LMFDB - Embedded newform 9196.2.a.j.1.5

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 = [6,0,-2,0,-7,0,4,0,6,0,0,0,1] 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:	$6$
Coefficient field:	6.6.34963625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 13x^{3} + 27x^{2} - 6x - 4 $ x^6 - 2x^5 - 10x^4 + 13x^3 + 27x^2 - 6*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.44304$ 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)

$f(q)$	$=$	$q+1.44304 q^{3} -4.29314 q^{5} -3.18498 q^{7} -0.917645 q^{9} +0.891846 q^{13} -6.19516 q^{15} -0.149898 q^{17} +1.00000 q^{19} -4.59605 q^{21} +1.71038 q^{23} +13.4310 q^{25} -5.65331 q^{27} +6.45632 q^{29} +6.04556 q^{31} +13.6736 q^{35} +5.91764 q^{37} +1.28697 q^{39} +7.88959 q^{41} -9.58112 q^{43} +3.93958 q^{45} -3.92668 q^{47} +3.14413 q^{49} -0.216308 q^{51} +12.1822 q^{53} +1.44304 q^{57} -2.15607 q^{59} +2.87920 q^{61} +2.92268 q^{63} -3.82882 q^{65} -0.0354569 q^{67} +2.46814 q^{69} -15.6871 q^{71} +3.90831 q^{73} +19.3815 q^{75} +2.55410 q^{79} -5.40499 q^{81} +1.23048 q^{83} +0.643533 q^{85} +9.31671 q^{87} -0.565663 q^{89} -2.84051 q^{91} +8.72396 q^{93} -4.29314 q^{95} -3.51675 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 7 q^{5} + 4 q^{7} + 6 q^{9} + q^{13} - 8 q^{15} - 9 q^{17} + 6 q^{19} + 5 q^{21} - 12 q^{23} + 17 q^{25} - 17 q^{27} - 8 q^{29} + 7 q^{31} + 19 q^{35} + 24 q^{37} - 17 q^{39} - 15 q^{41}+ \cdots - 13 q^{97}+O(q^{100}) $ 6 * q - 2 * q^3 - 7 * q^5 + 4 * q^7 + 6 * q^9 + q^13 - 8 * q^15 - 9 * q^17 + 6 * q^19 + 5 * q^21 - 12 * q^23 + 17 * q^25 - 17 * q^27 - 8 * q^29 + 7 * q^31 + 19 * q^35 + 24 * q^37 - 17 * q^39 - 15 * q^41 - 23 * q^43 + 8 * q^45 - 10 * q^47 + 10 * q^49 - 10 * q^51 + 16 * q^53 - 2 * q^57 - 27 * q^59 - 7 * q^61 + 16 * q^63 + 14 * q^65 - 11 * q^67 - 16 * q^71 + 6 * q^73 + 5 * q^75 + 34 * q^79 + 2 * q^81 - 2 * q^83 - 18 * q^85 - 18 * q^87 + 13 * q^89 - 25 * q^91 + 2 * q^93 - 7 * q^95 - 13 * 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
$3$	1.44304	0.833138	0.416569	−	0.909104i	$-0.363232\pi$
$3$	1.44304	0.833138	0.416569	+	0.909104i	$0.363232\pi$
$4$	0	0
$5$	−4.29314	−1.91995	−0.959975	−	0.280086i	$-0.909637\pi$
$5$	−4.29314	−1.91995	−0.959975	+	0.280086i	$0.909637\pi$
$6$	0	0
$7$	−3.18498	−1.20381	−0.601905	−	0.798567i	$-0.705592\pi$
$7$	−3.18498	−1.20381	−0.601905	+	0.798567i	$0.705592\pi$
$8$	0	0
$9$	−0.917645	−0.305882
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.891846	0.247354	0.123677	−	0.992323i	$-0.460531\pi$
$13$	0.891846	0.247354	0.123677	+	0.992323i	$0.460531\pi$
$14$	0	0
$15$	−6.19516	−1.59958
$16$	0	0
$17$	−0.149898	−0.0363556	−0.0181778	−	0.999835i	$-0.505786\pi$
$17$	−0.149898	−0.0363556	−0.0181778	+	0.999835i	$0.505786\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−4.59605	−1.00294
$22$	0	0
$23$	1.71038	0.356638	0.178319	−	0.983973i	$-0.442934\pi$
$23$	1.71038	0.356638	0.178319	+	0.983973i	$0.442934\pi$
$24$	0	0
$25$	13.4310	2.68621
$26$	0	0
$27$	−5.65331	−1.08798
$28$	0	0
$29$	6.45632	1.19891	0.599455	−	0.800409i	$-0.295384\pi$
$29$	6.45632	1.19891	0.599455	+	0.800409i	$0.295384\pi$
$30$	0	0
$31$	6.04556	1.08581	0.542907	−	0.839793i	$-0.317324\pi$
$31$	6.04556	1.08581	0.542907	+	0.839793i	$0.317324\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	13.6736	2.31126
$36$	0	0
$37$	5.91764	0.972855	0.486427	−	0.873721i	$-0.338300\pi$
$37$	5.91764	0.972855	0.486427	+	0.873721i	$0.338300\pi$
$38$	0	0
$39$	1.28697	0.206080
$40$	0	0
$41$	7.88959	1.23215	0.616073	−	0.787689i	$-0.288722\pi$
$41$	7.88959	1.23215	0.616073	+	0.787689i	$0.288722\pi$
$42$	0	0
$43$	−9.58112	−1.46111	−0.730554	−	0.682855i	$-0.760738\pi$
$43$	−9.58112	−1.46111	−0.730554	+	0.682855i	$0.760738\pi$
$44$	0	0
$45$	3.93958	0.587277
$46$	0	0
$47$	−3.92668	−0.572766	−0.286383	−	0.958115i	$-0.592453\pi$
$47$	−3.92668	−0.572766	−0.286383	+	0.958115i	$0.592453\pi$
$48$	0	0
$49$	3.14413	0.449161
$50$	0	0
$51$	−0.216308	−0.0302892
$52$	0	0
$53$	12.1822	1.67336	0.836678	−	0.547695i	$-0.184495\pi$
$53$	12.1822	1.67336	0.836678	+	0.547695i	$0.184495\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.44304	0.191135
$58$	0	0
$59$	−2.15607	−0.280696	−0.140348	−	0.990102i	$-0.544822\pi$
$59$	−2.15607	−0.280696	−0.140348	+	0.990102i	$0.544822\pi$
$60$	0	0
$61$	2.87920	0.368644	0.184322	−	0.982866i	$-0.440991\pi$
$61$	2.87920	0.368644	0.184322	+	0.982866i	$0.440991\pi$
$62$	0	0
$63$	2.92268	0.368224
$64$	0	0
$65$	−3.82882	−0.474906
$66$	0	0
$67$	−0.0354569	−0.00433175	−0.00216588	−	0.999998i	$-0.500689\pi$
$67$	−0.0354569	−0.00433175	−0.00216588	+	0.999998i	$0.500689\pi$
$68$	0	0
$69$	2.46814	0.297129
$70$	0	0
$71$	−15.6871	−1.86171	−0.930855	−	0.365388i	$-0.880936\pi$
$71$	−15.6871	−1.86171	−0.930855	+	0.365388i	$0.880936\pi$
$72$	0	0
$73$	3.90831	0.457433	0.228716	−	0.973493i	$-0.426547\pi$
$73$	3.90831	0.457433	0.228716	+	0.973493i	$0.426547\pi$
$74$	0	0
$75$	19.3815	2.23798
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.55410	0.287358	0.143679	−	0.989624i	$-0.454107\pi$
$79$	2.55410	0.287358	0.143679	+	0.989624i	$0.454107\pi$
$80$	0	0
$81$	−5.40499	−0.600555
$82$	0	0
$83$	1.23048	0.135063	0.0675314	−	0.997717i	$-0.478488\pi$
$83$	1.23048	0.135063	0.0675314	+	0.997717i	$0.478488\pi$
$84$	0	0
$85$	0.643533	0.0698010
$86$	0	0
$87$	9.31671	0.998856
$88$	0	0
$89$	−0.565663	−0.0599602	−0.0299801	−	0.999550i	$-0.509544\pi$
$89$	−0.565663	−0.0599602	−0.0299801	+	0.999550i	$0.509544\pi$
$90$	0	0
$91$	−2.84051	−0.297767
$92$	0	0
$93$	8.72396	0.904633
$94$	0	0
$95$	−4.29314	−0.440467
$96$	0	0
$97$	−3.51675	−0.357072	−0.178536	−	0.983933i	$-0.557136\pi$
$97$	−3.51675	−0.357072	−0.178536	+	0.983933i	$0.557136\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−12.4443	−1.23825	−0.619125	−	0.785292i	$-0.712513\pi$
$101$	−12.4443	−1.23825	−0.619125	+	0.785292i	$0.712513\pi$
$102$	0	0
$103$	−8.83314	−0.870355	−0.435177	−	0.900345i	$-0.643314\pi$
$103$	−8.83314	−0.870355	−0.435177	+	0.900345i	$0.643314\pi$
$104$	0	0
$105$	19.7315	1.92560
$106$	0	0
$107$	−5.71645	−0.552630	−0.276315	−	0.961067i	$-0.589113\pi$
$107$	−5.71645	−0.552630	−0.276315	+	0.961067i	$0.589113\pi$
$108$	0	0
$109$	2.19072	0.209833	0.104917	−	0.994481i	$-0.466542\pi$
$109$	2.19072	0.209833	0.104917	+	0.994481i	$0.466542\pi$
$110$	0	0
$111$	8.53938	0.810522
$112$	0	0
$113$	−13.5547	−1.27512	−0.637560	−	0.770401i	$-0.720056\pi$
$113$	−13.5547	−1.27512	−0.637560	+	0.770401i	$0.720056\pi$
$114$	0	0
$115$	−7.34288	−0.684727
$116$	0	0
$117$	−0.818398	−0.0756609
$118$	0	0
$119$	0.477423	0.0437653
$120$	0	0
$121$	0	0
$122$	0	0
$123$	11.3850	1.02655
$124$	0	0
$125$	−36.1956	−3.23743
$126$	0	0
$127$	−1.06419	−0.0944312	−0.0472156	−	0.998885i	$-0.515035\pi$
$127$	−1.06419	−0.0944312	−0.0472156	+	0.998885i	$0.515035\pi$
$128$	0	0
$129$	−13.8259	−1.21730
$130$	0	0
$131$	−22.0727	−1.92850	−0.964252	−	0.264985i	$-0.914633\pi$
$131$	−22.0727	−1.92850	−0.964252	+	0.264985i	$0.914633\pi$
$132$	0	0
$133$	−3.18498	−0.276173
$134$	0	0
$135$	24.2704	2.08887
$136$	0	0
$137$	−7.31607	−0.625054	−0.312527	−	0.949909i	$-0.601175\pi$
$137$	−7.31607	−0.625054	−0.312527	+	0.949909i	$0.601175\pi$
$138$	0	0
$139$	−10.5496	−0.894807	−0.447403	−	0.894332i	$-0.647651\pi$
$139$	−10.5496	−0.894807	−0.447403	+	0.894332i	$0.647651\pi$
$140$	0	0
$141$	−5.66635	−0.477193
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−27.7179	−2.30185
$146$	0	0
$147$	4.53709	0.374213
$148$	0	0
$149$	20.4974	1.67922	0.839608	−	0.543192i	$-0.182785\pi$
$149$	20.4974	1.67922	0.839608	+	0.543192i	$0.182785\pi$
$150$	0	0
$151$	2.38438	0.194038	0.0970190	−	0.995283i	$-0.469069\pi$
$151$	2.38438	0.194038	0.0970190	+	0.995283i	$0.469069\pi$
$152$	0	0
$153$	0.137553	0.0111205
$154$	0	0
$155$	−25.9544	−2.08471
$156$	0	0
$157$	−0.0864746	−0.00690143	−0.00345071	−	0.999994i	$-0.501098\pi$
$157$	−0.0864746	−0.00690143	−0.00345071	+	0.999994i	$0.501098\pi$
$158$	0	0
$159$	17.5794	1.39414
$160$	0	0
$161$	−5.44752	−0.429325
$162$	0	0
$163$	−16.1912	−1.26819	−0.634095	−	0.773255i	$-0.718627\pi$
$163$	−16.1912	−1.26819	−0.634095	+	0.773255i	$0.718627\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	9.97124	0.771598	0.385799	−	0.922583i	$-0.373926\pi$
$167$	9.97124	0.771598	0.385799	+	0.922583i	$0.373926\pi$
$168$	0	0
$169$	−12.2046	−0.938816
$170$	0	0
$171$	−0.917645	−0.0701741
$172$	0	0
$173$	24.9842	1.89952	0.949758	−	0.312987i	$-0.101329\pi$
$173$	24.9842	1.89952	0.949758	+	0.312987i	$0.101329\pi$
$174$	0	0
$175$	−42.7777	−3.23369
$176$	0	0
$177$	−3.11129	−0.233859
$178$	0	0
$179$	−8.65082	−0.646593	−0.323296	−	0.946298i	$-0.604791\pi$
$179$	−8.65082	−0.646593	−0.323296	+	0.946298i	$0.604791\pi$
$180$	0	0
$181$	−16.2875	−1.21064	−0.605319	−	0.795983i	$-0.706954\pi$
$181$	−16.2875	−1.21064	−0.605319	+	0.795983i	$0.706954\pi$
$182$	0	0
$183$	4.15479	0.307131
$184$	0	0
$185$	−25.4053	−1.86783
$186$	0	0
$187$	0	0
$188$	0	0
$189$	18.0057	1.30972
$190$	0	0
$191$	−10.6153	−0.768100	−0.384050	−	0.923312i	$-0.625471\pi$
$191$	−10.6153	−0.768100	−0.384050	+	0.923312i	$0.625471\pi$
$192$	0	0
$193$	−7.10324	−0.511303	−0.255651	−	0.966769i	$-0.582290\pi$
$193$	−7.10324	−0.511303	−0.255651	+	0.966769i	$0.582290\pi$
$194$	0	0
$195$	−5.52512	−0.395662
$196$	0	0
$197$	0.640780	0.0456537	0.0228269	−	0.999739i	$-0.492733\pi$
$197$	0.640780	0.0456537	0.0228269	+	0.999739i	$0.492733\pi$
$198$	0	0
$199$	21.7365	1.54086	0.770431	−	0.637524i	$-0.220041\pi$
$199$	21.7365	1.54086	0.770431	+	0.637524i	$0.220041\pi$
$200$	0	0
$201$	−0.0511656	−0.00360895
$202$	0	0
$203$	−20.5633	−1.44326
$204$	0	0
$205$	−33.8711	−2.36566
$206$	0	0
$207$	−1.56952	−0.109089
$208$	0	0
$209$	0	0
$210$	0	0
$211$	25.5972	1.76218	0.881090	−	0.472948i	$-0.156810\pi$
$211$	25.5972	1.76218	0.881090	+	0.472948i	$0.156810\pi$
$212$	0	0
$213$	−22.6370	−1.55106
$214$	0	0
$215$	41.1331	2.80525
$216$	0	0
$217$	−19.2550	−1.30712
$218$	0	0
$219$	5.63983	0.381104
$220$	0	0
$221$	−0.133686	−0.00899269
$222$	0	0
$223$	13.0847	0.876216	0.438108	−	0.898922i	$-0.355649\pi$
$223$	13.0847	0.876216	0.438108	+	0.898922i	$0.355649\pi$
$224$	0	0
$225$	−12.3249	−0.821662
$226$	0	0
$227$	−15.5174	−1.02993	−0.514963	−	0.857212i	$-0.672194\pi$
$227$	−15.5174	−1.02993	−0.514963	+	0.857212i	$0.672194\pi$
$228$	0	0
$229$	6.46707	0.427356	0.213678	−	0.976904i	$-0.431456\pi$
$229$	6.46707	0.427356	0.213678	+	0.976904i	$0.431456\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.5723	1.34774	0.673868	−	0.738852i	$-0.264632\pi$
$233$	20.5723	1.34774	0.673868	+	0.738852i	$0.264632\pi$
$234$	0	0
$235$	16.8578	1.09968
$236$	0	0
$237$	3.68565	0.239409
$238$	0	0
$239$	−22.3155	−1.44347	−0.721736	−	0.692169i	$-0.756655\pi$
$239$	−22.3155	−1.44347	−0.721736	+	0.692169i	$0.756655\pi$
$240$	0	0
$241$	28.7968	1.85497	0.927484	−	0.373864i	$-0.121967\pi$
$241$	28.7968	1.85497	0.927484	+	0.373864i	$0.121967\pi$
$242$	0	0
$243$	9.16031	0.587634
$244$	0	0
$245$	−13.4982	−0.862366
$246$	0	0
$247$	0.891846	0.0567468
$248$	0	0
$249$	1.77563	0.112526
$250$	0	0
$251$	−11.5389	−0.728331	−0.364165	−	0.931334i	$-0.618646\pi$
$251$	−11.5389	−0.728331	−0.364165	+	0.931334i	$0.618646\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.928642	0.0581538
$256$	0	0
$257$	24.6390	1.53694	0.768470	−	0.639886i	$-0.221018\pi$
$257$	24.6390	1.53694	0.768470	+	0.639886i	$0.221018\pi$
$258$	0	0
$259$	−18.8476	−1.17113
$260$	0	0
$261$	−5.92461	−0.366724
$262$	0	0
$263$	22.1840	1.36792	0.683961	−	0.729519i	$-0.260256\pi$
$263$	22.1840	1.36792	0.683961	+	0.729519i	$0.260256\pi$
$264$	0	0
$265$	−52.2999	−3.21276
$266$	0	0
$267$	−0.816273	−0.0499551
$268$	0	0
$269$	14.5304	0.885936	0.442968	−	0.896537i	$-0.353925\pi$
$269$	14.5304	0.885936	0.442968	+	0.896537i	$0.353925\pi$
$270$	0	0
$271$	10.0944	0.613189	0.306594	−	0.951840i	$-0.400810\pi$
$271$	10.0944	0.613189	0.306594	+	0.951840i	$0.400810\pi$
$272$	0	0
$273$	−4.09897	−0.248081
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−11.0488	−0.663856	−0.331928	−	0.943305i	$-0.607699\pi$
$277$	−11.0488	−0.663856	−0.331928	+	0.943305i	$0.607699\pi$
$278$	0	0
$279$	−5.54768	−0.332131
$280$	0	0
$281$	0.162238	0.00967831	0.00483915	−	0.999988i	$-0.498460\pi$
$281$	0.162238	0.00967831	0.00483915	+	0.999988i	$0.498460\pi$
$282$	0	0
$283$	−19.7147	−1.17191	−0.585957	−	0.810342i	$-0.699281\pi$
$283$	−19.7147	−1.17191	−0.585957	+	0.810342i	$0.699281\pi$
$284$	0	0
$285$	−6.19516	−0.366969
$286$	0	0
$287$	−25.1282	−1.48327
$288$	0	0
$289$	−16.9775	−0.998678
$290$	0	0
$291$	−5.07480	−0.297490
$292$	0	0
$293$	9.63486	0.562874	0.281437	−	0.959580i	$-0.409189\pi$
$293$	9.63486	0.562874	0.281437	+	0.959580i	$0.409189\pi$
$294$	0	0
$295$	9.25631	0.538923
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.52539	0.0882157
$300$	0	0
$301$	30.5157	1.75890
$302$	0	0
$303$	−17.9575	−1.03163
$304$	0	0
$305$	−12.3608	−0.707778
$306$	0	0
$307$	16.9356	0.966564	0.483282	−	0.875465i	$-0.339445\pi$
$307$	16.9356	0.966564	0.483282	+	0.875465i	$0.339445\pi$
$308$	0	0
$309$	−12.7465	−0.725125
$310$	0	0
$311$	−9.69990	−0.550031	−0.275016	−	0.961440i	$-0.588683\pi$
$311$	−9.69990	−0.550031	−0.275016	+	0.961440i	$0.588683\pi$
$312$	0	0
$313$	−21.1064	−1.19300	−0.596501	−	0.802612i	$-0.703443\pi$
$313$	−21.1064	−1.19300	−0.596501	+	0.802612i	$0.703443\pi$
$314$	0	0
$315$	−12.5475	−0.706971
$316$	0	0
$317$	−10.1639	−0.570862	−0.285431	−	0.958399i	$-0.592137\pi$
$317$	−10.1639	−0.570862	−0.285431	+	0.958399i	$0.592137\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−8.24905	−0.460417
$322$	0	0
$323$	−0.149898	−0.00834055
$324$	0	0
$325$	11.9784	0.664443
$326$	0	0
$327$	3.16130	0.174820
$328$	0	0
$329$	12.5064	0.689502
$330$	0	0
$331$	5.41881	0.297845	0.148922	−	0.988849i	$-0.452420\pi$
$331$	5.41881	0.297845	0.148922	+	0.988849i	$0.452420\pi$
$332$	0	0
$333$	−5.43030	−0.297578
$334$	0	0
$335$	0.152221	0.00831675
$336$	0	0
$337$	35.4590	1.93158	0.965788	−	0.259331i	$-0.0835022\pi$
$337$	35.4590	1.93158	0.965788	+	0.259331i	$0.0835022\pi$
$338$	0	0
$339$	−19.5599	−1.06235
$340$	0	0
$341$	0	0
$342$	0	0
$343$	12.2809	0.663106
$344$	0	0
$345$	−10.5960	−0.570472
$346$	0	0
$347$	−31.1448	−1.67194	−0.835970	−	0.548775i	$-0.815094\pi$
$347$	−31.1448	−1.67194	−0.835970	+	0.548775i	$0.815094\pi$
$348$	0	0
$349$	6.44944	0.345230	0.172615	−	0.984989i	$-0.444778\pi$
$349$	6.44944	0.345230	0.172615	+	0.984989i	$0.444778\pi$
$350$	0	0
$351$	−5.04188	−0.269115
$352$	0	0
$353$	−21.0750	−1.12171	−0.560855	−	0.827914i	$-0.689528\pi$
$353$	−21.0750	−1.12171	−0.560855	+	0.827914i	$0.689528\pi$
$354$	0	0
$355$	67.3467	3.57439
$356$	0	0
$357$	0.688939	0.0364625
$358$	0	0
$359$	−29.1246	−1.53713	−0.768567	−	0.639769i	$-0.779030\pi$
$359$	−29.1246	−1.53713	−0.768567	+	0.639769i	$0.779030\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−16.7789	−0.878248
$366$	0	0
$367$	21.2444	1.10895	0.554474	−	0.832201i	$-0.312920\pi$
$367$	21.2444	1.10895	0.554474	+	0.832201i	$0.312920\pi$
$368$	0	0
$369$	−7.23984	−0.376891
$370$	0	0
$371$	−38.8002	−2.01440
$372$	0	0
$373$	20.0676	1.03906	0.519529	−	0.854453i	$-0.326107\pi$
$373$	20.0676	1.03906	0.519529	+	0.854453i	$0.326107\pi$
$374$	0	0
$375$	−52.2316	−2.69723
$376$	0	0
$377$	5.75805	0.296554
$378$	0	0
$379$	17.0472	0.875658	0.437829	−	0.899058i	$-0.355748\pi$
$379$	17.0472	0.875658	0.437829	+	0.899058i	$0.355748\pi$
$380$	0	0
$381$	−1.53566	−0.0786742
$382$	0	0
$383$	−6.54074	−0.334216	−0.167108	−	0.985939i	$-0.553443\pi$
$383$	−6.54074	−0.334216	−0.167108	+	0.985939i	$0.553443\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.79207	0.446926
$388$	0	0
$389$	22.0327	1.11710	0.558550	−	0.829471i	$-0.311358\pi$
$389$	22.0327	1.11710	0.558550	+	0.829471i	$0.311358\pi$
$390$	0	0
$391$	−0.256382	−0.0129658
$392$	0	0
$393$	−31.8518	−1.60671
$394$	0	0
$395$	−10.9651	−0.551713
$396$	0	0
$397$	−0.619105	−0.0310720	−0.0155360	−	0.999879i	$-0.504945\pi$
$397$	−0.619105	−0.0310720	−0.0155360	+	0.999879i	$0.504945\pi$
$398$	0	0
$399$	−4.59605	−0.230090
$400$	0	0
$401$	−29.2904	−1.46269	−0.731346	−	0.682007i	$-0.761107\pi$
$401$	−29.2904	−1.46269	−0.731346	+	0.682007i	$0.761107\pi$
$402$	0	0
$403$	5.39171	0.268580
$404$	0	0
$405$	23.2044	1.15304
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−32.9569	−1.62962	−0.814808	−	0.579731i	$-0.803157\pi$
$409$	−32.9569	−1.62962	−0.814808	+	0.579731i	$0.803157\pi$
$410$	0	0
$411$	−10.5574	−0.520756
$412$	0	0
$413$	6.86705	0.337905
$414$	0	0
$415$	−5.28263	−0.259314
$416$	0	0
$417$	−15.2235	−0.745497
$418$	0	0
$419$	21.5622	1.05338	0.526692	−	0.850056i	$-0.323432\pi$
$419$	21.5622	1.05338	0.526692	+	0.850056i	$0.323432\pi$
$420$	0	0
$421$	−36.6290	−1.78519	−0.892594	−	0.450861i	$-0.851117\pi$
$421$	−36.6290	−1.78519	−0.892594	+	0.450861i	$0.851117\pi$
$422$	0	0
$423$	3.60330	0.175199
$424$	0	0
$425$	−2.01329	−0.0976588
$426$	0	0
$427$	−9.17021	−0.443778
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.81884	−0.376620	−0.188310	−	0.982110i	$-0.560301\pi$
$431$	−7.81884	−0.376620	−0.188310	+	0.982110i	$0.560301\pi$
$432$	0	0
$433$	−14.6422	−0.703660	−0.351830	−	0.936064i	$-0.614441\pi$
$433$	−14.6422	−0.703660	−0.351830	+	0.936064i	$0.614441\pi$
$434$	0	0
$435$	−39.9979	−1.91775
$436$	0	0
$437$	1.71038	0.0818184
$438$	0	0
$439$	14.7358	0.703300	0.351650	−	0.936132i	$-0.385621\pi$
$439$	14.7358	0.703300	0.351650	+	0.936132i	$0.385621\pi$
$440$	0	0
$441$	−2.88519	−0.137390
$442$	0	0
$443$	−13.0339	−0.619260	−0.309630	−	0.950857i	$-0.600205\pi$
$443$	−13.0339	−0.619260	−0.309630	+	0.950857i	$0.600205\pi$
$444$	0	0
$445$	2.42847	0.115121
$446$	0	0
$447$	29.5786	1.39902
$448$	0	0
$449$	−31.6701	−1.49460	−0.747302	−	0.664484i	$-0.768651\pi$
$449$	−31.6701	−1.49460	−0.747302	+	0.664484i	$0.768651\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.44075	0.161660
$454$	0	0
$455$	12.1947	0.571697
$456$	0	0
$457$	35.4503	1.65830	0.829148	−	0.559029i	$-0.188826\pi$
$457$	35.4503	1.65830	0.829148	+	0.559029i	$0.188826\pi$
$458$	0	0
$459$	0.847420	0.0395542
$460$	0	0
$461$	−38.2849	−1.78311	−0.891553	−	0.452916i	$-0.850384\pi$
$461$	−38.2849	−1.78311	−0.891553	+	0.452916i	$0.850384\pi$
$462$	0	0
$463$	−36.1313	−1.67916	−0.839581	−	0.543234i	$-0.817200\pi$
$463$	−36.1313	−1.67916	−0.839581	+	0.543234i	$0.817200\pi$
$464$	0	0
$465$	−37.4532	−1.73685
$466$	0	0
$467$	28.7931	1.33239	0.666194	−	0.745779i	$-0.267922\pi$
$467$	28.7931	1.33239	0.666194	+	0.745779i	$0.267922\pi$
$468$	0	0
$469$	0.112930	0.00521461
$470$	0	0
$471$	−0.124786	−0.00574984
$472$	0	0
$473$	0	0
$474$	0	0
$475$	13.4310	0.616258
$476$	0	0
$477$	−11.1789	−0.511849
$478$	0	0
$479$	−18.1375	−0.828723	−0.414361	−	0.910112i	$-0.635995\pi$
$479$	−18.1375	−0.828723	−0.414361	+	0.910112i	$0.635995\pi$
$480$	0	0
$481$	5.27763	0.240639
$482$	0	0
$483$	−7.86097	−0.357687
$484$	0	0
$485$	15.0979	0.685561
$486$	0	0
$487$	−3.09958	−0.140455	−0.0702276	−	0.997531i	$-0.522373\pi$
$487$	−3.09958	−0.140455	−0.0702276	+	0.997531i	$0.522373\pi$
$488$	0	0
$489$	−23.3644	−1.05658
$490$	0	0
$491$	−18.6828	−0.843144	−0.421572	−	0.906795i	$-0.638521\pi$
$491$	−18.6828	−0.843144	−0.421572	+	0.906795i	$0.638521\pi$
$492$	0	0
$493$	−0.967791	−0.0435871
$494$	0	0
$495$	0	0
$496$	0	0
$497$	49.9630	2.24115
$498$	0	0
$499$	−18.9834	−0.849813	−0.424906	−	0.905237i	$-0.639693\pi$
$499$	−18.9834	−0.849813	−0.424906	+	0.905237i	$0.639693\pi$
$500$	0	0
$501$	14.3889	0.642847
$502$	0	0
$503$	1.24512	0.0555170	0.0277585	−	0.999615i	$-0.491163\pi$
$503$	1.24512	0.0555170	0.0277585	+	0.999615i	$0.491163\pi$
$504$	0	0
$505$	53.4250	2.37738
$506$	0	0
$507$	−17.6117	−0.782163
$508$	0	0
$509$	−18.1680	−0.805282	−0.402641	−	0.915358i	$-0.631908\pi$
$509$	−18.1680	−0.805282	−0.402641	+	0.915358i	$0.631908\pi$
$510$	0	0
$511$	−12.4479	−0.550662
$512$	0	0
$513$	−5.65331	−0.249600
$514$	0	0
$515$	37.9219	1.67104
$516$	0	0
$517$	0	0
$518$	0	0
$519$	36.0531	1.58256
$520$	0	0
$521$	−21.1319	−0.925806	−0.462903	−	0.886409i	$-0.653192\pi$
$521$	−21.1319	−0.925806	−0.462903	+	0.886409i	$0.653192\pi$
$522$	0	0
$523$	5.83699	0.255234	0.127617	−	0.991824i	$-0.459267\pi$
$523$	5.83699	0.255234	0.127617	+	0.991824i	$0.459267\pi$
$524$	0	0
$525$	−61.7297	−2.69411
$526$	0	0
$527$	−0.906218	−0.0394755
$528$	0	0
$529$	−20.0746	−0.872809
$530$	0	0
$531$	1.97851	0.0858599
$532$	0	0
$533$	7.03630	0.304776
$534$	0	0
$535$	24.5415	1.06102
$536$	0	0
$537$	−12.4835	−0.538701
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−36.1044	−1.55225	−0.776125	−	0.630579i	$-0.782817\pi$
$541$	−36.1044	−1.55225	−0.776125	+	0.630579i	$0.782817\pi$
$542$	0	0
$543$	−23.5034	−1.00863
$544$	0	0
$545$	−9.40508	−0.402869
$546$	0	0
$547$	23.6154	1.00972	0.504860	−	0.863201i	$-0.331544\pi$
$547$	23.6154	1.00972	0.504860	+	0.863201i	$0.331544\pi$
$548$	0	0
$549$	−2.64208	−0.112761
$550$	0	0
$551$	6.45632	0.275049
$552$	0	0
$553$	−8.13476	−0.345925
$554$	0	0
$555$	−36.6607	−1.55616
$556$	0	0
$557$	18.2180	0.771921	0.385961	−	0.922515i	$-0.373870\pi$
$557$	18.2180	0.771921	0.385961	+	0.922515i	$0.373870\pi$
$558$	0	0
$559$	−8.54488	−0.361410
$560$	0	0
$561$	0	0
$562$	0	0
$563$	30.5165	1.28612	0.643058	−	0.765817i	$-0.277665\pi$
$563$	30.5165	1.28612	0.643058	+	0.765817i	$0.277665\pi$
$564$	0	0
$565$	58.1922	2.44817
$566$	0	0
$567$	17.2148	0.722954
$568$	0	0
$569$	31.9283	1.33850	0.669252	−	0.743035i	$-0.266615\pi$
$569$	31.9283	1.33850	0.669252	+	0.743035i	$0.266615\pi$
$570$	0	0
$571$	22.2999	0.933222	0.466611	−	0.884463i	$-0.345475\pi$
$571$	22.2999	0.933222	0.466611	+	0.884463i	$0.345475\pi$
$572$	0	0
$573$	−15.3183	−0.639933
$574$	0	0
$575$	22.9721	0.958004
$576$	0	0
$577$	−15.4682	−0.643951	−0.321976	−	0.946748i	$-0.604347\pi$
$577$	−15.4682	−0.643951	−0.321976	+	0.946748i	$0.604347\pi$
$578$	0	0
$579$	−10.2502	−0.425986
$580$	0	0
$581$	−3.91906	−0.162590
$582$	0	0
$583$	0	0
$584$	0	0
$585$	3.51349	0.145265
$586$	0	0
$587$	−7.93758	−0.327619	−0.163810	−	0.986492i	$-0.552378\pi$
$587$	−7.93758	−0.327619	−0.163810	+	0.986492i	$0.552378\pi$
$588$	0	0
$589$	6.04556	0.249103
$590$	0	0
$591$	0.924670	0.0380358
$592$	0	0
$593$	6.09113	0.250133	0.125066	−	0.992148i	$-0.460086\pi$
$593$	6.09113	0.250133	0.125066	+	0.992148i	$0.460086\pi$
$594$	0	0
$595$	−2.04964	−0.0840272
$596$	0	0
$597$	31.3666	1.28375
$598$	0	0
$599$	−18.6986	−0.764004	−0.382002	−	0.924162i	$-0.624765\pi$
$599$	−18.6986	−0.764004	−0.382002	+	0.924162i	$0.624765\pi$
$600$	0	0
$601$	3.82057	0.155844	0.0779222	−	0.996959i	$-0.475171\pi$
$601$	3.82057	0.155844	0.0779222	+	0.996959i	$0.475171\pi$
$602$	0	0
$603$	0.0325369	0.00132500
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−46.9999	−1.90767	−0.953833	−	0.300338i	$-0.902901\pi$
$607$	−46.9999	−1.90767	−0.953833	+	0.300338i	$0.902901\pi$
$608$	0	0
$609$	−29.6736	−1.20243
$610$	0	0
$611$	−3.50200	−0.141676
$612$	0	0
$613$	3.70436	0.149618	0.0748089	−	0.997198i	$-0.476165\pi$
$613$	3.70436	0.149618	0.0748089	+	0.997198i	$0.476165\pi$
$614$	0	0
$615$	−48.8772	−1.97092
$616$	0	0
$617$	−40.9171	−1.64726	−0.823631	−	0.567127i	$-0.808055\pi$
$617$	−40.9171	−1.64726	−0.823631	+	0.567127i	$0.808055\pi$
$618$	0	0
$619$	−11.3657	−0.456824	−0.228412	−	0.973565i	$-0.573353\pi$
$619$	−11.3657	−0.456824	−0.228412	+	0.973565i	$0.573353\pi$
$620$	0	0
$621$	−9.66928	−0.388015
$622$	0	0
$623$	1.80163	0.0721807
$624$	0	0
$625$	88.2376	3.52951
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.887044	−0.0353687
$630$	0	0
$631$	25.1913	1.00285	0.501425	−	0.865201i	$-0.332809\pi$
$631$	25.1913	1.00285	0.501425	+	0.865201i	$0.332809\pi$
$632$	0	0
$633$	36.9376	1.46814
$634$	0	0
$635$	4.56870	0.181303
$636$	0	0
$637$	2.80408	0.111102
$638$	0	0
$639$	14.3951	0.569463
$640$	0	0
$641$	−42.6015	−1.68266	−0.841330	−	0.540522i	$-0.818227\pi$
$641$	−42.6015	−1.68266	−0.841330	+	0.540522i	$0.818227\pi$
$642$	0	0
$643$	4.06828	0.160437	0.0802187	−	0.996777i	$-0.474438\pi$
$643$	4.06828	0.160437	0.0802187	+	0.996777i	$0.474438\pi$
$644$	0	0
$645$	59.3566	2.33716
$646$	0	0
$647$	−1.08891	−0.0428094	−0.0214047	−	0.999771i	$-0.506814\pi$
$647$	−1.08891	−0.0428094	−0.0214047	+	0.999771i	$0.506814\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−27.7857	−1.08901
$652$	0	0
$653$	20.1627	0.789026	0.394513	−	0.918890i	$-0.370913\pi$
$653$	20.1627	0.789026	0.394513	+	0.918890i	$0.370913\pi$
$654$	0	0
$655$	94.7614	3.70263
$656$	0	0
$657$	−3.58644	−0.139920
$658$	0	0
$659$	24.0501	0.936859	0.468430	−	0.883501i	$-0.344820\pi$
$659$	24.0501	0.936859	0.468430	+	0.883501i	$0.344820\pi$
$660$	0	0
$661$	−16.4673	−0.640503	−0.320252	−	0.947333i	$-0.603767\pi$
$661$	−16.4673	−0.640503	−0.320252	+	0.947333i	$0.603767\pi$
$662$	0	0
$663$	−0.192914	−0.00749215
$664$	0	0
$665$	13.6736	0.530239
$666$	0	0
$667$	11.0427	0.427577
$668$	0	0
$669$	18.8817	0.730009
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−28.7742	−1.10917	−0.554583	−	0.832129i	$-0.687122\pi$
$673$	−28.7742	−1.10917	−0.554583	+	0.832129i	$0.687122\pi$
$674$	0	0
$675$	−75.9298	−2.92254
$676$	0	0
$677$	−35.1961	−1.35270	−0.676348	−	0.736582i	$-0.736439\pi$
$677$	−35.1961	−1.35270	−0.676348	+	0.736582i	$0.736439\pi$
$678$	0	0
$679$	11.2008	0.429847
$680$	0	0
$681$	−22.3922	−0.858070
$682$	0	0
$683$	−28.5789	−1.09354	−0.546770	−	0.837283i	$-0.684143\pi$
$683$	−28.5789	−1.09354	−0.546770	+	0.837283i	$0.684143\pi$
$684$	0	0
$685$	31.4089	1.20007
$686$	0	0
$687$	9.33223	0.356047
$688$	0	0
$689$	10.8647	0.413910
$690$	0	0
$691$	18.7354	0.712727	0.356364	−	0.934347i	$-0.384016\pi$
$691$	18.7354	0.712727	0.356364	+	0.934347i	$0.384016\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	45.2910	1.71798
$696$	0	0
$697$	−1.18263	−0.0447955
$698$	0	0
$699$	29.6866	1.12285
$700$	0	0
$701$	−7.66391	−0.289462	−0.144731	−	0.989471i	$-0.546232\pi$
$701$	−7.66391	−0.289462	−0.144731	+	0.989471i	$0.546232\pi$
$702$	0	0
$703$	5.91764	0.223188
$704$	0	0
$705$	24.3264	0.916186
$706$	0	0
$707$	39.6348	1.49062
$708$	0	0
$709$	−3.10440	−0.116588	−0.0582941	−	0.998299i	$-0.518566\pi$
$709$	−3.10440	−0.116588	−0.0582941	+	0.998299i	$0.518566\pi$
$710$	0	0
$711$	−2.34375	−0.0878976
$712$	0	0
$713$	10.3402	0.387243
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−32.2021	−1.20261
$718$	0	0
$719$	−14.1597	−0.528069	−0.264034	−	0.964513i	$-0.585053\pi$
$719$	−14.1597	−0.528069	−0.264034	+	0.964513i	$0.585053\pi$
$720$	0	0
$721$	28.1334	1.04774
$722$	0	0
$723$	41.5549	1.54544
$724$	0	0
$725$	86.7151	3.22052
$726$	0	0
$727$	18.8580	0.699404	0.349702	−	0.936861i	$-0.386283\pi$
$727$	18.8580	0.699404	0.349702	+	0.936861i	$0.386283\pi$
$728$	0	0
$729$	29.4336	1.09014
$730$	0	0
$731$	1.43619	0.0531195
$732$	0	0
$733$	−21.0001	−0.775656	−0.387828	−	0.921732i	$-0.626775\pi$
$733$	−21.0001	−0.775656	−0.387828	+	0.921732i	$0.626775\pi$
$734$	0	0
$735$	−19.4784	−0.718470
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−33.6752	−1.23876	−0.619381	−	0.785091i	$-0.712616\pi$
$739$	−33.6752	−1.23876	−0.619381	+	0.785091i	$0.712616\pi$
$740$	0	0
$741$	1.28697	0.0472779
$742$	0	0
$743$	−24.0620	−0.882749	−0.441374	−	0.897323i	$-0.645509\pi$
$743$	−24.0620	−0.882749	−0.441374	+	0.897323i	$0.645509\pi$
$744$	0	0
$745$	−87.9984	−3.22401
$746$	0	0
$747$	−1.12914	−0.0413132
$748$	0	0
$749$	18.2068	0.665262
$750$	0	0
$751$	28.1906	1.02869	0.514346	−	0.857583i	$-0.328035\pi$
$751$	28.1906	1.02869	0.514346	+	0.857583i	$0.328035\pi$
$752$	0	0
$753$	−16.6511	−0.606800
$754$	0	0
$755$	−10.2365	−0.372543
$756$	0	0
$757$	15.5458	0.565023	0.282511	−	0.959264i	$-0.408833\pi$
$757$	15.5458	0.565023	0.282511	+	0.959264i	$0.408833\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12.5679	−0.455584	−0.227792	−	0.973710i	$-0.573151\pi$
$761$	−12.5679	−0.455584	−0.227792	+	0.973710i	$0.573151\pi$
$762$	0	0
$763$	−6.97742	−0.252600
$764$	0	0
$765$	−0.590535	−0.0213508
$766$	0	0
$767$	−1.92288	−0.0694313
$768$	0	0
$769$	19.0550	0.687139	0.343570	−	0.939127i	$-0.388364\pi$
$769$	19.0550	0.687139	0.343570	+	0.939127i	$0.388364\pi$
$770$	0	0
$771$	35.5550	1.28048
$772$	0	0
$773$	29.5088	1.06136	0.530678	−	0.847573i	$-0.321937\pi$
$773$	29.5088	1.06136	0.530678	+	0.847573i	$0.321937\pi$
$774$	0	0
$775$	81.1981	2.91672
$776$	0	0
$777$	−27.1978	−0.975715
$778$	0	0
$779$	7.88959	0.282674
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−36.4996	−1.30439
$784$	0	0
$785$	0.371248	0.0132504
$786$	0	0
$787$	−34.6089	−1.23368	−0.616838	−	0.787090i	$-0.711586\pi$
$787$	−34.6089	−1.23368	−0.616838	+	0.787090i	$0.711586\pi$
$788$	0	0
$789$	32.0123	1.13967
$790$	0	0
$791$	43.1715	1.53500
$792$	0	0
$793$	2.56780	0.0911854
$794$	0	0
$795$	−75.4707	−2.67667
$796$	0	0
$797$	13.8827	0.491752	0.245876	−	0.969301i	$-0.420924\pi$
$797$	13.8827	0.491752	0.245876	+	0.969301i	$0.420924\pi$
$798$	0	0
$799$	0.588603	0.0208233
$800$	0	0
$801$	0.519078	0.0183407
$802$	0	0
$803$	0	0
$804$	0	0
$805$	23.3870	0.824282
$806$	0	0
$807$	20.9680	0.738107
$808$	0	0
$809$	−38.6407	−1.35853	−0.679267	−	0.733891i	$-0.737702\pi$
$809$	−38.6407	−1.35853	−0.679267	+	0.733891i	$0.737702\pi$
$810$	0	0
$811$	−19.0764	−0.669863	−0.334931	−	0.942243i	$-0.608713\pi$
$811$	−19.0764	−0.669863	−0.334931	+	0.942243i	$0.608713\pi$
$812$	0	0
$813$	14.5665	0.510871
$814$	0	0
$815$	69.5109	2.43486
$816$	0	0
$817$	−9.58112	−0.335201
$818$	0	0
$819$	2.60658	0.0910814
$820$	0	0
$821$	−12.8223	−0.447500	−0.223750	−	0.974647i	$-0.571830\pi$
$821$	−12.8223	−0.447500	−0.223750	+	0.974647i	$0.571830\pi$
$822$	0	0
$823$	44.6103	1.55502	0.777508	−	0.628873i	$-0.216483\pi$
$823$	44.6103	1.55502	0.777508	+	0.628873i	$0.216483\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.9500	0.589411	0.294705	−	0.955588i	$-0.404778\pi$
$827$	16.9500	0.589411	0.294705	+	0.955588i	$0.404778\pi$
$828$	0	0
$829$	−40.2151	−1.39673	−0.698363	−	0.715743i	$-0.746088\pi$
$829$	−40.2151	−1.39673	−0.698363	+	0.715743i	$0.746088\pi$
$830$	0	0
$831$	−15.9438	−0.553084
$832$	0	0
$833$	−0.471298	−0.0163295
$834$	0	0
$835$	−42.8079	−1.48143
$836$	0	0
$837$	−34.1774	−1.18134
$838$	0	0
$839$	−16.8539	−0.581860	−0.290930	−	0.956744i	$-0.593965\pi$
$839$	−16.8539	−0.581860	−0.290930	+	0.956744i	$0.593965\pi$
$840$	0	0
$841$	12.6841	0.437383
$842$	0	0
$843$	0.234115	0.00806336
$844$	0	0
$845$	52.3961	1.80248
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−28.4490	−0.976366
$850$	0	0
$851$	10.1214	0.346957
$852$	0	0
$853$	17.1077	0.585757	0.292879	−	0.956150i	$-0.405387\pi$
$853$	17.1077	0.585757	0.292879	+	0.956150i	$0.405387\pi$
$854$	0	0
$855$	3.93958	0.134731
$856$	0	0
$857$	−20.8283	−0.711482	−0.355741	−	0.934585i	$-0.615772\pi$
$857$	−20.8283	−0.711482	−0.355741	+	0.934585i	$0.615772\pi$
$858$	0	0
$859$	1.69054	0.0576806	0.0288403	−	0.999584i	$-0.490819\pi$
$859$	1.69054	0.0576806	0.0288403	+	0.999584i	$0.490819\pi$
$860$	0	0
$861$	−36.2609	−1.23577
$862$	0	0
$863$	−15.7743	−0.536963	−0.268481	−	0.963285i	$-0.586522\pi$
$863$	−15.7743	−0.536963	−0.268481	+	0.963285i	$0.586522\pi$
$864$	0	0
$865$	−107.261	−3.64697
$866$	0	0
$867$	−24.4992	−0.832036
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−0.0316221	−0.00107147
$872$	0	0
$873$	3.22713	0.109222
$874$	0	0
$875$	115.282	3.89726
$876$	0	0
$877$	−14.1378	−0.477398	−0.238699	−	0.971094i	$-0.576721\pi$
$877$	−14.1378	−0.477398	−0.238699	+	0.971094i	$0.576721\pi$
$878$	0	0
$879$	13.9035	0.468952
$880$	0	0
$881$	23.4093	0.788680	0.394340	−	0.918965i	$-0.370973\pi$
$881$	23.4093	0.788680	0.394340	+	0.918965i	$0.370973\pi$
$882$	0	0
$883$	−29.8628	−1.00496	−0.502482	−	0.864588i	$-0.667580\pi$
$883$	−29.8628	−1.00496	−0.502482	+	0.864588i	$0.667580\pi$
$884$	0	0
$885$	13.3572	0.448997
$886$	0	0
$887$	−40.4679	−1.35878	−0.679390	−	0.733777i	$-0.737756\pi$
$887$	−40.4679	−1.35878	−0.679390	+	0.733777i	$0.737756\pi$
$888$	0	0
$889$	3.38941	0.113677
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.92668	−0.131401
$894$	0	0
$895$	37.1392	1.24143
$896$	0	0
$897$	2.20120	0.0734958
$898$	0	0
$899$	39.0321	1.30179
$900$	0	0
$901$	−1.82609	−0.0608359
$902$	0	0
$903$	44.0353	1.46540
$904$	0	0
$905$	69.9244	2.32436
$906$	0	0
$907$	−16.7806	−0.557191	−0.278595	−	0.960409i	$-0.589869\pi$
$907$	−16.7806	−0.557191	−0.278595	+	0.960409i	$0.589869\pi$
$908$	0	0
$909$	11.4194	0.378758
$910$	0	0
$911$	−58.6056	−1.94169	−0.970845	−	0.239707i	$-0.922949\pi$
$911$	−58.6056	−1.94169	−0.970845	+	0.239707i	$0.922949\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−17.8371	−0.589676
$916$	0	0
$917$	70.3014	2.32156
$918$	0	0
$919$	47.8554	1.57860	0.789301	−	0.614006i	$-0.210443\pi$
$919$	47.8554	1.57860	0.789301	+	0.614006i	$0.210443\pi$
$920$	0	0
$921$	24.4386	0.805281
$922$	0	0
$923$	−13.9904	−0.460501
$924$	0	0
$925$	79.4801	2.61329
$926$	0	0
$927$	8.10568	0.266226
$928$	0	0
$929$	−19.1317	−0.627690	−0.313845	−	0.949474i	$-0.601617\pi$
$929$	−19.1317	−0.627690	−0.313845	+	0.949474i	$0.601617\pi$
$930$	0	0
$931$	3.14413	0.103045
$932$	0	0
$933$	−13.9973	−0.458252
$934$	0	0
$935$	0	0
$936$	0	0
$937$	57.1084	1.86565	0.932825	−	0.360330i	$-0.117336\pi$
$937$	57.1084	1.86565	0.932825	+	0.360330i	$0.117336\pi$
$938$	0	0
$939$	−30.4573	−0.993935
$940$	0	0
$941$	−31.0790	−1.01314	−0.506572	−	0.862197i	$-0.669088\pi$
$941$	−31.0790	−1.01314	−0.506572	+	0.862197i	$0.669088\pi$
$942$	0	0
$943$	13.4942	0.439430
$944$	0	0
$945$	−77.3009	−2.51460
$946$	0	0
$947$	−43.3148	−1.40754	−0.703771	−	0.710427i	$-0.748502\pi$
$947$	−43.3148	−1.40754	−0.703771	+	0.710427i	$0.748502\pi$
$948$	0	0
$949$	3.48561	0.113148
$950$	0	0
$951$	−14.6669	−0.475607
$952$	0	0
$953$	−40.7978	−1.32157	−0.660786	−	0.750574i	$-0.729777\pi$
$953$	−40.7978	−1.32157	−0.660786	+	0.750574i	$0.729777\pi$
$954$	0	0
$955$	45.5732	1.47471
$956$	0	0
$957$	0	0
$958$	0	0
$959$	23.3016	0.752447
$960$	0	0
$961$	5.54878	0.178993
$962$	0	0
$963$	5.24567	0.169039
$964$	0	0
$965$	30.4952	0.981676
$966$	0	0
$967$	−38.1966	−1.22832	−0.614159	−	0.789182i	$-0.710505\pi$
$967$	−38.1966	−1.22832	−0.614159	+	0.789182i	$0.710505\pi$
$968$	0	0
$969$	−0.216308	−0.00694883
$970$	0	0
$971$	46.0351	1.47734	0.738668	−	0.674069i	$-0.235455\pi$
$971$	46.0351	1.47734	0.738668	+	0.674069i	$0.235455\pi$
$972$	0	0
$973$	33.6004	1.07718
$974$	0	0
$975$	17.2853	0.553572
$976$	0	0
$977$	38.6494	1.23650	0.618252	−	0.785980i	$-0.287841\pi$
$977$	38.6494	1.23650	0.618252	+	0.785980i	$0.287841\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−2.01031	−0.0641842
$982$	0	0
$983$	21.4558	0.684332	0.342166	−	0.939639i	$-0.388839\pi$
$983$	21.4558	0.684332	0.342166	+	0.939639i	$0.388839\pi$
$984$	0	0
$985$	−2.75096	−0.0876528
$986$	0	0
$987$	18.0472	0.574450
$988$	0	0
$989$	−16.3873	−0.521087
$990$	0	0
$991$	−24.0223	−0.763092	−0.381546	−	0.924350i	$-0.624608\pi$
$991$	−24.0223	−0.763092	−0.381546	+	0.924350i	$0.624608\pi$
$992$	0	0
$993$	7.81955	0.248146
$994$	0	0
$995$	−93.3179	−2.95838
$996$	0	0
$997$	0.509091	0.0161231	0.00806154	−	0.999968i	$-0.497434\pi$
$997$	0.509091	0.0161231	0.00806154	+	0.999968i	$0.497434\pi$
$998$	0	0
$999$	−33.4543	−1.05845

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.j.1.5		6
11.3	even	5		836.2.j.a.229.1	✓	12
11.4	even	5		836.2.j.a.533.1	yes	12
11.10	odd	2		9196.2.a.i.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.j.a.229.1	✓	12	11.3	even	5
836.2.j.a.533.1	yes	12	11.4	even	5
9196.2.a.i.1.5		6	11.10	odd	2
9196.2.a.j.1.5		6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.44304 q^{3} -4.29314 q^{5} -3.18498 q^{7} -0.917645 q^{9} +0.891846 q^{13} -6.19516 q^{15} -0.149898 q^{17} +1.00000 q^{19} -4.59605 q^{21} +1.71038 q^{23} +13.4310 q^{25} -5.65331 q^{27} +6.45632 q^{29} +6.04556 q^{31} +13.6736 q^{35} +5.91764 q^{37} +1.28697 q^{39} +7.88959 q^{41} -9.58112 q^{43} +3.93958 q^{45} -3.92668 q^{47} +3.14413 q^{49} -0.216308 q^{51} +12.1822 q^{53} +1.44304 q^{57} -2.15607 q^{59} +2.87920 q^{61} +2.92268 q^{63} -3.82882 q^{65} -0.0354569 q^{67} +2.46814 q^{69} -15.6871 q^{71} +3.90831 q^{73} +19.3815 q^{75} +2.55410 q^{79} -5.40499 q^{81} +1.23048 q^{83} +0.643533 q^{85} +9.31671 q^{87} -0.565663 q^{89} -2.84051 q^{91} +8.72396 q^{93} -4.29314 q^{95} -3.51675 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 7 q^{5} + 4 q^{7} + 6 q^{9} + q^{13} - 8 q^{15} - 9 q^{17} + 6 q^{19} + 5 q^{21} - 12 q^{23} + 17 q^{25} - 17 q^{27} - 8 q^{29} + 7 q^{31} + 19 q^{35} + 24 q^{37} - 17 q^{39} - 15 q^{41}+ \cdots - 13 q^{97}+O(q^{100}) \) 6 * q - 2 * q^3 - 7 * q^5 + 4 * q^7 + 6 * q^9 + q^13 - 8 * q^15 - 9 * q^17 + 6 * q^19 + 5 * q^21 - 12 * q^23 + 17 * q^25 - 17 * q^27 - 8 * q^29 + 7 * q^31 + 19 * q^35 + 24 * q^37 - 17 * q^39 - 15 * q^41 - 23 * q^43 + 8 * q^45 - 10 * q^47 + 10 * q^49 - 10 * q^51 + 16 * q^53 - 2 * q^57 - 27 * q^59 - 7 * q^61 + 16 * q^63 + 14 * q^65 - 11 * q^67 - 16 * q^71 + 6 * q^73 + 5 * q^75 + 34 * q^79 + 2 * q^81 - 2 * q^83 - 18 * q^85 - 18 * q^87 + 13 * q^89 - 25 * q^91 + 2 * q^93 - 7 * q^95 - 13 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more