LMFDB - Embedded newform 3465.2.a.bl.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:	$4$
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			1.2
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)

$f(q)$	$=$	$q-0.546295 q^{2} -1.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.02214 q^{8} +O(q^{10})$	\(q-0.546295 q^{2} -1.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.02214 q^{8} -0.546295 q^{10} +1.00000 q^{11} -0.568438 q^{13} +0.546295 q^{14} +2.29844 q^{16} -2.70156 q^{17} +6.62049 q^{19} -1.70156 q^{20} -0.546295 q^{22} -8.93103 q^{23} +1.00000 q^{25} +0.310535 q^{26} +1.70156 q^{28} -5.27000 q^{29} +9.66103 q^{31} -5.29991 q^{32} +1.47585 q^{34} -1.00000 q^{35} -2.75362 q^{37} -3.61674 q^{38} +2.02214 q^{40} +10.9831 q^{41} -0.0520550 q^{43} -1.70156 q^{44} +4.87897 q^{46} +10.1567 q^{47} +1.00000 q^{49} -0.546295 q^{50} +0.967233 q^{52} -1.56469 q^{53} +1.00000 q^{55} -2.02214 q^{56} +2.87897 q^{58} -6.36259 q^{59} -3.71308 q^{61} -5.27777 q^{62} -1.70156 q^{64} -0.568438 q^{65} -10.4146 q^{67} +4.59688 q^{68} +0.546295 q^{70} -2.56844 q^{71} +2.26625 q^{73} +1.50429 q^{74} -11.2652 q^{76} -1.00000 q^{77} -2.56844 q^{79} +2.29844 q^{80} -6.00000 q^{82} -17.1605 q^{83} -2.70156 q^{85} +0.0284374 q^{86} +2.02214 q^{88} +10.7772 q^{89} +0.568438 q^{91} +15.1967 q^{92} -5.54857 q^{94} +6.62049 q^{95} +17.9952 q^{97} -0.546295 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.561869\pi$
$2$	−0.546295	−0.386289	−0.193144	+	0.981170i	$0.561869\pi$
$3$	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
$13$	−0.568438	−0.157656	−0.0788282	−	0.996888i	$-0.525118\pi$
$13$	−0.568438	−0.157656	−0.0788282	+	0.996888i	$0.525118\pi$
$14$	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
$19$	6.62049	1.51885	0.759423	−	0.650598i	$-0.225482\pi$
$19$	6.62049	1.51885	0.759423	+	0.650598i	$0.225482\pi$
$20$	−1.70156	−0.380481
$21$	0	0
$22$	−0.546295	−0.116470
$23$	−8.93103	−1.86225	−0.931124	−	0.364703i	$-0.881171\pi$
$23$	−8.93103	−1.86225	−0.931124	+	0.364703i	$0.881171\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.310535	0.0609009
$27$	0	0
$28$	1.70156	0.321565
$29$	−5.27000	−0.978615	−0.489307	−	0.872111i	$-0.662750\pi$
$29$	−5.27000	−0.978615	−0.489307	+	0.872111i	$0.662750\pi$
$30$	0	0
$31$	9.66103	1.73517	0.867586	−	0.497287i	$-0.165671\pi$
$31$	9.66103	1.73517	0.867586	+	0.497287i	$0.165671\pi$
$32$	−5.29991	−0.936901
$33$	0	0
$34$	1.47585	0.253106
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−2.75362	−0.452692	−0.226346	−	0.974047i	$-0.572678\pi$
$37$	−2.75362	−0.452692	−0.226346	+	0.974047i	$0.572678\pi$
$38$	−3.61674	−0.586713
$39$	0	0
$40$	2.02214	0.319729
$41$	10.9831	1.71527	0.857635	−	0.514259i	$-0.171933\pi$
$41$	10.9831	1.71527	0.857635	+	0.514259i	$0.171933\pi$
$42$	0	0
$43$	−0.0520550	−0.00793831	−0.00396916	−	0.999992i	$-0.501263\pi$
$43$	−0.0520550	−0.00793831	−0.00396916	+	0.999992i	$0.501263\pi$
$44$	−1.70156	−0.256520
$45$	0	0
$46$	4.87897	0.719365
$47$	10.1567	1.48151	0.740756	−	0.671774i	$-0.234467\pi$
$47$	10.1567	1.48151	0.740756	+	0.671774i	$0.234467\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−0.546295	−0.0772577
$51$	0	0
$52$	0.967233	0.134131
$53$	−1.56469	−0.214926	−0.107463	−	0.994209i	$-0.534273\pi$
$53$	−1.56469	−0.214926	−0.107463	+	0.994209i	$0.534273\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−2.02214	−0.270220
$57$	0	0
$58$	2.87897	0.378028
$59$	−6.36259	−0.828339	−0.414169	−	0.910200i	$-0.635928\pi$
$59$	−6.36259	−0.828339	−0.414169	+	0.910200i	$0.635928\pi$
$60$	0	0
$61$	−3.71308	−0.475412	−0.237706	−	0.971337i	$-0.576395\pi$
$61$	−3.71308	−0.475412	−0.237706	+	0.971337i	$0.576395\pi$
$62$	−5.27777	−0.670277
$63$	0	0
$64$	−1.70156	−0.212695
$65$	−0.568438	−0.0705061
$66$	0	0
$67$	−10.4146	−1.27235	−0.636176	−	0.771544i	$-0.719485\pi$
$67$	−10.4146	−1.27235	−0.636176	+	0.771544i	$0.719485\pi$
$68$	4.59688	0.557453
$69$	0	0
$70$	0.546295	0.0652947
$71$	−2.56844	−0.304818	−0.152409	−	0.988318i	$-0.548703\pi$
$71$	−2.56844	−0.304818	−0.152409	+	0.988318i	$0.548703\pi$
$72$	0	0
$73$	2.26625	0.265244	0.132622	−	0.991167i	$-0.457660\pi$
$73$	2.26625	0.265244	0.132622	+	0.991167i	$0.457660\pi$
$74$	1.50429	0.174870
$75$	0	0
$76$	−11.2652	−1.29220
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−2.56844	−0.288972	−0.144486	−	0.989507i	$-0.546153\pi$
$79$	−2.56844	−0.288972	−0.144486	+	0.989507i	$0.546153\pi$
$80$	2.29844	0.256973
$81$	0	0
$82$	−6.00000	−0.662589
$83$	−17.1605	−1.88361	−0.941804	−	0.336161i	$-0.890871\pi$
$83$	−17.1605	−1.88361	−0.941804	+	0.336161i	$0.890871\pi$
$84$	0	0
$85$	−2.70156	−0.293026
$86$	0.0284374	0.00306648
$87$	0	0
$88$	2.02214	0.215561
$89$	10.7772	1.14238	0.571192	−	0.820816i	$-0.306481\pi$
$89$	10.7772	1.14238	0.571192	+	0.820816i	$0.306481\pi$
$90$	0	0
$91$	0.568438	0.0595885
$92$	15.1967	1.58437
$93$	0	0
$94$	−5.54857	−0.572292
$95$	6.62049	0.679248
$96$	0	0
$97$	17.9952	1.82713	0.913567	−	0.406689i	$-0.133317\pi$
$97$	17.9952	1.82713	0.913567	+	0.406689i	$0.133317\pi$
$98$	−0.546295	−0.0551841
$99$	0	0
$100$	−1.70156	−0.170156
$101$	−0.826342	−0.0822241	−0.0411120	−	0.999155i	$-0.513090\pi$
$101$	−0.826342	−0.0822241	−0.0411120	+	0.999155i	$0.513090\pi$
$102$	0	0
$103$	10.5921	1.04367	0.521833	−	0.853048i	$-0.325248\pi$
$103$	10.5921	1.04367	0.521833	+	0.853048i	$0.325248\pi$
$104$	−1.14946	−0.112714
$105$	0	0
$106$	0.854779	0.0830235
$107$	13.7864	1.33278	0.666390	−	0.745603i	$-0.267839\pi$
$107$	13.7864	1.33278	0.666390	+	0.745603i	$0.267839\pi$
$108$	0	0
$109$	19.8905	1.90516	0.952582	−	0.304282i	$-0.0984166\pi$
$109$	19.8905	1.90516	0.952582	+	0.304282i	$0.0984166\pi$
$110$	−0.546295	−0.0520872
$111$	0	0
$112$	−2.29844	−0.217182
$113$	−6.43531	−0.605383	−0.302692	−	0.953089i	$-0.597885\pi$
$113$	−6.43531	−0.605383	−0.302692	+	0.953089i	$0.597885\pi$
$114$	0	0
$115$	−8.93103	−0.832823
$116$	8.96723	0.832587
$117$	0	0
$118$	3.47585	0.319978
$119$	2.70156	0.247652
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.02844	0.183646
$123$	0	0
$124$	−16.4388	−1.47625
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.05206	0.359562	0.179781	−	0.983707i	$-0.442461\pi$
$127$	4.05206	0.359562	0.179781	+	0.983707i	$0.442461\pi$
$128$	11.5294	1.01906
$129$	0	0
$130$	0.310535	0.0272357
$131$	5.69781	0.497820	0.248910	−	0.968527i	$-0.419928\pi$
$131$	5.69781	0.497820	0.248910	+	0.968527i	$0.419928\pi$
$132$	0	0
$133$	−6.62049	−0.574070
$134$	5.68947	0.491495
$135$	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
$139$	−8.04724	−0.682558	−0.341279	−	0.939962i	$-0.610860\pi$
$139$	−8.04724	−0.682558	−0.341279	+	0.939962i	$0.610860\pi$
$140$	1.70156	0.143808
$141$	0	0
$142$	1.40312	0.117748
$143$	−0.568438	−0.0475352
$144$	0	0
$145$	−5.27000	−0.437650
$146$	−1.23804	−0.102461
$147$	0	0
$148$	4.68545	0.385142
$149$	−7.50723	−0.615017	−0.307508	−	0.951545i	$-0.599495\pi$
$149$	−7.50723	−0.615017	−0.307508	+	0.951545i	$0.599495\pi$
$150$	0	0
$151$	3.21795	0.261873	0.130936	−	0.991391i	$-0.458202\pi$
$151$	3.21795	0.261873	0.130936	+	0.991391i	$0.458202\pi$
$152$	13.3876	1.08588
$153$	0	0
$154$	0.546295	0.0440217
$155$	9.66103	0.775992
$156$	0	0
$157$	−3.35107	−0.267444	−0.133722	−	0.991019i	$-0.542693\pi$
$157$	−3.35107	−0.267444	−0.133722	+	0.991019i	$0.542693\pi$
$158$	1.40312	0.111627
$159$	0	0
$160$	−5.29991	−0.418995
$161$	8.93103	0.703864
$162$	0	0
$163$	10.9600	0.858457	0.429228	−	0.903196i	$-0.358786\pi$
$163$	10.9600	0.858457	0.429228	+	0.903196i	$0.358786\pi$
$164$	−18.6884	−1.45932
$165$	0	0
$166$	9.37469	0.727617
$167$	7.58830	0.587201	0.293600	−	0.955928i	$-0.405147\pi$
$167$	7.58830	0.587201	0.293600	+	0.955928i	$0.405147\pi$
$168$	0	0
$169$	−12.6769	−0.975144
$170$	1.47585	0.113192
$171$	0	0
$172$	0.0885748	0.00675377
$173$	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
$178$	−5.88755	−0.441290
$179$	−4.75362	−0.355302	−0.177651	−	0.984094i	$-0.556850\pi$
$179$	−4.75362	−0.355302	−0.177651	+	0.984094i	$0.556850\pi$
$180$	0	0
$181$	25.2652	1.87795	0.938973	−	0.343991i	$-0.111779\pi$
$181$	25.2652	1.87795	0.938973	+	0.343991i	$0.111779\pi$
$182$	−0.310535	−0.0230184
$183$	0	0
$184$	−18.0598	−1.33139
$185$	−2.75362	−0.202450
$186$	0	0
$187$	−2.70156	−0.197558
$188$	−17.2823	−1.26044
$189$	0	0
$190$	−3.61674	−0.262386
$191$	17.4504	1.26266	0.631332	−	0.775513i	$-0.282509\pi$
$191$	17.4504	1.26266	0.631332	+	0.775513i	$0.282509\pi$
$192$	0	0
$193$	22.9546	1.65231	0.826156	−	0.563442i	$-0.190523\pi$
$193$	22.9546	1.65231	0.826156	+	0.563442i	$0.190523\pi$
$194$	−9.83067	−0.705801
$195$	0	0
$196$	−1.70156	−0.121540
$197$	3.36634	0.239842	0.119921	−	0.992783i	$-0.461736\pi$
$197$	3.36634	0.239842	0.119921	+	0.992783i	$0.461736\pi$
$198$	0	0
$199$	9.54402	0.676557	0.338279	−	0.941046i	$-0.390155\pi$
$199$	9.54402	0.676557	0.338279	+	0.941046i	$0.390155\pi$
$200$	2.02214	0.142987
$201$	0	0
$202$	0.451426	0.0317622
$203$	5.27000	0.369882
$204$	0	0
$205$	10.9831	0.767092
$206$	−5.78638	−0.403156
$207$	0	0
$208$	−1.30652	−0.0905909
$209$	6.62049	0.457949
$210$	0	0
$211$	−7.73375	−0.532413	−0.266207	−	0.963916i	$-0.585770\pi$
$211$	−7.73375	−0.532413	−0.266207	+	0.963916i	$0.585770\pi$
$212$	2.66241	0.182855
$213$	0	0
$214$	−7.53143	−0.514838
$215$	−0.0520550	−0.00355012
$216$	0	0
$217$	−9.66103	−0.655833
$218$	−10.8661	−0.735943
$219$	0	0
$220$	−1.70156	−0.114719
$221$	1.53567	0.103300
$222$	0	0
$223$	−13.9141	−0.931758	−0.465879	−	0.884848i	$-0.654262\pi$
$223$	−13.9141	−0.931758	−0.465879	+	0.884848i	$0.654262\pi$
$224$	5.29991	0.354115
$225$	0	0
$226$	3.51558	0.233853
$227$	−0.597452	−0.0396543	−0.0198271	−	0.999803i	$-0.506312\pi$
$227$	−0.597452	−0.0396543	−0.0198271	+	0.999803i	$0.506312\pi$
$228$	0	0
$229$	21.1895	1.40024	0.700121	−	0.714024i	$-0.253129\pi$
$229$	21.1895	1.40024	0.700121	+	0.714024i	$0.253129\pi$
$230$	4.87897	0.321710
$231$	0	0
$232$	−10.6567	−0.699647
$233$	20.2821	1.32872	0.664362	−	0.747411i	$-0.268703\pi$
$233$	20.2821	1.32872	0.664362	+	0.747411i	$0.268703\pi$
$234$	0	0
$235$	10.1567	0.662553
$236$	10.8263	0.704735
$237$	0	0
$238$	−1.47585	−0.0956651
$239$	−2.23723	−0.144715	−0.0723573	−	0.997379i	$-0.523052\pi$
$239$	−2.23723	−0.144715	−0.0723573	+	0.997379i	$0.523052\pi$
$240$	0	0
$241$	8.93045	0.575261	0.287630	−	0.957741i	$-0.407133\pi$
$241$	8.93045	0.575261	0.287630	+	0.957741i	$0.407133\pi$
$242$	−0.546295	−0.0351172
$243$	0	0
$244$	6.31804	0.404471
$245$	1.00000	0.0638877
$246$	0	0
$247$	−3.76334	−0.239456
$248$	19.5360	1.24054
$249$	0	0
$250$	−0.546295	−0.0345507
$251$	−3.35884	−0.212008	−0.106004	−	0.994366i	$-0.533806\pi$
$251$	−3.35884	−0.212008	−0.106004	+	0.994366i	$0.533806\pi$
$252$	0	0
$253$	−8.93103	−0.561489
$254$	−2.21362	−0.138895
$255$	0	0
$256$	−2.89531	−0.180957
$257$	−5.80943	−0.362382	−0.181191	−	0.983448i	$-0.557995\pi$
$257$	−5.80943	−0.362382	−0.181191	+	0.983448i	$0.557995\pi$
$258$	0	0
$259$	2.75362	0.171101
$260$	0.967233	0.0599853
$261$	0	0
$262$	−3.11268	−0.192302
$263$	23.3221	1.43810	0.719050	−	0.694959i	$-0.244577\pi$
$263$	23.3221	1.43810	0.719050	+	0.694959i	$0.244577\pi$
$264$	0	0
$265$	−1.56469	−0.0961179
$266$	3.61674	0.221757
$267$	0	0
$268$	17.7212	1.08249
$269$	−29.1246	−1.77576	−0.887878	−	0.460080i	$-0.847821\pi$
$269$	−29.1246	−1.77576	−0.887878	+	0.460080i	$0.847821\pi$
$270$	0	0
$271$	31.2416	1.89779	0.948895	−	0.315592i	$-0.102203\pi$
$271$	31.2416	1.89779	0.948895	+	0.315592i	$0.102203\pi$
$272$	−6.20937	−0.376499
$273$	0	0
$274$	−1.85911	−0.112313
$275$	1.00000	0.0603023
$276$	0	0
$277$	−15.8988	−0.955269	−0.477634	−	0.878559i	$-0.658506\pi$
$277$	−15.8988	−0.955269	−0.477634	+	0.878559i	$0.658506\pi$
$278$	4.39616	0.263664
$279$	0	0
$280$	−2.02214	−0.120846
$281$	14.1041	0.841381	0.420690	−	0.907204i	$-0.361788\pi$
$281$	14.1041	0.841381	0.420690	+	0.907204i	$0.361788\pi$
$282$	0	0
$283$	−1.32745	−0.0789088	−0.0394544	−	0.999221i	$-0.512562\pi$
$283$	−1.32745	−0.0789088	−0.0394544	+	0.999221i	$0.512562\pi$
$284$	4.37036	0.259333
$285$	0	0
$286$	0.310535	0.0183623
$287$	−10.9831	−0.648311
$288$	0	0
$289$	−9.70156	−0.570680
$290$	2.87897	0.169059
$291$	0	0
$292$	−3.85616	−0.225665
$293$	−18.3784	−1.07368	−0.536840	−	0.843684i	$-0.680382\pi$
$293$	−18.3784	−1.07368	−0.536840	+	0.843684i	$0.680382\pi$
$294$	0	0
$295$	−6.36259	−0.370444
$296$	−5.56821	−0.323646
$297$	0	0
$298$	4.10116	0.237574
$299$	5.07674	0.293595
$300$	0	0
$301$	0.0520550	0.00300040
$302$	−1.75795	−0.101158
$303$	0	0
$304$	15.2168	0.872743
$305$	−3.71308	−0.212611
$306$	0	0
$307$	6.45143	0.368202	0.184101	−	0.982907i	$-0.441063\pi$
$307$	6.45143	0.368202	0.184101	+	0.982907i	$0.441063\pi$
$308$	1.70156	0.0969555
$309$	0	0
$310$	−5.27777	−0.299757
$311$	29.4136	1.66789	0.833946	−	0.551847i	$-0.186077\pi$
$311$	29.4136	1.66789	0.833946	+	0.551847i	$0.186077\pi$
$312$	0	0
$313$	−24.7542	−1.39919	−0.699595	−	0.714540i	$-0.746636\pi$
$313$	−24.7542	−1.39919	−0.699595	+	0.714540i	$0.746636\pi$
$314$	1.83067	0.103311
$315$	0	0
$316$	4.37036	0.245852
$317$	15.0955	0.847850	0.423925	−	0.905697i	$-0.360652\pi$
$317$	15.0955	0.847850	0.423925	+	0.905697i	$0.360652\pi$
$318$	0	0
$319$	−5.27000	−0.295063
$320$	−1.70156	−0.0951202
$321$	0	0
$322$	−4.87897	−0.271895
$323$	−17.8857	−0.995186
$324$	0	0
$325$	−0.568438	−0.0315313
$326$	−5.98741	−0.331612
$327$	0	0
$328$	22.2094	1.22631
$329$	−10.1567	−0.559959
$330$	0	0
$331$	6.78263	0.372807	0.186404	−	0.982473i	$-0.440317\pi$
$331$	6.78263	0.372807	0.186404	+	0.982473i	$0.440317\pi$
$332$	29.1996	1.60254
$333$	0	0
$334$	−4.14545	−0.226829
$335$	−10.4146	−0.569013
$336$	0	0
$337$	−31.9509	−1.74048	−0.870238	−	0.492631i	$-0.836035\pi$
$337$	−31.9509	−1.74048	−0.870238	+	0.492631i	$0.836035\pi$
$338$	6.92531	0.376687
$339$	0	0
$340$	4.59688	0.249301
$341$	9.66103	0.523174
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−0.105263	−0.00567538
$345$	0	0
$346$	−9.94768	−0.534791
$347$	13.6694	0.733810	0.366905	−	0.930258i	$-0.380417\pi$
$347$	13.6694	0.733810	0.366905	+	0.930258i	$0.380417\pi$
$348$	0	0
$349$	16.1720	0.865668	0.432834	−	0.901474i	$-0.357514\pi$
$349$	16.1720	0.865668	0.432834	+	0.901474i	$0.357514\pi$
$350$	0.546295	0.0292007
$351$	0	0
$352$	−5.29991	−0.282486
$353$	9.94313	0.529219	0.264610	−	0.964356i	$-0.414757\pi$
$353$	9.94313	0.529219	0.264610	+	0.964356i	$0.414757\pi$
$354$	0	0
$355$	−2.56844	−0.136319
$356$	−18.3381	−0.971919
$357$	0	0
$358$	2.59688	0.137249
$359$	12.5845	0.664187	0.332094	−	0.943246i	$-0.392245\pi$
$359$	12.5845	0.664187	0.332094	+	0.943246i	$0.392245\pi$
$360$	0	0
$361$	24.8309	1.30689
$362$	−13.8022	−0.725429
$363$	0	0
$364$	−0.967233	−0.0506968
$365$	2.26625	0.118621
$366$	0	0
$367$	36.9699	1.92981	0.964907	−	0.262592i	$-0.0845772\pi$
$367$	36.9699	1.92981	0.964907	+	0.262592i	$0.0845772\pi$
$368$	−20.5274	−1.07007
$369$	0	0
$370$	1.50429	0.0782041
$371$	1.56469	0.0812344
$372$	0	0
$373$	−13.4109	−0.694390	−0.347195	−	0.937793i	$-0.612866\pi$
$373$	−13.4109	−0.694390	−0.347195	+	0.937793i	$0.612866\pi$
$374$	1.47585	0.0763143
$375$	0	0
$376$	20.5384	1.05919
$377$	2.99567	0.154285
$378$	0	0
$379$	−0.805672	−0.0413846	−0.0206923	−	0.999786i	$-0.506587\pi$
$379$	−0.805672	−0.0413846	−0.0206923	+	0.999786i	$0.506587\pi$
$380$	−11.2652	−0.577892
$381$	0	0
$382$	−9.53304	−0.487753
$383$	22.3494	1.14200	0.571001	−	0.820949i	$-0.306555\pi$
$383$	22.3494	1.14200	0.571001	+	0.820949i	$0.306555\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	−12.5400	−0.638269
$387$	0	0
$388$	−30.6199	−1.55449
$389$	29.4863	1.49501	0.747507	−	0.664253i	$-0.231250\pi$
$389$	29.4863	1.49501	0.747507	+	0.664253i	$0.231250\pi$
$390$	0	0
$391$	24.1277	1.22019
$392$	2.02214	0.102134
$393$	0	0
$394$	−1.83902	−0.0926482
$395$	−2.56844	−0.129232
$396$	0	0
$397$	4.45143	0.223411	0.111705	−	0.993741i	$-0.464369\pi$
$397$	4.45143	0.223411	0.111705	+	0.993741i	$0.464369\pi$
$398$	−5.21384	−0.261346
$399$	0	0
$400$	2.29844	0.114922
$401$	−22.7123	−1.13420	−0.567099	−	0.823650i	$-0.691934\pi$
$401$	−22.7123	−1.13420	−0.567099	+	0.823650i	$0.691934\pi$
$402$	0	0
$403$	−5.49170	−0.273561
$404$	1.40607	0.0699547
$405$	0	0
$406$	−2.87897	−0.142881
$407$	−2.75362	−0.136492
$408$	0	0
$409$	5.09259	0.251812	0.125906	−	0.992042i	$-0.459816\pi$
$409$	5.09259	0.251812	0.125906	+	0.992042i	$0.459816\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	−18.0230	−0.887932
$413$	6.36259	0.313083
$414$	0	0
$415$	−17.1605	−0.842376
$416$	3.01267	0.147708
$417$	0	0
$418$	−3.61674	−0.176901
$419$	2.86286	0.139860	0.0699300	−	0.997552i	$-0.477722\pi$
$419$	2.86286	0.139860	0.0699300	+	0.997552i	$0.477722\pi$
$420$	0	0
$421$	−16.1047	−0.784894	−0.392447	−	0.919775i	$-0.628371\pi$
$421$	−16.1047	−0.784894	−0.392447	+	0.919775i	$0.628371\pi$
$422$	4.22491	0.205665
$423$	0	0
$424$	−3.16402	−0.153658
$425$	−2.70156	−0.131045
$426$	0	0
$427$	3.71308	0.179689
$428$	−23.4584	−1.13390
$429$	0	0
$430$	0.0284374	0.00137137
$431$	4.67794	0.225329	0.112664	−	0.993633i	$-0.464062\pi$
$431$	4.67794	0.225329	0.112664	+	0.993633i	$0.464062\pi$
$432$	0	0
$433$	−27.4337	−1.31838	−0.659189	−	0.751977i	$-0.729100\pi$
$433$	−27.4337	−1.31838	−0.659189	+	0.751977i	$0.729100\pi$
$434$	5.27777	0.253341
$435$	0	0
$436$	−33.8449	−1.62088
$437$	−59.1278	−2.82847
$438$	0	0
$439$	5.73433	0.273685	0.136842	−	0.990593i	$-0.456305\pi$
$439$	5.73433	0.273685	0.136842	+	0.990593i	$0.456305\pi$
$440$	2.02214	0.0964019
$441$	0	0
$442$	−0.838929	−0.0399038
$443$	9.58536	0.455414	0.227707	−	0.973730i	$-0.426877\pi$
$443$	9.58536	0.455414	0.227707	+	0.973730i	$0.426877\pi$
$444$	0	0
$445$	10.7772	0.510890
$446$	7.60121	0.359927
$447$	0	0
$448$	1.70156	0.0803913
$449$	−30.9463	−1.46045	−0.730223	−	0.683209i	$-0.760584\pi$
$449$	−30.9463	−1.46045	−0.730223	+	0.683209i	$0.760584\pi$
$450$	0	0
$451$	10.9831	0.517173
$452$	10.9501	0.515049
$453$	0	0
$454$	0.326385	0.0153180
$455$	0.568438	0.0266488
$456$	0	0
$457$	−42.2246	−1.97519	−0.987593	−	0.157036i	$-0.949806\pi$
$457$	−42.2246	−1.97519	−0.987593	+	0.157036i	$0.949806\pi$
$458$	−11.5757	−0.540898
$459$	0	0
$460$	15.1967	0.708550
$461$	−9.87464	−0.459908	−0.229954	−	0.973201i	$-0.573858\pi$
$461$	−9.87464	−0.459908	−0.229954	+	0.973201i	$0.573858\pi$
$462$	0	0
$463$	2.56009	0.118978	0.0594888	−	0.998229i	$-0.481053\pi$
$463$	2.56009	0.118978	0.0594888	+	0.998229i	$0.481053\pi$
$464$	−12.1128	−0.562321
$465$	0	0
$466$	−11.0800	−0.513271
$467$	19.6924	0.911256	0.455628	−	0.890170i	$-0.349415\pi$
$467$	19.6924	0.911256	0.455628	+	0.890170i	$0.349415\pi$
$468$	0	0
$469$	10.4146	0.480904
$470$	−5.54857	−0.255937
$471$	0	0
$472$	−12.8661	−0.592209
$473$	−0.0520550	−0.00239349
$474$	0	0
$475$	6.62049	0.303769
$476$	−4.59688	−0.210697
$477$	0	0
$478$	1.22219	0.0559016
$479$	−15.3672	−0.702144	−0.351072	−	0.936348i	$-0.614183\pi$
$479$	−15.3672	−0.702144	−0.351072	+	0.936348i	$0.614183\pi$
$480$	0	0
$481$	1.56526	0.0713698
$482$	−4.87866	−0.222217
$483$	0	0
$484$	−1.70156	−0.0773437
$485$	17.9952	0.817119
$486$	0	0
$487$	−1.52848	−0.0692621	−0.0346310	−	0.999400i	$-0.511026\pi$
$487$	−1.52848	−0.0692621	−0.0346310	+	0.999400i	$0.511026\pi$
$488$	−7.50839	−0.339889
$489$	0	0
$490$	−0.546295	−0.0246791
$491$	40.7359	1.83839	0.919193	−	0.393808i	$-0.128843\pi$
$491$	40.7359	1.83839	0.919193	+	0.393808i	$0.128843\pi$
$492$	0	0
$493$	14.2372	0.641213
$494$	2.05589	0.0924990
$495$	0	0
$496$	22.2053	0.997046
$497$	2.56844	0.115210
$498$	0	0
$499$	−20.3312	−0.910150	−0.455075	−	0.890453i	$-0.650388\pi$
$499$	−20.3312	−0.910150	−0.455075	+	0.890453i	$0.650388\pi$
$500$	−1.70156	−0.0760962
$501$	0	0
$502$	1.83491	0.0818963
$503$	23.5952	1.05206	0.526030	−	0.850466i	$-0.323680\pi$
$503$	23.5952	1.05206	0.526030	+	0.850466i	$0.323680\pi$
$504$	0	0
$505$	−0.826342	−0.0367717
$506$	4.87897	0.216897
$507$	0	0
$508$	−6.89482	−0.305908
$509$	−38.0993	−1.68872	−0.844361	−	0.535775i	$-0.820019\pi$
$509$	−38.0993	−1.68872	−0.844361	+	0.535775i	$0.820019\pi$
$510$	0	0
$511$	−2.26625	−0.100253
$512$	−21.4771	−0.949161
$513$	0	0
$514$	3.17366	0.139984
$515$	10.5921	0.466742
$516$	0	0
$517$	10.1567	0.446693
$518$	−1.50429	−0.0660945
$519$	0	0
$520$	−1.14946	−0.0504073
$521$	−33.9952	−1.48936	−0.744678	−	0.667424i	$-0.767397\pi$
$521$	−33.9952	−1.48936	−0.744678	+	0.667424i	$0.767397\pi$
$522$	0	0
$523$	10.9517	0.478884	0.239442	−	0.970911i	$-0.423035\pi$
$523$	10.9517	0.478884	0.239442	+	0.970911i	$0.423035\pi$
$524$	−9.69518	−0.423536
$525$	0	0
$526$	−12.7407	−0.555522
$527$	−26.0999	−1.13693
$528$	0	0
$529$	56.7633	2.46797
$530$	0.854779	0.0371292
$531$	0	0
$532$	11.2652	0.488408
$533$	−6.24321	−0.270423
$534$	0	0
$535$	13.7864	0.596037
$536$	−21.0599	−0.909650
$537$	0	0
$538$	15.9106	0.685954
$539$	1.00000	0.0430730
$540$	0	0
$541$	13.4971	0.580285	0.290143	−	0.956983i	$-0.406297\pi$
$541$	13.4971	0.580285	0.290143	+	0.956983i	$0.406297\pi$
$542$	−17.0671	−0.733095
$543$	0	0
$544$	14.3180	0.613881
$545$	19.8905	0.852015
$546$	0	0
$547$	−42.0182	−1.79657	−0.898285	−	0.439414i	$-0.855186\pi$
$547$	−42.0182	−1.79657	−0.898285	+	0.439414i	$0.855186\pi$
$548$	−5.79063	−0.247363
$549$	0	0
$550$	−0.546295	−0.0232941
$551$	−34.8900	−1.48636
$552$	0	0
$553$	2.56844	0.109221
$554$	8.68545	0.369009
$555$	0	0
$556$	13.6929	0.580707
$557$	27.9461	1.18411	0.592057	−	0.805896i	$-0.298316\pi$
$557$	27.9461	1.18411	0.592057	+	0.805896i	$0.298316\pi$
$558$	0	0
$559$	0.0295901	0.00125153
$560$	−2.29844	−0.0971267
$561$	0	0
$562$	−7.70500	−0.325016
$563$	0.266247	0.0112210	0.00561050	−	0.999984i	$-0.498214\pi$
$563$	0.266247	0.0112210	0.00561050	+	0.999984i	$0.498214\pi$
$564$	0	0
$565$	−6.43531	−0.270736
$566$	0.725180	0.0304816
$567$	0	0
$568$	−5.19375	−0.217925
$569$	35.3983	1.48397	0.741987	−	0.670414i	$-0.233884\pi$
$569$	35.3983	1.48397	0.741987	+	0.670414i	$0.233884\pi$
$570$	0	0
$571$	−0.440134	−0.0184191	−0.00920953	−	0.999958i	$-0.502932\pi$
$571$	−0.440134	−0.0184191	−0.00920953	+	0.999958i	$0.502932\pi$
$572$	0.967233	0.0404421
$573$	0	0
$574$	6.00000	0.250435
$575$	−8.93103	−0.372450
$576$	0	0
$577$	−25.1030	−1.04505	−0.522527	−	0.852623i	$-0.675010\pi$
$577$	−25.1030	−1.04505	−0.522527	+	0.852623i	$0.675010\pi$
$578$	5.29991	0.220447
$579$	0	0
$580$	8.96723	0.372344
$581$	17.1605	0.711937
$582$	0	0
$583$	−1.56469	−0.0648026
$584$	4.58268	0.189633
$585$	0	0
$586$	10.0400	0.414750
$587$	−3.24638	−0.133993	−0.0669963	−	0.997753i	$-0.521342\pi$
$587$	−3.24638	−0.133993	−0.0669963	+	0.997753i	$0.521342\pi$
$588$	0	0
$589$	63.9608	2.63546
$590$	3.47585	0.143098
$591$	0	0
$592$	−6.32902	−0.260121
$593$	−10.7252	−0.440430	−0.220215	−	0.975451i	$-0.570676\pi$
$593$	−10.7252	−0.440430	−0.220215	+	0.975451i	$0.570676\pi$
$594$	0	0
$595$	2.70156	0.110753
$596$	12.7740	0.523244
$597$	0	0
$598$	−2.77340	−0.113413
$599$	−36.3607	−1.48566	−0.742829	−	0.669481i	$-0.766517\pi$
$599$	−36.3607	−1.48566	−0.742829	+	0.669481i	$0.766517\pi$
$600$	0	0
$601$	−39.7034	−1.61954	−0.809769	−	0.586749i	$-0.800407\pi$
$601$	−39.7034	−1.61954	−0.809769	+	0.586749i	$0.800407\pi$
$602$	−0.0284374	−0.00115902
$603$	0	0
$604$	−5.47553	−0.222796
$605$	1.00000	0.0406558
$606$	0	0
$607$	−9.91893	−0.402597	−0.201299	−	0.979530i	$-0.564516\pi$
$607$	−9.91893	−0.402597	−0.201299	+	0.979530i	$0.564516\pi$
$608$	−35.0880	−1.42301
$609$	0	0
$610$	2.02844	0.0821290
$611$	−5.77348	−0.233570
$612$	0	0
$613$	39.2681	1.58602	0.793012	−	0.609206i	$-0.208512\pi$
$613$	39.2681	1.58602	0.793012	+	0.609206i	$0.208512\pi$
$614$	−3.52438	−0.142232
$615$	0	0
$616$	−2.02214	−0.0814745
$617$	−31.3462	−1.26195	−0.630976	−	0.775802i	$-0.717345\pi$
$617$	−31.3462	−1.26195	−0.630976	+	0.775802i	$0.717345\pi$
$618$	0	0
$619$	−12.3916	−0.498061	−0.249030	−	0.968496i	$-0.580112\pi$
$619$	−12.3916	−0.498061	−0.249030	+	0.968496i	$0.580112\pi$
$620$	−16.4388	−0.660200
$621$	0	0
$622$	−16.0685	−0.644287
$623$	−10.7772	−0.431781
$624$	0	0
$625$	1.00000	0.0400000
$626$	13.5231	0.540491
$627$	0	0
$628$	5.70205	0.227537
$629$	7.43907	0.296615
$630$	0	0
$631$	33.2233	1.32260	0.661299	−	0.750123i	$-0.270006\pi$
$631$	33.2233	1.32260	0.661299	+	0.750123i	$0.270006\pi$
$632$	−5.19375	−0.206596
$633$	0	0
$634$	−8.24661	−0.327515
$635$	4.05206	0.160801
$636$	0	0
$637$	−0.568438	−0.0225223
$638$	2.87897	0.113980
$639$	0	0
$640$	11.5294	0.455739
$641$	−18.9179	−0.747211	−0.373605	−	0.927588i	$-0.621879\pi$
$641$	−18.9179	−0.747211	−0.373605	+	0.927588i	$0.621879\pi$
$642$	0	0
$643$	−5.82554	−0.229737	−0.114868	−	0.993381i	$-0.536645\pi$
$643$	−5.82554	−0.229737	−0.114868	+	0.993381i	$0.536645\pi$
$644$	−15.1967	−0.598834
$645$	0	0
$646$	9.77085	0.384429
$647$	−8.30759	−0.326605	−0.163302	−	0.986576i	$-0.552215\pi$
$647$	−8.30759	−0.326605	−0.163302	+	0.986576i	$0.552215\pi$
$648$	0	0
$649$	−6.36259	−0.249753
$650$	0.310535	0.0121802
$651$	0	0
$652$	−18.6492	−0.730359
$653$	−44.4182	−1.73822	−0.869109	−	0.494621i	$-0.835307\pi$
$653$	−44.4182	−1.73822	−0.869109	+	0.494621i	$0.835307\pi$
$654$	0	0
$655$	5.69781	0.222632
$656$	25.2439	0.985610
$657$	0	0
$658$	5.54857	0.216306
$659$	13.4965	0.525750	0.262875	−	0.964830i	$-0.415329\pi$
$659$	13.4965	0.525750	0.262875	+	0.964830i	$0.415329\pi$
$660$	0	0
$661$	31.7450	1.23474	0.617370	−	0.786673i	$-0.288198\pi$
$661$	31.7450	1.23474	0.617370	+	0.786673i	$0.288198\pi$
$662$	−3.70532	−0.144011
$663$	0	0
$664$	−34.7010	−1.34666
$665$	−6.62049	−0.256732
$666$	0	0
$667$	47.0665	1.82242
$668$	−12.9120	−0.499579
$669$	0	0
$670$	5.68947	0.219803
$671$	−3.71308	−0.143342
$672$	0	0
$673$	5.81295	0.224073	0.112036	−	0.993704i	$-0.464263\pi$
$673$	5.81295	0.224073	0.112036	+	0.993704i	$0.464263\pi$
$674$	17.4546	0.672326
$675$	0	0
$676$	21.5705	0.829634
$677$	23.3699	0.898177	0.449088	−	0.893487i	$-0.351749\pi$
$677$	23.3699	0.898177	0.449088	+	0.893487i	$0.351749\pi$
$678$	0	0
$679$	−17.9952	−0.690592
$680$	−5.46295	−0.209494
$681$	0	0
$682$	−5.27777	−0.202096
$683$	−13.4705	−0.515433	−0.257716	−	0.966221i	$-0.582970\pi$
$683$	−13.4705	−0.515433	−0.257716	+	0.966221i	$0.582970\pi$
$684$	0	0
$685$	3.40312	0.130027
$686$	0.546295	0.0208576
$687$	0	0
$688$	−0.119645	−0.00456143
$689$	0.889427	0.0338845
$690$	0	0
$691$	9.79358	0.372565	0.186283	−	0.982496i	$-0.440356\pi$
$691$	9.79358	0.372565	0.186283	+	0.982496i	$0.440356\pi$
$692$	−30.9844	−1.17785
$693$	0	0
$694$	−7.46751	−0.283463
$695$	−8.04724	−0.305249
$696$	0	0
$697$	−29.6715	−1.12389
$698$	−8.83469	−0.334398
$699$	0	0
$700$	1.70156	0.0643130
$701$	9.27000	0.350123	0.175062	−	0.984557i	$-0.443988\pi$
$701$	9.27000	0.350123	0.175062	+	0.984557i	$0.443988\pi$
$702$	0	0
$703$	−18.2303	−0.687569
$704$	−1.70156	−0.0641300
$705$	0	0
$706$	−5.43188	−0.204431
$707$	0.826342	0.0310778
$708$	0	0
$709$	−19.9898	−0.750732	−0.375366	−	0.926877i	$-0.622483\pi$
$709$	−19.9898	−0.750732	−0.375366	+	0.926877i	$0.622483\pi$
$710$	1.40312	0.0526583
$711$	0	0
$712$	21.7931	0.816732
$713$	−86.2829	−3.23132
$714$	0	0
$715$	−0.568438	−0.0212584
$716$	8.08857	0.302284
$717$	0	0
$718$	−6.87487	−0.256568
$719$	26.7174	0.996391	0.498196	−	0.867065i	$-0.333996\pi$
$719$	26.7174	0.996391	0.498196	+	0.867065i	$0.333996\pi$
$720$	0	0
$721$	−10.5921	−0.394469
$722$	−13.5650	−0.504837
$723$	0	0
$724$	−42.9903	−1.59772
$725$	−5.27000	−0.195723
$726$	0	0
$727$	16.4955	0.611782	0.305891	−	0.952066i	$-0.401046\pi$
$727$	16.4955	0.611782	0.305891	+	0.952066i	$0.401046\pi$
$728$	1.14946	0.0426020
$729$	0	0
$730$	−1.23804	−0.0458219
$731$	0.140630	0.00520138
$732$	0	0
$733$	−22.6838	−0.837847	−0.418923	−	0.908022i	$-0.637592\pi$
$733$	−22.6838	−0.837847	−0.418923	+	0.908022i	$0.637592\pi$
$734$	−20.1965	−0.745465
$735$	0	0
$736$	47.3337	1.74474
$737$	−10.4146	−0.383628
$738$	0	0
$739$	−41.7166	−1.53457	−0.767285	−	0.641306i	$-0.778393\pi$
$739$	−41.7166	−1.53457	−0.767285	+	0.641306i	$0.778393\pi$
$740$	4.68545	0.172241
$741$	0	0
$742$	−0.854779	−0.0313799
$743$	26.4664	0.970959	0.485480	−	0.874248i	$-0.338645\pi$
$743$	26.4664	0.970959	0.485480	+	0.874248i	$0.338645\pi$
$744$	0	0
$745$	−7.50723	−0.275044
$746$	7.32630	0.268235
$747$	0	0
$748$	4.59688	0.168078
$749$	−13.7864	−0.503744
$750$	0	0
$751$	−36.2802	−1.32388	−0.661942	−	0.749555i	$-0.730268\pi$
$751$	−36.2802	−1.32388	−0.661942	+	0.749555i	$0.730268\pi$
$752$	23.3446	0.851291
$753$	0	0
$754$	−1.63652	−0.0595985
$755$	3.21795	0.117113
$756$	0	0
$757$	27.3451	0.993874	0.496937	−	0.867786i	$-0.334458\pi$
$757$	27.3451	0.993874	0.496937	+	0.867786i	$0.334458\pi$
$758$	0.440134	0.0159864
$759$	0	0
$760$	13.3876	0.485619
$761$	16.2537	0.589195	0.294597	−	0.955621i	$-0.404814\pi$
$761$	16.2537	0.589195	0.294597	+	0.955621i	$0.404814\pi$
$762$	0	0
$763$	−19.8905	−0.720084
$764$	−29.6929	−1.07425
$765$	0	0
$766$	−12.2094	−0.441143
$767$	3.61674	0.130593
$768$	0	0
$769$	−38.2698	−1.38004	−0.690022	−	0.723789i	$-0.742399\pi$
$769$	−38.2698	−1.38004	−0.690022	+	0.723789i	$0.742399\pi$
$770$	0.546295	0.0196871
$771$	0	0
$772$	−39.0588	−1.40576
$773$	−6.60438	−0.237543	−0.118772	−	0.992922i	$-0.537896\pi$
$773$	−6.60438	−0.237543	−0.118772	+	0.992922i	$0.537896\pi$
$774$	0	0
$775$	9.66103	0.347034
$776$	36.3888	1.30628
$777$	0	0
$778$	−16.1082	−0.577507
$779$	72.7134	2.60523
$780$	0	0
$781$	−2.56844	−0.0919060
$782$	−13.1808	−0.471346
$783$	0	0
$784$	2.29844	0.0820871
$785$	−3.35107	−0.119605
$786$	0	0
$787$	−23.1916	−0.826692	−0.413346	−	0.910574i	$-0.635640\pi$
$787$	−23.1916	−0.826692	−0.413346	+	0.910574i	$0.635640\pi$
$788$	−5.72804	−0.204053
$789$	0	0
$790$	1.40312	0.0499209
$791$	6.43531	0.228813
$792$	0	0
$793$	2.11066	0.0749517
$794$	−2.43179	−0.0863010
$795$	0	0
$796$	−16.2397	−0.575602
$797$	−32.5872	−1.15430	−0.577150	−	0.816638i	$-0.695835\pi$
$797$	−32.5872	−1.15430	−0.577150	+	0.816638i	$0.695835\pi$
$798$	0	0
$799$	−27.4391	−0.970724
$800$	−5.29991	−0.187380
$801$	0	0
$802$	12.4076	0.438127
$803$	2.26625	0.0799741
$804$	0	0
$805$	8.93103	0.314777
$806$	3.00009	0.105674
$807$	0	0
$808$	−1.67098	−0.0587849
$809$	−13.5303	−0.475699	−0.237850	−	0.971302i	$-0.576443\pi$
$809$	−13.5303	−0.475699	−0.237850	+	0.971302i	$0.576443\pi$
$810$	0	0
$811$	27.7241	0.973525	0.486763	−	0.873534i	$-0.338178\pi$
$811$	27.7241	0.973525	0.486763	+	0.873534i	$0.338178\pi$
$812$	−8.96723	−0.314688
$813$	0	0
$814$	1.50429	0.0527252
$815$	10.9600	0.383914
$816$	0	0
$817$	−0.344630	−0.0120571
$818$	−2.78205	−0.0972723
$819$	0	0
$820$	−18.6884	−0.652627
$821$	−40.9774	−1.43012	−0.715061	−	0.699062i	$-0.753601\pi$
$821$	−40.9774	−1.43012	−0.715061	+	0.699062i	$0.753601\pi$
$822$	0	0
$823$	3.27352	0.114108	0.0570539	−	0.998371i	$-0.481829\pi$
$823$	3.27352	0.114108	0.0570539	+	0.998371i	$0.481829\pi$
$824$	21.4187	0.746154
$825$	0	0
$826$	−3.47585	−0.120940
$827$	36.5217	1.26998	0.634992	−	0.772519i	$-0.281003\pi$
$827$	36.5217	1.26998	0.634992	+	0.772519i	$0.281003\pi$
$828$	0	0
$829$	21.2335	0.737469	0.368735	−	0.929535i	$-0.379791\pi$
$829$	21.2335	0.737469	0.368735	+	0.929535i	$0.379791\pi$
$830$	9.37469	0.325400
$831$	0	0
$832$	0.967233	0.0335328
$833$	−2.70156	−0.0936036
$834$	0	0
$835$	7.58830	0.262604
$836$	−11.2652	−0.389614
$837$	0	0
$838$	−1.56397	−0.0540263
$839$	−20.7571	−0.716616	−0.358308	−	0.933603i	$-0.616646\pi$
$839$	−20.7571	−0.716616	−0.358308	+	0.933603i	$0.616646\pi$
$840$	0	0
$841$	−1.22709	−0.0423136
$842$	8.79790	0.303196
$843$	0	0
$844$	13.1595	0.452967
$845$	−12.6769	−0.436098
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	−3.59633	−0.123499
$849$	0	0
$850$	1.47585	0.0506212
$851$	24.5926	0.843025
$852$	0	0
$853$	−31.3333	−1.07283	−0.536417	−	0.843953i	$-0.680222\pi$
$853$	−31.3333	−1.07283	−0.536417	+	0.843953i	$0.680222\pi$
$854$	−2.02844	−0.0694117
$855$	0	0
$856$	27.8780	0.952852
$857$	−13.5411	−0.462554	−0.231277	−	0.972888i	$-0.574290\pi$
$857$	−13.5411	−0.462554	−0.231277	+	0.972888i	$0.574290\pi$
$858$	0	0
$859$	−19.5644	−0.667530	−0.333765	−	0.942656i	$-0.608319\pi$
$859$	−19.5644	−0.667530	−0.333765	+	0.942656i	$0.608319\pi$
$860$	0.0885748	0.00302038
$861$	0	0
$862$	−2.55554	−0.0870419
$863$	8.91434	0.303448	0.151724	−	0.988423i	$-0.451518\pi$
$863$	8.91434	0.303448	0.151724	+	0.988423i	$0.451518\pi$
$864$	0	0
$865$	18.2094	0.619137
$866$	14.9869	0.509275
$867$	0	0
$868$	16.4388	0.557971
$869$	−2.56844	−0.0871283
$870$	0	0
$871$	5.92008	0.200594
$872$	40.2214	1.36207
$873$	0	0
$874$	32.3012	1.09260
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	6.17626	0.208557	0.104279	−	0.994548i	$-0.466747\pi$
$877$	6.17626	0.208557	0.104279	+	0.994548i	$0.466747\pi$
$878$	−3.13263	−0.105721
$879$	0	0
$880$	2.29844	0.0774803
$881$	−19.3511	−0.651954	−0.325977	−	0.945378i	$-0.605693\pi$
$881$	−19.3511	−0.651954	−0.325977	+	0.945378i	$0.605693\pi$
$882$	0	0
$883$	28.7925	0.968945	0.484473	−	0.874806i	$-0.339012\pi$
$883$	28.7925	0.968945	0.484473	+	0.874806i	$0.339012\pi$
$884$	−2.61304	−0.0878861
$885$	0	0
$886$	−5.23643	−0.175921
$887$	43.4970	1.46049	0.730243	−	0.683187i	$-0.239407\pi$
$887$	43.4970	1.46049	0.730243	+	0.683187i	$0.239407\pi$
$888$	0	0
$889$	−4.05206	−0.135902
$890$	−5.88755	−0.197351
$891$	0	0
$892$	23.6757	0.792722
$893$	67.2426	2.25019
$894$	0	0
$895$	−4.75362	−0.158896
$896$	−11.5294	−0.385169
$897$	0	0
$898$	16.9058	0.564154
$899$	−50.9136	−1.69806
$900$	0	0
$901$	4.22709	0.140825
$902$	−6.00000	−0.199778
$903$	0	0
$904$	−13.0131	−0.432810
$905$	25.2652	0.839843
$906$	0	0
$907$	−9.20210	−0.305551	−0.152775	−	0.988261i	$-0.548821\pi$
$907$	−9.20210	−0.305551	−0.152775	+	0.988261i	$0.548821\pi$
$908$	1.01660	0.0337371
$909$	0	0
$910$	−0.310535	−0.0102941
$911$	−7.98331	−0.264499	−0.132249	−	0.991216i	$-0.542220\pi$
$911$	−7.98331	−0.264499	−0.132249	+	0.991216i	$0.542220\pi$
$912$	0	0
$913$	−17.1605	−0.567929
$914$	23.0671	0.762992
$915$	0	0
$916$	−36.0553	−1.19130
$917$	−5.69781	−0.188158
$918$	0	0
$919$	−36.4176	−1.20131	−0.600653	−	0.799510i	$-0.705093\pi$
$919$	−36.4176	−1.20131	−0.600653	+	0.799510i	$0.705093\pi$
$920$	−18.0598	−0.595415
$921$	0	0
$922$	5.39447	0.177657
$923$	1.46000	0.0480565
$924$	0	0
$925$	−2.75362	−0.0905384
$926$	−1.39857	−0.0459597
$927$	0	0
$928$	27.9305	0.916865
$929$	−1.06660	−0.0349940	−0.0174970	−	0.999847i	$-0.505570\pi$
$929$	−1.06660	−0.0349940	−0.0174970	+	0.999847i	$0.505570\pi$
$930$	0	0
$931$	6.62049	0.216978
$932$	−34.5112	−1.13045
$933$	0	0
$934$	−10.7579	−0.352008
$935$	−2.70156	−0.0883505
$936$	0	0
$937$	−4.74611	−0.155049	−0.0775243	−	0.996990i	$-0.524702\pi$
$937$	−4.74611	−0.155049	−0.0775243	+	0.996990i	$0.524702\pi$
$938$	−5.68947	−0.185768
$939$	0	0
$940$	−17.2823	−0.563687
$941$	−33.1988	−1.08225	−0.541125	−	0.840942i	$-0.682001\pi$
$941$	−33.1988	−1.08225	−0.541125	+	0.840942i	$0.682001\pi$
$942$	0	0
$943$	−98.0902	−3.19426
$944$	−14.6240	−0.475971
$945$	0	0
$946$	0.0284374	0.000924579 0
$947$	−6.30361	−0.204840	−0.102420	−	0.994741i	$-0.532659\pi$
$947$	−6.30361	−0.204840	−0.102420	+	0.994741i	$0.532659\pi$
$948$	0	0
$949$	−1.28822	−0.0418175
$950$	−3.61674	−0.117343
$951$	0	0
$952$	5.46295	0.177055
$953$	39.4705	1.27857	0.639287	−	0.768968i	$-0.279230\pi$
$953$	39.4705	1.27857	0.639287	+	0.768968i	$0.279230\pi$
$954$	0	0
$955$	17.4504	0.564680
$956$	3.80679	0.123120
$957$	0	0
$958$	8.39501	0.271230
$959$	−3.40312	−0.109893
$960$	0	0
$961$	62.3355	2.01082
$962$	−0.855094	−0.0275693
$963$	0	0
$964$	−15.1957	−0.489421
$965$	22.9546	0.738936
$966$	0	0
$967$	−20.7192	−0.666285	−0.333142	−	0.942877i	$-0.608109\pi$
$967$	−20.7192	−0.666285	−0.333142	+	0.942877i	$0.608109\pi$
$968$	2.02214	0.0649942
$969$	0	0
$970$	−9.83067	−0.315644
$971$	−0.281521	−0.00903444	−0.00451722	−	0.999990i	$-0.501438\pi$
$971$	−0.281521	−0.00903444	−0.00451722	+	0.999990i	$0.501438\pi$
$972$	0	0
$973$	8.04724	0.257983
$974$	0.835001	0.0267551
$975$	0	0
$976$	−8.53429	−0.273176
$977$	−53.1267	−1.69967	−0.849836	−	0.527047i	$-0.823299\pi$
$977$	−53.1267	−1.69967	−0.849836	+	0.527047i	$0.823299\pi$
$978$	0	0
$979$	10.7772	0.344442
$980$	−1.70156	−0.0543544
$981$	0	0
$982$	−22.2538	−0.710147
$983$	47.2072	1.50567	0.752837	−	0.658207i	$-0.228685\pi$
$983$	47.2072	1.50567	0.752837	+	0.658207i	$0.228685\pi$
$984$	0	0
$985$	3.36634	0.107261
$986$	−7.77773	−0.247693
$987$	0	0
$988$	6.40356	0.203724
$989$	0.464905	0.0147831
$990$	0	0
$991$	−30.4901	−0.968549	−0.484274	−	0.874916i	$-0.660916\pi$
$991$	−30.4901	−0.968549	−0.484274	+	0.874916i	$0.660916\pi$
$992$	−51.2026	−1.62568
$993$	0	0
$994$	−1.40312	−0.0445044
$995$	9.54402	0.302566
$996$	0	0
$997$	6.62107	0.209691	0.104846	−	0.994489i	$-0.466565\pi$
$997$	6.62107	0.209691	0.104846	+	0.994489i	$0.466565\pi$
$998$	11.1068	0.351581
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-0.546295 q^{2} -1.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.02214 q^{8} +O(q^{10})\)	\(q-0.546295 q^{2} -1.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} +2.02214 q^{8} -0.546295 q^{10} +1.00000 q^{11} -0.568438 q^{13} +0.546295 q^{14} +2.29844 q^{16} -2.70156 q^{17} +6.62049 q^{19} -1.70156 q^{20} -0.546295 q^{22} -8.93103 q^{23} +1.00000 q^{25} +0.310535 q^{26} +1.70156 q^{28} -5.27000 q^{29} +9.66103 q^{31} -5.29991 q^{32} +1.47585 q^{34} -1.00000 q^{35} -2.75362 q^{37} -3.61674 q^{38} +2.02214 q^{40} +10.9831 q^{41} -0.0520550 q^{43} -1.70156 q^{44} +4.87897 q^{46} +10.1567 q^{47} +1.00000 q^{49} -0.546295 q^{50} +0.967233 q^{52} -1.56469 q^{53} +1.00000 q^{55} -2.02214 q^{56} +2.87897 q^{58} -6.36259 q^{59} -3.71308 q^{61} -5.27777 q^{62} -1.70156 q^{64} -0.568438 q^{65} -10.4146 q^{67} +4.59688 q^{68} +0.546295 q^{70} -2.56844 q^{71} +2.26625 q^{73} +1.50429 q^{74} -11.2652 q^{76} -1.00000 q^{77} -2.56844 q^{79} +2.29844 q^{80} -6.00000 q^{82} -17.1605 q^{83} -2.70156 q^{85} +0.0284374 q^{86} +2.02214 q^{88} +10.7772 q^{89} +0.568438 q^{91} +15.1967 q^{92} -5.54857 q^{94} +6.62049 q^{95} +17.9952 q^{97} -0.546295 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} + 4 q^{5} - 4 q^{7}+O(q^{10}) \) 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

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Database

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