LMFDB - Embedded newform 1456.2.a.u.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(1,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.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:	$11.6262185343$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.183064.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 6x + 8 $ x^4 - x^3 - 10x^2 + 6x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.87183$ of defining polynomial
Character	$\chi$	$=$	1456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.87183 q^{3} +2.47899 q^{5} -1.00000 q^{7} +5.24743 q^{9} +O(q^{10})$	$q+2.87183 q^{3} +2.47899 q^{5} -1.00000 q^{7} +5.24743 q^{9} +2.87183 q^{11} +1.00000 q^{13} +7.11926 q^{15} +0.624408 q^{17} +0.478994 q^{19} -2.87183 q^{21} -3.35083 q^{23} +1.14541 q^{25} +6.45423 q^{27} -4.64026 q^{29} -9.51210 q^{31} +8.24743 q^{33} -2.47899 q^{35} +4.87183 q^{37} +2.87183 q^{39} +1.28944 q^{41} -1.85459 q^{43} +13.0083 q^{45} -9.51210 q^{47} +1.00000 q^{49} +1.79320 q^{51} +8.64026 q^{53} +7.11926 q^{55} +1.37559 q^{57} -10.7017 q^{59} +12.4542 q^{61} -5.24743 q^{63} +2.47899 q^{65} +9.99109 q^{67} -9.62302 q^{69} -4.36807 q^{71} -2.72642 q^{73} +3.28944 q^{75} -2.87183 q^{77} -0.136505 q^{79} +2.79320 q^{81} +14.2227 q^{83} +1.54790 q^{85} -13.3261 q^{87} -13.5032 q^{89} -1.00000 q^{91} -27.3172 q^{93} +1.18742 q^{95} +17.2558 q^{97} +15.0697 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - 4 q^{7} + 9 q^{9}+O(q^{10}) $ 4 * q - q^3 - 4 * q^7 + 9 * q^9	$ 4 q - q^{3} - 4 q^{7} + 9 q^{9} - q^{11} + 4 q^{13} + 4 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} + 9 q^{23} + 14 q^{25} - 7 q^{27} - 4 q^{29} - 11 q^{31} + 21 q^{33} + 7 q^{37} - q^{39} + 13 q^{41} + 2 q^{43} + 12 q^{45} - 11 q^{47} + 4 q^{49} + 28 q^{51} + 20 q^{53} + 4 q^{55} + 6 q^{57} + 2 q^{59} + 17 q^{61} - 9 q^{63} + 3 q^{67} - 27 q^{69} + 8 q^{71} + 11 q^{73} + 21 q^{75} + q^{77} + 27 q^{79} + 32 q^{81} + 22 q^{83} - 8 q^{85} - 8 q^{87} + 10 q^{89} - 4 q^{91} - 27 q^{93} + 34 q^{95} + 17 q^{97} - 10 q^{99}+O(q^{100}) $ 4 * q - q^3 - 4 * q^7 + 9 * q^9 - q^11 + 4 * q^13 + 4 * q^15 + 2 * q^17 - 8 * q^19 + q^21 + 9 * q^23 + 14 * q^25 - 7 * q^27 - 4 * q^29 - 11 * q^31 + 21 * q^33 + 7 * q^37 - q^39 + 13 * q^41 + 2 * q^43 + 12 * q^45 - 11 * q^47 + 4 * q^49 + 28 * q^51 + 20 * q^53 + 4 * q^55 + 6 * q^57 + 2 * q^59 + 17 * q^61 - 9 * q^63 + 3 * q^67 - 27 * q^69 + 8 * q^71 + 11 * q^73 + 21 * q^75 + q^77 + 27 * q^79 + 32 * q^81 + 22 * q^83 - 8 * q^85 - 8 * q^87 + 10 * q^89 - 4 * q^91 - 27 * q^93 + 34 * q^95 + 17 * q^97 - 10 * 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
$3$	2.87183	1.65805	0.829027	−	0.559209i	$-0.188895\pi$
$3$	2.87183	1.65805	0.829027	+	0.559209i	$0.188895\pi$
$4$	0	0
$5$	2.47899	1.10864	0.554320	−	0.832304i	$-0.312978\pi$
$5$	2.47899	1.10864	0.554320	+	0.832304i	$0.312978\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	5.24743	1.74914
$10$	0	0
$11$	2.87183	0.865890	0.432945	−	0.901420i	$-0.357474\pi$
$11$	2.87183	0.865890	0.432945	+	0.901420i	$0.357474\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	7.11926	1.83818
$16$	0	0
$17$	0.624408	0.151441	0.0757206	−	0.997129i	$-0.475874\pi$
$17$	0.624408	0.151441	0.0757206	+	0.997129i	$0.475874\pi$
$18$	0	0
$19$	0.478994	0.109889	0.0549444	−	0.998489i	$-0.482502\pi$
$19$	0.478994	0.109889	0.0549444	+	0.998489i	$0.482502\pi$
$20$	0	0
$21$	−2.87183	−0.626685
$22$	0	0
$23$	−3.35083	−0.698696	−0.349348	−	0.936993i	$-0.613597\pi$
$23$	−3.35083	−0.698696	−0.349348	+	0.936993i	$0.613597\pi$
$24$	0	0
$25$	1.14541	0.229083
$26$	0	0
$27$	6.45423	1.24212
$28$	0	0
$29$	−4.64026	−0.861675	−0.430838	−	0.902429i	$-0.641782\pi$
$29$	−4.64026	−0.861675	−0.430838	+	0.902429i	$0.641782\pi$
$30$	0	0
$31$	−9.51210	−1.70842	−0.854212	−	0.519926i	$-0.825960\pi$
$31$	−9.51210	−1.70842	−0.854212	+	0.519926i	$0.825960\pi$
$32$	0	0
$33$	8.24743	1.43569
$34$	0	0
$35$	−2.47899	−0.419027
$36$	0	0
$37$	4.87183	0.800924	0.400462	−	0.916313i	$-0.368850\pi$
$37$	4.87183	0.800924	0.400462	+	0.916313i	$0.368850\pi$
$38$	0	0
$39$	2.87183	0.459861
$40$	0	0
$41$	1.28944	0.201376	0.100688	−	0.994918i	$-0.467896\pi$
$41$	1.28944	0.201376	0.100688	+	0.994918i	$0.467896\pi$
$42$	0	0
$43$	−1.85459	−0.282822	−0.141411	−	0.989951i	$-0.545164\pi$
$43$	−1.85459	−0.282822	−0.141411	+	0.989951i	$0.545164\pi$
$44$	0	0
$45$	13.0083	1.93917
$46$	0	0
$47$	−9.51210	−1.38748	−0.693741	−	0.720225i	$-0.744039\pi$
$47$	−9.51210	−1.38748	−0.693741	+	0.720225i	$0.744039\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.79320	0.251098
$52$	0	0
$53$	8.64026	1.18683	0.593416	−	0.804896i	$-0.297779\pi$
$53$	8.64026	1.18683	0.593416	+	0.804896i	$0.297779\pi$
$54$	0	0
$55$	7.11926	0.959961
$56$	0	0
$57$	1.37559	0.182202
$58$	0	0
$59$	−10.7017	−1.39324	−0.696618	−	0.717442i	$-0.745313\pi$
$59$	−10.7017	−1.39324	−0.696618	+	0.717442i	$0.745313\pi$
$60$	0	0
$61$	12.4542	1.59460	0.797300	−	0.603583i	$-0.206261\pi$
$61$	12.4542	1.59460	0.797300	+	0.603583i	$0.206261\pi$
$62$	0	0
$63$	−5.24743	−0.661113
$64$	0	0
$65$	2.47899	0.307481
$66$	0	0
$67$	9.99109	1.22061	0.610303	−	0.792168i	$-0.291048\pi$
$67$	9.99109	1.22061	0.610303	+	0.792168i	$0.291048\pi$
$68$	0	0
$69$	−9.62302	−1.15848
$70$	0	0
$71$	−4.36807	−0.518395	−0.259198	−	0.965824i	$-0.583458\pi$
$71$	−4.36807	−0.518395	−0.259198	+	0.965824i	$0.583458\pi$
$72$	0	0
$73$	−2.72642	−0.319103	−0.159552	−	0.987190i	$-0.551005\pi$
$73$	−2.72642	−0.319103	−0.159552	+	0.987190i	$0.551005\pi$
$74$	0	0
$75$	3.28944	0.379831
$76$	0	0
$77$	−2.87183	−0.327276
$78$	0	0
$79$	−0.136505	−0.0153580	−0.00767899	−	0.999971i	$-0.502444\pi$
$79$	−0.136505	−0.0153580	−0.00767899	+	0.999971i	$0.502444\pi$
$80$	0	0
$81$	2.79320	0.310355
$82$	0	0
$83$	14.2227	1.56114	0.780570	−	0.625068i	$-0.214929\pi$
$83$	14.2227	1.56114	0.780570	+	0.625068i	$0.214929\pi$
$84$	0	0
$85$	1.54790	0.167894
$86$	0	0
$87$	−13.3261	−1.42870
$88$	0	0
$89$	−13.5032	−1.43134	−0.715668	−	0.698441i	$-0.753877\pi$
$89$	−13.5032	−1.43134	−0.715668	+	0.698441i	$0.753877\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−27.3172	−2.83266
$94$	0	0
$95$	1.18742	0.121827
$96$	0	0
$97$	17.2558	1.75206	0.876029	−	0.482259i	$-0.160184\pi$
$97$	17.2558	1.75206	0.876029	+	0.482259i	$0.160184\pi$
$98$	0	0
$99$	15.0697	1.51456
$100$	0	0
$101$	11.1103	1.10552	0.552761	−	0.833340i	$-0.313574\pi$
$101$	11.1103	1.10552	0.552761	+	0.833340i	$0.313574\pi$
$102$	0	0
$103$	−19.9822	−1.96890	−0.984451	−	0.175657i	$-0.943795\pi$
$103$	−19.9822	−1.96890	−0.984451	+	0.175657i	$0.943795\pi$
$104$	0	0
$105$	−7.11926	−0.694768
$106$	0	0
$107$	3.04201	0.294082	0.147041	−	0.989130i	$-0.453025\pi$
$107$	3.04201	0.294082	0.147041	+	0.989130i	$0.453025\pi$
$108$	0	0
$109$	−20.6066	−1.97375	−0.986877	−	0.161476i	$-0.948375\pi$
$109$	−20.6066	−1.97375	−0.986877	+	0.161476i	$0.948375\pi$
$110$	0	0
$111$	13.9911	1.32798
$112$	0	0
$113$	−1.35083	−0.127075	−0.0635376	−	0.997979i	$-0.520238\pi$
$113$	−1.35083	−0.127075	−0.0635376	+	0.997979i	$0.520238\pi$
$114$	0	0
$115$	−8.30668	−0.774602
$116$	0	0
$117$	5.24743	0.485125
$118$	0	0
$119$	−0.624408	−0.0572394
$120$	0	0
$121$	−2.75257	−0.250234
$122$	0	0
$123$	3.70305	0.333892
$124$	0	0
$125$	−9.55550	−0.854670
$126$	0	0
$127$	−9.81878	−0.871276	−0.435638	−	0.900122i	$-0.643477\pi$
$127$	−9.81878	−0.871276	−0.435638	+	0.900122i	$0.643477\pi$
$128$	0	0
$129$	−5.32606	−0.468934
$130$	0	0
$131$	16.6672	1.45622	0.728108	−	0.685462i	$-0.240400\pi$
$131$	16.6672	1.45622	0.728108	+	0.685462i	$0.240400\pi$
$132$	0	0
$133$	−0.478994	−0.0415341
$134$	0	0
$135$	16.0000	1.37706
$136$	0	0
$137$	1.08755	0.0929154	0.0464577	−	0.998920i	$-0.485207\pi$
$137$	1.08755	0.0929154	0.0464577	+	0.998920i	$0.485207\pi$
$138$	0	0
$139$	17.3578	1.47227	0.736134	−	0.676836i	$-0.236649\pi$
$139$	17.3578	1.47227	0.736134	+	0.676836i	$0.236649\pi$
$140$	0	0
$141$	−27.3172	−2.30052
$142$	0	0
$143$	2.87183	0.240155
$144$	0	0
$145$	−11.5032	−0.955288
$146$	0	0
$147$	2.87183	0.236865
$148$	0	0
$149$	−6.83734	−0.560137	−0.280068	−	0.959980i	$-0.590357\pi$
$149$	−6.83734	−0.560137	−0.280068	+	0.959980i	$0.590357\pi$
$150$	0	0
$151$	7.11926	0.579357	0.289679	−	0.957124i	$-0.406452\pi$
$151$	7.11926	0.579357	0.289679	+	0.957124i	$0.406452\pi$
$152$	0	0
$153$	3.27653	0.264892
$154$	0	0
$155$	−23.5804	−1.89403
$156$	0	0
$157$	14.3247	1.14323	0.571617	−	0.820521i	$-0.306316\pi$
$157$	14.3247	1.14323	0.571617	+	0.820521i	$0.306316\pi$
$158$	0	0
$159$	24.8134	1.96783
$160$	0	0
$161$	3.35083	0.264082
$162$	0	0
$163$	−8.32254	−0.651872	−0.325936	−	0.945392i	$-0.605679\pi$
$163$	−8.32254	−0.651872	−0.325936	+	0.945392i	$0.605679\pi$
$164$	0	0
$165$	20.4453	1.59167
$166$	0	0
$167$	−13.9318	−1.07808	−0.539039	−	0.842281i	$-0.681212\pi$
$167$	−13.9318	−1.07808	−0.539039	+	0.842281i	$0.681212\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.51349	0.192211
$172$	0	0
$173$	−4.79672	−0.364688	−0.182344	−	0.983235i	$-0.558368\pi$
$173$	−4.79672	−0.364688	−0.182344	+	0.983235i	$0.558368\pi$
$174$	0	0
$175$	−1.14541	−0.0865851
$176$	0	0
$177$	−30.7334	−2.31006
$178$	0	0
$179$	−4.84707	−0.362287	−0.181143	−	0.983457i	$-0.557980\pi$
$179$	−4.84707	−0.362287	−0.181143	+	0.983457i	$0.557980\pi$
$180$	0	0
$181$	−0.376983	−0.0280209	−0.0140105	−	0.999902i	$-0.504460\pi$
$181$	−0.376983	−0.0280209	−0.0140105	+	0.999902i	$0.504460\pi$
$182$	0	0
$183$	35.7665	2.64393
$184$	0	0
$185$	12.0772	0.887937
$186$	0	0
$187$	1.79320	0.131131
$188$	0	0
$189$	−6.45423	−0.469476
$190$	0	0
$191$	−0.0317117	−0.00229458	−0.00114729	−	0.999999i	$-0.500365\pi$
$191$	−0.0317117	−0.00229458	−0.00114729	+	0.999999i	$0.500365\pi$
$192$	0	0
$193$	4.25633	0.306378	0.153189	−	0.988197i	$-0.451046\pi$
$193$	4.25633	0.306378	0.153189	+	0.988197i	$0.451046\pi$
$194$	0	0
$195$	7.11926	0.509821
$196$	0	0
$197$	−6.88288	−0.490385	−0.245192	−	0.969474i	$-0.578851\pi$
$197$	−6.88288	−0.490385	−0.245192	+	0.969474i	$0.578851\pi$
$198$	0	0
$199$	9.64860	0.683971	0.341986	−	0.939705i	$-0.388901\pi$
$199$	9.64860	0.683971	0.341986	+	0.939705i	$0.388901\pi$
$200$	0	0
$201$	28.6927	2.02383
$202$	0	0
$203$	4.64026	0.325683
$204$	0	0
$205$	3.19651	0.223254
$206$	0	0
$207$	−17.5832	−1.22212
$208$	0	0
$209$	1.37559	0.0951517
$210$	0	0
$211$	−16.5562	−1.13978	−0.569889	−	0.821721i	$-0.693014\pi$
$211$	−16.5562	−1.13978	−0.569889	+	0.821721i	$0.693014\pi$
$212$	0	0
$213$	−12.5444	−0.859527
$214$	0	0
$215$	−4.59751	−0.313548
$216$	0	0
$217$	9.51210	0.645723
$218$	0	0
$219$	−7.82982	−0.529090
$220$	0	0
$221$	0.624408	0.0420022
$222$	0	0
$223$	−25.3936	−1.70048	−0.850240	−	0.526395i	$-0.823543\pi$
$223$	−25.3936	−1.70048	−0.850240	+	0.526395i	$0.823543\pi$
$224$	0	0
$225$	6.01047	0.400698
$226$	0	0
$227$	−9.24882	−0.613865	−0.306933	−	0.951731i	$-0.599303\pi$
$227$	−9.24882	−0.613865	−0.306933	+	0.951731i	$0.599303\pi$
$228$	0	0
$229$	15.4528	1.02115	0.510576	−	0.859833i	$-0.329432\pi$
$229$	15.4528	1.02115	0.510576	+	0.859833i	$0.329432\pi$
$230$	0	0
$231$	−8.24743	−0.542641
$232$	0	0
$233$	7.14402	0.468020	0.234010	−	0.972234i	$-0.424815\pi$
$233$	7.14402	0.468020	0.234010	+	0.972234i	$0.424815\pi$
$234$	0	0
$235$	−23.5804	−1.53822
$236$	0	0
$237$	−0.392019	−0.0254644
$238$	0	0
$239$	20.1723	1.30484	0.652419	−	0.757858i	$-0.273754\pi$
$239$	20.1723	1.30484	0.652419	+	0.757858i	$0.273754\pi$
$240$	0	0
$241$	−12.5452	−0.808107	−0.404054	−	0.914735i	$-0.632399\pi$
$241$	−12.5452	−0.808107	−0.404054	+	0.914735i	$0.632399\pi$
$242$	0	0
$243$	−11.3411	−0.727532
$244$	0	0
$245$	2.47899	0.158377
$246$	0	0
$247$	0.478994	0.0304777
$248$	0	0
$249$	40.8451	2.58845
$250$	0	0
$251$	31.3172	1.97672	0.988361	−	0.152129i	$-0.0486130\pi$
$251$	31.3172	1.97672	0.988361	+	0.152129i	$0.0486130\pi$
$252$	0	0
$253$	−9.62302	−0.604994
$254$	0	0
$255$	4.44532	0.278377
$256$	0	0
$257$	8.65612	0.539954	0.269977	−	0.962867i	$-0.412984\pi$
$257$	8.65612	0.539954	0.269977	+	0.962867i	$0.412984\pi$
$258$	0	0
$259$	−4.87183	−0.302721
$260$	0	0
$261$	−24.3494	−1.50719
$262$	0	0
$263$	30.4006	1.87458	0.937291	−	0.348548i	$-0.113325\pi$
$263$	30.4006	1.87458	0.937291	+	0.348548i	$0.113325\pi$
$264$	0	0
$265$	21.4192	1.31577
$266$	0	0
$267$	−38.7789	−2.37323
$268$	0	0
$269$	−9.90707	−0.604045	−0.302022	−	0.953301i	$-0.597662\pi$
$269$	−9.90707	−0.604045	−0.302022	+	0.953301i	$0.597662\pi$
$270$	0	0
$271$	13.5280	0.821765	0.410882	−	0.911688i	$-0.365221\pi$
$271$	13.5280	0.821765	0.410882	+	0.911688i	$0.365221\pi$
$272$	0	0
$273$	−2.87183	−0.173811
$274$	0	0
$275$	3.28944	0.198360
$276$	0	0
$277$	18.7065	1.12396	0.561981	−	0.827150i	$-0.310039\pi$
$277$	18.7065	1.12396	0.561981	+	0.827150i	$0.310039\pi$
$278$	0	0
$279$	−49.9140	−2.98827
$280$	0	0
$281$	2.19576	0.130988	0.0654941	−	0.997853i	$-0.479138\pi$
$281$	2.19576	0.130988	0.0654941	+	0.997853i	$0.479138\pi$
$282$	0	0
$283$	5.70026	0.338846	0.169423	−	0.985543i	$-0.445810\pi$
$283$	5.70026	0.338846	0.169423	+	0.985543i	$0.445810\pi$
$284$	0	0
$285$	3.41009	0.201996
$286$	0	0
$287$	−1.28944	−0.0761130
$288$	0	0
$289$	−16.6101	−0.977066
$290$	0	0
$291$	49.5557	2.90500
$292$	0	0
$293$	−24.8898	−1.45408	−0.727039	−	0.686596i	$-0.759104\pi$
$293$	−24.8898	−1.45408	−0.727039	+	0.686596i	$0.759104\pi$
$294$	0	0
$295$	−26.5293	−1.54460
$296$	0	0
$297$	18.5355	1.07554
$298$	0	0
$299$	−3.35083	−0.193783
$300$	0	0
$301$	1.85459	0.106897
$302$	0	0
$303$	31.9071	1.83301
$304$	0	0
$305$	30.8740	1.76784
$306$	0	0
$307$	−29.2647	−1.67022	−0.835112	−	0.550081i	$-0.814597\pi$
$307$	−29.2647	−1.67022	−0.835112	+	0.550081i	$0.814597\pi$
$308$	0	0
$309$	−57.3855	−3.26455
$310$	0	0
$311$	14.4343	0.818493	0.409246	−	0.912424i	$-0.365792\pi$
$311$	14.4343	0.818493	0.409246	+	0.912424i	$0.365792\pi$
$312$	0	0
$313$	1.08755	0.0614718	0.0307359	−	0.999528i	$-0.490215\pi$
$313$	1.08755	0.0614718	0.0307359	+	0.999528i	$0.490215\pi$
$314$	0	0
$315$	−13.0083	−0.732937
$316$	0	0
$317$	−29.7348	−1.67007	−0.835035	−	0.550197i	$-0.814553\pi$
$317$	−29.7348	−1.67007	−0.835035	+	0.550197i	$0.814553\pi$
$318$	0	0
$319$	−13.3261	−0.746116
$320$	0	0
$321$	8.73615	0.487604
$322$	0	0
$323$	0.299088	0.0166417
$324$	0	0
$325$	1.14541	0.0635361
$326$	0	0
$327$	−59.1787	−3.27259
$328$	0	0
$329$	9.51210	0.524419
$330$	0	0
$331$	14.3136	0.786748	0.393374	−	0.919378i	$-0.371308\pi$
$331$	14.3136	0.786748	0.393374	+	0.919378i	$0.371308\pi$
$332$	0	0
$333$	25.5646	1.40093
$334$	0	0
$335$	24.7679	1.35321
$336$	0	0
$337$	−29.8279	−1.62483	−0.812414	−	0.583082i	$-0.801847\pi$
$337$	−29.8279	−1.62483	−0.812414	+	0.583082i	$0.801847\pi$
$338$	0	0
$339$	−3.87935	−0.210697
$340$	0	0
$341$	−27.3172	−1.47931
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−23.8554	−1.28433
$346$	0	0
$347$	25.4033	1.36372	0.681861	−	0.731482i	$-0.261171\pi$
$347$	25.4033	1.36372	0.681861	+	0.731482i	$0.261171\pi$
$348$	0	0
$349$	−28.9905	−1.55183	−0.775913	−	0.630839i	$-0.782711\pi$
$349$	−28.9905	−1.55183	−0.775913	+	0.630839i	$0.782711\pi$
$350$	0	0
$351$	6.45423	0.344501
$352$	0	0
$353$	−34.9658	−1.86104	−0.930520	−	0.366242i	$-0.880644\pi$
$353$	−34.9658	−1.86104	−0.930520	+	0.366242i	$0.880644\pi$
$354$	0	0
$355$	−10.8284	−0.574713
$356$	0	0
$357$	−1.79320	−0.0949060
$358$	0	0
$359$	−7.11926	−0.375740	−0.187870	−	0.982194i	$-0.560158\pi$
$359$	−7.11926	−0.375740	−0.187870	+	0.982194i	$0.560158\pi$
$360$	0	0
$361$	−18.7706	−0.987924
$362$	0	0
$363$	−7.90494	−0.414902
$364$	0	0
$365$	−6.75878	−0.353771
$366$	0	0
$367$	−14.1296	−0.737557	−0.368778	−	0.929517i	$-0.620224\pi$
$367$	−14.1296	−0.737557	−0.368778	+	0.929517i	$0.620224\pi$
$368$	0	0
$369$	6.76622	0.352235
$370$	0	0
$371$	−8.64026	−0.448580
$372$	0	0
$373$	27.8664	1.44287	0.721435	−	0.692482i	$-0.243483\pi$
$373$	27.8664	1.44287	0.721435	+	0.692482i	$0.243483\pi$
$374$	0	0
$375$	−27.4418	−1.41709
$376$	0	0
$377$	−4.64026	−0.238986
$378$	0	0
$379$	−13.8387	−0.710848	−0.355424	−	0.934705i	$-0.615663\pi$
$379$	−13.8387	−0.710848	−0.355424	+	0.934705i	$0.615663\pi$
$380$	0	0
$381$	−28.1979	−1.44462
$382$	0	0
$383$	24.1801	1.23554	0.617772	−	0.786357i	$-0.288035\pi$
$383$	24.1801	1.23554	0.617772	+	0.786357i	$0.288035\pi$
$384$	0	0
$385$	−7.11926	−0.362831
$386$	0	0
$387$	−9.73180	−0.494695
$388$	0	0
$389$	17.2143	0.872801	0.436400	−	0.899753i	$-0.356253\pi$
$389$	17.2143	0.872801	0.436400	+	0.899753i	$0.356253\pi$
$390$	0	0
$391$	−2.09228	−0.105811
$392$	0	0
$393$	47.8653	2.41449
$394$	0	0
$395$	−0.338395	−0.0170265
$396$	0	0
$397$	−6.60178	−0.331334	−0.165667	−	0.986182i	$-0.552978\pi$
$397$	−6.60178	−0.331334	−0.165667	+	0.986182i	$0.552978\pi$
$398$	0	0
$399$	−1.37559	−0.0688657
$400$	0	0
$401$	15.1303	0.755571	0.377786	−	0.925893i	$-0.376686\pi$
$401$	15.1303	0.755571	0.377786	+	0.925893i	$0.376686\pi$
$402$	0	0
$403$	−9.51210	−0.473831
$404$	0	0
$405$	6.92432	0.344072
$406$	0	0
$407$	13.9911	0.693513
$408$	0	0
$409$	21.4025	1.05829	0.529143	−	0.848533i	$-0.322514\pi$
$409$	21.4025	1.05829	0.529143	+	0.848533i	$0.322514\pi$
$410$	0	0
$411$	3.12325	0.154059
$412$	0	0
$413$	10.7017	0.526594
$414$	0	0
$415$	35.2579	1.73074
$416$	0	0
$417$	49.8486	2.44110
$418$	0	0
$419$	−23.4674	−1.14646	−0.573228	−	0.819396i	$-0.694309\pi$
$419$	−23.4674	−1.14646	−0.573228	+	0.819396i	$0.694309\pi$
$420$	0	0
$421$	12.1869	0.593951	0.296975	−	0.954885i	$-0.404022\pi$
$421$	12.1869	0.593951	0.296975	+	0.954885i	$0.404022\pi$
$422$	0	0
$423$	−49.9140	−2.42690
$424$	0	0
$425$	0.715205	0.0346925
$426$	0	0
$427$	−12.4542	−0.602702
$428$	0	0
$429$	8.24743	0.398189
$430$	0	0
$431$	11.2143	0.540175	0.270087	−	0.962836i	$-0.412947\pi$
$431$	11.2143	0.540175	0.270087	+	0.962836i	$0.412947\pi$
$432$	0	0
$433$	−21.1965	−1.01864	−0.509320	−	0.860577i	$-0.670103\pi$
$433$	−21.1965	−1.01864	−0.509320	+	0.860577i	$0.670103\pi$
$434$	0	0
$435$	−33.0352	−1.58392
$436$	0	0
$437$	−1.60503	−0.0767789
$438$	0	0
$439$	−7.88148	−0.376163	−0.188081	−	0.982153i	$-0.560227\pi$
$439$	−7.88148	−0.376163	−0.188081	+	0.982153i	$0.560227\pi$
$440$	0	0
$441$	5.24743	0.249877
$442$	0	0
$443$	−9.97737	−0.474039	−0.237020	−	0.971505i	$-0.576171\pi$
$443$	−9.97737	−0.474039	−0.237020	+	0.971505i	$0.576171\pi$
$444$	0	0
$445$	−33.4743	−1.58684
$446$	0	0
$447$	−19.6357	−0.928737
$448$	0	0
$449$	29.8637	1.40935	0.704677	−	0.709528i	$-0.251092\pi$
$449$	29.8637	1.40935	0.704677	+	0.709528i	$0.251092\pi$
$450$	0	0
$451$	3.70305	0.174370
$452$	0	0
$453$	20.4453	0.960605
$454$	0	0
$455$	−2.47899	−0.116217
$456$	0	0
$457$	3.00352	0.140499	0.0702495	−	0.997529i	$-0.477620\pi$
$457$	3.00352	0.140499	0.0702495	+	0.997529i	$0.477620\pi$
$458$	0	0
$459$	4.03007	0.188108
$460$	0	0
$461$	10.8174	0.503816	0.251908	−	0.967751i	$-0.418942\pi$
$461$	10.8174	0.503816	0.251908	+	0.967751i	$0.418942\pi$
$462$	0	0
$463$	−0.267376	−0.0124260	−0.00621301	−	0.999981i	$-0.501978\pi$
$463$	−0.267376	−0.0124260	−0.00621301	+	0.999981i	$0.501978\pi$
$464$	0	0
$465$	−67.7191	−3.14040
$466$	0	0
$467$	9.20328	0.425877	0.212939	−	0.977066i	$-0.431697\pi$
$467$	9.20328	0.425877	0.212939	+	0.977066i	$0.431697\pi$
$468$	0	0
$469$	−9.99109	−0.461346
$470$	0	0
$471$	41.1381	1.89554
$472$	0	0
$473$	−5.32606	−0.244893
$474$	0	0
$475$	0.548647	0.0251736
$476$	0	0
$477$	45.3391	2.07594
$478$	0	0
$479$	18.2544	0.834064	0.417032	−	0.908892i	$-0.363070\pi$
$479$	18.2544	0.834064	0.417032	+	0.908892i	$0.363070\pi$
$480$	0	0
$481$	4.87183	0.222136
$482$	0	0
$483$	9.62302	0.437862
$484$	0	0
$485$	42.7769	1.94240
$486$	0	0
$487$	24.4275	1.10692	0.553458	−	0.832877i	$-0.313308\pi$
$487$	24.4275	1.10692	0.553458	+	0.832877i	$0.313308\pi$
$488$	0	0
$489$	−23.9009	−1.08084
$490$	0	0
$491$	−4.76786	−0.215171	−0.107585	−	0.994196i	$-0.534312\pi$
$491$	−4.76786	−0.215171	−0.107585	+	0.994196i	$0.534312\pi$
$492$	0	0
$493$	−2.89742	−0.130493
$494$	0	0
$495$	37.3578	1.67911
$496$	0	0
$497$	4.36807	0.195935
$498$	0	0
$499$	−27.9843	−1.25275	−0.626375	−	0.779522i	$-0.715462\pi$
$499$	−27.9843	−1.25275	−0.626375	+	0.779522i	$0.715462\pi$
$500$	0	0
$501$	−40.0099	−1.78751
$502$	0	0
$503$	20.6989	0.922917	0.461459	−	0.887162i	$-0.347326\pi$
$503$	20.6989	0.922917	0.461459	+	0.887162i	$0.347326\pi$
$504$	0	0
$505$	27.5425	1.22562
$506$	0	0
$507$	2.87183	0.127543
$508$	0	0
$509$	−39.7734	−1.76293	−0.881463	−	0.472253i	$-0.843441\pi$
$509$	−39.7734	−1.76293	−0.881463	+	0.472253i	$0.843441\pi$
$510$	0	0
$511$	2.72642	0.120610
$512$	0	0
$513$	3.09154	0.136495
$514$	0	0
$515$	−49.5357	−2.18280
$516$	0	0
$517$	−27.3172	−1.20141
$518$	0	0
$519$	−13.7754	−0.604672
$520$	0	0
$521$	6.63545	0.290704	0.145352	−	0.989380i	$-0.453568\pi$
$521$	6.63545	0.290704	0.145352	+	0.989380i	$0.453568\pi$
$522$	0	0
$523$	−9.30611	−0.406928	−0.203464	−	0.979082i	$-0.565220\pi$
$523$	−9.30611	−0.406928	−0.203464	+	0.979082i	$0.565220\pi$
$524$	0	0
$525$	−3.28944	−0.143563
$526$	0	0
$527$	−5.93943	−0.258726
$528$	0	0
$529$	−11.7720	−0.511824
$530$	0	0
$531$	−56.1561	−2.43697
$532$	0	0
$533$	1.28944	0.0558517
$534$	0	0
$535$	7.54113	0.326031
$536$	0	0
$537$	−13.9200	−0.600691
$538$	0	0
$539$	2.87183	0.123699
$540$	0	0
$541$	38.0416	1.63554	0.817768	−	0.575548i	$-0.195211\pi$
$541$	38.0416	1.63554	0.817768	+	0.575548i	$0.195211\pi$
$542$	0	0
$543$	−1.08263	−0.0464602
$544$	0	0
$545$	−51.0836	−2.18818
$546$	0	0
$547$	−38.4227	−1.64284	−0.821418	−	0.570327i	$-0.806817\pi$
$547$	−38.4227	−1.64284	−0.821418	+	0.570327i	$0.806817\pi$
$548$	0	0
$549$	65.3526	2.78918
$550$	0	0
$551$	−2.22266	−0.0946885
$552$	0	0
$553$	0.136505	0.00580477
$554$	0	0
$555$	34.6838	1.47225
$556$	0	0
$557$	13.7526	0.582715	0.291358	−	0.956614i	$-0.405893\pi$
$557$	13.7526	0.582715	0.291358	+	0.956614i	$0.405893\pi$
$558$	0	0
$559$	−1.85459	−0.0784407
$560$	0	0
$561$	5.14976	0.217423
$562$	0	0
$563$	36.5204	1.53915	0.769576	−	0.638555i	$-0.220467\pi$
$563$	36.5204	1.53915	0.769576	+	0.638555i	$0.220467\pi$
$564$	0	0
$565$	−3.34869	−0.140881
$566$	0	0
$567$	−2.79320	−0.117303
$568$	0	0
$569$	7.72994	0.324056	0.162028	−	0.986786i	$-0.448196\pi$
$569$	7.72994	0.324056	0.162028	+	0.986786i	$0.448196\pi$
$570$	0	0
$571$	11.1212	0.465409	0.232704	−	0.972548i	$-0.425243\pi$
$571$	11.1212	0.465409	0.232704	+	0.972548i	$0.425243\pi$
$572$	0	0
$573$	−0.0910708	−0.00380454
$574$	0	0
$575$	−3.83808	−0.160059
$576$	0	0
$577$	−23.9160	−0.995635	−0.497818	−	0.867282i	$-0.665865\pi$
$577$	−23.9160	−0.995635	−0.497818	+	0.867282i	$0.665865\pi$
$578$	0	0
$579$	12.2235	0.507991
$580$	0	0
$581$	−14.2227	−0.590055
$582$	0	0
$583$	24.8134	1.02767
$584$	0	0
$585$	13.0083	0.537829
$586$	0	0
$587$	−7.52101	−0.310425	−0.155213	−	0.987881i	$-0.549606\pi$
$587$	−7.52101	−0.310425	−0.155213	+	0.987881i	$0.549606\pi$
$588$	0	0
$589$	−4.55624	−0.187737
$590$	0	0
$591$	−19.7665	−0.813084
$592$	0	0
$593$	30.7837	1.26414	0.632068	−	0.774913i	$-0.282206\pi$
$593$	30.7837	1.26414	0.632068	+	0.774913i	$0.282206\pi$
$594$	0	0
$595$	−1.54790	−0.0634579
$596$	0	0
$597$	27.7092	1.13406
$598$	0	0
$599$	−20.6135	−0.842246	−0.421123	−	0.907003i	$-0.638364\pi$
$599$	−20.6135	−0.842246	−0.421123	+	0.907003i	$0.638364\pi$
$600$	0	0
$601$	−10.7582	−0.438837	−0.219419	−	0.975631i	$-0.570416\pi$
$601$	−10.7582	−0.438837	−0.219419	+	0.975631i	$0.570416\pi$
$602$	0	0
$603$	52.4275	2.13501
$604$	0	0
$605$	−6.82362	−0.277420
$606$	0	0
$607$	−16.5404	−0.671354	−0.335677	−	0.941977i	$-0.608965\pi$
$607$	−16.5404	−0.671354	−0.335677	+	0.941977i	$0.608965\pi$
$608$	0	0
$609$	13.3261	0.539999
$610$	0	0
$611$	−9.51210	−0.384818
$612$	0	0
$613$	7.23713	0.292305	0.146152	−	0.989262i	$-0.453311\pi$
$613$	7.23713	0.292305	0.146152	+	0.989262i	$0.453311\pi$
$614$	0	0
$615$	9.17983	0.370166
$616$	0	0
$617$	8.08402	0.325450	0.162725	−	0.986671i	$-0.447972\pi$
$617$	8.08402	0.325450	0.162725	+	0.986671i	$0.447972\pi$
$618$	0	0
$619$	−9.99722	−0.401822	−0.200911	−	0.979609i	$-0.564390\pi$
$619$	−9.99722	−0.401822	−0.200911	+	0.979609i	$0.564390\pi$
$620$	0	0
$621$	−21.6270	−0.867862
$622$	0	0
$623$	13.5032	0.540994
$624$	0	0
$625$	−29.4151	−1.17660
$626$	0	0
$627$	3.95047	0.157767
$628$	0	0
$629$	3.04201	0.121293
$630$	0	0
$631$	−21.1303	−0.841184	−0.420592	−	0.907250i	$-0.638178\pi$
$631$	−21.1303	−0.841184	−0.420592	+	0.907250i	$0.638178\pi$
$632$	0	0
$633$	−47.5468	−1.88981
$634$	0	0
$635$	−24.3407	−0.965931
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−22.9211	−0.906746
$640$	0	0
$641$	6.27710	0.247931	0.123965	−	0.992287i	$-0.460439\pi$
$641$	6.27710	0.247931	0.123965	+	0.992287i	$0.460439\pi$
$642$	0	0
$643$	11.8099	0.465736	0.232868	−	0.972508i	$-0.425189\pi$
$643$	11.8099	0.465736	0.232868	+	0.972508i	$0.425189\pi$
$644$	0	0
$645$	−13.2033	−0.519879
$646$	0	0
$647$	−11.6319	−0.457298	−0.228649	−	0.973509i	$-0.573431\pi$
$647$	−11.6319	−0.457298	−0.228649	+	0.973509i	$0.573431\pi$
$648$	0	0
$649$	−30.7334	−1.20639
$650$	0	0
$651$	27.3172	1.07064
$652$	0	0
$653$	26.9085	1.05301	0.526505	−	0.850172i	$-0.323502\pi$
$653$	26.9085	1.05301	0.526505	+	0.850172i	$0.323502\pi$
$654$	0	0
$655$	41.3178	1.61442
$656$	0	0
$657$	−14.3067	−0.558157
$658$	0	0
$659$	46.6046	1.81546	0.907729	−	0.419556i	$-0.137814\pi$
$659$	46.6046	1.81546	0.907729	+	0.419556i	$0.137814\pi$
$660$	0	0
$661$	31.6605	1.23145	0.615725	−	0.787961i	$-0.288863\pi$
$661$	31.6605	1.23145	0.615725	+	0.787961i	$0.288863\pi$
$662$	0	0
$663$	1.79320	0.0696419
$664$	0	0
$665$	−1.18742	−0.0460463
$666$	0	0
$667$	15.5487	0.602049
$668$	0	0
$669$	−72.9261	−2.81949
$670$	0	0
$671$	35.7665	1.38075
$672$	0	0
$673$	−40.7569	−1.57106	−0.785532	−	0.618821i	$-0.787611\pi$
$673$	−40.7569	−1.57106	−0.785532	+	0.618821i	$0.787611\pi$
$674$	0	0
$675$	7.39276	0.284548
$676$	0	0
$677$	49.1587	1.88932	0.944662	−	0.328046i	$-0.106390\pi$
$677$	49.1587	1.88932	0.944662	+	0.328046i	$0.106390\pi$
$678$	0	0
$679$	−17.2558	−0.662215
$680$	0	0
$681$	−26.5611	−1.01782
$682$	0	0
$683$	27.3406	1.04616	0.523080	−	0.852284i	$-0.324783\pi$
$683$	27.3406	1.04616	0.523080	+	0.852284i	$0.324783\pi$
$684$	0	0
$685$	2.69602	0.103010
$686$	0	0
$687$	44.3780	1.69313
$688$	0	0
$689$	8.64026	0.329168
$690$	0	0
$691$	−21.2964	−0.810153	−0.405076	−	0.914283i	$-0.632755\pi$
$691$	−21.2964	−0.810153	−0.405076	+	0.914283i	$0.632755\pi$
$692$	0	0
$693$	−15.0697	−0.572452
$694$	0	0
$695$	43.0298	1.63221
$696$	0	0
$697$	0.805134	0.0304966
$698$	0	0
$699$	20.5164	0.776003
$700$	0	0
$701$	−30.9795	−1.17008	−0.585039	−	0.811005i	$-0.698921\pi$
$701$	−30.9795	−1.17008	−0.585039	+	0.811005i	$0.698921\pi$
$702$	0	0
$703$	2.33358	0.0880127
$704$	0	0
$705$	−67.7191	−2.55045
$706$	0	0
$707$	−11.1103	−0.417848
$708$	0	0
$709$	38.2256	1.43559	0.717797	−	0.696253i	$-0.245151\pi$
$709$	38.2256	1.43559	0.717797	+	0.696253i	$0.245151\pi$
$710$	0	0
$711$	−0.716298	−0.0268633
$712$	0	0
$713$	31.8734	1.19367
$714$	0	0
$715$	7.11926	0.266245
$716$	0	0
$717$	57.9315	2.16349
$718$	0	0
$719$	13.7892	0.514250	0.257125	−	0.966378i	$-0.417225\pi$
$719$	13.7892	0.514250	0.257125	+	0.966378i	$0.417225\pi$
$720$	0	0
$721$	19.9822	0.744175
$722$	0	0
$723$	−36.0277	−1.33989
$724$	0	0
$725$	−5.31502	−0.197395
$726$	0	0
$727$	47.2477	1.75232	0.876160	−	0.482021i	$-0.160097\pi$
$727$	47.2477	1.75232	0.876160	+	0.482021i	$0.160097\pi$
$728$	0	0
$729$	−40.9493	−1.51664
$730$	0	0
$731$	−1.15802	−0.0428309
$732$	0	0
$733$	31.7595	1.17306	0.586532	−	0.809926i	$-0.300493\pi$
$733$	31.7595	1.17306	0.586532	+	0.809926i	$0.300493\pi$
$734$	0	0
$735$	7.11926	0.262598
$736$	0	0
$737$	28.6927	1.05691
$738$	0	0
$739$	26.3158	0.968041	0.484021	−	0.875057i	$-0.339176\pi$
$739$	26.3158	0.968041	0.484021	+	0.875057i	$0.339176\pi$
$740$	0	0
$741$	1.37559	0.0505336
$742$	0	0
$743$	24.2456	0.889484	0.444742	−	0.895659i	$-0.353295\pi$
$743$	24.2456	0.889484	0.444742	+	0.895659i	$0.353295\pi$
$744$	0	0
$745$	−16.9497	−0.620990
$746$	0	0
$747$	74.6323	2.73065
$748$	0	0
$749$	−3.04201	−0.111153
$750$	0	0
$751$	48.9044	1.78455	0.892273	−	0.451497i	$-0.149110\pi$
$751$	48.9044	1.78455	0.892273	+	0.451497i	$0.149110\pi$
$752$	0	0
$753$	89.9376	3.27751
$754$	0	0
$755$	17.6486	0.642298
$756$	0	0
$757$	23.4647	0.852839	0.426420	−	0.904525i	$-0.359775\pi$
$757$	23.4647	0.852839	0.426420	+	0.904525i	$0.359775\pi$
$758$	0	0
$759$	−27.6357	−1.00311
$760$	0	0
$761$	9.75061	0.353459	0.176730	−	0.984259i	$-0.443448\pi$
$761$	9.75061	0.353459	0.176730	+	0.984259i	$0.443448\pi$
$762$	0	0
$763$	20.6066	0.746009
$764$	0	0
$765$	8.12251	0.293670
$766$	0	0
$767$	−10.7017	−0.386414
$768$	0	0
$769$	14.4534	0.521203	0.260602	−	0.965446i	$-0.416079\pi$
$769$	14.4534	0.521203	0.260602	+	0.965446i	$0.416079\pi$
$770$	0	0
$771$	24.8589	0.895273
$772$	0	0
$773$	5.16479	0.185765	0.0928824	−	0.995677i	$-0.470392\pi$
$773$	5.16479	0.185765	0.0928824	+	0.995677i	$0.470392\pi$
$774$	0	0
$775$	−10.8953	−0.391370
$776$	0	0
$777$	−13.9911	−0.501928
$778$	0	0
$779$	0.617633	0.0221290
$780$	0	0
$781$	−12.5444	−0.448873
$782$	0	0
$783$	−29.9493	−1.07030
$784$	0	0
$785$	35.5108	1.26743
$786$	0	0
$787$	5.17360	0.184419	0.0922095	−	0.995740i	$-0.470607\pi$
$787$	5.17360	0.184419	0.0922095	+	0.995740i	$0.470607\pi$
$788$	0	0
$789$	87.3055	3.10816
$790$	0	0
$791$	1.35083	0.0480299
$792$	0	0
$793$	12.4542	0.442263
$794$	0	0
$795$	61.5123	2.18162
$796$	0	0
$797$	−31.0843	−1.10106	−0.550531	−	0.834815i	$-0.685575\pi$
$797$	−31.0843	−1.10106	−0.550531	+	0.834815i	$0.685575\pi$
$798$	0	0
$799$	−5.93943	−0.210122
$800$	0	0
$801$	−70.8570	−2.50361
$802$	0	0
$803$	−7.82982	−0.276308
$804$	0	0
$805$	8.30668	0.292772
$806$	0	0
$807$	−28.4514	−1.00154
$808$	0	0
$809$	30.0614	1.05690	0.528451	−	0.848964i	$-0.322773\pi$
$809$	30.0614	1.05690	0.528451	+	0.848964i	$0.322773\pi$
$810$	0	0
$811$	−30.8066	−1.08177	−0.540883	−	0.841098i	$-0.681910\pi$
$811$	−30.8066	−1.08177	−0.540883	+	0.841098i	$0.681910\pi$
$812$	0	0
$813$	38.8500	1.36253
$814$	0	0
$815$	−20.6315	−0.722691
$816$	0	0
$817$	−0.888337	−0.0310790
$818$	0	0
$819$	−5.24743	−0.183360
$820$	0	0
$821$	7.48028	0.261064	0.130532	−	0.991444i	$-0.458332\pi$
$821$	7.48028	0.261064	0.130532	+	0.991444i	$0.458332\pi$
$822$	0	0
$823$	5.00139	0.174338	0.0871688	−	0.996194i	$-0.472218\pi$
$823$	5.00139	0.174338	0.0871688	+	0.996194i	$0.472218\pi$
$824$	0	0
$825$	9.44671	0.328892
$826$	0	0
$827$	40.6315	1.41290	0.706448	−	0.707765i	$-0.250296\pi$
$827$	40.6315	1.41290	0.706448	+	0.707765i	$0.250296\pi$
$828$	0	0
$829$	−16.5059	−0.573273	−0.286637	−	0.958039i	$-0.592537\pi$
$829$	−16.5059	−0.573273	−0.286637	+	0.958039i	$0.592537\pi$
$830$	0	0
$831$	53.7219	1.86359
$832$	0	0
$833$	0.624408	0.0216345
$834$	0	0
$835$	−34.5369	−1.19520
$836$	0	0
$837$	−61.3933	−2.12206
$838$	0	0
$839$	−35.2716	−1.21771	−0.608856	−	0.793281i	$-0.708371\pi$
$839$	−35.2716	−1.21771	−0.608856	+	0.793281i	$0.708371\pi$
$840$	0	0
$841$	−7.46795	−0.257516
$842$	0	0
$843$	6.30586	0.217185
$844$	0	0
$845$	2.47899	0.0852800
$846$	0	0
$847$	2.75257	0.0945796
$848$	0	0
$849$	16.3702	0.561824
$850$	0	0
$851$	−16.3247	−0.559603
$852$	0	0
$853$	−37.6399	−1.28876	−0.644382	−	0.764704i	$-0.722885\pi$
$853$	−37.6399	−1.28876	−0.644382	+	0.764704i	$0.722885\pi$
$854$	0	0
$855$	6.23092	0.213093
$856$	0	0
$857$	30.6672	1.04757	0.523785	−	0.851850i	$-0.324519\pi$
$857$	30.6672	1.04757	0.523785	+	0.851850i	$0.324519\pi$
$858$	0	0
$859$	25.1047	0.856562	0.428281	−	0.903646i	$-0.359119\pi$
$859$	25.1047	0.856562	0.428281	+	0.903646i	$0.359119\pi$
$860$	0	0
$861$	−3.70305	−0.126199
$862$	0	0
$863$	0.926277	0.0315308	0.0157654	−	0.999876i	$-0.494982\pi$
$863$	0.926277	0.0315308	0.0157654	+	0.999876i	$0.494982\pi$
$864$	0	0
$865$	−11.8910	−0.404308
$866$	0	0
$867$	−47.7015	−1.62003
$868$	0	0
$869$	−0.392019	−0.0132983
$870$	0	0
$871$	9.99109	0.338535
$872$	0	0
$873$	90.5483	3.06460
$874$	0	0
$875$	9.55550	0.323035
$876$	0	0
$877$	2.27088	0.0766820	0.0383410	−	0.999265i	$-0.487793\pi$
$877$	2.27088	0.0766820	0.0383410	+	0.999265i	$0.487793\pi$
$878$	0	0
$879$	−71.4794	−2.41094
$880$	0	0
$881$	8.58314	0.289173	0.144587	−	0.989492i	$-0.453815\pi$
$881$	8.58314	0.289173	0.144587	+	0.989492i	$0.453815\pi$
$882$	0	0
$883$	37.2627	1.25399	0.626995	−	0.779023i	$-0.284285\pi$
$883$	37.2627	1.25399	0.626995	+	0.779023i	$0.284285\pi$
$884$	0	0
$885$	−76.1878	−2.56103
$886$	0	0
$887$	0.684979	0.0229994	0.0114997	−	0.999934i	$-0.496339\pi$
$887$	0.684979	0.0229994	0.0114997	+	0.999934i	$0.496339\pi$
$888$	0	0
$889$	9.81878	0.329311
$890$	0	0
$891$	8.02159	0.268733
$892$	0	0
$893$	−4.55624	−0.152469
$894$	0	0
$895$	−12.0159	−0.401646
$896$	0	0
$897$	−9.62302	−0.321303
$898$	0	0
$899$	44.1386	1.47211
$900$	0	0
$901$	5.39505	0.179735
$902$	0	0
$903$	5.32606	0.177240
$904$	0	0
$905$	−0.934538	−0.0310651
$906$	0	0
$907$	19.9208	0.661459	0.330730	−	0.943726i	$-0.392705\pi$
$907$	19.9208	0.661459	0.330730	+	0.943726i	$0.392705\pi$
$908$	0	0
$909$	58.3007	1.93371
$910$	0	0
$911$	−6.62245	−0.219411	−0.109706	−	0.993964i	$-0.534991\pi$
$911$	−6.62245	−0.219411	−0.109706	+	0.993964i	$0.534991\pi$
$912$	0	0
$913$	40.8451	1.35178
$914$	0	0
$915$	88.6649	2.93117
$916$	0	0
$917$	−16.6672	−0.550398
$918$	0	0
$919$	−0.175811	−0.00579948	−0.00289974	−	0.999996i	$-0.500923\pi$
$919$	−0.175811	−0.00579948	−0.00289974	+	0.999996i	$0.500923\pi$
$920$	0	0
$921$	−84.0433	−2.76932
$922$	0	0
$923$	−4.36807	−0.143777
$924$	0	0
$925$	5.58026	0.183478
$926$	0	0
$927$	−104.855	−3.44389
$928$	0	0
$929$	−4.45227	−0.146074	−0.0730371	−	0.997329i	$-0.523269\pi$
$929$	−4.45227	−0.146074	−0.0730371	+	0.997329i	$0.523269\pi$
$930$	0	0
$931$	0.478994	0.0156984
$932$	0	0
$933$	41.4528	1.35711
$934$	0	0
$935$	4.44532	0.145378
$936$	0	0
$937$	−36.9334	−1.20656	−0.603281	−	0.797529i	$-0.706140\pi$
$937$	−36.9334	−1.20656	−0.603281	+	0.797529i	$0.706140\pi$
$938$	0	0
$939$	3.12325	0.101923
$940$	0	0
$941$	30.0492	0.979576	0.489788	−	0.871842i	$-0.337074\pi$
$941$	30.0492	0.979576	0.489788	+	0.871842i	$0.337074\pi$
$942$	0	0
$943$	−4.32068	−0.140701
$944$	0	0
$945$	−16.0000	−0.520480
$946$	0	0
$947$	−56.7899	−1.84543	−0.922713	−	0.385489i	$-0.874033\pi$
$947$	−56.7899	−1.84543	−0.922713	+	0.385489i	$0.874033\pi$
$948$	0	0
$949$	−2.72642	−0.0885033
$950$	0	0
$951$	−85.3933	−2.76907
$952$	0	0
$953$	31.9774	1.03585	0.517924	−	0.855426i	$-0.326705\pi$
$953$	31.9774	1.03585	0.517924	+	0.855426i	$0.326705\pi$
$954$	0	0
$955$	−0.0786132	−0.00254386
$956$	0	0
$957$	−38.2702	−1.23710
$958$	0	0
$959$	−1.08755	−0.0351187
$960$	0	0
$961$	59.4800	1.91871
$962$	0	0
$963$	15.9627	0.514392
$964$	0	0
$965$	10.5514	0.339662
$966$	0	0
$967$	−24.3185	−0.782032	−0.391016	−	0.920384i	$-0.627876\pi$
$967$	−24.3185	−0.782032	−0.391016	+	0.920384i	$0.627876\pi$
$968$	0	0
$969$	0.858931	0.0275928
$970$	0	0
$971$	−27.5624	−0.884521	−0.442260	−	0.896887i	$-0.645823\pi$
$971$	−27.5624	−0.884521	−0.442260	+	0.896887i	$0.645823\pi$
$972$	0	0
$973$	−17.3578	−0.556465
$974$	0	0
$975$	3.28944	0.105346
$976$	0	0
$977$	−7.61689	−0.243686	−0.121843	−	0.992549i	$-0.538880\pi$
$977$	−7.61689	−0.243686	−0.121843	+	0.992549i	$0.538880\pi$
$978$	0	0
$979$	−38.7789	−1.23938
$980$	0	0
$981$	−108.132	−3.45237
$982$	0	0
$983$	−10.1909	−0.325041	−0.162520	−	0.986705i	$-0.551962\pi$
$983$	−10.1909	−0.325041	−0.162520	+	0.986705i	$0.551962\pi$
$984$	0	0
$985$	−17.0626	−0.543660
$986$	0	0
$987$	27.3172	0.869515
$988$	0	0
$989$	6.21440	0.197606
$990$	0	0
$991$	46.1968	1.46749	0.733744	−	0.679426i	$-0.237771\pi$
$991$	46.1968	1.46749	0.733744	+	0.679426i	$0.237771\pi$
$992$	0	0
$993$	41.1064	1.30447
$994$	0	0
$995$	23.9188	0.758278
$996$	0	0
$997$	31.3282	0.992174	0.496087	−	0.868273i	$-0.334770\pi$
$997$	31.3282	0.992174	0.496087	+	0.868273i	$0.334770\pi$
$998$	0	0
$999$	31.4439	0.994842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.u.1.4		4
4.3	odd	2		728.2.a.h.1.1	✓	4
8.3	odd	2		5824.2.a.cc.1.4		4
8.5	even	2		5824.2.a.cf.1.1		4
12.11	even	2		6552.2.a.bt.1.2		4
28.27	even	2		5096.2.a.t.1.4		4
52.51	odd	2		9464.2.a.ba.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.h.1.1	✓	4	4.3	odd	2
1456.2.a.u.1.4		4	1.1	even	1	trivial
5096.2.a.t.1.4		4	28.27	even	2
5824.2.a.cc.1.4		4	8.3	odd	2
5824.2.a.cf.1.1		4	8.5	even	2
6552.2.a.bt.1.2		4	12.11	even	2
9464.2.a.ba.1.1		4	52.51	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

\(f(q)\)	\(=\)	\(q+2.87183 q^{3} +2.47899 q^{5} -1.00000 q^{7} +5.24743 q^{9} +O(q^{10})\)	\(q+2.87183 q^{3} +2.47899 q^{5} -1.00000 q^{7} +5.24743 q^{9} +2.87183 q^{11} +1.00000 q^{13} +7.11926 q^{15} +0.624408 q^{17} +0.478994 q^{19} -2.87183 q^{21} -3.35083 q^{23} +1.14541 q^{25} +6.45423 q^{27} -4.64026 q^{29} -9.51210 q^{31} +8.24743 q^{33} -2.47899 q^{35} +4.87183 q^{37} +2.87183 q^{39} +1.28944 q^{41} -1.85459 q^{43} +13.0083 q^{45} -9.51210 q^{47} +1.00000 q^{49} +1.79320 q^{51} +8.64026 q^{53} +7.11926 q^{55} +1.37559 q^{57} -10.7017 q^{59} +12.4542 q^{61} -5.24743 q^{63} +2.47899 q^{65} +9.99109 q^{67} -9.62302 q^{69} -4.36807 q^{71} -2.72642 q^{73} +3.28944 q^{75} -2.87183 q^{77} -0.136505 q^{79} +2.79320 q^{81} +14.2227 q^{83} +1.54790 q^{85} -13.3261 q^{87} -13.5032 q^{89} -1.00000 q^{91} -27.3172 q^{93} +1.18742 q^{95} +17.2558 q^{97} +15.0697 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - 4 q^{7} + 9 q^{9}+O(q^{10}) \) 4 * q - q^3 - 4 * q^7 + 9 * q^9	\( 4 q - q^{3} - 4 q^{7} + 9 q^{9} - q^{11} + 4 q^{13} + 4 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} + 9 q^{23} + 14 q^{25} - 7 q^{27} - 4 q^{29} - 11 q^{31} + 21 q^{33} + 7 q^{37} - q^{39} + 13 q^{41} + 2 q^{43} + 12 q^{45} - 11 q^{47} + 4 q^{49} + 28 q^{51} + 20 q^{53} + 4 q^{55} + 6 q^{57} + 2 q^{59} + 17 q^{61} - 9 q^{63} + 3 q^{67} - 27 q^{69} + 8 q^{71} + 11 q^{73} + 21 q^{75} + q^{77} + 27 q^{79} + 32 q^{81} + 22 q^{83} - 8 q^{85} - 8 q^{87} + 10 q^{89} - 4 q^{91} - 27 q^{93} + 34 q^{95} + 17 q^{97} - 10 q^{99}+O(q^{100}) \) 4 * q - q^3 - 4 * q^7 + 9 * q^9 - q^11 + 4 * q^13 + 4 * q^15 + 2 * q^17 - 8 * q^19 + q^21 + 9 * q^23 + 14 * q^25 - 7 * q^27 - 4 * q^29 - 11 * q^31 + 21 * q^33 + 7 * q^37 - q^39 + 13 * q^41 + 2 * q^43 + 12 * q^45 - 11 * q^47 + 4 * q^49 + 28 * q^51 + 20 * q^53 + 4 * q^55 + 6 * q^57 + 2 * q^59 + 17 * q^61 - 9 * q^63 + 3 * q^67 - 27 * q^69 + 8 * q^71 + 11 * q^73 + 21 * q^75 + q^77 + 27 * q^79 + 32 * q^81 + 22 * q^83 - 8 * q^85 - 8 * q^87 + 10 * q^89 - 4 * q^91 - 27 * q^93 + 34 * q^95 + 17 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

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