LMFDB - Embedded newform 1148.2.a.c.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,2,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1148.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:	$9.16682615204$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.470117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 8x^{2} + 7x + 1 $ x^5 - 2x^4 - 6x^3 + 8x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.189142$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.189142 q^{3} -1.51194 q^{5} -1.00000 q^{7} -2.96423 q^{9} +O(q^{10})$	$q+0.189142 q^{3} -1.51194 q^{5} -1.00000 q^{7} -2.96423 q^{9} +5.99368 q^{11} -2.57300 q^{13} -0.285972 q^{15} -2.45228 q^{17} +7.57962 q^{19} -0.189142 q^{21} -4.57857 q^{23} -2.71403 q^{25} -1.12809 q^{27} -7.11620 q^{29} -6.03822 q^{31} +1.13366 q^{33} +1.51194 q^{35} -4.63823 q^{37} -0.486663 q^{39} +1.00000 q^{41} -12.9042 q^{43} +4.48174 q^{45} -8.18840 q^{47} +1.00000 q^{49} -0.463830 q^{51} -2.35121 q^{53} -9.06211 q^{55} +1.43363 q^{57} +6.08600 q^{59} +9.42174 q^{61} +2.96423 q^{63} +3.89023 q^{65} -8.16869 q^{67} -0.866001 q^{69} -16.1895 q^{71} -0.0799738 q^{73} -0.513337 q^{75} -5.99368 q^{77} +7.86526 q^{79} +8.67931 q^{81} +13.0912 q^{83} +3.70771 q^{85} -1.34597 q^{87} -5.44776 q^{89} +2.57300 q^{91} -1.14208 q^{93} -11.4600 q^{95} -11.4936 q^{97} -17.7666 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} - q^{5} - 5 q^{7} + q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 - q^5 - 5 * q^7 + q^9	$ 5 q - 2 q^{3} - q^{5} - 5 q^{7} + q^{9} - 2 q^{11} - q^{13} - 9 q^{15} - 3 q^{17} - 4 q^{19} + 2 q^{21} - 8 q^{23} - 6 q^{25} - 8 q^{27} - 9 q^{29} - 11 q^{31} + 5 q^{33} + q^{35} - 11 q^{37} - 17 q^{39} + 5 q^{41} - 27 q^{43} - 3 q^{45} - 3 q^{47} + 5 q^{49} - 3 q^{51} - 19 q^{53} - 13 q^{55} - 11 q^{57} - 15 q^{59} - q^{63} + 7 q^{65} - 21 q^{67} + 14 q^{69} - 16 q^{71} - 10 q^{73} + 12 q^{75} + 2 q^{77} - 14 q^{79} - 7 q^{81} - 2 q^{83} - 21 q^{85} - 36 q^{87} + 6 q^{89} + q^{91} + 17 q^{93} + 9 q^{95} + 20 q^{97} - 17 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 - q^5 - 5 * q^7 + q^9 - 2 * q^11 - q^13 - 9 * q^15 - 3 * q^17 - 4 * q^19 + 2 * q^21 - 8 * q^23 - 6 * q^25 - 8 * q^27 - 9 * q^29 - 11 * q^31 + 5 * q^33 + q^35 - 11 * q^37 - 17 * q^39 + 5 * q^41 - 27 * q^43 - 3 * q^45 - 3 * q^47 + 5 * q^49 - 3 * q^51 - 19 * q^53 - 13 * q^55 - 11 * q^57 - 15 * q^59 - q^63 + 7 * q^65 - 21 * q^67 + 14 * q^69 - 16 * q^71 - 10 * q^73 + 12 * q^75 + 2 * q^77 - 14 * q^79 - 7 * q^81 - 2 * q^83 - 21 * q^85 - 36 * q^87 + 6 * q^89 + q^91 + 17 * q^93 + 9 * q^95 + 20 * q^97 - 17 * 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
$3$	0.189142	0.109201	0.0546007	−	0.998508i	$-0.482611\pi$
$3$	0.189142	0.109201	0.0546007	+	0.998508i	$0.482611\pi$
$4$	0	0
$5$	−1.51194	−0.676162	−0.338081	−	0.941117i	$-0.609778\pi$
$5$	−1.51194	−0.676162	−0.338081	+	0.941117i	$0.609778\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.96423	−0.988075
$10$	0	0
$11$	5.99368	1.80716	0.903582	−	0.428416i	$-0.140928\pi$
$11$	5.99368	1.80716	0.903582	+	0.428416i	$0.140928\pi$
$12$	0	0
$13$	−2.57300	−0.713621	−0.356811	−	0.934177i	$-0.616136\pi$
$13$	−2.57300	−0.713621	−0.356811	+	0.934177i	$0.616136\pi$
$14$	0	0
$15$	−0.285972	−0.0738377
$16$	0	0
$17$	−2.45228	−0.594766	−0.297383	−	0.954758i	$-0.596114\pi$
$17$	−2.45228	−0.594766	−0.297383	+	0.954758i	$0.596114\pi$
$18$	0	0
$19$	7.57962	1.73889	0.869443	−	0.494034i	$-0.164478\pi$
$19$	7.57962	1.73889	0.869443	+	0.494034i	$0.164478\pi$
$20$	0	0
$21$	−0.189142	−0.0412742
$22$	0	0
$23$	−4.57857	−0.954698	−0.477349	−	0.878714i	$-0.658402\pi$
$23$	−4.57857	−0.954698	−0.477349	+	0.878714i	$0.658402\pi$
$24$	0	0
$25$	−2.71403	−0.542806
$26$	0	0
$27$	−1.12809	−0.217100
$28$	0	0
$29$	−7.11620	−1.32144	−0.660722	−	0.750630i	$-0.729750\pi$
$29$	−7.11620	−1.32144	−0.660722	+	0.750630i	$0.729750\pi$
$30$	0	0
$31$	−6.03822	−1.08450	−0.542248	−	0.840218i	$-0.682427\pi$
$31$	−6.03822	−1.08450	−0.542248	+	0.840218i	$0.682427\pi$
$32$	0	0
$33$	1.13366	0.197345
$34$	0	0
$35$	1.51194	0.255565
$36$	0	0
$37$	−4.63823	−0.762521	−0.381260	−	0.924468i	$-0.624510\pi$
$37$	−4.63823	−0.762521	−0.381260	+	0.924468i	$0.624510\pi$
$38$	0	0
$39$	−0.486663	−0.0779284
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−12.9042	−1.96788	−0.983938	−	0.178511i	$-0.942872\pi$
$43$	−12.9042	−1.96788	−0.983938	+	0.178511i	$0.942872\pi$
$44$	0	0
$45$	4.48174	0.668098
$46$	0	0
$47$	−8.18840	−1.19440	−0.597200	−	0.802092i	$-0.703720\pi$
$47$	−8.18840	−1.19440	−0.597200	+	0.802092i	$0.703720\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.463830	−0.0649492
$52$	0	0
$53$	−2.35121	−0.322963	−0.161481	−	0.986876i	$-0.551627\pi$
$53$	−2.35121	−0.322963	−0.161481	+	0.986876i	$0.551627\pi$
$54$	0	0
$55$	−9.06211	−1.22193
$56$	0	0
$57$	1.43363	0.189889
$58$	0	0
$59$	6.08600	0.792329	0.396165	−	0.918180i	$-0.370341\pi$
$59$	6.08600	0.792329	0.396165	+	0.918180i	$0.370341\pi$
$60$	0	0
$61$	9.42174	1.20633	0.603165	−	0.797616i	$-0.293906\pi$
$61$	9.42174	1.20633	0.603165	+	0.797616i	$0.293906\pi$
$62$	0	0
$63$	2.96423	0.373457
$64$	0	0
$65$	3.89023	0.482523
$66$	0	0
$67$	−8.16869	−0.997964	−0.498982	−	0.866612i	$-0.666293\pi$
$67$	−8.16869	−0.997964	−0.498982	+	0.866612i	$0.666293\pi$
$68$	0	0
$69$	−0.866001	−0.104254
$70$	0	0
$71$	−16.1895	−1.92133	−0.960667	−	0.277702i	$-0.910427\pi$
$71$	−16.1895	−1.92133	−0.960667	+	0.277702i	$0.910427\pi$
$72$	0	0
$73$	−0.0799738	−0.00936023	−0.00468011	−	0.999989i	$-0.501490\pi$
$73$	−0.0799738	−0.00936023	−0.00468011	+	0.999989i	$0.501490\pi$
$74$	0	0
$75$	−0.513337	−0.0592751
$76$	0	0
$77$	−5.99368	−0.683044
$78$	0	0
$79$	7.86526	0.884910	0.442455	−	0.896791i	$-0.354108\pi$
$79$	7.86526	0.884910	0.442455	+	0.896791i	$0.354108\pi$
$80$	0	0
$81$	8.67931	0.964367
$82$	0	0
$83$	13.0912	1.43695	0.718474	−	0.695553i	$-0.244841\pi$
$83$	13.0912	1.43695	0.718474	+	0.695553i	$0.244841\pi$
$84$	0	0
$85$	3.70771	0.402158
$86$	0	0
$87$	−1.34597	−0.144304
$88$	0	0
$89$	−5.44776	−0.577462	−0.288731	−	0.957410i	$-0.593233\pi$
$89$	−5.44776	−0.577462	−0.288731	+	0.957410i	$0.593233\pi$
$90$	0	0
$91$	2.57300	0.269724
$92$	0	0
$93$	−1.14208	−0.118428
$94$	0	0
$95$	−11.4600	−1.17577
$96$	0	0
$97$	−11.4936	−1.16700	−0.583499	−	0.812114i	$-0.698317\pi$
$97$	−11.4936	−1.16700	−0.583499	+	0.812114i	$0.698317\pi$
$98$	0	0
$99$	−17.7666	−1.78561
$100$	0	0
$101$	11.1713	1.11158	0.555792	−	0.831322i	$-0.312415\pi$
$101$	11.1713	1.11158	0.555792	+	0.831322i	$0.312415\pi$
$102$	0	0
$103$	−2.61580	−0.257743	−0.128871	−	0.991661i	$-0.541135\pi$
$103$	−2.61580	−0.257743	−0.128871	+	0.991661i	$0.541135\pi$
$104$	0	0
$105$	0.285972	0.0279080
$106$	0	0
$107$	8.67625	0.838765	0.419383	−	0.907810i	$-0.362247\pi$
$107$	8.67625	0.838765	0.419383	+	0.907810i	$0.362247\pi$
$108$	0	0
$109$	−0.708796	−0.0678903	−0.0339452	−	0.999424i	$-0.510807\pi$
$109$	−0.708796	−0.0678903	−0.0339452	+	0.999424i	$0.510807\pi$
$110$	0	0
$111$	−0.877286	−0.0832683
$112$	0	0
$113$	12.4140	1.16781	0.583907	−	0.811821i	$-0.301523\pi$
$113$	12.4140	1.16781	0.583907	+	0.811821i	$0.301523\pi$
$114$	0	0
$115$	6.92254	0.645530
$116$	0	0
$117$	7.62695	0.705112
$118$	0	0
$119$	2.45228	0.224800
$120$	0	0
$121$	24.9242	2.26584
$122$	0	0
$123$	0.189142	0.0170544
$124$	0	0
$125$	11.6632	1.04319
$126$	0	0
$127$	0.390482	0.0346497	0.0173248	−	0.999850i	$-0.494485\pi$
$127$	0.390482	0.0346497	0.0173248	+	0.999850i	$0.494485\pi$
$128$	0	0
$129$	−2.44073	−0.214895
$130$	0	0
$131$	0.937891	0.0819439	0.0409720	−	0.999160i	$-0.486955\pi$
$131$	0.937891	0.0819439	0.0409720	+	0.999160i	$0.486955\pi$
$132$	0	0
$133$	−7.57962	−0.657237
$134$	0	0
$135$	1.70560	0.146795
$136$	0	0
$137$	−20.0992	−1.71719	−0.858597	−	0.512651i	$-0.828664\pi$
$137$	−20.0992	−1.71719	−0.858597	+	0.512651i	$0.828664\pi$
$138$	0	0
$139$	−15.2022	−1.28943	−0.644717	−	0.764422i	$-0.723025\pi$
$139$	−15.2022	−1.28943	−0.644717	+	0.764422i	$0.723025\pi$
$140$	0	0
$141$	−1.54877	−0.130430
$142$	0	0
$143$	−15.4217	−1.28963
$144$	0	0
$145$	10.7593	0.893510
$146$	0	0
$147$	0.189142	0.0156002
$148$	0	0
$149$	−8.58520	−0.703327	−0.351663	−	0.936127i	$-0.614384\pi$
$149$	−8.58520	−0.703327	−0.351663	+	0.936127i	$0.614384\pi$
$150$	0	0
$151$	5.88272	0.478729	0.239364	−	0.970930i	$-0.423061\pi$
$151$	5.88272	0.478729	0.239364	+	0.970930i	$0.423061\pi$
$152$	0	0
$153$	7.26912	0.587673
$154$	0	0
$155$	9.12945	0.733295
$156$	0	0
$157$	−9.93083	−0.792566	−0.396283	−	0.918128i	$-0.629700\pi$
$157$	−9.93083	−0.792566	−0.396283	+	0.918128i	$0.629700\pi$
$158$	0	0
$159$	−0.444712	−0.0352680
$160$	0	0
$161$	4.57857	0.360842
$162$	0	0
$163$	0.479942	0.0375919	0.0187960	−	0.999823i	$-0.494017\pi$
$163$	0.479942	0.0375919	0.0187960	+	0.999823i	$0.494017\pi$
$164$	0	0
$165$	−1.71403	−0.133437
$166$	0	0
$167$	13.5498	1.04852	0.524258	−	0.851559i	$-0.324343\pi$
$167$	13.5498	1.04852	0.524258	+	0.851559i	$0.324343\pi$
$168$	0	0
$169$	−6.37968	−0.490745
$170$	0	0
$171$	−22.4677	−1.71815
$172$	0	0
$173$	−0.684951	−0.0520758	−0.0260379	−	0.999661i	$-0.508289\pi$
$173$	−0.684951	−0.0520758	−0.0260379	+	0.999661i	$0.508289\pi$
$174$	0	0
$175$	2.71403	0.205161
$176$	0	0
$177$	1.15112	0.0865234
$178$	0	0
$179$	0.0838879	0.00627008	0.00313504	−	0.999995i	$-0.499002\pi$
$179$	0.0838879	0.00627008	0.00313504	+	0.999995i	$0.499002\pi$
$180$	0	0
$181$	12.8162	0.952618	0.476309	−	0.879278i	$-0.341974\pi$
$181$	12.8162	0.952618	0.476309	+	0.879278i	$0.341974\pi$
$182$	0	0
$183$	1.78205	0.131733
$184$	0	0
$185$	7.01274	0.515587
$186$	0	0
$187$	−14.6982	−1.07484
$188$	0	0
$189$	1.12809	0.0820563
$190$	0	0
$191$	23.6007	1.70769	0.853844	−	0.520528i	$-0.174265\pi$
$191$	23.6007	1.70769	0.853844	+	0.520528i	$0.174265\pi$
$192$	0	0
$193$	−19.5769	−1.40917	−0.704587	−	0.709617i	$-0.748868\pi$
$193$	−19.5769	−1.40917	−0.704587	+	0.709617i	$0.748868\pi$
$194$	0	0
$195$	0.735806	0.0526922
$196$	0	0
$197$	0.723514	0.0515482	0.0257741	−	0.999668i	$-0.491795\pi$
$197$	0.723514	0.0515482	0.0257741	+	0.999668i	$0.491795\pi$
$198$	0	0
$199$	−26.1583	−1.85431	−0.927157	−	0.374674i	$-0.877755\pi$
$199$	−26.1583	−1.85431	−0.927157	+	0.374674i	$0.877755\pi$
$200$	0	0
$201$	−1.54504	−0.108979
$202$	0	0
$203$	7.11620	0.499459
$204$	0	0
$205$	−1.51194	−0.105599
$206$	0	0
$207$	13.5719	0.943313
$208$	0	0
$209$	45.4299	3.14245
$210$	0	0
$211$	−6.14912	−0.423323	−0.211661	−	0.977343i	$-0.567887\pi$
$211$	−6.14912	−0.423323	−0.211661	+	0.977343i	$0.567887\pi$
$212$	0	0
$213$	−3.06211	−0.209812
$214$	0	0
$215$	19.5105	1.33060
$216$	0	0
$217$	6.03822	0.409901
$218$	0	0
$219$	−0.0151264	−0.00102215
$220$	0	0
$221$	6.30972	0.424438
$222$	0	0
$223$	16.0367	1.07390	0.536950	−	0.843614i	$-0.319577\pi$
$223$	16.0367	1.07390	0.536950	+	0.843614i	$0.319577\pi$
$224$	0	0
$225$	8.04499	0.536333
$226$	0	0
$227$	25.8470	1.71552	0.857762	−	0.514046i	$-0.171854\pi$
$227$	25.8470	1.71552	0.857762	+	0.514046i	$0.171854\pi$
$228$	0	0
$229$	−8.68209	−0.573729	−0.286864	−	0.957971i	$-0.592613\pi$
$229$	−8.68209	−0.573729	−0.286864	+	0.957971i	$0.592613\pi$
$230$	0	0
$231$	−1.13366	−0.0745893
$232$	0	0
$233$	16.8859	1.10623	0.553116	−	0.833104i	$-0.313438\pi$
$233$	16.8859	1.10623	0.553116	+	0.833104i	$0.313438\pi$
$234$	0	0
$235$	12.3804	0.807608
$236$	0	0
$237$	1.48765	0.0966334
$238$	0	0
$239$	−12.2054	−0.789501	−0.394750	−	0.918788i	$-0.629169\pi$
$239$	−12.2054	−0.789501	−0.394750	+	0.918788i	$0.629169\pi$
$240$	0	0
$241$	−1.10393	−0.0711103	−0.0355552	−	0.999368i	$-0.511320\pi$
$241$	−1.10393	−0.0711103	−0.0355552	+	0.999368i	$0.511320\pi$
$242$	0	0
$243$	5.02588	0.322411
$244$	0	0
$245$	−1.51194	−0.0965945
$246$	0	0
$247$	−19.5024	−1.24091
$248$	0	0
$249$	2.47610	0.156917
$250$	0	0
$251$	−20.2063	−1.27541	−0.637705	−	0.770281i	$-0.720116\pi$
$251$	−20.2063	−1.27541	−0.637705	+	0.770281i	$0.720116\pi$
$252$	0	0
$253$	−27.4425	−1.72530
$254$	0	0
$255$	0.701285	0.0439162
$256$	0	0
$257$	3.52848	0.220101	0.110050	−	0.993926i	$-0.464899\pi$
$257$	3.52848	0.220101	0.110050	+	0.993926i	$0.464899\pi$
$258$	0	0
$259$	4.63823	0.288206
$260$	0	0
$261$	21.0940	1.30569
$262$	0	0
$263$	6.85883	0.422934	0.211467	−	0.977385i	$-0.432176\pi$
$263$	6.85883	0.422934	0.211467	+	0.977385i	$0.432176\pi$
$264$	0	0
$265$	3.55489	0.218375
$266$	0	0
$267$	−1.03040	−0.0630596
$268$	0	0
$269$	1.40104	0.0854232	0.0427116	−	0.999087i	$-0.486400\pi$
$269$	1.40104	0.0854232	0.0427116	+	0.999087i	$0.486400\pi$
$270$	0	0
$271$	19.5833	1.18960	0.594799	−	0.803874i	$-0.297231\pi$
$271$	19.5833	1.18960	0.594799	+	0.803874i	$0.297231\pi$
$272$	0	0
$273$	0.486663	0.0294542
$274$	0	0
$275$	−16.2670	−0.980938
$276$	0	0
$277$	3.29060	0.197713	0.0988565	−	0.995102i	$-0.468482\pi$
$277$	3.29060	0.197713	0.0988565	+	0.995102i	$0.468482\pi$
$278$	0	0
$279$	17.8987	1.07156
$280$	0	0
$281$	19.9937	1.19272	0.596362	−	0.802716i	$-0.296612\pi$
$281$	19.9937	1.19272	0.596362	+	0.802716i	$0.296612\pi$
$282$	0	0
$283$	−19.0681	−1.13348	−0.566739	−	0.823897i	$-0.691795\pi$
$283$	−19.0681	−1.13348	−0.566739	+	0.823897i	$0.691795\pi$
$284$	0	0
$285$	−2.16756	−0.128395
$286$	0	0
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	−10.9863	−0.646254
$290$	0	0
$291$	−2.17392	−0.127438
$292$	0	0
$293$	1.20077	0.0701496	0.0350748	−	0.999385i	$-0.488833\pi$
$293$	1.20077	0.0701496	0.0350748	+	0.999385i	$0.488833\pi$
$294$	0	0
$295$	−9.20168	−0.535743
$296$	0	0
$297$	−6.76140	−0.392336
$298$	0	0
$299$	11.7807	0.681293
$300$	0	0
$301$	12.9042	0.743787
$302$	0	0
$303$	2.11296	0.121386
$304$	0	0
$305$	−14.2451	−0.815674
$306$	0	0
$307$	−32.7439	−1.86880	−0.934398	−	0.356232i	$-0.884061\pi$
$307$	−32.7439	−1.86880	−0.934398	+	0.356232i	$0.884061\pi$
$308$	0	0
$309$	−0.494759	−0.0281459
$310$	0	0
$311$	19.2651	1.09242	0.546211	−	0.837647i	$-0.316070\pi$
$311$	19.2651	1.09242	0.546211	+	0.837647i	$0.316070\pi$
$312$	0	0
$313$	29.9633	1.69362	0.846812	−	0.531893i	$-0.178519\pi$
$313$	29.9633	1.69362	0.846812	+	0.531893i	$0.178519\pi$
$314$	0	0
$315$	−4.48174	−0.252517
$316$	0	0
$317$	−11.7544	−0.660191	−0.330095	−	0.943948i	$-0.607081\pi$
$317$	−11.7544	−0.660191	−0.330095	+	0.943948i	$0.607081\pi$
$318$	0	0
$319$	−42.6522	−2.38807
$320$	0	0
$321$	1.64105	0.0915943
$322$	0	0
$323$	−18.5874	−1.03423
$324$	0	0
$325$	6.98319	0.387358
$326$	0	0
$327$	−0.134063	−0.00741372
$328$	0	0
$329$	8.18840	0.451441
$330$	0	0
$331$	−14.5852	−0.801675	−0.400837	−	0.916149i	$-0.631281\pi$
$331$	−14.5852	−0.801675	−0.400837	+	0.916149i	$0.631281\pi$
$332$	0	0
$333$	13.7488	0.753428
$334$	0	0
$335$	12.3506	0.674785
$336$	0	0
$337$	4.20359	0.228984	0.114492	−	0.993424i	$-0.463476\pi$
$337$	4.20359	0.228984	0.114492	+	0.993424i	$0.463476\pi$
$338$	0	0
$339$	2.34802	0.127527
$340$	0	0
$341$	−36.1912	−1.95986
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.30934	0.0704927
$346$	0	0
$347$	21.6744	1.16354	0.581771	−	0.813353i	$-0.302360\pi$
$347$	21.6744	1.16354	0.581771	+	0.813353i	$0.302360\pi$
$348$	0	0
$349$	11.3736	0.608813	0.304407	−	0.952542i	$-0.401542\pi$
$349$	11.3736	0.608813	0.304407	+	0.952542i	$0.401542\pi$
$350$	0	0
$351$	2.90257	0.154928
$352$	0	0
$353$	13.5784	0.722707	0.361354	−	0.932429i	$-0.382315\pi$
$353$	13.5784	0.722707	0.361354	+	0.932429i	$0.382315\pi$
$354$	0	0
$355$	24.4775	1.29913
$356$	0	0
$357$	0.463830	0.0245485
$358$	0	0
$359$	−5.12110	−0.270281	−0.135141	−	0.990826i	$-0.543149\pi$
$359$	−5.12110	−0.270281	−0.135141	+	0.990826i	$0.543149\pi$
$360$	0	0
$361$	38.4507	2.02372
$362$	0	0
$363$	4.71423	0.247433
$364$	0	0
$365$	0.120916	0.00632903
$366$	0	0
$367$	−19.1708	−1.00071	−0.500353	−	0.865822i	$-0.666796\pi$
$367$	−19.1708	−1.00071	−0.500353	+	0.865822i	$0.666796\pi$
$368$	0	0
$369$	−2.96423	−0.154311
$370$	0	0
$371$	2.35121	0.122069
$372$	0	0
$373$	−11.7642	−0.609130	−0.304565	−	0.952492i	$-0.598511\pi$
$373$	−11.7642	−0.609130	−0.304565	+	0.952492i	$0.598511\pi$
$374$	0	0
$375$	2.20600	0.113917
$376$	0	0
$377$	18.3100	0.943011
$378$	0	0
$379$	−1.09400	−0.0561951	−0.0280975	−	0.999605i	$-0.508945\pi$
$379$	−1.09400	−0.0561951	−0.0280975	+	0.999605i	$0.508945\pi$
$380$	0	0
$381$	0.0738566	0.00378379
$382$	0	0
$383$	36.0148	1.84027	0.920134	−	0.391604i	$-0.128080\pi$
$383$	36.0148	1.84027	0.920134	+	0.391604i	$0.128080\pi$
$384$	0	0
$385$	9.06211	0.461848
$386$	0	0
$387$	38.2510	1.94441
$388$	0	0
$389$	−7.34101	−0.372204	−0.186102	−	0.982530i	$-0.559585\pi$
$389$	−7.34101	−0.372204	−0.186102	+	0.982530i	$0.559585\pi$
$390$	0	0
$391$	11.2279	0.567822
$392$	0	0
$393$	0.177395	0.00894838
$394$	0	0
$395$	−11.8918	−0.598342
$396$	0	0
$397$	28.5849	1.43464	0.717318	−	0.696746i	$-0.245370\pi$
$397$	28.5849	1.43464	0.717318	+	0.696746i	$0.245370\pi$
$398$	0	0
$399$	−1.43363	−0.0717711
$400$	0	0
$401$	−11.1836	−0.558483	−0.279241	−	0.960221i	$-0.590083\pi$
$401$	−11.1836	−0.558483	−0.279241	+	0.960221i	$0.590083\pi$
$402$	0	0
$403$	15.5363	0.773920
$404$	0	0
$405$	−13.1226	−0.652068
$406$	0	0
$407$	−27.8001	−1.37800
$408$	0	0
$409$	−0.810658	−0.0400845	−0.0200422	−	0.999799i	$-0.506380\pi$
$409$	−0.810658	−0.0400845	−0.0200422	+	0.999799i	$0.506380\pi$
$410$	0	0
$411$	−3.80162	−0.187520
$412$	0	0
$413$	−6.08600	−0.299472
$414$	0	0
$415$	−19.7932	−0.971610
$416$	0	0
$417$	−2.87538	−0.140808
$418$	0	0
$419$	1.81425	0.0886319	0.0443159	−	0.999018i	$-0.485889\pi$
$419$	1.81425	0.0886319	0.0443159	+	0.999018i	$0.485889\pi$
$420$	0	0
$421$	−30.3311	−1.47825	−0.739124	−	0.673569i	$-0.764760\pi$
$421$	−30.3311	−1.47825	−0.739124	+	0.673569i	$0.764760\pi$
$422$	0	0
$423$	24.2723	1.18016
$424$	0	0
$425$	6.65556	0.322842
$426$	0	0
$427$	−9.42174	−0.455950
$428$	0	0
$429$	−2.91690	−0.140829
$430$	0	0
$431$	12.7769	0.615444	0.307722	−	0.951476i	$-0.400433\pi$
$431$	12.7769	0.615444	0.307722	+	0.951476i	$0.400433\pi$
$432$	0	0
$433$	13.1695	0.632885	0.316442	−	0.948612i	$-0.397512\pi$
$433$	13.1695	0.632885	0.316442	+	0.948612i	$0.397512\pi$
$434$	0	0
$435$	2.03504	0.0975725
$436$	0	0
$437$	−34.7038	−1.66011
$438$	0	0
$439$	−26.8576	−1.28184	−0.640922	−	0.767606i	$-0.721448\pi$
$439$	−26.8576	−1.28184	−0.640922	+	0.767606i	$0.721448\pi$
$440$	0	0
$441$	−2.96423	−0.141154
$442$	0	0
$443$	−0.791018	−0.0375824	−0.0187912	−	0.999823i	$-0.505982\pi$
$443$	−0.791018	−0.0375824	−0.0187912	+	0.999823i	$0.505982\pi$
$444$	0	0
$445$	8.23671	0.390457
$446$	0	0
$447$	−1.62382	−0.0768042
$448$	0	0
$449$	−15.0146	−0.708582	−0.354291	−	0.935135i	$-0.615278\pi$
$449$	−15.0146	−0.708582	−0.354291	+	0.935135i	$0.615278\pi$
$450$	0	0
$451$	5.99368	0.282232
$452$	0	0
$453$	1.11267	0.0522778
$454$	0	0
$455$	−3.89023	−0.182377
$456$	0	0
$457$	23.9943	1.12241	0.561203	−	0.827678i	$-0.310339\pi$
$457$	23.9943	1.12241	0.561203	+	0.827678i	$0.310339\pi$
$458$	0	0
$459$	2.76639	0.129124
$460$	0	0
$461$	−0.689846	−0.0321294	−0.0160647	−	0.999871i	$-0.505114\pi$
$461$	−0.689846	−0.0321294	−0.0160647	+	0.999871i	$0.505114\pi$
$462$	0	0
$463$	−19.7359	−0.917206	−0.458603	−	0.888641i	$-0.651650\pi$
$463$	−19.7359	−0.917206	−0.458603	+	0.888641i	$0.651650\pi$
$464$	0	0
$465$	1.72676	0.0800768
$466$	0	0
$467$	−18.0069	−0.833262	−0.416631	−	0.909076i	$-0.636789\pi$
$467$	−18.0069	−0.833262	−0.416631	+	0.909076i	$0.636789\pi$
$468$	0	0
$469$	8.16869	0.377195
$470$	0	0
$471$	−1.87834	−0.0865493
$472$	0	0
$473$	−77.3438	−3.55627
$474$	0	0
$475$	−20.5713	−0.943876
$476$	0	0
$477$	6.96950	0.319112
$478$	0	0
$479$	−21.8707	−0.999297	−0.499648	−	0.866228i	$-0.666537\pi$
$479$	−21.8707	−0.999297	−0.499648	+	0.866228i	$0.666537\pi$
$480$	0	0
$481$	11.9342	0.544151
$482$	0	0
$483$	0.866001	0.0394044
$484$	0	0
$485$	17.3777	0.789079
$486$	0	0
$487$	11.9588	0.541906	0.270953	−	0.962593i	$-0.412661\pi$
$487$	11.9588	0.541906	0.270953	+	0.962593i	$0.412661\pi$
$488$	0	0
$489$	0.0907772	0.00410509
$490$	0	0
$491$	−39.6236	−1.78819	−0.894094	−	0.447879i	$-0.852179\pi$
$491$	−39.6236	−1.78819	−0.894094	+	0.447879i	$0.852179\pi$
$492$	0	0
$493$	17.4509	0.785950
$494$	0	0
$495$	26.8621	1.20736
$496$	0	0
$497$	16.1895	0.726196
$498$	0	0
$499$	−36.5734	−1.63725	−0.818624	−	0.574329i	$-0.805263\pi$
$499$	−36.5734	−1.63725	−0.818624	+	0.574329i	$0.805263\pi$
$500$	0	0
$501$	2.56284	0.114499
$502$	0	0
$503$	−38.0403	−1.69614	−0.848068	−	0.529888i	$-0.822234\pi$
$503$	−38.0403	−1.69614	−0.848068	+	0.529888i	$0.822234\pi$
$504$	0	0
$505$	−16.8903	−0.751610
$506$	0	0
$507$	−1.20667	−0.0535900
$508$	0	0
$509$	7.25323	0.321494	0.160747	−	0.986996i	$-0.448610\pi$
$509$	7.25323	0.321494	0.160747	+	0.986996i	$0.448610\pi$
$510$	0	0
$511$	0.0799738	0.00353783
$512$	0	0
$513$	−8.55048	−0.377513
$514$	0	0
$515$	3.95495	0.174276
$516$	0	0
$517$	−49.0787	−2.15848
$518$	0	0
$519$	−0.129553	−0.00568675
$520$	0	0
$521$	−10.8304	−0.474490	−0.237245	−	0.971450i	$-0.576244\pi$
$521$	−10.8304	−0.474490	−0.237245	+	0.971450i	$0.576244\pi$
$522$	0	0
$523$	19.4652	0.851155	0.425578	−	0.904922i	$-0.360071\pi$
$523$	19.4652	0.851155	0.425578	+	0.904922i	$0.360071\pi$
$524$	0	0
$525$	0.513337	0.0224039
$526$	0	0
$527$	14.8074	0.645022
$528$	0	0
$529$	−2.03669	−0.0885519
$530$	0	0
$531$	−18.0403	−0.782881
$532$	0	0
$533$	−2.57300	−0.111449
$534$	0	0
$535$	−13.1180	−0.567141
$536$	0	0
$537$	0.0158667	0.000684701 0
$538$	0	0
$539$	5.99368	0.258166
$540$	0	0
$541$	27.8232	1.19621	0.598106	−	0.801417i	$-0.295920\pi$
$541$	27.8232	1.19621	0.598106	+	0.801417i	$0.295920\pi$
$542$	0	0
$543$	2.42408	0.104027
$544$	0	0
$545$	1.07166	0.0459048
$546$	0	0
$547$	40.0668	1.71313	0.856565	−	0.516039i	$-0.172594\pi$
$547$	40.0668	1.71313	0.856565	+	0.516039i	$0.172594\pi$
$548$	0	0
$549$	−27.9282	−1.19194
$550$	0	0
$551$	−53.9381	−2.29784
$552$	0	0
$553$	−7.86526	−0.334465
$554$	0	0
$555$	1.32641	0.0563028
$556$	0	0
$557$	9.47042	0.401275	0.200637	−	0.979666i	$-0.435699\pi$
$557$	9.47042	0.401275	0.200637	+	0.979666i	$0.435699\pi$
$558$	0	0
$559$	33.2025	1.40432
$560$	0	0
$561$	−2.78005	−0.117374
$562$	0	0
$563$	−8.28311	−0.349092	−0.174546	−	0.984649i	$-0.555846\pi$
$563$	−8.28311	−0.349092	−0.174546	+	0.984649i	$0.555846\pi$
$564$	0	0
$565$	−18.7693	−0.789631
$566$	0	0
$567$	−8.67931	−0.364497
$568$	0	0
$569$	7.73238	0.324158	0.162079	−	0.986778i	$-0.448180\pi$
$569$	7.73238	0.324158	0.162079	+	0.986778i	$0.448180\pi$
$570$	0	0
$571$	−13.0410	−0.545750	−0.272875	−	0.962050i	$-0.587974\pi$
$571$	−13.0410	−0.545750	−0.272875	+	0.962050i	$0.587974\pi$
$572$	0	0
$573$	4.46389	0.186482
$574$	0	0
$575$	12.4264	0.518215
$576$	0	0
$577$	−0.227296	−0.00946246	−0.00473123	−	0.999989i	$-0.501506\pi$
$577$	−0.227296	−0.00946246	−0.00473123	+	0.999989i	$0.501506\pi$
$578$	0	0
$579$	−3.70282	−0.153884
$580$	0	0
$581$	−13.0912	−0.543116
$582$	0	0
$583$	−14.0924	−0.583647
$584$	0	0
$585$	−11.5315	−0.476769
$586$	0	0
$587$	34.7688	1.43506	0.717531	−	0.696527i	$-0.245272\pi$
$587$	34.7688	1.43506	0.717531	+	0.696527i	$0.245272\pi$
$588$	0	0
$589$	−45.7675	−1.88582
$590$	0	0
$591$	0.136847	0.00562913
$592$	0	0
$593$	−2.70910	−0.111249	−0.0556247	−	0.998452i	$-0.517715\pi$
$593$	−2.70910	−0.111249	−0.0556247	+	0.998452i	$0.517715\pi$
$594$	0	0
$595$	−3.70771	−0.152001
$596$	0	0
$597$	−4.94764	−0.202493
$598$	0	0
$599$	39.0649	1.59615	0.798074	−	0.602559i	$-0.205852\pi$
$599$	39.0649	1.59615	0.798074	+	0.602559i	$0.205852\pi$
$600$	0	0
$601$	−17.0323	−0.694761	−0.347381	−	0.937724i	$-0.612929\pi$
$601$	−17.0323	−0.694761	−0.347381	+	0.937724i	$0.612929\pi$
$602$	0	0
$603$	24.2138	0.986064
$604$	0	0
$605$	−37.6840	−1.53207
$606$	0	0
$607$	−37.4924	−1.52177	−0.760885	−	0.648887i	$-0.775235\pi$
$607$	−37.4924	−1.52177	−0.760885	+	0.648887i	$0.775235\pi$
$608$	0	0
$609$	1.34597	0.0545416
$610$	0	0
$611$	21.0687	0.852350
$612$	0	0
$613$	1.39616	0.0563905	0.0281953	−	0.999602i	$-0.491024\pi$
$613$	1.39616	0.0563905	0.0281953	+	0.999602i	$0.491024\pi$
$614$	0	0
$615$	−0.285972	−0.0115315
$616$	0	0
$617$	−8.47566	−0.341217	−0.170609	−	0.985339i	$-0.554573\pi$
$617$	−8.47566	−0.341217	−0.170609	+	0.985339i	$0.554573\pi$
$618$	0	0
$619$	−17.0607	−0.685727	−0.342863	−	0.939385i	$-0.611397\pi$
$619$	−17.0607	−0.685727	−0.342863	+	0.939385i	$0.611397\pi$
$620$	0	0
$621$	5.16503	0.207265
$622$	0	0
$623$	5.44776	0.218260
$624$	0	0
$625$	−4.06392	−0.162557
$626$	0	0
$627$	8.59271	0.343160
$628$	0	0
$629$	11.3743	0.453521
$630$	0	0
$631$	2.70985	0.107877	0.0539387	−	0.998544i	$-0.482822\pi$
$631$	2.70985	0.107877	0.0539387	+	0.998544i	$0.482822\pi$
$632$	0	0
$633$	−1.16306	−0.0462274
$634$	0	0
$635$	−0.590387	−0.0234288
$636$	0	0
$637$	−2.57300	−0.101946
$638$	0	0
$639$	47.9892	1.89842
$640$	0	0
$641$	−37.9859	−1.50035	−0.750177	−	0.661237i	$-0.770032\pi$
$641$	−37.9859	−1.50035	−0.750177	+	0.661237i	$0.770032\pi$
$642$	0	0
$643$	−34.5744	−1.36348	−0.681741	−	0.731594i	$-0.738777\pi$
$643$	−34.5744	−1.36348	−0.681741	+	0.731594i	$0.738777\pi$
$644$	0	0
$645$	3.69025	0.145303
$646$	0	0
$647$	−2.34308	−0.0921158	−0.0460579	−	0.998939i	$-0.514666\pi$
$647$	−2.34308	−0.0921158	−0.0460579	+	0.998939i	$0.514666\pi$
$648$	0	0
$649$	36.4775	1.43187
$650$	0	0
$651$	1.14208	0.0447618
$652$	0	0
$653$	−24.2508	−0.949009	−0.474504	−	0.880253i	$-0.657373\pi$
$653$	−24.2508	−0.949009	−0.474504	+	0.880253i	$0.657373\pi$
$654$	0	0
$655$	−1.41804	−0.0554073
$656$	0	0
$657$	0.237060	0.00924861
$658$	0	0
$659$	0.292606	0.0113983	0.00569915	−	0.999984i	$-0.498186\pi$
$659$	0.292606	0.0113983	0.00569915	+	0.999984i	$0.498186\pi$
$660$	0	0
$661$	−46.1353	−1.79446	−0.897228	−	0.441568i	$-0.854422\pi$
$661$	−46.1353	−1.79446	−0.897228	+	0.441568i	$0.854422\pi$
$662$	0	0
$663$	1.19343	0.0463491
$664$	0	0
$665$	11.4600	0.444398
$666$	0	0
$667$	32.5820	1.26158
$668$	0	0
$669$	3.03322	0.117271
$670$	0	0
$671$	56.4709	2.18004
$672$	0	0
$673$	17.4042	0.670883	0.335441	−	0.942061i	$-0.391115\pi$
$673$	17.4042	0.670883	0.335441	+	0.942061i	$0.391115\pi$
$674$	0	0
$675$	3.06166	0.117843
$676$	0	0
$677$	23.8391	0.916210	0.458105	−	0.888898i	$-0.348528\pi$
$677$	23.8391	0.916210	0.458105	+	0.888898i	$0.348528\pi$
$678$	0	0
$679$	11.4936	0.441084
$680$	0	0
$681$	4.88876	0.187338
$682$	0	0
$683$	−26.0894	−0.998282	−0.499141	−	0.866521i	$-0.666351\pi$
$683$	−26.0894	−0.998282	−0.499141	+	0.866521i	$0.666351\pi$
$684$	0	0
$685$	30.3889	1.16110
$686$	0	0
$687$	−1.64215	−0.0626520
$688$	0	0
$689$	6.04965	0.230473
$690$	0	0
$691$	19.1131	0.727097	0.363548	−	0.931575i	$-0.381565\pi$
$691$	19.1131	0.727097	0.363548	+	0.931575i	$0.381565\pi$
$692$	0	0
$693$	17.7666	0.674898
$694$	0	0
$695$	22.9849	0.871865
$696$	0	0
$697$	−2.45228	−0.0928868
$698$	0	0
$699$	3.19384	0.120802
$700$	0	0
$701$	14.0707	0.531442	0.265721	−	0.964050i	$-0.414390\pi$
$701$	14.0707	0.531442	0.265721	+	0.964050i	$0.414390\pi$
$702$	0	0
$703$	−35.1561	−1.32594
$704$	0	0
$705$	2.34166	0.0881918
$706$	0	0
$707$	−11.1713	−0.420139
$708$	0	0
$709$	−49.0189	−1.84094	−0.920472	−	0.390808i	$-0.872196\pi$
$709$	−49.0189	−1.84094	−0.920472	+	0.390808i	$0.872196\pi$
$710$	0	0
$711$	−23.3144	−0.874358
$712$	0	0
$713$	27.6464	1.03537
$714$	0	0
$715$	23.3168	0.871999
$716$	0	0
$717$	−2.30855	−0.0862145
$718$	0	0
$719$	50.7846	1.89394	0.946972	−	0.321315i	$-0.104125\pi$
$719$	50.7846	1.89394	0.946972	+	0.321315i	$0.104125\pi$
$720$	0	0
$721$	2.61580	0.0974176
$722$	0	0
$723$	−0.208800	−0.00776534
$724$	0	0
$725$	19.3136	0.717288
$726$	0	0
$727$	−20.4041	−0.756745	−0.378373	−	0.925653i	$-0.623516\pi$
$727$	−20.4041	−0.756745	−0.378373	+	0.925653i	$0.623516\pi$
$728$	0	0
$729$	−25.0873	−0.929160
$730$	0	0
$731$	31.6448	1.17043
$732$	0	0
$733$	−10.0457	−0.371048	−0.185524	−	0.982640i	$-0.559398\pi$
$733$	−10.0457	−0.371048	−0.185524	+	0.982640i	$0.559398\pi$
$734$	0	0
$735$	−0.285972	−0.0105482
$736$	0	0
$737$	−48.9605	−1.80348
$738$	0	0
$739$	−33.7459	−1.24136	−0.620681	−	0.784063i	$-0.713144\pi$
$739$	−33.7459	−1.24136	−0.620681	+	0.784063i	$0.713144\pi$
$740$	0	0
$741$	−3.68872	−0.135509
$742$	0	0
$743$	−45.0786	−1.65377	−0.826886	−	0.562369i	$-0.809890\pi$
$743$	−45.0786	−1.65377	−0.826886	+	0.562369i	$0.809890\pi$
$744$	0	0
$745$	12.9803	0.475562
$746$	0	0
$747$	−38.8053	−1.41981
$748$	0	0
$749$	−8.67625	−0.317023
$750$	0	0
$751$	36.3502	1.32644	0.663218	−	0.748426i	$-0.269190\pi$
$751$	36.3502	1.32644	0.663218	+	0.748426i	$0.269190\pi$
$752$	0	0
$753$	−3.82186	−0.139276
$754$	0	0
$755$	−8.89433	−0.323698
$756$	0	0
$757$	−49.6889	−1.80598	−0.902988	−	0.429667i	$-0.858631\pi$
$757$	−49.6889	−1.80598	−0.902988	+	0.429667i	$0.858631\pi$
$758$	0	0
$759$	−5.19054	−0.188405
$760$	0	0
$761$	−10.0592	−0.364646	−0.182323	−	0.983239i	$-0.558362\pi$
$761$	−10.0592	−0.364646	−0.182323	+	0.983239i	$0.558362\pi$
$762$	0	0
$763$	0.708796	0.0256601
$764$	0	0
$765$	−10.9905	−0.397362
$766$	0	0
$767$	−15.6593	−0.565423
$768$	0	0
$769$	45.9624	1.65745	0.828723	−	0.559659i	$-0.189068\pi$
$769$	45.9624	1.65745	0.828723	+	0.559659i	$0.189068\pi$
$770$	0	0
$771$	0.667385	0.0240353
$772$	0	0
$773$	4.23830	0.152441	0.0762206	−	0.997091i	$-0.475715\pi$
$773$	4.23830	0.152441	0.0762206	+	0.997091i	$0.475715\pi$
$774$	0	0
$775$	16.3879	0.588671
$776$	0	0
$777$	0.877286	0.0314724
$778$	0	0
$779$	7.57962	0.271568
$780$	0	0
$781$	−97.0344	−3.47217
$782$	0	0
$783$	8.02769	0.286886
$784$	0	0
$785$	15.0149	0.535903
$786$	0	0
$787$	32.0054	1.14087	0.570434	−	0.821343i	$-0.306775\pi$
$787$	32.0054	1.14087	0.570434	+	0.821343i	$0.306775\pi$
$788$	0	0
$789$	1.29729	0.0461849
$790$	0	0
$791$	−12.4140	−0.441392
$792$	0	0
$793$	−24.2421	−0.860863
$794$	0	0
$795$	0.672380	0.0238469
$796$	0	0
$797$	−41.0575	−1.45433	−0.727165	−	0.686463i	$-0.759163\pi$
$797$	−41.0575	−1.45433	−0.727165	+	0.686463i	$0.759163\pi$
$798$	0	0
$799$	20.0803	0.710389
$800$	0	0
$801$	16.1484	0.570576
$802$	0	0
$803$	−0.479338	−0.0169155
$804$	0	0
$805$	−6.92254	−0.243987
$806$	0	0
$807$	0.264997	0.00932833
$808$	0	0
$809$	39.2825	1.38110	0.690549	−	0.723285i	$-0.257369\pi$
$809$	39.2825	1.38110	0.690549	+	0.723285i	$0.257369\pi$
$810$	0	0
$811$	0.408316	0.0143379	0.00716895	−	0.999974i	$-0.497718\pi$
$811$	0.408316	0.0143379	0.00716895	+	0.999974i	$0.497718\pi$
$812$	0	0
$813$	3.70402	0.129906
$814$	0	0
$815$	−0.725645	−0.0254182
$816$	0	0
$817$	−97.8092	−3.42191
$818$	0	0
$819$	−7.62695	−0.266507
$820$	0	0
$821$	26.8315	0.936425	0.468212	−	0.883616i	$-0.344898\pi$
$821$	26.8315	0.936425	0.468212	+	0.883616i	$0.344898\pi$
$822$	0	0
$823$	−27.9696	−0.974959	−0.487480	−	0.873134i	$-0.662084\pi$
$823$	−27.9696	−0.974959	−0.487480	+	0.873134i	$0.662084\pi$
$824$	0	0
$825$	−3.07678	−0.107120
$826$	0	0
$827$	2.18680	0.0760425	0.0380213	−	0.999277i	$-0.487895\pi$
$827$	2.18680	0.0760425	0.0380213	+	0.999277i	$0.487895\pi$
$828$	0	0
$829$	26.5207	0.921103	0.460551	−	0.887633i	$-0.347652\pi$
$829$	26.5207	0.921103	0.460551	+	0.887633i	$0.347652\pi$
$830$	0	0
$831$	0.622391	0.0215905
$832$	0	0
$833$	−2.45228	−0.0849665
$834$	0	0
$835$	−20.4866	−0.708967
$836$	0	0
$837$	6.81164	0.235445
$838$	0	0
$839$	−41.5129	−1.43319	−0.716593	−	0.697492i	$-0.754299\pi$
$839$	−41.5129	−1.43319	−0.716593	+	0.697492i	$0.754299\pi$
$840$	0	0
$841$	21.6403	0.746217
$842$	0	0
$843$	3.78165	0.130247
$844$	0	0
$845$	9.64571	0.331823
$846$	0	0
$847$	−24.9242	−0.856407
$848$	0	0
$849$	−3.60658	−0.123777
$850$	0	0
$851$	21.2365	0.727977
$852$	0	0
$853$	32.8885	1.12608	0.563040	−	0.826430i	$-0.309632\pi$
$853$	32.8885	1.12608	0.563040	+	0.826430i	$0.309632\pi$
$854$	0	0
$855$	33.9699	1.16175
$856$	0	0
$857$	48.2059	1.64668	0.823341	−	0.567548i	$-0.192108\pi$
$857$	48.2059	1.64668	0.823341	+	0.567548i	$0.192108\pi$
$858$	0	0
$859$	−49.7378	−1.69703	−0.848515	−	0.529171i	$-0.822503\pi$
$859$	−49.7378	−1.69703	−0.848515	+	0.529171i	$0.822503\pi$
$860$	0	0
$861$	−0.189142	−0.00644595
$862$	0	0
$863$	−4.93209	−0.167890	−0.0839451	−	0.996470i	$-0.526752\pi$
$863$	−4.93209	−0.167890	−0.0839451	+	0.996470i	$0.526752\pi$
$864$	0	0
$865$	1.03561	0.0352117
$866$	0	0
$867$	−2.07798	−0.0705718
$868$	0	0
$869$	47.1419	1.59918
$870$	0	0
$871$	21.0180	0.712169
$872$	0	0
$873$	34.0696	1.15308
$874$	0	0
$875$	−11.6632	−0.394287
$876$	0	0
$877$	5.09426	0.172021	0.0860104	−	0.996294i	$-0.472588\pi$
$877$	5.09426	0.172021	0.0860104	+	0.996294i	$0.472588\pi$
$878$	0	0
$879$	0.227116	0.00766042
$880$	0	0
$881$	6.60204	0.222429	0.111214	−	0.993796i	$-0.464526\pi$
$881$	6.60204	0.222429	0.111214	+	0.993796i	$0.464526\pi$
$882$	0	0
$883$	−8.16388	−0.274737	−0.137368	−	0.990520i	$-0.543864\pi$
$883$	−8.16388	−0.274737	−0.137368	+	0.990520i	$0.543864\pi$
$884$	0	0
$885$	−1.74043	−0.0585038
$886$	0	0
$887$	10.2461	0.344031	0.172015	−	0.985094i	$-0.444972\pi$
$887$	10.2461	0.344031	0.172015	+	0.985094i	$0.444972\pi$
$888$	0	0
$889$	−0.390482	−0.0130963
$890$	0	0
$891$	52.0210	1.74277
$892$	0	0
$893$	−62.0650	−2.07693
$894$	0	0
$895$	−0.126834	−0.00423958
$896$	0	0
$897$	2.22822	0.0743981
$898$	0	0
$899$	42.9692	1.43310
$900$	0	0
$901$	5.76582	0.192087
$902$	0	0
$903$	2.44073	0.0812225
$904$	0	0
$905$	−19.3773	−0.644123
$906$	0	0
$907$	48.5478	1.61200	0.806002	−	0.591913i	$-0.201627\pi$
$907$	48.5478	1.61200	0.806002	+	0.591913i	$0.201627\pi$
$908$	0	0
$909$	−33.1142	−1.09833
$910$	0	0
$911$	−5.66134	−0.187569	−0.0937843	−	0.995593i	$-0.529896\pi$
$911$	−5.66134	−0.187569	−0.0937843	+	0.995593i	$0.529896\pi$
$912$	0	0
$913$	78.4647	2.59680
$914$	0	0
$915$	−2.69436	−0.0890727
$916$	0	0
$917$	−0.937891	−0.0309719
$918$	0	0
$919$	20.0271	0.660635	0.330317	−	0.943870i	$-0.392844\pi$
$919$	20.0271	0.660635	0.330317	+	0.943870i	$0.392844\pi$
$920$	0	0
$921$	−6.19326	−0.204075
$922$	0	0
$923$	41.6554	1.37111
$924$	0	0
$925$	12.5883	0.413900
$926$	0	0
$927$	7.75383	0.254669
$928$	0	0
$929$	15.7004	0.515115	0.257557	−	0.966263i	$-0.417082\pi$
$929$	15.7004	0.515115	0.257557	+	0.966263i	$0.417082\pi$
$930$	0	0
$931$	7.57962	0.248412
$932$	0	0
$933$	3.64384	0.119294
$934$	0	0
$935$	22.2228	0.726765
$936$	0	0
$937$	−11.6447	−0.380414	−0.190207	−	0.981744i	$-0.560916\pi$
$937$	−11.6447	−0.380414	−0.190207	+	0.981744i	$0.560916\pi$
$938$	0	0
$939$	5.66732	0.184946
$940$	0	0
$941$	15.6441	0.509984	0.254992	−	0.966943i	$-0.417927\pi$
$941$	15.6441	0.509984	0.254992	+	0.966943i	$0.417927\pi$
$942$	0	0
$943$	−4.57857	−0.149099
$944$	0	0
$945$	−1.70560	−0.0554833
$946$	0	0
$947$	7.35951	0.239152	0.119576	−	0.992825i	$-0.461846\pi$
$947$	7.35951	0.239152	0.119576	+	0.992825i	$0.461846\pi$
$948$	0	0
$949$	0.205773	0.00667966
$950$	0	0
$951$	−2.22325	−0.0720937
$952$	0	0
$953$	−35.4898	−1.14963	−0.574814	−	0.818284i	$-0.694926\pi$
$953$	−35.4898	−1.14963	−0.574814	+	0.818284i	$0.694926\pi$
$954$	0	0
$955$	−35.6830	−1.15467
$956$	0	0
$957$	−8.06734	−0.260780
$958$	0	0
$959$	20.0992	0.649039
$960$	0	0
$961$	5.46013	0.176133
$962$	0	0
$963$	−25.7184	−0.828763
$964$	0	0
$965$	29.5991	0.952830
$966$	0	0
$967$	−47.1692	−1.51686	−0.758429	−	0.651755i	$-0.774033\pi$
$967$	−47.1692	−1.51686	−0.758429	+	0.651755i	$0.774033\pi$
$968$	0	0
$969$	−3.51566	−0.112939
$970$	0	0
$971$	30.2619	0.971150	0.485575	−	0.874195i	$-0.338610\pi$
$971$	30.2619	0.971150	0.485575	+	0.874195i	$0.338610\pi$
$972$	0	0
$973$	15.2022	0.487360
$974$	0	0
$975$	1.32082	0.0423000
$976$	0	0
$977$	43.8390	1.40253	0.701267	−	0.712899i	$-0.252618\pi$
$977$	43.8390	1.40253	0.701267	+	0.712899i	$0.252618\pi$
$978$	0	0
$979$	−32.6522	−1.04357
$980$	0	0
$981$	2.10103	0.0670808
$982$	0	0
$983$	−23.3235	−0.743904	−0.371952	−	0.928252i	$-0.621311\pi$
$983$	−23.3235	−0.743904	−0.371952	+	0.928252i	$0.621311\pi$
$984$	0	0
$985$	−1.09391	−0.0348549
$986$	0	0
$987$	1.54877	0.0492980
$988$	0	0
$989$	59.0829	1.87873
$990$	0	0
$991$	35.6864	1.13361	0.566807	−	0.823850i	$-0.308178\pi$
$991$	35.6864	1.13361	0.566807	+	0.823850i	$0.308178\pi$
$992$	0	0
$993$	−2.75868	−0.0875440
$994$	0	0
$995$	39.5499	1.25382
$996$	0	0
$997$	46.7142	1.47945	0.739727	−	0.672907i	$-0.234955\pi$
$997$	46.7142	1.47945	0.739727	+	0.672907i	$0.234955\pi$
$998$	0	0
$999$	5.23233	0.165544

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.a.c.1.3	✓	5
4.3	odd	2		4592.2.a.be.1.3		5
7.6	odd	2		8036.2.a.l.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.c.1.3	✓	5	1.1	even	1	trivial
4592.2.a.be.1.3		5	4.3	odd	2
8036.2.a.l.1.3		5	7.6	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q+0.189142 q^{3} -1.51194 q^{5} -1.00000 q^{7} -2.96423 q^{9} +O(q^{10})\)	\(q+0.189142 q^{3} -1.51194 q^{5} -1.00000 q^{7} -2.96423 q^{9} +5.99368 q^{11} -2.57300 q^{13} -0.285972 q^{15} -2.45228 q^{17} +7.57962 q^{19} -0.189142 q^{21} -4.57857 q^{23} -2.71403 q^{25} -1.12809 q^{27} -7.11620 q^{29} -6.03822 q^{31} +1.13366 q^{33} +1.51194 q^{35} -4.63823 q^{37} -0.486663 q^{39} +1.00000 q^{41} -12.9042 q^{43} +4.48174 q^{45} -8.18840 q^{47} +1.00000 q^{49} -0.463830 q^{51} -2.35121 q^{53} -9.06211 q^{55} +1.43363 q^{57} +6.08600 q^{59} +9.42174 q^{61} +2.96423 q^{63} +3.89023 q^{65} -8.16869 q^{67} -0.866001 q^{69} -16.1895 q^{71} -0.0799738 q^{73} -0.513337 q^{75} -5.99368 q^{77} +7.86526 q^{79} +8.67931 q^{81} +13.0912 q^{83} +3.70771 q^{85} -1.34597 q^{87} -5.44776 q^{89} +2.57300 q^{91} -1.14208 q^{93} -11.4600 q^{95} -11.4936 q^{97} -17.7666 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} - q^{5} - 5 q^{7} + q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 - q^5 - 5 * q^7 + q^9	\( 5 q - 2 q^{3} - q^{5} - 5 q^{7} + q^{9} - 2 q^{11} - q^{13} - 9 q^{15} - 3 q^{17} - 4 q^{19} + 2 q^{21} - 8 q^{23} - 6 q^{25} - 8 q^{27} - 9 q^{29} - 11 q^{31} + 5 q^{33} + q^{35} - 11 q^{37} - 17 q^{39} + 5 q^{41} - 27 q^{43} - 3 q^{45} - 3 q^{47} + 5 q^{49} - 3 q^{51} - 19 q^{53} - 13 q^{55} - 11 q^{57} - 15 q^{59} - q^{63} + 7 q^{65} - 21 q^{67} + 14 q^{69} - 16 q^{71} - 10 q^{73} + 12 q^{75} + 2 q^{77} - 14 q^{79} - 7 q^{81} - 2 q^{83} - 21 q^{85} - 36 q^{87} + 6 q^{89} + q^{91} + 17 q^{93} + 9 q^{95} + 20 q^{97} - 17 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 - q^5 - 5 * q^7 + q^9 - 2 * q^11 - q^13 - 9 * q^15 - 3 * q^17 - 4 * q^19 + 2 * q^21 - 8 * q^23 - 6 * q^25 - 8 * q^27 - 9 * q^29 - 11 * q^31 + 5 * q^33 + q^35 - 11 * q^37 - 17 * q^39 + 5 * q^41 - 27 * q^43 - 3 * q^45 - 3 * q^47 + 5 * q^49 - 3 * q^51 - 19 * q^53 - 13 * q^55 - 11 * q^57 - 15 * q^59 - q^63 + 7 * q^65 - 21 * q^67 + 14 * q^69 - 16 * q^71 - 10 * q^73 + 12 * q^75 + 2 * q^77 - 14 * q^79 - 7 * q^81 - 2 * q^83 - 21 * q^85 - 36 * q^87 + 6 * q^89 + q^91 + 17 * q^93 + 9 * q^95 + 20 * q^97 - 17 * q^99

Introduction

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