LMFDB - Embedded newform 400.10.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,10,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("400.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	400.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:	$206.014334466$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.49740556.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 45x^{2} + 304 $ x^4 - 45*x^2 + 304
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.05982$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-179.263 q^{3} -8712.99 q^{7} +12452.2 q^{9} +O(q^{10})$	$q-179.263 q^{3} -8712.99 q^{7} +12452.2 q^{9} -44557.8 q^{11} -21430.4 q^{13} +300220. q^{17} -565385. q^{19} +1.56192e6 q^{21} -950727. q^{23} +1.29622e6 q^{27} -803167. q^{29} +1.99843e6 q^{31} +7.98755e6 q^{33} +9.53656e6 q^{37} +3.84168e6 q^{39} -2.54355e7 q^{41} +2.32830e7 q^{43} +3.77353e7 q^{47} +3.55626e7 q^{49} -5.38183e7 q^{51} -4.79297e7 q^{53} +1.01353e8 q^{57} -7.00069e7 q^{59} +1.26942e8 q^{61} -1.08496e8 q^{63} +2.66595e8 q^{67} +1.70430e8 q^{69} -6.59169e7 q^{71} +1.47516e7 q^{73} +3.88231e8 q^{77} +4.66498e7 q^{79} -4.77460e8 q^{81} -2.01840e8 q^{83} +1.43978e8 q^{87} +5.54039e8 q^{89} +1.86723e8 q^{91} -3.58244e8 q^{93} +3.39489e8 q^{97} -5.54841e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 11628 q^{9}+O(q^{10}) $ 4 * q - 11628 * q^9	$ 4 q - 11628 q^{9} - 109968 q^{11} - 636880 q^{19} + 3523968 q^{21} + 3531720 q^{29} + 10587712 q^{31} + 1686816 q^{39} - 16788552 q^{41} + 46921028 q^{49} - 84017088 q^{51} - 460829040 q^{59} + 360490568 q^{61} + 286524864 q^{69} + 47611872 q^{71} - 728043520 q^{79} - 343387836 q^{81} + 1582700760 q^{89} - 473322528 q^{91} - 728787024 q^{99}+O(q^{100}) $ 4 * q - 11628 * q^9 - 109968 * q^11 - 636880 * q^19 + 3523968 * q^21 + 3531720 * q^29 + 10587712 * q^31 + 1686816 * q^39 - 16788552 * q^41 + 46921028 * q^49 - 84017088 * q^51 - 460829040 * q^59 + 360490568 * q^61 + 286524864 * q^69 + 47611872 * q^71 - 728043520 * q^79 - 343387836 * q^81 + 1582700760 * q^89 - 473322528 * q^91 - 728787024 * 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
$3$	−179.263	−1.27775	−0.638873	−	0.769312i	$-0.720599\pi$
$3$	−179.263	−1.27775	−0.638873	+	0.769312i	$0.720599\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−8712.99	−1.37160	−0.685798	−	0.727792i	$-0.740547\pi$
$7$	−8712.99	−1.37160	−0.685798	+	0.727792i	$0.740547\pi$
$8$	0	0
$9$	12452.2	0.632636
$10$	0	0
$11$	−44557.8	−0.917606	−0.458803	−	0.888538i	$-0.651722\pi$
$11$	−44557.8	−0.917606	−0.458803	+	0.888538i	$0.651722\pi$
$12$	0	0
$13$	−21430.4	−0.208107	−0.104053	−	0.994572i	$-0.533181\pi$
$13$	−21430.4	−0.208107	−0.104053	+	0.994572i	$0.533181\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	300220.	0.871805	0.435903	−	0.899994i	$-0.356429\pi$
$17$	300220.	0.871805	0.435903	+	0.899994i	$0.356429\pi$
$18$	0	0
$19$	−565385.	−0.995298	−0.497649	−	0.867379i	$-0.665803\pi$
$19$	−565385.	−0.995298	−0.497649	+	0.867379i	$0.665803\pi$
$20$	0	0
$21$	1.56192e6	1.75255
$22$	0	0
$23$	−950727.	−0.708403	−0.354202	−	0.935169i	$-0.615247\pi$
$23$	−950727.	−0.708403	−0.354202	+	0.935169i	$0.615247\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.29622e6	0.469398
$28$	0	0
$29$	−803167.	−0.210870	−0.105435	−	0.994426i	$-0.533623\pi$
$29$	−803167.	−0.210870	−0.105435	+	0.994426i	$0.533623\pi$
$30$	0	0
$31$	1.99843e6	0.388652	0.194326	−	0.980937i	$-0.437748\pi$
$31$	1.99843e6	0.388652	0.194326	+	0.980937i	$0.437748\pi$
$32$	0	0
$33$	7.98755e6	1.17247
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.53656e6	0.836535	0.418267	−	0.908324i	$-0.362637\pi$
$37$	9.53656e6	0.836535	0.418267	+	0.908324i	$0.362637\pi$
$38$	0	0
$39$	3.84168e6	0.265908
$40$	0	0
$41$	−2.54355e7	−1.40576	−0.702882	−	0.711306i	$-0.748104\pi$
$41$	−2.54355e7	−1.40576	−0.702882	+	0.711306i	$0.748104\pi$
$42$	0	0
$43$	2.32830e7	1.03856	0.519279	−	0.854605i	$-0.326200\pi$
$43$	2.32830e7	1.03856	0.519279	+	0.854605i	$0.326200\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.77353e7	1.12800	0.563998	−	0.825776i	$-0.309262\pi$
$47$	3.77353e7	1.12800	0.563998	+	0.825776i	$0.309262\pi$
$48$	0	0
$49$	3.55626e7	0.881274
$50$	0	0
$51$	−5.38183e7	−1.11395
$52$	0	0
$53$	−4.79297e7	−0.834379	−0.417190	−	0.908819i	$-0.636985\pi$
$53$	−4.79297e7	−0.834379	−0.417190	+	0.908819i	$0.636985\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.01353e8	1.27174
$58$	0	0
$59$	−7.00069e7	−0.752154	−0.376077	−	0.926588i	$-0.622727\pi$
$59$	−7.00069e7	−0.752154	−0.376077	+	0.926588i	$0.622727\pi$
$60$	0	0
$61$	1.26942e8	1.17387	0.586936	−	0.809633i	$-0.300334\pi$
$61$	1.26942e8	1.17387	0.586936	+	0.809633i	$0.300334\pi$
$62$	0	0
$63$	−1.08496e8	−0.867721
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.66595e8	1.61628	0.808138	−	0.588993i	$-0.200475\pi$
$67$	2.66595e8	1.61628	0.808138	+	0.588993i	$0.200475\pi$
$68$	0	0
$69$	1.70430e8	0.905160
$70$	0	0
$71$	−6.59169e7	−0.307846	−0.153923	−	0.988083i	$-0.549191\pi$
$71$	−6.59169e7	−0.307846	−0.153923	+	0.988083i	$0.549191\pi$
$72$	0	0
$73$	1.47516e7	0.0607977	0.0303989	−	0.999538i	$-0.490322\pi$
$73$	1.47516e7	0.0607977	0.0303989	+	0.999538i	$0.490322\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.88231e8	1.25858
$78$	0	0
$79$	4.66498e7	0.134750	0.0673748	−	0.997728i	$-0.478538\pi$
$79$	4.66498e7	0.134750	0.0673748	+	0.997728i	$0.478538\pi$
$80$	0	0
$81$	−4.77460e8	−1.23241
$82$	0	0
$83$	−2.01840e8	−0.466826	−0.233413	−	0.972378i	$-0.574989\pi$
$83$	−2.01840e8	−0.466826	−0.233413	+	0.972378i	$0.574989\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.43978e8	0.269438
$88$	0	0
$89$	5.54039e8	0.936020	0.468010	−	0.883723i	$-0.344971\pi$
$89$	5.54039e8	0.936020	0.468010	+	0.883723i	$0.344971\pi$
$90$	0	0
$91$	1.86723e8	0.285438
$92$	0	0
$93$	−3.58244e8	−0.496599
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.39489e8	0.389361	0.194681	−	0.980867i	$-0.437633\pi$
$97$	3.39489e8	0.389361	0.194681	+	0.980867i	$0.437633\pi$
$98$	0	0
$99$	−5.54841e8	−0.580511
$100$	0	0
$101$	1.33921e9	1.28056	0.640282	−	0.768140i	$-0.278817\pi$
$101$	1.33921e9	1.28056	0.640282	+	0.768140i	$0.278817\pi$
$102$	0	0
$103$	3.84306e8	0.336442	0.168221	−	0.985749i	$-0.446198\pi$
$103$	3.84306e8	0.336442	0.168221	+	0.985749i	$0.446198\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.97379e8	0.588082	0.294041	−	0.955793i	$-0.405000\pi$
$107$	7.97379e8	0.588082	0.294041	+	0.955793i	$0.405000\pi$
$108$	0	0
$109$	−6.63230e8	−0.450034	−0.225017	−	0.974355i	$-0.572244\pi$
$109$	−6.63230e8	−0.450034	−0.225017	+	0.974355i	$0.572244\pi$
$110$	0	0
$111$	−1.70955e9	−1.06888
$112$	0	0
$113$	1.48164e9	0.854847	0.427424	−	0.904051i	$-0.359421\pi$
$113$	1.48164e9	0.854847	0.427424	+	0.904051i	$0.359421\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.66856e8	−0.131656
$118$	0	0
$119$	−2.61581e9	−1.19576
$120$	0	0
$121$	−3.72554e8	−0.157999
$122$	0	0
$123$	4.55964e9	1.79621
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.28772e9	0.780344	0.390172	−	0.920742i	$-0.372416\pi$
$127$	2.28772e9	0.780344	0.390172	+	0.920742i	$0.372416\pi$
$128$	0	0
$129$	−4.17378e9	−1.32701
$130$	0	0
$131$	3.83999e9	1.13922	0.569612	−	0.821914i	$-0.307094\pi$
$131$	3.83999e9	1.13922	0.569612	+	0.821914i	$0.307094\pi$
$132$	0	0
$133$	4.92619e9	1.36515
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.82666e9	1.41311	0.706556	−	0.707657i	$-0.250248\pi$
$137$	5.82666e9	1.41311	0.706556	+	0.707657i	$0.250248\pi$
$138$	0	0
$139$	5.89895e9	1.34032	0.670159	−	0.742217i	$-0.266226\pi$
$139$	5.89895e9	1.34032	0.670159	+	0.742217i	$0.266226\pi$
$140$	0	0
$141$	−6.76455e9	−1.44129
$142$	0	0
$143$	9.54892e8	0.190960
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.37505e9	−1.12604
$148$	0	0
$149$	5.39333e9	0.896436	0.448218	−	0.893924i	$-0.352059\pi$
$149$	5.39333e9	0.896436	0.448218	+	0.893924i	$0.352059\pi$
$150$	0	0
$151$	−7.92204e8	−0.124005	−0.0620027	−	0.998076i	$-0.519749\pi$
$151$	−7.92204e8	−0.124005	−0.0620027	+	0.998076i	$0.519749\pi$
$152$	0	0
$153$	3.73839e9	0.551536
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1.18606e10	−1.55797	−0.778985	−	0.627042i	$-0.784265\pi$
$157$	−1.18606e10	−1.55797	−0.778985	+	0.627042i	$0.784265\pi$
$158$	0	0
$159$	8.59202e9	1.06613
$160$	0	0
$161$	8.28367e9	0.971642
$162$	0	0
$163$	−3.99906e9	−0.443724	−0.221862	−	0.975078i	$-0.571213\pi$
$163$	−3.99906e9	−0.443724	−0.221862	+	0.975078i	$0.571213\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.09118e10	1.08560	0.542802	−	0.839861i	$-0.317363\pi$
$167$	1.09118e10	1.08560	0.542802	+	0.839861i	$0.317363\pi$
$168$	0	0
$169$	−1.01452e10	−0.956692
$170$	0	0
$171$	−7.04028e9	−0.629662
$172$	0	0
$173$	−1.89328e10	−1.60696	−0.803482	−	0.595329i	$-0.797022\pi$
$173$	−1.89328e10	−1.60696	−0.803482	+	0.595329i	$0.797022\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.25496e10	0.961062
$178$	0	0
$179$	−2.09763e10	−1.52718	−0.763592	−	0.645699i	$-0.776566\pi$
$179$	−2.09763e10	−1.52718	−0.763592	+	0.645699i	$0.776566\pi$
$180$	0	0
$181$	−7.16950e9	−0.496518	−0.248259	−	0.968694i	$-0.579858\pi$
$181$	−7.16950e9	−0.496518	−0.248259	+	0.968694i	$0.579858\pi$
$182$	0	0
$183$	−2.27560e10	−1.49991
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.33771e10	−0.799974
$188$	0	0
$189$	−1.12939e10	−0.643824
$190$	0	0
$191$	−1.75301e10	−0.953091	−0.476545	−	0.879150i	$-0.658111\pi$
$191$	−1.75301e10	−0.953091	−0.476545	+	0.879150i	$0.658111\pi$
$192$	0	0
$193$	−3.13528e10	−1.62655	−0.813277	−	0.581877i	$-0.802319\pi$
$193$	−3.13528e10	−1.62655	−0.813277	+	0.581877i	$0.802319\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.85971e10	−0.879725	−0.439862	−	0.898065i	$-0.644973\pi$
$197$	−1.85971e10	−0.879725	−0.439862	+	0.898065i	$0.644973\pi$
$198$	0	0
$199$	1.26662e10	0.572544	0.286272	−	0.958148i	$-0.407584\pi$
$199$	1.26662e10	0.572544	0.286272	+	0.958148i	$0.407584\pi$
$200$	0	0
$201$	−4.77906e10	−2.06519
$202$	0	0
$203$	6.99798e9	0.289228
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.18386e10	−0.448161
$208$	0	0
$209$	2.51923e10	0.913291
$210$	0	0
$211$	6.20579e9	0.215539	0.107770	−	0.994176i	$-0.465629\pi$
$211$	6.20579e9	0.215539	0.107770	+	0.994176i	$0.465629\pi$
$212$	0	0
$213$	1.18164e10	0.393350
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.74123e10	−0.533074
$218$	0	0
$219$	−2.64442e9	−0.0776841
$220$	0	0
$221$	−6.43385e9	−0.181428
$222$	0	0
$223$	−4.30855e10	−1.16670	−0.583350	−	0.812221i	$-0.698258\pi$
$223$	−4.30855e10	−1.16670	−0.583350	+	0.812221i	$0.698258\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.28857e10	−0.572068	−0.286034	−	0.958219i	$-0.592337\pi$
$227$	−2.28857e10	−0.572068	−0.286034	+	0.958219i	$0.592337\pi$
$228$	0	0
$229$	−5.26747e9	−0.126573	−0.0632867	−	0.997995i	$-0.520158\pi$
$229$	−5.26747e9	−0.126573	−0.0632867	+	0.997995i	$0.520158\pi$
$230$	0	0
$231$	−6.95955e10	−1.60815
$232$	0	0
$233$	3.55179e10	0.789488	0.394744	−	0.918791i	$-0.370833\pi$
$233$	3.55179e10	0.789488	0.394744	+	0.918791i	$0.370833\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.36257e9	−0.172176
$238$	0	0
$239$	−5.72471e10	−1.13491	−0.567457	−	0.823403i	$-0.692073\pi$
$239$	−5.72471e10	−1.13491	−0.567457	+	0.823403i	$0.692073\pi$
$240$	0	0
$241$	3.89830e10	0.744386	0.372193	−	0.928155i	$-0.378606\pi$
$241$	3.89830e10	0.744386	0.372193	+	0.928155i	$0.378606\pi$
$242$	0	0
$243$	6.00774e10	1.10531
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.21164e10	0.207128
$248$	0	0
$249$	3.61824e10	0.596486
$250$	0	0
$251$	−4.56895e10	−0.726581	−0.363291	−	0.931676i	$-0.618347\pi$
$251$	−4.56895e10	−0.726581	−0.363291	+	0.931676i	$0.618347\pi$
$252$	0	0
$253$	4.23622e10	0.650035
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.29955e10	−0.185821	−0.0929104	−	0.995674i	$-0.529617\pi$
$257$	−1.29955e10	−0.185821	−0.0929104	+	0.995674i	$0.529617\pi$
$258$	0	0
$259$	−8.30920e10	−1.14739
$260$	0	0
$261$	−1.00012e10	−0.133404
$262$	0	0
$263$	4.80103e10	0.618776	0.309388	−	0.950936i	$-0.399876\pi$
$263$	4.80103e10	0.618776	0.309388	+	0.950936i	$0.399876\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.93185e10	−1.19600
$268$	0	0
$269$	−9.98823e10	−1.16306	−0.581531	−	0.813524i	$-0.697546\pi$
$269$	−9.98823e10	−1.16306	−0.581531	+	0.813524i	$0.697546\pi$
$270$	0	0
$271$	4.80217e10	0.540849	0.270424	−	0.962741i	$-0.412836\pi$
$271$	4.80217e10	0.540849	0.270424	+	0.962741i	$0.412836\pi$
$272$	0	0
$273$	−3.34725e10	−0.364718
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.77838e10	0.181495	0.0907477	−	0.995874i	$-0.471074\pi$
$277$	1.77838e10	0.181495	0.0907477	+	0.995874i	$0.471074\pi$
$278$	0	0
$279$	2.48848e10	0.245875
$280$	0	0
$281$	−1.52044e11	−1.45476	−0.727379	−	0.686236i	$-0.759262\pi$
$281$	−1.52044e11	−1.45476	−0.727379	+	0.686236i	$0.759262\pi$
$282$	0	0
$283$	1.86733e11	1.73055	0.865273	−	0.501301i	$-0.167145\pi$
$283$	1.86733e11	1.73055	0.865273	+	0.501301i	$0.167145\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.21619e11	1.92814
$288$	0	0
$289$	−2.84558e10	−0.239955
$290$	0	0
$291$	−6.08577e10	−0.497505
$292$	0	0
$293$	1.28928e11	1.02198	0.510991	−	0.859586i	$-0.329278\pi$
$293$	1.28928e11	1.02198	0.510991	+	0.859586i	$0.329278\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.77565e10	−0.430722
$298$	0	0
$299$	2.03745e10	0.147423
$300$	0	0
$301$	−2.02865e11	−1.42448
$302$	0	0
$303$	−2.40070e11	−1.63624
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−8.78331e10	−0.564333	−0.282167	−	0.959365i	$-0.591053\pi$
$307$	−8.78331e10	−0.564333	−0.282167	+	0.959365i	$0.591053\pi$
$308$	0	0
$309$	−6.88919e10	−0.429887
$310$	0	0
$311$	−2.76204e11	−1.67420	−0.837101	−	0.547049i	$-0.815751\pi$
$311$	−2.76204e11	−1.67420	−0.837101	+	0.547049i	$0.815751\pi$
$312$	0	0
$313$	−1.84107e11	−1.08423	−0.542115	−	0.840304i	$-0.682376\pi$
$313$	−1.84107e11	−1.08423	−0.542115	+	0.840304i	$0.682376\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.48865e11	1.94040	0.970198	−	0.242313i	$-0.0779060\pi$
$317$	3.48865e11	1.94040	0.970198	+	0.242313i	$0.0779060\pi$
$318$	0	0
$319$	3.57873e10	0.193496
$320$	0	0
$321$	−1.42940e11	−0.751419
$322$	0	0
$323$	−1.69740e11	−0.867706
$324$	0	0
$325$	0	0
$326$	0	0
$327$	1.18893e11	0.575029
$328$	0	0
$329$	−3.28788e11	−1.54716
$330$	0	0
$331$	−2.88744e11	−1.32217	−0.661084	−	0.750312i	$-0.729903\pi$
$331$	−2.88744e11	−1.32217	−0.661084	+	0.750312i	$0.729903\pi$
$332$	0	0
$333$	1.18751e11	0.529222
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.25882e11	−0.531654	−0.265827	−	0.964021i	$-0.585645\pi$
$337$	−1.25882e11	−0.531654	−0.265827	+	0.964021i	$0.585645\pi$
$338$	0	0
$339$	−2.65602e11	−1.09228
$340$	0	0
$341$	−8.90455e10	−0.356630
$342$	0	0
$343$	4.17441e10	0.162844
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.95822e11	1.09534	0.547669	−	0.836695i	$-0.315515\pi$
$347$	2.95822e11	1.09534	0.547669	+	0.836695i	$0.315515\pi$
$348$	0	0
$349$	3.73474e11	1.34755	0.673777	−	0.738934i	$-0.264671\pi$
$349$	3.73474e11	1.34755	0.673777	+	0.738934i	$0.264671\pi$
$350$	0	0
$351$	−2.77785e10	−0.0976848
$352$	0	0
$353$	−4.89872e11	−1.67918	−0.839589	−	0.543223i	$-0.817204\pi$
$353$	−4.89872e11	−1.67918	−0.839589	+	0.543223i	$0.817204\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.68918e11	1.52788
$358$	0	0
$359$	4.27559e11	1.35854	0.679268	−	0.733891i	$-0.262298\pi$
$359$	4.27559e11	1.35854	0.679268	+	0.733891i	$0.262298\pi$
$360$	0	0
$361$	−3.02753e9	−0.00938223
$362$	0	0
$363$	6.67851e10	0.201883
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−2.54403e10	−0.0732023	−0.0366012	−	0.999330i	$-0.511653\pi$
$367$	−2.54403e10	−0.0732023	−0.0366012	+	0.999330i	$0.511653\pi$
$368$	0	0
$369$	−3.16727e11	−0.889337
$370$	0	0
$371$	4.17611e11	1.14443
$372$	0	0
$373$	7.26003e11	1.94200	0.970998	−	0.239087i	$-0.0768481\pi$
$373$	7.26003e11	1.94200	0.970998	+	0.239087i	$0.0768481\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.72122e10	0.0438834
$378$	0	0
$379$	2.76699e11	0.688859	0.344430	−	0.938812i	$-0.388072\pi$
$379$	2.76699e11	0.688859	0.344430	+	0.938812i	$0.388072\pi$
$380$	0	0
$381$	−4.10103e11	−0.997082
$382$	0	0
$383$	1.71143e11	0.406410	0.203205	−	0.979136i	$-0.434864\pi$
$383$	1.71143e11	0.406410	0.203205	+	0.979136i	$0.434864\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.89924e11	0.657030
$388$	0	0
$389$	3.92384e10	0.0868837	0.0434419	−	0.999056i	$-0.486168\pi$
$389$	3.92384e10	0.0868837	0.0434419	+	0.999056i	$0.486168\pi$
$390$	0	0
$391$	−2.85427e11	−0.617590
$392$	0	0
$393$	−6.88367e11	−1.45564
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3.91381e11	−0.790756	−0.395378	−	0.918519i	$-0.629386\pi$
$397$	−3.91381e11	−0.790756	−0.395378	+	0.918519i	$0.629386\pi$
$398$	0	0
$399$	−8.83084e11	−1.74431
$400$	0	0
$401$	−5.04969e11	−0.975248	−0.487624	−	0.873054i	$-0.662136\pi$
$401$	−5.04969e11	−0.975248	−0.487624	+	0.873054i	$0.662136\pi$
$402$	0	0
$403$	−4.28272e10	−0.0808811
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.24928e11	−0.767609
$408$	0	0
$409$	9.44998e9	0.0166985	0.00834923	−	0.999965i	$-0.497342\pi$
$409$	9.44998e9	0.0166985	0.00834923	+	0.999965i	$0.497342\pi$
$410$	0	0
$411$	−1.04450e12	−1.80560
$412$	0	0
$413$	6.09969e11	1.03165
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.05746e12	−1.71259
$418$	0	0
$419$	5.77328e10	0.0915081	0.0457541	−	0.998953i	$-0.485431\pi$
$419$	5.77328e10	0.0915081	0.0457541	+	0.998953i	$0.485431\pi$
$420$	0	0
$421$	4.38976e9	0.00681038	0.00340519	−	0.999994i	$-0.498916\pi$
$421$	4.38976e9	0.00681038	0.00340519	+	0.999994i	$0.498916\pi$
$422$	0	0
$423$	4.69887e11	0.713612
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.10604e12	−1.61008
$428$	0	0
$429$	−1.71177e11	−0.243998
$430$	0	0
$431$	−9.94476e11	−1.38818	−0.694091	−	0.719887i	$-0.744193\pi$
$431$	−9.94476e11	−1.38818	−0.694091	+	0.719887i	$0.744193\pi$
$432$	0	0
$433$	1.91618e11	0.261963	0.130982	−	0.991385i	$-0.458187\pi$
$433$	1.91618e11	0.261963	0.130982	+	0.991385i	$0.458187\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.37527e11	0.705072
$438$	0	0
$439$	−1.38202e12	−1.77592	−0.887962	−	0.459917i	$-0.847879\pi$
$439$	−1.38202e12	−1.77592	−0.887962	+	0.459917i	$0.847879\pi$
$440$	0	0
$441$	4.42832e11	0.557526
$442$	0	0
$443$	−3.05660e11	−0.377070	−0.188535	−	0.982066i	$-0.560374\pi$
$443$	−3.05660e11	−0.377070	−0.188535	+	0.982066i	$0.560374\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−9.66825e11	−1.14542
$448$	0	0
$449$	1.49518e12	1.73614	0.868069	−	0.496444i	$-0.165361\pi$
$449$	1.49518e12	1.73614	0.868069	+	0.496444i	$0.165361\pi$
$450$	0	0
$451$	1.13335e12	1.28994
$452$	0	0
$453$	1.42013e11	0.158447
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.17977e12	−1.26525	−0.632624	−	0.774459i	$-0.718022\pi$
$457$	−1.17977e12	−1.26525	−0.632624	+	0.774459i	$0.718022\pi$
$458$	0	0
$459$	3.89151e11	0.409223
$460$	0	0
$461$	1.54415e12	1.59234	0.796170	−	0.605074i	$-0.206856\pi$
$461$	1.54415e12	1.59234	0.796170	+	0.605074i	$0.206856\pi$
$462$	0	0
$463$	−4.55769e11	−0.460926	−0.230463	−	0.973081i	$-0.574024\pi$
$463$	−4.55769e11	−0.460926	−0.230463	+	0.973081i	$0.574024\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.97342e11	0.678454	0.339227	−	0.940705i	$-0.389835\pi$
$467$	6.97342e11	0.678454	0.339227	+	0.940705i	$0.389835\pi$
$468$	0	0
$469$	−2.32284e12	−2.21688
$470$	0	0
$471$	2.12617e12	1.99069
$472$	0	0
$473$	−1.03744e12	−0.952987
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.96829e11	−0.527858
$478$	0	0
$479$	−2.43471e11	−0.211318	−0.105659	−	0.994402i	$-0.533695\pi$
$479$	−2.43471e11	−0.211318	−0.105659	+	0.994402i	$0.533695\pi$
$480$	0	0
$481$	−2.04373e11	−0.174088
$482$	0	0
$483$	−1.48495e12	−1.24151
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.56961e12	−1.26448	−0.632239	−	0.774773i	$-0.717864\pi$
$487$	−1.56961e12	−1.26448	−0.632239	+	0.774773i	$0.717864\pi$
$488$	0	0
$489$	7.16882e11	0.566967
$490$	0	0
$491$	−6.52870e11	−0.506944	−0.253472	−	0.967343i	$-0.581573\pi$
$491$	−6.52870e11	−0.506944	−0.253472	+	0.967343i	$0.581573\pi$
$492$	0	0
$493$	−2.41127e11	−0.183838
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.74333e11	0.422241
$498$	0	0
$499$	7.51465e11	0.542571	0.271285	−	0.962499i	$-0.412551\pi$
$499$	7.51465e11	0.542571	0.271285	+	0.962499i	$0.412551\pi$
$500$	0	0
$501$	−1.95608e12	−1.38713
$502$	0	0
$503$	1.31120e12	0.913299	0.456649	−	0.889647i	$-0.349049\pi$
$503$	1.31120e12	0.913299	0.456649	+	0.889647i	$0.349049\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1.81866e12	1.22241
$508$	0	0
$509$	1.78629e12	1.17957	0.589783	−	0.807561i	$-0.299213\pi$
$509$	1.78629e12	1.17957	0.589783	+	0.807561i	$0.299213\pi$
$510$	0	0
$511$	−1.28531e11	−0.0833899
$512$	0	0
$513$	−7.32862e11	−0.467191
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.68140e12	−1.03506
$518$	0	0
$519$	3.39394e12	2.05329
$520$	0	0
$521$	5.88627e11	0.350002	0.175001	−	0.984568i	$-0.444007\pi$
$521$	5.88627e11	0.350002	0.175001	+	0.984568i	$0.444007\pi$
$522$	0	0
$523$	6.01202e11	0.351369	0.175684	−	0.984447i	$-0.443786\pi$
$523$	6.01202e11	0.351369	0.175684	+	0.984447i	$0.443786\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.99968e11	0.338829
$528$	0	0
$529$	−8.97272e11	−0.498165
$530$	0	0
$531$	−8.71739e11	−0.475840
$532$	0	0
$533$	5.45093e11	0.292549
$534$	0	0
$535$	0	0
$536$	0	0
$537$	3.76028e12	1.95135
$538$	0	0
$539$	−1.58459e12	−0.808662
$540$	0	0
$541$	1.40863e12	0.706982	0.353491	−	0.935438i	$-0.384995\pi$
$541$	1.40863e12	0.706982	0.353491	+	0.935438i	$0.384995\pi$
$542$	0	0
$543$	1.28522e12	0.634424
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−8.05933e10	−0.0384907	−0.0192454	−	0.999815i	$-0.506126\pi$
$547$	−8.05933e10	−0.0384907	−0.0192454	+	0.999815i	$0.506126\pi$
$548$	0	0
$549$	1.58070e12	0.742635
$550$	0	0
$551$	4.54098e11	0.209878
$552$	0	0
$553$	−4.06459e11	−0.184822
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.92066e12	1.72588	0.862941	−	0.505305i	$-0.168620\pi$
$557$	3.92066e12	1.72588	0.862941	+	0.505305i	$0.168620\pi$
$558$	0	0
$559$	−4.98965e11	−0.216131
$560$	0	0
$561$	2.39802e12	1.02216
$562$	0	0
$563$	−3.86627e12	−1.62182	−0.810912	−	0.585168i	$-0.801029\pi$
$563$	−3.86627e12	−1.62182	−0.810912	+	0.585168i	$0.801029\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.16010e12	1.69036
$568$	0	0
$569$	−1.87990e12	−0.751849	−0.375924	−	0.926650i	$-0.622675\pi$
$569$	−1.87990e12	−0.751849	−0.375924	+	0.926650i	$0.622675\pi$
$570$	0	0
$571$	3.44726e12	1.35710	0.678550	−	0.734554i	$-0.262609\pi$
$571$	3.44726e12	1.35710	0.678550	+	0.734554i	$0.262609\pi$
$572$	0	0
$573$	3.14250e12	1.21781
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.68433e12	0.632608	0.316304	−	0.948658i	$-0.397558\pi$
$577$	1.68433e12	0.632608	0.316304	+	0.948658i	$0.397558\pi$
$578$	0	0
$579$	5.62039e12	2.07832
$580$	0	0
$581$	1.75863e12	0.640297
$582$	0	0
$583$	2.13564e12	0.765631
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.04715e12	1.75458	0.877292	−	0.479956i	$-0.159348\pi$
$587$	5.04715e12	1.75458	0.877292	+	0.479956i	$0.159348\pi$
$588$	0	0
$589$	−1.12988e12	−0.386825
$590$	0	0
$591$	3.33377e12	1.12407
$592$	0	0
$593$	6.66924e11	0.221478	0.110739	−	0.993850i	$-0.464678\pi$
$593$	6.66924e11	0.221478	0.110739	+	0.993850i	$0.464678\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−2.27059e12	−0.731566
$598$	0	0
$599$	−3.36479e12	−1.06792	−0.533958	−	0.845511i	$-0.679296\pi$
$599$	−3.36479e12	−1.06792	−0.533958	+	0.845511i	$0.679296\pi$
$600$	0	0
$601$	−2.79891e12	−0.875092	−0.437546	−	0.899196i	$-0.644152\pi$
$601$	−2.79891e12	−0.875092	−0.437546	+	0.899196i	$0.644152\pi$
$602$	0	0
$603$	3.31969e12	1.02251
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.38683e12	−1.01261	−0.506307	−	0.862353i	$-0.668990\pi$
$607$	−3.38683e12	−1.01261	−0.506307	+	0.862353i	$0.668990\pi$
$608$	0	0
$609$	−1.25448e12	−0.369560
$610$	0	0
$611$	−8.08685e11	−0.234744
$612$	0	0
$613$	7.53866e11	0.215636	0.107818	−	0.994171i	$-0.465614\pi$
$613$	7.53866e11	0.215636	0.107818	+	0.994171i	$0.465614\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.42132e11	−0.0394827	−0.0197414	−	0.999805i	$-0.506284\pi$
$617$	−1.42132e11	−0.0394827	−0.0197414	+	0.999805i	$0.506284\pi$
$618$	0	0
$619$	1.03073e12	0.282186	0.141093	−	0.989996i	$-0.454938\pi$
$619$	1.03073e12	0.282186	0.141093	+	0.989996i	$0.454938\pi$
$620$	0	0
$621$	−1.23235e12	−0.332523
$622$	0	0
$623$	−4.82733e12	−1.28384
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.51604e12	−1.16695
$628$	0	0
$629$	2.86307e12	0.729296
$630$	0	0
$631$	4.61780e12	1.15959	0.579793	−	0.814764i	$-0.303133\pi$
$631$	4.61780e12	1.15959	0.579793	+	0.814764i	$0.303133\pi$
$632$	0	0
$633$	−1.11247e12	−0.275404
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.62122e11	−0.183399
$638$	0	0
$639$	−8.20809e11	−0.194755
$640$	0	0
$641$	7.51099e12	1.75726	0.878630	−	0.477503i	$-0.158458\pi$
$641$	7.51099e12	1.75726	0.878630	+	0.477503i	$0.158458\pi$
$642$	0	0
$643$	4.42841e12	1.02164	0.510821	−	0.859687i	$-0.329341\pi$
$643$	4.42841e12	1.02164	0.510821	+	0.859687i	$0.329341\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.09396e12	−0.245432	−0.122716	−	0.992442i	$-0.539160\pi$
$647$	−1.09396e12	−0.245432	−0.122716	+	0.992442i	$0.539160\pi$
$648$	0	0
$649$	3.11935e12	0.690181
$650$	0	0
$651$	3.12138e12	0.681133
$652$	0	0
$653$	−8.24881e12	−1.77534	−0.887671	−	0.460477i	$-0.847678\pi$
$653$	−8.24881e12	−1.77534	−0.887671	+	0.460477i	$0.847678\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.83690e11	0.0384628
$658$	0	0
$659$	2.64086e12	0.545458	0.272729	−	0.962091i	$-0.412074\pi$
$659$	2.64086e12	0.545458	0.272729	+	0.962091i	$0.412074\pi$
$660$	0	0
$661$	8.94654e12	1.82284	0.911420	−	0.411478i	$-0.134987\pi$
$661$	8.94654e12	1.82284	0.911420	+	0.411478i	$0.134987\pi$
$662$	0	0
$663$	1.15335e12	0.231820
$664$	0	0
$665$	0	0
$666$	0	0
$667$	7.63592e11	0.149381
$668$	0	0
$669$	7.72362e12	1.49075
$670$	0	0
$671$	−5.65625e12	−1.07715
$672$	0	0
$673$	6.40541e12	1.20359	0.601796	−	0.798650i	$-0.294452\pi$
$673$	6.40541e12	1.20359	0.601796	+	0.798650i	$0.294452\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.41491e12	−1.17366	−0.586829	−	0.809711i	$-0.699624\pi$
$677$	−6.41491e12	−1.17366	−0.586829	+	0.809711i	$0.699624\pi$
$678$	0	0
$679$	−2.95796e12	−0.534046
$680$	0	0
$681$	4.10256e12	0.730958
$682$	0	0
$683$	2.15592e12	0.379088	0.189544	−	0.981872i	$-0.439299\pi$
$683$	2.15592e12	0.379088	0.189544	+	0.981872i	$0.439299\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.44262e11	0.161729
$688$	0	0
$689$	1.02715e12	0.173640
$690$	0	0
$691$	4.58074e12	0.764336	0.382168	−	0.924093i	$-0.375178\pi$
$691$	4.58074e12	0.764336	0.382168	+	0.924093i	$0.375178\pi$
$692$	0	0
$693$	4.83433e12	0.796226
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−7.63624e12	−1.22555
$698$	0	0
$699$	−6.36704e12	−1.00877
$700$	0	0
$701$	−1.22474e12	−0.191564	−0.0957819	−	0.995402i	$-0.530535\pi$
$701$	−1.22474e12	−0.191564	−0.0957819	+	0.995402i	$0.530535\pi$
$702$	0	0
$703$	−5.39183e12	−0.832601
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.16685e13	−1.75642
$708$	0	0
$709$	−5.65887e12	−0.841049	−0.420525	−	0.907281i	$-0.638154\pi$
$709$	−5.65887e12	−0.841049	−0.420525	+	0.907281i	$0.638154\pi$
$710$	0	0
$711$	5.80891e11	0.0852475
$712$	0	0
$713$	−1.89996e12	−0.275322
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.02623e13	1.45013
$718$	0	0
$719$	6.02904e12	0.841333	0.420667	−	0.907215i	$-0.361796\pi$
$719$	6.02904e12	0.841333	0.420667	+	0.907215i	$0.361796\pi$
$720$	0	0
$721$	−3.34846e12	−0.461462
$722$	0	0
$723$	−6.98820e12	−0.951137
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.63469e12	0.349805	0.174902	−	0.984586i	$-0.444039\pi$
$727$	2.63469e12	0.349805	0.174902	+	0.984586i	$0.444039\pi$
$728$	0	0
$729$	−1.37180e12	−0.179894
$730$	0	0
$731$	6.99002e12	0.905421
$732$	0	0
$733$	−5.28609e12	−0.676343	−0.338171	−	0.941085i	$-0.609808\pi$
$733$	−5.28609e12	−0.676343	−0.338171	+	0.941085i	$0.609808\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.18789e13	−1.48310
$738$	0	0
$739$	−2.51810e12	−0.310580	−0.155290	−	0.987869i	$-0.549631\pi$
$739$	−2.51810e12	−0.310580	−0.155290	+	0.987869i	$0.549631\pi$
$740$	0	0
$741$	−2.17203e12	−0.264657
$742$	0	0
$743$	7.02537e11	0.0845706	0.0422853	−	0.999106i	$-0.486536\pi$
$743$	7.02537e11	0.0845706	0.0422853	+	0.999106i	$0.486536\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.51335e12	−0.295331
$748$	0	0
$749$	−6.94755e12	−0.806610
$750$	0	0
$751$	−4.60108e12	−0.527813	−0.263907	−	0.964548i	$-0.585011\pi$
$751$	−4.60108e12	−0.527813	−0.263907	+	0.964548i	$0.585011\pi$
$752$	0	0
$753$	8.19042e12	0.928387
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.05764e12	0.338419	0.169210	−	0.985580i	$-0.445878\pi$
$757$	3.05764e12	0.338419	0.169210	+	0.985580i	$0.445878\pi$
$758$	0	0
$759$	−7.59398e12	−0.830580
$760$	0	0
$761$	−9.86753e12	−1.06654	−0.533270	−	0.845945i	$-0.679037\pi$
$761$	−9.86753e12	−1.06654	−0.533270	+	0.845945i	$0.679037\pi$
$762$	0	0
$763$	5.77872e12	0.617264
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.50028e12	0.156528
$768$	0	0
$769$	1.65694e13	1.70859	0.854294	−	0.519790i	$-0.173990\pi$
$769$	1.65694e13	1.70859	0.854294	+	0.519790i	$0.173990\pi$
$770$	0	0
$771$	2.32961e12	0.237432
$772$	0	0
$773$	−1.10674e13	−1.11490	−0.557451	−	0.830210i	$-0.688221\pi$
$773$	−1.10674e13	−1.11490	−0.557451	+	0.830210i	$0.688221\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.48953e13	1.46607
$778$	0	0
$779$	1.43808e13	1.39915
$780$	0	0
$781$	2.93711e12	0.282482
$782$	0	0
$783$	−1.04108e12	−0.0989819
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1.67201e12	−0.155365	−0.0776825	−	0.996978i	$-0.524752\pi$
$787$	−1.67201e12	−0.155365	−0.0776825	+	0.996978i	$0.524752\pi$
$788$	0	0
$789$	−8.60646e12	−0.790638
$790$	0	0
$791$	−1.29095e13	−1.17250
$792$	0	0
$793$	−2.72042e12	−0.244291
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.97864e12	−0.612644	−0.306322	−	0.951928i	$-0.599098\pi$
$797$	−6.97864e12	−0.612644	−0.306322	+	0.951928i	$0.599098\pi$
$798$	0	0
$799$	1.13289e13	0.983394
$800$	0	0
$801$	6.89899e12	0.592160
$802$	0	0
$803$	−6.57300e11	−0.0557884
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.79052e13	1.48610
$808$	0	0
$809$	5.27673e12	0.433108	0.216554	−	0.976271i	$-0.430518\pi$
$809$	5.27673e12	0.433108	0.216554	+	0.976271i	$0.430518\pi$
$810$	0	0
$811$	1.43675e12	0.116624	0.0583118	−	0.998298i	$-0.481428\pi$
$811$	1.43675e12	0.116624	0.0583118	+	0.998298i	$0.481428\pi$
$812$	0	0
$813$	−8.60851e12	−0.691068
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−1.31639e13	−1.03367
$818$	0	0
$819$	2.32511e12	0.180578
$820$	0	0
$821$	9.44899e12	0.725840	0.362920	−	0.931820i	$-0.381780\pi$
$821$	9.44899e12	0.725840	0.362920	+	0.931820i	$0.381780\pi$
$822$	0	0
$823$	−1.20237e13	−0.913562	−0.456781	−	0.889579i	$-0.650998\pi$
$823$	−1.20237e13	−0.913562	−0.456781	+	0.889579i	$0.650998\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7.86848e12	−0.584947	−0.292473	−	0.956274i	$-0.594478\pi$
$827$	−7.86848e12	−0.584947	−0.292473	+	0.956274i	$0.594478\pi$
$828$	0	0
$829$	4.86184e12	0.357524	0.178762	−	0.983892i	$-0.442791\pi$
$829$	4.86184e12	0.357524	0.178762	+	0.983892i	$0.442791\pi$
$830$	0	0
$831$	−3.18798e12	−0.231905
$832$	0	0
$833$	1.06766e13	0.768299
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.59040e12	0.182432
$838$	0	0
$839$	−1.76025e13	−1.22644	−0.613218	−	0.789914i	$-0.710125\pi$
$839$	−1.76025e13	−1.22644	−0.613218	+	0.789914i	$0.710125\pi$
$840$	0	0
$841$	−1.38621e13	−0.955534
$842$	0	0
$843$	2.72559e13	1.85881
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.24606e12	0.216711
$848$	0	0
$849$	−3.34744e13	−2.21120
$850$	0	0
$851$	−9.06666e12	−0.592604
$852$	0	0
$853$	−9.20458e11	−0.0595296	−0.0297648	−	0.999557i	$-0.509476\pi$
$853$	−9.20458e11	−0.0595296	−0.0297648	+	0.999557i	$0.509476\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.35047e13	1.48847	0.744237	−	0.667915i	$-0.232813\pi$
$857$	2.35047e13	1.48847	0.744237	+	0.667915i	$0.232813\pi$
$858$	0	0
$859$	2.02385e13	1.26826	0.634132	−	0.773225i	$-0.281357\pi$
$859$	2.02385e13	1.26826	0.634132	+	0.773225i	$0.281357\pi$
$860$	0	0
$861$	−3.97281e13	−2.46367
$862$	0	0
$863$	−3.07364e13	−1.88627	−0.943136	−	0.332406i	$-0.892140\pi$
$863$	−3.07364e13	−1.88627	−0.943136	+	0.332406i	$0.892140\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.10107e12	0.306602
$868$	0	0
$869$	−2.07861e12	−0.123647
$870$	0	0
$871$	−5.71325e12	−0.336358
$872$	0	0
$873$	4.22737e12	0.246324
$874$	0	0
$875$	0	0
$876$	0	0
$877$	6.65840e12	0.380077	0.190039	−	0.981777i	$-0.439139\pi$
$877$	6.65840e12	0.380077	0.190039	+	0.981777i	$0.439139\pi$
$878$	0	0
$879$	−2.31120e13	−1.30584
$880$	0	0
$881$	3.41842e12	0.191176	0.0955882	−	0.995421i	$-0.469527\pi$
$881$	3.41842e12	0.191176	0.0955882	+	0.995421i	$0.469527\pi$
$882$	0	0
$883$	−1.92500e13	−1.06563	−0.532815	−	0.846231i	$-0.678866\pi$
$883$	−1.92500e13	−1.06563	−0.532815	+	0.846231i	$0.678866\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.48977e13	−0.808095	−0.404048	−	0.914738i	$-0.632397\pi$
$887$	−1.48977e13	−0.808095	−0.404048	+	0.914738i	$0.632397\pi$
$888$	0	0
$889$	−1.99329e13	−1.07032
$890$	0	0
$891$	2.12745e13	1.13086
$892$	0	0
$893$	−2.13350e13	−1.12269
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−3.65239e12	−0.188370
$898$	0	0
$899$	−1.60507e12	−0.0819551
$900$	0	0
$901$	−1.43895e13	−0.727416
$902$	0	0
$903$	3.63661e13	1.82013
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.11529e13	0.547210	0.273605	−	0.961842i	$-0.411784\pi$
$907$	1.11529e13	0.547210	0.273605	+	0.961842i	$0.411784\pi$
$908$	0	0
$909$	1.66760e13	0.810132
$910$	0	0
$911$	2.25248e12	0.108350	0.0541750	−	0.998531i	$-0.482747\pi$
$911$	2.25248e12	0.108350	0.0541750	+	0.998531i	$0.482747\pi$
$912$	0	0
$913$	8.99353e12	0.428363
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.34578e13	−1.56255
$918$	0	0
$919$	−1.40633e13	−0.650380	−0.325190	−	0.945649i	$-0.605428\pi$
$919$	−1.40633e13	−0.650380	−0.325190	+	0.945649i	$0.605428\pi$
$920$	0	0
$921$	1.57452e13	0.721075
$922$	0	0
$923$	1.41263e12	0.0640649
$924$	0	0
$925$	0	0
$926$	0	0
$927$	4.78545e12	0.212845
$928$	0	0
$929$	1.04912e13	0.462118	0.231059	−	0.972940i	$-0.425781\pi$
$929$	1.04912e13	0.462118	0.231059	+	0.972940i	$0.425781\pi$
$930$	0	0
$931$	−2.01066e13	−0.877130
$932$	0	0
$933$	4.95131e13	2.13920
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.77841e13	1.60133	0.800666	−	0.599111i	$-0.204479\pi$
$937$	3.77841e13	1.60133	0.800666	+	0.599111i	$0.204479\pi$
$938$	0	0
$939$	3.30036e13	1.38537
$940$	0	0
$941$	1.11142e13	0.462088	0.231044	−	0.972943i	$-0.425786\pi$
$941$	1.11142e13	0.462088	0.231044	+	0.972943i	$0.425786\pi$
$942$	0	0
$943$	2.41822e13	0.995847
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.96563e12	−0.200631	−0.100316	−	0.994956i	$-0.531985\pi$
$947$	−4.96563e12	−0.200631	−0.100316	+	0.994956i	$0.531985\pi$
$948$	0	0
$949$	−3.16134e11	−0.0126524
$950$	0	0
$951$	−6.25385e13	−2.47933
$952$	0	0
$953$	3.02750e13	1.18896	0.594479	−	0.804111i	$-0.297358\pi$
$953$	3.02750e13	1.18896	0.594479	+	0.804111i	$0.297358\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6.41533e12	−0.247238
$958$	0	0
$959$	−5.07676e13	−1.93822
$960$	0	0
$961$	−2.24459e13	−0.848949
$962$	0	0
$963$	9.92910e12	0.372042
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−2.48487e13	−0.913869	−0.456934	−	0.889500i	$-0.651053\pi$
$967$	−2.48487e13	−0.913869	−0.456934	+	0.889500i	$0.651053\pi$
$968$	0	0
$969$	3.04281e13	1.10871
$970$	0	0
$971$	−4.43278e13	−1.60026	−0.800128	−	0.599829i	$-0.795235\pi$
$971$	−4.43278e13	−1.60026	−0.800128	+	0.599829i	$0.795235\pi$
$972$	0	0
$973$	−5.13975e13	−1.83837
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.73746e11	−0.0341917	−0.0170958	−	0.999854i	$-0.505442\pi$
$977$	−9.73746e11	−0.0341917	−0.0170958	+	0.999854i	$0.505442\pi$
$978$	0	0
$979$	−2.46867e13	−0.858897
$980$	0	0
$981$	−8.25866e12	−0.284708
$982$	0	0
$983$	2.85792e13	0.976247	0.488123	−	0.872775i	$-0.337682\pi$
$983$	2.85792e13	0.976247	0.488123	+	0.872775i	$0.337682\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	5.89394e13	1.97687
$988$	0	0
$989$	−2.21358e13	−0.735718
$990$	0	0
$991$	3.49036e13	1.14958	0.574789	−	0.818302i	$-0.305084\pi$
$991$	3.49036e13	1.14958	0.574789	+	0.818302i	$0.305084\pi$
$992$	0	0
$993$	5.17610e13	1.68940
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−8.78834e12	−0.281695	−0.140847	−	0.990031i	$-0.544983\pi$
$997$	−8.78834e12	−0.281695	−0.140847	+	0.990031i	$0.544983\pi$
$998$	0	0
$999$	1.23615e13	0.392668

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.10.a.ba.1.1		4
4.3	odd	2		25.10.a.e.1.2		4
5.2	odd	4		80.10.c.c.49.4		4
5.3	odd	4		80.10.c.c.49.1		4
5.4	even	2	inner	400.10.a.ba.1.4		4
12.11	even	2		225.10.a.s.1.3		4
20.3	even	4		5.10.b.a.4.3	yes	4
20.7	even	4		5.10.b.a.4.2	✓	4
20.19	odd	2		25.10.a.e.1.3		4
60.23	odd	4		45.10.b.b.19.2		4
60.47	odd	4		45.10.b.b.19.3		4
60.59	even	2		225.10.a.s.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.10.b.a.4.2	✓	4	20.7	even	4
5.10.b.a.4.3	yes	4	20.3	even	4
25.10.a.e.1.2		4	4.3	odd	2
25.10.a.e.1.3		4	20.19	odd	2
45.10.b.b.19.2		4	60.23	odd	4
45.10.b.b.19.3		4	60.47	odd	4
80.10.c.c.49.1		4	5.3	odd	4
80.10.c.c.49.4		4	5.2	odd	4
225.10.a.s.1.2		4	60.59	even	2
225.10.a.s.1.3		4	12.11	even	2
400.10.a.ba.1.1		4	1.1	even	1	trivial
400.10.a.ba.1.4		4	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q-179.263 q^{3} -8712.99 q^{7} +12452.2 q^{9} +O(q^{10})\)	\(q-179.263 q^{3} -8712.99 q^{7} +12452.2 q^{9} -44557.8 q^{11} -21430.4 q^{13} +300220. q^{17} -565385. q^{19} +1.56192e6 q^{21} -950727. q^{23} +1.29622e6 q^{27} -803167. q^{29} +1.99843e6 q^{31} +7.98755e6 q^{33} +9.53656e6 q^{37} +3.84168e6 q^{39} -2.54355e7 q^{41} +2.32830e7 q^{43} +3.77353e7 q^{47} +3.55626e7 q^{49} -5.38183e7 q^{51} -4.79297e7 q^{53} +1.01353e8 q^{57} -7.00069e7 q^{59} +1.26942e8 q^{61} -1.08496e8 q^{63} +2.66595e8 q^{67} +1.70430e8 q^{69} -6.59169e7 q^{71} +1.47516e7 q^{73} +3.88231e8 q^{77} +4.66498e7 q^{79} -4.77460e8 q^{81} -2.01840e8 q^{83} +1.43978e8 q^{87} +5.54039e8 q^{89} +1.86723e8 q^{91} -3.58244e8 q^{93} +3.39489e8 q^{97} -5.54841e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 11628 q^{9}+O(q^{10}) \) 4 * q - 11628 * q^9	\( 4 q - 11628 q^{9} - 109968 q^{11} - 636880 q^{19} + 3523968 q^{21} + 3531720 q^{29} + 10587712 q^{31} + 1686816 q^{39} - 16788552 q^{41} + 46921028 q^{49} - 84017088 q^{51} - 460829040 q^{59} + 360490568 q^{61} + 286524864 q^{69} + 47611872 q^{71} - 728043520 q^{79} - 343387836 q^{81} + 1582700760 q^{89} - 473322528 q^{91} - 728787024 q^{99}+O(q^{100}) \) 4 * q - 11628 * q^9 - 109968 * q^11 - 636880 * q^19 + 3523968 * q^21 + 3531720 * q^29 + 10587712 * q^31 + 1686816 * q^39 - 16788552 * q^41 + 46921028 * q^49 - 84017088 * q^51 - 460829040 * q^59 + 360490568 * q^61 + 286524864 * q^69 + 47611872 * q^71 - 728043520 * q^79 - 343387836 * q^81 + 1582700760 * q^89 - 473322528 * q^91 - 728787024 * q^99

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