LMFDB - Embedded newform 3483.2.a.s.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3483,2,Mod(1,3483)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3483, 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("3483.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3483 = 3^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3483.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:	$27.8118950240$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 22 x^{17} + 100 x^{16} + 181 x^{15} - 1020 x^{14} - 619 x^{13} + 5458 x^{12} + \cdots + 9 $ x^19 - 4x^18 - 22x^17 + 100x^16 + 181x^15 - 1020x^14 - 619x^13 + 5458x^12 + 145x^11 - 16363x^10 + 4943x^9 + 27069x^8 - 13490x^7 - 22501x^6 + 13478x^5 + 7567x^4 - 4310x^3 - 855x^2 + 225x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 387)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.29864$ of defining polynomial
Character	$\chi$	$=$	3483.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.29864 q^{2} -0.313529 q^{4} -2.76332 q^{5} +1.03232 q^{7} +3.00445 q^{8} +O(q^{10})$	\(q-1.29864 q^{2} -0.313529 q^{4} -2.76332 q^{5} +1.03232 q^{7} +3.00445 q^{8} +3.58856 q^{10} +2.94444 q^{11} -3.37124 q^{13} -1.34061 q^{14} -3.27464 q^{16} +1.55106 q^{17} +0.820840 q^{19} +0.866380 q^{20} -3.82377 q^{22} +8.59169 q^{23} +2.63594 q^{25} +4.37804 q^{26} -0.323661 q^{28} -1.24679 q^{29} -1.92547 q^{31} -1.75630 q^{32} -2.01428 q^{34} -2.85262 q^{35} -4.44739 q^{37} -1.06598 q^{38} -8.30225 q^{40} +6.27239 q^{41} +1.00000 q^{43} -0.923165 q^{44} -11.1575 q^{46} -10.1011 q^{47} -5.93432 q^{49} -3.42314 q^{50} +1.05698 q^{52} +8.79973 q^{53} -8.13642 q^{55} +3.10154 q^{56} +1.61914 q^{58} +6.63191 q^{59} -0.0697410 q^{61} +2.50049 q^{62} +8.83009 q^{64} +9.31583 q^{65} -11.5296 q^{67} -0.486303 q^{68} +3.70454 q^{70} -5.88849 q^{71} +1.89767 q^{73} +5.77557 q^{74} -0.257357 q^{76} +3.03959 q^{77} +6.99127 q^{79} +9.04889 q^{80} -8.14560 q^{82} +3.01827 q^{83} -4.28609 q^{85} -1.29864 q^{86} +8.84640 q^{88} -15.2655 q^{89} -3.48019 q^{91} -2.69374 q^{92} +13.1177 q^{94} -2.26824 q^{95} -3.87436 q^{97} +7.70656 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 4 q^{2} + 22 q^{4} + 9 q^{5} + 7 q^{7} + 12 q^{8}+O(q^{10}) $ 19 * q + 4 * q^2 + 22 * q^4 + 9 * q^5 + 7 * q^7 + 12 * q^8	$ 19 q + 4 q^{2} + 22 q^{4} + 9 q^{5} + 7 q^{7} + 12 q^{8} - 7 q^{10} + 5 q^{11} - 5 q^{13} + 17 q^{14} + 24 q^{16} + 21 q^{17} - 4 q^{19} + 21 q^{20} - 20 q^{22} + 22 q^{23} + 10 q^{25} + 17 q^{26} - q^{28} + 30 q^{29} - 5 q^{31} + 48 q^{32} - 6 q^{34} + 53 q^{35} - q^{37} + 21 q^{38} + 16 q^{40} + 29 q^{41} + 19 q^{43} + 29 q^{44} + 32 q^{47} - 10 q^{49} - 11 q^{50} + q^{52} + 38 q^{53} + 2 q^{55} + 46 q^{56} + 30 q^{58} + 30 q^{59} - 10 q^{61} + 25 q^{62} + 14 q^{64} + 8 q^{65} + 3 q^{67} + 47 q^{68} + 56 q^{70} + 21 q^{71} + 8 q^{73} + 28 q^{74} - 36 q^{76} + 49 q^{77} + 4 q^{79} + 70 q^{80} - 4 q^{82} + 29 q^{83} - 4 q^{85} + 4 q^{86} - 47 q^{88} + 54 q^{89} + 4 q^{91} + 12 q^{92} - 23 q^{94} + 33 q^{95} - 4 q^{97} + 49 q^{98}+O(q^{100}) $ 19 * q + 4 * q^2 + 22 * q^4 + 9 * q^5 + 7 * q^7 + 12 * q^8 - 7 * q^10 + 5 * q^11 - 5 * q^13 + 17 * q^14 + 24 * q^16 + 21 * q^17 - 4 * q^19 + 21 * q^20 - 20 * q^22 + 22 * q^23 + 10 * q^25 + 17 * q^26 - q^28 + 30 * q^29 - 5 * q^31 + 48 * q^32 - 6 * q^34 + 53 * q^35 - q^37 + 21 * q^38 + 16 * q^40 + 29 * q^41 + 19 * q^43 + 29 * q^44 + 32 * q^47 - 10 * q^49 - 11 * q^50 + q^52 + 38 * q^53 + 2 * q^55 + 46 * q^56 + 30 * q^58 + 30 * q^59 - 10 * q^61 + 25 * q^62 + 14 * q^64 + 8 * q^65 + 3 * q^67 + 47 * q^68 + 56 * q^70 + 21 * q^71 + 8 * q^73 + 28 * q^74 - 36 * q^76 + 49 * q^77 + 4 * q^79 + 70 * q^80 - 4 * q^82 + 29 * q^83 - 4 * q^85 + 4 * q^86 - 47 * q^88 + 54 * q^89 + 4 * q^91 + 12 * q^92 - 23 * q^94 + 33 * q^95 - 4 * q^97 + 49 * q^98

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 $

$2$	−1.29864	−0.918279	−0.459139	−	0.888364i	$-0.651842\pi$
$2$	−1.29864	−0.918279	−0.459139	+	0.888364i	$0.651842\pi$
$3$	0	0
$3$	0	0
$4$	−0.313529	−0.156764
$5$	−2.76332	−1.23579	−0.617897	−	0.786259i	$-0.712015\pi$
$5$	−2.76332	−1.23579	−0.617897	+	0.786259i	$0.712015\pi$
$6$	0	0
$7$	1.03232	0.390179	0.195090	−	0.980785i	$-0.437500\pi$
$7$	1.03232	0.390179	0.195090	+	0.980785i	$0.437500\pi$
$8$	3.00445	1.06223
$9$	0	0
$10$	3.58856	1.13480
$11$	2.94444	0.887781	0.443890	−	0.896081i	$-0.353598\pi$
$11$	2.94444	0.887781	0.443890	+	0.896081i	$0.353598\pi$
$12$	0	0
$13$	−3.37124	−0.935015	−0.467507	−	0.883989i	$-0.654848\pi$
$13$	−3.37124	−0.935015	−0.467507	+	0.883989i	$0.654848\pi$
$14$	−1.34061	−0.358293
$15$	0	0
$16$	−3.27464	−0.818661
$17$	1.55106	0.376188	0.188094	−	0.982151i	$-0.439769\pi$
$17$	1.55106	0.376188	0.188094	+	0.982151i	$0.439769\pi$
$18$	0	0
$19$	0.820840	0.188314	0.0941568	−	0.995557i	$-0.469984\pi$
$19$	0.820840	0.188314	0.0941568	+	0.995557i	$0.469984\pi$
$20$	0.866380	0.193729
$21$	0	0
$22$	−3.82377	−0.815230
$23$	8.59169	1.79149	0.895745	−	0.444568i	$-0.146643\pi$
$23$	8.59169	1.79149	0.895745	+	0.444568i	$0.146643\pi$
$24$	0	0
$25$	2.63594	0.527188
$26$	4.37804	0.858604
$27$	0	0
$28$	−0.323661	−0.0611662
$29$	−1.24679	−0.231523	−0.115762	−	0.993277i	$-0.536931\pi$
$29$	−1.24679	−0.231523	−0.115762	+	0.993277i	$0.536931\pi$
$30$	0	0
$31$	−1.92547	−0.345824	−0.172912	−	0.984937i	$-0.555318\pi$
$31$	−1.92547	−0.345824	−0.172912	+	0.984937i	$0.555318\pi$
$32$	−1.75630	−0.310473
$33$	0	0
$34$	−2.01428	−0.345446
$35$	−2.85262	−0.482181
$36$	0	0
$37$	−4.44739	−0.731147	−0.365574	−	0.930782i	$-0.619127\pi$
$37$	−4.44739	−0.731147	−0.365574	+	0.930782i	$0.619127\pi$
$38$	−1.06598	−0.172924
$39$	0	0
$40$	−8.30225	−1.31270
$41$	6.27239	0.979583	0.489792	−	0.871839i	$-0.337073\pi$
$41$	6.27239	0.979583	0.489792	+	0.871839i	$0.337073\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	−0.923165	−0.139172
$45$	0	0
$46$	−11.1575	−1.64509
$47$	−10.1011	−1.47340	−0.736698	−	0.676222i	$-0.763616\pi$
$47$	−10.1011	−1.47340	−0.736698	+	0.676222i	$0.763616\pi$
$48$	0	0
$49$	−5.93432	−0.847760
$50$	−3.42314	−0.484105
$51$	0	0
$52$	1.05698	0.146577
$53$	8.79973	1.20874	0.604368	−	0.796705i	$-0.293426\pi$
$53$	8.79973	1.20874	0.604368	+	0.796705i	$0.293426\pi$
$54$	0	0
$55$	−8.13642	−1.09711
$56$	3.10154	0.414461
$57$	0	0
$58$	1.61914	0.212603
$59$	6.63191	0.863402	0.431701	−	0.902017i	$-0.357914\pi$
$59$	6.63191	0.863402	0.431701	+	0.902017i	$0.357914\pi$
$60$	0	0
$61$	−0.0697410	−0.00892942	−0.00446471	−	0.999990i	$-0.501421\pi$
$61$	−0.0697410	−0.00892942	−0.00446471	+	0.999990i	$0.501421\pi$
$62$	2.50049	0.317563
$63$	0	0
$64$	8.83009	1.10376
$65$	9.31583	1.15549
$66$	0	0
$67$	−11.5296	−1.40857	−0.704283	−	0.709919i	$-0.748731\pi$
$67$	−11.5296	−1.40857	−0.704283	+	0.709919i	$0.748731\pi$
$68$	−0.486303	−0.0589729
$69$	0	0
$70$	3.70454	0.442777
$71$	−5.88849	−0.698835	−0.349418	−	0.936967i	$-0.613621\pi$
$71$	−5.88849	−0.698835	−0.349418	+	0.936967i	$0.613621\pi$
$72$	0	0
$73$	1.89767	0.222105	0.111053	−	0.993815i	$-0.464578\pi$
$73$	1.89767	0.222105	0.111053	+	0.993815i	$0.464578\pi$
$74$	5.77557	0.671397
$75$	0	0
$76$	−0.257357	−0.0295209
$77$	3.03959	0.346394
$78$	0	0
$79$	6.99127	0.786579	0.393290	−	0.919415i	$-0.371337\pi$
$79$	6.99127	0.786579	0.393290	+	0.919415i	$0.371337\pi$
$80$	9.04889	1.01170
$81$	0	0
$82$	−8.14560	−0.899531
$83$	3.01827	0.331298	0.165649	−	0.986185i	$-0.447028\pi$
$83$	3.01827	0.331298	0.165649	+	0.986185i	$0.447028\pi$
$84$	0	0
$85$	−4.28609	−0.464891
$86$	−1.29864	−0.140036
$87$	0	0
$88$	8.84640	0.943029
$89$	−15.2655	−1.61814	−0.809071	−	0.587711i	$-0.800029\pi$
$89$	−15.2655	−1.61814	−0.809071	+	0.587711i	$0.800029\pi$
$90$	0	0
$91$	−3.48019	−0.364823
$92$	−2.69374	−0.280842
$93$	0	0
$94$	13.1177	1.35299
$95$	−2.26824	−0.232717
$96$	0	0
$97$	−3.87436	−0.393382	−0.196691	−	0.980466i	$-0.563020\pi$
$97$	−3.87436	−0.393382	−0.196691	+	0.980466i	$0.563020\pi$
$98$	7.70656	0.778480
$99$	0	0
$100$	−0.826443	−0.0826443
$101$	−10.2298	−1.01790	−0.508952	−	0.860795i	$-0.669967\pi$
$101$	−10.2298	−1.01790	−0.508952	+	0.860795i	$0.669967\pi$
$102$	0	0
$103$	12.6795	1.24935	0.624676	−	0.780884i	$-0.285231\pi$
$103$	12.6795	1.24935	0.624676	+	0.780884i	$0.285231\pi$
$104$	−10.1287	−0.993203
$105$	0	0
$106$	−11.4277	−1.10996
$107$	10.2054	0.986597	0.493298	−	0.869860i	$-0.335791\pi$
$107$	10.2054	0.986597	0.493298	+	0.869860i	$0.335791\pi$
$108$	0	0
$109$	−4.68029	−0.448290	−0.224145	−	0.974556i	$-0.571959\pi$
$109$	−4.68029	−0.448290	−0.224145	+	0.974556i	$0.571959\pi$
$110$	10.5663	1.00746
$111$	0	0
$112$	−3.38047	−0.319424
$113$	−6.87782	−0.647011	−0.323505	−	0.946226i	$-0.604861\pi$
$113$	−6.87782	−0.647011	−0.323505	+	0.946226i	$0.604861\pi$
$114$	0	0
$115$	−23.7416	−2.21391
$116$	0.390905	0.0362946
$117$	0	0
$118$	−8.61248	−0.792843
$119$	1.60119	0.146781
$120$	0	0
$121$	−2.33030	−0.211846
$122$	0.0905686	0.00819969
$123$	0	0
$124$	0.603690	0.0542129
$125$	6.53266	0.584299
$126$	0	0
$127$	6.24851	0.554466	0.277233	−	0.960803i	$-0.410583\pi$
$127$	6.24851	0.554466	0.277233	+	0.960803i	$0.410583\pi$
$128$	−7.95453	−0.703087
$129$	0	0
$130$	−12.0979	−1.06106
$131$	11.5538	1.00946	0.504730	−	0.863277i	$-0.331592\pi$
$131$	11.5538	1.00946	0.504730	+	0.863277i	$0.331592\pi$
$132$	0	0
$133$	0.847368	0.0734761
$134$	14.9728	1.29346
$135$	0	0
$136$	4.66009	0.399599
$137$	−20.9113	−1.78657	−0.893287	−	0.449487i	$-0.851607\pi$
$137$	−20.9113	−1.78657	−0.893287	+	0.449487i	$0.851607\pi$
$138$	0	0
$139$	22.2623	1.88826	0.944131	−	0.329571i	$-0.106904\pi$
$139$	22.2623	1.88826	0.944131	+	0.329571i	$0.106904\pi$
$140$	0.894379	0.0755888
$141$	0	0
$142$	7.64704	0.641725
$143$	−9.92641	−0.830088
$144$	0	0
$145$	3.44528	0.286115
$146$	−2.46439	−0.203955
$147$	0	0
$148$	1.39439	0.114618
$149$	12.6588	1.03705	0.518526	−	0.855062i	$-0.326481\pi$
$149$	12.6588	1.03705	0.518526	+	0.855062i	$0.326481\pi$
$150$	0	0
$151$	−3.77805	−0.307453	−0.153727	−	0.988113i	$-0.549128\pi$
$151$	−3.77805	−0.307453	−0.153727	+	0.988113i	$0.549128\pi$
$152$	2.46617	0.200033
$153$	0	0
$154$	−3.94734	−0.318086
$155$	5.32069	0.427368
$156$	0	0
$157$	3.27623	0.261472	0.130736	−	0.991417i	$-0.458266\pi$
$157$	3.27623	0.261472	0.130736	+	0.991417i	$0.458266\pi$
$158$	−9.07916	−0.722299
$159$	0	0
$160$	4.85323	0.383681
$161$	8.86935	0.699002
$162$	0	0
$163$	−2.94951	−0.231023	−0.115512	−	0.993306i	$-0.536851\pi$
$163$	−2.94951	−0.231023	−0.115512	+	0.993306i	$0.536851\pi$
$164$	−1.96658	−0.153564
$165$	0	0
$166$	−3.91965	−0.304224
$167$	23.9161	1.85068	0.925342	−	0.379134i	$-0.123778\pi$
$167$	23.9161	1.85068	0.925342	+	0.379134i	$0.123778\pi$
$168$	0	0
$169$	−1.63471	−0.125747
$170$	5.56609	0.426900
$171$	0	0
$172$	−0.313529	−0.0239063
$173$	21.8077	1.65801	0.829004	−	0.559243i	$-0.188908\pi$
$173$	21.8077	1.65801	0.829004	+	0.559243i	$0.188908\pi$
$174$	0	0
$175$	2.72113	0.205698
$176$	−9.64197	−0.726791
$177$	0	0
$178$	19.8245	1.48591
$179$	6.92878	0.517882	0.258941	−	0.965893i	$-0.416627\pi$
$179$	6.92878	0.517882	0.258941	+	0.965893i	$0.416627\pi$
$180$	0	0
$181$	−11.0279	−0.819698	−0.409849	−	0.912153i	$-0.634419\pi$
$181$	−11.0279	−0.819698	−0.409849	+	0.912153i	$0.634419\pi$
$182$	4.51953	0.335010
$183$	0	0
$184$	25.8133	1.90298
$185$	12.2896	0.903548
$186$	0	0
$187$	4.56701	0.333973
$188$	3.16698	0.230976
$189$	0	0
$190$	2.94564	0.213699
$191$	25.3055	1.83104	0.915522	−	0.402269i	$-0.131778\pi$
$191$	25.3055	1.83104	0.915522	+	0.402269i	$0.131778\pi$
$192$	0	0
$193$	−21.7156	−1.56313	−0.781563	−	0.623826i	$-0.785577\pi$
$193$	−21.7156	−1.56313	−0.781563	+	0.623826i	$0.785577\pi$
$194$	5.03141	0.361234
$195$	0	0
$196$	1.86058	0.132899
$197$	6.79869	0.484386	0.242193	−	0.970228i	$-0.422133\pi$
$197$	6.79869	0.484386	0.242193	+	0.970228i	$0.422133\pi$
$198$	0	0
$199$	27.2806	1.93387	0.966935	−	0.255022i	$-0.0820827\pi$
$199$	27.2806	1.93387	0.966935	+	0.255022i	$0.0820827\pi$
$200$	7.91954	0.559996
$201$	0	0
$202$	13.2849	0.934719
$203$	−1.28708	−0.0903356
$204$	0	0
$205$	−17.3326	−1.21056
$206$	−16.4662	−1.14725
$207$	0	0
$208$	11.0396	0.765460
$209$	2.41691	0.167181
$210$	0	0
$211$	15.7701	1.08566	0.542828	−	0.839844i	$-0.317354\pi$
$211$	15.7701	1.08566	0.542828	+	0.839844i	$0.317354\pi$
$212$	−2.75897	−0.189487
$213$	0	0
$214$	−13.2532	−0.905971
$215$	−2.76332	−0.188457
$216$	0	0
$217$	−1.98769	−0.134933
$218$	6.07802	0.411655
$219$	0	0
$220$	2.55100	0.171988
$221$	−5.22902	−0.351742
$222$	0	0
$223$	21.1911	1.41906	0.709531	−	0.704674i	$-0.248907\pi$
$223$	21.1911	1.41906	0.709531	+	0.704674i	$0.248907\pi$
$224$	−1.81306	−0.121140
$225$	0	0
$226$	8.93183	0.594136
$227$	1.55400	0.103142	0.0515712	−	0.998669i	$-0.483577\pi$
$227$	1.55400	0.103142	0.0515712	+	0.998669i	$0.483577\pi$
$228$	0	0
$229$	1.98087	0.130899	0.0654496	−	0.997856i	$-0.479152\pi$
$229$	1.98087	0.130899	0.0654496	+	0.997856i	$0.479152\pi$
$230$	30.8318	2.03299
$231$	0	0
$232$	−3.74592	−0.245932
$233$	−25.5390	−1.67311	−0.836557	−	0.547880i	$-0.815435\pi$
$233$	−25.5390	−1.67311	−0.836557	+	0.547880i	$0.815435\pi$
$234$	0	0
$235$	27.9126	1.82081
$236$	−2.07930	−0.135351
$237$	0	0
$238$	−2.07937	−0.134786
$239$	−15.9764	−1.03343	−0.516714	−	0.856158i	$-0.672845\pi$
$239$	−15.9764	−1.03343	−0.516714	+	0.856158i	$0.672845\pi$
$240$	0	0
$241$	−9.71522	−0.625812	−0.312906	−	0.949784i	$-0.601303\pi$
$241$	−9.71522	−0.625812	−0.312906	+	0.949784i	$0.601303\pi$
$242$	3.02623	0.194533
$243$	0	0
$244$	0.0218658	0.00139981
$245$	16.3984	1.04766
$246$	0	0
$247$	−2.76725	−0.176076
$248$	−5.78497	−0.367346
$249$	0	0
$250$	−8.48358	−0.536549
$251$	5.91101	0.373100	0.186550	−	0.982445i	$-0.440269\pi$
$251$	5.91101	0.373100	0.186550	+	0.982445i	$0.440269\pi$
$252$	0	0
$253$	25.2977	1.59045
$254$	−8.11458	−0.509154
$255$	0	0
$256$	−7.33011	−0.458132
$257$	28.3893	1.77088	0.885439	−	0.464756i	$-0.153858\pi$
$257$	28.3893	1.77088	0.885439	+	0.464756i	$0.153858\pi$
$258$	0	0
$259$	−4.59112	−0.285278
$260$	−2.92078	−0.181139
$261$	0	0
$262$	−15.0043	−0.926966
$263$	26.2293	1.61737	0.808683	−	0.588245i	$-0.200181\pi$
$263$	26.2293	1.61737	0.808683	+	0.588245i	$0.200181\pi$
$264$	0	0
$265$	−24.3165	−1.49375
$266$	−1.10043	−0.0674715
$267$	0	0
$268$	3.61486	0.220813
$269$	16.9061	1.03078	0.515392	−	0.856954i	$-0.327646\pi$
$269$	16.9061	1.03078	0.515392	+	0.856954i	$0.327646\pi$
$270$	0	0
$271$	14.9590	0.908693	0.454347	−	0.890825i	$-0.349873\pi$
$271$	14.9590	0.908693	0.454347	+	0.890825i	$0.349873\pi$
$272$	−5.07918	−0.307971
$273$	0	0
$274$	27.1563	1.64057
$275$	7.76135	0.468027
$276$	0	0
$277$	0.928685	0.0557993	0.0278996	−	0.999611i	$-0.491118\pi$
$277$	0.928685	0.0557993	0.0278996	+	0.999611i	$0.491118\pi$
$278$	−28.9107	−1.73395
$279$	0	0
$280$	−8.57055	−0.512188
$281$	17.7982	1.06175	0.530876	−	0.847449i	$-0.321863\pi$
$281$	17.7982	1.06175	0.530876	+	0.847449i	$0.321863\pi$
$282$	0	0
$283$	1.75401	0.104265	0.0521326	−	0.998640i	$-0.483398\pi$
$283$	1.75401	0.104265	0.0521326	+	0.998640i	$0.483398\pi$
$284$	1.84621	0.109552
$285$	0	0
$286$	12.8909	0.762252
$287$	6.47510	0.382213
$288$	0	0
$289$	−14.5942	−0.858482
$290$	−4.47419	−0.262734
$291$	0	0
$292$	−0.594973	−0.0348182
$293$	−7.50072	−0.438197	−0.219098	−	0.975703i	$-0.570312\pi$
$293$	−7.50072	−0.438197	−0.219098	+	0.975703i	$0.570312\pi$
$294$	0	0
$295$	−18.3261	−1.06699
$296$	−13.3620	−0.776648
$297$	0	0
$298$	−16.4393	−0.952303
$299$	−28.9647	−1.67507
$300$	0	0
$301$	1.03232	0.0595018
$302$	4.90634	0.282328
$303$	0	0
$304$	−2.68796	−0.154165
$305$	0.192717	0.0110349
$306$	0	0
$307$	24.5577	1.40158	0.700792	−	0.713365i	$-0.252830\pi$
$307$	24.5577	1.40158	0.700792	+	0.713365i	$0.252830\pi$
$308$	−0.952999	−0.0543022
$309$	0	0
$310$	−6.90967	−0.392443
$311$	−18.7217	−1.06161	−0.530806	−	0.847494i	$-0.678111\pi$
$311$	−18.7217	−1.06161	−0.530806	+	0.847494i	$0.678111\pi$
$312$	0	0
$313$	−1.09855	−0.0620935	−0.0310468	−	0.999518i	$-0.509884\pi$
$313$	−1.09855	−0.0620935	−0.0310468	+	0.999518i	$0.509884\pi$
$314$	−4.25466	−0.240104
$315$	0	0
$316$	−2.19196	−0.123308
$317$	2.51719	0.141379	0.0706896	−	0.997498i	$-0.477480\pi$
$317$	2.51719	0.141379	0.0706896	+	0.997498i	$0.477480\pi$
$318$	0	0
$319$	−3.67110	−0.205542
$320$	−24.4004	−1.36402
$321$	0	0
$322$	−11.5181	−0.641879
$323$	1.27318	0.0708414
$324$	0	0
$325$	−8.88639	−0.492928
$326$	3.83035	0.212144
$327$	0	0
$328$	18.8451	1.04054
$329$	−10.4275	−0.574889
$330$	0	0
$331$	−13.5399	−0.744222	−0.372111	−	0.928188i	$-0.621366\pi$
$331$	−13.5399	−0.744222	−0.372111	+	0.928188i	$0.621366\pi$
$332$	−0.946314	−0.0519357
$333$	0	0
$334$	−31.0585	−1.69944
$335$	31.8600	1.74070
$336$	0	0
$337$	−2.44159	−0.133002	−0.0665009	−	0.997786i	$-0.521184\pi$
$337$	−2.44159	−0.133002	−0.0665009	+	0.997786i	$0.521184\pi$
$338$	2.12291	0.115471
$339$	0	0
$340$	1.34381	0.0728784
$341$	−5.66942	−0.307016
$342$	0	0
$343$	−13.3523	−0.720958
$344$	3.00445	0.161989
$345$	0	0
$346$	−28.3204	−1.52251
$347$	1.87938	0.100890	0.0504451	−	0.998727i	$-0.483936\pi$
$347$	1.87938	0.100890	0.0504451	+	0.998727i	$0.483936\pi$
$348$	0	0
$349$	24.1368	1.29201	0.646006	−	0.763332i	$-0.276438\pi$
$349$	24.1368	1.29201	0.646006	+	0.763332i	$0.276438\pi$
$350$	−3.53377	−0.188888
$351$	0	0
$352$	−5.17132	−0.275632
$353$	−13.5543	−0.721424	−0.360712	−	0.932677i	$-0.617466\pi$
$353$	−13.5543	−0.721424	−0.360712	+	0.932677i	$0.617466\pi$
$354$	0	0
$355$	16.2718	0.863616
$356$	4.78618	0.253667
$357$	0	0
$358$	−8.99801	−0.475560
$359$	−3.49289	−0.184348	−0.0921739	−	0.995743i	$-0.529382\pi$
$359$	−3.49289	−0.184348	−0.0921739	+	0.995743i	$0.529382\pi$
$360$	0	0
$361$	−18.3262	−0.964538
$362$	14.3213	0.752711
$363$	0	0
$364$	1.09114	0.0571913
$365$	−5.24387	−0.274476
$366$	0	0
$367$	29.1952	1.52398	0.761989	−	0.647590i	$-0.224223\pi$
$367$	29.1952	1.52398	0.761989	+	0.647590i	$0.224223\pi$
$368$	−28.1347	−1.46662
$369$	0	0
$370$	−15.9598	−0.829708
$371$	9.08412	0.471624
$372$	0	0
$373$	−5.91592	−0.306315	−0.153157	−	0.988202i	$-0.548944\pi$
$373$	−5.91592	−0.306315	−0.153157	+	0.988202i	$0.548944\pi$
$374$	−5.93091	−0.306680
$375$	0	0
$376$	−30.3482	−1.56509
$377$	4.20324	0.216478
$378$	0	0
$379$	8.06235	0.414135	0.207067	−	0.978327i	$-0.433608\pi$
$379$	8.06235	0.414135	0.207067	+	0.978327i	$0.433608\pi$
$380$	0.711160	0.0364817
$381$	0	0
$382$	−32.8628	−1.68141
$383$	4.63823	0.237003	0.118501	−	0.992954i	$-0.462191\pi$
$383$	4.63823	0.237003	0.118501	+	0.992954i	$0.462191\pi$
$384$	0	0
$385$	−8.39936	−0.428071
$386$	28.2008	1.43539
$387$	0	0
$388$	1.21472	0.0616683
$389$	−18.1399	−0.919727	−0.459864	−	0.887990i	$-0.652102\pi$
$389$	−18.1399	−0.919727	−0.459864	+	0.887990i	$0.652102\pi$
$390$	0	0
$391$	13.3263	0.673938
$392$	−17.8293	−0.900518
$393$	0	0
$394$	−8.82906	−0.444802
$395$	−19.3191	−0.972050
$396$	0	0
$397$	32.4912	1.63069	0.815343	−	0.578979i	$-0.196549\pi$
$397$	32.4912	1.63069	0.815343	+	0.578979i	$0.196549\pi$
$398$	−35.4277	−1.77583
$399$	0	0
$400$	−8.63176	−0.431588
$401$	4.95415	0.247399	0.123699	−	0.992320i	$-0.460524\pi$
$401$	4.95415	0.247399	0.123699	+	0.992320i	$0.460524\pi$
$402$	0	0
$403$	6.49122	0.323351
$404$	3.20734	0.159571
$405$	0	0
$406$	1.67146	0.0829533
$407$	−13.0951	−0.649098
$408$	0	0
$409$	27.1671	1.34333	0.671664	−	0.740856i	$-0.265580\pi$
$409$	27.1671	1.34333	0.671664	+	0.740856i	$0.265580\pi$
$410$	22.5089	1.11163
$411$	0	0
$412$	−3.97540	−0.195854
$413$	6.84624	0.336881
$414$	0	0
$415$	−8.34044	−0.409416
$416$	5.92093	0.290297
$417$	0	0
$418$	−3.13870	−0.153519
$419$	1.18237	0.0577625	0.0288812	−	0.999583i	$-0.490806\pi$
$419$	1.18237	0.0577625	0.0288812	+	0.999583i	$0.490806\pi$
$420$	0	0
$421$	25.7739	1.25614	0.628072	−	0.778156i	$-0.283844\pi$
$421$	25.7739	1.25614	0.628072	+	0.778156i	$0.283844\pi$
$422$	−20.4797	−0.996935
$423$	0	0
$424$	26.4383	1.28396
$425$	4.08851	0.198322
$426$	0	0
$427$	−0.0719948	−0.00348407
$428$	−3.19970	−0.154663
$429$	0	0
$430$	3.58856	0.173056
$431$	−9.39722	−0.452648	−0.226324	−	0.974052i	$-0.572671\pi$
$431$	−9.39722	−0.452648	−0.226324	+	0.974052i	$0.572671\pi$
$432$	0	0
$433$	−21.5740	−1.03678	−0.518391	−	0.855144i	$-0.673469\pi$
$433$	−21.5740	−1.03678	−0.518391	+	0.855144i	$0.673469\pi$
$434$	2.58130	0.123907
$435$	0	0
$436$	1.46740	0.0702759
$437$	7.05240	0.337362
$438$	0	0
$439$	2.09855	0.100158	0.0500790	−	0.998745i	$-0.484053\pi$
$439$	2.09855	0.100158	0.0500790	+	0.998745i	$0.484053\pi$
$440$	−24.4454	−1.16539
$441$	0	0
$442$	6.79062	0.322997
$443$	36.4126	1.73001	0.865007	−	0.501760i	$-0.167314\pi$
$443$	36.4126	1.73001	0.865007	+	0.501760i	$0.167314\pi$
$444$	0	0
$445$	42.1835	1.99969
$446$	−27.5197	−1.30309
$447$	0	0
$448$	9.11546	0.430665
$449$	7.78784	0.367531	0.183765	−	0.982970i	$-0.441171\pi$
$449$	7.78784	0.367531	0.183765	+	0.982970i	$0.441171\pi$
$450$	0	0
$451$	18.4687	0.869655
$452$	2.15639	0.101428
$453$	0	0
$454$	−2.01809	−0.0947135
$455$	9.61689	0.450847
$456$	0	0
$457$	−23.6015	−1.10403	−0.552016	−	0.833834i	$-0.686141\pi$
$457$	−23.6015	−1.10403	−0.552016	+	0.833834i	$0.686141\pi$
$458$	−2.57243	−0.120202
$459$	0	0
$460$	7.44367	0.347063
$461$	0.354367	0.0165045	0.00825227	−	0.999966i	$-0.497373\pi$
$461$	0.354367	0.0165045	0.00825227	+	0.999966i	$0.497373\pi$
$462$	0	0
$463$	−10.9981	−0.511123	−0.255561	−	0.966793i	$-0.582260\pi$
$463$	−10.9981	−0.511123	−0.255561	+	0.966793i	$0.582260\pi$
$464$	4.08280	0.189539
$465$	0	0
$466$	33.1660	1.53638
$467$	0.429641	0.0198814	0.00994072	−	0.999951i	$-0.496836\pi$
$467$	0.429641	0.0198814	0.00994072	+	0.999951i	$0.496836\pi$
$468$	0	0
$469$	−11.9022	−0.549593
$470$	−36.2484	−1.67202
$471$	0	0
$472$	19.9252	0.917133
$473$	2.94444	0.135385
$474$	0	0
$475$	2.16369	0.0992767
$476$	−0.502019	−0.0230100
$477$	0	0
$478$	20.7476	0.948975
$479$	−1.33246	−0.0608818	−0.0304409	−	0.999537i	$-0.509691\pi$
$479$	−1.33246	−0.0608818	−0.0304409	+	0.999537i	$0.509691\pi$
$480$	0	0
$481$	14.9933	0.683633
$482$	12.6166	0.574670
$483$	0	0
$484$	0.730616	0.0332098
$485$	10.7061	0.486139
$486$	0	0
$487$	−17.8776	−0.810113	−0.405057	−	0.914292i	$-0.632748\pi$
$487$	−17.8776	−0.810113	−0.405057	+	0.914292i	$0.632748\pi$
$488$	−0.209533	−0.00948511
$489$	0	0
$490$	−21.2957	−0.962041
$491$	−22.9561	−1.03600	−0.517998	−	0.855382i	$-0.673323\pi$
$491$	−22.9561	−1.03600	−0.517998	+	0.855382i	$0.673323\pi$
$492$	0	0
$493$	−1.93385	−0.0870964
$494$	3.59367	0.161687
$495$	0	0
$496$	6.30522	0.283113
$497$	−6.07879	−0.272671
$498$	0	0
$499$	9.44432	0.422786	0.211393	−	0.977401i	$-0.432200\pi$
$499$	9.44432	0.422786	0.211393	+	0.977401i	$0.432200\pi$
$500$	−2.04818	−0.0915972
$501$	0	0
$502$	−7.67629	−0.342610
$503$	44.4176	1.98048	0.990241	−	0.139366i	$-0.0445066\pi$
$503$	44.4176	1.98048	0.990241	+	0.139366i	$0.0445066\pi$
$504$	0	0
$505$	28.2682	1.25792
$506$	−32.8526	−1.46048
$507$	0	0
$508$	−1.95909	−0.0869205
$509$	42.6788	1.89170	0.945852	−	0.324599i	$-0.105229\pi$
$509$	42.6788	1.89170	0.945852	+	0.324599i	$0.105229\pi$
$510$	0	0
$511$	1.95900	0.0866609
$512$	25.4282	1.12378
$513$	0	0
$514$	−36.8676	−1.62616
$515$	−35.0376	−1.54394
$516$	0	0
$517$	−29.7420	−1.30805
$518$	5.96222	0.261965
$519$	0	0
$520$	27.9889	1.22739
$521$	−30.0291	−1.31560	−0.657799	−	0.753194i	$-0.728512\pi$
$521$	−30.0291	−1.31560	−0.657799	+	0.753194i	$0.728512\pi$
$522$	0	0
$523$	−20.0700	−0.877599	−0.438799	−	0.898585i	$-0.644596\pi$
$523$	−20.0700	−0.877599	−0.438799	+	0.898585i	$0.644596\pi$
$524$	−3.62245	−0.158247
$525$	0	0
$526$	−34.0624	−1.48519
$527$	−2.98652	−0.130095
$528$	0	0
$529$	50.8171	2.20944
$530$	31.5784	1.37168
$531$	0	0
$532$	−0.265674	−0.0115184
$533$	−21.1458	−0.915925
$534$	0	0
$535$	−28.2009	−1.21923
$536$	−34.6401	−1.49622
$537$	0	0
$538$	−21.9550	−0.946547
$539$	−17.4732	−0.752625
$540$	0	0
$541$	25.4769	1.09534	0.547669	−	0.836695i	$-0.315515\pi$
$541$	25.4769	1.09534	0.547669	+	0.836695i	$0.315515\pi$
$542$	−19.4264	−0.834434
$543$	0	0
$544$	−2.72414	−0.116796
$545$	12.9331	0.553994
$546$	0	0
$547$	−25.4459	−1.08799	−0.543994	−	0.839089i	$-0.683089\pi$
$547$	−25.4459	−1.08799	−0.543994	+	0.839089i	$0.683089\pi$
$548$	6.55629	0.280071
$549$	0	0
$550$	−10.0792	−0.429779
$551$	−1.02342	−0.0435990
$552$	0	0
$553$	7.21721	0.306907
$554$	−1.20603	−0.0512393
$555$	0	0
$556$	−6.97986	−0.296012
$557$	43.3451	1.83659	0.918295	−	0.395898i	$-0.129567\pi$
$557$	43.3451	1.83659	0.918295	+	0.395898i	$0.129567\pi$
$558$	0	0
$559$	−3.37124	−0.142588
$560$	9.34132	0.394743
$561$	0	0
$562$	−23.1135	−0.974985
$563$	−38.3393	−1.61581	−0.807905	−	0.589312i	$-0.799399\pi$
$563$	−38.3393	−1.61581	−0.807905	+	0.589312i	$0.799399\pi$
$564$	0	0
$565$	19.0056	0.799572
$566$	−2.27783	−0.0957445
$567$	0	0
$568$	−17.6916	−0.742325
$569$	−5.70303	−0.239083	−0.119542	−	0.992829i	$-0.538143\pi$
$569$	−5.70303	−0.239083	−0.119542	+	0.992829i	$0.538143\pi$
$570$	0	0
$571$	43.1923	1.80754	0.903771	−	0.428017i	$-0.140788\pi$
$571$	43.1923	1.80754	0.903771	+	0.428017i	$0.140788\pi$
$572$	3.11221	0.130128
$573$	0	0
$574$	−8.40884	−0.350978
$575$	22.6472	0.944452
$576$	0	0
$577$	1.62734	0.0677469	0.0338735	−	0.999426i	$-0.489216\pi$
$577$	1.62734	0.0677469	0.0338735	+	0.999426i	$0.489216\pi$
$578$	18.9526	0.788326
$579$	0	0
$580$	−1.08020	−0.0448527
$581$	3.11581	0.129266
$582$	0	0
$583$	25.9102	1.07309
$584$	5.70144	0.235927
$585$	0	0
$586$	9.74075	0.402387
$587$	−45.7807	−1.88957	−0.944786	−	0.327688i	$-0.893731\pi$
$587$	−45.7807	−1.88957	−0.944786	+	0.327688i	$0.893731\pi$
$588$	0	0
$589$	−1.58050	−0.0651235
$590$	23.7990	0.979791
$591$	0	0
$592$	14.5636	0.598561
$593$	22.1612	0.910051	0.455026	−	0.890478i	$-0.349630\pi$
$593$	22.1612	0.910051	0.455026	+	0.890478i	$0.349630\pi$
$594$	0	0
$595$	−4.42460	−0.181391
$596$	−3.96891	−0.162573
$597$	0	0
$598$	37.6147	1.53818
$599$	−32.6031	−1.33213	−0.666063	−	0.745895i	$-0.732022\pi$
$599$	−32.6031	−1.33213	−0.666063	+	0.745895i	$0.732022\pi$
$600$	0	0
$601$	1.63641	0.0667507	0.0333753	−	0.999443i	$-0.489374\pi$
$601$	1.63641	0.0667507	0.0333753	+	0.999443i	$0.489374\pi$
$602$	−1.34061	−0.0546392
$603$	0	0
$604$	1.18453	0.0481977
$605$	6.43937	0.261798
$606$	0	0
$607$	9.91603	0.402479	0.201240	−	0.979542i	$-0.435503\pi$
$607$	9.91603	0.402479	0.201240	+	0.979542i	$0.435503\pi$
$608$	−1.44164	−0.0584664
$609$	0	0
$610$	−0.250270	−0.0101331
$611$	34.0533	1.37765
$612$	0	0
$613$	26.3963	1.06614	0.533068	−	0.846072i	$-0.321039\pi$
$613$	26.3963	1.06614	0.533068	+	0.846072i	$0.321039\pi$
$614$	−31.8917	−1.28705
$615$	0	0
$616$	9.13229	0.367950
$617$	−19.9747	−0.804152	−0.402076	−	0.915606i	$-0.631711\pi$
$617$	−19.9747	−0.804152	−0.402076	+	0.915606i	$0.631711\pi$
$618$	0	0
$619$	36.0843	1.45035	0.725174	−	0.688565i	$-0.241759\pi$
$619$	36.0843	1.45035	0.725174	+	0.688565i	$0.241759\pi$
$620$	−1.66819	−0.0669960
$621$	0	0
$622$	24.3128	0.974855
$623$	−15.7589	−0.631366
$624$	0	0
$625$	−31.2315	−1.24926
$626$	1.42662	0.0570191
$627$	0	0
$628$	−1.02719	−0.0409895
$629$	−6.89819	−0.275049
$630$	0	0
$631$	−46.6716	−1.85797	−0.928984	−	0.370120i	$-0.879317\pi$
$631$	−46.6716	−1.85797	−0.928984	+	0.370120i	$0.879317\pi$
$632$	21.0049	0.835530
$633$	0	0
$634$	−3.26892	−0.129826
$635$	−17.2666	−0.685206
$636$	0	0
$637$	20.0060	0.792668
$638$	4.76744	0.188745
$639$	0	0
$640$	21.9809	0.868871
$641$	−0.767578	−0.0303175	−0.0151588	−	0.999885i	$-0.504825\pi$
$641$	−0.767578	−0.0303175	−0.0151588	+	0.999885i	$0.504825\pi$
$642$	0	0
$643$	28.8106	1.13618	0.568090	−	0.822967i	$-0.307683\pi$
$643$	28.8106	1.13618	0.568090	+	0.822967i	$0.307683\pi$
$644$	−2.78079	−0.109579
$645$	0	0
$646$	−1.65340	−0.0650521
$647$	−5.46805	−0.214971	−0.107486	−	0.994207i	$-0.534280\pi$
$647$	−5.46805	−0.214971	−0.107486	+	0.994207i	$0.534280\pi$
$648$	0	0
$649$	19.5272	0.766511
$650$	11.5402	0.452646
$651$	0	0
$652$	0.924755	0.0362162
$653$	5.71517	0.223652	0.111826	−	0.993728i	$-0.464330\pi$
$653$	5.71517	0.223652	0.111826	+	0.993728i	$0.464330\pi$
$654$	0	0
$655$	−31.9269	−1.24749
$656$	−20.5398	−0.801946
$657$	0	0
$658$	13.5416	0.527908
$659$	36.4298	1.41910	0.709552	−	0.704653i	$-0.248897\pi$
$659$	36.4298	1.41910	0.709552	+	0.704653i	$0.248897\pi$
$660$	0	0
$661$	−19.7081	−0.766555	−0.383278	−	0.923633i	$-0.625205\pi$
$661$	−19.7081	−0.766555	−0.383278	+	0.923633i	$0.625205\pi$
$662$	17.5835	0.683403
$663$	0	0
$664$	9.06822	0.351915
$665$	−2.34155	−0.0908013
$666$	0	0
$667$	−10.7120	−0.414772
$668$	−7.49838	−0.290121
$669$	0	0
$670$	−41.3747	−1.59845
$671$	−0.205348	−0.00792736
$672$	0	0
$673$	−3.37995	−0.130287	−0.0651437	−	0.997876i	$-0.520751\pi$
$673$	−3.37995	−0.130287	−0.0651437	+	0.997876i	$0.520751\pi$
$674$	3.17075	0.122133
$675$	0	0
$676$	0.512530	0.0197127
$677$	−16.9779	−0.652515	−0.326258	−	0.945281i	$-0.605788\pi$
$677$	−16.9779	−0.652515	−0.326258	+	0.945281i	$0.605788\pi$
$678$	0	0
$679$	−3.99957	−0.153489
$680$	−12.8773	−0.493823
$681$	0	0
$682$	7.36254	0.281926
$683$	−10.3202	−0.394890	−0.197445	−	0.980314i	$-0.563264\pi$
$683$	−10.3202	−0.394890	−0.197445	+	0.980314i	$0.563264\pi$
$684$	0	0
$685$	57.7846	2.20784
$686$	17.3399	0.662040
$687$	0	0
$688$	−3.27464	−0.124845
$689$	−29.6660	−1.13019
$690$	0	0
$691$	4.23841	0.161237	0.0806184	−	0.996745i	$-0.474310\pi$
$691$	4.23841	0.161237	0.0806184	+	0.996745i	$0.474310\pi$
$692$	−6.83733	−0.259916
$693$	0	0
$694$	−2.44064	−0.0926453
$695$	−61.5178	−2.33350
$696$	0	0
$697$	9.72889	0.368508
$698$	−31.3450	−1.18643
$699$	0	0
$700$	−0.853151	−0.0322461
$701$	−4.89904	−0.185034	−0.0925171	−	0.995711i	$-0.529491\pi$
$701$	−4.89904	−0.185034	−0.0925171	+	0.995711i	$0.529491\pi$
$702$	0	0
$703$	−3.65060	−0.137685
$704$	25.9996	0.979898
$705$	0	0
$706$	17.6022	0.662469
$707$	−10.5604	−0.397165
$708$	0	0
$709$	−34.3250	−1.28910	−0.644552	−	0.764561i	$-0.722956\pi$
$709$	−34.3250	−1.28910	−0.644552	+	0.764561i	$0.722956\pi$
$710$	−21.1312	−0.793041
$711$	0	0
$712$	−45.8644	−1.71884
$713$	−16.5430	−0.619541
$714$	0	0
$715$	27.4298	1.02582
$716$	−2.17237	−0.0811854
$717$	0	0
$718$	4.53602	0.169283
$719$	−13.6554	−0.509260	−0.254630	−	0.967039i	$-0.581954\pi$
$719$	−13.6554	−0.509260	−0.254630	+	0.967039i	$0.581954\pi$
$720$	0	0
$721$	13.0893	0.487471
$722$	23.7992	0.885715
$723$	0	0
$724$	3.45756	0.128499
$725$	−3.28647	−0.122056
$726$	0	0
$727$	38.3239	1.42135	0.710677	−	0.703518i	$-0.248389\pi$
$727$	38.3239	1.42135	0.710677	+	0.703518i	$0.248389\pi$
$728$	−10.4561	−0.387527
$729$	0	0
$730$	6.80990	0.252046
$731$	1.55106	0.0573682
$732$	0	0
$733$	4.26395	0.157493	0.0787463	−	0.996895i	$-0.474908\pi$
$733$	4.26395	0.157493	0.0787463	+	0.996895i	$0.474908\pi$
$734$	−37.9142	−1.39944
$735$	0	0
$736$	−15.0896	−0.556210
$737$	−33.9482	−1.25050
$738$	0	0
$739$	−33.6314	−1.23715	−0.618576	−	0.785725i	$-0.712290\pi$
$739$	−33.6314	−1.23715	−0.618576	+	0.785725i	$0.712290\pi$
$740$	−3.85313	−0.141644
$741$	0	0
$742$	−11.7970	−0.433082
$743$	16.9173	0.620637	0.310318	−	0.950633i	$-0.399564\pi$
$743$	16.9173	0.620637	0.310318	+	0.950633i	$0.399564\pi$
$744$	0	0
$745$	−34.9804	−1.28158
$746$	7.68266	0.281282
$747$	0	0
$748$	−1.43189	−0.0523550
$749$	10.5352	0.384950
$750$	0	0
$751$	−9.28470	−0.338804	−0.169402	−	0.985547i	$-0.554184\pi$
$751$	−9.28470	−0.338804	−0.169402	+	0.985547i	$0.554184\pi$
$752$	33.0775	1.20621
$753$	0	0
$754$	−5.45850	−0.198787
$755$	10.4400	0.379949
$756$	0	0
$757$	−34.5024	−1.25401	−0.627005	−	0.779015i	$-0.715719\pi$
$757$	−34.5024	−1.25401	−0.627005	+	0.779015i	$0.715719\pi$
$758$	−10.4701	−0.380291
$759$	0	0
$760$	−6.81482	−0.247199
$761$	−15.9585	−0.578494	−0.289247	−	0.957254i	$-0.593405\pi$
$761$	−15.9585	−0.578494	−0.289247	+	0.957254i	$0.593405\pi$
$762$	0	0
$763$	−4.83154	−0.174913
$764$	−7.93401	−0.287042
$765$	0	0
$766$	−6.02341	−0.217635
$767$	−22.3578	−0.807293
$768$	0	0
$769$	−15.7181	−0.566810	−0.283405	−	0.959000i	$-0.591464\pi$
$769$	−15.7181	−0.566810	−0.283405	+	0.959000i	$0.591464\pi$
$770$	10.9078	0.393089
$771$	0	0
$772$	6.80848	0.245042
$773$	0.818514	0.0294399	0.0147200	−	0.999892i	$-0.495314\pi$
$773$	0.818514	0.0294399	0.0147200	+	0.999892i	$0.495314\pi$
$774$	0	0
$775$	−5.07542	−0.182314
$776$	−11.6403	−0.417863
$777$	0	0
$778$	23.5572	0.844566
$779$	5.14863	0.184469
$780$	0	0
$781$	−17.3383	−0.620412
$782$	−17.3060	−0.618863
$783$	0	0
$784$	19.4328	0.694028
$785$	−9.05328	−0.323126
$786$	0	0
$787$	36.6420	1.30615	0.653074	−	0.757294i	$-0.273479\pi$
$787$	36.6420	1.30615	0.653074	+	0.757294i	$0.273479\pi$
$788$	−2.13158	−0.0759345
$789$	0	0
$790$	25.0886	0.892613
$791$	−7.10009	−0.252450
$792$	0	0
$793$	0.235114	0.00834914
$794$	−42.1944	−1.49742
$795$	0	0
$796$	−8.55325	−0.303162
$797$	12.0354	0.426316	0.213158	−	0.977018i	$-0.431625\pi$
$797$	12.0354	0.426316	0.213158	+	0.977018i	$0.431625\pi$
$798$	0	0
$799$	−15.6674	−0.554274
$800$	−4.62951	−0.163678
$801$	0	0
$802$	−6.43367	−0.227181
$803$	5.58756	0.197181
$804$	0	0
$805$	−24.5088	−0.863823
$806$	−8.42978	−0.296926
$807$	0	0
$808$	−30.7349	−1.08125
$809$	−8.87248	−0.311940	−0.155970	−	0.987762i	$-0.549850\pi$
$809$	−8.87248	−0.311940	−0.155970	+	0.987762i	$0.549850\pi$
$810$	0	0
$811$	9.69230	0.340343	0.170171	−	0.985414i	$-0.445568\pi$
$811$	9.69230	0.340343	0.170171	+	0.985414i	$0.445568\pi$
$812$	0.403538	0.0141614
$813$	0	0
$814$	17.0058	0.596053
$815$	8.15043	0.285497
$816$	0	0
$817$	0.820840	0.0287176
$818$	−35.2804	−1.23355
$819$	0	0
$820$	5.43428	0.189773
$821$	10.2797	0.358764	0.179382	−	0.983779i	$-0.442590\pi$
$821$	10.2797	0.358764	0.179382	+	0.983779i	$0.442590\pi$
$822$	0	0
$823$	41.9173	1.46115	0.730573	−	0.682835i	$-0.239253\pi$
$823$	41.9173	1.46115	0.730573	+	0.682835i	$0.239253\pi$
$824$	38.0950	1.32710
$825$	0	0
$826$	−8.89081	−0.309351
$827$	27.5477	0.957927	0.478963	−	0.877835i	$-0.341013\pi$
$827$	27.5477	0.957927	0.478963	+	0.877835i	$0.341013\pi$
$828$	0	0
$829$	−23.3262	−0.810154	−0.405077	−	0.914283i	$-0.632755\pi$
$829$	−23.3262	−0.810154	−0.405077	+	0.914283i	$0.632755\pi$
$830$	10.8312	0.375958
$831$	0	0
$832$	−29.7684	−1.03203
$833$	−9.20451	−0.318917
$834$	0	0
$835$	−66.0879	−2.28706
$836$	−0.757771	−0.0262081
$837$	0	0
$838$	−1.53547	−0.0530421
$839$	8.62159	0.297650	0.148825	−	0.988864i	$-0.452451\pi$
$839$	8.62159	0.297650	0.148825	+	0.988864i	$0.452451\pi$
$840$	0	0
$841$	−27.4455	−0.946397
$842$	−33.4711	−1.15349
$843$	0	0
$844$	−4.94437	−0.170192
$845$	4.51724	0.155398
$846$	0	0
$847$	−2.40561	−0.0826577
$848$	−28.8160	−0.989545
$849$	0	0
$850$	−5.30951	−0.182115
$851$	−38.2106	−1.30984
$852$	0	0
$853$	42.7715	1.46447	0.732235	−	0.681052i	$-0.238477\pi$
$853$	42.7715	1.46447	0.732235	+	0.681052i	$0.238477\pi$
$854$	0.0934955	0.00319935
$855$	0	0
$856$	30.6617	1.04799
$857$	44.5419	1.52152	0.760762	−	0.649031i	$-0.224826\pi$
$857$	44.5419	1.52152	0.760762	+	0.649031i	$0.224826\pi$
$858$	0	0
$859$	−31.9703	−1.09081	−0.545406	−	0.838172i	$-0.683625\pi$
$859$	−31.9703	−1.09081	−0.545406	+	0.838172i	$0.683625\pi$
$860$	0.866380	0.0295433
$861$	0	0
$862$	12.2036	0.415657
$863$	−39.0037	−1.32770	−0.663851	−	0.747865i	$-0.731079\pi$
$863$	−39.0037	−1.32770	−0.663851	+	0.747865i	$0.731079\pi$
$864$	0	0
$865$	−60.2616	−2.04896
$866$	28.0170	0.952055
$867$	0	0
$868$	0.623199	0.0211528
$869$	20.5853	0.698310
$870$	0	0
$871$	38.8691	1.31703
$872$	−14.0617	−0.476188
$873$	0	0
$874$	−9.15855	−0.309792
$875$	6.74377	0.227981
$876$	0	0
$877$	−50.1121	−1.69216	−0.846082	−	0.533053i	$-0.821045\pi$
$877$	−50.1121	−1.69216	−0.846082	+	0.533053i	$0.821045\pi$
$878$	−2.72526	−0.0919730
$879$	0	0
$880$	26.6439	0.898164
$881$	−30.9354	−1.04224	−0.521120	−	0.853483i	$-0.674486\pi$
$881$	−30.9354	−1.04224	−0.521120	+	0.853483i	$0.674486\pi$
$882$	0	0
$883$	35.3840	1.19077	0.595383	−	0.803442i	$-0.297000\pi$
$883$	35.3840	1.19077	0.595383	+	0.803442i	$0.297000\pi$
$884$	1.63945	0.0551405
$885$	0	0
$886$	−47.2869	−1.58864
$887$	48.9466	1.64347	0.821733	−	0.569873i	$-0.193008\pi$
$887$	48.9466	1.64347	0.821733	+	0.569873i	$0.193008\pi$
$888$	0	0
$889$	6.45045	0.216341
$890$	−54.7813	−1.83627
$891$	0	0
$892$	−6.64402	−0.222458
$893$	−8.29139	−0.277461
$894$	0	0
$895$	−19.1464	−0.639995
$896$	−8.21159	−0.274330
$897$	0	0
$898$	−10.1136	−0.337496
$899$	2.40066	0.0800664
$900$	0	0
$901$	13.6489	0.454712
$902$	−23.9842	−0.798586
$903$	0	0
$904$	−20.6640	−0.687276
$905$	30.4736	1.01298
$906$	0	0
$907$	28.3838	0.942469	0.471234	−	0.882008i	$-0.343809\pi$
$907$	28.3838	0.942469	0.471234	+	0.882008i	$0.343809\pi$
$908$	−0.487223	−0.0161691
$909$	0	0
$910$	−12.4889	−0.414003
$911$	5.79864	0.192118	0.0960588	−	0.995376i	$-0.469376\pi$
$911$	5.79864	0.192118	0.0960588	+	0.995376i	$0.469376\pi$
$912$	0	0
$913$	8.88710	0.294120
$914$	30.6499	1.01381
$915$	0	0
$916$	−0.621058	−0.0205203
$917$	11.9272	0.393871
$918$	0	0
$919$	−46.2912	−1.52701	−0.763503	−	0.645804i	$-0.776522\pi$
$919$	−46.2912	−1.52701	−0.763503	+	0.645804i	$0.776522\pi$
$920$	−71.3303	−2.35169
$921$	0	0
$922$	−0.460196	−0.0151558
$923$	19.8515	0.653421
$924$	0	0
$925$	−11.7231	−0.385452
$926$	14.2825	0.469353
$927$	0	0
$928$	2.18974	0.0718819
$929$	42.2402	1.38586	0.692928	−	0.721007i	$-0.256321\pi$
$929$	42.2402	1.38586	0.692928	+	0.721007i	$0.256321\pi$
$930$	0	0
$931$	−4.87113	−0.159645
$932$	8.00720	0.262285
$933$	0	0
$934$	−0.557950	−0.0182567
$935$	−12.6201	−0.412722
$936$	0	0
$937$	34.2767	1.11977	0.559886	−	0.828570i	$-0.310845\pi$
$937$	34.2767	1.11977	0.559886	+	0.828570i	$0.310845\pi$
$938$	15.4567	0.504680
$939$	0	0
$940$	−8.75139	−0.285439
$941$	−30.4043	−0.991151	−0.495576	−	0.868565i	$-0.665043\pi$
$941$	−30.4043	−0.991151	−0.495576	+	0.868565i	$0.665043\pi$
$942$	0	0
$943$	53.8904	1.75491
$944$	−21.7171	−0.706833
$945$	0	0
$946$	−3.82377	−0.124321
$947$	36.2690	1.17859	0.589293	−	0.807920i	$-0.299406\pi$
$947$	36.2690	1.17859	0.589293	+	0.807920i	$0.299406\pi$
$948$	0	0
$949$	−6.39750	−0.207672
$950$	−2.80985	−0.0911637
$951$	0	0
$952$	4.81069	0.155915
$953$	−21.1687	−0.685722	−0.342861	−	0.939386i	$-0.611396\pi$
$953$	−21.1687	−0.685722	−0.342861	+	0.939386i	$0.611396\pi$
$954$	0	0
$955$	−69.9273	−2.26279
$956$	5.00906	0.162005
$957$	0	0
$958$	1.73039	0.0559064
$959$	−21.5871	−0.697084
$960$	0	0
$961$	−27.2926	−0.880406
$962$	−19.4709	−0.627766
$963$	0	0
$964$	3.04600	0.0981051
$965$	60.0073	1.93170
$966$	0	0
$967$	−3.18099	−0.102294	−0.0511468	−	0.998691i	$-0.516288\pi$
$967$	−3.18099	−0.102294	−0.0511468	+	0.998691i	$0.516288\pi$
$968$	−7.00126	−0.225029
$969$	0	0
$970$	−13.9034	−0.446411
$971$	−23.4998	−0.754145	−0.377072	−	0.926184i	$-0.623069\pi$
$971$	−23.4998	−0.754145	−0.377072	+	0.926184i	$0.623069\pi$
$972$	0	0
$973$	22.9817	0.736760
$974$	23.2167	0.743910
$975$	0	0
$976$	0.228377	0.00731016
$977$	40.4628	1.29452	0.647260	−	0.762270i	$-0.275915\pi$
$977$	40.4628	1.29452	0.647260	+	0.762270i	$0.275915\pi$
$978$	0	0
$979$	−44.9484	−1.43656
$980$	−5.14138	−0.164235
$981$	0	0
$982$	29.8118	0.951333
$983$	41.8639	1.33525	0.667626	−	0.744497i	$-0.267310\pi$
$983$	41.8639	1.33525	0.667626	+	0.744497i	$0.267310\pi$
$984$	0	0
$985$	−18.7870	−0.598602
$986$	2.51138	0.0799787
$987$	0	0
$988$	0.867613	0.0276025
$989$	8.59169	0.273200
$990$	0	0
$991$	−18.0897	−0.574638	−0.287319	−	0.957835i	$-0.592764\pi$
$991$	−18.0897	−0.574638	−0.287319	+	0.957835i	$0.592764\pi$
$992$	3.38171	0.107369
$993$	0	0
$994$	7.89417	0.250388
$995$	−75.3850	−2.38987
$996$	0	0
$997$	−37.3396	−1.18256	−0.591278	−	0.806468i	$-0.701376\pi$
$997$	−37.3396	−1.18256	−0.591278	+	0.806468i	$0.701376\pi$
$998$	−12.2648	−0.388235
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3483.2.a.s.1.6		19
3.2	odd	2		3483.2.a.r.1.14		19
9.2	odd	6		1161.2.f.c.388.6		38
9.4	even	3		387.2.f.c.259.14	yes	38
9.5	odd	6		1161.2.f.c.775.6		38
9.7	even	3		387.2.f.c.130.14	✓	38

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
387.2.f.c.130.14	✓	38	9.7	even	3
387.2.f.c.259.14	yes	38	9.4	even	3
1161.2.f.c.388.6		38	9.2	odd	6
1161.2.f.c.775.6		38	9.5	odd	6
3483.2.a.r.1.14		19	3.2	odd	2
3483.2.a.s.1.6		19	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:	\( N \)	\(=\)	\( 3483 = 3^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3483.a (trivial)

\(f(q)\)	\(=\)	\(q-1.29864 q^{2} -0.313529 q^{4} -2.76332 q^{5} +1.03232 q^{7} +3.00445 q^{8} +O(q^{10})\)	\(q-1.29864 q^{2} -0.313529 q^{4} -2.76332 q^{5} +1.03232 q^{7} +3.00445 q^{8} +3.58856 q^{10} +2.94444 q^{11} -3.37124 q^{13} -1.34061 q^{14} -3.27464 q^{16} +1.55106 q^{17} +0.820840 q^{19} +0.866380 q^{20} -3.82377 q^{22} +8.59169 q^{23} +2.63594 q^{25} +4.37804 q^{26} -0.323661 q^{28} -1.24679 q^{29} -1.92547 q^{31} -1.75630 q^{32} -2.01428 q^{34} -2.85262 q^{35} -4.44739 q^{37} -1.06598 q^{38} -8.30225 q^{40} +6.27239 q^{41} +1.00000 q^{43} -0.923165 q^{44} -11.1575 q^{46} -10.1011 q^{47} -5.93432 q^{49} -3.42314 q^{50} +1.05698 q^{52} +8.79973 q^{53} -8.13642 q^{55} +3.10154 q^{56} +1.61914 q^{58} +6.63191 q^{59} -0.0697410 q^{61} +2.50049 q^{62} +8.83009 q^{64} +9.31583 q^{65} -11.5296 q^{67} -0.486303 q^{68} +3.70454 q^{70} -5.88849 q^{71} +1.89767 q^{73} +5.77557 q^{74} -0.257357 q^{76} +3.03959 q^{77} +6.99127 q^{79} +9.04889 q^{80} -8.14560 q^{82} +3.01827 q^{83} -4.28609 q^{85} -1.29864 q^{86} +8.84640 q^{88} -15.2655 q^{89} -3.48019 q^{91} -2.69374 q^{92} +13.1177 q^{94} -2.26824 q^{95} -3.87436 q^{97} +7.70656 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 4 q^{2} + 22 q^{4} + 9 q^{5} + 7 q^{7} + 12 q^{8}+O(q^{10}) \) 19 * q + 4 * q^2 + 22 * q^4 + 9 * q^5 + 7 * q^7 + 12 * q^8	\( 19 q + 4 q^{2} + 22 q^{4} + 9 q^{5} + 7 q^{7} + 12 q^{8} - 7 q^{10} + 5 q^{11} - 5 q^{13} + 17 q^{14} + 24 q^{16} + 21 q^{17} - 4 q^{19} + 21 q^{20} - 20 q^{22} + 22 q^{23} + 10 q^{25} + 17 q^{26} - q^{28} + 30 q^{29} - 5 q^{31} + 48 q^{32} - 6 q^{34} + 53 q^{35} - q^{37} + 21 q^{38} + 16 q^{40} + 29 q^{41} + 19 q^{43} + 29 q^{44} + 32 q^{47} - 10 q^{49} - 11 q^{50} + q^{52} + 38 q^{53} + 2 q^{55} + 46 q^{56} + 30 q^{58} + 30 q^{59} - 10 q^{61} + 25 q^{62} + 14 q^{64} + 8 q^{65} + 3 q^{67} + 47 q^{68} + 56 q^{70} + 21 q^{71} + 8 q^{73} + 28 q^{74} - 36 q^{76} + 49 q^{77} + 4 q^{79} + 70 q^{80} - 4 q^{82} + 29 q^{83} - 4 q^{85} + 4 q^{86} - 47 q^{88} + 54 q^{89} + 4 q^{91} + 12 q^{92} - 23 q^{94} + 33 q^{95} - 4 q^{97} + 49 q^{98}+O(q^{100}) \) 19 * q + 4 * q^2 + 22 * q^4 + 9 * q^5 + 7 * q^7 + 12 * q^8 - 7 * q^10 + 5 * q^11 - 5 * q^13 + 17 * q^14 + 24 * q^16 + 21 * q^17 - 4 * q^19 + 21 * q^20 - 20 * q^22 + 22 * q^23 + 10 * q^25 + 17 * q^26 - q^28 + 30 * q^29 - 5 * q^31 + 48 * q^32 - 6 * q^34 + 53 * q^35 - q^37 + 21 * q^38 + 16 * q^40 + 29 * q^41 + 19 * q^43 + 29 * q^44 + 32 * q^47 - 10 * q^49 - 11 * q^50 + q^52 + 38 * q^53 + 2 * q^55 + 46 * q^56 + 30 * q^58 + 30 * q^59 - 10 * q^61 + 25 * q^62 + 14 * q^64 + 8 * q^65 + 3 * q^67 + 47 * q^68 + 56 * q^70 + 21 * q^71 + 8 * q^73 + 28 * q^74 - 36 * q^76 + 49 * q^77 + 4 * q^79 + 70 * q^80 - 4 * q^82 + 29 * q^83 - 4 * q^85 + 4 * q^86 - 47 * q^88 + 54 * q^89 + 4 * q^91 + 12 * q^92 - 23 * q^94 + 33 * q^95 - 4 * q^97 + 49 * q^98

Introduction

L-functions

Modular forms

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Database

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