LMFDB - Embedded newform 1225.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1225,2,Mod(1,1225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1225, 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("1225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1225 = 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1225.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:	$9.78167424761$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.28825$ of defining polynomial
Character	$\chi$	$=$	1225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803 q^{2} +2.28825 q^{3} +4.85410 q^{4} -5.99070 q^{6} -7.47214 q^{8} +2.23607 q^{9} +O(q^{10})$	\(q-2.61803 q^{2} +2.28825 q^{3} +4.85410 q^{4} -5.99070 q^{6} -7.47214 q^{8} +2.23607 q^{9} -2.23607 q^{11} +11.1074 q^{12} -5.45052 q^{13} +9.85410 q^{16} -1.08036 q^{17} -5.85410 q^{18} +4.24264 q^{19} +5.85410 q^{22} -8.23607 q^{23} -17.0981 q^{24} +14.2697 q^{26} -1.74806 q^{27} -2.23607 q^{29} -2.62210 q^{31} -10.8541 q^{32} -5.11667 q^{33} +2.82843 q^{34} +10.8541 q^{36} +2.70820 q^{37} -11.1074 q^{38} -12.4721 q^{39} -7.94510 q^{41} -5.00000 q^{43} -10.8541 q^{44} +21.5623 q^{46} -9.48683 q^{47} +22.5486 q^{48} -2.47214 q^{51} -26.4574 q^{52} +3.70820 q^{53} +4.57649 q^{54} +9.70820 q^{57} +5.85410 q^{58} +8.94665 q^{59} -12.5216 q^{61} +6.86474 q^{62} +8.70820 q^{64} +13.3956 q^{66} +4.70820 q^{67} -5.24419 q^{68} -18.8461 q^{69} +1.47214 q^{71} -16.7082 q^{72} +6.86474 q^{73} -7.09017 q^{74} +20.5942 q^{76} +32.6525 q^{78} +8.70820 q^{79} -10.7082 q^{81} +20.8005 q^{82} +2.62210 q^{83} +13.0902 q^{86} -5.11667 q^{87} +16.7082 q^{88} +4.78282 q^{89} -39.9787 q^{92} -6.00000 q^{93} +24.8369 q^{94} -24.8369 q^{96} +19.5927 q^{97} -5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{2} + 6 q^{4} - 12 q^{8}+O(q^{10}) $ 4 * q - 6 * q^2 + 6 * q^4 - 12 * q^8	$ 4 q - 6 q^{2} + 6 q^{4} - 12 q^{8} + 26 q^{16} - 10 q^{18} + 10 q^{22} - 24 q^{23} - 30 q^{32} + 30 q^{36} - 16 q^{37} - 32 q^{39} - 20 q^{43} - 30 q^{44} + 46 q^{46} + 8 q^{51} - 12 q^{53} + 12 q^{57} + 10 q^{58} + 8 q^{64} - 8 q^{67} - 12 q^{71} - 40 q^{72} - 6 q^{74} + 68 q^{78} + 8 q^{79} - 16 q^{81} + 30 q^{86} + 40 q^{88} - 66 q^{92} - 24 q^{93} - 20 q^{99}+O(q^{100}) $ 4 * q - 6 * q^2 + 6 * q^4 - 12 * q^8 + 26 * q^16 - 10 * q^18 + 10 * q^22 - 24 * q^23 - 30 * q^32 + 30 * q^36 - 16 * q^37 - 32 * q^39 - 20 * q^43 - 30 * q^44 + 46 * q^46 + 8 * q^51 - 12 * q^53 + 12 * q^57 + 10 * q^58 + 8 * q^64 - 8 * q^67 - 12 * q^71 - 40 * q^72 - 6 * q^74 + 68 * q^78 + 8 * q^79 - 16 * q^81 + 30 * q^86 + 40 * q^88 - 66 * q^92 - 24 * q^93 - 20 * 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 $

$2$	−2.61803	−1.85123	−0.925615	−	0.378467i	$-0.876451\pi$
$2$	−2.61803	−1.85123	−0.925615	+	0.378467i	$0.876451\pi$
$3$	2.28825	1.32112	0.660560	−	0.750774i	$-0.270319\pi$
$3$	2.28825	1.32112	0.660560	+	0.750774i	$0.270319\pi$
$4$	4.85410	2.42705
$5$	0	0
$5$	0	0
$6$	−5.99070	−2.44569
$7$	0	0
$7$	0	0
$8$	−7.47214	−2.64180
$9$	2.23607	0.745356
$10$	0	0
$11$	−2.23607	−0.674200	−0.337100	−	0.941469i	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	11.1074	3.20642
$13$	−5.45052	−1.51170	−0.755852	−	0.654743i	$-0.772777\pi$
$13$	−5.45052	−1.51170	−0.755852	+	0.654743i	$0.772777\pi$
$14$	0	0
$15$	0	0
$16$	9.85410	2.46353
$17$	−1.08036	−0.262027	−0.131013	−	0.991381i	$-0.541823\pi$
$17$	−1.08036	−0.262027	−0.131013	+	0.991381i	$0.541823\pi$
$18$	−5.85410	−1.37983
$19$	4.24264	0.973329	0.486664	−	0.873589i	$-0.338214\pi$
$19$	4.24264	0.973329	0.486664	+	0.873589i	$0.338214\pi$
$20$	0	0
$21$	0	0
$22$	5.85410	1.24810
$23$	−8.23607	−1.71734	−0.858669	−	0.512530i	$-0.828708\pi$
$23$	−8.23607	−1.71734	−0.858669	+	0.512530i	$0.828708\pi$
$24$	−17.0981	−3.49013
$25$	0	0
$26$	14.2697	2.79851
$27$	−1.74806	−0.336415
$28$	0	0
$29$	−2.23607	−0.415227	−0.207614	−	0.978211i	$-0.566570\pi$
$29$	−2.23607	−0.415227	−0.207614	+	0.978211i	$0.566570\pi$
$30$	0	0
$31$	−2.62210	−0.470942	−0.235471	−	0.971881i	$-0.575663\pi$
$31$	−2.62210	−0.470942	−0.235471	+	0.971881i	$0.575663\pi$
$32$	−10.8541	−1.91875
$33$	−5.11667	−0.890698
$34$	2.82843	0.485071
$35$	0	0
$36$	10.8541	1.80902
$37$	2.70820	0.445226	0.222613	−	0.974907i	$-0.428541\pi$
$37$	2.70820	0.445226	0.222613	+	0.974907i	$0.428541\pi$
$38$	−11.1074	−1.80185
$39$	−12.4721	−1.99714
$40$	0	0
$41$	−7.94510	−1.24082	−0.620408	−	0.784279i	$-0.713033\pi$
$41$	−7.94510	−1.24082	−0.620408	+	0.784279i	$0.713033\pi$
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	−10.8541	−1.63632
$45$	0	0
$46$	21.5623	3.17919
$47$	−9.48683	−1.38380	−0.691898	−	0.721995i	$-0.743225\pi$
$47$	−9.48683	−1.38380	−0.691898	+	0.721995i	$0.743225\pi$
$48$	22.5486	3.25461
$49$	0	0
$50$	0	0
$51$	−2.47214	−0.346168
$52$	−26.4574	−3.66898
$53$	3.70820	0.509361	0.254680	−	0.967025i	$-0.418030\pi$
$53$	3.70820	0.509361	0.254680	+	0.967025i	$0.418030\pi$
$54$	4.57649	0.622782
$55$	0	0
$56$	0	0
$57$	9.70820	1.28588
$58$	5.85410	0.768681
$59$	8.94665	1.16475	0.582377	−	0.812919i	$-0.302123\pi$
$59$	8.94665	1.16475	0.582377	+	0.812919i	$0.302123\pi$
$60$	0	0
$61$	−12.5216	−1.60323	−0.801613	−	0.597844i	$-0.796024\pi$
$61$	−12.5216	−1.60323	−0.801613	+	0.597844i	$0.796024\pi$
$62$	6.86474	0.871822
$63$	0	0
$64$	8.70820	1.08853
$65$	0	0
$66$	13.3956	1.64889
$67$	4.70820	0.575199	0.287599	−	0.957751i	$-0.407143\pi$
$67$	4.70820	0.575199	0.287599	+	0.957751i	$0.407143\pi$
$68$	−5.24419	−0.635952
$69$	−18.8461	−2.26881
$70$	0	0
$71$	1.47214	0.174710	0.0873552	−	0.996177i	$-0.472158\pi$
$71$	1.47214	0.174710	0.0873552	+	0.996177i	$0.472158\pi$
$72$	−16.7082	−1.96908
$73$	6.86474	0.803457	0.401728	−	0.915759i	$-0.368410\pi$
$73$	6.86474	0.803457	0.401728	+	0.915759i	$0.368410\pi$
$74$	−7.09017	−0.824216
$75$	0	0
$76$	20.5942	2.36232
$77$	0	0
$78$	32.6525	3.69716
$79$	8.70820	0.979749	0.489875	−	0.871793i	$-0.337043\pi$
$79$	8.70820	0.979749	0.489875	+	0.871793i	$0.337043\pi$
$80$	0	0
$81$	−10.7082	−1.18980
$82$	20.8005	2.29704
$83$	2.62210	0.287812	0.143906	−	0.989591i	$-0.454034\pi$
$83$	2.62210	0.287812	0.143906	+	0.989591i	$0.454034\pi$
$84$	0	0
$85$	0	0
$86$	13.0902	1.41155
$87$	−5.11667	−0.548565
$88$	16.7082	1.78110
$89$	4.78282	0.506978	0.253489	−	0.967338i	$-0.418422\pi$
$89$	4.78282	0.506978	0.253489	+	0.967338i	$0.418422\pi$
$90$	0	0
$91$	0	0
$92$	−39.9787	−4.16807
$93$	−6.00000	−0.622171
$94$	24.8369	2.56173
$95$	0	0
$96$	−24.8369	−2.53490
$97$	19.5927	1.98933	0.994667	−	0.103143i	$-0.0328899\pi$
$97$	19.5927	1.98933	0.994667	+	0.103143i	$0.0328899\pi$
$98$	0	0
$99$	−5.00000	−0.502519
$100$	0	0
$101$	1.62054	0.161250	0.0806251	−	0.996744i	$-0.474308\pi$
$101$	1.62054	0.161250	0.0806251	+	0.996744i	$0.474308\pi$
$102$	6.47214	0.640837
$103$	9.48683	0.934765	0.467383	−	0.884055i	$-0.345197\pi$
$103$	9.48683	0.934765	0.467383	+	0.884055i	$0.345197\pi$
$104$	40.7271	3.99362
$105$	0	0
$106$	−9.70820	−0.942944
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−8.48528	−0.816497
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	0	0
$111$	6.19704	0.588197
$112$	0	0
$113$	−14.2361	−1.33922	−0.669608	−	0.742714i	$-0.733538\pi$
$113$	−14.2361	−1.33922	−0.669608	+	0.742714i	$0.733538\pi$
$114$	−25.4164	−2.38046
$115$	0	0
$116$	−10.8541	−1.00778
$117$	−12.1877	−1.12676
$118$	−23.4226	−2.15623
$119$	0	0
$120$	0	0
$121$	−6.00000	−0.545455
$122$	32.7820	2.96794
$123$	−18.1803	−1.63927
$124$	−12.7279	−1.14300
$125$	0	0
$126$	0	0
$127$	−7.00000	−0.621150	−0.310575	−	0.950549i	$-0.600522\pi$
$127$	−7.00000	−0.621150	−0.310575	+	0.950549i	$0.600522\pi$
$128$	−1.09017	−0.0963583
$129$	−11.4412	−1.00734
$130$	0	0
$131$	−2.16073	−0.188784	−0.0943918	−	0.995535i	$-0.530091\pi$
$131$	−2.16073	−0.188784	−0.0943918	+	0.995535i	$0.530091\pi$
$132$	−24.8369	−2.16177
$133$	0	0
$134$	−12.3262	−1.06482
$135$	0	0
$136$	8.07262	0.692221
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	49.3399	4.20009
$139$	−10.1058	−0.857165	−0.428582	−	0.903503i	$-0.640987\pi$
$139$	−10.1058	−0.857165	−0.428582	+	0.903503i	$0.640987\pi$
$140$	0	0
$141$	−21.7082	−1.82816
$142$	−3.85410	−0.323429
$143$	12.1877	1.01919
$144$	22.0344	1.83620
$145$	0	0
$146$	−17.9721	−1.48738
$147$	0	0
$148$	13.1459	1.08059
$149$	−21.6525	−1.77384	−0.886920	−	0.461923i	$-0.847160\pi$
$149$	−21.6525	−1.77384	−0.886920	+	0.461923i	$0.847160\pi$
$150$	0	0
$151$	−18.4164	−1.49871	−0.749353	−	0.662171i	$-0.769635\pi$
$151$	−18.4164	−1.49871	−0.749353	+	0.662171i	$0.769635\pi$
$152$	−31.7016	−2.57134
$153$	−2.41577	−0.195303
$154$	0	0
$155$	0	0
$156$	−60.5410	−4.84716
$157$	−4.03631	−0.322133	−0.161066	−	0.986944i	$-0.551493\pi$
$157$	−4.03631	−0.322133	−0.161066	+	0.986944i	$0.551493\pi$
$158$	−22.7984	−1.81374
$159$	8.48528	0.672927
$160$	0	0
$161$	0	0
$162$	28.0344	2.20259
$163$	15.7082	1.23036	0.615181	−	0.788386i	$-0.289083\pi$
$163$	15.7082	1.23036	0.615181	+	0.788386i	$0.289083\pi$
$164$	−38.5663	−3.01152
$165$	0	0
$166$	−6.86474	−0.532807
$167$	1.54173	0.119303	0.0596514	−	0.998219i	$-0.481001\pi$
$167$	1.54173	0.119303	0.0596514	+	0.998219i	$0.481001\pi$
$168$	0	0
$169$	16.7082	1.28525
$170$	0	0
$171$	9.48683	0.725476
$172$	−24.2705	−1.85061
$173$	−4.24264	−0.322562	−0.161281	−	0.986909i	$-0.551563\pi$
$173$	−4.24264	−0.322562	−0.161281	+	0.986909i	$0.551563\pi$
$174$	13.3956	1.01552
$175$	0	0
$176$	−22.0344	−1.66091
$177$	20.4721	1.53878
$178$	−12.5216	−0.938533
$179$	−9.70820	−0.725625	−0.362813	−	0.931862i	$-0.618183\pi$
$179$	−9.70820	−0.725625	−0.362813	+	0.931862i	$0.618183\pi$
$180$	0	0
$181$	1.62054	0.120454	0.0602271	−	0.998185i	$-0.480818\pi$
$181$	1.62054	0.120454	0.0602271	+	0.998185i	$0.480818\pi$
$182$	0	0
$183$	−28.6525	−2.11805
$184$	61.5410	4.53686
$185$	0	0
$186$	15.7082	1.15178
$187$	2.41577	0.176658
$188$	−46.0501	−3.35855
$189$	0	0
$190$	0	0
$191$	17.1246	1.23909	0.619547	−	0.784960i	$-0.287316\pi$
$191$	17.1246	1.23909	0.619547	+	0.784960i	$0.287316\pi$
$192$	19.9265	1.43807
$193$	−2.70820	−0.194941	−0.0974704	−	0.995238i	$-0.531075\pi$
$193$	−2.70820	−0.194941	−0.0974704	+	0.995238i	$0.531075\pi$
$194$	−51.2942	−3.68271
$195$	0	0
$196$	0	0
$197$	−11.9443	−0.850994	−0.425497	−	0.904960i	$-0.639901\pi$
$197$	−11.9443	−0.850994	−0.425497	+	0.904960i	$0.639901\pi$
$198$	13.0902	0.930278
$199$	15.7627	1.11739	0.558693	−	0.829374i	$-0.311303\pi$
$199$	15.7627	1.11739	0.558693	+	0.829374i	$0.311303\pi$
$200$	0	0
$201$	10.7735	0.759906
$202$	−4.24264	−0.298511
$203$	0	0
$204$	−12.0000	−0.840168
$205$	0	0
$206$	−24.8369	−1.73047
$207$	−18.4164	−1.28003
$208$	−53.7100	−3.72412
$209$	−9.48683	−0.656218
$210$	0	0
$211$	−19.4164	−1.33668	−0.668340	−	0.743856i	$-0.732995\pi$
$211$	−19.4164	−1.33668	−0.668340	+	0.743856i	$0.732995\pi$
$212$	18.0000	1.23625
$213$	3.36861	0.230813
$214$	15.7082	1.07379
$215$	0	0
$216$	13.0618	0.888741
$217$	0	0
$218$	2.61803	0.177316
$219$	15.7082	1.06146
$220$	0	0
$221$	5.88854	0.396106
$222$	−16.2241	−1.08889
$223$	15.3500	1.02791	0.513957	−	0.857816i	$-0.328179\pi$
$223$	15.3500	1.02791	0.513957	+	0.857816i	$0.328179\pi$
$224$	0	0
$225$	0	0
$226$	37.2705	2.47920
$227$	7.48373	0.496713	0.248356	−	0.968669i	$-0.420110\pi$
$227$	7.48373	0.496713	0.248356	+	0.968669i	$0.420110\pi$
$228$	47.1246	3.12090
$229$	−7.07107	−0.467269	−0.233635	−	0.972324i	$-0.575062\pi$
$229$	−7.07107	−0.467269	−0.233635	+	0.972324i	$0.575062\pi$
$230$	0	0
$231$	0	0
$232$	16.7082	1.09695
$233$	−13.4721	−0.882589	−0.441294	−	0.897362i	$-0.645481\pi$
$233$	−13.4721	−0.882589	−0.441294	+	0.897362i	$0.645481\pi$
$234$	31.9079	2.08589
$235$	0	0
$236$	43.4280	2.82692
$237$	19.9265	1.29437
$238$	0	0
$239$	−11.1246	−0.719591	−0.359796	−	0.933031i	$-0.617154\pi$
$239$	−11.1246	−0.719591	−0.359796	+	0.933031i	$0.617154\pi$
$240$	0	0
$241$	23.2163	1.49549	0.747747	−	0.663984i	$-0.231136\pi$
$241$	23.2163	1.49549	0.747747	+	0.663984i	$0.231136\pi$
$242$	15.7082	1.00976
$243$	−19.2588	−1.23545
$244$	−60.7811	−3.89111
$245$	0	0
$246$	47.5967	3.03466
$247$	−23.1246	−1.47138
$248$	19.5927	1.24414
$249$	6.00000	0.380235
$250$	0	0
$251$	20.0540	1.26580	0.632900	−	0.774234i	$-0.281865\pi$
$251$	20.0540	1.26580	0.632900	+	0.774234i	$0.281865\pi$
$252$	0	0
$253$	18.4164	1.15783
$254$	18.3262	1.14989
$255$	0	0
$256$	−14.5623	−0.910144
$257$	16.3516	1.01998	0.509991	−	0.860179i	$-0.329649\pi$
$257$	16.3516	1.01998	0.509991	+	0.860179i	$0.329649\pi$
$258$	29.9535	1.86482
$259$	0	0
$260$	0	0
$261$	−5.00000	−0.309492
$262$	5.65685	0.349482
$263$	−12.0557	−0.743388	−0.371694	−	0.928355i	$-0.621223\pi$
$263$	−12.0557	−0.743388	−0.371694	+	0.928355i	$0.621223\pi$
$264$	38.2325	2.35305
$265$	0	0
$266$	0	0
$267$	10.9443	0.669779
$268$	22.8541	1.39604
$269$	−5.86319	−0.357485	−0.178742	−	0.983896i	$-0.557203\pi$
$269$	−5.86319	−0.357485	−0.178742	+	0.983896i	$0.557203\pi$
$270$	0	0
$271$	−13.9358	−0.846540	−0.423270	−	0.906004i	$-0.639118\pi$
$271$	−13.9358	−0.846540	−0.423270	+	0.906004i	$0.639118\pi$
$272$	−10.6460	−0.645509
$273$	0	0
$274$	−15.7082	−0.948967
$275$	0	0
$276$	−91.4811	−5.50652
$277$	24.0000	1.44202	0.721010	−	0.692925i	$-0.243678\pi$
$277$	24.0000	1.44202	0.721010	+	0.692925i	$0.243678\pi$
$278$	26.4574	1.58681
$279$	−5.86319	−0.351020
$280$	0	0
$281$	24.5967	1.46732	0.733659	−	0.679517i	$-0.237811\pi$
$281$	24.5967	1.46732	0.733659	+	0.679517i	$0.237811\pi$
$282$	56.8328	3.38434
$283$	1.82688	0.108596	0.0542982	−	0.998525i	$-0.482708\pi$
$283$	1.82688	0.108596	0.0542982	+	0.998525i	$0.482708\pi$
$284$	7.14590	0.424031
$285$	0	0
$286$	−31.9079	−1.88675
$287$	0	0
$288$	−24.2705	−1.43015
$289$	−15.8328	−0.931342
$290$	0	0
$291$	44.8328	2.62815
$292$	33.3221	1.95003
$293$	22.6761	1.32475	0.662377	−	0.749171i	$-0.269548\pi$
$293$	22.6761	1.32475	0.662377	+	0.749171i	$0.269548\pi$
$294$	0	0
$295$	0	0
$296$	−20.2361	−1.17620
$297$	3.90879	0.226811
$298$	56.6869	3.28378
$299$	44.8909	2.59611
$300$	0	0
$301$	0	0
$302$	48.2148	2.77445
$303$	3.70820	0.213031
$304$	41.8074	2.39782
$305$	0	0
$306$	6.32456	0.361551
$307$	−33.5285	−1.91357	−0.956785	−	0.290795i	$-0.906080\pi$
$307$	−33.5285	−1.91357	−0.956785	+	0.290795i	$0.906080\pi$
$308$	0	0
$309$	21.7082	1.23494
$310$	0	0
$311$	−15.8902	−0.901051	−0.450525	−	0.892764i	$-0.648763\pi$
$311$	−15.8902	−0.901051	−0.450525	+	0.892764i	$0.648763\pi$
$312$	93.1935	5.27604
$313$	−21.2132	−1.19904	−0.599521	−	0.800359i	$-0.704642\pi$
$313$	−21.2132	−1.19904	−0.599521	+	0.800359i	$0.704642\pi$
$314$	10.5672	0.596341
$315$	0	0
$316$	42.2705	2.37790
$317$	−5.94427	−0.333864	−0.166932	−	0.985968i	$-0.553386\pi$
$317$	−5.94427	−0.333864	−0.166932	+	0.985968i	$0.553386\pi$
$318$	−22.2148	−1.24574
$319$	5.00000	0.279946
$320$	0	0
$321$	−13.7295	−0.766304
$322$	0	0
$323$	−4.58359	−0.255038
$324$	−51.9787	−2.88771
$325$	0	0
$326$	−41.1246	−2.27768
$327$	−2.28825	−0.126540
$328$	59.3669	3.27799
$329$	0	0
$330$	0	0
$331$	−14.7082	−0.808436	−0.404218	−	0.914663i	$-0.632456\pi$
$331$	−14.7082	−0.808436	−0.404218	+	0.914663i	$0.632456\pi$
$332$	12.7279	0.698535
$333$	6.05573	0.331852
$334$	−4.03631	−0.220857
$335$	0	0
$336$	0	0
$337$	−17.1246	−0.932837	−0.466419	−	0.884564i	$-0.654456\pi$
$337$	−17.1246	−0.932837	−0.466419	+	0.884564i	$0.654456\pi$
$338$	−43.7426	−2.37929
$339$	−32.5756	−1.76926
$340$	0	0
$341$	5.86319	0.317509
$342$	−24.8369	−1.34302
$343$	0	0
$344$	37.3607	2.01435
$345$	0	0
$346$	11.1074	0.597136
$347$	3.65248	0.196075	0.0980376	−	0.995183i	$-0.468743\pi$
$347$	3.65248	0.196075	0.0980376	+	0.995183i	$0.468743\pi$
$348$	−24.8369	−1.33139
$349$	17.9721	0.962025	0.481013	−	0.876714i	$-0.340269\pi$
$349$	17.9721	0.962025	0.481013	+	0.876714i	$0.340269\pi$
$350$	0	0
$351$	9.52786	0.508560
$352$	24.2705	1.29362
$353$	−16.3516	−0.870306	−0.435153	−	0.900356i	$-0.643306\pi$
$353$	−16.3516	−0.870306	−0.435153	+	0.900356i	$0.643306\pi$
$354$	−53.5967	−2.84864
$355$	0	0
$356$	23.2163	1.23046
$357$	0	0
$358$	25.4164	1.34330
$359$	17.9443	0.947062	0.473531	−	0.880777i	$-0.342979\pi$
$359$	17.9443	0.947062	0.473531	+	0.880777i	$0.342979\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	−4.24264	−0.222988
$363$	−13.7295	−0.720610
$364$	0	0
$365$	0	0
$366$	75.0132	3.92100
$367$	18.1784	0.948907	0.474454	−	0.880281i	$-0.342646\pi$
$367$	18.1784	0.948907	0.474454	+	0.880281i	$0.342646\pi$
$368$	−81.1591	−4.23071
$369$	−17.7658	−0.924850
$370$	0	0
$371$	0	0
$372$	−29.1246	−1.51004
$373$	−24.1246	−1.24913	−0.624563	−	0.780975i	$-0.714723\pi$
$373$	−24.1246	−1.24913	−0.624563	+	0.780975i	$0.714723\pi$
$374$	−6.32456	−0.327035
$375$	0	0
$376$	70.8869	3.65571
$377$	12.1877	0.627701
$378$	0	0
$379$	−19.0000	−0.975964	−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000	−0.975964	−0.487982	+	0.872854i	$0.662267\pi$
$380$	0	0
$381$	−16.0177	−0.820613
$382$	−44.8328	−2.29385
$383$	−8.94665	−0.457153	−0.228576	−	0.973526i	$-0.573407\pi$
$383$	−8.94665	−0.457153	−0.228576	+	0.973526i	$0.573407\pi$
$384$	−2.49458	−0.127301
$385$	0	0
$386$	7.09017	0.360880
$387$	−11.1803	−0.568329
$388$	95.1048	4.82821
$389$	32.2361	1.63443	0.817217	−	0.576330i	$-0.195516\pi$
$389$	32.2361	1.63443	0.817217	+	0.576330i	$0.195516\pi$
$390$	0	0
$391$	8.89794	0.449988
$392$	0	0
$393$	−4.94427	−0.249406
$394$	31.2705	1.57539
$395$	0	0
$396$	−24.2705	−1.21964
$397$	−9.89949	−0.496841	−0.248421	−	0.968652i	$-0.579912\pi$
$397$	−9.89949	−0.496841	−0.248421	+	0.968652i	$0.579912\pi$
$398$	−41.2672	−2.06854
$399$	0	0
$400$	0	0
$401$	−35.0689	−1.75126	−0.875628	−	0.482986i	$-0.839552\pi$
$401$	−35.0689	−1.75126	−0.875628	+	0.482986i	$0.839552\pi$
$402$	−28.2055	−1.40676
$403$	14.2918	0.711925
$404$	7.86629	0.391362
$405$	0	0
$406$	0	0
$407$	−6.05573	−0.300171
$408$	18.4721	0.914507
$409$	23.2163	1.14797	0.573986	−	0.818865i	$-0.305396\pi$
$409$	23.2163	1.14797	0.573986	+	0.818865i	$0.305396\pi$
$410$	0	0
$411$	13.7295	0.677225
$412$	46.0501	2.26872
$413$	0	0
$414$	48.2148	2.36963
$415$	0	0
$416$	59.1605	2.90058
$417$	−23.1246	−1.13242
$418$	24.8369	1.21481
$419$	33.3221	1.62789	0.813946	−	0.580940i	$-0.197315\pi$
$419$	33.3221	1.62789	0.813946	+	0.580940i	$0.197315\pi$
$420$	0	0
$421$	−22.7082	−1.10673	−0.553365	−	0.832939i	$-0.686657\pi$
$421$	−22.7082	−1.10673	−0.553365	+	0.832939i	$0.686657\pi$
$422$	50.8328	2.47450
$423$	−21.2132	−1.03142
$424$	−27.7082	−1.34563
$425$	0	0
$426$	−8.81913	−0.427288
$427$	0	0
$428$	−29.1246	−1.40779
$429$	27.8885	1.34647
$430$	0	0
$431$	11.2361	0.541222	0.270611	−	0.962689i	$-0.412774\pi$
$431$	11.2361	0.541222	0.270611	+	0.962689i	$0.412774\pi$
$432$	−17.2256	−0.828767
$433$	−24.2179	−1.16384	−0.581918	−	0.813247i	$-0.697698\pi$
$433$	−24.2179	−1.16384	−0.581918	+	0.813247i	$0.697698\pi$
$434$	0	0
$435$	0	0
$436$	−4.85410	−0.232469
$437$	−34.9427	−1.67153
$438$	−41.1246	−1.96501
$439$	−10.6947	−0.510431	−0.255215	−	0.966884i	$-0.582146\pi$
$439$	−10.6947	−0.510431	−0.255215	+	0.966884i	$0.582146\pi$
$440$	0	0
$441$	0	0
$442$	−15.4164	−0.733284
$443$	2.18034	0.103591	0.0517955	−	0.998658i	$-0.483506\pi$
$443$	2.18034	0.103591	0.0517955	+	0.998658i	$0.483506\pi$
$444$	30.0810	1.42758
$445$	0	0
$446$	−40.1869	−1.90290
$447$	−49.5462	−2.34345
$448$	0	0
$449$	36.5967	1.72711	0.863554	−	0.504257i	$-0.168234\pi$
$449$	36.5967	1.72711	0.863554	+	0.504257i	$0.168234\pi$
$450$	0	0
$451$	17.7658	0.836558
$452$	−69.1033	−3.25035
$453$	−42.1413	−1.97997
$454$	−19.5927	−0.919529
$455$	0	0
$456$	−72.5410	−3.39704
$457$	−20.7082	−0.968689	−0.484344	−	0.874877i	$-0.660942\pi$
$457$	−20.7082	−0.968689	−0.484344	+	0.874877i	$0.660942\pi$
$458$	18.5123	0.865023
$459$	1.88854	0.0881497
$460$	0	0
$461$	−30.2387	−1.40836	−0.704178	−	0.710024i	$-0.748684\pi$
$461$	−30.2387	−1.40836	−0.704178	+	0.710024i	$0.748684\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	−22.0344	−1.02292
$465$	0	0
$466$	35.2705	1.63387
$467$	41.2672	1.90962	0.954810	−	0.297217i	$-0.0960586\pi$
$467$	41.2672	1.90962	0.954810	+	0.297217i	$0.0960586\pi$
$468$	−59.1605	−2.73470
$469$	0	0
$470$	0	0
$471$	−9.23607	−0.425576
$472$	−66.8506	−3.07705
$473$	11.1803	0.514073
$474$	−52.1683	−2.39617
$475$	0	0
$476$	0	0
$477$	8.29180	0.379655
$478$	29.1246	1.33213
$479$	−12.1877	−0.556872	−0.278436	−	0.960455i	$-0.589816\pi$
$479$	−12.1877	−0.556872	−0.278436	+	0.960455i	$0.589816\pi$
$480$	0	0
$481$	−14.7611	−0.673050
$482$	−60.7811	−2.76850
$483$	0	0
$484$	−29.1246	−1.32385
$485$	0	0
$486$	50.4202	2.28711
$487$	−15.2918	−0.692937	−0.346469	−	0.938062i	$-0.612619\pi$
$487$	−15.2918	−0.692937	−0.346469	+	0.938062i	$0.612619\pi$
$488$	93.5630	4.23540
$489$	35.9442	1.62545
$490$	0	0
$491$	−1.47214	−0.0664366	−0.0332183	−	0.999448i	$-0.510576\pi$
$491$	−1.47214	−0.0664366	−0.0332183	+	0.999448i	$0.510576\pi$
$492$	−88.2492	−3.97858
$493$	2.41577	0.108801
$494$	60.5410	2.72387
$495$	0	0
$496$	−25.8384	−1.16018
$497$	0	0
$498$	−15.7082	−0.703901
$499$	1.41641	0.0634071	0.0317036	−	0.999497i	$-0.489907\pi$
$499$	1.41641	0.0634071	0.0317036	+	0.999497i	$0.489907\pi$
$500$	0	0
$501$	3.52786	0.157613
$502$	−52.5021	−2.34328
$503$	8.40647	0.374826	0.187413	−	0.982281i	$-0.439990\pi$
$503$	8.40647	0.374826	0.187413	+	0.982281i	$0.439990\pi$
$504$	0	0
$505$	0	0
$506$	−48.2148	−2.14341
$507$	38.2325	1.69796
$508$	−33.9787	−1.50756
$509$	−10.0270	−0.444440	−0.222220	−	0.974997i	$-0.571330\pi$
$509$	−10.0270	−0.444440	−0.222220	+	0.974997i	$0.571330\pi$
$510$	0	0
$511$	0	0
$512$	40.3050	1.78124
$513$	−7.41641	−0.327442
$514$	−42.8090	−1.88822
$515$	0	0
$516$	−55.5369	−2.44488
$517$	21.2132	0.932956
$518$	0	0
$519$	−9.70820	−0.426143
$520$	0	0
$521$	23.3739	1.02403	0.512015	−	0.858976i	$-0.328899\pi$
$521$	23.3739	1.02403	0.512015	+	0.858976i	$0.328899\pi$
$522$	13.0902	0.572941
$523$	22.2148	0.971383	0.485692	−	0.874130i	$-0.338568\pi$
$523$	22.2148	0.971383	0.485692	+	0.874130i	$0.338568\pi$
$524$	−10.4884	−0.458187
$525$	0	0
$526$	31.5623	1.37618
$527$	2.83282	0.123399
$528$	−50.4202	−2.19426
$529$	44.8328	1.94925
$530$	0	0
$531$	20.0053	0.868157
$532$	0	0
$533$	43.3050	1.87575
$534$	−28.6525	−1.23991
$535$	0	0
$536$	−35.1803	−1.51956
$537$	−22.2148	−0.958637
$538$	15.3500	0.661786
$539$	0	0
$540$	0	0
$541$	4.70820	0.202421	0.101211	−	0.994865i	$-0.467728\pi$
$541$	4.70820	0.202421	0.101211	+	0.994865i	$0.467728\pi$
$542$	36.4844	1.56714
$543$	3.70820	0.159134
$544$	11.7264	0.502764
$545$	0	0
$546$	0	0
$547$	−29.0000	−1.23995	−0.619975	−	0.784621i	$-0.712857\pi$
$547$	−29.0000	−1.23995	−0.619975	+	0.784621i	$0.712857\pi$
$548$	29.1246	1.24414
$549$	−27.9991	−1.19497
$550$	0	0
$551$	−9.48683	−0.404153
$552$	140.821	5.99374
$553$	0	0
$554$	−62.8328	−2.66951
$555$	0	0
$556$	−49.0547	−2.08038
$557$	24.5967	1.04220	0.521099	−	0.853496i	$-0.325522\pi$
$557$	24.5967	1.04220	0.521099	+	0.853496i	$0.325522\pi$
$558$	15.3500	0.649818
$559$	27.2526	1.15266
$560$	0	0
$561$	5.52786	0.233387
$562$	−64.3951	−2.71634
$563$	20.1328	0.848498	0.424249	−	0.905546i	$-0.360538\pi$
$563$	20.1328	0.848498	0.424249	+	0.905546i	$0.360538\pi$
$564$	−105.374	−4.43704
$565$	0	0
$566$	−4.78282	−0.201037
$567$	0	0
$568$	−11.0000	−0.461550
$569$	−12.0557	−0.505402	−0.252701	−	0.967544i	$-0.581319\pi$
$569$	−12.0557	−0.505402	−0.252701	+	0.967544i	$0.581319\pi$
$570$	0	0
$571$	−0.416408	−0.0174261	−0.00871306	−	0.999962i	$-0.502773\pi$
$571$	−0.416408	−0.0174261	−0.00871306	+	0.999962i	$0.502773\pi$
$572$	59.1605	2.47363
$573$	39.1853	1.63699
$574$	0	0
$575$	0	0
$576$	19.4721	0.811339
$577$	33.7047	1.40314	0.701572	−	0.712598i	$-0.252482\pi$
$577$	33.7047	1.40314	0.701572	+	0.712598i	$0.252482\pi$
$578$	41.4508	1.72413
$579$	−6.19704	−0.257540
$580$	0	0
$581$	0	0
$582$	−117.374	−4.86530
$583$	−8.29180	−0.343411
$584$	−51.2942	−2.12257
$585$	0	0
$586$	−59.3669	−2.45242
$587$	−15.3500	−0.633563	−0.316782	−	0.948499i	$-0.602602\pi$
$587$	−15.3500	−0.633563	−0.316782	+	0.948499i	$0.602602\pi$
$588$	0	0
$589$	−11.1246	−0.458382
$590$	0	0
$591$	−27.3314	−1.12426
$592$	26.6869	1.09683
$593$	11.7264	0.481544	0.240772	−	0.970582i	$-0.422599\pi$
$593$	11.7264	0.481544	0.240772	+	0.970582i	$0.422599\pi$
$594$	−10.2333	−0.419879
$595$	0	0
$596$	−105.103	−4.30520
$597$	36.0689	1.47620
$598$	−117.526	−4.80599
$599$	−29.1803	−1.19228	−0.596138	−	0.802882i	$-0.703299\pi$
$599$	−29.1803	−1.19228	−0.596138	+	0.802882i	$0.703299\pi$
$600$	0	0
$601$	46.4326	1.89403	0.947013	−	0.321196i	$-0.104085\pi$
$601$	46.4326	1.89403	0.947013	+	0.321196i	$0.104085\pi$
$602$	0	0
$603$	10.5279	0.428728
$604$	−89.3951	−3.63744
$605$	0	0
$606$	−9.70820	−0.394369
$607$	−21.1831	−0.859796	−0.429898	−	0.902878i	$-0.641450\pi$
$607$	−21.1831	−0.859796	−0.429898	+	0.902878i	$0.641450\pi$
$608$	−46.0501	−1.86758
$609$	0	0
$610$	0	0
$611$	51.7082	2.09189
$612$	−11.7264	−0.474010
$613$	12.4164	0.501494	0.250747	−	0.968053i	$-0.419324\pi$
$613$	12.4164	0.501494	0.250747	+	0.968053i	$0.419324\pi$
$614$	87.7787	3.54246
$615$	0	0
$616$	0	0
$617$	25.3607	1.02098	0.510491	−	0.859883i	$-0.329464\pi$
$617$	25.3607	1.02098	0.510491	+	0.859883i	$0.329464\pi$
$618$	−56.8328	−2.28615
$619$	19.1800	0.770909	0.385455	−	0.922727i	$-0.374045\pi$
$619$	19.1800	0.770909	0.385455	+	0.922727i	$0.374045\pi$
$620$	0	0
$621$	14.3972	0.577739
$622$	41.6011	1.66805
$623$	0	0
$624$	−122.902	−4.92001
$625$	0	0
$626$	55.5369	2.21970
$627$	−21.7082	−0.866942
$628$	−19.5927	−0.781832
$629$	−2.92584	−0.116661
$630$	0	0
$631$	12.1246	0.482673	0.241337	−	0.970441i	$-0.422414\pi$
$631$	12.1246	0.482673	0.241337	+	0.970441i	$0.422414\pi$
$632$	−65.0689	−2.58830
$633$	−44.4295	−1.76591
$634$	15.5623	0.618058
$635$	0	0
$636$	41.1884	1.63323
$637$	0	0
$638$	−13.0902	−0.518245
$639$	3.29180	0.130221
$640$	0	0
$641$	−25.4721	−1.00609	−0.503044	−	0.864261i	$-0.667787\pi$
$641$	−25.4721	−1.00609	−0.503044	+	0.864261i	$0.667787\pi$
$642$	35.9442	1.41861
$643$	−3.24109	−0.127816	−0.0639080	−	0.997956i	$-0.520356\pi$
$643$	−3.24109	−0.127816	−0.0639080	+	0.997956i	$0.520356\pi$
$644$	0	0
$645$	0	0
$646$	12.0000	0.472134
$647$	−6.40337	−0.251742	−0.125871	−	0.992047i	$-0.540173\pi$
$647$	−6.40337	−0.251742	−0.125871	+	0.992047i	$0.540173\pi$
$648$	80.0132	3.14321
$649$	−20.0053	−0.785278
$650$	0	0
$651$	0	0
$652$	76.2492	2.98615
$653$	25.4164	0.994621	0.497310	−	0.867573i	$-0.334321\pi$
$653$	25.4164	0.994621	0.497310	+	0.867573i	$0.334321\pi$
$654$	5.99070	0.234255
$655$	0	0
$656$	−78.2918	−3.05678
$657$	15.3500	0.598861
$658$	0	0
$659$	44.8328	1.74644	0.873219	−	0.487328i	$-0.162028\pi$
$659$	44.8328	1.74644	0.873219	+	0.487328i	$0.162028\pi$
$660$	0	0
$661$	−36.9458	−1.43702	−0.718512	−	0.695514i	$-0.755177\pi$
$661$	−36.9458	−1.43702	−0.718512	+	0.695514i	$0.755177\pi$
$662$	38.5066	1.49660
$663$	13.4744	0.523304
$664$	−19.5927	−0.760343
$665$	0	0
$666$	−15.8541	−0.614334
$667$	18.4164	0.713086
$668$	7.48373	0.289554
$669$	35.1246	1.35800
$670$	0	0
$671$	27.9991	1.08089
$672$	0	0
$673$	0.875388	0.0337437	0.0168719	−	0.999858i	$-0.494629\pi$
$673$	0.875388	0.0337437	0.0168719	+	0.999858i	$0.494629\pi$
$674$	44.8328	1.72690
$675$	0	0
$676$	81.1033	3.11936
$677$	20.1328	0.773768	0.386884	−	0.922128i	$-0.373551\pi$
$677$	20.1328	0.773768	0.386884	+	0.922128i	$0.373551\pi$
$678$	85.2841	3.27532
$679$	0	0
$680$	0	0
$681$	17.1246	0.656217
$682$	−15.3500	−0.587783
$683$	17.0689	0.653123	0.326561	−	0.945176i	$-0.394110\pi$
$683$	17.0689	0.653123	0.326561	+	0.945176i	$0.394110\pi$
$684$	46.0501	1.76077
$685$	0	0
$686$	0	0
$687$	−16.1803	−0.617318
$688$	−49.2705	−1.87842
$689$	−20.2117	−0.770003
$690$	0	0
$691$	−45.6374	−1.73613	−0.868064	−	0.496452i	$-0.834636\pi$
$691$	−45.6374	−1.73613	−0.868064	+	0.496452i	$0.834636\pi$
$692$	−20.5942	−0.782874
$693$	0	0
$694$	−9.56231	−0.362980
$695$	0	0
$696$	38.2325	1.44920
$697$	8.58359	0.325127
$698$	−47.0516	−1.78093
$699$	−30.8276	−1.16601
$700$	0	0
$701$	4.58359	0.173120	0.0865599	−	0.996247i	$-0.472413\pi$
$701$	4.58359	0.173120	0.0865599	+	0.996247i	$0.472413\pi$
$702$	−24.9443	−0.941461
$703$	11.4899	0.433351
$704$	−19.4721	−0.733884
$705$	0	0
$706$	42.8090	1.61114
$707$	0	0
$708$	99.3738	3.73470
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	0	0
$711$	19.4721	0.730262
$712$	−35.7379	−1.33933
$713$	21.5958	0.808768
$714$	0	0
$715$	0	0
$716$	−47.1246	−1.76113
$717$	−25.4558	−0.950666
$718$	−46.9787	−1.75323
$719$	−14.2697	−0.532168	−0.266084	−	0.963950i	$-0.585730\pi$
$719$	−14.2697	−0.532168	−0.266084	+	0.963950i	$0.585730\pi$
$720$	0	0
$721$	0	0
$722$	2.61803	0.0974331
$723$	53.1246	1.97573
$724$	7.86629	0.292348
$725$	0	0
$726$	35.9442	1.33402
$727$	−5.65685	−0.209801	−0.104901	−	0.994483i	$-0.533452\pi$
$727$	−5.65685	−0.209801	−0.104901	+	0.994483i	$0.533452\pi$
$728$	0	0
$729$	−11.9443	−0.442380
$730$	0	0
$731$	5.40182	0.199793
$732$	−139.082	−5.14062
$733$	44.4295	1.64104	0.820521	−	0.571617i	$-0.193684\pi$
$733$	44.4295	1.64104	0.820521	+	0.571617i	$0.193684\pi$
$734$	−47.5918	−1.75664
$735$	0	0
$736$	89.3951	3.29515
$737$	−10.5279	−0.387799
$738$	46.5114	1.71211
$739$	−20.7082	−0.761764	−0.380882	−	0.924624i	$-0.624380\pi$
$739$	−20.7082	−0.761764	−0.380882	+	0.924624i	$0.624380\pi$
$740$	0	0
$741$	−52.9148	−1.94387
$742$	0	0
$743$	−20.2918	−0.744434	−0.372217	−	0.928146i	$-0.621402\pi$
$743$	−20.2918	−0.744434	−0.372217	+	0.928146i	$0.621402\pi$
$744$	44.8328	1.64365
$745$	0	0
$746$	63.1591	2.31242
$747$	5.86319	0.214523
$748$	11.7264	0.428759
$749$	0	0
$750$	0	0
$751$	−26.8328	−0.979143	−0.489572	−	0.871963i	$-0.662847\pi$
$751$	−26.8328	−0.979143	−0.489572	+	0.871963i	$0.662847\pi$
$752$	−93.4842	−3.40902
$753$	45.8885	1.67227
$754$	−31.9079	−1.16202
$755$	0	0
$756$	0	0
$757$	−14.4164	−0.523973	−0.261987	−	0.965072i	$-0.584378\pi$
$757$	−14.4164	−0.523973	−0.261987	+	0.965072i	$0.584378\pi$
$758$	49.7426	1.80673
$759$	42.1413	1.52963
$760$	0	0
$761$	14.9675	0.542570	0.271285	−	0.962499i	$-0.412551\pi$
$761$	14.9675	0.542570	0.271285	+	0.962499i	$0.412551\pi$
$762$	41.9349	1.51914
$763$	0	0
$764$	83.1246	3.00734
$765$	0	0
$766$	23.4226	0.846294
$767$	−48.7639	−1.76076
$768$	−33.3221	−1.20241
$769$	18.1784	0.655532	0.327766	−	0.944759i	$-0.393704\pi$
$769$	18.1784	0.655532	0.327766	+	0.944759i	$0.393704\pi$
$770$	0	0
$771$	37.4164	1.34752
$772$	−13.1459	−0.473131
$773$	−5.86319	−0.210884	−0.105442	−	0.994425i	$-0.533626\pi$
$773$	−5.86319	−0.210884	−0.105442	+	0.994425i	$0.533626\pi$
$774$	29.2705	1.05211
$775$	0	0
$776$	−146.399	−5.25542
$777$	0	0
$778$	−84.3951	−3.02571
$779$	−33.7082	−1.20772
$780$	0	0
$781$	−3.29180	−0.117790
$782$	−23.2951	−0.833032
$783$	3.90879	0.139689
$784$	0	0
$785$	0	0
$786$	12.9443	0.461707
$787$	−12.1089	−0.431637	−0.215818	−	0.976434i	$-0.569242\pi$
$787$	−12.1089	−0.431637	−0.215818	+	0.976434i	$0.569242\pi$
$788$	−57.9787	−2.06541
$789$	−27.5865	−0.982104
$790$	0	0
$791$	0	0
$792$	37.3607	1.32755
$793$	68.2492	2.42360
$794$	25.9172	0.919768
$795$	0	0
$796$	76.5137	2.71195
$797$	−30.5424	−1.08187	−0.540934	−	0.841065i	$-0.681929\pi$
$797$	−30.5424	−1.08187	−0.540934	+	0.841065i	$0.681929\pi$
$798$	0	0
$799$	10.2492	0.362591
$800$	0	0
$801$	10.6947	0.377879
$802$	91.8115	3.24198
$803$	−15.3500	−0.541690
$804$	52.2958	1.84433
$805$	0	0
$806$	−37.4164	−1.31794
$807$	−13.4164	−0.472280
$808$	−12.1089	−0.425991
$809$	−29.0689	−1.02201	−0.511004	−	0.859578i	$-0.670726\pi$
$809$	−29.0689	−1.02201	−0.511004	+	0.859578i	$0.670726\pi$
$810$	0	0
$811$	7.07107	0.248299	0.124149	−	0.992264i	$-0.460380\pi$
$811$	7.07107	0.248299	0.124149	+	0.992264i	$0.460380\pi$
$812$	0	0
$813$	−31.8885	−1.11838
$814$	15.8541	0.555686
$815$	0	0
$816$	−24.3607	−0.852794
$817$	−21.2132	−0.742156
$818$	−60.7811	−2.12516
$819$	0	0
$820$	0	0
$821$	37.4164	1.30584	0.652921	−	0.757426i	$-0.273543\pi$
$821$	37.4164	1.30584	0.652921	+	0.757426i	$0.273543\pi$
$822$	−35.9442	−1.25370
$823$	−37.5410	−1.30860	−0.654299	−	0.756236i	$-0.727036\pi$
$823$	−37.5410	−1.30860	−0.654299	+	0.756236i	$0.727036\pi$
$824$	−70.8869	−2.46946
$825$	0	0
$826$	0	0
$827$	−7.47214	−0.259832	−0.129916	−	0.991525i	$-0.541471\pi$
$827$	−7.47214	−0.259832	−0.129916	+	0.991525i	$0.541471\pi$
$828$	−89.3951	−3.10670
$829$	−46.8453	−1.62700	−0.813502	−	0.581562i	$-0.802442\pi$
$829$	−46.8453	−1.62700	−0.813502	+	0.581562i	$0.802442\pi$
$830$	0	0
$831$	54.9179	1.90508
$832$	−47.4643	−1.64553
$833$	0	0
$834$	60.5410	2.09636
$835$	0	0
$836$	−46.0501	−1.59267
$837$	4.58359	0.158432
$838$	−87.2385	−3.01360
$839$	−40.8059	−1.40877	−0.704387	−	0.709816i	$-0.748778\pi$
$839$	−40.8059	−1.40877	−0.704387	+	0.709816i	$0.748778\pi$
$840$	0	0
$841$	−24.0000	−0.827586
$842$	59.4508	2.04881
$843$	56.2834	1.93850
$844$	−94.2492	−3.24419
$845$	0	0
$846$	55.5369	1.90940
$847$	0	0
$848$	36.5410	1.25482
$849$	4.18034	0.143469
$850$	0	0
$851$	−22.3050	−0.764604
$852$	16.3516	0.560196
$853$	3.24109	0.110973	0.0554864	−	0.998459i	$-0.482329\pi$
$853$	3.24109	0.110973	0.0554864	+	0.998459i	$0.482329\pi$
$854$	0	0
$855$	0	0
$856$	44.8328	1.53235
$857$	−17.5107	−0.598156	−0.299078	−	0.954229i	$-0.596679\pi$
$857$	−17.5107	−0.598156	−0.299078	+	0.954229i	$0.596679\pi$
$858$	−73.0132	−2.49263
$859$	−17.1468	−0.585041	−0.292520	−	0.956259i	$-0.594494\pi$
$859$	−17.1468	−0.585041	−0.292520	+	0.956259i	$0.594494\pi$
$860$	0	0
$861$	0	0
$862$	−29.4164	−1.00193
$863$	−43.3607	−1.47601	−0.738007	−	0.674793i	$-0.764233\pi$
$863$	−43.3607	−1.47601	−0.738007	+	0.674793i	$0.764233\pi$
$864$	18.9737	0.645497
$865$	0	0
$866$	63.4032	2.15453
$867$	−36.2294	−1.23041
$868$	0	0
$869$	−19.4721	−0.660547
$870$	0	0
$871$	−25.6622	−0.869530
$872$	7.47214	0.253038
$873$	43.8105	1.48276
$874$	91.4811	3.09439
$875$	0	0
$876$	76.2492	2.57622
$877$	51.7082	1.74606	0.873031	−	0.487665i	$-0.162151\pi$
$877$	51.7082	1.74606	0.873031	+	0.487665i	$0.162151\pi$
$878$	27.9991	0.944925
$879$	51.8885	1.75016
$880$	0	0
$881$	−11.1074	−0.374217	−0.187109	−	0.982339i	$-0.559912\pi$
$881$	−11.1074	−0.374217	−0.187109	+	0.982339i	$0.559912\pi$
$882$	0	0
$883$	19.2918	0.649221	0.324610	−	0.945848i	$-0.394767\pi$
$883$	19.2918	0.649221	0.324610	+	0.945848i	$0.394767\pi$
$884$	28.5836	0.961370
$885$	0	0
$886$	−5.70820	−0.191771
$887$	47.1304	1.58248	0.791242	−	0.611503i	$-0.209435\pi$
$887$	47.1304	1.58248	0.791242	+	0.611503i	$0.209435\pi$
$888$	−46.3051	−1.55390
$889$	0	0
$890$	0	0
$891$	23.9443	0.802163
$892$	74.5106	2.49480
$893$	−40.2492	−1.34689
$894$	129.714	4.33827
$895$	0	0
$896$	0	0
$897$	102.721	3.42977
$898$	−95.8115	−3.19727
$899$	5.86319	0.195548
$900$	0	0
$901$	−4.00621	−0.133466
$902$	−46.5114	−1.54866
$903$	0	0
$904$	106.374	3.53794
$905$	0	0
$906$	110.327	3.66538
$907$	−38.8328	−1.28942	−0.644711	−	0.764426i	$-0.723022\pi$
$907$	−38.8328	−1.28942	−0.644711	+	0.764426i	$0.723022\pi$
$908$	36.3268	1.20555
$909$	3.62365	0.120189
$910$	0	0
$911$	26.2361	0.869240	0.434620	−	0.900614i	$-0.356883\pi$
$911$	26.2361	0.869240	0.434620	+	0.900614i	$0.356883\pi$
$912$	95.6656	3.16781
$913$	−5.86319	−0.194043
$914$	54.2148	1.79327
$915$	0	0
$916$	−34.3237	−1.13409
$917$	0	0
$918$	−4.94427	−0.163185
$919$	−21.8328	−0.720198	−0.360099	−	0.932914i	$-0.617257\pi$
$919$	−21.8328	−0.720198	−0.360099	+	0.932914i	$0.617257\pi$
$920$	0	0
$921$	−76.7214	−2.52805
$922$	79.1659	2.60719
$923$	−8.02391	−0.264110
$924$	0	0
$925$	0	0
$926$	−36.6525	−1.20448
$927$	21.2132	0.696733
$928$	24.2705	0.796719
$929$	0.382559	0.0125513	0.00627567	−	0.999980i	$-0.498002\pi$
$929$	0.382559	0.0125513	0.00627567	+	0.999980i	$0.498002\pi$
$930$	0	0
$931$	0	0
$932$	−65.3951	−2.14209
$933$	−36.3607	−1.19040
$934$	−108.039	−3.53514
$935$	0	0
$936$	91.0685	2.97667
$937$	37.9774	1.24067	0.620334	−	0.784337i	$-0.286997\pi$
$937$	37.9774	1.24067	0.620334	+	0.784337i	$0.286997\pi$
$938$	0	0
$939$	−48.5410	−1.58408
$940$	0	0
$941$	−24.8369	−0.809658	−0.404829	−	0.914392i	$-0.632669\pi$
$941$	−24.8369	−0.809658	−0.404829	+	0.914392i	$0.632669\pi$
$942$	24.1803	0.787838
$943$	65.4364	2.13090
$944$	88.1612	2.86940
$945$	0	0
$946$	−29.2705	−0.951666
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	96.7253	3.14149
$949$	−37.4164	−1.21459
$950$	0	0
$951$	−13.6020	−0.441074
$952$	0	0
$953$	−12.5967	−0.408049	−0.204024	−	0.978966i	$-0.565402\pi$
$953$	−12.5967	−0.408049	−0.204024	+	0.978966i	$0.565402\pi$
$954$	−21.7082	−0.702829
$955$	0	0
$956$	−54.0000	−1.74648
$957$	11.4412	0.369842
$958$	31.9079	1.03090
$959$	0	0
$960$	0	0
$961$	−24.1246	−0.778213
$962$	38.6451	1.24597
$963$	−13.4164	−0.432338
$964$	112.694	3.62964
$965$	0	0
$966$	0	0
$967$	−34.2492	−1.10138	−0.550690	−	0.834710i	$-0.685635\pi$
$967$	−34.2492	−1.10138	−0.550690	+	0.834710i	$0.685635\pi$
$968$	44.8328	1.44098
$969$	−10.4884	−0.336935
$970$	0	0
$971$	−12.7279	−0.408458	−0.204229	−	0.978923i	$-0.565469\pi$
$971$	−12.7279	−0.408458	−0.204229	+	0.978923i	$0.565469\pi$
$972$	−93.4842	−2.99851
$973$	0	0
$974$	40.0344	1.28279
$975$	0	0
$976$	−123.389	−3.94959
$977$	32.2361	1.03132	0.515662	−	0.856792i	$-0.327546\pi$
$977$	32.2361	1.03132	0.515662	+	0.856792i	$0.327546\pi$
$978$	−94.1032	−3.00909
$979$	−10.6947	−0.341805
$980$	0	0
$981$	−2.23607	−0.0713922
$982$	3.85410	0.122989
$983$	−55.5369	−1.77135	−0.885676	−	0.464304i	$-0.846304\pi$
$983$	−55.5369	−1.77135	−0.885676	+	0.464304i	$0.846304\pi$
$984$	135.846	4.33061
$985$	0	0
$986$	−6.32456	−0.201415
$987$	0	0
$988$	−112.249	−3.57112
$989$	41.1803	1.30946
$990$	0	0
$991$	−5.00000	−0.158830	−0.0794151	−	0.996842i	$-0.525305\pi$
$991$	−5.00000	−0.158830	−0.0794151	+	0.996842i	$0.525305\pi$
$992$	28.4605	0.903622
$993$	−33.6560	−1.06804
$994$	0	0
$995$	0	0
$996$	29.1246	0.922849
$997$	10.7248	0.339658	0.169829	−	0.985474i	$-0.445678\pi$
$997$	10.7248	0.339658	0.169829	+	0.985474i	$0.445678\pi$
$998$	−3.70820	−0.117381
$999$	−4.73411	−0.149781

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1225.2.a.ba.1.2	yes	4
5.2	odd	4		1225.2.b.n.99.1		8
5.3	odd	4		1225.2.b.n.99.8		8
5.4	even	2		1225.2.a.bc.1.3	yes	4
7.6	odd	2	inner	1225.2.a.ba.1.1	✓	4
35.13	even	4		1225.2.b.n.99.7		8
35.27	even	4		1225.2.b.n.99.2		8
35.34	odd	2		1225.2.a.bc.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1225.2.a.ba.1.1	✓	4	7.6	odd	2	inner
1225.2.a.ba.1.2	yes	4	1.1	even	1	trivial
1225.2.a.bc.1.3	yes	4	5.4	even	2
1225.2.a.bc.1.4	yes	4	35.34	odd	2
1225.2.b.n.99.1		8	5.2	odd	4
1225.2.b.n.99.2		8	35.27	even	4
1225.2.b.n.99.7		8	35.13	even	4
1225.2.b.n.99.8		8	5.3	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:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61803 q^{2} +2.28825 q^{3} +4.85410 q^{4} -5.99070 q^{6} -7.47214 q^{8} +2.23607 q^{9} +O(q^{10})\)	\(q-2.61803 q^{2} +2.28825 q^{3} +4.85410 q^{4} -5.99070 q^{6} -7.47214 q^{8} +2.23607 q^{9} -2.23607 q^{11} +11.1074 q^{12} -5.45052 q^{13} +9.85410 q^{16} -1.08036 q^{17} -5.85410 q^{18} +4.24264 q^{19} +5.85410 q^{22} -8.23607 q^{23} -17.0981 q^{24} +14.2697 q^{26} -1.74806 q^{27} -2.23607 q^{29} -2.62210 q^{31} -10.8541 q^{32} -5.11667 q^{33} +2.82843 q^{34} +10.8541 q^{36} +2.70820 q^{37} -11.1074 q^{38} -12.4721 q^{39} -7.94510 q^{41} -5.00000 q^{43} -10.8541 q^{44} +21.5623 q^{46} -9.48683 q^{47} +22.5486 q^{48} -2.47214 q^{51} -26.4574 q^{52} +3.70820 q^{53} +4.57649 q^{54} +9.70820 q^{57} +5.85410 q^{58} +8.94665 q^{59} -12.5216 q^{61} +6.86474 q^{62} +8.70820 q^{64} +13.3956 q^{66} +4.70820 q^{67} -5.24419 q^{68} -18.8461 q^{69} +1.47214 q^{71} -16.7082 q^{72} +6.86474 q^{73} -7.09017 q^{74} +20.5942 q^{76} +32.6525 q^{78} +8.70820 q^{79} -10.7082 q^{81} +20.8005 q^{82} +2.62210 q^{83} +13.0902 q^{86} -5.11667 q^{87} +16.7082 q^{88} +4.78282 q^{89} -39.9787 q^{92} -6.00000 q^{93} +24.8369 q^{94} -24.8369 q^{96} +19.5927 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{2} + 6 q^{4} - 12 q^{8}+O(q^{10}) \) 4 * q - 6 * q^2 + 6 * q^4 - 12 * q^8	\( 4 q - 6 q^{2} + 6 q^{4} - 12 q^{8} + 26 q^{16} - 10 q^{18} + 10 q^{22} - 24 q^{23} - 30 q^{32} + 30 q^{36} - 16 q^{37} - 32 q^{39} - 20 q^{43} - 30 q^{44} + 46 q^{46} + 8 q^{51} - 12 q^{53} + 12 q^{57} + 10 q^{58} + 8 q^{64} - 8 q^{67} - 12 q^{71} - 40 q^{72} - 6 q^{74} + 68 q^{78} + 8 q^{79} - 16 q^{81} + 30 q^{86} + 40 q^{88} - 66 q^{92} - 24 q^{93} - 20 q^{99}+O(q^{100}) \) 4 * q - 6 * q^2 + 6 * q^4 - 12 * q^8 + 26 * q^16 - 10 * q^18 + 10 * q^22 - 24 * q^23 - 30 * q^32 + 30 * q^36 - 16 * q^37 - 32 * q^39 - 20 * q^43 - 30 * q^44 + 46 * q^46 + 8 * q^51 - 12 * q^53 + 12 * q^57 + 10 * q^58 + 8 * q^64 - 8 * q^67 - 12 * q^71 - 40 * q^72 - 6 * q^74 + 68 * q^78 + 8 * q^79 - 16 * q^81 + 30 * q^86 + 40 * q^88 - 66 * q^92 - 24 * q^93 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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