.151233
$932$	0	0
$933$	−38.0834	−1.24679
$934$	0	0
$935$	−8.62292	−0.281999
$936$	0	0
$937$	48.7060	1.59116	0.795578	−	0.605851i	$-0.207167\pi$
$937$	48.7060	1.59116	0.795578	+	0.605851i	$0.207167\pi$
$938$	0	0
$939$	33.2446	1.08490
$940$	0	0
$941$	23.3910	0.762525	0.381263	−	0.924467i	$-0.375489\pi$
$941$	23.3910	0.762525	0.381263	+	0.924467i	$0.375489\pi$
$942$	0	0
$943$	5.54270	0.180495
$944$	0	0
$945$	11.2466	0.365852
$946$	0	0
$947$	27.9401	0.907932	0.453966	−	0.891019i	$-0.350009\pi$
$947$	27.9401	0.907932	0.453966	+	0.891019i	$0.350009\pi$
$948$	0	0
$949$	−20.5410	−0.666790
$950$	0	0
$951$	10.4645	0.339335
$952$	0	0
$953$	−39.6415	−1.28412	−0.642058	−	0.766656i	$-0.721919\pi$
$953$	−39.6415	−1.28412	−0.642058	+	0.766656i	$0.721919\pi$
$954$	0	0
$955$	−28.5611	−0.924216
$956$	0	0
$957$	42.8031	1.38363
$958$	0	0
$959$	9.94228	0.321053
$960$	0	0
$961$	46.7132	1.50688
$962$	0	0
$963$	3.74895	0.120808
$964$	0	0
$965$	−15.8221	−0.509332
$966$	0	0
$967$	−13.5629	−0.436155	−0.218077	−	0.975931i	$-0.569979\pi$
$967$	−13.5629	−0.436155	−0.218077	+	0.975931i	$0.569979\pi$
$968$	0	0
$969$	4.44649	0.142842
$970$	0	0
$971$	12.6345	0.405461	0.202730	−	0.979235i	$-0.435019\pi$
$971$	12.6345	0.405461	0.202730	+	0.979235i	$0.435019\pi$
$972$	0	0
$973$	29.3166	0.939848
$974$	0	0
$975$	−8.12523	−0.260216
$976$	0	0
$977$	−29.4667	−0.942725	−0.471362	−	0.881940i	$-0.656238\pi$
$977$	−29.4667	−0.942725	−0.471362	+	0.881940i	$0.656238\pi$
$978$	0	0
$979$	−0.200181	−0.00639782
$980$	0	0
$981$	7.09786	0.226617
$982$	0	0
$983$	16.8277	0.536721	0.268360	−	0.963319i	$-0.413518\pi$
$983$	16.8277	0.536721	0.268360	+	0.963319i	$0.413518\pi$
$984$	0	0
$985$	−9.76778	−0.311227
$986$	0	0
$987$	−3.67909	−0.117107
$988$	0	0
$989$	−1.50470	−0.0478466
$990$	0	0
$991$	−27.7334	−0.880979	−0.440490	−	0.897758i	$-0.645195\pi$
$991$	−27.7334	−0.880979	−0.440490	+	0.897758i	$0.645195\pi$
$992$	0	0
$993$	13.4121	0.425621
$994$	0	0
$995$	−15.2036	−0.481985
$996$	0	0
$997$	45.5922	1.44392	0.721960	−	0.691935i	$-0.243241\pi$
$997$	45.5922	1.44392	0.721960	+	0.691935i	$0.243241\pi$
$998$	0	0
$999$	−10.8868	−0.344444

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.g.1.7	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.g.1.7	✓	27	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)

\(f(q)\)	\(=\)	\(q-1.94547 q^{3} -1.68966 q^{5} -1.54452 q^{7} +0.784840 q^{9} +O(q^{10})\)	\(q-1.94547 q^{3} -1.68966 q^{5} -1.54452 q^{7} +0.784840 q^{9} -2.23286 q^{11} -1.94704 q^{13} +3.28718 q^{15} -2.28556 q^{17} +1.00000 q^{19} +3.00481 q^{21} +0.628320 q^{23} -2.14505 q^{25} +4.30952 q^{27} +9.85349 q^{29} +8.81551 q^{31} +4.34395 q^{33} +2.60971 q^{35} -2.52622 q^{37} +3.78791 q^{39} +8.82145 q^{41} -2.39479 q^{43} -1.32611 q^{45} -1.22440 q^{47} -4.61447 q^{49} +4.44649 q^{51} +5.98586 q^{53} +3.77277 q^{55} -1.94547 q^{57} -8.90536 q^{59} -0.227978 q^{61} -1.21220 q^{63} +3.28984 q^{65} +10.6449 q^{67} -1.22238 q^{69} -11.5083 q^{71} +10.5499 q^{73} +4.17311 q^{75} +3.44869 q^{77} -1.00000 q^{79} -10.7385 q^{81} +11.1289 q^{83} +3.86183 q^{85} -19.1696 q^{87} +0.0896524 q^{89} +3.00724 q^{91} -17.1503 q^{93} -1.68966 q^{95} -6.86913 q^{97} -1.75244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9}+O(q^{10}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9	\( 27 q - 4 q^{3} - 10 q^{5} - 8 q^{7} + 19 q^{9} + 3 q^{11} - 5 q^{13} - 11 q^{15} - 17 q^{17} + 27 q^{19} - 28 q^{21} - 11 q^{23} + 13 q^{25} - 7 q^{27} - 39 q^{29} - 27 q^{31} - 18 q^{33} - 5 q^{35} - q^{37} - 22 q^{39} - 36 q^{41} - 2 q^{43} - 18 q^{45} - 12 q^{47} + 15 q^{49} + 4 q^{51} - 28 q^{53} + 5 q^{55} - 4 q^{57} - 30 q^{59} - 6 q^{61} - 4 q^{63} - 32 q^{65} + 13 q^{67} - 27 q^{69} - 59 q^{71} - 30 q^{73} - 21 q^{75} - 39 q^{77} - 27 q^{79} - 5 q^{81} + 4 q^{83} - 3 q^{85} + 22 q^{87} - 56 q^{89} - 8 q^{91} - 38 q^{93} - 10 q^{95} - 30 q^{97} + 31 q^{99}+O(q^{100}) \) 27 * q - 4 * q^3 - 10 * q^5 - 8 * q^7 + 19 * q^9 + 3 * q^11 - 5 * q^13 - 11 * q^15 - 17 * q^17 + 27 * q^19 - 28 * q^21 - 11 * q^23 + 13 * q^25 - 7 * q^27 - 39 * q^29 - 27 * q^31 - 18 * q^33 - 5 * q^35 - q^37 - 22 * q^39 - 36 * q^41 - 2 * q^43 - 18 * q^45 - 12 * q^47 + 15 * q^49 + 4 * q^51 - 28 * q^53 + 5 * q^55 - 4 * q^57 - 30 * q^59 - 6 * q^61 - 4 * q^63 - 32 * q^65 + 13 * q^67 - 27 * q^69 - 59 * q^71 - 30 * q^73 - 21 * q^75 - 39 * q^77 - 27 * q^79 - 5 * q^81 + 4 * q^83 - 3 * q^85 + 22 * q^87 - 56 * q^89 - 8 * q^91 - 38 * q^93 - 10 * q^95 - 30 * q^97 + 31 * q^99

Introduction

L-functions

Modular forms

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Database

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