LMFDB - Embedded newform 5355.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5355,2,Mod(1,5355)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5355, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5355.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5355 = 3^{2} \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5355.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:	$42.7598902824$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1785)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	5355.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.73205 q^{8} +O(q^{10})$	$q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.73205 q^{8} +1.73205 q^{10} +2.00000 q^{11} +1.73205 q^{14} -5.00000 q^{16} -1.00000 q^{17} -5.46410 q^{19} -1.00000 q^{20} -3.46410 q^{22} +1.00000 q^{25} -1.00000 q^{28} -1.46410 q^{29} -2.00000 q^{31} +5.19615 q^{32} +1.73205 q^{34} +1.00000 q^{35} -4.92820 q^{37} +9.46410 q^{38} -1.73205 q^{40} +10.0000 q^{41} -8.00000 q^{43} +2.00000 q^{44} +6.92820 q^{47} +1.00000 q^{49} -1.73205 q^{50} +6.92820 q^{53} -2.00000 q^{55} -1.73205 q^{56} +2.53590 q^{58} -6.92820 q^{59} +6.53590 q^{61} +3.46410 q^{62} +1.00000 q^{64} -1.00000 q^{68} -1.73205 q^{70} +8.92820 q^{71} +11.4641 q^{73} +8.53590 q^{74} -5.46410 q^{76} -2.00000 q^{77} -4.00000 q^{79} +5.00000 q^{80} -17.3205 q^{82} -17.8564 q^{83} +1.00000 q^{85} +13.8564 q^{86} +3.46410 q^{88} -11.8564 q^{89} -12.0000 q^{94} +5.46410 q^{95} +0.535898 q^{97} -1.73205 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7	$ 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{11} - 10 q^{16} - 2 q^{17} - 4 q^{19} - 2 q^{20} + 2 q^{25} - 2 q^{28} + 4 q^{29} - 4 q^{31} + 2 q^{35} + 4 q^{37} + 12 q^{38} + 20 q^{41} - 16 q^{43} + 4 q^{44} + 2 q^{49} - 4 q^{55} + 12 q^{58} + 20 q^{61} + 2 q^{64} - 2 q^{68} + 4 q^{71} + 16 q^{73} + 24 q^{74} - 4 q^{76} - 4 q^{77} - 8 q^{79} + 10 q^{80} - 8 q^{83} + 2 q^{85} + 4 q^{89} - 24 q^{94} + 4 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^11 - 10 * q^16 - 2 * q^17 - 4 * q^19 - 2 * q^20 + 2 * q^25 - 2 * q^28 + 4 * q^29 - 4 * q^31 + 2 * q^35 + 4 * q^37 + 12 * q^38 + 20 * q^41 - 16 * q^43 + 4 * q^44 + 2 * q^49 - 4 * q^55 + 12 * q^58 + 20 * q^61 + 2 * q^64 - 2 * q^68 + 4 * q^71 + 16 * q^73 + 24 * q^74 - 4 * q^76 - 4 * q^77 - 8 * q^79 + 10 * q^80 - 8 * q^83 + 2 * q^85 + 4 * q^89 - 24 * q^94 + 4 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.73205	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−1.73205	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.73205	0.612372
$9$	0	0
$10$	1.73205	0.547723
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	1.73205	0.462910
$15$	0	0
$16$	−5.00000	−1.25000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−5.46410	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$19$	−5.46410	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−3.46410	−0.738549
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−1.46410	−0.271877	−0.135938	−	0.990717i	$-0.543405\pi$
$29$	−1.46410	−0.271877	−0.135938	+	0.990717i	$0.543405\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	5.19615	0.918559
$33$	0	0
$34$	1.73205	0.297044
$35$	1.00000	0.169031
$36$	0	0
$37$	−4.92820	−0.810192	−0.405096	−	0.914274i	$-0.632762\pi$
$37$	−4.92820	−0.810192	−0.405096	+	0.914274i	$0.632762\pi$
$38$	9.46410	1.53528
$39$	0	0
$40$	−1.73205	−0.273861
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.73205	−0.244949
$51$	0	0
$52$	0	0
$53$	6.92820	0.951662	0.475831	−	0.879537i	$-0.342147\pi$
$53$	6.92820	0.951662	0.475831	+	0.879537i	$0.342147\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	−1.73205	−0.231455
$57$	0	0
$58$	2.53590	0.332980
$59$	−6.92820	−0.901975	−0.450988	−	0.892530i	$-0.648928\pi$
$59$	−6.92820	−0.901975	−0.450988	+	0.892530i	$0.648928\pi$
$60$	0	0
$61$	6.53590	0.836836	0.418418	−	0.908255i	$-0.362585\pi$
$61$	6.53590	0.836836	0.418418	+	0.908255i	$0.362585\pi$
$62$	3.46410	0.439941
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	−1.00000	−0.121268
$69$	0	0
$70$	−1.73205	−0.207020
$71$	8.92820	1.05958	0.529791	−	0.848128i	$-0.322270\pi$
$71$	8.92820	1.05958	0.529791	+	0.848128i	$0.322270\pi$
$72$	0	0
$73$	11.4641	1.34177	0.670886	−	0.741561i	$-0.265914\pi$
$73$	11.4641	1.34177	0.670886	+	0.741561i	$0.265914\pi$
$74$	8.53590	0.992278
$75$	0	0
$76$	−5.46410	−0.626775
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	5.00000	0.559017
$81$	0	0
$82$	−17.3205	−1.91273
$83$	−17.8564	−1.96000	−0.979998	−	0.199009i	$-0.936228\pi$
$83$	−17.8564	−1.96000	−0.979998	+	0.199009i	$0.936228\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	13.8564	1.49417
$87$	0	0
$88$	3.46410	0.369274
$89$	−11.8564	−1.25678	−0.628388	−	0.777900i	$-0.716285\pi$
$89$	−11.8564	−1.25678	−0.628388	+	0.777900i	$0.716285\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−12.0000	−1.23771
$95$	5.46410	0.560605
$96$	0	0
$97$	0.535898	0.0544122	0.0272061	−	0.999630i	$-0.491339\pi$
$97$	0.535898	0.0544122	0.0272061	+	0.999630i	$0.491339\pi$
$98$	−1.73205	−0.174964
$99$	0	0
$100$	1.00000	0.100000
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	−18.3923	−1.81225	−0.906124	−	0.423013i	$-0.860973\pi$
$103$	−18.3923	−1.81225	−0.906124	+	0.423013i	$0.860973\pi$
$104$	0	0
$105$	0	0
$106$	−12.0000	−1.16554
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	−4.92820	−0.472036	−0.236018	−	0.971749i	$-0.575842\pi$
$109$	−4.92820	−0.472036	−0.236018	+	0.971749i	$0.575842\pi$
$110$	3.46410	0.330289
$111$	0	0
$112$	5.00000	0.472456
$113$	−10.3923	−0.977626	−0.488813	−	0.872389i	$-0.662570\pi$
$113$	−10.3923	−0.977626	−0.488813	+	0.872389i	$0.662570\pi$
$114$	0	0
$115$	0	0
$116$	−1.46410	−0.135938
$117$	0	0
$118$	12.0000	1.10469
$119$	1.00000	0.0916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−11.3205	−1.02491
$123$	0	0
$124$	−2.00000	−0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	−12.1244	−1.07165
$129$	0	0
$130$	0	0
$131$	6.92820	0.605320	0.302660	−	0.953099i	$-0.402125\pi$
$131$	6.92820	0.605320	0.302660	+	0.953099i	$0.402125\pi$
$132$	0	0
$133$	5.46410	0.473798
$134$	0	0
$135$	0	0
$136$	−1.73205	−0.148522
$137$	8.00000	0.683486	0.341743	−	0.939793i	$-0.388983\pi$
$137$	8.00000	0.683486	0.341743	+	0.939793i	$0.388983\pi$
$138$	0	0
$139$	7.85641	0.666372	0.333186	−	0.942861i	$-0.391876\pi$
$139$	7.85641	0.666372	0.333186	+	0.942861i	$0.391876\pi$
$140$	1.00000	0.0845154
$141$	0	0
$142$	−15.4641	−1.29772
$143$	0	0
$144$	0	0
$145$	1.46410	0.121587
$146$	−19.8564	−1.64333
$147$	0	0
$148$	−4.92820	−0.405096
$149$	7.07180	0.579344	0.289672	−	0.957126i	$-0.406454\pi$
$149$	7.07180	0.579344	0.289672	+	0.957126i	$0.406454\pi$
$150$	0	0
$151$	−21.8564	−1.77865	−0.889325	−	0.457277i	$-0.848825\pi$
$151$	−21.8564	−1.77865	−0.889325	+	0.457277i	$0.848825\pi$
$152$	−9.46410	−0.767640
$153$	0	0
$154$	3.46410	0.279145
$155$	2.00000	0.160644
$156$	0	0
$157$	2.92820	0.233696	0.116848	−	0.993150i	$-0.462721\pi$
$157$	2.92820	0.233696	0.116848	+	0.993150i	$0.462721\pi$
$158$	6.92820	0.551178
$159$	0	0
$160$	−5.19615	−0.410792
$161$	0	0
$162$	0	0
$163$	−1.07180	−0.0839496	−0.0419748	−	0.999119i	$-0.513365\pi$
$163$	−1.07180	−0.0839496	−0.0419748	+	0.999119i	$0.513365\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	30.9282	2.40049
$167$	−16.7846	−1.29883	−0.649416	−	0.760433i	$-0.724987\pi$
$167$	−16.7846	−1.29883	−0.649416	+	0.760433i	$0.724987\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−1.73205	−0.132842
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−10.0000	−0.753778
$177$	0	0
$178$	20.5359	1.53923
$179$	6.53590	0.488516	0.244258	−	0.969710i	$-0.421456\pi$
$179$	6.53590	0.488516	0.244258	+	0.969710i	$0.421456\pi$
$180$	0	0
$181$	24.3923	1.81307	0.906533	−	0.422135i	$-0.138719\pi$
$181$	24.3923	1.81307	0.906533	+	0.422135i	$0.138719\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	4.92820	0.362329
$186$	0	0
$187$	−2.00000	−0.146254
$188$	6.92820	0.505291
$189$	0	0
$190$	−9.46410	−0.686598
$191$	2.53590	0.183491	0.0917456	−	0.995782i	$-0.470755\pi$
$191$	2.53590	0.183491	0.0917456	+	0.995782i	$0.470755\pi$
$192$	0	0
$193$	8.92820	0.642666	0.321333	−	0.946966i	$-0.395869\pi$
$193$	8.92820	0.642666	0.321333	+	0.946966i	$0.395869\pi$
$194$	−0.928203	−0.0666411
$195$	0	0
$196$	1.00000	0.0714286
$197$	−27.4641	−1.95674	−0.978368	−	0.206872i	$-0.933672\pi$
$197$	−27.4641	−1.95674	−0.978368	+	0.206872i	$0.933672\pi$
$198$	0	0
$199$	12.9282	0.916456	0.458228	−	0.888835i	$-0.348484\pi$
$199$	12.9282	0.916456	0.458228	+	0.888835i	$0.348484\pi$
$200$	1.73205	0.122474
$201$	0	0
$202$	−24.2487	−1.70613
$203$	1.46410	0.102760
$204$	0	0
$205$	−10.0000	−0.698430
$206$	31.8564	2.21954
$207$	0	0
$208$	0	0
$209$	−10.9282	−0.755920
$210$	0	0
$211$	17.8564	1.22929	0.614643	−	0.788806i	$-0.289300\pi$
$211$	17.8564	1.22929	0.614643	+	0.788806i	$0.289300\pi$
$212$	6.92820	0.475831
$213$	0	0
$214$	−13.8564	−0.947204
$215$	8.00000	0.545595
$216$	0	0
$217$	2.00000	0.135769
$218$	8.53590	0.578124
$219$	0	0
$220$	−2.00000	−0.134840
$221$	0	0
$222$	0	0
$223$	22.3923	1.49950	0.749750	−	0.661721i	$-0.230174\pi$
$223$	22.3923	1.49950	0.749750	+	0.661721i	$0.230174\pi$
$224$	−5.19615	−0.347183
$225$	0	0
$226$	18.0000	1.19734
$227$	22.9282	1.52180	0.760899	−	0.648870i	$-0.224758\pi$
$227$	22.9282	1.52180	0.760899	+	0.648870i	$0.224758\pi$
$228$	0	0
$229$	20.9282	1.38297	0.691487	−	0.722389i	$-0.256956\pi$
$229$	20.9282	1.38297	0.691487	+	0.722389i	$0.256956\pi$
$230$	0	0
$231$	0	0
$232$	−2.53590	−0.166490
$233$	3.46410	0.226941	0.113470	−	0.993541i	$-0.463803\pi$
$233$	3.46410	0.226941	0.113470	+	0.993541i	$0.463803\pi$
$234$	0	0
$235$	−6.92820	−0.451946
$236$	−6.92820	−0.450988
$237$	0	0
$238$	−1.73205	−0.112272
$239$	7.60770	0.492101	0.246050	−	0.969257i	$-0.420867\pi$
$239$	7.60770	0.492101	0.246050	+	0.969257i	$0.420867\pi$
$240$	0	0
$241$	19.3205	1.24454	0.622272	−	0.782801i	$-0.286210\pi$
$241$	19.3205	1.24454	0.622272	+	0.782801i	$0.286210\pi$
$242$	12.1244	0.779383
$243$	0	0
$244$	6.53590	0.418418
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	−3.46410	−0.219971
$249$	0	0
$250$	1.73205	0.109545
$251$	14.9282	0.942260	0.471130	−	0.882064i	$-0.343846\pi$
$251$	14.9282	0.942260	0.471130	+	0.882064i	$0.343846\pi$
$252$	0	0
$253$	0	0
$254$	−6.92820	−0.434714
$255$	0	0
$256$	19.0000	1.18750
$257$	18.7846	1.17175	0.585876	−	0.810401i	$-0.300751\pi$
$257$	18.7846	1.17175	0.585876	+	0.810401i	$0.300751\pi$
$258$	0	0
$259$	4.92820	0.306224
$260$	0	0
$261$	0	0
$262$	−12.0000	−0.741362
$263$	−17.3205	−1.06803	−0.534014	−	0.845476i	$-0.679317\pi$
$263$	−17.3205	−1.06803	−0.534014	+	0.845476i	$0.679317\pi$
$264$	0	0
$265$	−6.92820	−0.425596
$266$	−9.46410	−0.580281
$267$	0	0
$268$	0	0
$269$	12.9282	0.788246	0.394123	−	0.919058i	$-0.371048\pi$
$269$	12.9282	0.788246	0.394123	+	0.919058i	$0.371048\pi$
$270$	0	0
$271$	−13.4641	−0.817886	−0.408943	−	0.912560i	$-0.634103\pi$
$271$	−13.4641	−0.817886	−0.408943	+	0.912560i	$0.634103\pi$
$272$	5.00000	0.303170
$273$	0	0
$274$	−13.8564	−0.837096
$275$	2.00000	0.120605
$276$	0	0
$277$	−7.07180	−0.424903	−0.212452	−	0.977172i	$-0.568145\pi$
$277$	−7.07180	−0.424903	−0.212452	+	0.977172i	$0.568145\pi$
$278$	−13.6077	−0.816135
$279$	0	0
$280$	1.73205	0.103510
$281$	16.9282	1.00985	0.504926	−	0.863163i	$-0.331520\pi$
$281$	16.9282	1.00985	0.504926	+	0.863163i	$0.331520\pi$
$282$	0	0
$283$	17.8564	1.06145	0.530727	−	0.847543i	$-0.321919\pi$
$283$	17.8564	1.06145	0.530727	+	0.847543i	$0.321919\pi$
$284$	8.92820	0.529791
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	1.00000	0.0588235
$290$	−2.53590	−0.148913
$291$	0	0
$292$	11.4641	0.670886
$293$	3.07180	0.179456	0.0897281	−	0.995966i	$-0.471400\pi$
$293$	3.07180	0.179456	0.0897281	+	0.995966i	$0.471400\pi$
$294$	0	0
$295$	6.92820	0.403376
$296$	−8.53590	−0.496139
$297$	0	0
$298$	−12.2487	−0.709549
$299$	0	0
$300$	0	0
$301$	8.00000	0.461112
$302$	37.8564	2.17839
$303$	0	0
$304$	27.3205	1.56694
$305$	−6.53590	−0.374244
$306$	0	0
$307$	18.3923	1.04970	0.524852	−	0.851193i	$-0.324121\pi$
$307$	18.3923	1.04970	0.524852	+	0.851193i	$0.324121\pi$
$308$	−2.00000	−0.113961
$309$	0	0
$310$	−3.46410	−0.196748
$311$	−10.9282	−0.619682	−0.309841	−	0.950788i	$-0.600276\pi$
$311$	−10.9282	−0.619682	−0.309841	+	0.950788i	$0.600276\pi$
$312$	0	0
$313$	−28.2487	−1.59671	−0.798356	−	0.602186i	$-0.794297\pi$
$313$	−28.2487	−1.59671	−0.798356	+	0.602186i	$0.794297\pi$
$314$	−5.07180	−0.286218
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−7.46410	−0.419226	−0.209613	−	0.977784i	$-0.567220\pi$
$317$	−7.46410	−0.419226	−0.209613	+	0.977784i	$0.567220\pi$
$318$	0	0
$319$	−2.92820	−0.163948
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	5.46410	0.304031
$324$	0	0
$325$	0	0
$326$	1.85641	0.102817
$327$	0	0
$328$	17.3205	0.956365
$329$	−6.92820	−0.381964
$330$	0	0
$331$	9.07180	0.498631	0.249316	−	0.968422i	$-0.419794\pi$
$331$	9.07180	0.498631	0.249316	+	0.968422i	$0.419794\pi$
$332$	−17.8564	−0.979998
$333$	0	0
$334$	29.0718	1.59074
$335$	0	0
$336$	0	0
$337$	−11.8564	−0.645860	−0.322930	−	0.946423i	$-0.604668\pi$
$337$	−11.8564	−0.645860	−0.322930	+	0.946423i	$0.604668\pi$
$338$	22.5167	1.22474
$339$	0	0
$340$	1.00000	0.0542326
$341$	−4.00000	−0.216612
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−13.8564	−0.747087
$345$	0	0
$346$	31.1769	1.67608
$347$	8.78461	0.471583	0.235791	−	0.971804i	$-0.424232\pi$
$347$	8.78461	0.471583	0.235791	+	0.971804i	$0.424232\pi$
$348$	0	0
$349$	−7.85641	−0.420544	−0.210272	−	0.977643i	$-0.567435\pi$
$349$	−7.85641	−0.420544	−0.210272	+	0.977643i	$0.567435\pi$
$350$	1.73205	0.0925820
$351$	0	0
$352$	10.3923	0.553912
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	−8.92820	−0.473860
$356$	−11.8564	−0.628388
$357$	0	0
$358$	−11.3205	−0.598307
$359$	16.3923	0.865153	0.432576	−	0.901597i	$-0.357605\pi$
$359$	16.3923	0.865153	0.432576	+	0.901597i	$0.357605\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	−42.2487	−2.22054
$363$	0	0
$364$	0	0
$365$	−11.4641	−0.600059
$366$	0	0
$367$	−17.8564	−0.932097	−0.466048	−	0.884759i	$-0.654323\pi$
$367$	−17.8564	−0.932097	−0.466048	+	0.884759i	$0.654323\pi$
$368$	0	0
$369$	0	0
$370$	−8.53590	−0.443760
$371$	−6.92820	−0.359694
$372$	0	0
$373$	36.6410	1.89720	0.948600	−	0.316478i	$-0.102500\pi$
$373$	36.6410	1.89720	0.948600	+	0.316478i	$0.102500\pi$
$374$	3.46410	0.179124
$375$	0	0
$376$	12.0000	0.618853
$377$	0	0
$378$	0	0
$379$	−2.14359	−0.110109	−0.0550545	−	0.998483i	$-0.517533\pi$
$379$	−2.14359	−0.110109	−0.0550545	+	0.998483i	$0.517533\pi$
$380$	5.46410	0.280302
$381$	0	0
$382$	−4.39230	−0.224730
$383$	−12.7846	−0.653263	−0.326632	−	0.945152i	$-0.605914\pi$
$383$	−12.7846	−0.653263	−0.326632	+	0.945152i	$0.605914\pi$
$384$	0	0
$385$	2.00000	0.101929
$386$	−15.4641	−0.787102
$387$	0	0
$388$	0.535898	0.0272061
$389$	−8.92820	−0.452678	−0.226339	−	0.974049i	$-0.572676\pi$
$389$	−8.92820	−0.452678	−0.226339	+	0.974049i	$0.572676\pi$
$390$	0	0
$391$	0	0
$392$	1.73205	0.0874818
$393$	0	0
$394$	47.5692	2.39650
$395$	4.00000	0.201262
$396$	0	0
$397$	−1.32051	−0.0662744	−0.0331372	−	0.999451i	$-0.510550\pi$
$397$	−1.32051	−0.0662744	−0.0331372	+	0.999451i	$0.510550\pi$
$398$	−22.3923	−1.12242
$399$	0	0
$400$	−5.00000	−0.250000
$401$	29.1769	1.45703	0.728513	−	0.685032i	$-0.240212\pi$
$401$	29.1769	1.45703	0.728513	+	0.685032i	$0.240212\pi$
$402$	0	0
$403$	0	0
$404$	14.0000	0.696526
$405$	0	0
$406$	−2.53590	−0.125855
$407$	−9.85641	−0.488564
$408$	0	0
$409$	−11.8564	−0.586262	−0.293131	−	0.956072i	$-0.594697\pi$
$409$	−11.8564	−0.586262	−0.293131	+	0.956072i	$0.594697\pi$
$410$	17.3205	0.855399
$411$	0	0
$412$	−18.3923	−0.906124
$413$	6.92820	0.340915
$414$	0	0
$415$	17.8564	0.876537
$416$	0	0
$417$	0	0
$418$	18.9282	0.925809
$419$	28.7846	1.40622	0.703110	−	0.711081i	$-0.251794\pi$
$419$	28.7846	1.40622	0.703110	+	0.711081i	$0.251794\pi$
$420$	0	0
$421$	−19.8564	−0.967742	−0.483871	−	0.875139i	$-0.660770\pi$
$421$	−19.8564	−0.967742	−0.483871	+	0.875139i	$0.660770\pi$
$422$	−30.9282	−1.50556
$423$	0	0
$424$	12.0000	0.582772
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−6.53590	−0.316294
$428$	8.00000	0.386695
$429$	0	0
$430$	−13.8564	−0.668215
$431$	29.7128	1.43122	0.715608	−	0.698502i	$-0.246150\pi$
$431$	29.7128	1.43122	0.715608	+	0.698502i	$0.246150\pi$
$432$	0	0
$433$	31.7128	1.52402	0.762010	−	0.647565i	$-0.224213\pi$
$433$	31.7128	1.52402	0.762010	+	0.647565i	$0.224213\pi$
$434$	−3.46410	−0.166282
$435$	0	0
$436$	−4.92820	−0.236018
$437$	0	0
$438$	0	0
$439$	−7.85641	−0.374966	−0.187483	−	0.982268i	$-0.560033\pi$
$439$	−7.85641	−0.374966	−0.187483	+	0.982268i	$0.560033\pi$
$440$	−3.46410	−0.165145
$441$	0	0
$442$	0	0
$443$	−13.6077	−0.646521	−0.323261	−	0.946310i	$-0.604779\pi$
$443$	−13.6077	−0.646521	−0.323261	+	0.946310i	$0.604779\pi$
$444$	0	0
$445$	11.8564	0.562048
$446$	−38.7846	−1.83650
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−11.3205	−0.534248	−0.267124	−	0.963662i	$-0.586073\pi$
$449$	−11.3205	−0.534248	−0.267124	+	0.963662i	$0.586073\pi$
$450$	0	0
$451$	20.0000	0.941763
$452$	−10.3923	−0.488813
$453$	0	0
$454$	−39.7128	−1.86381
$455$	0	0
$456$	0	0
$457$	−4.92820	−0.230532	−0.115266	−	0.993335i	$-0.536772\pi$
$457$	−4.92820	−0.230532	−0.115266	+	0.993335i	$0.536772\pi$
$458$	−36.2487	−1.69379
$459$	0	0
$460$	0	0
$461$	25.7128	1.19757	0.598783	−	0.800912i	$-0.295651\pi$
$461$	25.7128	1.19757	0.598783	+	0.800912i	$0.295651\pi$
$462$	0	0
$463$	−13.0718	−0.607498	−0.303749	−	0.952752i	$-0.598238\pi$
$463$	−13.0718	−0.607498	−0.303749	+	0.952752i	$0.598238\pi$
$464$	7.32051	0.339846
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−1.85641	−0.0859042	−0.0429521	−	0.999077i	$-0.513676\pi$
$467$	−1.85641	−0.0859042	−0.0429521	+	0.999077i	$0.513676\pi$
$468$	0	0
$469$	0	0
$470$	12.0000	0.553519
$471$	0	0
$472$	−12.0000	−0.552345
$473$	−16.0000	−0.735681
$474$	0	0
$475$	−5.46410	−0.250710
$476$	1.00000	0.0458349
$477$	0	0
$478$	−13.1769	−0.602698
$479$	1.07180	0.0489716	0.0244858	−	0.999700i	$-0.492205\pi$
$479$	1.07180	0.0489716	0.0244858	+	0.999700i	$0.492205\pi$
$480$	0	0
$481$	0	0
$482$	−33.4641	−1.52425
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−0.535898	−0.0243339
$486$	0	0
$487$	24.7846	1.12310	0.561549	−	0.827444i	$-0.310206\pi$
$487$	24.7846	1.12310	0.561549	+	0.827444i	$0.310206\pi$
$488$	11.3205	0.512455
$489$	0	0
$490$	1.73205	0.0782461
$491$	18.2487	0.823553	0.411776	−	0.911285i	$-0.364908\pi$
$491$	18.2487	0.823553	0.411776	+	0.911285i	$0.364908\pi$
$492$	0	0
$493$	1.46410	0.0659398
$494$	0	0
$495$	0	0
$496$	10.0000	0.449013
$497$	−8.92820	−0.400485
$498$	0	0
$499$	17.8564	0.799363	0.399681	−	0.916654i	$-0.369121\pi$
$499$	17.8564	0.799363	0.399681	+	0.916654i	$0.369121\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−25.8564	−1.15403
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	0	0
$508$	4.00000	0.177471
$509$	43.8564	1.94390	0.971951	−	0.235185i	$-0.0755697\pi$
$509$	43.8564	1.94390	0.971951	+	0.235185i	$0.0755697\pi$
$510$	0	0
$511$	−11.4641	−0.507142
$512$	−8.66025	−0.382733
$513$	0	0
$514$	−32.5359	−1.43510
$515$	18.3923	0.810462
$516$	0	0
$517$	13.8564	0.609404
$518$	−8.53590	−0.375046
$519$	0	0
$520$	0	0
$521$	14.7846	0.647726	0.323863	−	0.946104i	$-0.395018\pi$
$521$	14.7846	0.647726	0.323863	+	0.946104i	$0.395018\pi$
$522$	0	0
$523$	−11.4641	−0.501290	−0.250645	−	0.968079i	$-0.580643\pi$
$523$	−11.4641	−0.501290	−0.250645	+	0.968079i	$0.580643\pi$
$524$	6.92820	0.302660
$525$	0	0
$526$	30.0000	1.30806
$527$	2.00000	0.0871214
$528$	0	0
$529$	−23.0000	−1.00000
$530$	12.0000	0.521247
$531$	0	0
$532$	5.46410	0.236899
$533$	0	0
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	0	0
$538$	−22.3923	−0.965401
$539$	2.00000	0.0861461
$540$	0	0
$541$	18.7846	0.807613	0.403807	−	0.914844i	$-0.367687\pi$
$541$	18.7846	0.807613	0.403807	+	0.914844i	$0.367687\pi$
$542$	23.3205	1.00170
$543$	0	0
$544$	−5.19615	−0.222783
$545$	4.92820	0.211101
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	8.00000	0.341743
$549$	0	0
$550$	−3.46410	−0.147710
$551$	8.00000	0.340811
$552$	0	0
$553$	4.00000	0.170097
$554$	12.2487	0.520398
$555$	0	0
$556$	7.85641	0.333186
$557$	28.7846	1.21964	0.609822	−	0.792539i	$-0.291241\pi$
$557$	28.7846	1.21964	0.609822	+	0.792539i	$0.291241\pi$
$558$	0	0
$559$	0	0
$560$	−5.00000	−0.211289
$561$	0	0
$562$	−29.3205	−1.23681
$563$	23.7128	0.999376	0.499688	−	0.866205i	$-0.333448\pi$
$563$	23.7128	0.999376	0.499688	+	0.866205i	$0.333448\pi$
$564$	0	0
$565$	10.3923	0.437208
$566$	−30.9282	−1.30001
$567$	0	0
$568$	15.4641	0.648859
$569$	−38.7846	−1.62594	−0.812968	−	0.582309i	$-0.802149\pi$
$569$	−38.7846	−1.62594	−0.812968	+	0.582309i	$0.802149\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	0	0
$574$	17.3205	0.722944
$575$	0	0
$576$	0	0
$577$	−2.92820	−0.121903	−0.0609513	−	0.998141i	$-0.519413\pi$
$577$	−2.92820	−0.121903	−0.0609513	+	0.998141i	$0.519413\pi$
$578$	−1.73205	−0.0720438
$579$	0	0
$580$	1.46410	0.0607935
$581$	17.8564	0.740809
$582$	0	0
$583$	13.8564	0.573874
$584$	19.8564	0.821664
$585$	0	0
$586$	−5.32051	−0.219788
$587$	−2.92820	−0.120860	−0.0604299	−	0.998172i	$-0.519247\pi$
$587$	−2.92820	−0.120860	−0.0604299	+	0.998172i	$0.519247\pi$
$588$	0	0
$589$	10.9282	0.450289
$590$	−12.0000	−0.494032
$591$	0	0
$592$	24.6410	1.01274
$593$	46.7846	1.92121	0.960607	−	0.277911i	$-0.0896420\pi$
$593$	46.7846	1.92121	0.960607	+	0.277911i	$0.0896420\pi$
$594$	0	0
$595$	−1.00000	−0.0409960
$596$	7.07180	0.289672
$597$	0	0
$598$	0	0
$599$	33.4641	1.36731	0.683653	−	0.729807i	$-0.260390\pi$
$599$	33.4641	1.36731	0.683653	+	0.729807i	$0.260390\pi$
$600$	0	0
$601$	19.3205	0.788100	0.394050	−	0.919089i	$-0.371074\pi$
$601$	19.3205	0.788100	0.394050	+	0.919089i	$0.371074\pi$
$602$	−13.8564	−0.564745
$603$	0	0
$604$	−21.8564	−0.889325
$605$	7.00000	0.284590
$606$	0	0
$607$	−16.7846	−0.681266	−0.340633	−	0.940196i	$-0.610641\pi$
$607$	−16.7846	−0.681266	−0.340633	+	0.940196i	$0.610641\pi$
$608$	−28.3923	−1.15146
$609$	0	0
$610$	11.3205	0.458354
$611$	0	0
$612$	0	0
$613$	−11.0718	−0.447186	−0.223593	−	0.974683i	$-0.571779\pi$
$613$	−11.0718	−0.447186	−0.223593	+	0.974683i	$0.571779\pi$
$614$	−31.8564	−1.28562
$615$	0	0
$616$	−3.46410	−0.139573
$617$	−16.2487	−0.654148	−0.327074	−	0.944999i	$-0.606063\pi$
$617$	−16.2487	−0.654148	−0.327074	+	0.944999i	$0.606063\pi$
$618$	0	0
$619$	−10.7846	−0.433470	−0.216735	−	0.976230i	$-0.569541\pi$
$619$	−10.7846	−0.433470	−0.216735	+	0.976230i	$0.569541\pi$
$620$	2.00000	0.0803219
$621$	0	0
$622$	18.9282	0.758952
$623$	11.8564	0.475017
$624$	0	0
$625$	1.00000	0.0400000
$626$	48.9282	1.95556
$627$	0	0
$628$	2.92820	0.116848
$629$	4.92820	0.196500
$630$	0	0
$631$	−40.7846	−1.62361	−0.811805	−	0.583929i	$-0.801515\pi$
$631$	−40.7846	−1.62361	−0.811805	+	0.583929i	$0.801515\pi$
$632$	−6.92820	−0.275589
$633$	0	0
$634$	12.9282	0.513445
$635$	−4.00000	−0.158735
$636$	0	0
$637$	0	0
$638$	5.07180	0.200794
$639$	0	0
$640$	12.1244	0.479257
$641$	12.3923	0.489467	0.244733	−	0.969590i	$-0.421300\pi$
$641$	12.3923	0.489467	0.244733	+	0.969590i	$0.421300\pi$
$642$	0	0
$643$	24.0000	0.946468	0.473234	−	0.880937i	$-0.343087\pi$
$643$	24.0000	0.946468	0.473234	+	0.880937i	$0.343087\pi$
$644$	0	0
$645$	0	0
$646$	−9.46410	−0.372360
$647$	30.9282	1.21591	0.607957	−	0.793970i	$-0.291989\pi$
$647$	30.9282	1.21591	0.607957	+	0.793970i	$0.291989\pi$
$648$	0	0
$649$	−13.8564	−0.543912
$650$	0	0
$651$	0	0
$652$	−1.07180	−0.0419748
$653$	−24.2487	−0.948925	−0.474463	−	0.880276i	$-0.657358\pi$
$653$	−24.2487	−0.948925	−0.474463	+	0.880276i	$0.657358\pi$
$654$	0	0
$655$	−6.92820	−0.270707
$656$	−50.0000	−1.95217
$657$	0	0
$658$	12.0000	0.467809
$659$	10.5359	0.410420	0.205210	−	0.978718i	$-0.434212\pi$
$659$	10.5359	0.410420	0.205210	+	0.978718i	$0.434212\pi$
$660$	0	0
$661$	27.8564	1.08349	0.541744	−	0.840543i	$-0.317764\pi$
$661$	27.8564	1.08349	0.541744	+	0.840543i	$0.317764\pi$
$662$	−15.7128	−0.610696
$663$	0	0
$664$	−30.9282	−1.20025
$665$	−5.46410	−0.211889
$666$	0	0
$667$	0	0
$668$	−16.7846	−0.649416
$669$	0	0
$670$	0	0
$671$	13.0718	0.504631
$672$	0	0
$673$	−16.9282	−0.652534	−0.326267	−	0.945278i	$-0.605791\pi$
$673$	−16.9282	−0.652534	−0.326267	+	0.945278i	$0.605791\pi$
$674$	20.5359	0.791013
$675$	0	0
$676$	−13.0000	−0.500000
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	0	0
$679$	−0.535898	−0.0205659
$680$	1.73205	0.0664211
$681$	0	0
$682$	6.92820	0.265295
$683$	6.14359	0.235078	0.117539	−	0.993068i	$-0.462499\pi$
$683$	6.14359	0.235078	0.117539	+	0.993068i	$0.462499\pi$
$684$	0	0
$685$	−8.00000	−0.305664
$686$	1.73205	0.0661300
$687$	0	0
$688$	40.0000	1.52499
$689$	0	0
$690$	0	0
$691$	12.9282	0.491812	0.245906	−	0.969294i	$-0.420915\pi$
$691$	12.9282	0.491812	0.245906	+	0.969294i	$0.420915\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	−15.2154	−0.577568
$695$	−7.85641	−0.298010
$696$	0	0
$697$	−10.0000	−0.378777
$698$	13.6077	0.515059
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	15.8564	0.598888	0.299444	−	0.954114i	$-0.403199\pi$
$701$	15.8564	0.598888	0.299444	+	0.954114i	$0.403199\pi$
$702$	0	0
$703$	26.9282	1.01562
$704$	2.00000	0.0753778
$705$	0	0
$706$	−10.3923	−0.391120
$707$	−14.0000	−0.526524
$708$	0	0
$709$	30.0000	1.12667	0.563337	−	0.826227i	$-0.309517\pi$
$709$	30.0000	1.12667	0.563337	+	0.826227i	$0.309517\pi$
$710$	15.4641	0.580357
$711$	0	0
$712$	−20.5359	−0.769615
$713$	0	0
$714$	0	0
$715$	0	0
$716$	6.53590	0.244258
$717$	0	0
$718$	−28.3923	−1.05959
$719$	18.9282	0.705903	0.352951	−	0.935642i	$-0.385178\pi$
$719$	18.9282	0.705903	0.352951	+	0.935642i	$0.385178\pi$
$720$	0	0
$721$	18.3923	0.684965
$722$	−18.8038	−0.699807
$723$	0	0
$724$	24.3923	0.906533
$725$	−1.46410	−0.0543754
$726$	0	0
$727$	23.4641	0.870235	0.435118	−	0.900374i	$-0.356707\pi$
$727$	23.4641	0.870235	0.435118	+	0.900374i	$0.356707\pi$
$728$	0	0
$729$	0	0
$730$	19.8564	0.734919
$731$	8.00000	0.295891
$732$	0	0
$733$	−19.7128	−0.728109	−0.364055	−	0.931378i	$-0.618608\pi$
$733$	−19.7128	−0.728109	−0.364055	+	0.931378i	$0.618608\pi$
$734$	30.9282	1.14158
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	9.85641	0.362574	0.181287	−	0.983430i	$-0.441974\pi$
$739$	9.85641	0.362574	0.181287	+	0.983430i	$0.441974\pi$
$740$	4.92820	0.181164
$741$	0	0
$742$	12.0000	0.440534
$743$	−5.07180	−0.186066	−0.0930331	−	0.995663i	$-0.529656\pi$
$743$	−5.07180	−0.186066	−0.0930331	+	0.995663i	$0.529656\pi$
$744$	0	0
$745$	−7.07180	−0.259091
$746$	−63.4641	−2.32359
$747$	0	0
$748$	−2.00000	−0.0731272
$749$	−8.00000	−0.292314
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	−34.6410	−1.26323
$753$	0	0
$754$	0	0
$755$	21.8564	0.795436
$756$	0	0
$757$	8.92820	0.324501	0.162251	−	0.986750i	$-0.448125\pi$
$757$	8.92820	0.324501	0.162251	+	0.986750i	$0.448125\pi$
$758$	3.71281	0.134855
$759$	0	0
$760$	9.46410	0.343299
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	4.92820	0.178413
$764$	2.53590	0.0917456
$765$	0	0
$766$	22.1436	0.800081
$767$	0	0
$768$	0	0
$769$	44.9282	1.62015	0.810076	−	0.586325i	$-0.199426\pi$
$769$	44.9282	1.62015	0.810076	+	0.586325i	$0.199426\pi$
$770$	−3.46410	−0.124838
$771$	0	0
$772$	8.92820	0.321333
$773$	33.7128	1.21257	0.606283	−	0.795249i	$-0.292660\pi$
$773$	33.7128	1.21257	0.606283	+	0.795249i	$0.292660\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0.928203	0.0333206
$777$	0	0
$778$	15.4641	0.554415
$779$	−54.6410	−1.95772
$780$	0	0
$781$	17.8564	0.638952
$782$	0	0
$783$	0	0
$784$	−5.00000	−0.178571
$785$	−2.92820	−0.104512
$786$	0	0
$787$	−9.07180	−0.323375	−0.161687	−	0.986842i	$-0.551694\pi$
$787$	−9.07180	−0.323375	−0.161687	+	0.986842i	$0.551694\pi$
$788$	−27.4641	−0.978368
$789$	0	0
$790$	−6.92820	−0.246494
$791$	10.3923	0.369508
$792$	0	0
$793$	0	0
$794$	2.28719	0.0811692
$795$	0	0
$796$	12.9282	0.458228
$797$	−36.9282	−1.30806	−0.654032	−	0.756467i	$-0.726924\pi$
$797$	−36.9282	−1.30806	−0.654032	+	0.756467i	$0.726924\pi$
$798$	0	0
$799$	−6.92820	−0.245102
$800$	5.19615	0.183712
$801$	0	0
$802$	−50.5359	−1.78448
$803$	22.9282	0.809119
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	24.2487	0.853067
$809$	48.1051	1.69129	0.845643	−	0.533749i	$-0.179217\pi$
$809$	48.1051	1.69129	0.845643	+	0.533749i	$0.179217\pi$
$810$	0	0
$811$	−27.8564	−0.978171	−0.489085	−	0.872236i	$-0.662669\pi$
$811$	−27.8564	−0.978171	−0.489085	+	0.872236i	$0.662669\pi$
$812$	1.46410	0.0513799
$813$	0	0
$814$	17.0718	0.598366
$815$	1.07180	0.0375434
$816$	0	0
$817$	43.7128	1.52932
$818$	20.5359	0.718021
$819$	0	0
$820$	−10.0000	−0.349215
$821$	36.3923	1.27010	0.635050	−	0.772471i	$-0.280979\pi$
$821$	36.3923	1.27010	0.635050	+	0.772471i	$0.280979\pi$
$822$	0	0
$823$	11.7128	0.408283	0.204141	−	0.978941i	$-0.434560\pi$
$823$	11.7128	0.408283	0.204141	+	0.978941i	$0.434560\pi$
$824$	−31.8564	−1.10977
$825$	0	0
$826$	−12.0000	−0.417533
$827$	−16.0000	−0.556375	−0.278187	−	0.960527i	$-0.589734\pi$
$827$	−16.0000	−0.556375	−0.278187	+	0.960527i	$0.589734\pi$
$828$	0	0
$829$	40.9282	1.42150	0.710748	−	0.703447i	$-0.248357\pi$
$829$	40.9282	1.42150	0.710748	+	0.703447i	$0.248357\pi$
$830$	−30.9282	−1.07353
$831$	0	0
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	16.7846	0.580855
$836$	−10.9282	−0.377960
$837$	0	0
$838$	−49.8564	−1.72226
$839$	−7.21539	−0.249103	−0.124551	−	0.992213i	$-0.539749\pi$
$839$	−7.21539	−0.249103	−0.124551	+	0.992213i	$0.539749\pi$
$840$	0	0
$841$	−26.8564	−0.926083
$842$	34.3923	1.18524
$843$	0	0
$844$	17.8564	0.614643
$845$	13.0000	0.447214
$846$	0	0
$847$	7.00000	0.240523
$848$	−34.6410	−1.18958
$849$	0	0
$850$	1.73205	0.0594089
$851$	0	0
$852$	0	0
$853$	−14.6795	−0.502616	−0.251308	−	0.967907i	$-0.580861\pi$
$853$	−14.6795	−0.502616	−0.251308	+	0.967907i	$0.580861\pi$
$854$	11.3205	0.387380
$855$	0	0
$856$	13.8564	0.473602
$857$	−33.7128	−1.15161	−0.575804	−	0.817588i	$-0.695311\pi$
$857$	−33.7128	−1.15161	−0.575804	+	0.817588i	$0.695311\pi$
$858$	0	0
$859$	−53.1769	−1.81437	−0.907186	−	0.420729i	$-0.861774\pi$
$859$	−53.1769	−1.81437	−0.907186	+	0.420729i	$0.861774\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−51.4641	−1.75287
$863$	34.1051	1.16095	0.580476	−	0.814277i	$-0.302867\pi$
$863$	34.1051	1.16095	0.580476	+	0.814277i	$0.302867\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	−54.9282	−1.86654
$867$	0	0
$868$	2.00000	0.0678844
$869$	−8.00000	−0.271381
$870$	0	0
$871$	0	0
$872$	−8.53590	−0.289062
$873$	0	0
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	40.9282	1.38205	0.691024	−	0.722832i	$-0.257160\pi$
$877$	40.9282	1.38205	0.691024	+	0.722832i	$0.257160\pi$
$878$	13.6077	0.459237
$879$	0	0
$880$	10.0000	0.337100
$881$	−47.8564	−1.61232	−0.806162	−	0.591695i	$-0.798459\pi$
$881$	−47.8564	−1.61232	−0.806162	+	0.591695i	$0.798459\pi$
$882$	0	0
$883$	−42.6410	−1.43498	−0.717492	−	0.696567i	$-0.754710\pi$
$883$	−42.6410	−1.43498	−0.717492	+	0.696567i	$0.754710\pi$
$884$	0	0
$885$	0	0
$886$	23.5692	0.791823
$887$	−29.0718	−0.976135	−0.488068	−	0.872806i	$-0.662298\pi$
$887$	−29.0718	−0.976135	−0.488068	+	0.872806i	$0.662298\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	−20.5359	−0.688365
$891$	0	0
$892$	22.3923	0.749750
$893$	−37.8564	−1.26682
$894$	0	0
$895$	−6.53590	−0.218471
$896$	12.1244	0.405046
$897$	0	0
$898$	19.6077	0.654317
$899$	2.92820	0.0976610
$900$	0	0
$901$	−6.92820	−0.230812
$902$	−34.6410	−1.15342
$903$	0	0
$904$	−18.0000	−0.598671
$905$	−24.3923	−0.810828
$906$	0	0
$907$	14.9282	0.495683	0.247841	−	0.968801i	$-0.420279\pi$
$907$	14.9282	0.495683	0.247841	+	0.968801i	$0.420279\pi$
$908$	22.9282	0.760899
$909$	0	0
$910$	0	0
$911$	−13.7128	−0.454326	−0.227163	−	0.973857i	$-0.572945\pi$
$911$	−13.7128	−0.454326	−0.227163	+	0.973857i	$0.572945\pi$
$912$	0	0
$913$	−35.7128	−1.18192
$914$	8.53590	0.282342
$915$	0	0
$916$	20.9282	0.691487
$917$	−6.92820	−0.228789
$918$	0	0
$919$	2.92820	0.0965925	0.0482963	−	0.998833i	$-0.484621\pi$
$919$	2.92820	0.0965925	0.0482963	+	0.998833i	$0.484621\pi$
$920$	0	0
$921$	0	0
$922$	−44.5359	−1.46671
$923$	0	0
$924$	0	0
$925$	−4.92820	−0.162038
$926$	22.6410	0.744030
$927$	0	0
$928$	−7.60770	−0.249735
$929$	−12.6410	−0.414738	−0.207369	−	0.978263i	$-0.566490\pi$
$929$	−12.6410	−0.414738	−0.207369	+	0.978263i	$0.566490\pi$
$930$	0	0
$931$	−5.46410	−0.179079
$932$	3.46410	0.113470
$933$	0	0
$934$	3.21539	0.105211
$935$	2.00000	0.0654070
$936$	0	0
$937$	−4.78461	−0.156306	−0.0781532	−	0.996941i	$-0.524902\pi$
$937$	−4.78461	−0.156306	−0.0781532	+	0.996941i	$0.524902\pi$
$938$	0	0
$939$	0	0
$940$	−6.92820	−0.225973
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	0	0
$944$	34.6410	1.12747
$945$	0	0
$946$	27.7128	0.901021
$947$	−1.85641	−0.0603251	−0.0301626	−	0.999545i	$-0.509602\pi$
$947$	−1.85641	−0.0603251	−0.0301626	+	0.999545i	$0.509602\pi$
$948$	0	0
$949$	0	0
$950$	9.46410	0.307056
$951$	0	0
$952$	1.73205	0.0561361
$953$	44.0000	1.42530	0.712650	−	0.701520i	$-0.247495\pi$
$953$	44.0000	1.42530	0.712650	+	0.701520i	$0.247495\pi$
$954$	0	0
$955$	−2.53590	−0.0820597
$956$	7.60770	0.246050
$957$	0	0
$958$	−1.85641	−0.0599778
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	19.3205	0.622272
$965$	−8.92820	−0.287409
$966$	0	0
$967$	7.21539	0.232031	0.116016	−	0.993247i	$-0.462988\pi$
$967$	7.21539	0.232031	0.116016	+	0.993247i	$0.462988\pi$
$968$	−12.1244	−0.389692
$969$	0	0
$970$	0.928203	0.0298028
$971$	50.6410	1.62515	0.812574	−	0.582858i	$-0.198066\pi$
$971$	50.6410	1.62515	0.812574	+	0.582858i	$0.198066\pi$
$972$	0	0
$973$	−7.85641	−0.251865
$974$	−42.9282	−1.37551
$975$	0	0
$976$	−32.6795	−1.04605
$977$	−39.7128	−1.27053	−0.635263	−	0.772296i	$-0.719108\pi$
$977$	−39.7128	−1.27053	−0.635263	+	0.772296i	$0.719108\pi$
$978$	0	0
$979$	−23.7128	−0.757865
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	−31.6077	−1.00864
$983$	−35.7128	−1.13906	−0.569531	−	0.821970i	$-0.692875\pi$
$983$	−35.7128	−1.13906	−0.569531	+	0.821970i	$0.692875\pi$
$984$	0	0
$985$	27.4641	0.875079
$986$	−2.53590	−0.0807595
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	35.7128	1.13445	0.567227	−	0.823561i	$-0.308016\pi$
$991$	35.7128	1.13445	0.567227	+	0.823561i	$0.308016\pi$
$992$	−10.3923	−0.329956
$993$	0	0
$994$	15.4641	0.490492
$995$	−12.9282	−0.409852
$996$	0	0
$997$	−34.3923	−1.08922	−0.544608	−	0.838691i	$-0.683321\pi$
$997$	−34.3923	−1.08922	−0.544608	+	0.838691i	$0.683321\pi$
$998$	−30.9282	−0.979015
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5355.2.a.w.1.1		2
3.2	odd	2		1785.2.a.q.1.2	✓	2
15.14	odd	2		8925.2.a.bh.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1785.2.a.q.1.2	✓	2	3.2	odd	2
5355.2.a.w.1.1		2	1.1	even	1	trivial
8925.2.a.bh.1.1		2	15.14	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:	\( N \)	\(=\)	\( 5355 = 3^{2} \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5355.a (trivial)

\(f(q)\)	\(=\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.73205 q^{8} +O(q^{10})\)	\(q-1.73205 q^{2} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{7} +1.73205 q^{8} +1.73205 q^{10} +2.00000 q^{11} +1.73205 q^{14} -5.00000 q^{16} -1.00000 q^{17} -5.46410 q^{19} -1.00000 q^{20} -3.46410 q^{22} +1.00000 q^{25} -1.00000 q^{28} -1.46410 q^{29} -2.00000 q^{31} +5.19615 q^{32} +1.73205 q^{34} +1.00000 q^{35} -4.92820 q^{37} +9.46410 q^{38} -1.73205 q^{40} +10.0000 q^{41} -8.00000 q^{43} +2.00000 q^{44} +6.92820 q^{47} +1.00000 q^{49} -1.73205 q^{50} +6.92820 q^{53} -2.00000 q^{55} -1.73205 q^{56} +2.53590 q^{58} -6.92820 q^{59} +6.53590 q^{61} +3.46410 q^{62} +1.00000 q^{64} -1.00000 q^{68} -1.73205 q^{70} +8.92820 q^{71} +11.4641 q^{73} +8.53590 q^{74} -5.46410 q^{76} -2.00000 q^{77} -4.00000 q^{79} +5.00000 q^{80} -17.3205 q^{82} -17.8564 q^{83} +1.00000 q^{85} +13.8564 q^{86} +3.46410 q^{88} -11.8564 q^{89} -12.0000 q^{94} +5.46410 q^{95} +0.535898 q^{97} -1.73205 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7	\( 2 q + 2 q^{4} - 2 q^{5} - 2 q^{7} + 4 q^{11} - 10 q^{16} - 2 q^{17} - 4 q^{19} - 2 q^{20} + 2 q^{25} - 2 q^{28} + 4 q^{29} - 4 q^{31} + 2 q^{35} + 4 q^{37} + 12 q^{38} + 20 q^{41} - 16 q^{43} + 4 q^{44} + 2 q^{49} - 4 q^{55} + 12 q^{58} + 20 q^{61} + 2 q^{64} - 2 q^{68} + 4 q^{71} + 16 q^{73} + 24 q^{74} - 4 q^{76} - 4 q^{77} - 8 q^{79} + 10 q^{80} - 8 q^{83} + 2 q^{85} + 4 q^{89} - 24 q^{94} + 4 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^4 - 2 * q^5 - 2 * q^7 + 4 * q^11 - 10 * q^16 - 2 * q^17 - 4 * q^19 - 2 * q^20 + 2 * q^25 - 2 * q^28 + 4 * q^29 - 4 * q^31 + 2 * q^35 + 4 * q^37 + 12 * q^38 + 20 * q^41 - 16 * q^43 + 4 * q^44 + 2 * q^49 - 4 * q^55 + 12 * q^58 + 20 * q^61 + 2 * q^64 - 2 * q^68 + 4 * q^71 + 16 * q^73 + 24 * q^74 - 4 * q^76 - 4 * q^77 - 8 * q^79 + 10 * q^80 - 8 * q^83 + 2 * q^85 + 4 * q^89 - 24 * q^94 + 4 * q^95 + 8 * q^97

Introduction

L-functions

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