LMFDB - Embedded newform 7248.2.a.bm.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7248,2,Mod(1,7248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7248, 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("7248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7248 = 2^{4} \cdot 3 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7248.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:	$57.8755713850$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 27x^{8} + 45x^{7} + 258x^{6} - 289x^{5} - 1133x^{4} + 510x^{3} + 2070x^{2} + 341x - 500 $ x^10 - 2x^9 - 27x^8 + 45x^7 + 258x^6 - 289x^5 - 1133x^4 + 510x^3 + 2070x^2 + 341*x - 500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3624)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.78540$ of defining polynomial
Character	$\chi$	$=$	7248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.78540 q^{5} -2.60389 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.78540 q^{5} -2.60389 q^{7} +1.00000 q^{9} -2.13465 q^{11} -1.66555 q^{13} -2.78540 q^{15} +0.462851 q^{17} +1.45443 q^{19} +2.60389 q^{21} +1.82714 q^{23} +2.75846 q^{25} -1.00000 q^{27} -2.76975 q^{29} +1.47639 q^{31} +2.13465 q^{33} -7.25287 q^{35} +11.2371 q^{37} +1.66555 q^{39} +4.20221 q^{41} -10.5552 q^{43} +2.78540 q^{45} -9.69535 q^{47} -0.219775 q^{49} -0.462851 q^{51} +1.64863 q^{53} -5.94586 q^{55} -1.45443 q^{57} +11.6565 q^{59} -2.46099 q^{61} -2.60389 q^{63} -4.63923 q^{65} -3.36755 q^{67} -1.82714 q^{69} -0.392399 q^{71} -1.20872 q^{73} -2.75846 q^{75} +5.55839 q^{77} +13.1740 q^{79} +1.00000 q^{81} +7.14774 q^{83} +1.28923 q^{85} +2.76975 q^{87} +3.77596 q^{89} +4.33691 q^{91} -1.47639 q^{93} +4.05116 q^{95} -5.41288 q^{97} -2.13465 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9	$ 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9} - 7 q^{11} + 6 q^{13} - 2 q^{15} + 7 q^{17} + 8 q^{21} - 25 q^{23} + 8 q^{25} - 10 q^{27} + 12 q^{29} - 11 q^{31} + 7 q^{33} - 9 q^{35} - 3 q^{37} - 6 q^{39} + 12 q^{41} + 2 q^{45} - 31 q^{47} + 14 q^{49} - 7 q^{51} + q^{53} - 9 q^{55} - 19 q^{59} + 24 q^{61} - 8 q^{63} + 20 q^{65} + q^{67} + 25 q^{69} - 34 q^{71} - 18 q^{73} - 8 q^{75} + 27 q^{77} - 25 q^{79} + 10 q^{81} - 14 q^{83} - 3 q^{85} - 12 q^{87} + 20 q^{89} + 12 q^{91} + 11 q^{93} - 48 q^{95} - 15 q^{97} - 7 q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9 - 7 * q^11 + 6 * q^13 - 2 * q^15 + 7 * q^17 + 8 * q^21 - 25 * q^23 + 8 * q^25 - 10 * q^27 + 12 * q^29 - 11 * q^31 + 7 * q^33 - 9 * q^35 - 3 * q^37 - 6 * q^39 + 12 * q^41 + 2 * q^45 - 31 * q^47 + 14 * q^49 - 7 * q^51 + q^53 - 9 * q^55 - 19 * q^59 + 24 * q^61 - 8 * q^63 + 20 * q^65 + q^67 + 25 * q^69 - 34 * q^71 - 18 * q^73 - 8 * q^75 + 27 * q^77 - 25 * q^79 + 10 * q^81 - 14 * q^83 - 3 * q^85 - 12 * q^87 + 20 * q^89 + 12 * q^91 + 11 * q^93 - 48 * q^95 - 15 * q^97 - 7 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.78540	1.24567	0.622835	−	0.782353i	$-0.285981\pi$
$5$	2.78540	1.24567	0.622835	+	0.782353i	$0.285981\pi$
$6$	0	0
$7$	−2.60389	−0.984177	−0.492088	−	0.870545i	$-0.663766\pi$
$7$	−2.60389	−0.984177	−0.492088	+	0.870545i	$0.663766\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.13465	−0.643621	−0.321811	−	0.946804i	$-0.604291\pi$
$11$	−2.13465	−0.643621	−0.321811	+	0.946804i	$0.604291\pi$
$12$	0	0
$13$	−1.66555	−0.461941	−0.230970	−	0.972961i	$-0.574190\pi$
$13$	−1.66555	−0.461941	−0.230970	+	0.972961i	$0.574190\pi$
$14$	0	0
$15$	−2.78540	−0.719188
$16$	0	0
$17$	0.462851	0.112258	0.0561289	−	0.998424i	$-0.482124\pi$
$17$	0.462851	0.112258	0.0561289	+	0.998424i	$0.482124\pi$
$18$	0	0
$19$	1.45443	0.333668	0.166834	−	0.985985i	$-0.446646\pi$
$19$	1.45443	0.333668	0.166834	+	0.985985i	$0.446646\pi$
$20$	0	0
$21$	2.60389	0.568215
$22$	0	0
$23$	1.82714	0.380985	0.190493	−	0.981689i	$-0.438991\pi$
$23$	1.82714	0.380985	0.190493	+	0.981689i	$0.438991\pi$
$24$	0	0
$25$	2.75846	0.551692
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.76975	−0.514329	−0.257165	−	0.966368i	$-0.582788\pi$
$29$	−2.76975	−0.514329	−0.257165	+	0.966368i	$0.582788\pi$
$30$	0	0
$31$	1.47639	0.265167	0.132583	−	0.991172i	$-0.457673\pi$
$31$	1.47639	0.265167	0.132583	+	0.991172i	$0.457673\pi$
$32$	0	0
$33$	2.13465	0.371595
$34$	0	0
$35$	−7.25287	−1.22596
$36$	0	0
$37$	11.2371	1.84737	0.923683	−	0.383157i	$-0.125163\pi$
$37$	11.2371	1.84737	0.923683	+	0.383157i	$0.125163\pi$
$38$	0	0
$39$	1.66555	0.266702
$40$	0	0
$41$	4.20221	0.656276	0.328138	−	0.944630i	$-0.393579\pi$
$41$	4.20221	0.656276	0.328138	+	0.944630i	$0.393579\pi$
$42$	0	0
$43$	−10.5552	−1.60965	−0.804827	−	0.593510i	$-0.797742\pi$
$43$	−10.5552	−1.60965	−0.804827	+	0.593510i	$0.797742\pi$
$44$	0	0
$45$	2.78540	0.415223
$46$	0	0
$47$	−9.69535	−1.41421	−0.707106	−	0.707107i	$-0.750000\pi$
$47$	−9.69535	−1.41421	−0.707106	+	0.707107i	$0.750000\pi$
$48$	0	0
$49$	−0.219775	−0.0313964
$50$	0	0
$51$	−0.462851	−0.0648121
$52$	0	0
$53$	1.64863	0.226457	0.113228	−	0.993569i	$-0.463881\pi$
$53$	1.64863	0.226457	0.113228	+	0.993569i	$0.463881\pi$
$54$	0	0
$55$	−5.94586	−0.801739
$56$	0	0
$57$	−1.45443	−0.192643
$58$	0	0
$59$	11.6565	1.51754	0.758771	−	0.651358i	$-0.225800\pi$
$59$	11.6565	1.51754	0.758771	+	0.651358i	$0.225800\pi$
$60$	0	0
$61$	−2.46099	−0.315097	−0.157548	−	0.987511i	$-0.550359\pi$
$61$	−2.46099	−0.315097	−0.157548	+	0.987511i	$0.550359\pi$
$62$	0	0
$63$	−2.60389	−0.328059
$64$	0	0
$65$	−4.63923	−0.575426
$66$	0	0
$67$	−3.36755	−0.411411	−0.205706	−	0.978614i	$-0.565949\pi$
$67$	−3.36755	−0.411411	−0.205706	+	0.978614i	$0.565949\pi$
$68$	0	0
$69$	−1.82714	−0.219962
$70$	0	0
$71$	−0.392399	−0.0465692	−0.0232846	−	0.999729i	$-0.507412\pi$
$71$	−0.392399	−0.0465692	−0.0232846	+	0.999729i	$0.507412\pi$
$72$	0	0
$73$	−1.20872	−0.141470	−0.0707352	−	0.997495i	$-0.522535\pi$
$73$	−1.20872	−0.141470	−0.0707352	+	0.997495i	$0.522535\pi$
$74$	0	0
$75$	−2.75846	−0.318520
$76$	0	0
$77$	5.55839	0.633437
$78$	0	0
$79$	13.1740	1.48219	0.741094	−	0.671401i	$-0.234307\pi$
$79$	13.1740	1.48219	0.741094	+	0.671401i	$0.234307\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.14774	0.784566	0.392283	−	0.919844i	$-0.371685\pi$
$83$	7.14774	0.784566	0.392283	+	0.919844i	$0.371685\pi$
$84$	0	0
$85$	1.28923	0.139836
$86$	0	0
$87$	2.76975	0.296948
$88$	0	0
$89$	3.77596	0.400251	0.200125	−	0.979770i	$-0.435865\pi$
$89$	3.77596	0.400251	0.200125	+	0.979770i	$0.435865\pi$
$90$	0	0
$91$	4.33691	0.454631
$92$	0	0
$93$	−1.47639	−0.153094
$94$	0	0
$95$	4.05116	0.415640
$96$	0	0
$97$	−5.41288	−0.549595	−0.274798	−	0.961502i	$-0.588611\pi$
$97$	−5.41288	−0.549595	−0.274798	+	0.961502i	$0.588611\pi$
$98$	0	0
$99$	−2.13465	−0.214540
$100$	0	0
$101$	−3.16213	−0.314644	−0.157322	−	0.987547i	$-0.550286\pi$
$101$	−3.16213	−0.314644	−0.157322	+	0.987547i	$0.550286\pi$
$102$	0	0
$103$	−13.5144	−1.33162	−0.665809	−	0.746122i	$-0.731913\pi$
$103$	−13.5144	−1.33162	−0.665809	+	0.746122i	$0.731913\pi$
$104$	0	0
$105$	7.25287	0.707808
$106$	0	0
$107$	−9.45590	−0.914137	−0.457068	−	0.889432i	$-0.651101\pi$
$107$	−9.45590	−0.914137	−0.457068	+	0.889432i	$0.651101\pi$
$108$	0	0
$109$	−15.6724	−1.50114	−0.750572	−	0.660788i	$-0.770222\pi$
$109$	−15.6724	−1.50114	−0.750572	+	0.660788i	$0.770222\pi$
$110$	0	0
$111$	−11.2371	−1.06658
$112$	0	0
$113$	−6.14101	−0.577698	−0.288849	−	0.957375i	$-0.593272\pi$
$113$	−6.14101	−0.577698	−0.288849	+	0.957375i	$0.593272\pi$
$114$	0	0
$115$	5.08932	0.474582
$116$	0	0
$117$	−1.66555	−0.153980
$118$	0	0
$119$	−1.20521	−0.110482
$120$	0	0
$121$	−6.44327	−0.585752
$122$	0	0
$123$	−4.20221	−0.378901
$124$	0	0
$125$	−6.24358	−0.558443
$126$	0	0
$127$	−15.6776	−1.39116	−0.695582	−	0.718447i	$-0.744853\pi$
$127$	−15.6776	−1.39116	−0.695582	+	0.718447i	$0.744853\pi$
$128$	0	0
$129$	10.5552	0.929334
$130$	0	0
$131$	−18.5483	−1.62057	−0.810285	−	0.586036i	$-0.800688\pi$
$131$	−18.5483	−1.62057	−0.810285	+	0.586036i	$0.800688\pi$
$132$	0	0
$133$	−3.78716	−0.328388
$134$	0	0
$135$	−2.78540	−0.239729
$136$	0	0
$137$	−1.60318	−0.136969	−0.0684843	−	0.997652i	$-0.521816\pi$
$137$	−1.60318	−0.136969	−0.0684843	+	0.997652i	$0.521816\pi$
$138$	0	0
$139$	3.55874	0.301849	0.150924	−	0.988545i	$-0.451775\pi$
$139$	3.55874	0.301849	0.150924	+	0.988545i	$0.451775\pi$
$140$	0	0
$141$	9.69535	0.816496
$142$	0	0
$143$	3.55537	0.297315
$144$	0	0
$145$	−7.71486	−0.640684
$146$	0	0
$147$	0.219775	0.0181267
$148$	0	0
$149$	9.21234	0.754704	0.377352	−	0.926070i	$-0.376835\pi$
$149$	9.21234	0.754704	0.377352	+	0.926070i	$0.376835\pi$
$150$	0	0
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	0	0
$153$	0.462851	0.0374193
$154$	0	0
$155$	4.11233	0.330310
$156$	0	0
$157$	−13.2429	−1.05690	−0.528450	−	0.848964i	$-0.677227\pi$
$157$	−13.2429	−1.05690	−0.528450	+	0.848964i	$0.677227\pi$
$158$	0	0
$159$	−1.64863	−0.130745
$160$	0	0
$161$	−4.75767	−0.374957
$162$	0	0
$163$	−16.3838	−1.28328	−0.641639	−	0.767006i	$-0.721745\pi$
$163$	−16.3838	−1.28328	−0.641639	+	0.767006i	$0.721745\pi$
$164$	0	0
$165$	5.94586	0.462884
$166$	0	0
$167$	0.327406	0.0253354	0.0126677	−	0.999920i	$-0.495968\pi$
$167$	0.327406	0.0253354	0.0126677	+	0.999920i	$0.495968\pi$
$168$	0	0
$169$	−10.2259	−0.786611
$170$	0	0
$171$	1.45443	0.111223
$172$	0	0
$173$	−18.6317	−1.41654	−0.708270	−	0.705942i	$-0.750524\pi$
$173$	−18.6317	−1.41654	−0.708270	+	0.705942i	$0.750524\pi$
$174$	0	0
$175$	−7.18272	−0.542963
$176$	0	0
$177$	−11.6565	−0.876153
$178$	0	0
$179$	6.85648	0.512478	0.256239	−	0.966614i	$-0.417517\pi$
$179$	6.85648	0.512478	0.256239	+	0.966614i	$0.417517\pi$
$180$	0	0
$181$	−14.2712	−1.06077	−0.530385	−	0.847757i	$-0.677953\pi$
$181$	−14.2712	−1.06077	−0.530385	+	0.847757i	$0.677953\pi$
$182$	0	0
$183$	2.46099	0.181921
$184$	0	0
$185$	31.2998	2.30121
$186$	0	0
$187$	−0.988025	−0.0722515
$188$	0	0
$189$	2.60389	0.189405
$190$	0	0
$191$	9.68421	0.700725	0.350362	−	0.936614i	$-0.386058\pi$
$191$	9.68421	0.700725	0.350362	+	0.936614i	$0.386058\pi$
$192$	0	0
$193$	0.506013	0.0364236	0.0182118	−	0.999834i	$-0.494203\pi$
$193$	0.506013	0.0364236	0.0182118	+	0.999834i	$0.494203\pi$
$194$	0	0
$195$	4.63923	0.332222
$196$	0	0
$197$	11.4524	0.815946	0.407973	−	0.912994i	$-0.366236\pi$
$197$	11.4524	0.815946	0.407973	+	0.912994i	$0.366236\pi$
$198$	0	0
$199$	8.59401	0.609213	0.304606	−	0.952478i	$-0.401475\pi$
$199$	8.59401	0.609213	0.304606	+	0.952478i	$0.401475\pi$
$200$	0	0
$201$	3.36755	0.237528
$202$	0	0
$203$	7.21211	0.506191
$204$	0	0
$205$	11.7049	0.817503
$206$	0	0
$207$	1.82714	0.126995
$208$	0	0
$209$	−3.10469	−0.214756
$210$	0	0
$211$	13.1306	0.903950	0.451975	−	0.892031i	$-0.350720\pi$
$211$	13.1306	0.903950	0.451975	+	0.892031i	$0.350720\pi$
$212$	0	0
$213$	0.392399	0.0268867
$214$	0	0
$215$	−29.4005	−2.00510
$216$	0	0
$217$	−3.84434	−0.260971
$218$	0	0
$219$	1.20872	0.0816780
$220$	0	0
$221$	−0.770902	−0.0518565
$222$	0	0
$223$	19.8360	1.32832	0.664159	−	0.747592i	$-0.268790\pi$
$223$	19.8360	1.32832	0.664159	+	0.747592i	$0.268790\pi$
$224$	0	0
$225$	2.75846	0.183897
$226$	0	0
$227$	3.51495	0.233295	0.116648	−	0.993173i	$-0.462785\pi$
$227$	3.51495	0.233295	0.116648	+	0.993173i	$0.462785\pi$
$228$	0	0
$229$	−9.03806	−0.597252	−0.298626	−	0.954370i	$-0.596528\pi$
$229$	−9.03806	−0.597252	−0.298626	+	0.954370i	$0.596528\pi$
$230$	0	0
$231$	−5.55839	−0.365715
$232$	0	0
$233$	12.5643	0.823118	0.411559	−	0.911383i	$-0.364984\pi$
$233$	12.5643	0.823118	0.411559	+	0.911383i	$0.364984\pi$
$234$	0	0
$235$	−27.0054	−1.76164
$236$	0	0
$237$	−13.1740	−0.855742
$238$	0	0
$239$	−0.0566575	−0.00366487	−0.00183243	−	0.999998i	$-0.500583\pi$
$239$	−0.0566575	−0.00366487	−0.00183243	+	0.999998i	$0.500583\pi$
$240$	0	0
$241$	15.7476	1.01439	0.507197	−	0.861830i	$-0.330682\pi$
$241$	15.7476	1.01439	0.507197	+	0.861830i	$0.330682\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.612162	−0.0391096
$246$	0	0
$247$	−2.42242	−0.154135
$248$	0	0
$249$	−7.14774	−0.452970
$250$	0	0
$251$	−18.0804	−1.14123	−0.570613	−	0.821219i	$-0.693294\pi$
$251$	−18.0804	−1.14123	−0.570613	+	0.821219i	$0.693294\pi$
$252$	0	0
$253$	−3.90031	−0.245210
$254$	0	0
$255$	−1.28923	−0.0807344
$256$	0	0
$257$	2.38813	0.148968	0.0744838	−	0.997222i	$-0.476269\pi$
$257$	2.38813	0.148968	0.0744838	+	0.997222i	$0.476269\pi$
$258$	0	0
$259$	−29.2601	−1.81814
$260$	0	0
$261$	−2.76975	−0.171443
$262$	0	0
$263$	−14.2171	−0.876661	−0.438331	−	0.898814i	$-0.644430\pi$
$263$	−14.2171	−0.876661	−0.438331	+	0.898814i	$0.644430\pi$
$264$	0	0
$265$	4.59209	0.282090
$266$	0	0
$267$	−3.77596	−0.231085
$268$	0	0
$269$	−3.62418	−0.220970	−0.110485	−	0.993878i	$-0.535240\pi$
$269$	−3.62418	−0.220970	−0.110485	+	0.993878i	$0.535240\pi$
$270$	0	0
$271$	4.32494	0.262722	0.131361	−	0.991335i	$-0.458065\pi$
$271$	4.32494	0.262722	0.131361	+	0.991335i	$0.458065\pi$
$272$	0	0
$273$	−4.33691	−0.262482
$274$	0	0
$275$	−5.88835	−0.355081
$276$	0	0
$277$	12.0874	0.726263	0.363131	−	0.931738i	$-0.381708\pi$
$277$	12.0874	0.726263	0.363131	+	0.931738i	$0.381708\pi$
$278$	0	0
$279$	1.47639	0.0883890
$280$	0	0
$281$	5.68742	0.339283	0.169641	−	0.985506i	$-0.445739\pi$
$281$	5.68742	0.339283	0.169641	+	0.985506i	$0.445739\pi$
$282$	0	0
$283$	1.92239	0.114275	0.0571373	−	0.998366i	$-0.481803\pi$
$283$	1.92239	0.114275	0.0571373	+	0.998366i	$0.481803\pi$
$284$	0	0
$285$	−4.05116	−0.239970
$286$	0	0
$287$	−10.9421	−0.645891
$288$	0	0
$289$	−16.7858	−0.987398
$290$	0	0
$291$	5.41288	0.317309
$292$	0	0
$293$	16.4356	0.960178	0.480089	−	0.877220i	$-0.340604\pi$
$293$	16.4356	0.960178	0.480089	+	0.877220i	$0.340604\pi$
$294$	0	0
$295$	32.4679	1.89035
$296$	0	0
$297$	2.13465	0.123865
$298$	0	0
$299$	−3.04320	−0.175993
$300$	0	0
$301$	27.4846	1.58418
$302$	0	0
$303$	3.16213	0.181660
$304$	0	0
$305$	−6.85483	−0.392507
$306$	0	0
$307$	20.0560	1.14465	0.572327	−	0.820026i	$-0.306041\pi$
$307$	20.0560	1.14465	0.572327	+	0.820026i	$0.306041\pi$
$308$	0	0
$309$	13.5144	0.768810
$310$	0	0
$311$	−3.49270	−0.198053	−0.0990263	−	0.995085i	$-0.531573\pi$
$311$	−3.49270	−0.198053	−0.0990263	+	0.995085i	$0.531573\pi$
$312$	0	0
$313$	−5.35968	−0.302947	−0.151473	−	0.988461i	$-0.548402\pi$
$313$	−5.35968	−0.302947	−0.151473	+	0.988461i	$0.548402\pi$
$314$	0	0
$315$	−7.25287	−0.408653
$316$	0	0
$317$	−21.1669	−1.18885	−0.594426	−	0.804150i	$-0.702621\pi$
$317$	−21.1669	−1.18885	−0.594426	+	0.804150i	$0.702621\pi$
$318$	0	0
$319$	5.91244	0.331033
$320$	0	0
$321$	9.45590	0.527777
$322$	0	0
$323$	0.673182	0.0374569
$324$	0	0
$325$	−4.59436	−0.254849
$326$	0	0
$327$	15.6724	0.866686
$328$	0	0
$329$	25.2456	1.39183
$330$	0	0
$331$	−10.9322	−0.600886	−0.300443	−	0.953800i	$-0.597135\pi$
$331$	−10.9322	−0.600886	−0.300443	+	0.953800i	$0.597135\pi$
$332$	0	0
$333$	11.2371	0.615789
$334$	0	0
$335$	−9.37997	−0.512482
$336$	0	0
$337$	1.10565	0.0602286	0.0301143	−	0.999546i	$-0.490413\pi$
$337$	1.10565	0.0602286	0.0301143	+	0.999546i	$0.490413\pi$
$338$	0	0
$339$	6.14101	0.333534
$340$	0	0
$341$	−3.15157	−0.170667
$342$	0	0
$343$	18.7995	1.01508
$344$	0	0
$345$	−5.08932	−0.274000
$346$	0	0
$347$	19.0954	1.02509	0.512547	−	0.858659i	$-0.328702\pi$
$347$	19.0954	1.02509	0.512547	+	0.858659i	$0.328702\pi$
$348$	0	0
$349$	7.19436	0.385105	0.192553	−	0.981287i	$-0.438323\pi$
$349$	7.19436	0.385105	0.192553	+	0.981287i	$0.438323\pi$
$350$	0	0
$351$	1.66555	0.0889006
$352$	0	0
$353$	−25.3376	−1.34858	−0.674291	−	0.738466i	$-0.735551\pi$
$353$	−25.3376	−1.34858	−0.674291	+	0.738466i	$0.735551\pi$
$354$	0	0
$355$	−1.09299	−0.0580098
$356$	0	0
$357$	1.20521	0.0637865
$358$	0	0
$359$	−34.2279	−1.80648	−0.903240	−	0.429135i	$-0.858818\pi$
$359$	−34.2279	−1.80648	−0.903240	+	0.429135i	$0.858818\pi$
$360$	0	0
$361$	−16.8846	−0.888666
$362$	0	0
$363$	6.44327	0.338184
$364$	0	0
$365$	−3.36678	−0.176225
$366$	0	0
$367$	−27.5828	−1.43981	−0.719905	−	0.694073i	$-0.755815\pi$
$367$	−27.5828	−1.43981	−0.719905	+	0.694073i	$0.755815\pi$
$368$	0	0
$369$	4.20221	0.218759
$370$	0	0
$371$	−4.29284	−0.222873
$372$	0	0
$373$	5.87892	0.304399	0.152199	−	0.988350i	$-0.451364\pi$
$373$	5.87892	0.304399	0.152199	+	0.988350i	$0.451364\pi$
$374$	0	0
$375$	6.24358	0.322417
$376$	0	0
$377$	4.61316	0.237590
$378$	0	0
$379$	−22.9491	−1.17881	−0.589407	−	0.807836i	$-0.700639\pi$
$379$	−22.9491	−1.17881	−0.589407	+	0.807836i	$0.700639\pi$
$380$	0	0
$381$	15.6776	0.803189
$382$	0	0
$383$	−10.8494	−0.554376	−0.277188	−	0.960816i	$-0.589402\pi$
$383$	−10.8494	−0.554376	−0.277188	+	0.960816i	$0.589402\pi$
$384$	0	0
$385$	15.4823	0.789053
$386$	0	0
$387$	−10.5552	−0.536551
$388$	0	0
$389$	−13.3476	−0.676750	−0.338375	−	0.941011i	$-0.609877\pi$
$389$	−13.3476	−0.676750	−0.338375	+	0.941011i	$0.609877\pi$
$390$	0	0
$391$	0.845694	0.0427686
$392$	0	0
$393$	18.5483	0.935637
$394$	0	0
$395$	36.6948	1.84632
$396$	0	0
$397$	−27.0205	−1.35612	−0.678060	−	0.735007i	$-0.737179\pi$
$397$	−27.0205	−1.35612	−0.678060	+	0.735007i	$0.737179\pi$
$398$	0	0
$399$	3.78716	0.189595
$400$	0	0
$401$	−15.9993	−0.798965	−0.399482	−	0.916741i	$-0.630810\pi$
$401$	−15.9993	−0.798965	−0.399482	+	0.916741i	$0.630810\pi$
$402$	0	0
$403$	−2.45900	−0.122491
$404$	0	0
$405$	2.78540	0.138408
$406$	0	0
$407$	−23.9873	−1.18900
$408$	0	0
$409$	−19.0210	−0.940527	−0.470263	−	0.882526i	$-0.655841\pi$
$409$	−19.0210	−0.940527	−0.470263	+	0.882526i	$0.655841\pi$
$410$	0	0
$411$	1.60318	0.0790788
$412$	0	0
$413$	−30.3521	−1.49353
$414$	0	0
$415$	19.9093	0.977310
$416$	0	0
$417$	−3.55874	−0.174272
$418$	0	0
$419$	−20.7971	−1.01601	−0.508003	−	0.861355i	$-0.669616\pi$
$419$	−20.7971	−1.01601	−0.508003	+	0.861355i	$0.669616\pi$
$420$	0	0
$421$	38.9449	1.89806	0.949030	−	0.315185i	$-0.102066\pi$
$421$	38.9449	1.89806	0.949030	+	0.315185i	$0.102066\pi$
$422$	0	0
$423$	−9.69535	−0.471404
$424$	0	0
$425$	1.27676	0.0619318
$426$	0	0
$427$	6.40813	0.310111
$428$	0	0
$429$	−3.55537	−0.171655
$430$	0	0
$431$	−15.8240	−0.762214	−0.381107	−	0.924531i	$-0.624457\pi$
$431$	−15.8240	−0.762214	−0.381107	+	0.924531i	$0.624457\pi$
$432$	0	0
$433$	−24.7181	−1.18788	−0.593938	−	0.804511i	$-0.702428\pi$
$433$	−24.7181	−1.18788	−0.593938	+	0.804511i	$0.702428\pi$
$434$	0	0
$435$	7.71486	0.369899
$436$	0	0
$437$	2.65744	0.127123
$438$	0	0
$439$	21.1324	1.00860	0.504298	−	0.863530i	$-0.331751\pi$
$439$	21.1324	1.00860	0.504298	+	0.863530i	$0.331751\pi$
$440$	0	0
$441$	−0.219775	−0.0104655
$442$	0	0
$443$	2.75355	0.130825	0.0654126	−	0.997858i	$-0.479164\pi$
$443$	2.75355	0.130825	0.0654126	+	0.997858i	$0.479164\pi$
$444$	0	0
$445$	10.5176	0.498580
$446$	0	0
$447$	−9.21234	−0.435729
$448$	0	0
$449$	−11.9520	−0.564052	−0.282026	−	0.959407i	$-0.591006\pi$
$449$	−11.9520	−0.564052	−0.282026	+	0.959407i	$0.591006\pi$
$450$	0	0
$451$	−8.97026	−0.422393
$452$	0	0
$453$	1.00000	0.0469841
$454$	0	0
$455$	12.0800	0.566321
$456$	0	0
$457$	−28.9306	−1.35332	−0.676658	−	0.736297i	$-0.736573\pi$
$457$	−28.9306	−1.35332	−0.676658	+	0.736297i	$0.736573\pi$
$458$	0	0
$459$	−0.462851	−0.0216040
$460$	0	0
$461$	14.9603	0.696773	0.348386	−	0.937351i	$-0.386730\pi$
$461$	14.9603	0.696773	0.348386	+	0.937351i	$0.386730\pi$
$462$	0	0
$463$	20.5746	0.956182	0.478091	−	0.878310i	$-0.341329\pi$
$463$	20.5746	0.956182	0.478091	+	0.878310i	$0.341329\pi$
$464$	0	0
$465$	−4.11233	−0.190705
$466$	0	0
$467$	15.3511	0.710363	0.355181	−	0.934797i	$-0.384419\pi$
$467$	15.3511	0.710363	0.355181	+	0.934797i	$0.384419\pi$
$468$	0	0
$469$	8.76871	0.404901
$470$	0	0
$471$	13.2429	0.610201
$472$	0	0
$473$	22.5317	1.03601
$474$	0	0
$475$	4.01198	0.184082
$476$	0	0
$477$	1.64863	0.0754855
$478$	0	0
$479$	−14.7804	−0.675334	−0.337667	−	0.941266i	$-0.609638\pi$
$479$	−14.7804	−0.675334	−0.337667	+	0.941266i	$0.609638\pi$
$480$	0	0
$481$	−18.7160	−0.853374
$482$	0	0
$483$	4.75767	0.216482
$484$	0	0
$485$	−15.0771	−0.684614
$486$	0	0
$487$	−17.0506	−0.772638	−0.386319	−	0.922365i	$-0.626254\pi$
$487$	−17.0506	−0.772638	−0.386319	+	0.922365i	$0.626254\pi$
$488$	0	0
$489$	16.3838	0.740901
$490$	0	0
$491$	−29.3654	−1.32524	−0.662621	−	0.748955i	$-0.730556\pi$
$491$	−29.3654	−1.32524	−0.662621	+	0.748955i	$0.730556\pi$
$492$	0	0
$493$	−1.28198	−0.0577375
$494$	0	0
$495$	−5.94586	−0.267246
$496$	0	0
$497$	1.02176	0.0458323
$498$	0	0
$499$	−6.63816	−0.297165	−0.148582	−	0.988900i	$-0.547471\pi$
$499$	−6.63816	−0.297165	−0.148582	+	0.988900i	$0.547471\pi$
$500$	0	0
$501$	−0.327406	−0.0146274
$502$	0	0
$503$	−36.7476	−1.63850	−0.819248	−	0.573440i	$-0.805609\pi$
$503$	−36.7476	−1.63850	−0.819248	+	0.573440i	$0.805609\pi$
$504$	0	0
$505$	−8.80781	−0.391942
$506$	0	0
$507$	10.2259	0.454150
$508$	0	0
$509$	−0.925961	−0.0410425	−0.0205212	−	0.999789i	$-0.506533\pi$
$509$	−0.925961	−0.0410425	−0.0205212	+	0.999789i	$0.506533\pi$
$510$	0	0
$511$	3.14738	0.139232
$512$	0	0
$513$	−1.45443	−0.0642145
$514$	0	0
$515$	−37.6432	−1.65876
$516$	0	0
$517$	20.6962	0.910217
$518$	0	0
$519$	18.6317	0.817840
$520$	0	0
$521$	−31.9914	−1.40157	−0.700784	−	0.713373i	$-0.747167\pi$
$521$	−31.9914	−1.40157	−0.700784	+	0.713373i	$0.747167\pi$
$522$	0	0
$523$	17.4523	0.763136	0.381568	−	0.924341i	$-0.375384\pi$
$523$	17.4523	0.763136	0.381568	+	0.924341i	$0.375384\pi$
$524$	0	0
$525$	7.18272	0.313480
$526$	0	0
$527$	0.683347	0.0297671
$528$	0	0
$529$	−19.6616	−0.854850
$530$	0	0
$531$	11.6565	0.505847
$532$	0	0
$533$	−6.99901	−0.303161
$534$	0	0
$535$	−26.3385	−1.13871
$536$	0	0
$537$	−6.85648	−0.295879
$538$	0	0
$539$	0.469143	0.0202074
$540$	0	0
$541$	9.69108	0.416652	0.208326	−	0.978059i	$-0.433198\pi$
$541$	9.69108	0.416652	0.208326	+	0.978059i	$0.433198\pi$
$542$	0	0
$543$	14.2712	0.612436
$544$	0	0
$545$	−43.6540	−1.86993
$546$	0	0
$547$	26.4621	1.13144	0.565718	−	0.824599i	$-0.308599\pi$
$547$	26.4621	1.13144	0.565718	+	0.824599i	$0.308599\pi$
$548$	0	0
$549$	−2.46099	−0.105032
$550$	0	0
$551$	−4.02839	−0.171615
$552$	0	0
$553$	−34.3036	−1.45874
$554$	0	0
$555$	−31.2998	−1.32860
$556$	0	0
$557$	24.5914	1.04197	0.520986	−	0.853565i	$-0.325565\pi$
$557$	24.5914	1.04197	0.520986	+	0.853565i	$0.325565\pi$
$558$	0	0
$559$	17.5802	0.743565
$560$	0	0
$561$	0.988025	0.0417144
$562$	0	0
$563$	5.70566	0.240465	0.120232	−	0.992746i	$-0.461636\pi$
$563$	5.70566	0.240465	0.120232	+	0.992746i	$0.461636\pi$
$564$	0	0
$565$	−17.1052	−0.719620
$566$	0	0
$567$	−2.60389	−0.109353
$568$	0	0
$569$	26.2456	1.10027	0.550137	−	0.835075i	$-0.314576\pi$
$569$	26.2456	1.10027	0.550137	+	0.835075i	$0.314576\pi$
$570$	0	0
$571$	17.6912	0.740355	0.370178	−	0.928961i	$-0.379297\pi$
$571$	17.6912	0.740355	0.370178	+	0.928961i	$0.379297\pi$
$572$	0	0
$573$	−9.68421	−0.404564
$574$	0	0
$575$	5.04010	0.210187
$576$	0	0
$577$	−13.2017	−0.549592	−0.274796	−	0.961503i	$-0.588610\pi$
$577$	−13.2017	−0.549592	−0.274796	+	0.961503i	$0.588610\pi$
$578$	0	0
$579$	−0.506013	−0.0210292
$580$	0	0
$581$	−18.6119	−0.772152
$582$	0	0
$583$	−3.51925	−0.145752
$584$	0	0
$585$	−4.63923	−0.191809
$586$	0	0
$587$	−6.84675	−0.282596	−0.141298	−	0.989967i	$-0.545128\pi$
$587$	−6.84675	−0.282596	−0.141298	+	0.989967i	$0.545128\pi$
$588$	0	0
$589$	2.14730	0.0884778
$590$	0	0
$591$	−11.4524	−0.471087
$592$	0	0
$593$	15.2172	0.624895	0.312447	−	0.949935i	$-0.398851\pi$
$593$	15.2172	0.624895	0.312447	+	0.949935i	$0.398851\pi$
$594$	0	0
$595$	−3.35700	−0.137623
$596$	0	0
$597$	−8.59401	−0.351729
$598$	0	0
$599$	15.3047	0.625333	0.312667	−	0.949863i	$-0.398778\pi$
$599$	15.3047	0.625333	0.312667	+	0.949863i	$0.398778\pi$
$600$	0	0
$601$	−0.496568	−0.0202554	−0.0101277	−	0.999949i	$-0.503224\pi$
$601$	−0.496568	−0.0202554	−0.0101277	+	0.999949i	$0.503224\pi$
$602$	0	0
$603$	−3.36755	−0.137137
$604$	0	0
$605$	−17.9471	−0.729653
$606$	0	0
$607$	−14.9605	−0.607228	−0.303614	−	0.952795i	$-0.598193\pi$
$607$	−14.9605	−0.607228	−0.303614	+	0.952795i	$0.598193\pi$
$608$	0	0
$609$	−7.21211	−0.292249
$610$	0	0
$611$	16.1481	0.653283
$612$	0	0
$613$	−23.4356	−0.946556	−0.473278	−	0.880913i	$-0.656929\pi$
$613$	−23.4356	−0.946556	−0.473278	+	0.880913i	$0.656929\pi$
$614$	0	0
$615$	−11.7049	−0.471985
$616$	0	0
$617$	11.0740	0.445822	0.222911	−	0.974839i	$-0.428444\pi$
$617$	11.0740	0.445822	0.222911	+	0.974839i	$0.428444\pi$
$618$	0	0
$619$	21.0721	0.846958	0.423479	−	0.905906i	$-0.360809\pi$
$619$	21.0721	0.846958	0.423479	+	0.905906i	$0.360809\pi$
$620$	0	0
$621$	−1.82714	−0.0733207
$622$	0	0
$623$	−9.83217	−0.393917
$624$	0	0
$625$	−31.1832	−1.24733
$626$	0	0
$627$	3.10469	0.123989
$628$	0	0
$629$	5.20110	0.207381
$630$	0	0
$631$	1.34352	0.0534849	0.0267424	−	0.999642i	$-0.491487\pi$
$631$	1.34352	0.0534849	0.0267424	+	0.999642i	$0.491487\pi$
$632$	0	0
$633$	−13.1306	−0.521896
$634$	0	0
$635$	−43.6685	−1.73293
$636$	0	0
$637$	0.366047	0.0145033
$638$	0	0
$639$	−0.392399	−0.0155231
$640$	0	0
$641$	23.4731	0.927133	0.463567	−	0.886062i	$-0.346569\pi$
$641$	23.4731	0.927133	0.463567	+	0.886062i	$0.346569\pi$
$642$	0	0
$643$	16.5571	0.652947	0.326474	−	0.945206i	$-0.394140\pi$
$643$	16.5571	0.652947	0.326474	+	0.945206i	$0.394140\pi$
$644$	0	0
$645$	29.4005	1.15764
$646$	0	0
$647$	12.1918	0.479309	0.239654	−	0.970858i	$-0.422966\pi$
$647$	12.1918	0.479309	0.239654	+	0.970858i	$0.422966\pi$
$648$	0	0
$649$	−24.8825	−0.976722
$650$	0	0
$651$	3.84434	0.150672
$652$	0	0
$653$	−9.20482	−0.360212	−0.180106	−	0.983647i	$-0.557644\pi$
$653$	−9.20482	−0.360212	−0.180106	+	0.983647i	$0.557644\pi$
$654$	0	0
$655$	−51.6644	−2.01870
$656$	0	0
$657$	−1.20872	−0.0471568
$658$	0	0
$659$	35.1555	1.36946	0.684731	−	0.728795i	$-0.259920\pi$
$659$	35.1555	1.36946	0.684731	+	0.728795i	$0.259920\pi$
$660$	0	0
$661$	−13.2697	−0.516130	−0.258065	−	0.966128i	$-0.583085\pi$
$661$	−13.2697	−0.516130	−0.258065	+	0.966128i	$0.583085\pi$
$662$	0	0
$663$	0.770902	0.0299394
$664$	0	0
$665$	−10.5488	−0.409063
$666$	0	0
$667$	−5.06072	−0.195952
$668$	0	0
$669$	−19.8360	−0.766904
$670$	0	0
$671$	5.25334	0.202803
$672$	0	0
$673$	−3.75157	−0.144612	−0.0723062	−	0.997382i	$-0.523036\pi$
$673$	−3.75157	−0.144612	−0.0723062	+	0.997382i	$0.523036\pi$
$674$	0	0
$675$	−2.75846	−0.106173
$676$	0	0
$677$	−39.3139	−1.51096	−0.755478	−	0.655174i	$-0.772595\pi$
$677$	−39.3139	−1.51096	−0.755478	+	0.655174i	$0.772595\pi$
$678$	0	0
$679$	14.0945	0.540899
$680$	0	0
$681$	−3.51495	−0.134693
$682$	0	0
$683$	−29.4203	−1.12574	−0.562869	−	0.826546i	$-0.690303\pi$
$683$	−29.4203	−1.12574	−0.562869	+	0.826546i	$0.690303\pi$
$684$	0	0
$685$	−4.46549	−0.170618
$686$	0	0
$687$	9.03806	0.344824
$688$	0	0
$689$	−2.74588	−0.104610
$690$	0	0
$691$	32.6555	1.24227	0.621136	−	0.783702i	$-0.286671\pi$
$691$	32.6555	1.24227	0.621136	+	0.783702i	$0.286671\pi$
$692$	0	0
$693$	5.55839	0.211146
$694$	0	0
$695$	9.91253	0.376004
$696$	0	0
$697$	1.94500	0.0736721
$698$	0	0
$699$	−12.5643	−0.475227
$700$	0	0
$701$	−28.7962	−1.08762	−0.543808	−	0.839210i	$-0.683018\pi$
$701$	−28.7962	−1.08762	−0.543808	+	0.839210i	$0.683018\pi$
$702$	0	0
$703$	16.3435	0.616408
$704$	0	0
$705$	27.0054	1.01708
$706$	0	0
$707$	8.23384	0.309665
$708$	0	0
$709$	−5.45498	−0.204866	−0.102433	−	0.994740i	$-0.532663\pi$
$709$	−5.45498	−0.204866	−0.102433	+	0.994740i	$0.532663\pi$
$710$	0	0
$711$	13.1740	0.494063
$712$	0	0
$713$	2.69757	0.101025
$714$	0	0
$715$	9.90314	0.370356
$716$	0	0
$717$	0.0566575	0.00211591
$718$	0	0
$719$	−43.3253	−1.61576	−0.807880	−	0.589348i	$-0.799385\pi$
$719$	−43.3253	−1.61576	−0.807880	+	0.589348i	$0.799385\pi$
$720$	0	0
$721$	35.1901	1.31055
$722$	0	0
$723$	−15.7476	−0.585661
$724$	0	0
$725$	−7.64024	−0.283751
$726$	0	0
$727$	−5.22629	−0.193832	−0.0969161	−	0.995293i	$-0.530898\pi$
$727$	−5.22629	−0.193832	−0.0969161	+	0.995293i	$0.530898\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.88549	−0.180696
$732$	0	0
$733$	22.1564	0.818365	0.409182	−	0.912453i	$-0.365814\pi$
$733$	22.1564	0.818365	0.409182	+	0.912453i	$0.365814\pi$
$734$	0	0
$735$	0.612162	0.0225799
$736$	0	0
$737$	7.18853	0.264793
$738$	0	0
$739$	41.7524	1.53589	0.767944	−	0.640517i	$-0.221280\pi$
$739$	41.7524	1.53589	0.767944	+	0.640517i	$0.221280\pi$
$740$	0	0
$741$	2.42242	0.0889899
$742$	0	0
$743$	6.24848	0.229235	0.114617	−	0.993410i	$-0.463436\pi$
$743$	6.24848	0.229235	0.114617	+	0.993410i	$0.463436\pi$
$744$	0	0
$745$	25.6601	0.940112
$746$	0	0
$747$	7.14774	0.261522
$748$	0	0
$749$	24.6221	0.899672
$750$	0	0
$751$	15.3900	0.561588	0.280794	−	0.959768i	$-0.409402\pi$
$751$	15.3900	0.561588	0.280794	+	0.959768i	$0.409402\pi$
$752$	0	0
$753$	18.0804	0.658887
$754$	0	0
$755$	−2.78540	−0.101371
$756$	0	0
$757$	14.3255	0.520670	0.260335	−	0.965518i	$-0.416167\pi$
$757$	14.3255	0.520670	0.260335	+	0.965518i	$0.416167\pi$
$758$	0	0
$759$	3.90031	0.141572
$760$	0	0
$761$	24.5584	0.890241	0.445121	−	0.895471i	$-0.353161\pi$
$761$	24.5584	0.890241	0.445121	+	0.895471i	$0.353161\pi$
$762$	0	0
$763$	40.8092	1.47739
$764$	0	0
$765$	1.28923	0.0466120
$766$	0	0
$767$	−19.4144	−0.701014
$768$	0	0
$769$	16.5628	0.597269	0.298635	−	0.954367i	$-0.403469\pi$
$769$	16.5628	0.597269	0.298635	+	0.954367i	$0.403469\pi$
$770$	0	0
$771$	−2.38813	−0.0860065
$772$	0	0
$773$	−11.0031	−0.395752	−0.197876	−	0.980227i	$-0.563404\pi$
$773$	−11.0031	−0.395752	−0.197876	+	0.980227i	$0.563404\pi$
$774$	0	0
$775$	4.07256	0.146291
$776$	0	0
$777$	29.2601	1.04970
$778$	0	0
$779$	6.11181	0.218978
$780$	0	0
$781$	0.837635	0.0299729
$782$	0	0
$783$	2.76975	0.0989827
$784$	0	0
$785$	−36.8868	−1.31655
$786$	0	0
$787$	48.9142	1.74360	0.871801	−	0.489861i	$-0.162952\pi$
$787$	48.9142	1.74360	0.871801	+	0.489861i	$0.162952\pi$
$788$	0	0
$789$	14.2171	0.506141
$790$	0	0
$791$	15.9905	0.568556
$792$	0	0
$793$	4.09890	0.145556
$794$	0	0
$795$	−4.59209	−0.162865
$796$	0	0
$797$	−9.87946	−0.349948	−0.174974	−	0.984573i	$-0.555984\pi$
$797$	−9.87946	−0.349948	−0.174974	+	0.984573i	$0.555984\pi$
$798$	0	0
$799$	−4.48750	−0.158756
$800$	0	0
$801$	3.77596	0.133417
$802$	0	0
$803$	2.58020	0.0910534
$804$	0	0
$805$	−13.2520	−0.467072
$806$	0	0
$807$	3.62418	0.127577
$808$	0	0
$809$	8.00829	0.281556	0.140778	−	0.990041i	$-0.455040\pi$
$809$	8.00829	0.281556	0.140778	+	0.990041i	$0.455040\pi$
$810$	0	0
$811$	29.6501	1.04115	0.520577	−	0.853815i	$-0.325717\pi$
$811$	29.6501	1.04115	0.520577	+	0.853815i	$0.325717\pi$
$812$	0	0
$813$	−4.32494	−0.151682
$814$	0	0
$815$	−45.6355	−1.59854
$816$	0	0
$817$	−15.3518	−0.537090
$818$	0	0
$819$	4.33691	0.151544
$820$	0	0
$821$	−32.8666	−1.14705	−0.573526	−	0.819188i	$-0.694425\pi$
$821$	−32.8666	−1.14705	−0.573526	+	0.819188i	$0.694425\pi$
$822$	0	0
$823$	−6.46219	−0.225258	−0.112629	−	0.993637i	$-0.535927\pi$
$823$	−6.46219	−0.225258	−0.112629	+	0.993637i	$0.535927\pi$
$824$	0	0
$825$	5.88835	0.205006
$826$	0	0
$827$	−9.82682	−0.341712	−0.170856	−	0.985296i	$-0.554653\pi$
$827$	−9.82682	−0.341712	−0.170856	+	0.985296i	$0.554653\pi$
$828$	0	0
$829$	−28.9463	−1.00535	−0.502673	−	0.864477i	$-0.667650\pi$
$829$	−28.9463	−1.00535	−0.502673	+	0.864477i	$0.667650\pi$
$830$	0	0
$831$	−12.0874	−0.419308
$832$	0	0
$833$	−0.101723	−0.00352449
$834$	0	0
$835$	0.911957	0.0315596
$836$	0	0
$837$	−1.47639	−0.0510314
$838$	0	0
$839$	−7.71104	−0.266215	−0.133107	−	0.991102i	$-0.542496\pi$
$839$	−7.71104	−0.266215	−0.133107	+	0.991102i	$0.542496\pi$
$840$	0	0
$841$	−21.3285	−0.735466
$842$	0	0
$843$	−5.68742	−0.195885
$844$	0	0
$845$	−28.4833	−0.979857
$846$	0	0
$847$	16.7775	0.576483
$848$	0	0
$849$	−1.92239	−0.0659764
$850$	0	0
$851$	20.5318	0.703820
$852$	0	0
$853$	−0.790147	−0.0270541	−0.0135271	−	0.999909i	$-0.504306\pi$
$853$	−0.790147	−0.0270541	−0.0135271	+	0.999909i	$0.504306\pi$
$854$	0	0
$855$	4.05116	0.138547
$856$	0	0
$857$	−7.35035	−0.251083	−0.125542	−	0.992088i	$-0.540067\pi$
$857$	−7.35035	−0.251083	−0.125542	+	0.992088i	$0.540067\pi$
$858$	0	0
$859$	12.3626	0.421807	0.210904	−	0.977507i	$-0.432359\pi$
$859$	12.3626	0.421807	0.210904	+	0.977507i	$0.432359\pi$
$860$	0	0
$861$	10.9421	0.372905
$862$	0	0
$863$	−5.83397	−0.198591	−0.0992953	−	0.995058i	$-0.531659\pi$
$863$	−5.83397	−0.198591	−0.0992953	+	0.995058i	$0.531659\pi$
$864$	0	0
$865$	−51.8967	−1.76454
$866$	0	0
$867$	16.7858	0.570075
$868$	0	0
$869$	−28.1218	−0.953968
$870$	0	0
$871$	5.60882	0.190048
$872$	0	0
$873$	−5.41288	−0.183198
$874$	0	0
$875$	16.2576	0.549607
$876$	0	0
$877$	22.1491	0.747923	0.373961	−	0.927444i	$-0.377999\pi$
$877$	22.1491	0.747923	0.373961	+	0.927444i	$0.377999\pi$
$878$	0	0
$879$	−16.4356	−0.554359
$880$	0	0
$881$	13.2474	0.446316	0.223158	−	0.974782i	$-0.428363\pi$
$881$	13.2474	0.446316	0.223158	+	0.974782i	$0.428363\pi$
$882$	0	0
$883$	−7.12004	−0.239608	−0.119804	−	0.992798i	$-0.538227\pi$
$883$	−7.12004	−0.239608	−0.119804	+	0.992798i	$0.538227\pi$
$884$	0	0
$885$	−32.4679	−1.09140
$886$	0	0
$887$	−19.8725	−0.667252	−0.333626	−	0.942706i	$-0.608272\pi$
$887$	−19.8725	−0.667252	−0.333626	+	0.942706i	$0.608272\pi$
$888$	0	0
$889$	40.8227	1.36915
$890$	0	0
$891$	−2.13465	−0.0715135
$892$	0	0
$893$	−14.1012	−0.471878
$894$	0	0
$895$	19.0981	0.638378
$896$	0	0
$897$	3.04320	0.101609
$898$	0	0
$899$	−4.08922	−0.136383
$900$	0	0
$901$	0.763069	0.0254215
$902$	0	0
$903$	−27.4846	−0.914629
$904$	0	0
$905$	−39.7510	−1.32137
$906$	0	0
$907$	4.54642	0.150962	0.0754808	−	0.997147i	$-0.475951\pi$
$907$	4.54642	0.150962	0.0754808	+	0.997147i	$0.475951\pi$
$908$	0	0
$909$	−3.16213	−0.104881
$910$	0	0
$911$	−46.8388	−1.55184	−0.775919	−	0.630832i	$-0.782714\pi$
$911$	−46.8388	−1.55184	−0.775919	+	0.630832i	$0.782714\pi$
$912$	0	0
$913$	−15.2579	−0.504964
$914$	0	0
$915$	6.85483	0.226614
$916$	0	0
$917$	48.2976	1.59493
$918$	0	0
$919$	35.8331	1.18203	0.591013	−	0.806662i	$-0.298728\pi$
$919$	35.8331	1.18203	0.591013	+	0.806662i	$0.298728\pi$
$920$	0	0
$921$	−20.0560	−0.660866
$922$	0	0
$923$	0.653561	0.0215122
$924$	0	0
$925$	30.9971	1.01918
$926$	0	0
$927$	−13.5144	−0.443873
$928$	0	0
$929$	24.3864	0.800091	0.400046	−	0.916495i	$-0.368994\pi$
$929$	24.3864	0.800091	0.400046	+	0.916495i	$0.368994\pi$
$930$	0	0
$931$	−0.319646	−0.0104760
$932$	0	0
$933$	3.49270	0.114346
$934$	0	0
$935$	−2.75205	−0.0900015
$936$	0	0
$937$	−19.5168	−0.637586	−0.318793	−	0.947824i	$-0.603277\pi$
$937$	−19.5168	−0.637586	−0.318793	+	0.947824i	$0.603277\pi$
$938$	0	0
$939$	5.35968	0.174906
$940$	0	0
$941$	−4.94688	−0.161264	−0.0806319	−	0.996744i	$-0.525694\pi$
$941$	−4.94688	−0.161264	−0.0806319	+	0.996744i	$0.525694\pi$
$942$	0	0
$943$	7.67804	0.250031
$944$	0	0
$945$	7.25287	0.235936
$946$	0	0
$947$	−3.66501	−0.119097	−0.0595484	−	0.998225i	$-0.518966\pi$
$947$	−3.66501	−0.119097	−0.0595484	+	0.998225i	$0.518966\pi$
$948$	0	0
$949$	2.01319	0.0653510
$950$	0	0
$951$	21.1669	0.686384
$952$	0	0
$953$	−43.2657	−1.40151	−0.700757	−	0.713400i	$-0.747154\pi$
$953$	−43.2657	−1.40151	−0.700757	+	0.713400i	$0.747154\pi$
$954$	0	0
$955$	26.9744	0.872871
$956$	0	0
$957$	−5.91244	−0.191122
$958$	0	0
$959$	4.17449	0.134801
$960$	0	0
$961$	−28.8203	−0.929687
$962$	0	0
$963$	−9.45590	−0.304712
$964$	0	0
$965$	1.40945	0.0453718
$966$	0	0
$967$	−38.8303	−1.24870	−0.624349	−	0.781146i	$-0.714636\pi$
$967$	−38.8303	−1.24870	−0.624349	+	0.781146i	$0.714636\pi$
$968$	0	0
$969$	−0.673182	−0.0216257
$970$	0	0
$971$	−43.5200	−1.39662	−0.698312	−	0.715794i	$-0.746065\pi$
$971$	−43.5200	−1.39662	−0.698312	+	0.715794i	$0.746065\pi$
$972$	0	0
$973$	−9.26657	−0.297072
$974$	0	0
$975$	4.59436	0.147137
$976$	0	0
$977$	−39.6311	−1.26791	−0.633956	−	0.773369i	$-0.718570\pi$
$977$	−39.6311	−1.26791	−0.633956	+	0.773369i	$0.718570\pi$
$978$	0	0
$979$	−8.06035	−0.257610
$980$	0	0
$981$	−15.6724	−0.500382
$982$	0	0
$983$	31.6352	1.00901	0.504503	−	0.863410i	$-0.331676\pi$
$983$	31.6352	1.00901	0.504503	+	0.863410i	$0.331676\pi$
$984$	0	0
$985$	31.8994	1.01640
$986$	0	0
$987$	−25.2456	−0.803576
$988$	0	0
$989$	−19.2859	−0.613255
$990$	0	0
$991$	8.38403	0.266328	0.133164	−	0.991094i	$-0.457486\pi$
$991$	8.38403	0.266328	0.133164	+	0.991094i	$0.457486\pi$
$992$	0	0
$993$	10.9322	0.346922
$994$	0	0
$995$	23.9378	0.758878
$996$	0	0
$997$	27.1509	0.859878	0.429939	−	0.902858i	$-0.358535\pi$
$997$	27.1509	0.859878	0.429939	+	0.902858i	$0.358535\pi$
$998$	0	0
$999$	−11.2371	−0.355526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7248.2.a.bm.1.8		10
4.3	odd	2		3624.2.a.l.1.8	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3624.2.a.l.1.8	✓	10	4.3	odd	2
7248.2.a.bm.1.8		10	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:	\( N \)	\(=\)	\( 7248 = 2^{4} \cdot 3 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7248.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.78540 q^{5} -2.60389 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.78540 q^{5} -2.60389 q^{7} +1.00000 q^{9} -2.13465 q^{11} -1.66555 q^{13} -2.78540 q^{15} +0.462851 q^{17} +1.45443 q^{19} +2.60389 q^{21} +1.82714 q^{23} +2.75846 q^{25} -1.00000 q^{27} -2.76975 q^{29} +1.47639 q^{31} +2.13465 q^{33} -7.25287 q^{35} +11.2371 q^{37} +1.66555 q^{39} +4.20221 q^{41} -10.5552 q^{43} +2.78540 q^{45} -9.69535 q^{47} -0.219775 q^{49} -0.462851 q^{51} +1.64863 q^{53} -5.94586 q^{55} -1.45443 q^{57} +11.6565 q^{59} -2.46099 q^{61} -2.60389 q^{63} -4.63923 q^{65} -3.36755 q^{67} -1.82714 q^{69} -0.392399 q^{71} -1.20872 q^{73} -2.75846 q^{75} +5.55839 q^{77} +13.1740 q^{79} +1.00000 q^{81} +7.14774 q^{83} +1.28923 q^{85} +2.76975 q^{87} +3.77596 q^{89} +4.33691 q^{91} -1.47639 q^{93} +4.05116 q^{95} -5.41288 q^{97} -2.13465 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9	\( 10 q - 10 q^{3} + 2 q^{5} - 8 q^{7} + 10 q^{9} - 7 q^{11} + 6 q^{13} - 2 q^{15} + 7 q^{17} + 8 q^{21} - 25 q^{23} + 8 q^{25} - 10 q^{27} + 12 q^{29} - 11 q^{31} + 7 q^{33} - 9 q^{35} - 3 q^{37} - 6 q^{39} + 12 q^{41} + 2 q^{45} - 31 q^{47} + 14 q^{49} - 7 q^{51} + q^{53} - 9 q^{55} - 19 q^{59} + 24 q^{61} - 8 q^{63} + 20 q^{65} + q^{67} + 25 q^{69} - 34 q^{71} - 18 q^{73} - 8 q^{75} + 27 q^{77} - 25 q^{79} + 10 q^{81} - 14 q^{83} - 3 q^{85} - 12 q^{87} + 20 q^{89} + 12 q^{91} + 11 q^{93} - 48 q^{95} - 15 q^{97} - 7 q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 + 2 * q^5 - 8 * q^7 + 10 * q^9 - 7 * q^11 + 6 * q^13 - 2 * q^15 + 7 * q^17 + 8 * q^21 - 25 * q^23 + 8 * q^25 - 10 * q^27 + 12 * q^29 - 11 * q^31 + 7 * q^33 - 9 * q^35 - 3 * q^37 - 6 * q^39 + 12 * q^41 + 2 * q^45 - 31 * q^47 + 14 * q^49 - 7 * q^51 + q^53 - 9 * q^55 - 19 * q^59 + 24 * q^61 - 8 * q^63 + 20 * q^65 + q^67 + 25 * q^69 - 34 * q^71 - 18 * q^73 - 8 * q^75 + 27 * q^77 - 25 * q^79 + 10 * q^81 - 14 * q^83 - 3 * q^85 - 12 * q^87 + 20 * q^89 + 12 * q^91 + 11 * q^93 - 48 * q^95 - 15 * q^97 - 7 * q^99

Introduction

L-functions

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