LMFDB - Embedded newform 968.6.a.p.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,6,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, 6, names="a")

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

chi := DirichletCharacter("968.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$155.251537579$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 2635 x^{14} + 10644 x^{13} + 2721739 x^{12} - 11107836 x^{11} + \cdots + 53\!\cdots\!20 $ x^16 - 4x^15 - 2635x^14 + 10644x^13 + 2721739x^12 - 11107836x^11 - 1388443249x^10 + 5594180788x^9 + 362606893936x^8 - 1331049359080x^7 - 46300877961632x^6 + 122447405593024x^5 + 2801116058142464x^4 - 3524304397180928x^3 - 71736438677110016x^2 + 11226318210301440*x + 533879212527569920
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{26}\cdot 11^{8} $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-10.4080$ 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)

$f(q)$	$=$	$q-11.0260 q^{3} -30.6908 q^{5} -62.4872 q^{7} -121.427 q^{9} +767.115 q^{13} +338.397 q^{15} -186.822 q^{17} +2764.74 q^{19} +688.983 q^{21} -1140.13 q^{23} -2183.07 q^{25} +4018.18 q^{27} -4715.77 q^{29} -5483.97 q^{31} +1917.78 q^{35} -3010.93 q^{37} -8458.21 q^{39} -8174.20 q^{41} -4238.82 q^{43} +3726.70 q^{45} -21699.7 q^{47} -12902.4 q^{49} +2059.90 q^{51} +13600.3 q^{53} -30484.0 q^{57} -23965.5 q^{59} +47317.7 q^{61} +7587.65 q^{63} -23543.4 q^{65} -18613.7 q^{67} +12571.1 q^{69} -23483.0 q^{71} +7444.27 q^{73} +24070.6 q^{75} -8499.94 q^{79} -14797.6 q^{81} +58176.0 q^{83} +5733.72 q^{85} +51996.0 q^{87} +58147.0 q^{89} -47934.9 q^{91} +60466.2 q^{93} -84852.1 q^{95} -142469. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{3} + 81 q^{5} - 47 q^{7} + 1416 q^{9} + 859 q^{13} - 738 q^{15} + 1226 q^{17} + 616 q^{19} + 1141 q^{21} + 2258 q^{23} + 10307 q^{25} + 564 q^{27} + 1613 q^{29} + 18511 q^{31} - 23544 q^{35}+ \cdots + 171314 q^{97}+O(q^{100}) $ 16 * q + 12 * q^3 + 81 * q^5 - 47 * q^7 + 1416 * q^9 + 859 * q^13 - 738 * q^15 + 1226 * q^17 + 616 * q^19 + 1141 * q^21 + 2258 * q^23 + 10307 * q^25 + 564 * q^27 + 1613 * q^29 + 18511 * q^31 - 23544 * q^35 + 8553 * q^37 + 12020 * q^39 + 25514 * q^41 - 18931 * q^43 + 23278 * q^45 + 38781 * q^47 + 94523 * q^49 - 68584 * q^51 + 779 * q^53 + 19293 * q^57 + 50316 * q^59 + 53153 * q^61 - 54911 * q^63 + 51517 * q^65 + 10491 * q^67 - 95356 * q^69 - 91677 * q^71 + 97660 * q^73 - 3771 * q^75 - 96003 * q^79 + 120400 * q^81 + 4552 * q^83 - 129679 * q^85 + 77219 * q^87 + 6697 * q^89 + 217380 * q^91 + 132586 * q^93 - 54168 * q^95 + 171314 * 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
$3$	−11.0260	−0.707318	−0.353659	−	0.935374i	$-0.615063\pi$
$3$	−11.0260	−0.707318	−0.353659	+	0.935374i	$0.615063\pi$
$4$	0	0
$5$	−30.6908	−0.549014	−0.274507	−	0.961585i	$-0.588515\pi$
$5$	−30.6908	−0.549014	−0.274507	+	0.961585i	$0.588515\pi$
$6$	0	0
$7$	−62.4872	−0.481998	−0.240999	−	0.970525i	$-0.577475\pi$
$7$	−62.4872	−0.481998	−0.240999	+	0.970525i	$0.577475\pi$
$8$	0	0
$9$	−121.427	−0.499701
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	767.115	1.25893	0.629466	−	0.777028i	$-0.283274\pi$
$13$	767.115	1.25893	0.629466	+	0.777028i	$0.283274\pi$
$14$	0	0
$15$	338.397	0.388328
$16$	0	0
$17$	−186.822	−0.156786	−0.0783928	−	0.996923i	$-0.524979\pi$
$17$	−186.822	−0.156786	−0.0783928	+	0.996923i	$0.524979\pi$
$18$	0	0
$19$	2764.74	1.75699	0.878497	−	0.477748i	$-0.158547\pi$
$19$	2764.74	1.75699	0.878497	+	0.477748i	$0.158547\pi$
$20$	0	0
$21$	688.983	0.340926
$22$	0	0
$23$	−1140.13	−0.449402	−0.224701	−	0.974428i	$-0.572141\pi$
$23$	−1140.13	−0.449402	−0.224701	+	0.974428i	$0.572141\pi$
$24$	0	0
$25$	−2183.07	−0.698584
$26$	0	0
$27$	4018.18	1.06077
$28$	0	0
$29$	−4715.77	−1.04126	−0.520628	−	0.853784i	$-0.674302\pi$
$29$	−4715.77	−1.04126	−0.520628	+	0.853784i	$0.674302\pi$
$30$	0	0
$31$	−5483.97	−1.02492	−0.512461	−	0.858711i	$-0.671266\pi$
$31$	−5483.97	−1.02492	−0.512461	+	0.858711i	$0.671266\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1917.78	0.264624
$36$	0	0
$37$	−3010.93	−0.361573	−0.180787	−	0.983522i	$-0.557864\pi$
$37$	−3010.93	−0.361573	−0.180787	+	0.983522i	$0.557864\pi$
$38$	0	0
$39$	−8458.21	−0.890466
$40$	0	0
$41$	−8174.20	−0.759426	−0.379713	−	0.925104i	$-0.623977\pi$
$41$	−8174.20	−0.759426	−0.379713	+	0.925104i	$0.623977\pi$
$42$	0	0
$43$	−4238.82	−0.349602	−0.174801	−	0.984604i	$-0.555928\pi$
$43$	−4238.82	−0.349602	−0.174801	+	0.984604i	$0.555928\pi$
$44$	0	0
$45$	3726.70	0.274343
$46$	0	0
$47$	−21699.7	−1.43288	−0.716440	−	0.697648i	$-0.754230\pi$
$47$	−21699.7	−1.43288	−0.716440	+	0.697648i	$0.754230\pi$
$48$	0	0
$49$	−12902.4	−0.767677
$50$	0	0
$51$	2059.90	0.110897
$52$	0	0
$53$	13600.3	0.665054	0.332527	−	0.943094i	$-0.392099\pi$
$53$	13600.3	0.665054	0.332527	+	0.943094i	$0.392099\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−30484.0	−1.24275
$58$	0	0
$59$	−23965.5	−0.896307	−0.448154	−	0.893957i	$-0.647918\pi$
$59$	−23965.5	−0.896307	−0.448154	+	0.893957i	$0.647918\pi$
$60$	0	0
$61$	47317.7	1.62817	0.814084	−	0.580747i	$-0.197239\pi$
$61$	47317.7	1.62817	0.814084	+	0.580747i	$0.197239\pi$
$62$	0	0
$63$	7587.65	0.240855
$64$	0	0
$65$	−23543.4	−0.691171
$66$	0	0
$67$	−18613.7	−0.506577	−0.253288	−	0.967391i	$-0.581512\pi$
$67$	−18613.7	−0.506577	−0.253288	+	0.967391i	$0.581512\pi$
$68$	0	0
$69$	12571.1	0.317870
$70$	0	0
$71$	−23483.0	−0.552850	−0.276425	−	0.961036i	$-0.589150\pi$
$71$	−23483.0	−0.552850	−0.276425	+	0.961036i	$0.589150\pi$
$72$	0	0
$73$	7444.27	0.163499	0.0817494	−	0.996653i	$-0.473949\pi$
$73$	7444.27	0.163499	0.0817494	+	0.996653i	$0.473949\pi$
$74$	0	0
$75$	24070.6	0.494121
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8499.94	−0.153231	−0.0766157	−	0.997061i	$-0.524411\pi$
$79$	−8499.94	−0.153231	−0.0766157	+	0.997061i	$0.524411\pi$
$80$	0	0
$81$	−14797.6	−0.250598
$82$	0	0
$83$	58176.0	0.926934	0.463467	−	0.886114i	$-0.346605\pi$
$83$	58176.0	0.926934	0.463467	+	0.886114i	$0.346605\pi$
$84$	0	0
$85$	5733.72	0.0860775
$86$	0	0
$87$	51996.0	0.736499
$88$	0	0
$89$	58147.0	0.778130	0.389065	−	0.921210i	$-0.372798\pi$
$89$	58147.0	0.778130	0.389065	+	0.921210i	$0.372798\pi$
$90$	0	0
$91$	−47934.9	−0.606803
$92$	0	0
$93$	60466.2	0.724946
$94$	0	0
$95$	−84852.1	−0.964614
$96$	0	0
$97$	−142469.	−1.53741	−0.768706	−	0.639603i	$-0.779099\pi$
$97$	−142469.	−1.53741	−0.768706	+	0.639603i	$0.779099\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−95236.9	−0.928970	−0.464485	−	0.885581i	$-0.653761\pi$
$101$	−95236.9	−0.928970	−0.464485	+	0.885581i	$0.653761\pi$
$102$	0	0
$103$	67353.9	0.625561	0.312780	−	0.949825i	$-0.398740\pi$
$103$	67353.9	0.625561	0.312780	+	0.949825i	$0.398740\pi$
$104$	0	0
$105$	−21145.5	−0.187173
$106$	0	0
$107$	−23295.4	−0.196703	−0.0983514	−	0.995152i	$-0.531357\pi$
$107$	−23295.4	−0.196703	−0.0983514	+	0.995152i	$0.531357\pi$
$108$	0	0
$109$	235745.	1.90054	0.950270	−	0.311427i	$-0.100807\pi$
$109$	235745.	1.90054	0.950270	+	0.311427i	$0.100807\pi$
$110$	0	0
$111$	33198.5	0.255747
$112$	0	0
$113$	−117664.	−0.866855	−0.433427	−	0.901188i	$-0.642696\pi$
$113$	−117664.	−0.866855	−0.433427	+	0.901188i	$0.642696\pi$
$114$	0	0
$115$	34991.6	0.246728
$116$	0	0
$117$	−93148.8	−0.629090
$118$	0	0
$119$	11674.0	0.0755704
$120$	0	0
$121$	0	0
$122$	0	0
$123$	90128.7	0.537156
$124$	0	0
$125$	162909.	0.932546
$126$	0	0
$127$	167746.	0.922874	0.461437	−	0.887173i	$-0.347334\pi$
$127$	167746.	0.922874	0.461437	+	0.887173i	$0.347334\pi$
$128$	0	0
$129$	46737.2	0.247280
$130$	0	0
$131$	−153627.	−0.782146	−0.391073	−	0.920360i	$-0.627896\pi$
$131$	−153627.	−0.782146	−0.391073	+	0.920360i	$0.627896\pi$
$132$	0	0
$133$	−172761.	−0.846869
$134$	0	0
$135$	−123321.	−0.582375
$136$	0	0
$137$	−94232.1	−0.428941	−0.214470	−	0.976730i	$-0.568803\pi$
$137$	−94232.1	−0.428941	−0.214470	+	0.976730i	$0.568803\pi$
$138$	0	0
$139$	−31302.9	−0.137419	−0.0687096	−	0.997637i	$-0.521888\pi$
$139$	−31302.9	−0.137419	−0.0687096	+	0.997637i	$0.521888\pi$
$140$	0	0
$141$	239261.	1.01350
$142$	0	0
$143$	0	0
$144$	0	0
$145$	144731.	0.571664
$146$	0	0
$147$	142261.	0.542992
$148$	0	0
$149$	158925.	0.586443	0.293222	−	0.956044i	$-0.405273\pi$
$149$	158925.	0.586443	0.293222	+	0.956044i	$0.405273\pi$
$150$	0	0
$151$	−309966.	−1.10630	−0.553149	−	0.833083i	$-0.686574\pi$
$151$	−309966.	−1.10630	−0.553149	+	0.833083i	$0.686574\pi$
$152$	0	0
$153$	22685.3	0.0783459
$154$	0	0
$155$	168307.	0.562696
$156$	0	0
$157$	−153535.	−0.497115	−0.248558	−	0.968617i	$-0.579956\pi$
$157$	−153535.	−0.497115	−0.248558	+	0.968617i	$0.579956\pi$
$158$	0	0
$159$	−149956.	−0.470405
$160$	0	0
$161$	71243.6	0.216611
$162$	0	0
$163$	545142.	1.60709	0.803545	−	0.595244i	$-0.202944\pi$
$163$	545142.	1.60709	0.803545	+	0.595244i	$0.202944\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−216305.	−0.600172	−0.300086	−	0.953912i	$-0.597015\pi$
$167$	−216305.	−0.600172	−0.300086	+	0.953912i	$0.597015\pi$
$168$	0	0
$169$	217173.	0.584910
$170$	0	0
$171$	−335715.	−0.877972
$172$	0	0
$173$	−710453.	−1.80476	−0.902381	−	0.430939i	$-0.858182\pi$
$173$	−710453.	−1.80476	−0.902381	+	0.430939i	$0.858182\pi$
$174$	0	0
$175$	136414.	0.336716
$176$	0	0
$177$	264244.	0.633974
$178$	0	0
$179$	543848.	1.26866	0.634329	−	0.773063i	$-0.281276\pi$
$179$	543848.	1.26866	0.634329	+	0.773063i	$0.281276\pi$
$180$	0	0
$181$	699884.	1.58792	0.793961	−	0.607968i	$-0.208015\pi$
$181$	699884.	1.58792	0.793961	+	0.607968i	$0.208015\pi$
$182$	0	0
$183$	−521725.	−1.15163
$184$	0	0
$185$	92407.8	0.198509
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−251084.	−0.511287
$190$	0	0
$191$	−544537.	−1.08005	−0.540025	−	0.841649i	$-0.681585\pi$
$191$	−544537.	−1.08005	−0.540025	+	0.841649i	$0.681585\pi$
$192$	0	0
$193$	757661.	1.46414	0.732069	−	0.681231i	$-0.238555\pi$
$193$	757661.	1.46414	0.732069	+	0.681231i	$0.238555\pi$
$194$	0	0
$195$	259589.	0.488878
$196$	0	0
$197$	393716.	0.722798	0.361399	−	0.932411i	$-0.382299\pi$
$197$	393716.	0.722798	0.361399	+	0.932411i	$0.382299\pi$
$198$	0	0
$199$	202030.	0.361646	0.180823	−	0.983516i	$-0.442124\pi$
$199$	202030.	0.361646	0.180823	+	0.983516i	$0.442124\pi$
$200$	0	0
$201$	205234.	0.358311
$202$	0	0
$203$	294675.	0.501884
$204$	0	0
$205$	250873.	0.416935
$206$	0	0
$207$	138443.	0.224567
$208$	0	0
$209$	0	0
$210$	0	0
$211$	640321.	0.990128	0.495064	−	0.868857i	$-0.335145\pi$
$211$	640321.	0.990128	0.495064	+	0.868857i	$0.335145\pi$
$212$	0	0
$213$	258923.	0.391041
$214$	0	0
$215$	130093.	0.191936
$216$	0	0
$217$	342678.	0.494011
$218$	0	0
$219$	−82080.5	−0.115646
$220$	0	0
$221$	−143314.	−0.197382
$222$	0	0
$223$	−574919.	−0.774184	−0.387092	−	0.922041i	$-0.626520\pi$
$223$	−574919.	−0.774184	−0.387092	+	0.922041i	$0.626520\pi$
$224$	0	0
$225$	265085.	0.349083
$226$	0	0
$227$	−1.15336e6	−1.48560	−0.742799	−	0.669514i	$-0.766502\pi$
$227$	−1.15336e6	−1.48560	−0.742799	+	0.669514i	$0.766502\pi$
$228$	0	0
$229$	1.44965e6	1.82673	0.913366	−	0.407139i	$-0.133474\pi$
$229$	1.44965e6	1.82673	0.913366	+	0.407139i	$0.133474\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.00916e6	1.21778	0.608892	−	0.793253i	$-0.291614\pi$
$233$	1.00916e6	1.21778	0.608892	+	0.793253i	$0.291614\pi$
$234$	0	0
$235$	665983.	0.786671
$236$	0	0
$237$	93720.3	0.108383
$238$	0	0
$239$	46802.2	0.0529994	0.0264997	−	0.999649i	$-0.491564\pi$
$239$	46802.2	0.0529994	0.0264997	+	0.999649i	$0.491564\pi$
$240$	0	0
$241$	−247446.	−0.274434	−0.137217	−	0.990541i	$-0.543816\pi$
$241$	−247446.	−0.274434	−0.137217	+	0.990541i	$0.543816\pi$
$242$	0	0
$243$	−813259.	−0.883513
$244$	0	0
$245$	395984.	0.421466
$246$	0	0
$247$	2.12087e6	2.21194
$248$	0	0
$249$	−641449.	−0.655637
$250$	0	0
$251$	1.56390e6	1.56684	0.783421	−	0.621491i	$-0.213473\pi$
$251$	1.56390e6	1.56684	0.783421	+	0.621491i	$0.213473\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−63220.0	−0.0608842
$256$	0	0
$257$	1.25939e6	1.18939	0.594697	−	0.803950i	$-0.297272\pi$
$257$	1.25939e6	1.18939	0.594697	+	0.803950i	$0.297272\pi$
$258$	0	0
$259$	188144.	0.174278
$260$	0	0
$261$	572623.	0.520316
$262$	0	0
$263$	−167697.	−0.149498	−0.0747492	−	0.997202i	$-0.523816\pi$
$263$	−167697.	−0.149498	−0.0747492	+	0.997202i	$0.523816\pi$
$264$	0	0
$265$	−417403.	−0.365124
$266$	0	0
$267$	−641128.	−0.550385
$268$	0	0
$269$	−1.04180e6	−0.877818	−0.438909	−	0.898532i	$-0.644635\pi$
$269$	−1.04180e6	−0.877818	−0.438909	+	0.898532i	$0.644635\pi$
$270$	0	0
$271$	−967496.	−0.800250	−0.400125	−	0.916461i	$-0.631033\pi$
$271$	−967496.	−0.800250	−0.400125	+	0.916461i	$0.631033\pi$
$272$	0	0
$273$	528530.	0.429203
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−359462.	−0.281484	−0.140742	−	0.990046i	$-0.544949\pi$
$277$	−359462.	−0.281484	−0.140742	+	0.990046i	$0.544949\pi$
$278$	0	0
$279$	665904.	0.512155
$280$	0	0
$281$	2.15740e6	1.62991	0.814956	−	0.579523i	$-0.196761\pi$
$281$	2.15740e6	1.62991	0.814956	+	0.579523i	$0.196761\pi$
$282$	0	0
$283$	−2.20021e6	−1.63304	−0.816522	−	0.577314i	$-0.804101\pi$
$283$	−2.20021e6	−1.63304	−0.816522	+	0.577314i	$0.804101\pi$
$284$	0	0
$285$	935579.	0.682289
$286$	0	0
$287$	510782.	0.366042
$288$	0	0
$289$	−1.38495e6	−0.975418
$290$	0	0
$291$	1.57086e6	1.08744
$292$	0	0
$293$	1.18632e6	0.807294	0.403647	−	0.914915i	$-0.367742\pi$
$293$	1.18632e6	0.807294	0.403647	+	0.914915i	$0.367742\pi$
$294$	0	0
$295$	735521.	0.492085
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−874612.	−0.565767
$300$	0	0
$301$	264872.	0.168507
$302$	0	0
$303$	1.05008e6	0.657077
$304$	0	0
$305$	−1.45222e6	−0.893887
$306$	0	0
$307$	−1.47791e6	−0.894957	−0.447478	−	0.894295i	$-0.647678\pi$
$307$	−1.47791e6	−0.894957	−0.447478	+	0.894295i	$0.647678\pi$
$308$	0	0
$309$	−742644.	−0.442471
$310$	0	0
$311$	1.46422e6	0.858434	0.429217	−	0.903201i	$-0.358790\pi$
$311$	1.46422e6	0.858434	0.429217	+	0.903201i	$0.358790\pi$
$312$	0	0
$313$	1.45447e6	0.839158	0.419579	−	0.907719i	$-0.362178\pi$
$313$	1.45447e6	0.839158	0.419579	+	0.907719i	$0.362178\pi$
$314$	0	0
$315$	−232871.	−0.132233
$316$	0	0
$317$	749716.	0.419033	0.209517	−	0.977805i	$-0.432811\pi$
$317$	749716.	0.419033	0.209517	+	0.977805i	$0.432811\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	256855.	0.139131
$322$	0	0
$323$	−516515.	−0.275471
$324$	0	0
$325$	−1.67467e6	−0.879470
$326$	0	0
$327$	−2.59933e6	−1.34429
$328$	0	0
$329$	1.35596e6	0.690646
$330$	0	0
$331$	212396.	0.106556	0.0532778	−	0.998580i	$-0.483033\pi$
$331$	212396.	0.106556	0.0532778	+	0.998580i	$0.483033\pi$
$332$	0	0
$333$	365609.	0.180678
$334$	0	0
$335$	571269.	0.278118
$336$	0	0
$337$	−748105.	−0.358829	−0.179414	−	0.983774i	$-0.557420\pi$
$337$	−748105.	−0.358829	−0.179414	+	0.983774i	$0.557420\pi$
$338$	0	0
$339$	1.29736e6	0.613142
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.85645e6	0.852018
$344$	0	0
$345$	−385817.	−0.174515
$346$	0	0
$347$	4.17224e6	1.86014	0.930070	−	0.367381i	$-0.119746\pi$
$347$	4.17224e6	1.86014	0.930070	+	0.367381i	$0.119746\pi$
$348$	0	0
$349$	1.92470e6	0.845864	0.422932	−	0.906161i	$-0.361001\pi$
$349$	1.92470e6	0.845864	0.422932	+	0.906161i	$0.361001\pi$
$350$	0	0
$351$	3.08240e6	1.33543
$352$	0	0
$353$	2.76647e6	1.18165	0.590825	−	0.806800i	$-0.298803\pi$
$353$	2.76647e6	1.18165	0.590825	+	0.806800i	$0.298803\pi$
$354$	0	0
$355$	720711.	0.303522
$356$	0	0
$357$	−128717.	−0.0534523
$358$	0	0
$359$	−2.88561e6	−1.18168	−0.590842	−	0.806787i	$-0.701204\pi$
$359$	−2.88561e6	−1.18168	−0.590842	+	0.806787i	$0.701204\pi$
$360$	0	0
$361$	5.16769e6	2.08703
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−228471.	−0.0897631
$366$	0	0
$367$	1.20578e6	0.467309	0.233654	−	0.972320i	$-0.424932\pi$
$367$	1.20578e6	0.467309	0.233654	+	0.972320i	$0.424932\pi$
$368$	0	0
$369$	992571.	0.379486
$370$	0	0
$371$	−849841.	−0.320555
$372$	0	0
$373$	2.45110e6	0.912197	0.456098	−	0.889929i	$-0.349247\pi$
$373$	2.45110e6	0.912197	0.456098	+	0.889929i	$0.349247\pi$
$374$	0	0
$375$	−1.79624e6	−0.659607
$376$	0	0
$377$	−3.61754e6	−1.31087
$378$	0	0
$379$	−2.74431e6	−0.981374	−0.490687	−	0.871336i	$-0.663254\pi$
$379$	−2.74431e6	−0.981374	−0.490687	+	0.871336i	$0.663254\pi$
$380$	0	0
$381$	−1.84956e6	−0.652765
$382$	0	0
$383$	−1.54697e6	−0.538870	−0.269435	−	0.963019i	$-0.586837\pi$
$383$	−1.54697e6	−0.538870	−0.269435	+	0.963019i	$0.586837\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	514708.	0.174696
$388$	0	0
$389$	4.54806e6	1.52388	0.761942	−	0.647645i	$-0.224246\pi$
$389$	4.54806e6	1.52388	0.761942	+	0.647645i	$0.224246\pi$
$390$	0	0
$391$	213002.	0.0704598
$392$	0	0
$393$	1.69389e6	0.553226
$394$	0	0
$395$	260870.	0.0841262
$396$	0	0
$397$	5.44927e6	1.73525	0.867625	−	0.497219i	$-0.165645\pi$
$397$	5.44927e6	1.73525	0.867625	+	0.497219i	$0.165645\pi$
$398$	0	0
$399$	1.90486e6	0.599006
$400$	0	0
$401$	2.28068e6	0.708277	0.354139	−	0.935193i	$-0.384774\pi$
$401$	2.28068e6	0.708277	0.354139	+	0.935193i	$0.384774\pi$
$402$	0	0
$403$	−4.20684e6	−1.29031
$404$	0	0
$405$	454149.	0.137582
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.63755e6	−0.484045	−0.242023	−	0.970271i	$-0.577811\pi$
$409$	−1.63755e6	−0.484045	−0.242023	+	0.970271i	$0.577811\pi$
$410$	0	0
$411$	1.03900e6	0.303398
$412$	0	0
$413$	1.49754e6	0.432019
$414$	0	0
$415$	−1.78547e6	−0.508900
$416$	0	0
$417$	345145.	0.0971991
$418$	0	0
$419$	−2.44185e6	−0.679490	−0.339745	−	0.940518i	$-0.610341\pi$
$419$	−2.44185e6	−0.679490	−0.339745	+	0.940518i	$0.610341\pi$
$420$	0	0
$421$	−5.02281e6	−1.38115	−0.690576	−	0.723260i	$-0.742643\pi$
$421$	−5.02281e6	−1.38115	−0.690576	+	0.723260i	$0.742643\pi$
$422$	0	0
$423$	2.63494e6	0.716012
$424$	0	0
$425$	407847.	0.109528
$426$	0	0
$427$	−2.95675e6	−0.784775
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4.19628e6	−1.08811	−0.544053	−	0.839051i	$-0.683111\pi$
$431$	−4.19628e6	−1.08811	−0.544053	+	0.839051i	$0.683111\pi$
$432$	0	0
$433$	−1.94669e6	−0.498972	−0.249486	−	0.968378i	$-0.580262\pi$
$433$	−1.94669e6	−0.498972	−0.249486	+	0.968378i	$0.580262\pi$
$434$	0	0
$435$	−1.59580e6	−0.404348
$436$	0	0
$437$	−3.15217e6	−0.789597
$438$	0	0
$439$	−3.81550e6	−0.944910	−0.472455	−	0.881355i	$-0.656632\pi$
$439$	−3.81550e6	−0.944910	−0.472455	+	0.881355i	$0.656632\pi$
$440$	0	0
$441$	1.56670e6	0.383609
$442$	0	0
$443$	−6.84054e6	−1.65608	−0.828040	−	0.560669i	$-0.810544\pi$
$443$	−6.84054e6	−1.65608	−0.828040	+	0.560669i	$0.810544\pi$
$444$	0	0
$445$	−1.78458e6	−0.427204
$446$	0	0
$447$	−1.75231e6	−0.414802
$448$	0	0
$449$	1.23343e6	0.288734	0.144367	−	0.989524i	$-0.453885\pi$
$449$	1.23343e6	0.288734	0.144367	+	0.989524i	$0.453885\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.41769e6	0.782504
$454$	0	0
$455$	1.47116e6	0.333143
$456$	0	0
$457$	−2.13534e6	−0.478273	−0.239136	−	0.970986i	$-0.576864\pi$
$457$	−2.13534e6	−0.478273	−0.239136	+	0.970986i	$0.576864\pi$
$458$	0	0
$459$	−750684.	−0.166313
$460$	0	0
$461$	2.79275e6	0.612040	0.306020	−	0.952025i	$-0.401002\pi$
$461$	2.79275e6	0.612040	0.306020	+	0.952025i	$0.401002\pi$
$462$	0	0
$463$	−2.67460e6	−0.579837	−0.289918	−	0.957051i	$-0.593628\pi$
$463$	−2.67460e6	−0.579837	−0.289918	+	0.957051i	$0.593628\pi$
$464$	0	0
$465$	−1.85576e6	−0.398005
$466$	0	0
$467$	4.92341e6	1.04466	0.522328	−	0.852745i	$-0.325064\pi$
$467$	4.92341e6	1.04466	0.522328	+	0.852745i	$0.325064\pi$
$468$	0	0
$469$	1.16312e6	0.244169
$470$	0	0
$471$	1.69287e6	0.351618
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−6.03563e6	−1.22741
$476$	0	0
$477$	−1.65144e6	−0.332328
$478$	0	0
$479$	3.94596e6	0.785803	0.392901	−	0.919581i	$-0.371471\pi$
$479$	3.94596e6	0.785803	0.392901	+	0.919581i	$0.371471\pi$
$480$	0	0
$481$	−2.30973e6	−0.455196
$482$	0	0
$483$	−785532.	−0.153213
$484$	0	0
$485$	4.37248e6	0.844060
$486$	0	0
$487$	8.86385e6	1.69356	0.846778	−	0.531946i	$-0.178539\pi$
$487$	8.86385e6	1.69356	0.846778	+	0.531946i	$0.178539\pi$
$488$	0	0
$489$	−6.01073e6	−1.13672
$490$	0	0
$491$	−178845.	−0.0334791	−0.0167395	−	0.999860i	$-0.505329\pi$
$491$	−178845.	−0.0334791	−0.0167395	+	0.999860i	$0.505329\pi$
$492$	0	0
$493$	881010.	0.163254
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.46738e6	0.266473
$498$	0	0
$499$	7.61576e6	1.36918	0.684592	−	0.728927i	$-0.259980\pi$
$499$	7.61576e6	1.36918	0.684592	+	0.728927i	$0.259980\pi$
$500$	0	0
$501$	2.38498e6	0.424513
$502$	0	0
$503$	8.42363e6	1.48450	0.742249	−	0.670125i	$-0.233759\pi$
$503$	8.42363e6	1.48450	0.742249	+	0.670125i	$0.233759\pi$
$504$	0	0
$505$	2.92290e6	0.510017
$506$	0	0
$507$	−2.39455e6	−0.413718
$508$	0	0
$509$	293702.	0.0502473	0.0251237	−	0.999684i	$-0.492002\pi$
$509$	293702.	0.0502473	0.0251237	+	0.999684i	$0.492002\pi$
$510$	0	0
$511$	−465171.	−0.0788062
$512$	0	0
$513$	1.11092e7	1.86376
$514$	0	0
$515$	−2.06714e6	−0.343442
$516$	0	0
$517$	0	0
$518$	0	0
$519$	7.83345e6	1.27654
$520$	0	0
$521$	−1.13382e7	−1.82999	−0.914996	−	0.403462i	$-0.867807\pi$
$521$	−1.13382e7	−1.82999	−0.914996	+	0.403462i	$0.867807\pi$
$522$	0	0
$523$	1.11445e7	1.78158	0.890790	−	0.454416i	$-0.150152\pi$
$523$	1.11445e7	1.78158	0.890790	+	0.454416i	$0.150152\pi$
$524$	0	0
$525$	−1.50410e6	−0.238166
$526$	0	0
$527$	1.02453e6	0.160693
$528$	0	0
$529$	−5.13644e6	−0.798038
$530$	0	0
$531$	2.91007e6	0.447886
$532$	0	0
$533$	−6.27055e6	−0.956066
$534$	0	0
$535$	714954.	0.107993
$536$	0	0
$537$	−5.99647e6	−0.897345
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−6.29022e6	−0.924002	−0.462001	−	0.886880i	$-0.652868\pi$
$541$	−6.29022e6	−0.924002	−0.462001	+	0.886880i	$0.652868\pi$
$542$	0	0
$543$	−7.71692e6	−1.12317
$544$	0	0
$545$	−7.23522e6	−1.04342
$546$	0	0
$547$	−4.54009e6	−0.648778	−0.324389	−	0.945924i	$-0.605159\pi$
$547$	−4.54009e6	−0.648778	−0.324389	+	0.945924i	$0.605159\pi$
$548$	0	0
$549$	−5.74567e6	−0.813597
$550$	0	0
$551$	−1.30379e7	−1.82948
$552$	0	0
$553$	531137.	0.0738573
$554$	0	0
$555$	−1.01889e6	−0.140409
$556$	0	0
$557$	−246032.	−0.0336011	−0.0168006	−	0.999859i	$-0.505348\pi$
$557$	−246032.	−0.0336011	−0.0168006	+	0.999859i	$0.505348\pi$
$558$	0	0
$559$	−3.25166e6	−0.440125
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−332746.	−0.0442427	−0.0221213	−	0.999755i	$-0.507042\pi$
$563$	−332746.	−0.0442427	−0.0221213	+	0.999755i	$0.507042\pi$
$564$	0	0
$565$	3.61119e6	0.475915
$566$	0	0
$567$	924657.	0.120788
$568$	0	0
$569$	1.26937e7	1.64365	0.821823	−	0.569743i	$-0.192957\pi$
$569$	1.26937e7	1.64365	0.821823	+	0.569743i	$0.192957\pi$
$570$	0	0
$571$	−4.43068e6	−0.568696	−0.284348	−	0.958721i	$-0.591777\pi$
$571$	−4.43068e6	−0.568696	−0.284348	+	0.958721i	$0.591777\pi$
$572$	0	0
$573$	6.00406e6	0.763939
$574$	0	0
$575$	2.48899e6	0.313945
$576$	0	0
$577$	2.81065e6	0.351453	0.175726	−	0.984439i	$-0.443773\pi$
$577$	2.81065e6	0.351453	0.175726	+	0.984439i	$0.443773\pi$
$578$	0	0
$579$	−8.35397e6	−1.03561
$580$	0	0
$581$	−3.63525e6	−0.446781
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.85881e6	0.345379
$586$	0	0
$587$	−8.61761e6	−1.03227	−0.516133	−	0.856508i	$-0.672629\pi$
$587$	−8.61761e6	−1.03227	−0.516133	+	0.856508i	$0.672629\pi$
$588$	0	0
$589$	−1.51618e7	−1.80078
$590$	0	0
$591$	−4.34111e6	−0.511248
$592$	0	0
$593$	3.31303e6	0.386891	0.193445	−	0.981111i	$-0.438034\pi$
$593$	3.31303e6	0.386891	0.193445	+	0.981111i	$0.438034\pi$
$594$	0	0
$595$	−358284.	−0.0414892
$596$	0	0
$597$	−2.22758e6	−0.255799
$598$	0	0
$599$	8.47569e6	0.965179	0.482589	−	0.875847i	$-0.339696\pi$
$599$	8.47569e6	0.965179	0.482589	+	0.875847i	$0.339696\pi$
$600$	0	0
$601$	1.70659e7	1.92727	0.963637	−	0.267215i	$-0.0861035\pi$
$601$	1.70659e7	1.92727	0.963637	+	0.267215i	$0.0861035\pi$
$602$	0	0
$603$	2.26021e6	0.253137
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2.20372e6	0.242764	0.121382	−	0.992606i	$-0.461267\pi$
$607$	2.20372e6	0.242764	0.121382	+	0.992606i	$0.461267\pi$
$608$	0	0
$609$	−3.24909e6	−0.354991
$610$	0	0
$611$	−1.66462e7	−1.80390
$612$	0	0
$613$	1.04961e7	1.12818	0.564090	−	0.825713i	$-0.309227\pi$
$613$	1.04961e7	1.12818	0.564090	+	0.825713i	$0.309227\pi$
$614$	0	0
$615$	−2.76612e6	−0.294906
$616$	0	0
$617$	1.40583e7	1.48669	0.743345	−	0.668908i	$-0.233238\pi$
$617$	1.40583e7	1.48669	0.743345	+	0.668908i	$0.233238\pi$
$618$	0	0
$619$	1.68713e7	1.76979	0.884895	−	0.465791i	$-0.154230\pi$
$619$	1.68713e7	1.76979	0.884895	+	0.465791i	$0.154230\pi$
$620$	0	0
$621$	−4.58125e6	−0.476711
$622$	0	0
$623$	−3.63344e6	−0.375057
$624$	0	0
$625$	1.82229e6	0.186603
$626$	0	0
$627$	0	0
$628$	0	0
$629$	562508.	0.0566894
$630$	0	0
$631$	4.06595e6	0.406526	0.203263	−	0.979124i	$-0.434845\pi$
$631$	4.06595e6	0.406526	0.203263	+	0.979124i	$0.434845\pi$
$632$	0	0
$633$	−7.06017e6	−0.700335
$634$	0	0
$635$	−5.14825e6	−0.506670
$636$	0	0
$637$	−9.89760e6	−0.966454
$638$	0	0
$639$	2.85147e6	0.276259
$640$	0	0
$641$	−1.28667e7	−1.23686	−0.618431	−	0.785839i	$-0.712232\pi$
$641$	−1.28667e7	−1.23686	−0.618431	+	0.785839i	$0.712232\pi$
$642$	0	0
$643$	−2.99428e6	−0.285604	−0.142802	−	0.989751i	$-0.545611\pi$
$643$	−2.99428e6	−0.285604	−0.142802	+	0.989751i	$0.545611\pi$
$644$	0	0
$645$	−1.43440e6	−0.135760
$646$	0	0
$647$	−1.72635e7	−1.62132	−0.810658	−	0.585521i	$-0.800890\pi$
$647$	−1.72635e7	−1.62132	−0.810658	+	0.585521i	$0.800890\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−3.77836e6	−0.349423
$652$	0	0
$653$	1.53789e7	1.41138	0.705689	−	0.708522i	$-0.250638\pi$
$653$	1.53789e7	1.41138	0.705689	+	0.708522i	$0.250638\pi$
$654$	0	0
$655$	4.71492e6	0.429409
$656$	0	0
$657$	−903937.	−0.0817005
$658$	0	0
$659$	−2.01583e7	−1.80818	−0.904088	−	0.427346i	$-0.859449\pi$
$659$	−2.01583e7	−1.80818	−0.904088	+	0.427346i	$0.859449\pi$
$660$	0	0
$661$	−7.61165e6	−0.677602	−0.338801	−	0.940858i	$-0.610021\pi$
$661$	−7.61165e6	−0.677602	−0.338801	+	0.940858i	$0.610021\pi$
$662$	0	0
$663$	1.58018e6	0.139612
$664$	0	0
$665$	5.30217e6	0.464943
$666$	0	0
$667$	5.37659e6	0.467943
$668$	0	0
$669$	6.33905e6	0.547594
$670$	0	0
$671$	0	0
$672$	0	0
$673$	9.02219e6	0.767846	0.383923	−	0.923365i	$-0.374573\pi$
$673$	9.02219e6	0.767846	0.383923	+	0.923365i	$0.374573\pi$
$674$	0	0
$675$	−8.77198e6	−0.741034
$676$	0	0
$677$	−8.13999e6	−0.682578	−0.341289	−	0.939958i	$-0.610864\pi$
$677$	−8.13999e6	−0.682578	−0.341289	+	0.939958i	$0.610864\pi$
$678$	0	0
$679$	8.90246e6	0.741030
$680$	0	0
$681$	1.27170e7	1.05079
$682$	0	0
$683$	−1.63028e6	−0.133725	−0.0668623	−	0.997762i	$-0.521299\pi$
$683$	−1.63028e6	−0.133725	−0.0668623	+	0.997762i	$0.521299\pi$
$684$	0	0
$685$	2.89206e6	0.235494
$686$	0	0
$687$	−1.59839e7	−1.29208
$688$	0	0
$689$	1.04330e7	0.837258
$690$	0	0
$691$	−8.36827e6	−0.666715	−0.333358	−	0.942800i	$-0.608182\pi$
$691$	−8.36827e6	−0.666715	−0.333358	+	0.942800i	$0.608182\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	960711.	0.0754450
$696$	0	0
$697$	1.52712e6	0.119067
$698$	0	0
$699$	−1.11270e7	−0.861361
$700$	0	0
$701$	1.87217e7	1.43896	0.719482	−	0.694511i	$-0.244379\pi$
$701$	1.87217e7	1.43896	0.719482	+	0.694511i	$0.244379\pi$
$702$	0	0
$703$	−8.32443e6	−0.635282
$704$	0	0
$705$	−7.34312e6	−0.556427
$706$	0	0
$707$	5.95108e6	0.447762
$708$	0	0
$709$	−1.40544e7	−1.05002	−0.525011	−	0.851096i	$-0.675939\pi$
$709$	−1.40544e7	−1.05002	−0.525011	+	0.851096i	$0.675939\pi$
$710$	0	0
$711$	1.03212e6	0.0765699
$712$	0	0
$713$	6.25245e6	0.460602
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−516041.	−0.0374875
$718$	0	0
$719$	−5.24149e6	−0.378122	−0.189061	−	0.981965i	$-0.560544\pi$
$719$	−5.24149e6	−0.378122	−0.189061	+	0.981965i	$0.560544\pi$
$720$	0	0
$721$	−4.20875e6	−0.301519
$722$	0	0
$723$	2.72834e6	0.194112
$724$	0	0
$725$	1.02949e7	0.727404
$726$	0	0
$727$	4.25207e6	0.298376	0.149188	−	0.988809i	$-0.452334\pi$
$727$	4.25207e6	0.298376	0.149188	+	0.988809i	$0.452334\pi$
$728$	0	0
$729$	1.25628e7	0.875523
$730$	0	0
$731$	791905.	0.0548125
$732$	0	0
$733$	1.84911e7	1.27117	0.635584	−	0.772032i	$-0.280759\pi$
$733$	1.84911e7	1.27117	0.635584	+	0.772032i	$0.280759\pi$
$734$	0	0
$735$	−4.36612e6	−0.298110
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1.31783e7	0.887663	0.443831	−	0.896110i	$-0.353619\pi$
$739$	1.31783e7	0.887663	0.443831	+	0.896110i	$0.353619\pi$
$740$	0	0
$741$	−2.33848e7	−1.56454
$742$	0	0
$743$	−1.41289e7	−0.938937	−0.469468	−	0.882949i	$-0.655554\pi$
$743$	−1.41289e7	−0.938937	−0.469468	+	0.882949i	$0.655554\pi$
$744$	0	0
$745$	−4.87753e6	−0.321966
$746$	0	0
$747$	−7.06416e6	−0.463190
$748$	0	0
$749$	1.45566e6	0.0948104
$750$	0	0
$751$	1.63912e7	1.06050	0.530251	−	0.847841i	$-0.322098\pi$
$751$	1.63912e7	1.06050	0.530251	+	0.847841i	$0.322098\pi$
$752$	0	0
$753$	−1.72436e7	−1.10826
$754$	0	0
$755$	9.51311e6	0.607372
$756$	0	0
$757$	−2.51059e7	−1.59234	−0.796172	−	0.605071i	$-0.793145\pi$
$757$	−2.51059e7	−1.59234	−0.796172	+	0.605071i	$0.793145\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.92504e6	−0.245687	−0.122843	−	0.992426i	$-0.539201\pi$
$761$	−3.92504e6	−0.245687	−0.122843	+	0.992426i	$0.539201\pi$
$762$	0	0
$763$	−1.47311e7	−0.916057
$764$	0	0
$765$	−696231.	−0.0430130
$766$	0	0
$767$	−1.83843e7	−1.12839
$768$	0	0
$769$	3.83427e6	0.233812	0.116906	−	0.993143i	$-0.462702\pi$
$769$	3.83427e6	0.233812	0.116906	+	0.993143i	$0.462702\pi$
$770$	0	0
$771$	−1.38860e7	−0.841280
$772$	0	0
$773$	1.15946e7	0.697922	0.348961	−	0.937137i	$-0.386535\pi$
$773$	1.15946e7	0.697922	0.348961	+	0.937137i	$0.386535\pi$
$774$	0	0
$775$	1.19719e7	0.715994
$776$	0	0
$777$	−2.07448e6	−0.123270
$778$	0	0
$779$	−2.25995e7	−1.33431
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1.89488e7	−1.10453
$784$	0	0
$785$	4.71210e6	0.272923
$786$	0	0
$787$	−1.80618e7	−1.03950	−0.519749	−	0.854319i	$-0.673975\pi$
$787$	−1.80618e7	−1.03950	−0.519749	+	0.854319i	$0.673975\pi$
$788$	0	0
$789$	1.84903e6	0.105743
$790$	0	0
$791$	7.35247e6	0.417823
$792$	0	0
$793$	3.62982e7	2.04975
$794$	0	0
$795$	4.60228e6	0.258259
$796$	0	0
$797$	−1.74588e7	−0.973572	−0.486786	−	0.873521i	$-0.661831\pi$
$797$	−1.74588e7	−0.973572	−0.486786	+	0.873521i	$0.661831\pi$
$798$	0	0
$799$	4.05399e6	0.224655
$800$	0	0
$801$	−7.06063e6	−0.388832
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2.18652e6	−0.118923
$806$	0	0
$807$	1.14869e7	0.620897
$808$	0	0
$809$	3.45131e7	1.85401	0.927007	−	0.375045i	$-0.122373\pi$
$809$	3.45131e7	1.85401	0.927007	+	0.375045i	$0.122373\pi$
$810$	0	0
$811$	−1.15510e7	−0.616690	−0.308345	−	0.951275i	$-0.599775\pi$
$811$	−1.15510e7	−0.616690	−0.308345	+	0.951275i	$0.599775\pi$
$812$	0	0
$813$	1.06676e7	0.566032
$814$	0	0
$815$	−1.67308e7	−0.882315
$816$	0	0
$817$	−1.17192e7	−0.614248
$818$	0	0
$819$	5.82060e6	0.303220
$820$	0	0
$821$	2.89284e7	1.49784	0.748921	−	0.662660i	$-0.230572\pi$
$821$	2.89284e7	1.49784	0.748921	+	0.662660i	$0.230572\pi$
$822$	0	0
$823$	−2.19999e7	−1.13220	−0.566098	−	0.824338i	$-0.691547\pi$
$823$	−2.19999e7	−1.13220	−0.566098	+	0.824338i	$0.691547\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−6.72489e6	−0.341917	−0.170959	−	0.985278i	$-0.554686\pi$
$827$	−6.72489e6	−0.341917	−0.170959	+	0.985278i	$0.554686\pi$
$828$	0	0
$829$	−1.41455e7	−0.714878	−0.357439	−	0.933937i	$-0.616350\pi$
$829$	−1.41455e7	−0.714878	−0.357439	+	0.933937i	$0.616350\pi$
$830$	0	0
$831$	3.96343e6	0.199099
$832$	0	0
$833$	2.41045e6	0.120361
$834$	0	0
$835$	6.63858e6	0.329503
$836$	0	0
$837$	−2.20356e7	−1.08720
$838$	0	0
$839$	−1.85571e7	−0.910133	−0.455066	−	0.890457i	$-0.650385\pi$
$839$	−1.85571e7	−0.910133	−0.455066	+	0.890457i	$0.650385\pi$
$840$	0	0
$841$	1.72731e6	0.0842134
$842$	0	0
$843$	−2.37875e7	−1.15287
$844$	0	0
$845$	−6.66522e6	−0.321124
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.42595e7	1.15508
$850$	0	0
$851$	3.43285e6	0.162492
$852$	0	0
$853$	−1.33810e7	−0.629672	−0.314836	−	0.949146i	$-0.601950\pi$
$853$	−1.33810e7	−0.629672	−0.314836	+	0.949146i	$0.601950\pi$
$854$	0	0
$855$	1.03034e7	0.482019
$856$	0	0
$857$	−2.96898e7	−1.38088	−0.690439	−	0.723391i	$-0.742582\pi$
$857$	−2.96898e7	−1.38088	−0.690439	+	0.723391i	$0.742582\pi$
$858$	0	0
$859$	1.96436e6	0.0908318	0.0454159	−	0.998968i	$-0.485539\pi$
$859$	1.96436e6	0.0908318	0.0454159	+	0.998968i	$0.485539\pi$
$860$	0	0
$861$	−5.63189e6	−0.258908
$862$	0	0
$863$	−2.55237e7	−1.16658	−0.583292	−	0.812263i	$-0.698236\pi$
$863$	−2.55237e7	−1.16658	−0.583292	+	0.812263i	$0.698236\pi$
$864$	0	0
$865$	2.18044e7	0.990839
$866$	0	0
$867$	1.52705e7	0.689931
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−1.42788e7	−0.637746
$872$	0	0
$873$	1.72996e7	0.768246
$874$	0	0
$875$	−1.01797e7	−0.449486
$876$	0	0
$877$	3.35461e7	1.47280	0.736399	−	0.676548i	$-0.236525\pi$
$877$	3.35461e7	1.47280	0.736399	+	0.676548i	$0.236525\pi$
$878$	0	0
$879$	−1.30803e7	−0.571013
$880$	0	0
$881$	−7.70609e6	−0.334499	−0.167249	−	0.985915i	$-0.553489\pi$
$881$	−7.70609e6	−0.334499	−0.167249	+	0.985915i	$0.553489\pi$
$882$	0	0
$883$	−3.80156e7	−1.64082	−0.820409	−	0.571777i	$-0.806254\pi$
$883$	−3.80156e7	−1.64082	−0.820409	+	0.571777i	$0.806254\pi$
$884$	0	0
$885$	−8.10986e6	−0.348061
$886$	0	0
$887$	−2.60108e7	−1.11006	−0.555028	−	0.831832i	$-0.687292\pi$
$887$	−2.60108e7	−1.11006	−0.555028	+	0.831832i	$0.687292\pi$
$888$	0	0
$889$	−1.04820e7	−0.444824
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.99942e7	−2.51756
$894$	0	0
$895$	−1.66911e7	−0.696511
$896$	0	0
$897$	9.64348e6	0.400177
$898$	0	0
$899$	2.58611e7	1.06721
$900$	0	0
$901$	−2.54083e6	−0.104271
$902$	0	0
$903$	−2.92047e6	−0.119188
$904$	0	0
$905$	−2.14800e7	−0.871792
$906$	0	0
$907$	1.61230e7	0.650771	0.325385	−	0.945582i	$-0.394506\pi$
$907$	1.61230e7	0.650771	0.325385	+	0.945582i	$0.394506\pi$
$908$	0	0
$909$	1.15644e7	0.464207
$910$	0	0
$911$	5.69067e6	0.227179	0.113589	−	0.993528i	$-0.463765\pi$
$911$	5.69067e6	0.227179	0.113589	+	0.993528i	$0.463765\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	1.60122e7	0.632263
$916$	0	0
$917$	9.59968e6	0.376993
$918$	0	0
$919$	5.98582e6	0.233795	0.116897	−	0.993144i	$-0.462705\pi$
$919$	5.98582e6	0.233795	0.116897	+	0.993144i	$0.462705\pi$
$920$	0	0
$921$	1.62954e7	0.633019
$922$	0	0
$923$	−1.80141e7	−0.696000
$924$	0	0
$925$	6.57308e6	0.252589
$926$	0	0
$927$	−8.17860e6	−0.312593
$928$	0	0
$929$	−4.08661e6	−0.155354	−0.0776772	−	0.996979i	$-0.524750\pi$
$929$	−4.08661e6	−0.155354	−0.0776772	+	0.996979i	$0.524750\pi$
$930$	0	0
$931$	−3.56717e7	−1.34880
$932$	0	0
$933$	−1.61445e7	−0.607186
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.29957e7	0.855654	0.427827	−	0.903861i	$-0.359279\pi$
$937$	2.29957e7	0.855654	0.427827	+	0.903861i	$0.359279\pi$
$938$	0	0
$939$	−1.60370e7	−0.593552
$940$	0	0
$941$	1.35157e6	0.0497583	0.0248791	−	0.999690i	$-0.492080\pi$
$941$	1.35157e6	0.0497583	0.0248791	+	0.999690i	$0.492080\pi$
$942$	0	0
$943$	9.31966e6	0.341288
$944$	0	0
$945$	7.70598e6	0.280704
$946$	0	0
$947$	4.10415e7	1.48713	0.743564	−	0.668664i	$-0.233134\pi$
$947$	4.10415e7	1.48713	0.743564	+	0.668664i	$0.233134\pi$
$948$	0	0
$949$	5.71061e6	0.205834
$950$	0	0
$951$	−8.26637e6	−0.296390
$952$	0	0
$953$	−350815.	−0.0125126	−0.00625628	−	0.999980i	$-0.501991\pi$
$953$	−350815.	−0.0125126	−0.00625628	+	0.999980i	$0.501991\pi$
$954$	0	0
$955$	1.67123e7	0.592962
$956$	0	0
$957$	0	0
$958$	0	0
$959$	5.88829e6	0.206749
$960$	0	0
$961$	1.44477e6	0.0504651
$962$	0	0
$963$	2.82870e6	0.0982925
$964$	0	0
$965$	−2.32532e7	−0.803832
$966$	0	0
$967$	1.88773e7	0.649192	0.324596	−	0.945853i	$-0.394772\pi$
$967$	1.88773e7	0.649192	0.324596	+	0.945853i	$0.394772\pi$
$968$	0	0
$969$	5.69509e6	0.194846
$970$	0	0
$971$	4.91221e7	1.67197	0.835985	−	0.548752i	$-0.184897\pi$
$971$	4.91221e7	1.67197	0.835985	+	0.548752i	$0.184897\pi$
$972$	0	0
$973$	1.95603e6	0.0662358
$974$	0	0
$975$	1.84649e7	0.622065
$976$	0	0
$977$	−1.85542e7	−0.621877	−0.310939	−	0.950430i	$-0.600643\pi$
$977$	−1.85542e7	−0.621877	−0.310939	+	0.950430i	$0.600643\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−2.86259e7	−0.949702
$982$	0	0
$983$	−7.81995e6	−0.258119	−0.129060	−	0.991637i	$-0.541196\pi$
$983$	−7.81995e6	−0.258119	−0.129060	+	0.991637i	$0.541196\pi$
$984$	0	0
$985$	−1.20835e7	−0.396826
$986$	0	0
$987$	−1.49508e7	−0.488507
$988$	0	0
$989$	4.83281e6	0.157112
$990$	0	0
$991$	−9.04999e6	−0.292728	−0.146364	−	0.989231i	$-0.546757\pi$
$991$	−9.04999e6	−0.292728	−0.146364	+	0.989231i	$0.546757\pi$
$992$	0	0
$993$	−2.34188e6	−0.0753688
$994$	0	0
$995$	−6.20047e6	−0.198549
$996$	0	0
$997$	−4.76118e7	−1.51697	−0.758484	−	0.651692i	$-0.774060\pi$
$997$	−4.76118e7	−1.51697	−0.758484	+	0.651692i	$0.774060\pi$
$998$	0	0
$999$	−1.20984e7	−0.383544

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.6.a.p.1.5		16
11.2	odd	10		88.6.i.b.81.3	yes	32
11.6	odd	10		88.6.i.b.25.3	✓	32
11.10	odd	2		968.6.a.q.1.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.6.i.b.25.3	✓	32	11.6	odd	10
88.6.i.b.81.3	yes	32	11.2	odd	10
968.6.a.p.1.5		16	1.1	even	1	trivial
968.6.a.q.1.5		16	11.10	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)

\(f(q)\)	\(=\)	\(q-11.0260 q^{3} -30.6908 q^{5} -62.4872 q^{7} -121.427 q^{9} +767.115 q^{13} +338.397 q^{15} -186.822 q^{17} +2764.74 q^{19} +688.983 q^{21} -1140.13 q^{23} -2183.07 q^{25} +4018.18 q^{27} -4715.77 q^{29} -5483.97 q^{31} +1917.78 q^{35} -3010.93 q^{37} -8458.21 q^{39} -8174.20 q^{41} -4238.82 q^{43} +3726.70 q^{45} -21699.7 q^{47} -12902.4 q^{49} +2059.90 q^{51} +13600.3 q^{53} -30484.0 q^{57} -23965.5 q^{59} +47317.7 q^{61} +7587.65 q^{63} -23543.4 q^{65} -18613.7 q^{67} +12571.1 q^{69} -23483.0 q^{71} +7444.27 q^{73} +24070.6 q^{75} -8499.94 q^{79} -14797.6 q^{81} +58176.0 q^{83} +5733.72 q^{85} +51996.0 q^{87} +58147.0 q^{89} -47934.9 q^{91} +60466.2 q^{93} -84852.1 q^{95} -142469. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{3} + 81 q^{5} - 47 q^{7} + 1416 q^{9} + 859 q^{13} - 738 q^{15} + 1226 q^{17} + 616 q^{19} + 1141 q^{21} + 2258 q^{23} + 10307 q^{25} + 564 q^{27} + 1613 q^{29} + 18511 q^{31} - 23544 q^{35}+ \cdots + 171314 q^{97}+O(q^{100}) \) 16 * q + 12 * q^3 + 81 * q^5 - 47 * q^7 + 1416 * q^9 + 859 * q^13 - 738 * q^15 + 1226 * q^17 + 616 * q^19 + 1141 * q^21 + 2258 * q^23 + 10307 * q^25 + 564 * q^27 + 1613 * q^29 + 18511 * q^31 - 23544 * q^35 + 8553 * q^37 + 12020 * q^39 + 25514 * q^41 - 18931 * q^43 + 23278 * q^45 + 38781 * q^47 + 94523 * q^49 - 68584 * q^51 + 779 * q^53 + 19293 * q^57 + 50316 * q^59 + 53153 * q^61 - 54911 * q^63 + 51517 * q^65 + 10491 * q^67 - 95356 * q^69 - 91677 * q^71 + 97660 * q^73 - 3771 * q^75 - 96003 * q^79 + 120400 * q^81 + 4552 * q^83 - 129679 * q^85 + 77219 * q^87 + 6697 * q^89 + 217380 * q^91 + 132586 * q^93 - 54168 * q^95 + 171314 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more