LMFDB - Embedded newform 7600.2.a.bw.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7600,2,Mod(1,7600)] 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("7600.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,2,0,0,0,0,0,-3,0,-1,0,-5] 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:	$60.6863055362$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.554958 q^{3} -3.04892 q^{7} -2.69202 q^{9} +2.93900 q^{11} -3.24698 q^{13} +2.15883 q^{17} +1.00000 q^{19} -1.69202 q^{21} +1.19806 q^{23} -3.15883 q^{27} -1.77479 q^{29} +9.34481 q^{31} +1.63102 q^{33} +1.15883 q^{37} -1.80194 q^{39} +8.57002 q^{41} +5.27413 q^{43} +2.35690 q^{47} +2.29590 q^{49} +1.19806 q^{51} -8.82371 q^{53} +0.554958 q^{57} +5.70171 q^{59} -9.96077 q^{61} +8.20775 q^{63} +4.98254 q^{67} +0.664874 q^{69} -2.70171 q^{71} -13.7778 q^{73} -8.96077 q^{77} -5.66487 q^{79} +6.32304 q^{81} -3.00969 q^{83} -0.984935 q^{87} -10.2838 q^{89} +9.89977 q^{91} +5.18598 q^{93} -3.24698 q^{97} -7.91185 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{3} - 3 q^{9} - q^{11} - 5 q^{13} - 2 q^{17} + 3 q^{19} + 8 q^{23} - q^{27} - 7 q^{29} + 5 q^{31} - 10 q^{33} - 5 q^{37} - q^{39} + q^{41} + 5 q^{43} + 3 q^{47} - 7 q^{49} + 8 q^{51} - 19 q^{53}+ \cdots - 20 q^{99}+O(q^{100}) $ 3 * q + 2 * q^3 - 3 * q^9 - q^11 - 5 * q^13 - 2 * q^17 + 3 * q^19 + 8 * q^23 - q^27 - 7 * q^29 + 5 * q^31 - 10 * q^33 - 5 * q^37 - q^39 + q^41 + 5 * q^43 + 3 * q^47 - 7 * q^49 + 8 * q^51 - 19 * q^53 + 2 * q^57 - 10 * q^59 - 17 * q^61 + 7 * q^63 - q^67 + 3 * q^69 + 19 * q^71 + q^73 - 14 * q^77 - 18 * q^79 - q^81 + 13 * q^83 - 28 * q^87 + 2 * q^89 + 7 * q^91 + q^93 - 5 * q^97 - 20 * 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
$3$	0.554958	0.320405	0.160203	−	0.987084i	$-0.448785\pi$
$3$	0.554958	0.320405	0.160203	+	0.987084i	$0.448785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.04892	−1.15238	−0.576191	−	0.817315i	$-0.695462\pi$
$7$	−3.04892	−1.15238	−0.576191	+	0.817315i	$0.695462\pi$
$8$	0	0
$9$	−2.69202	−0.897340
$10$	0	0
$11$	2.93900	0.886142	0.443071	−	0.896486i	$-0.353889\pi$
$11$	2.93900	0.886142	0.443071	+	0.896486i	$0.353889\pi$
$12$	0	0
$13$	−3.24698	−0.900550	−0.450275	−	0.892890i	$-0.648674\pi$
$13$	−3.24698	−0.900550	−0.450275	+	0.892890i	$0.648674\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.15883	0.523594	0.261797	−	0.965123i	$-0.415685\pi$
$17$	2.15883	0.523594	0.261797	+	0.965123i	$0.415685\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−1.69202	−0.369229
$22$	0	0
$23$	1.19806	0.249813	0.124907	−	0.992169i	$-0.460137\pi$
$23$	1.19806	0.249813	0.124907	+	0.992169i	$0.460137\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.15883	−0.607918
$28$	0	0
$29$	−1.77479	−0.329570	−0.164785	−	0.986329i	$-0.552693\pi$
$29$	−1.77479	−0.329570	−0.164785	+	0.986329i	$0.552693\pi$
$30$	0	0
$31$	9.34481	1.67838	0.839189	−	0.543840i	$-0.183030\pi$
$31$	9.34481	1.67838	0.839189	+	0.543840i	$0.183030\pi$
$32$	0	0
$33$	1.63102	0.283925
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.15883	0.190511	0.0952555	−	0.995453i	$-0.469633\pi$
$37$	1.15883	0.190511	0.0952555	+	0.995453i	$0.469633\pi$
$38$	0	0
$39$	−1.80194	−0.288541
$40$	0	0
$41$	8.57002	1.33841	0.669206	−	0.743077i	$-0.266634\pi$
$41$	8.57002	1.33841	0.669206	+	0.743077i	$0.266634\pi$
$42$	0	0
$43$	5.27413	0.804297	0.402148	−	0.915575i	$-0.368264\pi$
$43$	5.27413	0.804297	0.402148	+	0.915575i	$0.368264\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.35690	0.343789	0.171894	−	0.985115i	$-0.445011\pi$
$47$	2.35690	0.343789	0.171894	+	0.985115i	$0.445011\pi$
$48$	0	0
$49$	2.29590	0.327985
$50$	0	0
$51$	1.19806	0.167762
$52$	0	0
$53$	−8.82371	−1.21203	−0.606015	−	0.795453i	$-0.707233\pi$
$53$	−8.82371	−1.21203	−0.606015	+	0.795453i	$0.707233\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0.554958	0.0735060
$58$	0	0
$59$	5.70171	0.742299	0.371150	−	0.928573i	$-0.378964\pi$
$59$	5.70171	0.742299	0.371150	+	0.928573i	$0.378964\pi$
$60$	0	0
$61$	−9.96077	−1.27535	−0.637673	−	0.770307i	$-0.720103\pi$
$61$	−9.96077	−1.27535	−0.637673	+	0.770307i	$0.720103\pi$
$62$	0	0
$63$	8.20775	1.03408
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.98254	0.608714	0.304357	−	0.952558i	$-0.401558\pi$
$67$	4.98254	0.608714	0.304357	+	0.952558i	$0.401558\pi$
$68$	0	0
$69$	0.664874	0.0800415
$70$	0	0
$71$	−2.70171	−0.320634	−0.160317	−	0.987066i	$-0.551252\pi$
$71$	−2.70171	−0.320634	−0.160317	+	0.987066i	$0.551252\pi$
$72$	0	0
$73$	−13.7778	−1.61257	−0.806283	−	0.591530i	$-0.798524\pi$
$73$	−13.7778	−1.61257	−0.806283	+	0.591530i	$0.798524\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.96077	−1.02117
$78$	0	0
$79$	−5.66487	−0.637348	−0.318674	−	0.947864i	$-0.603238\pi$
$79$	−5.66487	−0.637348	−0.318674	+	0.947864i	$0.603238\pi$
$80$	0	0
$81$	6.32304	0.702560
$82$	0	0
$83$	−3.00969	−0.330356	−0.165178	−	0.986264i	$-0.552820\pi$
$83$	−3.00969	−0.330356	−0.165178	+	0.986264i	$0.552820\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.984935	−0.105596
$88$	0	0
$89$	−10.2838	−1.09008	−0.545041	−	0.838409i	$-0.683486\pi$
$89$	−10.2838	−1.09008	−0.545041	+	0.838409i	$0.683486\pi$
$90$	0	0
$91$	9.89977	1.03778
$92$	0	0
$93$	5.18598	0.537761
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.24698	−0.329681	−0.164840	−	0.986320i	$-0.552711\pi$
$97$	−3.24698	−0.329681	−0.164840	+	0.986320i	$0.552711\pi$
$98$	0	0
$99$	−7.91185	−0.795171
$100$	0	0
$101$	5.09246	0.506718	0.253359	−	0.967372i	$-0.418465\pi$
$101$	5.09246	0.506718	0.253359	+	0.967372i	$0.418465\pi$
$102$	0	0
$103$	−14.3110	−1.41010	−0.705051	−	0.709157i	$-0.749076\pi$
$103$	−14.3110	−1.41010	−0.705051	+	0.709157i	$0.749076\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.18598	−0.791369	−0.395684	−	0.918387i	$-0.629493\pi$
$107$	−8.18598	−0.791369	−0.395684	+	0.918387i	$0.629493\pi$
$108$	0	0
$109$	−11.3274	−1.08496	−0.542482	−	0.840067i	$-0.682515\pi$
$109$	−11.3274	−1.08496	−0.542482	+	0.840067i	$0.682515\pi$
$110$	0	0
$111$	0.643104	0.0610407
$112$	0	0
$113$	−3.43535	−0.323171	−0.161585	−	0.986859i	$-0.551661\pi$
$113$	−3.43535	−0.323171	−0.161585	+	0.986859i	$0.551661\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	8.74094	0.808100
$118$	0	0
$119$	−6.58211	−0.603381
$120$	0	0
$121$	−2.36227	−0.214752
$122$	0	0
$123$	4.75600	0.428834
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1.46144	0.129681	0.0648407	−	0.997896i	$-0.479346\pi$
$127$	1.46144	0.129681	0.0648407	+	0.997896i	$0.479346\pi$
$128$	0	0
$129$	2.92692	0.257701
$130$	0	0
$131$	13.2295	1.15587	0.577934	−	0.816083i	$-0.303859\pi$
$131$	13.2295	1.15587	0.577934	+	0.816083i	$0.303859\pi$
$132$	0	0
$133$	−3.04892	−0.264375
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.1739	−1.38183	−0.690915	−	0.722936i	$-0.742792\pi$
$137$	−16.1739	−1.38183	−0.690915	+	0.722936i	$0.742792\pi$
$138$	0	0
$139$	−13.0978	−1.11094	−0.555472	−	0.831535i	$-0.687462\pi$
$139$	−13.0978	−1.11094	−0.555472	+	0.831535i	$0.687462\pi$
$140$	0	0
$141$	1.30798	0.110152
$142$	0	0
$143$	−9.54288	−0.798015
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.27413	0.105088
$148$	0	0
$149$	11.0640	0.906397	0.453198	−	0.891410i	$-0.350283\pi$
$149$	11.0640	0.906397	0.453198	+	0.891410i	$0.350283\pi$
$150$	0	0
$151$	−10.4426	−0.849811	−0.424905	−	0.905238i	$-0.639693\pi$
$151$	−10.4426	−0.849811	−0.424905	+	0.905238i	$0.639693\pi$
$152$	0	0
$153$	−5.81163	−0.469842
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2.48427	−0.198266	−0.0991332	−	0.995074i	$-0.531607\pi$
$157$	−2.48427	−0.198266	−0.0991332	+	0.995074i	$0.531607\pi$
$158$	0	0
$159$	−4.89679	−0.388341
$160$	0	0
$161$	−3.65279	−0.287880
$162$	0	0
$163$	3.16852	0.248178	0.124089	−	0.992271i	$-0.460399\pi$
$163$	3.16852	0.248178	0.124089	+	0.992271i	$0.460399\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.74632	0.367281	0.183640	−	0.982993i	$-0.441212\pi$
$167$	4.74632	0.367281	0.183640	+	0.982993i	$0.441212\pi$
$168$	0	0
$169$	−2.45712	−0.189009
$170$	0	0
$171$	−2.69202	−0.205864
$172$	0	0
$173$	−3.96316	−0.301314	−0.150657	−	0.988586i	$-0.548139\pi$
$173$	−3.96316	−0.301314	−0.150657	+	0.988586i	$0.548139\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3.16421	0.237837
$178$	0	0
$179$	14.0858	1.05282	0.526409	−	0.850231i	$-0.323538\pi$
$179$	14.0858	1.05282	0.526409	+	0.850231i	$0.323538\pi$
$180$	0	0
$181$	−12.2513	−0.910631	−0.455316	−	0.890330i	$-0.650474\pi$
$181$	−12.2513	−0.910631	−0.455316	+	0.890330i	$0.650474\pi$
$182$	0	0
$183$	−5.52781	−0.408628
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.34481	0.463979
$188$	0	0
$189$	9.63102	0.700554
$190$	0	0
$191$	−20.1468	−1.45777	−0.728884	−	0.684637i	$-0.759961\pi$
$191$	−20.1468	−1.45777	−0.728884	+	0.684637i	$0.759961\pi$
$192$	0	0
$193$	9.52781	0.685827	0.342913	−	0.939367i	$-0.388586\pi$
$193$	9.52781	0.685827	0.342913	+	0.939367i	$0.388586\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.96316	−0.353611	−0.176805	−	0.984246i	$-0.556576\pi$
$197$	−4.96316	−0.353611	−0.176805	+	0.984246i	$0.556576\pi$
$198$	0	0
$199$	−5.42221	−0.384370	−0.192185	−	0.981359i	$-0.561557\pi$
$199$	−5.42221	−0.384370	−0.192185	+	0.981359i	$0.561557\pi$
$200$	0	0
$201$	2.76510	0.195035
$202$	0	0
$203$	5.41119	0.379791
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.22521	−0.224168
$208$	0	0
$209$	2.93900	0.203295
$210$	0	0
$211$	−15.9638	−1.09899	−0.549495	−	0.835497i	$-0.685180\pi$
$211$	−15.9638	−1.09899	−0.549495	+	0.835497i	$0.685180\pi$
$212$	0	0
$213$	−1.49934	−0.102733
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−28.4916	−1.93413
$218$	0	0
$219$	−7.64609	−0.516675
$220$	0	0
$221$	−7.00969	−0.471523
$222$	0	0
$223$	−23.2150	−1.55459	−0.777297	−	0.629134i	$-0.783410\pi$
$223$	−23.2150	−1.55459	−0.777297	+	0.629134i	$0.783410\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−17.4795	−1.16015	−0.580077	−	0.814562i	$-0.696978\pi$
$227$	−17.4795	−1.16015	−0.580077	+	0.814562i	$0.696978\pi$
$228$	0	0
$229$	−2.32544	−0.153669	−0.0768346	−	0.997044i	$-0.524481\pi$
$229$	−2.32544	−0.153669	−0.0768346	+	0.997044i	$0.524481\pi$
$230$	0	0
$231$	−4.97285	−0.327190
$232$	0	0
$233$	18.9342	1.24042	0.620211	−	0.784435i	$-0.287047\pi$
$233$	18.9342	1.24042	0.620211	+	0.784435i	$0.287047\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3.14377	−0.204210
$238$	0	0
$239$	−11.1685	−0.722432	−0.361216	−	0.932482i	$-0.617638\pi$
$239$	−11.1685	−0.722432	−0.361216	+	0.932482i	$0.617638\pi$
$240$	0	0
$241$	7.42998	0.478607	0.239303	−	0.970945i	$-0.423081\pi$
$241$	7.42998	0.478607	0.239303	+	0.970945i	$0.423081\pi$
$242$	0	0
$243$	12.9855	0.833022
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.24698	−0.206600
$248$	0	0
$249$	−1.67025	−0.105848
$250$	0	0
$251$	14.7409	0.930440	0.465220	−	0.885195i	$-0.345975\pi$
$251$	14.7409	0.930440	0.465220	+	0.885195i	$0.345975\pi$
$252$	0	0
$253$	3.52111	0.221370
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.34721	−0.208793	−0.104397	−	0.994536i	$-0.533291\pi$
$257$	−3.34721	−0.208793	−0.104397	+	0.994536i	$0.533291\pi$
$258$	0	0
$259$	−3.53319	−0.219542
$260$	0	0
$261$	4.77777	0.295737
$262$	0	0
$263$	−16.6853	−1.02886	−0.514430	−	0.857532i	$-0.671997\pi$
$263$	−16.6853	−1.02886	−0.514430	+	0.857532i	$0.671997\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−5.70709	−0.349268
$268$	0	0
$269$	15.5961	0.950911	0.475456	−	0.879740i	$-0.342283\pi$
$269$	15.5961	0.950911	0.475456	+	0.879740i	$0.342283\pi$
$270$	0	0
$271$	13.2131	0.802640	0.401320	−	0.915938i	$-0.368551\pi$
$271$	13.2131	0.802640	0.401320	+	0.915938i	$0.368551\pi$
$272$	0	0
$273$	5.49396	0.332510
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−12.9758	−0.779642	−0.389821	−	0.920891i	$-0.627463\pi$
$277$	−12.9758	−0.779642	−0.389821	+	0.920891i	$0.627463\pi$
$278$	0	0
$279$	−25.1564	−1.50608
$280$	0	0
$281$	24.8901	1.48482	0.742409	−	0.669947i	$-0.233683\pi$
$281$	24.8901	1.48482	0.742409	+	0.669947i	$0.233683\pi$
$282$	0	0
$283$	−13.5821	−0.807372	−0.403686	−	0.914898i	$-0.632271\pi$
$283$	−13.5821	−0.807372	−0.403686	+	0.914898i	$0.632271\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−26.1293	−1.54236
$288$	0	0
$289$	−12.3394	−0.725849
$290$	0	0
$291$	−1.80194	−0.105631
$292$	0	0
$293$	−1.27652	−0.0745751	−0.0372875	−	0.999305i	$-0.511872\pi$
$293$	−1.27652	−0.0745751	−0.0372875	+	0.999305i	$0.511872\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−9.28382	−0.538702
$298$	0	0
$299$	−3.89008	−0.224969
$300$	0	0
$301$	−16.0804	−0.926857
$302$	0	0
$303$	2.82610	0.162355
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−32.2295	−1.83944	−0.919718	−	0.392580i	$-0.871583\pi$
$307$	−32.2295	−1.83944	−0.919718	+	0.392580i	$0.871583\pi$
$308$	0	0
$309$	−7.94198	−0.451804
$310$	0	0
$311$	−12.4983	−0.708712	−0.354356	−	0.935111i	$-0.615300\pi$
$311$	−12.4983	−0.708712	−0.354356	+	0.935111i	$0.615300\pi$
$312$	0	0
$313$	20.9390	1.18354	0.591771	−	0.806106i	$-0.298429\pi$
$313$	20.9390	1.18354	0.591771	+	0.806106i	$0.298429\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.8562	−0.890575	−0.445287	−	0.895388i	$-0.646898\pi$
$317$	−15.8562	−0.890575	−0.445287	+	0.895388i	$0.646898\pi$
$318$	0	0
$319$	−5.21611	−0.292046
$320$	0	0
$321$	−4.54288	−0.253559
$322$	0	0
$323$	2.15883	0.120121
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−6.28621	−0.347628
$328$	0	0
$329$	−7.18598	−0.396176
$330$	0	0
$331$	13.2349	0.727456	0.363728	−	0.931505i	$-0.381504\pi$
$331$	13.2349	0.727456	0.363728	+	0.931505i	$0.381504\pi$
$332$	0	0
$333$	−3.11960	−0.170953
$334$	0	0
$335$	0	0
$336$	0	0
$337$	26.0780	1.42056	0.710279	−	0.703920i	$-0.248569\pi$
$337$	26.0780	1.42056	0.710279	+	0.703920i	$0.248569\pi$
$338$	0	0
$339$	−1.90648	−0.103546
$340$	0	0
$341$	27.4644	1.48728
$342$	0	0
$343$	14.3424	0.774418
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.98254	0.482208	0.241104	−	0.970499i	$-0.422490\pi$
$347$	8.98254	0.482208	0.241104	+	0.970499i	$0.422490\pi$
$348$	0	0
$349$	0.599564	0.0320939	0.0160470	−	0.999871i	$-0.494892\pi$
$349$	0.599564	0.0320939	0.0160470	+	0.999871i	$0.494892\pi$
$350$	0	0
$351$	10.2567	0.547460
$352$	0	0
$353$	−31.8896	−1.69731	−0.848656	−	0.528945i	$-0.822588\pi$
$353$	−31.8896	−1.69731	−0.848656	+	0.528945i	$0.822588\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3.65279	−0.193326
$358$	0	0
$359$	18.3763	0.969863	0.484931	−	0.874552i	$-0.338845\pi$
$359$	18.3763	0.969863	0.484931	+	0.874552i	$0.338845\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−1.31096	−0.0688077
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−14.6377	−0.764083	−0.382042	−	0.924145i	$-0.624779\pi$
$367$	−14.6377	−0.764083	−0.382042	+	0.924145i	$0.624779\pi$
$368$	0	0
$369$	−23.0707	−1.20101
$370$	0	0
$371$	26.9028	1.39672
$372$	0	0
$373$	1.02715	0.0531837	0.0265918	−	0.999646i	$-0.491535\pi$
$373$	1.02715	0.0531837	0.0265918	+	0.999646i	$0.491535\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5.76271	0.296795
$378$	0	0
$379$	22.5284	1.15721	0.578603	−	0.815609i	$-0.303598\pi$
$379$	22.5284	1.15721	0.578603	+	0.815609i	$0.303598\pi$
$380$	0	0
$381$	0.811035	0.0415506
$382$	0	0
$383$	32.6698	1.66935	0.834674	−	0.550745i	$-0.185656\pi$
$383$	32.6698	1.66935	0.834674	+	0.550745i	$0.185656\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−14.1981	−0.721728
$388$	0	0
$389$	−8.16421	−0.413942	−0.206971	−	0.978347i	$-0.566361\pi$
$389$	−8.16421	−0.413942	−0.206971	+	0.978347i	$0.566361\pi$
$390$	0	0
$391$	2.58642	0.130801
$392$	0	0
$393$	7.34183	0.370346
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−37.5478	−1.88447	−0.942235	−	0.334954i	$-0.891279\pi$
$397$	−37.5478	−1.88447	−0.942235	+	0.334954i	$0.891279\pi$
$398$	0	0
$399$	−1.69202	−0.0847070
$400$	0	0
$401$	32.2519	1.61058	0.805291	−	0.592880i	$-0.202009\pi$
$401$	32.2519	1.61058	0.805291	+	0.592880i	$0.202009\pi$
$402$	0	0
$403$	−30.3424	−1.51146
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.40581	0.168820
$408$	0	0
$409$	−31.5459	−1.55984	−0.779921	−	0.625878i	$-0.784741\pi$
$409$	−31.5459	−1.55984	−0.779921	+	0.625878i	$0.784741\pi$
$410$	0	0
$411$	−8.97584	−0.442745
$412$	0	0
$413$	−17.3840	−0.855413
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−7.26875	−0.355952
$418$	0	0
$419$	14.6866	0.717490	0.358745	−	0.933436i	$-0.383205\pi$
$419$	14.6866	0.717490	0.358745	+	0.933436i	$0.383205\pi$
$420$	0	0
$421$	15.1142	0.736622	0.368311	−	0.929703i	$-0.379936\pi$
$421$	15.1142	0.736622	0.368311	+	0.929703i	$0.379936\pi$
$422$	0	0
$423$	−6.34481	−0.308495
$424$	0	0
$425$	0	0
$426$	0	0
$427$	30.3696	1.46969
$428$	0	0
$429$	−5.29590	−0.255688
$430$	0	0
$431$	−1.67696	−0.0807761	−0.0403881	−	0.999184i	$-0.512859\pi$
$431$	−1.67696	−0.0807761	−0.0403881	+	0.999184i	$0.512859\pi$
$432$	0	0
$433$	13.1545	0.632166	0.316083	−	0.948732i	$-0.397632\pi$
$433$	13.1545	0.632166	0.316083	+	0.948732i	$0.397632\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.19806	0.0573111
$438$	0	0
$439$	−6.45580	−0.308118	−0.154059	−	0.988062i	$-0.549235\pi$
$439$	−6.45580	−0.308118	−0.154059	+	0.988062i	$0.549235\pi$
$440$	0	0
$441$	−6.18060	−0.294314
$442$	0	0
$443$	−25.3709	−1.20541	−0.602704	−	0.797965i	$-0.705910\pi$
$443$	−25.3709	−1.20541	−0.602704	+	0.797965i	$0.705910\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.14005	0.290414
$448$	0	0
$449$	−3.59956	−0.169874	−0.0849370	−	0.996386i	$-0.527069\pi$
$449$	−3.59956	−0.169874	−0.0849370	+	0.996386i	$0.527069\pi$
$450$	0	0
$451$	25.1873	1.18602
$452$	0	0
$453$	−5.79523	−0.272284
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.2784	0.574361	0.287181	−	0.957876i	$-0.407282\pi$
$457$	12.2784	0.574361	0.287181	+	0.957876i	$0.407282\pi$
$458$	0	0
$459$	−6.81940	−0.318302
$460$	0	0
$461$	39.4935	1.83939	0.919697	−	0.392628i	$-0.128434\pi$
$461$	39.4935	1.83939	0.919697	+	0.392628i	$0.128434\pi$
$462$	0	0
$463$	18.3991	0.855079	0.427540	−	0.903997i	$-0.359380\pi$
$463$	18.3991	0.855079	0.427540	+	0.903997i	$0.359380\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−36.2282	−1.67644	−0.838220	−	0.545332i	$-0.816404\pi$
$467$	−36.2282	−1.67644	−0.838220	+	0.545332i	$0.816404\pi$
$468$	0	0
$469$	−15.1914	−0.701472
$470$	0	0
$471$	−1.37867	−0.0635256
$472$	0	0
$473$	15.5007	0.712721
$474$	0	0
$475$	0	0
$476$	0	0
$477$	23.7536	1.08760
$478$	0	0
$479$	34.3129	1.56780	0.783898	−	0.620890i	$-0.213229\pi$
$479$	34.3129	1.56780	0.783898	+	0.620890i	$0.213229\pi$
$480$	0	0
$481$	−3.76271	−0.171565
$482$	0	0
$483$	−2.02715	−0.0922384
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−24.4722	−1.10894	−0.554470	−	0.832203i	$-0.687079\pi$
$487$	−24.4722	−1.10894	−0.554470	+	0.832203i	$0.687079\pi$
$488$	0	0
$489$	1.75840	0.0795175
$490$	0	0
$491$	−18.3067	−0.826168	−0.413084	−	0.910693i	$-0.635548\pi$
$491$	−18.3067	−0.826168	−0.413084	+	0.910693i	$0.635548\pi$
$492$	0	0
$493$	−3.83148	−0.172561
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.23729	0.369493
$498$	0	0
$499$	25.3424	1.13448	0.567241	−	0.823552i	$-0.308011\pi$
$499$	25.3424	1.13448	0.567241	+	0.823552i	$0.308011\pi$
$500$	0	0
$501$	2.63401	0.117679
$502$	0	0
$503$	14.1105	0.629156	0.314578	−	0.949232i	$-0.398137\pi$
$503$	14.1105	0.629156	0.314578	+	0.949232i	$0.398137\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.36360	−0.0605596
$508$	0	0
$509$	−17.8019	−0.789057	−0.394529	−	0.918884i	$-0.629092\pi$
$509$	−17.8019	−0.789057	−0.394529	+	0.918884i	$0.629092\pi$
$510$	0	0
$511$	42.0073	1.85829
$512$	0	0
$513$	−3.15883	−0.139466
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.92692	0.304646
$518$	0	0
$519$	−2.19939	−0.0965425
$520$	0	0
$521$	13.3948	0.586837	0.293418	−	0.955984i	$-0.405207\pi$
$521$	13.3948	0.586837	0.293418	+	0.955984i	$0.405207\pi$
$522$	0	0
$523$	−1.46921	−0.0642439	−0.0321219	−	0.999484i	$-0.510226\pi$
$523$	−1.46921	−0.0642439	−0.0321219	+	0.999484i	$0.510226\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.1739	0.878789
$528$	0	0
$529$	−21.5646	−0.937593
$530$	0	0
$531$	−15.3491	−0.666095
$532$	0	0
$533$	−27.8267	−1.20531
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.81700	0.337329
$538$	0	0
$539$	6.74764	0.290642
$540$	0	0
$541$	−42.7144	−1.83643	−0.918217	−	0.396077i	$-0.870371\pi$
$541$	−42.7144	−1.83643	−0.918217	+	0.396077i	$0.870371\pi$
$542$	0	0
$543$	−6.79895	−0.291771
$544$	0	0
$545$	0	0
$546$	0	0
$547$	10.6866	0.456928	0.228464	−	0.973552i	$-0.426630\pi$
$547$	10.6866	0.456928	0.228464	+	0.973552i	$0.426630\pi$
$548$	0	0
$549$	26.8146	1.14442
$550$	0	0
$551$	−1.77479	−0.0756086
$552$	0	0
$553$	17.2717	0.734469
$554$	0	0
$555$	0	0
$556$	0	0
$557$	34.2083	1.44945	0.724727	−	0.689036i	$-0.241966\pi$
$557$	34.2083	1.44945	0.724727	+	0.689036i	$0.241966\pi$
$558$	0	0
$559$	−17.1250	−0.724310
$560$	0	0
$561$	3.52111	0.148661
$562$	0	0
$563$	−5.46788	−0.230444	−0.115222	−	0.993340i	$-0.536758\pi$
$563$	−5.46788	−0.230444	−0.115222	+	0.993340i	$0.536758\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−19.2784	−0.809618
$568$	0	0
$569$	−17.4819	−0.732878	−0.366439	−	0.930442i	$-0.619423\pi$
$569$	−17.4819	−0.732878	−0.366439	+	0.930442i	$0.619423\pi$
$570$	0	0
$571$	7.21877	0.302096	0.151048	−	0.988526i	$-0.451735\pi$
$571$	7.21877	0.302096	0.151048	+	0.988526i	$0.451735\pi$
$572$	0	0
$573$	−11.1806	−0.467076
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.80625	−0.0751952	−0.0375976	−	0.999293i	$-0.511971\pi$
$577$	−1.80625	−0.0751952	−0.0375976	+	0.999293i	$0.511971\pi$
$578$	0	0
$579$	5.28754	0.219743
$580$	0	0
$581$	9.17629	0.380697
$582$	0	0
$583$	−25.9329	−1.07403
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.39075	0.0986767	0.0493384	−	0.998782i	$-0.484289\pi$
$587$	2.39075	0.0986767	0.0493384	+	0.998782i	$0.484289\pi$
$588$	0	0
$589$	9.34481	0.385046
$590$	0	0
$591$	−2.75435	−0.113299
$592$	0	0
$593$	42.2650	1.73562	0.867808	−	0.496899i	$-0.165528\pi$
$593$	42.2650	1.73562	0.867808	+	0.496899i	$0.165528\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.00910	−0.123154
$598$	0	0
$599$	−8.95838	−0.366029	−0.183015	−	0.983110i	$-0.558586\pi$
$599$	−8.95838	−0.366029	−0.183015	+	0.983110i	$0.558586\pi$
$600$	0	0
$601$	−19.8998	−0.811729	−0.405864	−	0.913933i	$-0.633029\pi$
$601$	−19.8998	−0.811729	−0.405864	+	0.913933i	$0.633029\pi$
$602$	0	0
$603$	−13.4131	−0.546224
$604$	0	0
$605$	0	0
$606$	0	0
$607$	26.0084	1.05565	0.527823	−	0.849354i	$-0.323008\pi$
$607$	26.0084	1.05565	0.527823	+	0.849354i	$0.323008\pi$
$608$	0	0
$609$	3.00298	0.121687
$610$	0	0
$611$	−7.65279	−0.309599
$612$	0	0
$613$	−41.3763	−1.67117	−0.835586	−	0.549360i	$-0.814872\pi$
$613$	−41.3763	−1.67117	−0.835586	+	0.549360i	$0.814872\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.9845	−1.77075	−0.885374	−	0.464880i	$-0.846098\pi$
$617$	−43.9845	−1.77075	−0.885374	+	0.464880i	$0.846098\pi$
$618$	0	0
$619$	−23.4553	−0.942749	−0.471374	−	0.881933i	$-0.656242\pi$
$619$	−23.4553	−0.942749	−0.471374	+	0.881933i	$0.656242\pi$
$620$	0	0
$621$	−3.78448	−0.151866
$622$	0	0
$623$	31.3545	1.25619
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.63102	0.0651368
$628$	0	0
$629$	2.50173	0.0997505
$630$	0	0
$631$	34.8877	1.38886	0.694429	−	0.719562i	$-0.255657\pi$
$631$	34.8877	1.38886	0.694429	+	0.719562i	$0.255657\pi$
$632$	0	0
$633$	−8.85922	−0.352122
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−7.45473	−0.295367
$638$	0	0
$639$	7.27306	0.287718
$640$	0	0
$641$	−38.5314	−1.52190	−0.760949	−	0.648812i	$-0.775266\pi$
$641$	−38.5314	−1.52190	−0.760949	+	0.648812i	$0.775266\pi$
$642$	0	0
$643$	−18.6353	−0.734906	−0.367453	−	0.930042i	$-0.619770\pi$
$643$	−18.6353	−0.734906	−0.367453	+	0.930042i	$0.619770\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.2282	0.755938	0.377969	−	0.925818i	$-0.376623\pi$
$647$	19.2282	0.755938	0.377969	+	0.925818i	$0.376623\pi$
$648$	0	0
$649$	16.7573	0.657783
$650$	0	0
$651$	−15.8116	−0.619706
$652$	0	0
$653$	3.98553	0.155966	0.0779828	−	0.996955i	$-0.475152\pi$
$653$	3.98553	0.155966	0.0779828	+	0.996955i	$0.475152\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	37.0901	1.44702
$658$	0	0
$659$	−33.5719	−1.30778	−0.653889	−	0.756591i	$-0.726864\pi$
$659$	−33.5719	−1.30778	−0.653889	+	0.756591i	$0.726864\pi$
$660$	0	0
$661$	36.9909	1.43878	0.719390	−	0.694607i	$-0.244422\pi$
$661$	36.9909	1.43878	0.719390	+	0.694607i	$0.244422\pi$
$662$	0	0
$663$	−3.89008	−0.151078
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2.12631	−0.0823310
$668$	0	0
$669$	−12.8834	−0.498100
$670$	0	0
$671$	−29.2747	−1.13014
$672$	0	0
$673$	−12.0747	−0.465447	−0.232723	−	0.972543i	$-0.574764\pi$
$673$	−12.0747	−0.465447	−0.232723	+	0.972543i	$0.574764\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−41.9135	−1.61087	−0.805434	−	0.592686i	$-0.798067\pi$
$677$	−41.9135	−1.61087	−0.805434	+	0.592686i	$0.798067\pi$
$678$	0	0
$679$	9.89977	0.379918
$680$	0	0
$681$	−9.70038	−0.371719
$682$	0	0
$683$	9.17092	0.350915	0.175458	−	0.984487i	$-0.443859\pi$
$683$	9.17092	0.350915	0.175458	+	0.984487i	$0.443859\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.29052	−0.0492364
$688$	0	0
$689$	28.6504	1.09149
$690$	0	0
$691$	0.111244	0.00423193	0.00211596	−	0.999998i	$-0.499326\pi$
$691$	0.111244	0.00423193	0.00211596	+	0.999998i	$0.499326\pi$
$692$	0	0
$693$	24.1226	0.916341
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.5013	0.700785
$698$	0	0
$699$	10.5077	0.397438
$700$	0	0
$701$	−27.7832	−1.04936	−0.524678	−	0.851301i	$-0.675814\pi$
$701$	−27.7832	−1.04936	−0.524678	+	0.851301i	$0.675814\pi$
$702$	0	0
$703$	1.15883	0.0437062
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−15.5265	−0.583933
$708$	0	0
$709$	37.4668	1.40710	0.703548	−	0.710648i	$-0.251598\pi$
$709$	37.4668	1.40710	0.703548	+	0.710648i	$0.251598\pi$
$710$	0	0
$711$	15.2500	0.571918
$712$	0	0
$713$	11.1957	0.419281
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.19806	−0.231471
$718$	0	0
$719$	21.0513	0.785081	0.392541	−	0.919735i	$-0.371596\pi$
$719$	21.0513	0.785081	0.392541	+	0.919735i	$0.371596\pi$
$720$	0	0
$721$	43.6329	1.62498
$722$	0	0
$723$	4.12333	0.153348
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−44.7023	−1.65792	−0.828958	−	0.559310i	$-0.811066\pi$
$727$	−44.7023	−1.65792	−0.828958	+	0.559310i	$0.811066\pi$
$728$	0	0
$729$	−11.7627	−0.435656
$730$	0	0
$731$	11.3860	0.421125
$732$	0	0
$733$	−25.0301	−0.924509	−0.462254	−	0.886747i	$-0.652959\pi$
$733$	−25.0301	−0.924509	−0.462254	+	0.886747i	$0.652959\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14.6437	0.539407
$738$	0	0
$739$	23.9982	0.882788	0.441394	−	0.897313i	$-0.354484\pi$
$739$	23.9982	0.882788	0.441394	+	0.897313i	$0.354484\pi$
$740$	0	0
$741$	−1.80194	−0.0661958
$742$	0	0
$743$	10.4601	0.383744	0.191872	−	0.981420i	$-0.438544\pi$
$743$	10.4601	0.383744	0.191872	+	0.981420i	$0.438544\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8.10215	0.296442
$748$	0	0
$749$	24.9584	0.911959
$750$	0	0
$751$	29.9420	1.09260	0.546299	−	0.837590i	$-0.316036\pi$
$751$	29.9420	1.09260	0.546299	+	0.837590i	$0.316036\pi$
$752$	0	0
$753$	8.18060	0.298118
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−27.2174	−0.989235	−0.494617	−	0.869111i	$-0.664692\pi$
$757$	−27.2174	−0.989235	−0.494617	+	0.869111i	$0.664692\pi$
$758$	0	0
$759$	1.95407	0.0709281
$760$	0	0
$761$	18.6813	0.677195	0.338598	−	0.940931i	$-0.390047\pi$
$761$	18.6813	0.677195	0.338598	+	0.940931i	$0.390047\pi$
$762$	0	0
$763$	34.5362	1.25029
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−18.5133	−0.668478
$768$	0	0
$769$	6.34780	0.228907	0.114454	−	0.993429i	$-0.463488\pi$
$769$	6.34780	0.228907	0.114454	+	0.993429i	$0.463488\pi$
$770$	0	0
$771$	−1.85756	−0.0668984
$772$	0	0
$773$	−49.7851	−1.79064	−0.895322	−	0.445419i	$-0.853055\pi$
$773$	−49.7851	−1.79064	−0.895322	+	0.445419i	$0.853055\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1.96077	−0.0703423
$778$	0	0
$779$	8.57002	0.307053
$780$	0	0
$781$	−7.94033	−0.284127
$782$	0	0
$783$	5.60627	0.200352
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−9.19865	−0.327897	−0.163948	−	0.986469i	$-0.552423\pi$
$787$	−9.19865	−0.327897	−0.163948	+	0.986469i	$0.552423\pi$
$788$	0	0
$789$	−9.25965	−0.329652
$790$	0	0
$791$	10.4741	0.372416
$792$	0	0
$793$	32.3424	1.14851
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13.5084	−0.478493	−0.239247	−	0.970959i	$-0.576900\pi$
$797$	−13.5084	−0.478493	−0.239247	+	0.970959i	$0.576900\pi$
$798$	0	0
$799$	5.08815	0.180006
$800$	0	0
$801$	27.6843	0.978175
$802$	0	0
$803$	−40.4929	−1.42896
$804$	0	0
$805$	0	0
$806$	0	0
$807$	8.65519	0.304677
$808$	0	0
$809$	31.9560	1.12351	0.561756	−	0.827303i	$-0.310126\pi$
$809$	31.9560	1.12351	0.561756	+	0.827303i	$0.310126\pi$
$810$	0	0
$811$	−40.7012	−1.42921	−0.714607	−	0.699526i	$-0.753394\pi$
$811$	−40.7012	−1.42921	−0.714607	+	0.699526i	$0.753394\pi$
$812$	0	0
$813$	7.33273	0.257170
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.27413	0.184518
$818$	0	0
$819$	−26.6504	−0.931240
$820$	0	0
$821$	−13.4300	−0.468709	−0.234355	−	0.972151i	$-0.575298\pi$
$821$	−13.4300	−0.468709	−0.234355	+	0.972151i	$0.575298\pi$
$822$	0	0
$823$	−26.7275	−0.931663	−0.465832	−	0.884873i	$-0.654245\pi$
$823$	−26.7275	−0.931663	−0.465832	+	0.884873i	$0.654245\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33.3435	1.15947	0.579733	−	0.814806i	$-0.303157\pi$
$827$	33.3435	1.15947	0.579733	+	0.814806i	$0.303157\pi$
$828$	0	0
$829$	12.2849	0.426672	0.213336	−	0.976979i	$-0.431567\pi$
$829$	12.2849	0.426672	0.213336	+	0.976979i	$0.431567\pi$
$830$	0	0
$831$	−7.20105	−0.249802
$832$	0	0
$833$	4.95646	0.171731
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−29.5187	−1.02032
$838$	0	0
$839$	−36.9922	−1.27711	−0.638557	−	0.769575i	$-0.720468\pi$
$839$	−36.9922	−1.27711	−0.638557	+	0.769575i	$0.720468\pi$
$840$	0	0
$841$	−25.8501	−0.891383
$842$	0	0
$843$	13.8130	0.475743
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.20237	0.247477
$848$	0	0
$849$	−7.53750	−0.258686
$850$	0	0
$851$	1.38835	0.0475922
$852$	0	0
$853$	−7.16959	−0.245482	−0.122741	−	0.992439i	$-0.539168\pi$
$853$	−7.16959	−0.245482	−0.122741	+	0.992439i	$0.539168\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.55124	−0.0871486	−0.0435743	−	0.999050i	$-0.513875\pi$
$857$	−2.55124	−0.0871486	−0.0435743	+	0.999050i	$0.513875\pi$
$858$	0	0
$859$	−53.9493	−1.84073	−0.920363	−	0.391065i	$-0.872107\pi$
$859$	−53.9493	−1.84073	−0.920363	+	0.391065i	$0.872107\pi$
$860$	0	0
$861$	−14.5007	−0.494181
$862$	0	0
$863$	21.4698	0.730840	0.365420	−	0.930843i	$-0.380925\pi$
$863$	21.4698	0.730840	0.365420	+	0.930843i	$0.380925\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−6.84787	−0.232566
$868$	0	0
$869$	−16.6491	−0.564781
$870$	0	0
$871$	−16.1782	−0.548178
$872$	0	0
$873$	8.74094	0.295836
$874$	0	0
$875$	0	0
$876$	0	0
$877$	38.9638	1.31571	0.657856	−	0.753144i	$-0.271463\pi$
$877$	38.9638	1.31571	0.657856	+	0.753144i	$0.271463\pi$
$878$	0	0
$879$	−0.708415	−0.0238942
$880$	0	0
$881$	−13.4373	−0.452713	−0.226357	−	0.974045i	$-0.572681\pi$
$881$	−13.4373	−0.452713	−0.226357	+	0.974045i	$0.572681\pi$
$882$	0	0
$883$	−27.8799	−0.938234	−0.469117	−	0.883136i	$-0.655428\pi$
$883$	−27.8799	−0.938234	−0.469117	+	0.883136i	$0.655428\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.9135	−0.970821	−0.485410	−	0.874286i	$-0.661330\pi$
$887$	−28.9135	−0.970821	−0.485410	+	0.874286i	$0.661330\pi$
$888$	0	0
$889$	−4.45580	−0.149443
$890$	0	0
$891$	18.5834	0.622568
$892$	0	0
$893$	2.35690	0.0788705
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.15883	−0.0720814
$898$	0	0
$899$	−16.5851	−0.553144
$900$	0	0
$901$	−19.0489	−0.634611
$902$	0	0
$903$	−8.92394	−0.296970
$904$	0	0
$905$	0	0
$906$	0	0
$907$	30.7127	1.01980	0.509900	−	0.860234i	$-0.329683\pi$
$907$	30.7127	1.01980	0.509900	+	0.860234i	$0.329683\pi$
$908$	0	0
$909$	−13.7090	−0.454699
$910$	0	0
$911$	−39.3690	−1.30435	−0.652176	−	0.758067i	$-0.726144\pi$
$911$	−39.3690	−1.30435	−0.652176	+	0.758067i	$0.726144\pi$
$912$	0	0
$913$	−8.84548	−0.292743
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−40.3357	−1.33200
$918$	0	0
$919$	7.87023	0.259615	0.129808	−	0.991539i	$-0.458564\pi$
$919$	7.87023	0.259615	0.129808	+	0.991539i	$0.458564\pi$
$920$	0	0
$921$	−17.8860	−0.589365
$922$	0	0
$923$	8.77240	0.288747
$924$	0	0
$925$	0	0
$926$	0	0
$927$	38.5254	1.26534
$928$	0	0
$929$	39.2924	1.28914	0.644572	−	0.764544i	$-0.277036\pi$
$929$	39.2924	1.28914	0.644572	+	0.764544i	$0.277036\pi$
$930$	0	0
$931$	2.29590	0.0752450
$932$	0	0
$933$	−6.93602	−0.227075
$934$	0	0
$935$	0	0
$936$	0	0
$937$	26.7554	0.874061	0.437031	−	0.899447i	$-0.356030\pi$
$937$	26.7554	0.874061	0.437031	+	0.899447i	$0.356030\pi$
$938$	0	0
$939$	11.6203	0.379213
$940$	0	0
$941$	−41.9191	−1.36653	−0.683263	−	0.730173i	$-0.739440\pi$
$941$	−41.9191	−1.36653	−0.683263	+	0.730173i	$0.739440\pi$
$942$	0	0
$943$	10.2674	0.334353
$944$	0	0
$945$	0	0
$946$	0	0
$947$	55.4868	1.80308	0.901539	−	0.432698i	$-0.142438\pi$
$947$	55.4868	1.80308	0.901539	+	0.432698i	$0.142438\pi$
$948$	0	0
$949$	44.7362	1.45220
$950$	0	0
$951$	−8.79954	−0.285345
$952$	0	0
$953$	−8.48081	−0.274720	−0.137360	−	0.990521i	$-0.543862\pi$
$953$	−8.48081	−0.274720	−0.137360	+	0.990521i	$0.543862\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.89472	−0.0935731
$958$	0	0
$959$	49.3129	1.59240
$960$	0	0
$961$	56.3256	1.81695
$962$	0	0
$963$	22.0368	0.710127
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−7.63102	−0.245397	−0.122699	−	0.992444i	$-0.539155\pi$
$967$	−7.63102	−0.245397	−0.122699	+	0.992444i	$0.539155\pi$
$968$	0	0
$969$	1.19806	0.0384873
$970$	0	0
$971$	19.3599	0.621288	0.310644	−	0.950526i	$-0.399455\pi$
$971$	19.3599	0.621288	0.310644	+	0.950526i	$0.399455\pi$
$972$	0	0
$973$	39.9342	1.28023
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−55.8883	−1.78802	−0.894012	−	0.448042i	$-0.852121\pi$
$977$	−55.8883	−1.78802	−0.894012	+	0.448042i	$0.852121\pi$
$978$	0	0
$979$	−30.2241	−0.965968
$980$	0	0
$981$	30.4935	0.973582
$982$	0	0
$983$	−4.58450	−0.146223	−0.0731114	−	0.997324i	$-0.523293\pi$
$983$	−4.58450	−0.146223	−0.0731114	+	0.997324i	$0.523293\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.98792	−0.126937
$988$	0	0
$989$	6.31873	0.200924
$990$	0	0
$991$	−27.5666	−0.875681	−0.437840	−	0.899053i	$-0.644257\pi$
$991$	−27.5666	−0.875681	−0.437840	+	0.899053i	$0.644257\pi$
$992$	0	0
$993$	7.34481	0.233081
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−15.9119	−0.503933	−0.251967	−	0.967736i	$-0.581077\pi$
$997$	−15.9119	−0.503933	−0.251967	+	0.967736i	$0.581077\pi$
$998$	0	0
$999$	−3.66056	−0.115815

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.bw.1.2		3
4.3	odd	2		475.2.a.d.1.1	✓	3
5.4	even	2		7600.2.a.bn.1.2		3
12.11	even	2		4275.2.a.bn.1.3		3
20.3	even	4		475.2.b.c.324.6		6
20.7	even	4		475.2.b.c.324.1		6
20.19	odd	2		475.2.a.h.1.3	yes	3
60.59	even	2		4275.2.a.z.1.1		3
76.75	even	2		9025.2.a.be.1.3		3
380.379	even	2		9025.2.a.w.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.d.1.1	✓	3	4.3	odd	2
475.2.a.h.1.3	yes	3	20.19	odd	2
475.2.b.c.324.1		6	20.7	even	4
475.2.b.c.324.6		6	20.3	even	4
4275.2.a.z.1.1		3	60.59	even	2
4275.2.a.bn.1.3		3	12.11	even	2
7600.2.a.bn.1.2		3	5.4	even	2
7600.2.a.bw.1.2		3	1.1	even	1	trivial
9025.2.a.w.1.1		3	380.379	even	2
9025.2.a.be.1.3		3	76.75	even	2

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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+0.554958 q^{3} -3.04892 q^{7} -2.69202 q^{9} +2.93900 q^{11} -3.24698 q^{13} +2.15883 q^{17} +1.00000 q^{19} -1.69202 q^{21} +1.19806 q^{23} -3.15883 q^{27} -1.77479 q^{29} +9.34481 q^{31} +1.63102 q^{33} +1.15883 q^{37} -1.80194 q^{39} +8.57002 q^{41} +5.27413 q^{43} +2.35690 q^{47} +2.29590 q^{49} +1.19806 q^{51} -8.82371 q^{53} +0.554958 q^{57} +5.70171 q^{59} -9.96077 q^{61} +8.20775 q^{63} +4.98254 q^{67} +0.664874 q^{69} -2.70171 q^{71} -13.7778 q^{73} -8.96077 q^{77} -5.66487 q^{79} +6.32304 q^{81} -3.00969 q^{83} -0.984935 q^{87} -10.2838 q^{89} +9.89977 q^{91} +5.18598 q^{93} -3.24698 q^{97} -7.91185 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{3} - 3 q^{9} - q^{11} - 5 q^{13} - 2 q^{17} + 3 q^{19} + 8 q^{23} - q^{27} - 7 q^{29} + 5 q^{31} - 10 q^{33} - 5 q^{37} - q^{39} + q^{41} + 5 q^{43} + 3 q^{47} - 7 q^{49} + 8 q^{51} - 19 q^{53}+ \cdots - 20 q^{99}+O(q^{100}) \) 3 * q + 2 * q^3 - 3 * q^9 - q^11 - 5 * q^13 - 2 * q^17 + 3 * q^19 + 8 * q^23 - q^27 - 7 * q^29 + 5 * q^31 - 10 * q^33 - 5 * q^37 - q^39 + q^41 + 5 * q^43 + 3 * q^47 - 7 * q^49 + 8 * q^51 - 19 * q^53 + 2 * q^57 - 10 * q^59 - 17 * q^61 + 7 * q^63 - q^67 + 3 * q^69 + 19 * q^71 + q^73 - 14 * q^77 - 18 * q^79 - q^81 + 13 * q^83 - 28 * q^87 + 2 * q^89 + 7 * q^91 + q^93 - 5 * q^97 - 20 * q^99

Introduction

L-functions

Modular forms

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