LMFDB - Embedded newform 6025.2.a.p.1.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6025,2,Mod(1,6025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6025, 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("6025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6025 = 5^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6025.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:	$48.1098672178$
Analytic rank:	$1$
Dimension:	$46$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.28
Character	$\chi$	$=$	6025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.636788 q^{2} -2.24729 q^{3} -1.59450 q^{4} -1.43105 q^{6} -1.14493 q^{7} -2.28894 q^{8} +2.05030 q^{9} +O(q^{10})$	\(q+0.636788 q^{2} -2.24729 q^{3} -1.59450 q^{4} -1.43105 q^{6} -1.14493 q^{7} -2.28894 q^{8} +2.05030 q^{9} -0.440443 q^{11} +3.58330 q^{12} -0.0742578 q^{13} -0.729079 q^{14} +1.73143 q^{16} +3.83474 q^{17} +1.30561 q^{18} -4.81267 q^{19} +2.57299 q^{21} -0.280469 q^{22} -6.24519 q^{23} +5.14390 q^{24} -0.0472865 q^{26} +2.13425 q^{27} +1.82559 q^{28} +2.54550 q^{29} -0.429517 q^{31} +5.68043 q^{32} +0.989801 q^{33} +2.44192 q^{34} -3.26920 q^{36} +10.9158 q^{37} -3.06465 q^{38} +0.166878 q^{39} -4.91543 q^{41} +1.63845 q^{42} +1.72125 q^{43} +0.702286 q^{44} -3.97686 q^{46} -0.485128 q^{47} -3.89103 q^{48} -5.68913 q^{49} -8.61777 q^{51} +0.118404 q^{52} +2.29994 q^{53} +1.35907 q^{54} +2.62068 q^{56} +10.8155 q^{57} +1.62094 q^{58} -1.98580 q^{59} +8.96972 q^{61} -0.273511 q^{62} -2.34745 q^{63} +0.154367 q^{64} +0.630294 q^{66} -1.53915 q^{67} -6.11450 q^{68} +14.0347 q^{69} +14.1657 q^{71} -4.69300 q^{72} +15.2643 q^{73} +6.95108 q^{74} +7.67381 q^{76} +0.504277 q^{77} +0.106266 q^{78} +1.84814 q^{79} -10.9472 q^{81} -3.13009 q^{82} +1.08755 q^{83} -4.10263 q^{84} +1.09608 q^{86} -5.72046 q^{87} +1.00815 q^{88} +1.27098 q^{89} +0.0850201 q^{91} +9.95795 q^{92} +0.965247 q^{93} -0.308924 q^{94} -12.7656 q^{96} +14.7657 q^{97} -3.62277 q^{98} -0.903038 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	$ 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) $ 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.636788	0.450277	0.225139	−	0.974327i	$-0.427716\pi$
$2$	0.636788	0.450277	0.225139	+	0.974327i	$0.427716\pi$
$3$	−2.24729	−1.29747	−0.648736	−	0.761014i	$-0.724702\pi$
$3$	−2.24729	−1.29747	−0.648736	+	0.761014i	$0.724702\pi$
$4$	−1.59450	−0.797250
$5$	0	0
$5$	0	0
$6$	−1.43105	−0.584222
$7$	−1.14493	−0.432743	−0.216372	−	0.976311i	$-0.569422\pi$
$7$	−1.14493	−0.432743	−0.216372	+	0.976311i	$0.569422\pi$
$8$	−2.28894	−0.809261
$9$	2.05030	0.683433
$10$	0	0
$11$	−0.440443	−0.132798	−0.0663992	−	0.997793i	$-0.521151\pi$
$11$	−0.440443	−0.132798	−0.0663992	+	0.997793i	$0.521151\pi$
$12$	3.58330	1.03441
$13$	−0.0742578	−0.0205954	−0.0102977	−	0.999947i	$-0.503278\pi$
$13$	−0.0742578	−0.0205954	−0.0102977	+	0.999947i	$0.503278\pi$
$14$	−0.729079	−0.194855
$15$	0	0
$16$	1.73143	0.432858
$17$	3.83474	0.930062	0.465031	−	0.885294i	$-0.346043\pi$
$17$	3.83474	0.930062	0.465031	+	0.885294i	$0.346043\pi$
$18$	1.30561	0.307734
$19$	−4.81267	−1.10410	−0.552051	−	0.833810i	$-0.686155\pi$
$19$	−4.81267	−1.10410	−0.552051	+	0.833810i	$0.686155\pi$
$20$	0	0
$21$	2.57299	0.561472
$22$	−0.280469	−0.0597961
$23$	−6.24519	−1.30221	−0.651106	−	0.758987i	$-0.725695\pi$
$23$	−6.24519	−1.30221	−0.651106	+	0.758987i	$0.725695\pi$
$24$	5.14390	1.04999
$25$	0	0
$26$	−0.0472865	−0.00927364
$27$	2.13425	0.410737
$28$	1.82559	0.345005
$29$	2.54550	0.472687	0.236343	−	0.971670i	$-0.424051\pi$
$29$	2.54550	0.472687	0.236343	+	0.971670i	$0.424051\pi$
$30$	0	0
$31$	−0.429517	−0.0771435	−0.0385717	−	0.999256i	$-0.512281\pi$
$31$	−0.429517	−0.0771435	−0.0385717	+	0.999256i	$0.512281\pi$
$32$	5.68043	1.00417
$33$	0.989801	0.172302
$34$	2.44192	0.418786
$35$	0	0
$36$	−3.26920	−0.544867
$37$	10.9158	1.79455	0.897276	−	0.441470i	$-0.145543\pi$
$37$	10.9158	1.79455	0.897276	+	0.441470i	$0.145543\pi$
$38$	−3.06465	−0.497153
$39$	0.166878	0.0267219
$40$	0	0
$41$	−4.91543	−0.767662	−0.383831	−	0.923403i	$-0.625395\pi$
$41$	−4.91543	−0.767662	−0.383831	+	0.923403i	$0.625395\pi$
$42$	1.63845	0.252818
$43$	1.72125	0.262489	0.131244	−	0.991350i	$-0.458103\pi$
$43$	1.72125	0.262489	0.131244	+	0.991350i	$0.458103\pi$
$44$	0.702286	0.105874
$45$	0	0
$46$	−3.97686	−0.586356
$47$	−0.485128	−0.0707631	−0.0353816	−	0.999374i	$-0.511265\pi$
$47$	−0.485128	−0.0707631	−0.0353816	+	0.999374i	$0.511265\pi$
$48$	−3.89103	−0.561621
$49$	−5.68913	−0.812733
$50$	0	0
$51$	−8.61777	−1.20673
$52$	0.118404	0.0164197
$53$	2.29994	0.315921	0.157960	−	0.987445i	$-0.449508\pi$
$53$	2.29994	0.315921	0.157960	+	0.987445i	$0.449508\pi$
$54$	1.35907	0.184946
$55$	0	0
$56$	2.62068	0.350203
$57$	10.8155	1.43254
$58$	1.62094	0.212840
$59$	−1.98580	−0.258530	−0.129265	−	0.991610i	$-0.541262\pi$
$59$	−1.98580	−0.258530	−0.129265	+	0.991610i	$0.541262\pi$
$60$	0	0
$61$	8.96972	1.14846	0.574228	−	0.818696i	$-0.305302\pi$
$61$	8.96972	1.14846	0.574228	+	0.818696i	$0.305302\pi$
$62$	−0.273511	−0.0347360
$63$	−2.34745	−0.295751
$64$	0.154367	0.0192959
$65$	0	0
$66$	0.630294	0.0775838
$67$	−1.53915	−0.188038	−0.0940188	−	0.995570i	$-0.529971\pi$
$67$	−1.53915	−0.188038	−0.0940188	+	0.995570i	$0.529971\pi$
$68$	−6.11450	−0.741492
$69$	14.0347	1.68958
$70$	0	0
$71$	14.1657	1.68116	0.840579	−	0.541688i	$-0.182215\pi$
$71$	14.1657	1.68116	0.840579	+	0.541688i	$0.182215\pi$
$72$	−4.69300	−0.553075
$73$	15.2643	1.78656	0.893278	−	0.449504i	$-0.148399\pi$
$73$	15.2643	1.78656	0.893278	+	0.449504i	$0.148399\pi$
$74$	6.95108	0.808046
$75$	0	0
$76$	7.67381	0.880246
$77$	0.504277	0.0574677
$78$	0.106266	0.0120323
$79$	1.84814	0.207932	0.103966	−	0.994581i	$-0.466847\pi$
$79$	1.84814	0.207932	0.103966	+	0.994581i	$0.466847\pi$
$80$	0	0
$81$	−10.9472	−1.21635
$82$	−3.13009	−0.345661
$83$	1.08755	0.119375	0.0596873	−	0.998217i	$-0.480990\pi$
$83$	1.08755	0.119375	0.0596873	+	0.998217i	$0.480990\pi$
$84$	−4.10263	−0.447634
$85$	0	0
$86$	1.09608	0.118193
$87$	−5.72046	−0.613298
$88$	1.00815	0.107469
$89$	1.27098	0.134723	0.0673617	−	0.997729i	$-0.478542\pi$
$89$	1.27098	0.134723	0.0673617	+	0.997729i	$0.478542\pi$
$90$	0	0
$91$	0.0850201	0.00891252
$92$	9.95795	1.03819
$93$	0.965247	0.100091
$94$	−0.308924	−0.0318630
$95$	0	0
$96$	−12.7656	−1.30288
$97$	14.7657	1.49923	0.749614	−	0.661876i	$-0.230239\pi$
$97$	14.7657	1.49923	0.749614	+	0.661876i	$0.230239\pi$
$98$	−3.62277	−0.365955
$99$	−0.903038	−0.0907588
$100$	0	0
$101$	−5.30890	−0.528255	−0.264127	−	0.964488i	$-0.585084\pi$
$101$	−5.30890	−0.528255	−0.264127	+	0.964488i	$0.585084\pi$
$102$	−5.48770	−0.543363
$103$	3.19134	0.314452	0.157226	−	0.987563i	$-0.449745\pi$
$103$	3.19134	0.314452	0.157226	+	0.987563i	$0.449745\pi$
$104$	0.169971	0.0166671
$105$	0	0
$106$	1.46457	0.142252
$107$	−9.30857	−0.899893	−0.449947	−	0.893055i	$-0.648557\pi$
$107$	−9.30857	−0.899893	−0.449947	+	0.893055i	$0.648557\pi$
$108$	−3.40307	−0.327460
$109$	3.00453	0.287782	0.143891	−	0.989594i	$-0.454039\pi$
$109$	3.00453	0.287782	0.143891	+	0.989594i	$0.454039\pi$
$110$	0	0
$111$	−24.5310	−2.32838
$112$	−1.98237	−0.187317
$113$	5.77781	0.543530	0.271765	−	0.962364i	$-0.412393\pi$
$113$	5.77781	0.543530	0.271765	+	0.962364i	$0.412393\pi$
$114$	6.88716	0.645041
$115$	0	0
$116$	−4.05879	−0.376850
$117$	−0.152251	−0.0140756
$118$	−1.26454	−0.116410
$119$	−4.39052	−0.402478
$120$	0	0
$121$	−10.8060	−0.982365
$122$	5.71182	0.517124
$123$	11.0464	0.996019
$124$	0.684865	0.0615027
$125$	0	0
$126$	−1.49483	−0.133170
$127$	−19.2286	−1.70627	−0.853133	−	0.521694i	$-0.825300\pi$
$127$	−19.2286	−1.70627	−0.853133	+	0.521694i	$0.825300\pi$
$128$	−11.2626	−0.995479
$129$	−3.86815	−0.340572
$130$	0	0
$131$	−15.0685	−1.31654	−0.658271	−	0.752781i	$-0.728712\pi$
$131$	−15.0685	−1.31654	−0.658271	+	0.752781i	$0.728712\pi$
$132$	−1.57824	−0.137368
$133$	5.51018	0.477793
$134$	−0.980116	−0.0846691
$135$	0	0
$136$	−8.77749	−0.752663
$137$	10.3778	0.886638	0.443319	−	0.896364i	$-0.353801\pi$
$137$	10.3778	0.886638	0.443319	+	0.896364i	$0.353801\pi$
$138$	8.93715	0.760781
$139$	−2.30756	−0.195725	−0.0978623	−	0.995200i	$-0.531200\pi$
$139$	−2.30756	−0.195725	−0.0978623	+	0.995200i	$0.531200\pi$
$140$	0	0
$141$	1.09022	0.0918132
$142$	9.02055	0.756988
$143$	0.0327063	0.00273504
$144$	3.54995	0.295829
$145$	0	0
$146$	9.72016	0.804446
$147$	12.7851	1.05450
$148$	−17.4053	−1.43071
$149$	−11.1945	−0.917092	−0.458546	−	0.888671i	$-0.651630\pi$
$149$	−11.1945	−0.917092	−0.458546	+	0.888671i	$0.651630\pi$
$150$	0	0
$151$	−6.66973	−0.542775	−0.271387	−	0.962470i	$-0.587482\pi$
$151$	−6.66973	−0.542775	−0.271387	+	0.962470i	$0.587482\pi$
$152$	11.0159	0.893507
$153$	7.86237	0.635635
$154$	0.321118	0.0258764
$155$	0	0
$156$	−0.266088	−0.0213041
$157$	−22.8522	−1.82380	−0.911901	−	0.410410i	$-0.865386\pi$
$157$	−22.8522	−1.82380	−0.911901	+	0.410410i	$0.865386\pi$
$158$	1.17687	0.0936269
$159$	−5.16862	−0.409898
$160$	0	0
$161$	7.15031	0.563523
$162$	−6.97103	−0.547696
$163$	6.90573	0.540898	0.270449	−	0.962734i	$-0.412828\pi$
$163$	6.90573	0.540898	0.270449	+	0.962734i	$0.412828\pi$
$164$	7.83766	0.612018
$165$	0	0
$166$	0.692542	0.0537517
$167$	2.49550	0.193108	0.0965538	−	0.995328i	$-0.469218\pi$
$167$	2.49550	0.193108	0.0965538	+	0.995328i	$0.469218\pi$
$168$	−5.88941	−0.454378
$169$	−12.9945	−0.999576
$170$	0	0
$171$	−9.86741	−0.754580
$172$	−2.74454	−0.209269
$173$	−0.427640	−0.0325129	−0.0162565	−	0.999868i	$-0.505175\pi$
$173$	−0.427640	−0.0325129	−0.0162565	+	0.999868i	$0.505175\pi$
$174$	−3.64272	−0.276154
$175$	0	0
$176$	−0.762597	−0.0574829
$177$	4.46267	0.335435
$178$	0.809344	0.0606629
$179$	6.01228	0.449379	0.224689	−	0.974430i	$-0.427863\pi$
$179$	6.01228	0.449379	0.224689	+	0.974430i	$0.427863\pi$
$180$	0	0
$181$	15.2285	1.13193	0.565963	−	0.824431i	$-0.308505\pi$
$181$	15.2285	1.13193	0.565963	+	0.824431i	$0.308505\pi$
$182$	0.0541398	0.00401311
$183$	−20.1575	−1.49009
$184$	14.2948	1.05383
$185$	0	0
$186$	0.614658	0.0450689
$187$	−1.68898	−0.123511
$188$	0.773536	0.0564159
$189$	−2.44357	−0.177744
$190$	0	0
$191$	−0.578755	−0.0418772	−0.0209386	−	0.999781i	$-0.506665\pi$
$191$	−0.578755	−0.0418772	−0.0209386	+	0.999781i	$0.506665\pi$
$192$	−0.346907	−0.0250359
$193$	−2.26365	−0.162941	−0.0814706	−	0.996676i	$-0.525962\pi$
$193$	−2.26365	−0.162941	−0.0814706	+	0.996676i	$0.525962\pi$
$194$	9.40261	0.675068
$195$	0	0
$196$	9.07132	0.647952
$197$	12.3730	0.881542	0.440771	−	0.897620i	$-0.354705\pi$
$197$	12.3730	0.881542	0.440771	+	0.897620i	$0.354705\pi$
$198$	−0.575044	−0.0408666
$199$	11.8775	0.841976	0.420988	−	0.907066i	$-0.361684\pi$
$199$	11.8775	0.841976	0.420988	+	0.907066i	$0.361684\pi$
$200$	0	0
$201$	3.45892	0.243974
$202$	−3.38064	−0.237861
$203$	−2.91442	−0.204552
$204$	13.7410	0.962065
$205$	0	0
$206$	2.03221	0.141590
$207$	−12.8045	−0.889974
$208$	−0.128572	−0.00891489
$209$	2.11971	0.146623
$210$	0	0
$211$	−11.3488	−0.781281	−0.390641	−	0.920543i	$-0.627746\pi$
$211$	−11.3488	−0.781281	−0.390641	+	0.920543i	$0.627746\pi$
$212$	−3.66725	−0.251868
$213$	−31.8344	−2.18126
$214$	−5.92759	−0.405202
$215$	0	0
$216$	−4.88517	−0.332394
$217$	0.491767	0.0333833
$218$	1.91325	0.129582
$219$	−34.3034	−2.31801
$220$	0	0
$221$	−0.284760	−0.0191550
$222$	−15.6211	−1.04842
$223$	−3.01780	−0.202087	−0.101043	−	0.994882i	$-0.532218\pi$
$223$	−3.01780	−0.202087	−0.101043	+	0.994882i	$0.532218\pi$
$224$	−6.50370	−0.434547
$225$	0	0
$226$	3.67924	0.244739
$227$	14.3773	0.954257	0.477129	−	0.878833i	$-0.341678\pi$
$227$	14.3773	0.954257	0.477129	+	0.878833i	$0.341678\pi$
$228$	−17.2452	−1.14209
$229$	−26.5382	−1.75369	−0.876847	−	0.480769i	$-0.840358\pi$
$229$	−26.5382	−1.75369	−0.876847	+	0.480769i	$0.840358\pi$
$230$	0	0
$231$	−1.13325	−0.0745627
$232$	−5.82648	−0.382527
$233$	15.2500	0.999061	0.499530	−	0.866296i	$-0.333506\pi$
$233$	15.2500	0.999061	0.499530	+	0.866296i	$0.333506\pi$
$234$	−0.0969514	−0.00633791
$235$	0	0
$236$	3.16636	0.206113
$237$	−4.15329	−0.269785
$238$	−2.79583	−0.181227
$239$	−9.98370	−0.645792	−0.322896	−	0.946434i	$-0.604656\pi$
$239$	−9.98370	−0.645792	−0.322896	+	0.946434i	$0.604656\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−6.88114	−0.442337
$243$	18.1987	1.16745
$244$	−14.3022	−0.915606
$245$	0	0
$246$	7.03421	0.448485
$247$	0.357378	0.0227394
$248$	0.983137	0.0624292
$249$	−2.44405	−0.154885
$250$	0	0
$251$	−19.2151	−1.21285	−0.606424	−	0.795141i	$-0.707397\pi$
$251$	−19.2151	−1.21285	−0.606424	+	0.795141i	$0.707397\pi$
$252$	3.74301	0.235788
$253$	2.75065	0.172932
$254$	−12.2446	−0.768293
$255$	0	0
$256$	−7.48060	−0.467538
$257$	−29.6396	−1.84887	−0.924435	−	0.381339i	$-0.875463\pi$
$257$	−29.6396	−1.84887	−0.924435	+	0.381339i	$0.875463\pi$
$258$	−2.46320	−0.153352
$259$	−12.4979	−0.776580
$260$	0	0
$261$	5.21903	0.323050
$262$	−9.59545	−0.592809
$263$	−24.3550	−1.50180	−0.750898	−	0.660418i	$-0.770379\pi$
$263$	−24.3550	−1.50180	−0.750898	+	0.660418i	$0.770379\pi$
$264$	−2.26559	−0.139438
$265$	0	0
$266$	3.50882	0.215139
$267$	−2.85625	−0.174800
$268$	2.45418	0.149913
$269$	−15.0768	−0.919248	−0.459624	−	0.888114i	$-0.652016\pi$
$269$	−15.0768	−0.919248	−0.459624	+	0.888114i	$0.652016\pi$
$270$	0	0
$271$	23.5456	1.43029	0.715146	−	0.698975i	$-0.246360\pi$
$271$	23.5456	1.43029	0.715146	+	0.698975i	$0.246360\pi$
$272$	6.63960	0.402585
$273$	−0.191064	−0.0115637
$274$	6.60848	0.399233
$275$	0	0
$276$	−22.3784	−1.34702
$277$	4.70682	0.282806	0.141403	−	0.989952i	$-0.454839\pi$
$277$	4.70682	0.282806	0.141403	+	0.989952i	$0.454839\pi$
$278$	−1.46943	−0.0881304
$279$	−0.880637	−0.0527224
$280$	0	0
$281$	2.94319	0.175576	0.0877879	−	0.996139i	$-0.472020\pi$
$281$	2.94319	0.175576	0.0877879	+	0.996139i	$0.472020\pi$
$282$	0.694240	0.0413414
$283$	12.3920	0.736628	0.368314	−	0.929702i	$-0.379935\pi$
$283$	12.3920	0.736628	0.368314	+	0.929702i	$0.379935\pi$
$284$	−22.5872	−1.34030
$285$	0	0
$286$	0.0208270	0.00123153
$287$	5.62783	0.332201
$288$	11.6466	0.686281
$289$	−2.29474	−0.134985
$290$	0	0
$291$	−33.1827	−1.94521
$292$	−24.3390	−1.42433
$293$	−11.0669	−0.646532	−0.323266	−	0.946308i	$-0.604781\pi$
$293$	−11.0669	−0.646532	−0.323266	+	0.946308i	$0.604781\pi$
$294$	8.14141	0.474817
$295$	0	0
$296$	−24.9856	−1.45226
$297$	−0.940016	−0.0545453
$298$	−7.12855	−0.412946
$299$	0.463754	0.0268196
$300$	0	0
$301$	−1.97072	−0.113590
$302$	−4.24720	−0.244399
$303$	11.9306	0.685396
$304$	−8.33282	−0.477920
$305$	0	0
$306$	5.00666	0.286212
$307$	−19.5529	−1.11595	−0.557973	−	0.829859i	$-0.688421\pi$
$307$	−19.5529	−1.11595	−0.557973	+	0.829859i	$0.688421\pi$
$308$	−0.804069	−0.0458161
$309$	−7.17184	−0.407992
$310$	0	0
$311$	6.79509	0.385314	0.192657	−	0.981266i	$-0.438289\pi$
$311$	6.79509	0.385314	0.192657	+	0.981266i	$0.438289\pi$
$312$	−0.381974	−0.0216250
$313$	4.82490	0.272719	0.136360	−	0.990659i	$-0.456460\pi$
$313$	4.82490	0.272719	0.136360	+	0.990659i	$0.456460\pi$
$314$	−14.5520	−0.821217
$315$	0	0
$316$	−2.94686	−0.165774
$317$	4.07296	0.228760	0.114380	−	0.993437i	$-0.463512\pi$
$317$	4.07296	0.228760	0.114380	+	0.993437i	$0.463512\pi$
$318$	−3.29132	−0.184568
$319$	−1.12115	−0.0627721
$320$	0	0
$321$	20.9190	1.16759
$322$	4.55324	0.253742
$323$	−18.4554	−1.02688
$324$	17.4553	0.969737
$325$	0	0
$326$	4.39749	0.243554
$327$	−6.75204	−0.373389
$328$	11.2511	0.621239
$329$	0.555438	0.0306223
$330$	0	0
$331$	−29.2742	−1.60906	−0.804528	−	0.593915i	$-0.797582\pi$
$331$	−29.2742	−1.60906	−0.804528	+	0.593915i	$0.797582\pi$
$332$	−1.73411	−0.0951714
$333$	22.3807	1.22645
$334$	1.58911	0.0869520
$335$	0	0
$336$	4.45496	0.243038
$337$	−3.70484	−0.201816	−0.100908	−	0.994896i	$-0.532175\pi$
$337$	−3.70484	−0.201816	−0.100908	+	0.994896i	$0.532175\pi$
$338$	−8.27474	−0.450086
$339$	−12.9844	−0.705215
$340$	0	0
$341$	0.189178	0.0102445
$342$	−6.28345	−0.339770
$343$	14.5282	0.784448
$344$	−3.93984	−0.212422
$345$	0	0
$346$	−0.272317	−0.0146398
$347$	−20.8963	−1.12177	−0.560886	−	0.827893i	$-0.689540\pi$
$347$	−20.8963	−1.12177	−0.560886	+	0.827893i	$0.689540\pi$
$348$	9.12128	0.488952
$349$	−28.4273	−1.52168	−0.760840	−	0.648939i	$-0.775213\pi$
$349$	−28.4273	−1.52168	−0.760840	+	0.648939i	$0.775213\pi$
$350$	0	0
$351$	−0.158485	−0.00845930
$352$	−2.50190	−0.133352
$353$	11.3118	0.602067	0.301033	−	0.953614i	$-0.402668\pi$
$353$	11.3118	0.602067	0.301033	+	0.953614i	$0.402668\pi$
$354$	2.84178	0.151039
$355$	0	0
$356$	−2.02657	−0.107408
$357$	9.86676	0.522204
$358$	3.82855	0.202345
$359$	10.1228	0.534263	0.267132	−	0.963660i	$-0.413924\pi$
$359$	10.1228	0.534263	0.267132	+	0.963660i	$0.413924\pi$
$360$	0	0
$361$	4.16181	0.219043
$362$	9.69733	0.509680
$363$	24.2842	1.27459
$364$	−0.135565	−0.00710551
$365$	0	0
$366$	−12.8361	−0.670953
$367$	3.11637	0.162673	0.0813366	−	0.996687i	$-0.474081\pi$
$367$	3.11637	0.162673	0.0813366	+	0.996687i	$0.474081\pi$
$368$	−10.8131	−0.563673
$369$	−10.0781	−0.524645
$370$	0	0
$371$	−2.63327	−0.136713
$372$	−1.53909	−0.0797980
$373$	6.89555	0.357038	0.178519	−	0.983936i	$-0.442869\pi$
$373$	6.89555	0.357038	0.178519	+	0.983936i	$0.442869\pi$
$374$	−1.07553	−0.0556141
$375$	0	0
$376$	1.11043	0.0572659
$377$	−0.189023	−0.00973517
$378$	−1.55604	−0.0800341
$379$	26.3335	1.35266	0.676330	−	0.736599i	$-0.263569\pi$
$379$	26.3335	1.35266	0.676330	+	0.736599i	$0.263569\pi$
$380$	0	0
$381$	43.2123	2.21383
$382$	−0.368545	−0.0188564
$383$	38.3736	1.96080	0.980400	−	0.197019i	$-0.0631262\pi$
$383$	38.3736	1.96080	0.980400	+	0.197019i	$0.0631262\pi$
$384$	25.3102	1.29161
$385$	0	0
$386$	−1.44147	−0.0733688
$387$	3.52908	0.179393
$388$	−23.5439	−1.19526
$389$	7.49361	0.379941	0.189970	−	0.981790i	$-0.439161\pi$
$389$	7.49361	0.379941	0.189970	+	0.981790i	$0.439161\pi$
$390$	0	0
$391$	−23.9487	−1.21114
$392$	13.0221	0.657713
$393$	33.8633	1.70818
$394$	7.87901	0.396939
$395$	0	0
$396$	1.43990	0.0723575
$397$	5.36992	0.269509	0.134754	−	0.990879i	$-0.456975\pi$
$397$	5.36992	0.269509	0.134754	+	0.990879i	$0.456975\pi$
$398$	7.56347	0.379123
$399$	−12.3830	−0.619923
$400$	0	0
$401$	−24.6951	−1.23321	−0.616606	−	0.787272i	$-0.711493\pi$
$401$	−24.6951	−1.23321	−0.616606	+	0.787272i	$0.711493\pi$
$402$	2.20260	0.109856
$403$	0.0318950	0.00158880
$404$	8.46504	0.421151
$405$	0	0
$406$	−1.85587	−0.0921052
$407$	−4.80780	−0.238314
$408$	19.7255	0.976559
$409$	−23.7785	−1.17577	−0.587885	−	0.808945i	$-0.700039\pi$
$409$	−23.7785	−1.17577	−0.587885	+	0.808945i	$0.700039\pi$
$410$	0	0
$411$	−23.3220	−1.15039
$412$	−5.08859	−0.250697
$413$	2.27361	0.111877
$414$	−8.15375	−0.400735
$415$	0	0
$416$	−0.421816	−0.0206812
$417$	5.18575	0.253947
$418$	1.34980	0.0660211
$419$	−31.0372	−1.51627	−0.758133	−	0.652100i	$-0.773888\pi$
$419$	−31.0372	−1.51627	−0.758133	+	0.652100i	$0.773888\pi$
$420$	0	0
$421$	−2.89741	−0.141211	−0.0706055	−	0.997504i	$-0.522493\pi$
$421$	−2.89741	−0.141211	−0.0706055	+	0.997504i	$0.522493\pi$
$422$	−7.22676	−0.351793
$423$	−0.994656	−0.0483618
$424$	−5.26441	−0.255663
$425$	0	0
$426$	−20.2718	−0.982170
$427$	−10.2697	−0.496987
$428$	14.8425	0.717440
$429$	−0.0735004	−0.00354863
$430$	0	0
$431$	−30.6994	−1.47874	−0.739370	−	0.673299i	$-0.764877\pi$
$431$	−30.6994	−1.47874	−0.739370	+	0.673299i	$0.764877\pi$
$432$	3.69532	0.177791
$433$	−26.8147	−1.28863	−0.644316	−	0.764759i	$-0.722858\pi$
$433$	−26.8147	−1.28863	−0.644316	+	0.764759i	$0.722858\pi$
$434$	0.313152	0.0150318
$435$	0	0
$436$	−4.79073	−0.229434
$437$	30.0560	1.43777
$438$	−21.8440	−1.04375
$439$	−16.8791	−0.805594	−0.402797	−	0.915289i	$-0.631962\pi$
$439$	−16.8791	−0.805594	−0.402797	+	0.915289i	$0.631962\pi$
$440$	0	0
$441$	−11.6644	−0.555448
$442$	−0.181332	−0.00862506
$443$	−16.8323	−0.799729	−0.399864	−	0.916574i	$-0.630943\pi$
$443$	−16.8323	−0.799729	−0.399864	+	0.916574i	$0.630943\pi$
$444$	39.1147	1.85630
$445$	0	0
$446$	−1.92170	−0.0909951
$447$	25.1573	1.18990
$448$	−0.176740	−0.00835017
$449$	14.1492	0.667741	0.333870	−	0.942619i	$-0.391645\pi$
$449$	14.1492	0.667741	0.333870	+	0.942619i	$0.391645\pi$
$450$	0	0
$451$	2.16497	0.101944
$452$	−9.21272	−0.433330
$453$	14.9888	0.704235
$454$	9.15532	0.429680
$455$	0	0
$456$	−24.7559	−1.15930
$457$	4.43915	0.207655	0.103827	−	0.994595i	$-0.466891\pi$
$457$	4.43915	0.207655	0.103827	+	0.994595i	$0.466891\pi$
$458$	−16.8992	−0.789649
$459$	8.18432	0.382011
$460$	0	0
$461$	0.00973326	0.000453323 0	0.000226661	−	1.00000i	$-0.499928\pi$
$461$	0.00973326	0.000453323 0	0.000226661		1.00000i	$0.499928\pi$
$462$	−0.721643	−0.0335739
$463$	−21.6957	−1.00829	−0.504143	−	0.863620i	$-0.668191\pi$
$463$	−21.6957	−1.00829	−0.504143	+	0.863620i	$0.668191\pi$
$464$	4.40735	0.204606
$465$	0	0
$466$	9.71103	0.449855
$467$	−24.0487	−1.11284	−0.556422	−	0.830900i	$-0.687826\pi$
$467$	−24.0487	−1.11284	−0.556422	+	0.830900i	$0.687826\pi$
$468$	0.242764	0.0112217
$469$	1.76223	0.0813721
$470$	0	0
$471$	51.3554	2.36633
$472$	4.54538	0.209218
$473$	−0.758114	−0.0348581
$474$	−2.64477	−0.121478
$475$	0	0
$476$	7.00068	0.320876
$477$	4.71556	0.215911
$478$	−6.35751	−0.290786
$479$	19.0129	0.868720	0.434360	−	0.900739i	$-0.356975\pi$
$479$	19.0129	0.868720	0.434360	+	0.900739i	$0.356975\pi$
$480$	0	0
$481$	−0.810585	−0.0369595
$482$	0.636788	0.0290049
$483$	−16.0688	−0.731156
$484$	17.2302	0.783190
$485$	0	0
$486$	11.5887	0.525674
$487$	23.1741	1.05012	0.525058	−	0.851066i	$-0.324044\pi$
$487$	23.1741	1.05012	0.525058	+	0.851066i	$0.324044\pi$
$488$	−20.5311	−0.929401
$489$	−15.5191	−0.701800
$490$	0	0
$491$	17.8038	0.803473	0.401736	−	0.915755i	$-0.368407\pi$
$491$	17.8038	0.803473	0.401736	+	0.915755i	$0.368407\pi$
$492$	−17.6135	−0.794076
$493$	9.76133	0.439628
$494$	0.227574	0.0102391
$495$	0	0
$496$	−0.743679	−0.0333922
$497$	−16.2188	−0.727510
$498$	−1.55634	−0.0697413
$499$	−13.6724	−0.612060	−0.306030	−	0.952022i	$-0.599001\pi$
$499$	−13.6724	−0.612060	−0.306030	+	0.952022i	$0.599001\pi$
$500$	0	0
$501$	−5.60810	−0.250552
$502$	−12.2360	−0.546118
$503$	13.1075	0.584433	0.292216	−	0.956352i	$-0.405607\pi$
$503$	13.1075	0.584433	0.292216	+	0.956352i	$0.405607\pi$
$504$	5.37317	0.239340
$505$	0	0
$506$	1.75158	0.0778672
$507$	29.2023	1.29692
$508$	30.6601	1.36032
$509$	−14.6586	−0.649731	−0.324865	−	0.945760i	$-0.605319\pi$
$509$	−14.6586	−0.649731	−0.324865	+	0.945760i	$0.605319\pi$
$510$	0	0
$511$	−17.4766	−0.773121
$512$	17.7616	0.784957
$513$	−10.2715	−0.453496
$514$	−18.8742	−0.832505
$515$	0	0
$516$	6.16777	0.271521
$517$	0.213671	0.00939723
$518$	−7.95851	−0.349677
$519$	0.961031	0.0421846
$520$	0	0
$521$	5.24664	0.229859	0.114930	−	0.993374i	$-0.463336\pi$
$521$	5.24664	0.229859	0.114930	+	0.993374i	$0.463336\pi$
$522$	3.32342	0.145462
$523$	−26.4312	−1.15575	−0.577877	−	0.816124i	$-0.696119\pi$
$523$	−26.4312	−1.15575	−0.577877	+	0.816124i	$0.696119\pi$
$524$	24.0267	1.04961
$525$	0	0
$526$	−15.5090	−0.676225
$527$	−1.64709	−0.0717482
$528$	1.71377	0.0745824
$529$	16.0023	0.695754
$530$	0	0
$531$	−4.07149	−0.176687
$532$	−8.78598	−0.380921
$533$	0.365009	0.0158103
$534$	−1.81883	−0.0787083
$535$	0	0
$536$	3.52303	0.152172
$537$	−13.5113	−0.583056
$538$	−9.60072	−0.413916
$539$	2.50574	0.107930
$540$	0	0
$541$	40.6899	1.74940	0.874698	−	0.484668i	$-0.161060\pi$
$541$	40.6899	1.74940	0.874698	+	0.484668i	$0.161060\pi$
$542$	14.9936	0.644028
$543$	−34.2228	−1.46864
$544$	21.7830	0.933938
$545$	0	0
$546$	−0.121668	−0.00520689
$547$	−16.6696	−0.712740	−0.356370	−	0.934345i	$-0.615986\pi$
$547$	−16.6696	−0.712740	−0.356370	+	0.934345i	$0.615986\pi$
$548$	−16.5475	−0.706872
$549$	18.3906	0.784892
$550$	0	0
$551$	−12.2506	−0.521895
$552$	−32.1246	−1.36731
$553$	−2.11599	−0.0899811
$554$	2.99725	0.127341
$555$	0	0
$556$	3.67940	0.156042
$557$	10.8329	0.459006	0.229503	−	0.973308i	$-0.426290\pi$
$557$	10.8329	0.459006	0.229503	+	0.973308i	$0.426290\pi$
$558$	−0.560780	−0.0237397
$559$	−0.127817	−0.00540606
$560$	0	0
$561$	3.79563	0.160252
$562$	1.87419	0.0790578
$563$	28.6816	1.20879	0.604393	−	0.796686i	$-0.293416\pi$
$563$	28.6816	1.20879	0.604393	+	0.796686i	$0.293416\pi$
$564$	−1.73836	−0.0731981
$565$	0	0
$566$	7.89108	0.331687
$567$	12.5338	0.526369
$568$	−32.4244	−1.36050
$569$	12.8372	0.538161	0.269081	−	0.963118i	$-0.413280\pi$
$569$	12.8372	0.538161	0.269081	+	0.963118i	$0.413280\pi$
$570$	0	0
$571$	2.68924	0.112541	0.0562707	−	0.998416i	$-0.482079\pi$
$571$	2.68924	0.112541	0.0562707	+	0.998416i	$0.482079\pi$
$572$	−0.0521502	−0.00218051
$573$	1.30063	0.0543345
$574$	3.58374	0.149582
$575$	0	0
$576$	0.316498	0.0131874
$577$	−3.19429	−0.132980	−0.0664900	−	0.997787i	$-0.521180\pi$
$577$	−3.19429	−0.132980	−0.0664900	+	0.997787i	$0.521180\pi$
$578$	−1.46126	−0.0607806
$579$	5.08708	0.211412
$580$	0	0
$581$	−1.24518	−0.0516586
$582$	−21.1304	−0.875882
$583$	−1.01299	−0.0419538
$584$	−34.9391	−1.44579
$585$	0	0
$586$	−7.04724	−0.291119
$587$	−9.55051	−0.394192	−0.197096	−	0.980384i	$-0.563151\pi$
$587$	−9.55051	−0.394192	−0.197096	+	0.980384i	$0.563151\pi$
$588$	−20.3859	−0.840699
$589$	2.06712	0.0851743
$590$	0	0
$591$	−27.8058	−1.14378
$592$	18.9000	0.776786
$593$	21.1508	0.868559	0.434280	−	0.900778i	$-0.357003\pi$
$593$	21.1508	0.868559	0.434280	+	0.900778i	$0.357003\pi$
$594$	−0.598591	−0.0245605
$595$	0	0
$596$	17.8497	0.731152
$597$	−26.6922	−1.09244
$598$	0.295313	0.0120762
$599$	2.77929	0.113559	0.0567793	−	0.998387i	$-0.481917\pi$
$599$	2.77929	0.113559	0.0567793	+	0.998387i	$0.481917\pi$
$600$	0	0
$601$	−18.5051	−0.754837	−0.377418	−	0.926043i	$-0.623188\pi$
$601$	−18.5051	−0.754837	−0.377418	+	0.926043i	$0.623188\pi$
$602$	−1.25493	−0.0511472
$603$	−3.15572	−0.128511
$604$	10.6349	0.432727
$605$	0	0
$606$	7.59728	0.308618
$607$	14.5288	0.589706	0.294853	−	0.955543i	$-0.404729\pi$
$607$	14.5288	0.589706	0.294853	+	0.955543i	$0.404729\pi$
$608$	−27.3380	−1.10870
$609$	6.54954	0.265401
$610$	0	0
$611$	0.0360245	0.00145739
$612$	−12.5365	−0.506760
$613$	14.9699	0.604627	0.302314	−	0.953209i	$-0.402241\pi$
$613$	14.9699	0.604627	0.302314	+	0.953209i	$0.402241\pi$
$614$	−12.4511	−0.502485
$615$	0	0
$616$	−1.15426	−0.0465063
$617$	0.478194	0.0192514	0.00962568	−	0.999954i	$-0.496936\pi$
$617$	0.478194	0.0192514	0.00962568	+	0.999954i	$0.496936\pi$
$618$	−4.56695	−0.183710
$619$	29.2455	1.17548	0.587738	−	0.809052i	$-0.300019\pi$
$619$	29.2455	1.17548	0.587738	+	0.809052i	$0.300019\pi$
$620$	0	0
$621$	−13.3288	−0.534867
$622$	4.32704	0.173498
$623$	−1.45518	−0.0583006
$624$	0.288939	0.0115668
$625$	0	0
$626$	3.07244	0.122799
$627$	−4.76359	−0.190239
$628$	36.4378	1.45403
$629$	41.8594	1.66904
$630$	0	0
$631$	25.1784	1.00234	0.501168	−	0.865350i	$-0.332904\pi$
$631$	25.1784	1.00234	0.501168	+	0.865350i	$0.332904\pi$
$632$	−4.23027	−0.168271
$633$	25.5039	1.01369
$634$	2.59362	0.103006
$635$	0	0
$636$	8.24137	0.326792
$637$	0.422462	0.0167386
$638$	−0.713932	−0.0282648
$639$	29.0439	1.14896
$640$	0	0
$641$	−47.7764	−1.88705	−0.943527	−	0.331297i	$-0.892514\pi$
$641$	−47.7764	−1.88705	−0.943527	+	0.331297i	$0.892514\pi$
$642$	13.3210	0.525738
$643$	1.76572	0.0696332	0.0348166	−	0.999394i	$-0.488915\pi$
$643$	1.76572	0.0696332	0.0348166	+	0.999394i	$0.488915\pi$
$644$	−11.4012	−0.449269
$645$	0	0
$646$	−11.7522	−0.462383
$647$	−17.7016	−0.695921	−0.347961	−	0.937509i	$-0.613126\pi$
$647$	−17.7016	−0.695921	−0.347961	+	0.937509i	$0.613126\pi$
$648$	25.0574	0.984347
$649$	0.874632	0.0343323
$650$	0	0
$651$	−1.10514	−0.0433139
$652$	−11.0112	−0.431231
$653$	36.0403	1.41037	0.705183	−	0.709025i	$-0.250865\pi$
$653$	36.0403	1.41037	0.705183	+	0.709025i	$0.250865\pi$
$654$	−4.29962	−0.168129
$655$	0	0
$656$	−8.51074	−0.332289
$657$	31.2965	1.22099
$658$	0.353696	0.0137885
$659$	−31.7767	−1.23784	−0.618922	−	0.785452i	$-0.712431\pi$
$659$	−31.7767	−1.23784	−0.618922	+	0.785452i	$0.712431\pi$
$660$	0	0
$661$	34.0434	1.32413	0.662067	−	0.749445i	$-0.269679\pi$
$661$	34.0434	1.32413	0.662067	+	0.749445i	$0.269679\pi$
$662$	−18.6415	−0.724521
$663$	0.639936	0.0248531
$664$	−2.48934	−0.0966052
$665$	0	0
$666$	14.2518	0.552245
$667$	−15.8971	−0.615538
$668$	−3.97907	−0.153955
$669$	6.78186	0.262202
$670$	0	0
$671$	−3.95065	−0.152513
$672$	14.6157	0.563812
$673$	−36.3663	−1.40182	−0.700909	−	0.713251i	$-0.747222\pi$
$673$	−36.3663	−1.40182	−0.700909	+	0.713251i	$0.747222\pi$
$674$	−2.35920	−0.0908731
$675$	0	0
$676$	20.7197	0.796912
$677$	−27.7344	−1.06592	−0.532959	−	0.846141i	$-0.678920\pi$
$677$	−27.7344	−1.06592	−0.532959	+	0.846141i	$0.678920\pi$
$678$	−8.26831	−0.317543
$679$	−16.9057	−0.648781
$680$	0	0
$681$	−32.3100	−1.23812
$682$	0.120466	0.00461288
$683$	31.2794	1.19687	0.598437	−	0.801170i	$-0.295789\pi$
$683$	31.2794	1.19687	0.598437	+	0.801170i	$0.295789\pi$
$684$	15.7336	0.601589
$685$	0	0
$686$	9.25138	0.353219
$687$	59.6390	2.27537
$688$	2.98024	0.113620
$689$	−0.170788	−0.00650652
$690$	0	0
$691$	48.1424	1.83142	0.915712	−	0.401836i	$-0.131628\pi$
$691$	48.1424	1.83142	0.915712	+	0.401836i	$0.131628\pi$
$692$	0.681873	0.0259209
$693$	1.03392	0.0392753
$694$	−13.3065	−0.505109
$695$	0	0
$696$	13.0938	0.496318
$697$	−18.8494	−0.713973
$698$	−18.1022	−0.685179
$699$	−34.2711	−1.29625
$700$	0	0
$701$	23.3483	0.881852	0.440926	−	0.897543i	$-0.354650\pi$
$701$	23.3483	0.881852	0.440926	+	0.897543i	$0.354650\pi$
$702$	−0.100921	−0.00380903
$703$	−52.5343	−1.98137
$704$	−0.0679898	−0.00256246
$705$	0	0
$706$	7.20323	0.271097
$707$	6.07832	0.228599
$708$	−7.11573	−0.267425
$709$	−39.2625	−1.47453	−0.737267	−	0.675601i	$-0.763884\pi$
$709$	−39.2625	−1.47453	−0.737267	+	0.675601i	$0.763884\pi$
$710$	0	0
$711$	3.78923	0.142107
$712$	−2.90919	−0.109026
$713$	2.68241	0.100457
$714$	6.28304	0.235137
$715$	0	0
$716$	−9.58658	−0.358267
$717$	22.4362	0.837897
$718$	6.44611	0.240567
$719$	2.50667	0.0934830	0.0467415	−	0.998907i	$-0.485116\pi$
$719$	2.50667	0.0934830	0.0467415	+	0.998907i	$0.485116\pi$
$720$	0	0
$721$	−3.65386	−0.136077
$722$	2.65019	0.0986299
$723$	−2.24729	−0.0835775
$724$	−24.2818	−0.902428
$725$	0	0
$726$	15.4639	0.573919
$727$	−17.3890	−0.644922	−0.322461	−	0.946583i	$-0.604510\pi$
$727$	−17.3890	−0.644922	−0.322461	+	0.946583i	$0.604510\pi$
$728$	−0.194606	−0.00721256
$729$	−8.05612	−0.298375
$730$	0	0
$731$	6.60057	0.244131
$732$	32.1412	1.18797
$733$	12.6365	0.466739	0.233370	−	0.972388i	$-0.425025\pi$
$733$	12.6365	0.466739	0.233370	+	0.972388i	$0.425025\pi$
$734$	1.98447	0.0732481
$735$	0	0
$736$	−35.4753	−1.30764
$737$	0.677909	0.0249711
$738$	−6.41762	−0.236236
$739$	49.2503	1.81170	0.905850	−	0.423598i	$-0.139233\pi$
$739$	49.2503	1.81170	0.905850	+	0.423598i	$0.139233\pi$
$740$	0	0
$741$	−0.803131	−0.0295038
$742$	−1.67684	−0.0615586
$743$	−42.2847	−1.55127	−0.775637	−	0.631179i	$-0.782571\pi$
$743$	−42.2847	−1.55127	−0.775637	+	0.631179i	$0.782571\pi$
$744$	−2.20939	−0.0810002
$745$	0	0
$746$	4.39101	0.160766
$747$	2.22981	0.0815845
$748$	2.69309	0.0984690
$749$	10.6577	0.389423
$750$	0	0
$751$	−44.6401	−1.62894	−0.814471	−	0.580204i	$-0.802973\pi$
$751$	−44.6401	−1.62894	−0.814471	+	0.580204i	$0.802973\pi$
$752$	−0.839966	−0.0306304
$753$	43.1819	1.57364
$754$	−0.120368	−0.00438353
$755$	0	0
$756$	3.89628	0.141706
$757$	28.3936	1.03198	0.515992	−	0.856593i	$-0.327423\pi$
$757$	28.3936	1.03198	0.515992	+	0.856593i	$0.327423\pi$
$758$	16.7689	0.609072
$759$	−6.18149	−0.224374
$760$	0	0
$761$	27.7195	1.00483	0.502415	−	0.864627i	$-0.332445\pi$
$761$	27.7195	1.00483	0.502415	+	0.864627i	$0.332445\pi$
$762$	27.5171	0.996838
$763$	−3.43998	−0.124536
$764$	0.922825	0.0333866
$765$	0	0
$766$	24.4359	0.882904
$767$	0.147461	0.00532452
$768$	16.8111	0.606617
$769$	35.1939	1.26912	0.634562	−	0.772872i	$-0.281181\pi$
$769$	35.1939	1.26912	0.634562	+	0.772872i	$0.281181\pi$
$770$	0	0
$771$	66.6088	2.39886
$772$	3.60939	0.129905
$773$	0.165321	0.00594618	0.00297309	−	0.999996i	$-0.499054\pi$
$773$	0.165321	0.00594618	0.00297309	+	0.999996i	$0.499054\pi$
$774$	2.24728	0.0807768
$775$	0	0
$776$	−33.7977	−1.21327
$777$	28.0863	1.00759
$778$	4.77184	0.171079
$779$	23.6564	0.847577
$780$	0	0
$781$	−6.23918	−0.223255
$782$	−15.2502	−0.545348
$783$	5.43273	0.194150
$784$	−9.85035	−0.351798
$785$	0	0
$786$	21.5637	0.769153
$787$	−22.9720	−0.818863	−0.409431	−	0.912341i	$-0.634273\pi$
$787$	−22.9720	−0.818863	−0.409431	+	0.912341i	$0.634273\pi$
$788$	−19.7288	−0.702810
$789$	54.7328	1.94854
$790$	0	0
$791$	−6.61519	−0.235209
$792$	2.06700	0.0734476
$793$	−0.666072	−0.0236529
$794$	3.41950	0.121354
$795$	0	0
$796$	−18.9387	−0.671265
$797$	−32.6066	−1.15499	−0.577493	−	0.816395i	$-0.695969\pi$
$797$	−32.6066	−1.15499	−0.577493	+	0.816395i	$0.695969\pi$
$798$	−7.88532	−0.279137
$799$	−1.86034	−0.0658141
$800$	0	0
$801$	2.60588	0.0920743
$802$	−15.7255	−0.555288
$803$	−6.72307	−0.237252
$804$	−5.51525	−0.194508
$805$	0	0
$806$	0.0203103	0.000715401 0
$807$	33.8818	1.19270
$808$	12.1517	0.427496
$809$	19.3356	0.679802	0.339901	−	0.940461i	$-0.389606\pi$
$809$	19.3356	0.679802	0.339901	+	0.940461i	$0.389606\pi$
$810$	0	0
$811$	−45.2010	−1.58722	−0.793612	−	0.608425i	$-0.791802\pi$
$811$	−45.2010	−1.58722	−0.793612	+	0.608425i	$0.791802\pi$
$812$	4.64704	0.163079
$813$	−52.9137	−1.85576
$814$	−3.06155	−0.107307
$815$	0	0
$816$	−14.9211	−0.522342
$817$	−8.28383	−0.289815
$818$	−15.1419	−0.529423
$819$	0.174316	0.00609111
$820$	0	0
$821$	29.4554	1.02800	0.514001	−	0.857790i	$-0.328163\pi$
$821$	29.4554	1.02800	0.514001	+	0.857790i	$0.328163\pi$
$822$	−14.8512	−0.517994
$823$	49.6297	1.72998	0.864991	−	0.501787i	$-0.167324\pi$
$823$	49.6297	1.72998	0.864991	+	0.501787i	$0.167324\pi$
$824$	−7.30476	−0.254473
$825$	0	0
$826$	1.44781	0.0503757
$827$	−8.61123	−0.299442	−0.149721	−	0.988728i	$-0.547838\pi$
$827$	−8.61123	−0.299442	−0.149721	+	0.988728i	$0.547838\pi$
$828$	20.4168	0.709532
$829$	−9.81975	−0.341054	−0.170527	−	0.985353i	$-0.554547\pi$
$829$	−9.81975	−0.341054	−0.170527	+	0.985353i	$0.554547\pi$
$830$	0	0
$831$	−10.5776	−0.366932
$832$	−0.0114630	−0.000397406 0
$833$	−21.8164	−0.755892
$834$	3.30222	0.114347
$835$	0	0
$836$	−3.37987	−0.116895
$837$	−0.916698	−0.0316857
$838$	−19.7641	−0.682740
$839$	−5.45749	−0.188414	−0.0942068	−	0.995553i	$-0.530031\pi$
$839$	−5.45749	−0.188414	−0.0942068	+	0.995553i	$0.530031\pi$
$840$	0	0
$841$	−22.5204	−0.776567
$842$	−1.84504	−0.0635841
$843$	−6.61419	−0.227805
$844$	18.0956	0.622876
$845$	0	0
$846$	−0.633386	−0.0217762
$847$	12.3721	0.425112
$848$	3.98219	0.136749
$849$	−27.8484	−0.955753
$850$	0	0
$851$	−68.1714	−2.33689
$852$	50.7599	1.73901
$853$	−1.89562	−0.0649050	−0.0324525	−	0.999473i	$-0.510332\pi$
$853$	−1.89562	−0.0649050	−0.0324525	+	0.999473i	$0.510332\pi$
$854$	−6.53964	−0.223782
$855$	0	0
$856$	21.3067	0.728249
$857$	40.1087	1.37009	0.685044	−	0.728502i	$-0.259783\pi$
$857$	40.1087	1.37009	0.685044	+	0.728502i	$0.259783\pi$
$858$	−0.0468042	−0.00159787
$859$	56.6142	1.93165	0.965825	−	0.259195i	$-0.0834573\pi$
$859$	56.6142	1.93165	0.965825	+	0.259195i	$0.0834573\pi$
$860$	0	0
$861$	−12.6474	−0.431021
$862$	−19.5491	−0.665843
$863$	−15.4130	−0.524664	−0.262332	−	0.964978i	$-0.584492\pi$
$863$	−15.4130	−0.524664	−0.262332	+	0.964978i	$0.584492\pi$
$864$	12.1235	0.412449
$865$	0	0
$866$	−17.0753	−0.580242
$867$	5.15694	0.175139
$868$	−0.784123	−0.0266149
$869$	−0.813998	−0.0276130
$870$	0	0
$871$	0.114294	0.00387271
$872$	−6.87718	−0.232891
$873$	30.2740	1.02462
$874$	19.1393	0.647398
$875$	0	0
$876$	54.6967	1.84803
$877$	−2.57230	−0.0868603	−0.0434301	−	0.999056i	$-0.513829\pi$
$877$	−2.57230	−0.0868603	−0.0434301	+	0.999056i	$0.513829\pi$
$878$	−10.7484	−0.362741
$879$	24.8704	0.838857
$880$	0	0
$881$	−50.0812	−1.68728	−0.843639	−	0.536910i	$-0.819591\pi$
$881$	−50.0812	−1.68728	−0.843639	+	0.536910i	$0.819591\pi$
$882$	−7.42776	−0.250106
$883$	−29.8568	−1.00476	−0.502381	−	0.864646i	$-0.667543\pi$
$883$	−29.8568	−1.00476	−0.502381	+	0.864646i	$0.667543\pi$
$884$	0.454049	0.0152713
$885$	0	0
$886$	−10.7186	−0.360100
$887$	−24.9301	−0.837072	−0.418536	−	0.908200i	$-0.637457\pi$
$887$	−24.9301	−0.837072	−0.418536	+	0.908200i	$0.637457\pi$
$888$	56.1499	1.88427
$889$	22.0155	0.738375
$890$	0	0
$891$	4.82160	0.161530
$892$	4.81188	0.161114
$893$	2.33476	0.0781298
$894$	16.0199	0.535786
$895$	0	0
$896$	12.8949	0.430787
$897$	−1.04219	−0.0347976
$898$	9.01003	0.300669
$899$	−1.09333	−0.0364647
$900$	0	0
$901$	8.81968	0.293826
$902$	1.37863	0.0459032
$903$	4.42877	0.147380
$904$	−13.2250	−0.439858
$905$	0	0
$906$	9.54469	0.317101
$907$	27.8785	0.925689	0.462844	−	0.886440i	$-0.346829\pi$
$907$	27.8785	0.925689	0.462844	+	0.886440i	$0.346829\pi$
$908$	−22.9247	−0.760782
$909$	−10.8848	−0.361027
$910$	0	0
$911$	−12.1116	−0.401275	−0.200638	−	0.979666i	$-0.564301\pi$
$911$	−12.1116	−0.401275	−0.200638	+	0.979666i	$0.564301\pi$
$912$	18.7262	0.620087
$913$	−0.479005	−0.0158528
$914$	2.82680	0.0935022
$915$	0	0
$916$	42.3152	1.39813
$917$	17.2524	0.569725
$918$	5.21168	0.172011
$919$	−15.4282	−0.508928	−0.254464	−	0.967082i	$-0.581899\pi$
$919$	−15.4282	−0.508928	−0.254464	+	0.967082i	$0.581899\pi$
$920$	0	0
$921$	43.9411	1.44791
$922$	0.00619803	0.000204121 0
$923$	−1.05191	−0.0346241
$924$	1.80697	0.0594451
$925$	0	0
$926$	−13.8156	−0.454008
$927$	6.54319	0.214906
$928$	14.4595	0.474657
$929$	−17.1882	−0.563925	−0.281963	−	0.959425i	$-0.590985\pi$
$929$	−17.1882	−0.563925	−0.281963	+	0.959425i	$0.590985\pi$
$930$	0	0
$931$	27.3799	0.897341
$932$	−24.3161	−0.796501
$933$	−15.2705	−0.499935
$934$	−15.3140	−0.501088
$935$	0	0
$936$	0.348492	0.0113908
$937$	−55.7620	−1.82167	−0.910833	−	0.412775i	$-0.864560\pi$
$937$	−55.7620	−1.82167	−0.910833	+	0.412775i	$0.864560\pi$
$938$	1.12217	0.0366400
$939$	−10.8429	−0.353846
$940$	0	0
$941$	16.8053	0.547836	0.273918	−	0.961753i	$-0.411680\pi$
$941$	16.8053	0.547836	0.273918	+	0.961753i	$0.411680\pi$
$942$	32.7025	1.06551
$943$	30.6978	0.999658
$944$	−3.43828	−0.111907
$945$	0	0
$946$	−0.482758	−0.0156958
$947$	13.5446	0.440139	0.220069	−	0.975484i	$-0.429372\pi$
$947$	13.5446	0.440139	0.220069	+	0.975484i	$0.429372\pi$
$948$	6.62243	0.215087
$949$	−1.13350	−0.0367949
$950$	0	0
$951$	−9.15312	−0.296810
$952$	10.0496	0.325710
$953$	25.8635	0.837802	0.418901	−	0.908032i	$-0.362416\pi$
$953$	25.8635	0.837802	0.418901	+	0.908032i	$0.362416\pi$
$954$	3.00281	0.0972197
$955$	0	0
$956$	15.9190	0.514858
$957$	2.51953	0.0814450
$958$	12.1072	0.391165
$959$	−11.8819	−0.383687
$960$	0	0
$961$	−30.8155	−0.994049
$962$	−0.516171	−0.0166420
$963$	−19.0853	−0.615016
$964$	−1.59450	−0.0513554
$965$	0	0
$966$	−10.2324	−0.329223
$967$	2.77504	0.0892392	0.0446196	−	0.999004i	$-0.485792\pi$
$967$	2.77504	0.0892392	0.0446196	+	0.999004i	$0.485792\pi$
$968$	24.7343	0.794990
$969$	41.4745	1.33235
$970$	0	0
$971$	50.6358	1.62498	0.812490	−	0.582975i	$-0.198111\pi$
$971$	50.6358	1.62498	0.812490	+	0.582975i	$0.198111\pi$
$972$	−29.0178	−0.930746
$973$	2.64200	0.0846986
$974$	14.7570	0.472844
$975$	0	0
$976$	15.5305	0.497118
$977$	45.4183	1.45306	0.726529	−	0.687135i	$-0.241132\pi$
$977$	45.4183	1.45306	0.726529	+	0.687135i	$0.241132\pi$
$978$	−9.88241	−0.316005
$979$	−0.559792	−0.0178910
$980$	0	0
$981$	6.16018	0.196680
$982$	11.3372	0.361786
$983$	−22.6856	−0.723559	−0.361780	−	0.932264i	$-0.617831\pi$
$983$	−22.6856	−0.723559	−0.361780	+	0.932264i	$0.617831\pi$
$984$	−25.2845	−0.806040
$985$	0	0
$986$	6.21590	0.197955
$987$	−1.24823	−0.0397315
$988$	−0.569840	−0.0181290
$989$	−10.7496	−0.341816
$990$	0	0
$991$	−42.6339	−1.35431	−0.677155	−	0.735841i	$-0.736787\pi$
$991$	−42.6339	−1.35431	−0.677155	+	0.735841i	$0.736787\pi$
$992$	−2.43984	−0.0774650
$993$	65.7875	2.08770
$994$	−10.3279	−0.327582
$995$	0	0
$996$	3.89703	0.123482
$997$	−48.2019	−1.52657	−0.763285	−	0.646062i	$-0.776415\pi$
$997$	−48.2019	−1.52657	−0.763285	+	0.646062i	$0.776415\pi$
$998$	−8.70642	−0.275597
$999$	23.2971	0.737089

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.p.1.28		46
5.2	odd	4		1205.2.b.c.724.28	yes	46
5.3	odd	4		1205.2.b.c.724.19	✓	46
5.4	even	2	inner	6025.2.a.p.1.19		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.c.724.19	✓	46	5.3	odd	4
1205.2.b.c.724.28	yes	46	5.2	odd	4
6025.2.a.p.1.19		46	5.4	even	2	inner
6025.2.a.p.1.28		46	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:	\( N \)	\(=\)	\( 6025 = 5^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.636788 q^{2} -2.24729 q^{3} -1.59450 q^{4} -1.43105 q^{6} -1.14493 q^{7} -2.28894 q^{8} +2.05030 q^{9} +O(q^{10})\)	\(q+0.636788 q^{2} -2.24729 q^{3} -1.59450 q^{4} -1.43105 q^{6} -1.14493 q^{7} -2.28894 q^{8} +2.05030 q^{9} -0.440443 q^{11} +3.58330 q^{12} -0.0742578 q^{13} -0.729079 q^{14} +1.73143 q^{16} +3.83474 q^{17} +1.30561 q^{18} -4.81267 q^{19} +2.57299 q^{21} -0.280469 q^{22} -6.24519 q^{23} +5.14390 q^{24} -0.0472865 q^{26} +2.13425 q^{27} +1.82559 q^{28} +2.54550 q^{29} -0.429517 q^{31} +5.68043 q^{32} +0.989801 q^{33} +2.44192 q^{34} -3.26920 q^{36} +10.9158 q^{37} -3.06465 q^{38} +0.166878 q^{39} -4.91543 q^{41} +1.63845 q^{42} +1.72125 q^{43} +0.702286 q^{44} -3.97686 q^{46} -0.485128 q^{47} -3.89103 q^{48} -5.68913 q^{49} -8.61777 q^{51} +0.118404 q^{52} +2.29994 q^{53} +1.35907 q^{54} +2.62068 q^{56} +10.8155 q^{57} +1.62094 q^{58} -1.98580 q^{59} +8.96972 q^{61} -0.273511 q^{62} -2.34745 q^{63} +0.154367 q^{64} +0.630294 q^{66} -1.53915 q^{67} -6.11450 q^{68} +14.0347 q^{69} +14.1657 q^{71} -4.69300 q^{72} +15.2643 q^{73} +6.95108 q^{74} +7.67381 q^{76} +0.504277 q^{77} +0.106266 q^{78} +1.84814 q^{79} -10.9472 q^{81} -3.13009 q^{82} +1.08755 q^{83} -4.10263 q^{84} +1.09608 q^{86} -5.72046 q^{87} +1.00815 q^{88} +1.27098 q^{89} +0.0850201 q^{91} +9.95795 q^{92} +0.965247 q^{93} -0.308924 q^{94} -12.7656 q^{96} +14.7657 q^{97} -3.62277 q^{98} -0.903038 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9}+O(q^{10}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9	\( 46 q + 34 q^{4} - 4 q^{6} + 34 q^{9} - 64 q^{11} - 66 q^{14} + 22 q^{16} - 14 q^{21} - 50 q^{24} - 60 q^{26} - 36 q^{29} - 36 q^{31} + 12 q^{34} - 34 q^{36} - 88 q^{39} - 76 q^{41} - 100 q^{44} - 12 q^{46} + 22 q^{49} - 112 q^{51} - 26 q^{54} - 120 q^{56} - 84 q^{59} - 78 q^{61} + 28 q^{64} - 2 q^{66} - 24 q^{69} - 172 q^{71} - 16 q^{74} - 18 q^{76} - 54 q^{79} - 42 q^{81} + 44 q^{84} - 80 q^{86} - 86 q^{89} - 88 q^{91} + 4 q^{94} - 122 q^{96} - 148 q^{99}+O(q^{100}) \) 46 * q + 34 * q^4 - 4 * q^6 + 34 * q^9 - 64 * q^11 - 66 * q^14 + 22 * q^16 - 14 * q^21 - 50 * q^24 - 60 * q^26 - 36 * q^29 - 36 * q^31 + 12 * q^34 - 34 * q^36 - 88 * q^39 - 76 * q^41 - 100 * q^44 - 12 * q^46 + 22 * q^49 - 112 * q^51 - 26 * q^54 - 120 * q^56 - 84 * q^59 - 78 * q^61 + 28 * q^64 - 2 * q^66 - 24 * q^69 - 172 * q^71 - 16 * q^74 - 18 * q^76 - 54 * q^79 - 42 * q^81 + 44 * q^84 - 80 * q^86 - 86 * q^89 - 88 * q^91 + 4 * q^94 - 122 * q^96 - 148 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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