LMFDB - Embedded newform 275.2.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,2,Mod(1,275)] 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("275.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	275.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,1,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.19588605559$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	275.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.618034 q^{2} +0.381966 q^{3} -1.61803 q^{4} -0.236068 q^{6} +3.85410 q^{7} +2.23607 q^{8} -2.85410 q^{9} +1.00000 q^{11} -0.618034 q^{12} +1.76393 q^{13} -2.38197 q^{14} +1.85410 q^{16} +1.61803 q^{17} +1.76393 q^{18} +6.70820 q^{19} +1.47214 q^{21} -0.618034 q^{22} +7.09017 q^{23} +0.854102 q^{24} -1.09017 q^{26} -2.23607 q^{27} -6.23607 q^{28} -3.61803 q^{29} -3.00000 q^{31} -5.61803 q^{32} +0.381966 q^{33} -1.00000 q^{34} +4.61803 q^{36} +5.76393 q^{37} -4.14590 q^{38} +0.673762 q^{39} -3.00000 q^{41} -0.909830 q^{42} -6.00000 q^{43} -1.61803 q^{44} -4.38197 q^{46} -5.94427 q^{47} +0.708204 q^{48} +7.85410 q^{49} +0.618034 q^{51} -2.85410 q^{52} -6.32624 q^{53} +1.38197 q^{54} +8.61803 q^{56} +2.56231 q^{57} +2.23607 q^{58} +9.47214 q^{59} -11.0902 q^{61} +1.85410 q^{62} -11.0000 q^{63} -0.236068 q^{64} -0.236068 q^{66} +8.00000 q^{67} -2.61803 q^{68} +2.70820 q^{69} -14.1803 q^{71} -6.38197 q^{72} +12.6180 q^{73} -3.56231 q^{74} -10.8541 q^{76} +3.85410 q^{77} -0.416408 q^{78} -0.854102 q^{79} +7.70820 q^{81} +1.85410 q^{82} -16.8541 q^{83} -2.38197 q^{84} +3.70820 q^{86} -1.38197 q^{87} +2.23607 q^{88} -18.0902 q^{89} +6.79837 q^{91} -11.4721 q^{92} -1.14590 q^{93} +3.67376 q^{94} -2.14590 q^{96} -0.618034 q^{97} -4.85410 q^{98} -2.85410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 3 q^{3} - q^{4} + 4 q^{6} + q^{7} + q^{9} + 2 q^{11} + q^{12} + 8 q^{13} - 7 q^{14} - 3 q^{16} + q^{17} + 8 q^{18} - 6 q^{21} + q^{22} + 3 q^{23} - 5 q^{24} + 9 q^{26} - 8 q^{28} - 5 q^{29}+ \cdots + q^{99}+O(q^{100}) $ 2 * q + q^2 + 3 * q^3 - q^4 + 4 * q^6 + q^7 + q^9 + 2 * q^11 + q^12 + 8 * q^13 - 7 * q^14 - 3 * q^16 + q^17 + 8 * q^18 - 6 * q^21 + q^22 + 3 * q^23 - 5 * q^24 + 9 * q^26 - 8 * q^28 - 5 * q^29 - 6 * q^31 - 9 * q^32 + 3 * q^33 - 2 * q^34 + 7 * q^36 + 16 * q^37 - 15 * q^38 + 17 * q^39 - 6 * q^41 - 13 * q^42 - 12 * q^43 - q^44 - 11 * q^46 + 6 * q^47 - 12 * q^48 + 9 * q^49 - q^51 + q^52 + 3 * q^53 + 5 * q^54 + 15 * q^56 - 15 * q^57 + 10 * q^59 - 11 * q^61 - 3 * q^62 - 22 * q^63 + 4 * q^64 + 4 * q^66 + 16 * q^67 - 3 * q^68 - 8 * q^69 - 6 * q^71 - 15 * q^72 + 23 * q^73 + 13 * q^74 - 15 * q^76 + q^77 + 26 * q^78 + 5 * q^79 + 2 * q^81 - 3 * q^82 - 27 * q^83 - 7 * q^84 - 6 * q^86 - 5 * q^87 - 25 * q^89 - 11 * q^91 - 14 * q^92 - 9 * q^93 + 23 * q^94 - 11 * q^96 + q^97 - 3 * q^98 + q^99

Coefficient data

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

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

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

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0.381966	0.220528	0.110264	−	0.993902i	$-0.464830\pi$
$3$	0.381966	0.220528	0.110264	+	0.993902i	$0.464830\pi$
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	−0.236068	−0.0963743
$7$	3.85410	1.45671	0.728357	−	0.685198i	$-0.240284\pi$
$7$	3.85410	1.45671	0.728357	+	0.685198i	$0.240284\pi$
$8$	2.23607	0.790569
$9$	−2.85410	−0.951367
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−0.618034	−0.178411
$13$	1.76393	0.489227	0.244613	−	0.969621i	$-0.421339\pi$
$13$	1.76393	0.489227	0.244613	+	0.969621i	$0.421339\pi$
$14$	−2.38197	−0.636607
$15$	0	0
$16$	1.85410	0.463525
$17$	1.61803	0.392431	0.196215	−	0.980561i	$-0.437135\pi$
$17$	1.61803	0.392431	0.196215	+	0.980561i	$0.437135\pi$
$18$	1.76393	0.415763
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	0	0
$21$	1.47214	0.321246
$22$	−0.618034	−0.131765
$23$	7.09017	1.47840	0.739201	−	0.673485i	$-0.235203\pi$
$23$	7.09017	1.47840	0.739201	+	0.673485i	$0.235203\pi$
$24$	0.854102	0.174343
$25$	0	0
$26$	−1.09017	−0.213800
$27$	−2.23607	−0.430331
$28$	−6.23607	−1.17851
$29$	−3.61803	−0.671852	−0.335926	−	0.941888i	$-0.609049\pi$
$29$	−3.61803	−0.671852	−0.335926	+	0.941888i	$0.609049\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−5.61803	−0.993137
$33$	0.381966	0.0664917
$34$	−1.00000	−0.171499
$35$	0	0
$36$	4.61803	0.769672
$37$	5.76393	0.947585	0.473792	−	0.880637i	$-0.342885\pi$
$37$	5.76393	0.947585	0.473792	+	0.880637i	$0.342885\pi$
$38$	−4.14590	−0.672553
$39$	0.673762	0.107888
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	−0.909830	−0.140390
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	−1.61803	−0.243928
$45$	0	0
$46$	−4.38197	−0.646086
$47$	−5.94427	−0.867061	−0.433531	−	0.901139i	$-0.642732\pi$
$47$	−5.94427	−0.867061	−0.433531	+	0.901139i	$0.642732\pi$
$48$	0.708204	0.102220
$49$	7.85410	1.12201
$50$	0	0
$51$	0.618034	0.0865421
$52$	−2.85410	−0.395793
$53$	−6.32624	−0.868976	−0.434488	−	0.900678i	$-0.643071\pi$
$53$	−6.32624	−0.868976	−0.434488	+	0.900678i	$0.643071\pi$
$54$	1.38197	0.188062
$55$	0	0
$56$	8.61803	1.15163
$57$	2.56231	0.339386
$58$	2.23607	0.293610
$59$	9.47214	1.23317	0.616584	−	0.787289i	$-0.288516\pi$
$59$	9.47214	1.23317	0.616584	+	0.787289i	$0.288516\pi$
$60$	0	0
$61$	−11.0902	−1.41995	−0.709975	−	0.704226i	$-0.751294\pi$
$61$	−11.0902	−1.41995	−0.709975	+	0.704226i	$0.751294\pi$
$62$	1.85410	0.235471
$63$	−11.0000	−1.38587
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	−0.236068	−0.0290580
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−2.61803	−0.317483
$69$	2.70820	0.326029
$70$	0	0
$71$	−14.1803	−1.68290	−0.841448	−	0.540338i	$-0.818297\pi$
$71$	−14.1803	−1.68290	−0.841448	+	0.540338i	$0.818297\pi$
$72$	−6.38197	−0.752122
$73$	12.6180	1.47683	0.738415	−	0.674347i	$-0.235575\pi$
$73$	12.6180	1.47683	0.738415	+	0.674347i	$0.235575\pi$
$74$	−3.56231	−0.414110
$75$	0	0
$76$	−10.8541	−1.24505
$77$	3.85410	0.439216
$78$	−0.416408	−0.0471489
$79$	−0.854102	−0.0960940	−0.0480470	−	0.998845i	$-0.515300\pi$
$79$	−0.854102	−0.0960940	−0.0480470	+	0.998845i	$0.515300\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	1.85410	0.204751
$83$	−16.8541	−1.84998	−0.924989	−	0.379994i	$-0.875926\pi$
$83$	−16.8541	−1.84998	−0.924989	+	0.379994i	$0.875926\pi$
$84$	−2.38197	−0.259894
$85$	0	0
$86$	3.70820	0.399866
$87$	−1.38197	−0.148162
$88$	2.23607	0.238366
$89$	−18.0902	−1.91755	−0.958777	−	0.284159i	$-0.908286\pi$
$89$	−18.0902	−1.91755	−0.958777	+	0.284159i	$0.908286\pi$
$90$	0	0
$91$	6.79837	0.712663
$92$	−11.4721	−1.19605
$93$	−1.14590	−0.118824
$94$	3.67376	0.378920
$95$	0	0
$96$	−2.14590	−0.219015
$97$	−0.618034	−0.0627518	−0.0313759	−	0.999508i	$-0.509989\pi$
$97$	−0.618034	−0.0627518	−0.0313759	+	0.999508i	$0.509989\pi$
$98$	−4.85410	−0.490338
$99$	−2.85410	−0.286848
$100$	0	0
$101$	5.09017	0.506491	0.253245	−	0.967402i	$-0.418502\pi$
$101$	5.09017	0.506491	0.253245	+	0.967402i	$0.418502\pi$
$102$	−0.381966	−0.0378203
$103$	7.61803	0.750627	0.375314	−	0.926898i	$-0.377535\pi$
$103$	7.61803	0.750627	0.375314	+	0.926898i	$0.377535\pi$
$104$	3.94427	0.386768
$105$	0	0
$106$	3.90983	0.379756
$107$	0.236068	0.0228216	0.0114108	−	0.999935i	$-0.496368\pi$
$107$	0.236068	0.0228216	0.0114108	+	0.999935i	$0.496368\pi$
$108$	3.61803	0.348145
$109$	8.09017	0.774898	0.387449	−	0.921891i	$-0.373356\pi$
$109$	8.09017	0.774898	0.387449	+	0.921891i	$0.373356\pi$
$110$	0	0
$111$	2.20163	0.208969
$112$	7.14590	0.675224
$113$	19.6525	1.84875	0.924375	−	0.381486i	$-0.124587\pi$
$113$	19.6525	1.84875	0.924375	+	0.381486i	$0.124587\pi$
$114$	−1.58359	−0.148317
$115$	0	0
$116$	5.85410	0.543540
$117$	−5.03444	−0.465434
$118$	−5.85410	−0.538914
$119$	6.23607	0.571659
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.85410	0.620541
$123$	−1.14590	−0.103322
$124$	4.85410	0.435911
$125$	0	0
$126$	6.79837	0.605647
$127$	1.61803	0.143577	0.0717886	−	0.997420i	$-0.477129\pi$
$127$	1.61803	0.143577	0.0717886	+	0.997420i	$0.477129\pi$
$128$	11.3820	1.00603
$129$	−2.29180	−0.201781
$130$	0	0
$131$	−1.09017	−0.0952486	−0.0476243	−	0.998865i	$-0.515165\pi$
$131$	−1.09017	−0.0952486	−0.0476243	+	0.998865i	$0.515165\pi$
$132$	−0.618034	−0.0537930
$133$	25.8541	2.24183
$134$	−4.94427	−0.427120
$135$	0	0
$136$	3.61803	0.310244
$137$	−14.5623	−1.24414	−0.622071	−	0.782961i	$-0.713708\pi$
$137$	−14.5623	−1.24414	−0.622071	+	0.782961i	$0.713708\pi$
$138$	−1.67376	−0.142480
$139$	−16.7082	−1.41717	−0.708586	−	0.705625i	$-0.750666\pi$
$139$	−16.7082	−1.41717	−0.708586	+	0.705625i	$0.750666\pi$
$140$	0	0
$141$	−2.27051	−0.191211
$142$	8.76393	0.735453
$143$	1.76393	0.147507
$144$	−5.29180	−0.440983
$145$	0	0
$146$	−7.79837	−0.645398
$147$	3.00000	0.247436
$148$	−9.32624	−0.766612
$149$	8.94427	0.732743	0.366372	−	0.930469i	$-0.380600\pi$
$149$	8.94427	0.732743	0.366372	+	0.930469i	$0.380600\pi$
$150$	0	0
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	15.0000	1.21666
$153$	−4.61803	−0.373346
$154$	−2.38197	−0.191944
$155$	0	0
$156$	−1.09017	−0.0872835
$157$	21.4164	1.70922	0.854608	−	0.519274i	$-0.173798\pi$
$157$	21.4164	1.70922	0.854608	+	0.519274i	$0.173798\pi$
$158$	0.527864	0.0419946
$159$	−2.41641	−0.191634
$160$	0	0
$161$	27.3262	2.15361
$162$	−4.76393	−0.374290
$163$	−0.145898	−0.0114276	−0.00571381	−	0.999984i	$-0.501819\pi$
$163$	−0.145898	−0.0114276	−0.00571381	+	0.999984i	$0.501819\pi$
$164$	4.85410	0.379042
$165$	0	0
$166$	10.4164	0.808470
$167$	−18.7082	−1.44768	−0.723842	−	0.689966i	$-0.757626\pi$
$167$	−18.7082	−1.44768	−0.723842	+	0.689966i	$0.757626\pi$
$168$	3.29180	0.253968
$169$	−9.88854	−0.760657
$170$	0	0
$171$	−19.1459	−1.46412
$172$	9.70820	0.740244
$173$	3.47214	0.263982	0.131991	−	0.991251i	$-0.457863\pi$
$173$	3.47214	0.263982	0.131991	+	0.991251i	$0.457863\pi$
$174$	0.854102	0.0647493
$175$	0	0
$176$	1.85410	0.139758
$177$	3.61803	0.271948
$178$	11.1803	0.838002
$179$	−11.3820	−0.850728	−0.425364	−	0.905022i	$-0.639854\pi$
$179$	−11.3820	−0.850728	−0.425364	+	0.905022i	$0.639854\pi$
$180$	0	0
$181$	5.09017	0.378349	0.189175	−	0.981943i	$-0.439419\pi$
$181$	5.09017	0.378349	0.189175	+	0.981943i	$0.439419\pi$
$182$	−4.20163	−0.311445
$183$	−4.23607	−0.313139
$184$	15.8541	1.16878
$185$	0	0
$186$	0.708204	0.0519280
$187$	1.61803	0.118322
$188$	9.61803	0.701467
$189$	−8.61803	−0.626870
$190$	0	0
$191$	−9.90983	−0.717050	−0.358525	−	0.933520i	$-0.616720\pi$
$191$	−9.90983	−0.717050	−0.358525	+	0.933520i	$0.616720\pi$
$192$	−0.0901699	−0.00650746
$193$	−4.94427	−0.355896	−0.177948	−	0.984040i	$-0.556946\pi$
$193$	−4.94427	−0.355896	−0.177948	+	0.984040i	$0.556946\pi$
$194$	0.381966	0.0274236
$195$	0	0
$196$	−12.7082	−0.907729
$197$	−8.90983	−0.634799	−0.317400	−	0.948292i	$-0.602810\pi$
$197$	−8.90983	−0.634799	−0.317400	+	0.948292i	$0.602810\pi$
$198$	1.76393	0.125357
$199$	8.09017	0.573497	0.286748	−	0.958006i	$-0.407426\pi$
$199$	8.09017	0.573497	0.286748	+	0.958006i	$0.407426\pi$
$200$	0	0
$201$	3.05573	0.215534
$202$	−3.14590	−0.221345
$203$	−13.9443	−0.978696
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	−4.70820	−0.328036
$207$	−20.2361	−1.40650
$208$	3.27051	0.226769
$209$	6.70820	0.464016
$210$	0	0
$211$	17.0000	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$212$	10.2361	0.703016
$213$	−5.41641	−0.371126
$214$	−0.145898	−0.00997338
$215$	0	0
$216$	−5.00000	−0.340207
$217$	−11.5623	−0.784900
$218$	−5.00000	−0.338643
$219$	4.81966	0.325682
$220$	0	0
$221$	2.85410	0.191988
$222$	−1.36068	−0.0913228
$223$	−18.8885	−1.26487	−0.632435	−	0.774613i	$-0.717945\pi$
$223$	−18.8885	−1.26487	−0.632435	+	0.774613i	$0.717945\pi$
$224$	−21.6525	−1.44672
$225$	0	0
$226$	−12.1459	−0.807933
$227$	5.03444	0.334148	0.167074	−	0.985944i	$-0.446568\pi$
$227$	5.03444	0.334148	0.167074	+	0.985944i	$0.446568\pi$
$228$	−4.14590	−0.274569
$229$	−17.0344	−1.12567	−0.562834	−	0.826570i	$-0.690289\pi$
$229$	−17.0344	−1.12567	−0.562834	+	0.826570i	$0.690289\pi$
$230$	0	0
$231$	1.47214	0.0968594
$232$	−8.09017	−0.531146
$233$	−22.5066	−1.47445	−0.737227	−	0.675645i	$-0.763865\pi$
$233$	−22.5066	−1.47445	−0.737227	+	0.675645i	$0.763865\pi$
$234$	3.11146	0.203402
$235$	0	0
$236$	−15.3262	−0.997653
$237$	−0.326238	−0.0211914
$238$	−3.85410	−0.249824
$239$	8.61803	0.557454	0.278727	−	0.960370i	$-0.410087\pi$
$239$	8.61803	0.557454	0.278727	+	0.960370i	$0.410087\pi$
$240$	0	0
$241$	−12.2705	−0.790413	−0.395207	−	0.918592i	$-0.629327\pi$
$241$	−12.2705	−0.790413	−0.395207	+	0.918592i	$0.629327\pi$
$242$	−0.618034	−0.0397287
$243$	9.65248	0.619207
$244$	17.9443	1.14876
$245$	0	0
$246$	0.708204	0.0451534
$247$	11.8328	0.752904
$248$	−6.70820	−0.425971
$249$	−6.43769	−0.407972
$250$	0	0
$251$	6.27051	0.395791	0.197896	−	0.980223i	$-0.436589\pi$
$251$	6.27051	0.395791	0.197896	+	0.980223i	$0.436589\pi$
$252$	17.7984	1.12119
$253$	7.09017	0.445755
$254$	−1.00000	−0.0627456
$255$	0	0
$256$	−6.56231	−0.410144
$257$	6.94427	0.433172	0.216586	−	0.976264i	$-0.430508\pi$
$257$	6.94427	0.433172	0.216586	+	0.976264i	$0.430508\pi$
$258$	1.41641	0.0881817
$259$	22.2148	1.38036
$260$	0	0
$261$	10.3262	0.639178
$262$	0.673762	0.0416252
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	0.854102	0.0525663
$265$	0	0
$266$	−15.9787	−0.979718
$267$	−6.90983	−0.422875
$268$	−12.9443	−0.790697
$269$	14.6738	0.894675	0.447338	−	0.894365i	$-0.352372\pi$
$269$	14.6738	0.894675	0.447338	+	0.894365i	$0.352372\pi$
$270$	0	0
$271$	−9.18034	−0.557666	−0.278833	−	0.960340i	$-0.589948\pi$
$271$	−9.18034	−0.557666	−0.278833	+	0.960340i	$0.589948\pi$
$272$	3.00000	0.181902
$273$	2.59675	0.157162
$274$	9.00000	0.543710
$275$	0	0
$276$	−4.38197	−0.263763
$277$	−2.52786	−0.151885	−0.0759423	−	0.997112i	$-0.524196\pi$
$277$	−2.52786	−0.151885	−0.0759423	+	0.997112i	$0.524196\pi$
$278$	10.3262	0.619327
$279$	8.56231	0.512612
$280$	0	0
$281$	19.3607	1.15496	0.577481	−	0.816404i	$-0.304036\pi$
$281$	19.3607	1.15496	0.577481	+	0.816404i	$0.304036\pi$
$282$	1.40325	0.0835625
$283$	−9.61803	−0.571733	−0.285866	−	0.958269i	$-0.592281\pi$
$283$	−9.61803	−0.571733	−0.285866	+	0.958269i	$0.592281\pi$
$284$	22.9443	1.36149
$285$	0	0
$286$	−1.09017	−0.0644631
$287$	−11.5623	−0.682501
$288$	16.0344	0.944839
$289$	−14.3820	−0.845998
$290$	0	0
$291$	−0.236068	−0.0138385
$292$	−20.4164	−1.19478
$293$	11.8885	0.694536	0.347268	−	0.937766i	$-0.387109\pi$
$293$	11.8885	0.694536	0.347268	+	0.937766i	$0.387109\pi$
$294$	−1.85410	−0.108133
$295$	0	0
$296$	12.8885	0.749131
$297$	−2.23607	−0.129750
$298$	−5.52786	−0.320221
$299$	12.5066	0.723274
$300$	0	0
$301$	−23.1246	−1.33288
$302$	1.85410	0.106692
$303$	1.94427	0.111696
$304$	12.4377	0.713351
$305$	0	0
$306$	2.85410	0.163158
$307$	−22.4508	−1.28134	−0.640669	−	0.767817i	$-0.721343\pi$
$307$	−22.4508	−1.28134	−0.640669	+	0.767817i	$0.721343\pi$
$308$	−6.23607	−0.355333
$309$	2.90983	0.165534
$310$	0	0
$311$	3.18034	0.180341	0.0901703	−	0.995926i	$-0.471259\pi$
$311$	3.18034	0.180341	0.0901703	+	0.995926i	$0.471259\pi$
$312$	1.50658	0.0852932
$313$	1.23607	0.0698667	0.0349333	−	0.999390i	$-0.488878\pi$
$313$	1.23607	0.0698667	0.0349333	+	0.999390i	$0.488878\pi$
$314$	−13.2361	−0.746955
$315$	0	0
$316$	1.38197	0.0777417
$317$	14.3820	0.807772	0.403886	−	0.914809i	$-0.367659\pi$
$317$	14.3820	0.807772	0.403886	+	0.914809i	$0.367659\pi$
$318$	1.49342	0.0837470
$319$	−3.61803	−0.202571
$320$	0	0
$321$	0.0901699	0.00503280
$322$	−16.8885	−0.941162
$323$	10.8541	0.603938
$324$	−12.4721	−0.692896
$325$	0	0
$326$	0.0901699	0.00499405
$327$	3.09017	0.170887
$328$	−6.70820	−0.370399
$329$	−22.9098	−1.26306
$330$	0	0
$331$	3.18034	0.174807	0.0874036	−	0.996173i	$-0.472143\pi$
$331$	3.18034	0.174807	0.0874036	+	0.996173i	$0.472143\pi$
$332$	27.2705	1.49666
$333$	−16.4508	−0.901501
$334$	11.5623	0.632661
$335$	0	0
$336$	2.72949	0.148906
$337$	−25.4164	−1.38452	−0.692260	−	0.721648i	$-0.743385\pi$
$337$	−25.4164	−1.38452	−0.692260	+	0.721648i	$0.743385\pi$
$338$	6.11146	0.332419
$339$	7.50658	0.407701
$340$	0	0
$341$	−3.00000	−0.162459
$342$	11.8328	0.639845
$343$	3.29180	0.177740
$344$	−13.4164	−0.723364
$345$	0	0
$346$	−2.14590	−0.115364
$347$	0.437694	0.0234967	0.0117483	−	0.999931i	$-0.496260\pi$
$347$	0.437694	0.0234967	0.0117483	+	0.999931i	$0.496260\pi$
$348$	2.23607	0.119866
$349$	−31.8328	−1.70397	−0.851986	−	0.523565i	$-0.824602\pi$
$349$	−31.8328	−1.70397	−0.851986	+	0.523565i	$0.824602\pi$
$350$	0	0
$351$	−3.94427	−0.210530
$352$	−5.61803	−0.299442
$353$	−23.3607	−1.24336	−0.621682	−	0.783270i	$-0.713550\pi$
$353$	−23.3607	−1.24336	−0.621682	+	0.783270i	$0.713550\pi$
$354$	−2.23607	−0.118846
$355$	0	0
$356$	29.2705	1.55133
$357$	2.38197	0.126067
$358$	7.03444	0.371782
$359$	4.47214	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$359$	4.47214	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	−3.14590	−0.165345
$363$	0.381966	0.0200480
$364$	−11.0000	−0.576557
$365$	0	0
$366$	2.61803	0.136847
$367$	13.8541	0.723178	0.361589	−	0.932338i	$-0.382234\pi$
$367$	13.8541	0.723178	0.361589	+	0.932338i	$0.382234\pi$
$368$	13.1459	0.685277
$369$	8.56231	0.445736
$370$	0	0
$371$	−24.3820	−1.26585
$372$	1.85410	0.0961307
$373$	25.1803	1.30379	0.651894	−	0.758310i	$-0.273975\pi$
$373$	25.1803	1.30379	0.651894	+	0.758310i	$0.273975\pi$
$374$	−1.00000	−0.0517088
$375$	0	0
$376$	−13.2918	−0.685472
$377$	−6.38197	−0.328688
$378$	5.32624	0.273952
$379$	−22.2361	−1.14219	−0.571095	−	0.820884i	$-0.693481\pi$
$379$	−22.2361	−1.14219	−0.571095	+	0.820884i	$0.693481\pi$
$380$	0	0
$381$	0.618034	0.0316628
$382$	6.12461	0.313362
$383$	22.9443	1.17240	0.586199	−	0.810167i	$-0.300624\pi$
$383$	22.9443	1.17240	0.586199	+	0.810167i	$0.300624\pi$
$384$	4.34752	0.221859
$385$	0	0
$386$	3.05573	0.155532
$387$	17.1246	0.870493
$388$	1.00000	0.0507673
$389$	14.4721	0.733766	0.366883	−	0.930267i	$-0.380425\pi$
$389$	14.4721	0.733766	0.366883	+	0.930267i	$0.380425\pi$
$390$	0	0
$391$	11.4721	0.580171
$392$	17.5623	0.887030
$393$	−0.416408	−0.0210050
$394$	5.50658	0.277417
$395$	0	0
$396$	4.61803	0.232065
$397$	26.0902	1.30943	0.654714	−	0.755877i	$-0.272789\pi$
$397$	26.0902	1.30943	0.654714	+	0.755877i	$0.272789\pi$
$398$	−5.00000	−0.250627
$399$	9.87539	0.494388
$400$	0	0
$401$	−10.3607	−0.517388	−0.258694	−	0.965959i	$-0.583292\pi$
$401$	−10.3607	−0.517388	−0.258694	+	0.965959i	$0.583292\pi$
$402$	−1.88854	−0.0941920
$403$	−5.29180	−0.263603
$404$	−8.23607	−0.409760
$405$	0	0
$406$	8.61803	0.427706
$407$	5.76393	0.285708
$408$	1.38197	0.0684175
$409$	−20.1246	−0.995098	−0.497549	−	0.867436i	$-0.665767\pi$
$409$	−20.1246	−0.995098	−0.497549	+	0.867436i	$0.665767\pi$
$410$	0	0
$411$	−5.56231	−0.274368
$412$	−12.3262	−0.607270
$413$	36.5066	1.79637
$414$	12.5066	0.614665
$415$	0	0
$416$	−9.90983	−0.485869
$417$	−6.38197	−0.312526
$418$	−4.14590	−0.202783
$419$	−1.18034	−0.0576634	−0.0288317	−	0.999584i	$-0.509179\pi$
$419$	−1.18034	−0.0576634	−0.0288317	+	0.999584i	$0.509179\pi$
$420$	0	0
$421$	21.2705	1.03666	0.518331	−	0.855180i	$-0.326554\pi$
$421$	21.2705	1.03666	0.518331	+	0.855180i	$0.326554\pi$
$422$	−10.5066	−0.511452
$423$	16.9656	0.824894
$424$	−14.1459	−0.686986
$425$	0	0
$426$	3.34752	0.162188
$427$	−42.7426	−2.06846
$428$	−0.381966	−0.0184630
$429$	0.673762	0.0325295
$430$	0	0
$431$	23.1803	1.11656	0.558279	−	0.829653i	$-0.311462\pi$
$431$	23.1803	1.11656	0.558279	+	0.829653i	$0.311462\pi$
$432$	−4.14590	−0.199470
$433$	16.8885	0.811612	0.405806	−	0.913959i	$-0.366991\pi$
$433$	16.8885	0.811612	0.405806	+	0.913959i	$0.366991\pi$
$434$	7.14590	0.343014
$435$	0	0
$436$	−13.0902	−0.626905
$437$	47.5623	2.27521
$438$	−2.97871	−0.142328
$439$	−34.2705	−1.63564	−0.817821	−	0.575473i	$-0.804818\pi$
$439$	−34.2705	−1.63564	−0.817821	+	0.575473i	$0.804818\pi$
$440$	0	0
$441$	−22.4164	−1.06745
$442$	−1.76393	−0.0839017
$443$	−5.34752	−0.254069	−0.127034	−	0.991898i	$-0.540546\pi$
$443$	−5.34752	−0.254069	−0.127034	+	0.991898i	$0.540546\pi$
$444$	−3.56231	−0.169060
$445$	0	0
$446$	11.6738	0.552769
$447$	3.41641	0.161591
$448$	−0.909830	−0.0429854
$449$	0.326238	0.0153961	0.00769806	−	0.999970i	$-0.497550\pi$
$449$	0.326238	0.0153961	0.00769806	+	0.999970i	$0.497550\pi$
$450$	0	0
$451$	−3.00000	−0.141264
$452$	−31.7984	−1.49567
$453$	−1.14590	−0.0538390
$454$	−3.11146	−0.146028
$455$	0	0
$456$	5.72949	0.268308
$457$	38.9787	1.82335	0.911674	−	0.410915i	$-0.134791\pi$
$457$	38.9787	1.82335	0.911674	+	0.410915i	$0.134791\pi$
$458$	10.5279	0.491935
$459$	−3.61803	−0.168875
$460$	0	0
$461$	13.1803	0.613870	0.306935	−	0.951731i	$-0.400697\pi$
$461$	13.1803	0.613870	0.306935	+	0.951731i	$0.400697\pi$
$462$	−0.909830	−0.0423291
$463$	−33.3607	−1.55040	−0.775201	−	0.631714i	$-0.782352\pi$
$463$	−33.3607	−1.55040	−0.775201	+	0.631714i	$0.782352\pi$
$464$	−6.70820	−0.311421
$465$	0	0
$466$	13.9098	0.644360
$467$	28.5279	1.32011	0.660056	−	0.751216i	$-0.270533\pi$
$467$	28.5279	1.32011	0.660056	+	0.751216i	$0.270533\pi$
$468$	8.14590	0.376544
$469$	30.8328	1.42373
$470$	0	0
$471$	8.18034	0.376930
$472$	21.1803	0.974904
$473$	−6.00000	−0.275880
$474$	0.201626	0.00926099
$475$	0	0
$476$	−10.0902	−0.462482
$477$	18.0557	0.826715
$478$	−5.32624	−0.243616
$479$	1.58359	0.0723562	0.0361781	−	0.999345i	$-0.488482\pi$
$479$	1.58359	0.0723562	0.0361781	+	0.999345i	$0.488482\pi$
$480$	0	0
$481$	10.1672	0.463584
$482$	7.58359	0.345423
$483$	10.4377	0.474932
$484$	−1.61803	−0.0735470
$485$	0	0
$486$	−5.96556	−0.270603
$487$	−3.58359	−0.162388	−0.0811940	−	0.996698i	$-0.525873\pi$
$487$	−3.58359	−0.162388	−0.0811940	+	0.996698i	$0.525873\pi$
$488$	−24.7984	−1.12257
$489$	−0.0557281	−0.00252011
$490$	0	0
$491$	−29.1803	−1.31689	−0.658445	−	0.752629i	$-0.728786\pi$
$491$	−29.1803	−1.31689	−0.658445	+	0.752629i	$0.728786\pi$
$492$	1.85410	0.0835894
$493$	−5.85410	−0.263655
$494$	−7.31308	−0.329031
$495$	0	0
$496$	−5.56231	−0.249755
$497$	−54.6525	−2.45150
$498$	3.97871	0.178290
$499$	5.20163	0.232857	0.116428	−	0.993199i	$-0.462855\pi$
$499$	5.20163	0.232857	0.116428	+	0.993199i	$0.462855\pi$
$500$	0	0
$501$	−7.14590	−0.319255
$502$	−3.87539	−0.172967
$503$	24.6525	1.09920	0.549600	−	0.835428i	$-0.314780\pi$
$503$	24.6525	1.09920	0.549600	+	0.835428i	$0.314780\pi$
$504$	−24.5967	−1.09563
$505$	0	0
$506$	−4.38197	−0.194802
$507$	−3.77709	−0.167746
$508$	−2.61803	−0.116156
$509$	18.6180	0.825230	0.412615	−	0.910906i	$-0.364616\pi$
$509$	18.6180	0.825230	0.412615	+	0.910906i	$0.364616\pi$
$510$	0	0
$511$	48.6312	2.15132
$512$	−18.7082	−0.826794
$513$	−15.0000	−0.662266
$514$	−4.29180	−0.189303
$515$	0	0
$516$	3.70820	0.163245
$517$	−5.94427	−0.261429
$518$	−13.7295	−0.603239
$519$	1.32624	0.0582154
$520$	0	0
$521$	−24.1803	−1.05936	−0.529680	−	0.848198i	$-0.677688\pi$
$521$	−24.1803	−1.05936	−0.529680	+	0.848198i	$0.677688\pi$
$522$	−6.38197	−0.279331
$523$	5.05573	0.221072	0.110536	−	0.993872i	$-0.464743\pi$
$523$	5.05573	0.221072	0.110536	+	0.993872i	$0.464743\pi$
$524$	1.76393	0.0770577
$525$	0	0
$526$	12.9787	0.565899
$527$	−4.85410	−0.211448
$528$	0.708204	0.0308206
$529$	27.2705	1.18567
$530$	0	0
$531$	−27.0344	−1.17319
$532$	−41.8328	−1.81368
$533$	−5.29180	−0.229213
$534$	4.27051	0.184803
$535$	0	0
$536$	17.8885	0.772667
$537$	−4.34752	−0.187610
$538$	−9.06888	−0.390987
$539$	7.85410	0.338300
$540$	0	0
$541$	−8.72949	−0.375310	−0.187655	−	0.982235i	$-0.560089\pi$
$541$	−8.72949	−0.375310	−0.187655	+	0.982235i	$0.560089\pi$
$542$	5.67376	0.243709
$543$	1.94427	0.0834367
$544$	−9.09017	−0.389738
$545$	0	0
$546$	−1.60488	−0.0686825
$547$	−27.8541	−1.19096	−0.595478	−	0.803372i	$-0.703037\pi$
$547$	−27.8541	−1.19096	−0.595478	+	0.803372i	$0.703037\pi$
$548$	23.5623	1.00653
$549$	31.6525	1.35089
$550$	0	0
$551$	−24.2705	−1.03396
$552$	6.05573	0.257749
$553$	−3.29180	−0.139981
$554$	1.56231	0.0663760
$555$	0	0
$556$	27.0344	1.14652
$557$	−14.2361	−0.603202	−0.301601	−	0.953434i	$-0.597521\pi$
$557$	−14.2361	−0.603202	−0.301601	+	0.953434i	$0.597521\pi$
$558$	−5.29180	−0.224020
$559$	−10.5836	−0.447638
$560$	0	0
$561$	0.618034	0.0260934
$562$	−11.9656	−0.504737
$563$	16.0344	0.675771	0.337886	−	0.941187i	$-0.390288\pi$
$563$	16.0344	0.675771	0.337886	+	0.941187i	$0.390288\pi$
$564$	3.67376	0.154693
$565$	0	0
$566$	5.94427	0.249856
$567$	29.7082	1.24763
$568$	−31.7082	−1.33045
$569$	28.6180	1.19973	0.599865	−	0.800101i	$-0.295221\pi$
$569$	28.6180	1.19973	0.599865	+	0.800101i	$0.295221\pi$
$570$	0	0
$571$	2.72949	0.114226	0.0571128	−	0.998368i	$-0.481811\pi$
$571$	2.72949	0.114226	0.0571128	+	0.998368i	$0.481811\pi$
$572$	−2.85410	−0.119336
$573$	−3.78522	−0.158130
$574$	7.14590	0.298264
$575$	0	0
$576$	0.673762	0.0280734
$577$	−4.56231	−0.189931	−0.0949656	−	0.995481i	$-0.530274\pi$
$577$	−4.56231	−0.189931	−0.0949656	+	0.995481i	$0.530274\pi$
$578$	8.88854	0.369715
$579$	−1.88854	−0.0784852
$580$	0	0
$581$	−64.9574	−2.69489
$582$	0.145898	0.00604767
$583$	−6.32624	−0.262006
$584$	28.2148	1.16754
$585$	0	0
$586$	−7.34752	−0.303523
$587$	25.0344	1.03328	0.516641	−	0.856202i	$-0.327182\pi$
$587$	25.0344	1.03328	0.516641	+	0.856202i	$0.327182\pi$
$588$	−4.85410	−0.200180
$589$	−20.1246	−0.829220
$590$	0	0
$591$	−3.40325	−0.139991
$592$	10.6869	0.439230
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	1.38197	0.0567028
$595$	0	0
$596$	−14.4721	−0.592802
$597$	3.09017	0.126472
$598$	−7.72949	−0.316082
$599$	−15.3262	−0.626213	−0.313107	−	0.949718i	$-0.601370\pi$
$599$	−15.3262	−0.626213	−0.313107	+	0.949718i	$0.601370\pi$
$600$	0	0
$601$	11.2705	0.459734	0.229867	−	0.973222i	$-0.426171\pi$
$601$	11.2705	0.459734	0.229867	+	0.973222i	$0.426171\pi$
$602$	14.2918	0.582490
$603$	−22.8328	−0.929824
$604$	4.85410	0.197511
$605$	0	0
$606$	−1.20163	−0.0488127
$607$	12.4721	0.506228	0.253114	−	0.967436i	$-0.418545\pi$
$607$	12.4721	0.506228	0.253114	+	0.967436i	$0.418545\pi$
$608$	−37.6869	−1.52841
$609$	−5.32624	−0.215830
$610$	0	0
$611$	−10.4853	−0.424189
$612$	7.47214	0.302043
$613$	26.5623	1.07284	0.536421	−	0.843951i	$-0.319776\pi$
$613$	26.5623	1.07284	0.536421	+	0.843951i	$0.319776\pi$
$614$	13.8754	0.559965
$615$	0	0
$616$	8.61803	0.347230
$617$	−15.8197	−0.636876	−0.318438	−	0.947944i	$-0.603158\pi$
$617$	−15.8197	−0.636876	−0.318438	+	0.947944i	$0.603158\pi$
$618$	−1.79837	−0.0723412
$619$	12.8885	0.518034	0.259017	−	0.965873i	$-0.416601\pi$
$619$	12.8885	0.518034	0.259017	+	0.965873i	$0.416601\pi$
$620$	0	0
$621$	−15.8541	−0.636203
$622$	−1.96556	−0.0788117
$623$	−69.7214	−2.79333
$624$	1.24922	0.0500090
$625$	0	0
$626$	−0.763932	−0.0305329
$627$	2.56231	0.102329
$628$	−34.6525	−1.38278
$629$	9.32624	0.371861
$630$	0	0
$631$	2.72949	0.108659	0.0543296	−	0.998523i	$-0.482698\pi$
$631$	2.72949	0.108659	0.0543296	+	0.998523i	$0.482698\pi$
$632$	−1.90983	−0.0759690
$633$	6.49342	0.258090
$634$	−8.88854	−0.353009
$635$	0	0
$636$	3.90983	0.155035
$637$	13.8541	0.548920
$638$	2.23607	0.0885268
$639$	40.4721	1.60105
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	−0.0557281	−0.00219941
$643$	37.4164	1.47556	0.737780	−	0.675042i	$-0.235874\pi$
$643$	37.4164	1.47556	0.737780	+	0.675042i	$0.235874\pi$
$644$	−44.2148	−1.74231
$645$	0	0
$646$	−6.70820	−0.263931
$647$	43.6525	1.71616	0.858078	−	0.513519i	$-0.171659\pi$
$647$	43.6525	1.71616	0.858078	+	0.513519i	$0.171659\pi$
$648$	17.2361	0.677097
$649$	9.47214	0.371814
$650$	0	0
$651$	−4.41641	−0.173093
$652$	0.236068	0.00924514
$653$	−34.7426	−1.35958	−0.679792	−	0.733405i	$-0.737930\pi$
$653$	−34.7426	−1.35958	−0.679792	+	0.733405i	$0.737930\pi$
$654$	−1.90983	−0.0746803
$655$	0	0
$656$	−5.56231	−0.217172
$657$	−36.0132	−1.40501
$658$	14.1591	0.551977
$659$	−24.2705	−0.945445	−0.472722	−	0.881211i	$-0.656729\pi$
$659$	−24.2705	−0.945445	−0.472722	+	0.881211i	$0.656729\pi$
$660$	0	0
$661$	−26.8197	−1.04316	−0.521582	−	0.853201i	$-0.674658\pi$
$661$	−26.8197	−1.04316	−0.521582	+	0.853201i	$0.674658\pi$
$662$	−1.96556	−0.0763936
$663$	1.09017	0.0423387
$664$	−37.6869	−1.46254
$665$	0	0
$666$	10.1672	0.393970
$667$	−25.6525	−0.993268
$668$	30.2705	1.17120
$669$	−7.21478	−0.278940
$670$	0	0
$671$	−11.0902	−0.428131
$672$	−8.27051	−0.319042
$673$	−24.4164	−0.941183	−0.470592	−	0.882351i	$-0.655960\pi$
$673$	−24.4164	−0.941183	−0.470592	+	0.882351i	$0.655960\pi$
$674$	15.7082	0.605057
$675$	0	0
$676$	16.0000	0.615385
$677$	−2.65248	−0.101943	−0.0509715	−	0.998700i	$-0.516232\pi$
$677$	−2.65248	−0.101943	−0.0509715	+	0.998700i	$0.516232\pi$
$678$	−4.63932	−0.178172
$679$	−2.38197	−0.0914115
$680$	0	0
$681$	1.92299	0.0736890
$682$	1.85410	0.0709972
$683$	1.36068	0.0520650	0.0260325	−	0.999661i	$-0.491713\pi$
$683$	1.36068	0.0520650	0.0260325	+	0.999661i	$0.491713\pi$
$684$	30.9787	1.18450
$685$	0	0
$686$	−2.03444	−0.0776754
$687$	−6.50658	−0.248241
$688$	−11.1246	−0.424122
$689$	−11.1591	−0.425126
$690$	0	0
$691$	18.9098	0.719364	0.359682	−	0.933075i	$-0.382885\pi$
$691$	18.9098	0.719364	0.359682	+	0.933075i	$0.382885\pi$
$692$	−5.61803	−0.213566
$693$	−11.0000	−0.417855
$694$	−0.270510	−0.0102684
$695$	0	0
$696$	−3.09017	−0.117133
$697$	−4.85410	−0.183862
$698$	19.6738	0.744663
$699$	−8.59675	−0.325159
$700$	0	0
$701$	44.3607	1.67548	0.837740	−	0.546070i	$-0.183877\pi$
$701$	44.3607	1.67548	0.837740	+	0.546070i	$0.183877\pi$
$702$	2.43769	0.0920048
$703$	38.6656	1.45830
$704$	−0.236068	−0.00889715
$705$	0	0
$706$	14.4377	0.543370
$707$	19.6180	0.737812
$708$	−5.85410	−0.220011
$709$	21.3050	0.800124	0.400062	−	0.916488i	$-0.368989\pi$
$709$	21.3050	0.800124	0.400062	+	0.916488i	$0.368989\pi$
$710$	0	0
$711$	2.43769	0.0914207
$712$	−40.4508	−1.51596
$713$	−21.2705	−0.796587
$714$	−1.47214	−0.0550933
$715$	0	0
$716$	18.4164	0.688253
$717$	3.29180	0.122934
$718$	−2.76393	−0.103149
$719$	15.6525	0.583739	0.291869	−	0.956458i	$-0.405723\pi$
$719$	15.6525	0.583739	0.291869	+	0.956458i	$0.405723\pi$
$720$	0	0
$721$	29.3607	1.09345
$722$	−16.0689	−0.598022
$723$	−4.68692	−0.174308
$724$	−8.23607	−0.306091
$725$	0	0
$726$	−0.236068	−0.00876130
$727$	−7.72949	−0.286671	−0.143335	−	0.989674i	$-0.545783\pi$
$727$	−7.72949	−0.286671	−0.143335	+	0.989674i	$0.545783\pi$
$728$	15.2016	0.563410
$729$	−19.4377	−0.719915
$730$	0	0
$731$	−9.70820	−0.359071
$732$	6.85410	0.253335
$733$	−7.83282	−0.289312	−0.144656	−	0.989482i	$-0.546207\pi$
$733$	−7.83282	−0.289312	−0.144656	+	0.989482i	$0.546207\pi$
$734$	−8.56231	−0.316040
$735$	0	0
$736$	−39.8328	−1.46826
$737$	8.00000	0.294684
$738$	−5.29180	−0.194794
$739$	−35.8541	−1.31891	−0.659457	−	0.751742i	$-0.729214\pi$
$739$	−35.8541	−1.31891	−0.659457	+	0.751742i	$0.729214\pi$
$740$	0	0
$741$	4.51973	0.166037
$742$	15.0689	0.553196
$743$	34.3262	1.25931	0.629654	−	0.776876i	$-0.283197\pi$
$743$	34.3262	1.25931	0.629654	+	0.776876i	$0.283197\pi$
$744$	−2.56231	−0.0939387
$745$	0	0
$746$	−15.5623	−0.569777
$747$	48.1033	1.76001
$748$	−2.61803	−0.0957248
$749$	0.909830	0.0332445
$750$	0	0
$751$	−22.2705	−0.812662	−0.406331	−	0.913726i	$-0.633192\pi$
$751$	−22.2705	−0.812662	−0.406331	+	0.913726i	$0.633192\pi$
$752$	−11.0213	−0.401905
$753$	2.39512	0.0872831
$754$	3.94427	0.143642
$755$	0	0
$756$	13.9443	0.507148
$757$	−12.5279	−0.455333	−0.227666	−	0.973739i	$-0.573110\pi$
$757$	−12.5279	−0.455333	−0.227666	+	0.973739i	$0.573110\pi$
$758$	13.7426	0.499155
$759$	2.70820	0.0983016
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−0.381966	−0.0138372
$763$	31.1803	1.12880
$764$	16.0344	0.580106
$765$	0	0
$766$	−14.1803	−0.512357
$767$	16.7082	0.603298
$768$	−2.50658	−0.0904483
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	2.65248	0.0955266
$772$	8.00000	0.287926
$773$	9.20163	0.330959	0.165480	−	0.986213i	$-0.447083\pi$
$773$	9.20163	0.330959	0.165480	+	0.986213i	$0.447083\pi$
$774$	−10.5836	−0.380419
$775$	0	0
$776$	−1.38197	−0.0496097
$777$	8.48529	0.304408
$778$	−8.94427	−0.320668
$779$	−20.1246	−0.721039
$780$	0	0
$781$	−14.1803	−0.507412
$782$	−7.09017	−0.253544
$783$	8.09017	0.289119
$784$	14.5623	0.520082
$785$	0	0
$786$	0.257354	0.00917952
$787$	34.7082	1.23721	0.618607	−	0.785701i	$-0.287697\pi$
$787$	34.7082	1.23721	0.618607	+	0.785701i	$0.287697\pi$
$788$	14.4164	0.513563
$789$	−8.02129	−0.285565
$790$	0	0
$791$	75.7426	2.69310
$792$	−6.38197	−0.226773
$793$	−19.5623	−0.694678
$794$	−16.1246	−0.572241
$795$	0	0
$796$	−13.0902	−0.463969
$797$	27.7984	0.984669	0.492334	−	0.870406i	$-0.336144\pi$
$797$	27.7984	0.984669	0.492334	+	0.870406i	$0.336144\pi$
$798$	−6.10333	−0.216055
$799$	−9.61803	−0.340262
$800$	0	0
$801$	51.6312	1.82430
$802$	6.40325	0.226107
$803$	12.6180	0.445281
$804$	−4.94427	−0.174371
$805$	0	0
$806$	3.27051	0.115199
$807$	5.60488	0.197301
$808$	11.3820	0.400416
$809$	−43.4164	−1.52644	−0.763220	−	0.646139i	$-0.776383\pi$
$809$	−43.4164	−1.52644	−0.763220	+	0.646139i	$0.776383\pi$
$810$	0	0
$811$	17.0000	0.596951	0.298475	−	0.954417i	$-0.403522\pi$
$811$	17.0000	0.596951	0.298475	+	0.954417i	$0.403522\pi$
$812$	22.5623	0.791782
$813$	−3.50658	−0.122981
$814$	−3.56231	−0.124859
$815$	0	0
$816$	1.14590	0.0401145
$817$	−40.2492	−1.40814
$818$	12.4377	0.434874
$819$	−19.4033	−0.678005
$820$	0	0
$821$	−40.3607	−1.40860	−0.704299	−	0.709904i	$-0.748738\pi$
$821$	−40.3607	−1.40860	−0.704299	+	0.709904i	$0.748738\pi$
$822$	3.43769	0.119903
$823$	0.583592	0.0203427	0.0101714	−	0.999948i	$-0.496762\pi$
$823$	0.583592	0.0203427	0.0101714	+	0.999948i	$0.496762\pi$
$824$	17.0344	0.593423
$825$	0	0
$826$	−22.5623	−0.785043
$827$	−26.0689	−0.906504	−0.453252	−	0.891382i	$-0.649736\pi$
$827$	−26.0689	−0.906504	−0.453252	+	0.891382i	$0.649736\pi$
$828$	32.7426	1.13789
$829$	51.1033	1.77489	0.887446	−	0.460912i	$-0.152478\pi$
$829$	51.1033	1.77489	0.887446	+	0.460912i	$0.152478\pi$
$830$	0	0
$831$	−0.965558	−0.0334948
$832$	−0.416408	−0.0144363
$833$	12.7082	0.440313
$834$	3.94427	0.136579
$835$	0	0
$836$	−10.8541	−0.375397
$837$	6.70820	0.231869
$838$	0.729490	0.0251998
$839$	43.3394	1.49624	0.748121	−	0.663562i	$-0.230956\pi$
$839$	43.3394	1.49624	0.748121	+	0.663562i	$0.230956\pi$
$840$	0	0
$841$	−15.9098	−0.548615
$842$	−13.1459	−0.453038
$843$	7.39512	0.254702
$844$	−27.5066	−0.946815
$845$	0	0
$846$	−10.4853	−0.360492
$847$	3.85410	0.132429
$848$	−11.7295	−0.402792
$849$	−3.67376	−0.126083
$850$	0	0
$851$	40.8673	1.40091
$852$	8.76393	0.300247
$853$	43.1459	1.47729	0.738644	−	0.674096i	$-0.235467\pi$
$853$	43.1459	1.47729	0.738644	+	0.674096i	$0.235467\pi$
$854$	26.4164	0.903951
$855$	0	0
$856$	0.527864	0.0180420
$857$	−30.8197	−1.05278	−0.526390	−	0.850243i	$-0.676455\pi$
$857$	−30.8197	−1.05278	−0.526390	+	0.850243i	$0.676455\pi$
$858$	−0.416408	−0.0142159
$859$	−6.18034	−0.210870	−0.105435	−	0.994426i	$-0.533624\pi$
$859$	−6.18034	−0.210870	−0.105435	+	0.994426i	$0.533624\pi$
$860$	0	0
$861$	−4.41641	−0.150511
$862$	−14.3262	−0.487954
$863$	48.5967	1.65425	0.827126	−	0.562016i	$-0.189974\pi$
$863$	48.5967	1.65425	0.827126	+	0.562016i	$0.189974\pi$
$864$	12.5623	0.427378
$865$	0	0
$866$	−10.4377	−0.354687
$867$	−5.49342	−0.186566
$868$	18.7082	0.634998
$869$	−0.854102	−0.0289734
$870$	0	0
$871$	14.1115	0.478148
$872$	18.0902	0.612610
$873$	1.76393	0.0597001
$874$	−29.3951	−0.994305
$875$	0	0
$876$	−7.79837	−0.263483
$877$	31.4164	1.06086	0.530428	−	0.847730i	$-0.322031\pi$
$877$	31.4164	1.06086	0.530428	+	0.847730i	$0.322031\pi$
$878$	21.1803	0.714802
$879$	4.54102	0.153165
$880$	0	0
$881$	−1.09017	−0.0367288	−0.0183644	−	0.999831i	$-0.505846\pi$
$881$	−1.09017	−0.0367288	−0.0183644	+	0.999831i	$0.505846\pi$
$882$	13.8541	0.466492
$883$	−0.347524	−0.0116951	−0.00584756	−	0.999983i	$-0.501861\pi$
$883$	−0.347524	−0.0116951	−0.00584756	+	0.999983i	$0.501861\pi$
$884$	−4.61803	−0.155321
$885$	0	0
$886$	3.30495	0.111032
$887$	−42.7771	−1.43631	−0.718157	−	0.695881i	$-0.755014\pi$
$887$	−42.7771	−1.43631	−0.718157	+	0.695881i	$0.755014\pi$
$888$	4.92299	0.165205
$889$	6.23607	0.209151
$890$	0	0
$891$	7.70820	0.258235
$892$	30.5623	1.02330
$893$	−39.8754	−1.33438
$894$	−2.11146	−0.0706177
$895$	0	0
$896$	43.8673	1.46550
$897$	4.77709	0.159502
$898$	−0.201626	−0.00672835
$899$	10.8541	0.362005
$900$	0	0
$901$	−10.2361	−0.341013
$902$	1.85410	0.0617348
$903$	−8.83282	−0.293938
$904$	43.9443	1.46156
$905$	0	0
$906$	0.708204	0.0235285
$907$	−38.8328	−1.28942	−0.644711	−	0.764426i	$-0.723022\pi$
$907$	−38.8328	−1.28942	−0.644711	+	0.764426i	$0.723022\pi$
$908$	−8.14590	−0.270331
$909$	−14.5279	−0.481859
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	4.75078	0.157314
$913$	−16.8541	−0.557789
$914$	−24.0902	−0.796832
$915$	0	0
$916$	27.5623	0.910684
$917$	−4.20163	−0.138750
$918$	2.23607	0.0738012
$919$	−3.41641	−0.112697	−0.0563484	−	0.998411i	$-0.517946\pi$
$919$	−3.41641	−0.112697	−0.0563484	+	0.998411i	$0.517946\pi$
$920$	0	0
$921$	−8.57546	−0.282571
$922$	−8.14590	−0.268271
$923$	−25.0132	−0.823318
$924$	−2.38197	−0.0783609
$925$	0	0
$926$	20.6180	0.677551
$927$	−21.7426	−0.714122
$928$	20.3262	0.667241
$929$	5.40325	0.177275	0.0886375	−	0.996064i	$-0.471749\pi$
$929$	5.40325	0.177275	0.0886375	+	0.996064i	$0.471749\pi$
$930$	0	0
$931$	52.6869	1.72674
$932$	36.4164	1.19286
$933$	1.21478	0.0397702
$934$	−17.6312	−0.576910
$935$	0	0
$936$	−11.2574	−0.367958
$937$	14.8328	0.484567	0.242283	−	0.970206i	$-0.422104\pi$
$937$	14.8328	0.484567	0.242283	+	0.970206i	$0.422104\pi$
$938$	−19.0557	−0.622192
$939$	0.472136	0.0154076
$940$	0	0
$941$	−47.7214	−1.55567	−0.777836	−	0.628467i	$-0.783683\pi$
$941$	−47.7214	−1.55567	−0.777836	+	0.628467i	$0.783683\pi$
$942$	−5.05573	−0.164725
$943$	−21.2705	−0.692663
$944$	17.5623	0.571604
$945$	0	0
$946$	3.70820	0.120564
$947$	5.36068	0.174199	0.0870993	−	0.996200i	$-0.472240\pi$
$947$	5.36068	0.174199	0.0870993	+	0.996200i	$0.472240\pi$
$948$	0.527864	0.0171442
$949$	22.2574	0.722504
$950$	0	0
$951$	5.49342	0.178136
$952$	13.9443	0.451936
$953$	−0.472136	−0.0152940	−0.00764699	−	0.999971i	$-0.502434\pi$
$953$	−0.472136	−0.0152940	−0.00764699	+	0.999971i	$0.502434\pi$
$954$	−11.1591	−0.361288
$955$	0	0
$956$	−13.9443	−0.450990
$957$	−1.38197	−0.0446726
$958$	−0.978714	−0.0316208
$959$	−56.1246	−1.81236
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−6.28367	−0.202594
$963$	−0.673762	−0.0217117
$964$	19.8541	0.639458
$965$	0	0
$966$	−6.45085	−0.207553
$967$	3.72949	0.119932	0.0599662	−	0.998200i	$-0.480901\pi$
$967$	3.72949	0.119932	0.0599662	+	0.998200i	$0.480901\pi$
$968$	2.23607	0.0718699
$969$	4.14590	0.133185
$970$	0	0
$971$	35.0902	1.12610	0.563049	−	0.826424i	$-0.309628\pi$
$971$	35.0902	1.12610	0.563049	+	0.826424i	$0.309628\pi$
$972$	−15.6180	−0.500949
$973$	−64.3951	−2.06441
$974$	2.21478	0.0709662
$975$	0	0
$976$	−20.5623	−0.658183
$977$	−1.06888	−0.0341966	−0.0170983	−	0.999854i	$-0.505443\pi$
$977$	−1.06888	−0.0341966	−0.0170983	+	0.999854i	$0.505443\pi$
$978$	0.0344419	0.00110133
$979$	−18.0902	−0.578164
$980$	0	0
$981$	−23.0902	−0.737212
$982$	18.0344	0.575502
$983$	−10.4721	−0.334009	−0.167005	−	0.985956i	$-0.553409\pi$
$983$	−10.4721	−0.334009	−0.167005	+	0.985956i	$0.553409\pi$
$984$	−2.56231	−0.0816833
$985$	0	0
$986$	3.61803	0.115222
$987$	−8.75078	−0.278540
$988$	−19.1459	−0.609112
$989$	−42.5410	−1.35273
$990$	0	0
$991$	−12.2705	−0.389786	−0.194893	−	0.980825i	$-0.562436\pi$
$991$	−12.2705	−0.389786	−0.194893	+	0.980825i	$0.562436\pi$
$992$	16.8541	0.535118
$993$	1.21478	0.0385499
$994$	33.7771	1.07134
$995$	0	0
$996$	10.4164	0.330057
$997$	38.8541	1.23052	0.615261	−	0.788324i	$-0.289051\pi$
$997$	38.8541	1.23052	0.615261	+	0.788324i	$0.289051\pi$
$998$	−3.21478	−0.101762
$999$	−12.8885	−0.407775

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	275.2.a.g.1.1	yes	2
3.2	odd	2		2475.2.a.n.1.2		2
4.3	odd	2		4400.2.a.bg.1.2		2
5.2	odd	4		275.2.b.e.199.2		4
5.3	odd	4		275.2.b.e.199.3		4
5.4	even	2		275.2.a.d.1.2	✓	2
11.10	odd	2		3025.2.a.i.1.2		2
15.2	even	4		2475.2.c.p.199.3		4
15.8	even	4		2475.2.c.p.199.2		4
15.14	odd	2		2475.2.a.s.1.1		2
20.3	even	4		4400.2.b.x.4049.2		4
20.7	even	4		4400.2.b.x.4049.3		4
20.19	odd	2		4400.2.a.bv.1.1		2
55.54	odd	2		3025.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.d.1.2	✓	2	5.4	even	2
275.2.a.g.1.1	yes	2	1.1	even	1	trivial
275.2.b.e.199.2		4	5.2	odd	4
275.2.b.e.199.3		4	5.3	odd	4
2475.2.a.n.1.2		2	3.2	odd	2
2475.2.a.s.1.1		2	15.14	odd	2
2475.2.c.p.199.2		4	15.8	even	4
2475.2.c.p.199.3		4	15.2	even	4
3025.2.a.i.1.2		2	11.10	odd	2
3025.2.a.m.1.1		2	55.54	odd	2
4400.2.a.bg.1.2		2	4.3	odd	2
4400.2.a.bv.1.1		2	20.19	odd	2
4400.2.b.x.4049.2		4	20.3	even	4
4400.2.b.x.4049.3		4	20.7	even	4

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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} +0.381966 q^{3} -1.61803 q^{4} -0.236068 q^{6} +3.85410 q^{7} +2.23607 q^{8} -2.85410 q^{9} +1.00000 q^{11} -0.618034 q^{12} +1.76393 q^{13} -2.38197 q^{14} +1.85410 q^{16} +1.61803 q^{17} +1.76393 q^{18} +6.70820 q^{19} +1.47214 q^{21} -0.618034 q^{22} +7.09017 q^{23} +0.854102 q^{24} -1.09017 q^{26} -2.23607 q^{27} -6.23607 q^{28} -3.61803 q^{29} -3.00000 q^{31} -5.61803 q^{32} +0.381966 q^{33} -1.00000 q^{34} +4.61803 q^{36} +5.76393 q^{37} -4.14590 q^{38} +0.673762 q^{39} -3.00000 q^{41} -0.909830 q^{42} -6.00000 q^{43} -1.61803 q^{44} -4.38197 q^{46} -5.94427 q^{47} +0.708204 q^{48} +7.85410 q^{49} +0.618034 q^{51} -2.85410 q^{52} -6.32624 q^{53} +1.38197 q^{54} +8.61803 q^{56} +2.56231 q^{57} +2.23607 q^{58} +9.47214 q^{59} -11.0902 q^{61} +1.85410 q^{62} -11.0000 q^{63} -0.236068 q^{64} -0.236068 q^{66} +8.00000 q^{67} -2.61803 q^{68} +2.70820 q^{69} -14.1803 q^{71} -6.38197 q^{72} +12.6180 q^{73} -3.56231 q^{74} -10.8541 q^{76} +3.85410 q^{77} -0.416408 q^{78} -0.854102 q^{79} +7.70820 q^{81} +1.85410 q^{82} -16.8541 q^{83} -2.38197 q^{84} +3.70820 q^{86} -1.38197 q^{87} +2.23607 q^{88} -18.0902 q^{89} +6.79837 q^{91} -11.4721 q^{92} -1.14590 q^{93} +3.67376 q^{94} -2.14590 q^{96} -0.618034 q^{97} -4.85410 q^{98} -2.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{3} - q^{4} + 4 q^{6} + q^{7} + q^{9} + 2 q^{11} + q^{12} + 8 q^{13} - 7 q^{14} - 3 q^{16} + q^{17} + 8 q^{18} - 6 q^{21} + q^{22} + 3 q^{23} - 5 q^{24} + 9 q^{26} - 8 q^{28} - 5 q^{29}+ \cdots + q^{99}+O(q^{100}) \) 2 * q + q^2 + 3 * q^3 - q^4 + 4 * q^6 + q^7 + q^9 + 2 * q^11 + q^12 + 8 * q^13 - 7 * q^14 - 3 * q^16 + q^17 + 8 * q^18 - 6 * q^21 + q^22 + 3 * q^23 - 5 * q^24 + 9 * q^26 - 8 * q^28 - 5 * q^29 - 6 * q^31 - 9 * q^32 + 3 * q^33 - 2 * q^34 + 7 * q^36 + 16 * q^37 - 15 * q^38 + 17 * q^39 - 6 * q^41 - 13 * q^42 - 12 * q^43 - q^44 - 11 * q^46 + 6 * q^47 - 12 * q^48 + 9 * q^49 - q^51 + q^52 + 3 * q^53 + 5 * q^54 + 15 * q^56 - 15 * q^57 + 10 * q^59 - 11 * q^61 - 3 * q^62 - 22 * q^63 + 4 * q^64 + 4 * q^66 + 16 * q^67 - 3 * q^68 - 8 * q^69 - 6 * q^71 - 15 * q^72 + 23 * q^73 + 13 * q^74 - 15 * q^76 + q^77 + 26 * q^78 + 5 * q^79 + 2 * q^81 - 3 * q^82 - 27 * q^83 - 7 * q^84 - 6 * q^86 - 5 * q^87 - 25 * q^89 - 11 * q^91 - 14 * q^92 - 9 * q^93 + 23 * q^94 - 11 * q^96 + q^97 - 3 * q^98 + q^99

Introduction

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