LMFDB - Embedded newform 4000.2.d.c.2001.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4000,2,Mod(2001,4000)] 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("4000.2001"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4000, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4000 = 2^{5} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4000.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [40,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$31.9401608085$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 1000)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2001.16
Character	$\chi$	$=$	4000.2001
Dual form			4000.2.d.c.2001.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.83526i q^{3} +1.31564 q^{7} -0.368171 q^{9} +6.60274i q^{11} -2.96705i q^{13} -2.69502 q^{17} -4.97013i q^{19} +2.41454i q^{21} +3.73739 q^{23} +4.83008i q^{27} +5.52846i q^{29} -8.36121 q^{31} -12.1177 q^{33} +7.19517i q^{37} +5.44530 q^{39} -3.77169 q^{41} +3.07951i q^{43} +8.77645 q^{47} -5.26908 q^{49} -4.94606i q^{51} +0.0464228i q^{53} +9.12147 q^{57} +1.02084i q^{59} -5.23191i q^{61} -0.484382 q^{63} +10.8187i q^{67} +6.85908i q^{69} -9.35643 q^{71} +12.4143 q^{73} +8.68684i q^{77} -1.43842 q^{79} -9.96896 q^{81} +13.0280i q^{83} -10.1462 q^{87} +3.94185 q^{89} -3.90357i q^{91} -15.3450i q^{93} +1.00691 q^{97} -2.43094i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 24 q^{9} - 48 q^{31} - 8 q^{39} + 44 q^{41} + 12 q^{49} - 96 q^{71} - 96 q^{79} - 56 q^{81} - 44 q^{89}+O(q^{100}) $ 40 * q - 24 * q^9 - 48 * q^31 - 8 * q^39 + 44 * q^41 + 12 * q^49 - 96 * q^71 - 96 * q^79 - 56 * q^81 - 44 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times$.

$n$	$1377$	$2501$	$2751$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.83526i	1.05959i	0.848127	+	0.529793i	$0.177730\pi$
$3$	1.83526i	1.05959i	−0.848127	+	0.529793i	$0.822270\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.31564	0.497266	0.248633	−	0.968598i	$-0.420019\pi$
$7$	1.31564	0.497266	0.248633	+	0.968598i	$0.420019\pi$
$8$	0	0
$9$	−0.368171	−0.122724
$10$	0	0
$11$	6.60274i	1.99080i	0.0958055	+	0.995400i	$0.469457\pi$
$11$	6.60274i	1.99080i	−0.0958055	+	0.995400i	$0.530543\pi$
$12$	0	0
$13$	− 2.96705i	− 0.822911i	−0.911430	−	0.411456i	$-0.865021\pi$
$13$	− 2.96705i	− 0.822911i	0.911430	−	0.411456i	$-0.134979\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.69502	−0.653639	−0.326819	−	0.945087i	$-0.605977\pi$
$17$	−2.69502	−0.653639	−0.326819	+	0.945087i	$0.605977\pi$
$18$	0	0
$19$	− 4.97013i	− 1.14023i	−0.821566	−	0.570113i	$-0.806899\pi$
$19$	− 4.97013i	− 1.14023i	0.821566	−	0.570113i	$-0.193101\pi$
$20$	0	0
$21$	2.41454i	0.526897i
$22$	0	0
$23$	3.73739	0.779301	0.389650	−	0.920963i	$-0.372596\pi$
$23$	3.73739	0.779301	0.389650	+	0.920963i	$0.372596\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.83008i	0.929550i
$28$	0	0
$29$	5.52846i	1.02661i	0.858206	+	0.513305i	$0.171579\pi$
$29$	5.52846i	1.02661i	−0.858206	+	0.513305i	$0.828421\pi$
$30$	0	0
$31$	−8.36121	−1.50172	−0.750859	−	0.660462i	$-0.770360\pi$
$31$	−8.36121	−1.50172	−0.750859	+	0.660462i	$0.770360\pi$
$32$	0	0
$33$	−12.1177	−2.10943
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.19517i	1.18288i	0.806349	+	0.591439i	$0.201440\pi$
$37$	7.19517i	1.18288i	−0.806349	+	0.591439i	$0.798560\pi$
$38$	0	0
$39$	5.44530	0.871945
$40$	0	0
$41$	−3.77169	−0.589040	−0.294520	−	0.955645i	$-0.595160\pi$
$41$	−3.77169	−0.589040	−0.294520	+	0.955645i	$0.595160\pi$
$42$	0	0
$43$	3.07951i	0.469621i	0.972041	+	0.234810i	$0.0754469\pi$
$43$	3.07951i	0.469621i	−0.972041	+	0.234810i	$0.924553\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.77645	1.28018	0.640088	−	0.768301i	$-0.278898\pi$
$47$	8.77645	1.28018	0.640088	+	0.768301i	$0.278898\pi$
$48$	0	0
$49$	−5.26908	−0.752726
$50$	0	0
$51$	− 4.94606i	− 0.692587i
$52$	0	0
$53$	0.0464228i	0.00637666i	0.999995	+	0.00318833i	$0.00101488\pi$
$53$	0.0464228i	0.00637666i	−0.999995	+	0.00318833i	$0.998985\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	9.12147	1.20817
$58$	0	0
$59$	1.02084i	0.132902i	0.997790	+	0.0664510i	$0.0211676\pi$
$59$	1.02084i	0.132902i	−0.997790	+	0.0664510i	$0.978832\pi$
$60$	0	0
$61$	− 5.23191i	− 0.669877i	−0.942240	−	0.334938i	$-0.891284\pi$
$61$	− 5.23191i	− 0.669877i	0.942240	−	0.334938i	$-0.108716\pi$
$62$	0	0
$63$	−0.484382	−0.0610264
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.8187i	1.32172i	0.750511	+	0.660858i	$0.229807\pi$
$67$	10.8187i	1.32172i	−0.750511	+	0.660858i	$0.770193\pi$
$68$	0	0
$69$	6.85908i	0.825737i
$70$	0	0
$71$	−9.35643	−1.11040	−0.555202	−	0.831715i	$-0.687359\pi$
$71$	−9.35643	−1.11040	−0.555202	+	0.831715i	$0.687359\pi$
$72$	0	0
$73$	12.4143	1.45298	0.726490	−	0.687177i	$-0.241150\pi$
$73$	12.4143	1.45298	0.726490	+	0.687177i	$0.241150\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8.68684i	0.989958i
$78$	0	0
$79$	−1.43842	−0.161835	−0.0809174	−	0.996721i	$-0.525785\pi$
$79$	−1.43842	−0.161835	−0.0809174	+	0.996721i	$0.525785\pi$
$80$	0	0
$81$	−9.96896	−1.10766
$82$	0	0
$83$	13.0280i	1.43001i	0.699120	+	0.715005i	$0.253575\pi$
$83$	13.0280i	1.43001i	−0.699120	+	0.715005i	$0.746425\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−10.1462	−1.08778
$88$	0	0
$89$	3.94185	0.417835	0.208917	−	0.977933i	$-0.433006\pi$
$89$	3.94185	0.417835	0.208917	+	0.977933i	$0.433006\pi$
$90$	0	0
$91$	− 3.90357i	− 0.409206i
$92$	0	0
$93$	− 15.3450i	− 1.59120i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.00691	0.102236	0.0511181	−	0.998693i	$-0.483722\pi$
$97$	1.00691	0.102236	0.0511181	+	0.998693i	$0.483722\pi$
$98$	0	0
$99$	− 2.43094i	− 0.244318i
$100$	0	0
$101$	2.50175i	0.248933i	0.992224	+	0.124466i	$0.0397219\pi$
$101$	2.50175i	0.248933i	−0.992224	+	0.124466i	$0.960278\pi$
$102$	0	0
$103$	5.34336	0.526497	0.263248	−	0.964728i	$-0.415206\pi$
$103$	5.34336	0.526497	0.263248	+	0.964728i	$0.415206\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.971619i	0.0939299i	0.998897	+	0.0469650i	$0.0149549\pi$
$107$	0.971619i	0.0939299i	−0.998897	+	0.0469650i	$0.985045\pi$
$108$	0	0
$109$	− 18.8454i	− 1.80506i	−0.430626	−	0.902531i	$-0.641707\pi$
$109$	− 18.8454i	− 1.80506i	0.430626	−	0.902531i	$-0.358293\pi$
$110$	0	0
$111$	−13.2050	−1.25336
$112$	0	0
$113$	−12.3812	−1.16472	−0.582361	−	0.812930i	$-0.697871\pi$
$113$	−12.3812	−1.16472	−0.582361	+	0.812930i	$0.697871\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.09238i	0.100991i
$118$	0	0
$119$	−3.54568	−0.325032
$120$	0	0
$121$	−32.5961	−2.96329
$122$	0	0
$123$	− 6.92203i	− 0.624139i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.8083	−1.04782	−0.523909	−	0.851774i	$-0.675527\pi$
$127$	−11.8083	−1.04782	−0.523909	+	0.851774i	$0.675527\pi$
$128$	0	0
$129$	−5.65169	−0.497604
$130$	0	0
$131$	12.6735i	1.10729i	0.832752	+	0.553646i	$0.186764\pi$
$131$	12.6735i	1.10729i	−0.832752	+	0.553646i	$0.813236\pi$
$132$	0	0
$133$	− 6.53892i	− 0.566996i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.33562	0.370417	0.185209	−	0.982699i	$-0.440704\pi$
$137$	4.33562	0.370417	0.185209	+	0.982699i	$0.440704\pi$
$138$	0	0
$139$	− 3.48791i	− 0.295841i	−0.988999	−	0.147920i	$-0.952742\pi$
$139$	− 3.48791i	− 0.295841i	0.988999	−	0.147920i	$-0.0472579\pi$
$140$	0	0
$141$	16.1070i	1.35646i
$142$	0	0
$143$	19.5906	1.63825
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 9.67013i	− 0.797579i
$148$	0	0
$149$	9.11838i	0.747007i	0.927629	+	0.373503i	$0.121844\pi$
$149$	9.11838i	0.747007i	−0.927629	+	0.373503i	$0.878156\pi$
$150$	0	0
$151$	10.5180	0.855944	0.427972	−	0.903792i	$-0.359228\pi$
$151$	10.5180	0.855944	0.427972	+	0.903792i	$0.359228\pi$
$152$	0	0
$153$	0.992229	0.0802170
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.59017i	− 0.446144i	−0.974802	−	0.223072i	$-0.928392\pi$
$157$	− 5.59017i	− 0.446144i	0.974802	−	0.223072i	$-0.0716085\pi$
$158$	0	0
$159$	−0.0851977	−0.00675662
$160$	0	0
$161$	4.91708	0.387520
$162$	0	0
$163$	5.48367i	0.429514i	0.976667	+	0.214757i	$0.0688960\pi$
$163$	5.48367i	0.429514i	−0.976667	+	0.214757i	$0.931104\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.4503	1.50511	0.752556	−	0.658528i	$-0.228821\pi$
$167$	19.4503	1.50511	0.752556	+	0.658528i	$0.228821\pi$
$168$	0	0
$169$	4.19663	0.322817
$170$	0	0
$171$	1.82986i	0.139933i
$172$	0	0
$173$	11.9677i	0.909885i	0.890521	+	0.454942i	$0.150340\pi$
$173$	11.9677i	0.909885i	−0.890521	+	0.454942i	$0.849660\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.87350	−0.140821
$178$	0	0
$179$	11.0142i	0.823244i	0.911355	+	0.411622i	$0.135038\pi$
$179$	11.0142i	0.823244i	−0.911355	+	0.411622i	$0.864962\pi$
$180$	0	0
$181$	18.6306i	1.38480i	0.721511	+	0.692402i	$0.243448\pi$
$181$	18.6306i	1.38480i	−0.721511	+	0.692402i	$0.756552\pi$
$182$	0	0
$183$	9.60190	0.709793
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 17.7945i	− 1.30126i
$188$	0	0
$189$	6.35466i	0.462234i
$190$	0	0
$191$	−10.6266	−0.768912	−0.384456	−	0.923143i	$-0.625611\pi$
$191$	−10.6266	−0.768912	−0.384456	+	0.923143i	$0.625611\pi$
$192$	0	0
$193$	−17.9777	−1.29406	−0.647031	−	0.762464i	$-0.723989\pi$
$193$	−17.9777	−1.29406	−0.647031	+	0.762464i	$0.723989\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 23.6813i	− 1.68722i	−0.536956	−	0.843610i	$-0.680426\pi$
$197$	− 23.6813i	− 1.68722i	0.536956	−	0.843610i	$-0.319574\pi$
$198$	0	0
$199$	−17.9303	−1.27104	−0.635522	−	0.772082i	$-0.719215\pi$
$199$	−17.9303	−1.27104	−0.635522	+	0.772082i	$0.719215\pi$
$200$	0	0
$201$	−19.8551	−1.40047
$202$	0	0
$203$	7.27348i	0.510498i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.37600	−0.0956387
$208$	0	0
$209$	32.8165	2.26996
$210$	0	0
$211$	15.0943i	1.03913i	0.854430	+	0.519566i	$0.173906\pi$
$211$	15.0943i	1.03913i	−0.854430	+	0.519566i	$0.826094\pi$
$212$	0	0
$213$	− 17.1715i	− 1.17657i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−11.0004	−0.746754
$218$	0	0
$219$	22.7834i	1.53956i
$220$	0	0
$221$	7.99626i	0.537886i
$222$	0	0
$223$	−21.0583	−1.41017	−0.705083	−	0.709125i	$-0.749090\pi$
$223$	−21.0583	−1.41017	−0.705083	+	0.709125i	$0.749090\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.94442i	0.460917i	0.973082	+	0.230459i	$0.0740226\pi$
$227$	6.94442i	0.460917i	−0.973082	+	0.230459i	$0.925977\pi$
$228$	0	0
$229$	− 19.8805i	− 1.31374i	−0.754003	−	0.656871i	$-0.771879\pi$
$229$	− 19.8805i	− 1.31374i	0.754003	−	0.656871i	$-0.228121\pi$
$230$	0	0
$231$	−15.9426	−1.04895
$232$	0	0
$233$	9.43816	0.618315	0.309157	−	0.951011i	$-0.399953\pi$
$233$	9.43816	0.618315	0.309157	+	0.951011i	$0.399953\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 2.63987i	− 0.171478i
$238$	0	0
$239$	21.3865	1.38338	0.691688	−	0.722197i	$-0.256868\pi$
$239$	21.3865	1.38338	0.691688	+	0.722197i	$0.256868\pi$
$240$	0	0
$241$	11.4117	0.735094	0.367547	−	0.930005i	$-0.380198\pi$
$241$	11.4117	0.735094	0.367547	+	0.930005i	$0.380198\pi$
$242$	0	0
$243$	− 3.80536i	− 0.244114i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−14.7466	−0.938305
$248$	0	0
$249$	−23.9097	−1.51522
$250$	0	0
$251$	− 2.76130i	− 0.174292i	−0.996196	−	0.0871459i	$-0.972225\pi$
$251$	− 2.76130i	− 0.174292i	0.996196	−	0.0871459i	$-0.0277746\pi$
$252$	0	0
$253$	24.6770i	1.55143i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.1685	−0.759052	−0.379526	−	0.925181i	$-0.623913\pi$
$257$	−12.1685	−0.759052	−0.379526	+	0.925181i	$0.623913\pi$
$258$	0	0
$259$	9.46627i	0.588206i
$260$	0	0
$261$	− 2.03542i	− 0.125989i
$262$	0	0
$263$	−22.4088	−1.38179	−0.690894	−	0.722956i	$-0.742783\pi$
$263$	−22.4088	−1.38179	−0.690894	+	0.722956i	$0.742783\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.23430i	0.442732i
$268$	0	0
$269$	− 29.7546i	− 1.81417i	−0.420946	−	0.907086i	$-0.638302\pi$
$269$	− 29.7546i	− 1.81417i	0.420946	−	0.907086i	$-0.361698\pi$
$270$	0	0
$271$	−15.1173	−0.918308	−0.459154	−	0.888357i	$-0.651847\pi$
$271$	−15.1173	−0.918308	−0.459154	+	0.888357i	$0.651847\pi$
$272$	0	0
$273$	7.16407	0.433589
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.51304i	0.571583i	0.958292	+	0.285792i	$0.0922565\pi$
$277$	9.51304i	0.571583i	−0.958292	+	0.285792i	$0.907743\pi$
$278$	0	0
$279$	3.07836	0.184296
$280$	0	0
$281$	−17.5890	−1.04927	−0.524637	−	0.851326i	$-0.675799\pi$
$281$	−17.5890	−1.04927	−0.524637	+	0.851326i	$0.675799\pi$
$282$	0	0
$283$	− 15.3245i	− 0.910944i	−0.890250	−	0.455472i	$-0.849471\pi$
$283$	− 15.3245i	− 0.910944i	0.890250	−	0.455472i	$-0.150529\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.96220	−0.292909
$288$	0	0
$289$	−9.73686	−0.572756
$290$	0	0
$291$	1.84794i	0.108328i
$292$	0	0
$293$	3.18687i	0.186179i	0.995658	+	0.0930894i	$0.0296742\pi$
$293$	3.18687i	0.186179i	−0.995658	+	0.0930894i	$0.970326\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−31.8918	−1.85055
$298$	0	0
$299$	− 11.0890i	− 0.641295i
$300$	0	0
$301$	4.05153i	0.233526i
$302$	0	0
$303$	−4.59135	−0.263766
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 28.0569i	− 1.60129i	−0.599138	−	0.800645i	$-0.704490\pi$
$307$	− 28.0569i	− 1.60129i	0.599138	−	0.800645i	$-0.295510\pi$
$308$	0	0
$309$	9.80644i	0.557869i
$310$	0	0
$311$	−27.7970	−1.57622	−0.788112	−	0.615532i	$-0.788941\pi$
$311$	−27.7970	−1.57622	−0.788112	+	0.615532i	$0.788941\pi$
$312$	0	0
$313$	21.6443	1.22341	0.611703	−	0.791088i	$-0.290485\pi$
$313$	21.6443	1.22341	0.611703	+	0.791088i	$0.290485\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 16.0032i	− 0.898827i	−0.893324	−	0.449414i	$-0.851633\pi$
$317$	− 16.0032i	− 0.898827i	0.893324	−	0.449414i	$-0.148367\pi$
$318$	0	0
$319$	−36.5030	−2.04377
$320$	0	0
$321$	−1.78317	−0.0995269
$322$	0	0
$323$	13.3946i	0.745296i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	34.5862	1.91262
$328$	0	0
$329$	11.5467	0.636589
$330$	0	0
$331$	− 5.39927i	− 0.296771i	−0.988930	−	0.148385i	$-0.952592\pi$
$331$	− 5.39927i	− 0.296771i	0.988930	−	0.148385i	$-0.0474076\pi$
$332$	0	0
$333$	− 2.64905i	− 0.145167i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.3679	1.43635	0.718176	−	0.695861i	$-0.244977\pi$
$337$	26.3679	1.43635	0.718176	+	0.695861i	$0.244977\pi$
$338$	0	0
$339$	− 22.7226i	− 1.23412i
$340$	0	0
$341$	− 55.2069i	− 2.98962i
$342$	0	0
$343$	−16.1417	−0.871571
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 8.70069i	− 0.467077i	−0.972348	−	0.233539i	$-0.924969\pi$
$347$	− 8.70069i	− 0.467077i	0.972348	−	0.233539i	$-0.0750306\pi$
$348$	0	0
$349$	14.7757i	0.790923i	0.918483	+	0.395462i	$0.129415\pi$
$349$	14.7757i	0.790923i	−0.918483	+	0.395462i	$0.870585\pi$
$350$	0	0
$351$	14.3311	0.764937
$352$	0	0
$353$	17.4259	0.927485	0.463743	−	0.885970i	$-0.346506\pi$
$353$	17.4259	0.927485	0.463743	+	0.885970i	$0.346506\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 6.50725i	− 0.344400i
$358$	0	0
$359$	12.3632	0.652506	0.326253	−	0.945282i	$-0.394214\pi$
$359$	12.3632	0.652506	0.326253	+	0.945282i	$0.394214\pi$
$360$	0	0
$361$	−5.70221	−0.300116
$362$	0	0
$363$	− 59.8223i	− 3.13986i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	4.15969	0.217134	0.108567	−	0.994089i	$-0.465374\pi$
$367$	4.15969	0.217134	0.108567	+	0.994089i	$0.465374\pi$
$368$	0	0
$369$	1.38863	0.0722891
$370$	0	0
$371$	0.0610758i	0.00317090i
$372$	0	0
$373$	33.9114i	1.75586i	0.478786	+	0.877932i	$0.341077\pi$
$373$	33.9114i	1.75586i	−0.478786	+	0.877932i	$0.658923\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.4032	0.844808
$378$	0	0
$379$	− 8.17397i	− 0.419869i	−0.977715	−	0.209934i	$-0.932675\pi$
$379$	− 8.17397i	− 0.419869i	0.977715	−	0.209934i	$-0.0673250\pi$
$380$	0	0
$381$	− 21.6713i	− 1.11025i
$382$	0	0
$383$	3.27290	0.167238	0.0836188	−	0.996498i	$-0.473352\pi$
$383$	3.27290	0.167238	0.0836188	+	0.996498i	$0.473352\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.13379i	− 0.0576336i
$388$	0	0
$389$	2.07560i	0.105237i	0.998615	+	0.0526186i	$0.0167568\pi$
$389$	2.07560i	0.105237i	−0.998615	+	0.0526186i	$0.983243\pi$
$390$	0	0
$391$	−10.0724	−0.509381
$392$	0	0
$393$	−23.2592	−1.17327
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 6.23114i	− 0.312732i	−0.987699	−	0.156366i	$-0.950022\pi$
$397$	− 6.23114i	− 0.312732i	0.987699	−	0.156366i	$-0.0499779\pi$
$398$	0	0
$399$	12.0006	0.600781
$400$	0	0
$401$	20.9959	1.04849	0.524243	−	0.851569i	$-0.324348\pi$
$401$	20.9959	1.04849	0.524243	+	0.851569i	$0.324348\pi$
$402$	0	0
$403$	24.8081i	1.23578i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−47.5078	−2.35488
$408$	0	0
$409$	6.61521	0.327101	0.163551	−	0.986535i	$-0.447705\pi$
$409$	6.61521	0.327101	0.163551	+	0.986535i	$0.447705\pi$
$410$	0	0
$411$	7.95698i	0.392489i
$412$	0	0
$413$	1.34306i	0.0660876i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6.40121	0.313469
$418$	0	0
$419$	17.6296i	0.861262i	0.902528	+	0.430631i	$0.141709\pi$
$419$	17.6296i	0.861262i	−0.902528	+	0.430631i	$0.858291\pi$
$420$	0	0
$421$	18.0508i	0.879742i	0.898061	+	0.439871i	$0.144976\pi$
$421$	18.0508i	0.879742i	−0.898061	+	0.439871i	$0.855024\pi$
$422$	0	0
$423$	−3.23124	−0.157108
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 6.88332i	− 0.333107i
$428$	0	0
$429$	35.9539i	1.73587i
$430$	0	0
$431$	−4.44104	−0.213917	−0.106959	−	0.994263i	$-0.534111\pi$
$431$	−4.44104	−0.213917	−0.106959	+	0.994263i	$0.534111\pi$
$432$	0	0
$433$	24.5345	1.17905	0.589526	−	0.807749i	$-0.299315\pi$
$433$	24.5345	1.17905	0.589526	+	0.807749i	$0.299315\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 18.5753i	− 0.888579i
$438$	0	0
$439$	26.1063	1.24598	0.622992	−	0.782228i	$-0.285917\pi$
$439$	26.1063	1.24598	0.622992	+	0.782228i	$0.285917\pi$
$440$	0	0
$441$	1.93993	0.0923774
$442$	0	0
$443$	− 29.4964i	− 1.40141i	−0.713449	−	0.700707i	$-0.752868\pi$
$443$	− 29.4964i	− 1.40141i	0.713449	−	0.700707i	$-0.247132\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−16.7346	−0.791518
$448$	0	0
$449$	−18.7321	−0.884021	−0.442010	−	0.897010i	$-0.645735\pi$
$449$	−18.7321	−0.884021	−0.442010	+	0.897010i	$0.645735\pi$
$450$	0	0
$451$	− 24.9035i	− 1.17266i
$452$	0	0
$453$	19.3033i	0.906947i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−0.111216	−0.00520245	−0.00260123	−	0.999997i	$-0.500828\pi$
$457$	−0.111216	−0.00520245	−0.00260123	+	0.999997i	$0.500828\pi$
$458$	0	0
$459$	− 13.0172i	− 0.607590i
$460$	0	0
$461$	− 3.71668i	− 0.173103i	−0.996247	−	0.0865516i	$-0.972415\pi$
$461$	− 3.71668i	− 0.173103i	0.996247	−	0.0865516i	$-0.0275847\pi$
$462$	0	0
$463$	11.9601	0.555831	0.277915	−	0.960606i	$-0.410356\pi$
$463$	11.9601	0.555831	0.277915	+	0.960606i	$0.410356\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.1232i	0.792365i	0.918172	+	0.396183i	$0.129665\pi$
$467$	17.1232i	0.792365i	−0.918172	+	0.396183i	$0.870335\pi$
$468$	0	0
$469$	14.2336i	0.657244i
$470$	0	0
$471$	10.2594	0.472728
$472$	0	0
$473$	−20.3332	−0.934921
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.0170915i	0 0.000782567i
$478$	0	0
$479$	−7.70565	−0.352080	−0.176040	−	0.984383i	$-0.556329\pi$
$479$	−7.70565	−0.352080	−0.176040	+	0.984383i	$0.556329\pi$
$480$	0	0
$481$	21.3484	0.973404
$482$	0	0
$483$	9.02410i	0.410611i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−0.471068	−0.0213461	−0.0106731	−	0.999943i	$-0.503397\pi$
$487$	−0.471068	−0.0213461	−0.0106731	+	0.999943i	$0.503397\pi$
$488$	0	0
$489$	−10.0639	−0.455107
$490$	0	0
$491$	42.9128i	1.93663i	0.249735	+	0.968314i	$0.419657\pi$
$491$	42.9128i	1.93663i	−0.249735	+	0.968314i	$0.580343\pi$
$492$	0	0
$493$	− 14.8993i	− 0.671032i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.3097	−0.552167
$498$	0	0
$499$	20.7158i	0.927365i	0.886001	+	0.463683i	$0.153472\pi$
$499$	20.7158i	0.927365i	−0.886001	+	0.463683i	$0.846528\pi$
$500$	0	0
$501$	35.6964i	1.59480i
$502$	0	0
$503$	6.72612	0.299903	0.149951	−	0.988693i	$-0.452088\pi$
$503$	6.72612	0.299903	0.149951	+	0.988693i	$0.452088\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.70189i	0.342053i
$508$	0	0
$509$	− 28.6335i	− 1.26916i	−0.772858	−	0.634579i	$-0.781174\pi$
$509$	− 28.6335i	− 1.26916i	0.772858	−	0.634579i	$-0.218826\pi$
$510$	0	0
$511$	16.3327	0.722518
$512$	0	0
$513$	24.0062	1.05990
$514$	0	0
$515$	0	0
$516$	0	0
$517$	57.9486i	2.54858i
$518$	0	0
$519$	−21.9637	−0.964102
$520$	0	0
$521$	−23.0885	−1.01152	−0.505762	−	0.862673i	$-0.668789\pi$
$521$	−23.0885	−1.01152	−0.505762	+	0.862673i	$0.668789\pi$
$522$	0	0
$523$	11.3256i	0.495236i	0.968858	+	0.247618i	$0.0796478\pi$
$523$	11.3256i	0.495236i	−0.968858	+	0.247618i	$0.920352\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.5337	0.981581
$528$	0	0
$529$	−9.03188	−0.392690
$530$	0	0
$531$	− 0.375844i	− 0.0163102i
$532$	0	0
$533$	11.1908i	0.484727i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−20.2140	−0.872298
$538$	0	0
$539$	− 34.7904i	− 1.49853i
$540$	0	0
$541$	28.1159i	1.20880i	0.796683	+	0.604398i	$0.206586\pi$
$541$	28.1159i	1.20880i	−0.796683	+	0.604398i	$0.793414\pi$
$542$	0	0
$543$	−34.1920	−1.46732
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 25.7268i	− 1.10000i	−0.835166	−	0.549998i	$-0.814628\pi$
$547$	− 25.7268i	− 1.10000i	0.835166	−	0.549998i	$-0.185372\pi$
$548$	0	0
$549$	1.92624i	0.0822098i
$550$	0	0
$551$	27.4772	1.17057
$552$	0	0
$553$	−1.89245	−0.0804750
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11.3682i	0.481685i	0.970564	+	0.240842i	$0.0774237\pi$
$557$	11.3682i	0.481685i	−0.970564	+	0.240842i	$0.922576\pi$
$558$	0	0
$559$	9.13705	0.386456
$560$	0	0
$561$	32.6575	1.37880
$562$	0	0
$563$	− 6.10463i	− 0.257279i	−0.991691	−	0.128640i	$-0.958939\pi$
$563$	− 6.10463i	− 0.257279i	0.991691	−	0.128640i	$-0.0410611\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−13.1156	−0.550803
$568$	0	0
$569$	20.0788	0.841749	0.420875	−	0.907119i	$-0.361723\pi$
$569$	20.0788	0.841749	0.420875	+	0.907119i	$0.361723\pi$
$570$	0	0
$571$	− 12.1145i	− 0.506975i	−0.967339	−	0.253487i	$-0.918422\pi$
$571$	− 12.1145i	− 0.506975i	0.967339	−	0.253487i	$-0.0815776\pi$
$572$	0	0
$573$	− 19.5025i	− 0.814729i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	37.1530	1.54670	0.773351	−	0.633979i	$-0.218579\pi$
$577$	37.1530	1.54670	0.773351	+	0.633979i	$0.218579\pi$
$578$	0	0
$579$	− 32.9937i	− 1.37117i
$580$	0	0
$581$	17.1402i	0.711095i
$582$	0	0
$583$	−0.306517	−0.0126946
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.6112i	0.603071i	0.953455	+	0.301535i	$0.0974991\pi$
$587$	14.6112i	0.603071i	−0.953455	+	0.301535i	$0.902501\pi$
$588$	0	0
$589$	41.5563i	1.71230i
$590$	0	0
$591$	43.4612	1.78776
$592$	0	0
$593$	43.9414	1.80446	0.902229	−	0.431257i	$-0.141930\pi$
$593$	43.9414	1.80446	0.902229	+	0.431257i	$0.141930\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 32.9067i	− 1.34678i
$598$	0	0
$599$	19.8918	0.812756	0.406378	−	0.913705i	$-0.366792\pi$
$599$	19.8918	0.812756	0.406378	+	0.913705i	$0.366792\pi$
$600$	0	0
$601$	−7.20689	−0.293975	−0.146988	−	0.989138i	$-0.546958\pi$
$601$	−7.20689	−0.293975	−0.146988	+	0.989138i	$0.546958\pi$
$602$	0	0
$603$	− 3.98314i	− 0.162206i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	6.17988	0.250834	0.125417	−	0.992104i	$-0.459973\pi$
$607$	6.17988	0.250834	0.125417	+	0.992104i	$0.459973\pi$
$608$	0	0
$609$	−13.3487	−0.540917
$610$	0	0
$611$	− 26.0401i	− 1.05347i
$612$	0	0
$613$	− 14.3999i	− 0.581605i	−0.956783	−	0.290803i	$-0.906078\pi$
$613$	− 14.3999i	− 0.581605i	0.956783	−	0.290803i	$-0.0939223\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.5229	1.55087	0.775437	−	0.631425i	$-0.217530\pi$
$617$	38.5229	1.55087	0.775437	+	0.631425i	$0.217530\pi$
$618$	0	0
$619$	4.77140i	0.191779i	0.995392	+	0.0958894i	$0.0305695\pi$
$619$	4.77140i	0.191779i	−0.995392	+	0.0958894i	$0.969430\pi$
$620$	0	0
$621$	18.0519i	0.724399i
$622$	0	0
$623$	5.18606	0.207775
$624$	0	0
$625$	0	0
$626$	0	0
$627$	60.2267i	2.40522i
$628$	0	0
$629$	− 19.3911i	− 0.773175i
$630$	0	0
$631$	16.4566	0.655127	0.327563	−	0.944829i	$-0.393772\pi$
$631$	16.4566	0.655127	0.327563	+	0.944829i	$0.393772\pi$
$632$	0	0
$633$	−27.7019	−1.10105
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15.6336i	0.619427i
$638$	0	0
$639$	3.44477	0.136273
$640$	0	0
$641$	37.6381	1.48662	0.743309	−	0.668949i	$-0.233255\pi$
$641$	37.6381	1.48662	0.743309	+	0.668949i	$0.233255\pi$
$642$	0	0
$643$	0.647618i	0.0255396i	0.999918	+	0.0127698i	$0.00406486\pi$
$643$	0.647618i	0.0255396i	−0.999918	+	0.0127698i	$0.995935\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	40.6274	1.59723	0.798613	−	0.601845i	$-0.205567\pi$
$647$	40.6274	1.59723	0.798613	+	0.601845i	$0.205567\pi$
$648$	0	0
$649$	−6.74033	−0.264581
$650$	0	0
$651$	− 20.1885i	− 0.791250i
$652$	0	0
$653$	21.7949i	0.852900i	0.904511	+	0.426450i	$0.140236\pi$
$653$	21.7949i	0.852900i	−0.904511	+	0.426450i	$0.859764\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−4.57057	−0.178315
$658$	0	0
$659$	− 10.4657i	− 0.407684i	−0.979004	−	0.203842i	$-0.934657\pi$
$659$	− 10.4657i	− 0.407684i	0.979004	−	0.203842i	$-0.0653429\pi$
$660$	0	0
$661$	− 23.8297i	− 0.926870i	−0.886131	−	0.463435i	$-0.846617\pi$
$661$	− 23.8297i	− 0.926870i	0.886131	−	0.463435i	$-0.153383\pi$
$662$	0	0
$663$	−14.6752	−0.569937
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.6620i	0.800037i
$668$	0	0
$669$	− 38.6474i	− 1.49419i
$670$	0	0
$671$	34.5449	1.33359
$672$	0	0
$673$	18.0022	0.693935	0.346968	−	0.937877i	$-0.387211\pi$
$673$	18.0022	0.693935	0.346968	+	0.937877i	$0.387211\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.0024i	1.46055i	0.683152	+	0.730276i	$0.260609\pi$
$677$	38.0024i	1.46055i	−0.683152	+	0.730276i	$0.739391\pi$
$678$	0	0
$679$	1.32473	0.0508386
$680$	0	0
$681$	−12.7448	−0.488382
$682$	0	0
$683$	− 44.0579i	− 1.68583i	−0.538046	−	0.842915i	$-0.680837\pi$
$683$	− 44.0579i	− 1.68583i	0.538046	−	0.842915i	$-0.319163\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	36.4859	1.39202
$688$	0	0
$689$	0.137739	0.00524742
$690$	0	0
$691$	− 6.13682i	− 0.233456i	−0.993164	−	0.116728i	$-0.962759\pi$
$691$	− 6.13682i	− 0.233456i	0.993164	−	0.116728i	$-0.0372405\pi$
$692$	0	0
$693$	− 3.19824i	− 0.121491i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10.1648	0.385019
$698$	0	0
$699$	17.3215i	0.655158i
$700$	0	0
$701$	− 41.6486i	− 1.57305i	−0.617561	−	0.786523i	$-0.711879\pi$
$701$	− 41.6486i	− 1.57305i	0.617561	−	0.786523i	$-0.288121\pi$
$702$	0	0
$703$	35.7609	1.34875
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.29140i	0.123786i
$708$	0	0
$709$	46.6032i	1.75022i	0.483925	+	0.875109i	$0.339211\pi$
$709$	46.6032i	1.75022i	−0.483925	+	0.875109i	$0.660789\pi$
$710$	0	0
$711$	0.529584	0.0198610
$712$	0	0
$713$	−31.2492	−1.17029
$714$	0	0
$715$	0	0
$716$	0	0
$717$	39.2497i	1.46581i
$718$	0	0
$719$	33.3634	1.24424	0.622122	−	0.782920i	$-0.286271\pi$
$719$	33.3634	1.24424	0.622122	+	0.782920i	$0.286271\pi$
$720$	0	0
$721$	7.02995	0.261809
$722$	0	0
$723$	20.9435i	0.778895i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−34.3672	−1.27461	−0.637305	−	0.770612i	$-0.719951\pi$
$727$	−34.3672	−1.27461	−0.637305	+	0.770612i	$0.719951\pi$
$728$	0	0
$729$	−22.9231	−0.849002
$730$	0	0
$731$	− 8.29934i	− 0.306962i
$732$	0	0
$733$	40.7127i	1.50376i	0.659302	+	0.751878i	$0.270852\pi$
$733$	40.7127i	1.50376i	−0.659302	+	0.751878i	$0.729148\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−71.4331	−2.63127
$738$	0	0
$739$	22.9998i	0.846061i	0.906115	+	0.423030i	$0.139034\pi$
$739$	22.9998i	0.846061i	−0.906115	+	0.423030i	$0.860966\pi$
$740$	0	0
$741$	− 27.0638i	− 0.994215i
$742$	0	0
$743$	14.7518	0.541192	0.270596	−	0.962693i	$-0.412779\pi$
$743$	14.7518	0.541192	0.270596	+	0.962693i	$0.412779\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4.79654i	− 0.175496i
$748$	0	0
$749$	1.27830i	0.0467082i
$750$	0	0
$751$	19.6790	0.718099	0.359049	−	0.933319i	$-0.383101\pi$
$751$	19.6790	0.718099	0.359049	+	0.933319i	$0.383101\pi$
$752$	0	0
$753$	5.06770	0.184677
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3.22189i	0.117102i	0.998284	+	0.0585508i	$0.0186480\pi$
$757$	3.22189i	0.117102i	−0.998284	+	0.0585508i	$0.981352\pi$
$758$	0	0
$759$	−45.2887	−1.64388
$760$	0	0
$761$	21.5082	0.779672	0.389836	−	0.920884i	$-0.372532\pi$
$761$	21.5082	0.779672	0.389836	+	0.920884i	$0.372532\pi$
$762$	0	0
$763$	− 24.7938i	− 0.897596i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.02888	0.109366
$768$	0	0
$769$	16.0974	0.580487	0.290243	−	0.956953i	$-0.406264\pi$
$769$	16.0974	0.580487	0.290243	+	0.956953i	$0.406264\pi$
$770$	0	0
$771$	− 22.3324i	− 0.804281i
$772$	0	0
$773$	2.13036i	0.0766237i	0.999266	+	0.0383118i	$0.0121980\pi$
$773$	2.13036i	0.0766237i	−0.999266	+	0.0383118i	$0.987802\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−17.3731	−0.623255
$778$	0	0
$779$	18.7458i	0.671639i
$780$	0	0
$781$	− 61.7781i	− 2.21059i
$782$	0	0
$783$	−26.7029	−0.954285
$784$	0	0
$785$	0	0
$786$	0	0
$787$	11.4913i	0.409619i	0.978802	+	0.204810i	$0.0656575\pi$
$787$	11.4913i	0.409619i	−0.978802	+	0.204810i	$0.934342\pi$
$788$	0	0
$789$	− 41.1260i	− 1.46412i
$790$	0	0
$791$	−16.2892	−0.579177
$792$	0	0
$793$	−15.5233	−0.551249
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 14.3363i	− 0.507818i	−0.967228	−	0.253909i	$-0.918284\pi$
$797$	− 14.3363i	− 0.507818i	0.967228	−	0.253909i	$-0.0817164\pi$
$798$	0	0
$799$	−23.6527	−0.836773
$800$	0	0
$801$	−1.45127	−0.0512782
$802$	0	0
$803$	81.9681i	2.89259i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	54.6074	1.92227
$808$	0	0
$809$	43.6620	1.53508	0.767538	−	0.641004i	$-0.221482\pi$
$809$	43.6620	1.53508	0.767538	+	0.641004i	$0.221482\pi$
$810$	0	0
$811$	− 18.8353i	− 0.661396i	−0.943737	−	0.330698i	$-0.892716\pi$
$811$	− 18.8353i	− 0.661396i	0.943737	−	0.330698i	$-0.107284\pi$
$812$	0	0
$813$	− 27.7441i	− 0.973027i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	15.3056	0.535474
$818$	0	0
$819$	1.43718i	0.0502193i
$820$	0	0
$821$	1.64016i	0.0572421i	0.999590	+	0.0286210i	$0.00911160\pi$
$821$	1.64016i	0.0572421i	−0.999590	+	0.0286210i	$0.990888\pi$
$822$	0	0
$823$	43.7219	1.52405	0.762024	−	0.647549i	$-0.224206\pi$
$823$	43.7219	1.52405	0.762024	+	0.647549i	$0.224206\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.2651i	1.12197i	0.827827	+	0.560983i	$0.189577\pi$
$827$	32.2651i	1.12197i	−0.827827	+	0.560983i	$0.810423\pi$
$828$	0	0
$829$	6.83658i	0.237444i	0.992928	+	0.118722i	$0.0378798\pi$
$829$	6.83658i	0.237444i	−0.992928	+	0.118722i	$0.962120\pi$
$830$	0	0
$831$	−17.4589	−0.605642
$832$	0	0
$833$	14.2003	0.492011
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 40.3854i	− 1.39592i
$838$	0	0
$839$	−12.3983	−0.428037	−0.214018	−	0.976830i	$-0.568655\pi$
$839$	−12.3983	−0.428037	−0.214018	+	0.976830i	$0.568655\pi$
$840$	0	0
$841$	−1.56388	−0.0539268
$842$	0	0
$843$	− 32.2804i	− 1.11180i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−42.8849	−1.47354
$848$	0	0
$849$	28.1243	0.965224
$850$	0	0
$851$	26.8912i	0.921818i
$852$	0	0
$853$	− 10.2780i	− 0.351911i	−0.984398	−	0.175955i	$-0.943699\pi$
$853$	− 10.2780i	− 0.351911i	0.984398	−	0.175955i	$-0.0563015\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.1493	0.380852	0.190426	−	0.981702i	$-0.439013\pi$
$857$	11.1493	0.380852	0.190426	+	0.981702i	$0.439013\pi$
$858$	0	0
$859$	30.3826i	1.03664i	0.855186	+	0.518321i	$0.173443\pi$
$859$	30.3826i	1.03664i	−0.855186	+	0.518321i	$0.826557\pi$
$860$	0	0
$861$	− 9.10692i	− 0.310363i
$862$	0	0
$863$	−46.2899	−1.57573	−0.787863	−	0.615851i	$-0.788812\pi$
$863$	−46.2899	−1.57573	−0.787863	+	0.615851i	$0.788812\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 17.8696i	− 0.606885i
$868$	0	0
$869$	− 9.49750i	− 0.322181i
$870$	0	0
$871$	32.0996	1.08765
$872$	0	0
$873$	−0.370715	−0.0125468
$874$	0	0
$875$	0	0
$876$	0	0
$877$	39.8781i	1.34659i	0.739376	+	0.673293i	$0.235121\pi$
$877$	39.8781i	1.34659i	−0.739376	+	0.673293i	$0.764879\pi$
$878$	0	0
$879$	−5.84872	−0.197273
$880$	0	0
$881$	38.6037	1.30059	0.650297	−	0.759680i	$-0.274645\pi$
$881$	38.6037	1.30059	0.650297	+	0.759680i	$0.274645\pi$
$882$	0	0
$883$	57.2207i	1.92563i	0.270164	+	0.962814i	$0.412922\pi$
$883$	57.2207i	1.92563i	−0.270164	+	0.962814i	$0.587078\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	11.8099	0.396537	0.198269	−	0.980148i	$-0.436468\pi$
$887$	11.8099	0.396537	0.198269	+	0.980148i	$0.436468\pi$
$888$	0	0
$889$	−15.5355	−0.521045
$890$	0	0
$891$	− 65.8224i	− 2.20513i
$892$	0	0
$893$	− 43.6201i	− 1.45969i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20.3512	0.679508
$898$	0	0
$899$	− 46.2246i	− 1.54168i
$900$	0	0
$901$	− 0.125110i	− 0.00416803i
$902$	0	0
$903$	−7.43561	−0.247441
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 36.7370i	− 1.21983i	−0.792466	−	0.609917i	$-0.791203\pi$
$907$	− 36.7370i	− 1.21983i	0.792466	−	0.609917i	$-0.208797\pi$
$908$	0	0
$909$	− 0.921071i	− 0.0305500i
$910$	0	0
$911$	16.4318	0.544409	0.272205	−	0.962239i	$-0.412247\pi$
$911$	16.4318	0.544409	0.272205	+	0.962239i	$0.412247\pi$
$912$	0	0
$913$	−86.0205	−2.84686
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.6738i	0.550618i
$918$	0	0
$919$	30.9480	1.02088	0.510440	−	0.859913i	$-0.329482\pi$
$919$	30.9480	1.02088	0.510440	+	0.859913i	$0.329482\pi$
$920$	0	0
$921$	51.4916	1.69671
$922$	0	0
$923$	27.7610i	0.913764i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.96727	−0.0646136
$928$	0	0
$929$	11.4537	0.375784	0.187892	−	0.982190i	$-0.439835\pi$
$929$	11.4537	0.375784	0.187892	+	0.982190i	$0.439835\pi$
$930$	0	0
$931$	26.1880i	0.858278i
$932$	0	0
$933$	− 51.0147i	− 1.67015i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	29.7274	0.971151	0.485575	−	0.874195i	$-0.338610\pi$
$937$	29.7274	0.971151	0.485575	+	0.874195i	$0.338610\pi$
$938$	0	0
$939$	39.7228i	1.29630i
$940$	0	0
$941$	22.9601i	0.748477i	0.927332	+	0.374239i	$0.122096\pi$
$941$	22.9601i	0.748477i	−0.927332	+	0.374239i	$0.877904\pi$
$942$	0	0
$943$	−14.0963	−0.459039
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 35.4333i	− 1.15143i	−0.817651	−	0.575714i	$-0.804724\pi$
$947$	− 35.4333i	− 1.15143i	0.817651	−	0.575714i	$-0.195276\pi$
$948$	0	0
$949$	− 36.8337i	− 1.19567i
$950$	0	0
$951$	29.3699	0.952385
$952$	0	0
$953$	−1.35496	−0.0438914	−0.0219457	−	0.999759i	$-0.506986\pi$
$953$	−1.35496	−0.0438914	−0.0219457	+	0.999759i	$0.506986\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 66.9924i	− 2.16556i
$958$	0	0
$959$	5.70413	0.184196
$960$	0	0
$961$	38.9099	1.25516
$962$	0	0
$963$	− 0.357722i	− 0.0115274i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−19.6245	−0.631080	−0.315540	−	0.948912i	$-0.602186\pi$
$967$	−19.6245	−0.631080	−0.315540	+	0.948912i	$0.602186\pi$
$968$	0	0
$969$	−24.5826	−0.789706
$970$	0	0
$971$	1.55749i	0.0499822i	0.999688	+	0.0249911i	$0.00795575\pi$
$971$	1.55749i	0.0499822i	−0.999688	+	0.0249911i	$0.992044\pi$
$972$	0	0
$973$	− 4.58884i	− 0.147112i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−49.4288	−1.58137	−0.790683	−	0.612226i	$-0.790274\pi$
$977$	−49.4288	−1.58137	−0.790683	+	0.612226i	$0.790274\pi$
$978$	0	0
$979$	26.0270i	0.831825i
$980$	0	0
$981$	6.93833i	0.221524i
$982$	0	0
$983$	−18.1091	−0.577591	−0.288795	−	0.957391i	$-0.593255\pi$
$983$	−18.1091	−0.577591	−0.288795	+	0.957391i	$0.593255\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	21.1911i	0.674521i
$988$	0	0
$989$	11.5093i	0.365976i
$990$	0	0
$991$	−4.82935	−0.153409	−0.0767046	−	0.997054i	$-0.524440\pi$
$991$	−4.82935	−0.153409	−0.0767046	+	0.997054i	$0.524440\pi$
$992$	0	0
$993$	9.90905	0.314454
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 18.3807i	− 0.582122i	−0.956705	−	0.291061i	$-0.905992\pi$
$997$	− 18.3807i	− 0.582122i	0.956705	−	0.291061i	$-0.0940082\pi$
$998$	0	0
$999$	−34.7533	−1.09955

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4000.2.d.c.2001.16		40
4.3	odd	2		1000.2.d.c.501.23	yes	40
5.2	odd	4		4000.2.f.d.3249.16		20
5.3	odd	4		4000.2.f.c.3249.5		20
5.4	even	2	inner	4000.2.d.c.2001.25		40
8.3	odd	2		1000.2.d.c.501.24	yes	40
8.5	even	2	inner	4000.2.d.c.2001.15		40
20.3	even	4		1000.2.f.c.749.1		20
20.7	even	4		1000.2.f.d.749.20		20
20.19	odd	2		1000.2.d.c.501.18	yes	40
40.3	even	4		1000.2.f.d.749.19		20
40.13	odd	4		4000.2.f.d.3249.15		20
40.19	odd	2		1000.2.d.c.501.17	✓	40
40.27	even	4		1000.2.f.c.749.2		20
40.29	even	2	inner	4000.2.d.c.2001.26		40
40.37	odd	4		4000.2.f.c.3249.6		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.d.c.501.17	✓	40	40.19	odd	2
1000.2.d.c.501.18	yes	40	20.19	odd	2
1000.2.d.c.501.23	yes	40	4.3	odd	2
1000.2.d.c.501.24	yes	40	8.3	odd	2
1000.2.f.c.749.1		20	20.3	even	4
1000.2.f.c.749.2		20	40.27	even	4
1000.2.f.d.749.19		20	40.3	even	4
1000.2.f.d.749.20		20	20.7	even	4
4000.2.d.c.2001.15		40	8.5	even	2	inner
4000.2.d.c.2001.16		40	1.1	even	1	trivial
4000.2.d.c.2001.25		40	5.4	even	2	inner
4000.2.d.c.2001.26		40	40.29	even	2	inner
4000.2.f.c.3249.5		20	5.3	odd	4
4000.2.f.c.3249.6		20	40.37	odd	4
4000.2.f.d.3249.15		20	40.13	odd	4
4000.2.f.d.3249.16		20	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:	\( N \)	\(=\)	\( 4000 = 2^{5} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4000.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.83526i q^{3} +1.31564 q^{7} -0.368171 q^{9} +6.60274i q^{11} -2.96705i q^{13} -2.69502 q^{17} -4.97013i q^{19} +2.41454i q^{21} +3.73739 q^{23} +4.83008i q^{27} +5.52846i q^{29} -8.36121 q^{31} -12.1177 q^{33} +7.19517i q^{37} +5.44530 q^{39} -3.77169 q^{41} +3.07951i q^{43} +8.77645 q^{47} -5.26908 q^{49} -4.94606i q^{51} +0.0464228i q^{53} +9.12147 q^{57} +1.02084i q^{59} -5.23191i q^{61} -0.484382 q^{63} +10.8187i q^{67} +6.85908i q^{69} -9.35643 q^{71} +12.4143 q^{73} +8.68684i q^{77} -1.43842 q^{79} -9.96896 q^{81} +13.0280i q^{83} -10.1462 q^{87} +3.94185 q^{89} -3.90357i q^{91} -15.3450i q^{93} +1.00691 q^{97} -2.43094i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 24 q^{9} - 48 q^{31} - 8 q^{39} + 44 q^{41} + 12 q^{49} - 96 q^{71} - 96 q^{79} - 56 q^{81} - 44 q^{89}+O(q^{100}) \) 40 * q - 24 * q^9 - 48 * q^31 - 8 * q^39 + 44 * q^41 + 12 * q^49 - 96 * q^71 - 96 * q^79 - 56 * q^81 - 44 * q^89

Introduction

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