LMFDB - Embedded newform 1456.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(1,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, 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("1456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.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:	$11.6262185343$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.64268.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 6x + 32 $ x^4 - x^3 - 12x^2 + 6x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 728)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.72110$ of defining polynomial
Character	$\chi$	$=$	1456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.72110 q^{3} +3.13311 q^{5} +1.00000 q^{7} -0.0378271 q^{9} +O(q^{10})$	$q+1.72110 q^{3} +3.13311 q^{5} +1.00000 q^{7} -0.0378271 q^{9} -4.35456 q^{11} -1.00000 q^{13} +5.39238 q^{15} +5.31673 q^{17} +2.49965 q^{19} +1.72110 q^{21} +2.85421 q^{23} +4.81638 q^{25} -5.22839 q^{27} +3.62581 q^{29} +6.17094 q^{31} -7.49461 q^{33} +3.13311 q^{35} +9.79675 q^{37} -1.72110 q^{39} +10.6706 q^{41} +0.449841 q^{43} -0.118516 q^{45} -11.8794 q^{47} +1.00000 q^{49} +9.15061 q^{51} +5.89203 q^{53} -13.6433 q^{55} +4.30214 q^{57} +7.80943 q^{59} -15.3797 q^{61} -0.0378271 q^{63} -3.13311 q^{65} -5.84656 q^{67} +4.91236 q^{69} -11.2914 q^{71} -4.17094 q^{73} +8.28945 q^{75} -4.35456 q^{77} -9.48767 q^{79} -8.88509 q^{81} -14.8415 q^{83} +16.6579 q^{85} +6.24037 q^{87} -5.57601 q^{89} -1.00000 q^{91} +10.6208 q^{93} +7.83168 q^{95} -0.979669 q^{97} +0.164720 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 2 q^{5} + 4 q^{7} + 13 q^{9}+O(q^{10}) $ 4 * q - q^3 + 2 * q^5 + 4 * q^7 + 13 * q^9	$ 4 q - q^{3} + 2 q^{5} + 4 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} - 10 q^{15} + 16 q^{17} + 10 q^{19} - q^{21} - 7 q^{23} + 14 q^{25} - 13 q^{27} + 4 q^{29} + q^{31} - q^{33} + 2 q^{35} + 5 q^{37} + q^{39} + 19 q^{41} - 14 q^{43} + 10 q^{45} + 13 q^{47} + 4 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 26 q^{59} - q^{61} + 13 q^{63} - 2 q^{65} - 5 q^{67} + 17 q^{69} + 18 q^{71} + 7 q^{73} - q^{75} + q^{77} - 9 q^{79} - 4 q^{81} - 12 q^{83} + 14 q^{85} + 46 q^{87} + 4 q^{89} - 4 q^{91} + 3 q^{93} + 26 q^{95} + 25 q^{97} + 48 q^{99}+O(q^{100}) $ 4 * q - q^3 + 2 * q^5 + 4 * q^7 + 13 * q^9 + q^11 - 4 * q^13 - 10 * q^15 + 16 * q^17 + 10 * q^19 - q^21 - 7 * q^23 + 14 * q^25 - 13 * q^27 + 4 * q^29 + q^31 - q^33 + 2 * q^35 + 5 * q^37 + q^39 + 19 * q^41 - 14 * q^43 + 10 * q^45 + 13 * q^47 + 4 * q^49 - 16 * q^51 - 8 * q^53 - 2 * q^55 + 12 * q^57 + 26 * q^59 - q^61 + 13 * q^63 - 2 * q^65 - 5 * q^67 + 17 * q^69 + 18 * q^71 + 7 * q^73 - q^75 + q^77 - 9 * q^79 - 4 * q^81 - 12 * q^83 + 14 * q^85 + 46 * q^87 + 4 * q^89 - 4 * q^91 + 3 * q^93 + 26 * q^95 + 25 * q^97 + 48 * 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
$3$	1.72110	0.993675	0.496838	−	0.867843i	$-0.334494\pi$
$3$	1.72110	0.993675	0.496838	+	0.867843i	$0.334494\pi$
$4$	0	0
$5$	3.13311	1.40117	0.700585	−	0.713569i	$-0.252923\pi$
$5$	3.13311	1.40117	0.700585	+	0.713569i	$0.252923\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.0378271	−0.0126090
$10$	0	0
$11$	−4.35456	−1.31295	−0.656474	−	0.754348i	$-0.727953\pi$
$11$	−4.35456	−1.31295	−0.656474	+	0.754348i	$0.727953\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	5.39238	1.39231
$16$	0	0
$17$	5.31673	1.28950	0.644748	−	0.764395i	$-0.276962\pi$
$17$	5.31673	1.28950	0.644748	+	0.764395i	$0.276962\pi$
$18$	0	0
$19$	2.49965	0.573459	0.286729	−	0.958012i	$-0.407432\pi$
$19$	2.49965	0.573459	0.286729	+	0.958012i	$0.407432\pi$
$20$	0	0
$21$	1.72110	0.375574
$22$	0	0
$23$	2.85421	0.595143	0.297572	−	0.954700i	$-0.403823\pi$
$23$	2.85421	0.595143	0.297572	+	0.954700i	$0.403823\pi$
$24$	0	0
$25$	4.81638	0.963276
$26$	0	0
$27$	−5.22839	−1.00620
$28$	0	0
$29$	3.62581	0.673297	0.336648	−	0.941630i	$-0.390707\pi$
$29$	3.62581	0.673297	0.336648	+	0.941630i	$0.390707\pi$
$30$	0	0
$31$	6.17094	1.10833	0.554167	−	0.832406i	$-0.313037\pi$
$31$	6.17094	1.10833	0.554167	+	0.832406i	$0.313037\pi$
$32$	0	0
$33$	−7.49461	−1.30464
$34$	0	0
$35$	3.13311	0.529592
$36$	0	0
$37$	9.79675	1.61058	0.805288	−	0.592884i	$-0.202011\pi$
$37$	9.79675	1.61058	0.805288	+	0.592884i	$0.202011\pi$
$38$	0	0
$39$	−1.72110	−0.275596
$40$	0	0
$41$	10.6706	1.66647	0.833233	−	0.552922i	$-0.186487\pi$
$41$	10.6706	1.66647	0.833233	+	0.552922i	$0.186487\pi$
$42$	0	0
$43$	0.449841	0.0686001	0.0343000	−	0.999412i	$-0.489080\pi$
$43$	0.449841	0.0686001	0.0343000	+	0.999412i	$0.489080\pi$
$44$	0	0
$45$	−0.118516	−0.0176674
$46$	0	0
$47$	−11.8794	−1.73278	−0.866391	−	0.499367i	$-0.833566\pi$
$47$	−11.8794	−1.73278	−0.866391	+	0.499367i	$0.833566\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	9.15061	1.28134
$52$	0	0
$53$	5.89203	0.809333	0.404667	−	0.914464i	$-0.367388\pi$
$53$	5.89203	0.809333	0.404667	+	0.914464i	$0.367388\pi$
$54$	0	0
$55$	−13.6433	−1.83966
$56$	0	0
$57$	4.30214	0.569832
$58$	0	0
$59$	7.80943	1.01670	0.508351	−	0.861150i	$-0.330255\pi$
$59$	7.80943	1.01670	0.508351	+	0.861150i	$0.330255\pi$
$60$	0	0
$61$	−15.3797	−1.96917	−0.984585	−	0.174910i	$-0.944037\pi$
$61$	−15.3797	−1.96917	−0.984585	+	0.174910i	$0.944037\pi$
$62$	0	0
$63$	−0.0378271	−0.00476576
$64$	0	0
$65$	−3.13311	−0.388614
$66$	0	0
$67$	−5.84656	−0.714271	−0.357135	−	0.934053i	$-0.616247\pi$
$67$	−5.84656	−0.714271	−0.357135	+	0.934053i	$0.616247\pi$
$68$	0	0
$69$	4.91236	0.591379
$70$	0	0
$71$	−11.2914	−1.34004	−0.670019	−	0.742344i	$-0.733714\pi$
$71$	−11.2914	−1.34004	−0.670019	+	0.742344i	$0.733714\pi$
$72$	0	0
$73$	−4.17094	−0.488171	−0.244086	−	0.969754i	$-0.578488\pi$
$73$	−4.17094	−0.488171	−0.244086	+	0.969754i	$0.578488\pi$
$74$	0	0
$75$	8.28945	0.957184
$76$	0	0
$77$	−4.35456	−0.496248
$78$	0	0
$79$	−9.48767	−1.06745	−0.533723	−	0.845659i	$-0.679208\pi$
$79$	−9.48767	−1.06745	−0.533723	+	0.845659i	$0.679208\pi$
$80$	0	0
$81$	−8.88509	−0.987232
$82$	0	0
$83$	−14.8415	−1.62907	−0.814534	−	0.580115i	$-0.803008\pi$
$83$	−14.8415	−1.62907	−0.814534	+	0.580115i	$0.803008\pi$
$84$	0	0
$85$	16.6579	1.80680
$86$	0	0
$87$	6.24037	0.669038
$88$	0	0
$89$	−5.57601	−0.591055	−0.295528	−	0.955334i	$-0.595495\pi$
$89$	−5.57601	−0.591055	−0.295528	+	0.955334i	$0.595495\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	10.6208	1.10132
$94$	0	0
$95$	7.83168	0.803513
$96$	0	0
$97$	−0.979669	−0.0994703	−0.0497352	−	0.998762i	$-0.515838\pi$
$97$	−0.979669	−0.0994703	−0.0497352	+	0.998762i	$0.515838\pi$
$98$	0	0
$99$	0.164720	0.0165550
$100$	0	0
$101$	−17.0623	−1.69776	−0.848880	−	0.528586i	$-0.822722\pi$
$101$	−17.0623	−1.69776	−0.848880	+	0.528586i	$0.822722\pi$
$102$	0	0
$103$	−3.19127	−0.314445	−0.157223	−	0.987563i	$-0.550254\pi$
$103$	−3.19127	−0.314445	−0.157223	+	0.987563i	$0.550254\pi$
$104$	0	0
$105$	5.39238	0.526243
$106$	0	0
$107$	4.89968	0.473670	0.236835	−	0.971550i	$-0.423890\pi$
$107$	4.89968	0.473670	0.236835	+	0.971550i	$0.423890\pi$
$108$	0	0
$109$	−2.12546	−0.203582	−0.101791	−	0.994806i	$-0.532457\pi$
$109$	−2.12546	−0.203582	−0.101791	+	0.994806i	$0.532457\pi$
$110$	0	0
$111$	16.8612	1.60039
$112$	0	0
$113$	−2.03018	−0.190983	−0.0954916	−	0.995430i	$-0.530442\pi$
$113$	−2.03018	−0.190983	−0.0954916	+	0.995430i	$0.530442\pi$
$114$	0	0
$115$	8.94254	0.833897
$116$	0	0
$117$	0.0378271	0.00349711
$118$	0	0
$119$	5.31673	0.487384
$120$	0	0
$121$	7.96217	0.723834
$122$	0	0
$123$	18.3651	1.65593
$124$	0	0
$125$	−0.575303	−0.0514567
$126$	0	0
$127$	9.57027	0.849224	0.424612	−	0.905375i	$-0.360411\pi$
$127$	9.57027	0.849224	0.424612	+	0.905375i	$0.360411\pi$
$128$	0	0
$129$	0.774219	0.0681662
$130$	0	0
$131$	18.7848	1.64123	0.820616	−	0.571479i	$-0.193630\pi$
$131$	18.7848	1.64123	0.820616	+	0.571479i	$0.193630\pi$
$132$	0	0
$133$	2.49965	0.216747
$134$	0	0
$135$	−16.3811	−1.40986
$136$	0	0
$137$	−2.87384	−0.245528	−0.122764	−	0.992436i	$-0.539176\pi$
$137$	−2.87384	−0.245528	−0.122764	+	0.992436i	$0.539176\pi$
$138$	0	0
$139$	−0.125462	−0.0106416	−0.00532078	−	0.999986i	$-0.501694\pi$
$139$	−0.125462	−0.0106416	−0.00532078	+	0.999986i	$0.501694\pi$
$140$	0	0
$141$	−20.4455	−1.72182
$142$	0	0
$143$	4.35456	0.364146
$144$	0	0
$145$	11.3601	0.943403
$146$	0	0
$147$	1.72110	0.141954
$148$	0	0
$149$	−20.8717	−1.70988	−0.854938	−	0.518730i	$-0.826405\pi$
$149$	−20.8717	−1.70988	−0.854938	+	0.518730i	$0.826405\pi$
$150$	0	0
$151$	−1.49200	−0.121417	−0.0607087	−	0.998156i	$-0.519336\pi$
$151$	−1.49200	−0.121417	−0.0607087	+	0.998156i	$0.519336\pi$
$152$	0	0
$153$	−0.201116	−0.0162593
$154$	0	0
$155$	19.3342	1.55296
$156$	0	0
$157$	−10.9727	−0.875719	−0.437859	−	0.899044i	$-0.644263\pi$
$157$	−10.9727	−0.875719	−0.437859	+	0.899044i	$0.644263\pi$
$158$	0	0
$159$	10.1408	0.804214
$160$	0	0
$161$	2.85421	0.224943
$162$	0	0
$163$	14.0503	1.10050	0.550252	−	0.834999i	$-0.314532\pi$
$163$	14.0503	1.10050	0.550252	+	0.834999i	$0.314532\pi$
$164$	0	0
$165$	−23.4815	−1.82803
$166$	0	0
$167$	10.9578	0.847943	0.423972	−	0.905676i	$-0.360636\pi$
$167$	10.9578	0.847943	0.423972	+	0.905676i	$0.360636\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.0945544	−0.00723076
$172$	0	0
$173$	−12.7589	−0.970043	−0.485021	−	0.874502i	$-0.661188\pi$
$173$	−12.7589	−0.970043	−0.485021	+	0.874502i	$0.661188\pi$
$174$	0	0
$175$	4.81638	0.364084
$176$	0	0
$177$	13.4408	1.01027
$178$	0	0
$179$	−6.89274	−0.515187	−0.257594	−	0.966253i	$-0.582930\pi$
$179$	−6.89274	−0.515187	−0.257594	+	0.966253i	$0.582930\pi$
$180$	0	0
$181$	−10.9124	−0.811110	−0.405555	−	0.914071i	$-0.632922\pi$
$181$	−10.9124	−0.811110	−0.405555	+	0.914071i	$0.632922\pi$
$182$	0	0
$183$	−26.4699	−1.95671
$184$	0	0
$185$	30.6943	2.25669
$186$	0	0
$187$	−23.1520	−1.69304
$188$	0	0
$189$	−5.22839	−0.380310
$190$	0	0
$191$	−8.89968	−0.643958	−0.321979	−	0.946747i	$-0.604348\pi$
$191$	−8.89968	−0.643958	−0.321979	+	0.946747i	$0.604348\pi$
$192$	0	0
$193$	2.70912	0.195006	0.0975032	−	0.995235i	$-0.468914\pi$
$193$	2.70912	0.195006	0.0975032	+	0.995235i	$0.468914\pi$
$194$	0	0
$195$	−5.39238	−0.386157
$196$	0	0
$197$	19.0728	1.35888	0.679441	−	0.733730i	$-0.262222\pi$
$197$	19.0728	1.35888	0.679441	+	0.733730i	$0.262222\pi$
$198$	0	0
$199$	−17.0251	−1.20688	−0.603440	−	0.797408i	$-0.706204\pi$
$199$	−17.0251	−1.20688	−0.603440	+	0.797408i	$0.706204\pi$
$200$	0	0
$201$	−10.0625	−0.709753
$202$	0	0
$203$	3.62581	0.254482
$204$	0	0
$205$	33.4321	2.33500
$206$	0	0
$207$	−0.107966	−0.00750418
$208$	0	0
$209$	−10.8849	−0.752922
$210$	0	0
$211$	−16.4245	−1.13071	−0.565354	−	0.824849i	$-0.691260\pi$
$211$	−16.4245	−1.13071	−0.565354	+	0.824849i	$0.691260\pi$
$212$	0	0
$213$	−19.4335	−1.33156
$214$	0	0
$215$	1.40940	0.0961203
$216$	0	0
$217$	6.17094	0.418911
$218$	0	0
$219$	−7.17859	−0.485084
$220$	0	0
$221$	−5.31673	−0.357642
$222$	0	0
$223$	10.5128	0.703990	0.351995	−	0.936002i	$-0.385503\pi$
$223$	10.5128	0.703990	0.351995	+	0.936002i	$0.385503\pi$
$224$	0	0
$225$	−0.182190	−0.0121460
$226$	0	0
$227$	12.2815	0.815153	0.407576	−	0.913171i	$-0.366374\pi$
$227$	12.2815	0.815153	0.407576	+	0.913171i	$0.366374\pi$
$228$	0	0
$229$	−4.82403	−0.318781	−0.159390	−	0.987216i	$-0.550953\pi$
$229$	−4.82403	−0.318781	−0.159390	+	0.987216i	$0.550953\pi$
$230$	0	0
$231$	−7.49461	−0.493109
$232$	0	0
$233$	23.1808	1.51862	0.759312	−	0.650727i	$-0.225536\pi$
$233$	23.1808	1.51862	0.759312	+	0.650727i	$0.225536\pi$
$234$	0	0
$235$	−37.2193	−2.42792
$236$	0	0
$237$	−16.3292	−1.06069
$238$	0	0
$239$	−4.55781	−0.294820	−0.147410	−	0.989075i	$-0.547094\pi$
$239$	−4.55781	−0.294820	−0.147410	+	0.989075i	$0.547094\pi$
$240$	0	0
$241$	−7.59130	−0.488999	−0.244499	−	0.969649i	$-0.578624\pi$
$241$	−7.59130	−0.488999	−0.244499	+	0.969649i	$0.578624\pi$
$242$	0	0
$243$	0.393087	0.0252165
$244$	0	0
$245$	3.13311	0.200167
$246$	0	0
$247$	−2.49965	−0.159049
$248$	0	0
$249$	−25.5437	−1.61877
$250$	0	0
$251$	5.10293	0.322094	0.161047	−	0.986947i	$-0.448513\pi$
$251$	5.10293	0.322094	0.161047	+	0.986947i	$0.448513\pi$
$252$	0	0
$253$	−12.4288	−0.781392
$254$	0	0
$255$	28.6699	1.79538
$256$	0	0
$257$	22.4673	1.40147	0.700737	−	0.713420i	$-0.252855\pi$
$257$	22.4673	1.40147	0.700737	+	0.713420i	$0.252855\pi$
$258$	0	0
$259$	9.79675	0.608740
$260$	0	0
$261$	−0.137154	−0.00848961
$262$	0	0
$263$	2.81568	0.173622	0.0868111	−	0.996225i	$-0.472332\pi$
$263$	2.81568	0.173622	0.0868111	+	0.996225i	$0.472332\pi$
$264$	0	0
$265$	18.4604	1.13401
$266$	0	0
$267$	−9.59684	−0.587317
$268$	0	0
$269$	−27.9979	−1.70706	−0.853530	−	0.521044i	$-0.825543\pi$
$269$	−27.9979	−1.70706	−0.853530	+	0.521044i	$0.825543\pi$
$270$	0	0
$271$	14.9229	0.906503	0.453251	−	0.891383i	$-0.350264\pi$
$271$	14.9229	0.906503	0.453251	+	0.891383i	$0.350264\pi$
$272$	0	0
$273$	−1.72110	−0.104165
$274$	0	0
$275$	−20.9732	−1.26473
$276$	0	0
$277$	9.71536	0.583739	0.291870	−	0.956458i	$-0.405723\pi$
$277$	9.71536	0.583739	0.291870	+	0.956458i	$0.405723\pi$
$278$	0	0
$279$	−0.233429	−0.0139750
$280$	0	0
$281$	18.5837	1.10861	0.554304	−	0.832314i	$-0.312985\pi$
$281$	18.5837	1.10861	0.554304	+	0.832314i	$0.312985\pi$
$282$	0	0
$283$	−8.67059	−0.515413	−0.257706	−	0.966223i	$-0.582967\pi$
$283$	−8.67059	−0.515413	−0.257706	+	0.966223i	$0.582967\pi$
$284$	0	0
$285$	13.4791	0.798431
$286$	0	0
$287$	10.6706	0.629865
$288$	0	0
$289$	11.2676	0.662801
$290$	0	0
$291$	−1.68610	−0.0988412
$292$	0	0
$293$	8.46039	0.494261	0.247131	−	0.968982i	$-0.420512\pi$
$293$	8.46039	0.494261	0.247131	+	0.968982i	$0.420512\pi$
$294$	0	0
$295$	24.4678	1.42457
$296$	0	0
$297$	22.7673	1.32110
$298$	0	0
$299$	−2.85421	−0.165063
$300$	0	0
$301$	0.449841	0.0259284
$302$	0	0
$303$	−29.3658	−1.68702
$304$	0	0
$305$	−48.1863	−2.75914
$306$	0	0
$307$	−10.6902	−0.610123	−0.305061	−	0.952333i	$-0.598677\pi$
$307$	−10.6902	−0.610123	−0.305061	+	0.952333i	$0.598677\pi$
$308$	0	0
$309$	−5.49248	−0.312456
$310$	0	0
$311$	17.2157	0.976213	0.488107	−	0.872784i	$-0.337688\pi$
$311$	17.2157	0.976213	0.488107	+	0.872784i	$0.337688\pi$
$312$	0	0
$313$	−12.2921	−0.694789	−0.347394	−	0.937719i	$-0.612933\pi$
$313$	−12.2921	−0.694789	−0.347394	+	0.937719i	$0.612933\pi$
$314$	0	0
$315$	−0.118516	−0.00667764
$316$	0	0
$317$	−27.1638	−1.52567	−0.762835	−	0.646594i	$-0.776193\pi$
$317$	−27.1638	−1.52567	−0.762835	+	0.646594i	$0.776193\pi$
$318$	0	0
$319$	−15.7888	−0.884004
$320$	0	0
$321$	8.43282	0.470674
$322$	0	0
$323$	13.2900	0.739473
$324$	0	0
$325$	−4.81638	−0.267165
$326$	0	0
$327$	−3.65813	−0.202295
$328$	0	0
$329$	−11.8794	−0.654930
$330$	0	0
$331$	−17.3936	−0.956038	−0.478019	−	0.878349i	$-0.658645\pi$
$331$	−17.3936	−0.956038	−0.478019	+	0.878349i	$0.658645\pi$
$332$	0	0
$333$	−0.370582	−0.0203078
$334$	0	0
$335$	−18.3179	−1.00081
$336$	0	0
$337$	−2.03018	−0.110591	−0.0552955	−	0.998470i	$-0.517610\pi$
$337$	−2.03018	−0.110591	−0.0552955	+	0.998470i	$0.517610\pi$
$338$	0	0
$339$	−3.49413	−0.189775
$340$	0	0
$341$	−26.8717	−1.45518
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	15.3910	0.828623
$346$	0	0
$347$	−10.1499	−0.544875	−0.272438	−	0.962173i	$-0.587830\pi$
$347$	−10.1499	−0.544875	−0.272438	+	0.962173i	$0.587830\pi$
$348$	0	0
$349$	24.3747	1.30475	0.652373	−	0.757898i	$-0.273774\pi$
$349$	24.3747	1.30475	0.652373	+	0.757898i	$0.273774\pi$
$350$	0	0
$351$	5.22839	0.279071
$352$	0	0
$353$	11.1374	0.592786	0.296393	−	0.955066i	$-0.404216\pi$
$353$	11.1374	0.592786	0.296393	+	0.955066i	$0.404216\pi$
$354$	0	0
$355$	−35.3771	−1.87762
$356$	0	0
$357$	9.15061	0.484301
$358$	0	0
$359$	−0.859944	−0.0453861	−0.0226931	−	0.999742i	$-0.507224\pi$
$359$	−0.859944	−0.0453861	−0.0226931	+	0.999742i	$0.507224\pi$
$360$	0	0
$361$	−12.7518	−0.671145
$362$	0	0
$363$	13.7037	0.719256
$364$	0	0
$365$	−13.0680	−0.684011
$366$	0	0
$367$	−19.5423	−1.02010	−0.510050	−	0.860145i	$-0.670373\pi$
$367$	−19.5423	−1.02010	−0.510050	+	0.860145i	$0.670373\pi$
$368$	0	0
$369$	−0.403637	−0.0210125
$370$	0	0
$371$	5.89203	0.305899
$372$	0	0
$373$	−8.44290	−0.437157	−0.218578	−	0.975819i	$-0.570142\pi$
$373$	−8.44290	−0.437157	−0.218578	+	0.975819i	$0.570142\pi$
$374$	0	0
$375$	−0.990152	−0.0511312
$376$	0	0
$377$	−3.62581	−0.186739
$378$	0	0
$379$	36.3424	1.86678	0.933391	−	0.358862i	$-0.116835\pi$
$379$	36.3424	1.86678	0.933391	+	0.358862i	$0.116835\pi$
$380$	0	0
$381$	16.4714	0.843853
$382$	0	0
$383$	19.4037	0.991481	0.495740	−	0.868471i	$-0.334897\pi$
$383$	19.4037	0.991481	0.495740	+	0.868471i	$0.334897\pi$
$384$	0	0
$385$	−13.6433	−0.695327
$386$	0	0
$387$	−0.0170162	−0.000864980 0
$388$	0	0
$389$	−24.8757	−1.26125	−0.630625	−	0.776088i	$-0.717201\pi$
$389$	−24.8757	−1.26125	−0.630625	+	0.776088i	$0.717201\pi$
$390$	0	0
$391$	15.1750	0.767435
$392$	0	0
$393$	32.3304	1.63085
$394$	0	0
$395$	−29.7259	−1.49567
$396$	0	0
$397$	−25.7019	−1.28994	−0.644972	−	0.764206i	$-0.723131\pi$
$397$	−25.7019	−1.28994	−0.644972	+	0.764206i	$0.723131\pi$
$398$	0	0
$399$	4.30214	0.215376
$400$	0	0
$401$	−6.44149	−0.321673	−0.160836	−	0.986981i	$-0.551419\pi$
$401$	−6.44149	−0.321673	−0.160836	+	0.986981i	$0.551419\pi$
$402$	0	0
$403$	−6.17094	−0.307396
$404$	0	0
$405$	−27.8380	−1.38328
$406$	0	0
$407$	−42.6605	−2.11460
$408$	0	0
$409$	2.04286	0.101013	0.0505065	−	0.998724i	$-0.483916\pi$
$409$	2.04286	0.101013	0.0505065	+	0.998724i	$0.483916\pi$
$410$	0	0
$411$	−4.94615	−0.243975
$412$	0	0
$413$	7.80943	0.384277
$414$	0	0
$415$	−46.5001	−2.28260
$416$	0	0
$417$	−0.215933	−0.0105743
$418$	0	0
$419$	3.73639	0.182535	0.0912674	−	0.995826i	$-0.470908\pi$
$419$	3.73639	0.182535	0.0912674	+	0.995826i	$0.470908\pi$
$420$	0	0
$421$	39.0383	1.90261	0.951305	−	0.308250i	$-0.0997434\pi$
$421$	39.0383	1.90261	0.951305	+	0.308250i	$0.0997434\pi$
$422$	0	0
$423$	0.449361	0.0218487
$424$	0	0
$425$	25.6074	1.24214
$426$	0	0
$427$	−15.3797	−0.744276
$428$	0	0
$429$	7.49461	0.361843
$430$	0	0
$431$	40.7441	1.96257	0.981287	−	0.192549i	$-0.0616755\pi$
$431$	40.7441	1.96257	0.981287	+	0.192549i	$0.0616755\pi$
$432$	0	0
$433$	11.6342	0.559102	0.279551	−	0.960131i	$-0.409814\pi$
$433$	11.6342	0.559102	0.279551	+	0.960131i	$0.409814\pi$
$434$	0	0
$435$	19.5518	0.937436
$436$	0	0
$437$	7.13451	0.341290
$438$	0	0
$439$	−7.07495	−0.337669	−0.168835	−	0.985644i	$-0.554000\pi$
$439$	−7.07495	−0.337669	−0.168835	+	0.985644i	$0.554000\pi$
$440$	0	0
$441$	−0.0378271	−0.00180129
$442$	0	0
$443$	10.3349	0.491027	0.245514	−	0.969393i	$-0.421043\pi$
$443$	10.3349	0.491027	0.245514	+	0.969393i	$0.421043\pi$
$444$	0	0
$445$	−17.4702	−0.828169
$446$	0	0
$447$	−35.9222	−1.69906
$448$	0	0
$449$	30.8001	1.45354	0.726772	−	0.686878i	$-0.241019\pi$
$449$	30.8001	1.45354	0.726772	+	0.686878i	$0.241019\pi$
$450$	0	0
$451$	−46.4657	−2.18798
$452$	0	0
$453$	−2.56788	−0.120649
$454$	0	0
$455$	−3.13311	−0.146882
$456$	0	0
$457$	−32.0900	−1.50111	−0.750554	−	0.660809i	$-0.770213\pi$
$457$	−32.0900	−1.50111	−0.750554	+	0.660809i	$0.770213\pi$
$458$	0	0
$459$	−27.7980	−1.29750
$460$	0	0
$461$	26.4765	1.23313	0.616566	−	0.787303i	$-0.288523\pi$
$461$	26.4765	1.23313	0.616566	+	0.787303i	$0.288523\pi$
$462$	0	0
$463$	32.8949	1.52876	0.764379	−	0.644768i	$-0.223046\pi$
$463$	32.8949	1.52876	0.764379	+	0.644768i	$0.223046\pi$
$464$	0	0
$465$	33.2761	1.54314
$466$	0	0
$467$	1.92483	0.0890703	0.0445352	−	0.999008i	$-0.485819\pi$
$467$	1.92483	0.0890703	0.0445352	+	0.999008i	$0.485819\pi$
$468$	0	0
$469$	−5.84656	−0.269969
$470$	0	0
$471$	−18.8851	−0.870180
$472$	0	0
$473$	−1.95886	−0.0900684
$474$	0	0
$475$	12.0393	0.552399
$476$	0	0
$477$	−0.222878	−0.0102049
$478$	0	0
$479$	32.6409	1.49140	0.745700	−	0.666282i	$-0.232115\pi$
$479$	32.6409	1.49140	0.745700	+	0.666282i	$0.232115\pi$
$480$	0	0
$481$	−9.79675	−0.446693
$482$	0	0
$483$	4.91236	0.223520
$484$	0	0
$485$	−3.06941	−0.139375
$486$	0	0
$487$	−23.4283	−1.06164	−0.530819	−	0.847485i	$-0.678116\pi$
$487$	−23.4283	−1.06164	−0.530819	+	0.847485i	$0.678116\pi$
$488$	0	0
$489$	24.1819	1.09354
$490$	0	0
$491$	23.0007	1.03801	0.519004	−	0.854772i	$-0.326303\pi$
$491$	23.0007	1.03801	0.519004	+	0.854772i	$0.326303\pi$
$492$	0	0
$493$	19.2775	0.868214
$494$	0	0
$495$	0.516086	0.0231964
$496$	0	0
$497$	−11.2914	−0.506487
$498$	0	0
$499$	−14.8971	−0.666884	−0.333442	−	0.942771i	$-0.608210\pi$
$499$	−14.8971	−0.666884	−0.333442	+	0.942771i	$0.608210\pi$
$500$	0	0
$501$	18.8595	0.842580
$502$	0	0
$503$	19.2516	0.858388	0.429194	−	0.903212i	$-0.358798\pi$
$503$	19.2516	0.858388	0.429194	+	0.903212i	$0.358798\pi$
$504$	0	0
$505$	−53.4580	−2.37885
$506$	0	0
$507$	1.72110	0.0764366
$508$	0	0
$509$	29.1230	1.29086	0.645428	−	0.763821i	$-0.276679\pi$
$509$	29.1230	1.29086	0.645428	+	0.763821i	$0.276679\pi$
$510$	0	0
$511$	−4.17094	−0.184511
$512$	0	0
$513$	−13.0691	−0.577017
$514$	0	0
$515$	−9.99860	−0.440591
$516$	0	0
$517$	51.7293	2.27505
$518$	0	0
$519$	−21.9593	−0.963908
$520$	0	0
$521$	4.51714	0.197900	0.0989499	−	0.995092i	$-0.468452\pi$
$521$	4.51714	0.197900	0.0989499	+	0.995092i	$0.468452\pi$
$522$	0	0
$523$	34.7812	1.52088	0.760439	−	0.649410i	$-0.224984\pi$
$523$	34.7812	1.52088	0.760439	+	0.649410i	$0.224984\pi$
$524$	0	0
$525$	8.28945	0.361781
$526$	0	0
$527$	32.8092	1.42919
$528$	0	0
$529$	−14.8535	−0.645805
$530$	0	0
$531$	−0.295408	−0.0128196
$532$	0	0
$533$	−10.6706	−0.462194
$534$	0	0
$535$	15.3512	0.663692
$536$	0	0
$537$	−11.8631	−0.511929
$538$	0	0
$539$	−4.35456	−0.187564
$540$	0	0
$541$	−33.1750	−1.42631	−0.713153	−	0.701008i	$-0.752734\pi$
$541$	−33.1750	−1.42631	−0.713153	+	0.701008i	$0.752734\pi$
$542$	0	0
$543$	−18.7812	−0.805980
$544$	0	0
$545$	−6.65931	−0.285253
$546$	0	0
$547$	−2.08400	−0.0891056	−0.0445528	−	0.999007i	$-0.514186\pi$
$547$	−2.08400	−0.0891056	−0.0445528	+	0.999007i	$0.514186\pi$
$548$	0	0
$549$	0.581769	0.0248293
$550$	0	0
$551$	9.06326	0.386108
$552$	0	0
$553$	−9.48767	−0.403457
$554$	0	0
$555$	52.8278	2.24242
$556$	0	0
$557$	−40.8118	−1.72925	−0.864626	−	0.502416i	$-0.832445\pi$
$557$	−40.8118	−1.72925	−0.864626	+	0.502416i	$0.832445\pi$
$558$	0	0
$559$	−0.449841	−0.0190262
$560$	0	0
$561$	−39.8468	−1.68234
$562$	0	0
$563$	−8.41966	−0.354846	−0.177423	−	0.984135i	$-0.556776\pi$
$563$	−8.41966	−0.354846	−0.177423	+	0.984135i	$0.556776\pi$
$564$	0	0
$565$	−6.36077	−0.267600
$566$	0	0
$567$	−8.88509	−0.373139
$568$	0	0
$569$	28.8542	1.20963	0.604816	−	0.796366i	$-0.293247\pi$
$569$	28.8542	1.20963	0.604816	+	0.796366i	$0.293247\pi$
$570$	0	0
$571$	−39.1787	−1.63958	−0.819788	−	0.572667i	$-0.805909\pi$
$571$	−39.1787	−1.63958	−0.819788	+	0.572667i	$0.805909\pi$
$572$	0	0
$573$	−15.3172	−0.639886
$574$	0	0
$575$	13.7469	0.573287
$576$	0	0
$577$	6.15131	0.256082	0.128041	−	0.991769i	$-0.459131\pi$
$577$	6.15131	0.256082	0.128041	+	0.991769i	$0.459131\pi$
$578$	0	0
$579$	4.66265	0.193773
$580$	0	0
$581$	−14.8415	−0.615730
$582$	0	0
$583$	−25.6572	−1.06261
$584$	0	0
$585$	0.118516	0.00490005
$586$	0	0
$587$	−47.4999	−1.96053	−0.980265	−	0.197686i	$-0.936657\pi$
$587$	−47.4999	−1.96053	−0.980265	+	0.197686i	$0.936657\pi$
$588$	0	0
$589$	15.4252	0.635583
$590$	0	0
$591$	32.8262	1.35029
$592$	0	0
$593$	31.6402	1.29931	0.649653	−	0.760231i	$-0.274914\pi$
$593$	31.6402	1.29931	0.649653	+	0.760231i	$0.274914\pi$
$594$	0	0
$595$	16.6579	0.682907
$596$	0	0
$597$	−29.3019	−1.19925
$598$	0	0
$599$	−6.95382	−0.284125	−0.142063	−	0.989858i	$-0.545373\pi$
$599$	−6.95382	−0.284125	−0.142063	+	0.989858i	$0.545373\pi$
$600$	0	0
$601$	−12.3825	−0.505094	−0.252547	−	0.967585i	$-0.581268\pi$
$601$	−12.3825	−0.505094	−0.252547	+	0.967585i	$0.581268\pi$
$602$	0	0
$603$	0.221158	0.00900626
$604$	0	0
$605$	24.9464	1.01421
$606$	0	0
$607$	−42.3830	−1.72027	−0.860137	−	0.510063i	$-0.829622\pi$
$607$	−42.3830	−1.72027	−0.860137	+	0.510063i	$0.829622\pi$
$608$	0	0
$609$	6.24037	0.252873
$610$	0	0
$611$	11.8794	0.480587
$612$	0	0
$613$	21.3644	0.862900	0.431450	−	0.902137i	$-0.358002\pi$
$613$	21.3644	0.862900	0.431450	+	0.902137i	$0.358002\pi$
$614$	0	0
$615$	57.5399	2.32023
$616$	0	0
$617$	−21.6189	−0.870343	−0.435171	−	0.900348i	$-0.643312\pi$
$617$	−21.6189	−0.870343	−0.435171	+	0.900348i	$0.643312\pi$
$618$	0	0
$619$	40.2256	1.61680	0.808401	−	0.588632i	$-0.200333\pi$
$619$	40.2256	1.61680	0.808401	+	0.588632i	$0.200333\pi$
$620$	0	0
$621$	−14.9229	−0.598836
$622$	0	0
$623$	−5.57601	−0.223398
$624$	0	0
$625$	−25.8844	−1.03538
$626$	0	0
$627$	−18.7339	−0.748160
$628$	0	0
$629$	52.0867	2.07683
$630$	0	0
$631$	12.2902	0.489264	0.244632	−	0.969616i	$-0.421333\pi$
$631$	12.2902	0.489264	0.244632	+	0.969616i	$0.421333\pi$
$632$	0	0
$633$	−28.2681	−1.12356
$634$	0	0
$635$	29.9847	1.18991
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	0.427119	0.0168966
$640$	0	0
$641$	32.4338	1.28106	0.640529	−	0.767934i	$-0.278715\pi$
$641$	32.4338	1.28106	0.640529	+	0.767934i	$0.278715\pi$
$642$	0	0
$643$	1.71989	0.0678258	0.0339129	−	0.999425i	$-0.489203\pi$
$643$	1.71989	0.0678258	0.0339129	+	0.999425i	$0.489203\pi$
$644$	0	0
$645$	2.42571	0.0955124
$646$	0	0
$647$	5.52259	0.217116	0.108558	−	0.994090i	$-0.465377\pi$
$647$	5.52259	0.217116	0.108558	+	0.994090i	$0.465377\pi$
$648$	0	0
$649$	−34.0066	−1.33488
$650$	0	0
$651$	10.6208	0.416261
$652$	0	0
$653$	−18.2256	−0.713221	−0.356611	−	0.934253i	$-0.616068\pi$
$653$	−18.2256	−0.713221	−0.356611	+	0.934253i	$0.616068\pi$
$654$	0	0
$655$	58.8548	2.29965
$656$	0	0
$657$	0.157774	0.00615536
$658$	0	0
$659$	−49.8260	−1.94095	−0.970473	−	0.241211i	$-0.922456\pi$
$659$	−49.8260	−1.94095	−0.970473	+	0.241211i	$0.922456\pi$
$660$	0	0
$661$	−26.8009	−1.04243	−0.521216	−	0.853425i	$-0.674522\pi$
$661$	−26.8009	−1.04243	−0.521216	+	0.853425i	$0.674522\pi$
$662$	0	0
$663$	−9.15061	−0.355380
$664$	0	0
$665$	7.83168	0.303699
$666$	0	0
$667$	10.3488	0.400708
$668$	0	0
$669$	18.0936	0.699538
$670$	0	0
$671$	66.9718	2.58542
$672$	0	0
$673$	16.7800	0.646820	0.323410	−	0.946259i	$-0.395171\pi$
$673$	16.7800	0.646820	0.323410	+	0.946259i	$0.395171\pi$
$674$	0	0
$675$	−25.1819	−0.969253
$676$	0	0
$677$	39.9241	1.53441	0.767204	−	0.641403i	$-0.221647\pi$
$677$	39.9241	1.53441	0.767204	+	0.641403i	$0.221647\pi$
$678$	0	0
$679$	−0.979669	−0.0375962
$680$	0	0
$681$	21.1377	0.809997
$682$	0	0
$683$	29.7862	1.13974	0.569869	−	0.821736i	$-0.306994\pi$
$683$	29.7862	1.13974	0.569869	+	0.821736i	$0.306994\pi$
$684$	0	0
$685$	−9.00404	−0.344027
$686$	0	0
$687$	−8.30262	−0.316765
$688$	0	0
$689$	−5.89203	−0.224469
$690$	0	0
$691$	−29.2074	−1.11110	−0.555550	−	0.831483i	$-0.687492\pi$
$691$	−29.2074	−1.11110	−0.555550	+	0.831483i	$0.687492\pi$
$692$	0	0
$693$	0.164720	0.00625720
$694$	0	0
$695$	−0.393087	−0.0149106
$696$	0	0
$697$	56.7326	2.14890
$698$	0	0
$699$	39.8964	1.50902
$700$	0	0
$701$	−20.8927	−0.789108	−0.394554	−	0.918873i	$-0.629101\pi$
$701$	−20.8927	−0.789108	−0.394554	+	0.918873i	$0.629101\pi$
$702$	0	0
$703$	24.4884	0.923599
$704$	0	0
$705$	−64.0580	−2.41256
$706$	0	0
$707$	−17.0623	−0.641693
$708$	0	0
$709$	−35.4281	−1.33053	−0.665265	−	0.746607i	$-0.731682\pi$
$709$	−35.4281	−1.33053	−0.665265	+	0.746607i	$0.731682\pi$
$710$	0	0
$711$	0.358891	0.0134595
$712$	0	0
$713$	17.6131	0.659617
$714$	0	0
$715$	13.6433	0.510231
$716$	0	0
$717$	−7.84443	−0.292955
$718$	0	0
$719$	−4.19971	−0.156623	−0.0783114	−	0.996929i	$-0.524953\pi$
$719$	−4.19971	−0.156623	−0.0783114	+	0.996929i	$0.524953\pi$
$720$	0	0
$721$	−3.19127	−0.118849
$722$	0	0
$723$	−13.0654	−0.485906
$724$	0	0
$725$	17.4633	0.648570
$726$	0	0
$727$	−7.17738	−0.266194	−0.133097	−	0.991103i	$-0.542492\pi$
$727$	−7.17738	−0.266194	−0.133097	+	0.991103i	$0.542492\pi$
$728$	0	0
$729$	27.3318	1.01229
$730$	0	0
$731$	2.39168	0.0884596
$732$	0	0
$733$	1.73158	0.0639574	0.0319787	−	0.999489i	$-0.489819\pi$
$733$	1.73158	0.0639574	0.0319787	+	0.999489i	$0.489819\pi$
$734$	0	0
$735$	5.39238	0.198901
$736$	0	0
$737$	25.4592	0.937801
$738$	0	0
$739$	28.5933	1.05182	0.525910	−	0.850540i	$-0.323725\pi$
$739$	28.5933	1.05182	0.525910	+	0.850540i	$0.323725\pi$
$740$	0	0
$741$	−4.30214	−0.158043
$742$	0	0
$743$	27.5375	1.01025	0.505127	−	0.863045i	$-0.331446\pi$
$743$	27.5375	1.01025	0.505127	+	0.863045i	$0.331446\pi$
$744$	0	0
$745$	−65.3933	−2.39583
$746$	0	0
$747$	0.561411	0.0205410
$748$	0	0
$749$	4.89968	0.179030
$750$	0	0
$751$	12.3582	0.450956	0.225478	−	0.974248i	$-0.427606\pi$
$751$	12.3582	0.450956	0.225478	+	0.974248i	$0.427606\pi$
$752$	0	0
$753$	8.78264	0.320057
$754$	0	0
$755$	−4.67460	−0.170126
$756$	0	0
$757$	8.21861	0.298711	0.149355	−	0.988784i	$-0.452280\pi$
$757$	8.21861	0.298711	0.149355	+	0.988784i	$0.452280\pi$
$758$	0	0
$759$	−21.3912	−0.776451
$760$	0	0
$761$	12.8145	0.464524	0.232262	−	0.972653i	$-0.425387\pi$
$761$	12.8145	0.464524	0.232262	+	0.972653i	$0.425387\pi$
$762$	0	0
$763$	−2.12546	−0.0769469
$764$	0	0
$765$	−0.630120	−0.0227820
$766$	0	0
$767$	−7.80943	−0.281982
$768$	0	0
$769$	−9.13524	−0.329425	−0.164713	−	0.986342i	$-0.552670\pi$
$769$	−9.13524	−0.329425	−0.164713	+	0.986342i	$0.552670\pi$
$770$	0	0
$771$	38.6685	1.39261
$772$	0	0
$773$	42.9347	1.54425	0.772126	−	0.635469i	$-0.219193\pi$
$773$	42.9347	1.54425	0.772126	+	0.635469i	$0.219193\pi$
$774$	0	0
$775$	29.7216	1.06763
$776$	0	0
$777$	16.8612	0.604890
$778$	0	0
$779$	26.6727	0.955649
$780$	0	0
$781$	49.1689	1.75940
$782$	0	0
$783$	−18.9572	−0.677474
$784$	0	0
$785$	−34.3788	−1.22703
$786$	0	0
$787$	−9.97243	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$787$	−9.97243	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$788$	0	0
$789$	4.84605	0.172524
$790$	0	0
$791$	−2.03018	−0.0721849
$792$	0	0
$793$	15.3797	0.546149
$794$	0	0
$795$	31.7721	1.12684
$796$	0	0
$797$	36.0635	1.27743	0.638716	−	0.769442i	$-0.279466\pi$
$797$	36.0635	1.27743	0.638716	+	0.769442i	$0.279466\pi$
$798$	0	0
$799$	−63.1593	−2.23442
$800$	0	0
$801$	0.210924	0.00745263
$802$	0	0
$803$	18.1626	0.640944
$804$	0	0
$805$	8.94254	0.315183
$806$	0	0
$807$	−48.1870	−1.69626
$808$	0	0
$809$	−16.0027	−0.562624	−0.281312	−	0.959616i	$-0.590770\pi$
$809$	−16.0027	−0.562624	−0.281312	+	0.959616i	$0.590770\pi$
$810$	0	0
$811$	51.2176	1.79849	0.899246	−	0.437442i	$-0.144116\pi$
$811$	51.2176	1.79849	0.899246	+	0.437442i	$0.144116\pi$
$812$	0	0
$813$	25.6838	0.900769
$814$	0	0
$815$	44.0211	1.54199
$816$	0	0
$817$	1.12444	0.0393393
$818$	0	0
$819$	0.0378271	0.00132178
$820$	0	0
$821$	−14.4022	−0.502641	−0.251321	−	0.967904i	$-0.580865\pi$
$821$	−14.4022	−0.502641	−0.251321	+	0.967904i	$0.580865\pi$
$822$	0	0
$823$	−38.2488	−1.33327	−0.666635	−	0.745385i	$-0.732266\pi$
$823$	−38.2488	−1.33327	−0.666635	+	0.745385i	$0.732266\pi$
$824$	0	0
$825$	−36.0969	−1.25673
$826$	0	0
$827$	38.3529	1.33366	0.666831	−	0.745209i	$-0.267650\pi$
$827$	38.3529	1.33366	0.666831	+	0.745209i	$0.267650\pi$
$828$	0	0
$829$	−7.65720	−0.265946	−0.132973	−	0.991120i	$-0.542452\pi$
$829$	−7.65720	−0.265946	−0.132973	+	0.991120i	$0.542452\pi$
$830$	0	0
$831$	16.7211	0.580047
$832$	0	0
$833$	5.31673	0.184214
$834$	0	0
$835$	34.3321	1.18811
$836$	0	0
$837$	−32.2641	−1.11521
$838$	0	0
$839$	27.5070	0.949649	0.474824	−	0.880081i	$-0.342512\pi$
$839$	27.5070	0.949649	0.474824	+	0.880081i	$0.342512\pi$
$840$	0	0
$841$	−15.8535	−0.546672
$842$	0	0
$843$	31.9843	1.10160
$844$	0	0
$845$	3.13311	0.107782
$846$	0	0
$847$	7.96217	0.273583
$848$	0	0
$849$	−14.9229	−0.512153
$850$	0	0
$851$	27.9620	0.958523
$852$	0	0
$853$	−27.0685	−0.926807	−0.463404	−	0.886147i	$-0.653372\pi$
$853$	−27.0685	−0.926807	−0.463404	+	0.886147i	$0.653372\pi$
$854$	0	0
$855$	−0.296249	−0.0101315
$856$	0	0
$857$	4.10102	0.140088	0.0700441	−	0.997544i	$-0.477686\pi$
$857$	4.10102	0.140088	0.0700441	+	0.997544i	$0.477686\pi$
$858$	0	0
$859$	−39.5463	−1.34930	−0.674651	−	0.738137i	$-0.735706\pi$
$859$	−39.5463	−1.34930	−0.674651	+	0.738137i	$0.735706\pi$
$860$	0	0
$861$	18.3651	0.625881
$862$	0	0
$863$	14.1499	0.481668	0.240834	−	0.970566i	$-0.422579\pi$
$863$	14.1499	0.481668	0.240834	+	0.970566i	$0.422579\pi$
$864$	0	0
$865$	−39.9751	−1.35919
$866$	0	0
$867$	19.3927	0.658610
$868$	0	0
$869$	41.3146	1.40150
$870$	0	0
$871$	5.84656	0.198103
$872$	0	0
$873$	0.0370580	0.00125422
$874$	0	0
$875$	−0.575303	−0.0194488
$876$	0	0
$877$	12.9363	0.436829	0.218414	−	0.975856i	$-0.429912\pi$
$877$	12.9363	0.436829	0.218414	+	0.975856i	$0.429912\pi$
$878$	0	0
$879$	14.5611	0.491135
$880$	0	0
$881$	42.5535	1.43367	0.716833	−	0.697245i	$-0.245591\pi$
$881$	42.5535	1.43367	0.716833	+	0.697245i	$0.245591\pi$
$882$	0	0
$883$	−38.4918	−1.29535	−0.647676	−	0.761916i	$-0.724259\pi$
$883$	−38.4918	−1.29535	−0.647676	+	0.761916i	$0.724259\pi$
$884$	0	0
$885$	42.1115	1.41556
$886$	0	0
$887$	3.06106	0.102780	0.0513902	−	0.998679i	$-0.483635\pi$
$887$	3.06106	0.102780	0.0513902	+	0.998679i	$0.483635\pi$
$888$	0	0
$889$	9.57027	0.320976
$890$	0	0
$891$	38.6906	1.29618
$892$	0	0
$893$	−29.6942	−0.993679
$894$	0	0
$895$	−21.5957	−0.721865
$896$	0	0
$897$	−4.91236	−0.164019
$898$	0	0
$899$	22.3747	0.746237
$900$	0	0
$901$	31.3264	1.04363
$902$	0	0
$903$	0.774219	0.0257644
$904$	0	0
$905$	−34.1896	−1.13650
$906$	0	0
$907$	−24.4752	−0.812686	−0.406343	−	0.913721i	$-0.633196\pi$
$907$	−24.4752	−0.812686	−0.406343	+	0.913721i	$0.633196\pi$
$908$	0	0
$909$	0.645416	0.0214071
$910$	0	0
$911$	−23.1322	−0.766404	−0.383202	−	0.923665i	$-0.625179\pi$
$911$	−23.1322	−0.766404	−0.383202	+	0.923665i	$0.625179\pi$
$912$	0	0
$913$	64.6283	2.13888
$914$	0	0
$915$	−82.9333	−2.74169
$916$	0	0
$917$	18.7848	0.620328
$918$	0	0
$919$	32.9478	1.08685	0.543424	−	0.839458i	$-0.317127\pi$
$919$	32.9478	1.08685	0.543424	+	0.839458i	$0.317127\pi$
$920$	0	0
$921$	−18.3989	−0.606264
$922$	0	0
$923$	11.2914	0.371660
$924$	0	0
$925$	47.1849	1.55143
$926$	0	0
$927$	0.120716	0.00396485
$928$	0	0
$929$	34.2816	1.12474	0.562371	−	0.826885i	$-0.309889\pi$
$929$	34.2816	1.12474	0.562371	+	0.826885i	$0.309889\pi$
$930$	0	0
$931$	2.49965	0.0819227
$932$	0	0
$933$	29.6299	0.970039
$934$	0	0
$935$	−72.5378	−2.37224
$936$	0	0
$937$	−47.0855	−1.53822	−0.769108	−	0.639119i	$-0.779299\pi$
$937$	−47.0855	−1.53822	−0.769108	+	0.639119i	$0.779299\pi$
$938$	0	0
$939$	−21.1558	−0.690394
$940$	0	0
$941$	−5.79194	−0.188812	−0.0944059	−	0.995534i	$-0.530095\pi$
$941$	−5.79194	−0.188812	−0.0944059	+	0.995534i	$0.530095\pi$
$942$	0	0
$943$	30.4561	0.991786
$944$	0	0
$945$	−16.3811	−0.532878
$946$	0	0
$947$	9.73378	0.316305	0.158153	−	0.987415i	$-0.449446\pi$
$947$	9.73378	0.316305	0.158153	+	0.987415i	$0.449446\pi$
$948$	0	0
$949$	4.17094	0.135394
$950$	0	0
$951$	−46.7515	−1.51602
$952$	0	0
$953$	1.53627	0.0497646	0.0248823	−	0.999690i	$-0.492079\pi$
$953$	1.53627	0.0497646	0.0248823	+	0.999690i	$0.492079\pi$
$954$	0	0
$955$	−27.8837	−0.902295
$956$	0	0
$957$	−27.1741	−0.878413
$958$	0	0
$959$	−2.87384	−0.0928010
$960$	0	0
$961$	7.08047	0.228402
$962$	0	0
$963$	−0.185341	−0.00597252
$964$	0	0
$965$	8.48796	0.273237
$966$	0	0
$967$	18.4965	0.594808	0.297404	−	0.954752i	$-0.403879\pi$
$967$	18.4965	0.594808	0.297404	+	0.954752i	$0.403879\pi$
$968$	0	0
$969$	22.8733	0.734796
$970$	0	0
$971$	−40.1506	−1.28849	−0.644247	−	0.764818i	$-0.722829\pi$
$971$	−40.1506	−1.28849	−0.644247	+	0.764818i	$0.722829\pi$
$972$	0	0
$973$	−0.125462	−0.00402213
$974$	0	0
$975$	−8.28945	−0.265475
$976$	0	0
$977$	16.3271	0.522349	0.261174	−	0.965292i	$-0.415890\pi$
$977$	16.3271	0.522349	0.261174	+	0.965292i	$0.415890\pi$
$978$	0	0
$979$	24.2810	0.776025
$980$	0	0
$981$	0.0804000	0.00256698
$982$	0	0
$983$	−36.8946	−1.17676	−0.588378	−	0.808586i	$-0.700233\pi$
$983$	−36.8946	−1.17676	−0.588378	+	0.808586i	$0.700233\pi$
$984$	0	0
$985$	59.7572	1.90402
$986$	0	0
$987$	−20.4455	−0.650788
$988$	0	0
$989$	1.28394	0.0408269
$990$	0	0
$991$	−6.15344	−0.195471	−0.0977353	−	0.995212i	$-0.531160\pi$
$991$	−6.15344	−0.195471	−0.0977353	+	0.995212i	$0.531160\pi$
$992$	0	0
$993$	−29.9361	−0.949992
$994$	0	0
$995$	−53.3417	−1.69104
$996$	0	0
$997$	−44.7094	−1.41596	−0.707980	−	0.706232i	$-0.750394\pi$
$997$	−44.7094	−1.41596	−0.707980	+	0.706232i	$0.750394\pi$
$998$	0	0
$999$	−51.2213	−1.62057

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.v.1.3		4
4.3	odd	2		728.2.a.i.1.2	✓	4
8.3	odd	2		5824.2.a.cb.1.3		4
8.5	even	2		5824.2.a.ce.1.2		4
12.11	even	2		6552.2.a.br.1.2		4
28.27	even	2		5096.2.a.s.1.3		4
52.51	odd	2		9464.2.a.z.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.i.1.2	✓	4	4.3	odd	2
1456.2.a.v.1.3		4	1.1	even	1	trivial
5096.2.a.s.1.3		4	28.27	even	2
5824.2.a.cb.1.3		4	8.3	odd	2
5824.2.a.ce.1.2		4	8.5	even	2
6552.2.a.br.1.2		4	12.11	even	2
9464.2.a.z.1.2		4	52.51	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

\(f(q)\)	\(=\)	\(q+1.72110 q^{3} +3.13311 q^{5} +1.00000 q^{7} -0.0378271 q^{9} +O(q^{10})\)	\(q+1.72110 q^{3} +3.13311 q^{5} +1.00000 q^{7} -0.0378271 q^{9} -4.35456 q^{11} -1.00000 q^{13} +5.39238 q^{15} +5.31673 q^{17} +2.49965 q^{19} +1.72110 q^{21} +2.85421 q^{23} +4.81638 q^{25} -5.22839 q^{27} +3.62581 q^{29} +6.17094 q^{31} -7.49461 q^{33} +3.13311 q^{35} +9.79675 q^{37} -1.72110 q^{39} +10.6706 q^{41} +0.449841 q^{43} -0.118516 q^{45} -11.8794 q^{47} +1.00000 q^{49} +9.15061 q^{51} +5.89203 q^{53} -13.6433 q^{55} +4.30214 q^{57} +7.80943 q^{59} -15.3797 q^{61} -0.0378271 q^{63} -3.13311 q^{65} -5.84656 q^{67} +4.91236 q^{69} -11.2914 q^{71} -4.17094 q^{73} +8.28945 q^{75} -4.35456 q^{77} -9.48767 q^{79} -8.88509 q^{81} -14.8415 q^{83} +16.6579 q^{85} +6.24037 q^{87} -5.57601 q^{89} -1.00000 q^{91} +10.6208 q^{93} +7.83168 q^{95} -0.979669 q^{97} +0.164720 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 2 q^{5} + 4 q^{7} + 13 q^{9}+O(q^{10}) \) 4 * q - q^3 + 2 * q^5 + 4 * q^7 + 13 * q^9	\( 4 q - q^{3} + 2 q^{5} + 4 q^{7} + 13 q^{9} + q^{11} - 4 q^{13} - 10 q^{15} + 16 q^{17} + 10 q^{19} - q^{21} - 7 q^{23} + 14 q^{25} - 13 q^{27} + 4 q^{29} + q^{31} - q^{33} + 2 q^{35} + 5 q^{37} + q^{39} + 19 q^{41} - 14 q^{43} + 10 q^{45} + 13 q^{47} + 4 q^{49} - 16 q^{51} - 8 q^{53} - 2 q^{55} + 12 q^{57} + 26 q^{59} - q^{61} + 13 q^{63} - 2 q^{65} - 5 q^{67} + 17 q^{69} + 18 q^{71} + 7 q^{73} - q^{75} + q^{77} - 9 q^{79} - 4 q^{81} - 12 q^{83} + 14 q^{85} + 46 q^{87} + 4 q^{89} - 4 q^{91} + 3 q^{93} + 26 q^{95} + 25 q^{97} + 48 q^{99}+O(q^{100}) \) 4 * q - q^3 + 2 * q^5 + 4 * q^7 + 13 * q^9 + q^11 - 4 * q^13 - 10 * q^15 + 16 * q^17 + 10 * q^19 - q^21 - 7 * q^23 + 14 * q^25 - 13 * q^27 + 4 * q^29 + q^31 - q^33 + 2 * q^35 + 5 * q^37 + q^39 + 19 * q^41 - 14 * q^43 + 10 * q^45 + 13 * q^47 + 4 * q^49 - 16 * q^51 - 8 * q^53 - 2 * q^55 + 12 * q^57 + 26 * q^59 - q^61 + 13 * q^63 - 2 * q^65 - 5 * q^67 + 17 * q^69 + 18 * q^71 + 7 * q^73 - q^75 + q^77 - 9 * q^79 - 4 * q^81 - 12 * q^83 + 14 * q^85 + 46 * q^87 + 4 * q^89 - 4 * q^91 + 3 * q^93 + 26 * q^95 + 25 * q^97 + 48 * q^99

Introduction

L-functions

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