LMFDB - Embedded newform 4400.2.b.x.4049.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(4049,4400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4400.4049"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-2,0,-4,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.1
Root			$-1.61803i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.x.4049.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803i q^{3} -2.85410i q^{7} -3.85410 q^{9} -1.00000 q^{11} +6.23607i q^{13} +0.618034i q^{17} -6.70820 q^{19} -7.47214 q^{21} +4.09017i q^{23} +2.23607i q^{27} +1.38197 q^{29} +3.00000 q^{31} +2.61803i q^{33} -10.2361i q^{37} +16.3262 q^{39} -3.00000 q^{41} +6.00000i q^{43} +11.9443i q^{47} -1.14590 q^{49} +1.61803 q^{51} +9.32624i q^{53} +17.5623i q^{57} +0.527864 q^{59} +0.0901699 q^{61} +11.0000i q^{63} +8.00000i q^{67} +10.7082 q^{69} -8.18034 q^{71} +10.3820i q^{73} +2.85410i q^{77} +5.85410 q^{79} -5.70820 q^{81} +10.1459i q^{83} -3.61803i q^{87} +6.90983 q^{89} +17.7984 q^{91} -7.85410i q^{93} -1.61803i q^{97} +3.85410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{9} - 4 q^{11} - 12 q^{21} + 10 q^{29} + 12 q^{31} + 34 q^{39} - 12 q^{41} - 18 q^{49} + 2 q^{51} + 20 q^{59} - 22 q^{61} + 16 q^{69} + 12 q^{71} + 10 q^{79} + 4 q^{81} + 50 q^{89} + 22 q^{91}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q - 2 * q^9 - 4 * q^11 - 12 * q^21 + 10 * q^29 + 12 * q^31 + 34 * q^39 - 12 * q^41 - 18 * q^49 + 2 * q^51 + 20 * q^59 - 22 * q^61 + 16 * q^69 + 12 * q^71 + 10 * q^79 + 4 * q^81 + 50 * q^89 + 22 * q^91 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	− 2.61803i	− 1.51152i	−0.654847	−	0.755761i	$-0.727267\pi$
$3$	− 2.61803i	− 1.51152i	0.654847	−	0.755761i	$-0.272733\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.85410i	− 1.07875i	−0.842066	−	0.539375i	$-0.818661\pi$
$7$	− 2.85410i	− 1.07875i	0.842066	−	0.539375i	$-0.181339\pi$
$8$	0	0
$9$	−3.85410	−1.28470
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.23607i	1.72957i	0.502139	+	0.864787i	$0.332547\pi$
$13$	6.23607i	1.72957i	−0.502139	+	0.864787i	$0.667453\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.618034i	0.149895i	0.997187	+	0.0749476i	$0.0238790\pi$
$17$	0.618034i	0.149895i	−0.997187	+	0.0749476i	$0.976121\pi$
$18$	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
$21$	−7.47214	−1.63055
$22$	0	0
$23$	4.09017i	0.852859i	0.904521	+	0.426430i	$0.140229\pi$
$23$	4.09017i	0.852859i	−0.904521	+	0.426430i	$0.859771\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2.23607i	0.430331i
$28$	0	0
$29$	1.38197	0.256625	0.128312	−	0.991734i	$-0.459044\pi$
$29$	1.38197	0.256625	0.128312	+	0.991734i	$0.459044\pi$
$30$	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
$33$	2.61803i	0.455741i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 10.2361i	− 1.68280i	−0.540413	−	0.841400i	$-0.681732\pi$
$37$	− 10.2361i	− 1.68280i	0.540413	−	0.841400i	$-0.318268\pi$
$38$	0	0
$39$	16.3262	2.61429
$40$	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
$43$	6.00000i	0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$	6.00000i	0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.9443i	1.74225i	0.491060	+	0.871126i	$0.336609\pi$
$47$	11.9443i	1.74225i	−0.491060	+	0.871126i	$0.663391\pi$
$48$	0	0
$49$	−1.14590	−0.163700
$50$	0	0
$51$	1.61803	0.226570
$52$	0	0
$53$	9.32624i	1.28106i	0.767934	+	0.640529i	$0.221285\pi$
$53$	9.32624i	1.28106i	−0.767934	+	0.640529i	$0.778715\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	17.5623i	2.32618i
$58$	0	0
$59$	0.527864	0.0687220	0.0343610	−	0.999409i	$-0.489060\pi$
$59$	0.527864	0.0687220	0.0343610	+	0.999409i	$0.489060\pi$
$60$	0	0
$61$	0.0901699	0.0115451	0.00577254	−	0.999983i	$-0.498163\pi$
$61$	0.0901699	0.0115451	0.00577254	+	0.999983i	$0.498163\pi$
$62$	0	0
$63$	11.0000i	1.38587i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000i	0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$	8.00000i	0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$	0	0
$69$	10.7082	1.28912
$70$	0	0
$71$	−8.18034	−0.970828	−0.485414	−	0.874284i	$-0.661331\pi$
$71$	−8.18034	−0.970828	−0.485414	+	0.874284i	$0.661331\pi$
$72$	0	0
$73$	10.3820i	1.21512i	0.794275	+	0.607559i	$0.207851\pi$
$73$	10.3820i	1.21512i	−0.794275	+	0.607559i	$0.792149\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.85410i	0.325255i
$78$	0	0
$79$	5.85410	0.658638	0.329319	−	0.944219i	$-0.393181\pi$
$79$	5.85410	0.658638	0.329319	+	0.944219i	$0.393181\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	10.1459i	1.11366i	0.830627	+	0.556828i	$0.187982\pi$
$83$	10.1459i	1.11366i	−0.830627	+	0.556828i	$0.812018\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.61803i	− 0.387894i
$88$	0	0
$89$	6.90983	0.732441	0.366220	−	0.930528i	$-0.380652\pi$
$89$	6.90983	0.732441	0.366220	+	0.930528i	$0.380652\pi$
$90$	0	0
$91$	17.7984	1.86578
$92$	0	0
$93$	− 7.85410i	− 0.814432i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.61803i	− 0.164286i	−0.996621	−	0.0821432i	$-0.973824\pi$
$97$	− 1.61803i	− 0.164286i	0.996621	−	0.0821432i	$-0.0261765\pi$
$98$	0	0
$99$	3.85410	0.387352
$100$	0	0
$101$	−6.09017	−0.605995	−0.302997	−	0.952991i	$-0.597987\pi$
$101$	−6.09017	−0.605995	−0.302997	+	0.952991i	$0.597987\pi$
$102$	0	0
$103$	− 5.38197i	− 0.530301i	−0.964207	−	0.265150i	$-0.914578\pi$
$103$	− 5.38197i	− 0.530301i	0.964207	−	0.265150i	$-0.0854216\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.23607i	− 0.409516i	−0.978813	−	0.204758i	$-0.934359\pi$
$107$	− 4.23607i	− 0.409516i	0.978813	−	0.204758i	$-0.0656408\pi$
$108$	0	0
$109$	3.09017	0.295985	0.147992	−	0.988989i	$-0.452719\pi$
$109$	3.09017	0.295985	0.147992	+	0.988989i	$0.452719\pi$
$110$	0	0
$111$	−26.7984	−2.54359
$112$	0	0
$113$	− 11.6525i	− 1.09617i	−0.836422	−	0.548086i	$-0.815357\pi$
$113$	− 11.6525i	− 1.09617i	0.836422	−	0.548086i	$-0.184643\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 24.0344i	− 2.22198i
$118$	0	0
$119$	1.76393	0.161699
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	7.85410i	0.708181i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 0.618034i	− 0.0548416i	−0.999624	−	0.0274208i	$-0.991271\pi$
$127$	− 0.618034i	− 0.0548416i	0.999624	−	0.0274208i	$-0.00872941\pi$
$128$	0	0
$129$	15.7082	1.38303
$130$	0	0
$131$	−10.0902	−0.881582	−0.440791	−	0.897610i	$-0.645302\pi$
$131$	−10.0902	−0.881582	−0.440791	+	0.897610i	$0.645302\pi$
$132$	0	0
$133$	19.1459i	1.66016i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.56231i	− 0.475220i	−0.971361	−	0.237610i	$-0.923636\pi$
$137$	− 5.56231i	− 0.475220i	0.971361	−	0.237610i	$-0.0763640\pi$
$138$	0	0
$139$	−3.29180	−0.279206	−0.139603	−	0.990208i	$-0.544583\pi$
$139$	−3.29180	−0.279206	−0.139603	+	0.990208i	$0.544583\pi$
$140$	0	0
$141$	31.2705	2.63345
$142$	0	0
$143$	− 6.23607i	− 0.521486i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.00000i	0.247436i
$148$	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
$153$	− 2.38197i	− 0.192571i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.41641i	0.432276i	0.976363	+	0.216138i	$0.0693462\pi$
$157$	5.41641i	0.432276i	−0.976363	+	0.216138i	$0.930654\pi$
$158$	0	0
$159$	24.4164	1.93635
$160$	0	0
$161$	11.6738	0.920021
$162$	0	0
$163$	6.85410i	0.536855i	0.963300	+	0.268427i	$0.0865039\pi$
$163$	6.85410i	0.536855i	−0.963300	+	0.268427i	$0.913496\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.29180i	− 0.409491i	−0.978815	−	0.204746i	$-0.934363\pi$
$167$	− 5.29180i	− 0.409491i	0.978815	−	0.204746i	$-0.0656368\pi$
$168$	0	0
$169$	−25.8885	−1.99143
$170$	0	0
$171$	25.8541	1.97711
$172$	0	0
$173$	− 5.47214i	− 0.416039i	−0.978125	−	0.208019i	$-0.933298\pi$
$173$	− 5.47214i	− 0.416039i	0.978125	−	0.208019i	$-0.0667017\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1.38197i	− 0.103875i
$178$	0	0
$179$	−13.6180	−1.01786	−0.508930	−	0.860808i	$-0.669959\pi$
$179$	−13.6180	−1.01786	−0.508930	+	0.860808i	$0.669959\pi$
$180$	0	0
$181$	−6.09017	−0.452679	−0.226339	−	0.974049i	$-0.572676\pi$
$181$	−6.09017	−0.452679	−0.226339	+	0.974049i	$0.572676\pi$
$182$	0	0
$183$	− 0.236068i	− 0.0174506i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 0.618034i	− 0.0451951i
$188$	0	0
$189$	6.38197	0.464220
$190$	0	0
$191$	21.0902	1.52603	0.763016	−	0.646380i	$-0.223718\pi$
$191$	21.0902	1.52603	0.763016	+	0.646380i	$0.223718\pi$
$192$	0	0
$193$	12.9443i	0.931749i	0.884851	+	0.465875i	$0.154260\pi$
$193$	12.9443i	0.931749i	−0.884851	+	0.465875i	$0.845740\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.0902i	1.43137i	0.698426	+	0.715683i	$0.253884\pi$
$197$	20.0902i	1.43137i	−0.698426	+	0.715683i	$0.746116\pi$
$198$	0	0
$199$	−3.09017	−0.219056	−0.109528	−	0.993984i	$-0.534934\pi$
$199$	−3.09017	−0.219056	−0.109528	+	0.993984i	$0.534934\pi$
$200$	0	0
$201$	20.9443	1.47730
$202$	0	0
$203$	− 3.94427i	− 0.276834i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 15.7639i	− 1.09567i
$208$	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
$213$	21.4164i	1.46743i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 8.56231i	− 0.581247i
$218$	0	0
$219$	27.1803	1.83668
$220$	0	0
$221$	−3.85410	−0.259255
$222$	0	0
$223$	− 16.8885i	− 1.13094i	−0.824769	−	0.565470i	$-0.808695\pi$
$223$	− 16.8885i	− 1.13094i	0.824769	−	0.565470i	$-0.191305\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 24.0344i	− 1.59522i	−0.603172	−	0.797611i	$-0.706097\pi$
$227$	− 24.0344i	− 1.59522i	0.603172	−	0.797611i	$-0.293903\pi$
$228$	0	0
$229$	−12.0344	−0.795258	−0.397629	−	0.917546i	$-0.630167\pi$
$229$	−12.0344	−0.795258	−0.397629	+	0.917546i	$0.630167\pi$
$230$	0	0
$231$	7.47214	0.491630
$232$	0	0
$233$	15.5066i	1.01587i	0.861395	+	0.507935i	$0.169591\pi$
$233$	15.5066i	1.01587i	−0.861395	+	0.507935i	$0.830409\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 15.3262i	− 0.995546i
$238$	0	0
$239$	6.38197	0.412815	0.206408	−	0.978466i	$-0.433823\pi$
$239$	6.38197	0.412815	0.206408	+	0.978466i	$0.433823\pi$
$240$	0	0
$241$	21.2705	1.37015	0.685077	−	0.728471i	$-0.259769\pi$
$241$	21.2705	1.37015	0.685077	+	0.728471i	$0.259769\pi$
$242$	0	0
$243$	21.6525i	1.38901i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 41.8328i	− 2.66176i
$248$	0	0
$249$	26.5623	1.68332
$250$	0	0
$251$	27.2705	1.72130	0.860650	−	0.509198i	$-0.170058\pi$
$251$	27.2705	1.72130	0.860650	+	0.509198i	$0.170058\pi$
$252$	0	0
$253$	− 4.09017i	− 0.257147i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.9443i	0.682685i	0.939939	+	0.341342i	$0.110882\pi$
$257$	10.9443i	0.682685i	−0.939939	+	0.341342i	$0.889118\pi$
$258$	0	0
$259$	−29.2148	−1.81532
$260$	0	0
$261$	−5.32624	−0.329686
$262$	0	0
$263$	21.0000i	1.29492i	0.762101	+	0.647458i	$0.224168\pi$
$263$	21.0000i	1.29492i	−0.762101	+	0.647458i	$0.775832\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 18.0902i	− 1.10710i
$268$	0	0
$269$	−30.3262	−1.84902	−0.924512	−	0.381154i	$-0.875527\pi$
$269$	−30.3262	−1.84902	−0.924512	+	0.381154i	$0.875527\pi$
$270$	0	0
$271$	−13.1803	−0.800649	−0.400324	−	0.916374i	$-0.631103\pi$
$271$	−13.1803	−0.800649	−0.400324	+	0.916374i	$0.631103\pi$
$272$	0	0
$273$	− 46.5967i	− 2.82016i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	11.4721i	0.689294i	0.938732	+	0.344647i	$0.112001\pi$
$277$	11.4721i	0.689294i	−0.938732	+	0.344647i	$0.887999\pi$
$278$	0	0
$279$	−11.5623	−0.692217
$280$	0	0
$281$	−25.3607	−1.51289	−0.756446	−	0.654057i	$-0.773066\pi$
$281$	−25.3607	−1.51289	−0.756446	+	0.654057i	$0.773066\pi$
$282$	0	0
$283$	7.38197i	0.438812i	0.975634	+	0.219406i	$0.0704120\pi$
$283$	7.38197i	0.438812i	−0.975634	+	0.219406i	$0.929588\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.56231i	0.505417i
$288$	0	0
$289$	16.6180	0.977531
$290$	0	0
$291$	−4.23607	−0.248323
$292$	0	0
$293$	− 23.8885i	− 1.39558i	−0.716301	−	0.697792i	$-0.754166\pi$
$293$	− 23.8885i	− 1.39558i	0.716301	−	0.697792i	$-0.245834\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.23607i	− 0.129750i
$298$	0	0
$299$	−25.5066	−1.47508
$300$	0	0
$301$	17.1246	0.987046
$302$	0	0
$303$	15.9443i	0.915974i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	33.4508i	1.90914i	0.297985	+	0.954570i	$0.403685\pi$
$307$	33.4508i	1.90914i	−0.297985	+	0.954570i	$0.596315\pi$
$308$	0	0
$309$	−14.0902	−0.801562
$310$	0	0
$311$	19.1803	1.08762	0.543809	−	0.839209i	$-0.316982\pi$
$311$	19.1803	1.08762	0.543809	+	0.839209i	$0.316982\pi$
$312$	0	0
$313$	− 3.23607i	− 0.182913i	−0.995809	−	0.0914567i	$-0.970848\pi$
$313$	− 3.23607i	− 0.182913i	0.995809	−	0.0914567i	$-0.0291523\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 16.6180i	− 0.933362i	−0.884426	−	0.466681i	$-0.845450\pi$
$317$	− 16.6180i	− 0.933362i	0.884426	−	0.466681i	$-0.154550\pi$
$318$	0	0
$319$	−1.38197	−0.0773752
$320$	0	0
$321$	−11.0902	−0.618993
$322$	0	0
$323$	− 4.14590i	− 0.230684i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 8.09017i	− 0.447387i
$328$	0	0
$329$	34.0902	1.87945
$330$	0	0
$331$	19.1803	1.05425	0.527123	−	0.849789i	$-0.323271\pi$
$331$	19.1803	1.05425	0.527123	+	0.849789i	$0.323271\pi$
$332$	0	0
$333$	39.4508i	2.16189i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 1.41641i	− 0.0771567i	−0.999256	−	0.0385783i	$-0.987717\pi$
$337$	− 1.41641i	− 0.0771567i	0.999256	−	0.0385783i	$-0.0122829\pi$
$338$	0	0
$339$	−30.5066	−1.65689
$340$	0	0
$341$	−3.00000	−0.162459
$342$	0	0
$343$	− 16.7082i	− 0.902158i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.5623i	1.10384i	0.833896	+	0.551921i	$0.186105\pi$
$347$	20.5623i	1.10384i	−0.833896	+	0.551921i	$0.813895\pi$
$348$	0	0
$349$	−21.8328	−1.16868	−0.584342	−	0.811508i	$-0.698647\pi$
$349$	−21.8328	−1.16868	−0.584342	+	0.811508i	$0.698647\pi$
$350$	0	0
$351$	−13.9443	−0.744290
$352$	0	0
$353$	21.3607i	1.13691i	0.822713	+	0.568457i	$0.192459\pi$
$353$	21.3607i	1.13691i	−0.822713	+	0.568457i	$0.807541\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 4.61803i	− 0.244412i
$358$	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
$363$	− 2.61803i	− 0.137411i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.14590i	0.373013i	0.982454	+	0.186506i	$0.0597165\pi$
$367$	7.14590i	0.373013i	−0.982454	+	0.186506i	$0.940283\pi$
$368$	0	0
$369$	11.5623	0.601910
$370$	0	0
$371$	26.6180	1.38194
$372$	0	0
$373$	2.81966i	0.145996i	0.997332	+	0.0729982i	$0.0232567\pi$
$373$	2.81966i	0.145996i	−0.997332	+	0.0729982i	$0.976743\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.61803i	0.443851i
$378$	0	0
$379$	−17.7639	−0.912472	−0.456236	−	0.889859i	$-0.650803\pi$
$379$	−17.7639	−0.912472	−0.456236	+	0.889859i	$0.650803\pi$
$380$	0	0
$381$	−1.61803	−0.0828944
$382$	0	0
$383$	− 5.05573i	− 0.258336i	−0.991623	−	0.129168i	$-0.958769\pi$
$383$	− 5.05573i	− 0.258336i	0.991623	−	0.129168i	$-0.0412306\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 23.1246i	− 1.17549i
$388$	0	0
$389$	−5.52786	−0.280274	−0.140137	−	0.990132i	$-0.544754\pi$
$389$	−5.52786	−0.280274	−0.140137	+	0.990132i	$0.544754\pi$
$390$	0	0
$391$	−2.52786	−0.127840
$392$	0	0
$393$	26.4164i	1.33253i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 14.9098i	− 0.748303i	−0.927368	−	0.374151i	$-0.877934\pi$
$397$	− 14.9098i	− 0.748303i	0.927368	−	0.374151i	$-0.122066\pi$
$398$	0	0
$399$	50.1246	2.50937
$400$	0	0
$401$	34.3607	1.71589	0.857945	−	0.513741i	$-0.171741\pi$
$401$	34.3607	1.71589	0.857945	+	0.513741i	$0.171741\pi$
$402$	0	0
$403$	18.7082i	0.931922i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.2361i	0.507383i
$408$	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
$411$	−14.5623	−0.718306
$412$	0	0
$413$	− 1.50658i	− 0.0741338i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	8.61803i	0.422027i
$418$	0	0
$419$	21.1803	1.03473	0.517364	−	0.855766i	$-0.326913\pi$
$419$	21.1803	1.03473	0.517364	+	0.855766i	$0.326913\pi$
$420$	0	0
$421$	−12.2705	−0.598028	−0.299014	−	0.954249i	$-0.596658\pi$
$421$	−12.2705	−0.598028	−0.299014	+	0.954249i	$0.596658\pi$
$422$	0	0
$423$	− 46.0344i	− 2.23827i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 0.257354i	− 0.0124542i
$428$	0	0
$429$	−16.3262	−0.788238
$430$	0	0
$431$	−0.819660	−0.0394816	−0.0197408	−	0.999805i	$-0.506284\pi$
$431$	−0.819660	−0.0394816	−0.0197408	+	0.999805i	$0.506284\pi$
$432$	0	0
$433$	− 18.8885i	− 0.907725i	−0.891072	−	0.453863i	$-0.850046\pi$
$433$	− 18.8885i	− 0.907725i	0.891072	−	0.453863i	$-0.149954\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 27.4377i	− 1.31252i
$438$	0	0
$439$	−0.729490	−0.0348167	−0.0174083	−	0.999848i	$-0.505542\pi$
$439$	−0.729490	−0.0348167	−0.0174083	+	0.999848i	$0.505542\pi$
$440$	0	0
$441$	4.41641	0.210305
$442$	0	0
$443$	36.6525i	1.74141i	0.491804	+	0.870706i	$0.336338\pi$
$443$	36.6525i	1.74141i	−0.491804	+	0.870706i	$0.663662\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 23.4164i	− 1.10756i
$448$	0	0
$449$	15.3262	0.723290	0.361645	−	0.932316i	$-0.382215\pi$
$449$	15.3262	0.723290	0.361645	+	0.932316i	$0.382215\pi$
$450$	0	0
$451$	3.00000	0.141264
$452$	0	0
$453$	− 7.85410i	− 0.369018i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	7.97871i	0.373228i	0.982433	+	0.186614i	$0.0597514\pi$
$457$	7.97871i	0.373228i	−0.982433	+	0.186614i	$0.940249\pi$
$458$	0	0
$459$	−1.38197	−0.0645046
$460$	0	0
$461$	−9.18034	−0.427571	−0.213786	−	0.976881i	$-0.568579\pi$
$461$	−9.18034	−0.427571	−0.213786	+	0.976881i	$0.568579\pi$
$462$	0	0
$463$	− 11.3607i	− 0.527976i	−0.964526	−	0.263988i	$-0.914962\pi$
$463$	− 11.3607i	− 0.527976i	0.964526	−	0.263988i	$-0.0850379\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	37.4721i	1.73400i	0.498305	+	0.867002i	$0.333956\pi$
$467$	37.4721i	1.73400i	−0.498305	+	0.867002i	$0.666044\pi$
$468$	0	0
$469$	22.8328	1.05432
$470$	0	0
$471$	14.1803	0.653396
$472$	0	0
$473$	− 6.00000i	− 0.275880i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 35.9443i	− 1.64578i
$478$	0	0
$479$	28.4164	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$479$	28.4164	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$480$	0	0
$481$	63.8328	2.91053
$482$	0	0
$483$	− 30.5623i	− 1.39063i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 30.4164i	− 1.37830i	−0.724619	−	0.689150i	$-0.757984\pi$
$487$	− 30.4164i	− 1.37830i	0.724619	−	0.689150i	$-0.242016\pi$
$488$	0	0
$489$	17.9443	0.811468
$490$	0	0
$491$	6.81966	0.307767	0.153883	−	0.988089i	$-0.450822\pi$
$491$	6.81966	0.307767	0.153883	+	0.988089i	$0.450822\pi$
$492$	0	0
$493$	0.854102i	0.0384668i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	23.3475i	1.04728i
$498$	0	0
$499$	29.7984	1.33396	0.666979	−	0.745076i	$-0.267587\pi$
$499$	29.7984	1.33396	0.666979	+	0.745076i	$0.267587\pi$
$500$	0	0
$501$	−13.8541	−0.618956
$502$	0	0
$503$	6.65248i	0.296619i	0.988941	+	0.148310i	$0.0473832\pi$
$503$	6.65248i	0.296619i	−0.988941	+	0.148310i	$0.952617\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	67.7771i	3.01009i
$508$	0	0
$509$	−16.3820	−0.726118	−0.363059	−	0.931766i	$-0.618268\pi$
$509$	−16.3820	−0.726118	−0.363059	+	0.931766i	$0.618268\pi$
$510$	0	0
$511$	29.6312	1.31081
$512$	0	0
$513$	− 15.0000i	− 0.662266i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 11.9443i	− 0.525308i
$518$	0	0
$519$	−14.3262	−0.628852
$520$	0	0
$521$	−1.81966	−0.0797208	−0.0398604	−	0.999205i	$-0.512691\pi$
$521$	−1.81966	−0.0797208	−0.0398604	+	0.999205i	$0.512691\pi$
$522$	0	0
$523$	− 22.9443i	− 1.00328i	−0.865076	−	0.501641i	$-0.832730\pi$
$523$	− 22.9443i	− 1.00328i	0.865076	−	0.501641i	$-0.167270\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.85410i	0.0807660i
$528$	0	0
$529$	6.27051	0.272631
$530$	0	0
$531$	−2.03444	−0.0882873
$532$	0	0
$533$	− 18.7082i	− 0.810342i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	35.6525i	1.53852i
$538$	0	0
$539$	1.14590	0.0493573
$540$	0	0
$541$	−42.2705	−1.81735	−0.908676	−	0.417503i	$-0.862905\pi$
$541$	−42.2705	−1.81735	−0.908676	+	0.417503i	$0.862905\pi$
$542$	0	0
$543$	15.9443i	0.684234i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 21.1459i	− 0.904133i	−0.891984	−	0.452067i	$-0.850687\pi$
$547$	− 21.1459i	− 0.904133i	0.891984	−	0.452067i	$-0.149313\pi$
$548$	0	0
$549$	−0.347524	−0.0148320
$550$	0	0
$551$	−9.27051	−0.394937
$552$	0	0
$553$	− 16.7082i	− 0.710505i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.76393i	0.413711i	0.978371	+	0.206856i	$0.0663230\pi$
$557$	9.76393i	0.413711i	−0.978371	+	0.206856i	$0.933677\pi$
$558$	0	0
$559$	−37.4164	−1.58255
$560$	0	0
$561$	−1.61803	−0.0683134
$562$	0	0
$563$	13.0344i	0.549336i	0.961539	+	0.274668i	$0.0885680\pi$
$563$	13.0344i	0.549336i	−0.961539	+	0.274668i	$0.911432\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	16.2918i	0.684191i
$568$	0	0
$569$	−26.3820	−1.10599	−0.552995	−	0.833185i	$-0.686515\pi$
$569$	−26.3820	−1.10599	−0.552995	+	0.833185i	$0.686515\pi$
$570$	0	0
$571$	−36.2705	−1.51787	−0.758937	−	0.651164i	$-0.774281\pi$
$571$	−36.2705	−1.51787	−0.758937	+	0.651164i	$0.774281\pi$
$572$	0	0
$573$	− 55.2148i	− 2.30663i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 15.5623i	− 0.647867i	−0.946080	−	0.323934i	$-0.894995\pi$
$577$	− 15.5623i	− 0.647867i	0.946080	−	0.323934i	$-0.105005\pi$
$578$	0	0
$579$	33.8885	1.40836
$580$	0	0
$581$	28.9574	1.20136
$582$	0	0
$583$	− 9.32624i	− 0.386253i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 4.03444i	− 0.166519i	−0.996528	−	0.0832596i	$-0.973467\pi$
$587$	− 4.03444i	− 0.166519i	0.996528	−	0.0832596i	$-0.0265331\pi$
$588$	0	0
$589$	−20.1246	−0.829220
$590$	0	0
$591$	52.5967	2.16354
$592$	0	0
$593$	9.00000i	0.369586i	0.982777	+	0.184793i	$0.0591614\pi$
$593$	9.00000i	0.369586i	−0.982777	+	0.184793i	$0.940839\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.09017i	0.331109i
$598$	0	0
$599$	0.326238	0.0133297	0.00666486	−	0.999978i	$-0.497878\pi$
$599$	0.326238	0.0133297	0.00666486	+	0.999978i	$0.497878\pi$
$600$	0	0
$601$	−22.2705	−0.908433	−0.454217	−	0.890891i	$-0.650081\pi$
$601$	−22.2705	−0.908433	−0.454217	+	0.890891i	$0.650081\pi$
$602$	0	0
$603$	− 30.8328i	− 1.25561i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.52786i	0.143192i	0.997434	+	0.0715958i	$0.0228092\pi$
$607$	3.52786i	0.143192i	−0.997434	+	0.0715958i	$0.977191\pi$
$608$	0	0
$609$	−10.3262	−0.418440
$610$	0	0
$611$	−74.4853	−3.01335
$612$	0	0
$613$	6.43769i	0.260016i	0.991513	+	0.130008i	$0.0415003\pi$
$613$	6.43769i	0.260016i	−0.991513	+	0.130008i	$0.958500\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.1803i	1.53708i	0.639800	+	0.768541i	$0.279017\pi$
$617$	38.1803i	1.53708i	−0.639800	+	0.768541i	$0.720983\pi$
$618$	0	0
$619$	−22.8885	−0.919968	−0.459984	−	0.887927i	$-0.652145\pi$
$619$	−22.8885	−0.919968	−0.459984	+	0.887927i	$0.652145\pi$
$620$	0	0
$621$	−9.14590	−0.367012
$622$	0	0
$623$	− 19.7214i	− 0.790120i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 17.5623i	− 0.701371i
$628$	0	0
$629$	6.32624	0.252244
$630$	0	0
$631$	−36.2705	−1.44391	−0.721953	−	0.691942i	$-0.756755\pi$
$631$	−36.2705	−1.44391	−0.721953	+	0.691942i	$0.756755\pi$
$632$	0	0
$633$	44.5066i	1.76898i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 7.14590i	− 0.283131i
$638$	0	0
$639$	31.5279	1.24722
$640$	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
$643$	− 10.5836i	− 0.417376i	−0.977982	−	0.208688i	$-0.933081\pi$
$643$	− 10.5836i	− 0.417376i	0.977982	−	0.208688i	$-0.0669193\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.3475i	0.485431i	0.970097	+	0.242716i	$0.0780382\pi$
$647$	12.3475i	0.485431i	−0.970097	+	0.242716i	$0.921962\pi$
$648$	0	0
$649$	−0.527864	−0.0207205
$650$	0	0
$651$	−22.4164	−0.878568
$652$	0	0
$653$	7.74265i	0.302993i	0.988458	+	0.151497i	$0.0484093\pi$
$653$	7.74265i	0.302993i	−0.988458	+	0.151497i	$0.951591\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 40.0132i	− 1.56106i
$658$	0	0
$659$	9.27051	0.361128	0.180564	−	0.983563i	$-0.442208\pi$
$659$	9.27051	0.361128	0.180564	+	0.983563i	$0.442208\pi$
$660$	0	0
$661$	−49.1803	−1.91289	−0.956447	−	0.291907i	$-0.905710\pi$
$661$	−49.1803	−1.91289	−0.956447	+	0.291907i	$0.905710\pi$
$662$	0	0
$663$	10.0902i	0.391870i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	5.65248i	0.218865i
$668$	0	0
$669$	−44.2148	−1.70944
$670$	0	0
$671$	−0.0901699	−0.00348097
$672$	0	0
$673$	2.41641i	0.0931457i	0.998915	+	0.0465728i	$0.0148300\pi$
$673$	2.41641i	0.0931457i	−0.998915	+	0.0465728i	$0.985170\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 28.6525i	− 1.10120i	−0.834768	−	0.550602i	$-0.814398\pi$
$677$	− 28.6525i	− 1.10120i	0.834768	−	0.550602i	$-0.185602\pi$
$678$	0	0
$679$	−4.61803	−0.177224
$680$	0	0
$681$	−62.9230	−2.41121
$682$	0	0
$683$	43.3607i	1.65915i	0.558395	+	0.829575i	$0.311417\pi$
$683$	43.3607i	1.65915i	−0.558395	+	0.829575i	$0.688583\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	31.5066i	1.20205i
$688$	0	0
$689$	−58.1591	−2.21568
$690$	0	0
$691$	−30.0902	−1.14468	−0.572342	−	0.820015i	$-0.693965\pi$
$691$	−30.0902	−1.14468	−0.572342	+	0.820015i	$0.693965\pi$
$692$	0	0
$693$	− 11.0000i	− 0.417855i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.85410i	− 0.0702291i
$698$	0	0
$699$	40.5967	1.53551
$700$	0	0
$701$	−0.360680	−0.0136227	−0.00681134	−	0.999977i	$-0.502168\pi$
$701$	−0.360680	−0.0136227	−0.00681134	+	0.999977i	$0.502168\pi$
$702$	0	0
$703$	68.6656i	2.58977i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	17.3820i	0.653716i
$708$	0	0
$709$	41.3050	1.55124	0.775620	−	0.631200i	$-0.217437\pi$
$709$	41.3050	1.55124	0.775620	+	0.631200i	$0.217437\pi$
$710$	0	0
$711$	−22.5623	−0.846153
$712$	0	0
$713$	12.2705i	0.459534i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 16.7082i	− 0.623979i
$718$	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
$721$	−15.3607	−0.572062
$722$	0	0
$723$	− 55.6869i	− 2.07102i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 41.2705i	− 1.53064i	−0.643651	−	0.765319i	$-0.722581\pi$
$727$	− 41.2705i	− 1.53064i	0.643651	−	0.765319i	$-0.277419\pi$
$728$	0	0
$729$	39.5623	1.46527
$730$	0	0
$731$	−3.70820	−0.137153
$732$	0	0
$733$	45.8328i	1.69287i	0.532489	+	0.846437i	$0.321257\pi$
$733$	45.8328i	1.69287i	−0.532489	+	0.846437i	$0.678743\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 8.00000i	− 0.294684i
$738$	0	0
$739$	−29.1459	−1.07215	−0.536075	−	0.844171i	$-0.680093\pi$
$739$	−29.1459	−1.07215	−0.536075	+	0.844171i	$0.680093\pi$
$740$	0	0
$741$	−109.520	−4.02331
$742$	0	0
$743$	− 18.6738i	− 0.685074i	−0.939504	−	0.342537i	$-0.888714\pi$
$743$	− 18.6738i	− 0.685074i	0.939504	−	0.342537i	$-0.111286\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 39.1033i	− 1.43072i
$748$	0	0
$749$	−12.0902	−0.441765
$750$	0	0
$751$	−11.2705	−0.411267	−0.205633	−	0.978629i	$-0.565925\pi$
$751$	−11.2705	−0.411267	−0.205633	+	0.978629i	$0.565925\pi$
$752$	0	0
$753$	− 71.3951i	− 2.60178i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	21.4721i	0.780418i	0.920726	+	0.390209i	$0.127597\pi$
$757$	21.4721i	0.780418i	−0.920726	+	0.390209i	$0.872403\pi$
$758$	0	0
$759$	−10.7082	−0.388683
$760$	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
$763$	− 8.81966i	− 0.319293i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.29180i	0.118860i
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	28.6525	1.03189
$772$	0	0
$773$	33.7984i	1.21564i	0.794074	+	0.607822i	$0.207956\pi$
$773$	33.7984i	1.21564i	−0.794074	+	0.607822i	$0.792044\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	76.4853i	2.74389i
$778$	0	0
$779$	20.1246	0.721039
$780$	0	0
$781$	8.18034	0.292716
$782$	0	0
$783$	3.09017i	0.110434i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	21.2918i	0.758971i	0.925198	+	0.379485i	$0.123899\pi$
$787$	21.2918i	0.758971i	−0.925198	+	0.379485i	$0.876101\pi$
$788$	0	0
$789$	54.9787	1.95729
$790$	0	0
$791$	−33.2574	−1.18250
$792$	0	0
$793$	0.562306i	0.0199681i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.20163i	− 0.113407i	−0.998391	−	0.0567037i	$-0.981941\pi$
$797$	− 3.20163i	− 0.113407i	0.998391	−	0.0567037i	$-0.0180590\pi$
$798$	0	0
$799$	−7.38197	−0.261155
$800$	0	0
$801$	−26.6312	−0.940967
$802$	0	0
$803$	− 10.3820i	− 0.366372i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	79.3951i	2.79484i
$808$	0	0
$809$	16.5836	0.583048	0.291524	−	0.956564i	$-0.405838\pi$
$809$	16.5836	0.583048	0.291524	+	0.956564i	$0.405838\pi$
$810$	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
$813$	34.5066i	1.21020i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 40.2492i	− 1.40814i
$818$	0	0
$819$	−68.5967	−2.39696
$820$	0	0
$821$	4.36068	0.152189	0.0760944	−	0.997101i	$-0.475755\pi$
$821$	4.36068	0.152189	0.0760944	+	0.997101i	$0.475755\pi$
$822$	0	0
$823$	− 27.4164i	− 0.955676i	−0.878448	−	0.477838i	$-0.841421\pi$
$823$	− 27.4164i	− 0.955676i	0.878448	−	0.477838i	$-0.158579\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.0689i	1.11514i	0.830128	+	0.557572i	$0.188267\pi$
$827$	32.0689i	1.11514i	−0.830128	+	0.557572i	$0.811733\pi$
$828$	0	0
$829$	36.1033	1.25392	0.626960	−	0.779051i	$-0.284299\pi$
$829$	36.1033	1.25392	0.626960	+	0.779051i	$0.284299\pi$
$830$	0	0
$831$	30.0344	1.04188
$832$	0	0
$833$	− 0.708204i	− 0.0245378i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	6.70820i	0.231869i
$838$	0	0
$839$	−48.3394	−1.66886	−0.834431	−	0.551113i	$-0.814203\pi$
$839$	−48.3394	−1.66886	−0.834431	+	0.551113i	$0.814203\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	0	0
$843$	66.3951i	2.28677i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.85410i	− 0.0980681i
$848$	0	0
$849$	19.3262	0.663275
$850$	0	0
$851$	41.8673	1.43519
$852$	0	0
$853$	49.8541i	1.70697i	0.521116	+	0.853486i	$0.325516\pi$
$853$	49.8541i	1.70697i	−0.521116	+	0.853486i	$0.674484\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	53.1803i	1.81661i	0.418313	+	0.908303i	$0.362621\pi$
$857$	53.1803i	1.81661i	−0.418313	+	0.908303i	$0.637379\pi$
$858$	0	0
$859$	16.1803	0.552066	0.276033	−	0.961148i	$-0.410980\pi$
$859$	16.1803	0.552066	0.276033	+	0.961148i	$0.410980\pi$
$860$	0	0
$861$	22.4164	0.763949
$862$	0	0
$863$	0.596748i	0.0203135i	0.999948	+	0.0101568i	$0.00323305\pi$
$863$	0.596748i	0.0203135i	−0.999948	+	0.0101568i	$0.996767\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 43.5066i	− 1.47756i
$868$	0	0
$869$	−5.85410	−0.198587
$870$	0	0
$871$	−49.8885	−1.69041
$872$	0	0
$873$	6.23607i	0.211059i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 4.58359i	− 0.154777i	−0.997001	−	0.0773885i	$-0.975342\pi$
$877$	− 4.58359i	− 0.154777i	0.997001	−	0.0773885i	$-0.0246582\pi$
$878$	0	0
$879$	−62.5410	−2.10946
$880$	0	0
$881$	10.0902	0.339946	0.169973	−	0.985449i	$-0.445632\pi$
$881$	10.0902	0.339946	0.169973	+	0.985449i	$0.445632\pi$
$882$	0	0
$883$	31.6525i	1.06519i	0.846370	+	0.532595i	$0.178783\pi$
$883$	31.6525i	1.06519i	−0.846370	+	0.532595i	$0.821217\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	28.7771i	0.966240i	0.875554	+	0.483120i	$0.160497\pi$
$887$	28.7771i	0.966240i	−0.875554	+	0.483120i	$0.839503\pi$
$888$	0	0
$889$	−1.76393	−0.0591604
$890$	0	0
$891$	5.70820	0.191232
$892$	0	0
$893$	− 80.1246i	− 2.68127i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	66.7771i	2.22962i
$898$	0	0
$899$	4.14590	0.138273
$900$	0	0
$901$	−5.76393	−0.192024
$902$	0	0
$903$	− 44.8328i	− 1.49194i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	14.8328i	0.492516i	0.969204	+	0.246258i	$0.0792010\pi$
$907$	14.8328i	0.492516i	−0.969204	+	0.246258i	$0.920799\pi$
$908$	0	0
$909$	23.4721	0.778522
$910$	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
$913$	− 10.1459i	− 0.335780i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	28.7984i	0.951006i
$918$	0	0
$919$	23.4164	0.772436	0.386218	−	0.922408i	$-0.373781\pi$
$919$	23.4164	0.772436	0.386218	+	0.922408i	$0.373781\pi$
$920$	0	0
$921$	87.5755	2.88571
$922$	0	0
$923$	− 51.0132i	− 1.67912i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	20.7426i	0.681278i
$928$	0	0
$929$	−54.5967	−1.79126	−0.895631	−	0.444799i	$-0.853275\pi$
$929$	−54.5967	−1.79126	−0.895631	+	0.444799i	$0.853275\pi$
$930$	0	0
$931$	7.68692	0.251929
$932$	0	0
$933$	− 50.2148i	− 1.64396i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.8328i	1.26861i	0.773082	+	0.634306i	$0.218714\pi$
$937$	38.8328i	1.26861i	−0.773082	+	0.634306i	$0.781286\pi$
$938$	0	0
$939$	−8.47214	−0.276478
$940$	0	0
$941$	41.7214	1.36008	0.680039	−	0.733176i	$-0.261963\pi$
$941$	41.7214	1.36008	0.680039	+	0.733176i	$0.261963\pi$
$942$	0	0
$943$	− 12.2705i	− 0.399583i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 39.3607i	− 1.27905i	−0.768770	−	0.639525i	$-0.779131\pi$
$947$	− 39.3607i	− 1.27905i	0.768770	−	0.639525i	$-0.220869\pi$
$948$	0	0
$949$	−64.7426	−2.10164
$950$	0	0
$951$	−43.5066	−1.41080
$952$	0	0
$953$	8.47214i	0.274439i	0.990541	+	0.137220i	$0.0438166\pi$
$953$	8.47214i	0.274439i	−0.990541	+	0.137220i	$0.956183\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.61803i	0.116954i
$958$	0	0
$959$	−15.8754	−0.512643
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	16.3262i	0.526106i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	37.2705i	1.19854i	0.800547	+	0.599269i	$0.204542\pi$
$967$	37.2705i	1.19854i	−0.800547	+	0.599269i	$0.795458\pi$
$968$	0	0
$969$	−10.8541	−0.348684
$970$	0	0
$971$	−23.9098	−0.767303	−0.383651	−	0.923478i	$-0.625334\pi$
$971$	−23.9098	−0.767303	−0.383651	+	0.923478i	$0.625334\pi$
$972$	0	0
$973$	9.39512i	0.301194i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 57.0689i	− 1.82580i	−0.408188	−	0.912898i	$-0.633839\pi$
$977$	− 57.0689i	− 1.82580i	0.408188	−	0.912898i	$-0.366161\pi$
$978$	0	0
$979$	−6.90983	−0.220839
$980$	0	0
$981$	−11.9098	−0.380252
$982$	0	0
$983$	1.52786i	0.0487313i	0.999703	+	0.0243656i	$0.00775659\pi$
$983$	1.52786i	0.0487313i	−0.999703	+	0.0243656i	$0.992243\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 89.2492i	− 2.84083i
$988$	0	0
$989$	−24.5410	−0.780359
$990$	0	0
$991$	−21.2705	−0.675680	−0.337840	−	0.941204i	$-0.609696\pi$
$991$	−21.2705	−0.675680	−0.337840	+	0.941204i	$0.609696\pi$
$992$	0	0
$993$	− 50.2148i	− 1.59352i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 32.1459i	− 1.01807i	−0.860746	−	0.509035i	$-0.830002\pi$
$997$	− 32.1459i	− 1.01807i	0.860746	−	0.509035i	$-0.169998\pi$
$998$	0	0
$999$	22.8885	0.724161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.x.4049.1		4
4.3	odd	2		275.2.b.e.199.1		4
5.2	odd	4		4400.2.a.bg.1.1		2
5.3	odd	4		4400.2.a.bv.1.2		2
5.4	even	2	inner	4400.2.b.x.4049.4		4
12.11	even	2		2475.2.c.p.199.4		4
20.3	even	4		275.2.a.d.1.1	✓	2
20.7	even	4		275.2.a.g.1.2	yes	2
20.19	odd	2		275.2.b.e.199.4		4
60.23	odd	4		2475.2.a.s.1.2		2
60.47	odd	4		2475.2.a.n.1.1		2
60.59	even	2		2475.2.c.p.199.1		4
220.43	odd	4		3025.2.a.m.1.2		2
220.87	odd	4		3025.2.a.i.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.d.1.1	✓	2	20.3	even	4
275.2.a.g.1.2	yes	2	20.7	even	4
275.2.b.e.199.1		4	4.3	odd	2
275.2.b.e.199.4		4	20.19	odd	2
2475.2.a.n.1.1		2	60.47	odd	4
2475.2.a.s.1.2		2	60.23	odd	4
2475.2.c.p.199.1		4	60.59	even	2
2475.2.c.p.199.4		4	12.11	even	2
3025.2.a.i.1.1		2	220.87	odd	4
3025.2.a.m.1.2		2	220.43	odd	4
4400.2.a.bg.1.1		2	5.2	odd	4
4400.2.a.bv.1.2		2	5.3	odd	4
4400.2.b.x.4049.1		4	1.1	even	1	trivial
4400.2.b.x.4049.4		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.61803i q^{3} -2.85410i q^{7} -3.85410 q^{9} -1.00000 q^{11} +6.23607i q^{13} +0.618034i q^{17} -6.70820 q^{19} -7.47214 q^{21} +4.09017i q^{23} +2.23607i q^{27} +1.38197 q^{29} +3.00000 q^{31} +2.61803i q^{33} -10.2361i q^{37} +16.3262 q^{39} -3.00000 q^{41} +6.00000i q^{43} +11.9443i q^{47} -1.14590 q^{49} +1.61803 q^{51} +9.32624i q^{53} +17.5623i q^{57} +0.527864 q^{59} +0.0901699 q^{61} +11.0000i q^{63} +8.00000i q^{67} +10.7082 q^{69} -8.18034 q^{71} +10.3820i q^{73} +2.85410i q^{77} +5.85410 q^{79} -5.70820 q^{81} +10.1459i q^{83} -3.61803i q^{87} +6.90983 q^{89} +17.7984 q^{91} -7.85410i q^{93} -1.61803i q^{97} +3.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{9} - 4 q^{11} - 12 q^{21} + 10 q^{29} + 12 q^{31} + 34 q^{39} - 12 q^{41} - 18 q^{49} + 2 q^{51} + 20 q^{59} - 22 q^{61} + 16 q^{69} + 12 q^{71} + 10 q^{79} + 4 q^{81} + 50 q^{89} + 22 q^{91}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q - 2 * q^9 - 4 * q^11 - 12 * q^21 + 10 * q^29 + 12 * q^31 + 34 * q^39 - 12 * q^41 - 18 * q^49 + 2 * q^51 + 20 * q^59 - 22 * q^61 + 16 * q^69 + 12 * q^71 + 10 * q^79 + 4 * q^81 + 50 * q^89 + 22 * q^91 + 2 * q^99

Introduction

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