LMFDB - Embedded newform 5202.2.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5202,2,Mod(1,5202)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5202, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("5202.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5202 = 2 \cdot 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5202.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:	$41.5381791315$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	5202.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.87939 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -3.87939 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.87939 q^{10} -1.65270 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.06418 q^{19} -3.87939 q^{20} +1.65270 q^{22} -5.06418 q^{23} +10.0496 q^{25} -2.00000 q^{26} +1.00000 q^{28} -3.16250 q^{29} -8.35504 q^{31} -1.00000 q^{32} -3.87939 q^{35} -4.69459 q^{37} -7.06418 q^{38} +3.87939 q^{40} +11.2763 q^{41} -9.88713 q^{43} -1.65270 q^{44} +5.06418 q^{46} -3.51754 q^{47} -6.00000 q^{49} -10.0496 q^{50} +2.00000 q^{52} +10.8452 q^{53} +6.41147 q^{55} -1.00000 q^{56} +3.16250 q^{58} +8.33275 q^{59} +12.9067 q^{61} +8.35504 q^{62} +1.00000 q^{64} -7.75877 q^{65} -5.75877 q^{67} +3.87939 q^{70} +4.73917 q^{71} +13.0496 q^{73} +4.69459 q^{74} +7.06418 q^{76} -1.65270 q^{77} +5.32770 q^{79} -3.87939 q^{80} -11.2763 q^{82} +12.8425 q^{83} +9.88713 q^{86} +1.65270 q^{88} +13.9709 q^{89} +2.00000 q^{91} -5.06418 q^{92} +3.51754 q^{94} -27.4047 q^{95} -3.86484 q^{97} +6.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 3 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} - 3 q^{14} + 3 q^{16} + 12 q^{19} - 6 q^{20} + 6 q^{22} - 6 q^{23} + 3 q^{25} - 6 q^{26} + 3 q^{28} - 12 q^{29} - 3 q^{32} - 6 q^{35} - 12 q^{37} - 12 q^{38} + 6 q^{40} - 6 q^{44} + 6 q^{46} + 12 q^{47} - 18 q^{49} - 3 q^{50} + 6 q^{52} + 6 q^{53} + 9 q^{55} - 3 q^{56} + 12 q^{58} + 6 q^{59} + 12 q^{61} + 3 q^{64} - 12 q^{65} - 6 q^{67} + 6 q^{70} + 12 q^{73} + 12 q^{74} + 12 q^{76} - 6 q^{77} + 12 q^{79} - 6 q^{80} + 21 q^{83} + 6 q^{88} + 6 q^{89} + 6 q^{91} - 6 q^{92} - 12 q^{94} - 30 q^{95} + 12 q^{97} + 18 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 3 * q^8 + 6 * q^10 - 6 * q^11 + 6 * q^13 - 3 * q^14 + 3 * q^16 + 12 * q^19 - 6 * q^20 + 6 * q^22 - 6 * q^23 + 3 * q^25 - 6 * q^26 + 3 * q^28 - 12 * q^29 - 3 * q^32 - 6 * q^35 - 12 * q^37 - 12 * q^38 + 6 * q^40 - 6 * q^44 + 6 * q^46 + 12 * q^47 - 18 * q^49 - 3 * q^50 + 6 * q^52 + 6 * q^53 + 9 * q^55 - 3 * q^56 + 12 * q^58 + 6 * q^59 + 12 * q^61 + 3 * q^64 - 12 * q^65 - 6 * q^67 + 6 * q^70 + 12 * q^73 + 12 * q^74 + 12 * q^76 - 6 * q^77 + 12 * q^79 - 6 * q^80 + 21 * q^83 + 6 * q^88 + 6 * q^89 + 6 * q^91 - 6 * q^92 - 12 * q^94 - 30 * q^95 + 12 * q^97 + 18 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.87939	−1.73491	−0.867457	−	0.497512i	$-0.834247\pi$
$5$	−3.87939	−1.73491	−0.867457	+	0.497512i	$0.834247\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.87939	1.22677
$11$	−1.65270	−0.498309	−0.249154	−	0.968464i	$-0.580153\pi$
$11$	−1.65270	−0.498309	−0.249154	+	0.968464i	$0.580153\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	7.06418	1.62063	0.810317	−	0.585992i	$-0.199295\pi$
$19$	7.06418	1.62063	0.810317	+	0.585992i	$0.199295\pi$
$20$	−3.87939	−0.867457
$21$	0	0
$22$	1.65270	0.352358
$23$	−5.06418	−1.05595	−0.527977	−	0.849259i	$-0.677049\pi$
$23$	−5.06418	−1.05595	−0.527977	+	0.849259i	$0.677049\pi$
$24$	0	0
$25$	10.0496	2.00993
$26$	−2.00000	−0.392232
$27$	0	0
$28$	1.00000	0.188982
$29$	−3.16250	−0.587262	−0.293631	−	0.955919i	$-0.594864\pi$
$29$	−3.16250	−0.587262	−0.293631	+	0.955919i	$0.594864\pi$
$30$	0	0
$31$	−8.35504	−1.50061	−0.750304	−	0.661092i	$-0.770093\pi$
$31$	−8.35504	−1.50061	−0.750304	+	0.661092i	$0.770093\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0	0
$35$	−3.87939	−0.655736
$36$	0	0
$37$	−4.69459	−0.771786	−0.385893	−	0.922543i	$-0.626107\pi$
$37$	−4.69459	−0.771786	−0.385893	+	0.922543i	$0.626107\pi$
$38$	−7.06418	−1.14596
$39$	0	0
$40$	3.87939	0.613385
$41$	11.2763	1.76106	0.880532	−	0.473987i	$-0.157186\pi$
$41$	11.2763	1.76106	0.880532	+	0.473987i	$0.157186\pi$
$42$	0	0
$43$	−9.88713	−1.50777	−0.753886	−	0.657005i	$-0.771823\pi$
$43$	−9.88713	−1.50777	−0.753886	+	0.657005i	$0.771823\pi$
$44$	−1.65270	−0.249154
$45$	0	0
$46$	5.06418	0.746672
$47$	−3.51754	−0.513086	−0.256543	−	0.966533i	$-0.582584\pi$
$47$	−3.51754	−0.513086	−0.256543	+	0.966533i	$0.582584\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−10.0496	−1.42123
$51$	0	0
$52$	2.00000	0.277350
$53$	10.8452	1.48971	0.744854	−	0.667228i	$-0.232519\pi$
$53$	10.8452	1.48971	0.744854	+	0.667228i	$0.232519\pi$
$54$	0	0
$55$	6.41147	0.864523
$56$	−1.00000	−0.133631
$57$	0	0
$58$	3.16250	0.415257
$59$	8.33275	1.08483	0.542416	−	0.840110i	$-0.317510\pi$
$59$	8.33275	1.08483	0.542416	+	0.840110i	$0.317510\pi$
$60$	0	0
$61$	12.9067	1.65254	0.826268	−	0.563276i	$-0.190459\pi$
$61$	12.9067	1.65254	0.826268	+	0.563276i	$0.190459\pi$
$62$	8.35504	1.06109
$63$	0	0
$64$	1.00000	0.125000
$65$	−7.75877	−0.962357
$66$	0	0
$67$	−5.75877	−0.703546	−0.351773	−	0.936085i	$-0.614421\pi$
$67$	−5.75877	−0.703546	−0.351773	+	0.936085i	$0.614421\pi$
$68$	0	0
$69$	0	0
$70$	3.87939	0.463675
$71$	4.73917	0.562436	0.281218	−	0.959644i	$-0.409262\pi$
$71$	4.73917	0.562436	0.281218	+	0.959644i	$0.409262\pi$
$72$	0	0
$73$	13.0496	1.52734	0.763672	−	0.645605i	$-0.223395\pi$
$73$	13.0496	1.52734	0.763672	+	0.645605i	$0.223395\pi$
$74$	4.69459	0.545735
$75$	0	0
$76$	7.06418	0.810317
$77$	−1.65270	−0.188343
$78$	0	0
$79$	5.32770	0.599413	0.299706	−	0.954032i	$-0.403111\pi$
$79$	5.32770	0.599413	0.299706	+	0.954032i	$0.403111\pi$
$80$	−3.87939	−0.433728
$81$	0	0
$82$	−11.2763	−1.24526
$83$	12.8425	1.40965	0.704826	−	0.709380i	$-0.251025\pi$
$83$	12.8425	1.40965	0.704826	+	0.709380i	$0.251025\pi$
$84$	0	0
$85$	0	0
$86$	9.88713	1.06616
$87$	0	0
$88$	1.65270	0.176179
$89$	13.9709	1.48091	0.740456	−	0.672104i	$-0.234609\pi$
$89$	13.9709	1.48091	0.740456	+	0.672104i	$0.234609\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	−5.06418	−0.527977
$93$	0	0
$94$	3.51754	0.362807
$95$	−27.4047	−2.81166
$96$	0	0
$97$	−3.86484	−0.392415	−0.196207	−	0.980562i	$-0.562863\pi$
$97$	−3.86484	−0.392415	−0.196207	+	0.980562i	$0.562863\pi$
$98$	6.00000	0.606092
$99$	0	0
$100$	10.0496	1.00496
$101$	−12.7861	−1.27227	−0.636133	−	0.771580i	$-0.719467\pi$
$101$	−12.7861	−1.27227	−0.636133	+	0.771580i	$0.719467\pi$
$102$	0	0
$103$	−6.51249	−0.641695	−0.320847	−	0.947131i	$-0.603968\pi$
$103$	−6.51249	−0.641695	−0.320847	+	0.947131i	$0.603968\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−10.8452	−1.05338
$107$	−11.2071	−1.08343	−0.541715	−	0.840562i	$-0.682225\pi$
$107$	−11.2071	−1.08343	−0.541715	+	0.840562i	$0.682225\pi$
$108$	0	0
$109$	−4.36959	−0.418530	−0.209265	−	0.977859i	$-0.567107\pi$
$109$	−4.36959	−0.418530	−0.209265	+	0.977859i	$0.567107\pi$
$110$	−6.41147	−0.611310
$111$	0	0
$112$	1.00000	0.0944911
$113$	−4.85204	−0.456442	−0.228221	−	0.973609i	$-0.573291\pi$
$113$	−4.85204	−0.456442	−0.228221	+	0.973609i	$0.573291\pi$
$114$	0	0
$115$	19.6459	1.83199
$116$	−3.16250	−0.293631
$117$	0	0
$118$	−8.33275	−0.767092
$119$	0	0
$120$	0	0
$121$	−8.26857	−0.751688
$122$	−12.9067	−1.16852
$123$	0	0
$124$	−8.35504	−0.750304
$125$	−19.5895	−1.75213
$126$	0	0
$127$	4.38919	0.389477	0.194739	−	0.980855i	$-0.437614\pi$
$127$	4.38919	0.389477	0.194739	+	0.980855i	$0.437614\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	7.75877	0.680489
$131$	9.71007	0.848373	0.424187	−	0.905575i	$-0.360560\pi$
$131$	9.71007	0.848373	0.424187	+	0.905575i	$0.360560\pi$
$132$	0	0
$133$	7.06418	0.612542
$134$	5.75877	0.497482
$135$	0	0
$136$	0	0
$137$	2.98040	0.254633	0.127316	−	0.991862i	$-0.459364\pi$
$137$	2.98040	0.254633	0.127316	+	0.991862i	$0.459364\pi$
$138$	0	0
$139$	11.6459	0.987792	0.493896	−	0.869521i	$-0.335572\pi$
$139$	11.6459	0.987792	0.493896	+	0.869521i	$0.335572\pi$
$140$	−3.87939	−0.327868
$141$	0	0
$142$	−4.73917	−0.397702
$143$	−3.30541	−0.276412
$144$	0	0
$145$	12.2686	1.01885
$146$	−13.0496	−1.08000
$147$	0	0
$148$	−4.69459	−0.385893
$149$	−17.8803	−1.46481	−0.732406	−	0.680868i	$-0.761603\pi$
$149$	−17.8803	−1.46481	−0.732406	+	0.680868i	$0.761603\pi$
$150$	0	0
$151$	−5.97771	−0.486459	−0.243230	−	0.969969i	$-0.578207\pi$
$151$	−5.97771	−0.486459	−0.243230	+	0.969969i	$0.578207\pi$
$152$	−7.06418	−0.572980
$153$	0	0
$154$	1.65270	0.133179
$155$	32.4124	2.60343
$156$	0	0
$157$	4.04458	0.322792	0.161396	−	0.986890i	$-0.448400\pi$
$157$	4.04458	0.322792	0.161396	+	0.986890i	$0.448400\pi$
$158$	−5.32770	−0.423849
$159$	0	0
$160$	3.87939	0.306692
$161$	−5.06418	−0.399113
$162$	0	0
$163$	−23.1925	−1.81658	−0.908290	−	0.418342i	$-0.862611\pi$
$163$	−23.1925	−1.81658	−0.908290	+	0.418342i	$0.862611\pi$
$164$	11.2763	0.880532
$165$	0	0
$166$	−12.8425	−0.996775
$167$	2.25671	0.174630	0.0873148	−	0.996181i	$-0.472171\pi$
$167$	2.25671	0.174630	0.0873148	+	0.996181i	$0.472171\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−9.88713	−0.753886
$173$	−8.18984	−0.622662	−0.311331	−	0.950301i	$-0.600775\pi$
$173$	−8.18984	−0.622662	−0.311331	+	0.950301i	$0.600775\pi$
$174$	0	0
$175$	10.0496	0.759681
$176$	−1.65270	−0.124577
$177$	0	0
$178$	−13.9709	−1.04716
$179$	−7.41416	−0.554161	−0.277080	−	0.960847i	$-0.589367\pi$
$179$	−7.41416	−0.554161	−0.277080	+	0.960847i	$0.589367\pi$
$180$	0	0
$181$	1.30541	0.0970302	0.0485151	−	0.998822i	$-0.484551\pi$
$181$	1.30541	0.0970302	0.0485151	+	0.998822i	$0.484551\pi$
$182$	−2.00000	−0.148250
$183$	0	0
$184$	5.06418	0.373336
$185$	18.2121	1.33898
$186$	0	0
$187$	0	0
$188$	−3.51754	−0.256543
$189$	0	0
$190$	27.4047	1.98814
$191$	−6.28581	−0.454825	−0.227413	−	0.973799i	$-0.573027\pi$
$191$	−6.28581	−0.454825	−0.227413	+	0.973799i	$0.573027\pi$
$192$	0	0
$193$	−2.78611	−0.200549	−0.100274	−	0.994960i	$-0.531972\pi$
$193$	−2.78611	−0.200549	−0.100274	+	0.994960i	$0.531972\pi$
$194$	3.86484	0.277479
$195$	0	0
$196$	−6.00000	−0.428571
$197$	−1.02910	−0.0733200	−0.0366600	−	0.999328i	$-0.511672\pi$
$197$	−1.02910	−0.0733200	−0.0366600	+	0.999328i	$0.511672\pi$
$198$	0	0
$199$	−25.2003	−1.78640	−0.893200	−	0.449660i	$-0.851545\pi$
$199$	−25.2003	−1.78640	−0.893200	+	0.449660i	$0.851545\pi$
$200$	−10.0496	−0.710616
$201$	0	0
$202$	12.7861	0.899628
$203$	−3.16250	−0.221964
$204$	0	0
$205$	−43.7452	−3.05529
$206$	6.51249	0.453747
$207$	0	0
$208$	2.00000	0.138675
$209$	−11.6750	−0.807576
$210$	0	0
$211$	4.49794	0.309651	0.154826	−	0.987942i	$-0.450518\pi$
$211$	4.49794	0.309651	0.154826	+	0.987942i	$0.450518\pi$
$212$	10.8452	0.744854
$213$	0	0
$214$	11.2071	0.766100
$215$	38.3560	2.61586
$216$	0	0
$217$	−8.35504	−0.567177
$218$	4.36959	0.295946
$219$	0	0
$220$	6.41147	0.432261
$221$	0	0
$222$	0	0
$223$	−25.2249	−1.68919	−0.844593	−	0.535409i	$-0.820158\pi$
$223$	−25.2249	−1.68919	−0.844593	+	0.535409i	$0.820158\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	4.85204	0.322753
$227$	−25.8384	−1.71496	−0.857478	−	0.514520i	$-0.827970\pi$
$227$	−25.8384	−1.71496	−0.857478	+	0.514520i	$0.827970\pi$
$228$	0	0
$229$	−6.65539	−0.439801	−0.219900	−	0.975522i	$-0.570573\pi$
$229$	−6.65539	−0.439801	−0.219900	+	0.975522i	$0.570573\pi$
$230$	−19.6459	−1.29541
$231$	0	0
$232$	3.16250	0.207629
$233$	−20.9067	−1.36965	−0.684823	−	0.728710i	$-0.740120\pi$
$233$	−20.9067	−1.36965	−0.684823	+	0.728710i	$0.740120\pi$
$234$	0	0
$235$	13.6459	0.890160
$236$	8.33275	0.542416
$237$	0	0
$238$	0	0
$239$	−16.9905	−1.09902	−0.549512	−	0.835486i	$-0.685186\pi$
$239$	−16.9905	−1.09902	−0.549512	+	0.835486i	$0.685186\pi$
$240$	0	0
$241$	−6.37733	−0.410800	−0.205400	−	0.978678i	$-0.565849\pi$
$241$	−6.37733	−0.410800	−0.205400	+	0.978678i	$0.565849\pi$
$242$	8.26857	0.531524
$243$	0	0
$244$	12.9067	0.826268
$245$	23.2763	1.48707
$246$	0	0
$247$	14.1284	0.898966
$248$	8.35504	0.530545
$249$	0	0
$250$	19.5895	1.23895
$251$	27.6604	1.74591	0.872956	−	0.487799i	$-0.162200\pi$
$251$	27.6604	1.74591	0.872956	+	0.487799i	$0.162200\pi$
$252$	0	0
$253$	8.36959	0.526191
$254$	−4.38919	−0.275402
$255$	0	0
$256$	1.00000	0.0625000
$257$	−5.71419	−0.356442	−0.178221	−	0.983991i	$-0.557034\pi$
$257$	−5.71419	−0.356442	−0.178221	+	0.983991i	$0.557034\pi$
$258$	0	0
$259$	−4.69459	−0.291708
$260$	−7.75877	−0.481179
$261$	0	0
$262$	−9.71007	−0.599890
$263$	6.71007	0.413761	0.206880	−	0.978366i	$-0.433669\pi$
$263$	6.71007	0.413761	0.206880	+	0.978366i	$0.433669\pi$
$264$	0	0
$265$	−42.0729	−2.58451
$266$	−7.06418	−0.433133
$267$	0	0
$268$	−5.75877	−0.351773
$269$	−15.0128	−0.915346	−0.457673	−	0.889121i	$-0.651317\pi$
$269$	−15.0128	−0.915346	−0.457673	+	0.889121i	$0.651317\pi$
$270$	0	0
$271$	2.96585	0.180163	0.0900813	−	0.995934i	$-0.471287\pi$
$271$	2.96585	0.180163	0.0900813	+	0.995934i	$0.471287\pi$
$272$	0	0
$273$	0	0
$274$	−2.98040	−0.180053
$275$	−16.6091	−1.00156
$276$	0	0
$277$	−2.93582	−0.176396	−0.0881982	−	0.996103i	$-0.528111\pi$
$277$	−2.93582	−0.176396	−0.0881982	+	0.996103i	$0.528111\pi$
$278$	−11.6459	−0.698474
$279$	0	0
$280$	3.87939	0.231838
$281$	−28.9513	−1.72709	−0.863545	−	0.504272i	$-0.831761\pi$
$281$	−28.9513	−1.72709	−0.863545	+	0.504272i	$0.831761\pi$
$282$	0	0
$283$	−10.4825	−0.623118	−0.311559	−	0.950227i	$-0.600851\pi$
$283$	−10.4825	−0.623118	−0.311559	+	0.950227i	$0.600851\pi$
$284$	4.73917	0.281218
$285$	0	0
$286$	3.30541	0.195453
$287$	11.2763	0.665620
$288$	0	0
$289$	0	0
$290$	−12.2686	−0.720435
$291$	0	0
$292$	13.0496	0.763672
$293$	8.54488	0.499197	0.249599	−	0.968349i	$-0.419701\pi$
$293$	8.54488	0.499197	0.249599	+	0.968349i	$0.419701\pi$
$294$	0	0
$295$	−32.3259	−1.88209
$296$	4.69459	0.272868
$297$	0	0
$298$	17.8803	1.03578
$299$	−10.1284	−0.585738
$300$	0	0
$301$	−9.88713	−0.569884
$302$	5.97771	0.343979
$303$	0	0
$304$	7.06418	0.405158
$305$	−50.0702	−2.86701
$306$	0	0
$307$	12.4534	0.710751	0.355375	−	0.934724i	$-0.384353\pi$
$307$	12.4534	0.710751	0.355375	+	0.934724i	$0.384353\pi$
$308$	−1.65270	−0.0941715
$309$	0	0
$310$	−32.4124	−1.84090
$311$	−23.2371	−1.31766	−0.658828	−	0.752294i	$-0.728947\pi$
$311$	−23.2371	−1.31766	−0.658828	+	0.752294i	$0.728947\pi$
$312$	0	0
$313$	21.7743	1.23075	0.615377	−	0.788233i	$-0.289004\pi$
$313$	21.7743	1.23075	0.615377	+	0.788233i	$0.289004\pi$
$314$	−4.04458	−0.228249
$315$	0	0
$316$	5.32770	0.299706
$317$	14.4884	0.813752	0.406876	−	0.913483i	$-0.366618\pi$
$317$	14.4884	0.813752	0.406876	+	0.913483i	$0.366618\pi$
$318$	0	0
$319$	5.22668	0.292638
$320$	−3.87939	−0.216864
$321$	0	0
$322$	5.06418	0.282216
$323$	0	0
$324$	0	0
$325$	20.0993	1.11491
$326$	23.1925	1.28452
$327$	0	0
$328$	−11.2763	−0.622630
$329$	−3.51754	−0.193928
$330$	0	0
$331$	−19.8425	−1.09065	−0.545323	−	0.838226i	$-0.683593\pi$
$331$	−19.8425	−1.09065	−0.545323	+	0.838226i	$0.683593\pi$
$332$	12.8425	0.704826
$333$	0	0
$334$	−2.25671	−0.123482
$335$	22.3405	1.22059
$336$	0	0
$337$	17.5526	0.956152	0.478076	−	0.878318i	$-0.341334\pi$
$337$	17.5526	0.956152	0.478076	+	0.878318i	$0.341334\pi$
$338$	9.00000	0.489535
$339$	0	0
$340$	0	0
$341$	13.8084	0.747767
$342$	0	0
$343$	−13.0000	−0.701934
$344$	9.88713	0.533078
$345$	0	0
$346$	8.18984	0.440289
$347$	−20.3327	−1.09152	−0.545760	−	0.837942i	$-0.683759\pi$
$347$	−20.3327	−1.09152	−0.545760	+	0.837942i	$0.683759\pi$
$348$	0	0
$349$	−22.9222	−1.22700	−0.613499	−	0.789696i	$-0.710238\pi$
$349$	−22.9222	−1.22700	−0.613499	+	0.789696i	$0.710238\pi$
$350$	−10.0496	−0.537175
$351$	0	0
$352$	1.65270	0.0880894
$353$	−22.5134	−1.19827	−0.599134	−	0.800649i	$-0.704488\pi$
$353$	−22.5134	−1.19827	−0.599134	+	0.800649i	$0.704488\pi$
$354$	0	0
$355$	−18.3851	−0.975778
$356$	13.9709	0.740456
$357$	0	0
$358$	7.41416	0.391851
$359$	2.04458	0.107909	0.0539543	−	0.998543i	$-0.482817\pi$
$359$	2.04458	0.107909	0.0539543	+	0.998543i	$0.482817\pi$
$360$	0	0
$361$	30.9026	1.62645
$362$	−1.30541	−0.0686107
$363$	0	0
$364$	2.00000	0.104828
$365$	−50.6245	−2.64981
$366$	0	0
$367$	25.8658	1.35018	0.675091	−	0.737734i	$-0.264104\pi$
$367$	25.8658	1.35018	0.675091	+	0.737734i	$0.264104\pi$
$368$	−5.06418	−0.263989
$369$	0	0
$370$	−18.2121	−0.946804
$371$	10.8452	0.563057
$372$	0	0
$373$	−8.01548	−0.415026	−0.207513	−	0.978232i	$-0.566537\pi$
$373$	−8.01548	−0.415026	−0.207513	+	0.978232i	$0.566537\pi$
$374$	0	0
$375$	0	0
$376$	3.51754	0.181403
$377$	−6.32501	−0.325754
$378$	0	0
$379$	20.7648	1.06661	0.533307	−	0.845922i	$-0.320949\pi$
$379$	20.7648	1.06661	0.533307	+	0.845922i	$0.320949\pi$
$380$	−27.4047	−1.40583
$381$	0	0
$382$	6.28581	0.321610
$383$	−19.9709	−1.02047	−0.510233	−	0.860036i	$-0.670441\pi$
$383$	−19.9709	−1.02047	−0.510233	+	0.860036i	$0.670441\pi$
$384$	0	0
$385$	6.41147	0.326759
$386$	2.78611	0.141809
$387$	0	0
$388$	−3.86484	−0.196207
$389$	−6.68004	−0.338692	−0.169346	−	0.985557i	$-0.554165\pi$
$389$	−6.68004	−0.338692	−0.169346	+	0.985557i	$0.554165\pi$
$390$	0	0
$391$	0	0
$392$	6.00000	0.303046
$393$	0	0
$394$	1.02910	0.0518451
$395$	−20.6682	−1.03993
$396$	0	0
$397$	14.3013	0.717761	0.358881	−	0.933383i	$-0.383159\pi$
$397$	14.3013	0.717761	0.358881	+	0.933383i	$0.383159\pi$
$398$	25.2003	1.26318
$399$	0	0
$400$	10.0496	0.502481
$401$	−9.55674	−0.477241	−0.238620	−	0.971113i	$-0.576695\pi$
$401$	−9.55674	−0.477241	−0.238620	+	0.971113i	$0.576695\pi$
$402$	0	0
$403$	−16.7101	−0.832388
$404$	−12.7861	−0.636133
$405$	0	0
$406$	3.16250	0.156952
$407$	7.75877	0.384588
$408$	0	0
$409$	21.0283	1.03978	0.519891	−	0.854233i	$-0.325973\pi$
$409$	21.0283	1.03978	0.519891	+	0.854233i	$0.325973\pi$
$410$	43.7452	2.16042
$411$	0	0
$412$	−6.51249	−0.320847
$413$	8.33275	0.410028
$414$	0	0
$415$	−49.8212	−2.44563
$416$	−2.00000	−0.0980581
$417$	0	0
$418$	11.6750	0.571043
$419$	4.09152	0.199884	0.0999419	−	0.994993i	$-0.468134\pi$
$419$	4.09152	0.199884	0.0999419	+	0.994993i	$0.468134\pi$
$420$	0	0
$421$	26.9614	1.31402	0.657009	−	0.753882i	$-0.271821\pi$
$421$	26.9614	1.31402	0.657009	+	0.753882i	$0.271821\pi$
$422$	−4.49794	−0.218956
$423$	0	0
$424$	−10.8452	−0.526691
$425$	0	0
$426$	0	0
$427$	12.9067	0.624600
$428$	−11.2071	−0.541715
$429$	0	0
$430$	−38.3560	−1.84969
$431$	30.4243	1.46549	0.732743	−	0.680506i	$-0.238240\pi$
$431$	30.4243	1.46549	0.732743	+	0.680506i	$0.238240\pi$
$432$	0	0
$433$	−4.24897	−0.204192	−0.102096	−	0.994775i	$-0.532555\pi$
$433$	−4.24897	−0.204192	−0.102096	+	0.994775i	$0.532555\pi$
$434$	8.35504	0.401055
$435$	0	0
$436$	−4.36959	−0.209265
$437$	−35.7743	−1.71131
$438$	0	0
$439$	−10.8675	−0.518679	−0.259339	−	0.965786i	$-0.583505\pi$
$439$	−10.8675	−0.518679	−0.259339	+	0.965786i	$0.583505\pi$
$440$	−6.41147	−0.305655
$441$	0	0
$442$	0	0
$443$	31.8384	1.51269	0.756345	−	0.654173i	$-0.226983\pi$
$443$	31.8384	1.51269	0.756345	+	0.654173i	$0.226983\pi$
$444$	0	0
$445$	−54.1985	−2.56926
$446$	25.2249	1.19443
$447$	0	0
$448$	1.00000	0.0472456
$449$	6.25133	0.295019	0.147509	−	0.989061i	$-0.452874\pi$
$449$	6.25133	0.295019	0.147509	+	0.989061i	$0.452874\pi$
$450$	0	0
$451$	−18.6364	−0.877554
$452$	−4.85204	−0.228221
$453$	0	0
$454$	25.8384	1.21266
$455$	−7.75877	−0.363737
$456$	0	0
$457$	−35.3164	−1.65203	−0.826017	−	0.563645i	$-0.809398\pi$
$457$	−35.3164	−1.65203	−0.826017	+	0.563645i	$0.809398\pi$
$458$	6.65539	0.310986
$459$	0	0
$460$	19.6459	0.915995
$461$	2.22668	0.103707	0.0518535	−	0.998655i	$-0.483487\pi$
$461$	2.22668	0.103707	0.0518535	+	0.998655i	$0.483487\pi$
$462$	0	0
$463$	6.09152	0.283097	0.141548	−	0.989931i	$-0.454792\pi$
$463$	6.09152	0.283097	0.141548	+	0.989931i	$0.454792\pi$
$464$	−3.16250	−0.146816
$465$	0	0
$466$	20.9067	0.968485
$467$	36.5672	1.69213	0.846063	−	0.533082i	$-0.178966\pi$
$467$	36.5672	1.69213	0.846063	+	0.533082i	$0.178966\pi$
$468$	0	0
$469$	−5.75877	−0.265915
$470$	−13.6459	−0.629438
$471$	0	0
$472$	−8.33275	−0.383546
$473$	16.3405	0.751336
$474$	0	0
$475$	70.9924	3.25735
$476$	0	0
$477$	0	0
$478$	16.9905	0.777128
$479$	−37.2472	−1.70187	−0.850934	−	0.525272i	$-0.823964\pi$
$479$	−37.2472	−1.70187	−0.850934	+	0.525272i	$0.823964\pi$
$480$	0	0
$481$	−9.38919	−0.428110
$482$	6.37733	0.290479
$483$	0	0
$484$	−8.26857	−0.375844
$485$	14.9932	0.680806
$486$	0	0
$487$	−16.4456	−0.745222	−0.372611	−	0.927988i	$-0.621537\pi$
$487$	−16.4456	−0.745222	−0.372611	+	0.927988i	$0.621537\pi$
$488$	−12.9067	−0.584260
$489$	0	0
$490$	−23.2763	−1.05152
$491$	30.4611	1.37469	0.687345	−	0.726331i	$-0.258776\pi$
$491$	30.4611	1.37469	0.687345	+	0.726331i	$0.258776\pi$
$492$	0	0
$493$	0	0
$494$	−14.1284	−0.635665
$495$	0	0
$496$	−8.35504	−0.375152
$497$	4.73917	0.212581
$498$	0	0
$499$	3.16344	0.141615	0.0708075	−	0.997490i	$-0.477442\pi$
$499$	3.16344	0.141615	0.0708075	+	0.997490i	$0.477442\pi$
$500$	−19.5895	−0.876067
$501$	0	0
$502$	−27.6604	−1.23455
$503$	−22.6263	−1.00886	−0.504428	−	0.863454i	$-0.668297\pi$
$503$	−22.6263	−1.00886	−0.504428	+	0.863454i	$0.668297\pi$
$504$	0	0
$505$	49.6023	2.20727
$506$	−8.36959	−0.372073
$507$	0	0
$508$	4.38919	0.194739
$509$	−27.2226	−1.20662	−0.603309	−	0.797507i	$-0.706152\pi$
$509$	−27.2226	−1.20662	−0.603309	+	0.797507i	$0.706152\pi$
$510$	0	0
$511$	13.0496	0.577282
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	5.71419	0.252042
$515$	25.2645	1.11328
$516$	0	0
$517$	5.81345	0.255675
$518$	4.69459	0.206269
$519$	0	0
$520$	7.75877	0.340245
$521$	1.83244	0.0802808	0.0401404	−	0.999194i	$-0.487219\pi$
$521$	1.83244	0.0802808	0.0401404	+	0.999194i	$0.487219\pi$
$522$	0	0
$523$	−9.84793	−0.430620	−0.215310	−	0.976546i	$-0.569076\pi$
$523$	−9.84793	−0.430620	−0.215310	+	0.976546i	$0.569076\pi$
$524$	9.71007	0.424187
$525$	0	0
$526$	−6.71007	−0.292573
$527$	0	0
$528$	0	0
$529$	2.64590	0.115039
$530$	42.0729	1.82753
$531$	0	0
$532$	7.06418	0.306271
$533$	22.5526	0.976863
$534$	0	0
$535$	43.4766	1.87966
$536$	5.75877	0.248741
$537$	0	0
$538$	15.0128	0.647247
$539$	9.91622	0.427122
$540$	0	0
$541$	−7.17705	−0.308566	−0.154283	−	0.988027i	$-0.549307\pi$
$541$	−7.17705	−0.308566	−0.154283	+	0.988027i	$0.549307\pi$
$542$	−2.96585	−0.127394
$543$	0	0
$544$	0	0
$545$	16.9513	0.726114
$546$	0	0
$547$	−42.1884	−1.80385	−0.901923	−	0.431897i	$-0.857845\pi$
$547$	−42.1884	−1.80385	−0.901923	+	0.431897i	$0.857845\pi$
$548$	2.98040	0.127316
$549$	0	0
$550$	16.6091	0.708213
$551$	−22.3405	−0.951737
$552$	0	0
$553$	5.32770	0.226557
$554$	2.93582	0.124731
$555$	0	0
$556$	11.6459	0.493896
$557$	13.4142	0.568376	0.284188	−	0.958769i	$-0.408276\pi$
$557$	13.4142	0.568376	0.284188	+	0.958769i	$0.408276\pi$
$558$	0	0
$559$	−19.7743	−0.836362
$560$	−3.87939	−0.163934
$561$	0	0
$562$	28.9513	1.22124
$563$	−22.0455	−0.929108	−0.464554	−	0.885545i	$-0.653785\pi$
$563$	−22.0455	−0.929108	−0.464554	+	0.885545i	$0.653785\pi$
$564$	0	0
$565$	18.8229	0.791887
$566$	10.4825	0.440611
$567$	0	0
$568$	−4.73917	−0.198851
$569$	14.7939	0.620191	0.310095	−	0.950705i	$-0.399639\pi$
$569$	14.7939	0.620191	0.310095	+	0.950705i	$0.399639\pi$
$570$	0	0
$571$	−2.32501	−0.0972985	−0.0486493	−	0.998816i	$-0.515492\pi$
$571$	−2.32501	−0.0972985	−0.0486493	+	0.998816i	$0.515492\pi$
$572$	−3.30541	−0.138206
$573$	0	0
$574$	−11.2763	−0.470664
$575$	−50.8931	−2.12239
$576$	0	0
$577$	−20.6067	−0.857868	−0.428934	−	0.903336i	$-0.641111\pi$
$577$	−20.6067	−0.857868	−0.428934	+	0.903336i	$0.641111\pi$
$578$	0	0
$579$	0	0
$580$	12.2686	0.509425
$581$	12.8425	0.532799
$582$	0	0
$583$	−17.9240	−0.742335
$584$	−13.0496	−0.539998
$585$	0	0
$586$	−8.54488	−0.352986
$587$	21.5544	0.889644	0.444822	−	0.895619i	$-0.353267\pi$
$587$	21.5544	0.889644	0.444822	+	0.895619i	$0.353267\pi$
$588$	0	0
$589$	−59.0215	−2.43194
$590$	32.3259	1.33084
$591$	0	0
$592$	−4.69459	−0.192947
$593$	−16.9649	−0.696666	−0.348333	−	0.937371i	$-0.613252\pi$
$593$	−16.9649	−0.696666	−0.348333	+	0.937371i	$0.613252\pi$
$594$	0	0
$595$	0	0
$596$	−17.8803	−0.732406
$597$	0	0
$598$	10.1284	0.414179
$599$	−13.6851	−0.559158	−0.279579	−	0.960123i	$-0.590195\pi$
$599$	−13.6851	−0.559158	−0.279579	+	0.960123i	$0.590195\pi$
$600$	0	0
$601$	3.77870	0.154136	0.0770681	−	0.997026i	$-0.475444\pi$
$601$	3.77870	0.154136	0.0770681	+	0.997026i	$0.475444\pi$
$602$	9.88713	0.402969
$603$	0	0
$604$	−5.97771	−0.243230
$605$	32.0770	1.30411
$606$	0	0
$607$	−22.2267	−0.902153	−0.451077	−	0.892485i	$-0.648960\pi$
$607$	−22.2267	−0.902153	−0.451077	+	0.892485i	$0.648960\pi$
$608$	−7.06418	−0.286490
$609$	0	0
$610$	50.0702	2.02728
$611$	−7.03508	−0.284609
$612$	0	0
$613$	23.3655	0.943722	0.471861	−	0.881673i	$-0.343582\pi$
$613$	23.3655	0.943722	0.471861	+	0.881673i	$0.343582\pi$
$614$	−12.4534	−0.502577
$615$	0	0
$616$	1.65270	0.0665893
$617$	−20.9067	−0.841673	−0.420837	−	0.907136i	$-0.638263\pi$
$617$	−20.9067	−0.841673	−0.420837	+	0.907136i	$0.638263\pi$
$618$	0	0
$619$	2.48784	0.0999946	0.0499973	−	0.998749i	$-0.484079\pi$
$619$	2.48784	0.0999946	0.0499973	+	0.998749i	$0.484079\pi$
$620$	32.4124	1.30171
$621$	0	0
$622$	23.2371	0.931723
$623$	13.9709	0.559732
$624$	0	0
$625$	25.7469	1.02988
$626$	−21.7743	−0.870274
$627$	0	0
$628$	4.04458	0.161396
$629$	0	0
$630$	0	0
$631$	17.7733	0.707545	0.353772	−	0.935332i	$-0.384899\pi$
$631$	17.7733	0.707545	0.353772	+	0.935332i	$0.384899\pi$
$632$	−5.32770	−0.211924
$633$	0	0
$634$	−14.4884	−0.575410
$635$	−17.0273	−0.675709
$636$	0	0
$637$	−12.0000	−0.475457
$638$	−5.22668	−0.206926
$639$	0	0
$640$	3.87939	0.153346
$641$	−6.21213	−0.245365	−0.122682	−	0.992446i	$-0.539150\pi$
$641$	−6.21213	−0.245365	−0.122682	+	0.992446i	$0.539150\pi$
$642$	0	0
$643$	−23.1533	−0.913078	−0.456539	−	0.889703i	$-0.650911\pi$
$643$	−23.1533	−0.913078	−0.456539	+	0.889703i	$0.650911\pi$
$644$	−5.06418	−0.199557
$645$	0	0
$646$	0	0
$647$	−38.9276	−1.53040	−0.765201	−	0.643792i	$-0.777360\pi$
$647$	−38.9276	−1.53040	−0.765201	+	0.643792i	$0.777360\pi$
$648$	0	0
$649$	−13.7716	−0.540581
$650$	−20.0993	−0.788358
$651$	0	0
$652$	−23.1925	−0.908290
$653$	−5.42839	−0.212429	−0.106215	−	0.994343i	$-0.533873\pi$
$653$	−5.42839	−0.212429	−0.106215	+	0.994343i	$0.533873\pi$
$654$	0	0
$655$	−37.6691	−1.47185
$656$	11.2763	0.440266
$657$	0	0
$658$	3.51754	0.137128
$659$	37.8512	1.47447	0.737237	−	0.675634i	$-0.236130\pi$
$659$	37.8512	1.47447	0.737237	+	0.675634i	$0.236130\pi$
$660$	0	0
$661$	−28.8485	−1.12208	−0.561039	−	0.827789i	$-0.689598\pi$
$661$	−28.8485	−1.12208	−0.561039	+	0.827789i	$0.689598\pi$
$662$	19.8425	0.771203
$663$	0	0
$664$	−12.8425	−0.498388
$665$	−27.4047	−1.06271
$666$	0	0
$667$	16.0155	0.620122
$668$	2.25671	0.0873148
$669$	0	0
$670$	−22.3405	−0.863088
$671$	−21.3310	−0.823474
$672$	0	0
$673$	33.7743	1.30190	0.650951	−	0.759120i	$-0.274370\pi$
$673$	33.7743	1.30190	0.650951	+	0.759120i	$0.274370\pi$
$674$	−17.5526	−0.676102
$675$	0	0
$676$	−9.00000	−0.346154
$677$	2.10338	0.0808394	0.0404197	−	0.999183i	$-0.487131\pi$
$677$	2.10338	0.0808394	0.0404197	+	0.999183i	$0.487131\pi$
$678$	0	0
$679$	−3.86484	−0.148319
$680$	0	0
$681$	0	0
$682$	−13.8084	−0.528751
$683$	25.4962	0.975584	0.487792	−	0.872960i	$-0.337802\pi$
$683$	25.4962	0.975584	0.487792	+	0.872960i	$0.337802\pi$
$684$	0	0
$685$	−11.5621	−0.441766
$686$	13.0000	0.496342
$687$	0	0
$688$	−9.88713	−0.376943
$689$	21.6905	0.826341
$690$	0	0
$691$	−7.67499	−0.291970	−0.145985	−	0.989287i	$-0.546635\pi$
$691$	−7.67499	−0.291970	−0.145985	+	0.989287i	$0.546635\pi$
$692$	−8.18984	−0.311331
$693$	0	0
$694$	20.3327	0.771821
$695$	−45.1789	−1.71373
$696$	0	0
$697$	0	0
$698$	22.9222	0.867618
$699$	0	0
$700$	10.0496	0.379840
$701$	−19.9641	−0.754034	−0.377017	−	0.926206i	$-0.623050\pi$
$701$	−19.9641	−0.754034	−0.377017	+	0.926206i	$0.623050\pi$
$702$	0	0
$703$	−33.1634	−1.25078
$704$	−1.65270	−0.0622886
$705$	0	0
$706$	22.5134	0.847304
$707$	−12.7861	−0.480871
$708$	0	0
$709$	20.3797	0.765375	0.382688	−	0.923878i	$-0.374999\pi$
$709$	20.3797	0.765375	0.382688	+	0.923878i	$0.374999\pi$
$710$	18.3851	0.689979
$711$	0	0
$712$	−13.9709	−0.523582
$713$	42.3114	1.58457
$714$	0	0
$715$	12.8229	0.479551
$716$	−7.41416	−0.277080
$717$	0	0
$718$	−2.04458	−0.0763030
$719$	11.5757	0.431702	0.215851	−	0.976426i	$-0.430747\pi$
$719$	11.5757	0.431702	0.215851	+	0.976426i	$0.430747\pi$
$720$	0	0
$721$	−6.51249	−0.242538
$722$	−30.9026	−1.15008
$723$	0	0
$724$	1.30541	0.0485151
$725$	−31.7820	−1.18035
$726$	0	0
$727$	−7.05644	−0.261709	−0.130854	−	0.991402i	$-0.541772\pi$
$727$	−7.05644	−0.261709	−0.130854	+	0.991402i	$0.541772\pi$
$728$	−2.00000	−0.0741249
$729$	0	0
$730$	50.6245	1.87370
$731$	0	0
$732$	0	0
$733$	19.3364	0.714205	0.357103	−	0.934065i	$-0.383765\pi$
$733$	19.3364	0.714205	0.357103	+	0.934065i	$0.383765\pi$
$734$	−25.8658	−0.954723
$735$	0	0
$736$	5.06418	0.186668
$737$	9.51754	0.350583
$738$	0	0
$739$	−16.6108	−0.611039	−0.305519	−	0.952186i	$-0.598830\pi$
$739$	−16.6108	−0.611039	−0.305519	+	0.952186i	$0.598830\pi$
$740$	18.2121	0.669491
$741$	0	0
$742$	−10.8452	−0.398141
$743$	5.92633	0.217416	0.108708	−	0.994074i	$-0.465329\pi$
$743$	5.92633	0.217416	0.108708	+	0.994074i	$0.465329\pi$
$744$	0	0
$745$	69.3646	2.54132
$746$	8.01548	0.293468
$747$	0	0
$748$	0	0
$749$	−11.2071	−0.409498
$750$	0	0
$751$	13.8658	0.505969	0.252985	−	0.967470i	$-0.418588\pi$
$751$	13.8658	0.505969	0.252985	+	0.967470i	$0.418588\pi$
$752$	−3.51754	−0.128272
$753$	0	0
$754$	6.32501	0.230343
$755$	23.1898	0.843965
$756$	0	0
$757$	49.0898	1.78420	0.892099	−	0.451840i	$-0.149232\pi$
$757$	49.0898	1.78420	0.892099	+	0.451840i	$0.149232\pi$
$758$	−20.7648	−0.754210
$759$	0	0
$760$	27.4047	0.994072
$761$	−13.9845	−0.506938	−0.253469	−	0.967343i	$-0.581572\pi$
$761$	−13.9845	−0.506938	−0.253469	+	0.967343i	$0.581572\pi$
$762$	0	0
$763$	−4.36959	−0.158190
$764$	−6.28581	−0.227413
$765$	0	0
$766$	19.9709	0.721578
$767$	16.6655	0.601756
$768$	0	0
$769$	8.74691	0.315422	0.157711	−	0.987485i	$-0.449589\pi$
$769$	8.74691	0.315422	0.157711	+	0.987485i	$0.449589\pi$
$770$	−6.41147	−0.231053
$771$	0	0
$772$	−2.78611	−0.100274
$773$	31.7520	1.14204	0.571019	−	0.820937i	$-0.306548\pi$
$773$	31.7520	1.14204	0.571019	+	0.820937i	$0.306548\pi$
$774$	0	0
$775$	−83.9650	−3.01611
$776$	3.86484	0.138740
$777$	0	0
$778$	6.68004	0.239491
$779$	79.6579	2.85404
$780$	0	0
$781$	−7.83244	−0.280267
$782$	0	0
$783$	0	0
$784$	−6.00000	−0.214286
$785$	−15.6905	−0.560017
$786$	0	0
$787$	−0.393304	−0.0140198	−0.00700989	−	0.999975i	$-0.502231\pi$
$787$	−0.393304	−0.0140198	−0.00700989	+	0.999975i	$0.502231\pi$
$788$	−1.02910	−0.0366600
$789$	0	0
$790$	20.6682	0.735341
$791$	−4.85204	−0.172519
$792$	0	0
$793$	25.8135	0.916663
$794$	−14.3013	−0.507534
$795$	0	0
$796$	−25.2003	−0.893200
$797$	−46.5586	−1.64919	−0.824595	−	0.565723i	$-0.808597\pi$
$797$	−46.5586	−1.64919	−0.824595	+	0.565723i	$0.808597\pi$
$798$	0	0
$799$	0	0
$800$	−10.0496	−0.355308
$801$	0	0
$802$	9.55674	0.337460
$803$	−21.5672	−0.761089
$804$	0	0
$805$	19.6459	0.692427
$806$	16.7101	0.588587
$807$	0	0
$808$	12.7861	0.449814
$809$	46.3797	1.63062	0.815312	−	0.579023i	$-0.196566\pi$
$809$	46.3797	1.63062	0.815312	+	0.579023i	$0.196566\pi$
$810$	0	0
$811$	−45.4201	−1.59492	−0.797459	−	0.603374i	$-0.793823\pi$
$811$	−45.4201	−1.59492	−0.797459	+	0.603374i	$0.793823\pi$
$812$	−3.16250	−0.110982
$813$	0	0
$814$	−7.75877	−0.271945
$815$	89.9728	3.15161
$816$	0	0
$817$	−69.8444	−2.44355
$818$	−21.0283	−0.735236
$819$	0	0
$820$	−43.7452	−1.52765
$821$	−8.29591	−0.289529	−0.144765	−	0.989466i	$-0.546243\pi$
$821$	−8.29591	−0.289529	−0.144765	+	0.989466i	$0.546243\pi$
$822$	0	0
$823$	22.7033	0.791386	0.395693	−	0.918383i	$-0.370504\pi$
$823$	22.7033	0.791386	0.395693	+	0.918383i	$0.370504\pi$
$824$	6.51249	0.226873
$825$	0	0
$826$	−8.33275	−0.289933
$827$	−40.9401	−1.42363	−0.711813	−	0.702369i	$-0.752125\pi$
$827$	−40.9401	−1.42363	−0.711813	+	0.702369i	$0.752125\pi$
$828$	0	0
$829$	−6.18304	−0.214746	−0.107373	−	0.994219i	$-0.534244\pi$
$829$	−6.18304	−0.214746	−0.107373	+	0.994219i	$0.534244\pi$
$830$	49.8212	1.72932
$831$	0	0
$832$	2.00000	0.0693375
$833$	0	0
$834$	0	0
$835$	−8.75465	−0.302967
$836$	−11.6750	−0.403788
$837$	0	0
$838$	−4.09152	−0.141339
$839$	41.2972	1.42574	0.712868	−	0.701298i	$-0.247396\pi$
$839$	41.2972	1.42574	0.712868	+	0.701298i	$0.247396\pi$
$840$	0	0
$841$	−18.9986	−0.655123
$842$	−26.9614	−0.929152
$843$	0	0
$844$	4.49794	0.154826
$845$	34.9145	1.20109
$846$	0	0
$847$	−8.26857	−0.284111
$848$	10.8452	0.372427
$849$	0	0
$850$	0	0
$851$	23.7743	0.814971
$852$	0	0
$853$	11.9845	0.410342	0.205171	−	0.978726i	$-0.434225\pi$
$853$	11.9845	0.410342	0.205171	+	0.978726i	$0.434225\pi$
$854$	−12.9067	−0.441659
$855$	0	0
$856$	11.2071	0.383050
$857$	3.26621	0.111571	0.0557857	−	0.998443i	$-0.482234\pi$
$857$	3.26621	0.111571	0.0557857	+	0.998443i	$0.482234\pi$
$858$	0	0
$859$	−22.7885	−0.777533	−0.388766	−	0.921336i	$-0.627099\pi$
$859$	−22.7885	−0.777533	−0.388766	+	0.921336i	$0.627099\pi$
$860$	38.3560	1.30793
$861$	0	0
$862$	−30.4243	−1.03625
$863$	−13.6595	−0.464975	−0.232488	−	0.972599i	$-0.574687\pi$
$863$	−13.6595	−0.464975	−0.232488	+	0.972599i	$0.574687\pi$
$864$	0	0
$865$	31.7716	1.08027
$866$	4.24897	0.144386
$867$	0	0
$868$	−8.35504	−0.283588
$869$	−8.80510	−0.298693
$870$	0	0
$871$	−11.5175	−0.390257
$872$	4.36959	0.147973
$873$	0	0
$874$	35.7743	1.21008
$875$	−19.5895	−0.662245
$876$	0	0
$877$	−8.57161	−0.289443	−0.144721	−	0.989472i	$-0.546229\pi$
$877$	−8.57161	−0.289443	−0.144721	+	0.989472i	$0.546229\pi$
$878$	10.8675	0.366761
$879$	0	0
$880$	6.41147	0.216131
$881$	−26.8539	−0.904731	−0.452366	−	0.891833i	$-0.649420\pi$
$881$	−26.8539	−0.904731	−0.452366	+	0.891833i	$0.649420\pi$
$882$	0	0
$883$	−18.3351	−0.617026	−0.308513	−	0.951220i	$-0.599831\pi$
$883$	−18.3351	−0.617026	−0.308513	+	0.951220i	$0.599831\pi$
$884$	0	0
$885$	0	0
$886$	−31.8384	−1.06963
$887$	36.3506	1.22053	0.610267	−	0.792196i	$-0.291062\pi$
$887$	36.3506	1.22053	0.610267	+	0.792196i	$0.291062\pi$
$888$	0	0
$889$	4.38919	0.147209
$890$	54.1985	1.81674
$891$	0	0
$892$	−25.2249	−0.844593
$893$	−24.8485	−0.831525
$894$	0	0
$895$	28.7624	0.961421
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−6.25133	−0.208610
$899$	26.4228	0.881251
$900$	0	0
$901$	0	0
$902$	18.6364	0.620524
$903$	0	0
$904$	4.85204	0.161377
$905$	−5.06418	−0.168339
$906$	0	0
$907$	−40.0310	−1.32921	−0.664603	−	0.747197i	$-0.731399\pi$
$907$	−40.0310	−1.32921	−0.664603	+	0.747197i	$0.731399\pi$
$908$	−25.8384	−0.857478
$909$	0	0
$910$	7.75877	0.257201
$911$	34.0155	1.12698	0.563492	−	0.826122i	$-0.309458\pi$
$911$	34.0155	1.12698	0.563492	+	0.826122i	$0.309458\pi$
$912$	0	0
$913$	−21.2249	−0.702443
$914$	35.3164	1.16816
$915$	0	0
$916$	−6.65539	−0.219900
$917$	9.71007	0.320655
$918$	0	0
$919$	−8.83244	−0.291355	−0.145678	−	0.989332i	$-0.546536\pi$
$919$	−8.83244	−0.291355	−0.145678	+	0.989332i	$0.546536\pi$
$920$	−19.6459	−0.647706
$921$	0	0
$922$	−2.22668	−0.0733319
$923$	9.47834	0.311983
$924$	0	0
$925$	−47.1789	−1.55123
$926$	−6.09152	−0.200180
$927$	0	0
$928$	3.16250	0.103814
$929$	−50.8795	−1.66930	−0.834651	−	0.550779i	$-0.814331\pi$
$929$	−50.8795	−1.66930	−0.834651	+	0.550779i	$0.814331\pi$
$930$	0	0
$931$	−42.3851	−1.38911
$932$	−20.9067	−0.684823
$933$	0	0
$934$	−36.5672	−1.19651
$935$	0	0
$936$	0	0
$937$	−53.3969	−1.74440	−0.872201	−	0.489148i	$-0.837308\pi$
$937$	−53.3969	−1.74440	−0.872201	+	0.489148i	$0.837308\pi$
$938$	5.75877	0.188031
$939$	0	0
$940$	13.6459	0.445080
$941$	−10.9649	−0.357446	−0.178723	−	0.983899i	$-0.557197\pi$
$941$	−10.9649	−0.357446	−0.178723	+	0.983899i	$0.557197\pi$
$942$	0	0
$943$	−57.1052	−1.85960
$944$	8.33275	0.271208
$945$	0	0
$946$	−16.3405	−0.531275
$947$	−11.5553	−0.375497	−0.187749	−	0.982217i	$-0.560119\pi$
$947$	−11.5553	−0.375497	−0.187749	+	0.982217i	$0.560119\pi$
$948$	0	0
$949$	26.0993	0.847218
$950$	−70.9924	−2.30330
$951$	0	0
$952$	0	0
$953$	−41.8343	−1.35515	−0.677573	−	0.735455i	$-0.736968\pi$
$953$	−41.8343	−1.35515	−0.677573	+	0.735455i	$0.736968\pi$
$954$	0	0
$955$	24.3851	0.789082
$956$	−16.9905	−0.549512
$957$	0	0
$958$	37.2472	1.20340
$959$	2.98040	0.0962421
$960$	0	0
$961$	38.8066	1.25183
$962$	9.38919	0.302719
$963$	0	0
$964$	−6.37733	−0.205400
$965$	10.8084	0.347935
$966$	0	0
$967$	30.2900	0.974062	0.487031	−	0.873385i	$-0.338080\pi$
$967$	30.2900	0.974062	0.487031	+	0.873385i	$0.338080\pi$
$968$	8.26857	0.265762
$969$	0	0
$970$	−14.9932	−0.481402
$971$	−40.9309	−1.31353	−0.656767	−	0.754093i	$-0.728077\pi$
$971$	−40.9309	−1.31353	−0.656767	+	0.754093i	$0.728077\pi$
$972$	0	0
$973$	11.6459	0.373350
$974$	16.4456	0.526952
$975$	0	0
$976$	12.9067	0.413134
$977$	−23.5621	−0.753819	−0.376909	−	0.926250i	$-0.623013\pi$
$977$	−23.5621	−0.753819	−0.376909	+	0.926250i	$0.623013\pi$
$978$	0	0
$979$	−23.0898	−0.737952
$980$	23.2763	0.743534
$981$	0	0
$982$	−30.4611	−0.972053
$983$	−48.2586	−1.53921	−0.769605	−	0.638521i	$-0.779547\pi$
$983$	−48.2586	−1.53921	−0.769605	+	0.638521i	$0.779547\pi$
$984$	0	0
$985$	3.99226	0.127204
$986$	0	0
$987$	0	0
$988$	14.1284	0.449483
$989$	50.0702	1.59214
$990$	0	0
$991$	33.7202	1.07116	0.535578	−	0.844486i	$-0.320094\pi$
$991$	33.7202	1.07116	0.535578	+	0.844486i	$0.320094\pi$
$992$	8.35504	0.265273
$993$	0	0
$994$	−4.73917	−0.150317
$995$	97.7616	3.09925
$996$	0	0
$997$	19.9554	0.631995	0.315997	−	0.948760i	$-0.397661\pi$
$997$	19.9554	0.631995	0.315997	+	0.948760i	$0.397661\pi$
$998$	−3.16344	−0.100137
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5202.2.a.bh.1.1	✓	3
3.2	odd	2		5202.2.a.bq.1.3	yes	3
17.16	even	2		5202.2.a.bj.1.3	yes	3
51.50	odd	2		5202.2.a.bl.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5202.2.a.bh.1.1	✓	3	1.1	even	1	trivial
5202.2.a.bj.1.3	yes	3	17.16	even	2
5202.2.a.bl.1.1	yes	3	51.50	odd	2
5202.2.a.bq.1.3	yes	3	3.2	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:	\( N \)	\(=\)	\( 5202 = 2 \cdot 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5202.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.87939 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.87939 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.87939 q^{10} -1.65270 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.06418 q^{19} -3.87939 q^{20} +1.65270 q^{22} -5.06418 q^{23} +10.0496 q^{25} -2.00000 q^{26} +1.00000 q^{28} -3.16250 q^{29} -8.35504 q^{31} -1.00000 q^{32} -3.87939 q^{35} -4.69459 q^{37} -7.06418 q^{38} +3.87939 q^{40} +11.2763 q^{41} -9.88713 q^{43} -1.65270 q^{44} +5.06418 q^{46} -3.51754 q^{47} -6.00000 q^{49} -10.0496 q^{50} +2.00000 q^{52} +10.8452 q^{53} +6.41147 q^{55} -1.00000 q^{56} +3.16250 q^{58} +8.33275 q^{59} +12.9067 q^{61} +8.35504 q^{62} +1.00000 q^{64} -7.75877 q^{65} -5.75877 q^{67} +3.87939 q^{70} +4.73917 q^{71} +13.0496 q^{73} +4.69459 q^{74} +7.06418 q^{76} -1.65270 q^{77} +5.32770 q^{79} -3.87939 q^{80} -11.2763 q^{82} +12.8425 q^{83} +9.88713 q^{86} +1.65270 q^{88} +13.9709 q^{89} +2.00000 q^{91} -5.06418 q^{92} +3.51754 q^{94} -27.4047 q^{95} -3.86484 q^{97} +6.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} - 6 q^{5} + 3 q^{7} - 3 q^{8} + 6 q^{10} - 6 q^{11} + 6 q^{13} - 3 q^{14} + 3 q^{16} + 12 q^{19} - 6 q^{20} + 6 q^{22} - 6 q^{23} + 3 q^{25} - 6 q^{26} + 3 q^{28} - 12 q^{29} - 3 q^{32} - 6 q^{35} - 12 q^{37} - 12 q^{38} + 6 q^{40} - 6 q^{44} + 6 q^{46} + 12 q^{47} - 18 q^{49} - 3 q^{50} + 6 q^{52} + 6 q^{53} + 9 q^{55} - 3 q^{56} + 12 q^{58} + 6 q^{59} + 12 q^{61} + 3 q^{64} - 12 q^{65} - 6 q^{67} + 6 q^{70} + 12 q^{73} + 12 q^{74} + 12 q^{76} - 6 q^{77} + 12 q^{79} - 6 q^{80} + 21 q^{83} + 6 q^{88} + 6 q^{89} + 6 q^{91} - 6 q^{92} - 12 q^{94} - 30 q^{95} + 12 q^{97} + 18 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 6 * q^5 + 3 * q^7 - 3 * q^8 + 6 * q^10 - 6 * q^11 + 6 * q^13 - 3 * q^14 + 3 * q^16 + 12 * q^19 - 6 * q^20 + 6 * q^22 - 6 * q^23 + 3 * q^25 - 6 * q^26 + 3 * q^28 - 12 * q^29 - 3 * q^32 - 6 * q^35 - 12 * q^37 - 12 * q^38 + 6 * q^40 - 6 * q^44 + 6 * q^46 + 12 * q^47 - 18 * q^49 - 3 * q^50 + 6 * q^52 + 6 * q^53 + 9 * q^55 - 3 * q^56 + 12 * q^58 + 6 * q^59 + 12 * q^61 + 3 * q^64 - 12 * q^65 - 6 * q^67 + 6 * q^70 + 12 * q^73 + 12 * q^74 + 12 * q^76 - 6 * q^77 + 12 * q^79 - 6 * q^80 + 21 * q^83 + 6 * q^88 + 6 * q^89 + 6 * q^91 - 6 * q^92 - 12 * q^94 - 30 * q^95 + 12 * q^97 + 18 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more