LMFDB - Embedded newform 1456.2.a.u.1.2

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:	4.4.183064.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} + 6x + 8 $ x^4 - x^3 - 10x^2 + 6x + 8
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			1.2
Root			$1.27274$ 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)

$f(q)$	$=$	$q-1.27274 q^{3} +2.87007 q^{5} -1.00000 q^{7} -1.38012 q^{9} +O(q^{10})$	$q-1.27274 q^{3} +2.87007 q^{5} -1.00000 q^{7} -1.38012 q^{9} -1.27274 q^{11} +1.00000 q^{13} -3.65287 q^{15} +3.10738 q^{17} +0.870071 q^{19} +1.27274 q^{21} +0.402673 q^{23} +3.23731 q^{25} +5.57478 q^{27} +6.52294 q^{29} +5.79568 q^{31} +1.61988 q^{33} -2.87007 q^{35} +0.727256 q^{37} -1.27274 q^{39} -6.12026 q^{41} +0.237307 q^{43} -3.96105 q^{45} +5.79568 q^{47} +1.00000 q^{49} -3.95490 q^{51} -2.52294 q^{53} -3.65287 q^{55} -1.10738 q^{57} -3.19465 q^{59} +11.5748 q^{61} +1.38012 q^{63} +2.87007 q^{65} -4.92561 q^{67} -0.512500 q^{69} +1.43811 q^{71} +3.51005 q^{73} -4.12026 q^{75} +1.27274 q^{77} +12.6883 q^{79} -2.95490 q^{81} +6.32458 q^{83} +8.91839 q^{85} -8.30203 q^{87} +16.7213 q^{89} -1.00000 q^{91} -7.37642 q^{93} +2.49717 q^{95} -6.34117 q^{97} +1.75654 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - 4 q^{7} + 9 q^{9}+O(q^{10}) $ 4 * q - q^3 - 4 * q^7 + 9 * q^9	$ 4 q - q^{3} - 4 q^{7} + 9 q^{9} - q^{11} + 4 q^{13} + 4 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} + 9 q^{23} + 14 q^{25} - 7 q^{27} - 4 q^{29} - 11 q^{31} + 21 q^{33} + 7 q^{37} - q^{39} + 13 q^{41} + 2 q^{43} + 12 q^{45} - 11 q^{47} + 4 q^{49} + 28 q^{51} + 20 q^{53} + 4 q^{55} + 6 q^{57} + 2 q^{59} + 17 q^{61} - 9 q^{63} + 3 q^{67} - 27 q^{69} + 8 q^{71} + 11 q^{73} + 21 q^{75} + q^{77} + 27 q^{79} + 32 q^{81} + 22 q^{83} - 8 q^{85} - 8 q^{87} + 10 q^{89} - 4 q^{91} - 27 q^{93} + 34 q^{95} + 17 q^{97} - 10 q^{99}+O(q^{100}) $ 4 * q - q^3 - 4 * q^7 + 9 * q^9 - q^11 + 4 * q^13 + 4 * q^15 + 2 * q^17 - 8 * q^19 + q^21 + 9 * q^23 + 14 * q^25 - 7 * q^27 - 4 * q^29 - 11 * q^31 + 21 * q^33 + 7 * q^37 - q^39 + 13 * q^41 + 2 * q^43 + 12 * q^45 - 11 * q^47 + 4 * q^49 + 28 * q^51 + 20 * q^53 + 4 * q^55 + 6 * q^57 + 2 * q^59 + 17 * q^61 - 9 * q^63 + 3 * q^67 - 27 * q^69 + 8 * q^71 + 11 * q^73 + 21 * q^75 + q^77 + 27 * q^79 + 32 * q^81 + 22 * q^83 - 8 * q^85 - 8 * q^87 + 10 * q^89 - 4 * q^91 - 27 * q^93 + 34 * q^95 + 17 * q^97 - 10 * 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
$3$	−1.27274	−0.734819	−0.367410	−	0.930059i	$-0.619755\pi$
$3$	−1.27274	−0.734819	−0.367410	+	0.930059i	$0.619755\pi$
$4$	0	0
$5$	2.87007	1.28353	0.641767	−	0.766899i	$-0.278201\pi$
$5$	2.87007	1.28353	0.641767	+	0.766899i	$0.278201\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.38012	−0.460041
$10$	0	0
$11$	−1.27274	−0.383747	−0.191873	−	0.981420i	$-0.561456\pi$
$11$	−1.27274	−0.383747	−0.191873	+	0.981420i	$0.561456\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−3.65287	−0.943166
$16$	0	0
$17$	3.10738	0.753650	0.376825	−	0.926285i	$-0.377016\pi$
$17$	3.10738	0.753650	0.376825	+	0.926285i	$0.377016\pi$
$18$	0	0
$19$	0.870071	0.199608	0.0998040	−	0.995007i	$-0.468178\pi$
$19$	0.870071	0.199608	0.0998040	+	0.995007i	$0.468178\pi$
$20$	0	0
$21$	1.27274	0.277736
$22$	0	0
$23$	0.402673	0.0839632	0.0419816	−	0.999118i	$-0.486633\pi$
$23$	0.402673	0.0839632	0.0419816	+	0.999118i	$0.486633\pi$
$24$	0	0
$25$	3.23731	0.647461
$26$	0	0
$27$	5.57478	1.07287
$28$	0	0
$29$	6.52294	1.21128	0.605640	−	0.795739i	$-0.292917\pi$
$29$	6.52294	1.21128	0.605640	+	0.795739i	$0.292917\pi$
$30$	0	0
$31$	5.79568	1.04094	0.520468	−	0.853881i	$-0.325758\pi$
$31$	5.79568	1.04094	0.520468	+	0.853881i	$0.325758\pi$
$32$	0	0
$33$	1.61988	0.281985
$34$	0	0
$35$	−2.87007	−0.485131
$36$	0	0
$37$	0.727256	0.119560	0.0597801	−	0.998212i	$-0.480960\pi$
$37$	0.727256	0.119560	0.0597801	+	0.998212i	$0.480960\pi$
$38$	0	0
$39$	−1.27274	−0.203802
$40$	0	0
$41$	−6.12026	−0.955825	−0.477912	−	0.878408i	$-0.658606\pi$
$41$	−6.12026	−0.955825	−0.477912	+	0.878408i	$0.658606\pi$
$42$	0	0
$43$	0.237307	0.0361890	0.0180945	−	0.999836i	$-0.494240\pi$
$43$	0.237307	0.0361890	0.0180945	+	0.999836i	$0.494240\pi$
$44$	0	0
$45$	−3.96105	−0.590478
$46$	0	0
$47$	5.79568	0.845387	0.422694	−	0.906273i	$-0.361085\pi$
$47$	5.79568	0.845387	0.422694	+	0.906273i	$0.361085\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.95490	−0.553796
$52$	0	0
$53$	−2.52294	−0.346552	−0.173276	−	0.984873i	$-0.555435\pi$
$53$	−2.52294	−0.346552	−0.173276	+	0.984873i	$0.555435\pi$
$54$	0	0
$55$	−3.65287	−0.492552
$56$	0	0
$57$	−1.10738	−0.146676
$58$	0	0
$59$	−3.19465	−0.415908	−0.207954	−	0.978139i	$-0.566681\pi$
$59$	−3.19465	−0.415908	−0.207954	+	0.978139i	$0.566681\pi$
$60$	0	0
$61$	11.5748	1.48200	0.740999	−	0.671506i	$-0.234352\pi$
$61$	11.5748	1.48200	0.740999	+	0.671506i	$0.234352\pi$
$62$	0	0
$63$	1.38012	0.173879
$64$	0	0
$65$	2.87007	0.355988
$66$	0	0
$67$	−4.92561	−0.601759	−0.300880	−	0.953662i	$-0.597280\pi$
$67$	−4.92561	−0.601759	−0.300880	+	0.953662i	$0.597280\pi$
$68$	0	0
$69$	−0.512500	−0.0616978
$70$	0	0
$71$	1.43811	0.170672	0.0853362	−	0.996352i	$-0.472804\pi$
$71$	1.43811	0.170672	0.0853362	+	0.996352i	$0.472804\pi$
$72$	0	0
$73$	3.51005	0.410820	0.205410	−	0.978676i	$-0.434147\pi$
$73$	3.51005	0.410820	0.205410	+	0.978676i	$0.434147\pi$
$74$	0	0
$75$	−4.12026	−0.475767
$76$	0	0
$77$	1.27274	0.145043
$78$	0	0
$79$	12.6883	1.42755	0.713773	−	0.700377i	$-0.246985\pi$
$79$	12.6883	1.42755	0.713773	+	0.700377i	$0.246985\pi$
$80$	0	0
$81$	−2.95490	−0.328322
$82$	0	0
$83$	6.32458	0.694213	0.347107	−	0.937826i	$-0.387164\pi$
$83$	6.32458	0.694213	0.347107	+	0.937826i	$0.387164\pi$
$84$	0	0
$85$	8.91839	0.967336
$86$	0	0
$87$	−8.30203	−0.890071
$88$	0	0
$89$	16.7213	1.77245	0.886227	−	0.463252i	$-0.153317\pi$
$89$	16.7213	1.77245	0.886227	+	0.463252i	$0.153317\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−7.37642	−0.764899
$94$	0	0
$95$	2.49717	0.256204
$96$	0	0
$97$	−6.34117	−0.643848	−0.321924	−	0.946765i	$-0.604330\pi$
$97$	−6.34117	−0.643848	−0.321924	+	0.946765i	$0.604330\pi$
$98$	0	0
$99$	1.75654	0.176539
$100$	0	0
$101$	−14.5785	−1.45061	−0.725306	−	0.688426i	$-0.758302\pi$
$101$	−14.5785	−1.45061	−0.725306	+	0.688426i	$0.758302\pi$
$102$	0	0
$103$	9.85122	0.970670	0.485335	−	0.874328i	$-0.338698\pi$
$103$	9.85122	0.970670	0.485335	+	0.874328i	$0.338698\pi$
$104$	0	0
$105$	3.65287	0.356483
$106$	0	0
$107$	2.25986	0.218469	0.109234	−	0.994016i	$-0.465160\pi$
$107$	2.25986	0.218469	0.109234	+	0.994016i	$0.465160\pi$
$108$	0	0
$109$	6.74384	0.645943	0.322972	−	0.946409i	$-0.395318\pi$
$109$	6.74384	0.645943	0.322972	+	0.946409i	$0.395318\pi$
$110$	0	0
$111$	−0.925611	−0.0878551
$112$	0	0
$113$	2.40267	0.226025	0.113012	−	0.993594i	$-0.463950\pi$
$113$	2.40267	0.226025	0.113012	+	0.993594i	$0.463950\pi$
$114$	0	0
$115$	1.15570	0.107770
$116$	0	0
$117$	−1.38012	−0.127592
$118$	0	0
$119$	−3.10738	−0.284853
$120$	0	0
$121$	−9.38012	−0.852738
$122$	0	0
$123$	7.78953	0.702358
$124$	0	0
$125$	−5.05905	−0.452496
$126$	0	0
$127$	14.9514	1.32672	0.663360	−	0.748300i	$-0.269130\pi$
$127$	14.9514	1.32672	0.663360	+	0.748300i	$0.269130\pi$
$128$	0	0
$129$	−0.302031	−0.0265923
$130$	0	0
$131$	13.2655	1.15901	0.579507	−	0.814967i	$-0.303245\pi$
$131$	13.2655	1.15901	0.579507	+	0.814967i	$0.303245\pi$
$132$	0	0
$133$	−0.870071	−0.0754447
$134$	0	0
$135$	16.0000	1.37706
$136$	0	0
$137$	17.6078	1.50433	0.752166	−	0.658973i	$-0.229009\pi$
$137$	17.6078	1.50433	0.752166	+	0.658973i	$0.229009\pi$
$138$	0	0
$139$	−14.9586	−1.26877	−0.634386	−	0.773017i	$-0.718747\pi$
$139$	−14.9586	−1.26877	−0.634386	+	0.773017i	$0.718747\pi$
$140$	0	0
$141$	−7.37642	−0.621207
$142$	0	0
$143$	−1.27274	−0.106432
$144$	0	0
$145$	18.7213	1.55472
$146$	0	0
$147$	−1.27274	−0.104974
$148$	0	0
$149$	−6.79813	−0.556925	−0.278462	−	0.960447i	$-0.589825\pi$
$149$	−6.79813	−0.556925	−0.278462	+	0.960447i	$0.589825\pi$
$150$	0	0
$151$	−3.65287	−0.297266	−0.148633	−	0.988892i	$-0.547487\pi$
$151$	−3.65287	−0.297266	−0.148633	+	0.988892i	$0.547487\pi$
$152$	0	0
$153$	−4.28856	−0.346710
$154$	0	0
$155$	16.6340	1.33608
$156$	0	0
$157$	−2.29285	−0.182989	−0.0914945	−	0.995806i	$-0.529164\pi$
$157$	−2.29285	−0.182989	−0.0914945	+	0.995806i	$0.529164\pi$
$158$	0	0
$159$	3.21105	0.254653
$160$	0	0
$161$	−0.402673	−0.0317351
$162$	0	0
$163$	14.7860	1.15813	0.579065	−	0.815281i	$-0.303418\pi$
$163$	14.7860	1.15813	0.579065	+	0.815281i	$0.303418\pi$
$164$	0	0
$165$	4.64916	0.361937
$166$	0	0
$167$	−1.84997	−0.143155	−0.0715774	−	0.997435i	$-0.522803\pi$
$167$	−1.84997	−0.143155	−0.0715774	+	0.997435i	$0.522803\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.20080	−0.0918278
$172$	0	0
$173$	−17.1331	−1.30261	−0.651305	−	0.758816i	$-0.725778\pi$
$173$	−17.1331	−1.30261	−0.651305	+	0.758816i	$0.725778\pi$
$174$	0	0
$175$	−3.23731	−0.244717
$176$	0	0
$177$	4.06598	0.305618
$178$	0	0
$179$	0.568040	0.0424573	0.0212286	−	0.999775i	$-0.493242\pi$
$179$	0.568040	0.0424573	0.0212286	+	0.999775i	$0.493242\pi$
$180$	0	0
$181$	−9.48750	−0.705200	−0.352600	−	0.935774i	$-0.614702\pi$
$181$	−9.48750	−0.705200	−0.352600	+	0.935774i	$0.614702\pi$
$182$	0	0
$183$	−14.7317	−1.08900
$184$	0	0
$185$	2.08728	0.153460
$186$	0	0
$187$	−3.95490	−0.289211
$188$	0	0
$189$	−5.57478	−0.405505
$190$	0	0
$191$	27.2606	1.97251	0.986255	−	0.165231i	$-0.0528368\pi$
$191$	27.2606	1.97251	0.986255	+	0.165231i	$0.0528368\pi$
$192$	0	0
$193$	12.5455	0.903044	0.451522	−	0.892260i	$-0.350881\pi$
$193$	12.5455	0.903044	0.451522	+	0.892260i	$0.350881\pi$
$194$	0	0
$195$	−3.65287	−0.261587
$196$	0	0
$197$	−24.1460	−1.72033	−0.860167	−	0.510013i	$-0.829641\pi$
$197$	−24.1460	−1.72033	−0.860167	+	0.510013i	$0.829641\pi$
$198$	0	0
$199$	−18.4840	−1.31029	−0.655147	−	0.755501i	$-0.727394\pi$
$199$	−18.4840	−1.31029	−0.655147	+	0.755501i	$0.727394\pi$
$200$	0	0
$201$	6.26904	0.442184
$202$	0	0
$203$	−6.52294	−0.457820
$204$	0	0
$205$	−17.5656	−1.22683
$206$	0	0
$207$	−0.555738	−0.0386265
$208$	0	0
$209$	−1.10738	−0.0765989
$210$	0	0
$211$	−6.95735	−0.478963	−0.239482	−	0.970901i	$-0.576978\pi$
$211$	−6.95735	−0.478963	−0.239482	+	0.970901i	$0.576978\pi$
$212$	0	0
$213$	−1.83035	−0.125413
$214$	0	0
$215$	0.681087	0.0464498
$216$	0	0
$217$	−5.79568	−0.393436
$218$	0	0
$219$	−4.46740	−0.301879
$220$	0	0
$221$	3.10738	0.209025
$222$	0	0
$223$	−15.7555	−1.05506	−0.527532	−	0.849535i	$-0.676883\pi$
$223$	−15.7555	−1.05506	−0.527532	+	0.849535i	$0.676883\pi$
$224$	0	0
$225$	−4.46788	−0.297859
$226$	0	0
$227$	−14.2148	−0.943466	−0.471733	−	0.881741i	$-0.656371\pi$
$227$	−14.2148	−0.943466	−0.471733	+	0.881741i	$0.656371\pi$
$228$	0	0
$229$	2.97990	0.196917	0.0984586	−	0.995141i	$-0.468609\pi$
$229$	2.97990	0.196917	0.0984586	+	0.995141i	$0.468609\pi$
$230$	0	0
$231$	−1.61988	−0.106580
$232$	0	0
$233$	−2.35757	−0.154450	−0.0772248	−	0.997014i	$-0.524606\pi$
$233$	−2.35757	−0.154450	−0.0772248	+	0.997014i	$0.524606\pi$
$234$	0	0
$235$	16.6340	1.08508
$236$	0	0
$237$	−16.1490	−1.04899
$238$	0	0
$239$	30.0258	1.94221	0.971103	−	0.238661i	$-0.0767086\pi$
$239$	30.0258	1.94221	0.971103	+	0.238661i	$0.0767086\pi$
$240$	0	0
$241$	18.4614	1.18921	0.594603	−	0.804020i	$-0.297309\pi$
$241$	18.4614	1.18921	0.594603	+	0.804020i	$0.297309\pi$
$242$	0	0
$243$	−12.9635	−0.831609
$244$	0	0
$245$	2.87007	0.183362
$246$	0	0
$247$	0.870071	0.0553613
$248$	0	0
$249$	−8.04958	−0.510121
$250$	0	0
$251$	11.3764	0.718073	0.359037	−	0.933323i	$-0.383105\pi$
$251$	11.3764	0.718073	0.359037	+	0.933323i	$0.383105\pi$
$252$	0	0
$253$	−0.512500	−0.0322206
$254$	0	0
$255$	−11.3508	−0.710817
$256$	0	0
$257$	−16.1533	−1.00761	−0.503806	−	0.863817i	$-0.668067\pi$
$257$	−16.1533	−1.00761	−0.503806	+	0.863817i	$0.668067\pi$
$258$	0	0
$259$	−0.727256	−0.0451895
$260$	0	0
$261$	−9.00245	−0.557238
$262$	0	0
$263$	−22.9905	−1.41766	−0.708828	−	0.705381i	$-0.750776\pi$
$263$	−22.9905	−1.41766	−0.708828	+	0.705381i	$0.750776\pi$
$264$	0	0
$265$	−7.24101	−0.444812
$266$	0	0
$267$	−21.2819	−1.30243
$268$	0	0
$269$	3.44533	0.210065	0.105033	−	0.994469i	$-0.466505\pi$
$269$	3.44533	0.210065	0.105033	+	0.994469i	$0.466505\pi$
$270$	0	0
$271$	−15.4260	−0.937063	−0.468531	−	0.883447i	$-0.655217\pi$
$271$	−15.4260	−0.937063	−0.468531	+	0.883447i	$0.655217\pi$
$272$	0	0
$273$	1.27274	0.0770300
$274$	0	0
$275$	−4.12026	−0.248461
$276$	0	0
$277$	−23.8544	−1.43327	−0.716637	−	0.697446i	$-0.754320\pi$
$277$	−23.8544	−1.43327	−0.716637	+	0.697446i	$0.754320\pi$
$278$	0	0
$279$	−7.99875	−0.478872
$280$	0	0
$281$	−13.4639	−0.803188	−0.401594	−	0.915818i	$-0.631544\pi$
$281$	−13.4639	−0.803188	−0.401594	+	0.915818i	$0.631544\pi$
$282$	0	0
$283$	−13.4002	−0.796561	−0.398280	−	0.917264i	$-0.630393\pi$
$283$	−13.4002	−0.796561	−0.398280	+	0.917264i	$0.630393\pi$
$284$	0	0
$285$	−3.17825	−0.188263
$286$	0	0
$287$	6.12026	0.361268
$288$	0	0
$289$	−7.34420	−0.432012
$290$	0	0
$291$	8.07069	0.473112
$292$	0	0
$293$	−13.5901	−0.793943	−0.396971	−	0.917831i	$-0.629939\pi$
$293$	−13.5901	−0.793943	−0.396971	+	0.917831i	$0.629939\pi$
$294$	0	0
$295$	−9.16888	−0.533833
$296$	0	0
$297$	−7.09526	−0.411709
$298$	0	0
$299$	0.402673	0.0232872
$300$	0	0
$301$	−0.237307	−0.0136781
$302$	0	0
$303$	18.5547	1.06594
$304$	0	0
$305$	33.2204	1.90220
$306$	0	0
$307$	−20.5844	−1.17482	−0.587408	−	0.809291i	$-0.699852\pi$
$307$	−20.5844	−1.17482	−0.587408	+	0.809291i	$0.699852\pi$
$308$	0	0
$309$	−12.5381	−0.713267
$310$	0	0
$311$	−22.7696	−1.29115	−0.645573	−	0.763698i	$-0.723381\pi$
$311$	−22.7696	−1.29115	−0.645573	+	0.763698i	$0.723381\pi$
$312$	0	0
$313$	17.6078	0.995250	0.497625	−	0.867392i	$-0.334206\pi$
$313$	17.6078	0.995250	0.497625	+	0.867392i	$0.334206\pi$
$314$	0	0
$315$	3.96105	0.223180
$316$	0	0
$317$	−6.52890	−0.366700	−0.183350	−	0.983048i	$-0.558694\pi$
$317$	−6.52890	−0.366700	−0.183350	+	0.983048i	$0.558694\pi$
$318$	0	0
$319$	−8.30203	−0.464824
$320$	0	0
$321$	−2.87622	−0.160535
$322$	0	0
$323$	2.70364	0.150434
$324$	0	0
$325$	3.23731	0.179573
$326$	0	0
$327$	−8.58319	−0.474651
$328$	0	0
$329$	−5.79568	−0.319526
$330$	0	0
$331$	−23.7116	−1.30331	−0.651654	−	0.758516i	$-0.725925\pi$
$331$	−23.7116	−1.30331	−0.651654	+	0.758516i	$0.725925\pi$
$332$	0	0
$333$	−1.00370	−0.0550025
$334$	0	0
$335$	−14.1369	−0.772379
$336$	0	0
$337$	17.0141	0.926819	0.463410	−	0.886144i	$-0.346626\pi$
$337$	17.0141	0.926819	0.463410	+	0.886144i	$0.346626\pi$
$338$	0	0
$339$	−3.05799	−0.166087
$340$	0	0
$341$	−7.37642	−0.399456
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.47091	−0.0791912
$346$	0	0
$347$	10.3893	0.557727	0.278864	−	0.960331i	$-0.410042\pi$
$347$	10.3893	0.557727	0.278864	+	0.960331i	$0.410042\pi$
$348$	0	0
$349$	17.8123	0.953469	0.476734	−	0.879047i	$-0.341820\pi$
$349$	17.8123	0.953469	0.476734	+	0.879047i	$0.341820\pi$
$350$	0	0
$351$	5.57478	0.297559
$352$	0	0
$353$	13.1076	0.697645	0.348823	−	0.937189i	$-0.386582\pi$
$353$	13.1076	0.697645	0.348823	+	0.937189i	$0.386582\pi$
$354$	0	0
$355$	4.12748	0.219064
$356$	0	0
$357$	3.95490	0.209315
$358$	0	0
$359$	3.65287	0.192791	0.0963955	−	0.995343i	$-0.469269\pi$
$359$	3.65287	0.192791	0.0963955	+	0.995343i	$0.469269\pi$
$360$	0	0
$361$	−18.2430	−0.960157
$362$	0	0
$363$	11.9385	0.626609
$364$	0	0
$365$	10.0741	0.527302
$366$	0	0
$367$	−29.8676	−1.55908	−0.779539	−	0.626354i	$-0.784546\pi$
$367$	−29.8676	−1.55908	−0.779539	+	0.626354i	$0.784546\pi$
$368$	0	0
$369$	8.44671	0.439718
$370$	0	0
$371$	2.52294	0.130984
$372$	0	0
$373$	26.8897	1.39230	0.696148	−	0.717899i	$-0.254896\pi$
$373$	26.8897	1.39230	0.696148	+	0.717899i	$0.254896\pi$
$374$	0	0
$375$	6.43888	0.332502
$376$	0	0
$377$	6.52294	0.335948
$378$	0	0
$379$	−25.3930	−1.30435	−0.652176	−	0.758068i	$-0.726144\pi$
$379$	−25.3930	−1.30435	−0.652176	+	0.758068i	$0.726144\pi$
$380$	0	0
$381$	−19.0293	−0.974900
$382$	0	0
$383$	−14.8219	−0.757365	−0.378683	−	0.925527i	$-0.623623\pi$
$383$	−14.8219	−0.757365	−0.378683	+	0.925527i	$0.623623\pi$
$384$	0	0
$385$	3.65287	0.186167
$386$	0	0
$387$	−0.327512	−0.0166484
$388$	0	0
$389$	26.2856	1.33273	0.666367	−	0.745624i	$-0.267848\pi$
$389$	26.2856	1.33273	0.666367	+	0.745624i	$0.267848\pi$
$390$	0	0
$391$	1.25126	0.0632789
$392$	0	0
$393$	−16.8836	−0.851666
$394$	0	0
$395$	36.4163	1.83230
$396$	0	0
$397$	−14.3053	−0.717960	−0.358980	−	0.933345i	$-0.616875\pi$
$397$	−14.3053	−0.717960	−0.358980	+	0.933345i	$0.616875\pi$
$398$	0	0
$399$	1.10738	0.0554382
$400$	0	0
$401$	25.7659	1.28669	0.643344	−	0.765577i	$-0.277546\pi$
$401$	25.7659	1.28669	0.643344	+	0.765577i	$0.277546\pi$
$402$	0	0
$403$	5.79568	0.288703
$404$	0	0
$405$	−8.48076	−0.421413
$406$	0	0
$407$	−0.925611	−0.0458808
$408$	0	0
$409$	26.6811	1.31929	0.659647	−	0.751575i	$-0.270706\pi$
$409$	26.6811	1.31929	0.659647	+	0.751575i	$0.270706\pi$
$410$	0	0
$411$	−22.4102	−1.10541
$412$	0	0
$413$	3.19465	0.157199
$414$	0	0
$415$	18.1520	0.891047
$416$	0	0
$417$	19.0385	0.932318
$418$	0	0
$419$	29.4354	1.43801	0.719006	−	0.695004i	$-0.244597\pi$
$419$	29.4354	1.43801	0.719006	+	0.695004i	$0.244597\pi$
$420$	0	0
$421$	−18.3895	−0.896249	−0.448125	−	0.893971i	$-0.647908\pi$
$421$	−18.3895	−0.896249	−0.448125	+	0.893971i	$0.647908\pi$
$422$	0	0
$423$	−7.99875	−0.388912
$424$	0	0
$425$	10.0595	0.487959
$426$	0	0
$427$	−11.5748	−0.560143
$428$	0	0
$429$	1.61988	0.0782084
$430$	0	0
$431$	20.2856	0.977124	0.488562	−	0.872529i	$-0.337522\pi$
$431$	20.2856	0.977124	0.488562	+	0.872529i	$0.337522\pi$
$432$	0	0
$433$	−0.434409	−0.0208764	−0.0104382	−	0.999946i	$-0.503323\pi$
$433$	−0.434409	−0.0208764	−0.0104382	+	0.999946i	$0.503323\pi$
$434$	0	0
$435$	−23.8274	−1.14244
$436$	0	0
$437$	0.350354	0.0167597
$438$	0	0
$439$	−13.5512	−0.646762	−0.323381	−	0.946269i	$-0.604819\pi$
$439$	−13.5512	−0.646762	−0.323381	+	0.946269i	$0.604819\pi$
$440$	0	0
$441$	−1.38012	−0.0657201
$442$	0	0
$443$	−15.1979	−0.722073	−0.361036	−	0.932552i	$-0.617577\pi$
$443$	−15.1979	−0.722073	−0.361036	+	0.932552i	$0.617577\pi$
$444$	0	0
$445$	47.9913	2.27501
$446$	0	0
$447$	8.65228	0.409239
$448$	0	0
$449$	5.69994	0.268997	0.134498	−	0.990914i	$-0.457058\pi$
$449$	5.69994	0.268997	0.134498	+	0.990914i	$0.457058\pi$
$450$	0	0
$451$	7.78953	0.366795
$452$	0	0
$453$	4.64916	0.218437
$454$	0	0
$455$	−2.87007	−0.134551
$456$	0	0
$457$	21.0880	0.986457	0.493229	−	0.869900i	$-0.335817\pi$
$457$	21.0880	0.986457	0.493229	+	0.869900i	$0.335817\pi$
$458$	0	0
$459$	17.3229	0.808565
$460$	0	0
$461$	−25.5463	−1.18981	−0.594904	−	0.803797i	$-0.702810\pi$
$461$	−25.5463	−1.18981	−0.594904	+	0.803797i	$0.702810\pi$
$462$	0	0
$463$	−29.9643	−1.39256	−0.696279	−	0.717771i	$-0.745162\pi$
$463$	−29.9643	−1.39256	−0.696279	+	0.717771i	$0.745162\pi$
$464$	0	0
$465$	−21.1708	−0.981775
$466$	0	0
$467$	−3.13315	−0.144985	−0.0724924	−	0.997369i	$-0.523095\pi$
$467$	−3.13315	−0.144985	−0.0724924	+	0.997369i	$0.523095\pi$
$468$	0	0
$469$	4.92561	0.227444
$470$	0	0
$471$	2.91821	0.134464
$472$	0	0
$473$	−0.302031	−0.0138874
$474$	0	0
$475$	2.81669	0.129238
$476$	0	0
$477$	3.48196	0.159428
$478$	0	0
$479$	−16.9360	−0.773828	−0.386914	−	0.922116i	$-0.626459\pi$
$479$	−16.9360	−0.773828	−0.386914	+	0.922116i	$0.626459\pi$
$480$	0	0
$481$	0.727256	0.0331600
$482$	0	0
$483$	0.512500	0.0233196
$484$	0	0
$485$	−18.1996	−0.826402
$486$	0	0
$487$	−21.2021	−0.960757	−0.480378	−	0.877061i	$-0.659501\pi$
$487$	−21.2021	−0.960757	−0.480378	+	0.877061i	$0.659501\pi$
$488$	0	0
$489$	−18.8188	−0.851016
$490$	0	0
$491$	34.1369	1.54057	0.770287	−	0.637697i	$-0.220113\pi$
$491$	34.1369	1.54057	0.770287	+	0.637697i	$0.220113\pi$
$492$	0	0
$493$	20.2692	0.912880
$494$	0	0
$495$	5.04140	0.226594
$496$	0	0
$497$	−1.43811	−0.0645081
$498$	0	0
$499$	−4.64195	−0.207802	−0.103901	−	0.994588i	$-0.533133\pi$
$499$	−4.64195	−0.207802	−0.103901	+	0.994588i	$0.533133\pi$
$500$	0	0
$501$	2.35454	0.105193
$502$	0	0
$503$	−9.99510	−0.445660	−0.222830	−	0.974857i	$-0.571529\pi$
$503$	−9.99510	−0.445660	−0.222830	+	0.974857i	$0.571529\pi$
$504$	0	0
$505$	−41.8413	−1.86191
$506$	0	0
$507$	−1.27274	−0.0565246
$508$	0	0
$509$	39.2877	1.74139	0.870697	−	0.491819i	$-0.163668\pi$
$509$	39.2877	1.74139	0.870697	+	0.491819i	$0.163668\pi$
$510$	0	0
$511$	−3.51005	−0.155276
$512$	0	0
$513$	4.85045	0.214153
$514$	0	0
$515$	28.2737	1.24589
$516$	0	0
$517$	−7.37642	−0.324415
$518$	0	0
$519$	21.8061	0.957182
$520$	0	0
$521$	30.5262	1.33738	0.668688	−	0.743543i	$-0.266856\pi$
$521$	30.5262	1.33738	0.668688	+	0.743543i	$0.266856\pi$
$522$	0	0
$523$	32.0424	1.40111	0.700557	−	0.713596i	$-0.252935\pi$
$523$	32.0424	1.40111	0.700557	+	0.713596i	$0.252935\pi$
$524$	0	0
$525$	4.12026	0.179823
$526$	0	0
$527$	18.0094	0.784501
$528$	0	0
$529$	−22.8379	−0.992950
$530$	0	0
$531$	4.40901	0.191335
$532$	0	0
$533$	−6.12026	−0.265098
$534$	0	0
$535$	6.48595	0.280412
$536$	0	0
$537$	−0.722970	−0.0311984
$538$	0	0
$539$	−1.27274	−0.0548210
$540$	0	0
$541$	−31.6152	−1.35924	−0.679621	−	0.733563i	$-0.737856\pi$
$541$	−31.6152	−1.35924	−0.679621	+	0.733563i	$0.737856\pi$
$542$	0	0
$543$	12.0752	0.518195
$544$	0	0
$545$	19.3553	0.829090
$546$	0	0
$547$	−27.8470	−1.19065	−0.595327	−	0.803484i	$-0.702977\pi$
$547$	−27.8470	−1.19065	−0.595327	+	0.803484i	$0.702977\pi$
$548$	0	0
$549$	−15.9746	−0.681779
$550$	0	0
$551$	5.67542	0.241781
$552$	0	0
$553$	−12.6883	−0.539562
$554$	0	0
$555$	−2.65657	−0.112765
$556$	0	0
$557$	20.3801	0.863533	0.431767	−	0.901985i	$-0.357890\pi$
$557$	20.3801	0.863533	0.431767	+	0.901985i	$0.357890\pi$
$558$	0	0
$559$	0.237307	0.0100370
$560$	0	0
$561$	5.03357	0.212518
$562$	0	0
$563$	4.24327	0.178833	0.0894163	−	0.995994i	$-0.471500\pi$
$563$	4.24327	0.178833	0.0894163	+	0.995994i	$0.471500\pi$
$564$	0	0
$565$	6.89584	0.290110
$566$	0	0
$567$	2.95490	0.124094
$568$	0	0
$569$	19.5780	0.820752	0.410376	−	0.911916i	$-0.365397\pi$
$569$	19.5780	0.820752	0.410376	+	0.911916i	$0.365397\pi$
$570$	0	0
$571$	43.8287	1.83417	0.917086	−	0.398689i	$-0.130535\pi$
$571$	43.8287	1.83417	0.917086	+	0.398689i	$0.130535\pi$
$572$	0	0
$573$	−34.6958	−1.44944
$574$	0	0
$575$	1.30358	0.0543629
$576$	0	0
$577$	−25.4803	−1.06076	−0.530379	−	0.847761i	$-0.677950\pi$
$577$	−25.4803	−1.06076	−0.530379	+	0.847761i	$0.677950\pi$
$578$	0	0
$579$	−15.9672	−0.663574
$580$	0	0
$581$	−6.32458	−0.262388
$582$	0	0
$583$	3.21105	0.132988
$584$	0	0
$585$	−3.96105	−0.163769
$586$	0	0
$587$	−7.12993	−0.294284	−0.147142	−	0.989115i	$-0.547007\pi$
$587$	−7.12993	−0.294284	−0.147142	+	0.989115i	$0.547007\pi$
$588$	0	0
$589$	5.04265	0.207779
$590$	0	0
$591$	30.7317	1.26413
$592$	0	0
$593$	−21.7672	−0.893870	−0.446935	−	0.894566i	$-0.647485\pi$
$593$	−21.7672	−0.893870	−0.446935	+	0.894566i	$0.647485\pi$
$594$	0	0
$595$	−8.91839	−0.365619
$596$	0	0
$597$	23.5254	0.962830
$598$	0	0
$599$	35.2998	1.44231	0.721155	−	0.692774i	$-0.243612\pi$
$599$	35.2998	1.44231	0.721155	+	0.692774i	$0.243612\pi$
$600$	0	0
$601$	−41.9613	−1.71164	−0.855819	−	0.517275i	$-0.826947\pi$
$601$	−41.9613	−1.71164	−0.855819	+	0.517275i	$0.826947\pi$
$602$	0	0
$603$	6.79794	0.276834
$604$	0	0
$605$	−26.9216	−1.09452
$606$	0	0
$607$	−20.5877	−0.835627	−0.417814	−	0.908533i	$-0.637204\pi$
$607$	−20.5877	−0.835627	−0.417814	+	0.908533i	$0.637204\pi$
$608$	0	0
$609$	8.30203	0.336415
$610$	0	0
$611$	5.79568	0.234468
$612$	0	0
$613$	−25.9006	−1.04612	−0.523058	−	0.852297i	$-0.675209\pi$
$613$	−25.9006	−1.04612	−0.523058	+	0.852297i	$0.675209\pi$
$614$	0	0
$615$	22.3565	0.901501
$616$	0	0
$617$	6.51972	0.262474	0.131237	−	0.991351i	$-0.458105\pi$
$617$	6.51972	0.262474	0.131237	+	0.991351i	$0.458105\pi$
$618$	0	0
$619$	13.1898	0.530141	0.265071	−	0.964229i	$-0.414605\pi$
$619$	13.1898	0.530141	0.265071	+	0.964229i	$0.414605\pi$
$620$	0	0
$621$	2.24481	0.0900813
$622$	0	0
$623$	−16.7213	−0.669924
$624$	0	0
$625$	−30.7064	−1.22826
$626$	0	0
$627$	1.40941	0.0562864
$628$	0	0
$629$	2.25986	0.0901064
$630$	0	0
$631$	−31.7659	−1.26458	−0.632291	−	0.774731i	$-0.717885\pi$
$631$	−31.7659	−1.26458	−0.632291	+	0.774731i	$0.717885\pi$
$632$	0	0
$633$	8.85492	0.351952
$634$	0	0
$635$	42.9115	1.70289
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−1.98477	−0.0785162
$640$	0	0
$641$	30.5981	1.20855	0.604276	−	0.796775i	$-0.293462\pi$
$641$	30.5981	1.20855	0.604276	+	0.796775i	$0.293462\pi$
$642$	0	0
$643$	−27.8770	−1.09936	−0.549681	−	0.835375i	$-0.685251\pi$
$643$	−27.8770	−1.09936	−0.549681	+	0.835375i	$0.685251\pi$
$644$	0	0
$645$	−0.866850	−0.0341322
$646$	0	0
$647$	−17.4381	−0.685563	−0.342781	−	0.939415i	$-0.611369\pi$
$647$	−17.4381	−0.685563	−0.342781	+	0.939415i	$0.611369\pi$
$648$	0	0
$649$	4.06598	0.159604
$650$	0	0
$651$	7.37642	0.289105
$652$	0	0
$653$	25.1496	0.984178	0.492089	−	0.870545i	$-0.336233\pi$
$653$	25.1496	0.984178	0.492089	+	0.870545i	$0.336233\pi$
$654$	0	0
$655$	38.0730	1.48764
$656$	0	0
$657$	−4.84430	−0.188994
$658$	0	0
$659$	−24.2254	−0.943687	−0.471843	−	0.881682i	$-0.656411\pi$
$659$	−24.2254	−0.943687	−0.471843	+	0.881682i	$0.656411\pi$
$660$	0	0
$661$	4.64301	0.180592	0.0902962	−	0.995915i	$-0.471219\pi$
$661$	4.64301	0.180592	0.0902962	+	0.995915i	$0.471219\pi$
$662$	0	0
$663$	−3.95490	−0.153595
$664$	0	0
$665$	−2.49717	−0.0968359
$666$	0	0
$667$	2.62661	0.101703
$668$	0	0
$669$	20.0527	0.775282
$670$	0	0
$671$	−14.7317	−0.568712
$672$	0	0
$673$	−45.1764	−1.74142	−0.870711	−	0.491795i	$-0.836341\pi$
$673$	−45.1764	−1.74142	−0.870711	+	0.491795i	$0.836341\pi$
$674$	0	0
$675$	18.0473	0.694639
$676$	0	0
$677$	−37.7612	−1.45128	−0.725640	−	0.688074i	$-0.758456\pi$
$677$	−37.7612	−1.45128	−0.725640	+	0.688074i	$0.758456\pi$
$678$	0	0
$679$	6.34117	0.243352
$680$	0	0
$681$	18.0917	0.693277
$682$	0	0
$683$	−18.1132	−0.693084	−0.346542	−	0.938035i	$-0.612644\pi$
$683$	−18.1132	−0.693084	−0.346542	+	0.938035i	$0.612644\pi$
$684$	0	0
$685$	50.5355	1.93086
$686$	0	0
$687$	−3.79265	−0.144699
$688$	0	0
$689$	−2.52294	−0.0961163
$690$	0	0
$691$	14.6762	0.558309	0.279154	−	0.960246i	$-0.409946\pi$
$691$	14.6762	0.558309	0.279154	+	0.960246i	$0.409946\pi$
$692$	0	0
$693$	−1.75654	−0.0667255
$694$	0	0
$695$	−42.9322	−1.62851
$696$	0	0
$697$	−19.0180	−0.720357
$698$	0	0
$699$	3.00058	0.113493
$700$	0	0
$701$	37.2310	1.40620	0.703099	−	0.711092i	$-0.251799\pi$
$701$	37.2310	1.40620	0.703099	+	0.711092i	$0.251799\pi$
$702$	0	0
$703$	0.632764	0.0238651
$704$	0	0
$705$	−21.1708	−0.797340
$706$	0	0
$707$	14.5785	0.548280
$708$	0	0
$709$	16.5260	0.620646	0.310323	−	0.950631i	$-0.399563\pi$
$709$	16.5260	0.620646	0.310323	+	0.950631i	$0.399563\pi$
$710$	0	0
$711$	−17.5114	−0.656729
$712$	0	0
$713$	2.33377	0.0874003
$714$	0	0
$715$	−3.65287	−0.136609
$716$	0	0
$717$	−38.2151	−1.42717
$718$	0	0
$719$	22.8024	0.850387	0.425193	−	0.905103i	$-0.360206\pi$
$719$	22.8024	0.850387	0.425193	+	0.905103i	$0.360206\pi$
$720$	0	0
$721$	−9.85122	−0.366879
$722$	0	0
$723$	−23.4967	−0.873851
$724$	0	0
$725$	21.1167	0.784256
$726$	0	0
$727$	−11.5586	−0.428683	−0.214342	−	0.976759i	$-0.568761\pi$
$727$	−11.5586	−0.428683	−0.214342	+	0.976759i	$0.568761\pi$
$728$	0	0
$729$	25.3639	0.939404
$730$	0	0
$731$	0.737402	0.0272738
$732$	0	0
$733$	9.82420	0.362865	0.181432	−	0.983403i	$-0.441927\pi$
$733$	9.82420	0.362865	0.181432	+	0.983403i	$0.441927\pi$
$734$	0	0
$735$	−3.65287	−0.134738
$736$	0	0
$737$	6.26904	0.230923
$738$	0	0
$739$	−5.21846	−0.191964	−0.0959820	−	0.995383i	$-0.530599\pi$
$739$	−5.21846	−0.191964	−0.0959820	+	0.995383i	$0.530599\pi$
$740$	0	0
$741$	−1.10738	−0.0406805
$742$	0	0
$743$	38.8704	1.42602	0.713008	−	0.701156i	$-0.247332\pi$
$743$	38.8704	1.42602	0.713008	+	0.701156i	$0.247332\pi$
$744$	0	0
$745$	−19.5111	−0.714832
$746$	0	0
$747$	−8.72870	−0.319366
$748$	0	0
$749$	−2.25986	−0.0825734
$750$	0	0
$751$	−2.82516	−0.103091	−0.0515457	−	0.998671i	$-0.516415\pi$
$751$	−2.82516	−0.103091	−0.0515457	+	0.998671i	$0.516415\pi$
$752$	0	0
$753$	−14.4793	−0.527654
$754$	0	0
$755$	−10.4840	−0.381551
$756$	0	0
$757$	12.1069	0.440033	0.220016	−	0.975496i	$-0.429389\pi$
$757$	12.1069	0.440033	0.220016	+	0.975496i	$0.429389\pi$
$758$	0	0
$759$	0.652282	0.0236763
$760$	0	0
$761$	−27.1014	−0.982425	−0.491213	−	0.871040i	$-0.663446\pi$
$761$	−27.1014	−0.982425	−0.491213	+	0.871040i	$0.663446\pi$
$762$	0	0
$763$	−6.74384	−0.244144
$764$	0	0
$765$	−12.3085	−0.445014
$766$	0	0
$767$	−3.19465	−0.115352
$768$	0	0
$769$	33.8666	1.22126	0.610630	−	0.791916i	$-0.290916\pi$
$769$	33.8666	1.22126	0.610630	+	0.791916i	$0.290916\pi$
$770$	0	0
$771$	20.5590	0.740413
$772$	0	0
$773$	11.6950	0.420641	0.210321	−	0.977632i	$-0.432549\pi$
$773$	11.6950	0.420641	0.210321	+	0.977632i	$0.432549\pi$
$774$	0	0
$775$	18.7624	0.673965
$776$	0	0
$777$	0.925611	0.0332061
$778$	0	0
$779$	−5.32506	−0.190790
$780$	0	0
$781$	−1.83035	−0.0654950
$782$	0	0
$783$	36.3639	1.29954
$784$	0	0
$785$	−6.58063	−0.234873
$786$	0	0
$787$	−38.1114	−1.35852	−0.679262	−	0.733896i	$-0.737700\pi$
$787$	−38.1114	−1.35852	−0.679262	+	0.733896i	$0.737700\pi$
$788$	0	0
$789$	29.2611	1.04172
$790$	0	0
$791$	−2.40267	−0.0854292
$792$	0	0
$793$	11.5748	0.411032
$794$	0	0
$795$	9.21595	0.326856
$796$	0	0
$797$	22.6587	0.802613	0.401307	−	0.915944i	$-0.368556\pi$
$797$	22.6587	0.802613	0.401307	+	0.915944i	$0.368556\pi$
$798$	0	0
$799$	18.0094	0.637126
$800$	0	0
$801$	−23.0774	−0.815401
$802$	0	0
$803$	−4.46740	−0.157651
$804$	0	0
$805$	−1.15570	−0.0407331
$806$	0	0
$807$	−4.38502	−0.154360
$808$	0	0
$809$	33.7176	1.18545	0.592724	−	0.805406i	$-0.298053\pi$
$809$	33.7176	1.18545	0.592724	+	0.805406i	$0.298053\pi$
$810$	0	0
$811$	−0.778613	−0.0273408	−0.0136704	−	0.999907i	$-0.504352\pi$
$811$	−0.778613	−0.0273408	−0.0136704	+	0.999907i	$0.504352\pi$
$812$	0	0
$813$	19.6333	0.688572
$814$	0	0
$815$	42.4369	1.48650
$816$	0	0
$817$	0.206474	0.00722360
$818$	0	0
$819$	1.38012	0.0482254
$820$	0	0
$821$	−45.2671	−1.57983	−0.789916	−	0.613215i	$-0.789876\pi$
$821$	−45.2671	−1.57983	−0.789916	+	0.613215i	$0.789876\pi$
$822$	0	0
$823$	16.5949	0.578461	0.289231	−	0.957259i	$-0.406601\pi$
$823$	16.5949	0.578461	0.289231	+	0.957259i	$0.406601\pi$
$824$	0	0
$825$	5.24404	0.182574
$826$	0	0
$827$	−22.4369	−0.780208	−0.390104	−	0.920771i	$-0.627561\pi$
$827$	−22.4369	−0.780208	−0.390104	+	0.920771i	$0.627561\pi$
$828$	0	0
$829$	−24.6585	−0.856426	−0.428213	−	0.903678i	$-0.640857\pi$
$829$	−24.6585	−0.856426	−0.428213	+	0.903678i	$0.640857\pi$
$830$	0	0
$831$	30.3606	1.05320
$832$	0	0
$833$	3.10738	0.107664
$834$	0	0
$835$	−5.30954	−0.183744
$836$	0	0
$837$	32.3096	1.11678
$838$	0	0
$839$	1.97148	0.0680632	0.0340316	−	0.999421i	$-0.489165\pi$
$839$	1.97148	0.0680632	0.0340316	+	0.999421i	$0.489165\pi$
$840$	0	0
$841$	13.5487	0.467197
$842$	0	0
$843$	17.1361	0.590198
$844$	0	0
$845$	2.87007	0.0987334
$846$	0	0
$847$	9.38012	0.322305
$848$	0	0
$849$	17.0551	0.585328
$850$	0	0
$851$	0.292847	0.0100386
$852$	0	0
$853$	42.3980	1.45168	0.725839	−	0.687864i	$-0.241452\pi$
$853$	42.3980	1.45168	0.725839	+	0.687864i	$0.241452\pi$
$854$	0	0
$855$	−3.44639	−0.117864
$856$	0	0
$857$	27.2655	0.931373	0.465686	−	0.884950i	$-0.345807\pi$
$857$	27.2655	0.931373	0.465686	+	0.884950i	$0.345807\pi$
$858$	0	0
$859$	54.7624	1.86847	0.934234	−	0.356659i	$-0.116084\pi$
$859$	54.7624	1.86847	0.934234	+	0.356659i	$0.116084\pi$
$860$	0	0
$861$	−7.78953	−0.265466
$862$	0	0
$863$	29.0008	0.987198	0.493599	−	0.869690i	$-0.335681\pi$
$863$	29.0008	0.987198	0.493599	+	0.869690i	$0.335681\pi$
$864$	0	0
$865$	−49.1734	−1.67194
$866$	0	0
$867$	9.34729	0.317451
$868$	0	0
$869$	−16.1490	−0.547816
$870$	0	0
$871$	−4.92561	−0.166898
$872$	0	0
$873$	8.75159	0.296196
$874$	0	0
$875$	5.05905	0.171027
$876$	0	0
$877$	−29.8698	−1.00863	−0.504315	−	0.863520i	$-0.668255\pi$
$877$	−29.8698	−1.00863	−0.504315	+	0.863520i	$0.668255\pi$
$878$	0	0
$879$	17.2967	0.583404
$880$	0	0
$881$	6.74581	0.227272	0.113636	−	0.993522i	$-0.463750\pi$
$881$	6.74581	0.227272	0.113636	+	0.993522i	$0.463750\pi$
$882$	0	0
$883$	−14.8971	−0.501327	−0.250664	−	0.968074i	$-0.580649\pi$
$883$	−14.8971	−0.501327	−0.250664	+	0.968074i	$0.580649\pi$
$884$	0	0
$885$	11.6696	0.392271
$886$	0	0
$887$	27.1167	0.910491	0.455246	−	0.890366i	$-0.349551\pi$
$887$	27.1167	0.910491	0.455246	+	0.890366i	$0.349551\pi$
$888$	0	0
$889$	−14.9514	−0.501453
$890$	0	0
$891$	3.76083	0.125992
$892$	0	0
$893$	5.04265	0.168746
$894$	0	0
$895$	1.63031	0.0544954
$896$	0	0
$897$	−0.512500	−0.0171119
$898$	0	0
$899$	37.8049	1.26086
$900$	0	0
$901$	−7.83972	−0.261179
$902$	0	0
$903$	0.302031	0.0100510
$904$	0	0
$905$	−27.2298	−0.905149
$906$	0	0
$907$	−13.5688	−0.450545	−0.225272	−	0.974296i	$-0.572327\pi$
$907$	−13.5688	−0.450545	−0.225272	+	0.974296i	$0.572327\pi$
$908$	0	0
$909$	20.1201	0.667341
$910$	0	0
$911$	34.3742	1.13887	0.569433	−	0.822038i	$-0.307163\pi$
$911$	34.3742	1.13887	0.569433	+	0.822038i	$0.307163\pi$
$912$	0	0
$913$	−8.04958	−0.266402
$914$	0	0
$915$	−42.2811	−1.39777
$916$	0	0
$917$	−13.2655	−0.438066
$918$	0	0
$919$	51.8083	1.70900	0.854499	−	0.519454i	$-0.173865\pi$
$919$	51.8083	1.70900	0.854499	+	0.519454i	$0.173865\pi$
$920$	0	0
$921$	26.1987	0.863277
$922$	0	0
$923$	1.43811	0.0473360
$924$	0	0
$925$	2.35435	0.0774105
$926$	0	0
$927$	−13.5959	−0.446548
$928$	0	0
$929$	39.9068	1.30930	0.654649	−	0.755933i	$-0.272816\pi$
$929$	39.9068	1.30930	0.654649	+	0.755933i	$0.272816\pi$
$930$	0	0
$931$	0.870071	0.0285154
$932$	0	0
$933$	28.9799	0.948759
$934$	0	0
$935$	−11.3508	−0.371212
$936$	0	0
$937$	0.543520	0.0177560	0.00887802	−	0.999961i	$-0.497174\pi$
$937$	0.543520	0.0177560	0.00887802	+	0.999961i	$0.497174\pi$
$938$	0	0
$939$	−22.4102	−0.731329
$940$	0	0
$941$	−51.4745	−1.67802	−0.839010	−	0.544115i	$-0.816865\pi$
$941$	−51.4745	−1.67802	−0.839010	+	0.544115i	$0.816865\pi$
$942$	0	0
$943$	−2.46447	−0.0802541
$944$	0	0
$945$	−16.0000	−0.520480
$946$	0	0
$947$	−60.7007	−1.97251	−0.986254	−	0.165236i	$-0.947161\pi$
$947$	−60.7007	−1.97251	−0.986254	+	0.165236i	$0.947161\pi$
$948$	0	0
$949$	3.51005	0.113941
$950$	0	0
$951$	8.30962	0.269458
$952$	0	0
$953$	37.1979	1.20496	0.602479	−	0.798135i	$-0.294180\pi$
$953$	37.1979	1.20496	0.602479	+	0.798135i	$0.294180\pi$
$954$	0	0
$955$	78.2399	2.53178
$956$	0	0
$957$	10.5664	0.341562
$958$	0	0
$959$	−17.6078	−0.568584
$960$	0	0
$961$	2.58992	0.0835459
$962$	0	0
$963$	−3.11888	−0.100505
$964$	0	0
$965$	36.0064	1.15909
$966$	0	0
$967$	−15.9713	−0.513602	−0.256801	−	0.966464i	$-0.582669\pi$
$967$	−15.9713	−0.513602	−0.256801	+	0.966464i	$0.582669\pi$
$968$	0	0
$969$	−3.44104	−0.110542
$970$	0	0
$971$	5.49687	0.176403	0.0882015	−	0.996103i	$-0.471888\pi$
$971$	5.49687	0.176403	0.0882015	+	0.996103i	$0.471888\pi$
$972$	0	0
$973$	14.9586	0.479551
$974$	0	0
$975$	−4.12026	−0.131954
$976$	0	0
$977$	−6.77664	−0.216804	−0.108402	−	0.994107i	$-0.534573\pi$
$977$	−6.77664	−0.216804	−0.108402	+	0.994107i	$0.534573\pi$
$978$	0	0
$979$	−21.2819	−0.680173
$980$	0	0
$981$	−9.30733	−0.297160
$982$	0	0
$983$	−29.5852	−0.943622	−0.471811	−	0.881700i	$-0.656399\pi$
$983$	−29.5852	−0.943622	−0.471811	+	0.881700i	$0.656399\pi$
$984$	0	0
$985$	−69.3008	−2.20811
$986$	0	0
$987$	7.37642	0.234794
$988$	0	0
$989$	0.0955572	0.00303854
$990$	0	0
$991$	−26.7440	−0.849552	−0.424776	−	0.905298i	$-0.639647\pi$
$991$	−26.7440	−0.849552	−0.424776	+	0.905298i	$0.639647\pi$
$992$	0	0
$993$	30.1788	0.957696
$994$	0	0
$995$	−53.0503	−1.68181
$996$	0	0
$997$	32.7952	1.03863	0.519317	−	0.854582i	$-0.326186\pi$
$997$	32.7952	1.03863	0.519317	+	0.854582i	$0.326186\pi$
$998$	0	0
$999$	4.05429	0.128272

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.u.1.2		4
4.3	odd	2		728.2.a.h.1.3	✓	4
8.3	odd	2		5824.2.a.cc.1.2		4
8.5	even	2		5824.2.a.cf.1.3		4
12.11	even	2		6552.2.a.bt.1.1		4
28.27	even	2		5096.2.a.t.1.2		4
52.51	odd	2		9464.2.a.ba.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.h.1.3	✓	4	4.3	odd	2
1456.2.a.u.1.2		4	1.1	even	1	trivial
5096.2.a.t.1.2		4	28.27	even	2
5824.2.a.cc.1.2		4	8.3	odd	2
5824.2.a.cf.1.3		4	8.5	even	2
6552.2.a.bt.1.1		4	12.11	even	2
9464.2.a.ba.1.3		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)

\(f(q)\)	\(=\)	\(q-1.27274 q^{3} +2.87007 q^{5} -1.00000 q^{7} -1.38012 q^{9} +O(q^{10})\)	\(q-1.27274 q^{3} +2.87007 q^{5} -1.00000 q^{7} -1.38012 q^{9} -1.27274 q^{11} +1.00000 q^{13} -3.65287 q^{15} +3.10738 q^{17} +0.870071 q^{19} +1.27274 q^{21} +0.402673 q^{23} +3.23731 q^{25} +5.57478 q^{27} +6.52294 q^{29} +5.79568 q^{31} +1.61988 q^{33} -2.87007 q^{35} +0.727256 q^{37} -1.27274 q^{39} -6.12026 q^{41} +0.237307 q^{43} -3.96105 q^{45} +5.79568 q^{47} +1.00000 q^{49} -3.95490 q^{51} -2.52294 q^{53} -3.65287 q^{55} -1.10738 q^{57} -3.19465 q^{59} +11.5748 q^{61} +1.38012 q^{63} +2.87007 q^{65} -4.92561 q^{67} -0.512500 q^{69} +1.43811 q^{71} +3.51005 q^{73} -4.12026 q^{75} +1.27274 q^{77} +12.6883 q^{79} -2.95490 q^{81} +6.32458 q^{83} +8.91839 q^{85} -8.30203 q^{87} +16.7213 q^{89} -1.00000 q^{91} -7.37642 q^{93} +2.49717 q^{95} -6.34117 q^{97} +1.75654 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - 4 q^{7} + 9 q^{9}+O(q^{10}) \) 4 * q - q^3 - 4 * q^7 + 9 * q^9	\( 4 q - q^{3} - 4 q^{7} + 9 q^{9} - q^{11} + 4 q^{13} + 4 q^{15} + 2 q^{17} - 8 q^{19} + q^{21} + 9 q^{23} + 14 q^{25} - 7 q^{27} - 4 q^{29} - 11 q^{31} + 21 q^{33} + 7 q^{37} - q^{39} + 13 q^{41} + 2 q^{43} + 12 q^{45} - 11 q^{47} + 4 q^{49} + 28 q^{51} + 20 q^{53} + 4 q^{55} + 6 q^{57} + 2 q^{59} + 17 q^{61} - 9 q^{63} + 3 q^{67} - 27 q^{69} + 8 q^{71} + 11 q^{73} + 21 q^{75} + q^{77} + 27 q^{79} + 32 q^{81} + 22 q^{83} - 8 q^{85} - 8 q^{87} + 10 q^{89} - 4 q^{91} - 27 q^{93} + 34 q^{95} + 17 q^{97} - 10 q^{99}+O(q^{100}) \) 4 * q - q^3 - 4 * q^7 + 9 * q^9 - q^11 + 4 * q^13 + 4 * q^15 + 2 * q^17 - 8 * q^19 + q^21 + 9 * q^23 + 14 * q^25 - 7 * q^27 - 4 * q^29 - 11 * q^31 + 21 * q^33 + 7 * q^37 - q^39 + 13 * q^41 + 2 * q^43 + 12 * q^45 - 11 * q^47 + 4 * q^49 + 28 * q^51 + 20 * q^53 + 4 * q^55 + 6 * q^57 + 2 * q^59 + 17 * q^61 - 9 * q^63 + 3 * q^67 - 27 * q^69 + 8 * q^71 + 11 * q^73 + 21 * q^75 + q^77 + 27 * q^79 + 32 * q^81 + 22 * q^83 - 8 * q^85 - 8 * q^87 + 10 * q^89 - 4 * q^91 - 27 * q^93 + 34 * q^95 + 17 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

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