LMFDB - Embedded newform 10000.2.a.bl.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:	$0$
Dimension:	$8$
Coefficient field:	8.8.3266578125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 6x^{6} + 23x^{5} + x^{4} - 45x^{3} + 25x^{2} + 10x - 5 $ x^8 - 3x^7 - 6x^6 + 23x^5 + x^4 - 45x^3 + 25x^2 + 10x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			$1.31796$ 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)

$f(q)$	$=$	$q-2.14505 q^{3} +2.05609 q^{7} +1.60124 q^{9} +O(q^{10})$	$q-2.14505 q^{3} +2.05609 q^{7} +1.60124 q^{9} -0.879900 q^{11} -2.89312 q^{13} +4.30391 q^{17} +6.14272 q^{19} -4.41043 q^{21} +4.27058 q^{23} +3.00040 q^{27} +2.60819 q^{29} +0.366119 q^{31} +1.88743 q^{33} +10.5820 q^{37} +6.20590 q^{39} -2.84443 q^{41} +7.77738 q^{43} -7.33281 q^{47} -2.77248 q^{49} -9.23210 q^{51} -9.25285 q^{53} -13.1765 q^{57} +11.0547 q^{59} -8.23086 q^{61} +3.29231 q^{63} -0.282587 q^{67} -9.16061 q^{69} +2.50237 q^{71} -1.11299 q^{73} -1.80916 q^{77} -12.6791 q^{79} -11.2397 q^{81} +16.9446 q^{83} -5.59470 q^{87} -17.1149 q^{89} -5.94853 q^{91} -0.785345 q^{93} +6.22744 q^{97} -1.40893 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) $ 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9	$ 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} - 5 q^{11} + 2 q^{13} + 7 q^{17} + 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} + 27 q^{31} - 3 q^{33} + 10 q^{37} + 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47} + 4 q^{49} + 10 q^{51} + 2 q^{53} + 2 q^{57} + 39 q^{59} - 18 q^{61} - 14 q^{63} - 12 q^{67} + 19 q^{69} + 13 q^{71} + 12 q^{73} + 41 q^{77} + 16 q^{79} - 28 q^{81} + 64 q^{83} - 4 q^{87} - 25 q^{89} + 26 q^{91} + 40 q^{93} - 8 q^{97} - 6 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 - 5 * q^11 + 2 * q^13 + 7 * q^17 + 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 + 27 * q^31 - 3 * q^33 + 10 * q^37 + 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 + 10 * q^51 + 2 * q^53 + 2 * q^57 + 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 + 13 * q^71 + 12 * q^73 + 41 * q^77 + 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 + 26 * q^91 + 40 * q^93 - 8 * q^97 - 6 * 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
$3$	−2.14505	−1.23845	−0.619223	−	0.785215i	$-0.712552\pi$
$3$	−2.14505	−1.23845	−0.619223	+	0.785215i	$0.712552\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.05609	0.777131	0.388565	−	0.921421i	$-0.372971\pi$
$7$	2.05609	0.777131	0.388565	+	0.921421i	$0.372971\pi$
$8$	0	0
$9$	1.60124	0.533748
$10$	0	0
$11$	−0.879900	−0.265300	−0.132650	−	0.991163i	$-0.542349\pi$
$11$	−0.879900	−0.265300	−0.132650	+	0.991163i	$0.542349\pi$
$12$	0	0
$13$	−2.89312	−0.802408	−0.401204	−	0.915989i	$-0.631408\pi$
$13$	−2.89312	−0.802408	−0.401204	+	0.915989i	$0.631408\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.30391	1.04385	0.521925	−	0.852991i	$-0.325214\pi$
$17$	4.30391	1.04385	0.521925	+	0.852991i	$0.325214\pi$
$18$	0	0
$19$	6.14272	1.40924	0.704619	−	0.709586i	$-0.251118\pi$
$19$	6.14272	1.40924	0.704619	+	0.709586i	$0.251118\pi$
$20$	0	0
$21$	−4.41043	−0.962434
$22$	0	0
$23$	4.27058	0.890477	0.445239	−	0.895412i	$-0.353119\pi$
$23$	4.27058	0.890477	0.445239	+	0.895412i	$0.353119\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.00040	0.577428
$28$	0	0
$29$	2.60819	0.484329	0.242164	−	0.970235i	$-0.422143\pi$
$29$	2.60819	0.484329	0.242164	+	0.970235i	$0.422143\pi$
$30$	0	0
$31$	0.366119	0.0657570	0.0328785	−	0.999459i	$-0.489533\pi$
$31$	0.366119	0.0657570	0.0328785	+	0.999459i	$0.489533\pi$
$32$	0	0
$33$	1.88743	0.328559
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.5820	1.73968	0.869838	−	0.493338i	$-0.164223\pi$
$37$	10.5820	1.73968	0.869838	+	0.493338i	$0.164223\pi$
$38$	0	0
$39$	6.20590	0.993739
$40$	0	0
$41$	−2.84443	−0.444225	−0.222113	−	0.975021i	$-0.571295\pi$
$41$	−2.84443	−0.444225	−0.222113	+	0.975021i	$0.571295\pi$
$42$	0	0
$43$	7.77738	1.18604	0.593019	−	0.805188i	$-0.297936\pi$
$43$	7.77738	1.18604	0.593019	+	0.805188i	$0.297936\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.33281	−1.06960	−0.534800	−	0.844979i	$-0.679613\pi$
$47$	−7.33281	−1.06960	−0.534800	+	0.844979i	$0.679613\pi$
$48$	0	0
$49$	−2.77248	−0.396068
$50$	0	0
$51$	−9.23210	−1.29275
$52$	0	0
$53$	−9.25285	−1.27098	−0.635488	−	0.772110i	$-0.719201\pi$
$53$	−9.25285	−1.27098	−0.635488	+	0.772110i	$0.719201\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−13.1765	−1.74526
$58$	0	0
$59$	11.0547	1.43919	0.719597	−	0.694392i	$-0.244327\pi$
$59$	11.0547	1.43919	0.719597	+	0.694392i	$0.244327\pi$
$60$	0	0
$61$	−8.23086	−1.05385	−0.526927	−	0.849911i	$-0.676656\pi$
$61$	−8.23086	−1.05385	−0.526927	+	0.849911i	$0.676656\pi$
$62$	0	0
$63$	3.29231	0.414792
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−0.282587	−0.0345235	−0.0172618	−	0.999851i	$-0.505495\pi$
$67$	−0.282587	−0.0345235	−0.0172618	+	0.999851i	$0.505495\pi$
$68$	0	0
$69$	−9.16061	−1.10281
$70$	0	0
$71$	2.50237	0.296977	0.148488	−	0.988914i	$-0.452559\pi$
$71$	2.50237	0.296977	0.148488	+	0.988914i	$0.452559\pi$
$72$	0	0
$73$	−1.11299	−0.130266	−0.0651331	−	0.997877i	$-0.520747\pi$
$73$	−1.11299	−0.130266	−0.0651331	+	0.997877i	$0.520747\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.80916	−0.206173
$78$	0	0
$79$	−12.6791	−1.42651	−0.713255	−	0.700905i	$-0.752780\pi$
$79$	−12.6791	−1.42651	−0.713255	+	0.700905i	$0.752780\pi$
$80$	0	0
$81$	−11.2397	−1.24886
$82$	0	0
$83$	16.9446	1.85991	0.929954	−	0.367675i	$-0.119846\pi$
$83$	16.9446	1.85991	0.929954	+	0.367675i	$0.119846\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.59470	−0.599815
$88$	0	0
$89$	−17.1149	−1.81417	−0.907087	−	0.420943i	$-0.861699\pi$
$89$	−17.1149	−1.81417	−0.907087	+	0.420943i	$0.861699\pi$
$90$	0	0
$91$	−5.94853	−0.623576
$92$	0	0
$93$	−0.785345	−0.0814365
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.22744	0.632301	0.316151	−	0.948709i	$-0.397610\pi$
$97$	6.22744	0.632301	0.316151	+	0.948709i	$0.397610\pi$
$98$	0	0
$99$	−1.40893	−0.141603
$100$	0	0
$101$	6.36132	0.632975	0.316487	−	0.948597i	$-0.397497\pi$
$101$	6.36132	0.632975	0.316487	+	0.948597i	$0.397497\pi$
$102$	0	0
$103$	4.23876	0.417657	0.208829	−	0.977952i	$-0.433035\pi$
$103$	4.23876	0.417657	0.208829	+	0.977952i	$0.433035\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.6312	1.12443	0.562216	−	0.826991i	$-0.309949\pi$
$107$	11.6312	1.12443	0.562216	+	0.826991i	$0.309949\pi$
$108$	0	0
$109$	20.0255	1.91810	0.959049	−	0.283241i	$-0.0914095\pi$
$109$	20.0255	1.91810	0.959049	+	0.283241i	$0.0914095\pi$
$110$	0	0
$111$	−22.6990	−2.15449
$112$	0	0
$113$	−3.06624	−0.288447	−0.144224	−	0.989545i	$-0.546068\pi$
$113$	−3.06624	−0.288447	−0.144224	+	0.989545i	$0.546068\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.63259	−0.428283
$118$	0	0
$119$	8.84924	0.811208
$120$	0	0
$121$	−10.2258	−0.929616
$122$	0	0
$123$	6.10144	0.550149
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.9260	−0.969529	−0.484765	−	0.874645i	$-0.661095\pi$
$127$	−10.9260	−0.969529	−0.484765	+	0.874645i	$0.661095\pi$
$128$	0	0
$129$	−16.6829	−1.46884
$130$	0	0
$131$	10.6534	0.930788	0.465394	−	0.885104i	$-0.345913\pi$
$131$	10.6534	0.930788	0.465394	+	0.885104i	$0.345913\pi$
$132$	0	0
$133$	12.6300	1.09516
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.0312	1.11333	0.556664	−	0.830738i	$-0.312081\pi$
$137$	13.0312	1.11333	0.556664	+	0.830738i	$0.312081\pi$
$138$	0	0
$139$	10.1331	0.859477	0.429739	−	0.902953i	$-0.358606\pi$
$139$	10.1331	0.859477	0.429739	+	0.902953i	$0.358606\pi$
$140$	0	0
$141$	15.7292	1.32464
$142$	0	0
$143$	2.54566	0.212879
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.94710	0.490509
$148$	0	0
$149$	−6.24794	−0.511851	−0.255926	−	0.966697i	$-0.582380\pi$
$149$	−6.24794	−0.511851	−0.255926	+	0.966697i	$0.582380\pi$
$150$	0	0
$151$	−14.7390	−1.19944	−0.599722	−	0.800208i	$-0.704722\pi$
$151$	−14.7390	−1.19944	−0.599722	+	0.800208i	$0.704722\pi$
$152$	0	0
$153$	6.89160	0.557153
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−23.9213	−1.90913	−0.954565	−	0.298003i	$-0.903680\pi$
$157$	−23.9213	−1.90913	−0.954565	+	0.298003i	$0.903680\pi$
$158$	0	0
$159$	19.8478	1.57404
$160$	0	0
$161$	8.78071	0.692017
$162$	0	0
$163$	−7.41590	−0.580858	−0.290429	−	0.956896i	$-0.593798\pi$
$163$	−7.41590	−0.580858	−0.290429	+	0.956896i	$0.593798\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.5319	1.43404	0.717020	−	0.697053i	$-0.245506\pi$
$167$	18.5319	1.43404	0.717020	+	0.697053i	$0.245506\pi$
$168$	0	0
$169$	−4.62984	−0.356141
$170$	0	0
$171$	9.83599	0.752177
$172$	0	0
$173$	8.64659	0.657388	0.328694	−	0.944437i	$-0.393392\pi$
$173$	8.64659	0.657388	0.328694	+	0.944437i	$0.393392\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−23.7128	−1.78236
$178$	0	0
$179$	0.647796	0.0484186	0.0242093	−	0.999707i	$-0.492293\pi$
$179$	0.647796	0.0484186	0.0242093	+	0.999707i	$0.492293\pi$
$180$	0	0
$181$	−6.94173	−0.515975	−0.257987	−	0.966148i	$-0.583059\pi$
$181$	−6.94173	−0.515975	−0.257987	+	0.966148i	$0.583059\pi$
$182$	0	0
$183$	17.6556	1.30514
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.78701	−0.276933
$188$	0	0
$189$	6.16912	0.448737
$190$	0	0
$191$	−10.4434	−0.755655	−0.377828	−	0.925876i	$-0.623329\pi$
$191$	−10.4434	−0.755655	−0.377828	+	0.925876i	$0.623329\pi$
$192$	0	0
$193$	20.0977	1.44666	0.723332	−	0.690501i	$-0.242610\pi$
$193$	20.0977	1.44666	0.723332	+	0.690501i	$0.242610\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.15345	−0.580909	−0.290455	−	0.956889i	$-0.593807\pi$
$197$	−8.15345	−0.580909	−0.290455	+	0.956889i	$0.593807\pi$
$198$	0	0
$199$	7.21311	0.511324	0.255662	−	0.966766i	$-0.417707\pi$
$199$	7.21311	0.511324	0.255662	+	0.966766i	$0.417707\pi$
$200$	0	0
$201$	0.606164	0.0427555
$202$	0	0
$203$	5.36268	0.376387
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.83823	0.475290
$208$	0	0
$209$	−5.40498	−0.373870
$210$	0	0
$211$	18.1002	1.24607	0.623033	−	0.782195i	$-0.285900\pi$
$211$	18.1002	1.24607	0.623033	+	0.782195i	$0.285900\pi$
$212$	0	0
$213$	−5.36771	−0.367789
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.752776	0.0511018
$218$	0	0
$219$	2.38743	0.161327
$220$	0	0
$221$	−12.4517	−0.837594
$222$	0	0
$223$	−5.29657	−0.354685	−0.177342	−	0.984149i	$-0.556750\pi$
$223$	−5.29657	−0.354685	−0.177342	+	0.984149i	$0.556750\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.8593	−1.25174	−0.625868	−	0.779929i	$-0.715255\pi$
$227$	−18.8593	−1.25174	−0.625868	+	0.779929i	$0.715255\pi$
$228$	0	0
$229$	−26.2394	−1.73395	−0.866974	−	0.498354i	$-0.833938\pi$
$229$	−26.2394	−1.73395	−0.866974	+	0.498354i	$0.833938\pi$
$230$	0	0
$231$	3.88073	0.255333
$232$	0	0
$233$	9.29670	0.609047	0.304524	−	0.952505i	$-0.401503\pi$
$233$	9.29670	0.609047	0.304524	+	0.952505i	$0.401503\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	27.1973	1.76665
$238$	0	0
$239$	−6.44186	−0.416689	−0.208345	−	0.978056i	$-0.566808\pi$
$239$	−6.44186	−0.416689	−0.208345	+	0.978056i	$0.566808\pi$
$240$	0	0
$241$	−18.0604	−1.16337	−0.581687	−	0.813413i	$-0.697607\pi$
$241$	−18.0604	−1.16337	−0.581687	+	0.813413i	$0.697607\pi$
$242$	0	0
$243$	15.1086	0.969218
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−17.7717	−1.13078
$248$	0	0
$249$	−36.3470	−2.30340
$250$	0	0
$251$	−12.8374	−0.810292	−0.405146	−	0.914252i	$-0.632779\pi$
$251$	−12.8374	−0.810292	−0.405146	+	0.914252i	$0.632779\pi$
$252$	0	0
$253$	−3.75768	−0.236243
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.8739	1.17732	0.588662	−	0.808379i	$-0.299655\pi$
$257$	18.8739	1.17732	0.588662	+	0.808379i	$0.299655\pi$
$258$	0	0
$259$	21.7577	1.35196
$260$	0	0
$261$	4.17634	0.258509
$262$	0	0
$263$	−13.7763	−0.849484	−0.424742	−	0.905314i	$-0.639635\pi$
$263$	−13.7763	−0.849484	−0.424742	+	0.905314i	$0.639635\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	36.7123	2.24676
$268$	0	0
$269$	5.91693	0.360762	0.180381	−	0.983597i	$-0.442267\pi$
$269$	5.91693	0.360762	0.180381	+	0.983597i	$0.442267\pi$
$270$	0	0
$271$	6.11366	0.371378	0.185689	−	0.982609i	$-0.440548\pi$
$271$	6.11366	0.371378	0.185689	+	0.982609i	$0.440548\pi$
$272$	0	0
$273$	12.7599	0.772265
$274$	0	0
$275$	0	0
$276$	0	0
$277$	22.7920	1.36944	0.684719	−	0.728807i	$-0.259925\pi$
$277$	22.7920	1.36944	0.684719	+	0.728807i	$0.259925\pi$
$278$	0	0
$279$	0.586246	0.0350976
$280$	0	0
$281$	−29.5494	−1.76277	−0.881386	−	0.472397i	$-0.843389\pi$
$281$	−29.5494	−1.76277	−0.881386	+	0.472397i	$0.843389\pi$
$282$	0	0
$283$	−22.6118	−1.34413	−0.672067	−	0.740490i	$-0.734593\pi$
$283$	−22.6118	−1.34413	−0.672067	+	0.740490i	$0.734593\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.84841	−0.345221
$288$	0	0
$289$	1.52361	0.0896240
$290$	0	0
$291$	−13.3582	−0.783071
$292$	0	0
$293$	16.0002	0.934742	0.467371	−	0.884061i	$-0.345201\pi$
$293$	16.0002	0.934742	0.467371	+	0.884061i	$0.345201\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.64006	−0.153192
$298$	0	0
$299$	−12.3553	−0.714526
$300$	0	0
$301$	15.9910	0.921707
$302$	0	0
$303$	−13.6453	−0.783905
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−10.6779	−0.609418	−0.304709	−	0.952445i	$-0.598559\pi$
$307$	−10.6779	−0.609418	−0.304709	+	0.952445i	$0.598559\pi$
$308$	0	0
$309$	−9.09235	−0.517246
$310$	0	0
$311$	14.5417	0.824584	0.412292	−	0.911052i	$-0.364728\pi$
$311$	14.5417	0.824584	0.412292	+	0.911052i	$0.364728\pi$
$312$	0	0
$313$	11.4556	0.647507	0.323753	−	0.946141i	$-0.395055\pi$
$313$	11.4556	0.647507	0.323753	+	0.946141i	$0.395055\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.57627	0.0885323	0.0442661	−	0.999020i	$-0.485905\pi$
$317$	1.57627	0.0885323	0.0442661	+	0.999020i	$0.485905\pi$
$318$	0	0
$319$	−2.29494	−0.128492
$320$	0	0
$321$	−24.9495	−1.39255
$322$	0	0
$323$	26.4377	1.47103
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−42.9558	−2.37546
$328$	0	0
$329$	−15.0769	−0.831219
$330$	0	0
$331$	7.57270	0.416233	0.208117	−	0.978104i	$-0.433267\pi$
$331$	7.57270	0.416233	0.208117	+	0.978104i	$0.433267\pi$
$332$	0	0
$333$	16.9444	0.928548
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−7.98640	−0.435047	−0.217523	−	0.976055i	$-0.569798\pi$
$337$	−7.98640	−0.435047	−0.217523	+	0.976055i	$0.569798\pi$
$338$	0	0
$339$	6.57723	0.357226
$340$	0	0
$341$	−0.322148	−0.0174453
$342$	0	0
$343$	−20.0931	−1.08493
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.44265	0.131128	0.0655642	−	0.997848i	$-0.479115\pi$
$347$	2.44265	0.131128	0.0655642	+	0.997848i	$0.479115\pi$
$348$	0	0
$349$	6.72867	0.360177	0.180089	−	0.983650i	$-0.442362\pi$
$349$	6.72867	0.360177	0.180089	+	0.983650i	$0.442362\pi$
$350$	0	0
$351$	−8.68054	−0.463333
$352$	0	0
$353$	24.6776	1.31346	0.656728	−	0.754127i	$-0.271940\pi$
$353$	24.6776	1.31346	0.656728	+	0.754127i	$0.271940\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−18.9821	−1.00464
$358$	0	0
$359$	9.30614	0.491159	0.245580	−	0.969376i	$-0.421022\pi$
$359$	9.30614	0.491159	0.245580	+	0.969376i	$0.421022\pi$
$360$	0	0
$361$	18.7331	0.985950
$362$	0	0
$363$	21.9348	1.15128
$364$	0	0
$365$	0	0
$366$	0	0
$367$	23.7217	1.23827	0.619133	−	0.785286i	$-0.287484\pi$
$367$	23.7217	1.23827	0.619133	+	0.785286i	$0.287484\pi$
$368$	0	0
$369$	−4.55462	−0.237104
$370$	0	0
$371$	−19.0247	−0.987715
$372$	0	0
$373$	15.7703	0.816557	0.408278	−	0.912857i	$-0.366129\pi$
$373$	15.7703	0.816557	0.408278	+	0.912857i	$0.366129\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7.54581	−0.388629
$378$	0	0
$379$	21.8143	1.12053	0.560263	−	0.828315i	$-0.310700\pi$
$379$	21.8143	1.12053	0.560263	+	0.828315i	$0.310700\pi$
$380$	0	0
$381$	23.4369	1.20071
$382$	0	0
$383$	20.1387	1.02904	0.514519	−	0.857479i	$-0.327971\pi$
$383$	20.1387	1.02904	0.514519	+	0.857479i	$0.327971\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.4535	0.633045
$388$	0	0
$389$	35.9930	1.82492	0.912458	−	0.409171i	$-0.134182\pi$
$389$	35.9930	1.82492	0.912458	+	0.409171i	$0.134182\pi$
$390$	0	0
$391$	18.3802	0.929525
$392$	0	0
$393$	−22.8520	−1.15273
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.50238	−0.0754023	−0.0377011	−	0.999289i	$-0.512003\pi$
$397$	−1.50238	−0.0754023	−0.0377011	+	0.999289i	$0.512003\pi$
$398$	0	0
$399$	−27.0920	−1.35630
$400$	0	0
$401$	34.9812	1.74688	0.873438	−	0.486935i	$-0.161885\pi$
$401$	34.9812	1.74688	0.873438	+	0.486935i	$0.161885\pi$
$402$	0	0
$403$	−1.05923	−0.0527639
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.31113	−0.461535
$408$	0	0
$409$	16.0583	0.794031	0.397016	−	0.917812i	$-0.370046\pi$
$409$	16.0583	0.794031	0.397016	+	0.917812i	$0.370046\pi$
$410$	0	0
$411$	−27.9525	−1.37880
$412$	0	0
$413$	22.7294	1.11844
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−21.7360	−1.06442
$418$	0	0
$419$	−30.4184	−1.48603	−0.743017	−	0.669273i	$-0.766606\pi$
$419$	−30.4184	−1.48603	−0.743017	+	0.669273i	$0.766606\pi$
$420$	0	0
$421$	3.86503	0.188370	0.0941850	−	0.995555i	$-0.469975\pi$
$421$	3.86503	0.188370	0.0941850	+	0.995555i	$0.469975\pi$
$422$	0	0
$423$	−11.7416	−0.570896
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−16.9234	−0.818982
$428$	0	0
$429$	−5.46057	−0.263639
$430$	0	0
$431$	−27.9602	−1.34679	−0.673397	−	0.739281i	$-0.735166\pi$
$431$	−27.9602	−1.34679	−0.673397	+	0.739281i	$0.735166\pi$
$432$	0	0
$433$	5.38943	0.258999	0.129500	−	0.991579i	$-0.458663\pi$
$433$	5.38943	0.258999	0.129500	+	0.991579i	$0.458663\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.2330	1.25489
$438$	0	0
$439$	32.8325	1.56701	0.783504	−	0.621387i	$-0.213430\pi$
$439$	32.8325	1.56701	0.783504	+	0.621387i	$0.213430\pi$
$440$	0	0
$441$	−4.43941	−0.211400
$442$	0	0
$443$	10.1986	0.484550	0.242275	−	0.970208i	$-0.422106\pi$
$443$	10.1986	0.484550	0.242275	+	0.970208i	$0.422106\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.4021	0.633900
$448$	0	0
$449$	−40.4691	−1.90986	−0.954928	−	0.296837i	$-0.904068\pi$
$449$	−40.4691	−1.90986	−0.954928	+	0.296837i	$0.904068\pi$
$450$	0	0
$451$	2.50281	0.117853
$452$	0	0
$453$	31.6159	1.48545
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.8873	1.53840	0.769201	−	0.639007i	$-0.220655\pi$
$457$	32.8873	1.53840	0.769201	+	0.639007i	$0.220655\pi$
$458$	0	0
$459$	12.9135	0.602749
$460$	0	0
$461$	−23.3686	−1.08838	−0.544192	−	0.838961i	$-0.683164\pi$
$461$	−23.3686	−1.08838	−0.544192	+	0.838961i	$0.683164\pi$
$462$	0	0
$463$	29.0697	1.35098	0.675492	−	0.737367i	$-0.263931\pi$
$463$	29.0697	1.35098	0.675492	+	0.737367i	$0.263931\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15.5012	0.717308	0.358654	−	0.933471i	$-0.383236\pi$
$467$	15.5012	0.717308	0.358654	+	0.933471i	$0.383236\pi$
$468$	0	0
$469$	−0.581026	−0.0268293
$470$	0	0
$471$	51.3125	2.36435
$472$	0	0
$473$	−6.84331	−0.314656
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−14.8161	−0.678381
$478$	0	0
$479$	27.8045	1.27042	0.635211	−	0.772339i	$-0.280913\pi$
$479$	27.8045	1.27042	0.635211	+	0.772339i	$0.280913\pi$
$480$	0	0
$481$	−30.6151	−1.39593
$482$	0	0
$483$	−18.8351	−0.857026
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.00649	0.0456086	0.0228043	−	0.999740i	$-0.492741\pi$
$487$	1.00649	0.0456086	0.0228043	+	0.999740i	$0.492741\pi$
$488$	0	0
$489$	15.9075	0.719362
$490$	0	0
$491$	−12.6297	−0.569970	−0.284985	−	0.958532i	$-0.591989\pi$
$491$	−12.6297	−0.569970	−0.284985	+	0.958532i	$0.591989\pi$
$492$	0	0
$493$	11.2254	0.505567
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.14511	0.230790
$498$	0	0
$499$	2.96228	0.132610	0.0663050	−	0.997799i	$-0.478879\pi$
$499$	2.96228	0.132610	0.0663050	+	0.997799i	$0.478879\pi$
$500$	0	0
$501$	−39.7518	−1.77598
$502$	0	0
$503$	24.0211	1.07105	0.535523	−	0.844520i	$-0.320114\pi$
$503$	24.0211	1.07105	0.535523	+	0.844520i	$0.320114\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.93124	0.441062
$508$	0	0
$509$	−8.94073	−0.396291	−0.198146	−	0.980173i	$-0.563492\pi$
$509$	−8.94073	−0.396291	−0.198146	+	0.980173i	$0.563492\pi$
$510$	0	0
$511$	−2.28842	−0.101234
$512$	0	0
$513$	18.4307	0.813733
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.45213	0.283764
$518$	0	0
$519$	−18.5474	−0.814139
$520$	0	0
$521$	42.4850	1.86130	0.930650	−	0.365911i	$-0.119243\pi$
$521$	42.4850	1.86130	0.930650	+	0.365911i	$0.119243\pi$
$522$	0	0
$523$	2.00639	0.0877333	0.0438666	−	0.999037i	$-0.486032\pi$
$523$	2.00639	0.0877333	0.0438666	+	0.999037i	$0.486032\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.57574	0.0686405
$528$	0	0
$529$	−4.76216	−0.207050
$530$	0	0
$531$	17.7012	0.768166
$532$	0	0
$533$	8.22928	0.356450
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.38956	−0.0599638
$538$	0	0
$539$	2.43950	0.105077
$540$	0	0
$541$	28.1853	1.21178	0.605891	−	0.795548i	$-0.292817\pi$
$541$	28.1853	1.21178	0.605891	+	0.795548i	$0.292817\pi$
$542$	0	0
$543$	14.8904	0.639006
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−16.5381	−0.707116	−0.353558	−	0.935413i	$-0.615028\pi$
$547$	−16.5381	−0.707116	−0.353558	+	0.935413i	$0.615028\pi$
$548$	0	0
$549$	−13.1796	−0.562492
$550$	0	0
$551$	16.0214	0.682534
$552$	0	0
$553$	−26.0694	−1.10858
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.6671	−1.34178	−0.670889	−	0.741558i	$-0.734087\pi$
$557$	−31.6671	−1.34178	−0.670889	+	0.741558i	$0.734087\pi$
$558$	0	0
$559$	−22.5009	−0.951687
$560$	0	0
$561$	8.12332	0.342967
$562$	0	0
$563$	−23.8552	−1.00538	−0.502688	−	0.864468i	$-0.667655\pi$
$563$	−23.8552	−1.00538	−0.502688	+	0.864468i	$0.667655\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−23.1100	−0.970528
$568$	0	0
$569$	−15.7062	−0.658437	−0.329219	−	0.944254i	$-0.606785\pi$
$569$	−15.7062	−0.658437	−0.329219	+	0.944254i	$0.606785\pi$
$570$	0	0
$571$	−15.0827	−0.631191	−0.315595	−	0.948894i	$-0.602204\pi$
$571$	−15.0827	−0.631191	−0.315595	+	0.948894i	$0.602204\pi$
$572$	0	0
$573$	22.4015	0.935838
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−27.5107	−1.14529	−0.572644	−	0.819804i	$-0.694082\pi$
$577$	−27.5107	−1.14529	−0.572644	+	0.819804i	$0.694082\pi$
$578$	0	0
$579$	−43.1106	−1.79161
$580$	0	0
$581$	34.8396	1.44539
$582$	0	0
$583$	8.14158	0.337190
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.17213	0.337300	0.168650	−	0.985676i	$-0.446059\pi$
$587$	8.17213	0.337300	0.168650	+	0.985676i	$0.446059\pi$
$588$	0	0
$589$	2.24897	0.0926672
$590$	0	0
$591$	17.4896	0.719425
$592$	0	0
$593$	15.2102	0.624607	0.312304	−	0.949982i	$-0.398899\pi$
$593$	15.2102	0.624607	0.312304	+	0.949982i	$0.398899\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.4725	−0.633247
$598$	0	0
$599$	−13.5445	−0.553414	−0.276707	−	0.960954i	$-0.589243\pi$
$599$	−13.5445	−0.553414	−0.276707	+	0.960954i	$0.589243\pi$
$600$	0	0
$601$	−2.96392	−0.120901	−0.0604505	−	0.998171i	$-0.519254\pi$
$601$	−2.96392	−0.120901	−0.0604505	+	0.998171i	$0.519254\pi$
$602$	0	0
$603$	−0.452491	−0.0184269
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−30.6091	−1.24239	−0.621193	−	0.783658i	$-0.713352\pi$
$607$	−30.6091	−1.24239	−0.621193	+	0.783658i	$0.713352\pi$
$608$	0	0
$609$	−11.5032	−0.466134
$610$	0	0
$611$	21.2147	0.858255
$612$	0	0
$613$	23.6671	0.955905	0.477953	−	0.878386i	$-0.341379\pi$
$613$	23.6671	0.955905	0.477953	+	0.878386i	$0.341379\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	45.7699	1.84263	0.921313	−	0.388823i	$-0.127118\pi$
$617$	45.7699	1.84263	0.921313	+	0.388823i	$0.127118\pi$
$618$	0	0
$619$	7.10511	0.285578	0.142789	−	0.989753i	$-0.454393\pi$
$619$	7.10511	0.285578	0.142789	+	0.989753i	$0.454393\pi$
$620$	0	0
$621$	12.8135	0.514187
$622$	0	0
$623$	−35.1898	−1.40985
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.5940	0.463018
$628$	0	0
$629$	45.5441	1.81596
$630$	0	0
$631$	24.5053	0.975542	0.487771	−	0.872972i	$-0.337810\pi$
$631$	24.5053	0.975542	0.487771	+	0.872972i	$0.337810\pi$
$632$	0	0
$633$	−38.8258	−1.54319
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.02112	0.317808
$638$	0	0
$639$	4.00690	0.158511
$640$	0	0
$641$	−1.72771	−0.0682405	−0.0341202	−	0.999418i	$-0.510863\pi$
$641$	−1.72771	−0.0682405	−0.0341202	+	0.999418i	$0.510863\pi$
$642$	0	0
$643$	−9.61256	−0.379082	−0.189541	−	0.981873i	$-0.560700\pi$
$643$	−9.61256	−0.379082	−0.189541	+	0.981873i	$0.560700\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−7.77347	−0.305606	−0.152803	−	0.988257i	$-0.548830\pi$
$647$	−7.77347	−0.305606	−0.152803	+	0.988257i	$0.548830\pi$
$648$	0	0
$649$	−9.72699	−0.381818
$650$	0	0
$651$	−1.61474	−0.0632868
$652$	0	0
$653$	8.23519	0.322268	0.161134	−	0.986933i	$-0.448485\pi$
$653$	8.23519	0.322268	0.161134	+	0.986933i	$0.448485\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.78217	−0.0695292
$658$	0	0
$659$	−21.8377	−0.850675	−0.425338	−	0.905035i	$-0.639845\pi$
$659$	−21.8377	−0.850675	−0.425338	+	0.905035i	$0.639845\pi$
$660$	0	0
$661$	−28.0960	−1.09281	−0.546403	−	0.837522i	$-0.684003\pi$
$661$	−28.0960	−1.09281	−0.546403	+	0.837522i	$0.684003\pi$
$662$	0	0
$663$	26.7096	1.03731
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.1385	0.431284
$668$	0	0
$669$	11.3614	0.439258
$670$	0	0
$671$	7.24233	0.279587
$672$	0	0
$673$	−13.4089	−0.516874	−0.258437	−	0.966028i	$-0.583207\pi$
$673$	−13.4089	−0.516874	−0.258437	+	0.966028i	$0.583207\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.1770	0.506435	0.253217	−	0.967409i	$-0.418511\pi$
$677$	13.1770	0.506435	0.253217	+	0.967409i	$0.418511\pi$
$678$	0	0
$679$	12.8042	0.491381
$680$	0	0
$681$	40.4542	1.55021
$682$	0	0
$683$	−12.6250	−0.483084	−0.241542	−	0.970390i	$-0.577653\pi$
$683$	−12.6250	−0.483084	−0.241542	+	0.970390i	$0.577653\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	56.2848	2.14740
$688$	0	0
$689$	26.7696	1.01984
$690$	0	0
$691$	24.7940	0.943207	0.471603	−	0.881811i	$-0.343675\pi$
$691$	24.7940	0.943207	0.471603	+	0.881811i	$0.343675\pi$
$692$	0	0
$693$	−2.89690	−0.110044
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−12.2422	−0.463705
$698$	0	0
$699$	−19.9419	−0.754272
$700$	0	0
$701$	32.2564	1.21831	0.609154	−	0.793052i	$-0.291509\pi$
$701$	32.2564	1.21831	0.609154	+	0.793052i	$0.291509\pi$
$702$	0	0
$703$	65.0025	2.45162
$704$	0	0
$705$	0	0
$706$	0	0
$707$	13.0795	0.491904
$708$	0	0
$709$	41.1147	1.54409	0.772047	−	0.635566i	$-0.219233\pi$
$709$	41.1147	1.54409	0.772047	+	0.635566i	$0.219233\pi$
$710$	0	0
$711$	−20.3023	−0.761396
$712$	0	0
$713$	1.56354	0.0585551
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.8181	0.516047
$718$	0	0
$719$	40.3268	1.50393	0.751967	−	0.659200i	$-0.229105\pi$
$719$	40.3268	1.50393	0.751967	+	0.659200i	$0.229105\pi$
$720$	0	0
$721$	8.71528	0.324574
$722$	0	0
$723$	38.7405	1.44077
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−0.513233	−0.0190347	−0.00951737	−	0.999955i	$-0.503030\pi$
$727$	−0.513233	−0.0190347	−0.00951737	+	0.999955i	$0.503030\pi$
$728$	0	0
$729$	1.31049	0.0485368
$730$	0	0
$731$	33.4731	1.23805
$732$	0	0
$733$	−30.1586	−1.11393	−0.556966	−	0.830535i	$-0.688035\pi$
$733$	−30.1586	−1.11393	−0.556966	+	0.830535i	$0.688035\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.248648	0.00915908
$738$	0	0
$739$	44.0345	1.61983	0.809917	−	0.586544i	$-0.199512\pi$
$739$	44.0345	1.61983	0.809917	+	0.586544i	$0.199512\pi$
$740$	0	0
$741$	38.1211	1.40041
$742$	0	0
$743$	−21.2532	−0.779702	−0.389851	−	0.920878i	$-0.627474\pi$
$743$	−21.2532	−0.779702	−0.389851	+	0.920878i	$0.627474\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	27.1324	0.992722
$748$	0	0
$749$	23.9149	0.873830
$750$	0	0
$751$	52.3062	1.90868	0.954340	−	0.298724i	$-0.0965610\pi$
$751$	52.3062	1.90868	0.954340	+	0.298724i	$0.0965610\pi$
$752$	0	0
$753$	27.5370	1.00350
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−16.6588	−0.605475	−0.302737	−	0.953074i	$-0.597901\pi$
$757$	−16.6588	−0.605475	−0.302737	+	0.953074i	$0.597901\pi$
$758$	0	0
$759$	8.06042	0.292575
$760$	0	0
$761$	45.2275	1.63950	0.819748	−	0.572724i	$-0.194113\pi$
$761$	45.2275	1.63950	0.819748	+	0.572724i	$0.194113\pi$
$762$	0	0
$763$	41.1744	1.49061
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−31.9825	−1.15482
$768$	0	0
$769$	−13.5059	−0.487034	−0.243517	−	0.969897i	$-0.578301\pi$
$769$	−13.5059	−0.487034	−0.243517	+	0.969897i	$0.578301\pi$
$770$	0	0
$771$	−40.4856	−1.45805
$772$	0	0
$773$	−49.5219	−1.78118	−0.890590	−	0.454808i	$-0.849708\pi$
$773$	−49.5219	−1.78118	−0.890590	+	0.454808i	$0.849708\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−46.6713	−1.67432
$778$	0	0
$779$	−17.4725	−0.626019
$780$	0	0
$781$	−2.20183	−0.0787878
$782$	0	0
$783$	7.82562	0.279665
$784$	0	0
$785$	0	0
$786$	0	0
$787$	39.2860	1.40040	0.700198	−	0.713949i	$-0.253095\pi$
$787$	39.2860	1.40040	0.700198	+	0.713949i	$0.253095\pi$
$788$	0	0
$789$	29.5509	1.05204
$790$	0	0
$791$	−6.30447	−0.224161
$792$	0	0
$793$	23.8129	0.845620
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.38163	0.226049	0.113024	−	0.993592i	$-0.463946\pi$
$797$	6.38163	0.226049	0.113024	+	0.993592i	$0.463946\pi$
$798$	0	0
$799$	−31.5597	−1.11650
$800$	0	0
$801$	−27.4051	−0.968311
$802$	0	0
$803$	0.979323	0.0345596
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.6921	−0.446784
$808$	0	0
$809$	−10.4041	−0.365789	−0.182894	−	0.983133i	$-0.558547\pi$
$809$	−10.4041	−0.365789	−0.182894	+	0.983133i	$0.558547\pi$
$810$	0	0
$811$	46.8641	1.64562	0.822811	−	0.568316i	$-0.192405\pi$
$811$	46.8641	1.64562	0.822811	+	0.568316i	$0.192405\pi$
$812$	0	0
$813$	−13.1141	−0.459932
$814$	0	0
$815$	0	0
$816$	0	0
$817$	47.7743	1.67141
$818$	0	0
$819$	−9.52505	−0.332832
$820$	0	0
$821$	−14.4665	−0.504885	−0.252442	−	0.967612i	$-0.581234\pi$
$821$	−14.4665	−0.504885	−0.252442	+	0.967612i	$0.581234\pi$
$822$	0	0
$823$	30.1548	1.05113	0.525564	−	0.850754i	$-0.323854\pi$
$823$	30.1548	1.05113	0.525564	+	0.850754i	$0.323854\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.6990	1.34570	0.672848	−	0.739781i	$-0.265071\pi$
$827$	38.6990	1.34570	0.672848	+	0.739781i	$0.265071\pi$
$828$	0	0
$829$	36.4577	1.26623	0.633114	−	0.774059i	$-0.281777\pi$
$829$	36.4577	1.26623	0.633114	+	0.774059i	$0.281777\pi$
$830$	0	0
$831$	−48.8900	−1.69597
$832$	0	0
$833$	−11.9325	−0.413436
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.09851	0.0379699
$838$	0	0
$839$	38.7899	1.33917	0.669587	−	0.742733i	$-0.266471\pi$
$839$	38.7899	1.33917	0.669587	+	0.742733i	$0.266471\pi$
$840$	0	0
$841$	−22.1974	−0.765426
$842$	0	0
$843$	63.3851	2.18310
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−21.0252	−0.722433
$848$	0	0
$849$	48.5036	1.66464
$850$	0	0
$851$	45.1914	1.54914
$852$	0	0
$853$	−16.2077	−0.554942	−0.277471	−	0.960734i	$-0.589496\pi$
$853$	−16.2077	−0.554942	−0.277471	+	0.960734i	$0.589496\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.7493	0.640465	0.320233	−	0.947339i	$-0.396239\pi$
$857$	18.7493	0.640465	0.320233	+	0.947339i	$0.396239\pi$
$858$	0	0
$859$	−37.0954	−1.26568	−0.632839	−	0.774283i	$-0.718111\pi$
$859$	−37.0954	−1.26568	−0.632839	+	0.774283i	$0.718111\pi$
$860$	0	0
$861$	12.5451	0.427537
$862$	0	0
$863$	9.38354	0.319419	0.159710	−	0.987164i	$-0.448944\pi$
$863$	9.38354	0.319419	0.159710	+	0.987164i	$0.448944\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.26822	−0.110994
$868$	0	0
$869$	11.1563	0.378453
$870$	0	0
$871$	0.817560	0.0277020
$872$	0	0
$873$	9.97165	0.337489
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.9766	−0.438189	−0.219094	−	0.975704i	$-0.570310\pi$
$877$	−12.9766	−0.438189	−0.219094	+	0.975704i	$0.570310\pi$
$878$	0	0
$879$	−34.3212	−1.15763
$880$	0	0
$881$	21.8240	0.735270	0.367635	−	0.929970i	$-0.380168\pi$
$881$	21.8240	0.735270	0.367635	+	0.929970i	$0.380168\pi$
$882$	0	0
$883$	−35.9161	−1.20867	−0.604336	−	0.796730i	$-0.706561\pi$
$883$	−35.9161	−1.20867	−0.604336	+	0.796730i	$0.706561\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−29.2312	−0.981489	−0.490744	−	0.871304i	$-0.663275\pi$
$887$	−29.2312	−0.981489	−0.490744	+	0.871304i	$0.663275\pi$
$888$	0	0
$889$	−22.4650	−0.753451
$890$	0	0
$891$	9.88985	0.331322
$892$	0	0
$893$	−45.0434	−1.50732
$894$	0	0
$895$	0	0
$896$	0	0
$897$	26.5028	0.884902
$898$	0	0
$899$	0.954909	0.0318480
$900$	0	0
$901$	−39.8234	−1.32671
$902$	0	0
$903$	−34.3016	−1.14148
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9.09788	0.302090	0.151045	−	0.988527i	$-0.451736\pi$
$907$	9.09788	0.302090	0.151045	+	0.988527i	$0.451736\pi$
$908$	0	0
$909$	10.1860	0.337849
$910$	0	0
$911$	−3.71672	−0.123140	−0.0615702	−	0.998103i	$-0.519611\pi$
$911$	−3.71672	−0.123140	−0.0615702	+	0.998103i	$0.519611\pi$
$912$	0	0
$913$	−14.9095	−0.493433
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.9043	0.723344
$918$	0	0
$919$	−43.2514	−1.42673	−0.713366	−	0.700791i	$-0.752830\pi$
$919$	−43.2514	−1.42673	−0.713366	+	0.700791i	$0.752830\pi$
$920$	0	0
$921$	22.9046	0.754731
$922$	0	0
$923$	−7.23966	−0.238296
$924$	0	0
$925$	0	0
$926$	0	0
$927$	6.78728	0.222924
$928$	0	0
$929$	35.1807	1.15424	0.577121	−	0.816659i	$-0.304176\pi$
$929$	35.1807	1.15424	0.577121	+	0.816659i	$0.304176\pi$
$930$	0	0
$931$	−17.0306	−0.558154
$932$	0	0
$933$	−31.1927	−1.02120
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−28.7941	−0.940661	−0.470331	−	0.882490i	$-0.655865\pi$
$937$	−28.7941	−0.940661	−0.470331	+	0.882490i	$0.655865\pi$
$938$	0	0
$939$	−24.5728	−0.801902
$940$	0	0
$941$	−27.3593	−0.891888	−0.445944	−	0.895061i	$-0.647132\pi$
$941$	−27.3593	−0.891888	−0.445944	+	0.895061i	$0.647132\pi$
$942$	0	0
$943$	−12.1474	−0.395572
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11.7609	−0.382178	−0.191089	−	0.981573i	$-0.561202\pi$
$947$	−11.7609	−0.382178	−0.191089	+	0.981573i	$0.561202\pi$
$948$	0	0
$949$	3.22003	0.104527
$950$	0	0
$951$	−3.38118	−0.109642
$952$	0	0
$953$	12.3956	0.401534	0.200767	−	0.979639i	$-0.435657\pi$
$953$	12.3956	0.401534	0.200767	+	0.979639i	$0.435657\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	4.92277	0.159131
$958$	0	0
$959$	26.7933	0.865201
$960$	0	0
$961$	−30.8660	−0.995676
$962$	0	0
$963$	18.6244	0.600162
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−39.2090	−1.26088	−0.630438	−	0.776240i	$-0.717125\pi$
$967$	−39.2090	−1.26088	−0.630438	+	0.776240i	$0.717125\pi$
$968$	0	0
$969$	−56.7102	−1.82179
$970$	0	0
$971$	−36.6646	−1.17662	−0.588311	−	0.808634i	$-0.700207\pi$
$971$	−36.6646	−1.17662	−0.588311	+	0.808634i	$0.700207\pi$
$972$	0	0
$973$	20.8346	0.667926
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.3208	0.554143	0.277071	−	0.960849i	$-0.410636\pi$
$977$	17.3208	0.554143	0.277071	+	0.960849i	$0.410636\pi$
$978$	0	0
$979$	15.0594	0.481300
$980$	0	0
$981$	32.0657	1.02378
$982$	0	0
$983$	55.4332	1.76804	0.884022	−	0.467445i	$-0.154825\pi$
$983$	55.4332	1.76804	0.884022	+	0.467445i	$0.154825\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	32.3408	1.02942
$988$	0	0
$989$	33.2139	1.05614
$990$	0	0
$991$	22.3179	0.708951	0.354476	−	0.935065i	$-0.384659\pi$
$991$	22.3179	0.708951	0.354476	+	0.935065i	$0.384659\pi$
$992$	0	0
$993$	−16.2438	−0.515482
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−43.4633	−1.37650	−0.688249	−	0.725475i	$-0.741620\pi$
$997$	−43.4633	−1.37650	−0.688249	+	0.725475i	$0.741620\pi$
$998$	0	0
$999$	31.7504	1.00454

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.bl.1.1		8
4.3	odd	2		5000.2.a.k.1.8	✓	8
5.4	even	2		10000.2.a.bg.1.8		8
20.19	odd	2		5000.2.a.n.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5000.2.a.k.1.8	✓	8	4.3	odd	2
5000.2.a.n.1.1	yes	8	20.19	odd	2
10000.2.a.bg.1.8		8	5.4	even	2
10000.2.a.bl.1.1		8	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)

\(f(q)\)	\(=\)	\(q-2.14505 q^{3} +2.05609 q^{7} +1.60124 q^{9} +O(q^{10})\)	\(q-2.14505 q^{3} +2.05609 q^{7} +1.60124 q^{9} -0.879900 q^{11} -2.89312 q^{13} +4.30391 q^{17} +6.14272 q^{19} -4.41043 q^{21} +4.27058 q^{23} +3.00040 q^{27} +2.60819 q^{29} +0.366119 q^{31} +1.88743 q^{33} +10.5820 q^{37} +6.20590 q^{39} -2.84443 q^{41} +7.77738 q^{43} -7.33281 q^{47} -2.77248 q^{49} -9.23210 q^{51} -9.25285 q^{53} -13.1765 q^{57} +11.0547 q^{59} -8.23086 q^{61} +3.29231 q^{63} -0.282587 q^{67} -9.16061 q^{69} +2.50237 q^{71} -1.11299 q^{73} -1.80916 q^{77} -12.6791 q^{79} -11.2397 q^{81} +16.9446 q^{83} -5.59470 q^{87} -17.1149 q^{89} -5.94853 q^{91} -0.785345 q^{93} +6.22744 q^{97} -1.40893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9	\( 8 q + 3 q^{3} + 2 q^{7} - 3 q^{9} - 5 q^{11} + 2 q^{13} + 7 q^{17} + 10 q^{19} - 14 q^{21} - 2 q^{23} - 6 q^{27} - 10 q^{29} + 27 q^{31} - 3 q^{33} + 10 q^{37} + 18 q^{39} - 16 q^{41} + 4 q^{43} - 8 q^{47} + 4 q^{49} + 10 q^{51} + 2 q^{53} + 2 q^{57} + 39 q^{59} - 18 q^{61} - 14 q^{63} - 12 q^{67} + 19 q^{69} + 13 q^{71} + 12 q^{73} + 41 q^{77} + 16 q^{79} - 28 q^{81} + 64 q^{83} - 4 q^{87} - 25 q^{89} + 26 q^{91} + 40 q^{93} - 8 q^{97} - 6 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + 2 * q^7 - 3 * q^9 - 5 * q^11 + 2 * q^13 + 7 * q^17 + 10 * q^19 - 14 * q^21 - 2 * q^23 - 6 * q^27 - 10 * q^29 + 27 * q^31 - 3 * q^33 + 10 * q^37 + 18 * q^39 - 16 * q^41 + 4 * q^43 - 8 * q^47 + 4 * q^49 + 10 * q^51 + 2 * q^53 + 2 * q^57 + 39 * q^59 - 18 * q^61 - 14 * q^63 - 12 * q^67 + 19 * q^69 + 13 * q^71 + 12 * q^73 + 41 * q^77 + 16 * q^79 - 28 * q^81 + 64 * q^83 - 4 * q^87 - 25 * q^89 + 26 * q^91 + 40 * q^93 - 8 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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