LMFDB - Embedded newform 968.4.a.o.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,4,Mod(1,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("968.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	968.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.1138488856$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 87x^{6} + 12x^{5} + 2157x^{4} + 2939x^{3} - 5906x^{2} - 3030x + 45 $ x^8 - 2x^7 - 87x^6 + 12x^5 + 2157x^4 + 2939x^3 - 5906x^2 - 3030*x + 45
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 11 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$7.88904$ of defining polynomial
Character	$\chi$	$=$	968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.54031 q^{3} -6.21683 q^{5} +10.0967 q^{7} +3.69499 q^{9} -19.8412 q^{13} -34.4431 q^{15} -0.0283422 q^{17} -72.4162 q^{19} +55.9390 q^{21} +92.3700 q^{23} -86.3511 q^{25} -129.117 q^{27} -83.3142 q^{29} -172.056 q^{31} -62.7697 q^{35} +178.367 q^{37} -109.926 q^{39} +235.671 q^{41} +69.3945 q^{43} -22.9711 q^{45} -433.018 q^{47} -241.056 q^{49} -0.157025 q^{51} -599.884 q^{53} -401.208 q^{57} -365.211 q^{59} +685.581 q^{61} +37.3073 q^{63} +123.349 q^{65} +224.941 q^{67} +511.758 q^{69} -254.652 q^{71} +882.320 q^{73} -478.411 q^{75} -386.959 q^{79} -815.112 q^{81} -1214.30 q^{83} +0.176199 q^{85} -461.586 q^{87} -1236.77 q^{89} -200.332 q^{91} -953.241 q^{93} +450.199 q^{95} -891.507 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} - 13 q^{5} + 9 q^{7} + 36 q^{9} - 7 q^{13} - 66 q^{15} - 94 q^{17} + 92 q^{19} - 17 q^{21} - 46 q^{23} + 101 q^{25} + 124 q^{27} - 241 q^{29} - 265 q^{31} + 664 q^{35} - 469 q^{37} - 788 q^{39}+ \cdots - 4702 q^{97}+O(q^{100}) $ 8 * q - 8 * q^3 - 13 * q^5 + 9 * q^7 + 36 * q^9 - 7 * q^13 - 66 * q^15 - 94 * q^17 + 92 * q^19 - 17 * q^21 - 46 * q^23 + 101 * q^25 + 124 * q^27 - 241 * q^29 - 265 * q^31 + 664 * q^35 - 469 * q^37 - 788 * q^39 + 74 * q^41 - 35 * q^43 - 566 * q^45 + 197 * q^47 + 221 * q^49 + 608 * q^51 - 247 * q^53 + 979 * q^57 - 1208 * q^59 - 2061 * q^61 + 1617 * q^63 - 837 * q^65 - 829 * q^67 - 1860 * q^69 - 2005 * q^71 - 320 * q^73 - 2323 * q^75 + 1581 * q^79 - 3752 * q^81 - 692 * q^83 - 21 * q^85 + 131 * q^87 - 3205 * q^89 - 2724 * q^91 - 722 * q^93 + 432 * q^95 - 4702 * 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
$3$	5.54031	1.06623	0.533116	−	0.846042i	$-0.321021\pi$
$3$	5.54031	1.06623	0.533116	+	0.846042i	$0.321021\pi$
$4$	0	0
$5$	−6.21683	−0.556050	−0.278025	−	0.960574i	$-0.589680\pi$
$5$	−6.21683	−0.556050	−0.278025	+	0.960574i	$0.589680\pi$
$6$	0	0
$7$	10.0967	0.545173	0.272586	−	0.962131i	$-0.412121\pi$
$7$	10.0967	0.545173	0.272586	+	0.962131i	$0.412121\pi$
$8$	0	0
$9$	3.69499	0.136851
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−19.8412	−0.423305	−0.211652	−	0.977345i	$-0.567885\pi$
$13$	−19.8412	−0.423305	−0.211652	+	0.977345i	$0.567885\pi$
$14$	0	0
$15$	−34.4431	−0.592878
$16$	0	0
$17$	−0.0283422	−0.000404353 0	−0.000202176	−	1.00000i	$-0.500064\pi$
$17$	−0.0283422	−0.000404353 0	−0.000202176		1.00000i	$0.500064\pi$
$18$	0	0
$19$	−72.4162	−0.874391	−0.437195	−	0.899367i	$-0.644028\pi$
$19$	−72.4162	−0.874391	−0.437195	+	0.899367i	$0.644028\pi$
$20$	0	0
$21$	55.9390	0.581281
$22$	0	0
$23$	92.3700	0.837412	0.418706	−	0.908122i	$-0.362484\pi$
$23$	92.3700	0.837412	0.418706	+	0.908122i	$0.362484\pi$
$24$	0	0
$25$	−86.3511	−0.690808
$26$	0	0
$27$	−129.117	−0.920317
$28$	0	0
$29$	−83.3142	−0.533485	−0.266742	−	0.963768i	$-0.585947\pi$
$29$	−83.3142	−0.533485	−0.266742	+	0.963768i	$0.585947\pi$
$30$	0	0
$31$	−172.056	−0.996842	−0.498421	−	0.866935i	$-0.666087\pi$
$31$	−172.056	−0.996842	−0.498421	+	0.866935i	$0.666087\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−62.7697	−0.303143
$36$	0	0
$37$	178.367	0.792521	0.396260	−	0.918138i	$-0.370308\pi$
$37$	178.367	0.792521	0.396260	+	0.918138i	$0.370308\pi$
$38$	0	0
$39$	−109.926	−0.451341
$40$	0	0
$41$	235.671	0.897697	0.448849	−	0.893608i	$-0.351834\pi$
$41$	235.671	0.897697	0.448849	+	0.893608i	$0.351834\pi$
$42$	0	0
$43$	69.3945	0.246106	0.123053	−	0.992400i	$-0.460731\pi$
$43$	69.3945	0.246106	0.123053	+	0.992400i	$0.460731\pi$
$44$	0	0
$45$	−22.9711	−0.0760962
$46$	0	0
$47$	−433.018	−1.34388	−0.671938	−	0.740607i	$-0.734538\pi$
$47$	−433.018	−1.34388	−0.671938	+	0.740607i	$0.734538\pi$
$48$	0	0
$49$	−241.056	−0.702787
$50$	0	0
$51$	−0.157025	−0.000431134 0
$52$	0	0
$53$	−599.884	−1.55472	−0.777362	−	0.629054i	$-0.783443\pi$
$53$	−599.884	−1.55472	−0.777362	+	0.629054i	$0.783443\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−401.208	−0.932303
$58$	0	0
$59$	−365.211	−0.805871	−0.402935	−	0.915228i	$-0.632010\pi$
$59$	−365.211	−0.805871	−0.402935	+	0.915228i	$0.632010\pi$
$60$	0	0
$61$	685.581	1.43901	0.719506	−	0.694487i	$-0.244368\pi$
$61$	685.581	1.43901	0.719506	+	0.694487i	$0.244368\pi$
$62$	0	0
$63$	37.3073	0.0746076
$64$	0	0
$65$	123.349	0.235379
$66$	0	0
$67$	224.941	0.410164	0.205082	−	0.978745i	$-0.434254\pi$
$67$	224.941	0.410164	0.205082	+	0.978745i	$0.434254\pi$
$68$	0	0
$69$	511.758	0.892876
$70$	0	0
$71$	−254.652	−0.425657	−0.212828	−	0.977090i	$-0.568268\pi$
$71$	−254.652	−0.425657	−0.212828	+	0.977090i	$0.568268\pi$
$72$	0	0
$73$	882.320	1.41463	0.707313	−	0.706900i	$-0.249907\pi$
$73$	882.320	1.41463	0.707313	+	0.706900i	$0.249907\pi$
$74$	0	0
$75$	−478.411	−0.736562
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−386.959	−0.551092	−0.275546	−	0.961288i	$-0.588859\pi$
$79$	−386.959	−0.551092	−0.275546	+	0.961288i	$0.588859\pi$
$80$	0	0
$81$	−815.112	−1.11812
$82$	0	0
$83$	−1214.30	−1.60587	−0.802935	−	0.596067i	$-0.796729\pi$
$83$	−1214.30	−1.60587	−0.802935	+	0.596067i	$0.796729\pi$
$84$	0	0
$85$	0.176199	0.000224840 0
$86$	0	0
$87$	−461.586	−0.568819
$88$	0	0
$89$	−1236.77	−1.47300	−0.736500	−	0.676438i	$-0.763523\pi$
$89$	−1236.77	−1.47300	−0.736500	+	0.676438i	$0.763523\pi$
$90$	0	0
$91$	−200.332	−0.230774
$92$	0	0
$93$	−953.241	−1.06287
$94$	0	0
$95$	450.199	0.486205
$96$	0	0
$97$	−891.507	−0.933184	−0.466592	−	0.884473i	$-0.654518\pi$
$97$	−891.507	−0.933184	−0.466592	+	0.884473i	$0.654518\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	308.007	0.303444	0.151722	−	0.988423i	$-0.451518\pi$
$101$	308.007	0.303444	0.151722	+	0.988423i	$0.451518\pi$
$102$	0	0
$103$	−1112.92	−1.06465	−0.532325	−	0.846540i	$-0.678682\pi$
$103$	−1112.92	−1.06465	−0.532325	+	0.846540i	$0.678682\pi$
$104$	0	0
$105$	−347.763	−0.323221
$106$	0	0
$107$	1733.86	1.56653	0.783266	−	0.621687i	$-0.213552\pi$
$107$	1733.86	1.56653	0.783266	+	0.621687i	$0.213552\pi$
$108$	0	0
$109$	−1565.32	−1.37550	−0.687752	−	0.725946i	$-0.741402\pi$
$109$	−1565.32	−1.37550	−0.687752	+	0.725946i	$0.741402\pi$
$110$	0	0
$111$	988.205	0.845011
$112$	0	0
$113$	656.139	0.546233	0.273117	−	0.961981i	$-0.411945\pi$
$113$	656.139	0.546233	0.273117	+	0.961981i	$0.411945\pi$
$114$	0	0
$115$	−574.249	−0.465643
$116$	0	0
$117$	−73.3130	−0.0579298
$118$	0	0
$119$	−0.286164	−0.000220442 0
$120$	0	0
$121$	0	0
$122$	0	0
$123$	1305.69	0.957154
$124$	0	0
$125$	1313.93	0.940174
$126$	0	0
$127$	−2668.84	−1.86474	−0.932368	−	0.361510i	$-0.882261\pi$
$127$	−2668.84	−1.86474	−0.932368	+	0.361510i	$0.882261\pi$
$128$	0	0
$129$	384.467	0.262406
$130$	0	0
$131$	−694.820	−0.463410	−0.231705	−	0.972786i	$-0.574430\pi$
$131$	−694.820	−0.463410	−0.231705	+	0.972786i	$0.574430\pi$
$132$	0	0
$133$	−731.168	−0.476694
$134$	0	0
$135$	802.697	0.511742
$136$	0	0
$137$	2477.03	1.54472	0.772361	−	0.635184i	$-0.219076\pi$
$137$	2477.03	1.54472	0.772361	+	0.635184i	$0.219076\pi$
$138$	0	0
$139$	2196.35	1.34023	0.670115	−	0.742257i	$-0.266245\pi$
$139$	2196.35	1.34023	0.670115	+	0.742257i	$0.266245\pi$
$140$	0	0
$141$	−2399.05	−1.43288
$142$	0	0
$143$	0	0
$144$	0	0
$145$	517.950	0.296644
$146$	0	0
$147$	−1335.52	−0.749334
$148$	0	0
$149$	−3340.74	−1.83681	−0.918403	−	0.395645i	$-0.870521\pi$
$149$	−3340.74	−1.83681	−0.918403	+	0.395645i	$0.870521\pi$
$150$	0	0
$151$	728.502	0.392613	0.196307	−	0.980543i	$-0.437105\pi$
$151$	728.502	0.392613	0.196307	+	0.980543i	$0.437105\pi$
$152$	0	0
$153$	−0.104724	−5.53362e−5 0
$154$	0	0
$155$	1069.64	0.554294
$156$	0	0
$157$	2954.83	1.50204	0.751022	−	0.660277i	$-0.229561\pi$
$157$	2954.83	1.50204	0.751022	+	0.660277i	$0.229561\pi$
$158$	0	0
$159$	−3323.54	−1.65770
$160$	0	0
$161$	932.636	0.456534
$162$	0	0
$163$	−6.35358	−0.00305307	−0.00152654	−	0.999999i	$-0.500486\pi$
$163$	−6.35358	−0.00305307	−0.00152654	+	0.999999i	$0.500486\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	733.870	0.340051	0.170026	−	0.985440i	$-0.445615\pi$
$167$	733.870	0.340051	0.170026	+	0.985440i	$0.445615\pi$
$168$	0	0
$169$	−1803.33	−0.820813
$170$	0	0
$171$	−267.577	−0.119661
$172$	0	0
$173$	−329.366	−0.144747	−0.0723736	−	0.997378i	$-0.523057\pi$
$173$	−329.366	−0.144747	−0.0723736	+	0.997378i	$0.523057\pi$
$174$	0	0
$175$	−871.864	−0.376610
$176$	0	0
$177$	−2023.38	−0.859245
$178$	0	0
$179$	2671.42	1.11548	0.557741	−	0.830015i	$-0.311668\pi$
$179$	2671.42	1.11548	0.557741	+	0.830015i	$0.311668\pi$
$180$	0	0
$181$	−1545.63	−0.634727	−0.317364	−	0.948304i	$-0.602798\pi$
$181$	−1545.63	−0.634727	−0.317364	+	0.948304i	$0.602798\pi$
$182$	0	0
$183$	3798.33	1.53432
$184$	0	0
$185$	−1108.87	−0.440681
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1303.66	−0.501732
$190$	0	0
$191$	1185.87	0.449248	0.224624	−	0.974445i	$-0.427885\pi$
$191$	1185.87	0.449248	0.224624	+	0.974445i	$0.427885\pi$
$192$	0	0
$193$	1340.94	0.500117	0.250059	−	0.968231i	$-0.419550\pi$
$193$	1340.94	0.500117	0.250059	+	0.968231i	$0.419550\pi$
$194$	0	0
$195$	683.394	0.250968
$196$	0	0
$197$	−797.183	−0.288309	−0.144155	−	0.989555i	$-0.546046\pi$
$197$	−797.183	−0.288309	−0.144155	+	0.989555i	$0.546046\pi$
$198$	0	0
$199$	119.756	0.0426597	0.0213299	−	0.999772i	$-0.493210\pi$
$199$	119.756	0.0426597	0.0213299	+	0.999772i	$0.493210\pi$
$200$	0	0
$201$	1246.24	0.437330
$202$	0	0
$203$	−841.202	−0.290841
$204$	0	0
$205$	−1465.12	−0.499164
$206$	0	0
$207$	341.306	0.114601
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1059.12	−0.345559	−0.172780	−	0.984961i	$-0.555275\pi$
$211$	−1059.12	−0.345559	−0.172780	+	0.984961i	$0.555275\pi$
$212$	0	0
$213$	−1410.85	−0.453849
$214$	0	0
$215$	−431.414	−0.136847
$216$	0	0
$217$	−1737.20	−0.543451
$218$	0	0
$219$	4888.32	1.50832
$220$	0	0
$221$	0.562344	0.000171165 0
$222$	0	0
$223$	−6175.28	−1.85438	−0.927192	−	0.374587i	$-0.877785\pi$
$223$	−6175.28	−1.85438	−0.927192	+	0.374587i	$0.877785\pi$
$224$	0	0
$225$	−319.066	−0.0945380
$226$	0	0
$227$	4220.64	1.23407	0.617034	−	0.786936i	$-0.288334\pi$
$227$	4220.64	1.23407	0.617034	+	0.786936i	$0.288334\pi$
$228$	0	0
$229$	−2459.73	−0.709798	−0.354899	−	0.934905i	$-0.615485\pi$
$229$	−2459.73	−0.709798	−0.354899	+	0.934905i	$0.615485\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1244.39	0.349884	0.174942	−	0.984579i	$-0.444026\pi$
$233$	1244.39	0.349884	0.174942	+	0.984579i	$0.444026\pi$
$234$	0	0
$235$	2692.00	0.747263
$236$	0	0
$237$	−2143.87	−0.587592
$238$	0	0
$239$	3484.12	0.942967	0.471483	−	0.881875i	$-0.343719\pi$
$239$	3484.12	0.942967	0.471483	+	0.881875i	$0.343719\pi$
$240$	0	0
$241$	−6857.39	−1.83288	−0.916439	−	0.400174i	$-0.868950\pi$
$241$	−6857.39	−1.83288	−0.916439	+	0.400174i	$0.868950\pi$
$242$	0	0
$243$	−1029.81	−0.271862
$244$	0	0
$245$	1498.60	0.390784
$246$	0	0
$247$	1436.83	0.370134
$248$	0	0
$249$	−6727.61	−1.71223
$250$	0	0
$251$	−5654.75	−1.42201	−0.711005	−	0.703187i	$-0.751760\pi$
$251$	−5654.75	−1.42201	−0.711005	+	0.703187i	$0.751760\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.976195	0.000239732 0
$256$	0	0
$257$	−3870.41	−0.939416	−0.469708	−	0.882822i	$-0.655641\pi$
$257$	−3870.41	−0.939416	−0.469708	+	0.882822i	$0.655641\pi$
$258$	0	0
$259$	1800.92	0.432061
$260$	0	0
$261$	−307.845	−0.0730081
$262$	0	0
$263$	1016.12	0.238239	0.119119	−	0.992880i	$-0.461993\pi$
$263$	1016.12	0.238239	0.119119	+	0.992880i	$0.461993\pi$
$264$	0	0
$265$	3729.37	0.864504
$266$	0	0
$267$	−6852.06	−1.57056
$268$	0	0
$269$	452.110	0.102474	0.0512372	−	0.998687i	$-0.483684\pi$
$269$	452.110	0.102474	0.0512372	+	0.998687i	$0.483684\pi$
$270$	0	0
$271$	−444.082	−0.0995428	−0.0497714	−	0.998761i	$-0.515849\pi$
$271$	−444.082	−0.0995428	−0.0497714	+	0.998761i	$0.515849\pi$
$272$	0	0
$273$	−1109.90	−0.246059
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7800.35	−1.69198	−0.845988	−	0.533202i	$-0.820989\pi$
$277$	−7800.35	−1.69198	−0.845988	+	0.533202i	$0.820989\pi$
$278$	0	0
$279$	−635.743	−0.136419
$280$	0	0
$281$	4754.72	1.00941	0.504703	−	0.863293i	$-0.331602\pi$
$281$	4754.72	1.00941	0.504703	+	0.863293i	$0.331602\pi$
$282$	0	0
$283$	5405.83	1.13549	0.567744	−	0.823205i	$-0.307816\pi$
$283$	5405.83	1.13549	0.567744	+	0.823205i	$0.307816\pi$
$284$	0	0
$285$	2494.24	0.518407
$286$	0	0
$287$	2379.51	0.489400
$288$	0	0
$289$	−4913.00	−1.00000
$290$	0	0
$291$	−4939.22	−0.994991
$292$	0	0
$293$	−5578.19	−1.11222	−0.556112	−	0.831107i	$-0.687707\pi$
$293$	−5578.19	−1.11222	−0.556112	+	0.831107i	$0.687707\pi$
$294$	0	0
$295$	2270.45	0.448104
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1832.73	−0.354481
$300$	0	0
$301$	700.658	0.134170
$302$	0	0
$303$	1706.45	0.323541
$304$	0	0
$305$	−4262.14	−0.800162
$306$	0	0
$307$	7248.81	1.34759	0.673797	−	0.738916i	$-0.264662\pi$
$307$	7248.81	1.34759	0.673797	+	0.738916i	$0.264662\pi$
$308$	0	0
$309$	−6165.90	−1.13517
$310$	0	0
$311$	953.186	0.173795	0.0868975	−	0.996217i	$-0.472305\pi$
$311$	953.186	0.173795	0.0868975	+	0.996217i	$0.472305\pi$
$312$	0	0
$313$	3365.71	0.607800	0.303900	−	0.952704i	$-0.401711\pi$
$313$	3365.71	0.607800	0.303900	+	0.952704i	$0.401711\pi$
$314$	0	0
$315$	−231.933	−0.0414856
$316$	0	0
$317$	8802.93	1.55969	0.779845	−	0.625972i	$-0.215298\pi$
$317$	8802.93	1.55969	0.779845	+	0.625972i	$0.215298\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	9606.13	1.67029
$322$	0	0
$323$	2.05244	0.000353562 0
$324$	0	0
$325$	1713.31	0.292423
$326$	0	0
$327$	−8672.32	−1.46661
$328$	0	0
$329$	−4372.07	−0.732645
$330$	0	0
$331$	1884.62	0.312955	0.156478	−	0.987682i	$-0.449986\pi$
$331$	1884.62	0.312955	0.156478	+	0.987682i	$0.449986\pi$
$332$	0	0
$333$	659.062	0.108458
$334$	0	0
$335$	−1398.42	−0.228071
$336$	0	0
$337$	−1979.74	−0.320009	−0.160005	−	0.987116i	$-0.551151\pi$
$337$	−1979.74	−0.320009	−0.160005	+	0.987116i	$0.551151\pi$
$338$	0	0
$339$	3635.21	0.582412
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−5897.06	−0.928313
$344$	0	0
$345$	−3181.51	−0.496484
$346$	0	0
$347$	8538.82	1.32100	0.660502	−	0.750825i	$-0.270344\pi$
$347$	8538.82	1.32100	0.660502	+	0.750825i	$0.270344\pi$
$348$	0	0
$349$	944.572	0.144876	0.0724381	−	0.997373i	$-0.476922\pi$
$349$	944.572	0.144876	0.0724381	+	0.997373i	$0.476922\pi$
$350$	0	0
$351$	2561.84	0.389575
$352$	0	0
$353$	3709.02	0.559239	0.279619	−	0.960111i	$-0.409792\pi$
$353$	3709.02	0.559239	0.279619	+	0.960111i	$0.409792\pi$
$354$	0	0
$355$	1583.13	0.236687
$356$	0	0
$357$	−1.58544	−0.000235043 0
$358$	0	0
$359$	9466.81	1.39175	0.695876	−	0.718162i	$-0.255016\pi$
$359$	9466.81	1.39175	0.695876	+	0.718162i	$0.255016\pi$
$360$	0	0
$361$	−1614.89	−0.235441
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5485.23	−0.786603
$366$	0	0
$367$	2353.26	0.334712	0.167356	−	0.985897i	$-0.446477\pi$
$367$	2353.26	0.334712	0.167356	+	0.985897i	$0.446477\pi$
$368$	0	0
$369$	870.800	0.122851
$370$	0	0
$371$	−6056.87	−0.847593
$372$	0	0
$373$	−1447.21	−0.200894	−0.100447	−	0.994942i	$-0.532027\pi$
$373$	−1447.21	−0.200894	−0.100447	+	0.994942i	$0.532027\pi$
$374$	0	0
$375$	7279.59	1.00244
$376$	0	0
$377$	1653.06	0.225827
$378$	0	0
$379$	8402.10	1.13875	0.569376	−	0.822078i	$-0.307185\pi$
$379$	8402.10	1.13875	0.569376	+	0.822078i	$0.307185\pi$
$380$	0	0
$381$	−14786.2	−1.98824
$382$	0	0
$383$	7136.19	0.952069	0.476035	−	0.879427i	$-0.342074\pi$
$383$	7136.19	0.952069	0.476035	+	0.879427i	$0.342074\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	256.412	0.0336799
$388$	0	0
$389$	11712.7	1.52663	0.763316	−	0.646025i	$-0.223570\pi$
$389$	11712.7	1.52663	0.763316	+	0.646025i	$0.223570\pi$
$390$	0	0
$391$	−2.61797	−0.000338610 0
$392$	0	0
$393$	−3849.52	−0.494103
$394$	0	0
$395$	2405.66	0.306435
$396$	0	0
$397$	−13000.5	−1.64352	−0.821759	−	0.569835i	$-0.807007\pi$
$397$	−13000.5	−1.64352	−0.821759	+	0.569835i	$0.807007\pi$
$398$	0	0
$399$	−4050.89	−0.508267
$400$	0	0
$401$	11061.4	1.37750	0.688750	−	0.724999i	$-0.258160\pi$
$401$	11061.4	1.37750	0.688750	+	0.724999i	$0.258160\pi$
$402$	0	0
$403$	3413.79	0.421968
$404$	0	0
$405$	5067.41	0.621732
$406$	0	0
$407$	0	0
$408$	0	0
$409$	9340.89	1.12928	0.564642	−	0.825336i	$-0.309014\pi$
$409$	9340.89	1.12928	0.564642	+	0.825336i	$0.309014\pi$
$410$	0	0
$411$	13723.5	1.64703
$412$	0	0
$413$	−3687.44	−0.439339
$414$	0	0
$415$	7549.12	0.892944
$416$	0	0
$417$	12168.5	1.42900
$418$	0	0
$419$	2181.92	0.254400	0.127200	−	0.991877i	$-0.459401\pi$
$419$	2181.92	0.254400	0.127200	+	0.991877i	$0.459401\pi$
$420$	0	0
$421$	3508.41	0.406151	0.203076	−	0.979163i	$-0.434906\pi$
$421$	3508.41	0.406151	0.203076	+	0.979163i	$0.434906\pi$
$422$	0	0
$423$	−1600.00	−0.183911
$424$	0	0
$425$	2.44738	0.000279330 0
$426$	0	0
$427$	6922.14	0.784510
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12861.8	1.43742	0.718712	−	0.695308i	$-0.244732\pi$
$431$	12861.8	1.43742	0.718712	+	0.695308i	$0.244732\pi$
$432$	0	0
$433$	8152.45	0.904808	0.452404	−	0.891813i	$-0.350567\pi$
$433$	8152.45	0.904808	0.452404	+	0.891813i	$0.350567\pi$
$434$	0	0
$435$	2869.60	0.316292
$436$	0	0
$437$	−6689.09	−0.732225
$438$	0	0
$439$	−1617.22	−0.175821	−0.0879107	−	0.996128i	$-0.528019\pi$
$439$	−1617.22	−0.175821	−0.0879107	+	0.996128i	$0.528019\pi$
$440$	0	0
$441$	−890.698	−0.0961773
$442$	0	0
$443$	4746.12	0.509018	0.254509	−	0.967070i	$-0.418086\pi$
$443$	4746.12	0.509018	0.254509	+	0.967070i	$0.418086\pi$
$444$	0	0
$445$	7688.76	0.819061
$446$	0	0
$447$	−18508.7	−1.95846
$448$	0	0
$449$	6696.84	0.703883	0.351941	−	0.936022i	$-0.385522\pi$
$449$	6696.84	0.703883	0.351941	+	0.936022i	$0.385522\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	4036.12	0.418617
$454$	0	0
$455$	1245.43	0.128322
$456$	0	0
$457$	10921.4	1.11790	0.558952	−	0.829200i	$-0.311204\pi$
$457$	10921.4	1.11790	0.558952	+	0.829200i	$0.311204\pi$
$458$	0	0
$459$	3.65946	0.000372133 0
$460$	0	0
$461$	−10627.0	−1.07365	−0.536823	−	0.843695i	$-0.680376\pi$
$461$	−10627.0	−1.07365	−0.536823	+	0.843695i	$0.680376\pi$
$462$	0	0
$463$	3046.02	0.305746	0.152873	−	0.988246i	$-0.451147\pi$
$463$	3046.02	0.305746	0.152873	+	0.988246i	$0.451147\pi$
$464$	0	0
$465$	5926.13	0.591006
$466$	0	0
$467$	7881.07	0.780926	0.390463	−	0.920619i	$-0.372315\pi$
$467$	7881.07	0.780926	0.390463	+	0.920619i	$0.372315\pi$
$468$	0	0
$469$	2271.18	0.223610
$470$	0	0
$471$	16370.6	1.60153
$472$	0	0
$473$	0	0
$474$	0	0
$475$	6253.22	0.604036
$476$	0	0
$477$	−2216.56	−0.212766
$478$	0	0
$479$	−12763.1	−1.21746	−0.608730	−	0.793378i	$-0.708321\pi$
$479$	−12763.1	−1.21746	−0.608730	+	0.793378i	$0.708321\pi$
$480$	0	0
$481$	−3539.01	−0.335478
$482$	0	0
$483$	5167.09	0.486772
$484$	0	0
$485$	5542.35	0.518897
$486$	0	0
$487$	−4287.14	−0.398909	−0.199455	−	0.979907i	$-0.563917\pi$
$487$	−4287.14	−0.398909	−0.199455	+	0.979907i	$0.563917\pi$
$488$	0	0
$489$	−35.2008	−0.00325528
$490$	0	0
$491$	1729.93	0.159003	0.0795016	−	0.996835i	$-0.474667\pi$
$491$	1729.93	0.159003	0.0795016	+	0.996835i	$0.474667\pi$
$492$	0	0
$493$	2.36131	0.000215716 0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2571.16	−0.232057
$498$	0	0
$499$	−13257.4	−1.18934	−0.594670	−	0.803970i	$-0.702717\pi$
$499$	−13257.4	−1.18934	−0.594670	+	0.803970i	$0.702717\pi$
$500$	0	0
$501$	4065.87	0.362574
$502$	0	0
$503$	−131.981	−0.0116993	−0.00584964	−	0.999983i	$-0.501862\pi$
$503$	−131.981	−0.0116993	−0.00584964	+	0.999983i	$0.501862\pi$
$504$	0	0
$505$	−1914.82	−0.168730
$506$	0	0
$507$	−9990.98	−0.875177
$508$	0	0
$509$	8169.00	0.711365	0.355682	−	0.934607i	$-0.384249\pi$
$509$	8169.00	0.711365	0.355682	+	0.934607i	$0.384249\pi$
$510$	0	0
$511$	8908.56	0.771216
$512$	0	0
$513$	9350.16	0.804717
$514$	0	0
$515$	6918.82	0.591999
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−1824.79	−0.154334
$520$	0	0
$521$	2530.99	0.212831	0.106415	−	0.994322i	$-0.466063\pi$
$521$	2530.99	0.212831	0.106415	+	0.994322i	$0.466063\pi$
$522$	0	0
$523$	1559.52	0.130388	0.0651940	−	0.997873i	$-0.479233\pi$
$523$	1559.52	0.130388	0.0651940	+	0.997873i	$0.479233\pi$
$524$	0	0
$525$	−4830.39	−0.401554
$526$	0	0
$527$	4.87644	0.000403076 0
$528$	0	0
$529$	−3634.78	−0.298741
$530$	0	0
$531$	−1349.45	−0.110284
$532$	0	0
$533$	−4675.99	−0.380000
$534$	0	0
$535$	−10779.1	−0.871070
$536$	0	0
$537$	14800.5	1.18936
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−14840.8	−1.17940	−0.589700	−	0.807622i	$-0.700754\pi$
$541$	−14840.8	−1.17940	−0.589700	+	0.807622i	$0.700754\pi$
$542$	0	0
$543$	−8563.25	−0.676767
$544$	0	0
$545$	9731.29	0.764849
$546$	0	0
$547$	−4484.16	−0.350510	−0.175255	−	0.984523i	$-0.556075\pi$
$547$	−4484.16	−0.350510	−0.175255	+	0.984523i	$0.556075\pi$
$548$	0	0
$549$	2533.21	0.196931
$550$	0	0
$551$	6033.30	0.466474
$552$	0	0
$553$	−3907.02	−0.300440
$554$	0	0
$555$	−6143.50	−0.469869
$556$	0	0
$557$	−5315.84	−0.404379	−0.202190	−	0.979346i	$-0.564806\pi$
$557$	−5315.84	−0.404379	−0.202190	+	0.979346i	$0.564806\pi$
$558$	0	0
$559$	−1376.87	−0.104178
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−4602.65	−0.344544	−0.172272	−	0.985049i	$-0.555111\pi$
$563$	−4602.65	−0.344544	−0.172272	+	0.985049i	$0.555111\pi$
$564$	0	0
$565$	−4079.10	−0.303733
$566$	0	0
$567$	−8229.97	−0.609570
$568$	0	0
$569$	−7732.30	−0.569692	−0.284846	−	0.958573i	$-0.591942\pi$
$569$	−7732.30	−0.569692	−0.284846	+	0.958573i	$0.591942\pi$
$570$	0	0
$571$	−18830.2	−1.38007	−0.690033	−	0.723778i	$-0.742404\pi$
$571$	−18830.2	−1.38007	−0.690033	+	0.723778i	$0.742404\pi$
$572$	0	0
$573$	6570.07	0.479003
$574$	0	0
$575$	−7976.25	−0.578491
$576$	0	0
$577$	13904.5	1.00321	0.501605	−	0.865097i	$-0.332743\pi$
$577$	13904.5	1.00321	0.501605	+	0.865097i	$0.332743\pi$
$578$	0	0
$579$	7429.19	0.533241
$580$	0	0
$581$	−12260.5	−0.875476
$582$	0	0
$583$	0	0
$584$	0	0
$585$	455.774	0.0322119
$586$	0	0
$587$	25593.2	1.79956	0.899782	−	0.436340i	$-0.143726\pi$
$587$	25593.2	1.79956	0.899782	+	0.436340i	$0.143726\pi$
$588$	0	0
$589$	12459.6	0.871629
$590$	0	0
$591$	−4416.64	−0.307405
$592$	0	0
$593$	14766.2	1.02256	0.511279	−	0.859415i	$-0.329172\pi$
$593$	14766.2	1.02256	0.511279	+	0.859415i	$0.329172\pi$
$594$	0	0
$595$	1.77903	0.000122577 0
$596$	0	0
$597$	663.485	0.0454852
$598$	0	0
$599$	−1873.15	−0.127771	−0.0638856	−	0.997957i	$-0.520349\pi$
$599$	−1873.15	−0.127771	−0.0638856	+	0.997957i	$0.520349\pi$
$600$	0	0
$601$	1711.60	0.116169	0.0580845	−	0.998312i	$-0.481501\pi$
$601$	1711.60	0.116169	0.0580845	+	0.998312i	$0.481501\pi$
$602$	0	0
$603$	831.155	0.0561314
$604$	0	0
$605$	0	0
$606$	0	0
$607$	13265.7	0.887046	0.443523	−	0.896263i	$-0.353728\pi$
$607$	13265.7	0.887046	0.443523	+	0.896263i	$0.353728\pi$
$608$	0	0
$609$	−4660.52	−0.310105
$610$	0	0
$611$	8591.61	0.568870
$612$	0	0
$613$	25182.4	1.65923	0.829615	−	0.558336i	$-0.188560\pi$
$613$	25182.4	1.65923	0.829615	+	0.558336i	$0.188560\pi$
$614$	0	0
$615$	−8117.23	−0.532225
$616$	0	0
$617$	−4896.84	−0.319513	−0.159756	−	0.987156i	$-0.551071\pi$
$617$	−4896.84	−0.319513	−0.159756	+	0.987156i	$0.551071\pi$
$618$	0	0
$619$	−2943.21	−0.191111	−0.0955554	−	0.995424i	$-0.530463\pi$
$619$	−2943.21	−0.191111	−0.0955554	+	0.995424i	$0.530463\pi$
$620$	0	0
$621$	−11926.5	−0.770685
$622$	0	0
$623$	−12487.3	−0.803039
$624$	0	0
$625$	2625.39	0.168025
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.05530	−0.000320458 0
$630$	0	0
$631$	−30497.6	−1.92408	−0.962038	−	0.272916i	$-0.912012\pi$
$631$	−30497.6	−1.92408	−0.962038	+	0.272916i	$0.912012\pi$
$632$	0	0
$633$	−5867.86	−0.368446
$634$	0	0
$635$	16591.7	1.03689
$636$	0	0
$637$	4782.84	0.297493
$638$	0	0
$639$	−940.936	−0.0582517
$640$	0	0
$641$	−25166.2	−1.55071	−0.775354	−	0.631527i	$-0.782429\pi$
$641$	−25166.2	−1.55071	−0.775354	+	0.631527i	$0.782429\pi$
$642$	0	0
$643$	−6165.04	−0.378111	−0.189056	−	0.981966i	$-0.560543\pi$
$643$	−6165.04	−0.378111	−0.189056	+	0.981966i	$0.560543\pi$
$644$	0	0
$645$	−2390.16	−0.145911
$646$	0	0
$647$	−13006.1	−0.790295	−0.395148	−	0.918618i	$-0.629307\pi$
$647$	−13006.1	−0.790295	−0.395148	+	0.918618i	$0.629307\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−9624.63	−0.579445
$652$	0	0
$653$	18408.0	1.10316	0.551579	−	0.834123i	$-0.314026\pi$
$653$	18408.0	1.10316	0.551579	+	0.834123i	$0.314026\pi$
$654$	0	0
$655$	4319.58	0.257679
$656$	0	0
$657$	3260.16	0.193593
$658$	0	0
$659$	13817.0	0.816745	0.408372	−	0.912815i	$-0.366096\pi$
$659$	13817.0	0.816745	0.408372	+	0.912815i	$0.366096\pi$
$660$	0	0
$661$	−4130.31	−0.243042	−0.121521	−	0.992589i	$-0.538777\pi$
$661$	−4130.31	−0.243042	−0.121521	+	0.992589i	$0.538777\pi$
$662$	0	0
$663$	3.11556	0.000182501 0
$664$	0	0
$665$	4545.54	0.265066
$666$	0	0
$667$	−7695.74	−0.446747
$668$	0	0
$669$	−34213.0	−1.97720
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−9193.43	−0.526569	−0.263284	−	0.964718i	$-0.584806\pi$
$673$	−9193.43	−0.526569	−0.263284	+	0.964718i	$0.584806\pi$
$674$	0	0
$675$	11149.4	0.635763
$676$	0	0
$677$	−4453.79	−0.252841	−0.126420	−	0.991977i	$-0.540349\pi$
$677$	−4453.79	−0.252841	−0.126420	+	0.991977i	$0.540349\pi$
$678$	0	0
$679$	−9001.32	−0.508747
$680$	0	0
$681$	23383.6	1.31580
$682$	0	0
$683$	−12898.6	−0.722620	−0.361310	−	0.932446i	$-0.617670\pi$
$683$	−12898.6	−0.722620	−0.361310	+	0.932446i	$0.617670\pi$
$684$	0	0
$685$	−15399.3	−0.858942
$686$	0	0
$687$	−13627.7	−0.756810
$688$	0	0
$689$	11902.4	0.658122
$690$	0	0
$691$	−29150.8	−1.60485	−0.802424	−	0.596754i	$-0.796457\pi$
$691$	−29150.8	−1.60485	−0.802424	+	0.596754i	$0.796457\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13654.3	−0.745235
$696$	0	0
$697$	−6.67943	−0.000362986 0
$698$	0	0
$699$	6894.32	0.373058
$700$	0	0
$701$	154.490	0.00832382	0.00416191	−	0.999991i	$-0.498675\pi$
$701$	154.490	0.00832382	0.00416191	+	0.999991i	$0.498675\pi$
$702$	0	0
$703$	−12916.6	−0.692973
$704$	0	0
$705$	14914.5	0.796755
$706$	0	0
$707$	3109.86	0.165429
$708$	0	0
$709$	−18780.0	−0.994780	−0.497390	−	0.867527i	$-0.665708\pi$
$709$	−18780.0	−0.994780	−0.497390	+	0.867527i	$0.665708\pi$
$710$	0	0
$711$	−1429.81	−0.0754176
$712$	0	0
$713$	−15892.8	−0.834768
$714$	0	0
$715$	0	0
$716$	0	0
$717$	19303.1	1.00542
$718$	0	0
$719$	−10916.4	−0.566222	−0.283111	−	0.959087i	$-0.591366\pi$
$719$	−10916.4	−0.566222	−0.283111	+	0.959087i	$0.591366\pi$
$720$	0	0
$721$	−11236.8	−0.580419
$722$	0	0
$723$	−37992.1	−1.95427
$724$	0	0
$725$	7194.27	0.368536
$726$	0	0
$727$	19277.1	0.983424	0.491712	−	0.870758i	$-0.336371\pi$
$727$	19277.1	0.983424	0.491712	+	0.870758i	$0.336371\pi$
$728$	0	0
$729$	16302.5	0.828255
$730$	0	0
$731$	−1.96679	−9.95137e−5 0
$732$	0	0
$733$	34562.4	1.74160	0.870798	−	0.491641i	$-0.163603\pi$
$733$	34562.4	1.74160	0.870798	+	0.491641i	$0.163603\pi$
$734$	0	0
$735$	8302.71	0.416667
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−19140.3	−0.952757	−0.476378	−	0.879240i	$-0.658051\pi$
$739$	−19140.3	−0.952757	−0.476378	+	0.879240i	$0.658051\pi$
$740$	0	0
$741$	7960.46	0.394649
$742$	0	0
$743$	−18198.4	−0.898564	−0.449282	−	0.893390i	$-0.648320\pi$
$743$	−18198.4	−0.898564	−0.449282	+	0.893390i	$0.648320\pi$
$744$	0	0
$745$	20768.8	1.02136
$746$	0	0
$747$	−4486.84	−0.219765
$748$	0	0
$749$	17506.4	0.854031
$750$	0	0
$751$	25102.0	1.21969	0.609845	−	0.792521i	$-0.291232\pi$
$751$	25102.0	1.21969	0.609845	+	0.792521i	$0.291232\pi$
$752$	0	0
$753$	−31329.0	−1.51619
$754$	0	0
$755$	−4528.97	−0.218313
$756$	0	0
$757$	30787.1	1.47817	0.739086	−	0.673611i	$-0.235258\pi$
$757$	30787.1	1.47817	0.739086	+	0.673611i	$0.235258\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22966.0	−1.09398	−0.546990	−	0.837139i	$-0.684226\pi$
$761$	−22966.0	−1.09398	−0.546990	+	0.837139i	$0.684226\pi$
$762$	0	0
$763$	−15804.6	−0.749888
$764$	0	0
$765$	0.651052	3.07697e−5 0
$766$	0	0
$767$	7246.22	0.341129
$768$	0	0
$769$	−28729.3	−1.34721	−0.673605	−	0.739091i	$-0.735255\pi$
$769$	−28729.3	−1.34721	−0.673605	+	0.739091i	$0.735255\pi$
$770$	0	0
$771$	−21443.3	−1.00164
$772$	0	0
$773$	−19244.6	−0.895446	−0.447723	−	0.894172i	$-0.647765\pi$
$773$	−19244.6	−0.895446	−0.447723	+	0.894172i	$0.647765\pi$
$774$	0	0
$775$	14857.2	0.688627
$776$	0	0
$777$	9977.65	0.460677
$778$	0	0
$779$	−17066.4	−0.784938
$780$	0	0
$781$	0	0
$782$	0	0
$783$	10757.3	0.490975
$784$	0	0
$785$	−18369.6	−0.835211
$786$	0	0
$787$	29271.2	1.32580	0.662901	−	0.748707i	$-0.269325\pi$
$787$	29271.2	1.32580	0.662901	+	0.748707i	$0.269325\pi$
$788$	0	0
$789$	5629.63	0.254018
$790$	0	0
$791$	6624.87	0.297792
$792$	0	0
$793$	−13602.8	−0.609141
$794$	0	0
$795$	20661.9	0.921762
$796$	0	0
$797$	−12532.5	−0.556994	−0.278497	−	0.960437i	$-0.589836\pi$
$797$	−12532.5	−0.556994	−0.278497	+	0.960437i	$0.589836\pi$
$798$	0	0
$799$	12.2727	0.000543400 0
$800$	0	0
$801$	−4569.83	−0.201582
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−5798.04	−0.253856
$806$	0	0
$807$	2504.83	0.109262
$808$	0	0
$809$	−35942.4	−1.56201	−0.781006	−	0.624523i	$-0.785293\pi$
$809$	−35942.4	−1.56201	−0.781006	+	0.624523i	$0.785293\pi$
$810$	0	0
$811$	21630.0	0.936537	0.468269	−	0.883586i	$-0.344878\pi$
$811$	21630.0	0.936537	0.468269	+	0.883586i	$0.344878\pi$
$812$	0	0
$813$	−2460.35	−0.106136
$814$	0	0
$815$	39.4991	0.00169766
$816$	0	0
$817$	−5025.29	−0.215193
$818$	0	0
$819$	−740.223	−0.0315818
$820$	0	0
$821$	31467.1	1.33765	0.668824	−	0.743421i	$-0.266798\pi$
$821$	31467.1	1.33765	0.668824	+	0.743421i	$0.266798\pi$
$822$	0	0
$823$	−35233.1	−1.49228	−0.746142	−	0.665787i	$-0.768096\pi$
$823$	−35233.1	−1.49228	−0.746142	+	0.665787i	$0.768096\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9490.61	0.399058	0.199529	−	0.979892i	$-0.436059\pi$
$827$	9490.61	0.399058	0.199529	+	0.979892i	$0.436059\pi$
$828$	0	0
$829$	−38433.6	−1.61020	−0.805099	−	0.593140i	$-0.797888\pi$
$829$	−38433.6	−1.61020	−0.805099	+	0.593140i	$0.797888\pi$
$830$	0	0
$831$	−43216.3	−1.80404
$832$	0	0
$833$	6.83206	0.000284174 0
$834$	0	0
$835$	−4562.34	−0.189086
$836$	0	0
$837$	22215.3	0.917411
$838$	0	0
$839$	9689.22	0.398700	0.199350	−	0.979928i	$-0.436117\pi$
$839$	9689.22	0.398700	0.199350	+	0.979928i	$0.436117\pi$
$840$	0	0
$841$	−17447.7	−0.715394
$842$	0	0
$843$	26342.6	1.07626
$844$	0	0
$845$	11211.0	0.456413
$846$	0	0
$847$	0	0
$848$	0	0
$849$	29950.0	1.21069
$850$	0	0
$851$	16475.7	0.663667
$852$	0	0
$853$	−24698.1	−0.991380	−0.495690	−	0.868499i	$-0.665085\pi$
$853$	−24698.1	−0.991380	−0.495690	+	0.868499i	$0.665085\pi$
$854$	0	0
$855$	1663.48	0.0665378
$856$	0	0
$857$	−15732.3	−0.627079	−0.313539	−	0.949575i	$-0.601515\pi$
$857$	−15732.3	−0.627079	−0.313539	+	0.949575i	$0.601515\pi$
$858$	0	0
$859$	−34609.5	−1.37469	−0.687347	−	0.726329i	$-0.741225\pi$
$859$	−34609.5	−1.37469	−0.687347	+	0.726329i	$0.741225\pi$
$860$	0	0
$861$	13183.2	0.521814
$862$	0	0
$863$	−2726.72	−0.107553	−0.0537767	−	0.998553i	$-0.517126\pi$
$863$	−2726.72	−0.107553	−0.0537767	+	0.998553i	$0.517126\pi$
$864$	0	0
$865$	2047.61	0.0804867
$866$	0	0
$867$	−27219.5	−1.06623
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−4463.11	−0.173624
$872$	0	0
$873$	−3294.11	−0.127707
$874$	0	0
$875$	13266.4	0.512557
$876$	0	0
$877$	−2460.98	−0.0947564	−0.0473782	−	0.998877i	$-0.515087\pi$
$877$	−2460.98	−0.0947564	−0.0473782	+	0.998877i	$0.515087\pi$
$878$	0	0
$879$	−30904.9	−1.18589
$880$	0	0
$881$	25163.9	0.962307	0.481153	−	0.876636i	$-0.340218\pi$
$881$	25163.9	0.962307	0.481153	+	0.876636i	$0.340218\pi$
$882$	0	0
$883$	8621.41	0.328577	0.164289	−	0.986412i	$-0.447467\pi$
$883$	8621.41	0.328577	0.164289	+	0.986412i	$0.447467\pi$
$884$	0	0
$885$	12579.0	0.477783
$886$	0	0
$887$	−42583.1	−1.61195	−0.805975	−	0.591950i	$-0.798358\pi$
$887$	−42583.1	−1.61195	−0.805975	+	0.591950i	$0.798358\pi$
$888$	0	0
$889$	−26946.6	−1.01660
$890$	0	0
$891$	0	0
$892$	0	0
$893$	31357.5	1.17507
$894$	0	0
$895$	−16607.8	−0.620264
$896$	0	0
$897$	−10153.9	−0.377959
$898$	0	0
$899$	14334.7	0.531800
$900$	0	0
$901$	17.0020	0.000628657 0
$902$	0	0
$903$	3881.86	0.143057
$904$	0	0
$905$	9608.90	0.352940
$906$	0	0
$907$	9507.78	0.348071	0.174036	−	0.984739i	$-0.444319\pi$
$907$	9507.78	0.348071	0.174036	+	0.984739i	$0.444319\pi$
$908$	0	0
$909$	1138.08	0.0415267
$910$	0	0
$911$	11475.1	0.417330	0.208665	−	0.977987i	$-0.433088\pi$
$911$	11475.1	0.417330	0.208665	+	0.977987i	$0.433088\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−23613.6	−0.853159
$916$	0	0
$917$	−7015.42	−0.252639
$918$	0	0
$919$	−42341.9	−1.51984	−0.759918	−	0.650019i	$-0.774761\pi$
$919$	−42341.9	−1.51984	−0.759918	+	0.650019i	$0.774761\pi$
$920$	0	0
$921$	40160.6	1.43685
$922$	0	0
$923$	5052.61	0.180183
$924$	0	0
$925$	−15402.1	−0.547480
$926$	0	0
$927$	−4112.21	−0.145699
$928$	0	0
$929$	−42583.6	−1.50390	−0.751950	−	0.659221i	$-0.770886\pi$
$929$	−42583.6	−1.50390	−0.751950	+	0.659221i	$0.770886\pi$
$930$	0	0
$931$	17456.4	0.614510
$932$	0	0
$933$	5280.94	0.185306
$934$	0	0
$935$	0	0
$936$	0	0
$937$	8318.19	0.290014	0.145007	−	0.989431i	$-0.453680\pi$
$937$	8318.19	0.290014	0.145007	+	0.989431i	$0.453680\pi$
$938$	0	0
$939$	18647.1	0.648056
$940$	0	0
$941$	−32022.6	−1.10936	−0.554679	−	0.832064i	$-0.687159\pi$
$941$	−32022.6	−1.10936	−0.554679	+	0.832064i	$0.687159\pi$
$942$	0	0
$943$	21768.9	0.751743
$944$	0	0
$945$	8104.63	0.278988
$946$	0	0
$947$	−28430.6	−0.975575	−0.487788	−	0.872962i	$-0.662196\pi$
$947$	−28430.6	−0.975575	−0.487788	+	0.872962i	$0.662196\pi$
$948$	0	0
$949$	−17506.3	−0.598818
$950$	0	0
$951$	48770.9	1.66299
$952$	0	0
$953$	6805.74	0.231332	0.115666	−	0.993288i	$-0.463100\pi$
$953$	6805.74	0.231332	0.115666	+	0.993288i	$0.463100\pi$
$954$	0	0
$955$	−7372.34	−0.249804
$956$	0	0
$957$	0	0
$958$	0	0
$959$	25009.9	0.842140
$960$	0	0
$961$	−187.857	−0.00630584
$962$	0	0
$963$	6406.60	0.214382
$964$	0	0
$965$	−8336.36	−0.278090
$966$	0	0
$967$	−41912.2	−1.39380	−0.696901	−	0.717167i	$-0.745438\pi$
$967$	−41912.2	−1.39380	−0.696901	+	0.717167i	$0.745438\pi$
$968$	0	0
$969$	11.3711	0.000376980 0
$970$	0	0
$971$	50663.6	1.67443	0.837216	−	0.546873i	$-0.184182\pi$
$971$	50663.6	1.67443	0.837216	+	0.546873i	$0.184182\pi$
$972$	0	0
$973$	22176.0	0.730657
$974$	0	0
$975$	9492.26	0.311790
$976$	0	0
$977$	8475.61	0.277542	0.138771	−	0.990324i	$-0.455685\pi$
$977$	8475.61	0.277542	0.138771	+	0.990324i	$0.455685\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−5783.82	−0.188240
$982$	0	0
$983$	−25920.5	−0.841034	−0.420517	−	0.907285i	$-0.638151\pi$
$983$	−25920.5	−0.841034	−0.420517	+	0.907285i	$0.638151\pi$
$984$	0	0
$985$	4955.95	0.160314
$986$	0	0
$987$	−24222.6	−0.781170
$988$	0	0
$989$	6409.97	0.206092
$990$	0	0
$991$	−52359.2	−1.67835	−0.839174	−	0.543863i	$-0.816961\pi$
$991$	−52359.2	−1.67835	−0.839174	+	0.543863i	$0.816961\pi$
$992$	0	0
$993$	10441.4	0.333683
$994$	0	0
$995$	−744.503	−0.0237209
$996$	0	0
$997$	15452.3	0.490852	0.245426	−	0.969415i	$-0.421072\pi$
$997$	15452.3	0.490852	0.245426	+	0.969415i	$0.421072\pi$
$998$	0	0
$999$	−23030.1	−0.729370

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.4.a.o.1.7		8
4.3	odd	2		1936.4.a.bv.1.2		8
11.2	odd	10		88.4.i.a.81.4	yes	16
11.6	odd	10		88.4.i.a.25.4	✓	16
11.10	odd	2		968.4.a.n.1.7		8
44.35	even	10		176.4.m.e.81.1		16
44.39	even	10		176.4.m.e.113.1		16
44.43	even	2		1936.4.a.bw.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.a.25.4	✓	16	11.6	odd	10
88.4.i.a.81.4	yes	16	11.2	odd	10
176.4.m.e.81.1		16	44.35	even	10
176.4.m.e.113.1		16	44.39	even	10
968.4.a.n.1.7		8	11.10	odd	2
968.4.a.o.1.7		8	1.1	even	1	trivial
1936.4.a.bv.1.2		8	4.3	odd	2
1936.4.a.bw.1.2		8	44.43	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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)

\(f(q)\)	\(=\)	\(q+5.54031 q^{3} -6.21683 q^{5} +10.0967 q^{7} +3.69499 q^{9} -19.8412 q^{13} -34.4431 q^{15} -0.0283422 q^{17} -72.4162 q^{19} +55.9390 q^{21} +92.3700 q^{23} -86.3511 q^{25} -129.117 q^{27} -83.3142 q^{29} -172.056 q^{31} -62.7697 q^{35} +178.367 q^{37} -109.926 q^{39} +235.671 q^{41} +69.3945 q^{43} -22.9711 q^{45} -433.018 q^{47} -241.056 q^{49} -0.157025 q^{51} -599.884 q^{53} -401.208 q^{57} -365.211 q^{59} +685.581 q^{61} +37.3073 q^{63} +123.349 q^{65} +224.941 q^{67} +511.758 q^{69} -254.652 q^{71} +882.320 q^{73} -478.411 q^{75} -386.959 q^{79} -815.112 q^{81} -1214.30 q^{83} +0.176199 q^{85} -461.586 q^{87} -1236.77 q^{89} -200.332 q^{91} -953.241 q^{93} +450.199 q^{95} -891.507 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} - 13 q^{5} + 9 q^{7} + 36 q^{9} - 7 q^{13} - 66 q^{15} - 94 q^{17} + 92 q^{19} - 17 q^{21} - 46 q^{23} + 101 q^{25} + 124 q^{27} - 241 q^{29} - 265 q^{31} + 664 q^{35} - 469 q^{37} - 788 q^{39}+ \cdots - 4702 q^{97}+O(q^{100}) \) 8 * q - 8 * q^3 - 13 * q^5 + 9 * q^7 + 36 * q^9 - 7 * q^13 - 66 * q^15 - 94 * q^17 + 92 * q^19 - 17 * q^21 - 46 * q^23 + 101 * q^25 + 124 * q^27 - 241 * q^29 - 265 * q^31 + 664 * q^35 - 469 * q^37 - 788 * q^39 + 74 * q^41 - 35 * q^43 - 566 * q^45 + 197 * q^47 + 221 * q^49 + 608 * q^51 - 247 * q^53 + 979 * q^57 - 1208 * q^59 - 2061 * q^61 + 1617 * q^63 - 837 * q^65 - 829 * q^67 - 1860 * q^69 - 2005 * q^71 - 320 * q^73 - 2323 * q^75 + 1581 * q^79 - 3752 * q^81 - 692 * q^83 - 21 * q^85 + 131 * q^87 - 3205 * q^89 - 2724 * q^91 - 722 * q^93 + 432 * q^95 - 4702 * q^97

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