LMFDB - Embedded newform 8036.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + \cdots + 3381 $ x^15 - x^14 - 37x^13 + 39x^12 + 537x^11 - 616x^10 - 3853x^9 + 4929x^8 + 13971x^7 - 20311x^6 - 22309x^5 + 38415x^4 + 8429x^3 - 22584x^2 - 399*x + 3381
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.47735$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47735 q^{3} +3.49844 q^{5} +3.13725 q^{9} +O(q^{10})$	$q-2.47735 q^{3} +3.49844 q^{5} +3.13725 q^{9} -0.886832 q^{11} +6.85228 q^{13} -8.66684 q^{15} -3.02899 q^{17} -6.07400 q^{19} -4.52270 q^{23} +7.23905 q^{25} -0.340013 q^{27} -7.22633 q^{29} -9.72069 q^{31} +2.19699 q^{33} +7.05671 q^{37} -16.9755 q^{39} +1.00000 q^{41} +4.25996 q^{43} +10.9755 q^{45} +11.7610 q^{47} +7.50385 q^{51} +6.23725 q^{53} -3.10252 q^{55} +15.0474 q^{57} -0.453451 q^{59} +1.80500 q^{61} +23.9722 q^{65} +0.386390 q^{67} +11.2043 q^{69} +0.406313 q^{71} -2.47266 q^{73} -17.9336 q^{75} +14.1588 q^{79} -8.56942 q^{81} -7.79158 q^{83} -10.5967 q^{85} +17.9021 q^{87} +5.14645 q^{89} +24.0815 q^{93} -21.2495 q^{95} +11.4082 q^{97} -2.78221 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * q^99

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
$3$	−2.47735	−1.43030	−0.715149	−	0.698973i	$-0.753641\pi$
$3$	−2.47735	−1.43030	−0.715149	+	0.698973i	$0.753641\pi$
$4$	0	0
$5$	3.49844	1.56455	0.782274	−	0.622935i	$-0.214060\pi$
$5$	3.49844	1.56455	0.782274	+	0.622935i	$0.214060\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.13725	1.04575
$10$	0	0
$11$	−0.886832	−0.267390	−0.133695	−	0.991023i	$-0.542684\pi$
$11$	−0.886832	−0.267390	−0.133695	+	0.991023i	$0.542684\pi$
$12$	0	0
$13$	6.85228	1.90048	0.950240	−	0.311519i	$-0.100838\pi$
$13$	6.85228	1.90048	0.950240	+	0.311519i	$0.100838\pi$
$14$	0	0
$15$	−8.66684	−2.23777
$16$	0	0
$17$	−3.02899	−0.734637	−0.367318	−	0.930095i	$-0.619724\pi$
$17$	−3.02899	−0.734637	−0.367318	+	0.930095i	$0.619724\pi$
$18$	0	0
$19$	−6.07400	−1.39347	−0.696735	−	0.717328i	$-0.745365\pi$
$19$	−6.07400	−1.39347	−0.696735	+	0.717328i	$0.745365\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.52270	−0.943048	−0.471524	−	0.881853i	$-0.656296\pi$
$23$	−4.52270	−0.943048	−0.471524	+	0.881853i	$0.656296\pi$
$24$	0	0
$25$	7.23905	1.44781
$26$	0	0
$27$	−0.340013	−0.0654355
$28$	0	0
$29$	−7.22633	−1.34189	−0.670947	−	0.741505i	$-0.734112\pi$
$29$	−7.22633	−1.34189	−0.670947	+	0.741505i	$0.734112\pi$
$30$	0	0
$31$	−9.72069	−1.74589	−0.872944	−	0.487821i	$-0.837792\pi$
$31$	−9.72069	−1.74589	−0.872944	+	0.487821i	$0.837792\pi$
$32$	0	0
$33$	2.19699	0.382447
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.05671	1.16012	0.580058	−	0.814575i	$-0.303030\pi$
$37$	7.05671	1.16012	0.580058	+	0.814575i	$0.303030\pi$
$38$	0	0
$39$	−16.9755	−2.71825
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	4.25996	0.649638	0.324819	−	0.945776i	$-0.394697\pi$
$43$	4.25996	0.649638	0.324819	+	0.945776i	$0.394697\pi$
$44$	0	0
$45$	10.9755	1.63613
$46$	0	0
$47$	11.7610	1.71551	0.857756	−	0.514056i	$-0.171858\pi$
$47$	11.7610	1.71551	0.857756	+	0.514056i	$0.171858\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	7.50385	1.05075
$52$	0	0
$53$	6.23725	0.856752	0.428376	−	0.903601i	$-0.359086\pi$
$53$	6.23725	0.856752	0.428376	+	0.903601i	$0.359086\pi$
$54$	0	0
$55$	−3.10252	−0.418344
$56$	0	0
$57$	15.0474	1.99308
$58$	0	0
$59$	−0.453451	−0.0590343	−0.0295171	−	0.999564i	$-0.509397\pi$
$59$	−0.453451	−0.0590343	−0.0295171	+	0.999564i	$0.509397\pi$
$60$	0	0
$61$	1.80500	0.231106	0.115553	−	0.993301i	$-0.463136\pi$
$61$	1.80500	0.231106	0.115553	+	0.993301i	$0.463136\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	23.9722	2.97339
$66$	0	0
$67$	0.386390	0.0472050	0.0236025	−	0.999721i	$-0.492486\pi$
$67$	0.386390	0.0472050	0.0236025	+	0.999721i	$0.492486\pi$
$68$	0	0
$69$	11.2043	1.34884
$70$	0	0
$71$	0.406313	0.0482205	0.0241103	−	0.999709i	$-0.492325\pi$
$71$	0.406313	0.0482205	0.0241103	+	0.999709i	$0.492325\pi$
$72$	0	0
$73$	−2.47266	−0.289403	−0.144701	−	0.989475i	$-0.546222\pi$
$73$	−2.47266	−0.289403	−0.144701	+	0.989475i	$0.546222\pi$
$74$	0	0
$75$	−17.9336	−2.07080
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.1588	1.59299	0.796497	−	0.604643i	$-0.206684\pi$
$79$	14.1588	1.59299	0.796497	+	0.604643i	$0.206684\pi$
$80$	0	0
$81$	−8.56942	−0.952157
$82$	0	0
$83$	−7.79158	−0.855237	−0.427618	−	0.903959i	$-0.640647\pi$
$83$	−7.79158	−0.855237	−0.427618	+	0.903959i	$0.640647\pi$
$84$	0	0
$85$	−10.5967	−1.14937
$86$	0	0
$87$	17.9021	1.91931
$88$	0	0
$89$	5.14645	0.545523	0.272761	−	0.962082i	$-0.412063\pi$
$89$	5.14645	0.545523	0.272761	+	0.962082i	$0.412063\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	24.0815	2.49714
$94$	0	0
$95$	−21.2495	−2.18015
$96$	0	0
$97$	11.4082	1.15833	0.579165	−	0.815211i	$-0.303379\pi$
$97$	11.4082	1.15833	0.579165	+	0.815211i	$0.303379\pi$
$98$	0	0
$99$	−2.78221	−0.279623
$100$	0	0
$101$	−7.23469	−0.719878	−0.359939	−	0.932976i	$-0.617203\pi$
$101$	−7.23469	−0.719878	−0.359939	+	0.932976i	$0.617203\pi$
$102$	0	0
$103$	9.81614	0.967213	0.483606	−	0.875286i	$-0.339327\pi$
$103$	9.81614	0.967213	0.483606	+	0.875286i	$0.339327\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.91514	0.958533	0.479266	−	0.877669i	$-0.340903\pi$
$107$	9.91514	0.958533	0.479266	+	0.877669i	$0.340903\pi$
$108$	0	0
$109$	4.44277	0.425540	0.212770	−	0.977102i	$-0.431751\pi$
$109$	4.44277	0.425540	0.212770	+	0.977102i	$0.431751\pi$
$110$	0	0
$111$	−17.4819	−1.65931
$112$	0	0
$113$	13.0359	1.22631	0.613157	−	0.789961i	$-0.289899\pi$
$113$	13.0359	1.22631	0.613157	+	0.789961i	$0.289899\pi$
$114$	0	0
$115$	−15.8224	−1.47544
$116$	0	0
$117$	21.4973	1.98743
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.2135	−0.928503
$122$	0	0
$123$	−2.47735	−0.223375
$124$	0	0
$125$	7.83317	0.700620
$126$	0	0
$127$	21.6259	1.91899	0.959496	−	0.281722i	$-0.0909057\pi$
$127$	21.6259	1.91899	0.959496	+	0.281722i	$0.0909057\pi$
$128$	0	0
$129$	−10.5534	−0.929175
$130$	0	0
$131$	−2.13139	−0.186221	−0.0931104	−	0.995656i	$-0.529681\pi$
$131$	−2.13139	−0.186221	−0.0931104	+	0.995656i	$0.529681\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.18951	−0.102377
$136$	0	0
$137$	2.35667	0.201344	0.100672	−	0.994920i	$-0.467901\pi$
$137$	2.35667	0.201344	0.100672	+	0.994920i	$0.467901\pi$
$138$	0	0
$139$	−12.9520	−1.09858	−0.549288	−	0.835633i	$-0.685101\pi$
$139$	−12.9520	−1.09858	−0.549288	+	0.835633i	$0.685101\pi$
$140$	0	0
$141$	−29.1360	−2.45369
$142$	0	0
$143$	−6.07682	−0.508169
$144$	0	0
$145$	−25.2808	−2.09946
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.5047	−1.18827	−0.594137	−	0.804364i	$-0.702506\pi$
$149$	−14.5047	−1.18827	−0.594137	+	0.804364i	$0.702506\pi$
$150$	0	0
$151$	11.6316	0.946569	0.473285	−	0.880910i	$-0.343068\pi$
$151$	11.6316	0.946569	0.473285	+	0.880910i	$0.343068\pi$
$152$	0	0
$153$	−9.50268	−0.768246
$154$	0	0
$155$	−34.0072	−2.73152
$156$	0	0
$157$	−5.60289	−0.447160	−0.223580	−	0.974686i	$-0.571774\pi$
$157$	−5.60289	−0.447160	−0.223580	+	0.974686i	$0.571774\pi$
$158$	0	0
$159$	−15.4518	−1.22541
$160$	0	0
$161$	0	0
$162$	0	0
$163$	7.42331	0.581439	0.290719	−	0.956808i	$-0.406105\pi$
$163$	7.42331	0.581439	0.290719	+	0.956808i	$0.406105\pi$
$164$	0	0
$165$	7.68603	0.598356
$166$	0	0
$167$	−17.5163	−1.35545	−0.677726	−	0.735314i	$-0.737035\pi$
$167$	−17.5163	−1.35545	−0.677726	+	0.735314i	$0.737035\pi$
$168$	0	0
$169$	33.9537	2.61182
$170$	0	0
$171$	−19.0556	−1.45722
$172$	0	0
$173$	−9.59063	−0.729162	−0.364581	−	0.931172i	$-0.618788\pi$
$173$	−9.59063	−0.729162	−0.364581	+	0.931172i	$0.618788\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.12336	0.0844366
$178$	0	0
$179$	24.6408	1.84174	0.920870	−	0.389871i	$-0.127480\pi$
$179$	24.6408	1.84174	0.920870	+	0.389871i	$0.127480\pi$
$180$	0	0
$181$	3.27455	0.243395	0.121698	−	0.992567i	$-0.461166\pi$
$181$	3.27455	0.243395	0.121698	+	0.992567i	$0.461166\pi$
$182$	0	0
$183$	−4.47160	−0.330551
$184$	0	0
$185$	24.6874	1.81506
$186$	0	0
$187$	2.68620	0.196434
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.00730	0.0728855	0.0364428	−	0.999336i	$-0.488397\pi$
$191$	1.00730	0.0728855	0.0364428	+	0.999336i	$0.488397\pi$
$192$	0	0
$193$	19.9178	1.43371	0.716857	−	0.697220i	$-0.245580\pi$
$193$	19.9178	1.43371	0.716857	+	0.697220i	$0.245580\pi$
$194$	0	0
$195$	−59.3876	−4.25283
$196$	0	0
$197$	7.96044	0.567158	0.283579	−	0.958949i	$-0.408478\pi$
$197$	7.96044	0.567158	0.283579	+	0.958949i	$0.408478\pi$
$198$	0	0
$199$	16.1963	1.14813	0.574064	−	0.818811i	$-0.305366\pi$
$199$	16.1963	1.14813	0.574064	+	0.818811i	$0.305366\pi$
$200$	0	0
$201$	−0.957221	−0.0675172
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3.49844	0.244341
$206$	0	0
$207$	−14.1888	−0.986193
$208$	0	0
$209$	5.38661	0.372600
$210$	0	0
$211$	10.1084	0.695894	0.347947	−	0.937514i	$-0.386879\pi$
$211$	10.1084	0.695894	0.347947	+	0.937514i	$0.386879\pi$
$212$	0	0
$213$	−1.00658	−0.0689696
$214$	0	0
$215$	14.9032	1.01639
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.12564	0.413932
$220$	0	0
$221$	−20.7554	−1.39616
$222$	0	0
$223$	−5.60799	−0.375539	−0.187770	−	0.982213i	$-0.560126\pi$
$223$	−5.60799	−0.375539	−0.187770	+	0.982213i	$0.560126\pi$
$224$	0	0
$225$	22.7107	1.51405
$226$	0	0
$227$	5.16589	0.342872	0.171436	−	0.985195i	$-0.445159\pi$
$227$	5.16589	0.342872	0.171436	+	0.985195i	$0.445159\pi$
$228$	0	0
$229$	−6.39496	−0.422591	−0.211296	−	0.977422i	$-0.567768\pi$
$229$	−6.39496	−0.422591	−0.211296	+	0.977422i	$0.567768\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.1923	0.929771	0.464885	−	0.885371i	$-0.346096\pi$
$233$	14.1923	0.929771	0.464885	+	0.885371i	$0.346096\pi$
$234$	0	0
$235$	41.1450	2.68400
$236$	0	0
$237$	−35.0763	−2.27845
$238$	0	0
$239$	−15.2318	−0.985266	−0.492633	−	0.870237i	$-0.663966\pi$
$239$	−15.2318	−0.985266	−0.492633	+	0.870237i	$0.663966\pi$
$240$	0	0
$241$	6.90521	0.444804	0.222402	−	0.974955i	$-0.428610\pi$
$241$	6.90521	0.444804	0.222402	+	0.974955i	$0.428610\pi$
$242$	0	0
$243$	22.2495	1.42730
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−41.6207	−2.64826
$248$	0	0
$249$	19.3024	1.22324
$250$	0	0
$251$	21.2459	1.34103	0.670515	−	0.741896i	$-0.266073\pi$
$251$	21.2459	1.34103	0.670515	+	0.741896i	$0.266073\pi$
$252$	0	0
$253$	4.01088	0.252162
$254$	0	0
$255$	26.2517	1.64395
$256$	0	0
$257$	6.06200	0.378137	0.189068	−	0.981964i	$-0.439453\pi$
$257$	6.06200	0.378137	0.189068	+	0.981964i	$0.439453\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−22.6708	−1.40329
$262$	0	0
$263$	−15.1984	−0.937171	−0.468586	−	0.883418i	$-0.655236\pi$
$263$	−15.1984	−0.937171	−0.468586	+	0.883418i	$0.655236\pi$
$264$	0	0
$265$	21.8206	1.34043
$266$	0	0
$267$	−12.7495	−0.780260
$268$	0	0
$269$	−18.8933	−1.15195	−0.575973	−	0.817469i	$-0.695377\pi$
$269$	−18.8933	−1.15195	−0.575973	+	0.817469i	$0.695377\pi$
$270$	0	0
$271$	−14.9578	−0.908621	−0.454310	−	0.890843i	$-0.650114\pi$
$271$	−14.9578	−0.908621	−0.454310	+	0.890843i	$0.650114\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.41982	−0.387130
$276$	0	0
$277$	−13.0971	−0.786931	−0.393465	−	0.919339i	$-0.628724\pi$
$277$	−13.0971	−0.786931	−0.393465	+	0.919339i	$0.628724\pi$
$278$	0	0
$279$	−30.4962	−1.82576
$280$	0	0
$281$	−21.2844	−1.26972	−0.634860	−	0.772627i	$-0.718942\pi$
$281$	−21.2844	−1.26972	−0.634860	+	0.772627i	$0.718942\pi$
$282$	0	0
$283$	−22.7683	−1.35343	−0.676717	−	0.736243i	$-0.736598\pi$
$283$	−22.7683	−1.35343	−0.676717	+	0.736243i	$0.736598\pi$
$284$	0	0
$285$	52.6424	3.11826
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−7.82525	−0.460309
$290$	0	0
$291$	−28.2621	−1.65675
$292$	0	0
$293$	−4.51752	−0.263916	−0.131958	−	0.991255i	$-0.542126\pi$
$293$	−4.51752	−0.263916	−0.131958	+	0.991255i	$0.542126\pi$
$294$	0	0
$295$	−1.58637	−0.0923620
$296$	0	0
$297$	0.301534	0.0174968
$298$	0	0
$299$	−30.9908	−1.79224
$300$	0	0
$301$	0	0
$302$	0	0
$303$	17.9228	1.02964
$304$	0	0
$305$	6.31467	0.361577
$306$	0	0
$307$	−4.60882	−0.263039	−0.131520	−	0.991314i	$-0.541986\pi$
$307$	−4.60882	−0.263039	−0.131520	+	0.991314i	$0.541986\pi$
$308$	0	0
$309$	−24.3180	−1.38340
$310$	0	0
$311$	15.9788	0.906077	0.453039	−	0.891491i	$-0.350340\pi$
$311$	15.9788	0.906077	0.453039	+	0.891491i	$0.350340\pi$
$312$	0	0
$313$	32.4129	1.83208	0.916042	−	0.401082i	$-0.131366\pi$
$313$	32.4129	1.83208	0.916042	+	0.401082i	$0.131366\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−13.1162	−0.736679	−0.368340	−	0.929691i	$-0.620074\pi$
$317$	−13.1162	−0.736679	−0.368340	+	0.929691i	$0.620074\pi$
$318$	0	0
$319$	6.40854	0.358809
$320$	0	0
$321$	−24.5632	−1.37099
$322$	0	0
$323$	18.3981	1.02370
$324$	0	0
$325$	49.6040	2.75153
$326$	0	0
$327$	−11.0063	−0.608649
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.6358	1.46403	0.732017	−	0.681286i	$-0.238579\pi$
$331$	26.6358	1.46403	0.732017	+	0.681286i	$0.238579\pi$
$332$	0	0
$333$	22.1387	1.21319
$334$	0	0
$335$	1.35176	0.0738545
$336$	0	0
$337$	−22.0648	−1.20194	−0.600972	−	0.799270i	$-0.705220\pi$
$337$	−22.0648	−1.20194	−0.600972	+	0.799270i	$0.705220\pi$
$338$	0	0
$339$	−32.2945	−1.75399
$340$	0	0
$341$	8.62062	0.466833
$342$	0	0
$343$	0	0
$344$	0	0
$345$	39.1975	2.11032
$346$	0	0
$347$	29.4858	1.58288	0.791440	−	0.611246i	$-0.209332\pi$
$347$	29.4858	1.58288	0.791440	+	0.611246i	$0.209332\pi$
$348$	0	0
$349$	14.9842	0.802084	0.401042	−	0.916060i	$-0.368648\pi$
$349$	14.9842	0.802084	0.401042	+	0.916060i	$0.368648\pi$
$350$	0	0
$351$	−2.32986	−0.124359
$352$	0	0
$353$	−18.2650	−0.972149	−0.486075	−	0.873917i	$-0.661572\pi$
$353$	−18.2650	−0.972149	−0.486075	+	0.873917i	$0.661572\pi$
$354$	0	0
$355$	1.42146	0.0754433
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.1774	1.06492	0.532462	−	0.846454i	$-0.321267\pi$
$359$	20.1774	1.06492	0.532462	+	0.846454i	$0.321267\pi$
$360$	0	0
$361$	17.8935	0.941761
$362$	0	0
$363$	25.3025	1.32803
$364$	0	0
$365$	−8.65044	−0.452785
$366$	0	0
$367$	−17.2336	−0.899585	−0.449792	−	0.893133i	$-0.648502\pi$
$367$	−17.2336	−0.899585	−0.449792	+	0.893133i	$0.648502\pi$
$368$	0	0
$369$	3.13725	0.163319
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.7019	−0.554125	−0.277062	−	0.960852i	$-0.589361\pi$
$373$	−10.7019	−0.554125	−0.277062	+	0.960852i	$0.589361\pi$
$374$	0	0
$375$	−19.4055	−1.00209
$376$	0	0
$377$	−49.5168	−2.55024
$378$	0	0
$379$	24.9770	1.28298	0.641492	−	0.767130i	$-0.278316\pi$
$379$	24.9770	1.28298	0.641492	+	0.767130i	$0.278316\pi$
$380$	0	0
$381$	−53.5750	−2.74473
$382$	0	0
$383$	4.22751	0.216016	0.108008	−	0.994150i	$-0.465553\pi$
$383$	4.22751	0.216016	0.108008	+	0.994150i	$0.465553\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	13.3646	0.679359
$388$	0	0
$389$	25.5041	1.29311	0.646555	−	0.762867i	$-0.276209\pi$
$389$	25.5041	1.29311	0.646555	+	0.762867i	$0.276209\pi$
$390$	0	0
$391$	13.6992	0.692798
$392$	0	0
$393$	5.28020	0.266351
$394$	0	0
$395$	49.5338	2.49231
$396$	0	0
$397$	30.2426	1.51783	0.758917	−	0.651187i	$-0.225729\pi$
$397$	30.2426	1.51783	0.758917	+	0.651187i	$0.225729\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.81709	0.290491	0.145246	−	0.989396i	$-0.453603\pi$
$401$	5.81709	0.290491	0.145246	+	0.989396i	$0.453603\pi$
$402$	0	0
$403$	−66.6089	−3.31802
$404$	0	0
$405$	−29.9796	−1.48970
$406$	0	0
$407$	−6.25812	−0.310203
$408$	0	0
$409$	0.436763	0.0215965	0.0107983	−	0.999942i	$-0.496563\pi$
$409$	0.436763	0.0215965	0.0107983	+	0.999942i	$0.496563\pi$
$410$	0	0
$411$	−5.83829	−0.287982
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−27.2583	−1.33806
$416$	0	0
$417$	32.0867	1.57129
$418$	0	0
$419$	7.44035	0.363485	0.181743	−	0.983346i	$-0.441826\pi$
$419$	7.44035	0.363485	0.181743	+	0.983346i	$0.441826\pi$
$420$	0	0
$421$	22.7010	1.10638	0.553189	−	0.833056i	$-0.313411\pi$
$421$	22.7010	1.10638	0.553189	+	0.833056i	$0.313411\pi$
$422$	0	0
$423$	36.8971	1.79400
$424$	0	0
$425$	−21.9270	−1.06361
$426$	0	0
$427$	0	0
$428$	0	0
$429$	15.0544	0.726833
$430$	0	0
$431$	33.3117	1.60457	0.802285	−	0.596942i	$-0.203618\pi$
$431$	33.3117	1.60457	0.802285	+	0.596942i	$0.203618\pi$
$432$	0	0
$433$	13.8756	0.666816	0.333408	−	0.942783i	$-0.391801\pi$
$433$	13.8756	0.666816	0.333408	+	0.942783i	$0.391801\pi$
$434$	0	0
$435$	62.6294	3.00285
$436$	0	0
$437$	27.4709	1.31411
$438$	0	0
$439$	−25.0594	−1.19602	−0.598010	−	0.801488i	$-0.704042\pi$
$439$	−25.0594	−1.19602	−0.598010	+	0.801488i	$0.704042\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.6739	0.602154	0.301077	−	0.953600i	$-0.402654\pi$
$443$	12.6739	0.602154	0.301077	+	0.953600i	$0.402654\pi$
$444$	0	0
$445$	18.0045	0.853497
$446$	0	0
$447$	35.9333	1.69959
$448$	0	0
$449$	−35.8184	−1.69037	−0.845187	−	0.534470i	$-0.820511\pi$
$449$	−35.8184	−1.69037	−0.845187	+	0.534470i	$0.820511\pi$
$450$	0	0
$451$	−0.886832	−0.0417593
$452$	0	0
$453$	−28.8156	−1.35388
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−21.1037	−0.987190	−0.493595	−	0.869692i	$-0.664318\pi$
$457$	−21.1037	−0.987190	−0.493595	+	0.869692i	$0.664318\pi$
$458$	0	0
$459$	1.02989	0.0480713
$460$	0	0
$461$	33.8364	1.57592	0.787958	−	0.615729i	$-0.211138\pi$
$461$	33.8364	1.57592	0.787958	+	0.615729i	$0.211138\pi$
$462$	0	0
$463$	31.5226	1.46498	0.732490	−	0.680778i	$-0.238358\pi$
$463$	31.5226	1.46498	0.732490	+	0.680778i	$0.238358\pi$
$464$	0	0
$465$	84.2477	3.90689
$466$	0	0
$467$	−33.1549	−1.53422	−0.767112	−	0.641513i	$-0.778307\pi$
$467$	−33.1549	−1.53422	−0.767112	+	0.641513i	$0.778307\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	13.8803	0.639571
$472$	0	0
$473$	−3.77787	−0.173707
$474$	0	0
$475$	−43.9700	−2.01748
$476$	0	0
$477$	19.5678	0.895948
$478$	0	0
$479$	7.80316	0.356536	0.178268	−	0.983982i	$-0.442951\pi$
$479$	7.80316	0.356536	0.178268	+	0.983982i	$0.442951\pi$
$480$	0	0
$481$	48.3545	2.20478
$482$	0	0
$483$	0	0
$484$	0	0
$485$	39.9109	1.81226
$486$	0	0
$487$	11.9654	0.542202	0.271101	−	0.962551i	$-0.412612\pi$
$487$	11.9654	0.542202	0.271101	+	0.962551i	$0.412612\pi$
$488$	0	0
$489$	−18.3901	−0.831630
$490$	0	0
$491$	4.00762	0.180861	0.0904307	−	0.995903i	$-0.471176\pi$
$491$	4.00762	0.180861	0.0904307	+	0.995903i	$0.471176\pi$
$492$	0	0
$493$	21.8884	0.985806
$494$	0	0
$495$	−9.73339	−0.437483
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−11.7405	−0.525578	−0.262789	−	0.964853i	$-0.584642\pi$
$499$	−11.7405	−0.525578	−0.262789	+	0.964853i	$0.584642\pi$
$500$	0	0
$501$	43.3940	1.93870
$502$	0	0
$503$	5.10200	0.227487	0.113743	−	0.993510i	$-0.463716\pi$
$503$	5.10200	0.227487	0.113743	+	0.993510i	$0.463716\pi$
$504$	0	0
$505$	−25.3101	−1.12628
$506$	0	0
$507$	−84.1151	−3.73568
$508$	0	0
$509$	22.1952	0.983784	0.491892	−	0.870656i	$-0.336305\pi$
$509$	22.1952	0.983784	0.491892	+	0.870656i	$0.336305\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.06524	0.0911824
$514$	0	0
$515$	34.3411	1.51325
$516$	0	0
$517$	−10.4300	−0.458711
$518$	0	0
$519$	23.7593	1.04292
$520$	0	0
$521$	26.9053	1.17874	0.589370	−	0.807863i	$-0.299376\pi$
$521$	26.9053	1.17874	0.589370	+	0.807863i	$0.299376\pi$
$522$	0	0
$523$	−28.5119	−1.24674	−0.623369	−	0.781928i	$-0.714236\pi$
$523$	−28.5119	−1.24674	−0.623369	+	0.781928i	$0.714236\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	29.4438	1.28259
$528$	0	0
$529$	−2.54517	−0.110660
$530$	0	0
$531$	−1.42259	−0.0617351
$532$	0	0
$533$	6.85228	0.296805
$534$	0	0
$535$	34.6875	1.49967
$536$	0	0
$537$	−61.0438	−2.63423
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−13.1630	−0.565920	−0.282960	−	0.959132i	$-0.591316\pi$
$541$	−13.1630	−0.565920	−0.282960	+	0.959132i	$0.591316\pi$
$542$	0	0
$543$	−8.11220	−0.348128
$544$	0	0
$545$	15.5427	0.665778
$546$	0	0
$547$	−13.8254	−0.591133	−0.295567	−	0.955322i	$-0.595508\pi$
$547$	−13.8254	−0.591133	−0.295567	+	0.955322i	$0.595508\pi$
$548$	0	0
$549$	5.66273	0.241679
$550$	0	0
$551$	43.8927	1.86989
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−61.1594	−2.59607
$556$	0	0
$557$	−34.3090	−1.45372	−0.726861	−	0.686785i	$-0.759021\pi$
$557$	−34.3090	−1.45372	−0.726861	+	0.686785i	$0.759021\pi$
$558$	0	0
$559$	29.1904	1.23462
$560$	0	0
$561$	−6.65465	−0.280960
$562$	0	0
$563$	13.4964	0.568806	0.284403	−	0.958705i	$-0.408205\pi$
$563$	13.4964	0.568806	0.284403	+	0.958705i	$0.408205\pi$
$564$	0	0
$565$	45.6053	1.91863
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.21418	−0.302434	−0.151217	−	0.988501i	$-0.548319\pi$
$569$	−7.21418	−0.302434	−0.151217	+	0.988501i	$0.548319\pi$
$570$	0	0
$571$	−9.09658	−0.380680	−0.190340	−	0.981718i	$-0.560959\pi$
$571$	−9.09658	−0.380680	−0.190340	+	0.981718i	$0.560959\pi$
$572$	0	0
$573$	−2.49543	−0.104248
$574$	0	0
$575$	−32.7401	−1.36536
$576$	0	0
$577$	25.5931	1.06545	0.532727	−	0.846287i	$-0.321167\pi$
$577$	25.5931	1.06545	0.532727	+	0.846287i	$0.321167\pi$
$578$	0	0
$579$	−49.3433	−2.05064
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−5.53139	−0.229087
$584$	0	0
$585$	75.2069	3.10942
$586$	0	0
$587$	−14.8861	−0.614416	−0.307208	−	0.951642i	$-0.599395\pi$
$587$	−14.8861	−0.614416	−0.307208	+	0.951642i	$0.599395\pi$
$588$	0	0
$589$	59.0435	2.43284
$590$	0	0
$591$	−19.7208	−0.811204
$592$	0	0
$593$	−9.85853	−0.404841	−0.202421	−	0.979299i	$-0.564881\pi$
$593$	−9.85853	−0.404841	−0.202421	+	0.979299i	$0.564881\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−40.1239	−1.64216
$598$	0	0
$599$	11.2816	0.460953	0.230476	−	0.973078i	$-0.425972\pi$
$599$	11.2816	0.460953	0.230476	+	0.973078i	$0.425972\pi$
$600$	0	0
$601$	−4.84169	−0.197497	−0.0987483	−	0.995112i	$-0.531484\pi$
$601$	−4.84169	−0.197497	−0.0987483	+	0.995112i	$0.531484\pi$
$602$	0	0
$603$	1.21220	0.0493646
$604$	0	0
$605$	−35.7314	−1.45269
$606$	0	0
$607$	28.9727	1.17596	0.587982	−	0.808874i	$-0.299923\pi$
$607$	28.9727	1.17596	0.587982	+	0.808874i	$0.299923\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	80.5894	3.26030
$612$	0	0
$613$	4.05986	0.163976	0.0819881	−	0.996633i	$-0.473873\pi$
$613$	4.05986	0.163976	0.0819881	+	0.996633i	$0.473873\pi$
$614$	0	0
$615$	−8.66684	−0.349481
$616$	0	0
$617$	31.0323	1.24931	0.624657	−	0.780900i	$-0.285239\pi$
$617$	31.0323	1.24931	0.624657	+	0.780900i	$0.285239\pi$
$618$	0	0
$619$	12.8594	0.516862	0.258431	−	0.966030i	$-0.416794\pi$
$619$	12.8594	0.516862	0.258431	+	0.966030i	$0.416794\pi$
$620$	0	0
$621$	1.53778	0.0617088
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−8.79140	−0.351656
$626$	0	0
$627$	−13.3445	−0.532929
$628$	0	0
$629$	−21.3747	−0.852264
$630$	0	0
$631$	−28.0819	−1.11792	−0.558961	−	0.829194i	$-0.688800\pi$
$631$	−28.0819	−1.11792	−0.558961	+	0.829194i	$0.688800\pi$
$632$	0	0
$633$	−25.0421	−0.995335
$634$	0	0
$635$	75.6570	3.00235
$636$	0	0
$637$	0	0
$638$	0	0
$639$	1.27471	0.0504266
$640$	0	0
$641$	−29.1285	−1.15051	−0.575253	−	0.817976i	$-0.695096\pi$
$641$	−29.1285	−1.15051	−0.575253	+	0.817976i	$0.695096\pi$
$642$	0	0
$643$	34.0791	1.34395	0.671974	−	0.740575i	$-0.265447\pi$
$643$	34.0791	1.34395	0.671974	+	0.740575i	$0.265447\pi$
$644$	0	0
$645$	−36.9204	−1.45374
$646$	0	0
$647$	−7.04612	−0.277012	−0.138506	−	0.990362i	$-0.544230\pi$
$647$	−7.04612	−0.277012	−0.138506	+	0.990362i	$0.544230\pi$
$648$	0	0
$649$	0.402135	0.0157852
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13.5790	0.531388	0.265694	−	0.964057i	$-0.414399\pi$
$653$	13.5790	0.531388	0.265694	+	0.964057i	$0.414399\pi$
$654$	0	0
$655$	−7.45654	−0.291351
$656$	0	0
$657$	−7.75735	−0.302643
$658$	0	0
$659$	38.7358	1.50893	0.754466	−	0.656339i	$-0.227896\pi$
$659$	38.7358	1.50893	0.754466	+	0.656339i	$0.227896\pi$
$660$	0	0
$661$	−26.6698	−1.03734	−0.518668	−	0.854976i	$-0.673572\pi$
$661$	−26.6698	−1.03734	−0.518668	+	0.854976i	$0.673572\pi$
$662$	0	0
$663$	51.4184	1.99693
$664$	0	0
$665$	0	0
$666$	0	0
$667$	32.6825	1.26547
$668$	0	0
$669$	13.8929	0.537132
$670$	0	0
$671$	−1.60073	−0.0617955
$672$	0	0
$673$	−43.4682	−1.67558	−0.837789	−	0.545995i	$-0.816152\pi$
$673$	−43.4682	−1.67558	−0.837789	+	0.545995i	$0.816152\pi$
$674$	0	0
$675$	−2.46137	−0.0947381
$676$	0	0
$677$	37.2749	1.43259	0.716296	−	0.697797i	$-0.245836\pi$
$677$	37.2749	1.43259	0.716296	+	0.697797i	$0.245836\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.7977	−0.490409
$682$	0	0
$683$	−27.7986	−1.06368	−0.531842	−	0.846844i	$-0.678500\pi$
$683$	−27.7986	−1.06368	−0.531842	+	0.846844i	$0.678500\pi$
$684$	0	0
$685$	8.24466	0.315012
$686$	0	0
$687$	15.8425	0.604431
$688$	0	0
$689$	42.7394	1.62824
$690$	0	0
$691$	−31.1281	−1.18417	−0.592084	−	0.805876i	$-0.701695\pi$
$691$	−31.1281	−1.18417	−0.592084	+	0.805876i	$0.701695\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−45.3118	−1.71877
$696$	0	0
$697$	−3.02899	−0.114731
$698$	0	0
$699$	−35.1593	−1.32985
$700$	0	0
$701$	−12.0547	−0.455301	−0.227651	−	0.973743i	$-0.573104\pi$
$701$	−12.0547	−0.455301	−0.227651	+	0.973743i	$0.573104\pi$
$702$	0	0
$703$	−42.8625	−1.61659
$704$	0	0
$705$	−101.930	−3.83892
$706$	0	0
$707$	0	0
$708$	0	0
$709$	46.4903	1.74598	0.872989	−	0.487739i	$-0.162178\pi$
$709$	46.4903	1.74598	0.872989	+	0.487739i	$0.162178\pi$
$710$	0	0
$711$	44.4198	1.66587
$712$	0	0
$713$	43.9638	1.64646
$714$	0	0
$715$	−21.2594	−0.795055
$716$	0	0
$717$	37.7346	1.40922
$718$	0	0
$719$	−12.5884	−0.469467	−0.234734	−	0.972060i	$-0.575422\pi$
$719$	−12.5884	−0.469467	−0.234734	+	0.972060i	$0.575422\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−17.1066	−0.636201
$724$	0	0
$725$	−52.3117	−1.94281
$726$	0	0
$727$	6.55299	0.243037	0.121519	−	0.992589i	$-0.461224\pi$
$727$	6.55299	0.243037	0.121519	+	0.992589i	$0.461224\pi$
$728$	0	0
$729$	−29.4114	−1.08931
$730$	0	0
$731$	−12.9034	−0.477248
$732$	0	0
$733$	−6.10776	−0.225595	−0.112798	−	0.993618i	$-0.535981\pi$
$733$	−6.10776	−0.225595	−0.112798	+	0.993618i	$0.535981\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−0.342663	−0.0126221
$738$	0	0
$739$	22.0038	0.809421	0.404711	−	0.914445i	$-0.367372\pi$
$739$	22.0038	0.809421	0.404711	+	0.914445i	$0.367372\pi$
$740$	0	0
$741$	103.109	3.78780
$742$	0	0
$743$	34.6353	1.27065	0.635323	−	0.772247i	$-0.280867\pi$
$743$	34.6353	1.27065	0.635323	+	0.772247i	$0.280867\pi$
$744$	0	0
$745$	−50.7439	−1.85911
$746$	0	0
$747$	−24.4441	−0.894363
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−20.3539	−0.742725	−0.371363	−	0.928488i	$-0.621109\pi$
$751$	−20.3539	−0.742725	−0.371363	+	0.928488i	$0.621109\pi$
$752$	0	0
$753$	−52.6335	−1.91807
$754$	0	0
$755$	40.6925	1.48095
$756$	0	0
$757$	29.9426	1.08828	0.544141	−	0.838994i	$-0.316856\pi$
$757$	29.9426	1.08828	0.544141	+	0.838994i	$0.316856\pi$
$758$	0	0
$759$	−9.93633	−0.360666
$760$	0	0
$761$	−53.3438	−1.93371	−0.966855	−	0.255326i	$-0.917817\pi$
$761$	−53.3438	−1.93371	−0.966855	+	0.255326i	$0.917817\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−33.2445	−1.20196
$766$	0	0
$767$	−3.10717	−0.112193
$768$	0	0
$769$	−0.577705	−0.0208326	−0.0104163	−	0.999946i	$-0.503316\pi$
$769$	−0.577705	−0.0208326	−0.0104163	+	0.999946i	$0.503316\pi$
$770$	0	0
$771$	−15.0177	−0.540848
$772$	0	0
$773$	7.91026	0.284512	0.142256	−	0.989830i	$-0.454564\pi$
$773$	7.91026	0.284512	0.142256	+	0.989830i	$0.454564\pi$
$774$	0	0
$775$	−70.3686	−2.52771
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.07400	−0.217624
$780$	0	0
$781$	−0.360331	−0.0128937
$782$	0	0
$783$	2.45704	0.0878075
$784$	0	0
$785$	−19.6014	−0.699603
$786$	0	0
$787$	24.6891	0.880072	0.440036	−	0.897980i	$-0.354966\pi$
$787$	24.6891	0.880072	0.440036	+	0.897980i	$0.354966\pi$
$788$	0	0
$789$	37.6516	1.34043
$790$	0	0
$791$	0	0
$792$	0	0
$793$	12.3683	0.439213
$794$	0	0
$795$	−54.0572	−1.91721
$796$	0	0
$797$	11.1868	0.396258	0.198129	−	0.980176i	$-0.436514\pi$
$797$	11.1868	0.396258	0.198129	+	0.980176i	$0.436514\pi$
$798$	0	0
$799$	−35.6238	−1.26028
$800$	0	0
$801$	16.1457	0.570480
$802$	0	0
$803$	2.19283	0.0773834
$804$	0	0
$805$	0	0
$806$	0	0
$807$	46.8053	1.64763
$808$	0	0
$809$	4.29964	0.151167	0.0755836	−	0.997139i	$-0.475918\pi$
$809$	4.29964	0.151167	0.0755836	+	0.997139i	$0.475918\pi$
$810$	0	0
$811$	−19.4277	−0.682199	−0.341100	−	0.940027i	$-0.610799\pi$
$811$	−19.4277	−0.682199	−0.341100	+	0.940027i	$0.610799\pi$
$812$	0	0
$813$	37.0556	1.29960
$814$	0	0
$815$	25.9700	0.909689
$816$	0	0
$817$	−25.8750	−0.905252
$818$	0	0
$819$	0	0
$820$	0	0
$821$	47.1341	1.64499	0.822495	−	0.568772i	$-0.192581\pi$
$821$	47.1341	1.64499	0.822495	+	0.568772i	$0.192581\pi$
$822$	0	0
$823$	5.94967	0.207392	0.103696	−	0.994609i	$-0.466933\pi$
$823$	5.94967	0.207392	0.103696	+	0.994609i	$0.466933\pi$
$824$	0	0
$825$	15.9041	0.553710
$826$	0	0
$827$	4.01224	0.139519	0.0697596	−	0.997564i	$-0.477777\pi$
$827$	4.01224	0.139519	0.0697596	+	0.997564i	$0.477777\pi$
$828$	0	0
$829$	−3.09097	−0.107354	−0.0536769	−	0.998558i	$-0.517094\pi$
$829$	−3.09097	−0.107354	−0.0536769	+	0.998558i	$0.517094\pi$
$830$	0	0
$831$	32.4462	1.12555
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−61.2797	−2.12067
$836$	0	0
$837$	3.30516	0.114243
$838$	0	0
$839$	−35.1218	−1.21254	−0.606270	−	0.795259i	$-0.707335\pi$
$839$	−35.1218	−1.21254	−0.606270	+	0.795259i	$0.707335\pi$
$840$	0	0
$841$	23.2198	0.800682
$842$	0	0
$843$	52.7288	1.81608
$844$	0	0
$845$	118.785	4.08632
$846$	0	0
$847$	0	0
$848$	0	0
$849$	56.4050	1.93581
$850$	0	0
$851$	−31.9154	−1.09405
$852$	0	0
$853$	21.6536	0.741405	0.370702	−	0.928752i	$-0.379117\pi$
$853$	21.6536	0.741405	0.370702	+	0.928752i	$0.379117\pi$
$854$	0	0
$855$	−66.6649	−2.27989
$856$	0	0
$857$	54.7616	1.87062	0.935310	−	0.353828i	$-0.115120\pi$
$857$	54.7616	1.87062	0.935310	+	0.353828i	$0.115120\pi$
$858$	0	0
$859$	−0.263155	−0.00897874	−0.00448937	−	0.999990i	$-0.501429\pi$
$859$	−0.263155	−0.00897874	−0.00448937	+	0.999990i	$0.501429\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−54.0124	−1.83860	−0.919302	−	0.393552i	$-0.871246\pi$
$863$	−54.0124	−1.83860	−0.919302	+	0.393552i	$0.871246\pi$
$864$	0	0
$865$	−33.5522	−1.14081
$866$	0	0
$867$	19.3859	0.658378
$868$	0	0
$869$	−12.5565	−0.425950
$870$	0	0
$871$	2.64765	0.0897122
$872$	0	0
$873$	35.7904	1.21132
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.54025	0.0520105	0.0260053	−	0.999662i	$-0.491721\pi$
$877$	1.54025	0.0520105	0.0260053	+	0.999662i	$0.491721\pi$
$878$	0	0
$879$	11.1915	0.377478
$880$	0	0
$881$	−6.28992	−0.211913	−0.105956	−	0.994371i	$-0.533790\pi$
$881$	−6.28992	−0.211913	−0.105956	+	0.994371i	$0.533790\pi$
$882$	0	0
$883$	−6.57266	−0.221188	−0.110594	−	0.993866i	$-0.535275\pi$
$883$	−6.57266	−0.221188	−0.110594	+	0.993866i	$0.535275\pi$
$884$	0	0
$885$	3.92999	0.132105
$886$	0	0
$887$	24.6143	0.826468	0.413234	−	0.910625i	$-0.364399\pi$
$887$	24.6143	0.826468	0.413234	+	0.910625i	$0.364399\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	7.59963	0.254597
$892$	0	0
$893$	−71.4361	−2.39052
$894$	0	0
$895$	86.2042	2.88149
$896$	0	0
$897$	76.7750	2.56344
$898$	0	0
$899$	70.2449	2.34280
$900$	0	0
$901$	−18.8925	−0.629402
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.4558	0.380804
$906$	0	0
$907$	−21.4285	−0.711522	−0.355761	−	0.934577i	$-0.615778\pi$
$907$	−21.4285	−0.711522	−0.355761	+	0.934577i	$0.615778\pi$
$908$	0	0
$909$	−22.6970	−0.752812
$910$	0	0
$911$	20.6989	0.685786	0.342893	−	0.939374i	$-0.388593\pi$
$911$	20.6989	0.685786	0.342893	+	0.939374i	$0.388593\pi$
$912$	0	0
$913$	6.90982	0.228682
$914$	0	0
$915$	−15.6436	−0.517162
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−13.5900	−0.448292	−0.224146	−	0.974556i	$-0.571959\pi$
$919$	−13.5900	−0.448292	−0.224146	+	0.974556i	$0.571959\pi$
$920$	0	0
$921$	11.4176	0.376224
$922$	0	0
$923$	2.78417	0.0916421
$924$	0	0
$925$	51.0839	1.67963
$926$	0	0
$927$	30.7957	1.01146
$928$	0	0
$929$	42.8414	1.40558	0.702790	−	0.711398i	$-0.251937\pi$
$929$	42.8414	1.40558	0.702790	+	0.711398i	$0.251937\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−39.5852	−1.29596
$934$	0	0
$935$	9.39750	0.307331
$936$	0	0
$937$	17.0062	0.555568	0.277784	−	0.960644i	$-0.410400\pi$
$937$	17.0062	0.555568	0.277784	+	0.960644i	$0.410400\pi$
$938$	0	0
$939$	−80.2979	−2.62042
$940$	0	0
$941$	−43.8180	−1.42843	−0.714213	−	0.699928i	$-0.753215\pi$
$941$	−43.8180	−1.42843	−0.714213	+	0.699928i	$0.753215\pi$
$942$	0	0
$943$	−4.52270	−0.147279
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.2023	−0.494008	−0.247004	−	0.969014i	$-0.579446\pi$
$947$	−15.2023	−0.494008	−0.247004	+	0.969014i	$0.579446\pi$
$948$	0	0
$949$	−16.9434	−0.550004
$950$	0	0
$951$	32.4934	1.05367
$952$	0	0
$953$	19.0486	0.617044	0.308522	−	0.951217i	$-0.400166\pi$
$953$	19.0486	0.617044	0.308522	+	0.951217i	$0.400166\pi$
$954$	0	0
$955$	3.52397	0.114033
$956$	0	0
$957$	−15.8762	−0.513204
$958$	0	0
$959$	0	0
$960$	0	0
$961$	63.4918	2.04812
$962$	0	0
$963$	31.1063	1.00239
$964$	0	0
$965$	69.6812	2.24312
$966$	0	0
$967$	−12.3278	−0.396435	−0.198218	−	0.980158i	$-0.563515\pi$
$967$	−12.3278	−0.396435	−0.198218	+	0.980158i	$0.563515\pi$
$968$	0	0
$969$	−45.5784	−1.46419
$970$	0	0
$971$	0.959044	0.0307772	0.0153886	−	0.999882i	$-0.495101\pi$
$971$	0.959044	0.0307772	0.0153886	+	0.999882i	$0.495101\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−122.886	−3.93551
$976$	0	0
$977$	−37.7183	−1.20672	−0.603358	−	0.797471i	$-0.706171\pi$
$977$	−37.7183	−1.20672	−0.603358	+	0.797471i	$0.706171\pi$
$978$	0	0
$979$	−4.56404	−0.145867
$980$	0	0
$981$	13.9381	0.445008
$982$	0	0
$983$	32.2162	1.02754	0.513768	−	0.857929i	$-0.328249\pi$
$983$	32.2162	1.02754	0.513768	+	0.857929i	$0.328249\pi$
$984$	0	0
$985$	27.8491	0.887345
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−19.2665	−0.612640
$990$	0	0
$991$	11.3658	0.361048	0.180524	−	0.983571i	$-0.442221\pi$
$991$	11.3658	0.361048	0.180524	+	0.983571i	$0.442221\pi$
$992$	0	0
$993$	−65.9861	−2.09400
$994$	0	0
$995$	56.6618	1.79630
$996$	0	0
$997$	10.3504	0.327799	0.163900	−	0.986477i	$-0.447593\pi$
$997$	10.3504	0.327799	0.163900	+	0.986477i	$0.447593\pi$
$998$	0	0
$999$	−2.39937	−0.0759127

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.r.1.4		15
7.3	odd	6		1148.2.i.e.821.4	yes	30
7.5	odd	6		1148.2.i.e.165.4	✓	30
7.6	odd	2		8036.2.a.q.1.12		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.4	✓	30	7.5	odd	6
1148.2.i.e.821.4	yes	30	7.3	odd	6
8036.2.a.q.1.12		15	7.6	odd	2
8036.2.a.r.1.4		15	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47735 q^{3} +3.49844 q^{5} +3.13725 q^{9} +O(q^{10})\)	\(q-2.47735 q^{3} +3.49844 q^{5} +3.13725 q^{9} -0.886832 q^{11} +6.85228 q^{13} -8.66684 q^{15} -3.02899 q^{17} -6.07400 q^{19} -4.52270 q^{23} +7.23905 q^{25} -0.340013 q^{27} -7.22633 q^{29} -9.72069 q^{31} +2.19699 q^{33} +7.05671 q^{37} -16.9755 q^{39} +1.00000 q^{41} +4.25996 q^{43} +10.9755 q^{45} +11.7610 q^{47} +7.50385 q^{51} +6.23725 q^{53} -3.10252 q^{55} +15.0474 q^{57} -0.453451 q^{59} +1.80500 q^{61} +23.9722 q^{65} +0.386390 q^{67} +11.2043 q^{69} +0.406313 q^{71} -2.47266 q^{73} -17.9336 q^{75} +14.1588 q^{79} -8.56942 q^{81} -7.79158 q^{83} -10.5967 q^{85} +17.9021 q^{87} +5.14645 q^{89} +24.0815 q^{93} -21.2495 q^{95} +11.4082 q^{97} -2.78221 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * q^99

Introduction

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