LMFDB - Embedded newform 968.4.a.r.1.8

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:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4 x^{9} - 193 x^{8} + 670 x^{7} + 10959 x^{6} - 33408 x^{5} - 177207 x^{4} + 365822 x^{3} + \cdots - 781744 $ x^10 - 4x^9 - 193x^8 + 670x^7 + 10959x^6 - 33408x^5 - 177207x^4 + 365822x^3 + 990744x^2 - 836120*x - 781744
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$4.92098$ 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)

$f(q)$	$=$	$q+6.53902 q^{3} +21.4485 q^{5} +30.2596 q^{7} +15.7588 q^{9} +O(q^{10})$	$q+6.53902 q^{3} +21.4485 q^{5} +30.2596 q^{7} +15.7588 q^{9} -33.6548 q^{13} +140.252 q^{15} -4.20297 q^{17} -50.2109 q^{19} +197.868 q^{21} -27.2296 q^{23} +335.037 q^{25} -73.5067 q^{27} +24.3115 q^{29} +163.374 q^{31} +649.021 q^{35} +310.877 q^{37} -220.070 q^{39} -401.421 q^{41} -216.635 q^{43} +338.001 q^{45} +101.959 q^{47} +572.641 q^{49} -27.4833 q^{51} +235.683 q^{53} -328.330 q^{57} -126.474 q^{59} -255.197 q^{61} +476.853 q^{63} -721.845 q^{65} -332.902 q^{67} -178.055 q^{69} +337.823 q^{71} -539.223 q^{73} +2190.81 q^{75} +47.9109 q^{79} -906.148 q^{81} +24.3993 q^{83} -90.1473 q^{85} +158.974 q^{87} -1497.82 q^{89} -1018.38 q^{91} +1068.31 q^{93} -1076.95 q^{95} -895.266 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9}+O(q^{10}) $ 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9	$ 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9} - 45 q^{13} + 120 q^{15} + 17 q^{17} + 147 q^{19} - 131 q^{21} + 164 q^{23} + 439 q^{25} + 420 q^{27} - 177 q^{29} + 275 q^{31} + 220 q^{35} + 745 q^{37} + 524 q^{39} - 967 q^{41} + 380 q^{43} - 44 q^{45} + 769 q^{47} + 503 q^{49} + 956 q^{51} + 701 q^{53} - 1293 q^{57} + 1291 q^{59} + 1359 q^{61} - 929 q^{63} - 173 q^{65} + 2260 q^{67} + 1988 q^{69} + 465 q^{71} - 111 q^{73} + 4584 q^{75} - 1827 q^{79} + 6874 q^{81} + 4947 q^{83} - 2609 q^{85} - 1303 q^{87} + 446 q^{89} + 2176 q^{91} + 4204 q^{93} - 108 q^{95} + 3511 q^{97}+O(q^{100}) $ 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9 - 45 * q^13 + 120 * q^15 + 17 * q^17 + 147 * q^19 - 131 * q^21 + 164 * q^23 + 439 * q^25 + 420 * q^27 - 177 * q^29 + 275 * q^31 + 220 * q^35 + 745 * q^37 + 524 * q^39 - 967 * q^41 + 380 * q^43 - 44 * q^45 + 769 * q^47 + 503 * q^49 + 956 * q^51 + 701 * q^53 - 1293 * q^57 + 1291 * q^59 + 1359 * q^61 - 929 * q^63 - 173 * q^65 + 2260 * q^67 + 1988 * q^69 + 465 * q^71 - 111 * q^73 + 4584 * q^75 - 1827 * q^79 + 6874 * q^81 + 4947 * q^83 - 2609 * q^85 - 1303 * q^87 + 446 * q^89 + 2176 * q^91 + 4204 * q^93 - 108 * q^95 + 3511 * q^97

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
$3$	6.53902	1.25843	0.629217	−	0.777229i	$-0.283376\pi$
$3$	6.53902	1.25843	0.629217	+	0.777229i	$0.283376\pi$
$4$	0	0
$5$	21.4485	1.91841	0.959205	−	0.282711i	$-0.0912339\pi$
$5$	21.4485	1.91841	0.959205	+	0.282711i	$0.0912339\pi$
$6$	0	0
$7$	30.2596	1.63386	0.816931	−	0.576735i	$-0.195673\pi$
$7$	30.2596	1.63386	0.816931	+	0.576735i	$0.195673\pi$
$8$	0	0
$9$	15.7588	0.583658
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−33.6548	−0.718013	−0.359007	−	0.933335i	$-0.616884\pi$
$13$	−33.6548	−0.718013	−0.359007	+	0.933335i	$0.616884\pi$
$14$	0	0
$15$	140.252	2.41419
$16$	0	0
$17$	−4.20297	−0.0599629	−0.0299815	−	0.999550i	$-0.509545\pi$
$17$	−4.20297	−0.0599629	−0.0299815	+	0.999550i	$0.509545\pi$
$18$	0	0
$19$	−50.2109	−0.606272	−0.303136	−	0.952947i	$-0.598034\pi$
$19$	−50.2109	−0.606272	−0.303136	+	0.952947i	$0.598034\pi$
$20$	0	0
$21$	197.868	2.05611
$22$	0	0
$23$	−27.2296	−0.246859	−0.123430	−	0.992353i	$-0.539389\pi$
$23$	−27.2296	−0.246859	−0.123430	+	0.992353i	$0.539389\pi$
$24$	0	0
$25$	335.037	2.68030
$26$	0	0
$27$	−73.5067	−0.523940
$28$	0	0
$29$	24.3115	0.155674	0.0778369	−	0.996966i	$-0.475199\pi$
$29$	24.3115	0.155674	0.0778369	+	0.996966i	$0.475199\pi$
$30$	0	0
$31$	163.374	0.946545	0.473272	−	0.880916i	$-0.343073\pi$
$31$	163.374	0.946545	0.473272	+	0.880916i	$0.343073\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	649.021	3.13442
$36$	0	0
$37$	310.877	1.38129	0.690647	−	0.723192i	$-0.257326\pi$
$37$	310.877	1.38129	0.690647	+	0.723192i	$0.257326\pi$
$38$	0	0
$39$	−220.070	−0.903573
$40$	0	0
$41$	−401.421	−1.52906	−0.764530	−	0.644589i	$-0.777029\pi$
$41$	−401.421	−1.52906	−0.764530	+	0.644589i	$0.777029\pi$
$42$	0	0
$43$	−216.635	−0.768292	−0.384146	−	0.923272i	$-0.625504\pi$
$43$	−216.635	−0.768292	−0.384146	+	0.923272i	$0.625504\pi$
$44$	0	0
$45$	338.001	1.11969
$46$	0	0
$47$	101.959	0.316430	0.158215	−	0.987405i	$-0.449426\pi$
$47$	101.959	0.316430	0.158215	+	0.987405i	$0.449426\pi$
$48$	0	0
$49$	572.641	1.66951
$50$	0	0
$51$	−27.4833	−0.0754594
$52$	0	0
$53$	235.683	0.610822	0.305411	−	0.952221i	$-0.401206\pi$
$53$	235.683	0.610822	0.305411	+	0.952221i	$0.401206\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−328.330	−0.762954
$58$	0	0
$59$	−126.474	−0.279076	−0.139538	−	0.990217i	$-0.544562\pi$
$59$	−126.474	−0.279076	−0.139538	+	0.990217i	$0.544562\pi$
$60$	0	0
$61$	−255.197	−0.535650	−0.267825	−	0.963468i	$-0.586305\pi$
$61$	−255.197	−0.535650	−0.267825	+	0.963468i	$0.586305\pi$
$62$	0	0
$63$	476.853	0.953616
$64$	0	0
$65$	−721.845	−1.37744
$66$	0	0
$67$	−332.902	−0.607023	−0.303511	−	0.952828i	$-0.598159\pi$
$67$	−332.902	−0.607023	−0.303511	+	0.952828i	$0.598159\pi$
$68$	0	0
$69$	−178.055	−0.310656
$70$	0	0
$71$	337.823	0.564679	0.282340	−	0.959314i	$-0.408889\pi$
$71$	337.823	0.564679	0.282340	+	0.959314i	$0.408889\pi$
$72$	0	0
$73$	−539.223	−0.864538	−0.432269	−	0.901745i	$-0.642287\pi$
$73$	−539.223	−0.864538	−0.432269	+	0.901745i	$0.642287\pi$
$74$	0	0
$75$	2190.81	3.37298
$76$	0	0
$77$	0	0
$78$	0	0
$79$	47.9109	0.0682329	0.0341165	−	0.999418i	$-0.489138\pi$
$79$	47.9109	0.0682329	0.0341165	+	0.999418i	$0.489138\pi$
$80$	0	0
$81$	−906.148	−1.24300
$82$	0	0
$83$	24.3993	0.0322671	0.0161335	−	0.999870i	$-0.494864\pi$
$83$	24.3993	0.0322671	0.0161335	+	0.999870i	$0.494864\pi$
$84$	0	0
$85$	−90.1473	−0.115033
$86$	0	0
$87$	158.974	0.195905
$88$	0	0
$89$	−1497.82	−1.78392	−0.891958	−	0.452119i	$-0.850668\pi$
$89$	−1497.82	−1.78392	−0.891958	+	0.452119i	$0.850668\pi$
$90$	0	0
$91$	−1018.38	−1.17313
$92$	0	0
$93$	1068.31	1.19116
$94$	0	0
$95$	−1076.95	−1.16308
$96$	0	0
$97$	−895.266	−0.937118	−0.468559	−	0.883432i	$-0.655227\pi$
$97$	−895.266	−0.937118	−0.468559	+	0.883432i	$0.655227\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	250.780	0.247065	0.123532	−	0.992341i	$-0.460578\pi$
$101$	250.780	0.247065	0.123532	+	0.992341i	$0.460578\pi$
$102$	0	0
$103$	−1155.31	−1.10520	−0.552602	−	0.833445i	$-0.686365\pi$
$103$	−1155.31	−1.10520	−0.552602	+	0.833445i	$0.686365\pi$
$104$	0	0
$105$	4243.96	3.94446
$106$	0	0
$107$	195.343	0.176491	0.0882453	−	0.996099i	$-0.471874\pi$
$107$	195.343	0.176491	0.0882453	+	0.996099i	$0.471874\pi$
$108$	0	0
$109$	2257.53	1.98378	0.991891	−	0.127094i	$-0.0405649\pi$
$109$	2257.53	1.98378	0.991891	+	0.127094i	$0.0405649\pi$
$110$	0	0
$111$	2032.83	1.73827
$112$	0	0
$113$	−264.921	−0.220546	−0.110273	−	0.993901i	$-0.535173\pi$
$113$	−264.921	−0.220546	−0.110273	+	0.993901i	$0.535173\pi$
$114$	0	0
$115$	−584.034	−0.473578
$116$	0	0
$117$	−530.358	−0.419074
$118$	0	0
$119$	−127.180	−0.0979712
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−2624.90	−1.92422
$124$	0	0
$125$	4504.98	3.22350
$126$	0	0
$127$	1737.69	1.21413	0.607066	−	0.794651i	$-0.292346\pi$
$127$	1737.69	1.21413	0.607066	+	0.794651i	$0.292346\pi$
$128$	0	0
$129$	−1416.58	−0.966846
$130$	0	0
$131$	−894.597	−0.596651	−0.298326	−	0.954464i	$-0.596428\pi$
$131$	−894.597	−0.596651	−0.298326	+	0.954464i	$0.596428\pi$
$132$	0	0
$133$	−1519.36	−0.990565
$134$	0	0
$135$	−1576.61	−1.00513
$136$	0	0
$137$	622.038	0.387915	0.193957	−	0.981010i	$-0.437868\pi$
$137$	622.038	0.387915	0.193957	+	0.981010i	$0.437868\pi$
$138$	0	0
$139$	−473.354	−0.288844	−0.144422	−	0.989516i	$-0.546132\pi$
$139$	−473.354	−0.288844	−0.144422	+	0.989516i	$0.546132\pi$
$140$	0	0
$141$	666.709	0.398206
$142$	0	0
$143$	0	0
$144$	0	0
$145$	521.445	0.298646
$146$	0	0
$147$	3744.51	2.10097
$148$	0	0
$149$	369.186	0.202986	0.101493	−	0.994836i	$-0.467638\pi$
$149$	369.186	0.202986	0.101493	+	0.994836i	$0.467638\pi$
$150$	0	0
$151$	−1761.54	−0.949353	−0.474677	−	0.880160i	$-0.657435\pi$
$151$	−1761.54	−0.949353	−0.474677	+	0.880160i	$0.657435\pi$
$152$	0	0
$153$	−66.2336	−0.0349978
$154$	0	0
$155$	3504.13	1.81586
$156$	0	0
$157$	306.217	0.155661	0.0778306	−	0.996967i	$-0.475201\pi$
$157$	306.217	0.155661	0.0778306	+	0.996967i	$0.475201\pi$
$158$	0	0
$159$	1541.14	0.768680
$160$	0	0
$161$	−823.956	−0.403334
$162$	0	0
$163$	702.496	0.337569	0.168785	−	0.985653i	$-0.446016\pi$
$163$	702.496	0.337569	0.168785	+	0.985653i	$0.446016\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2816.77	1.30520	0.652598	−	0.757704i	$-0.273679\pi$
$167$	2816.77	1.30520	0.652598	+	0.757704i	$0.273679\pi$
$168$	0	0
$169$	−1064.35	−0.484457
$170$	0	0
$171$	−791.261	−0.353855
$172$	0	0
$173$	−3033.13	−1.33298	−0.666488	−	0.745516i	$-0.732203\pi$
$173$	−3033.13	−1.33298	−0.666488	+	0.745516i	$0.732203\pi$
$174$	0	0
$175$	10138.1	4.37924
$176$	0	0
$177$	−827.014	−0.351199
$178$	0	0
$179$	−2840.96	−1.18628	−0.593138	−	0.805101i	$-0.702111\pi$
$179$	−2840.96	−1.18628	−0.593138	+	0.805101i	$0.702111\pi$
$180$	0	0
$181$	−3466.71	−1.42364	−0.711819	−	0.702363i	$-0.752129\pi$
$181$	−3466.71	−1.42364	−0.711819	+	0.702363i	$0.752129\pi$
$182$	0	0
$183$	−1668.74	−0.674080
$184$	0	0
$185$	6667.84	2.64989
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2224.28	−0.856045
$190$	0	0
$191$	−2496.56	−0.945782	−0.472891	−	0.881121i	$-0.656790\pi$
$191$	−2496.56	−0.945782	−0.472891	+	0.881121i	$0.656790\pi$
$192$	0	0
$193$	600.212	0.223856	0.111928	−	0.993716i	$-0.464297\pi$
$193$	600.212	0.223856	0.111928	+	0.993716i	$0.464297\pi$
$194$	0	0
$195$	−4720.16	−1.73342
$196$	0	0
$197$	−1918.58	−0.693872	−0.346936	−	0.937889i	$-0.612778\pi$
$197$	−1918.58	−0.693872	−0.346936	+	0.937889i	$0.612778\pi$
$198$	0	0
$199$	3261.98	1.16199	0.580994	−	0.813907i	$-0.302664\pi$
$199$	3261.98	1.16199	0.580994	+	0.813907i	$0.302664\pi$
$200$	0	0
$201$	−2176.86	−0.763898
$202$	0	0
$203$	735.656	0.254349
$204$	0	0
$205$	−8609.87	−2.93336
$206$	0	0
$207$	−429.105	−0.144081
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−773.508	−0.252372	−0.126186	−	0.992007i	$-0.540274\pi$
$211$	−773.508	−0.252372	−0.126186	+	0.992007i	$0.540274\pi$
$212$	0	0
$213$	2209.03	0.710612
$214$	0	0
$215$	−4646.50	−1.47390
$216$	0	0
$217$	4943.63	1.54652
$218$	0	0
$219$	−3525.99	−1.08796
$220$	0	0
$221$	141.450	0.0430542
$222$	0	0
$223$	2916.32	0.875747	0.437873	−	0.899037i	$-0.355732\pi$
$223$	2916.32	0.875747	0.437873	+	0.899037i	$0.355732\pi$
$224$	0	0
$225$	5279.77	1.56438
$226$	0	0
$227$	6438.84	1.88265	0.941324	−	0.337505i	$-0.109583\pi$
$227$	6438.84	1.88265	0.941324	+	0.337505i	$0.109583\pi$
$228$	0	0
$229$	−1223.60	−0.353091	−0.176545	−	0.984292i	$-0.556492\pi$
$229$	−1223.60	−0.353091	−0.176545	+	0.984292i	$0.556492\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4479.16	1.25940	0.629699	−	0.776839i	$-0.283178\pi$
$233$	4479.16	1.25940	0.629699	+	0.776839i	$0.283178\pi$
$234$	0	0
$235$	2186.86	0.607042
$236$	0	0
$237$	313.290	0.0858667
$238$	0	0
$239$	4946.72	1.33882	0.669408	−	0.742895i	$-0.266548\pi$
$239$	4946.72	1.33882	0.669408	+	0.742895i	$0.266548\pi$
$240$	0	0
$241$	−4331.64	−1.15778	−0.578891	−	0.815405i	$-0.696514\pi$
$241$	−4331.64	−1.15778	−0.578891	+	0.815405i	$0.696514\pi$
$242$	0	0
$243$	−3940.64	−1.04030
$244$	0	0
$245$	12282.3	3.20280
$246$	0	0
$247$	1689.84	0.435311
$248$	0	0
$249$	159.547	0.0406060
$250$	0	0
$251$	5739.23	1.44325	0.721627	−	0.692282i	$-0.243395\pi$
$251$	5739.23	1.44325	0.721627	+	0.692282i	$0.243395\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−589.475	−0.144762
$256$	0	0
$257$	−5995.36	−1.45518	−0.727588	−	0.686014i	$-0.759359\pi$
$257$	−5995.36	−1.45518	−0.727588	+	0.686014i	$0.759359\pi$
$258$	0	0
$259$	9407.00	2.25684
$260$	0	0
$261$	383.119	0.0908602
$262$	0	0
$263$	−4497.32	−1.05444	−0.527218	−	0.849730i	$-0.676765\pi$
$263$	−4497.32	−1.05444	−0.527218	+	0.849730i	$0.676765\pi$
$264$	0	0
$265$	5055.04	1.17181
$266$	0	0
$267$	−9794.26	−2.24494
$268$	0	0
$269$	3407.29	0.772290	0.386145	−	0.922438i	$-0.373806\pi$
$269$	3407.29	0.772290	0.386145	+	0.922438i	$0.373806\pi$
$270$	0	0
$271$	7863.92	1.76273	0.881364	−	0.472439i	$-0.156626\pi$
$271$	7863.92	1.76273	0.881364	+	0.472439i	$0.156626\pi$
$272$	0	0
$273$	−6659.21	−1.47631
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−957.275	−0.207643	−0.103821	−	0.994596i	$-0.533107\pi$
$277$	−957.275	−0.207643	−0.103821	+	0.994596i	$0.533107\pi$
$278$	0	0
$279$	2574.58	0.552458
$280$	0	0
$281$	322.048	0.0683693	0.0341846	−	0.999416i	$-0.489117\pi$
$281$	322.048	0.0683693	0.0341846	+	0.999416i	$0.489117\pi$
$282$	0	0
$283$	1421.95	0.298680	0.149340	−	0.988786i	$-0.452285\pi$
$283$	1421.95	0.298680	0.149340	+	0.988786i	$0.452285\pi$
$284$	0	0
$285$	−7042.18	−1.46366
$286$	0	0
$287$	−12146.8	−2.49827
$288$	0	0
$289$	−4895.34	−0.996404
$290$	0	0
$291$	−5854.16	−1.17930
$292$	0	0
$293$	−5781.45	−1.15275	−0.576376	−	0.817185i	$-0.695534\pi$
$293$	−5781.45	−1.15275	−0.576376	+	0.817185i	$0.695534\pi$
$294$	0	0
$295$	−2712.67	−0.535382
$296$	0	0
$297$	0	0
$298$	0	0
$299$	916.408	0.177248
$300$	0	0
$301$	−6555.29	−1.25528
$302$	0	0
$303$	1639.85	0.310915
$304$	0	0
$305$	−5473.59	−1.02760
$306$	0	0
$307$	6085.59	1.13134	0.565672	−	0.824630i	$-0.308617\pi$
$307$	6085.59	1.13134	0.565672	+	0.824630i	$0.308617\pi$
$308$	0	0
$309$	−7554.59	−1.39083
$310$	0	0
$311$	9548.05	1.74090	0.870451	−	0.492256i	$-0.163828\pi$
$311$	9548.05	1.74090	0.870451	+	0.492256i	$0.163828\pi$
$312$	0	0
$313$	6451.03	1.16496	0.582482	−	0.812843i	$-0.302081\pi$
$313$	6451.03	1.16496	0.582482	+	0.812843i	$0.302081\pi$
$314$	0	0
$315$	10227.8	1.82943
$316$	0	0
$317$	4084.56	0.723696	0.361848	−	0.932237i	$-0.382146\pi$
$317$	4084.56	0.723696	0.361848	+	0.932237i	$0.382146\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	1277.35	0.222102
$322$	0	0
$323$	211.035	0.0363538
$324$	0	0
$325$	−11275.6	−1.92449
$326$	0	0
$327$	14762.0	2.49646
$328$	0	0
$329$	3085.22	0.517003
$330$	0	0
$331$	−3655.20	−0.606973	−0.303486	−	0.952836i	$-0.598151\pi$
$331$	−3655.20	−0.606973	−0.303486	+	0.952836i	$0.598151\pi$
$332$	0	0
$333$	4899.03	0.806202
$334$	0	0
$335$	−7140.25	−1.16452
$336$	0	0
$337$	−2213.07	−0.357725	−0.178863	−	0.983874i	$-0.557242\pi$
$337$	−2213.07	−0.357725	−0.178863	+	0.983874i	$0.557242\pi$
$338$	0	0
$339$	−1732.33	−0.277543
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6948.84	1.09388
$344$	0	0
$345$	−3819.01	−0.595966
$346$	0	0
$347$	−10578.2	−1.63651	−0.818257	−	0.574853i	$-0.805059\pi$
$347$	−10578.2	−1.63651	−0.818257	+	0.574853i	$0.805059\pi$
$348$	0	0
$349$	−9580.45	−1.46943	−0.734713	−	0.678378i	$-0.762683\pi$
$349$	−9580.45	−1.46943	−0.734713	+	0.678378i	$0.762683\pi$
$350$	0	0
$351$	2473.86	0.376196
$352$	0	0
$353$	−8280.00	−1.24844	−0.624221	−	0.781248i	$-0.714584\pi$
$353$	−8280.00	−1.24844	−0.624221	+	0.781248i	$0.714584\pi$
$354$	0	0
$355$	7245.79	1.08329
$356$	0	0
$357$	−831.632	−0.123290
$358$	0	0
$359$	186.533	0.0274230	0.0137115	−	0.999906i	$-0.495635\pi$
$359$	186.533	0.0274230	0.0137115	+	0.999906i	$0.495635\pi$
$360$	0	0
$361$	−4337.87	−0.632434
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11565.5	−1.65854
$366$	0	0
$367$	3485.77	0.495792	0.247896	−	0.968787i	$-0.420261\pi$
$367$	3485.77	0.495792	0.247896	+	0.968787i	$0.420261\pi$
$368$	0	0
$369$	−6325.90	−0.892447
$370$	0	0
$371$	7131.67	0.997999
$372$	0	0
$373$	−6408.45	−0.889589	−0.444795	−	0.895633i	$-0.646723\pi$
$373$	−6408.45	−0.889589	−0.444795	+	0.895633i	$0.646723\pi$
$374$	0	0
$375$	29458.1	4.05656
$376$	0	0
$377$	−818.201	−0.111776
$378$	0	0
$379$	−6670.66	−0.904087	−0.452043	−	0.891996i	$-0.649305\pi$
$379$	−6670.66	−0.904087	−0.452043	+	0.891996i	$0.649305\pi$
$380$	0	0
$381$	11362.8	1.52791
$382$	0	0
$383$	1228.27	0.163869	0.0819343	−	0.996638i	$-0.473890\pi$
$383$	1228.27	0.163869	0.0819343	+	0.996638i	$0.473890\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−3413.90	−0.448420
$388$	0	0
$389$	248.294	0.0323625	0.0161813	−	0.999869i	$-0.494849\pi$
$389$	248.294	0.0323625	0.0161813	+	0.999869i	$0.494849\pi$
$390$	0	0
$391$	114.445	0.0148024
$392$	0	0
$393$	−5849.79	−0.750847
$394$	0	0
$395$	1027.62	0.130899
$396$	0	0
$397$	8861.58	1.12028	0.560139	−	0.828399i	$-0.310748\pi$
$397$	8861.58	1.12028	0.560139	+	0.828399i	$0.310748\pi$
$398$	0	0
$399$	−9935.12	−1.24656
$400$	0	0
$401$	934.697	0.116400	0.0582002	−	0.998305i	$-0.481464\pi$
$401$	934.697	0.116400	0.0582002	+	0.998305i	$0.481464\pi$
$402$	0	0
$403$	−5498.33	−0.679632
$404$	0	0
$405$	−19435.5	−2.38459
$406$	0	0
$407$	0	0
$408$	0	0
$409$	11069.4	1.33825	0.669127	−	0.743148i	$-0.266668\pi$
$409$	11069.4	1.33825	0.669127	+	0.743148i	$0.266668\pi$
$410$	0	0
$411$	4067.52	0.488165
$412$	0	0
$413$	−3827.04	−0.455971
$414$	0	0
$415$	523.327	0.0619015
$416$	0	0
$417$	−3095.27	−0.363492
$418$	0	0
$419$	9689.20	1.12971	0.564855	−	0.825190i	$-0.308932\pi$
$419$	9689.20	1.12971	0.564855	+	0.825190i	$0.308932\pi$
$420$	0	0
$421$	−8134.70	−0.941712	−0.470856	−	0.882210i	$-0.656055\pi$
$421$	−8134.70	−0.941712	−0.470856	+	0.882210i	$0.656055\pi$
$422$	0	0
$423$	1606.74	0.184687
$424$	0	0
$425$	−1408.15	−0.160718
$426$	0	0
$427$	−7722.15	−0.875178
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6663.03	0.744656	0.372328	−	0.928101i	$-0.378560\pi$
$431$	6663.03	0.744656	0.372328	+	0.928101i	$0.378560\pi$
$432$	0	0
$433$	4564.49	0.506595	0.253297	−	0.967388i	$-0.418485\pi$
$433$	4564.49	0.506595	0.253297	+	0.967388i	$0.418485\pi$
$434$	0	0
$435$	3409.74	0.375826
$436$	0	0
$437$	1367.22	0.149664
$438$	0	0
$439$	2583.73	0.280899	0.140450	−	0.990088i	$-0.455145\pi$
$439$	2583.73	0.280899	0.140450	+	0.990088i	$0.455145\pi$
$440$	0	0
$441$	9024.11	0.974421
$442$	0	0
$443$	3657.24	0.392236	0.196118	−	0.980580i	$-0.437166\pi$
$443$	3657.24	0.392236	0.196118	+	0.980580i	$0.437166\pi$
$444$	0	0
$445$	−32125.9	−3.42228
$446$	0	0
$447$	2414.12	0.255445
$448$	0	0
$449$	−7359.64	−0.773548	−0.386774	−	0.922175i	$-0.626411\pi$
$449$	−7359.64	−0.773548	−0.386774	+	0.922175i	$0.626411\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−11518.8	−1.19470
$454$	0	0
$455$	−21842.7	−2.25055
$456$	0	0
$457$	−10646.4	−1.08976	−0.544879	−	0.838514i	$-0.683425\pi$
$457$	−10646.4	−1.08976	−0.544879	+	0.838514i	$0.683425\pi$
$458$	0	0
$459$	308.946	0.0314170
$460$	0	0
$461$	8558.37	0.864649	0.432324	−	0.901718i	$-0.357694\pi$
$461$	8558.37	0.864649	0.432324	+	0.901718i	$0.357694\pi$
$462$	0	0
$463$	−10142.5	−1.01806	−0.509032	−	0.860748i	$-0.669996\pi$
$463$	−10142.5	−1.01806	−0.509032	+	0.860748i	$0.669996\pi$
$464$	0	0
$465$	22913.6	2.28514
$466$	0	0
$467$	−11968.3	−1.18593	−0.592963	−	0.805230i	$-0.702042\pi$
$467$	−11968.3	−1.18593	−0.592963	+	0.805230i	$0.702042\pi$
$468$	0	0
$469$	−10073.5	−0.991791
$470$	0	0
$471$	2002.36	0.195889
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−16822.5	−1.62499
$476$	0	0
$477$	3714.07	0.356511
$478$	0	0
$479$	−12205.5	−1.16427	−0.582135	−	0.813092i	$-0.697783\pi$
$479$	−12205.5	−1.16427	−0.582135	+	0.813092i	$0.697783\pi$
$480$	0	0
$481$	−10462.5	−0.991787
$482$	0	0
$483$	−5387.86	−0.507570
$484$	0	0
$485$	−19202.1	−1.79778
$486$	0	0
$487$	−19155.9	−1.78242	−0.891210	−	0.453591i	$-0.850143\pi$
$487$	−19155.9	−1.78242	−0.891210	+	0.453591i	$0.850143\pi$
$488$	0	0
$489$	4593.64	0.424809
$490$	0	0
$491$	14080.2	1.29416	0.647079	−	0.762423i	$-0.275990\pi$
$491$	14080.2	1.29416	0.647079	+	0.762423i	$0.275990\pi$
$492$	0	0
$493$	−102.181	−0.00933465
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10222.4	0.922609
$498$	0	0
$499$	5800.44	0.520367	0.260184	−	0.965559i	$-0.416217\pi$
$499$	5800.44	0.520367	0.260184	+	0.965559i	$0.416217\pi$
$500$	0	0
$501$	18418.9	1.64250
$502$	0	0
$503$	−13528.6	−1.19923	−0.599615	−	0.800289i	$-0.704680\pi$
$503$	−13528.6	−1.19923	−0.599615	+	0.800289i	$0.704680\pi$
$504$	0	0
$505$	5378.84	0.473971
$506$	0	0
$507$	−6959.82	−0.609658
$508$	0	0
$509$	−18419.2	−1.60396	−0.801981	−	0.597349i	$-0.796221\pi$
$509$	−18419.2	−1.60396	−0.801981	+	0.597349i	$0.796221\pi$
$510$	0	0
$511$	−16316.7	−1.41254
$512$	0	0
$513$	3690.84	0.317650
$514$	0	0
$515$	−24779.6	−2.12024
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−19833.7	−1.67746
$520$	0	0
$521$	−599.441	−0.0504069	−0.0252035	−	0.999682i	$-0.508023\pi$
$521$	−599.441	−0.0504069	−0.0252035	+	0.999682i	$0.508023\pi$
$522$	0	0
$523$	1409.23	0.117823	0.0589114	−	0.998263i	$-0.481237\pi$
$523$	1409.23	0.117823	0.0589114	+	0.998263i	$0.481237\pi$
$524$	0	0
$525$	66293.1	5.51098
$526$	0	0
$527$	−686.657	−0.0567576
$528$	0	0
$529$	−11425.5	−0.939060
$530$	0	0
$531$	−1993.07	−0.162885
$532$	0	0
$533$	13509.8	1.09788
$534$	0	0
$535$	4189.80	0.338581
$536$	0	0
$537$	−18577.1	−1.49285
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5085.37	−0.404135	−0.202068	−	0.979372i	$-0.564766\pi$
$541$	−5085.37	−0.404135	−0.202068	+	0.979372i	$0.564766\pi$
$542$	0	0
$543$	−22668.9	−1.79156
$544$	0	0
$545$	48420.6	3.80571
$546$	0	0
$547$	−9052.03	−0.707563	−0.353781	−	0.935328i	$-0.615104\pi$
$547$	−9052.03	−0.707563	−0.353781	+	0.935328i	$0.615104\pi$
$548$	0	0
$549$	−4021.59	−0.312636
$550$	0	0
$551$	−1220.70	−0.0943806
$552$	0	0
$553$	1449.76	0.111483
$554$	0	0
$555$	43601.1	3.33471
$556$	0	0
$557$	18831.3	1.43251	0.716253	−	0.697841i	$-0.245856\pi$
$557$	18831.3	1.43251	0.716253	+	0.697841i	$0.245856\pi$
$558$	0	0
$559$	7290.82	0.551644
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11389.6	0.852598	0.426299	−	0.904582i	$-0.359817\pi$
$563$	11389.6	0.852598	0.426299	+	0.904582i	$0.359817\pi$
$564$	0	0
$565$	−5682.16	−0.423098
$566$	0	0
$567$	−27419.6	−2.03089
$568$	0	0
$569$	7176.33	0.528730	0.264365	−	0.964423i	$-0.414838\pi$
$569$	7176.33	0.528730	0.264365	+	0.964423i	$0.414838\pi$
$570$	0	0
$571$	10932.3	0.801233	0.400617	−	0.916246i	$-0.368796\pi$
$571$	10932.3	0.801233	0.400617	+	0.916246i	$0.368796\pi$
$572$	0	0
$573$	−16325.0	−1.19021
$574$	0	0
$575$	−9122.93	−0.661656
$576$	0	0
$577$	−7379.46	−0.532428	−0.266214	−	0.963914i	$-0.585773\pi$
$577$	−7379.46	−0.532428	−0.266214	+	0.963914i	$0.585773\pi$
$578$	0	0
$579$	3924.80	0.281708
$580$	0	0
$581$	738.311	0.0527200
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−11375.4	−0.803955
$586$	0	0
$587$	−17224.2	−1.21110	−0.605551	−	0.795806i	$-0.707047\pi$
$587$	−17224.2	−1.21110	−0.605551	+	0.795806i	$0.707047\pi$
$588$	0	0
$589$	−8203.17	−0.573864
$590$	0	0
$591$	−12545.6	−0.873193
$592$	0	0
$593$	−12587.2	−0.871659	−0.435829	−	0.900029i	$-0.643545\pi$
$593$	−12587.2	−0.871659	−0.435829	+	0.900029i	$0.643545\pi$
$594$	0	0
$595$	−2727.82	−0.187949
$596$	0	0
$597$	21330.2	1.46229
$598$	0	0
$599$	−2913.74	−0.198752	−0.0993758	−	0.995050i	$-0.531685\pi$
$599$	−2913.74	−0.198752	−0.0993758	+	0.995050i	$0.531685\pi$
$600$	0	0
$601$	28320.1	1.92213	0.961065	−	0.276321i	$-0.0891152\pi$
$601$	28320.1	1.92213	0.961065	+	0.276321i	$0.0891152\pi$
$602$	0	0
$603$	−5246.13	−0.354293
$604$	0	0
$605$	0	0
$606$	0	0
$607$	4333.01	0.289739	0.144870	−	0.989451i	$-0.453724\pi$
$607$	4333.01	0.289739	0.144870	+	0.989451i	$0.453724\pi$
$608$	0	0
$609$	4810.47	0.320082
$610$	0	0
$611$	−3431.40	−0.227201
$612$	0	0
$613$	−15825.9	−1.04275	−0.521373	−	0.853329i	$-0.674580\pi$
$613$	−15825.9	−1.04275	−0.521373	+	0.853329i	$0.674580\pi$
$614$	0	0
$615$	−56300.1	−3.69144
$616$	0	0
$617$	22079.8	1.44068	0.720341	−	0.693620i	$-0.243985\pi$
$617$	22079.8	1.44068	0.720341	+	0.693620i	$0.243985\pi$
$618$	0	0
$619$	28516.6	1.85166	0.925830	−	0.377940i	$-0.123367\pi$
$619$	28516.6	1.85166	0.925830	+	0.377940i	$0.123367\pi$
$620$	0	0
$621$	2001.56	0.129339
$622$	0	0
$623$	−45323.3	−2.91467
$624$	0	0
$625$	54745.2	3.50369
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−1306.61	−0.0828264
$630$	0	0
$631$	26406.3	1.66596	0.832979	−	0.553304i	$-0.186633\pi$
$631$	26406.3	1.66596	0.832979	+	0.553304i	$0.186633\pi$
$632$	0	0
$633$	−5057.99	−0.317594
$634$	0	0
$635$	37270.7	2.32920
$636$	0	0
$637$	−19272.1	−1.19873
$638$	0	0
$639$	5323.67	0.329579
$640$	0	0
$641$	−18599.1	−1.14605	−0.573026	−	0.819537i	$-0.694231\pi$
$641$	−18599.1	−1.14605	−0.573026	+	0.819537i	$0.694231\pi$
$642$	0	0
$643$	1577.64	0.0967590	0.0483795	−	0.998829i	$-0.484594\pi$
$643$	1577.64	0.0967590	0.0483795	+	0.998829i	$0.484594\pi$
$644$	0	0
$645$	−30383.5	−1.85481
$646$	0	0
$647$	−15069.3	−0.915664	−0.457832	−	0.889039i	$-0.651374\pi$
$647$	−15069.3	−0.915664	−0.457832	+	0.889039i	$0.651374\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	32326.5	1.94620
$652$	0	0
$653$	5529.26	0.331358	0.165679	−	0.986180i	$-0.447019\pi$
$653$	5529.26	0.331358	0.165679	+	0.986180i	$0.447019\pi$
$654$	0	0
$655$	−19187.7	−1.14462
$656$	0	0
$657$	−8497.48	−0.504594
$658$	0	0
$659$	30631.0	1.81065	0.905323	−	0.424723i	$-0.139628\pi$
$659$	30631.0	1.81065	0.905323	+	0.424723i	$0.139628\pi$
$660$	0	0
$661$	9811.77	0.577358	0.288679	−	0.957426i	$-0.406784\pi$
$661$	9811.77	0.577358	0.288679	+	0.957426i	$0.406784\pi$
$662$	0	0
$663$	924.945	0.0541809
$664$	0	0
$665$	−32587.9	−1.90031
$666$	0	0
$667$	−661.993	−0.0384295
$668$	0	0
$669$	19069.9	1.10207
$670$	0	0
$671$	0	0
$672$	0	0
$673$	18169.9	1.04071	0.520355	−	0.853950i	$-0.325799\pi$
$673$	18169.9	1.04071	0.520355	+	0.853950i	$0.325799\pi$
$674$	0	0
$675$	−24627.5	−1.40431
$676$	0	0
$677$	−19059.8	−1.08202	−0.541011	−	0.841016i	$-0.681958\pi$
$677$	−19059.8	−1.08202	−0.541011	+	0.841016i	$0.681958\pi$
$678$	0	0
$679$	−27090.3	−1.53112
$680$	0	0
$681$	42103.7	2.36919
$682$	0	0
$683$	−4036.99	−0.226166	−0.113083	−	0.993586i	$-0.536073\pi$
$683$	−4036.99	−0.226166	−0.113083	+	0.993586i	$0.536073\pi$
$684$	0	0
$685$	13341.8	0.744179
$686$	0	0
$687$	−8001.15	−0.444342
$688$	0	0
$689$	−7931.88	−0.438578
$690$	0	0
$691$	562.205	0.0309512	0.0154756	−	0.999880i	$-0.495074\pi$
$691$	562.205	0.0309512	0.0154756	+	0.999880i	$0.495074\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10152.7	−0.554122
$696$	0	0
$697$	1687.16	0.0916869
$698$	0	0
$699$	29289.3	1.58487
$700$	0	0
$701$	−18038.5	−0.971903	−0.485952	−	0.873986i	$-0.661527\pi$
$701$	−18038.5	−0.971903	−0.485952	+	0.873986i	$0.661527\pi$
$702$	0	0
$703$	−15609.4	−0.837439
$704$	0	0
$705$	14299.9	0.763922
$706$	0	0
$707$	7588.49	0.403670
$708$	0	0
$709$	30547.4	1.61810	0.809048	−	0.587742i	$-0.199983\pi$
$709$	30547.4	1.61810	0.809048	+	0.587742i	$0.199983\pi$
$710$	0	0
$711$	755.017	0.0398247
$712$	0	0
$713$	−4448.62	−0.233663
$714$	0	0
$715$	0	0
$716$	0	0
$717$	32346.7	1.68481
$718$	0	0
$719$	12126.7	0.628999	0.314500	−	0.949258i	$-0.398163\pi$
$719$	12126.7	0.628999	0.314500	+	0.949258i	$0.398163\pi$
$720$	0	0
$721$	−34959.2	−1.80575
$722$	0	0
$723$	−28324.7	−1.45699
$724$	0	0
$725$	8145.27	0.417252
$726$	0	0
$727$	20713.8	1.05671	0.528357	−	0.849022i	$-0.322808\pi$
$727$	20713.8	1.05671	0.528357	+	0.849022i	$0.322808\pi$
$728$	0	0
$729$	−1301.90	−0.0661434
$730$	0	0
$731$	910.511	0.0460691
$732$	0	0
$733$	26708.4	1.34583	0.672917	−	0.739718i	$-0.265041\pi$
$733$	26708.4	1.34583	0.672917	+	0.739718i	$0.265041\pi$
$734$	0	0
$735$	80314.0	4.03051
$736$	0	0
$737$	0	0
$738$	0	0
$739$	20691.0	1.02995	0.514973	−	0.857206i	$-0.327802\pi$
$739$	20691.0	1.02995	0.514973	+	0.857206i	$0.327802\pi$
$740$	0	0
$741$	11049.9	0.547811
$742$	0	0
$743$	11020.6	0.544154	0.272077	−	0.962275i	$-0.412289\pi$
$743$	11020.6	0.544154	0.272077	+	0.962275i	$0.412289\pi$
$744$	0	0
$745$	7918.48	0.389410
$746$	0	0
$747$	384.502	0.0188329
$748$	0	0
$749$	5910.99	0.288361
$750$	0	0
$751$	15313.0	0.744047	0.372023	−	0.928223i	$-0.378664\pi$
$751$	15313.0	0.744047	0.372023	+	0.928223i	$0.378664\pi$
$752$	0	0
$753$	37528.9	1.81624
$754$	0	0
$755$	−37782.4	−1.82125
$756$	0	0
$757$	−4954.60	−0.237884	−0.118942	−	0.992901i	$-0.537950\pi$
$757$	−4954.60	−0.237884	−0.118942	+	0.992901i	$0.537950\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−17854.3	−0.850482	−0.425241	−	0.905080i	$-0.639811\pi$
$761$	−17854.3	−0.850482	−0.425241	+	0.905080i	$0.639811\pi$
$762$	0	0
$763$	68311.9	3.24123
$764$	0	0
$765$	−1420.61	−0.0671402
$766$	0	0
$767$	4256.45	0.200380
$768$	0	0
$769$	10833.4	0.508012	0.254006	−	0.967203i	$-0.418252\pi$
$769$	10833.4	0.508012	0.254006	+	0.967203i	$0.418252\pi$
$770$	0	0
$771$	−39203.8	−1.83124
$772$	0	0
$773$	−5064.93	−0.235670	−0.117835	−	0.993033i	$-0.537595\pi$
$773$	−5064.93	−0.235670	−0.117835	+	0.993033i	$0.537595\pi$
$774$	0	0
$775$	54736.4	2.53702
$776$	0	0
$777$	61512.5	2.84009
$778$	0	0
$779$	20155.7	0.927026
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1787.06	−0.0815636
$784$	0	0
$785$	6567.90	0.298622
$786$	0	0
$787$	11217.2	0.508067	0.254034	−	0.967195i	$-0.418243\pi$
$787$	11217.2	0.508067	0.254034	+	0.967195i	$0.418243\pi$
$788$	0	0
$789$	−29408.1	−1.32694
$790$	0	0
$791$	−8016.40	−0.360342
$792$	0	0
$793$	8588.61	0.384604
$794$	0	0
$795$	33055.0	1.47464
$796$	0	0
$797$	4416.89	0.196304	0.0981520	−	0.995171i	$-0.468707\pi$
$797$	4416.89	0.196304	0.0981520	+	0.995171i	$0.468707\pi$
$798$	0	0
$799$	−428.529	−0.0189740
$800$	0	0
$801$	−23603.8	−1.04120
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−17672.6	−0.773761
$806$	0	0
$807$	22280.3	0.971877
$808$	0	0
$809$	6470.17	0.281186	0.140593	−	0.990068i	$-0.455099\pi$
$809$	6470.17	0.281186	0.140593	+	0.990068i	$0.455099\pi$
$810$	0	0
$811$	24603.9	1.06530	0.532651	−	0.846335i	$-0.321196\pi$
$811$	24603.9	1.06530	0.532651	+	0.846335i	$0.321196\pi$
$812$	0	0
$813$	51422.3	2.21828
$814$	0	0
$815$	15067.5	0.647596
$816$	0	0
$817$	10877.5	0.465794
$818$	0	0
$819$	−16048.4	−0.684709
$820$	0	0
$821$	−38448.2	−1.63441	−0.817205	−	0.576347i	$-0.804478\pi$
$821$	−38448.2	−1.63441	−0.817205	+	0.576347i	$0.804478\pi$
$822$	0	0
$823$	−731.282	−0.0309731	−0.0154866	−	0.999880i	$-0.504930\pi$
$823$	−731.282	−0.0309731	−0.0154866	+	0.999880i	$0.504930\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36191.2	−1.52176	−0.760878	−	0.648895i	$-0.775231\pi$
$827$	−36191.2	−1.52176	−0.760878	+	0.648895i	$0.775231\pi$
$828$	0	0
$829$	28430.8	1.19113	0.595563	−	0.803309i	$-0.296929\pi$
$829$	28430.8	1.19113	0.595563	+	0.803309i	$0.296929\pi$
$830$	0	0
$831$	−6259.64	−0.261305
$832$	0	0
$833$	−2406.79	−0.100109
$834$	0	0
$835$	60415.3	2.50390
$836$	0	0
$837$	−12009.1	−0.495932
$838$	0	0
$839$	−32932.5	−1.35513	−0.677566	−	0.735462i	$-0.736965\pi$
$839$	−32932.5	−1.35513	−0.677566	+	0.735462i	$0.736965\pi$
$840$	0	0
$841$	−23797.9	−0.975766
$842$	0	0
$843$	2105.88	0.0860383
$844$	0	0
$845$	−22828.7	−0.929387
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9298.17	0.375869
$850$	0	0
$851$	−8465.06	−0.340985
$852$	0	0
$853$	31536.2	1.26586	0.632930	−	0.774209i	$-0.281852\pi$
$853$	31536.2	1.26586	0.632930	+	0.774209i	$0.281852\pi$
$854$	0	0
$855$	−16971.3	−0.678839
$856$	0	0
$857$	−16030.3	−0.638955	−0.319478	−	0.947594i	$-0.603507\pi$
$857$	−16030.3	−0.638955	−0.319478	+	0.947594i	$0.603507\pi$
$858$	0	0
$859$	−31445.6	−1.24902	−0.624511	−	0.781016i	$-0.714702\pi$
$859$	−31445.6	−1.24902	−0.624511	+	0.781016i	$0.714702\pi$
$860$	0	0
$861$	−79428.3	−3.14391
$862$	0	0
$863$	−17831.0	−0.703331	−0.351666	−	0.936126i	$-0.614385\pi$
$863$	−17831.0	−0.703331	−0.351666	+	0.936126i	$0.614385\pi$
$864$	0	0
$865$	−65056.1	−2.55719
$866$	0	0
$867$	−32010.7	−1.25391
$868$	0	0
$869$	0	0
$870$	0	0
$871$	11203.8	0.435850
$872$	0	0
$873$	−14108.3	−0.546956
$874$	0	0
$875$	136319.	5.26675
$876$	0	0
$877$	−45044.8	−1.73439	−0.867193	−	0.497972i	$-0.834078\pi$
$877$	−45044.8	−1.73439	−0.867193	+	0.497972i	$0.834078\pi$
$878$	0	0
$879$	−37805.0	−1.45066
$880$	0	0
$881$	−7783.25	−0.297644	−0.148822	−	0.988864i	$-0.547548\pi$
$881$	−7783.25	−0.297644	−0.148822	+	0.988864i	$0.547548\pi$
$882$	0	0
$883$	30397.0	1.15848	0.579242	−	0.815156i	$-0.303349\pi$
$883$	30397.0	1.15848	0.579242	+	0.815156i	$0.303349\pi$
$884$	0	0
$885$	−17738.2	−0.673743
$886$	0	0
$887$	8845.47	0.334838	0.167419	−	0.985886i	$-0.446457\pi$
$887$	8845.47	0.334838	0.167419	+	0.985886i	$0.446457\pi$
$888$	0	0
$889$	52581.7	1.98373
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5119.43	−0.191842
$894$	0	0
$895$	−60934.3	−2.27576
$896$	0	0
$897$	5992.41	0.223055
$898$	0	0
$899$	3971.88	0.147352
$900$	0	0
$901$	−990.569	−0.0366267
$902$	0	0
$903$	−42865.2	−1.57969
$904$	0	0
$905$	−74355.7	−2.73112
$906$	0	0
$907$	−8358.99	−0.306015	−0.153008	−	0.988225i	$-0.548896\pi$
$907$	−8358.99	−0.306015	−0.153008	+	0.988225i	$0.548896\pi$
$908$	0	0
$909$	3951.98	0.144201
$910$	0	0
$911$	28296.4	1.02909	0.514545	−	0.857464i	$-0.327961\pi$
$911$	28296.4	1.02909	0.514545	+	0.857464i	$0.327961\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−35791.9	−1.29316
$916$	0	0
$917$	−27070.1	−0.974846
$918$	0	0
$919$	36397.5	1.30647	0.653234	−	0.757156i	$-0.273412\pi$
$919$	36397.5	1.30647	0.653234	+	0.757156i	$0.273412\pi$
$920$	0	0
$921$	39793.8	1.42372
$922$	0	0
$923$	−11369.4	−0.405447
$924$	0	0
$925$	104155.	3.70228
$926$	0	0
$927$	−18206.2	−0.645061
$928$	0	0
$929$	20398.7	0.720410	0.360205	−	0.932873i	$-0.382707\pi$
$929$	20398.7	0.720410	0.360205	+	0.932873i	$0.382707\pi$
$930$	0	0
$931$	−28752.8	−1.01218
$932$	0	0
$933$	62434.9	2.19081
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3597.39	−0.125423	−0.0627116	−	0.998032i	$-0.519975\pi$
$937$	−3597.39	−0.125423	−0.0627116	+	0.998032i	$0.519975\pi$
$938$	0	0
$939$	42183.4	1.46603
$940$	0	0
$941$	−10513.7	−0.364227	−0.182114	−	0.983278i	$-0.558294\pi$
$941$	−10513.7	−0.364227	−0.182114	+	0.983278i	$0.558294\pi$
$942$	0	0
$943$	10930.5	0.377463
$944$	0	0
$945$	−47707.4	−1.64225
$946$	0	0
$947$	−25100.8	−0.861317	−0.430658	−	0.902515i	$-0.641719\pi$
$947$	−25100.8	−0.861317	−0.430658	+	0.902515i	$0.641719\pi$
$948$	0	0
$949$	18147.5	0.620750
$950$	0	0
$951$	26709.0	0.910724
$952$	0	0
$953$	−10578.1	−0.359557	−0.179778	−	0.983707i	$-0.557538\pi$
$953$	−10578.1	−0.359557	−0.179778	+	0.983707i	$0.557538\pi$
$954$	0	0
$955$	−53547.3	−1.81440
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18822.6	0.633799
$960$	0	0
$961$	−3099.85	−0.104053
$962$	0	0
$963$	3078.36	0.103010
$964$	0	0
$965$	12873.6	0.429448
$966$	0	0
$967$	25384.3	0.844162	0.422081	−	0.906558i	$-0.361300\pi$
$967$	25384.3	0.844162	0.422081	+	0.906558i	$0.361300\pi$
$968$	0	0
$969$	1379.96	0.0457489
$970$	0	0
$971$	−15659.2	−0.517536	−0.258768	−	0.965940i	$-0.583316\pi$
$971$	−15659.2	−0.517536	−0.258768	+	0.965940i	$0.583316\pi$
$972$	0	0
$973$	−14323.5	−0.471932
$974$	0	0
$975$	−73731.5	−2.42184
$976$	0	0
$977$	−42614.0	−1.39544	−0.697719	−	0.716372i	$-0.745801\pi$
$977$	−42614.0	−1.39544	−0.697719	+	0.716372i	$0.745801\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	35575.9	1.15785
$982$	0	0
$983$	−23964.4	−0.777565	−0.388782	−	0.921330i	$-0.627104\pi$
$983$	−23964.4	−0.777565	−0.388782	+	0.921330i	$0.627104\pi$
$984$	0	0
$985$	−41150.5	−1.33113
$986$	0	0
$987$	20174.3	0.650614
$988$	0	0
$989$	5898.89	0.189660
$990$	0	0
$991$	14914.9	0.478089	0.239044	−	0.971009i	$-0.423166\pi$
$991$	14914.9	0.478089	0.239044	+	0.971009i	$0.423166\pi$
$992$	0	0
$993$	−23901.4	−0.763835
$994$	0	0
$995$	69964.6	2.22917
$996$	0	0
$997$	51018.1	1.62062	0.810311	−	0.586000i	$-0.199298\pi$
$997$	51018.1	1.62062	0.810311	+	0.586000i	$0.199298\pi$
$998$	0	0
$999$	−22851.5	−0.723714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.4.a.r.1.8		10
4.3	odd	2		1936.4.a.by.1.3		10
11.2	odd	10		88.4.i.b.81.4	yes	20
11.6	odd	10		88.4.i.b.25.4	✓	20
11.10	odd	2		968.4.a.s.1.8		10
44.35	even	10		176.4.m.f.81.2		20
44.39	even	10		176.4.m.f.113.2		20
44.43	even	2		1936.4.a.bx.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.b.25.4	✓	20	11.6	odd	10
88.4.i.b.81.4	yes	20	11.2	odd	10
176.4.m.f.81.2		20	44.35	even	10
176.4.m.f.113.2		20	44.39	even	10
968.4.a.r.1.8		10	1.1	even	1	trivial
968.4.a.s.1.8		10	11.10	odd	2
1936.4.a.bx.1.3		10	44.43	even	2
1936.4.a.by.1.3		10	4.3	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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)

\(f(q)\)	\(=\)	\(q+6.53902 q^{3} +21.4485 q^{5} +30.2596 q^{7} +15.7588 q^{9} +O(q^{10})\)	\(q+6.53902 q^{3} +21.4485 q^{5} +30.2596 q^{7} +15.7588 q^{9} -33.6548 q^{13} +140.252 q^{15} -4.20297 q^{17} -50.2109 q^{19} +197.868 q^{21} -27.2296 q^{23} +335.037 q^{25} -73.5067 q^{27} +24.3115 q^{29} +163.374 q^{31} +649.021 q^{35} +310.877 q^{37} -220.070 q^{39} -401.421 q^{41} -216.635 q^{43} +338.001 q^{45} +101.959 q^{47} +572.641 q^{49} -27.4833 q^{51} +235.683 q^{53} -328.330 q^{57} -126.474 q^{59} -255.197 q^{61} +476.853 q^{63} -721.845 q^{65} -332.902 q^{67} -178.055 q^{69} +337.823 q^{71} -539.223 q^{73} +2190.81 q^{75} +47.9109 q^{79} -906.148 q^{81} +24.3993 q^{83} -90.1473 q^{85} +158.974 q^{87} -1497.82 q^{89} -1018.38 q^{91} +1068.31 q^{93} -1076.95 q^{95} -895.266 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9}+O(q^{10}) \) 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9	\( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9} - 45 q^{13} + 120 q^{15} + 17 q^{17} + 147 q^{19} - 131 q^{21} + 164 q^{23} + 439 q^{25} + 420 q^{27} - 177 q^{29} + 275 q^{31} + 220 q^{35} + 745 q^{37} + 524 q^{39} - 967 q^{41} + 380 q^{43} - 44 q^{45} + 769 q^{47} + 503 q^{49} + 956 q^{51} + 701 q^{53} - 1293 q^{57} + 1291 q^{59} + 1359 q^{61} - 929 q^{63} - 173 q^{65} + 2260 q^{67} + 1988 q^{69} + 465 q^{71} - 111 q^{73} + 4584 q^{75} - 1827 q^{79} + 6874 q^{81} + 4947 q^{83} - 2609 q^{85} - 1303 q^{87} + 446 q^{89} + 2176 q^{91} + 4204 q^{93} - 108 q^{95} + 3511 q^{97}+O(q^{100}) \) 10 * q + 9 * q^3 + 13 * q^5 - 3 * q^7 + 141 * q^9 - 45 * q^13 + 120 * q^15 + 17 * q^17 + 147 * q^19 - 131 * q^21 + 164 * q^23 + 439 * q^25 + 420 * q^27 - 177 * q^29 + 275 * q^31 + 220 * q^35 + 745 * q^37 + 524 * q^39 - 967 * q^41 + 380 * q^43 - 44 * q^45 + 769 * q^47 + 503 * q^49 + 956 * q^51 + 701 * q^53 - 1293 * q^57 + 1291 * q^59 + 1359 * q^61 - 929 * q^63 - 173 * q^65 + 2260 * q^67 + 1988 * q^69 + 465 * q^71 - 111 * q^73 + 4584 * q^75 - 1827 * q^79 + 6874 * q^81 + 4947 * q^83 - 2609 * q^85 - 1303 * q^87 + 446 * q^89 + 2176 * q^91 + 4204 * q^93 - 108 * q^95 + 3511 * q^97

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