LMFDB - Embedded newform 4024.2.a.g.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.14143 q^{3} +2.09880 q^{5} +3.79183 q^{7} +1.58570 q^{9} +O(q^{10})$	$q-2.14143 q^{3} +2.09880 q^{5} +3.79183 q^{7} +1.58570 q^{9} +1.14326 q^{11} +1.19641 q^{13} -4.49442 q^{15} +0.537255 q^{17} +6.83130 q^{19} -8.11993 q^{21} -1.54695 q^{23} -0.595048 q^{25} +3.02861 q^{27} +2.80657 q^{29} +5.34282 q^{31} -2.44821 q^{33} +7.95829 q^{35} +0.325571 q^{37} -2.56203 q^{39} +11.9253 q^{41} -1.80114 q^{43} +3.32807 q^{45} -10.1003 q^{47} +7.37800 q^{49} -1.15049 q^{51} -10.0676 q^{53} +2.39947 q^{55} -14.6287 q^{57} +10.7309 q^{59} -7.07687 q^{61} +6.01272 q^{63} +2.51103 q^{65} -6.68983 q^{67} +3.31268 q^{69} +14.8568 q^{71} +4.74311 q^{73} +1.27425 q^{75} +4.33505 q^{77} +0.815987 q^{79} -11.2427 q^{81} +14.7342 q^{83} +1.12759 q^{85} -6.01006 q^{87} -1.94781 q^{89} +4.53659 q^{91} -11.4413 q^{93} +14.3375 q^{95} -10.1984 q^{97} +1.81287 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * 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
$3$	−2.14143	−1.23635	−0.618176	−	0.786040i	$-0.712128\pi$
$3$	−2.14143	−1.23635	−0.618176	+	0.786040i	$0.712128\pi$
$4$	0	0
$5$	2.09880	0.938611	0.469305	−	0.883036i	$-0.344504\pi$
$5$	2.09880	0.938611	0.469305	+	0.883036i	$0.344504\pi$
$6$	0	0
$7$	3.79183	1.43318	0.716589	−	0.697496i	$-0.245702\pi$
$7$	3.79183	1.43318	0.716589	+	0.697496i	$0.245702\pi$
$8$	0	0
$9$	1.58570	0.528567
$10$	0	0
$11$	1.14326	0.344706	0.172353	−	0.985035i	$-0.444863\pi$
$11$	1.14326	0.344706	0.172353	+	0.985035i	$0.444863\pi$
$12$	0	0
$13$	1.19641	0.331825	0.165912	−	0.986140i	$-0.446943\pi$
$13$	1.19641	0.331825	0.165912	+	0.986140i	$0.446943\pi$
$14$	0	0
$15$	−4.49442	−1.16045
$16$	0	0
$17$	0.537255	0.130304	0.0651518	−	0.997875i	$-0.479247\pi$
$17$	0.537255	0.130304	0.0651518	+	0.997875i	$0.479247\pi$
$18$	0	0
$19$	6.83130	1.56721	0.783604	−	0.621261i	$-0.213379\pi$
$19$	6.83130	1.56721	0.783604	+	0.621261i	$0.213379\pi$
$20$	0	0
$21$	−8.11993	−1.77191
$22$	0	0
$23$	−1.54695	−0.322562	−0.161281	−	0.986909i	$-0.551562\pi$
$23$	−1.54695	−0.322562	−0.161281	+	0.986909i	$0.551562\pi$
$24$	0	0
$25$	−0.595048	−0.119010
$26$	0	0
$27$	3.02861	0.582857
$28$	0	0
$29$	2.80657	0.521167	0.260584	−	0.965451i	$-0.416085\pi$
$29$	2.80657	0.521167	0.260584	+	0.965451i	$0.416085\pi$
$30$	0	0
$31$	5.34282	0.959599	0.479800	−	0.877378i	$-0.340709\pi$
$31$	5.34282	0.959599	0.479800	+	0.877378i	$0.340709\pi$
$32$	0	0
$33$	−2.44821	−0.426178
$34$	0	0
$35$	7.95829	1.34520
$36$	0	0
$37$	0.325571	0.0535236	0.0267618	−	0.999642i	$-0.491480\pi$
$37$	0.325571	0.0535236	0.0267618	+	0.999642i	$0.491480\pi$
$38$	0	0
$39$	−2.56203	−0.410253
$40$	0	0
$41$	11.9253	1.86241	0.931207	−	0.364491i	$-0.118757\pi$
$41$	11.9253	1.86241	0.931207	+	0.364491i	$0.118757\pi$
$42$	0	0
$43$	−1.80114	−0.274672	−0.137336	−	0.990525i	$-0.543854\pi$
$43$	−1.80114	−0.274672	−0.137336	+	0.990525i	$0.543854\pi$
$44$	0	0
$45$	3.32807	0.496119
$46$	0	0
$47$	−10.1003	−1.47328	−0.736640	−	0.676285i	$-0.763589\pi$
$47$	−10.1003	−1.47328	−0.736640	+	0.676285i	$0.763589\pi$
$48$	0	0
$49$	7.37800	1.05400
$50$	0	0
$51$	−1.15049	−0.161101
$52$	0	0
$53$	−10.0676	−1.38289	−0.691445	−	0.722429i	$-0.743026\pi$
$53$	−10.0676	−1.38289	−0.691445	+	0.722429i	$0.743026\pi$
$54$	0	0
$55$	2.39947	0.323545
$56$	0	0
$57$	−14.6287	−1.93762
$58$	0	0
$59$	10.7309	1.39704	0.698521	−	0.715590i	$-0.253842\pi$
$59$	10.7309	1.39704	0.698521	+	0.715590i	$0.253842\pi$
$60$	0	0
$61$	−7.07687	−0.906100	−0.453050	−	0.891485i	$-0.649664\pi$
$61$	−7.07687	−0.906100	−0.453050	+	0.891485i	$0.649664\pi$
$62$	0	0
$63$	6.01272	0.757531
$64$	0	0
$65$	2.51103	0.311455
$66$	0	0
$67$	−6.68983	−0.817293	−0.408646	−	0.912693i	$-0.633999\pi$
$67$	−6.68983	−0.817293	−0.408646	+	0.912693i	$0.633999\pi$
$68$	0	0
$69$	3.31268	0.398800
$70$	0	0
$71$	14.8568	1.76318	0.881592	−	0.472013i	$-0.156473\pi$
$71$	14.8568	1.76318	0.881592	+	0.472013i	$0.156473\pi$
$72$	0	0
$73$	4.74311	0.555139	0.277570	−	0.960706i	$-0.410471\pi$
$73$	4.74311	0.555139	0.277570	+	0.960706i	$0.410471\pi$
$74$	0	0
$75$	1.27425	0.147138
$76$	0	0
$77$	4.33505	0.494025
$78$	0	0
$79$	0.815987	0.0918057	0.0459029	−	0.998946i	$-0.485384\pi$
$79$	0.815987	0.0918057	0.0459029	+	0.998946i	$0.485384\pi$
$80$	0	0
$81$	−11.2427	−1.24918
$82$	0	0
$83$	14.7342	1.61729	0.808643	−	0.588299i	$-0.200202\pi$
$83$	14.7342	1.61729	0.808643	+	0.588299i	$0.200202\pi$
$84$	0	0
$85$	1.12759	0.122304
$86$	0	0
$87$	−6.01006	−0.644346
$88$	0	0
$89$	−1.94781	−0.206467	−0.103233	−	0.994657i	$-0.532919\pi$
$89$	−1.94781	−0.206467	−0.103233	+	0.994657i	$0.532919\pi$
$90$	0	0
$91$	4.53659	0.475564
$92$	0	0
$93$	−11.4413	−1.18640
$94$	0	0
$95$	14.3375	1.47100
$96$	0	0
$97$	−10.1984	−1.03549	−0.517745	−	0.855535i	$-0.673228\pi$
$97$	−10.1984	−1.03549	−0.517745	+	0.855535i	$0.673228\pi$
$98$	0	0
$99$	1.81287	0.182200
$100$	0	0
$101$	−19.3728	−1.92767	−0.963833	−	0.266509i	$-0.914130\pi$
$101$	−19.3728	−1.92767	−0.963833	+	0.266509i	$0.914130\pi$
$102$	0	0
$103$	−7.94070	−0.782421	−0.391210	−	0.920301i	$-0.627944\pi$
$103$	−7.94070	−0.782421	−0.391210	+	0.920301i	$0.627944\pi$
$104$	0	0
$105$	−17.0421	−1.66314
$106$	0	0
$107$	−3.65559	−0.353399	−0.176700	−	0.984265i	$-0.556542\pi$
$107$	−3.65559	−0.353399	−0.176700	+	0.984265i	$0.556542\pi$
$108$	0	0
$109$	−17.1625	−1.64387	−0.821935	−	0.569581i	$-0.807105\pi$
$109$	−17.1625	−1.64387	−0.821935	+	0.569581i	$0.807105\pi$
$110$	0	0
$111$	−0.697186	−0.0661740
$112$	0	0
$113$	2.47789	0.233101	0.116550	−	0.993185i	$-0.462816\pi$
$113$	2.47789	0.233101	0.116550	+	0.993185i	$0.462816\pi$
$114$	0	0
$115$	−3.24674	−0.302760
$116$	0	0
$117$	1.89715	0.175392
$118$	0	0
$119$	2.03718	0.186748
$120$	0	0
$121$	−9.69295	−0.881178
$122$	0	0
$123$	−25.5371	−2.30260
$124$	0	0
$125$	−11.7429	−1.05031
$126$	0	0
$127$	−13.0199	−1.15533	−0.577666	−	0.816273i	$-0.696036\pi$
$127$	−13.0199	−1.15533	−0.577666	+	0.816273i	$0.696036\pi$
$128$	0	0
$129$	3.85702	0.339591
$130$	0	0
$131$	18.7752	1.64040	0.820201	−	0.572076i	$-0.193862\pi$
$131$	18.7752	1.64040	0.820201	+	0.572076i	$0.193862\pi$
$132$	0	0
$133$	25.9031	2.24609
$134$	0	0
$135$	6.35645	0.547076
$136$	0	0
$137$	13.2844	1.13496	0.567480	−	0.823387i	$-0.307918\pi$
$137$	13.2844	1.13496	0.567480	+	0.823387i	$0.307918\pi$
$138$	0	0
$139$	−11.7308	−0.994990	−0.497495	−	0.867467i	$-0.665747\pi$
$139$	−11.7308	−0.994990	−0.497495	+	0.867467i	$0.665747\pi$
$140$	0	0
$141$	21.6290	1.82149
$142$	0	0
$143$	1.36781	0.114382
$144$	0	0
$145$	5.89042	0.489173
$146$	0	0
$147$	−15.7994	−1.30312
$148$	0	0
$149$	0.555974	0.0455471	0.0227736	−	0.999741i	$-0.492750\pi$
$149$	0.555974	0.0455471	0.0227736	+	0.999741i	$0.492750\pi$
$150$	0	0
$151$	14.1431	1.15095	0.575474	−	0.817820i	$-0.304817\pi$
$151$	14.1431	1.15095	0.575474	+	0.817820i	$0.304817\pi$
$152$	0	0
$153$	0.851927	0.0688742
$154$	0	0
$155$	11.2135	0.900691
$156$	0	0
$157$	8.26127	0.659321	0.329661	−	0.944100i	$-0.393066\pi$
$157$	8.26127	0.659321	0.329661	+	0.944100i	$0.393066\pi$
$158$	0	0
$159$	21.5590	1.70974
$160$	0	0
$161$	−5.86578	−0.462288
$162$	0	0
$163$	9.25871	0.725198	0.362599	−	0.931945i	$-0.381889\pi$
$163$	9.25871	0.725198	0.362599	+	0.931945i	$0.381889\pi$
$164$	0	0
$165$	−5.13829	−0.400016
$166$	0	0
$167$	4.31718	0.334074	0.167037	−	0.985951i	$-0.446580\pi$
$167$	4.31718	0.334074	0.167037	+	0.985951i	$0.446580\pi$
$168$	0	0
$169$	−11.5686	−0.889892
$170$	0	0
$171$	10.8324	0.828375
$172$	0	0
$173$	4.65467	0.353888	0.176944	−	0.984221i	$-0.443379\pi$
$173$	4.65467	0.353888	0.176944	+	0.984221i	$0.443379\pi$
$174$	0	0
$175$	−2.25632	−0.170562
$176$	0	0
$177$	−22.9794	−1.72724
$178$	0	0
$179$	−18.3952	−1.37492	−0.687462	−	0.726220i	$-0.741275\pi$
$179$	−18.3952	−1.37492	−0.687462	+	0.726220i	$0.741275\pi$
$180$	0	0
$181$	−11.6301	−0.864459	−0.432229	−	0.901764i	$-0.642273\pi$
$181$	−11.6301	−0.864459	−0.432229	+	0.901764i	$0.642273\pi$
$182$	0	0
$183$	15.1546	1.12026
$184$	0	0
$185$	0.683308	0.0502378
$186$	0	0
$187$	0.614223	0.0449164
$188$	0	0
$189$	11.4840	0.835338
$190$	0	0
$191$	3.25338	0.235407	0.117703	−	0.993049i	$-0.462447\pi$
$191$	3.25338	0.235407	0.117703	+	0.993049i	$0.462447\pi$
$192$	0	0
$193$	24.4037	1.75662	0.878309	−	0.478094i	$-0.158672\pi$
$193$	24.4037	1.75662	0.878309	+	0.478094i	$0.158672\pi$
$194$	0	0
$195$	−5.37718	−0.385068
$196$	0	0
$197$	−14.8798	−1.06014	−0.530069	−	0.847954i	$-0.677834\pi$
$197$	−14.8798	−1.06014	−0.530069	+	0.847954i	$0.677834\pi$
$198$	0	0
$199$	20.3714	1.44409	0.722046	−	0.691845i	$-0.243202\pi$
$199$	20.3714	1.44409	0.722046	+	0.691845i	$0.243202\pi$
$200$	0	0
$201$	14.3258	1.01046
$202$	0	0
$203$	10.6420	0.746925
$204$	0	0
$205$	25.0287	1.74808
$206$	0	0
$207$	−2.45300	−0.170496
$208$	0	0
$209$	7.80996	0.540226
$210$	0	0
$211$	−0.0374307	−0.00257684	−0.00128842	−	0.999999i	$-0.500410\pi$
$211$	−0.0374307	−0.00257684	−0.00128842	+	0.999999i	$0.500410\pi$
$212$	0	0
$213$	−31.8148	−2.17992
$214$	0	0
$215$	−3.78024	−0.257810
$216$	0	0
$217$	20.2591	1.37528
$218$	0	0
$219$	−10.1570	−0.686347
$220$	0	0
$221$	0.642779	0.0432380
$222$	0	0
$223$	3.37003	0.225674	0.112837	−	0.993614i	$-0.464006\pi$
$223$	3.37003	0.225674	0.112837	+	0.993614i	$0.464006\pi$
$224$	0	0
$225$	−0.943568	−0.0629045
$226$	0	0
$227$	10.5598	0.700882	0.350441	−	0.936585i	$-0.386032\pi$
$227$	10.5598	0.700882	0.350441	+	0.936585i	$0.386032\pi$
$228$	0	0
$229$	16.0769	1.06239	0.531196	−	0.847249i	$-0.321743\pi$
$229$	16.0769	1.06239	0.531196	+	0.847249i	$0.321743\pi$
$230$	0	0
$231$	−9.28320	−0.610789
$232$	0	0
$233$	−10.5821	−0.693255	−0.346627	−	0.938003i	$-0.612673\pi$
$233$	−10.5821	−0.693255	−0.346627	+	0.938003i	$0.612673\pi$
$234$	0	0
$235$	−21.1985	−1.38284
$236$	0	0
$237$	−1.74738	−0.113504
$238$	0	0
$239$	4.33190	0.280207	0.140104	−	0.990137i	$-0.455256\pi$
$239$	4.33190	0.280207	0.140104	+	0.990137i	$0.455256\pi$
$240$	0	0
$241$	19.4081	1.25018	0.625092	−	0.780551i	$-0.285061\pi$
$241$	19.4081	1.25018	0.625092	+	0.780551i	$0.285061\pi$
$242$	0	0
$243$	14.9895	0.961575
$244$	0	0
$245$	15.4849	0.989296
$246$	0	0
$247$	8.17305	0.520039
$248$	0	0
$249$	−31.5522	−1.99954
$250$	0	0
$251$	15.7569	0.994565	0.497283	−	0.867589i	$-0.334331\pi$
$251$	15.7569	0.994565	0.497283	+	0.867589i	$0.334331\pi$
$252$	0	0
$253$	−1.76857	−0.111189
$254$	0	0
$255$	−2.41465	−0.151211
$256$	0	0
$257$	9.16690	0.571816	0.285908	−	0.958257i	$-0.407705\pi$
$257$	9.16690	0.571816	0.285908	+	0.958257i	$0.407705\pi$
$258$	0	0
$259$	1.23451	0.0767088
$260$	0	0
$261$	4.45039	0.275472
$262$	0	0
$263$	28.0001	1.72656	0.863279	−	0.504727i	$-0.168407\pi$
$263$	28.0001	1.72656	0.863279	+	0.504727i	$0.168407\pi$
$264$	0	0
$265$	−21.1298	−1.29800
$266$	0	0
$267$	4.17108	0.255266
$268$	0	0
$269$	5.55853	0.338909	0.169455	−	0.985538i	$-0.445799\pi$
$269$	5.55853	0.338909	0.169455	+	0.985538i	$0.445799\pi$
$270$	0	0
$271$	21.2839	1.29291	0.646453	−	0.762954i	$-0.276252\pi$
$271$	21.2839	1.29291	0.646453	+	0.762954i	$0.276252\pi$
$272$	0	0
$273$	−9.71478	−0.587965
$274$	0	0
$275$	−0.680295	−0.0410233
$276$	0	0
$277$	−11.5345	−0.693042	−0.346521	−	0.938042i	$-0.612637\pi$
$277$	−11.5345	−0.693042	−0.346521	+	0.938042i	$0.612637\pi$
$278$	0	0
$279$	8.47213	0.507213
$280$	0	0
$281$	26.9575	1.60815	0.804076	−	0.594527i	$-0.202661\pi$
$281$	26.9575	1.60815	0.804076	+	0.594527i	$0.202661\pi$
$282$	0	0
$283$	−10.7910	−0.641459	−0.320730	−	0.947171i	$-0.603928\pi$
$283$	−10.7910	−0.641459	−0.320730	+	0.947171i	$0.603928\pi$
$284$	0	0
$285$	−30.7027	−1.81867
$286$	0	0
$287$	45.2186	2.66917
$288$	0	0
$289$	−16.7114	−0.983021
$290$	0	0
$291$	21.8391	1.28023
$292$	0	0
$293$	23.9188	1.39735	0.698675	−	0.715439i	$-0.253773\pi$
$293$	23.9188	1.39735	0.698675	+	0.715439i	$0.253773\pi$
$294$	0	0
$295$	22.5219	1.31128
$296$	0	0
$297$	3.46250	0.200914
$298$	0	0
$299$	−1.85079	−0.107034
$300$	0	0
$301$	−6.82964	−0.393654
$302$	0	0
$303$	41.4854	2.38327
$304$	0	0
$305$	−14.8529	−0.850476
$306$	0	0
$307$	−9.49861	−0.542114	−0.271057	−	0.962563i	$-0.587373\pi$
$307$	−9.49861	−0.542114	−0.271057	+	0.962563i	$0.587373\pi$
$308$	0	0
$309$	17.0044	0.967348
$310$	0	0
$311$	8.01607	0.454550	0.227275	−	0.973831i	$-0.427018\pi$
$311$	8.01607	0.454550	0.227275	+	0.973831i	$0.427018\pi$
$312$	0	0
$313$	−1.03187	−0.0583249	−0.0291624	−	0.999575i	$-0.509284\pi$
$313$	−1.03187	−0.0583249	−0.0291624	+	0.999575i	$0.509284\pi$
$314$	0	0
$315$	12.6195	0.711027
$316$	0	0
$317$	−26.2396	−1.47376	−0.736881	−	0.676023i	$-0.763702\pi$
$317$	−26.2396	−1.47376	−0.736881	+	0.676023i	$0.763702\pi$
$318$	0	0
$319$	3.20864	0.179650
$320$	0	0
$321$	7.82817	0.436926
$322$	0	0
$323$	3.67015	0.204213
$324$	0	0
$325$	−0.711922	−0.0394903
$326$	0	0
$327$	36.7522	2.03240
$328$	0	0
$329$	−38.2987	−2.11147
$330$	0	0
$331$	−34.2428	−1.88215	−0.941077	−	0.338193i	$-0.890184\pi$
$331$	−34.2428	−1.88215	−0.941077	+	0.338193i	$0.890184\pi$
$332$	0	0
$333$	0.516259	0.0282908
$334$	0	0
$335$	−14.0406	−0.767120
$336$	0	0
$337$	3.77353	0.205557	0.102779	−	0.994704i	$-0.467227\pi$
$337$	3.77353	0.205557	0.102779	+	0.994704i	$0.467227\pi$
$338$	0	0
$339$	−5.30622	−0.288195
$340$	0	0
$341$	6.10824	0.330780
$342$	0	0
$343$	1.43330	0.0773911
$344$	0	0
$345$	6.95265	0.374318
$346$	0	0
$347$	−17.2459	−0.925807	−0.462903	−	0.886409i	$-0.653192\pi$
$347$	−17.2459	−0.925807	−0.462903	+	0.886409i	$0.653192\pi$
$348$	0	0
$349$	22.9203	1.22690	0.613449	−	0.789734i	$-0.289782\pi$
$349$	22.9203	1.22690	0.613449	+	0.789734i	$0.289782\pi$
$350$	0	0
$351$	3.62347	0.193406
$352$	0	0
$353$	29.2348	1.55601	0.778006	−	0.628257i	$-0.216231\pi$
$353$	29.2348	1.55601	0.778006	+	0.628257i	$0.216231\pi$
$354$	0	0
$355$	31.1815	1.65494
$356$	0	0
$357$	−4.36247	−0.230887
$358$	0	0
$359$	−17.3591	−0.916179	−0.458089	−	0.888906i	$-0.651466\pi$
$359$	−17.3591	−0.916179	−0.458089	+	0.888906i	$0.651466\pi$
$360$	0	0
$361$	27.6667	1.45614
$362$	0	0
$363$	20.7567	1.08945
$364$	0	0
$365$	9.95483	0.521060
$366$	0	0
$367$	−13.0378	−0.680568	−0.340284	−	0.940323i	$-0.610523\pi$
$367$	−13.0378	−0.680568	−0.340284	+	0.940323i	$0.610523\pi$
$368$	0	0
$369$	18.9099	0.984411
$370$	0	0
$371$	−38.1746	−1.98193
$372$	0	0
$373$	−9.46001	−0.489821	−0.244910	−	0.969546i	$-0.578759\pi$
$373$	−9.46001	−0.489821	−0.244910	+	0.969546i	$0.578759\pi$
$374$	0	0
$375$	25.1465	1.29856
$376$	0	0
$377$	3.35781	0.172936
$378$	0	0
$379$	−11.8303	−0.607682	−0.303841	−	0.952723i	$-0.598269\pi$
$379$	−11.8303	−0.607682	−0.303841	+	0.952723i	$0.598269\pi$
$380$	0	0
$381$	27.8812	1.42840
$382$	0	0
$383$	−5.27671	−0.269627	−0.134814	−	0.990871i	$-0.543044\pi$
$383$	−5.27671	−0.269627	−0.134814	+	0.990871i	$0.543044\pi$
$384$	0	0
$385$	9.09840	0.463698
$386$	0	0
$387$	−2.85608	−0.145183
$388$	0	0
$389$	8.45523	0.428697	0.214349	−	0.976757i	$-0.431237\pi$
$389$	8.45523	0.428697	0.214349	+	0.976757i	$0.431237\pi$
$390$	0	0
$391$	−0.831108	−0.0420309
$392$	0	0
$393$	−40.2058	−2.02811
$394$	0	0
$395$	1.71259	0.0861698
$396$	0	0
$397$	−14.8927	−0.747441	−0.373720	−	0.927541i	$-0.621918\pi$
$397$	−14.8927	−0.747441	−0.373720	+	0.927541i	$0.621918\pi$
$398$	0	0
$399$	−55.4697	−2.77696
$400$	0	0
$401$	−8.43927	−0.421437	−0.210718	−	0.977547i	$-0.567580\pi$
$401$	−8.43927	−0.421437	−0.210718	+	0.977547i	$0.567580\pi$
$402$	0	0
$403$	6.39222	0.318419
$404$	0	0
$405$	−23.5961	−1.17250
$406$	0	0
$407$	0.372213	0.0184499
$408$	0	0
$409$	18.2460	0.902209	0.451104	−	0.892471i	$-0.351030\pi$
$409$	18.2460	0.902209	0.451104	+	0.892471i	$0.351030\pi$
$410$	0	0
$411$	−28.4475	−1.40321
$412$	0	0
$413$	40.6897	2.00221
$414$	0	0
$415$	30.9241	1.51800
$416$	0	0
$417$	25.1205	1.23016
$418$	0	0
$419$	32.6421	1.59467	0.797335	−	0.603537i	$-0.206243\pi$
$419$	32.6421	1.59467	0.797335	+	0.603537i	$0.206243\pi$
$420$	0	0
$421$	8.79104	0.428449	0.214225	−	0.976784i	$-0.431278\pi$
$421$	8.79104	0.428449	0.214225	+	0.976784i	$0.431278\pi$
$422$	0	0
$423$	−16.0161	−0.778728
$424$	0	0
$425$	−0.319692	−0.0155074
$426$	0	0
$427$	−26.8343	−1.29860
$428$	0	0
$429$	−2.92907	−0.141417
$430$	0	0
$431$	−11.0589	−0.532690	−0.266345	−	0.963878i	$-0.585816\pi$
$431$	−11.0589	−0.532690	−0.266345	+	0.963878i	$0.585816\pi$
$432$	0	0
$433$	3.96183	0.190394	0.0951968	−	0.995458i	$-0.469652\pi$
$433$	3.96183	0.190394	0.0951968	+	0.995458i	$0.469652\pi$
$434$	0	0
$435$	−12.6139	−0.604790
$436$	0	0
$437$	−10.5677	−0.505521
$438$	0	0
$439$	19.8957	0.949569	0.474784	−	0.880102i	$-0.342526\pi$
$439$	19.8957	0.949569	0.474784	+	0.880102i	$0.342526\pi$
$440$	0	0
$441$	11.6993	0.557110
$442$	0	0
$443$	24.1742	1.14855	0.574275	−	0.818662i	$-0.305284\pi$
$443$	24.1742	1.14855	0.574275	+	0.818662i	$0.305284\pi$
$444$	0	0
$445$	−4.08805	−0.193792
$446$	0	0
$447$	−1.19058	−0.0563123
$448$	0	0
$449$	0.990302	0.0467352	0.0233676	−	0.999727i	$-0.492561\pi$
$449$	0.990302	0.0467352	0.0233676	+	0.999727i	$0.492561\pi$
$450$	0	0
$451$	13.6337	0.641986
$452$	0	0
$453$	−30.2864	−1.42298
$454$	0	0
$455$	9.52139	0.446370
$456$	0	0
$457$	36.3671	1.70118	0.850591	−	0.525828i	$-0.176245\pi$
$457$	36.3671	1.70118	0.850591	+	0.525828i	$0.176245\pi$
$458$	0	0
$459$	1.62714	0.0759483
$460$	0	0
$461$	29.0421	1.35263	0.676314	−	0.736614i	$-0.263576\pi$
$461$	29.0421	1.35263	0.676314	+	0.736614i	$0.263576\pi$
$462$	0	0
$463$	0.333862	0.0155159	0.00775794	−	0.999970i	$-0.497531\pi$
$463$	0.333862	0.0155159	0.00775794	+	0.999970i	$0.497531\pi$
$464$	0	0
$465$	−24.0129	−1.11357
$466$	0	0
$467$	−30.4739	−1.41016	−0.705082	−	0.709126i	$-0.749090\pi$
$467$	−30.4739	−1.41016	−0.705082	+	0.709126i	$0.749090\pi$
$468$	0	0
$469$	−25.3667	−1.17133
$470$	0	0
$471$	−17.6909	−0.815153
$472$	0	0
$473$	−2.05918	−0.0946811
$474$	0	0
$475$	−4.06495	−0.186513
$476$	0	0
$477$	−15.9642	−0.730950
$478$	0	0
$479$	−19.6929	−0.899790	−0.449895	−	0.893081i	$-0.648539\pi$
$479$	−19.6929	−0.899790	−0.449895	+	0.893081i	$0.648539\pi$
$480$	0	0
$481$	0.389517	0.0177605
$482$	0	0
$483$	12.5611	0.571551
$484$	0	0
$485$	−21.4044	−0.971922
$486$	0	0
$487$	1.83184	0.0830088	0.0415044	−	0.999138i	$-0.486785\pi$
$487$	1.83184	0.0830088	0.0415044	+	0.999138i	$0.486785\pi$
$488$	0	0
$489$	−19.8268	−0.896600
$490$	0	0
$491$	−27.2460	−1.22960	−0.614798	−	0.788685i	$-0.710762\pi$
$491$	−27.2460	−1.22960	−0.614798	+	0.788685i	$0.710762\pi$
$492$	0	0
$493$	1.50785	0.0679099
$494$	0	0
$495$	3.80485	0.171015
$496$	0	0
$497$	56.3347	2.52696
$498$	0	0
$499$	−31.5063	−1.41041	−0.705207	−	0.709001i	$-0.749146\pi$
$499$	−31.5063	−1.41041	−0.705207	+	0.709001i	$0.749146\pi$
$500$	0	0
$501$	−9.24492	−0.413033
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−40.6596	−1.80933
$506$	0	0
$507$	24.7733	1.10022
$508$	0	0
$509$	−20.1541	−0.893317	−0.446658	−	0.894705i	$-0.647386\pi$
$509$	−20.1541	−0.893317	−0.446658	+	0.894705i	$0.647386\pi$
$510$	0	0
$511$	17.9851	0.795613
$512$	0	0
$513$	20.6894	0.913458
$514$	0	0
$515$	−16.6659	−0.734389
$516$	0	0
$517$	−11.5473	−0.507849
$518$	0	0
$519$	−9.96762	−0.437530
$520$	0	0
$521$	−28.1739	−1.23432	−0.617160	−	0.786838i	$-0.711717\pi$
$521$	−28.1739	−1.23432	−0.617160	+	0.786838i	$0.711717\pi$
$522$	0	0
$523$	−39.9826	−1.74832	−0.874160	−	0.485639i	$-0.838587\pi$
$523$	−39.9826	−1.74832	−0.874160	+	0.485639i	$0.838587\pi$
$524$	0	0
$525$	4.83174	0.210875
$526$	0	0
$527$	2.87046	0.125039
$528$	0	0
$529$	−20.6069	−0.895954
$530$	0	0
$531$	17.0160	0.738430
$532$	0	0
$533$	14.2675	0.617995
$534$	0	0
$535$	−7.67234	−0.331704
$536$	0	0
$537$	39.3920	1.69989
$538$	0	0
$539$	8.43498	0.363320
$540$	0	0
$541$	−33.8827	−1.45673	−0.728365	−	0.685189i	$-0.759719\pi$
$541$	−33.8827	−1.45673	−0.728365	+	0.685189i	$0.759719\pi$
$542$	0	0
$543$	24.9050	1.06878
$544$	0	0
$545$	−36.0206	−1.54295
$546$	0	0
$547$	−35.3197	−1.51016	−0.755080	−	0.655633i	$-0.772402\pi$
$547$	−35.3197	−1.51016	−0.755080	+	0.655633i	$0.772402\pi$
$548$	0	0
$549$	−11.2218	−0.478935
$550$	0	0
$551$	19.1725	0.816777
$552$	0	0
$553$	3.09409	0.131574
$554$	0	0
$555$	−1.46325	−0.0621116
$556$	0	0
$557$	18.9503	0.802951	0.401476	−	0.915870i	$-0.368497\pi$
$557$	18.9503	0.802951	0.401476	+	0.915870i	$0.368497\pi$
$558$	0	0
$559$	−2.15491	−0.0911430
$560$	0	0
$561$	−1.31531	−0.0555325
$562$	0	0
$563$	−11.4140	−0.481045	−0.240522	−	0.970644i	$-0.577319\pi$
$563$	−11.4140	−0.481045	−0.240522	+	0.970644i	$0.577319\pi$
$564$	0	0
$565$	5.20060	0.218791
$566$	0	0
$567$	−42.6303	−1.79030
$568$	0	0
$569$	−36.0287	−1.51040	−0.755201	−	0.655494i	$-0.772461\pi$
$569$	−36.0287	−1.51040	−0.755201	+	0.655494i	$0.772461\pi$
$570$	0	0
$571$	10.7077	0.448104	0.224052	−	0.974577i	$-0.428071\pi$
$571$	10.7077	0.448104	0.224052	+	0.974577i	$0.428071\pi$
$572$	0	0
$573$	−6.96688	−0.291046
$574$	0	0
$575$	0.920509	0.0383879
$576$	0	0
$577$	−7.45607	−0.310400	−0.155200	−	0.987883i	$-0.549602\pi$
$577$	−7.45607	−0.310400	−0.155200	+	0.987883i	$0.549602\pi$
$578$	0	0
$579$	−52.2587	−2.17180
$580$	0	0
$581$	55.8696	2.31786
$582$	0	0
$583$	−11.5099	−0.476691
$584$	0	0
$585$	3.98174	0.164625
$586$	0	0
$587$	36.1452	1.49187	0.745936	−	0.666017i	$-0.232002\pi$
$587$	36.1452	1.49187	0.745936	+	0.666017i	$0.232002\pi$
$588$	0	0
$589$	36.4984	1.50389
$590$	0	0
$591$	31.8639	1.31070
$592$	0	0
$593$	−27.3577	−1.12345	−0.561724	−	0.827325i	$-0.689862\pi$
$593$	−27.3577	−1.12345	−0.561724	+	0.827325i	$0.689862\pi$
$594$	0	0
$595$	4.27563	0.175284
$596$	0	0
$597$	−43.6239	−1.78541
$598$	0	0
$599$	−9.46657	−0.386793	−0.193397	−	0.981121i	$-0.561950\pi$
$599$	−9.46657	−0.386793	−0.193397	+	0.981121i	$0.561950\pi$
$600$	0	0
$601$	−13.9191	−0.567772	−0.283886	−	0.958858i	$-0.591624\pi$
$601$	−13.9191	−0.567772	−0.283886	+	0.958858i	$0.591624\pi$
$602$	0	0
$603$	−10.6081	−0.431994
$604$	0	0
$605$	−20.3436	−0.827083
$606$	0	0
$607$	−16.0580	−0.651776	−0.325888	−	0.945408i	$-0.605663\pi$
$607$	−16.0580	−0.651776	−0.325888	+	0.945408i	$0.605663\pi$
$608$	0	0
$609$	−22.7891	−0.923463
$610$	0	0
$611$	−12.0841	−0.488871
$612$	0	0
$613$	40.5306	1.63702	0.818508	−	0.574495i	$-0.194801\pi$
$613$	40.5306	1.63702	0.818508	+	0.574495i	$0.194801\pi$
$614$	0	0
$615$	−53.5972	−2.16125
$616$	0	0
$617$	4.31378	0.173666	0.0868331	−	0.996223i	$-0.472325\pi$
$617$	4.31378	0.173666	0.0868331	+	0.996223i	$0.472325\pi$
$618$	0	0
$619$	42.0360	1.68957	0.844784	−	0.535107i	$-0.179729\pi$
$619$	42.0360	1.68957	0.844784	+	0.535107i	$0.179729\pi$
$620$	0	0
$621$	−4.68512	−0.188007
$622$	0	0
$623$	−7.38575	−0.295904
$624$	0	0
$625$	−21.6707	−0.866827
$626$	0	0
$627$	−16.7244	−0.667910
$628$	0	0
$629$	0.174915	0.00697431
$630$	0	0
$631$	10.3189	0.410788	0.205394	−	0.978679i	$-0.434152\pi$
$631$	10.3189	0.410788	0.205394	+	0.978679i	$0.434152\pi$
$632$	0	0
$633$	0.0801551	0.00318588
$634$	0	0
$635$	−27.3262	−1.08441
$636$	0	0
$637$	8.82712	0.349743
$638$	0	0
$639$	23.5585	0.931961
$640$	0	0
$641$	27.5707	1.08898	0.544489	−	0.838768i	$-0.316724\pi$
$641$	27.5707	1.08898	0.544489	+	0.838768i	$0.316724\pi$
$642$	0	0
$643$	13.0724	0.515526	0.257763	−	0.966208i	$-0.417015\pi$
$643$	13.0724	0.515526	0.257763	+	0.966208i	$0.417015\pi$
$644$	0	0
$645$	8.09510	0.318744
$646$	0	0
$647$	−24.9111	−0.979358	−0.489679	−	0.871903i	$-0.662886\pi$
$647$	−24.9111	−0.979358	−0.489679	+	0.871903i	$0.662886\pi$
$648$	0	0
$649$	12.2682	0.481569
$650$	0	0
$651$	−43.3833	−1.70033
$652$	0	0
$653$	−4.97978	−0.194874	−0.0974370	−	0.995242i	$-0.531064\pi$
$653$	−4.97978	−0.194874	−0.0974370	+	0.995242i	$0.531064\pi$
$654$	0	0
$655$	39.4055	1.53970
$656$	0	0
$657$	7.52116	0.293428
$658$	0	0
$659$	−34.5849	−1.34724	−0.673619	−	0.739079i	$-0.735261\pi$
$659$	−34.5849	−1.34724	−0.673619	+	0.739079i	$0.735261\pi$
$660$	0	0
$661$	17.4709	0.679539	0.339770	−	0.940509i	$-0.389651\pi$
$661$	17.4709	0.679539	0.339770	+	0.940509i	$0.389651\pi$
$662$	0	0
$663$	−1.37646	−0.0534574
$664$	0	0
$665$	54.3655	2.10820
$666$	0	0
$667$	−4.34163	−0.168108
$668$	0	0
$669$	−7.21666	−0.279012
$670$	0	0
$671$	−8.09071	−0.312338
$672$	0	0
$673$	40.8340	1.57404	0.787018	−	0.616930i	$-0.211624\pi$
$673$	40.8340	1.57404	0.787018	+	0.616930i	$0.211624\pi$
$674$	0	0
$675$	−1.80217	−0.0693655
$676$	0	0
$677$	−5.39605	−0.207387	−0.103693	−	0.994609i	$-0.533066\pi$
$677$	−5.39605	−0.207387	−0.103693	+	0.994609i	$0.533066\pi$
$678$	0	0
$679$	−38.6706	−1.48404
$680$	0	0
$681$	−22.6131	−0.866537
$682$	0	0
$683$	49.0906	1.87840	0.939199	−	0.343373i	$-0.111570\pi$
$683$	49.0906	1.87840	0.939199	+	0.343373i	$0.111570\pi$
$684$	0	0
$685$	27.8812	1.06529
$686$	0	0
$687$	−34.4275	−1.31349
$688$	0	0
$689$	−12.0450	−0.458877
$690$	0	0
$691$	−38.8646	−1.47848	−0.739240	−	0.673442i	$-0.764815\pi$
$691$	−38.8646	−1.47848	−0.739240	+	0.673442i	$0.764815\pi$
$692$	0	0
$693$	6.87411	0.261126
$694$	0	0
$695$	−24.6205	−0.933908
$696$	0	0
$697$	6.40691	0.242679
$698$	0	0
$699$	22.6607	0.857107
$700$	0	0
$701$	−4.65960	−0.175991	−0.0879953	−	0.996121i	$-0.528046\pi$
$701$	−4.65960	−0.175991	−0.0879953	+	0.996121i	$0.528046\pi$
$702$	0	0
$703$	2.22407	0.0838825
$704$	0	0
$705$	45.3950	1.70967
$706$	0	0
$707$	−73.4584	−2.76269
$708$	0	0
$709$	−8.51493	−0.319785	−0.159893	−	0.987134i	$-0.551115\pi$
$709$	−8.51493	−0.319785	−0.159893	+	0.987134i	$0.551115\pi$
$710$	0	0
$711$	1.29391	0.0485255
$712$	0	0
$713$	−8.26509	−0.309530
$714$	0	0
$715$	2.87076	0.107360
$716$	0	0
$717$	−9.27643	−0.346435
$718$	0	0
$719$	−2.34959	−0.0876249	−0.0438125	−	0.999040i	$-0.513950\pi$
$719$	−2.34959	−0.0876249	−0.0438125	+	0.999040i	$0.513950\pi$
$720$	0	0
$721$	−30.1098	−1.12135
$722$	0	0
$723$	−41.5610	−1.54567
$724$	0	0
$725$	−1.67004	−0.0620238
$726$	0	0
$727$	5.15160	0.191062	0.0955312	−	0.995426i	$-0.469545\pi$
$727$	5.15160	0.191062	0.0955312	+	0.995426i	$0.469545\pi$
$728$	0	0
$729$	1.62914	0.0603387
$730$	0	0
$731$	−0.967675	−0.0357907
$732$	0	0
$733$	−36.5961	−1.35171	−0.675854	−	0.737035i	$-0.736225\pi$
$733$	−36.5961	−1.35171	−0.675854	+	0.737035i	$0.736225\pi$
$734$	0	0
$735$	−33.1598	−1.22312
$736$	0	0
$737$	−7.64822	−0.281726
$738$	0	0
$739$	36.2274	1.33265	0.666323	−	0.745663i	$-0.267867\pi$
$739$	36.2274	1.33265	0.666323	+	0.745663i	$0.267867\pi$
$740$	0	0
$741$	−17.5020	−0.642951
$742$	0	0
$743$	−10.5919	−0.388578	−0.194289	−	0.980944i	$-0.562240\pi$
$743$	−10.5919	−0.388578	−0.194289	+	0.980944i	$0.562240\pi$
$744$	0	0
$745$	1.16688	0.0427510
$746$	0	0
$747$	23.3640	0.854845
$748$	0	0
$749$	−13.8614	−0.506484
$750$	0	0
$751$	11.5251	0.420558	0.210279	−	0.977641i	$-0.432563\pi$
$751$	11.5251	0.420558	0.210279	+	0.977641i	$0.432563\pi$
$752$	0	0
$753$	−33.7422	−1.22963
$754$	0	0
$755$	29.6835	1.08029
$756$	0	0
$757$	−25.5807	−0.929747	−0.464874	−	0.885377i	$-0.653900\pi$
$757$	−25.5807	−0.929747	−0.464874	+	0.885377i	$0.653900\pi$
$758$	0	0
$759$	3.78726	0.137469
$760$	0	0
$761$	−9.20991	−0.333859	−0.166929	−	0.985969i	$-0.553385\pi$
$761$	−9.20991	−0.333859	−0.166929	+	0.985969i	$0.553385\pi$
$762$	0	0
$763$	−65.0774	−2.35596
$764$	0	0
$765$	1.78802	0.0646461
$766$	0	0
$767$	12.8385	0.463573
$768$	0	0
$769$	31.9598	1.15250	0.576250	−	0.817273i	$-0.304515\pi$
$769$	31.9598	1.15250	0.576250	+	0.817273i	$0.304515\pi$
$770$	0	0
$771$	−19.6302	−0.706966
$772$	0	0
$773$	−16.6321	−0.598215	−0.299107	−	0.954219i	$-0.596689\pi$
$773$	−16.6321	−0.598215	−0.299107	+	0.954219i	$0.596689\pi$
$774$	0	0
$775$	−3.17923	−0.114201
$776$	0	0
$777$	−2.64361	−0.0948391
$778$	0	0
$779$	81.4651	2.91879
$780$	0	0
$781$	16.9853	0.607780
$782$	0	0
$783$	8.50002	0.303766
$784$	0	0
$785$	17.3387	0.618846
$786$	0	0
$787$	−49.1287	−1.75125	−0.875625	−	0.482991i	$-0.839550\pi$
$787$	−49.1287	−1.75125	−0.875625	+	0.482991i	$0.839550\pi$
$788$	0	0
$789$	−59.9600	−2.13463
$790$	0	0
$791$	9.39576	0.334075
$792$	0	0
$793$	−8.46685	−0.300667
$794$	0	0
$795$	45.2480	1.60478
$796$	0	0
$797$	26.8162	0.949878	0.474939	−	0.880019i	$-0.342470\pi$
$797$	26.8162	0.949878	0.474939	+	0.880019i	$0.342470\pi$
$798$	0	0
$799$	−5.42644	−0.191974
$800$	0	0
$801$	−3.08864	−0.109132
$802$	0	0
$803$	5.42261	0.191360
$804$	0	0
$805$	−12.3111	−0.433909
$806$	0	0
$807$	−11.9032	−0.419011
$808$	0	0
$809$	6.01093	0.211333	0.105667	−	0.994402i	$-0.466302\pi$
$809$	6.01093	0.211333	0.105667	+	0.994402i	$0.466302\pi$
$810$	0	0
$811$	−9.00767	−0.316302	−0.158151	−	0.987415i	$-0.550553\pi$
$811$	−9.00767	−0.316302	−0.158151	+	0.987415i	$0.550553\pi$
$812$	0	0
$813$	−45.5779	−1.59849
$814$	0	0
$815$	19.4322	0.680679
$816$	0	0
$817$	−12.3042	−0.430468
$818$	0	0
$819$	7.19369	0.251368
$820$	0	0
$821$	−2.03151	−0.0709002	−0.0354501	−	0.999371i	$-0.511286\pi$
$821$	−2.03151	−0.0709002	−0.0354501	+	0.999371i	$0.511286\pi$
$822$	0	0
$823$	30.1864	1.05223	0.526116	−	0.850413i	$-0.323648\pi$
$823$	30.1864	1.05223	0.526116	+	0.850413i	$0.323648\pi$
$824$	0	0
$825$	1.45680	0.0507193
$826$	0	0
$827$	−7.81638	−0.271802	−0.135901	−	0.990722i	$-0.543393\pi$
$827$	−7.81638	−0.271802	−0.135901	+	0.990722i	$0.543393\pi$
$828$	0	0
$829$	−14.0104	−0.486602	−0.243301	−	0.969951i	$-0.578230\pi$
$829$	−14.0104	−0.486602	−0.243301	+	0.969951i	$0.578230\pi$
$830$	0	0
$831$	24.7003	0.856844
$832$	0	0
$833$	3.96387	0.137340
$834$	0	0
$835$	9.06089	0.313565
$836$	0	0
$837$	16.1813	0.559309
$838$	0	0
$839$	3.26528	0.112730	0.0563650	−	0.998410i	$-0.482049\pi$
$839$	3.26528	0.112730	0.0563650	+	0.998410i	$0.482049\pi$
$840$	0	0
$841$	−21.1232	−0.728385
$842$	0	0
$843$	−57.7275	−1.98824
$844$	0	0
$845$	−24.2801	−0.835263
$846$	0	0
$847$	−36.7541	−1.26288
$848$	0	0
$849$	23.1082	0.793070
$850$	0	0
$851$	−0.503642	−0.0172646
$852$	0	0
$853$	−49.3196	−1.68867	−0.844335	−	0.535816i	$-0.820004\pi$
$853$	−49.3196	−1.68867	−0.844335	+	0.535816i	$0.820004\pi$
$854$	0	0
$855$	22.7350	0.777522
$856$	0	0
$857$	−36.2747	−1.23912	−0.619560	−	0.784949i	$-0.712689\pi$
$857$	−36.2747	−1.23912	−0.619560	+	0.784949i	$0.712689\pi$
$858$	0	0
$859$	13.2566	0.452308	0.226154	−	0.974092i	$-0.427385\pi$
$859$	13.2566	0.452308	0.226154	+	0.974092i	$0.427385\pi$
$860$	0	0
$861$	−96.8323	−3.30004
$862$	0	0
$863$	18.6444	0.634662	0.317331	−	0.948315i	$-0.397213\pi$
$863$	18.6444	0.634662	0.317331	+	0.948315i	$0.397213\pi$
$864$	0	0
$865$	9.76921	0.332163
$866$	0	0
$867$	35.7861	1.21536
$868$	0	0
$869$	0.932886	0.0316460
$870$	0	0
$871$	−8.00379	−0.271198
$872$	0	0
$873$	−16.1716	−0.547326
$874$	0	0
$875$	−44.5270	−1.50529
$876$	0	0
$877$	−5.66263	−0.191214	−0.0956068	−	0.995419i	$-0.530479\pi$
$877$	−5.66263	−0.191214	−0.0956068	+	0.995419i	$0.530479\pi$
$878$	0	0
$879$	−51.2203	−1.72762
$880$	0	0
$881$	45.9689	1.54873	0.774367	−	0.632737i	$-0.218069\pi$
$881$	45.9689	1.54873	0.774367	+	0.632737i	$0.218069\pi$
$882$	0	0
$883$	42.7924	1.44008	0.720039	−	0.693933i	$-0.244124\pi$
$883$	42.7924	1.44008	0.720039	+	0.693933i	$0.244124\pi$
$884$	0	0
$885$	−48.2291	−1.62120
$886$	0	0
$887$	−16.9906	−0.570488	−0.285244	−	0.958455i	$-0.592075\pi$
$887$	−16.9906	−0.570488	−0.285244	+	0.958455i	$0.592075\pi$
$888$	0	0
$889$	−49.3694	−1.65580
$890$	0	0
$891$	−12.8533	−0.430601
$892$	0	0
$893$	−68.9982	−2.30894
$894$	0	0
$895$	−38.6079	−1.29052
$896$	0	0
$897$	3.96333	0.132332
$898$	0	0
$899$	14.9950	0.500112
$900$	0	0
$901$	−5.40887	−0.180195
$902$	0	0
$903$	14.6252	0.486695
$904$	0	0
$905$	−24.4092	−0.811390
$906$	0	0
$907$	14.3565	0.476701	0.238351	−	0.971179i	$-0.423393\pi$
$907$	14.3565	0.476701	0.238351	+	0.971179i	$0.423393\pi$
$908$	0	0
$909$	−30.7195	−1.01890
$910$	0	0
$911$	−35.8051	−1.18628	−0.593138	−	0.805101i	$-0.702111\pi$
$911$	−35.8051	−1.18628	−0.593138	+	0.805101i	$0.702111\pi$
$912$	0	0
$913$	16.8450	0.557489
$914$	0	0
$915$	31.8064	1.05149
$916$	0	0
$917$	71.1926	2.35099
$918$	0	0
$919$	19.7644	0.651969	0.325984	−	0.945375i	$-0.394304\pi$
$919$	19.7644	0.651969	0.325984	+	0.945375i	$0.394304\pi$
$920$	0	0
$921$	20.3406	0.670244
$922$	0	0
$923$	17.7749	0.585068
$924$	0	0
$925$	−0.193730	−0.00636981
$926$	0	0
$927$	−12.5916	−0.413562
$928$	0	0
$929$	−37.4746	−1.22950	−0.614750	−	0.788722i	$-0.710743\pi$
$929$	−37.4746	−1.22950	−0.614750	+	0.788722i	$0.710743\pi$
$930$	0	0
$931$	50.4013	1.65184
$932$	0	0
$933$	−17.1658	−0.561984
$934$	0	0
$935$	1.28913	0.0421591
$936$	0	0
$937$	−4.97265	−0.162450	−0.0812248	−	0.996696i	$-0.525883\pi$
$937$	−4.97265	−0.162450	−0.0812248	+	0.996696i	$0.525883\pi$
$938$	0	0
$939$	2.20968	0.0721101
$940$	0	0
$941$	27.9822	0.912193	0.456096	−	0.889930i	$-0.349247\pi$
$941$	27.9822	0.912193	0.456096	+	0.889930i	$0.349247\pi$
$942$	0	0
$943$	−18.4478	−0.600743
$944$	0	0
$945$	24.1026	0.784057
$946$	0	0
$947$	13.2730	0.431315	0.215657	−	0.976469i	$-0.430811\pi$
$947$	13.2730	0.431315	0.215657	+	0.976469i	$0.430811\pi$
$948$	0	0
$949$	5.67471	0.184209
$950$	0	0
$951$	56.1901	1.82209
$952$	0	0
$953$	−34.6398	−1.12209	−0.561046	−	0.827784i	$-0.689601\pi$
$953$	−34.6398	−1.12209	−0.561046	+	0.827784i	$0.689601\pi$
$954$	0	0
$955$	6.82819	0.220955
$956$	0	0
$957$	−6.87107	−0.222110
$958$	0	0
$959$	50.3721	1.62660
$960$	0	0
$961$	−2.45424	−0.0791689
$962$	0	0
$963$	−5.79668	−0.186795
$964$	0	0
$965$	51.2184	1.64878
$966$	0	0
$967$	−25.5287	−0.820948	−0.410474	−	0.911872i	$-0.634637\pi$
$967$	−25.5287	−0.820948	−0.410474	+	0.911872i	$0.634637\pi$
$968$	0	0
$969$	−7.85936	−0.252479
$970$	0	0
$971$	12.4275	0.398816	0.199408	−	0.979917i	$-0.436098\pi$
$971$	12.4275	0.398816	0.199408	+	0.979917i	$0.436098\pi$
$972$	0	0
$973$	−44.4811	−1.42600
$974$	0	0
$975$	1.52453	0.0488240
$976$	0	0
$977$	48.2271	1.54292	0.771461	−	0.636277i	$-0.219526\pi$
$977$	48.2271	1.54292	0.771461	+	0.636277i	$0.219526\pi$
$978$	0	0
$979$	−2.22685	−0.0711704
$980$	0	0
$981$	−27.2146	−0.868896
$982$	0	0
$983$	−40.0874	−1.27859	−0.639295	−	0.768961i	$-0.720774\pi$
$983$	−40.0874	−1.27859	−0.639295	+	0.768961i	$0.720774\pi$
$984$	0	0
$985$	−31.2296	−0.995058
$986$	0	0
$987$	82.0137	2.61053
$988$	0	0
$989$	2.78628	0.0885986
$990$	0	0
$991$	12.7583	0.405280	0.202640	−	0.979253i	$-0.435048\pi$
$991$	12.7583	0.405280	0.202640	+	0.979253i	$0.435048\pi$
$992$	0	0
$993$	73.3284	2.32701
$994$	0	0
$995$	42.7555	1.35544
$996$	0	0
$997$	−32.3421	−1.02428	−0.512142	−	0.858901i	$-0.671148\pi$
$997$	−32.3421	−1.02428	−0.512142	+	0.858901i	$0.671148\pi$
$998$	0	0
$999$	0.986029	0.0311966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.7	✓	33
4.3	odd	2		8048.2.a.x.1.27		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.7	✓	33	1.1	even	1	trivial
8048.2.a.x.1.27		33	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.14143 q^{3} +2.09880 q^{5} +3.79183 q^{7} +1.58570 q^{9} +O(q^{10})\)	\(q-2.14143 q^{3} +2.09880 q^{5} +3.79183 q^{7} +1.58570 q^{9} +1.14326 q^{11} +1.19641 q^{13} -4.49442 q^{15} +0.537255 q^{17} +6.83130 q^{19} -8.11993 q^{21} -1.54695 q^{23} -0.595048 q^{25} +3.02861 q^{27} +2.80657 q^{29} +5.34282 q^{31} -2.44821 q^{33} +7.95829 q^{35} +0.325571 q^{37} -2.56203 q^{39} +11.9253 q^{41} -1.80114 q^{43} +3.32807 q^{45} -10.1003 q^{47} +7.37800 q^{49} -1.15049 q^{51} -10.0676 q^{53} +2.39947 q^{55} -14.6287 q^{57} +10.7309 q^{59} -7.07687 q^{61} +6.01272 q^{63} +2.51103 q^{65} -6.68983 q^{67} +3.31268 q^{69} +14.8568 q^{71} +4.74311 q^{73} +1.27425 q^{75} +4.33505 q^{77} +0.815987 q^{79} -11.2427 q^{81} +14.7342 q^{83} +1.12759 q^{85} -6.01006 q^{87} -1.94781 q^{89} +4.53659 q^{91} -11.4413 q^{93} +14.3375 q^{95} -10.1984 q^{97} +1.81287 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more