LMFDB - Embedded newform 7225.2.a.bo.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$0$
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	$-0.856909\pi$
$3$	−3.11994	−1.80130	−0.900650	+	0.434545i	$0.856909\pi$
$4$	−1.50423	−0.752114
$5$	0	0
$5$	0	0
$6$	−2.19679	−0.896834
$7$	4.06346	1.53584	0.767922	−	0.640544i	$-0.221291\pi$
$7$	4.06346	1.53584	0.767922	+	0.640544i	$0.221291\pi$
$8$	−2.46736	−0.872345
$9$	6.73405	2.24468
$10$	0	0
$11$	−0.231031	−0.0696583	−0.0348292	−	0.999393i	$-0.511089\pi$
$11$	−0.231031	−0.0696583	−0.0348292	+	0.999393i	$0.511089\pi$
$12$	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	$-0.437291\pi$
$23$	1.87741	0.391467	0.195734	+	0.980657i	$0.437291\pi$
$24$	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	$-0.377306\pi$
$29$	4.04943	0.751961	0.375980	+	0.926628i	$0.377306\pi$
$30$	0	0
$31$	0.130455	0.0234304	0.0117152	−	0.999931i	$-0.496271\pi$
$31$	0.130455	0.0234304	0.0117152	+	0.999931i	$0.496271\pi$
$32$	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	$-0.341149\pi$
$37$	5.82227	0.957175	0.478587	+	0.878040i	$0.341149\pi$
$38$	5.20096	0.843707
$39$	7.34835	1.17668
$40$	0	0
$41$	8.10305	1.26548	0.632742	−	0.774363i	$-0.281929\pi$
$41$	8.10305	1.26548	0.632742	+	0.774363i	$0.281929\pi$
$42$	−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	$-0.306342\pi$
$61$	8.92792	1.14310	0.571552	+	0.820566i	$0.306342\pi$
$62$	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	$-0.273461\pi$
$71$	11.0065	1.30623	0.653117	+	0.757257i	$0.273461\pi$
$72$	−16.6154	−1.95814
$73$	−0.471385	−0.0551714	−0.0275857	−	0.999619i	$-0.508782\pi$
$73$	−0.471385	−0.0551714	−0.0275857	+	0.999619i	$0.508782\pi$
$74$	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	$-0.578076\pi$
$79$	−4.31667	−0.485663	−0.242831	+	0.970069i	$0.578076\pi$
$80$	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	$-0.651155\pi$
$97$	−9.00621	−0.914442	−0.457221	+	0.889353i	$0.651155\pi$
$98$	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	$-0.339365\pi$
$107$	10.0027	0.966999	0.483500	+	0.875345i	$0.339365\pi$
$108$	17.5243	1.68628
$109$	−8.14200	−0.779863	−0.389931	−	0.920844i	$-0.627501\pi$
$109$	−8.14200	−0.779863	−0.389931	+	0.920844i	$0.627501\pi$
$110$	0	0
$111$	−18.1651	−1.72416
$112$	5.16530	0.488075
$113$	13.0656	1.22911	0.614556	−	0.788873i	$-0.289335\pi$
$113$	13.0656	1.22911	0.614556	+	0.788873i	$0.289335\pi$
$114$	−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	$-0.420245\pi$
$131$	5.67574	0.495891	0.247946	+	0.968774i	$0.420245\pi$
$132$	−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	$-0.576980\pi$
$139$	−5.64706	−0.478977	−0.239489	+	0.970899i	$0.576980\pi$
$140$	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	$-0.660352\pi$
$163$	−12.3259	−0.965443	−0.482721	+	0.875774i	$0.660352\pi$
$164$	−12.1888	−0.951788
$165$	0	0
$166$	−10.8600	−0.842902
$167$	−2.10123	−0.162598	−0.0812990	−	0.996690i	$-0.525907\pi$
$167$	−2.10123	−0.162598	−0.0812990	+	0.996690i	$0.525907\pi$
$168$	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	$-0.471059\pi$
$173$	2.38845	0.181590	0.0907952	+	0.995870i	$0.471059\pi$
$174$	−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	$-0.622450\pi$
$181$	−10.0975	−0.750542	−0.375271	+	0.926915i	$0.622450\pi$
$182$	−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	$-0.421204\pi$
$193$	6.80795	0.490047	0.245023	+	0.969517i	$0.421204\pi$
$194$	−6.34137	−0.455284
$195$	0	0
$196$	−14.3078	−1.02198
$197$	6.05995	0.431753	0.215877	−	0.976421i	$-0.430739\pi$
$197$	6.05995	0.431753	0.215877	+	0.976421i	$0.430739\pi$
$198$	−1.09543	−0.0778492
$199$	11.3467	0.804343	0.402172	−	0.915564i	$-0.368255\pi$
$199$	11.3467	0.804343	0.402172	+	0.915564i	$0.368255\pi$
$200$	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	$-0.656746\pi$
$211$	−13.7348	−0.945543	−0.472772	+	0.881185i	$0.656746\pi$
$212$	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	$-0.217843\pi$
$227$	23.3475	1.54963	0.774816	+	0.632187i	$0.217843\pi$
$228$	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	$-0.481629\pi$
$233$	1.76097	0.115365	0.0576824	+	0.998335i	$0.481629\pi$
$234$	−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	$-0.394135\pi$
$241$	10.1369	0.652972	0.326486	+	0.945202i	$0.394135\pi$
$242$	−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	$-0.369625\pi$
$269$	13.0629	0.796457	0.398229	+	0.917286i	$0.369625\pi$
$270$	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	$-0.606724\pi$
$277$	−10.9525	−0.658075	−0.329037	+	0.944317i	$0.606724\pi$
$278$	−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	$-0.504442\pi$
$283$	−0.469462	−0.0279066	−0.0139533	+	0.999903i	$0.504442\pi$
$284$	−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	$-0.744226\pi$
$311$	−24.4834	−1.38833	−0.694163	+	0.719817i	$0.744226\pi$
$312$	−18.1310	−1.02647
$313$	2.87540	0.162527	0.0812635	−	0.996693i	$-0.474104\pi$
$313$	2.87540	0.162527	0.0812635	+	0.996693i	$0.474104\pi$
$314$	5.05790	0.285434
$315$	0	0
$316$	6.49325	0.365274
$317$	23.5740	1.32405	0.662025	−	0.749482i	$-0.269697\pi$
$317$	23.5740	1.32405	0.662025	+	0.749482i	$0.269697\pi$
$318$	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	$-0.351855\pi$
$337$	16.4774	0.897581	0.448790	+	0.893637i	$0.351855\pi$
$338$	−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	$-0.348906\pi$
$347$	17.0278	0.914100	0.457050	+	0.889441i	$0.348906\pi$
$348$	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	$-0.345292\pi$
$367$	17.8974	0.934237	0.467119	+	0.884195i	$0.345292\pi$
$368$	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	$-0.621579\pi$
$379$	−14.5126	−0.745464	−0.372732	+	0.927939i	$0.621579\pi$
$380$	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	$-0.845908\pi$
$397$	−35.2709	−1.77019	−0.885097	+	0.465406i	$0.845908\pi$
$398$	7.98930	0.400468
$399$	−93.6452	−4.68812
$400$	0	0
$401$	−33.6355	−1.67968	−0.839838	−	0.542837i	$-0.817350\pi$
$401$	−33.6355	−1.67968	−0.839838	+	0.542837i	$0.817350\pi$
$402$	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	$-0.549546\pi$
$419$	−6.34658	−0.310051	−0.155025	+	0.987910i	$0.549546\pi$
$420$	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	$-0.210410\pi$
$431$	32.7753	1.57873	0.789365	+	0.613924i	$0.210410\pi$
$432$	−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	$-0.302637\pi$
$439$	24.3493	1.16213	0.581063	+	0.813858i	$0.302637\pi$
$440$	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	$-0.178348\pi$
$449$	35.8993	1.69419	0.847097	+	0.531438i	$0.178348\pi$
$450$	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	$-0.392916\pi$
$479$	14.4494	0.660209	0.330104	+	0.943944i	$0.392916\pi$
$480$	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	$-0.204434\pi$
$487$	35.3420	1.60150	0.800750	+	0.598998i	$0.204434\pi$
$488$	−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	$-0.693572\pi$
$499$	−25.5250	−1.14266	−0.571329	+	0.820721i	$0.693572\pi$
$500$	0	0
$501$	6.55572	0.292888
$502$	12.3896	0.552973
$503$	−37.8180	−1.68622	−0.843110	−	0.537742i	$-0.819278\pi$
$503$	−37.8180	−1.68622	−0.843110	+	0.537742i	$0.819278\pi$
$504$	−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	$-0.373526\pi$
$521$	17.6650	0.773916	0.386958	+	0.922097i	$0.373526\pi$
$522$	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	$-0.584610\pi$
$541$	−12.2201	−0.525382	−0.262691	+	0.964880i	$0.584610\pi$
$542$	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	$-0.602566\pi$
$547$	−14.8127	−0.633345	−0.316672	+	0.948535i	$0.602566\pi$
$548$	−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	$-0.101074\pi$
$571$	45.4021	1.90002	0.950009	+	0.312224i	$0.101074\pi$
$572$	−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	$-0.409721\pi$
$601$	13.7204	0.559667	0.279833	+	0.960049i	$0.409721\pi$
$602$	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	$-0.681341\pi$
$607$	−26.5777	−1.07876	−0.539379	+	0.842063i	$0.681341\pi$
$608$	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	$-0.550567\pi$
$617$	−7.85884	−0.316385	−0.158193	+	0.987408i	$0.550567\pi$
$618$	−2.29394	−0.0922757
$619$	−32.7760	−1.31738	−0.658689	−	0.752416i	$-0.728889\pi$
$619$	−32.7760	−1.31738	−0.658689	+	0.752416i	$0.728889\pi$
$620$	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	$-0.625395\pi$
$641$	−19.4356	−0.767660	−0.383830	+	0.923404i	$0.625395\pi$
$642$	−21.9738	−0.867238
$643$	−16.0075	−0.631276	−0.315638	−	0.948880i	$-0.602218\pi$
$643$	−16.0075	−0.631276	−0.315638	+	0.948880i	$0.602218\pi$
$644$	−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	$-0.532549\pi$
$653$	−5.21693	−0.204154	−0.102077	+	0.994776i	$0.532549\pi$
$654$	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	$-0.581467\pi$
$673$	−13.1346	−0.506302	−0.253151	+	0.967427i	$0.581467\pi$
$674$	11.6019	0.446889
$675$	0	0
$676$	11.2105	0.431172
$677$	−18.9820	−0.729539	−0.364769	−	0.931098i	$-0.618852\pi$
$677$	−18.9820	−0.729539	−0.364769	+	0.931098i	$0.618852\pi$
$678$	−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	$-0.407617\pi$
$683$	14.9579	0.572347	0.286173	+	0.958178i	$0.407617\pi$
$684$	−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	$-0.430987\pi$
$691$	11.3095	0.430233	0.215116	+	0.976588i	$0.430987\pi$
$692$	−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	$-0.867105\pi$
$709$	−48.6798	−1.82821	−0.914105	+	0.405479i	$0.867105\pi$
$710$	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	$-0.702171\pi$
$719$	−31.8171	−1.18658	−0.593289	+	0.804990i	$0.702171\pi$
$720$	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	$-0.487583\pi$
$743$	2.12614	0.0780004	0.0390002	+	0.999239i	$0.487583\pi$
$744$	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	$-0.442986\pi$
$751$	9.76463	0.356316	0.178158	+	0.984002i	$0.442986\pi$
$752$	−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	$-0.600660\pi$
$787$	−17.4487	−0.621979	−0.310990	+	0.950413i	$0.600660\pi$
$788$	−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	$-0.612285\pi$
$809$	−19.6531	−0.690966	−0.345483	+	0.938425i	$0.612285\pi$
$810$	0	0
$811$	30.7782	1.08077	0.540385	−	0.841418i	$-0.318279\pi$
$811$	30.7782	1.08077	0.540385	+	0.841418i	$0.318279\pi$
$812$	−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	$-0.290646\pi$
$821$	35.0315	1.22261	0.611303	+	0.791397i	$0.290646\pi$
$822$	−43.2437	−1.50830
$823$	30.1072	1.04947	0.524736	−	0.851265i	$-0.324164\pi$
$823$	30.1072	1.04947	0.524736	+	0.851265i	$0.324164\pi$
$824$	−2.57648	−0.0897560
$825$	0	0
$826$	6.57384	0.228733
$827$	−20.7366	−0.721082	−0.360541	−	0.932743i	$-0.617408\pi$
$827$	−20.7366	−0.721082	−0.360541	+	0.932743i	$0.617408\pi$
$828$	−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	$-0.440610\pi$
$839$	10.7461	0.370997	0.185498	+	0.982645i	$0.440610\pi$
$840$	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	$-0.230446\pi$
$853$	43.7616	1.49837	0.749184	+	0.662362i	$0.230446\pi$
$854$	25.5439	0.874094
$855$	0	0
$856$	−24.6803	−0.843557
$857$	−0.857197	−0.0292813	−0.0146406	−	0.999893i	$-0.504660\pi$
$857$	−0.857197	−0.0292813	−0.0146406	+	0.999893i	$0.504660\pi$
$858$	−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	$-0.350572\pi$
$877$	26.7943	0.904780	0.452390	+	0.891820i	$0.350572\pi$
$878$	17.1446	0.578601
$879$	−64.5686	−2.17785
$880$	0	0
$881$	21.6063	0.727933	0.363967	−	0.931412i	$-0.381422\pi$
$881$	21.6063	0.727933	0.363967	+	0.931412i	$0.381422\pi$
$882$	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	$-0.307355\pi$
$887$	33.8887	1.13787	0.568936	+	0.822382i	$0.307355\pi$
$888$	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	$-0.406979\pi$
$907$	17.3527	0.576187	0.288093	+	0.957602i	$0.406979\pi$
$908$	−35.1200	−1.16550
$909$	−46.7720	−1.55133
$910$	0	0
$911$	−44.2282	−1.46535	−0.732673	−	0.680581i	$-0.761727\pi$
$911$	−44.2282	−1.46535	−0.732673	+	0.680581i	$0.761727\pi$
$912$	−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	$-0.552417\pi$
$929$	−9.99294	−0.327858	−0.163929	+	0.986472i	$0.552417\pi$
$930$	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	$-0.619062\pi$
$941$	−22.4168	−0.730767	−0.365384	+	0.930857i	$0.619062\pi$
$942$	−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	$-0.246183\pi$
$947$	44.0389	1.43107	0.715535	+	0.698577i	$0.246183\pi$
$948$	−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	$-0.493124\pi$
$983$	1.35448	0.0432013	0.0216007	+	0.999767i	$0.493124\pi$
$984$	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	$-0.211153\pi$
$991$	49.6083	1.57586	0.787930	+	0.615765i	$0.211153\pi$
$992$	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	$-0.437506\pi$
$997$	12.3189	0.390143	0.195072	+	0.980789i	$0.437506\pi$
$998$	−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.bo.1.8		12
5.4	even	2		1445.2.a.r.1.5	✓	12
17.16	even	2		7225.2.a.bn.1.8		12
85.4	even	4		1445.2.d.i.866.16		24
85.64	even	4		1445.2.d.i.866.15		24
85.84	even	2		1445.2.a.s.1.5	yes	12

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

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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
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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