LMFDB - Embedded newform 10000.2.a.a.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:	$2$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-0.381966 q^{3} -3.00000 q^{7} -2.85410 q^{9} +O(q^{10})$	$q-0.381966 q^{3} -3.00000 q^{7} -2.85410 q^{9} -4.23607 q^{11} -1.00000 q^{13} -1.14590 q^{17} -5.85410 q^{19} +1.14590 q^{21} -1.76393 q^{23} +2.23607 q^{27} -9.47214 q^{29} +0.236068 q^{31} +1.61803 q^{33} +8.32624 q^{37} +0.381966 q^{39} +1.47214 q^{41} -6.23607 q^{43} -11.9443 q^{47} +2.00000 q^{49} +0.437694 q^{51} -10.4721 q^{53} +2.23607 q^{57} +4.47214 q^{59} -8.85410 q^{61} +8.56231 q^{63} -10.2361 q^{67} +0.673762 q^{69} +3.00000 q^{71} -7.70820 q^{73} +12.7082 q^{77} +7.23607 q^{79} +7.70820 q^{81} -4.52786 q^{83} +3.61803 q^{87} +4.47214 q^{89} +3.00000 q^{91} -0.0901699 q^{93} -9.56231 q^{97} +12.0902 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 6 q^{7} + q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 6 * q^7 + q^9	$ 2 q - 3 q^{3} - 6 q^{7} + q^{9} - 4 q^{11} - 2 q^{13} - 9 q^{17} - 5 q^{19} + 9 q^{21} - 8 q^{23} - 10 q^{29} - 4 q^{31} + q^{33} + q^{37} + 3 q^{39} - 6 q^{41} - 8 q^{43} - 6 q^{47} + 4 q^{49} + 21 q^{51} - 12 q^{53} - 11 q^{61} - 3 q^{63} - 16 q^{67} + 17 q^{69} + 6 q^{71} - 2 q^{73} + 12 q^{77} + 10 q^{79} + 2 q^{81} - 18 q^{83} + 5 q^{87} + 6 q^{91} + 11 q^{93} + q^{97} + 13 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 6 * q^7 + q^9 - 4 * q^11 - 2 * q^13 - 9 * q^17 - 5 * q^19 + 9 * q^21 - 8 * q^23 - 10 * q^29 - 4 * q^31 + q^33 + q^37 + 3 * q^39 - 6 * q^41 - 8 * q^43 - 6 * q^47 + 4 * q^49 + 21 * q^51 - 12 * q^53 - 11 * q^61 - 3 * q^63 - 16 * q^67 + 17 * q^69 + 6 * q^71 - 2 * q^73 + 12 * q^77 + 10 * q^79 + 2 * q^81 - 18 * q^83 + 5 * q^87 + 6 * q^91 + 11 * q^93 + q^97 + 13 * 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
$3$	−0.381966	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$3$	−0.381966	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	0	0
$9$	−2.85410	−0.951367
$10$	0	0
$11$	−4.23607	−1.27722	−0.638611	−	0.769529i	$-0.720491\pi$
$11$	−4.23607	−1.27722	−0.638611	+	0.769529i	$0.720491\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.14590	−0.277921	−0.138961	−	0.990298i	$-0.544376\pi$
$17$	−1.14590	−0.277921	−0.138961	+	0.990298i	$0.544376\pi$
$18$	0	0
$19$	−5.85410	−1.34302	−0.671512	−	0.740994i	$-0.734355\pi$
$19$	−5.85410	−1.34302	−0.671512	+	0.740994i	$0.734355\pi$
$20$	0	0
$21$	1.14590	0.250055
$22$	0	0
$23$	−1.76393	−0.367805	−0.183903	−	0.982944i	$-0.558873\pi$
$23$	−1.76393	−0.367805	−0.183903	+	0.982944i	$0.558873\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	−9.47214	−1.75893	−0.879466	−	0.475962i	$-0.842100\pi$
$29$	−9.47214	−1.75893	−0.879466	+	0.475962i	$0.842100\pi$
$30$	0	0
$31$	0.236068	0.0423991	0.0211995	−	0.999775i	$-0.493251\pi$
$31$	0.236068	0.0423991	0.0211995	+	0.999775i	$0.493251\pi$
$32$	0	0
$33$	1.61803	0.281664
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.32624	1.36883	0.684413	−	0.729095i	$-0.260059\pi$
$37$	8.32624	1.36883	0.684413	+	0.729095i	$0.260059\pi$
$38$	0	0
$39$	0.381966	0.0611635
$40$	0	0
$41$	1.47214	0.229909	0.114955	−	0.993371i	$-0.463328\pi$
$41$	1.47214	0.229909	0.114955	+	0.993371i	$0.463328\pi$
$42$	0	0
$43$	−6.23607	−0.950991	−0.475496	−	0.879718i	$-0.657731\pi$
$43$	−6.23607	−0.950991	−0.475496	+	0.879718i	$0.657731\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.9443	−1.74225	−0.871126	−	0.491060i	$-0.836609\pi$
$47$	−11.9443	−1.74225	−0.871126	+	0.491060i	$0.836609\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	0.437694	0.0612894
$52$	0	0
$53$	−10.4721	−1.43846	−0.719229	−	0.694773i	$-0.755505\pi$
$53$	−10.4721	−1.43846	−0.719229	+	0.694773i	$0.755505\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.23607	0.296174
$58$	0	0
$59$	4.47214	0.582223	0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214	0.582223	0.291111	+	0.956689i	$0.405975\pi$
$60$	0	0
$61$	−8.85410	−1.13365	−0.566826	−	0.823838i	$-0.691829\pi$
$61$	−8.85410	−1.13365	−0.566826	+	0.823838i	$0.691829\pi$
$62$	0	0
$63$	8.56231	1.07875
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.2361	−1.25053	−0.625267	−	0.780411i	$-0.715010\pi$
$67$	−10.2361	−1.25053	−0.625267	+	0.780411i	$0.715010\pi$
$68$	0	0
$69$	0.673762	0.0811114
$70$	0	0
$71$	3.00000	0.356034	0.178017	−	0.984027i	$-0.443032\pi$
$71$	3.00000	0.356034	0.178017	+	0.984027i	$0.443032\pi$
$72$	0	0
$73$	−7.70820	−0.902177	−0.451089	−	0.892479i	$-0.648964\pi$
$73$	−7.70820	−0.902177	−0.451089	+	0.892479i	$0.648964\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	12.7082	1.44823
$78$	0	0
$79$	7.23607	0.814121	0.407061	−	0.913401i	$-0.366554\pi$
$79$	7.23607	0.814121	0.407061	+	0.913401i	$0.366554\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	0	0
$83$	−4.52786	−0.496998	−0.248499	−	0.968632i	$-0.579937\pi$
$83$	−4.52786	−0.496998	−0.248499	+	0.968632i	$0.579937\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.61803	0.387894
$88$	0	0
$89$	4.47214	0.474045	0.237023	−	0.971504i	$-0.423828\pi$
$89$	4.47214	0.474045	0.237023	+	0.971504i	$0.423828\pi$
$90$	0	0
$91$	3.00000	0.314485
$92$	0	0
$93$	−0.0901699	−0.00935019
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.56231	−0.970905	−0.485453	−	0.874263i	$-0.661345\pi$
$97$	−9.56231	−0.970905	−0.485453	+	0.874263i	$0.661345\pi$
$98$	0	0
$99$	12.0902	1.21511
$100$	0	0
$101$	0.618034	0.0614967	0.0307483	−	0.999527i	$-0.490211\pi$
$101$	0.618034	0.0614967	0.0307483	+	0.999527i	$0.490211\pi$
$102$	0	0
$103$	−13.1459	−1.29530	−0.647652	−	0.761936i	$-0.724249\pi$
$103$	−13.1459	−1.29530	−0.647652	+	0.761936i	$0.724249\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.09017	−0.105391	−0.0526954	−	0.998611i	$-0.516781\pi$
$107$	−1.09017	−0.105391	−0.0526954	+	0.998611i	$0.516781\pi$
$108$	0	0
$109$	−15.0000	−1.43674	−0.718370	−	0.695662i	$-0.755111\pi$
$109$	−15.0000	−1.43674	−0.718370	+	0.695662i	$0.755111\pi$
$110$	0	0
$111$	−3.18034	−0.301865
$112$	0	0
$113$	−3.76393	−0.354081	−0.177040	−	0.984204i	$-0.556652\pi$
$113$	−3.76393	−0.354081	−0.177040	+	0.984204i	$0.556652\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.85410	0.263862
$118$	0	0
$119$	3.43769	0.315133
$120$	0	0
$121$	6.94427	0.631297
$122$	0	0
$123$	−0.562306	−0.0507014
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−3.52786	−0.313047	−0.156524	−	0.987674i	$-0.550029\pi$
$127$	−3.52786	−0.313047	−0.156524	+	0.987674i	$0.550029\pi$
$128$	0	0
$129$	2.38197	0.209720
$130$	0	0
$131$	16.4164	1.43431	0.717154	−	0.696915i	$-0.245444\pi$
$131$	16.4164	1.43431	0.717154	+	0.696915i	$0.245444\pi$
$132$	0	0
$133$	17.5623	1.52285
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.09017	−0.434883	−0.217441	−	0.976073i	$-0.569771\pi$
$137$	−5.09017	−0.434883	−0.217441	+	0.976073i	$0.569771\pi$
$138$	0	0
$139$	5.32624	0.451766	0.225883	−	0.974154i	$-0.427473\pi$
$139$	5.32624	0.451766	0.225883	+	0.974154i	$0.427473\pi$
$140$	0	0
$141$	4.56231	0.384215
$142$	0	0
$143$	4.23607	0.354238
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.763932	−0.0630081
$148$	0	0
$149$	2.23607	0.183186	0.0915929	−	0.995797i	$-0.470804\pi$
$149$	2.23607	0.183186	0.0915929	+	0.995797i	$0.470804\pi$
$150$	0	0
$151$	9.70820	0.790042	0.395021	−	0.918672i	$-0.370737\pi$
$151$	9.70820	0.790042	0.395021	+	0.918672i	$0.370737\pi$
$152$	0	0
$153$	3.27051	0.264405
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.56231	−0.364112	−0.182056	−	0.983288i	$-0.558275\pi$
$157$	−4.56231	−0.364112	−0.182056	+	0.983288i	$0.558275\pi$
$158$	0	0
$159$	4.00000	0.317221
$160$	0	0
$161$	5.29180	0.417052
$162$	0	0
$163$	1.85410	0.145224	0.0726122	−	0.997360i	$-0.476866\pi$
$163$	1.85410	0.145224	0.0726122	+	0.997360i	$0.476866\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.3820	1.03553	0.517764	−	0.855524i	$-0.326765\pi$
$167$	13.3820	1.03553	0.517764	+	0.855524i	$0.326765\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	16.7082	1.27771
$172$	0	0
$173$	1.23607	0.0939765	0.0469883	−	0.998895i	$-0.485038\pi$
$173$	1.23607	0.0939765	0.0469883	+	0.998895i	$0.485038\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.70820	−0.128396
$178$	0	0
$179$	−11.1803	−0.835658	−0.417829	−	0.908526i	$-0.637209\pi$
$179$	−11.1803	−0.835658	−0.417829	+	0.908526i	$0.637209\pi$
$180$	0	0
$181$	5.41641	0.402598	0.201299	−	0.979530i	$-0.435484\pi$
$181$	5.41641	0.402598	0.201299	+	0.979530i	$0.435484\pi$
$182$	0	0
$183$	3.38197	0.250002
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.85410	0.354967
$188$	0	0
$189$	−6.70820	−0.487950
$190$	0	0
$191$	13.6525	0.987858	0.493929	−	0.869502i	$-0.335560\pi$
$191$	13.6525	0.987858	0.493929	+	0.869502i	$0.335560\pi$
$192$	0	0
$193$	14.6525	1.05471	0.527354	−	0.849646i	$-0.323184\pi$
$193$	14.6525	1.05471	0.527354	+	0.849646i	$0.323184\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	−2.56231	−0.181637	−0.0908185	−	0.995867i	$-0.528948\pi$
$199$	−2.56231	−0.181637	−0.0908185	+	0.995867i	$0.528948\pi$
$200$	0	0
$201$	3.90983	0.275778
$202$	0	0
$203$	28.4164	1.99444
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.03444	0.349918
$208$	0	0
$209$	24.7984	1.71534
$210$	0	0
$211$	22.2705	1.53317	0.766583	−	0.642146i	$-0.221956\pi$
$211$	22.2705	1.53317	0.766583	+	0.642146i	$0.221956\pi$
$212$	0	0
$213$	−1.14590	−0.0785156
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.708204	−0.0480760
$218$	0	0
$219$	2.94427	0.198955
$220$	0	0
$221$	1.14590	0.0770814
$222$	0	0
$223$	−9.00000	−0.602685	−0.301342	−	0.953516i	$-0.597435\pi$
$223$	−9.00000	−0.602685	−0.301342	+	0.953516i	$0.597435\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	29.3607	1.94874	0.974368	−	0.224958i	$-0.0722246\pi$
$227$	29.3607	1.94874	0.974368	+	0.224958i	$0.0722246\pi$
$228$	0	0
$229$	−2.56231	−0.169322	−0.0846610	−	0.996410i	$-0.526981\pi$
$229$	−2.56231	−0.169322	−0.0846610	+	0.996410i	$0.526981\pi$
$230$	0	0
$231$	−4.85410	−0.319376
$232$	0	0
$233$	−14.2918	−0.936287	−0.468143	−	0.883653i	$-0.655077\pi$
$233$	−14.2918	−0.936287	−0.468143	+	0.883653i	$0.655077\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2.76393	−0.179537
$238$	0	0
$239$	6.90983	0.446960	0.223480	−	0.974709i	$-0.428258\pi$
$239$	6.90983	0.446960	0.223480	+	0.974709i	$0.428258\pi$
$240$	0	0
$241$	15.6180	1.00605	0.503023	−	0.864273i	$-0.332221\pi$
$241$	15.6180	1.00605	0.503023	+	0.864273i	$0.332221\pi$
$242$	0	0
$243$	−9.65248	−0.619207
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.85410	0.372488
$248$	0	0
$249$	1.72949	0.109602
$250$	0	0
$251$	−0.819660	−0.0517365	−0.0258682	−	0.999665i	$-0.508235\pi$
$251$	−0.819660	−0.0517365	−0.0258682	+	0.999665i	$0.508235\pi$
$252$	0	0
$253$	7.47214	0.469769
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−25.7426	−1.60578	−0.802891	−	0.596126i	$-0.796706\pi$
$257$	−25.7426	−1.60578	−0.802891	+	0.596126i	$0.796706\pi$
$258$	0	0
$259$	−24.9787	−1.55210
$260$	0	0
$261$	27.0344	1.67339
$262$	0	0
$263$	4.61803	0.284760	0.142380	−	0.989812i	$-0.454524\pi$
$263$	4.61803	0.284760	0.142380	+	0.989812i	$0.454524\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.70820	−0.104540
$268$	0	0
$269$	1.90983	0.116444	0.0582222	−	0.998304i	$-0.481457\pi$
$269$	1.90983	0.116444	0.0582222	+	0.998304i	$0.481457\pi$
$270$	0	0
$271$	−28.5066	−1.73165	−0.865826	−	0.500346i	$-0.833206\pi$
$271$	−28.5066	−1.73165	−0.865826	+	0.500346i	$0.833206\pi$
$272$	0	0
$273$	−1.14590	−0.0693529
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−0.291796	−0.0175323	−0.00876616	−	0.999962i	$-0.502790\pi$
$277$	−0.291796	−0.0175323	−0.00876616	+	0.999962i	$0.502790\pi$
$278$	0	0
$279$	−0.673762	−0.0403371
$280$	0	0
$281$	3.18034	0.189723	0.0948616	−	0.995490i	$-0.469759\pi$
$281$	3.18034	0.189723	0.0948616	+	0.995490i	$0.469759\pi$
$282$	0	0
$283$	−12.2918	−0.730671	−0.365336	−	0.930876i	$-0.619046\pi$
$283$	−12.2918	−0.730671	−0.365336	+	0.930876i	$0.619046\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.41641	−0.260692
$288$	0	0
$289$	−15.6869	−0.922760
$290$	0	0
$291$	3.65248	0.214112
$292$	0	0
$293$	−8.56231	−0.500215	−0.250108	−	0.968218i	$-0.580466\pi$
$293$	−8.56231	−0.500215	−0.250108	+	0.968218i	$0.580466\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−9.47214	−0.549629
$298$	0	0
$299$	1.76393	0.102011
$300$	0	0
$301$	18.7082	1.07832
$302$	0	0
$303$	−0.236068	−0.0135618
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−23.1246	−1.31979	−0.659896	−	0.751357i	$-0.729400\pi$
$307$	−23.1246	−1.31979	−0.659896	+	0.751357i	$0.729400\pi$
$308$	0	0
$309$	5.02129	0.285651
$310$	0	0
$311$	9.90983	0.561935	0.280967	−	0.959717i	$-0.409345\pi$
$311$	9.90983	0.561935	0.280967	+	0.959717i	$0.409345\pi$
$312$	0	0
$313$	16.5623	0.936157	0.468078	−	0.883687i	$-0.344946\pi$
$313$	16.5623	0.936157	0.468078	+	0.883687i	$0.344946\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.38197	−0.470778	−0.235389	−	0.971901i	$-0.575636\pi$
$317$	−8.38197	−0.470778	−0.235389	+	0.971901i	$0.575636\pi$
$318$	0	0
$319$	40.1246	2.24655
$320$	0	0
$321$	0.416408	0.0232416
$322$	0	0
$323$	6.70820	0.373254
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.72949	0.316842
$328$	0	0
$329$	35.8328	1.97553
$330$	0	0
$331$	15.5623	0.855382	0.427691	−	0.903925i	$-0.359327\pi$
$331$	15.5623	0.855382	0.427691	+	0.903925i	$0.359327\pi$
$332$	0	0
$333$	−23.7639	−1.30226
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.58359	0.249684	0.124842	−	0.992177i	$-0.460158\pi$
$337$	4.58359	0.249684	0.124842	+	0.992177i	$0.460158\pi$
$338$	0	0
$339$	1.43769	0.0780848
$340$	0	0
$341$	−1.00000	−0.0541530
$342$	0	0
$343$	15.0000	0.809924
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.0902	−1.13218	−0.566090	−	0.824344i	$-0.691544\pi$
$347$	−21.0902	−1.13218	−0.566090	+	0.824344i	$0.691544\pi$
$348$	0	0
$349$	−17.7639	−0.950881	−0.475441	−	0.879748i	$-0.657711\pi$
$349$	−17.7639	−0.950881	−0.475441	+	0.879748i	$0.657711\pi$
$350$	0	0
$351$	−2.23607	−0.119352
$352$	0	0
$353$	21.2361	1.13028	0.565141	−	0.824994i	$-0.308822\pi$
$353$	21.2361	1.13028	0.565141	+	0.824994i	$0.308822\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.31308	−0.0694957
$358$	0	0
$359$	8.09017	0.426983	0.213491	−	0.976945i	$-0.431516\pi$
$359$	8.09017	0.426983	0.213491	+	0.976945i	$0.431516\pi$
$360$	0	0
$361$	15.2705	0.803711
$362$	0	0
$363$	−2.65248	−0.139219
$364$	0	0
$365$	0	0
$366$	0	0
$367$	35.5410	1.85523	0.927613	−	0.373543i	$-0.121857\pi$
$367$	35.5410	1.85523	0.927613	+	0.373543i	$0.121857\pi$
$368$	0	0
$369$	−4.20163	−0.218728
$370$	0	0
$371$	31.4164	1.63106
$372$	0	0
$373$	11.7639	0.609113	0.304557	−	0.952494i	$-0.401492\pi$
$373$	11.7639	0.609113	0.304557	+	0.952494i	$0.401492\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.47214	0.487840
$378$	0	0
$379$	−9.47214	−0.486551	−0.243275	−	0.969957i	$-0.578222\pi$
$379$	−9.47214	−0.486551	−0.243275	+	0.969957i	$0.578222\pi$
$380$	0	0
$381$	1.34752	0.0690358
$382$	0	0
$383$	11.3262	0.578744	0.289372	−	0.957217i	$-0.406554\pi$
$383$	11.3262	0.578744	0.289372	+	0.957217i	$0.406554\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	17.7984	0.904742
$388$	0	0
$389$	18.2148	0.923526	0.461763	−	0.887003i	$-0.347217\pi$
$389$	18.2148	0.923526	0.461763	+	0.887003i	$0.347217\pi$
$390$	0	0
$391$	2.02129	0.102221
$392$	0	0
$393$	−6.27051	−0.316305
$394$	0	0
$395$	0	0
$396$	0	0
$397$	10.2361	0.513734	0.256867	−	0.966447i	$-0.417310\pi$
$397$	10.2361	0.513734	0.256867	+	0.966447i	$0.417310\pi$
$398$	0	0
$399$	−6.70820	−0.335830
$400$	0	0
$401$	−14.1803	−0.708132	−0.354066	−	0.935220i	$-0.615201\pi$
$401$	−14.1803	−0.708132	−0.354066	+	0.935220i	$0.615201\pi$
$402$	0	0
$403$	−0.236068	−0.0117594
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−35.2705	−1.74829
$408$	0	0
$409$	−6.70820	−0.331699	−0.165850	−	0.986151i	$-0.553037\pi$
$409$	−6.70820	−0.331699	−0.165850	+	0.986151i	$0.553037\pi$
$410$	0	0
$411$	1.94427	0.0959039
$412$	0	0
$413$	−13.4164	−0.660178
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.03444	−0.0996270
$418$	0	0
$419$	−22.7639	−1.11209	−0.556045	−	0.831152i	$-0.687682\pi$
$419$	−22.7639	−1.11209	−0.556045	+	0.831152i	$0.687682\pi$
$420$	0	0
$421$	21.2705	1.03666	0.518331	−	0.855180i	$-0.326554\pi$
$421$	21.2705	1.03666	0.518331	+	0.855180i	$0.326554\pi$
$422$	0	0
$423$	34.0902	1.65752
$424$	0	0
$425$	0	0
$426$	0	0
$427$	26.5623	1.28544
$428$	0	0
$429$	−1.61803	−0.0781194
$430$	0	0
$431$	−26.0689	−1.25569	−0.627847	−	0.778337i	$-0.716064\pi$
$431$	−26.0689	−1.25569	−0.627847	+	0.778337i	$0.716064\pi$
$432$	0	0
$433$	21.3607	1.02653	0.513264	−	0.858231i	$-0.328436\pi$
$433$	21.3607	1.02653	0.513264	+	0.858231i	$0.328436\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.3262	0.493971
$438$	0	0
$439$	−25.6525	−1.22433	−0.612163	−	0.790732i	$-0.709700\pi$
$439$	−25.6525	−1.22433	−0.612163	+	0.790732i	$0.709700\pi$
$440$	0	0
$441$	−5.70820	−0.271819
$442$	0	0
$443$	19.4164	0.922501	0.461251	−	0.887270i	$-0.347401\pi$
$443$	19.4164	0.922501	0.461251	+	0.887270i	$0.347401\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−0.854102	−0.0403976
$448$	0	0
$449$	−3.94427	−0.186142	−0.0930709	−	0.995659i	$-0.529668\pi$
$449$	−3.94427	−0.186142	−0.0930709	+	0.995659i	$0.529668\pi$
$450$	0	0
$451$	−6.23607	−0.293645
$452$	0	0
$453$	−3.70820	−0.174227
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−40.2148	−1.88117	−0.940584	−	0.339561i	$-0.889722\pi$
$457$	−40.2148	−1.88117	−0.940584	+	0.339561i	$0.889722\pi$
$458$	0	0
$459$	−2.56231	−0.119598
$460$	0	0
$461$	5.81966	0.271049	0.135524	−	0.990774i	$-0.456728\pi$
$461$	5.81966	0.271049	0.135524	+	0.990774i	$0.456728\pi$
$462$	0	0
$463$	−34.9787	−1.62560	−0.812799	−	0.582544i	$-0.802057\pi$
$463$	−34.9787	−1.62560	−0.812799	+	0.582544i	$0.802057\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.1803	−0.887560	−0.443780	−	0.896136i	$-0.646363\pi$
$467$	−19.1803	−0.887560	−0.443780	+	0.896136i	$0.646363\pi$
$468$	0	0
$469$	30.7082	1.41797
$470$	0	0
$471$	1.74265	0.0802969
$472$	0	0
$473$	26.4164	1.21463
$474$	0	0
$475$	0	0
$476$	0	0
$477$	29.8885	1.36850
$478$	0	0
$479$	15.2016	0.694580	0.347290	−	0.937758i	$-0.387102\pi$
$479$	15.2016	0.694580	0.347290	+	0.937758i	$0.387102\pi$
$480$	0	0
$481$	−8.32624	−0.379644
$482$	0	0
$483$	−2.02129	−0.0919717
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.1246	1.00256	0.501281	−	0.865285i	$-0.332862\pi$
$487$	22.1246	1.00256	0.501281	+	0.865285i	$0.332862\pi$
$488$	0	0
$489$	−0.708204	−0.0320261
$490$	0	0
$491$	−19.2361	−0.868112	−0.434056	−	0.900886i	$-0.642918\pi$
$491$	−19.2361	−0.868112	−0.434056	+	0.900886i	$0.642918\pi$
$492$	0	0
$493$	10.8541	0.488844
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9.00000	−0.403705
$498$	0	0
$499$	−34.1459	−1.52858	−0.764290	−	0.644873i	$-0.776910\pi$
$499$	−34.1459	−1.52858	−0.764290	+	0.644873i	$0.776910\pi$
$500$	0	0
$501$	−5.11146	−0.228363
$502$	0	0
$503$	−30.5066	−1.36022	−0.680111	−	0.733110i	$-0.738068\pi$
$503$	−30.5066	−1.36022	−0.680111	+	0.733110i	$0.738068\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4.58359	0.203564
$508$	0	0
$509$	−1.90983	−0.0846517	−0.0423259	−	0.999104i	$-0.513477\pi$
$509$	−1.90983	−0.0846517	−0.0423259	+	0.999104i	$0.513477\pi$
$510$	0	0
$511$	23.1246	1.02297
$512$	0	0
$513$	−13.0902	−0.577945
$514$	0	0
$515$	0	0
$516$	0	0
$517$	50.5967	2.22524
$518$	0	0
$519$	−0.472136	−0.0207245
$520$	0	0
$521$	−23.4508	−1.02740	−0.513700	−	0.857970i	$-0.671726\pi$
$521$	−23.4508	−1.02740	−0.513700	+	0.857970i	$0.671726\pi$
$522$	0	0
$523$	−4.65248	−0.203439	−0.101719	−	0.994813i	$-0.532434\pi$
$523$	−4.65248	−0.203439	−0.101719	+	0.994813i	$0.532434\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−0.270510	−0.0117836
$528$	0	0
$529$	−19.8885	−0.864719
$530$	0	0
$531$	−12.7639	−0.553907
$532$	0	0
$533$	−1.47214	−0.0637653
$534$	0	0
$535$	0	0
$536$	0	0
$537$	4.27051	0.184286
$538$	0	0
$539$	−8.47214	−0.364921
$540$	0	0
$541$	15.2918	0.657446	0.328723	−	0.944426i	$-0.393382\pi$
$541$	15.2918	0.657446	0.328723	+	0.944426i	$0.393382\pi$
$542$	0	0
$543$	−2.06888	−0.0887843
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.38197	0.358387	0.179193	−	0.983814i	$-0.442651\pi$
$547$	8.38197	0.358387	0.179193	+	0.983814i	$0.442651\pi$
$548$	0	0
$549$	25.2705	1.07852
$550$	0	0
$551$	55.4508	2.36229
$552$	0	0
$553$	−21.7082	−0.923127
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−24.8885	−1.05456	−0.527281	−	0.849691i	$-0.676788\pi$
$557$	−24.8885	−1.05456	−0.527281	+	0.849691i	$0.676788\pi$
$558$	0	0
$559$	6.23607	0.263758
$560$	0	0
$561$	−1.85410	−0.0782802
$562$	0	0
$563$	−28.1459	−1.18621	−0.593104	−	0.805126i	$-0.702098\pi$
$563$	−28.1459	−1.18621	−0.593104	+	0.805126i	$0.702098\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−23.1246	−0.971142
$568$	0	0
$569$	17.7639	0.744703	0.372351	−	0.928092i	$-0.378552\pi$
$569$	17.7639	0.744703	0.372351	+	0.928092i	$0.378552\pi$
$570$	0	0
$571$	−17.3262	−0.725080	−0.362540	−	0.931968i	$-0.618090\pi$
$571$	−17.3262	−0.725080	−0.362540	+	0.931968i	$0.618090\pi$
$572$	0	0
$573$	−5.21478	−0.217851
$574$	0	0
$575$	0	0
$576$	0	0
$577$	43.0000	1.79011	0.895057	−	0.445952i	$-0.147135\pi$
$577$	43.0000	1.79011	0.895057	+	0.445952i	$0.147135\pi$
$578$	0	0
$579$	−5.59675	−0.232593
$580$	0	0
$581$	13.5836	0.563542
$582$	0	0
$583$	44.3607	1.83723
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−34.1803	−1.41077	−0.705387	−	0.708823i	$-0.749227\pi$
$587$	−34.1803	−1.41077	−0.705387	+	0.708823i	$0.749227\pi$
$588$	0	0
$589$	−1.38197	−0.0569429
$590$	0	0
$591$	4.58359	0.188544
$592$	0	0
$593$	−29.0132	−1.19143	−0.595714	−	0.803197i	$-0.703131\pi$
$593$	−29.0132	−1.19143	−0.595714	+	0.803197i	$0.703131\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.978714	0.0400561
$598$	0	0
$599$	−8.94427	−0.365453	−0.182727	−	0.983164i	$-0.558492\pi$
$599$	−8.94427	−0.365453	−0.182727	+	0.983164i	$0.558492\pi$
$600$	0	0
$601$	38.8328	1.58402	0.792012	−	0.610506i	$-0.209034\pi$
$601$	38.8328	1.58402	0.792012	+	0.610506i	$0.209034\pi$
$602$	0	0
$603$	29.2148	1.18972
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−33.8541	−1.37410	−0.687048	−	0.726612i	$-0.741094\pi$
$607$	−33.8541	−1.37410	−0.687048	+	0.726612i	$0.741094\pi$
$608$	0	0
$609$	−10.8541	−0.439830
$610$	0	0
$611$	11.9443	0.483214
$612$	0	0
$613$	21.4377	0.865860	0.432930	−	0.901427i	$-0.357480\pi$
$613$	21.4377	0.865860	0.432930	+	0.901427i	$0.357480\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14.3607	−0.578139	−0.289070	−	0.957308i	$-0.593346\pi$
$617$	−14.3607	−0.578139	−0.289070	+	0.957308i	$0.593346\pi$
$618$	0	0
$619$	−46.9574	−1.88738	−0.943689	−	0.330833i	$-0.892670\pi$
$619$	−46.9574	−1.88738	−0.943689	+	0.330833i	$0.892670\pi$
$620$	0	0
$621$	−3.94427	−0.158278
$622$	0	0
$623$	−13.4164	−0.537517
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−9.47214	−0.378281
$628$	0	0
$629$	−9.54102	−0.380425
$630$	0	0
$631$	44.8328	1.78477	0.892383	−	0.451279i	$-0.149032\pi$
$631$	44.8328	1.78477	0.892383	+	0.451279i	$0.149032\pi$
$632$	0	0
$633$	−8.50658	−0.338106
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.00000	−0.0792429
$638$	0	0
$639$	−8.56231	−0.338720
$640$	0	0
$641$	−30.3607	−1.19917	−0.599587	−	0.800309i	$-0.704669\pi$
$641$	−30.3607	−1.19917	−0.599587	+	0.800309i	$0.704669\pi$
$642$	0	0
$643$	−31.7639	−1.25265	−0.626324	−	0.779563i	$-0.715441\pi$
$643$	−31.7639	−1.25265	−0.626324	+	0.779563i	$0.715441\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.6525	−0.929875	−0.464937	−	0.885344i	$-0.653923\pi$
$647$	−23.6525	−0.929875	−0.464937	+	0.885344i	$0.653923\pi$
$648$	0	0
$649$	−18.9443	−0.743628
$650$	0	0
$651$	0.270510	0.0106021
$652$	0	0
$653$	8.59675	0.336417	0.168208	−	0.985751i	$-0.446202\pi$
$653$	8.59675	0.336417	0.168208	+	0.985751i	$0.446202\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	22.0000	0.858302
$658$	0	0
$659$	25.6525	0.999279	0.499639	−	0.866234i	$-0.333466\pi$
$659$	25.6525	0.999279	0.499639	+	0.866234i	$0.333466\pi$
$660$	0	0
$661$	−27.2705	−1.06070	−0.530350	−	0.847779i	$-0.677939\pi$
$661$	−27.2705	−1.06070	−0.530350	+	0.847779i	$0.677939\pi$
$662$	0	0
$663$	−0.437694	−0.0169986
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.7082	0.646944
$668$	0	0
$669$	3.43769	0.132909
$670$	0	0
$671$	37.5066	1.44793
$672$	0	0
$673$	−3.23607	−0.124741	−0.0623706	−	0.998053i	$-0.519866\pi$
$673$	−3.23607	−0.124741	−0.0623706	+	0.998053i	$0.519866\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.38197	−0.322145	−0.161073	−	0.986943i	$-0.551495\pi$
$677$	−8.38197	−0.322145	−0.161073	+	0.986943i	$0.551495\pi$
$678$	0	0
$679$	28.6869	1.10090
$680$	0	0
$681$	−11.2148	−0.429751
$682$	0	0
$683$	−11.3607	−0.434704	−0.217352	−	0.976093i	$-0.569742\pi$
$683$	−11.3607	−0.434704	−0.217352	+	0.976093i	$0.569742\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0.978714	0.0373403
$688$	0	0
$689$	10.4721	0.398957
$690$	0	0
$691$	32.2705	1.22763	0.613814	−	0.789451i	$-0.289634\pi$
$691$	32.2705	1.22763	0.613814	+	0.789451i	$0.289634\pi$
$692$	0	0
$693$	−36.2705	−1.37780
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.68692	−0.0638966
$698$	0	0
$699$	5.45898	0.206478
$700$	0	0
$701$	18.1803	0.686662	0.343331	−	0.939214i	$-0.388445\pi$
$701$	18.1803	0.686662	0.343331	+	0.939214i	$0.388445\pi$
$702$	0	0
$703$	−48.7426	−1.83836
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.85410	−0.0697307
$708$	0	0
$709$	12.2361	0.459535	0.229768	−	0.973246i	$-0.426203\pi$
$709$	12.2361	0.459535	0.229768	+	0.973246i	$0.426203\pi$
$710$	0	0
$711$	−20.6525	−0.774528
$712$	0	0
$713$	−0.416408	−0.0155946
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.63932	−0.0985672
$718$	0	0
$719$	−11.5066	−0.429123	−0.214561	−	0.976710i	$-0.568832\pi$
$719$	−11.5066	−0.429123	−0.214561	+	0.976710i	$0.568832\pi$
$720$	0	0
$721$	39.4377	1.46874
$722$	0	0
$723$	−5.96556	−0.221861
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.65248	0.0983749	0.0491874	−	0.998790i	$-0.484337\pi$
$727$	2.65248	0.0983749	0.0491874	+	0.998790i	$0.484337\pi$
$728$	0	0
$729$	−19.4377	−0.719915
$730$	0	0
$731$	7.14590	0.264301
$732$	0	0
$733$	−31.7771	−1.17371	−0.586857	−	0.809691i	$-0.699635\pi$
$733$	−31.7771	−1.17371	−0.586857	+	0.809691i	$0.699635\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	43.3607	1.59721
$738$	0	0
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	0	0
$741$	−2.23607	−0.0821440
$742$	0	0
$743$	15.2705	0.560221	0.280110	−	0.959968i	$-0.409629\pi$
$743$	15.2705	0.560221	0.280110	+	0.959968i	$0.409629\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.9230	0.472827
$748$	0	0
$749$	3.27051	0.119502
$750$	0	0
$751$	−7.85410	−0.286600	−0.143300	−	0.989679i	$-0.545771\pi$
$751$	−7.85410	−0.286600	−0.143300	+	0.989679i	$0.545771\pi$
$752$	0	0
$753$	0.313082	0.0114094
$754$	0	0
$755$	0	0
$756$	0	0
$757$	16.4164	0.596664	0.298332	−	0.954462i	$-0.403570\pi$
$757$	16.4164	0.596664	0.298332	+	0.954462i	$0.403570\pi$
$758$	0	0
$759$	−2.85410	−0.103597
$760$	0	0
$761$	26.9230	0.975957	0.487979	−	0.872856i	$-0.337734\pi$
$761$	26.9230	0.975957	0.487979	+	0.872856i	$0.337734\pi$
$762$	0	0
$763$	45.0000	1.62911
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.47214	−0.161479
$768$	0	0
$769$	3.41641	0.123199	0.0615994	−	0.998101i	$-0.480380\pi$
$769$	3.41641	0.123199	0.0615994	+	0.998101i	$0.480380\pi$
$770$	0	0
$771$	9.83282	0.354120
$772$	0	0
$773$	−30.7984	−1.10774	−0.553870	−	0.832603i	$-0.686850\pi$
$773$	−30.7984	−1.10774	−0.553870	+	0.832603i	$0.686850\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	9.54102	0.342282
$778$	0	0
$779$	−8.61803	−0.308773
$780$	0	0
$781$	−12.7082	−0.454735
$782$	0	0
$783$	−21.1803	−0.756924
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−11.4164	−0.406951	−0.203475	−	0.979080i	$-0.565224\pi$
$787$	−11.4164	−0.406951	−0.203475	+	0.979080i	$0.565224\pi$
$788$	0	0
$789$	−1.76393	−0.0627976
$790$	0	0
$791$	11.2918	0.401490
$792$	0	0
$793$	8.85410	0.314418
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.61803	−0.199001	−0.0995005	−	0.995038i	$-0.531724\pi$
$797$	−5.61803	−0.199001	−0.0995005	+	0.995038i	$0.531724\pi$
$798$	0	0
$799$	13.6869	0.484208
$800$	0	0
$801$	−12.7639	−0.450991
$802$	0	0
$803$	32.6525	1.15228
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.729490	−0.0256793
$808$	0	0
$809$	−2.23607	−0.0786160	−0.0393080	−	0.999227i	$-0.512515\pi$
$809$	−2.23607	−0.0786160	−0.0393080	+	0.999227i	$0.512515\pi$
$810$	0	0
$811$	−40.9443	−1.43775	−0.718874	−	0.695140i	$-0.755342\pi$
$811$	−40.9443	−1.43775	−0.718874	+	0.695140i	$0.755342\pi$
$812$	0	0
$813$	10.8885	0.381878
$814$	0	0
$815$	0	0
$816$	0	0
$817$	36.5066	1.27720
$818$	0	0
$819$	−8.56231	−0.299191
$820$	0	0
$821$	39.1591	1.36666	0.683330	−	0.730109i	$-0.260531\pi$
$821$	39.1591	1.36666	0.683330	+	0.730109i	$0.260531\pi$
$822$	0	0
$823$	−20.8328	−0.726186	−0.363093	−	0.931753i	$-0.618279\pi$
$823$	−20.8328	−0.726186	−0.363093	+	0.931753i	$0.618279\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.03444	−0.175065	−0.0875323	−	0.996162i	$-0.527898\pi$
$827$	−5.03444	−0.175065	−0.0875323	+	0.996162i	$0.527898\pi$
$828$	0	0
$829$	53.4164	1.85523	0.927614	−	0.373540i	$-0.121856\pi$
$829$	53.4164	1.85523	0.927614	+	0.373540i	$0.121856\pi$
$830$	0	0
$831$	0.111456	0.00386637
$832$	0	0
$833$	−2.29180	−0.0794060
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0.527864	0.0182457
$838$	0	0
$839$	55.2492	1.90742	0.953708	−	0.300736i	$-0.0972322\pi$
$839$	55.2492	1.90742	0.953708	+	0.300736i	$0.0972322\pi$
$840$	0	0
$841$	60.7214	2.09384
$842$	0	0
$843$	−1.21478	−0.0418393
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−20.8328	−0.715824
$848$	0	0
$849$	4.69505	0.161134
$850$	0	0
$851$	−14.6869	−0.503461
$852$	0	0
$853$	18.1459	0.621304	0.310652	−	0.950524i	$-0.399453\pi$
$853$	18.1459	0.621304	0.310652	+	0.950524i	$0.399453\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	56.3394	1.92452	0.962259	−	0.272137i	$-0.0877304\pi$
$857$	56.3394	1.92452	0.962259	+	0.272137i	$0.0877304\pi$
$858$	0	0
$859$	−34.3951	−1.17355	−0.586773	−	0.809751i	$-0.699602\pi$
$859$	−34.3951	−1.17355	−0.586773	+	0.809751i	$0.699602\pi$
$860$	0	0
$861$	1.68692	0.0574900
$862$	0	0
$863$	−11.2361	−0.382480	−0.191240	−	0.981543i	$-0.561251\pi$
$863$	−11.2361	−0.382480	−0.191240	+	0.981543i	$0.561251\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.99187	0.203495
$868$	0	0
$869$	−30.6525	−1.03981
$870$	0	0
$871$	10.2361	0.346836
$872$	0	0
$873$	27.2918	0.923687
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−4.43769	−0.149850	−0.0749251	−	0.997189i	$-0.523872\pi$
$877$	−4.43769	−0.149850	−0.0749251	+	0.997189i	$0.523872\pi$
$878$	0	0
$879$	3.27051	0.110312
$880$	0	0
$881$	38.1803	1.28633	0.643164	−	0.765728i	$-0.277621\pi$
$881$	38.1803	1.28633	0.643164	+	0.765728i	$0.277621\pi$
$882$	0	0
$883$	−41.6869	−1.40288	−0.701438	−	0.712730i	$-0.747458\pi$
$883$	−41.6869	−1.40288	−0.701438	+	0.712730i	$0.747458\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33.1803	1.11409	0.557043	−	0.830483i	$-0.311936\pi$
$887$	33.1803	1.11409	0.557043	+	0.830483i	$0.311936\pi$
$888$	0	0
$889$	10.5836	0.354962
$890$	0	0
$891$	−32.6525	−1.09390
$892$	0	0
$893$	69.9230	2.33988
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−0.673762	−0.0224963
$898$	0	0
$899$	−2.23607	−0.0745770
$900$	0	0
$901$	12.0000	0.399778
$902$	0	0
$903$	−7.14590	−0.237801
$904$	0	0
$905$	0	0
$906$	0	0
$907$	25.6180	0.850633	0.425316	−	0.905045i	$-0.360163\pi$
$907$	25.6180	0.850633	0.425316	+	0.905045i	$0.360163\pi$
$908$	0	0
$909$	−1.76393	−0.0585059
$910$	0	0
$911$	16.4164	0.543900	0.271950	−	0.962311i	$-0.412331\pi$
$911$	16.4164	0.543900	0.271950	+	0.962311i	$0.412331\pi$
$912$	0	0
$913$	19.1803	0.634777
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−49.2492	−1.62635
$918$	0	0
$919$	5.65248	0.186458	0.0932290	−	0.995645i	$-0.470281\pi$
$919$	5.65248	0.186458	0.0932290	+	0.995645i	$0.470281\pi$
$920$	0	0
$921$	8.83282	0.291051
$922$	0	0
$923$	−3.00000	−0.0987462
$924$	0	0
$925$	0	0
$926$	0	0
$927$	37.5197	1.23231
$928$	0	0
$929$	−52.8885	−1.73522	−0.867608	−	0.497248i	$-0.834344\pi$
$929$	−52.8885	−1.73522	−0.867608	+	0.497248i	$0.834344\pi$
$930$	0	0
$931$	−11.7082	−0.383721
$932$	0	0
$933$	−3.78522	−0.123922
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−41.9230	−1.36956	−0.684782	−	0.728748i	$-0.740103\pi$
$937$	−41.9230	−1.36956	−0.684782	+	0.728748i	$0.740103\pi$
$938$	0	0
$939$	−6.32624	−0.206449
$940$	0	0
$941$	−9.38197	−0.305843	−0.152922	−	0.988238i	$-0.548868\pi$
$941$	−9.38197	−0.305843	−0.152922	+	0.988238i	$0.548868\pi$
$942$	0	0
$943$	−2.59675	−0.0845617
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.3820	−1.27974	−0.639871	−	0.768482i	$-0.721012\pi$
$947$	−39.3820	−1.27974	−0.639871	+	0.768482i	$0.721012\pi$
$948$	0	0
$949$	7.70820	0.250219
$950$	0	0
$951$	3.20163	0.103820
$952$	0	0
$953$	−44.6180	−1.44532	−0.722660	−	0.691204i	$-0.757081\pi$
$953$	−44.6180	−1.44532	−0.722660	+	0.691204i	$0.757081\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−15.3262	−0.495427
$958$	0	0
$959$	15.2705	0.493110
$960$	0	0
$961$	−30.9443	−0.998202
$962$	0	0
$963$	3.11146	0.100265
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−13.1246	−0.422059	−0.211030	−	0.977480i	$-0.567682\pi$
$967$	−13.1246	−0.422059	−0.211030	+	0.977480i	$0.567682\pi$
$968$	0	0
$969$	−2.56231	−0.0823131
$970$	0	0
$971$	−6.06888	−0.194760	−0.0973799	−	0.995247i	$-0.531046\pi$
$971$	−6.06888	−0.194760	−0.0973799	+	0.995247i	$0.531046\pi$
$972$	0	0
$973$	−15.9787	−0.512254
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.0132	−1.60006	−0.800031	−	0.599958i	$-0.795184\pi$
$977$	−50.0132	−1.60006	−0.800031	+	0.599958i	$0.795184\pi$
$978$	0	0
$979$	−18.9443	−0.605462
$980$	0	0
$981$	42.8115	1.36687
$982$	0	0
$983$	38.4853	1.22749	0.613745	−	0.789504i	$-0.289662\pi$
$983$	38.4853	1.22749	0.613745	+	0.789504i	$0.289662\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−13.6869	−0.435659
$988$	0	0
$989$	11.0000	0.349780
$990$	0	0
$991$	3.45085	0.109620	0.0548099	−	0.998497i	$-0.482545\pi$
$991$	3.45085	0.109620	0.0548099	+	0.998497i	$0.482545\pi$
$992$	0	0
$993$	−5.94427	−0.188636
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−27.2492	−0.862992	−0.431496	−	0.902115i	$-0.642014\pi$
$997$	−27.2492	−0.862992	−0.431496	+	0.902115i	$0.642014\pi$
$998$	0	0
$999$	18.6180	0.589049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.a.1.2		2
4.3	odd	2		1250.2.a.d.1.1		2
5.4	even	2		10000.2.a.n.1.1		2
20.3	even	4		1250.2.b.b.1249.1		4
20.7	even	4		1250.2.b.b.1249.4		4
20.19	odd	2		1250.2.a.a.1.2		2
25.9	even	10		400.2.u.c.81.1		4
25.14	even	10		400.2.u.c.321.1		4
100.11	odd	10		250.2.d.a.101.1		4
100.23	even	20		250.2.e.b.149.1		8
100.27	even	20		250.2.e.b.149.2		8
100.39	odd	10		50.2.d.a.21.1	✓	4
100.59	odd	10		50.2.d.a.31.1	yes	4
100.63	even	20		250.2.e.b.99.2		8
100.87	even	20		250.2.e.b.99.1		8
100.91	odd	10		250.2.d.a.151.1		4
300.59	even	10		450.2.h.a.181.1		4
300.239	even	10		450.2.h.a.271.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.d.a.21.1	✓	4	100.39	odd	10
50.2.d.a.31.1	yes	4	100.59	odd	10
250.2.d.a.101.1		4	100.11	odd	10
250.2.d.a.151.1		4	100.91	odd	10
250.2.e.b.99.1		8	100.87	even	20
250.2.e.b.99.2		8	100.63	even	20
250.2.e.b.149.1		8	100.23	even	20
250.2.e.b.149.2		8	100.27	even	20
400.2.u.c.81.1		4	25.9	even	10
400.2.u.c.321.1		4	25.14	even	10
450.2.h.a.181.1		4	300.59	even	10
450.2.h.a.271.1		4	300.239	even	10
1250.2.a.a.1.2		2	20.19	odd	2
1250.2.a.d.1.1		2	4.3	odd	2
1250.2.b.b.1249.1		4	20.3	even	4
1250.2.b.b.1249.4		4	20.7	even	4
10000.2.a.a.1.2		2	1.1	even	1	trivial
10000.2.a.n.1.1		2	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)

\(f(q)\)	\(=\)	\(q-0.381966 q^{3} -3.00000 q^{7} -2.85410 q^{9} +O(q^{10})\)	\(q-0.381966 q^{3} -3.00000 q^{7} -2.85410 q^{9} -4.23607 q^{11} -1.00000 q^{13} -1.14590 q^{17} -5.85410 q^{19} +1.14590 q^{21} -1.76393 q^{23} +2.23607 q^{27} -9.47214 q^{29} +0.236068 q^{31} +1.61803 q^{33} +8.32624 q^{37} +0.381966 q^{39} +1.47214 q^{41} -6.23607 q^{43} -11.9443 q^{47} +2.00000 q^{49} +0.437694 q^{51} -10.4721 q^{53} +2.23607 q^{57} +4.47214 q^{59} -8.85410 q^{61} +8.56231 q^{63} -10.2361 q^{67} +0.673762 q^{69} +3.00000 q^{71} -7.70820 q^{73} +12.7082 q^{77} +7.23607 q^{79} +7.70820 q^{81} -4.52786 q^{83} +3.61803 q^{87} +4.47214 q^{89} +3.00000 q^{91} -0.0901699 q^{93} -9.56231 q^{97} +12.0902 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 6 q^{7} + q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 6 * q^7 + q^9	\( 2 q - 3 q^{3} - 6 q^{7} + q^{9} - 4 q^{11} - 2 q^{13} - 9 q^{17} - 5 q^{19} + 9 q^{21} - 8 q^{23} - 10 q^{29} - 4 q^{31} + q^{33} + q^{37} + 3 q^{39} - 6 q^{41} - 8 q^{43} - 6 q^{47} + 4 q^{49} + 21 q^{51} - 12 q^{53} - 11 q^{61} - 3 q^{63} - 16 q^{67} + 17 q^{69} + 6 q^{71} - 2 q^{73} + 12 q^{77} + 10 q^{79} + 2 q^{81} - 18 q^{83} + 5 q^{87} + 6 q^{91} + 11 q^{93} + q^{97} + 13 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 6 * q^7 + q^9 - 4 * q^11 - 2 * q^13 - 9 * q^17 - 5 * q^19 + 9 * q^21 - 8 * q^23 - 10 * q^29 - 4 * q^31 + q^33 + q^37 + 3 * q^39 - 6 * q^41 - 8 * q^43 - 6 * q^47 + 4 * q^49 + 21 * q^51 - 12 * q^53 - 11 * q^61 - 3 * q^63 - 16 * q^67 + 17 * q^69 + 6 * q^71 - 2 * q^73 + 12 * q^77 + 10 * q^79 + 2 * q^81 - 18 * q^83 + 5 * q^87 + 6 * q^91 + 11 * q^93 + q^97 + 13 * q^99

Introduction

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