LMFDB - Embedded newform 9248.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9248,2,Mod(1,9248)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9248, 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("9248.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9248 = 2^{5} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9248.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:	$73.8456517893$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.45968.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 17 $ x^4 - 9*x^2 + 17
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			1.1
Root			$-2.51053$ of defining polynomial
Character	$\chi$	$=$	9248.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.51053 q^{3} +0.302776 q^{5} +3.27066 q^{7} +3.30278 q^{9} +0.760128 q^{11} +5.90833 q^{13} -0.760128 q^{15} +2.51053 q^{19} -8.21110 q^{21} +3.27066 q^{23} -4.90833 q^{25} -0.760128 q^{27} -7.30278 q^{29} -0.760128 q^{31} -1.90833 q^{33} +0.990277 q^{35} -4.00000 q^{37} -14.8330 q^{39} -5.21110 q^{41} -10.8023 q^{43} +1.00000 q^{45} +9.28200 q^{47} +3.69722 q^{49} -2.09167 q^{53} +0.230148 q^{55} -6.30278 q^{57} -13.0826 q^{59} -14.2111 q^{61} +10.8023 q^{63} +1.78890 q^{65} -10.0421 q^{67} -8.21110 q^{69} -10.0421 q^{71} -12.2111 q^{73} +12.3225 q^{75} +2.48612 q^{77} -8.00000 q^{81} +12.5527 q^{83} +18.3339 q^{87} +9.21110 q^{89} +19.3241 q^{91} +1.90833 q^{93} +0.760128 q^{95} +1.51388 q^{97} +2.51053 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5} + 6 q^{9} + 2 q^{13} - 4 q^{21} + 2 q^{25} - 22 q^{29} + 14 q^{33} - 16 q^{37} + 8 q^{41} + 4 q^{45} + 22 q^{49} - 30 q^{53} - 18 q^{57} - 28 q^{61} + 36 q^{65} - 4 q^{69} - 20 q^{73} + 46 q^{77}+ \cdots - 30 q^{97}+O(q^{100}) $ 4 * q - 6 * q^5 + 6 * q^9 + 2 * q^13 - 4 * q^21 + 2 * q^25 - 22 * q^29 + 14 * q^33 - 16 * q^37 + 8 * q^41 + 4 * q^45 + 22 * q^49 - 30 * q^53 - 18 * q^57 - 28 * q^61 + 36 * q^65 - 4 * q^69 - 20 * q^73 + 46 * q^77 - 32 * q^81 + 8 * q^89 - 14 * q^93 - 30 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.51053	−1.44946	−0.724728	−	0.689035i	$-0.758035\pi$
$3$	−2.51053	−1.44946	−0.724728	+	0.689035i	$0.758035\pi$
$4$	0	0
$5$	0.302776	0.135405	0.0677027	−	0.997706i	$-0.478433\pi$
$5$	0.302776	0.135405	0.0677027	+	0.997706i	$0.478433\pi$
$6$	0	0
$7$	3.27066	1.23619	0.618097	−	0.786102i	$-0.287904\pi$
$7$	3.27066	1.23619	0.618097	+	0.786102i	$0.287904\pi$
$8$	0	0
$9$	3.30278	1.10093
$10$	0	0
$11$	0.760128	0.229187	0.114594	−	0.993412i	$-0.463443\pi$
$11$	0.760128	0.229187	0.114594	+	0.993412i	$0.463443\pi$
$12$	0	0
$13$	5.90833	1.63868	0.819338	−	0.573311i	$-0.194341\pi$
$13$	5.90833	1.63868	0.819338	+	0.573311i	$0.194341\pi$
$14$	0	0
$15$	−0.760128	−0.196264
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	2.51053	0.575956	0.287978	−	0.957637i	$-0.407017\pi$
$19$	2.51053	0.575956	0.287978	+	0.957637i	$0.407017\pi$
$20$	0	0
$21$	−8.21110	−1.79181
$22$	0	0
$23$	3.27066	0.681980	0.340990	−	0.940067i	$-0.389238\pi$
$23$	3.27066	0.681980	0.340990	+	0.940067i	$0.389238\pi$
$24$	0	0
$25$	−4.90833	−0.981665
$26$	0	0
$27$	−0.760128	−0.146287
$28$	0	0
$29$	−7.30278	−1.35609	−0.678046	−	0.735020i	$-0.737173\pi$
$29$	−7.30278	−1.35609	−0.678046	+	0.735020i	$0.737173\pi$
$30$	0	0
$31$	−0.760128	−0.136523	−0.0682615	−	0.997667i	$-0.521745\pi$
$31$	−0.760128	−0.136523	−0.0682615	+	0.997667i	$0.521745\pi$
$32$	0	0
$33$	−1.90833	−0.332197
$34$	0	0
$35$	0.990277	0.167387
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	−14.8330	−2.37519
$40$	0	0
$41$	−5.21110	−0.813837	−0.406919	−	0.913464i	$-0.633397\pi$
$41$	−5.21110	−0.813837	−0.406919	+	0.913464i	$0.633397\pi$
$42$	0	0
$43$	−10.8023	−1.64733	−0.823665	−	0.567077i	$-0.808074\pi$
$43$	−10.8023	−1.64733	−0.823665	+	0.567077i	$0.808074\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	9.28200	1.35392	0.676960	−	0.736020i	$-0.263297\pi$
$47$	9.28200	1.35392	0.676960	+	0.736020i	$0.263297\pi$
$48$	0	0
$49$	3.69722	0.528175
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.09167	−0.287313	−0.143657	−	0.989628i	$-0.545886\pi$
$53$	−2.09167	−0.287313	−0.143657	+	0.989628i	$0.545886\pi$
$54$	0	0
$55$	0.230148	0.0310332
$56$	0	0
$57$	−6.30278	−0.834823
$58$	0	0
$59$	−13.0826	−1.70322	−0.851608	−	0.524180i	$-0.824372\pi$
$59$	−13.0826	−1.70322	−0.851608	+	0.524180i	$0.824372\pi$
$60$	0	0
$61$	−14.2111	−1.81955	−0.909773	−	0.415107i	$-0.863744\pi$
$61$	−14.2111	−1.81955	−0.909773	+	0.415107i	$0.863744\pi$
$62$	0	0
$63$	10.8023	1.36096
$64$	0	0
$65$	1.78890	0.221885
$66$	0	0
$67$	−10.0421	−1.22684	−0.613421	−	0.789756i	$-0.710207\pi$
$67$	−10.0421	−1.22684	−0.613421	+	0.789756i	$0.710207\pi$
$68$	0	0
$69$	−8.21110	−0.988501
$70$	0	0
$71$	−10.0421	−1.19178	−0.595891	−	0.803065i	$-0.703201\pi$
$71$	−10.0421	−1.19178	−0.595891	+	0.803065i	$0.703201\pi$
$72$	0	0
$73$	−12.2111	−1.42920	−0.714601	−	0.699533i	$-0.753392\pi$
$73$	−12.2111	−1.42920	−0.714601	+	0.699533i	$0.753392\pi$
$74$	0	0
$75$	12.3225	1.42288
$76$	0	0
$77$	2.48612	0.283320
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−8.00000	−0.888889
$82$	0	0
$83$	12.5527	1.37783	0.688917	−	0.724840i	$-0.258086\pi$
$83$	12.5527	1.37783	0.688917	+	0.724840i	$0.258086\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	18.3339	1.96560
$88$	0	0
$89$	9.21110	0.976375	0.488187	−	0.872739i	$-0.337658\pi$
$89$	9.21110	0.976375	0.488187	+	0.872739i	$0.337658\pi$
$90$	0	0
$91$	19.3241	2.02572
$92$	0	0
$93$	1.90833	0.197884
$94$	0	0
$95$	0.760128	0.0779875
$96$	0	0
$97$	1.51388	0.153711	0.0768555	−	0.997042i	$-0.475512\pi$
$97$	1.51388	0.153711	0.0768555	+	0.997042i	$0.475512\pi$
$98$	0	0
$99$	2.51053	0.252318
$100$	0	0
$101$	3.00000	0.298511	0.149256	−	0.988799i	$-0.452312\pi$
$101$	3.00000	0.298511	0.149256	+	0.988799i	$0.452312\pi$
$102$	0	0
$103$	13.0826	1.28907	0.644536	−	0.764574i	$-0.277051\pi$
$103$	13.0826	1.28907	0.644536	+	0.764574i	$0.277051\pi$
$104$	0	0
$105$	−2.48612	−0.242621
$106$	0	0
$107$	−9.28200	−0.897325	−0.448663	−	0.893701i	$-0.648099\pi$
$107$	−9.28200	−0.897325	−0.448663	+	0.893701i	$0.648099\pi$
$108$	0	0
$109$	−9.42221	−0.902484	−0.451242	−	0.892402i	$-0.649019\pi$
$109$	−9.42221	−0.902484	−0.451242	+	0.892402i	$0.649019\pi$
$110$	0	0
$111$	10.0421	0.953157
$112$	0	0
$113$	−12.9083	−1.21431	−0.607157	−	0.794582i	$-0.707690\pi$
$113$	−12.9083	−1.21431	−0.607157	+	0.794582i	$0.707690\pi$
$114$	0	0
$115$	0.990277	0.0923438
$116$	0	0
$117$	19.5139	1.80406
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.4222	−0.947473
$122$	0	0
$123$	13.0826	1.17962
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	10.8023	0.958546	0.479273	−	0.877666i	$-0.340900\pi$
$127$	10.8023	0.958546	0.479273	+	0.877666i	$0.340900\pi$
$128$	0	0
$129$	27.1194	2.38773
$130$	0	0
$131$	−10.0421	−0.877385	−0.438693	−	0.898637i	$-0.644558\pi$
$131$	−10.0421	−0.877385	−0.438693	+	0.898637i	$0.644558\pi$
$132$	0	0
$133$	8.21110	0.711993
$134$	0	0
$135$	−0.230148	−0.0198080
$136$	0	0
$137$	−17.2111	−1.47044	−0.735222	−	0.677827i	$-0.762922\pi$
$137$	−17.2111	−1.47044	−0.735222	+	0.677827i	$0.762922\pi$
$138$	0	0
$139$	−19.8541	−1.68400	−0.842002	−	0.539474i	$-0.818623\pi$
$139$	−19.8541	−1.68400	−0.842002	+	0.539474i	$0.818623\pi$
$140$	0	0
$141$	−23.3028	−1.96245
$142$	0	0
$143$	4.49109	0.375563
$144$	0	0
$145$	−2.21110	−0.183622
$146$	0	0
$147$	−9.28200	−0.765567
$148$	0	0
$149$	−5.21110	−0.426910	−0.213455	−	0.976953i	$-0.568472\pi$
$149$	−5.21110	−0.426910	−0.213455	+	0.976953i	$0.568472\pi$
$150$	0	0
$151$	−0.760128	−0.0618584	−0.0309292	−	0.999522i	$-0.509847\pi$
$151$	−0.760128	−0.0618584	−0.0309292	+	0.999522i	$0.509847\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−0.230148	−0.0184860
$156$	0	0
$157$	−3.09167	−0.246742	−0.123371	−	0.992361i	$-0.539371\pi$
$157$	−3.09167	−0.246742	−0.123371	+	0.992361i	$0.539371\pi$
$158$	0	0
$159$	5.25121	0.416448
$160$	0	0
$161$	10.6972	0.843059
$162$	0	0
$163$	9.28200	0.727023	0.363511	−	0.931590i	$-0.381578\pi$
$163$	9.28200	0.727023	0.363511	+	0.931590i	$0.381578\pi$
$164$	0	0
$165$	−0.577795	−0.0449813
$166$	0	0
$167$	7.53160	0.582813	0.291406	−	0.956599i	$-0.405877\pi$
$167$	7.53160	0.582813	0.291406	+	0.956599i	$0.405877\pi$
$168$	0	0
$169$	21.9083	1.68526
$170$	0	0
$171$	8.29173	0.634084
$172$	0	0
$173$	6.21110	0.472221	0.236111	−	0.971726i	$-0.424127\pi$
$173$	6.21110	0.472221	0.236111	+	0.971726i	$0.424127\pi$
$174$	0	0
$175$	−16.0535	−1.21353
$176$	0	0
$177$	32.8444	2.46874
$178$	0	0
$179$	−23.3549	−1.74563	−0.872815	−	0.488052i	$-0.837708\pi$
$179$	−23.3549	−1.74563	−0.872815	+	0.488052i	$0.837708\pi$
$180$	0	0
$181$	−3.78890	−0.281627	−0.140813	−	0.990036i	$-0.544972\pi$
$181$	−3.78890	−0.281627	−0.140813	+	0.990036i	$0.544972\pi$
$182$	0	0
$183$	35.6774	2.63735
$184$	0	0
$185$	−1.21110	−0.0890420
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.48612	−0.180839
$190$	0	0
$191$	17.5737	1.27159	0.635795	−	0.771858i	$-0.280672\pi$
$191$	17.5737	1.27159	0.635795	+	0.771858i	$0.280672\pi$
$192$	0	0
$193$	−5.00000	−0.359908	−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000	−0.359908	−0.179954	+	0.983675i	$0.557595\pi$
$194$	0	0
$195$	−4.49109	−0.321613
$196$	0	0
$197$	23.9361	1.70538	0.852688	−	0.522421i	$-0.174971\pi$
$197$	23.9361	1.70538	0.852688	+	0.522421i	$0.174971\pi$
$198$	0	0
$199$	20.8444	1.47762	0.738810	−	0.673914i	$-0.235388\pi$
$199$	20.8444	1.47762	0.738810	+	0.673914i	$0.235388\pi$
$200$	0	0
$201$	25.2111	1.77825
$202$	0	0
$203$	−23.8849	−1.67639
$204$	0	0
$205$	−1.57779	−0.110198
$206$	0	0
$207$	10.8023	0.750809
$208$	0	0
$209$	1.90833	0.132002
$210$	0	0
$211$	−16.3533	−1.12581	−0.562904	−	0.826522i	$-0.690316\pi$
$211$	−16.3533	−1.12581	−0.562904	+	0.826522i	$0.690316\pi$
$212$	0	0
$213$	25.2111	1.72744
$214$	0	0
$215$	−3.27066	−0.223057
$216$	0	0
$217$	−2.48612	−0.168769
$218$	0	0
$219$	30.6564	2.07157
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−9.81198	−0.657059	−0.328530	−	0.944494i	$-0.606553\pi$
$223$	−9.81198	−0.657059	−0.328530	+	0.944494i	$0.606553\pi$
$224$	0	0
$225$	−16.2111	−1.08074
$226$	0	0
$227$	7.76175	0.515165	0.257583	−	0.966256i	$-0.417074\pi$
$227$	7.76175	0.515165	0.257583	+	0.966256i	$0.417074\pi$
$228$	0	0
$229$	−21.2111	−1.40167	−0.700835	−	0.713324i	$-0.747189\pi$
$229$	−21.2111	−1.40167	−0.700835	+	0.713324i	$0.747189\pi$
$230$	0	0
$231$	−6.24149	−0.410660
$232$	0	0
$233$	−6.42221	−0.420733	−0.210366	−	0.977623i	$-0.567466\pi$
$233$	−6.42221	−0.420733	−0.210366	+	0.977623i	$0.567466\pi$
$234$	0	0
$235$	2.81036	0.183328
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.8428	−0.895415	−0.447707	−	0.894180i	$-0.647759\pi$
$239$	−13.8428	−0.895415	−0.447707	+	0.894180i	$0.647759\pi$
$240$	0	0
$241$	−3.09167	−0.199152	−0.0995761	−	0.995030i	$-0.531749\pi$
$241$	−3.09167	−0.199152	−0.0995761	+	0.995030i	$0.531749\pi$
$242$	0	0
$243$	22.3646	1.43469
$244$	0	0
$245$	1.11943	0.0715177
$246$	0	0
$247$	14.8330	0.943804
$248$	0	0
$249$	−31.5139	−1.99711
$250$	0	0
$251$	10.5721	0.667306	0.333653	−	0.942696i	$-0.391719\pi$
$251$	10.5721	0.667306	0.333653	+	0.942696i	$0.391719\pi$
$252$	0	0
$253$	2.48612	0.156301
$254$	0	0
$255$	0	0
$256$	0	0
$257$	29.5139	1.84103	0.920513	−	0.390713i	$-0.127771\pi$
$257$	29.5139	1.84103	0.920513	+	0.390713i	$0.127771\pi$
$258$	0	0
$259$	−13.0826	−0.812916
$260$	0	0
$261$	−24.1194	−1.49296
$262$	0	0
$263$	30.6564	1.89035	0.945177	−	0.326560i	$-0.105889\pi$
$263$	30.6564	1.89035	0.945177	+	0.326560i	$0.105889\pi$
$264$	0	0
$265$	−0.633308	−0.0389038
$266$	0	0
$267$	−23.1248	−1.41521
$268$	0	0
$269$	17.9083	1.09189	0.545945	−	0.837821i	$-0.316171\pi$
$269$	17.9083	1.09189	0.545945	+	0.837821i	$0.316171\pi$
$270$	0	0
$271$	−6.77147	−0.411338	−0.205669	−	0.978622i	$-0.565937\pi$
$271$	−6.77147	−0.411338	−0.205669	+	0.978622i	$0.565937\pi$
$272$	0	0
$273$	−48.5139	−2.93619
$274$	0	0
$275$	−3.73096	−0.224985
$276$	0	0
$277$	−10.7889	−0.648242	−0.324121	−	0.946016i	$-0.605069\pi$
$277$	−10.7889	−0.648242	−0.324121	+	0.946016i	$0.605069\pi$
$278$	0	0
$279$	−2.51053	−0.150302
$280$	0	0
$281$	20.9083	1.24729	0.623643	−	0.781709i	$-0.285652\pi$
$281$	20.9083	1.24729	0.623643	+	0.781709i	$0.285652\pi$
$282$	0	0
$283$	−26.1653	−1.55536	−0.777682	−	0.628657i	$-0.783605\pi$
$283$	−26.1653	−1.55536	−0.777682	+	0.628657i	$0.783605\pi$
$284$	0	0
$285$	−1.90833	−0.113040
$286$	0	0
$287$	−17.0438	−1.00606
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−3.80064	−0.222798
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	−3.96111	−0.230625
$296$	0	0
$297$	−0.577795	−0.0335271
$298$	0	0
$299$	19.3241	1.11754
$300$	0	0
$301$	−35.3305	−2.03642
$302$	0	0
$303$	−7.53160	−0.432679
$304$	0	0
$305$	−4.30278	−0.246376
$306$	0	0
$307$	9.81198	0.559999	0.280000	−	0.960000i	$-0.409666\pi$
$307$	9.81198	0.559999	0.280000	+	0.960000i	$0.409666\pi$
$308$	0	0
$309$	−32.8444	−1.86845
$310$	0	0
$311$	15.5932	0.884208	0.442104	−	0.896964i	$-0.354232\pi$
$311$	15.5932	0.884208	0.442104	+	0.896964i	$0.354232\pi$
$312$	0	0
$313$	−10.4222	−0.589098	−0.294549	−	0.955636i	$-0.595169\pi$
$313$	−10.4222	−0.589098	−0.294549	+	0.955636i	$0.595169\pi$
$314$	0	0
$315$	3.27066	0.184281
$316$	0	0
$317$	−13.4222	−0.753866	−0.376933	−	0.926240i	$-0.623021\pi$
$317$	−13.4222	−0.753866	−0.376933	+	0.926240i	$0.623021\pi$
$318$	0	0
$319$	−5.55105	−0.310799
$320$	0	0
$321$	23.3028	1.30063
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−29.0000	−1.60863
$326$	0	0
$327$	23.6548	1.30811
$328$	0	0
$329$	30.3583	1.67371
$330$	0	0
$331$	12.8525	0.706437	0.353219	−	0.935541i	$-0.385087\pi$
$331$	12.8525	0.706437	0.353219	+	0.935541i	$0.385087\pi$
$332$	0	0
$333$	−13.2111	−0.723964
$334$	0	0
$335$	−3.04051	−0.166121
$336$	0	0
$337$	−7.11943	−0.387820	−0.193910	−	0.981019i	$-0.562117\pi$
$337$	−7.11943	−0.387820	−0.193910	+	0.981019i	$0.562117\pi$
$338$	0	0
$339$	32.4068	1.76009
$340$	0	0
$341$	−0.577795	−0.0312893
$342$	0	0
$343$	−10.8023	−0.583267
$344$	0	0
$345$	−2.48612	−0.133848
$346$	0	0
$347$	2.51053	0.134772	0.0673862	−	0.997727i	$-0.478534\pi$
$347$	2.51053	0.134772	0.0673862	+	0.997727i	$0.478534\pi$
$348$	0	0
$349$	−2.57779	−0.137986	−0.0689931	−	0.997617i	$-0.521979\pi$
$349$	−2.57779	−0.137986	−0.0689931	+	0.997617i	$0.521979\pi$
$350$	0	0
$351$	−4.49109	−0.239716
$352$	0	0
$353$	−24.6333	−1.31110	−0.655549	−	0.755152i	$-0.727563\pi$
$353$	−24.6333	−1.31110	−0.655549	+	0.755152i	$0.727563\pi$
$354$	0	0
$355$	−3.04051	−0.161374
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13.3128	0.702622	0.351311	−	0.936259i	$-0.385736\pi$
$359$	13.3128	0.702622	0.351311	+	0.936259i	$0.385736\pi$
$360$	0	0
$361$	−12.6972	−0.668275
$362$	0	0
$363$	26.1653	1.37332
$364$	0	0
$365$	−3.69722	−0.193522
$366$	0	0
$367$	−26.1653	−1.36582	−0.682908	−	0.730504i	$-0.739285\pi$
$367$	−26.1653	−1.36582	−0.682908	+	0.730504i	$0.739285\pi$
$368$	0	0
$369$	−17.2111	−0.895974
$370$	0	0
$371$	−6.84115	−0.355175
$372$	0	0
$373$	6.21110	0.321599	0.160799	−	0.986987i	$-0.448593\pi$
$373$	6.21110	0.321599	0.160799	+	0.986987i	$0.448593\pi$
$374$	0	0
$375$	7.53160	0.388930
$376$	0	0
$377$	−43.1472	−2.22219
$378$	0	0
$379$	−0.760128	−0.0390452	−0.0195226	−	0.999809i	$-0.506215\pi$
$379$	−0.760128	−0.0390452	−0.0195226	+	0.999809i	$0.506215\pi$
$380$	0	0
$381$	−27.1194	−1.38937
$382$	0	0
$383$	27.6159	1.41110	0.705552	−	0.708658i	$-0.250699\pi$
$383$	27.6159	1.41110	0.705552	+	0.708658i	$0.250699\pi$
$384$	0	0
$385$	0.752737	0.0383630
$386$	0	0
$387$	−35.6774	−1.81359
$388$	0	0
$389$	17.0000	0.861934	0.430967	−	0.902368i	$-0.358172\pi$
$389$	17.0000	0.861934	0.430967	+	0.902368i	$0.358172\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	25.2111	1.27173
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−28.2111	−1.41587	−0.707937	−	0.706275i	$-0.750374\pi$
$397$	−28.2111	−1.41587	−0.707937	+	0.706275i	$0.750374\pi$
$398$	0	0
$399$	−20.6142	−1.03200
$400$	0	0
$401$	−11.3305	−0.565820	−0.282910	−	0.959146i	$-0.591300\pi$
$401$	−11.3305	−0.565820	−0.282910	+	0.959146i	$0.591300\pi$
$402$	0	0
$403$	−4.49109	−0.223717
$404$	0	0
$405$	−2.42221	−0.120360
$406$	0	0
$407$	−3.04051	−0.150713
$408$	0	0
$409$	−19.7889	−0.978498	−0.489249	−	0.872144i	$-0.662729\pi$
$409$	−19.7889	−0.978498	−0.489249	+	0.872144i	$0.662729\pi$
$410$	0	0
$411$	43.2090	2.13134
$412$	0	0
$413$	−42.7889	−2.10550
$414$	0	0
$415$	3.80064	0.186566
$416$	0	0
$417$	49.8444	2.44089
$418$	0	0
$419$	30.8865	1.50891	0.754453	−	0.656354i	$-0.227902\pi$
$419$	30.8865	1.50891	0.754453	+	0.656354i	$0.227902\pi$
$420$	0	0
$421$	37.5139	1.82831	0.914157	−	0.405360i	$-0.132854\pi$
$421$	37.5139	1.82831	0.914157	+	0.405360i	$0.132854\pi$
$422$	0	0
$423$	30.6564	1.49056
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−46.4797	−2.24931
$428$	0	0
$429$	−11.2750	−0.544363
$430$	0	0
$431$	−36.4376	−1.75514	−0.877568	−	0.479452i	$-0.840835\pi$
$431$	−36.4376	−1.75514	−0.877568	+	0.479452i	$0.840835\pi$
$432$	0	0
$433$	25.2111	1.21157	0.605784	−	0.795629i	$-0.292860\pi$
$433$	25.2111	1.21157	0.605784	+	0.795629i	$0.292860\pi$
$434$	0	0
$435$	5.55105	0.266152
$436$	0	0
$437$	8.21110	0.392790
$438$	0	0
$439$	40.1685	1.91714	0.958570	−	0.284858i	$-0.0919466\pi$
$439$	40.1685	1.91714	0.958570	+	0.284858i	$0.0919466\pi$
$440$	0	0
$441$	12.2111	0.581481
$442$	0	0
$443$	−12.3225	−0.585460	−0.292730	−	0.956195i	$-0.594564\pi$
$443$	−12.3225	−0.585460	−0.292730	+	0.956195i	$0.594564\pi$
$444$	0	0
$445$	2.78890	0.132206
$446$	0	0
$447$	13.0826	0.618788
$448$	0	0
$449$	−9.90833	−0.467603	−0.233801	−	0.972284i	$-0.575117\pi$
$449$	−9.90833	−0.467603	−0.233801	+	0.972284i	$0.575117\pi$
$450$	0	0
$451$	−3.96111	−0.186521
$452$	0	0
$453$	1.90833	0.0896610
$454$	0	0
$455$	5.85088	0.274293
$456$	0	0
$457$	14.7250	0.688806	0.344403	−	0.938822i	$-0.388081\pi$
$457$	14.7250	0.688806	0.344403	+	0.938822i	$0.388081\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−27.6333	−1.28701	−0.643506	−	0.765441i	$-0.722521\pi$
$461$	−27.6333	−1.28701	−0.643506	+	0.765441i	$0.722521\pi$
$462$	0	0
$463$	−20.0843	−0.933395	−0.466697	−	0.884417i	$-0.654556\pi$
$463$	−20.0843	−0.933395	−0.466697	+	0.884417i	$0.654556\pi$
$464$	0	0
$465$	0.577795	0.0267946
$466$	0	0
$467$	−6.31117	−0.292046	−0.146023	−	0.989281i	$-0.546647\pi$
$467$	−6.31117	−0.292046	−0.146023	+	0.989281i	$0.546647\pi$
$468$	0	0
$469$	−32.8444	−1.51661
$470$	0	0
$471$	7.76175	0.357642
$472$	0	0
$473$	−8.21110	−0.377547
$474$	0	0
$475$	−12.3225	−0.565396
$476$	0	0
$477$	−6.90833	−0.316311
$478$	0	0
$479$	22.3646	1.02187	0.510933	−	0.859620i	$-0.329300\pi$
$479$	22.3646	1.02187	0.510933	+	0.859620i	$0.329300\pi$
$480$	0	0
$481$	−23.6333	−1.07759
$482$	0	0
$483$	−26.8557	−1.22198
$484$	0	0
$485$	0.458365	0.0208133
$486$	0	0
$487$	9.51215	0.431037	0.215518	−	0.976500i	$-0.430856\pi$
$487$	9.51215	0.431037	0.215518	+	0.976500i	$0.430856\pi$
$488$	0	0
$489$	−23.3028	−1.05379
$490$	0	0
$491$	−36.4376	−1.64440	−0.822202	−	0.569195i	$-0.807255\pi$
$491$	−36.4376	−1.64440	−0.822202	+	0.569195i	$0.807255\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0.760128	0.0341652
$496$	0	0
$497$	−32.8444	−1.47327
$498$	0	0
$499$	−33.3971	−1.49506	−0.747529	−	0.664229i	$-0.768760\pi$
$499$	−33.3971	−1.49506	−0.747529	+	0.664229i	$0.768760\pi$
$500$	0	0
$501$	−18.9083	−0.844762
$502$	0	0
$503$	0.760128	0.0338924	0.0169462	−	0.999856i	$-0.494606\pi$
$503$	0.760128	0.0338924	0.0169462	+	0.999856i	$0.494606\pi$
$504$	0	0
$505$	0.908327	0.0404200
$506$	0	0
$507$	−55.0016	−2.44271
$508$	0	0
$509$	18.2111	0.807193	0.403596	−	0.914937i	$-0.367760\pi$
$509$	18.2111	0.807193	0.403596	+	0.914937i	$0.367760\pi$
$510$	0	0
$511$	−39.9384	−1.76677
$512$	0	0
$513$	−1.90833	−0.0842547
$514$	0	0
$515$	3.96111	0.174547
$516$	0	0
$517$	7.05551	0.310301
$518$	0	0
$519$	−15.5932	−0.684465
$520$	0	0
$521$	5.21110	0.228303	0.114151	−	0.993463i	$-0.463585\pi$
$521$	5.21110	0.228303	0.114151	+	0.993463i	$0.463585\pi$
$522$	0	0
$523$	−9.28200	−0.405874	−0.202937	−	0.979192i	$-0.565049\pi$
$523$	−9.28200	−0.405874	−0.202937	+	0.979192i	$0.565049\pi$
$524$	0	0
$525$	40.3028	1.75896
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12.3028	−0.534903
$530$	0	0
$531$	−43.2090	−1.87511
$532$	0	0
$533$	−30.7889	−1.33362
$534$	0	0
$535$	−2.81036	−0.121503
$536$	0	0
$537$	58.6333	2.53021
$538$	0	0
$539$	2.81036	0.121051
$540$	0	0
$541$	7.30278	0.313971	0.156985	−	0.987601i	$-0.449822\pi$
$541$	7.30278	0.313971	0.156985	+	0.987601i	$0.449822\pi$
$542$	0	0
$543$	9.51215	0.408206
$544$	0	0
$545$	−2.85281	−0.122201
$546$	0	0
$547$	−3.80064	−0.162504	−0.0812518	−	0.996694i	$-0.525892\pi$
$547$	−3.80064	−0.162504	−0.0812518	+	0.996694i	$0.525892\pi$
$548$	0	0
$549$	−46.9361	−2.00318
$550$	0	0
$551$	−18.3339	−0.781049
$552$	0	0
$553$	0	0
$554$	0	0
$555$	3.04051	0.129063
$556$	0	0
$557$	2.78890	0.118169	0.0590847	−	0.998253i	$-0.481182\pi$
$557$	2.78890	0.118169	0.0590847	+	0.998253i	$0.481182\pi$
$558$	0	0
$559$	−63.8233	−2.69944
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.1248	0.974593	0.487297	−	0.873237i	$-0.337983\pi$
$563$	23.1248	0.974593	0.487297	+	0.873237i	$0.337983\pi$
$564$	0	0
$565$	−3.90833	−0.164425
$566$	0	0
$567$	−26.1653	−1.09884
$568$	0	0
$569$	−36.2111	−1.51805	−0.759024	−	0.651062i	$-0.774324\pi$
$569$	−36.2111	−1.51805	−0.759024	+	0.651062i	$0.774324\pi$
$570$	0	0
$571$	10.5721	0.442429	0.221215	−	0.975225i	$-0.428998\pi$
$571$	10.5721	0.442429	0.221215	+	0.975225i	$0.428998\pi$
$572$	0	0
$573$	−44.1194	−1.84312
$574$	0	0
$575$	−16.0535	−0.669476
$576$	0	0
$577$	−10.6333	−0.442670	−0.221335	−	0.975198i	$-0.571041\pi$
$577$	−10.6333	−0.442670	−0.221335	+	0.975198i	$0.571041\pi$
$578$	0	0
$579$	12.5527	0.521671
$580$	0	0
$581$	41.0555	1.70327
$582$	0	0
$583$	−1.58994	−0.0658486
$584$	0	0
$585$	5.90833	0.244279
$586$	0	0
$587$	9.51215	0.392609	0.196304	−	0.980543i	$-0.437106\pi$
$587$	9.51215	0.392609	0.196304	+	0.980543i	$0.437106\pi$
$588$	0	0
$589$	−1.90833	−0.0786312
$590$	0	0
$591$	−60.0923	−2.47187
$592$	0	0
$593$	−7.78890	−0.319852	−0.159926	−	0.987129i	$-0.551126\pi$
$593$	−7.78890	−0.319852	−0.159926	+	0.987129i	$0.551126\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−52.3305	−2.14175
$598$	0	0
$599$	−7.76175	−0.317136	−0.158568	−	0.987348i	$-0.550688\pi$
$599$	−7.76175	−0.317136	−0.158568	+	0.987348i	$0.550688\pi$
$600$	0	0
$601$	−38.7527	−1.58076	−0.790379	−	0.612619i	$-0.790116\pi$
$601$	−38.7527	−1.58076	−0.790379	+	0.612619i	$0.790116\pi$
$602$	0	0
$603$	−33.1669	−1.35066
$604$	0	0
$605$	−3.15559	−0.128293
$606$	0	0
$607$	−7.76175	−0.315040	−0.157520	−	0.987516i	$-0.550350\pi$
$607$	−7.76175	−0.315040	−0.157520	+	0.987516i	$0.550350\pi$
$608$	0	0
$609$	59.9638	2.42986
$610$	0	0
$611$	54.8411	2.21863
$612$	0	0
$613$	25.2111	1.01827	0.509133	−	0.860688i	$-0.329966\pi$
$613$	25.2111	1.01827	0.509133	+	0.860688i	$0.329966\pi$
$614$	0	0
$615$	3.96111	0.159727
$616$	0	0
$617$	25.0555	1.00870	0.504348	−	0.863500i	$-0.331733\pi$
$617$	25.0555	1.00870	0.504348	+	0.863500i	$0.331733\pi$
$618$	0	0
$619$	10.8023	0.434179	0.217090	−	0.976152i	$-0.430344\pi$
$619$	10.8023	0.434179	0.217090	+	0.976152i	$0.430344\pi$
$620$	0	0
$621$	−2.48612	−0.0997646
$622$	0	0
$623$	30.1264	1.20699
$624$	0	0
$625$	23.6333	0.945332
$626$	0	0
$627$	−4.79092	−0.191331
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−2.51053	−0.0999427	−0.0499714	−	0.998751i	$-0.515913\pi$
$631$	−2.51053	−0.0999427	−0.0499714	+	0.998751i	$0.515913\pi$
$632$	0	0
$633$	41.0555	1.63181
$634$	0	0
$635$	3.27066	0.129792
$636$	0	0
$637$	21.8444	0.865507
$638$	0	0
$639$	−33.1669	−1.31206
$640$	0	0
$641$	−9.72498	−0.384114	−0.192057	−	0.981384i	$-0.561516\pi$
$641$	−9.72498	−0.384114	−0.192057	+	0.981384i	$0.561516\pi$
$642$	0	0
$643$	33.1669	1.30798	0.653988	−	0.756505i	$-0.273095\pi$
$643$	33.1669	1.30798	0.653988	+	0.756505i	$0.273095\pi$
$644$	0	0
$645$	8.21110	0.323312
$646$	0	0
$647$	−20.8444	−0.819478	−0.409739	−	0.912203i	$-0.634380\pi$
$647$	−20.8444	−0.819478	−0.409739	+	0.912203i	$0.634380\pi$
$648$	0	0
$649$	−9.94449	−0.390355
$650$	0	0
$651$	6.24149	0.244623
$652$	0	0
$653$	27.4222	1.07311	0.536557	−	0.843864i	$-0.319725\pi$
$653$	27.4222	1.07311	0.536557	+	0.843864i	$0.319725\pi$
$654$	0	0
$655$	−3.04051	−0.118803
$656$	0	0
$657$	−40.3305	−1.57344
$658$	0	0
$659$	−20.8444	−0.811982	−0.405991	−	0.913877i	$-0.633074\pi$
$659$	−20.8444	−0.811982	−0.405991	+	0.913877i	$0.633074\pi$
$660$	0	0
$661$	14.4861	0.563445	0.281722	−	0.959496i	$-0.409094\pi$
$661$	14.4861	0.563445	0.281722	+	0.959496i	$0.409094\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.48612	0.0964077
$666$	0	0
$667$	−23.8849	−0.924827
$668$	0	0
$669$	24.6333	0.952379
$670$	0	0
$671$	−10.8023	−0.417017
$672$	0	0
$673$	14.4222	0.555935	0.277968	−	0.960590i	$-0.410339\pi$
$673$	14.4222	0.555935	0.277968	+	0.960590i	$0.410339\pi$
$674$	0	0
$675$	3.73096	0.143605
$676$	0	0
$677$	40.5139	1.55707	0.778537	−	0.627598i	$-0.215962\pi$
$677$	40.5139	1.55707	0.778537	+	0.627598i	$0.215962\pi$
$678$	0	0
$679$	4.95138	0.190017
$680$	0	0
$681$	−19.4861	−0.746710
$682$	0	0
$683$	−31.7163	−1.21359	−0.606796	−	0.794858i	$-0.707545\pi$
$683$	−31.7163	−1.21359	−0.606796	+	0.794858i	$0.707545\pi$
$684$	0	0
$685$	−5.21110	−0.199106
$686$	0	0
$687$	53.2512	2.03166
$688$	0	0
$689$	−12.3583	−0.470813
$690$	0	0
$691$	24.6450	0.937541	0.468771	−	0.883320i	$-0.344697\pi$
$691$	24.6450	0.937541	0.468771	+	0.883320i	$0.344697\pi$
$692$	0	0
$693$	8.21110	0.311914
$694$	0	0
$695$	−6.01134	−0.228023
$696$	0	0
$697$	0	0
$698$	0	0
$699$	16.1232	0.609834
$700$	0	0
$701$	−9.78890	−0.369722	−0.184861	−	0.982765i	$-0.559183\pi$
$701$	−9.78890	−0.369722	−0.184861	+	0.982765i	$0.559183\pi$
$702$	0	0
$703$	−10.0421	−0.378746
$704$	0	0
$705$	−7.05551	−0.265726
$706$	0	0
$707$	9.81198	0.369018
$708$	0	0
$709$	−6.09167	−0.228778	−0.114389	−	0.993436i	$-0.536491\pi$
$709$	−6.09167	−0.228778	−0.114389	+	0.993436i	$0.536491\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.48612	−0.0931060
$714$	0	0
$715$	1.35979	0.0508533
$716$	0	0
$717$	34.7527	1.29786
$718$	0	0
$719$	25.4052	0.947453	0.473726	−	0.880672i	$-0.342909\pi$
$719$	25.4052	0.947453	0.473726	+	0.880672i	$0.342909\pi$
$720$	0	0
$721$	42.7889	1.59354
$722$	0	0
$723$	7.76175	0.288662
$724$	0	0
$725$	35.8444	1.33123
$726$	0	0
$727$	−49.2901	−1.82807	−0.914034	−	0.405638i	$-0.867049\pi$
$727$	−49.2901	−1.82807	−0.914034	+	0.405638i	$0.867049\pi$
$728$	0	0
$729$	−32.1472	−1.19064
$730$	0	0
$731$	0	0
$732$	0	0
$733$	5.90833	0.218229	0.109115	−	0.994029i	$-0.465198\pi$
$733$	5.90833	0.218229	0.109115	+	0.994029i	$0.465198\pi$
$734$	0	0
$735$	−2.81036	−0.103662
$736$	0	0
$737$	−7.63331	−0.281177
$738$	0	0
$739$	40.9287	1.50558	0.752792	−	0.658258i	$-0.228706\pi$
$739$	40.9287	1.50558	0.752792	+	0.658258i	$0.228706\pi$
$740$	0	0
$741$	−37.2389	−1.36800
$742$	0	0
$743$	−17.8039	−0.653161	−0.326580	−	0.945169i	$-0.605896\pi$
$743$	−17.8039	−0.653161	−0.326580	+	0.945169i	$0.605896\pi$
$744$	0	0
$745$	−1.57779	−0.0578059
$746$	0	0
$747$	41.4586	1.51689
$748$	0	0
$749$	−30.3583	−1.10927
$750$	0	0
$751$	12.3225	0.449655	0.224828	−	0.974399i	$-0.427818\pi$
$751$	12.3225	0.449655	0.224828	+	0.974399i	$0.427818\pi$
$752$	0	0
$753$	−26.5416	−0.967231
$754$	0	0
$755$	−0.230148	−0.00837595
$756$	0	0
$757$	−7.78890	−0.283092	−0.141546	−	0.989932i	$-0.545207\pi$
$757$	−7.78890	−0.283092	−0.141546	+	0.989932i	$0.545207\pi$
$758$	0	0
$759$	−6.24149	−0.226552
$760$	0	0
$761$	−19.7889	−0.717347	−0.358673	−	0.933463i	$-0.616771\pi$
$761$	−19.7889	−0.717347	−0.358673	+	0.933463i	$0.616771\pi$
$762$	0	0
$763$	−30.8168	−1.11564
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−77.2965	−2.79102
$768$	0	0
$769$	28.3305	1.02163	0.510813	−	0.859692i	$-0.329345\pi$
$769$	28.3305	1.02163	0.510813	+	0.859692i	$0.329345\pi$
$770$	0	0
$771$	−74.0956	−2.66849
$772$	0	0
$773$	−24.6333	−0.885998	−0.442999	−	0.896522i	$-0.646086\pi$
$773$	−24.6333	−0.885998	−0.442999	+	0.896522i	$0.646086\pi$
$774$	0	0
$775$	3.73096	0.134020
$776$	0	0
$777$	32.8444	1.17829
$778$	0	0
$779$	−13.0826	−0.468734
$780$	0	0
$781$	−7.63331	−0.273141
$782$	0	0
$783$	5.55105	0.198378
$784$	0	0
$785$	−0.936083	−0.0334102
$786$	0	0
$787$	6.08103	0.216765	0.108383	−	0.994109i	$-0.465433\pi$
$787$	6.08103	0.216765	0.108383	+	0.994109i	$0.465433\pi$
$788$	0	0
$789$	−76.9638	−2.73999
$790$	0	0
$791$	−42.2188	−1.50113
$792$	0	0
$793$	−83.9638	−2.98164
$794$	0	0
$795$	1.58994	0.0563894
$796$	0	0
$797$	33.8444	1.19883	0.599415	−	0.800438i	$-0.295400\pi$
$797$	33.8444	1.19883	0.599415	+	0.800438i	$0.295400\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	30.4222	1.07492
$802$	0	0
$803$	−9.28200	−0.327555
$804$	0	0
$805$	3.23886	0.114155
$806$	0	0
$807$	−44.9594	−1.58265
$808$	0	0
$809$	47.6333	1.67470	0.837349	−	0.546669i	$-0.184104\pi$
$809$	47.6333	1.67470	0.837349	+	0.546669i	$0.184104\pi$
$810$	0	0
$811$	13.0826	0.459394	0.229697	−	0.973262i	$-0.426227\pi$
$811$	13.0826	0.459394	0.229697	+	0.973262i	$0.426227\pi$
$812$	0	0
$813$	17.0000	0.596216
$814$	0	0
$815$	2.81036	0.0984428
$816$	0	0
$817$	−27.1194	−0.948789
$818$	0	0
$819$	63.8233	2.23017
$820$	0	0
$821$	−31.4222	−1.09664	−0.548321	−	0.836268i	$-0.684733\pi$
$821$	−31.4222	−1.09664	−0.548321	+	0.836268i	$0.684733\pi$
$822$	0	0
$823$	−26.3258	−0.917658	−0.458829	−	0.888525i	$-0.651731\pi$
$823$	−26.3258	−0.917658	−0.458829	+	0.888525i	$0.651731\pi$
$824$	0	0
$825$	9.36669	0.326106
$826$	0	0
$827$	26.9254	0.936289	0.468144	−	0.883652i	$-0.344923\pi$
$827$	26.9254	0.936289	0.468144	+	0.883652i	$0.344923\pi$
$828$	0	0
$829$	−2.42221	−0.0841267	−0.0420633	−	0.999115i	$-0.513393\pi$
$829$	−2.42221	−0.0841267	−0.0420633	+	0.999115i	$0.513393\pi$
$830$	0	0
$831$	27.0859	0.939599
$832$	0	0
$833$	0	0
$834$	0	0
$835$	2.28038	0.0789160
$836$	0	0
$837$	0.577795	0.0199715
$838$	0	0
$839$	−37.1280	−1.28180	−0.640901	−	0.767624i	$-0.721439\pi$
$839$	−37.1280	−1.28180	−0.640901	+	0.767624i	$0.721439\pi$
$840$	0	0
$841$	24.3305	0.838984
$842$	0	0
$843$	−52.4910	−1.80789
$844$	0	0
$845$	6.63331	0.228193
$846$	0	0
$847$	−34.0875	−1.17126
$848$	0	0
$849$	65.6888	2.25443
$850$	0	0
$851$	−13.0826	−0.448467
$852$	0	0
$853$	−34.2111	−1.17137	−0.585683	−	0.810540i	$-0.699174\pi$
$853$	−34.2111	−1.17137	−0.585683	+	0.810540i	$0.699174\pi$
$854$	0	0
$855$	2.51053	0.0858584
$856$	0	0
$857$	14.5778	0.497968	0.248984	−	0.968508i	$-0.419903\pi$
$857$	14.5778	0.497968	0.248984	+	0.968508i	$0.419903\pi$
$858$	0	0
$859$	3.96111	0.135151	0.0675756	−	0.997714i	$-0.478474\pi$
$859$	3.96111	0.135151	0.0675756	+	0.997714i	$0.478474\pi$
$860$	0	0
$861$	42.7889	1.45824
$862$	0	0
$863$	6.31117	0.214835	0.107417	−	0.994214i	$-0.465742\pi$
$863$	6.31117	0.214835	0.107417	+	0.994214i	$0.465742\pi$
$864$	0	0
$865$	1.88057	0.0639413
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−59.3322	−2.01039
$872$	0	0
$873$	5.00000	0.169224
$874$	0	0
$875$	−9.81198	−0.331706
$876$	0	0
$877$	7.33053	0.247534	0.123767	−	0.992311i	$-0.460502\pi$
$877$	7.33053	0.247534	0.123767	+	0.992311i	$0.460502\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−33.3583	−1.12387	−0.561935	−	0.827182i	$-0.689943\pi$
$881$	−33.3583	−1.12387	−0.561935	+	0.827182i	$0.689943\pi$
$882$	0	0
$883$	−15.5932	−0.524752	−0.262376	−	0.964966i	$-0.584506\pi$
$883$	−15.5932	−0.524752	−0.262376	+	0.964966i	$0.584506\pi$
$884$	0	0
$885$	9.94449	0.334280
$886$	0	0
$887$	6.31117	0.211908	0.105954	−	0.994371i	$-0.466210\pi$
$887$	6.31117	0.211908	0.105954	+	0.994371i	$0.466210\pi$
$888$	0	0
$889$	35.3305	1.18495
$890$	0	0
$891$	−6.08103	−0.203722
$892$	0	0
$893$	23.3028	0.779798
$894$	0	0
$895$	−7.07130	−0.236368
$896$	0	0
$897$	−48.5139	−1.61983
$898$	0	0
$899$	5.55105	0.185138
$900$	0	0
$901$	0	0
$902$	0	0
$903$	88.6985	2.95170
$904$	0	0
$905$	−1.14719	−0.0381337
$906$	0	0
$907$	32.6369	1.08369	0.541846	−	0.840478i	$-0.317726\pi$
$907$	32.6369	1.08369	0.541846	+	0.840478i	$0.317726\pi$
$908$	0	0
$909$	9.90833	0.328638
$910$	0	0
$911$	26.3954	0.874520	0.437260	−	0.899335i	$-0.355949\pi$
$911$	26.3954	0.874520	0.437260	+	0.899335i	$0.355949\pi$
$912$	0	0
$913$	9.54163	0.315782
$914$	0	0
$915$	10.8023	0.357112
$916$	0	0
$917$	−32.8444	−1.08462
$918$	0	0
$919$	29.5964	0.976296	0.488148	−	0.872761i	$-0.337673\pi$
$919$	29.5964	0.976296	0.488148	+	0.872761i	$0.337673\pi$
$920$	0	0
$921$	−24.6333	−0.811695
$922$	0	0
$923$	−59.3322	−1.95294
$924$	0	0
$925$	19.6333	0.645539
$926$	0	0
$927$	43.2090	1.41917
$928$	0	0
$929$	21.9083	0.718789	0.359394	−	0.933186i	$-0.382983\pi$
$929$	21.9083	0.718789	0.359394	+	0.933186i	$0.382983\pi$
$930$	0	0
$931$	9.28200	0.304205
$932$	0	0
$933$	−39.1472	−1.28162
$934$	0	0
$935$	0	0
$936$	0	0
$937$	25.1472	0.821523	0.410761	−	0.911743i	$-0.365263\pi$
$937$	25.1472	0.821523	0.410761	+	0.911743i	$0.365263\pi$
$938$	0	0
$939$	26.1653	0.853872
$940$	0	0
$941$	31.3305	1.02135	0.510673	−	0.859775i	$-0.329396\pi$
$941$	31.3305	1.02135	0.510673	+	0.859775i	$0.329396\pi$
$942$	0	0
$943$	−17.0438	−0.555021
$944$	0	0
$945$	−0.752737	−0.0244865
$946$	0	0
$947$	4.72123	0.153419	0.0767097	−	0.997053i	$-0.475559\pi$
$947$	4.72123	0.153419	0.0767097	+	0.997053i	$0.475559\pi$
$948$	0	0
$949$	−72.1472	−2.34200
$950$	0	0
$951$	33.6969	1.09270
$952$	0	0
$953$	14.4861	0.469252	0.234626	−	0.972086i	$-0.424614\pi$
$953$	14.4861	0.469252	0.234626	+	0.972086i	$0.424614\pi$
$954$	0	0
$955$	5.32090	0.172180
$956$	0	0
$957$	13.9361	0.450490
$958$	0	0
$959$	−56.2917	−1.81775
$960$	0	0
$961$	−30.4222	−0.981361
$962$	0	0
$963$	−30.6564	−0.987888
$964$	0	0
$965$	−1.51388	−0.0487335
$966$	0	0
$967$	−20.3144	−0.653267	−0.326634	−	0.945151i	$-0.605914\pi$
$967$	−20.3144	−0.653267	−0.326634	+	0.945151i	$0.605914\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−16.8136	−0.539574	−0.269787	−	0.962920i	$-0.586953\pi$
$971$	−16.8136	−0.539574	−0.269787	+	0.962920i	$0.586953\pi$
$972$	0	0
$973$	−64.9361	−2.08176
$974$	0	0
$975$	72.8055	2.33164
$976$	0	0
$977$	31.3028	1.00146	0.500732	−	0.865602i	$-0.333064\pi$
$977$	31.3028	1.00146	0.500732	+	0.865602i	$0.333064\pi$
$978$	0	0
$979$	7.00162	0.223773
$980$	0	0
$981$	−31.1194	−0.993567
$982$	0	0
$983$	23.1248	0.737566	0.368783	−	0.929516i	$-0.379775\pi$
$983$	23.1248	0.737566	0.368783	+	0.929516i	$0.379775\pi$
$984$	0	0
$985$	7.24726	0.230917
$986$	0	0
$987$	−76.2155	−2.42597
$988$	0	0
$989$	−35.3305	−1.12345
$990$	0	0
$991$	44.6596	1.41866	0.709330	−	0.704877i	$-0.248998\pi$
$991$	44.6596	1.41866	0.709330	+	0.704877i	$0.248998\pi$
$992$	0	0
$993$	−32.2666	−1.02395
$994$	0	0
$995$	6.31117	0.200078
$996$	0	0
$997$	−28.6972	−0.908850	−0.454425	−	0.890785i	$-0.650155\pi$
$997$	−28.6972	−0.908850	−0.454425	+	0.890785i	$0.650155\pi$
$998$	0	0
$999$	3.04051	0.0961976

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9248.2.a.z.1.1	✓	4
4.3	odd	2	inner	9248.2.a.z.1.4	yes	4
17.16	even	2		9248.2.a.bi.1.4	yes	4
68.67	odd	2		9248.2.a.bi.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9248.2.a.z.1.1	✓	4	1.1	even	1	trivial
9248.2.a.z.1.4	yes	4	4.3	odd	2	inner
9248.2.a.bi.1.1	yes	4	68.67	odd	2
9248.2.a.bi.1.4	yes	4	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9248 = 2^{5} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9248.a (trivial)

\(f(q)\)	\(=\)	\(q-2.51053 q^{3} +0.302776 q^{5} +3.27066 q^{7} +3.30278 q^{9} +0.760128 q^{11} +5.90833 q^{13} -0.760128 q^{15} +2.51053 q^{19} -8.21110 q^{21} +3.27066 q^{23} -4.90833 q^{25} -0.760128 q^{27} -7.30278 q^{29} -0.760128 q^{31} -1.90833 q^{33} +0.990277 q^{35} -4.00000 q^{37} -14.8330 q^{39} -5.21110 q^{41} -10.8023 q^{43} +1.00000 q^{45} +9.28200 q^{47} +3.69722 q^{49} -2.09167 q^{53} +0.230148 q^{55} -6.30278 q^{57} -13.0826 q^{59} -14.2111 q^{61} +10.8023 q^{63} +1.78890 q^{65} -10.0421 q^{67} -8.21110 q^{69} -10.0421 q^{71} -12.2111 q^{73} +12.3225 q^{75} +2.48612 q^{77} -8.00000 q^{81} +12.5527 q^{83} +18.3339 q^{87} +9.21110 q^{89} +19.3241 q^{91} +1.90833 q^{93} +0.760128 q^{95} +1.51388 q^{97} +2.51053 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5} + 6 q^{9} + 2 q^{13} - 4 q^{21} + 2 q^{25} - 22 q^{29} + 14 q^{33} - 16 q^{37} + 8 q^{41} + 4 q^{45} + 22 q^{49} - 30 q^{53} - 18 q^{57} - 28 q^{61} + 36 q^{65} - 4 q^{69} - 20 q^{73} + 46 q^{77}+ \cdots - 30 q^{97}+O(q^{100}) \) 4 * q - 6 * q^5 + 6 * q^9 + 2 * q^13 - 4 * q^21 + 2 * q^25 - 22 * q^29 + 14 * q^33 - 16 * q^37 + 8 * q^41 + 4 * q^45 + 22 * q^49 - 30 * q^53 - 18 * q^57 - 28 * q^61 + 36 * q^65 - 4 * q^69 - 20 * q^73 + 46 * q^77 - 32 * q^81 + 8 * q^89 - 14 * q^93 - 30 * q^97

Introduction

L-functions

Modular forms

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Database

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