LMFDB - Embedded newform 7225.2.a.bn.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7225,2,Mod(1,7225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [12,-3,-3,21,0,-9,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.6919154604$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 18 x^{10} + 55 x^{9} + 114 x^{8} - 354 x^{7} - 309 x^{6} + 936 x^{5} + 396 x^{4} + \cdots + 9 $ x^12 - 3x^11 - 18x^10 + 55x^9 + 114x^8 - 354x^7 - 309x^6 + 936x^5 + 396x^4 - 899x^3 - 297x^2 + 153*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1445)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-0.704111$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.704111 q^{2} +3.11994 q^{3} -1.50423 q^{4} +2.19679 q^{6} -4.06346 q^{7} -2.46736 q^{8} +6.73405 q^{9} +0.231031 q^{11} -4.69311 q^{12} -2.35528 q^{13} -2.86112 q^{14} +1.27116 q^{16} +4.74152 q^{18} +7.38657 q^{19} -12.6778 q^{21} +0.162671 q^{22} -1.87741 q^{23} -7.69804 q^{24} -1.65838 q^{26} +11.6500 q^{27} +6.11237 q^{28} -4.04943 q^{29} -0.130455 q^{31} +5.82977 q^{32} +0.720802 q^{33} -10.1295 q^{36} -5.82227 q^{37} +5.20096 q^{38} -7.34835 q^{39} -8.10305 q^{41} -8.92655 q^{42} +3.69785 q^{43} -0.347523 q^{44} -1.32190 q^{46} -1.30633 q^{47} +3.96595 q^{48} +9.51170 q^{49} +3.54288 q^{52} -8.26017 q^{53} +8.20291 q^{54} +10.0260 q^{56} +23.0457 q^{57} -2.85125 q^{58} +2.29764 q^{59} -8.92792 q^{61} -0.0918548 q^{62} -27.3635 q^{63} +1.56248 q^{64} +0.507524 q^{66} -14.5995 q^{67} -5.85742 q^{69} -11.0065 q^{71} -16.6154 q^{72} +0.471385 q^{73} -4.09952 q^{74} -11.1111 q^{76} -0.938783 q^{77} -5.17405 q^{78} +4.31667 q^{79} +16.1453 q^{81} -5.70544 q^{82} -15.4238 q^{83} +19.0703 q^{84} +2.60370 q^{86} -12.6340 q^{87} -0.570036 q^{88} +3.68913 q^{89} +9.57059 q^{91} +2.82405 q^{92} -0.407012 q^{93} -0.919804 q^{94} +18.1885 q^{96} +9.00621 q^{97} +6.69729 q^{98} +1.55577 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} - 3 q^{3} + 21 q^{4} - 9 q^{6} - 6 q^{7} - 12 q^{8} + 21 q^{9} - 6 q^{11} - 6 q^{12} - 9 q^{13} - 18 q^{14} + 39 q^{16} + 9 q^{18} + 27 q^{19} + 6 q^{21} - 15 q^{22} - 18 q^{23} - 36 q^{24}+ \cdots - 6 q^{99}+O(q^{100}) $ 12 * q - 3 * q^2 - 3 * q^3 + 21 * q^4 - 9 * q^6 - 6 * q^7 - 12 * q^8 + 21 * q^9 - 6 * q^11 - 6 * q^12 - 9 * q^13 - 18 * q^14 + 39 * q^16 + 9 * q^18 + 27 * q^19 + 6 * q^21 - 15 * q^22 - 18 * q^23 - 36 * q^24 + 6 * q^26 - 12 * q^27 - 39 * q^28 - 6 * q^31 - 12 * q^32 + 48 * q^36 - 21 * q^37 + 3 * q^38 - 9 * q^39 - 15 * q^41 - 30 * q^42 - 45 * q^43 - 42 * q^44 - 15 * q^46 - 3 * q^47 + 63 * q^48 + 12 * q^49 - 51 * q^52 + 12 * q^53 - 27 * q^54 + 3 * q^56 - 6 * q^57 + 51 * q^58 + 36 * q^59 + 3 * q^61 - 36 * q^62 - 72 * q^63 + 42 * q^64 - 39 * q^66 - 78 * q^67 - 66 * q^69 - 30 * q^71 + 12 * q^72 + 33 * q^73 + 9 * q^74 + 27 * q^76 + 39 * q^77 + 24 * q^78 - 3 * q^79 + 12 * q^81 + 51 * q^82 - 27 * q^83 + 27 * q^84 + 18 * q^86 - 24 * q^87 - 36 * q^88 + 6 * q^89 - 24 * q^91 + 3 * q^92 + 12 * q^93 + 21 * q^94 - 39 * q^96 + 6 * q^97 + 42 * q^98 - 6 * q^99

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 $

$2$	0.704111	0.497881	0.248941	−	0.968519i	$-0.419918\pi$
$2$	0.704111	0.497881	0.248941	+	0.968519i	$0.419918\pi$
$3$	3.11994	1.80130	0.900650	−	0.434545i	$-0.143091\pi$
$3$	3.11994	1.80130	0.900650	+	0.434545i	$0.143091\pi$
$4$	−1.50423	−0.752114
$5$	0	0
$5$	0	0
$6$	2.19679	0.896834
$7$	−4.06346	−1.53584	−0.767922	−	0.640544i	$-0.778709\pi$
$7$	−4.06346	−1.53584	−0.767922	+	0.640544i	$0.778709\pi$
$8$	−2.46736	−0.872345
$9$	6.73405	2.24468
$10$	0	0
$11$	0.231031	0.0696583	0.0348292	−	0.999393i	$-0.488911\pi$
$11$	0.231031	0.0696583	0.0348292	+	0.999393i	$0.488911\pi$
$12$	−4.69311	−1.35478
$13$	−2.35528	−0.653238	−0.326619	−	0.945156i	$-0.605909\pi$
$13$	−2.35528	−0.653238	−0.326619	+	0.945156i	$0.605909\pi$
$14$	−2.86112	−0.764668
$15$	0	0
$16$	1.27116	0.317790
$17$	0	0
$17$	0	0
$18$	4.74152	1.11759
$19$	7.38657	1.69460	0.847298	−	0.531118i	$-0.178228\pi$
$19$	7.38657	1.69460	0.847298	+	0.531118i	$0.178228\pi$
$20$	0	0
$21$	−12.6778	−2.76651
$22$	0.162671	0.0346816
$23$	−1.87741	−0.391467	−0.195734	−	0.980657i	$-0.562709\pi$
$23$	−1.87741	−0.391467	−0.195734	+	0.980657i	$0.562709\pi$
$24$	−7.69804	−1.57136
$25$	0	0
$26$	−1.65838	−0.325235
$27$	11.6500	2.24205
$28$	6.11237	1.15513
$29$	−4.04943	−0.751961	−0.375980	−	0.926628i	$-0.622694\pi$
$29$	−4.04943	−0.751961	−0.375980	+	0.926628i	$0.622694\pi$
$30$	0	0
$31$	−0.130455	−0.0234304	−0.0117152	−	0.999931i	$-0.503729\pi$
$31$	−0.130455	−0.0234304	−0.0117152	+	0.999931i	$0.503729\pi$
$32$	5.82977	1.03057
$33$	0.720802	0.125476
$34$	0	0
$35$	0	0
$36$	−10.1295	−1.68826
$37$	−5.82227	−0.957175	−0.478587	−	0.878040i	$-0.658851\pi$
$37$	−5.82227	−0.957175	−0.478587	+	0.878040i	$0.658851\pi$
$38$	5.20096	0.843707
$39$	−7.34835	−1.17668
$40$	0	0
$41$	−8.10305	−1.26548	−0.632742	−	0.774363i	$-0.718071\pi$
$41$	−8.10305	−1.26548	−0.632742	+	0.774363i	$0.718071\pi$
$42$	−8.92655	−1.37740
$43$	3.69785	0.563917	0.281959	−	0.959427i	$-0.409016\pi$
$43$	3.69785	0.563917	0.281959	+	0.959427i	$0.409016\pi$
$44$	−0.347523	−0.0523910
$45$	0	0
$46$	−1.32190	−0.194904
$47$	−1.30633	−0.190549	−0.0952743	−	0.995451i	$-0.530373\pi$
$47$	−1.30633	−0.190549	−0.0952743	+	0.995451i	$0.530373\pi$
$48$	3.96595	0.572435
$49$	9.51170	1.35881
$50$	0	0
$51$	0	0
$52$	3.54288	0.491309
$53$	−8.26017	−1.13462	−0.567311	−	0.823504i	$-0.692016\pi$
$53$	−8.26017	−1.13462	−0.567311	+	0.823504i	$0.692016\pi$
$54$	8.20291	1.11627
$55$	0	0
$56$	10.0260	1.33978
$57$	23.0457	3.05248
$58$	−2.85125	−0.374387
$59$	2.29764	0.299128	0.149564	−	0.988752i	$-0.452213\pi$
$59$	2.29764	0.299128	0.149564	+	0.988752i	$0.452213\pi$
$60$	0	0
$61$	−8.92792	−1.14310	−0.571552	−	0.820566i	$-0.693658\pi$
$61$	−8.92792	−1.14310	−0.571552	+	0.820566i	$0.693658\pi$
$62$	−0.0918548	−0.0116656
$63$	−27.3635	−3.44748
$64$	1.56248	0.195310
$65$	0	0
$66$	0.507524	0.0624719
$67$	−14.5995	−1.78361	−0.891805	−	0.452419i	$-0.850561\pi$
$67$	−14.5995	−1.78361	−0.891805	+	0.452419i	$0.850561\pi$
$68$	0	0
$69$	−5.85742	−0.705150
$70$	0	0
$71$	−11.0065	−1.30623	−0.653117	−	0.757257i	$-0.726539\pi$
$71$	−11.0065	−1.30623	−0.653117	+	0.757257i	$0.726539\pi$
$72$	−16.6154	−1.95814
$73$	0.471385	0.0551714	0.0275857	−	0.999619i	$-0.491218\pi$
$73$	0.471385	0.0551714	0.0275857	+	0.999619i	$0.491218\pi$
$74$	−4.09952	−0.476560
$75$	0	0
$76$	−11.1111	−1.27453
$77$	−0.938783	−0.106984
$78$	−5.17405	−0.585846
$79$	4.31667	0.485663	0.242831	−	0.970069i	$-0.421924\pi$
$79$	4.31667	0.485663	0.242831	+	0.970069i	$0.421924\pi$
$80$	0	0
$81$	16.1453	1.79392
$82$	−5.70544	−0.630061
$83$	−15.4238	−1.69298	−0.846489	−	0.532407i	$-0.821288\pi$
$83$	−15.4238	−1.69298	−0.846489	+	0.532407i	$0.821288\pi$
$84$	19.0703	2.08073
$85$	0	0
$86$	2.60370	0.280764
$87$	−12.6340	−1.35451
$88$	−0.570036	−0.0607661
$89$	3.68913	0.391047	0.195523	−	0.980699i	$-0.437359\pi$
$89$	3.68913	0.391047	0.195523	+	0.980699i	$0.437359\pi$
$90$	0	0
$91$	9.57059	1.00327
$92$	2.82405	0.294428
$93$	−0.407012	−0.0422052
$94$	−0.919804	−0.0948706
$95$	0	0
$96$	18.1885	1.85636
$97$	9.00621	0.914442	0.457221	−	0.889353i	$-0.348845\pi$
$97$	9.00621	0.914442	0.457221	+	0.889353i	$0.348845\pi$
$98$	6.69729	0.676528
$99$	1.55577	0.156361
$100$	0	0
$101$	−6.94560	−0.691113	−0.345556	−	0.938398i	$-0.612310\pi$
$101$	−6.94560	−0.691113	−0.345556	+	0.938398i	$0.612310\pi$
$102$	0	0
$103$	1.04422	0.102891	0.0514453	−	0.998676i	$-0.483617\pi$
$103$	1.04422	0.102891	0.0514453	+	0.998676i	$0.483617\pi$
$104$	5.81134	0.569849
$105$	0	0
$106$	−5.81607	−0.564907
$107$	−10.0027	−0.966999	−0.483500	−	0.875345i	$-0.660635\pi$
$107$	−10.0027	−0.966999	−0.483500	+	0.875345i	$0.660635\pi$
$108$	−17.5243	−1.68628
$109$	8.14200	0.779863	0.389931	−	0.920844i	$-0.372499\pi$
$109$	8.14200	0.779863	0.389931	+	0.920844i	$0.372499\pi$
$110$	0	0
$111$	−18.1651	−1.72416
$112$	−5.16530	−0.488075
$113$	−13.0656	−1.22911	−0.614556	−	0.788873i	$-0.710665\pi$
$113$	−13.0656	−1.22911	−0.614556	+	0.788873i	$0.710665\pi$
$114$	16.2267	1.51977
$115$	0	0
$116$	6.09127	0.565560
$117$	−15.8606	−1.46631
$118$	1.61779	0.148930
$119$	0	0
$120$	0	0
$121$	−10.9466	−0.995148
$122$	−6.28624	−0.569130
$123$	−25.2811	−2.27952
$124$	0.196234	0.0176223
$125$	0	0
$126$	−19.2670	−1.71644
$127$	−0.745900	−0.0661879	−0.0330939	−	0.999452i	$-0.510536\pi$
$127$	−0.745900	−0.0661879	−0.0330939	+	0.999452i	$0.510536\pi$
$128$	−10.5594	−0.933325
$129$	11.5371	1.01578
$130$	0	0
$131$	−5.67574	−0.495891	−0.247946	−	0.968774i	$-0.579755\pi$
$131$	−5.67574	−0.495891	−0.247946	+	0.968774i	$0.579755\pi$
$132$	−1.08425	−0.0943720
$133$	−30.0150	−2.60263
$134$	−10.2797	−0.888027
$135$	0	0
$136$	0	0
$137$	19.6850	1.68180	0.840900	−	0.541190i	$-0.182026\pi$
$137$	19.6850	1.68180	0.840900	+	0.541190i	$0.182026\pi$
$138$	−4.12427	−0.351081
$139$	5.64706	0.478977	0.239489	−	0.970899i	$-0.423020\pi$
$139$	5.64706	0.478977	0.239489	+	0.970899i	$0.423020\pi$
$140$	0	0
$141$	−4.07569	−0.343235
$142$	−7.74981	−0.650350
$143$	−0.544142	−0.0455034
$144$	8.56005	0.713338
$145$	0	0
$146$	0.331907	0.0274688
$147$	29.6760	2.44763
$148$	8.75802	0.719905
$149$	−19.0000	−1.55654	−0.778269	−	0.627931i	$-0.783902\pi$
$149$	−19.0000	−1.55654	−0.778269	+	0.627931i	$0.783902\pi$
$150$	0	0
$151$	6.95236	0.565775	0.282887	−	0.959153i	$-0.408708\pi$
$151$	6.95236	0.565775	0.282887	+	0.959153i	$0.408708\pi$
$152$	−18.2254	−1.47827
$153$	0	0
$154$	−0.661007	−0.0532655
$155$	0	0
$156$	11.0536	0.884996
$157$	7.18338	0.573296	0.286648	−	0.958036i	$-0.407459\pi$
$157$	7.18338	0.573296	0.286648	+	0.958036i	$0.407459\pi$
$158$	3.03941	0.241802
$159$	−25.7713	−2.04379
$160$	0	0
$161$	7.62878	0.601232
$162$	11.3681	0.893159
$163$	12.3259	0.965443	0.482721	−	0.875774i	$-0.339648\pi$
$163$	12.3259	0.965443	0.482721	+	0.875774i	$0.339648\pi$
$164$	12.1888	0.951788
$165$	0	0
$166$	−10.8600	−0.842902
$167$	2.10123	0.162598	0.0812990	−	0.996690i	$-0.474093\pi$
$167$	2.10123	0.162598	0.0812990	+	0.996690i	$0.474093\pi$
$168$	31.2807	2.41336
$169$	−7.45265	−0.573281
$170$	0	0
$171$	49.7415	3.80383
$172$	−5.56242	−0.424130
$173$	−2.38845	−0.181590	−0.0907952	−	0.995870i	$-0.528941\pi$
$173$	−2.38845	−0.181590	−0.0907952	+	0.995870i	$0.528941\pi$
$174$	−8.89573	−0.674384
$175$	0	0
$176$	0.293677	0.0221367
$177$	7.16852	0.538819
$178$	2.59755	0.194695
$179$	−2.06564	−0.154394	−0.0771968	−	0.997016i	$-0.524597\pi$
$179$	−2.06564	−0.154394	−0.0771968	+	0.997016i	$0.524597\pi$
$180$	0	0
$181$	10.0975	0.750542	0.375271	−	0.926915i	$-0.377550\pi$
$181$	10.0975	0.750542	0.375271	+	0.926915i	$0.377550\pi$
$182$	6.73875	0.499510
$183$	−27.8546	−2.05907
$184$	4.63225	0.341494
$185$	0	0
$186$	−0.286582	−0.0210132
$187$	0	0
$188$	1.96503	0.143314
$189$	−47.3394	−3.44343
$190$	0	0
$191$	−26.5500	−1.92109	−0.960547	−	0.278119i	$-0.910289\pi$
$191$	−26.5500	−1.92109	−0.960547	+	0.278119i	$0.910289\pi$
$192$	4.87485	0.351812
$193$	−6.80795	−0.490047	−0.245023	−	0.969517i	$-0.578796\pi$
$193$	−6.80795	−0.490047	−0.245023	+	0.969517i	$0.578796\pi$
$194$	6.34137	0.455284
$195$	0	0
$196$	−14.3078	−1.02198
$197$	−6.05995	−0.431753	−0.215877	−	0.976421i	$-0.569261\pi$
$197$	−6.05995	−0.431753	−0.215877	+	0.976421i	$0.569261\pi$
$198$	1.09543	0.0778492
$199$	−11.3467	−0.804343	−0.402172	−	0.915564i	$-0.631745\pi$
$199$	−11.3467	−0.804343	−0.402172	+	0.915564i	$0.631745\pi$
$200$	0	0
$201$	−45.5496	−3.21282
$202$	−4.89047	−0.344092
$203$	16.4547	1.15489
$204$	0	0
$205$	0	0
$206$	0.735250	0.0512273
$207$	−12.6426	−0.878720
$208$	−2.99394	−0.207592
$209$	1.70652	0.118043
$210$	0	0
$211$	13.7348	0.945543	0.472772	−	0.881185i	$-0.343254\pi$
$211$	13.7348	0.945543	0.472772	+	0.881185i	$0.343254\pi$
$212$	12.4252	0.853365
$213$	−34.3397	−2.35292
$214$	−7.04302	−0.481451
$215$	0	0
$216$	−28.7449	−1.95584
$217$	0.530099	0.0359854
$218$	5.73287	0.388279
$219$	1.47069	0.0993802
$220$	0	0
$221$	0	0
$222$	−12.7903	−0.858427
$223$	3.57068	0.239110	0.119555	−	0.992828i	$-0.461853\pi$
$223$	3.57068	0.239110	0.119555	+	0.992828i	$0.461853\pi$
$224$	−23.6890	−1.58279
$225$	0	0
$226$	−9.19966	−0.611952
$227$	−23.3475	−1.54963	−0.774816	−	0.632187i	$-0.782157\pi$
$227$	−23.3475	−1.54963	−0.774816	+	0.632187i	$0.782157\pi$
$228$	−34.6660	−2.29581
$229$	25.2530	1.66877	0.834384	−	0.551184i	$-0.185824\pi$
$229$	25.2530	1.66877	0.834384	+	0.551184i	$0.185824\pi$
$230$	0	0
$231$	−2.92895	−0.192711
$232$	9.99142	0.655969
$233$	−1.76097	−0.115365	−0.0576824	−	0.998335i	$-0.518371\pi$
$233$	−1.76097	−0.115365	−0.0576824	+	0.998335i	$0.518371\pi$
$234$	−11.1676	−0.730049
$235$	0	0
$236$	−3.45618	−0.224978
$237$	13.4678	0.874825
$238$	0	0
$239$	21.2916	1.37724	0.688621	−	0.725122i	$-0.258216\pi$
$239$	21.2916	1.37724	0.688621	+	0.725122i	$0.258216\pi$
$240$	0	0
$241$	−10.1369	−0.652972	−0.326486	−	0.945202i	$-0.605865\pi$
$241$	−10.1369	−0.652972	−0.326486	+	0.945202i	$0.605865\pi$
$242$	−7.70763	−0.495465
$243$	15.4223	0.989340
$244$	13.4296	0.859744
$245$	0	0
$246$	−17.8007	−1.13493
$247$	−17.3975	−1.10697
$248$	0.321880	0.0204394
$249$	−48.1213	−3.04956
$250$	0	0
$251$	17.5960	1.11065	0.555326	−	0.831633i	$-0.312593\pi$
$251$	17.5960	1.11065	0.555326	+	0.831633i	$0.312593\pi$
$252$	41.1610	2.59290
$253$	−0.433739	−0.0272689
$254$	−0.525196	−0.0329537
$255$	0	0
$256$	−10.5599	−0.659995
$257$	−10.0995	−0.629990	−0.314995	−	0.949093i	$-0.602003\pi$
$257$	−10.0995	−0.629990	−0.314995	+	0.949093i	$0.602003\pi$
$258$	8.12339	0.505740
$259$	23.6585	1.47007
$260$	0	0
$261$	−27.2691	−1.68791
$262$	−3.99635	−0.246895
$263$	16.0060	0.986973	0.493487	−	0.869753i	$-0.335722\pi$
$263$	16.0060	0.986973	0.493487	+	0.869753i	$0.335722\pi$
$264$	−1.77848	−0.109458
$265$	0	0
$266$	−21.1339	−1.29580
$267$	11.5099	0.704393
$268$	21.9610	1.34148
$269$	−13.0629	−0.796457	−0.398229	−	0.917286i	$-0.630375\pi$
$269$	−13.0629	−0.796457	−0.398229	+	0.917286i	$0.630375\pi$
$270$	0	0
$271$	17.3388	1.05325	0.526627	−	0.850096i	$-0.323456\pi$
$271$	17.3388	1.05325	0.526627	+	0.850096i	$0.323456\pi$
$272$	0	0
$273$	29.8597	1.80719
$274$	13.8604	0.837337
$275$	0	0
$276$	8.81089	0.530353
$277$	10.9525	0.658075	0.329037	−	0.944317i	$-0.393276\pi$
$277$	10.9525	0.658075	0.329037	+	0.944317i	$0.393276\pi$
$278$	3.97615	0.238474
$279$	−0.878491	−0.0525939
$280$	0	0
$281$	−3.09350	−0.184543	−0.0922714	−	0.995734i	$-0.529413\pi$
$281$	−3.09350	−0.184543	−0.0922714	+	0.995734i	$0.529413\pi$
$282$	−2.86974	−0.170890
$283$	0.469462	0.0279066	0.0139533	−	0.999903i	$-0.495558\pi$
$283$	0.469462	0.0279066	0.0139533	+	0.999903i	$0.495558\pi$
$284$	16.5563	0.982437
$285$	0	0
$286$	−0.383136	−0.0226553
$287$	32.9264	1.94358
$288$	39.2579	2.31330
$289$	0	0
$290$	0	0
$291$	28.0989	1.64718
$292$	−0.709070	−0.0414952
$293$	20.6954	1.20904	0.604520	−	0.796590i	$-0.293365\pi$
$293$	20.6954	1.20904	0.604520	+	0.796590i	$0.293365\pi$
$294$	20.8952	1.21863
$295$	0	0
$296$	14.3657	0.834987
$297$	2.69151	0.156177
$298$	−13.3781	−0.774971
$299$	4.42183	0.255721
$300$	0	0
$301$	−15.0261	−0.866088
$302$	4.89523	0.281689
$303$	−21.6699	−1.24490
$304$	9.38951	0.538525
$305$	0	0
$306$	0	0
$307$	−21.9900	−1.25504	−0.627519	−	0.778601i	$-0.715929\pi$
$307$	−21.9900	−1.25504	−0.627519	+	0.778601i	$0.715929\pi$
$308$	1.41214	0.0804644
$309$	3.25792	0.185337
$310$	0	0
$311$	24.4834	1.38833	0.694163	−	0.719817i	$-0.255774\pi$
$311$	24.4834	1.38833	0.694163	+	0.719817i	$0.255774\pi$
$312$	18.1310	1.02647
$313$	−2.87540	−0.162527	−0.0812635	−	0.996693i	$-0.525896\pi$
$313$	−2.87540	−0.162527	−0.0812635	+	0.996693i	$0.525896\pi$
$314$	5.05790	0.285434
$315$	0	0
$316$	−6.49325	−0.365274
$317$	−23.5740	−1.32405	−0.662025	−	0.749482i	$-0.730303\pi$
$317$	−23.5740	−1.32405	−0.662025	+	0.749482i	$0.730303\pi$
$318$	−18.1458	−1.01757
$319$	−0.935542	−0.0523803
$320$	0	0
$321$	−31.2079	−1.74186
$322$	5.37150	0.299342
$323$	0	0
$324$	−24.2862	−1.34923
$325$	0	0
$326$	8.67883	0.480676
$327$	25.4026	1.40477
$328$	19.9932	1.10394
$329$	5.30824	0.292653
$330$	0	0
$331$	25.0816	1.37861	0.689306	−	0.724471i	$-0.257916\pi$
$331$	25.0816	1.37861	0.689306	+	0.724471i	$0.257916\pi$
$332$	23.2009	1.27331
$333$	−39.2074	−2.14855
$334$	1.47950	0.0809545
$335$	0	0
$336$	−16.1155	−0.879171
$337$	−16.4774	−0.897581	−0.448790	−	0.893637i	$-0.648145\pi$
$337$	−16.4774	−0.897581	−0.448790	+	0.893637i	$0.648145\pi$
$338$	−5.24749	−0.285426
$339$	−40.7641	−2.21400
$340$	0	0
$341$	−0.0301391	−0.00163212
$342$	35.0235	1.89386
$343$	−10.2062	−0.551082
$344$	−9.12395	−0.491930
$345$	0	0
$346$	−1.68173	−0.0904105
$347$	−17.0278	−0.914100	−0.457050	−	0.889441i	$-0.651094\pi$
$347$	−17.0278	−0.914100	−0.457050	+	0.889441i	$0.651094\pi$
$348$	19.0044	1.01874
$349$	−6.72216	−0.359829	−0.179915	−	0.983682i	$-0.557582\pi$
$349$	−6.72216	−0.359829	−0.179915	+	0.983682i	$0.557582\pi$
$350$	0	0
$351$	−27.4391	−1.46459
$352$	1.34685	0.0717875
$353$	−2.22223	−0.118277	−0.0591386	−	0.998250i	$-0.518835\pi$
$353$	−2.22223	−0.118277	−0.0591386	+	0.998250i	$0.518835\pi$
$354$	5.04743	0.268268
$355$	0	0
$356$	−5.54929	−0.294112
$357$	0	0
$358$	−1.45444	−0.0768696
$359$	16.7305	0.883000	0.441500	−	0.897261i	$-0.354447\pi$
$359$	16.7305	0.883000	0.441500	+	0.897261i	$0.354447\pi$
$360$	0	0
$361$	35.5614	1.87165
$362$	7.10977	0.373681
$363$	−34.1529	−1.79256
$364$	−14.3964	−0.754574
$365$	0	0
$366$	−19.6127	−1.02517
$367$	−17.8974	−0.934237	−0.467119	−	0.884195i	$-0.654708\pi$
$367$	−17.8974	−0.934237	−0.467119	+	0.884195i	$0.654708\pi$
$368$	−2.38649	−0.124404
$369$	−54.5664	−2.84061
$370$	0	0
$371$	33.5648	1.74260
$372$	0.612240	0.0317431
$373$	−16.3386	−0.845983	−0.422991	−	0.906134i	$-0.639020\pi$
$373$	−16.3386	−0.845983	−0.422991	+	0.906134i	$0.639020\pi$
$374$	0	0
$375$	0	0
$376$	3.22320	0.166224
$377$	9.53755	0.491209
$378$	−33.3322	−1.71442
$379$	14.5126	0.745464	0.372732	−	0.927939i	$-0.378421\pi$
$379$	14.5126	0.745464	0.372732	+	0.927939i	$0.378421\pi$
$380$	0	0
$381$	−2.32717	−0.119224
$382$	−18.6942	−0.956476
$383$	1.18611	0.0606074	0.0303037	−	0.999541i	$-0.490353\pi$
$383$	1.18611	0.0606074	0.0303037	+	0.999541i	$0.490353\pi$
$384$	−32.9446	−1.68120
$385$	0	0
$386$	−4.79355	−0.243985
$387$	24.9015	1.26582
$388$	−13.5474	−0.687765
$389$	−23.1832	−1.17543	−0.587717	−	0.809066i	$-0.699973\pi$
$389$	−23.1832	−1.17543	−0.587717	+	0.809066i	$0.699973\pi$
$390$	0	0
$391$	0	0
$392$	−23.4688	−1.18535
$393$	−17.7080	−0.893249
$394$	−4.26687	−0.214962
$395$	0	0
$396$	−2.34023	−0.117601
$397$	35.2709	1.77019	0.885097	−	0.465406i	$-0.154092\pi$
$397$	35.2709	1.77019	0.885097	+	0.465406i	$0.154092\pi$
$398$	−7.98930	−0.400468
$399$	−93.6452	−4.68812
$400$	0	0
$401$	33.6355	1.67968	0.839838	−	0.542837i	$-0.182650\pi$
$401$	33.6355	1.67968	0.839838	+	0.542837i	$0.182650\pi$
$402$	−32.0719	−1.59960
$403$	0.307258	0.0153056
$404$	10.4478	0.519796
$405$	0	0
$406$	11.5859	0.575000
$407$	−1.34512	−0.0666752
$408$	0	0
$409$	18.0019	0.890135	0.445068	−	0.895497i	$-0.353180\pi$
$409$	18.0019	0.890135	0.445068	+	0.895497i	$0.353180\pi$
$410$	0	0
$411$	61.4160	3.02943
$412$	−1.57075	−0.0773854
$413$	−9.33638	−0.459413
$414$	−8.90177	−0.437498
$415$	0	0
$416$	−13.7307	−0.673205
$417$	17.6185	0.862782
$418$	1.20158	0.0587712
$419$	6.34658	0.310051	0.155025	−	0.987910i	$-0.450454\pi$
$419$	6.34658	0.310051	0.155025	+	0.987910i	$0.450454\pi$
$420$	0	0
$421$	−22.0829	−1.07625	−0.538127	−	0.842864i	$-0.680868\pi$
$421$	−22.0829	−1.07625	−0.538127	+	0.842864i	$0.680868\pi$
$422$	9.67082	0.470768
$423$	−8.79692	−0.427721
$424$	20.3808	0.989781
$425$	0	0
$426$	−24.1790	−1.17148
$427$	36.2782	1.75563
$428$	15.0464	0.727294
$429$	−1.69769	−0.0819654
$430$	0	0
$431$	−32.7753	−1.57873	−0.789365	−	0.613924i	$-0.789590\pi$
$431$	−32.7753	−1.57873	−0.789365	+	0.613924i	$0.789590\pi$
$432$	14.8090	0.712500
$433$	−2.69414	−0.129472	−0.0647361	−	0.997902i	$-0.520621\pi$
$433$	−2.69414	−0.129472	−0.0647361	+	0.997902i	$0.520621\pi$
$434$	0.373248	0.0179165
$435$	0	0
$436$	−12.2474	−0.586546
$437$	−13.8676	−0.663378
$438$	1.03553	0.0494796
$439$	−24.3493	−1.16213	−0.581063	−	0.813858i	$-0.697363\pi$
$439$	−24.3493	−1.16213	−0.581063	+	0.813858i	$0.697363\pi$
$440$	0	0
$441$	64.0522	3.05011
$442$	0	0
$443$	31.9645	1.51868	0.759341	−	0.650693i	$-0.225522\pi$
$443$	31.9645	1.51868	0.759341	+	0.650693i	$0.225522\pi$
$444$	27.3245	1.29676
$445$	0	0
$446$	2.51415	0.119048
$447$	−59.2788	−2.80379
$448$	−6.34907	−0.299965
$449$	−35.8993	−1.69419	−0.847097	−	0.531438i	$-0.821652\pi$
$449$	−35.8993	−1.69419	−0.847097	+	0.531438i	$0.821652\pi$
$450$	0	0
$451$	−1.87205	−0.0881515
$452$	19.6537	0.924433
$453$	21.6910	1.01913
$454$	−16.4393	−0.771532
$455$	0	0
$456$	−56.8621	−2.66281
$457$	−42.0589	−1.96743	−0.983717	−	0.179724i	$-0.942480\pi$
$457$	−42.0589	−1.96743	−0.983717	+	0.179724i	$0.942480\pi$
$458$	17.7809	0.830848
$459$	0	0
$460$	0	0
$461$	−24.5172	−1.14188	−0.570940	−	0.820992i	$-0.693421\pi$
$461$	−24.5172	−1.14188	−0.570940	+	0.820992i	$0.693421\pi$
$462$	−2.06230	−0.0959471
$463$	−12.9662	−0.602590	−0.301295	−	0.953531i	$-0.597419\pi$
$463$	−12.9662	−0.602590	−0.301295	+	0.953531i	$0.597419\pi$
$464$	−5.14747	−0.238966
$465$	0	0
$466$	−1.23992	−0.0574380
$467$	32.3125	1.49524	0.747622	−	0.664124i	$-0.231195\pi$
$467$	32.3125	1.49524	0.747622	+	0.664124i	$0.231195\pi$
$468$	23.8579	1.10283
$469$	59.3244	2.73935
$470$	0	0
$471$	22.4118	1.03268
$472$	−5.66912	−0.260942
$473$	0.854317	0.0392815
$474$	9.48279	0.435559
$475$	0	0
$476$	0	0
$477$	−55.6244	−2.54687
$478$	14.9917	0.685703
$479$	−14.4494	−0.660209	−0.330104	−	0.943944i	$-0.607084\pi$
$479$	−14.4494	−0.660209	−0.330104	+	0.943944i	$0.607084\pi$
$480$	0	0
$481$	13.7131	0.625263
$482$	−7.13747	−0.325103
$483$	23.8014	1.08300
$484$	16.4662	0.748465
$485$	0	0
$486$	10.8590	0.492574
$487$	−35.3420	−1.60150	−0.800750	−	0.598998i	$-0.795566\pi$
$487$	−35.3420	−1.60150	−0.800750	+	0.598998i	$0.795566\pi$
$488$	22.0284	0.997181
$489$	38.4563	1.73905
$490$	0	0
$491$	38.2821	1.72765	0.863825	−	0.503793i	$-0.168063\pi$
$491$	38.2821	1.72765	0.863825	+	0.503793i	$0.168063\pi$
$492$	38.0285	1.71446
$493$	0	0
$494$	−12.2497	−0.551141
$495$	0	0
$496$	−0.165829	−0.00744595
$497$	44.7246	2.00617
$498$	−33.8827	−1.51832
$499$	25.5250	1.14266	0.571329	−	0.820721i	$-0.306428\pi$
$499$	25.5250	1.14266	0.571329	+	0.820721i	$0.306428\pi$
$500$	0	0
$501$	6.55572	0.292888
$502$	12.3896	0.552973
$503$	37.8180	1.68622	0.843110	−	0.537742i	$-0.180722\pi$
$503$	37.8180	1.68622	0.843110	+	0.537742i	$0.180722\pi$
$504$	67.5158	3.00739
$505$	0	0
$506$	−0.305400	−0.0135767
$507$	−23.2518	−1.03265
$508$	1.12200	0.0497809
$509$	16.2321	0.719476	0.359738	−	0.933053i	$-0.382866\pi$
$509$	16.2321	0.719476	0.359738	+	0.933053i	$0.382866\pi$
$510$	0	0
$511$	−1.91545	−0.0847346
$512$	13.6834	0.604726
$513$	86.0537	3.79937
$514$	−7.11117	−0.313660
$515$	0	0
$516$	−17.3544	−0.763986
$517$	−0.301803	−0.0132733
$518$	16.6582	0.731921
$519$	−7.45183	−0.327099
$520$	0	0
$521$	−17.6650	−0.773916	−0.386958	−	0.922097i	$-0.626474\pi$
$521$	−17.6650	−0.773916	−0.386958	+	0.922097i	$0.626474\pi$
$522$	−19.2004	−0.840381
$523$	−9.51295	−0.415972	−0.207986	−	0.978132i	$-0.566691\pi$
$523$	−9.51295	−0.415972	−0.207986	+	0.978132i	$0.566691\pi$
$524$	8.53760	0.372967
$525$	0	0
$526$	11.2700	0.491396
$527$	0	0
$528$	0.916255	0.0398749
$529$	−19.4753	−0.846753
$530$	0	0
$531$	15.4724	0.671447
$532$	45.1494	1.95748
$533$	19.0850	0.826662
$534$	8.10422	0.350704
$535$	0	0
$536$	36.0222	1.55592
$537$	−6.44469	−0.278109
$538$	−9.19770	−0.396541
$539$	2.19749	0.0946527
$540$	0	0
$541$	12.2201	0.525382	0.262691	−	0.964880i	$-0.415390\pi$
$541$	12.2201	0.525382	0.262691	+	0.964880i	$0.415390\pi$
$542$	12.2084	0.524396
$543$	31.5037	1.35195
$544$	0	0
$545$	0	0
$546$	21.0245	0.899767
$547$	14.8127	0.633345	0.316672	−	0.948535i	$-0.397434\pi$
$547$	14.8127	0.633345	0.316672	+	0.948535i	$0.397434\pi$
$548$	−29.6107	−1.26491
$549$	−60.1211	−2.56591
$550$	0	0
$551$	−29.9114	−1.27427
$552$	14.4524	0.615134
$553$	−17.5406	−0.745902
$554$	7.71180	0.327643
$555$	0	0
$556$	−8.49447	−0.360246
$557$	−18.6501	−0.790230	−0.395115	−	0.918632i	$-0.629295\pi$
$557$	−18.6501	−0.790230	−0.395115	+	0.918632i	$0.629295\pi$
$558$	−0.618555	−0.0261855
$559$	−8.70949	−0.368372
$560$	0	0
$561$	0	0
$562$	−2.17817	−0.0918804
$563$	12.5501	0.528923	0.264461	−	0.964396i	$-0.414806\pi$
$563$	12.5501	0.528923	0.264461	+	0.964396i	$0.414806\pi$
$564$	6.13077	0.258152
$565$	0	0
$566$	0.330553	0.0138942
$567$	−65.6057	−2.75518
$568$	27.1571	1.13949
$569$	13.8442	0.580377	0.290189	−	0.956969i	$-0.406282\pi$
$569$	13.8442	0.580377	0.290189	+	0.956969i	$0.406282\pi$
$570$	0	0
$571$	−45.4021	−1.90002	−0.950009	−	0.312224i	$-0.898926\pi$
$571$	−45.4021	−1.90002	−0.950009	+	0.312224i	$0.898926\pi$
$572$	0.818514	0.0342238
$573$	−82.8346	−3.46047
$574$	23.1838	0.967675
$575$	0	0
$576$	10.5218	0.438409
$577$	18.7645	0.781177	0.390588	−	0.920565i	$-0.372272\pi$
$577$	18.7645	0.781177	0.390588	+	0.920565i	$0.372272\pi$
$578$	0	0
$579$	−21.2404	−0.882722
$580$	0	0
$581$	62.6738	2.60015
$582$	19.7847	0.820103
$583$	−1.90835	−0.0790358
$584$	−1.16308	−0.0481285
$585$	0	0
$586$	14.5719	0.601959
$587$	26.4900	1.09336	0.546679	−	0.837342i	$-0.315892\pi$
$587$	26.4900	1.09336	0.546679	+	0.837342i	$0.315892\pi$
$588$	−44.6394	−1.84090
$589$	−0.963615	−0.0397051
$590$	0	0
$591$	−18.9067	−0.777717
$592$	−7.40103	−0.304181
$593$	33.8360	1.38948	0.694740	−	0.719261i	$-0.255520\pi$
$593$	33.8360	1.38948	0.694740	+	0.719261i	$0.255520\pi$
$594$	1.89512	0.0777578
$595$	0	0
$596$	28.5803	1.17069
$597$	−35.4009	−1.44886
$598$	3.11346	0.127319
$599$	10.9189	0.446133	0.223067	−	0.974803i	$-0.428393\pi$
$599$	10.9189	0.446133	0.223067	+	0.974803i	$0.428393\pi$
$600$	0	0
$601$	−13.7204	−0.559667	−0.279833	−	0.960049i	$-0.590279\pi$
$601$	−13.7204	−0.559667	−0.279833	+	0.960049i	$0.590279\pi$
$602$	−10.5800	−0.431209
$603$	−98.3137	−4.00364
$604$	−10.4579	−0.425527
$605$	0	0
$606$	−15.2580	−0.619813
$607$	26.5777	1.07876	0.539379	−	0.842063i	$-0.318659\pi$
$607$	26.5777	1.07876	0.539379	+	0.842063i	$0.318659\pi$
$608$	43.0620	1.74639
$609$	51.3377	2.08031
$610$	0	0
$611$	3.07679	0.124473
$612$	0	0
$613$	−11.9043	−0.480812	−0.240406	−	0.970672i	$-0.577281\pi$
$613$	−11.9043	−0.480812	−0.240406	+	0.970672i	$0.577281\pi$
$614$	−15.4834	−0.624860
$615$	0	0
$616$	2.31632	0.0933272
$617$	7.85884	0.316385	0.158193	−	0.987408i	$-0.449433\pi$
$617$	7.85884	0.316385	0.158193	+	0.987408i	$0.449433\pi$
$618$	2.29394	0.0922757
$619$	32.7760	1.31738	0.658689	−	0.752416i	$-0.271111\pi$
$619$	32.7760	1.31738	0.658689	+	0.752416i	$0.271111\pi$
$620$	0	0
$621$	−21.8719	−0.877688
$622$	17.2390	0.691222
$623$	−14.9906	−0.600587
$624$	−9.34092	−0.373936
$625$	0	0
$626$	−2.02460	−0.0809192
$627$	5.32426	0.212630
$628$	−10.8054	−0.431184
$629$	0	0
$630$	0	0
$631$	−12.5294	−0.498789	−0.249395	−	0.968402i	$-0.580232\pi$
$631$	−12.5294	−0.498789	−0.249395	+	0.968402i	$0.580232\pi$
$632$	−10.6508	−0.423666
$633$	42.8518	1.70321
$634$	−16.5987	−0.659220
$635$	0	0
$636$	38.7659	1.53717
$637$	−22.4027	−0.887628
$638$	−0.658725	−0.0260792
$639$	−74.1185	−2.93208
$640$	0	0
$641$	19.4356	0.767660	0.383830	−	0.923404i	$-0.374605\pi$
$641$	19.4356	0.767660	0.383830	+	0.923404i	$0.374605\pi$
$642$	−21.9738	−0.867238
$643$	16.0075	0.631276	0.315638	−	0.948880i	$-0.397782\pi$
$643$	16.0075	0.631276	0.315638	+	0.948880i	$0.397782\pi$
$644$	−11.4754	−0.452195
$645$	0	0
$646$	0	0
$647$	41.3567	1.62590	0.812949	−	0.582335i	$-0.197861\pi$
$647$	41.3567	1.62590	0.812949	+	0.582335i	$0.197861\pi$
$648$	−39.8363	−1.56492
$649$	0.530826	0.0208367
$650$	0	0
$651$	1.65388	0.0648206
$652$	−18.5410	−0.726123
$653$	5.21693	0.204154	0.102077	−	0.994776i	$-0.467451\pi$
$653$	5.21693	0.204154	0.102077	+	0.994776i	$0.467451\pi$
$654$	17.8862	0.699407
$655$	0	0
$656$	−10.3003	−0.402158
$657$	3.17433	0.123842
$658$	3.73759	0.145706
$659$	−10.7781	−0.419854	−0.209927	−	0.977717i	$-0.567323\pi$
$659$	−10.7781	−0.419854	−0.209927	+	0.977717i	$0.567323\pi$
$660$	0	0
$661$	−10.3851	−0.403935	−0.201968	−	0.979392i	$-0.564734\pi$
$661$	−10.3851	−0.403935	−0.201968	+	0.979392i	$0.564734\pi$
$662$	17.6602	0.686385
$663$	0	0
$664$	38.0560	1.47686
$665$	0	0
$666$	−27.6064	−1.06973
$667$	7.60245	0.294368
$668$	−3.16073	−0.122292
$669$	11.1403	0.430709
$670$	0	0
$671$	−2.06262	−0.0796267
$672$	−73.9084	−2.85108
$673$	13.1346	0.506302	0.253151	−	0.967427i	$-0.418533\pi$
$673$	13.1346	0.506302	0.253151	+	0.967427i	$0.418533\pi$
$674$	−11.6019	−0.446889
$675$	0	0
$676$	11.2105	0.431172
$677$	18.9820	0.729539	0.364769	−	0.931098i	$-0.381148\pi$
$677$	18.9820	0.729539	0.364769	+	0.931098i	$0.381148\pi$
$678$	−28.7024	−1.10231
$679$	−36.5964	−1.40444
$680$	0	0
$681$	−72.8430	−2.79135
$682$	−0.0212213	−0.000812604 0
$683$	−14.9579	−0.572347	−0.286173	−	0.958178i	$-0.592383\pi$
$683$	−14.9579	−0.572347	−0.286173	+	0.958178i	$0.592383\pi$
$684$	−74.8226	−2.86091
$685$	0	0
$686$	−7.18628	−0.274373
$687$	78.7880	3.00595
$688$	4.70056	0.179207
$689$	19.4550	0.741177
$690$	0	0
$691$	−11.3095	−0.430233	−0.215116	−	0.976588i	$-0.569013\pi$
$691$	−11.3095	−0.430233	−0.215116	+	0.976588i	$0.569013\pi$
$692$	3.59277	0.136577
$693$	−6.32181	−0.240146
$694$	−11.9895	−0.455113
$695$	0	0
$696$	31.1727	1.18160
$697$	0	0
$698$	−4.73315	−0.179152
$699$	−5.49412	−0.207807
$700$	0	0
$701$	−13.0707	−0.493674	−0.246837	−	0.969057i	$-0.579391\pi$
$701$	−13.0707	−0.493674	−0.246837	+	0.969057i	$0.579391\pi$
$702$	−19.3202	−0.729192
$703$	−43.0066	−1.62202
$704$	0.360980	0.0136050
$705$	0	0
$706$	−1.56469	−0.0588880
$707$	28.2232	1.06144
$708$	−10.7831	−0.405253
$709$	48.6798	1.82821	0.914105	−	0.405479i	$-0.132895\pi$
$709$	48.6798	1.82821	0.914105	+	0.405479i	$0.132895\pi$
$710$	0	0
$711$	29.0686	1.09016
$712$	−9.10242	−0.341128
$713$	0.244918	0.00917224
$714$	0	0
$715$	0	0
$716$	3.10720	0.116122
$717$	66.4287	2.48083
$718$	11.7801	0.439629
$719$	31.8171	1.18658	0.593289	−	0.804990i	$-0.297829\pi$
$719$	31.8171	1.18658	0.593289	+	0.804990i	$0.297829\pi$
$720$	0	0
$721$	−4.24316	−0.158024
$722$	25.0392	0.931861
$723$	−31.6264	−1.17620
$724$	−15.1890	−0.564493
$725$	0	0
$726$	−24.0474	−0.892482
$727$	−6.69618	−0.248348	−0.124174	−	0.992260i	$-0.539628\pi$
$727$	−6.69618	−0.248348	−0.124174	+	0.992260i	$0.539628\pi$
$728$	−23.6141	−0.875198
$729$	−0.319170	−0.0118211
$730$	0	0
$731$	0	0
$732$	41.8997	1.54866
$733$	38.7110	1.42982	0.714912	−	0.699215i	$-0.246467\pi$
$733$	38.7110	1.42982	0.714912	+	0.699215i	$0.246467\pi$
$734$	−12.6018	−0.465139
$735$	0	0
$736$	−10.9449	−0.403433
$737$	−3.37293	−0.124243
$738$	−38.4207	−1.41429
$739$	19.1997	0.706271	0.353136	−	0.935572i	$-0.385115\pi$
$739$	19.1997	0.706271	0.353136	+	0.935572i	$0.385115\pi$
$740$	0	0
$741$	−54.2791	−1.99399
$742$	23.6334	0.867608
$743$	−2.12614	−0.0780004	−0.0390002	−	0.999239i	$-0.512417\pi$
$743$	−2.12614	−0.0780004	−0.0390002	+	0.999239i	$0.512417\pi$
$744$	1.00425	0.0368175
$745$	0	0
$746$	−11.5042	−0.421199
$747$	−103.864	−3.80020
$748$	0	0
$749$	40.6456	1.48516
$750$	0	0
$751$	−9.76463	−0.356316	−0.178158	−	0.984002i	$-0.557014\pi$
$751$	−9.76463	−0.356316	−0.178158	+	0.984002i	$0.557014\pi$
$752$	−1.66056	−0.0605544
$753$	54.8987	2.00062
$754$	6.71549	0.244564
$755$	0	0
$756$	71.2093	2.58986
$757$	29.2324	1.06247	0.531236	−	0.847224i	$-0.321728\pi$
$757$	29.2324	1.06247	0.531236	+	0.847224i	$0.321728\pi$
$758$	10.2185	0.371153
$759$	−1.35324	−0.0491196
$760$	0	0
$761$	−11.3975	−0.413161	−0.206580	−	0.978430i	$-0.566234\pi$
$761$	−11.3975	−0.413161	−0.206580	+	0.978430i	$0.566234\pi$
$762$	−1.63858	−0.0593595
$763$	−33.0847	−1.19775
$764$	39.9373	1.44488
$765$	0	0
$766$	0.835153	0.0301753
$767$	−5.41160	−0.195401
$768$	−32.9464	−1.18885
$769$	14.6414	0.527984	0.263992	−	0.964525i	$-0.414961\pi$
$769$	14.6414	0.527984	0.263992	+	0.964525i	$0.414961\pi$
$770$	0	0
$771$	−31.5099	−1.13480
$772$	10.2407	0.368571
$773$	−16.8387	−0.605646	−0.302823	−	0.953047i	$-0.597929\pi$
$773$	−16.8387	−0.605646	−0.302823	+	0.953047i	$0.597929\pi$
$774$	17.5334	0.630226
$775$	0	0
$776$	−22.2216	−0.797709
$777$	73.8133	2.64804
$778$	−16.3235	−0.585227
$779$	−59.8538	−2.14448
$780$	0	0
$781$	−2.54284	−0.0909901
$782$	0	0
$783$	−47.1760	−1.68593
$784$	12.0909	0.431817
$785$	0	0
$786$	−12.4684	−0.444732
$787$	17.4487	0.621979	0.310990	−	0.950413i	$-0.399340\pi$
$787$	17.4487	0.621979	0.310990	+	0.950413i	$0.399340\pi$
$788$	9.11554	0.324728
$789$	49.9379	1.77784
$790$	0	0
$791$	53.0917	1.88772
$792$	−3.83865	−0.136401
$793$	21.0278	0.746718
$794$	24.8346	0.881347
$795$	0	0
$796$	17.0680	0.604958
$797$	−25.0208	−0.886282	−0.443141	−	0.896452i	$-0.646136\pi$
$797$	−25.0208	−0.886282	−0.443141	+	0.896452i	$0.646136\pi$
$798$	−65.9366	−2.33413
$799$	0	0
$800$	0	0
$801$	24.8428	0.877776
$802$	23.6831	0.836279
$803$	0.108904	0.00384315
$804$	68.5170	2.41641
$805$	0	0
$806$	0.216344	0.00762039
$807$	−40.7554	−1.43466
$808$	17.1373	0.602889
$809$	19.6531	0.690966	0.345483	−	0.938425i	$-0.387715\pi$
$809$	19.6531	0.690966	0.345483	+	0.938425i	$0.387715\pi$
$810$	0	0
$811$	−30.7782	−1.08077	−0.540385	−	0.841418i	$-0.681721\pi$
$811$	−30.7782	−1.08077	−0.540385	+	0.841418i	$0.681721\pi$
$812$	−24.7516	−0.868612
$813$	54.0960	1.89723
$814$	−0.947114	−0.0331963
$815$	0	0
$816$	0	0
$817$	27.3144	0.955612
$818$	12.6753	0.443182
$819$	64.4488	2.25202
$820$	0	0
$821$	−35.0315	−1.22261	−0.611303	−	0.791397i	$-0.709354\pi$
$821$	−35.0315	−1.22261	−0.611303	+	0.791397i	$0.709354\pi$
$822$	43.2437	1.50830
$823$	−30.1072	−1.04947	−0.524736	−	0.851265i	$-0.675836\pi$
$823$	−30.1072	−1.04947	−0.524736	+	0.851265i	$0.675836\pi$
$824$	−2.57648	−0.0897560
$825$	0	0
$826$	−6.57384	−0.228733
$827$	20.7366	0.721082	0.360541	−	0.932743i	$-0.382592\pi$
$827$	20.7366	0.721082	0.360541	+	0.932743i	$0.382592\pi$
$828$	19.0173	0.660898
$829$	54.3221	1.88669	0.943343	−	0.331820i	$-0.107663\pi$
$829$	54.3221	1.88669	0.943343	+	0.331820i	$0.107663\pi$
$830$	0	0
$831$	34.1713	1.18539
$832$	−3.68008	−0.127584
$833$	0	0
$834$	12.4054	0.429563
$835$	0	0
$836$	−2.56700	−0.0887816
$837$	−1.51980	−0.0525321
$838$	4.46870	0.154369
$839$	−10.7461	−0.370997	−0.185498	−	0.982645i	$-0.559390\pi$
$839$	−10.7461	−0.370997	−0.185498	+	0.982645i	$0.559390\pi$
$840$	0	0
$841$	−12.6021	−0.434555
$842$	−15.5488	−0.535847
$843$	−9.65155	−0.332417
$844$	−20.6603	−0.711156
$845$	0	0
$846$	−6.19401	−0.212954
$847$	44.4812	1.52839
$848$	−10.5000	−0.360571
$849$	1.46469	0.0502682
$850$	0	0
$851$	10.9308	0.374703
$852$	51.6548	1.76966
$853$	−43.7616	−1.49837	−0.749184	−	0.662362i	$-0.769554\pi$
$853$	−43.7616	−1.49837	−0.749184	+	0.662362i	$0.769554\pi$
$854$	25.5439	0.874094
$855$	0	0
$856$	24.6803	0.843557
$857$	0.857197	0.0292813	0.0146406	−	0.999893i	$-0.495340\pi$
$857$	0.857197	0.0292813	0.0146406	+	0.999893i	$0.495340\pi$
$858$	−1.19536	−0.0408090
$859$	−38.5994	−1.31700	−0.658498	−	0.752583i	$-0.728808\pi$
$859$	−38.5994	−1.31700	−0.658498	+	0.752583i	$0.728808\pi$
$860$	0	0
$861$	102.729	3.50098
$862$	−23.0774	−0.786021
$863$	−18.0234	−0.613522	−0.306761	−	0.951787i	$-0.599245\pi$
$863$	−18.0234	−0.613522	−0.306761	+	0.951787i	$0.599245\pi$
$864$	67.9169	2.31058
$865$	0	0
$866$	−1.89697	−0.0644618
$867$	0	0
$868$	−0.797389	−0.0270652
$869$	0.997281	0.0338305
$870$	0	0
$871$	34.3859	1.16512
$872$	−20.0893	−0.680309
$873$	60.6483	2.05263
$874$	−9.76434	−0.330284
$875$	0	0
$876$	−2.21226	−0.0747453
$877$	−26.7943	−0.904780	−0.452390	−	0.891820i	$-0.649428\pi$
$877$	−26.7943	−0.904780	−0.452390	+	0.891820i	$0.649428\pi$
$878$	−17.1446	−0.578601
$879$	64.5686	2.17785
$880$	0	0
$881$	−21.6063	−0.727933	−0.363967	−	0.931412i	$-0.618578\pi$
$881$	−21.6063	−0.727933	−0.363967	+	0.931412i	$0.618578\pi$
$882$	45.0999	1.51859
$883$	−20.8987	−0.703297	−0.351649	−	0.936132i	$-0.614379\pi$
$883$	−20.8987	−0.703297	−0.351649	+	0.936132i	$0.614379\pi$
$884$	0	0
$885$	0	0
$886$	22.5066	0.756123
$887$	−33.8887	−1.13787	−0.568936	−	0.822382i	$-0.692645\pi$
$887$	−33.8887	−1.13787	−0.568936	+	0.822382i	$0.692645\pi$
$888$	44.8200	1.50406
$889$	3.03093	0.101654
$890$	0	0
$891$	3.73005	0.124961
$892$	−5.37111	−0.179838
$893$	−9.64933	−0.322903
$894$	−41.7388	−1.39596
$895$	0	0
$896$	42.9076	1.43344
$897$	13.7959	0.460631
$898$	−25.2771	−0.843508
$899$	0.528269	0.0176187
$900$	0	0
$901$	0	0
$902$	−1.31813	−0.0438890
$903$	−46.8805	−1.56009
$904$	32.2377	1.07221
$905$	0	0
$906$	15.2728	0.507406
$907$	−17.3527	−0.576187	−0.288093	−	0.957602i	$-0.593021\pi$
$907$	−17.3527	−0.576187	−0.288093	+	0.957602i	$0.593021\pi$
$908$	35.1200	1.16550
$909$	−46.7720	−1.55133
$910$	0	0
$911$	44.2282	1.46535	0.732673	−	0.680581i	$-0.238273\pi$
$911$	44.2282	1.46535	0.732673	+	0.680581i	$0.238273\pi$
$912$	29.2947	0.970046
$913$	−3.56336	−0.117930
$914$	−29.6141	−0.979549
$915$	0	0
$916$	−37.9863	−1.25510
$917$	23.0631	0.761611
$918$	0	0
$919$	4.31977	0.142496	0.0712480	−	0.997459i	$-0.477302\pi$
$919$	4.31977	0.142496	0.0712480	+	0.997459i	$0.477302\pi$
$920$	0	0
$921$	−68.6077	−2.26070
$922$	−17.2628	−0.568521
$923$	25.9235	0.853282
$924$	4.40581	0.144941
$925$	0	0
$926$	−9.12963	−0.300018
$927$	7.03186	0.230957
$928$	−23.6072	−0.774945
$929$	9.99294	0.327858	0.163929	−	0.986472i	$-0.447583\pi$
$929$	9.99294	0.327858	0.163929	+	0.986472i	$0.447583\pi$
$930$	0	0
$931$	70.2588	2.30264
$932$	2.64890	0.0867676
$933$	76.3869	2.50079
$934$	22.7516	0.744454
$935$	0	0
$936$	39.1338	1.27913
$937$	−41.6190	−1.35964	−0.679818	−	0.733381i	$-0.737941\pi$
$937$	−41.6190	−1.35964	−0.679818	+	0.733381i	$0.737941\pi$
$938$	41.7709	1.36387
$939$	−8.97108	−0.292760
$940$	0	0
$941$	22.4168	0.730767	0.365384	−	0.930857i	$-0.380938\pi$
$941$	22.4168	0.730767	0.365384	+	0.930857i	$0.380938\pi$
$942$	15.7804	0.514152
$943$	15.2128	0.495395
$944$	2.92067	0.0950597
$945$	0	0
$946$	0.601533	0.0195575
$947$	−44.0389	−1.43107	−0.715535	−	0.698577i	$-0.753817\pi$
$947$	−44.0389	−1.43107	−0.715535	+	0.698577i	$0.753817\pi$
$948$	−20.2586	−0.657968
$949$	−1.11024	−0.0360400
$950$	0	0
$951$	−73.5497	−2.38501
$952$	0	0
$953$	43.8375	1.42004	0.710019	−	0.704183i	$-0.248686\pi$
$953$	43.8375	1.42004	0.710019	+	0.704183i	$0.248686\pi$
$954$	−39.1657	−1.26804
$955$	0	0
$956$	−32.0275	−1.03584
$957$	−2.91884	−0.0943527
$958$	−10.1740	−0.328706
$959$	−79.9891	−2.58298
$960$	0	0
$961$	−30.9830	−0.999451
$962$	9.65553	0.311307
$963$	−67.3588	−2.17061
$964$	15.2481	0.491110
$965$	0	0
$966$	16.7588	0.539205
$967$	9.41429	0.302743	0.151372	−	0.988477i	$-0.451631\pi$
$967$	9.41429	0.302743	0.151372	+	0.988477i	$0.451631\pi$
$968$	27.0093	0.868112
$969$	0	0
$970$	0	0
$971$	−14.2960	−0.458780	−0.229390	−	0.973335i	$-0.573673\pi$
$971$	−14.2960	−0.458780	−0.229390	+	0.973335i	$0.573673\pi$
$972$	−23.1986	−0.744097
$973$	−22.9466	−0.735634
$974$	−24.8847	−0.797357
$975$	0	0
$976$	−11.3488	−0.363267
$977$	2.92222	0.0934900	0.0467450	−	0.998907i	$-0.485115\pi$
$977$	2.92222	0.0934900	0.0467450	+	0.998907i	$0.485115\pi$
$978$	27.0775	0.865842
$979$	0.852301	0.0272397
$980$	0	0
$981$	54.8287	1.75054
$982$	26.9549	0.860164
$983$	−1.35448	−0.0432013	−0.0216007	−	0.999767i	$-0.506876\pi$
$983$	−1.35448	−0.0432013	−0.0216007	+	0.999767i	$0.506876\pi$
$984$	62.3776	1.98853
$985$	0	0
$986$	0	0
$987$	16.5614	0.527155
$988$	26.1697	0.832571
$989$	−6.94239	−0.220755
$990$	0	0
$991$	−49.6083	−1.57586	−0.787930	−	0.615765i	$-0.788847\pi$
$991$	−49.6083	−1.57586	−0.787930	+	0.615765i	$0.788847\pi$
$992$	−0.760522	−0.0241466
$993$	78.2533	2.48329
$994$	31.4910	0.998835
$995$	0	0
$996$	72.3854	2.29362
$997$	−12.3189	−0.390143	−0.195072	−	0.980789i	$-0.562494\pi$
$997$	−12.3189	−0.390143	−0.195072	+	0.980789i	$0.562494\pi$
$998$	17.9724	0.568908
$999$	−67.8296	−2.14603

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bn.1.8		12
5.4	even	2		1445.2.a.s.1.5	yes	12
17.16	even	2		7225.2.a.bo.1.8		12
85.4	even	4		1445.2.d.i.866.15		24
85.64	even	4		1445.2.d.i.866.16		24
85.84	even	2		1445.2.a.r.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1445.2.a.r.1.5	✓	12	85.84	even	2
1445.2.a.s.1.5	yes	12	5.4	even	2
1445.2.d.i.866.15		24	85.4	even	4
1445.2.d.i.866.16		24	85.64	even	4
7225.2.a.bn.1.8		12	1.1	even	1	trivial
7225.2.a.bo.1.8		12	17.16	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q+0.704111 q^{2} +3.11994 q^{3} -1.50423 q^{4} +2.19679 q^{6} -4.06346 q^{7} -2.46736 q^{8} +6.73405 q^{9} +0.231031 q^{11} -4.69311 q^{12} -2.35528 q^{13} -2.86112 q^{14} +1.27116 q^{16} +4.74152 q^{18} +7.38657 q^{19} -12.6778 q^{21} +0.162671 q^{22} -1.87741 q^{23} -7.69804 q^{24} -1.65838 q^{26} +11.6500 q^{27} +6.11237 q^{28} -4.04943 q^{29} -0.130455 q^{31} +5.82977 q^{32} +0.720802 q^{33} -10.1295 q^{36} -5.82227 q^{37} +5.20096 q^{38} -7.34835 q^{39} -8.10305 q^{41} -8.92655 q^{42} +3.69785 q^{43} -0.347523 q^{44} -1.32190 q^{46} -1.30633 q^{47} +3.96595 q^{48} +9.51170 q^{49} +3.54288 q^{52} -8.26017 q^{53} +8.20291 q^{54} +10.0260 q^{56} +23.0457 q^{57} -2.85125 q^{58} +2.29764 q^{59} -8.92792 q^{61} -0.0918548 q^{62} -27.3635 q^{63} +1.56248 q^{64} +0.507524 q^{66} -14.5995 q^{67} -5.85742 q^{69} -11.0065 q^{71} -16.6154 q^{72} +0.471385 q^{73} -4.09952 q^{74} -11.1111 q^{76} -0.938783 q^{77} -5.17405 q^{78} +4.31667 q^{79} +16.1453 q^{81} -5.70544 q^{82} -15.4238 q^{83} +19.0703 q^{84} +2.60370 q^{86} -12.6340 q^{87} -0.570036 q^{88} +3.68913 q^{89} +9.57059 q^{91} +2.82405 q^{92} -0.407012 q^{93} -0.919804 q^{94} +18.1885 q^{96} +9.00621 q^{97} +6.69729 q^{98} +1.55577 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} - 3 q^{3} + 21 q^{4} - 9 q^{6} - 6 q^{7} - 12 q^{8} + 21 q^{9} - 6 q^{11} - 6 q^{12} - 9 q^{13} - 18 q^{14} + 39 q^{16} + 9 q^{18} + 27 q^{19} + 6 q^{21} - 15 q^{22} - 18 q^{23} - 36 q^{24}+ \cdots - 6 q^{99}+O(q^{100}) \) 12 * q - 3 * q^2 - 3 * q^3 + 21 * q^4 - 9 * q^6 - 6 * q^7 - 12 * q^8 + 21 * q^9 - 6 * q^11 - 6 * q^12 - 9 * q^13 - 18 * q^14 + 39 * q^16 + 9 * q^18 + 27 * q^19 + 6 * q^21 - 15 * q^22 - 18 * q^23 - 36 * q^24 + 6 * q^26 - 12 * q^27 - 39 * q^28 - 6 * q^31 - 12 * q^32 + 48 * q^36 - 21 * q^37 + 3 * q^38 - 9 * q^39 - 15 * q^41 - 30 * q^42 - 45 * q^43 - 42 * q^44 - 15 * q^46 - 3 * q^47 + 63 * q^48 + 12 * q^49 - 51 * q^52 + 12 * q^53 - 27 * q^54 + 3 * q^56 - 6 * q^57 + 51 * q^58 + 36 * q^59 + 3 * q^61 - 36 * q^62 - 72 * q^63 + 42 * q^64 - 39 * q^66 - 78 * q^67 - 66 * q^69 - 30 * q^71 + 12 * q^72 + 33 * q^73 + 9 * q^74 + 27 * q^76 + 39 * q^77 + 24 * q^78 - 3 * q^79 + 12 * q^81 + 51 * q^82 - 27 * q^83 + 27 * q^84 + 18 * q^86 - 24 * q^87 - 36 * q^88 + 6 * q^89 - 24 * q^91 + 3 * q^92 + 12 * q^93 + 21 * q^94 - 39 * q^96 + 6 * q^97 + 42 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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