LMFDB - Embedded newform 225.10.a.p.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:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 652x + 4000 $ x^3 - x^2 - 652*x + 4000
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2\cdot 5 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$22.2334$ 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)

$f(q)$	$=$	$q+21.4187 q^{2} -53.2406 q^{4} -9905.49 q^{7} -12106.7 q^{8} +O(q^{10})$	$q+21.4187 q^{2} -53.2406 q^{4} -9905.49 q^{7} -12106.7 q^{8} -36453.6 q^{11} -164867. q^{13} -212162. q^{14} -232050. q^{16} +82357.1 q^{17} -609617. q^{19} -780787. q^{22} +1.88578e6 q^{23} -3.53123e6 q^{26} +527374. q^{28} -339235. q^{29} +547314. q^{31} +1.22842e6 q^{32} +1.76398e6 q^{34} -5.25687e6 q^{37} -1.30572e7 q^{38} -2.05812e6 q^{41} +6.76158e6 q^{43} +1.94081e6 q^{44} +4.03909e7 q^{46} +3.15241e7 q^{47} +5.77651e7 q^{49} +8.77761e6 q^{52} -4.89593e7 q^{53} +1.19923e8 q^{56} -7.26597e6 q^{58} -8.77960e7 q^{59} +3.84654e7 q^{61} +1.17227e7 q^{62} +1.45121e8 q^{64} +1.36116e8 q^{67} -4.38474e6 q^{68} -3.49218e8 q^{71} -1.61345e8 q^{73} -1.12595e8 q^{74} +3.24564e7 q^{76} +3.61091e8 q^{77} -1.26975e8 q^{79} -4.40822e7 q^{82} -2.87494e8 q^{83} +1.44824e8 q^{86} +4.41333e8 q^{88} +5.63133e8 q^{89} +1.63309e9 q^{91} -1.00400e8 q^{92} +6.75205e8 q^{94} +4.71704e8 q^{97} +1.23725e9 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 33 q^{2} + 341 q^{4} - 5258 q^{7} - 105 q^{8}+O(q^{10}) $ 3 * q + 33 * q^2 + 341 * q^4 - 5258 * q^7 - 105 * q^8	$ 3 q + 33 q^{2} + 341 q^{4} - 5258 q^{7} - 105 q^{8} + 54699 q^{11} - 215884 q^{13} - 272922 q^{14} - 699247 q^{16} + 334983 q^{17} + 818845 q^{19} - 72761 q^{22} + 3526854 q^{23} + 1280004 q^{26} + 428134 q^{28} - 2175480 q^{29} + 4274066 q^{31} - 9464577 q^{32} - 6838963 q^{34} + 10305042 q^{37} - 24180495 q^{38} - 5926311 q^{41} + 24429956 q^{43} + 21995703 q^{44} + 14223246 q^{46} + 66858708 q^{47} - 6453929 q^{49} + 57862932 q^{52} + 132620514 q^{53} + 169538130 q^{56} - 201908320 q^{58} - 5670960 q^{59} + 125306926 q^{61} - 39831174 q^{62} + 167542401 q^{64} - 88829483 q^{67} - 71162559 q^{68} - 297550596 q^{71} - 181321729 q^{73} + 251507358 q^{74} + 89414865 q^{76} + 561214086 q^{77} - 310025170 q^{79} + 1368322979 q^{82} - 731088801 q^{83} - 33947196 q^{86} + 943671285 q^{88} + 1103860035 q^{89} + 1183187656 q^{91} - 190024242 q^{92} + 1727891132 q^{94} + 332236842 q^{97} + 457927581 q^{98}+O(q^{100}) $ 3 * q + 33 * q^2 + 341 * q^4 - 5258 * q^7 - 105 * q^8 + 54699 * q^11 - 215884 * q^13 - 272922 * q^14 - 699247 * q^16 + 334983 * q^17 + 818845 * q^19 - 72761 * q^22 + 3526854 * q^23 + 1280004 * q^26 + 428134 * q^28 - 2175480 * q^29 + 4274066 * q^31 - 9464577 * q^32 - 6838963 * q^34 + 10305042 * q^37 - 24180495 * q^38 - 5926311 * q^41 + 24429956 * q^43 + 21995703 * q^44 + 14223246 * q^46 + 66858708 * q^47 - 6453929 * q^49 + 57862932 * q^52 + 132620514 * q^53 + 169538130 * q^56 - 201908320 * q^58 - 5670960 * q^59 + 125306926 * q^61 - 39831174 * q^62 + 167542401 * q^64 - 88829483 * q^67 - 71162559 * q^68 - 297550596 * q^71 - 181321729 * q^73 + 251507358 * q^74 + 89414865 * q^76 + 561214086 * q^77 - 310025170 * q^79 + 1368322979 * q^82 - 731088801 * q^83 - 33947196 * q^86 + 943671285 * q^88 + 1103860035 * q^89 + 1183187656 * q^91 - 190024242 * q^92 + 1727891132 * q^94 + 332236842 * q^97 + 457927581 * q^98

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 $

$2$	21.4187	0.946580	0.473290	−	0.880907i	$-0.343066\pi$
$2$	21.4187	0.946580	0.473290	+	0.880907i	$0.343066\pi$
$3$	0	0
$3$	0	0
$4$	−53.2406	−0.103986
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−9905.49	−1.55932	−0.779659	−	0.626204i	$-0.784608\pi$
$7$	−9905.49	−1.55932	−0.779659	+	0.626204i	$0.784608\pi$
$8$	−12106.7	−1.04501
$9$	0	0
$10$	0	0
$11$	−36453.6	−0.750712	−0.375356	−	0.926881i	$-0.622480\pi$
$11$	−36453.6	−0.750712	−0.375356	+	0.926881i	$0.622480\pi$
$12$	0	0
$13$	−164867.	−1.60099	−0.800494	−	0.599341i	$-0.795429\pi$
$13$	−164867.	−1.60099	−0.800494	+	0.599341i	$0.795429\pi$
$14$	−212162.	−1.47602
$15$	0	0
$16$	−232050.	−0.885201
$17$	82357.1	0.239156	0.119578	−	0.992825i	$-0.461846\pi$
$17$	82357.1	0.239156	0.119578	+	0.992825i	$0.461846\pi$
$18$	0	0
$19$	−609617.	−1.07316	−0.536582	−	0.843848i	$-0.680285\pi$
$19$	−609617.	−1.07316	−0.536582	+	0.843848i	$0.680285\pi$
$20$	0	0
$21$	0	0
$22$	−780787.	−0.710609
$23$	1.88578e6	1.40513	0.702564	−	0.711620i	$-0.252038\pi$
$23$	1.88578e6	1.40513	0.702564	+	0.711620i	$0.252038\pi$
$24$	0	0
$25$	0	0
$26$	−3.53123e6	−1.51546
$27$	0	0
$28$	527374.	0.162147
$29$	−339235.	−0.0890657	−0.0445328	−	0.999008i	$-0.514180\pi$
$29$	−339235.	−0.0890657	−0.0445328	+	0.999008i	$0.514180\pi$
$30$	0	0
$31$	547314.	0.106441	0.0532205	−	0.998583i	$-0.483051\pi$
$31$	547314.	0.106441	0.0532205	+	0.998583i	$0.483051\pi$
$32$	1.22842e6	0.207097
$33$	0	0
$34$	1.76398e6	0.226380
$35$	0	0
$36$	0	0
$37$	−5.25687e6	−0.461126	−0.230563	−	0.973057i	$-0.574057\pi$
$37$	−5.25687e6	−0.461126	−0.230563	+	0.973057i	$0.574057\pi$
$38$	−1.30572e7	−1.01584
$39$	0	0
$40$	0	0
$41$	−2.05812e6	−0.113748	−0.0568739	−	0.998381i	$-0.518113\pi$
$41$	−2.05812e6	−0.113748	−0.0568739	+	0.998381i	$0.518113\pi$
$42$	0	0
$43$	6.76158e6	0.301606	0.150803	−	0.988564i	$-0.451814\pi$
$43$	6.76158e6	0.301606	0.150803	+	0.988564i	$0.451814\pi$
$44$	1.94081e6	0.0780632
$45$	0	0
$46$	4.03909e7	1.33007
$47$	3.15241e7	0.942330	0.471165	−	0.882045i	$-0.343834\pi$
$47$	3.15241e7	0.942330	0.471165	+	0.882045i	$0.343834\pi$
$48$	0	0
$49$	5.77651e7	1.43147
$50$	0	0
$51$	0	0
$52$	8.77761e6	0.166480
$53$	−4.89593e7	−0.852303	−0.426151	−	0.904652i	$-0.640131\pi$
$53$	−4.89593e7	−0.852303	−0.426151	+	0.904652i	$0.640131\pi$
$54$	0	0
$55$	0	0
$56$	1.19923e8	1.62950
$57$	0	0
$58$	−7.26597e6	−0.0843078
$59$	−8.77960e7	−0.943281	−0.471640	−	0.881791i	$-0.656338\pi$
$59$	−8.77960e7	−0.943281	−0.471640	+	0.881791i	$0.656338\pi$
$60$	0	0
$61$	3.84654e7	0.355701	0.177851	−	0.984057i	$-0.443086\pi$
$61$	3.84654e7	0.355701	0.177851	+	0.984057i	$0.443086\pi$
$62$	1.17227e7	0.100755
$63$	0	0
$64$	1.45121e8	1.08124
$65$	0	0
$66$	0	0
$67$	1.36116e8	0.825227	0.412613	−	0.910906i	$-0.364616\pi$
$67$	1.36116e8	0.825227	0.412613	+	0.910906i	$0.364616\pi$
$68$	−4.38474e6	−0.0248687
$69$	0	0
$70$	0	0
$71$	−3.49218e8	−1.63092	−0.815462	−	0.578810i	$-0.803517\pi$
$71$	−3.49218e8	−1.63092	−0.815462	+	0.578810i	$0.803517\pi$
$72$	0	0
$73$	−1.61345e8	−0.664971	−0.332486	−	0.943108i	$-0.607887\pi$
$73$	−1.61345e8	−0.664971	−0.332486	+	0.943108i	$0.607887\pi$
$74$	−1.12595e8	−0.436492
$75$	0	0
$76$	3.24564e7	0.111594
$77$	3.61091e8	1.17060
$78$	0	0
$79$	−1.26975e8	−0.366772	−0.183386	−	0.983041i	$-0.558706\pi$
$79$	−1.26975e8	−0.366772	−0.183386	+	0.983041i	$0.558706\pi$
$80$	0	0
$81$	0	0
$82$	−4.40822e7	−0.107671
$83$	−2.87494e8	−0.664931	−0.332466	−	0.943115i	$-0.607881\pi$
$83$	−2.87494e8	−0.664931	−0.332466	+	0.943115i	$0.607881\pi$
$84$	0	0
$85$	0	0
$86$	1.44824e8	0.285494
$87$	0	0
$88$	4.41333e8	0.784502
$89$	5.63133e8	0.951385	0.475692	−	0.879612i	$-0.342198\pi$
$89$	5.63133e8	0.951385	0.475692	+	0.879612i	$0.342198\pi$
$90$	0	0
$91$	1.63309e9	2.49645
$92$	−1.00400e8	−0.146113
$93$	0	0
$94$	6.75205e8	0.891991
$95$	0	0
$96$	0	0
$97$	4.71704e8	0.541000	0.270500	−	0.962720i	$-0.412811\pi$
$97$	4.71704e8	0.541000	0.270500	+	0.962720i	$0.412811\pi$
$98$	1.23725e9	1.35500
$99$	0	0
$100$	0	0
$101$	1.25673e9	1.20170	0.600851	−	0.799361i	$-0.294828\pi$
$101$	1.25673e9	1.20170	0.600851	+	0.799361i	$0.294828\pi$
$102$	0	0
$103$	−1.95398e9	−1.71062	−0.855309	−	0.518119i	$-0.826633\pi$
$103$	−1.95398e9	−1.71062	−0.855309	+	0.518119i	$0.826633\pi$
$104$	1.99599e9	1.67305
$105$	0	0
$106$	−1.04864e9	−0.806773
$107$	2.12306e9	1.56580	0.782899	−	0.622148i	$-0.213740\pi$
$107$	2.12306e9	1.56580	0.782899	+	0.622148i	$0.213740\pi$
$108$	0	0
$109$	−1.18863e8	−0.0806544	−0.0403272	−	0.999187i	$-0.512840\pi$
$109$	−1.18863e8	−0.0806544	−0.0403272	+	0.999187i	$0.512840\pi$
$110$	0	0
$111$	0	0
$112$	2.29857e9	1.38031
$113$	−1.85455e9	−1.07001	−0.535003	−	0.844850i	$-0.679689\pi$
$113$	−1.85455e9	−1.07001	−0.535003	+	0.844850i	$0.679689\pi$
$114$	0	0
$115$	0	0
$116$	1.80611e7	0.00926154
$117$	0	0
$118$	−1.88047e9	−0.892891
$119$	−8.15787e8	−0.372920
$120$	0	0
$121$	−1.02908e9	−0.436432
$122$	8.23877e8	0.336700
$123$	0	0
$124$	−2.91393e7	−0.0110683
$125$	0	0
$126$	0	0
$127$	5.39208e9	1.83924	0.919622	−	0.392805i	$-0.128495\pi$
$127$	5.39208e9	1.83924	0.919622	+	0.392805i	$0.128495\pi$
$128$	2.47934e9	0.816379
$129$	0	0
$130$	0	0
$131$	−3.83690e9	−1.13831	−0.569154	−	0.822231i	$-0.692729\pi$
$131$	−3.83690e9	−1.13831	−0.569154	+	0.822231i	$0.692729\pi$
$132$	0	0
$133$	6.03856e9	1.67340
$134$	2.91543e9	0.781143
$135$	0	0
$136$	−9.97072e8	−0.249920
$137$	−2.55408e9	−0.619430	−0.309715	−	0.950829i	$-0.600234\pi$
$137$	−2.55408e9	−0.619430	−0.309715	+	0.950829i	$0.600234\pi$
$138$	0	0
$139$	−4.09121e9	−0.929577	−0.464789	−	0.885422i	$-0.653870\pi$
$139$	−4.09121e9	−0.929577	−0.464789	+	0.885422i	$0.653870\pi$
$140$	0	0
$141$	0	0
$142$	−7.47978e9	−1.54380
$143$	6.00999e9	1.20188
$144$	0	0
$145$	0	0
$146$	−3.45580e9	−0.629449
$147$	0	0
$148$	2.79879e8	0.0479504
$149$	6.13643e9	1.01995	0.509974	−	0.860190i	$-0.329655\pi$
$149$	6.13643e9	1.01995	0.509974	+	0.860190i	$0.329655\pi$
$150$	0	0
$151$	−5.83754e9	−0.913763	−0.456882	−	0.889528i	$-0.651034\pi$
$151$	−5.83754e9	−0.913763	−0.456882	+	0.889528i	$0.651034\pi$
$152$	7.38046e9	1.12147
$153$	0	0
$154$	7.73408e9	1.10807
$155$	0	0
$156$	0	0
$157$	1.35031e9	0.177371	0.0886857	−	0.996060i	$-0.471733\pi$
$157$	1.35031e9	0.177371	0.0886857	+	0.996060i	$0.471733\pi$
$158$	−2.71963e9	−0.347179
$159$	0	0
$160$	0	0
$161$	−1.86796e10	−2.19104
$162$	0	0
$163$	4.50806e9	0.500202	0.250101	−	0.968220i	$-0.419536\pi$
$163$	4.50806e9	0.500202	0.250101	+	0.968220i	$0.419536\pi$
$164$	1.09575e8	0.0118281
$165$	0	0
$166$	−6.15773e9	−0.629411
$167$	1.81836e10	1.80908	0.904538	−	0.426393i	$-0.140216\pi$
$167$	1.81836e10	1.80908	0.904538	+	0.426393i	$0.140216\pi$
$168$	0	0
$169$	1.65766e10	1.56316
$170$	0	0
$171$	0	0
$172$	−3.59991e8	−0.0313627
$173$	9.77143e9	0.829375	0.414687	−	0.909964i	$-0.363891\pi$
$173$	9.77143e9	0.829375	0.414687	+	0.909964i	$0.363891\pi$
$174$	0	0
$175$	0	0
$176$	8.45906e9	0.664531
$177$	0	0
$178$	1.20616e10	0.900562
$179$	−6.65866e9	−0.484784	−0.242392	−	0.970178i	$-0.577932\pi$
$179$	−6.65866e9	−0.484784	−0.242392	+	0.970178i	$0.577932\pi$
$180$	0	0
$181$	−7.32873e9	−0.507546	−0.253773	−	0.967264i	$-0.581672\pi$
$181$	−7.32873e9	−0.507546	−0.253773	+	0.967264i	$0.581672\pi$
$182$	3.49785e10	2.36309
$183$	0	0
$184$	−2.28306e10	−1.46837
$185$	0	0
$186$	0	0
$187$	−3.00221e9	−0.179537
$188$	−1.67836e9	−0.0979887
$189$	0	0
$190$	0	0
$191$	−8.06956e9	−0.438732	−0.219366	−	0.975643i	$-0.570399\pi$
$191$	−8.06956e9	−0.438732	−0.219366	+	0.975643i	$0.570399\pi$
$192$	0	0
$193$	−3.12603e9	−0.162176	−0.0810878	−	0.996707i	$-0.525839\pi$
$193$	−3.12603e9	−0.162176	−0.0810878	+	0.996707i	$0.525839\pi$
$194$	1.01033e10	0.512100
$195$	0	0
$196$	−3.07545e9	−0.148853
$197$	−2.40133e10	−1.13594	−0.567968	−	0.823051i	$-0.692270\pi$
$197$	−2.40133e10	−1.13594	−0.567968	+	0.823051i	$0.692270\pi$
$198$	0	0
$199$	9.73605e9	0.440093	0.220046	−	0.975489i	$-0.429379\pi$
$199$	9.73605e9	0.440093	0.220046	+	0.975489i	$0.429379\pi$
$200$	0	0
$201$	0	0
$202$	2.69175e10	1.13751
$203$	3.36029e9	0.138882
$204$	0	0
$205$	0	0
$206$	−4.18517e10	−1.61924
$207$	0	0
$208$	3.82574e10	1.41720
$209$	2.22227e10	0.805637
$210$	0	0
$211$	−2.39305e10	−0.831151	−0.415576	−	0.909559i	$-0.636420\pi$
$211$	−2.39305e10	−0.831151	−0.415576	+	0.909559i	$0.636420\pi$
$212$	2.60662e9	0.0886272
$213$	0	0
$214$	4.54732e10	1.48215
$215$	0	0
$216$	0	0
$217$	−5.42141e9	−0.165975
$218$	−2.54589e9	−0.0763459
$219$	0	0
$220$	0	0
$221$	−1.35779e10	−0.382885
$222$	0	0
$223$	−2.02289e10	−0.547773	−0.273886	−	0.961762i	$-0.588309\pi$
$223$	−2.02289e10	−0.547773	−0.273886	+	0.961762i	$0.588309\pi$
$224$	−1.21681e10	−0.322930
$225$	0	0
$226$	−3.97220e10	−1.01285
$227$	4.42152e10	1.10524	0.552618	−	0.833434i	$-0.313629\pi$
$227$	4.42152e10	1.10524	0.552618	+	0.833434i	$0.313629\pi$
$228$	0	0
$229$	−4.89483e10	−1.17619	−0.588096	−	0.808791i	$-0.700122\pi$
$229$	−4.89483e10	−1.17619	−0.588096	+	0.808791i	$0.700122\pi$
$230$	0	0
$231$	0	0
$232$	4.10702e9	0.0930746
$233$	6.68132e10	1.48512	0.742559	−	0.669781i	$-0.233612\pi$
$233$	6.68132e10	1.48512	0.742559	+	0.669781i	$0.233612\pi$
$234$	0	0
$235$	0	0
$236$	4.67431e9	0.0980876
$237$	0	0
$238$	−1.74731e10	−0.352999
$239$	2.21153e10	0.438431	0.219216	−	0.975676i	$-0.429650\pi$
$239$	2.21153e10	0.438431	0.219216	+	0.975676i	$0.429650\pi$
$240$	0	0
$241$	9.97637e10	1.90500	0.952502	−	0.304532i	$-0.0985002\pi$
$241$	9.97637e10	1.90500	0.952502	+	0.304532i	$0.0985002\pi$
$242$	−2.20416e10	−0.413118
$243$	0	0
$244$	−2.04792e9	−0.0369878
$245$	0	0
$246$	0	0
$247$	1.00506e11	1.71812
$248$	−6.62616e9	−0.111232
$249$	0	0
$250$	0	0
$251$	4.97276e10	0.790799	0.395399	−	0.918509i	$-0.370606\pi$
$251$	4.97276e10	0.790799	0.395399	+	0.918509i	$0.370606\pi$
$252$	0	0
$253$	−6.87435e10	−1.05485
$254$	1.15491e11	1.74099
$255$	0	0
$256$	−2.11977e10	−0.308467
$257$	5.81308e10	0.831204	0.415602	−	0.909547i	$-0.363571\pi$
$257$	5.81308e10	0.831204	0.415602	+	0.909547i	$0.363571\pi$
$258$	0	0
$259$	5.20718e10	0.719041
$260$	0	0
$261$	0	0
$262$	−8.21814e10	−1.07750
$263$	−3.23476e10	−0.416909	−0.208454	−	0.978032i	$-0.566843\pi$
$263$	−3.23476e10	−0.416909	−0.208454	+	0.978032i	$0.566843\pi$
$264$	0	0
$265$	0	0
$266$	1.29338e11	1.58401
$267$	0	0
$268$	−7.24691e9	−0.0858116
$269$	−6.24220e9	−0.0726863	−0.0363431	−	0.999339i	$-0.511571\pi$
$269$	−6.24220e9	−0.0726863	−0.0363431	+	0.999339i	$0.511571\pi$
$270$	0	0
$271$	8.05816e10	0.907557	0.453779	−	0.891114i	$-0.350076\pi$
$271$	8.05816e10	0.907557	0.453779	+	0.891114i	$0.350076\pi$
$272$	−1.91110e10	−0.211701
$273$	0	0
$274$	−5.47050e10	−0.586340
$275$	0	0
$276$	0	0
$277$	7.07976e10	0.722536	0.361268	−	0.932462i	$-0.382344\pi$
$277$	7.07976e10	0.722536	0.361268	+	0.932462i	$0.382344\pi$
$278$	−8.76284e10	−0.879920
$279$	0	0
$280$	0	0
$281$	−8.15619e10	−0.780385	−0.390193	−	0.920733i	$-0.627592\pi$
$281$	−8.15619e10	−0.780385	−0.390193	+	0.920733i	$0.627592\pi$
$282$	0	0
$283$	−1.89301e11	−1.75434	−0.877168	−	0.480184i	$-0.840570\pi$
$283$	−1.89301e11	−1.75434	−0.877168	+	0.480184i	$0.840570\pi$
$284$	1.85926e10	0.169593
$285$	0	0
$286$	1.28726e11	1.13768
$287$	2.03867e10	0.177369
$288$	0	0
$289$	−1.11805e11	−0.942805
$290$	0	0
$291$	0	0
$292$	8.59011e9	0.0691474
$293$	−5.39489e10	−0.427640	−0.213820	−	0.976873i	$-0.568591\pi$
$293$	−5.39489e10	−0.427640	−0.213820	+	0.976873i	$0.568591\pi$
$294$	0	0
$295$	0	0
$296$	6.36433e10	0.481881
$297$	0	0
$298$	1.31434e11	0.965462
$299$	−3.10903e11	−2.24959
$300$	0	0
$301$	−6.69768e10	−0.470300
$302$	−1.25032e11	−0.864950
$303$	0	0
$304$	1.41462e11	0.949966
$305$	0	0
$306$	0	0
$307$	1.30037e11	0.835496	0.417748	−	0.908563i	$-0.362819\pi$
$307$	1.30037e11	0.835496	0.417748	+	0.908563i	$0.362819\pi$
$308$	−1.92247e10	−0.121725
$309$	0	0
$310$	0	0
$311$	−2.09580e9	−0.0127036	−0.00635182	−	0.999980i	$-0.502022\pi$
$311$	−2.09580e9	−0.0127036	−0.00635182	+	0.999980i	$0.502022\pi$
$312$	0	0
$313$	2.43788e11	1.43570	0.717848	−	0.696200i	$-0.245127\pi$
$313$	2.43788e11	1.43570	0.717848	+	0.696200i	$0.245127\pi$
$314$	2.89217e10	0.167896
$315$	0	0
$316$	6.76022e9	0.0381390
$317$	−3.46483e11	−1.92715	−0.963573	−	0.267445i	$-0.913821\pi$
$317$	−3.46483e11	−1.92715	−0.963573	+	0.267445i	$0.913821\pi$
$318$	0	0
$319$	1.23664e10	0.0668626
$320$	0	0
$321$	0	0
$322$	−4.00092e11	−2.07400
$323$	−5.02063e10	−0.256653
$324$	0	0
$325$	0	0
$326$	9.65566e10	0.473481
$327$	0	0
$328$	2.49170e10	0.118868
$329$	−3.12262e11	−1.46939
$330$	0	0
$331$	−1.43303e11	−0.656189	−0.328094	−	0.944645i	$-0.606406\pi$
$331$	−1.43303e11	−0.656189	−0.328094	+	0.944645i	$0.606406\pi$
$332$	1.53063e10	0.0691433
$333$	0	0
$334$	3.89470e11	1.71244
$335$	0	0
$336$	0	0
$337$	−3.34820e11	−1.41409	−0.707044	−	0.707170i	$-0.749972\pi$
$337$	−3.34820e11	−1.41409	−0.707044	+	0.707170i	$0.749972\pi$
$338$	3.55048e11	1.47966
$339$	0	0
$340$	0	0
$341$	−1.99515e10	−0.0799064
$342$	0	0
$343$	−1.72469e11	−0.672804
$344$	−8.18604e10	−0.315182
$345$	0	0
$346$	2.09291e11	0.785070
$347$	−9.22101e10	−0.341425	−0.170713	−	0.985321i	$-0.554607\pi$
$347$	−9.22101e10	−0.341425	−0.170713	+	0.985321i	$0.554607\pi$
$348$	0	0
$349$	−4.71738e11	−1.70211	−0.851053	−	0.525080i	$-0.824035\pi$
$349$	−4.71738e11	−1.70211	−0.851053	+	0.525080i	$0.824035\pi$
$350$	0	0
$351$	0	0
$352$	−4.47805e10	−0.155470
$353$	−4.01029e11	−1.37464	−0.687322	−	0.726353i	$-0.741214\pi$
$353$	−4.01029e11	−1.37464	−0.687322	+	0.726353i	$0.741214\pi$
$354$	0	0
$355$	0	0
$356$	−2.99816e10	−0.0989303
$357$	0	0
$358$	−1.42620e11	−0.458887
$359$	2.36110e10	0.0750220	0.0375110	−	0.999296i	$-0.488057\pi$
$359$	2.36110e10	0.0750220	0.0375110	+	0.999296i	$0.488057\pi$
$360$	0	0
$361$	4.89457e10	0.151681
$362$	−1.56972e11	−0.480433
$363$	0	0
$364$	−8.69465e10	−0.259595
$365$	0	0
$366$	0	0
$367$	−1.23035e11	−0.354022	−0.177011	−	0.984209i	$-0.556643\pi$
$367$	−1.23035e11	−0.354022	−0.177011	+	0.984209i	$0.556643\pi$
$368$	−4.37596e11	−1.24382
$369$	0	0
$370$	0	0
$371$	4.84966e11	1.32901
$372$	0	0
$373$	2.39248e11	0.639968	0.319984	−	0.947423i	$-0.396322\pi$
$373$	2.39248e11	0.639968	0.319984	+	0.947423i	$0.396322\pi$
$374$	−6.43034e10	−0.169946
$375$	0	0
$376$	−3.81653e11	−0.984745
$377$	5.59287e10	0.142593
$378$	0	0
$379$	−5.90629e11	−1.47041	−0.735205	−	0.677845i	$-0.762914\pi$
$379$	−5.90629e11	−1.47041	−0.735205	+	0.677845i	$0.762914\pi$
$380$	0	0
$381$	0	0
$382$	−1.72839e11	−0.415295
$383$	−5.30525e10	−0.125983	−0.0629914	−	0.998014i	$-0.520064\pi$
$383$	−5.30525e10	−0.125983	−0.0629914	+	0.998014i	$0.520064\pi$
$384$	0	0
$385$	0	0
$386$	−6.69554e10	−0.153512
$387$	0	0
$388$	−2.51138e10	−0.0562562
$389$	−3.10395e11	−0.687293	−0.343646	−	0.939099i	$-0.611662\pi$
$389$	−3.10395e11	−0.687293	−0.343646	+	0.939099i	$0.611662\pi$
$390$	0	0
$391$	1.55307e11	0.336045
$392$	−6.99345e11	−1.49591
$393$	0	0
$394$	−5.14333e11	−1.07525
$395$	0	0
$396$	0	0
$397$	2.09351e11	0.422979	0.211489	−	0.977380i	$-0.432169\pi$
$397$	2.09351e11	0.422979	0.211489	+	0.977380i	$0.432169\pi$
$398$	2.08533e11	0.416583
$399$	0	0
$400$	0	0
$401$	−4.43868e11	−0.857244	−0.428622	−	0.903484i	$-0.641001\pi$
$401$	−4.43868e11	−0.857244	−0.428622	+	0.903484i	$0.641001\pi$
$402$	0	0
$403$	−9.02338e10	−0.170411
$404$	−6.69092e10	−0.124960
$405$	0	0
$406$	7.19730e10	0.131463
$407$	1.91632e11	0.346172
$408$	0	0
$409$	4.56031e11	0.805823	0.402912	−	0.915239i	$-0.367998\pi$
$409$	4.56031e11	0.805823	0.402912	+	0.915239i	$0.367998\pi$
$410$	0	0
$411$	0	0
$412$	1.04031e11	0.177880
$413$	8.69663e11	1.47087
$414$	0	0
$415$	0	0
$416$	−2.02526e11	−0.331559
$417$	0	0
$418$	4.75982e11	0.762600
$419$	1.12392e12	1.78144	0.890720	−	0.454552i	$-0.150201\pi$
$419$	1.12392e12	1.78144	0.890720	+	0.454552i	$0.150201\pi$
$420$	0	0
$421$	−1.02822e12	−1.59521	−0.797605	−	0.603180i	$-0.793900\pi$
$421$	−1.02822e12	−1.59521	−0.797605	+	0.603180i	$0.793900\pi$
$422$	−5.12559e11	−0.786751
$423$	0	0
$424$	5.92736e11	0.890666
$425$	0	0
$426$	0	0
$427$	−3.81018e11	−0.554652
$428$	−1.13033e11	−0.162820
$429$	0	0
$430$	0	0
$431$	1.37714e12	1.92234	0.961171	−	0.275953i	$-0.0889934\pi$
$431$	1.37714e12	1.92234	0.961171	+	0.275953i	$0.0889934\pi$
$432$	0	0
$433$	−1.94790e11	−0.266299	−0.133150	−	0.991096i	$-0.542509\pi$
$433$	−1.94790e11	−0.266299	−0.133150	+	0.991096i	$0.542509\pi$
$434$	−1.16119e11	−0.157109
$435$	0	0
$436$	6.32835e9	0.00838689
$437$	−1.14961e12	−1.50793
$438$	0	0
$439$	−5.32533e11	−0.684315	−0.342157	−	0.939643i	$-0.611158\pi$
$439$	−5.32533e11	−0.684315	−0.342157	+	0.939643i	$0.611158\pi$
$440$	0	0
$441$	0	0
$442$	−2.90821e11	−0.362432
$443$	7.26633e11	0.896393	0.448196	−	0.893935i	$-0.352067\pi$
$443$	7.26633e11	0.896393	0.448196	+	0.893935i	$0.352067\pi$
$444$	0	0
$445$	0	0
$446$	−4.33276e11	−0.518511
$447$	0	0
$448$	−1.43749e12	−1.68599
$449$	−9.34830e11	−1.08549	−0.542743	−	0.839899i	$-0.682614\pi$
$449$	−9.34830e11	−1.08549	−0.542743	+	0.839899i	$0.682614\pi$
$450$	0	0
$451$	7.50258e10	0.0853918
$452$	9.87374e10	0.111265
$453$	0	0
$454$	9.47031e11	1.04620
$455$	0	0
$456$	0	0
$457$	1.09190e12	1.17101	0.585503	−	0.810670i	$-0.300897\pi$
$457$	1.09190e12	1.17101	0.585503	+	0.810670i	$0.300897\pi$
$458$	−1.04841e12	−1.11336
$459$	0	0
$460$	0	0
$461$	−9.93683e11	−1.02469	−0.512347	−	0.858779i	$-0.671224\pi$
$461$	−9.93683e11	−1.02469	−0.512347	+	0.858779i	$0.671224\pi$
$462$	0	0
$463$	6.95734e11	0.703605	0.351802	−	0.936074i	$-0.385569\pi$
$463$	6.95734e11	0.703605	0.351802	+	0.936074i	$0.385569\pi$
$464$	7.87197e10	0.0788411
$465$	0	0
$466$	1.43105e12	1.40578
$467$	1.87957e12	1.82866	0.914328	−	0.404975i	$-0.132720\pi$
$467$	1.87957e12	1.82866	0.914328	+	0.404975i	$0.132720\pi$
$468$	0	0
$469$	−1.34830e12	−1.28679
$470$	0	0
$471$	0	0
$472$	1.06292e12	0.985739
$473$	−2.46484e11	−0.226419
$474$	0	0
$475$	0	0
$476$	4.34330e10	0.0387783
$477$	0	0
$478$	4.73679e11	0.415010
$479$	2.29011e11	0.198768	0.0993838	−	0.995049i	$-0.468313\pi$
$479$	2.29011e11	0.198768	0.0993838	+	0.995049i	$0.468313\pi$
$480$	0	0
$481$	8.66683e11	0.738256
$482$	2.13681e12	1.80324
$483$	0	0
$484$	5.47890e10	0.0453826
$485$	0	0
$486$	0	0
$487$	4.81410e11	0.387824	0.193912	−	0.981019i	$-0.437882\pi$
$487$	4.81410e11	0.387824	0.193912	+	0.981019i	$0.437882\pi$
$488$	−4.65689e11	−0.371712
$489$	0	0
$490$	0	0
$491$	1.70320e12	1.32251	0.661257	−	0.750160i	$-0.270023\pi$
$491$	1.70320e12	1.32251	0.661257	+	0.750160i	$0.270023\pi$
$492$	0	0
$493$	−2.79384e10	−0.0213006
$494$	2.15270e12	1.62634
$495$	0	0
$496$	−1.27004e11	−0.0942217
$497$	3.45917e12	2.54313
$498$	0	0
$499$	2.06075e12	1.48789	0.743947	−	0.668238i	$-0.232951\pi$
$499$	2.06075e12	1.48789	0.743947	+	0.668238i	$0.232951\pi$
$500$	0	0
$501$	0	0
$502$	1.06510e12	0.748555
$503$	−5.97493e11	−0.416176	−0.208088	−	0.978110i	$-0.566724\pi$
$503$	−5.97493e11	−0.416176	−0.208088	+	0.978110i	$0.566724\pi$
$504$	0	0
$505$	0	0
$506$	−1.47239e12	−0.998497
$507$	0	0
$508$	−2.87077e11	−0.191255
$509$	8.46938e11	0.559270	0.279635	−	0.960106i	$-0.409787\pi$
$509$	8.46938e11	0.559270	0.279635	+	0.960106i	$0.409787\pi$
$510$	0	0
$511$	1.59820e12	1.03690
$512$	−1.72345e12	−1.10837
$513$	0	0
$514$	1.24508e12	0.786801
$515$	0	0
$516$	0	0
$517$	−1.14917e12	−0.707418
$518$	1.11531e12	0.680630
$519$	0	0
$520$	0	0
$521$	−7.31013e11	−0.434666	−0.217333	−	0.976098i	$-0.569736\pi$
$521$	−7.31013e11	−0.434666	−0.217333	+	0.976098i	$0.569736\pi$
$522$	0	0
$523$	−1.04453e12	−0.610467	−0.305233	−	0.952278i	$-0.598734\pi$
$523$	−1.04453e12	−0.610467	−0.305233	+	0.952278i	$0.598734\pi$
$524$	2.04279e11	0.118368
$525$	0	0
$526$	−6.92842e11	−0.394638
$527$	4.50751e10	0.0254559
$528$	0	0
$529$	1.75502e12	0.974386
$530$	0	0
$531$	0	0
$532$	−3.21496e11	−0.174010
$533$	3.39315e11	0.182109
$534$	0	0
$535$	0	0
$536$	−1.64792e12	−0.862371
$537$	0	0
$538$	−1.33700e11	−0.0688034
$539$	−2.10575e12	−1.07462
$540$	0	0
$541$	−1.88034e12	−0.943733	−0.471866	−	0.881670i	$-0.656420\pi$
$541$	−1.88034e12	−0.943733	−0.471866	+	0.881670i	$0.656420\pi$
$542$	1.72595e12	0.859076
$543$	0	0
$544$	1.01169e11	0.0495284
$545$	0	0
$546$	0	0
$547$	1.08624e12	0.518780	0.259390	−	0.965773i	$-0.416479\pi$
$547$	1.08624e12	0.518780	0.259390	+	0.965773i	$0.416479\pi$
$548$	1.35981e11	0.0644117
$549$	0	0
$550$	0	0
$551$	2.06804e11	0.0955821
$552$	0	0
$553$	1.25775e12	0.571914
$554$	1.51639e12	0.683938
$555$	0	0
$556$	2.17819e11	0.0966626
$557$	2.11965e12	0.933072	0.466536	−	0.884502i	$-0.345502\pi$
$557$	2.11965e12	0.933072	0.466536	+	0.884502i	$0.345502\pi$
$558$	0	0
$559$	−1.11476e12	−0.482868
$560$	0	0
$561$	0	0
$562$	−1.74695e12	−0.738697
$563$	3.04489e12	1.27727	0.638636	−	0.769509i	$-0.279499\pi$
$563$	3.04489e12	1.27727	0.638636	+	0.769509i	$0.279499\pi$
$564$	0	0
$565$	0	0
$566$	−4.05456e12	−1.66062
$567$	0	0
$568$	4.22788e12	1.70433
$569$	−3.25275e12	−1.30090	−0.650452	−	0.759548i	$-0.725420\pi$
$569$	−3.25275e12	−1.30090	−0.650452	+	0.759548i	$0.725420\pi$
$570$	0	0
$571$	−1.35916e12	−0.535066	−0.267533	−	0.963549i	$-0.586208\pi$
$571$	−1.35916e12	−0.535066	−0.267533	+	0.963549i	$0.586208\pi$
$572$	−3.19975e11	−0.124978
$573$	0	0
$574$	4.36655e11	0.167894
$575$	0	0
$576$	0	0
$577$	2.35782e12	0.885564	0.442782	−	0.896629i	$-0.353992\pi$
$577$	2.35782e12	0.885564	0.442782	+	0.896629i	$0.353992\pi$
$578$	−2.39472e12	−0.892440
$579$	0	0
$580$	0	0
$581$	2.84777e12	1.03684
$582$	0	0
$583$	1.78474e12	0.639834
$584$	1.95336e12	0.694902
$585$	0	0
$586$	−1.15551e12	−0.404796
$587$	−2.37065e12	−0.824130	−0.412065	−	0.911154i	$-0.635192\pi$
$587$	−2.37065e12	−0.824130	−0.412065	+	0.911154i	$0.635192\pi$
$588$	0	0
$589$	−3.33652e11	−0.114229
$590$	0	0
$591$	0	0
$592$	1.21986e12	0.408189
$593$	4.56707e12	1.51667	0.758336	−	0.651864i	$-0.226013\pi$
$593$	4.56707e12	1.51667	0.758336	+	0.651864i	$0.226013\pi$
$594$	0	0
$595$	0	0
$596$	−3.26707e11	−0.106060
$597$	0	0
$598$	−6.65912e12	−2.12942
$599$	5.30493e12	1.68368	0.841839	−	0.539729i	$-0.181473\pi$
$599$	5.30493e12	1.68368	0.841839	+	0.539729i	$0.181473\pi$
$600$	0	0
$601$	−2.71307e12	−0.848255	−0.424128	−	0.905602i	$-0.639419\pi$
$601$	−2.71307e12	−0.848255	−0.424128	+	0.905602i	$0.639419\pi$
$602$	−1.43455e12	−0.445177
$603$	0	0
$604$	3.10794e11	0.0950182
$605$	0	0
$606$	0	0
$607$	4.00734e12	1.19814	0.599069	−	0.800698i	$-0.295538\pi$
$607$	4.00734e12	1.19814	0.599069	+	0.800698i	$0.295538\pi$
$608$	−7.48868e11	−0.222249
$609$	0	0
$610$	0	0
$611$	−5.19728e12	−1.50866
$612$	0	0
$613$	2.24954e12	0.643459	0.321730	−	0.946832i	$-0.395736\pi$
$613$	2.24954e12	0.643459	0.321730	+	0.946832i	$0.395736\pi$
$614$	2.78522e12	0.790864
$615$	0	0
$616$	−4.37162e12	−1.22329
$617$	1.19664e12	0.332414	0.166207	−	0.986091i	$-0.446848\pi$
$617$	1.19664e12	0.332414	0.166207	+	0.986091i	$0.446848\pi$
$618$	0	0
$619$	−4.36723e12	−1.19563	−0.597817	−	0.801633i	$-0.703965\pi$
$619$	−4.36723e12	−1.19563	−0.597817	+	0.801633i	$0.703965\pi$
$620$	0	0
$621$	0	0
$622$	−4.48893e10	−0.0120250
$623$	−5.57811e12	−1.48351
$624$	0	0
$625$	0	0
$626$	5.22161e12	1.35900
$627$	0	0
$628$	−7.18911e10	−0.0184441
$629$	−4.32940e11	−0.110281
$630$	0	0
$631$	2.78790e12	0.700077	0.350039	−	0.936735i	$-0.386168\pi$
$631$	2.78790e12	0.700077	0.350039	+	0.936735i	$0.386168\pi$
$632$	1.53725e12	0.383281
$633$	0	0
$634$	−7.42120e12	−1.82420
$635$	0	0
$636$	0	0
$637$	−9.52355e12	−2.29177
$638$	2.64871e11	0.0632909
$639$	0	0
$640$	0	0
$641$	2.28295e12	0.534115	0.267058	−	0.963681i	$-0.413949\pi$
$641$	2.28295e12	0.534115	0.267058	+	0.963681i	$0.413949\pi$
$642$	0	0
$643$	2.29995e12	0.530603	0.265301	−	0.964166i	$-0.414529\pi$
$643$	2.29995e12	0.530603	0.265301	+	0.964166i	$0.414529\pi$
$644$	9.94512e11	0.227837
$645$	0	0
$646$	−1.07535e12	−0.242943
$647$	1.85647e12	0.416503	0.208251	−	0.978075i	$-0.433223\pi$
$647$	1.85647e12	0.416503	0.208251	+	0.978075i	$0.433223\pi$
$648$	0	0
$649$	3.20048e12	0.708132
$650$	0	0
$651$	0	0
$652$	−2.40012e11	−0.0520138
$653$	7.75397e12	1.66884	0.834420	−	0.551129i	$-0.185803\pi$
$653$	7.75397e12	1.66884	0.834420	+	0.551129i	$0.185803\pi$
$654$	0	0
$655$	0	0
$656$	4.77587e11	0.100690
$657$	0	0
$658$	−6.68824e12	−1.39090
$659$	−7.42166e12	−1.53291	−0.766455	−	0.642298i	$-0.777981\pi$
$659$	−7.42166e12	−1.53291	−0.766455	+	0.642298i	$0.777981\pi$
$660$	0	0
$661$	5.79783e12	1.18130	0.590648	−	0.806929i	$-0.298872\pi$
$661$	5.79783e12	1.18130	0.590648	+	0.806929i	$0.298872\pi$
$662$	−3.06936e12	−0.621135
$663$	0	0
$664$	3.48060e12	0.694861
$665$	0	0
$666$	0	0
$667$	−6.39724e11	−0.125149
$668$	−9.68108e11	−0.188118
$669$	0	0
$670$	0	0
$671$	−1.40220e12	−0.267029
$672$	0	0
$673$	6.32456e12	1.18840	0.594200	−	0.804318i	$-0.297469\pi$
$673$	6.32456e12	1.18840	0.594200	+	0.804318i	$0.297469\pi$
$674$	−7.17139e12	−1.33855
$675$	0	0
$676$	−8.82546e11	−0.162546
$677$	−4.95066e12	−0.905762	−0.452881	−	0.891571i	$-0.649604\pi$
$677$	−4.95066e12	−0.905762	−0.452881	+	0.891571i	$0.649604\pi$
$678$	0	0
$679$	−4.67246e12	−0.843591
$680$	0	0
$681$	0	0
$682$	−4.27336e11	−0.0756379
$683$	7.32611e12	1.28819	0.644095	−	0.764945i	$-0.277234\pi$
$683$	7.32611e12	1.28819	0.644095	+	0.764945i	$0.277234\pi$
$684$	0	0
$685$	0	0
$686$	−3.69406e12	−0.636863
$687$	0	0
$688$	−1.56903e12	−0.266982
$689$	8.07176e12	1.36453
$690$	0	0
$691$	−5.66806e12	−0.945765	−0.472882	−	0.881126i	$-0.656786\pi$
$691$	−5.66806e12	−0.945765	−0.472882	+	0.881126i	$0.656786\pi$
$692$	−5.20237e11	−0.0862430
$693$	0	0
$694$	−1.97502e12	−0.323187
$695$	0	0
$696$	0	0
$697$	−1.69501e11	−0.0272034
$698$	−1.01040e13	−1.61118
$699$	0	0
$700$	0	0
$701$	4.20985e12	0.658470	0.329235	−	0.944248i	$-0.393209\pi$
$701$	4.20985e12	0.658470	0.329235	+	0.944248i	$0.393209\pi$
$702$	0	0
$703$	3.20468e12	0.494863
$704$	−5.29018e12	−0.811696
$705$	0	0
$706$	−8.58952e12	−1.30121
$707$	−1.24486e13	−1.87384
$708$	0	0
$709$	1.19654e13	1.77835	0.889176	−	0.457566i	$-0.151279\pi$
$709$	1.19654e13	1.77835	0.889176	+	0.457566i	$0.151279\pi$
$710$	0	0
$711$	0	0
$712$	−6.81769e12	−0.994208
$713$	1.03211e12	0.149563
$714$	0	0
$715$	0	0
$716$	3.54511e11	0.0504105
$717$	0	0
$718$	5.05715e11	0.0710143
$719$	4.07118e11	0.0568120	0.0284060	−	0.999596i	$-0.490957\pi$
$719$	4.07118e11	0.0568120	0.0284060	+	0.999596i	$0.490957\pi$
$720$	0	0
$721$	1.93551e13	2.66740
$722$	1.04835e12	0.143579
$723$	0	0
$724$	3.90186e11	0.0527774
$725$	0	0
$726$	0	0
$727$	−6.44424e12	−0.855592	−0.427796	−	0.903875i	$-0.640710\pi$
$727$	−6.44424e12	−0.855592	−0.427796	+	0.903875i	$0.640710\pi$
$728$	−1.97713e13	−2.60882
$729$	0	0
$730$	0	0
$731$	5.56864e11	0.0721308
$732$	0	0
$733$	−5.46122e12	−0.698749	−0.349375	−	0.936983i	$-0.613606\pi$
$733$	−5.46122e12	−0.698749	−0.349375	+	0.936983i	$0.613606\pi$
$734$	−2.63524e12	−0.335110
$735$	0	0
$736$	2.31654e12	0.290998
$737$	−4.96192e12	−0.619507
$738$	0	0
$739$	−1.88201e12	−0.232125	−0.116062	−	0.993242i	$-0.537027\pi$
$739$	−1.88201e12	−0.232125	−0.116062	+	0.993242i	$0.537027\pi$
$740$	0	0
$741$	0	0
$742$	1.03873e13	1.25802
$743$	1.36704e13	1.64563	0.822813	−	0.568312i	$-0.192403\pi$
$743$	1.36704e13	1.64563	0.822813	+	0.568312i	$0.192403\pi$
$744$	0	0
$745$	0	0
$746$	5.12437e12	0.605781
$747$	0	0
$748$	1.59840e11	0.0186693
$749$	−2.10300e13	−2.44158
$750$	0	0
$751$	8.52585e12	0.978043	0.489021	−	0.872272i	$-0.337354\pi$
$751$	8.52585e12	0.978043	0.489021	+	0.872272i	$0.337354\pi$
$752$	−7.31518e12	−0.834152
$753$	0	0
$754$	1.19792e12	0.134976
$755$	0	0
$756$	0	0
$757$	7.50792e12	0.830975	0.415487	−	0.909599i	$-0.363611\pi$
$757$	7.50792e12	0.830975	0.415487	+	0.909599i	$0.363611\pi$
$758$	−1.26505e13	−1.39186
$759$	0	0
$760$	0	0
$761$	1.61560e13	1.74624	0.873118	−	0.487509i	$-0.162094\pi$
$761$	1.61560e13	1.74624	0.873118	+	0.487509i	$0.162094\pi$
$762$	0	0
$763$	1.17740e12	0.125766
$764$	4.29628e11	0.0456218
$765$	0	0
$766$	−1.13631e12	−0.119253
$767$	1.44747e13	1.51018
$768$	0	0
$769$	1.11830e12	0.115317	0.0576583	−	0.998336i	$-0.481637\pi$
$769$	1.11830e12	0.115317	0.0576583	+	0.998336i	$0.481637\pi$
$770$	0	0
$771$	0	0
$772$	1.66432e11	0.0168639
$773$	−1.73165e13	−1.74442	−0.872212	−	0.489129i	$-0.837315\pi$
$773$	−1.73165e13	−1.74442	−0.872212	+	0.489129i	$0.837315\pi$
$774$	0	0
$775$	0	0
$776$	−5.71078e12	−0.565351
$777$	0	0
$778$	−6.64825e12	−0.650578
$779$	1.25466e12	0.122070
$780$	0	0
$781$	1.27302e13	1.22435
$782$	3.32648e12	0.318093
$783$	0	0
$784$	−1.34044e13	−1.26714
$785$	0	0
$786$	0	0
$787$	3.88515e11	0.0361012	0.0180506	−	0.999837i	$-0.494254\pi$
$787$	3.88515e11	0.0361012	0.0180506	+	0.999837i	$0.494254\pi$
$788$	1.27848e12	0.118121
$789$	0	0
$790$	0	0
$791$	1.83702e13	1.66848
$792$	0	0
$793$	−6.34166e12	−0.569474
$794$	4.48403e12	0.400383
$795$	0	0
$796$	−5.18353e11	−0.0457633
$797$	−1.08432e12	−0.0951910	−0.0475955	−	0.998867i	$-0.515156\pi$
$797$	−1.08432e12	−0.0951910	−0.0475955	+	0.998867i	$0.515156\pi$
$798$	0	0
$799$	2.59624e12	0.225363
$800$	0	0
$801$	0	0
$802$	−9.50706e12	−0.811450
$803$	5.88161e12	0.499202
$804$	0	0
$805$	0	0
$806$	−1.93269e12	−0.161307
$807$	0	0
$808$	−1.52149e13	−1.25579
$809$	−4.06803e12	−0.333900	−0.166950	−	0.985965i	$-0.553392\pi$
$809$	−4.06803e12	−0.333900	−0.166950	+	0.985965i	$0.553392\pi$
$810$	0	0
$811$	4.87468e12	0.395688	0.197844	−	0.980234i	$-0.436606\pi$
$811$	4.87468e12	0.395688	0.197844	+	0.980234i	$0.436606\pi$
$812$	−1.78904e11	−0.0144417
$813$	0	0
$814$	4.10449e12	0.327680
$815$	0	0
$816$	0	0
$817$	−4.12198e12	−0.323673
$818$	9.76758e12	0.762777
$819$	0	0
$820$	0	0
$821$	1.03829e12	0.0797582	0.0398791	−	0.999205i	$-0.487303\pi$
$821$	1.03829e12	0.0797582	0.0398791	+	0.999205i	$0.487303\pi$
$822$	0	0
$823$	−1.10923e13	−0.842794	−0.421397	−	0.906876i	$-0.638460\pi$
$823$	−1.10923e13	−0.842794	−0.421397	+	0.906876i	$0.638460\pi$
$824$	2.36563e13	1.78761
$825$	0	0
$826$	1.86270e13	1.39230
$827$	−1.45526e13	−1.08184	−0.540922	−	0.841073i	$-0.681925\pi$
$827$	−1.45526e13	−1.08184	−0.540922	+	0.841073i	$0.681925\pi$
$828$	0	0
$829$	1.37985e13	1.01470	0.507348	−	0.861741i	$-0.330626\pi$
$829$	1.37985e13	1.01470	0.507348	+	0.861741i	$0.330626\pi$
$830$	0	0
$831$	0	0
$832$	−2.39256e13	−1.73104
$833$	4.75736e12	0.342345
$834$	0	0
$835$	0	0
$836$	−1.18315e12	−0.0837746
$837$	0	0
$838$	2.40728e13	1.68628
$839$	−9.20234e11	−0.0641165	−0.0320582	−	0.999486i	$-0.510206\pi$
$839$	−9.20234e11	−0.0641165	−0.0320582	+	0.999486i	$0.510206\pi$
$840$	0	0
$841$	−1.43921e13	−0.992067
$842$	−2.20232e13	−1.50999
$843$	0	0
$844$	1.27407e12	0.0864277
$845$	0	0
$846$	0	0
$847$	1.01936e13	0.680536
$848$	1.13610e13	0.754460
$849$	0	0
$850$	0	0
$851$	−9.91330e12	−0.647941
$852$	0	0
$853$	7.46964e12	0.483091	0.241545	−	0.970390i	$-0.422346\pi$
$853$	7.46964e12	0.483091	0.241545	+	0.970390i	$0.422346\pi$
$854$	−8.16090e12	−0.525022
$855$	0	0
$856$	−2.57033e13	−1.63628
$857$	2.05568e13	1.30179	0.650897	−	0.759166i	$-0.274393\pi$
$857$	2.05568e13	1.30179	0.650897	+	0.759166i	$0.274393\pi$
$858$	0	0
$859$	−1.05923e12	−0.0663775	−0.0331888	−	0.999449i	$-0.510566\pi$
$859$	−1.05923e12	−0.0663775	−0.0331888	+	0.999449i	$0.510566\pi$
$860$	0	0
$861$	0	0
$862$	2.94965e13	1.81965
$863$	1.53023e13	0.939092	0.469546	−	0.882908i	$-0.344418\pi$
$863$	1.53023e13	0.939092	0.469546	+	0.882908i	$0.344418\pi$
$864$	0	0
$865$	0	0
$866$	−4.17213e12	−0.252074
$867$	0	0
$868$	2.88639e11	0.0172590
$869$	4.62869e12	0.275340
$870$	0	0
$871$	−2.24410e13	−1.32118
$872$	1.43904e12	0.0842847
$873$	0	0
$874$	−2.46230e13	−1.42738
$875$	0	0
$876$	0	0
$877$	1.74558e11	0.00996416	0.00498208	−	0.999988i	$-0.498414\pi$
$877$	1.74558e11	0.00996416	0.00498208	+	0.999988i	$0.498414\pi$
$878$	−1.14061e13	−0.647759
$879$	0	0
$880$	0	0
$881$	−8.59039e12	−0.480420	−0.240210	−	0.970721i	$-0.577216\pi$
$881$	−8.59039e12	−0.480420	−0.240210	+	0.970721i	$0.577216\pi$
$882$	0	0
$883$	1.70227e13	0.942336	0.471168	−	0.882043i	$-0.343833\pi$
$883$	1.70227e13	0.942336	0.471168	+	0.882043i	$0.343833\pi$
$884$	7.22898e11	0.0398145
$885$	0	0
$886$	1.55635e13	0.848508
$887$	−2.01246e13	−1.09162	−0.545810	−	0.837909i	$-0.683778\pi$
$887$	−2.01246e13	−1.09162	−0.545810	+	0.837909i	$0.683778\pi$
$888$	0	0
$889$	−5.34112e13	−2.86797
$890$	0	0
$891$	0	0
$892$	1.07700e12	0.0569604
$893$	−1.92177e13	−1.01127
$894$	0	0
$895$	0	0
$896$	−2.45591e13	−1.27299
$897$	0	0
$898$	−2.00228e13	−1.02750
$899$	−1.85668e11	−0.00948023
$900$	0	0
$901$	−4.03214e12	−0.203833
$902$	1.60695e12	0.0808302
$903$	0	0
$904$	2.24525e13	1.11817
$905$	0	0
$906$	0	0
$907$	2.10130e13	1.03099	0.515495	−	0.856893i	$-0.327608\pi$
$907$	2.10130e13	1.03099	0.515495	+	0.856893i	$0.327608\pi$
$908$	−2.35404e12	−0.114929
$909$	0	0
$910$	0	0
$911$	−1.95345e12	−0.0939656	−0.0469828	−	0.998896i	$-0.514961\pi$
$911$	−1.95345e12	−0.0939656	−0.0469828	+	0.998896i	$0.514961\pi$
$912$	0	0
$913$	1.04802e13	0.499172
$914$	2.33870e13	1.10845
$915$	0	0
$916$	2.60604e12	0.122307
$917$	3.80064e13	1.77499
$918$	0	0
$919$	−2.47656e12	−0.114533	−0.0572663	−	0.998359i	$-0.518238\pi$
$919$	−2.47656e12	−0.114533	−0.0572663	+	0.998359i	$0.518238\pi$
$920$	0	0
$921$	0	0
$922$	−2.12834e13	−0.969954
$923$	5.75744e13	2.61109
$924$	0	0
$925$	0	0
$926$	1.49017e13	0.666018
$927$	0	0
$928$	−4.16725e11	−0.0184452
$929$	−4.40844e13	−1.94184	−0.970922	−	0.239395i	$-0.923051\pi$
$929$	−4.40844e13	−1.94184	−0.970922	+	0.239395i	$0.923051\pi$
$930$	0	0
$931$	−3.52146e13	−1.53621
$932$	−3.55718e12	−0.154431
$933$	0	0
$934$	4.02578e13	1.73097
$935$	0	0
$936$	0	0
$937$	−4.92250e12	−0.208621	−0.104310	−	0.994545i	$-0.533264\pi$
$937$	−4.92250e12	−0.208621	−0.104310	+	0.994545i	$0.533264\pi$
$938$	−2.88787e13	−1.21805
$939$	0	0
$940$	0	0
$941$	1.54680e13	0.643103	0.321552	−	0.946892i	$-0.395796\pi$
$941$	1.54680e13	0.643103	0.321552	+	0.946892i	$0.395796\pi$
$942$	0	0
$943$	−3.88116e12	−0.159830
$944$	2.03731e13	0.834993
$945$	0	0
$946$	−5.27936e12	−0.214324
$947$	5.49107e12	0.221862	0.110931	−	0.993828i	$-0.464617\pi$
$947$	5.49107e12	0.221862	0.110931	+	0.993828i	$0.464617\pi$
$948$	0	0
$949$	2.66004e13	1.06461
$950$	0	0
$951$	0	0
$952$	9.87649e12	0.389705
$953$	−1.76932e13	−0.694845	−0.347422	−	0.937709i	$-0.612943\pi$
$953$	−1.76932e13	−0.694845	−0.347422	+	0.937709i	$0.612943\pi$
$954$	0	0
$955$	0	0
$956$	−1.17743e12	−0.0455905
$957$	0	0
$958$	4.90510e12	0.188150
$959$	2.52994e13	0.965888
$960$	0	0
$961$	−2.61401e13	−0.988670
$962$	1.85632e13	0.698819
$963$	0	0
$964$	−5.31148e12	−0.198093
$965$	0	0
$966$	0	0
$967$	−2.89231e13	−1.06372	−0.531858	−	0.846834i	$-0.678506\pi$
$967$	−2.89231e13	−1.06372	−0.531858	+	0.846834i	$0.678506\pi$
$968$	1.24588e13	0.456076
$969$	0	0
$970$	0	0
$971$	−2.69482e13	−0.972843	−0.486422	−	0.873724i	$-0.661698\pi$
$971$	−2.69482e13	−0.972843	−0.486422	+	0.873724i	$0.661698\pi$
$972$	0	0
$973$	4.05255e13	1.44951
$974$	1.03112e13	0.367107
$975$	0	0
$976$	−8.92590e12	−0.314867
$977$	3.90598e13	1.37153	0.685764	−	0.727824i	$-0.259468\pi$
$977$	3.90598e13	1.37153	0.685764	+	0.727824i	$0.259468\pi$
$978$	0	0
$979$	−2.05282e13	−0.714216
$980$	0	0
$981$	0	0
$982$	3.64804e13	1.25187
$983$	9.03830e12	0.308742	0.154371	−	0.988013i	$-0.450665\pi$
$983$	9.03830e12	0.308742	0.154371	+	0.988013i	$0.450665\pi$
$984$	0	0
$985$	0	0
$986$	−5.98404e11	−0.0201627
$987$	0	0
$988$	−5.35098e12	−0.178660
$989$	1.27509e13	0.423795
$990$	0	0
$991$	4.89096e13	1.61088	0.805439	−	0.592678i	$-0.201929\pi$
$991$	4.89096e13	1.61088	0.805439	+	0.592678i	$0.201929\pi$
$992$	6.72333e11	0.0220436
$993$	0	0
$994$	7.40909e13	2.40728
$995$	0	0
$996$	0	0
$997$	−4.01445e13	−1.28676	−0.643380	−	0.765547i	$-0.722468\pi$
$997$	−4.01445e13	−1.28676	−0.643380	+	0.765547i	$0.722468\pi$
$998$	4.41384e13	1.40841
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.10.a.p.1.2		3
3.2	odd	2		25.10.a.c.1.2	✓	3
5.2	odd	4		225.10.b.m.199.5		6
5.3	odd	4		225.10.b.m.199.2		6
5.4	even	2		225.10.a.m.1.2		3
12.11	even	2		400.10.a.y.1.1		3
15.2	even	4		25.10.b.c.24.2		6
15.8	even	4		25.10.b.c.24.5		6
15.14	odd	2		25.10.a.d.1.2	yes	3
60.23	odd	4		400.10.c.q.49.2		6
60.47	odd	4		400.10.c.q.49.5		6
60.59	even	2		400.10.a.u.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.10.a.c.1.2	✓	3	3.2	odd	2
25.10.a.d.1.2	yes	3	15.14	odd	2
25.10.b.c.24.2		6	15.2	even	4
25.10.b.c.24.5		6	15.8	even	4
225.10.a.m.1.2		3	5.4	even	2
225.10.a.p.1.2		3	1.1	even	1	trivial
225.10.b.m.199.2		6	5.3	odd	4
225.10.b.m.199.5		6	5.2	odd	4
400.10.a.u.1.3		3	60.59	even	2
400.10.a.y.1.1		3	12.11	even	2
400.10.c.q.49.2		6	60.23	odd	4
400.10.c.q.49.5		6	60.47	odd	4

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

\(f(q)\)	\(=\)	\(q+21.4187 q^{2} -53.2406 q^{4} -9905.49 q^{7} -12106.7 q^{8} +O(q^{10})\)	\(q+21.4187 q^{2} -53.2406 q^{4} -9905.49 q^{7} -12106.7 q^{8} -36453.6 q^{11} -164867. q^{13} -212162. q^{14} -232050. q^{16} +82357.1 q^{17} -609617. q^{19} -780787. q^{22} +1.88578e6 q^{23} -3.53123e6 q^{26} +527374. q^{28} -339235. q^{29} +547314. q^{31} +1.22842e6 q^{32} +1.76398e6 q^{34} -5.25687e6 q^{37} -1.30572e7 q^{38} -2.05812e6 q^{41} +6.76158e6 q^{43} +1.94081e6 q^{44} +4.03909e7 q^{46} +3.15241e7 q^{47} +5.77651e7 q^{49} +8.77761e6 q^{52} -4.89593e7 q^{53} +1.19923e8 q^{56} -7.26597e6 q^{58} -8.77960e7 q^{59} +3.84654e7 q^{61} +1.17227e7 q^{62} +1.45121e8 q^{64} +1.36116e8 q^{67} -4.38474e6 q^{68} -3.49218e8 q^{71} -1.61345e8 q^{73} -1.12595e8 q^{74} +3.24564e7 q^{76} +3.61091e8 q^{77} -1.26975e8 q^{79} -4.40822e7 q^{82} -2.87494e8 q^{83} +1.44824e8 q^{86} +4.41333e8 q^{88} +5.63133e8 q^{89} +1.63309e9 q^{91} -1.00400e8 q^{92} +6.75205e8 q^{94} +4.71704e8 q^{97} +1.23725e9 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 33 q^{2} + 341 q^{4} - 5258 q^{7} - 105 q^{8}+O(q^{10}) \) 3 * q + 33 * q^2 + 341 * q^4 - 5258 * q^7 - 105 * q^8	\( 3 q + 33 q^{2} + 341 q^{4} - 5258 q^{7} - 105 q^{8} + 54699 q^{11} - 215884 q^{13} - 272922 q^{14} - 699247 q^{16} + 334983 q^{17} + 818845 q^{19} - 72761 q^{22} + 3526854 q^{23} + 1280004 q^{26} + 428134 q^{28} - 2175480 q^{29} + 4274066 q^{31} - 9464577 q^{32} - 6838963 q^{34} + 10305042 q^{37} - 24180495 q^{38} - 5926311 q^{41} + 24429956 q^{43} + 21995703 q^{44} + 14223246 q^{46} + 66858708 q^{47} - 6453929 q^{49} + 57862932 q^{52} + 132620514 q^{53} + 169538130 q^{56} - 201908320 q^{58} - 5670960 q^{59} + 125306926 q^{61} - 39831174 q^{62} + 167542401 q^{64} - 88829483 q^{67} - 71162559 q^{68} - 297550596 q^{71} - 181321729 q^{73} + 251507358 q^{74} + 89414865 q^{76} + 561214086 q^{77} - 310025170 q^{79} + 1368322979 q^{82} - 731088801 q^{83} - 33947196 q^{86} + 943671285 q^{88} + 1103860035 q^{89} + 1183187656 q^{91} - 190024242 q^{92} + 1727891132 q^{94} + 332236842 q^{97} + 457927581 q^{98}+O(q^{100}) \) 3 * q + 33 * q^2 + 341 * q^4 - 5258 * q^7 - 105 * q^8 + 54699 * q^11 - 215884 * q^13 - 272922 * q^14 - 699247 * q^16 + 334983 * q^17 + 818845 * q^19 - 72761 * q^22 + 3526854 * q^23 + 1280004 * q^26 + 428134 * q^28 - 2175480 * q^29 + 4274066 * q^31 - 9464577 * q^32 - 6838963 * q^34 + 10305042 * q^37 - 24180495 * q^38 - 5926311 * q^41 + 24429956 * q^43 + 21995703 * q^44 + 14223246 * q^46 + 66858708 * q^47 - 6453929 * q^49 + 57862932 * q^52 + 132620514 * q^53 + 169538130 * q^56 - 201908320 * q^58 - 5670960 * q^59 + 125306926 * q^61 - 39831174 * q^62 + 167542401 * q^64 - 88829483 * q^67 - 71162559 * q^68 - 297550596 * q^71 - 181321729 * q^73 + 251507358 * q^74 + 89414865 * q^76 + 561214086 * q^77 - 310025170 * q^79 + 1368322979 * q^82 - 731088801 * q^83 - 33947196 * q^86 + 943671285 * q^88 + 1103860035 * q^89 + 1183187656 * q^91 - 190024242 * q^92 + 1727891132 * q^94 + 332236842 * q^97 + 457927581 * q^98

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