LMFDB - Embedded newform 8024.2.a.z.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:	$1$
Dimension:	$24$
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)

$f(q)$	$=$	$q-2.15097 q^{3} -2.56894 q^{5} +2.78287 q^{7} +1.62668 q^{9} +O(q^{10})$	$q-2.15097 q^{3} -2.56894 q^{5} +2.78287 q^{7} +1.62668 q^{9} +4.07020 q^{11} -4.02428 q^{13} +5.52572 q^{15} +1.00000 q^{17} -3.97202 q^{19} -5.98588 q^{21} +1.07237 q^{23} +1.59946 q^{25} +2.95397 q^{27} -3.05094 q^{29} +6.82405 q^{31} -8.75488 q^{33} -7.14903 q^{35} +0.708241 q^{37} +8.65612 q^{39} -0.0957297 q^{41} -8.14529 q^{43} -4.17885 q^{45} +5.46484 q^{47} +0.744371 q^{49} -2.15097 q^{51} -10.9777 q^{53} -10.4561 q^{55} +8.54371 q^{57} +1.00000 q^{59} +4.62155 q^{61} +4.52684 q^{63} +10.3381 q^{65} +10.0665 q^{67} -2.30665 q^{69} -3.08299 q^{71} -4.20207 q^{73} -3.44040 q^{75} +11.3268 q^{77} +7.75788 q^{79} -11.2340 q^{81} -8.88992 q^{83} -2.56894 q^{85} +6.56249 q^{87} -0.468545 q^{89} -11.1991 q^{91} -14.6784 q^{93} +10.2039 q^{95} -4.48393 q^{97} +6.62092 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	$ 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) $ 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * 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
$3$	−2.15097	−1.24186	−0.620932	−	0.783864i	$-0.713246\pi$
$3$	−2.15097	−1.24186	−0.620932	+	0.783864i	$0.713246\pi$
$4$	0	0
$5$	−2.56894	−1.14887	−0.574433	−	0.818552i	$-0.694777\pi$
$5$	−2.56894	−1.14887	−0.574433	+	0.818552i	$0.694777\pi$
$6$	0	0
$7$	2.78287	1.05183	0.525913	−	0.850538i	$-0.323724\pi$
$7$	2.78287	1.05183	0.525913	+	0.850538i	$0.323724\pi$
$8$	0	0
$9$	1.62668	0.542227
$10$	0	0
$11$	4.07020	1.22721	0.613605	−	0.789613i	$-0.289719\pi$
$11$	4.07020	1.22721	0.613605	+	0.789613i	$0.289719\pi$
$12$	0	0
$13$	−4.02428	−1.11614	−0.558068	−	0.829796i	$-0.688457\pi$
$13$	−4.02428	−1.11614	−0.558068	+	0.829796i	$0.688457\pi$
$14$	0	0
$15$	5.52572	1.42674
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−3.97202	−0.911245	−0.455622	−	0.890173i	$-0.650583\pi$
$19$	−3.97202	−0.911245	−0.455622	+	0.890173i	$0.650583\pi$
$20$	0	0
$21$	−5.98588	−1.30623
$22$	0	0
$23$	1.07237	0.223605	0.111803	−	0.993730i	$-0.464338\pi$
$23$	1.07237	0.223605	0.111803	+	0.993730i	$0.464338\pi$
$24$	0	0
$25$	1.59946	0.319892
$26$	0	0
$27$	2.95397	0.568492
$28$	0	0
$29$	−3.05094	−0.566546	−0.283273	−	0.959039i	$-0.591420\pi$
$29$	−3.05094	−0.566546	−0.283273	+	0.959039i	$0.591420\pi$
$30$	0	0
$31$	6.82405	1.22564	0.612818	−	0.790224i	$-0.290036\pi$
$31$	6.82405	1.22564	0.612818	+	0.790224i	$0.290036\pi$
$32$	0	0
$33$	−8.75488	−1.52403
$34$	0	0
$35$	−7.14903	−1.20841
$36$	0	0
$37$	0.708241	0.116434	0.0582171	−	0.998304i	$-0.481458\pi$
$37$	0.708241	0.116434	0.0582171	+	0.998304i	$0.481458\pi$
$38$	0	0
$39$	8.65612	1.38609
$40$	0	0
$41$	−0.0957297	−0.0149505	−0.00747523	−	0.999972i	$-0.502379\pi$
$41$	−0.0957297	−0.0149505	−0.00747523	+	0.999972i	$0.502379\pi$
$42$	0	0
$43$	−8.14529	−1.24214	−0.621072	−	0.783753i	$-0.713303\pi$
$43$	−8.14529	−1.24214	−0.621072	+	0.783753i	$0.713303\pi$
$44$	0	0
$45$	−4.17885	−0.622946
$46$	0	0
$47$	5.46484	0.797129	0.398565	−	0.917140i	$-0.369509\pi$
$47$	5.46484	0.797129	0.398565	+	0.917140i	$0.369509\pi$
$48$	0	0
$49$	0.744371	0.106339
$50$	0	0
$51$	−2.15097	−0.301196
$52$	0	0
$53$	−10.9777	−1.50791	−0.753954	−	0.656927i	$-0.771856\pi$
$53$	−10.9777	−1.50791	−0.753954	+	0.656927i	$0.771856\pi$
$54$	0	0
$55$	−10.4561	−1.40990
$56$	0	0
$57$	8.54371	1.13164
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	4.62155	0.591729	0.295864	−	0.955230i	$-0.404392\pi$
$61$	4.62155	0.591729	0.295864	+	0.955230i	$0.404392\pi$
$62$	0	0
$63$	4.52684	0.570329
$64$	0	0
$65$	10.3381	1.28229
$66$	0	0
$67$	10.0665	1.22982	0.614912	−	0.788596i	$-0.289192\pi$
$67$	10.0665	1.22982	0.614912	+	0.788596i	$0.289192\pi$
$68$	0	0
$69$	−2.30665	−0.277687
$70$	0	0
$71$	−3.08299	−0.365884	−0.182942	−	0.983124i	$-0.558562\pi$
$71$	−3.08299	−0.365884	−0.182942	+	0.983124i	$0.558562\pi$
$72$	0	0
$73$	−4.20207	−0.491815	−0.245907	−	0.969293i	$-0.579086\pi$
$73$	−4.20207	−0.491815	−0.245907	+	0.969293i	$0.579086\pi$
$74$	0	0
$75$	−3.44040	−0.397263
$76$	0	0
$77$	11.3268	1.29081
$78$	0	0
$79$	7.75788	0.872829	0.436415	−	0.899746i	$-0.356248\pi$
$79$	7.75788	0.872829	0.436415	+	0.899746i	$0.356248\pi$
$80$	0	0
$81$	−11.2340	−1.24822
$82$	0	0
$83$	−8.88992	−0.975795	−0.487897	−	0.872901i	$-0.662236\pi$
$83$	−8.88992	−0.975795	−0.487897	+	0.872901i	$0.662236\pi$
$84$	0	0
$85$	−2.56894	−0.278641
$86$	0	0
$87$	6.56249	0.703573
$88$	0	0
$89$	−0.468545	−0.0496657	−0.0248328	−	0.999692i	$-0.507905\pi$
$89$	−0.468545	−0.0496657	−0.0248328	+	0.999692i	$0.507905\pi$
$90$	0	0
$91$	−11.1991	−1.17398
$92$	0	0
$93$	−14.6784	−1.52207
$94$	0	0
$95$	10.2039	1.04690
$96$	0	0
$97$	−4.48393	−0.455274	−0.227637	−	0.973746i	$-0.573100\pi$
$97$	−4.48393	−0.455274	−0.227637	+	0.973746i	$0.573100\pi$
$98$	0	0
$99$	6.62092	0.665427
$100$	0	0
$101$	−12.8394	−1.27757	−0.638784	−	0.769386i	$-0.720562\pi$
$101$	−12.8394	−1.27757	−0.638784	+	0.769386i	$0.720562\pi$
$102$	0	0
$103$	14.3350	1.41247	0.706237	−	0.707976i	$-0.250391\pi$
$103$	14.3350	1.41247	0.706237	+	0.707976i	$0.250391\pi$
$104$	0	0
$105$	15.3774	1.50068
$106$	0	0
$107$	1.22541	0.118464	0.0592322	−	0.998244i	$-0.481135\pi$
$107$	1.22541	0.118464	0.0592322	+	0.998244i	$0.481135\pi$
$108$	0	0
$109$	13.9381	1.33503	0.667516	−	0.744596i	$-0.267358\pi$
$109$	13.9381	1.33503	0.667516	+	0.744596i	$0.267358\pi$
$110$	0	0
$111$	−1.52341	−0.144595
$112$	0	0
$113$	15.7881	1.48522	0.742608	−	0.669727i	$-0.233589\pi$
$113$	15.7881	1.48522	0.742608	+	0.669727i	$0.233589\pi$
$114$	0	0
$115$	−2.75486	−0.256892
$116$	0	0
$117$	−6.54622	−0.605199
$118$	0	0
$119$	2.78287	0.255105
$120$	0	0
$121$	5.56651	0.506047
$122$	0	0
$123$	0.205912	0.0185664
$124$	0	0
$125$	8.73579	0.781352
$126$	0	0
$127$	3.20105	0.284047	0.142024	−	0.989863i	$-0.454639\pi$
$127$	3.20105	0.284047	0.142024	+	0.989863i	$0.454639\pi$
$128$	0	0
$129$	17.5203	1.54257
$130$	0	0
$131$	0.0484156	0.00423009	0.00211504	−	0.999998i	$-0.499327\pi$
$131$	0.0484156	0.00423009	0.00211504	+	0.999998i	$0.499327\pi$
$132$	0	0
$133$	−11.0536	−0.958472
$134$	0	0
$135$	−7.58858	−0.653121
$136$	0	0
$137$	10.9158	0.932598	0.466299	−	0.884627i	$-0.345587\pi$
$137$	10.9158	0.932598	0.466299	+	0.884627i	$0.345587\pi$
$138$	0	0
$139$	−5.99858	−0.508793	−0.254397	−	0.967100i	$-0.581877\pi$
$139$	−5.99858	−0.508793	−0.254397	+	0.967100i	$0.581877\pi$
$140$	0	0
$141$	−11.7547	−0.989926
$142$	0	0
$143$	−16.3796	−1.36973
$144$	0	0
$145$	7.83769	0.650885
$146$	0	0
$147$	−1.60112	−0.132058
$148$	0	0
$149$	8.09817	0.663428	0.331714	−	0.943380i	$-0.392373\pi$
$149$	8.09817	0.663428	0.331714	+	0.943380i	$0.392373\pi$
$150$	0	0
$151$	−15.6644	−1.27475	−0.637376	−	0.770553i	$-0.719980\pi$
$151$	−15.6644	−1.27475	−0.637376	+	0.770553i	$0.719980\pi$
$152$	0	0
$153$	1.62668	0.131509
$154$	0	0
$155$	−17.5306	−1.40809
$156$	0	0
$157$	7.32214	0.584371	0.292185	−	0.956362i	$-0.405618\pi$
$157$	7.32214	0.584371	0.292185	+	0.956362i	$0.405618\pi$
$158$	0	0
$159$	23.6128	1.87262
$160$	0	0
$161$	2.98428	0.235194
$162$	0	0
$163$	8.45914	0.662571	0.331285	−	0.943531i	$-0.392518\pi$
$163$	8.45914	0.662571	0.331285	+	0.943531i	$0.392518\pi$
$164$	0	0
$165$	22.4908	1.75091
$166$	0	0
$167$	1.59368	0.123323	0.0616614	−	0.998097i	$-0.480360\pi$
$167$	1.59368	0.123323	0.0616614	+	0.998097i	$0.480360\pi$
$168$	0	0
$169$	3.19485	0.245757
$170$	0	0
$171$	−6.46122	−0.494102
$172$	0	0
$173$	0.616053	0.0468376	0.0234188	−	0.999726i	$-0.492545\pi$
$173$	0.616053	0.0468376	0.0234188	+	0.999726i	$0.492545\pi$
$174$	0	0
$175$	4.45109	0.336471
$176$	0	0
$177$	−2.15097	−0.161677
$178$	0	0
$179$	14.5045	1.08412	0.542060	−	0.840340i	$-0.317644\pi$
$179$	14.5045	1.08412	0.542060	+	0.840340i	$0.317644\pi$
$180$	0	0
$181$	−6.68708	−0.497047	−0.248523	−	0.968626i	$-0.579945\pi$
$181$	−6.68708	−0.497047	−0.248523	+	0.968626i	$0.579945\pi$
$182$	0	0
$183$	−9.94082	−0.734847
$184$	0	0
$185$	−1.81943	−0.133767
$186$	0	0
$187$	4.07020	0.297642
$188$	0	0
$189$	8.22052	0.597955
$190$	0	0
$191$	−6.72089	−0.486306	−0.243153	−	0.969988i	$-0.578182\pi$
$191$	−6.72089	−0.486306	−0.243153	+	0.969988i	$0.578182\pi$
$192$	0	0
$193$	−8.91815	−0.641943	−0.320971	−	0.947089i	$-0.604009\pi$
$193$	−8.91815	−0.641943	−0.320971	+	0.947089i	$0.604009\pi$
$194$	0	0
$195$	−22.2371	−1.59243
$196$	0	0
$197$	25.3412	1.80548	0.902741	−	0.430184i	$-0.141551\pi$
$197$	25.3412	1.80548	0.902741	+	0.430184i	$0.141551\pi$
$198$	0	0
$199$	−14.6444	−1.03812	−0.519058	−	0.854739i	$-0.673717\pi$
$199$	−14.6444	−1.03812	−0.519058	+	0.854739i	$0.673717\pi$
$200$	0	0
$201$	−21.6528	−1.52727
$202$	0	0
$203$	−8.49038	−0.595908
$204$	0	0
$205$	0.245924	0.0171761
$206$	0	0
$207$	1.74441	0.121245
$208$	0	0
$209$	−16.1669	−1.11829
$210$	0	0
$211$	−5.33063	−0.366976	−0.183488	−	0.983022i	$-0.558739\pi$
$211$	−5.33063	−0.366976	−0.183488	+	0.983022i	$0.558739\pi$
$212$	0	0
$213$	6.63143	0.454378
$214$	0	0
$215$	20.9248	1.42706
$216$	0	0
$217$	18.9905	1.28916
$218$	0	0
$219$	9.03853	0.610767
$220$	0	0
$221$	−4.02428	−0.270703
$222$	0	0
$223$	−4.77529	−0.319777	−0.159889	−	0.987135i	$-0.551113\pi$
$223$	−4.77529	−0.319777	−0.159889	+	0.987135i	$0.551113\pi$
$224$	0	0
$225$	2.60181	0.173454
$226$	0	0
$227$	−11.6027	−0.770096	−0.385048	−	0.922897i	$-0.625815\pi$
$227$	−11.6027	−0.770096	−0.385048	+	0.922897i	$0.625815\pi$
$228$	0	0
$229$	−15.1569	−1.00160	−0.500798	−	0.865564i	$-0.666960\pi$
$229$	−15.1569	−1.00160	−0.500798	+	0.865564i	$0.666960\pi$
$230$	0	0
$231$	−24.3637	−1.60301
$232$	0	0
$233$	5.39287	0.353299	0.176649	−	0.984274i	$-0.443474\pi$
$233$	5.39287	0.353299	0.176649	+	0.984274i	$0.443474\pi$
$234$	0	0
$235$	−14.0389	−0.915794
$236$	0	0
$237$	−16.6870	−1.08394
$238$	0	0
$239$	5.33850	0.345319	0.172659	−	0.984982i	$-0.444764\pi$
$239$	5.33850	0.345319	0.172659	+	0.984982i	$0.444764\pi$
$240$	0	0
$241$	4.69745	0.302589	0.151295	−	0.988489i	$-0.451656\pi$
$241$	4.69745	0.302589	0.151295	+	0.988489i	$0.451656\pi$
$242$	0	0
$243$	15.3020	0.981624
$244$	0	0
$245$	−1.91225	−0.122169
$246$	0	0
$247$	15.9845	1.01707
$248$	0	0
$249$	19.1220	1.21180
$250$	0	0
$251$	−12.1482	−0.766786	−0.383393	−	0.923585i	$-0.625245\pi$
$251$	−12.1482	−0.766786	−0.383393	+	0.923585i	$0.625245\pi$
$252$	0	0
$253$	4.36477	0.274411
$254$	0	0
$255$	5.52572	0.346034
$256$	0	0
$257$	−8.30428	−0.518007	−0.259003	−	0.965876i	$-0.583394\pi$
$257$	−8.30428	−0.518007	−0.259003	+	0.965876i	$0.583394\pi$
$258$	0	0
$259$	1.97094	0.122468
$260$	0	0
$261$	−4.96291	−0.307196
$262$	0	0
$263$	−2.14654	−0.132362	−0.0661808	−	0.997808i	$-0.521081\pi$
$263$	−2.14654	−0.132362	−0.0661808	+	0.997808i	$0.521081\pi$
$264$	0	0
$265$	28.2012	1.73238
$266$	0	0
$267$	1.00783	0.0616780
$268$	0	0
$269$	−20.6029	−1.25618	−0.628091	−	0.778140i	$-0.716163\pi$
$269$	−20.6029	−1.25618	−0.628091	+	0.778140i	$0.716163\pi$
$270$	0	0
$271$	3.33103	0.202346	0.101173	−	0.994869i	$-0.467740\pi$
$271$	3.33103	0.202346	0.101173	+	0.994869i	$0.467740\pi$
$272$	0	0
$273$	24.0889	1.45792
$274$	0	0
$275$	6.51012	0.392575
$276$	0	0
$277$	−8.75015	−0.525746	−0.262873	−	0.964831i	$-0.584670\pi$
$277$	−8.75015	−0.525746	−0.262873	+	0.964831i	$0.584670\pi$
$278$	0	0
$279$	11.1006	0.664573
$280$	0	0
$281$	9.05315	0.540065	0.270033	−	0.962851i	$-0.412965\pi$
$281$	9.05315	0.540065	0.270033	+	0.962851i	$0.412965\pi$
$282$	0	0
$283$	14.1001	0.838164	0.419082	−	0.907948i	$-0.362352\pi$
$283$	14.1001	0.838164	0.419082	+	0.907948i	$0.362352\pi$
$284$	0	0
$285$	−21.9483	−1.30011
$286$	0	0
$287$	−0.266403	−0.0157253
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	9.64481	0.565389
$292$	0	0
$293$	18.8080	1.09878	0.549388	−	0.835567i	$-0.314861\pi$
$293$	18.8080	1.09878	0.549388	+	0.835567i	$0.314861\pi$
$294$	0	0
$295$	−2.56894	−0.149570
$296$	0	0
$297$	12.0232	0.697659
$298$	0	0
$299$	−4.31553	−0.249574
$300$	0	0
$301$	−22.6673	−1.30652
$302$	0	0
$303$	27.6172	1.58657
$304$	0	0
$305$	−11.8725	−0.679817
$306$	0	0
$307$	−24.3972	−1.39242	−0.696212	−	0.717836i	$-0.745133\pi$
$307$	−24.3972	−1.39242	−0.696212	+	0.717836i	$0.745133\pi$
$308$	0	0
$309$	−30.8343	−1.75410
$310$	0	0
$311$	−28.4022	−1.61054	−0.805271	−	0.592907i	$-0.797980\pi$
$311$	−28.4022	−1.61054	−0.805271	+	0.592907i	$0.797980\pi$
$312$	0	0
$313$	24.5672	1.38862	0.694311	−	0.719675i	$-0.255709\pi$
$313$	24.5672	1.38862	0.694311	+	0.719675i	$0.255709\pi$
$314$	0	0
$315$	−11.6292	−0.655231
$316$	0	0
$317$	−21.5375	−1.20967	−0.604834	−	0.796352i	$-0.706760\pi$
$317$	−21.5375	−1.20967	−0.604834	+	0.796352i	$0.706760\pi$
$318$	0	0
$319$	−12.4179	−0.695271
$320$	0	0
$321$	−2.63581	−0.147117
$322$	0	0
$323$	−3.97202	−0.221009
$324$	0	0
$325$	−6.43668	−0.357043
$326$	0	0
$327$	−29.9805	−1.65793
$328$	0	0
$329$	15.2079	0.838441
$330$	0	0
$331$	−16.9863	−0.933650	−0.466825	−	0.884350i	$-0.654602\pi$
$331$	−16.9863	−0.933650	−0.466825	+	0.884350i	$0.654602\pi$
$332$	0	0
$333$	1.15208	0.0631337
$334$	0	0
$335$	−25.8603	−1.41290
$336$	0	0
$337$	−6.47855	−0.352909	−0.176455	−	0.984309i	$-0.556463\pi$
$337$	−6.47855	−0.352909	−0.176455	+	0.984309i	$0.556463\pi$
$338$	0	0
$339$	−33.9597	−1.84444
$340$	0	0
$341$	27.7753	1.50411
$342$	0	0
$343$	−17.4086	−0.939977
$344$	0	0
$345$	5.92564	0.319026
$346$	0	0
$347$	−28.6790	−1.53957	−0.769785	−	0.638304i	$-0.779636\pi$
$347$	−28.6790	−1.53957	−0.769785	+	0.638304i	$0.779636\pi$
$348$	0	0
$349$	−0.264216	−0.0141431	−0.00707157	−	0.999975i	$-0.502251\pi$
$349$	−0.264216	−0.0141431	−0.00707157	+	0.999975i	$0.502251\pi$
$350$	0	0
$351$	−11.8876	−0.634514
$352$	0	0
$353$	2.66269	0.141721	0.0708604	−	0.997486i	$-0.477426\pi$
$353$	2.66269	0.141721	0.0708604	+	0.997486i	$0.477426\pi$
$354$	0	0
$355$	7.92003	0.420352
$356$	0	0
$357$	−5.98588	−0.316806
$358$	0	0
$359$	1.46953	0.0775586	0.0387793	−	0.999248i	$-0.487653\pi$
$359$	1.46953	0.0775586	0.0387793	+	0.999248i	$0.487653\pi$
$360$	0	0
$361$	−3.22302	−0.169633
$362$	0	0
$363$	−11.9734	−0.628441
$364$	0	0
$365$	10.7949	0.565029
$366$	0	0
$367$	−26.9519	−1.40688	−0.703439	−	0.710755i	$-0.748353\pi$
$367$	−26.9519	−1.40688	−0.703439	+	0.710755i	$0.748353\pi$
$368$	0	0
$369$	−0.155722	−0.00810654
$370$	0	0
$371$	−30.5496	−1.58606
$372$	0	0
$373$	−33.5541	−1.73736	−0.868682	−	0.495371i	$-0.835032\pi$
$373$	−33.5541	−1.73736	−0.868682	+	0.495371i	$0.835032\pi$
$374$	0	0
$375$	−18.7904	−0.970334
$376$	0	0
$377$	12.2778	0.632341
$378$	0	0
$379$	23.4439	1.20423	0.602117	−	0.798408i	$-0.294324\pi$
$379$	23.4439	1.20423	0.602117	+	0.798408i	$0.294324\pi$
$380$	0	0
$381$	−6.88536	−0.352748
$382$	0	0
$383$	−32.8606	−1.67910	−0.839548	−	0.543286i	$-0.817180\pi$
$383$	−32.8606	−1.67910	−0.839548	+	0.543286i	$0.817180\pi$
$384$	0	0
$385$	−29.0980	−1.48297
$386$	0	0
$387$	−13.2498	−0.673524
$388$	0	0
$389$	6.73379	0.341417	0.170708	−	0.985322i	$-0.445394\pi$
$389$	6.73379	0.341417	0.170708	+	0.985322i	$0.445394\pi$
$390$	0	0
$391$	1.07237	0.0542323
$392$	0	0
$393$	−0.104141	−0.00525320
$394$	0	0
$395$	−19.9295	−1.00276
$396$	0	0
$397$	−35.8963	−1.80158	−0.900792	−	0.434252i	$-0.857013\pi$
$397$	−35.8963	−1.80158	−0.900792	+	0.434252i	$0.857013\pi$
$398$	0	0
$399$	23.7761	1.19029
$400$	0	0
$401$	15.2178	0.759941	0.379971	−	0.924999i	$-0.375934\pi$
$401$	15.2178	0.759941	0.379971	+	0.924999i	$0.375934\pi$
$402$	0	0
$403$	−27.4619	−1.36798
$404$	0	0
$405$	28.8594	1.43403
$406$	0	0
$407$	2.88268	0.142889
$408$	0	0
$409$	−2.85208	−0.141026	−0.0705131	−	0.997511i	$-0.522464\pi$
$409$	−2.85208	−0.141026	−0.0705131	+	0.997511i	$0.522464\pi$
$410$	0	0
$411$	−23.4795	−1.15816
$412$	0	0
$413$	2.78287	0.136936
$414$	0	0
$415$	22.8377	1.12106
$416$	0	0
$417$	12.9028	0.631852
$418$	0	0
$419$	1.55117	0.0757794	0.0378897	−	0.999282i	$-0.487936\pi$
$419$	1.55117	0.0757794	0.0378897	+	0.999282i	$0.487936\pi$
$420$	0	0
$421$	−33.1554	−1.61589	−0.807947	−	0.589255i	$-0.799421\pi$
$421$	−33.1554	−1.61589	−0.807947	+	0.589255i	$0.799421\pi$
$422$	0	0
$423$	8.88955	0.432225
$424$	0	0
$425$	1.59946	0.0775853
$426$	0	0
$427$	12.8612	0.622396
$428$	0	0
$429$	35.2321	1.70102
$430$	0	0
$431$	2.74959	0.132443	0.0662215	−	0.997805i	$-0.478906\pi$
$431$	2.74959	0.132443	0.0662215	+	0.997805i	$0.478906\pi$
$432$	0	0
$433$	−17.6690	−0.849117	−0.424558	−	0.905401i	$-0.639571\pi$
$433$	−17.6690	−0.849117	−0.424558	+	0.905401i	$0.639571\pi$
$434$	0	0
$435$	−16.8587	−0.808311
$436$	0	0
$437$	−4.25949	−0.203759
$438$	0	0
$439$	−15.8393	−0.755969	−0.377984	−	0.925812i	$-0.623383\pi$
$439$	−15.8393	−0.755969	−0.377984	+	0.925812i	$0.623383\pi$
$440$	0	0
$441$	1.21085	0.0576597
$442$	0	0
$443$	24.7351	1.17520	0.587601	−	0.809151i	$-0.300072\pi$
$443$	24.7351	1.17520	0.587601	+	0.809151i	$0.300072\pi$
$444$	0	0
$445$	1.20366	0.0570592
$446$	0	0
$447$	−17.4189	−0.823887
$448$	0	0
$449$	−10.3057	−0.486356	−0.243178	−	0.969982i	$-0.578190\pi$
$449$	−10.3057	−0.486356	−0.243178	+	0.969982i	$0.578190\pi$
$450$	0	0
$451$	−0.389639	−0.0183474
$452$	0	0
$453$	33.6937	1.58307
$454$	0	0
$455$	28.7697	1.34875
$456$	0	0
$457$	−2.62412	−0.122751	−0.0613755	−	0.998115i	$-0.519549\pi$
$457$	−2.62412	−0.122751	−0.0613755	+	0.998115i	$0.519549\pi$
$458$	0	0
$459$	2.95397	0.137880
$460$	0	0
$461$	−1.99620	−0.0929724	−0.0464862	−	0.998919i	$-0.514802\pi$
$461$	−1.99620	−0.0929724	−0.0464862	+	0.998919i	$0.514802\pi$
$462$	0	0
$463$	−15.9922	−0.743221	−0.371611	−	0.928389i	$-0.621194\pi$
$463$	−15.9922	−0.743221	−0.371611	+	0.928389i	$0.621194\pi$
$464$	0	0
$465$	37.7078	1.74866
$466$	0	0
$467$	−4.81043	−0.222600	−0.111300	−	0.993787i	$-0.535501\pi$
$467$	−4.81043	−0.222600	−0.111300	+	0.993787i	$0.535501\pi$
$468$	0	0
$469$	28.0139	1.29356
$470$	0	0
$471$	−15.7497	−0.725709
$472$	0	0
$473$	−33.1529	−1.52437
$474$	0	0
$475$	−6.35310	−0.291500
$476$	0	0
$477$	−17.8573	−0.817629
$478$	0	0
$479$	−30.8510	−1.40962	−0.704808	−	0.709398i	$-0.748967\pi$
$479$	−30.8510	−1.40962	−0.704808	+	0.709398i	$0.748967\pi$
$480$	0	0
$481$	−2.85016	−0.129956
$482$	0	0
$483$	−6.41910	−0.292079
$484$	0	0
$485$	11.5190	0.523049
$486$	0	0
$487$	−6.02841	−0.273173	−0.136587	−	0.990628i	$-0.543613\pi$
$487$	−6.02841	−0.273173	−0.136587	+	0.990628i	$0.543613\pi$
$488$	0	0
$489$	−18.1954	−0.822823
$490$	0	0
$491$	24.4426	1.10308	0.551541	−	0.834148i	$-0.314040\pi$
$491$	24.4426	1.10308	0.551541	+	0.834148i	$0.314040\pi$
$492$	0	0
$493$	−3.05094	−0.137408
$494$	0	0
$495$	−17.0087	−0.764486
$496$	0	0
$497$	−8.57957	−0.384846
$498$	0	0
$499$	−23.1493	−1.03630	−0.518152	−	0.855288i	$-0.673380\pi$
$499$	−23.1493	−1.03630	−0.518152	+	0.855288i	$0.673380\pi$
$500$	0	0
$501$	−3.42796	−0.153150
$502$	0	0
$503$	−7.54696	−0.336502	−0.168251	−	0.985744i	$-0.553812\pi$
$503$	−7.54696	−0.336502	−0.168251	+	0.985744i	$0.553812\pi$
$504$	0	0
$505$	32.9837	1.46775
$506$	0	0
$507$	−6.87202	−0.305197
$508$	0	0
$509$	−32.7985	−1.45377	−0.726885	−	0.686760i	$-0.759032\pi$
$509$	−32.7985	−1.45377	−0.726885	+	0.686760i	$0.759032\pi$
$510$	0	0
$511$	−11.6938	−0.517304
$512$	0	0
$513$	−11.7332	−0.518035
$514$	0	0
$515$	−36.8259	−1.62274
$516$	0	0
$517$	22.2430	0.978245
$518$	0	0
$519$	−1.32511	−0.0581660
$520$	0	0
$521$	−25.9842	−1.13839	−0.569194	−	0.822204i	$-0.692745\pi$
$521$	−25.9842	−1.13839	−0.569194	+	0.822204i	$0.692745\pi$
$522$	0	0
$523$	−17.9022	−0.782807	−0.391403	−	0.920219i	$-0.628010\pi$
$523$	−17.9022	−0.782807	−0.391403	+	0.920219i	$0.628010\pi$
$524$	0	0
$525$	−9.57418	−0.417851
$526$	0	0
$527$	6.82405	0.297260
$528$	0	0
$529$	−21.8500	−0.950001
$530$	0	0
$531$	1.62668	0.0705920
$532$	0	0
$533$	0.385243	0.0166867
$534$	0	0
$535$	−3.14799	−0.136100
$536$	0	0
$537$	−31.1989	−1.34633
$538$	0	0
$539$	3.02974	0.130500
$540$	0	0
$541$	23.9569	1.02999	0.514994	−	0.857194i	$-0.327794\pi$
$541$	23.9569	1.02999	0.514994	+	0.857194i	$0.327794\pi$
$542$	0	0
$543$	14.3837	0.617265
$544$	0	0
$545$	−35.8063	−1.53377
$546$	0	0
$547$	0.788372	0.0337083	0.0168542	−	0.999858i	$-0.494635\pi$
$547$	0.788372	0.0337083	0.0168542	+	0.999858i	$0.494635\pi$
$548$	0	0
$549$	7.51779	0.320851
$550$	0	0
$551$	12.1184	0.516262
$552$	0	0
$553$	21.5892	0.918065
$554$	0	0
$555$	3.91354	0.166121
$556$	0	0
$557$	−0.163360	−0.00692180	−0.00346090	−	0.999994i	$-0.501102\pi$
$557$	−0.163360	−0.00692180	−0.00346090	+	0.999994i	$0.501102\pi$
$558$	0	0
$559$	32.7789	1.38640
$560$	0	0
$561$	−8.75488	−0.369631
$562$	0	0
$563$	1.22643	0.0516877	0.0258438	−	0.999666i	$-0.491773\pi$
$563$	1.22643	0.0516877	0.0258438	+	0.999666i	$0.491773\pi$
$564$	0	0
$565$	−40.5586	−1.70631
$566$	0	0
$567$	−31.2626	−1.31291
$568$	0	0
$569$	35.9616	1.50759	0.753794	−	0.657111i	$-0.228222\pi$
$569$	35.9616	1.50759	0.753794	+	0.657111i	$0.228222\pi$
$570$	0	0
$571$	−4.93994	−0.206730	−0.103365	−	0.994643i	$-0.532961\pi$
$571$	−4.93994	−0.206730	−0.103365	+	0.994643i	$0.532961\pi$
$572$	0	0
$573$	14.4564	0.603926
$574$	0	0
$575$	1.71522	0.0715296
$576$	0	0
$577$	25.2170	1.04980	0.524899	−	0.851165i	$-0.324103\pi$
$577$	25.2170	1.04980	0.524899	+	0.851165i	$0.324103\pi$
$578$	0	0
$579$	19.1827	0.797206
$580$	0	0
$581$	−24.7395	−1.02637
$582$	0	0
$583$	−44.6816	−1.85052
$584$	0	0
$585$	16.8169	0.695292
$586$	0	0
$587$	40.4401	1.66914	0.834571	−	0.550901i	$-0.185716\pi$
$587$	40.4401	1.66914	0.834571	+	0.550901i	$0.185716\pi$
$588$	0	0
$589$	−27.1053	−1.11686
$590$	0	0
$591$	−54.5081	−2.24216
$592$	0	0
$593$	−37.9140	−1.55694	−0.778471	−	0.627680i	$-0.784005\pi$
$593$	−37.9140	−1.55694	−0.778471	+	0.627680i	$0.784005\pi$
$594$	0	0
$595$	−7.14903	−0.293082
$596$	0	0
$597$	31.4998	1.28920
$598$	0	0
$599$	20.6532	0.843865	0.421933	−	0.906627i	$-0.361352\pi$
$599$	20.6532	0.843865	0.421933	+	0.906627i	$0.361352\pi$
$600$	0	0
$601$	33.8849	1.38219	0.691097	−	0.722762i	$-0.257128\pi$
$601$	33.8849	1.38219	0.691097	+	0.722762i	$0.257128\pi$
$602$	0	0
$603$	16.3750	0.666843
$604$	0	0
$605$	−14.3001	−0.581380
$606$	0	0
$607$	3.77053	0.153041	0.0765205	−	0.997068i	$-0.475619\pi$
$607$	3.77053	0.153041	0.0765205	+	0.997068i	$0.475619\pi$
$608$	0	0
$609$	18.2626	0.740036
$610$	0	0
$611$	−21.9921	−0.889704
$612$	0	0
$613$	29.9125	1.20815	0.604077	−	0.796926i	$-0.293542\pi$
$613$	29.9125	1.20815	0.604077	+	0.796926i	$0.293542\pi$
$614$	0	0
$615$	−0.528975	−0.0213304
$616$	0	0
$617$	11.5907	0.466625	0.233313	−	0.972402i	$-0.425043\pi$
$617$	11.5907	0.466625	0.233313	+	0.972402i	$0.425043\pi$
$618$	0	0
$619$	−18.1090	−0.727864	−0.363932	−	0.931425i	$-0.618566\pi$
$619$	−18.1090	−0.727864	−0.363932	+	0.931425i	$0.618566\pi$
$620$	0	0
$621$	3.16776	0.127118
$622$	0	0
$623$	−1.30390	−0.0522397
$624$	0	0
$625$	−30.4390	−1.21756
$626$	0	0
$627$	34.7746	1.38876
$628$	0	0
$629$	0.708241	0.0282394
$630$	0	0
$631$	−31.9147	−1.27050	−0.635252	−	0.772305i	$-0.719104\pi$
$631$	−31.9147	−1.27050	−0.635252	+	0.772305i	$0.719104\pi$
$632$	0	0
$633$	11.4660	0.455734
$634$	0	0
$635$	−8.22330	−0.326332
$636$	0	0
$637$	−2.99556	−0.118688
$638$	0	0
$639$	−5.01505	−0.198392
$640$	0	0
$641$	−29.8271	−1.17810	−0.589049	−	0.808097i	$-0.700498\pi$
$641$	−29.8271	−1.17810	−0.589049	+	0.808097i	$0.700498\pi$
$642$	0	0
$643$	−5.31042	−0.209423	−0.104711	−	0.994503i	$-0.533392\pi$
$643$	−5.31042	−0.209423	−0.104711	+	0.994503i	$0.533392\pi$
$644$	0	0
$645$	−45.0086	−1.77221
$646$	0	0
$647$	9.70838	0.381676	0.190838	−	0.981622i	$-0.438879\pi$
$647$	9.70838	0.381676	0.190838	+	0.981622i	$0.438879\pi$
$648$	0	0
$649$	4.07020	0.159769
$650$	0	0
$651$	−40.8480	−1.60096
$652$	0	0
$653$	34.4421	1.34782	0.673912	−	0.738811i	$-0.264613\pi$
$653$	34.4421	1.34782	0.673912	+	0.738811i	$0.264613\pi$
$654$	0	0
$655$	−0.124377	−0.00485980
$656$	0	0
$657$	−6.83543	−0.266675
$658$	0	0
$659$	−1.93496	−0.0753753	−0.0376876	−	0.999290i	$-0.511999\pi$
$659$	−1.93496	−0.0753753	−0.0376876	+	0.999290i	$0.511999\pi$
$660$	0	0
$661$	−28.7033	−1.11643	−0.558214	−	0.829697i	$-0.688513\pi$
$661$	−28.7033	−1.11643	−0.558214	+	0.829697i	$0.688513\pi$
$662$	0	0
$663$	8.65612	0.336176
$664$	0	0
$665$	28.3961	1.10115
$666$	0	0
$667$	−3.27175	−0.126683
$668$	0	0
$669$	10.2715	0.397120
$670$	0	0
$671$	18.8106	0.726176
$672$	0	0
$673$	−5.02085	−0.193540	−0.0967698	−	0.995307i	$-0.530851\pi$
$673$	−5.02085	−0.193540	−0.0967698	+	0.995307i	$0.530851\pi$
$674$	0	0
$675$	4.72476	0.181856
$676$	0	0
$677$	−8.04055	−0.309023	−0.154512	−	0.987991i	$-0.549380\pi$
$677$	−8.04055	−0.309023	−0.154512	+	0.987991i	$0.549380\pi$
$678$	0	0
$679$	−12.4782	−0.478869
$680$	0	0
$681$	24.9570	0.956355
$682$	0	0
$683$	6.14061	0.234964	0.117482	−	0.993075i	$-0.462518\pi$
$683$	6.14061	0.234964	0.117482	+	0.993075i	$0.462518\pi$
$684$	0	0
$685$	−28.0420	−1.07143
$686$	0	0
$687$	32.6021	1.24385
$688$	0	0
$689$	44.1775	1.68303
$690$	0	0
$691$	−7.77780	−0.295881	−0.147941	−	0.988996i	$-0.547264\pi$
$691$	−7.77780	−0.295881	−0.147941	+	0.988996i	$0.547264\pi$
$692$	0	0
$693$	18.4252	0.699914
$694$	0	0
$695$	15.4100	0.584535
$696$	0	0
$697$	−0.0957297	−0.00362602
$698$	0	0
$699$	−11.5999	−0.438749
$700$	0	0
$701$	−25.3310	−0.956739	−0.478370	−	0.878159i	$-0.658772\pi$
$701$	−25.3310	−0.956739	−0.478370	+	0.878159i	$0.658772\pi$
$702$	0	0
$703$	−2.81315	−0.106100
$704$	0	0
$705$	30.1972	1.13729
$706$	0	0
$707$	−35.7304	−1.34378
$708$	0	0
$709$	−26.8701	−1.00913	−0.504564	−	0.863374i	$-0.668347\pi$
$709$	−26.8701	−1.00913	−0.504564	+	0.863374i	$0.668347\pi$
$710$	0	0
$711$	12.6196	0.473272
$712$	0	0
$713$	7.31793	0.274059
$714$	0	0
$715$	42.0783	1.57364
$716$	0	0
$717$	−11.4830	−0.428839
$718$	0	0
$719$	0.872600	0.0325425	0.0162712	−	0.999868i	$-0.494820\pi$
$719$	0.872600	0.0325425	0.0162712	+	0.999868i	$0.494820\pi$
$720$	0	0
$721$	39.8926	1.48568
$722$	0	0
$723$	−10.1041	−0.375775
$724$	0	0
$725$	−4.87986	−0.181234
$726$	0	0
$727$	8.18342	0.303506	0.151753	−	0.988418i	$-0.451508\pi$
$727$	8.18342	0.303506	0.151753	+	0.988418i	$0.451508\pi$
$728$	0	0
$729$	0.787667	0.0291728
$730$	0	0
$731$	−8.14529	−0.301264
$732$	0	0
$733$	−35.5065	−1.31146	−0.655731	−	0.754995i	$-0.727639\pi$
$733$	−35.5065	−1.31146	−0.655731	+	0.754995i	$0.727639\pi$
$734$	0	0
$735$	4.11319	0.151717
$736$	0	0
$737$	40.9728	1.50925
$738$	0	0
$739$	8.43988	0.310466	0.155233	−	0.987878i	$-0.450387\pi$
$739$	8.43988	0.310466	0.155233	+	0.987878i	$0.450387\pi$
$740$	0	0
$741$	−34.3823	−1.26307
$742$	0	0
$743$	35.4900	1.30200	0.651002	−	0.759076i	$-0.274349\pi$
$743$	35.4900	1.30200	0.651002	+	0.759076i	$0.274349\pi$
$744$	0	0
$745$	−20.8037	−0.762189
$746$	0	0
$747$	−14.4611	−0.529102
$748$	0	0
$749$	3.41014	0.124604
$750$	0	0
$751$	−26.1853	−0.955516	−0.477758	−	0.878491i	$-0.658550\pi$
$751$	−26.1853	−0.955516	−0.477758	+	0.878491i	$0.658550\pi$
$752$	0	0
$753$	26.1304	0.952245
$754$	0	0
$755$	40.2410	1.46452
$756$	0	0
$757$	28.6123	1.03993	0.519966	−	0.854187i	$-0.325945\pi$
$757$	28.6123	1.03993	0.519966	+	0.854187i	$0.325945\pi$
$758$	0	0
$759$	−9.38850	−0.340781
$760$	0	0
$761$	48.2377	1.74862	0.874308	−	0.485371i	$-0.161315\pi$
$761$	48.2377	1.74862	0.874308	+	0.485371i	$0.161315\pi$
$762$	0	0
$763$	38.7880	1.40422
$764$	0	0
$765$	−4.17885	−0.151087
$766$	0	0
$767$	−4.02428	−0.145308
$768$	0	0
$769$	−41.0100	−1.47886	−0.739429	−	0.673234i	$-0.764905\pi$
$769$	−41.0100	−1.47886	−0.739429	+	0.673234i	$0.764905\pi$
$770$	0	0
$771$	17.8623	0.643294
$772$	0	0
$773$	23.8455	0.857662	0.428831	−	0.903385i	$-0.358926\pi$
$773$	23.8455	0.857662	0.428831	+	0.903385i	$0.358926\pi$
$774$	0	0
$775$	10.9148	0.392072
$776$	0	0
$777$	−4.23944	−0.152089
$778$	0	0
$779$	0.380241	0.0136235
$780$	0	0
$781$	−12.5484	−0.449017
$782$	0	0
$783$	−9.01239	−0.322077
$784$	0	0
$785$	−18.8102	−0.671363
$786$	0	0
$787$	−8.14523	−0.290346	−0.145173	−	0.989406i	$-0.546374\pi$
$787$	−8.14523	−0.290346	−0.145173	+	0.989406i	$0.546374\pi$
$788$	0	0
$789$	4.61715	0.164375
$790$	0	0
$791$	43.9361	1.56219
$792$	0	0
$793$	−18.5984	−0.660449
$794$	0	0
$795$	−60.6600	−2.15139
$796$	0	0
$797$	−20.2895	−0.718692	−0.359346	−	0.933204i	$-0.617000\pi$
$797$	−20.2895	−0.718692	−0.359346	+	0.933204i	$0.617000\pi$
$798$	0	0
$799$	5.46484	0.193332
$800$	0	0
$801$	−0.762173	−0.0269301
$802$	0	0
$803$	−17.1033	−0.603561
$804$	0	0
$805$	−7.66643	−0.270206
$806$	0	0
$807$	44.3163	1.56001
$808$	0	0
$809$	2.08834	0.0734223	0.0367111	−	0.999326i	$-0.488312\pi$
$809$	2.08834	0.0734223	0.0367111	+	0.999326i	$0.488312\pi$
$810$	0	0
$811$	5.60880	0.196952	0.0984759	−	0.995139i	$-0.468603\pi$
$811$	5.60880	0.196952	0.0984759	+	0.995139i	$0.468603\pi$
$812$	0	0
$813$	−7.16496	−0.251286
$814$	0	0
$815$	−21.7310	−0.761205
$816$	0	0
$817$	32.3533	1.13190
$818$	0	0
$819$	−18.2173	−0.636564
$820$	0	0
$821$	47.9842	1.67466	0.837331	−	0.546697i	$-0.184115\pi$
$821$	47.9842	1.67466	0.837331	+	0.546697i	$0.184115\pi$
$822$	0	0
$823$	35.8135	1.24838	0.624190	−	0.781272i	$-0.285429\pi$
$823$	35.8135	1.24838	0.624190	+	0.781272i	$0.285429\pi$
$824$	0	0
$825$	−14.0031	−0.487525
$826$	0	0
$827$	−14.9734	−0.520675	−0.260338	−	0.965518i	$-0.583834\pi$
$827$	−14.9734	−0.520675	−0.260338	+	0.965518i	$0.583834\pi$
$828$	0	0
$829$	−6.64225	−0.230695	−0.115347	−	0.993325i	$-0.536798\pi$
$829$	−6.64225	−0.230695	−0.115347	+	0.993325i	$0.536798\pi$
$830$	0	0
$831$	18.8213	0.652905
$832$	0	0
$833$	0.744371	0.0257909
$834$	0	0
$835$	−4.09407	−0.141681
$836$	0	0
$837$	20.1581	0.696764
$838$	0	0
$839$	−37.6825	−1.30095	−0.650473	−	0.759530i	$-0.725429\pi$
$839$	−37.6825	−1.30095	−0.650473	+	0.759530i	$0.725429\pi$
$840$	0	0
$841$	−19.6918	−0.679026
$842$	0	0
$843$	−19.4731	−0.670688
$844$	0	0
$845$	−8.20737	−0.282342
$846$	0	0
$847$	15.4909	0.532273
$848$	0	0
$849$	−30.3289	−1.04089
$850$	0	0
$851$	0.759499	0.0260353
$852$	0	0
$853$	6.14586	0.210430	0.105215	−	0.994449i	$-0.466447\pi$
$853$	6.14586	0.210430	0.105215	+	0.994449i	$0.466447\pi$
$854$	0	0
$855$	16.5985	0.567656
$856$	0	0
$857$	25.0039	0.854117	0.427058	−	0.904224i	$-0.359550\pi$
$857$	25.0039	0.854117	0.427058	+	0.904224i	$0.359550\pi$
$858$	0	0
$859$	18.1445	0.619083	0.309541	−	0.950886i	$-0.399825\pi$
$859$	18.1445	0.619083	0.309541	+	0.950886i	$0.399825\pi$
$860$	0	0
$861$	0.573026	0.0195287
$862$	0	0
$863$	10.9697	0.373413	0.186706	−	0.982416i	$-0.440219\pi$
$863$	10.9697	0.373413	0.186706	+	0.982416i	$0.440219\pi$
$864$	0	0
$865$	−1.58260	−0.0538101
$866$	0	0
$867$	−2.15097	−0.0730508
$868$	0	0
$869$	31.5761	1.07115
$870$	0	0
$871$	−40.5106	−1.37265
$872$	0	0
$873$	−7.29393	−0.246862
$874$	0	0
$875$	24.3106	0.821847
$876$	0	0
$877$	−41.8914	−1.41457	−0.707287	−	0.706927i	$-0.750081\pi$
$877$	−41.8914	−1.41457	−0.707287	+	0.706927i	$0.750081\pi$
$878$	0	0
$879$	−40.4555	−1.36453
$880$	0	0
$881$	−24.0269	−0.809487	−0.404744	−	0.914430i	$-0.632639\pi$
$881$	−24.0269	−0.809487	−0.404744	+	0.914430i	$0.632639\pi$
$882$	0	0
$883$	−29.1539	−0.981108	−0.490554	−	0.871411i	$-0.663206\pi$
$883$	−29.1539	−0.981108	−0.490554	+	0.871411i	$0.663206\pi$
$884$	0	0
$885$	5.52572	0.185745
$886$	0	0
$887$	−20.6956	−0.694889	−0.347445	−	0.937700i	$-0.612951\pi$
$887$	−20.6956	−0.694889	−0.347445	+	0.937700i	$0.612951\pi$
$888$	0	0
$889$	8.90810	0.298768
$890$	0	0
$891$	−45.7244	−1.53183
$892$	0	0
$893$	−21.7065	−0.726380
$894$	0	0
$895$	−37.2613	−1.24551
$896$	0	0
$897$	9.28259	0.309937
$898$	0	0
$899$	−20.8198	−0.694379
$900$	0	0
$901$	−10.9777	−0.365722
$902$	0	0
$903$	48.7567	1.62252
$904$	0	0
$905$	17.1787	0.571040
$906$	0	0
$907$	−19.2230	−0.638289	−0.319144	−	0.947706i	$-0.603395\pi$
$907$	−19.2230	−0.638289	−0.319144	+	0.947706i	$0.603395\pi$
$908$	0	0
$909$	−20.8856	−0.692732
$910$	0	0
$911$	50.4760	1.67234	0.836172	−	0.548468i	$-0.184789\pi$
$911$	50.4760	1.67234	0.836172	+	0.548468i	$0.184789\pi$
$912$	0	0
$913$	−36.1837	−1.19751
$914$	0	0
$915$	25.5374	0.844240
$916$	0	0
$917$	0.134734	0.00444932
$918$	0	0
$919$	45.2306	1.49202	0.746009	−	0.665936i	$-0.231967\pi$
$919$	45.2306	1.49202	0.746009	+	0.665936i	$0.231967\pi$
$920$	0	0
$921$	52.4778	1.72920
$922$	0	0
$923$	12.4068	0.408376
$924$	0	0
$925$	1.13280	0.0372464
$926$	0	0
$927$	23.3185	0.765881
$928$	0	0
$929$	−17.9569	−0.589145	−0.294573	−	0.955629i	$-0.595177\pi$
$929$	−17.9569	−0.589145	−0.294573	+	0.955629i	$0.595177\pi$
$930$	0	0
$931$	−2.95666	−0.0969006
$932$	0	0
$933$	61.0924	2.00007
$934$	0	0
$935$	−10.4561	−0.341951
$936$	0	0
$937$	−30.8128	−1.00661	−0.503305	−	0.864109i	$-0.667883\pi$
$937$	−30.8128	−1.00661	−0.503305	+	0.864109i	$0.667883\pi$
$938$	0	0
$939$	−52.8435	−1.72448
$940$	0	0
$941$	−7.92858	−0.258464	−0.129232	−	0.991614i	$-0.541251\pi$
$941$	−7.92858	−0.258464	−0.129232	+	0.991614i	$0.541251\pi$
$942$	0	0
$943$	−0.102658	−0.00334300
$944$	0	0
$945$	−21.1180	−0.686970
$946$	0	0
$947$	2.85955	0.0929228	0.0464614	−	0.998920i	$-0.485206\pi$
$947$	2.85955	0.0929228	0.0464614	+	0.998920i	$0.485206\pi$
$948$	0	0
$949$	16.9103	0.548932
$950$	0	0
$951$	46.3266	1.50224
$952$	0	0
$953$	20.5248	0.664864	0.332432	−	0.943127i	$-0.392131\pi$
$953$	20.5248	0.664864	0.332432	+	0.943127i	$0.392131\pi$
$954$	0	0
$955$	17.2656	0.558701
$956$	0	0
$957$	26.7106	0.863432
$958$	0	0
$959$	30.3772	0.980931
$960$	0	0
$961$	15.5677	0.502185
$962$	0	0
$963$	1.99334	0.0642346
$964$	0	0
$965$	22.9102	0.737506
$966$	0	0
$967$	44.7566	1.43927	0.719637	−	0.694351i	$-0.244308\pi$
$967$	44.7566	1.43927	0.719637	+	0.694351i	$0.244308\pi$
$968$	0	0
$969$	8.54371	0.274464
$970$	0	0
$971$	17.5468	0.563103	0.281552	−	0.959546i	$-0.409151\pi$
$971$	17.5468	0.563103	0.281552	+	0.959546i	$0.409151\pi$
$972$	0	0
$973$	−16.6933	−0.535162
$974$	0	0
$975$	13.8451	0.443399
$976$	0	0
$977$	−6.75312	−0.216052	−0.108026	−	0.994148i	$-0.534453\pi$
$977$	−6.75312	−0.216052	−0.108026	+	0.994148i	$0.534453\pi$
$978$	0	0
$979$	−1.90707	−0.0609503
$980$	0	0
$981$	22.6729	0.723890
$982$	0	0
$983$	−10.7230	−0.342012	−0.171006	−	0.985270i	$-0.554702\pi$
$983$	−10.7230	−0.342012	−0.171006	+	0.985270i	$0.554702\pi$
$984$	0	0
$985$	−65.1000	−2.07426
$986$	0	0
$987$	−32.7119	−1.04123
$988$	0	0
$989$	−8.73479	−0.277750
$990$	0	0
$991$	−54.2504	−1.72332	−0.861660	−	0.507487i	$-0.830575\pi$
$991$	−54.2504	−1.72332	−0.861660	+	0.507487i	$0.830575\pi$
$992$	0	0
$993$	36.5370	1.15947
$994$	0	0
$995$	37.6207	1.19266
$996$	0	0
$997$	−41.6793	−1.32000	−0.659998	−	0.751268i	$-0.729443\pi$
$997$	−41.6793	−1.32000	−0.659998	+	0.751268i	$0.729443\pi$
$998$	0	0
$999$	2.09212	0.0661918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.z.1.6	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.z.1.6	✓	24	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15097 q^{3} -2.56894 q^{5} +2.78287 q^{7} +1.62668 q^{9} +O(q^{10})\)	\(q-2.15097 q^{3} -2.56894 q^{5} +2.78287 q^{7} +1.62668 q^{9} +4.07020 q^{11} -4.02428 q^{13} +5.52572 q^{15} +1.00000 q^{17} -3.97202 q^{19} -5.98588 q^{21} +1.07237 q^{23} +1.59946 q^{25} +2.95397 q^{27} -3.05094 q^{29} +6.82405 q^{31} -8.75488 q^{33} -7.14903 q^{35} +0.708241 q^{37} +8.65612 q^{39} -0.0957297 q^{41} -8.14529 q^{43} -4.17885 q^{45} +5.46484 q^{47} +0.744371 q^{49} -2.15097 q^{51} -10.9777 q^{53} -10.4561 q^{55} +8.54371 q^{57} +1.00000 q^{59} +4.62155 q^{61} +4.52684 q^{63} +10.3381 q^{65} +10.0665 q^{67} -2.30665 q^{69} -3.08299 q^{71} -4.20207 q^{73} -3.44040 q^{75} +11.3268 q^{77} +7.75788 q^{79} -11.2340 q^{81} -8.88992 q^{83} -2.56894 q^{85} +6.56249 q^{87} -0.468545 q^{89} -11.1991 q^{91} -14.6784 q^{93} +10.2039 q^{95} -4.48393 q^{97} +6.62092 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9}+O(q^{10}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9	\( 24 q - 5 q^{3} - 7 q^{5} - 12 q^{7} + 33 q^{9} - 15 q^{11} - 17 q^{13} - 7 q^{15} + 24 q^{17} - q^{19} + 10 q^{21} - 21 q^{23} + 19 q^{25} - 26 q^{27} - 18 q^{29} - 23 q^{31} + 10 q^{33} - 5 q^{35} - 29 q^{37} - 44 q^{39} + 14 q^{41} - 23 q^{43} - 24 q^{45} - 17 q^{47} + 46 q^{49} - 5 q^{51} + 7 q^{53} - 33 q^{55} - 2 q^{57} + 24 q^{59} - 9 q^{61} - 52 q^{63} - 15 q^{65} + 2 q^{67} + 2 q^{69} - 51 q^{71} - 18 q^{73} + 7 q^{75} + 14 q^{77} - 36 q^{79} + 12 q^{81} - 39 q^{83} - 7 q^{85} - 5 q^{87} + 46 q^{89} - 52 q^{91} - 20 q^{93} - 52 q^{95} + 50 q^{97} - 53 q^{99}+O(q^{100}) \) 24 * q - 5 * q^3 - 7 * q^5 - 12 * q^7 + 33 * q^9 - 15 * q^11 - 17 * q^13 - 7 * q^15 + 24 * q^17 - q^19 + 10 * q^21 - 21 * q^23 + 19 * q^25 - 26 * q^27 - 18 * q^29 - 23 * q^31 + 10 * q^33 - 5 * q^35 - 29 * q^37 - 44 * q^39 + 14 * q^41 - 23 * q^43 - 24 * q^45 - 17 * q^47 + 46 * q^49 - 5 * q^51 + 7 * q^53 - 33 * q^55 - 2 * q^57 + 24 * q^59 - 9 * q^61 - 52 * q^63 - 15 * q^65 + 2 * q^67 + 2 * q^69 - 51 * q^71 - 18 * q^73 + 7 * q^75 + 14 * q^77 - 36 * q^79 + 12 * q^81 - 39 * q^83 - 7 * q^85 - 5 * q^87 + 46 * q^89 - 52 * q^91 - 20 * q^93 - 52 * q^95 + 50 * q^97 - 53 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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