LMFDB - Embedded newform 10000.2.a.r.1.1

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.108625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 34x^{2} + 9x + 261 $ x^4 - x^3 - 34x^2 + 9x + 261
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5000)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.71963$ 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)

$f(q)$	$=$	$q-1.61803 q^{3} -4.33766 q^{7} -0.381966 q^{9} +O(q^{10})$	$q-1.61803 q^{3} -4.33766 q^{7} -0.381966 q^{9} +3.10159 q^{11} -0.680822 q^{13} +4.95569 q^{17} -2.93721 q^{19} +7.01848 q^{21} -5.57373 q^{23} +5.47214 q^{27} -0.763932 q^{29} +4.47214 q^{31} -5.01848 q^{33} +7.15296 q^{37} +1.10159 q^{39} -9.63652 q^{41} -10.4278 q^{43} -9.01848 q^{47} +11.8153 q^{49} -8.01848 q^{51} -0.472136 q^{53} +4.75251 q^{57} +9.71963 q^{59} +9.65684 q^{61} +1.65684 q^{63} +3.40045 q^{67} +9.01848 q^{69} +9.49062 q^{71} +9.00706 q^{73} -13.4537 q^{77} +0.680822 q^{79} -7.70820 q^{81} -8.00000 q^{83} +1.23607 q^{87} +8.97418 q^{89} +2.95317 q^{91} -7.23607 q^{93} -16.7381 q^{97} -1.18470 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 3 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 3 * q^7 - 6 * q^9	$ 4 q - 2 q^{3} + 3 q^{7} - 6 q^{9} + q^{11} + 4 q^{13} - 5 q^{17} - 14 q^{19} + q^{21} + 7 q^{23} + 4 q^{27} - 12 q^{29} + 7 q^{33} + 4 q^{37} - 7 q^{39} - 7 q^{41} + q^{43} - 9 q^{47} + 43 q^{49} - 5 q^{51} + 16 q^{53} + 17 q^{57} + 23 q^{59} + 25 q^{61} - 7 q^{63} - 9 q^{67} + 9 q^{69} - 7 q^{71} - 2 q^{73} - 63 q^{77} - 4 q^{79} - 4 q^{81} - 32 q^{83} - 4 q^{87} - 16 q^{89} - 17 q^{91} - 20 q^{93} - 24 q^{97} - 9 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 3 * q^7 - 6 * q^9 + q^11 + 4 * q^13 - 5 * q^17 - 14 * q^19 + q^21 + 7 * q^23 + 4 * q^27 - 12 * q^29 + 7 * q^33 + 4 * q^37 - 7 * q^39 - 7 * q^41 + q^43 - 9 * q^47 + 43 * q^49 - 5 * q^51 + 16 * q^53 + 17 * q^57 + 23 * q^59 + 25 * q^61 - 7 * q^63 - 9 * q^67 + 9 * q^69 - 7 * q^71 - 2 * q^73 - 63 * q^77 - 4 * q^79 - 4 * q^81 - 32 * q^83 - 4 * q^87 - 16 * q^89 - 17 * q^91 - 20 * q^93 - 24 * q^97 - 9 * 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
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.33766	−1.63948	−0.819741	−	0.572735i	$-0.805883\pi$
$7$	−4.33766	−1.63948	−0.819741	+	0.572735i	$0.805883\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	3.10159	0.935165	0.467583	−	0.883949i	$-0.345125\pi$
$11$	3.10159	0.935165	0.467583	+	0.883949i	$0.345125\pi$
$12$	0	0
$13$	−0.680822	−0.188826	−0.0944130	−	0.995533i	$-0.530097\pi$
$13$	−0.680822	−0.188826	−0.0944130	+	0.995533i	$0.530097\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.95569	1.20193	0.600966	−	0.799274i	$-0.294783\pi$
$17$	4.95569	1.20193	0.600966	+	0.799274i	$0.294783\pi$
$18$	0	0
$19$	−2.93721	−0.673843	−0.336921	−	0.941533i	$-0.609386\pi$
$19$	−2.93721	−0.673843	−0.336921	+	0.941533i	$0.609386\pi$
$20$	0	0
$21$	7.01848	1.53156
$22$	0	0
$23$	−5.57373	−1.16220	−0.581101	−	0.813831i	$-0.697378\pi$
$23$	−5.57373	−1.16220	−0.581101	+	0.813831i	$0.697378\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.47214	1.05311
$28$	0	0
$29$	−0.763932	−0.141859	−0.0709293	−	0.997481i	$-0.522596\pi$
$29$	−0.763932	−0.141859	−0.0709293	+	0.997481i	$0.522596\pi$
$30$	0	0
$31$	4.47214	0.803219	0.401610	−	0.915811i	$-0.368451\pi$
$31$	4.47214	0.803219	0.401610	+	0.915811i	$0.368451\pi$
$32$	0	0
$33$	−5.01848	−0.873606
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.15296	1.17594	0.587969	−	0.808883i	$-0.299928\pi$
$37$	7.15296	1.17594	0.587969	+	0.808883i	$0.299928\pi$
$38$	0	0
$39$	1.10159	0.176396
$40$	0	0
$41$	−9.63652	−1.50497	−0.752485	−	0.658609i	$-0.771145\pi$
$41$	−9.63652	−1.50497	−0.752485	+	0.658609i	$0.771145\pi$
$42$	0	0
$43$	−10.4278	−1.59023	−0.795115	−	0.606459i	$-0.792589\pi$
$43$	−10.4278	−1.59023	−0.795115	+	0.606459i	$0.792589\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.01848	−1.31548	−0.657740	−	0.753245i	$-0.728488\pi$
$47$	−9.01848	−1.31548	−0.657740	+	0.753245i	$0.728488\pi$
$48$	0	0
$49$	11.8153	1.68790
$50$	0	0
$51$	−8.01848	−1.12281
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.75251	0.629485
$58$	0	0
$59$	9.71963	1.26539	0.632694	−	0.774402i	$-0.281949\pi$
$59$	9.71963	1.26539	0.632694	+	0.774402i	$0.281949\pi$
$60$	0	0
$61$	9.65684	1.23643	0.618216	−	0.786008i	$-0.287856\pi$
$61$	9.65684	1.23643	0.618216	+	0.786008i	$0.287856\pi$
$62$	0	0
$63$	1.65684	0.208742
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.40045	0.415431	0.207715	−	0.978189i	$-0.433397\pi$
$67$	3.40045	0.415431	0.207715	+	0.978189i	$0.433397\pi$
$68$	0	0
$69$	9.01848	1.08570
$70$	0	0
$71$	9.49062	1.12633	0.563165	−	0.826345i	$-0.309584\pi$
$71$	9.49062	1.12633	0.563165	+	0.826345i	$0.309584\pi$
$72$	0	0
$73$	9.00706	1.05420	0.527098	−	0.849804i	$-0.323280\pi$
$73$	9.00706	1.05420	0.527098	+	0.849804i	$0.323280\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.4537	−1.53319
$78$	0	0
$79$	0.680822	0.0765984	0.0382992	−	0.999266i	$-0.487806\pi$
$79$	0.680822	0.0765984	0.0382992	+	0.999266i	$0.487806\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.23607	0.132520
$88$	0	0
$89$	8.97418	0.951261	0.475630	−	0.879645i	$-0.342220\pi$
$89$	8.97418	0.951261	0.475630	+	0.879645i	$0.342220\pi$
$90$	0	0
$91$	2.95317	0.309577
$92$	0	0
$93$	−7.23607	−0.750345
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.7381	−1.69950	−0.849749	−	0.527188i	$-0.823246\pi$
$97$	−16.7381	−1.69950	−0.849749	+	0.527188i	$0.823246\pi$
$98$	0	0
$99$	−1.18470	−0.119067
$100$	0	0
$101$	6.55525	0.652271	0.326136	−	0.945323i	$-0.394253\pi$
$101$	6.55525	0.652271	0.326136	+	0.945323i	$0.394253\pi$
$102$	0	0
$103$	−8.42077	−0.829723	−0.414862	−	0.909885i	$-0.636170\pi$
$103$	−8.42077	−0.829723	−0.414862	+	0.909885i	$0.636170\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.414849	−0.0401050	−0.0200525	−	0.999799i	$-0.506383\pi$
$107$	−0.414849	−0.0401050	−0.0200525	+	0.999799i	$0.506383\pi$
$108$	0	0
$109$	17.4851	1.67477	0.837385	−	0.546613i	$-0.184083\pi$
$109$	17.4851	1.67477	0.837385	+	0.546613i	$0.184083\pi$
$110$	0	0
$111$	−11.5737	−1.09853
$112$	0	0
$113$	14.0255	1.31941	0.659706	−	0.751524i	$-0.270681\pi$
$113$	14.0255	1.31941	0.659706	+	0.751524i	$0.270681\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0.260051	0.0240417
$118$	0	0
$119$	−21.4961	−1.97055
$120$	0	0
$121$	−1.38012	−0.125466
$122$	0	0
$123$	15.5922	1.40590
$124$	0	0
$125$	0	0
$126$	0	0
$127$	18.7267	1.66172	0.830862	−	0.556478i	$-0.187848\pi$
$127$	18.7267	1.66172	0.830862	+	0.556478i	$0.187848\pi$
$128$	0	0
$129$	16.8726	1.48555
$130$	0	0
$131$	13.6639	1.19382	0.596910	−	0.802308i	$-0.296395\pi$
$131$	13.6639	1.19382	0.596910	+	0.802308i	$0.296395\pi$
$132$	0	0
$133$	12.7406	1.10475
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.27853	−0.450975	−0.225488	−	0.974246i	$-0.572398\pi$
$137$	−5.27853	−0.450975	−0.225488	+	0.974246i	$0.572398\pi$
$138$	0	0
$139$	−16.1918	−1.37337	−0.686684	−	0.726956i	$-0.740934\pi$
$139$	−16.1918	−1.37337	−0.686684	+	0.726956i	$0.740934\pi$
$140$	0	0
$141$	14.5922	1.22889
$142$	0	0
$143$	−2.11163	−0.176583
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−19.1176	−1.57679
$148$	0	0
$149$	20.3322	1.66568	0.832838	−	0.553517i	$-0.186715\pi$
$149$	20.3322	1.66568	0.832838	+	0.553517i	$0.186715\pi$
$150$	0	0
$151$	7.35614	0.598634	0.299317	−	0.954154i	$-0.403241\pi$
$151$	7.35614	0.598634	0.299317	+	0.954154i	$0.403241\pi$
$152$	0	0
$153$	−1.89291	−0.153032
$154$	0	0
$155$	0	0
$156$	0	0
$157$	21.7027	1.73206	0.866032	−	0.499988i	$-0.166662\pi$
$157$	21.7027	1.73206	0.866032	+	0.499988i	$0.166662\pi$
$158$	0	0
$159$	0.763932	0.0605838
$160$	0	0
$161$	24.1769	1.90541
$162$	0	0
$163$	−10.9043	−0.854093	−0.427046	−	0.904230i	$-0.640446\pi$
$163$	−10.9043	−0.854093	−0.427046	+	0.904230i	$0.640446\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.02738	0.234266	0.117133	−	0.993116i	$-0.462630\pi$
$167$	3.02738	0.234266	0.117133	+	0.993116i	$0.462630\pi$
$168$	0	0
$169$	−12.5365	−0.964345
$170$	0	0
$171$	1.12192	0.0857950
$172$	0	0
$173$	−15.0927	−1.14748	−0.573738	−	0.819039i	$-0.694507\pi$
$173$	−15.0927	−1.14748	−0.573738	+	0.819039i	$0.694507\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−15.7267	−1.18209
$178$	0	0
$179$	−3.35022	−0.250407	−0.125204	−	0.992131i	$-0.539958\pi$
$179$	−3.35022	−0.250407	−0.125204	+	0.992131i	$0.539958\pi$
$180$	0	0
$181$	−2.81530	−0.209259	−0.104630	−	0.994511i	$-0.533366\pi$
$181$	−2.81530	−0.209259	−0.104630	+	0.994511i	$0.533366\pi$
$182$	0	0
$183$	−15.6251	−1.15504
$184$	0	0
$185$	0	0
$186$	0	0
$187$	15.3705	1.12401
$188$	0	0
$189$	−23.7363	−1.72656
$190$	0	0
$191$	16.2228	1.17384	0.586920	−	0.809645i	$-0.300340\pi$
$191$	16.2228	1.17384	0.586920	+	0.809645i	$0.300340\pi$
$192$	0	0
$193$	−18.4278	−1.32646	−0.663232	−	0.748414i	$-0.730816\pi$
$193$	−18.4278	−1.32646	−0.663232	+	0.748414i	$0.730816\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.3835	1.16728	0.583639	−	0.812013i	$-0.301628\pi$
$197$	16.3835	1.16728	0.583639	+	0.812013i	$0.301628\pi$
$198$	0	0
$199$	15.4020	1.09182	0.545910	−	0.837844i	$-0.316184\pi$
$199$	15.4020	1.09182	0.545910	+	0.837844i	$0.316184\pi$
$200$	0	0
$201$	−5.50204	−0.388084
$202$	0	0
$203$	3.31368	0.232575
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.12897	0.147974
$208$	0	0
$209$	−9.11004	−0.630154
$210$	0	0
$211$	−25.0200	−1.72245	−0.861225	−	0.508223i	$-0.830302\pi$
$211$	−25.0200	−1.72245	−0.861225	+	0.508223i	$0.830302\pi$
$212$	0	0
$213$	−15.3561	−1.05219
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−19.3986	−1.31686
$218$	0	0
$219$	−14.5737	−0.984801
$220$	0	0
$221$	−3.37394	−0.226956
$222$	0	0
$223$	−9.37054	−0.627498	−0.313749	−	0.949506i	$-0.601585\pi$
$223$	−9.37054	−0.627498	−0.313749	+	0.949506i	$0.601585\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.719626	0.0477633	0.0238816	−	0.999715i	$-0.492398\pi$
$227$	0.719626	0.0477633	0.0238816	+	0.999715i	$0.492398\pi$
$228$	0	0
$229$	−6.41187	−0.423708	−0.211854	−	0.977301i	$-0.567950\pi$
$229$	−6.41187	−0.423708	−0.211854	+	0.977301i	$0.567950\pi$
$230$	0	0
$231$	21.7685	1.43226
$232$	0	0
$233$	−29.5107	−1.93331	−0.966654	−	0.256087i	$-0.917567\pi$
$233$	−29.5107	−1.93331	−0.966654	+	0.256087i	$0.917567\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1.10159	−0.0715561
$238$	0	0
$239$	−21.8885	−1.41585	−0.707926	−	0.706287i	$-0.750369\pi$
$239$	−21.8885	−1.41585	−0.707926	+	0.706287i	$0.750369\pi$
$240$	0	0
$241$	−25.5294	−1.64449	−0.822247	−	0.569130i	$-0.807280\pi$
$241$	−25.5294	−1.64449	−0.822247	+	0.569130i	$0.807280\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.99972	0.127239
$248$	0	0
$249$	12.9443	0.820310
$250$	0	0
$251$	1.10865	0.0699775	0.0349887	−	0.999388i	$-0.488860\pi$
$251$	1.10865	0.0699775	0.0349887	+	0.999388i	$0.488860\pi$
$252$	0	0
$253$	−17.2874	−1.08685
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.58629	0.597976	0.298988	−	0.954257i	$-0.403351\pi$
$257$	9.58629	0.597976	0.298988	+	0.954257i	$0.403351\pi$
$258$	0	0
$259$	−31.0271	−1.92793
$260$	0	0
$261$	0.291796	0.0180617
$262$	0	0
$263$	15.4678	0.953784	0.476892	−	0.878962i	$-0.341763\pi$
$263$	15.4678	0.953784	0.476892	+	0.878962i	$0.341763\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−14.5205	−0.888642
$268$	0	0
$269$	−25.5420	−1.55732	−0.778661	−	0.627445i	$-0.784101\pi$
$269$	−25.5420	−1.55732	−0.778661	+	0.627445i	$0.784101\pi$
$270$	0	0
$271$	−18.7954	−1.14174	−0.570869	−	0.821041i	$-0.693394\pi$
$271$	−18.7954	−1.14174	−0.570869	+	0.821041i	$0.693394\pi$
$272$	0	0
$273$	−4.77833	−0.289198
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−24.2228	−1.45541	−0.727704	−	0.685892i	$-0.759412\pi$
$277$	−24.2228	−1.45541	−0.727704	+	0.685892i	$0.759412\pi$
$278$	0	0
$279$	−1.70820	−0.102267
$280$	0	0
$281$	−3.59405	−0.214403	−0.107202	−	0.994237i	$-0.534189\pi$
$281$	−3.59405	−0.214403	−0.107202	+	0.994237i	$0.534189\pi$
$282$	0	0
$283$	5.13603	0.305306	0.152653	−	0.988280i	$-0.451218\pi$
$283$	5.13603	0.305306	0.152653	+	0.988280i	$0.451218\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	41.7999	2.46737
$288$	0	0
$289$	7.55891	0.444642
$290$	0	0
$291$	27.0828	1.58762
$292$	0	0
$293$	9.30478	0.543591	0.271795	−	0.962355i	$-0.412383\pi$
$293$	9.30478	0.543591	0.271795	+	0.962355i	$0.412383\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.9723	0.984835
$298$	0	0
$299$	3.79471	0.219454
$300$	0	0
$301$	45.2324	2.60715
$302$	0	0
$303$	−10.6066	−0.609334
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−18.7326	−1.06913	−0.534563	−	0.845129i	$-0.679524\pi$
$307$	−18.7326	−1.06913	−0.534563	+	0.845129i	$0.679524\pi$
$308$	0	0
$309$	13.6251	0.775104
$310$	0	0
$311$	13.3472	0.756853	0.378426	−	0.925631i	$-0.376465\pi$
$311$	13.3472	0.756853	0.378426	+	0.925631i	$0.376465\pi$
$312$	0	0
$313$	6.68238	0.377710	0.188855	−	0.982005i	$-0.439522\pi$
$313$	6.68238	0.377710	0.188855	+	0.982005i	$0.439522\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.3705	−0.975627	−0.487813	−	0.872948i	$-0.662205\pi$
$317$	−17.3705	−0.975627	−0.487813	+	0.872948i	$0.662205\pi$
$318$	0	0
$319$	−2.36941	−0.132661
$320$	0	0
$321$	0.671240	0.0374650
$322$	0	0
$323$	−14.5559	−0.809913
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−28.2915	−1.56452
$328$	0	0
$329$	39.1191	2.15671
$330$	0	0
$331$	−17.4278	−0.957920	−0.478960	−	0.877837i	$-0.658986\pi$
$331$	−17.4278	−0.957920	−0.478960	+	0.877837i	$0.658986\pi$
$332$	0	0
$333$	−2.73219	−0.149723
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.4146	0.785211	0.392606	−	0.919707i	$-0.371574\pi$
$337$	14.4146	0.785211	0.392606	+	0.919707i	$0.371574\pi$
$338$	0	0
$339$	−22.6938	−1.23256
$340$	0	0
$341$	13.8707	0.751143
$342$	0	0
$343$	−20.8871	−1.12780
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−25.9753	−1.39443	−0.697214	−	0.716863i	$-0.745577\pi$
$347$	−25.9753	−1.39443	−0.697214	+	0.716863i	$0.745577\pi$
$348$	0	0
$349$	−20.6292	−1.10425	−0.552127	−	0.833760i	$-0.686184\pi$
$349$	−20.6292	−1.10425	−0.552127	+	0.833760i	$0.686184\pi$
$350$	0	0
$351$	−3.72555	−0.198855
$352$	0	0
$353$	−2.46621	−0.131263	−0.0656317	−	0.997844i	$-0.520906\pi$
$353$	−2.46621	−0.131263	−0.0656317	+	0.997844i	$0.520906\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	34.7815	1.84083
$358$	0	0
$359$	−16.3411	−0.862448	−0.431224	−	0.902245i	$-0.641918\pi$
$359$	−16.3411	−0.862448	−0.431224	+	0.902245i	$0.641918\pi$
$360$	0	0
$361$	−10.3728	−0.545936
$362$	0	0
$363$	2.23309	0.117207
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.56075	0.238069	0.119035	−	0.992890i	$-0.462020\pi$
$367$	4.56075	0.238069	0.119035	+	0.992890i	$0.462020\pi$
$368$	0	0
$369$	3.68082	0.191616
$370$	0	0
$371$	2.04797	0.106325
$372$	0	0
$373$	17.4020	0.901042	0.450521	−	0.892766i	$-0.351238\pi$
$373$	17.4020	0.901042	0.450521	+	0.892766i	$0.351238\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.520101	0.0267866
$378$	0	0
$379$	4.14704	0.213019	0.106509	−	0.994312i	$-0.466033\pi$
$379$	4.14704	0.213019	0.106509	+	0.994312i	$0.466033\pi$
$380$	0	0
$381$	−30.3004	−1.55234
$382$	0	0
$383$	29.8020	1.52281	0.761406	−	0.648275i	$-0.224509\pi$
$383$	29.8020	1.52281	0.761406	+	0.648275i	$0.224509\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.98308	0.202471
$388$	0	0
$389$	−26.3924	−1.33815	−0.669075	−	0.743195i	$-0.733309\pi$
$389$	−26.3924	−1.33815	−0.669075	+	0.743195i	$0.733309\pi$
$390$	0	0
$391$	−27.6217	−1.39689
$392$	0	0
$393$	−22.1087	−1.11523
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.4721	0.525581	0.262791	−	0.964853i	$-0.415357\pi$
$397$	10.4721	0.525581	0.262791	+	0.964853i	$0.415357\pi$
$398$	0	0
$399$	−20.6148	−1.03203
$400$	0	0
$401$	−2.36462	−0.118084	−0.0590418	−	0.998256i	$-0.518805\pi$
$401$	−2.36462	−0.118084	−0.0590418	+	0.998256i	$0.518805\pi$
$402$	0	0
$403$	−3.04473	−0.151669
$404$	0	0
$405$	0	0
$406$	0	0
$407$	22.1856	1.09970
$408$	0	0
$409$	34.5091	1.70637	0.853183	−	0.521612i	$-0.174669\pi$
$409$	34.5091	1.70637	0.853183	+	0.521612i	$0.174669\pi$
$410$	0	0
$411$	8.54085	0.421289
$412$	0	0
$413$	−42.1604	−2.07458
$414$	0	0
$415$	0	0
$416$	0	0
$417$	26.1988	1.28296
$418$	0	0
$419$	−23.7337	−1.15947	−0.579735	−	0.814805i	$-0.696844\pi$
$419$	−23.7337	−1.15947	−0.579735	+	0.814805i	$0.696844\pi$
$420$	0	0
$421$	−10.4438	−0.508999	−0.254500	−	0.967073i	$-0.581911\pi$
$421$	−10.4438	−0.508999	−0.254500	+	0.967073i	$0.581911\pi$
$422$	0	0
$423$	3.44475	0.167490
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−41.8881	−2.02711
$428$	0	0
$429$	3.41669	0.164959
$430$	0	0
$431$	25.0182	1.20508	0.602542	−	0.798087i	$-0.294155\pi$
$431$	25.0182	1.20508	0.602542	+	0.798087i	$0.294155\pi$
$432$	0	0
$433$	−29.1121	−1.39904	−0.699518	−	0.714615i	$-0.746602\pi$
$433$	−29.1121	−1.39904	−0.699518	+	0.714615i	$0.746602\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.3712	0.783142
$438$	0	0
$439$	−23.9573	−1.14342	−0.571709	−	0.820457i	$-0.693719\pi$
$439$	−23.9573	−1.14342	−0.571709	+	0.820457i	$0.693719\pi$
$440$	0	0
$441$	−4.51304	−0.214907
$442$	0	0
$443$	12.7292	0.604783	0.302391	−	0.953184i	$-0.402215\pi$
$443$	12.7292	0.604783	0.302391	+	0.953184i	$0.402215\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−32.8981	−1.55603
$448$	0	0
$449$	−13.4552	−0.634991	−0.317495	−	0.948260i	$-0.602842\pi$
$449$	−13.4552	−0.634991	−0.317495	+	0.948260i	$0.602842\pi$
$450$	0	0
$451$	−29.8885	−1.40740
$452$	0	0
$453$	−11.9025	−0.559228
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.10865	0.238973	0.119486	−	0.992836i	$-0.461875\pi$
$457$	5.10865	0.238973	0.119486	+	0.992836i	$0.461875\pi$
$458$	0	0
$459$	27.1182	1.26577
$460$	0	0
$461$	5.88837	0.274249	0.137124	−	0.990554i	$-0.456214\pi$
$461$	5.88837	0.274249	0.137124	+	0.990554i	$0.456214\pi$
$462$	0	0
$463$	−18.5180	−0.860604	−0.430302	−	0.902685i	$-0.641593\pi$
$463$	−18.5180	−0.860604	−0.430302	+	0.902685i	$0.641593\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.08155	−0.281421	−0.140710	−	0.990051i	$-0.544939\pi$
$467$	−6.08155	−0.281421	−0.140710	+	0.990051i	$0.544939\pi$
$468$	0	0
$469$	−14.7500	−0.681091
$470$	0	0
$471$	−35.1157	−1.61805
$472$	0	0
$473$	−32.3429	−1.48713
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.180340	0.00825720
$478$	0	0
$479$	0.907308	0.0414560	0.0207280	−	0.999785i	$-0.493402\pi$
$479$	0.907308	0.0414560	0.0207280	+	0.999785i	$0.493402\pi$
$480$	0	0
$481$	−4.86989	−0.222048
$482$	0	0
$483$	−39.1191	−1.77998
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.9269	0.585775	0.292888	−	0.956147i	$-0.405384\pi$
$487$	12.9269	0.585775	0.292888	+	0.956147i	$0.405384\pi$
$488$	0	0
$489$	17.6436	0.797870
$490$	0	0
$491$	37.6004	1.69688	0.848441	−	0.529289i	$-0.177541\pi$
$491$	37.6004	1.69688	0.848441	+	0.529289i	$0.177541\pi$
$492$	0	0
$493$	−3.78581	−0.170504
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−41.1671	−1.84660
$498$	0	0
$499$	−8.67716	−0.388443	−0.194222	−	0.980958i	$-0.562218\pi$
$499$	−8.67716	−0.388443	−0.194222	+	0.980958i	$0.562218\pi$
$500$	0	0
$501$	−4.89841	−0.218845
$502$	0	0
$503$	3.05023	0.136003	0.0680015	−	0.997685i	$-0.478338\pi$
$503$	3.05023	0.136003	0.0680015	+	0.997685i	$0.478338\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	20.2845	0.900864
$508$	0	0
$509$	−10.4034	−0.461124	−0.230562	−	0.973058i	$-0.574056\pi$
$509$	−10.4034	−0.461124	−0.230562	+	0.973058i	$0.574056\pi$
$510$	0	0
$511$	−39.0696	−1.72834
$512$	0	0
$513$	−16.0728	−0.709633
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−27.9717	−1.23019
$518$	0	0
$519$	24.4205	1.07194
$520$	0	0
$521$	14.4703	0.633955	0.316978	−	0.948433i	$-0.397332\pi$
$521$	14.4703	0.633955	0.316978	+	0.948433i	$0.397332\pi$
$522$	0	0
$523$	−10.3694	−0.453422	−0.226711	−	0.973962i	$-0.572797\pi$
$523$	−10.3694	−0.453422	−0.226711	+	0.973962i	$0.572797\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.1625	0.965415
$528$	0	0
$529$	8.06645	0.350715
$530$	0	0
$531$	−3.71257	−0.161112
$532$	0	0
$533$	6.56075	0.284178
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.42077	0.233924
$538$	0	0
$539$	36.6462	1.57847
$540$	0	0
$541$	3.07353	0.132141	0.0660707	−	0.997815i	$-0.478954\pi$
$541$	3.07353	0.132141	0.0660707	+	0.997815i	$0.478954\pi$
$542$	0	0
$543$	4.55525	0.195484
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−18.3411	−0.784207	−0.392104	−	0.919921i	$-0.628253\pi$
$547$	−18.3411	−0.784207	−0.392104	+	0.919921i	$0.628253\pi$
$548$	0	0
$549$	−3.68858	−0.157425
$550$	0	0
$551$	2.24383	0.0955904
$552$	0	0
$553$	−2.95317	−0.125582
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−18.8043	−0.796764	−0.398382	−	0.917220i	$-0.630428\pi$
$557$	−18.8043	−0.796764	−0.398382	+	0.917220i	$0.630428\pi$
$558$	0	0
$559$	7.09949	0.300276
$560$	0	0
$561$	−24.8701	−1.05001
$562$	0	0
$563$	−28.6091	−1.20573	−0.602866	−	0.797843i	$-0.705974\pi$
$563$	−28.6091	−1.20573	−0.602866	+	0.797843i	$0.705974\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	33.4356	1.40416
$568$	0	0
$569$	−0.136997	−0.00574323	−0.00287162	−	0.999996i	$-0.500914\pi$
$569$	−0.136997	−0.00574323	−0.00287162	+	0.999996i	$0.500914\pi$
$570$	0	0
$571$	39.3392	1.64630	0.823148	−	0.567828i	$-0.192216\pi$
$571$	39.3392	1.64630	0.823148	+	0.567828i	$0.192216\pi$
$572$	0	0
$573$	−26.2490	−1.09657
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.97165	0.0820810	0.0410405	−	0.999157i	$-0.486933\pi$
$577$	1.97165	0.0820810	0.0410405	+	0.999157i	$0.486933\pi$
$578$	0	0
$579$	29.8169	1.23915
$580$	0	0
$581$	34.7013	1.43965
$582$	0	0
$583$	−1.46437	−0.0606481
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.01848	0.0420372	0.0210186	−	0.999779i	$-0.493309\pi$
$587$	1.01848	0.0420372	0.0210186	+	0.999779i	$0.493309\pi$
$588$	0	0
$589$	−13.1356	−0.541244
$590$	0	0
$591$	−26.5091	−1.09044
$592$	0	0
$593$	17.1973	0.706207	0.353103	−	0.935584i	$-0.385126\pi$
$593$	17.1973	0.706207	0.353103	+	0.935584i	$0.385126\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−24.9210	−1.01995
$598$	0	0
$599$	16.9493	0.692530	0.346265	−	0.938137i	$-0.387450\pi$
$599$	16.9493	0.692530	0.346265	+	0.938137i	$0.387450\pi$
$600$	0	0
$601$	−42.3214	−1.72633	−0.863163	−	0.504925i	$-0.831520\pi$
$601$	−42.3214	−1.72633	−0.863163	+	0.504925i	$0.831520\pi$
$602$	0	0
$603$	−1.29886	−0.0528935
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4.38721	0.178071	0.0890356	−	0.996028i	$-0.471622\pi$
$607$	4.38721	0.178071	0.0890356	+	0.996028i	$0.471622\pi$
$608$	0	0
$609$	−5.36164	−0.217265
$610$	0	0
$611$	6.13998	0.248397
$612$	0	0
$613$	21.6196	0.873207	0.436604	−	0.899654i	$-0.356181\pi$
$613$	21.6196	0.873207	0.436604	+	0.899654i	$0.356181\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18.3075	0.737031	0.368516	−	0.929622i	$-0.379866\pi$
$617$	18.3075	0.737031	0.368516	+	0.929622i	$0.379866\pi$
$618$	0	0
$619$	19.5557	0.786009	0.393004	−	0.919537i	$-0.371436\pi$
$619$	19.5557	0.786009	0.393004	+	0.919537i	$0.371436\pi$
$620$	0	0
$621$	−30.5002	−1.22393
$622$	0	0
$623$	−38.9269	−1.55957
$624$	0	0
$625$	0	0
$626$	0	0
$627$	14.7403	0.588673
$628$	0	0
$629$	35.4479	1.41340
$630$	0	0
$631$	−30.6063	−1.21842	−0.609209	−	0.793009i	$-0.708513\pi$
$631$	−30.6063	−1.21842	−0.609209	+	0.793009i	$0.708513\pi$
$632$	0	0
$633$	40.4833	1.60907
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−8.04411	−0.318719
$638$	0	0
$639$	−3.62509	−0.143406
$640$	0	0
$641$	−4.87627	−0.192601	−0.0963005	−	0.995352i	$-0.530701\pi$
$641$	−4.87627	−0.192601	−0.0963005	+	0.995352i	$0.530701\pi$
$642$	0	0
$643$	8.97942	0.354114	0.177057	−	0.984201i	$-0.443342\pi$
$643$	8.97942	0.354114	0.177057	+	0.984201i	$0.443342\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	30.8981	1.21473	0.607365	−	0.794423i	$-0.292226\pi$
$647$	30.8981	1.21473	0.607365	+	0.794423i	$0.292226\pi$
$648$	0	0
$649$	30.1463	1.18335
$650$	0	0
$651$	31.3876	1.23018
$652$	0	0
$653$	1.20982	0.0473440	0.0236720	−	0.999720i	$-0.492464\pi$
$653$	1.20982	0.0473440	0.0236720	+	0.999720i	$0.492464\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−3.44039	−0.134222
$658$	0	0
$659$	−23.6975	−0.923122	−0.461561	−	0.887108i	$-0.652710\pi$
$659$	−23.6975	−0.923122	−0.461561	+	0.887108i	$0.652710\pi$
$660$	0	0
$661$	6.90363	0.268520	0.134260	−	0.990946i	$-0.457134\pi$
$661$	6.90363	0.268520	0.134260	+	0.990946i	$0.457134\pi$
$662$	0	0
$663$	5.45915	0.212016
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.25795	0.164868
$668$	0	0
$669$	15.1619	0.586191
$670$	0	0
$671$	29.9516	1.15627
$672$	0	0
$673$	5.11965	0.197348	0.0986741	−	0.995120i	$-0.468540\pi$
$673$	5.11965	0.197348	0.0986741	+	0.995120i	$0.468540\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.9495	−1.11262	−0.556310	−	0.830975i	$-0.687783\pi$
$677$	−28.9495	−1.11262	−0.556310	+	0.830975i	$0.687783\pi$
$678$	0	0
$679$	72.6042	2.78629
$680$	0	0
$681$	−1.16438	−0.0446191
$682$	0	0
$683$	−8.78057	−0.335979	−0.167990	−	0.985789i	$-0.553728\pi$
$683$	−8.78057	−0.335979	−0.167990	+	0.985789i	$0.553728\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	10.3746	0.395817
$688$	0	0
$689$	0.321440	0.0122459
$690$	0	0
$691$	−31.5365	−1.19970	−0.599852	−	0.800111i	$-0.704774\pi$
$691$	−31.5365	−1.19970	−0.599852	+	0.800111i	$0.704774\pi$
$692$	0	0
$693$	5.13884	0.195208
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−47.7556	−1.80887
$698$	0	0
$699$	47.7492	1.80604
$700$	0	0
$701$	−37.7219	−1.42474	−0.712368	−	0.701807i	$-0.752377\pi$
$701$	−37.7219	−1.42474	−0.712368	+	0.701807i	$0.752377\pi$
$702$	0	0
$703$	−21.0098	−0.792398
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−28.4344	−1.06939
$708$	0	0
$709$	−39.6282	−1.48827	−0.744134	−	0.668030i	$-0.767138\pi$
$709$	−39.6282	−1.48827	−0.744134	+	0.668030i	$0.767138\pi$
$710$	0	0
$711$	−0.260051	−0.00975266
$712$	0	0
$713$	−24.9265	−0.933504
$714$	0	0
$715$	0	0
$716$	0	0
$717$	35.4164	1.32265
$718$	0	0
$719$	−27.5025	−1.02567	−0.512834	−	0.858488i	$-0.671405\pi$
$719$	−27.5025	−1.02567	−0.512834	+	0.858488i	$0.671405\pi$
$720$	0	0
$721$	36.5264	1.36032
$722$	0	0
$723$	41.3075	1.53624
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21.9431	0.813826	0.406913	−	0.913467i	$-0.366605\pi$
$727$	21.9431	0.813826	0.406913	+	0.913467i	$0.366605\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	−51.6771	−1.91135
$732$	0	0
$733$	−28.1201	−1.03864	−0.519319	−	0.854580i	$-0.673814\pi$
$733$	−28.1201	−1.03864	−0.519319	+	0.854580i	$0.673814\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.5468	0.388496
$738$	0	0
$739$	−20.1466	−0.741104	−0.370552	−	0.928812i	$-0.620831\pi$
$739$	−20.1466	−0.741104	−0.370552	+	0.928812i	$0.620831\pi$
$740$	0	0
$741$	−3.23561	−0.118863
$742$	0	0
$743$	−9.20250	−0.337607	−0.168804	−	0.985650i	$-0.553990\pi$
$743$	−9.20250	−0.337607	−0.168804	+	0.985650i	$0.553990\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	3.05573	0.111803
$748$	0	0
$749$	1.79948	0.0657514
$750$	0	0
$751$	11.5139	0.420149	0.210074	−	0.977685i	$-0.432629\pi$
$751$	11.5139	0.420149	0.210074	+	0.977685i	$0.432629\pi$
$752$	0	0
$753$	−1.79384	−0.0653710
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−16.1429	−0.586724	−0.293362	−	0.956001i	$-0.594774\pi$
$757$	−16.1429	−0.586724	−0.293362	+	0.956001i	$0.594774\pi$
$758$	0	0
$759$	27.9717	1.01531
$760$	0	0
$761$	46.5750	1.68834	0.844171	−	0.536074i	$-0.180093\pi$
$761$	46.5750	1.68834	0.844171	+	0.536074i	$0.180093\pi$
$762$	0	0
$763$	−75.8445	−2.74576
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.61733	−0.238938
$768$	0	0
$769$	3.15662	0.113831	0.0569153	−	0.998379i	$-0.481874\pi$
$769$	3.15662	0.113831	0.0569153	+	0.998379i	$0.481874\pi$
$770$	0	0
$771$	−15.5109	−0.558613
$772$	0	0
$773$	−43.2853	−1.55687	−0.778433	−	0.627728i	$-0.783985\pi$
$773$	−43.2853	−1.55687	−0.778433	+	0.627728i	$0.783985\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	50.2029	1.80102
$778$	0	0
$779$	28.3045	1.01411
$780$	0	0
$781$	29.4360	1.05330
$782$	0	0
$783$	−4.18034	−0.149393
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7.02738	−0.250499	−0.125250	−	0.992125i	$-0.539973\pi$
$787$	−7.02738	−0.250499	−0.125250	+	0.992125i	$0.539973\pi$
$788$	0	0
$789$	−25.0274	−0.890998
$790$	0	0
$791$	−60.8380	−2.16315
$792$	0	0
$793$	−6.57458	−0.233470
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19.8176	−0.701974	−0.350987	−	0.936380i	$-0.614154\pi$
$797$	−19.8176	−0.701974	−0.350987	+	0.936380i	$0.614154\pi$
$798$	0	0
$799$	−44.6928	−1.58112
$800$	0	0
$801$	−3.42783	−0.121116
$802$	0	0
$803$	27.9362	0.985848
$804$	0	0
$805$	0	0
$806$	0	0
$807$	41.3278	1.45481
$808$	0	0
$809$	−29.3787	−1.03290	−0.516451	−	0.856317i	$-0.672747\pi$
$809$	−29.3787	−1.03290	−0.516451	+	0.856317i	$0.672747\pi$
$810$	0	0
$811$	47.4470	1.66609	0.833045	−	0.553206i	$-0.186596\pi$
$811$	47.4470	1.66609	0.833045	+	0.553206i	$0.186596\pi$
$812$	0	0
$813$	30.4116	1.06658
$814$	0	0
$815$	0	0
$816$	0	0
$817$	30.6288	1.07156
$818$	0	0
$819$	−1.12801	−0.0394159
$820$	0	0
$821$	−27.2612	−0.951422	−0.475711	−	0.879602i	$-0.657809\pi$
$821$	−27.2612	−0.951422	−0.475711	+	0.879602i	$0.657809\pi$
$822$	0	0
$823$	−37.6799	−1.31344	−0.656719	−	0.754136i	$-0.728056\pi$
$823$	−37.6799	−1.31344	−0.656719	+	0.754136i	$0.728056\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.4333	−0.884404	−0.442202	−	0.896916i	$-0.645802\pi$
$827$	−25.4333	−0.884404	−0.442202	+	0.896916i	$0.645802\pi$
$828$	0	0
$829$	−18.6296	−0.647034	−0.323517	−	0.946222i	$-0.604865\pi$
$829$	−18.6296	−0.647034	−0.323517	+	0.946222i	$0.604865\pi$
$830$	0	0
$831$	39.1933	1.35960
$832$	0	0
$833$	58.5530	2.02874
$834$	0	0
$835$	0	0
$836$	0	0
$837$	24.4721	0.845881
$838$	0	0
$839$	27.9541	0.965084	0.482542	−	0.875873i	$-0.339714\pi$
$839$	27.9541	0.965084	0.482542	+	0.875873i	$0.339714\pi$
$840$	0	0
$841$	−28.4164	−0.979876
$842$	0	0
$843$	5.81530	0.200289
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.98651	0.205699
$848$	0	0
$849$	−8.31028	−0.285208
$850$	0	0
$851$	−39.8686	−1.36668
$852$	0	0
$853$	22.8523	0.782447	0.391223	−	0.920296i	$-0.372052\pi$
$853$	22.8523	0.782447	0.391223	+	0.920296i	$0.372052\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.4758	−0.699440	−0.349720	−	0.936854i	$-0.613723\pi$
$857$	−20.4758	−0.699440	−0.349720	+	0.936854i	$0.613723\pi$
$858$	0	0
$859$	36.7858	1.25511	0.627557	−	0.778570i	$-0.284055\pi$
$859$	36.7858	1.25511	0.627557	+	0.778570i	$0.284055\pi$
$860$	0	0
$861$	−67.6337	−2.30495
$862$	0	0
$863$	37.2675	1.26860	0.634301	−	0.773086i	$-0.281288\pi$
$863$	37.2675	1.26860	0.634301	+	0.773086i	$0.281288\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−12.2306	−0.415372
$868$	0	0
$869$	2.11163	0.0716322
$870$	0	0
$871$	−2.31510	−0.0784441
$872$	0	0
$873$	6.39339	0.216383
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.29248	0.111179	0.0555895	−	0.998454i	$-0.482296\pi$
$877$	3.29248	0.111179	0.0555895	+	0.998454i	$0.482296\pi$
$878$	0	0
$879$	−15.0554	−0.507808
$880$	0	0
$881$	−28.1290	−0.947688	−0.473844	−	0.880609i	$-0.657134\pi$
$881$	−28.1290	−0.947688	−0.473844	+	0.880609i	$0.657134\pi$
$882$	0	0
$883$	5.84066	0.196554	0.0982770	−	0.995159i	$-0.468667\pi$
$883$	5.84066	0.196554	0.0982770	+	0.995159i	$0.468667\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−41.4497	−1.39174	−0.695872	−	0.718166i	$-0.744982\pi$
$887$	−41.4497	−1.39174	−0.695872	+	0.718166i	$0.744982\pi$
$888$	0	0
$889$	−81.2300	−2.72437
$890$	0	0
$891$	−23.9077	−0.800938
$892$	0	0
$893$	26.4892	0.886427
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.13998	−0.205008
$898$	0	0
$899$	−3.41641	−0.113944
$900$	0	0
$901$	−2.33976	−0.0779488
$902$	0	0
$903$	−73.1875	−2.43553
$904$	0	0
$905$	0	0
$906$	0	0
$907$	55.6481	1.84776	0.923882	−	0.382677i	$-0.124998\pi$
$907$	55.6481	1.84776	0.923882	+	0.382677i	$0.124998\pi$
$908$	0	0
$909$	−2.50388	−0.0830485
$910$	0	0
$911$	47.2557	1.56565	0.782825	−	0.622241i	$-0.213778\pi$
$911$	47.2557	1.56565	0.782825	+	0.622241i	$0.213778\pi$
$912$	0	0
$913$	−24.8127	−0.821182
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−59.2693	−1.95725
$918$	0	0
$919$	24.3233	0.802350	0.401175	−	0.916001i	$-0.368602\pi$
$919$	24.3233	0.802350	0.401175	+	0.916001i	$0.368602\pi$
$920$	0	0
$921$	30.3100	0.998748
$922$	0	0
$923$	−6.46142	−0.212680
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.21645	0.105642
$928$	0	0
$929$	−46.8908	−1.53844	−0.769218	−	0.638986i	$-0.779354\pi$
$929$	−46.8908	−1.53844	−0.769218	+	0.638986i	$0.779354\pi$
$930$	0	0
$931$	−34.7040	−1.13738
$932$	0	0
$933$	−21.5963	−0.707031
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39.3082	1.28414	0.642071	−	0.766645i	$-0.278076\pi$
$937$	39.3082	1.28414	0.642071	+	0.766645i	$0.278076\pi$
$938$	0	0
$939$	−10.8123	−0.352847
$940$	0	0
$941$	−15.3014	−0.498811	−0.249405	−	0.968399i	$-0.580235\pi$
$941$	−15.3014	−0.498811	−0.249405	+	0.968399i	$0.580235\pi$
$942$	0	0
$943$	53.7113	1.74908
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10.6506	−0.346099	−0.173050	−	0.984913i	$-0.555362\pi$
$947$	−10.6506	−0.346099	−0.173050	+	0.984913i	$0.555362\pi$
$948$	0	0
$949$	−6.13220	−0.199060
$950$	0	0
$951$	28.1061	0.911404
$952$	0	0
$953$	4.07917	0.132137	0.0660686	−	0.997815i	$-0.478954\pi$
$953$	4.07917	0.132137	0.0660686	+	0.997815i	$0.478954\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.83378	0.123928
$958$	0	0
$959$	22.8965	0.739366
$960$	0	0
$961$	−11.0000	−0.354839
$962$	0	0
$963$	0.158458	0.00510625
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−39.8314	−1.28089	−0.640446	−	0.768003i	$-0.721250\pi$
$967$	−39.8314	−1.28089	−0.640446	+	0.768003i	$0.721250\pi$
$968$	0	0
$969$	23.5520	0.756599
$970$	0	0
$971$	−5.84492	−0.187572	−0.0937862	−	0.995592i	$-0.529897\pi$
$971$	−5.84492	−0.187572	−0.0937862	+	0.995592i	$0.529897\pi$
$972$	0	0
$973$	70.2344	2.25161
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.5510	1.61727	0.808635	−	0.588310i	$-0.200207\pi$
$977$	50.5510	1.61727	0.808635	+	0.588310i	$0.200207\pi$
$978$	0	0
$979$	27.8342	0.889586
$980$	0	0
$981$	−6.67872	−0.213235
$982$	0	0
$983$	0.243831	0.00777699	0.00388850	−	0.999992i	$-0.498762\pi$
$983$	0.243831	0.00777699	0.00388850	+	0.999992i	$0.498762\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−63.2961	−2.01474
$988$	0	0
$989$	58.1219	1.84817
$990$	0	0
$991$	2.59403	0.0824020	0.0412010	−	0.999151i	$-0.486882\pi$
$991$	2.59403	0.0824020	0.0412010	+	0.999151i	$0.486882\pi$
$992$	0	0
$993$	28.1988	0.894863
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.62492	0.114802	0.0574012	−	0.998351i	$-0.481719\pi$
$997$	3.62492	0.114802	0.0574012	+	0.998351i	$0.481719\pi$
$998$	0	0
$999$	39.1420	1.23840

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.r.1.1		4
4.3	odd	2		5000.2.a.h.1.4	yes	4
5.4	even	2		10000.2.a.y.1.4		4
20.19	odd	2		5000.2.a.g.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.g.1.1	✓	4	20.19	odd	2
5000.2.a.h.1.4	yes	4	4.3	odd	2
10000.2.a.r.1.1		4	1.1	even	1	trivial
10000.2.a.y.1.4		4	5.4	even	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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{3} -4.33766 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q-1.61803 q^{3} -4.33766 q^{7} -0.381966 q^{9} +3.10159 q^{11} -0.680822 q^{13} +4.95569 q^{17} -2.93721 q^{19} +7.01848 q^{21} -5.57373 q^{23} +5.47214 q^{27} -0.763932 q^{29} +4.47214 q^{31} -5.01848 q^{33} +7.15296 q^{37} +1.10159 q^{39} -9.63652 q^{41} -10.4278 q^{43} -9.01848 q^{47} +11.8153 q^{49} -8.01848 q^{51} -0.472136 q^{53} +4.75251 q^{57} +9.71963 q^{59} +9.65684 q^{61} +1.65684 q^{63} +3.40045 q^{67} +9.01848 q^{69} +9.49062 q^{71} +9.00706 q^{73} -13.4537 q^{77} +0.680822 q^{79} -7.70820 q^{81} -8.00000 q^{83} +1.23607 q^{87} +8.97418 q^{89} +2.95317 q^{91} -7.23607 q^{93} -16.7381 q^{97} -1.18470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 3 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 3 * q^7 - 6 * q^9	\( 4 q - 2 q^{3} + 3 q^{7} - 6 q^{9} + q^{11} + 4 q^{13} - 5 q^{17} - 14 q^{19} + q^{21} + 7 q^{23} + 4 q^{27} - 12 q^{29} + 7 q^{33} + 4 q^{37} - 7 q^{39} - 7 q^{41} + q^{43} - 9 q^{47} + 43 q^{49} - 5 q^{51} + 16 q^{53} + 17 q^{57} + 23 q^{59} + 25 q^{61} - 7 q^{63} - 9 q^{67} + 9 q^{69} - 7 q^{71} - 2 q^{73} - 63 q^{77} - 4 q^{79} - 4 q^{81} - 32 q^{83} - 4 q^{87} - 16 q^{89} - 17 q^{91} - 20 q^{93} - 24 q^{97} - 9 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 3 * q^7 - 6 * q^9 + q^11 + 4 * q^13 - 5 * q^17 - 14 * q^19 + q^21 + 7 * q^23 + 4 * q^27 - 12 * q^29 + 7 * q^33 + 4 * q^37 - 7 * q^39 - 7 * q^41 + q^43 - 9 * q^47 + 43 * q^49 - 5 * q^51 + 16 * q^53 + 17 * q^57 + 23 * q^59 + 25 * q^61 - 7 * q^63 - 9 * q^67 + 9 * q^69 - 7 * q^71 - 2 * q^73 - 63 * q^77 - 4 * q^79 - 4 * q^81 - 32 * q^83 - 4 * q^87 - 16 * q^89 - 17 * q^91 - 20 * q^93 - 24 * q^97 - 9 * q^99

Introduction

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