LMFDB - Embedded newform 10000.2.a.x.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, 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("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.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:	$79.8504020213$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 4x + 16 $ x^4 - x^3 - 9x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.71472$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.71472 q^{3} -2.77447 q^{7} -0.0597522 q^{9} +O(q^{10})$	$q+1.71472 q^{3} -2.77447 q^{7} -0.0597522 q^{9} -2.77447 q^{11} -5.67779 q^{13} -5.15643 q^{17} -1.41238 q^{19} -4.75742 q^{21} -0.654963 q^{23} -5.24660 q^{27} +4.09668 q^{29} +7.12710 q^{31} -4.75742 q^{33} +1.04746 q^{37} -9.73579 q^{39} +9.10722 q^{41} +9.24660 q^{43} -2.77447 q^{47} +0.697669 q^{49} -8.84181 q^{51} +0.526111 q^{53} -2.42184 q^{57} +3.78206 q^{59} -10.8325 q^{61} +0.165781 q^{63} -4.32340 q^{67} -1.12307 q^{69} +13.2466 q^{71} -4.21324 q^{73} +7.69767 q^{77} +9.90157 q^{79} -8.81717 q^{81} +4.67603 q^{83} +7.02464 q^{87} -9.18401 q^{89} +15.7528 q^{91} +12.2209 q^{93} +0.0901699 q^{97} +0.165781 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 2 q^{7} + 7 q^{9}+O(q^{10}) $ 4 * q + q^3 + 2 * q^7 + 7 * q^9	$ 4 q + q^{3} + 2 q^{7} + 7 q^{9} + 2 q^{11} - 11 q^{13} - 12 q^{17} + 5 q^{19} - 7 q^{21} - 4 q^{23} + 10 q^{27} + 15 q^{29} + 12 q^{31} - 7 q^{33} - 12 q^{37} + 11 q^{39} + 13 q^{41} + 6 q^{43} + 2 q^{47} - 2 q^{49} - 13 q^{51} - 11 q^{53} + 8 q^{61} + 21 q^{63} + 22 q^{67} - 31 q^{69} + 22 q^{71} - 21 q^{73} + 26 q^{77} + 10 q^{79} - 16 q^{81} - 24 q^{83} + 25 q^{87} - 5 q^{89} + 12 q^{91} + 23 q^{93} - 22 q^{97} + 21 q^{99}+O(q^{100}) $ 4 * q + q^3 + 2 * q^7 + 7 * q^9 + 2 * q^11 - 11 * q^13 - 12 * q^17 + 5 * q^19 - 7 * q^21 - 4 * q^23 + 10 * q^27 + 15 * q^29 + 12 * q^31 - 7 * q^33 - 12 * q^37 + 11 * q^39 + 13 * q^41 + 6 * q^43 + 2 * q^47 - 2 * q^49 - 13 * q^51 - 11 * q^53 + 8 * q^61 + 21 * q^63 + 22 * q^67 - 31 * q^69 + 22 * q^71 - 21 * q^73 + 26 * q^77 + 10 * q^79 - 16 * q^81 - 24 * q^83 + 25 * q^87 - 5 * q^89 + 12 * q^91 + 23 * q^93 - 22 * q^97 + 21 * 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
$3$	1.71472	0.989991	0.494996	−	0.868895i	$-0.335170\pi$
$3$	1.71472	0.989991	0.494996	+	0.868895i	$0.335170\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.77447	−1.04865	−0.524325	−	0.851518i	$-0.675682\pi$
$7$	−2.77447	−1.04865	−0.524325	+	0.851518i	$0.675682\pi$
$8$	0	0
$9$	−0.0597522	−0.0199174
$10$	0	0
$11$	−2.77447	−0.836533	−0.418267	−	0.908324i	$-0.637362\pi$
$11$	−2.77447	−0.836533	−0.418267	+	0.908324i	$0.637362\pi$
$12$	0	0
$13$	−5.67779	−1.57473	−0.787367	−	0.616484i	$-0.788556\pi$
$13$	−5.67779	−1.57473	−0.787367	+	0.616484i	$0.788556\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.15643	−1.25062	−0.625309	−	0.780377i	$-0.715027\pi$
$17$	−5.15643	−1.25062	−0.625309	+	0.780377i	$0.715027\pi$
$18$	0	0
$19$	−1.41238	−0.324023	−0.162012	−	0.986789i	$-0.551798\pi$
$19$	−1.41238	−0.324023	−0.162012	+	0.986789i	$0.551798\pi$
$20$	0	0
$21$	−4.75742	−1.03815
$22$	0	0
$23$	−0.654963	−0.136569	−0.0682846	−	0.997666i	$-0.521753\pi$
$23$	−0.654963	−0.136569	−0.0682846	+	0.997666i	$0.521753\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.24660	−1.00971
$28$	0	0
$29$	4.09668	0.760735	0.380367	−	0.924836i	$-0.375798\pi$
$29$	4.09668	0.760735	0.380367	+	0.924836i	$0.375798\pi$
$30$	0	0
$31$	7.12710	1.28006	0.640032	−	0.768348i	$-0.278921\pi$
$31$	7.12710	1.28006	0.640032	+	0.768348i	$0.278921\pi$
$32$	0	0
$33$	−4.75742	−0.828161
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.04746	0.172202	0.0861010	−	0.996286i	$-0.472559\pi$
$37$	1.04746	0.172202	0.0861010	+	0.996286i	$0.472559\pi$
$38$	0	0
$39$	−9.73579	−1.55897
$40$	0	0
$41$	9.10722	1.42231	0.711154	−	0.703036i	$-0.248173\pi$
$41$	9.10722	1.42231	0.711154	+	0.703036i	$0.248173\pi$
$42$	0	0
$43$	9.24660	1.41009	0.705047	−	0.709161i	$-0.250926\pi$
$43$	9.24660	1.41009	0.705047	+	0.709161i	$0.250926\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.77447	−0.404698	−0.202349	−	0.979314i	$-0.564857\pi$
$47$	−2.77447	−0.404698	−0.202349	+	0.979314i	$0.564857\pi$
$48$	0	0
$49$	0.697669	0.0996670
$50$	0	0
$51$	−8.84181	−1.23810
$52$	0	0
$53$	0.526111	0.0722669	0.0361335	−	0.999347i	$-0.488496\pi$
$53$	0.526111	0.0722669	0.0361335	+	0.999347i	$0.488496\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.42184	−0.320780
$58$	0	0
$59$	3.78206	0.492382	0.246191	−	0.969221i	$-0.420821\pi$
$59$	3.78206	0.492382	0.246191	+	0.969221i	$0.420821\pi$
$60$	0	0
$61$	−10.8325	−1.38696	−0.693478	−	0.720478i	$-0.743922\pi$
$61$	−10.8325	−1.38696	−0.693478	+	0.720478i	$0.743922\pi$
$62$	0	0
$63$	0.165781	0.0208864
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.32340	−0.528188	−0.264094	−	0.964497i	$-0.585073\pi$
$67$	−4.32340	−0.528188	−0.264094	+	0.964497i	$0.585073\pi$
$68$	0	0
$69$	−1.12307	−0.135202
$70$	0	0
$71$	13.2466	1.57208	0.786041	−	0.618174i	$-0.212127\pi$
$71$	13.2466	1.57208	0.786041	+	0.618174i	$0.212127\pi$
$72$	0	0
$73$	−4.21324	−0.493123	−0.246561	−	0.969127i	$-0.579301\pi$
$73$	−4.21324	−0.493123	−0.246561	+	0.969127i	$0.579301\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.69767	0.877231
$78$	0	0
$79$	9.90157	1.11401	0.557007	−	0.830508i	$-0.311950\pi$
$79$	9.90157	1.11401	0.557007	+	0.830508i	$0.311950\pi$
$80$	0	0
$81$	−8.81717	−0.979686
$82$	0	0
$83$	4.67603	0.513261	0.256631	−	0.966510i	$-0.417388\pi$
$83$	4.67603	0.513261	0.256631	+	0.966510i	$0.417388\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.02464	0.753121
$88$	0	0
$89$	−9.18401	−0.973504	−0.486752	−	0.873540i	$-0.661818\pi$
$89$	−9.18401	−0.973504	−0.486752	+	0.873540i	$0.661818\pi$
$90$	0	0
$91$	15.7528	1.65135
$92$	0	0
$93$	12.2209	1.26725
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.0901699	0.00915537	0.00457769	−	0.999990i	$-0.498543\pi$
$97$	0.0901699	0.00915537	0.00457769	+	0.999990i	$0.498543\pi$
$98$	0	0
$99$	0.165781	0.0166616
$100$	0	0
$101$	−7.90632	−0.786709	−0.393354	−	0.919387i	$-0.628685\pi$
$101$	−7.90632	−0.786709	−0.393354	+	0.919387i	$0.628685\pi$
$102$	0	0
$103$	7.83422	0.771929	0.385964	−	0.922514i	$-0.373869\pi$
$103$	7.83422	0.771929	0.385964	+	0.922514i	$0.373869\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.2220	−0.988194	−0.494097	−	0.869407i	$-0.664501\pi$
$107$	−10.2220	−0.988194	−0.494097	+	0.869407i	$0.664501\pi$
$108$	0	0
$109$	3.91091	0.374598	0.187299	−	0.982303i	$-0.440027\pi$
$109$	3.91091	0.374598	0.187299	+	0.982303i	$0.440027\pi$
$110$	0	0
$111$	1.79610	0.170479
$112$	0	0
$113$	6.37545	0.599752	0.299876	−	0.953978i	$-0.403055\pi$
$113$	6.37545	0.599752	0.299876	+	0.953978i	$0.403055\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.339260	0.0313646
$118$	0	0
$119$	14.3064	1.31146
$120$	0	0
$121$	−3.30233	−0.300212
$122$	0	0
$123$	15.6163	1.40807
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.54893	0.847331	0.423665	−	0.905819i	$-0.360743\pi$
$127$	9.54893	0.847331	0.423665	+	0.905819i	$0.360743\pi$
$128$	0	0
$129$	15.8553	1.39598
$130$	0	0
$131$	−5.59923	−0.489207	−0.244604	−	0.969623i	$-0.578658\pi$
$131$	−5.59923	−0.489207	−0.244604	+	0.969623i	$0.578658\pi$
$132$	0	0
$133$	3.91861	0.339787
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.98414	−0.682131	−0.341066	−	0.940039i	$-0.610788\pi$
$137$	−7.98414	−0.682131	−0.341066	+	0.940039i	$0.610788\pi$
$138$	0	0
$139$	22.0884	1.87352	0.936758	−	0.349979i	$-0.113811\pi$
$139$	22.0884	1.87352	0.936758	+	0.349979i	$0.113811\pi$
$140$	0	0
$141$	−4.75742	−0.400647
$142$	0	0
$143$	15.7528	1.31732
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.19630	0.0986694
$148$	0	0
$149$	4.18401	0.342768	0.171384	−	0.985204i	$-0.445176\pi$
$149$	4.18401	0.342768	0.171384	+	0.985204i	$0.445176\pi$
$150$	0	0
$151$	0.331561	0.0269821	0.0134910	−	0.999909i	$-0.495706\pi$
$151$	0.331561	0.0269821	0.0134910	+	0.999909i	$0.495706\pi$
$152$	0	0
$153$	0.308108	0.0249091
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.60750	−0.287910	−0.143955	−	0.989584i	$-0.545982\pi$
$157$	−3.60750	−0.287910	−0.143955	+	0.989584i	$0.545982\pi$
$158$	0	0
$159$	0.902131	0.0715436
$160$	0	0
$161$	1.81717	0.143213
$162$	0	0
$163$	5.09385	0.398981	0.199490	−	0.979900i	$-0.436071\pi$
$163$	5.09385	0.398981	0.199490	+	0.979900i	$0.436071\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.8629	1.61442	0.807209	−	0.590265i	$-0.200977\pi$
$167$	20.8629	1.61442	0.807209	+	0.590265i	$0.200977\pi$
$168$	0	0
$169$	19.2373	1.47979
$170$	0	0
$171$	0.0843930	0.00645370
$172$	0	0
$173$	11.9877	0.911409	0.455704	−	0.890131i	$-0.349387\pi$
$173$	11.9877	0.911409	0.455704	+	0.890131i	$0.349387\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.48516	0.487454
$178$	0	0
$179$	8.54134	0.638410	0.319205	−	0.947686i	$-0.396584\pi$
$179$	8.54134	0.638410	0.319205	+	0.947686i	$0.396584\pi$
$180$	0	0
$181$	7.82844	0.581884	0.290942	−	0.956741i	$-0.406031\pi$
$181$	7.82844	0.581884	0.290942	+	0.956741i	$0.406031\pi$
$182$	0	0
$183$	−18.5746	−1.37307
$184$	0	0
$185$	0	0
$186$	0	0
$187$	14.3064	1.04618
$188$	0	0
$189$	14.5565	1.05883
$190$	0	0
$191$	−3.36611	−0.243563	−0.121782	−	0.992557i	$-0.538861\pi$
$191$	−3.36611	−0.243563	−0.121782	+	0.992557i	$0.538861\pi$
$192$	0	0
$193$	15.4211	1.11004	0.555018	−	0.831839i	$-0.312712\pi$
$193$	15.4211	1.11004	0.555018	+	0.831839i	$0.312712\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.80489	−0.627322	−0.313661	−	0.949535i	$-0.601556\pi$
$197$	−8.80489	−0.627322	−0.313661	+	0.949535i	$0.601556\pi$
$198$	0	0
$199$	17.6222	1.24920	0.624601	−	0.780944i	$-0.285262\pi$
$199$	17.6222	1.24920	0.624601	+	0.780944i	$0.285262\pi$
$200$	0	0
$201$	−7.41340	−0.522901
$202$	0	0
$203$	−11.3661	−0.797744
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0.0391355	0.00272010
$208$	0	0
$209$	3.91861	0.271056
$210$	0	0
$211$	−6.10144	−0.420040	−0.210020	−	0.977697i	$-0.567353\pi$
$211$	−6.10144	−0.420040	−0.210020	+	0.977697i	$0.567353\pi$
$212$	0	0
$213$	22.7142	1.55635
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−19.7739	−1.34234
$218$	0	0
$219$	−7.22451	−0.488187
$220$	0	0
$221$	29.2771	1.96939
$222$	0	0
$223$	−5.02866	−0.336744	−0.168372	−	0.985723i	$-0.553851\pi$
$223$	−5.02866	−0.336744	−0.168372	+	0.985723i	$0.553851\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.8382	−1.18397	−0.591983	−	0.805951i	$-0.701655\pi$
$227$	−17.8382	−1.18397	−0.591983	+	0.805951i	$0.701655\pi$
$228$	0	0
$229$	−21.5566	−1.42450	−0.712251	−	0.701925i	$-0.752324\pi$
$229$	−21.5566	−1.42450	−0.712251	+	0.701925i	$0.752324\pi$
$230$	0	0
$231$	13.1993	0.868451
$232$	0	0
$233$	−14.7065	−0.963452	−0.481726	−	0.876322i	$-0.659990\pi$
$233$	−14.7065	−0.963452	−0.481726	+	0.876322i	$0.659990\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	16.9784	1.10286
$238$	0	0
$239$	−6.94025	−0.448927	−0.224464	−	0.974482i	$-0.572063\pi$
$239$	−6.94025	−0.448927	−0.224464	+	0.974482i	$0.572063\pi$
$240$	0	0
$241$	3.31003	0.213218	0.106609	−	0.994301i	$-0.466001\pi$
$241$	3.31003	0.213218	0.106609	+	0.994301i	$0.466001\pi$
$242$	0	0
$243$	0.620870	0.0398288
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.01921	0.510250
$248$	0	0
$249$	8.01806	0.508124
$250$	0	0
$251$	9.46454	0.597397	0.298698	−	0.954348i	$-0.403448\pi$
$251$	9.46454	0.597397	0.298698	+	0.954348i	$0.403448\pi$
$252$	0	0
$253$	1.81717	0.114245
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.07963	0.441615	0.220808	−	0.975317i	$-0.429131\pi$
$257$	7.07963	0.441615	0.220808	+	0.975317i	$0.429131\pi$
$258$	0	0
$259$	−2.90615	−0.180580
$260$	0	0
$261$	−0.244786	−0.0151519
$262$	0	0
$263$	1.43345	0.0883906	0.0441953	−	0.999023i	$-0.485928\pi$
$263$	1.43345	0.0883906	0.0441953	+	0.999023i	$0.485928\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−15.7480	−0.963760
$268$	0	0
$269$	−14.2830	−0.870848	−0.435424	−	0.900226i	$-0.643402\pi$
$269$	−14.2830	−0.870848	−0.435424	+	0.900226i	$0.643402\pi$
$270$	0	0
$271$	−30.7855	−1.87009	−0.935044	−	0.354533i	$-0.884640\pi$
$271$	−30.7855	−1.87009	−0.935044	+	0.354533i	$0.884640\pi$
$272$	0	0
$273$	27.0116	1.63482
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3.18969	−0.191650	−0.0958249	−	0.995398i	$-0.530549\pi$
$277$	−3.18969	−0.191650	−0.0958249	+	0.995398i	$0.530549\pi$
$278$	0	0
$279$	−0.425860	−0.0254956
$280$	0	0
$281$	20.3889	1.21630	0.608151	−	0.793822i	$-0.291912\pi$
$281$	20.3889	1.21630	0.608151	+	0.793822i	$0.291912\pi$
$282$	0	0
$283$	−13.2969	−0.790419	−0.395209	−	0.918591i	$-0.629328\pi$
$283$	−13.2969	−0.790419	−0.395209	+	0.918591i	$0.629328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−25.2677	−1.49150
$288$	0	0
$289$	9.58880	0.564047
$290$	0	0
$291$	0.154616	0.00906374
$292$	0	0
$293$	26.0420	1.52139	0.760696	−	0.649108i	$-0.224858\pi$
$293$	26.0420	1.52139	0.760696	+	0.649108i	$0.224858\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	14.5565	0.844655
$298$	0	0
$299$	3.71874	0.215060
$300$	0	0
$301$	−25.6544	−1.47869
$302$	0	0
$303$	−13.5571	−0.778835
$304$	0	0
$305$	0	0
$306$	0	0
$307$	25.1000	1.43253	0.716267	−	0.697826i	$-0.245849\pi$
$307$	25.1000	1.43253	0.716267	+	0.697826i	$0.245849\pi$
$308$	0	0
$309$	13.4335	0.764203
$310$	0	0
$311$	−0.723328	−0.0410162	−0.0205081	−	0.999790i	$-0.506528\pi$
$311$	−0.723328	−0.0410162	−0.0205081	+	0.999790i	$0.506528\pi$
$312$	0	0
$313$	−21.6163	−1.22182	−0.610912	−	0.791698i	$-0.709197\pi$
$313$	−21.6163	−1.22182	−0.610912	+	0.791698i	$0.709197\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.8337	1.05780	0.528902	−	0.848683i	$-0.322604\pi$
$317$	18.8337	1.05780	0.528902	+	0.848683i	$0.322604\pi$
$318$	0	0
$319$	−11.3661	−0.636380
$320$	0	0
$321$	−17.5278	−0.978304
$322$	0	0
$323$	7.28286	0.405229
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.70610	0.370848
$328$	0	0
$329$	7.69767	0.424386
$330$	0	0
$331$	−16.8759	−0.927584	−0.463792	−	0.885944i	$-0.653511\pi$
$331$	−16.8759	−0.927584	−0.463792	+	0.885944i	$0.653511\pi$
$332$	0	0
$333$	−0.0625883	−0.00342982
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−19.3521	−1.05417	−0.527087	−	0.849811i	$-0.676716\pi$
$337$	−19.3521	−1.05417	−0.527087	+	0.849811i	$0.676716\pi$
$338$	0	0
$339$	10.9321	0.593750
$340$	0	0
$341$	−19.7739	−1.07082
$342$	0	0
$343$	17.4856	0.944134
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.03868	−0.485222	−0.242611	−	0.970124i	$-0.578004\pi$
$347$	−9.03868	−0.485222	−0.242611	+	0.970124i	$0.578004\pi$
$348$	0	0
$349$	36.7305	1.96614	0.983068	−	0.183240i	$-0.0586584\pi$
$349$	36.7305	1.96614	0.983068	+	0.183240i	$0.0586584\pi$
$350$	0	0
$351$	29.7891	1.59002
$352$	0	0
$353$	−18.3737	−0.977933	−0.488967	−	0.872302i	$-0.662626\pi$
$353$	−18.3737	−0.977933	−0.488967	+	0.872302i	$0.662626\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.5313	1.29834
$358$	0	0
$359$	24.4259	1.28915	0.644574	−	0.764542i	$-0.277035\pi$
$359$	24.4259	1.28915	0.644574	+	0.764542i	$0.277035\pi$
$360$	0	0
$361$	−17.0052	−0.895009
$362$	0	0
$363$	−5.66256	−0.297207
$364$	0	0
$365$	0	0
$366$	0	0
$367$	14.5224	0.758065	0.379032	−	0.925383i	$-0.376257\pi$
$367$	14.5224	0.758065	0.379032	+	0.925383i	$0.376257\pi$
$368$	0	0
$369$	−0.544176	−0.0283287
$370$	0	0
$371$	−1.45968	−0.0757827
$372$	0	0
$373$	23.2887	1.20585	0.602923	−	0.797800i	$-0.294003\pi$
$373$	23.2887	1.20585	0.602923	+	0.797800i	$0.294003\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−23.2601	−1.19796
$378$	0	0
$379$	−28.6176	−1.46999	−0.734993	−	0.678075i	$-0.762815\pi$
$379$	−28.6176	−1.46999	−0.734993	+	0.678075i	$0.762815\pi$
$380$	0	0
$381$	16.3737	0.838850
$382$	0	0
$383$	−16.6157	−0.849023	−0.424512	−	0.905422i	$-0.639554\pi$
$383$	−16.6157	−0.849023	−0.424512	+	0.905422i	$0.639554\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.552505	−0.0280854
$388$	0	0
$389$	26.3767	1.33735	0.668677	−	0.743553i	$-0.266861\pi$
$389$	26.3767	1.33735	0.668677	+	0.743553i	$0.266861\pi$
$390$	0	0
$391$	3.37727	0.170796
$392$	0	0
$393$	−9.60109	−0.484311
$394$	0	0
$395$	0	0
$396$	0	0
$397$	17.9086	0.898807	0.449403	−	0.893329i	$-0.351637\pi$
$397$	17.9086	0.898807	0.449403	+	0.893329i	$0.351637\pi$
$398$	0	0
$399$	6.71930	0.336386
$400$	0	0
$401$	32.0164	1.59882	0.799411	−	0.600785i	$-0.205145\pi$
$401$	32.0164	1.59882	0.799411	+	0.600785i	$0.205145\pi$
$402$	0	0
$403$	−40.4661	−2.01576
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.90615	−0.144053
$408$	0	0
$409$	18.0344	0.891743	0.445871	−	0.895097i	$-0.352894\pi$
$409$	18.0344	0.891743	0.445871	+	0.895097i	$0.352894\pi$
$410$	0	0
$411$	−13.6905	−0.675304
$412$	0	0
$413$	−10.4932	−0.516337
$414$	0	0
$415$	0	0
$416$	0	0
$417$	37.8753	1.85476
$418$	0	0
$419$	19.5300	0.954104	0.477052	−	0.878875i	$-0.341705\pi$
$419$	19.5300	0.954104	0.477052	+	0.878875i	$0.341705\pi$
$420$	0	0
$421$	−19.5537	−0.952989	−0.476494	−	0.879178i	$-0.658093\pi$
$421$	−19.5537	−0.952989	−0.476494	+	0.879178i	$0.658093\pi$
$422$	0	0
$423$	0.165781	0.00806052
$424$	0	0
$425$	0	0
$426$	0	0
$427$	30.0543	1.45443
$428$	0	0
$429$	27.0116	1.30413
$430$	0	0
$431$	21.1963	1.02099	0.510495	−	0.859881i	$-0.329462\pi$
$431$	21.1963	1.02099	0.510495	+	0.859881i	$0.329462\pi$
$432$	0	0
$433$	−8.54967	−0.410871	−0.205435	−	0.978671i	$-0.565861\pi$
$433$	−8.54967	−0.410871	−0.205435	+	0.978671i	$0.565861\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.925059	0.0442516
$438$	0	0
$439$	28.0714	1.33977	0.669887	−	0.742463i	$-0.266343\pi$
$439$	28.0714	1.33977	0.669887	+	0.742463i	$0.266343\pi$
$440$	0	0
$441$	−0.0416872	−0.00198511
$442$	0	0
$443$	−33.0546	−1.57047	−0.785236	−	0.619197i	$-0.787458\pi$
$443$	−33.0546	−1.57047	−0.785236	+	0.619197i	$0.787458\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	7.17439	0.339337
$448$	0	0
$449$	−37.1628	−1.75382	−0.876911	−	0.480652i	$-0.840400\pi$
$449$	−37.1628	−1.75382	−0.876911	+	0.480652i	$0.840400\pi$
$450$	0	0
$451$	−25.2677	−1.18981
$452$	0	0
$453$	0.568533	0.0267120
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−29.3139	−1.37125	−0.685624	−	0.727956i	$-0.740471\pi$
$457$	−29.3139	−1.37125	−0.685624	+	0.727956i	$0.740471\pi$
$458$	0	0
$459$	27.0538	1.26276
$460$	0	0
$461$	8.90741	0.414859	0.207430	−	0.978250i	$-0.433490\pi$
$461$	8.90741	0.414859	0.207430	+	0.978250i	$0.433490\pi$
$462$	0	0
$463$	−18.2771	−0.849410	−0.424705	−	0.905332i	$-0.639622\pi$
$463$	−18.2771	−0.849410	−0.424705	+	0.905332i	$0.639622\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.6841	−1.69754	−0.848768	−	0.528765i	$-0.822655\pi$
$467$	−36.6841	−1.69754	−0.848768	+	0.528765i	$0.822655\pi$
$468$	0	0
$469$	11.9951	0.553884
$470$	0	0
$471$	−6.18583	−0.285028
$472$	0	0
$473$	−25.6544	−1.17959
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.0314363	−0.00143937
$478$	0	0
$479$	12.5898	0.575242	0.287621	−	0.957744i	$-0.407136\pi$
$479$	12.5898	0.575242	0.287621	+	0.957744i	$0.407136\pi$
$480$	0	0
$481$	−5.94728	−0.271173
$482$	0	0
$483$	3.11593	0.141780
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−18.3256	−0.830411	−0.415205	−	0.909728i	$-0.636290\pi$
$487$	−18.3256	−0.830411	−0.415205	+	0.909728i	$0.636290\pi$
$488$	0	0
$489$	8.73449	0.394987
$490$	0	0
$491$	−21.7047	−0.979519	−0.489760	−	0.871857i	$-0.662915\pi$
$491$	−21.7047	−0.979519	−0.489760	+	0.871857i	$0.662915\pi$
$492$	0	0
$493$	−21.1243	−0.951389
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−36.7523	−1.64856
$498$	0	0
$499$	−15.8391	−0.709055	−0.354527	−	0.935046i	$-0.615358\pi$
$499$	−15.8391	−0.709055	−0.354527	+	0.935046i	$0.615358\pi$
$500$	0	0
$501$	35.7739	1.59826
$502$	0	0
$503$	27.5530	1.22853	0.614263	−	0.789101i	$-0.289453\pi$
$503$	27.5530	1.22853	0.614263	+	0.789101i	$0.289453\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	32.9864	1.46498
$508$	0	0
$509$	−27.6120	−1.22388	−0.611941	−	0.790903i	$-0.709611\pi$
$509$	−27.6120	−1.22388	−0.611941	+	0.790903i	$0.709611\pi$
$510$	0	0
$511$	11.6895	0.517113
$512$	0	0
$513$	7.41022	0.327169
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.69767	0.338543
$518$	0	0
$519$	20.5555	0.902287
$520$	0	0
$521$	17.0892	0.748689	0.374345	−	0.927290i	$-0.377868\pi$
$521$	17.0892	0.748689	0.374345	+	0.927290i	$0.377868\pi$
$522$	0	0
$523$	36.3152	1.58795	0.793977	−	0.607947i	$-0.208007\pi$
$523$	36.3152	1.58795	0.793977	+	0.607947i	$0.208007\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−36.7504	−1.60087
$528$	0	0
$529$	−22.5710	−0.981349
$530$	0	0
$531$	−0.225987	−0.00980698
$532$	0	0
$533$	−51.7088	−2.23976
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14.6460	0.632020
$538$	0	0
$539$	−1.93566	−0.0833747
$540$	0	0
$541$	24.9366	1.07211	0.536055	−	0.844183i	$-0.319914\pi$
$541$	24.9366	1.07211	0.536055	+	0.844183i	$0.319914\pi$
$542$	0	0
$543$	13.4235	0.576060
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6.67201	0.285275	0.142637	−	0.989775i	$-0.454442\pi$
$547$	6.67201	0.285275	0.142637	+	0.989775i	$0.454442\pi$
$548$	0	0
$549$	0.647264	0.0276245
$550$	0	0
$551$	−5.78609	−0.246496
$552$	0	0
$553$	−27.4716	−1.16821
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.4328	0.865765	0.432882	−	0.901450i	$-0.357497\pi$
$557$	20.4328	0.865765	0.432882	+	0.901450i	$0.357497\pi$
$558$	0	0
$559$	−52.5002	−2.22052
$560$	0	0
$561$	24.5313	1.03571
$562$	0	0
$563$	−0.602805	−0.0254052	−0.0127026	−	0.999919i	$-0.504043\pi$
$563$	−0.602805	−0.0254052	−0.0127026	+	0.999919i	$0.504043\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	24.4630	1.02735
$568$	0	0
$569$	33.8776	1.42022	0.710112	−	0.704089i	$-0.248644\pi$
$569$	33.8776	1.42022	0.710112	+	0.704089i	$0.248644\pi$
$570$	0	0
$571$	6.23012	0.260722	0.130361	−	0.991467i	$-0.458386\pi$
$571$	6.23012	0.260722	0.130361	+	0.991467i	$0.458386\pi$
$572$	0	0
$573$	−5.77192	−0.241125
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.78104	−0.157407	−0.0787034	−	0.996898i	$-0.525078\pi$
$577$	−3.78104	−0.157407	−0.0787034	+	0.996898i	$0.525078\pi$
$578$	0	0
$579$	26.4428	1.09893
$580$	0	0
$581$	−12.9735	−0.538232
$582$	0	0
$583$	−1.45968	−0.0604537
$584$	0	0
$585$	0	0
$586$	0	0
$587$	16.3526	0.674945	0.337473	−	0.941335i	$-0.390428\pi$
$587$	16.3526	0.674945	0.337473	+	0.941335i	$0.390428\pi$
$588$	0	0
$589$	−10.0662	−0.414770
$590$	0	0
$591$	−15.0979	−0.621043
$592$	0	0
$593$	−9.53314	−0.391479	−0.195740	−	0.980656i	$-0.562711\pi$
$593$	−9.53314	−0.391479	−0.195740	+	0.980656i	$0.562711\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	30.2170	1.23670
$598$	0	0
$599$	16.0682	0.656528	0.328264	−	0.944586i	$-0.393536\pi$
$599$	16.0682	0.656528	0.328264	+	0.944586i	$0.393536\pi$
$600$	0	0
$601$	15.0380	0.613413	0.306707	−	0.951804i	$-0.400773\pi$
$601$	15.0380	0.613413	0.306707	+	0.951804i	$0.400773\pi$
$602$	0	0
$603$	0.258333	0.0105201
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−44.8348	−1.81979	−0.909894	−	0.414840i	$-0.863837\pi$
$607$	−44.8348	−1.81979	−0.909894	+	0.414840i	$0.863837\pi$
$608$	0	0
$609$	−19.4896	−0.789760
$610$	0	0
$611$	15.7528	0.637291
$612$	0	0
$613$	15.5488	0.628011	0.314006	−	0.949421i	$-0.398329\pi$
$613$	15.5488	0.628011	0.314006	+	0.949421i	$0.398329\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17.5994	−0.708525	−0.354263	−	0.935146i	$-0.615268\pi$
$617$	−17.5994	−0.708525	−0.354263	+	0.935146i	$0.615268\pi$
$618$	0	0
$619$	37.2887	1.49876	0.749381	−	0.662140i	$-0.230351\pi$
$619$	37.2887	1.49876	0.749381	+	0.662140i	$0.230351\pi$
$620$	0	0
$621$	3.43633	0.137895
$622$	0	0
$623$	25.4807	1.02086
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.71930	0.268343
$628$	0	0
$629$	−5.40118	−0.215359
$630$	0	0
$631$	−8.55551	−0.340589	−0.170295	−	0.985393i	$-0.554472\pi$
$631$	−8.55551	−0.340589	−0.170295	+	0.985393i	$0.554472\pi$
$632$	0	0
$633$	−10.4622	−0.415836
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−3.96121	−0.156949
$638$	0	0
$639$	−0.791514	−0.0313118
$640$	0	0
$641$	−22.8156	−0.901162	−0.450581	−	0.892736i	$-0.648783\pi$
$641$	−22.8156	−0.901162	−0.450581	+	0.892736i	$0.648783\pi$
$642$	0	0
$643$	34.7745	1.37137	0.685686	−	0.727898i	$-0.259503\pi$
$643$	34.7745	1.37137	0.685686	+	0.727898i	$0.259503\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.4581	−0.647034	−0.323517	−	0.946222i	$-0.604865\pi$
$647$	−16.4581	−0.647034	−0.323517	+	0.946222i	$0.604865\pi$
$648$	0	0
$649$	−10.4932	−0.411894
$650$	0	0
$651$	−33.9066	−1.32890
$652$	0	0
$653$	−30.2092	−1.18218	−0.591088	−	0.806607i	$-0.701301\pi$
$653$	−30.2092	−1.18218	−0.591088	+	0.806607i	$0.701301\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0.251751	0.00982173
$658$	0	0
$659$	28.4942	1.10998	0.554989	−	0.831858i	$-0.312723\pi$
$659$	28.4942	1.10998	0.554989	+	0.831858i	$0.312723\pi$
$660$	0	0
$661$	50.7342	1.97333	0.986666	−	0.162759i	$-0.0520393\pi$
$661$	50.7342	1.97333	0.986666	+	0.162759i	$0.0520393\pi$
$662$	0	0
$663$	50.2019	1.94968
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.68317	−0.103893
$668$	0	0
$669$	−8.62273	−0.333374
$670$	0	0
$671$	30.0543	1.16023
$672$	0	0
$673$	20.4447	0.788084	0.394042	−	0.919093i	$-0.371077\pi$
$673$	20.4447	0.788084	0.394042	+	0.919093i	$0.371077\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.53360	−0.327973	−0.163986	−	0.986463i	$-0.552435\pi$
$677$	−8.53360	−0.327973	−0.163986	+	0.986463i	$0.552435\pi$
$678$	0	0
$679$	−0.250174	−0.00960078
$680$	0	0
$681$	−30.5875	−1.17212
$682$	0	0
$683$	−48.5684	−1.85842	−0.929210	−	0.369553i	$-0.879511\pi$
$683$	−48.5684	−1.85842	−0.929210	+	0.369553i	$0.879511\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−36.9635	−1.41024
$688$	0	0
$689$	−2.98715	−0.113801
$690$	0	0
$691$	−44.8076	−1.70456	−0.852281	−	0.523084i	$-0.824781\pi$
$691$	−44.8076	−1.70456	−0.852281	+	0.523084i	$0.824781\pi$
$692$	0	0
$693$	−0.459953	−0.0174722
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−46.9608	−1.77877
$698$	0	0
$699$	−25.2174	−0.953809
$700$	0	0
$701$	2.21913	0.0838152	0.0419076	−	0.999121i	$-0.486656\pi$
$701$	2.21913	0.0838152	0.0419076	+	0.999121i	$0.486656\pi$
$702$	0	0
$703$	−1.47942	−0.0557974
$704$	0	0
$705$	0	0
$706$	0	0
$707$	21.9358	0.824982
$708$	0	0
$709$	34.3587	1.29037	0.645184	−	0.764027i	$-0.276781\pi$
$709$	34.3587	1.29037	0.645184	+	0.764027i	$0.276781\pi$
$710$	0	0
$711$	−0.591640	−0.0221882
$712$	0	0
$713$	−4.66799	−0.174817
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.9005	−0.444434
$718$	0	0
$719$	−11.9900	−0.447151	−0.223575	−	0.974687i	$-0.571773\pi$
$719$	−11.9900	−0.447151	−0.223575	+	0.974687i	$0.571773\pi$
$720$	0	0
$721$	−21.7358	−0.809483
$722$	0	0
$723$	5.67576	0.211084
$724$	0	0
$725$	0	0
$726$	0	0
$727$	42.1827	1.56447	0.782235	−	0.622983i	$-0.214080\pi$
$727$	42.1827	1.56447	0.782235	+	0.622983i	$0.214080\pi$
$728$	0	0
$729$	27.5161	1.01912
$730$	0	0
$731$	−47.6795	−1.76349
$732$	0	0
$733$	0.322383	0.0119075	0.00595375	−	0.999982i	$-0.498105\pi$
$733$	0.322383	0.0119075	0.00595375	+	0.999982i	$0.498105\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	11.9951	0.441847
$738$	0	0
$739$	5.49978	0.202313	0.101156	−	0.994871i	$-0.467746\pi$
$739$	5.49978	0.202313	0.101156	+	0.994871i	$0.467746\pi$
$740$	0	0
$741$	13.7507	0.505143
$742$	0	0
$743$	−6.03812	−0.221517	−0.110759	−	0.993847i	$-0.535328\pi$
$743$	−6.03812	−0.221517	−0.110759	+	0.993847i	$0.535328\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−0.279403	−0.0102228
$748$	0	0
$749$	28.3605	1.03627
$750$	0	0
$751$	13.5719	0.495244	0.247622	−	0.968857i	$-0.420351\pi$
$751$	13.5719	0.495244	0.247622	+	0.968857i	$0.420351\pi$
$752$	0	0
$753$	16.2290	0.591417
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−33.0661	−1.20181	−0.600904	−	0.799321i	$-0.705193\pi$
$757$	−33.0661	−1.20181	−0.600904	+	0.799321i	$0.705193\pi$
$758$	0	0
$759$	3.11593	0.113101
$760$	0	0
$761$	5.18860	0.188087	0.0940434	−	0.995568i	$-0.470021\pi$
$761$	5.18860	0.188087	0.0940434	+	0.995568i	$0.470021\pi$
$762$	0	0
$763$	−10.8507	−0.392822
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−21.4737	−0.775372
$768$	0	0
$769$	33.0698	1.19253	0.596264	−	0.802789i	$-0.296651\pi$
$769$	33.0698	1.19253	0.596264	+	0.802789i	$0.296651\pi$
$770$	0	0
$771$	12.1396	0.437195
$772$	0	0
$773$	1.13095	0.0406776	0.0203388	−	0.999793i	$-0.493526\pi$
$773$	1.13095	0.0406776	0.0203388	+	0.999793i	$0.493526\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.98323	−0.178772
$778$	0	0
$779$	−12.8629	−0.460861
$780$	0	0
$781$	−36.7523	−1.31510
$782$	0	0
$783$	−21.4937	−0.768121
$784$	0	0
$785$	0	0
$786$	0	0
$787$	15.7610	0.561819	0.280909	−	0.959734i	$-0.409364\pi$
$787$	15.7610	0.561819	0.280909	+	0.959734i	$0.409364\pi$
$788$	0	0
$789$	2.45797	0.0875059
$790$	0	0
$791$	−17.6885	−0.628930
$792$	0	0
$793$	61.5044	2.18409
$794$	0	0
$795$	0	0
$796$	0	0
$797$	23.1397	0.819649	0.409824	−	0.912164i	$-0.365590\pi$
$797$	23.1397	0.819649	0.409824	+	0.912164i	$0.365590\pi$
$798$	0	0
$799$	14.3064	0.506122
$800$	0	0
$801$	0.548765	0.0193897
$802$	0	0
$803$	11.6895	0.412514
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−24.4912	−0.862132
$808$	0	0
$809$	33.5400	1.17921	0.589603	−	0.807694i	$-0.299284\pi$
$809$	33.5400	1.17921	0.589603	+	0.807694i	$0.299284\pi$
$810$	0	0
$811$	11.9626	0.420064	0.210032	−	0.977695i	$-0.432643\pi$
$811$	11.9626	0.420064	0.210032	+	0.977695i	$0.432643\pi$
$812$	0	0
$813$	−52.7884	−1.85137
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−13.0598	−0.456903
$818$	0	0
$819$	−0.941266	−0.0328905
$820$	0	0
$821$	−10.6941	−0.373227	−0.186613	−	0.982433i	$-0.559751\pi$
$821$	−10.6941	−0.373227	−0.186613	+	0.982433i	$0.559751\pi$
$822$	0	0
$823$	−12.0211	−0.419028	−0.209514	−	0.977806i	$-0.567188\pi$
$823$	−12.0211	−0.419028	−0.209514	+	0.977806i	$0.567188\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.13974	−0.143953	−0.0719764	−	0.997406i	$-0.522931\pi$
$827$	−4.13974	−0.143953	−0.0719764	+	0.997406i	$0.522931\pi$
$828$	0	0
$829$	−20.3503	−0.706796	−0.353398	−	0.935473i	$-0.614974\pi$
$829$	−20.3503	−0.706796	−0.353398	+	0.935473i	$0.614974\pi$
$830$	0	0
$831$	−5.46940	−0.189732
$832$	0	0
$833$	−3.59748	−0.124645
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−37.3931	−1.29249
$838$	0	0
$839$	−3.13515	−0.108237	−0.0541186	−	0.998535i	$-0.517235\pi$
$839$	−3.13515	−0.108237	−0.0541186	+	0.998535i	$0.517235\pi$
$840$	0	0
$841$	−12.2172	−0.421283
$842$	0	0
$843$	34.9612	1.20413
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.16221	0.314817
$848$	0	0
$849$	−22.8004	−0.782508
$850$	0	0
$851$	−0.686050	−0.0235175
$852$	0	0
$853$	−23.3471	−0.799388	−0.399694	−	0.916649i	$-0.630884\pi$
$853$	−23.3471	−0.799388	−0.399694	+	0.916649i	$0.630884\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−15.3036	−0.522762	−0.261381	−	0.965236i	$-0.584178\pi$
$857$	−15.3036	−0.522762	−0.261381	+	0.965236i	$0.584178\pi$
$858$	0	0
$859$	2.96132	0.101039	0.0505194	−	0.998723i	$-0.483912\pi$
$859$	2.96132	0.101039	0.0505194	+	0.998723i	$0.483912\pi$
$860$	0	0
$861$	−43.3269	−1.47658
$862$	0	0
$863$	−37.8220	−1.28748	−0.643739	−	0.765246i	$-0.722618\pi$
$863$	−37.8220	−1.28748	−0.643739	+	0.765246i	$0.722618\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	16.4421	0.558402
$868$	0	0
$869$	−27.4716	−0.931909
$870$	0	0
$871$	24.5474	0.831755
$872$	0	0
$873$	−0.00538785	−0.000182351 0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.6011	0.526810	0.263405	−	0.964685i	$-0.415154\pi$
$877$	15.6011	0.526810	0.263405	+	0.964685i	$0.415154\pi$
$878$	0	0
$879$	44.6547	1.50616
$880$	0	0
$881$	1.16438	0.0392289	0.0196144	−	0.999808i	$-0.493756\pi$
$881$	1.16438	0.0392289	0.0196144	+	0.999808i	$0.493756\pi$
$882$	0	0
$883$	48.8851	1.64511	0.822557	−	0.568682i	$-0.192547\pi$
$883$	48.8851	1.64511	0.822557	+	0.568682i	$0.192547\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.62892	0.289731	0.144865	−	0.989451i	$-0.453725\pi$
$887$	8.62892	0.289731	0.144865	+	0.989451i	$0.453725\pi$
$888$	0	0
$889$	−26.4932	−0.888554
$890$	0	0
$891$	24.4630	0.819540
$892$	0	0
$893$	3.91861	0.131131
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.37658	0.212908
$898$	0	0
$899$	29.1975	0.973790
$900$	0	0
$901$	−2.71286	−0.0903784
$902$	0	0
$903$	−43.9900	−1.46389
$904$	0	0
$905$	0	0
$906$	0	0
$907$	29.5085	0.979815	0.489907	−	0.871774i	$-0.337031\pi$
$907$	29.5085	0.979815	0.489907	+	0.871774i	$0.337031\pi$
$908$	0	0
$909$	0.472420	0.0156692
$910$	0	0
$911$	38.4088	1.27254	0.636270	−	0.771466i	$-0.280476\pi$
$911$	38.4088	1.27254	0.636270	+	0.771466i	$0.280476\pi$
$912$	0	0
$913$	−12.9735	−0.429360
$914$	0	0
$915$	0	0
$916$	0	0
$917$	15.5349	0.513007
$918$	0	0
$919$	−41.6905	−1.37524	−0.687622	−	0.726069i	$-0.741345\pi$
$919$	−41.6905	−1.37524	−0.687622	+	0.726069i	$0.741345\pi$
$920$	0	0
$921$	43.0394	1.41820
$922$	0	0
$923$	−75.2114	−2.47561
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−0.468112	−0.0153748
$928$	0	0
$929$	28.1047	0.922086	0.461043	−	0.887378i	$-0.347475\pi$
$929$	28.1047	0.922086	0.461043	+	0.887378i	$0.347475\pi$
$930$	0	0
$931$	−0.985376	−0.0322944
$932$	0	0
$933$	−1.24030	−0.0406056
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.8754	0.420620	0.210310	−	0.977635i	$-0.432553\pi$
$937$	12.8754	0.420620	0.210310	+	0.977635i	$0.432553\pi$
$938$	0	0
$939$	−37.0658	−1.20960
$940$	0	0
$941$	10.1510	0.330913	0.165457	−	0.986217i	$-0.447090\pi$
$941$	10.1510	0.330913	0.165457	+	0.986217i	$0.447090\pi$
$942$	0	0
$943$	−5.96489	−0.194243
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−57.3750	−1.86444	−0.932218	−	0.361896i	$-0.882129\pi$
$947$	−57.3750	−1.86444	−0.932218	+	0.361896i	$0.882129\pi$
$948$	0	0
$949$	23.9219	0.776538
$950$	0	0
$951$	32.2944	1.04722
$952$	0	0
$953$	−5.17264	−0.167558	−0.0837791	−	0.996484i	$-0.526699\pi$
$953$	−5.17264	−0.167558	−0.0837791	+	0.996484i	$0.526699\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−19.4896	−0.630010
$958$	0	0
$959$	22.1517	0.715317
$960$	0	0
$961$	19.7955	0.638566
$962$	0	0
$963$	0.610785	0.0196823
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−41.3979	−1.33127	−0.665634	−	0.746279i	$-0.731839\pi$
$967$	−41.3979	−1.33127	−0.665634	+	0.746279i	$0.731839\pi$
$968$	0	0
$969$	12.4880	0.401173
$970$	0	0
$971$	−43.0848	−1.38266	−0.691329	−	0.722540i	$-0.742975\pi$
$971$	−43.0848	−1.38266	−0.691329	+	0.722540i	$0.742975\pi$
$972$	0	0
$973$	−61.2836	−1.96466
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40.4614	−1.29448	−0.647238	−	0.762288i	$-0.724076\pi$
$977$	−40.4614	−1.29448	−0.647238	+	0.762288i	$0.724076\pi$
$978$	0	0
$979$	25.4807	0.814368
$980$	0	0
$981$	−0.233686	−0.00746101
$982$	0	0
$983$	40.3786	1.28788	0.643938	−	0.765078i	$-0.277299\pi$
$983$	40.3786	1.28788	0.643938	+	0.765078i	$0.277299\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	13.1993	0.420139
$988$	0	0
$989$	−6.05618	−0.192575
$990$	0	0
$991$	58.1435	1.84699	0.923494	−	0.383613i	$-0.125320\pi$
$991$	58.1435	1.84699	0.923494	+	0.383613i	$0.125320\pi$
$992$	0	0
$993$	−28.9374	−0.918300
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−39.2968	−1.24454	−0.622271	−	0.782802i	$-0.713790\pi$
$997$	−39.2968	−1.24454	−0.622271	+	0.782802i	$0.713790\pi$
$998$	0	0
$999$	−5.49563	−0.173874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.x.1.3		4
4.3	odd	2		1250.2.a.f.1.2		4
5.4	even	2		10000.2.a.t.1.2		4
20.3	even	4		1250.2.b.e.1249.6		8
20.7	even	4		1250.2.b.e.1249.3		8
20.19	odd	2		1250.2.a.l.1.3		4
25.4	even	10		400.2.u.d.241.1		8
25.19	even	10		400.2.u.d.161.1		8
100.3	even	20		250.2.e.c.49.4		16
100.19	odd	10		50.2.d.b.11.2	✓	8
100.31	odd	10		250.2.d.d.51.1		8
100.47	even	20		250.2.e.c.49.1		16
100.67	even	20		250.2.e.c.199.4		16
100.71	odd	10		250.2.d.d.201.1		8
100.79	odd	10		50.2.d.b.41.2	yes	8
100.83	even	20		250.2.e.c.199.1		16
300.119	even	10		450.2.h.e.361.1		8
300.179	even	10		450.2.h.e.91.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.d.b.11.2	✓	8	100.19	odd	10
50.2.d.b.41.2	yes	8	100.79	odd	10
250.2.d.d.51.1		8	100.31	odd	10
250.2.d.d.201.1		8	100.71	odd	10
250.2.e.c.49.1		16	100.47	even	20
250.2.e.c.49.4		16	100.3	even	20
250.2.e.c.199.1		16	100.83	even	20
250.2.e.c.199.4		16	100.67	even	20
400.2.u.d.161.1		8	25.19	even	10
400.2.u.d.241.1		8	25.4	even	10
450.2.h.e.91.1		8	300.179	even	10
450.2.h.e.361.1		8	300.119	even	10
1250.2.a.f.1.2		4	4.3	odd	2
1250.2.a.l.1.3		4	20.19	odd	2
1250.2.b.e.1249.3		8	20.7	even	4
1250.2.b.e.1249.6		8	20.3	even	4
10000.2.a.t.1.2		4	5.4	even	2
10000.2.a.x.1.3		4	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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q+1.71472 q^{3} -2.77447 q^{7} -0.0597522 q^{9} +O(q^{10})\)	\(q+1.71472 q^{3} -2.77447 q^{7} -0.0597522 q^{9} -2.77447 q^{11} -5.67779 q^{13} -5.15643 q^{17} -1.41238 q^{19} -4.75742 q^{21} -0.654963 q^{23} -5.24660 q^{27} +4.09668 q^{29} +7.12710 q^{31} -4.75742 q^{33} +1.04746 q^{37} -9.73579 q^{39} +9.10722 q^{41} +9.24660 q^{43} -2.77447 q^{47} +0.697669 q^{49} -8.84181 q^{51} +0.526111 q^{53} -2.42184 q^{57} +3.78206 q^{59} -10.8325 q^{61} +0.165781 q^{63} -4.32340 q^{67} -1.12307 q^{69} +13.2466 q^{71} -4.21324 q^{73} +7.69767 q^{77} +9.90157 q^{79} -8.81717 q^{81} +4.67603 q^{83} +7.02464 q^{87} -9.18401 q^{89} +15.7528 q^{91} +12.2209 q^{93} +0.0901699 q^{97} +0.165781 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 2 q^{7} + 7 q^{9}+O(q^{10}) \) 4 * q + q^3 + 2 * q^7 + 7 * q^9	\( 4 q + q^{3} + 2 q^{7} + 7 q^{9} + 2 q^{11} - 11 q^{13} - 12 q^{17} + 5 q^{19} - 7 q^{21} - 4 q^{23} + 10 q^{27} + 15 q^{29} + 12 q^{31} - 7 q^{33} - 12 q^{37} + 11 q^{39} + 13 q^{41} + 6 q^{43} + 2 q^{47} - 2 q^{49} - 13 q^{51} - 11 q^{53} + 8 q^{61} + 21 q^{63} + 22 q^{67} - 31 q^{69} + 22 q^{71} - 21 q^{73} + 26 q^{77} + 10 q^{79} - 16 q^{81} - 24 q^{83} + 25 q^{87} - 5 q^{89} + 12 q^{91} + 23 q^{93} - 22 q^{97} + 21 q^{99}+O(q^{100}) \) 4 * q + q^3 + 2 * q^7 + 7 * q^9 + 2 * q^11 - 11 * q^13 - 12 * q^17 + 5 * q^19 - 7 * q^21 - 4 * q^23 + 10 * q^27 + 15 * q^29 + 12 * q^31 - 7 * q^33 - 12 * q^37 + 11 * q^39 + 13 * q^41 + 6 * q^43 + 2 * q^47 - 2 * q^49 - 13 * q^51 - 11 * q^53 + 8 * q^61 + 21 * q^63 + 22 * q^67 - 31 * q^69 + 22 * q^71 - 21 * q^73 + 26 * q^77 + 10 * q^79 - 16 * q^81 - 24 * q^83 + 25 * q^87 - 5 * q^89 + 12 * q^91 + 23 * q^93 - 22 * q^97 + 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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