LMFDB - Embedded newform 4304.2.a.i.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4304,2,Mod(1,4304)] 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("4304.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4304 = 2^{4} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4304.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.3676130300$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 7x^{4} + 27x^{3} - 15x^{2} - 20x + 12 $ x^7 - x^6 - 10x^5 + 7x^4 + 27x^3 - 15x^2 - 20*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 538)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.870832$ of defining polynomial
Character	$\chi$	$=$	4304.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.79016 q^{3} +2.36234 q^{5} +3.88429 q^{7} +4.78499 q^{9} -2.24531 q^{11} -0.152297 q^{13} +6.59131 q^{15} -0.664994 q^{17} -1.76305 q^{19} +10.8378 q^{21} -1.84606 q^{23} +0.580659 q^{25} +4.98041 q^{27} -5.78775 q^{29} +8.74754 q^{31} -6.26478 q^{33} +9.17602 q^{35} +1.17330 q^{37} -0.424932 q^{39} -2.82985 q^{41} +10.8695 q^{43} +11.3038 q^{45} +5.51807 q^{47} +8.08771 q^{49} -1.85544 q^{51} -10.2524 q^{53} -5.30420 q^{55} -4.91920 q^{57} +10.1961 q^{59} +9.64218 q^{61} +18.5863 q^{63} -0.359777 q^{65} +5.11712 q^{67} -5.15081 q^{69} +6.15492 q^{71} -5.99070 q^{73} +1.62013 q^{75} -8.72145 q^{77} -1.58845 q^{79} -0.458837 q^{81} -14.5243 q^{83} -1.57094 q^{85} -16.1487 q^{87} -5.61030 q^{89} -0.591565 q^{91} +24.4070 q^{93} -4.16494 q^{95} +9.43184 q^{97} -10.7438 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{3} - 6 q^{5} + 3 q^{7} + 7 q^{9} + 12 q^{11} + 3 q^{13} + 6 q^{15} - 8 q^{17} + 7 q^{19} - 5 q^{21} + 22 q^{23} + 9 q^{25} + 22 q^{27} - 7 q^{29} + 3 q^{31} - 19 q^{33} + 25 q^{35} - 13 q^{37}+ \cdots + 9 q^{99}+O(q^{100}) $ 7 * q + 4 * q^3 - 6 * q^5 + 3 * q^7 + 7 * q^9 + 12 * q^11 + 3 * q^13 + 6 * q^15 - 8 * q^17 + 7 * q^19 - 5 * q^21 + 22 * q^23 + 9 * q^25 + 22 * q^27 - 7 * q^29 + 3 * q^31 - 19 * q^33 + 25 * q^35 - 13 * q^37 - q^39 - 11 * q^41 + 14 * q^43 - 17 * q^45 + 23 * q^47 + 6 * q^49 + 21 * q^51 - 21 * q^53 - 15 * q^55 - 14 * q^57 + 20 * q^59 + 13 * q^61 + 12 * q^63 - 29 * q^65 + 30 * q^67 + 13 * q^69 + 23 * q^71 + 9 * q^73 - 3 * q^75 + 8 * q^77 - 11 * q^79 + 43 * q^81 + 3 * q^83 + 21 * q^85 - 28 * q^87 + 16 * q^89 + 10 * q^91 - 13 * q^93 + 21 * q^97 + 9 * 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
$3$	2.79016	1.61090	0.805450	−	0.592664i	$-0.201924\pi$
$3$	2.79016	1.61090	0.805450	+	0.592664i	$0.201924\pi$
$4$	0	0
$5$	2.36234	1.05647	0.528236	−	0.849098i	$-0.322854\pi$
$5$	2.36234	1.05647	0.528236	+	0.849098i	$0.322854\pi$
$6$	0	0
$7$	3.88429	1.46812	0.734062	−	0.679083i	$-0.237622\pi$
$7$	3.88429	1.46812	0.734062	+	0.679083i	$0.237622\pi$
$8$	0	0
$9$	4.78499	1.59500
$10$	0	0
$11$	−2.24531	−0.676987	−0.338494	−	0.940969i	$-0.609917\pi$
$11$	−2.24531	−0.676987	−0.338494	+	0.940969i	$0.609917\pi$
$12$	0	0
$13$	−0.152297	−0.0422395	−0.0211198	−	0.999777i	$-0.506723\pi$
$13$	−0.152297	−0.0422395	−0.0211198	+	0.999777i	$0.506723\pi$
$14$	0	0
$15$	6.59131	1.70187
$16$	0	0
$17$	−0.664994	−0.161285	−0.0806424	−	0.996743i	$-0.525697\pi$
$17$	−0.664994	−0.161285	−0.0806424	+	0.996743i	$0.525697\pi$
$18$	0	0
$19$	−1.76305	−0.404473	−0.202236	−	0.979337i	$-0.564821\pi$
$19$	−1.76305	−0.404473	−0.202236	+	0.979337i	$0.564821\pi$
$20$	0	0
$21$	10.8378	2.36500
$22$	0	0
$23$	−1.84606	−0.384931	−0.192465	−	0.981304i	$-0.561648\pi$
$23$	−1.84606	−0.384931	−0.192465	+	0.981304i	$0.561648\pi$
$24$	0	0
$25$	0.580659	0.116132
$26$	0	0
$27$	4.98041	0.958480
$28$	0	0
$29$	−5.78775	−1.07476	−0.537379	−	0.843341i	$-0.680585\pi$
$29$	−5.78775	−1.07476	−0.537379	+	0.843341i	$0.680585\pi$
$30$	0	0
$31$	8.74754	1.57110	0.785552	−	0.618795i	$-0.212379\pi$
$31$	8.74754	1.57110	0.785552	+	0.618795i	$0.212379\pi$
$32$	0	0
$33$	−6.26478	−1.09056
$34$	0	0
$35$	9.17602	1.55103
$36$	0	0
$37$	1.17330	0.192889	0.0964444	−	0.995338i	$-0.469253\pi$
$37$	1.17330	0.192889	0.0964444	+	0.995338i	$0.469253\pi$
$38$	0	0
$39$	−0.424932	−0.0680436
$40$	0	0
$41$	−2.82985	−0.441948	−0.220974	−	0.975280i	$-0.570924\pi$
$41$	−2.82985	−0.441948	−0.220974	+	0.975280i	$0.570924\pi$
$42$	0	0
$43$	10.8695	1.65759	0.828794	−	0.559554i	$-0.189028\pi$
$43$	10.8695	1.65759	0.828794	+	0.559554i	$0.189028\pi$
$44$	0	0
$45$	11.3038	1.68507
$46$	0	0
$47$	5.51807	0.804893	0.402447	−	0.915443i	$-0.368160\pi$
$47$	5.51807	0.804893	0.402447	+	0.915443i	$0.368160\pi$
$48$	0	0
$49$	8.08771	1.15539
$50$	0	0
$51$	−1.85544	−0.259813
$52$	0	0
$53$	−10.2524	−1.40828	−0.704139	−	0.710062i	$-0.748667\pi$
$53$	−10.2524	−1.40828	−0.704139	+	0.710062i	$0.748667\pi$
$54$	0	0
$55$	−5.30420	−0.715218
$56$	0	0
$57$	−4.91920	−0.651565
$58$	0	0
$59$	10.1961	1.32742	0.663711	−	0.747989i	$-0.268981\pi$
$59$	10.1961	1.32742	0.663711	+	0.747989i	$0.268981\pi$
$60$	0	0
$61$	9.64218	1.23455	0.617277	−	0.786746i	$-0.288236\pi$
$61$	9.64218	1.23455	0.617277	+	0.786746i	$0.288236\pi$
$62$	0	0
$63$	18.5863	2.34165
$64$	0	0
$65$	−0.359777	−0.0446248
$66$	0	0
$67$	5.11712	0.625155	0.312578	−	0.949892i	$-0.398808\pi$
$67$	5.11712	0.625155	0.312578	+	0.949892i	$0.398808\pi$
$68$	0	0
$69$	−5.15081	−0.620085
$70$	0	0
$71$	6.15492	0.730454	0.365227	−	0.930918i	$-0.380991\pi$
$71$	6.15492	0.730454	0.365227	+	0.930918i	$0.380991\pi$
$72$	0	0
$73$	−5.99070	−0.701158	−0.350579	−	0.936533i	$-0.614015\pi$
$73$	−5.99070	−0.701158	−0.350579	+	0.936533i	$0.614015\pi$
$74$	0	0
$75$	1.62013	0.187077
$76$	0	0
$77$	−8.72145	−0.993901
$78$	0	0
$79$	−1.58845	−0.178714	−0.0893572	−	0.996000i	$-0.528481\pi$
$79$	−1.58845	−0.178714	−0.0893572	+	0.996000i	$0.528481\pi$
$80$	0	0
$81$	−0.458837	−0.0509819
$82$	0	0
$83$	−14.5243	−1.59425	−0.797125	−	0.603814i	$-0.793647\pi$
$83$	−14.5243	−1.59425	−0.797125	+	0.603814i	$0.793647\pi$
$84$	0	0
$85$	−1.57094	−0.170393
$86$	0	0
$87$	−16.1487	−1.73133
$88$	0	0
$89$	−5.61030	−0.594691	−0.297345	−	0.954770i	$-0.596101\pi$
$89$	−5.61030	−0.594691	−0.297345	+	0.954770i	$0.596101\pi$
$90$	0	0
$91$	−0.591565	−0.0620128
$92$	0	0
$93$	24.4070	2.53089
$94$	0	0
$95$	−4.16494	−0.427314
$96$	0	0
$97$	9.43184	0.957658	0.478829	−	0.877908i	$-0.341061\pi$
$97$	9.43184	0.957658	0.478829	+	0.877908i	$0.341061\pi$
$98$	0	0
$99$	−10.7438	−1.07979
$100$	0	0
$101$	18.9376	1.88436	0.942180	−	0.335106i	$-0.108772\pi$
$101$	18.9376	1.88436	0.942180	+	0.335106i	$0.108772\pi$
$102$	0	0
$103$	−11.3673	−1.12005	−0.560026	−	0.828475i	$-0.689209\pi$
$103$	−11.3673	−1.12005	−0.560026	+	0.828475i	$0.689209\pi$
$104$	0	0
$105$	25.6026	2.49855
$106$	0	0
$107$	14.2088	1.37361	0.686806	−	0.726841i	$-0.259012\pi$
$107$	14.2088	1.37361	0.686806	+	0.726841i	$0.259012\pi$
$108$	0	0
$109$	13.2500	1.26912	0.634560	−	0.772873i	$-0.281181\pi$
$109$	13.2500	1.26912	0.634560	+	0.772873i	$0.281181\pi$
$110$	0	0
$111$	3.27368	0.310724
$112$	0	0
$113$	−17.5143	−1.64761	−0.823803	−	0.566876i	$-0.808152\pi$
$113$	−17.5143	−1.64761	−0.823803	+	0.566876i	$0.808152\pi$
$114$	0	0
$115$	−4.36103	−0.406668
$116$	0	0
$117$	−0.728738	−0.0673719
$118$	0	0
$119$	−2.58303	−0.236786
$120$	0	0
$121$	−5.95857	−0.541688
$122$	0	0
$123$	−7.89573	−0.711934
$124$	0	0
$125$	−10.4400	−0.933781
$126$	0	0
$127$	−14.4012	−1.27790	−0.638949	−	0.769249i	$-0.720630\pi$
$127$	−14.4012	−1.27790	−0.638949	+	0.769249i	$0.720630\pi$
$128$	0	0
$129$	30.3277	2.67021
$130$	0	0
$131$	0.204126	0.0178346	0.00891730	−	0.999960i	$-0.497161\pi$
$131$	0.204126	0.0178346	0.00891730	+	0.999960i	$0.497161\pi$
$132$	0	0
$133$	−6.84822	−0.593816
$134$	0	0
$135$	11.7654	1.01261
$136$	0	0
$137$	−9.93816	−0.849074	−0.424537	−	0.905411i	$-0.639563\pi$
$137$	−9.93816	−0.849074	−0.424537	+	0.905411i	$0.639563\pi$
$138$	0	0
$139$	6.11879	0.518989	0.259495	−	0.965745i	$-0.416444\pi$
$139$	6.11879	0.518989	0.259495	+	0.965745i	$0.416444\pi$
$140$	0	0
$141$	15.3963	1.29660
$142$	0	0
$143$	0.341954	0.0285956
$144$	0	0
$145$	−13.6726	−1.13545
$146$	0	0
$147$	22.5660	1.86121
$148$	0	0
$149$	−13.1330	−1.07590	−0.537949	−	0.842977i	$-0.680801\pi$
$149$	−13.1330	−1.07590	−0.537949	+	0.842977i	$0.680801\pi$
$150$	0	0
$151$	−6.57750	−0.535270	−0.267635	−	0.963520i	$-0.586242\pi$
$151$	−6.57750	−0.535270	−0.267635	+	0.963520i	$0.586242\pi$
$152$	0	0
$153$	−3.18199	−0.257249
$154$	0	0
$155$	20.6647	1.65983
$156$	0	0
$157$	4.45872	0.355845	0.177922	−	0.984045i	$-0.443062\pi$
$157$	4.45872	0.355845	0.177922	+	0.984045i	$0.443062\pi$
$158$	0	0
$159$	−28.6059	−2.26860
$160$	0	0
$161$	−7.17065	−0.565126
$162$	0	0
$163$	−8.57083	−0.671319	−0.335660	−	0.941983i	$-0.608959\pi$
$163$	−8.57083	−0.671319	−0.335660	+	0.941983i	$0.608959\pi$
$164$	0	0
$165$	−14.7996	−1.15214
$166$	0	0
$167$	−1.45022	−0.112221	−0.0561105	−	0.998425i	$-0.517870\pi$
$167$	−1.45022	−0.112221	−0.0561105	+	0.998425i	$0.517870\pi$
$168$	0	0
$169$	−12.9768	−0.998216
$170$	0	0
$171$	−8.43620	−0.645132
$172$	0	0
$173$	−13.8967	−1.05655	−0.528274	−	0.849074i	$-0.677161\pi$
$173$	−13.8967	−1.05655	−0.528274	+	0.849074i	$0.677161\pi$
$174$	0	0
$175$	2.25545	0.170496
$176$	0	0
$177$	28.4488	2.13834
$178$	0	0
$179$	1.93736	0.144805	0.0724025	−	0.997375i	$-0.476933\pi$
$179$	1.93736	0.144805	0.0724025	+	0.997375i	$0.476933\pi$
$180$	0	0
$181$	−21.8586	−1.62473	−0.812367	−	0.583146i	$-0.801822\pi$
$181$	−21.8586	−1.62473	−0.812367	+	0.583146i	$0.801822\pi$
$182$	0	0
$183$	26.9032	1.98874
$184$	0	0
$185$	2.77173	0.203781
$186$	0	0
$187$	1.49312	0.109188
$188$	0	0
$189$	19.3454	1.40717
$190$	0	0
$191$	−20.6013	−1.49066	−0.745329	−	0.666697i	$-0.767707\pi$
$191$	−20.6013	−1.49066	−0.745329	+	0.666697i	$0.767707\pi$
$192$	0	0
$193$	15.9686	1.14945	0.574724	−	0.818347i	$-0.305110\pi$
$193$	15.9686	1.14945	0.574724	+	0.818347i	$0.305110\pi$
$194$	0	0
$195$	−1.00383	−0.0718861
$196$	0	0
$197$	−21.2617	−1.51483	−0.757417	−	0.652932i	$-0.773539\pi$
$197$	−21.2617	−1.51483	−0.757417	+	0.652932i	$0.773539\pi$
$198$	0	0
$199$	15.4590	1.09586	0.547930	−	0.836524i	$-0.315416\pi$
$199$	15.4590	1.09586	0.547930	+	0.836524i	$0.315416\pi$
$200$	0	0
$201$	14.2776	1.00706
$202$	0	0
$203$	−22.4813	−1.57788
$204$	0	0
$205$	−6.68507	−0.466906
$206$	0	0
$207$	−8.83340	−0.613963
$208$	0	0
$209$	3.95861	0.273823
$210$	0	0
$211$	−6.05713	−0.416990	−0.208495	−	0.978023i	$-0.566857\pi$
$211$	−6.05713	−0.416990	−0.208495	+	0.978023i	$0.566857\pi$
$212$	0	0
$213$	17.1732	1.17669
$214$	0	0
$215$	25.6775	1.75119
$216$	0	0
$217$	33.9780	2.30658
$218$	0	0
$219$	−16.7150	−1.12950
$220$	0	0
$221$	0.101276	0.00681259
$222$	0	0
$223$	−23.7788	−1.59234	−0.796172	−	0.605070i	$-0.793145\pi$
$223$	−23.7788	−1.59234	−0.796172	+	0.605070i	$0.793145\pi$
$224$	0	0
$225$	2.77845	0.185230
$226$	0	0
$227$	2.91310	0.193349	0.0966746	−	0.995316i	$-0.469179\pi$
$227$	2.91310	0.193349	0.0966746	+	0.995316i	$0.469179\pi$
$228$	0	0
$229$	−23.4623	−1.55043	−0.775216	−	0.631696i	$-0.782359\pi$
$229$	−23.4623	−1.55043	−0.775216	+	0.631696i	$0.782359\pi$
$230$	0	0
$231$	−24.3342	−1.60108
$232$	0	0
$233$	27.7100	1.81535	0.907673	−	0.419679i	$-0.137857\pi$
$233$	27.7100	1.81535	0.907673	+	0.419679i	$0.137857\pi$
$234$	0	0
$235$	13.0356	0.850347
$236$	0	0
$237$	−4.43202	−0.287891
$238$	0	0
$239$	−10.2652	−0.664000	−0.332000	−	0.943279i	$-0.607723\pi$
$239$	−10.2652	−0.664000	−0.332000	+	0.943279i	$0.607723\pi$
$240$	0	0
$241$	12.6345	0.813860	0.406930	−	0.913459i	$-0.366599\pi$
$241$	12.6345	0.813860	0.406930	+	0.913459i	$0.366599\pi$
$242$	0	0
$243$	−16.2215	−1.04061
$244$	0	0
$245$	19.1059	1.22063
$246$	0	0
$247$	0.268507	0.0170847
$248$	0	0
$249$	−40.5252	−2.56818
$250$	0	0
$251$	27.4666	1.73368	0.866839	−	0.498589i	$-0.166148\pi$
$251$	27.4666	1.73368	0.866839	+	0.498589i	$0.166148\pi$
$252$	0	0
$253$	4.14499	0.260593
$254$	0	0
$255$	−4.38318	−0.274486
$256$	0	0
$257$	1.72998	0.107913	0.0539565	−	0.998543i	$-0.482817\pi$
$257$	1.72998	0.107913	0.0539565	+	0.998543i	$0.482817\pi$
$258$	0	0
$259$	4.55743	0.283185
$260$	0	0
$261$	−27.6943	−1.71423
$262$	0	0
$263$	−22.4198	−1.38246	−0.691231	−	0.722633i	$-0.742931\pi$
$263$	−22.4198	−1.38246	−0.691231	+	0.722633i	$0.742931\pi$
$264$	0	0
$265$	−24.2197	−1.48781
$266$	0	0
$267$	−15.6536	−0.957987
$268$	0	0
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	5.05568	0.307111	0.153555	−	0.988140i	$-0.450928\pi$
$271$	5.05568	0.307111	0.153555	+	0.988140i	$0.450928\pi$
$272$	0	0
$273$	−1.65056	−0.0998964
$274$	0	0
$275$	−1.30376	−0.0786198
$276$	0	0
$277$	−17.0935	−1.02705	−0.513525	−	0.858074i	$-0.671661\pi$
$277$	−17.0935	−1.02705	−0.513525	+	0.858074i	$0.671661\pi$
$278$	0	0
$279$	41.8569	2.50591
$280$	0	0
$281$	24.4306	1.45741	0.728704	−	0.684829i	$-0.240123\pi$
$281$	24.4306	1.45741	0.728704	+	0.684829i	$0.240123\pi$
$282$	0	0
$283$	−12.5843	−0.748057	−0.374028	−	0.927417i	$-0.622024\pi$
$283$	−12.5843	−0.748057	−0.374028	+	0.927417i	$0.622024\pi$
$284$	0	0
$285$	−11.6208	−0.688359
$286$	0	0
$287$	−10.9920	−0.648835
$288$	0	0
$289$	−16.5578	−0.973987
$290$	0	0
$291$	26.3163	1.54269
$292$	0	0
$293$	−5.03934	−0.294401	−0.147201	−	0.989107i	$-0.547026\pi$
$293$	−5.03934	−0.294401	−0.147201	+	0.989107i	$0.547026\pi$
$294$	0	0
$295$	24.0867	1.40238
$296$	0	0
$297$	−11.1826	−0.648879
$298$	0	0
$299$	0.281149	0.0162593
$300$	0	0
$301$	42.2204	2.43354
$302$	0	0
$303$	52.8389	3.03552
$304$	0	0
$305$	22.7781	1.30427
$306$	0	0
$307$	19.6450	1.12120	0.560601	−	0.828086i	$-0.310570\pi$
$307$	19.6450	1.12120	0.560601	+	0.828086i	$0.310570\pi$
$308$	0	0
$309$	−31.7165	−1.80429
$310$	0	0
$311$	−26.7504	−1.51687	−0.758437	−	0.651746i	$-0.774037\pi$
$311$	−26.7504	−1.51687	−0.758437	+	0.651746i	$0.774037\pi$
$312$	0	0
$313$	28.2351	1.59594	0.797971	−	0.602696i	$-0.205907\pi$
$313$	28.2351	1.59594	0.797971	+	0.602696i	$0.205907\pi$
$314$	0	0
$315$	43.9072	2.47389
$316$	0	0
$317$	−34.9342	−1.96210	−0.981049	−	0.193761i	$-0.937931\pi$
$317$	−34.9342	−1.96210	−0.981049	+	0.193761i	$0.937931\pi$
$318$	0	0
$319$	12.9953	0.727597
$320$	0	0
$321$	39.6447	2.21275
$322$	0	0
$323$	1.17242	0.0652352
$324$	0	0
$325$	−0.0884325	−0.00490535
$326$	0	0
$327$	36.9696	2.04443
$328$	0	0
$329$	21.4338	1.18168
$330$	0	0
$331$	6.94889	0.381946	0.190973	−	0.981595i	$-0.438836\pi$
$331$	6.94889	0.381946	0.190973	+	0.981595i	$0.438836\pi$
$332$	0	0
$333$	5.61421	0.307657
$334$	0	0
$335$	12.0884	0.660459
$336$	0	0
$337$	−10.2113	−0.556246	−0.278123	−	0.960546i	$-0.589712\pi$
$337$	−10.2113	−0.556246	−0.278123	+	0.960546i	$0.589712\pi$
$338$	0	0
$339$	−48.8677	−2.65413
$340$	0	0
$341$	−19.6410	−1.06362
$342$	0	0
$343$	4.22500	0.228129
$344$	0	0
$345$	−12.1680	−0.655102
$346$	0	0
$347$	−1.35144	−0.0725492	−0.0362746	−	0.999342i	$-0.511549\pi$
$347$	−1.35144	−0.0725492	−0.0362746	+	0.999342i	$0.511549\pi$
$348$	0	0
$349$	18.8717	1.01018	0.505089	−	0.863067i	$-0.331460\pi$
$349$	18.8717	1.01018	0.505089	+	0.863067i	$0.331460\pi$
$350$	0	0
$351$	−0.758500	−0.0404857
$352$	0	0
$353$	23.1371	1.23146	0.615732	−	0.787956i	$-0.288860\pi$
$353$	23.1371	1.23146	0.615732	+	0.787956i	$0.288860\pi$
$354$	0	0
$355$	14.5400	0.771704
$356$	0	0
$357$	−7.20707	−0.381438
$358$	0	0
$359$	3.97007	0.209532	0.104766	−	0.994497i	$-0.466591\pi$
$359$	3.97007	0.209532	0.104766	+	0.994497i	$0.466591\pi$
$360$	0	0
$361$	−15.8916	−0.836402
$362$	0	0
$363$	−16.6254	−0.872605
$364$	0	0
$365$	−14.1521	−0.740754
$366$	0	0
$367$	−36.6793	−1.91464	−0.957322	−	0.289025i	$-0.906669\pi$
$367$	−36.6793	−1.91464	−0.957322	+	0.289025i	$0.906669\pi$
$368$	0	0
$369$	−13.5408	−0.704906
$370$	0	0
$371$	−39.8234	−2.06753
$372$	0	0
$373$	−5.11979	−0.265092	−0.132546	−	0.991177i	$-0.542315\pi$
$373$	−5.11979	−0.265092	−0.132546	+	0.991177i	$0.542315\pi$
$374$	0	0
$375$	−29.1292	−1.50423
$376$	0	0
$377$	0.881454	0.0453972
$378$	0	0
$379$	14.3927	0.739301	0.369650	−	0.929171i	$-0.379477\pi$
$379$	14.3927	0.739301	0.369650	+	0.929171i	$0.379477\pi$
$380$	0	0
$381$	−40.1816	−2.05856
$382$	0	0
$383$	7.15093	0.365395	0.182698	−	0.983169i	$-0.441517\pi$
$383$	7.15093	0.365395	0.182698	+	0.983169i	$0.441517\pi$
$384$	0	0
$385$	−20.6030	−1.05003
$386$	0	0
$387$	52.0106	2.64385
$388$	0	0
$389$	1.63839	0.0830694	0.0415347	−	0.999137i	$-0.486775\pi$
$389$	1.63839	0.0830694	0.0415347	+	0.999137i	$0.486775\pi$
$390$	0	0
$391$	1.22762	0.0620835
$392$	0	0
$393$	0.569545	0.0287297
$394$	0	0
$395$	−3.75246	−0.188807
$396$	0	0
$397$	9.16066	0.459760	0.229880	−	0.973219i	$-0.426167\pi$
$397$	9.16066	0.459760	0.229880	+	0.973219i	$0.426167\pi$
$398$	0	0
$399$	−19.1076	−0.956578
$400$	0	0
$401$	−23.5962	−1.17834	−0.589170	−	0.808009i	$-0.700545\pi$
$401$	−23.5962	−1.17834	−0.589170	+	0.808009i	$0.700545\pi$
$402$	0	0
$403$	−1.33222	−0.0663627
$404$	0	0
$405$	−1.08393	−0.0538609
$406$	0	0
$407$	−2.63442	−0.130583
$408$	0	0
$409$	−2.71755	−0.134374	−0.0671870	−	0.997740i	$-0.521402\pi$
$409$	−2.71755	−0.134374	−0.0671870	+	0.997740i	$0.521402\pi$
$410$	0	0
$411$	−27.7291	−1.36777
$412$	0	0
$413$	39.6047	1.94882
$414$	0	0
$415$	−34.3114	−1.68428
$416$	0	0
$417$	17.0724	0.836039
$418$	0	0
$419$	27.9763	1.36673	0.683366	−	0.730076i	$-0.260516\pi$
$419$	27.9763	1.36673	0.683366	+	0.730076i	$0.260516\pi$
$420$	0	0
$421$	20.3928	0.993883	0.496941	−	0.867784i	$-0.334456\pi$
$421$	20.3928	0.993883	0.496941	+	0.867784i	$0.334456\pi$
$422$	0	0
$423$	26.4039	1.28380
$424$	0	0
$425$	−0.386135	−0.0187303
$426$	0	0
$427$	37.4530	1.81248
$428$	0	0
$429$	0.954106	0.0460647
$430$	0	0
$431$	9.56842	0.460894	0.230447	−	0.973085i	$-0.425981\pi$
$431$	9.56842	0.460894	0.230447	+	0.973085i	$0.425981\pi$
$432$	0	0
$433$	−25.7870	−1.23924	−0.619622	−	0.784901i	$-0.712714\pi$
$433$	−25.7870	−1.23924	−0.619622	+	0.784901i	$0.712714\pi$
$434$	0	0
$435$	−38.1488	−1.82910
$436$	0	0
$437$	3.25471	0.155694
$438$	0	0
$439$	−10.5835	−0.505124	−0.252562	−	0.967581i	$-0.581273\pi$
$439$	−10.5835	−0.505124	−0.252562	+	0.967581i	$0.581273\pi$
$440$	0	0
$441$	38.6996	1.84284
$442$	0	0
$443$	7.91213	0.375917	0.187958	−	0.982177i	$-0.439813\pi$
$443$	7.91213	0.375917	0.187958	+	0.982177i	$0.439813\pi$
$444$	0	0
$445$	−13.2534	−0.628274
$446$	0	0
$447$	−36.6432	−1.73316
$448$	0	0
$449$	−6.03733	−0.284919	−0.142460	−	0.989801i	$-0.545501\pi$
$449$	−6.03733	−0.284919	−0.142460	+	0.989801i	$0.545501\pi$
$450$	0	0
$451$	6.35390	0.299193
$452$	0	0
$453$	−18.3523	−0.862266
$454$	0	0
$455$	−1.39748	−0.0655148
$456$	0	0
$457$	−4.72498	−0.221025	−0.110513	−	0.993875i	$-0.535249\pi$
$457$	−4.72498	−0.221025	−0.110513	+	0.993875i	$0.535249\pi$
$458$	0	0
$459$	−3.31194	−0.154588
$460$	0	0
$461$	25.3520	1.18076	0.590381	−	0.807125i	$-0.298978\pi$
$461$	25.3520	1.18076	0.590381	+	0.807125i	$0.298978\pi$
$462$	0	0
$463$	35.4324	1.64668	0.823341	−	0.567547i	$-0.192107\pi$
$463$	35.4324	1.64668	0.823341	+	0.567547i	$0.192107\pi$
$464$	0	0
$465$	57.6578	2.67381
$466$	0	0
$467$	−20.1572	−0.932765	−0.466383	−	0.884583i	$-0.654443\pi$
$467$	−20.1572	−0.932765	−0.466383	+	0.884583i	$0.654443\pi$
$468$	0	0
$469$	19.8764	0.917806
$470$	0	0
$471$	12.4405	0.573230
$472$	0	0
$473$	−24.4055	−1.12217
$474$	0	0
$475$	−1.02373	−0.0469722
$476$	0	0
$477$	−49.0578	−2.24620
$478$	0	0
$479$	21.2364	0.970314	0.485157	−	0.874427i	$-0.338762\pi$
$479$	21.2364	0.970314	0.485157	+	0.874427i	$0.338762\pi$
$480$	0	0
$481$	−0.178689	−0.00814753
$482$	0	0
$483$	−20.0072	−0.910361
$484$	0	0
$485$	22.2812	1.01174
$486$	0	0
$487$	−1.46670	−0.0664624	−0.0332312	−	0.999448i	$-0.510580\pi$
$487$	−1.46670	−0.0664624	−0.0332312	+	0.999448i	$0.510580\pi$
$488$	0	0
$489$	−23.9140	−1.08143
$490$	0	0
$491$	−14.3389	−0.647105	−0.323553	−	0.946210i	$-0.604877\pi$
$491$	−14.3389	−0.647105	−0.323553	+	0.946210i	$0.604877\pi$
$492$	0	0
$493$	3.84882	0.173342
$494$	0	0
$495$	−25.3805	−1.14077
$496$	0	0
$497$	23.9075	1.07240
$498$	0	0
$499$	13.7751	0.616659	0.308329	−	0.951280i	$-0.400230\pi$
$499$	13.7751	0.616659	0.308329	+	0.951280i	$0.400230\pi$
$500$	0	0
$501$	−4.04633	−0.180777
$502$	0	0
$503$	16.4029	0.731370	0.365685	−	0.930739i	$-0.380835\pi$
$503$	16.4029	0.731370	0.365685	+	0.930739i	$0.380835\pi$
$504$	0	0
$505$	44.7371	1.99077
$506$	0	0
$507$	−36.2074	−1.60803
$508$	0	0
$509$	14.6232	0.648160	0.324080	−	0.946030i	$-0.394945\pi$
$509$	14.6232	0.648160	0.324080	+	0.946030i	$0.394945\pi$
$510$	0	0
$511$	−23.2696	−1.02939
$512$	0	0
$513$	−8.78073	−0.387679
$514$	0	0
$515$	−26.8534	−1.18330
$516$	0	0
$517$	−12.3898	−0.544902
$518$	0	0
$519$	−38.7741	−1.70199
$520$	0	0
$521$	13.6453	0.597811	0.298905	−	0.954283i	$-0.403378\pi$
$521$	13.6453	0.597811	0.298905	+	0.954283i	$0.403378\pi$
$522$	0	0
$523$	−2.36730	−0.103515	−0.0517574	−	0.998660i	$-0.516482\pi$
$523$	−2.36730	−0.103515	−0.0517574	+	0.998660i	$0.516482\pi$
$524$	0	0
$525$	6.29307	0.274652
$526$	0	0
$527$	−5.81706	−0.253395
$528$	0	0
$529$	−19.5921	−0.851828
$530$	0	0
$531$	48.7883	2.11723
$532$	0	0
$533$	0.430977	0.0186677
$534$	0	0
$535$	33.5659	1.45118
$536$	0	0
$537$	5.40554	0.233266
$538$	0	0
$539$	−18.1595	−0.782183
$540$	0	0
$541$	8.38943	0.360690	0.180345	−	0.983603i	$-0.442279\pi$
$541$	8.38943	0.360690	0.180345	+	0.983603i	$0.442279\pi$
$542$	0	0
$543$	−60.9889	−2.61728
$544$	0	0
$545$	31.3011	1.34079
$546$	0	0
$547$	−4.07641	−0.174295	−0.0871473	−	0.996195i	$-0.527775\pi$
$547$	−4.07641	−0.174295	−0.0871473	+	0.996195i	$0.527775\pi$
$548$	0	0
$549$	46.1377	1.96911
$550$	0	0
$551$	10.2041	0.434710
$552$	0	0
$553$	−6.16999	−0.262375
$554$	0	0
$555$	7.73356	0.328271
$556$	0	0
$557$	−17.9458	−0.760387	−0.380193	−	0.924907i	$-0.624143\pi$
$557$	−17.9458	−0.760387	−0.380193	+	0.924907i	$0.624143\pi$
$558$	0	0
$559$	−1.65539	−0.0700157
$560$	0	0
$561$	4.16604	0.175890
$562$	0	0
$563$	−36.1794	−1.52478	−0.762391	−	0.647117i	$-0.775975\pi$
$563$	−36.1794	−1.52478	−0.762391	+	0.647117i	$0.775975\pi$
$564$	0	0
$565$	−41.3748	−1.74065
$566$	0	0
$567$	−1.78226	−0.0748477
$568$	0	0
$569$	20.7037	0.867945	0.433973	−	0.900926i	$-0.357111\pi$
$569$	20.7037	0.867945	0.433973	+	0.900926i	$0.357111\pi$
$570$	0	0
$571$	−11.8564	−0.496175	−0.248087	−	0.968738i	$-0.579802\pi$
$571$	−11.8564	−0.496175	−0.248087	+	0.968738i	$0.579802\pi$
$572$	0	0
$573$	−57.4809	−2.40130
$574$	0	0
$575$	−1.07193	−0.0447027
$576$	0	0
$577$	34.0176	1.41617	0.708086	−	0.706127i	$-0.249559\pi$
$577$	34.0176	1.41617	0.708086	+	0.706127i	$0.249559\pi$
$578$	0	0
$579$	44.5550	1.85164
$580$	0	0
$581$	−56.4167	−2.34056
$582$	0	0
$583$	23.0199	0.953387
$584$	0	0
$585$	−1.72153	−0.0711765
$586$	0	0
$587$	−20.2066	−0.834015	−0.417008	−	0.908903i	$-0.636921\pi$
$587$	−20.2066	−0.834015	−0.417008	+	0.908903i	$0.636921\pi$
$588$	0	0
$589$	−15.4224	−0.635469
$590$	0	0
$591$	−59.3235	−2.44024
$592$	0	0
$593$	−27.6602	−1.13587	−0.567934	−	0.823074i	$-0.692257\pi$
$593$	−27.6602	−1.13587	−0.567934	+	0.823074i	$0.692257\pi$
$594$	0	0
$595$	−6.10200	−0.250158
$596$	0	0
$597$	43.1331	1.76532
$598$	0	0
$599$	11.2617	0.460141	0.230070	−	0.973174i	$-0.426104\pi$
$599$	11.2617	0.460141	0.230070	+	0.973174i	$0.426104\pi$
$600$	0	0
$601$	−3.94629	−0.160972	−0.0804862	−	0.996756i	$-0.525647\pi$
$601$	−3.94629	−0.160972	−0.0804862	+	0.996756i	$0.525647\pi$
$602$	0	0
$603$	24.4854	0.997121
$604$	0	0
$605$	−14.0762	−0.572278
$606$	0	0
$607$	35.3920	1.43652	0.718258	−	0.695777i	$-0.244940\pi$
$607$	35.3920	1.43652	0.718258	+	0.695777i	$0.244940\pi$
$608$	0	0
$609$	−62.7264	−2.54180
$610$	0	0
$611$	−0.840384	−0.0339983
$612$	0	0
$613$	23.7537	0.959401	0.479701	−	0.877432i	$-0.340745\pi$
$613$	23.7537	0.959401	0.479701	+	0.877432i	$0.340745\pi$
$614$	0	0
$615$	−18.6524	−0.752138
$616$	0	0
$617$	−5.90000	−0.237525	−0.118762	−	0.992923i	$-0.537893\pi$
$617$	−5.90000	−0.237525	−0.118762	+	0.992923i	$0.537893\pi$
$618$	0	0
$619$	20.7014	0.832059	0.416029	−	0.909351i	$-0.363421\pi$
$619$	20.7014	0.832059	0.416029	+	0.909351i	$0.363421\pi$
$620$	0	0
$621$	−9.19415	−0.368949
$622$	0	0
$623$	−21.7920	−0.873080
$624$	0	0
$625$	−27.5661	−1.10265
$626$	0	0
$627$	11.0452	0.441101
$628$	0	0
$629$	−0.780235	−0.0311100
$630$	0	0
$631$	−5.36235	−0.213472	−0.106736	−	0.994287i	$-0.534040\pi$
$631$	−5.36235	−0.213472	−0.106736	+	0.994287i	$0.534040\pi$
$632$	0	0
$633$	−16.9004	−0.671729
$634$	0	0
$635$	−34.0205	−1.35006
$636$	0	0
$637$	−1.23173	−0.0488030
$638$	0	0
$639$	29.4512	1.16507
$640$	0	0
$641$	−26.7918	−1.05821	−0.529107	−	0.848555i	$-0.677473\pi$
$641$	−26.7918	−1.05821	−0.529107	+	0.848555i	$0.677473\pi$
$642$	0	0
$643$	−8.07707	−0.318529	−0.159264	−	0.987236i	$-0.550912\pi$
$643$	−8.07707	−0.318529	−0.159264	+	0.987236i	$0.550912\pi$
$644$	0	0
$645$	71.6445	2.82100
$646$	0	0
$647$	13.4690	0.529523	0.264761	−	0.964314i	$-0.414707\pi$
$647$	13.4690	0.529523	0.264761	+	0.964314i	$0.414707\pi$
$648$	0	0
$649$	−22.8935	−0.898647
$650$	0	0
$651$	94.8040	3.71566
$652$	0	0
$653$	−15.6835	−0.613744	−0.306872	−	0.951751i	$-0.599282\pi$
$653$	−15.6835	−0.613744	−0.306872	+	0.951751i	$0.599282\pi$
$654$	0	0
$655$	0.482216	0.0188417
$656$	0	0
$657$	−28.6654	−1.11835
$658$	0	0
$659$	13.5401	0.527447	0.263724	−	0.964598i	$-0.415049\pi$
$659$	13.5401	0.527447	0.263724	+	0.964598i	$0.415049\pi$
$660$	0	0
$661$	50.0074	1.94506	0.972531	−	0.232775i	$-0.0747806\pi$
$661$	50.0074	1.94506	0.972531	+	0.232775i	$0.0747806\pi$
$662$	0	0
$663$	0.282577	0.0109744
$664$	0	0
$665$	−16.1778	−0.627349
$666$	0	0
$667$	10.6845	0.413707
$668$	0	0
$669$	−66.3466	−2.56511
$670$	0	0
$671$	−21.6497	−0.835778
$672$	0	0
$673$	16.4521	0.634180	0.317090	−	0.948395i	$-0.397294\pi$
$673$	16.4521	0.634180	0.317090	+	0.948395i	$0.397294\pi$
$674$	0	0
$675$	2.89192	0.111310
$676$	0	0
$677$	42.6665	1.63981	0.819903	−	0.572503i	$-0.194027\pi$
$677$	42.6665	1.63981	0.819903	+	0.572503i	$0.194027\pi$
$678$	0	0
$679$	36.6360	1.40596
$680$	0	0
$681$	8.12801	0.311466
$682$	0	0
$683$	43.3205	1.65761	0.828807	−	0.559534i	$-0.189020\pi$
$683$	43.3205	1.65761	0.828807	+	0.559534i	$0.189020\pi$
$684$	0	0
$685$	−23.4773	−0.897023
$686$	0	0
$687$	−65.4636	−2.49759
$688$	0	0
$689$	1.56141	0.0594850
$690$	0	0
$691$	−19.8389	−0.754707	−0.377354	−	0.926069i	$-0.623166\pi$
$691$	−19.8389	−0.754707	−0.377354	+	0.926069i	$0.623166\pi$
$692$	0	0
$693$	−41.7321	−1.58527
$694$	0	0
$695$	14.4547	0.548297
$696$	0	0
$697$	1.88183	0.0712795
$698$	0	0
$699$	77.3154	2.92434
$700$	0	0
$701$	−0.479263	−0.0181015	−0.00905077	−	0.999959i	$-0.502881\pi$
$701$	−0.479263	−0.0181015	−0.00905077	+	0.999959i	$0.502881\pi$
$702$	0	0
$703$	−2.06859	−0.0780182
$704$	0	0
$705$	36.3713	1.36982
$706$	0	0
$707$	73.5591	2.76647
$708$	0	0
$709$	37.2475	1.39886	0.699429	−	0.714702i	$-0.253438\pi$
$709$	37.2475	1.39886	0.699429	+	0.714702i	$0.253438\pi$
$710$	0	0
$711$	−7.60071	−0.285049
$712$	0	0
$713$	−16.1485	−0.604767
$714$	0	0
$715$	0.807812	0.0302104
$716$	0	0
$717$	−28.6415	−1.06964
$718$	0	0
$719$	20.0777	0.748772	0.374386	−	0.927273i	$-0.377853\pi$
$719$	20.0777	0.748772	0.374386	+	0.927273i	$0.377853\pi$
$720$	0	0
$721$	−44.1538	−1.64437
$722$	0	0
$723$	35.2523	1.31105
$724$	0	0
$725$	−3.36071	−0.124814
$726$	0	0
$727$	−36.5795	−1.35666	−0.678329	−	0.734758i	$-0.737296\pi$
$727$	−36.5795	−1.35666	−0.678329	+	0.734758i	$0.737296\pi$
$728$	0	0
$729$	−43.8839	−1.62533
$730$	0	0
$731$	−7.22817	−0.267344
$732$	0	0
$733$	18.6190	0.687710	0.343855	−	0.939023i	$-0.388267\pi$
$733$	18.6190	0.687710	0.343855	+	0.939023i	$0.388267\pi$
$734$	0	0
$735$	53.3086	1.96632
$736$	0	0
$737$	−11.4895	−0.423222
$738$	0	0
$739$	−28.8894	−1.06271	−0.531356	−	0.847149i	$-0.678317\pi$
$739$	−28.8894	−1.06271	−0.531356	+	0.847149i	$0.678317\pi$
$740$	0	0
$741$	0.749179	0.0275218
$742$	0	0
$743$	11.5934	0.425320	0.212660	−	0.977126i	$-0.431787\pi$
$743$	11.5934	0.425320	0.212660	+	0.977126i	$0.431787\pi$
$744$	0	0
$745$	−31.0247	−1.13666
$746$	0	0
$747$	−69.4987	−2.54282
$748$	0	0
$749$	55.1909	2.01663
$750$	0	0
$751$	−54.6438	−1.99398	−0.996990	−	0.0775296i	$-0.975297\pi$
$751$	−54.6438	−1.99398	−0.996990	+	0.0775296i	$0.975297\pi$
$752$	0	0
$753$	76.6362	2.79278
$754$	0	0
$755$	−15.5383	−0.565497
$756$	0	0
$757$	37.1233	1.34927	0.674634	−	0.738153i	$-0.264302\pi$
$757$	37.1233	1.34927	0.674634	+	0.738153i	$0.264302\pi$
$758$	0	0
$759$	11.5652	0.419790
$760$	0	0
$761$	38.3748	1.39109	0.695543	−	0.718485i	$-0.255164\pi$
$761$	38.3748	1.39109	0.695543	+	0.718485i	$0.255164\pi$
$762$	0	0
$763$	51.4669	1.86323
$764$	0	0
$765$	−7.51695	−0.271776
$766$	0	0
$767$	−1.55283	−0.0560696
$768$	0	0
$769$	30.1412	1.08692	0.543460	−	0.839435i	$-0.317114\pi$
$769$	30.1412	1.08692	0.543460	+	0.839435i	$0.317114\pi$
$770$	0	0
$771$	4.82691	0.173837
$772$	0	0
$773$	14.7449	0.530336	0.265168	−	0.964202i	$-0.414573\pi$
$773$	14.7449	0.530336	0.265168	+	0.964202i	$0.414573\pi$
$774$	0	0
$775$	5.07934	0.182455
$776$	0	0
$777$	12.7159	0.456182
$778$	0	0
$779$	4.98918	0.178756
$780$	0	0
$781$	−13.8197	−0.494508
$782$	0	0
$783$	−28.8253	−1.03013
$784$	0	0
$785$	10.5330	0.375940
$786$	0	0
$787$	44.0842	1.57143	0.785715	−	0.618588i	$-0.212295\pi$
$787$	44.0842	1.57143	0.785715	+	0.618588i	$0.212295\pi$
$788$	0	0
$789$	−62.5548	−2.22701
$790$	0	0
$791$	−68.0306	−2.41889
$792$	0	0
$793$	−1.46847	−0.0521470
$794$	0	0
$795$	−67.5769	−2.39671
$796$	0	0
$797$	38.7280	1.37181	0.685907	−	0.727689i	$-0.259405\pi$
$797$	38.7280	1.37181	0.685907	+	0.727689i	$0.259405\pi$
$798$	0	0
$799$	−3.66948	−0.129817
$800$	0	0
$801$	−26.8452	−0.948530
$802$	0	0
$803$	13.4510	0.474675
$804$	0	0
$805$	−16.9395	−0.597040
$806$	0	0
$807$	−2.79016	−0.0982183
$808$	0	0
$809$	12.7907	0.449697	0.224848	−	0.974394i	$-0.427811\pi$
$809$	12.7907	0.449697	0.224848	+	0.974394i	$0.427811\pi$
$810$	0	0
$811$	6.26400	0.219959	0.109979	−	0.993934i	$-0.464921\pi$
$811$	6.26400	0.219959	0.109979	+	0.993934i	$0.464921\pi$
$812$	0	0
$813$	14.1061	0.494724
$814$	0	0
$815$	−20.2472	−0.709230
$816$	0	0
$817$	−19.1636	−0.670449
$818$	0	0
$819$	−2.83063	−0.0989103
$820$	0	0
$821$	−31.8437	−1.11135	−0.555676	−	0.831399i	$-0.687540\pi$
$821$	−31.8437	−1.11135	−0.555676	+	0.831399i	$0.687540\pi$
$822$	0	0
$823$	−23.6039	−0.822781	−0.411391	−	0.911459i	$-0.634957\pi$
$823$	−23.6039	−0.822781	−0.411391	+	0.911459i	$0.634957\pi$
$824$	0	0
$825$	−3.63770	−0.126649
$826$	0	0
$827$	23.1364	0.804532	0.402266	−	0.915523i	$-0.368223\pi$
$827$	23.1364	0.804532	0.402266	+	0.915523i	$0.368223\pi$
$828$	0	0
$829$	5.12273	0.177920	0.0889598	−	0.996035i	$-0.471646\pi$
$829$	5.12273	0.177920	0.0889598	+	0.996035i	$0.471646\pi$
$830$	0	0
$831$	−47.6937	−1.65448
$832$	0	0
$833$	−5.37828	−0.186346
$834$	0	0
$835$	−3.42591	−0.118558
$836$	0	0
$837$	43.5663	1.50587
$838$	0	0
$839$	5.94481	0.205238	0.102619	−	0.994721i	$-0.467278\pi$
$839$	5.94481	0.205238	0.102619	+	0.994721i	$0.467278\pi$
$840$	0	0
$841$	4.49799	0.155103
$842$	0	0
$843$	68.1653	2.34774
$844$	0	0
$845$	−30.6557	−1.05459
$846$	0	0
$847$	−23.1448	−0.795265
$848$	0	0
$849$	−35.1121	−1.20504
$850$	0	0
$851$	−2.16598	−0.0742488
$852$	0	0
$853$	−38.0526	−1.30290	−0.651448	−	0.758693i	$-0.725838\pi$
$853$	−38.0526	−1.30290	−0.651448	+	0.758693i	$0.725838\pi$
$854$	0	0
$855$	−19.9292	−0.681564
$856$	0	0
$857$	−23.5758	−0.805335	−0.402667	−	0.915346i	$-0.631917\pi$
$857$	−23.5758	−0.805335	−0.402667	+	0.915346i	$0.631917\pi$
$858$	0	0
$859$	9.42527	0.321586	0.160793	−	0.986988i	$-0.448595\pi$
$859$	9.42527	0.321586	0.160793	+	0.986988i	$0.448595\pi$
$860$	0	0
$861$	−30.6693	−1.04521
$862$	0	0
$863$	28.1396	0.957884	0.478942	−	0.877846i	$-0.341020\pi$
$863$	28.1396	0.957884	0.478942	+	0.877846i	$0.341020\pi$
$864$	0	0
$865$	−32.8288	−1.11621
$866$	0	0
$867$	−46.1989	−1.56900
$868$	0	0
$869$	3.56656	0.120987
$870$	0	0
$871$	−0.779320	−0.0264063
$872$	0	0
$873$	45.1313	1.52746
$874$	0	0
$875$	−40.5520	−1.37091
$876$	0	0
$877$	30.7237	1.03747	0.518733	−	0.854936i	$-0.326404\pi$
$877$	30.7237	1.03747	0.518733	+	0.854936i	$0.326404\pi$
$878$	0	0
$879$	−14.0606	−0.474251
$880$	0	0
$881$	−26.6012	−0.896217	−0.448108	−	0.893979i	$-0.647902\pi$
$881$	−26.6012	−0.896217	−0.448108	+	0.893979i	$0.647902\pi$
$882$	0	0
$883$	39.0409	1.31383	0.656915	−	0.753965i	$-0.271861\pi$
$883$	39.0409	1.31383	0.656915	+	0.753965i	$0.271861\pi$
$884$	0	0
$885$	67.2058	2.25910
$886$	0	0
$887$	39.9648	1.34189	0.670943	−	0.741509i	$-0.265889\pi$
$887$	39.9648	1.34189	0.670943	+	0.741509i	$0.265889\pi$
$888$	0	0
$889$	−55.9383	−1.87611
$890$	0	0
$891$	1.03023	0.0345141
$892$	0	0
$893$	−9.72866	−0.325557
$894$	0	0
$895$	4.57670	0.152982
$896$	0	0
$897$	0.784452	0.0261921
$898$	0	0
$899$	−50.6285	−1.68856
$900$	0	0
$901$	6.81780	0.227134
$902$	0	0
$903$	117.802	3.92019
$904$	0	0
$905$	−51.6374	−1.71649
$906$	0	0
$907$	41.5798	1.38064	0.690318	−	0.723506i	$-0.257471\pi$
$907$	41.5798	1.38064	0.690318	+	0.723506i	$0.257471\pi$
$908$	0	0
$909$	90.6162	3.00555
$910$	0	0
$911$	0.891781	0.0295460	0.0147730	−	0.999891i	$-0.495297\pi$
$911$	0.891781	0.0295460	0.0147730	+	0.999891i	$0.495297\pi$
$912$	0	0
$913$	32.6116	1.07929
$914$	0	0
$915$	63.5546	2.10105
$916$	0	0
$917$	0.792886	0.0261834
$918$	0	0
$919$	9.62868	0.317621	0.158810	−	0.987309i	$-0.449234\pi$
$919$	9.62868	0.317621	0.158810	+	0.987309i	$0.449234\pi$
$920$	0	0
$921$	54.8128	1.80614
$922$	0	0
$923$	−0.937373	−0.0308540
$924$	0	0
$925$	0.681286	0.0224005
$926$	0	0
$927$	−54.3923	−1.78648
$928$	0	0
$929$	0.396563	0.0130108	0.00650540	−	0.999979i	$-0.497929\pi$
$929$	0.396563	0.0130108	0.00650540	+	0.999979i	$0.497929\pi$
$930$	0	0
$931$	−14.2591	−0.467323
$932$	0	0
$933$	−74.6378	−2.44353
$934$	0	0
$935$	3.52726	0.115354
$936$	0	0
$937$	15.4786	0.505664	0.252832	−	0.967510i	$-0.418638\pi$
$937$	15.4786	0.505664	0.252832	+	0.967510i	$0.418638\pi$
$938$	0	0
$939$	78.7804	2.57090
$940$	0	0
$941$	−29.4435	−0.959832	−0.479916	−	0.877315i	$-0.659333\pi$
$941$	−29.4435	−0.959832	−0.479916	+	0.877315i	$0.659333\pi$
$942$	0	0
$943$	5.22408	0.170120
$944$	0	0
$945$	45.7003	1.48663
$946$	0	0
$947$	−1.41962	−0.0461316	−0.0230658	−	0.999734i	$-0.507343\pi$
$947$	−1.41962	−0.0461316	−0.0230658	+	0.999734i	$0.507343\pi$
$948$	0	0
$949$	0.912363	0.0296166
$950$	0	0
$951$	−97.4719	−3.16074
$952$	0	0
$953$	21.4840	0.695936	0.347968	−	0.937506i	$-0.386872\pi$
$953$	21.4840	0.695936	0.347968	+	0.937506i	$0.386872\pi$
$954$	0	0
$955$	−48.6673	−1.57484
$956$	0	0
$957$	36.2590	1.17209
$958$	0	0
$959$	−38.6027	−1.24655
$960$	0	0
$961$	45.5194	1.46837
$962$	0	0
$963$	67.9888	2.19091
$964$	0	0
$965$	37.7234	1.21436
$966$	0	0
$967$	−40.8566	−1.31386	−0.656929	−	0.753952i	$-0.728145\pi$
$967$	−40.8566	−1.31386	−0.656929	+	0.753952i	$0.728145\pi$
$968$	0	0
$969$	3.27124	0.105087
$970$	0	0
$971$	−32.3471	−1.03807	−0.519034	−	0.854754i	$-0.673708\pi$
$971$	−32.3471	−1.03807	−0.519034	+	0.854754i	$0.673708\pi$
$972$	0	0
$973$	23.7672	0.761941
$974$	0	0
$975$	−0.246741	−0.00790203
$976$	0	0
$977$	−33.6227	−1.07569	−0.537843	−	0.843045i	$-0.680761\pi$
$977$	−33.6227	−1.07569	−0.537843	+	0.843045i	$0.680761\pi$
$978$	0	0
$979$	12.5969	0.402598
$980$	0	0
$981$	63.4012	2.02424
$982$	0	0
$983$	−47.1311	−1.50325	−0.751624	−	0.659592i	$-0.770729\pi$
$983$	−47.1311	−1.50325	−0.751624	+	0.659592i	$0.770729\pi$
$984$	0	0
$985$	−50.2274	−1.60038
$986$	0	0
$987$	59.8037	1.90357
$988$	0	0
$989$	−20.0658	−0.638057
$990$	0	0
$991$	−59.6574	−1.89508	−0.947541	−	0.319635i	$-0.896440\pi$
$991$	−59.6574	−1.89508	−0.947541	+	0.319635i	$0.896440\pi$
$992$	0	0
$993$	19.3885	0.615276
$994$	0	0
$995$	36.5195	1.15774
$996$	0	0
$997$	−51.0492	−1.61674	−0.808371	−	0.588673i	$-0.799651\pi$
$997$	−51.0492	−1.61674	−0.808371	+	0.588673i	$0.799651\pi$
$998$	0	0
$999$	5.84350	0.184880

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4304.2.a.i.1.6		7
4.3	odd	2		538.2.a.d.1.2	✓	7
12.11	even	2		4842.2.a.o.1.2		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
538.2.a.d.1.2	✓	7	4.3	odd	2
4304.2.a.i.1.6		7	1.1	even	1	trivial
4842.2.a.o.1.2		7	12.11	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:	\( N \)	\(=\)	\( 4304 = 2^{4} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4304.a (trivial)

\(f(q)\)	\(=\)	\(q+2.79016 q^{3} +2.36234 q^{5} +3.88429 q^{7} +4.78499 q^{9} -2.24531 q^{11} -0.152297 q^{13} +6.59131 q^{15} -0.664994 q^{17} -1.76305 q^{19} +10.8378 q^{21} -1.84606 q^{23} +0.580659 q^{25} +4.98041 q^{27} -5.78775 q^{29} +8.74754 q^{31} -6.26478 q^{33} +9.17602 q^{35} +1.17330 q^{37} -0.424932 q^{39} -2.82985 q^{41} +10.8695 q^{43} +11.3038 q^{45} +5.51807 q^{47} +8.08771 q^{49} -1.85544 q^{51} -10.2524 q^{53} -5.30420 q^{55} -4.91920 q^{57} +10.1961 q^{59} +9.64218 q^{61} +18.5863 q^{63} -0.359777 q^{65} +5.11712 q^{67} -5.15081 q^{69} +6.15492 q^{71} -5.99070 q^{73} +1.62013 q^{75} -8.72145 q^{77} -1.58845 q^{79} -0.458837 q^{81} -14.5243 q^{83} -1.57094 q^{85} -16.1487 q^{87} -5.61030 q^{89} -0.591565 q^{91} +24.4070 q^{93} -4.16494 q^{95} +9.43184 q^{97} -10.7438 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{3} - 6 q^{5} + 3 q^{7} + 7 q^{9} + 12 q^{11} + 3 q^{13} + 6 q^{15} - 8 q^{17} + 7 q^{19} - 5 q^{21} + 22 q^{23} + 9 q^{25} + 22 q^{27} - 7 q^{29} + 3 q^{31} - 19 q^{33} + 25 q^{35} - 13 q^{37}+ \cdots + 9 q^{99}+O(q^{100}) \) 7 * q + 4 * q^3 - 6 * q^5 + 3 * q^7 + 7 * q^9 + 12 * q^11 + 3 * q^13 + 6 * q^15 - 8 * q^17 + 7 * q^19 - 5 * q^21 + 22 * q^23 + 9 * q^25 + 22 * q^27 - 7 * q^29 + 3 * q^31 - 19 * q^33 + 25 * q^35 - 13 * q^37 - q^39 - 11 * q^41 + 14 * q^43 - 17 * q^45 + 23 * q^47 + 6 * q^49 + 21 * q^51 - 21 * q^53 - 15 * q^55 - 14 * q^57 + 20 * q^59 + 13 * q^61 + 12 * q^63 - 29 * q^65 + 30 * q^67 + 13 * q^69 + 23 * q^71 + 9 * q^73 - 3 * q^75 + 8 * q^77 - 11 * q^79 + 43 * q^81 + 3 * q^83 + 21 * q^85 - 28 * q^87 + 16 * q^89 + 10 * q^91 - 13 * q^93 + 21 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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