LMFDB - Embedded newform 725.2.a.j.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [725,2,Mod(1,725)] 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("725.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,6,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.294577.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 7x^{3} - x^{2} + 7x + 3 $ x^5 - 7x^3 - x^2 + 7x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.35417$ of defining polynomial
Character	$\chi$	$=$	725.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.35417 q^{2} +1.80364 q^{3} +3.54210 q^{4} +4.24606 q^{6} -0.0127981 q^{7} +3.63036 q^{8} +0.253109 q^{9} +0.615705 q^{11} +6.38866 q^{12} +3.05573 q^{13} -0.0301289 q^{14} +1.46227 q^{16} -6.63473 q^{17} +0.595861 q^{18} -3.54210 q^{19} -0.0230832 q^{21} +1.44947 q^{22} -1.27619 q^{23} +6.54785 q^{24} +7.19369 q^{26} -4.95440 q^{27} -0.0453322 q^{28} -1.00000 q^{29} +6.68608 q^{31} -3.81829 q^{32} +1.11051 q^{33} -15.6193 q^{34} +0.896537 q^{36} +6.22381 q^{37} -8.33869 q^{38} +5.51142 q^{39} +7.00863 q^{41} -0.0543416 q^{42} +5.31124 q^{43} +2.18089 q^{44} -3.00437 q^{46} +3.22196 q^{47} +2.63740 q^{48} -6.99984 q^{49} -11.9666 q^{51} +10.8237 q^{52} -3.77788 q^{53} -11.6635 q^{54} -0.0464618 q^{56} -6.38866 q^{57} -2.35417 q^{58} -8.04970 q^{59} -0.250232 q^{61} +15.7402 q^{62} -0.00323932 q^{63} -11.9134 q^{64} +2.61432 q^{66} +2.90821 q^{67} -23.5009 q^{68} -2.30179 q^{69} -14.0268 q^{71} +0.918877 q^{72} +8.75366 q^{73} +14.6519 q^{74} -12.5465 q^{76} -0.00787986 q^{77} +12.9748 q^{78} -8.35000 q^{79} -9.69526 q^{81} +16.4995 q^{82} +16.2457 q^{83} -0.0817629 q^{84} +12.5035 q^{86} -1.80364 q^{87} +2.23523 q^{88} +15.2351 q^{89} -0.0391075 q^{91} -4.52040 q^{92} +12.0593 q^{93} +7.58503 q^{94} -6.88681 q^{96} -9.90841 q^{97} -16.4788 q^{98} +0.155840 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 6 q^{3} + 4 q^{4} - q^{6} + 10 q^{7} - 3 q^{8} + 7 q^{9} + 2 q^{11} + 18 q^{12} + 8 q^{13} - 13 q^{14} + 6 q^{16} + 3 q^{17} - 16 q^{18} - 4 q^{19} + 15 q^{21} + 16 q^{22} + 3 q^{23} - 19 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) $ 5 * q + 6 * q^3 + 4 * q^4 - q^6 + 10 * q^7 - 3 * q^8 + 7 * q^9 + 2 * q^11 + 18 * q^12 + 8 * q^13 - 13 * q^14 + 6 * q^16 + 3 * q^17 - 16 * q^18 - 4 * q^19 + 15 * q^21 + 16 * q^22 + 3 * q^23 - 19 * q^24 - 4 * q^26 + 21 * q^27 + 10 * q^28 - 5 * q^29 + 5 * q^31 + 4 * q^32 + 3 * q^33 - 9 * q^34 + 35 * q^36 + 4 * q^37 + 3 * q^38 + 4 * q^39 - 3 * q^41 - 16 * q^42 + 12 * q^43 - 19 * q^44 - q^47 + 20 * q^48 + 13 * q^49 - 7 * q^51 + 26 * q^52 + 12 * q^53 - 58 * q^54 - 8 * q^56 - 18 * q^57 - 15 * q^59 - 7 * q^61 + 10 * q^62 + 33 * q^63 - 9 * q^64 + 35 * q^66 + 17 * q^67 - 43 * q^68 + 18 * q^69 - 17 * q^71 - 49 * q^72 + 10 * q^73 + 12 * q^74 - 34 * q^76 - 8 * q^77 - 39 * q^78 - 2 * q^79 + 21 * q^81 + 31 * q^82 + 19 * q^83 + 38 * q^84 + 31 * q^86 - 6 * q^87 - 3 * q^88 + 14 * q^89 - 15 * q^91 - q^92 - 15 * q^93 + 27 * q^94 + 6 * q^96 - 9 * q^97 - 51 * q^98 - 12 * 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 $

$2$	2.35417	1.66465	0.832324	−	0.554290i	$-0.187010\pi$
$2$	2.35417	1.66465	0.832324	+	0.554290i	$0.187010\pi$
$3$	1.80364	1.04133	0.520665	−	0.853761i	$-0.325684\pi$
$3$	1.80364	1.04133	0.520665	+	0.853761i	$0.325684\pi$
$4$	3.54210	1.77105
$5$	0	0
$5$	0	0
$6$	4.24606	1.73345
$7$	−0.0127981	−0.00483723	−0.00241862	−	0.999997i	$-0.500770\pi$
$7$	−0.0127981	−0.00483723	−0.00241862	+	0.999997i	$0.500770\pi$
$8$	3.63036	1.28353
$9$	0.253109	0.0843697
$10$	0	0
$11$	0.615705	0.185642	0.0928210	−	0.995683i	$-0.470412\pi$
$11$	0.615705	0.185642	0.0928210	+	0.995683i	$0.470412\pi$
$12$	6.38866	1.84425
$13$	3.05573	0.847506	0.423753	−	0.905778i	$-0.360713\pi$
$13$	3.05573	0.847506	0.423753	+	0.905778i	$0.360713\pi$
$14$	−0.0301289	−0.00805229
$15$	0	0
$16$	1.46227	0.365567
$17$	−6.63473	−1.60916	−0.804579	−	0.593846i	$-0.797609\pi$
$17$	−6.63473	−1.60916	−0.804579	+	0.593846i	$0.797609\pi$
$18$	0.595861	0.140446
$19$	−3.54210	−0.812613	−0.406307	−	0.913737i	$-0.633184\pi$
$19$	−3.54210	−0.812613	−0.406307	+	0.913737i	$0.633184\pi$
$20$	0	0
$21$	−0.0230832	−0.00503716
$22$	1.44947	0.309028
$23$	−1.27619	−0.266104	−0.133052	−	0.991109i	$-0.542478\pi$
$23$	−1.27619	−0.266104	−0.133052	+	0.991109i	$0.542478\pi$
$24$	6.54785	1.33657
$25$	0	0
$26$	7.19369	1.41080
$27$	−4.95440	−0.953474
$28$	−0.0453322	−0.00856698
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.68608	1.20086	0.600428	−	0.799679i	$-0.294997\pi$
$31$	6.68608	1.20086	0.600428	+	0.799679i	$0.294997\pi$
$32$	−3.81829	−0.674985
$33$	1.11051	0.193315
$34$	−15.6193	−2.67868
$35$	0	0
$36$	0.896537	0.149423
$37$	6.22381	1.02319	0.511594	−	0.859227i	$-0.329055\pi$
$37$	6.22381	1.02319	0.511594	+	0.859227i	$0.329055\pi$
$38$	−8.33869	−1.35271
$39$	5.51142	0.882534
$40$	0	0
$41$	7.00863	1.09456	0.547282	−	0.836948i	$-0.315663\pi$
$41$	7.00863	1.09456	0.547282	+	0.836948i	$0.315663\pi$
$42$	−0.0543416	−0.00838509
$43$	5.31124	0.809956	0.404978	−	0.914326i	$-0.367279\pi$
$43$	5.31124	0.809956	0.404978	+	0.914326i	$0.367279\pi$
$44$	2.18089	0.328781
$45$	0	0
$46$	−3.00437	−0.442970
$47$	3.22196	0.469971	0.234985	−	0.971999i	$-0.424496\pi$
$47$	3.22196	0.469971	0.234985	+	0.971999i	$0.424496\pi$
$48$	2.63740	0.380677
$49$	−6.99984	−0.999977
$50$	0	0
$51$	−11.9666	−1.67567
$52$	10.8237	1.50097
$53$	−3.77788	−0.518931	−0.259466	−	0.965752i	$-0.583546\pi$
$53$	−3.77788	−0.518931	−0.259466	+	0.965752i	$0.583546\pi$
$54$	−11.6635	−1.58720
$55$	0	0
$56$	−0.0464618	−0.00620871
$57$	−6.38866	−0.846199
$58$	−2.35417	−0.309117
$59$	−8.04970	−1.04798	−0.523991	−	0.851724i	$-0.675557\pi$
$59$	−8.04970	−1.04798	−0.523991	+	0.851724i	$0.675557\pi$
$60$	0	0
$61$	−0.250232	−0.0320389	−0.0160194	−	0.999872i	$-0.505099\pi$
$61$	−0.250232	−0.0320389	−0.0160194	+	0.999872i	$0.505099\pi$
$62$	15.7402	1.99900
$63$	−0.00323932	−0.000408116 0
$64$	−11.9134	−1.48918
$65$	0	0
$66$	2.61432	0.321801
$67$	2.90821	0.355294	0.177647	−	0.984094i	$-0.443151\pi$
$67$	2.90821	0.355294	0.177647	+	0.984094i	$0.443151\pi$
$68$	−23.5009	−2.84990
$69$	−2.30179	−0.277103
$70$	0	0
$71$	−14.0268	−1.66468	−0.832339	−	0.554268i	$-0.812998\pi$
$71$	−14.0268	−1.66468	−0.832339	+	0.554268i	$0.812998\pi$
$72$	0.918877	0.108291
$73$	8.75366	1.02454	0.512269	−	0.858825i	$-0.328805\pi$
$73$	8.75366	1.02454	0.512269	+	0.858825i	$0.328805\pi$
$74$	14.6519	1.70325
$75$	0	0
$76$	−12.5465	−1.43918
$77$	−0.00787986	−0.000897993 0
$78$	12.9748	1.46911
$79$	−8.35000	−0.939448	−0.469724	−	0.882813i	$-0.655647\pi$
$79$	−8.35000	−0.939448	−0.469724	+	0.882813i	$0.655647\pi$
$80$	0	0
$81$	−9.69526	−1.07725
$82$	16.4995	1.82206
$83$	16.2457	1.78320	0.891599	−	0.452825i	$-0.149584\pi$
$83$	16.2457	1.78320	0.891599	+	0.452825i	$0.149584\pi$
$84$	−0.0817629	−0.00892106
$85$	0	0
$86$	12.5035	1.34829
$87$	−1.80364	−0.193370
$88$	2.23523	0.238276
$89$	15.2351	1.61491	0.807457	−	0.589927i	$-0.200843\pi$
$89$	15.2351	1.61491	0.807457	+	0.589927i	$0.200843\pi$
$90$	0	0
$91$	−0.0391075	−0.00409958
$92$	−4.52040	−0.471284
$93$	12.0593	1.25049
$94$	7.58503	0.782336
$95$	0	0
$96$	−6.88681	−0.702883
$97$	−9.90841	−1.00605	−0.503023	−	0.864273i	$-0.667779\pi$
$97$	−9.90841	−1.00605	−0.503023	+	0.864273i	$0.667779\pi$
$98$	−16.4788	−1.66461
$99$	0.155840	0.0156626
$100$	0	0
$101$	−14.5937	−1.45212	−0.726062	−	0.687630i	$-0.758651\pi$
$101$	−14.5937	−1.45212	−0.726062	+	0.687630i	$0.758651\pi$
$102$	−28.1715	−2.78939
$103$	8.34137	0.821899	0.410950	−	0.911658i	$-0.365197\pi$
$103$	8.34137	0.821899	0.410950	+	0.911658i	$0.365197\pi$
$104$	11.0934	1.08780
$105$	0	0
$106$	−8.89375	−0.863838
$107$	6.71964	0.649612	0.324806	−	0.945781i	$-0.394701\pi$
$107$	6.71964	0.649612	0.324806	+	0.945781i	$0.394701\pi$
$108$	−17.5490	−1.68865
$109$	13.2479	1.26892	0.634460	−	0.772955i	$-0.281222\pi$
$109$	13.2479	1.26892	0.634460	+	0.772955i	$0.281222\pi$
$110$	0	0
$111$	11.2255	1.06548
$112$	−0.0187143	−0.00176833
$113$	−8.24441	−0.775569	−0.387784	−	0.921750i	$-0.626759\pi$
$113$	−8.24441	−0.775569	−0.387784	+	0.921750i	$0.626759\pi$
$114$	−15.0400	−1.40862
$115$	0	0
$116$	−3.54210	−0.328876
$117$	0.773432	0.0715038
$118$	−18.9503	−1.74452
$119$	0.0849120	0.00778387
$120$	0	0
$121$	−10.6209	−0.965537
$122$	−0.589087	−0.0533334
$123$	12.6410	1.13980
$124$	23.6828	2.12678
$125$	0	0
$126$	−0.00762590	−0.000679369 0
$127$	11.9709	1.06225	0.531123	−	0.847295i	$-0.321770\pi$
$127$	11.9709	1.06225	0.531123	+	0.847295i	$0.321770\pi$
$128$	−20.4096	−1.80397
$129$	9.57955	0.843433
$130$	0	0
$131$	−7.31459	−0.639078	−0.319539	−	0.947573i	$-0.603528\pi$
$131$	−7.31459	−0.639078	−0.319539	+	0.947573i	$0.603528\pi$
$132$	3.93353	0.342370
$133$	0.0453322	0.00393080
$134$	6.84640	0.591439
$135$	0	0
$136$	−24.0864	−2.06540
$137$	18.5756	1.58702	0.793511	−	0.608556i	$-0.208251\pi$
$137$	18.5756	1.58702	0.793511	+	0.608556i	$0.208251\pi$
$138$	−5.41879	−0.461278
$139$	−13.0723	−1.10877	−0.554387	−	0.832259i	$-0.687047\pi$
$139$	−13.0723	−1.10877	−0.554387	+	0.832259i	$0.687047\pi$
$140$	0	0
$141$	5.81125	0.489395
$142$	−33.0215	−2.77110
$143$	1.88142	0.157333
$144$	0.370114	0.0308428
$145$	0	0
$146$	20.6076	1.70550
$147$	−12.6252	−1.04131
$148$	22.0454	1.81212
$149$	15.7013	1.28630	0.643150	−	0.765740i	$-0.277627\pi$
$149$	15.7013	1.28630	0.643150	+	0.765740i	$0.277627\pi$
$150$	0	0
$151$	14.5106	1.18086	0.590428	−	0.807090i	$-0.298959\pi$
$151$	14.5106	1.18086	0.590428	+	0.807090i	$0.298959\pi$
$152$	−12.8591	−1.04301
$153$	−1.67931	−0.135764
$154$	−0.0185505	−0.00149484
$155$	0	0
$156$	19.5220	1.56301
$157$	11.7705	0.939386	0.469693	−	0.882830i	$-0.344365\pi$
$157$	11.7705	0.939386	0.469693	+	0.882830i	$0.344365\pi$
$158$	−19.6573	−1.56385
$159$	−6.81392	−0.540379
$160$	0	0
$161$	0.0163329	0.00128721
$162$	−22.8243	−1.79324
$163$	−11.4716	−0.898523	−0.449261	−	0.893400i	$-0.648313\pi$
$163$	−11.4716	−0.898523	−0.449261	+	0.893400i	$0.648313\pi$
$164$	24.8253	1.93853
$165$	0	0
$166$	38.2451	2.96840
$167$	−17.1940	−1.33051	−0.665256	−	0.746616i	$-0.731677\pi$
$167$	−17.1940	−1.33051	−0.665256	+	0.746616i	$0.731677\pi$
$168$	−0.0838002	−0.00646532
$169$	−3.66254	−0.281734
$170$	0	0
$171$	−0.896537	−0.0685599
$172$	18.8129	1.43447
$173$	−12.5883	−0.957068	−0.478534	−	0.878069i	$-0.658832\pi$
$173$	−12.5883	−0.957068	−0.478534	+	0.878069i	$0.658832\pi$
$174$	−4.24606	−0.321893
$175$	0	0
$176$	0.900326	0.0678646
$177$	−14.5187	−1.09130
$178$	35.8659	2.68826
$179$	−13.9297	−1.04115	−0.520576	−	0.853815i	$-0.674283\pi$
$179$	−13.9297	−1.04115	−0.520576	+	0.853815i	$0.674283\pi$
$180$	0	0
$181$	−2.34991	−0.174667	−0.0873335	−	0.996179i	$-0.527835\pi$
$181$	−2.34991	−0.174667	−0.0873335	+	0.996179i	$0.527835\pi$
$182$	−0.0920656	−0.00682436
$183$	−0.451327	−0.0333631
$184$	−4.63304	−0.341552
$185$	0	0
$186$	28.3895	2.08162
$187$	−4.08503	−0.298727
$188$	11.4125	0.832342
$189$	0.0634069	0.00461218
$190$	0	0
$191$	6.08289	0.440143	0.220071	−	0.975484i	$-0.429371\pi$
$191$	6.08289	0.440143	0.220071	+	0.975484i	$0.429371\pi$
$192$	−21.4875	−1.55073
$193$	4.22212	0.303915	0.151958	−	0.988387i	$-0.451442\pi$
$193$	4.22212	0.303915	0.151958	+	0.988387i	$0.451442\pi$
$194$	−23.3260	−1.67471
$195$	0	0
$196$	−24.7941	−1.77101
$197$	14.2067	1.01218	0.506092	−	0.862479i	$-0.331090\pi$
$197$	14.2067	1.01218	0.506092	+	0.862479i	$0.331090\pi$
$198$	0.366874	0.0260726
$199$	17.9466	1.27220	0.636102	−	0.771605i	$-0.280546\pi$
$199$	17.9466	1.27220	0.636102	+	0.771605i	$0.280546\pi$
$200$	0	0
$201$	5.24535	0.369978
$202$	−34.3559	−2.41727
$203$	0.0127981	0.000898252 0
$204$	−42.3870	−2.96769
$205$	0	0
$206$	19.6370	1.36817
$207$	−0.323016	−0.0224512
$208$	4.46829	0.309820
$209$	−2.18089	−0.150855
$210$	0	0
$211$	5.57966	0.384120	0.192060	−	0.981383i	$-0.438483\pi$
$211$	5.57966	0.384120	0.192060	+	0.981383i	$0.438483\pi$
$212$	−13.3816	−0.919053
$213$	−25.2993	−1.73348
$214$	15.8191	1.08137
$215$	0	0
$216$	−17.9862	−1.22381
$217$	−0.0855693	−0.00580882
$218$	31.1878	2.11230
$219$	15.7884	1.06688
$220$	0	0
$221$	−20.2739	−1.36377
$222$	26.4267	1.77364
$223$	28.0322	1.87718	0.938588	−	0.345041i	$-0.112135\pi$
$223$	28.0322	1.87718	0.938588	+	0.345041i	$0.112135\pi$
$224$	0.0488669	0.00326506
$225$	0	0
$226$	−19.4087	−1.29105
$227$	24.6498	1.63606	0.818031	−	0.575174i	$-0.195066\pi$
$227$	24.6498	1.63606	0.818031	+	0.575174i	$0.195066\pi$
$228$	−22.6293	−1.49866
$229$	−13.8635	−0.916123	−0.458062	−	0.888920i	$-0.651456\pi$
$229$	−13.8635	−0.916123	−0.458062	+	0.888920i	$0.651456\pi$
$230$	0	0
$231$	−0.0142124	−0.000935108 0
$232$	−3.63036	−0.238345
$233$	−3.13422	−0.205330	−0.102665	−	0.994716i	$-0.532737\pi$
$233$	−3.13422	−0.205330	−0.102665	+	0.994716i	$0.532737\pi$
$234$	1.82079	0.119029
$235$	0	0
$236$	−28.5128	−1.85603
$237$	−15.0604	−0.978276
$238$	0.199897	0.0129574
$239$	−16.8652	−1.09092	−0.545460	−	0.838137i	$-0.683645\pi$
$239$	−16.8652	−1.09092	−0.545460	+	0.838137i	$0.683645\pi$
$240$	0	0
$241$	1.59078	0.102471	0.0512355	−	0.998687i	$-0.483684\pi$
$241$	1.59078	0.102471	0.0512355	+	0.998687i	$0.483684\pi$
$242$	−25.0034	−1.60728
$243$	−2.62355	−0.168301
$244$	−0.886345	−0.0567424
$245$	0	0
$246$	29.7591	1.89737
$247$	−10.8237	−0.688694
$248$	24.2729	1.54133
$249$	29.3014	1.85690
$250$	0	0
$251$	0.138433	0.00873779	0.00436889	−	0.999990i	$-0.498609\pi$
$251$	0.138433	0.00873779	0.00436889	+	0.999990i	$0.498609\pi$
$252$	−0.0114740	−0.000722793 0
$253$	−0.785758	−0.0494002
$254$	28.1815	1.76826
$255$	0	0
$256$	−24.2208	−1.51380
$257$	27.6954	1.72759	0.863796	−	0.503842i	$-0.168080\pi$
$257$	27.6954	1.72759	0.863796	+	0.503842i	$0.168080\pi$
$258$	22.5519	1.40402
$259$	−0.0796531	−0.00494940
$260$	0	0
$261$	−0.253109	−0.0156671
$262$	−17.2198	−1.06384
$263$	−8.79890	−0.542564	−0.271282	−	0.962500i	$-0.587448\pi$
$263$	−8.79890	−0.542564	−0.271282	+	0.962500i	$0.587448\pi$
$264$	4.03154	0.248124
$265$	0	0
$266$	0.106720	0.00654339
$267$	27.4785	1.68166
$268$	10.3012	0.629243
$269$	0.657325	0.0400778	0.0200389	−	0.999799i	$-0.493621\pi$
$269$	0.657325	0.0400778	0.0200389	+	0.999799i	$0.493621\pi$
$270$	0	0
$271$	−12.3389	−0.749536	−0.374768	−	0.927119i	$-0.622278\pi$
$271$	−12.3389	−0.749536	−0.374768	+	0.927119i	$0.622278\pi$
$272$	−9.70176	−0.588256
$273$	−0.0705358	−0.00426902
$274$	43.7301	2.64183
$275$	0	0
$276$	−8.15316	−0.490763
$277$	−25.4620	−1.52986	−0.764930	−	0.644113i	$-0.777227\pi$
$277$	−25.4620	−1.52986	−0.764930	+	0.644113i	$0.777227\pi$
$278$	−30.7743	−1.84572
$279$	1.69231	0.101316
$280$	0	0
$281$	11.0301	0.658003	0.329002	−	0.944329i	$-0.393288\pi$
$281$	11.0301	0.658003	0.329002	+	0.944329i	$0.393288\pi$
$282$	13.6806	0.814670
$283$	−3.85338	−0.229059	−0.114530	−	0.993420i	$-0.536536\pi$
$283$	−3.85338	−0.229059	−0.114530	+	0.993420i	$0.536536\pi$
$284$	−49.6844	−2.94823
$285$	0	0
$286$	4.42919	0.261903
$287$	−0.0896973	−0.00529466
$288$	−0.966444	−0.0569483
$289$	27.0196	1.58939
$290$	0	0
$291$	−17.8712	−1.04763
$292$	31.0064	1.81451
$293$	−21.0271	−1.22842	−0.614209	−	0.789143i	$-0.710525\pi$
$293$	−21.0271	−1.22842	−0.614209	+	0.789143i	$0.710525\pi$
$294$	−29.7217	−1.73341
$295$	0	0
$296$	22.5947	1.31329
$297$	−3.05045	−0.177005
$298$	36.9634	2.14123
$299$	−3.89969	−0.225525
$300$	0	0
$301$	−0.0679739	−0.00391795
$302$	34.1604	1.96571
$303$	−26.3217	−1.51214
$304$	−5.17950	−0.297065
$305$	0	0
$306$	−3.95337	−0.225999
$307$	15.4208	0.880113	0.440057	−	0.897970i	$-0.354958\pi$
$307$	15.4208	0.880113	0.440057	+	0.897970i	$0.354958\pi$
$308$	−0.0279112	−0.00159039
$309$	15.0448	0.855869
$310$	0	0
$311$	9.65112	0.547265	0.273632	−	0.961834i	$-0.411775\pi$
$311$	9.65112	0.547265	0.273632	+	0.961834i	$0.411775\pi$
$312$	20.0084	1.13275
$313$	4.20630	0.237754	0.118877	−	0.992909i	$-0.462071\pi$
$313$	4.20630	0.237754	0.118877	+	0.992909i	$0.462071\pi$
$314$	27.7096	1.56375
$315$	0	0
$316$	−29.5765	−1.66381
$317$	−19.6625	−1.10436	−0.552179	−	0.833725i	$-0.686204\pi$
$317$	−19.6625	−1.10436	−0.552179	+	0.833725i	$0.686204\pi$
$318$	−16.0411	−0.899541
$319$	−0.615705	−0.0344728
$320$	0	0
$321$	12.1198	0.676461
$322$	0.0384503	0.00214275
$323$	23.5009	1.30762
$324$	−34.3416	−1.90787
$325$	0	0
$326$	−27.0060	−1.49572
$327$	23.8944	1.32137
$328$	25.4438	1.40490
$329$	−0.0412350	−0.00227336
$330$	0	0
$331$	10.3331	0.567960	0.283980	−	0.958830i	$-0.408345\pi$
$331$	10.3331	0.567960	0.283980	+	0.958830i	$0.408345\pi$
$332$	57.5439	3.15813
$333$	1.57530	0.0863261
$334$	−40.4775	−2.21483
$335$	0	0
$336$	−0.0337538	−0.00184142
$337$	−5.22967	−0.284879	−0.142439	−	0.989804i	$-0.545495\pi$
$337$	−5.22967	−0.284879	−0.142439	+	0.989804i	$0.545495\pi$
$338$	−8.62224	−0.468988
$339$	−14.8699	−0.807623
$340$	0	0
$341$	4.11665	0.222929
$342$	−2.11060	−0.114128
$343$	0.179172	0.00967435
$344$	19.2817	1.03960
$345$	0	0
$346$	−29.6349	−1.59318
$347$	−1.83303	−0.0984020	−0.0492010	−	0.998789i	$-0.515667\pi$
$347$	−1.83303	−0.0984020	−0.0492010	+	0.998789i	$0.515667\pi$
$348$	−6.38866	−0.342468
$349$	−20.3449	−1.08904	−0.544519	−	0.838748i	$-0.683288\pi$
$349$	−20.3449	−1.08904	−0.544519	+	0.838748i	$0.683288\pi$
$350$	0	0
$351$	−15.1393	−0.808075
$352$	−2.35094	−0.125306
$353$	14.2365	0.757734	0.378867	−	0.925451i	$-0.376314\pi$
$353$	14.2365	0.757734	0.378867	+	0.925451i	$0.376314\pi$
$354$	−34.1795	−1.81662
$355$	0	0
$356$	53.9641	2.86009
$357$	0.153151	0.00810558
$358$	−32.7927	−1.73315
$359$	−10.9779	−0.579392	−0.289696	−	0.957119i	$-0.593554\pi$
$359$	−10.9779	−0.579392	−0.289696	+	0.957119i	$0.593554\pi$
$360$	0	0
$361$	−6.45353	−0.339660
$362$	−5.53207	−0.290759
$363$	−19.1563	−1.00544
$364$	−0.138523	−0.00726056
$365$	0	0
$366$	−1.06250	−0.0555377
$367$	30.9755	1.61691	0.808453	−	0.588561i	$-0.200305\pi$
$367$	30.9755	1.61691	0.808453	+	0.588561i	$0.200305\pi$
$368$	−1.86614	−0.0972791
$369$	1.77395	0.0923480
$370$	0	0
$371$	0.0483497	0.00251019
$372$	42.7151	2.21468
$373$	−31.6787	−1.64026	−0.820130	−	0.572177i	$-0.806099\pi$
$373$	−31.6787	−1.64026	−0.820130	+	0.572177i	$0.806099\pi$
$374$	−9.61685	−0.497275
$375$	0	0
$376$	11.6969	0.603220
$377$	−3.05573	−0.157378
$378$	0.149270	0.00767765
$379$	−7.98024	−0.409917	−0.204959	−	0.978771i	$-0.565706\pi$
$379$	−7.98024	−0.409917	−0.204959	+	0.978771i	$0.565706\pi$
$380$	0	0
$381$	21.5912	1.10615
$382$	14.3201	0.732682
$383$	−19.0485	−0.973334	−0.486667	−	0.873588i	$-0.661787\pi$
$383$	−19.0485	−0.973334	−0.486667	+	0.873588i	$0.661787\pi$
$384$	−36.8116	−1.87853
$385$	0	0
$386$	9.93958	0.505911
$387$	1.34432	0.0683358
$388$	−35.0966	−1.78176
$389$	13.3610	0.677432	0.338716	−	0.940889i	$-0.390007\pi$
$389$	13.3610	0.677432	0.338716	+	0.940889i	$0.390007\pi$
$390$	0	0
$391$	8.46719	0.428204
$392$	−25.4119	−1.28350
$393$	−13.1929	−0.665492
$394$	33.4449	1.68493
$395$	0	0
$396$	0.552002	0.0277392
$397$	14.3046	0.717925	0.358963	−	0.933352i	$-0.383131\pi$
$397$	14.3046	0.717925	0.358963	+	0.933352i	$0.383131\pi$
$398$	42.2494	2.11777
$399$	0.0817629	0.00409326
$400$	0	0
$401$	3.71660	0.185598	0.0927991	−	0.995685i	$-0.470419\pi$
$401$	3.71660	0.185598	0.0927991	+	0.995685i	$0.470419\pi$
$402$	12.3484	0.615884
$403$	20.4308	1.01773
$404$	−51.6922	−2.57178
$405$	0	0
$406$	0.0301289	0.00149527
$407$	3.83203	0.189947
$408$	−43.4432	−2.15076
$409$	31.0905	1.53733	0.768664	−	0.639653i	$-0.220922\pi$
$409$	31.0905	1.53733	0.768664	+	0.639653i	$0.220922\pi$
$410$	0	0
$411$	33.5037	1.65261
$412$	29.5460	1.45562
$413$	0.103021	0.00506933
$414$	−0.760433	−0.0373732
$415$	0	0
$416$	−11.6676	−0.572054
$417$	−23.5776	−1.15460
$418$	−5.13417	−0.251121
$419$	26.8273	1.31060	0.655301	−	0.755368i	$-0.272542\pi$
$419$	26.8273	1.31060	0.655301	+	0.755368i	$0.272542\pi$
$420$	0	0
$421$	−26.6460	−1.29865	−0.649323	−	0.760513i	$-0.724948\pi$
$421$	−26.6460	−1.29865	−0.649323	+	0.760513i	$0.724948\pi$
$422$	13.1354	0.639424
$423$	0.815507	0.0396513
$424$	−13.7151	−0.666062
$425$	0	0
$426$	−59.5588	−2.88563
$427$	0.00320249	0.000154979 0
$428$	23.8016	1.15050
$429$	3.39341	0.163835
$430$	0	0
$431$	12.6953	0.611513	0.305756	−	0.952110i	$-0.401091\pi$
$431$	12.6953	0.611513	0.305756	+	0.952110i	$0.401091\pi$
$432$	−7.24466	−0.348559
$433$	−5.59059	−0.268667	−0.134333	−	0.990936i	$-0.542889\pi$
$433$	−5.59059	−0.268667	−0.134333	+	0.990936i	$0.542889\pi$
$434$	−0.201444	−0.00966964
$435$	0	0
$436$	46.9254	2.24732
$437$	4.52040	0.216240
$438$	37.1686	1.77599
$439$	−33.7319	−1.60994	−0.804969	−	0.593317i	$-0.797818\pi$
$439$	−33.7319	−1.60994	−0.804969	+	0.593317i	$0.797818\pi$
$440$	0	0
$441$	−1.77172	−0.0843677
$442$	−47.7281	−2.27020
$443$	−16.1981	−0.769593	−0.384796	−	0.923001i	$-0.625728\pi$
$443$	−16.1981	−0.769593	−0.384796	+	0.923001i	$0.625728\pi$
$444$	39.7619	1.88701
$445$	0	0
$446$	65.9925	3.12483
$447$	28.3194	1.33946
$448$	0.152469	0.00720351
$449$	5.47981	0.258608	0.129304	−	0.991605i	$-0.458726\pi$
$449$	5.47981	0.258608	0.129304	+	0.991605i	$0.458726\pi$
$450$	0	0
$451$	4.31525	0.203197
$452$	−29.2025	−1.37357
$453$	26.1719	1.22966
$454$	58.0297	2.72347
$455$	0	0
$456$	−23.1931	−1.08612
$457$	−35.1916	−1.64619	−0.823097	−	0.567901i	$-0.807756\pi$
$457$	−35.1916	−1.64619	−0.823097	+	0.567901i	$0.807756\pi$
$458$	−32.6369	−1.52502
$459$	32.8711	1.53429
$460$	0	0
$461$	−13.6103	−0.633896	−0.316948	−	0.948443i	$-0.602658\pi$
$461$	−13.6103	−0.633896	−0.316948	+	0.948443i	$0.602658\pi$
$462$	−0.0334584	−0.00155663
$463$	14.7376	0.684912	0.342456	−	0.939534i	$-0.388741\pi$
$463$	14.7376	0.684912	0.342456	+	0.939534i	$0.388741\pi$
$464$	−1.46227	−0.0678842
$465$	0	0
$466$	−7.37849	−0.341802
$467$	−3.30836	−0.153093	−0.0765464	−	0.997066i	$-0.524389\pi$
$467$	−3.30836	−0.153093	−0.0765464	+	0.997066i	$0.524389\pi$
$468$	2.73957	0.126637
$469$	−0.0372196	−0.00171864
$470$	0	0
$471$	21.2297	0.978211
$472$	−29.2233	−1.34511
$473$	3.27016	0.150362
$474$	−35.4546	−1.62848
$475$	0	0
$476$	0.300767	0.0137856
$477$	−0.956215	−0.0437821
$478$	−39.7035	−1.81600
$479$	1.23504	0.0564306	0.0282153	−	0.999602i	$-0.491018\pi$
$479$	1.23504	0.0564306	0.0282153	+	0.999602i	$0.491018\pi$
$480$	0	0
$481$	19.0183	0.867158
$482$	3.74496	0.170578
$483$	0.0294586	0.00134041
$484$	−37.6203	−1.71001
$485$	0	0
$486$	−6.17628	−0.280162
$487$	−21.6520	−0.981145	−0.490573	−	0.871400i	$-0.663212\pi$
$487$	−21.6520	−0.981145	−0.490573	+	0.871400i	$0.663212\pi$
$488$	−0.908430	−0.0411227
$489$	−20.6906	−0.935659
$490$	0	0
$491$	−24.5870	−1.10959	−0.554797	−	0.831986i	$-0.687204\pi$
$491$	−24.5870	−1.10959	−0.554797	+	0.831986i	$0.687204\pi$
$492$	44.7758	2.01865
$493$	6.63473	0.298813
$494$	−25.4807	−1.14643
$495$	0	0
$496$	9.77686	0.438994
$497$	0.179517	0.00805243
$498$	68.9803	3.09108
$499$	20.9955	0.939888	0.469944	−	0.882696i	$-0.344274\pi$
$499$	20.9955	0.939888	0.469944	+	0.882696i	$0.344274\pi$
$500$	0	0
$501$	−31.0117	−1.38550
$502$	0.325893	0.0145453
$503$	−1.14978	−0.0512661	−0.0256331	−	0.999671i	$-0.508160\pi$
$503$	−1.14978	−0.0512661	−0.0256331	+	0.999671i	$0.508160\pi$
$504$	−0.0117599	−0.000523827 0
$505$	0	0
$506$	−1.84980	−0.0822338
$507$	−6.60590	−0.293378
$508$	42.4021	1.88129
$509$	10.1966	0.451955	0.225977	−	0.974133i	$-0.427442\pi$
$509$	10.1966	0.451955	0.225977	+	0.974133i	$0.427442\pi$
$510$	0	0
$511$	−0.112030	−0.00495593
$512$	−16.2005	−0.715968
$513$	17.5490	0.774806
$514$	65.1996	2.87583
$515$	0	0
$516$	33.9317	1.49376
$517$	1.98378	0.0872463
$518$	−0.187517	−0.00823901
$519$	−22.7047	−0.996624
$520$	0	0
$521$	5.03932	0.220777	0.110388	−	0.993889i	$-0.464791\pi$
$521$	5.03932	0.220777	0.110388	+	0.993889i	$0.464791\pi$
$522$	−0.595861	−0.0260801
$523$	37.4208	1.63630	0.818149	−	0.575006i	$-0.195000\pi$
$523$	37.4208	1.63630	0.818149	+	0.575006i	$0.195000\pi$
$524$	−25.9090	−1.13184
$525$	0	0
$526$	−20.7141	−0.903177
$527$	−44.3603	−1.93237
$528$	1.62386	0.0706695
$529$	−21.3713	−0.929188
$530$	0	0
$531$	−2.03745	−0.0884179
$532$	0.160571	0.00696164
$533$	21.4164	0.927649
$534$	64.6891	2.79937
$535$	0	0
$536$	10.5578	0.456029
$537$	−25.1241	−1.08418
$538$	1.54745	0.0667154
$539$	−4.30983	−0.185638
$540$	0	0
$541$	−3.79343	−0.163092	−0.0815461	−	0.996670i	$-0.525986\pi$
$541$	−3.79343	−0.163092	−0.0815461	+	0.996670i	$0.525986\pi$
$542$	−29.0479	−1.24771
$543$	−4.23838	−0.181886
$544$	25.3333	1.08616
$545$	0	0
$546$	−0.166053	−0.00710641
$547$	21.8706	0.935119	0.467560	−	0.883962i	$-0.345133\pi$
$547$	21.8706	0.935119	0.467560	+	0.883962i	$0.345133\pi$
$548$	65.7967	2.81069
$549$	−0.0633359	−0.00270311
$550$	0	0
$551$	3.54210	0.150899
$552$	−8.35632	−0.355668
$553$	0.106864	0.00454433
$554$	−59.9417	−2.54668
$555$	0	0
$556$	−46.3032	−1.96369
$557$	−30.5287	−1.29354	−0.646771	−	0.762684i	$-0.723881\pi$
$557$	−30.5287	−1.29354	−0.646771	+	0.762684i	$0.723881\pi$
$558$	3.98398	0.168655
$559$	16.2297	0.686443
$560$	0	0
$561$	−7.36792	−0.311074
$562$	25.9668	1.09534
$563$	−26.7529	−1.12750	−0.563750	−	0.825945i	$-0.690642\pi$
$563$	−26.7529	−1.12750	−0.563750	+	0.825945i	$0.690642\pi$
$564$	20.5840	0.866743
$565$	0	0
$566$	−9.07149	−0.381303
$567$	0.124081	0.00521092
$568$	−50.9224	−2.13666
$569$	−32.5806	−1.36585	−0.682924	−	0.730489i	$-0.739292\pi$
$569$	−32.5806	−1.36585	−0.682924	+	0.730489i	$0.739292\pi$
$570$	0	0
$571$	44.1692	1.84842	0.924211	−	0.381881i	$-0.124724\pi$
$571$	44.1692	1.84842	0.924211	+	0.381881i	$0.124724\pi$
$572$	6.66419	0.278644
$573$	10.9713	0.458334
$574$	−0.211162	−0.00881374
$575$	0	0
$576$	−3.01540	−0.125642
$577$	30.7443	1.27990	0.639950	−	0.768417i	$-0.278955\pi$
$577$	30.7443	1.27990	0.639950	+	0.768417i	$0.278955\pi$
$578$	63.6087	2.64577
$579$	7.61518	0.316476
$580$	0	0
$581$	−0.207915	−0.00862575
$582$	−42.0717	−1.74393
$583$	−2.32606	−0.0963354
$584$	31.7789	1.31502
$585$	0	0
$586$	−49.5014	−2.04488
$587$	5.45253	0.225050	0.112525	−	0.993649i	$-0.464106\pi$
$587$	5.45253	0.225050	0.112525	+	0.993649i	$0.464106\pi$
$588$	−44.7196	−1.84421
$589$	−23.6828	−0.975832
$590$	0	0
$591$	25.6237	1.05402
$592$	9.10089	0.374044
$593$	−11.9686	−0.491491	−0.245745	−	0.969334i	$-0.579033\pi$
$593$	−11.9686	−0.491491	−0.245745	+	0.969334i	$0.579033\pi$
$594$	−7.18126	−0.294650
$595$	0	0
$596$	55.6155	2.27810
$597$	32.3692	1.32478
$598$	−9.18053	−0.375420
$599$	−36.2858	−1.48260	−0.741300	−	0.671174i	$-0.765790\pi$
$599$	−36.2858	−1.48260	−0.741300	+	0.671174i	$0.765790\pi$
$600$	0	0
$601$	46.2257	1.88559	0.942793	−	0.333380i	$-0.108189\pi$
$601$	46.2257	1.88559	0.942793	+	0.333380i	$0.108189\pi$
$602$	−0.160022	−0.00652200
$603$	0.736093	0.0299760
$604$	51.3980	2.09135
$605$	0	0
$606$	−61.9656	−2.51718
$607$	0.329296	0.0133657	0.00668286	−	0.999978i	$-0.497873\pi$
$607$	0.329296	0.0133657	0.00668286	+	0.999978i	$0.497873\pi$
$608$	13.5248	0.548502
$609$	0.0230832	0.000935377 0
$610$	0	0
$611$	9.84542	0.398303
$612$	−5.94828	−0.240445
$613$	−30.4610	−1.23031	−0.615154	−	0.788407i	$-0.710906\pi$
$613$	−30.4610	−1.23031	−0.615154	+	0.788407i	$0.710906\pi$
$614$	36.3032	1.46508
$615$	0	0
$616$	−0.0286067	−0.00115260
$617$	−44.5426	−1.79322	−0.896609	−	0.442823i	$-0.853977\pi$
$617$	−44.5426	−1.79322	−0.896609	+	0.442823i	$0.853977\pi$
$618$	35.4180	1.42472
$619$	25.8857	1.04043	0.520216	−	0.854035i	$-0.325851\pi$
$619$	25.8857	1.04043	0.520216	+	0.854035i	$0.325851\pi$
$620$	0	0
$621$	6.32276	0.253724
$622$	22.7203	0.911002
$623$	−0.194980	−0.00781171
$624$	8.05918	0.322626
$625$	0	0
$626$	9.90233	0.395777
$627$	−3.93353	−0.157090
$628$	41.6922	1.66370
$629$	−41.2933	−1.64647
$630$	0	0
$631$	38.4771	1.53175	0.765874	−	0.642991i	$-0.222307\pi$
$631$	38.4771	1.53175	0.765874	+	0.642991i	$0.222307\pi$
$632$	−30.3135	−1.20581
$633$	10.0637	0.399996
$634$	−46.2889	−1.83837
$635$	0	0
$636$	−24.1356	−0.957038
$637$	−21.3896	−0.847486
$638$	−1.44947	−0.0573851
$639$	−3.55031	−0.140448
$640$	0	0
$641$	−48.4274	−1.91277	−0.956384	−	0.292111i	$-0.905642\pi$
$641$	−48.4274	−1.91277	−0.956384	+	0.292111i	$0.905642\pi$
$642$	28.5320	1.12607
$643$	43.5976	1.71932	0.859660	−	0.510867i	$-0.170676\pi$
$643$	43.5976	1.71932	0.859660	+	0.510867i	$0.170676\pi$
$644$	0.0578526	0.00227971
$645$	0	0
$646$	55.3249	2.17673
$647$	16.8934	0.664148	0.332074	−	0.943253i	$-0.392252\pi$
$647$	16.8934	0.664148	0.332074	+	0.943253i	$0.392252\pi$
$648$	−35.1973	−1.38268
$649$	−4.95624	−0.194549
$650$	0	0
$651$	−0.154336	−0.00604890
$652$	−40.6334	−1.59133
$653$	−11.2128	−0.438790	−0.219395	−	0.975636i	$-0.570408\pi$
$653$	−11.2128	−0.438790	−0.219395	+	0.975636i	$0.570408\pi$
$654$	56.2515	2.19961
$655$	0	0
$656$	10.2485	0.400137
$657$	2.21563	0.0864400
$658$	−0.0970740	−0.00378434
$659$	−1.82415	−0.0710588	−0.0355294	−	0.999369i	$-0.511312\pi$
$659$	−1.82415	−0.0710588	−0.0355294	+	0.999369i	$0.511312\pi$
$660$	0	0
$661$	1.57157	0.0611269	0.0305635	−	0.999533i	$-0.490270\pi$
$661$	1.57157	0.0611269	0.0305635	+	0.999533i	$0.490270\pi$
$662$	24.3259	0.945453
$663$	−36.5668	−1.42014
$664$	58.9778	2.28878
$665$	0	0
$666$	3.70853	0.143703
$667$	1.27619	0.0494144
$668$	−60.9028	−2.35640
$669$	50.5599	1.95476
$670$	0	0
$671$	−0.154069	−0.00594776
$672$	0.0881383	0.00340001
$673$	−37.6250	−1.45034	−0.725169	−	0.688571i	$-0.758238\pi$
$673$	−37.6250	−1.45034	−0.725169	+	0.688571i	$0.758238\pi$
$674$	−12.3115	−0.474222
$675$	0	0
$676$	−12.9731	−0.498965
$677$	14.6714	0.563869	0.281934	−	0.959434i	$-0.409024\pi$
$677$	14.6714	0.563869	0.281934	+	0.959434i	$0.409024\pi$
$678$	−35.0063	−1.34441
$679$	0.126809	0.00486648
$680$	0	0
$681$	44.4592	1.70368
$682$	9.69129	0.371099
$683$	0.255891	0.00979141	0.00489570	−	0.999988i	$-0.498442\pi$
$683$	0.255891	0.00979141	0.00489570	+	0.999988i	$0.498442\pi$
$684$	−3.17562	−0.121423
$685$	0	0
$686$	0.421800	0.0161044
$687$	−25.0047	−0.953987
$688$	7.76646	0.296094
$689$	−11.5442	−0.439797
$690$	0	0
$691$	−23.9201	−0.909963	−0.454982	−	0.890501i	$-0.650354\pi$
$691$	−23.9201	−0.909963	−0.454982	+	0.890501i	$0.650354\pi$
$692$	−44.5889	−1.69502
$693$	−0.00199446	−7.57634e−5 0
$694$	−4.31525	−0.163805
$695$	0	0
$696$	−6.54785	−0.248196
$697$	−46.5003	−1.76133
$698$	−47.8953	−1.81286
$699$	−5.65301	−0.213816
$700$	0	0
$701$	−50.2201	−1.89679	−0.948393	−	0.317097i	$-0.897292\pi$
$701$	−50.2201	−1.89679	−0.948393	+	0.317097i	$0.897292\pi$
$702$	−35.6404	−1.34516
$703$	−22.0454	−0.831457
$704$	−7.33516	−0.276454
$705$	0	0
$706$	33.5152	1.26136
$707$	0.186771	0.00702426
$708$	−51.4268	−1.93274
$709$	−13.1055	−0.492189	−0.246095	−	0.969246i	$-0.579147\pi$
$709$	−13.1055	−0.492189	−0.246095	+	0.969246i	$0.579147\pi$
$710$	0	0
$711$	−2.11346	−0.0792610
$712$	55.3088	2.07278
$713$	−8.53273	−0.319553
$714$	0.360542	0.0134929
$715$	0	0
$716$	−49.3402	−1.84393
$717$	−30.4188	−1.13601
$718$	−25.8438	−0.964483
$719$	−37.4625	−1.39712	−0.698558	−	0.715554i	$-0.746174\pi$
$719$	−37.4625	−1.39712	−0.698558	+	0.715554i	$0.746174\pi$
$720$	0	0
$721$	−0.106754	−0.00397572
$722$	−15.1927	−0.565413
$723$	2.86919	0.106706
$724$	−8.32360	−0.309344
$725$	0	0
$726$	−45.0970	−1.67371
$727$	21.1218	0.783363	0.391682	−	0.920101i	$-0.371893\pi$
$727$	21.1218	0.783363	0.391682	+	0.920101i	$0.371893\pi$
$728$	−0.141974	−0.00526192
$729$	24.3538	0.901994
$730$	0	0
$731$	−35.2386	−1.30335
$732$	−1.59865	−0.0590876
$733$	−14.4913	−0.535250	−0.267625	−	0.963523i	$-0.586239\pi$
$733$	−14.4913	−0.535250	−0.267625	+	0.963523i	$0.586239\pi$
$734$	72.9214	2.69158
$735$	0	0
$736$	4.87287	0.179617
$737$	1.79060	0.0659575
$738$	4.17617	0.153727
$739$	−39.5124	−1.45349	−0.726744	−	0.686908i	$-0.758967\pi$
$739$	−39.5124	−1.45349	−0.726744	+	0.686908i	$0.758967\pi$
$740$	0	0
$741$	−19.5220	−0.717159
$742$	0.113823	0.00417858
$743$	−39.9742	−1.46651	−0.733255	−	0.679954i	$-0.762000\pi$
$743$	−39.9742	−1.46651	−0.733255	+	0.679954i	$0.762000\pi$
$744$	43.7795	1.60503
$745$	0	0
$746$	−74.5769	−2.73045
$747$	4.11194	0.150448
$748$	−14.4696	−0.529061
$749$	−0.0859987	−0.00314233
$750$	0	0
$751$	23.6980	0.864752	0.432376	−	0.901693i	$-0.357675\pi$
$751$	23.6980	0.864752	0.432376	+	0.901693i	$0.357675\pi$
$752$	4.71137	0.171806
$753$	0.249682	0.00909893
$754$	−7.19369	−0.261979
$755$	0	0
$756$	0.224594	0.00816839
$757$	36.0636	1.31075	0.655377	−	0.755302i	$-0.272510\pi$
$757$	36.0636	1.31075	0.655377	+	0.755302i	$0.272510\pi$
$758$	−18.7868	−0.682368
$759$	−1.41722	−0.0514419
$760$	0	0
$761$	−23.7429	−0.860679	−0.430340	−	0.902667i	$-0.641606\pi$
$761$	−23.7429	−0.860679	−0.430340	+	0.902667i	$0.641606\pi$
$762$	50.8292	1.84135
$763$	−0.169548	−0.00613806
$764$	21.5462	0.779515
$765$	0	0
$766$	−44.8434	−1.62026
$767$	−24.5977	−0.888171
$768$	−43.6855	−1.57636
$769$	−44.3789	−1.60035	−0.800173	−	0.599770i	$-0.795259\pi$
$769$	−44.3789	−1.60035	−0.800173	+	0.599770i	$0.795259\pi$
$770$	0	0
$771$	49.9525	1.79899
$772$	14.9552	0.538249
$773$	−25.0988	−0.902742	−0.451371	−	0.892336i	$-0.649065\pi$
$773$	−25.0988	−0.902742	−0.451371	+	0.892336i	$0.649065\pi$
$774$	3.16476	0.113755
$775$	0	0
$776$	−35.9711	−1.29129
$777$	−0.143665	−0.00515397
$778$	31.4541	1.12769
$779$	−24.8253	−0.889457
$780$	0	0
$781$	−8.63638	−0.309034
$782$	19.9332	0.712809
$783$	4.95440	0.177056
$784$	−10.2356	−0.365559
$785$	0	0
$786$	−31.0582	−1.10781
$787$	−30.0739	−1.07202	−0.536009	−	0.844213i	$-0.680069\pi$
$787$	−30.0739	−1.07202	−0.536009	+	0.844213i	$0.680069\pi$
$788$	50.3215	1.79263
$789$	−15.8700	−0.564988
$790$	0	0
$791$	0.105513	0.00375161
$792$	0.565757	0.0201033
$793$	−0.764639	−0.0271531
$794$	33.6753	1.19509
$795$	0	0
$796$	63.5688	2.25314
$797$	−47.2484	−1.67363	−0.836813	−	0.547489i	$-0.815584\pi$
$797$	−47.2484	−1.67363	−0.836813	+	0.547489i	$0.815584\pi$
$798$	0.192483	0.00681384
$799$	−21.3768	−0.756257
$800$	0	0
$801$	3.85613	0.136250
$802$	8.74950	0.308955
$803$	5.38967	0.190197
$804$	18.5796	0.655250
$805$	0	0
$806$	48.0976	1.69417
$807$	1.18558	0.0417343
$808$	−52.9802	−1.86384
$809$	6.39951	0.224995	0.112497	−	0.993652i	$-0.464115\pi$
$809$	6.39951	0.224995	0.112497	+	0.993652i	$0.464115\pi$
$810$	0	0
$811$	−0.587673	−0.0206360	−0.0103180	−	0.999947i	$-0.503284\pi$
$811$	−0.587673	−0.0206360	−0.0103180	+	0.999947i	$0.503284\pi$
$812$	0.0453322	0.00159085
$813$	−22.2549	−0.780515
$814$	9.02124	0.316194
$815$	0	0
$816$	−17.4985	−0.612569
$817$	−18.8129	−0.658181
$818$	73.1923	2.55911
$819$	−0.00989847	−0.000345881 0
$820$	0	0
$821$	−49.1599	−1.71569	−0.857846	−	0.513907i	$-0.828198\pi$
$821$	−49.1599	−1.71569	−0.857846	+	0.513907i	$0.828198\pi$
$822$	78.8732	2.75102
$823$	−34.0871	−1.18820	−0.594100	−	0.804391i	$-0.702492\pi$
$823$	−34.0871	−1.18820	−0.594100	+	0.804391i	$0.702492\pi$
$824$	30.2822	1.05493
$825$	0	0
$826$	0.242529	0.00843865
$827$	49.1111	1.70776	0.853881	−	0.520468i	$-0.174243\pi$
$827$	49.1111	1.70776	0.853881	+	0.520468i	$0.174243\pi$
$828$	−1.14415	−0.0397621
$829$	−46.1300	−1.60216	−0.801081	−	0.598556i	$-0.795741\pi$
$829$	−46.1300	−1.60216	−0.801081	+	0.598556i	$0.795741\pi$
$830$	0	0
$831$	−45.9241	−1.59309
$832$	−36.4042	−1.26209
$833$	46.4420	1.60912
$834$	−55.5056	−1.92200
$835$	0	0
$836$	−7.72492	−0.267172
$837$	−33.1255	−1.14499
$838$	63.1560	2.18169
$839$	16.6944	0.576356	0.288178	−	0.957577i	$-0.406951\pi$
$839$	16.6944	0.576356	0.288178	+	0.957577i	$0.406951\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−62.7291	−2.16179
$843$	19.8944	0.685199
$844$	19.7637	0.680295
$845$	0	0
$846$	1.91984	0.0660054
$847$	0.135928	0.00467053
$848$	−5.52428	−0.189704
$849$	−6.95009	−0.238527
$850$	0	0
$851$	−7.94278	−0.272275
$852$	−89.6126	−3.07008
$853$	12.5126	0.428424	0.214212	−	0.976787i	$-0.431282\pi$
$853$	12.5126	0.428424	0.214212	+	0.976787i	$0.431282\pi$
$854$	0.00753920	0.000257986 0
$855$	0	0
$856$	24.3947	0.833794
$857$	−6.82871	−0.233264	−0.116632	−	0.993175i	$-0.537210\pi$
$857$	−6.82871	−0.233264	−0.116632	+	0.993175i	$0.537210\pi$
$858$	7.98865	0.272728
$859$	18.0611	0.616236	0.308118	−	0.951348i	$-0.400301\pi$
$859$	18.0611	0.616236	0.308118	+	0.951348i	$0.400301\pi$
$860$	0	0
$861$	−0.161781	−0.00551349
$862$	29.8869	1.01795
$863$	47.8533	1.62894	0.814472	−	0.580203i	$-0.197027\pi$
$863$	47.8533	1.62894	0.814472	+	0.580203i	$0.197027\pi$
$864$	18.9173	0.643581
$865$	0	0
$866$	−13.1612	−0.447235
$867$	48.7336	1.65508
$868$	−0.303095	−0.0102877
$869$	−5.14113	−0.174401
$870$	0	0
$871$	8.88668	0.301114
$872$	48.0947	1.62869
$873$	−2.50791	−0.0848798
$874$	10.6418	0.359963
$875$	0	0
$876$	55.9242	1.88950
$877$	20.4812	0.691602	0.345801	−	0.938308i	$-0.387607\pi$
$877$	20.4812	0.691602	0.345801	+	0.938308i	$0.387607\pi$
$878$	−79.4106	−2.67998
$879$	−37.9253	−1.27919
$880$	0	0
$881$	−30.9655	−1.04325	−0.521627	−	0.853174i	$-0.674675\pi$
$881$	−30.9655	−1.04325	−0.521627	+	0.853174i	$0.674675\pi$
$882$	−4.17093	−0.140442
$883$	13.2897	0.447235	0.223618	−	0.974677i	$-0.428213\pi$
$883$	13.2897	0.447235	0.223618	+	0.974677i	$0.428213\pi$
$884$	−71.8122	−2.41531
$885$	0	0
$886$	−38.1329	−1.28110
$887$	33.9462	1.13980	0.569900	−	0.821714i	$-0.306982\pi$
$887$	33.9462	1.13980	0.569900	+	0.821714i	$0.306982\pi$
$888$	40.7526	1.36757
$889$	−0.153205	−0.00513833
$890$	0	0
$891$	−5.96942	−0.199983
$892$	99.2929	3.32457
$893$	−11.4125	−0.381905
$894$	66.6687	2.22973
$895$	0	0
$896$	0.261205	0.00872624
$897$	−7.03363	−0.234846
$898$	12.9004	0.430492
$899$	−6.68608	−0.222993
$900$	0	0
$901$	25.0652	0.835043
$902$	10.1588	0.338251
$903$	−0.122600	−0.00407988
$904$	−29.9302	−0.995462
$905$	0	0
$906$	61.6129	2.04695
$907$	20.0608	0.666107	0.333053	−	0.942908i	$-0.391921\pi$
$907$	20.0608	0.666107	0.333053	+	0.942908i	$0.391921\pi$
$908$	87.3119	2.89755
$909$	−3.69379	−0.122515
$910$	0	0
$911$	−25.3431	−0.839655	−0.419827	−	0.907604i	$-0.637909\pi$
$911$	−25.3431	−0.839655	−0.419827	+	0.907604i	$0.637909\pi$
$912$	−9.34195	−0.309343
$913$	10.0026	0.331037
$914$	−82.8469	−2.74033
$915$	0	0
$916$	−49.1058	−1.62250
$917$	0.0936129	0.00309137
$918$	77.3840	2.55405
$919$	20.7223	0.683566	0.341783	−	0.939779i	$-0.388969\pi$
$919$	20.7223	0.683566	0.341783	+	0.939779i	$0.388969\pi$
$920$	0	0
$921$	27.8136	0.916489
$922$	−32.0409	−1.05521
$923$	−42.8621	−1.41082
$924$	−0.0503418	−0.00165612
$925$	0	0
$926$	34.6947	1.14014
$927$	2.11128	0.0693434
$928$	3.81829	0.125342
$929$	14.2042	0.466024	0.233012	−	0.972474i	$-0.425142\pi$
$929$	14.2042	0.466024	0.233012	+	0.972474i	$0.425142\pi$
$930$	0	0
$931$	24.7941	0.812594
$932$	−11.1017	−0.363649
$933$	17.4071	0.569883
$934$	−7.78843	−0.254845
$935$	0	0
$936$	2.80783	0.0917769
$937$	38.5926	1.26077	0.630383	−	0.776284i	$-0.282898\pi$
$937$	38.5926	1.26077	0.630383	+	0.776284i	$0.282898\pi$
$938$	−0.0876210	−0.00286093
$939$	7.58664	0.247581
$940$	0	0
$941$	−2.07239	−0.0675578	−0.0337789	−	0.999429i	$-0.510754\pi$
$941$	−2.07239	−0.0675578	−0.0337789	+	0.999429i	$0.510754\pi$
$942$	49.9782	1.62838
$943$	−8.94436	−0.291268
$944$	−11.7708	−0.383108
$945$	0	0
$946$	7.69849	0.250299
$947$	17.6870	0.574751	0.287376	−	0.957818i	$-0.407217\pi$
$947$	17.6870	0.574751	0.287376	+	0.957818i	$0.407217\pi$
$948$	−53.3453	−1.73258
$949$	26.7488	0.868303
$950$	0	0
$951$	−35.4641	−1.15000
$952$	0.308261	0.00999080
$953$	26.5946	0.861485	0.430742	−	0.902475i	$-0.358252\pi$
$953$	26.5946	0.861485	0.430742	+	0.902475i	$0.358252\pi$
$954$	−2.25109	−0.0728817
$955$	0	0
$956$	−59.7383	−1.93207
$957$	−1.11051	−0.0358976
$958$	2.90750	0.0939370
$959$	−0.237733	−0.00767679
$960$	0	0
$961$	13.7037	0.442055
$962$	44.7722	1.44351
$963$	1.70080	0.0548076
$964$	5.63470	0.181481
$965$	0	0
$966$	0.0693503	0.00223131
$967$	11.9643	0.384745	0.192372	−	0.981322i	$-0.438382\pi$
$967$	11.9643	0.384745	0.192372	+	0.981322i	$0.438382\pi$
$968$	−38.5577	−1.23929
$969$	42.3870	1.36167
$970$	0	0
$971$	−18.6333	−0.597970	−0.298985	−	0.954258i	$-0.596648\pi$
$971$	−18.6333	−0.597970	−0.298985	+	0.954258i	$0.596648\pi$
$972$	−9.29289	−0.298070
$973$	0.167300	0.00536340
$974$	−50.9724	−1.63326
$975$	0	0
$976$	−0.365906	−0.0117124
$977$	−12.5708	−0.402175	−0.201088	−	0.979573i	$-0.564448\pi$
$977$	−12.5708	−0.402175	−0.201088	+	0.979573i	$0.564448\pi$
$978$	−48.7090	−1.55754
$979$	9.38030	0.299796
$980$	0	0
$981$	3.35317	0.107058
$982$	−57.8818	−1.84708
$983$	22.4214	0.715131	0.357565	−	0.933888i	$-0.383607\pi$
$983$	22.4214	0.715131	0.357565	+	0.933888i	$0.383607\pi$
$984$	45.8915	1.46297
$985$	0	0
$986$	15.6193	0.497418
$987$	−0.0743730	−0.00236732
$988$	−38.3386	−1.21971
$989$	−6.77816	−0.215533
$990$	0	0
$991$	−42.0680	−1.33633	−0.668167	−	0.744011i	$-0.732921\pi$
$991$	−42.0680	−1.33633	−0.668167	+	0.744011i	$0.732921\pi$
$992$	−25.5294	−0.810560
$993$	18.6372	0.591434
$994$	0.422613	0.0134045
$995$	0	0
$996$	103.788	3.28866
$997$	21.6692	0.686269	0.343134	−	0.939286i	$-0.388511\pi$
$997$	21.6692	0.686269	0.343134	+	0.939286i	$0.388511\pi$
$998$	49.4269	1.56458
$999$	−30.8352	−0.975584

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.a.j.1.5	yes	5
3.2	odd	2		6525.2.a.bp.1.1		5
5.2	odd	4		725.2.b.g.349.9		10
5.3	odd	4		725.2.b.g.349.2		10
5.4	even	2		725.2.a.i.1.1	✓	5
15.14	odd	2		6525.2.a.bo.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
725.2.a.i.1.1	✓	5	5.4	even	2
725.2.a.j.1.5	yes	5	1.1	even	1	trivial
725.2.b.g.349.2		10	5.3	odd	4
725.2.b.g.349.9		10	5.2	odd	4
6525.2.a.bo.1.5		5	15.14	odd	2
6525.2.a.bp.1.1		5	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35417 q^{2} +1.80364 q^{3} +3.54210 q^{4} +4.24606 q^{6} -0.0127981 q^{7} +3.63036 q^{8} +0.253109 q^{9} +0.615705 q^{11} +6.38866 q^{12} +3.05573 q^{13} -0.0301289 q^{14} +1.46227 q^{16} -6.63473 q^{17} +0.595861 q^{18} -3.54210 q^{19} -0.0230832 q^{21} +1.44947 q^{22} -1.27619 q^{23} +6.54785 q^{24} +7.19369 q^{26} -4.95440 q^{27} -0.0453322 q^{28} -1.00000 q^{29} +6.68608 q^{31} -3.81829 q^{32} +1.11051 q^{33} -15.6193 q^{34} +0.896537 q^{36} +6.22381 q^{37} -8.33869 q^{38} +5.51142 q^{39} +7.00863 q^{41} -0.0543416 q^{42} +5.31124 q^{43} +2.18089 q^{44} -3.00437 q^{46} +3.22196 q^{47} +2.63740 q^{48} -6.99984 q^{49} -11.9666 q^{51} +10.8237 q^{52} -3.77788 q^{53} -11.6635 q^{54} -0.0464618 q^{56} -6.38866 q^{57} -2.35417 q^{58} -8.04970 q^{59} -0.250232 q^{61} +15.7402 q^{62} -0.00323932 q^{63} -11.9134 q^{64} +2.61432 q^{66} +2.90821 q^{67} -23.5009 q^{68} -2.30179 q^{69} -14.0268 q^{71} +0.918877 q^{72} +8.75366 q^{73} +14.6519 q^{74} -12.5465 q^{76} -0.00787986 q^{77} +12.9748 q^{78} -8.35000 q^{79} -9.69526 q^{81} +16.4995 q^{82} +16.2457 q^{83} -0.0817629 q^{84} +12.5035 q^{86} -1.80364 q^{87} +2.23523 q^{88} +15.2351 q^{89} -0.0391075 q^{91} -4.52040 q^{92} +12.0593 q^{93} +7.58503 q^{94} -6.88681 q^{96} -9.90841 q^{97} -16.4788 q^{98} +0.155840 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 6 q^{3} + 4 q^{4} - q^{6} + 10 q^{7} - 3 q^{8} + 7 q^{9} + 2 q^{11} + 18 q^{12} + 8 q^{13} - 13 q^{14} + 6 q^{16} + 3 q^{17} - 16 q^{18} - 4 q^{19} + 15 q^{21} + 16 q^{22} + 3 q^{23} - 19 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) \) 5 * q + 6 * q^3 + 4 * q^4 - q^6 + 10 * q^7 - 3 * q^8 + 7 * q^9 + 2 * q^11 + 18 * q^12 + 8 * q^13 - 13 * q^14 + 6 * q^16 + 3 * q^17 - 16 * q^18 - 4 * q^19 + 15 * q^21 + 16 * q^22 + 3 * q^23 - 19 * q^24 - 4 * q^26 + 21 * q^27 + 10 * q^28 - 5 * q^29 + 5 * q^31 + 4 * q^32 + 3 * q^33 - 9 * q^34 + 35 * q^36 + 4 * q^37 + 3 * q^38 + 4 * q^39 - 3 * q^41 - 16 * q^42 + 12 * q^43 - 19 * q^44 - q^47 + 20 * q^48 + 13 * q^49 - 7 * q^51 + 26 * q^52 + 12 * q^53 - 58 * q^54 - 8 * q^56 - 18 * q^57 - 15 * q^59 - 7 * q^61 + 10 * q^62 + 33 * q^63 - 9 * q^64 + 35 * q^66 + 17 * q^67 - 43 * q^68 + 18 * q^69 - 17 * q^71 - 49 * q^72 + 10 * q^73 + 12 * q^74 - 34 * q^76 - 8 * q^77 - 39 * q^78 - 2 * q^79 + 21 * q^81 + 31 * q^82 + 19 * q^83 + 38 * q^84 + 31 * q^86 - 6 * q^87 - 3 * q^88 + 14 * q^89 - 15 * q^91 - q^92 - 15 * q^93 + 27 * q^94 + 6 * q^96 - 9 * q^97 - 51 * q^98 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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