LMFDB - Embedded newform 3465.2.a.bi.1.2

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			1.2
Root			$2.17009$ 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)

$f(q)$	$=$	$q+1.53919 q^{2} +0.369102 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.51026 q^{8} +O(q^{10})$	\(q+1.53919 q^{2} +0.369102 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.51026 q^{8} +1.53919 q^{10} -1.00000 q^{11} -0.829914 q^{13} +1.53919 q^{14} -4.60197 q^{16} +2.82991 q^{17} +6.49693 q^{19} +0.369102 q^{20} -1.53919 q^{22} +6.97107 q^{23} +1.00000 q^{25} -1.27739 q^{26} +0.369102 q^{28} -3.26180 q^{29} -10.2979 q^{31} -2.06278 q^{32} +4.35577 q^{34} +1.00000 q^{35} +11.3112 q^{37} +10.0000 q^{38} -2.51026 q^{40} +0.199016 q^{41} -0.447480 q^{43} -0.369102 q^{44} +10.7298 q^{46} +6.82991 q^{47} +1.00000 q^{49} +1.53919 q^{50} -0.306323 q^{52} +7.31124 q^{53} -1.00000 q^{55} -2.51026 q^{56} -5.02052 q^{58} +7.95774 q^{59} -6.87936 q^{61} -15.8504 q^{62} +6.02893 q^{64} -0.829914 q^{65} +6.20620 q^{67} +1.04453 q^{68} +1.53919 q^{70} +11.4186 q^{71} +5.66701 q^{73} +17.4101 q^{74} +2.39803 q^{76} -1.00000 q^{77} -14.5464 q^{79} -4.60197 q^{80} +0.306323 q^{82} -2.89496 q^{83} +2.82991 q^{85} -0.688756 q^{86} +2.51026 q^{88} +0.581449 q^{89} -0.829914 q^{91} +2.57304 q^{92} +10.5125 q^{94} +6.49693 q^{95} -2.15676 q^{97} +1.53919 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{13} + 3 q^{14} + 5 q^{16} + 14 q^{17} + 2 q^{19} + 5 q^{20} - 3 q^{22} + 6 q^{23} + 3 q^{25} - 10 q^{26} + 5 q^{28} - 2 q^{29} - 4 q^{31} + 11 q^{32} + 16 q^{34} + 3 q^{35} + 8 q^{37} + 30 q^{38} + 9 q^{40} + 10 q^{41} - 2 q^{43} - 5 q^{44} - 8 q^{46} + 26 q^{47} + 3 q^{49} + 3 q^{50} - 22 q^{52} - 4 q^{53} - 3 q^{55} + 9 q^{56} + 18 q^{58} + 8 q^{59} - 8 q^{61} - 20 q^{62} + 33 q^{64} - 8 q^{65} - 6 q^{67} + 32 q^{68} + 3 q^{70} + 20 q^{71} - 6 q^{73} - 10 q^{74} + 26 q^{76} - 3 q^{77} - 8 q^{79} + 5 q^{80} + 22 q^{82} - 10 q^{83} + 14 q^{85} - 28 q^{86} - 9 q^{88} + 16 q^{89} - 8 q^{91} - 26 q^{92} + 28 q^{94} + 2 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 - 8 * q^13 + 3 * q^14 + 5 * q^16 + 14 * q^17 + 2 * q^19 + 5 * q^20 - 3 * q^22 + 6 * q^23 + 3 * q^25 - 10 * q^26 + 5 * q^28 - 2 * q^29 - 4 * q^31 + 11 * q^32 + 16 * q^34 + 3 * q^35 + 8 * q^37 + 30 * q^38 + 9 * q^40 + 10 * q^41 - 2 * q^43 - 5 * q^44 - 8 * q^46 + 26 * q^47 + 3 * q^49 + 3 * q^50 - 22 * q^52 - 4 * q^53 - 3 * q^55 + 9 * q^56 + 18 * q^58 + 8 * q^59 - 8 * q^61 - 20 * q^62 + 33 * q^64 - 8 * q^65 - 6 * q^67 + 32 * q^68 + 3 * q^70 + 20 * q^71 - 6 * q^73 - 10 * q^74 + 26 * q^76 - 3 * q^77 - 8 * q^79 + 5 * q^80 + 22 * q^82 - 10 * q^83 + 14 * q^85 - 28 * q^86 - 9 * q^88 + 16 * q^89 - 8 * q^91 - 26 * q^92 + 28 * q^94 + 2 * 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 $

$2$	1.53919	1.08837	0.544185	−	0.838965i	$-0.316839\pi$
$2$	1.53919	1.08837	0.544185	+	0.838965i	$0.316839\pi$
$3$	0	0
$3$	0	0
$4$	0.369102	0.184551
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.51026	−0.887511
$9$	0	0
$10$	1.53919	0.486734
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.829914	−0.230177	−0.115088	−	0.993355i	$-0.536715\pi$
$13$	−0.829914	−0.230177	−0.115088	+	0.993355i	$0.536715\pi$
$14$	1.53919	0.411366
$15$	0	0
$16$	−4.60197	−1.15049
$17$	2.82991	0.686355	0.343177	−	0.939271i	$-0.388497\pi$
$17$	2.82991	0.686355	0.343177	+	0.939271i	$0.388497\pi$
$18$	0	0
$19$	6.49693	1.49050	0.745249	−	0.666786i	$-0.232331\pi$
$19$	6.49693	1.49050	0.745249	+	0.666786i	$0.232331\pi$
$20$	0.369102	0.0825338
$21$	0	0
$22$	−1.53919	−0.328156
$23$	6.97107	1.45357	0.726784	−	0.686866i	$-0.241014\pi$
$23$	6.97107	1.45357	0.726784	+	0.686866i	$0.241014\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.27739	−0.250518
$27$	0	0
$28$	0.369102	0.0697538
$29$	−3.26180	−0.605700	−0.302850	−	0.953038i	$-0.597938\pi$
$29$	−3.26180	−0.605700	−0.302850	+	0.953038i	$0.597938\pi$
$30$	0	0
$31$	−10.2979	−1.84956	−0.924780	−	0.380503i	$-0.875751\pi$
$31$	−10.2979	−1.84956	−0.924780	+	0.380503i	$0.875751\pi$
$32$	−2.06278	−0.364651
$33$	0	0
$34$	4.35577	0.747009
$35$	1.00000	0.169031
$36$	0	0
$37$	11.3112	1.85956	0.929778	−	0.368119i	$-0.119998\pi$
$37$	11.3112	1.85956	0.929778	+	0.368119i	$0.119998\pi$
$38$	10.0000	1.62221
$39$	0	0
$40$	−2.51026	−0.396907
$41$	0.199016	0.0310811	0.0155405	−	0.999879i	$-0.495053\pi$
$41$	0.199016	0.0310811	0.0155405	+	0.999879i	$0.495053\pi$
$42$	0	0
$43$	−0.447480	−0.0682401	−0.0341200	−	0.999418i	$-0.510863\pi$
$43$	−0.447480	−0.0682401	−0.0341200	+	0.999418i	$0.510863\pi$
$44$	−0.369102	−0.0556443
$45$	0	0
$46$	10.7298	1.58202
$47$	6.82991	0.996245	0.498123	−	0.867107i	$-0.334023\pi$
$47$	6.82991	0.996245	0.498123	+	0.867107i	$0.334023\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.53919	0.217674
$51$	0	0
$52$	−0.306323	−0.0424794
$53$	7.31124	1.00428	0.502138	−	0.864787i	$-0.332547\pi$
$53$	7.31124	1.00428	0.502138	+	0.864787i	$0.332547\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−2.51026	−0.335448
$57$	0	0
$58$	−5.02052	−0.659226
$59$	7.95774	1.03601	0.518005	−	0.855378i	$-0.326675\pi$
$59$	7.95774	1.03601	0.518005	+	0.855378i	$0.326675\pi$
$60$	0	0
$61$	−6.87936	−0.880812	−0.440406	−	0.897799i	$-0.645165\pi$
$61$	−6.87936	−0.880812	−0.440406	+	0.897799i	$0.645165\pi$
$62$	−15.8504	−2.01301
$63$	0	0
$64$	6.02893	0.753616
$65$	−0.829914	−0.102938
$66$	0	0
$67$	6.20620	0.758208	0.379104	−	0.925354i	$-0.376232\pi$
$67$	6.20620	0.758208	0.379104	+	0.925354i	$0.376232\pi$
$68$	1.04453	0.126668
$69$	0	0
$70$	1.53919	0.183968
$71$	11.4186	1.35513	0.677566	−	0.735462i	$-0.263035\pi$
$71$	11.4186	1.35513	0.677566	+	0.735462i	$0.263035\pi$
$72$	0	0
$73$	5.66701	0.663274	0.331637	−	0.943407i	$-0.392399\pi$
$73$	5.66701	0.663274	0.331637	+	0.943407i	$0.392399\pi$
$74$	17.4101	2.02389
$75$	0	0
$76$	2.39803	0.275073
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−14.5464	−1.63660	−0.818298	−	0.574795i	$-0.805082\pi$
$79$	−14.5464	−1.63660	−0.818298	+	0.574795i	$0.805082\pi$
$80$	−4.60197	−0.514516
$81$	0	0
$82$	0.306323	0.0338277
$83$	−2.89496	−0.317763	−0.158882	−	0.987298i	$-0.550789\pi$
$83$	−2.89496	−0.317763	−0.158882	+	0.987298i	$0.550789\pi$
$84$	0	0
$85$	2.82991	0.306947
$86$	−0.688756	−0.0742705
$87$	0	0
$88$	2.51026	0.267595
$89$	0.581449	0.0616335	0.0308168	−	0.999525i	$-0.490189\pi$
$89$	0.581449	0.0616335	0.0308168	+	0.999525i	$0.490189\pi$
$90$	0	0
$91$	−0.829914	−0.0869986
$92$	2.57304	0.268258
$93$	0	0
$94$	10.5125	1.08428
$95$	6.49693	0.666571
$96$	0	0
$97$	−2.15676	−0.218985	−0.109493	−	0.993988i	$-0.534923\pi$
$97$	−2.15676	−0.218985	−0.109493	+	0.993988i	$0.534923\pi$
$98$	1.53919	0.155482
$99$	0	0
$100$	0.369102	0.0369102
$101$	7.21953	0.718371	0.359185	−	0.933266i	$-0.383055\pi$
$101$	7.21953	0.718371	0.359185	+	0.933266i	$0.383055\pi$
$102$	0	0
$103$	3.66701	0.361322	0.180661	−	0.983545i	$-0.442176\pi$
$103$	3.66701	0.361322	0.180661	+	0.983545i	$0.442176\pi$
$104$	2.08330	0.204284
$105$	0	0
$106$	11.2534	1.09303
$107$	9.26180	0.895372	0.447686	−	0.894191i	$-0.352248\pi$
$107$	9.26180	0.895372	0.447686	+	0.894191i	$0.352248\pi$
$108$	0	0
$109$	6.86376	0.657429	0.328715	−	0.944429i	$-0.393385\pi$
$109$	6.86376	0.657429	0.328715	+	0.944429i	$0.393385\pi$
$110$	−1.53919	−0.146756
$111$	0	0
$112$	−4.60197	−0.434845
$113$	3.75872	0.353591	0.176795	−	0.984248i	$-0.443427\pi$
$113$	3.75872	0.353591	0.176795	+	0.984248i	$0.443427\pi$
$114$	0	0
$115$	6.97107	0.650056
$116$	−1.20394	−0.111783
$117$	0	0
$118$	12.2485	1.12756
$119$	2.82991	0.259418
$120$	0	0
$121$	1.00000	0.0909091
$122$	−10.5886	−0.958650
$123$	0	0
$124$	−3.80098	−0.341338
$125$	1.00000	0.0894427
$126$	0	0
$127$	−21.5441	−1.91173	−0.955865	−	0.293805i	$-0.905078\pi$
$127$	−21.5441	−1.91173	−0.955865	+	0.293805i	$0.905078\pi$
$128$	13.4052	1.18487
$129$	0	0
$130$	−1.27739	−0.112035
$131$	12.6803	1.10789	0.553943	−	0.832554i	$-0.313122\pi$
$131$	12.6803	1.10789	0.553943	+	0.832554i	$0.313122\pi$
$132$	0	0
$133$	6.49693	0.563355
$134$	9.55252	0.825212
$135$	0	0
$136$	−7.10382	−0.609147
$137$	−19.6248	−1.67666	−0.838328	−	0.545166i	$-0.816467\pi$
$137$	−19.6248	−1.67666	−0.838328	+	0.545166i	$0.816467\pi$
$138$	0	0
$139$	−0.496928	−0.0421489	−0.0210745	−	0.999778i	$-0.506709\pi$
$139$	−0.496928	−0.0421489	−0.0210745	+	0.999778i	$0.506709\pi$
$140$	0.369102	0.0311948
$141$	0	0
$142$	17.5753	1.47489
$143$	0.829914	0.0694009
$144$	0	0
$145$	−3.26180	−0.270877
$146$	8.72261	0.721888
$147$	0	0
$148$	4.17501	0.343183
$149$	13.3340	1.09237	0.546183	−	0.837666i	$-0.316080\pi$
$149$	13.3340	1.09237	0.546183	+	0.837666i	$0.316080\pi$
$150$	0	0
$151$	−1.05559	−0.0859028	−0.0429514	−	0.999077i	$-0.513676\pi$
$151$	−1.05559	−0.0859028	−0.0429514	+	0.999077i	$0.513676\pi$
$152$	−16.3090	−1.32283
$153$	0	0
$154$	−1.53919	−0.124031
$155$	−10.2979	−0.827148
$156$	0	0
$157$	−0.496928	−0.0396592	−0.0198296	−	0.999803i	$-0.506312\pi$
$157$	−0.496928	−0.0396592	−0.0198296	+	0.999803i	$0.506312\pi$
$158$	−22.3896	−1.78122
$159$	0	0
$160$	−2.06278	−0.163077
$161$	6.97107	0.549397
$162$	0	0
$163$	1.31124	0.102705	0.0513523	−	0.998681i	$-0.483647\pi$
$163$	1.31124	0.102705	0.0513523	+	0.998681i	$0.483647\pi$
$164$	0.0734572	0.00573605
$165$	0	0
$166$	−4.45589	−0.345844
$167$	−3.17727	−0.245865	−0.122932	−	0.992415i	$-0.539230\pi$
$167$	−3.17727	−0.245865	−0.122932	+	0.992415i	$0.539230\pi$
$168$	0	0
$169$	−12.3112	−0.947019
$170$	4.35577	0.334072
$171$	0	0
$172$	−0.165166	−0.0125938
$173$	17.9493	1.36466	0.682331	−	0.731043i	$-0.260966\pi$
$173$	17.9493	1.36466	0.682331	+	0.731043i	$0.260966\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	4.60197	0.346886
$177$	0	0
$178$	0.894960	0.0670801
$179$	−15.7321	−1.17587	−0.587935	−	0.808908i	$-0.700059\pi$
$179$	−15.7321	−1.17587	−0.587935	+	0.808908i	$0.700059\pi$
$180$	0	0
$181$	−16.8104	−1.24951	−0.624755	−	0.780821i	$-0.714801\pi$
$181$	−16.8104	−1.24951	−0.624755	+	0.780821i	$0.714801\pi$
$182$	−1.27739	−0.0946867
$183$	0	0
$184$	−17.4992	−1.29006
$185$	11.3112	0.831619
$186$	0	0
$187$	−2.82991	−0.206944
$188$	2.52094	0.183858
$189$	0	0
$190$	10.0000	0.725476
$191$	2.92162	0.211401	0.105701	−	0.994398i	$-0.466291\pi$
$191$	2.92162	0.211401	0.105701	+	0.994398i	$0.466291\pi$
$192$	0	0
$193$	−22.8287	−1.64325	−0.821623	−	0.570032i	$-0.806931\pi$
$193$	−22.8287	−1.64325	−0.821623	+	0.570032i	$0.806931\pi$
$194$	−3.31965	−0.238337
$195$	0	0
$196$	0.369102	0.0263645
$197$	−19.7237	−1.40525	−0.702626	−	0.711559i	$-0.747989\pi$
$197$	−19.7237	−1.40525	−0.702626	+	0.711559i	$0.747989\pi$
$198$	0	0
$199$	−10.6381	−0.754114	−0.377057	−	0.926190i	$-0.623064\pi$
$199$	−10.6381	−0.754114	−0.377057	+	0.926190i	$0.623064\pi$
$200$	−2.51026	−0.177502
$201$	0	0
$202$	11.1122	0.781854
$203$	−3.26180	−0.228933
$204$	0	0
$205$	0.199016	0.0138999
$206$	5.64423	0.393252
$207$	0	0
$208$	3.81924	0.264816
$209$	−6.49693	−0.449402
$210$	0	0
$211$	7.50307	0.516533	0.258266	−	0.966074i	$-0.416849\pi$
$211$	7.50307	0.516533	0.258266	+	0.966074i	$0.416849\pi$
$212$	2.69860	0.185340
$213$	0	0
$214$	14.2557	0.974496
$215$	−0.447480	−0.0305179
$216$	0	0
$217$	−10.2979	−0.699068
$218$	10.5646	0.715527
$219$	0	0
$220$	−0.369102	−0.0248849
$221$	−2.34858	−0.157983
$222$	0	0
$223$	24.0338	1.60943	0.804713	−	0.593664i	$-0.202319\pi$
$223$	24.0338	1.60943	0.804713	+	0.593664i	$0.202319\pi$
$224$	−2.06278	−0.137825
$225$	0	0
$226$	5.78539	0.384838
$227$	−1.47641	−0.0979927	−0.0489964	−	0.998799i	$-0.515602\pi$
$227$	−1.47641	−0.0979927	−0.0489964	+	0.998799i	$0.515602\pi$
$228$	0	0
$229$	8.07223	0.533428	0.266714	−	0.963776i	$-0.414062\pi$
$229$	8.07223	0.533428	0.266714	+	0.963776i	$0.414062\pi$
$230$	10.7298	0.707502
$231$	0	0
$232$	8.18795	0.537565
$233$	6.44748	0.422388	0.211194	−	0.977444i	$-0.432265\pi$
$233$	6.44748	0.422388	0.211194	+	0.977444i	$0.432265\pi$
$234$	0	0
$235$	6.82991	0.445534
$236$	2.93722	0.191197
$237$	0	0
$238$	4.35577	0.282343
$239$	8.18342	0.529341	0.264671	−	0.964339i	$-0.414737\pi$
$239$	8.18342	0.529341	0.264671	+	0.964339i	$0.414737\pi$
$240$	0	0
$241$	10.9060	0.702519	0.351259	−	0.936278i	$-0.385754\pi$
$241$	10.9060	0.702519	0.351259	+	0.936278i	$0.385754\pi$
$242$	1.53919	0.0989428
$243$	0	0
$244$	−2.53919	−0.162555
$245$	1.00000	0.0638877
$246$	0	0
$247$	−5.39189	−0.343078
$248$	25.8504	1.64150
$249$	0	0
$250$	1.53919	0.0973469
$251$	−18.7948	−1.18632	−0.593160	−	0.805085i	$-0.702120\pi$
$251$	−18.7948	−1.18632	−0.593160	+	0.805085i	$0.702120\pi$
$252$	0	0
$253$	−6.97107	−0.438267
$254$	−33.1605	−2.08067
$255$	0	0
$256$	8.57531	0.535957
$257$	13.0784	0.815807	0.407903	−	0.913025i	$-0.366260\pi$
$257$	13.0784	0.815807	0.407903	+	0.913025i	$0.366260\pi$
$258$	0	0
$259$	11.3112	0.702846
$260$	−0.306323	−0.0189973
$261$	0	0
$262$	19.5174	1.20579
$263$	10.0267	0.618270	0.309135	−	0.951018i	$-0.399960\pi$
$263$	10.0267	0.618270	0.309135	+	0.951018i	$0.399960\pi$
$264$	0	0
$265$	7.31124	0.449126
$266$	10.0000	0.613139
$267$	0	0
$268$	2.29072	0.139928
$269$	11.9155	0.726500	0.363250	−	0.931692i	$-0.381667\pi$
$269$	11.9155	0.726500	0.363250	+	0.931692i	$0.381667\pi$
$270$	0	0
$271$	25.1773	1.52941	0.764705	−	0.644380i	$-0.222885\pi$
$271$	25.1773	1.52941	0.764705	+	0.644380i	$0.222885\pi$
$272$	−13.0232	−0.789646
$273$	0	0
$274$	−30.2062	−1.82482
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−5.33791	−0.320724	−0.160362	−	0.987058i	$-0.551266\pi$
$277$	−5.33791	−0.320724	−0.160362	+	0.987058i	$0.551266\pi$
$278$	−0.764867	−0.0458737
$279$	0	0
$280$	−2.51026	−0.150017
$281$	−16.4703	−0.982534	−0.491267	−	0.871009i	$-0.663466\pi$
$281$	−16.4703	−0.982534	−0.491267	+	0.871009i	$0.663466\pi$
$282$	0	0
$283$	−0.948284	−0.0563696	−0.0281848	−	0.999603i	$-0.508973\pi$
$283$	−0.948284	−0.0563696	−0.0281848	+	0.999603i	$0.508973\pi$
$284$	4.21461	0.250091
$285$	0	0
$286$	1.27739	0.0755339
$287$	0.199016	0.0117475
$288$	0	0
$289$	−8.99159	−0.528917
$290$	−5.02052	−0.294815
$291$	0	0
$292$	2.09171	0.122408
$293$	7.72487	0.451292	0.225646	−	0.974209i	$-0.427551\pi$
$293$	7.72487	0.451292	0.225646	+	0.974209i	$0.427551\pi$
$294$	0	0
$295$	7.95774	0.463318
$296$	−28.3942	−1.65038
$297$	0	0
$298$	20.5236	1.18890
$299$	−5.78539	−0.334577
$300$	0	0
$301$	−0.447480	−0.0257923
$302$	−1.62475	−0.0934941
$303$	0	0
$304$	−29.8987	−1.71481
$305$	−6.87936	−0.393911
$306$	0	0
$307$	−32.7526	−1.86929	−0.934644	−	0.355584i	$-0.884282\pi$
$307$	−32.7526	−1.86929	−0.934644	+	0.355584i	$0.884282\pi$
$308$	−0.369102	−0.0210316
$309$	0	0
$310$	−15.8504	−0.900244
$311$	−22.9204	−1.29970	−0.649848	−	0.760064i	$-0.725168\pi$
$311$	−22.9204	−1.29970	−0.649848	+	0.760064i	$0.725168\pi$
$312$	0	0
$313$	−11.5753	−0.654275	−0.327137	−	0.944977i	$-0.606084\pi$
$313$	−11.5753	−0.654275	−0.327137	+	0.944977i	$0.606084\pi$
$314$	−0.764867	−0.0431639
$315$	0	0
$316$	−5.36910	−0.302036
$317$	−12.5236	−0.703395	−0.351697	−	0.936114i	$-0.614395\pi$
$317$	−12.5236	−0.703395	−0.351697	+	0.936114i	$0.614395\pi$
$318$	0	0
$319$	3.26180	0.182625
$320$	6.02893	0.337027
$321$	0	0
$322$	10.7298	0.597948
$323$	18.3857	1.02301
$324$	0	0
$325$	−0.829914	−0.0460353
$326$	2.01825	0.111781
$327$	0	0
$328$	−0.499582	−0.0275848
$329$	6.82991	0.376545
$330$	0	0
$331$	−18.1711	−0.998776	−0.499388	−	0.866379i	$-0.666442\pi$
$331$	−18.1711	−0.998776	−0.499388	+	0.866379i	$0.666442\pi$
$332$	−1.06854	−0.0586436
$333$	0	0
$334$	−4.89043	−0.267592
$335$	6.20620	0.339081
$336$	0	0
$337$	28.0905	1.53019	0.765093	−	0.643920i	$-0.222693\pi$
$337$	28.0905	1.53019	0.765093	+	0.643920i	$0.222693\pi$
$338$	−18.9493	−1.03071
$339$	0	0
$340$	1.04453	0.0566475
$341$	10.2979	0.557663
$342$	0	0
$343$	1.00000	0.0539949
$344$	1.12329	0.0605638
$345$	0	0
$346$	27.6274	1.48526
$347$	−22.0761	−1.18511	−0.592554	−	0.805531i	$-0.701880\pi$
$347$	−22.0761	−1.18511	−0.592554	+	0.805531i	$0.701880\pi$
$348$	0	0
$349$	18.2134	0.974941	0.487470	−	0.873140i	$-0.337920\pi$
$349$	18.2134	0.974941	0.487470	+	0.873140i	$0.337920\pi$
$350$	1.53919	0.0822731
$351$	0	0
$352$	2.06278	0.109947
$353$	1.41855	0.0755018	0.0377509	−	0.999287i	$-0.487981\pi$
$353$	1.41855	0.0755018	0.0377509	+	0.999287i	$0.487981\pi$
$354$	0	0
$355$	11.4186	0.606034
$356$	0.214614	0.0113745
$357$	0	0
$358$	−24.2146	−1.27978
$359$	−13.0700	−0.689806	−0.344903	−	0.938638i	$-0.612088\pi$
$359$	−13.0700	−0.689806	−0.344903	+	0.938638i	$0.612088\pi$
$360$	0	0
$361$	23.2101	1.22158
$362$	−25.8744	−1.35993
$363$	0	0
$364$	−0.306323	−0.0160557
$365$	5.66701	0.296625
$366$	0	0
$367$	19.7926	1.03316	0.516582	−	0.856238i	$-0.327204\pi$
$367$	19.7926	1.03316	0.516582	+	0.856238i	$0.327204\pi$
$368$	−32.0806	−1.67232
$369$	0	0
$370$	17.4101	0.905110
$371$	7.31124	0.379581
$372$	0	0
$373$	−31.4101	−1.62636	−0.813178	−	0.582015i	$-0.802264\pi$
$373$	−31.4101	−1.62636	−0.813178	+	0.582015i	$0.802264\pi$
$374$	−4.35577	−0.225232
$375$	0	0
$376$	−17.1449	−0.884178
$377$	2.70701	0.139418
$378$	0	0
$379$	21.9421	1.12709	0.563546	−	0.826085i	$-0.309437\pi$
$379$	21.9421	1.12709	0.563546	+	0.826085i	$0.309437\pi$
$380$	2.39803	0.123016
$381$	0	0
$382$	4.49693	0.230083
$383$	−6.95547	−0.355408	−0.177704	−	0.984084i	$-0.556867\pi$
$383$	−6.95547	−0.355408	−0.177704	+	0.984084i	$0.556867\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−35.1377	−1.78846
$387$	0	0
$388$	−0.796064	−0.0404140
$389$	−2.76487	−0.140184	−0.0700922	−	0.997541i	$-0.522329\pi$
$389$	−2.76487	−0.140184	−0.0700922	+	0.997541i	$0.522329\pi$
$390$	0	0
$391$	19.7275	0.997664
$392$	−2.51026	−0.126787
$393$	0	0
$394$	−30.3584	−1.52944
$395$	−14.5464	−0.731908
$396$	0	0
$397$	6.92162	0.347386	0.173693	−	0.984800i	$-0.444430\pi$
$397$	6.92162	0.347386	0.173693	+	0.984800i	$0.444430\pi$
$398$	−16.3740	−0.820756
$399$	0	0
$400$	−4.60197	−0.230098
$401$	6.68035	0.333601	0.166800	−	0.985991i	$-0.446656\pi$
$401$	6.68035	0.333601	0.166800	+	0.985991i	$0.446656\pi$
$402$	0	0
$403$	8.54638	0.425725
$404$	2.66475	0.132576
$405$	0	0
$406$	−5.02052	−0.249164
$407$	−11.3112	−0.560678
$408$	0	0
$409$	15.0772	0.745517	0.372759	−	0.927928i	$-0.378412\pi$
$409$	15.0772	0.745517	0.372759	+	0.927928i	$0.378412\pi$
$410$	0.306323	0.0151282
$411$	0	0
$412$	1.35350	0.0666824
$413$	7.95774	0.391575
$414$	0	0
$415$	−2.89496	−0.142108
$416$	1.71193	0.0839342
$417$	0	0
$418$	−10.0000	−0.489116
$419$	9.54533	0.466320	0.233160	−	0.972438i	$-0.425093\pi$
$419$	9.54533	0.466320	0.233160	+	0.972438i	$0.425093\pi$
$420$	0	0
$421$	−32.7442	−1.59585	−0.797927	−	0.602755i	$-0.794070\pi$
$421$	−32.7442	−1.59585	−0.797927	+	0.602755i	$0.794070\pi$
$422$	11.5486	0.562179
$423$	0	0
$424$	−18.3531	−0.891306
$425$	2.82991	0.137271
$426$	0	0
$427$	−6.87936	−0.332916
$428$	3.41855	0.165242
$429$	0	0
$430$	−0.688756	−0.0332148
$431$	23.0289	1.10926	0.554632	−	0.832096i	$-0.312859\pi$
$431$	23.0289	1.10926	0.554632	+	0.832096i	$0.312859\pi$
$432$	0	0
$433$	−34.3279	−1.64969	−0.824846	−	0.565357i	$-0.808738\pi$
$433$	−34.3279	−1.64969	−0.824846	+	0.565357i	$0.808738\pi$
$434$	−15.8504	−0.760845
$435$	0	0
$436$	2.53343	0.121329
$437$	45.2905	2.16654
$438$	0	0
$439$	−16.5958	−0.792076	−0.396038	−	0.918234i	$-0.629615\pi$
$439$	−16.5958	−0.792076	−0.396038	+	0.918234i	$0.629615\pi$
$440$	2.51026	0.119672
$441$	0	0
$442$	−3.61491	−0.171944
$443$	−10.9177	−0.518718	−0.259359	−	0.965781i	$-0.583511\pi$
$443$	−10.9177	−0.518718	−0.259359	+	0.965781i	$0.583511\pi$
$444$	0	0
$445$	0.581449	0.0275633
$446$	36.9926	1.75165
$447$	0	0
$448$	6.02893	0.284840
$449$	24.5997	1.16093	0.580466	−	0.814285i	$-0.302870\pi$
$449$	24.5997	1.16093	0.580466	+	0.814285i	$0.302870\pi$
$450$	0	0
$451$	−0.199016	−0.00937129
$452$	1.38735	0.0652556
$453$	0	0
$454$	−2.27247	−0.106652
$455$	−0.829914	−0.0389069
$456$	0	0
$457$	−36.9854	−1.73011	−0.865053	−	0.501680i	$-0.832715\pi$
$457$	−36.9854	−1.73011	−0.865053	+	0.501680i	$0.832715\pi$
$458$	12.4247	0.580568
$459$	0	0
$460$	2.57304	0.119969
$461$	−37.3595	−1.74000	−0.870002	−	0.493048i	$-0.835883\pi$
$461$	−37.3595	−1.74000	−0.870002	+	0.493048i	$0.835883\pi$
$462$	0	0
$463$	7.05559	0.327901	0.163951	−	0.986469i	$-0.447576\pi$
$463$	7.05559	0.327901	0.163951	+	0.986469i	$0.447576\pi$
$464$	15.0107	0.696853
$465$	0	0
$466$	9.92389	0.459715
$467$	19.8648	0.919234	0.459617	−	0.888117i	$-0.347987\pi$
$467$	19.8648	0.919234	0.459617	+	0.888117i	$0.347987\pi$
$468$	0	0
$469$	6.20620	0.286576
$470$	10.5125	0.484907
$471$	0	0
$472$	−19.9760	−0.919470
$473$	0.447480	0.0205752
$474$	0	0
$475$	6.49693	0.298100
$476$	1.04453	0.0478759
$477$	0	0
$478$	12.5958	0.576120
$479$	−6.39803	−0.292334	−0.146167	−	0.989260i	$-0.546694\pi$
$479$	−6.39803	−0.292334	−0.146167	+	0.989260i	$0.546694\pi$
$480$	0	0
$481$	−9.38735	−0.428026
$482$	16.7864	0.764601
$483$	0	0
$484$	0.369102	0.0167774
$485$	−2.15676	−0.0979332
$486$	0	0
$487$	−25.8082	−1.16948	−0.584740	−	0.811221i	$-0.698803\pi$
$487$	−25.8082	−1.16948	−0.584740	+	0.811221i	$0.698803\pi$
$488$	17.2690	0.781730
$489$	0	0
$490$	1.53919	0.0695335
$491$	17.1689	0.774820	0.387410	−	0.921908i	$-0.373370\pi$
$491$	17.1689	0.774820	0.387410	+	0.921908i	$0.373370\pi$
$492$	0	0
$493$	−9.23060	−0.415725
$494$	−8.29914	−0.373396
$495$	0	0
$496$	47.3907	2.12790
$497$	11.4186	0.512192
$498$	0	0
$499$	−13.1629	−0.589252	−0.294626	−	0.955613i	$-0.595195\pi$
$499$	−13.1629	−0.589252	−0.294626	+	0.955613i	$0.595195\pi$
$500$	0.369102	0.0165068
$501$	0	0
$502$	−28.9288	−1.29116
$503$	30.7259	1.37000	0.685000	−	0.728543i	$-0.259802\pi$
$503$	30.7259	1.37000	0.685000	+	0.728543i	$0.259802\pi$
$504$	0	0
$505$	7.21953	0.321265
$506$	−10.7298	−0.476998
$507$	0	0
$508$	−7.95198	−0.352812
$509$	−25.2183	−1.11778	−0.558891	−	0.829241i	$-0.688773\pi$
$509$	−25.2183	−1.11778	−0.558891	+	0.829241i	$0.688773\pi$
$510$	0	0
$511$	5.66701	0.250694
$512$	−13.6114	−0.601546
$513$	0	0
$514$	20.1301	0.887900
$515$	3.66701	0.161588
$516$	0	0
$517$	−6.82991	−0.300379
$518$	17.4101	0.764958
$519$	0	0
$520$	2.08330	0.0913587
$521$	−0.183417	−0.00803567	−0.00401783	−	0.999992i	$-0.501279\pi$
$521$	−0.183417	−0.00803567	−0.00401783	+	0.999992i	$0.501279\pi$
$522$	0	0
$523$	−4.86830	−0.212876	−0.106438	−	0.994319i	$-0.533945\pi$
$523$	−4.86830	−0.212876	−0.106438	+	0.994319i	$0.533945\pi$
$524$	4.68035	0.204462
$525$	0	0
$526$	15.4329	0.672908
$527$	−29.1422	−1.26945
$528$	0	0
$529$	25.5958	1.11286
$530$	11.2534	0.488816
$531$	0	0
$532$	2.39803	0.103968
$533$	−0.165166	−0.00715413
$534$	0	0
$535$	9.26180	0.400422
$536$	−15.5792	−0.672918
$537$	0	0
$538$	18.3402	0.790701
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−45.7009	−1.96483	−0.982417	−	0.186701i	$-0.940221\pi$
$541$	−45.7009	−1.96483	−0.982417	+	0.186701i	$0.940221\pi$
$542$	38.7526	1.66457
$543$	0	0
$544$	−5.83749	−0.250280
$545$	6.86376	0.294011
$546$	0	0
$547$	−24.5958	−1.05164	−0.525821	−	0.850595i	$-0.676242\pi$
$547$	−24.5958	−1.05164	−0.525821	+	0.850595i	$0.676242\pi$
$548$	−7.24354	−0.309429
$549$	0	0
$550$	−1.53919	−0.0656312
$551$	−21.1917	−0.902795
$552$	0	0
$553$	−14.5464	−0.618575
$554$	−8.21604	−0.349066
$555$	0	0
$556$	−0.183417	−0.00777863
$557$	−33.6514	−1.42586	−0.712928	−	0.701237i	$-0.752631\pi$
$557$	−33.6514	−1.42586	−0.712928	+	0.701237i	$0.752631\pi$
$558$	0	0
$559$	0.371370	0.0157073
$560$	−4.60197	−0.194469
$561$	0	0
$562$	−25.3509	−1.06936
$563$	32.2290	1.35829	0.679145	−	0.734004i	$-0.262351\pi$
$563$	32.2290	1.35829	0.679145	+	0.734004i	$0.262351\pi$
$564$	0	0
$565$	3.75872	0.158131
$566$	−1.45959	−0.0613511
$567$	0	0
$568$	−28.6635	−1.20269
$569$	−5.33403	−0.223614	−0.111807	−	0.993730i	$-0.535664\pi$
$569$	−5.33403	−0.223614	−0.111807	+	0.993730i	$0.535664\pi$
$570$	0	0
$571$	−29.5441	−1.23638	−0.618191	−	0.786028i	$-0.712134\pi$
$571$	−29.5441	−1.23638	−0.618191	+	0.786028i	$0.712134\pi$
$572$	0.306323	0.0128080
$573$	0	0
$574$	0.306323	0.0127857
$575$	6.97107	0.290714
$576$	0	0
$577$	3.78539	0.157588	0.0787938	−	0.996891i	$-0.474893\pi$
$577$	3.78539	0.157588	0.0787938	+	0.996891i	$0.474893\pi$
$578$	−13.8398	−0.575658
$579$	0	0
$580$	−1.20394	−0.0499907
$581$	−2.89496	−0.120103
$582$	0	0
$583$	−7.31124	−0.302801
$584$	−14.2257	−0.588663
$585$	0	0
$586$	11.8900	0.491173
$587$	−14.5353	−0.599937	−0.299968	−	0.953949i	$-0.596976\pi$
$587$	−14.5353	−0.599937	−0.299968	+	0.953949i	$0.596976\pi$
$588$	0	0
$589$	−66.9048	−2.75676
$590$	12.2485	0.504261
$591$	0	0
$592$	−52.0540	−2.13941
$593$	−43.6814	−1.79378	−0.896890	−	0.442254i	$-0.854179\pi$
$593$	−43.6814	−1.79378	−0.896890	+	0.442254i	$0.854179\pi$
$594$	0	0
$595$	2.82991	0.116015
$596$	4.92162	0.201598
$597$	0	0
$598$	−8.90480	−0.364144
$599$	22.8371	0.933099	0.466549	−	0.884495i	$-0.345497\pi$
$599$	22.8371	0.933099	0.466549	+	0.884495i	$0.345497\pi$
$600$	0	0
$601$	−27.8687	−1.13679	−0.568394	−	0.822757i	$-0.692435\pi$
$601$	−27.8687	−1.13679	−0.568394	+	0.822757i	$0.692435\pi$
$602$	−0.688756	−0.0280716
$603$	0	0
$604$	−0.389621	−0.0158535
$605$	1.00000	0.0406558
$606$	0	0
$607$	29.8987	1.21355	0.606775	−	0.794874i	$-0.292463\pi$
$607$	29.8987	1.21355	0.606775	+	0.794874i	$0.292463\pi$
$608$	−13.4017	−0.543512
$609$	0	0
$610$	−10.5886	−0.428721
$611$	−5.66824	−0.229312
$612$	0	0
$613$	21.0700	0.851008	0.425504	−	0.904957i	$-0.360097\pi$
$613$	21.0700	0.851008	0.425504	+	0.904957i	$0.360097\pi$
$614$	−50.4124	−2.03448
$615$	0	0
$616$	2.51026	0.101141
$617$	−4.10731	−0.165354	−0.0826770	−	0.996576i	$-0.526347\pi$
$617$	−4.10731	−0.165354	−0.0826770	+	0.996576i	$0.526347\pi$
$618$	0	0
$619$	−14.5536	−0.584957	−0.292479	−	0.956272i	$-0.594480\pi$
$619$	−14.5536	−0.584957	−0.292479	+	0.956272i	$0.594480\pi$
$620$	−3.80098	−0.152651
$621$	0	0
$622$	−35.2788	−1.41455
$623$	0.581449	0.0232953
$624$	0	0
$625$	1.00000	0.0400000
$626$	−17.8166	−0.712094
$627$	0	0
$628$	−0.183417	−0.00731915
$629$	32.0098	1.27632
$630$	0	0
$631$	12.9939	0.517277	0.258639	−	0.965974i	$-0.416726\pi$
$631$	12.9939	0.517277	0.258639	+	0.965974i	$0.416726\pi$
$632$	36.5152	1.45250
$633$	0	0
$634$	−19.2762	−0.765555
$635$	−21.5441	−0.854952
$636$	0	0
$637$	−0.829914	−0.0328824
$638$	5.02052	0.198764
$639$	0	0
$640$	13.4052	0.529888
$641$	−8.54638	−0.337562	−0.168781	−	0.985654i	$-0.553983\pi$
$641$	−8.54638	−0.337562	−0.168781	+	0.985654i	$0.553983\pi$
$642$	0	0
$643$	17.7659	0.700619	0.350310	−	0.936634i	$-0.386076\pi$
$643$	17.7659	0.700619	0.350310	+	0.936634i	$0.386076\pi$
$644$	2.57304	0.101392
$645$	0	0
$646$	28.2991	1.11341
$647$	−44.9698	−1.76795	−0.883974	−	0.467537i	$-0.845142\pi$
$647$	−44.9698	−1.76795	−0.883974	+	0.467537i	$0.845142\pi$
$648$	0	0
$649$	−7.95774	−0.312369
$650$	−1.27739	−0.0501035
$651$	0	0
$652$	0.483983	0.0189542
$653$	4.75258	0.185983	0.0929914	−	0.995667i	$-0.470357\pi$
$653$	4.75258	0.185983	0.0929914	+	0.995667i	$0.470357\pi$
$654$	0	0
$655$	12.6803	0.495462
$656$	−0.915865	−0.0357585
$657$	0	0
$658$	10.5125	0.409821
$659$	17.2990	0.673872	0.336936	−	0.941528i	$-0.390609\pi$
$659$	17.2990	0.673872	0.336936	+	0.941528i	$0.390609\pi$
$660$	0	0
$661$	29.1629	1.13431	0.567153	−	0.823613i	$-0.308045\pi$
$661$	29.1629	1.13431	0.567153	+	0.823613i	$0.308045\pi$
$662$	−27.9688	−1.08704
$663$	0	0
$664$	7.26710	0.282018
$665$	6.49693	0.251940
$666$	0	0
$667$	−22.7382	−0.880427
$668$	−1.17274	−0.0453747
$669$	0	0
$670$	9.55252	0.369046
$671$	6.87936	0.265575
$672$	0	0
$673$	36.0905	1.39119	0.695593	−	0.718436i	$-0.255142\pi$
$673$	36.0905	1.39119	0.695593	+	0.718436i	$0.255142\pi$
$674$	43.2366	1.66541
$675$	0	0
$676$	−4.54411	−0.174773
$677$	12.7310	0.489293	0.244646	−	0.969612i	$-0.421328\pi$
$677$	12.7310	0.489293	0.244646	+	0.969612i	$0.421328\pi$
$678$	0	0
$679$	−2.15676	−0.0827687
$680$	−7.10382	−0.272419
$681$	0	0
$682$	15.8504	0.606944
$683$	5.47253	0.209401	0.104700	−	0.994504i	$-0.466612\pi$
$683$	5.47253	0.209401	0.104700	+	0.994504i	$0.466612\pi$
$684$	0	0
$685$	−19.6248	−0.749823
$686$	1.53919	0.0587665
$687$	0	0
$688$	2.05929	0.0785097
$689$	−6.06770	−0.231161
$690$	0	0
$691$	30.7226	1.16874	0.584372	−	0.811486i	$-0.301341\pi$
$691$	30.7226	1.16874	0.584372	+	0.811486i	$0.301341\pi$
$692$	6.62514	0.251850
$693$	0	0
$694$	−33.9793	−1.28984
$695$	−0.496928	−0.0188496
$696$	0	0
$697$	0.563198	0.0213326
$698$	28.0338	1.06110
$699$	0	0
$700$	0.369102	0.0139508
$701$	14.7649	0.557661	0.278831	−	0.960340i	$-0.410053\pi$
$701$	14.7649	0.557661	0.278831	+	0.960340i	$0.410053\pi$
$702$	0	0
$703$	73.4883	2.77167
$704$	−6.02893	−0.227224
$705$	0	0
$706$	2.18342	0.0821740
$707$	7.21953	0.271519
$708$	0	0
$709$	−5.63317	−0.211558	−0.105779	−	0.994390i	$-0.533734\pi$
$709$	−5.63317	−0.211558	−0.105779	+	0.994390i	$0.533734\pi$
$710$	17.5753	0.659589
$711$	0	0
$712$	−1.45959	−0.0547004
$713$	−71.7875	−2.68846
$714$	0	0
$715$	0.829914	0.0310370
$716$	−5.80674	−0.217008
$717$	0	0
$718$	−20.1171	−0.750765
$719$	−35.0505	−1.30716	−0.653581	−	0.756856i	$-0.726734\pi$
$719$	−35.0505	−1.30716	−0.653581	+	0.756856i	$0.726734\pi$
$720$	0	0
$721$	3.66701	0.136567
$722$	35.7247	1.32954
$723$	0	0
$724$	−6.20477	−0.230599
$725$	−3.26180	−0.121140
$726$	0	0
$727$	−1.80486	−0.0669385	−0.0334693	−	0.999440i	$-0.510656\pi$
$727$	−1.80486	−0.0669385	−0.0334693	+	0.999440i	$0.510656\pi$
$728$	2.08330	0.0772122
$729$	0	0
$730$	8.72261	0.322838
$731$	−1.26633	−0.0468369
$732$	0	0
$733$	−34.0338	−1.25707	−0.628534	−	0.777782i	$-0.716345\pi$
$733$	−34.0338	−1.25707	−0.628534	+	0.777782i	$0.716345\pi$
$734$	30.4645	1.12447
$735$	0	0
$736$	−14.3798	−0.530046
$737$	−6.20620	−0.228608
$738$	0	0
$739$	−14.1568	−0.520765	−0.260382	−	0.965506i	$-0.583849\pi$
$739$	−14.1568	−0.520765	−0.260382	+	0.965506i	$0.583849\pi$
$740$	4.17501	0.153476
$741$	0	0
$742$	11.2534	0.413125
$743$	−6.62249	−0.242955	−0.121478	−	0.992594i	$-0.538763\pi$
$743$	−6.62249	−0.242955	−0.121478	+	0.992594i	$0.538763\pi$
$744$	0	0
$745$	13.3340	0.488521
$746$	−48.3461	−1.77008
$747$	0	0
$748$	−1.04453	−0.0381917
$749$	9.26180	0.338419
$750$	0	0
$751$	28.6947	1.04709	0.523543	−	0.851999i	$-0.324610\pi$
$751$	28.6947	1.04709	0.523543	+	0.851999i	$0.324610\pi$
$752$	−31.4310	−1.14617
$753$	0	0
$754$	4.16660	0.151738
$755$	−1.05559	−0.0384169
$756$	0	0
$757$	13.3874	0.486572	0.243286	−	0.969955i	$-0.421775\pi$
$757$	13.3874	0.486572	0.243286	+	0.969955i	$0.421775\pi$
$758$	33.7731	1.22669
$759$	0	0
$760$	−16.3090	−0.591589
$761$	−37.3041	−1.35227	−0.676135	−	0.736777i	$-0.736347\pi$
$761$	−37.3041	−1.35227	−0.676135	+	0.736777i	$0.736347\pi$
$762$	0	0
$763$	6.86376	0.248485
$764$	1.07838	0.0390143
$765$	0	0
$766$	−10.7058	−0.386816
$767$	−6.60424	−0.238465
$768$	0	0
$769$	−12.0156	−0.433294	−0.216647	−	0.976250i	$-0.569512\pi$
$769$	−12.0156	−0.433294	−0.216647	+	0.976250i	$0.569512\pi$
$770$	−1.53919	−0.0554685
$771$	0	0
$772$	−8.42612	−0.303263
$773$	−3.20394	−0.115238	−0.0576188	−	0.998339i	$-0.518351\pi$
$773$	−3.20394	−0.115238	−0.0576188	+	0.998339i	$0.518351\pi$
$774$	0	0
$775$	−10.2979	−0.369912
$776$	5.41402	0.194352
$777$	0	0
$778$	−4.25565	−0.152573
$779$	1.29299	0.0463262
$780$	0	0
$781$	−11.4186	−0.408588
$782$	30.3644	1.08583
$783$	0	0
$784$	−4.60197	−0.164356
$785$	−0.496928	−0.0177361
$786$	0	0
$787$	−5.13170	−0.182925	−0.0914627	−	0.995809i	$-0.529154\pi$
$787$	−5.13170	−0.182925	−0.0914627	+	0.995809i	$0.529154\pi$
$788$	−7.28005	−0.259341
$789$	0	0
$790$	−22.3896	−0.796587
$791$	3.75872	0.133645
$792$	0	0
$793$	5.70928	0.202742
$794$	10.6537	0.378085
$795$	0	0
$796$	−3.92654	−0.139173
$797$	−8.86376	−0.313971	−0.156985	−	0.987601i	$-0.550178\pi$
$797$	−8.86376	−0.313971	−0.156985	+	0.987601i	$0.550178\pi$
$798$	0	0
$799$	19.3281	0.683778
$800$	−2.06278	−0.0729303
$801$	0	0
$802$	10.2823	0.363081
$803$	−5.66701	−0.199985
$804$	0	0
$805$	6.97107	0.245698
$806$	13.1545	0.463347
$807$	0	0
$808$	−18.1229	−0.637562
$809$	−16.7915	−0.590359	−0.295179	−	0.955442i	$-0.595379\pi$
$809$	−16.7915	−0.590359	−0.295179	+	0.955442i	$0.595379\pi$
$810$	0	0
$811$	23.9421	0.840722	0.420361	−	0.907357i	$-0.361903\pi$
$811$	23.9421	0.840722	0.420361	+	0.907357i	$0.361903\pi$
$812$	−1.20394	−0.0422499
$813$	0	0
$814$	−17.4101	−0.610225
$815$	1.31124	0.0459309
$816$	0	0
$817$	−2.90725	−0.101712
$818$	23.2066	0.811399
$819$	0	0
$820$	0.0734572	0.00256524
$821$	−15.7054	−0.548122	−0.274061	−	0.961712i	$-0.588367\pi$
$821$	−15.7054	−0.548122	−0.274061	+	0.961712i	$0.588367\pi$
$822$	0	0
$823$	−40.5152	−1.41227	−0.706135	−	0.708077i	$-0.749563\pi$
$823$	−40.5152	−1.41227	−0.706135	+	0.708077i	$0.749563\pi$
$824$	−9.20516	−0.320677
$825$	0	0
$826$	12.2485	0.426179
$827$	−29.6391	−1.03065	−0.515327	−	0.856994i	$-0.672329\pi$
$827$	−29.6391	−1.03065	−0.515327	+	0.856994i	$0.672329\pi$
$828$	0	0
$829$	28.7838	0.999702	0.499851	−	0.866111i	$-0.333388\pi$
$829$	28.7838	0.999702	0.499851	+	0.866111i	$0.333388\pi$
$830$	−4.45589	−0.154666
$831$	0	0
$832$	−5.00349	−0.173465
$833$	2.82991	0.0980507
$834$	0	0
$835$	−3.17727	−0.109954
$836$	−2.39803	−0.0829377
$837$	0	0
$838$	14.6921	0.507529
$839$	11.1350	0.384423	0.192212	−	0.981353i	$-0.438434\pi$
$839$	11.1350	0.384423	0.192212	+	0.981353i	$0.438434\pi$
$840$	0	0
$841$	−18.3607	−0.633127
$842$	−50.3995	−1.73688
$843$	0	0
$844$	2.76940	0.0953267
$845$	−12.3112	−0.423520
$846$	0	0
$847$	1.00000	0.0343604
$848$	−33.6461	−1.15541
$849$	0	0
$850$	4.35577	0.149402
$851$	78.8515	2.70299
$852$	0	0
$853$	2.00719	0.0687248	0.0343624	−	0.999409i	$-0.489060\pi$
$853$	2.00719	0.0687248	0.0343624	+	0.999409i	$0.489060\pi$
$854$	−10.5886	−0.362336
$855$	0	0
$856$	−23.2495	−0.794652
$857$	−21.2013	−0.724222	−0.362111	−	0.932135i	$-0.617944\pi$
$857$	−21.2013	−0.724222	−0.362111	+	0.932135i	$0.617944\pi$
$858$	0	0
$859$	38.6959	1.32029	0.660144	−	0.751139i	$-0.270495\pi$
$859$	38.6959	1.32029	0.660144	+	0.751139i	$0.270495\pi$
$860$	−0.165166	−0.00563211
$861$	0	0
$862$	35.4459	1.20729
$863$	−38.3728	−1.30623	−0.653113	−	0.757261i	$-0.726537\pi$
$863$	−38.3728	−1.30623	−0.653113	+	0.757261i	$0.726537\pi$
$864$	0	0
$865$	17.9493	0.610295
$866$	−52.8371	−1.79548
$867$	0	0
$868$	−3.80098	−0.129014
$869$	14.5464	0.493452
$870$	0	0
$871$	−5.15061	−0.174522
$872$	−17.2298	−0.583476
$873$	0	0
$874$	69.7107	2.35800
$875$	1.00000	0.0338062
$876$	0	0
$877$	−32.3195	−1.09135	−0.545676	−	0.837997i	$-0.683727\pi$
$877$	−32.3195	−1.09135	−0.545676	+	0.837997i	$0.683727\pi$
$878$	−25.5441	−0.862072
$879$	0	0
$880$	4.60197	0.155132
$881$	45.3217	1.52693	0.763464	−	0.645850i	$-0.223497\pi$
$881$	45.3217	1.52693	0.763464	+	0.645850i	$0.223497\pi$
$882$	0	0
$883$	42.5197	1.43090	0.715451	−	0.698663i	$-0.246221\pi$
$883$	42.5197	1.43090	0.715451	+	0.698663i	$0.246221\pi$
$884$	−0.866868	−0.0291559
$885$	0	0
$886$	−16.8045	−0.564557
$887$	44.6947	1.50070	0.750351	−	0.661040i	$-0.229885\pi$
$887$	44.6947	1.50070	0.750351	+	0.661040i	$0.229885\pi$
$888$	0	0
$889$	−21.5441	−0.722566
$890$	0.894960	0.0299991
$891$	0	0
$892$	8.87095	0.297021
$893$	44.3735	1.48490
$894$	0	0
$895$	−15.7321	−0.525865
$896$	13.4052	0.447837
$897$	0	0
$898$	37.8636	1.26352
$899$	33.5897	1.12028
$900$	0	0
$901$	20.6902	0.689290
$902$	−0.306323	−0.0101994
$903$	0	0
$904$	−9.43537	−0.313816
$905$	−16.8104	−0.558798
$906$	0	0
$907$	5.28912	0.175622	0.0878111	−	0.996137i	$-0.472013\pi$
$907$	5.28912	0.175622	0.0878111	+	0.996137i	$0.472013\pi$
$908$	−0.544946	−0.0180847
$909$	0	0
$910$	−1.27739	−0.0423452
$911$	47.7321	1.58143	0.790717	−	0.612182i	$-0.209708\pi$
$911$	47.7321	1.58143	0.790717	+	0.612182i	$0.209708\pi$
$912$	0	0
$913$	2.89496	0.0958092
$914$	−56.9276	−1.88300
$915$	0	0
$916$	2.97948	0.0984448
$917$	12.6803	0.418742
$918$	0	0
$919$	45.6886	1.50713	0.753564	−	0.657375i	$-0.228333\pi$
$919$	45.6886	1.50713	0.753564	+	0.657375i	$0.228333\pi$
$920$	−17.4992	−0.576931
$921$	0	0
$922$	−57.5033	−1.89377
$923$	−9.47641	−0.311920
$924$	0	0
$925$	11.3112	0.371911
$926$	10.8599	0.356878
$927$	0	0
$928$	6.72836	0.220869
$929$	−35.3874	−1.16102	−0.580511	−	0.814253i	$-0.697147\pi$
$929$	−35.3874	−1.16102	−0.580511	+	0.814253i	$0.697147\pi$
$930$	0	0
$931$	6.49693	0.212928
$932$	2.37978	0.0779523
$933$	0	0
$934$	30.5757	1.00047
$935$	−2.82991	−0.0925481
$936$	0	0
$937$	−23.2378	−0.759145	−0.379573	−	0.925162i	$-0.623929\pi$
$937$	−23.2378	−0.759145	−0.379573	+	0.925162i	$0.623929\pi$
$938$	9.55252	0.311901
$939$	0	0
$940$	2.52094	0.0822239
$941$	−29.2651	−0.954015	−0.477008	−	0.878899i	$-0.658279\pi$
$941$	−29.2651	−0.954015	−0.477008	+	0.878899i	$0.658279\pi$
$942$	0	0
$943$	1.38735	0.0451785
$944$	−36.6213	−1.19192
$945$	0	0
$946$	0.688756	0.0223934
$947$	5.55705	0.180580	0.0902900	−	0.995916i	$-0.471221\pi$
$947$	5.55705	0.180580	0.0902900	+	0.995916i	$0.471221\pi$
$948$	0	0
$949$	−4.70313	−0.152670
$950$	10.0000	0.324443
$951$	0	0
$952$	−7.10382	−0.230236
$953$	11.7548	0.380777	0.190388	−	0.981709i	$-0.439025\pi$
$953$	11.7548	0.380777	0.190388	+	0.981709i	$0.439025\pi$
$954$	0	0
$955$	2.92162	0.0945415
$956$	3.02052	0.0976906
$957$	0	0
$958$	−9.84778	−0.318167
$959$	−19.6248	−0.633716
$960$	0	0
$961$	75.0470	2.42087
$962$	−14.4489	−0.465852
$963$	0	0
$964$	4.02544	0.129651
$965$	−22.8287	−0.734882
$966$	0	0
$967$	33.2267	1.06850	0.534250	−	0.845327i	$-0.320594\pi$
$967$	33.2267	1.06850	0.534250	+	0.845327i	$0.320594\pi$
$968$	−2.51026	−0.0806828
$969$	0	0
$970$	−3.31965	−0.106588
$971$	−3.16621	−0.101609	−0.0508043	−	0.998709i	$-0.516178\pi$
$971$	−3.16621	−0.101609	−0.0508043	+	0.998709i	$0.516178\pi$
$972$	0	0
$973$	−0.496928	−0.0159308
$974$	−39.7237	−1.27283
$975$	0	0
$976$	31.6586	1.01337
$977$	32.4391	1.03782	0.518909	−	0.854830i	$-0.326338\pi$
$977$	32.4391	1.03782	0.518909	+	0.854830i	$0.326338\pi$
$978$	0	0
$979$	−0.581449	−0.0185832
$980$	0.369102	0.0117905
$981$	0	0
$982$	26.4261	0.843292
$983$	30.1496	0.961622	0.480811	−	0.876824i	$-0.340342\pi$
$983$	30.1496	0.961622	0.480811	+	0.876824i	$0.340342\pi$
$984$	0	0
$985$	−19.7237	−0.628448
$986$	−14.2076	−0.452463
$987$	0	0
$988$	−1.99016	−0.0633154
$989$	−3.11942	−0.0991916
$990$	0	0
$991$	−21.4452	−0.681230	−0.340615	−	0.940203i	$-0.610635\pi$
$991$	−21.4452	−0.681230	−0.340615	+	0.940203i	$0.610635\pi$
$992$	21.2423	0.674444
$993$	0	0
$994$	17.5753	0.557455
$995$	−10.6381	−0.337250
$996$	0	0
$997$	39.6547	1.25588	0.627939	−	0.778263i	$-0.283899\pi$
$997$	39.6547	1.25588	0.627939	+	0.778263i	$0.283899\pi$
$998$	−20.2602	−0.641325
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3465.2.a.bi.1.2		3
3.2	odd	2		385.2.a.e.1.2	✓	3
12.11	even	2		6160.2.a.bo.1.3		3
15.2	even	4		1925.2.b.m.1849.2		6
15.8	even	4		1925.2.b.m.1849.5		6
15.14	odd	2		1925.2.a.w.1.2		3
21.20	even	2		2695.2.a.h.1.2		3
33.32	even	2		4235.2.a.p.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
385.2.a.e.1.2	✓	3	3.2	odd	2
1925.2.a.w.1.2		3	15.14	odd	2
1925.2.b.m.1849.2		6	15.2	even	4
1925.2.b.m.1849.5		6	15.8	even	4
2695.2.a.h.1.2		3	21.20	even	2
3465.2.a.bi.1.2		3	1.1	even	1	trivial
4235.2.a.p.1.2		3	33.32	even	2
6160.2.a.bo.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)

\(f(q)\)	\(=\)	\(q+1.53919 q^{2} +0.369102 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.51026 q^{8} +O(q^{10})\)	\(q+1.53919 q^{2} +0.369102 q^{4} +1.00000 q^{5} +1.00000 q^{7} -2.51026 q^{8} +1.53919 q^{10} -1.00000 q^{11} -0.829914 q^{13} +1.53919 q^{14} -4.60197 q^{16} +2.82991 q^{17} +6.49693 q^{19} +0.369102 q^{20} -1.53919 q^{22} +6.97107 q^{23} +1.00000 q^{25} -1.27739 q^{26} +0.369102 q^{28} -3.26180 q^{29} -10.2979 q^{31} -2.06278 q^{32} +4.35577 q^{34} +1.00000 q^{35} +11.3112 q^{37} +10.0000 q^{38} -2.51026 q^{40} +0.199016 q^{41} -0.447480 q^{43} -0.369102 q^{44} +10.7298 q^{46} +6.82991 q^{47} +1.00000 q^{49} +1.53919 q^{50} -0.306323 q^{52} +7.31124 q^{53} -1.00000 q^{55} -2.51026 q^{56} -5.02052 q^{58} +7.95774 q^{59} -6.87936 q^{61} -15.8504 q^{62} +6.02893 q^{64} -0.829914 q^{65} +6.20620 q^{67} +1.04453 q^{68} +1.53919 q^{70} +11.4186 q^{71} +5.66701 q^{73} +17.4101 q^{74} +2.39803 q^{76} -1.00000 q^{77} -14.5464 q^{79} -4.60197 q^{80} +0.306323 q^{82} -2.89496 q^{83} +2.82991 q^{85} -0.688756 q^{86} +2.51026 q^{88} +0.581449 q^{89} -0.829914 q^{91} +2.57304 q^{92} +10.5125 q^{94} +6.49693 q^{95} -2.15676 q^{97} +1.53919 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} + 3 q^{7} + 9 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 + 3 * q^7 + 9 * q^8	\( 3 q + 3 q^{2} + 5 q^{4} + 3 q^{5} + 3 q^{7} + 9 q^{8} + 3 q^{10} - 3 q^{11} - 8 q^{13} + 3 q^{14} + 5 q^{16} + 14 q^{17} + 2 q^{19} + 5 q^{20} - 3 q^{22} + 6 q^{23} + 3 q^{25} - 10 q^{26} + 5 q^{28} - 2 q^{29} - 4 q^{31} + 11 q^{32} + 16 q^{34} + 3 q^{35} + 8 q^{37} + 30 q^{38} + 9 q^{40} + 10 q^{41} - 2 q^{43} - 5 q^{44} - 8 q^{46} + 26 q^{47} + 3 q^{49} + 3 q^{50} - 22 q^{52} - 4 q^{53} - 3 q^{55} + 9 q^{56} + 18 q^{58} + 8 q^{59} - 8 q^{61} - 20 q^{62} + 33 q^{64} - 8 q^{65} - 6 q^{67} + 32 q^{68} + 3 q^{70} + 20 q^{71} - 6 q^{73} - 10 q^{74} + 26 q^{76} - 3 q^{77} - 8 q^{79} + 5 q^{80} + 22 q^{82} - 10 q^{83} + 14 q^{85} - 28 q^{86} - 9 q^{88} + 16 q^{89} - 8 q^{91} - 26 q^{92} + 28 q^{94} + 2 q^{95} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 5 * q^4 + 3 * q^5 + 3 * q^7 + 9 * q^8 + 3 * q^10 - 3 * q^11 - 8 * q^13 + 3 * q^14 + 5 * q^16 + 14 * q^17 + 2 * q^19 + 5 * q^20 - 3 * q^22 + 6 * q^23 + 3 * q^25 - 10 * q^26 + 5 * q^28 - 2 * q^29 - 4 * q^31 + 11 * q^32 + 16 * q^34 + 3 * q^35 + 8 * q^37 + 30 * q^38 + 9 * q^40 + 10 * q^41 - 2 * q^43 - 5 * q^44 - 8 * q^46 + 26 * q^47 + 3 * q^49 + 3 * q^50 - 22 * q^52 - 4 * q^53 - 3 * q^55 + 9 * q^56 + 18 * q^58 + 8 * q^59 - 8 * q^61 - 20 * q^62 + 33 * q^64 - 8 * q^65 - 6 * q^67 + 32 * q^68 + 3 * q^70 + 20 * q^71 - 6 * q^73 - 10 * q^74 + 26 * q^76 - 3 * q^77 - 8 * q^79 + 5 * q^80 + 22 * q^82 - 10 * q^83 + 14 * q^85 - 28 * q^86 - 9 * q^88 + 16 * q^89 - 8 * q^91 - 26 * q^92 + 28 * q^94 + 2 * q^95 + 3 * q^98

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