LMFDB - Embedded newform 3344.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.13676.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 7x + 1 $ x^4 - x^3 - 6x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.48425$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.48425 q^{3} -1.91023 q^{5} +3.48425 q^{7} +3.17148 q^{9} +O(q^{10})$	$q-2.48425 q^{3} -1.91023 q^{5} +3.48425 q^{7} +3.17148 q^{9} -1.00000 q^{11} -1.66378 q^{13} +4.74549 q^{15} -1.94847 q^{17} -1.00000 q^{19} -8.65572 q^{21} +5.70725 q^{23} -1.35101 q^{25} -0.425988 q^{27} -1.22300 q^{29} +0.402537 q^{31} +2.48425 q^{33} -6.65572 q^{35} -9.65572 q^{37} +4.13324 q^{39} +8.40121 q^{41} -0.648991 q^{43} -6.05826 q^{45} +5.64093 q^{47} +5.13997 q^{49} +4.84048 q^{51} -2.17953 q^{53} +1.91023 q^{55} +2.48425 q^{57} +5.65572 q^{59} +6.29142 q^{61} +11.0502 q^{63} +3.17821 q^{65} +9.71398 q^{67} -14.1782 q^{69} -1.73532 q^{71} -0.672442 q^{73} +3.35624 q^{75} -3.48425 q^{77} -8.73743 q^{79} -8.45617 q^{81} +7.97192 q^{83} +3.72204 q^{85} +3.03824 q^{87} -7.20972 q^{89} -5.79702 q^{91} -1.00000 q^{93} +1.91023 q^{95} -11.1332 q^{97} -3.17148 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9}+O(q^{10}) $ 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9	$ 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9} - 4 q^{11} - 5 q^{13} + q^{15} - 4 q^{19} - 12 q^{21} + 8 q^{23} - 3 q^{25} - 8 q^{27} - q^{29} + 7 q^{31} - q^{33} - 4 q^{35} - 16 q^{37} + 8 q^{39} - 7 q^{41} - 5 q^{43} - 7 q^{45} + 4 q^{47} - 13 q^{49} - 4 q^{51} - 18 q^{53} + 3 q^{55} - q^{57} - 6 q^{61} + 6 q^{63} - 24 q^{65} - q^{67} - 20 q^{69} + q^{71} - 6 q^{73} + q^{75} - 3 q^{77} + 4 q^{79} - 16 q^{81} + 25 q^{83} - 4 q^{85} + 9 q^{87} - 14 q^{89} - 13 q^{91} - 4 q^{93} + 3 q^{95} - 36 q^{97} - q^{99}+O(q^{100}) $ 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9 - 4 * q^11 - 5 * q^13 + q^15 - 4 * q^19 - 12 * q^21 + 8 * q^23 - 3 * q^25 - 8 * q^27 - q^29 + 7 * q^31 - q^33 - 4 * q^35 - 16 * q^37 + 8 * q^39 - 7 * q^41 - 5 * q^43 - 7 * q^45 + 4 * q^47 - 13 * q^49 - 4 * q^51 - 18 * q^53 + 3 * q^55 - q^57 - 6 * q^61 + 6 * q^63 - 24 * q^65 - q^67 - 20 * q^69 + q^71 - 6 * q^73 + q^75 - 3 * q^77 + 4 * q^79 - 16 * q^81 + 25 * q^83 - 4 * q^85 + 9 * q^87 - 14 * q^89 - 13 * q^91 - 4 * q^93 + 3 * q^95 - 36 * q^97 - q^99

Display 100 coefficients 10 coefficients

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
$3$	−2.48425	−1.43428	−0.717140	−	0.696929i	$-0.754549\pi$
$3$	−2.48425	−1.43428	−0.717140	+	0.696929i	$0.754549\pi$
$4$	0	0
$5$	−1.91023	−0.854282	−0.427141	−	0.904185i	$-0.640479\pi$
$5$	−1.91023	−0.854282	−0.427141	+	0.904185i	$0.640479\pi$
$6$	0	0
$7$	3.48425	1.31692	0.658461	−	0.752615i	$-0.271208\pi$
$7$	3.48425	1.31692	0.658461	+	0.752615i	$0.271208\pi$
$8$	0	0
$9$	3.17148	1.05716
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.66378	−0.461449	−0.230725	−	0.973019i	$-0.574110\pi$
$13$	−1.66378	−0.461449	−0.230725	+	0.973019i	$0.574110\pi$
$14$	0	0
$15$	4.74549	1.22528
$16$	0	0
$17$	−1.94847	−0.472574	−0.236287	−	0.971683i	$-0.575931\pi$
$17$	−1.94847	−0.472574	−0.236287	+	0.971683i	$0.575931\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−8.65572	−1.88883
$22$	0	0
$23$	5.70725	1.19004	0.595022	−	0.803710i	$-0.297143\pi$
$23$	5.70725	1.19004	0.595022	+	0.803710i	$0.297143\pi$
$24$	0	0
$25$	−1.35101	−0.270202
$26$	0	0
$27$	−0.425988	−0.0819814
$28$	0	0
$29$	−1.22300	−0.227106	−0.113553	−	0.993532i	$-0.536223\pi$
$29$	−1.22300	−0.227106	−0.113553	+	0.993532i	$0.536223\pi$
$30$	0	0
$31$	0.402537	0.0722977	0.0361489	−	0.999346i	$-0.488491\pi$
$31$	0.402537	0.0722977	0.0361489	+	0.999346i	$0.488491\pi$
$32$	0	0
$33$	2.48425	0.432452
$34$	0	0
$35$	−6.65572	−1.12502
$36$	0	0
$37$	−9.65572	−1.58739	−0.793695	−	0.608315i	$-0.791846\pi$
$37$	−9.65572	−1.58739	−0.793695	+	0.608315i	$0.791846\pi$
$38$	0	0
$39$	4.13324	0.661847
$40$	0	0
$41$	8.40121	1.31205	0.656024	−	0.754740i	$-0.272237\pi$
$41$	8.40121	1.31205	0.656024	+	0.754740i	$0.272237\pi$
$42$	0	0
$43$	−0.648991	−0.0989701	−0.0494851	−	0.998775i	$-0.515758\pi$
$43$	−0.648991	−0.0989701	−0.0494851	+	0.998775i	$0.515758\pi$
$44$	0	0
$45$	−6.05826	−0.903112
$46$	0	0
$47$	5.64093	0.822815	0.411407	−	0.911452i	$-0.365037\pi$
$47$	5.64093	0.822815	0.411407	+	0.911452i	$0.365037\pi$
$48$	0	0
$49$	5.13997	0.734281
$50$	0	0
$51$	4.84048	0.677803
$52$	0	0
$53$	−2.17953	−0.299382	−0.149691	−	0.988733i	$-0.547828\pi$
$53$	−2.17953	−0.299382	−0.149691	+	0.988733i	$0.547828\pi$
$54$	0	0
$55$	1.91023	0.257576
$56$	0	0
$57$	2.48425	0.329046
$58$	0	0
$59$	5.65572	0.736312	0.368156	−	0.929764i	$-0.379989\pi$
$59$	5.65572	0.736312	0.368156	+	0.929764i	$0.379989\pi$
$60$	0	0
$61$	6.29142	0.805534	0.402767	−	0.915302i	$-0.368048\pi$
$61$	6.29142	0.805534	0.402767	+	0.915302i	$0.368048\pi$
$62$	0	0
$63$	11.0502	1.39219
$64$	0	0
$65$	3.17821	0.394208
$66$	0	0
$67$	9.71398	1.18675	0.593376	−	0.804926i	$-0.297795\pi$
$67$	9.71398	1.18675	0.593376	+	0.804926i	$0.297795\pi$
$68$	0	0
$69$	−14.1782	−1.70686
$70$	0	0
$71$	−1.73532	−0.205945	−0.102973	−	0.994684i	$-0.532835\pi$
$71$	−1.73532	−0.205945	−0.102973	+	0.994684i	$0.532835\pi$
$72$	0	0
$73$	−0.672442	−0.0787033	−0.0393517	−	0.999225i	$-0.512529\pi$
$73$	−0.672442	−0.0787033	−0.0393517	+	0.999225i	$0.512529\pi$
$74$	0	0
$75$	3.35624	0.387545
$76$	0	0
$77$	−3.48425	−0.397067
$78$	0	0
$79$	−8.73743	−0.983038	−0.491519	−	0.870867i	$-0.663558\pi$
$79$	−8.73743	−0.983038	−0.491519	+	0.870867i	$0.663558\pi$
$80$	0	0
$81$	−8.45617	−0.939574
$82$	0	0
$83$	7.97192	0.875032	0.437516	−	0.899211i	$-0.355858\pi$
$83$	7.97192	0.875032	0.437516	+	0.899211i	$0.355858\pi$
$84$	0	0
$85$	3.72204	0.403712
$86$	0	0
$87$	3.03824	0.325733
$88$	0	0
$89$	−7.20972	−0.764228	−0.382114	−	0.924115i	$-0.624804\pi$
$89$	−7.20972	−0.764228	−0.382114	+	0.924115i	$0.624804\pi$
$90$	0	0
$91$	−5.79702	−0.607692
$92$	0	0
$93$	−1.00000	−0.103695
$94$	0	0
$95$	1.91023	0.195986
$96$	0	0
$97$	−11.1332	−1.13041	−0.565204	−	0.824951i	$-0.691203\pi$
$97$	−11.1332	−1.13041	−0.565204	+	0.824951i	$0.691203\pi$
$98$	0	0
$99$	−3.17148	−0.318745
$100$	0	0
$101$	1.42599	0.141891	0.0709455	−	0.997480i	$-0.477398\pi$
$101$	1.42599	0.141891	0.0709455	+	0.997480i	$0.477398\pi$
$102$	0	0
$103$	−12.6329	−1.24475	−0.622377	−	0.782718i	$-0.713833\pi$
$103$	−12.6329	−1.24475	−0.622377	+	0.782718i	$0.713833\pi$
$104$	0	0
$105$	16.5344	1.61360
$106$	0	0
$107$	−2.89201	−0.279581	−0.139791	−	0.990181i	$-0.544643\pi$
$107$	−2.89201	−0.279581	−0.139791	+	0.990181i	$0.544643\pi$
$108$	0	0
$109$	7.93698	0.760225	0.380112	−	0.924940i	$-0.375885\pi$
$109$	7.93698	0.760225	0.380112	+	0.924940i	$0.375885\pi$
$110$	0	0
$111$	23.9872	2.27676
$112$	0	0
$113$	0.178206	0.0167642	0.00838212	−	0.999965i	$-0.497332\pi$
$113$	0.178206	0.0167642	0.00838212	+	0.999965i	$0.497332\pi$
$114$	0	0
$115$	−10.9022	−1.01663
$116$	0	0
$117$	−5.27664	−0.487825
$118$	0	0
$119$	−6.78896	−0.622343
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−20.8707	−1.88184
$124$	0	0
$125$	12.1319	1.08511
$126$	0	0
$127$	5.32293	0.472334	0.236167	−	0.971712i	$-0.424109\pi$
$127$	5.32293	0.472334	0.236167	+	0.971712i	$0.424109\pi$
$128$	0	0
$129$	1.61225	0.141951
$130$	0	0
$131$	−6.13474	−0.535994	−0.267997	−	0.963420i	$-0.586362\pi$
$131$	−6.13474	−0.535994	−0.267997	+	0.963420i	$0.586362\pi$
$132$	0	0
$133$	−3.48425	−0.302122
$134$	0	0
$135$	0.813736	0.0700352
$136$	0	0
$137$	10.0789	0.861097	0.430548	−	0.902567i	$-0.358320\pi$
$137$	10.0789	0.861097	0.430548	+	0.902567i	$0.358320\pi$
$138$	0	0
$139$	−6.82390	−0.578796	−0.289398	−	0.957209i	$-0.593455\pi$
$139$	−6.82390	−0.578796	−0.289398	+	0.957209i	$0.593455\pi$
$140$	0	0
$141$	−14.0135	−1.18015
$142$	0	0
$143$	1.66378	0.139132
$144$	0	0
$145$	2.33622	0.194013
$146$	0	0
$147$	−12.7689	−1.05316
$148$	0	0
$149$	−3.86737	−0.316827	−0.158414	−	0.987373i	$-0.550638\pi$
$149$	−3.86737	−0.316827	−0.158414	+	0.987373i	$0.550638\pi$
$150$	0	0
$151$	−17.0850	−1.39036	−0.695179	−	0.718837i	$-0.744675\pi$
$151$	−17.0850	−1.39036	−0.695179	+	0.718837i	$0.744675\pi$
$152$	0	0
$153$	−6.17953	−0.499586
$154$	0	0
$155$	−0.768939	−0.0617627
$156$	0	0
$157$	−12.1298	−0.968064	−0.484032	−	0.875050i	$-0.660828\pi$
$157$	−12.1298	−0.968064	−0.484032	+	0.875050i	$0.660828\pi$
$158$	0	0
$159$	5.41450	0.429398
$160$	0	0
$161$	19.8855	1.56719
$162$	0	0
$163$	−18.8024	−1.47272	−0.736360	−	0.676590i	$-0.763457\pi$
$163$	−18.8024	−1.47272	−0.736360	+	0.676590i	$0.763457\pi$
$164$	0	0
$165$	−4.74549	−0.369436
$166$	0	0
$167$	−5.02002	−0.388461	−0.194230	−	0.980956i	$-0.562221\pi$
$167$	−5.02002	−0.388461	−0.194230	+	0.980956i	$0.562221\pi$
$168$	0	0
$169$	−10.2318	−0.787065
$170$	0	0
$171$	−3.17148	−0.242529
$172$	0	0
$173$	−21.8324	−1.65989	−0.829944	−	0.557846i	$-0.811628\pi$
$173$	−21.8324	−1.65989	−0.829944	+	0.557846i	$0.811628\pi$
$174$	0	0
$175$	−4.70725	−0.355835
$176$	0	0
$177$	−14.0502	−1.05608
$178$	0	0
$179$	−8.56595	−0.640249	−0.320125	−	0.947375i	$-0.603725\pi$
$179$	−8.56595	−0.640249	−0.320125	+	0.947375i	$0.603725\pi$
$180$	0	0
$181$	8.58418	0.638057	0.319028	−	0.947745i	$-0.396643\pi$
$181$	8.58418	0.638057	0.319028	+	0.947745i	$0.396643\pi$
$182$	0	0
$183$	−15.6294	−1.15536
$184$	0	0
$185$	18.4447	1.35608
$186$	0	0
$187$	1.94847	0.142486
$188$	0	0
$189$	−1.48425	−0.107963
$190$	0	0
$191$	−18.4696	−1.33642	−0.668208	−	0.743975i	$-0.732938\pi$
$191$	−18.4696	−1.33642	−0.668208	+	0.743975i	$0.732938\pi$
$192$	0	0
$193$	9.73400	0.700668	0.350334	−	0.936625i	$-0.386068\pi$
$193$	9.73400	0.700668	0.350334	+	0.936625i	$0.386068\pi$
$194$	0	0
$195$	−7.89544	−0.565405
$196$	0	0
$197$	−7.51293	−0.535274	−0.267637	−	0.963520i	$-0.586243\pi$
$197$	−7.51293	−0.535274	−0.267637	+	0.963520i	$0.586243\pi$
$198$	0	0
$199$	14.4434	1.02386	0.511931	−	0.859027i	$-0.328930\pi$
$199$	14.4434	1.02386	0.511931	+	0.859027i	$0.328930\pi$
$200$	0	0
$201$	−24.1319	−1.70213
$202$	0	0
$203$	−4.26124	−0.299081
$204$	0	0
$205$	−16.0483	−1.12086
$206$	0	0
$207$	18.1004	1.25806
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−25.9255	−1.78478	−0.892392	−	0.451261i	$-0.850974\pi$
$211$	−25.9255	−1.78478	−0.892392	+	0.451261i	$0.850974\pi$
$212$	0	0
$213$	4.31097	0.295383
$214$	0	0
$215$	1.23972	0.0845484
$216$	0	0
$217$	1.40254	0.0952104
$218$	0	0
$219$	1.67051	0.112883
$220$	0	0
$221$	3.24183	0.218069
$222$	0	0
$223$	−26.9041	−1.80163	−0.900817	−	0.434198i	$-0.857032\pi$
$223$	−26.9041	−1.80163	−0.900817	+	0.434198i	$0.857032\pi$
$224$	0	0
$225$	−4.28469	−0.285646
$226$	0	0
$227$	19.5560	1.29797	0.648987	−	0.760799i	$-0.275193\pi$
$227$	19.5560	1.29797	0.648987	+	0.760799i	$0.275193\pi$
$228$	0	0
$229$	−28.6913	−1.89597	−0.947987	−	0.318308i	$-0.896885\pi$
$229$	−28.6913	−1.89597	−0.947987	+	0.318308i	$0.896885\pi$
$230$	0	0
$231$	8.65572	0.569505
$232$	0	0
$233$	−8.06499	−0.528355	−0.264177	−	0.964474i	$-0.585100\pi$
$233$	−8.06499	−0.528355	−0.264177	+	0.964474i	$0.585100\pi$
$234$	0	0
$235$	−10.7755	−0.702916
$236$	0	0
$237$	21.7059	1.40995
$238$	0	0
$239$	3.76744	0.243695	0.121848	−	0.992549i	$-0.461118\pi$
$239$	3.76744	0.243695	0.121848	+	0.992549i	$0.461118\pi$
$240$	0	0
$241$	−16.9450	−1.09153	−0.545763	−	0.837940i	$-0.683760\pi$
$241$	−16.9450	−1.09153	−0.545763	+	0.837940i	$0.683760\pi$
$242$	0	0
$243$	22.2852	1.42959
$244$	0	0
$245$	−9.81853	−0.627283
$246$	0	0
$247$	1.66378	0.105864
$248$	0	0
$249$	−19.8042	−1.25504
$250$	0	0
$251$	5.15819	0.325582	0.162791	−	0.986661i	$-0.447950\pi$
$251$	5.15819	0.325582	0.162791	+	0.986661i	$0.447950\pi$
$252$	0	0
$253$	−5.70725	−0.358812
$254$	0	0
$255$	−9.24645	−0.579035
$256$	0	0
$257$	−4.60942	−0.287528	−0.143764	−	0.989612i	$-0.545921\pi$
$257$	−4.60942	−0.287528	−0.143764	+	0.989612i	$0.545921\pi$
$258$	0	0
$259$	−33.6429	−2.09047
$260$	0	0
$261$	−3.87872	−0.240087
$262$	0	0
$263$	3.33755	0.205802	0.102901	−	0.994692i	$-0.467188\pi$
$263$	3.33755	0.205802	0.102901	+	0.994692i	$0.467188\pi$
$264$	0	0
$265$	4.16342	0.255757
$266$	0	0
$267$	17.9107	1.09612
$268$	0	0
$269$	26.8129	1.63481	0.817405	−	0.576063i	$-0.195412\pi$
$269$	26.8129	1.63481	0.817405	+	0.576063i	$0.195412\pi$
$270$	0	0
$271$	2.84722	0.172956	0.0864780	−	0.996254i	$-0.472439\pi$
$271$	2.84722	0.172956	0.0864780	+	0.996254i	$0.472439\pi$
$272$	0	0
$273$	14.4012	0.871601
$274$	0	0
$275$	1.35101	0.0814689
$276$	0	0
$277$	−19.6810	−1.18251	−0.591257	−	0.806483i	$-0.701368\pi$
$277$	−19.6810	−1.18251	−0.591257	+	0.806483i	$0.701368\pi$
$278$	0	0
$279$	1.27664	0.0764302
$280$	0	0
$281$	−24.1534	−1.44087	−0.720436	−	0.693521i	$-0.756058\pi$
$281$	−24.1534	−1.44087	−0.720436	+	0.693521i	$0.756058\pi$
$282$	0	0
$283$	−1.05436	−0.0626749	−0.0313375	−	0.999509i	$-0.509977\pi$
$283$	−1.05436	−0.0626749	−0.0313375	+	0.999509i	$0.509977\pi$
$284$	0	0
$285$	−4.74549	−0.281098
$286$	0	0
$287$	29.2719	1.72786
$288$	0	0
$289$	−13.2035	−0.776674
$290$	0	0
$291$	27.6577	1.62132
$292$	0	0
$293$	−26.1071	−1.52519	−0.762597	−	0.646873i	$-0.776076\pi$
$293$	−26.1071	−1.52519	−0.762597	+	0.646873i	$0.776076\pi$
$294$	0	0
$295$	−10.8037	−0.629018
$296$	0	0
$297$	0.425988	0.0247183
$298$	0	0
$299$	−9.49560	−0.549145
$300$	0	0
$301$	−2.26124	−0.130336
$302$	0	0
$303$	−3.54250	−0.203511
$304$	0	0
$305$	−12.0181	−0.688154
$306$	0	0
$307$	−4.34758	−0.248129	−0.124065	−	0.992274i	$-0.539593\pi$
$307$	−4.34758	−0.248129	−0.124065	+	0.992274i	$0.539593\pi$
$308$	0	0
$309$	31.3832	1.78533
$310$	0	0
$311$	16.9685	0.962195	0.481097	−	0.876667i	$-0.340238\pi$
$311$	16.9685	0.962195	0.481097	+	0.876667i	$0.340238\pi$
$312$	0	0
$313$	19.9342	1.12674	0.563372	−	0.826203i	$-0.309504\pi$
$313$	19.9342	1.12674	0.563372	+	0.826203i	$0.309504\pi$
$314$	0	0
$315$	−21.1085	−1.18933
$316$	0	0
$317$	−0.534272	−0.0300077	−0.0150038	−	0.999887i	$-0.504776\pi$
$317$	−0.534272	−0.0300077	−0.0150038	+	0.999887i	$0.504776\pi$
$318$	0	0
$319$	1.22300	0.0684750
$320$	0	0
$321$	7.18447	0.400998
$322$	0	0
$323$	1.94847	0.108416
$324$	0	0
$325$	2.24778	0.124684
$326$	0	0
$327$	−19.7174	−1.09038
$328$	0	0
$329$	19.6544	1.08358
$330$	0	0
$331$	−8.57684	−0.471426	−0.235713	−	0.971823i	$-0.575742\pi$
$331$	−8.57684	−0.471426	−0.235713	+	0.971823i	$0.575742\pi$
$332$	0	0
$333$	−30.6229	−1.67812
$334$	0	0
$335$	−18.5560	−1.01382
$336$	0	0
$337$	13.4842	0.734534	0.367267	−	0.930116i	$-0.380294\pi$
$337$	13.4842	0.734534	0.367267	+	0.930116i	$0.380294\pi$
$338$	0	0
$339$	−0.442708	−0.0240446
$340$	0	0
$341$	−0.402537	−0.0217986
$342$	0	0
$343$	−6.48081	−0.349931
$344$	0	0
$345$	27.0837	1.45814
$346$	0	0
$347$	8.16804	0.438484	0.219242	−	0.975671i	$-0.429642\pi$
$347$	8.16804	0.438484	0.219242	+	0.975671i	$0.429642\pi$
$348$	0	0
$349$	30.9570	1.65709	0.828545	−	0.559922i	$-0.189169\pi$
$349$	30.9570	1.65709	0.828545	+	0.559922i	$0.189169\pi$
$350$	0	0
$351$	0.708749	0.0378302
$352$	0	0
$353$	−22.0810	−1.17525	−0.587626	−	0.809133i	$-0.699937\pi$
$353$	−22.0810	−1.17525	−0.587626	+	0.809133i	$0.699937\pi$
$354$	0	0
$355$	3.31487	0.175935
$356$	0	0
$357$	16.8654	0.892614
$358$	0	0
$359$	3.98178	0.210150	0.105075	−	0.994464i	$-0.466492\pi$
$359$	3.98178	0.210150	0.105075	+	0.994464i	$0.466492\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.48425	−0.130389
$364$	0	0
$365$	1.28452	0.0672349
$366$	0	0
$367$	11.7004	0.610755	0.305378	−	0.952231i	$-0.401217\pi$
$367$	11.7004	0.610755	0.305378	+	0.952231i	$0.401217\pi$
$368$	0	0
$369$	26.6442	1.38704
$370$	0	0
$371$	−7.59403	−0.394262
$372$	0	0
$373$	1.49410	0.0773616	0.0386808	−	0.999252i	$-0.487684\pi$
$373$	1.49410	0.0773616	0.0386808	+	0.999252i	$0.487684\pi$
$374$	0	0
$375$	−30.1386	−1.55635
$376$	0	0
$377$	2.03481	0.104798
$378$	0	0
$379$	−20.5955	−1.05792	−0.528961	−	0.848646i	$-0.677418\pi$
$379$	−20.5955	−1.05792	−0.528961	+	0.848646i	$0.677418\pi$
$380$	0	0
$381$	−13.2235	−0.677459
$382$	0	0
$383$	11.8400	0.604997	0.302498	−	0.953150i	$-0.402179\pi$
$383$	11.8400	0.604997	0.302498	+	0.953150i	$0.402179\pi$
$384$	0	0
$385$	6.65572	0.339207
$386$	0	0
$387$	−2.05826	−0.104627
$388$	0	0
$389$	13.2778	0.673213	0.336606	−	0.941645i	$-0.390721\pi$
$389$	13.2778	0.673213	0.336606	+	0.941645i	$0.390721\pi$
$390$	0	0
$391$	−11.1204	−0.562384
$392$	0	0
$393$	15.2402	0.768766
$394$	0	0
$395$	16.6905	0.839792
$396$	0	0
$397$	23.3297	1.17088	0.585441	−	0.810715i	$-0.300921\pi$
$397$	23.3297	1.17088	0.585441	+	0.810715i	$0.300921\pi$
$398$	0	0
$399$	8.65572	0.433328
$400$	0	0
$401$	−27.6840	−1.38247	−0.691236	−	0.722629i	$-0.742933\pi$
$401$	−27.6840	−1.38247	−0.691236	+	0.722629i	$0.742933\pi$
$402$	0	0
$403$	−0.669732	−0.0333617
$404$	0	0
$405$	16.1533	0.802662
$406$	0	0
$407$	9.65572	0.478616
$408$	0	0
$409$	−21.0809	−1.04238	−0.521190	−	0.853440i	$-0.674512\pi$
$409$	−21.0809	−1.04238	−0.521190	+	0.853440i	$0.674512\pi$
$410$	0	0
$411$	−25.0384	−1.23505
$412$	0	0
$413$	19.7059	0.969665
$414$	0	0
$415$	−15.2282	−0.747525
$416$	0	0
$417$	16.9522	0.830155
$418$	0	0
$419$	−7.34488	−0.358821	−0.179411	−	0.983774i	$-0.557419\pi$
$419$	−7.34488	−0.358821	−0.179411	+	0.983774i	$0.557419\pi$
$420$	0	0
$421$	−8.80507	−0.429133	−0.214567	−	0.976709i	$-0.568834\pi$
$421$	−8.80507	−0.429133	−0.214567	+	0.976709i	$0.568834\pi$
$422$	0	0
$423$	17.8901	0.869845
$424$	0	0
$425$	2.63240	0.127690
$426$	0	0
$427$	21.9209	1.06083
$428$	0	0
$429$	−4.13324	−0.199555
$430$	0	0
$431$	16.2035	0.780493	0.390246	−	0.920710i	$-0.372390\pi$
$431$	16.2035	0.780493	0.390246	+	0.920710i	$0.372390\pi$
$432$	0	0
$433$	34.5185	1.65885	0.829427	−	0.558616i	$-0.188667\pi$
$433$	34.5185	1.65885	0.829427	+	0.558616i	$0.188667\pi$
$434$	0	0
$435$	−5.80375	−0.278268
$436$	0	0
$437$	−5.70725	−0.273015
$438$	0	0
$439$	−29.4072	−1.40353	−0.701765	−	0.712409i	$-0.747604\pi$
$439$	−29.4072	−1.40353	−0.701765	+	0.712409i	$0.747604\pi$
$440$	0	0
$441$	16.3013	0.776251
$442$	0	0
$443$	22.0798	1.04904	0.524521	−	0.851398i	$-0.324244\pi$
$443$	22.0798	1.04904	0.524521	+	0.851398i	$0.324244\pi$
$444$	0	0
$445$	13.7722	0.652867
$446$	0	0
$447$	9.60749	0.454419
$448$	0	0
$449$	11.6384	0.549250	0.274625	−	0.961551i	$-0.411446\pi$
$449$	11.6384	0.549250	0.274625	+	0.961551i	$0.411446\pi$
$450$	0	0
$451$	−8.40121	−0.395597
$452$	0	0
$453$	42.4434	1.99416
$454$	0	0
$455$	11.0737	0.519141
$456$	0	0
$457$	−18.0581	−0.844723	−0.422362	−	0.906427i	$-0.638799\pi$
$457$	−18.0581	−0.844723	−0.422362	+	0.906427i	$0.638799\pi$
$458$	0	0
$459$	0.830025	0.0387423
$460$	0	0
$461$	−22.3899	−1.04280	−0.521400	−	0.853313i	$-0.674590\pi$
$461$	−22.3899	−1.04280	−0.521400	+	0.853313i	$0.674590\pi$
$462$	0	0
$463$	28.2432	1.31257	0.656286	−	0.754512i	$-0.272126\pi$
$463$	28.2432	1.31257	0.656286	+	0.754512i	$0.272126\pi$
$464$	0	0
$465$	1.91023	0.0885849
$466$	0	0
$467$	30.9760	1.43340	0.716699	−	0.697382i	$-0.245652\pi$
$467$	30.9760	1.43340	0.716699	+	0.697382i	$0.245652\pi$
$468$	0	0
$469$	33.8459	1.56286
$470$	0	0
$471$	30.1334	1.38847
$472$	0	0
$473$	0.648991	0.0298406
$474$	0	0
$475$	1.35101	0.0619886
$476$	0	0
$477$	−6.91234	−0.316494
$478$	0	0
$479$	28.0468	1.28149	0.640745	−	0.767754i	$-0.278626\pi$
$479$	28.0468	1.28149	0.640745	+	0.767754i	$0.278626\pi$
$480$	0	0
$481$	16.0650	0.732500
$482$	0	0
$483$	−49.4004	−2.24779
$484$	0	0
$485$	21.2671	0.965688
$486$	0	0
$487$	−22.8511	−1.03548	−0.517741	−	0.855537i	$-0.673227\pi$
$487$	−22.8511	−1.03548	−0.517741	+	0.855537i	$0.673227\pi$
$488$	0	0
$489$	46.7098	2.11229
$490$	0	0
$491$	−6.52922	−0.294659	−0.147330	−	0.989087i	$-0.547068\pi$
$491$	−6.52922	−0.294659	−0.147330	+	0.989087i	$0.547068\pi$
$492$	0	0
$493$	2.38299	0.107324
$494$	0	0
$495$	6.05826	0.272298
$496$	0	0
$497$	−6.04630	−0.271214
$498$	0	0
$499$	28.9590	1.29638	0.648191	−	0.761478i	$-0.275526\pi$
$499$	28.9590	1.29638	0.648191	+	0.761478i	$0.275526\pi$
$500$	0	0
$501$	12.4710	0.557161
$502$	0	0
$503$	−20.7563	−0.925476	−0.462738	−	0.886495i	$-0.653133\pi$
$503$	−20.7563	−0.925476	−0.462738	+	0.886495i	$0.653133\pi$
$504$	0	0
$505$	−2.72397	−0.121215
$506$	0	0
$507$	25.4184	1.12887
$508$	0	0
$509$	12.5785	0.557533	0.278767	−	0.960359i	$-0.410074\pi$
$509$	12.5785	0.557533	0.278767	+	0.960359i	$0.410074\pi$
$510$	0	0
$511$	−2.34295	−0.103646
$512$	0	0
$513$	0.425988	0.0188078
$514$	0	0
$515$	24.1317	1.06337
$516$	0	0
$517$	−5.64093	−0.248088
$518$	0	0
$519$	54.2371	2.38074
$520$	0	0
$521$	−5.69576	−0.249536	−0.124768	−	0.992186i	$-0.539819\pi$
$521$	−5.69576	−0.249536	−0.124768	+	0.992186i	$0.539819\pi$
$522$	0	0
$523$	−13.3571	−0.584067	−0.292033	−	0.956408i	$-0.594332\pi$
$523$	−13.3571	−0.584067	−0.292033	+	0.956408i	$0.594332\pi$
$524$	0	0
$525$	11.6940	0.510366
$526$	0	0
$527$	−0.784332	−0.0341660
$528$	0	0
$529$	9.57268	0.416204
$530$	0	0
$531$	17.9370	0.778399
$532$	0	0
$533$	−13.9778	−0.605444
$534$	0	0
$535$	5.52442	0.238841
$536$	0	0
$537$	21.2799	0.918297
$538$	0	0
$539$	−5.13997	−0.221394
$540$	0	0
$541$	−39.3603	−1.69223	−0.846116	−	0.532999i	$-0.821065\pi$
$541$	−39.3603	−1.69223	−0.846116	+	0.532999i	$0.821065\pi$
$542$	0	0
$543$	−21.3252	−0.915152
$544$	0	0
$545$	−15.1615	−0.649447
$546$	0	0
$547$	−44.6774	−1.91027	−0.955134	−	0.296174i	$-0.904289\pi$
$547$	−44.6774	−1.91027	−0.955134	+	0.296174i	$0.904289\pi$
$548$	0	0
$549$	19.9531	0.851577
$550$	0	0
$551$	1.22300	0.0521017
$552$	0	0
$553$	−30.4434	−1.29458
$554$	0	0
$555$	−45.8211	−1.94500
$556$	0	0
$557$	3.26154	0.138196	0.0690979	−	0.997610i	$-0.477988\pi$
$557$	3.26154	0.138196	0.0690979	+	0.997610i	$0.477988\pi$
$558$	0	0
$559$	1.07978	0.0456697
$560$	0	0
$561$	−4.84048	−0.204365
$562$	0	0
$563$	−43.2931	−1.82459	−0.912293	−	0.409538i	$-0.865690\pi$
$563$	−43.2931	−1.82459	−0.912293	+	0.409538i	$0.865690\pi$
$564$	0	0
$565$	−0.340415	−0.0143214
$566$	0	0
$567$	−29.4634	−1.23735
$568$	0	0
$569$	−1.33815	−0.0560983	−0.0280491	−	0.999607i	$-0.508929\pi$
$569$	−1.33815	−0.0560983	−0.0280491	+	0.999607i	$0.508929\pi$
$570$	0	0
$571$	32.1206	1.34420	0.672102	−	0.740458i	$-0.265391\pi$
$571$	32.1206	1.34420	0.672102	+	0.740458i	$0.265391\pi$
$572$	0	0
$573$	45.8831	1.91679
$574$	0	0
$575$	−7.71055	−0.321552
$576$	0	0
$577$	−5.44288	−0.226590	−0.113295	−	0.993561i	$-0.536141\pi$
$577$	−5.44288	−0.226590	−0.113295	+	0.993561i	$0.536141\pi$
$578$	0	0
$579$	−24.1816	−1.00495
$580$	0	0
$581$	27.7761	1.15235
$582$	0	0
$583$	2.17953	0.0902671
$584$	0	0
$585$	10.0796	0.416740
$586$	0	0
$587$	0.356412	0.0147107	0.00735536	−	0.999973i	$-0.497659\pi$
$587$	0.356412	0.0147107	0.00735536	+	0.999973i	$0.497659\pi$
$588$	0	0
$589$	−0.402537	−0.0165862
$590$	0	0
$591$	18.6640	0.767732
$592$	0	0
$593$	−30.6124	−1.25710	−0.628551	−	0.777769i	$-0.716352\pi$
$593$	−30.6124	−1.25710	−0.628551	+	0.777769i	$0.716352\pi$
$594$	0	0
$595$	12.9685	0.531656
$596$	0	0
$597$	−35.8808	−1.46850
$598$	0	0
$599$	−32.6155	−1.33264	−0.666318	−	0.745668i	$-0.732130\pi$
$599$	−32.6155	−1.33264	−0.666318	+	0.745668i	$0.732130\pi$
$600$	0	0
$601$	15.0487	0.613849	0.306925	−	0.951734i	$-0.400700\pi$
$601$	15.0487	0.613849	0.306925	+	0.951734i	$0.400700\pi$
$602$	0	0
$603$	30.8076	1.25458
$604$	0	0
$605$	−1.91023	−0.0776620
$606$	0	0
$607$	40.8175	1.65673	0.828366	−	0.560188i	$-0.189271\pi$
$607$	40.8175	1.65673	0.828366	+	0.560188i	$0.189271\pi$
$608$	0	0
$609$	10.5860	0.428965
$610$	0	0
$611$	−9.38527	−0.379687
$612$	0	0
$613$	1.83855	0.0742584	0.0371292	−	0.999310i	$-0.488179\pi$
$613$	1.83855	0.0742584	0.0371292	+	0.999310i	$0.488179\pi$
$614$	0	0
$615$	39.8678	1.60763
$616$	0	0
$617$	6.58074	0.264931	0.132465	−	0.991188i	$-0.457711\pi$
$617$	6.58074	0.264931	0.132465	+	0.991188i	$0.457711\pi$
$618$	0	0
$619$	27.6177	1.11005	0.555024	−	0.831835i	$-0.312709\pi$
$619$	27.6177	1.11005	0.555024	+	0.831835i	$0.312709\pi$
$620$	0	0
$621$	−2.43122	−0.0975614
$622$	0	0
$623$	−25.1204	−1.00643
$624$	0	0
$625$	−16.4197	−0.656789
$626$	0	0
$627$	−2.48425	−0.0992112
$628$	0	0
$629$	18.8139	0.750160
$630$	0	0
$631$	4.57955	0.182309	0.0911545	−	0.995837i	$-0.470944\pi$
$631$	4.57955	0.182309	0.0911545	+	0.995837i	$0.470944\pi$
$632$	0	0
$633$	64.4053	2.55988
$634$	0	0
$635$	−10.1680	−0.403506
$636$	0	0
$637$	−8.55177	−0.338833
$638$	0	0
$639$	−5.50354	−0.217717
$640$	0	0
$641$	−33.6796	−1.33027	−0.665133	−	0.746725i	$-0.731625\pi$
$641$	−33.6796	−1.33027	−0.665133	+	0.746725i	$0.731625\pi$
$642$	0	0
$643$	−35.4608	−1.39844	−0.699219	−	0.714907i	$-0.746469\pi$
$643$	−35.4608	−1.39844	−0.699219	+	0.714907i	$0.746469\pi$
$644$	0	0
$645$	−3.07978	−0.121266
$646$	0	0
$647$	0.784451	0.0308399	0.0154200	−	0.999881i	$-0.495091\pi$
$647$	0.784451	0.0308399	0.0154200	+	0.999881i	$0.495091\pi$
$648$	0	0
$649$	−5.65572	−0.222006
$650$	0	0
$651$	−3.48425	−0.136558
$652$	0	0
$653$	−18.5810	−0.727132	−0.363566	−	0.931568i	$-0.618441\pi$
$653$	−18.5810	−0.727132	−0.363566	+	0.931568i	$0.618441\pi$
$654$	0	0
$655$	11.7188	0.457891
$656$	0	0
$657$	−2.13263	−0.0832019
$658$	0	0
$659$	44.3200	1.72646	0.863230	−	0.504810i	$-0.168437\pi$
$659$	44.3200	1.72646	0.863230	+	0.504810i	$0.168437\pi$
$660$	0	0
$661$	−31.3663	−1.22001	−0.610003	−	0.792399i	$-0.708832\pi$
$661$	−31.3663	−1.22001	−0.610003	+	0.792399i	$0.708832\pi$
$662$	0	0
$663$	−8.05350	−0.312772
$664$	0	0
$665$	6.65572	0.258098
$666$	0	0
$667$	−6.97998	−0.270266
$668$	0	0
$669$	66.8365	2.58405
$670$	0	0
$671$	−6.29142	−0.242878
$672$	0	0
$673$	−32.2456	−1.24298	−0.621488	−	0.783424i	$-0.713472\pi$
$673$	−32.2456	−1.24298	−0.621488	+	0.783424i	$0.713472\pi$
$674$	0	0
$675$	0.575513	0.0221515
$676$	0	0
$677$	11.4571	0.440331	0.220165	−	0.975463i	$-0.429340\pi$
$677$	11.4571	0.440331	0.220165	+	0.975463i	$0.429340\pi$
$678$	0	0
$679$	−38.7909	−1.48866
$680$	0	0
$681$	−48.5818	−1.86166
$682$	0	0
$683$	−5.32190	−0.203637	−0.101818	−	0.994803i	$-0.532466\pi$
$683$	−5.32190	−0.203637	−0.101818	+	0.994803i	$0.532466\pi$
$684$	0	0
$685$	−19.2530	−0.735620
$686$	0	0
$687$	71.2762	2.71936
$688$	0	0
$689$	3.62626	0.138150
$690$	0	0
$691$	−20.8595	−0.793532	−0.396766	−	0.917920i	$-0.629868\pi$
$691$	−20.8595	−0.793532	−0.396766	+	0.917920i	$0.629868\pi$
$692$	0	0
$693$	−11.0502	−0.419762
$694$	0	0
$695$	13.0352	0.494455
$696$	0	0
$697$	−16.3695	−0.620040
$698$	0	0
$699$	20.0354	0.757809
$700$	0	0
$701$	−16.2534	−0.613881	−0.306940	−	0.951729i	$-0.599305\pi$
$701$	−16.2534	−0.613881	−0.306940	+	0.951729i	$0.599305\pi$
$702$	0	0
$703$	9.65572	0.364172
$704$	0	0
$705$	26.7690	1.00818
$706$	0	0
$707$	4.96849	0.186859
$708$	0	0
$709$	−38.3657	−1.44085	−0.720427	−	0.693531i	$-0.756054\pi$
$709$	−38.3657	−1.44085	−0.720427	+	0.693531i	$0.756054\pi$
$710$	0	0
$711$	−27.7105	−1.03923
$712$	0	0
$713$	2.29738	0.0860374
$714$	0	0
$715$	−3.17821	−0.118858
$716$	0	0
$717$	−9.35924	−0.349527
$718$	0	0
$719$	−6.53890	−0.243860	−0.121930	−	0.992539i	$-0.538908\pi$
$719$	−6.53890	−0.243860	−0.121930	+	0.992539i	$0.538908\pi$
$720$	0	0
$721$	−44.0160	−1.63924
$722$	0	0
$723$	42.0956	1.56555
$724$	0	0
$725$	1.65229	0.0613645
$726$	0	0
$727$	13.0348	0.483434	0.241717	−	0.970347i	$-0.422289\pi$
$727$	13.0348	0.483434	0.241717	+	0.970347i	$0.422289\pi$
$728$	0	0
$729$	−29.9933	−1.11086
$730$	0	0
$731$	1.26454	0.0467707
$732$	0	0
$733$	30.4214	1.12364	0.561820	−	0.827259i	$-0.310101\pi$
$733$	30.4214	1.12364	0.561820	+	0.827259i	$0.310101\pi$
$734$	0	0
$735$	24.3917	0.899700
$736$	0	0
$737$	−9.71398	−0.357819
$738$	0	0
$739$	−29.9089	−1.10022	−0.550108	−	0.835093i	$-0.685414\pi$
$739$	−29.9089	−1.10022	−0.550108	+	0.835093i	$0.685414\pi$
$740$	0	0
$741$	−4.13324	−0.151838
$742$	0	0
$743$	17.1461	0.629028	0.314514	−	0.949253i	$-0.398159\pi$
$743$	17.1461	0.629028	0.314514	+	0.949253i	$0.398159\pi$
$744$	0	0
$745$	7.38757	0.270660
$746$	0	0
$747$	25.2828	0.925048
$748$	0	0
$749$	−10.0765	−0.368187
$750$	0	0
$751$	−14.4354	−0.526756	−0.263378	−	0.964693i	$-0.584837\pi$
$751$	−14.4354	−0.526756	−0.263378	+	0.964693i	$0.584837\pi$
$752$	0	0
$753$	−12.8142	−0.466976
$754$	0	0
$755$	32.6363	1.18776
$756$	0	0
$757$	53.9121	1.95947	0.979734	−	0.200303i	$-0.0641926\pi$
$757$	53.9121	1.95947	0.979734	+	0.200303i	$0.0641926\pi$
$758$	0	0
$759$	14.1782	0.514636
$760$	0	0
$761$	−32.1631	−1.16591	−0.582957	−	0.812503i	$-0.698104\pi$
$761$	−32.1631	−1.16591	−0.582957	+	0.812503i	$0.698104\pi$
$762$	0	0
$763$	27.6544	1.00116
$764$	0	0
$765$	11.8044	0.426787
$766$	0	0
$767$	−9.40987	−0.339771
$768$	0	0
$769$	29.2792	1.05584	0.527918	−	0.849296i	$-0.322973\pi$
$769$	29.2792	1.05584	0.527918	+	0.849296i	$0.322973\pi$
$770$	0	0
$771$	11.4509	0.412396
$772$	0	0
$773$	39.3987	1.41707	0.708537	−	0.705674i	$-0.249356\pi$
$773$	39.3987	1.41707	0.708537	+	0.705674i	$0.249356\pi$
$774$	0	0
$775$	−0.543831	−0.0195350
$776$	0	0
$777$	83.5772	2.99832
$778$	0	0
$779$	−8.40121	−0.301005
$780$	0	0
$781$	1.73532	0.0620948
$782$	0	0
$783$	0.520984	0.0186185
$784$	0	0
$785$	23.1708	0.827000
$786$	0	0
$787$	43.1681	1.53878	0.769388	−	0.638782i	$-0.220561\pi$
$787$	43.1681	1.53878	0.769388	+	0.638782i	$0.220561\pi$
$788$	0	0
$789$	−8.29129	−0.295178
$790$	0	0
$791$	0.620914	0.0220772
$792$	0	0
$793$	−10.4675	−0.371713
$794$	0	0
$795$	−10.3430	−0.366827
$796$	0	0
$797$	11.7400	0.415852	0.207926	−	0.978145i	$-0.433329\pi$
$797$	11.7400	0.415852	0.207926	+	0.978145i	$0.433329\pi$
$798$	0	0
$799$	−10.9912	−0.388841
$800$	0	0
$801$	−22.8654	−0.807910
$802$	0	0
$803$	0.672442	0.0237299
$804$	0	0
$805$	−37.9859	−1.33883
$806$	0	0
$807$	−66.6098	−2.34477
$808$	0	0
$809$	−50.1877	−1.76451	−0.882253	−	0.470776i	$-0.843974\pi$
$809$	−50.1877	−1.76451	−0.882253	+	0.470776i	$0.843974\pi$
$810$	0	0
$811$	−38.8743	−1.36506	−0.682530	−	0.730857i	$-0.739120\pi$
$811$	−38.8743	−1.36506	−0.682530	+	0.730857i	$0.739120\pi$
$812$	0	0
$813$	−7.07318	−0.248067
$814$	0	0
$815$	35.9170	1.25812
$816$	0	0
$817$	0.648991	0.0227053
$818$	0	0
$819$	−18.3851	−0.642427
$820$	0	0
$821$	22.7270	0.793177	0.396588	−	0.917997i	$-0.370194\pi$
$821$	22.7270	0.793177	0.396588	+	0.917997i	$0.370194\pi$
$822$	0	0
$823$	38.8418	1.35394	0.676970	−	0.736010i	$-0.263293\pi$
$823$	38.8418	1.35394	0.676970	+	0.736010i	$0.263293\pi$
$824$	0	0
$825$	−3.35624	−0.116849
$826$	0	0
$827$	−21.6632	−0.753304	−0.376652	−	0.926355i	$-0.622925\pi$
$827$	−21.6632	−0.753304	−0.376652	+	0.926355i	$0.622925\pi$
$828$	0	0
$829$	−8.77417	−0.304739	−0.152370	−	0.988324i	$-0.548690\pi$
$829$	−8.77417	−0.304739	−0.152370	+	0.988324i	$0.548690\pi$
$830$	0	0
$831$	48.8924	1.69606
$832$	0	0
$833$	−10.0151	−0.347002
$834$	0	0
$835$	9.58941	0.331855
$836$	0	0
$837$	−0.171476	−0.00592707
$838$	0	0
$839$	48.2017	1.66411	0.832053	−	0.554696i	$-0.187165\pi$
$839$	48.2017	1.66411	0.832053	+	0.554696i	$0.187165\pi$
$840$	0	0
$841$	−27.5043	−0.948423
$842$	0	0
$843$	60.0030	2.06661
$844$	0	0
$845$	19.5452	0.672375
$846$	0	0
$847$	3.48425	0.119720
$848$	0	0
$849$	2.61928	0.0898934
$850$	0	0
$851$	−55.1076	−1.88906
$852$	0	0
$853$	36.3065	1.24311	0.621556	−	0.783370i	$-0.286501\pi$
$853$	36.3065	1.24311	0.621556	+	0.783370i	$0.286501\pi$
$854$	0	0
$855$	6.05826	0.207188
$856$	0	0
$857$	5.28500	0.180532	0.0902661	−	0.995918i	$-0.471228\pi$
$857$	5.28500	0.180532	0.0902661	+	0.995918i	$0.471228\pi$
$858$	0	0
$859$	4.73997	0.161726	0.0808628	−	0.996725i	$-0.474232\pi$
$859$	4.73997	0.161726	0.0808628	+	0.996725i	$0.474232\pi$
$860$	0	0
$861$	−72.7185	−2.47824
$862$	0	0
$863$	51.4510	1.75141	0.875706	−	0.482845i	$-0.160397\pi$
$863$	51.4510	1.75141	0.875706	+	0.482845i	$0.160397\pi$
$864$	0	0
$865$	41.7050	1.41801
$866$	0	0
$867$	32.8006	1.11397
$868$	0	0
$869$	8.73743	0.296397
$870$	0	0
$871$	−16.1619	−0.547626
$872$	0	0
$873$	−35.3088	−1.19502
$874$	0	0
$875$	42.2705	1.42901
$876$	0	0
$877$	−30.0504	−1.01473	−0.507365	−	0.861732i	$-0.669380\pi$
$877$	−30.0504	−1.01473	−0.507365	+	0.861732i	$0.669380\pi$
$878$	0	0
$879$	64.8565	2.18756
$880$	0	0
$881$	−47.6586	−1.60566	−0.802829	−	0.596209i	$-0.796673\pi$
$881$	−47.6586	−1.60566	−0.802829	+	0.596209i	$0.796673\pi$
$882$	0	0
$883$	−39.4883	−1.32889	−0.664444	−	0.747338i	$-0.731332\pi$
$883$	−39.4883	−1.32889	−0.664444	+	0.747338i	$0.731332\pi$
$884$	0	0
$885$	26.8392	0.902188
$886$	0	0
$887$	14.6754	0.492753	0.246377	−	0.969174i	$-0.420760\pi$
$887$	14.6754	0.492753	0.246377	+	0.969174i	$0.420760\pi$
$888$	0	0
$889$	18.5464	0.622026
$890$	0	0
$891$	8.45617	0.283292
$892$	0	0
$893$	−5.64093	−0.188767
$894$	0	0
$895$	16.3630	0.546954
$896$	0	0
$897$	23.5894	0.787627
$898$	0	0
$899$	−0.492304	−0.0164192
$900$	0	0
$901$	4.24676	0.141480
$902$	0	0
$903$	5.61748	0.186938
$904$	0	0
$905$	−16.3978	−0.545081
$906$	0	0
$907$	9.83393	0.326530	0.163265	−	0.986582i	$-0.447797\pi$
$907$	9.83393	0.326530	0.163265	+	0.986582i	$0.447797\pi$
$908$	0	0
$909$	4.52249	0.150001
$910$	0	0
$911$	56.0977	1.85860	0.929301	−	0.369324i	$-0.120411\pi$
$911$	56.0977	1.85860	0.929301	+	0.369324i	$0.120411\pi$
$912$	0	0
$913$	−7.97192	−0.263832
$914$	0	0
$915$	29.8559	0.987005
$916$	0	0
$917$	−21.3749	−0.705862
$918$	0	0
$919$	27.2672	0.899464	0.449732	−	0.893164i	$-0.351520\pi$
$919$	27.2672	0.899464	0.449732	+	0.893164i	$0.351520\pi$
$920$	0	0
$921$	10.8004	0.355887
$922$	0	0
$923$	2.88720	0.0950333
$924$	0	0
$925$	13.0450	0.428916
$926$	0	0
$927$	−40.0649	−1.31590
$928$	0	0
$929$	−54.7586	−1.79657	−0.898285	−	0.439413i	$-0.855186\pi$
$929$	−54.7586	−1.79657	−0.898285	+	0.439413i	$0.855186\pi$
$930$	0	0
$931$	−5.13997	−0.168456
$932$	0	0
$933$	−42.1539	−1.38006
$934$	0	0
$935$	−3.72204	−0.121724
$936$	0	0
$937$	38.9866	1.27364	0.636818	−	0.771014i	$-0.280250\pi$
$937$	38.9866	1.27364	0.636818	+	0.771014i	$0.280250\pi$
$938$	0	0
$939$	−49.5213	−1.61607
$940$	0	0
$941$	−34.6983	−1.13113	−0.565567	−	0.824703i	$-0.691342\pi$
$941$	−34.6983	−1.13113	−0.565567	+	0.824703i	$0.691342\pi$
$942$	0	0
$943$	47.9478	1.56139
$944$	0	0
$945$	2.83525	0.0922308
$946$	0	0
$947$	−27.4190	−0.890998	−0.445499	−	0.895282i	$-0.646974\pi$
$947$	−27.4190	−0.890998	−0.445499	+	0.895282i	$0.646974\pi$
$948$	0	0
$949$	1.11879	0.0363176
$950$	0	0
$951$	1.32726	0.0430394
$952$	0	0
$953$	35.6632	1.15525	0.577623	−	0.816304i	$-0.303981\pi$
$953$	35.6632	1.15525	0.577623	+	0.816304i	$0.303981\pi$
$954$	0	0
$955$	35.2813	1.14168
$956$	0	0
$957$	−3.03824	−0.0982123
$958$	0	0
$959$	35.1173	1.13400
$960$	0	0
$961$	−30.8380	−0.994773
$962$	0	0
$963$	−9.17195	−0.295562
$964$	0	0
$965$	−18.5942	−0.598569
$966$	0	0
$967$	30.0020	0.964798	0.482399	−	0.875952i	$-0.339765\pi$
$967$	30.0020	0.964798	0.482399	+	0.875952i	$0.339765\pi$
$968$	0	0
$969$	−4.84048	−0.155499
$970$	0	0
$971$	9.49843	0.304819	0.152410	−	0.988317i	$-0.451297\pi$
$971$	9.49843	0.304819	0.152410	+	0.988317i	$0.451297\pi$
$972$	0	0
$973$	−23.7761	−0.762228
$974$	0	0
$975$	−5.58404	−0.178832
$976$	0	0
$977$	−52.1848	−1.66954	−0.834770	−	0.550599i	$-0.814399\pi$
$977$	−52.1848	−1.66954	−0.834770	+	0.550599i	$0.814399\pi$
$978$	0	0
$979$	7.20972	0.230423
$980$	0	0
$981$	25.1719	0.803678
$982$	0	0
$983$	−16.6651	−0.531536	−0.265768	−	0.964037i	$-0.585625\pi$
$983$	−16.6651	−0.531536	−0.265768	+	0.964037i	$0.585625\pi$
$984$	0	0
$985$	14.3514	0.457275
$986$	0	0
$987$	−48.8263	−1.55416
$988$	0	0
$989$	−3.70395	−0.117779
$990$	0	0
$991$	−11.2562	−0.357565	−0.178782	−	0.983889i	$-0.557216\pi$
$991$	−11.2562	−0.357565	−0.178782	+	0.983889i	$0.557216\pi$
$992$	0	0
$993$	21.3070	0.676157
$994$	0	0
$995$	−27.5902	−0.874667
$996$	0	0
$997$	57.7391	1.82862	0.914308	−	0.405020i	$-0.132736\pi$
$997$	57.7391	1.82862	0.914308	+	0.405020i	$0.132736\pi$
$998$	0	0
$999$	4.11322	0.130136

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.s.1.1		4
4.3	odd	2		1672.2.a.f.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.f.1.4	✓	4	4.3	odd	2
3344.2.a.s.1.1		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-2.48425 q^{3} -1.91023 q^{5} +3.48425 q^{7} +3.17148 q^{9} +O(q^{10})\)	\(q-2.48425 q^{3} -1.91023 q^{5} +3.48425 q^{7} +3.17148 q^{9} -1.00000 q^{11} -1.66378 q^{13} +4.74549 q^{15} -1.94847 q^{17} -1.00000 q^{19} -8.65572 q^{21} +5.70725 q^{23} -1.35101 q^{25} -0.425988 q^{27} -1.22300 q^{29} +0.402537 q^{31} +2.48425 q^{33} -6.65572 q^{35} -9.65572 q^{37} +4.13324 q^{39} +8.40121 q^{41} -0.648991 q^{43} -6.05826 q^{45} +5.64093 q^{47} +5.13997 q^{49} +4.84048 q^{51} -2.17953 q^{53} +1.91023 q^{55} +2.48425 q^{57} +5.65572 q^{59} +6.29142 q^{61} +11.0502 q^{63} +3.17821 q^{65} +9.71398 q^{67} -14.1782 q^{69} -1.73532 q^{71} -0.672442 q^{73} +3.35624 q^{75} -3.48425 q^{77} -8.73743 q^{79} -8.45617 q^{81} +7.97192 q^{83} +3.72204 q^{85} +3.03824 q^{87} -7.20972 q^{89} -5.79702 q^{91} -1.00000 q^{93} +1.91023 q^{95} -11.1332 q^{97} -3.17148 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9}+O(q^{10}) \) 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9	\( 4 q + q^{3} - 3 q^{5} + 3 q^{7} + q^{9} - 4 q^{11} - 5 q^{13} + q^{15} - 4 q^{19} - 12 q^{21} + 8 q^{23} - 3 q^{25} - 8 q^{27} - q^{29} + 7 q^{31} - q^{33} - 4 q^{35} - 16 q^{37} + 8 q^{39} - 7 q^{41} - 5 q^{43} - 7 q^{45} + 4 q^{47} - 13 q^{49} - 4 q^{51} - 18 q^{53} + 3 q^{55} - q^{57} - 6 q^{61} + 6 q^{63} - 24 q^{65} - q^{67} - 20 q^{69} + q^{71} - 6 q^{73} + q^{75} - 3 q^{77} + 4 q^{79} - 16 q^{81} + 25 q^{83} - 4 q^{85} + 9 q^{87} - 14 q^{89} - 13 q^{91} - 4 q^{93} + 3 q^{95} - 36 q^{97} - q^{99}+O(q^{100}) \) 4 * q + q^3 - 3 * q^5 + 3 * q^7 + q^9 - 4 * q^11 - 5 * q^13 + q^15 - 4 * q^19 - 12 * q^21 + 8 * q^23 - 3 * q^25 - 8 * q^27 - q^29 + 7 * q^31 - q^33 - 4 * q^35 - 16 * q^37 + 8 * q^39 - 7 * q^41 - 5 * q^43 - 7 * q^45 + 4 * q^47 - 13 * q^49 - 4 * q^51 - 18 * q^53 + 3 * q^55 - q^57 - 6 * q^61 + 6 * q^63 - 24 * q^65 - q^67 - 20 * q^69 + q^71 - 6 * q^73 + q^75 - 3 * q^77 + 4 * q^79 - 16 * q^81 + 25 * q^83 - 4 * q^85 + 9 * q^87 - 14 * q^89 - 13 * q^91 - 4 * q^93 + 3 * q^95 - 36 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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