LMFDB - Embedded newform 968.4.a.r.1.5

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			1.5
Root			$1.28082$ 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)

$f(q)$	$=$	$q+0.662787 q^{3} -9.39213 q^{5} -19.7277 q^{7} -26.5607 q^{9} +O(q^{10})$	$q+0.662787 q^{3} -9.39213 q^{5} -19.7277 q^{7} -26.5607 q^{9} -85.6945 q^{13} -6.22498 q^{15} +52.9559 q^{17} -108.331 q^{19} -13.0752 q^{21} -47.7420 q^{23} -36.7879 q^{25} -35.4994 q^{27} +289.458 q^{29} +11.3979 q^{31} +185.285 q^{35} +20.3463 q^{37} -56.7972 q^{39} -49.4445 q^{41} -441.725 q^{43} +249.462 q^{45} +273.911 q^{47} +46.1807 q^{49} +35.0985 q^{51} -92.8754 q^{53} -71.8006 q^{57} +670.029 q^{59} +141.691 q^{61} +523.981 q^{63} +804.854 q^{65} +791.545 q^{67} -31.6428 q^{69} -255.560 q^{71} -513.138 q^{73} -24.3826 q^{75} -1170.83 q^{79} +693.611 q^{81} +656.977 q^{83} -497.369 q^{85} +191.849 q^{87} -609.733 q^{89} +1690.55 q^{91} +7.55438 q^{93} +1017.46 q^{95} +664.571 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.662787	0.127553	0.0637767	−	0.997964i	$-0.479685\pi$
$3$	0.662787	0.127553	0.0637767	+	0.997964i	$0.479685\pi$
$4$	0	0
$5$	−9.39213	−0.840057	−0.420029	−	0.907511i	$-0.637980\pi$
$5$	−9.39213	−0.840057	−0.420029	+	0.907511i	$0.637980\pi$
$6$	0	0
$7$	−19.7277	−1.06519	−0.532597	−	0.846369i	$-0.678784\pi$
$7$	−19.7277	−1.06519	−0.532597	+	0.846369i	$0.678784\pi$
$8$	0	0
$9$	−26.5607	−0.983730
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−85.6945	−1.82826	−0.914130	−	0.405421i	$-0.867125\pi$
$13$	−85.6945	−1.82826	−0.914130	+	0.405421i	$0.867125\pi$
$14$	0	0
$15$	−6.22498	−0.107152
$16$	0	0
$17$	52.9559	0.755512	0.377756	−	0.925905i	$-0.376696\pi$
$17$	52.9559	0.755512	0.377756	+	0.925905i	$0.376696\pi$
$18$	0	0
$19$	−108.331	−1.30805	−0.654024	−	0.756474i	$-0.726920\pi$
$19$	−108.331	−1.30805	−0.654024	+	0.756474i	$0.726920\pi$
$20$	0	0
$21$	−13.0752	−0.135869
$22$	0	0
$23$	−47.7420	−0.432821	−0.216411	−	0.976302i	$-0.569435\pi$
$23$	−47.7420	−0.432821	−0.216411	+	0.976302i	$0.569435\pi$
$24$	0	0
$25$	−36.7879	−0.294304
$26$	0	0
$27$	−35.4994	−0.253032
$28$	0	0
$29$	289.458	1.85348	0.926742	−	0.375699i	$-0.122598\pi$
$29$	289.458	1.85348	0.926742	+	0.375699i	$0.122598\pi$
$30$	0	0
$31$	11.3979	0.0660362	0.0330181	−	0.999455i	$-0.489488\pi$
$31$	11.3979	0.0660362	0.0330181	+	0.999455i	$0.489488\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	185.285	0.894824
$36$	0	0
$37$	20.3463	0.0904029	0.0452015	−	0.998978i	$-0.485607\pi$
$37$	20.3463	0.0904029	0.0452015	+	0.998978i	$0.485607\pi$
$38$	0	0
$39$	−56.7972	−0.233201
$40$	0	0
$41$	−49.4445	−0.188340	−0.0941699	−	0.995556i	$-0.530020\pi$
$41$	−49.4445	−0.188340	−0.0941699	+	0.995556i	$0.530020\pi$
$42$	0	0
$43$	−441.725	−1.56657	−0.783284	−	0.621664i	$-0.786457\pi$
$43$	−441.725	−1.56657	−0.783284	+	0.621664i	$0.786457\pi$
$44$	0	0
$45$	249.462	0.826390
$46$	0	0
$47$	273.911	0.850085	0.425043	−	0.905173i	$-0.360259\pi$
$47$	273.911	0.850085	0.425043	+	0.905173i	$0.360259\pi$
$48$	0	0
$49$	46.1807	0.134638
$50$	0	0
$51$	35.0985	0.0963682
$52$	0	0
$53$	−92.8754	−0.240706	−0.120353	−	0.992731i	$-0.538403\pi$
$53$	−92.8754	−0.240706	−0.120353	+	0.992731i	$0.538403\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−71.8006	−0.166846
$58$	0	0
$59$	670.029	1.47848	0.739240	−	0.673442i	$-0.235185\pi$
$59$	670.029	1.47848	0.739240	+	0.673442i	$0.235185\pi$
$60$	0	0
$61$	141.691	0.297404	0.148702	−	0.988882i	$-0.452490\pi$
$61$	141.691	0.297404	0.148702	+	0.988882i	$0.452490\pi$
$62$	0	0
$63$	523.981	1.04786
$64$	0	0
$65$	804.854	1.53584
$66$	0	0
$67$	791.545	1.44332	0.721661	−	0.692247i	$-0.243379\pi$
$67$	791.545	1.44332	0.721661	+	0.692247i	$0.243379\pi$
$68$	0	0
$69$	−31.6428	−0.0552079
$70$	0	0
$71$	−255.560	−0.427174	−0.213587	−	0.976924i	$-0.568515\pi$
$71$	−255.560	−0.427174	−0.213587	+	0.976924i	$0.568515\pi$
$72$	0	0
$73$	−513.138	−0.822715	−0.411358	−	0.911474i	$-0.634945\pi$
$73$	−513.138	−0.822715	−0.411358	+	0.911474i	$0.634945\pi$
$74$	0	0
$75$	−24.3826	−0.0375394
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1170.83	−1.66744	−0.833722	−	0.552184i	$-0.813795\pi$
$79$	−1170.83	−1.66744	−0.833722	+	0.552184i	$0.813795\pi$
$80$	0	0
$81$	693.611	0.951455
$82$	0	0
$83$	656.977	0.868827	0.434413	−	0.900714i	$-0.356956\pi$
$83$	656.977	0.868827	0.434413	+	0.900714i	$0.356956\pi$
$84$	0	0
$85$	−497.369	−0.634673
$86$	0	0
$87$	191.849	0.236418
$88$	0	0
$89$	−609.733	−0.726198	−0.363099	−	0.931751i	$-0.618281\pi$
$89$	−609.733	−0.726198	−0.363099	+	0.931751i	$0.618281\pi$
$90$	0	0
$91$	1690.55	1.94745
$92$	0	0
$93$	7.55438	0.00842315
$94$	0	0
$95$	1017.46	1.09884
$96$	0	0
$97$	664.571	0.695639	0.347820	−	0.937562i	$-0.386922\pi$
$97$	664.571	0.695639	0.347820	+	0.937562i	$0.386922\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−932.640	−0.918824	−0.459412	−	0.888223i	$-0.651940\pi$
$101$	−932.640	−0.918824	−0.459412	+	0.888223i	$0.651940\pi$
$102$	0	0
$103$	−742.064	−0.709881	−0.354940	−	0.934889i	$-0.615499\pi$
$103$	−742.064	−0.709881	−0.354940	+	0.934889i	$0.615499\pi$
$104$	0	0
$105$	122.804	0.114138
$106$	0	0
$107$	152.304	0.137606	0.0688029	−	0.997630i	$-0.478082\pi$
$107$	152.304	0.137606	0.0688029	+	0.997630i	$0.478082\pi$
$108$	0	0
$109$	−1731.58	−1.52160	−0.760802	−	0.648984i	$-0.775194\pi$
$109$	−1731.58	−1.52160	−0.760802	+	0.648984i	$0.775194\pi$
$110$	0	0
$111$	13.4853	0.0115312
$112$	0	0
$113$	2167.95	1.80481	0.902404	−	0.430890i	$-0.141800\pi$
$113$	2167.95	1.80481	0.902404	+	0.430890i	$0.141800\pi$
$114$	0	0
$115$	448.399	0.363595
$116$	0	0
$117$	2276.11	1.79851
$118$	0	0
$119$	−1044.70	−0.804766
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−32.7712	−0.0240234
$124$	0	0
$125$	1519.53	1.08729
$126$	0	0
$127$	1019.80	0.712542	0.356271	−	0.934383i	$-0.384048\pi$
$127$	1019.80	0.712542	0.356271	+	0.934383i	$0.384048\pi$
$128$	0	0
$129$	−292.770	−0.199821
$130$	0	0
$131$	−538.711	−0.359293	−0.179646	−	0.983731i	$-0.557495\pi$
$131$	−538.711	−0.359293	−0.179646	+	0.983731i	$0.557495\pi$
$132$	0	0
$133$	2137.12	1.39332
$134$	0	0
$135$	333.415	0.212561
$136$	0	0
$137$	415.023	0.258816	0.129408	−	0.991591i	$-0.458692\pi$
$137$	415.023	0.258816	0.129408	+	0.991591i	$0.458692\pi$
$138$	0	0
$139$	753.789	0.459968	0.229984	−	0.973194i	$-0.426133\pi$
$139$	753.789	0.459968	0.229984	+	0.973194i	$0.426133\pi$
$140$	0	0
$141$	181.545	0.108431
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2718.63	−1.55703
$146$	0	0
$147$	30.6080	0.0171735
$148$	0	0
$149$	−1896.98	−1.04300	−0.521499	−	0.853252i	$-0.674627\pi$
$149$	−1896.98	−1.04300	−0.521499	+	0.853252i	$0.674627\pi$
$150$	0	0
$151$	−264.760	−0.142688	−0.0713440	−	0.997452i	$-0.522729\pi$
$151$	−264.760	−0.142688	−0.0713440	+	0.997452i	$0.522729\pi$
$152$	0	0
$153$	−1406.55	−0.743220
$154$	0	0
$155$	−107.051	−0.0554742
$156$	0	0
$157$	−1779.71	−0.904692	−0.452346	−	0.891842i	$-0.649413\pi$
$157$	−1779.71	−0.904692	−0.452346	+	0.891842i	$0.649413\pi$
$158$	0	0
$159$	−61.5567	−0.0307029
$160$	0	0
$161$	941.837	0.461038
$162$	0	0
$163$	−701.252	−0.336971	−0.168485	−	0.985704i	$-0.553888\pi$
$163$	−701.252	−0.336971	−0.168485	+	0.985704i	$0.553888\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1064.01	−0.493026	−0.246513	−	0.969140i	$-0.579285\pi$
$167$	−1064.01	−0.493026	−0.246513	+	0.969140i	$0.579285\pi$
$168$	0	0
$169$	5146.55	2.34253
$170$	0	0
$171$	2877.36	1.28677
$172$	0	0
$173$	−910.835	−0.400286	−0.200143	−	0.979767i	$-0.564141\pi$
$173$	−910.835	−0.400286	−0.200143	+	0.979767i	$0.564141\pi$
$174$	0	0
$175$	725.740	0.313490
$176$	0	0
$177$	444.087	0.188585
$178$	0	0
$179$	3226.60	1.34730	0.673652	−	0.739049i	$-0.264725\pi$
$179$	3226.60	1.34730	0.673652	+	0.739049i	$0.264725\pi$
$180$	0	0
$181$	−3464.47	−1.42272	−0.711359	−	0.702829i	$-0.751920\pi$
$181$	−3464.47	−1.42272	−0.711359	+	0.702829i	$0.751920\pi$
$182$	0	0
$183$	93.9108	0.0379349
$184$	0	0
$185$	−191.095	−0.0759436
$186$	0	0
$187$	0	0
$188$	0	0
$189$	700.319	0.269528
$190$	0	0
$191$	1165.49	0.441529	0.220765	−	0.975327i	$-0.429145\pi$
$191$	1165.49	0.441529	0.220765	+	0.975327i	$0.429145\pi$
$192$	0	0
$193$	−777.864	−0.290113	−0.145057	−	0.989423i	$-0.546336\pi$
$193$	−777.864	−0.290113	−0.145057	+	0.989423i	$0.546336\pi$
$194$	0	0
$195$	533.447	0.195902
$196$	0	0
$197$	902.817	0.326513	0.163256	−	0.986584i	$-0.447800\pi$
$197$	902.817	0.326513	0.163256	+	0.986584i	$0.447800\pi$
$198$	0	0
$199$	4874.92	1.73655	0.868276	−	0.496082i	$-0.165229\pi$
$199$	4874.92	1.73655	0.868276	+	0.496082i	$0.165229\pi$
$200$	0	0
$201$	524.626	0.184101
$202$	0	0
$203$	−5710.33	−1.97432
$204$	0	0
$205$	464.389	0.158216
$206$	0	0
$207$	1268.06	0.425779
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2015.91	0.657729	0.328865	−	0.944377i	$-0.393334\pi$
$211$	2015.91	0.657729	0.328865	+	0.944377i	$0.393334\pi$
$212$	0	0
$213$	−169.382	−0.0544876
$214$	0	0
$215$	4148.74	1.31601
$216$	0	0
$217$	−224.854	−0.0703414
$218$	0	0
$219$	−340.101	−0.104940
$220$	0	0
$221$	−4538.03	−1.38127
$222$	0	0
$223$	3037.84	0.912236	0.456118	−	0.889919i	$-0.349239\pi$
$223$	3037.84	0.912236	0.456118	+	0.889919i	$0.349239\pi$
$224$	0	0
$225$	977.114	0.289515
$226$	0	0
$227$	5567.12	1.62776	0.813882	−	0.581030i	$-0.197350\pi$
$227$	5567.12	1.62776	0.813882	+	0.581030i	$0.197350\pi$
$228$	0	0
$229$	−2949.37	−0.851091	−0.425545	−	0.904937i	$-0.639918\pi$
$229$	−2949.37	−0.851091	−0.425545	+	0.904937i	$0.639918\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2405.55	−0.676364	−0.338182	−	0.941081i	$-0.609812\pi$
$233$	−2405.55	−0.676364	−0.338182	+	0.941081i	$0.609812\pi$
$234$	0	0
$235$	−2572.61	−0.714120
$236$	0	0
$237$	−776.008	−0.212688
$238$	0	0
$239$	3069.96	0.830875	0.415438	−	0.909622i	$-0.363628\pi$
$239$	3069.96	0.830875	0.415438	+	0.909622i	$0.363628\pi$
$240$	0	0
$241$	3897.95	1.04186	0.520931	−	0.853599i	$-0.325585\pi$
$241$	3897.95	1.04186	0.520931	+	0.853599i	$0.325585\pi$
$242$	0	0
$243$	1418.20	0.374393
$244$	0	0
$245$	−433.735	−0.113103
$246$	0	0
$247$	9283.40	2.39145
$248$	0	0
$249$	435.436	0.110822
$250$	0	0
$251$	2116.97	0.532358	0.266179	−	0.963924i	$-0.414239\pi$
$251$	2116.97	0.532358	0.266179	+	0.963924i	$0.414239\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−329.650	−0.0809548
$256$	0	0
$257$	−4031.64	−0.978548	−0.489274	−	0.872130i	$-0.662738\pi$
$257$	−4031.64	−0.978548	−0.489274	+	0.872130i	$0.662738\pi$
$258$	0	0
$259$	−401.385	−0.0962966
$260$	0	0
$261$	−7688.21	−1.82333
$262$	0	0
$263$	−6138.32	−1.43918	−0.719591	−	0.694398i	$-0.755671\pi$
$263$	−6138.32	−1.43918	−0.719591	+	0.694398i	$0.755671\pi$
$264$	0	0
$265$	872.298	0.202207
$266$	0	0
$267$	−404.123	−0.0926290
$268$	0	0
$269$	2635.88	0.597443	0.298722	−	0.954340i	$-0.403440\pi$
$269$	2635.88	0.597443	0.298722	+	0.954340i	$0.403440\pi$
$270$	0	0
$271$	−6739.49	−1.51068	−0.755342	−	0.655331i	$-0.772529\pi$
$271$	−6739.49	−1.51068	−0.755342	+	0.655331i	$0.772529\pi$
$272$	0	0
$273$	1120.48	0.248404
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5878.23	−1.27505	−0.637525	−	0.770430i	$-0.720042\pi$
$277$	−5878.23	−1.27505	−0.637525	+	0.770430i	$0.720042\pi$
$278$	0	0
$279$	−302.736	−0.0649618
$280$	0	0
$281$	−7225.36	−1.53391	−0.766955	−	0.641701i	$-0.778229\pi$
$281$	−7225.36	−1.53391	−0.766955	+	0.641701i	$0.778229\pi$
$282$	0	0
$283$	8634.99	1.81377	0.906885	−	0.421379i	$-0.138454\pi$
$283$	8634.99	1.81377	0.906885	+	0.421379i	$0.138454\pi$
$284$	0	0
$285$	674.361	0.140160
$286$	0	0
$287$	975.424	0.200618
$288$	0	0
$289$	−2108.67	−0.429202
$290$	0	0
$291$	440.469	0.0887312
$292$	0	0
$293$	−4218.83	−0.841183	−0.420591	−	0.907250i	$-0.638177\pi$
$293$	−4218.83	−0.841183	−0.420591	+	0.907250i	$0.638177\pi$
$294$	0	0
$295$	−6292.99	−1.24201
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4091.22	0.791310
$300$	0	0
$301$	8714.20	1.66870
$302$	0	0
$303$	−618.142	−0.117199
$304$	0	0
$305$	−1330.78	−0.249836
$306$	0	0
$307$	1711.65	0.318206	0.159103	−	0.987262i	$-0.449140\pi$
$307$	1711.65	0.318206	0.159103	+	0.987262i	$0.449140\pi$
$308$	0	0
$309$	−491.830	−0.0905477
$310$	0	0
$311$	606.380	0.110562	0.0552808	−	0.998471i	$-0.482395\pi$
$311$	606.380	0.110562	0.0552808	+	0.998471i	$0.482395\pi$
$312$	0	0
$313$	5035.03	0.909255	0.454628	−	0.890682i	$-0.349772\pi$
$313$	5035.03	0.909255	0.454628	+	0.890682i	$0.349772\pi$
$314$	0	0
$315$	−4921.29	−0.880265
$316$	0	0
$317$	9480.55	1.67975	0.839875	−	0.542780i	$-0.182628\pi$
$317$	9480.55	1.67975	0.839875	+	0.542780i	$0.182628\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	100.945	0.0175521
$322$	0	0
$323$	−5736.79	−0.988246
$324$	0	0
$325$	3152.52	0.538063
$326$	0	0
$327$	−1147.67	−0.194086
$328$	0	0
$329$	−5403.62	−0.905505
$330$	0	0
$331$	−10140.2	−1.68385	−0.841926	−	0.539593i	$-0.818578\pi$
$331$	−10140.2	−1.68385	−0.841926	+	0.539593i	$0.818578\pi$
$332$	0	0
$333$	−540.412	−0.0889321
$334$	0	0
$335$	−7434.29	−1.21247
$336$	0	0
$337$	1982.26	0.320417	0.160208	−	0.987083i	$-0.448783\pi$
$337$	1982.26	0.320417	0.160208	+	0.987083i	$0.448783\pi$
$338$	0	0
$339$	1436.89	0.230210
$340$	0	0
$341$	0	0
$342$	0	0
$343$	5855.55	0.921779
$344$	0	0
$345$	297.193	0.0463778
$346$	0	0
$347$	4798.11	0.742294	0.371147	−	0.928574i	$-0.378965\pi$
$347$	4798.11	0.742294	0.371147	+	0.928574i	$0.378965\pi$
$348$	0	0
$349$	−729.644	−0.111911	−0.0559555	−	0.998433i	$-0.517820\pi$
$349$	−729.644	−0.111911	−0.0559555	+	0.998433i	$0.517820\pi$
$350$	0	0
$351$	3042.10	0.462608
$352$	0	0
$353$	−4096.07	−0.617597	−0.308799	−	0.951127i	$-0.599927\pi$
$353$	−4096.07	−0.617597	−0.308799	+	0.951127i	$0.599927\pi$
$354$	0	0
$355$	2400.25	0.358851
$356$	0	0
$357$	−692.412	−0.102651
$358$	0	0
$359$	−1355.25	−0.199240	−0.0996200	−	0.995026i	$-0.531763\pi$
$359$	−1355.25	−0.199240	−0.0996200	+	0.995026i	$0.531763\pi$
$360$	0	0
$361$	4876.68	0.710990
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4819.45	0.691128
$366$	0	0
$367$	−7790.26	−1.10803	−0.554017	−	0.832506i	$-0.686906\pi$
$367$	−7790.26	−1.10803	−0.554017	+	0.832506i	$0.686906\pi$
$368$	0	0
$369$	1313.28	0.185276
$370$	0	0
$371$	1832.21	0.256399
$372$	0	0
$373$	7121.13	0.988520	0.494260	−	0.869314i	$-0.335439\pi$
$373$	7121.13	0.988520	0.494260	+	0.869314i	$0.335439\pi$
$374$	0	0
$375$	1007.13	0.138688
$376$	0	0
$377$	−24805.0	−3.38865
$378$	0	0
$379$	−1840.59	−0.249458	−0.124729	−	0.992191i	$-0.539806\pi$
$379$	−1840.59	−0.249458	−0.124729	+	0.992191i	$0.539806\pi$
$380$	0	0
$381$	675.912	0.0908872
$382$	0	0
$383$	−5323.39	−0.710216	−0.355108	−	0.934825i	$-0.615556\pi$
$383$	−5323.39	−0.710216	−0.355108	+	0.934825i	$0.615556\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11732.5	1.54108
$388$	0	0
$389$	−5912.37	−0.770615	−0.385308	−	0.922788i	$-0.625905\pi$
$389$	−5912.37	−0.770615	−0.385308	+	0.922788i	$0.625905\pi$
$390$	0	0
$391$	−2528.22	−0.327002
$392$	0	0
$393$	−357.051	−0.0458291
$394$	0	0
$395$	10996.5	1.40075
$396$	0	0
$397$	−4609.55	−0.582737	−0.291369	−	0.956611i	$-0.594111\pi$
$397$	−4609.55	−0.582737	−0.291369	+	0.956611i	$0.594111\pi$
$398$	0	0
$399$	1416.46	0.177723
$400$	0	0
$401$	−5072.94	−0.631747	−0.315873	−	0.948801i	$-0.602297\pi$
$401$	−5072.94	−0.631747	−0.315873	+	0.948801i	$0.602297\pi$
$402$	0	0
$403$	−976.737	−0.120731
$404$	0	0
$405$	−6514.48	−0.799277
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10370.6	1.25378	0.626889	−	0.779109i	$-0.284328\pi$
$409$	10370.6	1.25378	0.626889	+	0.779109i	$0.284328\pi$
$410$	0	0
$411$	275.072	0.0330129
$412$	0	0
$413$	−13218.1	−1.57487
$414$	0	0
$415$	−6170.41	−0.729864
$416$	0	0
$417$	499.602	0.0586705
$418$	0	0
$419$	16721.4	1.94963	0.974816	−	0.223010i	$-0.0715882\pi$
$419$	16721.4	1.94963	0.974816	+	0.223010i	$0.0715882\pi$
$420$	0	0
$421$	−7306.91	−0.845883	−0.422942	−	0.906157i	$-0.639003\pi$
$421$	−7306.91	−0.845883	−0.422942	+	0.906157i	$0.639003\pi$
$422$	0	0
$423$	−7275.27	−0.836254
$424$	0	0
$425$	−1948.14	−0.222350
$426$	0	0
$427$	−2795.23	−0.316793
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7705.77	−0.861193	−0.430596	−	0.902545i	$-0.641697\pi$
$431$	−7705.77	−0.861193	−0.430596	+	0.902545i	$0.641697\pi$
$432$	0	0
$433$	−6991.97	−0.776011	−0.388005	−	0.921657i	$-0.626836\pi$
$433$	−6991.97	−0.776011	−0.388005	+	0.921657i	$0.626836\pi$
$434$	0	0
$435$	−1801.87	−0.198605
$436$	0	0
$437$	5171.95	0.566151
$438$	0	0
$439$	−11732.1	−1.27550	−0.637750	−	0.770243i	$-0.720135\pi$
$439$	−11732.1	−1.27550	−0.637750	+	0.770243i	$0.720135\pi$
$440$	0	0
$441$	−1226.59	−0.132447
$442$	0	0
$443$	−10327.8	−1.10765	−0.553824	−	0.832634i	$-0.686832\pi$
$443$	−10327.8	−1.10765	−0.553824	+	0.832634i	$0.686832\pi$
$444$	0	0
$445$	5726.69	0.610048
$446$	0	0
$447$	−1257.29	−0.133038
$448$	0	0
$449$	−10419.2	−1.09513	−0.547563	−	0.836764i	$-0.684444\pi$
$449$	−10419.2	−1.09513	−0.547563	+	0.836764i	$0.684444\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−175.480	−0.0182003
$454$	0	0
$455$	−15877.9	−1.63597
$456$	0	0
$457$	−13853.3	−1.41801	−0.709006	−	0.705202i	$-0.750856\pi$
$457$	−13853.3	−1.41801	−0.709006	+	0.705202i	$0.750856\pi$
$458$	0	0
$459$	−1879.90	−0.191168
$460$	0	0
$461$	289.586	0.0292568	0.0146284	−	0.999893i	$-0.495343\pi$
$461$	289.586	0.0292568	0.0146284	+	0.999893i	$0.495343\pi$
$462$	0	0
$463$	−13404.9	−1.34552	−0.672762	−	0.739859i	$-0.734892\pi$
$463$	−13404.9	−1.34552	−0.672762	+	0.739859i	$0.734892\pi$
$464$	0	0
$465$	−70.9517	−0.00707593
$466$	0	0
$467$	7087.76	0.702318	0.351159	−	0.936316i	$-0.385788\pi$
$467$	7087.76	0.702318	0.351159	+	0.936316i	$0.385788\pi$
$468$	0	0
$469$	−15615.3	−1.53742
$470$	0	0
$471$	−1179.57	−0.115397
$472$	0	0
$473$	0	0
$474$	0	0
$475$	3985.29	0.384963
$476$	0	0
$477$	2466.84	0.236790
$478$	0	0
$479$	1210.55	0.115473	0.0577366	−	0.998332i	$-0.481612\pi$
$479$	1210.55	0.115473	0.0577366	+	0.998332i	$0.481612\pi$
$480$	0	0
$481$	−1743.56	−0.165280
$482$	0	0
$483$	624.238	0.0588071
$484$	0	0
$485$	−6241.74	−0.584377
$486$	0	0
$487$	−396.389	−0.0368832	−0.0184416	−	0.999830i	$-0.505870\pi$
$487$	−396.389	−0.0368832	−0.0184416	+	0.999830i	$0.505870\pi$
$488$	0	0
$489$	−464.781	−0.0429818
$490$	0	0
$491$	−9228.46	−0.848217	−0.424108	−	0.905611i	$-0.639412\pi$
$491$	−9228.46	−0.848217	−0.424108	+	0.905611i	$0.639412\pi$
$492$	0	0
$493$	15328.5	1.40033
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5041.60	0.455023
$498$	0	0
$499$	−4886.09	−0.438340	−0.219170	−	0.975687i	$-0.570335\pi$
$499$	−4886.09	−0.438340	−0.219170	+	0.975687i	$0.570335\pi$
$500$	0	0
$501$	−705.210	−0.0628872
$502$	0	0
$503$	12745.9	1.12985	0.564923	−	0.825143i	$-0.308906\pi$
$503$	12745.9	1.12985	0.564923	+	0.825143i	$0.308906\pi$
$504$	0	0
$505$	8759.48	0.771865
$506$	0	0
$507$	3411.07	0.298798
$508$	0	0
$509$	635.548	0.0553442	0.0276721	−	0.999617i	$-0.491191\pi$
$509$	635.548	0.0553442	0.0276721	+	0.999617i	$0.491191\pi$
$510$	0	0
$511$	10123.0	0.876351
$512$	0	0
$513$	3845.69	0.330978
$514$	0	0
$515$	6969.56	0.596340
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−603.690	−0.0510579
$520$	0	0
$521$	14720.2	1.23782	0.618910	−	0.785462i	$-0.287575\pi$
$521$	14720.2	1.23782	0.618910	+	0.785462i	$0.287575\pi$
$522$	0	0
$523$	15460.0	1.29258	0.646288	−	0.763094i	$-0.276320\pi$
$523$	15460.0	1.29258	0.646288	+	0.763094i	$0.276320\pi$
$524$	0	0
$525$	481.011	0.0399868
$526$	0	0
$527$	603.587	0.0498912
$528$	0	0
$529$	−9887.70	−0.812666
$530$	0	0
$531$	−17796.4	−1.45442
$532$	0	0
$533$	4237.12	0.344334
$534$	0	0
$535$	−1430.46	−0.115597
$536$	0	0
$537$	2138.55	0.171853
$538$	0	0
$539$	0	0
$540$	0	0
$541$	5308.88	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$541$	5308.88	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$542$	0	0
$543$	−2296.21	−0.181473
$544$	0	0
$545$	16263.2	1.27823
$546$	0	0
$547$	4284.55	0.334907	0.167454	−	0.985880i	$-0.446446\pi$
$547$	4284.55	0.334907	0.167454	+	0.985880i	$0.446446\pi$
$548$	0	0
$549$	−3763.41	−0.292565
$550$	0	0
$551$	−31357.4	−2.42445
$552$	0	0
$553$	23097.6	1.77615
$554$	0	0
$555$	−126.655	−0.00968687
$556$	0	0
$557$	−2174.03	−0.165380	−0.0826900	−	0.996575i	$-0.526351\pi$
$557$	−2174.03	−0.165380	−0.0826900	+	0.996575i	$0.526351\pi$
$558$	0	0
$559$	37853.4	2.86409
$560$	0	0
$561$	0	0
$562$	0	0
$563$	13272.1	0.993520	0.496760	−	0.867888i	$-0.334523\pi$
$563$	13272.1	0.993520	0.496760	+	0.867888i	$0.334523\pi$
$564$	0	0
$565$	−20361.6	−1.51614
$566$	0	0
$567$	−13683.3	−1.01348
$568$	0	0
$569$	−19569.5	−1.44182	−0.720911	−	0.693028i	$-0.756276\pi$
$569$	−19569.5	−1.44182	−0.720911	+	0.693028i	$0.756276\pi$
$570$	0	0
$571$	863.969	0.0633205	0.0316602	−	0.999499i	$-0.489921\pi$
$571$	863.969	0.0633205	0.0316602	+	0.999499i	$0.489921\pi$
$572$	0	0
$573$	772.474	0.0563186
$574$	0	0
$575$	1756.33	0.127381
$576$	0	0
$577$	−9788.27	−0.706224	−0.353112	−	0.935581i	$-0.614876\pi$
$577$	−9788.27	−0.706224	−0.353112	+	0.935581i	$0.614876\pi$
$578$	0	0
$579$	−515.558	−0.0370050
$580$	0	0
$581$	−12960.6	−0.925469
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−21377.5	−1.51086
$586$	0	0
$587$	19959.0	1.40340	0.701700	−	0.712472i	$-0.252425\pi$
$587$	19959.0	1.40340	0.701700	+	0.712472i	$0.252425\pi$
$588$	0	0
$589$	−1234.75	−0.0863786
$590$	0	0
$591$	598.375	0.0416478
$592$	0	0
$593$	−7003.79	−0.485010	−0.242505	−	0.970150i	$-0.577969\pi$
$593$	−7003.79	−0.485010	−0.242505	+	0.970150i	$0.577969\pi$
$594$	0	0
$595$	9811.93	0.676050
$596$	0	0
$597$	3231.03	0.221503
$598$	0	0
$599$	23664.4	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$599$	23664.4	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$600$	0	0
$601$	−11377.7	−0.772222	−0.386111	−	0.922452i	$-0.626182\pi$
$601$	−11377.7	−0.772222	−0.386111	+	0.922452i	$0.626182\pi$
$602$	0	0
$603$	−21024.0	−1.41984
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8946.34	−0.598222	−0.299111	−	0.954218i	$-0.596690\pi$
$607$	−8946.34	−0.598222	−0.299111	+	0.954218i	$0.596690\pi$
$608$	0	0
$609$	−3784.74	−0.251831
$610$	0	0
$611$	−23472.7	−1.55418
$612$	0	0
$613$	22141.7	1.45888	0.729441	−	0.684043i	$-0.239780\pi$
$613$	22141.7	1.45888	0.729441	+	0.684043i	$0.239780\pi$
$614$	0	0
$615$	307.791	0.0201810
$616$	0	0
$617$	13951.1	0.910289	0.455144	−	0.890418i	$-0.349588\pi$
$617$	13951.1	0.910289	0.455144	+	0.890418i	$0.349588\pi$
$618$	0	0
$619$	−3194.61	−0.207435	−0.103718	−	0.994607i	$-0.533074\pi$
$619$	−3194.61	−0.207435	−0.103718	+	0.994607i	$0.533074\pi$
$620$	0	0
$621$	1694.81	0.109517
$622$	0	0
$623$	12028.6	0.773541
$624$	0	0
$625$	−9673.15	−0.619082
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1077.46	0.0683005
$630$	0	0
$631$	−740.119	−0.0466936	−0.0233468	−	0.999727i	$-0.507432\pi$
$631$	−740.119	−0.0466936	−0.0233468	+	0.999727i	$0.507432\pi$
$632$	0	0
$633$	1336.12	0.0838957
$634$	0	0
$635$	−9578.11	−0.598576
$636$	0	0
$637$	−3957.43	−0.246152
$638$	0	0
$639$	6787.85	0.420224
$640$	0	0
$641$	301.293	0.0185653	0.00928264	−	0.999957i	$-0.497045\pi$
$641$	301.293	0.0185653	0.00928264	+	0.999957i	$0.497045\pi$
$642$	0	0
$643$	−10320.0	−0.632941	−0.316471	−	0.948602i	$-0.602498\pi$
$643$	−10320.0	−0.632941	−0.316471	+	0.948602i	$0.602498\pi$
$644$	0	0
$645$	2749.73	0.167861
$646$	0	0
$647$	−16812.8	−1.02161	−0.510804	−	0.859697i	$-0.670652\pi$
$647$	−16812.8	−1.02161	−0.510804	+	0.859697i	$0.670652\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−149.030	−0.00897229
$652$	0	0
$653$	−13434.0	−0.805076	−0.402538	−	0.915403i	$-0.631872\pi$
$653$	−13434.0	−0.805076	−0.402538	+	0.915403i	$0.631872\pi$
$654$	0	0
$655$	5059.64	0.301827
$656$	0	0
$657$	13629.3	0.809330
$658$	0	0
$659$	27135.2	1.60400	0.802002	−	0.597321i	$-0.203768\pi$
$659$	27135.2	1.60400	0.802002	+	0.597321i	$0.203768\pi$
$660$	0	0
$661$	31849.6	1.87414	0.937070	−	0.349141i	$-0.113526\pi$
$661$	31849.6	1.87414	0.937070	+	0.349141i	$0.113526\pi$
$662$	0	0
$663$	−3007.75	−0.176186
$664$	0	0
$665$	−20072.1	−1.17047
$666$	0	0
$667$	−13819.3	−0.802227
$668$	0	0
$669$	2013.44	0.116359
$670$	0	0
$671$	0	0
$672$	0	0
$673$	18223.9	1.04381	0.521903	−	0.853005i	$-0.325222\pi$
$673$	18223.9	1.04381	0.521903	+	0.853005i	$0.325222\pi$
$674$	0	0
$675$	1305.95	0.0744681
$676$	0	0
$677$	5659.76	0.321303	0.160652	−	0.987011i	$-0.448640\pi$
$677$	5659.76	0.321303	0.160652	+	0.987011i	$0.448640\pi$
$678$	0	0
$679$	−13110.4	−0.740990
$680$	0	0
$681$	3689.81	0.207627
$682$	0	0
$683$	24421.5	1.36817	0.684086	−	0.729402i	$-0.260201\pi$
$683$	24421.5	1.36817	0.684086	+	0.729402i	$0.260201\pi$
$684$	0	0
$685$	−3897.95	−0.217421
$686$	0	0
$687$	−1954.80	−0.108560
$688$	0	0
$689$	7958.91	0.440073
$690$	0	0
$691$	−6423.62	−0.353641	−0.176821	−	0.984243i	$-0.556581\pi$
$691$	−6423.62	−0.353641	−0.176821	+	0.984243i	$0.556581\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7079.68	−0.386400
$696$	0	0
$697$	−2618.38	−0.142293
$698$	0	0
$699$	−1594.37	−0.0862726
$700$	0	0
$701$	−9495.02	−0.511586	−0.255793	−	0.966732i	$-0.582337\pi$
$701$	−9495.02	−0.511586	−0.255793	+	0.966732i	$0.582337\pi$
$702$	0	0
$703$	−2204.14	−0.118251
$704$	0	0
$705$	−1705.09	−0.0910885
$706$	0	0
$707$	18398.8	0.978725
$708$	0	0
$709$	11232.7	0.594999	0.297499	−	0.954722i	$-0.403847\pi$
$709$	11232.7	0.594999	0.297499	+	0.954722i	$0.403847\pi$
$710$	0	0
$711$	31098.0	1.64032
$712$	0	0
$713$	−544.158	−0.0285819
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2034.73	0.105981
$718$	0	0
$719$	32337.5	1.67731	0.838654	−	0.544664i	$-0.183343\pi$
$719$	32337.5	1.67731	0.838654	+	0.544664i	$0.183343\pi$
$720$	0	0
$721$	14639.2	0.756160
$722$	0	0
$723$	2583.51	0.132893
$724$	0	0
$725$	−10648.6	−0.545487
$726$	0	0
$727$	−21638.4	−1.10388	−0.551941	−	0.833883i	$-0.686113\pi$
$727$	−21638.4	−1.10388	−0.551941	+	0.833883i	$0.686113\pi$
$728$	0	0
$729$	−17787.5	−0.903700
$730$	0	0
$731$	−23392.0	−1.18356
$732$	0	0
$733$	23941.8	1.20643	0.603213	−	0.797580i	$-0.293887\pi$
$733$	23941.8	1.20643	0.603213	+	0.797580i	$0.293887\pi$
$734$	0	0
$735$	−287.474	−0.0144267
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−27877.2	−1.38766	−0.693828	−	0.720140i	$-0.744077\pi$
$739$	−27877.2	−1.38766	−0.693828	+	0.720140i	$0.744077\pi$
$740$	0	0
$741$	6152.92	0.305038
$742$	0	0
$743$	34443.3	1.70068	0.850339	−	0.526235i	$-0.176397\pi$
$743$	34443.3	1.70068	0.850339	+	0.526235i	$0.176397\pi$
$744$	0	0
$745$	17816.7	0.876178
$746$	0	0
$747$	−17449.8	−0.854691
$748$	0	0
$749$	−3004.61	−0.146577
$750$	0	0
$751$	13655.1	0.663491	0.331746	−	0.943369i	$-0.392362\pi$
$751$	13655.1	0.663491	0.331746	+	0.943369i	$0.392362\pi$
$752$	0	0
$753$	1403.10	0.0679041
$754$	0	0
$755$	2486.66	0.119866
$756$	0	0
$757$	19534.1	0.937886	0.468943	−	0.883228i	$-0.344635\pi$
$757$	19534.1	0.937886	0.468943	+	0.883228i	$0.344635\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3020.95	0.143902	0.0719509	−	0.997408i	$-0.477078\pi$
$761$	3020.95	0.143902	0.0719509	+	0.997408i	$0.477078\pi$
$762$	0	0
$763$	34159.9	1.62080
$764$	0	0
$765$	13210.5	0.624347
$766$	0	0
$767$	−57417.8	−2.70305
$768$	0	0
$769$	8422.74	0.394970	0.197485	−	0.980306i	$-0.436723\pi$
$769$	8422.74	0.394970	0.197485	+	0.980306i	$0.436723\pi$
$770$	0	0
$771$	−2672.12	−0.124817
$772$	0	0
$773$	30115.2	1.40125	0.700627	−	0.713528i	$-0.252904\pi$
$773$	30115.2	1.40125	0.700627	+	0.713528i	$0.252904\pi$
$774$	0	0
$775$	−419.305	−0.0194347
$776$	0	0
$777$	−266.033	−0.0122830
$778$	0	0
$779$	5356.39	0.246358
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−10275.6	−0.468990
$784$	0	0
$785$	16715.3	0.759993
$786$	0	0
$787$	−37179.3	−1.68399	−0.841993	−	0.539488i	$-0.818618\pi$
$787$	−37179.3	−1.68399	−0.841993	+	0.539488i	$0.818618\pi$
$788$	0	0
$789$	−4068.40	−0.183573
$790$	0	0
$791$	−42768.6	−1.92247
$792$	0	0
$793$	−12142.1	−0.543732
$794$	0	0
$795$	578.148	0.0257922
$796$	0	0
$797$	−23122.0	−1.02763	−0.513816	−	0.857900i	$-0.671769\pi$
$797$	−23122.0	−1.02763	−0.513816	+	0.857900i	$0.671769\pi$
$798$	0	0
$799$	14505.2	0.642250
$800$	0	0
$801$	16194.9	0.714382
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−8845.86	−0.387299
$806$	0	0
$807$	1747.03	0.0762059
$808$	0	0
$809$	19005.8	0.825969	0.412984	−	0.910738i	$-0.364486\pi$
$809$	19005.8	0.825969	0.412984	+	0.910738i	$0.364486\pi$
$810$	0	0
$811$	35071.2	1.51852	0.759258	−	0.650790i	$-0.225562\pi$
$811$	35071.2	1.51852	0.759258	+	0.650790i	$0.225562\pi$
$812$	0	0
$813$	−4466.85	−0.192693
$814$	0	0
$815$	6586.24	0.283075
$816$	0	0
$817$	47852.6	2.04915
$818$	0	0
$819$	−44902.3	−1.91577
$820$	0	0
$821$	11374.8	0.483538	0.241769	−	0.970334i	$-0.422272\pi$
$821$	11374.8	0.483538	0.241769	+	0.970334i	$0.422272\pi$
$822$	0	0
$823$	21938.6	0.929201	0.464601	−	0.885520i	$-0.346198\pi$
$823$	21938.6	0.929201	0.464601	+	0.885520i	$0.346198\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4404.93	0.185217	0.0926084	−	0.995703i	$-0.470480\pi$
$827$	4404.93	0.185217	0.0926084	+	0.995703i	$0.470480\pi$
$828$	0	0
$829$	29327.1	1.22868	0.614338	−	0.789043i	$-0.289423\pi$
$829$	29327.1	1.22868	0.614338	+	0.789043i	$0.289423\pi$
$830$	0	0
$831$	−3896.02	−0.162637
$832$	0	0
$833$	2445.54	0.101720
$834$	0	0
$835$	9993.29	0.414170
$836$	0	0
$837$	−404.618	−0.0167093
$838$	0	0
$839$	−20773.4	−0.854801	−0.427401	−	0.904062i	$-0.640571\pi$
$839$	−20773.4	−0.854801	−0.427401	+	0.904062i	$0.640571\pi$
$840$	0	0
$841$	59397.0	2.43540
$842$	0	0
$843$	−4788.88	−0.195656
$844$	0	0
$845$	−48337.0	−1.96786
$846$	0	0
$847$	0	0
$848$	0	0
$849$	5723.16	0.231353
$850$	0	0
$851$	−971.371	−0.0391283
$852$	0	0
$853$	30933.7	1.24168	0.620838	−	0.783939i	$-0.286792\pi$
$853$	30933.7	1.24168	0.620838	+	0.783939i	$0.286792\pi$
$854$	0	0
$855$	−27024.5	−1.08096
$856$	0	0
$857$	−15472.6	−0.616727	−0.308364	−	0.951269i	$-0.599781\pi$
$857$	−15472.6	−0.616727	−0.308364	+	0.951269i	$0.599781\pi$
$858$	0	0
$859$	8000.77	0.317791	0.158896	−	0.987295i	$-0.449207\pi$
$859$	8000.77	0.317791	0.158896	+	0.987295i	$0.449207\pi$
$860$	0	0
$861$	646.499	0.0255896
$862$	0	0
$863$	41827.3	1.64985	0.824923	−	0.565245i	$-0.191218\pi$
$863$	41827.3	1.64985	0.824923	+	0.565245i	$0.191218\pi$
$864$	0	0
$865$	8554.68	0.336263
$866$	0	0
$867$	−1397.60	−0.0547462
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−67831.0	−2.63877
$872$	0	0
$873$	−17651.5	−0.684321
$874$	0	0
$875$	−29976.8	−1.15817
$876$	0	0
$877$	18620.2	0.716944	0.358472	−	0.933540i	$-0.383298\pi$
$877$	18620.2	0.716944	0.358472	+	0.933540i	$0.383298\pi$
$878$	0	0
$879$	−2796.19	−0.107296
$880$	0	0
$881$	45254.9	1.73062	0.865311	−	0.501236i	$-0.167121\pi$
$881$	45254.9	1.73062	0.865311	+	0.501236i	$0.167121\pi$
$882$	0	0
$883$	10629.7	0.405117	0.202559	−	0.979270i	$-0.435074\pi$
$883$	10629.7	0.405117	0.202559	+	0.979270i	$0.435074\pi$
$884$	0	0
$885$	−4170.92	−0.158422
$886$	0	0
$887$	1256.63	0.0475687	0.0237844	−	0.999717i	$-0.492428\pi$
$887$	1256.63	0.0475687	0.0237844	+	0.999717i	$0.492428\pi$
$888$	0	0
$889$	−20118.3	−0.758995
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−29673.1	−1.11195
$894$	0	0
$895$	−30304.6	−1.13181
$896$	0	0
$897$	2711.61	0.100934
$898$	0	0
$899$	3299.21	0.122397
$900$	0	0
$901$	−4918.30	−0.181856
$902$	0	0
$903$	5775.66	0.212848
$904$	0	0
$905$	32538.7	1.19516
$906$	0	0
$907$	13155.9	0.481626	0.240813	−	0.970572i	$-0.422586\pi$
$907$	13155.9	0.481626	0.240813	+	0.970572i	$0.422586\pi$
$908$	0	0
$909$	24771.6	0.903874
$910$	0	0
$911$	−2297.90	−0.0835707	−0.0417853	−	0.999127i	$-0.513305\pi$
$911$	−2297.90	−0.0835707	−0.0417853	+	0.999127i	$0.513305\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−882.022	−0.0318675
$916$	0	0
$917$	10627.5	0.382716
$918$	0	0
$919$	−13249.6	−0.475585	−0.237793	−	0.971316i	$-0.576424\pi$
$919$	−13249.6	−0.475585	−0.237793	+	0.971316i	$0.576424\pi$
$920$	0	0
$921$	1134.46	0.0405883
$922$	0	0
$923$	21900.1	0.780985
$924$	0	0
$925$	−748.498	−0.0266059
$926$	0	0
$927$	19709.7	0.698331
$928$	0	0
$929$	7890.86	0.278677	0.139338	−	0.990245i	$-0.455502\pi$
$929$	7890.86	0.278677	0.139338	+	0.990245i	$0.455502\pi$
$930$	0	0
$931$	−5002.82	−0.176112
$932$	0	0
$933$	401.901	0.0141025
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−33392.0	−1.16421	−0.582107	−	0.813112i	$-0.697771\pi$
$937$	−33392.0	−1.16421	−0.582107	+	0.813112i	$0.697771\pi$
$938$	0	0
$939$	3337.16	0.115979
$940$	0	0
$941$	26778.6	0.927690	0.463845	−	0.885916i	$-0.346470\pi$
$941$	26778.6	0.927690	0.463845	+	0.885916i	$0.346470\pi$
$942$	0	0
$943$	2360.58	0.0815175
$944$	0	0
$945$	−6577.49	−0.226419
$946$	0	0
$947$	20360.5	0.698656	0.349328	−	0.937000i	$-0.386410\pi$
$947$	20360.5	0.698656	0.349328	+	0.937000i	$0.386410\pi$
$948$	0	0
$949$	43973.1	1.50414
$950$	0	0
$951$	6283.59	0.214258
$952$	0	0
$953$	1584.56	0.0538603	0.0269301	−	0.999637i	$-0.491427\pi$
$953$	1584.56	0.0538603	0.0269301	+	0.999637i	$0.491427\pi$
$954$	0	0
$955$	−10946.5	−0.370910
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8187.44	−0.275690
$960$	0	0
$961$	−29661.1	−0.995639
$962$	0	0
$963$	−4045.31	−0.135367
$964$	0	0
$965$	7305.80	0.243712
$966$	0	0
$967$	15443.0	0.513561	0.256780	−	0.966470i	$-0.417338\pi$
$967$	15443.0	0.513561	0.256780	+	0.966470i	$0.417338\pi$
$968$	0	0
$969$	−3802.27	−0.126054
$970$	0	0
$971$	−49914.7	−1.64968	−0.824840	−	0.565367i	$-0.808735\pi$
$971$	−49914.7	−1.64968	−0.824840	+	0.565367i	$0.808735\pi$
$972$	0	0
$973$	−14870.5	−0.489955
$974$	0	0
$975$	2089.45	0.0686319
$976$	0	0
$977$	31478.4	1.03079	0.515396	−	0.856952i	$-0.327645\pi$
$977$	31478.4	1.03079	0.515396	+	0.856952i	$0.327645\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	45991.9	1.49685
$982$	0	0
$983$	34484.1	1.11889	0.559446	−	0.828866i	$-0.311014\pi$
$983$	34484.1	1.11889	0.559446	+	0.828866i	$0.311014\pi$
$984$	0	0
$985$	−8479.37	−0.274289
$986$	0	0
$987$	−3581.45	−0.115500
$988$	0	0
$989$	21088.8	0.678044
$990$	0	0
$991$	−19116.9	−0.612785	−0.306392	−	0.951905i	$-0.599122\pi$
$991$	−19116.9	−0.612785	−0.306392	+	0.951905i	$0.599122\pi$
$992$	0	0
$993$	−6720.79	−0.214781
$994$	0	0
$995$	−45785.8	−1.45880
$996$	0	0
$997$	−28970.4	−0.920261	−0.460131	−	0.887851i	$-0.652197\pi$
$997$	−28970.4	−0.920261	−0.460131	+	0.887851i	$0.652197\pi$
$998$	0	0
$999$	−722.280	−0.0228748

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.4.a.r.1.5		10
4.3	odd	2		1936.4.a.by.1.6		10
11.7	odd	10		88.4.i.b.49.3	yes	20
11.8	odd	10		88.4.i.b.9.3	✓	20
11.10	odd	2		968.4.a.s.1.5		10
44.7	even	10		176.4.m.f.49.3		20
44.19	even	10		176.4.m.f.97.3		20
44.43	even	2		1936.4.a.bx.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.4.i.b.9.3	✓	20	11.8	odd	10
88.4.i.b.49.3	yes	20	11.7	odd	10
176.4.m.f.49.3		20	44.7	even	10
176.4.m.f.97.3		20	44.19	even	10
968.4.a.r.1.5		10	1.1	even	1	trivial
968.4.a.s.1.5		10	11.10	odd	2
1936.4.a.bx.1.6		10	44.43	even	2
1936.4.a.by.1.6		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)

\(f(q)\)	\(=\)	\(q+0.662787 q^{3} -9.39213 q^{5} -19.7277 q^{7} -26.5607 q^{9} +O(q^{10})\)	\(q+0.662787 q^{3} -9.39213 q^{5} -19.7277 q^{7} -26.5607 q^{9} -85.6945 q^{13} -6.22498 q^{15} +52.9559 q^{17} -108.331 q^{19} -13.0752 q^{21} -47.7420 q^{23} -36.7879 q^{25} -35.4994 q^{27} +289.458 q^{29} +11.3979 q^{31} +185.285 q^{35} +20.3463 q^{37} -56.7972 q^{39} -49.4445 q^{41} -441.725 q^{43} +249.462 q^{45} +273.911 q^{47} +46.1807 q^{49} +35.0985 q^{51} -92.8754 q^{53} -71.8006 q^{57} +670.029 q^{59} +141.691 q^{61} +523.981 q^{63} +804.854 q^{65} +791.545 q^{67} -31.6428 q^{69} -255.560 q^{71} -513.138 q^{73} -24.3826 q^{75} -1170.83 q^{79} +693.611 q^{81} +656.977 q^{83} -497.369 q^{85} +191.849 q^{87} -609.733 q^{89} +1690.55 q^{91} +7.55438 q^{93} +1017.46 q^{95} +664.571 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 9 q^{3} + 13 q^{5} - 3 q^{7} + 141 q^{9}+O(q^{10}) \) 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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