LMFDB - Embedded newform 6004.2.a.h.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q-1.79377 q^{3} +4.30057 q^{5} +0.947969 q^{7} +0.217622 q^{9} +O(q^{10})$	$q-1.79377 q^{3} +4.30057 q^{5} +0.947969 q^{7} +0.217622 q^{9} +3.02341 q^{11} +3.61145 q^{13} -7.71425 q^{15} -3.28353 q^{17} -1.00000 q^{19} -1.70044 q^{21} -4.49358 q^{23} +13.4949 q^{25} +4.99095 q^{27} -6.42284 q^{29} -0.341594 q^{31} -5.42330 q^{33} +4.07681 q^{35} +2.28491 q^{37} -6.47812 q^{39} +10.1807 q^{41} +4.04078 q^{43} +0.935899 q^{45} +0.584425 q^{47} -6.10135 q^{49} +5.88991 q^{51} -0.0341910 q^{53} +13.0024 q^{55} +1.79377 q^{57} +12.0282 q^{59} +9.19029 q^{61} +0.206299 q^{63} +15.5313 q^{65} -15.4319 q^{67} +8.06046 q^{69} +0.662802 q^{71} +11.2868 q^{73} -24.2069 q^{75} +2.86609 q^{77} -1.00000 q^{79} -9.60551 q^{81} +4.85719 q^{83} -14.1211 q^{85} +11.5211 q^{87} +15.3631 q^{89} +3.42354 q^{91} +0.612743 q^{93} -4.30057 q^{95} -10.9427 q^{97} +0.657959 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * 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
$3$	−1.79377	−1.03564	−0.517818	−	0.855491i	$-0.673255\pi$
$3$	−1.79377	−1.03564	−0.517818	+	0.855491i	$0.673255\pi$
$4$	0	0
$5$	4.30057	1.92328	0.961638	−	0.274323i	$-0.0884538\pi$
$5$	4.30057	1.92328	0.961638	+	0.274323i	$0.0884538\pi$
$6$	0	0
$7$	0.947969	0.358299	0.179149	−	0.983822i	$-0.442665\pi$
$7$	0.947969	0.358299	0.179149	+	0.983822i	$0.442665\pi$
$8$	0	0
$9$	0.217622	0.0725406
$10$	0	0
$11$	3.02341	0.911591	0.455795	−	0.890085i	$-0.349355\pi$
$11$	3.02341	0.911591	0.455795	+	0.890085i	$0.349355\pi$
$12$	0	0
$13$	3.61145	1.00164	0.500818	−	0.865553i	$-0.333033\pi$
$13$	3.61145	1.00164	0.500818	+	0.865553i	$0.333033\pi$
$14$	0	0
$15$	−7.71425	−1.99181
$16$	0	0
$17$	−3.28353	−0.796373	−0.398187	−	0.917304i	$-0.630360\pi$
$17$	−3.28353	−0.796373	−0.398187	+	0.917304i	$0.630360\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.70044	−0.371067
$22$	0	0
$23$	−4.49358	−0.936976	−0.468488	−	0.883470i	$-0.655201\pi$
$23$	−4.49358	−0.936976	−0.468488	+	0.883470i	$0.655201\pi$
$24$	0	0
$25$	13.4949	2.69899
$26$	0	0
$27$	4.99095	0.960510
$28$	0	0
$29$	−6.42284	−1.19269	−0.596345	−	0.802728i	$-0.703381\pi$
$29$	−6.42284	−1.19269	−0.596345	+	0.802728i	$0.703381\pi$
$30$	0	0
$31$	−0.341594	−0.0613521	−0.0306761	−	0.999529i	$-0.509766\pi$
$31$	−0.341594	−0.0613521	−0.0306761	+	0.999529i	$0.509766\pi$
$32$	0	0
$33$	−5.42330	−0.944076
$34$	0	0
$35$	4.07681	0.689107
$36$	0	0
$37$	2.28491	0.375637	0.187819	−	0.982204i	$-0.439858\pi$
$37$	2.28491	0.375637	0.187819	+	0.982204i	$0.439858\pi$
$38$	0	0
$39$	−6.47812	−1.03733
$40$	0	0
$41$	10.1807	1.58996	0.794979	−	0.606636i	$-0.207482\pi$
$41$	10.1807	1.58996	0.794979	+	0.606636i	$0.207482\pi$
$42$	0	0
$43$	4.04078	0.616214	0.308107	−	0.951352i	$-0.400305\pi$
$43$	4.04078	0.616214	0.308107	+	0.951352i	$0.400305\pi$
$44$	0	0
$45$	0.935899	0.139516
$46$	0	0
$47$	0.584425	0.0852472	0.0426236	−	0.999091i	$-0.486428\pi$
$47$	0.584425	0.0852472	0.0426236	+	0.999091i	$0.486428\pi$
$48$	0	0
$49$	−6.10135	−0.871622
$50$	0	0
$51$	5.88991	0.824752
$52$	0	0
$53$	−0.0341910	−0.00469650	−0.00234825	−	0.999997i	$-0.500747\pi$
$53$	−0.0341910	−0.00469650	−0.00234825	+	0.999997i	$0.500747\pi$
$54$	0	0
$55$	13.0024	1.75324
$56$	0	0
$57$	1.79377	0.237591
$58$	0	0
$59$	12.0282	1.56594	0.782968	−	0.622062i	$-0.213705\pi$
$59$	12.0282	1.56594	0.782968	+	0.622062i	$0.213705\pi$
$60$	0	0
$61$	9.19029	1.17670	0.588348	−	0.808608i	$-0.299779\pi$
$61$	9.19029	1.17670	0.588348	+	0.808608i	$0.299779\pi$
$62$	0	0
$63$	0.206299	0.0259912
$64$	0	0
$65$	15.5313	1.92642
$66$	0	0
$67$	−15.4319	−1.88531	−0.942653	−	0.333775i	$-0.891677\pi$
$67$	−15.4319	−1.88531	−0.942653	+	0.333775i	$0.891677\pi$
$68$	0	0
$69$	8.06046	0.970366
$70$	0	0
$71$	0.662802	0.0786602	0.0393301	−	0.999226i	$-0.487478\pi$
$71$	0.662802	0.0786602	0.0393301	+	0.999226i	$0.487478\pi$
$72$	0	0
$73$	11.2868	1.32102	0.660512	−	0.750815i	$-0.270339\pi$
$73$	11.2868	1.32102	0.660512	+	0.750815i	$0.270339\pi$
$74$	0	0
$75$	−24.2069	−2.79517
$76$	0	0
$77$	2.86609	0.326622
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−9.60551	−1.06728
$82$	0	0
$83$	4.85719	0.533146	0.266573	−	0.963815i	$-0.414109\pi$
$83$	4.85719	0.533146	0.266573	+	0.963815i	$0.414109\pi$
$84$	0	0
$85$	−14.1211	−1.53164
$86$	0	0
$87$	11.5211	1.23519
$88$	0	0
$89$	15.3631	1.62848	0.814242	−	0.580526i	$-0.197153\pi$
$89$	15.3631	1.62848	0.814242	+	0.580526i	$0.197153\pi$
$90$	0	0
$91$	3.42354	0.358885
$92$	0	0
$93$	0.612743	0.0635385
$94$	0	0
$95$	−4.30057	−0.441230
$96$	0	0
$97$	−10.9427	−1.11107	−0.555533	−	0.831494i	$-0.687486\pi$
$97$	−10.9427	−1.11107	−0.555533	+	0.831494i	$0.687486\pi$
$98$	0	0
$99$	0.657959	0.0661274
$100$	0	0
$101$	10.7646	1.07111	0.535557	−	0.844499i	$-0.320102\pi$
$101$	10.7646	1.07111	0.535557	+	0.844499i	$0.320102\pi$
$102$	0	0
$103$	−8.93572	−0.880463	−0.440232	−	0.897884i	$-0.645104\pi$
$103$	−8.93572	−0.880463	−0.440232	+	0.897884i	$0.645104\pi$
$104$	0	0
$105$	−7.31287	−0.713663
$106$	0	0
$107$	−4.25159	−0.411017	−0.205509	−	0.978655i	$-0.565885\pi$
$107$	−4.25159	−0.411017	−0.205509	+	0.978655i	$0.565885\pi$
$108$	0	0
$109$	20.5325	1.96665	0.983327	−	0.181847i	$-0.0582075\pi$
$109$	20.5325	1.96665	0.983327	+	0.181847i	$0.0582075\pi$
$110$	0	0
$111$	−4.09861	−0.389023
$112$	0	0
$113$	−10.6893	−1.00557	−0.502784	−	0.864412i	$-0.667691\pi$
$113$	−10.6893	−1.00557	−0.502784	+	0.864412i	$0.667691\pi$
$114$	0	0
$115$	−19.3250	−1.80206
$116$	0	0
$117$	0.785931	0.0726593
$118$	0	0
$119$	−3.11268	−0.285339
$120$	0	0
$121$	−1.85902	−0.169002
$122$	0	0
$123$	−18.2619	−1.64662
$124$	0	0
$125$	36.5331	3.26762
$126$	0	0
$127$	6.32119	0.560915	0.280457	−	0.959867i	$-0.409514\pi$
$127$	6.32119	0.560915	0.280457	+	0.959867i	$0.409514\pi$
$128$	0	0
$129$	−7.24825	−0.638173
$130$	0	0
$131$	−9.04374	−0.790155	−0.395078	−	0.918648i	$-0.629282\pi$
$131$	−9.04374	−0.790155	−0.395078	+	0.918648i	$0.629282\pi$
$132$	0	0
$133$	−0.947969	−0.0821993
$134$	0	0
$135$	21.4640	1.84732
$136$	0	0
$137$	15.3078	1.30783	0.653916	−	0.756567i	$-0.273125\pi$
$137$	15.3078	1.30783	0.653916	+	0.756567i	$0.273125\pi$
$138$	0	0
$139$	−8.30936	−0.704791	−0.352395	−	0.935851i	$-0.614633\pi$
$139$	−8.30936	−0.704791	−0.352395	+	0.935851i	$0.614633\pi$
$140$	0	0
$141$	−1.04833	−0.0882850
$142$	0	0
$143$	10.9189	0.913083
$144$	0	0
$145$	−27.6219	−2.29387
$146$	0	0
$147$	10.9444	0.902683
$148$	0	0
$149$	4.67391	0.382902	0.191451	−	0.981502i	$-0.438681\pi$
$149$	4.67391	0.382902	0.191451	+	0.981502i	$0.438681\pi$
$150$	0	0
$151$	−15.5770	−1.26764	−0.633819	−	0.773481i	$-0.718514\pi$
$151$	−15.5770	−1.26764	−0.633819	+	0.773481i	$0.718514\pi$
$152$	0	0
$153$	−0.714568	−0.0577694
$154$	0	0
$155$	−1.46905	−0.117997
$156$	0	0
$157$	20.6502	1.64806	0.824032	−	0.566543i	$-0.191719\pi$
$157$	20.6502	1.64806	0.824032	+	0.566543i	$0.191719\pi$
$158$	0	0
$159$	0.0613309	0.00486386
$160$	0	0
$161$	−4.25977	−0.335717
$162$	0	0
$163$	18.0920	1.41707	0.708537	−	0.705673i	$-0.249355\pi$
$163$	18.0920	1.41707	0.708537	+	0.705673i	$0.249355\pi$
$164$	0	0
$165$	−23.3233	−1.81572
$166$	0	0
$167$	−18.7610	−1.45177	−0.725887	−	0.687814i	$-0.758570\pi$
$167$	−18.7610	−1.45177	−0.725887	+	0.687814i	$0.758570\pi$
$168$	0	0
$169$	0.0425791	0.00327532
$170$	0	0
$171$	−0.217622	−0.0166420
$172$	0	0
$173$	7.90427	0.600951	0.300475	−	0.953790i	$-0.402855\pi$
$173$	7.90427	0.600951	0.300475	+	0.953790i	$0.402855\pi$
$174$	0	0
$175$	12.7928	0.967043
$176$	0	0
$177$	−21.5758	−1.62174
$178$	0	0
$179$	−18.1879	−1.35943	−0.679714	−	0.733477i	$-0.737896\pi$
$179$	−18.1879	−1.35943	−0.679714	+	0.733477i	$0.737896\pi$
$180$	0	0
$181$	17.3064	1.28637	0.643187	−	0.765709i	$-0.277612\pi$
$181$	17.3064	1.28637	0.643187	+	0.765709i	$0.277612\pi$
$182$	0	0
$183$	−16.4853	−1.21863
$184$	0	0
$185$	9.82643	0.722454
$186$	0	0
$187$	−9.92744	−0.725966
$188$	0	0
$189$	4.73127	0.344149
$190$	0	0
$191$	−3.53813	−0.256010	−0.128005	−	0.991774i	$-0.540857\pi$
$191$	−3.53813	−0.256010	−0.128005	+	0.991774i	$0.540857\pi$
$192$	0	0
$193$	−27.4269	−1.97423	−0.987117	−	0.160001i	$-0.948850\pi$
$193$	−27.4269	−1.97423	−0.987117	+	0.160001i	$0.948850\pi$
$194$	0	0
$195$	−27.8596	−1.99507
$196$	0	0
$197$	11.5185	0.820661	0.410330	−	0.911937i	$-0.365413\pi$
$197$	11.5185	0.820661	0.410330	+	0.911937i	$0.365413\pi$
$198$	0	0
$199$	−0.919767	−0.0652006	−0.0326003	−	0.999468i	$-0.510379\pi$
$199$	−0.919767	−0.0652006	−0.0326003	+	0.999468i	$0.510379\pi$
$200$	0	0
$201$	27.6813	1.95249
$202$	0	0
$203$	−6.08865	−0.427340
$204$	0	0
$205$	43.7829	3.05793
$206$	0	0
$207$	−0.977901	−0.0679688
$208$	0	0
$209$	−3.02341	−0.209133
$210$	0	0
$211$	−20.0378	−1.37946	−0.689730	−	0.724067i	$-0.742271\pi$
$211$	−20.0378	−1.37946	−0.689730	+	0.724067i	$0.742271\pi$
$212$	0	0
$213$	−1.18892	−0.0814632
$214$	0	0
$215$	17.3777	1.18515
$216$	0	0
$217$	−0.323821	−0.0219824
$218$	0	0
$219$	−20.2460	−1.36810
$220$	0	0
$221$	−11.8583	−0.797676
$222$	0	0
$223$	−10.5539	−0.706743	−0.353371	−	0.935483i	$-0.614965\pi$
$223$	−10.5539	−0.706743	−0.353371	+	0.935483i	$0.614965\pi$
$224$	0	0
$225$	2.93679	0.195786
$226$	0	0
$227$	25.2651	1.67691	0.838453	−	0.544974i	$-0.183460\pi$
$227$	25.2651	1.67691	0.838453	+	0.544974i	$0.183460\pi$
$228$	0	0
$229$	6.09585	0.402825	0.201412	−	0.979507i	$-0.435447\pi$
$229$	6.09585	0.402825	0.201412	+	0.979507i	$0.435447\pi$
$230$	0	0
$231$	−5.14112	−0.338261
$232$	0	0
$233$	−24.1211	−1.58023	−0.790113	−	0.612961i	$-0.789978\pi$
$233$	−24.1211	−1.58023	−0.790113	+	0.612961i	$0.789978\pi$
$234$	0	0
$235$	2.51336	0.163954
$236$	0	0
$237$	1.79377	0.116518
$238$	0	0
$239$	−14.5942	−0.944020	−0.472010	−	0.881593i	$-0.656471\pi$
$239$	−14.5942	−0.944020	−0.472010	+	0.881593i	$0.656471\pi$
$240$	0	0
$241$	−2.92383	−0.188340	−0.0941701	−	0.995556i	$-0.530020\pi$
$241$	−2.92383	−0.188340	−0.0941701	+	0.995556i	$0.530020\pi$
$242$	0	0
$243$	2.25723	0.144802
$244$	0	0
$245$	−26.2393	−1.67637
$246$	0	0
$247$	−3.61145	−0.229791
$248$	0	0
$249$	−8.71270	−0.552145
$250$	0	0
$251$	23.0472	1.45473	0.727365	−	0.686251i	$-0.240745\pi$
$251$	23.0472	1.45473	0.727365	+	0.686251i	$0.240745\pi$
$252$	0	0
$253$	−13.5859	−0.854139
$254$	0	0
$255$	25.3300	1.58623
$256$	0	0
$257$	1.75262	0.109326	0.0546628	−	0.998505i	$-0.482592\pi$
$257$	1.75262	0.109326	0.0546628	+	0.998505i	$0.482592\pi$
$258$	0	0
$259$	2.16603	0.134590
$260$	0	0
$261$	−1.39775	−0.0865185
$262$	0	0
$263$	−16.0105	−0.987250	−0.493625	−	0.869675i	$-0.664329\pi$
$263$	−16.0105	−0.987250	−0.493625	+	0.869675i	$0.664329\pi$
$264$	0	0
$265$	−0.147041	−0.00903265
$266$	0	0
$267$	−27.5579	−1.68651
$268$	0	0
$269$	−31.8495	−1.94190	−0.970950	−	0.239280i	$-0.923088\pi$
$269$	−31.8495	−1.94190	−0.970950	+	0.239280i	$0.923088\pi$
$270$	0	0
$271$	9.03127	0.548610	0.274305	−	0.961643i	$-0.411552\pi$
$271$	9.03127	0.548610	0.274305	+	0.961643i	$0.411552\pi$
$272$	0	0
$273$	−6.14106	−0.371674
$274$	0	0
$275$	40.8007	2.46037
$276$	0	0
$277$	9.92657	0.596430	0.298215	−	0.954499i	$-0.403609\pi$
$277$	9.92657	0.596430	0.298215	+	0.954499i	$0.403609\pi$
$278$	0	0
$279$	−0.0743384	−0.00445052
$280$	0	0
$281$	12.9171	0.770570	0.385285	−	0.922798i	$-0.374103\pi$
$281$	12.9171	0.770570	0.385285	+	0.922798i	$0.374103\pi$
$282$	0	0
$283$	26.8122	1.59382	0.796909	−	0.604100i	$-0.206467\pi$
$283$	26.8122	1.59382	0.796909	+	0.604100i	$0.206467\pi$
$284$	0	0
$285$	7.71425	0.456953
$286$	0	0
$287$	9.65099	0.569680
$288$	0	0
$289$	−6.21843	−0.365790
$290$	0	0
$291$	19.6288	1.15066
$292$	0	0
$293$	−13.9815	−0.816807	−0.408404	−	0.912801i	$-0.633914\pi$
$293$	−13.9815	−0.816807	−0.408404	+	0.912801i	$0.633914\pi$
$294$	0	0
$295$	51.7281	3.01173
$296$	0	0
$297$	15.0897	0.875592
$298$	0	0
$299$	−16.2283	−0.938509
$300$	0	0
$301$	3.83054	0.220789
$302$	0	0
$303$	−19.3092	−1.10928
$304$	0	0
$305$	39.5235	2.26311
$306$	0	0
$307$	29.9321	1.70832	0.854159	−	0.520013i	$-0.174073\pi$
$307$	29.9321	1.70832	0.854159	+	0.520013i	$0.174073\pi$
$308$	0	0
$309$	16.0287	0.911839
$310$	0	0
$311$	−6.95052	−0.394128	−0.197064	−	0.980391i	$-0.563141\pi$
$311$	−6.95052	−0.394128	−0.197064	+	0.980391i	$0.563141\pi$
$312$	0	0
$313$	20.1727	1.14023	0.570115	−	0.821565i	$-0.306899\pi$
$313$	20.1727	1.14023	0.570115	+	0.821565i	$0.306899\pi$
$314$	0	0
$315$	0.887203	0.0499882
$316$	0	0
$317$	11.2755	0.633296	0.316648	−	0.948543i	$-0.397443\pi$
$317$	11.2755	0.633296	0.316648	+	0.948543i	$0.397443\pi$
$318$	0	0
$319$	−19.4188	−1.08725
$320$	0	0
$321$	7.62639	0.425664
$322$	0	0
$323$	3.28353	0.182700
$324$	0	0
$325$	48.7363	2.70340
$326$	0	0
$327$	−36.8306	−2.03674
$328$	0	0
$329$	0.554017	0.0305439
$330$	0	0
$331$	9.02783	0.496214	0.248107	−	0.968733i	$-0.420191\pi$
$331$	9.02783	0.496214	0.248107	+	0.968733i	$0.420191\pi$
$332$	0	0
$333$	0.497247	0.0272490
$334$	0	0
$335$	−66.3660	−3.62596
$336$	0	0
$337$	−15.5027	−0.844486	−0.422243	−	0.906483i	$-0.638757\pi$
$337$	−15.5027	−0.844486	−0.422243	+	0.906483i	$0.638757\pi$
$338$	0	0
$339$	19.1743	1.04140
$340$	0	0
$341$	−1.03278	−0.0559281
$342$	0	0
$343$	−12.4197	−0.670600
$344$	0	0
$345$	34.6646	1.86628
$346$	0	0
$347$	29.6669	1.59260	0.796301	−	0.604901i	$-0.206787\pi$
$347$	29.6669	1.59260	0.796301	+	0.604901i	$0.206787\pi$
$348$	0	0
$349$	−0.321335	−0.0172006	−0.00860032	−	0.999963i	$-0.502738\pi$
$349$	−0.321335	−0.0172006	−0.00860032	+	0.999963i	$0.502738\pi$
$350$	0	0
$351$	18.0246	0.962081
$352$	0	0
$353$	24.4451	1.30108	0.650539	−	0.759472i	$-0.274543\pi$
$353$	24.4451	1.30108	0.650539	+	0.759472i	$0.274543\pi$
$354$	0	0
$355$	2.85043	0.151285
$356$	0	0
$357$	5.58345	0.295508
$358$	0	0
$359$	5.62028	0.296627	0.148313	−	0.988940i	$-0.452616\pi$
$359$	5.62028	0.296627	0.148313	+	0.988940i	$0.452616\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.33466	0.175024
$364$	0	0
$365$	48.5399	2.54069
$366$	0	0
$367$	28.1612	1.47000	0.735001	−	0.678066i	$-0.237182\pi$
$367$	28.1612	1.47000	0.735001	+	0.678066i	$0.237182\pi$
$368$	0	0
$369$	2.21554	0.115337
$370$	0	0
$371$	−0.0324120	−0.00168275
$372$	0	0
$373$	−23.5279	−1.21823	−0.609115	−	0.793082i	$-0.708475\pi$
$373$	−23.5279	−1.21823	−0.609115	+	0.793082i	$0.708475\pi$
$374$	0	0
$375$	−65.5321	−3.38406
$376$	0	0
$377$	−23.1958	−1.19464
$378$	0	0
$379$	24.5158	1.25929	0.629647	−	0.776881i	$-0.283200\pi$
$379$	24.5158	1.25929	0.629647	+	0.776881i	$0.283200\pi$
$380$	0	0
$381$	−11.3388	−0.580903
$382$	0	0
$383$	−24.4099	−1.24729	−0.623643	−	0.781709i	$-0.714348\pi$
$383$	−24.4099	−1.24729	−0.623643	+	0.781709i	$0.714348\pi$
$384$	0	0
$385$	12.3259	0.628184
$386$	0	0
$387$	0.879363	0.0447005
$388$	0	0
$389$	28.5600	1.44805	0.724025	−	0.689773i	$-0.242290\pi$
$389$	28.5600	1.44805	0.724025	+	0.689773i	$0.242290\pi$
$390$	0	0
$391$	14.7548	0.746182
$392$	0	0
$393$	16.2224	0.818313
$394$	0	0
$395$	−4.30057	−0.216385
$396$	0	0
$397$	7.22405	0.362564	0.181282	−	0.983431i	$-0.441975\pi$
$397$	7.22405	0.362564	0.181282	+	0.983431i	$0.441975\pi$
$398$	0	0
$399$	1.70044	0.0851285
$400$	0	0
$401$	16.4183	0.819890	0.409945	−	0.912110i	$-0.365548\pi$
$401$	16.4183	0.819890	0.409945	+	0.912110i	$0.365548\pi$
$402$	0	0
$403$	−1.23365	−0.0614525
$404$	0	0
$405$	−41.3092	−2.05267
$406$	0	0
$407$	6.90821	0.342427
$408$	0	0
$409$	−7.93853	−0.392535	−0.196267	−	0.980550i	$-0.562882\pi$
$409$	−7.93853	−0.392535	−0.196267	+	0.980550i	$0.562882\pi$
$410$	0	0
$411$	−27.4587	−1.35444
$412$	0	0
$413$	11.4023	0.561073
$414$	0	0
$415$	20.8887	1.02539
$416$	0	0
$417$	14.9051	0.729906
$418$	0	0
$419$	−21.6309	−1.05674	−0.528368	−	0.849015i	$-0.677196\pi$
$419$	−21.6309	−1.05674	−0.528368	+	0.849015i	$0.677196\pi$
$420$	0	0
$421$	−24.6568	−1.20170	−0.600850	−	0.799362i	$-0.705171\pi$
$421$	−24.6568	−1.20170	−0.600850	+	0.799362i	$0.705171\pi$
$422$	0	0
$423$	0.127184	0.00618388
$424$	0	0
$425$	−44.3110	−2.14940
$426$	0	0
$427$	8.71211	0.421608
$428$	0	0
$429$	−19.5860	−0.945621
$430$	0	0
$431$	−0.412123	−0.0198512	−0.00992562	−	0.999951i	$-0.503159\pi$
$431$	−0.412123	−0.0198512	−0.00992562	+	0.999951i	$0.503159\pi$
$432$	0	0
$433$	−19.8261	−0.952784	−0.476392	−	0.879233i	$-0.658056\pi$
$433$	−19.8261	−0.952784	−0.476392	+	0.879233i	$0.658056\pi$
$434$	0	0
$435$	49.5474	2.37562
$436$	0	0
$437$	4.49358	0.214957
$438$	0	0
$439$	22.1289	1.05615	0.528076	−	0.849197i	$-0.322914\pi$
$439$	22.1289	1.05615	0.528076	+	0.849197i	$0.322914\pi$
$440$	0	0
$441$	−1.32779	−0.0632280
$442$	0	0
$443$	16.5799	0.787733	0.393866	−	0.919168i	$-0.371137\pi$
$443$	16.5799	0.787733	0.393866	+	0.919168i	$0.371137\pi$
$444$	0	0
$445$	66.0701	3.13202
$446$	0	0
$447$	−8.38393	−0.396546
$448$	0	0
$449$	−18.1840	−0.858155	−0.429077	−	0.903268i	$-0.641161\pi$
$449$	−18.1840	−0.858155	−0.429077	+	0.903268i	$0.641161\pi$
$450$	0	0
$451$	30.7804	1.44939
$452$	0	0
$453$	27.9416	1.31281
$454$	0	0
$455$	14.7232	0.690234
$456$	0	0
$457$	−2.08588	−0.0975735	−0.0487868	−	0.998809i	$-0.515535\pi$
$457$	−2.08588	−0.0975735	−0.0487868	+	0.998809i	$0.515535\pi$
$458$	0	0
$459$	−16.3880	−0.764924
$460$	0	0
$461$	36.9318	1.72009	0.860043	−	0.510221i	$-0.170436\pi$
$461$	36.9318	1.72009	0.860043	+	0.510221i	$0.170436\pi$
$462$	0	0
$463$	10.9516	0.508962	0.254481	−	0.967078i	$-0.418095\pi$
$463$	10.9516	0.508962	0.254481	+	0.967078i	$0.418095\pi$
$464$	0	0
$465$	2.63515	0.122202
$466$	0	0
$467$	−1.97975	−0.0916121	−0.0458061	−	0.998950i	$-0.514586\pi$
$467$	−1.97975	−0.0916121	−0.0458061	+	0.998950i	$0.514586\pi$
$468$	0	0
$469$	−14.6290	−0.675502
$470$	0	0
$471$	−37.0418	−1.70679
$472$	0	0
$473$	12.2169	0.561735
$474$	0	0
$475$	−13.4949	−0.619190
$476$	0	0
$477$	−0.00744071	−0.000340687 0
$478$	0	0
$479$	−23.0271	−1.05214	−0.526068	−	0.850443i	$-0.676334\pi$
$479$	−23.0271	−1.05214	−0.526068	+	0.850443i	$0.676334\pi$
$480$	0	0
$481$	8.25185	0.376252
$482$	0	0
$483$	7.64107	0.347681
$484$	0	0
$485$	−47.0601	−2.13689
$486$	0	0
$487$	30.9540	1.40266	0.701330	−	0.712837i	$-0.252590\pi$
$487$	30.9540	1.40266	0.701330	+	0.712837i	$0.252590\pi$
$488$	0	0
$489$	−32.4529	−1.46757
$490$	0	0
$491$	−3.96702	−0.179029	−0.0895146	−	0.995986i	$-0.528532\pi$
$491$	−3.96702	−0.179029	−0.0895146	+	0.995986i	$0.528532\pi$
$492$	0	0
$493$	21.0896	0.949827
$494$	0	0
$495$	2.82960	0.127181
$496$	0	0
$497$	0.628316	0.0281838
$498$	0	0
$499$	2.43369	0.108947	0.0544734	−	0.998515i	$-0.482652\pi$
$499$	2.43369	0.108947	0.0544734	+	0.998515i	$0.482652\pi$
$500$	0	0
$501$	33.6531	1.50351
$502$	0	0
$503$	20.0534	0.894135	0.447067	−	0.894500i	$-0.352468\pi$
$503$	20.0534	0.894135	0.447067	+	0.894500i	$0.352468\pi$
$504$	0	0
$505$	46.2938	2.06005
$506$	0	0
$507$	−0.0763773	−0.00339204
$508$	0	0
$509$	−6.21045	−0.275274	−0.137637	−	0.990483i	$-0.543951\pi$
$509$	−6.21045	−0.275274	−0.137637	+	0.990483i	$0.543951\pi$
$510$	0	0
$511$	10.6996	0.473321
$512$	0	0
$513$	−4.99095	−0.220356
$514$	0	0
$515$	−38.4287	−1.69337
$516$	0	0
$517$	1.76695	0.0777106
$518$	0	0
$519$	−14.1785	−0.622366
$520$	0	0
$521$	6.65277	0.291463	0.145732	−	0.989324i	$-0.453446\pi$
$521$	6.65277	0.291463	0.145732	+	0.989324i	$0.453446\pi$
$522$	0	0
$523$	−14.7892	−0.646686	−0.323343	−	0.946282i	$-0.604807\pi$
$523$	−14.7892	−0.646686	−0.323343	+	0.946282i	$0.604807\pi$
$524$	0	0
$525$	−22.9473	−1.00150
$526$	0	0
$527$	1.12164	0.0488592
$528$	0	0
$529$	−2.80774	−0.122076
$530$	0	0
$531$	2.61760	0.113594
$532$	0	0
$533$	36.7671	1.59256
$534$	0	0
$535$	−18.2843	−0.790499
$536$	0	0
$537$	32.6250	1.40787
$538$	0	0
$539$	−18.4469	−0.794563
$540$	0	0
$541$	−31.6999	−1.36289	−0.681443	−	0.731872i	$-0.738647\pi$
$541$	−31.6999	−1.36289	−0.681443	+	0.731872i	$0.738647\pi$
$542$	0	0
$543$	−31.0437	−1.33221
$544$	0	0
$545$	88.3014	3.78242
$546$	0	0
$547$	−14.3708	−0.614453	−0.307227	−	0.951636i	$-0.599401\pi$
$547$	−14.3708	−0.614453	−0.307227	+	0.951636i	$0.599401\pi$
$548$	0	0
$549$	2.00001	0.0853582
$550$	0	0
$551$	6.42284	0.273622
$552$	0	0
$553$	−0.947969	−0.0403117
$554$	0	0
$555$	−17.6264	−0.748199
$556$	0	0
$557$	−28.3489	−1.20118	−0.600590	−	0.799557i	$-0.705068\pi$
$557$	−28.3489	−1.20118	−0.600590	+	0.799557i	$0.705068\pi$
$558$	0	0
$559$	14.5931	0.617222
$560$	0	0
$561$	17.8076	0.751837
$562$	0	0
$563$	−46.5813	−1.96317	−0.981585	−	0.191027i	$-0.938818\pi$
$563$	−46.5813	−1.96317	−0.981585	+	0.191027i	$0.938818\pi$
$564$	0	0
$565$	−45.9703	−1.93398
$566$	0	0
$567$	−9.10572	−0.382404
$568$	0	0
$569$	3.14044	0.131654	0.0658271	−	0.997831i	$-0.479031\pi$
$569$	3.14044	0.131654	0.0658271	+	0.997831i	$0.479031\pi$
$570$	0	0
$571$	−46.9785	−1.96599	−0.982995	−	0.183634i	$-0.941214\pi$
$571$	−46.9785	−1.96599	−0.982995	+	0.183634i	$0.941214\pi$
$572$	0	0
$573$	6.34660	0.265133
$574$	0	0
$575$	−60.6406	−2.52889
$576$	0	0
$577$	14.6731	0.610849	0.305425	−	0.952216i	$-0.401202\pi$
$577$	14.6731	0.610849	0.305425	+	0.952216i	$0.401202\pi$
$578$	0	0
$579$	49.1977	2.04459
$580$	0	0
$581$	4.60447	0.191026
$582$	0	0
$583$	−0.103373	−0.00428128
$584$	0	0
$585$	3.37995	0.139744
$586$	0	0
$587$	−31.6657	−1.30698	−0.653491	−	0.756934i	$-0.726696\pi$
$587$	−31.6657	−1.30698	−0.653491	+	0.756934i	$0.726696\pi$
$588$	0	0
$589$	0.341594	0.0140751
$590$	0	0
$591$	−20.6616	−0.849905
$592$	0	0
$593$	44.2939	1.81893	0.909466	−	0.415778i	$-0.136491\pi$
$593$	44.2939	1.81893	0.909466	+	0.415778i	$0.136491\pi$
$594$	0	0
$595$	−13.3863	−0.548786
$596$	0	0
$597$	1.64985	0.0675240
$598$	0	0
$599$	27.6784	1.13091	0.565455	−	0.824779i	$-0.308701\pi$
$599$	27.6784	1.13091	0.565455	+	0.824779i	$0.308701\pi$
$600$	0	0
$601$	9.20370	0.375427	0.187713	−	0.982224i	$-0.439892\pi$
$601$	9.20370	0.375427	0.187713	+	0.982224i	$0.439892\pi$
$602$	0	0
$603$	−3.35832	−0.136761
$604$	0	0
$605$	−7.99486	−0.325037
$606$	0	0
$607$	−4.42556	−0.179628	−0.0898140	−	0.995959i	$-0.528627\pi$
$607$	−4.42556	−0.179628	−0.0898140	+	0.995959i	$0.528627\pi$
$608$	0	0
$609$	10.9217	0.442568
$610$	0	0
$611$	2.11062	0.0853867
$612$	0	0
$613$	−24.3405	−0.983102	−0.491551	−	0.870849i	$-0.663570\pi$
$613$	−24.3405	−0.983102	−0.491551	+	0.870849i	$0.663570\pi$
$614$	0	0
$615$	−78.5365	−3.16690
$616$	0	0
$617$	−15.7271	−0.633148	−0.316574	−	0.948568i	$-0.602533\pi$
$617$	−15.7271	−0.633148	−0.316574	+	0.948568i	$0.602533\pi$
$618$	0	0
$619$	3.45007	0.138670	0.0693351	−	0.997593i	$-0.477912\pi$
$619$	3.45007	0.138670	0.0693351	+	0.997593i	$0.477912\pi$
$620$	0	0
$621$	−22.4273	−0.899975
$622$	0	0
$623$	14.5637	0.583483
$624$	0	0
$625$	89.6386	3.58554
$626$	0	0
$627$	5.42330	0.216586
$628$	0	0
$629$	−7.50258	−0.299147
$630$	0	0
$631$	34.2610	1.36391	0.681955	−	0.731395i	$-0.261130\pi$
$631$	34.2610	1.36391	0.681955	+	0.731395i	$0.261130\pi$
$632$	0	0
$633$	35.9433	1.42862
$634$	0	0
$635$	27.1847	1.07879
$636$	0	0
$637$	−22.0347	−0.873048
$638$	0	0
$639$	0.144240	0.00570606
$640$	0	0
$641$	−19.3063	−0.762554	−0.381277	−	0.924461i	$-0.624516\pi$
$641$	−19.3063	−0.762554	−0.381277	+	0.924461i	$0.624516\pi$
$642$	0	0
$643$	44.4830	1.75424	0.877118	−	0.480274i	$-0.159463\pi$
$643$	44.4830	1.75424	0.877118	+	0.480274i	$0.159463\pi$
$644$	0	0
$645$	−31.1716	−1.22738
$646$	0	0
$647$	−21.5934	−0.848925	−0.424462	−	0.905446i	$-0.639537\pi$
$647$	−21.5934	−0.848925	−0.424462	+	0.905446i	$0.639537\pi$
$648$	0	0
$649$	36.3661	1.42749
$650$	0	0
$651$	0.580861	0.0227657
$652$	0	0
$653$	0.628942	0.0246124	0.0123062	−	0.999924i	$-0.496083\pi$
$653$	0.628942	0.0246124	0.0123062	+	0.999924i	$0.496083\pi$
$654$	0	0
$655$	−38.8933	−1.51969
$656$	0	0
$657$	2.45626	0.0958279
$658$	0	0
$659$	34.6172	1.34849	0.674246	−	0.738506i	$-0.264469\pi$
$659$	34.6172	1.34849	0.674246	+	0.738506i	$0.264469\pi$
$660$	0	0
$661$	25.8833	1.00674	0.503372	−	0.864070i	$-0.332093\pi$
$661$	25.8833	1.00674	0.503372	+	0.864070i	$0.332093\pi$
$662$	0	0
$663$	21.2711	0.826102
$664$	0	0
$665$	−4.07681	−0.158092
$666$	0	0
$667$	28.8615	1.11752
$668$	0	0
$669$	18.9313	0.731928
$670$	0	0
$671$	27.7860	1.07267
$672$	0	0
$673$	0.479837	0.0184964	0.00924818	−	0.999957i	$-0.497056\pi$
$673$	0.479837	0.0184964	0.00924818	+	0.999957i	$0.497056\pi$
$674$	0	0
$675$	67.3526	2.59240
$676$	0	0
$677$	5.70640	0.219315	0.109657	−	0.993969i	$-0.465025\pi$
$677$	5.70640	0.219315	0.109657	+	0.993969i	$0.465025\pi$
$678$	0	0
$679$	−10.3734	−0.398094
$680$	0	0
$681$	−45.3199	−1.73666
$682$	0	0
$683$	−23.8801	−0.913747	−0.456874	−	0.889532i	$-0.651031\pi$
$683$	−23.8801	−0.913747	−0.456874	+	0.889532i	$0.651031\pi$
$684$	0	0
$685$	65.8322	2.51532
$686$	0	0
$687$	−10.9346	−0.417180
$688$	0	0
$689$	−0.123479	−0.00470418
$690$	0	0
$691$	28.6692	1.09063	0.545314	−	0.838232i	$-0.316410\pi$
$691$	28.6692	1.09063	0.545314	+	0.838232i	$0.316410\pi$
$692$	0	0
$693$	0.623725	0.0236933
$694$	0	0
$695$	−35.7350	−1.35551
$696$	0	0
$697$	−33.4286	−1.26620
$698$	0	0
$699$	43.2678	1.63654
$700$	0	0
$701$	41.5450	1.56913	0.784566	−	0.620045i	$-0.212886\pi$
$701$	41.5450	1.56913	0.784566	+	0.620045i	$0.212886\pi$
$702$	0	0
$703$	−2.28491	−0.0861771
$704$	0	0
$705$	−4.50840	−0.169796
$706$	0	0
$707$	10.2045	0.383779
$708$	0	0
$709$	−11.5228	−0.432748	−0.216374	−	0.976311i	$-0.569423\pi$
$709$	−11.5228	−0.432748	−0.216374	+	0.976311i	$0.569423\pi$
$710$	0	0
$711$	−0.217622	−0.00816146
$712$	0	0
$713$	1.53498	0.0574855
$714$	0	0
$715$	46.9574	1.75611
$716$	0	0
$717$	26.1787	0.977660
$718$	0	0
$719$	19.6755	0.733774	0.366887	−	0.930266i	$-0.380424\pi$
$719$	19.6755	0.733774	0.366887	+	0.930266i	$0.380424\pi$
$720$	0	0
$721$	−8.47079	−0.315469
$722$	0	0
$723$	5.24468	0.195052
$724$	0	0
$725$	−86.6758	−3.21906
$726$	0	0
$727$	−42.5925	−1.57967	−0.789835	−	0.613320i	$-0.789834\pi$
$727$	−42.5925	−1.57967	−0.789835	+	0.613320i	$0.789834\pi$
$728$	0	0
$729$	24.7676	0.917317
$730$	0	0
$731$	−13.2680	−0.490736
$732$	0	0
$733$	−2.37085	−0.0875693	−0.0437847	−	0.999041i	$-0.513942\pi$
$733$	−2.37085	−0.0875693	−0.0437847	+	0.999041i	$0.513942\pi$
$734$	0	0
$735$	47.0674	1.73611
$736$	0	0
$737$	−46.6569	−1.71863
$738$	0	0
$739$	11.2725	0.414665	0.207333	−	0.978270i	$-0.433522\pi$
$739$	11.2725	0.414665	0.207333	+	0.978270i	$0.433522\pi$
$740$	0	0
$741$	6.47812	0.237980
$742$	0	0
$743$	32.3936	1.18841	0.594204	−	0.804315i	$-0.297467\pi$
$743$	32.3936	1.18841	0.594204	+	0.804315i	$0.297467\pi$
$744$	0	0
$745$	20.1005	0.736425
$746$	0	0
$747$	1.05703	0.0386748
$748$	0	0
$749$	−4.03038	−0.147267
$750$	0	0
$751$	42.7209	1.55891	0.779453	−	0.626461i	$-0.215497\pi$
$751$	42.7209	1.55891	0.779453	+	0.626461i	$0.215497\pi$
$752$	0	0
$753$	−41.3415	−1.50657
$754$	0	0
$755$	−66.9900	−2.43802
$756$	0	0
$757$	−7.21894	−0.262377	−0.131188	−	0.991357i	$-0.541879\pi$
$757$	−7.21894	−0.262377	−0.131188	+	0.991357i	$0.541879\pi$
$758$	0	0
$759$	24.3700	0.884577
$760$	0	0
$761$	51.0754	1.85148	0.925741	−	0.378157i	$-0.123442\pi$
$761$	51.0754	1.85148	0.925741	+	0.378157i	$0.123442\pi$
$762$	0	0
$763$	19.4641	0.704649
$764$	0	0
$765$	−3.07305	−0.111106
$766$	0	0
$767$	43.4392	1.56850
$768$	0	0
$769$	−27.7675	−1.00132	−0.500661	−	0.865643i	$-0.666910\pi$
$769$	−27.7675	−1.00132	−0.500661	+	0.865643i	$0.666910\pi$
$770$	0	0
$771$	−3.14381	−0.113221
$772$	0	0
$773$	33.2579	1.19620	0.598101	−	0.801421i	$-0.295922\pi$
$773$	33.2579	1.19620	0.598101	+	0.801421i	$0.295922\pi$
$774$	0	0
$775$	−4.60979	−0.165589
$776$	0	0
$777$	−3.88536	−0.139386
$778$	0	0
$779$	−10.1807	−0.364762
$780$	0	0
$781$	2.00392	0.0717059
$782$	0	0
$783$	−32.0561	−1.14559
$784$	0	0
$785$	88.8077	3.16968
$786$	0	0
$787$	−25.6081	−0.912829	−0.456415	−	0.889767i	$-0.650867\pi$
$787$	−25.6081	−0.912829	−0.456415	+	0.889767i	$0.650867\pi$
$788$	0	0
$789$	28.7192	1.02243
$790$	0	0
$791$	−10.1332	−0.360294
$792$	0	0
$793$	33.1903	1.17862
$794$	0	0
$795$	0.263758	0.00935454
$796$	0	0
$797$	−27.1310	−0.961029	−0.480514	−	0.876987i	$-0.659550\pi$
$797$	−27.1310	−0.961029	−0.480514	+	0.876987i	$0.659550\pi$
$798$	0	0
$799$	−1.91898	−0.0678885
$800$	0	0
$801$	3.34334	0.118131
$802$	0	0
$803$	34.1247	1.20423
$804$	0	0
$805$	−18.3195	−0.645677
$806$	0	0
$807$	57.1309	2.01110
$808$	0	0
$809$	−46.4747	−1.63396	−0.816982	−	0.576664i	$-0.804354\pi$
$809$	−46.4747	−1.63396	−0.816982	+	0.576664i	$0.804354\pi$
$810$	0	0
$811$	36.8879	1.29531	0.647655	−	0.761934i	$-0.275750\pi$
$811$	36.8879	1.29531	0.647655	+	0.761934i	$0.275750\pi$
$812$	0	0
$813$	−16.2000	−0.568160
$814$	0	0
$815$	77.8060	2.72542
$816$	0	0
$817$	−4.04078	−0.141369
$818$	0	0
$819$	0.745038	0.0260337
$820$	0	0
$821$	−19.5119	−0.680969	−0.340484	−	0.940250i	$-0.610591\pi$
$821$	−19.5119	−0.680969	−0.340484	+	0.940250i	$0.610591\pi$
$822$	0	0
$823$	−47.3508	−1.65055	−0.825273	−	0.564733i	$-0.808979\pi$
$823$	−47.3508	−1.65055	−0.825273	+	0.564733i	$0.808979\pi$
$824$	0	0
$825$	−73.1871	−2.54805
$826$	0	0
$827$	−21.0553	−0.732165	−0.366082	−	0.930582i	$-0.619301\pi$
$827$	−21.0553	−0.732165	−0.366082	+	0.930582i	$0.619301\pi$
$828$	0	0
$829$	−18.1777	−0.631339	−0.315670	−	0.948869i	$-0.602229\pi$
$829$	−18.1777	−0.631339	−0.315670	+	0.948869i	$0.602229\pi$
$830$	0	0
$831$	−17.8060	−0.617684
$832$	0	0
$833$	20.0340	0.694136
$834$	0	0
$835$	−80.6832	−2.79216
$836$	0	0
$837$	−1.70488	−0.0589293
$838$	0	0
$839$	9.31045	0.321433	0.160716	−	0.987001i	$-0.448620\pi$
$839$	9.31045	0.321433	0.160716	+	0.987001i	$0.448620\pi$
$840$	0	0
$841$	12.2528	0.422512
$842$	0	0
$843$	−23.1704	−0.798029
$844$	0	0
$845$	0.183115	0.00629934
$846$	0	0
$847$	−1.76229	−0.0605532
$848$	0	0
$849$	−48.0949	−1.65061
$850$	0	0
$851$	−10.2674	−0.351963
$852$	0	0
$853$	−34.4920	−1.18098	−0.590491	−	0.807044i	$-0.701066\pi$
$853$	−34.4920	−1.18098	−0.590491	+	0.807044i	$0.701066\pi$
$854$	0	0
$855$	−0.935899	−0.0320071
$856$	0	0
$857$	12.3753	0.422734	0.211367	−	0.977407i	$-0.432208\pi$
$857$	12.3753	0.422734	0.211367	+	0.977407i	$0.432208\pi$
$858$	0	0
$859$	−7.09012	−0.241912	−0.120956	−	0.992658i	$-0.538596\pi$
$859$	−7.09012	−0.241912	−0.120956	+	0.992658i	$0.538596\pi$
$860$	0	0
$861$	−17.3117	−0.589981
$862$	0	0
$863$	−18.2003	−0.619546	−0.309773	−	0.950811i	$-0.600253\pi$
$863$	−18.2003	−0.619546	−0.309773	+	0.950811i	$0.600253\pi$
$864$	0	0
$865$	33.9929	1.15579
$866$	0	0
$867$	11.1545	0.378825
$868$	0	0
$869$	−3.02341	−0.102562
$870$	0	0
$871$	−55.7315	−1.88839
$872$	0	0
$873$	−2.38138	−0.0805975
$874$	0	0
$875$	34.6322	1.17078
$876$	0	0
$877$	1.79027	0.0604532	0.0302266	−	0.999543i	$-0.490377\pi$
$877$	1.79027	0.0604532	0.0302266	+	0.999543i	$0.490377\pi$
$878$	0	0
$879$	25.0796	0.845914
$880$	0	0
$881$	26.6589	0.898161	0.449080	−	0.893491i	$-0.351752\pi$
$881$	26.6589	0.898161	0.449080	+	0.893491i	$0.351752\pi$
$882$	0	0
$883$	−8.42695	−0.283590	−0.141795	−	0.989896i	$-0.545287\pi$
$883$	−8.42695	−0.283590	−0.141795	+	0.989896i	$0.545287\pi$
$884$	0	0
$885$	−92.7885	−3.11905
$886$	0	0
$887$	−37.2037	−1.24918	−0.624589	−	0.780953i	$-0.714734\pi$
$887$	−37.2037	−1.24918	−0.624589	+	0.780953i	$0.714734\pi$
$888$	0	0
$889$	5.99229	0.200975
$890$	0	0
$891$	−29.0413	−0.972921
$892$	0	0
$893$	−0.584425	−0.0195570
$894$	0	0
$895$	−78.2185	−2.61455
$896$	0	0
$897$	29.1100	0.971953
$898$	0	0
$899$	2.19400	0.0731742
$900$	0	0
$901$	0.112267	0.00374016
$902$	0	0
$903$	−6.87112	−0.228656
$904$	0	0
$905$	74.4274	2.47405
$906$	0	0
$907$	−36.6231	−1.21605	−0.608026	−	0.793917i	$-0.708038\pi$
$907$	−36.6231	−1.21605	−0.608026	+	0.793917i	$0.708038\pi$
$908$	0	0
$909$	2.34260	0.0776993
$910$	0	0
$911$	42.4502	1.40644	0.703220	−	0.710973i	$-0.251745\pi$
$911$	42.4502	1.40644	0.703220	+	0.710973i	$0.251745\pi$
$912$	0	0
$913$	14.6853	0.486011
$914$	0	0
$915$	−70.8962	−2.34376
$916$	0	0
$917$	−8.57318	−0.283111
$918$	0	0
$919$	11.3951	0.375889	0.187945	−	0.982180i	$-0.439817\pi$
$919$	11.3951	0.375889	0.187945	+	0.982180i	$0.439817\pi$
$920$	0	0
$921$	−53.6915	−1.76919
$922$	0	0
$923$	2.39368	0.0787889
$924$	0	0
$925$	30.8347	1.01384
$926$	0	0
$927$	−1.94461	−0.0638693
$928$	0	0
$929$	−39.1159	−1.28335	−0.641675	−	0.766976i	$-0.721760\pi$
$929$	−39.1159	−1.28335	−0.641675	+	0.766976i	$0.721760\pi$
$930$	0	0
$931$	6.10135	0.199964
$932$	0	0
$933$	12.4677	0.408173
$934$	0	0
$935$	−42.6937	−1.39623
$936$	0	0
$937$	−49.0987	−1.60398	−0.801992	−	0.597335i	$-0.796226\pi$
$937$	−49.0987	−1.60398	−0.801992	+	0.597335i	$0.796226\pi$
$938$	0	0
$939$	−36.1853	−1.18086
$940$	0	0
$941$	−22.0572	−0.719045	−0.359523	−	0.933136i	$-0.617060\pi$
$941$	−22.0572	−0.719045	−0.359523	+	0.933136i	$0.617060\pi$
$942$	0	0
$943$	−45.7478	−1.48975
$944$	0	0
$945$	20.3472	0.661894
$946$	0	0
$947$	38.5954	1.25418	0.627092	−	0.778945i	$-0.284245\pi$
$947$	38.5954	1.25418	0.627092	+	0.778945i	$0.284245\pi$
$948$	0	0
$949$	40.7619	1.32319
$950$	0	0
$951$	−20.2257	−0.655864
$952$	0	0
$953$	−26.5622	−0.860433	−0.430216	−	0.902726i	$-0.641563\pi$
$953$	−26.5622	−0.860433	−0.430216	+	0.902726i	$0.641563\pi$
$954$	0	0
$955$	−15.2160	−0.492378
$956$	0	0
$957$	34.8330	1.12599
$958$	0	0
$959$	14.5113	0.468594
$960$	0	0
$961$	−30.8833	−0.996236
$962$	0	0
$963$	−0.925240	−0.0298154
$964$	0	0
$965$	−117.952	−3.79699
$966$	0	0
$967$	−22.0429	−0.708852	−0.354426	−	0.935084i	$-0.615324\pi$
$967$	−22.0429	−0.708852	−0.354426	+	0.935084i	$0.615324\pi$
$968$	0	0
$969$	−5.88991	−0.189211
$970$	0	0
$971$	25.6457	0.823011	0.411505	−	0.911407i	$-0.365003\pi$
$971$	25.6457	0.823011	0.411505	+	0.911407i	$0.365003\pi$
$972$	0	0
$973$	−7.87702	−0.252526
$974$	0	0
$975$	−87.4219	−2.79974
$976$	0	0
$977$	−8.04022	−0.257229	−0.128615	−	0.991695i	$-0.541053\pi$
$977$	−8.04022	−0.257229	−0.128615	+	0.991695i	$0.541053\pi$
$978$	0	0
$979$	46.4488	1.48451
$980$	0	0
$981$	4.46831	0.142662
$982$	0	0
$983$	−2.14404	−0.0683842	−0.0341921	−	0.999415i	$-0.510886\pi$
$983$	−2.14404	−0.0683842	−0.0341921	+	0.999415i	$0.510886\pi$
$984$	0	0
$985$	49.5363	1.57836
$986$	0	0
$987$	−0.993781	−0.0316324
$988$	0	0
$989$	−18.1576	−0.577378
$990$	0	0
$991$	−49.5339	−1.57350	−0.786748	−	0.617274i	$-0.788237\pi$
$991$	−49.5339	−1.57350	−0.786748	+	0.617274i	$0.788237\pi$
$992$	0	0
$993$	−16.1939	−0.513897
$994$	0	0
$995$	−3.95553	−0.125399
$996$	0	0
$997$	−13.6775	−0.433171	−0.216585	−	0.976264i	$-0.569492\pi$
$997$	−13.6775	−0.433171	−0.216585	+	0.976264i	$0.569492\pi$
$998$	0	0
$999$	11.4039	0.360803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.h.1.9	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.h.1.9	✓	31	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)

\(f(q)\)	\(=\)	\(q-1.79377 q^{3} +4.30057 q^{5} +0.947969 q^{7} +0.217622 q^{9} +O(q^{10})\)	\(q-1.79377 q^{3} +4.30057 q^{5} +0.947969 q^{7} +0.217622 q^{9} +3.02341 q^{11} +3.61145 q^{13} -7.71425 q^{15} -3.28353 q^{17} -1.00000 q^{19} -1.70044 q^{21} -4.49358 q^{23} +13.4949 q^{25} +4.99095 q^{27} -6.42284 q^{29} -0.341594 q^{31} -5.42330 q^{33} +4.07681 q^{35} +2.28491 q^{37} -6.47812 q^{39} +10.1807 q^{41} +4.04078 q^{43} +0.935899 q^{45} +0.584425 q^{47} -6.10135 q^{49} +5.88991 q^{51} -0.0341910 q^{53} +13.0024 q^{55} +1.79377 q^{57} +12.0282 q^{59} +9.19029 q^{61} +0.206299 q^{63} +15.5313 q^{65} -15.4319 q^{67} +8.06046 q^{69} +0.662802 q^{71} +11.2868 q^{73} -24.2069 q^{75} +2.86609 q^{77} -1.00000 q^{79} -9.60551 q^{81} +4.85719 q^{83} -14.1211 q^{85} +11.5211 q^{87} +15.3631 q^{89} +3.42354 q^{91} +0.612743 q^{93} -4.30057 q^{95} -10.9427 q^{97} +0.657959 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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