LMFDB - Embedded newform 9300.2.a.ba.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(1,9300)] 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("9300.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,6,0,0,0,4,0,6,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 22x^{4} + 30x^{3} + 112x^{2} - 64x - 64 $ x^6 - 2x^5 - 22x^4 + 30x^3 + 112x^2 - 64*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.562228$ of defining polynomial
Character	$\chi$	$=$	9300.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +4.71477 q^{7} +1.00000 q^{9} +1.34171 q^{11} +1.32539 q^{13} -4.78916 q^{17} -6.34219 q^{19} +4.71477 q^{21} +4.93203 q^{23} +1.00000 q^{27} -1.60070 q^{29} -1.00000 q^{31} +1.34171 q^{33} -10.3987 q^{37} +1.32539 q^{39} -2.04016 q^{41} +5.69845 q^{43} +12.4567 q^{47} +15.2291 q^{49} -4.78916 q^{51} +4.46617 q^{53} -6.34219 q^{57} -4.95487 q^{59} +2.40968 q^{61} +4.71477 q^{63} +12.6473 q^{67} +4.93203 q^{69} +10.6175 q^{71} +2.00000 q^{73} +6.32587 q^{77} +7.60711 q^{79} +1.00000 q^{81} +13.2763 q^{83} -1.60070 q^{87} +11.1150 q^{89} +6.24891 q^{91} -1.00000 q^{93} +8.51432 q^{97} +1.34171 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 4 q^{7} + 6 q^{9} - 3 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 4 q^{21} + 5 q^{23} + 6 q^{27} - 6 q^{31} - 3 q^{33} + 8 q^{37} + 2 q^{39} + 18 q^{41} + 15 q^{43} + q^{47} + 14 q^{49} + 4 q^{51}+ \cdots - 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 4 * q^7 + 6 * q^9 - 3 * q^11 + 2 * q^13 + 4 * q^17 - 3 * q^19 + 4 * q^21 + 5 * q^23 + 6 * q^27 - 6 * q^31 - 3 * q^33 + 8 * q^37 + 2 * q^39 + 18 * q^41 + 15 * q^43 + q^47 + 14 * q^49 + 4 * q^51 + 5 * q^53 - 3 * q^57 - 12 * q^59 + 28 * q^61 + 4 * q^63 + 3 * q^67 + 5 * q^69 + 25 * q^71 + 12 * q^73 + 8 * q^77 + 3 * q^79 + 6 * q^81 + 19 * q^83 + 4 * q^89 + 16 * q^91 - 6 * q^93 - 2 * q^97 - 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.71477	1.78202	0.891008	−	0.453987i	$-0.149999\pi$
$7$	4.71477	1.78202	0.891008	+	0.453987i	$0.149999\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.34171	0.404541	0.202271	−	0.979330i	$-0.435168\pi$
$11$	1.34171	0.404541	0.202271	+	0.979330i	$0.435168\pi$
$12$	0	0
$13$	1.32539	0.367597	0.183799	−	0.982964i	$-0.441161\pi$
$13$	1.32539	0.367597	0.183799	+	0.982964i	$0.441161\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.78916	−1.16154	−0.580770	−	0.814067i	$-0.697249\pi$
$17$	−4.78916	−1.16154	−0.580770	+	0.814067i	$0.697249\pi$
$18$	0	0
$19$	−6.34219	−1.45500	−0.727499	−	0.686109i	$-0.759317\pi$
$19$	−6.34219	−1.45500	−0.727499	+	0.686109i	$0.759317\pi$
$20$	0	0
$21$	4.71477	1.02885
$22$	0	0
$23$	4.93203	1.02840	0.514199	−	0.857671i	$-0.328089\pi$
$23$	4.93203	1.02840	0.514199	+	0.857671i	$0.328089\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.60070	−0.297243	−0.148622	−	0.988894i	$-0.547484\pi$
$29$	−1.60070	−0.297243	−0.148622	+	0.988894i	$0.547484\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	0	0
$33$	1.34171	0.233562
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.3987	−1.70953	−0.854766	−	0.519014i	$-0.826299\pi$
$37$	−10.3987	−1.70953	−0.854766	+	0.519014i	$0.826299\pi$
$38$	0	0
$39$	1.32539	0.212232
$40$	0	0
$41$	−2.04016	−0.318620	−0.159310	−	0.987229i	$-0.550927\pi$
$41$	−2.04016	−0.318620	−0.159310	+	0.987229i	$0.550927\pi$
$42$	0	0
$43$	5.69845	0.869006	0.434503	−	0.900670i	$-0.356924\pi$
$43$	5.69845	0.869006	0.434503	+	0.900670i	$0.356924\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.4567	1.81700	0.908501	−	0.417884i	$-0.137228\pi$
$47$	12.4567	1.81700	0.908501	+	0.417884i	$0.137228\pi$
$48$	0	0
$49$	15.2291	2.17558
$50$	0	0
$51$	−4.78916	−0.670616
$52$	0	0
$53$	4.46617	0.613475	0.306738	−	0.951794i	$-0.400763\pi$
$53$	4.46617	0.613475	0.306738	+	0.951794i	$0.400763\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.34219	−0.840043
$58$	0	0
$59$	−4.95487	−0.645070	−0.322535	−	0.946558i	$-0.604535\pi$
$59$	−4.95487	−0.645070	−0.322535	+	0.946558i	$0.604535\pi$
$60$	0	0
$61$	2.40968	0.308528	0.154264	−	0.988030i	$-0.450699\pi$
$61$	2.40968	0.308528	0.154264	+	0.988030i	$0.450699\pi$
$62$	0	0
$63$	4.71477	0.594006
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.6473	1.54511	0.772555	−	0.634947i	$-0.218978\pi$
$67$	12.6473	1.54511	0.772555	+	0.634947i	$0.218978\pi$
$68$	0	0
$69$	4.93203	0.593746
$70$	0	0
$71$	10.6175	1.26007	0.630033	−	0.776568i	$-0.283041\pi$
$71$	10.6175	1.26007	0.630033	+	0.776568i	$0.283041\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.32587	0.720899
$78$	0	0
$79$	7.60711	0.855867	0.427934	−	0.903810i	$-0.359242\pi$
$79$	7.60711	0.855867	0.427934	+	0.903810i	$0.359242\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.2763	1.45726	0.728631	−	0.684907i	$-0.240157\pi$
$83$	13.2763	1.45726	0.728631	+	0.684907i	$0.240157\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.60070	−0.171614
$88$	0	0
$89$	11.1150	1.17819	0.589095	−	0.808064i	$-0.299484\pi$
$89$	11.1150	1.17819	0.589095	+	0.808064i	$0.299484\pi$
$90$	0	0
$91$	6.24891	0.655064
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.51432	0.864498	0.432249	−	0.901754i	$-0.357720\pi$
$97$	8.51432	0.864498	0.432249	+	0.901754i	$0.357720\pi$
$98$	0	0
$99$	1.34171	0.134847
$100$	0	0
$101$	−13.6586	−1.35909	−0.679543	−	0.733636i	$-0.737822\pi$
$101$	−13.6586	−1.35909	−0.679543	+	0.733636i	$0.737822\pi$
$102$	0	0
$103$	−1.38188	−0.136160	−0.0680801	−	0.997680i	$-0.521687\pi$
$103$	−1.38188	−0.136160	−0.0680801	+	0.997680i	$0.521687\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−16.2480	−1.57075	−0.785376	−	0.619019i	$-0.787530\pi$
$107$	−16.2480	−1.57075	−0.785376	+	0.619019i	$0.787530\pi$
$108$	0	0
$109$	15.2128	1.45712	0.728559	−	0.684983i	$-0.240190\pi$
$109$	15.2128	1.45712	0.728559	+	0.684983i	$0.240190\pi$
$110$	0	0
$111$	−10.3987	−0.986998
$112$	0	0
$113$	6.38986	0.601107	0.300554	−	0.953765i	$-0.402829\pi$
$113$	6.38986	0.601107	0.300554	+	0.953765i	$0.402829\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.32539	0.122532
$118$	0	0
$119$	−22.5798	−2.06989
$120$	0	0
$121$	−9.19981	−0.836346
$122$	0	0
$123$	−2.04016	−0.183955
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.1210	1.25303	0.626516	−	0.779409i	$-0.284480\pi$
$127$	14.1210	1.25303	0.626516	+	0.779409i	$0.284480\pi$
$128$	0	0
$129$	5.69845	0.501721
$130$	0	0
$131$	7.12910	0.622872	0.311436	−	0.950267i	$-0.399190\pi$
$131$	7.12910	0.622872	0.311436	+	0.950267i	$0.399190\pi$
$132$	0	0
$133$	−29.9020	−2.59283
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.08367	0.776070	0.388035	−	0.921645i	$-0.373154\pi$
$137$	9.08367	0.776070	0.388035	+	0.921645i	$0.373154\pi$
$138$	0	0
$139$	16.6556	1.41271	0.706354	−	0.707859i	$-0.250339\pi$
$139$	16.6556	1.41271	0.706354	+	0.707859i	$0.250339\pi$
$140$	0	0
$141$	12.4567	1.04905
$142$	0	0
$143$	1.77829	0.148708
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.2291	1.25607
$148$	0	0
$149$	−23.8674	−1.95529	−0.977646	−	0.210256i	$-0.932570\pi$
$149$	−23.8674	−1.95529	−0.977646	+	0.210256i	$0.932570\pi$
$150$	0	0
$151$	12.9195	1.05138	0.525689	−	0.850677i	$-0.323808\pi$
$151$	12.9195	1.05138	0.525689	+	0.850677i	$0.323808\pi$
$152$	0	0
$153$	−4.78916	−0.387180
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−16.7241	−1.33473	−0.667363	−	0.744733i	$-0.732577\pi$
$157$	−16.7241	−1.33473	−0.667363	+	0.744733i	$0.732577\pi$
$158$	0	0
$159$	4.46617	0.354190
$160$	0	0
$161$	23.2534	1.83262
$162$	0	0
$163$	18.0813	1.41623	0.708117	−	0.706095i	$-0.249545\pi$
$163$	18.0813	1.41623	0.708117	+	0.706095i	$0.249545\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−17.2896	−1.33791	−0.668953	−	0.743305i	$-0.733257\pi$
$167$	−17.2896	−1.33791	−0.668953	+	0.743305i	$0.733257\pi$
$168$	0	0
$169$	−11.2433	−0.864872
$170$	0	0
$171$	−6.34219	−0.484999
$172$	0	0
$173$	25.1144	1.90941	0.954706	−	0.297550i	$-0.0961696\pi$
$173$	25.1144	1.90941	0.954706	+	0.297550i	$0.0961696\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.95487	−0.372431
$178$	0	0
$179$	−17.2504	−1.28935	−0.644677	−	0.764455i	$-0.723008\pi$
$179$	−17.2504	−1.28935	−0.644677	+	0.764455i	$0.723008\pi$
$180$	0	0
$181$	19.0287	1.41439	0.707195	−	0.707018i	$-0.249960\pi$
$181$	19.0287	1.41439	0.707195	+	0.707018i	$0.249960\pi$
$182$	0	0
$183$	2.40968	0.178129
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−6.42566	−0.469891
$188$	0	0
$189$	4.71477	0.342949
$190$	0	0
$191$	−9.34109	−0.675897	−0.337949	−	0.941165i	$-0.609733\pi$
$191$	−9.34109	−0.675897	−0.337949	+	0.941165i	$0.609733\pi$
$192$	0	0
$193$	17.8031	1.28149	0.640747	−	0.767752i	$-0.278625\pi$
$193$	17.8031	1.28149	0.640747	+	0.767752i	$0.278625\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.2029	−1.51064	−0.755321	−	0.655356i	$-0.772519\pi$
$197$	−21.2029	−1.51064	−0.755321	+	0.655356i	$0.772519\pi$
$198$	0	0
$199$	25.0752	1.77754	0.888768	−	0.458357i	$-0.151562\pi$
$199$	25.0752	1.77754	0.888768	+	0.458357i	$0.151562\pi$
$200$	0	0
$201$	12.6473	0.892070
$202$	0	0
$203$	−7.54696	−0.529693
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.93203	0.342800
$208$	0	0
$209$	−8.50939	−0.588607
$210$	0	0
$211$	9.91173	0.682352	0.341176	−	0.940000i	$-0.389175\pi$
$211$	9.91173	0.682352	0.341176	+	0.940000i	$0.389175\pi$
$212$	0	0
$213$	10.6175	0.727499
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.71477	−0.320060
$218$	0	0
$219$	2.00000	0.135147
$220$	0	0
$221$	−6.34750	−0.426979
$222$	0	0
$223$	−11.1616	−0.747436	−0.373718	−	0.927542i	$-0.621917\pi$
$223$	−11.1616	−0.747436	−0.373718	+	0.927542i	$0.621917\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.3406	−1.21731	−0.608655	−	0.793435i	$-0.708290\pi$
$227$	−18.3406	−1.21731	−0.608655	+	0.793435i	$0.708290\pi$
$228$	0	0
$229$	−23.7042	−1.56642	−0.783209	−	0.621759i	$-0.786418\pi$
$229$	−23.7042	−1.56642	−0.783209	+	0.621759i	$0.786418\pi$
$230$	0	0
$231$	6.32587	0.416211
$232$	0	0
$233$	−5.15581	−0.337768	−0.168884	−	0.985636i	$-0.554016\pi$
$233$	−5.15581	−0.337768	−0.168884	+	0.985636i	$0.554016\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.60711	0.494135
$238$	0	0
$239$	−6.96338	−0.450424	−0.225212	−	0.974310i	$-0.572307\pi$
$239$	−6.96338	−0.450424	−0.225212	+	0.974310i	$0.572307\pi$
$240$	0	0
$241$	−0.282553	−0.0182008	−0.00910041	−	0.999959i	$-0.502897\pi$
$241$	−0.282553	−0.0182008	−0.00910041	+	0.999959i	$0.502897\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−8.40587	−0.534853
$248$	0	0
$249$	13.2763	0.841350
$250$	0	0
$251$	18.9810	1.19807	0.599035	−	0.800723i	$-0.295551\pi$
$251$	18.9810	1.19807	0.599035	+	0.800723i	$0.295551\pi$
$252$	0	0
$253$	6.61736	0.416030
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.88159	−0.117370	−0.0586852	−	0.998277i	$-0.518691\pi$
$257$	−1.88159	−0.117370	−0.0586852	+	0.998277i	$0.518691\pi$
$258$	0	0
$259$	−49.0274	−3.04641
$260$	0	0
$261$	−1.60070	−0.0990811
$262$	0	0
$263$	−25.8074	−1.59135	−0.795676	−	0.605722i	$-0.792884\pi$
$263$	−25.8074	−1.59135	−0.795676	+	0.605722i	$0.792884\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.1150	0.680228
$268$	0	0
$269$	−9.15216	−0.558017	−0.279009	−	0.960289i	$-0.590006\pi$
$269$	−9.15216	−0.558017	−0.279009	+	0.960289i	$0.590006\pi$
$270$	0	0
$271$	21.3929	1.29953	0.649764	−	0.760136i	$-0.274868\pi$
$271$	21.3929	1.29953	0.649764	+	0.760136i	$0.274868\pi$
$272$	0	0
$273$	6.24891	0.378201
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.68161	0.101038	0.0505191	−	0.998723i	$-0.483912\pi$
$277$	1.68161	0.101038	0.0505191	+	0.998723i	$0.483912\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	4.26092	0.254185	0.127093	−	0.991891i	$-0.459435\pi$
$281$	4.26092	0.254185	0.127093	+	0.991891i	$0.459435\pi$
$282$	0	0
$283$	19.6468	1.16788	0.583939	−	0.811797i	$-0.301511\pi$
$283$	19.6468	1.16788	0.583939	+	0.811797i	$0.301511\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.61891	−0.567786
$288$	0	0
$289$	5.93601	0.349177
$290$	0	0
$291$	8.51432	0.499118
$292$	0	0
$293$	−32.1333	−1.87725	−0.938624	−	0.344942i	$-0.887899\pi$
$293$	−32.1333	−1.87725	−0.938624	+	0.344942i	$0.887899\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.34171	0.0778540
$298$	0	0
$299$	6.53686	0.378036
$300$	0	0
$301$	26.8669	1.54858
$302$	0	0
$303$	−13.6586	−0.784668
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6.60166	−0.376777	−0.188388	−	0.982095i	$-0.560326\pi$
$307$	−6.60166	−0.376777	−0.188388	+	0.982095i	$0.560326\pi$
$308$	0	0
$309$	−1.38188	−0.0786121
$310$	0	0
$311$	19.3887	1.09944	0.549718	−	0.835351i	$-0.314735\pi$
$311$	19.3887	1.09944	0.549718	+	0.835351i	$0.314735\pi$
$312$	0	0
$313$	−19.4393	−1.09877	−0.549387	−	0.835568i	$-0.685139\pi$
$313$	−19.4393	−1.09877	−0.549387	+	0.835568i	$0.685139\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−27.9705	−1.57098	−0.785489	−	0.618876i	$-0.787588\pi$
$317$	−27.9705	−1.57098	−0.785489	+	0.618876i	$0.787588\pi$
$318$	0	0
$319$	−2.14768	−0.120247
$320$	0	0
$321$	−16.2480	−0.906874
$322$	0	0
$323$	30.3737	1.69004
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.2128	0.841268
$328$	0	0
$329$	58.7307	3.23793
$330$	0	0
$331$	4.37804	0.240639	0.120319	−	0.992735i	$-0.461608\pi$
$331$	4.37804	0.240639	0.120319	+	0.992735i	$0.461608\pi$
$332$	0	0
$333$	−10.3987	−0.569844
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−3.71430	−0.202331	−0.101165	−	0.994870i	$-0.532257\pi$
$337$	−3.71430	−0.202331	−0.101165	+	0.994870i	$0.532257\pi$
$338$	0	0
$339$	6.38986	0.347050
$340$	0	0
$341$	−1.34171	−0.0726578
$342$	0	0
$343$	38.7983	2.09491
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−4.59865	−0.246869	−0.123434	−	0.992353i	$-0.539391\pi$
$347$	−4.59865	−0.246869	−0.123434	+	0.992353i	$0.539391\pi$
$348$	0	0
$349$	−2.80659	−0.150233	−0.0751166	−	0.997175i	$-0.523933\pi$
$349$	−2.80659	−0.150233	−0.0751166	+	0.997175i	$0.523933\pi$
$350$	0	0
$351$	1.32539	0.0707441
$352$	0	0
$353$	−33.5284	−1.78454	−0.892268	−	0.451506i	$-0.850887\pi$
$353$	−33.5284	−1.78454	−0.892268	+	0.451506i	$0.850887\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−22.5798	−1.19505
$358$	0	0
$359$	−12.6388	−0.667049	−0.333524	−	0.942741i	$-0.608238\pi$
$359$	−12.6388	−0.667049	−0.333524	+	0.942741i	$0.608238\pi$
$360$	0	0
$361$	21.2233	1.11702
$362$	0	0
$363$	−9.19981	−0.482865
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.69063	0.192650	0.0963248	−	0.995350i	$-0.469291\pi$
$367$	3.69063	0.192650	0.0963248	+	0.995350i	$0.469291\pi$
$368$	0	0
$369$	−2.04016	−0.106207
$370$	0	0
$371$	21.0570	1.09322
$372$	0	0
$373$	10.4764	0.542450	0.271225	−	0.962516i	$-0.412571\pi$
$373$	10.4764	0.542450	0.271225	+	0.962516i	$0.412571\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.12156	−0.109266
$378$	0	0
$379$	9.88245	0.507627	0.253814	−	0.967253i	$-0.418315\pi$
$379$	9.88245	0.507627	0.253814	+	0.967253i	$0.418315\pi$
$380$	0	0
$381$	14.1210	0.723438
$382$	0	0
$383$	−9.46526	−0.483652	−0.241826	−	0.970320i	$-0.577746\pi$
$383$	−9.46526	−0.483652	−0.241826	+	0.970320i	$0.577746\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5.69845	0.289669
$388$	0	0
$389$	−23.1338	−1.17293	−0.586465	−	0.809975i	$-0.699481\pi$
$389$	−23.1338	−1.17293	−0.586465	+	0.809975i	$0.699481\pi$
$390$	0	0
$391$	−23.6202	−1.19453
$392$	0	0
$393$	7.12910	0.359615
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−17.3468	−0.870613	−0.435307	−	0.900282i	$-0.643360\pi$
$397$	−17.3468	−0.870613	−0.435307	+	0.900282i	$0.643360\pi$
$398$	0	0
$399$	−29.9020	−1.49697
$400$	0	0
$401$	−0.833960	−0.0416460	−0.0208230	−	0.999783i	$-0.506629\pi$
$401$	−0.833960	−0.0416460	−0.0208230	+	0.999783i	$0.506629\pi$
$402$	0	0
$403$	−1.32539	−0.0660224
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.9520	−0.691576
$408$	0	0
$409$	1.91666	0.0947729	0.0473864	−	0.998877i	$-0.484911\pi$
$409$	1.91666	0.0947729	0.0473864	+	0.998877i	$0.484911\pi$
$410$	0	0
$411$	9.08367	0.448064
$412$	0	0
$413$	−23.3611	−1.14953
$414$	0	0
$415$	0	0
$416$	0	0
$417$	16.6556	0.815627
$418$	0	0
$419$	12.0263	0.587522	0.293761	−	0.955879i	$-0.405093\pi$
$419$	12.0263	0.587522	0.293761	+	0.955879i	$0.405093\pi$
$420$	0	0
$421$	−21.3605	−1.04105	−0.520523	−	0.853848i	$-0.674263\pi$
$421$	−21.3605	−1.04105	−0.520523	+	0.853848i	$0.674263\pi$
$422$	0	0
$423$	12.4567	0.605667
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11.3611	0.549803
$428$	0	0
$429$	1.77829	0.0858567
$430$	0	0
$431$	25.5711	1.23172	0.615859	−	0.787857i	$-0.288809\pi$
$431$	25.5711	1.23172	0.615859	+	0.787857i	$0.288809\pi$
$432$	0	0
$433$	15.3641	0.738353	0.369176	−	0.929359i	$-0.379640\pi$
$433$	15.3641	0.738353	0.369176	+	0.929359i	$0.379640\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−31.2798	−1.49632
$438$	0	0
$439$	−1.01649	−0.0485144	−0.0242572	−	0.999706i	$-0.507722\pi$
$439$	−1.01649	−0.0485144	−0.0242572	+	0.999706i	$0.507722\pi$
$440$	0	0
$441$	15.2291	0.725195
$442$	0	0
$443$	−32.0135	−1.52101	−0.760503	−	0.649335i	$-0.775047\pi$
$443$	−32.0135	−1.52101	−0.760503	+	0.649335i	$0.775047\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−23.8674	−1.12889
$448$	0	0
$449$	−19.1290	−0.902752	−0.451376	−	0.892334i	$-0.649067\pi$
$449$	−19.1290	−0.902752	−0.451376	+	0.892334i	$0.649067\pi$
$450$	0	0
$451$	−2.73731	−0.128895
$452$	0	0
$453$	12.9195	0.607013
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.4642	−1.19117	−0.595583	−	0.803294i	$-0.703079\pi$
$457$	−25.4642	−1.19117	−0.595583	+	0.803294i	$0.703079\pi$
$458$	0	0
$459$	−4.78916	−0.223539
$460$	0	0
$461$	29.1908	1.35955	0.679774	−	0.733421i	$-0.262078\pi$
$461$	29.1908	1.35955	0.679774	+	0.733421i	$0.262078\pi$
$462$	0	0
$463$	−4.91268	−0.228312	−0.114156	−	0.993463i	$-0.536416\pi$
$463$	−4.91268	−0.228312	−0.114156	+	0.993463i	$0.536416\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.3296	0.663093	0.331546	−	0.943439i	$-0.392430\pi$
$467$	14.3296	0.663093	0.331546	+	0.943439i	$0.392430\pi$
$468$	0	0
$469$	59.6291	2.75341
$470$	0	0
$471$	−16.7241	−0.770604
$472$	0	0
$473$	7.64568	0.351549
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.46617	0.204492
$478$	0	0
$479$	34.2683	1.56576	0.782880	−	0.622173i	$-0.213750\pi$
$479$	34.2683	1.56576	0.782880	+	0.622173i	$0.213750\pi$
$480$	0	0
$481$	−13.7823	−0.628419
$482$	0	0
$483$	23.2534	1.05807
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−12.3979	−0.561802	−0.280901	−	0.959737i	$-0.590633\pi$
$487$	−12.3979	−0.561802	−0.280901	+	0.959737i	$0.590633\pi$
$488$	0	0
$489$	18.0813	0.817663
$490$	0	0
$491$	−7.60659	−0.343281	−0.171640	−	0.985160i	$-0.554907\pi$
$491$	−7.60659	−0.343281	−0.171640	+	0.985160i	$0.554907\pi$
$492$	0	0
$493$	7.66602	0.345260
$494$	0	0
$495$	0	0
$496$	0	0
$497$	50.0591	2.24546
$498$	0	0
$499$	4.51463	0.202102	0.101051	−	0.994881i	$-0.467779\pi$
$499$	4.51463	0.202102	0.101051	+	0.994881i	$0.467779\pi$
$500$	0	0
$501$	−17.2896	−0.772440
$502$	0	0
$503$	25.5975	1.14134	0.570669	−	0.821180i	$-0.306684\pi$
$503$	25.5975	1.14134	0.570669	+	0.821180i	$0.306684\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−11.2433	−0.499334
$508$	0	0
$509$	14.4304	0.639615	0.319808	−	0.947483i	$-0.396382\pi$
$509$	14.4304	0.639615	0.319808	+	0.947483i	$0.396382\pi$
$510$	0	0
$511$	9.42955	0.417139
$512$	0	0
$513$	−6.34219	−0.280014
$514$	0	0
$515$	0	0
$516$	0	0
$517$	16.7133	0.735052
$518$	0	0
$519$	25.1144	1.10240
$520$	0	0
$521$	−3.56091	−0.156006	−0.0780032	−	0.996953i	$-0.524854\pi$
$521$	−3.56091	−0.156006	−0.0780032	+	0.996953i	$0.524854\pi$
$522$	0	0
$523$	−1.27209	−0.0556248	−0.0278124	−	0.999613i	$-0.508854\pi$
$523$	−1.27209	−0.0556248	−0.0278124	+	0.999613i	$0.508854\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.78916	0.208619
$528$	0	0
$529$	1.32490	0.0576044
$530$	0	0
$531$	−4.95487	−0.215023
$532$	0	0
$533$	−2.70401	−0.117124
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−17.2504	−0.744409
$538$	0	0
$539$	20.4330	0.880114
$540$	0	0
$541$	3.68995	0.158643	0.0793216	−	0.996849i	$-0.474725\pi$
$541$	3.68995	0.158643	0.0793216	+	0.996849i	$0.474725\pi$
$542$	0	0
$543$	19.0287	0.816599
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−1.73685	−0.0742622	−0.0371311	−	0.999310i	$-0.511822\pi$
$547$	−1.73685	−0.0742622	−0.0371311	+	0.999310i	$0.511822\pi$
$548$	0	0
$549$	2.40968	0.102843
$550$	0	0
$551$	10.1520	0.432488
$552$	0	0
$553$	35.8658	1.52517
$554$	0	0
$555$	0	0
$556$	0	0
$557$	21.7367	0.921013	0.460507	−	0.887656i	$-0.347668\pi$
$557$	21.7367	0.921013	0.460507	+	0.887656i	$0.347668\pi$
$558$	0	0
$559$	7.55267	0.319444
$560$	0	0
$561$	−6.42566	−0.271292
$562$	0	0
$563$	27.2352	1.14783	0.573913	−	0.818916i	$-0.305425\pi$
$563$	27.2352	1.14783	0.573913	+	0.818916i	$0.305425\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.71477	0.198002
$568$	0	0
$569$	1.51643	0.0635721	0.0317860	−	0.999495i	$-0.489880\pi$
$569$	1.51643	0.0635721	0.0317860	+	0.999495i	$0.489880\pi$
$570$	0	0
$571$	−35.2786	−1.47636	−0.738181	−	0.674602i	$-0.764315\pi$
$571$	−35.2786	−1.47636	−0.738181	+	0.674602i	$0.764315\pi$
$572$	0	0
$573$	−9.34109	−0.390230
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−13.4566	−0.560205	−0.280103	−	0.959970i	$-0.590368\pi$
$577$	−13.4566	−0.560205	−0.280103	+	0.959970i	$0.590368\pi$
$578$	0	0
$579$	17.8031	0.739871
$580$	0	0
$581$	62.5947	2.59687
$582$	0	0
$583$	5.99231	0.248176
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−30.7551	−1.26940	−0.634700	−	0.772759i	$-0.718876\pi$
$587$	−30.7551	−1.26940	−0.634700	+	0.772759i	$0.718876\pi$
$588$	0	0
$589$	6.34219	0.261325
$590$	0	0
$591$	−21.2029	−0.872169
$592$	0	0
$593$	15.2843	0.627651	0.313826	−	0.949481i	$-0.398389\pi$
$593$	15.2843	0.627651	0.313826	+	0.949481i	$0.398389\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	25.0752	1.02626
$598$	0	0
$599$	−27.6366	−1.12920	−0.564600	−	0.825365i	$-0.690969\pi$
$599$	−27.6366	−1.12920	−0.564600	+	0.825365i	$0.690969\pi$
$600$	0	0
$601$	−18.8757	−0.769957	−0.384978	−	0.922926i	$-0.625791\pi$
$601$	−18.8757	−0.769957	−0.384978	+	0.922926i	$0.625791\pi$
$602$	0	0
$603$	12.6473	0.515037
$604$	0	0
$605$	0	0
$606$	0	0
$607$	26.7275	1.08483	0.542417	−	0.840109i	$-0.317509\pi$
$607$	26.7275	1.08483	0.542417	+	0.840109i	$0.317509\pi$
$608$	0	0
$609$	−7.54696	−0.305818
$610$	0	0
$611$	16.5100	0.667924
$612$	0	0
$613$	4.89084	0.197539	0.0987695	−	0.995110i	$-0.468509\pi$
$613$	4.89084	0.197539	0.0987695	+	0.995110i	$0.468509\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.0234	1.53076	0.765382	−	0.643577i	$-0.222550\pi$
$617$	38.0234	1.53076	0.765382	+	0.643577i	$0.222550\pi$
$618$	0	0
$619$	13.0813	0.525781	0.262890	−	0.964826i	$-0.415324\pi$
$619$	13.0813	0.525781	0.262890	+	0.964826i	$0.415324\pi$
$620$	0	0
$621$	4.93203	0.197915
$622$	0	0
$623$	52.4048	2.09955
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−8.50939	−0.339832
$628$	0	0
$629$	49.8009	1.98569
$630$	0	0
$631$	11.1767	0.444936	0.222468	−	0.974940i	$-0.428589\pi$
$631$	11.1767	0.444936	0.222468	+	0.974940i	$0.428589\pi$
$632$	0	0
$633$	9.91173	0.393956
$634$	0	0
$635$	0	0
$636$	0	0
$637$	20.1845	0.799738
$638$	0	0
$639$	10.6175	0.420022
$640$	0	0
$641$	−25.7498	−1.01706	−0.508529	−	0.861045i	$-0.669811\pi$
$641$	−25.7498	−1.01706	−0.508529	+	0.861045i	$0.669811\pi$
$642$	0	0
$643$	−13.3612	−0.526916	−0.263458	−	0.964671i	$-0.584863\pi$
$643$	−13.3612	−0.526916	−0.263458	+	0.964671i	$0.584863\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−42.4657	−1.66950	−0.834749	−	0.550630i	$-0.814387\pi$
$647$	−42.4657	−1.66950	−0.834749	+	0.550630i	$0.814387\pi$
$648$	0	0
$649$	−6.64801	−0.260957
$650$	0	0
$651$	−4.71477	−0.184787
$652$	0	0
$653$	−19.1674	−0.750077	−0.375039	−	0.927009i	$-0.622371\pi$
$653$	−19.1674	−0.750077	−0.375039	+	0.927009i	$0.622371\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	1.29270	0.0503566	0.0251783	−	0.999683i	$-0.491985\pi$
$659$	1.29270	0.0503566	0.0251783	+	0.999683i	$0.491985\pi$
$660$	0	0
$661$	13.8806	0.539894	0.269947	−	0.962875i	$-0.412994\pi$
$661$	13.8806	0.539894	0.269947	+	0.962875i	$0.412994\pi$
$662$	0	0
$663$	−6.34750	−0.246516
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.89472	−0.305685
$668$	0	0
$669$	−11.1616	−0.431532
$670$	0	0
$671$	3.23310	0.124812
$672$	0	0
$673$	−3.49479	−0.134714	−0.0673571	−	0.997729i	$-0.521457\pi$
$673$	−3.49479	−0.134714	−0.0673571	+	0.997729i	$0.521457\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−38.3787	−1.47501	−0.737506	−	0.675340i	$-0.763997\pi$
$677$	−38.3787	−1.47501	−0.737506	+	0.675340i	$0.763997\pi$
$678$	0	0
$679$	40.1431	1.54055
$680$	0	0
$681$	−18.3406	−0.702814
$682$	0	0
$683$	25.2416	0.965844	0.482922	−	0.875663i	$-0.339575\pi$
$683$	25.2416	0.965844	0.482922	+	0.875663i	$0.339575\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−23.7042	−0.904371
$688$	0	0
$689$	5.91942	0.225512
$690$	0	0
$691$	7.89882	0.300485	0.150243	−	0.988649i	$-0.451995\pi$
$691$	7.89882	0.300485	0.150243	+	0.988649i	$0.451995\pi$
$692$	0	0
$693$	6.32587	0.240300
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.77066	0.370090
$698$	0	0
$699$	−5.15581	−0.195011
$700$	0	0
$701$	13.6212	0.514466	0.257233	−	0.966349i	$-0.417189\pi$
$701$	13.6212	0.514466	0.257233	+	0.966349i	$0.417189\pi$
$702$	0	0
$703$	65.9503	2.48736
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−64.3974	−2.42191
$708$	0	0
$709$	13.2132	0.496233	0.248117	−	0.968730i	$-0.420188\pi$
$709$	13.2132	0.496233	0.248117	+	0.968730i	$0.420188\pi$
$710$	0	0
$711$	7.60711	0.285289
$712$	0	0
$713$	−4.93203	−0.184706
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.96338	−0.260052
$718$	0	0
$719$	−18.7538	−0.699397	−0.349699	−	0.936862i	$-0.613716\pi$
$719$	−18.7538	−0.699397	−0.349699	+	0.936862i	$0.613716\pi$
$720$	0	0
$721$	−6.51523	−0.242640
$722$	0	0
$723$	−0.282553	−0.0105083
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−16.5390	−0.613397	−0.306699	−	0.951807i	$-0.599224\pi$
$727$	−16.5390	−0.613397	−0.306699	+	0.951807i	$0.599224\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−27.2908	−1.00939
$732$	0	0
$733$	−47.9906	−1.77257	−0.886287	−	0.463136i	$-0.846724\pi$
$733$	−47.9906	−1.77257	−0.886287	+	0.463136i	$0.846724\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	16.9690	0.625061
$738$	0	0
$739$	4.04590	0.148831	0.0744155	−	0.997227i	$-0.476291\pi$
$739$	4.04590	0.148831	0.0744155	+	0.997227i	$0.476291\pi$
$740$	0	0
$741$	−8.40587	−0.308797
$742$	0	0
$743$	−47.4317	−1.74010	−0.870050	−	0.492963i	$-0.835914\pi$
$743$	−47.4317	−1.74010	−0.870050	+	0.492963i	$0.835914\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.2763	0.485754
$748$	0	0
$749$	−76.6056	−2.79911
$750$	0	0
$751$	−30.0062	−1.09494	−0.547470	−	0.836825i	$-0.684409\pi$
$751$	−30.0062	−1.09494	−0.547470	+	0.836825i	$0.684409\pi$
$752$	0	0
$753$	18.9810	0.691706
$754$	0	0
$755$	0	0
$756$	0	0
$757$	5.82623	0.211758	0.105879	−	0.994379i	$-0.466234\pi$
$757$	5.82623	0.211758	0.105879	+	0.994379i	$0.466234\pi$
$758$	0	0
$759$	6.61736	0.240195
$760$	0	0
$761$	4.53588	0.164426	0.0822128	−	0.996615i	$-0.473801\pi$
$761$	4.53588	0.164426	0.0822128	+	0.996615i	$0.473801\pi$
$762$	0	0
$763$	71.7248	2.59661
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.56714	−0.237126
$768$	0	0
$769$	−2.20050	−0.0793522	−0.0396761	−	0.999213i	$-0.512633\pi$
$769$	−2.20050	−0.0793522	−0.0396761	+	0.999213i	$0.512633\pi$
$770$	0	0
$771$	−1.88159	−0.0677639
$772$	0	0
$773$	−42.6954	−1.53565	−0.767824	−	0.640661i	$-0.778660\pi$
$773$	−42.6954	−1.53565	−0.767824	+	0.640661i	$0.778660\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−49.0274	−1.75885
$778$	0	0
$779$	12.9391	0.463591
$780$	0	0
$781$	14.2456	0.509749
$782$	0	0
$783$	−1.60070	−0.0572045
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−8.02784	−0.286161	−0.143081	−	0.989711i	$-0.545701\pi$
$787$	−8.02784	−0.286161	−0.143081	+	0.989711i	$0.545701\pi$
$788$	0	0
$789$	−25.8074	−0.918768
$790$	0	0
$791$	30.1267	1.07118
$792$	0	0
$793$	3.19377	0.113414
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.72472	0.238202	0.119101	−	0.992882i	$-0.461999\pi$
$797$	6.72472	0.238202	0.119101	+	0.992882i	$0.461999\pi$
$798$	0	0
$799$	−59.6572	−2.11052
$800$	0	0
$801$	11.1150	0.392730
$802$	0	0
$803$	2.68342	0.0946959
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.15216	−0.322171
$808$	0	0
$809$	−25.6519	−0.901874	−0.450937	−	0.892556i	$-0.648910\pi$
$809$	−25.6519	−0.901874	−0.450937	+	0.892556i	$0.648910\pi$
$810$	0	0
$811$	−7.67112	−0.269370	−0.134685	−	0.990888i	$-0.543002\pi$
$811$	−7.67112	−0.269370	−0.134685	+	0.990888i	$0.543002\pi$
$812$	0	0
$813$	21.3929	0.750283
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−36.1407	−1.26440
$818$	0	0
$819$	6.24891	0.218355
$820$	0	0
$821$	40.9562	1.42938	0.714691	−	0.699440i	$-0.246567\pi$
$821$	40.9562	1.42938	0.714691	+	0.699440i	$0.246567\pi$
$822$	0	0
$823$	−47.8062	−1.66642	−0.833209	−	0.552958i	$-0.813499\pi$
$823$	−47.8062	−1.66642	−0.833209	+	0.552958i	$0.813499\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	6.86170	0.238605	0.119302	−	0.992858i	$-0.461934\pi$
$827$	6.86170	0.238605	0.119302	+	0.992858i	$0.461934\pi$
$828$	0	0
$829$	2.67942	0.0930600	0.0465300	−	0.998917i	$-0.485184\pi$
$829$	2.67942	0.0930600	0.0465300	+	0.998917i	$0.485184\pi$
$830$	0	0
$831$	1.68161	0.0583344
$832$	0	0
$833$	−72.9345	−2.52703
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	0	0
$839$	−3.84750	−0.132831	−0.0664153	−	0.997792i	$-0.521156\pi$
$839$	−3.84750	−0.132831	−0.0664153	+	0.997792i	$0.521156\pi$
$840$	0	0
$841$	−26.4377	−0.911646
$842$	0	0
$843$	4.26092	0.146754
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−43.3750	−1.49038
$848$	0	0
$849$	19.6468	0.674275
$850$	0	0
$851$	−51.2865	−1.75808
$852$	0	0
$853$	17.8417	0.610889	0.305445	−	0.952210i	$-0.401195\pi$
$853$	17.8417	0.610889	0.305445	+	0.952210i	$0.401195\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.9000	0.953047	0.476523	−	0.879162i	$-0.341897\pi$
$857$	27.9000	0.953047	0.476523	+	0.879162i	$0.341897\pi$
$858$	0	0
$859$	−0.466458	−0.0159154	−0.00795768	−	0.999968i	$-0.502533\pi$
$859$	−0.466458	−0.0159154	−0.00795768	+	0.999968i	$0.502533\pi$
$860$	0	0
$861$	−9.61891	−0.327812
$862$	0	0
$863$	18.5205	0.630444	0.315222	−	0.949018i	$-0.397921\pi$
$863$	18.5205	0.630444	0.315222	+	0.949018i	$0.397921\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	5.93601	0.201597
$868$	0	0
$869$	10.2066	0.346234
$870$	0	0
$871$	16.7626	0.567978
$872$	0	0
$873$	8.51432	0.288166
$874$	0	0
$875$	0	0
$876$	0	0
$877$	18.9253	0.639062	0.319531	−	0.947576i	$-0.396475\pi$
$877$	18.9253	0.639062	0.319531	+	0.947576i	$0.396475\pi$
$878$	0	0
$879$	−32.1333	−1.08383
$880$	0	0
$881$	24.9803	0.841609	0.420804	−	0.907151i	$-0.361748\pi$
$881$	24.9803	0.841609	0.420804	+	0.907151i	$0.361748\pi$
$882$	0	0
$883$	−34.1349	−1.14873	−0.574365	−	0.818599i	$-0.694751\pi$
$883$	−34.1349	−1.14873	−0.574365	+	0.818599i	$0.694751\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.4660	0.888642	0.444321	−	0.895868i	$-0.353445\pi$
$887$	26.4660	0.888642	0.444321	+	0.895868i	$0.353445\pi$
$888$	0	0
$889$	66.5771	2.23292
$890$	0	0
$891$	1.34171	0.0449490
$892$	0	0
$893$	−79.0029	−2.64373
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.53686	0.218259
$898$	0	0
$899$	1.60070	0.0533865
$900$	0	0
$901$	−21.3892	−0.712577
$902$	0	0
$903$	26.8669	0.894075
$904$	0	0
$905$	0	0
$906$	0	0
$907$	52.0256	1.72748	0.863741	−	0.503936i	$-0.168115\pi$
$907$	52.0256	1.72748	0.863741	+	0.503936i	$0.168115\pi$
$908$	0	0
$909$	−13.6586	−0.453028
$910$	0	0
$911$	12.6122	0.417861	0.208931	−	0.977930i	$-0.433002\pi$
$911$	12.6122	0.417861	0.208931	+	0.977930i	$0.433002\pi$
$912$	0	0
$913$	17.8129	0.589522
$914$	0	0
$915$	0	0
$916$	0	0
$917$	33.6121	1.10997
$918$	0	0
$919$	−9.73642	−0.321175	−0.160587	−	0.987022i	$-0.551339\pi$
$919$	−9.73642	−0.321175	−0.160587	+	0.987022i	$0.551339\pi$
$920$	0	0
$921$	−6.60166	−0.217532
$922$	0	0
$923$	14.0723	0.463196
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.38188	−0.0453867
$928$	0	0
$929$	46.8012	1.53550	0.767749	−	0.640750i	$-0.221377\pi$
$929$	46.8012	1.53550	0.767749	+	0.640750i	$0.221377\pi$
$930$	0	0
$931$	−96.5857	−3.16547
$932$	0	0
$933$	19.3887	0.634759
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−27.3246	−0.892656	−0.446328	−	0.894869i	$-0.647268\pi$
$937$	−27.3246	−0.892656	−0.446328	+	0.894869i	$0.647268\pi$
$938$	0	0
$939$	−19.4393	−0.634378
$940$	0	0
$941$	20.7003	0.674812	0.337406	−	0.941359i	$-0.390451\pi$
$941$	20.7003	0.674812	0.337406	+	0.941359i	$0.390451\pi$
$942$	0	0
$943$	−10.0621	−0.327669
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−27.9924	−0.909629	−0.454815	−	0.890586i	$-0.650294\pi$
$947$	−27.9924	−0.909629	−0.454815	+	0.890586i	$0.650294\pi$
$948$	0	0
$949$	2.65078	0.0860480
$950$	0	0
$951$	−27.9705	−0.907004
$952$	0	0
$953$	−24.8808	−0.805967	−0.402984	−	0.915207i	$-0.632027\pi$
$953$	−24.8808	−0.805967	−0.402984	+	0.915207i	$0.632027\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.14768	−0.0694248
$958$	0	0
$959$	42.8275	1.38297
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−16.2480	−0.523584
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−55.3041	−1.77846	−0.889229	−	0.457461i	$-0.848759\pi$
$967$	−55.3041	−1.77846	−0.889229	+	0.457461i	$0.848759\pi$
$968$	0	0
$969$	30.3737	0.975744
$970$	0	0
$971$	−15.7026	−0.503920	−0.251960	−	0.967738i	$-0.581075\pi$
$971$	−15.7026	−0.503920	−0.251960	+	0.967738i	$0.581075\pi$
$972$	0	0
$973$	78.5272	2.51747
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.4507	1.29413	0.647066	−	0.762434i	$-0.275996\pi$
$977$	40.4507	1.29413	0.647066	+	0.762434i	$0.275996\pi$
$978$	0	0
$979$	14.9132	0.476626
$980$	0	0
$981$	15.2128	0.485706
$982$	0	0
$983$	−34.3949	−1.09703	−0.548513	−	0.836142i	$-0.684806\pi$
$983$	−34.3949	−1.09703	−0.548513	+	0.836142i	$0.684806\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	58.7307	1.86942
$988$	0	0
$989$	28.1049	0.893685
$990$	0	0
$991$	−12.3399	−0.391990	−0.195995	−	0.980605i	$-0.562794\pi$
$991$	−12.3399	−0.391990	−0.195995	+	0.980605i	$0.562794\pi$
$992$	0	0
$993$	4.37804	0.138933
$994$	0	0
$995$	0	0
$996$	0	0
$997$	16.3448	0.517645	0.258823	−	0.965925i	$-0.416665\pi$
$997$	16.3448	0.517645	0.258823	+	0.965925i	$0.416665\pi$
$998$	0	0
$999$	−10.3987	−0.328999

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.a.ba.1.6	yes	6
5.2	odd	4		9300.2.g.t.3349.6		12
5.3	odd	4		9300.2.g.t.3349.7		12
5.4	even	2		9300.2.a.y.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9300.2.a.y.1.1	✓	6	5.4	even	2
9300.2.a.ba.1.6	yes	6	1.1	even	1	trivial
9300.2.g.t.3349.6		12	5.2	odd	4
9300.2.g.t.3349.7		12	5.3	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:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +4.71477 q^{7} +1.00000 q^{9} +1.34171 q^{11} +1.32539 q^{13} -4.78916 q^{17} -6.34219 q^{19} +4.71477 q^{21} +4.93203 q^{23} +1.00000 q^{27} -1.60070 q^{29} -1.00000 q^{31} +1.34171 q^{33} -10.3987 q^{37} +1.32539 q^{39} -2.04016 q^{41} +5.69845 q^{43} +12.4567 q^{47} +15.2291 q^{49} -4.78916 q^{51} +4.46617 q^{53} -6.34219 q^{57} -4.95487 q^{59} +2.40968 q^{61} +4.71477 q^{63} +12.6473 q^{67} +4.93203 q^{69} +10.6175 q^{71} +2.00000 q^{73} +6.32587 q^{77} +7.60711 q^{79} +1.00000 q^{81} +13.2763 q^{83} -1.60070 q^{87} +11.1150 q^{89} +6.24891 q^{91} -1.00000 q^{93} +8.51432 q^{97} +1.34171 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 4 q^{7} + 6 q^{9} - 3 q^{11} + 2 q^{13} + 4 q^{17} - 3 q^{19} + 4 q^{21} + 5 q^{23} + 6 q^{27} - 6 q^{31} - 3 q^{33} + 8 q^{37} + 2 q^{39} + 18 q^{41} + 15 q^{43} + q^{47} + 14 q^{49} + 4 q^{51}+ \cdots - 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 4 * q^7 + 6 * q^9 - 3 * q^11 + 2 * q^13 + 4 * q^17 - 3 * q^19 + 4 * q^21 + 5 * q^23 + 6 * q^27 - 6 * q^31 - 3 * q^33 + 8 * q^37 + 2 * q^39 + 18 * q^41 + 15 * q^43 + q^47 + 14 * q^49 + 4 * q^51 + 5 * q^53 - 3 * q^57 - 12 * q^59 + 28 * q^61 + 4 * q^63 + 3 * q^67 + 5 * q^69 + 25 * q^71 + 12 * q^73 + 8 * q^77 + 3 * q^79 + 6 * q^81 + 19 * q^83 + 4 * q^89 + 16 * q^91 - 6 * q^93 - 2 * q^97 - 3 * q^99

Introduction

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