LMFDB - Embedded newform 3969.2.a.bh.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 22 x^{10} + 92 x^{9} + 125 x^{8} - 620 x^{7} - 94 x^{6} + 1280 x^{5} - 234 x^{4} + \cdots - 7 $ x^12 - 4x^11 - 22x^10 + 92x^9 + 125x^8 - 620x^7 - 94x^6 + 1280x^5 - 234x^4 - 736x^3 + 96x^2 + 60*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 441)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$0.114341$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.29987 q^{2} -0.310333 q^{4} +3.52584 q^{5} -3.00314 q^{8} +O(q^{10})$	$q+1.29987 q^{2} -0.310333 q^{4} +3.52584 q^{5} -3.00314 q^{8} +4.58314 q^{10} +1.17853 q^{11} -3.22061 q^{13} -3.28303 q^{16} -4.90317 q^{17} -6.86637 q^{19} -1.09418 q^{20} +1.53194 q^{22} -4.29987 q^{23} +7.43156 q^{25} -4.18637 q^{26} -2.72281 q^{29} -1.92080 q^{31} +1.73876 q^{32} -6.37350 q^{34} -9.76457 q^{37} -8.92540 q^{38} -10.5886 q^{40} +6.65345 q^{41} -9.66881 q^{43} -0.365738 q^{44} -5.58928 q^{46} +0.633218 q^{47} +9.66008 q^{50} +0.999459 q^{52} -2.22756 q^{53} +4.15533 q^{55} -3.53930 q^{58} +8.21304 q^{59} -9.65916 q^{61} -2.49680 q^{62} +8.82622 q^{64} -11.3553 q^{65} +5.33301 q^{67} +1.52161 q^{68} -3.27719 q^{71} +1.03807 q^{73} -12.6927 q^{74} +2.13086 q^{76} +1.00408 q^{79} -11.5754 q^{80} +8.64864 q^{82} +7.31195 q^{83} -17.2878 q^{85} -12.5682 q^{86} -3.53930 q^{88} +12.0429 q^{89} +1.33439 q^{92} +0.823103 q^{94} -24.2097 q^{95} +10.9291 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} + 12 q^{4} - 12 q^{8}+O(q^{10}) $ 12 * q - 4 * q^2 + 12 * q^4 - 12 * q^8	$ 12 q - 4 q^{2} + 12 q^{4} - 12 q^{8} - 20 q^{11} + 12 q^{16} - 32 q^{23} + 12 q^{25} - 16 q^{29} - 48 q^{32} + 12 q^{37} - 56 q^{44} - 24 q^{46} + 4 q^{50} - 32 q^{53} + 48 q^{64} - 60 q^{65} + 12 q^{67} - 56 q^{71} - 68 q^{74} - 12 q^{79} - 12 q^{85} - 76 q^{86} - 16 q^{92} - 64 q^{95}+O(q^{100}) $ 12 * q - 4 * q^2 + 12 * q^4 - 12 * q^8 - 20 * q^11 + 12 * q^16 - 32 * q^23 + 12 * q^25 - 16 * q^29 - 48 * q^32 + 12 * q^37 - 56 * q^44 - 24 * q^46 + 4 * q^50 - 32 * q^53 + 48 * q^64 - 60 * q^65 + 12 * q^67 - 56 * q^71 - 68 * q^74 - 12 * q^79 - 12 * q^85 - 76 * q^86 - 16 * q^92 - 64 * q^95

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 $

$2$	1.29987	0.919148	0.459574	−	0.888139i	$-0.348002\pi$
$2$	1.29987	0.919148	0.459574	+	0.888139i	$0.348002\pi$
$3$	0	0
$3$	0	0
$4$	−0.310333	−0.155166
$5$	3.52584	1.57680	0.788402	−	0.615160i	$-0.210909\pi$
$5$	3.52584	1.57680	0.788402	+	0.615160i	$0.210909\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.00314	−1.06177
$9$	0	0
$10$	4.58314	1.44932
$11$	1.17853	0.355342	0.177671	−	0.984090i	$-0.443144\pi$
$11$	1.17853	0.355342	0.177671	+	0.984090i	$0.443144\pi$
$12$	0	0
$13$	−3.22061	−0.893235	−0.446618	−	0.894725i	$-0.647372\pi$
$13$	−3.22061	−0.893235	−0.446618	+	0.894725i	$0.647372\pi$
$14$	0	0
$15$	0	0
$16$	−3.28303	−0.820757
$17$	−4.90317	−1.18919	−0.594597	−	0.804024i	$-0.702688\pi$
$17$	−4.90317	−1.18919	−0.594597	+	0.804024i	$0.702688\pi$
$18$	0	0
$19$	−6.86637	−1.57525	−0.787627	−	0.616153i	$-0.788690\pi$
$19$	−6.86637	−1.57525	−0.787627	+	0.616153i	$0.788690\pi$
$20$	−1.09418	−0.244667
$21$	0	0
$22$	1.53194	0.326612
$23$	−4.29987	−0.896585	−0.448293	−	0.893887i	$-0.647968\pi$
$23$	−4.29987	−0.896585	−0.448293	+	0.893887i	$0.647968\pi$
$24$	0	0
$25$	7.43156	1.48631
$26$	−4.18637	−0.821016
$27$	0	0
$28$	0	0
$29$	−2.72281	−0.505613	−0.252806	−	0.967517i	$-0.581354\pi$
$29$	−2.72281	−0.505613	−0.252806	+	0.967517i	$0.581354\pi$
$30$	0	0
$31$	−1.92080	−0.344986	−0.172493	−	0.985011i	$-0.555182\pi$
$31$	−1.92080	−0.344986	−0.172493	+	0.985011i	$0.555182\pi$
$32$	1.73876	0.307372
$33$	0	0
$34$	−6.37350	−1.09305
$35$	0	0
$36$	0	0
$37$	−9.76457	−1.60529	−0.802643	−	0.596460i	$-0.796573\pi$
$37$	−9.76457	−1.60529	−0.802643	+	0.596460i	$0.796573\pi$
$38$	−8.92540	−1.44789
$39$	0	0
$40$	−10.5886	−1.67420
$41$	6.65345	1.03909	0.519547	−	0.854442i	$-0.326101\pi$
$41$	6.65345	1.03909	0.519547	+	0.854442i	$0.326101\pi$
$42$	0	0
$43$	−9.66881	−1.47448	−0.737240	−	0.675631i	$-0.763871\pi$
$43$	−9.66881	−1.47448	−0.737240	+	0.675631i	$0.763871\pi$
$44$	−0.365738	−0.0551370
$45$	0	0
$46$	−5.58928	−0.824095
$47$	0.633218	0.0923644	0.0461822	−	0.998933i	$-0.485295\pi$
$47$	0.633218	0.0923644	0.0461822	+	0.998933i	$0.485295\pi$
$48$	0	0
$49$	0	0
$50$	9.66008	1.36614
$51$	0	0
$52$	0.999459	0.138600
$53$	−2.22756	−0.305978	−0.152989	−	0.988228i	$-0.548890\pi$
$53$	−2.22756	−0.305978	−0.152989	+	0.988228i	$0.548890\pi$
$54$	0	0
$55$	4.15533	0.560304
$56$	0	0
$57$	0	0
$58$	−3.53930	−0.464733
$59$	8.21304	1.06925	0.534623	−	0.845091i	$-0.320454\pi$
$59$	8.21304	1.06925	0.534623	+	0.845091i	$0.320454\pi$
$60$	0	0
$61$	−9.65916	−1.23673	−0.618364	−	0.785892i	$-0.712204\pi$
$61$	−9.65916	−1.23673	−0.618364	+	0.785892i	$0.712204\pi$
$62$	−2.49680	−0.317093
$63$	0	0
$64$	8.82622	1.10328
$65$	−11.3553	−1.40846
$66$	0	0
$67$	5.33301	0.651531	0.325766	−	0.945451i	$-0.394378\pi$
$67$	5.33301	0.651531	0.325766	+	0.945451i	$0.394378\pi$
$68$	1.52161	0.184523
$69$	0	0
$70$	0	0
$71$	−3.27719	−0.388931	−0.194466	−	0.980909i	$-0.562297\pi$
$71$	−3.27719	−0.388931	−0.194466	+	0.980909i	$0.562297\pi$
$72$	0	0
$73$	1.03807	0.121497	0.0607486	−	0.998153i	$-0.480651\pi$
$73$	1.03807	0.121497	0.0607486	+	0.998153i	$0.480651\pi$
$74$	−12.6927	−1.47550
$75$	0	0
$76$	2.13086	0.244426
$77$	0	0
$78$	0	0
$79$	1.00408	0.112968	0.0564838	−	0.998404i	$-0.482011\pi$
$79$	1.00408	0.112968	0.0564838	+	0.998404i	$0.482011\pi$
$80$	−11.5754	−1.29417
$81$	0	0
$82$	8.64864	0.955082
$83$	7.31195	0.802591	0.401296	−	0.915949i	$-0.368560\pi$
$83$	7.31195	0.802591	0.401296	+	0.915949i	$0.368560\pi$
$84$	0	0
$85$	−17.2878	−1.87513
$86$	−12.5682	−1.35527
$87$	0	0
$88$	−3.53930	−0.377291
$89$	12.0429	1.27654	0.638271	−	0.769812i	$-0.279650\pi$
$89$	12.0429	1.27654	0.638271	+	0.769812i	$0.279650\pi$
$90$	0	0
$91$	0	0
$92$	1.33439	0.139120
$93$	0	0
$94$	0.823103	0.0848966
$95$	−24.2097	−2.48387
$96$	0	0
$97$	10.9291	1.10968	0.554840	−	0.831957i	$-0.312779\pi$
$97$	10.9291	1.10968	0.554840	+	0.831957i	$0.312779\pi$
$98$	0	0
$99$	0	0
$100$	−2.30626	−0.230626
$101$	−1.59509	−0.158718	−0.0793588	−	0.996846i	$-0.525287\pi$
$101$	−1.59509	−0.158718	−0.0793588	+	0.996846i	$0.525287\pi$
$102$	0	0
$103$	2.33556	0.230129	0.115065	−	0.993358i	$-0.463292\pi$
$103$	2.33556	0.230129	0.115065	+	0.993358i	$0.463292\pi$
$104$	9.67192	0.948410
$105$	0	0
$106$	−2.89554	−0.281240
$107$	−2.22362	−0.214966	−0.107483	−	0.994207i	$-0.534279\pi$
$107$	−2.22362	−0.214966	−0.107483	+	0.994207i	$0.534279\pi$
$108$	0	0
$109$	−0.919564	−0.0880782	−0.0440391	−	0.999030i	$-0.514023\pi$
$109$	−0.919564	−0.0880782	−0.0440391	+	0.999030i	$0.514023\pi$
$110$	5.40139	0.515003
$111$	0	0
$112$	0	0
$113$	−2.38655	−0.224508	−0.112254	−	0.993680i	$-0.535807\pi$
$113$	−2.38655	−0.224508	−0.112254	+	0.993680i	$0.535807\pi$
$114$	0	0
$115$	−15.1607	−1.41374
$116$	0.844976	0.0784540
$117$	0	0
$118$	10.6759	0.982796
$119$	0	0
$120$	0	0
$121$	−9.61106	−0.873732
$122$	−12.5557	−1.13674
$123$	0	0
$124$	0.596087	0.0535302
$125$	8.57330	0.766819
$126$	0	0
$127$	−3.04170	−0.269907	−0.134954	−	0.990852i	$-0.543089\pi$
$127$	−3.04170	−0.269907	−0.134954	+	0.990852i	$0.543089\pi$
$128$	7.99544	0.706704
$129$	0	0
$130$	−14.7605	−1.29458
$131$	−3.26176	−0.284981	−0.142490	−	0.989796i	$-0.545511\pi$
$131$	−3.26176	−0.284981	−0.142490	+	0.989796i	$0.545511\pi$
$132$	0	0
$133$	0	0
$134$	6.93223	0.598854
$135$	0	0
$136$	14.7249	1.26265
$137$	20.9338	1.78849	0.894246	−	0.447575i	$-0.147712\pi$
$137$	20.9338	1.78849	0.894246	+	0.447575i	$0.147712\pi$
$138$	0	0
$139$	16.6239	1.41002	0.705010	−	0.709197i	$-0.250942\pi$
$139$	16.6239	1.41002	0.705010	+	0.709197i	$0.250942\pi$
$140$	0	0
$141$	0	0
$142$	−4.25993	−0.357486
$143$	−3.79559	−0.317404
$144$	0	0
$145$	−9.60019	−0.797252
$146$	1.34936	0.111674
$147$	0	0
$148$	3.03026	0.249086
$149$	1.12844	0.0924456	0.0462228	−	0.998931i	$-0.485282\pi$
$149$	1.12844	0.0924456	0.0462228	+	0.998931i	$0.485282\pi$
$150$	0	0
$151$	−19.6295	−1.59743	−0.798714	−	0.601711i	$-0.794486\pi$
$151$	−19.6295	−1.59743	−0.798714	+	0.601711i	$0.794486\pi$
$152$	20.6206	1.67256
$153$	0	0
$154$	0	0
$155$	−6.77244	−0.543976
$156$	0	0
$157$	9.33237	0.744804	0.372402	−	0.928071i	$-0.378534\pi$
$157$	9.33237	0.744804	0.372402	+	0.928071i	$0.378534\pi$
$158$	1.30517	0.103834
$159$	0	0
$160$	6.13058	0.484665
$161$	0	0
$162$	0	0
$163$	16.9011	1.32380	0.661899	−	0.749593i	$-0.269751\pi$
$163$	16.9011	1.32380	0.661899	+	0.749593i	$0.269751\pi$
$164$	−2.06478	−0.161232
$165$	0	0
$166$	9.50460	0.737700
$167$	−5.14638	−0.398239	−0.199119	−	0.979975i	$-0.563808\pi$
$167$	−5.14638	−0.398239	−0.199119	+	0.979975i	$0.563808\pi$
$168$	0	0
$169$	−2.62770	−0.202131
$170$	−22.4719	−1.72352
$171$	0	0
$172$	3.00055	0.228790
$173$	9.73669	0.740266	0.370133	−	0.928979i	$-0.379312\pi$
$173$	9.73669	0.740266	0.370133	+	0.928979i	$0.379312\pi$
$174$	0	0
$175$	0	0
$176$	−3.86916	−0.291649
$177$	0	0
$178$	15.6542	1.17333
$179$	1.37598	0.102846	0.0514228	−	0.998677i	$-0.483624\pi$
$179$	1.37598	0.102846	0.0514228	+	0.998677i	$0.483624\pi$
$180$	0	0
$181$	−5.66560	−0.421120	−0.210560	−	0.977581i	$-0.567529\pi$
$181$	−5.66560	−0.421120	−0.210560	+	0.977581i	$0.567529\pi$
$182$	0	0
$183$	0	0
$184$	12.9131	0.951967
$185$	−34.4283	−2.53122
$186$	0	0
$187$	−5.77856	−0.422570
$188$	−0.196508	−0.0143318
$189$	0	0
$190$	−31.4696	−2.28304
$191$	−25.0129	−1.80987	−0.904936	−	0.425547i	$-0.860082\pi$
$191$	−25.0129	−1.80987	−0.904936	+	0.425547i	$0.860082\pi$
$192$	0	0
$193$	17.5338	1.26211	0.631054	−	0.775739i	$-0.282623\pi$
$193$	17.5338	1.26211	0.631054	+	0.775739i	$0.282623\pi$
$194$	14.2064	1.01996
$195$	0	0
$196$	0	0
$197$	−19.7540	−1.40741	−0.703707	−	0.710490i	$-0.748473\pi$
$197$	−19.7540	−1.40741	−0.703707	+	0.710490i	$0.748473\pi$
$198$	0	0
$199$	−19.0222	−1.34845	−0.674224	−	0.738527i	$-0.735522\pi$
$199$	−19.0222	−1.34845	−0.674224	+	0.738527i	$0.735522\pi$
$200$	−22.3180	−1.57812
$201$	0	0
$202$	−2.07342	−0.145885
$203$	0	0
$204$	0	0
$205$	23.4590	1.63845
$206$	3.03593	0.211523
$207$	0	0
$208$	10.5733	0.733129
$209$	−8.09225	−0.559753
$210$	0	0
$211$	−7.43619	−0.511928	−0.255964	−	0.966686i	$-0.582393\pi$
$211$	−7.43619	−0.511928	−0.255964	+	0.966686i	$0.582393\pi$
$212$	0.691283	0.0474775
$213$	0	0
$214$	−2.89043	−0.197585
$215$	−34.0907	−2.32497
$216$	0	0
$217$	0	0
$218$	−1.19532	−0.0809570
$219$	0	0
$220$	−1.28953	−0.0869403
$221$	15.7912	1.06223
$222$	0	0
$223$	−3.29129	−0.220401	−0.110201	−	0.993909i	$-0.535149\pi$
$223$	−3.29129	−0.220401	−0.110201	+	0.993909i	$0.535149\pi$
$224$	0	0
$225$	0	0
$226$	−3.10221	−0.206356
$227$	−18.0169	−1.19583	−0.597913	−	0.801561i	$-0.704003\pi$
$227$	−18.0169	−1.19583	−0.597913	+	0.801561i	$0.704003\pi$
$228$	0	0
$229$	4.25491	0.281173	0.140586	−	0.990068i	$-0.455101\pi$
$229$	4.25491	0.281173	0.140586	+	0.990068i	$0.455101\pi$
$230$	−19.7069	−1.29944
$231$	0	0
$232$	8.17696	0.536844
$233$	−14.7055	−0.963390	−0.481695	−	0.876339i	$-0.659979\pi$
$233$	−14.7055	−0.963390	−0.481695	+	0.876339i	$0.659979\pi$
$234$	0	0
$235$	2.23263	0.145641
$236$	−2.54877	−0.165911
$237$	0	0
$238$	0	0
$239$	−14.1637	−0.916176	−0.458088	−	0.888907i	$-0.651466\pi$
$239$	−14.1637	−0.916176	−0.458088	+	0.888907i	$0.651466\pi$
$240$	0	0
$241$	7.93503	0.511140	0.255570	−	0.966791i	$-0.417737\pi$
$241$	7.93503	0.511140	0.255570	+	0.966791i	$0.417737\pi$
$242$	−12.4931	−0.803090
$243$	0	0
$244$	2.99755	0.191899
$245$	0	0
$246$	0	0
$247$	22.1139	1.40707
$248$	5.76843	0.366296
$249$	0	0
$250$	11.1442	0.704821
$251$	8.05097	0.508173	0.254087	−	0.967181i	$-0.418225\pi$
$251$	8.05097	0.508173	0.254087	+	0.967181i	$0.418225\pi$
$252$	0	0
$253$	−5.06755	−0.318594
$254$	−3.95382	−0.248085
$255$	0	0
$256$	−7.25938	−0.453712
$257$	−17.5537	−1.09497	−0.547486	−	0.836815i	$-0.684415\pi$
$257$	−17.5537	−1.09497	−0.547486	+	0.836815i	$0.684415\pi$
$258$	0	0
$259$	0	0
$260$	3.52393	0.218545
$261$	0	0
$262$	−4.23986	−0.261940
$263$	−23.3486	−1.43973	−0.719867	−	0.694112i	$-0.755797\pi$
$263$	−23.3486	−1.43973	−0.719867	+	0.694112i	$0.755797\pi$
$264$	0	0
$265$	−7.85401	−0.482468
$266$	0	0
$267$	0	0
$268$	−1.65501	−0.101096
$269$	0.538488	0.0328322	0.0164161	−	0.999865i	$-0.494774\pi$
$269$	0.538488	0.0328322	0.0164161	+	0.999865i	$0.494774\pi$
$270$	0	0
$271$	−14.4150	−0.875648	−0.437824	−	0.899061i	$-0.644251\pi$
$271$	−14.4150	−0.875648	−0.437824	+	0.899061i	$0.644251\pi$
$272$	16.0973	0.976040
$273$	0	0
$274$	27.2112	1.64389
$275$	8.75835	0.528148
$276$	0	0
$277$	21.9066	1.31624	0.658121	−	0.752912i	$-0.271351\pi$
$277$	21.9066	1.31624	0.658121	+	0.752912i	$0.271351\pi$
$278$	21.6089	1.29602
$279$	0	0
$280$	0	0
$281$	1.55324	0.0926588	0.0463294	−	0.998926i	$-0.485248\pi$
$281$	1.55324	0.0926588	0.0463294	+	0.998926i	$0.485248\pi$
$282$	0	0
$283$	−2.65142	−0.157610	−0.0788051	−	0.996890i	$-0.525110\pi$
$283$	−2.65142	−0.157610	−0.0788051	+	0.996890i	$0.525110\pi$
$284$	1.01702	0.0603490
$285$	0	0
$286$	−4.93379	−0.291741
$287$	0	0
$288$	0	0
$289$	7.04111	0.414183
$290$	−12.4790	−0.732793
$291$	0	0
$292$	−0.322148	−0.0188523
$293$	10.3863	0.606773	0.303386	−	0.952868i	$-0.401883\pi$
$293$	10.3863	0.606773	0.303386	+	0.952868i	$0.401883\pi$
$294$	0	0
$295$	28.9579	1.68599
$296$	29.3243	1.70444
$297$	0	0
$298$	1.46683	0.0849712
$299$	13.8482	0.800861
$300$	0	0
$301$	0	0
$302$	−25.5159	−1.46827
$303$	0	0
$304$	22.5425	1.29290
$305$	−34.0567	−1.95008
$306$	0	0
$307$	−10.6425	−0.607400	−0.303700	−	0.952768i	$-0.598222\pi$
$307$	−10.6425	−0.607400	−0.303700	+	0.952768i	$0.598222\pi$
$308$	0	0
$309$	0	0
$310$	−8.80331	−0.499994
$311$	−13.7096	−0.777399	−0.388699	−	0.921365i	$-0.627075\pi$
$311$	−13.7096	−0.777399	−0.388699	+	0.921365i	$0.627075\pi$
$312$	0	0
$313$	21.2179	1.19931	0.599653	−	0.800260i	$-0.295305\pi$
$313$	21.2179	1.19931	0.599653	+	0.800260i	$0.295305\pi$
$314$	12.1309	0.684586
$315$	0	0
$316$	−0.311598	−0.0175288
$317$	3.57043	0.200535	0.100268	−	0.994961i	$-0.468030\pi$
$317$	3.57043	0.200535	0.100268	+	0.994961i	$0.468030\pi$
$318$	0	0
$319$	−3.20892	−0.179665
$320$	31.1198	1.73965
$321$	0	0
$322$	0	0
$323$	33.6670	1.87328
$324$	0	0
$325$	−23.9341	−1.32763
$326$	21.9693	1.21677
$327$	0	0
$328$	−19.9812	−1.10328
$329$	0	0
$330$	0	0
$331$	−23.9456	−1.31617	−0.658085	−	0.752944i	$-0.728633\pi$
$331$	−23.9456	−1.31617	−0.658085	+	0.752944i	$0.728633\pi$
$332$	−2.26914	−0.124535
$333$	0	0
$334$	−6.68963	−0.366040
$335$	18.8034	1.02734
$336$	0	0
$337$	27.4937	1.49768	0.748838	−	0.662753i	$-0.230612\pi$
$337$	27.4937	1.49768	0.748838	+	0.662753i	$0.230612\pi$
$338$	−3.41568	−0.185788
$339$	0	0
$340$	5.36497	0.290956
$341$	−2.26373	−0.122588
$342$	0	0
$343$	0	0
$344$	29.0368	1.56556
$345$	0	0
$346$	12.6564	0.680415
$347$	−5.12824	−0.275299	−0.137649	−	0.990481i	$-0.543955\pi$
$347$	−5.12824	−0.275299	−0.137649	+	0.990481i	$0.543955\pi$
$348$	0	0
$349$	15.1396	0.810404	0.405202	−	0.914227i	$-0.367201\pi$
$349$	15.1396	0.810404	0.405202	+	0.914227i	$0.367201\pi$
$350$	0	0
$351$	0	0
$352$	2.04918	0.109222
$353$	32.9757	1.75512	0.877559	−	0.479468i	$-0.159171\pi$
$353$	32.9757	1.75512	0.877559	+	0.479468i	$0.159171\pi$
$354$	0	0
$355$	−11.5549	−0.613269
$356$	−3.73730	−0.198076
$357$	0	0
$358$	1.78860	0.0945303
$359$	−24.0355	−1.26855	−0.634274	−	0.773109i	$-0.718701\pi$
$359$	−24.0355	−1.26855	−0.634274	+	0.773109i	$0.718701\pi$
$360$	0	0
$361$	28.1470	1.48142
$362$	−7.36455	−0.387072
$363$	0	0
$364$	0	0
$365$	3.66008	0.191577
$366$	0	0
$367$	−2.65501	−0.138590	−0.0692952	−	0.997596i	$-0.522075\pi$
$367$	−2.65501	−0.138590	−0.0692952	+	0.997596i	$0.522075\pi$
$368$	14.1166	0.735879
$369$	0	0
$370$	−44.7524	−2.32657
$371$	0	0
$372$	0	0
$373$	−31.9183	−1.65267	−0.826334	−	0.563181i	$-0.809577\pi$
$373$	−31.9183	−1.65267	−0.826334	+	0.563181i	$0.809577\pi$
$374$	−7.51139	−0.388405
$375$	0	0
$376$	−1.90164	−0.0980697
$377$	8.76909	0.451631
$378$	0	0
$379$	30.2681	1.55477	0.777384	−	0.629027i	$-0.216546\pi$
$379$	30.2681	1.55477	0.777384	+	0.629027i	$0.216546\pi$
$380$	7.51307	0.385412
$381$	0	0
$382$	−32.5136	−1.66354
$383$	1.73305	0.0885548	0.0442774	−	0.999019i	$-0.485901\pi$
$383$	1.73305	0.0885548	0.0442774	+	0.999019i	$0.485901\pi$
$384$	0	0
$385$	0	0
$386$	22.7917	1.16006
$387$	0	0
$388$	−3.39165	−0.172185
$389$	−11.0835	−0.561956	−0.280978	−	0.959714i	$-0.590659\pi$
$389$	−11.0835	−0.561956	−0.280978	+	0.959714i	$0.590659\pi$
$390$	0	0
$391$	21.0830	1.06621
$392$	0	0
$393$	0	0
$394$	−25.6777	−1.29362
$395$	3.54022	0.178128
$396$	0	0
$397$	25.3391	1.27173	0.635867	−	0.771799i	$-0.280643\pi$
$397$	25.3391	1.27173	0.635867	+	0.771799i	$0.280643\pi$
$398$	−24.7264	−1.23942
$399$	0	0
$400$	−24.3980	−1.21990
$401$	−34.8244	−1.73905	−0.869524	−	0.493890i	$-0.835575\pi$
$401$	−34.8244	−1.73905	−0.869524	+	0.493890i	$0.835575\pi$
$402$	0	0
$403$	6.18614	0.308154
$404$	0.495009	0.0246276
$405$	0	0
$406$	0	0
$407$	−11.5079	−0.570425
$408$	0	0
$409$	−18.2462	−0.902215	−0.451107	−	0.892470i	$-0.648971\pi$
$409$	−18.2462	−0.902215	−0.451107	+	0.892470i	$0.648971\pi$
$410$	30.4937	1.50598
$411$	0	0
$412$	−0.724799	−0.0357083
$413$	0	0
$414$	0	0
$415$	25.7808	1.26553
$416$	−5.59985	−0.274555
$417$	0	0
$418$	−10.5189	−0.514496
$419$	8.41439	0.411070	0.205535	−	0.978650i	$-0.434107\pi$
$419$	8.41439	0.411070	0.205535	+	0.978650i	$0.434107\pi$
$420$	0	0
$421$	−0.288582	−0.0140646	−0.00703230	−	0.999975i	$-0.502238\pi$
$421$	−0.288582	−0.0140646	−0.00703230	+	0.999975i	$0.502238\pi$
$422$	−9.66609	−0.470538
$423$	0	0
$424$	6.68966	0.324878
$425$	−36.4382	−1.76751
$426$	0	0
$427$	0	0
$428$	0.690062	0.0333554
$429$	0	0
$430$	−44.3136	−2.13699
$431$	−13.4959	−0.650075	−0.325037	−	0.945701i	$-0.605377\pi$
$431$	−13.4959	−0.650075	−0.325037	+	0.945701i	$0.605377\pi$
$432$	0	0
$433$	−4.85211	−0.233177	−0.116589	−	0.993180i	$-0.537196\pi$
$433$	−4.85211	−0.233177	−0.116589	+	0.993180i	$0.537196\pi$
$434$	0	0
$435$	0	0
$436$	0.285371	0.0136668
$437$	29.5245	1.41235
$438$	0	0
$439$	2.54793	0.121606	0.0608031	−	0.998150i	$-0.480634\pi$
$439$	2.54793	0.121606	0.0608031	+	0.998150i	$0.480634\pi$
$440$	−12.4790	−0.594914
$441$	0	0
$442$	20.5265	0.976347
$443$	−0.645506	−0.0306689	−0.0153345	−	0.999882i	$-0.504881\pi$
$443$	−0.645506	−0.0306689	−0.0153345	+	0.999882i	$0.504881\pi$
$444$	0	0
$445$	42.4613	2.01286
$446$	−4.27826	−0.202581
$447$	0	0
$448$	0	0
$449$	−5.22658	−0.246658	−0.123329	−	0.992366i	$-0.539357\pi$
$449$	−5.22658	−0.246658	−0.123329	+	0.992366i	$0.539357\pi$
$450$	0	0
$451$	7.84133	0.369234
$452$	0.740624	0.0348360
$453$	0	0
$454$	−23.4197	−1.09914
$455$	0	0
$456$	0	0
$457$	−2.86075	−0.133820	−0.0669101	−	0.997759i	$-0.521314\pi$
$457$	−2.86075	−0.133820	−0.0669101	+	0.997759i	$0.521314\pi$
$458$	5.53084	0.258439
$459$	0	0
$460$	4.70485	0.219365
$461$	−3.65248	−0.170113	−0.0850566	−	0.996376i	$-0.527107\pi$
$461$	−3.65248	−0.170113	−0.0850566	+	0.996376i	$0.527107\pi$
$462$	0	0
$463$	30.8103	1.43188	0.715939	−	0.698163i	$-0.245999\pi$
$463$	30.8103	1.43188	0.715939	+	0.698163i	$0.245999\pi$
$464$	8.93905	0.414985
$465$	0	0
$466$	−19.1153	−0.885498
$467$	20.5770	0.952191	0.476096	−	0.879393i	$-0.342052\pi$
$467$	20.5770	0.952191	0.476096	+	0.879393i	$0.342052\pi$
$468$	0	0
$469$	0	0
$470$	2.90213	0.133865
$471$	0	0
$472$	−24.6649	−1.13529
$473$	−11.3950	−0.523944
$474$	0	0
$475$	−51.0278	−2.34132
$476$	0	0
$477$	0	0
$478$	−18.4110	−0.842102
$479$	25.1832	1.15065	0.575325	−	0.817925i	$-0.304876\pi$
$479$	25.1832	1.15065	0.575325	+	0.817925i	$0.304876\pi$
$480$	0	0
$481$	31.4478	1.43390
$482$	10.3145	0.469814
$483$	0	0
$484$	2.98262	0.135574
$485$	38.5342	1.74975
$486$	0	0
$487$	−32.7615	−1.48456	−0.742282	−	0.670088i	$-0.766256\pi$
$487$	−32.7615	−1.48456	−0.742282	+	0.670088i	$0.766256\pi$
$488$	29.0078	1.31312
$489$	0	0
$490$	0	0
$491$	−3.52001	−0.158856	−0.0794278	−	0.996841i	$-0.525309\pi$
$491$	−3.52001	−0.158856	−0.0794278	+	0.996841i	$0.525309\pi$
$492$	0	0
$493$	13.3504	0.601272
$494$	28.7452	1.29331
$495$	0	0
$496$	6.30605	0.283150
$497$	0	0
$498$	0	0
$499$	15.6416	0.700216	0.350108	−	0.936709i	$-0.386145\pi$
$499$	15.6416	0.700216	0.350108	+	0.936709i	$0.386145\pi$
$500$	−2.66057	−0.118984
$501$	0	0
$502$	10.4652	0.467086
$503$	−36.5427	−1.62936	−0.814678	−	0.579913i	$-0.803086\pi$
$503$	−36.5427	−1.62936	−0.814678	+	0.579913i	$0.803086\pi$
$504$	0	0
$505$	−5.62404	−0.250267
$506$	−6.58716	−0.292835
$507$	0	0
$508$	0.943938	0.0418805
$509$	−37.6458	−1.66862	−0.834311	−	0.551294i	$-0.814134\pi$
$509$	−37.6458	−1.66862	−0.834311	+	0.551294i	$0.814134\pi$
$510$	0	0
$511$	0	0
$512$	−25.4272	−1.12373
$513$	0	0
$514$	−22.8176	−1.00644
$515$	8.23480	0.362869
$516$	0	0
$517$	0.746270	0.0328209
$518$	0	0
$519$	0	0
$520$	34.1017	1.49546
$521$	−14.3423	−0.628347	−0.314174	−	0.949366i	$-0.601727\pi$
$521$	−14.3423	−0.628347	−0.314174	+	0.949366i	$0.601727\pi$
$522$	0	0
$523$	10.4844	0.458453	0.229226	−	0.973373i	$-0.426380\pi$
$523$	10.4844	0.458453	0.229226	+	0.973373i	$0.426380\pi$
$524$	1.01223	0.0442194
$525$	0	0
$526$	−30.3501	−1.32333
$527$	9.41802	0.410256
$528$	0	0
$529$	−4.51110	−0.196135
$530$	−10.2092	−0.443460
$531$	0	0
$532$	0	0
$533$	−21.4281	−0.928156
$534$	0	0
$535$	−7.84014	−0.338959
$536$	−16.0158	−0.691776
$537$	0	0
$538$	0.699965	0.0301776
$539$	0	0
$540$	0	0
$541$	−46.0922	−1.98166	−0.990830	−	0.135118i	$-0.956859\pi$
$541$	−46.0922	−1.98166	−0.990830	+	0.135118i	$0.956859\pi$
$542$	−18.7376	−0.804851
$543$	0	0
$544$	−8.52542	−0.365525
$545$	−3.24224	−0.138882
$546$	0	0
$547$	24.3585	1.04149	0.520747	−	0.853711i	$-0.325653\pi$
$547$	24.3585	1.04149	0.520747	+	0.853711i	$0.325653\pi$
$548$	−6.49643	−0.277514
$549$	0	0
$550$	11.3847	0.485447
$551$	18.6958	0.796468
$552$	0	0
$553$	0	0
$554$	28.4758	1.20982
$555$	0	0
$556$	−5.15894	−0.218788
$557$	30.5775	1.29561	0.647806	−	0.761805i	$-0.275687\pi$
$557$	30.5775	1.29561	0.647806	+	0.761805i	$0.275687\pi$
$558$	0	0
$559$	31.1394	1.31706
$560$	0	0
$561$	0	0
$562$	2.01902	0.0851672
$563$	8.82714	0.372019	0.186010	−	0.982548i	$-0.440444\pi$
$563$	8.82714	0.372019	0.186010	+	0.982548i	$0.440444\pi$
$564$	0	0
$565$	−8.41459	−0.354005
$566$	−3.44650	−0.144867
$567$	0	0
$568$	9.84186	0.412955
$569$	7.12055	0.298509	0.149254	−	0.988799i	$-0.452313\pi$
$569$	7.12055	0.298509	0.149254	+	0.988799i	$0.452313\pi$
$570$	0	0
$571$	6.66361	0.278863	0.139432	−	0.990232i	$-0.455472\pi$
$571$	6.66361	0.278863	0.139432	+	0.990232i	$0.455472\pi$
$572$	1.17790	0.0492503
$573$	0	0
$574$	0	0
$575$	−31.9548	−1.33261
$576$	0	0
$577$	7.91259	0.329405	0.164703	−	0.986343i	$-0.447334\pi$
$577$	7.91259	0.329405	0.164703	+	0.986343i	$0.447334\pi$
$578$	9.15254	0.380696
$579$	0	0
$580$	2.97925	0.123707
$581$	0	0
$582$	0	0
$583$	−2.62525	−0.108727
$584$	−3.11747	−0.129002
$585$	0	0
$586$	13.5008	0.557714
$587$	18.2778	0.754406	0.377203	−	0.926131i	$-0.376886\pi$
$587$	18.2778	0.754406	0.377203	+	0.926131i	$0.376886\pi$
$588$	0	0
$589$	13.1889	0.543441
$590$	37.6415	1.54968
$591$	0	0
$592$	32.0574	1.31755
$593$	−28.3816	−1.16549	−0.582745	−	0.812655i	$-0.698022\pi$
$593$	−28.3816	−1.16549	−0.582745	+	0.812655i	$0.698022\pi$
$594$	0	0
$595$	0	0
$596$	−0.350192	−0.0143444
$597$	0	0
$598$	18.0009	0.736111
$599$	9.38902	0.383625	0.191813	−	0.981432i	$-0.438563\pi$
$599$	9.38902	0.383625	0.191813	+	0.981432i	$0.438563\pi$
$600$	0	0
$601$	12.6286	0.515133	0.257566	−	0.966261i	$-0.417079\pi$
$601$	12.6286	0.515133	0.257566	+	0.966261i	$0.417079\pi$
$602$	0	0
$603$	0	0
$604$	6.09168	0.247867
$605$	−33.8871	−1.37771
$606$	0	0
$607$	24.0265	0.975207	0.487604	−	0.873065i	$-0.337871\pi$
$607$	24.0265	0.975207	0.487604	+	0.873065i	$0.337871\pi$
$608$	−11.9389	−0.484188
$609$	0	0
$610$	−44.2693	−1.79241
$611$	−2.03935	−0.0825031
$612$	0	0
$613$	−28.5415	−1.15278	−0.576390	−	0.817175i	$-0.695539\pi$
$613$	−28.5415	−1.15278	−0.576390	+	0.817175i	$0.695539\pi$
$614$	−13.8339	−0.558291
$615$	0	0
$616$	0	0
$617$	12.1110	0.487570	0.243785	−	0.969829i	$-0.421611\pi$
$617$	12.1110	0.487570	0.243785	+	0.969829i	$0.421611\pi$
$618$	0	0
$619$	−26.5739	−1.06810	−0.534048	−	0.845454i	$-0.679330\pi$
$619$	−26.5739	−1.06810	−0.534048	+	0.845454i	$0.679330\pi$
$620$	2.10171	0.0844067
$621$	0	0
$622$	−17.8207	−0.714545
$623$	0	0
$624$	0	0
$625$	−6.92971	−0.277188
$626$	27.5806	1.10234
$627$	0	0
$628$	−2.89614	−0.115569
$629$	47.8774	1.90900
$630$	0	0
$631$	3.30962	0.131754	0.0658770	−	0.997828i	$-0.479015\pi$
$631$	3.30962	0.131754	0.0658770	+	0.997828i	$0.479015\pi$
$632$	−3.01538	−0.119945
$633$	0	0
$634$	4.64110	0.184322
$635$	−10.7245	−0.425591
$636$	0	0
$637$	0	0
$638$	−4.17119	−0.165139
$639$	0	0
$640$	28.1907	1.11433
$641$	32.5844	1.28701	0.643503	−	0.765443i	$-0.277480\pi$
$641$	32.5844	1.28701	0.643503	+	0.765443i	$0.277480\pi$
$642$	0	0
$643$	−43.0654	−1.69833	−0.849166	−	0.528126i	$-0.822895\pi$
$643$	−43.0654	−1.69833	−0.849166	+	0.528126i	$0.822895\pi$
$644$	0	0
$645$	0	0
$646$	43.7628	1.72182
$647$	46.1975	1.81621	0.908106	−	0.418739i	$-0.137528\pi$
$647$	46.1975	1.81621	0.908106	+	0.418739i	$0.137528\pi$
$648$	0	0
$649$	9.67935	0.379948
$650$	−31.1113	−1.22029
$651$	0	0
$652$	−5.24497	−0.205409
$653$	32.0005	1.25228	0.626138	−	0.779713i	$-0.284635\pi$
$653$	32.0005	1.25228	0.626138	+	0.779713i	$0.284635\pi$
$654$	0	0
$655$	−11.5004	−0.449359
$656$	−21.8435	−0.852845
$657$	0	0
$658$	0	0
$659$	−38.4139	−1.49639	−0.748197	−	0.663477i	$-0.769080\pi$
$659$	−38.4139	−1.49639	−0.748197	+	0.663477i	$0.769080\pi$
$660$	0	0
$661$	28.0260	1.09009	0.545043	−	0.838408i	$-0.316513\pi$
$661$	28.0260	1.09009	0.545043	+	0.838408i	$0.316513\pi$
$662$	−31.1262	−1.20976
$663$	0	0
$664$	−21.9588	−0.852167
$665$	0	0
$666$	0	0
$667$	11.7077	0.453325
$668$	1.59709	0.0617932
$669$	0	0
$670$	24.4420	0.944275
$671$	−11.3837	−0.439461
$672$	0	0
$673$	−1.59256	−0.0613888	−0.0306944	−	0.999529i	$-0.509772\pi$
$673$	−1.59256	−0.0613888	−0.0306944	+	0.999529i	$0.509772\pi$
$674$	35.7383	1.37659
$675$	0	0
$676$	0.815462	0.0313639
$677$	−42.0334	−1.61547	−0.807737	−	0.589543i	$-0.799308\pi$
$677$	−42.0334	−1.61547	−0.807737	+	0.589543i	$0.799308\pi$
$678$	0	0
$679$	0	0
$680$	51.9177	1.99095
$681$	0	0
$682$	−2.94256	−0.112676
$683$	35.7289	1.36713	0.683565	−	0.729890i	$-0.260429\pi$
$683$	35.7289	1.36713	0.683565	+	0.729890i	$0.260429\pi$
$684$	0	0
$685$	73.8092	2.82010
$686$	0	0
$687$	0	0
$688$	31.7430	1.21019
$689$	7.17408	0.273311
$690$	0	0
$691$	−51.1349	−1.94526	−0.972632	−	0.232351i	$-0.925358\pi$
$691$	−51.1349	−1.94526	−0.972632	+	0.232351i	$0.925358\pi$
$692$	−3.02161	−0.114864
$693$	0	0
$694$	−6.66606	−0.253040
$695$	58.6132	2.22333
$696$	0	0
$697$	−32.6230	−1.23569
$698$	19.6795	0.744881
$699$	0	0
$700$	0	0
$701$	−24.5761	−0.928226	−0.464113	−	0.885776i	$-0.653627\pi$
$701$	−24.5761	−0.928226	−0.464113	+	0.885776i	$0.653627\pi$
$702$	0	0
$703$	67.0472	2.52873
$704$	10.4020	0.392040
$705$	0	0
$706$	42.8642	1.61321
$707$	0	0
$708$	0	0
$709$	30.8976	1.16038	0.580192	−	0.814480i	$-0.302978\pi$
$709$	30.8976	1.16038	0.580192	+	0.814480i	$0.302978\pi$
$710$	−15.0198	−0.563685
$711$	0	0
$712$	−36.1664	−1.35539
$713$	8.25920	0.309310
$714$	0	0
$715$	−13.3827	−0.500483
$716$	−0.427011	−0.0159582
$717$	0	0
$718$	−31.2431	−1.16598
$719$	−6.11380	−0.228006	−0.114003	−	0.993480i	$-0.536367\pi$
$719$	−6.11380	−0.228006	−0.114003	+	0.993480i	$0.536367\pi$
$720$	0	0
$721$	0	0
$722$	36.5875	1.36165
$723$	0	0
$724$	1.75822	0.0653437
$725$	−20.2347	−0.751498
$726$	0	0
$727$	44.4983	1.65035	0.825176	−	0.564876i	$-0.191076\pi$
$727$	44.4983	1.65035	0.825176	+	0.564876i	$0.191076\pi$
$728$	0	0
$729$	0	0
$730$	4.75763	0.176088
$731$	47.4079	1.75344
$732$	0	0
$733$	−9.83708	−0.363341	−0.181670	−	0.983359i	$-0.558150\pi$
$733$	−9.83708	−0.363341	−0.181670	+	0.983359i	$0.558150\pi$
$734$	−3.45118	−0.127385
$735$	0	0
$736$	−7.47643	−0.275585
$737$	6.28514	0.231516
$738$	0	0
$739$	14.8493	0.546239	0.273120	−	0.961980i	$-0.411944\pi$
$739$	14.8493	0.546239	0.273120	+	0.961980i	$0.411944\pi$
$740$	10.6842	0.392760
$741$	0	0
$742$	0	0
$743$	6.08402	0.223201	0.111601	−	0.993753i	$-0.464402\pi$
$743$	6.08402	0.223201	0.111601	+	0.993753i	$0.464402\pi$
$744$	0	0
$745$	3.97871	0.145769
$746$	−41.4897	−1.51905
$747$	0	0
$748$	1.79328	0.0655686
$749$	0	0
$750$	0	0
$751$	22.2010	0.810127	0.405063	−	0.914289i	$-0.367249\pi$
$751$	22.2010	0.810127	0.405063	+	0.914289i	$0.367249\pi$
$752$	−2.07887	−0.0758087
$753$	0	0
$754$	11.3987	0.415116
$755$	−69.2106	−2.51883
$756$	0	0
$757$	25.0464	0.910329	0.455164	−	0.890407i	$-0.349581\pi$
$757$	25.0464	0.910329	0.455164	+	0.890407i	$0.349581\pi$
$758$	39.3446	1.42906
$759$	0	0
$760$	72.7051	2.63729
$761$	−6.75264	−0.244783	−0.122392	−	0.992482i	$-0.539056\pi$
$761$	−6.75264	−0.244783	−0.122392	+	0.992482i	$0.539056\pi$
$762$	0	0
$763$	0	0
$764$	7.76233	0.280831
$765$	0	0
$766$	2.25275	0.0813950
$767$	−26.4510	−0.955089
$768$	0	0
$769$	42.1610	1.52036	0.760182	−	0.649710i	$-0.225110\pi$
$769$	42.1610	1.52036	0.760182	+	0.649710i	$0.225110\pi$
$770$	0	0
$771$	0	0
$772$	−5.44130	−0.195837
$773$	−3.29852	−0.118639	−0.0593197	−	0.998239i	$-0.518893\pi$
$773$	−3.29852	−0.118639	−0.0593197	+	0.998239i	$0.518893\pi$
$774$	0	0
$775$	−14.2746	−0.512757
$776$	−32.8215	−1.17822
$777$	0	0
$778$	−14.4071	−0.516521
$779$	−45.6851	−1.63684
$780$	0	0
$781$	−3.86229	−0.138203
$782$	27.4052	0.980009
$783$	0	0
$784$	0	0
$785$	32.9045	1.17441
$786$	0	0
$787$	−6.72910	−0.239867	−0.119933	−	0.992782i	$-0.538268\pi$
$787$	−6.72910	−0.239867	−0.119933	+	0.992782i	$0.538268\pi$
$788$	6.13031	0.218383
$789$	0	0
$790$	4.60183	0.163726
$791$	0	0
$792$	0	0
$793$	31.1083	1.10469
$794$	32.9376	1.16891
$795$	0	0
$796$	5.90321	0.209234
$797$	17.7260	0.627888	0.313944	−	0.949441i	$-0.398349\pi$
$797$	17.7260	0.627888	0.313944	+	0.949441i	$0.398349\pi$
$798$	0	0
$799$	−3.10478	−0.109839
$800$	12.9217	0.456850
$801$	0	0
$802$	−45.2673	−1.59844
$803$	1.22340	0.0431730
$804$	0	0
$805$	0	0
$806$	8.04120	0.283239
$807$	0	0
$808$	4.79028	0.168521
$809$	39.0857	1.37418	0.687089	−	0.726573i	$-0.258888\pi$
$809$	39.0857	1.37418	0.687089	+	0.726573i	$0.258888\pi$
$810$	0	0
$811$	−13.9559	−0.490058	−0.245029	−	0.969516i	$-0.578797\pi$
$811$	−13.9559	−0.490058	−0.245029	+	0.969516i	$0.578797\pi$
$812$	0	0
$813$	0	0
$814$	−14.9588	−0.524305
$815$	59.5907	2.08737
$816$	0	0
$817$	66.3896	2.32268
$818$	−23.7177	−0.829269
$819$	0	0
$820$	−7.28010	−0.254232
$821$	−44.9966	−1.57039	−0.785196	−	0.619247i	$-0.787438\pi$
$821$	−44.9966	−1.57039	−0.785196	+	0.619247i	$0.787438\pi$
$822$	0	0
$823$	−55.2465	−1.92577	−0.962886	−	0.269909i	$-0.913006\pi$
$823$	−55.2465	−1.92577	−0.962886	+	0.269909i	$0.913006\pi$
$824$	−7.01400	−0.244344
$825$	0	0
$826$	0	0
$827$	−13.8901	−0.483005	−0.241502	−	0.970400i	$-0.577640\pi$
$827$	−13.8901	−0.483005	−0.241502	+	0.970400i	$0.577640\pi$
$828$	0	0
$829$	38.9792	1.35381	0.676903	−	0.736073i	$-0.263322\pi$
$829$	38.9792	1.35381	0.676903	+	0.736073i	$0.263322\pi$
$830$	33.5117	1.16321
$831$	0	0
$832$	−28.4258	−0.985486
$833$	0	0
$834$	0	0
$835$	−18.1453	−0.627944
$836$	2.51129	0.0868548
$837$	0	0
$838$	10.9376	0.377834
$839$	−38.9415	−1.34441	−0.672206	−	0.740365i	$-0.734653\pi$
$839$	−38.9415	−1.34441	−0.672206	+	0.740365i	$0.734653\pi$
$840$	0	0
$841$	−21.5863	−0.744356
$842$	−0.375119	−0.0129275
$843$	0	0
$844$	2.30769	0.0794340
$845$	−9.26487	−0.318721
$846$	0	0
$847$	0	0
$848$	7.31313	0.251134
$849$	0	0
$850$	−47.3650	−1.62461
$851$	41.9864	1.43928
$852$	0	0
$853$	−7.67779	−0.262883	−0.131441	−	0.991324i	$-0.541960\pi$
$853$	−7.67779	−0.262883	−0.131441	+	0.991324i	$0.541960\pi$
$854$	0	0
$855$	0	0
$856$	6.67784	0.228244
$857$	15.9639	0.545316	0.272658	−	0.962111i	$-0.412097\pi$
$857$	15.9639	0.545316	0.272658	+	0.962111i	$0.412097\pi$
$858$	0	0
$859$	−48.3291	−1.64897	−0.824483	−	0.565887i	$-0.808534\pi$
$859$	−48.3291	−1.64897	−0.824483	+	0.565887i	$0.808534\pi$
$860$	10.5795	0.360756
$861$	0	0
$862$	−17.5430	−0.597515
$863$	−34.3052	−1.16776	−0.583881	−	0.811839i	$-0.698467\pi$
$863$	−34.3052	−1.16776	−0.583881	+	0.811839i	$0.698467\pi$
$864$	0	0
$865$	34.3300	1.16726
$866$	−6.30712	−0.214325
$867$	0	0
$868$	0	0
$869$	1.18334	0.0401421
$870$	0	0
$871$	−17.1755	−0.581970
$872$	2.76158	0.0935188
$873$	0	0
$874$	38.3781	1.29816
$875$	0	0
$876$	0	0
$877$	14.1815	0.478876	0.239438	−	0.970912i	$-0.423037\pi$
$877$	14.1815	0.478876	0.239438	+	0.970912i	$0.423037\pi$
$878$	3.31198	0.111774
$879$	0	0
$880$	−13.6421	−0.459874
$881$	46.2822	1.55929	0.779643	−	0.626224i	$-0.215400\pi$
$881$	46.2822	1.55929	0.779643	+	0.626224i	$0.215400\pi$
$882$	0	0
$883$	−4.37483	−0.147225	−0.0736124	−	0.997287i	$-0.523453\pi$
$883$	−4.37483	−0.147225	−0.0736124	+	0.997287i	$0.523453\pi$
$884$	−4.90052	−0.164822
$885$	0	0
$886$	−0.839076	−0.0281893
$887$	19.1442	0.642798	0.321399	−	0.946944i	$-0.395847\pi$
$887$	19.1442	0.642798	0.321399	+	0.946944i	$0.395847\pi$
$888$	0	0
$889$	0	0
$890$	55.1942	1.85011
$891$	0	0
$892$	1.02139	0.0341988
$893$	−4.34791	−0.145497
$894$	0	0
$895$	4.85149	0.162167
$896$	0	0
$897$	0	0
$898$	−6.79389	−0.226715
$899$	5.22997	0.174429
$900$	0	0
$901$	10.9221	0.363868
$902$	10.1927	0.339380
$903$	0	0
$904$	7.16713	0.238375
$905$	−19.9760	−0.664024
$906$	0	0
$907$	39.8450	1.32303	0.661515	−	0.749932i	$-0.269914\pi$
$907$	39.8450	1.32303	0.661515	+	0.749932i	$0.269914\pi$
$908$	5.59124	0.185552
$909$	0	0
$910$	0	0
$911$	−28.7454	−0.952377	−0.476189	−	0.879343i	$-0.657982\pi$
$911$	−28.7454	−0.952377	−0.476189	+	0.879343i	$0.657982\pi$
$912$	0	0
$913$	8.61739	0.285194
$914$	−3.71861	−0.123001
$915$	0	0
$916$	−1.32044	−0.0436285
$917$	0	0
$918$	0	0
$919$	−16.0269	−0.528680	−0.264340	−	0.964430i	$-0.585154\pi$
$919$	−16.0269	−0.528680	−0.264340	+	0.964430i	$0.585154\pi$
$920$	45.5296	1.50107
$921$	0	0
$922$	−4.74776	−0.156359
$923$	10.5545	0.347407
$924$	0	0
$925$	−72.5660	−2.38596
$926$	40.0495	1.31611
$927$	0	0
$928$	−4.73430	−0.155411
$929$	−14.0159	−0.459847	−0.229924	−	0.973209i	$-0.573848\pi$
$929$	−14.0159	−0.459847	−0.229924	+	0.973209i	$0.573848\pi$
$930$	0	0
$931$	0	0
$932$	4.56360	0.149486
$933$	0	0
$934$	26.7475	0.875205
$935$	−20.3743	−0.666310
$936$	0	0
$937$	−51.5307	−1.68344	−0.841718	−	0.539918i	$-0.818455\pi$
$937$	−51.5307	−1.68344	−0.841718	+	0.539918i	$0.818455\pi$
$938$	0	0
$939$	0	0
$940$	−0.692857	−0.0225985
$941$	46.3128	1.50975	0.754877	−	0.655866i	$-0.227697\pi$
$941$	46.3128	1.50975	0.754877	+	0.655866i	$0.227697\pi$
$942$	0	0
$943$	−28.6090	−0.931637
$944$	−26.9636	−0.877592
$945$	0	0
$946$	−14.8121	−0.481582
$947$	20.0083	0.650181	0.325091	−	0.945683i	$-0.394605\pi$
$947$	20.0083	0.650181	0.325091	+	0.945683i	$0.394605\pi$
$948$	0	0
$949$	−3.34322	−0.108526
$950$	−66.3297	−2.15202
$951$	0	0
$952$	0	0
$953$	30.0109	0.972148	0.486074	−	0.873918i	$-0.338429\pi$
$953$	30.0109	0.972148	0.486074	+	0.873918i	$0.338429\pi$
$954$	0	0
$955$	−88.1917	−2.85382
$956$	4.39547	0.142160
$957$	0	0
$958$	32.7349	1.05762
$959$	0	0
$960$	0	0
$961$	−27.3105	−0.880985
$962$	40.8782	1.31796
$963$	0	0
$964$	−2.46250	−0.0793117
$965$	61.8213	1.99010
$966$	0	0
$967$	−33.1442	−1.06585	−0.532923	−	0.846164i	$-0.678906\pi$
$967$	−33.1442	−1.06585	−0.532923	+	0.846164i	$0.678906\pi$
$968$	28.8633	0.927702
$969$	0	0
$970$	50.0896	1.60828
$971$	−41.6469	−1.33651	−0.668256	−	0.743932i	$-0.732959\pi$
$971$	−41.6469	−1.33651	−0.668256	+	0.743932i	$0.732959\pi$
$972$	0	0
$973$	0	0
$974$	−42.5857	−1.36453
$975$	0	0
$976$	31.7113	1.01505
$977$	37.5181	1.20031	0.600154	−	0.799884i	$-0.295106\pi$
$977$	37.5181	1.20031	0.600154	+	0.799884i	$0.295106\pi$
$978$	0	0
$979$	14.1929	0.453608
$980$	0	0
$981$	0	0
$982$	−4.57556	−0.146012
$983$	−48.0758	−1.53338	−0.766690	−	0.642017i	$-0.778098\pi$
$983$	−48.0758	−1.53338	−0.766690	+	0.642017i	$0.778098\pi$
$984$	0	0
$985$	−69.6495	−2.21922
$986$	17.3538	0.552658
$987$	0	0
$988$	−6.86265	−0.218330
$989$	41.5747	1.32200
$990$	0	0
$991$	−34.1286	−1.08413	−0.542065	−	0.840337i	$-0.682357\pi$
$991$	−34.1286	−1.08413	−0.542065	+	0.840337i	$0.682357\pi$
$992$	−3.33981	−0.106039
$993$	0	0
$994$	0	0
$995$	−67.0692	−2.12624
$996$	0	0
$997$	44.0827	1.39611	0.698056	−	0.716043i	$-0.254049\pi$
$997$	44.0827	1.39611	0.698056	+	0.716043i	$0.254049\pi$
$998$	20.3321	0.643602
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.bh.1.10		12
3.2	odd	2		3969.2.a.bi.1.3		12
7.6	odd	2	inner	3969.2.a.bh.1.9		12
9.2	odd	6		1323.2.f.h.442.10		24
9.4	even	3		441.2.f.h.295.4	yes	24
9.5	odd	6		1323.2.f.h.883.10		24
9.7	even	3		441.2.f.h.148.4	yes	24
21.20	even	2		3969.2.a.bi.1.4		12
63.2	odd	6		1323.2.g.h.361.10		24
63.4	even	3		441.2.g.h.79.3		24
63.5	even	6		1323.2.h.h.802.4		24
63.11	odd	6		1323.2.h.h.226.3		24
63.13	odd	6		441.2.f.h.295.3	yes	24
63.16	even	3		441.2.g.h.67.3		24
63.20	even	6		1323.2.f.h.442.9		24
63.23	odd	6		1323.2.h.h.802.3		24
63.25	even	3		441.2.h.h.373.10		24
63.31	odd	6		441.2.g.h.79.4		24
63.32	odd	6		1323.2.g.h.667.10		24
63.34	odd	6		441.2.f.h.148.3	✓	24
63.38	even	6		1323.2.h.h.226.4		24
63.40	odd	6		441.2.h.h.214.9		24
63.41	even	6		1323.2.f.h.883.9		24
63.47	even	6		1323.2.g.h.361.9		24
63.52	odd	6		441.2.h.h.373.9		24
63.58	even	3		441.2.h.h.214.10		24
63.59	even	6		1323.2.g.h.667.9		24
63.61	odd	6		441.2.g.h.67.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
441.2.f.h.148.3	✓	24	63.34	odd	6
441.2.f.h.148.4	yes	24	9.7	even	3
441.2.f.h.295.3	yes	24	63.13	odd	6
441.2.f.h.295.4	yes	24	9.4	even	3
441.2.g.h.67.3		24	63.16	even	3
441.2.g.h.67.4		24	63.61	odd	6
441.2.g.h.79.3		24	63.4	even	3
441.2.g.h.79.4		24	63.31	odd	6
441.2.h.h.214.9		24	63.40	odd	6
441.2.h.h.214.10		24	63.58	even	3
441.2.h.h.373.9		24	63.52	odd	6
441.2.h.h.373.10		24	63.25	even	3
1323.2.f.h.442.9		24	63.20	even	6
1323.2.f.h.442.10		24	9.2	odd	6
1323.2.f.h.883.9		24	63.41	even	6
1323.2.f.h.883.10		24	9.5	odd	6
1323.2.g.h.361.9		24	63.47	even	6
1323.2.g.h.361.10		24	63.2	odd	6
1323.2.g.h.667.9		24	63.59	even	6
1323.2.g.h.667.10		24	63.32	odd	6
1323.2.h.h.226.3		24	63.11	odd	6
1323.2.h.h.226.4		24	63.38	even	6
1323.2.h.h.802.3		24	63.23	odd	6
1323.2.h.h.802.4		24	63.5	even	6
3969.2.a.bh.1.9		12	7.6	odd	2	inner
3969.2.a.bh.1.10		12	1.1	even	1	trivial
3969.2.a.bi.1.3		12	3.2	odd	2
3969.2.a.bi.1.4		12	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+1.29987 q^{2} -0.310333 q^{4} +3.52584 q^{5} -3.00314 q^{8} +O(q^{10})\)	\(q+1.29987 q^{2} -0.310333 q^{4} +3.52584 q^{5} -3.00314 q^{8} +4.58314 q^{10} +1.17853 q^{11} -3.22061 q^{13} -3.28303 q^{16} -4.90317 q^{17} -6.86637 q^{19} -1.09418 q^{20} +1.53194 q^{22} -4.29987 q^{23} +7.43156 q^{25} -4.18637 q^{26} -2.72281 q^{29} -1.92080 q^{31} +1.73876 q^{32} -6.37350 q^{34} -9.76457 q^{37} -8.92540 q^{38} -10.5886 q^{40} +6.65345 q^{41} -9.66881 q^{43} -0.365738 q^{44} -5.58928 q^{46} +0.633218 q^{47} +9.66008 q^{50} +0.999459 q^{52} -2.22756 q^{53} +4.15533 q^{55} -3.53930 q^{58} +8.21304 q^{59} -9.65916 q^{61} -2.49680 q^{62} +8.82622 q^{64} -11.3553 q^{65} +5.33301 q^{67} +1.52161 q^{68} -3.27719 q^{71} +1.03807 q^{73} -12.6927 q^{74} +2.13086 q^{76} +1.00408 q^{79} -11.5754 q^{80} +8.64864 q^{82} +7.31195 q^{83} -17.2878 q^{85} -12.5682 q^{86} -3.53930 q^{88} +12.0429 q^{89} +1.33439 q^{92} +0.823103 q^{94} -24.2097 q^{95} +10.9291 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} + 12 q^{4} - 12 q^{8}+O(q^{10}) \) 12 * q - 4 * q^2 + 12 * q^4 - 12 * q^8	\( 12 q - 4 q^{2} + 12 q^{4} - 12 q^{8} - 20 q^{11} + 12 q^{16} - 32 q^{23} + 12 q^{25} - 16 q^{29} - 48 q^{32} + 12 q^{37} - 56 q^{44} - 24 q^{46} + 4 q^{50} - 32 q^{53} + 48 q^{64} - 60 q^{65} + 12 q^{67} - 56 q^{71} - 68 q^{74} - 12 q^{79} - 12 q^{85} - 76 q^{86} - 16 q^{92} - 64 q^{95}+O(q^{100}) \) 12 * q - 4 * q^2 + 12 * q^4 - 12 * q^8 - 20 * q^11 + 12 * q^16 - 32 * q^23 + 12 * q^25 - 16 * q^29 - 48 * q^32 + 12 * q^37 - 56 * q^44 - 24 * q^46 + 4 * q^50 - 32 * q^53 + 48 * q^64 - 60 * q^65 + 12 * q^67 - 56 * q^71 - 68 * q^74 - 12 * q^79 - 12 * q^85 - 76 * q^86 - 16 * q^92 - 64 * q^95

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