LMFDB - Embedded newform 3465.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3465,2,Mod(1,3465)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3465, 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("3465.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3465.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:	$27.6681643004$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 385)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	3465.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.67513 q^{2} +5.15633 q^{4} +1.00000 q^{5} -1.00000 q^{7} +8.44358 q^{8} +O(q^{10})$	\(q+2.67513 q^{2} +5.15633 q^{4} +1.00000 q^{5} -1.00000 q^{7} +8.44358 q^{8} +2.67513 q^{10} +1.00000 q^{11} +5.83146 q^{13} -2.67513 q^{14} +12.2750 q^{16} -5.44358 q^{17} -1.35026 q^{19} +5.15633 q^{20} +2.67513 q^{22} +3.19394 q^{23} +1.00000 q^{25} +15.5999 q^{26} -5.15633 q^{28} +3.61213 q^{29} -5.28726 q^{31} +15.9502 q^{32} -14.5623 q^{34} -1.00000 q^{35} -8.54420 q^{37} -3.61213 q^{38} +8.44358 q^{40} +5.02539 q^{41} +5.89446 q^{43} +5.15633 q^{44} +8.54420 q^{46} +11.8315 q^{47} +1.00000 q^{49} +2.67513 q^{50} +30.0689 q^{52} +0.231548 q^{53} +1.00000 q^{55} -8.44358 q^{56} +9.66291 q^{58} -13.5999 q^{59} -1.41327 q^{61} -14.1441 q^{62} +18.1187 q^{64} +5.83146 q^{65} -10.8568 q^{67} -28.0689 q^{68} -2.67513 q^{70} +15.5369 q^{71} -11.3684 q^{73} -22.8568 q^{74} -6.96239 q^{76} -1.00000 q^{77} +1.96968 q^{79} +12.2750 q^{80} +13.4436 q^{82} -10.6253 q^{83} -5.44358 q^{85} +15.7685 q^{86} +8.44358 q^{88} -7.22425 q^{89} -5.83146 q^{91} +16.4690 q^{92} +31.6507 q^{94} -1.35026 q^{95} -0.836381 q^{97} +2.67513 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 3 * q^7 + 9 * q^8	$ 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8} + 3 q^{10} + 3 q^{11} + 2 q^{13} - 3 q^{14} + 5 q^{16} + 6 q^{19} + 5 q^{20} + 3 q^{22} + 10 q^{23} + 3 q^{25} + 20 q^{26} - 5 q^{28} + 10 q^{29} - 10 q^{31} + 11 q^{32} - 6 q^{34} - 3 q^{35} - 16 q^{37} - 10 q^{38} + 9 q^{40} - 2 q^{43} + 5 q^{44} + 16 q^{46} + 20 q^{47} + 3 q^{49} + 3 q^{50} + 32 q^{52} + 12 q^{53} + 3 q^{55} - 9 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} - 6 q^{62} + 33 q^{64} + 2 q^{65} - 2 q^{67} - 26 q^{68} - 3 q^{70} + 24 q^{71} + 4 q^{73} - 38 q^{74} - 10 q^{76} - 3 q^{77} + 8 q^{79} + 5 q^{80} + 24 q^{82} + 10 q^{83} + 36 q^{86} + 9 q^{88} - 20 q^{89} - 2 q^{91} + 18 q^{92} + 38 q^{94} + 6 q^{95} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 3 * q^7 + 9 * q^8 + 3 * q^10 + 3 * q^11 + 2 * q^13 - 3 * q^14 + 5 * q^16 + 6 * q^19 + 5 * q^20 + 3 * q^22 + 10 * q^23 + 3 * q^25 + 20 * q^26 - 5 * q^28 + 10 * q^29 - 10 * q^31 + 11 * q^32 - 6 * q^34 - 3 * q^35 - 16 * q^37 - 10 * q^38 + 9 * q^40 - 2 * q^43 + 5 * q^44 + 16 * q^46 + 20 * q^47 + 3 * q^49 + 3 * q^50 + 32 * q^52 + 12 * q^53 + 3 * q^55 - 9 * q^56 - 2 * q^58 - 14 * q^59 + 10 * q^61 - 6 * q^62 + 33 * q^64 + 2 * q^65 - 2 * q^67 - 26 * q^68 - 3 * q^70 + 24 * q^71 + 4 * q^73 - 38 * q^74 - 10 * q^76 - 3 * q^77 + 8 * q^79 + 5 * q^80 + 24 * q^82 + 10 * q^83 + 36 * q^86 + 9 * q^88 - 20 * q^89 - 2 * q^91 + 18 * q^92 + 38 * q^94 + 6 * q^95 + 3 * 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 $

$2$	2.67513	1.89160	0.945802	−	0.324745i	$-0.105279\pi$
$2$	2.67513	1.89160	0.945802	+	0.324745i	$0.105279\pi$
$3$	0	0
$3$	0	0
$4$	5.15633	2.57816
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	8.44358	2.98526
$9$	0	0
$10$	2.67513	0.845951
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.83146	1.61735	0.808677	−	0.588252i	$-0.200184\pi$
$13$	5.83146	1.61735	0.808677	+	0.588252i	$0.200184\pi$
$14$	−2.67513	−0.714959
$15$	0	0
$16$	12.2750	3.06876
$17$	−5.44358	−1.32026	−0.660131	−	0.751150i	$-0.729499\pi$
$17$	−5.44358	−1.32026	−0.660131	+	0.751150i	$0.729499\pi$
$18$	0	0
$19$	−1.35026	−0.309771	−0.154886	−	0.987932i	$-0.549501\pi$
$19$	−1.35026	−0.309771	−0.154886	+	0.987932i	$0.549501\pi$
$20$	5.15633	1.15299
$21$	0	0
$22$	2.67513	0.570340
$23$	3.19394	0.665982	0.332991	−	0.942930i	$-0.391942\pi$
$23$	3.19394	0.665982	0.332991	+	0.942930i	$0.391942\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	15.5999	3.05939
$27$	0	0
$28$	−5.15633	−0.974454
$29$	3.61213	0.670755	0.335378	−	0.942084i	$-0.391136\pi$
$29$	3.61213	0.670755	0.335378	+	0.942084i	$0.391136\pi$
$30$	0	0
$31$	−5.28726	−0.949620	−0.474810	−	0.880088i	$-0.657483\pi$
$31$	−5.28726	−0.949620	−0.474810	+	0.880088i	$0.657483\pi$
$32$	15.9502	2.81962
$33$	0	0
$34$	−14.5623	−2.49741
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−8.54420	−1.40466	−0.702329	−	0.711853i	$-0.747856\pi$
$37$	−8.54420	−1.40466	−0.702329	+	0.711853i	$0.747856\pi$
$38$	−3.61213	−0.585964
$39$	0	0
$40$	8.44358	1.33505
$41$	5.02539	0.784834	0.392417	−	0.919787i	$-0.371639\pi$
$41$	5.02539	0.784834	0.392417	+	0.919787i	$0.371639\pi$
$42$	0	0
$43$	5.89446	0.898897	0.449448	−	0.893306i	$-0.351621\pi$
$43$	5.89446	0.898897	0.449448	+	0.893306i	$0.351621\pi$
$44$	5.15633	0.777345
$45$	0	0
$46$	8.54420	1.25977
$47$	11.8315	1.72580	0.862898	−	0.505379i	$-0.168647\pi$
$47$	11.8315	1.72580	0.862898	+	0.505379i	$0.168647\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	2.67513	0.378321
$51$	0	0
$52$	30.0689	4.16980
$53$	0.231548	0.0318056	0.0159028	−	0.999874i	$-0.494938\pi$
$53$	0.231548	0.0318056	0.0159028	+	0.999874i	$0.494938\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−8.44358	−1.12832
$57$	0	0
$58$	9.66291	1.26880
$59$	−13.5999	−1.77056	−0.885279	−	0.465061i	$-0.846032\pi$
$59$	−13.5999	−1.77056	−0.885279	+	0.465061i	$0.846032\pi$
$60$	0	0
$61$	−1.41327	−0.180950	−0.0904751	−	0.995899i	$-0.528839\pi$
$61$	−1.41327	−0.180950	−0.0904751	+	0.995899i	$0.528839\pi$
$62$	−14.1441	−1.79630
$63$	0	0
$64$	18.1187	2.26484
$65$	5.83146	0.723303
$66$	0	0
$67$	−10.8568	−1.32638	−0.663188	−	0.748453i	$-0.730797\pi$
$67$	−10.8568	−1.32638	−0.663188	+	0.748453i	$0.730797\pi$
$68$	−28.0689	−3.40385
$69$	0	0
$70$	−2.67513	−0.319739
$71$	15.5369	1.84389	0.921946	−	0.387319i	$-0.126599\pi$
$71$	15.5369	1.84389	0.921946	+	0.387319i	$0.126599\pi$
$72$	0	0
$73$	−11.3684	−1.33057	−0.665283	−	0.746591i	$-0.731689\pi$
$73$	−11.3684	−1.33057	−0.665283	+	0.746591i	$0.731689\pi$
$74$	−22.8568	−2.65705
$75$	0	0
$76$	−6.96239	−0.798641
$77$	−1.00000	−0.113961
$78$	0	0
$79$	1.96968	0.221607	0.110803	−	0.993842i	$-0.464658\pi$
$79$	1.96968	0.221607	0.110803	+	0.993842i	$0.464658\pi$
$80$	12.2750	1.37239
$81$	0	0
$82$	13.4436	1.48460
$83$	−10.6253	−1.16628	−0.583139	−	0.812372i	$-0.698176\pi$
$83$	−10.6253	−1.16628	−0.583139	+	0.812372i	$0.698176\pi$
$84$	0	0
$85$	−5.44358	−0.590439
$86$	15.7685	1.70036
$87$	0	0
$88$	8.44358	0.900089
$89$	−7.22425	−0.765769	−0.382885	−	0.923796i	$-0.625069\pi$
$89$	−7.22425	−0.765769	−0.382885	+	0.923796i	$0.625069\pi$
$90$	0	0
$91$	−5.83146	−0.611303
$92$	16.4690	1.71701
$93$	0	0
$94$	31.6507	3.26452
$95$	−1.35026	−0.138534
$96$	0	0
$97$	−0.836381	−0.0849216	−0.0424608	−	0.999098i	$-0.513520\pi$
$97$	−0.836381	−0.0849216	−0.0424608	+	0.999098i	$0.513520\pi$
$98$	2.67513	0.270229
$99$	0	0
$100$	5.15633	0.515633
$101$	−7.41327	−0.737648	−0.368824	−	0.929499i	$-0.620239\pi$
$101$	−7.41327	−0.737648	−0.368824	+	0.929499i	$0.620239\pi$
$102$	0	0
$103$	4.21933	0.415743	0.207871	−	0.978156i	$-0.433346\pi$
$103$	4.21933	0.415743	0.207871	+	0.978156i	$0.433346\pi$
$104$	49.2384	4.82822
$105$	0	0
$106$	0.619421	0.0601635
$107$	11.5369	1.11531	0.557657	−	0.830071i	$-0.311700\pi$
$107$	11.5369	1.11531	0.557657	+	0.830071i	$0.311700\pi$
$108$	0	0
$109$	−2.18664	−0.209442	−0.104721	−	0.994502i	$-0.533395\pi$
$109$	−2.18664	−0.209442	−0.104721	+	0.994502i	$0.533395\pi$
$110$	2.67513	0.255064
$111$	0	0
$112$	−12.2750	−1.15988
$113$	9.35026	0.879599	0.439799	−	0.898096i	$-0.355050\pi$
$113$	9.35026	0.879599	0.439799	+	0.898096i	$0.355050\pi$
$114$	0	0
$115$	3.19394	0.297836
$116$	18.6253	1.72932
$117$	0	0
$118$	−36.3815	−3.34919
$119$	5.44358	0.499012
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.78067	−0.342286
$123$	0	0
$124$	−27.2628	−2.44827
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.9624	−1.50517	−0.752584	−	0.658496i	$-0.771193\pi$
$127$	−16.9624	−1.50517	−0.752584	+	0.658496i	$0.771193\pi$
$128$	16.5696	1.46456
$129$	0	0
$130$	15.5999	1.36820
$131$	9.92478	0.867132	0.433566	−	0.901122i	$-0.357255\pi$
$131$	9.92478	0.867132	0.433566	+	0.901122i	$0.357255\pi$
$132$	0	0
$133$	1.35026	0.117083
$134$	−29.0435	−2.50898
$135$	0	0
$136$	−45.9633	−3.94132
$137$	10.9927	0.939170	0.469585	−	0.882887i	$-0.344404\pi$
$137$	10.9927	0.939170	0.469585	+	0.882887i	$0.344404\pi$
$138$	0	0
$139$	6.88717	0.584162	0.292081	−	0.956394i	$-0.405652\pi$
$139$	6.88717	0.584162	0.292081	+	0.956394i	$0.405652\pi$
$140$	−5.15633	−0.435789
$141$	0	0
$142$	41.5633	3.48791
$143$	5.83146	0.487651
$144$	0	0
$145$	3.61213	0.299971
$146$	−30.4119	−2.51690
$147$	0	0
$148$	−44.0567	−3.62144
$149$	−22.8119	−1.86883	−0.934414	−	0.356190i	$-0.884076\pi$
$149$	−22.8119	−1.86883	−0.934414	+	0.356190i	$0.884076\pi$
$150$	0	0
$151$	−3.24472	−0.264052	−0.132026	−	0.991246i	$-0.542148\pi$
$151$	−3.24472	−0.264052	−0.132026	+	0.991246i	$0.542148\pi$
$152$	−11.4010	−0.924747
$153$	0	0
$154$	−2.67513	−0.215568
$155$	−5.28726	−0.424683
$156$	0	0
$157$	−5.42548	−0.433001	−0.216500	−	0.976283i	$-0.569464\pi$
$157$	−5.42548	−0.433001	−0.216500	+	0.976283i	$0.569464\pi$
$158$	5.26916	0.419192
$159$	0	0
$160$	15.9502	1.26097
$161$	−3.19394	−0.251717
$162$	0	0
$163$	3.38058	0.264787	0.132394	−	0.991197i	$-0.457734\pi$
$163$	3.38058	0.264787	0.132394	+	0.991197i	$0.457734\pi$
$164$	25.9126	2.02343
$165$	0	0
$166$	−28.4241	−2.20614
$167$	−11.2750	−0.872489	−0.436244	−	0.899828i	$-0.643692\pi$
$167$	−11.2750	−0.872489	−0.436244	+	0.899828i	$0.643692\pi$
$168$	0	0
$169$	21.0059	1.61584
$170$	−14.5623	−1.11688
$171$	0	0
$172$	30.3938	2.31750
$173$	8.98049	0.682774	0.341387	−	0.939923i	$-0.389103\pi$
$173$	8.98049	0.682774	0.341387	+	0.939923i	$0.389103\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	12.2750	0.925266
$177$	0	0
$178$	−19.3258	−1.44853
$179$	26.2374	1.96108	0.980539	−	0.196326i	$-0.0629010\pi$
$179$	26.2374	1.96108	0.980539	+	0.196326i	$0.0629010\pi$
$180$	0	0
$181$	−11.1998	−0.832476	−0.416238	−	0.909256i	$-0.636652\pi$
$181$	−11.1998	−0.832476	−0.416238	+	0.909256i	$0.636652\pi$
$182$	−15.5999	−1.15634
$183$	0	0
$184$	26.9683	1.98813
$185$	−8.54420	−0.628182
$186$	0	0
$187$	−5.44358	−0.398074
$188$	61.0068	4.44938
$189$	0	0
$190$	−3.61213	−0.262051
$191$	−11.1998	−0.810390	−0.405195	−	0.914230i	$-0.632796\pi$
$191$	−11.1998	−0.810390	−0.405195	+	0.914230i	$0.632796\pi$
$192$	0	0
$193$	0.604833	0.0435368	0.0217684	−	0.999763i	$-0.493070\pi$
$193$	0.604833	0.0435368	0.0217684	+	0.999763i	$0.493070\pi$
$194$	−2.23743	−0.160638
$195$	0	0
$196$	5.15633	0.368309
$197$	−15.3054	−1.09046	−0.545231	−	0.838286i	$-0.683558\pi$
$197$	−15.3054	−1.09046	−0.545231	+	0.838286i	$0.683558\pi$
$198$	0	0
$199$	−12.5623	−0.890518	−0.445259	−	0.895402i	$-0.646888\pi$
$199$	−12.5623	−0.890518	−0.445259	+	0.895402i	$0.646888\pi$
$200$	8.44358	0.597051
$201$	0	0
$202$	−19.8315	−1.39534
$203$	−3.61213	−0.253522
$204$	0	0
$205$	5.02539	0.350989
$206$	11.2873	0.786421
$207$	0	0
$208$	71.5814	4.96327
$209$	−1.35026	−0.0933996
$210$	0	0
$211$	−4.43866	−0.305570	−0.152785	−	0.988259i	$-0.548824\pi$
$211$	−4.43866	−0.305570	−0.152785	+	0.988259i	$0.548824\pi$
$212$	1.19394	0.0819999
$213$	0	0
$214$	30.8627	2.10973
$215$	5.89446	0.401999
$216$	0	0
$217$	5.28726	0.358922
$218$	−5.84955	−0.396182
$219$	0	0
$220$	5.15633	0.347639
$221$	−31.7440	−2.13533
$222$	0	0
$223$	−7.78067	−0.521032	−0.260516	−	0.965469i	$-0.583893\pi$
$223$	−7.78067	−0.521032	−0.260516	+	0.965469i	$0.583893\pi$
$224$	−15.9502	−1.06572
$225$	0	0
$226$	25.0132	1.66385
$227$	10.4485	0.693492	0.346746	−	0.937959i	$-0.387287\pi$
$227$	10.4485	0.693492	0.346746	+	0.937959i	$0.387287\pi$
$228$	0	0
$229$	−29.4518	−1.94623	−0.973116	−	0.230316i	$-0.926024\pi$
$229$	−29.4518	−1.94623	−0.973116	+	0.230316i	$0.926024\pi$
$230$	8.54420	0.563388
$231$	0	0
$232$	30.4993	2.00238
$233$	8.73084	0.571976	0.285988	−	0.958233i	$-0.407678\pi$
$233$	8.73084	0.571976	0.285988	+	0.958233i	$0.407678\pi$
$234$	0	0
$235$	11.8315	0.771799
$236$	−70.1255	−4.56478
$237$	0	0
$238$	14.5623	0.943933
$239$	21.2144	1.37225	0.686123	−	0.727486i	$-0.259311\pi$
$239$	21.2144	1.37225	0.686123	+	0.727486i	$0.259311\pi$
$240$	0	0
$241$	−9.33804	−0.601516	−0.300758	−	0.953700i	$-0.597240\pi$
$241$	−9.33804	−0.601516	−0.300758	+	0.953700i	$0.597240\pi$
$242$	2.67513	0.171964
$243$	0	0
$244$	−7.28726	−0.466519
$245$	1.00000	0.0638877
$246$	0	0
$247$	−7.87399	−0.501010
$248$	−44.6434	−2.83486
$249$	0	0
$250$	2.67513	0.169190
$251$	−1.87636	−0.118435	−0.0592174	−	0.998245i	$-0.518861\pi$
$251$	−1.87636	−0.118435	−0.0592174	+	0.998245i	$0.518861\pi$
$252$	0	0
$253$	3.19394	0.200801
$254$	−45.3766	−2.84718
$255$	0	0
$256$	8.08840	0.505525
$257$	−27.1392	−1.69290	−0.846448	−	0.532472i	$-0.821263\pi$
$257$	−27.1392	−1.69290	−0.846448	+	0.532472i	$0.821263\pi$
$258$	0	0
$259$	8.54420	0.530911
$260$	30.0689	1.86479
$261$	0	0
$262$	26.5501	1.64027
$263$	−12.8119	−0.790018	−0.395009	−	0.918677i	$-0.629259\pi$
$263$	−12.8119	−0.790018	−0.395009	+	0.918677i	$0.629259\pi$
$264$	0	0
$265$	0.231548	0.0142239
$266$	3.61213	0.221474
$267$	0	0
$268$	−55.9814	−3.41961
$269$	6.26187	0.381793	0.190896	−	0.981610i	$-0.438861\pi$
$269$	6.26187	0.381793	0.190896	+	0.981610i	$0.438861\pi$
$270$	0	0
$271$	−5.73813	−0.348567	−0.174283	−	0.984696i	$-0.555761\pi$
$271$	−5.73813	−0.348567	−0.174283	+	0.984696i	$0.555761\pi$
$272$	−66.8202	−4.05157
$273$	0	0
$274$	29.4069	1.77654
$275$	1.00000	0.0603023
$276$	0	0
$277$	−8.35756	−0.502157	−0.251078	−	0.967967i	$-0.580785\pi$
$277$	−8.35756	−0.502157	−0.251078	+	0.967967i	$0.580785\pi$
$278$	18.4241	1.10500
$279$	0	0
$280$	−8.44358	−0.504601
$281$	8.44851	0.503996	0.251998	−	0.967728i	$-0.418912\pi$
$281$	8.44851	0.503996	0.251998	+	0.967728i	$0.418912\pi$
$282$	0	0
$283$	0.836381	0.0497177	0.0248588	−	0.999691i	$-0.492086\pi$
$283$	0.836381	0.0497177	0.0248588	+	0.999691i	$0.492086\pi$
$284$	80.1133	4.75385
$285$	0	0
$286$	15.5999	0.922442
$287$	−5.02539	−0.296640
$288$	0	0
$289$	12.6326	0.743094
$290$	9.66291	0.567426
$291$	0	0
$292$	−58.6190	−3.43042
$293$	2.71862	0.158824	0.0794118	−	0.996842i	$-0.474696\pi$
$293$	2.71862	0.158824	0.0794118	+	0.996842i	$0.474696\pi$
$294$	0	0
$295$	−13.5999	−0.791817
$296$	−72.1436	−4.19326
$297$	0	0
$298$	−61.0249	−3.53508
$299$	18.6253	1.07713
$300$	0	0
$301$	−5.89446	−0.339751
$302$	−8.68006	−0.499481
$303$	0	0
$304$	−16.5745	−0.950614
$305$	−1.41327	−0.0809234
$306$	0	0
$307$	−8.36344	−0.477326	−0.238663	−	0.971102i	$-0.576709\pi$
$307$	−8.36344	−0.477326	−0.238663	+	0.971102i	$0.576709\pi$
$308$	−5.15633	−0.293809
$309$	0	0
$310$	−14.1441	−0.803331
$311$	−4.43629	−0.251559	−0.125779	−	0.992058i	$-0.540143\pi$
$311$	−4.43629	−0.251559	−0.125779	+	0.992058i	$0.540143\pi$
$312$	0	0
$313$	29.7889	1.68377	0.841885	−	0.539658i	$-0.181446\pi$
$313$	29.7889	1.68377	0.841885	+	0.539658i	$0.181446\pi$
$314$	−14.5139	−0.819066
$315$	0	0
$316$	10.1563	0.571338
$317$	−15.4010	−0.865009	−0.432504	−	0.901632i	$-0.642370\pi$
$317$	−15.4010	−0.865009	−0.432504	+	0.901632i	$0.642370\pi$
$318$	0	0
$319$	3.61213	0.202240
$320$	18.1187	1.01287
$321$	0	0
$322$	−8.54420	−0.476150
$323$	7.35026	0.408980
$324$	0	0
$325$	5.83146	0.323471
$326$	9.04349	0.500873
$327$	0	0
$328$	42.4323	2.34293
$329$	−11.8315	−0.652289
$330$	0	0
$331$	6.26187	0.344183	0.172092	−	0.985081i	$-0.444947\pi$
$331$	6.26187	0.344183	0.172092	+	0.985081i	$0.444947\pi$
$332$	−54.7875	−3.00685
$333$	0	0
$334$	−30.1622	−1.65040
$335$	−10.8568	−0.593173
$336$	0	0
$337$	−15.8700	−0.864495	−0.432248	−	0.901755i	$-0.642279\pi$
$337$	−15.8700	−0.864495	−0.432248	+	0.901755i	$0.642279\pi$
$338$	56.1935	3.05652
$339$	0	0
$340$	−28.0689	−1.52225
$341$	−5.28726	−0.286321
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	49.7704	2.68344
$345$	0	0
$346$	24.0240	1.29154
$347$	−6.79147	−0.364585	−0.182293	−	0.983244i	$-0.558352\pi$
$347$	−6.79147	−0.364585	−0.182293	+	0.983244i	$0.558352\pi$
$348$	0	0
$349$	26.7489	1.43184	0.715919	−	0.698183i	$-0.246008\pi$
$349$	26.7489	1.43184	0.715919	+	0.698183i	$0.246008\pi$
$350$	−2.67513	−0.142992
$351$	0	0
$352$	15.9502	0.850147
$353$	16.8627	0.897512	0.448756	−	0.893654i	$-0.351867\pi$
$353$	16.8627	0.897512	0.448756	+	0.893654i	$0.351867\pi$
$354$	0	0
$355$	15.5369	0.824613
$356$	−37.2506	−1.97428
$357$	0	0
$358$	70.1886	3.70958
$359$	3.79289	0.200181	0.100091	−	0.994978i	$-0.468087\pi$
$359$	3.79289	0.200181	0.100091	+	0.994978i	$0.468087\pi$
$360$	0	0
$361$	−17.1768	−0.904042
$362$	−29.9610	−1.57471
$363$	0	0
$364$	−30.0689	−1.57604
$365$	−11.3684	−0.595047
$366$	0	0
$367$	−6.36977	−0.332500	−0.166250	−	0.986084i	$-0.553166\pi$
$367$	−6.36977	−0.332500	−0.166250	+	0.986084i	$0.553166\pi$
$368$	39.2057	2.04374
$369$	0	0
$370$	−22.8568	−1.18827
$371$	−0.231548	−0.0120214
$372$	0	0
$373$	−21.3317	−1.10451	−0.552257	−	0.833674i	$-0.686233\pi$
$373$	−21.3317	−1.10451	−0.552257	+	0.833674i	$0.686233\pi$
$374$	−14.5623	−0.752998
$375$	0	0
$376$	99.8999	5.15194
$377$	21.0640	1.08485
$378$	0	0
$379$	24.7875	1.27325	0.636624	−	0.771174i	$-0.280330\pi$
$379$	24.7875	1.27325	0.636624	+	0.771174i	$0.280330\pi$
$380$	−6.96239	−0.357163
$381$	0	0
$382$	−29.9610	−1.53294
$383$	5.45817	0.278900	0.139450	−	0.990229i	$-0.455467\pi$
$383$	5.45817	0.278900	0.139450	+	0.990229i	$0.455467\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	1.61801	0.0823544
$387$	0	0
$388$	−4.31265	−0.218942
$389$	13.7235	0.695811	0.347906	−	0.937530i	$-0.386893\pi$
$389$	13.7235	0.695811	0.347906	+	0.937530i	$0.386893\pi$
$390$	0	0
$391$	−17.3865	−0.879271
$392$	8.44358	0.426465
$393$	0	0
$394$	−40.9438	−2.06272
$395$	1.96968	0.0991055
$396$	0	0
$397$	2.11142	0.105969	0.0529846	−	0.998595i	$-0.483127\pi$
$397$	2.11142	0.105969	0.0529846	+	0.998595i	$0.483127\pi$
$398$	−33.6058	−1.68451
$399$	0	0
$400$	12.2750	0.613752
$401$	−19.1490	−0.956257	−0.478128	−	0.878290i	$-0.658685\pi$
$401$	−19.1490	−0.956257	−0.478128	+	0.878290i	$0.658685\pi$
$402$	0	0
$403$	−30.8324	−1.53587
$404$	−38.2252	−1.90178
$405$	0	0
$406$	−9.66291	−0.479562
$407$	−8.54420	−0.423520
$408$	0	0
$409$	−18.6883	−0.924077	−0.462039	−	0.886860i	$-0.652882\pi$
$409$	−18.6883	−0.924077	−0.462039	+	0.886860i	$0.652882\pi$
$410$	13.4436	0.663931
$411$	0	0
$412$	21.7562	1.07185
$413$	13.5999	0.669208
$414$	0	0
$415$	−10.6253	−0.521575
$416$	93.0127	4.56032
$417$	0	0
$418$	−3.61213	−0.176675
$419$	0.773377	0.0377819	0.0188910	−	0.999822i	$-0.493986\pi$
$419$	0.773377	0.0377819	0.0188910	+	0.999822i	$0.493986\pi$
$420$	0	0
$421$	−10.5198	−0.512702	−0.256351	−	0.966584i	$-0.582520\pi$
$421$	−10.5198	−0.512702	−0.256351	+	0.966584i	$0.582520\pi$
$422$	−11.8740	−0.578017
$423$	0	0
$424$	1.95509	0.0949478
$425$	−5.44358	−0.264053
$426$	0	0
$427$	1.41327	0.0683927
$428$	59.4880	2.87546
$429$	0	0
$430$	15.7685	0.760422
$431$	24.7308	1.19124	0.595621	−	0.803265i	$-0.296906\pi$
$431$	24.7308	1.19124	0.595621	+	0.803265i	$0.296906\pi$
$432$	0	0
$433$	−18.5599	−0.891933	−0.445967	−	0.895050i	$-0.647140\pi$
$433$	−18.5599	−0.891933	−0.445967	+	0.895050i	$0.647140\pi$
$434$	14.1441	0.678939
$435$	0	0
$436$	−11.2750	−0.539976
$437$	−4.31265	−0.206302
$438$	0	0
$439$	−1.42548	−0.0680347	−0.0340173	−	0.999421i	$-0.510830\pi$
$439$	−1.42548	−0.0680347	−0.0340173	+	0.999421i	$0.510830\pi$
$440$	8.44358	0.402532
$441$	0	0
$442$	−84.9194	−4.03920
$443$	−40.1925	−1.90960	−0.954802	−	0.297242i	$-0.903933\pi$
$443$	−40.1925	−1.90960	−0.954802	+	0.297242i	$0.903933\pi$
$444$	0	0
$445$	−7.22425	−0.342462
$446$	−20.8143	−0.985586
$447$	0	0
$448$	−18.1187	−0.856029
$449$	−12.6556	−0.597256	−0.298628	−	0.954370i	$-0.596529\pi$
$449$	−12.6556	−0.597256	−0.298628	+	0.954370i	$0.596529\pi$
$450$	0	0
$451$	5.02539	0.236636
$452$	48.2130	2.26775
$453$	0	0
$454$	27.9511	1.31181
$455$	−5.83146	−0.273383
$456$	0	0
$457$	−0.544198	−0.0254565	−0.0127283	−	0.999919i	$-0.504052\pi$
$457$	−0.544198	−0.0254565	−0.0127283	+	0.999919i	$0.504052\pi$
$458$	−78.7875	−3.68150
$459$	0	0
$460$	16.4690	0.767870
$461$	11.5755	0.539123	0.269562	−	0.962983i	$-0.413121\pi$
$461$	11.5755	0.539123	0.269562	+	0.962983i	$0.413121\pi$
$462$	0	0
$463$	23.7948	1.10584	0.552919	−	0.833235i	$-0.313514\pi$
$463$	23.7948	1.10584	0.552919	+	0.833235i	$0.313514\pi$
$464$	44.3390	2.05839
$465$	0	0
$466$	23.3561	1.08195
$467$	2.66784	0.123453	0.0617264	−	0.998093i	$-0.480339\pi$
$467$	2.66784	0.123453	0.0617264	+	0.998093i	$0.480339\pi$
$468$	0	0
$469$	10.8568	0.501323
$470$	31.6507	1.45994
$471$	0	0
$472$	−114.832	−5.28557
$473$	5.89446	0.271028
$474$	0	0
$475$	−1.35026	−0.0619543
$476$	28.0689	1.28654
$477$	0	0
$478$	56.7513	2.59574
$479$	10.7104	0.489369	0.244685	−	0.969603i	$-0.421316\pi$
$479$	10.7104	0.489369	0.244685	+	0.969603i	$0.421316\pi$
$480$	0	0
$481$	−49.8251	−2.27183
$482$	−24.9805	−1.13783
$483$	0	0
$484$	5.15633	0.234378
$485$	−0.836381	−0.0379781
$486$	0	0
$487$	17.4314	0.789891	0.394945	−	0.918705i	$-0.370764\pi$
$487$	17.4314	0.789891	0.394945	+	0.918705i	$0.370764\pi$
$488$	−11.9330	−0.540183
$489$	0	0
$490$	2.67513	0.120850
$491$	28.3693	1.28029	0.640145	−	0.768254i	$-0.278874\pi$
$491$	28.3693	1.28029	0.640145	+	0.768254i	$0.278874\pi$
$492$	0	0
$493$	−19.6629	−0.885573
$494$	−21.0640	−0.947712
$495$	0	0
$496$	−64.9013	−2.91415
$497$	−15.5369	−0.696925
$498$	0	0
$499$	27.4763	1.23001	0.615003	−	0.788524i	$-0.289155\pi$
$499$	27.4763	1.23001	0.615003	+	0.788524i	$0.289155\pi$
$500$	5.15633	0.230598
$501$	0	0
$502$	−5.01951	−0.224032
$503$	−20.2981	−0.905046	−0.452523	−	0.891753i	$-0.649476\pi$
$503$	−20.2981	−0.905046	−0.452523	+	0.891753i	$0.649476\pi$
$504$	0	0
$505$	−7.41327	−0.329886
$506$	8.54420	0.379836
$507$	0	0
$508$	−87.4636	−3.88057
$509$	24.2619	1.07539	0.537694	−	0.843140i	$-0.319296\pi$
$509$	24.2619	1.07539	0.537694	+	0.843140i	$0.319296\pi$
$510$	0	0
$511$	11.3684	0.502907
$512$	−11.5017	−0.508306
$513$	0	0
$514$	−72.6009	−3.20229
$515$	4.21933	0.185926
$516$	0	0
$517$	11.8315	0.520347
$518$	22.8568	1.00427
$519$	0	0
$520$	49.2384	2.15925
$521$	−2.20123	−0.0964377	−0.0482188	−	0.998837i	$-0.515354\pi$
$521$	−2.20123	−0.0964377	−0.0482188	+	0.998837i	$0.515354\pi$
$522$	0	0
$523$	−22.1378	−0.968017	−0.484008	−	0.875063i	$-0.660820\pi$
$523$	−22.1378	−0.968017	−0.484008	+	0.875063i	$0.660820\pi$
$524$	51.1754	2.23561
$525$	0	0
$526$	−34.2736	−1.49440
$527$	28.7816	1.25375
$528$	0	0
$529$	−12.7988	−0.556468
$530$	0.619421	0.0269059
$531$	0	0
$532$	6.96239	0.301858
$533$	29.3054	1.26936
$534$	0	0
$535$	11.5369	0.498784
$536$	−91.6707	−3.95957
$537$	0	0
$538$	16.7513	0.722200
$539$	1.00000	0.0430730
$540$	0	0
$541$	23.0640	0.991597	0.495799	−	0.868438i	$-0.334875\pi$
$541$	23.0640	0.991597	0.495799	+	0.868438i	$0.334875\pi$
$542$	−15.3503	−0.659350
$543$	0	0
$544$	−86.8261	−3.72264
$545$	−2.18664	−0.0936655
$546$	0	0
$547$	21.3766	0.913998	0.456999	−	0.889467i	$-0.348924\pi$
$547$	21.3766	0.913998	0.456999	+	0.889467i	$0.348924\pi$
$548$	56.6820	2.42133
$549$	0	0
$550$	2.67513	0.114068
$551$	−4.87732	−0.207781
$552$	0	0
$553$	−1.96968	−0.0837594
$554$	−22.3576	−0.949882
$555$	0	0
$556$	35.5125	1.50606
$557$	9.19394	0.389560	0.194780	−	0.980847i	$-0.437601\pi$
$557$	9.19394	0.389560	0.194780	+	0.980847i	$0.437601\pi$
$558$	0	0
$559$	34.3733	1.45384
$560$	−12.2750	−0.518715
$561$	0	0
$562$	22.6009	0.953360
$563$	9.79877	0.412969	0.206484	−	0.978450i	$-0.433798\pi$
$563$	9.79877	0.412969	0.206484	+	0.978450i	$0.433798\pi$
$564$	0	0
$565$	9.35026	0.393368
$566$	2.23743	0.0940461
$567$	0	0
$568$	131.187	5.50449
$569$	33.5125	1.40492	0.702458	−	0.711725i	$-0.252086\pi$
$569$	33.5125	1.40492	0.702458	+	0.711725i	$0.252086\pi$
$570$	0	0
$571$	43.1392	1.80532	0.902659	−	0.430356i	$-0.141612\pi$
$571$	43.1392	1.80532	0.902659	+	0.430356i	$0.141612\pi$
$572$	30.0689	1.25724
$573$	0	0
$574$	−13.4436	−0.561124
$575$	3.19394	0.133196
$576$	0	0
$577$	−14.8510	−0.618254	−0.309127	−	0.951021i	$-0.600037\pi$
$577$	−14.8510	−0.618254	−0.309127	+	0.951021i	$0.600037\pi$
$578$	33.7938	1.40564
$579$	0	0
$580$	18.6253	0.773374
$581$	10.6253	0.440812
$582$	0	0
$583$	0.231548	0.00958974
$584$	−95.9897	−3.97208
$585$	0	0
$586$	7.27267	0.300431
$587$	−14.7938	−0.610607	−0.305304	−	0.952255i	$-0.598758\pi$
$587$	−14.7938	−0.610607	−0.305304	+	0.952255i	$0.598758\pi$
$588$	0	0
$589$	7.13918	0.294165
$590$	−36.3815	−1.49780
$591$	0	0
$592$	−104.880	−4.31056
$593$	27.4191	1.12597	0.562985	−	0.826467i	$-0.309653\pi$
$593$	27.4191	1.12597	0.562985	+	0.826467i	$0.309653\pi$
$594$	0	0
$595$	5.44358	0.223165
$596$	−117.626	−4.81814
$597$	0	0
$598$	49.8251	2.03750
$599$	−11.3258	−0.462761	−0.231380	−	0.972863i	$-0.574324\pi$
$599$	−11.3258	−0.462761	−0.231380	+	0.972863i	$0.574324\pi$
$600$	0	0
$601$	15.5393	0.633860	0.316930	−	0.948449i	$-0.397348\pi$
$601$	15.5393	0.633860	0.316930	+	0.948449i	$0.397348\pi$
$602$	−15.7685	−0.642674
$603$	0	0
$604$	−16.7308	−0.680768
$605$	1.00000	0.0406558
$606$	0	0
$607$	17.7235	0.719377	0.359688	−	0.933073i	$-0.382883\pi$
$607$	17.7235	0.719377	0.359688	+	0.933073i	$0.382883\pi$
$608$	−21.5369	−0.873437
$609$	0	0
$610$	−3.78067	−0.153075
$611$	68.9946	2.79122
$612$	0	0
$613$	22.2941	0.900450	0.450225	−	0.892915i	$-0.351344\pi$
$613$	22.2941	0.900450	0.450225	+	0.892915i	$0.351344\pi$
$614$	−22.3733	−0.902912
$615$	0	0
$616$	−8.44358	−0.340202
$617$	30.9438	1.24575	0.622876	−	0.782321i	$-0.285964\pi$
$617$	30.9438	1.24575	0.622876	+	0.782321i	$0.285964\pi$
$618$	0	0
$619$	32.4119	1.30274	0.651371	−	0.758759i	$-0.274194\pi$
$619$	32.4119	1.30274	0.651371	+	0.758759i	$0.274194\pi$
$620$	−27.2628	−1.09490
$621$	0	0
$622$	−11.8677	−0.475850
$623$	7.22425	0.289434
$624$	0	0
$625$	1.00000	0.0400000
$626$	79.6893	3.18502
$627$	0	0
$628$	−27.9756	−1.11635
$629$	46.5111	1.85452
$630$	0	0
$631$	−27.3258	−1.08782	−0.543912	−	0.839142i	$-0.683057\pi$
$631$	−27.3258	−1.08782	−0.543912	+	0.839142i	$0.683057\pi$
$632$	16.6312	0.661553
$633$	0	0
$634$	−41.1998	−1.63625
$635$	−16.9624	−0.673132
$636$	0	0
$637$	5.83146	0.231051
$638$	9.66291	0.382558
$639$	0	0
$640$	16.5696	0.654971
$641$	−19.4460	−0.768069	−0.384034	−	0.923319i	$-0.625466\pi$
$641$	−19.4460	−0.768069	−0.384034	+	0.923319i	$0.625466\pi$
$642$	0	0
$643$	5.29314	0.208741	0.104370	−	0.994538i	$-0.466717\pi$
$643$	5.29314	0.208741	0.104370	+	0.994538i	$0.466717\pi$
$644$	−16.4690	−0.648969
$645$	0	0
$646$	19.6629	0.773627
$647$	−35.0966	−1.37979	−0.689896	−	0.723909i	$-0.742344\pi$
$647$	−35.0966	−1.37979	−0.689896	+	0.723909i	$0.742344\pi$
$648$	0	0
$649$	−13.5999	−0.533843
$650$	15.5999	0.611879
$651$	0	0
$652$	17.4314	0.682665
$653$	−27.7988	−1.08785	−0.543925	−	0.839134i	$-0.683062\pi$
$653$	−27.7988	−1.08785	−0.543925	+	0.839134i	$0.683062\pi$
$654$	0	0
$655$	9.92478	0.387793
$656$	61.6869	2.40847
$657$	0	0
$658$	−31.6507	−1.23387
$659$	−19.6180	−0.764209	−0.382105	−	0.924119i	$-0.624801\pi$
$659$	−19.6180	−0.764209	−0.382105	+	0.924119i	$0.624801\pi$
$660$	0	0
$661$	21.5633	0.838713	0.419357	−	0.907822i	$-0.362256\pi$
$661$	21.5633	0.838713	0.419357	+	0.907822i	$0.362256\pi$
$662$	16.7513	0.651058
$663$	0	0
$664$	−89.7156	−3.48164
$665$	1.35026	0.0523609
$666$	0	0
$667$	11.5369	0.446711
$668$	−58.1378	−2.24942
$669$	0	0
$670$	−29.0435	−1.12205
$671$	−1.41327	−0.0545585
$672$	0	0
$673$	−21.0679	−0.812109	−0.406054	−	0.913849i	$-0.633096\pi$
$673$	−21.0679	−0.812109	−0.406054	+	0.913849i	$0.633096\pi$
$674$	−42.4544	−1.63528
$675$	0	0
$676$	108.313	4.16589
$677$	−34.5174	−1.32661	−0.663306	−	0.748349i	$-0.730847\pi$
$677$	−34.5174	−1.32661	−0.663306	+	0.748349i	$0.730847\pi$
$678$	0	0
$679$	0.836381	0.0320973
$680$	−45.9633	−1.76261
$681$	0	0
$682$	−14.1441	−0.541606
$683$	−33.7802	−1.29256	−0.646282	−	0.763099i	$-0.723677\pi$
$683$	−33.7802	−1.29256	−0.646282	+	0.763099i	$0.723677\pi$
$684$	0	0
$685$	10.9927	0.420010
$686$	−2.67513	−0.102137
$687$	0	0
$688$	72.3547	2.75850
$689$	1.35026	0.0514409
$690$	0	0
$691$	−13.8618	−0.527327	−0.263663	−	0.964615i	$-0.584931\pi$
$691$	−13.8618	−0.527327	−0.263663	+	0.964615i	$0.584931\pi$
$692$	46.3063	1.76030
$693$	0	0
$694$	−18.1681	−0.689651
$695$	6.88717	0.261245
$696$	0	0
$697$	−27.3561	−1.03619
$698$	71.5569	2.70847
$699$	0	0
$700$	−5.15633	−0.194891
$701$	−40.5256	−1.53063	−0.765316	−	0.643655i	$-0.777417\pi$
$701$	−40.5256	−1.53063	−0.765316	+	0.643655i	$0.777417\pi$
$702$	0	0
$703$	11.5369	0.435123
$704$	18.1187	0.682875
$705$	0	0
$706$	45.1100	1.69774
$707$	7.41327	0.278805
$708$	0	0
$709$	−0.850969	−0.0319588	−0.0159794	−	0.999872i	$-0.505087\pi$
$709$	−0.850969	−0.0319588	−0.0159794	+	0.999872i	$0.505087\pi$
$710$	41.5633	1.55984
$711$	0	0
$712$	−60.9986	−2.28602
$713$	−16.8872	−0.632429
$714$	0	0
$715$	5.83146	0.218084
$716$	135.289	5.05598
$717$	0	0
$718$	10.1465	0.378663
$719$	−22.5769	−0.841976	−0.420988	−	0.907066i	$-0.638317\pi$
$719$	−22.5769	−0.841976	−0.420988	+	0.907066i	$0.638317\pi$
$720$	0	0
$721$	−4.21933	−0.157136
$722$	−45.9502	−1.71009
$723$	0	0
$724$	−57.7499	−2.14626
$725$	3.61213	0.134151
$726$	0	0
$727$	12.5174	0.464244	0.232122	−	0.972687i	$-0.425433\pi$
$727$	12.5174	0.464244	0.232122	+	0.972687i	$0.425433\pi$
$728$	−49.2384	−1.82490
$729$	0	0
$730$	−30.4119	−1.12559
$731$	−32.0870	−1.18678
$732$	0	0
$733$	16.6678	0.615641	0.307820	−	0.951445i	$-0.400400\pi$
$733$	16.6678	0.615641	0.307820	+	0.951445i	$0.400400\pi$
$734$	−17.0400	−0.628957
$735$	0	0
$736$	50.9438	1.87781
$737$	−10.8568	−0.399917
$738$	0	0
$739$	42.7005	1.57076	0.785382	−	0.619011i	$-0.212467\pi$
$739$	42.7005	1.57076	0.785382	+	0.619011i	$0.212467\pi$
$740$	−44.0567	−1.61956
$741$	0	0
$742$	−0.619421	−0.0227397
$743$	19.6873	0.722259	0.361129	−	0.932516i	$-0.382391\pi$
$743$	19.6873	0.722259	0.361129	+	0.932516i	$0.382391\pi$
$744$	0	0
$745$	−22.8119	−0.835765
$746$	−57.0651	−2.08930
$747$	0	0
$748$	−28.0689	−1.02630
$749$	−11.5369	−0.421549
$750$	0	0
$751$	−5.85940	−0.213813	−0.106906	−	0.994269i	$-0.534094\pi$
$751$	−5.85940	−0.213813	−0.106906	+	0.994269i	$0.534094\pi$
$752$	145.232	5.29605
$753$	0	0
$754$	56.3488	2.05210
$755$	−3.24472	−0.118088
$756$	0	0
$757$	−40.5863	−1.47513	−0.737567	−	0.675274i	$-0.764025\pi$
$757$	−40.5863	−1.47513	−0.737567	+	0.675274i	$0.764025\pi$
$758$	66.3098	2.40848
$759$	0	0
$760$	−11.4010	−0.413559
$761$	−21.8472	−0.791960	−0.395980	−	0.918259i	$-0.629595\pi$
$761$	−21.8472	−0.791960	−0.395980	+	0.918259i	$0.629595\pi$
$762$	0	0
$763$	2.18664	0.0791618
$764$	−57.7499	−2.08932
$765$	0	0
$766$	14.6013	0.527567
$767$	−79.3073	−2.86362
$768$	0	0
$769$	45.2892	1.63317	0.816585	−	0.577226i	$-0.195865\pi$
$769$	45.2892	1.63317	0.816585	+	0.577226i	$0.195865\pi$
$770$	−2.67513	−0.0964050
$771$	0	0
$772$	3.11871	0.112245
$773$	33.8153	1.21625	0.608125	−	0.793841i	$-0.291922\pi$
$773$	33.8153	1.21625	0.608125	+	0.793841i	$0.291922\pi$
$774$	0	0
$775$	−5.28726	−0.189924
$776$	−7.06205	−0.253513
$777$	0	0
$778$	36.7123	1.31620
$779$	−6.78560	−0.243119
$780$	0	0
$781$	15.5369	0.555954
$782$	−46.5111	−1.66323
$783$	0	0
$784$	12.2750	0.438394
$785$	−5.42548	−0.193644
$786$	0	0
$787$	1.27504	0.0454502	0.0227251	−	0.999742i	$-0.492766\pi$
$787$	1.27504	0.0454502	0.0227251	+	0.999742i	$0.492766\pi$
$788$	−78.9194	−2.81139
$789$	0	0
$790$	5.26916	0.187468
$791$	−9.35026	−0.332457
$792$	0	0
$793$	−8.24140	−0.292661
$794$	5.64832	0.200452
$795$	0	0
$796$	−64.7753	−2.29590
$797$	42.5256	1.50634	0.753168	−	0.657829i	$-0.228525\pi$
$797$	42.5256	1.50634	0.753168	+	0.657829i	$0.228525\pi$
$798$	0	0
$799$	−64.4055	−2.27850
$800$	15.9502	0.563924
$801$	0	0
$802$	−51.2262	−1.80886
$803$	−11.3684	−0.401181
$804$	0	0
$805$	−3.19394	−0.112571
$806$	−82.4807	−2.90526
$807$	0	0
$808$	−62.5945	−2.20207
$809$	14.7151	0.517356	0.258678	−	0.965964i	$-0.416713\pi$
$809$	14.7151	0.517356	0.258678	+	0.965964i	$0.416713\pi$
$810$	0	0
$811$	51.7743	1.81804	0.909021	−	0.416750i	$-0.136831\pi$
$811$	51.7743	1.81804	0.909021	+	0.416750i	$0.136831\pi$
$812$	−18.6253	−0.653620
$813$	0	0
$814$	−22.8568	−0.801132
$815$	3.38058	0.118417
$816$	0	0
$817$	−7.95906	−0.278452
$818$	−49.9937	−1.74799
$819$	0	0
$820$	25.9126	0.904906
$821$	−2.64974	−0.0924765	−0.0462383	−	0.998930i	$-0.514723\pi$
$821$	−2.64974	−0.0924765	−0.0462383	+	0.998930i	$0.514723\pi$
$822$	0	0
$823$	5.76845	0.201076	0.100538	−	0.994933i	$-0.467944\pi$
$823$	5.76845	0.201076	0.100538	+	0.994933i	$0.467944\pi$
$824$	35.6263	1.24110
$825$	0	0
$826$	36.3815	1.26588
$827$	13.4920	0.469163	0.234581	−	0.972096i	$-0.424628\pi$
$827$	13.4920	0.469163	0.234581	+	0.972096i	$0.424628\pi$
$828$	0	0
$829$	−4.70052	−0.163256	−0.0816280	−	0.996663i	$-0.526012\pi$
$829$	−4.70052	−0.163256	−0.0816280	+	0.996663i	$0.526012\pi$
$830$	−28.4241	−0.986614
$831$	0	0
$832$	105.658	3.66305
$833$	−5.44358	−0.188609
$834$	0	0
$835$	−11.2750	−0.390189
$836$	−6.96239	−0.240799
$837$	0	0
$838$	2.06888	0.0714684
$839$	38.8045	1.33968	0.669839	−	0.742506i	$-0.266363\pi$
$839$	38.8045	1.33968	0.669839	+	0.742506i	$0.266363\pi$
$840$	0	0
$841$	−15.9525	−0.550088
$842$	−28.1417	−0.969828
$843$	0	0
$844$	−22.8872	−0.787809
$845$	21.0059	0.722624
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	2.84226	0.0976036
$849$	0	0
$850$	−14.5623	−0.499483
$851$	−27.2896	−0.935476
$852$	0	0
$853$	20.6824	0.708153	0.354076	−	0.935217i	$-0.384795\pi$
$853$	20.6824	0.708153	0.354076	+	0.935217i	$0.384795\pi$
$854$	3.78067	0.129372
$855$	0	0
$856$	97.4128	3.32950
$857$	−26.3453	−0.899940	−0.449970	−	0.893044i	$-0.648565\pi$
$857$	−26.3453	−0.899940	−0.449970	+	0.893044i	$0.648565\pi$
$858$	0	0
$859$	−8.51151	−0.290409	−0.145205	−	0.989402i	$-0.546384\pi$
$859$	−8.51151	−0.290409	−0.145205	+	0.989402i	$0.546384\pi$
$860$	30.3938	1.03642
$861$	0	0
$862$	66.1582	2.25336
$863$	−7.56722	−0.257591	−0.128796	−	0.991671i	$-0.541111\pi$
$863$	−7.56722	−0.257591	−0.128796	+	0.991671i	$0.541111\pi$
$864$	0	0
$865$	8.98049	0.305346
$866$	−49.6502	−1.68718
$867$	0	0
$868$	27.2628	0.925360
$869$	1.96968	0.0668169
$870$	0	0
$871$	−63.3112	−2.14522
$872$	−18.4631	−0.625239
$873$	0	0
$874$	−11.5369	−0.390242
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	17.2955	0.584028	0.292014	−	0.956414i	$-0.405675\pi$
$877$	17.2955	0.584028	0.292014	+	0.956414i	$0.405675\pi$
$878$	−3.81336	−0.128695
$879$	0	0
$880$	12.2750	0.413791
$881$	−20.4504	−0.688992	−0.344496	−	0.938788i	$-0.611950\pi$
$881$	−20.4504	−0.688992	−0.344496	+	0.938788i	$0.611950\pi$
$882$	0	0
$883$	−49.6589	−1.67116	−0.835578	−	0.549371i	$-0.814867\pi$
$883$	−49.6589	−1.67116	−0.835578	+	0.549371i	$0.814867\pi$
$884$	−163.682	−5.50524
$885$	0	0
$886$	−107.520	−3.61221
$887$	47.1100	1.58180	0.790900	−	0.611946i	$-0.209613\pi$
$887$	47.1100	1.58180	0.790900	+	0.611946i	$0.209613\pi$
$888$	0	0
$889$	16.9624	0.568900
$890$	−19.3258	−0.647803
$891$	0	0
$892$	−40.1197	−1.34331
$893$	−15.9756	−0.534602
$894$	0	0
$895$	26.2374	0.877020
$896$	−16.5696	−0.553551
$897$	0	0
$898$	−33.8554	−1.12977
$899$	−19.0982	−0.636962
$900$	0	0
$901$	−1.26045	−0.0419917
$902$	13.4436	0.447622
$903$	0	0
$904$	78.9497	2.62583
$905$	−11.1998	−0.372294
$906$	0	0
$907$	14.4591	0.480107	0.240054	−	0.970760i	$-0.422835\pi$
$907$	14.4591	0.480107	0.240054	+	0.970760i	$0.422835\pi$
$908$	53.8759	1.78793
$909$	0	0
$910$	−15.5999	−0.517132
$911$	31.5369	1.04486	0.522432	−	0.852681i	$-0.325025\pi$
$911$	31.5369	1.04486	0.522432	+	0.852681i	$0.325025\pi$
$912$	0	0
$913$	−10.6253	−0.351646
$914$	−1.45580	−0.0481536
$915$	0	0
$916$	−151.863	−5.01770
$917$	−9.92478	−0.327745
$918$	0	0
$919$	5.26328	0.173620	0.0868098	−	0.996225i	$-0.472333\pi$
$919$	5.26328	0.173620	0.0868098	+	0.996225i	$0.472333\pi$
$920$	26.9683	0.889117
$921$	0	0
$922$	30.9659	1.01981
$923$	90.6028	2.98223
$924$	0	0
$925$	−8.54420	−0.280932
$926$	63.6542	2.09181
$927$	0	0
$928$	57.6140	1.89127
$929$	−26.0508	−0.854699	−0.427349	−	0.904087i	$-0.640553\pi$
$929$	−26.0508	−0.854699	−0.427349	+	0.904087i	$0.640553\pi$
$930$	0	0
$931$	−1.35026	−0.0442530
$932$	45.0191	1.47465
$933$	0	0
$934$	7.13681	0.233524
$935$	−5.44358	−0.178024
$936$	0	0
$937$	−29.3439	−0.958624	−0.479312	−	0.877645i	$-0.659114\pi$
$937$	−29.3439	−0.958624	−0.479312	+	0.877645i	$0.659114\pi$
$938$	29.0435	0.948304
$939$	0	0
$940$	61.0068	1.98982
$941$	28.6375	0.933556	0.466778	−	0.884374i	$-0.345415\pi$
$941$	28.6375	0.933556	0.466778	+	0.884374i	$0.345415\pi$
$942$	0	0
$943$	16.0508	0.522685
$944$	−166.939	−5.43341
$945$	0	0
$946$	15.7685	0.512677
$947$	52.8178	1.71635	0.858174	−	0.513358i	$-0.171599\pi$
$947$	52.8178	1.71635	0.858174	+	0.513358i	$0.171599\pi$
$948$	0	0
$949$	−66.2941	−2.15200
$950$	−3.61213	−0.117193
$951$	0	0
$952$	45.9633	1.48968
$953$	−37.1939	−1.20483	−0.602415	−	0.798183i	$-0.705795\pi$
$953$	−37.1939	−1.20483	−0.602415	+	0.798183i	$0.705795\pi$
$954$	0	0
$955$	−11.1998	−0.362418
$956$	109.388	3.53787
$957$	0	0
$958$	28.6516	0.925693
$959$	−10.9927	−0.354973
$960$	0	0
$961$	−3.04491	−0.0982228
$962$	−133.289	−4.29740
$963$	0	0
$964$	−48.1500	−1.55081
$965$	0.604833	0.0194703
$966$	0	0
$967$	4.07125	0.130923	0.0654613	−	0.997855i	$-0.479148\pi$
$967$	4.07125	0.130923	0.0654613	+	0.997855i	$0.479148\pi$
$968$	8.44358	0.271387
$969$	0	0
$970$	−2.23743	−0.0718395
$971$	−0.773377	−0.0248188	−0.0124094	−	0.999923i	$-0.503950\pi$
$971$	−0.773377	−0.0248188	−0.0124094	+	0.999923i	$0.503950\pi$
$972$	0	0
$973$	−6.88717	−0.220792
$974$	46.6312	1.49416
$975$	0	0
$976$	−17.3479	−0.555292
$977$	37.8740	1.21170	0.605848	−	0.795580i	$-0.292834\pi$
$977$	37.8740	1.21170	0.605848	+	0.795580i	$0.292834\pi$
$978$	0	0
$979$	−7.22425	−0.230888
$980$	5.15633	0.164713
$981$	0	0
$982$	75.8916	2.42180
$983$	15.5794	0.496907	0.248453	−	0.968644i	$-0.420078\pi$
$983$	15.5794	0.496907	0.248453	+	0.968644i	$0.420078\pi$
$984$	0	0
$985$	−15.3054	−0.487669
$986$	−52.6009	−1.67515
$987$	0	0
$988$	−40.6009	−1.29169
$989$	18.8265	0.598649
$990$	0	0
$991$	27.0982	0.860804	0.430402	−	0.902637i	$-0.358372\pi$
$991$	27.0982	0.860804	0.430402	+	0.902637i	$0.358372\pi$
$992$	−84.3327	−2.67756
$993$	0	0
$994$	−41.5633	−1.31831
$995$	−12.5623	−0.398252
$996$	0	0
$997$	50.4060	1.59637	0.798187	−	0.602410i	$-0.205793\pi$
$997$	50.4060	1.59637	0.798187	+	0.602410i	$0.205793\pi$
$998$	73.5026	2.32668
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3465.2.a.bh.1.3		3
3.2	odd	2		385.2.a.f.1.1	✓	3
12.11	even	2		6160.2.a.bn.1.3		3
15.2	even	4		1925.2.b.n.1849.1		6
15.8	even	4		1925.2.b.n.1849.6		6
15.14	odd	2		1925.2.a.v.1.3		3
21.20	even	2		2695.2.a.g.1.1		3
33.32	even	2		4235.2.a.q.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.a.f.1.1	✓	3	3.2	odd	2
1925.2.a.v.1.3		3	15.14	odd	2
1925.2.b.n.1849.1		6	15.2	even	4
1925.2.b.n.1849.6		6	15.8	even	4
2695.2.a.g.1.1		3	21.20	even	2
3465.2.a.bh.1.3		3	1.1	even	1	trivial
4235.2.a.q.1.3		3	33.32	even	2
6160.2.a.bn.1.3		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3465.a (trivial)

\(f(q)\)	\(=\)	\(q+2.67513 q^{2} +5.15633 q^{4} +1.00000 q^{5} -1.00000 q^{7} +8.44358 q^{8} +O(q^{10})\)	\(q+2.67513 q^{2} +5.15633 q^{4} +1.00000 q^{5} -1.00000 q^{7} +8.44358 q^{8} +2.67513 q^{10} +1.00000 q^{11} +5.83146 q^{13} -2.67513 q^{14} +12.2750 q^{16} -5.44358 q^{17} -1.35026 q^{19} +5.15633 q^{20} +2.67513 q^{22} +3.19394 q^{23} +1.00000 q^{25} +15.5999 q^{26} -5.15633 q^{28} +3.61213 q^{29} -5.28726 q^{31} +15.9502 q^{32} -14.5623 q^{34} -1.00000 q^{35} -8.54420 q^{37} -3.61213 q^{38} +8.44358 q^{40} +5.02539 q^{41} +5.89446 q^{43} +5.15633 q^{44} +8.54420 q^{46} +11.8315 q^{47} +1.00000 q^{49} +2.67513 q^{50} +30.0689 q^{52} +0.231548 q^{53} +1.00000 q^{55} -8.44358 q^{56} +9.66291 q^{58} -13.5999 q^{59} -1.41327 q^{61} -14.1441 q^{62} +18.1187 q^{64} +5.83146 q^{65} -10.8568 q^{67} -28.0689 q^{68} -2.67513 q^{70} +15.5369 q^{71} -11.3684 q^{73} -22.8568 q^{74} -6.96239 q^{76} -1.00000 q^{77} +1.96968 q^{79} +12.2750 q^{80} +13.4436 q^{82} -10.6253 q^{83} -5.44358 q^{85} +15.7685 q^{86} +8.44358 q^{88} -7.22425 q^{89} -5.83146 q^{91} +16.4690 q^{92} +31.6507 q^{94} -1.35026 q^{95} -0.836381 q^{97} +2.67513 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 3 * q^7 + 9 * q^8	\( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} - 3 q^{7} + 9 q^{8} + 3 q^{10} + 3 q^{11} + 2 q^{13} - 3 q^{14} + 5 q^{16} + 6 q^{19} + 5 q^{20} + 3 q^{22} + 10 q^{23} + 3 q^{25} + 20 q^{26} - 5 q^{28} + 10 q^{29} - 10 q^{31} + 11 q^{32} - 6 q^{34} - 3 q^{35} - 16 q^{37} - 10 q^{38} + 9 q^{40} - 2 q^{43} + 5 q^{44} + 16 q^{46} + 20 q^{47} + 3 q^{49} + 3 q^{50} + 32 q^{52} + 12 q^{53} + 3 q^{55} - 9 q^{56} - 2 q^{58} - 14 q^{59} + 10 q^{61} - 6 q^{62} + 33 q^{64} + 2 q^{65} - 2 q^{67} - 26 q^{68} - 3 q^{70} + 24 q^{71} + 4 q^{73} - 38 q^{74} - 10 q^{76} - 3 q^{77} + 8 q^{79} + 5 q^{80} + 24 q^{82} + 10 q^{83} + 36 q^{86} + 9 q^{88} - 20 q^{89} - 2 q^{91} + 18 q^{92} + 38 q^{94} + 6 q^{95} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 - 3 * q^7 + 9 * q^8 + 3 * q^10 + 3 * q^11 + 2 * q^13 - 3 * q^14 + 5 * q^16 + 6 * q^19 + 5 * q^20 + 3 * q^22 + 10 * q^23 + 3 * q^25 + 20 * q^26 - 5 * q^28 + 10 * q^29 - 10 * q^31 + 11 * q^32 - 6 * q^34 - 3 * q^35 - 16 * q^37 - 10 * q^38 + 9 * q^40 - 2 * q^43 + 5 * q^44 + 16 * q^46 + 20 * q^47 + 3 * q^49 + 3 * q^50 + 32 * q^52 + 12 * q^53 + 3 * q^55 - 9 * q^56 - 2 * q^58 - 14 * q^59 + 10 * q^61 - 6 * q^62 + 33 * q^64 + 2 * q^65 - 2 * q^67 - 26 * q^68 - 3 * q^70 + 24 * q^71 + 4 * q^73 - 38 * q^74 - 10 * q^76 - 3 * q^77 + 8 * q^79 + 5 * q^80 + 24 * q^82 + 10 * q^83 + 36 * q^86 + 9 * q^88 - 20 * q^89 - 2 * q^91 + 18 * q^92 + 38 * q^94 + 6 * q^95 + 3 * q^98

Introduction

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