LMFDB - Embedded newform 8024.2.a.bc.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56116 q^{3} +0.330730 q^{5} -0.779452 q^{7} +3.55957 q^{9} +O(q^{10})$	$q-2.56116 q^{3} +0.330730 q^{5} -0.779452 q^{7} +3.55957 q^{9} -5.15394 q^{11} +5.72913 q^{13} -0.847053 q^{15} -1.00000 q^{17} -2.03432 q^{19} +1.99630 q^{21} -2.48300 q^{23} -4.89062 q^{25} -1.43314 q^{27} -7.11605 q^{29} -7.80592 q^{31} +13.2001 q^{33} -0.257788 q^{35} +2.75688 q^{37} -14.6733 q^{39} +3.44404 q^{41} +7.75707 q^{43} +1.17725 q^{45} +6.76229 q^{47} -6.39246 q^{49} +2.56116 q^{51} +11.0153 q^{53} -1.70456 q^{55} +5.21023 q^{57} -1.00000 q^{59} -1.33446 q^{61} -2.77451 q^{63} +1.89479 q^{65} -4.09046 q^{67} +6.35936 q^{69} -12.6098 q^{71} +7.56444 q^{73} +12.5257 q^{75} +4.01725 q^{77} -5.69070 q^{79} -7.00819 q^{81} -6.36843 q^{83} -0.330730 q^{85} +18.2254 q^{87} -1.57192 q^{89} -4.46558 q^{91} +19.9923 q^{93} -0.672811 q^{95} -1.93780 q^{97} -18.3458 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	$ 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) $ 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * 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
$3$	−2.56116	−1.47869	−0.739345	−	0.673327i	$-0.764865\pi$
$3$	−2.56116	−1.47869	−0.739345	+	0.673327i	$0.764865\pi$
$4$	0	0
$5$	0.330730	0.147907	0.0739534	−	0.997262i	$-0.476438\pi$
$5$	0.330730	0.147907	0.0739534	+	0.997262i	$0.476438\pi$
$6$	0	0
$7$	−0.779452	−0.294605	−0.147302	−	0.989091i	$-0.547059\pi$
$7$	−0.779452	−0.294605	−0.147302	+	0.989091i	$0.547059\pi$
$8$	0	0
$9$	3.55957	1.18652
$10$	0	0
$11$	−5.15394	−1.55397	−0.776986	−	0.629518i	$-0.783253\pi$
$11$	−5.15394	−1.55397	−0.776986	+	0.629518i	$0.783253\pi$
$12$	0	0
$13$	5.72913	1.58898	0.794488	−	0.607280i	$-0.207739\pi$
$13$	5.72913	1.58898	0.794488	+	0.607280i	$0.207739\pi$
$14$	0	0
$15$	−0.847053	−0.218708
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.03432	−0.466705	−0.233353	−	0.972392i	$-0.574970\pi$
$19$	−2.03432	−0.466705	−0.233353	+	0.972392i	$0.574970\pi$
$20$	0	0
$21$	1.99630	0.435629
$22$	0	0
$23$	−2.48300	−0.517740	−0.258870	−	0.965912i	$-0.583350\pi$
$23$	−2.48300	−0.517740	−0.258870	+	0.965912i	$0.583350\pi$
$24$	0	0
$25$	−4.89062	−0.978124
$26$	0	0
$27$	−1.43314	−0.275808
$28$	0	0
$29$	−7.11605	−1.32142	−0.660709	−	0.750642i	$-0.729744\pi$
$29$	−7.11605	−1.32142	−0.660709	+	0.750642i	$0.729744\pi$
$30$	0	0
$31$	−7.80592	−1.40199	−0.700993	−	0.713168i	$-0.747259\pi$
$31$	−7.80592	−1.40199	−0.700993	+	0.713168i	$0.747259\pi$
$32$	0	0
$33$	13.2001	2.29784
$34$	0	0
$35$	−0.257788	−0.0435741
$36$	0	0
$37$	2.75688	0.453229	0.226614	−	0.973985i	$-0.427234\pi$
$37$	2.75688	0.453229	0.226614	+	0.973985i	$0.427234\pi$
$38$	0	0
$39$	−14.6733	−2.34960
$40$	0	0
$41$	3.44404	0.537869	0.268934	−	0.963159i	$-0.413329\pi$
$41$	3.44404	0.537869	0.268934	+	0.963159i	$0.413329\pi$
$42$	0	0
$43$	7.75707	1.18294	0.591471	−	0.806326i	$-0.298548\pi$
$43$	7.75707	1.18294	0.591471	+	0.806326i	$0.298548\pi$
$44$	0	0
$45$	1.17725	0.175495
$46$	0	0
$47$	6.76229	0.986381	0.493190	−	0.869921i	$-0.335831\pi$
$47$	6.76229	0.986381	0.493190	+	0.869921i	$0.335831\pi$
$48$	0	0
$49$	−6.39246	−0.913208
$50$	0	0
$51$	2.56116	0.358635
$52$	0	0
$53$	11.0153	1.51306	0.756531	−	0.653957i	$-0.226892\pi$
$53$	11.0153	1.51306	0.756531	+	0.653957i	$0.226892\pi$
$54$	0	0
$55$	−1.70456	−0.229843
$56$	0	0
$57$	5.21023	0.690112
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−1.33446	−0.170860	−0.0854302	−	0.996344i	$-0.527226\pi$
$61$	−1.33446	−0.170860	−0.0854302	+	0.996344i	$0.527226\pi$
$62$	0	0
$63$	−2.77451	−0.349555
$64$	0	0
$65$	1.89479	0.235020
$66$	0	0
$67$	−4.09046	−0.499729	−0.249865	−	0.968281i	$-0.580386\pi$
$67$	−4.09046	−0.499729	−0.249865	+	0.968281i	$0.580386\pi$
$68$	0	0
$69$	6.35936	0.765577
$70$	0	0
$71$	−12.6098	−1.49651	−0.748253	−	0.663413i	$-0.769107\pi$
$71$	−12.6098	−1.49651	−0.748253	+	0.663413i	$0.769107\pi$
$72$	0	0
$73$	7.56444	0.885351	0.442675	−	0.896682i	$-0.354029\pi$
$73$	7.56444	0.885351	0.442675	+	0.896682i	$0.354029\pi$
$74$	0	0
$75$	12.5257	1.44634
$76$	0	0
$77$	4.01725	0.457808
$78$	0	0
$79$	−5.69070	−0.640253	−0.320127	−	0.947375i	$-0.603725\pi$
$79$	−5.69070	−0.640253	−0.320127	+	0.947375i	$0.603725\pi$
$80$	0	0
$81$	−7.00819	−0.778688
$82$	0	0
$83$	−6.36843	−0.699026	−0.349513	−	0.936931i	$-0.613653\pi$
$83$	−6.36843	−0.699026	−0.349513	+	0.936931i	$0.613653\pi$
$84$	0	0
$85$	−0.330730	−0.0358727
$86$	0	0
$87$	18.2254	1.95397
$88$	0	0
$89$	−1.57192	−0.166623	−0.0833116	−	0.996524i	$-0.526550\pi$
$89$	−1.57192	−0.166623	−0.0833116	+	0.996524i	$0.526550\pi$
$90$	0	0
$91$	−4.46558	−0.468120
$92$	0	0
$93$	19.9923	2.07310
$94$	0	0
$95$	−0.672811	−0.0690289
$96$	0	0
$97$	−1.93780	−0.196754	−0.0983770	−	0.995149i	$-0.531365\pi$
$97$	−1.93780	−0.196754	−0.0983770	+	0.995149i	$0.531365\pi$
$98$	0	0
$99$	−18.3458	−1.84382
$100$	0	0
$101$	−11.5089	−1.14517	−0.572587	−	0.819844i	$-0.694060\pi$
$101$	−11.5089	−1.14517	−0.572587	+	0.819844i	$0.694060\pi$
$102$	0	0
$103$	−16.4051	−1.61644	−0.808222	−	0.588878i	$-0.799570\pi$
$103$	−16.4051	−1.61644	−0.808222	+	0.588878i	$0.799570\pi$
$104$	0	0
$105$	0.660237	0.0644325
$106$	0	0
$107$	10.6161	1.02630	0.513148	−	0.858300i	$-0.328479\pi$
$107$	10.6161	1.02630	0.513148	+	0.858300i	$0.328479\pi$
$108$	0	0
$109$	−4.25776	−0.407820	−0.203910	−	0.978990i	$-0.565365\pi$
$109$	−4.25776	−0.407820	−0.203910	+	0.978990i	$0.565365\pi$
$110$	0	0
$111$	−7.06083	−0.670185
$112$	0	0
$113$	11.7476	1.10512	0.552561	−	0.833472i	$-0.313650\pi$
$113$	11.7476	1.10512	0.552561	+	0.833472i	$0.313650\pi$
$114$	0	0
$115$	−0.821200	−0.0765773
$116$	0	0
$117$	20.3932	1.88535
$118$	0	0
$119$	0.779452	0.0714522
$120$	0	0
$121$	15.5631	1.41483
$122$	0	0
$123$	−8.82075	−0.795341
$124$	0	0
$125$	−3.27112	−0.292578
$126$	0	0
$127$	16.0079	1.42047	0.710236	−	0.703964i	$-0.248588\pi$
$127$	16.0079	1.42047	0.710236	+	0.703964i	$0.248588\pi$
$128$	0	0
$129$	−19.8671	−1.74920
$130$	0	0
$131$	−12.7605	−1.11489	−0.557446	−	0.830213i	$-0.688219\pi$
$131$	−12.7605	−1.11489	−0.557446	+	0.830213i	$0.688219\pi$
$132$	0	0
$133$	1.58566	0.137494
$134$	0	0
$135$	−0.473982	−0.0407939
$136$	0	0
$137$	−0.848842	−0.0725215	−0.0362607	−	0.999342i	$-0.511545\pi$
$137$	−0.848842	−0.0725215	−0.0362607	+	0.999342i	$0.511545\pi$
$138$	0	0
$139$	−1.10356	−0.0936024	−0.0468012	−	0.998904i	$-0.514903\pi$
$139$	−1.10356	−0.0936024	−0.0468012	+	0.998904i	$0.514903\pi$
$140$	0	0
$141$	−17.3193	−1.45855
$142$	0	0
$143$	−29.5276	−2.46923
$144$	0	0
$145$	−2.35349	−0.195447
$146$	0	0
$147$	16.3721	1.35035
$148$	0	0
$149$	−3.47127	−0.284377	−0.142189	−	0.989840i	$-0.545414\pi$
$149$	−3.47127	−0.284377	−0.142189	+	0.989840i	$0.545414\pi$
$150$	0	0
$151$	19.4358	1.58166	0.790831	−	0.612034i	$-0.209648\pi$
$151$	19.4358	1.58166	0.790831	+	0.612034i	$0.209648\pi$
$152$	0	0
$153$	−3.55957	−0.287774
$154$	0	0
$155$	−2.58165	−0.207363
$156$	0	0
$157$	−15.3085	−1.22175	−0.610875	−	0.791727i	$-0.709182\pi$
$157$	−15.3085	−1.22175	−0.610875	+	0.791727i	$0.709182\pi$
$158$	0	0
$159$	−28.2119	−2.23735
$160$	0	0
$161$	1.93538	0.152529
$162$	0	0
$163$	−22.1733	−1.73675	−0.868373	−	0.495912i	$-0.834834\pi$
$163$	−22.1733	−1.73675	−0.868373	+	0.495912i	$0.834834\pi$
$164$	0	0
$165$	4.36566	0.339867
$166$	0	0
$167$	23.3487	1.80677	0.903387	−	0.428827i	$-0.141073\pi$
$167$	23.3487	1.80677	0.903387	+	0.428827i	$0.141073\pi$
$168$	0	0
$169$	19.8230	1.52484
$170$	0	0
$171$	−7.24130	−0.553756
$172$	0	0
$173$	−10.8988	−0.828620	−0.414310	−	0.910136i	$-0.635977\pi$
$173$	−10.8988	−0.828620	−0.414310	+	0.910136i	$0.635977\pi$
$174$	0	0
$175$	3.81200	0.288160
$176$	0	0
$177$	2.56116	0.192509
$178$	0	0
$179$	−14.4950	−1.08341	−0.541704	−	0.840570i	$-0.682220\pi$
$179$	−14.4950	−1.08341	−0.541704	+	0.840570i	$0.682220\pi$
$180$	0	0
$181$	−9.42242	−0.700363	−0.350182	−	0.936682i	$-0.613880\pi$
$181$	−9.42242	−0.700363	−0.350182	+	0.936682i	$0.613880\pi$
$182$	0	0
$183$	3.41778	0.252649
$184$	0	0
$185$	0.911783	0.0670356
$186$	0	0
$187$	5.15394	0.376894
$188$	0	0
$189$	1.11706	0.0812544
$190$	0	0
$191$	4.30446	0.311460	0.155730	−	0.987800i	$-0.450227\pi$
$191$	4.30446	0.311460	0.155730	+	0.987800i	$0.450227\pi$
$192$	0	0
$193$	14.0772	1.01330	0.506650	−	0.862152i	$-0.330884\pi$
$193$	14.0772	1.01330	0.506650	+	0.862152i	$0.330884\pi$
$194$	0	0
$195$	−4.85288	−0.347522
$196$	0	0
$197$	−9.71437	−0.692120	−0.346060	−	0.938212i	$-0.612481\pi$
$197$	−9.71437	−0.692120	−0.346060	+	0.938212i	$0.612481\pi$
$198$	0	0
$199$	14.4893	1.02712	0.513561	−	0.858053i	$-0.328326\pi$
$199$	14.4893	1.02712	0.513561	+	0.858053i	$0.328326\pi$
$200$	0	0
$201$	10.4763	0.738944
$202$	0	0
$203$	5.54662	0.389296
$204$	0	0
$205$	1.13905	0.0795544
$206$	0	0
$207$	−8.83839	−0.614310
$208$	0	0
$209$	10.4848	0.725248
$210$	0	0
$211$	−3.51527	−0.242001	−0.121000	−	0.992652i	$-0.538610\pi$
$211$	−3.51527	−0.242001	−0.121000	+	0.992652i	$0.538610\pi$
$212$	0	0
$213$	32.2957	2.21287
$214$	0	0
$215$	2.56549	0.174965
$216$	0	0
$217$	6.08434	0.413032
$218$	0	0
$219$	−19.3738	−1.30916
$220$	0	0
$221$	−5.72913	−0.385383
$222$	0	0
$223$	19.2189	1.28699	0.643496	−	0.765450i	$-0.277483\pi$
$223$	19.2189	1.28699	0.643496	+	0.765450i	$0.277483\pi$
$224$	0	0
$225$	−17.4085	−1.16057
$226$	0	0
$227$	−22.3637	−1.48433	−0.742164	−	0.670218i	$-0.766201\pi$
$227$	−22.3637	−1.48433	−0.742164	+	0.670218i	$0.766201\pi$
$228$	0	0
$229$	14.2906	0.944349	0.472174	−	0.881505i	$-0.343469\pi$
$229$	14.2906	0.944349	0.472174	+	0.881505i	$0.343469\pi$
$230$	0	0
$231$	−10.2888	−0.676956
$232$	0	0
$233$	−12.1297	−0.794641	−0.397321	−	0.917680i	$-0.630060\pi$
$233$	−12.1297	−0.794641	−0.397321	+	0.917680i	$0.630060\pi$
$234$	0	0
$235$	2.23649	0.145892
$236$	0	0
$237$	14.5748	0.946736
$238$	0	0
$239$	18.7994	1.21603	0.608015	−	0.793926i	$-0.291966\pi$
$239$	18.7994	1.21603	0.608015	+	0.793926i	$0.291966\pi$
$240$	0	0
$241$	−4.34517	−0.279897	−0.139948	−	0.990159i	$-0.544694\pi$
$241$	−4.34517	−0.279897	−0.139948	+	0.990159i	$0.544694\pi$
$242$	0	0
$243$	22.2485	1.42725
$244$	0	0
$245$	−2.11417	−0.135070
$246$	0	0
$247$	−11.6549	−0.741584
$248$	0	0
$249$	16.3106	1.03364
$250$	0	0
$251$	−21.7464	−1.37262	−0.686309	−	0.727310i	$-0.740770\pi$
$251$	−21.7464	−1.37262	−0.686309	+	0.727310i	$0.740770\pi$
$252$	0	0
$253$	12.7972	0.804555
$254$	0	0
$255$	0.847053	0.0530445
$256$	0	0
$257$	−6.11283	−0.381308	−0.190654	−	0.981657i	$-0.561061\pi$
$257$	−6.11283	−0.381308	−0.190654	+	0.981657i	$0.561061\pi$
$258$	0	0
$259$	−2.14886	−0.133523
$260$	0	0
$261$	−25.3300	−1.56789
$262$	0	0
$263$	13.7263	0.846398	0.423199	−	0.906037i	$-0.360907\pi$
$263$	13.7263	0.846398	0.423199	+	0.906037i	$0.360907\pi$
$264$	0	0
$265$	3.64307	0.223792
$266$	0	0
$267$	4.02595	0.246384
$268$	0	0
$269$	24.0581	1.46685	0.733425	−	0.679770i	$-0.237921\pi$
$269$	24.0581	1.46685	0.733425	+	0.679770i	$0.237921\pi$
$270$	0	0
$271$	29.6780	1.80281	0.901406	−	0.432974i	$-0.142536\pi$
$271$	29.6780	1.80281	0.901406	+	0.432974i	$0.142536\pi$
$272$	0	0
$273$	11.4371	0.692204
$274$	0	0
$275$	25.2060	1.51998
$276$	0	0
$277$	2.05214	0.123301	0.0616505	−	0.998098i	$-0.480364\pi$
$277$	2.05214	0.123301	0.0616505	+	0.998098i	$0.480364\pi$
$278$	0	0
$279$	−27.7857	−1.66349
$280$	0	0
$281$	6.21504	0.370758	0.185379	−	0.982667i	$-0.440649\pi$
$281$	6.21504	0.370758	0.185379	+	0.982667i	$0.440649\pi$
$282$	0	0
$283$	−16.3428	−0.971478	−0.485739	−	0.874104i	$-0.661450\pi$
$283$	−16.3428	−0.971478	−0.485739	+	0.874104i	$0.661450\pi$
$284$	0	0
$285$	1.72318	0.102072
$286$	0	0
$287$	−2.68446	−0.158459
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	4.96303	0.290938
$292$	0	0
$293$	−33.4965	−1.95689	−0.978444	−	0.206513i	$-0.933788\pi$
$293$	−33.4965	−1.95689	−0.978444	+	0.206513i	$0.933788\pi$
$294$	0	0
$295$	−0.330730	−0.0192558
$296$	0	0
$297$	7.38632	0.428598
$298$	0	0
$299$	−14.2254	−0.822677
$300$	0	0
$301$	−6.04626	−0.348500
$302$	0	0
$303$	29.4761	1.69336
$304$	0	0
$305$	−0.441346	−0.0252714
$306$	0	0
$307$	32.4951	1.85459	0.927296	−	0.374329i	$-0.122127\pi$
$307$	32.4951	1.85459	0.927296	+	0.374329i	$0.122127\pi$
$308$	0	0
$309$	42.0162	2.39022
$310$	0	0
$311$	−2.65204	−0.150383	−0.0751916	−	0.997169i	$-0.523957\pi$
$311$	−2.65204	−0.150383	−0.0751916	+	0.997169i	$0.523957\pi$
$312$	0	0
$313$	26.7491	1.51195	0.755975	−	0.654601i	$-0.227163\pi$
$313$	26.7491	1.51195	0.755975	+	0.654601i	$0.227163\pi$
$314$	0	0
$315$	−0.917612	−0.0517016
$316$	0	0
$317$	4.11861	0.231324	0.115662	−	0.993289i	$-0.463101\pi$
$317$	4.11861	0.231324	0.115662	+	0.993289i	$0.463101\pi$
$318$	0	0
$319$	36.6757	2.05345
$320$	0	0
$321$	−27.1896	−1.51757
$322$	0	0
$323$	2.03432	0.113193
$324$	0	0
$325$	−28.0190	−1.55421
$326$	0	0
$327$	10.9048	0.603039
$328$	0	0
$329$	−5.27087	−0.290593
$330$	0	0
$331$	7.91605	0.435105	0.217553	−	0.976049i	$-0.430193\pi$
$331$	7.91605	0.435105	0.217553	+	0.976049i	$0.430193\pi$
$332$	0	0
$333$	9.81331	0.537766
$334$	0	0
$335$	−1.35284	−0.0739133
$336$	0	0
$337$	7.55239	0.411405	0.205702	−	0.978615i	$-0.434052\pi$
$337$	7.55239	0.411405	0.205702	+	0.978615i	$0.434052\pi$
$338$	0	0
$339$	−30.0876	−1.63413
$340$	0	0
$341$	40.2313	2.17865
$342$	0	0
$343$	10.4388	0.563641
$344$	0	0
$345$	2.10323	0.113234
$346$	0	0
$347$	−13.2616	−0.711920	−0.355960	−	0.934501i	$-0.615846\pi$
$347$	−13.2616	−0.711920	−0.355960	+	0.934501i	$0.615846\pi$
$348$	0	0
$349$	15.9382	0.853153	0.426577	−	0.904451i	$-0.359719\pi$
$349$	15.9382	0.853153	0.426577	+	0.904451i	$0.359719\pi$
$350$	0	0
$351$	−8.21065	−0.438252
$352$	0	0
$353$	5.11401	0.272191	0.136096	−	0.990696i	$-0.456545\pi$
$353$	5.11401	0.272191	0.136096	+	0.990696i	$0.456545\pi$
$354$	0	0
$355$	−4.17043	−0.221343
$356$	0	0
$357$	−1.99630	−0.105656
$358$	0	0
$359$	34.7764	1.83543	0.917715	−	0.397241i	$-0.130032\pi$
$359$	34.7764	1.83543	0.917715	+	0.397241i	$0.130032\pi$
$360$	0	0
$361$	−14.8615	−0.782186
$362$	0	0
$363$	−39.8598	−2.09210
$364$	0	0
$365$	2.50179	0.130949
$366$	0	0
$367$	16.5925	0.866121	0.433061	−	0.901365i	$-0.357434\pi$
$367$	16.5925	0.866121	0.433061	+	0.901365i	$0.357434\pi$
$368$	0	0
$369$	12.2593	0.638193
$370$	0	0
$371$	−8.58586	−0.445756
$372$	0	0
$373$	−11.1068	−0.575090	−0.287545	−	0.957767i	$-0.592839\pi$
$373$	−11.1068	−0.575090	−0.287545	+	0.957767i	$0.592839\pi$
$374$	0	0
$375$	8.37788	0.432632
$376$	0	0
$377$	−40.7688	−2.09970
$378$	0	0
$379$	23.8738	1.22631	0.613157	−	0.789961i	$-0.289899\pi$
$379$	23.8738	1.22631	0.613157	+	0.789961i	$0.289899\pi$
$380$	0	0
$381$	−40.9989	−2.10044
$382$	0	0
$383$	−10.0375	−0.512891	−0.256445	−	0.966559i	$-0.582551\pi$
$383$	−10.0375	−0.512891	−0.256445	+	0.966559i	$0.582551\pi$
$384$	0	0
$385$	1.32862	0.0677129
$386$	0	0
$387$	27.6118	1.40359
$388$	0	0
$389$	11.4882	0.582473	0.291237	−	0.956651i	$-0.405933\pi$
$389$	11.4882	0.582473	0.291237	+	0.956651i	$0.405933\pi$
$390$	0	0
$391$	2.48300	0.125571
$392$	0	0
$393$	32.6818	1.64858
$394$	0	0
$395$	−1.88208	−0.0946978
$396$	0	0
$397$	22.9943	1.15405	0.577027	−	0.816725i	$-0.304213\pi$
$397$	22.9943	1.15405	0.577027	+	0.816725i	$0.304213\pi$
$398$	0	0
$399$	−4.06112	−0.203311
$400$	0	0
$401$	5.64807	0.282051	0.141026	−	0.990006i	$-0.454960\pi$
$401$	5.64807	0.282051	0.141026	+	0.990006i	$0.454960\pi$
$402$	0	0
$403$	−44.7212	−2.22772
$404$	0	0
$405$	−2.31782	−0.115173
$406$	0	0
$407$	−14.2088	−0.704305
$408$	0	0
$409$	−24.3958	−1.20630	−0.603148	−	0.797630i	$-0.706087\pi$
$409$	−24.3958	−1.20630	−0.603148	+	0.797630i	$0.706087\pi$
$410$	0	0
$411$	2.17403	0.107237
$412$	0	0
$413$	0.779452	0.0383543
$414$	0	0
$415$	−2.10623	−0.103391
$416$	0	0
$417$	2.82639	0.138409
$418$	0	0
$419$	16.5195	0.807030	0.403515	−	0.914973i	$-0.367788\pi$
$419$	16.5195	0.807030	0.403515	+	0.914973i	$0.367788\pi$
$420$	0	0
$421$	−5.15060	−0.251025	−0.125512	−	0.992092i	$-0.540058\pi$
$421$	−5.15060	−0.251025	−0.125512	+	0.992092i	$0.540058\pi$
$422$	0	0
$423$	24.0708	1.17036
$424$	0	0
$425$	4.89062	0.237230
$426$	0	0
$427$	1.04015	0.0503363
$428$	0	0
$429$	75.6251	3.65122
$430$	0	0
$431$	−31.9381	−1.53841	−0.769203	−	0.639004i	$-0.779347\pi$
$431$	−31.9381	−1.53841	−0.769203	+	0.639004i	$0.779347\pi$
$432$	0	0
$433$	−13.9442	−0.670117	−0.335059	−	0.942197i	$-0.608756\pi$
$433$	−13.9442	−0.670117	−0.335059	+	0.942197i	$0.608756\pi$
$434$	0	0
$435$	6.02767	0.289005
$436$	0	0
$437$	5.05121	0.241632
$438$	0	0
$439$	13.1893	0.629490	0.314745	−	0.949176i	$-0.398081\pi$
$439$	13.1893	0.629490	0.314745	+	0.949176i	$0.398081\pi$
$440$	0	0
$441$	−22.7544	−1.08354
$442$	0	0
$443$	4.69376	0.223007	0.111504	−	0.993764i	$-0.464433\pi$
$443$	4.69376	0.223007	0.111504	+	0.993764i	$0.464433\pi$
$444$	0	0
$445$	−0.519881	−0.0246447
$446$	0	0
$447$	8.89049	0.420506
$448$	0	0
$449$	−14.6667	−0.692165	−0.346083	−	0.938204i	$-0.612488\pi$
$449$	−14.6667	−0.692165	−0.346083	+	0.938204i	$0.612488\pi$
$450$	0	0
$451$	−17.7504	−0.835833
$452$	0	0
$453$	−49.7783	−2.33879
$454$	0	0
$455$	−1.47690	−0.0692382
$456$	0	0
$457$	36.2847	1.69733	0.848664	−	0.528932i	$-0.177407\pi$
$457$	36.2847	1.69733	0.848664	+	0.528932i	$0.177407\pi$
$458$	0	0
$459$	1.43314	0.0668932
$460$	0	0
$461$	24.2140	1.12776	0.563880	−	0.825857i	$-0.309308\pi$
$461$	24.2140	1.12776	0.563880	+	0.825857i	$0.309308\pi$
$462$	0	0
$463$	14.0166	0.651405	0.325702	−	0.945472i	$-0.394399\pi$
$463$	14.0166	0.651405	0.325702	+	0.945472i	$0.394399\pi$
$464$	0	0
$465$	6.61203	0.306626
$466$	0	0
$467$	37.8694	1.75238	0.876192	−	0.481962i	$-0.160076\pi$
$467$	37.8694	1.75238	0.876192	+	0.481962i	$0.160076\pi$
$468$	0	0
$469$	3.18832	0.147223
$470$	0	0
$471$	39.2075	1.80659
$472$	0	0
$473$	−39.9795	−1.83826
$474$	0	0
$475$	9.94909	0.456496
$476$	0	0
$477$	39.2095	1.79528
$478$	0	0
$479$	−36.0308	−1.64629	−0.823145	−	0.567831i	$-0.807783\pi$
$479$	−36.0308	−1.64629	−0.823145	+	0.567831i	$0.807783\pi$
$480$	0	0
$481$	15.7946	0.720170
$482$	0	0
$483$	−4.95681	−0.225543
$484$	0	0
$485$	−0.640888	−0.0291012
$486$	0	0
$487$	28.7231	1.30157	0.650785	−	0.759262i	$-0.274440\pi$
$487$	28.7231	1.30157	0.650785	+	0.759262i	$0.274440\pi$
$488$	0	0
$489$	56.7894	2.56811
$490$	0	0
$491$	7.22968	0.326271	0.163135	−	0.986604i	$-0.447839\pi$
$491$	7.22968	0.326271	0.163135	+	0.986604i	$0.447839\pi$
$492$	0	0
$493$	7.11605	0.320491
$494$	0	0
$495$	−6.06750	−0.272714
$496$	0	0
$497$	9.82872	0.440878
$498$	0	0
$499$	4.81808	0.215687	0.107843	−	0.994168i	$-0.465606\pi$
$499$	4.81808	0.215687	0.107843	+	0.994168i	$0.465606\pi$
$500$	0	0
$501$	−59.7998	−2.67166
$502$	0	0
$503$	−2.50528	−0.111705	−0.0558524	−	0.998439i	$-0.517788\pi$
$503$	−2.50528	−0.111705	−0.0558524	+	0.998439i	$0.517788\pi$
$504$	0	0
$505$	−3.80632	−0.169379
$506$	0	0
$507$	−50.7699	−2.25477
$508$	0	0
$509$	22.3734	0.991685	0.495843	−	0.868412i	$-0.334859\pi$
$509$	22.3734	0.991685	0.495843	+	0.868412i	$0.334859\pi$
$510$	0	0
$511$	−5.89612	−0.260829
$512$	0	0
$513$	2.91547	0.128721
$514$	0	0
$515$	−5.42566	−0.239083
$516$	0	0
$517$	−34.8524	−1.53281
$518$	0	0
$519$	27.9136	1.22527
$520$	0	0
$521$	27.8899	1.22188	0.610939	−	0.791678i	$-0.290792\pi$
$521$	27.8899	1.22188	0.610939	+	0.791678i	$0.290792\pi$
$522$	0	0
$523$	−2.20344	−0.0963498	−0.0481749	−	0.998839i	$-0.515340\pi$
$523$	−2.20344	−0.0963498	−0.0481749	+	0.998839i	$0.515340\pi$
$524$	0	0
$525$	−9.76316	−0.426099
$526$	0	0
$527$	7.80592	0.340031
$528$	0	0
$529$	−16.8347	−0.731945
$530$	0	0
$531$	−3.55957	−0.154472
$532$	0	0
$533$	19.7314	0.854660
$534$	0	0
$535$	3.51106	0.151796
$536$	0	0
$537$	37.1241	1.60202
$538$	0	0
$539$	32.9464	1.41910
$540$	0	0
$541$	35.5768	1.52957	0.764783	−	0.644288i	$-0.222846\pi$
$541$	35.5768	1.52957	0.764783	+	0.644288i	$0.222846\pi$
$542$	0	0
$543$	24.1324	1.03562
$544$	0	0
$545$	−1.40817	−0.0603193
$546$	0	0
$547$	−10.4438	−0.446546	−0.223273	−	0.974756i	$-0.571674\pi$
$547$	−10.4438	−0.446546	−0.223273	+	0.974756i	$0.571674\pi$
$548$	0	0
$549$	−4.75011	−0.202730
$550$	0	0
$551$	14.4763	0.616713
$552$	0	0
$553$	4.43562	0.188622
$554$	0	0
$555$	−2.33523	−0.0991248
$556$	0	0
$557$	−17.4734	−0.740372	−0.370186	−	0.928958i	$-0.620706\pi$
$557$	−17.4734	−0.740372	−0.370186	+	0.928958i	$0.620706\pi$
$558$	0	0
$559$	44.4413	1.87967
$560$	0	0
$561$	−13.2001	−0.557309
$562$	0	0
$563$	−12.6780	−0.534316	−0.267158	−	0.963653i	$-0.586085\pi$
$563$	−12.6780	−0.534316	−0.267158	+	0.963653i	$0.586085\pi$
$564$	0	0
$565$	3.88529	0.163455
$566$	0	0
$567$	5.46254	0.229405
$568$	0	0
$569$	−10.3866	−0.435429	−0.217714	−	0.976013i	$-0.569860\pi$
$569$	−10.3866	−0.435429	−0.217714	+	0.976013i	$0.569860\pi$
$570$	0	0
$571$	1.01663	0.0425445	0.0212722	−	0.999774i	$-0.493228\pi$
$571$	1.01663	0.0425445	0.0212722	+	0.999774i	$0.493228\pi$
$572$	0	0
$573$	−11.0244	−0.460552
$574$	0	0
$575$	12.1434	0.506414
$576$	0	0
$577$	6.61561	0.275412	0.137706	−	0.990473i	$-0.456027\pi$
$577$	6.61561	0.275412	0.137706	+	0.990473i	$0.456027\pi$
$578$	0	0
$579$	−36.0541	−1.49836
$580$	0	0
$581$	4.96388	0.205937
$582$	0	0
$583$	−56.7720	−2.35126
$584$	0	0
$585$	6.74464	0.278857
$586$	0	0
$587$	−38.3156	−1.58145	−0.790727	−	0.612169i	$-0.790297\pi$
$587$	−38.3156	−1.58145	−0.790727	+	0.612169i	$0.790297\pi$
$588$	0	0
$589$	15.8798	0.654314
$590$	0	0
$591$	24.8801	1.02343
$592$	0	0
$593$	17.1264	0.703298	0.351649	−	0.936132i	$-0.385621\pi$
$593$	17.1264	0.703298	0.351649	+	0.936132i	$0.385621\pi$
$594$	0	0
$595$	0.257788	0.0105683
$596$	0	0
$597$	−37.1096	−1.51879
$598$	0	0
$599$	−10.8591	−0.443691	−0.221845	−	0.975082i	$-0.571208\pi$
$599$	−10.8591	−0.443691	−0.221845	+	0.975082i	$0.571208\pi$
$600$	0	0
$601$	17.9862	0.733673	0.366837	−	0.930285i	$-0.380441\pi$
$601$	17.9862	0.733673	0.366837	+	0.930285i	$0.380441\pi$
$602$	0	0
$603$	−14.5603	−0.592940
$604$	0	0
$605$	5.14719	0.209263
$606$	0	0
$607$	−39.4889	−1.60281	−0.801403	−	0.598124i	$-0.795913\pi$
$607$	−39.4889	−1.60281	−0.801403	+	0.598124i	$0.795913\pi$
$608$	0	0
$609$	−14.2058	−0.575648
$610$	0	0
$611$	38.7420	1.56734
$612$	0	0
$613$	23.2401	0.938657	0.469329	−	0.883024i	$-0.344496\pi$
$613$	23.2401	0.938657	0.469329	+	0.883024i	$0.344496\pi$
$614$	0	0
$615$	−2.91728	−0.117636
$616$	0	0
$617$	−6.15363	−0.247736	−0.123868	−	0.992299i	$-0.539530\pi$
$617$	−6.15363	−0.247736	−0.123868	+	0.992299i	$0.539530\pi$
$618$	0	0
$619$	18.0030	0.723600	0.361800	−	0.932256i	$-0.382162\pi$
$619$	18.0030	0.723600	0.361800	+	0.932256i	$0.382162\pi$
$620$	0	0
$621$	3.55848	0.142797
$622$	0	0
$623$	1.22524	0.0490880
$624$	0	0
$625$	23.3712	0.934849
$626$	0	0
$627$	−26.8533	−1.07242
$628$	0	0
$629$	−2.75688	−0.109924
$630$	0	0
$631$	1.69130	0.0673295	0.0336648	−	0.999433i	$-0.489282\pi$
$631$	1.69130	0.0673295	0.0336648	+	0.999433i	$0.489282\pi$
$632$	0	0
$633$	9.00317	0.357844
$634$	0	0
$635$	5.29429	0.210097
$636$	0	0
$637$	−36.6232	−1.45107
$638$	0	0
$639$	−44.8854	−1.77564
$640$	0	0
$641$	22.3688	0.883516	0.441758	−	0.897134i	$-0.354355\pi$
$641$	22.3688	0.883516	0.441758	+	0.897134i	$0.354355\pi$
$642$	0	0
$643$	36.7240	1.44825	0.724127	−	0.689667i	$-0.242243\pi$
$643$	36.7240	1.44825	0.724127	+	0.689667i	$0.242243\pi$
$644$	0	0
$645$	−6.57065	−0.258719
$646$	0	0
$647$	−6.65817	−0.261760	−0.130880	−	0.991398i	$-0.541780\pi$
$647$	−6.65817	−0.261760	−0.130880	+	0.991398i	$0.541780\pi$
$648$	0	0
$649$	5.15394	0.202310
$650$	0	0
$651$	−15.5830	−0.610746
$652$	0	0
$653$	4.31877	0.169007	0.0845033	−	0.996423i	$-0.473070\pi$
$653$	4.31877	0.169007	0.0845033	+	0.996423i	$0.473070\pi$
$654$	0	0
$655$	−4.22029	−0.164900
$656$	0	0
$657$	26.9261	1.05049
$658$	0	0
$659$	−28.5011	−1.11024	−0.555122	−	0.831769i	$-0.687329\pi$
$659$	−28.5011	−1.11024	−0.555122	+	0.831769i	$0.687329\pi$
$660$	0	0
$661$	47.5495	1.84946	0.924731	−	0.380621i	$-0.124290\pi$
$661$	47.5495	1.84946	0.924731	+	0.380621i	$0.124290\pi$
$662$	0	0
$663$	14.6733	0.569862
$664$	0	0
$665$	0.524423	0.0203363
$666$	0	0
$667$	17.6691	0.684151
$668$	0	0
$669$	−49.2227	−1.90306
$670$	0	0
$671$	6.87774	0.265512
$672$	0	0
$673$	47.5008	1.83102	0.915511	−	0.402294i	$-0.131787\pi$
$673$	47.5008	1.83102	0.915511	+	0.402294i	$0.131787\pi$
$674$	0	0
$675$	7.00894	0.269774
$676$	0	0
$677$	48.7480	1.87354	0.936769	−	0.349949i	$-0.113801\pi$
$677$	48.7480	1.87354	0.936769	+	0.349949i	$0.113801\pi$
$678$	0	0
$679$	1.51042	0.0579647
$680$	0	0
$681$	57.2770	2.19486
$682$	0	0
$683$	13.5615	0.518918	0.259459	−	0.965754i	$-0.416456\pi$
$683$	13.5615	0.518918	0.259459	+	0.965754i	$0.416456\pi$
$684$	0	0
$685$	−0.280737	−0.0107264
$686$	0	0
$687$	−36.6006	−1.39640
$688$	0	0
$689$	63.1079	2.40422
$690$	0	0
$691$	−0.958224	−0.0364526	−0.0182263	−	0.999834i	$-0.505802\pi$
$691$	−0.958224	−0.0364526	−0.0182263	+	0.999834i	$0.505802\pi$
$692$	0	0
$693$	14.2997	0.543199
$694$	0	0
$695$	−0.364979	−0.0138444
$696$	0	0
$697$	−3.44404	−0.130452
$698$	0	0
$699$	31.0661	1.17503
$700$	0	0
$701$	19.5303	0.737651	0.368825	−	0.929499i	$-0.379760\pi$
$701$	19.5303	0.737651	0.368825	+	0.929499i	$0.379760\pi$
$702$	0	0
$703$	−5.60839	−0.211524
$704$	0	0
$705$	−5.72802	−0.215730
$706$	0	0
$707$	8.97060	0.337374
$708$	0	0
$709$	25.9278	0.973738	0.486869	−	0.873475i	$-0.338139\pi$
$709$	25.9278	0.973738	0.486869	+	0.873475i	$0.338139\pi$
$710$	0	0
$711$	−20.2564	−0.759675
$712$	0	0
$713$	19.3821	0.725865
$714$	0	0
$715$	−9.76566	−0.365215
$716$	0	0
$717$	−48.1482	−1.79813
$718$	0	0
$719$	−12.8114	−0.477784	−0.238892	−	0.971046i	$-0.576784\pi$
$719$	−12.8114	−0.477784	−0.238892	+	0.971046i	$0.576784\pi$
$720$	0	0
$721$	12.7870	0.476212
$722$	0	0
$723$	11.1287	0.413880
$724$	0	0
$725$	34.8019	1.29251
$726$	0	0
$727$	−35.2036	−1.30563	−0.652814	−	0.757518i	$-0.726412\pi$
$727$	−35.2036	−1.30563	−0.652814	+	0.757518i	$0.726412\pi$
$728$	0	0
$729$	−35.9576	−1.33176
$730$	0	0
$731$	−7.75707	−0.286905
$732$	0	0
$733$	21.9802	0.811858	0.405929	−	0.913905i	$-0.366948\pi$
$733$	21.9802	0.811858	0.405929	+	0.913905i	$0.366948\pi$
$734$	0	0
$735$	5.41475	0.199726
$736$	0	0
$737$	21.0820	0.776566
$738$	0	0
$739$	−3.85683	−0.141876	−0.0709378	−	0.997481i	$-0.522599\pi$
$739$	−3.85683	−0.141876	−0.0709378	+	0.997481i	$0.522599\pi$
$740$	0	0
$741$	29.8501	1.09657
$742$	0	0
$743$	−3.79516	−0.139231	−0.0696155	−	0.997574i	$-0.522177\pi$
$743$	−3.79516	−0.139231	−0.0696155	+	0.997574i	$0.522177\pi$
$744$	0	0
$745$	−1.14805	−0.0420613
$746$	0	0
$747$	−22.6689	−0.829410
$748$	0	0
$749$	−8.27473	−0.302352
$750$	0	0
$751$	12.6192	0.460482	0.230241	−	0.973134i	$-0.426049\pi$
$751$	12.6192	0.460482	0.230241	+	0.973134i	$0.426049\pi$
$752$	0	0
$753$	55.6960	2.02968
$754$	0	0
$755$	6.42799	0.233939
$756$	0	0
$757$	−2.60696	−0.0947516	−0.0473758	−	0.998877i	$-0.515086\pi$
$757$	−2.60696	−0.0947516	−0.0473758	+	0.998877i	$0.515086\pi$
$758$	0	0
$759$	−32.7758	−1.18969
$760$	0	0
$761$	−34.2670	−1.24218	−0.621089	−	0.783740i	$-0.713309\pi$
$761$	−34.2670	−1.24218	−0.621089	+	0.783740i	$0.713309\pi$
$762$	0	0
$763$	3.31872	0.120146
$764$	0	0
$765$	−1.17725	−0.0425637
$766$	0	0
$767$	−5.72913	−0.206867
$768$	0	0
$769$	−18.3023	−0.659998	−0.329999	−	0.943981i	$-0.607048\pi$
$769$	−18.3023	−0.659998	−0.329999	+	0.943981i	$0.607048\pi$
$770$	0	0
$771$	15.6560	0.563836
$772$	0	0
$773$	−10.7069	−0.385099	−0.192550	−	0.981287i	$-0.561676\pi$
$773$	−10.7069	−0.385099	−0.192550	+	0.981287i	$0.561676\pi$
$774$	0	0
$775$	38.1758	1.37131
$776$	0	0
$777$	5.50358	0.197440
$778$	0	0
$779$	−7.00629	−0.251026
$780$	0	0
$781$	64.9901	2.32553
$782$	0	0
$783$	10.1983	0.364457
$784$	0	0
$785$	−5.06297	−0.180705
$786$	0	0
$787$	−25.6304	−0.913624	−0.456812	−	0.889563i	$-0.651009\pi$
$787$	−25.6304	−0.913624	−0.456812	+	0.889563i	$0.651009\pi$
$788$	0	0
$789$	−35.1552	−1.25156
$790$	0	0
$791$	−9.15670	−0.325575
$792$	0	0
$793$	−7.64531	−0.271493
$794$	0	0
$795$	−9.33051	−0.330919
$796$	0	0
$797$	45.0004	1.59400	0.796998	−	0.603982i	$-0.206420\pi$
$797$	45.0004	1.59400	0.796998	+	0.603982i	$0.206420\pi$
$798$	0	0
$799$	−6.76229	−0.239233
$800$	0	0
$801$	−5.59535	−0.197702
$802$	0	0
$803$	−38.9867	−1.37581
$804$	0	0
$805$	0.640086	0.0225601
$806$	0	0
$807$	−61.6168	−2.16902
$808$	0	0
$809$	10.1359	0.356360	0.178180	−	0.983998i	$-0.442979\pi$
$809$	10.1359	0.356360	0.178180	+	0.983998i	$0.442979\pi$
$810$	0	0
$811$	48.0920	1.68874	0.844368	−	0.535763i	$-0.179976\pi$
$811$	48.0920	1.68874	0.844368	+	0.535763i	$0.179976\pi$
$812$	0	0
$813$	−76.0104	−2.66580
$814$	0	0
$815$	−7.33336	−0.256876
$816$	0	0
$817$	−15.7804	−0.552085
$818$	0	0
$819$	−15.8955	−0.555435
$820$	0	0
$821$	2.77764	0.0969404	0.0484702	−	0.998825i	$-0.484565\pi$
$821$	2.77764	0.0969404	0.0484702	+	0.998825i	$0.484565\pi$
$822$	0	0
$823$	48.5347	1.69181	0.845907	−	0.533331i	$-0.179060\pi$
$823$	48.5347	1.69181	0.845907	+	0.533331i	$0.179060\pi$
$824$	0	0
$825$	−64.5566	−2.24757
$826$	0	0
$827$	18.1497	0.631128	0.315564	−	0.948904i	$-0.397806\pi$
$827$	18.1497	0.631128	0.315564	+	0.948904i	$0.397806\pi$
$828$	0	0
$829$	−22.0471	−0.765727	−0.382864	−	0.923805i	$-0.625062\pi$
$829$	−22.0471	−0.765727	−0.382864	+	0.923805i	$0.625062\pi$
$830$	0	0
$831$	−5.25586	−0.182324
$832$	0	0
$833$	6.39246	0.221485
$834$	0	0
$835$	7.72209	0.267234
$836$	0	0
$837$	11.1870	0.386679
$838$	0	0
$839$	−37.6875	−1.30112	−0.650559	−	0.759456i	$-0.725465\pi$
$839$	−37.6875	−1.30112	−0.650559	+	0.759456i	$0.725465\pi$
$840$	0	0
$841$	21.6382	0.746144
$842$	0	0
$843$	−15.9177	−0.548236
$844$	0	0
$845$	6.55605	0.225535
$846$	0	0
$847$	−12.1307	−0.416816
$848$	0	0
$849$	41.8566	1.43651
$850$	0	0
$851$	−6.84533	−0.234655
$852$	0	0
$853$	−47.9052	−1.64024	−0.820122	−	0.572189i	$-0.806094\pi$
$853$	−47.9052	−1.64024	−0.820122	+	0.572189i	$0.806094\pi$
$854$	0	0
$855$	−2.39491	−0.0819043
$856$	0	0
$857$	41.3838	1.41364	0.706821	−	0.707392i	$-0.250128\pi$
$857$	41.3838	1.41364	0.706821	+	0.707392i	$0.250128\pi$
$858$	0	0
$859$	−46.9286	−1.60118	−0.800591	−	0.599211i	$-0.795481\pi$
$859$	−46.9286	−1.60118	−0.800591	+	0.599211i	$0.795481\pi$
$860$	0	0
$861$	6.87535	0.234311
$862$	0	0
$863$	30.7554	1.04693	0.523463	−	0.852048i	$-0.324640\pi$
$863$	30.7554	1.04693	0.523463	+	0.852048i	$0.324640\pi$
$864$	0	0
$865$	−3.60456	−0.122559
$866$	0	0
$867$	−2.56116	−0.0869817
$868$	0	0
$869$	29.3295	0.994936
$870$	0	0
$871$	−23.4348	−0.794058
$872$	0	0
$873$	−6.89773	−0.233453
$874$	0	0
$875$	2.54968	0.0861949
$876$	0	0
$877$	−44.8287	−1.51376	−0.756879	−	0.653555i	$-0.773277\pi$
$877$	−44.8287	−1.51376	−0.756879	+	0.653555i	$0.773277\pi$
$878$	0	0
$879$	85.7901	2.89363
$880$	0	0
$881$	−30.5679	−1.02986	−0.514930	−	0.857233i	$-0.672182\pi$
$881$	−30.5679	−1.02986	−0.514930	+	0.857233i	$0.672182\pi$
$882$	0	0
$883$	4.53952	0.152767	0.0763834	−	0.997079i	$-0.475663\pi$
$883$	4.53952	0.152767	0.0763834	+	0.997079i	$0.475663\pi$
$884$	0	0
$885$	0.847053	0.0284734
$886$	0	0
$887$	35.7094	1.19900	0.599501	−	0.800374i	$-0.295366\pi$
$887$	35.7094	1.19900	0.599501	+	0.800374i	$0.295366\pi$
$888$	0	0
$889$	−12.4774	−0.418478
$890$	0	0
$891$	36.1198	1.21006
$892$	0	0
$893$	−13.7567	−0.460349
$894$	0	0
$895$	−4.79393	−0.160243
$896$	0	0
$897$	36.4336	1.21648
$898$	0	0
$899$	55.5473	1.85261
$900$	0	0
$901$	−11.0153	−0.366972
$902$	0	0
$903$	15.4855	0.515324
$904$	0	0
$905$	−3.11627	−0.103588
$906$	0	0
$907$	−34.5099	−1.14588	−0.572942	−	0.819596i	$-0.694198\pi$
$907$	−34.5099	−1.14588	−0.572942	+	0.819596i	$0.694198\pi$
$908$	0	0
$909$	−40.9665	−1.35877
$910$	0	0
$911$	26.9432	0.892669	0.446335	−	0.894866i	$-0.352729\pi$
$911$	26.9432	0.892669	0.446335	+	0.894866i	$0.352729\pi$
$912$	0	0
$913$	32.8225	1.08627
$914$	0	0
$915$	1.13036	0.0373686
$916$	0	0
$917$	9.94621	0.328453
$918$	0	0
$919$	43.8528	1.44657	0.723285	−	0.690549i	$-0.242631\pi$
$919$	43.8528	1.44657	0.723285	+	0.690549i	$0.242631\pi$
$920$	0	0
$921$	−83.2252	−2.74236
$922$	0	0
$923$	−72.2432	−2.37791
$924$	0	0
$925$	−13.4829	−0.443314
$926$	0	0
$927$	−58.3951	−1.91795
$928$	0	0
$929$	−10.7943	−0.354149	−0.177074	−	0.984197i	$-0.556663\pi$
$929$	−10.7943	−0.354149	−0.177074	+	0.984197i	$0.556663\pi$
$930$	0	0
$931$	13.0043	0.426199
$932$	0	0
$933$	6.79231	0.222370
$934$	0	0
$935$	1.70456	0.0557451
$936$	0	0
$937$	−13.8736	−0.453229	−0.226615	−	0.973984i	$-0.572766\pi$
$937$	−13.8736	−0.453229	−0.226615	+	0.973984i	$0.572766\pi$
$938$	0	0
$939$	−68.5089	−2.23570
$940$	0	0
$941$	−32.3922	−1.05596	−0.527979	−	0.849258i	$-0.677050\pi$
$941$	−32.3922	−1.05596	−0.527979	+	0.849258i	$0.677050\pi$
$942$	0	0
$943$	−8.55154	−0.278476
$944$	0	0
$945$	0.369446	0.0120181
$946$	0	0
$947$	44.8817	1.45846	0.729230	−	0.684269i	$-0.239879\pi$
$947$	44.8817	1.45846	0.729230	+	0.684269i	$0.239879\pi$
$948$	0	0
$949$	43.3377	1.40680
$950$	0	0
$951$	−10.5484	−0.342057
$952$	0	0
$953$	−23.4966	−0.761130	−0.380565	−	0.924754i	$-0.624270\pi$
$953$	−23.4966	−0.761130	−0.380565	+	0.924754i	$0.624270\pi$
$954$	0	0
$955$	1.42361	0.0460670
$956$	0	0
$957$	−93.9326	−3.03641
$958$	0	0
$959$	0.661631	0.0213652
$960$	0	0
$961$	29.9324	0.965563
$962$	0	0
$963$	37.7887	1.21772
$964$	0	0
$965$	4.65575	0.149874
$966$	0	0
$967$	40.5011	1.30243	0.651214	−	0.758894i	$-0.274260\pi$
$967$	40.5011	1.30243	0.651214	+	0.758894i	$0.274260\pi$
$968$	0	0
$969$	−5.21023	−0.167377
$970$	0	0
$971$	36.1909	1.16142	0.580711	−	0.814110i	$-0.302775\pi$
$971$	36.1909	1.16142	0.580711	+	0.814110i	$0.302775\pi$
$972$	0	0
$973$	0.860169	0.0275757
$974$	0	0
$975$	71.7613	2.29820
$976$	0	0
$977$	−0.748739	−0.0239543	−0.0119771	−	0.999928i	$-0.503813\pi$
$977$	−0.748739	−0.0239543	−0.0119771	+	0.999928i	$0.503813\pi$
$978$	0	0
$979$	8.10159	0.258928
$980$	0	0
$981$	−15.1558	−0.483887
$982$	0	0
$983$	25.5515	0.814967	0.407483	−	0.913213i	$-0.366406\pi$
$983$	25.5515	0.814967	0.407483	+	0.913213i	$0.366406\pi$
$984$	0	0
$985$	−3.21283	−0.102369
$986$	0	0
$987$	13.4996	0.429696
$988$	0	0
$989$	−19.2608	−0.612457
$990$	0	0
$991$	−6.05413	−0.192316	−0.0961579	−	0.995366i	$-0.530655\pi$
$991$	−6.05413	−0.192316	−0.0961579	+	0.995366i	$0.530655\pi$
$992$	0	0
$993$	−20.2743	−0.643386
$994$	0	0
$995$	4.79205	0.151918
$996$	0	0
$997$	−14.1257	−0.447367	−0.223683	−	0.974662i	$-0.571808\pi$
$997$	−14.1257	−0.447367	−0.223683	+	0.974662i	$0.571808\pi$
$998$	0	0
$999$	−3.95100	−0.125004

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.bc.1.6	✓	33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.bc.1.6	✓	33	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56116 q^{3} +0.330730 q^{5} -0.779452 q^{7} +3.55957 q^{9} +O(q^{10})\)	\(q-2.56116 q^{3} +0.330730 q^{5} -0.779452 q^{7} +3.55957 q^{9} -5.15394 q^{11} +5.72913 q^{13} -0.847053 q^{15} -1.00000 q^{17} -2.03432 q^{19} +1.99630 q^{21} -2.48300 q^{23} -4.89062 q^{25} -1.43314 q^{27} -7.11605 q^{29} -7.80592 q^{31} +13.2001 q^{33} -0.257788 q^{35} +2.75688 q^{37} -14.6733 q^{39} +3.44404 q^{41} +7.75707 q^{43} +1.17725 q^{45} +6.76229 q^{47} -6.39246 q^{49} +2.56116 q^{51} +11.0153 q^{53} -1.70456 q^{55} +5.21023 q^{57} -1.00000 q^{59} -1.33446 q^{61} -2.77451 q^{63} +1.89479 q^{65} -4.09046 q^{67} +6.35936 q^{69} -12.6098 q^{71} +7.56444 q^{73} +12.5257 q^{75} +4.01725 q^{77} -5.69070 q^{79} -7.00819 q^{81} -6.36843 q^{83} -0.330730 q^{85} +18.2254 q^{87} -1.57192 q^{89} -4.46558 q^{91} +19.9923 q^{93} -0.672811 q^{95} -1.93780 q^{97} -18.3458 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9}+O(q^{10}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9	\( 33 q - 3 q^{3} + 3 q^{5} + 48 q^{9} - 7 q^{11} + 9 q^{13} - 7 q^{15} - 33 q^{17} + 5 q^{19} + 14 q^{21} - 19 q^{23} + 52 q^{25} - 24 q^{27} + 14 q^{29} + 19 q^{31} + 34 q^{33} + 15 q^{35} + 13 q^{37} + 26 q^{39} + 32 q^{41} + 11 q^{43} - 27 q^{47} + 93 q^{49} + 3 q^{51} - 7 q^{53} + 31 q^{55} + 6 q^{57} - 33 q^{59} + 33 q^{61} + 10 q^{63} + 21 q^{65} - 10 q^{67} + 78 q^{69} - 29 q^{71} + 30 q^{73} + 31 q^{75} + 6 q^{77} + 66 q^{79} + 97 q^{81} - 9 q^{83} - 3 q^{85} - 27 q^{87} + 68 q^{89} + 28 q^{91} + 34 q^{93} + 8 q^{95} + 62 q^{97} + 15 q^{99}+O(q^{100}) \) 33 * q - 3 * q^3 + 3 * q^5 + 48 * q^9 - 7 * q^11 + 9 * q^13 - 7 * q^15 - 33 * q^17 + 5 * q^19 + 14 * q^21 - 19 * q^23 + 52 * q^25 - 24 * q^27 + 14 * q^29 + 19 * q^31 + 34 * q^33 + 15 * q^35 + 13 * q^37 + 26 * q^39 + 32 * q^41 + 11 * q^43 - 27 * q^47 + 93 * q^49 + 3 * q^51 - 7 * q^53 + 31 * q^55 + 6 * q^57 - 33 * q^59 + 33 * q^61 + 10 * q^63 + 21 * q^65 - 10 * q^67 + 78 * q^69 - 29 * q^71 + 30 * q^73 + 31 * q^75 + 6 * q^77 + 66 * q^79 + 97 * q^81 - 9 * q^83 - 3 * q^85 - 27 * q^87 + 68 * q^89 + 28 * q^91 + 34 * q^93 + 8 * q^95 + 62 * q^97 + 15 * q^99

Introduction

L-functions

Modular forms

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