LMFDB - Embedded newform 10000.2.a.be.1.2

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:	$8$
Coefficient field:	8.8.6152203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 625)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.0573749$ 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)

$f(q)$	$=$	$q-3.02566 q^{3} -0.369971 q^{7} +6.15465 q^{9} +O(q^{10})$	$q-3.02566 q^{3} -0.369971 q^{7} +6.15465 q^{9} +1.74633 q^{11} +1.11622 q^{13} +5.48800 q^{17} +3.75219 q^{19} +1.11941 q^{21} -7.24619 q^{23} -9.54490 q^{27} +4.19284 q^{29} -0.305684 q^{31} -5.28380 q^{33} -9.21956 q^{37} -3.37731 q^{39} -4.18641 q^{41} -7.17118 q^{43} -0.810273 q^{47} -6.86312 q^{49} -16.6048 q^{51} -3.91508 q^{53} -11.3529 q^{57} +1.85738 q^{59} +9.68874 q^{61} -2.27704 q^{63} +12.4701 q^{67} +21.9245 q^{69} -11.6767 q^{71} +3.43137 q^{73} -0.646091 q^{77} -5.69346 q^{79} +10.4157 q^{81} -7.13371 q^{83} -12.6861 q^{87} +1.52999 q^{89} -0.412970 q^{91} +0.924896 q^{93} +6.39742 q^{97} +10.7480 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9	$ 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9} - q^{11} + 10 q^{13} + 15 q^{17} + 10 q^{19} - 14 q^{21} - 30 q^{23} - 20 q^{27} + 10 q^{29} + 9 q^{31} + 5 q^{33} - 10 q^{37} - 8 q^{39} - 4 q^{41} - 30 q^{47} - 4 q^{49} + 14 q^{51} + 10 q^{53} - 10 q^{57} + 5 q^{59} + 6 q^{61} - 10 q^{67} + 3 q^{69} + 9 q^{71} + 5 q^{77} + 20 q^{79} + 8 q^{81} - 40 q^{83} - 40 q^{87} - 5 q^{89} - 6 q^{91} - 40 q^{93} + 22 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9 - q^11 + 10 * q^13 + 15 * q^17 + 10 * q^19 - 14 * q^21 - 30 * q^23 - 20 * q^27 + 10 * q^29 + 9 * q^31 + 5 * q^33 - 10 * q^37 - 8 * q^39 - 4 * q^41 - 30 * q^47 - 4 * q^49 + 14 * q^51 + 10 * q^53 - 10 * q^57 + 5 * q^59 + 6 * q^61 - 10 * q^67 + 3 * q^69 + 9 * q^71 + 5 * q^77 + 20 * q^79 + 8 * q^81 - 40 * q^83 - 40 * q^87 - 5 * q^89 - 6 * q^91 - 40 * q^93 + 22 * 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
$3$	−3.02566	−1.74687	−0.873434	−	0.486942i	$-0.838112\pi$
$3$	−3.02566	−1.74687	−0.873434	+	0.486942i	$0.838112\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.369971	−0.139836	−0.0699180	−	0.997553i	$-0.522274\pi$
$7$	−0.369971	−0.139836	−0.0699180	+	0.997553i	$0.522274\pi$
$8$	0	0
$9$	6.15465	2.05155
$10$	0	0
$11$	1.74633	0.526538	0.263269	−	0.964722i	$-0.415199\pi$
$11$	1.74633	0.526538	0.263269	+	0.964722i	$0.415199\pi$
$12$	0	0
$13$	1.11622	0.309584	0.154792	−	0.987947i	$-0.450529\pi$
$13$	1.11622	0.309584	0.154792	+	0.987947i	$0.450529\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.48800	1.33103	0.665517	−	0.746382i	$-0.268211\pi$
$17$	5.48800	1.33103	0.665517	+	0.746382i	$0.268211\pi$
$18$	0	0
$19$	3.75219	0.860812	0.430406	−	0.902635i	$-0.358370\pi$
$19$	3.75219	0.860812	0.430406	+	0.902635i	$0.358370\pi$
$20$	0	0
$21$	1.11941	0.244275
$22$	0	0
$23$	−7.24619	−1.51094	−0.755468	−	0.655186i	$-0.772590\pi$
$23$	−7.24619	−1.51094	−0.755468	+	0.655186i	$0.772590\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−9.54490	−1.83692
$28$	0	0
$29$	4.19284	0.778590	0.389295	−	0.921113i	$-0.372719\pi$
$29$	4.19284	0.778590	0.389295	+	0.921113i	$0.372719\pi$
$30$	0	0
$31$	−0.305684	−0.0549024	−0.0274512	−	0.999623i	$-0.508739\pi$
$31$	−0.305684	−0.0549024	−0.0274512	+	0.999623i	$0.508739\pi$
$32$	0	0
$33$	−5.28380	−0.919792
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9.21956	−1.51569	−0.757843	−	0.652436i	$-0.773747\pi$
$37$	−9.21956	−1.51569	−0.757843	+	0.652436i	$0.773747\pi$
$38$	0	0
$39$	−3.37731	−0.540803
$40$	0	0
$41$	−4.18641	−0.653807	−0.326904	−	0.945058i	$-0.606005\pi$
$41$	−4.18641	−0.653807	−0.326904	+	0.945058i	$0.606005\pi$
$42$	0	0
$43$	−7.17118	−1.09359	−0.546797	−	0.837265i	$-0.684153\pi$
$43$	−7.17118	−1.09359	−0.546797	+	0.837265i	$0.684153\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.810273	−0.118190	−0.0590952	−	0.998252i	$-0.518822\pi$
$47$	−0.810273	−0.118190	−0.0590952	+	0.998252i	$0.518822\pi$
$48$	0	0
$49$	−6.86312	−0.980446
$50$	0	0
$51$	−16.6048	−2.32514
$52$	0	0
$53$	−3.91508	−0.537778	−0.268889	−	0.963171i	$-0.586657\pi$
$53$	−3.91508	−0.537778	−0.268889	+	0.963171i	$0.586657\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−11.3529	−1.50372
$58$	0	0
$59$	1.85738	0.241810	0.120905	−	0.992664i	$-0.461420\pi$
$59$	1.85738	0.241810	0.120905	+	0.992664i	$0.461420\pi$
$60$	0	0
$61$	9.68874	1.24052	0.620258	−	0.784398i	$-0.287028\pi$
$61$	9.68874	1.24052	0.620258	+	0.784398i	$0.287028\pi$
$62$	0	0
$63$	−2.27704	−0.286880
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.4701	1.52346	0.761730	−	0.647894i	$-0.224350\pi$
$67$	12.4701	1.52346	0.761730	+	0.647894i	$0.224350\pi$
$68$	0	0
$69$	21.9245	2.63941
$70$	0	0
$71$	−11.6767	−1.38577	−0.692885	−	0.721048i	$-0.743661\pi$
$71$	−11.6767	−1.38577	−0.692885	+	0.721048i	$0.743661\pi$
$72$	0	0
$73$	3.43137	0.401611	0.200806	−	0.979631i	$-0.435644\pi$
$73$	3.43137	0.401611	0.200806	+	0.979631i	$0.435644\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.646091	−0.0736290
$78$	0	0
$79$	−5.69346	−0.640564	−0.320282	−	0.947322i	$-0.603778\pi$
$79$	−5.69346	−0.640564	−0.320282	+	0.947322i	$0.603778\pi$
$80$	0	0
$81$	10.4157	1.15730
$82$	0	0
$83$	−7.13371	−0.783027	−0.391513	−	0.920172i	$-0.628048\pi$
$83$	−7.13371	−0.783027	−0.391513	+	0.920172i	$0.628048\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−12.6861	−1.36009
$88$	0	0
$89$	1.52999	0.162179	0.0810894	−	0.996707i	$-0.474160\pi$
$89$	1.52999	0.162179	0.0810894	+	0.996707i	$0.474160\pi$
$90$	0	0
$91$	−0.412970	−0.0432911
$92$	0	0
$93$	0.924896	0.0959072
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.39742	0.649559	0.324780	−	0.945790i	$-0.394710\pi$
$97$	6.39742	0.649559	0.324780	+	0.945790i	$0.394710\pi$
$98$	0	0
$99$	10.7480	1.08022
$100$	0	0
$101$	−12.2487	−1.21879	−0.609396	−	0.792866i	$-0.708588\pi$
$101$	−12.2487	−1.21879	−0.609396	+	0.792866i	$0.708588\pi$
$102$	0	0
$103$	7.66730	0.755482	0.377741	−	0.925911i	$-0.376701\pi$
$103$	7.66730	0.755482	0.377741	+	0.925911i	$0.376701\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.758003	0.0732789	0.0366394	−	0.999329i	$-0.488335\pi$
$107$	0.758003	0.0732789	0.0366394	+	0.999329i	$0.488335\pi$
$108$	0	0
$109$	−3.26839	−0.313055	−0.156528	−	0.987674i	$-0.550030\pi$
$109$	−3.26839	−0.313055	−0.156528	+	0.987674i	$0.550030\pi$
$110$	0	0
$111$	27.8953	2.64771
$112$	0	0
$113$	0.847957	0.0797691	0.0398846	−	0.999204i	$-0.487301\pi$
$113$	0.847957	0.0797691	0.0398846	+	0.999204i	$0.487301\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.86995	0.635128
$118$	0	0
$119$	−2.03040	−0.186127
$120$	0	0
$121$	−7.95034	−0.722758
$122$	0	0
$123$	12.6667	1.14212
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.3225	−1.27092	−0.635459	−	0.772135i	$-0.719189\pi$
$127$	−14.3225	−1.27092	−0.635459	+	0.772135i	$0.719189\pi$
$128$	0	0
$129$	21.6976	1.91036
$130$	0	0
$131$	16.6266	1.45267	0.726334	−	0.687342i	$-0.241222\pi$
$131$	16.6266	1.45267	0.726334	+	0.687342i	$0.241222\pi$
$132$	0	0
$133$	−1.38820	−0.120373
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.07189	−0.262449	−0.131224	−	0.991353i	$-0.541891\pi$
$137$	−3.07189	−0.262449	−0.131224	+	0.991353i	$0.541891\pi$
$138$	0	0
$139$	15.7200	1.33335	0.666677	−	0.745347i	$-0.267716\pi$
$139$	15.7200	1.33335	0.666677	+	0.745347i	$0.267716\pi$
$140$	0	0
$141$	2.45161	0.206463
$142$	0	0
$143$	1.94929	0.163008
$144$	0	0
$145$	0	0
$146$	0	0
$147$	20.7655	1.71271
$148$	0	0
$149$	14.8504	1.21660	0.608298	−	0.793709i	$-0.291853\pi$
$149$	14.8504	1.21660	0.608298	+	0.793709i	$0.291853\pi$
$150$	0	0
$151$	0.712013	0.0579428	0.0289714	−	0.999580i	$-0.490777\pi$
$151$	0.712013	0.0579428	0.0289714	+	0.999580i	$0.490777\pi$
$152$	0	0
$153$	33.7767	2.73068
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−22.0704	−1.76141	−0.880704	−	0.473667i	$-0.842930\pi$
$157$	−22.0704	−1.76141	−0.880704	+	0.473667i	$0.842930\pi$
$158$	0	0
$159$	11.8457	0.939428
$160$	0	0
$161$	2.68088	0.211283
$162$	0	0
$163$	13.1619	1.03092	0.515460	−	0.856914i	$-0.327621\pi$
$163$	13.1619	1.03092	0.515460	+	0.856914i	$0.327621\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.4081	−0.805403	−0.402702	−	0.915331i	$-0.631929\pi$
$167$	−10.4081	−0.805403	−0.402702	+	0.915331i	$0.631929\pi$
$168$	0	0
$169$	−11.7540	−0.904158
$170$	0	0
$171$	23.0934	1.76600
$172$	0	0
$173$	−9.19247	−0.698890	−0.349445	−	0.936957i	$-0.613630\pi$
$173$	−9.19247	−0.698890	−0.349445	+	0.936957i	$0.613630\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−5.61981	−0.422411
$178$	0	0
$179$	21.6873	1.62099	0.810494	−	0.585747i	$-0.199199\pi$
$179$	21.6873	1.62099	0.810494	+	0.585747i	$0.199199\pi$
$180$	0	0
$181$	10.7680	0.800378	0.400189	−	0.916433i	$-0.368944\pi$
$181$	10.7680	0.800378	0.400189	+	0.916433i	$0.368944\pi$
$182$	0	0
$183$	−29.3149	−2.16702
$184$	0	0
$185$	0	0
$186$	0	0
$187$	9.58385	0.700840
$188$	0	0
$189$	3.53134	0.256867
$190$	0	0
$191$	3.83941	0.277810	0.138905	−	0.990306i	$-0.455642\pi$
$191$	3.83941	0.277810	0.138905	+	0.990306i	$0.455642\pi$
$192$	0	0
$193$	−10.5334	−0.758208	−0.379104	−	0.925354i	$-0.623768\pi$
$193$	−10.5334	−0.758208	−0.379104	+	0.925354i	$0.623768\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.65302	−0.687749	−0.343875	−	0.939016i	$-0.611740\pi$
$197$	−9.65302	−0.687749	−0.343875	+	0.939016i	$0.611740\pi$
$198$	0	0
$199$	15.8462	1.12331	0.561654	−	0.827372i	$-0.310165\pi$
$199$	15.8462	1.12331	0.561654	+	0.827372i	$0.310165\pi$
$200$	0	0
$201$	−37.7302	−2.66129
$202$	0	0
$203$	−1.55123	−0.108875
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−44.5978	−3.09976
$208$	0	0
$209$	6.55256	0.453250
$210$	0	0
$211$	−7.25106	−0.499184	−0.249592	−	0.968351i	$-0.580296\pi$
$211$	−7.25106	−0.499184	−0.249592	+	0.968351i	$0.580296\pi$
$212$	0	0
$213$	35.3298	2.42076
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0.113094	0.00767733
$218$	0	0
$219$	−10.3822	−0.701562
$220$	0	0
$221$	6.12583	0.412068
$222$	0	0
$223$	−22.8653	−1.53118	−0.765588	−	0.643331i	$-0.777552\pi$
$223$	−22.8653	−1.53118	−0.765588	+	0.643331i	$0.777552\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.665418	0.0441654	0.0220827	−	0.999756i	$-0.492970\pi$
$227$	0.665418	0.0441654	0.0220827	+	0.999756i	$0.492970\pi$
$228$	0	0
$229$	−23.4732	−1.55115	−0.775576	−	0.631254i	$-0.782541\pi$
$229$	−23.4732	−1.55115	−0.775576	+	0.631254i	$0.782541\pi$
$230$	0	0
$231$	1.95486	0.128620
$232$	0	0
$233$	17.8293	1.16803	0.584017	−	0.811742i	$-0.301480\pi$
$233$	17.8293	1.16803	0.584017	+	0.811742i	$0.301480\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	17.2265	1.11898
$238$	0	0
$239$	23.8706	1.54406	0.772030	−	0.635587i	$-0.219242\pi$
$239$	23.8706	1.54406	0.772030	+	0.635587i	$0.219242\pi$
$240$	0	0
$241$	−4.21325	−0.271399	−0.135700	−	0.990750i	$-0.543328\pi$
$241$	−4.21325	−0.271399	−0.135700	+	0.990750i	$0.543328\pi$
$242$	0	0
$243$	−2.87982	−0.184741
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.18828	0.266494
$248$	0	0
$249$	21.5842	1.36784
$250$	0	0
$251$	−19.5741	−1.23551	−0.617755	−	0.786371i	$-0.711958\pi$
$251$	−19.5741	−1.23551	−0.617755	+	0.786371i	$0.711958\pi$
$252$	0	0
$253$	−12.6542	−0.795565
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.4169	−1.14881	−0.574407	−	0.818570i	$-0.694767\pi$
$257$	−18.4169	−1.14881	−0.574407	+	0.818570i	$0.694767\pi$
$258$	0	0
$259$	3.41097	0.211948
$260$	0	0
$261$	25.8054	1.59732
$262$	0	0
$263$	4.32450	0.266660	0.133330	−	0.991072i	$-0.457433\pi$
$263$	4.32450	0.266660	0.133330	+	0.991072i	$0.457433\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.62924	−0.283305
$268$	0	0
$269$	4.73895	0.288939	0.144469	−	0.989509i	$-0.453852\pi$
$269$	4.73895	0.288939	0.144469	+	0.989509i	$0.453852\pi$
$270$	0	0
$271$	−9.96528	−0.605348	−0.302674	−	0.953094i	$-0.597879\pi$
$271$	−9.96528	−0.605348	−0.302674	+	0.953094i	$0.597879\pi$
$272$	0	0
$273$	1.24951	0.0756238
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−17.6092	−1.05804	−0.529018	−	0.848611i	$-0.677440\pi$
$277$	−17.6092	−1.05804	−0.529018	+	0.848611i	$0.677440\pi$
$278$	0	0
$279$	−1.88137	−0.112635
$280$	0	0
$281$	−25.4964	−1.52099	−0.760494	−	0.649345i	$-0.775043\pi$
$281$	−25.4964	−1.52099	−0.760494	+	0.649345i	$0.775043\pi$
$282$	0	0
$283$	16.1004	0.957072	0.478536	−	0.878068i	$-0.341168\pi$
$283$	16.1004	0.957072	0.478536	+	0.878068i	$0.341168\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.54885	0.0914258
$288$	0	0
$289$	13.1181	0.771654
$290$	0	0
$291$	−19.3564	−1.13469
$292$	0	0
$293$	24.9049	1.45496	0.727481	−	0.686128i	$-0.240691\pi$
$293$	24.9049	1.45496	0.727481	+	0.686128i	$0.240691\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−16.6685	−0.967207
$298$	0	0
$299$	−8.08836	−0.467762
$300$	0	0
$301$	2.65313	0.152924
$302$	0	0
$303$	37.0605	2.12907
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−1.74743	−0.0997311	−0.0498655	−	0.998756i	$-0.515879\pi$
$307$	−1.74743	−0.0997311	−0.0498655	+	0.998756i	$0.515879\pi$
$308$	0	0
$309$	−23.1987	−1.31973
$310$	0	0
$311$	−18.3262	−1.03919	−0.519593	−	0.854414i	$-0.673916\pi$
$311$	−18.3262	−1.03919	−0.519593	+	0.854414i	$0.673916\pi$
$312$	0	0
$313$	3.30758	0.186955	0.0934777	−	0.995621i	$-0.470202\pi$
$313$	3.30758	0.186955	0.0934777	+	0.995621i	$0.470202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.999043	−0.0561118	−0.0280559	−	0.999606i	$-0.508932\pi$
$317$	−0.999043	−0.0561118	−0.0280559	+	0.999606i	$0.508932\pi$
$318$	0	0
$319$	7.32207	0.409957
$320$	0	0
$321$	−2.29346	−0.128009
$322$	0	0
$323$	20.5920	1.14577
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.88906	0.546866
$328$	0	0
$329$	0.299778	0.0165273
$330$	0	0
$331$	14.2009	0.780555	0.390277	−	0.920697i	$-0.372379\pi$
$331$	14.2009	0.780555	0.390277	+	0.920697i	$0.372379\pi$
$332$	0	0
$333$	−56.7432	−3.10951
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.1530	0.716489	0.358245	−	0.933628i	$-0.383375\pi$
$337$	13.1530	0.716489	0.358245	+	0.933628i	$0.383375\pi$
$338$	0	0
$339$	−2.56563	−0.139346
$340$	0	0
$341$	−0.533824	−0.0289082
$342$	0	0
$343$	5.12896	0.276938
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.5474	−0.727265	−0.363633	−	0.931542i	$-0.618464\pi$
$347$	−13.5474	−0.727265	−0.363633	+	0.931542i	$0.618464\pi$
$348$	0	0
$349$	−32.0976	−1.71814	−0.859072	−	0.511854i	$-0.828959\pi$
$349$	−32.0976	−1.71814	−0.859072	+	0.511854i	$0.828959\pi$
$350$	0	0
$351$	−10.6542	−0.568681
$352$	0	0
$353$	18.3122	0.974662	0.487331	−	0.873217i	$-0.337971\pi$
$353$	18.3122	0.974662	0.487331	+	0.873217i	$0.337971\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.14332	0.325139
$358$	0	0
$359$	14.4364	0.761927	0.380963	−	0.924590i	$-0.375592\pi$
$359$	14.4364	0.761927	0.380963	+	0.924590i	$0.375592\pi$
$360$	0	0
$361$	−4.92106	−0.259003
$362$	0	0
$363$	24.0551	1.26256
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.4130	−0.543555	−0.271777	−	0.962360i	$-0.587611\pi$
$367$	−10.4130	−0.543555	−0.271777	+	0.962360i	$0.587611\pi$
$368$	0	0
$369$	−25.7659	−1.34132
$370$	0	0
$371$	1.44847	0.0752008
$372$	0	0
$373$	10.0225	0.518944	0.259472	−	0.965751i	$-0.416451\pi$
$373$	10.0225	0.518944	0.259472	+	0.965751i	$0.416451\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.68014	0.241039
$378$	0	0
$379$	−14.2995	−0.734516	−0.367258	−	0.930119i	$-0.619703\pi$
$379$	−14.2995	−0.734516	−0.367258	+	0.930119i	$0.619703\pi$
$380$	0	0
$381$	43.3351	2.22013
$382$	0	0
$383$	−5.03705	−0.257381	−0.128690	−	0.991685i	$-0.541077\pi$
$383$	−5.03705	−0.257381	−0.128690	+	0.991685i	$0.541077\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−44.1361	−2.24356
$388$	0	0
$389$	10.1326	0.513743	0.256872	−	0.966446i	$-0.417308\pi$
$389$	10.1326	0.513743	0.256872	+	0.966446i	$0.417308\pi$
$390$	0	0
$391$	−39.7671	−2.01111
$392$	0	0
$393$	−50.3064	−2.53762
$394$	0	0
$395$	0	0
$396$	0	0
$397$	19.6679	0.987104	0.493552	−	0.869716i	$-0.335698\pi$
$397$	19.6679	0.987104	0.493552	+	0.869716i	$0.335698\pi$
$398$	0	0
$399$	4.20024	0.210275
$400$	0	0
$401$	−23.0931	−1.15321	−0.576606	−	0.817022i	$-0.695623\pi$
$401$	−23.0931	−1.15321	−0.576606	+	0.817022i	$0.695623\pi$
$402$	0	0
$403$	−0.341211	−0.0169969
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−16.1004	−0.798066
$408$	0	0
$409$	38.6338	1.91032	0.955159	−	0.296093i	$-0.0956838\pi$
$409$	38.6338	1.91032	0.955159	+	0.296093i	$0.0956838\pi$
$410$	0	0
$411$	9.29450	0.458464
$412$	0	0
$413$	−0.687177	−0.0338138
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−47.5635	−2.32919
$418$	0	0
$419$	10.7891	0.527082	0.263541	−	0.964648i	$-0.415110\pi$
$419$	10.7891	0.527082	0.263541	+	0.964648i	$0.415110\pi$
$420$	0	0
$421$	32.4550	1.58176	0.790880	−	0.611971i	$-0.209623\pi$
$421$	32.4550	1.58176	0.790880	+	0.611971i	$0.209623\pi$
$422$	0	0
$423$	−4.98694	−0.242473
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−3.58456	−0.173469
$428$	0	0
$429$	−5.89790	−0.284753
$430$	0	0
$431$	−13.7920	−0.664339	−0.332170	−	0.943220i	$-0.607781\pi$
$431$	−13.7920	−0.664339	−0.332170	+	0.943220i	$0.607781\pi$
$432$	0	0
$433$	−21.1120	−1.01458	−0.507290	−	0.861776i	$-0.669353\pi$
$433$	−21.1120	−1.01458	−0.507290	+	0.861776i	$0.669353\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−27.1891	−1.30063
$438$	0	0
$439$	−36.3457	−1.73468	−0.867342	−	0.497713i	$-0.834173\pi$
$439$	−36.3457	−1.73468	−0.867342	+	0.497713i	$0.834173\pi$
$440$	0	0
$441$	−42.2401	−2.01143
$442$	0	0
$443$	−6.38810	−0.303508	−0.151754	−	0.988418i	$-0.548492\pi$
$443$	−6.38810	−0.303508	−0.151754	+	0.988418i	$0.548492\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−44.9324	−2.12523
$448$	0	0
$449$	−35.1628	−1.65943	−0.829717	−	0.558185i	$-0.811498\pi$
$449$	−35.1628	−1.65943	−0.829717	+	0.558185i	$0.811498\pi$
$450$	0	0
$451$	−7.31084	−0.344254
$452$	0	0
$453$	−2.15431	−0.101218
$454$	0	0
$455$	0	0
$456$	0	0
$457$	22.2994	1.04312	0.521561	−	0.853214i	$-0.325350\pi$
$457$	22.2994	1.04312	0.521561	+	0.853214i	$0.325350\pi$
$458$	0	0
$459$	−52.3824	−2.44500
$460$	0	0
$461$	−1.88541	−0.0878122	−0.0439061	−	0.999036i	$-0.513980\pi$
$461$	−1.88541	−0.0878122	−0.0439061	+	0.999036i	$0.513980\pi$
$462$	0	0
$463$	12.9152	0.600218	0.300109	−	0.953905i	$-0.402977\pi$
$463$	12.9152	0.600218	0.300109	+	0.953905i	$0.402977\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.61816	0.121154	0.0605769	−	0.998164i	$-0.480706\pi$
$467$	2.61816	0.121154	0.0605769	+	0.998164i	$0.480706\pi$
$468$	0	0
$469$	−4.61357	−0.213035
$470$	0	0
$471$	66.7776	3.07695
$472$	0	0
$473$	−12.5232	−0.575819
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−24.0960	−1.10328
$478$	0	0
$479$	−6.73316	−0.307646	−0.153823	−	0.988098i	$-0.549159\pi$
$479$	−6.73316	−0.307646	−0.153823	+	0.988098i	$0.549159\pi$
$480$	0	0
$481$	−10.2911	−0.469233
$482$	0	0
$483$	−8.11146	−0.369084
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5.55795	0.251855	0.125927	−	0.992039i	$-0.459809\pi$
$487$	5.55795	0.251855	0.125927	+	0.992039i	$0.459809\pi$
$488$	0	0
$489$	−39.8235	−1.80088
$490$	0	0
$491$	5.55199	0.250558	0.125279	−	0.992122i	$-0.460017\pi$
$491$	5.55199	0.250558	0.125279	+	0.992122i	$0.460017\pi$
$492$	0	0
$493$	23.0103	1.03633
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.32005	0.193781
$498$	0	0
$499$	−19.2580	−0.862107	−0.431054	−	0.902326i	$-0.641858\pi$
$499$	−19.2580	−0.862107	−0.431054	+	0.902326i	$0.641858\pi$
$500$	0	0
$501$	31.4914	1.40693
$502$	0	0
$503$	−31.1565	−1.38920	−0.694600	−	0.719396i	$-0.744419\pi$
$503$	−31.1565	−1.38920	−0.694600	+	0.719396i	$0.744419\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	35.5638	1.57944
$508$	0	0
$509$	−21.2250	−0.940782	−0.470391	−	0.882458i	$-0.655887\pi$
$509$	−21.2250	−0.940782	−0.470391	+	0.882458i	$0.655887\pi$
$510$	0	0
$511$	−1.26951	−0.0561597
$512$	0	0
$513$	−35.8143	−1.58124
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.41500	−0.0622317
$518$	0	0
$519$	27.8133	1.22087
$520$	0	0
$521$	24.0095	1.05188	0.525938	−	0.850523i	$-0.323714\pi$
$521$	24.0095	1.05188	0.525938	+	0.850523i	$0.323714\pi$
$522$	0	0
$523$	−22.8190	−0.997804	−0.498902	−	0.866658i	$-0.666263\pi$
$523$	−22.8190	−0.997804	−0.498902	+	0.866658i	$0.666263\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.67759	−0.0730770
$528$	0	0
$529$	29.5073	1.28293
$530$	0	0
$531$	11.4315	0.496086
$532$	0	0
$533$	−4.67296	−0.202408
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−65.6186	−2.83165
$538$	0	0
$539$	−11.9853	−0.516242
$540$	0	0
$541$	27.5072	1.18263	0.591314	−	0.806442i	$-0.298609\pi$
$541$	27.5072	1.18263	0.591314	+	0.806442i	$0.298609\pi$
$542$	0	0
$543$	−32.5803	−1.39816
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−0.243224	−0.0103995	−0.00519975	−	0.999986i	$-0.501655\pi$
$547$	−0.243224	−0.0103995	−0.00519975	+	0.999986i	$0.501655\pi$
$548$	0	0
$549$	59.6308	2.54498
$550$	0	0
$551$	15.7323	0.670220
$552$	0	0
$553$	2.10642	0.0895739
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.7280	1.17487	0.587436	−	0.809271i	$-0.300138\pi$
$557$	27.7280	1.17487	0.587436	+	0.809271i	$0.300138\pi$
$558$	0	0
$559$	−8.00463	−0.338560
$560$	0	0
$561$	−28.9975	−1.22428
$562$	0	0
$563$	12.4049	0.522805	0.261403	−	0.965230i	$-0.415815\pi$
$563$	12.4049	0.522805	0.261403	+	0.965230i	$0.415815\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−3.85353	−0.161833
$568$	0	0
$569$	−26.6074	−1.11544	−0.557720	−	0.830029i	$-0.688324\pi$
$569$	−26.6074	−1.11544	−0.557720	+	0.830029i	$0.688324\pi$
$570$	0	0
$571$	−12.9219	−0.540765	−0.270382	−	0.962753i	$-0.587150\pi$
$571$	−12.9219	−0.540765	−0.270382	+	0.962753i	$0.587150\pi$
$572$	0	0
$573$	−11.6168	−0.485298
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.3432	−1.17994	−0.589971	−	0.807425i	$-0.700861\pi$
$577$	−28.3432	−1.17994	−0.589971	+	0.807425i	$0.700861\pi$
$578$	0	0
$579$	31.8704	1.32449
$580$	0	0
$581$	2.63927	0.109495
$582$	0	0
$583$	−6.83702	−0.283160
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−8.38885	−0.346245	−0.173123	−	0.984900i	$-0.555386\pi$
$587$	−8.38885	−0.346245	−0.173123	+	0.984900i	$0.555386\pi$
$588$	0	0
$589$	−1.14698	−0.0472606
$590$	0	0
$591$	29.2068	1.20141
$592$	0	0
$593$	−30.9031	−1.26904	−0.634518	−	0.772908i	$-0.718801\pi$
$593$	−30.9031	−1.26904	−0.634518	+	0.772908i	$0.718801\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−47.9454	−1.96227
$598$	0	0
$599$	−32.6384	−1.33357	−0.666784	−	0.745251i	$-0.732329\pi$
$599$	−32.6384	−1.33357	−0.666784	+	0.745251i	$0.732329\pi$
$600$	0	0
$601$	16.9351	0.690796	0.345398	−	0.938456i	$-0.387744\pi$
$601$	16.9351	0.690796	0.345398	+	0.938456i	$0.387744\pi$
$602$	0	0
$603$	76.7488	3.12545
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.3044	1.47355	0.736775	−	0.676138i	$-0.236348\pi$
$607$	36.3044	1.47355	0.736775	+	0.676138i	$0.236348\pi$
$608$	0	0
$609$	4.69350	0.190190
$610$	0	0
$611$	−0.904445	−0.0365899
$612$	0	0
$613$	30.7941	1.24376	0.621881	−	0.783112i	$-0.286369\pi$
$613$	30.7941	1.24376	0.621881	+	0.783112i	$0.286369\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.3238	1.42208	0.711042	−	0.703150i	$-0.248224\pi$
$617$	35.3238	1.42208	0.711042	+	0.703150i	$0.248224\pi$
$618$	0	0
$619$	−3.95098	−0.158803	−0.0794017	−	0.996843i	$-0.525301\pi$
$619$	−3.95098	−0.158803	−0.0794017	+	0.996843i	$0.525301\pi$
$620$	0	0
$621$	69.1642	2.77546
$622$	0	0
$623$	−0.566053	−0.0226784
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−19.8258	−0.791768
$628$	0	0
$629$	−50.5969	−2.01743
$630$	0	0
$631$	−13.5268	−0.538494	−0.269247	−	0.963071i	$-0.586775\pi$
$631$	−13.5268	−0.538494	−0.269247	+	0.963071i	$0.586775\pi$
$632$	0	0
$633$	21.9393	0.872008
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.66077	−0.303531
$638$	0	0
$639$	−71.8660	−2.84298
$640$	0	0
$641$	−36.1269	−1.42693	−0.713464	−	0.700692i	$-0.752875\pi$
$641$	−36.1269	−1.42693	−0.713464	+	0.700692i	$0.752875\pi$
$642$	0	0
$643$	7.35135	0.289909	0.144954	−	0.989438i	$-0.453696\pi$
$643$	7.35135	0.289909	0.144954	+	0.989438i	$0.453696\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−43.7004	−1.71804	−0.859020	−	0.511943i	$-0.828926\pi$
$647$	−43.7004	−1.71804	−0.859020	+	0.511943i	$0.828926\pi$
$648$	0	0
$649$	3.24359	0.127322
$650$	0	0
$651$	−0.342185	−0.0134113
$652$	0	0
$653$	−35.8445	−1.40270	−0.701352	−	0.712815i	$-0.747420\pi$
$653$	−35.8445	−1.40270	−0.701352	+	0.712815i	$0.747420\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	21.1189	0.823925
$658$	0	0
$659$	−0.0683150	−0.00266117	−0.00133059	−	0.999999i	$-0.500424\pi$
$659$	−0.0683150	−0.00266117	−0.00133059	+	0.999999i	$0.500424\pi$
$660$	0	0
$661$	−26.3573	−1.02518	−0.512590	−	0.858634i	$-0.671314\pi$
$661$	−26.3573	−1.02518	−0.512590	+	0.858634i	$0.671314\pi$
$662$	0	0
$663$	−18.5347	−0.719828
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−30.3821	−1.17640
$668$	0	0
$669$	69.1828	2.67476
$670$	0	0
$671$	16.9197	0.653179
$672$	0	0
$673$	−5.07703	−0.195705	−0.0978525	−	0.995201i	$-0.531197\pi$
$673$	−5.07703	−0.195705	−0.0978525	+	0.995201i	$0.531197\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.5508	1.48163	0.740814	−	0.671710i	$-0.234440\pi$
$677$	38.5508	1.48163	0.740814	+	0.671710i	$0.234440\pi$
$678$	0	0
$679$	−2.36686	−0.0908318
$680$	0	0
$681$	−2.01333	−0.0771511
$682$	0	0
$683$	−36.6133	−1.40097	−0.700484	−	0.713668i	$-0.747032\pi$
$683$	−36.6133	−1.40097	−0.700484	+	0.713668i	$0.747032\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	71.0220	2.70966
$688$	0	0
$689$	−4.37010	−0.166488
$690$	0	0
$691$	−29.3768	−1.11755	−0.558774	−	0.829320i	$-0.688728\pi$
$691$	−29.3768	−1.11755	−0.558774	+	0.829320i	$0.688728\pi$
$692$	0	0
$693$	−3.97647	−0.151053
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−22.9750	−0.870240
$698$	0	0
$699$	−53.9454	−2.04040
$700$	0	0
$701$	−0.566147	−0.0213831	−0.0106915	−	0.999943i	$-0.503403\pi$
$701$	−0.566147	−0.0213831	−0.0106915	+	0.999943i	$0.503403\pi$
$702$	0	0
$703$	−34.5936	−1.30472
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.53167	0.170431
$708$	0	0
$709$	0.657257	0.0246838	0.0123419	−	0.999924i	$-0.496071\pi$
$709$	0.657257	0.0246838	0.0123419	+	0.999924i	$0.496071\pi$
$710$	0	0
$711$	−35.0412	−1.31415
$712$	0	0
$713$	2.21504	0.0829540
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−72.2244	−2.69727
$718$	0	0
$719$	−35.1052	−1.30920	−0.654601	−	0.755975i	$-0.727163\pi$
$719$	−35.1052	−1.30920	−0.654601	+	0.755975i	$0.727163\pi$
$720$	0	0
$721$	−2.83668	−0.105644
$722$	0	0
$723$	12.7479	0.474099
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.2180	1.52869	0.764346	−	0.644807i	$-0.223062\pi$
$727$	41.2180	1.52869	0.764346	+	0.644807i	$0.223062\pi$
$728$	0	0
$729$	−22.5338	−0.834587
$730$	0	0
$731$	−39.3554	−1.45561
$732$	0	0
$733$	−40.1082	−1.48143	−0.740715	−	0.671820i	$-0.765513\pi$
$733$	−40.1082	−1.48143	−0.740715	+	0.671820i	$0.765513\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.7768	0.802160
$738$	0	0
$739$	28.5434	1.04999	0.524993	−	0.851106i	$-0.324068\pi$
$739$	28.5434	1.04999	0.524993	+	0.851106i	$0.324068\pi$
$740$	0	0
$741$	−12.6723	−0.465530
$742$	0	0
$743$	−29.6851	−1.08904	−0.544520	−	0.838748i	$-0.683288\pi$
$743$	−29.6851	−1.08904	−0.544520	+	0.838748i	$0.683288\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−43.9055	−1.60642
$748$	0	0
$749$	−0.280439	−0.0102470
$750$	0	0
$751$	−45.2113	−1.64978	−0.824892	−	0.565290i	$-0.808764\pi$
$751$	−45.2113	−1.64978	−0.824892	+	0.565290i	$0.808764\pi$
$752$	0	0
$753$	59.2248	2.15827
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.69813	0.207102	0.103551	−	0.994624i	$-0.466980\pi$
$757$	5.69813	0.207102	0.103551	+	0.994624i	$0.466980\pi$
$758$	0	0
$759$	38.2875	1.38975
$760$	0	0
$761$	−41.6303	−1.50910	−0.754548	−	0.656245i	$-0.772144\pi$
$761$	−41.6303	−1.50910	−0.754548	+	0.656245i	$0.772144\pi$
$762$	0	0
$763$	1.20921	0.0437764
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.07325	0.0748607
$768$	0	0
$769$	−15.9261	−0.574308	−0.287154	−	0.957884i	$-0.592709\pi$
$769$	−15.9261	−0.574308	−0.287154	+	0.957884i	$0.592709\pi$
$770$	0	0
$771$	55.7233	2.00683
$772$	0	0
$773$	−47.8320	−1.72040	−0.860198	−	0.509959i	$-0.829660\pi$
$773$	−47.8320	−1.72040	−0.860198	+	0.509959i	$0.829660\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−10.3205	−0.370245
$778$	0	0
$779$	−15.7082	−0.562805
$780$	0	0
$781$	−20.3914	−0.729661
$782$	0	0
$783$	−40.0202	−1.43021
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−40.5561	−1.44567	−0.722834	−	0.691021i	$-0.757161\pi$
$787$	−40.5561	−1.44567	−0.722834	+	0.691021i	$0.757161\pi$
$788$	0	0
$789$	−13.0845	−0.465821
$790$	0	0
$791$	−0.313720	−0.0111546
$792$	0	0
$793$	10.8148	0.384044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.8316	0.985847	0.492924	−	0.870073i	$-0.335928\pi$
$797$	27.8316	0.985847	0.492924	+	0.870073i	$0.335928\pi$
$798$	0	0
$799$	−4.44678	−0.157316
$800$	0	0
$801$	9.41656	0.332718
$802$	0	0
$803$	5.99230	0.211464
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−14.3385	−0.504738
$808$	0	0
$809$	−8.24706	−0.289951	−0.144976	−	0.989435i	$-0.546310\pi$
$809$	−8.24706	−0.289951	−0.144976	+	0.989435i	$0.546310\pi$
$810$	0	0
$811$	18.3560	0.644567	0.322283	−	0.946643i	$-0.395550\pi$
$811$	18.3560	0.644567	0.322283	+	0.946643i	$0.395550\pi$
$812$	0	0
$813$	30.1516	1.05746
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−26.9076	−0.941379
$818$	0	0
$819$	−2.54169	−0.0888137
$820$	0	0
$821$	−15.8635	−0.553640	−0.276820	−	0.960922i	$-0.589281\pi$
$821$	−15.8635	−0.553640	−0.276820	+	0.960922i	$0.589281\pi$
$822$	0	0
$823$	32.4705	1.13185	0.565925	−	0.824457i	$-0.308519\pi$
$823$	32.4705	1.13185	0.565925	+	0.824457i	$0.308519\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28.7946	−1.00129	−0.500643	−	0.865654i	$-0.666903\pi$
$827$	−28.7946	−1.00129	−0.500643	+	0.865654i	$0.666903\pi$
$828$	0	0
$829$	−8.59224	−0.298421	−0.149210	−	0.988805i	$-0.547673\pi$
$829$	−8.59224	−0.298421	−0.149210	+	0.988805i	$0.547673\pi$
$830$	0	0
$831$	53.2796	1.84825
$832$	0	0
$833$	−37.6648	−1.30501
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.91772	0.100851
$838$	0	0
$839$	47.0541	1.62449	0.812244	−	0.583318i	$-0.198246\pi$
$839$	47.0541	1.62449	0.812244	+	0.583318i	$0.198246\pi$
$840$	0	0
$841$	−11.4201	−0.393797
$842$	0	0
$843$	77.1436	2.65697
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.94140	0.101068
$848$	0	0
$849$	−48.7145	−1.67188
$850$	0	0
$851$	66.8067	2.29011
$852$	0	0
$853$	−18.6891	−0.639904	−0.319952	−	0.947434i	$-0.603667\pi$
$853$	−18.6891	−0.639904	−0.319952	+	0.947434i	$0.603667\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.06228	0.104606	0.0523028	−	0.998631i	$-0.483344\pi$
$857$	3.06228	0.104606	0.0523028	+	0.998631i	$0.483344\pi$
$858$	0	0
$859$	−16.9664	−0.578885	−0.289442	−	0.957195i	$-0.593470\pi$
$859$	−16.9664	−0.578885	−0.289442	+	0.957195i	$0.593470\pi$
$860$	0	0
$861$	−4.68630	−0.159709
$862$	0	0
$863$	21.8969	0.745380	0.372690	−	0.927956i	$-0.378435\pi$
$863$	21.8969	0.745380	0.372690	+	0.927956i	$0.378435\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−39.6910	−1.34798
$868$	0	0
$869$	−9.94264	−0.337281
$870$	0	0
$871$	13.9194	0.471640
$872$	0	0
$873$	39.3738	1.33260
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.8320	1.58141	0.790703	−	0.612200i	$-0.209715\pi$
$877$	46.8320	1.58141	0.790703	+	0.612200i	$0.209715\pi$
$878$	0	0
$879$	−75.3540	−2.54163
$880$	0	0
$881$	0.0281377	0.000947982 0	0.000473991	−	1.00000i	$-0.499849\pi$
$881$	0.0281377	0.000947982 0	0.000473991		1.00000i	$0.499849\pi$
$882$	0	0
$883$	29.2717	0.985070	0.492535	−	0.870293i	$-0.336070\pi$
$883$	29.2717	0.985070	0.492535	+	0.870293i	$0.336070\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−19.8797	−0.667494	−0.333747	−	0.942663i	$-0.608313\pi$
$887$	−19.8797	−0.667494	−0.333747	+	0.942663i	$0.608313\pi$
$888$	0	0
$889$	5.29892	0.177720
$890$	0	0
$891$	18.1893	0.609365
$892$	0	0
$893$	−3.04030	−0.101740
$894$	0	0
$895$	0	0
$896$	0	0
$897$	24.4727	0.817119
$898$	0	0
$899$	−1.28168	−0.0427465
$900$	0	0
$901$	−21.4860	−0.715801
$902$	0	0
$903$	−8.02748	−0.267138
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14.2958	0.474685	0.237343	−	0.971426i	$-0.423724\pi$
$907$	14.2958	0.474685	0.237343	+	0.971426i	$0.423724\pi$
$908$	0	0
$909$	−75.3865	−2.50041
$910$	0	0
$911$	−19.4005	−0.642766	−0.321383	−	0.946949i	$-0.604148\pi$
$911$	−19.4005	−0.642766	−0.321383	+	0.946949i	$0.604148\pi$
$912$	0	0
$913$	−12.4578	−0.412293
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.15135	−0.203135
$918$	0	0
$919$	−4.29914	−0.141816	−0.0709078	−	0.997483i	$-0.522590\pi$
$919$	−4.29914	−0.141816	−0.0709078	+	0.997483i	$0.522590\pi$
$920$	0	0
$921$	5.28714	0.174217
$922$	0	0
$923$	−13.0338	−0.429013
$924$	0	0
$925$	0	0
$926$	0	0
$927$	47.1895	1.54991
$928$	0	0
$929$	43.0317	1.41182	0.705912	−	0.708300i	$-0.250538\pi$
$929$	43.0317	1.41182	0.705912	+	0.708300i	$0.250538\pi$
$930$	0	0
$931$	−25.7517	−0.843979
$932$	0	0
$933$	55.4490	1.81532
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.9808	0.554740	0.277370	−	0.960763i	$-0.410537\pi$
$937$	16.9808	0.554740	0.277370	+	0.960763i	$0.410537\pi$
$938$	0	0
$939$	−10.0076	−0.326586
$940$	0	0
$941$	−18.0448	−0.588242	−0.294121	−	0.955768i	$-0.595027\pi$
$941$	−18.0448	−0.588242	−0.294121	+	0.955768i	$0.595027\pi$
$942$	0	0
$943$	30.3355	0.987860
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−7.20872	−0.234252	−0.117126	−	0.993117i	$-0.537368\pi$
$947$	−7.20872	−0.234252	−0.117126	+	0.993117i	$0.537368\pi$
$948$	0	0
$949$	3.83017	0.124333
$950$	0	0
$951$	3.02277	0.0980200
$952$	0	0
$953$	13.8000	0.447026	0.223513	−	0.974701i	$-0.428247\pi$
$953$	13.8000	0.447026	0.223513	+	0.974701i	$0.428247\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−22.1541	−0.716141
$958$	0	0
$959$	1.13651	0.0366998
$960$	0	0
$961$	−30.9066	−0.996986
$962$	0	0
$963$	4.66524	0.150335
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−14.8956	−0.479010	−0.239505	−	0.970895i	$-0.576985\pi$
$967$	−14.8956	−0.479010	−0.239505	+	0.970895i	$0.576985\pi$
$968$	0	0
$969$	−62.3045	−2.00151
$970$	0	0
$971$	−39.7938	−1.27704	−0.638522	−	0.769604i	$-0.720454\pi$
$971$	−39.7938	−1.27704	−0.638522	+	0.769604i	$0.720454\pi$
$972$	0	0
$973$	−5.81595	−0.186451
$974$	0	0
$975$	0	0
$976$	0	0
$977$	30.9627	0.990584	0.495292	−	0.868726i	$-0.335061\pi$
$977$	30.9627	0.990584	0.495292	+	0.868726i	$0.335061\pi$
$978$	0	0
$979$	2.67187	0.0853933
$980$	0	0
$981$	−20.1158	−0.642248
$982$	0	0
$983$	27.7549	0.885243	0.442621	−	0.896709i	$-0.354049\pi$
$983$	27.7549	0.885243	0.442621	+	0.896709i	$0.354049\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.907027	−0.0288710
$988$	0	0
$989$	51.9637	1.65235
$990$	0	0
$991$	−15.0180	−0.477062	−0.238531	−	0.971135i	$-0.576666\pi$
$991$	−15.0180	−0.477062	−0.238531	+	0.971135i	$0.576666\pi$
$992$	0	0
$993$	−42.9673	−1.36353
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−17.9138	−0.567335	−0.283668	−	0.958923i	$-0.591551\pi$
$997$	−17.9138	−0.567335	−0.283668	+	0.958923i	$0.591551\pi$
$998$	0	0
$999$	87.9999	2.78419

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.be.1.2		8
4.3	odd	2		625.2.a.g.1.1	yes	8
5.4	even	2		10000.2.a.bn.1.7		8
12.11	even	2		5625.2.a.s.1.8		8
20.3	even	4		625.2.b.d.624.13		16
20.7	even	4		625.2.b.d.624.4		16
20.19	odd	2		625.2.a.e.1.8	✓	8
60.59	even	2		5625.2.a.be.1.1		8
100.3	even	20		625.2.e.k.249.7		32
100.11	odd	10		625.2.d.m.501.1		16
100.19	odd	10		625.2.d.p.251.1		16
100.23	even	20		625.2.e.j.124.7		32
100.27	even	20		625.2.e.j.124.2		32
100.31	odd	10		625.2.d.n.251.4		16
100.39	odd	10		625.2.d.q.501.4		16
100.47	even	20		625.2.e.k.249.2		32
100.59	odd	10		625.2.d.q.126.4		16
100.63	even	20		625.2.e.j.499.2		32
100.67	even	20		625.2.e.k.374.7		32
100.71	odd	10		625.2.d.n.376.4		16
100.79	odd	10		625.2.d.p.376.1		16
100.83	even	20		625.2.e.k.374.2		32
100.87	even	20		625.2.e.j.499.7		32
100.91	odd	10		625.2.d.m.126.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.8	✓	8	20.19	odd	2
625.2.a.g.1.1	yes	8	4.3	odd	2
625.2.b.d.624.4		16	20.7	even	4
625.2.b.d.624.13		16	20.3	even	4
625.2.d.m.126.1		16	100.91	odd	10
625.2.d.m.501.1		16	100.11	odd	10
625.2.d.n.251.4		16	100.31	odd	10
625.2.d.n.376.4		16	100.71	odd	10
625.2.d.p.251.1		16	100.19	odd	10
625.2.d.p.376.1		16	100.79	odd	10
625.2.d.q.126.4		16	100.59	odd	10
625.2.d.q.501.4		16	100.39	odd	10
625.2.e.j.124.2		32	100.27	even	20
625.2.e.j.124.7		32	100.23	even	20
625.2.e.j.499.2		32	100.63	even	20
625.2.e.j.499.7		32	100.87	even	20
625.2.e.k.249.2		32	100.47	even	20
625.2.e.k.249.7		32	100.3	even	20
625.2.e.k.374.2		32	100.83	even	20
625.2.e.k.374.7		32	100.67	even	20
5625.2.a.s.1.8		8	12.11	even	2
5625.2.a.be.1.1		8	60.59	even	2
10000.2.a.be.1.2		8	1.1	even	1	trivial
10000.2.a.bn.1.7		8	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)

\(f(q)\)	\(=\)	\(q-3.02566 q^{3} -0.369971 q^{7} +6.15465 q^{9} +O(q^{10})\)	\(q-3.02566 q^{3} -0.369971 q^{7} +6.15465 q^{9} +1.74633 q^{11} +1.11622 q^{13} +5.48800 q^{17} +3.75219 q^{19} +1.11941 q^{21} -7.24619 q^{23} -9.54490 q^{27} +4.19284 q^{29} -0.305684 q^{31} -5.28380 q^{33} -9.21956 q^{37} -3.37731 q^{39} -4.18641 q^{41} -7.17118 q^{43} -0.810273 q^{47} -6.86312 q^{49} -16.6048 q^{51} -3.91508 q^{53} -11.3529 q^{57} +1.85738 q^{59} +9.68874 q^{61} -2.27704 q^{63} +12.4701 q^{67} +21.9245 q^{69} -11.6767 q^{71} +3.43137 q^{73} -0.646091 q^{77} -5.69346 q^{79} +10.4157 q^{81} -7.13371 q^{83} -12.6861 q^{87} +1.52999 q^{89} -0.412970 q^{91} +0.924896 q^{93} +6.39742 q^{97} +10.7480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9	\( 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9} - q^{11} + 10 q^{13} + 15 q^{17} + 10 q^{19} - 14 q^{21} - 30 q^{23} - 20 q^{27} + 10 q^{29} + 9 q^{31} + 5 q^{33} - 10 q^{37} - 8 q^{39} - 4 q^{41} - 30 q^{47} - 4 q^{49} + 14 q^{51} + 10 q^{53} - 10 q^{57} + 5 q^{59} + 6 q^{61} - 10 q^{67} + 3 q^{69} + 9 q^{71} + 5 q^{77} + 20 q^{79} + 8 q^{81} - 40 q^{83} - 40 q^{87} - 5 q^{89} - 6 q^{91} - 40 q^{93} + 22 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9 - q^11 + 10 * q^13 + 15 * q^17 + 10 * q^19 - 14 * q^21 - 30 * q^23 - 20 * q^27 + 10 * q^29 + 9 * q^31 + 5 * q^33 - 10 * q^37 - 8 * q^39 - 4 * q^41 - 30 * q^47 - 4 * q^49 + 14 * q^51 + 10 * q^53 - 10 * q^57 + 5 * q^59 + 6 * q^61 - 10 * q^67 + 3 * q^69 + 9 * q^71 + 5 * q^77 + 20 * q^79 + 8 * q^81 - 40 * q^83 - 40 * q^87 - 5 * q^89 - 6 * q^91 - 40 * q^93 + 22 * q^99

Introduction

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