LMFDB - Embedded newform 208.10.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,10,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, 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("208.1");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	208.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:	$107.127453922$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 1438x^{3} - 4164x^{2} + 396957x - 59580 $ x^5 - 1438x^3 - 4164x^2 + 396957*x - 59580
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$35.1685$ of defining polynomial
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-47.8784 q^{3} -109.762 q^{5} -5947.44 q^{7} -17390.7 q^{9} +O(q^{10})$	$q-47.8784 q^{3} -109.762 q^{5} -5947.44 q^{7} -17390.7 q^{9} +25205.7 q^{11} +28561.0 q^{13} +5255.24 q^{15} +109318. q^{17} +904609. q^{19} +284754. q^{21} +435749. q^{23} -1.94108e6 q^{25} +1.77503e6 q^{27} +6.44791e6 q^{29} -6.62308e6 q^{31} -1.20681e6 q^{33} +652804. q^{35} +4.14357e6 q^{37} -1.36745e6 q^{39} +1.49568e7 q^{41} -4.01789e7 q^{43} +1.90884e6 q^{45} -6.30151e6 q^{47} -4.98153e6 q^{49} -5.23397e6 q^{51} +1.53111e7 q^{53} -2.76663e6 q^{55} -4.33112e7 q^{57} +1.52760e8 q^{59} +8.66321e7 q^{61} +1.03430e8 q^{63} -3.13492e6 q^{65} +1.01034e8 q^{67} -2.08630e7 q^{69} -4.13122e8 q^{71} -3.14453e8 q^{73} +9.29357e7 q^{75} -1.49909e8 q^{77} +2.00580e8 q^{79} +2.57315e8 q^{81} -6.34578e7 q^{83} -1.19990e7 q^{85} -3.08715e8 q^{87} +3.47074e7 q^{89} -1.69865e8 q^{91} +3.17102e8 q^{93} -9.92918e7 q^{95} -1.25403e9 q^{97} -4.38343e8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9}+O(q^{10}) $ 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9	$ 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9} - 121746 q^{11} + 142805 q^{13} - 105973 q^{15} - 495669 q^{17} + 840738 q^{19} - 1599467 q^{21} + 592152 q^{23} + 1670362 q^{25} - 6847883 q^{27} + 10678182 q^{29} - 12885296 q^{31} + 17278298 q^{33} - 8380731 q^{35} + 7171823 q^{37} - 4598321 q^{39} + 9294012 q^{41} - 12831975 q^{43} + 26135198 q^{45} - 43354215 q^{47} + 25249488 q^{49} - 16905901 q^{51} + 93231780 q^{53} - 99448846 q^{55} + 90173382 q^{57} - 246496182 q^{59} - 132232612 q^{61} + 416955202 q^{63} + 51495483 q^{65} + 369388534 q^{67} - 579986760 q^{69} - 212150457 q^{71} - 252729806 q^{73} + 752457788 q^{75} + 449666118 q^{77} + 1247271728 q^{79} - 317713115 q^{81} - 1696894296 q^{83} - 775363765 q^{85} + 614530466 q^{87} - 753854382 q^{89} - 288437539 q^{91} - 892784668 q^{93} - 1442632962 q^{95} + 3824606 q^{97} - 3016199848 q^{99}+O(q^{100}) $ 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9 - 121746 * q^11 + 142805 * q^13 - 105973 * q^15 - 495669 * q^17 + 840738 * q^19 - 1599467 * q^21 + 592152 * q^23 + 1670362 * q^25 - 6847883 * q^27 + 10678182 * q^29 - 12885296 * q^31 + 17278298 * q^33 - 8380731 * q^35 + 7171823 * q^37 - 4598321 * q^39 + 9294012 * q^41 - 12831975 * q^43 + 26135198 * q^45 - 43354215 * q^47 + 25249488 * q^49 - 16905901 * q^51 + 93231780 * q^53 - 99448846 * q^55 + 90173382 * q^57 - 246496182 * q^59 - 132232612 * q^61 + 416955202 * q^63 + 51495483 * q^65 + 369388534 * q^67 - 579986760 * q^69 - 212150457 * q^71 - 252729806 * q^73 + 752457788 * q^75 + 449666118 * q^77 + 1247271728 * q^79 - 317713115 * q^81 - 1696894296 * q^83 - 775363765 * q^85 + 614530466 * q^87 - 753854382 * q^89 - 288437539 * q^91 - 892784668 * q^93 - 1442632962 * q^95 + 3824606 * q^97 - 3016199848 * 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
$3$	−47.8784	−0.341267	−0.170633	−	0.985335i	$-0.554581\pi$
$3$	−47.8784	−0.341267	−0.170633	+	0.985335i	$0.554581\pi$
$4$	0	0
$5$	−109.762	−0.0785394	−0.0392697	−	0.999229i	$-0.512503\pi$
$5$	−109.762	−0.0785394	−0.0392697	+	0.999229i	$0.512503\pi$
$6$	0	0
$7$	−5947.44	−0.936244	−0.468122	−	0.883664i	$-0.655069\pi$
$7$	−5947.44	−0.936244	−0.468122	+	0.883664i	$0.655069\pi$
$8$	0	0
$9$	−17390.7	−0.883537
$10$	0	0
$11$	25205.7	0.519076	0.259538	−	0.965733i	$-0.416430\pi$
$11$	25205.7	0.519076	0.259538	+	0.965733i	$0.416430\pi$
$12$	0	0
$13$	28561.0	0.277350
$13$	28561.0	0.277350
$14$	0	0
$15$	5255.24	0.0268029
$16$	0	0
$17$	109318.	0.317447	0.158724	−	0.987323i	$-0.449262\pi$
$17$	109318.	0.317447	0.158724	+	0.987323i	$0.449262\pi$
$18$	0	0
$19$	904609.	1.59246	0.796232	−	0.604992i	$-0.206824\pi$
$19$	904609.	1.59246	0.796232	+	0.604992i	$0.206824\pi$
$20$	0	0
$21$	284754.	0.319509
$22$	0	0
$23$	435749.	0.324684	0.162342	−	0.986735i	$-0.448095\pi$
$23$	435749.	0.324684	0.162342	+	0.986735i	$0.448095\pi$
$24$	0	0
$25$	−1.94108e6	−0.993832
$26$	0	0
$27$	1.77503e6	0.642789
$28$	0	0
$29$	6.44791e6	1.69289	0.846443	−	0.532479i	$-0.178740\pi$
$29$	6.44791e6	1.69289	0.846443	+	0.532479i	$0.178740\pi$
$30$	0	0
$31$	−6.62308e6	−1.28805	−0.644024	−	0.765005i	$-0.722736\pi$
$31$	−6.62308e6	−1.28805	−0.644024	+	0.765005i	$0.722736\pi$
$32$	0	0
$33$	−1.20681e6	−0.177143
$34$	0	0
$35$	652804.	0.0735320
$36$	0	0
$37$	4.14357e6	0.363469	0.181734	−	0.983348i	$-0.441829\pi$
$37$	4.14357e6	0.363469	0.181734	+	0.983348i	$0.441829\pi$
$38$	0	0
$39$	−1.36745e6	−0.0946504
$40$	0	0
$41$	1.49568e7	0.826632	0.413316	−	0.910588i	$-0.364370\pi$
$41$	1.49568e7	0.826632	0.413316	+	0.910588i	$0.364370\pi$
$42$	0	0
$43$	−4.01789e7	−1.79221	−0.896107	−	0.443838i	$-0.853616\pi$
$43$	−4.01789e7	−1.79221	−0.896107	+	0.443838i	$0.853616\pi$
$44$	0	0
$45$	1.90884e6	0.0693925
$46$	0	0
$47$	−6.30151e6	−0.188367	−0.0941834	−	0.995555i	$-0.530024\pi$
$47$	−6.30151e6	−0.188367	−0.0941834	+	0.995555i	$0.530024\pi$
$48$	0	0
$49$	−4.98153e6	−0.123447
$50$	0	0
$51$	−5.23397e6	−0.108334
$52$	0	0
$53$	1.53111e7	0.266542	0.133271	−	0.991080i	$-0.457452\pi$
$53$	1.53111e7	0.266542	0.133271	+	0.991080i	$0.457452\pi$
$54$	0	0
$55$	−2.76663e6	−0.0407679
$56$	0	0
$57$	−4.33112e7	−0.543455
$58$	0	0
$59$	1.52760e8	1.64126	0.820629	−	0.571462i	$-0.193624\pi$
$59$	1.52760e8	1.64126	0.820629	+	0.571462i	$0.193624\pi$
$60$	0	0
$61$	8.66321e7	0.801114	0.400557	−	0.916272i	$-0.368817\pi$
$61$	8.66321e7	0.801114	0.400557	+	0.916272i	$0.368817\pi$
$62$	0	0
$63$	1.03430e8	0.827206
$64$	0	0
$65$	−3.13492e6	−0.0217829
$66$	0	0
$67$	1.01034e8	0.612537	0.306268	−	0.951945i	$-0.400920\pi$
$67$	1.01034e8	0.612537	0.306268	+	0.951945i	$0.400920\pi$
$68$	0	0
$69$	−2.08630e7	−0.110804
$70$	0	0
$71$	−4.13122e8	−1.92937	−0.964685	−	0.263406i	$-0.915154\pi$
$71$	−4.13122e8	−1.92937	−0.964685	+	0.263406i	$0.915154\pi$
$72$	0	0
$73$	−3.14453e8	−1.29599	−0.647997	−	0.761643i	$-0.724393\pi$
$73$	−3.14453e8	−1.29599	−0.647997	+	0.761643i	$0.724393\pi$
$74$	0	0
$75$	9.29357e7	0.339162
$76$	0	0
$77$	−1.49909e8	−0.485982
$78$	0	0
$79$	2.00580e8	0.579383	0.289692	−	0.957120i	$-0.406447\pi$
$79$	2.00580e8	0.579383	0.289692	+	0.957120i	$0.406447\pi$
$80$	0	0
$81$	2.57315e8	0.664175
$82$	0	0
$83$	−6.34578e7	−0.146769	−0.0733843	−	0.997304i	$-0.523380\pi$
$83$	−6.34578e7	−0.146769	−0.0733843	+	0.997304i	$0.523380\pi$
$84$	0	0
$85$	−1.19990e7	−0.0249321
$86$	0	0
$87$	−3.08715e8	−0.577726
$88$	0	0
$89$	3.47074e7	0.0586364	0.0293182	−	0.999570i	$-0.490666\pi$
$89$	3.47074e7	0.0586364	0.0293182	+	0.999570i	$0.490666\pi$
$90$	0	0
$91$	−1.69865e8	−0.259667
$92$	0	0
$93$	3.17102e8	0.439568
$94$	0	0
$95$	−9.92918e7	−0.125071
$96$	0	0
$97$	−1.25403e9	−1.43825	−0.719127	−	0.694879i	$-0.755458\pi$
$97$	−1.25403e9	−1.43825	−0.719127	+	0.694879i	$0.755458\pi$
$98$	0	0
$99$	−4.38343e8	−0.458623
$100$	0	0
$101$	−9.06459e8	−0.866766	−0.433383	−	0.901210i	$-0.642680\pi$
$101$	−9.06459e8	−0.866766	−0.433383	+	0.901210i	$0.642680\pi$
$102$	0	0
$103$	4.17013e8	0.365075	0.182537	−	0.983199i	$-0.441569\pi$
$103$	4.17013e8	0.365075	0.182537	+	0.983199i	$0.441569\pi$
$104$	0	0
$105$	−3.12552e7	−0.0250940
$106$	0	0
$107$	−6.71636e8	−0.495344	−0.247672	−	0.968844i	$-0.579666\pi$
$107$	−6.71636e8	−0.495344	−0.247672	+	0.968844i	$0.579666\pi$
$108$	0	0
$109$	1.66748e9	1.13147	0.565734	−	0.824588i	$-0.308593\pi$
$109$	1.66748e9	1.13147	0.565734	+	0.824588i	$0.308593\pi$
$110$	0	0
$111$	−1.98387e8	−0.124040
$112$	0	0
$113$	−1.84580e9	−1.06496	−0.532478	−	0.846444i	$-0.678739\pi$
$113$	−1.84580e9	−1.06496	−0.532478	+	0.846444i	$0.678739\pi$
$114$	0	0
$115$	−4.78287e7	−0.0255005
$116$	0	0
$117$	−4.96695e8	−0.245049
$118$	0	0
$119$	−6.50162e8	−0.297208
$120$	0	0
$121$	−1.72262e9	−0.730560
$122$	0	0
$123$	−7.16109e8	−0.282102
$124$	0	0
$125$	4.27436e8	0.156594
$126$	0	0
$127$	−1.87922e9	−0.641006	−0.320503	−	0.947248i	$-0.603852\pi$
$127$	−1.87922e9	−0.641006	−0.320503	+	0.947248i	$0.603852\pi$
$128$	0	0
$129$	1.92370e9	0.611623
$130$	0	0
$131$	−3.46045e9	−1.02663	−0.513313	−	0.858201i	$-0.671582\pi$
$131$	−3.46045e9	−1.02663	−0.513313	+	0.858201i	$0.671582\pi$
$132$	0	0
$133$	−5.38011e9	−1.49093
$134$	0	0
$135$	−1.94831e8	−0.0504842
$136$	0	0
$137$	5.04786e9	1.22424	0.612118	−	0.790766i	$-0.290318\pi$
$137$	5.04786e9	1.22424	0.612118	+	0.790766i	$0.290318\pi$
$138$	0	0
$139$	−3.59716e9	−0.817322	−0.408661	−	0.912686i	$-0.634004\pi$
$139$	−3.59716e9	−0.817322	−0.408661	+	0.912686i	$0.634004\pi$
$140$	0	0
$141$	3.01706e8	0.0642833
$142$	0	0
$143$	7.19899e8	0.143966
$144$	0	0
$145$	−7.07736e8	−0.132958
$146$	0	0
$147$	2.38508e8	0.0421283
$148$	0	0
$149$	1.27561e9	0.212022	0.106011	−	0.994365i	$-0.466192\pi$
$149$	1.27561e9	0.212022	0.106011	+	0.994365i	$0.466192\pi$
$150$	0	0
$151$	−4.35273e9	−0.681342	−0.340671	−	0.940183i	$-0.610654\pi$
$151$	−4.35273e9	−0.681342	−0.340671	+	0.940183i	$0.610654\pi$
$152$	0	0
$153$	−1.90111e9	−0.280476
$154$	0	0
$155$	7.26963e8	0.101163
$156$	0	0
$157$	−1.41002e10	−1.85215	−0.926075	−	0.377339i	$-0.876839\pi$
$157$	−1.41002e10	−1.85215	−0.926075	+	0.377339i	$0.876839\pi$
$158$	0	0
$159$	−7.33072e8	−0.0909619
$160$	0	0
$161$	−2.59159e9	−0.303983
$162$	0	0
$163$	−7.24812e8	−0.0804231	−0.0402116	−	0.999191i	$-0.512803\pi$
$163$	−7.24812e8	−0.0804231	−0.0402116	+	0.999191i	$0.512803\pi$
$164$	0	0
$165$	1.32462e8	0.0139127
$166$	0	0
$167$	8.33031e8	0.0828776	0.0414388	−	0.999141i	$-0.486806\pi$
$167$	8.33031e8	0.0828776	0.0414388	+	0.999141i	$0.486806\pi$
$168$	0	0
$169$	8.15731e8	0.0769231
$170$	0	0
$171$	−1.57317e10	−1.40700
$172$	0	0
$173$	5.77955e8	0.0490553	0.0245277	−	0.999699i	$-0.492192\pi$
$173$	5.77955e8	0.0490553	0.0245277	+	0.999699i	$0.492192\pi$
$174$	0	0
$175$	1.15444e10	0.930469
$176$	0	0
$177$	−7.31392e9	−0.560106
$178$	0	0
$179$	1.56569e10	1.13990	0.569952	−	0.821678i	$-0.306962\pi$
$179$	1.56569e10	1.13990	0.569952	+	0.821678i	$0.306962\pi$
$180$	0	0
$181$	−2.18552e10	−1.51356	−0.756781	−	0.653668i	$-0.773229\pi$
$181$	−2.18552e10	−1.51356	−0.756781	+	0.653668i	$0.773229\pi$
$182$	0	0
$183$	−4.14780e9	−0.273394
$184$	0	0
$185$	−4.54807e8	−0.0285466
$186$	0	0
$187$	2.75543e9	0.164779
$188$	0	0
$189$	−1.05569e10	−0.601807
$190$	0	0
$191$	−1.67784e10	−0.912222	−0.456111	−	0.889923i	$-0.650758\pi$
$191$	−1.67784e10	−0.912222	−0.456111	+	0.889923i	$0.650758\pi$
$192$	0	0
$193$	−2.70036e10	−1.40092	−0.700462	−	0.713690i	$-0.747023\pi$
$193$	−2.70036e10	−1.40092	−0.700462	+	0.713690i	$0.747023\pi$
$194$	0	0
$195$	1.50095e8	0.00743378
$196$	0	0
$197$	1.58277e10	0.748720	0.374360	−	0.927283i	$-0.377862\pi$
$197$	1.58277e10	0.748720	0.374360	+	0.927283i	$0.377862\pi$
$198$	0	0
$199$	−8.80397e9	−0.397960	−0.198980	−	0.980004i	$-0.563763\pi$
$199$	−8.80397e9	−0.397960	−0.198980	+	0.980004i	$0.563763\pi$
$200$	0	0
$201$	−4.83736e9	−0.209038
$202$	0	0
$203$	−3.83485e10	−1.58495
$204$	0	0
$205$	−1.64169e9	−0.0649232
$206$	0	0
$207$	−7.57796e9	−0.286870
$208$	0	0
$209$	2.28013e10	0.826610
$210$	0	0
$211$	1.82054e10	0.632308	0.316154	−	0.948708i	$-0.397608\pi$
$211$	1.82054e10	0.632308	0.316154	+	0.948708i	$0.397608\pi$
$212$	0	0
$213$	1.97796e10	0.658430
$214$	0	0
$215$	4.41012e9	0.140759
$216$	0	0
$217$	3.93904e10	1.20593
$218$	0	0
$219$	1.50555e10	0.442280
$220$	0	0
$221$	3.12223e9	0.0880440
$222$	0	0
$223$	−1.53511e10	−0.415688	−0.207844	−	0.978162i	$-0.566645\pi$
$223$	−1.53511e10	−0.415688	−0.207844	+	0.978162i	$0.566645\pi$
$224$	0	0
$225$	3.37566e10	0.878087
$226$	0	0
$227$	−4.20620e10	−1.05141	−0.525707	−	0.850666i	$-0.676199\pi$
$227$	−4.20620e10	−1.05141	−0.525707	+	0.850666i	$0.676199\pi$
$228$	0	0
$229$	−6.68760e10	−1.60698	−0.803490	−	0.595318i	$-0.797026\pi$
$229$	−6.68760e10	−1.60698	−0.803490	+	0.595318i	$0.797026\pi$
$230$	0	0
$231$	7.17741e9	0.165849
$232$	0	0
$233$	−5.19268e10	−1.15422	−0.577112	−	0.816665i	$-0.695820\pi$
$233$	−5.19268e10	−1.15422	−0.577112	+	0.816665i	$0.695820\pi$
$234$	0	0
$235$	6.91668e8	0.0147942
$236$	0	0
$237$	−9.60345e9	−0.197724
$238$	0	0
$239$	−7.45881e10	−1.47870	−0.739348	−	0.673323i	$-0.764866\pi$
$239$	−7.45881e10	−1.47870	−0.739348	+	0.673323i	$0.764866\pi$
$240$	0	0
$241$	5.74852e10	1.09769	0.548845	−	0.835924i	$-0.315068\pi$
$241$	5.74852e10	1.09769	0.548845	+	0.835924i	$0.315068\pi$
$242$	0	0
$243$	−4.72577e10	−0.869449
$244$	0	0
$245$	5.46784e8	0.00969545
$246$	0	0
$247$	2.58365e10	0.441670
$248$	0	0
$249$	3.03826e9	0.0500873
$250$	0	0
$251$	1.07873e11	1.71546	0.857732	−	0.514097i	$-0.171873\pi$
$251$	1.07873e11	1.71546	0.857732	+	0.514097i	$0.171873\pi$
$252$	0	0
$253$	1.09833e10	0.168536
$254$	0	0
$255$	5.74492e8	0.00850850
$256$	0	0
$257$	−6.64074e10	−0.949550	−0.474775	−	0.880107i	$-0.657470\pi$
$257$	−6.64074e10	−0.949550	−0.474775	+	0.880107i	$0.657470\pi$
$258$	0	0
$259$	−2.46436e10	−0.340295
$260$	0	0
$261$	−1.12133e11	−1.49573
$262$	0	0
$263$	8.15356e10	1.05086	0.525432	−	0.850836i	$-0.323904\pi$
$263$	8.15356e10	1.05086	0.525432	+	0.850836i	$0.323904\pi$
$264$	0	0
$265$	−1.68058e9	−0.0209340
$266$	0	0
$267$	−1.66174e9	−0.0200107
$268$	0	0
$269$	1.00568e11	1.17105	0.585523	−	0.810656i	$-0.300889\pi$
$269$	1.00568e11	1.17105	0.585523	+	0.810656i	$0.300889\pi$
$270$	0	0
$271$	2.98757e10	0.336477	0.168239	−	0.985746i	$-0.446192\pi$
$271$	2.98757e10	0.336477	0.168239	+	0.985746i	$0.446192\pi$
$272$	0	0
$273$	8.13286e9	0.0886158
$274$	0	0
$275$	−4.89261e10	−0.515874
$276$	0	0
$277$	−3.17195e10	−0.323718	−0.161859	−	0.986814i	$-0.551749\pi$
$277$	−3.17195e10	−0.323718	−0.161859	+	0.986814i	$0.551749\pi$
$278$	0	0
$279$	1.15180e11	1.13804
$280$	0	0
$281$	9.86953e10	0.944318	0.472159	−	0.881513i	$-0.343475\pi$
$281$	9.86953e10	0.944318	0.472159	+	0.881513i	$0.343475\pi$
$282$	0	0
$283$	1.06164e11	0.983871	0.491935	−	0.870632i	$-0.336290\pi$
$283$	1.06164e11	0.983871	0.491935	+	0.870632i	$0.336290\pi$
$284$	0	0
$285$	4.75393e9	0.0426826
$286$	0	0
$287$	−8.89549e10	−0.773929
$288$	0	0
$289$	−1.06637e11	−0.899227
$290$	0	0
$291$	6.00410e10	0.490828
$292$	0	0
$293$	2.26355e11	1.79426	0.897132	−	0.441763i	$-0.145647\pi$
$293$	2.26355e11	1.79426	0.897132	+	0.441763i	$0.145647\pi$
$294$	0	0
$295$	−1.67673e10	−0.128903
$296$	0	0
$297$	4.47407e10	0.333656
$298$	0	0
$299$	1.24454e10	0.0900511
$300$	0	0
$301$	2.38962e11	1.67795
$302$	0	0
$303$	4.33998e10	0.295798
$304$	0	0
$305$	−9.50892e9	−0.0629190
$306$	0	0
$307$	2.23009e10	0.143285	0.0716424	−	0.997430i	$-0.477176\pi$
$307$	2.23009e10	0.143285	0.0716424	+	0.997430i	$0.477176\pi$
$308$	0	0
$309$	−1.99659e10	−0.124588
$310$	0	0
$311$	2.71805e11	1.64754	0.823770	−	0.566925i	$-0.191867\pi$
$311$	2.71805e11	1.64754	0.823770	+	0.566925i	$0.191867\pi$
$312$	0	0
$313$	−7.90775e10	−0.465697	−0.232849	−	0.972513i	$-0.574805\pi$
$313$	−7.90775e10	−0.465697	−0.232849	+	0.972513i	$0.574805\pi$
$314$	0	0
$315$	−1.13527e10	−0.0649683
$316$	0	0
$317$	2.03032e11	1.12927	0.564636	−	0.825340i	$-0.309017\pi$
$317$	2.03032e11	1.12927	0.564636	+	0.825340i	$0.309017\pi$
$318$	0	0
$319$	1.62524e11	0.878736
$320$	0	0
$321$	3.21568e10	0.169044
$322$	0	0
$323$	9.88899e10	0.505523
$324$	0	0
$325$	−5.54391e10	−0.275639
$326$	0	0
$327$	−7.98365e10	−0.386132
$328$	0	0
$329$	3.74779e10	0.176357
$330$	0	0
$331$	−3.24137e11	−1.48424	−0.742118	−	0.670270i	$-0.766178\pi$
$331$	−3.24137e11	−1.48424	−0.742118	+	0.670270i	$0.766178\pi$
$332$	0	0
$333$	−7.20594e10	−0.321138
$334$	0	0
$335$	−1.10897e10	−0.0481083
$336$	0	0
$337$	2.68510e11	1.13403	0.567017	−	0.823706i	$-0.308097\pi$
$337$	2.68510e11	1.13403	0.567017	+	0.823706i	$0.308097\pi$
$338$	0	0
$339$	8.83739e10	0.363434
$340$	0	0
$341$	−1.66939e11	−0.668595
$342$	0	0
$343$	2.69628e11	1.05182
$344$	0	0
$345$	2.28996e9	0.00870247
$346$	0	0
$347$	−2.74715e11	−1.01719	−0.508593	−	0.861007i	$-0.669834\pi$
$347$	−2.74715e11	−1.01719	−0.508593	+	0.861007i	$0.669834\pi$
$348$	0	0
$349$	−3.54006e11	−1.27731	−0.638655	−	0.769494i	$-0.720509\pi$
$349$	−3.54006e11	−1.27731	−0.638655	+	0.769494i	$0.720509\pi$
$350$	0	0
$351$	5.06966e10	0.178277
$352$	0	0
$353$	−2.25650e11	−0.773481	−0.386741	−	0.922189i	$-0.626399\pi$
$353$	−2.25650e11	−0.773481	−0.386741	+	0.922189i	$0.626399\pi$
$354$	0	0
$355$	4.53451e10	0.151532
$356$	0	0
$357$	3.11287e10	0.101427
$358$	0	0
$359$	−3.39022e10	−0.107722	−0.0538608	−	0.998548i	$-0.517153\pi$
$359$	−3.39022e10	−0.107722	−0.0538608	+	0.998548i	$0.517153\pi$
$360$	0	0
$361$	4.95629e11	1.53594
$362$	0	0
$363$	8.24764e10	0.249316
$364$	0	0
$365$	3.45150e10	0.101787
$366$	0	0
$367$	7.74387e10	0.222823	0.111412	−	0.993774i	$-0.464463\pi$
$367$	7.74387e10	0.222823	0.111412	+	0.993774i	$0.464463\pi$
$368$	0	0
$369$	−2.60109e11	−0.730360
$370$	0	0
$371$	−9.10620e10	−0.249548
$372$	0	0
$373$	6.50821e11	1.74089	0.870446	−	0.492264i	$-0.163831\pi$
$373$	6.50821e11	1.74089	0.870446	+	0.492264i	$0.163831\pi$
$374$	0	0
$375$	−2.04649e10	−0.0534404
$376$	0	0
$377$	1.84159e11	0.469522
$378$	0	0
$379$	−1.93989e11	−0.482948	−0.241474	−	0.970407i	$-0.577631\pi$
$379$	−1.93989e11	−0.482948	−0.241474	+	0.970407i	$0.577631\pi$
$380$	0	0
$381$	8.99743e10	0.218754
$382$	0	0
$383$	−4.44645e11	−1.05589	−0.527946	−	0.849278i	$-0.677037\pi$
$383$	−4.44645e11	−1.05589	−0.527946	+	0.849278i	$0.677037\pi$
$384$	0	0
$385$	1.64544e10	0.0381687
$386$	0	0
$387$	6.98737e11	1.58349
$388$	0	0
$389$	−1.69623e11	−0.375589	−0.187794	−	0.982208i	$-0.560134\pi$
$389$	−1.69623e11	−0.375589	−0.187794	+	0.982208i	$0.560134\pi$
$390$	0	0
$391$	4.76352e10	0.103070
$392$	0	0
$393$	1.65681e11	0.350353
$394$	0	0
$395$	−2.20161e10	−0.0455044
$396$	0	0
$397$	2.94468e11	0.594951	0.297476	−	0.954729i	$-0.403855\pi$
$397$	2.94468e11	0.594951	0.297476	+	0.954729i	$0.403855\pi$
$398$	0	0
$399$	2.57591e11	0.508806
$400$	0	0
$401$	4.43362e11	0.856265	0.428133	−	0.903716i	$-0.359172\pi$
$401$	4.43362e11	0.856265	0.428133	+	0.903716i	$0.359172\pi$
$402$	0	0
$403$	−1.89162e11	−0.357240
$404$	0	0
$405$	−2.82434e10	−0.0521639
$406$	0	0
$407$	1.04441e11	0.188668
$408$	0	0
$409$	−1.01253e11	−0.178918	−0.0894592	−	0.995990i	$-0.528514\pi$
$409$	−1.01253e11	−0.178918	−0.0894592	+	0.995990i	$0.528514\pi$
$410$	0	0
$411$	−2.41684e11	−0.417791
$412$	0	0
$413$	−9.08533e11	−1.53662
$414$	0	0
$415$	6.96526e9	0.0115271
$416$	0	0
$417$	1.72226e11	0.278925
$418$	0	0
$419$	−7.93982e10	−0.125848	−0.0629242	−	0.998018i	$-0.520043\pi$
$419$	−7.93982e10	−0.125848	−0.0629242	+	0.998018i	$0.520043\pi$
$420$	0	0
$421$	−6.06765e11	−0.941350	−0.470675	−	0.882307i	$-0.655990\pi$
$421$	−6.06765e11	−0.941350	−0.470675	+	0.882307i	$0.655990\pi$
$422$	0	0
$423$	1.09587e11	0.166429
$424$	0	0
$425$	−2.12195e11	−0.315489
$426$	0	0
$427$	−5.15239e11	−0.750038
$428$	0	0
$429$	−3.44676e10	−0.0491307
$430$	0	0
$431$	2.72544e11	0.380442	0.190221	−	0.981741i	$-0.439080\pi$
$431$	2.72544e11	0.380442	0.190221	+	0.981741i	$0.439080\pi$
$432$	0	0
$433$	−1.15522e12	−1.57931	−0.789655	−	0.613551i	$-0.789740\pi$
$433$	−1.15522e12	−1.57931	−0.789655	+	0.613551i	$0.789740\pi$
$434$	0	0
$435$	3.38853e10	0.0453742
$436$	0	0
$437$	3.94182e11	0.517047
$438$	0	0
$439$	−1.78053e11	−0.228801	−0.114401	−	0.993435i	$-0.536495\pi$
$439$	−1.78053e11	−0.228801	−0.114401	+	0.993435i	$0.536495\pi$
$440$	0	0
$441$	8.66321e10	0.109070
$442$	0	0
$443$	6.55260e10	0.0808345	0.0404173	−	0.999183i	$-0.487131\pi$
$443$	6.55260e10	0.0808345	0.0404173	+	0.999183i	$0.487131\pi$
$444$	0	0
$445$	−3.80956e9	−0.00460527
$446$	0	0
$447$	−6.10742e10	−0.0723559
$448$	0	0
$449$	−7.98150e11	−0.926779	−0.463390	−	0.886155i	$-0.653367\pi$
$449$	−7.98150e11	−0.926779	−0.463390	+	0.886155i	$0.653367\pi$
$450$	0	0
$451$	3.76997e11	0.429085
$452$	0	0
$453$	2.08402e11	0.232519
$454$	0	0
$455$	1.86447e10	0.0203941
$456$	0	0
$457$	−8.92736e11	−0.957415	−0.478708	−	0.877974i	$-0.658895\pi$
$457$	−8.92736e11	−0.957415	−0.478708	+	0.877974i	$0.658895\pi$
$458$	0	0
$459$	1.94042e11	0.204051
$460$	0	0
$461$	2.32490e11	0.239745	0.119872	−	0.992789i	$-0.461751\pi$
$461$	2.32490e11	0.239745	0.119872	+	0.992789i	$0.461751\pi$
$462$	0	0
$463$	1.54421e12	1.56168	0.780841	−	0.624730i	$-0.214791\pi$
$463$	1.54421e12	1.56168	0.780841	+	0.624730i	$0.214791\pi$
$464$	0	0
$465$	−3.48058e10	−0.0345234
$466$	0	0
$467$	−1.63317e12	−1.58893	−0.794465	−	0.607310i	$-0.792249\pi$
$467$	−1.63317e12	−1.58893	−0.794465	+	0.607310i	$0.792249\pi$
$468$	0	0
$469$	−6.00895e11	−0.573484
$470$	0	0
$471$	6.75094e11	0.632077
$472$	0	0
$473$	−1.01274e12	−0.930295
$474$	0	0
$475$	−1.75592e12	−1.58264
$476$	0	0
$477$	−2.66271e11	−0.235500
$478$	0	0
$479$	−1.54414e12	−1.34022	−0.670112	−	0.742260i	$-0.733754\pi$
$479$	−1.54414e12	−1.34022	−0.670112	+	0.742260i	$0.733754\pi$
$480$	0	0
$481$	1.18344e11	0.100808
$482$	0	0
$483$	1.24081e11	0.103739
$484$	0	0
$485$	1.37645e11	0.112960
$486$	0	0
$487$	−2.55439e11	−0.205782	−0.102891	−	0.994693i	$-0.532809\pi$
$487$	−2.55439e11	−0.205782	−0.102891	+	0.994693i	$0.532809\pi$
$488$	0	0
$489$	3.47028e10	0.0274457
$490$	0	0
$491$	−1.58407e12	−1.23000	−0.615002	−	0.788526i	$-0.710845\pi$
$491$	−1.58407e12	−1.23000	−0.615002	+	0.788526i	$0.710845\pi$
$492$	0	0
$493$	7.04872e11	0.537402
$494$	0	0
$495$	4.81135e10	0.0360200
$496$	0	0
$497$	2.45702e12	1.80636
$498$	0	0
$499$	6.29948e11	0.454833	0.227417	−	0.973798i	$-0.426972\pi$
$499$	6.29948e11	0.454833	0.227417	+	0.973798i	$0.426972\pi$
$500$	0	0
$501$	−3.98842e10	−0.0282834
$502$	0	0
$503$	−5.49263e11	−0.382582	−0.191291	−	0.981533i	$-0.561267\pi$
$503$	−5.49263e11	−0.382582	−0.191291	+	0.981533i	$0.561267\pi$
$504$	0	0
$505$	9.94949e10	0.0680753
$506$	0	0
$507$	−3.90559e10	−0.0262513
$508$	0	0
$509$	1.36923e11	0.0904165	0.0452083	−	0.998978i	$-0.485605\pi$
$509$	1.36923e11	0.0904165	0.0452083	+	0.998978i	$0.485605\pi$
$510$	0	0
$511$	1.87019e12	1.21337
$512$	0	0
$513$	1.60570e12	1.02362
$514$	0	0
$515$	−4.57722e10	−0.0286727
$516$	0	0
$517$	−1.58834e11	−0.0977767
$518$	0	0
$519$	−2.76715e10	−0.0167410
$520$	0	0
$521$	−1.66627e12	−0.990779	−0.495389	−	0.868671i	$-0.664975\pi$
$521$	−1.66627e12	−0.990779	−0.495389	+	0.868671i	$0.664975\pi$
$522$	0	0
$523$	−1.97187e12	−1.15244	−0.576222	−	0.817293i	$-0.695474\pi$
$523$	−1.97187e12	−1.15244	−0.576222	+	0.817293i	$0.695474\pi$
$524$	0	0
$525$	−5.52730e11	−0.317538
$526$	0	0
$527$	−7.24021e11	−0.408887
$528$	0	0
$529$	−1.61128e12	−0.894580
$530$	0	0
$531$	−2.65660e12	−1.45011
$532$	0	0
$533$	4.27182e11	0.229266
$534$	0	0
$535$	7.37202e10	0.0389040
$536$	0	0
$537$	−7.49629e11	−0.389011
$538$	0	0
$539$	−1.25563e11	−0.0640784
$540$	0	0
$541$	1.54916e12	0.777513	0.388756	−	0.921341i	$-0.372905\pi$
$541$	1.54916e12	0.777513	0.388756	+	0.921341i	$0.372905\pi$
$542$	0	0
$543$	1.04639e12	0.516529
$544$	0	0
$545$	−1.83027e11	−0.0888648
$546$	0	0
$547$	−1.98052e12	−0.945879	−0.472939	−	0.881095i	$-0.656807\pi$
$547$	−1.98052e12	−0.945879	−0.472939	+	0.881095i	$0.656807\pi$
$548$	0	0
$549$	−1.50659e12	−0.707814
$550$	0	0
$551$	5.83283e12	2.69586
$552$	0	0
$553$	−1.19294e12	−0.542444
$554$	0	0
$555$	2.17754e10	0.00974200
$556$	0	0
$557$	8.86577e11	0.390273	0.195136	−	0.980776i	$-0.437485\pi$
$557$	8.86577e11	0.390273	0.195136	+	0.980776i	$0.437485\pi$
$558$	0	0
$559$	−1.14755e12	−0.497071
$560$	0	0
$561$	−1.31926e11	−0.0562336
$562$	0	0
$563$	−3.89786e12	−1.63508	−0.817539	−	0.575873i	$-0.804662\pi$
$563$	−3.89786e12	−1.63508	−0.817539	+	0.575873i	$0.804662\pi$
$564$	0	0
$565$	2.02599e11	0.0836410
$566$	0	0
$567$	−1.53037e12	−0.621830
$568$	0	0
$569$	−2.26590e12	−0.906223	−0.453112	−	0.891454i	$-0.649686\pi$
$569$	−2.26590e12	−0.906223	−0.453112	+	0.891454i	$0.649686\pi$
$570$	0	0
$571$	3.09546e12	1.21860	0.609302	−	0.792938i	$-0.291450\pi$
$571$	3.09546e12	1.21860	0.609302	+	0.792938i	$0.291450\pi$
$572$	0	0
$573$	8.03323e11	0.311311
$574$	0	0
$575$	−8.45822e11	−0.322681
$576$	0	0
$577$	−3.23415e12	−1.21470	−0.607350	−	0.794434i	$-0.707767\pi$
$577$	−3.23415e12	−1.21470	−0.607350	+	0.794434i	$0.707767\pi$
$578$	0	0
$579$	1.29289e12	0.478089
$580$	0	0
$581$	3.77411e11	0.137411
$582$	0	0
$583$	3.85927e11	0.138356
$584$	0	0
$585$	5.45183e10	0.0192460
$586$	0	0
$587$	8.78491e11	0.305398	0.152699	−	0.988273i	$-0.451204\pi$
$587$	8.78491e11	0.305398	0.152699	+	0.988273i	$0.451204\pi$
$588$	0	0
$589$	−5.99129e12	−2.05117
$590$	0	0
$591$	−7.57805e11	−0.255513
$592$	0	0
$593$	−1.91730e12	−0.636712	−0.318356	−	0.947971i	$-0.603131\pi$
$593$	−1.91730e12	−0.636712	−0.318356	+	0.947971i	$0.603131\pi$
$594$	0	0
$595$	7.13632e10	0.0233425
$596$	0	0
$597$	4.21520e11	0.135811
$598$	0	0
$599$	5.10464e12	1.62011	0.810054	−	0.586355i	$-0.199438\pi$
$599$	5.10464e12	1.62011	0.810054	+	0.586355i	$0.199438\pi$
$600$	0	0
$601$	3.00419e12	0.939273	0.469636	−	0.882860i	$-0.344385\pi$
$601$	3.00419e12	0.939273	0.469636	+	0.882860i	$0.344385\pi$
$602$	0	0
$603$	−1.75705e12	−0.541199
$604$	0	0
$605$	1.89079e11	0.0573778
$606$	0	0
$607$	−1.18434e12	−0.354101	−0.177051	−	0.984202i	$-0.556656\pi$
$607$	−1.18434e12	−0.354101	−0.177051	+	0.984202i	$0.556656\pi$
$608$	0	0
$609$	1.83607e12	0.540892
$610$	0	0
$611$	−1.79978e11	−0.0522436
$612$	0	0
$613$	6.42597e11	0.183809	0.0919045	−	0.995768i	$-0.470705\pi$
$613$	6.42597e11	0.183809	0.0919045	+	0.995768i	$0.470705\pi$
$614$	0	0
$615$	7.86017e10	0.0221561
$616$	0	0
$617$	5.05474e12	1.40416	0.702078	−	0.712100i	$-0.252256\pi$
$617$	5.05474e12	1.40416	0.702078	+	0.712100i	$0.252256\pi$
$618$	0	0
$619$	3.09492e11	0.0847308	0.0423654	−	0.999102i	$-0.486511\pi$
$619$	3.09492e11	0.0847308	0.0423654	+	0.999102i	$0.486511\pi$
$620$	0	0
$621$	7.73466e11	0.208703
$622$	0	0
$623$	−2.06421e11	−0.0548980
$624$	0	0
$625$	3.74425e12	0.981533
$626$	0	0
$627$	−1.09169e12	−0.282094
$628$	0	0
$629$	4.52966e11	0.115382
$630$	0	0
$631$	7.94237e11	0.199443	0.0997214	−	0.995015i	$-0.468205\pi$
$631$	7.94237e11	0.199443	0.0997214	+	0.995015i	$0.468205\pi$
$632$	0	0
$633$	−8.71644e11	−0.215786
$634$	0	0
$635$	2.06268e11	0.0503442
$636$	0	0
$637$	−1.42278e11	−0.0342380
$638$	0	0
$639$	7.18446e12	1.70467
$640$	0	0
$641$	2.30298e12	0.538802	0.269401	−	0.963028i	$-0.413174\pi$
$641$	2.30298e12	0.538802	0.269401	+	0.963028i	$0.413174\pi$
$642$	0	0
$643$	2.54593e12	0.587350	0.293675	−	0.955905i	$-0.405122\pi$
$643$	2.54593e12	0.587350	0.293675	+	0.955905i	$0.405122\pi$
$644$	0	0
$645$	−2.11149e11	−0.0480365
$646$	0	0
$647$	2.74568e11	0.0616001	0.0308000	−	0.999526i	$-0.490194\pi$
$647$	2.74568e11	0.0616001	0.0308000	+	0.999526i	$0.490194\pi$
$648$	0	0
$649$	3.85043e12	0.851937
$650$	0	0
$651$	−1.88595e12	−0.411543
$652$	0	0
$653$	−6.05719e12	−1.30365	−0.651826	−	0.758369i	$-0.725997\pi$
$653$	−6.05719e12	−1.30365	−0.651826	+	0.758369i	$0.725997\pi$
$654$	0	0
$655$	3.79827e11	0.0806306
$656$	0	0
$657$	5.46855e12	1.14506
$658$	0	0
$659$	−4.97718e12	−1.02801	−0.514007	−	0.857786i	$-0.671840\pi$
$659$	−4.97718e12	−1.02801	−0.514007	+	0.857786i	$0.671840\pi$
$660$	0	0
$661$	1.79834e12	0.366409	0.183204	−	0.983075i	$-0.441353\pi$
$661$	1.79834e12	0.366409	0.183204	+	0.983075i	$0.441353\pi$
$662$	0	0
$663$	−1.49487e11	−0.0300465
$664$	0	0
$665$	5.90532e11	0.117097
$666$	0	0
$667$	2.80967e12	0.549653
$668$	0	0
$669$	7.34986e11	0.141860
$670$	0	0
$671$	2.18362e12	0.415839
$672$	0	0
$673$	1.37693e12	0.258728	0.129364	−	0.991597i	$-0.458706\pi$
$673$	1.37693e12	0.258728	0.129364	+	0.991597i	$0.458706\pi$
$674$	0	0
$675$	−3.44547e12	−0.638824
$676$	0	0
$677$	6.49642e11	0.118857	0.0594285	−	0.998233i	$-0.481072\pi$
$677$	6.49642e11	0.118857	0.0594285	+	0.998233i	$0.481072\pi$
$678$	0	0
$679$	7.45828e12	1.34656
$680$	0	0
$681$	2.01386e12	0.358813
$682$	0	0
$683$	1.10011e12	0.193438	0.0967190	−	0.995312i	$-0.469165\pi$
$683$	1.10011e12	0.193438	0.0967190	+	0.995312i	$0.469165\pi$
$684$	0	0
$685$	−5.54064e11	−0.0961507
$686$	0	0
$687$	3.20192e12	0.548409
$688$	0	0
$689$	4.37301e11	0.0739254
$690$	0	0
$691$	−7.72289e12	−1.28863	−0.644315	−	0.764760i	$-0.722858\pi$
$691$	−7.72289e12	−1.28863	−0.644315	+	0.764760i	$0.722858\pi$
$692$	0	0
$693$	2.60702e12	0.429383
$694$	0	0
$695$	3.94832e11	0.0641920
$696$	0	0
$697$	1.63505e12	0.262412
$698$	0	0
$699$	2.48617e12	0.393898
$700$	0	0
$701$	1.40436e12	0.219658	0.109829	−	0.993950i	$-0.464970\pi$
$701$	1.40436e12	0.219658	0.109829	+	0.993950i	$0.464970\pi$
$702$	0	0
$703$	3.74831e12	0.578810
$704$	0	0
$705$	−3.31159e10	−0.00504877
$706$	0	0
$707$	5.39111e12	0.811505
$708$	0	0
$709$	5.19764e12	0.772499	0.386250	−	0.922394i	$-0.373770\pi$
$709$	5.19764e12	0.772499	0.386250	+	0.922394i	$0.373770\pi$
$710$	0	0
$711$	−3.48822e12	−0.511907
$712$	0	0
$713$	−2.88600e12	−0.418209
$714$	0	0
$715$	−7.90176e10	−0.0113070
$716$	0	0
$717$	3.57116e12	0.504630
$718$	0	0
$719$	1.27042e13	1.77284	0.886418	−	0.462886i	$-0.153186\pi$
$719$	1.27042e13	1.77284	0.886418	+	0.462886i	$0.153186\pi$
$720$	0	0
$721$	−2.48016e12	−0.341799
$722$	0	0
$723$	−2.75230e12	−0.374605
$724$	0	0
$725$	−1.25159e13	−1.68244
$726$	0	0
$727$	−9.91998e11	−0.131706	−0.0658531	−	0.997829i	$-0.520977\pi$
$727$	−9.91998e11	−0.131706	−0.0658531	+	0.997829i	$0.520977\pi$
$728$	0	0
$729$	−2.80211e12	−0.367461
$730$	0	0
$731$	−4.39227e12	−0.568933
$732$	0	0
$733$	5.54849e12	0.709916	0.354958	−	0.934882i	$-0.384495\pi$
$733$	5.54849e12	0.709916	0.354958	+	0.934882i	$0.384495\pi$
$734$	0	0
$735$	−2.61791e10	−0.00330873
$736$	0	0
$737$	2.54663e12	0.317953
$738$	0	0
$739$	−8.78597e12	−1.08365	−0.541826	−	0.840491i	$-0.682267\pi$
$739$	−8.78597e12	−1.08365	−0.541826	+	0.840491i	$0.682267\pi$
$740$	0	0
$741$	−1.23701e12	−0.150727
$742$	0	0
$743$	−4.04326e12	−0.486724	−0.243362	−	0.969936i	$-0.578250\pi$
$743$	−4.04326e12	−0.486724	−0.243362	+	0.969936i	$0.578250\pi$
$744$	0	0
$745$	−1.40014e11	−0.0166521
$746$	0	0
$747$	1.10357e12	0.129676
$748$	0	0
$749$	3.99451e12	0.463763
$750$	0	0
$751$	2.52936e11	0.0290156	0.0145078	−	0.999895i	$-0.495382\pi$
$751$	2.52936e11	0.0290156	0.0145078	+	0.999895i	$0.495382\pi$
$752$	0	0
$753$	−5.16479e12	−0.585431
$754$	0	0
$755$	4.77765e11	0.0535122
$756$	0	0
$757$	9.99638e12	1.10640	0.553199	−	0.833049i	$-0.313407\pi$
$757$	9.99638e12	1.10640	0.553199	+	0.833049i	$0.313407\pi$
$758$	0	0
$759$	−5.25864e11	−0.0575156
$760$	0	0
$761$	−2.98848e12	−0.323013	−0.161507	−	0.986872i	$-0.551635\pi$
$761$	−2.98848e12	−0.323013	−0.161507	+	0.986872i	$0.551635\pi$
$762$	0	0
$763$	−9.91726e12	−1.05933
$764$	0	0
$765$	2.08670e11	0.0220284
$766$	0	0
$767$	4.36299e12	0.455203
$768$	0	0
$769$	1.54887e13	1.59715	0.798576	−	0.601894i	$-0.205587\pi$
$769$	1.54887e13	1.59715	0.798576	+	0.601894i	$0.205587\pi$
$770$	0	0
$771$	3.17948e12	0.324050
$772$	0	0
$773$	1.07589e13	1.08383	0.541915	−	0.840434i	$-0.317700\pi$
$773$	1.07589e13	1.08383	0.541915	+	0.840434i	$0.317700\pi$
$774$	0	0
$775$	1.28559e13	1.28010
$776$	0	0
$777$	1.17990e12	0.116131
$778$	0	0
$779$	1.35301e13	1.31638
$780$	0	0
$781$	−1.04130e13	−1.00149
$782$	0	0
$783$	1.14452e13	1.08817
$784$	0	0
$785$	1.54767e12	0.145467
$786$	0	0
$787$	−1.29304e13	−1.20151	−0.600755	−	0.799433i	$-0.705133\pi$
$787$	−1.29304e13	−1.20151	−0.600755	+	0.799433i	$0.705133\pi$
$788$	0	0
$789$	−3.90379e12	−0.358625
$790$	0	0
$791$	1.09778e13	0.997059
$792$	0	0
$793$	2.47430e12	0.222189
$794$	0	0
$795$	8.04635e10	0.00714409
$796$	0	0
$797$	1.24713e13	1.09484	0.547418	−	0.836859i	$-0.315611\pi$
$797$	1.24713e13	1.09484	0.547418	+	0.836859i	$0.315611\pi$
$798$	0	0
$799$	−6.88868e11	−0.0597965
$800$	0	0
$801$	−6.03585e11	−0.0518075
$802$	0	0
$803$	−7.92600e12	−0.672719
$804$	0	0
$805$	2.84459e11	0.0238747
$806$	0	0
$807$	−4.81502e12	−0.399639
$808$	0	0
$809$	−2.40763e13	−1.97616	−0.988079	−	0.153948i	$-0.950801\pi$
$809$	−2.40763e13	−1.97616	−0.988079	+	0.153948i	$0.950801\pi$
$810$	0	0
$811$	−1.14686e13	−0.930928	−0.465464	−	0.885067i	$-0.654112\pi$
$811$	−1.14686e13	−0.930928	−0.465464	+	0.885067i	$0.654112\pi$
$812$	0	0
$813$	−1.43040e12	−0.114829
$814$	0	0
$815$	7.95569e10	0.00631638
$816$	0	0
$817$	−3.63462e13	−2.85403
$818$	0	0
$819$	2.95406e12	0.229426
$820$	0	0
$821$	−2.50113e13	−1.92129	−0.960643	−	0.277786i	$-0.910399\pi$
$821$	−2.50113e13	−1.92129	−0.960643	+	0.277786i	$0.910399\pi$
$822$	0	0
$823$	−1.29351e13	−0.982808	−0.491404	−	0.870932i	$-0.663516\pi$
$823$	−1.29351e13	−0.982808	−0.491404	+	0.870932i	$0.663516\pi$
$824$	0	0
$825$	2.34250e12	0.176051
$826$	0	0
$827$	−1.00464e13	−0.746851	−0.373425	−	0.927660i	$-0.621817\pi$
$827$	−1.00464e13	−0.746851	−0.373425	+	0.927660i	$0.621817\pi$
$828$	0	0
$829$	8.77304e12	0.645141	0.322570	−	0.946545i	$-0.395453\pi$
$829$	8.77304e12	0.645141	0.322570	+	0.946545i	$0.395453\pi$
$830$	0	0
$831$	1.51868e12	0.110474
$832$	0	0
$833$	−5.44571e11	−0.0391879
$834$	0	0
$835$	−9.14353e10	−0.00650916
$836$	0	0
$837$	−1.17561e13	−0.827943
$838$	0	0
$839$	1.54259e13	1.07478	0.537392	−	0.843332i	$-0.319410\pi$
$839$	1.54259e13	1.07478	0.537392	+	0.843332i	$0.319410\pi$
$840$	0	0
$841$	2.70683e13	1.86586
$842$	0	0
$843$	−4.72537e12	−0.322264
$844$	0	0
$845$	−8.95364e10	−0.00604149
$846$	0	0
$847$	1.02452e13	0.683983
$848$	0	0
$849$	−5.08296e12	−0.335762
$850$	0	0
$851$	1.80556e12	0.118012
$852$	0	0
$853$	−1.84928e12	−0.119600	−0.0598002	−	0.998210i	$-0.519046\pi$
$853$	−1.84928e12	−0.119600	−0.0598002	+	0.998210i	$0.519046\pi$
$854$	0	0
$855$	1.72675e12	0.110505
$856$	0	0
$857$	−2.34715e13	−1.48637	−0.743185	−	0.669086i	$-0.766686\pi$
$857$	−2.34715e13	−1.48637	−0.743185	+	0.669086i	$0.766686\pi$
$858$	0	0
$859$	−1.25121e12	−0.0784079	−0.0392039	−	0.999231i	$-0.512482\pi$
$859$	−1.25121e12	−0.0784079	−0.0392039	+	0.999231i	$0.512482\pi$
$860$	0	0
$861$	4.25902e12	0.264116
$862$	0	0
$863$	−2.38217e13	−1.46192	−0.730961	−	0.682419i	$-0.760928\pi$
$863$	−2.38217e13	−1.46192	−0.730961	+	0.682419i	$0.760928\pi$
$864$	0	0
$865$	−6.34375e10	−0.00385278
$866$	0	0
$867$	5.10563e12	0.306876
$868$	0	0
$869$	5.05575e12	0.300744
$870$	0	0
$871$	2.88564e12	0.169887
$872$	0	0
$873$	2.18084e13	1.27075
$874$	0	0
$875$	−2.54215e12	−0.146611
$876$	0	0
$877$	−1.35717e13	−0.774707	−0.387353	−	0.921931i	$-0.626611\pi$
$877$	−1.35717e13	−0.774707	−0.387353	+	0.921931i	$0.626611\pi$
$878$	0	0
$879$	−1.08375e13	−0.612323
$880$	0	0
$881$	2.38436e12	0.133346	0.0666730	−	0.997775i	$-0.478762\pi$
$881$	2.38436e12	0.133346	0.0666730	+	0.997775i	$0.478762\pi$
$882$	0	0
$883$	5.99779e11	0.0332023	0.0166011	−	0.999862i	$-0.494715\pi$
$883$	5.99779e11	0.0332023	0.0166011	+	0.999862i	$0.494715\pi$
$884$	0	0
$885$	8.02792e11	0.0439904
$886$	0	0
$887$	−1.02483e13	−0.555897	−0.277949	−	0.960596i	$-0.589654\pi$
$887$	−1.02483e13	−0.555897	−0.277949	+	0.960596i	$0.589654\pi$
$888$	0	0
$889$	1.11766e13	0.600138
$890$	0	0
$891$	6.48579e12	0.344757
$892$	0	0
$893$	−5.70040e12	−0.299967
$894$	0	0
$895$	−1.71854e12	−0.0895274
$896$	0	0
$897$	−5.95867e11	−0.0307315
$898$	0	0
$899$	−4.27050e13	−2.18052
$900$	0	0
$901$	1.67378e12	0.0846130
$902$	0	0
$903$	−1.14411e13	−0.572628
$904$	0	0
$905$	2.39887e12	0.118874
$906$	0	0
$907$	2.43855e13	1.19646	0.598231	−	0.801323i	$-0.295870\pi$
$907$	2.43855e13	1.19646	0.598231	+	0.801323i	$0.295870\pi$
$908$	0	0
$909$	1.57639e13	0.765820
$910$	0	0
$911$	−2.90386e13	−1.39683	−0.698415	−	0.715693i	$-0.746111\pi$
$911$	−2.90386e13	−1.39683	−0.698415	+	0.715693i	$0.746111\pi$
$912$	0	0
$913$	−1.59949e12	−0.0761841
$914$	0	0
$915$	4.55272e11	0.0214722
$916$	0	0
$917$	2.05809e13	0.961173
$918$	0	0
$919$	1.41793e13	0.655746	0.327873	−	0.944722i	$-0.393668\pi$
$919$	1.41793e13	0.655746	0.327873	+	0.944722i	$0.393668\pi$
$920$	0	0
$921$	−1.06773e12	−0.0488984
$922$	0	0
$923$	−1.17992e13	−0.535111
$924$	0	0
$925$	−8.04299e12	−0.361226
$926$	0	0
$927$	−7.25212e12	−0.322557
$928$	0	0
$929$	−3.56246e13	−1.56920	−0.784601	−	0.620001i	$-0.787132\pi$
$929$	−3.56246e13	−1.56920	−0.784601	+	0.620001i	$0.787132\pi$
$930$	0	0
$931$	−4.50634e12	−0.196585
$932$	0	0
$933$	−1.30136e13	−0.562250
$934$	0	0
$935$	−3.02442e11	−0.0129417
$936$	0	0
$937$	9.35575e12	0.396506	0.198253	−	0.980151i	$-0.436473\pi$
$937$	9.35575e12	0.396506	0.198253	+	0.980151i	$0.436473\pi$
$938$	0	0
$939$	3.78610e12	0.158927
$940$	0	0
$941$	1.23413e13	0.513106	0.256553	−	0.966530i	$-0.417413\pi$
$941$	1.23413e13	0.513106	0.256553	+	0.966530i	$0.417413\pi$
$942$	0	0
$943$	6.51742e12	0.268394
$944$	0	0
$945$	1.15875e12	0.0472656
$946$	0	0
$947$	−2.93722e12	−0.118676	−0.0593379	−	0.998238i	$-0.518899\pi$
$947$	−2.93722e12	−0.118676	−0.0593379	+	0.998238i	$0.518899\pi$
$948$	0	0
$949$	−8.98110e12	−0.359444
$950$	0	0
$951$	−9.72086e12	−0.385383
$952$	0	0
$953$	−3.90544e12	−0.153374	−0.0766870	−	0.997055i	$-0.524434\pi$
$953$	−3.90544e12	−0.153374	−0.0766870	+	0.997055i	$0.524434\pi$
$954$	0	0
$955$	1.84163e12	0.0716454
$956$	0	0
$957$	−7.78137e12	−0.299883
$958$	0	0
$959$	−3.00219e13	−1.14618
$960$	0	0
$961$	1.74255e13	0.659069
$962$	0	0
$963$	1.16802e13	0.437655
$964$	0	0
$965$	2.96398e12	0.110028
$966$	0	0
$967$	1.06697e13	0.392405	0.196203	−	0.980563i	$-0.437139\pi$
$967$	1.06697e13	0.392405	0.196203	+	0.980563i	$0.437139\pi$
$968$	0	0
$969$	−4.73469e12	−0.172518
$970$	0	0
$971$	3.10264e12	0.112007	0.0560034	−	0.998431i	$-0.482164\pi$
$971$	3.10264e12	0.112007	0.0560034	+	0.998431i	$0.482164\pi$
$972$	0	0
$973$	2.13939e13	0.765213
$974$	0	0
$975$	2.65434e12	0.0940665
$976$	0	0
$977$	−4.66167e13	−1.63688	−0.818438	−	0.574594i	$-0.805160\pi$
$977$	−4.66167e13	−1.63688	−0.818438	+	0.574594i	$0.805160\pi$
$978$	0	0
$979$	8.74824e11	0.0304368
$980$	0	0
$981$	−2.89986e13	−0.999694
$982$	0	0
$983$	3.22316e13	1.10101	0.550505	−	0.834832i	$-0.314435\pi$
$983$	3.22316e13	1.10101	0.550505	+	0.834832i	$0.314435\pi$
$984$	0	0
$985$	−1.73728e12	−0.0588040
$986$	0	0
$987$	−1.79438e12	−0.0601849
$988$	0	0
$989$	−1.75079e13	−0.581903
$990$	0	0
$991$	2.25008e13	0.741081	0.370540	−	0.928816i	$-0.379172\pi$
$991$	2.25008e13	0.741081	0.370540	+	0.928816i	$0.379172\pi$
$992$	0	0
$993$	1.55192e13	0.506520
$994$	0	0
$995$	9.66342e11	0.0312556
$996$	0	0
$997$	−3.13756e13	−1.00569	−0.502844	−	0.864377i	$-0.667713\pi$
$997$	−3.13756e13	−1.00569	−0.502844	+	0.864377i	$0.667713\pi$
$998$	0	0
$999$	7.35495e12	0.233633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.10.a.h.1.3		5
4.3	odd	2		13.10.a.b.1.5	✓	5
12.11	even	2		117.10.a.e.1.1		5
20.19	odd	2		325.10.a.b.1.1		5
52.51	odd	2		169.10.a.b.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.10.a.b.1.5	✓	5	4.3	odd	2
117.10.a.e.1.1		5	12.11	even	2
169.10.a.b.1.1		5	52.51	odd	2
208.10.a.h.1.3		5	1.1	even	1	trivial
325.10.a.b.1.1		5	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

\(f(q)\)	\(=\)	\(q-47.8784 q^{3} -109.762 q^{5} -5947.44 q^{7} -17390.7 q^{9} +O(q^{10})\)	\(q-47.8784 q^{3} -109.762 q^{5} -5947.44 q^{7} -17390.7 q^{9} +25205.7 q^{11} +28561.0 q^{13} +5255.24 q^{15} +109318. q^{17} +904609. q^{19} +284754. q^{21} +435749. q^{23} -1.94108e6 q^{25} +1.77503e6 q^{27} +6.44791e6 q^{29} -6.62308e6 q^{31} -1.20681e6 q^{33} +652804. q^{35} +4.14357e6 q^{37} -1.36745e6 q^{39} +1.49568e7 q^{41} -4.01789e7 q^{43} +1.90884e6 q^{45} -6.30151e6 q^{47} -4.98153e6 q^{49} -5.23397e6 q^{51} +1.53111e7 q^{53} -2.76663e6 q^{55} -4.33112e7 q^{57} +1.52760e8 q^{59} +8.66321e7 q^{61} +1.03430e8 q^{63} -3.13492e6 q^{65} +1.01034e8 q^{67} -2.08630e7 q^{69} -4.13122e8 q^{71} -3.14453e8 q^{73} +9.29357e7 q^{75} -1.49909e8 q^{77} +2.00580e8 q^{79} +2.57315e8 q^{81} -6.34578e7 q^{83} -1.19990e7 q^{85} -3.08715e8 q^{87} +3.47074e7 q^{89} -1.69865e8 q^{91} +3.17102e8 q^{93} -9.92918e7 q^{95} -1.25403e9 q^{97} -4.38343e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9}+O(q^{10}) \) 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9	\( 5 q - 161 q^{3} + 1803 q^{5} - 10099 q^{7} + 61060 q^{9} - 121746 q^{11} + 142805 q^{13} - 105973 q^{15} - 495669 q^{17} + 840738 q^{19} - 1599467 q^{21} + 592152 q^{23} + 1670362 q^{25} - 6847883 q^{27} + 10678182 q^{29} - 12885296 q^{31} + 17278298 q^{33} - 8380731 q^{35} + 7171823 q^{37} - 4598321 q^{39} + 9294012 q^{41} - 12831975 q^{43} + 26135198 q^{45} - 43354215 q^{47} + 25249488 q^{49} - 16905901 q^{51} + 93231780 q^{53} - 99448846 q^{55} + 90173382 q^{57} - 246496182 q^{59} - 132232612 q^{61} + 416955202 q^{63} + 51495483 q^{65} + 369388534 q^{67} - 579986760 q^{69} - 212150457 q^{71} - 252729806 q^{73} + 752457788 q^{75} + 449666118 q^{77} + 1247271728 q^{79} - 317713115 q^{81} - 1696894296 q^{83} - 775363765 q^{85} + 614530466 q^{87} - 753854382 q^{89} - 288437539 q^{91} - 892784668 q^{93} - 1442632962 q^{95} + 3824606 q^{97} - 3016199848 q^{99}+O(q^{100}) \) 5 * q - 161 * q^3 + 1803 * q^5 - 10099 * q^7 + 61060 * q^9 - 121746 * q^11 + 142805 * q^13 - 105973 * q^15 - 495669 * q^17 + 840738 * q^19 - 1599467 * q^21 + 592152 * q^23 + 1670362 * q^25 - 6847883 * q^27 + 10678182 * q^29 - 12885296 * q^31 + 17278298 * q^33 - 8380731 * q^35 + 7171823 * q^37 - 4598321 * q^39 + 9294012 * q^41 - 12831975 * q^43 + 26135198 * q^45 - 43354215 * q^47 + 25249488 * q^49 - 16905901 * q^51 + 93231780 * q^53 - 99448846 * q^55 + 90173382 * q^57 - 246496182 * q^59 - 132232612 * q^61 + 416955202 * q^63 + 51495483 * q^65 + 369388534 * q^67 - 579986760 * q^69 - 212150457 * q^71 - 252729806 * q^73 + 752457788 * q^75 + 449666118 * q^77 + 1247271728 * q^79 - 317713115 * q^81 - 1696894296 * q^83 - 775363765 * q^85 + 614530466 * q^87 - 753854382 * q^89 - 288437539 * q^91 - 892784668 * q^93 - 1442632962 * q^95 + 3824606 * q^97 - 3016199848 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more