LMFDB - Embedded newform 9196.2.a.o.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9196,2,Mod(1,9196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9196, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("9196.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,-1,0,3,0,-2,0,8,0,0,0,0] 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:	$73.4304296988$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 14x^{5} + 8x^{4} + 46x^{3} - 2x^{2} - 48x - 21 $ x^7 - x^6 - 14x^5 + 8x^4 + 46x^3 - 2x^2 - 48*x - 21
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.26735$ of defining polynomial
Character	$\chi$	$=$	9196.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.26735 q^{3} +3.95060 q^{5} -4.74040 q^{7} -1.39382 q^{9} -3.37193 q^{13} +5.00680 q^{15} +1.77379 q^{17} -1.00000 q^{19} -6.00775 q^{21} +6.66797 q^{23} +10.6073 q^{25} -5.56851 q^{27} +2.49494 q^{29} +3.70606 q^{31} -18.7274 q^{35} -4.97929 q^{37} -4.27342 q^{39} -8.19238 q^{41} -6.63677 q^{43} -5.50644 q^{45} -5.30576 q^{47} +15.4714 q^{49} +2.24801 q^{51} -2.97697 q^{53} -1.26735 q^{57} -11.0538 q^{59} +7.58735 q^{61} +6.60727 q^{63} -13.3212 q^{65} -12.7655 q^{67} +8.45066 q^{69} -7.43497 q^{71} +8.29154 q^{73} +13.4431 q^{75} +3.05245 q^{79} -2.87580 q^{81} -5.54077 q^{83} +7.00753 q^{85} +3.16196 q^{87} -4.63855 q^{89} +15.9843 q^{91} +4.69688 q^{93} -3.95060 q^{95} -8.56613 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{3} + 3 q^{5} - 2 q^{7} + 8 q^{9} + 11 q^{15} - 12 q^{17} - 7 q^{19} - q^{21} - 5 q^{23} + 4 q^{25} - 13 q^{27} + 2 q^{29} + 5 q^{31} - 16 q^{35} + 8 q^{37} - 15 q^{39} - 31 q^{41} - 12 q^{43}+ \cdots - 25 q^{97}+O(q^{100}) $ 7 * q - q^3 + 3 * q^5 - 2 * q^7 + 8 * q^9 + 11 * q^15 - 12 * q^17 - 7 * q^19 - q^21 - 5 * q^23 + 4 * q^25 - 13 * q^27 + 2 * q^29 + 5 * q^31 - 16 * q^35 + 8 * q^37 - 15 * q^39 - 31 * q^41 - 12 * q^43 - 24 * q^45 + 10 * q^47 - 3 * q^49 - 5 * q^51 - 25 * q^53 + q^57 - 7 * q^59 - q^61 - 24 * q^63 - 13 * q^65 - 5 * q^67 + 31 * q^69 + q^71 + 12 * q^73 - 12 * q^75 - 20 * q^79 + 35 * q^81 - 14 * q^83 + 6 * q^85 - 28 * q^87 - 18 * q^89 + 25 * q^91 - 35 * q^93 - 3 * q^95 - 25 * q^97

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
$3$	1.26735	0.731705	0.365853	−	0.930673i	$-0.380777\pi$
$3$	1.26735	0.731705	0.365853	+	0.930673i	$0.380777\pi$
$4$	0	0
$5$	3.95060	1.76676	0.883382	−	0.468654i	$-0.155261\pi$
$5$	3.95060	1.76676	0.883382	+	0.468654i	$0.155261\pi$
$6$	0	0
$7$	−4.74040	−1.79170	−0.895852	−	0.444353i	$-0.853433\pi$
$7$	−4.74040	−1.79170	−0.895852	+	0.444353i	$0.853433\pi$
$8$	0	0
$9$	−1.39382	−0.464607
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.37193	−0.935206	−0.467603	−	0.883939i	$-0.654882\pi$
$13$	−3.37193	−0.935206	−0.467603	+	0.883939i	$0.654882\pi$
$14$	0	0
$15$	5.00680	1.29275
$16$	0	0
$17$	1.77379	0.430207	0.215103	−	0.976591i	$-0.430991\pi$
$17$	1.77379	0.430207	0.215103	+	0.976591i	$0.430991\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−6.00775	−1.31100
$22$	0	0
$23$	6.66797	1.39037	0.695184	−	0.718831i	$-0.255323\pi$
$23$	6.66797	1.39037	0.695184	+	0.718831i	$0.255323\pi$
$24$	0	0
$25$	10.6073	2.12145
$26$	0	0
$27$	−5.56851	−1.07166
$28$	0	0
$29$	2.49494	0.463299	0.231649	−	0.972799i	$-0.425588\pi$
$29$	2.49494	0.463299	0.231649	+	0.972799i	$0.425588\pi$
$30$	0	0
$31$	3.70606	0.665629	0.332814	−	0.942992i	$-0.392002\pi$
$31$	3.70606	0.665629	0.332814	+	0.942992i	$0.392002\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−18.7274	−3.16552
$36$	0	0
$37$	−4.97929	−0.818591	−0.409296	−	0.912402i	$-0.634225\pi$
$37$	−4.97929	−0.818591	−0.409296	+	0.912402i	$0.634225\pi$
$38$	0	0
$39$	−4.27342	−0.684295
$40$	0	0
$41$	−8.19238	−1.27944	−0.639718	−	0.768610i	$-0.720949\pi$
$41$	−8.19238	−1.27944	−0.639718	+	0.768610i	$0.720949\pi$
$42$	0	0
$43$	−6.63677	−1.01210	−0.506049	−	0.862505i	$-0.668894\pi$
$43$	−6.63677	−1.01210	−0.506049	+	0.862505i	$0.668894\pi$
$44$	0	0
$45$	−5.50644	−0.820851
$46$	0	0
$47$	−5.30576	−0.773924	−0.386962	−	0.922096i	$-0.626476\pi$
$47$	−5.30576	−0.773924	−0.386962	+	0.922096i	$0.626476\pi$
$48$	0	0
$49$	15.4714	2.21020
$50$	0	0
$51$	2.24801	0.314785
$52$	0	0
$53$	−2.97697	−0.408918	−0.204459	−	0.978875i	$-0.565544\pi$
$53$	−2.97697	−0.408918	−0.204459	+	0.978875i	$0.565544\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.26735	−0.167865
$58$	0	0
$59$	−11.0538	−1.43908	−0.719542	−	0.694449i	$-0.755648\pi$
$59$	−11.0538	−1.43908	−0.719542	+	0.694449i	$0.755648\pi$
$60$	0	0
$61$	7.58735	0.971460	0.485730	−	0.874109i	$-0.338554\pi$
$61$	7.58735	0.971460	0.485730	+	0.874109i	$0.338554\pi$
$62$	0	0
$63$	6.60727	0.832438
$64$	0	0
$65$	−13.3212	−1.65229
$66$	0	0
$67$	−12.7655	−1.55956	−0.779779	−	0.626054i	$-0.784669\pi$
$67$	−12.7655	−1.55956	−0.779779	+	0.626054i	$0.784669\pi$
$68$	0	0
$69$	8.45066	1.01734
$70$	0	0
$71$	−7.43497	−0.882368	−0.441184	−	0.897417i	$-0.645441\pi$
$71$	−7.43497	−0.882368	−0.441184	+	0.897417i	$0.645441\pi$
$72$	0	0
$73$	8.29154	0.970451	0.485226	−	0.874389i	$-0.338737\pi$
$73$	8.29154	0.970451	0.485226	+	0.874389i	$0.338737\pi$
$74$	0	0
$75$	13.4431	1.55228
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.05245	0.343428	0.171714	−	0.985147i	$-0.445070\pi$
$79$	3.05245	0.343428	0.171714	+	0.985147i	$0.445070\pi$
$80$	0	0
$81$	−2.87580	−0.319533
$82$	0	0
$83$	−5.54077	−0.608179	−0.304089	−	0.952643i	$-0.598352\pi$
$83$	−5.54077	−0.608179	−0.304089	+	0.952643i	$0.598352\pi$
$84$	0	0
$85$	7.00753	0.760074
$86$	0	0
$87$	3.16196	0.338998
$88$	0	0
$89$	−4.63855	−0.491686	−0.245843	−	0.969310i	$-0.579065\pi$
$89$	−4.63855	−0.491686	−0.245843	+	0.969310i	$0.579065\pi$
$90$	0	0
$91$	15.9843	1.67561
$92$	0	0
$93$	4.69688	0.487044
$94$	0	0
$95$	−3.95060	−0.405323
$96$	0	0
$97$	−8.56613	−0.869758	−0.434879	−	0.900489i	$-0.643209\pi$
$97$	−8.56613	−0.869758	−0.434879	+	0.900489i	$0.643209\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.4199	−1.13632	−0.568162	−	0.822917i	$-0.692345\pi$
$101$	−11.4199	−1.13632	−0.568162	+	0.822917i	$0.692345\pi$
$102$	0	0
$103$	−11.6580	−1.14870	−0.574350	−	0.818610i	$-0.694745\pi$
$103$	−11.6580	−1.14870	−0.574350	+	0.818610i	$0.694745\pi$
$104$	0	0
$105$	−23.7343	−2.31623
$106$	0	0
$107$	11.6641	1.12761	0.563804	−	0.825909i	$-0.309337\pi$
$107$	11.6641	1.12761	0.563804	+	0.825909i	$0.309337\pi$
$108$	0	0
$109$	−12.6240	−1.20916	−0.604580	−	0.796544i	$-0.706659\pi$
$109$	−12.6240	−1.20916	−0.604580	+	0.796544i	$0.706659\pi$
$110$	0	0
$111$	−6.31051	−0.598968
$112$	0	0
$113$	16.3065	1.53399	0.766995	−	0.641653i	$-0.221751\pi$
$113$	16.3065	1.53399	0.766995	+	0.641653i	$0.221751\pi$
$114$	0	0
$115$	26.3425	2.45645
$116$	0	0
$117$	4.69987	0.434503
$118$	0	0
$119$	−8.40847	−0.770803
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−10.3826	−0.936170
$124$	0	0
$125$	22.1521	1.98135
$126$	0	0
$127$	−9.77625	−0.867502	−0.433751	−	0.901033i	$-0.642810\pi$
$127$	−9.77625	−0.867502	−0.433751	+	0.901033i	$0.642810\pi$
$128$	0	0
$129$	−8.41112	−0.740558
$130$	0	0
$131$	3.27480	0.286120	0.143060	−	0.989714i	$-0.454306\pi$
$131$	3.27480	0.286120	0.143060	+	0.989714i	$0.454306\pi$
$132$	0	0
$133$	4.74040	0.411045
$134$	0	0
$135$	−21.9990	−1.89337
$136$	0	0
$137$	12.5617	1.07322	0.536611	−	0.843830i	$-0.319704\pi$
$137$	12.5617	1.07322	0.536611	+	0.843830i	$0.319704\pi$
$138$	0	0
$139$	14.7161	1.24820	0.624102	−	0.781343i	$-0.285465\pi$
$139$	14.7161	1.24820	0.624102	+	0.781343i	$0.285465\pi$
$140$	0	0
$141$	−6.72426	−0.566284
$142$	0	0
$143$	0	0
$144$	0	0
$145$	9.85652	0.818539
$146$	0	0
$147$	19.6077	1.61722
$148$	0	0
$149$	14.9914	1.22815	0.614073	−	0.789249i	$-0.289530\pi$
$149$	14.9914	1.22815	0.614073	+	0.789249i	$0.289530\pi$
$150$	0	0
$151$	−18.1159	−1.47425	−0.737125	−	0.675756i	$-0.763817\pi$
$151$	−18.1159	−1.47425	−0.737125	+	0.675756i	$0.763817\pi$
$152$	0	0
$153$	−2.47234	−0.199877
$154$	0	0
$155$	14.6412	1.17601
$156$	0	0
$157$	−5.10143	−0.407139	−0.203569	−	0.979061i	$-0.565254\pi$
$157$	−5.10143	−0.407139	−0.203569	+	0.979061i	$0.565254\pi$
$158$	0	0
$159$	−3.77286	−0.299208
$160$	0	0
$161$	−31.6089	−2.49113
$162$	0	0
$163$	−13.8051	−1.08130	−0.540650	−	0.841248i	$-0.681822\pi$
$163$	−13.8051	−1.08130	−0.540650	+	0.841248i	$0.681822\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.53772	−0.196375	−0.0981875	−	0.995168i	$-0.531304\pi$
$167$	−2.53772	−0.196375	−0.0981875	+	0.995168i	$0.531304\pi$
$168$	0	0
$169$	−1.63007	−0.125390
$170$	0	0
$171$	1.39382	0.106588
$172$	0	0
$173$	−21.0192	−1.59806	−0.799030	−	0.601291i	$-0.794653\pi$
$173$	−21.0192	−1.59806	−0.799030	+	0.601291i	$0.794653\pi$
$174$	0	0
$175$	−50.2827	−3.80102
$176$	0	0
$177$	−14.0091	−1.05299
$178$	0	0
$179$	−6.06440	−0.453275	−0.226637	−	0.973979i	$-0.572773\pi$
$179$	−6.06440	−0.453275	−0.226637	+	0.973979i	$0.572773\pi$
$180$	0	0
$181$	−26.2273	−1.94946	−0.974730	−	0.223388i	$-0.928288\pi$
$181$	−26.2273	−1.94946	−0.974730	+	0.223388i	$0.928288\pi$
$182$	0	0
$183$	9.61583	0.710823
$184$	0	0
$185$	−19.6712	−1.44626
$186$	0	0
$187$	0	0
$188$	0	0
$189$	26.3970	1.92010
$190$	0	0
$191$	16.8085	1.21622	0.608110	−	0.793853i	$-0.291928\pi$
$191$	16.8085	1.21622	0.608110	+	0.793853i	$0.291928\pi$
$192$	0	0
$193$	7.85730	0.565581	0.282791	−	0.959182i	$-0.408740\pi$
$193$	7.85730	0.565581	0.282791	+	0.959182i	$0.408740\pi$
$194$	0	0
$195$	−16.8826	−1.20899
$196$	0	0
$197$	16.2833	1.16014	0.580068	−	0.814568i	$-0.303026\pi$
$197$	16.2833	1.16014	0.580068	+	0.814568i	$0.303026\pi$
$198$	0	0
$199$	18.9502	1.34334	0.671672	−	0.740849i	$-0.265576\pi$
$199$	18.9502	1.34334	0.671672	+	0.740849i	$0.265576\pi$
$200$	0	0
$201$	−16.1784	−1.14114
$202$	0	0
$203$	−11.8270	−0.830093
$204$	0	0
$205$	−32.3649	−2.26046
$206$	0	0
$207$	−9.29397	−0.645975
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−11.5161	−0.792800	−0.396400	−	0.918078i	$-0.629741\pi$
$211$	−11.5161	−0.792800	−0.396400	+	0.918078i	$0.629741\pi$
$212$	0	0
$213$	−9.42271	−0.645634
$214$	0	0
$215$	−26.2192	−1.78814
$216$	0	0
$217$	−17.5682	−1.19261
$218$	0	0
$219$	10.5083	0.710084
$220$	0	0
$221$	−5.98109	−0.402332
$222$	0	0
$223$	−22.4362	−1.50244	−0.751219	−	0.660053i	$-0.770534\pi$
$223$	−22.4362	−1.50244	−0.751219	+	0.660053i	$0.770534\pi$
$224$	0	0
$225$	−14.7846	−0.985643
$226$	0	0
$227$	25.1061	1.66635	0.833174	−	0.553011i	$-0.186521\pi$
$227$	25.1061	1.66635	0.833174	+	0.553011i	$0.186521\pi$
$228$	0	0
$229$	−24.8026	−1.63900	−0.819501	−	0.573077i	$-0.805749\pi$
$229$	−24.8026	−1.63900	−0.819501	+	0.573077i	$0.805749\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.93408	0.519779	0.259889	−	0.965638i	$-0.416314\pi$
$233$	7.93408	0.519779	0.259889	+	0.965638i	$0.416314\pi$
$234$	0	0
$235$	−20.9609	−1.36734
$236$	0	0
$237$	3.86853	0.251288
$238$	0	0
$239$	−22.8546	−1.47834	−0.739169	−	0.673520i	$-0.764782\pi$
$239$	−22.8546	−1.47834	−0.739169	+	0.673520i	$0.764782\pi$
$240$	0	0
$241$	−23.7712	−1.53124	−0.765618	−	0.643296i	$-0.777567\pi$
$241$	−23.7712	−1.53124	−0.765618	+	0.643296i	$0.777567\pi$
$242$	0	0
$243$	13.0609	0.837857
$244$	0	0
$245$	61.1214	3.90490
$246$	0	0
$247$	3.37193	0.214551
$248$	0	0
$249$	−7.02210	−0.445008
$250$	0	0
$251$	−9.14370	−0.577145	−0.288573	−	0.957458i	$-0.593181\pi$
$251$	−9.14370	−0.577145	−0.288573	+	0.957458i	$0.593181\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	8.88100	0.556150
$256$	0	0
$257$	−4.60674	−0.287361	−0.143680	−	0.989624i	$-0.545894\pi$
$257$	−4.60674	−0.287361	−0.143680	+	0.989624i	$0.545894\pi$
$258$	0	0
$259$	23.6039	1.46667
$260$	0	0
$261$	−3.47750	−0.215252
$262$	0	0
$263$	−0.916322	−0.0565028	−0.0282514	−	0.999601i	$-0.508994\pi$
$263$	−0.916322	−0.0565028	−0.0282514	+	0.999601i	$0.508994\pi$
$264$	0	0
$265$	−11.7608	−0.722462
$266$	0	0
$267$	−5.87867	−0.359769
$268$	0	0
$269$	3.05438	0.186229	0.0931144	−	0.995655i	$-0.470318\pi$
$269$	3.05438	0.186229	0.0931144	+	0.995655i	$0.470318\pi$
$270$	0	0
$271$	−15.5757	−0.946155	−0.473078	−	0.881021i	$-0.656857\pi$
$271$	−15.5757	−0.946155	−0.473078	+	0.881021i	$0.656857\pi$
$272$	0	0
$273$	20.2577	1.22605
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.4770	0.629504	0.314752	−	0.949174i	$-0.398079\pi$
$277$	10.4770	0.629504	0.314752	+	0.949174i	$0.398079\pi$
$278$	0	0
$279$	−5.16559	−0.309256
$280$	0	0
$281$	−21.7029	−1.29469	−0.647343	−	0.762199i	$-0.724120\pi$
$281$	−21.7029	−1.29469	−0.647343	+	0.762199i	$0.724120\pi$
$282$	0	0
$283$	−10.7991	−0.641941	−0.320971	−	0.947089i	$-0.604009\pi$
$283$	−10.7991	−0.641941	−0.320971	+	0.947089i	$0.604009\pi$
$284$	0	0
$285$	−5.00680	−0.296577
$286$	0	0
$287$	38.8352	2.29237
$288$	0	0
$289$	−13.8537	−0.814922
$290$	0	0
$291$	−10.8563	−0.636407
$292$	0	0
$293$	16.5069	0.964343	0.482171	−	0.876077i	$-0.339848\pi$
$293$	16.5069	0.964343	0.482171	+	0.876077i	$0.339848\pi$
$294$	0	0
$295$	−43.6693	−2.54252
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−22.4840	−1.30028
$300$	0	0
$301$	31.4610	1.81338
$302$	0	0
$303$	−14.4730	−0.831454
$304$	0	0
$305$	29.9746	1.71634
$306$	0	0
$307$	31.1644	1.77865	0.889324	−	0.457277i	$-0.151175\pi$
$307$	31.1644	1.77865	0.889324	+	0.457277i	$0.151175\pi$
$308$	0	0
$309$	−14.7748	−0.840510
$310$	0	0
$311$	21.8797	1.24068	0.620342	−	0.784332i	$-0.286994\pi$
$311$	21.8797	1.24068	0.620342	+	0.784332i	$0.286994\pi$
$312$	0	0
$313$	−25.6800	−1.45152	−0.725758	−	0.687950i	$-0.758511\pi$
$313$	−25.6800	−1.45152	−0.725758	+	0.687950i	$0.758511\pi$
$314$	0	0
$315$	26.1027	1.47072
$316$	0	0
$317$	12.6850	0.712463	0.356231	−	0.934398i	$-0.384061\pi$
$317$	12.6850	0.712463	0.356231	+	0.934398i	$0.384061\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	14.7825	0.825077
$322$	0	0
$323$	−1.77379	−0.0986962
$324$	0	0
$325$	−35.7670	−1.98400
$326$	0	0
$327$	−15.9990	−0.884749
$328$	0	0
$329$	25.1514	1.38664
$330$	0	0
$331$	−9.06997	−0.498531	−0.249265	−	0.968435i	$-0.580189\pi$
$331$	−9.06997	−0.498531	−0.249265	+	0.968435i	$0.580189\pi$
$332$	0	0
$333$	6.94025	0.380323
$334$	0	0
$335$	−50.4316	−2.75537
$336$	0	0
$337$	6.72107	0.366120	0.183060	−	0.983102i	$-0.441400\pi$
$337$	6.72107	0.366120	0.183060	+	0.983102i	$0.441400\pi$
$338$	0	0
$339$	20.6661	1.12243
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−40.1578	−2.16832
$344$	0	0
$345$	33.3852	1.79740
$346$	0	0
$347$	−4.49308	−0.241201	−0.120601	−	0.992701i	$-0.538482\pi$
$347$	−4.49308	−0.241201	−0.120601	+	0.992701i	$0.538482\pi$
$348$	0	0
$349$	−14.2906	−0.764960	−0.382480	−	0.923964i	$-0.624930\pi$
$349$	−14.2906	−0.764960	−0.382480	+	0.923964i	$0.624930\pi$
$350$	0	0
$351$	18.7767	1.00222
$352$	0	0
$353$	1.61433	0.0859221	0.0429610	−	0.999077i	$-0.486321\pi$
$353$	1.61433	0.0859221	0.0429610	+	0.999077i	$0.486321\pi$
$354$	0	0
$355$	−29.3726	−1.55894
$356$	0	0
$357$	−10.6565	−0.564001
$358$	0	0
$359$	−36.9969	−1.95262	−0.976310	−	0.216375i	$-0.930577\pi$
$359$	−36.9969	−1.95262	−0.976310	+	0.216375i	$0.930577\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	32.7566	1.71456
$366$	0	0
$367$	12.2236	0.638067	0.319034	−	0.947743i	$-0.396642\pi$
$367$	12.2236	0.638067	0.319034	+	0.947743i	$0.396642\pi$
$368$	0	0
$369$	11.4187	0.594435
$370$	0	0
$371$	14.1120	0.732660
$372$	0	0
$373$	20.4230	1.05746	0.528732	−	0.848789i	$-0.322668\pi$
$373$	20.4230	1.05746	0.528732	+	0.848789i	$0.322668\pi$
$374$	0	0
$375$	28.0745	1.44976
$376$	0	0
$377$	−8.41277	−0.433279
$378$	0	0
$379$	5.03991	0.258883	0.129441	−	0.991587i	$-0.458682\pi$
$379$	5.03991	0.258883	0.129441	+	0.991587i	$0.458682\pi$
$380$	0	0
$381$	−12.3899	−0.634756
$382$	0	0
$383$	−8.35130	−0.426732	−0.213366	−	0.976972i	$-0.568443\pi$
$383$	−8.35130	−0.426732	−0.213366	+	0.976972i	$0.568443\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	9.25047	0.470228
$388$	0	0
$389$	34.6403	1.75633	0.878167	−	0.478355i	$-0.158767\pi$
$389$	34.6403	1.75633	0.878167	+	0.478355i	$0.158767\pi$
$390$	0	0
$391$	11.8276	0.598146
$392$	0	0
$393$	4.15032	0.209356
$394$	0	0
$395$	12.0590	0.606756
$396$	0	0
$397$	33.9984	1.70633	0.853167	−	0.521639i	$-0.174679\pi$
$397$	33.9984	1.70633	0.853167	+	0.521639i	$0.174679\pi$
$398$	0	0
$399$	6.00775	0.300764
$400$	0	0
$401$	28.8161	1.43901	0.719505	−	0.694488i	$-0.244369\pi$
$401$	28.8161	1.43901	0.719505	+	0.694488i	$0.244369\pi$
$402$	0	0
$403$	−12.4966	−0.622500
$404$	0	0
$405$	−11.3611	−0.564539
$406$	0	0
$407$	0	0
$408$	0	0
$409$	21.5764	1.06688	0.533442	−	0.845837i	$-0.320898\pi$
$409$	21.5764	1.06688	0.533442	+	0.845837i	$0.320898\pi$
$410$	0	0
$411$	15.9201	0.785282
$412$	0	0
$413$	52.3995	2.57841
$414$	0	0
$415$	−21.8894	−1.07451
$416$	0	0
$417$	18.6505	0.913318
$418$	0	0
$419$	−24.5709	−1.20037	−0.600184	−	0.799862i	$-0.704906\pi$
$419$	−24.5709	−1.20037	−0.600184	+	0.799862i	$0.704906\pi$
$420$	0	0
$421$	−21.5957	−1.05251	−0.526255	−	0.850327i	$-0.676404\pi$
$421$	−21.5957	−1.05251	−0.526255	+	0.850327i	$0.676404\pi$
$422$	0	0
$423$	7.39528	0.359571
$424$	0	0
$425$	18.8150	0.912664
$426$	0	0
$427$	−35.9671	−1.74057
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.4339	−1.17694	−0.588469	−	0.808520i	$-0.700269\pi$
$431$	−24.4339	−1.17694	−0.588469	+	0.808520i	$0.700269\pi$
$432$	0	0
$433$	8.73113	0.419591	0.209796	−	0.977745i	$-0.432720\pi$
$433$	8.73113	0.419591	0.209796	+	0.977745i	$0.432720\pi$
$434$	0	0
$435$	12.4917	0.598930
$436$	0	0
$437$	−6.66797	−0.318972
$438$	0	0
$439$	−8.38421	−0.400156	−0.200078	−	0.979780i	$-0.564120\pi$
$439$	−8.38421	−0.400156	−0.200078	+	0.979780i	$0.564120\pi$
$440$	0	0
$441$	−21.5644	−1.02687
$442$	0	0
$443$	16.6717	0.792097	0.396048	−	0.918230i	$-0.370381\pi$
$443$	16.6717	0.792097	0.396048	+	0.918230i	$0.370381\pi$
$444$	0	0
$445$	−18.3251	−0.868692
$446$	0	0
$447$	18.9994	0.898641
$448$	0	0
$449$	−0.650289	−0.0306890	−0.0153445	−	0.999882i	$-0.504885\pi$
$449$	−0.650289	−0.0306890	−0.0153445	+	0.999882i	$0.504885\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−22.9592	−1.07872
$454$	0	0
$455$	63.1477	2.96041
$456$	0	0
$457$	−14.6265	−0.684198	−0.342099	−	0.939664i	$-0.611138\pi$
$457$	−14.6265	−0.684198	−0.342099	+	0.939664i	$0.611138\pi$
$458$	0	0
$459$	−9.87736	−0.461036
$460$	0	0
$461$	29.0733	1.35408	0.677039	−	0.735947i	$-0.263263\pi$
$461$	29.0733	1.35408	0.677039	+	0.735947i	$0.263263\pi$
$462$	0	0
$463$	7.08958	0.329481	0.164740	−	0.986337i	$-0.447321\pi$
$463$	7.08958	0.329481	0.164740	+	0.986337i	$0.447321\pi$
$464$	0	0
$465$	18.5555	0.860492
$466$	0	0
$467$	−16.7081	−0.773160	−0.386580	−	0.922256i	$-0.626344\pi$
$467$	−16.7081	−0.773160	−0.386580	+	0.922256i	$0.626344\pi$
$468$	0	0
$469$	60.5138	2.79427
$470$	0	0
$471$	−6.46531	−0.297906
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−10.6073	−0.486695
$476$	0	0
$477$	4.14936	0.189986
$478$	0	0
$479$	29.4687	1.34646	0.673230	−	0.739433i	$-0.264906\pi$
$479$	29.4687	1.34646	0.673230	+	0.739433i	$0.264906\pi$
$480$	0	0
$481$	16.7898	0.765551
$482$	0	0
$483$	−40.0595	−1.82277
$484$	0	0
$485$	−33.8414	−1.53666
$486$	0	0
$487$	−32.1707	−1.45779	−0.728897	−	0.684624i	$-0.759967\pi$
$487$	−32.1707	−1.45779	−0.728897	+	0.684624i	$0.759967\pi$
$488$	0	0
$489$	−17.4959	−0.791193
$490$	0	0
$491$	−23.2604	−1.04973	−0.524865	−	0.851186i	$-0.675884\pi$
$491$	−23.2604	−1.04973	−0.524865	+	0.851186i	$0.675884\pi$
$492$	0	0
$493$	4.42549	0.199314
$494$	0	0
$495$	0	0
$496$	0	0
$497$	35.2447	1.58094
$498$	0	0
$499$	−9.30468	−0.416535	−0.208267	−	0.978072i	$-0.566782\pi$
$499$	−9.30468	−0.416535	−0.208267	+	0.978072i	$0.566782\pi$
$500$	0	0
$501$	−3.21619	−0.143689
$502$	0	0
$503$	6.43825	0.287067	0.143534	−	0.989645i	$-0.454153\pi$
$503$	6.43825	0.287067	0.143534	+	0.989645i	$0.454153\pi$
$504$	0	0
$505$	−45.1155	−2.00762
$506$	0	0
$507$	−2.06587	−0.0917485
$508$	0	0
$509$	−3.63925	−0.161307	−0.0806534	−	0.996742i	$-0.525701\pi$
$509$	−3.63925	−0.161307	−0.0806534	+	0.996742i	$0.525701\pi$
$510$	0	0
$511$	−39.3052	−1.73876
$512$	0	0
$513$	5.56851	0.245856
$514$	0	0
$515$	−46.0563	−2.02948
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−26.6387	−1.16931
$520$	0	0
$521$	−42.0264	−1.84121	−0.920605	−	0.390495i	$-0.872304\pi$
$521$	−42.0264	−1.84121	−0.920605	+	0.390495i	$0.872304\pi$
$522$	0	0
$523$	20.5021	0.896492	0.448246	−	0.893910i	$-0.352049\pi$
$523$	20.5021	0.896492	0.448246	+	0.893910i	$0.352049\pi$
$524$	0	0
$525$	−63.7259	−2.78122
$526$	0	0
$527$	6.57377	0.286358
$528$	0	0
$529$	21.4619	0.933125
$530$	0	0
$531$	15.4071	0.668609
$532$	0	0
$533$	27.6242	1.19654
$534$	0	0
$535$	46.0801	1.99222
$536$	0	0
$537$	−7.68573	−0.331664
$538$	0	0
$539$	0	0
$540$	0	0
$541$	4.15919	0.178817	0.0894087	−	0.995995i	$-0.471502\pi$
$541$	4.15919	0.178817	0.0894087	+	0.995995i	$0.471502\pi$
$542$	0	0
$543$	−33.2392	−1.42643
$544$	0	0
$545$	−49.8724	−2.13630
$546$	0	0
$547$	−29.1955	−1.24831	−0.624155	−	0.781300i	$-0.714557\pi$
$547$	−29.1955	−1.24831	−0.624155	+	0.781300i	$0.714557\pi$
$548$	0	0
$549$	−10.5754	−0.451347
$550$	0	0
$551$	−2.49494	−0.106288
$552$	0	0
$553$	−14.4698	−0.615321
$554$	0	0
$555$	−24.9303	−1.05823
$556$	0	0
$557$	8.04380	0.340827	0.170413	−	0.985373i	$-0.445490\pi$
$557$	8.04380	0.340827	0.170413	+	0.985373i	$0.445490\pi$
$558$	0	0
$559$	22.3787	0.946520
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8.16644	0.344174	0.172087	−	0.985082i	$-0.444949\pi$
$563$	8.16644	0.344174	0.172087	+	0.985082i	$0.444949\pi$
$564$	0	0
$565$	64.4207	2.71020
$566$	0	0
$567$	13.6324	0.572508
$568$	0	0
$569$	5.47550	0.229545	0.114772	−	0.993392i	$-0.463386\pi$
$569$	5.47550	0.229545	0.114772	+	0.993392i	$0.463386\pi$
$570$	0	0
$571$	−20.3843	−0.853055	−0.426527	−	0.904475i	$-0.640263\pi$
$571$	−20.3843	−0.853055	−0.426527	+	0.904475i	$0.640263\pi$
$572$	0	0
$573$	21.3023	0.889915
$574$	0	0
$575$	70.7290	2.94960
$576$	0	0
$577$	13.3538	0.555928	0.277964	−	0.960591i	$-0.410340\pi$
$577$	13.3538	0.555928	0.277964	+	0.960591i	$0.410340\pi$
$578$	0	0
$579$	9.95796	0.413839
$580$	0	0
$581$	26.2655	1.08968
$582$	0	0
$583$	0	0
$584$	0	0
$585$	18.5673	0.767665
$586$	0	0
$587$	−11.7290	−0.484109	−0.242055	−	0.970263i	$-0.577821\pi$
$587$	−11.7290	−0.484109	−0.242055	+	0.970263i	$0.577821\pi$
$588$	0	0
$589$	−3.70606	−0.152706
$590$	0	0
$591$	20.6366	0.848878
$592$	0	0
$593$	−6.42487	−0.263838	−0.131919	−	0.991261i	$-0.542114\pi$
$593$	−6.42487	−0.263838	−0.131919	+	0.991261i	$0.542114\pi$
$594$	0	0
$595$	−33.2185	−1.36183
$596$	0	0
$597$	24.0166	0.982932
$598$	0	0
$599$	38.8562	1.58762	0.793812	−	0.608164i	$-0.208094\pi$
$599$	38.8562	1.58762	0.793812	+	0.608164i	$0.208094\pi$
$600$	0	0
$601$	19.3623	0.789806	0.394903	−	0.918723i	$-0.370778\pi$
$601$	19.3623	0.789806	0.394903	+	0.918723i	$0.370778\pi$
$602$	0	0
$603$	17.7929	0.724582
$604$	0	0
$605$	0	0
$606$	0	0
$607$	9.86272	0.400315	0.200158	−	0.979764i	$-0.435855\pi$
$607$	9.86272	0.400315	0.200158	+	0.979764i	$0.435855\pi$
$608$	0	0
$609$	−14.9890	−0.607384
$610$	0	0
$611$	17.8907	0.723778
$612$	0	0
$613$	−29.7988	−1.20356	−0.601782	−	0.798660i	$-0.705542\pi$
$613$	−29.7988	−1.20356	−0.601782	+	0.798660i	$0.705542\pi$
$614$	0	0
$615$	−41.0176	−1.65399
$616$	0	0
$617$	−40.6885	−1.63806	−0.819029	−	0.573753i	$-0.805487\pi$
$617$	−40.6885	−1.63806	−0.819029	+	0.573753i	$0.805487\pi$
$618$	0	0
$619$	−11.7274	−0.471365	−0.235682	−	0.971830i	$-0.575732\pi$
$619$	−11.7274	−0.471365	−0.235682	+	0.971830i	$0.575732\pi$
$620$	0	0
$621$	−37.1307	−1.49000
$622$	0	0
$623$	21.9886	0.880955
$624$	0	0
$625$	34.4779	1.37911
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−8.83221	−0.352163
$630$	0	0
$631$	−5.78803	−0.230418	−0.115209	−	0.993341i	$-0.536754\pi$
$631$	−5.78803	−0.230418	−0.115209	+	0.993341i	$0.536754\pi$
$632$	0	0
$633$	−14.5949	−0.580096
$634$	0	0
$635$	−38.6221	−1.53267
$636$	0	0
$637$	−52.1685	−2.06699
$638$	0	0
$639$	10.3630	0.409955
$640$	0	0
$641$	19.1158	0.755030	0.377515	−	0.926003i	$-0.376779\pi$
$641$	19.1158	0.755030	0.377515	+	0.926003i	$0.376779\pi$
$642$	0	0
$643$	8.18171	0.322655	0.161328	−	0.986901i	$-0.448422\pi$
$643$	8.18171	0.322655	0.161328	+	0.986901i	$0.448422\pi$
$644$	0	0
$645$	−33.2290	−1.30839
$646$	0	0
$647$	12.7159	0.499915	0.249958	−	0.968257i	$-0.419583\pi$
$647$	12.7159	0.499915	0.249958	+	0.968257i	$0.419583\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−22.2651	−0.872639
$652$	0	0
$653$	49.2651	1.92789	0.963946	−	0.266099i	$-0.0857348\pi$
$653$	49.2651	1.92789	0.963946	+	0.266099i	$0.0857348\pi$
$654$	0	0
$655$	12.9374	0.505507
$656$	0	0
$657$	−11.5569	−0.450879
$658$	0	0
$659$	−0.546884	−0.0213036	−0.0106518	−	0.999943i	$-0.503391\pi$
$659$	−0.546884	−0.0213036	−0.0106518	+	0.999943i	$0.503391\pi$
$660$	0	0
$661$	26.1784	1.01822	0.509111	−	0.860701i	$-0.329974\pi$
$661$	26.1784	1.01822	0.509111	+	0.860701i	$0.329974\pi$
$662$	0	0
$663$	−7.58014	−0.294388
$664$	0	0
$665$	18.7274	0.726219
$666$	0	0
$667$	16.6362	0.644156
$668$	0	0
$669$	−28.4345	−1.09934
$670$	0	0
$671$	0	0
$672$	0	0
$673$	37.3458	1.43957	0.719787	−	0.694195i	$-0.244239\pi$
$673$	37.3458	1.43957	0.719787	+	0.694195i	$0.244239\pi$
$674$	0	0
$675$	−59.0667	−2.27348
$676$	0	0
$677$	5.87665	0.225858	0.112929	−	0.993603i	$-0.463977\pi$
$677$	5.87665	0.225858	0.112929	+	0.993603i	$0.463977\pi$
$678$	0	0
$679$	40.6069	1.55835
$680$	0	0
$681$	31.8182	1.21928
$682$	0	0
$683$	15.6552	0.599031	0.299516	−	0.954091i	$-0.403175\pi$
$683$	15.6552	0.599031	0.299516	+	0.954091i	$0.403175\pi$
$684$	0	0
$685$	49.6265	1.89613
$686$	0	0
$687$	−31.4336	−1.19927
$688$	0	0
$689$	10.0381	0.382423
$690$	0	0
$691$	−32.5630	−1.23875	−0.619377	−	0.785094i	$-0.712615\pi$
$691$	−32.5630	−1.23875	−0.619377	+	0.785094i	$0.712615\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	58.1375	2.20528
$696$	0	0
$697$	−14.5315	−0.550422
$698$	0	0
$699$	10.0553	0.380325
$700$	0	0
$701$	9.96344	0.376314	0.188157	−	0.982139i	$-0.439749\pi$
$701$	9.96344	0.376314	0.188157	+	0.982139i	$0.439749\pi$
$702$	0	0
$703$	4.97929	0.187798
$704$	0	0
$705$	−26.5649	−1.00049
$706$	0	0
$707$	54.1349	2.03595
$708$	0	0
$709$	41.2326	1.54852	0.774261	−	0.632867i	$-0.218122\pi$
$709$	41.2326	1.54852	0.774261	+	0.632867i	$0.218122\pi$
$710$	0	0
$711$	−4.25457	−0.159559
$712$	0	0
$713$	24.7119	0.925470
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−28.9647	−1.08171
$718$	0	0
$719$	45.7345	1.70561	0.852804	−	0.522231i	$-0.174900\pi$
$719$	45.7345	1.70561	0.852804	+	0.522231i	$0.174900\pi$
$720$	0	0
$721$	55.2638	2.05813
$722$	0	0
$723$	−30.1264	−1.12041
$724$	0	0
$725$	26.4645	0.982867
$726$	0	0
$727$	30.2567	1.12216	0.561080	−	0.827762i	$-0.310386\pi$
$727$	30.2567	1.12216	0.561080	+	0.827762i	$0.310386\pi$
$728$	0	0
$729$	25.1801	0.932598
$730$	0	0
$731$	−11.7722	−0.435411
$732$	0	0
$733$	47.3263	1.74803	0.874017	−	0.485894i	$-0.161506\pi$
$733$	47.3263	1.74803	0.874017	+	0.485894i	$0.161506\pi$
$734$	0	0
$735$	77.4623	2.85724
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−19.4166	−0.714251	−0.357126	−	0.934056i	$-0.616243\pi$
$739$	−19.4166	−0.714251	−0.357126	+	0.934056i	$0.616243\pi$
$740$	0	0
$741$	4.27342	0.156988
$742$	0	0
$743$	−44.6624	−1.63850	−0.819252	−	0.573434i	$-0.805611\pi$
$743$	−44.6624	−1.63850	−0.819252	+	0.573434i	$0.805611\pi$
$744$	0	0
$745$	59.2252	2.16984
$746$	0	0
$747$	7.72285	0.282564
$748$	0	0
$749$	−55.2924	−2.02034
$750$	0	0
$751$	20.4500	0.746230	0.373115	−	0.927785i	$-0.378290\pi$
$751$	20.4500	0.746230	0.373115	+	0.927785i	$0.378290\pi$
$752$	0	0
$753$	−11.5883	−0.422300
$754$	0	0
$755$	−71.5687	−2.60465
$756$	0	0
$757$	−26.4719	−0.962137	−0.481068	−	0.876683i	$-0.659751\pi$
$757$	−26.4719	−0.962137	−0.481068	+	0.876683i	$0.659751\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−11.6847	−0.423569	−0.211784	−	0.977316i	$-0.567927\pi$
$761$	−11.6847	−0.423569	−0.211784	+	0.977316i	$0.567927\pi$
$762$	0	0
$763$	59.8428	2.16646
$764$	0	0
$765$	−9.76725	−0.353136
$766$	0	0
$767$	37.2727	1.34584
$768$	0	0
$769$	47.8048	1.72388	0.861942	−	0.507006i	$-0.169248\pi$
$769$	47.8048	1.72388	0.861942	+	0.507006i	$0.169248\pi$
$770$	0	0
$771$	−5.83836	−0.210263
$772$	0	0
$773$	32.0729	1.15358	0.576791	−	0.816892i	$-0.304305\pi$
$773$	32.0729	1.15358	0.576791	+	0.816892i	$0.304305\pi$
$774$	0	0
$775$	39.3112	1.41210
$776$	0	0
$777$	29.9144	1.07317
$778$	0	0
$779$	8.19238	0.293523
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−13.8931	−0.496499
$784$	0	0
$785$	−20.1537	−0.719318
$786$	0	0
$787$	−29.3096	−1.04478	−0.522388	−	0.852708i	$-0.674959\pi$
$787$	−29.3096	−1.04478	−0.522388	+	0.852708i	$0.674959\pi$
$788$	0	0
$789$	−1.16130	−0.0413434
$790$	0	0
$791$	−77.2995	−2.74845
$792$	0	0
$793$	−25.5840	−0.908515
$794$	0	0
$795$	−14.9051	−0.528629
$796$	0	0
$797$	21.2144	0.751452	0.375726	−	0.926731i	$-0.377393\pi$
$797$	21.2144	0.751452	0.375726	+	0.926731i	$0.377393\pi$
$798$	0	0
$799$	−9.41129	−0.332947
$800$	0	0
$801$	6.46531	0.228441
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−124.874	−4.40124
$806$	0	0
$807$	3.87097	0.136265
$808$	0	0
$809$	23.0489	0.810356	0.405178	−	0.914238i	$-0.367209\pi$
$809$	23.0489	0.810356	0.405178	+	0.914238i	$0.367209\pi$
$810$	0	0
$811$	−10.2453	−0.359763	−0.179881	−	0.983688i	$-0.557571\pi$
$811$	−10.2453	−0.359763	−0.179881	+	0.983688i	$0.557571\pi$
$812$	0	0
$813$	−19.7399	−0.692307
$814$	0	0
$815$	−54.5385	−1.91040
$816$	0	0
$817$	6.63677	0.232191
$818$	0	0
$819$	−22.2793	−0.778501
$820$	0	0
$821$	46.3606	1.61800	0.808998	−	0.587812i	$-0.200010\pi$
$821$	46.3606	1.61800	0.808998	+	0.587812i	$0.200010\pi$
$822$	0	0
$823$	−12.1369	−0.423065	−0.211533	−	0.977371i	$-0.567845\pi$
$823$	−12.1369	−0.423065	−0.211533	+	0.977371i	$0.567845\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36.7261	−1.27709	−0.638546	−	0.769584i	$-0.720464\pi$
$827$	−36.7261	−1.27709	−0.638546	+	0.769584i	$0.720464\pi$
$828$	0	0
$829$	−39.2159	−1.36203	−0.681013	−	0.732271i	$-0.738460\pi$
$829$	−39.2159	−1.36203	−0.681013	+	0.732271i	$0.738460\pi$
$830$	0	0
$831$	13.2781	0.460612
$832$	0	0
$833$	27.4430	0.950843
$834$	0	0
$835$	−10.0255	−0.346948
$836$	0	0
$837$	−20.6373	−0.713328
$838$	0	0
$839$	−5.94469	−0.205234	−0.102617	−	0.994721i	$-0.532722\pi$
$839$	−5.94469	−0.205234	−0.102617	+	0.994721i	$0.532722\pi$
$840$	0	0
$841$	−22.7753	−0.785354
$842$	0	0
$843$	−27.5052	−0.947328
$844$	0	0
$845$	−6.43976	−0.221535
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−13.6863	−0.469712
$850$	0	0
$851$	−33.2018	−1.13814
$852$	0	0
$853$	−43.9703	−1.50551	−0.752757	−	0.658299i	$-0.771276\pi$
$853$	−43.9703	−1.50551	−0.752757	+	0.658299i	$0.771276\pi$
$854$	0	0
$855$	5.50644	0.188316
$856$	0	0
$857$	50.2226	1.71557	0.857785	−	0.514008i	$-0.171840\pi$
$857$	50.2226	1.71557	0.857785	+	0.514008i	$0.171840\pi$
$858$	0	0
$859$	1.33169	0.0454365	0.0227183	−	0.999742i	$-0.492768\pi$
$859$	1.33169	0.0454365	0.0227183	+	0.999742i	$0.492768\pi$
$860$	0	0
$861$	49.2178	1.67734
$862$	0	0
$863$	−11.9240	−0.405898	−0.202949	−	0.979189i	$-0.565053\pi$
$863$	−11.9240	−0.405898	−0.202949	+	0.979189i	$0.565053\pi$
$864$	0	0
$865$	−83.0385	−2.82340
$866$	0	0
$867$	−17.5575	−0.596283
$868$	0	0
$869$	0	0
$870$	0	0
$871$	43.0445	1.45851
$872$	0	0
$873$	11.9397	0.404096
$874$	0	0
$875$	−105.010	−3.54998
$876$	0	0
$877$	−31.1974	−1.05346	−0.526731	−	0.850032i	$-0.676583\pi$
$877$	−31.1974	−1.05346	−0.526731	+	0.850032i	$0.676583\pi$
$878$	0	0
$879$	20.9200	0.705615
$880$	0	0
$881$	−24.5987	−0.828751	−0.414376	−	0.910106i	$-0.636000\pi$
$881$	−24.5987	−0.828751	−0.414376	+	0.910106i	$0.636000\pi$
$882$	0	0
$883$	8.21839	0.276571	0.138285	−	0.990392i	$-0.455841\pi$
$883$	8.21839	0.276571	0.138285	+	0.990392i	$0.455841\pi$
$884$	0	0
$885$	−55.3443	−1.86038
$886$	0	0
$887$	34.9556	1.17369	0.586847	−	0.809698i	$-0.300369\pi$
$887$	34.9556	1.17369	0.586847	+	0.809698i	$0.300369\pi$
$888$	0	0
$889$	46.3433	1.55431
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.30576	0.177550
$894$	0	0
$895$	−23.9581	−0.800829
$896$	0	0
$897$	−28.4951	−0.951423
$898$	0	0
$899$	9.24640	0.308385
$900$	0	0
$901$	−5.28051	−0.175919
$902$	0	0
$903$	39.8721	1.32686
$904$	0	0
$905$	−103.614	−3.44423
$906$	0	0
$907$	42.4858	1.41072	0.705359	−	0.708851i	$-0.250786\pi$
$907$	42.4858	1.41072	0.705359	+	0.708851i	$0.250786\pi$
$908$	0	0
$909$	15.9173	0.527944
$910$	0	0
$911$	−44.8547	−1.48610	−0.743051	−	0.669235i	$-0.766622\pi$
$911$	−44.8547	−1.48610	−0.743051	+	0.669235i	$0.766622\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	37.9883	1.25586
$916$	0	0
$917$	−15.5239	−0.512643
$918$	0	0
$919$	−12.7521	−0.420654	−0.210327	−	0.977631i	$-0.567453\pi$
$919$	−12.7521	−0.420654	−0.210327	+	0.977631i	$0.567453\pi$
$920$	0	0
$921$	39.4963	1.30145
$922$	0	0
$923$	25.0702	0.825196
$924$	0	0
$925$	−52.8167	−1.73660
$926$	0	0
$927$	16.2492	0.533694
$928$	0	0
$929$	24.3234	0.798025	0.399012	−	0.916946i	$-0.369353\pi$
$929$	24.3234	0.798025	0.399012	+	0.916946i	$0.369353\pi$
$930$	0	0
$931$	−15.4714	−0.507055
$932$	0	0
$933$	27.7292	0.907815
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.4944	0.800197	0.400098	−	0.916472i	$-0.368976\pi$
$937$	24.4944	0.800197	0.400098	+	0.916472i	$0.368976\pi$
$938$	0	0
$939$	−32.5455	−1.06208
$940$	0	0
$941$	−27.9939	−0.912575	−0.456288	−	0.889832i	$-0.650821\pi$
$941$	−27.9939	−0.912575	−0.456288	+	0.889832i	$0.650821\pi$
$942$	0	0
$943$	−54.6266	−1.77889
$944$	0	0
$945$	104.284	3.39236
$946$	0	0
$947$	−7.07867	−0.230026	−0.115013	−	0.993364i	$-0.536691\pi$
$947$	−7.07867	−0.230026	−0.115013	+	0.993364i	$0.536691\pi$
$948$	0	0
$949$	−27.9585	−0.907572
$950$	0	0
$951$	16.0764	0.521313
$952$	0	0
$953$	−23.1909	−0.751228	−0.375614	−	0.926776i	$-0.622568\pi$
$953$	−23.1909	−0.751228	−0.375614	+	0.926776i	$0.622568\pi$
$954$	0	0
$955$	66.4037	2.14877
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−59.5477	−1.92289
$960$	0	0
$961$	−17.2651	−0.556938
$962$	0	0
$963$	−16.2576	−0.523895
$964$	0	0
$965$	31.0411	0.999248
$966$	0	0
$967$	−46.0591	−1.48116	−0.740580	−	0.671968i	$-0.765449\pi$
$967$	−46.0591	−1.48116	−0.740580	+	0.671968i	$0.765449\pi$
$968$	0	0
$969$	−2.24801	−0.0722165
$970$	0	0
$971$	46.5490	1.49383	0.746915	−	0.664920i	$-0.231534\pi$
$971$	46.5490	1.49383	0.746915	+	0.664920i	$0.231534\pi$
$972$	0	0
$973$	−69.7603	−2.23641
$974$	0	0
$975$	−45.3294	−1.45170
$976$	0	0
$977$	13.1105	0.419443	0.209721	−	0.977761i	$-0.432744\pi$
$977$	13.1105	0.419443	0.209721	+	0.977761i	$0.432744\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	17.5956	0.561785
$982$	0	0
$983$	36.7492	1.17212	0.586058	−	0.810269i	$-0.300679\pi$
$983$	36.7492	1.17212	0.586058	+	0.810269i	$0.300679\pi$
$984$	0	0
$985$	64.3288	2.04969
$986$	0	0
$987$	31.8757	1.01461
$988$	0	0
$989$	−44.2538	−1.40719
$990$	0	0
$991$	29.9778	0.952277	0.476139	−	0.879370i	$-0.342036\pi$
$991$	29.9778	0.952277	0.476139	+	0.879370i	$0.342036\pi$
$992$	0	0
$993$	−11.4948	−0.364778
$994$	0	0
$995$	74.8647	2.37337
$996$	0	0
$997$	1.94157	0.0614901	0.0307450	−	0.999527i	$-0.490212\pi$
$997$	1.94157	0.0614901	0.0307450	+	0.999527i	$0.490212\pi$
$998$	0	0
$999$	27.7273	0.877252

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.o.1.6	✓	7
11.10	odd	2		9196.2.a.p.1.6	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.o.1.6	✓	7	1.1	even	1	trivial
9196.2.a.p.1.6	yes	7	11.10	odd	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:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q+1.26735 q^{3} +3.95060 q^{5} -4.74040 q^{7} -1.39382 q^{9} -3.37193 q^{13} +5.00680 q^{15} +1.77379 q^{17} -1.00000 q^{19} -6.00775 q^{21} +6.66797 q^{23} +10.6073 q^{25} -5.56851 q^{27} +2.49494 q^{29} +3.70606 q^{31} -18.7274 q^{35} -4.97929 q^{37} -4.27342 q^{39} -8.19238 q^{41} -6.63677 q^{43} -5.50644 q^{45} -5.30576 q^{47} +15.4714 q^{49} +2.24801 q^{51} -2.97697 q^{53} -1.26735 q^{57} -11.0538 q^{59} +7.58735 q^{61} +6.60727 q^{63} -13.3212 q^{65} -12.7655 q^{67} +8.45066 q^{69} -7.43497 q^{71} +8.29154 q^{73} +13.4431 q^{75} +3.05245 q^{79} -2.87580 q^{81} -5.54077 q^{83} +7.00753 q^{85} +3.16196 q^{87} -4.63855 q^{89} +15.9843 q^{91} +4.69688 q^{93} -3.95060 q^{95} -8.56613 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{3} + 3 q^{5} - 2 q^{7} + 8 q^{9} + 11 q^{15} - 12 q^{17} - 7 q^{19} - q^{21} - 5 q^{23} + 4 q^{25} - 13 q^{27} + 2 q^{29} + 5 q^{31} - 16 q^{35} + 8 q^{37} - 15 q^{39} - 31 q^{41} - 12 q^{43}+ \cdots - 25 q^{97}+O(q^{100}) \) 7 * q - q^3 + 3 * q^5 - 2 * q^7 + 8 * q^9 + 11 * q^15 - 12 * q^17 - 7 * q^19 - q^21 - 5 * q^23 + 4 * q^25 - 13 * q^27 + 2 * q^29 + 5 * q^31 - 16 * q^35 + 8 * q^37 - 15 * q^39 - 31 * q^41 - 12 * q^43 - 24 * q^45 + 10 * q^47 - 3 * q^49 - 5 * q^51 - 25 * q^53 + q^57 - 7 * q^59 - q^61 - 24 * q^63 - 13 * q^65 - 5 * q^67 + 31 * q^69 + q^71 + 12 * q^73 - 12 * q^75 - 20 * q^79 + 35 * q^81 - 14 * q^83 + 6 * q^85 - 28 * q^87 - 18 * q^89 + 25 * q^91 - 35 * q^93 - 3 * q^95 - 25 * q^97

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