LMFDB - Embedded newform 6025.2.a.q.1.6

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:	$0$
Dimension:	$66$
Twist minimal:	no (minimal twist has level 1205)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
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)

$f(q)$	$=$	$q-2.48606 q^{2} -2.54168 q^{3} +4.18051 q^{4} +6.31876 q^{6} -3.51286 q^{7} -5.42088 q^{8} +3.46011 q^{9} +O(q^{10})$	\(q-2.48606 q^{2} -2.54168 q^{3} +4.18051 q^{4} +6.31876 q^{6} -3.51286 q^{7} -5.42088 q^{8} +3.46011 q^{9} -1.13852 q^{11} -10.6255 q^{12} +1.61484 q^{13} +8.73319 q^{14} +5.11563 q^{16} +0.605228 q^{17} -8.60206 q^{18} +5.94172 q^{19} +8.92855 q^{21} +2.83042 q^{22} +8.81659 q^{23} +13.7781 q^{24} -4.01459 q^{26} -1.16946 q^{27} -14.6855 q^{28} +3.56141 q^{29} +4.03229 q^{31} -1.87602 q^{32} +2.89374 q^{33} -1.50463 q^{34} +14.4650 q^{36} +2.61189 q^{37} -14.7715 q^{38} -4.10439 q^{39} +4.59420 q^{41} -22.1969 q^{42} +9.85605 q^{43} -4.75957 q^{44} -21.9186 q^{46} -7.19792 q^{47} -13.0023 q^{48} +5.34020 q^{49} -1.53829 q^{51} +6.75084 q^{52} +12.1481 q^{53} +2.90734 q^{54} +19.0428 q^{56} -15.1019 q^{57} -8.85388 q^{58} +7.01670 q^{59} -4.30294 q^{61} -10.0245 q^{62} -12.1549 q^{63} -5.56735 q^{64} -7.19401 q^{66} +6.53582 q^{67} +2.53016 q^{68} -22.4089 q^{69} +0.792019 q^{71} -18.7569 q^{72} -9.29872 q^{73} -6.49332 q^{74} +24.8394 q^{76} +3.99945 q^{77} +10.2038 q^{78} +3.53559 q^{79} -7.40796 q^{81} -11.4215 q^{82} +3.61819 q^{83} +37.3259 q^{84} -24.5028 q^{86} -9.05194 q^{87} +6.17176 q^{88} +6.34937 q^{89} -5.67270 q^{91} +36.8578 q^{92} -10.2488 q^{93} +17.8945 q^{94} +4.76825 q^{96} -0.677908 q^{97} -13.2761 q^{98} -3.93939 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	$ 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) $ 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.48606	−1.75791	−0.878956	−	0.476903i	$-0.841759\pi$
$2$	−2.48606	−1.75791	−0.878956	+	0.476903i	$0.841759\pi$
$3$	−2.54168	−1.46744	−0.733718	−	0.679454i	$-0.762217\pi$
$3$	−2.54168	−1.46744	−0.733718	+	0.679454i	$0.762217\pi$
$4$	4.18051	2.09025
$5$	0	0
$5$	0	0
$6$	6.31876	2.57962
$7$	−3.51286	−1.32774	−0.663868	−	0.747849i	$-0.731087\pi$
$7$	−3.51286	−1.32774	−0.663868	+	0.747849i	$0.731087\pi$
$8$	−5.42088	−1.91657
$9$	3.46011	1.15337
$10$	0	0
$11$	−1.13852	−0.343275	−0.171638	−	0.985160i	$-0.554906\pi$
$11$	−1.13852	−0.343275	−0.171638	+	0.985160i	$0.554906\pi$
$12$	−10.6255	−3.06732
$13$	1.61484	0.447875	0.223938	−	0.974603i	$-0.428109\pi$
$13$	1.61484	0.447875	0.223938	+	0.974603i	$0.428109\pi$
$14$	8.73319	2.33404
$15$	0	0
$16$	5.11563	1.27891
$17$	0.605228	0.146789	0.0733947	−	0.997303i	$-0.476617\pi$
$17$	0.605228	0.146789	0.0733947	+	0.997303i	$0.476617\pi$
$18$	−8.60206	−2.02752
$19$	5.94172	1.36312	0.681561	−	0.731761i	$-0.261301\pi$
$19$	5.94172	1.36312	0.681561	+	0.731761i	$0.261301\pi$
$20$	0	0
$21$	8.92855	1.94837
$22$	2.83042	0.603448
$23$	8.81659	1.83839	0.919193	−	0.393807i	$-0.128842\pi$
$23$	8.81659	1.83839	0.919193	+	0.393807i	$0.128842\pi$
$24$	13.7781	2.81245
$25$	0	0
$26$	−4.01459	−0.787325
$27$	−1.16946	−0.225062
$28$	−14.6855	−2.77531
$29$	3.56141	0.661337	0.330668	−	0.943747i	$-0.392726\pi$
$29$	3.56141	0.661337	0.330668	+	0.943747i	$0.392726\pi$
$30$	0	0
$31$	4.03229	0.724221	0.362111	−	0.932135i	$-0.382056\pi$
$31$	4.03229	0.724221	0.362111	+	0.932135i	$0.382056\pi$
$32$	−1.87602	−0.331637
$33$	2.89374	0.503735
$34$	−1.50463	−0.258043
$35$	0	0
$36$	14.4650	2.41084
$37$	2.61189	0.429392	0.214696	−	0.976681i	$-0.431124\pi$
$37$	2.61189	0.429392	0.214696	+	0.976681i	$0.431124\pi$
$38$	−14.7715	−2.39625
$39$	−4.10439	−0.657229
$40$	0	0
$41$	4.59420	0.717493	0.358747	−	0.933435i	$-0.383204\pi$
$41$	4.59420	0.717493	0.358747	+	0.933435i	$0.383204\pi$
$42$	−22.1969	−3.42506
$43$	9.85605	1.50303	0.751517	−	0.659714i	$-0.229323\pi$
$43$	9.85605	1.50303	0.751517	+	0.659714i	$0.229323\pi$
$44$	−4.75957	−0.717533
$45$	0	0
$46$	−21.9186	−3.23172
$47$	−7.19792	−1.04993	−0.524963	−	0.851125i	$-0.675921\pi$
$47$	−7.19792	−1.04993	−0.524963	+	0.851125i	$0.675921\pi$
$48$	−13.0023	−1.87672
$49$	5.34020	0.762885
$50$	0	0
$51$	−1.53829	−0.215404
$52$	6.75084	0.936173
$53$	12.1481	1.66867	0.834335	−	0.551257i	$-0.185852\pi$
$53$	12.1481	1.66867	0.834335	+	0.551257i	$0.185852\pi$
$54$	2.90734	0.395639
$55$	0	0
$56$	19.0428	2.54470
$57$	−15.1019	−2.00030
$58$	−8.85388	−1.16257
$59$	7.01670	0.913497	0.456749	−	0.889596i	$-0.349014\pi$
$59$	7.01670	0.913497	0.456749	+	0.889596i	$0.349014\pi$
$60$	0	0
$61$	−4.30294	−0.550935	−0.275468	−	0.961310i	$-0.588833\pi$
$61$	−4.30294	−0.550935	−0.275468	+	0.961310i	$0.588833\pi$
$62$	−10.0245	−1.27312
$63$	−12.1549	−1.53137
$64$	−5.56735	−0.695919
$65$	0	0
$66$	−7.19401	−0.885521
$67$	6.53582	0.798477	0.399239	−	0.916847i	$-0.369274\pi$
$67$	6.53582	0.798477	0.399239	+	0.916847i	$0.369274\pi$
$68$	2.53016	0.306827
$69$	−22.4089	−2.69772
$70$	0	0
$71$	0.792019	0.0939954	0.0469977	−	0.998895i	$-0.485035\pi$
$71$	0.792019	0.0939954	0.0469977	+	0.998895i	$0.485035\pi$
$72$	−18.7569	−2.21052
$73$	−9.29872	−1.08833	−0.544166	−	0.838977i	$-0.683154\pi$
$73$	−9.29872	−1.08833	−0.544166	+	0.838977i	$0.683154\pi$
$74$	−6.49332	−0.754833
$75$	0	0
$76$	24.8394	2.84927
$77$	3.99945	0.455779
$78$	10.2038	1.15535
$79$	3.53559	0.397785	0.198892	−	0.980021i	$-0.436266\pi$
$79$	3.53559	0.397785	0.198892	+	0.980021i	$0.436266\pi$
$80$	0	0
$81$	−7.40796	−0.823107
$82$	−11.4215	−1.26129
$83$	3.61819	0.397148	0.198574	−	0.980086i	$-0.436369\pi$
$83$	3.61819	0.397148	0.198574	+	0.980086i	$0.436369\pi$
$84$	37.3259	4.07259
$85$	0	0
$86$	−24.5028	−2.64220
$87$	−9.05194	−0.970470
$88$	6.17176	0.657911
$89$	6.34937	0.673032	0.336516	−	0.941678i	$-0.390751\pi$
$89$	6.34937	0.673032	0.336516	+	0.941678i	$0.390751\pi$
$90$	0	0
$91$	−5.67270	−0.594660
$92$	36.8578	3.84269
$93$	−10.2488	−1.06275
$94$	17.8945	1.84568
$95$	0	0
$96$	4.76825	0.486657
$97$	−0.677908	−0.0688311	−0.0344156	−	0.999408i	$-0.510957\pi$
$97$	−0.677908	−0.0688311	−0.0344156	+	0.999408i	$0.510957\pi$
$98$	−13.2761	−1.34109
$99$	−3.93939	−0.395924
$100$	0	0
$101$	4.30782	0.428645	0.214322	−	0.976763i	$-0.431246\pi$
$101$	4.30782	0.428645	0.214322	+	0.976763i	$0.431246\pi$
$102$	3.82429	0.378661
$103$	17.2316	1.69788	0.848942	−	0.528486i	$-0.177240\pi$
$103$	17.2316	1.69788	0.848942	+	0.528486i	$0.177240\pi$
$104$	−8.75384	−0.858385
$105$	0	0
$106$	−30.2009	−2.93338
$107$	−0.435102	−0.0420629	−0.0210314	−	0.999779i	$-0.506695\pi$
$107$	−0.435102	−0.0420629	−0.0210314	+	0.999779i	$0.506695\pi$
$108$	−4.88892	−0.470437
$109$	−0.431611	−0.0413408	−0.0206704	−	0.999786i	$-0.506580\pi$
$109$	−0.431611	−0.0413408	−0.0206704	+	0.999786i	$0.506580\pi$
$110$	0	0
$111$	−6.63858	−0.630106
$112$	−17.9705	−1.69805
$113$	−4.79972	−0.451519	−0.225760	−	0.974183i	$-0.572486\pi$
$113$	−4.79972	−0.451519	−0.225760	+	0.974183i	$0.572486\pi$
$114$	37.5443	3.51635
$115$	0	0
$116$	14.8885	1.38236
$117$	5.58752	0.516566
$118$	−17.4440	−1.60585
$119$	−2.12608	−0.194898
$120$	0	0
$121$	−9.70378	−0.882162
$122$	10.6974	0.968496
$123$	−11.6770	−1.05288
$124$	16.8570	1.51381
$125$	0	0
$126$	30.2178	2.69202
$127$	1.21249	0.107591	0.0537957	−	0.998552i	$-0.482868\pi$
$127$	1.21249	0.107591	0.0537957	+	0.998552i	$0.482868\pi$
$128$	17.5928	1.55500
$129$	−25.0509	−2.20561
$130$	0	0
$131$	17.8199	1.55693	0.778466	−	0.627686i	$-0.215998\pi$
$131$	17.8199	1.55693	0.778466	+	0.627686i	$0.215998\pi$
$132$	12.0973	1.05293
$133$	−20.8724	−1.80987
$134$	−16.2485	−1.40365
$135$	0	0
$136$	−3.28087	−0.281332
$137$	21.3771	1.82637	0.913183	−	0.407550i	$-0.133617\pi$
$137$	21.3771	1.82637	0.913183	+	0.407550i	$0.133617\pi$
$138$	55.7100	4.74235
$139$	−1.65274	−0.140183	−0.0700916	−	0.997541i	$-0.522329\pi$
$139$	−1.65274	−0.140183	−0.0700916	+	0.997541i	$0.522329\pi$
$140$	0	0
$141$	18.2948	1.54070
$142$	−1.96901	−0.165236
$143$	−1.83852	−0.153744
$144$	17.7007	1.47506
$145$	0	0
$146$	23.1172	1.91319
$147$	−13.5730	−1.11949
$148$	10.9190	0.897539
$149$	−8.53889	−0.699533	−0.349767	−	0.936837i	$-0.613739\pi$
$149$	−8.53889	−0.699533	−0.349767	+	0.936837i	$0.613739\pi$
$150$	0	0
$151$	19.7618	1.60819	0.804095	−	0.594501i	$-0.202650\pi$
$151$	19.7618	1.60819	0.804095	+	0.594501i	$0.202650\pi$
$152$	−32.2093	−2.61252
$153$	2.09416	0.169303
$154$	−9.94288	−0.801220
$155$	0	0
$156$	−17.1584	−1.37377
$157$	−12.5231	−0.999451	−0.499725	−	0.866184i	$-0.666566\pi$
$157$	−12.5231	−0.999451	−0.499725	+	0.866184i	$0.666566\pi$
$158$	−8.78969	−0.699271
$159$	−30.8765	−2.44867
$160$	0	0
$161$	−30.9715	−2.44089
$162$	18.4167	1.44695
$163$	19.2464	1.50750	0.753748	−	0.657163i	$-0.228244\pi$
$163$	19.2464	1.50750	0.753748	+	0.657163i	$0.228244\pi$
$164$	19.2061	1.49974
$165$	0	0
$166$	−8.99505	−0.698151
$167$	−10.3383	−0.800002	−0.400001	−	0.916515i	$-0.630990\pi$
$167$	−10.3383	−0.800002	−0.400001	+	0.916515i	$0.630990\pi$
$168$	−48.4006	−3.73419
$169$	−10.3923	−0.799408
$170$	0	0
$171$	20.5590	1.57219
$172$	41.2033	3.14172
$173$	−9.42560	−0.716615	−0.358307	−	0.933604i	$-0.616646\pi$
$173$	−9.42560	−0.716615	−0.358307	+	0.933604i	$0.616646\pi$
$174$	22.5037	1.70600
$175$	0	0
$176$	−5.82423	−0.439018
$177$	−17.8342	−1.34050
$178$	−15.7849	−1.18313
$179$	5.85667	0.437748	0.218874	−	0.975753i	$-0.429762\pi$
$179$	5.85667	0.437748	0.218874	+	0.975753i	$0.429762\pi$
$180$	0	0
$181$	9.90186	0.736000	0.368000	−	0.929826i	$-0.380043\pi$
$181$	9.90186	0.736000	0.368000	+	0.929826i	$0.380043\pi$
$182$	14.1027	1.04536
$183$	10.9367	0.808463
$184$	−47.7937	−3.52340
$185$	0	0
$186$	25.4791	1.86822
$187$	−0.689061	−0.0503892
$188$	−30.0910	−2.19461
$189$	4.10814	0.298823
$190$	0	0
$191$	22.4029	1.62102	0.810509	−	0.585726i	$-0.199190\pi$
$191$	22.4029	1.62102	0.810509	+	0.585726i	$0.199190\pi$
$192$	14.1504	1.02122
$193$	4.82046	0.346984	0.173492	−	0.984835i	$-0.444495\pi$
$193$	4.82046	0.346984	0.173492	+	0.984835i	$0.444495\pi$
$194$	1.68532	0.120999
$195$	0	0
$196$	22.3247	1.59462
$197$	9.40255	0.669904	0.334952	−	0.942235i	$-0.391280\pi$
$197$	9.40255	0.669904	0.334952	+	0.942235i	$0.391280\pi$
$198$	9.79357	0.695999
$199$	−19.8646	−1.40816	−0.704082	−	0.710118i	$-0.748641\pi$
$199$	−19.8646	−1.40816	−0.704082	+	0.710118i	$0.748641\pi$
$200$	0	0
$201$	−16.6119	−1.17172
$202$	−10.7095	−0.753519
$203$	−12.5107	−0.878081
$204$	−6.43085	−0.450249
$205$	0	0
$206$	−42.8390	−2.98473
$207$	30.5064	2.12034
$208$	8.26092	0.572791
$209$	−6.76473	−0.467926
$210$	0	0
$211$	−24.3041	−1.67317	−0.836583	−	0.547841i	$-0.815450\pi$
$211$	−24.3041	−1.67317	−0.836583	+	0.547841i	$0.815450\pi$
$212$	50.7853	3.48795
$213$	−2.01306	−0.137932
$214$	1.08169	0.0739428
$215$	0	0
$216$	6.33948	0.431347
$217$	−14.1649	−0.961575
$218$	1.07301	0.0726735
$219$	23.6343	1.59706
$220$	0	0
$221$	0.977345	0.0657433
$222$	16.5039	1.10767
$223$	−27.8002	−1.86164	−0.930819	−	0.365480i	$-0.880905\pi$
$223$	−27.8002	−1.86164	−0.930819	+	0.365480i	$0.880905\pi$
$224$	6.59022	0.440327
$225$	0	0
$226$	11.9324	0.793731
$227$	18.3354	1.21697	0.608483	−	0.793567i	$-0.291778\pi$
$227$	18.3354	1.21697	0.608483	+	0.793567i	$0.291778\pi$
$228$	−63.1337	−4.18113
$229$	−16.9464	−1.11985	−0.559926	−	0.828542i	$-0.689171\pi$
$229$	−16.9464	−1.11985	−0.559926	+	0.828542i	$0.689171\pi$
$230$	0	0
$231$	−10.1653	−0.668827
$232$	−19.3060	−1.26750
$233$	−28.2361	−1.84981	−0.924905	−	0.380198i	$-0.875856\pi$
$233$	−28.2361	−1.84981	−0.924905	+	0.380198i	$0.875856\pi$
$234$	−13.8909	−0.908078
$235$	0	0
$236$	29.3334	1.90944
$237$	−8.98632	−0.583724
$238$	5.28557	0.342613
$239$	1.81005	0.117082	0.0585411	−	0.998285i	$-0.481355\pi$
$239$	1.81005	0.117082	0.0585411	+	0.998285i	$0.481355\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	24.1242	1.55076
$243$	22.3370	1.43292
$244$	−17.9885	−1.15159
$245$	0	0
$246$	29.0296	1.85086
$247$	9.59490	0.610509
$248$	−21.8586	−1.38802
$249$	−9.19626	−0.582790
$250$	0	0
$251$	−13.9243	−0.878892	−0.439446	−	0.898269i	$-0.644825\pi$
$251$	−13.9243	−0.878892	−0.439446	+	0.898269i	$0.644825\pi$
$252$	−50.8136	−3.20096
$253$	−10.0378	−0.631073
$254$	−3.01433	−0.189136
$255$	0	0
$256$	−32.6022	−2.03764
$257$	−25.0100	−1.56008	−0.780039	−	0.625731i	$-0.784801\pi$
$257$	−25.0100	−1.56008	−0.780039	+	0.625731i	$0.784801\pi$
$258$	62.2781	3.87726
$259$	−9.17521	−0.570120
$260$	0	0
$261$	12.3229	0.762767
$262$	−44.3014	−2.73695
$263$	−29.4278	−1.81460	−0.907298	−	0.420488i	$-0.861859\pi$
$263$	−29.4278	−1.81460	−0.907298	+	0.420488i	$0.861859\pi$
$264$	−15.6866	−0.965443
$265$	0	0
$266$	51.8902	3.18159
$267$	−16.1380	−0.987631
$268$	27.3230	1.66902
$269$	15.6348	0.953271	0.476636	−	0.879101i	$-0.341856\pi$
$269$	15.6348	0.953271	0.476636	+	0.879101i	$0.341856\pi$
$270$	0	0
$271$	−9.16024	−0.556445	−0.278222	−	0.960517i	$-0.589745\pi$
$271$	−9.16024	−0.556445	−0.278222	+	0.960517i	$0.589745\pi$
$272$	3.09612	0.187730
$273$	14.4182	0.872627
$274$	−53.1447	−3.21059
$275$	0	0
$276$	−93.6806	−5.63891
$277$	23.4278	1.40764	0.703822	−	0.710377i	$-0.251475\pi$
$277$	23.4278	1.40764	0.703822	+	0.710377i	$0.251475\pi$
$278$	4.10881	0.246430
$279$	13.9522	0.835296
$280$	0	0
$281$	3.41128	0.203500	0.101750	−	0.994810i	$-0.467556\pi$
$281$	3.41128	0.203500	0.101750	+	0.994810i	$0.467556\pi$
$282$	−45.4820	−2.70841
$283$	−6.65665	−0.395697	−0.197848	−	0.980233i	$-0.563395\pi$
$283$	−6.65665	−0.395697	−0.197848	+	0.980233i	$0.563395\pi$
$284$	3.31104	0.196474
$285$	0	0
$286$	4.57067	0.270269
$287$	−16.1388	−0.952642
$288$	−6.49126	−0.382501
$289$	−16.6337	−0.978453
$290$	0	0
$291$	1.72302	0.101005
$292$	−38.8734	−2.27489
$293$	−21.0097	−1.22740	−0.613701	−	0.789539i	$-0.710320\pi$
$293$	−21.0097	−1.22740	−0.613701	+	0.789539i	$0.710320\pi$
$294$	33.7434	1.96796
$295$	0	0
$296$	−14.1587	−0.822960
$297$	1.33144	0.0772582
$298$	21.2282	1.22972
$299$	14.2374	0.823368
$300$	0	0
$301$	−34.6229	−1.99563
$302$	−49.1290	−2.82706
$303$	−10.9491	−0.629009
$304$	30.3956	1.74331
$305$	0	0
$306$	−5.20621	−0.297619
$307$	0.787116	0.0449231	0.0224615	−	0.999748i	$-0.492850\pi$
$307$	0.787116	0.0449231	0.0224615	+	0.999748i	$0.492850\pi$
$308$	16.7197	0.952694
$309$	−43.7972	−2.49154
$310$	0	0
$311$	6.67763	0.378654	0.189327	−	0.981914i	$-0.439369\pi$
$311$	6.67763	0.378654	0.189327	+	0.981914i	$0.439369\pi$
$312$	22.2494	1.25963
$313$	33.0650	1.86895	0.934473	−	0.356035i	$-0.115872\pi$
$313$	33.0650	1.86895	0.934473	+	0.356035i	$0.115872\pi$
$314$	31.1332	1.75695
$315$	0	0
$316$	14.7806	0.831471
$317$	30.6269	1.72018	0.860090	−	0.510143i	$-0.170407\pi$
$317$	30.6269	1.72018	0.860090	+	0.510143i	$0.170407\pi$
$318$	76.7610	4.30454
$319$	−4.05472	−0.227021
$320$	0	0
$321$	1.10589	0.0617246
$322$	76.9970	4.29088
$323$	3.59609	0.200092
$324$	−30.9690	−1.72050
$325$	0	0
$326$	−47.8478	−2.65005
$327$	1.09701	0.0606650
$328$	−24.9046	−1.37513
$329$	25.2853	1.39402
$330$	0	0
$331$	17.8708	0.982265	0.491133	−	0.871085i	$-0.336583\pi$
$331$	17.8708	0.982265	0.491133	+	0.871085i	$0.336583\pi$
$332$	15.1259	0.830140
$333$	9.03743	0.495248
$334$	25.7017	1.40633
$335$	0	0
$336$	45.6752	2.49179
$337$	10.0513	0.547528	0.273764	−	0.961797i	$-0.411731\pi$
$337$	10.0513	0.547528	0.273764	+	0.961797i	$0.411731\pi$
$338$	25.8359	1.40529
$339$	12.1993	0.662576
$340$	0	0
$341$	−4.59083	−0.248607
$342$	−51.1110	−2.76376
$343$	5.83066	0.314826
$344$	−53.4285	−2.88067
$345$	0	0
$346$	23.4326	1.25975
$347$	7.98608	0.428715	0.214358	−	0.976755i	$-0.431234\pi$
$347$	7.98608	0.428715	0.214358	+	0.976755i	$0.431234\pi$
$348$	−37.8417	−2.02853
$349$	32.1666	1.72184	0.860920	−	0.508740i	$-0.169889\pi$
$349$	32.1666	1.72184	0.860920	+	0.508740i	$0.169889\pi$
$350$	0	0
$351$	−1.88848	−0.100800
$352$	2.13588	0.113843
$353$	6.03479	0.321200	0.160600	−	0.987020i	$-0.448657\pi$
$353$	6.03479	0.321200	0.160600	+	0.987020i	$0.448657\pi$
$354$	44.3369	2.35648
$355$	0	0
$356$	26.5436	1.40681
$357$	5.40381	0.286000
$358$	−14.5601	−0.769523
$359$	2.89613	0.152852	0.0764259	−	0.997075i	$-0.475649\pi$
$359$	2.89613	0.152852	0.0764259	+	0.997075i	$0.475649\pi$
$360$	0	0
$361$	16.3040	0.858104
$362$	−24.6167	−1.29382
$363$	24.6639	1.29452
$364$	−23.7148	−1.24299
$365$	0	0
$366$	−27.1893	−1.42121
$367$	0.0128090	0.000668625 0	0.000334313	−	1.00000i	$-0.499894\pi$
$367$	0.0128090	0.000668625 0	0.000334313		1.00000i	$0.499894\pi$
$368$	45.1025	2.35113
$369$	15.8964	0.827536
$370$	0	0
$371$	−42.6746	−2.21556
$372$	−42.8451	−2.22142
$373$	−25.0094	−1.29494	−0.647470	−	0.762091i	$-0.724173\pi$
$373$	−25.0094	−1.29494	−0.647470	+	0.762091i	$0.724173\pi$
$374$	1.71305	0.0885797
$375$	0	0
$376$	39.0191	2.01226
$377$	5.75109	0.296196
$378$	−10.2131	−0.525305
$379$	−15.0753	−0.774367	−0.387184	−	0.922003i	$-0.626552\pi$
$379$	−15.0753	−0.774367	−0.387184	+	0.922003i	$0.626552\pi$
$380$	0	0
$381$	−3.08176	−0.157884
$382$	−55.6951	−2.84961
$383$	26.8907	1.37405	0.687025	−	0.726634i	$-0.258916\pi$
$383$	26.8907	1.37405	0.687025	+	0.726634i	$0.258916\pi$
$384$	−44.7153	−2.28187
$385$	0	0
$386$	−11.9840	−0.609968
$387$	34.1030	1.73355
$388$	−2.83400	−0.143875
$389$	−12.6043	−0.639064	−0.319532	−	0.947575i	$-0.603526\pi$
$389$	−12.6043	−0.639064	−0.319532	+	0.947575i	$0.603526\pi$
$390$	0	0
$391$	5.33605	0.269856
$392$	−28.9486	−1.46212
$393$	−45.2924	−2.28470
$394$	−23.3753	−1.17763
$395$	0	0
$396$	−16.4687	−0.827581
$397$	−14.8584	−0.745721	−0.372861	−	0.927887i	$-0.621623\pi$
$397$	−14.8584	−0.745721	−0.372861	+	0.927887i	$0.621623\pi$
$398$	49.3847	2.47543
$399$	53.0509	2.65587
$400$	0	0
$401$	29.6303	1.47967	0.739834	−	0.672790i	$-0.234904\pi$
$401$	29.6303	1.47967	0.739834	+	0.672790i	$0.234904\pi$
$402$	41.2983	2.05977
$403$	6.51150	0.324361
$404$	18.0089	0.895976
$405$	0	0
$406$	31.1025	1.54359
$407$	−2.97368	−0.147400
$408$	8.33890	0.412837
$409$	−37.5515	−1.85680	−0.928400	−	0.371581i	$-0.878816\pi$
$409$	−37.5515	−1.85680	−0.928400	+	0.371581i	$0.878816\pi$
$410$	0	0
$411$	−54.3336	−2.68008
$412$	72.0370	3.54901
$413$	−24.6487	−1.21288
$414$	−75.8408	−3.72737
$415$	0	0
$416$	−3.02947	−0.148532
$417$	4.20072	0.205710
$418$	16.8176	0.822573
$419$	11.0720	0.540903	0.270452	−	0.962734i	$-0.412827\pi$
$419$	11.0720	0.540903	0.270452	+	0.962734i	$0.412827\pi$
$420$	0	0
$421$	−23.7902	−1.15946	−0.579731	−	0.814808i	$-0.696842\pi$
$421$	−23.7902	−1.15946	−0.579731	+	0.814808i	$0.696842\pi$
$422$	60.4216	2.94128
$423$	−24.9056	−1.21095
$424$	−65.8534	−3.19812
$425$	0	0
$426$	5.00458	0.242473
$427$	15.1156	0.731497
$428$	−1.81895	−0.0879221
$429$	4.67291	0.225610
$430$	0	0
$431$	34.0014	1.63779	0.818896	−	0.573942i	$-0.194587\pi$
$431$	34.0014	1.63779	0.818896	+	0.573942i	$0.194587\pi$
$432$	−5.98251	−0.287834
$433$	−26.6020	−1.27841	−0.639205	−	0.769036i	$-0.720737\pi$
$433$	−26.6020	−1.27841	−0.639205	+	0.769036i	$0.720737\pi$
$434$	35.2148	1.69036
$435$	0	0
$436$	−1.80435	−0.0864128
$437$	52.3857	2.50595
$438$	−58.7564	−2.80749
$439$	−0.945536	−0.0451280	−0.0225640	−	0.999745i	$-0.507183\pi$
$439$	−0.945536	−0.0451280	−0.0225640	+	0.999745i	$0.507183\pi$
$440$	0	0
$441$	18.4777	0.879890
$442$	−2.42974	−0.115571
$443$	−13.7990	−0.655608	−0.327804	−	0.944746i	$-0.606309\pi$
$443$	−13.7990	−0.655608	−0.327804	+	0.944746i	$0.606309\pi$
$444$	−27.7526	−1.31708
$445$	0	0
$446$	69.1130	3.27260
$447$	21.7031	1.02652
$448$	19.5573	0.923997
$449$	1.63316	0.0770736	0.0385368	−	0.999257i	$-0.487730\pi$
$449$	1.63316	0.0770736	0.0385368	+	0.999257i	$0.487730\pi$
$450$	0	0
$451$	−5.23056	−0.246298
$452$	−20.0653	−0.943790
$453$	−50.2280	−2.35992
$454$	−45.5831	−2.13932
$455$	0	0
$456$	81.8657	3.83371
$457$	−27.7891	−1.29992	−0.649959	−	0.759969i	$-0.725214\pi$
$457$	−27.7891	−1.29992	−0.649959	+	0.759969i	$0.725214\pi$
$458$	42.1299	1.96860
$459$	−0.707788	−0.0330367
$460$	0	0
$461$	22.0285	1.02597	0.512985	−	0.858397i	$-0.328540\pi$
$461$	22.0285	1.02597	0.512985	+	0.858397i	$0.328540\pi$
$462$	25.2716	1.17574
$463$	29.3034	1.36184	0.680922	−	0.732356i	$-0.261579\pi$
$463$	29.3034	1.36184	0.680922	+	0.732356i	$0.261579\pi$
$464$	18.2189	0.845789
$465$	0	0
$466$	70.1968	3.25180
$467$	−20.4980	−0.948533	−0.474266	−	0.880381i	$-0.657287\pi$
$467$	−20.4980	−0.948533	−0.474266	+	0.880381i	$0.657287\pi$
$468$	23.3587	1.07975
$469$	−22.9594	−1.06017
$470$	0	0
$471$	31.8296	1.46663
$472$	−38.0367	−1.75078
$473$	−11.2213	−0.515954
$474$	22.3405	1.02614
$475$	0	0
$476$	−8.88810	−0.407386
$477$	42.0338	1.92460
$478$	−4.49989	−0.205820
$479$	35.9333	1.64183	0.820916	−	0.571049i	$-0.193463\pi$
$479$	35.9333	1.64183	0.820916	+	0.571049i	$0.193463\pi$
$480$	0	0
$481$	4.21778	0.192314
$482$	2.48606	0.113237
$483$	78.7194	3.58186
$484$	−40.5667	−1.84394
$485$	0	0
$486$	−55.5312	−2.51895
$487$	−24.5813	−1.11389	−0.556943	−	0.830551i	$-0.688026\pi$
$487$	−24.5813	−1.11389	−0.556943	+	0.830551i	$0.688026\pi$
$488$	23.3257	1.05591
$489$	−48.9182	−2.21216
$490$	0	0
$491$	−0.973185	−0.0439192	−0.0219596	−	0.999759i	$-0.506991\pi$
$491$	−0.973185	−0.0439192	−0.0219596	+	0.999759i	$0.506991\pi$
$492$	−48.8156	−2.20078
$493$	2.15546	0.0970772
$494$	−23.8535	−1.07322
$495$	0	0
$496$	20.6277	0.926213
$497$	−2.78225	−0.124801
$498$	22.8625	1.02449
$499$	10.6335	0.476022	0.238011	−	0.971262i	$-0.423505\pi$
$499$	10.6335	0.476022	0.238011	+	0.971262i	$0.423505\pi$
$500$	0	0
$501$	26.2766	1.17395
$502$	34.6166	1.54501
$503$	26.8280	1.19620	0.598100	−	0.801422i	$-0.295923\pi$
$503$	26.8280	1.19620	0.598100	+	0.801422i	$0.295923\pi$
$504$	65.8902	2.93498
$505$	0	0
$506$	24.9547	1.10937
$507$	26.4139	1.17308
$508$	5.06884	0.224893
$509$	−18.1531	−0.804623	−0.402311	−	0.915503i	$-0.631793\pi$
$509$	−18.1531	−0.804623	−0.402311	+	0.915503i	$0.631793\pi$
$510$	0	0
$511$	32.6651	1.44502
$512$	45.8654	2.02698
$513$	−6.94858	−0.306787
$514$	62.1763	2.74248
$515$	0	0
$516$	−104.725	−4.61028
$517$	8.19495	0.360413
$518$	22.8101	1.00222
$519$	23.9568	1.05159
$520$	0	0
$521$	22.4186	0.982176	0.491088	−	0.871110i	$-0.336599\pi$
$521$	22.4186	0.982176	0.491088	+	0.871110i	$0.336599\pi$
$522$	−30.6354	−1.34088
$523$	−4.11538	−0.179953	−0.0899764	−	0.995944i	$-0.528679\pi$
$523$	−4.11538	−0.179953	−0.0899764	+	0.995944i	$0.528679\pi$
$524$	74.4963	3.25439
$525$	0	0
$526$	73.1594	3.18990
$527$	2.44046	0.106308
$528$	14.8033	0.644231
$529$	54.7323	2.37966
$530$	0	0
$531$	24.2786	1.05360
$532$	−87.2573	−3.78309
$533$	7.41888	0.321347
$534$	40.1202	1.73617
$535$	0	0
$536$	−35.4299	−1.53034
$537$	−14.8858	−0.642368
$538$	−38.8691	−1.67577
$539$	−6.07990	−0.261880
$540$	0	0
$541$	−15.3050	−0.658013	−0.329006	−	0.944328i	$-0.606714\pi$
$541$	−15.3050	−0.658013	−0.329006	+	0.944328i	$0.606714\pi$
$542$	22.7729	0.978181
$543$	−25.1673	−1.08003
$544$	−1.13542	−0.0486808
$545$	0	0
$546$	−35.8444	−1.53400
$547$	−9.68504	−0.414102	−0.207051	−	0.978330i	$-0.566387\pi$
$547$	−9.68504	−0.414102	−0.207051	+	0.978330i	$0.566387\pi$
$548$	89.3670	3.81757
$549$	−14.8887	−0.635433
$550$	0	0
$551$	21.1609	0.901483
$552$	121.476	5.17036
$553$	−12.4200	−0.528153
$554$	−58.2431	−2.47451
$555$	0	0
$556$	−6.90928	−0.293019
$557$	−23.8770	−1.01170	−0.505850	−	0.862621i	$-0.668821\pi$
$557$	−23.8770	−1.01170	−0.505850	+	0.862621i	$0.668821\pi$
$558$	−34.6860	−1.46838
$559$	15.9159	0.673171
$560$	0	0
$561$	1.75137	0.0739429
$562$	−8.48066	−0.357735
$563$	−27.1255	−1.14320	−0.571602	−	0.820531i	$-0.693678\pi$
$563$	−27.1255	−1.14320	−0.571602	+	0.820531i	$0.693678\pi$
$564$	76.4815	3.22045
$565$	0	0
$566$	16.5489	0.695600
$567$	26.0231	1.09287
$568$	−4.29344	−0.180149
$569$	18.5561	0.777914	0.388957	−	0.921256i	$-0.372835\pi$
$569$	18.5561	0.777914	0.388957	+	0.921256i	$0.372835\pi$
$570$	0	0
$571$	−27.6667	−1.15781	−0.578907	−	0.815394i	$-0.696521\pi$
$571$	−27.6667	−1.15781	−0.578907	+	0.815394i	$0.696521\pi$
$572$	−7.68593	−0.321365
$573$	−56.9410	−2.37874
$574$	40.1220	1.67466
$575$	0	0
$576$	−19.2637	−0.802653
$577$	−12.6565	−0.526897	−0.263448	−	0.964673i	$-0.584860\pi$
$577$	−12.6565	−0.526897	−0.263448	+	0.964673i	$0.584860\pi$
$578$	41.3524	1.72003
$579$	−12.2520	−0.509178
$580$	0	0
$581$	−12.7102	−0.527308
$582$	−4.28354	−0.177558
$583$	−13.8308	−0.572813
$584$	50.4072	2.08587
$585$	0	0
$586$	52.2315	2.15766
$587$	5.40800	0.223212	0.111606	−	0.993753i	$-0.464401\pi$
$587$	5.40800	0.223212	0.111606	+	0.993753i	$0.464401\pi$
$588$	−56.7422	−2.34001
$589$	23.9587	0.987203
$590$	0	0
$591$	−23.8982	−0.983042
$592$	13.3615	0.549153
$593$	37.4447	1.53767	0.768834	−	0.639448i	$-0.220837\pi$
$593$	37.4447	1.53767	0.768834	+	0.639448i	$0.220837\pi$
$594$	−3.31005	−0.135813
$595$	0	0
$596$	−35.6969	−1.46220
$597$	50.4894	2.06639
$598$	−35.3950	−1.44741
$599$	−35.4989	−1.45045	−0.725223	−	0.688514i	$-0.758263\pi$
$599$	−35.4989	−1.45045	−0.725223	+	0.688514i	$0.758263\pi$
$600$	0	0
$601$	18.4649	0.753200	0.376600	−	0.926376i	$-0.377093\pi$
$601$	18.4649	0.753200	0.376600	+	0.926376i	$0.377093\pi$
$602$	86.0748	3.50815
$603$	22.6147	0.920941
$604$	82.6143	3.36153
$605$	0	0
$606$	27.2201	1.10574
$607$	−22.9185	−0.930235	−0.465118	−	0.885249i	$-0.653988\pi$
$607$	−22.9185	−0.930235	−0.465118	+	0.885249i	$0.653988\pi$
$608$	−11.1468	−0.452063
$609$	31.7982	1.28853
$610$	0	0
$611$	−11.6235	−0.470235
$612$	8.75464	0.353885
$613$	28.7490	1.16116	0.580581	−	0.814202i	$-0.302825\pi$
$613$	28.7490	1.16116	0.580581	+	0.814202i	$0.302825\pi$
$614$	−1.95682	−0.0789708
$615$	0	0
$616$	−21.6805	−0.873533
$617$	16.1808	0.651416	0.325708	−	0.945470i	$-0.394397\pi$
$617$	16.1808	0.651416	0.325708	+	0.945470i	$0.394397\pi$
$618$	108.883	4.37990
$619$	−0.0391287	−0.00157271	−0.000786357	−	1.00000i	$-0.500250\pi$
$619$	−0.0391287	−0.00157271	−0.000786357		1.00000i	$0.500250\pi$
$620$	0	0
$621$	−10.3106	−0.413751
$622$	−16.6010	−0.665640
$623$	−22.3044	−0.893609
$624$	−20.9966	−0.840535
$625$	0	0
$626$	−82.2017	−3.28544
$627$	17.1938	0.686652
$628$	−52.3529	−2.08911
$629$	1.58079	0.0630302
$630$	0	0
$631$	−0.783434	−0.0311880	−0.0155940	−	0.999878i	$-0.504964\pi$
$631$	−0.783434	−0.0311880	−0.0155940	+	0.999878i	$0.504964\pi$
$632$	−19.1660	−0.762383
$633$	61.7732	2.45526
$634$	−76.1405	−3.02392
$635$	0	0
$636$	−129.080	−5.11834
$637$	8.62355	0.341677
$638$	10.0803	0.399082
$639$	2.74048	0.108412
$640$	0	0
$641$	21.9104	0.865410	0.432705	−	0.901536i	$-0.357559\pi$
$641$	21.9104	0.865410	0.432705	+	0.901536i	$0.357559\pi$
$642$	−2.74931	−0.108506
$643$	48.8006	1.92451	0.962253	−	0.272156i	$-0.0877368\pi$
$643$	48.8006	1.92451	0.962253	+	0.272156i	$0.0877368\pi$
$644$	−129.476	−5.10209
$645$	0	0
$646$	−8.94011	−0.351744
$647$	7.78379	0.306012	0.153006	−	0.988225i	$-0.451105\pi$
$647$	7.78379	0.306012	0.153006	+	0.988225i	$0.451105\pi$
$648$	40.1577	1.57754
$649$	−7.98863	−0.313581
$650$	0	0
$651$	36.0026	1.41105
$652$	80.4598	3.15105
$653$	−4.36818	−0.170940	−0.0854700	−	0.996341i	$-0.527239\pi$
$653$	−4.36818	−0.170940	−0.0854700	+	0.996341i	$0.527239\pi$
$654$	−2.72725	−0.106644
$655$	0	0
$656$	23.5022	0.917608
$657$	−32.1746	−1.25525
$658$	−62.8609	−2.45057
$659$	−2.34291	−0.0912670	−0.0456335	−	0.998958i	$-0.514531\pi$
$659$	−2.34291	−0.0912670	−0.0456335	+	0.998958i	$0.514531\pi$
$660$	0	0
$661$	22.3568	0.869578	0.434789	−	0.900532i	$-0.356823\pi$
$661$	22.3568	0.869578	0.434789	+	0.900532i	$0.356823\pi$
$662$	−44.4278	−1.72674
$663$	−2.48409	−0.0964742
$664$	−19.6138	−0.761162
$665$	0	0
$666$	−22.4676	−0.870603
$667$	31.3995	1.21579
$668$	−43.2194	−1.67221
$669$	70.6591	2.73184
$670$	0	0
$671$	4.89897	0.189122
$672$	−16.7502	−0.646153
$673$	−33.3606	−1.28596	−0.642979	−	0.765884i	$-0.722302\pi$
$673$	−33.3606	−1.28596	−0.642979	+	0.765884i	$0.722302\pi$
$674$	−24.9881	−0.962506
$675$	0	0
$676$	−43.4451	−1.67097
$677$	21.4136	0.822993	0.411497	−	0.911411i	$-0.365006\pi$
$677$	21.4136	0.822993	0.411497	+	0.911411i	$0.365006\pi$
$678$	−30.3283	−1.16475
$679$	2.38140	0.0913896
$680$	0	0
$681$	−46.6028	−1.78582
$682$	11.4131	0.437030
$683$	−15.2657	−0.584125	−0.292063	−	0.956399i	$-0.594342\pi$
$683$	−15.2657	−0.584125	−0.292063	+	0.956399i	$0.594342\pi$
$684$	85.9471	3.28627
$685$	0	0
$686$	−14.4954	−0.553436
$687$	43.0724	1.64331
$688$	50.4199	1.92224
$689$	19.6172	0.747356
$690$	0	0
$691$	−30.5646	−1.16273	−0.581366	−	0.813642i	$-0.697482\pi$
$691$	−30.5646	−1.16273	−0.581366	+	0.813642i	$0.697482\pi$
$692$	−39.4038	−1.49791
$693$	13.8385	0.525682
$694$	−19.8539	−0.753644
$695$	0	0
$696$	49.0695	1.85997
$697$	2.78054	0.105320
$698$	−79.9683	−3.02684
$699$	71.7671	2.71448
$700$	0	0
$701$	13.1136	0.495296	0.247648	−	0.968850i	$-0.420342\pi$
$701$	13.1136	0.495296	0.247648	+	0.968850i	$0.420342\pi$
$702$	4.69488	0.177197
$703$	15.5191	0.585314
$704$	6.33851	0.238892
$705$	0	0
$706$	−15.0029	−0.564641
$707$	−15.1328	−0.569127
$708$	−74.5560	−2.80198
$709$	4.35194	0.163441	0.0817203	−	0.996655i	$-0.473959\pi$
$709$	4.35194	0.163441	0.0817203	+	0.996655i	$0.473959\pi$
$710$	0	0
$711$	12.2335	0.458793
$712$	−34.4192	−1.28991
$713$	35.5511	1.33140
$714$	−13.4342	−0.502763
$715$	0	0
$716$	24.4839	0.915005
$717$	−4.60055	−0.171811
$718$	−7.19996	−0.268700
$719$	12.0472	0.449285	0.224642	−	0.974441i	$-0.427879\pi$
$719$	12.0472	0.449285	0.224642	+	0.974441i	$0.427879\pi$
$720$	0	0
$721$	−60.5324	−2.25434
$722$	−40.5327	−1.50847
$723$	2.54168	0.0945259
$724$	41.3948	1.53843
$725$	0	0
$726$	−61.3159	−2.27565
$727$	19.5591	0.725408	0.362704	−	0.931904i	$-0.381854\pi$
$727$	19.5591	0.725408	0.362704	+	0.931904i	$0.381854\pi$
$728$	30.7510	1.13971
$729$	−34.5495	−1.27961
$730$	0	0
$731$	5.96516	0.220629
$732$	45.7209	1.68989
$733$	8.62248	0.318478	0.159239	−	0.987240i	$-0.449096\pi$
$733$	8.62248	0.318478	0.159239	+	0.987240i	$0.449096\pi$
$734$	−0.0318440	−0.00117538
$735$	0	0
$736$	−16.5401	−0.609678
$737$	−7.44113	−0.274098
$738$	−39.5195	−1.45473
$739$	35.8849	1.32005	0.660023	−	0.751245i	$-0.270546\pi$
$739$	35.8849	1.32005	0.660023	+	0.751245i	$0.270546\pi$
$740$	0	0
$741$	−24.3871	−0.895883
$742$	106.092	3.89475
$743$	10.1991	0.374169	0.187084	−	0.982344i	$-0.440096\pi$
$743$	10.1991	0.374169	0.187084	+	0.982344i	$0.440096\pi$
$744$	55.5574	2.03683
$745$	0	0
$746$	62.1750	2.27639
$747$	12.5193	0.458059
$748$	−2.88063	−0.105326
$749$	1.52845	0.0558484
$750$	0	0
$751$	−5.49326	−0.200452	−0.100226	−	0.994965i	$-0.531957\pi$
$751$	−5.49326	−0.200452	−0.100226	+	0.994965i	$0.531957\pi$
$752$	−36.8219	−1.34276
$753$	35.3910	1.28972
$754$	−14.2976	−0.520687
$755$	0	0
$756$	17.1741	0.624616
$757$	−51.8664	−1.88512	−0.942558	−	0.334043i	$-0.891587\pi$
$757$	−51.8664	−1.88512	−0.942558	+	0.334043i	$0.891587\pi$
$758$	37.4782	1.36127
$759$	25.5129	0.926059
$760$	0	0
$761$	13.0411	0.472739	0.236370	−	0.971663i	$-0.424042\pi$
$761$	13.0411	0.472739	0.236370	+	0.971663i	$0.424042\pi$
$762$	7.66146	0.277545
$763$	1.51619	0.0548897
$764$	93.6556	3.38834
$765$	0	0
$766$	−66.8520	−2.41546
$767$	11.3308	0.409133
$768$	82.8642	2.99010
$769$	12.1598	0.438493	0.219247	−	0.975669i	$-0.429640\pi$
$769$	12.1598	0.438493	0.219247	+	0.975669i	$0.429640\pi$
$770$	0	0
$771$	63.5672	2.28932
$772$	20.1520	0.725286
$773$	20.2445	0.728143	0.364072	−	0.931371i	$-0.381386\pi$
$773$	20.2445	0.728143	0.364072	+	0.931371i	$0.381386\pi$
$774$	−84.7823	−3.04744
$775$	0	0
$776$	3.67486	0.131920
$777$	23.3204	0.836615
$778$	31.3351	1.12342
$779$	27.2974	0.978031
$780$	0	0
$781$	−0.901726	−0.0322663
$782$	−13.2658	−0.474382
$783$	−4.16491	−0.148842
$784$	27.3185	0.975660
$785$	0	0
$786$	112.600	4.01630
$787$	−15.3104	−0.545758	−0.272879	−	0.962048i	$-0.587976\pi$
$787$	−15.3104	−0.545758	−0.272879	+	0.962048i	$0.587976\pi$
$788$	39.3074	1.40027
$789$	74.7959	2.66281
$790$	0	0
$791$	16.8607	0.599499
$792$	21.3550	0.758816
$793$	−6.94855	−0.246750
$794$	36.9389	1.31091
$795$	0	0
$796$	−83.0441	−2.94342
$797$	−10.8457	−0.384173	−0.192087	−	0.981378i	$-0.561525\pi$
$797$	−10.8457	−0.384173	−0.192087	+	0.981378i	$0.561525\pi$
$798$	−131.888	−4.66878
$799$	−4.35639	−0.154118
$800$	0	0
$801$	21.9695	0.776255
$802$	−73.6628	−2.60113
$803$	10.5867	0.373598
$804$	−69.4463	−2.44918
$805$	0	0
$806$	−16.1880	−0.570198
$807$	−39.7386	−1.39887
$808$	−23.3522	−0.821528
$809$	8.35659	0.293802	0.146901	−	0.989151i	$-0.453070\pi$
$809$	8.35659	0.293802	0.146901	+	0.989151i	$0.453070\pi$
$810$	0	0
$811$	−39.5988	−1.39050	−0.695251	−	0.718767i	$-0.744707\pi$
$811$	−39.5988	−1.39050	−0.695251	+	0.718767i	$0.744707\pi$
$812$	−52.3012	−1.83541
$813$	23.2824	0.816548
$814$	7.39275	0.259116
$815$	0	0
$816$	−7.86934	−0.275482
$817$	58.5618	2.04882
$818$	93.3553	3.26409
$819$	−19.6282	−0.685864
$820$	0	0
$821$	32.5787	1.13701	0.568503	−	0.822681i	$-0.307523\pi$
$821$	32.5787	1.13701	0.568503	+	0.822681i	$0.307523\pi$
$822$	135.077	4.71134
$823$	38.4107	1.33891	0.669456	−	0.742852i	$-0.266527\pi$
$823$	38.4107	1.33891	0.669456	+	0.742852i	$0.266527\pi$
$824$	−93.4107	−3.25412
$825$	0	0
$826$	61.2783	2.13214
$827$	−2.35569	−0.0819154	−0.0409577	−	0.999161i	$-0.513041\pi$
$827$	−2.35569	−0.0819154	−0.0409577	+	0.999161i	$0.513041\pi$
$828$	127.532	4.43205
$829$	−34.4683	−1.19713	−0.598567	−	0.801073i	$-0.704263\pi$
$829$	−34.4683	−1.19713	−0.598567	+	0.801073i	$0.704263\pi$
$830$	0	0
$831$	−59.5460	−2.06563
$832$	−8.99037	−0.311685
$833$	3.23204	0.111983
$834$	−10.4432	−0.361620
$835$	0	0
$836$	−28.2800	−0.978085
$837$	−4.71559	−0.162995
$838$	−27.5257	−0.950860
$839$	15.3715	0.530682	0.265341	−	0.964155i	$-0.414516\pi$
$839$	15.3715	0.530682	0.265341	+	0.964155i	$0.414516\pi$
$840$	0	0
$841$	−16.3164	−0.562634
$842$	59.1439	2.03823
$843$	−8.67037	−0.298624
$844$	−101.604	−3.49734
$845$	0	0
$846$	61.9170	2.12875
$847$	34.0880	1.17128
$848$	62.1452	2.13408
$849$	16.9190	0.580660
$850$	0	0
$851$	23.0280	0.789389
$852$	−8.41560	−0.288314
$853$	−4.42189	−0.151403	−0.0757013	−	0.997131i	$-0.524120\pi$
$853$	−4.42189	−0.151403	−0.0757013	+	0.997131i	$0.524120\pi$
$854$	−37.5784	−1.28591
$855$	0	0
$856$	2.35864	0.0806165
$857$	−39.0679	−1.33453	−0.667267	−	0.744819i	$-0.732536\pi$
$857$	−39.0679	−1.33453	−0.667267	+	0.744819i	$0.732536\pi$
$858$	−11.6172	−0.396603
$859$	−24.2696	−0.828067	−0.414034	−	0.910262i	$-0.635880\pi$
$859$	−24.2696	−0.828067	−0.414034	+	0.910262i	$0.635880\pi$
$860$	0	0
$861$	41.0195	1.39794
$862$	−84.5297	−2.87909
$863$	−5.99481	−0.204066	−0.102033	−	0.994781i	$-0.532535\pi$
$863$	−5.99481	−0.204066	−0.102033	+	0.994781i	$0.532535\pi$
$864$	2.19393	0.0746390
$865$	0	0
$866$	66.1343	2.24733
$867$	42.2775	1.43582
$868$	−59.2164	−2.00994
$869$	−4.02532	−0.136550
$870$	0	0
$871$	10.5543	0.357618
$872$	2.33971	0.0792326
$873$	−2.34564	−0.0793878
$874$	−130.234	−4.40523
$875$	0	0
$876$	98.8035	3.33826
$877$	14.3876	0.485835	0.242918	−	0.970047i	$-0.421895\pi$
$877$	14.3876	0.485835	0.242918	+	0.970047i	$0.421895\pi$
$878$	2.35066	0.0793310
$879$	53.3999	1.80113
$880$	0	0
$881$	−18.9648	−0.638939	−0.319470	−	0.947596i	$-0.603505\pi$
$881$	−18.9648	−0.638939	−0.319470	+	0.947596i	$0.603505\pi$
$882$	−45.9367	−1.54677
$883$	43.2919	1.45689	0.728445	−	0.685104i	$-0.240243\pi$
$883$	43.2919	1.45689	0.728445	+	0.685104i	$0.240243\pi$
$884$	4.08580	0.137420
$885$	0	0
$886$	34.3051	1.15250
$887$	25.8303	0.867298	0.433649	−	0.901082i	$-0.357226\pi$
$887$	25.8303	0.867298	0.433649	+	0.901082i	$0.357226\pi$
$888$	35.9869	1.20764
$889$	−4.25932	−0.142853
$890$	0	0
$891$	8.43407	0.282552
$892$	−116.219	−3.89130
$893$	−42.7680	−1.43118
$894$	−53.9553	−1.80453
$895$	0	0
$896$	−61.8012	−2.06463
$897$	−36.1867	−1.20824
$898$	−4.06014	−0.135489
$899$	14.3606	0.478954
$900$	0	0
$901$	7.35237	0.244943
$902$	13.0035	0.432970
$903$	88.0003	2.92847
$904$	26.0187	0.865369
$905$	0	0
$906$	124.870	4.14853
$907$	59.5457	1.97718	0.988592	−	0.150619i	$-0.0481267\pi$
$907$	59.5457	1.97718	0.988592	+	0.150619i	$0.0481267\pi$
$908$	76.6515	2.54377
$909$	14.9056	0.494386
$910$	0	0
$911$	−20.7916	−0.688856	−0.344428	−	0.938813i	$-0.611927\pi$
$911$	−20.7916	−0.688856	−0.344428	+	0.938813i	$0.611927\pi$
$912$	−77.2558	−2.55820
$913$	−4.11936	−0.136331
$914$	69.0854	2.28514
$915$	0	0
$916$	−70.8448	−2.34078
$917$	−62.5989	−2.06720
$918$	1.75960	0.0580756
$919$	−12.8058	−0.422426	−0.211213	−	0.977440i	$-0.567741\pi$
$919$	−12.8058	−0.422426	−0.211213	+	0.977440i	$0.567741\pi$
$920$	0	0
$921$	−2.00059	−0.0659217
$922$	−54.7643	−1.80357
$923$	1.27898	0.0420982
$924$	−42.4961	−1.39802
$925$	0	0
$926$	−72.8501	−2.39400
$927$	59.6234	1.95829
$928$	−6.68129	−0.219324
$929$	−15.1271	−0.496303	−0.248151	−	0.968721i	$-0.579823\pi$
$929$	−15.1271	−0.496303	−0.248151	+	0.968721i	$0.579823\pi$
$930$	0	0
$931$	31.7299	1.03991
$932$	−118.041	−3.86657
$933$	−16.9724	−0.555650
$934$	50.9592	1.66744
$935$	0	0
$936$	−30.2893	−0.990036
$937$	−24.0118	−0.784432	−0.392216	−	0.919873i	$-0.628292\pi$
$937$	−24.0118	−0.784432	−0.392216	+	0.919873i	$0.628292\pi$
$938$	57.0786	1.86368
$939$	−84.0405	−2.74256
$940$	0	0
$941$	−12.9166	−0.421070	−0.210535	−	0.977586i	$-0.567521\pi$
$941$	−12.9166	−0.421070	−0.210535	+	0.977586i	$0.567521\pi$
$942$	−79.1304	−2.57821
$943$	40.5052	1.31903
$944$	35.8949	1.16828
$945$	0	0
$946$	27.8968	0.907002
$947$	51.6692	1.67902	0.839511	−	0.543342i	$-0.182841\pi$
$947$	51.6692	1.67902	0.839511	+	0.543342i	$0.182841\pi$
$948$	−37.5674	−1.22013
$949$	−15.0159	−0.487437
$950$	0	0
$951$	−77.8437	−2.52425
$952$	11.5252	0.373535
$953$	−40.4194	−1.30931	−0.654657	−	0.755926i	$-0.727187\pi$
$953$	−40.4194	−1.30931	−0.654657	+	0.755926i	$0.727187\pi$
$954$	−104.499	−3.38327
$955$	0	0
$956$	7.56691	0.244731
$957$	10.3058	0.333138
$958$	−89.3323	−2.88620
$959$	−75.0947	−2.42493
$960$	0	0
$961$	−14.7406	−0.475503
$962$	−10.4857	−0.338071
$963$	−1.50550	−0.0485141
$964$	−4.18051	−0.134645
$965$	0	0
$966$	−195.701	−6.29659
$967$	−53.4789	−1.71977	−0.859883	−	0.510492i	$-0.829463\pi$
$967$	−53.4789	−1.71977	−0.859883	+	0.510492i	$0.829463\pi$
$968$	52.6031	1.69073
$969$	−9.14010	−0.293622
$970$	0	0
$971$	−21.3737	−0.685915	−0.342957	−	0.939351i	$-0.611429\pi$
$971$	−21.3737	−0.685915	−0.342957	+	0.939351i	$0.611429\pi$
$972$	93.3800	2.99516
$973$	5.80583	0.186126
$974$	61.1107	1.95811
$975$	0	0
$976$	−22.0123	−0.704596
$977$	30.5961	0.978855	0.489427	−	0.872044i	$-0.337206\pi$
$977$	30.5961	0.978855	0.489427	+	0.872044i	$0.337206\pi$
$978$	121.614	3.88877
$979$	−7.22885	−0.231035
$980$	0	0
$981$	−1.49342	−0.0476813
$982$	2.41940	0.0772061
$983$	−5.02215	−0.160182	−0.0800909	−	0.996788i	$-0.525521\pi$
$983$	−5.02215	−0.160182	−0.0800909	+	0.996788i	$0.525521\pi$
$984$	63.2994	2.01791
$985$	0	0
$986$	−5.35862	−0.170653
$987$	−64.2670	−2.04564
$988$	40.1116	1.27612
$989$	86.8968	2.76316
$990$	0	0
$991$	−44.6859	−1.41949	−0.709747	−	0.704457i	$-0.751191\pi$
$991$	−44.6859	−1.41949	−0.709747	+	0.704457i	$0.751191\pi$
$992$	−7.56468	−0.240179
$993$	−45.4217	−1.44141
$994$	6.91686	0.219389
$995$	0	0
$996$	−38.4451	−1.21818
$997$	1.76761	0.0559806	0.0279903	−	0.999608i	$-0.491089\pi$
$997$	1.76761	0.0559806	0.0279903	+	0.999608i	$0.491089\pi$
$998$	−26.4356	−0.836805
$999$	−3.05449	−0.0966398

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6025.2.a.q.1.6		66
5.2	odd	4		1205.2.b.d.724.6	✓	66
5.3	odd	4		1205.2.b.d.724.61	yes	66
5.4	even	2	inner	6025.2.a.q.1.61		66

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1205.2.b.d.724.6	✓	66	5.2	odd	4
1205.2.b.d.724.61	yes	66	5.3	odd	4
6025.2.a.q.1.6		66	1.1	even	1	trivial
6025.2.a.q.1.61		66	5.4	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.48606 q^{2} -2.54168 q^{3} +4.18051 q^{4} +6.31876 q^{6} -3.51286 q^{7} -5.42088 q^{8} +3.46011 q^{9} +O(q^{10})\)	\(q-2.48606 q^{2} -2.54168 q^{3} +4.18051 q^{4} +6.31876 q^{6} -3.51286 q^{7} -5.42088 q^{8} +3.46011 q^{9} -1.13852 q^{11} -10.6255 q^{12} +1.61484 q^{13} +8.73319 q^{14} +5.11563 q^{16} +0.605228 q^{17} -8.60206 q^{18} +5.94172 q^{19} +8.92855 q^{21} +2.83042 q^{22} +8.81659 q^{23} +13.7781 q^{24} -4.01459 q^{26} -1.16946 q^{27} -14.6855 q^{28} +3.56141 q^{29} +4.03229 q^{31} -1.87602 q^{32} +2.89374 q^{33} -1.50463 q^{34} +14.4650 q^{36} +2.61189 q^{37} -14.7715 q^{38} -4.10439 q^{39} +4.59420 q^{41} -22.1969 q^{42} +9.85605 q^{43} -4.75957 q^{44} -21.9186 q^{46} -7.19792 q^{47} -13.0023 q^{48} +5.34020 q^{49} -1.53829 q^{51} +6.75084 q^{52} +12.1481 q^{53} +2.90734 q^{54} +19.0428 q^{56} -15.1019 q^{57} -8.85388 q^{58} +7.01670 q^{59} -4.30294 q^{61} -10.0245 q^{62} -12.1549 q^{63} -5.56735 q^{64} -7.19401 q^{66} +6.53582 q^{67} +2.53016 q^{68} -22.4089 q^{69} +0.792019 q^{71} -18.7569 q^{72} -9.29872 q^{73} -6.49332 q^{74} +24.8394 q^{76} +3.99945 q^{77} +10.2038 q^{78} +3.53559 q^{79} -7.40796 q^{81} -11.4215 q^{82} +3.61819 q^{83} +37.3259 q^{84} -24.5028 q^{86} -9.05194 q^{87} +6.17176 q^{88} +6.34937 q^{89} -5.67270 q^{91} +36.8578 q^{92} -10.2488 q^{93} +17.8945 q^{94} +4.76825 q^{96} -0.677908 q^{97} -13.2761 q^{98} -3.93939 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9	\( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) \) 66 * q + 78 * q^4 + 16 * q^6 + 90 * q^9 + 48 * q^11 + 30 * q^14 + 98 * q^16 + 12 * q^19 + 18 * q^21 + 42 * q^24 + 48 * q^26 + 56 * q^29 + 48 * q^31 + 8 * q^34 + 158 * q^36 + 84 * q^39 + 56 * q^41 + 144 * q^44 + 36 * q^46 + 98 * q^49 + 44 * q^51 + 86 * q^54 + 104 * q^56 + 108 * q^59 + 22 * q^61 + 136 * q^64 + 74 * q^66 + 20 * q^69 + 212 * q^71 + 84 * q^74 + 6 * q^76 + 66 * q^79 + 162 * q^81 - 52 * q^84 + 100 * q^86 + 54 * q^89 + 72 * q^91 - 96 * q^94 + 122 * q^96 + 112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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