LMFDB - Embedded newform 4400.2.a.bg.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] 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("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-3,0,0,0,-1,0,1,0,-2,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.1341768894$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.381966 q^{3} -3.85410 q^{7} -2.85410 q^{9} -1.00000 q^{11} +1.76393 q^{13} +1.61803 q^{17} -6.70820 q^{19} +1.47214 q^{21} -7.09017 q^{23} +2.23607 q^{27} -3.61803 q^{29} +3.00000 q^{31} +0.381966 q^{33} +5.76393 q^{37} -0.673762 q^{39} -3.00000 q^{41} +6.00000 q^{43} +5.94427 q^{47} +7.85410 q^{49} -0.618034 q^{51} -6.32624 q^{53} +2.56231 q^{57} -9.47214 q^{59} -11.0902 q^{61} +11.0000 q^{63} -8.00000 q^{67} +2.70820 q^{69} +14.1803 q^{71} +12.6180 q^{73} +3.85410 q^{77} +0.854102 q^{79} +7.70820 q^{81} +16.8541 q^{83} +1.38197 q^{87} -18.0902 q^{89} -6.79837 q^{91} -1.14590 q^{93} -0.618034 q^{97} +2.85410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - q^{7} + q^{9} - 2 q^{11} + 8 q^{13} + q^{17} - 6 q^{21} - 3 q^{23} - 5 q^{29} + 6 q^{31} + 3 q^{33} + 16 q^{37} - 17 q^{39} - 6 q^{41} + 12 q^{43} - 6 q^{47} + 9 q^{49} + q^{51} + 3 q^{53}+ \cdots - q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - q^7 + q^9 - 2 * q^11 + 8 * q^13 + q^17 - 6 * q^21 - 3 * q^23 - 5 * q^29 + 6 * q^31 + 3 * q^33 + 16 * q^37 - 17 * q^39 - 6 * q^41 + 12 * q^43 - 6 * q^47 + 9 * q^49 + q^51 + 3 * q^53 - 15 * q^57 - 10 * q^59 - 11 * q^61 + 22 * q^63 - 16 * q^67 - 8 * q^69 + 6 * q^71 + 23 * q^73 + q^77 - 5 * q^79 + 2 * q^81 + 27 * q^83 + 5 * q^87 - 25 * q^89 + 11 * q^91 - 9 * q^93 + q^97 - 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	0
$2$	0	0
$3$	−0.381966	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$3$	−0.381966	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.85410	−1.45671	−0.728357	−	0.685198i	$-0.759716\pi$
$7$	−3.85410	−1.45671	−0.728357	+	0.685198i	$0.759716\pi$
$8$	0	0
$9$	−2.85410	−0.951367
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.76393	0.489227	0.244613	−	0.969621i	$-0.421339\pi$
$13$	1.76393	0.489227	0.244613	+	0.969621i	$0.421339\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.61803	0.392431	0.196215	−	0.980561i	$-0.437135\pi$
$17$	1.61803	0.392431	0.196215	+	0.980561i	$0.437135\pi$
$18$	0	0
$19$	−6.70820	−1.53897	−0.769484	−	0.638666i	$-0.779486\pi$
$19$	−6.70820	−1.53897	−0.769484	+	0.638666i	$0.779486\pi$
$20$	0	0
$21$	1.47214	0.321246
$22$	0	0
$23$	−7.09017	−1.47840	−0.739201	−	0.673485i	$-0.764797\pi$
$23$	−7.09017	−1.47840	−0.739201	+	0.673485i	$0.764797\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.23607	0.430331
$28$	0	0
$29$	−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	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	0.381966	0.0664917
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.76393	0.947585	0.473792	−	0.880637i	$-0.342885\pi$
$37$	5.76393	0.947585	0.473792	+	0.880637i	$0.342885\pi$
$38$	0	0
$39$	−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	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.94427	0.867061	0.433531	−	0.901139i	$-0.357268\pi$
$47$	5.94427	0.867061	0.433531	+	0.901139i	$0.357268\pi$
$48$	0	0
$49$	7.85410	1.12201
$50$	0	0
$51$	−0.618034	−0.0865421
$52$	0	0
$53$	−6.32624	−0.868976	−0.434488	−	0.900678i	$-0.643071\pi$
$53$	−6.32624	−0.868976	−0.434488	+	0.900678i	$0.643071\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.56231	0.339386
$58$	0	0
$59$	−9.47214	−1.23317	−0.616584	−	0.787289i	$-0.711484\pi$
$59$	−9.47214	−1.23317	−0.616584	+	0.787289i	$0.711484\pi$
$60$	0	0
$61$	−11.0902	−1.41995	−0.709975	−	0.704226i	$-0.751294\pi$
$61$	−11.0902	−1.41995	−0.709975	+	0.704226i	$0.751294\pi$
$62$	0	0
$63$	11.0000	1.38587
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	2.70820	0.326029
$70$	0	0
$71$	14.1803	1.68290	0.841448	−	0.540338i	$-0.181703\pi$
$71$	14.1803	1.68290	0.841448	+	0.540338i	$0.181703\pi$
$72$	0	0
$73$	12.6180	1.47683	0.738415	−	0.674347i	$-0.235575\pi$
$73$	12.6180	1.47683	0.738415	+	0.674347i	$0.235575\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.85410	0.439216
$78$	0	0
$79$	0.854102	0.0960940	0.0480470	−	0.998845i	$-0.484700\pi$
$79$	0.854102	0.0960940	0.0480470	+	0.998845i	$0.484700\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	0	0
$83$	16.8541	1.84998	0.924989	−	0.379994i	$-0.124074\pi$
$83$	16.8541	1.84998	0.924989	+	0.379994i	$0.124074\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.38197	0.148162
$88$	0	0
$89$	−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
$92$	0	0
$93$	−1.14590	−0.118824
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.618034	−0.0627518	−0.0313759	−	0.999508i	$-0.509989\pi$
$97$	−0.618034	−0.0627518	−0.0313759	+	0.999508i	$0.509989\pi$
$98$	0	0
$99$	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	0
$103$	−7.61803	−0.750627	−0.375314	−	0.926898i	$-0.622465\pi$
$103$	−7.61803	−0.750627	−0.375314	+	0.926898i	$0.622465\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.236068	−0.0228216	−0.0114108	−	0.999935i	$-0.503632\pi$
$107$	−0.236068	−0.0228216	−0.0114108	+	0.999935i	$0.503632\pi$
$108$	0	0
$109$	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
$112$	0	0
$113$	19.6525	1.84875	0.924375	−	0.381486i	$-0.124587\pi$
$113$	19.6525	1.84875	0.924375	+	0.381486i	$0.124587\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−5.03444	−0.465434
$118$	0	0
$119$	−6.23607	−0.571659
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.14590	0.103322
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1.61803	−0.143577	−0.0717886	−	0.997420i	$-0.522871\pi$
$127$	−1.61803	−0.143577	−0.0717886	+	0.997420i	$0.522871\pi$
$128$	0	0
$129$	−2.29180	−0.201781
$130$	0	0
$131$	1.09017	0.0952486	0.0476243	−	0.998865i	$-0.484835\pi$
$131$	1.09017	0.0952486	0.0476243	+	0.998865i	$0.484835\pi$
$132$	0	0
$133$	25.8541	2.24183
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.5623	−1.24414	−0.622071	−	0.782961i	$-0.713708\pi$
$137$	−14.5623	−1.24414	−0.622071	+	0.782961i	$0.713708\pi$
$138$	0	0
$139$	16.7082	1.41717	0.708586	−	0.705625i	$-0.249334\pi$
$139$	16.7082	1.41717	0.708586	+	0.705625i	$0.249334\pi$
$140$	0	0
$141$	−2.27051	−0.191211
$142$	0	0
$143$	−1.76393	−0.147507
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	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	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	0	0
$153$	−4.61803	−0.373346
$154$	0	0
$155$	0	0
$156$	0	0
$157$	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	0
$159$	2.41641	0.191634
$160$	0	0
$161$	27.3262	2.15361
$162$	0	0
$163$	0.145898	0.0114276	0.00571381	−	0.999984i	$-0.498181\pi$
$163$	0.145898	0.0114276	0.00571381	+	0.999984i	$0.498181\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.7082	1.44768	0.723842	−	0.689966i	$-0.242374\pi$
$167$	18.7082	1.44768	0.723842	+	0.689966i	$0.242374\pi$
$168$	0	0
$169$	−9.88854	−0.760657
$170$	0	0
$171$	19.1459	1.46412
$172$	0	0
$173$	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	0
$175$	0	0
$176$	0	0
$177$	3.61803	0.271948
$178$	0	0
$179$	11.3820	0.850728	0.425364	−	0.905022i	$-0.360146\pi$
$179$	11.3820	0.850728	0.425364	+	0.905022i	$0.360146\pi$
$180$	0	0
$181$	5.09017	0.378349	0.189175	−	0.981943i	$-0.439419\pi$
$181$	5.09017	0.378349	0.189175	+	0.981943i	$0.439419\pi$
$182$	0	0
$183$	4.23607	0.313139
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.61803	−0.118322
$188$	0	0
$189$	−8.61803	−0.626870
$190$	0	0
$191$	9.90983	0.717050	0.358525	−	0.933520i	$-0.383280\pi$
$191$	9.90983	0.717050	0.358525	+	0.933520i	$0.383280\pi$
$192$	0	0
$193$	−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	0
$195$	0	0
$196$	0	0
$197$	−8.90983	−0.634799	−0.317400	−	0.948292i	$-0.602810\pi$
$197$	−8.90983	−0.634799	−0.317400	+	0.948292i	$0.602810\pi$
$198$	0	0
$199$	−8.09017	−0.573497	−0.286748	−	0.958006i	$-0.592574\pi$
$199$	−8.09017	−0.573497	−0.286748	+	0.958006i	$0.592574\pi$
$200$	0	0
$201$	3.05573	0.215534
$202$	0	0
$203$	13.9443	0.978696
$204$	0	0
$205$	0	0
$206$	0	0
$207$	20.2361	1.40650
$208$	0	0
$209$	6.70820	0.464016
$210$	0	0
$211$	−17.0000	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$211$	−17.0000	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$212$	0	0
$213$	−5.41641	−0.371126
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−11.5623	−0.784900
$218$	0	0
$219$	−4.81966	−0.325682
$220$	0	0
$221$	2.85410	0.191988
$222$	0	0
$223$	18.8885	1.26487	0.632435	−	0.774613i	$-0.282055\pi$
$223$	18.8885	1.26487	0.632435	+	0.774613i	$0.282055\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.03444	−0.334148	−0.167074	−	0.985944i	$-0.553432\pi$
$227$	−5.03444	−0.334148	−0.167074	+	0.985944i	$0.553432\pi$
$228$	0	0
$229$	−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
$232$	0	0
$233$	−22.5066	−1.47445	−0.737227	−	0.675645i	$-0.763865\pi$
$233$	−22.5066	−1.47445	−0.737227	+	0.675645i	$0.763865\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−0.326238	−0.0211914
$238$	0	0
$239$	−8.61803	−0.557454	−0.278727	−	0.960370i	$-0.589913\pi$
$239$	−8.61803	−0.557454	−0.278727	+	0.960370i	$0.589913\pi$
$240$	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	0
$243$	−9.65248	−0.619207
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11.8328	−0.752904
$248$	0	0
$249$	−6.43769	−0.407972
$250$	0	0
$251$	−6.27051	−0.395791	−0.197896	−	0.980223i	$-0.563411\pi$
$251$	−6.27051	−0.395791	−0.197896	+	0.980223i	$0.563411\pi$
$252$	0	0
$253$	7.09017	0.445755
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6.94427	0.433172	0.216586	−	0.976264i	$-0.430508\pi$
$257$	6.94427	0.433172	0.216586	+	0.976264i	$0.430508\pi$
$258$	0	0
$259$	−22.2148	−1.38036
$260$	0	0
$261$	10.3262	0.639178
$262$	0	0
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.90983	0.422875
$268$	0	0
$269$	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	$-0.410052\pi$
$271$	9.18034	0.557666	0.278833	+	0.960340i	$0.410052\pi$
$272$	0	0
$273$	2.59675	0.157162
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−2.52786	−0.151885	−0.0759423	−	0.997112i	$-0.524196\pi$
$277$	−2.52786	−0.151885	−0.0759423	+	0.997112i	$0.524196\pi$
$278$	0	0
$279$	−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	$0.304036\pi$
$282$	0	0
$283$	9.61803	0.571733	0.285866	−	0.958269i	$-0.407719\pi$
$283$	9.61803	0.571733	0.285866	+	0.958269i	$0.407719\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.5623	0.682501
$288$	0	0
$289$	−14.3820	−0.845998
$290$	0	0
$291$	0.236068	0.0138385
$292$	0	0
$293$	11.8885	0.694536	0.347268	−	0.937766i	$-0.387109\pi$
$293$	11.8885	0.694536	0.347268	+	0.937766i	$0.387109\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.23607	−0.129750
$298$	0	0
$299$	−12.5066	−0.723274
$300$	0	0
$301$	−23.1246	−1.33288
$302$	0	0
$303$	−1.94427	−0.111696
$304$	0	0
$305$	0	0
$306$	0	0
$307$	22.4508	1.28134	0.640669	−	0.767817i	$-0.278657\pi$
$307$	22.4508	1.28134	0.640669	+	0.767817i	$0.278657\pi$
$308$	0	0
$309$	2.90983	0.165534
$310$	0	0
$311$	−3.18034	−0.180341	−0.0901703	−	0.995926i	$-0.528741\pi$
$311$	−3.18034	−0.180341	−0.0901703	+	0.995926i	$0.528741\pi$
$312$	0	0
$313$	1.23607	0.0698667	0.0349333	−	0.999390i	$-0.488878\pi$
$313$	1.23607	0.0698667	0.0349333	+	0.999390i	$0.488878\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.3820	0.807772	0.403886	−	0.914809i	$-0.367659\pi$
$317$	14.3820	0.807772	0.403886	+	0.914809i	$0.367659\pi$
$318$	0	0
$319$	3.61803	0.202571
$320$	0	0
$321$	0.0901699	0.00503280
$322$	0	0
$323$	−10.8541	−0.603938
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−3.09017	−0.170887
$328$	0	0
$329$	−22.9098	−1.26306
$330$	0	0
$331$	−3.18034	−0.174807	−0.0874036	−	0.996173i	$-0.527857\pi$
$331$	−3.18034	−0.174807	−0.0874036	+	0.996173i	$0.527857\pi$
$332$	0	0
$333$	−16.4508	−0.901501
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−25.4164	−1.38452	−0.692260	−	0.721648i	$-0.743385\pi$
$337$	−25.4164	−1.38452	−0.692260	+	0.721648i	$0.743385\pi$
$338$	0	0
$339$	−7.50658	−0.407701
$340$	0	0
$341$	−3.00000	−0.162459
$342$	0	0
$343$	−3.29180	−0.177740
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.437694	−0.0234967	−0.0117483	−	0.999931i	$-0.503740\pi$
$347$	−0.437694	−0.0234967	−0.0117483	+	0.999931i	$0.503740\pi$
$348$	0	0
$349$	−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
$352$	0	0
$353$	−23.3607	−1.24336	−0.621682	−	0.783270i	$-0.713550\pi$
$353$	−23.3607	−1.24336	−0.621682	+	0.783270i	$0.713550\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.38197	0.126067
$358$	0	0
$359$	−4.47214	−0.236030	−0.118015	−	0.993012i	$-0.537653\pi$
$359$	−4.47214	−0.236030	−0.118015	+	0.993012i	$0.537653\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0	0
$363$	−0.381966	−0.0200480
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−13.8541	−0.723178	−0.361589	−	0.932338i	$-0.617766\pi$
$367$	−13.8541	−0.723178	−0.361589	+	0.932338i	$0.617766\pi$
$368$	0	0
$369$	8.56231	0.445736
$370$	0	0
$371$	24.3820	1.26585
$372$	0	0
$373$	25.1803	1.30379	0.651894	−	0.758310i	$-0.273975\pi$
$373$	25.1803	1.30379	0.651894	+	0.758310i	$0.273975\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.38197	−0.328688
$378$	0	0
$379$	22.2361	1.14219	0.571095	−	0.820884i	$-0.306519\pi$
$379$	22.2361	1.14219	0.571095	+	0.820884i	$0.306519\pi$
$380$	0	0
$381$	0.618034	0.0316628
$382$	0	0
$383$	−22.9443	−1.17240	−0.586199	−	0.810167i	$-0.699376\pi$
$383$	−22.9443	−1.17240	−0.586199	+	0.810167i	$0.699376\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−17.1246	−0.870493
$388$	0	0
$389$	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
$392$	0	0
$393$	−0.416408	−0.0210050
$394$	0	0
$395$	0	0
$396$	0	0
$397$	26.0902	1.30943	0.654714	−	0.755877i	$-0.272789\pi$
$397$	26.0902	1.30943	0.654714	+	0.755877i	$0.272789\pi$
$398$	0	0
$399$	−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	$0.583292\pi$
$402$	0	0
$403$	5.29180	0.263603
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.76393	−0.285708
$408$	0	0
$409$	−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
$412$	0	0
$413$	36.5066	1.79637
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.38197	−0.312526
$418$	0	0
$419$	1.18034	0.0576634	0.0288317	−	0.999584i	$-0.490821\pi$
$419$	1.18034	0.0576634	0.0288317	+	0.999584i	$0.490821\pi$
$420$	0	0
$421$	21.2705	1.03666	0.518331	−	0.855180i	$-0.326554\pi$
$421$	21.2705	1.03666	0.518331	+	0.855180i	$0.326554\pi$
$422$	0	0
$423$	−16.9656	−0.824894
$424$	0	0
$425$	0	0
$426$	0	0
$427$	42.7426	2.06846
$428$	0	0
$429$	0.673762	0.0325295
$430$	0	0
$431$	−23.1803	−1.11656	−0.558279	−	0.829653i	$-0.688538\pi$
$431$	−23.1803	−1.11656	−0.558279	+	0.829653i	$0.688538\pi$
$432$	0	0
$433$	16.8885	0.811612	0.405806	−	0.913959i	$-0.366991\pi$
$433$	16.8885	0.811612	0.405806	+	0.913959i	$0.366991\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	47.5623	2.27521
$438$	0	0
$439$	34.2705	1.63564	0.817821	−	0.575473i	$-0.195182\pi$
$439$	34.2705	1.63564	0.817821	+	0.575473i	$0.195182\pi$
$440$	0	0
$441$	−22.4164	−1.06745
$442$	0	0
$443$	5.34752	0.254069	0.127034	−	0.991898i	$-0.459454\pi$
$443$	5.34752	0.254069	0.127034	+	0.991898i	$0.459454\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−3.41641	−0.161591
$448$	0	0
$449$	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
$452$	0	0
$453$	−1.14590	−0.0538390
$454$	0	0
$455$	0	0
$456$	0	0
$457$	38.9787	1.82335	0.911674	−	0.410915i	$-0.134791\pi$
$457$	38.9787	1.82335	0.911674	+	0.410915i	$0.134791\pi$
$458$	0	0
$459$	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	0
$463$	33.3607	1.55040	0.775201	−	0.631714i	$-0.217648\pi$
$463$	33.3607	1.55040	0.775201	+	0.631714i	$0.217648\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.5279	−1.32011	−0.660056	−	0.751216i	$-0.729467\pi$
$467$	−28.5279	−1.32011	−0.660056	+	0.751216i	$0.729467\pi$
$468$	0	0
$469$	30.8328	1.42373
$470$	0	0
$471$	−8.18034	−0.376930
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	0	0
$476$	0	0
$477$	18.0557	0.826715
$478$	0	0
$479$	−1.58359	−0.0723562	−0.0361781	−	0.999345i	$-0.511518\pi$
$479$	−1.58359	−0.0723562	−0.0361781	+	0.999345i	$0.511518\pi$
$480$	0	0
$481$	10.1672	0.463584
$482$	0	0
$483$	−10.4377	−0.474932
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3.58359	0.162388	0.0811940	−	0.996698i	$-0.474127\pi$
$487$	3.58359	0.162388	0.0811940	+	0.996698i	$0.474127\pi$
$488$	0	0
$489$	−0.0557281	−0.00252011
$490$	0	0
$491$	29.1803	1.31689	0.658445	−	0.752629i	$-0.271214\pi$
$491$	29.1803	1.31689	0.658445	+	0.752629i	$0.271214\pi$
$492$	0	0
$493$	−5.85410	−0.263655
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−54.6525	−2.45150
$498$	0	0
$499$	−5.20163	−0.232857	−0.116428	−	0.993199i	$-0.537145\pi$
$499$	−5.20163	−0.232857	−0.116428	+	0.993199i	$0.537145\pi$
$500$	0	0
$501$	−7.14590	−0.319255
$502$	0	0
$503$	−24.6525	−1.09920	−0.549600	−	0.835428i	$-0.685220\pi$
$503$	−24.6525	−1.09920	−0.549600	+	0.835428i	$0.685220\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.77709	0.167746
$508$	0	0
$509$	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
$512$	0	0
$513$	−15.0000	−0.662266
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.94427	−0.261429
$518$	0	0
$519$	−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	$0.677688\pi$
$522$	0	0
$523$	−5.05573	−0.221072	−0.110536	−	0.993872i	$-0.535257\pi$
$523$	−5.05573	−0.221072	−0.110536	+	0.993872i	$0.535257\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.85410	0.211448
$528$	0	0
$529$	27.2705	1.18567
$530$	0	0
$531$	27.0344	1.17319
$532$	0	0
$533$	−5.29180	−0.229213
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−4.34752	−0.187610
$538$	0	0
$539$	−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	$0.560089\pi$
$542$	0	0
$543$	−1.94427	−0.0834367
$544$	0	0
$545$	0	0
$546$	0	0
$547$	27.8541	1.19096	0.595478	−	0.803372i	$-0.296963\pi$
$547$	27.8541	1.19096	0.595478	+	0.803372i	$0.296963\pi$
$548$	0	0
$549$	31.6525	1.35089
$550$	0	0
$551$	24.2705	1.03396
$552$	0	0
$553$	−3.29180	−0.139981
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−14.2361	−0.603202	−0.301601	−	0.953434i	$-0.597521\pi$
$557$	−14.2361	−0.603202	−0.301601	+	0.953434i	$0.597521\pi$
$558$	0	0
$559$	10.5836	0.447638
$560$	0	0
$561$	0.618034	0.0260934
$562$	0	0
$563$	−16.0344	−0.675771	−0.337886	−	0.941187i	$-0.609712\pi$
$563$	−16.0344	−0.675771	−0.337886	+	0.941187i	$0.609712\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−29.7082	−1.24763
$568$	0	0
$569$	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	$-0.518189\pi$
$571$	−2.72949	−0.114226	−0.0571128	+	0.998368i	$0.518189\pi$
$572$	0	0
$573$	−3.78522	−0.158130
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.56231	−0.189931	−0.0949656	−	0.995481i	$-0.530274\pi$
$577$	−4.56231	−0.189931	−0.0949656	+	0.995481i	$0.530274\pi$
$578$	0	0
$579$	1.88854	0.0784852
$580$	0	0
$581$	−64.9574	−2.69489
$582$	0	0
$583$	6.32624	0.262006
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.0344	−1.03328	−0.516641	−	0.856202i	$-0.672818\pi$
$587$	−25.0344	−1.03328	−0.516641	+	0.856202i	$0.672818\pi$
$588$	0	0
$589$	−20.1246	−0.829220
$590$	0	0
$591$	3.40325	0.139991
$592$	0	0
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.09017	0.126472
$598$	0	0
$599$	15.3262	0.626213	0.313107	−	0.949718i	$-0.398630\pi$
$599$	15.3262	0.626213	0.313107	+	0.949718i	$0.398630\pi$
$600$	0	0
$601$	11.2705	0.459734	0.229867	−	0.973222i	$-0.426171\pi$
$601$	11.2705	0.459734	0.229867	+	0.973222i	$0.426171\pi$
$602$	0	0
$603$	22.8328	0.929824
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.4721	−0.506228	−0.253114	−	0.967436i	$-0.581455\pi$
$607$	−12.4721	−0.506228	−0.253114	+	0.967436i	$0.581455\pi$
$608$	0	0
$609$	−5.32624	−0.215830
$610$	0	0
$611$	10.4853	0.424189
$612$	0	0
$613$	26.5623	1.07284	0.536421	−	0.843951i	$-0.319776\pi$
$613$	26.5623	1.07284	0.536421	+	0.843951i	$0.319776\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−15.8197	−0.636876	−0.318438	−	0.947944i	$-0.603158\pi$
$617$	−15.8197	−0.636876	−0.318438	+	0.947944i	$0.603158\pi$
$618$	0	0
$619$	−12.8885	−0.518034	−0.259017	−	0.965873i	$-0.583399\pi$
$619$	−12.8885	−0.518034	−0.259017	+	0.965873i	$0.583399\pi$
$620$	0	0
$621$	−15.8541	−0.636203
$622$	0	0
$623$	69.7214	2.79333
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−2.56231	−0.102329
$628$	0	0
$629$	9.32624	0.371861
$630$	0	0
$631$	−2.72949	−0.108659	−0.0543296	−	0.998523i	$-0.517302\pi$
$631$	−2.72949	−0.108659	−0.0543296	+	0.998523i	$0.517302\pi$
$632$	0	0
$633$	6.49342	0.258090
$634$	0	0
$635$	0	0
$636$	0	0
$637$	13.8541	0.548920
$638$	0	0
$639$	−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	0
$643$	−37.4164	−1.47556	−0.737780	−	0.675042i	$-0.764126\pi$
$643$	−37.4164	−1.47556	−0.737780	+	0.675042i	$0.764126\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−43.6525	−1.71616	−0.858078	−	0.513519i	$-0.828341\pi$
$647$	−43.6525	−1.71616	−0.858078	+	0.513519i	$0.828341\pi$
$648$	0	0
$649$	9.47214	0.371814
$650$	0	0
$651$	4.41641	0.173093
$652$	0	0
$653$	−34.7426	−1.35958	−0.679792	−	0.733405i	$-0.737930\pi$
$653$	−34.7426	−1.35958	−0.679792	+	0.733405i	$0.737930\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−36.0132	−1.40501
$658$	0	0
$659$	24.2705	0.945445	0.472722	−	0.881211i	$-0.343271\pi$
$659$	24.2705	0.945445	0.472722	+	0.881211i	$0.343271\pi$
$660$	0	0
$661$	−26.8197	−1.04316	−0.521582	−	0.853201i	$-0.674658\pi$
$661$	−26.8197	−1.04316	−0.521582	+	0.853201i	$0.674658\pi$
$662$	0	0
$663$	−1.09017	−0.0423387
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25.6525	0.993268
$668$	0	0
$669$	−7.21478	−0.278940
$670$	0	0
$671$	11.0902	0.428131
$672$	0	0
$673$	−24.4164	−0.941183	−0.470592	−	0.882351i	$-0.655960\pi$
$673$	−24.4164	−0.941183	−0.470592	+	0.882351i	$0.655960\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.65248	−0.101943	−0.0509715	−	0.998700i	$-0.516232\pi$
$677$	−2.65248	−0.101943	−0.0509715	+	0.998700i	$0.516232\pi$
$678$	0	0
$679$	2.38197	0.0914115
$680$	0	0
$681$	1.92299	0.0736890
$682$	0	0
$683$	−1.36068	−0.0520650	−0.0260325	−	0.999661i	$-0.508287\pi$
$683$	−1.36068	−0.0520650	−0.0260325	+	0.999661i	$0.508287\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	6.50658	0.248241
$688$	0	0
$689$	−11.1591	−0.425126
$690$	0	0
$691$	−18.9098	−0.719364	−0.359682	−	0.933075i	$-0.617115\pi$
$691$	−18.9098	−0.719364	−0.359682	+	0.933075i	$0.617115\pi$
$692$	0	0
$693$	−11.0000	−0.417855
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.85410	−0.183862
$698$	0	0
$699$	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	$0.183877\pi$
$702$	0	0
$703$	−38.6656	−1.45830
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−19.6180	−0.737812
$708$	0	0
$709$	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
$712$	0	0
$713$	−21.2705	−0.796587
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.29180	0.122934
$718$	0	0
$719$	−15.6525	−0.583739	−0.291869	−	0.956458i	$-0.594277\pi$
$719$	−15.6525	−0.583739	−0.291869	+	0.956458i	$0.594277\pi$
$720$	0	0
$721$	29.3607	1.09345
$722$	0	0
$723$	4.68692	0.174308
$724$	0	0
$725$	0	0
$726$	0	0
$727$	7.72949	0.286671	0.143335	−	0.989674i	$-0.454217\pi$
$727$	7.72949	0.286671	0.143335	+	0.989674i	$0.454217\pi$
$728$	0	0
$729$	−19.4377	−0.719915
$730$	0	0
$731$	9.70820	0.359071
$732$	0	0
$733$	−7.83282	−0.289312	−0.144656	−	0.989482i	$-0.546207\pi$
$733$	−7.83282	−0.289312	−0.144656	+	0.989482i	$0.546207\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	35.8541	1.31891	0.659457	−	0.751742i	$-0.270786\pi$
$739$	35.8541	1.31891	0.659457	+	0.751742i	$0.270786\pi$
$740$	0	0
$741$	4.51973	0.166037
$742$	0	0
$743$	−34.3262	−1.25931	−0.629654	−	0.776876i	$-0.716803\pi$
$743$	−34.3262	−1.25931	−0.629654	+	0.776876i	$0.716803\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−48.1033	−1.76001
$748$	0	0
$749$	0.909830	0.0332445
$750$	0	0
$751$	22.2705	0.812662	0.406331	−	0.913726i	$-0.366808\pi$
$751$	22.2705	0.812662	0.406331	+	0.913726i	$0.366808\pi$
$752$	0	0
$753$	2.39512	0.0872831
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−12.5279	−0.455333	−0.227666	−	0.973739i	$-0.573110\pi$
$757$	−12.5279	−0.455333	−0.227666	+	0.973739i	$0.573110\pi$
$758$	0	0
$759$	−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	0
$763$	−31.1803	−1.12880
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16.7082	−0.603298
$768$	0	0
$769$	−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
$772$	0	0
$773$	9.20163	0.330959	0.165480	−	0.986213i	$-0.447083\pi$
$773$	9.20163	0.330959	0.165480	+	0.986213i	$0.447083\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.48529	0.304408
$778$	0	0
$779$	20.1246	0.721039
$780$	0	0
$781$	−14.1803	−0.507412
$782$	0	0
$783$	−8.09017	−0.289119
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−34.7082	−1.23721	−0.618607	−	0.785701i	$-0.712303\pi$
$787$	−34.7082	−1.23721	−0.618607	+	0.785701i	$0.712303\pi$
$788$	0	0
$789$	−8.02129	−0.285565
$790$	0	0
$791$	−75.7426	−2.69310
$792$	0	0
$793$	−19.5623	−0.694678
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.7984	0.984669	0.492334	−	0.870406i	$-0.336144\pi$
$797$	27.7984	0.984669	0.492334	+	0.870406i	$0.336144\pi$
$798$	0	0
$799$	9.61803	0.340262
$800$	0	0
$801$	51.6312	1.82430
$802$	0	0
$803$	−12.6180	−0.445281
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−5.60488	−0.197301
$808$	0	0
$809$	−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	$-0.596478\pi$
$811$	−17.0000	−0.596951	−0.298475	+	0.954417i	$0.596478\pi$
$812$	0	0
$813$	−3.50658	−0.122981
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−40.2492	−1.40814
$818$	0	0
$819$	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	$0.748738\pi$
$822$	0	0
$823$	−0.583592	−0.0203427	−0.0101714	−	0.999948i	$-0.503238\pi$
$823$	−0.583592	−0.0203427	−0.0101714	+	0.999948i	$0.503238\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.0689	0.906504	0.453252	−	0.891382i	$-0.350264\pi$
$827$	26.0689	0.906504	0.453252	+	0.891382i	$0.350264\pi$
$828$	0	0
$829$	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	0
$833$	12.7082	0.440313
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.70820	0.231869
$838$	0	0
$839$	−43.3394	−1.49624	−0.748121	−	0.663562i	$-0.769044\pi$
$839$	−43.3394	−1.49624	−0.748121	+	0.663562i	$0.769044\pi$
$840$	0	0
$841$	−15.9098	−0.548615
$842$	0	0
$843$	−7.39512	−0.254702
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.85410	−0.132429
$848$	0	0
$849$	−3.67376	−0.126083
$850$	0	0
$851$	−40.8673	−1.40091
$852$	0	0
$853$	43.1459	1.47729	0.738644	−	0.674096i	$-0.235467\pi$
$853$	43.1459	1.47729	0.738644	+	0.674096i	$0.235467\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−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	0
$859$	6.18034	0.210870	0.105435	−	0.994426i	$-0.466376\pi$
$859$	6.18034	0.210870	0.105435	+	0.994426i	$0.466376\pi$
$860$	0	0
$861$	−4.41641	−0.150511
$862$	0	0
$863$	−48.5967	−1.65425	−0.827126	−	0.562016i	$-0.810026\pi$
$863$	−48.5967	−1.65425	−0.827126	+	0.562016i	$0.810026\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.49342	0.186566
$868$	0	0
$869$	−0.854102	−0.0289734
$870$	0	0
$871$	−14.1115	−0.478148
$872$	0	0
$873$	1.76393	0.0597001
$874$	0	0
$875$	0	0
$876$	0	0
$877$	31.4164	1.06086	0.530428	−	0.847730i	$-0.322031\pi$
$877$	31.4164	1.06086	0.530428	+	0.847730i	$0.322031\pi$
$878$	0	0
$879$	−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	$0.505846\pi$
$882$	0	0
$883$	0.347524	0.0116951	0.00584756	−	0.999983i	$-0.498139\pi$
$883$	0.347524	0.0116951	0.00584756	+	0.999983i	$0.498139\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.7771	1.43631	0.718157	−	0.695881i	$-0.244986\pi$
$887$	42.7771	1.43631	0.718157	+	0.695881i	$0.244986\pi$
$888$	0	0
$889$	6.23607	0.209151
$890$	0	0
$891$	−7.70820	−0.258235
$892$	0	0
$893$	−39.8754	−1.33438
$894$	0	0
$895$	0	0
$896$	0	0
$897$	4.77709	0.159502
$898$	0	0
$899$	−10.8541	−0.362005
$900$	0	0
$901$	−10.2361	−0.341013
$902$	0	0
$903$	8.83282	0.293938
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38.8328	1.28942	0.644711	−	0.764426i	$-0.276978\pi$
$907$	38.8328	1.28942	0.644711	+	0.764426i	$0.276978\pi$
$908$	0	0
$909$	−14.5279	−0.481859
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	−16.8541	−0.557789
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.20163	−0.138750
$918$	0	0
$919$	3.41641	0.112697	0.0563484	−	0.998411i	$-0.482054\pi$
$919$	3.41641	0.112697	0.0563484	+	0.998411i	$0.482054\pi$
$920$	0	0
$921$	−8.57546	−0.282571
$922$	0	0
$923$	25.0132	0.823318
$924$	0	0
$925$	0	0
$926$	0	0
$927$	21.7426	0.714122
$928$	0	0
$929$	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
$932$	0	0
$933$	1.21478	0.0397702
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.8328	0.484567	0.242283	−	0.970206i	$-0.422104\pi$
$937$	14.8328	0.484567	0.242283	+	0.970206i	$0.422104\pi$
$938$	0	0
$939$	−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	$0.783683\pi$
$942$	0	0
$943$	21.2705	0.692663
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.36068	−0.174199	−0.0870993	−	0.996200i	$-0.527760\pi$
$947$	−5.36068	−0.174199	−0.0870993	+	0.996200i	$0.527760\pi$
$948$	0	0
$949$	22.2574	0.722504
$950$	0	0
$951$	−5.49342	−0.178136
$952$	0	0
$953$	−0.472136	−0.0152940	−0.00764699	−	0.999971i	$-0.502434\pi$
$953$	−0.472136	−0.0152940	−0.00764699	+	0.999971i	$0.502434\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.38197	−0.0446726
$958$	0	0
$959$	56.1246	1.81236
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	0.673762	0.0217117
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−3.72949	−0.119932	−0.0599662	−	0.998200i	$-0.519099\pi$
$967$	−3.72949	−0.119932	−0.0599662	+	0.998200i	$0.519099\pi$
$968$	0	0
$969$	4.14590	0.133185
$970$	0	0
$971$	−35.0902	−1.12610	−0.563049	−	0.826424i	$-0.690372\pi$
$971$	−35.0902	−1.12610	−0.563049	+	0.826424i	$0.690372\pi$
$972$	0	0
$973$	−64.3951	−2.06441
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−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	0
$979$	18.0902	0.578164
$980$	0	0
$981$	−23.0902	−0.737212
$982$	0	0
$983$	10.4721	0.334009	0.167005	−	0.985956i	$-0.446591\pi$
$983$	10.4721	0.334009	0.167005	+	0.985956i	$0.446591\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.75078	0.278540
$988$	0	0
$989$	−42.5410	−1.35273
$990$	0	0
$991$	12.2705	0.389786	0.194893	−	0.980825i	$-0.437564\pi$
$991$	12.2705	0.389786	0.194893	+	0.980825i	$0.437564\pi$
$992$	0	0
$993$	1.21478	0.0385499
$994$	0	0
$995$	0	0
$996$	0	0
$997$	38.8541	1.23052	0.615261	−	0.788324i	$-0.289051\pi$
$997$	38.8541	1.23052	0.615261	+	0.788324i	$0.289051\pi$
$998$	0	0
$999$	12.8885	0.407775

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bg.1.2		2
4.3	odd	2		275.2.a.g.1.1	yes	2
5.2	odd	4		4400.2.b.x.4049.3		4
5.3	odd	4		4400.2.b.x.4049.2		4
5.4	even	2		4400.2.a.bv.1.1		2
12.11	even	2		2475.2.a.n.1.2		2
20.3	even	4		275.2.b.e.199.3		4
20.7	even	4		275.2.b.e.199.2		4
20.19	odd	2		275.2.a.d.1.2	✓	2
44.43	even	2		3025.2.a.i.1.2		2
60.23	odd	4		2475.2.c.p.199.2		4
60.47	odd	4		2475.2.c.p.199.3		4
60.59	even	2		2475.2.a.s.1.1		2
220.219	even	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	20.19	odd	2
275.2.a.g.1.1	yes	2	4.3	odd	2
275.2.b.e.199.2		4	20.7	even	4
275.2.b.e.199.3		4	20.3	even	4
2475.2.a.n.1.2		2	12.11	even	2
2475.2.a.s.1.1		2	60.59	even	2
2475.2.c.p.199.2		4	60.23	odd	4
2475.2.c.p.199.3		4	60.47	odd	4
3025.2.a.i.1.2		2	44.43	even	2
3025.2.a.m.1.1		2	220.219	even	2
4400.2.a.bg.1.2		2	1.1	even	1	trivial
4400.2.a.bv.1.1		2	5.4	even	2
4400.2.b.x.4049.2		4	5.3	odd	4
4400.2.b.x.4049.3		4	5.2	odd	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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q-0.381966 q^{3} -3.85410 q^{7} -2.85410 q^{9} -1.00000 q^{11} +1.76393 q^{13} +1.61803 q^{17} -6.70820 q^{19} +1.47214 q^{21} -7.09017 q^{23} +2.23607 q^{27} -3.61803 q^{29} +3.00000 q^{31} +0.381966 q^{33} +5.76393 q^{37} -0.673762 q^{39} -3.00000 q^{41} +6.00000 q^{43} +5.94427 q^{47} +7.85410 q^{49} -0.618034 q^{51} -6.32624 q^{53} +2.56231 q^{57} -9.47214 q^{59} -11.0902 q^{61} +11.0000 q^{63} -8.00000 q^{67} +2.70820 q^{69} +14.1803 q^{71} +12.6180 q^{73} +3.85410 q^{77} +0.854102 q^{79} +7.70820 q^{81} +16.8541 q^{83} +1.38197 q^{87} -18.0902 q^{89} -6.79837 q^{91} -1.14590 q^{93} -0.618034 q^{97} +2.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - q^{7} + q^{9} - 2 q^{11} + 8 q^{13} + q^{17} - 6 q^{21} - 3 q^{23} - 5 q^{29} + 6 q^{31} + 3 q^{33} + 16 q^{37} - 17 q^{39} - 6 q^{41} + 12 q^{43} - 6 q^{47} + 9 q^{49} + q^{51} + 3 q^{53}+ \cdots - q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - q^7 + q^9 - 2 * q^11 + 8 * q^13 + q^17 - 6 * q^21 - 3 * q^23 - 5 * q^29 + 6 * q^31 + 3 * q^33 + 16 * q^37 - 17 * q^39 - 6 * q^41 + 12 * q^43 - 6 * q^47 + 9 * q^49 + q^51 + 3 * q^53 - 15 * q^57 - 10 * q^59 - 11 * q^61 + 22 * q^63 - 16 * q^67 - 8 * q^69 + 6 * q^71 + 23 * q^73 + q^77 - 5 * q^79 + 2 * q^81 + 27 * q^83 + 5 * q^87 - 25 * q^89 + 11 * q^91 - 9 * q^93 + q^97 - q^99

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