LMFDB - Embedded newform 6975.2.a.ck.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6975,2,Mod(1,6975)] 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("6975.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6975 = 3^{2} \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,0,0,12,0,0,0,0,0,0,0,0,0,0,0,44,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$55.6956554098$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 20x^{14} + 152x^{12} - 571x^{10} + 1130x^{8} - 1138x^{6} + 492x^{4} - 43x^{2} + 1 $ x^16 - 20x^14 + 152x^12 - 571x^10 + 1130x^8 - 1138x^6 + 492x^4 - 43*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 5 $
Twist minimal:	no (minimal twist has level 1395)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$1.11940$ of defining polynomial
Character	$\chi$	$=$	6975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.33304 q^{2} -0.223003 q^{4} -2.26227 q^{7} -2.96335 q^{8} -4.88845 q^{11} -6.07923 q^{13} -3.01569 q^{14} -3.50426 q^{16} +4.32765 q^{17} -7.22973 q^{19} -6.51650 q^{22} +4.12776 q^{23} -8.10386 q^{26} +0.504492 q^{28} -4.04025 q^{29} -1.00000 q^{31} +1.25538 q^{32} +5.76894 q^{34} +6.36881 q^{37} -9.63753 q^{38} +8.39034 q^{41} -5.73252 q^{43} +1.09014 q^{44} +5.50247 q^{46} +10.7510 q^{47} -1.88216 q^{49} +1.35569 q^{52} -9.22059 q^{53} +6.70389 q^{56} -5.38582 q^{58} +0.990591 q^{59} +10.2960 q^{61} -1.33304 q^{62} +8.68200 q^{64} -11.6127 q^{67} -0.965080 q^{68} -9.62456 q^{71} +1.35569 q^{73} +8.48988 q^{74} +1.61225 q^{76} +11.0590 q^{77} +0.231061 q^{79} +11.1847 q^{82} -4.96163 q^{83} -7.64169 q^{86} +14.4862 q^{88} +17.8808 q^{89} +13.7528 q^{91} -0.920502 q^{92} +14.3315 q^{94} +8.43207 q^{97} -2.50899 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{4} + 44 q^{16} - 16 q^{31} + 24 q^{34} + 88 q^{46} + 16 q^{49} + 64 q^{61} + 176 q^{64} - 12 q^{76} + 72 q^{79} - 16 q^{91} - 8 q^{94}+O(q^{100}) $ 16 * q + 12 * q^4 + 44 * q^16 - 16 * q^31 + 24 * q^34 + 88 * q^46 + 16 * q^49 + 64 * q^61 + 176 * q^64 - 12 * q^76 + 72 * q^79 - 16 * q^91 - 8 * q^94

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 $

$2$	1.33304	0.942602	0.471301	−	0.881972i	$-0.343785\pi$
$2$	1.33304	0.942602	0.471301	+	0.881972i	$0.343785\pi$
$3$	0	0
$3$	0	0
$4$	−0.223003	−0.111501
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.26227	−0.855056	−0.427528	−	0.904002i	$-0.640615\pi$
$7$	−2.26227	−0.855056	−0.427528	+	0.904002i	$0.640615\pi$
$8$	−2.96335	−1.04770
$9$	0	0
$10$	0	0
$11$	−4.88845	−1.47392	−0.736962	−	0.675935i	$-0.763740\pi$
$11$	−4.88845	−1.47392	−0.736962	+	0.675935i	$0.763740\pi$
$12$	0	0
$13$	−6.07923	−1.68607	−0.843037	−	0.537855i	$-0.819235\pi$
$13$	−6.07923	−1.68607	−0.843037	+	0.537855i	$0.819235\pi$
$14$	−3.01569	−0.805977
$15$	0	0
$16$	−3.50426	−0.876066
$17$	4.32765	1.04961	0.524805	−	0.851222i	$-0.324138\pi$
$17$	4.32765	1.04961	0.524805	+	0.851222i	$0.324138\pi$
$18$	0	0
$19$	−7.22973	−1.65861	−0.829307	−	0.558793i	$-0.811265\pi$
$19$	−7.22973	−1.65861	−0.829307	+	0.558793i	$0.811265\pi$
$20$	0	0
$21$	0	0
$22$	−6.51650	−1.38932
$23$	4.12776	0.860697	0.430348	−	0.902663i	$-0.358391\pi$
$23$	4.12776	0.860697	0.430348	+	0.902663i	$0.358391\pi$
$24$	0	0
$25$	0	0
$26$	−8.10386	−1.58930
$27$	0	0
$28$	0.504492	0.0953400
$29$	−4.04025	−0.750256	−0.375128	−	0.926973i	$-0.622401\pi$
$29$	−4.04025	−0.750256	−0.375128	+	0.926973i	$0.622401\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.25538	0.221922
$33$	0	0
$34$	5.76894	0.989365
$35$	0	0
$36$	0	0
$37$	6.36881	1.04703	0.523513	−	0.852018i	$-0.324621\pi$
$37$	6.36881	1.04703	0.523513	+	0.852018i	$0.324621\pi$
$38$	−9.63753	−1.56341
$39$	0	0
$40$	0	0
$41$	8.39034	1.31035	0.655175	−	0.755477i	$-0.272595\pi$
$41$	8.39034	1.31035	0.655175	+	0.755477i	$0.272595\pi$
$42$	0	0
$43$	−5.73252	−0.874202	−0.437101	−	0.899412i	$-0.643995\pi$
$43$	−5.73252	−0.874202	−0.437101	+	0.899412i	$0.643995\pi$
$44$	1.09014	0.164345
$45$	0	0
$46$	5.50247	0.811294
$47$	10.7510	1.56819	0.784094	−	0.620642i	$-0.213128\pi$
$47$	10.7510	1.56819	0.784094	+	0.620642i	$0.213128\pi$
$48$	0	0
$49$	−1.88216	−0.268880
$50$	0	0
$51$	0	0
$52$	1.35569	0.188000
$53$	−9.22059	−1.26655	−0.633273	−	0.773929i	$-0.718289\pi$
$53$	−9.22059	−1.26655	−0.633273	+	0.773929i	$0.718289\pi$
$54$	0	0
$55$	0	0
$56$	6.70389	0.895845
$57$	0	0
$58$	−5.38582	−0.707193
$59$	0.990591	0.128964	0.0644820	−	0.997919i	$-0.479460\pi$
$59$	0.990591	0.128964	0.0644820	+	0.997919i	$0.479460\pi$
$60$	0	0
$61$	10.2960	1.31827	0.659137	−	0.752023i	$-0.270922\pi$
$61$	10.2960	1.31827	0.659137	+	0.752023i	$0.270922\pi$
$62$	−1.33304	−0.169296
$63$	0	0
$64$	8.68200	1.08525
$65$	0	0
$66$	0	0
$67$	−11.6127	−1.41872	−0.709361	−	0.704845i	$-0.751016\pi$
$67$	−11.6127	−1.41872	−0.709361	+	0.704845i	$0.751016\pi$
$68$	−0.965080	−0.117033
$69$	0	0
$70$	0	0
$71$	−9.62456	−1.14223	−0.571113	−	0.820872i	$-0.693488\pi$
$71$	−9.62456	−1.14223	−0.571113	+	0.820872i	$0.693488\pi$
$72$	0	0
$73$	1.35569	0.158671	0.0793355	−	0.996848i	$-0.474720\pi$
$73$	1.35569	0.158671	0.0793355	+	0.996848i	$0.474720\pi$
$74$	8.48988	0.986929
$75$	0	0
$76$	1.61225	0.184938
$77$	11.0590	1.26029
$78$	0	0
$79$	0.231061	0.0259964	0.0129982	−	0.999916i	$-0.495862\pi$
$79$	0.231061	0.0259964	0.0129982	+	0.999916i	$0.495862\pi$
$80$	0	0
$81$	0	0
$82$	11.1847	1.23514
$83$	−4.96163	−0.544609	−0.272305	−	0.962211i	$-0.587786\pi$
$83$	−4.96163	−0.544609	−0.272305	+	0.962211i	$0.587786\pi$
$84$	0	0
$85$	0	0
$86$	−7.64169	−0.824024
$87$	0	0
$88$	14.4862	1.54423
$89$	17.8808	1.89536	0.947678	−	0.319227i	$-0.103423\pi$
$89$	17.8808	1.89536	0.947678	+	0.319227i	$0.103423\pi$
$90$	0	0
$91$	13.7528	1.44169
$92$	−0.920502	−0.0959690
$93$	0	0
$94$	14.3315	1.47818
$95$	0	0
$96$	0	0
$97$	8.43207	0.856147	0.428073	−	0.903744i	$-0.359193\pi$
$97$	8.43207	0.856147	0.428073	+	0.903744i	$0.359193\pi$
$98$	−2.50899	−0.253446
$99$	0	0
$100$	0	0
$101$	8.64222	0.859933	0.429967	−	0.902845i	$-0.358525\pi$
$101$	8.64222	0.859933	0.429967	+	0.902845i	$0.358525\pi$
$102$	0	0
$103$	5.17265	0.509676	0.254838	−	0.966984i	$-0.417978\pi$
$103$	5.17265	0.509676	0.254838	+	0.966984i	$0.417978\pi$
$104$	18.0149	1.76651
$105$	0	0
$106$	−12.2914	−1.19385
$107$	−4.66258	−0.450749	−0.225375	−	0.974272i	$-0.572361\pi$
$107$	−4.66258	−0.450749	−0.225375	+	0.974272i	$0.572361\pi$
$108$	0	0
$109$	11.1486	1.06784	0.533922	−	0.845533i	$-0.320717\pi$
$109$	11.1486	1.06784	0.533922	+	0.845533i	$0.320717\pi$
$110$	0	0
$111$	0	0
$112$	7.92757	0.749085
$113$	1.41328	0.132950	0.0664749	−	0.997788i	$-0.478825\pi$
$113$	1.41328	0.132950	0.0664749	+	0.997788i	$0.478825\pi$
$114$	0	0
$115$	0	0
$116$	0.900989	0.0836547
$117$	0	0
$118$	1.32050	0.121562
$119$	−9.79030	−0.897475
$120$	0	0
$121$	12.8969	1.17245
$122$	13.7251	1.24261
$123$	0	0
$124$	0.223003	0.0200263
$125$	0	0
$126$	0	0
$127$	0.918410	0.0814957	0.0407479	−	0.999169i	$-0.487026\pi$
$127$	0.918410	0.0814957	0.0407479	+	0.999169i	$0.487026\pi$
$128$	9.06270	0.801037
$129$	0	0
$130$	0	0
$131$	1.93833	0.169353	0.0846765	−	0.996408i	$-0.473014\pi$
$131$	1.93833	0.169353	0.0846765	+	0.996408i	$0.473014\pi$
$132$	0	0
$133$	16.3556	1.41821
$134$	−15.4803	−1.33729
$135$	0	0
$136$	−12.8244	−1.09968
$137$	−2.29555	−0.196122	−0.0980608	−	0.995180i	$-0.531264\pi$
$137$	−2.29555	−0.196122	−0.0980608	+	0.995180i	$0.531264\pi$
$138$	0	0
$139$	11.7174	0.993858	0.496929	−	0.867791i	$-0.334461\pi$
$139$	11.7174	0.993858	0.496929	+	0.867791i	$0.334461\pi$
$140$	0	0
$141$	0	0
$142$	−12.8299	−1.07666
$143$	29.7180	2.48514
$144$	0	0
$145$	0	0
$146$	1.80718	0.149564
$147$	0	0
$148$	−1.42026	−0.116745
$149$	−17.8708	−1.46403	−0.732017	−	0.681286i	$-0.761421\pi$
$149$	−17.8708	−1.46403	−0.732017	+	0.681286i	$0.761421\pi$
$150$	0	0
$151$	−14.0489	−1.14328	−0.571641	−	0.820504i	$-0.693693\pi$
$151$	−14.0489	−1.14328	−0.571641	+	0.820504i	$0.693693\pi$
$152$	21.4243	1.73774
$153$	0	0
$154$	14.7421	1.18795
$155$	0	0
$156$	0	0
$157$	−21.8631	−1.74486	−0.872432	−	0.488735i	$-0.837459\pi$
$157$	−21.8631	−1.74486	−0.872432	+	0.488735i	$0.837459\pi$
$158$	0.308014	0.0245043
$159$	0	0
$160$	0	0
$161$	−9.33808	−0.735944
$162$	0	0
$163$	−1.27928	−0.100201	−0.0501003	−	0.998744i	$-0.515954\pi$
$163$	−1.27928	−0.100201	−0.0501003	+	0.998744i	$0.515954\pi$
$164$	−1.87107	−0.146106
$165$	0	0
$166$	−6.61405	−0.513350
$167$	−4.95546	−0.383465	−0.191733	−	0.981447i	$-0.561411\pi$
$167$	−4.95546	−0.383465	−0.191733	+	0.981447i	$0.561411\pi$
$168$	0	0
$169$	23.9570	1.84285
$170$	0	0
$171$	0	0
$172$	1.27837	0.0974748
$173$	7.89629	0.600344	0.300172	−	0.953885i	$-0.402956\pi$
$173$	7.89629	0.600344	0.300172	+	0.953885i	$0.402956\pi$
$174$	0	0
$175$	0	0
$176$	17.1304	1.29125
$177$	0	0
$178$	23.8358	1.78657
$179$	2.80603	0.209733	0.104866	−	0.994486i	$-0.466559\pi$
$179$	2.80603	0.209733	0.104866	+	0.994486i	$0.466559\pi$
$180$	0	0
$181$	−18.6229	−1.38423	−0.692114	−	0.721788i	$-0.743320\pi$
$181$	−18.6229	−1.38423	−0.692114	+	0.721788i	$0.743320\pi$
$182$	18.3331	1.35894
$183$	0	0
$184$	−12.2320	−0.901755
$185$	0	0
$186$	0	0
$187$	−21.1555	−1.54705
$188$	−2.39750	−0.174855
$189$	0	0
$190$	0	0
$191$	6.26507	0.453324	0.226662	−	0.973973i	$-0.427219\pi$
$191$	6.26507	0.453324	0.226662	+	0.973973i	$0.427219\pi$
$192$	0	0
$193$	−23.7905	−1.71248	−0.856238	−	0.516581i	$-0.827205\pi$
$193$	−23.7905	−1.71248	−0.856238	+	0.516581i	$0.827205\pi$
$194$	11.2403	0.807005
$195$	0	0
$196$	0.419727	0.0299805
$197$	8.02750	0.571936	0.285968	−	0.958239i	$-0.407685\pi$
$197$	8.02750	0.571936	0.285968	+	0.958239i	$0.407685\pi$
$198$	0	0
$199$	−1.61898	−0.114767	−0.0573833	−	0.998352i	$-0.518276\pi$
$199$	−1.61898	−0.114767	−0.0573833	+	0.998352i	$0.518276\pi$
$200$	0	0
$201$	0	0
$202$	11.5204	0.810575
$203$	9.14013	0.641511
$204$	0	0
$205$	0	0
$206$	6.89535	0.480422
$207$	0	0
$208$	21.3032	1.47711
$209$	35.3422	2.44467
$210$	0	0
$211$	13.4962	0.929117	0.464559	−	0.885542i	$-0.346213\pi$
$211$	13.4962	0.929117	0.464559	+	0.885542i	$0.346213\pi$
$212$	2.05622	0.141222
$213$	0	0
$214$	−6.21541	−0.424877
$215$	0	0
$216$	0	0
$217$	2.26227	0.153573
$218$	14.8616	1.00655
$219$	0	0
$220$	0	0
$221$	−26.3088	−1.76972
$222$	0	0
$223$	−19.9475	−1.33579	−0.667893	−	0.744258i	$-0.732803\pi$
$223$	−19.9475	−1.33579	−0.667893	+	0.744258i	$0.732803\pi$
$224$	−2.84000	−0.189756
$225$	0	0
$226$	1.88395	0.125319
$227$	−10.0990	−0.670297	−0.335148	−	0.942165i	$-0.608786\pi$
$227$	−10.0990	−0.670297	−0.335148	+	0.942165i	$0.608786\pi$
$228$	0	0
$229$	20.2799	1.34014	0.670068	−	0.742300i	$-0.266265\pi$
$229$	20.2799	1.34014	0.670068	+	0.742300i	$0.266265\pi$
$230$	0	0
$231$	0	0
$232$	11.9727	0.786046
$233$	−7.32867	−0.480117	−0.240058	−	0.970758i	$-0.577167\pi$
$233$	−7.32867	−0.480117	−0.240058	+	0.970758i	$0.577167\pi$
$234$	0	0
$235$	0	0
$236$	−0.220905	−0.0143797
$237$	0	0
$238$	−13.0509	−0.845962
$239$	−25.1464	−1.62658	−0.813291	−	0.581857i	$-0.802327\pi$
$239$	−25.1464	−1.62658	−0.813291	+	0.581857i	$0.802327\pi$
$240$	0	0
$241$	16.7421	1.07845	0.539225	−	0.842162i	$-0.318717\pi$
$241$	16.7421	1.07845	0.539225	+	0.842162i	$0.318717\pi$
$242$	17.1922	1.10515
$243$	0	0
$244$	−2.29605	−0.146990
$245$	0	0
$246$	0	0
$247$	43.9512	2.79655
$248$	2.96335	0.188173
$249$	0	0
$250$	0	0
$251$	10.1763	0.642324	0.321162	−	0.947024i	$-0.395927\pi$
$251$	10.1763	0.642324	0.321162	+	0.947024i	$0.395927\pi$
$252$	0	0
$253$	−20.1783	−1.26860
$254$	1.22428	0.0768180
$255$	0	0
$256$	−5.28306	−0.330191
$257$	4.80511	0.299735	0.149867	−	0.988706i	$-0.452115\pi$
$257$	4.80511	0.299735	0.149867	+	0.988706i	$0.452115\pi$
$258$	0	0
$259$	−14.4079	−0.895266
$260$	0	0
$261$	0	0
$262$	2.58388	0.159633
$263$	24.3858	1.50370	0.751848	−	0.659337i	$-0.229163\pi$
$263$	24.3858	1.50370	0.751848	+	0.659337i	$0.229163\pi$
$264$	0	0
$265$	0	0
$266$	21.8026	1.33681
$267$	0	0
$268$	2.58968	0.158190
$269$	−5.68385	−0.346551	−0.173275	−	0.984873i	$-0.555435\pi$
$269$	−5.68385	−0.346551	−0.173275	+	0.984873i	$0.555435\pi$
$270$	0	0
$271$	−7.83393	−0.475877	−0.237939	−	0.971280i	$-0.576472\pi$
$271$	−7.83393	−0.475877	−0.237939	+	0.971280i	$0.576472\pi$
$272$	−15.1652	−0.919528
$273$	0	0
$274$	−3.06005	−0.184865
$275$	0	0
$276$	0	0
$277$	1.61928	0.0972928	0.0486464	−	0.998816i	$-0.484509\pi$
$277$	1.61928	0.0972928	0.0486464	+	0.998816i	$0.484509\pi$
$278$	15.6198	0.936812
$279$	0	0
$280$	0	0
$281$	−0.528423	−0.0315231	−0.0157615	−	0.999876i	$-0.505017\pi$
$281$	−0.528423	−0.0315231	−0.0157615	+	0.999876i	$0.505017\pi$
$282$	0	0
$283$	0.346702	0.0206093	0.0103046	−	0.999947i	$-0.496720\pi$
$283$	0.346702	0.0206093	0.0103046	+	0.999947i	$0.496720\pi$
$284$	2.14631	0.127360
$285$	0	0
$286$	39.6153	2.34250
$287$	−18.9812	−1.12042
$288$	0	0
$289$	1.72859	0.101682
$290$	0	0
$291$	0	0
$292$	−0.302322	−0.0176921
$293$	−3.47069	−0.202760	−0.101380	−	0.994848i	$-0.532326\pi$
$293$	−3.47069	−0.202760	−0.101380	+	0.994848i	$0.532326\pi$
$294$	0	0
$295$	0	0
$296$	−18.8730	−1.09697
$297$	0	0
$298$	−23.8225	−1.38000
$299$	−25.0936	−1.45120
$300$	0	0
$301$	12.9685	0.747491
$302$	−18.7277	−1.07766
$303$	0	0
$304$	25.3349	1.45306
$305$	0	0
$306$	0	0
$307$	27.2489	1.55518	0.777589	−	0.628773i	$-0.216443\pi$
$307$	27.2489	1.55518	0.777589	+	0.628773i	$0.216443\pi$
$308$	−2.46618	−0.140524
$309$	0	0
$310$	0	0
$311$	−5.76440	−0.326869	−0.163435	−	0.986554i	$-0.552257\pi$
$311$	−5.76440	−0.326869	−0.163435	+	0.986554i	$0.552257\pi$
$312$	0	0
$313$	0.238264	0.0134675	0.00673374	−	0.999977i	$-0.497857\pi$
$313$	0.238264	0.0134675	0.00673374	+	0.999977i	$0.497857\pi$
$314$	−29.1444	−1.64471
$315$	0	0
$316$	−0.0515273	−0.00289864
$317$	−19.9231	−1.11900	−0.559498	−	0.828832i	$-0.689006\pi$
$317$	−19.9231	−1.11900	−0.559498	+	0.828832i	$0.689006\pi$
$318$	0	0
$319$	19.7506	1.10582
$320$	0	0
$321$	0	0
$322$	−12.4480	−0.693702
$323$	−31.2878	−1.74090
$324$	0	0
$325$	0	0
$326$	−1.70533	−0.0944494
$327$	0	0
$328$	−24.8635	−1.37286
$329$	−24.3215	−1.34089
$330$	0	0
$331$	24.5579	1.34982	0.674912	−	0.737898i	$-0.264182\pi$
$331$	24.5579	1.34982	0.674912	+	0.737898i	$0.264182\pi$
$332$	1.10646	0.0607247
$333$	0	0
$334$	−6.60583	−0.361455
$335$	0	0
$336$	0	0
$337$	−12.6863	−0.691067	−0.345533	−	0.938406i	$-0.612302\pi$
$337$	−12.6863	−0.691067	−0.345533	+	0.938406i	$0.612302\pi$
$338$	31.9357	1.73707
$339$	0	0
$340$	0	0
$341$	4.88845	0.264724
$342$	0	0
$343$	20.0938	1.08496
$344$	16.9875	0.915904
$345$	0	0
$346$	10.5261	0.565886
$347$	−4.44092	−0.238401	−0.119201	−	0.992870i	$-0.538033\pi$
$347$	−4.44092	−0.238401	−0.119201	+	0.992870i	$0.538033\pi$
$348$	0	0
$349$	36.6583	1.96227	0.981137	−	0.193315i	$-0.0619239\pi$
$349$	36.6583	1.96227	0.981137	+	0.193315i	$0.0619239\pi$
$350$	0	0
$351$	0	0
$352$	−6.13686	−0.327096
$353$	−22.1918	−1.18115	−0.590575	−	0.806983i	$-0.701099\pi$
$353$	−22.1918	−1.18115	−0.590575	+	0.806983i	$0.701099\pi$
$354$	0	0
$355$	0	0
$356$	−3.98746	−0.211335
$357$	0	0
$358$	3.74055	0.197694
$359$	28.6249	1.51076	0.755382	−	0.655284i	$-0.227451\pi$
$359$	28.6249	1.51076	0.755382	+	0.655284i	$0.227451\pi$
$360$	0	0
$361$	33.2690	1.75100
$362$	−24.8251	−1.30478
$363$	0	0
$364$	−3.06692	−0.160750
$365$	0	0
$366$	0	0
$367$	−17.2622	−0.901077	−0.450539	−	0.892757i	$-0.648768\pi$
$367$	−17.2622	−0.901077	−0.450539	+	0.892757i	$0.648768\pi$
$368$	−14.4647	−0.754027
$369$	0	0
$370$	0	0
$371$	20.8594	1.08297
$372$	0	0
$373$	21.4593	1.11112	0.555560	−	0.831477i	$-0.312504\pi$
$373$	21.4593	1.11112	0.555560	+	0.831477i	$0.312504\pi$
$374$	−28.2012	−1.45825
$375$	0	0
$376$	−31.8589	−1.64300
$377$	24.5616	1.26499
$378$	0	0
$379$	−4.91234	−0.252330	−0.126165	−	0.992009i	$-0.540267\pi$
$379$	−4.91234	−0.252330	−0.126165	+	0.992009i	$0.540267\pi$
$380$	0	0
$381$	0	0
$382$	8.35159	0.427305
$383$	15.1473	0.773991	0.386995	−	0.922082i	$-0.373513\pi$
$383$	15.1473	0.773991	0.386995	+	0.922082i	$0.373513\pi$
$384$	0	0
$385$	0	0
$386$	−31.7137	−1.61418
$387$	0	0
$388$	−1.88038	−0.0954616
$389$	−17.4614	−0.885330	−0.442665	−	0.896687i	$-0.645967\pi$
$389$	−17.4614	−0.885330	−0.442665	+	0.896687i	$0.645967\pi$
$390$	0	0
$391$	17.8635	0.903396
$392$	5.57749	0.281706
$393$	0	0
$394$	10.7010	0.539108
$395$	0	0
$396$	0	0
$397$	−7.79511	−0.391225	−0.195613	−	0.980681i	$-0.562670\pi$
$397$	−7.79511	−0.391225	−0.195613	+	0.980681i	$0.562670\pi$
$398$	−2.15817	−0.108179
$399$	0	0
$400$	0	0
$401$	−5.18319	−0.258836	−0.129418	−	0.991590i	$-0.541311\pi$
$401$	−5.18319	−0.258836	−0.129418	+	0.991590i	$0.541311\pi$
$402$	0	0
$403$	6.07923	0.302828
$404$	−1.92724	−0.0958839
$405$	0	0
$406$	12.1842	0.604690
$407$	−31.1336	−1.54324
$408$	0	0
$409$	−27.6314	−1.36629	−0.683143	−	0.730285i	$-0.739387\pi$
$409$	−27.6314	−1.36629	−0.683143	+	0.730285i	$0.739387\pi$
$410$	0	0
$411$	0	0
$412$	−1.15352	−0.0568296
$413$	−2.24098	−0.110271
$414$	0	0
$415$	0	0
$416$	−7.63174	−0.374177
$417$	0	0
$418$	47.1126	2.30435
$419$	−36.1293	−1.76503	−0.882517	−	0.470281i	$-0.844153\pi$
$419$	−36.1293	−1.76503	−0.882517	+	0.470281i	$0.844153\pi$
$420$	0	0
$421$	18.6662	0.909734	0.454867	−	0.890559i	$-0.349687\pi$
$421$	18.6662	0.909734	0.454867	+	0.890559i	$0.349687\pi$
$422$	17.9910	0.875788
$423$	0	0
$424$	27.3239	1.32696
$425$	0	0
$426$	0	0
$427$	−23.2924	−1.12720
$428$	1.03977	0.0502592
$429$	0	0
$430$	0	0
$431$	38.0240	1.83155	0.915777	−	0.401688i	$-0.131576\pi$
$431$	38.0240	1.83155	0.915777	+	0.401688i	$0.131576\pi$
$432$	0	0
$433$	6.79863	0.326721	0.163361	−	0.986566i	$-0.447767\pi$
$433$	6.79863	0.326721	0.163361	+	0.986566i	$0.447767\pi$
$434$	3.01569	0.144758
$435$	0	0
$436$	−2.48618	−0.119066
$437$	−29.8426	−1.42756
$438$	0	0
$439$	36.7446	1.75373	0.876863	−	0.480741i	$-0.159632\pi$
$439$	36.7446	1.75373	0.876863	+	0.480741i	$0.159632\pi$
$440$	0	0
$441$	0	0
$442$	−35.0707	−1.66814
$443$	−15.2695	−0.725478	−0.362739	−	0.931891i	$-0.618158\pi$
$443$	−15.2695	−0.725478	−0.362739	+	0.931891i	$0.618158\pi$
$444$	0	0
$445$	0	0
$446$	−26.5909	−1.25911
$447$	0	0
$448$	−19.6410	−0.927949
$449$	23.2699	1.09817	0.549087	−	0.835765i	$-0.314976\pi$
$449$	23.2699	1.09817	0.549087	+	0.835765i	$0.314976\pi$
$450$	0	0
$451$	−41.0157	−1.93136
$452$	−0.315165	−0.0148241
$453$	0	0
$454$	−13.4624	−0.631823
$455$	0	0
$456$	0	0
$457$	35.2102	1.64707	0.823533	−	0.567268i	$-0.192000\pi$
$457$	35.2102	1.64707	0.823533	+	0.567268i	$0.192000\pi$
$458$	27.0340	1.26321
$459$	0	0
$460$	0	0
$461$	34.4127	1.60276	0.801378	−	0.598158i	$-0.204100\pi$
$461$	34.4127	1.60276	0.801378	+	0.598158i	$0.204100\pi$
$462$	0	0
$463$	−21.0471	−0.978141	−0.489071	−	0.872244i	$-0.662664\pi$
$463$	−21.0471	−0.978141	−0.489071	+	0.872244i	$0.662664\pi$
$464$	14.1581	0.657274
$465$	0	0
$466$	−9.76941	−0.452559
$467$	−8.48587	−0.392679	−0.196340	−	0.980536i	$-0.562905\pi$
$467$	−8.48587	−0.392679	−0.196340	+	0.980536i	$0.562905\pi$
$468$	0	0
$469$	26.2711	1.21309
$470$	0	0
$471$	0	0
$472$	−2.93547	−0.135116
$473$	28.0232	1.28851
$474$	0	0
$475$	0	0
$476$	2.18327	0.100070
$477$	0	0
$478$	−33.5211	−1.53322
$479$	−34.1647	−1.56102	−0.780512	−	0.625141i	$-0.785041\pi$
$479$	−34.1647	−1.56102	−0.780512	+	0.625141i	$0.785041\pi$
$480$	0	0
$481$	−38.7175	−1.76536
$482$	22.3178	1.01655
$483$	0	0
$484$	−2.87606	−0.130730
$485$	0	0
$486$	0	0
$487$	5.39095	0.244287	0.122144	−	0.992512i	$-0.461023\pi$
$487$	5.39095	0.244287	0.122144	+	0.992512i	$0.461023\pi$
$488$	−30.5108	−1.38116
$489$	0	0
$490$	0	0
$491$	19.9004	0.898094	0.449047	−	0.893508i	$-0.351764\pi$
$491$	19.9004	0.898094	0.449047	+	0.893508i	$0.351764\pi$
$492$	0	0
$493$	−17.4848	−0.787477
$494$	58.5887	2.63603
$495$	0	0
$496$	3.50426	0.157346
$497$	21.7733	0.976666
$498$	0	0
$499$	−27.6783	−1.23905	−0.619526	−	0.784976i	$-0.712675\pi$
$499$	−27.6783	−1.23905	−0.619526	+	0.784976i	$0.712675\pi$
$500$	0	0
$501$	0	0
$502$	13.5655	0.605456
$503$	−15.4069	−0.686960	−0.343480	−	0.939160i	$-0.611606\pi$
$503$	−15.4069	−0.686960	−0.343480	+	0.939160i	$0.611606\pi$
$504$	0	0
$505$	0	0
$506$	−26.8985	−1.19579
$507$	0	0
$508$	−0.204808	−0.00908689
$509$	31.1678	1.38149	0.690744	−	0.723099i	$-0.257283\pi$
$509$	31.1678	1.38149	0.690744	+	0.723099i	$0.257283\pi$
$510$	0	0
$511$	−3.06692	−0.135673
$512$	−25.1679	−1.11228
$513$	0	0
$514$	6.40541	0.282530
$515$	0	0
$516$	0	0
$517$	−52.5555	−2.31139
$518$	−19.2064	−0.843879
$519$	0	0
$520$	0	0
$521$	0.324182	0.0142027	0.00710134	−	0.999975i	$-0.497740\pi$
$521$	0.324182	0.0142027	0.00710134	+	0.999975i	$0.497740\pi$
$522$	0	0
$523$	35.3521	1.54584	0.772921	−	0.634503i	$-0.218795\pi$
$523$	35.3521	1.54584	0.772921	+	0.634503i	$0.218795\pi$
$524$	−0.432254	−0.0188831
$525$	0	0
$526$	32.5073	1.41739
$527$	−4.32765	−0.188516
$528$	0	0
$529$	−5.96163	−0.259201
$530$	0	0
$531$	0	0
$532$	−3.64734	−0.158132
$533$	−51.0068	−2.20935
$534$	0	0
$535$	0	0
$536$	34.4127	1.48640
$537$	0	0
$538$	−7.57681	−0.326659
$539$	9.20083	0.396308
$540$	0	0
$541$	32.1201	1.38095	0.690476	−	0.723355i	$-0.257401\pi$
$541$	32.1201	1.38095	0.690476	+	0.723355i	$0.257401\pi$
$542$	−10.4429	−0.448563
$543$	0	0
$544$	5.43285	0.232932
$545$	0	0
$546$	0	0
$547$	28.9253	1.23676	0.618378	−	0.785880i	$-0.287790\pi$
$547$	28.9253	1.23676	0.618378	+	0.785880i	$0.287790\pi$
$548$	0.511913	0.0218679
$549$	0	0
$550$	0	0
$551$	29.2100	1.24439
$552$	0	0
$553$	−0.522721	−0.0222284
$554$	2.15856	0.0917084
$555$	0	0
$556$	−2.61302	−0.110817
$557$	17.6366	0.747286	0.373643	−	0.927573i	$-0.378109\pi$
$557$	17.6366	0.747286	0.373643	+	0.927573i	$0.378109\pi$
$558$	0	0
$559$	34.8493	1.47397
$560$	0	0
$561$	0	0
$562$	−0.704410	−0.0297137
$563$	−17.8325	−0.751551	−0.375776	−	0.926711i	$-0.622624\pi$
$563$	−17.8325	−0.751551	−0.375776	+	0.926711i	$0.622624\pi$
$564$	0	0
$565$	0	0
$566$	0.462168	0.0194264
$567$	0	0
$568$	28.5210	1.19671
$569$	11.6624	0.488915	0.244458	−	0.969660i	$-0.421390\pi$
$569$	11.6624	0.488915	0.244458	+	0.969660i	$0.421390\pi$
$570$	0	0
$571$	−23.7640	−0.994494	−0.497247	−	0.867609i	$-0.665656\pi$
$571$	−23.7640	−0.994494	−0.497247	+	0.867609i	$0.665656\pi$
$572$	−6.62720	−0.277097
$573$	0	0
$574$	−25.3027	−1.05611
$575$	0	0
$576$	0	0
$577$	10.1226	0.421410	0.210705	−	0.977550i	$-0.432424\pi$
$577$	10.1226	0.421410	0.210705	+	0.977550i	$0.432424\pi$
$578$	2.30429	0.0958457
$579$	0	0
$580$	0	0
$581$	11.2245	0.465671
$582$	0	0
$583$	45.0744	1.86679
$584$	−4.01737	−0.166240
$585$	0	0
$586$	−4.62657	−0.191122
$587$	−38.7269	−1.59843	−0.799215	−	0.601045i	$-0.794751\pi$
$587$	−38.7269	−1.59843	−0.799215	+	0.601045i	$0.794751\pi$
$588$	0	0
$589$	7.22973	0.297896
$590$	0	0
$591$	0	0
$592$	−22.3180	−0.917264
$593$	6.62601	0.272097	0.136049	−	0.990702i	$-0.456560\pi$
$593$	6.62601	0.272097	0.136049	+	0.990702i	$0.456560\pi$
$594$	0	0
$595$	0	0
$596$	3.98524	0.163242
$597$	0	0
$598$	−33.4507	−1.36790
$599$	24.7616	1.01173	0.505866	−	0.862612i	$-0.331173\pi$
$599$	24.7616	1.01173	0.505866	+	0.862612i	$0.331173\pi$
$600$	0	0
$601$	26.0843	1.06400	0.532000	−	0.846744i	$-0.321441\pi$
$601$	26.0843	1.06400	0.532000	+	0.846744i	$0.321441\pi$
$602$	17.2875	0.704587
$603$	0	0
$604$	3.13294	0.127478
$605$	0	0
$606$	0	0
$607$	31.2003	1.26638	0.633191	−	0.773996i	$-0.281745\pi$
$607$	31.2003	1.26638	0.633191	+	0.773996i	$0.281745\pi$
$608$	−9.07607	−0.368083
$609$	0	0
$610$	0	0
$611$	−65.3575	−2.64408
$612$	0	0
$613$	5.85422	0.236450	0.118225	−	0.992987i	$-0.462280\pi$
$613$	5.85422	0.236450	0.118225	+	0.992987i	$0.462280\pi$
$614$	36.3239	1.46591
$615$	0	0
$616$	−32.7716	−1.32041
$617$	−18.4986	−0.744724	−0.372362	−	0.928088i	$-0.621452\pi$
$617$	−18.4986	−0.744724	−0.372362	+	0.928088i	$0.621452\pi$
$618$	0	0
$619$	13.6560	0.548882	0.274441	−	0.961604i	$-0.411507\pi$
$619$	13.6560	0.548882	0.274441	+	0.961604i	$0.411507\pi$
$620$	0	0
$621$	0	0
$622$	−7.68418	−0.308108
$623$	−40.4510	−1.62064
$624$	0	0
$625$	0	0
$626$	0.317616	0.0126945
$627$	0	0
$628$	4.87553	0.194555
$629$	27.5620	1.09897
$630$	0	0
$631$	17.6705	0.703452	0.351726	−	0.936103i	$-0.385595\pi$
$631$	17.6705	0.703452	0.351726	+	0.936103i	$0.385595\pi$
$632$	−0.684715	−0.0272365
$633$	0	0
$634$	−26.5584	−1.05477
$635$	0	0
$636$	0	0
$637$	11.4421	0.453351
$638$	26.3283	1.04235
$639$	0	0
$640$	0	0
$641$	−0.814903	−0.0321867	−0.0160934	−	0.999870i	$-0.505123\pi$
$641$	−0.814903	−0.0321867	−0.0160934	+	0.999870i	$0.505123\pi$
$642$	0	0
$643$	34.6000	1.36449	0.682245	−	0.731124i	$-0.261004\pi$
$643$	34.6000	1.36449	0.682245	+	0.731124i	$0.261004\pi$
$644$	2.08242	0.0820588
$645$	0	0
$646$	−41.7079	−1.64098
$647$	17.9892	0.707227	0.353614	−	0.935392i	$-0.384953\pi$
$647$	17.9892	0.707227	0.353614	+	0.935392i	$0.384953\pi$
$648$	0	0
$649$	−4.84245	−0.190083
$650$	0	0
$651$	0	0
$652$	0.285282	0.0111725
$653$	1.86147	0.0728450	0.0364225	−	0.999336i	$-0.488404\pi$
$653$	1.86147	0.0728450	0.0364225	+	0.999336i	$0.488404\pi$
$654$	0	0
$655$	0	0
$656$	−29.4020	−1.14795
$657$	0	0
$658$	−32.4216	−1.26392
$659$	−2.06422	−0.0804105	−0.0402053	−	0.999191i	$-0.512801\pi$
$659$	−2.06422	−0.0804105	−0.0402053	+	0.999191i	$0.512801\pi$
$660$	0	0
$661$	−7.31667	−0.284586	−0.142293	−	0.989825i	$-0.545447\pi$
$661$	−7.31667	−0.284586	−0.142293	+	0.989825i	$0.545447\pi$
$662$	32.7367	1.27235
$663$	0	0
$664$	14.7031	0.570589
$665$	0	0
$666$	0	0
$667$	−16.6772	−0.645743
$668$	1.10508	0.0427569
$669$	0	0
$670$	0	0
$671$	−50.3317	−1.94303
$672$	0	0
$673$	−23.9205	−0.922069	−0.461035	−	0.887382i	$-0.652522\pi$
$673$	−23.9205	−0.922069	−0.461035	+	0.887382i	$0.652522\pi$
$674$	−16.9114	−0.651401
$675$	0	0
$676$	−5.34248	−0.205480
$677$	−25.4349	−0.977543	−0.488772	−	0.872412i	$-0.662555\pi$
$677$	−25.4349	−0.977543	−0.488772	+	0.872412i	$0.662555\pi$
$678$	0	0
$679$	−19.0756	−0.732053
$680$	0	0
$681$	0	0
$682$	6.51650	0.249530
$683$	−6.87665	−0.263128	−0.131564	−	0.991308i	$-0.542000\pi$
$683$	−6.87665	−0.263128	−0.131564	+	0.991308i	$0.542000\pi$
$684$	0	0
$685$	0	0
$686$	26.7858	1.02269
$687$	0	0
$688$	20.0883	0.765858
$689$	56.0541	2.13549
$690$	0	0
$691$	−7.85648	−0.298875	−0.149437	−	0.988771i	$-0.547746\pi$
$691$	−7.85648	−0.298875	−0.149437	+	0.988771i	$0.547746\pi$
$692$	−1.76090	−0.0669393
$693$	0	0
$694$	−5.91993	−0.224717
$695$	0	0
$696$	0	0
$697$	36.3105	1.37536
$698$	48.8670	1.84964
$699$	0	0
$700$	0	0
$701$	−17.5943	−0.664527	−0.332263	−	0.943187i	$-0.607812\pi$
$701$	−17.5943	−0.664527	−0.332263	+	0.943187i	$0.607812\pi$
$702$	0	0
$703$	−46.0448	−1.73661
$704$	−42.4415	−1.59958
$705$	0	0
$706$	−29.5825	−1.11335
$707$	−19.5510	−0.735291
$708$	0	0
$709$	−25.7814	−0.968240	−0.484120	−	0.875002i	$-0.660860\pi$
$709$	−25.7814	−0.968240	−0.484120	+	0.875002i	$0.660860\pi$
$710$	0	0
$711$	0	0
$712$	−52.9870	−1.98577
$713$	−4.12776	−0.154586
$714$	0	0
$715$	0	0
$716$	−0.625753	−0.0233855
$717$	0	0
$718$	38.1582	1.42405
$719$	−4.52192	−0.168639	−0.0843196	−	0.996439i	$-0.526872\pi$
$719$	−4.52192	−0.168639	−0.0843196	+	0.996439i	$0.526872\pi$
$720$	0	0
$721$	−11.7019	−0.435802
$722$	44.3490	1.65050
$723$	0	0
$724$	4.15296	0.154343
$725$	0	0
$726$	0	0
$727$	−27.6527	−1.02558	−0.512792	−	0.858513i	$-0.671389\pi$
$727$	−27.6527	−1.02558	−0.512792	+	0.858513i	$0.671389\pi$
$728$	−40.7545	−1.51046
$729$	0	0
$730$	0	0
$731$	−24.8084	−0.917571
$732$	0	0
$733$	49.9649	1.84550	0.922748	−	0.385403i	$-0.125937\pi$
$733$	49.9649	1.84550	0.922748	+	0.385403i	$0.125937\pi$
$734$	−23.0111	−0.849357
$735$	0	0
$736$	5.18190	0.191008
$737$	56.7683	2.09109
$738$	0	0
$739$	17.2507	0.634576	0.317288	−	0.948329i	$-0.397228\pi$
$739$	17.2507	0.634576	0.317288	+	0.948329i	$0.397228\pi$
$740$	0	0
$741$	0	0
$742$	27.8065	1.02081
$743$	−23.2895	−0.854411	−0.427205	−	0.904155i	$-0.640502\pi$
$743$	−23.2895	−0.854411	−0.427205	+	0.904155i	$0.640502\pi$
$744$	0	0
$745$	0	0
$746$	28.6061	1.04734
$747$	0	0
$748$	4.71774	0.172498
$749$	10.5480	0.385416
$750$	0	0
$751$	−10.5346	−0.384412	−0.192206	−	0.981355i	$-0.561564\pi$
$751$	−10.5346	−0.384412	−0.192206	+	0.981355i	$0.561564\pi$
$752$	−37.6742	−1.37384
$753$	0	0
$754$	32.7416	1.19238
$755$	0	0
$756$	0	0
$757$	−8.00662	−0.291006	−0.145503	−	0.989358i	$-0.546480\pi$
$757$	−8.00662	−0.291006	−0.145503	+	0.989358i	$0.546480\pi$
$758$	−6.54835	−0.237847
$759$	0	0
$760$	0	0
$761$	19.9494	0.723165	0.361582	−	0.932340i	$-0.382237\pi$
$761$	19.9494	0.723165	0.361582	+	0.932340i	$0.382237\pi$
$762$	0	0
$763$	−25.2212	−0.913067
$764$	−1.39713	−0.0505464
$765$	0	0
$766$	20.1920	0.729565
$767$	−6.02203	−0.217443
$768$	0	0
$769$	51.5543	1.85910	0.929548	−	0.368700i	$-0.120197\pi$
$769$	51.5543	1.85910	0.929548	+	0.368700i	$0.120197\pi$
$770$	0	0
$771$	0	0
$772$	5.30535	0.190944
$773$	14.8224	0.533123	0.266562	−	0.963818i	$-0.414112\pi$
$773$	14.8224	0.533123	0.266562	+	0.963818i	$0.414112\pi$
$774$	0	0
$775$	0	0
$776$	−24.9872	−0.896988
$777$	0	0
$778$	−23.2768	−0.834514
$779$	−60.6599	−2.17337
$780$	0	0
$781$	47.0492	1.68355
$782$	23.8128	0.851543
$783$	0	0
$784$	6.59557	0.235556
$785$	0	0
$786$	0	0
$787$	−15.8528	−0.565092	−0.282546	−	0.959254i	$-0.591179\pi$
$787$	−15.8528	−0.565092	−0.282546	+	0.959254i	$0.591179\pi$
$788$	−1.79016	−0.0637717
$789$	0	0
$790$	0	0
$791$	−3.19720	−0.113679
$792$	0	0
$793$	−62.5920	−2.22271
$794$	−10.3912	−0.368770
$795$	0	0
$796$	0.361038	0.0127966
$797$	30.3453	1.07489	0.537444	−	0.843300i	$-0.319390\pi$
$797$	30.3453	1.07489	0.537444	+	0.843300i	$0.319390\pi$
$798$	0	0
$799$	46.5264	1.64599
$800$	0	0
$801$	0	0
$802$	−6.90940	−0.243979
$803$	−6.62720	−0.233869
$804$	0	0
$805$	0	0
$806$	8.10386	0.285446
$807$	0	0
$808$	−25.6100	−0.900955
$809$	−35.0670	−1.23289	−0.616444	−	0.787399i	$-0.711427\pi$
$809$	−35.0670	−1.23289	−0.616444	+	0.787399i	$0.711427\pi$
$810$	0	0
$811$	−3.53788	−0.124232	−0.0621158	−	0.998069i	$-0.519785\pi$
$811$	−3.53788	−0.124232	−0.0621158	+	0.998069i	$0.519785\pi$
$812$	−2.03828	−0.0715294
$813$	0	0
$814$	−41.5024	−1.45466
$815$	0	0
$816$	0	0
$817$	41.4446	1.44996
$818$	−36.8338	−1.28786
$819$	0	0
$820$	0	0
$821$	−45.7938	−1.59821	−0.799107	−	0.601189i	$-0.794694\pi$
$821$	−45.7938	−1.59821	−0.799107	+	0.601189i	$0.794694\pi$
$822$	0	0
$823$	9.82724	0.342556	0.171278	−	0.985223i	$-0.445210\pi$
$823$	9.82724	0.342556	0.171278	+	0.985223i	$0.445210\pi$
$824$	−15.3284	−0.533989
$825$	0	0
$826$	−2.98732	−0.103942
$827$	−29.5330	−1.02696	−0.513481	−	0.858101i	$-0.671644\pi$
$827$	−29.5330	−1.02696	−0.513481	+	0.858101i	$0.671644\pi$
$828$	0	0
$829$	−9.36400	−0.325225	−0.162613	−	0.986690i	$-0.551992\pi$
$829$	−9.36400	−0.325225	−0.162613	+	0.986690i	$0.551992\pi$
$830$	0	0
$831$	0	0
$832$	−52.7799	−1.82981
$833$	−8.14532	−0.282219
$834$	0	0
$835$	0	0
$836$	−7.88141	−0.272584
$837$	0	0
$838$	−48.1619	−1.66372
$839$	34.0552	1.17572	0.587858	−	0.808964i	$-0.299972\pi$
$839$	34.0552	1.17572	0.587858	+	0.808964i	$0.299972\pi$
$840$	0	0
$841$	−12.6763	−0.437115
$842$	24.8828	0.857517
$843$	0	0
$844$	−3.00969	−0.103598
$845$	0	0
$846$	0	0
$847$	−29.1763	−1.00251
$848$	32.3114	1.10958
$849$	0	0
$850$	0	0
$851$	26.2889	0.901172
$852$	0	0
$853$	12.2952	0.420980	0.210490	−	0.977596i	$-0.432494\pi$
$853$	12.2952	0.420980	0.210490	+	0.977596i	$0.432494\pi$
$854$	−31.0497	−1.06250
$855$	0	0
$856$	13.8169	0.472251
$857$	−46.3090	−1.58189	−0.790943	−	0.611890i	$-0.790410\pi$
$857$	−46.3090	−1.58189	−0.790943	+	0.611890i	$0.790410\pi$
$858$	0	0
$859$	−16.6495	−0.568072	−0.284036	−	0.958814i	$-0.591674\pi$
$859$	−16.6495	−0.568072	−0.284036	+	0.958814i	$0.591674\pi$
$860$	0	0
$861$	0	0
$862$	50.6876	1.72643
$863$	27.5142	0.936595	0.468298	−	0.883571i	$-0.344867\pi$
$863$	27.5142	0.936595	0.468298	+	0.883571i	$0.344867\pi$
$864$	0	0
$865$	0	0
$866$	9.06284	0.307968
$867$	0	0
$868$	−0.504492	−0.0171236
$869$	−1.12953	−0.0383167
$870$	0	0
$871$	70.5965	2.39207
$872$	−33.0373	−1.11878
$873$	0	0
$874$	−39.7814	−1.34562
$875$	0	0
$876$	0	0
$877$	24.4839	0.826762	0.413381	−	0.910558i	$-0.364348\pi$
$877$	24.4839	0.826762	0.413381	+	0.910558i	$0.364348\pi$
$878$	48.9821	1.65307
$879$	0	0
$880$	0	0
$881$	39.9948	1.34746	0.673730	−	0.738978i	$-0.264691\pi$
$881$	39.9948	1.34746	0.673730	+	0.738978i	$0.264691\pi$
$882$	0	0
$883$	41.9829	1.41284	0.706418	−	0.707795i	$-0.250310\pi$
$883$	41.9829	1.41284	0.706418	+	0.707795i	$0.250310\pi$
$884$	5.86694	0.197327
$885$	0	0
$886$	−20.3549	−0.683837
$887$	−47.6126	−1.59867	−0.799337	−	0.600883i	$-0.794816\pi$
$887$	−47.6126	−1.59867	−0.799337	+	0.600883i	$0.794816\pi$
$888$	0	0
$889$	−2.07769	−0.0696834
$890$	0	0
$891$	0	0
$892$	4.44836	0.148942
$893$	−77.7265	−2.60102
$894$	0	0
$895$	0	0
$896$	−20.5022	−0.684931
$897$	0	0
$898$	31.0197	1.03514
$899$	4.04025	0.134750
$900$	0	0
$901$	−39.9035	−1.32938
$902$	−54.6756	−1.82050
$903$	0	0
$904$	−4.18803	−0.139292
$905$	0	0
$906$	0	0
$907$	−44.5631	−1.47969	−0.739846	−	0.672776i	$-0.765102\pi$
$907$	−44.5631	−1.47969	−0.739846	+	0.672776i	$0.765102\pi$
$908$	2.25212	0.0747391
$909$	0	0
$910$	0	0
$911$	7.19070	0.238239	0.119119	−	0.992880i	$-0.461993\pi$
$911$	7.19070	0.238239	0.119119	+	0.992880i	$0.461993\pi$
$912$	0	0
$913$	24.2547	0.802712
$914$	46.9367	1.55253
$915$	0	0
$916$	−4.52249	−0.149427
$917$	−4.38503	−0.144806
$918$	0	0
$919$	−34.5933	−1.14113	−0.570564	−	0.821253i	$-0.693275\pi$
$919$	−34.5933	−1.14113	−0.570564	+	0.821253i	$0.693275\pi$
$920$	0	0
$921$	0	0
$922$	45.8735	1.51076
$923$	58.5099	1.92588
$924$	0	0
$925$	0	0
$926$	−28.0566	−0.921998
$927$	0	0
$928$	−5.07206	−0.166498
$929$	26.5992	0.872691	0.436345	−	0.899779i	$-0.356273\pi$
$929$	26.5992	0.872691	0.436345	+	0.899779i	$0.356273\pi$
$930$	0	0
$931$	13.6075	0.445968
$932$	1.63431	0.0535337
$933$	0	0
$934$	−11.3120	−0.370140
$935$	0	0
$936$	0	0
$937$	1.12555	0.0367702	0.0183851	−	0.999831i	$-0.494148\pi$
$937$	1.12555	0.0367702	0.0183851	+	0.999831i	$0.494148\pi$
$938$	35.0204	1.14346
$939$	0	0
$940$	0	0
$941$	−3.70702	−0.120845	−0.0604227	−	0.998173i	$-0.519245\pi$
$941$	−3.70702	−0.120845	−0.0604227	+	0.998173i	$0.519245\pi$
$942$	0	0
$943$	34.6333	1.12781
$944$	−3.47129	−0.112981
$945$	0	0
$946$	37.3560	1.21455
$947$	5.91493	0.192209	0.0961047	−	0.995371i	$-0.469362\pi$
$947$	5.91493	0.192209	0.0961047	+	0.995371i	$0.469362\pi$
$948$	0	0
$949$	−8.24152	−0.267531
$950$	0	0
$951$	0	0
$952$	29.0121	0.940288
$953$	−35.3239	−1.14425	−0.572127	−	0.820165i	$-0.693881\pi$
$953$	−35.3239	−1.14425	−0.572127	+	0.820165i	$0.693881\pi$
$954$	0	0
$955$	0	0
$956$	5.60771	0.181366
$957$	0	0
$958$	−45.5429	−1.47142
$959$	5.19313	0.167695
$960$	0	0
$961$	1.00000	0.0322581
$962$	−51.6119	−1.66404
$963$	0	0
$964$	−3.73353	−0.120249
$965$	0	0
$966$	0	0
$967$	−19.9475	−0.641469	−0.320735	−	0.947169i	$-0.603930\pi$
$967$	−19.9475	−0.641469	−0.320735	+	0.947169i	$0.603930\pi$
$968$	−38.2182	−1.22838
$969$	0	0
$970$	0	0
$971$	31.0305	0.995818	0.497909	−	0.867229i	$-0.334101\pi$
$971$	31.0305	0.995818	0.497909	+	0.867229i	$0.334101\pi$
$972$	0	0
$973$	−26.5079	−0.849804
$974$	7.18636	0.230266
$975$	0	0
$976$	−36.0801	−1.15489
$977$	17.1868	0.549853	0.274927	−	0.961465i	$-0.411346\pi$
$977$	17.1868	0.549853	0.274927	+	0.961465i	$0.411346\pi$
$978$	0	0
$979$	−87.4092	−2.79361
$980$	0	0
$981$	0	0
$982$	26.5281	0.846545
$983$	−36.9855	−1.17965	−0.589827	−	0.807530i	$-0.700804\pi$
$983$	−36.9855	−1.17965	−0.589827	+	0.807530i	$0.700804\pi$
$984$	0	0
$985$	0	0
$986$	−23.3080	−0.742277
$987$	0	0
$988$	−9.80125	−0.311819
$989$	−23.6625	−0.752423
$990$	0	0
$991$	34.4903	1.09562	0.547809	−	0.836603i	$-0.315462\pi$
$991$	34.4903	1.09562	0.547809	+	0.836603i	$0.315462\pi$
$992$	−1.25538	−0.0398584
$993$	0	0
$994$	29.0247	0.920608
$995$	0	0
$996$	0	0
$997$	−47.7422	−1.51201	−0.756005	−	0.654566i	$-0.772851\pi$
$997$	−47.7422	−1.51201	−0.756005	+	0.654566i	$0.772851\pi$
$998$	−36.8963	−1.16793
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6975.2.a.ck.1.13		16
3.2	odd	2	inner	6975.2.a.ck.1.3		16
5.2	odd	4		1395.2.c.g.559.13	yes	16
5.3	odd	4		1395.2.c.g.559.3	✓	16
5.4	even	2	inner	6975.2.a.ck.1.4		16
15.2	even	4		1395.2.c.g.559.4	yes	16
15.8	even	4		1395.2.c.g.559.14	yes	16
15.14	odd	2	inner	6975.2.a.ck.1.14		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1395.2.c.g.559.3	✓	16	5.3	odd	4
1395.2.c.g.559.4	yes	16	15.2	even	4
1395.2.c.g.559.13	yes	16	5.2	odd	4
1395.2.c.g.559.14	yes	16	15.8	even	4
6975.2.a.ck.1.3		16	3.2	odd	2	inner
6975.2.a.ck.1.4		16	5.4	even	2	inner
6975.2.a.ck.1.13		16	1.1	even	1	trivial
6975.2.a.ck.1.14		16	15.14	odd	2	inner

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:	\( N \)	\(=\)	\( 6975 = 3^{2} \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6975.a (trivial)

\(f(q)\)	\(=\)	\(q+1.33304 q^{2} -0.223003 q^{4} -2.26227 q^{7} -2.96335 q^{8} -4.88845 q^{11} -6.07923 q^{13} -3.01569 q^{14} -3.50426 q^{16} +4.32765 q^{17} -7.22973 q^{19} -6.51650 q^{22} +4.12776 q^{23} -8.10386 q^{26} +0.504492 q^{28} -4.04025 q^{29} -1.00000 q^{31} +1.25538 q^{32} +5.76894 q^{34} +6.36881 q^{37} -9.63753 q^{38} +8.39034 q^{41} -5.73252 q^{43} +1.09014 q^{44} +5.50247 q^{46} +10.7510 q^{47} -1.88216 q^{49} +1.35569 q^{52} -9.22059 q^{53} +6.70389 q^{56} -5.38582 q^{58} +0.990591 q^{59} +10.2960 q^{61} -1.33304 q^{62} +8.68200 q^{64} -11.6127 q^{67} -0.965080 q^{68} -9.62456 q^{71} +1.35569 q^{73} +8.48988 q^{74} +1.61225 q^{76} +11.0590 q^{77} +0.231061 q^{79} +11.1847 q^{82} -4.96163 q^{83} -7.64169 q^{86} +14.4862 q^{88} +17.8808 q^{89} +13.7528 q^{91} -0.920502 q^{92} +14.3315 q^{94} +8.43207 q^{97} -2.50899 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{4} + 44 q^{16} - 16 q^{31} + 24 q^{34} + 88 q^{46} + 16 q^{49} + 64 q^{61} + 176 q^{64} - 12 q^{76} + 72 q^{79} - 16 q^{91} - 8 q^{94}+O(q^{100}) \) 16 * q + 12 * q^4 + 44 * q^16 - 16 * q^31 + 24 * q^34 + 88 * q^46 + 16 * q^49 + 64 * q^61 + 176 * q^64 - 12 * q^76 + 72 * q^79 - 16 * q^91 - 8 * q^94

Introduction

L-functions

Modular forms

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Database

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