LMFDB - Embedded newform 6003.2.a.t.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.16689 q^{2} -0.638362 q^{4} -1.66410 q^{5} -0.970636 q^{7} -3.07868 q^{8} +O(q^{10})$	\(q+1.16689 q^{2} -0.638362 q^{4} -1.66410 q^{5} -0.970636 q^{7} -3.07868 q^{8} -1.94183 q^{10} +3.13024 q^{11} +3.36483 q^{13} -1.13263 q^{14} -2.31577 q^{16} -3.46748 q^{17} +3.42577 q^{19} +1.06230 q^{20} +3.65265 q^{22} +1.00000 q^{23} -2.23076 q^{25} +3.92639 q^{26} +0.619617 q^{28} +1.00000 q^{29} -6.67837 q^{31} +3.45511 q^{32} -4.04618 q^{34} +1.61524 q^{35} -3.48322 q^{37} +3.99751 q^{38} +5.12325 q^{40} +7.11748 q^{41} +5.93664 q^{43} -1.99822 q^{44} +1.16689 q^{46} +6.42522 q^{47} -6.05787 q^{49} -2.60305 q^{50} -2.14798 q^{52} +8.21082 q^{53} -5.20904 q^{55} +2.98828 q^{56} +1.16689 q^{58} -13.8303 q^{59} -1.90373 q^{61} -7.79294 q^{62} +8.66329 q^{64} -5.59943 q^{65} -0.345727 q^{67} +2.21351 q^{68} +1.88481 q^{70} -9.30966 q^{71} +2.86037 q^{73} -4.06455 q^{74} -2.18688 q^{76} -3.03832 q^{77} +3.43789 q^{79} +3.85368 q^{80} +8.30533 q^{82} +1.04752 q^{83} +5.77025 q^{85} +6.92743 q^{86} -9.63701 q^{88} -9.73586 q^{89} -3.26602 q^{91} -0.638362 q^{92} +7.49754 q^{94} -5.70084 q^{95} -8.02056 q^{97} -7.06888 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	$ 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) $ 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

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 $

$2$	1.16689	0.825118	0.412559	−	0.910931i	$-0.364635\pi$
$2$	1.16689	0.825118	0.412559	+	0.910931i	$0.364635\pi$
$3$	0	0
$3$	0	0
$4$	−0.638362	−0.319181
$5$	−1.66410	−0.744210	−0.372105	−	0.928191i	$-0.621364\pi$
$5$	−1.66410	−0.744210	−0.372105	+	0.928191i	$0.621364\pi$
$6$	0	0
$7$	−0.970636	−0.366866	−0.183433	−	0.983032i	$-0.558721\pi$
$7$	−0.970636	−0.366866	−0.183433	+	0.983032i	$0.558721\pi$
$8$	−3.07868	−1.08848
$9$	0	0
$10$	−1.94183	−0.614061
$11$	3.13024	0.943802	0.471901	−	0.881652i	$-0.343568\pi$
$11$	3.13024	0.943802	0.471901	+	0.881652i	$0.343568\pi$
$12$	0	0
$13$	3.36483	0.933236	0.466618	−	0.884459i	$-0.345472\pi$
$13$	3.36483	0.933236	0.466618	+	0.884459i	$0.345472\pi$
$14$	−1.13263	−0.302707
$15$	0	0
$16$	−2.31577	−0.578943
$17$	−3.46748	−0.840988	−0.420494	−	0.907295i	$-0.638143\pi$
$17$	−3.46748	−0.840988	−0.420494	+	0.907295i	$0.638143\pi$
$18$	0	0
$19$	3.42577	0.785926	0.392963	−	0.919554i	$-0.371450\pi$
$19$	3.42577	0.785926	0.392963	+	0.919554i	$0.371450\pi$
$20$	1.06230	0.237538
$21$	0	0
$22$	3.65265	0.778748
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.23076	−0.446151
$26$	3.92639	0.770029
$27$	0	0
$28$	0.619617	0.117097
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−6.67837	−1.19947	−0.599735	−	0.800199i	$-0.704727\pi$
$31$	−6.67837	−1.19947	−0.599735	+	0.800199i	$0.704727\pi$
$32$	3.45511	0.610784
$33$	0	0
$34$	−4.04618	−0.693914
$35$	1.61524	0.273025
$36$	0	0
$37$	−3.48322	−0.572638	−0.286319	−	0.958134i	$-0.592432\pi$
$37$	−3.48322	−0.572638	−0.286319	+	0.958134i	$0.592432\pi$
$38$	3.99751	0.648481
$39$	0	0
$40$	5.12325	0.810058
$41$	7.11748	1.11156	0.555782	−	0.831328i	$-0.312419\pi$
$41$	7.11748	1.11156	0.555782	+	0.831328i	$0.312419\pi$
$42$	0	0
$43$	5.93664	0.905330	0.452665	−	0.891681i	$-0.350473\pi$
$43$	5.93664	0.905330	0.452665	+	0.891681i	$0.350473\pi$
$44$	−1.99822	−0.301244
$45$	0	0
$46$	1.16689	0.172049
$47$	6.42522	0.937215	0.468607	−	0.883407i	$-0.344756\pi$
$47$	6.42522	0.937215	0.468607	+	0.883407i	$0.344756\pi$
$48$	0	0
$49$	−6.05787	−0.865409
$50$	−2.60305	−0.368127
$51$	0	0
$52$	−2.14798	−0.297871
$53$	8.21082	1.12784	0.563921	−	0.825828i	$-0.309292\pi$
$53$	8.21082	1.12784	0.563921	+	0.825828i	$0.309292\pi$
$54$	0	0
$55$	−5.20904	−0.702387
$56$	2.98828	0.399326
$57$	0	0
$58$	1.16689	0.153220
$59$	−13.8303	−1.80056	−0.900278	−	0.435315i	$-0.856637\pi$
$59$	−13.8303	−1.80056	−0.900278	+	0.435315i	$0.856637\pi$
$60$	0	0
$61$	−1.90373	−0.243748	−0.121874	−	0.992546i	$-0.538890\pi$
$61$	−1.90373	−0.243748	−0.121874	+	0.992546i	$0.538890\pi$
$62$	−7.79294	−0.989704
$63$	0	0
$64$	8.66329	1.08291
$65$	−5.59943	−0.694524
$66$	0	0
$67$	−0.345727	−0.0422372	−0.0211186	−	0.999777i	$-0.506723\pi$
$67$	−0.345727	−0.0422372	−0.0211186	+	0.999777i	$0.506723\pi$
$68$	2.21351	0.268427
$69$	0	0
$70$	1.88481	0.225278
$71$	−9.30966	−1.10485	−0.552427	−	0.833561i	$-0.686298\pi$
$71$	−9.30966	−1.10485	−0.552427	+	0.833561i	$0.686298\pi$
$72$	0	0
$73$	2.86037	0.334781	0.167390	−	0.985891i	$-0.446466\pi$
$73$	2.86037	0.334781	0.167390	+	0.985891i	$0.446466\pi$
$74$	−4.06455	−0.472494
$75$	0	0
$76$	−2.18688	−0.250853
$77$	−3.03832	−0.346249
$78$	0	0
$79$	3.43789	0.386793	0.193396	−	0.981121i	$-0.438050\pi$
$79$	3.43789	0.386793	0.193396	+	0.981121i	$0.438050\pi$
$80$	3.85368	0.430855
$81$	0	0
$82$	8.30533	0.917171
$83$	1.04752	0.114980	0.0574901	−	0.998346i	$-0.481690\pi$
$83$	1.04752	0.114980	0.0574901	+	0.998346i	$0.481690\pi$
$84$	0	0
$85$	5.77025	0.625872
$86$	6.92743	0.747004
$87$	0	0
$88$	−9.63701	−1.02731
$89$	−9.73586	−1.03200	−0.515999	−	0.856589i	$-0.672579\pi$
$89$	−9.73586	−1.03200	−0.515999	+	0.856589i	$0.672579\pi$
$90$	0	0
$91$	−3.26602	−0.342372
$92$	−0.638362	−0.0665538
$93$	0	0
$94$	7.49754	0.773312
$95$	−5.70084	−0.584894
$96$	0	0
$97$	−8.02056	−0.814364	−0.407182	−	0.913347i	$-0.633489\pi$
$97$	−8.02056	−0.814364	−0.407182	+	0.913347i	$0.633489\pi$
$98$	−7.06888	−0.714065
$99$	0	0
$100$	1.42403	0.142403
$101$	14.3892	1.43178	0.715889	−	0.698215i	$-0.246022\pi$
$101$	14.3892	1.43178	0.715889	+	0.698215i	$0.246022\pi$
$102$	0	0
$103$	−15.6662	−1.54364	−0.771818	−	0.635844i	$-0.780652\pi$
$103$	−15.6662	−1.54364	−0.771818	+	0.635844i	$0.780652\pi$
$104$	−10.3592	−1.01581
$105$	0	0
$106$	9.58114	0.930603
$107$	−6.90171	−0.667214	−0.333607	−	0.942712i	$-0.608266\pi$
$107$	−6.90171	−0.667214	−0.333607	+	0.942712i	$0.608266\pi$
$108$	0	0
$109$	−11.1779	−1.07065	−0.535323	−	0.844647i	$-0.679810\pi$
$109$	−11.1779	−1.07065	−0.535323	+	0.844647i	$0.679810\pi$
$110$	−6.07839	−0.579552
$111$	0	0
$112$	2.24777	0.212394
$113$	−11.3063	−1.06360	−0.531802	−	0.846868i	$-0.678485\pi$
$113$	−11.3063	−1.06360	−0.531802	+	0.846868i	$0.678485\pi$
$114$	0	0
$115$	−1.66410	−0.155179
$116$	−0.638362	−0.0592704
$117$	0	0
$118$	−16.1385	−1.48567
$119$	3.36566	0.308530
$120$	0	0
$121$	−1.20161	−0.109238
$122$	−2.22145	−0.201120
$123$	0	0
$124$	4.26322	0.382848
$125$	12.0327	1.07624
$126$	0	0
$127$	13.2943	1.17968	0.589840	−	0.807520i	$-0.299191\pi$
$127$	13.2943	1.17968	0.589840	+	0.807520i	$0.299191\pi$
$128$	3.19890	0.282745
$129$	0	0
$130$	−6.53393	−0.573064
$131$	7.78581	0.680250	0.340125	−	0.940380i	$-0.389531\pi$
$131$	7.78581	0.680250	0.340125	+	0.940380i	$0.389531\pi$
$132$	0	0
$133$	−3.32518	−0.288329
$134$	−0.403426	−0.0348507
$135$	0	0
$136$	10.6753	0.915398
$137$	−7.77414	−0.664189	−0.332095	−	0.943246i	$-0.607755\pi$
$137$	−7.77414	−0.664189	−0.332095	+	0.943246i	$0.607755\pi$
$138$	0	0
$139$	−22.7573	−1.93025	−0.965123	−	0.261798i	$-0.915684\pi$
$139$	−22.7573	−1.93025	−0.965123	+	0.261798i	$0.915684\pi$
$140$	−1.03111	−0.0871445
$141$	0	0
$142$	−10.8634	−0.911634
$143$	10.5327	0.880790
$144$	0	0
$145$	−1.66410	−0.138196
$146$	3.33774	0.276234
$147$	0	0
$148$	2.22356	0.182775
$149$	−12.8192	−1.05019	−0.525093	−	0.851045i	$-0.675970\pi$
$149$	−12.8192	−1.05019	−0.525093	+	0.851045i	$0.675970\pi$
$150$	0	0
$151$	−17.2566	−1.40432	−0.702159	−	0.712020i	$-0.747781\pi$
$151$	−17.2566	−1.40432	−0.702159	+	0.712020i	$0.747781\pi$
$152$	−10.5469	−0.855464
$153$	0	0
$154$	−3.54539	−0.285696
$155$	11.1135	0.892658
$156$	0	0
$157$	−6.10782	−0.487457	−0.243728	−	0.969844i	$-0.578371\pi$
$157$	−6.10782	−0.487457	−0.243728	+	0.969844i	$0.578371\pi$
$158$	4.01165	0.319150
$159$	0	0
$160$	−5.74967	−0.454551
$161$	−0.970636	−0.0764968
$162$	0	0
$163$	19.4145	1.52066	0.760330	−	0.649537i	$-0.225037\pi$
$163$	19.4145	1.52066	0.760330	+	0.649537i	$0.225037\pi$
$164$	−4.54353	−0.354790
$165$	0	0
$166$	1.22234	0.0948722
$167$	6.86540	0.531260	0.265630	−	0.964075i	$-0.414420\pi$
$167$	6.86540	0.531260	0.265630	+	0.964075i	$0.414420\pi$
$168$	0	0
$169$	−1.67792	−0.129071
$170$	6.73327	0.516418
$171$	0	0
$172$	−3.78973	−0.288964
$173$	−5.77618	−0.439155	−0.219577	−	0.975595i	$-0.570468\pi$
$173$	−5.77618	−0.439155	−0.219577	+	0.975595i	$0.570468\pi$
$174$	0	0
$175$	2.16525	0.163678
$176$	−7.24891	−0.546407
$177$	0	0
$178$	−11.3607	−0.851520
$179$	−19.9440	−1.49068	−0.745342	−	0.666682i	$-0.767714\pi$
$179$	−19.9440	−1.49068	−0.745342	+	0.666682i	$0.767714\pi$
$180$	0	0
$181$	−7.56445	−0.562261	−0.281130	−	0.959670i	$-0.590709\pi$
$181$	−7.56445	−0.562261	−0.281130	+	0.959670i	$0.590709\pi$
$182$	−3.81110	−0.282497
$183$	0	0
$184$	−3.07868	−0.226964
$185$	5.79645	0.426163
$186$	0	0
$187$	−10.8540	−0.793726
$188$	−4.10162	−0.299141
$189$	0	0
$190$	−6.65227	−0.482606
$191$	−0.388024	−0.0280764	−0.0140382	−	0.999901i	$-0.504469\pi$
$191$	−0.388024	−0.0280764	−0.0140382	+	0.999901i	$0.504469\pi$
$192$	0	0
$193$	−6.75873	−0.486504	−0.243252	−	0.969963i	$-0.578214\pi$
$193$	−6.75873	−0.486504	−0.243252	+	0.969963i	$0.578214\pi$
$194$	−9.35913	−0.671946
$195$	0	0
$196$	3.86711	0.276222
$197$	9.18467	0.654381	0.327190	−	0.944958i	$-0.393898\pi$
$197$	9.18467	0.654381	0.327190	+	0.944958i	$0.393898\pi$
$198$	0	0
$199$	−3.41182	−0.241857	−0.120929	−	0.992661i	$-0.538587\pi$
$199$	−3.41182	−0.241857	−0.120929	+	0.992661i	$0.538587\pi$
$200$	6.86779	0.485626
$201$	0	0
$202$	16.7906	1.18138
$203$	−0.970636	−0.0681253
$204$	0	0
$205$	−11.8442	−0.827237
$206$	−18.2808	−1.27368
$207$	0	0
$208$	−7.79217	−0.540290
$209$	10.7235	0.741759
$210$	0	0
$211$	−8.15205	−0.561210	−0.280605	−	0.959823i	$-0.590535\pi$
$211$	−8.15205	−0.561210	−0.280605	+	0.959823i	$0.590535\pi$
$212$	−5.24147	−0.359986
$213$	0	0
$214$	−8.05356	−0.550530
$215$	−9.87920	−0.673756
$216$	0	0
$217$	6.48226	0.440045
$218$	−13.0434	−0.883409
$219$	0	0
$220$	3.32525	0.224189
$221$	−11.6675	−0.784840
$222$	0	0
$223$	−18.2528	−1.22230	−0.611148	−	0.791516i	$-0.709292\pi$
$223$	−18.2528	−1.22230	−0.611148	+	0.791516i	$0.709292\pi$
$224$	−3.35366	−0.224076
$225$	0	0
$226$	−13.1932	−0.877599
$227$	−21.0299	−1.39581	−0.697903	−	0.716193i	$-0.745883\pi$
$227$	−21.0299	−1.39581	−0.697903	+	0.716193i	$0.745883\pi$
$228$	0	0
$229$	−2.19624	−0.145132	−0.0725659	−	0.997364i	$-0.523119\pi$
$229$	−2.19624	−0.145132	−0.0725659	+	0.997364i	$0.523119\pi$
$230$	−1.94183	−0.128041
$231$	0	0
$232$	−3.07868	−0.202126
$233$	−17.1062	−1.12067	−0.560334	−	0.828267i	$-0.689327\pi$
$233$	−17.1062	−1.12067	−0.560334	+	0.828267i	$0.689327\pi$
$234$	0	0
$235$	−10.6922	−0.697485
$236$	8.82876	0.574703
$237$	0	0
$238$	3.92737	0.254573
$239$	0.0237289	0.00153490	0.000767448	−	1.00000i	$-0.499756\pi$
$239$	0.0237289	0.00153490	0.000767448		1.00000i	$0.499756\pi$
$240$	0	0
$241$	−18.0526	−1.16287	−0.581437	−	0.813592i	$-0.697509\pi$
$241$	−18.0526	−1.16287	−0.581437	+	0.813592i	$0.697509\pi$
$242$	−1.40215	−0.0901338
$243$	0	0
$244$	1.21527	0.0777996
$245$	10.0809	0.644047
$246$	0	0
$247$	11.5271	0.733454
$248$	20.5606	1.30560
$249$	0	0
$250$	14.0409	0.888025
$251$	−18.9850	−1.19833	−0.599163	−	0.800627i	$-0.704500\pi$
$251$	−18.9850	−1.19833	−0.599163	+	0.800627i	$0.704500\pi$
$252$	0	0
$253$	3.13024	0.196796
$254$	15.5130	0.973375
$255$	0	0
$256$	−13.5938	−0.849613
$257$	−7.95157	−0.496005	−0.248003	−	0.968759i	$-0.579774\pi$
$257$	−7.95157	−0.496005	−0.248003	+	0.968759i	$0.579774\pi$
$258$	0	0
$259$	3.38094	0.210082
$260$	3.57446	0.221679
$261$	0	0
$262$	9.08521	0.561286
$263$	−6.77467	−0.417744	−0.208872	−	0.977943i	$-0.566979\pi$
$263$	−6.77467	−0.417744	−0.208872	+	0.977943i	$0.566979\pi$
$264$	0	0
$265$	−13.6637	−0.839352
$266$	−3.88012	−0.237906
$267$	0	0
$268$	0.220699	0.0134813
$269$	28.4683	1.73574	0.867872	−	0.496787i	$-0.165487\pi$
$269$	28.4683	1.73574	0.867872	+	0.496787i	$0.165487\pi$
$270$	0	0
$271$	−21.1005	−1.28177	−0.640883	−	0.767639i	$-0.721432\pi$
$271$	−21.1005	−1.28177	−0.640883	+	0.767639i	$0.721432\pi$
$272$	8.02989	0.486884
$273$	0	0
$274$	−9.07158	−0.548034
$275$	−6.98280	−0.421078
$276$	0	0
$277$	22.0874	1.32710	0.663551	−	0.748131i	$-0.269049\pi$
$277$	22.0874	1.32710	0.663551	+	0.748131i	$0.269049\pi$
$278$	−26.5553	−1.59268
$279$	0	0
$280$	−4.97281	−0.297182
$281$	21.4004	1.27664	0.638320	−	0.769771i	$-0.279629\pi$
$281$	21.4004	1.27664	0.638320	+	0.769771i	$0.279629\pi$
$282$	0	0
$283$	14.7098	0.874406	0.437203	−	0.899363i	$-0.355969\pi$
$283$	14.7098	0.874406	0.437203	+	0.899363i	$0.355969\pi$
$284$	5.94293	0.352648
$285$	0	0
$286$	12.2905	0.726755
$287$	−6.90848	−0.407795
$288$	0	0
$289$	−4.97656	−0.292739
$290$	−1.94183	−0.114028
$291$	0	0
$292$	−1.82595	−0.106856
$293$	24.7990	1.44877	0.724387	−	0.689394i	$-0.242123\pi$
$293$	24.7990	1.44877	0.724387	+	0.689394i	$0.242123\pi$
$294$	0	0
$295$	23.0151	1.33999
$296$	10.7237	0.623305
$297$	0	0
$298$	−14.9586	−0.866527
$299$	3.36483	0.194593
$300$	0	0
$301$	−5.76232	−0.332135
$302$	−20.1365	−1.15873
$303$	0	0
$304$	−7.93330	−0.455006
$305$	3.16801	0.181400
$306$	0	0
$307$	32.1800	1.83661	0.918304	−	0.395877i	$-0.129559\pi$
$307$	32.1800	1.83661	0.918304	+	0.395877i	$0.129559\pi$
$308$	1.93955	0.110516
$309$	0	0
$310$	12.9683	0.736548
$311$	−24.4553	−1.38673	−0.693366	−	0.720585i	$-0.743873\pi$
$311$	−24.4553	−1.38673	−0.693366	+	0.720585i	$0.743873\pi$
$312$	0	0
$313$	19.1018	1.07970	0.539848	−	0.841763i	$-0.318482\pi$
$313$	19.1018	1.07970	0.539848	+	0.841763i	$0.318482\pi$
$314$	−7.12717	−0.402209
$315$	0	0
$316$	−2.19462	−0.123457
$317$	−0.189764	−0.0106582	−0.00532909	−	0.999986i	$-0.501696\pi$
$317$	−0.189764	−0.0106582	−0.00532909	+	0.999986i	$0.501696\pi$
$318$	0	0
$319$	3.13024	0.175260
$320$	−14.4166	−0.805913
$321$	0	0
$322$	−1.13263	−0.0631189
$323$	−11.8788	−0.660954
$324$	0	0
$325$	−7.50611	−0.416364
$326$	22.6546	1.25472
$327$	0	0
$328$	−21.9125	−1.20991
$329$	−6.23655	−0.343832
$330$	0	0
$331$	1.51985	0.0835385	0.0417692	−	0.999127i	$-0.486701\pi$
$331$	1.51985	0.0835385	0.0417692	+	0.999127i	$0.486701\pi$
$332$	−0.668697	−0.0366995
$333$	0	0
$334$	8.01118	0.438352
$335$	0.575325	0.0314334
$336$	0	0
$337$	−14.8217	−0.807389	−0.403694	−	0.914894i	$-0.632274\pi$
$337$	−14.8217	−0.807389	−0.403694	+	0.914894i	$0.632274\pi$
$338$	−1.95796	−0.106499
$339$	0	0
$340$	−3.68351	−0.199766
$341$	−20.9049	−1.13206
$342$	0	0
$343$	12.6744	0.684355
$344$	−18.2771	−0.985433
$345$	0	0
$346$	−6.74018	−0.362355
$347$	0.764551	0.0410433	0.0205216	−	0.999789i	$-0.493467\pi$
$347$	0.764551	0.0410433	0.0205216	+	0.999789i	$0.493467\pi$
$348$	0	0
$349$	−18.8271	−1.00779	−0.503896	−	0.863764i	$-0.668101\pi$
$349$	−18.8271	−1.00779	−0.503896	+	0.863764i	$0.668101\pi$
$350$	2.52662	0.135053
$351$	0	0
$352$	10.8153	0.576459
$353$	19.6288	1.04473	0.522367	−	0.852721i	$-0.325049\pi$
$353$	19.6288	1.04473	0.522367	+	0.852721i	$0.325049\pi$
$354$	0	0
$355$	15.4922	0.822243
$356$	6.21500	0.329394
$357$	0	0
$358$	−23.2725	−1.22999
$359$	8.40034	0.443353	0.221677	−	0.975120i	$-0.428847\pi$
$359$	8.40034	0.443353	0.221677	+	0.975120i	$0.428847\pi$
$360$	0	0
$361$	−7.26409	−0.382321
$362$	−8.82690	−0.463931
$363$	0	0
$364$	2.08491	0.109279
$365$	−4.75995	−0.249147
$366$	0	0
$367$	−30.7870	−1.60707	−0.803534	−	0.595260i	$-0.797049\pi$
$367$	−30.7870	−1.60707	−0.803534	+	0.595260i	$0.797049\pi$
$368$	−2.31577	−0.120718
$369$	0	0
$370$	6.76383	0.351635
$371$	−7.96972	−0.413767
$372$	0	0
$373$	−3.79852	−0.196680	−0.0983399	−	0.995153i	$-0.531353\pi$
$373$	−3.79852	−0.196680	−0.0983399	+	0.995153i	$0.531353\pi$
$374$	−12.6655	−0.654918
$375$	0	0
$376$	−19.7812	−1.02014
$377$	3.36483	0.173298
$378$	0	0
$379$	10.4712	0.537867	0.268934	−	0.963159i	$-0.413329\pi$
$379$	10.4712	0.537867	0.268934	+	0.963159i	$0.413329\pi$
$380$	3.63920	0.186687
$381$	0	0
$382$	−0.452783	−0.0231664
$383$	−13.4398	−0.686740	−0.343370	−	0.939200i	$-0.611568\pi$
$383$	−13.4398	−0.686740	−0.343370	+	0.939200i	$0.611568\pi$
$384$	0	0
$385$	5.05608	0.257682
$386$	−7.88671	−0.401423
$387$	0	0
$388$	5.12002	0.259930
$389$	29.5183	1.49664	0.748319	−	0.663339i	$-0.230861\pi$
$389$	29.5183	1.49664	0.748319	+	0.663339i	$0.230861\pi$
$390$	0	0
$391$	−3.46748	−0.175358
$392$	18.6503	0.941980
$393$	0	0
$394$	10.7175	0.539941
$395$	−5.72101	−0.287855
$396$	0	0
$397$	−22.4511	−1.12679	−0.563394	−	0.826189i	$-0.690505\pi$
$397$	−22.4511	−1.12679	−0.563394	+	0.826189i	$0.690505\pi$
$398$	−3.98123	−0.199561
$399$	0	0
$400$	5.16592	0.258296
$401$	31.5457	1.57532	0.787659	−	0.616111i	$-0.211293\pi$
$401$	31.5457	1.57532	0.787659	+	0.616111i	$0.211293\pi$
$402$	0	0
$403$	−22.4716	−1.11939
$404$	−9.18551	−0.456996
$405$	0	0
$406$	−1.13263	−0.0562114
$407$	−10.9033	−0.540457
$408$	0	0
$409$	−36.5651	−1.80803	−0.904015	−	0.427501i	$-0.859394\pi$
$409$	−36.5651	−1.80803	−0.904015	+	0.427501i	$0.859394\pi$
$410$	−13.8209	−0.682568
$411$	0	0
$412$	10.0007	0.492699
$413$	13.4242	0.660563
$414$	0	0
$415$	−1.74318	−0.0855695
$416$	11.6259	0.570005
$417$	0	0
$418$	12.5131	0.612038
$419$	32.5009	1.58777	0.793887	−	0.608065i	$-0.208054\pi$
$419$	32.5009	1.58777	0.793887	+	0.608065i	$0.208054\pi$
$420$	0	0
$421$	22.1960	1.08177	0.540885	−	0.841097i	$-0.318090\pi$
$421$	22.1960	1.08177	0.540885	+	0.841097i	$0.318090\pi$
$422$	−9.51256	−0.463064
$423$	0	0
$424$	−25.2785	−1.22763
$425$	7.73511	0.375208
$426$	0	0
$427$	1.84783	0.0894227
$428$	4.40579	0.212962
$429$	0	0
$430$	−11.5280	−0.555928
$431$	−10.1047	−0.486726	−0.243363	−	0.969935i	$-0.578251\pi$
$431$	−10.1047	−0.486726	−0.243363	+	0.969935i	$0.578251\pi$
$432$	0	0
$433$	1.99683	0.0959617	0.0479808	−	0.998848i	$-0.484721\pi$
$433$	1.99683	0.0959617	0.0479808	+	0.998848i	$0.484721\pi$
$434$	7.56410	0.363089
$435$	0	0
$436$	7.13553	0.341730
$437$	3.42577	0.163877
$438$	0	0
$439$	−27.5731	−1.31599	−0.657997	−	0.753021i	$-0.728596\pi$
$439$	−27.5731	−1.31599	−0.657997	+	0.753021i	$0.728596\pi$
$440$	16.0370	0.764534
$441$	0	0
$442$	−13.6147	−0.647585
$443$	11.2234	0.533240	0.266620	−	0.963802i	$-0.414093\pi$
$443$	11.2234	0.533240	0.266620	+	0.963802i	$0.414093\pi$
$444$	0	0
$445$	16.2015	0.768024
$446$	−21.2990	−1.00854
$447$	0	0
$448$	−8.40890	−0.397283
$449$	25.2119	1.18982	0.594911	−	0.803791i	$-0.297187\pi$
$449$	25.2119	1.18982	0.594911	+	0.803791i	$0.297187\pi$
$450$	0	0
$451$	22.2794	1.04910
$452$	7.21749	0.339482
$453$	0	0
$454$	−24.5397	−1.15170
$455$	5.43501	0.254797
$456$	0	0
$457$	19.1195	0.894375	0.447187	−	0.894440i	$-0.352426\pi$
$457$	19.1195	0.894375	0.447187	+	0.894440i	$0.352426\pi$
$458$	−2.56278	−0.119751
$459$	0	0
$460$	1.06230	0.0495300
$461$	28.0742	1.30754	0.653772	−	0.756692i	$-0.273186\pi$
$461$	28.0742	1.30754	0.653772	+	0.756692i	$0.273186\pi$
$462$	0	0
$463$	−11.4780	−0.533430	−0.266715	−	0.963775i	$-0.585938\pi$
$463$	−11.4780	−0.533430	−0.266715	+	0.963775i	$0.585938\pi$
$464$	−2.31577	−0.107507
$465$	0	0
$466$	−19.9611	−0.924682
$467$	34.8739	1.61377	0.806886	−	0.590707i	$-0.201151\pi$
$467$	34.8739	1.61377	0.806886	+	0.590707i	$0.201151\pi$
$468$	0	0
$469$	0.335575	0.0154954
$470$	−12.4767	−0.575507
$471$	0	0
$472$	42.5792	1.95987
$473$	18.5831	0.854452
$474$	0	0
$475$	−7.64206	−0.350642
$476$	−2.14851	−0.0984769
$477$	0	0
$478$	0.0276891	0.00126647
$479$	3.69004	0.168602	0.0843011	−	0.996440i	$-0.473134\pi$
$479$	3.69004	0.168602	0.0843011	+	0.996440i	$0.473134\pi$
$480$	0	0
$481$	−11.7205	−0.534407
$482$	−21.0655	−0.959507
$483$	0	0
$484$	0.767064	0.0348665
$485$	13.3470	0.606058
$486$	0	0
$487$	−31.7082	−1.43684	−0.718418	−	0.695612i	$-0.755133\pi$
$487$	−31.7082	−1.43684	−0.718418	+	0.695612i	$0.755133\pi$
$488$	5.86098	0.265314
$489$	0	0
$490$	11.7634	0.531414
$491$	21.1652	0.955173	0.477587	−	0.878585i	$-0.341512\pi$
$491$	21.1652	0.955173	0.477587	+	0.878585i	$0.341512\pi$
$492$	0	0
$493$	−3.46748	−0.156168
$494$	13.4509	0.605186
$495$	0	0
$496$	15.4656	0.694424
$497$	9.03629	0.405333
$498$	0	0
$499$	40.2349	1.80116	0.900582	−	0.434687i	$-0.143141\pi$
$499$	40.2349	1.80116	0.900582	+	0.434687i	$0.143141\pi$
$500$	−7.68124	−0.343515
$501$	0	0
$502$	−22.1535	−0.988759
$503$	−5.43044	−0.242131	−0.121066	−	0.992644i	$-0.538631\pi$
$503$	−5.43044	−0.242131	−0.121066	+	0.992644i	$0.538631\pi$
$504$	0	0
$505$	−23.9451	−1.06554
$506$	3.65265	0.162380
$507$	0	0
$508$	−8.48659	−0.376531
$509$	−0.165579	−0.00733915	−0.00366958	−	0.999993i	$-0.501168\pi$
$509$	−0.165579	−0.00733915	−0.00366958	+	0.999993i	$0.501168\pi$
$510$	0	0
$511$	−2.77638	−0.122820
$512$	−22.2603	−0.983776
$513$	0	0
$514$	−9.27862	−0.409263
$515$	26.0702	1.14879
$516$	0	0
$517$	20.1125	0.884545
$518$	3.94520	0.173342
$519$	0	0
$520$	17.2389	0.755975
$521$	−2.11999	−0.0928786	−0.0464393	−	0.998921i	$-0.514787\pi$
$521$	−2.11999	−0.0928786	−0.0464393	+	0.998921i	$0.514787\pi$
$522$	0	0
$523$	−10.1662	−0.444538	−0.222269	−	0.974985i	$-0.571346\pi$
$523$	−10.1662	−0.444538	−0.222269	+	0.974985i	$0.571346\pi$
$524$	−4.97017	−0.217123
$525$	0	0
$526$	−7.90531	−0.344688
$527$	23.1571	1.00874
$528$	0	0
$529$	1.00000	0.0434783
$530$	−15.9440	−0.692564
$531$	0	0
$532$	2.12267	0.0920293
$533$	23.9491	1.03735
$534$	0	0
$535$	11.4852	0.496547
$536$	1.06438	0.0459743
$537$	0	0
$538$	33.2195	1.43219
$539$	−18.9626	−0.816775
$540$	0	0
$541$	−17.0049	−0.731100	−0.365550	−	0.930792i	$-0.619119\pi$
$541$	−17.0049	−0.731100	−0.365550	+	0.930792i	$0.619119\pi$
$542$	−24.6220	−1.05761
$543$	0	0
$544$	−11.9806	−0.513662
$545$	18.6011	0.796786
$546$	0	0
$547$	−5.79318	−0.247698	−0.123849	−	0.992301i	$-0.539524\pi$
$547$	−5.79318	−0.247698	−0.123849	+	0.992301i	$0.539524\pi$
$548$	4.96271	0.211997
$549$	0	0
$550$	−8.14817	−0.347439
$551$	3.42577	0.145943
$552$	0	0
$553$	−3.33694	−0.141901
$554$	25.7736	1.09502
$555$	0	0
$556$	14.5274	0.616097
$557$	−11.4176	−0.483779	−0.241890	−	0.970304i	$-0.577767\pi$
$557$	−11.4176	−0.483779	−0.241890	+	0.970304i	$0.577767\pi$
$558$	0	0
$559$	19.9758	0.844886
$560$	−3.74052	−0.158066
$561$	0	0
$562$	24.9720	1.05338
$563$	39.9407	1.68330	0.841650	−	0.540023i	$-0.181584\pi$
$563$	39.9407	1.68330	0.841650	+	0.540023i	$0.181584\pi$
$564$	0	0
$565$	18.8148	0.791546
$566$	17.1647	0.721488
$567$	0	0
$568$	28.6615	1.20261
$569$	−9.79116	−0.410467	−0.205233	−	0.978713i	$-0.565795\pi$
$569$	−9.79116	−0.410467	−0.205233	+	0.978713i	$0.565795\pi$
$570$	0	0
$571$	29.3432	1.22798	0.613988	−	0.789315i	$-0.289564\pi$
$571$	29.3432	1.22798	0.613988	+	0.789315i	$0.289564\pi$
$572$	−6.72368	−0.281131
$573$	0	0
$574$	−8.06145	−0.336479
$575$	−2.23076	−0.0930289
$576$	0	0
$577$	−27.0324	−1.12537	−0.562687	−	0.826670i	$-0.690233\pi$
$577$	−27.0324	−1.12537	−0.562687	+	0.826670i	$0.690233\pi$
$578$	−5.80711	−0.241544
$579$	0	0
$580$	1.06230	0.0441096
$581$	−1.01676	−0.0421823
$582$	0	0
$583$	25.7018	1.06446
$584$	−8.80618	−0.364402
$585$	0	0
$586$	28.9378	1.19541
$587$	−4.30700	−0.177769	−0.0888845	−	0.996042i	$-0.528330\pi$
$587$	−4.30700	−0.177769	−0.0888845	+	0.996042i	$0.528330\pi$
$588$	0	0
$589$	−22.8786	−0.942695
$590$	26.8562	1.10565
$591$	0	0
$592$	8.06635	0.331525
$593$	−16.2048	−0.665453	−0.332727	−	0.943023i	$-0.607969\pi$
$593$	−16.2048	−0.665453	−0.332727	+	0.943023i	$0.607969\pi$
$594$	0	0
$595$	−5.60082	−0.229611
$596$	8.18326	0.335200
$597$	0	0
$598$	3.92639	0.160562
$599$	14.9770	0.611945	0.305973	−	0.952040i	$-0.401018\pi$
$599$	14.9770	0.611945	0.305973	+	0.952040i	$0.401018\pi$
$600$	0	0
$601$	27.1354	1.10688	0.553438	−	0.832890i	$-0.313315\pi$
$601$	27.1354	1.10688	0.553438	+	0.832890i	$0.313315\pi$
$602$	−6.72401	−0.274050
$603$	0	0
$604$	11.0159	0.448232
$605$	1.99961	0.0812957
$606$	0	0
$607$	−4.67826	−0.189885	−0.0949423	−	0.995483i	$-0.530267\pi$
$607$	−4.67826	−0.189885	−0.0949423	+	0.995483i	$0.530267\pi$
$608$	11.8364	0.480031
$609$	0	0
$610$	3.69672	0.149676
$611$	21.6198	0.874642
$612$	0	0
$613$	−39.1493	−1.58122	−0.790612	−	0.612317i	$-0.790238\pi$
$613$	−39.1493	−1.58122	−0.790612	+	0.612317i	$0.790238\pi$
$614$	37.5506	1.51542
$615$	0	0
$616$	9.35403	0.376885
$617$	−5.78299	−0.232815	−0.116407	−	0.993202i	$-0.537138\pi$
$617$	−5.78299	−0.232815	−0.116407	+	0.993202i	$0.537138\pi$
$618$	0	0
$619$	49.3680	1.98427	0.992134	−	0.125181i	$-0.0399511\pi$
$619$	49.3680	1.98427	0.992134	+	0.125181i	$0.0399511\pi$
$620$	−7.09444	−0.284919
$621$	0	0
$622$	−28.5367	−1.14422
$623$	9.44997	0.378605
$624$	0	0
$625$	−8.86995	−0.354798
$626$	22.2897	0.890876
$627$	0	0
$628$	3.89900	0.155587
$629$	12.0780	0.481582
$630$	0	0
$631$	−16.3185	−0.649628	−0.324814	−	0.945778i	$-0.605302\pi$
$631$	−16.3185	−0.649628	−0.324814	+	0.945778i	$0.605302\pi$
$632$	−10.5842	−0.421016
$633$	0	0
$634$	−0.221434	−0.00879426
$635$	−22.1231	−0.877930
$636$	0	0
$637$	−20.3837	−0.807631
$638$	3.65265	0.144610
$639$	0	0
$640$	−5.32330	−0.210422
$641$	11.2290	0.443521	0.221760	−	0.975101i	$-0.428820\pi$
$641$	11.2290	0.443521	0.221760	+	0.975101i	$0.428820\pi$
$642$	0	0
$643$	21.6694	0.854556	0.427278	−	0.904120i	$-0.359472\pi$
$643$	21.6694	0.854556	0.427278	+	0.904120i	$0.359472\pi$
$644$	0.619617	0.0244163
$645$	0	0
$646$	−13.8613	−0.545365
$647$	−6.83356	−0.268655	−0.134327	−	0.990937i	$-0.542887\pi$
$647$	−6.83356	−0.268655	−0.134327	+	0.990937i	$0.542887\pi$
$648$	0	0
$649$	−43.2922	−1.69937
$650$	−8.75883	−0.343549
$651$	0	0
$652$	−12.3935	−0.485366
$653$	−41.1434	−1.61006	−0.805032	−	0.593231i	$-0.797852\pi$
$653$	−41.1434	−1.61006	−0.805032	+	0.593231i	$0.797852\pi$
$654$	0	0
$655$	−12.9564	−0.506249
$656$	−16.4824	−0.643531
$657$	0	0
$658$	−7.27738	−0.283702
$659$	−37.5297	−1.46195	−0.730974	−	0.682405i	$-0.760934\pi$
$659$	−37.5297	−1.46195	−0.730974	+	0.682405i	$0.760934\pi$
$660$	0	0
$661$	13.9108	0.541068	0.270534	−	0.962710i	$-0.412800\pi$
$661$	13.9108	0.541068	0.270534	+	0.962710i	$0.412800\pi$
$662$	1.77350	0.0689291
$663$	0	0
$664$	−3.22498	−0.125154
$665$	5.53344	0.214578
$666$	0	0
$667$	1.00000	0.0387202
$668$	−4.38261	−0.169568
$669$	0	0
$670$	0.671343	0.0259362
$671$	−5.95913	−0.230050
$672$	0	0
$673$	−40.5659	−1.56370	−0.781850	−	0.623467i	$-0.785724\pi$
$673$	−40.5659	−1.56370	−0.781850	+	0.623467i	$0.785724\pi$
$674$	−17.2953	−0.666191
$675$	0	0
$676$	1.07112	0.0411970
$677$	−1.71291	−0.0658325	−0.0329163	−	0.999458i	$-0.510479\pi$
$677$	−1.71291	−0.0658325	−0.0329163	+	0.999458i	$0.510479\pi$
$678$	0	0
$679$	7.78504	0.298762
$680$	−17.7648	−0.681249
$681$	0	0
$682$	−24.3937	−0.934085
$683$	−8.39156	−0.321094	−0.160547	−	0.987028i	$-0.551326\pi$
$683$	−8.39156	−0.321094	−0.160547	+	0.987028i	$0.551326\pi$
$684$	0	0
$685$	12.9370	0.494296
$686$	14.7897	0.564673
$687$	0	0
$688$	−13.7479	−0.524134
$689$	27.6280	1.05254
$690$	0	0
$691$	9.31311	0.354288	0.177144	−	0.984185i	$-0.443314\pi$
$691$	9.31311	0.354288	0.177144	+	0.984185i	$0.443314\pi$
$692$	3.68729	0.140170
$693$	0	0
$694$	0.892149	0.0338655
$695$	37.8705	1.43651
$696$	0	0
$697$	−24.6797	−0.934812
$698$	−21.9692	−0.831548
$699$	0	0
$700$	−1.38221	−0.0522428
$701$	12.1056	0.457222	0.228611	−	0.973518i	$-0.426582\pi$
$701$	12.1056	0.457222	0.228611	+	0.973518i	$0.426582\pi$
$702$	0	0
$703$	−11.9327	−0.450051
$704$	27.1181	1.02205
$705$	0	0
$706$	22.9046	0.862028
$707$	−13.9667	−0.525270
$708$	0	0
$709$	33.0136	1.23985	0.619925	−	0.784661i	$-0.287163\pi$
$709$	33.0136	1.23985	0.619925	+	0.784661i	$0.287163\pi$
$710$	18.0778	0.678447
$711$	0	0
$712$	29.9736	1.12331
$713$	−6.67837	−0.250107
$714$	0	0
$715$	−17.5275	−0.655493
$716$	12.7315	0.475798
$717$	0	0
$718$	9.80230	0.365818
$719$	17.2610	0.643727	0.321864	−	0.946786i	$-0.395691\pi$
$719$	17.2610	0.643727	0.321864	+	0.946786i	$0.395691\pi$
$720$	0	0
$721$	15.2062	0.566307
$722$	−8.47641	−0.315459
$723$	0	0
$724$	4.82885	0.179463
$725$	−2.23076	−0.0828482
$726$	0	0
$727$	−16.5001	−0.611955	−0.305977	−	0.952039i	$-0.598983\pi$
$727$	−16.5001	−0.611955	−0.305977	+	0.952039i	$0.598983\pi$
$728$	10.0551	0.372665
$729$	0	0
$730$	−5.55435	−0.205576
$731$	−20.5852	−0.761372
$732$	0	0
$733$	−21.7144	−0.802039	−0.401020	−	0.916069i	$-0.631344\pi$
$733$	−21.7144	−0.802039	−0.401020	+	0.916069i	$0.631344\pi$
$734$	−35.9251	−1.32602
$735$	0	0
$736$	3.45511	0.127357
$737$	−1.08221	−0.0398636
$738$	0	0
$739$	−15.4134	−0.566993	−0.283496	−	0.958973i	$-0.591494\pi$
$739$	−15.4134	−0.566993	−0.283496	+	0.958973i	$0.591494\pi$
$740$	−3.70023	−0.136023
$741$	0	0
$742$	−9.29980	−0.341406
$743$	47.2089	1.73193	0.865964	−	0.500106i	$-0.166706\pi$
$743$	47.2089	1.73193	0.865964	+	0.500106i	$0.166706\pi$
$744$	0	0
$745$	21.3324	0.781559
$746$	−4.43246	−0.162284
$747$	0	0
$748$	6.92881	0.253342
$749$	6.69905	0.244778
$750$	0	0
$751$	−1.92362	−0.0701939	−0.0350969	−	0.999384i	$-0.511174\pi$
$751$	−1.92362	−0.0701939	−0.0350969	+	0.999384i	$0.511174\pi$
$752$	−14.8793	−0.542594
$753$	0	0
$754$	3.92639	0.142991
$755$	28.7167	1.04511
$756$	0	0
$757$	−33.3233	−1.21116	−0.605578	−	0.795786i	$-0.707058\pi$
$757$	−33.3233	−1.21116	−0.605578	+	0.795786i	$0.707058\pi$
$758$	12.2187	0.443804
$759$	0	0
$760$	17.5511	0.636645
$761$	12.2967	0.445754	0.222877	−	0.974847i	$-0.428455\pi$
$761$	12.2967	0.445754	0.222877	+	0.974847i	$0.428455\pi$
$762$	0	0
$763$	10.8496	0.392783
$764$	0.247700	0.00896147
$765$	0	0
$766$	−15.6828	−0.566641
$767$	−46.5367	−1.68034
$768$	0	0
$769$	21.7078	0.782803	0.391401	−	0.920220i	$-0.371990\pi$
$769$	21.7078	0.782803	0.391401	+	0.920220i	$0.371990\pi$
$770$	5.89991	0.212618
$771$	0	0
$772$	4.31452	0.155283
$773$	−39.5976	−1.42423	−0.712114	−	0.702064i	$-0.752262\pi$
$773$	−39.5976	−1.42423	−0.712114	+	0.702064i	$0.752262\pi$
$774$	0	0
$775$	14.8978	0.535145
$776$	24.6928	0.886419
$777$	0	0
$778$	34.4447	1.23490
$779$	24.3829	0.873606
$780$	0	0
$781$	−29.1414	−1.04276
$782$	−4.04618	−0.144691
$783$	0	0
$784$	14.0286	0.501022
$785$	10.1640	0.362770
$786$	0	0
$787$	−35.7413	−1.27404	−0.637020	−	0.770847i	$-0.719833\pi$
$787$	−35.7413	−1.27404	−0.637020	+	0.770847i	$0.719833\pi$
$788$	−5.86314	−0.208866
$789$	0	0
$790$	−6.67580	−0.237514
$791$	10.9743	0.390200
$792$	0	0
$793$	−6.40573	−0.227474
$794$	−26.1980	−0.929732
$795$	0	0
$796$	2.17798	0.0771963
$797$	−42.4281	−1.50288	−0.751440	−	0.659801i	$-0.770640\pi$
$797$	−42.4281	−1.50288	−0.751440	+	0.659801i	$0.770640\pi$
$798$	0	0
$799$	−22.2793	−0.788187
$800$	−7.70752	−0.272502
$801$	0	0
$802$	36.8105	1.29982
$803$	8.95364	0.315967
$804$	0	0
$805$	1.61524	0.0569297
$806$	−26.2219	−0.923627
$807$	0	0
$808$	−44.2998	−1.55846
$809$	−44.7480	−1.57325	−0.786627	−	0.617428i	$-0.788175\pi$
$809$	−44.7480	−1.57325	−0.786627	+	0.617428i	$0.788175\pi$
$810$	0	0
$811$	−49.1461	−1.72575	−0.862877	−	0.505414i	$-0.831340\pi$
$811$	−49.1461	−1.72575	−0.862877	+	0.505414i	$0.831340\pi$
$812$	0.619617	0.0217443
$813$	0	0
$814$	−12.7230	−0.445941
$815$	−32.3078	−1.13169
$816$	0	0
$817$	20.3376	0.711522
$818$	−42.6676	−1.49184
$819$	0	0
$820$	7.56090	0.264038
$821$	−25.4417	−0.887921	−0.443960	−	0.896046i	$-0.646427\pi$
$821$	−25.4417	−0.887921	−0.443960	+	0.896046i	$0.646427\pi$
$822$	0	0
$823$	−4.39388	−0.153161	−0.0765804	−	0.997063i	$-0.524400\pi$
$823$	−4.39388	−0.153161	−0.0765804	+	0.997063i	$0.524400\pi$
$824$	48.2313	1.68022
$825$	0	0
$826$	15.6646	0.545042
$827$	−35.1132	−1.22101	−0.610503	−	0.792014i	$-0.709033\pi$
$827$	−35.1132	−1.22101	−0.610503	+	0.792014i	$0.709033\pi$
$828$	0	0
$829$	12.5291	0.435152	0.217576	−	0.976043i	$-0.430185\pi$
$829$	12.5291	0.435152	0.217576	+	0.976043i	$0.430185\pi$
$830$	−2.03411	−0.0706049
$831$	0	0
$832$	29.1505	1.01061
$833$	21.0055	0.727799
$834$	0	0
$835$	−11.4247	−0.395369
$836$	−6.84546	−0.236755
$837$	0	0
$838$	37.9251	1.31010
$839$	−1.22438	−0.0422703	−0.0211352	−	0.999777i	$-0.506728\pi$
$839$	−1.22438	−0.0422703	−0.0211352	+	0.999777i	$0.506728\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	25.9004	0.892587
$843$	0	0
$844$	5.20396	0.179128
$845$	2.79224	0.0960559
$846$	0	0
$847$	1.16633	0.0400755
$848$	−19.0144	−0.652956
$849$	0	0
$850$	9.02604	0.309591
$851$	−3.48322	−0.119403
$852$	0	0
$853$	32.0204	1.09636	0.548178	−	0.836361i	$-0.315322\pi$
$853$	32.0204	1.09636	0.548178	+	0.836361i	$0.315322\pi$
$854$	2.15622	0.0737842
$855$	0	0
$856$	21.2482	0.726248
$857$	31.7302	1.08388	0.541941	−	0.840416i	$-0.317690\pi$
$857$	31.7302	1.08388	0.541941	+	0.840416i	$0.317690\pi$
$858$	0	0
$859$	27.2799	0.930778	0.465389	−	0.885106i	$-0.345915\pi$
$859$	27.2799	0.930778	0.465389	+	0.885106i	$0.345915\pi$
$860$	6.30650	0.215050
$861$	0	0
$862$	−11.7911	−0.401606
$863$	−19.1687	−0.652512	−0.326256	−	0.945282i	$-0.605787\pi$
$863$	−19.1687	−0.652512	−0.326256	+	0.945282i	$0.605787\pi$
$864$	0	0
$865$	9.61217	0.326824
$866$	2.33009	0.0791797
$867$	0	0
$868$	−4.13803	−0.140454
$869$	10.7614	0.365056
$870$	0	0
$871$	−1.16331	−0.0394173
$872$	34.4131	1.16538
$873$	0	0
$874$	3.99751	0.135218
$875$	−11.6794	−0.394836
$876$	0	0
$877$	−18.5102	−0.625044	−0.312522	−	0.949911i	$-0.601174\pi$
$877$	−18.5102	−0.625044	−0.312522	+	0.949911i	$0.601174\pi$
$878$	−32.1749	−1.08585
$879$	0	0
$880$	12.0629	0.406642
$881$	3.89803	0.131328	0.0656640	−	0.997842i	$-0.479083\pi$
$881$	3.89803	0.131328	0.0656640	+	0.997842i	$0.479083\pi$
$882$	0	0
$883$	21.4260	0.721043	0.360522	−	0.932751i	$-0.382599\pi$
$883$	21.4260	0.721043	0.360522	+	0.932751i	$0.382599\pi$
$884$	7.44808	0.250506
$885$	0	0
$886$	13.0965	0.439986
$887$	−40.8605	−1.37196	−0.685981	−	0.727619i	$-0.740627\pi$
$887$	−40.8605	−1.37196	−0.685981	+	0.727619i	$0.740627\pi$
$888$	0	0
$889$	−12.9039	−0.432784
$890$	18.9054	0.633710
$891$	0	0
$892$	11.6519	0.390133
$893$	22.0113	0.736581
$894$	0	0
$895$	33.1889	1.10938
$896$	−3.10496	−0.103730
$897$	0	0
$898$	29.4196	0.981743
$899$	−6.67837	−0.222736
$900$	0	0
$901$	−28.4709	−0.948502
$902$	25.9977	0.865627
$903$	0	0
$904$	34.8085	1.15771
$905$	12.5880	0.418440
$906$	0	0
$907$	22.6638	0.752539	0.376269	−	0.926510i	$-0.377207\pi$
$907$	22.6638	0.752539	0.376269	+	0.926510i	$0.377207\pi$
$908$	13.4247	0.445515
$909$	0	0
$910$	6.34207	0.210238
$911$	−25.2415	−0.836290	−0.418145	−	0.908380i	$-0.637320\pi$
$911$	−25.2415	−0.836290	−0.418145	+	0.908380i	$0.637320\pi$
$912$	0	0
$913$	3.27899	0.108519
$914$	22.3105	0.737964
$915$	0	0
$916$	1.40200	0.0463233
$917$	−7.55719	−0.249560
$918$	0	0
$919$	−16.3262	−0.538552	−0.269276	−	0.963063i	$-0.586784\pi$
$919$	−16.3262	−0.538552	−0.269276	+	0.963063i	$0.586784\pi$
$920$	5.12325	0.168909
$921$	0	0
$922$	32.7595	1.07888
$923$	−31.3254	−1.03109
$924$	0	0
$925$	7.77022	0.255483
$926$	−13.3936	−0.440142
$927$	0	0
$928$	3.45511	0.113420
$929$	−17.1115	−0.561410	−0.280705	−	0.959794i	$-0.590568\pi$
$929$	−17.1115	−0.561410	−0.280705	+	0.959794i	$0.590568\pi$
$930$	0	0
$931$	−20.7529	−0.680148
$932$	10.9200	0.357696
$933$	0	0
$934$	40.6941	1.33155
$935$	18.0623	0.590699
$936$	0	0
$937$	−56.6451	−1.85051	−0.925257	−	0.379342i	$-0.876150\pi$
$937$	−56.6451	−1.85051	−0.925257	+	0.379342i	$0.876150\pi$
$938$	0.391579	0.0127855
$939$	0	0
$940$	6.82552	0.222624
$941$	17.5292	0.571435	0.285717	−	0.958314i	$-0.407768\pi$
$941$	17.5292	0.571435	0.285717	+	0.958314i	$0.407768\pi$
$942$	0	0
$943$	7.11748	0.231777
$944$	32.0279	1.04242
$945$	0	0
$946$	21.6845	0.705023
$947$	3.76309	0.122284	0.0611420	−	0.998129i	$-0.480526\pi$
$947$	3.76309	0.122284	0.0611420	+	0.998129i	$0.480526\pi$
$948$	0	0
$949$	9.62466	0.312430
$950$	−8.91746	−0.289321
$951$	0	0
$952$	−10.3618	−0.335828
$953$	49.0028	1.58736	0.793679	−	0.608337i	$-0.208163\pi$
$953$	49.0028	1.58736	0.793679	+	0.608337i	$0.208163\pi$
$954$	0	0
$955$	0.645713	0.0208948
$956$	−0.0151476	−0.000489910 0
$957$	0	0
$958$	4.30588	0.139117
$959$	7.54585	0.243668
$960$	0	0
$961$	13.6006	0.438729
$962$	−13.6765	−0.440948
$963$	0	0
$964$	11.5241	0.371167
$965$	11.2472	0.362061
$966$	0	0
$967$	−7.49183	−0.240921	−0.120461	−	0.992718i	$-0.538437\pi$
$967$	−7.49183	−0.240921	−0.120461	+	0.992718i	$0.538437\pi$
$968$	3.69939	0.118903
$969$	0	0
$970$	15.5746	0.500069
$971$	27.8867	0.894926	0.447463	−	0.894303i	$-0.352328\pi$
$971$	27.8867	0.894926	0.447463	+	0.894303i	$0.352328\pi$
$972$	0	0
$973$	22.0890	0.708141
$974$	−37.0001	−1.18556
$975$	0	0
$976$	4.40860	0.141116
$977$	−50.8283	−1.62614	−0.813071	−	0.582165i	$-0.802206\pi$
$977$	−50.8283	−1.62614	−0.813071	+	0.582165i	$0.802206\pi$
$978$	0	0
$979$	−30.4755	−0.974003
$980$	−6.43528	−0.205567
$981$	0	0
$982$	24.6975	0.788130
$983$	41.3134	1.31769	0.658847	−	0.752277i	$-0.271044\pi$
$983$	41.3134	1.31769	0.658847	+	0.752277i	$0.271044\pi$
$984$	0	0
$985$	−15.2843	−0.486997
$986$	−4.04618	−0.128857
$987$	0	0
$988$	−7.35849	−0.234105
$989$	5.93664	0.188774
$990$	0	0
$991$	−37.8775	−1.20322	−0.601610	−	0.798790i	$-0.705474\pi$
$991$	−37.8775	−1.20322	−0.601610	+	0.798790i	$0.705474\pi$
$992$	−23.0745	−0.732617
$993$	0	0
$994$	10.5444	0.334447
$995$	5.67762	0.179993
$996$	0	0
$997$	−52.4367	−1.66069	−0.830344	−	0.557252i	$-0.811856\pi$
$997$	−52.4367	−1.66069	−0.830344	+	0.557252i	$0.811856\pi$
$998$	46.9498	1.48617
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.t.1.17	✓	22
3.2	odd	2		6003.2.a.u.1.6	yes	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.t.1.17	✓	22	1.1	even	1	trivial
6003.2.a.u.1.6	yes	22	3.2	odd	2

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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+1.16689 q^{2} -0.638362 q^{4} -1.66410 q^{5} -0.970636 q^{7} -3.07868 q^{8} +O(q^{10})\)	\(q+1.16689 q^{2} -0.638362 q^{4} -1.66410 q^{5} -0.970636 q^{7} -3.07868 q^{8} -1.94183 q^{10} +3.13024 q^{11} +3.36483 q^{13} -1.13263 q^{14} -2.31577 q^{16} -3.46748 q^{17} +3.42577 q^{19} +1.06230 q^{20} +3.65265 q^{22} +1.00000 q^{23} -2.23076 q^{25} +3.92639 q^{26} +0.619617 q^{28} +1.00000 q^{29} -6.67837 q^{31} +3.45511 q^{32} -4.04618 q^{34} +1.61524 q^{35} -3.48322 q^{37} +3.99751 q^{38} +5.12325 q^{40} +7.11748 q^{41} +5.93664 q^{43} -1.99822 q^{44} +1.16689 q^{46} +6.42522 q^{47} -6.05787 q^{49} -2.60305 q^{50} -2.14798 q^{52} +8.21082 q^{53} -5.20904 q^{55} +2.98828 q^{56} +1.16689 q^{58} -13.8303 q^{59} -1.90373 q^{61} -7.79294 q^{62} +8.66329 q^{64} -5.59943 q^{65} -0.345727 q^{67} +2.21351 q^{68} +1.88481 q^{70} -9.30966 q^{71} +2.86037 q^{73} -4.06455 q^{74} -2.18688 q^{76} -3.03832 q^{77} +3.43789 q^{79} +3.85368 q^{80} +8.30533 q^{82} +1.04752 q^{83} +5.77025 q^{85} +6.92743 q^{86} -9.63701 q^{88} -9.73586 q^{89} -3.26602 q^{91} -0.638362 q^{92} +7.49754 q^{94} -5.70084 q^{95} -8.02056 q^{97} -7.06888 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8}+O(q^{10}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8	\( 22 q - 3 q^{2} + 17 q^{4} - 6 q^{7} - 6 q^{8} - 12 q^{10} - 28 q^{13} - q^{14} + 3 q^{16} - 10 q^{17} - 8 q^{19} - 11 q^{22} + 22 q^{23} + 11 q^{26} - 21 q^{28} + 22 q^{29} - 18 q^{31} + 5 q^{32} - 33 q^{34} + 2 q^{35} - 28 q^{37} + 14 q^{38} - 30 q^{40} - 10 q^{41} - 14 q^{43} + 37 q^{44} - 3 q^{46} - 18 q^{47} + 2 q^{49} + 7 q^{50} - 57 q^{52} + 20 q^{53} - 42 q^{55} - 2 q^{56} - 3 q^{58} - 20 q^{59} - 38 q^{61} + 4 q^{62} - 24 q^{64} + 12 q^{65} - 50 q^{67} + 11 q^{68} - 48 q^{70} + 12 q^{71} - 46 q^{73} - 6 q^{74} - 16 q^{76} - 14 q^{77} - 20 q^{79} - 58 q^{80} - 42 q^{82} + 22 q^{83} - 66 q^{85} + 22 q^{86} - 68 q^{88} - 14 q^{89} - 16 q^{91} + 17 q^{92} - 27 q^{94} - 20 q^{95} - 48 q^{97} - 28 q^{98}+O(q^{100}) \) 22 * q - 3 * q^2 + 17 * q^4 - 6 * q^7 - 6 * q^8 - 12 * q^10 - 28 * q^13 - q^14 + 3 * q^16 - 10 * q^17 - 8 * q^19 - 11 * q^22 + 22 * q^23 + 11 * q^26 - 21 * q^28 + 22 * q^29 - 18 * q^31 + 5 * q^32 - 33 * q^34 + 2 * q^35 - 28 * q^37 + 14 * q^38 - 30 * q^40 - 10 * q^41 - 14 * q^43 + 37 * q^44 - 3 * q^46 - 18 * q^47 + 2 * q^49 + 7 * q^50 - 57 * q^52 + 20 * q^53 - 42 * q^55 - 2 * q^56 - 3 * q^58 - 20 * q^59 - 38 * q^61 + 4 * q^62 - 24 * q^64 + 12 * q^65 - 50 * q^67 + 11 * q^68 - 48 * q^70 + 12 * q^71 - 46 * q^73 - 6 * q^74 - 16 * q^76 - 14 * q^77 - 20 * q^79 - 58 * q^80 - 42 * q^82 + 22 * q^83 - 66 * q^85 + 22 * q^86 - 68 * q^88 - 14 * q^89 - 16 * q^91 + 17 * q^92 - 27 * q^94 - 20 * q^95 - 48 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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