LMFDB - Embedded newform 3465.2.a.bl.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:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.546295$ 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)

$f(q)$	$=$	$q+0.546295 q^{2} -1.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.02214 q^{8} +O(q^{10})$	\(q+0.546295 q^{2} -1.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.02214 q^{8} +0.546295 q^{10} +1.00000 q^{11} +4.56844 q^{13} -0.546295 q^{14} +2.29844 q^{16} -2.70156 q^{17} -8.02362 q^{19} -1.70156 q^{20} +0.546295 q^{22} +3.52790 q^{23} +1.00000 q^{25} +2.49571 q^{26} +1.70156 q^{28} -0.133124 q^{29} +2.33897 q^{31} +5.29991 q^{32} -1.47585 q^{34} -1.00000 q^{35} +6.75362 q^{37} -4.38326 q^{38} -2.02214 q^{40} -10.9831 q^{41} +9.45518 q^{43} -1.70156 q^{44} +1.92728 q^{46} +0.649507 q^{47} +1.00000 q^{49} +0.546295 q^{50} -7.77348 q^{52} -11.8384 q^{53} +1.00000 q^{55} +2.02214 q^{56} -0.0727248 q^{58} +0.959466 q^{59} +13.1162 q^{61} +1.27777 q^{62} -1.70156 q^{64} +4.56844 q^{65} +6.41464 q^{67} +4.59688 q^{68} -0.546295 q^{70} +2.56844 q^{71} +12.5400 q^{73} +3.68947 q^{74} +13.6527 q^{76} -1.00000 q^{77} +2.56844 q^{79} +2.29844 q^{80} -6.00000 q^{82} +7.75737 q^{83} -2.70156 q^{85} +5.16531 q^{86} -2.02214 q^{88} -13.3741 q^{89} -4.56844 q^{91} -6.00295 q^{92} +0.354822 q^{94} -8.02362 q^{95} -1.78581 q^{97} +0.546295 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7}+O(q^{10}) $ 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7	$ 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7} + 4 q^{11} + 8 q^{13} + 22 q^{16} + 2 q^{17} + 10 q^{19} + 6 q^{20} + 2 q^{23} + 4 q^{25} - 20 q^{26} - 6 q^{28} + 2 q^{29} + 24 q^{31} - 4 q^{35} + 8 q^{37} - 16 q^{38} + 6 q^{43} + 6 q^{44} - 12 q^{46} - 4 q^{47} + 4 q^{49} + 12 q^{52} - 14 q^{53} + 4 q^{55} - 20 q^{58} + 2 q^{59} + 6 q^{61} - 8 q^{62} + 6 q^{64} + 8 q^{65} - 8 q^{67} + 44 q^{68} + 4 q^{73} + 36 q^{74} + 56 q^{76} - 4 q^{77} + 22 q^{80} - 24 q^{82} - 6 q^{83} + 2 q^{85} + 36 q^{86} - 18 q^{89} - 8 q^{91} + 44 q^{92} - 36 q^{94} + 10 q^{95} - 6 q^{97}+O(q^{100}) $ 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7 + 4 * q^11 + 8 * q^13 + 22 * q^16 + 2 * q^17 + 10 * q^19 + 6 * q^20 + 2 * q^23 + 4 * q^25 - 20 * q^26 - 6 * q^28 + 2 * q^29 + 24 * q^31 - 4 * q^35 + 8 * q^37 - 16 * q^38 + 6 * q^43 + 6 * q^44 - 12 * q^46 - 4 * q^47 + 4 * q^49 + 12 * q^52 - 14 * q^53 + 4 * q^55 - 20 * q^58 + 2 * q^59 + 6 * q^61 - 8 * q^62 + 6 * q^64 + 8 * q^65 - 8 * q^67 + 44 * q^68 + 4 * q^73 + 36 * q^74 + 56 * q^76 - 4 * q^77 + 22 * q^80 - 24 * q^82 - 6 * q^83 + 2 * q^85 + 36 * q^86 - 18 * q^89 - 8 * q^91 + 44 * q^92 - 36 * q^94 + 10 * q^95 - 6 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.546295	0.386289	0.193144	−	0.981170i	$-0.438131\pi$
$2$	0.546295	0.386289	0.193144	+	0.981170i	$0.438131\pi$
$3$	0	0
$3$	0	0
$4$	−1.70156	−0.850781
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.02214	−0.714936
$9$	0	0
$10$	0.546295	0.172754
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.56844	1.26706	0.633528	−	0.773719i	$-0.281606\pi$
$13$	4.56844	1.26706	0.633528	+	0.773719i	$0.281606\pi$
$14$	−0.546295	−0.146003
$15$	0	0
$16$	2.29844	0.574609
$17$	−2.70156	−0.655225	−0.327613	−	0.944812i	$-0.606244\pi$
$17$	−2.70156	−0.655225	−0.327613	+	0.944812i	$0.606244\pi$
$18$	0	0
$19$	−8.02362	−1.84074	−0.920372	−	0.391044i	$-0.872114\pi$
$19$	−8.02362	−1.84074	−0.920372	+	0.391044i	$0.872114\pi$
$20$	−1.70156	−0.380481
$21$	0	0
$22$	0.546295	0.116470
$23$	3.52790	0.735619	0.367809	−	0.929901i	$-0.380108\pi$
$23$	3.52790	0.735619	0.367809	+	0.929901i	$0.380108\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.49571	0.489450
$27$	0	0
$28$	1.70156	0.321565
$29$	−0.133124	−0.0247205	−0.0123602	−	0.999924i	$-0.503934\pi$
$29$	−0.133124	−0.0247205	−0.0123602	+	0.999924i	$0.503934\pi$
$30$	0	0
$31$	2.33897	0.420092	0.210046	−	0.977692i	$-0.432639\pi$
$31$	2.33897	0.420092	0.210046	+	0.977692i	$0.432639\pi$
$32$	5.29991	0.936901
$33$	0	0
$34$	−1.47585	−0.253106
$35$	−1.00000	−0.169031
$36$	0	0
$37$	6.75362	1.11029	0.555144	−	0.831754i	$-0.312663\pi$
$37$	6.75362	1.11029	0.555144	+	0.831754i	$0.312663\pi$
$38$	−4.38326	−0.711059
$39$	0	0
$40$	−2.02214	−0.319729
$41$	−10.9831	−1.71527	−0.857635	−	0.514259i	$-0.828067\pi$
$41$	−10.9831	−1.71527	−0.857635	+	0.514259i	$0.828067\pi$
$42$	0	0
$43$	9.45518	1.44190	0.720951	−	0.692986i	$-0.243705\pi$
$43$	9.45518	1.44190	0.720951	+	0.692986i	$0.243705\pi$
$44$	−1.70156	−0.256520
$45$	0	0
$46$	1.92728	0.284161
$47$	0.649507	0.0947403	0.0473702	−	0.998877i	$-0.484916\pi$
$47$	0.649507	0.0947403	0.0473702	+	0.998877i	$0.484916\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.546295	0.0772577
$51$	0	0
$52$	−7.77348	−1.07799
$53$	−11.8384	−1.62613	−0.813067	−	0.582170i	$-0.802204\pi$
$53$	−11.8384	−1.62613	−0.813067	+	0.582170i	$0.802204\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	2.02214	0.270220
$57$	0	0
$58$	−0.0727248	−0.00954923
$59$	0.959466	0.124912	0.0624559	−	0.998048i	$-0.480107\pi$
$59$	0.959466	0.124912	0.0624559	+	0.998048i	$0.480107\pi$
$60$	0	0
$61$	13.1162	1.67936	0.839679	−	0.543083i	$-0.182743\pi$
$61$	13.1162	1.67936	0.839679	+	0.543083i	$0.182743\pi$
$62$	1.27777	0.162277
$63$	0	0
$64$	−1.70156	−0.212695
$65$	4.56844	0.566645
$66$	0	0
$67$	6.41464	0.783674	0.391837	−	0.920035i	$-0.371840\pi$
$67$	6.41464	0.783674	0.391837	+	0.920035i	$0.371840\pi$
$68$	4.59688	0.557453
$69$	0	0
$70$	−0.546295	−0.0652947
$71$	2.56844	0.304818	0.152409	−	0.988318i	$-0.451297\pi$
$71$	2.56844	0.304818	0.152409	+	0.988318i	$0.451297\pi$
$72$	0	0
$73$	12.5400	1.46770	0.733848	−	0.679314i	$-0.237722\pi$
$73$	12.5400	1.46770	0.733848	+	0.679314i	$0.237722\pi$
$74$	3.68947	0.428892
$75$	0	0
$76$	13.6527	1.56607
$77$	−1.00000	−0.113961
$78$	0	0
$79$	2.56844	0.288972	0.144486	−	0.989507i	$-0.453847\pi$
$79$	2.56844	0.288972	0.144486	+	0.989507i	$0.453847\pi$
$80$	2.29844	0.256973
$81$	0	0
$82$	−6.00000	−0.662589
$83$	7.75737	0.851482	0.425741	−	0.904845i	$-0.360013\pi$
$83$	7.75737	0.851482	0.425741	+	0.904845i	$0.360013\pi$
$84$	0	0
$85$	−2.70156	−0.293026
$86$	5.16531	0.556990
$87$	0	0
$88$	−2.02214	−0.215561
$89$	−13.3741	−1.41765	−0.708826	−	0.705383i	$-0.750775\pi$
$89$	−13.3741	−1.41765	−0.708826	+	0.705383i	$0.750775\pi$
$90$	0	0
$91$	−4.56844	−0.478902
$92$	−6.00295	−0.625851
$93$	0	0
$94$	0.354822	0.0365971
$95$	−8.02362	−0.823206
$96$	0	0
$97$	−1.78581	−0.181321	−0.0906606	−	0.995882i	$-0.528898\pi$
$97$	−1.78581	−0.181321	−0.0906606	+	0.995882i	$0.528898\pi$
$98$	0.546295	0.0551841
$99$	0	0
$100$	−1.70156	−0.170156
$101$	11.6326	1.15749	0.578743	−	0.815510i	$-0.303543\pi$
$101$	11.6326	1.15749	0.578743	+	0.815510i	$0.303543\pi$
$102$	0	0
$103$	−9.18893	−0.905412	−0.452706	−	0.891660i	$-0.649541\pi$
$103$	−9.18893	−0.905412	−0.452706	+	0.891660i	$0.649541\pi$
$104$	−9.23804	−0.905864
$105$	0	0
$106$	−6.46728	−0.628157
$107$	13.0199	1.25868	0.629339	−	0.777131i	$-0.283326\pi$
$107$	13.0199	1.25868	0.629339	+	0.777131i	$0.283326\pi$
$108$	0	0
$109$	0.109506	0.0104888	0.00524439	−	0.999986i	$-0.498331\pi$
$109$	0.109506	0.0104888	0.00524439	+	0.999986i	$0.498331\pi$
$110$	0.546295	0.0520872
$111$	0	0
$112$	−2.29844	−0.217182
$113$	3.83844	0.361090	0.180545	−	0.983567i	$-0.442214\pi$
$113$	3.83844	0.361090	0.180545	+	0.983567i	$0.442214\pi$
$114$	0	0
$115$	3.52790	0.328979
$116$	0.226518	0.0210317
$117$	0	0
$118$	0.524151	0.0482520
$119$	2.70156	0.247652
$120$	0	0
$121$	1.00000	0.0909091
$122$	7.16531	0.648717
$123$	0	0
$124$	−3.97991	−0.357406
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.45518	−0.484069	−0.242034	−	0.970268i	$-0.577815\pi$
$127$	−5.45518	−0.484069	−0.242034	+	0.970268i	$0.577815\pi$
$128$	−11.5294	−1.01906
$129$	0	0
$130$	2.49571	0.218889
$131$	21.1084	1.84425	0.922126	−	0.386889i	$-0.126450\pi$
$131$	21.1084	1.84425	0.922126	+	0.386889i	$0.126450\pi$
$132$	0	0
$133$	8.02362	0.695736
$134$	3.50429	0.302724
$135$	0	0
$136$	5.46295	0.468444
$137$	3.40312	0.290749	0.145374	−	0.989377i	$-0.453561\pi$
$137$	3.40312	0.290749	0.145374	+	0.989377i	$0.453561\pi$
$138$	0	0
$139$	21.2410	1.80164	0.900818	−	0.434196i	$-0.142967\pi$
$139$	21.2410	1.80164	0.900818	+	0.434196i	$0.142967\pi$
$140$	1.70156	0.143808
$141$	0	0
$142$	1.40312	0.117748
$143$	4.56844	0.382032
$144$	0	0
$145$	−0.133124	−0.0110553
$146$	6.85054	0.566954
$147$	0	0
$148$	−11.4917	−0.944612
$149$	11.5072	0.942709	0.471355	−	0.881944i	$-0.343765\pi$
$149$	11.5072	0.942709	0.471355	+	0.881944i	$0.343765\pi$
$150$	0	0
$151$	7.58830	0.617527	0.308764	−	0.951139i	$-0.400085\pi$
$151$	7.58830	0.617527	0.308764	+	0.951139i	$0.400085\pi$
$152$	16.2249	1.31601
$153$	0	0
$154$	−0.546295	−0.0440217
$155$	2.33897	0.187871
$156$	0	0
$157$	−12.8583	−1.02620	−0.513102	−	0.858328i	$-0.671504\pi$
$157$	−12.8583	−1.02620	−0.513102	+	0.858328i	$0.671504\pi$
$158$	1.40312	0.111627
$159$	0	0
$160$	5.29991	0.418995
$161$	−3.52790	−0.278038
$162$	0	0
$163$	22.6525	1.77428	0.887139	−	0.461503i	$-0.152690\pi$
$163$	22.6525	1.77428	0.887139	+	0.461503i	$0.152690\pi$
$164$	18.6884	1.45932
$165$	0	0
$166$	4.23781	0.328918
$167$	3.21795	0.249012	0.124506	−	0.992219i	$-0.460265\pi$
$167$	3.21795	0.249012	0.124506	+	0.992219i	$0.460265\pi$
$168$	0	0
$169$	7.87063	0.605433
$170$	−1.47585	−0.113192
$171$	0	0
$172$	−16.0886	−1.22674
$173$	18.2094	1.38443	0.692216	−	0.721690i	$-0.256634\pi$
$173$	18.2094	1.38443	0.692216	+	0.721690i	$0.256634\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	2.29844	0.173251
$177$	0	0
$178$	−7.30621	−0.547623
$179$	4.75362	0.355302	0.177651	−	0.984094i	$-0.443150\pi$
$179$	4.75362	0.355302	0.177651	+	0.984094i	$0.443150\pi$
$180$	0	0
$181$	0.347316	0.0258158	0.0129079	−	0.999917i	$-0.495891\pi$
$181$	0.347316	0.0258158	0.0129079	+	0.999917i	$0.495891\pi$
$182$	−2.49571	−0.184995
$183$	0	0
$184$	−7.13393	−0.525920
$185$	6.75362	0.496536
$186$	0	0
$187$	−2.70156	−0.197558
$188$	−1.10518	−0.0806033
$189$	0	0
$190$	−4.38326	−0.317995
$191$	−11.8379	−0.856558	−0.428279	−	0.903647i	$-0.640880\pi$
$191$	−11.8379	−0.856558	−0.428279	+	0.903647i	$0.640880\pi$
$192$	0	0
$193$	−4.14840	−0.298608	−0.149304	−	0.988791i	$-0.547703\pi$
$193$	−4.14840	−0.298608	−0.149304	+	0.988791i	$0.547703\pi$
$194$	−0.975577	−0.0700424
$195$	0	0
$196$	−1.70156	−0.121540
$197$	−19.3663	−1.37979	−0.689897	−	0.723907i	$-0.742344\pi$
$197$	−19.3663	−1.37979	−0.689897	+	0.723907i	$0.742344\pi$
$198$	0	0
$199$	13.2622	0.940135	0.470067	−	0.882631i	$-0.344230\pi$
$199$	13.2622	0.940135	0.470067	+	0.882631i	$0.344230\pi$
$200$	−2.02214	−0.142987
$201$	0	0
$202$	6.35482	0.447124
$203$	0.133124	0.00934345
$204$	0	0
$205$	−10.9831	−0.767092
$206$	−5.01986	−0.349751
$207$	0	0
$208$	10.5003	0.728063
$209$	−8.02362	−0.555005
$210$	0	0
$211$	2.54000	0.174861	0.0874304	−	0.996171i	$-0.472134\pi$
$211$	2.54000	0.174861	0.0874304	+	0.996171i	$0.472134\pi$
$212$	20.1438	1.38348
$213$	0	0
$214$	7.11268	0.486213
$215$	9.45518	0.644838
$216$	0	0
$217$	−2.33897	−0.158780
$218$	0.0598226	0.00405170
$219$	0	0
$220$	−1.70156	−0.114719
$221$	−12.3419	−0.830207
$222$	0	0
$223$	20.5110	1.37352	0.686759	−	0.726886i	$-0.259033\pi$
$223$	20.5110	1.37352	0.686759	+	0.726886i	$0.259033\pi$
$224$	−5.29991	−0.354115
$225$	0	0
$226$	2.09692	0.139485
$227$	−19.6119	−1.30169	−0.650844	−	0.759211i	$-0.725585\pi$
$227$	−19.6119	−1.30169	−0.650844	+	0.759211i	$0.725585\pi$
$228$	0	0
$229$	20.4230	1.34959	0.674795	−	0.738006i	$-0.264232\pi$
$229$	20.4230	1.34959	0.674795	+	0.738006i	$0.264232\pi$
$230$	1.92728	0.127081
$231$	0	0
$232$	0.269195	0.0176735
$233$	17.3304	1.13535	0.567676	−	0.823252i	$-0.307843\pi$
$233$	17.3304	1.13535	0.567676	+	0.823252i	$0.307843\pi$
$234$	0	0
$235$	0.649507	0.0423692
$236$	−1.63259	−0.106273
$237$	0	0
$238$	1.47585	0.0956651
$239$	11.6404	0.752952	0.376476	−	0.926426i	$-0.377136\pi$
$239$	11.6404	0.752952	0.376476	+	0.926426i	$0.377136\pi$
$240$	0	0
$241$	−22.5429	−1.45212	−0.726059	−	0.687632i	$-0.758650\pi$
$241$	−22.5429	−1.45212	−0.726059	+	0.687632i	$0.758650\pi$
$242$	0.546295	0.0351172
$243$	0	0
$244$	−22.3180	−1.42877
$245$	1.00000	0.0638877
$246$	0	0
$247$	−36.6554	−2.33233
$248$	−4.72974	−0.300339
$249$	0	0
$250$	0.546295	0.0345507
$251$	−11.4474	−0.722554	−0.361277	−	0.932458i	$-0.617659\pi$
$251$	−11.4474	−0.722554	−0.361277	+	0.932458i	$0.617659\pi$
$252$	0	0
$253$	3.52790	0.221797
$254$	−2.98014	−0.186990
$255$	0	0
$256$	−2.89531	−0.180957
$257$	28.6157	1.78500	0.892498	−	0.451051i	$-0.148951\pi$
$257$	28.6157	1.78500	0.892498	+	0.451051i	$0.148951\pi$
$258$	0	0
$259$	−6.75362	−0.419649
$260$	−7.77348	−0.482091
$261$	0	0
$262$	11.5314	0.712414
$263$	8.67794	0.535105	0.267552	−	0.963543i	$-0.413785\pi$
$263$	8.67794	0.535105	0.267552	+	0.963543i	$0.413785\pi$
$264$	0	0
$265$	−11.8384	−0.727230
$266$	4.38326	0.268755
$267$	0	0
$268$	−10.9149	−0.666735
$269$	−29.8911	−1.82249	−0.911245	−	0.411864i	$-0.864878\pi$
$269$	−29.8911	−1.82249	−0.911245	+	0.411864i	$0.864878\pi$
$270$	0	0
$271$	20.9678	1.27370	0.636852	−	0.770986i	$-0.280236\pi$
$271$	20.9678	1.27370	0.636852	+	0.770986i	$0.280236\pi$
$272$	−6.20937	−0.376499
$273$	0	0
$274$	1.85911	0.112313
$275$	1.00000	0.0603023
$276$	0	0
$277$	−13.7137	−0.823974	−0.411987	−	0.911190i	$-0.635165\pi$
$277$	−13.7137	−0.823974	−0.411987	+	0.911190i	$0.635165\pi$
$278$	11.6038	0.695952
$279$	0	0
$280$	2.02214	0.120846
$281$	−4.91036	−0.292927	−0.146464	−	0.989216i	$-0.546789\pi$
$281$	−4.91036	−0.292927	−0.146464	+	0.989216i	$0.546789\pi$
$282$	0	0
$283$	−25.4788	−1.51456	−0.757279	−	0.653092i	$-0.773472\pi$
$283$	−25.4788	−1.51456	−0.757279	+	0.653092i	$0.773472\pi$
$284$	−4.37036	−0.259333
$285$	0	0
$286$	2.49571	0.147575
$287$	10.9831	0.648311
$288$	0	0
$289$	−9.70156	−0.570680
$290$	−0.0727248	−0.00427055
$291$	0	0
$292$	−21.3376	−1.24869
$293$	2.16907	0.126718	0.0633591	−	0.997991i	$-0.479819\pi$
$293$	2.16907	0.126718	0.0633591	+	0.997991i	$0.479819\pi$
$294$	0	0
$295$	0.959466	0.0558622
$296$	−13.6568	−0.793784
$297$	0	0
$298$	6.28634	0.364158
$299$	16.1170	0.932071
$300$	0	0
$301$	−9.45518	−0.544987
$302$	4.14545	0.238544
$303$	0	0
$304$	−18.4418	−1.05771
$305$	13.1162	0.751032
$306$	0	0
$307$	12.3548	0.705127	0.352563	−	0.935788i	$-0.385310\pi$
$307$	12.3548	0.705127	0.352563	+	0.935788i	$0.385310\pi$
$308$	1.70156	0.0969555
$309$	0	0
$310$	1.27777	0.0725724
$311$	−22.6073	−1.28194	−0.640972	−	0.767564i	$-0.721469\pi$
$311$	−22.6073	−1.28194	−0.640972	+	0.767564i	$0.721469\pi$
$312$	0	0
$313$	−34.2614	−1.93657	−0.968285	−	0.249848i	$-0.919620\pi$
$313$	−34.2614	−1.93657	−0.968285	+	0.249848i	$0.919620\pi$
$314$	−7.02442	−0.396411
$315$	0	0
$316$	−4.37036	−0.245852
$317$	−8.28929	−0.465573	−0.232786	−	0.972528i	$-0.574784\pi$
$317$	−8.28929	−0.465573	−0.232786	+	0.972528i	$0.574784\pi$
$318$	0	0
$319$	−0.133124	−0.00745350
$320$	−1.70156	−0.0951202
$321$	0	0
$322$	−1.92728	−0.107403
$323$	21.6763	1.20610
$324$	0	0
$325$	4.56844	0.253411
$326$	12.3749	0.685383
$327$	0	0
$328$	22.2094	1.22631
$329$	−0.649507	−0.0358085
$330$	0	0
$331$	21.4267	1.17772	0.588860	−	0.808235i	$-0.299577\pi$
$331$	21.4267	1.17772	0.588860	+	0.808235i	$0.299577\pi$
$332$	−13.1996	−0.724425
$333$	0	0
$334$	1.75795	0.0961906
$335$	6.41464	0.350469
$336$	0	0
$337$	−20.2585	−1.10355	−0.551775	−	0.833993i	$-0.686049\pi$
$337$	−20.2585	−1.10355	−0.551775	+	0.833993i	$0.686049\pi$
$338$	4.29968	0.233872
$339$	0	0
$340$	4.59688	0.249301
$341$	2.33897	0.126662
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−19.1197	−1.03087
$345$	0	0
$346$	9.94768	0.534791
$347$	23.9431	1.28533	0.642667	−	0.766146i	$-0.277828\pi$
$347$	23.9431	1.28533	0.642667	+	0.766146i	$0.277828\pi$
$348$	0	0
$349$	−25.5751	−1.36901	−0.684503	−	0.729010i	$-0.739981\pi$
$349$	−25.5751	−1.36901	−0.684503	+	0.729010i	$0.739981\pi$
$350$	−0.546295	−0.0292007
$351$	0	0
$352$	5.29991	0.282486
$353$	−0.330628	−0.0175976	−0.00879878	−	0.999961i	$-0.502801\pi$
$353$	−0.330628	−0.0175976	−0.00879878	+	0.999961i	$0.502801\pi$
$354$	0	0
$355$	2.56844	0.136319
$356$	22.7569	1.20611
$357$	0	0
$358$	2.59688	0.137249
$359$	23.6248	1.24687	0.623435	−	0.781875i	$-0.285737\pi$
$359$	23.6248	1.24687	0.623435	+	0.781875i	$0.285737\pi$
$360$	0	0
$361$	45.3784	2.38834
$362$	0.189737	0.00997235
$363$	0	0
$364$	7.77348	0.407441
$365$	12.5400	0.656374
$366$	0	0
$367$	−22.3730	−1.16786	−0.583932	−	0.811803i	$-0.698486\pi$
$367$	−22.3730	−1.16786	−0.583932	+	0.811803i	$0.698486\pi$
$368$	8.10867	0.422694
$369$	0	0
$370$	3.68947	0.191806
$371$	11.8384	0.614621
$372$	0	0
$373$	−11.9922	−0.620934	−0.310467	−	0.950584i	$-0.600485\pi$
$373$	−11.9922	−0.620934	−0.310467	+	0.950584i	$0.600485\pi$
$374$	−1.47585	−0.0763143
$375$	0	0
$376$	−1.31340	−0.0677333
$377$	−0.608168	−0.0313222
$378$	0	0
$379$	18.2088	0.935323	0.467662	−	0.883908i	$-0.345097\pi$
$379$	18.2088	0.935323	0.467662	+	0.883908i	$0.345097\pi$
$380$	13.6527	0.700368
$381$	0	0
$382$	−6.46696	−0.330879
$383$	−22.3494	−1.14200	−0.571001	−	0.820949i	$-0.693445\pi$
$383$	−22.3494	−1.14200	−0.571001	+	0.820949i	$0.693445\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−2.26625	−0.115349
$387$	0	0
$388$	3.03866	0.154265
$389$	−25.4863	−1.29221	−0.646103	−	0.763250i	$-0.723603\pi$
$389$	−25.4863	−1.29221	−0.646103	+	0.763250i	$0.723603\pi$
$390$	0	0
$391$	−9.53085	−0.481996
$392$	−2.02214	−0.102134
$393$	0	0
$394$	−10.5797	−0.532999
$395$	2.56844	0.129232
$396$	0	0
$397$	10.3548	0.519694	0.259847	−	0.965650i	$-0.416328\pi$
$397$	10.3548	0.519694	0.259847	+	0.965650i	$0.416328\pi$
$398$	7.24509	0.363163
$399$	0	0
$400$	2.29844	0.114922
$401$	−0.0939709	−0.00469268	−0.00234634	−	0.999997i	$-0.500747\pi$
$401$	−0.0939709	−0.00469268	−0.00234634	+	0.999997i	$0.500747\pi$
$402$	0	0
$403$	10.6855	0.532280
$404$	−19.7936	−0.984767
$405$	0	0
$406$	0.0727248	0.00360927
$407$	6.75362	0.334764
$408$	0	0
$409$	2.90741	0.143762	0.0718811	−	0.997413i	$-0.477100\pi$
$409$	2.90741	0.143762	0.0718811	+	0.997413i	$0.477100\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	15.6355	0.770308
$413$	−0.959466	−0.0472122
$414$	0	0
$415$	7.75737	0.380794
$416$	24.2123	1.18711
$417$	0	0
$418$	−4.38326	−0.214392
$419$	−16.2660	−0.794645	−0.397323	−	0.917679i	$-0.630061\pi$
$419$	−16.2660	−0.794645	−0.397323	+	0.917679i	$0.630061\pi$
$420$	0	0
$421$	−16.1047	−0.784894	−0.392447	−	0.919775i	$-0.628371\pi$
$421$	−16.1047	−0.784894	−0.392447	+	0.919775i	$0.628371\pi$
$422$	1.38759	0.0675468
$423$	0	0
$424$	23.9390	1.16258
$425$	−2.70156	−0.131045
$426$	0	0
$427$	−13.1162	−0.634738
$428$	−22.1541	−1.07086
$429$	0	0
$430$	5.16531	0.249094
$431$	19.3221	0.930711	0.465355	−	0.885124i	$-0.345927\pi$
$431$	19.3221	0.930711	0.465355	+	0.885124i	$0.345927\pi$
$432$	0	0
$433$	37.0462	1.78033	0.890163	−	0.455643i	$-0.150591\pi$
$433$	37.0462	1.78033	0.890163	+	0.455643i	$0.150591\pi$
$434$	−1.27777	−0.0613348
$435$	0	0
$436$	−0.186331	−0.00892366
$437$	−28.3066	−1.35409
$438$	0	0
$439$	14.4750	0.690856	0.345428	−	0.938445i	$-0.387734\pi$
$439$	14.4750	0.690856	0.345428	+	0.938445i	$0.387734\pi$
$440$	−2.02214	−0.0964019
$441$	0	0
$442$	−6.74233	−0.320700
$443$	26.4146	1.25500	0.627499	−	0.778618i	$-0.284079\pi$
$443$	26.4146	1.25500	0.627499	+	0.778618i	$0.284079\pi$
$444$	0	0
$445$	−13.3741	−0.633994
$446$	11.2050	0.530574
$447$	0	0
$448$	1.70156	0.0803913
$449$	13.7526	0.649023	0.324511	−	0.945882i	$-0.394800\pi$
$449$	13.7526	0.649023	0.324511	+	0.945882i	$0.394800\pi$
$450$	0	0
$451$	−10.9831	−0.517173
$452$	−6.53134	−0.307208
$453$	0	0
$454$	−10.7139	−0.502828
$455$	−4.56844	−0.214172
$456$	0	0
$457$	−9.98473	−0.467066	−0.233533	−	0.972349i	$-0.575029\pi$
$457$	−9.98473	−0.467066	−0.233533	+	0.972349i	$0.575029\pi$
$458$	11.1570	0.521331
$459$	0	0
$460$	−6.00295	−0.279889
$461$	−3.31911	−0.154586	−0.0772931	−	0.997008i	$-0.524628\pi$
$461$	−3.31911	−0.154586	−0.0772931	+	0.997008i	$0.524628\pi$
$462$	0	0
$463$	−20.1726	−0.937500	−0.468750	−	0.883331i	$-0.655295\pi$
$463$	−20.1726	−0.937500	−0.468750	+	0.883331i	$0.655295\pi$
$464$	−0.305977	−0.0142046
$465$	0	0
$466$	9.46751	0.438574
$467$	−3.69241	−0.170864	−0.0854322	−	0.996344i	$-0.527227\pi$
$467$	−3.69241	−0.170864	−0.0854322	+	0.996344i	$0.527227\pi$
$468$	0	0
$469$	−6.41464	−0.296201
$470$	0.354822	0.0163667
$471$	0	0
$472$	−1.94018	−0.0893039
$473$	9.45518	0.434750
$474$	0	0
$475$	−8.02362	−0.368149
$476$	−4.59688	−0.210697
$477$	0	0
$478$	6.35907	0.290857
$479$	−41.0516	−1.87569	−0.937847	−	0.347049i	$-0.887184\pi$
$479$	−41.0516	−1.87569	−0.937847	+	0.347049i	$0.887184\pi$
$480$	0	0
$481$	30.8535	1.40680
$482$	−12.3151	−0.560937
$483$	0	0
$484$	−1.70156	−0.0773437
$485$	−1.78581	−0.0810893
$486$	0	0
$487$	−8.08402	−0.366322	−0.183161	−	0.983083i	$-0.558633\pi$
$487$	−8.08402	−0.366322	−0.183161	+	0.983083i	$0.558633\pi$
$488$	−26.5229	−1.20063
$489$	0	0
$490$	0.546295	0.0246791
$491$	3.47348	0.156756	0.0783779	−	0.996924i	$-0.475026\pi$
$491$	3.47348	0.156756	0.0783779	+	0.996924i	$0.475026\pi$
$492$	0	0
$493$	0.359642	0.0161975
$494$	−20.0247	−0.900952
$495$	0	0
$496$	5.37598	0.241389
$497$	−2.56844	−0.115210
$498$	0	0
$499$	−29.0719	−1.30144	−0.650719	−	0.759319i	$-0.725532\pi$
$499$	−29.0719	−1.30144	−0.650719	+	0.759319i	$0.725532\pi$
$500$	−1.70156	−0.0760962
$501$	0	0
$502$	−6.25366	−0.279115
$503$	−30.6109	−1.36487	−0.682435	−	0.730946i	$-0.739079\pi$
$503$	−30.6109	−1.36487	−0.682435	+	0.730946i	$0.739079\pi$
$504$	0	0
$505$	11.6326	0.517643
$506$	1.92728	0.0856778
$507$	0	0
$508$	9.28233	0.411837
$509$	0.696166	0.0308570	0.0154285	−	0.999881i	$-0.495089\pi$
$509$	0.696166	0.0308570	0.0154285	+	0.999881i	$0.495089\pi$
$510$	0	0
$511$	−12.5400	−0.554737
$512$	21.4771	0.949161
$513$	0	0
$514$	15.6326	0.689524
$515$	−9.18893	−0.404913
$516$	0	0
$517$	0.649507	0.0285653
$518$	−3.68947	−0.162106
$519$	0	0
$520$	−9.23804	−0.405115
$521$	−14.2142	−0.622735	−0.311368	−	0.950290i	$-0.600787\pi$
$521$	−14.2142	−0.622735	−0.311368	+	0.950290i	$0.600787\pi$
$522$	0	0
$523$	5.04830	0.220747	0.110373	−	0.993890i	$-0.464795\pi$
$523$	5.04830	0.220747	0.110373	+	0.993890i	$0.464795\pi$
$524$	−35.9173	−1.56906
$525$	0	0
$526$	4.74071	0.206705
$527$	−6.31888	−0.275255
$528$	0	0
$529$	−10.5539	−0.458865
$530$	−6.46728	−0.280921
$531$	0	0
$532$	−13.6527	−0.591919
$533$	−50.1755	−2.17334
$534$	0	0
$535$	13.0199	0.562898
$536$	−12.9713	−0.560276
$537$	0	0
$538$	−16.3293	−0.704008
$539$	1.00000	0.0430730
$540$	0	0
$541$	36.1154	1.55272	0.776361	−	0.630288i	$-0.217063\pi$
$541$	36.1154	1.55272	0.776361	+	0.630288i	$0.217063\pi$
$542$	11.4546	0.492017
$543$	0	0
$544$	−14.3180	−0.613881
$545$	0.109506	0.00469073
$546$	0	0
$547$	11.4213	0.488341	0.244171	−	0.969732i	$-0.421484\pi$
$547$	11.4213	0.488341	0.244171	+	0.969732i	$0.421484\pi$
$548$	−5.79063	−0.247363
$549$	0	0
$550$	0.546295	0.0232941
$551$	1.06813	0.0455040
$552$	0	0
$553$	−2.56844	−0.109221
$554$	−7.49170	−0.318292
$555$	0	0
$556$	−36.1429	−1.53280
$557$	−3.52733	−0.149458	−0.0747288	−	0.997204i	$-0.523809\pi$
$557$	−3.52733	−0.149458	−0.0747288	+	0.997204i	$0.523809\pi$
$558$	0	0
$559$	43.1954	1.82697
$560$	−2.29844	−0.0971267
$561$	0	0
$562$	−2.68250	−0.113155
$563$	10.5400	0.444208	0.222104	−	0.975023i	$-0.428708\pi$
$563$	10.5400	0.444208	0.222104	+	0.975023i	$0.428708\pi$
$564$	0	0
$565$	3.83844	0.161484
$566$	−13.9189	−0.585056
$567$	0	0
$568$	−5.19375	−0.217925
$569$	15.6173	0.654712	0.327356	−	0.944901i	$-0.393842\pi$
$569$	15.6173	0.654712	0.327356	+	0.944901i	$0.393842\pi$
$570$	0	0
$571$	−9.94737	−0.416284	−0.208142	−	0.978099i	$-0.566742\pi$
$571$	−9.94737	−0.416284	−0.208142	+	0.978099i	$0.566742\pi$
$572$	−7.77348	−0.325026
$573$	0	0
$574$	6.00000	0.250435
$575$	3.52790	0.147124
$576$	0	0
$577$	29.1030	1.21158	0.605788	−	0.795626i	$-0.292858\pi$
$577$	29.1030	1.21158	0.605788	+	0.795626i	$0.292858\pi$
$578$	−5.29991	−0.220447
$579$	0	0
$580$	0.226518	0.00940566
$581$	−7.75737	−0.321830
$582$	0	0
$583$	−11.8384	−0.490298
$584$	−25.3577	−1.04931
$585$	0	0
$586$	1.18495	0.0489498
$587$	−12.7536	−0.526398	−0.263199	−	0.964742i	$-0.584778\pi$
$587$	−12.7536	−0.526398	−0.263199	+	0.964742i	$0.584778\pi$
$588$	0	0
$589$	−18.7670	−0.773282
$590$	0.524151	0.0215790
$591$	0	0
$592$	15.5228	0.637982
$593$	3.91893	0.160931	0.0804656	−	0.996757i	$-0.474359\pi$
$593$	3.91893	0.160931	0.0804656	+	0.996757i	$0.474359\pi$
$594$	0	0
$595$	2.70156	0.110753
$596$	−19.5803	−0.802039
$597$	0	0
$598$	8.80464	0.360048
$599$	11.9420	0.487936	0.243968	−	0.969783i	$-0.421551\pi$
$599$	11.9420	0.487936	0.243968	+	0.969783i	$0.421551\pi$
$600$	0	0
$601$	16.6878	0.680710	0.340355	−	0.940297i	$-0.389453\pi$
$601$	16.6878	0.680710	0.340355	+	0.940297i	$0.389453\pi$
$602$	−5.16531	−0.210522
$603$	0	0
$604$	−12.9120	−0.525381
$605$	1.00000	0.0406558
$606$	0	0
$607$	4.72518	0.191789	0.0958946	−	0.995391i	$-0.469429\pi$
$607$	4.72518	0.191789	0.0958946	+	0.995391i	$0.469429\pi$
$608$	−42.5245	−1.72459
$609$	0	0
$610$	7.16531	0.290115
$611$	2.96723	0.120041
$612$	0	0
$613$	−6.84938	−0.276644	−0.138322	−	0.990387i	$-0.544171\pi$
$613$	−6.84938	−0.276644	−0.138322	+	0.990387i	$0.544171\pi$
$614$	6.74937	0.272383
$615$	0	0
$616$	2.02214	0.0814745
$617$	−21.0725	−0.848347	−0.424173	−	0.905581i	$-0.639435\pi$
$617$	−21.0725	−0.848347	−0.424173	+	0.905581i	$0.639435\pi$
$618$	0	0
$619$	−29.2209	−1.17449	−0.587243	−	0.809410i	$-0.699787\pi$
$619$	−29.2209	−1.17449	−0.587243	+	0.809410i	$0.699787\pi$
$620$	−3.97991	−0.159837
$621$	0	0
$622$	−12.3503	−0.495200
$623$	13.3741	0.535822
$624$	0	0
$625$	1.00000	0.0400000
$626$	−18.7168	−0.748075
$627$	0	0
$628$	21.8792	0.873075
$629$	−18.2453	−0.727488
$630$	0	0
$631$	−23.8201	−0.948265	−0.474132	−	0.880454i	$-0.657238\pi$
$631$	−23.8201	−0.948265	−0.474132	+	0.880454i	$0.657238\pi$
$632$	−5.19375	−0.206596
$633$	0	0
$634$	−4.52839	−0.179846
$635$	−5.45518	−0.216482
$636$	0	0
$637$	4.56844	0.181008
$638$	−0.0727248	−0.00287920
$639$	0	0
$640$	−11.5294	−0.455739
$641$	30.9179	1.22118	0.610591	−	0.791946i	$-0.290932\pi$
$641$	30.9179	1.22118	0.610591	+	0.791946i	$0.290932\pi$
$642$	0	0
$643$	12.4224	0.489892	0.244946	−	0.969537i	$-0.421230\pi$
$643$	12.4224	0.489892	0.244946	+	0.969537i	$0.421230\pi$
$644$	6.00295	0.236549
$645$	0	0
$646$	11.8416	0.465903
$647$	−31.6924	−1.24596	−0.622979	−	0.782239i	$-0.714078\pi$
$647$	−31.6924	−1.24596	−0.622979	+	0.782239i	$0.714078\pi$
$648$	0	0
$649$	0.959466	0.0376623
$650$	2.49571	0.0978899
$651$	0	0
$652$	−38.5446	−1.50952
$653$	−25.4037	−0.994124	−0.497062	−	0.867715i	$-0.665588\pi$
$653$	−25.4037	−0.994124	−0.497062	+	0.867715i	$0.665588\pi$
$654$	0	0
$655$	21.1084	0.824775
$656$	−25.2439	−0.985610
$657$	0	0
$658$	−0.354822	−0.0138324
$659$	17.1004	0.666135	0.333068	−	0.942903i	$-0.391916\pi$
$659$	17.1004	0.666135	0.333068	+	0.942903i	$0.391916\pi$
$660$	0	0
$661$	17.8675	0.694963	0.347482	−	0.937687i	$-0.387037\pi$
$661$	17.8675	0.694963	0.347482	+	0.937687i	$0.387037\pi$
$662$	11.7053	0.454940
$663$	0	0
$664$	−15.6865	−0.608755
$665$	8.02362	0.311143
$666$	0	0
$667$	−0.469648	−0.0181848
$668$	−5.47553	−0.211855
$669$	0	0
$670$	3.50429	0.135382
$671$	13.1162	0.506346
$672$	0	0
$673$	−30.7973	−1.18715	−0.593575	−	0.804779i	$-0.702284\pi$
$673$	−30.7973	−1.18715	−0.593575	+	0.804779i	$0.702284\pi$
$674$	−11.0671	−0.426289
$675$	0	0
$676$	−13.3924	−0.515091
$677$	−1.54800	−0.0594944	−0.0297472	−	0.999557i	$-0.509470\pi$
$677$	−1.54800	−0.0594944	−0.0297472	+	0.999557i	$0.509470\pi$
$678$	0	0
$679$	1.78581	0.0685330
$680$	5.46295	0.209494
$681$	0	0
$682$	1.27777	0.0489283
$683$	28.2767	1.08198	0.540989	−	0.841030i	$-0.318050\pi$
$683$	28.2767	1.08198	0.540989	+	0.841030i	$0.318050\pi$
$684$	0	0
$685$	3.40312	0.130027
$686$	−0.546295	−0.0208576
$687$	0	0
$688$	21.7321	0.828530
$689$	−54.0832	−2.06041
$690$	0	0
$691$	−11.4061	−0.433907	−0.216954	−	0.976182i	$-0.569612\pi$
$691$	−11.4061	−0.433907	−0.216954	+	0.976182i	$0.569612\pi$
$692$	−30.9844	−1.17785
$693$	0	0
$694$	13.0800	0.496510
$695$	21.2410	0.805717
$696$	0	0
$697$	29.6715	1.12389
$698$	−13.9716	−0.528831
$699$	0	0
$700$	1.70156	0.0643130
$701$	4.13312	0.156106	0.0780530	−	0.996949i	$-0.475130\pi$
$701$	4.13312	0.156106	0.0780530	+	0.996949i	$0.475130\pi$
$702$	0	0
$703$	−54.1884	−2.04376
$704$	−1.70156	−0.0641300
$705$	0	0
$706$	−0.180620	−0.00679774
$707$	−11.6326	−0.437489
$708$	0	0
$709$	38.5867	1.44915	0.724576	−	0.689195i	$-0.242036\pi$
$709$	38.5867	1.44915	0.724576	+	0.689195i	$0.242036\pi$
$710$	1.40312	0.0526583
$711$	0	0
$712$	27.0444	1.01353
$713$	8.25167	0.309027
$714$	0	0
$715$	4.56844	0.170850
$716$	−8.08857	−0.302284
$717$	0	0
$718$	12.9061	0.481652
$719$	13.4920	0.503165	0.251583	−	0.967836i	$-0.419049\pi$
$719$	13.4920	0.503165	0.251583	+	0.967836i	$0.419049\pi$
$720$	0	0
$721$	9.18893	0.342214
$722$	24.7900	0.922588
$723$	0	0
$724$	−0.590980	−0.0219636
$725$	−0.133124	−0.00494409
$726$	0	0
$727$	−15.0923	−0.559743	−0.279872	−	0.960037i	$-0.590292\pi$
$727$	−15.0923	−0.559743	−0.279872	+	0.960037i	$0.590292\pi$
$728$	9.23804	0.342385
$729$	0	0
$730$	6.85054	0.253550
$731$	−25.5438	−0.944770
$732$	0	0
$733$	5.07134	0.187314	0.0936572	−	0.995605i	$-0.470144\pi$
$733$	5.07134	0.187314	0.0936572	+	0.995605i	$0.470144\pi$
$734$	−12.2223	−0.451132
$735$	0	0
$736$	18.6976	0.689202
$737$	6.41464	0.236286
$738$	0	0
$739$	−22.7021	−0.835112	−0.417556	−	0.908651i	$-0.637113\pi$
$739$	−22.7021	−0.835112	−0.417556	+	0.908651i	$0.637113\pi$
$740$	−11.4917	−0.422443
$741$	0	0
$742$	6.46728	0.237421
$743$	−29.2727	−1.07391	−0.536955	−	0.843611i	$-0.680426\pi$
$743$	−29.2727	−1.07391	−0.536955	+	0.843611i	$0.680426\pi$
$744$	0	0
$745$	11.5072	0.421592
$746$	−6.55129	−0.239860
$747$	0	0
$748$	4.59688	0.168078
$749$	−13.0199	−0.475735
$750$	0	0
$751$	7.65211	0.279229	0.139615	−	0.990206i	$-0.455414\pi$
$751$	7.65211	0.279229	0.139615	+	0.990206i	$0.455414\pi$
$752$	1.49285	0.0544387
$753$	0	0
$754$	−0.332239	−0.0120994
$755$	7.58830	0.276167
$756$	0	0
$757$	−20.9576	−0.761717	−0.380858	−	0.924633i	$-0.624371\pi$
$757$	−20.9576	−0.761717	−0.380858	+	0.924633i	$0.624371\pi$
$758$	9.94737	0.361305
$759$	0	0
$760$	16.2249	0.588539
$761$	8.16509	0.295984	0.147992	−	0.988989i	$-0.452719\pi$
$761$	8.16509	0.295984	0.147992	+	0.988989i	$0.452719\pi$
$762$	0	0
$763$	−0.109506	−0.00396439
$764$	20.1429	0.728743
$765$	0	0
$766$	−12.2094	−0.441143
$767$	4.38326	0.158270
$768$	0	0
$769$	−46.3583	−1.67172	−0.835862	−	0.548939i	$-0.815032\pi$
$769$	−46.3583	−1.67172	−0.835862	+	0.548939i	$0.815032\pi$
$770$	−0.546295	−0.0196871
$771$	0	0
$772$	7.05876	0.254050
$773$	24.2169	0.871021	0.435510	−	0.900184i	$-0.356568\pi$
$773$	24.2169	0.871021	0.435510	+	0.900184i	$0.356568\pi$
$774$	0	0
$775$	2.33897	0.0840184
$776$	3.61116	0.129633
$777$	0	0
$778$	−13.9230	−0.499165
$779$	88.1241	3.15737
$780$	0	0
$781$	2.56844	0.0919060
$782$	−5.20665	−0.186190
$783$	0	0
$784$	2.29844	0.0820871
$785$	−12.8583	−0.458933
$786$	0	0
$787$	47.1916	1.68220	0.841100	−	0.540880i	$-0.181909\pi$
$787$	47.1916	1.68220	0.841100	+	0.540880i	$0.181909\pi$
$788$	32.9530	1.17390
$789$	0	0
$790$	1.40312	0.0499209
$791$	−3.83844	−0.136479
$792$	0	0
$793$	59.9206	2.12784
$794$	5.65678	0.200752
$795$	0	0
$796$	−22.5665	−0.799849
$797$	6.97474	0.247058	0.123529	−	0.992341i	$-0.460579\pi$
$797$	6.97474	0.247058	0.123529	+	0.992341i	$0.460579\pi$
$798$	0	0
$799$	−1.75468	−0.0620763
$800$	5.29991	0.187380
$801$	0	0
$802$	−0.0513358	−0.00181273
$803$	12.5400	0.442527
$804$	0	0
$805$	−3.52790	−0.124342
$806$	5.83740	0.205614
$807$	0	0
$808$	−23.5228	−0.827528
$809$	39.1428	1.37619	0.688093	−	0.725622i	$-0.258448\pi$
$809$	39.1428	1.37619	0.688093	+	0.725622i	$0.258448\pi$
$810$	0	0
$811$	−22.1116	−0.776444	−0.388222	−	0.921566i	$-0.626911\pi$
$811$	−22.1116	−0.776444	−0.388222	+	0.921566i	$0.626911\pi$
$812$	−0.226518	−0.00794923
$813$	0	0
$814$	3.68947	0.129316
$815$	22.6525	0.793481
$816$	0	0
$817$	−75.8647	−2.65417
$818$	1.58830	0.0555337
$819$	0	0
$820$	18.6884	0.652627
$821$	49.1868	1.71663	0.858316	−	0.513122i	$-0.171511\pi$
$821$	49.1868	1.71663	0.858316	+	0.513122i	$0.171511\pi$
$822$	0	0
$823$	−4.04853	−0.141123	−0.0705615	−	0.997507i	$-0.522479\pi$
$823$	−4.04853	−0.141123	−0.0705615	+	0.997507i	$0.522479\pi$
$824$	18.5813	0.647312
$825$	0	0
$826$	−0.524151	−0.0182375
$827$	−20.5217	−0.713610	−0.356805	−	0.934179i	$-0.616134\pi$
$827$	−20.5217	−0.713610	−0.356805	+	0.934179i	$0.616134\pi$
$828$	0	0
$829$	22.7665	0.790714	0.395357	−	0.918528i	$-0.370621\pi$
$829$	22.7665	0.790714	0.395357	+	0.918528i	$0.370621\pi$
$830$	4.23781	0.147097
$831$	0	0
$832$	−7.77348	−0.269497
$833$	−2.70156	−0.0936036
$834$	0	0
$835$	3.21795	0.111362
$836$	13.6527	0.472188
$837$	0	0
$838$	−8.88602	−0.306963
$839$	−9.06473	−0.312949	−0.156475	−	0.987682i	$-0.550013\pi$
$839$	−9.06473	−0.312949	−0.156475	+	0.987682i	$0.550013\pi$
$840$	0	0
$841$	−28.9823	−0.999389
$842$	−8.79790	−0.303196
$843$	0	0
$844$	−4.32197	−0.148768
$845$	7.87063	0.270758
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−27.2099	−0.934392
$849$	0	0
$850$	−1.47585	−0.0506212
$851$	23.8261	0.816749
$852$	0	0
$853$	−13.0854	−0.448035	−0.224018	−	0.974585i	$-0.571917\pi$
$853$	−13.0854	−0.448035	−0.224018	+	0.974585i	$0.571917\pi$
$854$	−7.16531	−0.245192
$855$	0	0
$856$	−26.3280	−0.899874
$857$	−38.4589	−1.31373	−0.656866	−	0.754007i	$-0.728118\pi$
$857$	−38.4589	−1.31373	−0.656866	+	0.754007i	$0.728118\pi$
$858$	0	0
$859$	−0.435576	−0.0148617	−0.00743083	−	0.999972i	$-0.502365\pi$
$859$	−0.435576	−0.0148617	−0.00743083	+	0.999972i	$0.502365\pi$
$860$	−16.0886	−0.548616
$861$	0	0
$862$	10.5555	0.359523
$863$	−38.7362	−1.31860	−0.659298	−	0.751882i	$-0.729146\pi$
$863$	−38.7362	−1.31860	−0.659298	+	0.751882i	$0.729146\pi$
$864$	0	0
$865$	18.2094	0.619137
$866$	20.2381	0.687719
$867$	0	0
$868$	3.97991	0.135087
$869$	2.56844	0.0871283
$870$	0	0
$871$	29.3049	0.992959
$872$	−0.221437	−0.00749881
$873$	0	0
$874$	−15.4637	−0.523068
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−34.8044	−1.17526	−0.587630	−	0.809130i	$-0.699939\pi$
$877$	−34.8044	−1.17526	−0.587630	+	0.809130i	$0.699939\pi$
$878$	7.90764	0.266870
$879$	0	0
$880$	2.29844	0.0774803
$881$	−28.8583	−0.972261	−0.486130	−	0.873886i	$-0.661592\pi$
$881$	−28.8583	−0.972261	−0.486130	+	0.873886i	$0.661592\pi$
$882$	0	0
$883$	−27.5988	−0.928772	−0.464386	−	0.885633i	$-0.653725\pi$
$883$	−27.5988	−0.928772	−0.464386	+	0.885633i	$0.653725\pi$
$884$	21.0005	0.706325
$885$	0	0
$886$	14.4302	0.484791
$887$	−34.0939	−1.14476	−0.572380	−	0.819988i	$-0.693980\pi$
$887$	−34.0939	−1.14476	−0.572380	+	0.819988i	$0.693980\pi$
$888$	0	0
$889$	5.45518	0.182961
$890$	−7.30621	−0.244905
$891$	0	0
$892$	−34.9007	−1.16856
$893$	−5.21140	−0.174393
$894$	0	0
$895$	4.75362	0.158896
$896$	11.5294	0.385169
$897$	0	0
$898$	7.51295	0.250710
$899$	−0.311373	−0.0103849
$900$	0	0
$901$	31.9823	1.06548
$902$	−6.00000	−0.199778
$903$	0	0
$904$	−7.76188	−0.258156
$905$	0.347316	0.0115452
$906$	0	0
$907$	−26.7979	−0.889810	−0.444905	−	0.895578i	$-0.646763\pi$
$907$	−26.7979	−0.889810	−0.444905	+	0.895578i	$0.646763\pi$
$908$	33.3709	1.10745
$909$	0	0
$910$	−2.49571	−0.0827321
$911$	27.2083	0.901451	0.450726	−	0.892663i	$-0.351165\pi$
$911$	27.2083	0.901451	0.450726	+	0.892663i	$0.351165\pi$
$912$	0	0
$913$	7.75737	0.256731
$914$	−5.45460	−0.180422
$915$	0	0
$916$	−34.7510	−1.14820
$917$	−21.1084	−0.697062
$918$	0	0
$919$	1.61134	0.0531533	0.0265767	−	0.999647i	$-0.491539\pi$
$919$	1.61134	0.0531533	0.0265767	+	0.999647i	$0.491539\pi$
$920$	−7.13393	−0.235199
$921$	0	0
$922$	−1.81321	−0.0597149
$923$	11.7338	0.386221
$924$	0	0
$925$	6.75362	0.222058
$926$	−11.0202	−0.362146
$927$	0	0
$928$	−0.705544	−0.0231606
$929$	−53.7396	−1.76314	−0.881570	−	0.472053i	$-0.843513\pi$
$929$	−53.7396	−1.76314	−0.881570	+	0.472053i	$0.843513\pi$
$930$	0	0
$931$	−8.02362	−0.262963
$932$	−29.4888	−0.965936
$933$	0	0
$934$	−2.01715	−0.0660030
$935$	−2.70156	−0.0883505
$936$	0	0
$937$	−26.0601	−0.851348	−0.425674	−	0.904877i	$-0.639963\pi$
$937$	−26.0601	−0.851348	−0.425674	+	0.904877i	$0.639963\pi$
$938$	−3.50429	−0.114419
$939$	0	0
$940$	−1.10518	−0.0360469
$941$	57.6176	1.87828	0.939139	−	0.343537i	$-0.111625\pi$
$941$	57.6176	1.87828	0.939139	+	0.343537i	$0.111625\pi$
$942$	0	0
$943$	−38.7473	−1.26178
$944$	2.20527	0.0717755
$945$	0	0
$946$	5.16531	0.167939
$947$	−58.3245	−1.89529	−0.947646	−	0.319323i	$-0.896545\pi$
$947$	−58.3245	−1.89529	−0.947646	+	0.319323i	$0.896545\pi$
$948$	0	0
$949$	57.2882	1.85965
$950$	−4.38326	−0.142212
$951$	0	0
$952$	−5.46295	−0.177055
$953$	−2.27670	−0.0737496	−0.0368748	−	0.999320i	$-0.511740\pi$
$953$	−2.27670	−0.0737496	−0.0368748	+	0.999320i	$0.511740\pi$
$954$	0	0
$955$	−11.8379	−0.383064
$956$	−19.8068	−0.640597
$957$	0	0
$958$	−22.4263	−0.724559
$959$	−3.40312	−0.109893
$960$	0	0
$961$	−25.5292	−0.823523
$962$	16.8551	0.543430
$963$	0	0
$964$	38.3582	1.23544
$965$	−4.14840	−0.133542
$966$	0	0
$967$	51.7348	1.66368	0.831840	−	0.555016i	$-0.187288\pi$
$967$	51.7348	1.66368	0.831840	+	0.555016i	$0.187288\pi$
$968$	−2.02214	−0.0649942
$969$	0	0
$970$	−0.975577	−0.0313239
$971$	21.6846	0.695893	0.347947	−	0.937514i	$-0.386879\pi$
$971$	21.6846	0.695893	0.347947	+	0.937514i	$0.386879\pi$
$972$	0	0
$973$	−21.2410	−0.680955
$974$	−4.41626	−0.141506
$975$	0	0
$976$	30.1468	0.964975
$977$	15.7235	0.503041	0.251520	−	0.967852i	$-0.419069\pi$
$977$	15.7235	0.503041	0.251520	+	0.967852i	$0.419069\pi$
$978$	0	0
$979$	−13.3741	−0.427438
$980$	−1.70156	−0.0543544
$981$	0	0
$982$	1.89754	0.0605530
$983$	−26.0134	−0.829699	−0.414849	−	0.909890i	$-0.636166\pi$
$983$	−26.0134	−0.829699	−0.414849	+	0.909890i	$0.636166\pi$
$984$	0	0
$985$	−19.3663	−0.617063
$986$	0.196471	0.00625690
$987$	0	0
$988$	62.3714	1.98430
$989$	33.3570	1.06069
$990$	0	0
$991$	39.8932	1.26725	0.633624	−	0.773641i	$-0.281567\pi$
$991$	39.8932	1.26725	0.633624	+	0.773641i	$0.281567\pi$
$992$	12.3963	0.393584
$993$	0	0
$994$	−1.40312	−0.0445044
$995$	13.2622	0.420441
$996$	0	0
$997$	10.9914	0.348102	0.174051	−	0.984737i	$-0.444314\pi$
$997$	10.9914	0.348102	0.174051	+	0.984737i	$0.444314\pi$
$998$	−15.8818	−0.502731
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3465.2.a.bl.1.3		4
3.2	odd	2		1155.2.a.u.1.2	✓	4
15.14	odd	2		5775.2.a.bz.1.3		4
21.20	even	2		8085.2.a.bn.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.u.1.2	✓	4	3.2	odd	2
3465.2.a.bl.1.3		4	1.1	even	1	trivial
5775.2.a.bz.1.3		4	15.14	odd	2
8085.2.a.bn.1.2		4	21.20	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)

\(f(q)\)	\(=\)	\(q+0.546295 q^{2} -1.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.02214 q^{8} +O(q^{10})\)	\(q+0.546295 q^{2} -1.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.02214 q^{8} +0.546295 q^{10} +1.00000 q^{11} +4.56844 q^{13} -0.546295 q^{14} +2.29844 q^{16} -2.70156 q^{17} -8.02362 q^{19} -1.70156 q^{20} +0.546295 q^{22} +3.52790 q^{23} +1.00000 q^{25} +2.49571 q^{26} +1.70156 q^{28} -0.133124 q^{29} +2.33897 q^{31} +5.29991 q^{32} -1.47585 q^{34} -1.00000 q^{35} +6.75362 q^{37} -4.38326 q^{38} -2.02214 q^{40} -10.9831 q^{41} +9.45518 q^{43} -1.70156 q^{44} +1.92728 q^{46} +0.649507 q^{47} +1.00000 q^{49} +0.546295 q^{50} -7.77348 q^{52} -11.8384 q^{53} +1.00000 q^{55} +2.02214 q^{56} -0.0727248 q^{58} +0.959466 q^{59} +13.1162 q^{61} +1.27777 q^{62} -1.70156 q^{64} +4.56844 q^{65} +6.41464 q^{67} +4.59688 q^{68} -0.546295 q^{70} +2.56844 q^{71} +12.5400 q^{73} +3.68947 q^{74} +13.6527 q^{76} -1.00000 q^{77} +2.56844 q^{79} +2.29844 q^{80} -6.00000 q^{82} +7.75737 q^{83} -2.70156 q^{85} +5.16531 q^{86} -2.02214 q^{88} -13.3741 q^{89} -4.56844 q^{91} -6.00295 q^{92} +0.354822 q^{94} -8.02362 q^{95} -1.78581 q^{97} +0.546295 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7}+O(q^{10}) \) 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7	\( 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7} + 4 q^{11} + 8 q^{13} + 22 q^{16} + 2 q^{17} + 10 q^{19} + 6 q^{20} + 2 q^{23} + 4 q^{25} - 20 q^{26} - 6 q^{28} + 2 q^{29} + 24 q^{31} - 4 q^{35} + 8 q^{37} - 16 q^{38} + 6 q^{43} + 6 q^{44} - 12 q^{46} - 4 q^{47} + 4 q^{49} + 12 q^{52} - 14 q^{53} + 4 q^{55} - 20 q^{58} + 2 q^{59} + 6 q^{61} - 8 q^{62} + 6 q^{64} + 8 q^{65} - 8 q^{67} + 44 q^{68} + 4 q^{73} + 36 q^{74} + 56 q^{76} - 4 q^{77} + 22 q^{80} - 24 q^{82} - 6 q^{83} + 2 q^{85} + 36 q^{86} - 18 q^{89} - 8 q^{91} + 44 q^{92} - 36 q^{94} + 10 q^{95} - 6 q^{97}+O(q^{100}) \) 4 * q + 6 * q^4 + 4 * q^5 - 4 * q^7 + 4 * q^11 + 8 * q^13 + 22 * q^16 + 2 * q^17 + 10 * q^19 + 6 * q^20 + 2 * q^23 + 4 * q^25 - 20 * q^26 - 6 * q^28 + 2 * q^29 + 24 * q^31 - 4 * q^35 + 8 * q^37 - 16 * q^38 + 6 * q^43 + 6 * q^44 - 12 * q^46 - 4 * q^47 + 4 * q^49 + 12 * q^52 - 14 * q^53 + 4 * q^55 - 20 * q^58 + 2 * q^59 + 6 * q^61 - 8 * q^62 + 6 * q^64 + 8 * q^65 - 8 * q^67 + 44 * q^68 + 4 * q^73 + 36 * q^74 + 56 * q^76 - 4 * q^77 + 22 * q^80 - 24 * q^82 - 6 * q^83 + 2 * q^85 + 36 * q^86 - 18 * q^89 - 8 * q^91 + 44 * q^92 - 36 * q^94 + 10 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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