LMFDB - Embedded newform 296.2.a.d.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [296,2,Mod(1,296)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(296, 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("296.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 296 = 2^{3} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	296.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.36357189983$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.48389.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 8x^{2} + 3x + 10 $ x^4 - 2x^3 - 8x^2 + 3*x + 10
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.70606$ of defining polynomial
Character	$\chi$	$=$	296.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70606 q^{3} +2.87345 q^{5} -4.19623 q^{7} +4.32278 q^{9} +3.32278 q^{11} -4.90229 q^{13} +7.77575 q^{15} +2.00000 q^{17} -7.15903 q^{19} -11.3553 q^{21} -5.15539 q^{23} +3.25674 q^{25} +3.57952 q^{27} +3.49017 q^{29} +9.26592 q^{31} +8.99164 q^{33} -12.0577 q^{35} +1.00000 q^{37} -13.2659 q^{39} +7.35162 q^{41} -1.74691 q^{43} +12.4213 q^{45} -10.9628 q^{47} +10.6084 q^{49} +5.41213 q^{51} -0.784105 q^{53} +9.54785 q^{55} -19.3728 q^{57} +0.513481 q^{59} +4.62036 q^{61} -18.1394 q^{63} -14.0865 q^{65} -6.28558 q^{67} -13.9508 q^{69} +5.27357 q^{71} -2.17104 q^{73} +8.81295 q^{75} -13.9431 q^{77} -7.97964 q^{79} -3.28193 q^{81} +11.1766 q^{83} +5.74691 q^{85} +9.44462 q^{87} -4.76657 q^{89} +20.5712 q^{91} +25.0742 q^{93} -20.5712 q^{95} -0.431789 q^{97} +14.3636 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 5 q^{5} - q^{7} + 8 q^{9} + 4 q^{11} + 5 q^{13} + 8 q^{17} + 2 q^{19} + q^{21} - 9 q^{23} + 7 q^{25} - q^{27} + 7 q^{29} - q^{31} + 3 q^{33} - 12 q^{35} + 4 q^{37} - 15 q^{39} + 2 q^{41}+ \cdots + 44 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 5 * q^5 - q^7 + 8 * q^9 + 4 * q^11 + 5 * q^13 + 8 * q^17 + 2 * q^19 + q^21 - 9 * q^23 + 7 * q^25 - q^27 + 7 * q^29 - q^31 + 3 * q^33 - 12 * q^35 + 4 * q^37 - 15 * q^39 + 2 * q^41 + 6 * q^43 - 29 * q^47 + 9 * q^49 + 4 * q^51 - 5 * q^53 - 5 * q^55 - 32 * q^57 - 10 * q^59 - q^61 - 28 * q^63 - 2 * q^65 - q^67 - 27 * q^69 - 17 * q^71 + 8 * q^73 + 19 * q^75 - 27 * q^77 + 15 * q^79 - 8 * q^81 + 15 * q^83 + 10 * q^85 - 17 * q^87 - 20 * q^89 + 34 * q^91 - 6 * q^93 - 34 * q^95 + 2 * q^97 + 44 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.70606	1.56235	0.781173	−	0.624315i	$-0.214622\pi$
$3$	2.70606	1.56235	0.781173	+	0.624315i	$0.214622\pi$
$4$	0	0
$5$	2.87345	1.28505	0.642524	−	0.766266i	$-0.277887\pi$
$5$	2.87345	1.28505	0.642524	+	0.766266i	$0.277887\pi$
$6$	0	0
$7$	−4.19623	−1.58603	−0.793013	−	0.609204i	$-0.791489\pi$
$7$	−4.19623	−1.58603	−0.793013	+	0.609204i	$0.791489\pi$
$8$	0	0
$9$	4.32278	1.44093
$10$	0	0
$11$	3.32278	1.00185	0.500927	−	0.865489i	$-0.332992\pi$
$11$	3.32278	1.00185	0.500927	+	0.865489i	$0.332992\pi$
$12$	0	0
$13$	−4.90229	−1.35965	−0.679826	−	0.733374i	$-0.737945\pi$
$13$	−4.90229	−1.35965	−0.679826	+	0.733374i	$0.737945\pi$
$14$	0	0
$15$	7.77575	2.00769
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−7.15903	−1.64240	−0.821198	−	0.570644i	$-0.806694\pi$
$19$	−7.15903	−1.64240	−0.821198	+	0.570644i	$0.806694\pi$
$20$	0	0
$21$	−11.3553	−2.47792
$22$	0	0
$23$	−5.15539	−1.07497	−0.537486	−	0.843273i	$-0.680626\pi$
$23$	−5.15539	−1.07497	−0.537486	+	0.843273i	$0.680626\pi$
$24$	0	0
$25$	3.25674	0.651348
$26$	0	0
$27$	3.57952	0.688878
$28$	0	0
$29$	3.49017	0.648108	0.324054	−	0.946039i	$-0.394954\pi$
$29$	3.49017	0.648108	0.324054	+	0.946039i	$0.394954\pi$
$30$	0	0
$31$	9.26592	1.66421	0.832104	−	0.554620i	$-0.187136\pi$
$31$	9.26592	1.66421	0.832104	+	0.554620i	$0.187136\pi$
$32$	0	0
$33$	8.99164	1.56524
$34$	0	0
$35$	−12.0577	−2.03812
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−13.2659	−2.12425
$40$	0	0
$41$	7.35162	1.14813	0.574065	−	0.818810i	$-0.305366\pi$
$41$	7.35162	1.14813	0.574065	+	0.818810i	$0.305366\pi$
$42$	0	0
$43$	−1.74691	−0.266401	−0.133201	−	0.991089i	$-0.542525\pi$
$43$	−1.74691	−0.266401	−0.133201	+	0.991089i	$0.542525\pi$
$44$	0	0
$45$	12.4213	1.85166
$46$	0	0
$47$	−10.9628	−1.59909	−0.799545	−	0.600607i	$-0.794926\pi$
$47$	−10.9628	−1.59909	−0.799545	+	0.600607i	$0.794926\pi$
$48$	0	0
$49$	10.6084	1.51548
$50$	0	0
$51$	5.41213	0.757849
$52$	0	0
$53$	−0.784105	−0.107705	−0.0538526	−	0.998549i	$-0.517150\pi$
$53$	−0.784105	−0.107705	−0.0538526	+	0.998549i	$0.517150\pi$
$54$	0	0
$55$	9.54785	1.28743
$56$	0	0
$57$	−19.3728	−2.56599
$58$	0	0
$59$	0.513481	0.0668496	0.0334248	−	0.999441i	$-0.489359\pi$
$59$	0.513481	0.0668496	0.0334248	+	0.999441i	$0.489359\pi$
$60$	0	0
$61$	4.62036	0.591577	0.295788	−	0.955253i	$-0.404418\pi$
$61$	4.62036	0.591577	0.295788	+	0.955253i	$0.404418\pi$
$62$	0	0
$63$	−18.1394	−2.28535
$64$	0	0
$65$	−14.0865	−1.74722
$66$	0	0
$67$	−6.28558	−0.767906	−0.383953	−	0.923353i	$-0.625438\pi$
$67$	−6.28558	−0.767906	−0.383953	+	0.923353i	$0.625438\pi$
$68$	0	0
$69$	−13.9508	−1.67948
$70$	0	0
$71$	5.27357	0.625858	0.312929	−	0.949777i	$-0.398690\pi$
$71$	5.27357	0.625858	0.312929	+	0.949777i	$0.398690\pi$
$72$	0	0
$73$	−2.17104	−0.254101	−0.127051	−	0.991896i	$-0.540551\pi$
$73$	−2.17104	−0.254101	−0.127051	+	0.991896i	$0.540551\pi$
$74$	0	0
$75$	8.81295	1.01763
$76$	0	0
$77$	−13.9431	−1.58897
$78$	0	0
$79$	−7.97964	−0.897779	−0.448890	−	0.893587i	$-0.648180\pi$
$79$	−7.97964	−0.897779	−0.448890	+	0.893587i	$0.648180\pi$
$80$	0	0
$81$	−3.28193	−0.364659
$82$	0	0
$83$	11.1766	1.22679	0.613394	−	0.789777i	$-0.289804\pi$
$83$	11.1766	1.22679	0.613394	+	0.789777i	$0.289804\pi$
$84$	0	0
$85$	5.74691	0.623340
$86$	0	0
$87$	9.44462	1.01257
$88$	0	0
$89$	−4.76657	−0.505256	−0.252628	−	0.967564i	$-0.581295\pi$
$89$	−4.76657	−0.505256	−0.252628	+	0.967564i	$0.581295\pi$
$90$	0	0
$91$	20.5712	2.15644
$92$	0	0
$93$	25.0742	2.60007
$94$	0	0
$95$	−20.5712	−2.11056
$96$	0	0
$97$	−0.431789	−0.0438416	−0.0219208	−	0.999760i	$-0.506978\pi$
$97$	−0.431789	−0.0438416	−0.0219208	+	0.999760i	$0.506978\pi$
$98$	0	0
$99$	14.3636	1.44360
$100$	0	0
$101$	4.11454	0.409412	0.204706	−	0.978824i	$-0.434376\pi$
$101$	4.11454	0.409412	0.204706	+	0.978824i	$0.434376\pi$
$102$	0	0
$103$	0.480992	0.0473935	0.0236968	−	0.999719i	$-0.492456\pi$
$103$	0.480992	0.0473935	0.0236968	+	0.999719i	$0.492456\pi$
$104$	0	0
$105$	−32.6288	−3.18425
$106$	0	0
$107$	−1.49017	−0.144060	−0.0720300	−	0.997402i	$-0.522948\pi$
$107$	−1.49017	−0.144060	−0.0720300	+	0.997402i	$0.522948\pi$
$108$	0	0
$109$	−16.3181	−1.56299	−0.781494	−	0.623913i	$-0.785542\pi$
$109$	−16.3181	−1.56299	−0.781494	+	0.623913i	$0.785542\pi$
$110$	0	0
$111$	2.70606	0.256848
$112$	0	0
$113$	12.8243	1.20640	0.603202	−	0.797588i	$-0.293891\pi$
$113$	12.8243	1.20640	0.603202	+	0.797588i	$0.293891\pi$
$114$	0	0
$115$	−14.8138	−1.38139
$116$	0	0
$117$	−21.1915	−1.95916
$118$	0	0
$119$	−8.39246	−0.769336
$120$	0	0
$121$	0.0408460	0.00371327
$122$	0	0
$123$	19.8939	1.79378
$124$	0	0
$125$	−5.00918	−0.448034
$126$	0	0
$127$	16.7674	1.48787	0.743933	−	0.668255i	$-0.232958\pi$
$127$	16.7674	1.48787	0.743933	+	0.668255i	$0.232958\pi$
$128$	0	0
$129$	−4.72725	−0.416211
$130$	0	0
$131$	−8.90594	−0.778116	−0.389058	−	0.921213i	$-0.627199\pi$
$131$	−8.90594	−0.778116	−0.389058	+	0.921213i	$0.627199\pi$
$132$	0	0
$133$	30.0410	2.60488
$134$	0	0
$135$	10.2856	0.885242
$136$	0	0
$137$	11.3236	0.967440	0.483720	−	0.875223i	$-0.339285\pi$
$137$	11.3236	0.967440	0.483720	+	0.875223i	$0.339285\pi$
$138$	0	0
$139$	20.9840	1.77984	0.889919	−	0.456118i	$-0.150761\pi$
$139$	20.9840	1.77984	0.889919	+	0.456118i	$0.150761\pi$
$140$	0	0
$141$	−29.6660	−2.49833
$142$	0	0
$143$	−16.2892	−1.36217
$144$	0	0
$145$	10.0288	0.832850
$146$	0	0
$147$	28.7069	2.36770
$148$	0	0
$149$	−10.2539	−0.840033	−0.420017	−	0.907516i	$-0.637976\pi$
$149$	−10.2539	−0.840033	−0.420017	+	0.907516i	$0.637976\pi$
$150$	0	0
$151$	9.07734	0.738704	0.369352	−	0.929290i	$-0.379580\pi$
$151$	9.07734	0.738704	0.369352	+	0.929290i	$0.379580\pi$
$152$	0	0
$153$	8.64555	0.698952
$154$	0	0
$155$	26.6252	2.13859
$156$	0	0
$157$	13.1415	1.04881	0.524403	−	0.851470i	$-0.324289\pi$
$157$	13.1415	1.04881	0.524403	+	0.851470i	$0.324289\pi$
$158$	0	0
$159$	−2.12184	−0.168273
$160$	0	0
$161$	21.6332	1.70493
$162$	0	0
$163$	−4.05768	−0.317822	−0.158911	−	0.987293i	$-0.550798\pi$
$163$	−4.05768	−0.317822	−0.158911	+	0.987293i	$0.550798\pi$
$164$	0	0
$165$	25.8371	2.01141
$166$	0	0
$167$	−15.1265	−1.17053	−0.585264	−	0.810843i	$-0.699009\pi$
$167$	−15.1265	−1.17053	−0.585264	+	0.810843i	$0.699009\pi$
$168$	0	0
$169$	11.0325	0.848653
$170$	0	0
$171$	−30.9469	−2.36657
$172$	0	0
$173$	18.0401	1.37157	0.685783	−	0.727806i	$-0.259460\pi$
$173$	18.0401	1.37157	0.685783	+	0.727806i	$0.259460\pi$
$174$	0	0
$175$	−13.6660	−1.03306
$176$	0	0
$177$	1.38951	0.104442
$178$	0	0
$179$	10.1394	0.757852	0.378926	−	0.925427i	$-0.376293\pi$
$179$	10.1394	0.757852	0.378926	+	0.925427i	$0.376293\pi$
$180$	0	0
$181$	−5.51135	−0.409655	−0.204828	−	0.978798i	$-0.565663\pi$
$181$	−5.51135	−0.409655	−0.204828	+	0.978798i	$0.565663\pi$
$182$	0	0
$183$	12.5030	0.924248
$184$	0	0
$185$	2.87345	0.211261
$186$	0	0
$187$	6.64555	0.485971
$188$	0	0
$189$	−15.0205	−1.09258
$190$	0	0
$191$	18.2929	1.32363	0.661813	−	0.749669i	$-0.269787\pi$
$191$	18.2929	1.32363	0.661813	+	0.749669i	$0.269787\pi$
$192$	0	0
$193$	−21.9440	−1.57956	−0.789780	−	0.613390i	$-0.789806\pi$
$193$	−21.9440	−1.57956	−0.789780	+	0.613390i	$0.789806\pi$
$194$	0	0
$195$	−38.1190	−2.72976
$196$	0	0
$197$	18.4326	1.31327	0.656635	−	0.754209i	$-0.271979\pi$
$197$	18.4326	1.31327	0.656635	+	0.754209i	$0.271979\pi$
$198$	0	0
$199$	16.8986	1.19791	0.598957	−	0.800782i	$-0.295582\pi$
$199$	16.8986	1.19791	0.598957	+	0.800782i	$0.295582\pi$
$200$	0	0
$201$	−17.0092	−1.19973
$202$	0	0
$203$	−14.6456	−1.02792
$204$	0	0
$205$	21.1245	1.47540
$206$	0	0
$207$	−22.2856	−1.54895
$208$	0	0
$209$	−23.7879	−1.64544
$210$	0	0
$211$	7.89394	0.543441	0.271721	−	0.962376i	$-0.412407\pi$
$211$	7.89394	0.543441	0.271721	+	0.962376i	$0.412407\pi$
$212$	0	0
$213$	14.2706	0.977807
$214$	0	0
$215$	−5.01966	−0.342338
$216$	0	0
$217$	−38.8819	−2.63948
$218$	0	0
$219$	−5.87497	−0.396994
$220$	0	0
$221$	−9.80459	−0.659528
$222$	0	0
$223$	−4.76009	−0.318759	−0.159380	−	0.987217i	$-0.550949\pi$
$223$	−4.76009	−0.318759	−0.159380	+	0.987217i	$0.550949\pi$
$224$	0	0
$225$	14.0782	0.938544
$226$	0	0
$227$	2.41647	0.160387	0.0801935	−	0.996779i	$-0.474446\pi$
$227$	2.41647	0.160387	0.0801935	+	0.996779i	$0.474446\pi$
$228$	0	0
$229$	7.76444	0.513089	0.256544	−	0.966532i	$-0.417416\pi$
$229$	7.76444	0.513089	0.256544	+	0.966532i	$0.417416\pi$
$230$	0	0
$231$	−37.7310	−2.48252
$232$	0	0
$233$	−13.3524	−0.874747	−0.437374	−	0.899280i	$-0.644091\pi$
$233$	−13.3524	−0.874747	−0.437374	+	0.899280i	$0.644091\pi$
$234$	0	0
$235$	−31.5011	−2.05491
$236$	0	0
$237$	−21.5934	−1.40264
$238$	0	0
$239$	0.203888	0.0131885	0.00659423	−	0.999978i	$-0.497901\pi$
$239$	0.203888	0.0131885	0.00659423	+	0.999978i	$0.497901\pi$
$240$	0	0
$241$	−17.0453	−1.09799	−0.548993	−	0.835827i	$-0.684989\pi$
$241$	−17.0453	−1.09799	−0.548993	+	0.835827i	$0.684989\pi$
$242$	0	0
$243$	−19.6197	−1.25860
$244$	0	0
$245$	30.4826	1.94746
$246$	0	0
$247$	35.0957	2.23309
$248$	0	0
$249$	30.2445	1.91667
$250$	0	0
$251$	−5.96067	−0.376234	−0.188117	−	0.982147i	$-0.560238\pi$
$251$	−5.96067	−0.376234	−0.188117	+	0.982147i	$0.560238\pi$
$252$	0	0
$253$	−17.1302	−1.07697
$254$	0	0
$255$	15.5515	0.973873
$256$	0	0
$257$	−3.92561	−0.244873	−0.122436	−	0.992476i	$-0.539071\pi$
$257$	−3.92561	−0.244873	−0.122436	+	0.992476i	$0.539071\pi$
$258$	0	0
$259$	−4.19623	−0.260741
$260$	0	0
$261$	15.0872	0.933875
$262$	0	0
$263$	30.9644	1.90935	0.954675	−	0.297651i	$-0.0962032\pi$
$263$	30.9644	1.90935	0.954675	+	0.297651i	$0.0962032\pi$
$264$	0	0
$265$	−2.25309	−0.138406
$266$	0	0
$267$	−12.8986	−0.789384
$268$	0	0
$269$	−17.5515	−1.07013	−0.535067	−	0.844810i	$-0.679714\pi$
$269$	−17.5515	−1.07013	−0.535067	+	0.844810i	$0.679714\pi$
$270$	0	0
$271$	−8.58139	−0.521283	−0.260641	−	0.965436i	$-0.583934\pi$
$271$	−8.58139	−0.521283	−0.260641	+	0.965436i	$0.583934\pi$
$272$	0	0
$273$	55.6669	3.36911
$274$	0	0
$275$	10.8214	0.652556
$276$	0	0
$277$	2.87345	0.172649	0.0863246	−	0.996267i	$-0.472488\pi$
$277$	2.87345	0.172649	0.0863246	+	0.996267i	$0.472488\pi$
$278$	0	0
$279$	40.0545	2.39800
$280$	0	0
$281$	−13.1590	−0.785002	−0.392501	−	0.919752i	$-0.628390\pi$
$281$	−13.1590	−0.785002	−0.392501	+	0.919752i	$0.628390\pi$
$282$	0	0
$283$	28.6865	1.70524	0.852618	−	0.522534i	$-0.175013\pi$
$283$	28.6865	1.70524	0.852618	+	0.522534i	$0.175013\pi$
$284$	0	0
$285$	−55.6669	−3.29742
$286$	0	0
$287$	−30.8491	−1.82096
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−1.16845	−0.0684957
$292$	0	0
$293$	−7.13502	−0.416832	−0.208416	−	0.978040i	$-0.566831\pi$
$293$	−7.13502	−0.416832	−0.208416	+	0.978040i	$0.566831\pi$
$294$	0	0
$295$	1.47546	0.0859049
$296$	0	0
$297$	11.8939	0.690156
$298$	0	0
$299$	25.2732	1.46159
$300$	0	0
$301$	7.33043	0.422519
$302$	0	0
$303$	11.1342	0.639643
$304$	0	0
$305$	13.2764	0.760205
$306$	0	0
$307$	−16.6667	−0.951221	−0.475610	−	0.879656i	$-0.657773\pi$
$307$	−16.6667	−0.951221	−0.475610	+	0.879656i	$0.657773\pi$
$308$	0	0
$309$	1.30159	0.0740451
$310$	0	0
$311$	−5.82495	−0.330303	−0.165151	−	0.986268i	$-0.552811\pi$
$311$	−5.82495	−0.330303	−0.165151	+	0.986268i	$0.552811\pi$
$312$	0	0
$313$	−32.2547	−1.82315	−0.911573	−	0.411139i	$-0.865131\pi$
$313$	−32.2547	−1.82315	−0.911573	+	0.411139i	$0.865131\pi$
$314$	0	0
$315$	−52.1227	−2.93678
$316$	0	0
$317$	7.90299	0.443876	0.221938	−	0.975061i	$-0.428762\pi$
$317$	7.90299	0.443876	0.221938	+	0.975061i	$0.428762\pi$
$318$	0	0
$319$	11.5971	0.649310
$320$	0	0
$321$	−4.03249	−0.225072
$322$	0	0
$323$	−14.3181	−0.796679
$324$	0	0
$325$	−15.9655	−0.885607
$326$	0	0
$327$	−44.1577	−2.44193
$328$	0	0
$329$	46.0025	2.53620
$330$	0	0
$331$	17.1774	0.944155	0.472077	−	0.881557i	$-0.343504\pi$
$331$	17.1774	0.944155	0.472077	+	0.881557i	$0.343504\pi$
$332$	0	0
$333$	4.32278	0.236887
$334$	0	0
$335$	−18.0613	−0.986796
$336$	0	0
$337$	−3.96833	−0.216169	−0.108084	−	0.994142i	$-0.534472\pi$
$337$	−3.96833	−0.216169	−0.108084	+	0.994142i	$0.534472\pi$
$338$	0	0
$339$	34.7032	1.88482
$340$	0	0
$341$	30.7886	1.66729
$342$	0	0
$343$	−15.1415	−0.817564
$344$	0	0
$345$	−40.0870	−2.15821
$346$	0	0
$347$	−14.6456	−0.786215	−0.393107	−	0.919493i	$-0.628600\pi$
$347$	−14.6456	−0.786215	−0.393107	+	0.919493i	$0.628600\pi$
$348$	0	0
$349$	20.5895	1.10213	0.551066	−	0.834462i	$-0.314221\pi$
$349$	20.5895	1.10213	0.551066	+	0.834462i	$0.314221\pi$
$350$	0	0
$351$	−17.5478	−0.936635
$352$	0	0
$353$	32.1620	1.71181	0.855905	−	0.517133i	$-0.173001\pi$
$353$	32.1620	1.71181	0.855905	+	0.517133i	$0.173001\pi$
$354$	0	0
$355$	15.1534	0.804258
$356$	0	0
$357$	−22.7105	−1.20197
$358$	0	0
$359$	−5.55068	−0.292954	−0.146477	−	0.989214i	$-0.546793\pi$
$359$	−5.55068	−0.292954	−0.146477	+	0.989214i	$0.546793\pi$
$360$	0	0
$361$	32.2518	1.69746
$362$	0	0
$363$	0.110532	0.00580141
$364$	0	0
$365$	−6.23839	−0.326532
$366$	0	0
$367$	30.5895	1.59676	0.798380	−	0.602154i	$-0.205691\pi$
$367$	30.5895	1.59676	0.798380	+	0.602154i	$0.205691\pi$
$368$	0	0
$369$	31.7794	1.65437
$370$	0	0
$371$	3.29029	0.170823
$372$	0	0
$373$	−11.4523	−0.592976	−0.296488	−	0.955037i	$-0.595816\pi$
$373$	−11.4523	−0.592976	−0.296488	+	0.955037i	$0.595816\pi$
$374$	0	0
$375$	−13.5551	−0.699985
$376$	0	0
$377$	−17.1098	−0.881201
$378$	0	0
$379$	−7.99717	−0.410787	−0.205394	−	0.978679i	$-0.565847\pi$
$379$	−7.99717	−0.410787	−0.205394	+	0.978679i	$0.565847\pi$
$380$	0	0
$381$	45.3736	2.32456
$382$	0	0
$383$	−28.1154	−1.43663	−0.718314	−	0.695719i	$-0.755086\pi$
$383$	−28.1154	−1.43663	−0.718314	+	0.695719i	$0.755086\pi$
$384$	0	0
$385$	−40.0650	−2.04190
$386$	0	0
$387$	−7.55150	−0.383864
$388$	0	0
$389$	−16.1541	−0.819044	−0.409522	−	0.912300i	$-0.634305\pi$
$389$	−16.1541	−0.819044	−0.409522	+	0.912300i	$0.634305\pi$
$390$	0	0
$391$	−10.3108	−0.521438
$392$	0	0
$393$	−24.1000	−1.21569
$394$	0	0
$395$	−22.9291	−1.15369
$396$	0	0
$397$	−11.3720	−0.570743	−0.285372	−	0.958417i	$-0.592117\pi$
$397$	−11.3720	−0.570743	−0.285372	+	0.958417i	$0.592117\pi$
$398$	0	0
$399$	81.2927	4.06973
$400$	0	0
$401$	−26.9636	−1.34650	−0.673250	−	0.739415i	$-0.735102\pi$
$401$	−26.9636	−1.34650	−0.673250	+	0.739415i	$0.735102\pi$
$402$	0	0
$403$	−45.4243	−2.26274
$404$	0	0
$405$	−9.43048	−0.468604
$406$	0	0
$407$	3.32278	0.164704
$408$	0	0
$409$	20.9453	1.03568	0.517838	−	0.855478i	$-0.326737\pi$
$409$	20.9453	1.03568	0.517838	+	0.855478i	$0.326737\pi$
$410$	0	0
$411$	30.6424	1.51148
$412$	0	0
$413$	−2.15469	−0.106025
$414$	0	0
$415$	32.1154	1.57648
$416$	0	0
$417$	56.7840	2.78072
$418$	0	0
$419$	−29.0314	−1.41828	−0.709139	−	0.705069i	$-0.750916\pi$
$419$	−29.0314	−1.41828	−0.709139	+	0.705069i	$0.750916\pi$
$420$	0	0
$421$	−1.64485	−0.0801653	−0.0400826	−	0.999196i	$-0.512762\pi$
$421$	−1.64485	−0.0801653	−0.0400826	+	0.999196i	$0.512762\pi$
$422$	0	0
$423$	−47.3898	−2.30417
$424$	0	0
$425$	6.51348	0.315950
$426$	0	0
$427$	−19.3881	−0.938257
$428$	0	0
$429$	−44.0797	−2.12819
$430$	0	0
$431$	14.8243	0.714059	0.357030	−	0.934093i	$-0.383790\pi$
$431$	14.8243	0.714059	0.357030	+	0.934093i	$0.383790\pi$
$432$	0	0
$433$	19.7514	0.949191	0.474595	−	0.880204i	$-0.342594\pi$
$433$	19.7514	0.949191	0.474595	+	0.880204i	$0.342594\pi$
$434$	0	0
$435$	27.1387	1.30120
$436$	0	0
$437$	36.9076	1.76553
$438$	0	0
$439$	5.27640	0.251829	0.125915	−	0.992041i	$-0.459813\pi$
$439$	5.27640	0.251829	0.125915	+	0.992041i	$0.459813\pi$
$440$	0	0
$441$	45.8576	2.18369
$442$	0	0
$443$	6.35915	0.302132	0.151066	−	0.988524i	$-0.451729\pi$
$443$	6.35915	0.302132	0.151066	+	0.988524i	$0.451729\pi$
$444$	0	0
$445$	−13.6965	−0.649278
$446$	0	0
$447$	−27.7477	−1.31242
$448$	0	0
$449$	−1.49382	−0.0704976	−0.0352488	−	0.999379i	$-0.511222\pi$
$449$	−1.49382	−0.0704976	−0.0352488	+	0.999379i	$0.511222\pi$
$450$	0	0
$451$	24.4278	1.15026
$452$	0	0
$453$	24.5639	1.15411
$454$	0	0
$455$	59.1103	2.77113
$456$	0	0
$457$	−29.0030	−1.35670	−0.678350	−	0.734739i	$-0.737305\pi$
$457$	−29.0030	−1.35670	−0.678350	+	0.734739i	$0.737305\pi$
$458$	0	0
$459$	7.15903	0.334155
$460$	0	0
$461$	12.6959	0.591309	0.295654	−	0.955295i	$-0.404462\pi$
$461$	12.6959	0.591309	0.295654	+	0.955295i	$0.404462\pi$
$462$	0	0
$463$	−26.4465	−1.22907	−0.614536	−	0.788888i	$-0.710657\pi$
$463$	−26.4465	−1.22907	−0.614536	+	0.788888i	$0.710657\pi$
$464$	0	0
$465$	72.0494	3.34121
$466$	0	0
$467$	18.4318	0.852921	0.426461	−	0.904506i	$-0.359760\pi$
$467$	18.4318	0.852921	0.426461	+	0.904506i	$0.359760\pi$
$468$	0	0
$469$	26.3757	1.21792
$470$	0	0
$471$	35.5617	1.63860
$472$	0	0
$473$	−5.80459	−0.266895
$474$	0	0
$475$	−23.3151	−1.06977
$476$	0	0
$477$	−3.38951	−0.155195
$478$	0	0
$479$	−15.1234	−0.691004	−0.345502	−	0.938418i	$-0.612291\pi$
$479$	−15.1234	−0.691004	−0.345502	+	0.938418i	$0.612291\pi$
$480$	0	0
$481$	−4.90229	−0.223525
$482$	0	0
$483$	58.5408	2.66370
$484$	0	0
$485$	−1.24073	−0.0563385
$486$	0	0
$487$	−18.7105	−0.847855	−0.423928	−	0.905696i	$-0.639349\pi$
$487$	−18.7105	−0.847855	−0.423928	+	0.905696i	$0.639349\pi$
$488$	0	0
$489$	−10.9803	−0.496548
$490$	0	0
$491$	−22.5098	−1.01585	−0.507927	−	0.861400i	$-0.669588\pi$
$491$	−22.5098	−1.01585	−0.507927	+	0.861400i	$0.669588\pi$
$492$	0	0
$493$	6.98034	0.314379
$494$	0	0
$495$	41.2732	1.85509
$496$	0	0
$497$	−22.1291	−0.992627
$498$	0	0
$499$	0.513481	0.0229866	0.0114933	−	0.999934i	$-0.496341\pi$
$499$	0.513481	0.0229866	0.0114933	+	0.999934i	$0.496341\pi$
$500$	0	0
$501$	−40.9334	−1.82877
$502$	0	0
$503$	−17.5430	−0.782205	−0.391102	−	0.920347i	$-0.627906\pi$
$503$	−17.5430	−0.782205	−0.391102	+	0.920347i	$0.627906\pi$
$504$	0	0
$505$	11.8229	0.526114
$506$	0	0
$507$	29.8546	1.32589
$508$	0	0
$509$	5.82638	0.258250	0.129125	−	0.991628i	$-0.458783\pi$
$509$	5.82638	0.258250	0.129125	+	0.991628i	$0.458783\pi$
$510$	0	0
$511$	9.11019	0.403011
$512$	0	0
$513$	−25.6259	−1.13141
$514$	0	0
$515$	1.38211	0.0609030
$516$	0	0
$517$	−36.4269	−1.60206
$518$	0	0
$519$	48.8178	2.14286
$520$	0	0
$521$	−12.5070	−0.547942	−0.273971	−	0.961738i	$-0.588337\pi$
$521$	−12.5070	−0.547942	−0.273971	+	0.961738i	$0.588337\pi$
$522$	0	0
$523$	−28.1227	−1.22972	−0.614859	−	0.788637i	$-0.710787\pi$
$523$	−28.1227	−1.22972	−0.614859	+	0.788637i	$0.710787\pi$
$524$	0	0
$525$	−36.9812	−1.61399
$526$	0	0
$527$	18.5318	0.807259
$528$	0	0
$529$	3.57800	0.155565
$530$	0	0
$531$	2.21966	0.0963252
$532$	0	0
$533$	−36.0398	−1.56106
$534$	0	0
$535$	−4.28193	−0.185124
$536$	0	0
$537$	27.4378	1.18403
$538$	0	0
$539$	35.2492	1.51829
$540$	0	0
$541$	−44.9495	−1.93253	−0.966265	−	0.257551i	$-0.917084\pi$
$541$	−44.9495	−1.93253	−0.966265	+	0.257551i	$0.917084\pi$
$542$	0	0
$543$	−14.9141	−0.640024
$544$	0	0
$545$	−46.8892	−2.00851
$546$	0	0
$547$	2.87463	0.122910	0.0614552	−	0.998110i	$-0.480426\pi$
$547$	2.87463	0.122910	0.0614552	+	0.998110i	$0.480426\pi$
$548$	0	0
$549$	19.9728	0.852418
$550$	0	0
$551$	−24.9862	−1.06445
$552$	0	0
$553$	33.4844	1.42390
$554$	0	0
$555$	7.77575	0.330062
$556$	0	0
$557$	16.5066	0.699409	0.349704	−	0.936860i	$-0.386282\pi$
$557$	16.5066	0.699409	0.349704	+	0.936860i	$0.386282\pi$
$558$	0	0
$559$	8.56386	0.362213
$560$	0	0
$561$	17.9833	0.759255
$562$	0	0
$563$	15.5515	0.655417	0.327709	−	0.944779i	$-0.393724\pi$
$563$	15.5515	0.655417	0.327709	+	0.944779i	$0.393724\pi$
$564$	0	0
$565$	36.8499	1.55029
$566$	0	0
$567$	13.7717	0.578359
$568$	0	0
$569$	−28.2620	−1.18481	−0.592403	−	0.805642i	$-0.701821\pi$
$569$	−28.2620	−1.18481	−0.592403	+	0.805642i	$0.701821\pi$
$570$	0	0
$571$	−41.1802	−1.72334	−0.861669	−	0.507470i	$-0.830581\pi$
$571$	−41.1802	−1.72334	−0.861669	+	0.507470i	$0.830581\pi$
$572$	0	0
$573$	49.5017	2.06796
$574$	0	0
$575$	−16.7898	−0.700181
$576$	0	0
$577$	−25.9016	−1.07830	−0.539149	−	0.842211i	$-0.681254\pi$
$577$	−25.9016	−1.07830	−0.539149	+	0.842211i	$0.681254\pi$
$578$	0	0
$579$	−59.3817	−2.46782
$580$	0	0
$581$	−46.8995	−1.94572
$582$	0	0
$583$	−2.60541	−0.107905
$584$	0	0
$585$	−60.8929	−2.51761
$586$	0	0
$587$	−19.3224	−0.797522	−0.398761	−	0.917055i	$-0.630560\pi$
$587$	−19.3224	−0.797522	−0.398761	+	0.917055i	$0.630560\pi$
$588$	0	0
$589$	−66.3350	−2.73329
$590$	0	0
$591$	49.8798	2.05178
$592$	0	0
$593$	−5.33809	−0.219209	−0.109605	−	0.993975i	$-0.534958\pi$
$593$	−5.33809	−0.219209	−0.109605	+	0.993975i	$0.534958\pi$
$594$	0	0
$595$	−24.1154	−0.988633
$596$	0	0
$597$	45.7288	1.87156
$598$	0	0
$599$	−16.1329	−0.659172	−0.329586	−	0.944126i	$-0.606909\pi$
$599$	−16.1329	−0.659172	−0.329586	+	0.944126i	$0.606909\pi$
$600$	0	0
$601$	40.6147	1.65671	0.828354	−	0.560205i	$-0.189278\pi$
$601$	40.6147	1.65671	0.828354	+	0.560205i	$0.189278\pi$
$602$	0	0
$603$	−27.1712	−1.10650
$604$	0	0
$605$	0.117369	0.00477173
$606$	0	0
$607$	−1.16268	−0.0471919	−0.0235960	−	0.999722i	$-0.507512\pi$
$607$	−1.16268	−0.0471919	−0.0235960	+	0.999722i	$0.507512\pi$
$608$	0	0
$609$	−39.6318	−1.60596
$610$	0	0
$611$	53.7429	2.17420
$612$	0	0
$613$	5.29759	0.213968	0.106984	−	0.994261i	$-0.465881\pi$
$613$	5.29759	0.213968	0.106984	+	0.994261i	$0.465881\pi$
$614$	0	0
$615$	57.1643	2.30509
$616$	0	0
$617$	−38.6893	−1.55757	−0.778787	−	0.627288i	$-0.784165\pi$
$617$	−38.6893	−1.55757	−0.778787	+	0.627288i	$0.784165\pi$
$618$	0	0
$619$	28.7773	1.15666	0.578328	−	0.815804i	$-0.303705\pi$
$619$	28.7773	1.15666	0.578328	+	0.815804i	$0.303705\pi$
$620$	0	0
$621$	−18.4538	−0.740525
$622$	0	0
$623$	20.0016	0.801349
$624$	0	0
$625$	−30.6773	−1.22709
$626$	0	0
$627$	−64.3715	−2.57075
$628$	0	0
$629$	2.00000	0.0797452
$630$	0	0
$631$	−24.7646	−0.985862	−0.492931	−	0.870068i	$-0.664074\pi$
$631$	−24.7646	−0.985862	−0.492931	+	0.870068i	$0.664074\pi$
$632$	0	0
$633$	21.3615	0.849043
$634$	0	0
$635$	48.1803	1.91198
$636$	0	0
$637$	−52.0053	−2.06052
$638$	0	0
$639$	22.7965	0.901815
$640$	0	0
$641$	−1.77611	−0.0701520	−0.0350760	−	0.999385i	$-0.511167\pi$
$641$	−1.77611	−0.0701520	−0.0350760	+	0.999385i	$0.511167\pi$
$642$	0	0
$643$	−7.09406	−0.279762	−0.139881	−	0.990168i	$-0.544672\pi$
$643$	−7.09406	−0.279762	−0.139881	+	0.990168i	$0.544672\pi$
$644$	0	0
$645$	−13.5835	−0.534851
$646$	0	0
$647$	−25.1627	−0.989247	−0.494624	−	0.869107i	$-0.664694\pi$
$647$	−25.1627	−0.989247	−0.494624	+	0.869107i	$0.664694\pi$
$648$	0	0
$649$	1.70618	0.0669736
$650$	0	0
$651$	−105.217	−4.12378
$652$	0	0
$653$	34.5581	1.35236	0.676181	−	0.736735i	$-0.263634\pi$
$653$	34.5581	1.35236	0.676181	+	0.736735i	$0.263634\pi$
$654$	0	0
$655$	−25.5908	−0.999916
$656$	0	0
$657$	−9.38493	−0.366141
$658$	0	0
$659$	13.9284	0.542575	0.271287	−	0.962498i	$-0.412551\pi$
$659$	13.9284	0.542575	0.271287	+	0.962498i	$0.412551\pi$
$660$	0	0
$661$	−8.15973	−0.317377	−0.158688	−	0.987329i	$-0.550727\pi$
$661$	−8.15973	−0.317377	−0.158688	+	0.987329i	$0.550727\pi$
$662$	0	0
$663$	−26.5318	−1.03041
$664$	0	0
$665$	86.3213	3.34740
$666$	0	0
$667$	−17.9932	−0.696698
$668$	0	0
$669$	−12.8811	−0.498013
$670$	0	0
$671$	15.3524	0.592674
$672$	0	0
$673$	26.9608	1.03926	0.519631	−	0.854391i	$-0.326069\pi$
$673$	26.9608	1.03926	0.519631	+	0.854391i	$0.326069\pi$
$674$	0	0
$675$	11.6576	0.448700
$676$	0	0
$677$	45.3423	1.74265	0.871323	−	0.490710i	$-0.163262\pi$
$677$	45.3423	1.74265	0.871323	+	0.490710i	$0.163262\pi$
$678$	0	0
$679$	1.81189	0.0695339
$680$	0	0
$681$	6.53913	0.250580
$682$	0	0
$683$	27.2518	1.04276	0.521380	−	0.853324i	$-0.325417\pi$
$683$	27.2518	1.04276	0.521380	+	0.853324i	$0.325417\pi$
$684$	0	0
$685$	32.5378	1.24321
$686$	0	0
$687$	21.0111	0.801622
$688$	0	0
$689$	3.84392	0.146442
$690$	0	0
$691$	−4.11536	−0.156556	−0.0782778	−	0.996932i	$-0.524942\pi$
$691$	−4.11536	−0.156556	−0.0782778	+	0.996932i	$0.524942\pi$
$692$	0	0
$693$	−60.2731	−2.28959
$694$	0	0
$695$	60.2965	2.28718
$696$	0	0
$697$	14.7032	0.556925
$698$	0	0
$699$	−36.1325	−1.36666
$700$	0	0
$701$	49.3069	1.86230	0.931148	−	0.364642i	$-0.118809\pi$
$701$	49.3069	1.86230	0.931148	+	0.364642i	$0.118809\pi$
$702$	0	0
$703$	−7.15903	−0.270008
$704$	0	0
$705$	−85.2440	−3.21048
$706$	0	0
$707$	−17.2656	−0.649338
$708$	0	0
$709$	18.3039	0.687419	0.343709	−	0.939076i	$-0.388317\pi$
$709$	18.3039	0.687419	0.343709	+	0.939076i	$0.388317\pi$
$710$	0	0
$711$	−34.4942	−1.29363
$712$	0	0
$713$	−47.7694	−1.78898
$714$	0	0
$715$	−46.8064	−1.75046
$716$	0	0
$717$	0.551735	0.0206049
$718$	0	0
$719$	26.3933	0.984303	0.492152	−	0.870509i	$-0.336211\pi$
$719$	26.3933	0.984303	0.492152	+	0.870509i	$0.336211\pi$
$720$	0	0
$721$	−2.01835	−0.0751674
$722$	0	0
$723$	−46.1257	−1.71543
$724$	0	0
$725$	11.3666	0.422144
$726$	0	0
$727$	−32.3192	−1.19865	−0.599327	−	0.800504i	$-0.704565\pi$
$727$	−32.3192	−1.19865	−0.599327	+	0.800504i	$0.704565\pi$
$728$	0	0
$729$	−43.2463	−1.60171
$730$	0	0
$731$	−3.49382	−0.129224
$732$	0	0
$733$	−1.73077	−0.0639276	−0.0319638	−	0.999489i	$-0.510176\pi$
$733$	−1.73077	−0.0639276	−0.0319638	+	0.999489i	$0.510176\pi$
$734$	0	0
$735$	82.4879	3.04261
$736$	0	0
$737$	−20.8856	−0.769330
$738$	0	0
$739$	45.9196	1.68918	0.844590	−	0.535414i	$-0.179844\pi$
$739$	45.9196	1.68918	0.844590	+	0.535414i	$0.179844\pi$
$740$	0	0
$741$	94.9712	3.48885
$742$	0	0
$743$	30.9995	1.13726	0.568631	−	0.822593i	$-0.307473\pi$
$743$	30.9995	1.13726	0.568631	+	0.822593i	$0.307473\pi$
$744$	0	0
$745$	−29.4641	−1.07948
$746$	0	0
$747$	48.3138	1.76771
$748$	0	0
$749$	6.25309	0.228483
$750$	0	0
$751$	−12.2963	−0.448697	−0.224349	−	0.974509i	$-0.572025\pi$
$751$	−12.2963	−0.448697	−0.224349	+	0.974509i	$0.572025\pi$
$752$	0	0
$753$	−16.1300	−0.587808
$754$	0	0
$755$	26.0833	0.949270
$756$	0	0
$757$	5.30524	0.192822	0.0964112	−	0.995342i	$-0.469264\pi$
$757$	5.30524	0.192822	0.0964112	+	0.995342i	$0.469264\pi$
$758$	0	0
$759$	−46.3554	−1.68259
$760$	0	0
$761$	−1.98186	−0.0718422	−0.0359211	−	0.999355i	$-0.511437\pi$
$761$	−1.98186	−0.0718422	−0.0359211	+	0.999355i	$0.511437\pi$
$762$	0	0
$763$	68.4744	2.47894
$764$	0	0
$765$	24.8426	0.898186
$766$	0	0
$767$	−2.51724	−0.0908921
$768$	0	0
$769$	17.4458	0.629111	0.314556	−	0.949239i	$-0.398144\pi$
$769$	17.4458	0.629111	0.314556	+	0.949239i	$0.398144\pi$
$770$	0	0
$771$	−10.6229	−0.382576
$772$	0	0
$773$	−41.2008	−1.48189	−0.740945	−	0.671565i	$-0.765622\pi$
$773$	−41.2008	−1.48189	−0.740945	+	0.671565i	$0.765622\pi$
$774$	0	0
$775$	30.1767	1.08398
$776$	0	0
$777$	−11.3553	−0.407368
$778$	0	0
$779$	−52.6305	−1.88568
$780$	0	0
$781$	17.5229	0.627019
$782$	0	0
$783$	12.4931	0.446468
$784$	0	0
$785$	37.7615	1.34777
$786$	0	0
$787$	−7.25522	−0.258621	−0.129310	−	0.991604i	$-0.541276\pi$
$787$	−7.25522	−0.258621	−0.129310	+	0.991604i	$0.541276\pi$
$788$	0	0
$789$	83.7917	2.98306
$790$	0	0
$791$	−53.8135	−1.91339
$792$	0	0
$793$	−22.6504	−0.804339
$794$	0	0
$795$	−6.09701	−0.216239
$796$	0	0
$797$	−5.40118	−0.191320	−0.0956598	−	0.995414i	$-0.530496\pi$
$797$	−5.40118	−0.191320	−0.0956598	+	0.995414i	$0.530496\pi$
$798$	0	0
$799$	−21.9256	−0.775672
$800$	0	0
$801$	−20.6048	−0.728036
$802$	0	0
$803$	−7.21388	−0.254572
$804$	0	0
$805$	62.1620	2.19092
$806$	0	0
$807$	−47.4955	−1.67192
$808$	0	0
$809$	−13.5922	−0.477877	−0.238939	−	0.971035i	$-0.576799\pi$
$809$	−13.5922	−0.477877	−0.238939	+	0.971035i	$0.576799\pi$
$810$	0	0
$811$	23.2441	0.816212	0.408106	−	0.912935i	$-0.366189\pi$
$811$	23.2441	0.816212	0.408106	+	0.912935i	$0.366189\pi$
$812$	0	0
$813$	−23.2218	−0.814424
$814$	0	0
$815$	−11.6596	−0.408416
$816$	0	0
$817$	12.5062	0.437536
$818$	0	0
$819$	88.9245	3.10727
$820$	0	0
$821$	−44.3933	−1.54934	−0.774668	−	0.632368i	$-0.782083\pi$
$821$	−44.3933	−1.54934	−0.774668	+	0.632368i	$0.782083\pi$
$822$	0	0
$823$	−22.9876	−0.801299	−0.400649	−	0.916231i	$-0.631215\pi$
$823$	−22.9876	−0.801299	−0.400649	+	0.916231i	$0.631215\pi$
$824$	0	0
$825$	29.2835	1.01952
$826$	0	0
$827$	3.01540	0.104856	0.0524279	−	0.998625i	$-0.483304\pi$
$827$	3.01540	0.104856	0.0524279	+	0.998625i	$0.483304\pi$
$828$	0	0
$829$	38.2080	1.32702	0.663509	−	0.748168i	$-0.269066\pi$
$829$	38.2080	1.32702	0.663509	+	0.748168i	$0.269066\pi$
$830$	0	0
$831$	7.77575	0.269738
$832$	0	0
$833$	21.2167	0.735116
$834$	0	0
$835$	−43.4654	−1.50418
$836$	0	0
$837$	33.1675	1.14644
$838$	0	0
$839$	4.45580	0.153831	0.0769157	−	0.997038i	$-0.475493\pi$
$839$	4.45580	0.153831	0.0769157	+	0.997038i	$0.475493\pi$
$840$	0	0
$841$	−16.8187	−0.579956
$842$	0	0
$843$	−35.6092	−1.22645
$844$	0	0
$845$	31.7014	1.09056
$846$	0	0
$847$	−0.171399	−0.00588934
$848$	0	0
$849$	77.6275	2.66417
$850$	0	0
$851$	−5.15539	−0.176724
$852$	0	0
$853$	9.45403	0.323700	0.161850	−	0.986815i	$-0.448254\pi$
$853$	9.45403	0.323700	0.161850	+	0.986815i	$0.448254\pi$
$854$	0	0
$855$	−88.9245	−3.04116
$856$	0	0
$857$	6.74822	0.230515	0.115257	−	0.993336i	$-0.463231\pi$
$857$	6.74822	0.230515	0.115257	+	0.993336i	$0.463231\pi$
$858$	0	0
$859$	32.9709	1.12495	0.562477	−	0.826813i	$-0.309849\pi$
$859$	32.9709	1.12495	0.562477	+	0.826813i	$0.309849\pi$
$860$	0	0
$861$	−83.4796	−2.84498
$862$	0	0
$863$	17.4545	0.594158	0.297079	−	0.954853i	$-0.403988\pi$
$863$	17.4545	0.594158	0.297079	+	0.954853i	$0.403988\pi$
$864$	0	0
$865$	51.8375	1.76253
$866$	0	0
$867$	−35.1788	−1.19474
$868$	0	0
$869$	−26.5146	−0.899445
$870$	0	0
$871$	30.8138	1.04408
$872$	0	0
$873$	−1.86653	−0.0631724
$874$	0	0
$875$	21.0197	0.710594
$876$	0	0
$877$	6.77033	0.228618	0.114309	−	0.993445i	$-0.463535\pi$
$877$	6.77033	0.228618	0.114309	+	0.993445i	$0.463535\pi$
$878$	0	0
$879$	−19.3078	−0.651237
$880$	0	0
$881$	40.9217	1.37869	0.689344	−	0.724434i	$-0.257899\pi$
$881$	40.9217	1.37869	0.689344	+	0.724434i	$0.257899\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	0	0
$885$	3.99270	0.134213
$886$	0	0
$887$	−50.6143	−1.69946	−0.849732	−	0.527215i	$-0.823236\pi$
$887$	−50.6143	−1.69946	−0.849732	+	0.527215i	$0.823236\pi$
$888$	0	0
$889$	−70.3599	−2.35979
$890$	0	0
$891$	−10.9051	−0.365335
$892$	0	0
$893$	78.4831	2.62634
$894$	0	0
$895$	29.1350	0.973876
$896$	0	0
$897$	68.3909	2.28351
$898$	0	0
$899$	32.3396	1.07859
$900$	0	0
$901$	−1.56821	−0.0522447
$902$	0	0
$903$	19.8366	0.660121
$904$	0	0
$905$	−15.8366	−0.526427
$906$	0	0
$907$	45.8842	1.52356	0.761779	−	0.647837i	$-0.224326\pi$
$907$	45.8842	1.52356	0.761779	+	0.647837i	$0.224326\pi$
$908$	0	0
$909$	17.7862	0.589932
$910$	0	0
$911$	−43.2184	−1.43189	−0.715944	−	0.698158i	$-0.754003\pi$
$911$	−43.2184	−1.43189	−0.715944	+	0.698158i	$0.754003\pi$
$912$	0	0
$913$	37.1372	1.22906
$914$	0	0
$915$	35.9268	1.18770
$916$	0	0
$917$	37.3714	1.23411
$918$	0	0
$919$	22.5822	0.744919	0.372459	−	0.928049i	$-0.378515\pi$
$919$	22.5822	0.744919	0.372459	+	0.928049i	$0.378515\pi$
$920$	0	0
$921$	−45.1012	−1.48614
$922$	0	0
$923$	−25.8526	−0.850949
$924$	0	0
$925$	3.25674	0.107081
$926$	0	0
$927$	2.07922	0.0682906
$928$	0	0
$929$	7.10983	0.233266	0.116633	−	0.993175i	$-0.462790\pi$
$929$	7.10983	0.233266	0.116633	+	0.993175i	$0.462790\pi$
$930$	0	0
$931$	−75.9456	−2.48902
$932$	0	0
$933$	−15.7627	−0.516047
$934$	0	0
$935$	19.0957	0.624496
$936$	0	0
$937$	−58.4139	−1.90830	−0.954149	−	0.299331i	$-0.903236\pi$
$937$	−58.4139	−1.90830	−0.954149	+	0.299331i	$0.903236\pi$
$938$	0	0
$939$	−87.2833	−2.84838
$940$	0	0
$941$	−21.0957	−0.687700	−0.343850	−	0.939025i	$-0.611731\pi$
$941$	−21.0957	−0.687700	−0.343850	+	0.939025i	$0.611731\pi$
$942$	0	0
$943$	−37.9004	−1.23421
$944$	0	0
$945$	−43.1607	−1.40402
$946$	0	0
$947$	−11.6442	−0.378387	−0.189194	−	0.981940i	$-0.560587\pi$
$947$	−11.6442	−0.378387	−0.189194	+	0.981940i	$0.560587\pi$
$948$	0	0
$949$	10.6431	0.345489
$950$	0	0
$951$	21.3860	0.693489
$952$	0	0
$953$	−41.6665	−1.34971	−0.674855	−	0.737950i	$-0.735794\pi$
$953$	−41.6665	−1.34971	−0.674855	+	0.737950i	$0.735794\pi$
$954$	0	0
$955$	52.5638	1.70092
$956$	0	0
$957$	31.3823	1.01445
$958$	0	0
$959$	−47.5164	−1.53439
$960$	0	0
$961$	54.8572	1.76959
$962$	0	0
$963$	−6.44167	−0.207580
$964$	0	0
$965$	−63.0550	−2.02981
$966$	0	0
$967$	12.8254	0.412438	0.206219	−	0.978506i	$-0.433884\pi$
$967$	12.8254	0.412438	0.206219	+	0.978506i	$0.433884\pi$
$968$	0	0
$969$	−38.7456	−1.24469
$970$	0	0
$971$	31.3125	1.00487	0.502434	−	0.864616i	$-0.332438\pi$
$971$	31.3125	1.00487	0.502434	+	0.864616i	$0.332438\pi$
$972$	0	0
$973$	−88.0537	−2.82287
$974$	0	0
$975$	−43.2037	−1.38362
$976$	0	0
$977$	−43.0140	−1.37614	−0.688070	−	0.725644i	$-0.741542\pi$
$977$	−43.0140	−1.37614	−0.688070	+	0.725644i	$0.741542\pi$
$978$	0	0
$979$	−15.8383	−0.506193
$980$	0	0
$981$	−70.5394	−2.25215
$982$	0	0
$983$	16.6617	0.531425	0.265713	−	0.964052i	$-0.414393\pi$
$983$	16.6617	0.531425	0.265713	+	0.964052i	$0.414393\pi$
$984$	0	0
$985$	52.9653	1.68761
$986$	0	0
$987$	124.486	3.96242
$988$	0	0
$989$	9.00599	0.286374
$990$	0	0
$991$	40.5428	1.28788	0.643942	−	0.765074i	$-0.277298\pi$
$991$	40.5428	1.28788	0.643942	+	0.765074i	$0.277298\pi$
$992$	0	0
$993$	46.4831	1.47510
$994$	0	0
$995$	48.5575	1.53938
$996$	0	0
$997$	5.21507	0.165163	0.0825815	−	0.996584i	$-0.473684\pi$
$997$	5.21507	0.165163	0.0825815	+	0.996584i	$0.473684\pi$
$998$	0	0
$999$	3.57952	0.113251

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	296.2.a.d.1.4	✓	4
3.2	odd	2		2664.2.a.r.1.2		4
4.3	odd	2		592.2.a.j.1.1		4
5.4	even	2		7400.2.a.n.1.1		4
8.3	odd	2		2368.2.a.bh.1.4		4
8.5	even	2		2368.2.a.bg.1.1		4
12.11	even	2		5328.2.a.bp.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.a.d.1.4	✓	4	1.1	even	1	trivial
592.2.a.j.1.1		4	4.3	odd	2
2368.2.a.bg.1.1		4	8.5	even	2
2368.2.a.bh.1.4		4	8.3	odd	2
2664.2.a.r.1.2		4	3.2	odd	2
5328.2.a.bp.1.2		4	12.11	even	2
7400.2.a.n.1.1		4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 296 = 2^{3} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	296.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70606 q^{3} +2.87345 q^{5} -4.19623 q^{7} +4.32278 q^{9} +3.32278 q^{11} -4.90229 q^{13} +7.77575 q^{15} +2.00000 q^{17} -7.15903 q^{19} -11.3553 q^{21} -5.15539 q^{23} +3.25674 q^{25} +3.57952 q^{27} +3.49017 q^{29} +9.26592 q^{31} +8.99164 q^{33} -12.0577 q^{35} +1.00000 q^{37} -13.2659 q^{39} +7.35162 q^{41} -1.74691 q^{43} +12.4213 q^{45} -10.9628 q^{47} +10.6084 q^{49} +5.41213 q^{51} -0.784105 q^{53} +9.54785 q^{55} -19.3728 q^{57} +0.513481 q^{59} +4.62036 q^{61} -18.1394 q^{63} -14.0865 q^{65} -6.28558 q^{67} -13.9508 q^{69} +5.27357 q^{71} -2.17104 q^{73} +8.81295 q^{75} -13.9431 q^{77} -7.97964 q^{79} -3.28193 q^{81} +11.1766 q^{83} +5.74691 q^{85} +9.44462 q^{87} -4.76657 q^{89} +20.5712 q^{91} +25.0742 q^{93} -20.5712 q^{95} -0.431789 q^{97} +14.3636 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 5 q^{5} - q^{7} + 8 q^{9} + 4 q^{11} + 5 q^{13} + 8 q^{17} + 2 q^{19} + q^{21} - 9 q^{23} + 7 q^{25} - q^{27} + 7 q^{29} - q^{31} + 3 q^{33} - 12 q^{35} + 4 q^{37} - 15 q^{39} + 2 q^{41}+ \cdots + 44 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 5 * q^5 - q^7 + 8 * q^9 + 4 * q^11 + 5 * q^13 + 8 * q^17 + 2 * q^19 + q^21 - 9 * q^23 + 7 * q^25 - q^27 + 7 * q^29 - q^31 + 3 * q^33 - 12 * q^35 + 4 * q^37 - 15 * q^39 + 2 * q^41 + 6 * q^43 - 29 * q^47 + 9 * q^49 + 4 * q^51 - 5 * q^53 - 5 * q^55 - 32 * q^57 - 10 * q^59 - q^61 - 28 * q^63 - 2 * q^65 - q^67 - 27 * q^69 - 17 * q^71 + 8 * q^73 + 19 * q^75 - 27 * q^77 + 15 * q^79 - 8 * q^81 + 15 * q^83 + 10 * q^85 - 17 * q^87 - 20 * q^89 + 34 * q^91 - 6 * q^93 - 34 * q^95 + 2 * q^97 + 44 * q^99

Introduction

L-functions

Modular forms

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Database

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