LMFDB - Embedded newform 225.10.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,10,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("225.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	225.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:	$115.883063137$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\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			1.2
Root			$6.05982$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.843944 q^{2} -511.288 q^{4} -8712.99 q^{7} +863.597 q^{8} +O(q^{10})$	$q-0.843944 q^{2} -511.288 q^{4} -8712.99 q^{7} +863.597 q^{8} -44557.8 q^{11} +21430.4 q^{13} +7353.27 q^{14} +261051. q^{16} +300220. q^{17} +565385. q^{19} +37604.2 q^{22} +950727. q^{23} -18086.1 q^{26} +4.45485e6 q^{28} +803167. q^{29} -1.99843e6 q^{31} -662474. q^{32} -253369. q^{34} -9.53656e6 q^{37} -477153. q^{38} +2.54355e7 q^{41} +2.32830e7 q^{43} +2.27818e7 q^{44} -802360. q^{46} -3.77353e7 q^{47} +3.55626e7 q^{49} -1.09571e7 q^{52} -4.79297e7 q^{53} -7.52451e6 q^{56} -677827. q^{58} -7.00069e7 q^{59} +1.26942e8 q^{61} +1.68656e6 q^{62} -1.33099e8 q^{64} +2.66595e8 q^{67} -1.53499e8 q^{68} -6.59169e7 q^{71} -1.47516e7 q^{73} +8.04832e6 q^{74} -2.89074e8 q^{76} +3.88231e8 q^{77} -4.66498e7 q^{79} -2.14661e7 q^{82} +2.01840e8 q^{83} -1.96495e7 q^{86} -3.84800e7 q^{88} -5.54039e8 q^{89} -1.86723e8 q^{91} -4.86095e8 q^{92} +3.18465e7 q^{94} -3.39489e8 q^{97} -3.00128e7 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 1368 q^{4}+O(q^{10}) $ 4 * q + 1368 * q^4	$ 4 q + 1368 q^{4} - 109968 q^{11} - 424536 q^{14} + 1631264 q^{16} + 636880 q^{19} - 6618768 q^{26} - 3531720 q^{29} - 10587712 q^{31} - 26434624 q^{34} + 16788552 q^{41} + 20638944 q^{44} - 61250072 q^{46} + 46921028 q^{49} - 315178080 q^{56} - 460829040 q^{59} + 360490568 q^{61} - 134995072 q^{64} + 47611872 q^{71} + 1176861744 q^{74} - 1168489440 q^{76} + 728043520 q^{79} + 2375904552 q^{86} - 1582700760 q^{89} + 473322528 q^{91} - 3327101704 q^{94}+O(q^{100}) $ 4 * q + 1368 * q^4 - 109968 * q^11 - 424536 * q^14 + 1631264 * q^16 + 636880 * q^19 - 6618768 * q^26 - 3531720 * q^29 - 10587712 * q^31 - 26434624 * q^34 + 16788552 * q^41 + 20638944 * q^44 - 61250072 * q^46 + 46921028 * q^49 - 315178080 * q^56 - 460829040 * q^59 + 360490568 * q^61 - 134995072 * q^64 + 47611872 * q^71 + 1176861744 * q^74 - 1168489440 * q^76 + 728043520 * q^79 + 2375904552 * q^86 - 1582700760 * q^89 + 473322528 * q^91 - 3327101704 * q^94

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 $

$2$	−0.843944	−0.0372974	−0.0186487	−	0.999826i	$-0.505936\pi$
$2$	−0.843944	−0.0372974	−0.0186487	+	0.999826i	$0.505936\pi$
$3$	0	0
$3$	0	0
$4$	−511.288	−0.998609
$5$	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	$0.740547\pi$
$8$	863.597	0.0745429
$9$	0	0
$10$	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	$-0.466819\pi$
$13$	21430.4	0.208107	0.104053	+	0.994572i	$0.466819\pi$
$14$	7353.27	0.0511569
$15$	0	0
$16$	261051.	0.995829
$17$	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	$-0.334197\pi$
$19$	565385.	0.995298	0.497649	+	0.867379i	$0.334197\pi$
$20$	0	0
$21$	0	0
$22$	37604.2	0.0342243
$23$	950727.	0.708403	0.354202	−	0.935169i	$-0.384753\pi$
$23$	950727.	0.708403	0.354202	+	0.935169i	$0.384753\pi$
$24$	0	0
$25$	0	0
$26$	−18086.1	−0.00776183
$27$	0	0
$28$	4.45485e6	1.36969
$29$	803167.	0.210870	0.105435	−	0.994426i	$-0.466377\pi$
$29$	803167.	0.210870	0.105435	+	0.994426i	$0.466377\pi$
$30$	0	0
$31$	−1.99843e6	−0.388652	−0.194326	−	0.980937i	$-0.562252\pi$
$31$	−1.99843e6	−0.388652	−0.194326	+	0.980937i	$0.562252\pi$
$32$	−662474.	−0.111685
$33$	0	0
$34$	−253369.	−0.0325161
$35$	0	0
$36$	0	0
$37$	−9.53656e6	−0.836535	−0.418267	−	0.908324i	$-0.637363\pi$
$37$	−9.53656e6	−0.836535	−0.418267	+	0.908324i	$0.637363\pi$
$38$	−477153.	−0.0371220
$39$	0	0
$40$	0	0
$41$	2.54355e7	1.40576	0.702882	−	0.711306i	$-0.251896\pi$
$41$	2.54355e7	1.40576	0.702882	+	0.711306i	$0.251896\pi$
$42$	0	0
$43$	2.32830e7	1.03856	0.519279	−	0.854605i	$-0.326200\pi$
$43$	2.32830e7	1.03856	0.519279	+	0.854605i	$0.326200\pi$
$44$	2.27818e7	0.916329
$45$	0	0
$46$	−802360.	−0.0264216
$47$	−3.77353e7	−1.12800	−0.563998	−	0.825776i	$-0.690738\pi$
$47$	−3.77353e7	−1.12800	−0.563998	+	0.825776i	$0.690738\pi$
$48$	0	0
$49$	3.55626e7	0.881274
$50$	0	0
$51$	0	0
$52$	−1.09571e7	−0.207817
$53$	−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
$56$	−7.52451e6	−0.102243
$57$	0	0
$58$	−677827.	−0.00786490
$59$	−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	$0.300334\pi$
$62$	1.68656e6	0.0144957
$63$	0	0
$64$	−1.33099e8	−0.991663
$65$	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	$0.200475\pi$
$68$	−1.53499e8	−0.870593
$69$	0	0
$70$	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	$-0.509678\pi$
$73$	−1.47516e7	−0.0607977	−0.0303989	+	0.999538i	$0.509678\pi$
$74$	8.04832e6	0.0312006
$75$	0	0
$76$	−2.89074e8	−0.993913
$77$	3.88231e8	1.25858
$78$	0	0
$79$	−4.66498e7	−0.134750	−0.0673748	−	0.997728i	$-0.521462\pi$
$79$	−4.66498e7	−0.134750	−0.0673748	+	0.997728i	$0.521462\pi$
$80$	0	0
$81$	0	0
$82$	−2.14661e7	−0.0524313
$83$	2.01840e8	0.466826	0.233413	−	0.972378i	$-0.425011\pi$
$83$	2.01840e8	0.466826	0.233413	+	0.972378i	$0.425011\pi$
$84$	0	0
$85$	0	0
$86$	−1.96495e7	−0.0387355
$87$	0	0
$88$	−3.84800e7	−0.0684010
$89$	−5.54039e8	−0.936020	−0.468010	−	0.883723i	$-0.655029\pi$
$89$	−5.54039e8	−0.936020	−0.468010	+	0.883723i	$0.655029\pi$
$90$	0	0
$91$	−1.86723e8	−0.285438
$92$	−4.86095e8	−0.707418
$93$	0	0
$94$	3.18465e7	0.0420713
$95$	0	0
$96$	0	0
$97$	−3.39489e8	−0.389361	−0.194681	−	0.980867i	$-0.562367\pi$
$97$	−3.39489e8	−0.389361	−0.194681	+	0.980867i	$0.562367\pi$
$98$	−3.00128e7	−0.0328692
$99$	0	0
$100$	0	0
$101$	−1.33921e9	−1.28056	−0.640282	−	0.768140i	$-0.721183\pi$
$101$	−1.33921e9	−1.28056	−0.640282	+	0.768140i	$0.721183\pi$
$102$	0	0
$103$	3.84306e8	0.336442	0.168221	−	0.985749i	$-0.446198\pi$
$103$	3.84306e8	0.336442	0.168221	+	0.985749i	$0.446198\pi$
$104$	1.85073e7	0.0155129
$105$	0	0
$106$	4.04500e7	0.0311202
$107$	−7.97379e8	−0.588082	−0.294041	−	0.955793i	$-0.595000\pi$
$107$	−7.97379e8	−0.588082	−0.294041	+	0.955793i	$0.595000\pi$
$108$	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
$111$	0	0
$112$	−2.27453e9	−1.36587
$113$	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
$116$	−4.10649e8	−0.210577
$117$	0	0
$118$	5.90819e7	0.0280534
$119$	−2.61581e9	−1.19576
$120$	0	0
$121$	−3.72554e8	−0.157999
$122$	−1.07132e8	−0.0437824
$123$	0	0
$124$	1.02177e9	0.388112
$125$	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	$0.372416\pi$
$128$	4.51514e8	0.148671
$129$	0	0
$130$	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
$134$	−2.24991e8	−0.0602829
$135$	0	0
$136$	2.59269e8	0.0649869
$137$	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	$-0.733774\pi$
$139$	−5.89895e9	−1.34032	−0.670159	+	0.742217i	$0.733774\pi$
$140$	0	0
$141$	0	0
$142$	5.56301e7	0.0114819
$143$	−9.54892e8	−0.190960
$144$	0	0
$145$	0	0
$146$	1.24495e7	0.00226760
$147$	0	0
$148$	4.87593e9	0.835371
$149$	−5.39333e9	−0.896436	−0.448218	−	0.893924i	$-0.647941\pi$
$149$	−5.39333e9	−0.896436	−0.448218	+	0.893924i	$0.647941\pi$
$150$	0	0
$151$	7.92204e8	0.124005	0.0620027	−	0.998076i	$-0.480251\pi$
$151$	7.92204e8	0.124005	0.0620027	+	0.998076i	$0.480251\pi$
$152$	4.88265e8	0.0741924
$153$	0	0
$154$	−3.27645e8	−0.0469419
$155$	0	0
$156$	0	0
$157$	1.18606e10	1.55797	0.778985	−	0.627042i	$-0.215735\pi$
$157$	1.18606e10	1.55797	0.778985	+	0.627042i	$0.215735\pi$
$158$	3.93698e7	0.00502581
$159$	0	0
$160$	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	$0.571213\pi$
$164$	−1.30048e10	−1.40381
$165$	0	0
$166$	−1.70341e8	−0.0174114
$167$	−1.09118e10	−1.08560	−0.542802	−	0.839861i	$-0.682637\pi$
$167$	−1.09118e10	−1.08560	−0.542802	+	0.839861i	$0.682637\pi$
$168$	0	0
$169$	−1.01452e10	−0.956692
$170$	0	0
$171$	0	0
$172$	−1.19043e10	−1.03711
$173$	−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
$176$	−1.16318e10	−0.913778
$177$	0	0
$178$	4.67577e8	0.0349111
$179$	−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	$0.579858\pi$
$182$	1.57584e8	0.0106461
$183$	0	0
$184$	8.21045e8	0.0528064
$185$	0	0
$186$	0	0
$187$	−1.33771e10	−0.799974
$188$	1.92936e10	1.12643
$189$	0	0
$190$	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	$-0.197681\pi$
$193$	3.13528e10	1.62655	0.813277	+	0.581877i	$0.197681\pi$
$194$	2.86509e8	0.0145222
$195$	0	0
$196$	−1.81827e10	−0.880048
$197$	−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	$-0.592416\pi$
$199$	−1.26662e10	−0.572544	−0.286272	+	0.958148i	$0.592416\pi$
$200$	0	0
$201$	0	0
$202$	1.13021e9	0.0477617
$203$	−6.99798e9	−0.289228
$204$	0	0
$205$	0	0
$206$	−3.24333e8	−0.0125484
$207$	0	0
$208$	5.59443e9	0.207239
$209$	−2.51923e10	−0.913291
$210$	0	0
$211$	−6.20579e9	−0.215539	−0.107770	−	0.994176i	$-0.534371\pi$
$211$	−6.20579e9	−0.215539	−0.107770	+	0.994176i	$0.534371\pi$
$212$	2.45059e10	0.833218
$213$	0	0
$214$	6.72943e8	0.0219339
$215$	0	0
$216$	0	0
$217$	1.74123e10	0.533074
$218$	5.59729e8	0.0167851
$219$	0	0
$220$	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	$0.698258\pi$
$224$	5.77213e9	0.153186
$225$	0	0
$226$	−1.25042e9	−0.0318836
$227$	2.28857e10	0.572068	0.286034	−	0.958219i	$-0.407663\pi$
$227$	2.28857e10	0.572068	0.286034	+	0.958219i	$0.407663\pi$
$228$	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
$231$	0	0
$232$	6.93612e8	0.0157189
$233$	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
$236$	3.57937e10	0.751108
$237$	0	0
$238$	2.20760e9	0.0445989
$239$	−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	$0.378606\pi$
$242$	3.14415e8	0.00589296
$243$	0	0
$244$	−6.49039e10	−1.17224
$245$	0	0
$246$	0	0
$247$	1.21164e10	0.207128
$248$	−1.72584e9	−0.0289713
$249$	0	0
$250$	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
$254$	−1.93071e9	−0.0291048
$255$	0	0
$256$	6.77655e10	0.986118
$257$	−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
$261$	0	0
$262$	−3.24073e9	−0.0424901
$263$	−4.80103e10	−0.618776	−0.309388	−	0.950936i	$-0.600124\pi$
$263$	−4.80103e10	−0.618776	−0.309388	+	0.950936i	$0.600124\pi$
$264$	0	0
$265$	0	0
$266$	4.15743e9	0.0509164
$267$	0	0
$268$	−1.36307e11	−1.61403
$269$	9.98823e10	1.16306	0.581531	−	0.813524i	$-0.302454\pi$
$269$	9.98823e10	1.16306	0.581531	+	0.813524i	$0.302454\pi$
$270$	0	0
$271$	−4.80217e10	−0.540849	−0.270424	−	0.962741i	$-0.587164\pi$
$271$	−4.80217e10	−0.540849	−0.270424	+	0.962741i	$0.587164\pi$
$272$	7.83726e10	0.868169
$273$	0	0
$274$	−4.91737e9	−0.0527054
$275$	0	0
$276$	0	0
$277$	−1.77838e10	−0.181495	−0.0907477	−	0.995874i	$-0.528926\pi$
$277$	−1.77838e10	−0.181495	−0.0907477	+	0.995874i	$0.528926\pi$
$278$	4.97838e9	0.0499904
$279$	0	0
$280$	0	0
$281$	1.52044e11	1.45476	0.727379	−	0.686236i	$-0.240738\pi$
$281$	1.52044e11	1.45476	0.727379	+	0.686236i	$0.240738\pi$
$282$	0	0
$283$	1.86733e11	1.73055	0.865273	−	0.501301i	$-0.167145\pi$
$283$	1.86733e11	1.73055	0.865273	+	0.501301i	$0.167145\pi$
$284$	3.37025e10	0.307418
$285$	0	0
$286$	8.05875e8	0.00712230
$287$	−2.21619e11	−1.92814
$288$	0	0
$289$	−2.84558e10	−0.239955
$290$	0	0
$291$	0	0
$292$	7.54233e9	0.0607132
$293$	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
$296$	−8.23575e9	−0.0623577
$297$	0	0
$298$	4.55167e9	0.0334347
$299$	2.03745e10	0.147423
$300$	0	0
$301$	−2.02865e11	−1.42448
$302$	−6.68575e8	−0.00462508
$303$	0	0
$304$	1.47594e11	0.991146
$305$	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	$0.591053\pi$
$308$	−1.98498e11	−1.25683
$309$	0	0
$310$	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	$-0.317624\pi$
$313$	1.84107e11	1.08423	0.542115	+	0.840304i	$0.317624\pi$
$314$	−1.00097e10	−0.0581082
$315$	0	0
$316$	2.38515e10	0.134562
$317$	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
$321$	0	0
$322$	6.99095e9	0.0362397
$323$	1.69740e11	0.867706
$324$	0	0
$325$	0	0
$326$	3.37498e9	0.0165498
$327$	0	0
$328$	2.19660e10	0.104790
$329$	3.28788e11	1.54716
$330$	0	0
$331$	2.88744e11	1.32217	0.661084	−	0.750312i	$-0.270097\pi$
$331$	2.88744e11	1.32217	0.661084	+	0.750312i	$0.270097\pi$
$332$	−1.03198e11	−0.466177
$333$	0	0
$334$	9.20892e9	0.0404902
$335$	0	0
$336$	0	0
$337$	1.25882e11	0.531654	0.265827	−	0.964021i	$-0.414355\pi$
$337$	1.25882e11	0.531654	0.265827	+	0.964021i	$0.414355\pi$
$338$	8.56201e9	0.0356821
$339$	0	0
$340$	0	0
$341$	8.90455e10	0.356630
$342$	0	0
$343$	4.17441e10	0.162844
$344$	2.01071e10	0.0774172
$345$	0	0
$346$	1.59782e10	0.0599356
$347$	−2.95822e11	−1.09534	−0.547669	−	0.836695i	$-0.684485\pi$
$347$	−2.95822e11	−1.09534	−0.547669	+	0.836695i	$0.684485\pi$
$348$	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
$351$	0	0
$352$	2.95183e10	0.102483
$353$	−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
$356$	2.83273e11	0.934718
$357$	0	0
$358$	1.77029e10	0.0569600
$359$	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
$362$	6.05065e9	0.0185188
$363$	0	0
$364$	9.54693e10	0.285041
$365$	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	$0.511653\pi$
$368$	2.48188e11	0.705448
$369$	0	0
$370$	0	0
$371$	4.17611e11	1.14443
$372$	0	0
$373$	−7.26003e11	−1.94200	−0.970998	−	0.239087i	$-0.923152\pi$
$373$	−7.26003e11	−1.94200	−0.970998	+	0.239087i	$0.923152\pi$
$374$	1.12895e10	0.0298369
$375$	0	0
$376$	−3.25881e10	−0.0840842
$377$	1.72122e10	0.0438834
$378$	0	0
$379$	−2.76699e11	−0.688859	−0.344430	−	0.938812i	$-0.611928\pi$
$379$	−2.76699e11	−0.688859	−0.344430	+	0.938812i	$0.611928\pi$
$380$	0	0
$381$	0	0
$382$	1.47944e10	0.0355478
$383$	−1.71143e11	−0.406410	−0.203205	−	0.979136i	$-0.565136\pi$
$383$	−1.71143e11	−0.406410	−0.203205	+	0.979136i	$0.565136\pi$
$384$	0	0
$385$	0	0
$386$	−2.64600e10	−0.0606662
$387$	0	0
$388$	1.73576e11	0.388819
$389$	−3.92384e10	−0.0868837	−0.0434419	−	0.999056i	$-0.513832\pi$
$389$	−3.92384e10	−0.0868837	−0.0434419	+	0.999056i	$0.513832\pi$
$390$	0	0
$391$	2.85427e11	0.617590
$392$	3.07117e10	0.0656927
$393$	0	0
$394$	1.56949e10	0.0328114
$395$	0	0
$396$	0	0
$397$	3.91381e11	0.790756	0.395378	−	0.918519i	$-0.370614\pi$
$397$	3.91381e11	0.790756	0.395378	+	0.918519i	$0.370614\pi$
$398$	1.06896e10	0.0213544
$399$	0	0
$400$	0	0
$401$	5.04969e11	0.975248	0.487624	−	0.873054i	$-0.337864\pi$
$401$	5.04969e11	0.975248	0.487624	+	0.873054i	$0.337864\pi$
$402$	0	0
$403$	−4.28272e10	−0.0808811
$404$	6.84720e11	1.27878
$405$	0	0
$406$	5.90590e9	0.0107875
$407$	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
$411$	0	0
$412$	−1.96491e11	−0.335974
$413$	6.09969e11	1.03165
$414$	0	0
$415$	0	0
$416$	−1.41971e10	−0.0232423
$417$	0	0
$418$	2.12609e10	0.0340634
$419$	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	$0.498916\pi$
$422$	5.23734e9	0.00803905
$423$	0	0
$424$	−4.13920e10	−0.0621970
$425$	0	0
$426$	0	0
$427$	−1.10604e12	−1.61008
$428$	4.07690e11	0.587264
$429$	0	0
$430$	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	$-0.541813\pi$
$433$	−1.91618e11	−0.261963	−0.130982	+	0.991385i	$0.541813\pi$
$434$	−1.46950e10	−0.0198823
$435$	0	0
$436$	3.39101e11	0.449407
$437$	5.37527e11	0.705072
$438$	0	0
$439$	1.38202e12	1.77592	0.887962	−	0.459917i	$-0.152121\pi$
$439$	1.38202e12	1.77592	0.887962	+	0.459917i	$0.152121\pi$
$440$	0	0
$441$	0	0
$442$	−5.42980e9	−0.00676681
$443$	3.05660e11	0.377070	0.188535	−	0.982066i	$-0.439626\pi$
$443$	3.05660e11	0.377070	0.188535	+	0.982066i	$0.439626\pi$
$444$	0	0
$445$	0	0
$446$	3.63617e10	0.0435148
$447$	0	0
$448$	1.15969e12	1.36016
$449$	−1.49518e12	−1.73614	−0.868069	−	0.496444i	$-0.834639\pi$
$449$	−1.49518e12	−1.73614	−0.868069	+	0.496444i	$0.834639\pi$
$450$	0	0
$451$	−1.13335e12	−1.28994
$452$	−7.57542e11	−0.853658
$453$	0	0
$454$	−1.93142e10	−0.0213367
$455$	0	0
$456$	0	0
$457$	1.17977e12	1.26525	0.632624	−	0.774459i	$-0.281978\pi$
$457$	1.17977e12	1.26525	0.632624	+	0.774459i	$0.281978\pi$
$458$	4.44545e9	0.00472086
$459$	0	0
$460$	0	0
$461$	−1.54415e12	−1.59234	−0.796170	−	0.605074i	$-0.793144\pi$
$461$	−1.54415e12	−1.59234	−0.796170	+	0.605074i	$0.793144\pi$
$462$	0	0
$463$	−4.55769e11	−0.460926	−0.230463	−	0.973081i	$-0.574024\pi$
$463$	−4.55769e11	−0.460926	−0.230463	+	0.973081i	$0.574024\pi$
$464$	2.09667e11	0.209990
$465$	0	0
$466$	−2.99751e10	−0.0294458
$467$	−6.97342e11	−0.678454	−0.339227	−	0.940705i	$-0.610165\pi$
$467$	−6.97342e11	−0.678454	−0.339227	+	0.940705i	$0.610165\pi$
$468$	0	0
$469$	−2.32284e12	−2.21688
$470$	0	0
$471$	0	0
$472$	−6.04578e10	−0.0560677
$473$	−1.03744e12	−0.952987
$474$	0	0
$475$	0	0
$476$	1.33743e12	1.19410
$477$	0	0
$478$	4.83134e10	0.0423294
$479$	−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
$482$	−3.28994e10	−0.0277637
$483$	0	0
$484$	1.90482e11	0.157780
$485$	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	$0.717864\pi$
$488$	1.09627e11	0.0875039
$489$	0	0
$490$	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
$494$	−1.02256e10	−0.00772534
$495$	0	0
$496$	−5.21691e11	−0.387031
$497$	5.74333e11	0.422241
$498$	0	0
$499$	−7.51465e11	−0.542571	−0.271285	−	0.962499i	$-0.587449\pi$
$499$	−7.51465e11	−0.542571	−0.271285	+	0.962499i	$0.587449\pi$
$500$	0	0
$501$	0	0
$502$	3.85593e10	0.0270996
$503$	−1.31120e12	−0.913299	−0.456649	−	0.889647i	$-0.650951\pi$
$503$	−1.31120e12	−0.913299	−0.456649	+	0.889647i	$0.650951\pi$
$504$	0	0
$505$	0	0
$506$	3.57513e10	0.0242446
$507$	0	0
$508$	−1.16968e12	−0.779259
$509$	−1.78629e12	−1.17957	−0.589783	−	0.807561i	$-0.700787\pi$
$509$	−1.78629e12	−1.17957	−0.589783	+	0.807561i	$0.700787\pi$
$510$	0	0
$511$	1.28531e11	0.0833899
$512$	−2.88366e11	−0.185451
$513$	0	0
$514$	1.09675e10	0.00693063
$515$	0	0
$516$	0	0
$517$	1.68140e12	1.03506
$518$	−7.01249e10	−0.0427946
$519$	0	0
$520$	0	0
$521$	−5.88627e11	−0.350002	−0.175001	−	0.984568i	$-0.555993\pi$
$521$	−5.88627e11	−0.350002	−0.175001	+	0.984568i	$0.555993\pi$
$522$	0	0
$523$	6.01202e11	0.351369	0.175684	−	0.984447i	$-0.443786\pi$
$523$	6.01202e11	0.351369	0.175684	+	0.984447i	$0.443786\pi$
$524$	−1.96334e12	−1.13764
$525$	0	0
$526$	4.05180e10	0.0230787
$527$	−5.99968e11	−0.338829
$528$	0	0
$529$	−8.97272e11	−0.498165
$530$	0	0
$531$	0	0
$532$	2.51870e12	1.36325
$533$	5.45093e11	0.292549
$534$	0	0
$535$	0	0
$536$	2.30231e11	0.120482
$537$	0	0
$538$	−8.42950e10	−0.0433792
$539$	−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	$0.384995\pi$
$542$	4.05276e10	0.0201723
$543$	0	0
$544$	−1.98888e11	−0.0973673
$545$	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	$0.506126\pi$
$548$	−2.97910e12	−1.41115
$549$	0	0
$550$	0	0
$551$	4.54098e11	0.209878
$552$	0	0
$553$	4.06459e11	0.184822
$554$	1.50085e10	0.00676930
$555$	0	0
$556$	3.01606e12	1.33845
$557$	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
$561$	0	0
$562$	−1.28317e11	−0.0542587
$563$	3.86627e12	1.62182	0.810912	−	0.585168i	$-0.198971\pi$
$563$	3.86627e12	1.62182	0.810912	+	0.585168i	$0.198971\pi$
$564$	0	0
$565$	0	0
$566$	−1.57592e11	−0.0645448
$567$	0	0
$568$	−5.69256e10	−0.0229478
$569$	1.87990e12	0.751849	0.375924	−	0.926650i	$-0.377325\pi$
$569$	1.87990e12	0.751849	0.375924	+	0.926650i	$0.377325\pi$
$570$	0	0
$571$	−3.44726e12	−1.35710	−0.678550	−	0.734554i	$-0.737391\pi$
$571$	−3.44726e12	−1.35710	−0.678550	+	0.734554i	$0.737391\pi$
$572$	4.88225e11	0.190694
$573$	0	0
$574$	1.87034e11	0.0719146
$575$	0	0
$576$	0	0
$577$	−1.68433e12	−0.632608	−0.316304	−	0.948658i	$-0.602442\pi$
$577$	−1.68433e12	−0.632608	−0.316304	+	0.948658i	$0.602442\pi$
$578$	2.40151e10	0.00894971
$579$	0	0
$580$	0	0
$581$	−1.75863e12	−0.640297
$582$	0	0
$583$	2.13564e12	0.765631
$584$	−1.27395e10	−0.00453204
$585$	0	0
$586$	−1.08808e11	−0.0381173
$587$	−5.04715e12	−1.75458	−0.877292	−	0.479956i	$-0.840652\pi$
$587$	−5.04715e12	−1.75458	−0.877292	+	0.479956i	$0.840652\pi$
$588$	0	0
$589$	−1.12988e12	−0.386825
$590$	0	0
$591$	0	0
$592$	−2.48952e12	−0.833045
$593$	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
$596$	2.75755e12	0.895189
$597$	0	0
$598$	−1.71949e10	−0.00549851
$599$	−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	$0.644152\pi$
$602$	1.71206e11	0.0531295
$603$	0	0
$604$	−4.05044e11	−0.123833
$605$	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	$0.668990\pi$
$608$	−3.74553e11	−0.111160
$609$	0	0
$610$	0	0
$611$	−8.08685e11	−0.234744
$612$	0	0
$613$	−7.53866e11	−0.215636	−0.107818	−	0.994171i	$-0.534386\pi$
$613$	−7.53866e11	−0.215636	−0.107818	+	0.994171i	$0.534386\pi$
$614$	7.41262e10	0.0210482
$615$	0	0
$616$	3.35275e11	0.0938185
$617$	−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	$-0.545062\pi$
$619$	−1.03073e12	−0.282186	−0.141093	+	0.989996i	$0.545062\pi$
$620$	0	0
$621$	0	0
$622$	2.33100e11	0.0624433
$623$	4.82733e12	1.28384
$624$	0	0
$625$	0	0
$626$	−1.55376e11	−0.0404390
$627$	0	0
$628$	−6.06419e12	−1.55580
$629$	−2.86307e12	−0.729296
$630$	0	0
$631$	−4.61780e12	−1.15959	−0.579793	−	0.814764i	$-0.696867\pi$
$631$	−4.61780e12	−1.15959	−0.579793	+	0.814764i	$0.696867\pi$
$632$	−4.02866e10	−0.0100446
$633$	0	0
$634$	−2.94422e11	−0.0723717
$635$	0	0
$636$	0	0
$637$	7.62122e11	0.183399
$638$	3.02025e10	0.00721688
$639$	0	0
$640$	0	0
$641$	−7.51099e12	−1.75726	−0.878630	−	0.477503i	$-0.841542\pi$
$641$	−7.51099e12	−1.75726	−0.878630	+	0.477503i	$0.841542\pi$
$642$	0	0
$643$	4.42841e12	1.02164	0.510821	−	0.859687i	$-0.329341\pi$
$643$	4.42841e12	1.02164	0.510821	+	0.859687i	$0.329341\pi$
$644$	4.23534e12	0.970291
$645$	0	0
$646$	−1.43251e11	−0.0323632
$647$	1.09396e12	0.245432	0.122716	−	0.992442i	$-0.460840\pi$
$647$	1.09396e12	0.245432	0.122716	+	0.992442i	$0.460840\pi$
$648$	0	0
$649$	3.11935e12	0.690181
$650$	0	0
$651$	0	0
$652$	2.04467e12	0.443107
$653$	−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
$656$	6.63994e12	1.39990
$657$	0	0
$658$	−2.77478e11	−0.0577049
$659$	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	$0.134987\pi$
$662$	−2.43683e11	−0.0493134
$663$	0	0
$664$	1.74308e11	0.0347986
$665$	0	0
$666$	0	0
$667$	7.63592e11	0.149381
$668$	5.57906e12	1.08409
$669$	0	0
$670$	0	0
$671$	−5.65625e12	−1.07715
$672$	0	0
$673$	−6.40541e12	−1.20359	−0.601796	−	0.798650i	$-0.705548\pi$
$673$	−6.40541e12	−1.20359	−0.601796	+	0.798650i	$0.705548\pi$
$674$	−1.06237e11	−0.0198293
$675$	0	0
$676$	5.18713e12	0.955361
$677$	−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
$681$	0	0
$682$	−7.51494e10	−0.0133014
$683$	−2.15592e12	−0.379088	−0.189544	−	0.981872i	$-0.560701\pi$
$683$	−2.15592e12	−0.379088	−0.189544	+	0.981872i	$0.560701\pi$
$684$	0	0
$685$	0	0
$686$	−3.52297e10	−0.00607366
$687$	0	0
$688$	6.07804e12	1.03423
$689$	−1.02715e12	−0.173640
$690$	0	0
$691$	−4.58074e12	−0.764336	−0.382168	−	0.924093i	$-0.624822\pi$
$691$	−4.58074e12	−0.764336	−0.382168	+	0.924093i	$0.624822\pi$
$692$	9.68009e12	1.60473
$693$	0	0
$694$	2.49657e11	0.0408533
$695$	0	0
$696$	0	0
$697$	7.63624e12	1.22555
$698$	−3.15191e11	−0.0502603
$699$	0	0
$700$	0	0
$701$	1.22474e12	0.191564	0.0957819	−	0.995402i	$-0.469465\pi$
$701$	1.22474e12	0.191564	0.0957819	+	0.995402i	$0.469465\pi$
$702$	0	0
$703$	−5.39183e12	−0.832601
$704$	5.93058e12	0.909956
$705$	0	0
$706$	4.13424e11	0.0626289
$707$	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
$711$	0	0
$712$	−4.78466e11	−0.0697736
$713$	−1.89996e12	−0.275322
$714$	0	0
$715$	0	0
$716$	1.07250e13	1.52506
$717$	0	0
$718$	−3.60836e11	−0.0506698
$719$	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
$722$	2.55506e9	0.000349933 0
$723$	0	0
$724$	3.66568e12	0.495827
$725$	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	$0.444039\pi$
$728$	−1.61254e11	−0.0212774
$729$	0	0
$730$	0	0
$731$	6.99002e12	0.905421
$732$	0	0
$733$	5.28609e12	0.676343	0.338171	−	0.941085i	$-0.390192\pi$
$733$	5.28609e12	0.676343	0.338171	+	0.941085i	$0.390192\pi$
$734$	2.14702e10	0.00273026
$735$	0	0
$736$	−6.29831e11	−0.0791178
$737$	−1.18789e13	−1.48310
$738$	0	0
$739$	2.51810e12	0.310580	0.155290	−	0.987869i	$-0.450369\pi$
$739$	2.51810e12	0.310580	0.155290	+	0.987869i	$0.450369\pi$
$740$	0	0
$741$	0	0
$742$	−3.52440e11	−0.0426843
$743$	−7.02537e11	−0.0845706	−0.0422853	−	0.999106i	$-0.513464\pi$
$743$	−7.02537e11	−0.0845706	−0.0422853	+	0.999106i	$0.513464\pi$
$744$	0	0
$745$	0	0
$746$	6.12705e11	0.0724314
$747$	0	0
$748$	6.83956e12	0.798861
$749$	6.94755e12	0.806610
$750$	0	0
$751$	4.60108e12	0.527813	0.263907	−	0.964548i	$-0.414989\pi$
$751$	4.60108e12	0.527813	0.263907	+	0.964548i	$0.414989\pi$
$752$	−9.85083e12	−1.12329
$753$	0	0
$754$	−1.45261e10	−0.00163674
$755$	0	0
$756$	0	0
$757$	−3.05764e12	−0.338419	−0.169210	−	0.985580i	$-0.554122\pi$
$757$	−3.05764e12	−0.338419	−0.169210	+	0.985580i	$0.554122\pi$
$758$	2.33518e11	0.0256927
$759$	0	0
$760$	0	0
$761$	9.86753e12	1.06654	0.533270	−	0.845945i	$-0.320963\pi$
$761$	9.86753e12	1.06654	0.533270	+	0.845945i	$0.320963\pi$
$762$	0	0
$763$	5.77872e12	0.617264
$764$	8.96293e12	0.951765
$765$	0	0
$766$	1.44435e11	0.0151580
$767$	−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
$771$	0	0
$772$	−1.60303e13	−1.62429
$773$	−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
$776$	−2.93181e11	−0.0290241
$777$	0	0
$778$	3.31150e10	0.00324054
$779$	1.43808e13	1.39915
$780$	0	0
$781$	2.93711e12	0.282482
$782$	−2.40884e11	−0.0230345
$783$	0	0
$784$	9.28363e12	0.877598
$785$	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	$0.524752\pi$
$788$	9.50846e12	0.878501
$789$	0	0
$790$	0	0
$791$	−1.29095e13	−1.17250
$792$	0	0
$793$	2.72042e12	0.244291
$794$	−3.30303e11	−0.0294931
$795$	0	0
$796$	6.47609e12	0.571748
$797$	−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
$801$	0	0
$802$	−4.26165e11	−0.0363742
$803$	6.57300e11	0.0557884
$804$	0	0
$805$	0	0
$806$	3.61437e10	0.00301665
$807$	0	0
$808$	−1.15654e12	−0.0954570
$809$	−5.27673e12	−0.433108	−0.216554	−	0.976271i	$-0.569482\pi$
$809$	−5.27673e12	−0.433108	−0.216554	+	0.976271i	$0.569482\pi$
$810$	0	0
$811$	−1.43675e12	−0.116624	−0.0583118	−	0.998298i	$-0.518572\pi$
$811$	−1.43675e12	−0.116624	−0.0583118	+	0.998298i	$0.518572\pi$
$812$	3.57798e12	0.288826
$813$	0	0
$814$	−3.58615e11	−0.0286298
$815$	0	0
$816$	0	0
$817$	1.31639e13	1.03367
$818$	−7.97525e9	−0.000622809 0
$819$	0	0
$820$	0	0
$821$	−9.44899e12	−0.725840	−0.362920	−	0.931820i	$-0.618220\pi$
$821$	−9.44899e12	−0.725840	−0.362920	+	0.931820i	$0.618220\pi$
$822$	0	0
$823$	−1.20237e13	−0.913562	−0.456781	−	0.889579i	$-0.650998\pi$
$823$	−1.20237e13	−0.913562	−0.456781	+	0.889579i	$0.650998\pi$
$824$	3.31886e11	0.0250794
$825$	0	0
$826$	−5.14780e11	−0.0384779
$827$	7.86848e12	0.584947	0.292473	−	0.956274i	$-0.405522\pi$
$827$	7.86848e12	0.584947	0.292473	+	0.956274i	$0.405522\pi$
$828$	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
$831$	0	0
$832$	−2.85236e12	−0.206372
$833$	1.06766e13	0.768299
$834$	0	0
$835$	0	0
$836$	1.28805e13	0.912021
$837$	0	0
$838$	−4.87232e10	−0.00341301
$839$	−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
$842$	−3.70471e9	−0.000254009 0
$843$	0	0
$844$	3.17294e12	0.215239
$845$	0	0
$846$	0	0
$847$	3.24606e12	0.216711
$848$	−1.25121e13	−0.830899
$849$	0	0
$850$	0	0
$851$	−9.06666e12	−0.592604
$852$	0	0
$853$	9.20458e11	0.0595296	0.0297648	−	0.999557i	$-0.490524\pi$
$853$	9.20458e11	0.0595296	0.0297648	+	0.999557i	$0.490524\pi$
$854$	9.33439e11	0.0600517
$855$	0	0
$856$	−6.88614e11	−0.0438373
$857$	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	$-0.718643\pi$
$859$	−2.02385e13	−1.26826	−0.634132	+	0.773225i	$0.718643\pi$
$860$	0	0
$861$	0	0
$862$	8.39281e11	0.0517756
$863$	3.07364e13	1.88627	0.943136	−	0.332406i	$-0.107860\pi$
$863$	3.07364e13	1.88627	0.943136	+	0.332406i	$0.107860\pi$
$864$	0	0
$865$	0	0
$866$	1.61715e11	0.00977055
$867$	0	0
$868$	−8.90269e12	−0.532332
$869$	2.07861e12	0.123647
$870$	0	0
$871$	5.71325e12	0.336358
$872$	−5.72763e11	−0.0335468
$873$	0	0
$874$	−4.53642e11	−0.0262973
$875$	0	0
$876$	0	0
$877$	−6.65840e12	−0.380077	−0.190039	−	0.981777i	$-0.560861\pi$
$877$	−6.65840e12	−0.380077	−0.190039	+	0.981777i	$0.560861\pi$
$878$	−1.16635e12	−0.0662373
$879$	0	0
$880$	0	0
$881$	−3.41842e12	−0.191176	−0.0955882	−	0.995421i	$-0.530473\pi$
$881$	−3.41842e12	−0.191176	−0.0955882	+	0.995421i	$0.530473\pi$
$882$	0	0
$883$	−1.92500e13	−1.06563	−0.532815	−	0.846231i	$-0.678866\pi$
$883$	−1.92500e13	−1.06563	−0.532815	+	0.846231i	$0.678866\pi$
$884$	−3.28955e12	−0.181176
$885$	0	0
$886$	−2.57960e11	−0.0140637
$887$	1.48977e13	0.808095	0.404048	−	0.914738i	$-0.367603\pi$
$887$	1.48977e13	0.808095	0.404048	+	0.914738i	$0.367603\pi$
$888$	0	0
$889$	−1.99329e13	−1.07032
$890$	0	0
$891$	0	0
$892$	2.20291e13	1.16508
$893$	−2.13350e13	−1.12269
$894$	0	0
$895$	0	0
$896$	−3.93404e12	−0.203917
$897$	0	0
$898$	1.26185e12	0.0647534
$899$	−1.60507e12	−0.0819551
$900$	0	0
$901$	−1.43895e13	−0.727416
$902$	9.56481e11	0.0481113
$903$	0	0
$904$	1.27954e12	0.0637228
$905$	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	$0.411784\pi$
$908$	−1.17012e13	−0.571273
$909$	0	0
$910$	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
$914$	−9.95663e11	−0.0471905
$915$	0	0
$916$	2.69319e12	0.126397
$917$	−3.34578e13	−1.56255
$918$	0	0
$919$	1.40633e13	0.650380	0.325190	−	0.945649i	$-0.394572\pi$
$919$	1.40633e13	0.650380	0.325190	+	0.945649i	$0.394572\pi$
$920$	0	0
$921$	0	0
$922$	1.30318e12	0.0593901
$923$	−1.41263e12	−0.0640649
$924$	0	0
$925$	0	0
$926$	3.84644e11	0.0171913
$927$	0	0
$928$	−5.32077e11	−0.0235509
$929$	−1.04912e13	−0.462118	−0.231059	−	0.972940i	$-0.574219\pi$
$929$	−1.04912e13	−0.462118	−0.231059	+	0.972940i	$0.574219\pi$
$930$	0	0
$931$	2.01066e13	0.877130
$932$	−1.81599e13	−0.788389
$933$	0	0
$934$	5.88518e11	0.0253045
$935$	0	0
$936$	0	0
$937$	−3.77841e13	−1.60133	−0.800666	−	0.599111i	$-0.795521\pi$
$937$	−3.77841e13	−1.60133	−0.800666	+	0.599111i	$0.795521\pi$
$938$	1.96035e12	0.0826837
$939$	0	0
$940$	0	0
$941$	−1.11142e13	−0.462088	−0.231044	−	0.972943i	$-0.574214\pi$
$941$	−1.11142e13	−0.462088	−0.231044	+	0.972943i	$0.574214\pi$
$942$	0	0
$943$	2.41822e13	0.995847
$944$	−1.82753e13	−0.749017
$945$	0	0
$946$	8.75539e11	0.0355439
$947$	4.96563e12	0.200631	0.100316	−	0.994956i	$-0.468015\pi$
$947$	4.96563e12	0.200631	0.100316	+	0.994956i	$0.468015\pi$
$948$	0	0
$949$	−3.16134e11	−0.0126524
$950$	0	0
$951$	0	0
$952$	−2.25901e12	−0.0891357
$953$	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
$956$	2.92698e13	1.13334
$957$	0	0
$958$	2.05476e11	0.00788162
$959$	−5.07676e13	−1.93822
$960$	0	0
$961$	−2.24459e13	−0.848949
$962$	1.72479e11	0.00649304
$963$	0	0
$964$	−1.99315e13	−0.743351
$965$	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	$0.651053\pi$
$968$	−3.21737e11	−0.0117777
$969$	0	0
$970$	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
$974$	1.32466e12	0.0471618
$975$	0	0
$976$	3.31383e13	1.16898
$977$	−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
$981$	0	0
$982$	5.50985e11	0.0189077
$983$	−2.85792e13	−0.976247	−0.488123	−	0.872775i	$-0.662318\pi$
$983$	−2.85792e13	−0.976247	−0.488123	+	0.872775i	$0.662318\pi$
$984$	0	0
$985$	0	0
$986$	−2.03497e11	−0.00685666
$987$	0	0
$988$	−6.19499e12	−0.206840
$989$	2.21358e13	0.735718
$990$	0	0
$991$	−3.49036e13	−1.14958	−0.574789	−	0.818302i	$-0.694916\pi$
$991$	−3.49036e13	−1.14958	−0.574789	+	0.818302i	$0.694916\pi$
$992$	1.32391e12	0.0434065
$993$	0	0
$994$	−4.84705e11	−0.0157485
$995$	0	0
$996$	0	0
$997$	8.78834e12	0.281695	0.140847	−	0.990031i	$-0.455017\pi$
$997$	8.78834e12	0.281695	0.140847	+	0.990031i	$0.455017\pi$
$998$	6.34194e11	0.0202365
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.10.b.a.4.2	✓	4	15.8	even	4
5.10.b.a.4.3	yes	4	15.2	even	4
25.10.a.e.1.2		4	15.14	odd	2
25.10.a.e.1.3		4	3.2	odd	2
45.10.b.b.19.2		4	5.2	odd	4
45.10.b.b.19.3		4	5.3	odd	4
80.10.c.c.49.1		4	60.47	odd	4
80.10.c.c.49.4		4	60.23	odd	4
225.10.a.s.1.2		4	1.1	even	1	trivial
225.10.a.s.1.3		4	5.4	even	2	inner
400.10.a.ba.1.1		4	60.59	even	2
400.10.a.ba.1.4		4	12.11	even	2

Overview	Random
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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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.843944 q^{2} -511.288 q^{4} -8712.99 q^{7} +863.597 q^{8} +O(q^{10})\)	\(q-0.843944 q^{2} -511.288 q^{4} -8712.99 q^{7} +863.597 q^{8} -44557.8 q^{11} +21430.4 q^{13} +7353.27 q^{14} +261051. q^{16} +300220. q^{17} +565385. q^{19} +37604.2 q^{22} +950727. q^{23} -18086.1 q^{26} +4.45485e6 q^{28} +803167. q^{29} -1.99843e6 q^{31} -662474. q^{32} -253369. q^{34} -9.53656e6 q^{37} -477153. q^{38} +2.54355e7 q^{41} +2.32830e7 q^{43} +2.27818e7 q^{44} -802360. q^{46} -3.77353e7 q^{47} +3.55626e7 q^{49} -1.09571e7 q^{52} -4.79297e7 q^{53} -7.52451e6 q^{56} -677827. q^{58} -7.00069e7 q^{59} +1.26942e8 q^{61} +1.68656e6 q^{62} -1.33099e8 q^{64} +2.66595e8 q^{67} -1.53499e8 q^{68} -6.59169e7 q^{71} -1.47516e7 q^{73} +8.04832e6 q^{74} -2.89074e8 q^{76} +3.88231e8 q^{77} -4.66498e7 q^{79} -2.14661e7 q^{82} +2.01840e8 q^{83} -1.96495e7 q^{86} -3.84800e7 q^{88} -5.54039e8 q^{89} -1.86723e8 q^{91} -4.86095e8 q^{92} +3.18465e7 q^{94} -3.39489e8 q^{97} -3.00128e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 1368 q^{4}+O(q^{10}) \) 4 * q + 1368 * q^4	\( 4 q + 1368 q^{4} - 109968 q^{11} - 424536 q^{14} + 1631264 q^{16} + 636880 q^{19} - 6618768 q^{26} - 3531720 q^{29} - 10587712 q^{31} - 26434624 q^{34} + 16788552 q^{41} + 20638944 q^{44} - 61250072 q^{46} + 46921028 q^{49} - 315178080 q^{56} - 460829040 q^{59} + 360490568 q^{61} - 134995072 q^{64} + 47611872 q^{71} + 1176861744 q^{74} - 1168489440 q^{76} + 728043520 q^{79} + 2375904552 q^{86} - 1582700760 q^{89} + 473322528 q^{91} - 3327101704 q^{94}+O(q^{100}) \) 4 * q + 1368 * q^4 - 109968 * q^11 - 424536 * q^14 + 1631264 * q^16 + 636880 * q^19 - 6618768 * q^26 - 3531720 * q^29 - 10587712 * q^31 - 26434624 * q^34 + 16788552 * q^41 + 20638944 * q^44 - 61250072 * q^46 + 46921028 * q^49 - 315178080 * q^56 - 460829040 * q^59 + 360490568 * q^61 - 134995072 * q^64 + 47611872 * q^71 + 1176861744 * q^74 - 1168489440 * q^76 + 728043520 * q^79 + 2375904552 * q^86 - 1582700760 * q^89 + 473322528 * q^91 - 3327101704 * q^94

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