LMFDB - Embedded newform 6003.2.a.w.1.6

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:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-1.79065 q^{2} +1.20642 q^{4} +3.95479 q^{5} -4.84424 q^{7} +1.42102 q^{8} +O(q^{10})$	\(q-1.79065 q^{2} +1.20642 q^{4} +3.95479 q^{5} -4.84424 q^{7} +1.42102 q^{8} -7.08163 q^{10} +2.42853 q^{11} -0.0494338 q^{13} +8.67432 q^{14} -4.95739 q^{16} +6.29065 q^{17} -4.92728 q^{19} +4.77113 q^{20} -4.34863 q^{22} -1.00000 q^{23} +10.6403 q^{25} +0.0885186 q^{26} -5.84418 q^{28} +1.00000 q^{29} -0.565621 q^{31} +6.03489 q^{32} -11.2643 q^{34} -19.1579 q^{35} -0.214959 q^{37} +8.82301 q^{38} +5.61985 q^{40} +6.17753 q^{41} -1.79767 q^{43} +2.92982 q^{44} +1.79065 q^{46} +1.25536 q^{47} +16.4666 q^{49} -19.0531 q^{50} -0.0596379 q^{52} +0.794820 q^{53} +9.60430 q^{55} -6.88378 q^{56} -1.79065 q^{58} +6.36752 q^{59} -10.4253 q^{61} +1.01283 q^{62} -0.891582 q^{64} -0.195500 q^{65} -9.79770 q^{67} +7.58916 q^{68} +34.3051 q^{70} +13.9922 q^{71} +2.91945 q^{73} +0.384916 q^{74} -5.94436 q^{76} -11.7643 q^{77} -2.38557 q^{79} -19.6054 q^{80} -11.0618 q^{82} -13.7414 q^{83} +24.8782 q^{85} +3.21900 q^{86} +3.45099 q^{88} -12.9549 q^{89} +0.239469 q^{91} -1.20642 q^{92} -2.24791 q^{94} -19.4863 q^{95} +6.46228 q^{97} -29.4859 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	$ 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) $ 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * 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 $

$2$	−1.79065	−1.26618	−0.633090	−	0.774079i	$-0.718214\pi$
$2$	−1.79065	−1.26618	−0.633090	+	0.774079i	$0.718214\pi$
$3$	0	0
$3$	0	0
$4$	1.20642	0.603209
$5$	3.95479	1.76863	0.884317	−	0.466887i	$-0.154624\pi$
$5$	3.95479	1.76863	0.884317	+	0.466887i	$0.154624\pi$
$6$	0	0
$7$	−4.84424	−1.83095	−0.915475	−	0.402376i	$-0.868185\pi$
$7$	−4.84424	−1.83095	−0.915475	+	0.402376i	$0.868185\pi$
$8$	1.42102	0.502408
$9$	0	0
$10$	−7.08163	−2.23941
$11$	2.42853	0.732228	0.366114	−	0.930570i	$-0.380688\pi$
$11$	2.42853	0.732228	0.366114	+	0.930570i	$0.380688\pi$
$12$	0	0
$13$	−0.0494338	−0.0137105	−0.00685524	−	0.999977i	$-0.502182\pi$
$13$	−0.0494338	−0.0137105	−0.00685524	+	0.999977i	$0.502182\pi$
$14$	8.67432	2.31831
$15$	0	0
$16$	−4.95739	−1.23935
$17$	6.29065	1.52571	0.762854	−	0.646571i	$-0.223798\pi$
$17$	6.29065	1.52571	0.762854	+	0.646571i	$0.223798\pi$
$18$	0	0
$19$	−4.92728	−1.13039	−0.565197	−	0.824956i	$-0.691200\pi$
$19$	−4.92728	−1.13039	−0.565197	+	0.824956i	$0.691200\pi$
$20$	4.77113	1.06686
$21$	0	0
$22$	−4.34863	−0.927132
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	10.6403	2.12807
$26$	0.0885186	0.0173599
$27$	0	0
$28$	−5.84418	−1.10445
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.565621	−0.101589	−0.0507943	−	0.998709i	$-0.516175\pi$
$31$	−0.565621	−0.101589	−0.0507943	+	0.998709i	$0.516175\pi$
$32$	6.03489	1.06683
$33$	0	0
$34$	−11.2643	−1.93182
$35$	−19.1579	−3.23828
$36$	0	0
$37$	−0.214959	−0.0353390	−0.0176695	−	0.999844i	$-0.505625\pi$
$37$	−0.214959	−0.0353390	−0.0176695	+	0.999844i	$0.505625\pi$
$38$	8.82301	1.43128
$39$	0	0
$40$	5.61985	0.888576
$41$	6.17753	0.964768	0.482384	−	0.875960i	$-0.339771\pi$
$41$	6.17753	0.964768	0.482384	+	0.875960i	$0.339771\pi$
$42$	0	0
$43$	−1.79767	−0.274142	−0.137071	−	0.990561i	$-0.543769\pi$
$43$	−1.79767	−0.274142	−0.137071	+	0.990561i	$0.543769\pi$
$44$	2.92982	0.441687
$45$	0	0
$46$	1.79065	0.264017
$47$	1.25536	0.183113	0.0915567	−	0.995800i	$-0.470816\pi$
$47$	1.25536	0.183113	0.0915567	+	0.995800i	$0.470816\pi$
$48$	0	0
$49$	16.4666	2.35237
$50$	−19.0531	−2.69451
$51$	0	0
$52$	−0.0596379	−0.00827029
$53$	0.794820	0.109177	0.0545884	−	0.998509i	$-0.482615\pi$
$53$	0.794820	0.109177	0.0545884	+	0.998509i	$0.482615\pi$
$54$	0	0
$55$	9.60430	1.29504
$56$	−6.88378	−0.919883
$57$	0	0
$58$	−1.79065	−0.235124
$59$	6.36752	0.828981	0.414490	−	0.910054i	$-0.363960\pi$
$59$	6.36752	0.828981	0.414490	+	0.910054i	$0.363960\pi$
$60$	0	0
$61$	−10.4253	−1.33482	−0.667411	−	0.744689i	$-0.732598\pi$
$61$	−10.4253	−1.33482	−0.667411	+	0.744689i	$0.732598\pi$
$62$	1.01283	0.128629
$63$	0	0
$64$	−0.891582	−0.111448
$65$	−0.195500	−0.0242488
$66$	0	0
$67$	−9.79770	−1.19698	−0.598490	−	0.801130i	$-0.704232\pi$
$67$	−9.79770	−1.19698	−0.598490	+	0.801130i	$0.704232\pi$
$68$	7.58916	0.920321
$69$	0	0
$70$	34.3051	4.10024
$71$	13.9922	1.66057	0.830284	−	0.557340i	$-0.188178\pi$
$71$	13.9922	1.66057	0.830284	+	0.557340i	$0.188178\pi$
$72$	0	0
$73$	2.91945	0.341695	0.170848	−	0.985297i	$-0.445349\pi$
$73$	2.91945	0.341695	0.170848	+	0.985297i	$0.445349\pi$
$74$	0.384916	0.0447456
$75$	0	0
$76$	−5.94436	−0.681865
$77$	−11.7643	−1.34067
$78$	0	0
$79$	−2.38557	−0.268397	−0.134199	−	0.990954i	$-0.542846\pi$
$79$	−2.38557	−0.268397	−0.134199	+	0.990954i	$0.542846\pi$
$80$	−19.6054	−2.19195
$81$	0	0
$82$	−11.0618	−1.22157
$83$	−13.7414	−1.50832	−0.754160	−	0.656691i	$-0.771956\pi$
$83$	−13.7414	−1.50832	−0.754160	+	0.656691i	$0.771956\pi$
$84$	0	0
$85$	24.8782	2.69842
$86$	3.21900	0.347113
$87$	0	0
$88$	3.45099	0.367877
$89$	−12.9549	−1.37322	−0.686608	−	0.727028i	$-0.740901\pi$
$89$	−12.9549	−1.37322	−0.686608	+	0.727028i	$0.740901\pi$
$90$	0	0
$91$	0.239469	0.0251032
$92$	−1.20642	−0.125778
$93$	0	0
$94$	−2.24791	−0.231854
$95$	−19.4863	−1.99925
$96$	0	0
$97$	6.46228	0.656146	0.328073	−	0.944652i	$-0.393601\pi$
$97$	6.46228	0.656146	0.328073	+	0.944652i	$0.393601\pi$
$98$	−29.4859	−2.97853
$99$	0	0
$100$	12.8367	1.28367
$101$	−11.6194	−1.15618	−0.578089	−	0.815974i	$-0.696201\pi$
$101$	−11.6194	−1.15618	−0.578089	+	0.815974i	$0.696201\pi$
$102$	0	0
$103$	18.4239	1.81537	0.907683	−	0.419657i	$-0.137850\pi$
$103$	18.4239	1.81537	0.907683	+	0.419657i	$0.137850\pi$
$104$	−0.0702467	−0.00688825
$105$	0	0
$106$	−1.42324	−0.138237
$107$	13.3412	1.28974	0.644870	−	0.764292i	$-0.276911\pi$
$107$	13.3412	1.28974	0.644870	+	0.764292i	$0.276911\pi$
$108$	0	0
$109$	−6.61274	−0.633385	−0.316693	−	0.948528i	$-0.602572\pi$
$109$	−6.61274	−0.633385	−0.316693	+	0.948528i	$0.602572\pi$
$110$	−17.1979	−1.63976
$111$	0	0
$112$	24.0148	2.26918
$113$	19.3926	1.82430	0.912152	−	0.409851i	$-0.134419\pi$
$113$	19.3926	1.82430	0.912152	+	0.409851i	$0.134419\pi$
$114$	0	0
$115$	−3.95479	−0.368786
$116$	1.20642	0.112013
$117$	0	0
$118$	−11.4020	−1.04964
$119$	−30.4734	−2.79349
$120$	0	0
$121$	−5.10227	−0.463842
$122$	18.6680	1.69012
$123$	0	0
$124$	−0.682376	−0.0612792
$125$	22.3063	1.99514
$126$	0	0
$127$	−2.66127	−0.236149	−0.118075	−	0.993005i	$-0.537672\pi$
$127$	−2.66127	−0.236149	−0.118075	+	0.993005i	$0.537672\pi$
$128$	−10.4733	−0.925715
$129$	0	0
$130$	0.350072	0.0307033
$131$	13.7091	1.19777	0.598884	−	0.800836i	$-0.295611\pi$
$131$	13.7091	1.19777	0.598884	+	0.800836i	$0.295611\pi$
$132$	0	0
$133$	23.8689	2.06969
$134$	17.5442	1.51559
$135$	0	0
$136$	8.93917	0.766528
$137$	7.96978	0.680904	0.340452	−	0.940262i	$-0.389420\pi$
$137$	7.96978	0.680904	0.340452	+	0.940262i	$0.389420\pi$
$138$	0	0
$139$	15.1921	1.28858	0.644290	−	0.764781i	$-0.277153\pi$
$139$	15.1921	1.28858	0.644290	+	0.764781i	$0.277153\pi$
$140$	−23.1125	−1.95336
$141$	0	0
$142$	−25.0551	−2.10258
$143$	−0.120051	−0.0100392
$144$	0	0
$145$	3.95479	0.328427
$146$	−5.22770	−0.432647
$147$	0	0
$148$	−0.259331	−0.0213168
$149$	6.29023	0.515315	0.257658	−	0.966236i	$-0.417049\pi$
$149$	6.29023	0.515315	0.257658	+	0.966236i	$0.417049\pi$
$150$	0	0
$151$	9.50571	0.773564	0.386782	−	0.922171i	$-0.373587\pi$
$151$	9.50571	0.773564	0.386782	+	0.922171i	$0.373587\pi$
$152$	−7.00178	−0.567919
$153$	0	0
$154$	21.0658	1.69753
$155$	−2.23691	−0.179673
$156$	0	0
$157$	1.83092	0.146124	0.0730618	−	0.997327i	$-0.476723\pi$
$157$	1.83092	0.146124	0.0730618	+	0.997327i	$0.476723\pi$
$158$	4.27171	0.339839
$159$	0	0
$160$	23.8667	1.88683
$161$	4.84424	0.381779
$162$	0	0
$163$	−3.02888	−0.237240	−0.118620	−	0.992940i	$-0.537847\pi$
$163$	−3.02888	−0.237240	−0.118620	+	0.992940i	$0.537847\pi$
$164$	7.45269	0.581957
$165$	0	0
$166$	24.6061	1.90980
$167$	3.02253	0.233890	0.116945	−	0.993138i	$-0.462690\pi$
$167$	3.02253	0.233890	0.116945	+	0.993138i	$0.462690\pi$
$168$	0	0
$169$	−12.9976	−0.999812
$170$	−44.5481	−3.41668
$171$	0	0
$172$	−2.16875	−0.165365
$173$	22.6059	1.71869	0.859347	−	0.511393i	$-0.170870\pi$
$173$	22.6059	1.71869	0.859347	+	0.511393i	$0.170870\pi$
$174$	0	0
$175$	−51.5443	−3.89638
$176$	−12.0391	−0.907485
$177$	0	0
$178$	23.1977	1.73874
$179$	−17.3819	−1.29918	−0.649591	−	0.760284i	$-0.725060\pi$
$179$	−17.3819	−1.29918	−0.649591	+	0.760284i	$0.725060\pi$
$180$	0	0
$181$	17.5528	1.30469	0.652345	−	0.757922i	$-0.273785\pi$
$181$	17.5528	1.30469	0.652345	+	0.757922i	$0.273785\pi$
$182$	−0.428805	−0.0317851
$183$	0	0
$184$	−1.42102	−0.104759
$185$	−0.850117	−0.0625018
$186$	0	0
$187$	15.2770	1.11717
$188$	1.51449	0.110456
$189$	0	0
$190$	34.8931	2.53141
$191$	17.7181	1.28203	0.641017	−	0.767527i	$-0.278513\pi$
$191$	17.7181	1.28203	0.641017	+	0.767527i	$0.278513\pi$
$192$	0	0
$193$	−6.51056	−0.468641	−0.234320	−	0.972159i	$-0.575286\pi$
$193$	−6.51056	−0.468641	−0.234320	+	0.972159i	$0.575286\pi$
$194$	−11.5717	−0.830798
$195$	0	0
$196$	19.8656	1.41897
$197$	−7.98809	−0.569128	−0.284564	−	0.958657i	$-0.591849\pi$
$197$	−7.98809	−0.569128	−0.284564	+	0.958657i	$0.591849\pi$
$198$	0	0
$199$	−13.9966	−0.992194	−0.496097	−	0.868267i	$-0.665234\pi$
$199$	−13.9966	−0.992194	−0.496097	+	0.868267i	$0.665234\pi$
$200$	15.1202	1.06916
$201$	0	0
$202$	20.8063	1.46393
$203$	−4.84424	−0.339999
$204$	0	0
$205$	24.4308	1.70632
$206$	−32.9908	−2.29858
$207$	0	0
$208$	0.245063	0.0169921
$209$	−11.9660	−0.827706
$210$	0	0
$211$	−26.8271	−1.84685	−0.923427	−	0.383773i	$-0.874624\pi$
$211$	−26.8271	−1.84685	−0.923427	+	0.383773i	$0.874624\pi$
$212$	0.958885	0.0658565
$213$	0	0
$214$	−23.8894	−1.63304
$215$	−7.10941	−0.484858
$216$	0	0
$217$	2.74000	0.186004
$218$	11.8411	0.801979
$219$	0	0
$220$	11.5868	0.781182
$221$	−0.310971	−0.0209182
$222$	0	0
$223$	−29.1274	−1.95051	−0.975257	−	0.221075i	$-0.929043\pi$
$223$	−29.1274	−1.95051	−0.975257	+	0.221075i	$0.929043\pi$
$224$	−29.2344	−1.95331
$225$	0	0
$226$	−34.7254	−2.30990
$227$	5.67006	0.376335	0.188168	−	0.982137i	$-0.439745\pi$
$227$	5.67006	0.376335	0.188168	+	0.982137i	$0.439745\pi$
$228$	0	0
$229$	16.0102	1.05798	0.528990	−	0.848628i	$-0.322571\pi$
$229$	16.0102	1.05798	0.528990	+	0.848628i	$0.322571\pi$
$230$	7.08163	0.466949
$231$	0	0
$232$	1.42102	0.0932948
$233$	13.0641	0.855855	0.427927	−	0.903813i	$-0.359244\pi$
$233$	13.0641	0.855855	0.427927	+	0.903813i	$0.359244\pi$
$234$	0	0
$235$	4.96469	0.323861
$236$	7.68190	0.500049
$237$	0	0
$238$	54.5671	3.53706
$239$	14.0832	0.910964	0.455482	−	0.890245i	$-0.349467\pi$
$239$	14.0832	0.910964	0.455482	+	0.890245i	$0.349467\pi$
$240$	0	0
$241$	−6.20832	−0.399913	−0.199957	−	0.979805i	$-0.564080\pi$
$241$	−6.20832	−0.399913	−0.199957	+	0.979805i	$0.564080\pi$
$242$	9.13636	0.587308
$243$	0	0
$244$	−12.5773	−0.805178
$245$	65.1220	4.16049
$246$	0	0
$247$	0.243574	0.0154982
$248$	−0.803762	−0.0510389
$249$	0	0
$250$	−39.9428	−2.52620
$251$	6.01738	0.379814	0.189907	−	0.981802i	$-0.439181\pi$
$251$	6.01738	0.379814	0.189907	+	0.981802i	$0.439181\pi$
$252$	0	0
$253$	−2.42853	−0.152680
$254$	4.76540	0.299008
$255$	0	0
$256$	20.5371	1.28357
$257$	−19.8736	−1.23968	−0.619841	−	0.784728i	$-0.712803\pi$
$257$	−19.8736	−1.23968	−0.619841	+	0.784728i	$0.712803\pi$
$258$	0	0
$259$	1.04131	0.0647040
$260$	−0.235855	−0.0146271
$261$	0	0
$262$	−24.5481	−1.51659
$263$	21.5203	1.32700	0.663500	−	0.748177i	$-0.269070\pi$
$263$	21.5203	1.32700	0.663500	+	0.748177i	$0.269070\pi$
$264$	0	0
$265$	3.14334	0.193094
$266$	−42.7408	−2.62060
$267$	0	0
$268$	−11.8201	−0.722029
$269$	−8.46684	−0.516232	−0.258116	−	0.966114i	$-0.583102\pi$
$269$	−8.46684	−0.516232	−0.258116	+	0.966114i	$0.583102\pi$
$270$	0	0
$271$	−12.5378	−0.761617	−0.380809	−	0.924654i	$-0.624354\pi$
$271$	−12.5378	−0.761617	−0.380809	+	0.924654i	$0.624354\pi$
$272$	−31.1852	−1.89088
$273$	0	0
$274$	−14.2711	−0.862146
$275$	25.8403	1.55823
$276$	0	0
$277$	5.63697	0.338693	0.169346	−	0.985557i	$-0.445834\pi$
$277$	5.63697	0.338693	0.169346	+	0.985557i	$0.445834\pi$
$278$	−27.2038	−1.63157
$279$	0	0
$280$	−27.2239	−1.62694
$281$	27.7391	1.65477	0.827387	−	0.561632i	$-0.189826\pi$
$281$	27.7391	1.65477	0.827387	+	0.561632i	$0.189826\pi$
$282$	0	0
$283$	−8.95950	−0.532587	−0.266294	−	0.963892i	$-0.585799\pi$
$283$	−8.95950	−0.532587	−0.266294	+	0.963892i	$0.585799\pi$
$284$	16.8805	1.00167
$285$	0	0
$286$	0.214970	0.0127114
$287$	−29.9254	−1.76644
$288$	0	0
$289$	22.5723	1.32778
$290$	−7.08163	−0.415848
$291$	0	0
$292$	3.52207	0.206114
$293$	11.7695	0.687579	0.343790	−	0.939047i	$-0.388289\pi$
$293$	11.7695	0.687579	0.343790	+	0.939047i	$0.388289\pi$
$294$	0	0
$295$	25.1822	1.46616
$296$	−0.305462	−0.0177546
$297$	0	0
$298$	−11.2636	−0.652482
$299$	0.0494338	0.00285883
$300$	0	0
$301$	8.70835	0.501941
$302$	−17.0214	−0.979471
$303$	0	0
$304$	24.4264	1.40095
$305$	−41.2298	−2.36081
$306$	0	0
$307$	3.78933	0.216268	0.108134	−	0.994136i	$-0.465512\pi$
$307$	3.78933	0.216268	0.108134	+	0.994136i	$0.465512\pi$
$308$	−14.1927	−0.808706
$309$	0	0
$310$	4.00552	0.227498
$311$	−11.8415	−0.671470	−0.335735	−	0.941956i	$-0.608985\pi$
$311$	−11.8415	−0.671470	−0.335735	+	0.941956i	$0.608985\pi$
$312$	0	0
$313$	−10.3420	−0.584564	−0.292282	−	0.956332i	$-0.594415\pi$
$313$	−10.3420	−0.584564	−0.292282	+	0.956332i	$0.594415\pi$
$314$	−3.27854	−0.185019
$315$	0	0
$316$	−2.87799	−0.161900
$317$	6.24324	0.350655	0.175328	−	0.984510i	$-0.443902\pi$
$317$	6.24324	0.350655	0.175328	+	0.984510i	$0.443902\pi$
$318$	0	0
$319$	2.42853	0.135971
$320$	−3.52602	−0.197110
$321$	0	0
$322$	−8.67432	−0.483401
$323$	−30.9958	−1.72465
$324$	0	0
$325$	−0.525993	−0.0291768
$326$	5.42366	0.300388
$327$	0	0
$328$	8.77842	0.484707
$329$	−6.08127	−0.335271
$330$	0	0
$331$	−6.86402	−0.377280	−0.188640	−	0.982046i	$-0.560408\pi$
$331$	−6.86402	−0.377280	−0.188640	+	0.982046i	$0.560408\pi$
$332$	−16.5779	−0.909833
$333$	0	0
$334$	−5.41229	−0.296147
$335$	−38.7478	−2.11702
$336$	0	0
$337$	−3.56819	−0.194372	−0.0971858	−	0.995266i	$-0.530984\pi$
$337$	−3.56819	−0.194372	−0.0971858	+	0.995266i	$0.530984\pi$
$338$	23.2740	1.26594
$339$	0	0
$340$	30.0135	1.62771
$341$	−1.37363	−0.0743860
$342$	0	0
$343$	−45.8586	−2.47613
$344$	−2.55454	−0.137731
$345$	0	0
$346$	−40.4792	−2.17617
$347$	−23.5968	−1.26674	−0.633372	−	0.773847i	$-0.718330\pi$
$347$	−23.5968	−1.26674	−0.633372	+	0.773847i	$0.718330\pi$
$348$	0	0
$349$	−3.32035	−0.177734	−0.0888671	−	0.996043i	$-0.528325\pi$
$349$	−3.32035	−0.177734	−0.0888671	+	0.996043i	$0.528325\pi$
$350$	92.2977	4.93352
$351$	0	0
$352$	14.6559	0.781161
$353$	7.50225	0.399304	0.199652	−	0.979867i	$-0.436019\pi$
$353$	7.50225	0.399304	0.199652	+	0.979867i	$0.436019\pi$
$354$	0	0
$355$	55.3362	2.93694
$356$	−15.6290	−0.828337
$357$	0	0
$358$	31.1248	1.64500
$359$	−25.8373	−1.36364	−0.681820	−	0.731520i	$-0.738811\pi$
$359$	−25.8373	−1.36364	−0.681820	+	0.731520i	$0.738811\pi$
$360$	0	0
$361$	5.27804	0.277792
$362$	−31.4309	−1.65197
$363$	0	0
$364$	0.288900	0.0151425
$365$	11.5458	0.604334
$366$	0	0
$367$	1.87423	0.0978339	0.0489170	−	0.998803i	$-0.484423\pi$
$367$	1.87423	0.0978339	0.0489170	+	0.998803i	$0.484423\pi$
$368$	4.95739	0.258422
$369$	0	0
$370$	1.52226	0.0791385
$371$	−3.85029	−0.199897
$372$	0	0
$373$	19.0753	0.987681	0.493841	−	0.869552i	$-0.335593\pi$
$373$	19.0753	0.987681	0.493841	+	0.869552i	$0.335593\pi$
$374$	−27.3557	−1.41453
$375$	0	0
$376$	1.78390	0.0919977
$377$	−0.0494338	−0.00254597
$378$	0	0
$379$	0.873778	0.0448829	0.0224415	−	0.999748i	$-0.492856\pi$
$379$	0.873778	0.0448829	0.0224415	+	0.999748i	$0.492856\pi$
$380$	−23.5087	−1.20597
$381$	0	0
$382$	−31.7268	−1.62329
$383$	24.8827	1.27145	0.635724	−	0.771917i	$-0.280702\pi$
$383$	24.8827	1.27145	0.635724	+	0.771917i	$0.280702\pi$
$384$	0	0
$385$	−46.5255	−2.37116
$386$	11.6581	0.593383
$387$	0	0
$388$	7.79622	0.395793
$389$	12.9404	0.656103	0.328051	−	0.944660i	$-0.393608\pi$
$389$	12.9404	0.656103	0.328051	+	0.944660i	$0.393608\pi$
$390$	0	0
$391$	−6.29065	−0.318132
$392$	23.3995	1.18185
$393$	0	0
$394$	14.3039	0.720618
$395$	−9.43441	−0.474697
$396$	0	0
$397$	37.7049	1.89235	0.946176	−	0.323652i	$-0.104911\pi$
$397$	37.7049	1.89235	0.946176	+	0.323652i	$0.104911\pi$
$398$	25.0630	1.25629
$399$	0	0
$400$	−52.7483	−2.63742
$401$	22.6973	1.13345	0.566726	−	0.823907i	$-0.308210\pi$
$401$	22.6973	1.13345	0.566726	+	0.823907i	$0.308210\pi$
$402$	0	0
$403$	0.0279608	0.00139283
$404$	−14.0179	−0.697417
$405$	0	0
$406$	8.67432	0.430499
$407$	−0.522033	−0.0258762
$408$	0	0
$409$	−27.4556	−1.35759	−0.678795	−	0.734328i	$-0.737497\pi$
$409$	−27.4556	−1.35759	−0.678795	+	0.734328i	$0.737497\pi$
$410$	−43.7470	−2.16051
$411$	0	0
$412$	22.2270	1.09505
$413$	−30.8458	−1.51782
$414$	0	0
$415$	−54.3445	−2.66767
$416$	−0.298328	−0.0146267
$417$	0	0
$418$	21.4269	1.04802
$419$	−31.6202	−1.54475	−0.772374	−	0.635168i	$-0.780931\pi$
$419$	−31.6202	−1.54475	−0.772374	+	0.635168i	$0.780931\pi$
$420$	0	0
$421$	28.3905	1.38367	0.691835	−	0.722056i	$-0.256803\pi$
$421$	28.3905	1.38367	0.691835	+	0.722056i	$0.256803\pi$
$422$	48.0379	2.33845
$423$	0	0
$424$	1.12946	0.0548513
$425$	66.9347	3.24681
$426$	0	0
$427$	50.5026	2.44399
$428$	16.0951	0.777984
$429$	0	0
$430$	12.7304	0.613917
$431$	−0.297135	−0.0143125	−0.00715624	−	0.999974i	$-0.502278\pi$
$431$	−0.297135	−0.0143125	−0.00715624	+	0.999974i	$0.502278\pi$
$432$	0	0
$433$	40.5981	1.95102	0.975511	−	0.219952i	$-0.0705902\pi$
$433$	40.5981	1.95102	0.975511	+	0.219952i	$0.0705902\pi$
$434$	−4.90638	−0.235514
$435$	0	0
$436$	−7.97773	−0.382064
$437$	4.92728	0.235704
$438$	0	0
$439$	27.9036	1.33177	0.665883	−	0.746056i	$-0.268055\pi$
$439$	27.9036	1.33177	0.665883	+	0.746056i	$0.268055\pi$
$440$	13.6479	0.650640
$441$	0	0
$442$	0.556840	0.0264862
$443$	0.507901	0.0241311	0.0120655	−	0.999927i	$-0.496159\pi$
$443$	0.507901	0.0241311	0.0120655	+	0.999927i	$0.496159\pi$
$444$	0	0
$445$	−51.2339	−2.42872
$446$	52.1569	2.46970
$447$	0	0
$448$	4.31903	0.204055
$449$	32.9431	1.55468	0.777339	−	0.629081i	$-0.216569\pi$
$449$	32.9431	1.55468	0.777339	+	0.629081i	$0.216569\pi$
$450$	0	0
$451$	15.0023	0.706430
$452$	23.3956	1.10044
$453$	0	0
$454$	−10.1531	−0.476508
$455$	0.947049	0.0443984
$456$	0	0
$457$	−30.2709	−1.41601	−0.708006	−	0.706206i	$-0.750405\pi$
$457$	−30.2709	−1.41601	−0.708006	+	0.706206i	$0.750405\pi$
$458$	−28.6685	−1.33959
$459$	0	0
$460$	−4.77113	−0.222455
$461$	11.1106	0.517471	0.258735	−	0.965948i	$-0.416694\pi$
$461$	11.1106	0.517471	0.258735	+	0.965948i	$0.416694\pi$
$462$	0	0
$463$	10.9649	0.509582	0.254791	−	0.966996i	$-0.417993\pi$
$463$	10.9649	0.509582	0.254791	+	0.966996i	$0.417993\pi$
$464$	−4.95739	−0.230141
$465$	0	0
$466$	−23.3931	−1.08367
$467$	40.2868	1.86425	0.932126	−	0.362135i	$-0.117952\pi$
$467$	40.2868	1.86425	0.932126	+	0.362135i	$0.117952\pi$
$468$	0	0
$469$	47.4624	2.19161
$470$	−8.89001	−0.410066
$471$	0	0
$472$	9.04840	0.416487
$473$	−4.36569	−0.200735
$474$	0	0
$475$	−52.4279	−2.40556
$476$	−36.7637	−1.68506
$477$	0	0
$478$	−25.2180	−1.15344
$479$	18.2674	0.834661	0.417330	−	0.908755i	$-0.362966\pi$
$479$	18.2674	0.834661	0.417330	+	0.908755i	$0.362966\pi$
$480$	0	0
$481$	0.0106262	0.000484515 0
$482$	11.1169	0.506362
$483$	0	0
$484$	−6.15547	−0.279794
$485$	25.5570	1.16048
$486$	0	0
$487$	−9.84734	−0.446226	−0.223113	−	0.974793i	$-0.571622\pi$
$487$	−9.84734	−0.446226	−0.223113	+	0.974793i	$0.571622\pi$
$488$	−14.8146	−0.670626
$489$	0	0
$490$	−116.611	−5.26793
$491$	11.2422	0.507355	0.253678	−	0.967289i	$-0.418360\pi$
$491$	11.2422	0.507355	0.253678	+	0.967289i	$0.418360\pi$
$492$	0	0
$493$	6.29065	0.283317
$494$	−0.436155	−0.0196236
$495$	0	0
$496$	2.80401	0.125904
$497$	−67.7815	−3.04042
$498$	0	0
$499$	−27.9986	−1.25339	−0.626695	−	0.779264i	$-0.715593\pi$
$499$	−27.9986	−1.25339	−0.626695	+	0.779264i	$0.715593\pi$
$500$	26.9108	1.20349
$501$	0	0
$502$	−10.7750	−0.480912
$503$	−9.75474	−0.434943	−0.217471	−	0.976067i	$-0.569781\pi$
$503$	−9.75474	−0.434943	−0.217471	+	0.976067i	$0.569781\pi$
$504$	0	0
$505$	−45.9524	−2.04486
$506$	4.34863	0.193320
$507$	0	0
$508$	−3.21061	−0.142448
$509$	1.02475	0.0454213	0.0227107	−	0.999742i	$-0.492770\pi$
$509$	1.02475	0.0454213	0.0227107	+	0.999742i	$0.492770\pi$
$510$	0	0
$511$	−14.1425	−0.625627
$512$	−15.8282	−0.699513
$513$	0	0
$514$	35.5866	1.56966
$515$	72.8628	3.21072
$516$	0	0
$517$	3.04868	0.134081
$518$	−1.86462	−0.0819268
$519$	0	0
$520$	−0.277811	−0.0121828
$521$	−26.2578	−1.15037	−0.575187	−	0.818022i	$-0.695071\pi$
$521$	−26.2578	−1.15037	−0.575187	+	0.818022i	$0.695071\pi$
$522$	0	0
$523$	−1.33098	−0.0581995	−0.0290998	−	0.999577i	$-0.509264\pi$
$523$	−1.33098	−0.0581995	−0.0290998	+	0.999577i	$0.509264\pi$
$524$	16.5389	0.722505
$525$	0	0
$526$	−38.5353	−1.68022
$527$	−3.55813	−0.154994
$528$	0	0
$529$	1.00000	0.0434783
$530$	−5.62862	−0.244492
$531$	0	0
$532$	28.7959	1.24846
$533$	−0.305379	−0.0132274
$534$	0	0
$535$	52.7615	2.28108
$536$	−13.9228	−0.601372
$537$	0	0
$538$	15.1611	0.653642
$539$	39.9896	1.72247
$540$	0	0
$541$	−20.5378	−0.882990	−0.441495	−	0.897264i	$-0.645552\pi$
$541$	−20.5378	−0.882990	−0.441495	+	0.897264i	$0.645552\pi$
$542$	22.4508	0.964344
$543$	0	0
$544$	37.9634	1.62767
$545$	−26.1520	−1.12023
$546$	0	0
$547$	32.5224	1.39056	0.695279	−	0.718740i	$-0.255281\pi$
$547$	32.5224	1.39056	0.695279	+	0.718740i	$0.255281\pi$
$548$	9.61489	0.410728
$549$	0	0
$550$	−46.2709	−1.97300
$551$	−4.92728	−0.209909
$552$	0	0
$553$	11.5563	0.491422
$554$	−10.0938	−0.428845
$555$	0	0
$556$	18.3281	0.777284
$557$	−24.4826	−1.03736	−0.518681	−	0.854968i	$-0.673577\pi$
$557$	−24.4826	−1.03736	−0.518681	+	0.854968i	$0.673577\pi$
$558$	0	0
$559$	0.0888658	0.00375862
$560$	94.9733	4.01335
$561$	0	0
$562$	−49.6709	−2.09524
$563$	21.1259	0.890352	0.445176	−	0.895443i	$-0.353141\pi$
$563$	21.1259	0.890352	0.445176	+	0.895443i	$0.353141\pi$
$564$	0	0
$565$	76.6937	3.22653
$566$	16.0433	0.674351
$567$	0	0
$568$	19.8833	0.834283
$569$	−23.8455	−0.999654	−0.499827	−	0.866125i	$-0.666603\pi$
$569$	−23.8455	−0.999654	−0.499827	+	0.866125i	$0.666603\pi$
$570$	0	0
$571$	7.08548	0.296518	0.148259	−	0.988949i	$-0.452633\pi$
$571$	7.08548	0.296518	0.148259	+	0.988949i	$0.452633\pi$
$572$	−0.144832	−0.00605574
$573$	0	0
$574$	53.5859	2.23663
$575$	−10.6403	−0.443733
$576$	0	0
$577$	−20.2245	−0.841956	−0.420978	−	0.907071i	$-0.638313\pi$
$577$	−20.2245	−0.841956	−0.420978	+	0.907071i	$0.638313\pi$
$578$	−40.4191	−1.68121
$579$	0	0
$580$	4.77113	0.198110
$581$	66.5668	2.76166
$582$	0	0
$583$	1.93024	0.0799423
$584$	4.14860	0.171670
$585$	0	0
$586$	−21.0750	−0.870598
$587$	37.5943	1.55168	0.775842	−	0.630927i	$-0.217325\pi$
$587$	37.5943	1.55168	0.775842	+	0.630927i	$0.217325\pi$
$588$	0	0
$589$	2.78697	0.114835
$590$	−45.0924	−1.85643
$591$	0	0
$592$	1.06564	0.0437974
$593$	5.64090	0.231644	0.115822	−	0.993270i	$-0.463050\pi$
$593$	5.64090	0.231644	0.115822	+	0.993270i	$0.463050\pi$
$594$	0	0
$595$	−120.516	−4.94067
$596$	7.58865	0.310843
$597$	0	0
$598$	−0.0885186	−0.00361979
$599$	18.3395	0.749331	0.374666	−	0.927160i	$-0.377757\pi$
$599$	18.3395	0.749331	0.374666	+	0.927160i	$0.377757\pi$
$600$	0	0
$601$	22.1017	0.901547	0.450773	−	0.892638i	$-0.351148\pi$
$601$	22.1017	0.901547	0.450773	+	0.892638i	$0.351148\pi$
$602$	−15.5936	−0.635547
$603$	0	0
$604$	11.4679	0.466621
$605$	−20.1784	−0.820368
$606$	0	0
$607$	17.1659	0.696742	0.348371	−	0.937357i	$-0.386735\pi$
$607$	17.1659	0.696742	0.348371	+	0.937357i	$0.386735\pi$
$608$	−29.7356	−1.20594
$609$	0	0
$610$	73.8281	2.98921
$611$	−0.0620574	−0.00251057
$612$	0	0
$613$	4.63611	0.187251	0.0936254	−	0.995607i	$-0.470154\pi$
$613$	4.63611	0.187251	0.0936254	+	0.995607i	$0.470154\pi$
$614$	−6.78535	−0.273835
$615$	0	0
$616$	−16.7174	−0.673564
$617$	40.9456	1.64841	0.824204	−	0.566293i	$-0.191623\pi$
$617$	40.9456	1.64841	0.824204	+	0.566293i	$0.191623\pi$
$618$	0	0
$619$	−4.79469	−0.192715	−0.0963574	−	0.995347i	$-0.530719\pi$
$619$	−4.79469	−0.192715	−0.0963574	+	0.995347i	$0.530719\pi$
$620$	−2.69865	−0.108381
$621$	0	0
$622$	21.2040	0.850201
$623$	62.7566	2.51429
$624$	0	0
$625$	35.0151	1.40060
$626$	18.5188	0.740162
$627$	0	0
$628$	2.20886	0.0881431
$629$	−1.35223	−0.0539170
$630$	0	0
$631$	11.2837	0.449199	0.224599	−	0.974451i	$-0.427893\pi$
$631$	11.2837	0.449199	0.224599	+	0.974451i	$0.427893\pi$
$632$	−3.38995	−0.134845
$633$	0	0
$634$	−11.1794	−0.443993
$635$	−10.5248	−0.417662
$636$	0	0
$637$	−0.814008	−0.0322522
$638$	−4.34863	−0.172164
$639$	0	0
$640$	−41.4196	−1.63725
$641$	−15.8701	−0.626830	−0.313415	−	0.949616i	$-0.601473\pi$
$641$	−15.8701	−0.626830	−0.313415	+	0.949616i	$0.601473\pi$
$642$	0	0
$643$	27.6326	1.08972	0.544861	−	0.838526i	$-0.316582\pi$
$643$	27.6326	1.08972	0.544861	+	0.838526i	$0.316582\pi$
$644$	5.84418	0.230293
$645$	0	0
$646$	55.5025	2.18372
$647$	12.9242	0.508103	0.254051	−	0.967191i	$-0.418237\pi$
$647$	12.9242	0.508103	0.254051	+	0.967191i	$0.418237\pi$
$648$	0	0
$649$	15.4637	0.607003
$650$	0.941867	0.0369431
$651$	0	0
$652$	−3.65410	−0.143105
$653$	19.3955	0.759006	0.379503	−	0.925191i	$-0.376095\pi$
$653$	19.3955	0.759006	0.379503	+	0.925191i	$0.376095\pi$
$654$	0	0
$655$	54.2165	2.11841
$656$	−30.6244	−1.19568
$657$	0	0
$658$	10.8894	0.424514
$659$	42.7422	1.66500	0.832499	−	0.554026i	$-0.186909\pi$
$659$	42.7422	1.66500	0.832499	+	0.554026i	$0.186909\pi$
$660$	0	0
$661$	44.6973	1.73852	0.869262	−	0.494352i	$-0.164595\pi$
$661$	44.6973	1.73852	0.869262	+	0.494352i	$0.164595\pi$
$662$	12.2910	0.477705
$663$	0	0
$664$	−19.5269	−0.757792
$665$	94.3963	3.66053
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	3.64644	0.141085
$669$	0	0
$670$	69.3837	2.68053
$671$	−25.3181	−0.977394
$672$	0	0
$673$	−4.46749	−0.172209	−0.0861045	−	0.996286i	$-0.527442\pi$
$673$	−4.46749	−0.172209	−0.0861045	+	0.996286i	$0.527442\pi$
$674$	6.38937	0.246109
$675$	0	0
$676$	−15.6805	−0.603096
$677$	28.4322	1.09274	0.546370	−	0.837544i	$-0.316009\pi$
$677$	28.4322	1.09274	0.546370	+	0.837544i	$0.316009\pi$
$678$	0	0
$679$	−31.3048	−1.20137
$680$	35.3525	1.35571
$681$	0	0
$682$	2.45968	0.0941860
$683$	12.4906	0.477939	0.238970	−	0.971027i	$-0.423190\pi$
$683$	12.4906	0.477939	0.238970	+	0.971027i	$0.423190\pi$
$684$	0	0
$685$	31.5188	1.20427
$686$	82.1165	3.13522
$687$	0	0
$688$	8.91177	0.339758
$689$	−0.0392910	−0.00149687
$690$	0	0
$691$	34.0948	1.29703	0.648515	−	0.761202i	$-0.275391\pi$
$691$	34.0948	1.29703	0.648515	+	0.761202i	$0.275391\pi$
$692$	27.2722	1.03673
$693$	0	0
$694$	42.2536	1.60392
$695$	60.0817	2.27903
$696$	0	0
$697$	38.8607	1.47195
$698$	5.94558	0.225043
$699$	0	0
$700$	−62.1840	−2.35033
$701$	18.9945	0.717412	0.358706	−	0.933451i	$-0.383218\pi$
$701$	18.9945	0.717412	0.358706	+	0.933451i	$0.383218\pi$
$702$	0	0
$703$	1.05916	0.0399471
$704$	−2.16523	−0.0816052
$705$	0	0
$706$	−13.4339	−0.505591
$707$	56.2873	2.11690
$708$	0	0
$709$	−15.9134	−0.597640	−0.298820	−	0.954309i	$-0.596593\pi$
$709$	−15.9134	−0.597640	−0.298820	+	0.954309i	$0.596593\pi$
$710$	−99.0876	−3.71869
$711$	0	0
$712$	−18.4092	−0.689915
$713$	0.565621	0.0211827
$714$	0	0
$715$	−0.474777	−0.0177557
$716$	−20.9698	−0.783679
$717$	0	0
$718$	46.2655	1.72661
$719$	−10.6566	−0.397422	−0.198711	−	0.980058i	$-0.563676\pi$
$719$	−10.6566	−0.397422	−0.198711	+	0.980058i	$0.563676\pi$
$720$	0	0
$721$	−89.2499	−3.32384
$722$	−9.45112	−0.351734
$723$	0	0
$724$	21.1760	0.787002
$725$	10.6403	0.395172
$726$	0	0
$727$	−35.7978	−1.32767	−0.663834	−	0.747880i	$-0.731072\pi$
$727$	−35.7978	−1.32767	−0.663834	+	0.747880i	$0.731072\pi$
$728$	0.340291	0.0126120
$729$	0	0
$730$	−20.6744	−0.765195
$731$	−11.3085	−0.418261
$732$	0	0
$733$	3.91555	0.144624	0.0723121	−	0.997382i	$-0.476962\pi$
$733$	3.91555	0.144624	0.0723121	+	0.997382i	$0.476962\pi$
$734$	−3.35608	−0.123875
$735$	0	0
$736$	−6.03489	−0.222449
$737$	−23.7940	−0.876462
$738$	0	0
$739$	−13.6366	−0.501630	−0.250815	−	0.968035i	$-0.580699\pi$
$739$	−13.6366	−0.501630	−0.250815	+	0.968035i	$0.580699\pi$
$740$	−1.02560	−0.0377017
$741$	0	0
$742$	6.89452	0.253106
$743$	17.4027	0.638441	0.319221	−	0.947680i	$-0.396579\pi$
$743$	17.4027	0.638441	0.319221	+	0.947680i	$0.396579\pi$
$744$	0	0
$745$	24.8765	0.911405
$746$	−34.1571	−1.25058
$747$	0	0
$748$	18.4305	0.673885
$749$	−64.6278	−2.36145
$750$	0	0
$751$	14.7260	0.537361	0.268680	−	0.963229i	$-0.413412\pi$
$751$	14.7260	0.537361	0.268680	+	0.963229i	$0.413412\pi$
$752$	−6.22332	−0.226941
$753$	0	0
$754$	0.0885186	0.00322366
$755$	37.5931	1.36815
$756$	0	0
$757$	51.0829	1.85664	0.928319	−	0.371784i	$-0.121254\pi$
$757$	51.0829	1.85664	0.928319	+	0.371784i	$0.121254\pi$
$758$	−1.56463	−0.0568298
$759$	0	0
$760$	−27.6905	−1.00444
$761$	−30.5895	−1.10887	−0.554434	−	0.832228i	$-0.687065\pi$
$761$	−30.5895	−1.10887	−0.554434	+	0.832228i	$0.687065\pi$
$762$	0	0
$763$	32.0337	1.15970
$764$	21.3754	0.773335
$765$	0	0
$766$	−44.5562	−1.60988
$767$	−0.314771	−0.0113657
$768$	0	0
$769$	−3.27708	−0.118175	−0.0590873	−	0.998253i	$-0.518819\pi$
$769$	−3.27708	−0.118175	−0.0590873	+	0.998253i	$0.518819\pi$
$770$	83.3108	3.00231
$771$	0	0
$772$	−7.85447	−0.282688
$773$	−23.4943	−0.845030	−0.422515	−	0.906356i	$-0.638853\pi$
$773$	−23.4943	−0.845030	−0.422515	+	0.906356i	$0.638853\pi$
$774$	0	0
$775$	−6.01840	−0.216187
$776$	9.18306	0.329653
$777$	0	0
$778$	−23.1716	−0.830743
$779$	−30.4384	−1.09057
$780$	0	0
$781$	33.9804	1.21591
$782$	11.2643	0.402812
$783$	0	0
$784$	−81.6315	−2.91541
$785$	7.24091	0.258439
$786$	0	0
$787$	−5.98308	−0.213274	−0.106637	−	0.994298i	$-0.534008\pi$
$787$	−5.98308	−0.213274	−0.106637	+	0.994298i	$0.534008\pi$
$788$	−9.63699	−0.343303
$789$	0	0
$790$	16.8937	0.601051
$791$	−93.9424	−3.34021
$792$	0	0
$793$	0.515362	0.0183011
$794$	−67.5161	−2.39606
$795$	0	0
$796$	−16.8858	−0.598501
$797$	8.10954	0.287254	0.143627	−	0.989632i	$-0.454123\pi$
$797$	8.10954	0.287254	0.143627	+	0.989632i	$0.454123\pi$
$798$	0	0
$799$	7.89705	0.279378
$800$	64.2133	2.27028
$801$	0	0
$802$	−40.6430	−1.43515
$803$	7.08995	0.250199
$804$	0	0
$805$	19.1579	0.675228
$806$	−0.0500680	−0.00176357
$807$	0	0
$808$	−16.5115	−0.580873
$809$	−42.3498	−1.48894	−0.744470	−	0.667656i	$-0.767298\pi$
$809$	−42.3498	−1.48894	−0.744470	+	0.667656i	$0.767298\pi$
$810$	0	0
$811$	2.85674	0.100314	0.0501569	−	0.998741i	$-0.484028\pi$
$811$	2.85674	0.100314	0.0501569	+	0.998741i	$0.484028\pi$
$812$	−5.84418	−0.205090
$813$	0	0
$814$	0.934778	0.0327639
$815$	−11.9786	−0.419591
$816$	0	0
$817$	8.85763	0.309889
$818$	49.1632	1.71895
$819$	0	0
$820$	29.4738	1.02927
$821$	−6.13762	−0.214204	−0.107102	−	0.994248i	$-0.534157\pi$
$821$	−6.13762	−0.214204	−0.107102	+	0.994248i	$0.534157\pi$
$822$	0	0
$823$	57.2383	1.99520	0.997601	−	0.0692301i	$-0.0220543\pi$
$823$	57.2383	1.99520	0.997601	+	0.0692301i	$0.0220543\pi$
$824$	26.1809	0.912054
$825$	0	0
$826$	55.2339	1.92183
$827$	14.9731	0.520666	0.260333	−	0.965519i	$-0.416168\pi$
$827$	14.9731	0.520666	0.260333	+	0.965519i	$0.416168\pi$
$828$	0	0
$829$	23.8665	0.828916	0.414458	−	0.910068i	$-0.363971\pi$
$829$	23.8665	0.828916	0.414458	+	0.910068i	$0.363971\pi$
$830$	97.3118	3.37774
$831$	0	0
$832$	0.0440743	0.00152800
$833$	103.586	3.58904
$834$	0	0
$835$	11.9535	0.413667
$836$	−14.4360	−0.499280
$837$	0	0
$838$	56.6206	1.95593
$839$	−20.7204	−0.715346	−0.357673	−	0.933847i	$-0.616430\pi$
$839$	−20.7204	−0.715346	−0.357673	+	0.933847i	$0.616430\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−50.8374	−1.75197
$843$	0	0
$844$	−32.3647	−1.11404
$845$	−51.4026	−1.76830
$846$	0	0
$847$	24.7166	0.849272
$848$	−3.94023	−0.135308
$849$	0	0
$850$	−119.856	−4.11104
$851$	0.214959	0.00736870
$852$	0	0
$853$	17.6989	0.606000	0.303000	−	0.952991i	$-0.402012\pi$
$853$	17.6989	0.606000	0.303000	+	0.952991i	$0.402012\pi$
$854$	−90.4324	−3.09453
$855$	0	0
$856$	18.9581	0.647976
$857$	−36.3081	−1.24026	−0.620131	−	0.784498i	$-0.712920\pi$
$857$	−36.3081	−1.24026	−0.620131	+	0.784498i	$0.712920\pi$
$858$	0	0
$859$	−27.9935	−0.955127	−0.477564	−	0.878597i	$-0.658480\pi$
$859$	−27.9935	−0.955127	−0.477564	+	0.878597i	$0.658480\pi$
$860$	−8.57693	−0.292471
$861$	0	0
$862$	0.532064	0.0181222
$863$	−14.8824	−0.506603	−0.253301	−	0.967387i	$-0.581516\pi$
$863$	−14.8824	−0.506603	−0.253301	+	0.967387i	$0.581516\pi$
$864$	0	0
$865$	89.4015	3.03974
$866$	−72.6969	−2.47034
$867$	0	0
$868$	3.30559	0.112199
$869$	−5.79341	−0.196528
$870$	0	0
$871$	0.484338	0.0164112
$872$	−9.39686	−0.318218
$873$	0	0
$874$	−8.82301	−0.298443
$875$	−108.057	−3.65300
$876$	0	0
$877$	−35.3740	−1.19450	−0.597248	−	0.802057i	$-0.703739\pi$
$877$	−35.3740	−1.19450	−0.597248	+	0.802057i	$0.703739\pi$
$878$	−49.9655	−1.68626
$879$	0	0
$880$	−47.6123	−1.60501
$881$	−12.3559	−0.416281	−0.208140	−	0.978099i	$-0.566741\pi$
$881$	−12.3559	−0.416281	−0.208140	+	0.978099i	$0.566741\pi$
$882$	0	0
$883$	38.6094	1.29931	0.649655	−	0.760229i	$-0.274913\pi$
$883$	38.6094	1.29931	0.649655	+	0.760229i	$0.274913\pi$
$884$	−0.375161	−0.0126180
$885$	0	0
$886$	−0.909471	−0.0305543
$887$	35.9859	1.20829	0.604145	−	0.796875i	$-0.293515\pi$
$887$	35.9859	1.20829	0.604145	+	0.796875i	$0.293515\pi$
$888$	0	0
$889$	12.8918	0.432378
$890$	91.7418	3.07519
$891$	0	0
$892$	−35.1398	−1.17657
$893$	−6.18552	−0.206990
$894$	0	0
$895$	−68.7416	−2.29778
$896$	50.7350	1.69494
$897$	0	0
$898$	−58.9894	−1.96850
$899$	−0.565621	−0.0188645
$900$	0	0
$901$	4.99993	0.166572
$902$	−26.8638	−0.894467
$903$	0	0
$904$	27.5574	0.916545
$905$	69.4176	2.30752
$906$	0	0
$907$	−51.2057	−1.70026	−0.850129	−	0.526575i	$-0.823476\pi$
$907$	−51.2057	−1.70026	−0.850129	+	0.526575i	$0.823476\pi$
$908$	6.84047	0.227009
$909$	0	0
$910$	−1.69583	−0.0562163
$911$	−23.2835	−0.771418	−0.385709	−	0.922621i	$-0.626043\pi$
$911$	−23.2835	−0.771418	−0.385709	+	0.922621i	$0.626043\pi$
$912$	0	0
$913$	−33.3715	−1.10443
$914$	54.2045	1.79293
$915$	0	0
$916$	19.3150	0.638184
$917$	−66.4100	−2.19305
$918$	0	0
$919$	56.0808	1.84994	0.924968	−	0.380046i	$-0.124092\pi$
$919$	56.0808	1.84994	0.924968	+	0.380046i	$0.124092\pi$
$920$	−5.61985	−0.185281
$921$	0	0
$922$	−19.8951	−0.655211
$923$	−0.691688	−0.0227672
$924$	0	0
$925$	−2.28724	−0.0752039
$926$	−19.6342	−0.645221
$927$	0	0
$928$	6.03489	0.198105
$929$	29.1165	0.955282	0.477641	−	0.878555i	$-0.341492\pi$
$929$	29.1165	0.955282	0.477641	+	0.878555i	$0.341492\pi$
$930$	0	0
$931$	−81.1356	−2.65911
$932$	15.7607	0.516260
$933$	0	0
$934$	−72.1395	−2.36048
$935$	60.4173	1.97586
$936$	0	0
$937$	−41.1307	−1.34368	−0.671841	−	0.740695i	$-0.734496\pi$
$937$	−41.1307	−1.34368	−0.671841	+	0.740695i	$0.734496\pi$
$938$	−84.9884	−2.77497
$939$	0	0
$940$	5.98950	0.195356
$941$	−56.0895	−1.82846	−0.914232	−	0.405191i	$-0.867205\pi$
$941$	−56.0895	−1.82846	−0.914232	+	0.405191i	$0.867205\pi$
$942$	0	0
$943$	−6.17753	−0.201168
$944$	−31.5663	−1.02740
$945$	0	0
$946$	7.81742	0.254166
$947$	15.0925	0.490440	0.245220	−	0.969467i	$-0.421140\pi$
$947$	15.0925	0.490440	0.245220	+	0.969467i	$0.421140\pi$
$948$	0	0
$949$	−0.144319	−0.00468480
$950$	93.8798	3.04586
$951$	0	0
$952$	−43.3035	−1.40347
$953$	−43.9646	−1.42415	−0.712077	−	0.702101i	$-0.752245\pi$
$953$	−43.9646	−1.42415	−0.712077	+	0.702101i	$0.752245\pi$
$954$	0	0
$955$	70.0712	2.26745
$956$	16.9902	0.549502
$957$	0	0
$958$	−32.7106	−1.05683
$959$	−38.6075	−1.24670
$960$	0	0
$961$	−30.6801	−0.989680
$962$	−0.0190279	−0.000613483 0
$963$	0	0
$964$	−7.48984	−0.241231
$965$	−25.7479	−0.828854
$966$	0	0
$967$	29.9713	0.963811	0.481905	−	0.876223i	$-0.339945\pi$
$967$	29.9713	0.963811	0.481905	+	0.876223i	$0.339945\pi$
$968$	−7.25044	−0.233038
$969$	0	0
$970$	−45.7635	−1.46938
$971$	44.8838	1.44039	0.720194	−	0.693772i	$-0.244053\pi$
$971$	44.8838	1.44039	0.720194	+	0.693772i	$0.244053\pi$
$972$	0	0
$973$	−73.5943	−2.35932
$974$	17.6331	0.565002
$975$	0	0
$976$	51.6823	1.65431
$977$	−58.0461	−1.85706	−0.928530	−	0.371257i	$-0.878927\pi$
$977$	−58.0461	−1.85706	−0.928530	+	0.371257i	$0.878927\pi$
$978$	0	0
$979$	−31.4613	−1.00551
$980$	78.5644	2.50965
$981$	0	0
$982$	−20.1309	−0.642403
$983$	6.63594	0.211654	0.105827	−	0.994385i	$-0.466251\pi$
$983$	6.63594	0.211654	0.105827	+	0.994385i	$0.466251\pi$
$984$	0	0
$985$	−31.5912	−1.00658
$986$	−11.2643	−0.358730
$987$	0	0
$988$	0.293852	0.00934869
$989$	1.79767	0.0571627
$990$	0	0
$991$	29.0184	0.921799	0.460899	−	0.887452i	$-0.347527\pi$
$991$	29.0184	0.921799	0.460899	+	0.887452i	$0.347527\pi$
$992$	−3.41346	−0.108378
$993$	0	0
$994$	121.373	3.84971
$995$	−55.3536	−1.75483
$996$	0	0
$997$	−16.5119	−0.522936	−0.261468	−	0.965212i	$-0.584207\pi$
$997$	−16.5119	−0.522936	−0.261468	+	0.965212i	$0.584207\pi$
$998$	50.1357	1.58702
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.w.1.6	yes	30
3.2	odd	2		6003.2.a.v.1.25	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6003.2.a.v.1.25	✓	30	3.2	odd	2
6003.2.a.w.1.6	yes	30	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79065 q^{2} +1.20642 q^{4} +3.95479 q^{5} -4.84424 q^{7} +1.42102 q^{8} +O(q^{10})\)	\(q-1.79065 q^{2} +1.20642 q^{4} +3.95479 q^{5} -4.84424 q^{7} +1.42102 q^{8} -7.08163 q^{10} +2.42853 q^{11} -0.0494338 q^{13} +8.67432 q^{14} -4.95739 q^{16} +6.29065 q^{17} -4.92728 q^{19} +4.77113 q^{20} -4.34863 q^{22} -1.00000 q^{23} +10.6403 q^{25} +0.0885186 q^{26} -5.84418 q^{28} +1.00000 q^{29} -0.565621 q^{31} +6.03489 q^{32} -11.2643 q^{34} -19.1579 q^{35} -0.214959 q^{37} +8.82301 q^{38} +5.61985 q^{40} +6.17753 q^{41} -1.79767 q^{43} +2.92982 q^{44} +1.79065 q^{46} +1.25536 q^{47} +16.4666 q^{49} -19.0531 q^{50} -0.0596379 q^{52} +0.794820 q^{53} +9.60430 q^{55} -6.88378 q^{56} -1.79065 q^{58} +6.36752 q^{59} -10.4253 q^{61} +1.01283 q^{62} -0.891582 q^{64} -0.195500 q^{65} -9.79770 q^{67} +7.58916 q^{68} +34.3051 q^{70} +13.9922 q^{71} +2.91945 q^{73} +0.384916 q^{74} -5.94436 q^{76} -11.7643 q^{77} -2.38557 q^{79} -19.6054 q^{80} -11.0618 q^{82} -13.7414 q^{83} +24.8782 q^{85} +3.21900 q^{86} +3.45099 q^{88} -12.9549 q^{89} +0.239469 q^{91} -1.20642 q^{92} -2.24791 q^{94} -19.4863 q^{95} +6.46228 q^{97} -29.4859 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8}+O(q^{10}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8	\( 30 q + q^{2} + 37 q^{4} + 10 q^{7} + 6 q^{8} + 8 q^{10} + 36 q^{13} + 7 q^{14} + 47 q^{16} + 18 q^{17} + 16 q^{19} + 25 q^{22} - 30 q^{23} + 56 q^{25} + 11 q^{26} + 27 q^{28} + 30 q^{29} + 14 q^{31} - 7 q^{32} + 3 q^{34} - 22 q^{35} + 40 q^{37} + 6 q^{38} + 30 q^{40} + 14 q^{41} + 34 q^{43} + 5 q^{44} - q^{46} - 2 q^{47} + 74 q^{49} - 21 q^{50} + 71 q^{52} + 16 q^{53} + 22 q^{55} + 14 q^{56} + q^{58} - 32 q^{59} + 46 q^{61} + 20 q^{62} + 68 q^{64} + 12 q^{65} + 14 q^{67} + 27 q^{68} - 32 q^{71} + 50 q^{73} - 26 q^{74} + 56 q^{76} + 34 q^{77} + 16 q^{79} + 2 q^{80} + 38 q^{82} - 14 q^{83} + 38 q^{85} + 10 q^{86} + 40 q^{88} - 2 q^{89} + 32 q^{91} - 37 q^{92} + 29 q^{94} - 28 q^{95} + 56 q^{97} + 8 q^{98}+O(q^{100}) \) 30 * q + q^2 + 37 * q^4 + 10 * q^7 + 6 * q^8 + 8 * q^10 + 36 * q^13 + 7 * q^14 + 47 * q^16 + 18 * q^17 + 16 * q^19 + 25 * q^22 - 30 * q^23 + 56 * q^25 + 11 * q^26 + 27 * q^28 + 30 * q^29 + 14 * q^31 - 7 * q^32 + 3 * q^34 - 22 * q^35 + 40 * q^37 + 6 * q^38 + 30 * q^40 + 14 * q^41 + 34 * q^43 + 5 * q^44 - q^46 - 2 * q^47 + 74 * q^49 - 21 * q^50 + 71 * q^52 + 16 * q^53 + 22 * q^55 + 14 * q^56 + q^58 - 32 * q^59 + 46 * q^61 + 20 * q^62 + 68 * q^64 + 12 * q^65 + 14 * q^67 + 27 * q^68 - 32 * q^71 + 50 * q^73 - 26 * q^74 + 56 * q^76 + 34 * q^77 + 16 * q^79 + 2 * q^80 + 38 * q^82 - 14 * q^83 + 38 * q^85 + 10 * q^86 + 40 * q^88 - 2 * q^89 + 32 * q^91 - 37 * q^92 + 29 * q^94 - 28 * q^95 + 56 * q^97 + 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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