LMFDB - Embedded newform 9405.2.a.v.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, 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("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9405.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:	$75.0993031010$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\Q(\zeta_{22})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 4x^{3} + 3x^{2} + 3x - 1 $ x^5 - x^4 - 4x^3 + 3x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.91899$ of defining polynomial
Character	$\chi$	$=$	9405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.39788 q^{2} +3.74982 q^{4} -1.00000 q^{5} -2.89389 q^{7} +4.19584 q^{8} +O(q^{10})$	\(q+2.39788 q^{2} +3.74982 q^{4} -1.00000 q^{5} -2.89389 q^{7} +4.19584 q^{8} -2.39788 q^{10} +1.00000 q^{11} +4.73780 q^{13} -6.93920 q^{14} +2.56149 q^{16} +5.65389 q^{17} +1.00000 q^{19} -3.74982 q^{20} +2.39788 q^{22} +4.00714 q^{23} +1.00000 q^{25} +11.3607 q^{26} -10.8516 q^{28} -9.32825 q^{29} -6.60270 q^{31} -2.24955 q^{32} +13.5573 q^{34} +2.89389 q^{35} +6.07686 q^{37} +2.39788 q^{38} -4.19584 q^{40} -5.47333 q^{41} +10.9515 q^{43} +3.74982 q^{44} +9.60863 q^{46} +0.295438 q^{47} +1.37462 q^{49} +2.39788 q^{50} +17.7659 q^{52} +3.81149 q^{53} -1.00000 q^{55} -12.1423 q^{56} -22.3680 q^{58} +5.54910 q^{59} -1.01018 q^{61} -15.8325 q^{62} -10.5171 q^{64} -4.73780 q^{65} +6.98807 q^{67} +21.2010 q^{68} +6.93920 q^{70} +1.02085 q^{71} -0.202033 q^{73} +14.5716 q^{74} +3.74982 q^{76} -2.89389 q^{77} +7.28690 q^{79} -2.56149 q^{80} -13.1244 q^{82} +13.7041 q^{83} -5.65389 q^{85} +26.2603 q^{86} +4.19584 q^{88} +15.8152 q^{89} -13.7107 q^{91} +15.0260 q^{92} +0.708423 q^{94} -1.00000 q^{95} +4.81308 q^{97} +3.29618 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8}+O(q^{10}) $ 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8	$ 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} - 3 q^{10} + 5 q^{11} + q^{13} - 3 q^{16} + 3 q^{17} + 5 q^{19} - 5 q^{20} + 3 q^{22} + 8 q^{23} + 5 q^{25} + 16 q^{26} - 22 q^{28} - 11 q^{29} - 5 q^{31} + 2 q^{32} + 4 q^{34} + 11 q^{35} - 9 q^{37} + 3 q^{38} + 3 q^{40} - 15 q^{41} - 13 q^{43} + 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{49} + 3 q^{50} + q^{52} + 5 q^{53} - 5 q^{55} - 33 q^{58} + 17 q^{59} + 3 q^{61} - 14 q^{62} - 17 q^{64} - q^{65} - 28 q^{67} + 25 q^{68} + 6 q^{71} - 16 q^{73} + 21 q^{74} + 5 q^{76} - 11 q^{77} + 3 q^{79} + 3 q^{80} + 2 q^{82} + 33 q^{83} - 3 q^{85} - 10 q^{86} - 3 q^{88} + 16 q^{89} - 22 q^{91} + 19 q^{92} - 10 q^{94} - 5 q^{95} - 14 q^{97} - 10 q^{98}+O(q^{100}) $ 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 - 3 * q^10 + 5 * q^11 + q^13 - 3 * q^16 + 3 * q^17 + 5 * q^19 - 5 * q^20 + 3 * q^22 + 8 * q^23 + 5 * q^25 + 16 * q^26 - 22 * q^28 - 11 * q^29 - 5 * q^31 + 2 * q^32 + 4 * q^34 + 11 * q^35 - 9 * q^37 + 3 * q^38 + 3 * q^40 - 15 * q^41 - 13 * q^43 + 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^49 + 3 * q^50 + q^52 + 5 * q^53 - 5 * q^55 - 33 * q^58 + 17 * q^59 + 3 * q^61 - 14 * q^62 - 17 * q^64 - q^65 - 28 * q^67 + 25 * q^68 + 6 * q^71 - 16 * q^73 + 21 * q^74 + 5 * q^76 - 11 * q^77 + 3 * q^79 + 3 * q^80 + 2 * q^82 + 33 * q^83 - 3 * q^85 - 10 * q^86 - 3 * q^88 + 16 * q^89 - 22 * q^91 + 19 * q^92 - 10 * q^94 - 5 * q^95 - 14 * q^97 - 10 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.39788	1.69556	0.847778	−	0.530352i	$-0.177940\pi$
$2$	2.39788	1.69556	0.847778	+	0.530352i	$0.177940\pi$
$3$	0	0
$3$	0	0
$4$	3.74982	1.87491
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.89389	−1.09379	−0.546895	−	0.837201i	$-0.684190\pi$
$7$	−2.89389	−1.09379	−0.546895	+	0.837201i	$0.684190\pi$
$8$	4.19584	1.48345
$9$	0	0
$10$	−2.39788	−0.758275
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.73780	1.31403	0.657015	−	0.753878i	$-0.271819\pi$
$13$	4.73780	1.31403	0.657015	+	0.753878i	$0.271819\pi$
$14$	−6.93920	−1.85458
$15$	0	0
$16$	2.56149	0.640372
$17$	5.65389	1.37127	0.685635	−	0.727946i	$-0.259525\pi$
$17$	5.65389	1.37127	0.685635	+	0.727946i	$0.259525\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−3.74982	−0.838484
$21$	0	0
$22$	2.39788	0.511229
$23$	4.00714	0.835547	0.417773	−	0.908551i	$-0.362811\pi$
$23$	4.00714	0.835547	0.417773	+	0.908551i	$0.362811\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	11.3607	2.22801
$27$	0	0
$28$	−10.8516	−2.05075
$29$	−9.32825	−1.73221	−0.866106	−	0.499860i	$-0.833385\pi$
$29$	−9.32825	−1.73221	−0.866106	+	0.499860i	$0.833385\pi$
$30$	0	0
$31$	−6.60270	−1.18588	−0.592940	−	0.805247i	$-0.702033\pi$
$31$	−6.60270	−1.18588	−0.592940	+	0.805247i	$0.702033\pi$
$32$	−2.24955	−0.397669
$33$	0	0
$34$	13.5573	2.32506
$35$	2.89389	0.489157
$36$	0	0
$37$	6.07686	0.999029	0.499515	−	0.866305i	$-0.333512\pi$
$37$	6.07686	0.999029	0.499515	+	0.866305i	$0.333512\pi$
$38$	2.39788	0.388987
$39$	0	0
$40$	−4.19584	−0.663421
$41$	−5.47333	−0.854791	−0.427395	−	0.904065i	$-0.640569\pi$
$41$	−5.47333	−0.854791	−0.427395	+	0.904065i	$0.640569\pi$
$42$	0	0
$43$	10.9515	1.67009	0.835043	−	0.550185i	$-0.185443\pi$
$43$	10.9515	1.67009	0.835043	+	0.550185i	$0.185443\pi$
$44$	3.74982	0.565306
$45$	0	0
$46$	9.60863	1.41672
$47$	0.295438	0.0430940	0.0215470	−	0.999768i	$-0.493141\pi$
$47$	0.295438	0.0430940	0.0215470	+	0.999768i	$0.493141\pi$
$48$	0	0
$49$	1.37462	0.196375
$50$	2.39788	0.339111
$51$	0	0
$52$	17.7659	2.46368
$53$	3.81149	0.523549	0.261774	−	0.965129i	$-0.415692\pi$
$53$	3.81149	0.523549	0.261774	+	0.965129i	$0.415692\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	−12.1423	−1.62259
$57$	0	0
$58$	−22.3680	−2.93706
$59$	5.54910	0.722431	0.361215	−	0.932482i	$-0.382362\pi$
$59$	5.54910	0.722431	0.361215	+	0.932482i	$0.382362\pi$
$60$	0	0
$61$	−1.01018	−0.129340	−0.0646700	−	0.997907i	$-0.520599\pi$
$61$	−1.01018	−0.129340	−0.0646700	+	0.997907i	$0.520599\pi$
$62$	−15.8325	−2.01073
$63$	0	0
$64$	−10.5171	−1.31464
$65$	−4.73780	−0.587652
$66$	0	0
$67$	6.98807	0.853729	0.426864	−	0.904316i	$-0.359618\pi$
$67$	6.98807	0.853729	0.426864	+	0.904316i	$0.359618\pi$
$68$	21.2010	2.57100
$69$	0	0
$70$	6.93920	0.829393
$71$	1.02085	0.121152	0.0605761	−	0.998164i	$-0.480706\pi$
$71$	1.02085	0.121152	0.0605761	+	0.998164i	$0.480706\pi$
$72$	0	0
$73$	−0.202033	−0.0236462	−0.0118231	−	0.999930i	$-0.503763\pi$
$73$	−0.202033	−0.0236462	−0.0118231	+	0.999930i	$0.503763\pi$
$74$	14.5716	1.69391
$75$	0	0
$76$	3.74982	0.430133
$77$	−2.89389	−0.329790
$78$	0	0
$79$	7.28690	0.819840	0.409920	−	0.912121i	$-0.365557\pi$
$79$	7.28690	0.819840	0.409920	+	0.912121i	$0.365557\pi$
$80$	−2.56149	−0.286383
$81$	0	0
$82$	−13.1244	−1.44935
$83$	13.7041	1.50422	0.752109	−	0.659039i	$-0.229037\pi$
$83$	13.7041	1.50422	0.752109	+	0.659039i	$0.229037\pi$
$84$	0	0
$85$	−5.65389	−0.613250
$86$	26.2603	2.83172
$87$	0	0
$88$	4.19584	0.447278
$89$	15.8152	1.67641	0.838203	−	0.545359i	$-0.183606\pi$
$89$	15.8152	1.67641	0.838203	+	0.545359i	$0.183606\pi$
$90$	0	0
$91$	−13.7107	−1.43727
$92$	15.0260	1.56657
$93$	0	0
$94$	0.708423	0.0730683
$95$	−1.00000	−0.102598
$96$	0	0
$97$	4.81308	0.488694	0.244347	−	0.969688i	$-0.421426\pi$
$97$	4.81308	0.488694	0.244347	+	0.969688i	$0.421426\pi$
$98$	3.29618	0.332964
$99$	0	0
$100$	3.74982	0.374982
$101$	−1.76221	−0.175346	−0.0876730	−	0.996149i	$-0.527943\pi$
$101$	−1.76221	−0.175346	−0.0876730	+	0.996149i	$0.527943\pi$
$102$	0	0
$103$	−5.53571	−0.545450	−0.272725	−	0.962092i	$-0.587925\pi$
$103$	−5.53571	−0.545450	−0.272725	+	0.962092i	$0.587925\pi$
$104$	19.8791	1.94930
$105$	0	0
$106$	9.13949	0.887706
$107$	4.32073	0.417701	0.208851	−	0.977948i	$-0.433028\pi$
$107$	4.32073	0.417701	0.208851	+	0.977948i	$0.433028\pi$
$108$	0	0
$109$	−13.0039	−1.24555	−0.622776	−	0.782400i	$-0.713995\pi$
$109$	−13.0039	−1.24555	−0.622776	+	0.782400i	$0.713995\pi$
$110$	−2.39788	−0.228629
$111$	0	0
$112$	−7.41267	−0.700432
$113$	17.0047	1.59967	0.799833	−	0.600223i	$-0.204922\pi$
$113$	17.0047	1.59967	0.799833	+	0.600223i	$0.204922\pi$
$114$	0	0
$115$	−4.00714	−0.373668
$116$	−34.9792	−3.24774
$117$	0	0
$118$	13.3061	1.22492
$119$	−16.3618	−1.49988
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.42228	−0.219303
$123$	0	0
$124$	−24.7589	−2.22342
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−9.80245	−0.869826	−0.434913	−	0.900472i	$-0.643221\pi$
$127$	−9.80245	−0.869826	−0.434913	+	0.900472i	$0.643221\pi$
$128$	−20.7197	−1.83138
$129$	0	0
$130$	−11.3607	−0.996396
$131$	13.4383	1.17411	0.587053	−	0.809549i	$-0.300288\pi$
$131$	13.4383	1.17411	0.587053	+	0.809549i	$0.300288\pi$
$132$	0	0
$133$	−2.89389	−0.250932
$134$	16.7565	1.44754
$135$	0	0
$136$	23.7228	2.03422
$137$	16.2122	1.38510	0.692549	−	0.721371i	$-0.256488\pi$
$137$	16.2122	1.38510	0.692549	+	0.721371i	$0.256488\pi$
$138$	0	0
$139$	19.0373	1.61472	0.807359	−	0.590060i	$-0.200896\pi$
$139$	19.0373	1.61472	0.807359	+	0.590060i	$0.200896\pi$
$140$	10.8516	0.917125
$141$	0	0
$142$	2.44787	0.205420
$143$	4.73780	0.396195
$144$	0	0
$145$	9.32825	0.774669
$146$	−0.484451	−0.0400934
$147$	0	0
$148$	22.7871	1.87309
$149$	15.3234	1.25534	0.627670	−	0.778480i	$-0.284009\pi$
$149$	15.3234	1.25534	0.627670	+	0.778480i	$0.284009\pi$
$150$	0	0
$151$	−1.25936	−0.102485	−0.0512426	−	0.998686i	$-0.516318\pi$
$151$	−1.25936	−0.102485	−0.0512426	+	0.998686i	$0.516318\pi$
$152$	4.19584	0.340328
$153$	0	0
$154$	−6.93920	−0.559177
$155$	6.60270	0.530342
$156$	0	0
$157$	12.1861	0.972559	0.486279	−	0.873803i	$-0.338354\pi$
$157$	12.1861	0.972559	0.486279	+	0.873803i	$0.338354\pi$
$158$	17.4731	1.39008
$159$	0	0
$160$	2.24955	0.177843
$161$	−11.5962	−0.913912
$162$	0	0
$163$	6.09722	0.477571	0.238786	−	0.971072i	$-0.423251\pi$
$163$	6.09722	0.477571	0.238786	+	0.971072i	$0.423251\pi$
$164$	−20.5240	−1.60265
$165$	0	0
$166$	32.8607	2.55048
$167$	−1.06720	−0.0825826	−0.0412913	−	0.999147i	$-0.513147\pi$
$167$	−1.06720	−0.0825826	−0.0412913	+	0.999147i	$0.513147\pi$
$168$	0	0
$169$	9.44675	0.726673
$170$	−13.5573	−1.03980
$171$	0	0
$172$	41.0661	3.13126
$173$	13.0289	0.990571	0.495286	−	0.868730i	$-0.335063\pi$
$173$	13.0289	0.990571	0.495286	+	0.868730i	$0.335063\pi$
$174$	0	0
$175$	−2.89389	−0.218758
$176$	2.56149	0.193079
$177$	0	0
$178$	37.9229	2.84244
$179$	−20.6273	−1.54176	−0.770878	−	0.636983i	$-0.780182\pi$
$179$	−20.6273	−1.54176	−0.770878	+	0.636983i	$0.780182\pi$
$180$	0	0
$181$	−20.5908	−1.53050	−0.765251	−	0.643732i	$-0.777385\pi$
$181$	−20.5908	−1.53050	−0.765251	+	0.643732i	$0.777385\pi$
$182$	−32.8766	−2.43697
$183$	0	0
$184$	16.8133	1.23950
$185$	−6.07686	−0.446779
$186$	0	0
$187$	5.65389	0.413453
$188$	1.10784	0.0807973
$189$	0	0
$190$	−2.39788	−0.173960
$191$	−8.40047	−0.607837	−0.303918	−	0.952698i	$-0.598295\pi$
$191$	−8.40047	−0.607837	−0.303918	+	0.952698i	$0.598295\pi$
$192$	0	0
$193$	11.4982	0.827661	0.413830	−	0.910354i	$-0.364191\pi$
$193$	11.4982	0.827661	0.413830	+	0.910354i	$0.364191\pi$
$194$	11.5412	0.828607
$195$	0	0
$196$	5.15458	0.368185
$197$	−27.0376	−1.92635	−0.963173	−	0.268883i	$-0.913345\pi$
$197$	−27.0376	−1.92635	−0.963173	+	0.268883i	$0.913345\pi$
$198$	0	0
$199$	−5.18460	−0.367526	−0.183763	−	0.982971i	$-0.558828\pi$
$199$	−5.18460	−0.367526	−0.183763	+	0.982971i	$0.558828\pi$
$200$	4.19584	0.296691
$201$	0	0
$202$	−4.22555	−0.297309
$203$	26.9950	1.89468
$204$	0	0
$205$	5.47333	0.382274
$206$	−13.2740	−0.924840
$207$	0	0
$208$	12.1358	0.841467
$209$	1.00000	0.0691714
$210$	0	0
$211$	−21.5427	−1.48306	−0.741529	−	0.670921i	$-0.765899\pi$
$211$	−21.5427	−1.48306	−0.741529	+	0.670921i	$0.765899\pi$
$212$	14.2924	0.981606
$213$	0	0
$214$	10.3606	0.708235
$215$	−10.9515	−0.746885
$216$	0	0
$217$	19.1075	1.29710
$218$	−31.1818	−2.11190
$219$	0	0
$220$	−3.74982	−0.252813
$221$	26.7870	1.80189
$222$	0	0
$223$	−26.9600	−1.80537	−0.902687	−	0.430298i	$-0.858408\pi$
$223$	−26.9600	−1.80537	−0.902687	+	0.430298i	$0.858408\pi$
$224$	6.50997	0.434966
$225$	0	0
$226$	40.7751	2.71232
$227$	−9.02570	−0.599057	−0.299528	−	0.954087i	$-0.596829\pi$
$227$	−9.02570	−0.599057	−0.299528	+	0.954087i	$0.596829\pi$
$228$	0	0
$229$	−13.7697	−0.909926	−0.454963	−	0.890510i	$-0.650347\pi$
$229$	−13.7697	−0.909926	−0.454963	+	0.890510i	$0.650347\pi$
$230$	−9.60863	−0.633575
$231$	0	0
$232$	−39.1399	−2.56966
$233$	2.45587	0.160890	0.0804448	−	0.996759i	$-0.474366\pi$
$233$	2.45587	0.160890	0.0804448	+	0.996759i	$0.474366\pi$
$234$	0	0
$235$	−0.295438	−0.0192722
$236$	20.8081	1.35449
$237$	0	0
$238$	−39.2335	−2.54313
$239$	22.8439	1.47765	0.738825	−	0.673897i	$-0.235381\pi$
$239$	22.8439	1.47765	0.738825	+	0.673897i	$0.235381\pi$
$240$	0	0
$241$	16.1181	1.03826	0.519129	−	0.854696i	$-0.326256\pi$
$241$	16.1181	1.03826	0.519129	+	0.854696i	$0.326256\pi$
$242$	2.39788	0.154141
$243$	0	0
$244$	−3.78798	−0.242501
$245$	−1.37462	−0.0878215
$246$	0	0
$247$	4.73780	0.301459
$248$	−27.7039	−1.75920
$249$	0	0
$250$	−2.39788	−0.151655
$251$	18.8758	1.19143	0.595716	−	0.803195i	$-0.296868\pi$
$251$	18.8758	1.19143	0.595716	+	0.803195i	$0.296868\pi$
$252$	0	0
$253$	4.00714	0.251927
$254$	−23.5051	−1.47484
$255$	0	0
$256$	−28.6490	−1.79056
$257$	−7.48029	−0.466608	−0.233304	−	0.972404i	$-0.574954\pi$
$257$	−7.48029	−0.466608	−0.233304	+	0.972404i	$0.574954\pi$
$258$	0	0
$259$	−17.5858	−1.09273
$260$	−17.7659	−1.10179
$261$	0	0
$262$	32.2233	1.99076
$263$	17.7703	1.09577	0.547883	−	0.836555i	$-0.315434\pi$
$263$	17.7703	1.09577	0.547883	+	0.836555i	$0.315434\pi$
$264$	0	0
$265$	−3.81149	−0.234138
$266$	−6.93920	−0.425470
$267$	0	0
$268$	26.2040	1.60066
$269$	−25.9320	−1.58110	−0.790550	−	0.612398i	$-0.790205\pi$
$269$	−25.9320	−1.58110	−0.790550	+	0.612398i	$0.790205\pi$
$270$	0	0
$271$	−16.2686	−0.988248	−0.494124	−	0.869391i	$-0.664511\pi$
$271$	−16.2686	−0.988248	−0.494124	+	0.869391i	$0.664511\pi$
$272$	14.4824	0.878122
$273$	0	0
$274$	38.8748	2.34851
$275$	1.00000	0.0603023
$276$	0	0
$277$	−17.9217	−1.07681	−0.538405	−	0.842686i	$-0.680973\pi$
$277$	−17.9217	−1.07681	−0.538405	+	0.842686i	$0.680973\pi$
$278$	45.6490	2.73784
$279$	0	0
$280$	12.1423	0.725643
$281$	28.8563	1.72142	0.860712	−	0.509093i	$-0.170019\pi$
$281$	28.8563	1.72142	0.860712	+	0.509093i	$0.170019\pi$
$282$	0	0
$283$	−10.7620	−0.639736	−0.319868	−	0.947462i	$-0.603639\pi$
$283$	−10.7620	−0.639736	−0.319868	+	0.947462i	$0.603639\pi$
$284$	3.82799	0.227149
$285$	0	0
$286$	11.3607	0.671770
$287$	15.8392	0.934961
$288$	0	0
$289$	14.9665	0.880380
$290$	22.3680	1.31349
$291$	0	0
$292$	−0.757587	−0.0443345
$293$	−5.14307	−0.300462	−0.150231	−	0.988651i	$-0.548002\pi$
$293$	−5.14307	−0.300462	−0.150231	+	0.988651i	$0.548002\pi$
$294$	0	0
$295$	−5.54910	−0.323081
$296$	25.4975	1.48201
$297$	0	0
$298$	36.7436	2.12850
$299$	18.9850	1.09793
$300$	0	0
$301$	−31.6924	−1.82672
$302$	−3.01979	−0.173769
$303$	0	0
$304$	2.56149	0.146911
$305$	1.01018	0.0578427
$306$	0	0
$307$	28.1017	1.60385	0.801925	−	0.597425i	$-0.203809\pi$
$307$	28.1017	1.60385	0.801925	+	0.597425i	$0.203809\pi$
$308$	−10.8516	−0.618326
$309$	0	0
$310$	15.8325	0.899224
$311$	14.1909	0.804692	0.402346	−	0.915488i	$-0.368195\pi$
$311$	14.1909	0.804692	0.402346	+	0.915488i	$0.368195\pi$
$312$	0	0
$313$	−9.46083	−0.534758	−0.267379	−	0.963591i	$-0.586158\pi$
$313$	−9.46083	−0.534758	−0.267379	+	0.963591i	$0.586158\pi$
$314$	29.2208	1.64903
$315$	0	0
$316$	27.3245	1.53712
$317$	23.8673	1.34052	0.670260	−	0.742126i	$-0.266182\pi$
$317$	23.8673	1.34052	0.670260	+	0.742126i	$0.266182\pi$
$318$	0	0
$319$	−9.32825	−0.522282
$320$	10.5171	0.587926
$321$	0	0
$322$	−27.8064	−1.54959
$323$	5.65389	0.314591
$324$	0	0
$325$	4.73780	0.262806
$326$	14.6204	0.809748
$327$	0	0
$328$	−22.9652	−1.26804
$329$	−0.854966	−0.0471358
$330$	0	0
$331$	−13.1843	−0.724674	−0.362337	−	0.932047i	$-0.618021\pi$
$331$	−13.1843	−0.724674	−0.362337	+	0.932047i	$0.618021\pi$
$332$	51.3877	2.82027
$333$	0	0
$334$	−2.55902	−0.140023
$335$	−6.98807	−0.381799
$336$	0	0
$337$	17.0148	0.926854	0.463427	−	0.886135i	$-0.346620\pi$
$337$	17.0148	0.926854	0.463427	+	0.886135i	$0.346620\pi$
$338$	22.6521	1.23211
$339$	0	0
$340$	−21.2010	−1.14979
$341$	−6.60270	−0.357556
$342$	0	0
$343$	16.2792	0.878997
$344$	45.9507	2.47750
$345$	0	0
$346$	31.2418	1.67957
$347$	30.2606	1.62447	0.812236	−	0.583329i	$-0.198250\pi$
$347$	30.2606	1.62447	0.812236	+	0.583329i	$0.198250\pi$
$348$	0	0
$349$	5.13463	0.274851	0.137425	−	0.990512i	$-0.456117\pi$
$349$	5.13463	0.274851	0.137425	+	0.990512i	$0.456117\pi$
$350$	−6.93920	−0.370916
$351$	0	0
$352$	−2.24955	−0.119902
$353$	10.5661	0.562376	0.281188	−	0.959653i	$-0.409272\pi$
$353$	10.5661	0.562376	0.281188	+	0.959653i	$0.409272\pi$
$354$	0	0
$355$	−1.02085	−0.0541809
$356$	59.3040	3.14311
$357$	0	0
$358$	−49.4617	−2.61413
$359$	−16.5178	−0.871778	−0.435889	−	0.900000i	$-0.643566\pi$
$359$	−16.5178	−0.871778	−0.435889	+	0.900000i	$0.643566\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−49.3742	−2.59505
$363$	0	0
$364$	−51.4126	−2.69475
$365$	0.202033	0.0105749
$366$	0	0
$367$	19.8761	1.03752	0.518762	−	0.854919i	$-0.326393\pi$
$367$	19.8761	1.03752	0.518762	+	0.854919i	$0.326393\pi$
$368$	10.2642	0.535061
$369$	0	0
$370$	−14.5716	−0.757539
$371$	−11.0301	−0.572652
$372$	0	0
$373$	−11.4151	−0.591052	−0.295526	−	0.955335i	$-0.595495\pi$
$373$	−11.4151	−0.591052	−0.295526	+	0.955335i	$0.595495\pi$
$374$	13.5573	0.701033
$375$	0	0
$376$	1.23961	0.0639280
$377$	−44.1954	−2.27618
$378$	0	0
$379$	1.99925	0.102695	0.0513474	−	0.998681i	$-0.483648\pi$
$379$	1.99925	0.102695	0.0513474	+	0.998681i	$0.483648\pi$
$380$	−3.74982	−0.192362
$381$	0	0
$382$	−20.1433	−1.03062
$383$	7.07007	0.361264	0.180632	−	0.983551i	$-0.442186\pi$
$383$	7.07007	0.361264	0.180632	+	0.983551i	$0.442186\pi$
$384$	0	0
$385$	2.89389	0.147487
$386$	27.5713	1.40334
$387$	0	0
$388$	18.0481	0.916256
$389$	−17.0237	−0.863135	−0.431568	−	0.902081i	$-0.642039\pi$
$389$	−17.0237	−0.863135	−0.431568	+	0.902081i	$0.642039\pi$
$390$	0	0
$391$	22.6559	1.14576
$392$	5.76771	0.291313
$393$	0	0
$394$	−64.8327	−3.26623
$395$	−7.28690	−0.366644
$396$	0	0
$397$	−25.3590	−1.27273	−0.636366	−	0.771387i	$-0.719563\pi$
$397$	−25.3590	−1.27273	−0.636366	+	0.771387i	$0.719563\pi$
$398$	−12.4320	−0.623161
$399$	0	0
$400$	2.56149	0.128074
$401$	−19.6449	−0.981019	−0.490509	−	0.871436i	$-0.663189\pi$
$401$	−19.6449	−0.981019	−0.490509	+	0.871436i	$0.663189\pi$
$402$	0	0
$403$	−31.2823	−1.55828
$404$	−6.60795	−0.328758
$405$	0	0
$406$	64.7306	3.21253
$407$	6.07686	0.301219
$408$	0	0
$409$	−28.6527	−1.41678	−0.708392	−	0.705819i	$-0.750579\pi$
$409$	−28.6527	−1.41678	−0.708392	+	0.705819i	$0.750579\pi$
$410$	13.1244	0.648167
$411$	0	0
$412$	−20.7579	−1.02267
$413$	−16.0585	−0.790187
$414$	0	0
$415$	−13.7041	−0.672706
$416$	−10.6579	−0.522548
$417$	0	0
$418$	2.39788	0.117284
$419$	2.80518	0.137042	0.0685211	−	0.997650i	$-0.478172\pi$
$419$	2.80518	0.137042	0.0685211	+	0.997650i	$0.478172\pi$
$420$	0	0
$421$	3.26165	0.158963	0.0794815	−	0.996836i	$-0.474674\pi$
$421$	3.26165	0.158963	0.0794815	+	0.996836i	$0.474674\pi$
$422$	−51.6566	−2.51461
$423$	0	0
$424$	15.9924	0.776661
$425$	5.65389	0.274254
$426$	0	0
$427$	2.92335	0.141471
$428$	16.2020	0.783151
$429$	0	0
$430$	−26.2603	−1.26639
$431$	−32.7606	−1.57802	−0.789012	−	0.614378i	$-0.789407\pi$
$431$	−32.7606	−1.57802	−0.789012	+	0.614378i	$0.789407\pi$
$432$	0	0
$433$	−21.8656	−1.05079	−0.525396	−	0.850858i	$-0.676083\pi$
$433$	−21.8656	−1.05079	−0.525396	+	0.850858i	$0.676083\pi$
$434$	45.8175	2.19931
$435$	0	0
$436$	−48.7624	−2.33529
$437$	4.00714	0.191688
$438$	0	0
$439$	30.6280	1.46180	0.730898	−	0.682486i	$-0.239101\pi$
$439$	30.6280	1.46180	0.730898	+	0.682486i	$0.239101\pi$
$440$	−4.19584	−0.200029
$441$	0	0
$442$	64.2319	3.05520
$443$	−4.54914	−0.216136	−0.108068	−	0.994144i	$-0.534466\pi$
$443$	−4.54914	−0.216136	−0.108068	+	0.994144i	$0.534466\pi$
$444$	0	0
$445$	−15.8152	−0.749711
$446$	−64.6467	−3.06111
$447$	0	0
$448$	30.4355	1.43794
$449$	−11.5308	−0.544172	−0.272086	−	0.962273i	$-0.587714\pi$
$449$	−11.5308	−0.544172	−0.272086	+	0.962273i	$0.587714\pi$
$450$	0	0
$451$	−5.47333	−0.257729
$452$	63.7644	2.99923
$453$	0	0
$454$	−21.6425	−1.01573
$455$	13.7107	0.642767
$456$	0	0
$457$	11.3778	0.532231	0.266115	−	0.963941i	$-0.414260\pi$
$457$	11.3778	0.532231	0.266115	+	0.963941i	$0.414260\pi$
$458$	−33.0180	−1.54283
$459$	0	0
$460$	−15.0260	−0.700593
$461$	−4.37265	−0.203655	−0.101827	−	0.994802i	$-0.532469\pi$
$461$	−4.37265	−0.203655	−0.101827	+	0.994802i	$0.532469\pi$
$462$	0	0
$463$	−9.31982	−0.433129	−0.216564	−	0.976268i	$-0.569485\pi$
$463$	−9.31982	−0.433129	−0.216564	+	0.976268i	$0.569485\pi$
$464$	−23.8942	−1.10926
$465$	0	0
$466$	5.88888	0.272797
$467$	28.0711	1.29897	0.649487	−	0.760372i	$-0.274984\pi$
$467$	28.0711	1.29897	0.649487	+	0.760372i	$0.274984\pi$
$468$	0	0
$469$	−20.2227	−0.933799
$470$	−0.708423	−0.0326771
$471$	0	0
$472$	23.2831	1.07169
$473$	10.9515	0.503550
$474$	0	0
$475$	1.00000	0.0458831
$476$	−61.3536	−2.81214
$477$	0	0
$478$	54.7769	2.50544
$479$	5.65184	0.258239	0.129120	−	0.991629i	$-0.458785\pi$
$479$	5.65184	0.258239	0.129120	+	0.991629i	$0.458785\pi$
$480$	0	0
$481$	28.7909	1.31275
$482$	38.6492	1.76042
$483$	0	0
$484$	3.74982	0.170446
$485$	−4.81308	−0.218550
$486$	0	0
$487$	−6.09349	−0.276122	−0.138061	−	0.990424i	$-0.544087\pi$
$487$	−6.09349	−0.276122	−0.138061	+	0.990424i	$0.544087\pi$
$488$	−4.23855	−0.191870
$489$	0	0
$490$	−3.29618	−0.148906
$491$	−8.56286	−0.386436	−0.193218	−	0.981156i	$-0.561893\pi$
$491$	−8.56286	−0.386436	−0.193218	+	0.981156i	$0.561893\pi$
$492$	0	0
$493$	−52.7409	−2.37533
$494$	11.3607	0.511140
$495$	0	0
$496$	−16.9127	−0.759404
$497$	−2.95422	−0.132515
$498$	0	0
$499$	−1.35537	−0.0606749	−0.0303374	−	0.999540i	$-0.509658\pi$
$499$	−1.35537	−0.0606749	−0.0303374	+	0.999540i	$0.509658\pi$
$500$	−3.74982	−0.167697
$501$	0	0
$502$	45.2619	2.02014
$503$	−13.9030	−0.619903	−0.309952	−	0.950752i	$-0.600313\pi$
$503$	−13.9030	−0.619903	−0.309952	+	0.950752i	$0.600313\pi$
$504$	0	0
$505$	1.76221	0.0784172
$506$	9.60863	0.427156
$507$	0	0
$508$	−36.7574	−1.63084
$509$	3.21978	0.142714	0.0713572	−	0.997451i	$-0.477267\pi$
$509$	3.21978	0.142714	0.0713572	+	0.997451i	$0.477267\pi$
$510$	0	0
$511$	0.584663	0.0258640
$512$	−27.2574	−1.20462
$513$	0	0
$514$	−17.9368	−0.791160
$515$	5.53571	0.243933
$516$	0	0
$517$	0.295438	0.0129933
$518$	−42.1686	−1.85278
$519$	0	0
$520$	−19.8791	−0.871755
$521$	−3.09750	−0.135704	−0.0678520	−	0.997695i	$-0.521615\pi$
$521$	−3.09750	−0.135704	−0.0678520	+	0.997695i	$0.521615\pi$
$522$	0	0
$523$	13.2007	0.577228	0.288614	−	0.957446i	$-0.406806\pi$
$523$	13.2007	0.577228	0.288614	+	0.957446i	$0.406806\pi$
$524$	50.3910	2.20134
$525$	0	0
$526$	42.6111	1.85793
$527$	−37.3309	−1.62616
$528$	0	0
$529$	−6.94281	−0.301861
$530$	−9.13949	−0.396994
$531$	0	0
$532$	−10.8516	−0.470475
$533$	−25.9316	−1.12322
$534$	0	0
$535$	−4.32073	−0.186802
$536$	29.3209	1.26647
$537$	0	0
$538$	−62.1817	−2.68084
$539$	1.37462	0.0592092
$540$	0	0
$541$	−6.57608	−0.282728	−0.141364	−	0.989958i	$-0.545149\pi$
$541$	−6.57608	−0.282728	−0.141364	+	0.989958i	$0.545149\pi$
$542$	−39.0102	−1.67563
$543$	0	0
$544$	−12.7187	−0.545311
$545$	13.0039	0.557028
$546$	0	0
$547$	−23.9559	−1.02428	−0.512139	−	0.858902i	$-0.671147\pi$
$547$	−23.9559	−1.02428	−0.512139	+	0.858902i	$0.671147\pi$
$548$	60.7926	2.59693
$549$	0	0
$550$	2.39788	0.102246
$551$	−9.32825	−0.397397
$552$	0	0
$553$	−21.0875	−0.896732
$554$	−42.9740	−1.82579
$555$	0	0
$556$	71.3862	3.02745
$557$	11.2404	0.476271	0.238135	−	0.971232i	$-0.423464\pi$
$557$	11.2404	0.476271	0.238135	+	0.971232i	$0.423464\pi$
$558$	0	0
$559$	51.8860	2.19454
$560$	7.41267	0.313243
$561$	0	0
$562$	69.1939	2.91877
$563$	−44.9880	−1.89602	−0.948009	−	0.318242i	$-0.896907\pi$
$563$	−44.9880	−1.89602	−0.948009	+	0.318242i	$0.896907\pi$
$564$	0	0
$565$	−17.0047	−0.715392
$566$	−25.8060	−1.08471
$567$	0	0
$568$	4.28331	0.179724
$569$	−33.2099	−1.39223	−0.696115	−	0.717930i	$-0.745090\pi$
$569$	−33.2099	−1.39223	−0.696115	+	0.717930i	$0.745090\pi$
$570$	0	0
$571$	−44.6440	−1.86829	−0.934146	−	0.356891i	$-0.883837\pi$
$571$	−44.6440	−1.86829	−0.934146	+	0.356891i	$0.883837\pi$
$572$	17.7659	0.742829
$573$	0	0
$574$	37.9806	1.58528
$575$	4.00714	0.167109
$576$	0	0
$577$	−45.6776	−1.90158	−0.950792	−	0.309831i	$-0.899728\pi$
$577$	−45.6776	−1.90158	−0.950792	+	0.309831i	$0.899728\pi$
$578$	35.8877	1.49273
$579$	0	0
$580$	34.9792	1.45243
$581$	−39.6581	−1.64530
$582$	0	0
$583$	3.81149	0.157856
$584$	−0.847700	−0.0350781
$585$	0	0
$586$	−12.3325	−0.509449
$587$	17.5933	0.726155	0.363078	−	0.931759i	$-0.381726\pi$
$587$	17.5933	0.726155	0.363078	+	0.931759i	$0.381726\pi$
$588$	0	0
$589$	−6.60270	−0.272060
$590$	−13.3061	−0.547802
$591$	0	0
$592$	15.5658	0.639750
$593$	−32.4709	−1.33342	−0.666710	−	0.745317i	$-0.732298\pi$
$593$	−32.4709	−1.33342	−0.666710	+	0.745317i	$0.732298\pi$
$594$	0	0
$595$	16.3618	0.670767
$596$	57.4598	2.35365
$597$	0	0
$598$	45.5238	1.86161
$599$	−1.31594	−0.0537679	−0.0268839	−	0.999639i	$-0.508558\pi$
$599$	−1.31594	−0.0537679	−0.0268839	+	0.999639i	$0.508558\pi$
$600$	0	0
$601$	−4.53331	−0.184918	−0.0924588	−	0.995717i	$-0.529473\pi$
$601$	−4.53331	−0.184918	−0.0924588	+	0.995717i	$0.529473\pi$
$602$	−75.9946	−3.09731
$603$	0	0
$604$	−4.72236	−0.192150
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−14.2654	−0.579016	−0.289508	−	0.957176i	$-0.593492\pi$
$607$	−14.2654	−0.579016	−0.289508	+	0.957176i	$0.593492\pi$
$608$	−2.24955	−0.0912315
$609$	0	0
$610$	2.42228	0.0980754
$611$	1.39972	0.0566268
$612$	0	0
$613$	5.06662	0.204639	0.102319	−	0.994752i	$-0.467374\pi$
$613$	5.06662	0.204639	0.102319	+	0.994752i	$0.467374\pi$
$614$	67.3845	2.71942
$615$	0	0
$616$	−12.1423	−0.489228
$617$	−19.5460	−0.786893	−0.393446	−	0.919348i	$-0.628717\pi$
$617$	−19.5460	−0.786893	−0.393446	+	0.919348i	$0.628717\pi$
$618$	0	0
$619$	8.78654	0.353161	0.176580	−	0.984286i	$-0.443496\pi$
$619$	8.78654	0.353161	0.176580	+	0.984286i	$0.443496\pi$
$620$	24.7589	0.994342
$621$	0	0
$622$	34.0280	1.36440
$623$	−45.7675	−1.83363
$624$	0	0
$625$	1.00000	0.0400000
$626$	−22.6859	−0.906712
$627$	0	0
$628$	45.6957	1.82346
$629$	34.3579	1.36994
$630$	0	0
$631$	−0.0559751	−0.00222833	−0.00111417	−	0.999999i	$-0.500355\pi$
$631$	−0.0559751	−0.00222833	−0.00111417	+	0.999999i	$0.500355\pi$
$632$	30.5747	1.21620
$633$	0	0
$634$	57.2308	2.27293
$635$	9.80245	0.388998
$636$	0	0
$637$	6.51269	0.258042
$638$	−22.3680	−0.885558
$639$	0	0
$640$	20.7197	0.819017
$641$	−4.49262	−0.177448	−0.0887239	−	0.996056i	$-0.528279\pi$
$641$	−4.49262	−0.177448	−0.0887239	+	0.996056i	$0.528279\pi$
$642$	0	0
$643$	−5.48669	−0.216374	−0.108187	−	0.994131i	$-0.534505\pi$
$643$	−5.48669	−0.216374	−0.108187	+	0.994131i	$0.534505\pi$
$644$	−43.4838	−1.71350
$645$	0	0
$646$	13.5573	0.533406
$647$	34.5798	1.35947	0.679736	−	0.733456i	$-0.262094\pi$
$647$	34.5798	1.35947	0.679736	+	0.733456i	$0.262094\pi$
$648$	0	0
$649$	5.54910	0.217821
$650$	11.3607	0.445602
$651$	0	0
$652$	22.8635	0.895402
$653$	−11.3537	−0.444304	−0.222152	−	0.975012i	$-0.571308\pi$
$653$	−11.3537	−0.444304	−0.222152	+	0.975012i	$0.571308\pi$
$654$	0	0
$655$	−13.4383	−0.525076
$656$	−14.0199	−0.547384
$657$	0	0
$658$	−2.05010	−0.0799213
$659$	−27.1448	−1.05741	−0.528706	−	0.848805i	$-0.677323\pi$
$659$	−27.1448	−1.05741	−0.528706	+	0.848805i	$0.677323\pi$
$660$	0	0
$661$	5.01901	0.195217	0.0976084	−	0.995225i	$-0.468881\pi$
$661$	5.01901	0.195217	0.0976084	+	0.995225i	$0.468881\pi$
$662$	−31.6143	−1.22873
$663$	0	0
$664$	57.5002	2.23144
$665$	2.89389	0.112220
$666$	0	0
$667$	−37.3796	−1.44734
$668$	−4.00181	−0.154835
$669$	0	0
$670$	−16.7565	−0.647362
$671$	−1.01018	−0.0389975
$672$	0	0
$673$	−38.5035	−1.48420	−0.742100	−	0.670289i	$-0.766170\pi$
$673$	−38.5035	−1.48420	−0.742100	+	0.670289i	$0.766170\pi$
$674$	40.7994	1.57153
$675$	0	0
$676$	35.4236	1.36244
$677$	−17.8660	−0.686647	−0.343324	−	0.939217i	$-0.611553\pi$
$677$	−17.8660	−0.686647	−0.343324	+	0.939217i	$0.611553\pi$
$678$	0	0
$679$	−13.9285	−0.534528
$680$	−23.7228	−0.909729
$681$	0	0
$682$	−15.8325	−0.606257
$683$	−44.9512	−1.72001	−0.860005	−	0.510286i	$-0.829540\pi$
$683$	−44.9512	−1.72001	−0.860005	+	0.510286i	$0.829540\pi$
$684$	0	0
$685$	−16.2122	−0.619435
$686$	39.0356	1.49039
$687$	0	0
$688$	28.0521	1.06948
$689$	18.0581	0.687958
$690$	0	0
$691$	24.1309	0.917983	0.458991	−	0.888441i	$-0.348211\pi$
$691$	24.1309	0.917983	0.458991	+	0.888441i	$0.348211\pi$
$692$	48.8561	1.85723
$693$	0	0
$694$	72.5611	2.75438
$695$	−19.0373	−0.722124
$696$	0	0
$697$	−30.9456	−1.17215
$698$	12.3122	0.466024
$699$	0	0
$700$	−10.8516	−0.410151
$701$	1.53772	0.0580789	0.0290395	−	0.999578i	$-0.490755\pi$
$701$	1.53772	0.0580789	0.0290395	+	0.999578i	$0.490755\pi$
$702$	0	0
$703$	6.07686	0.229193
$704$	−10.5171	−0.396379
$705$	0	0
$706$	25.3362	0.953540
$707$	5.09964	0.191792
$708$	0	0
$709$	5.19348	0.195045	0.0975226	−	0.995233i	$-0.468908\pi$
$709$	5.19348	0.195045	0.0975226	+	0.995233i	$0.468908\pi$
$710$	−2.44787	−0.0918668
$711$	0	0
$712$	66.3580	2.48687
$713$	−26.4580	−0.990858
$714$	0	0
$715$	−4.73780	−0.177184
$716$	−77.3485	−2.89065
$717$	0	0
$718$	−39.6077	−1.47815
$719$	37.4841	1.39792	0.698960	−	0.715160i	$-0.253646\pi$
$719$	37.4841	1.39792	0.698960	+	0.715160i	$0.253646\pi$
$720$	0	0
$721$	16.0198	0.596607
$722$	2.39788	0.0892398
$723$	0	0
$724$	−77.2117	−2.86955
$725$	−9.32825	−0.346443
$726$	0	0
$727$	28.1916	1.04557	0.522785	−	0.852464i	$-0.324893\pi$
$727$	28.1916	1.04557	0.522785	+	0.852464i	$0.324893\pi$
$728$	−57.5279	−2.13213
$729$	0	0
$730$	0.484451	0.0179303
$731$	61.9185	2.29014
$732$	0	0
$733$	−18.1906	−0.671884	−0.335942	−	0.941883i	$-0.609055\pi$
$733$	−18.1906	−0.671884	−0.335942	+	0.941883i	$0.609055\pi$
$734$	47.6604	1.75918
$735$	0	0
$736$	−9.01428	−0.332271
$737$	6.98807	0.257409
$738$	0	0
$739$	5.65484	0.208017	0.104008	−	0.994576i	$-0.466833\pi$
$739$	5.65484	0.208017	0.104008	+	0.994576i	$0.466833\pi$
$740$	−22.7871	−0.837670
$741$	0	0
$742$	−26.4487	−0.970963
$743$	44.5565	1.63462	0.817310	−	0.576198i	$-0.195464\pi$
$743$	44.5565	1.63462	0.817310	+	0.576198i	$0.195464\pi$
$744$	0	0
$745$	−15.3234	−0.561405
$746$	−27.3720	−1.00216
$747$	0	0
$748$	21.2010	0.775187
$749$	−12.5037	−0.456877
$750$	0	0
$751$	−18.8205	−0.686769	−0.343384	−	0.939195i	$-0.611573\pi$
$751$	−18.8205	−0.686769	−0.343384	+	0.939195i	$0.611573\pi$
$752$	0.756760	0.0275962
$753$	0	0
$754$	−105.975	−3.85939
$755$	1.25936	0.0458328
$756$	0	0
$757$	2.97780	0.108230	0.0541150	−	0.998535i	$-0.482766\pi$
$757$	2.97780	0.108230	0.0541150	+	0.998535i	$0.482766\pi$
$758$	4.79397	0.174125
$759$	0	0
$760$	−4.19584	−0.152199
$761$	41.9642	1.52120	0.760601	−	0.649219i	$-0.224904\pi$
$761$	41.9642	1.52120	0.760601	+	0.649219i	$0.224904\pi$
$762$	0	0
$763$	37.6320	1.36237
$764$	−31.5002	−1.13964
$765$	0	0
$766$	16.9532	0.612542
$767$	26.2905	0.949295
$768$	0	0
$769$	−2.33683	−0.0842683	−0.0421341	−	0.999112i	$-0.513416\pi$
$769$	−2.33683	−0.0842683	−0.0421341	+	0.999112i	$0.513416\pi$
$770$	6.93920	0.250072
$771$	0	0
$772$	43.1162	1.55179
$773$	−4.40125	−0.158302	−0.0791509	−	0.996863i	$-0.525221\pi$
$773$	−4.40125	−0.158302	−0.0791509	+	0.996863i	$0.525221\pi$
$774$	0	0
$775$	−6.60270	−0.237176
$776$	20.1949	0.724955
$777$	0	0
$778$	−40.8207	−1.46349
$779$	−5.47333	−0.196102
$780$	0	0
$781$	1.02085	0.0365288
$782$	54.3262	1.94270
$783$	0	0
$784$	3.52108	0.125753
$785$	−12.1861	−0.434942
$786$	0	0
$787$	−13.1902	−0.470180	−0.235090	−	0.971974i	$-0.575539\pi$
$787$	−13.1902	−0.470180	−0.235090	+	0.971974i	$0.575539\pi$
$788$	−101.386	−3.61172
$789$	0	0
$790$	−17.4731	−0.621664
$791$	−49.2097	−1.74970
$792$	0	0
$793$	−4.78602	−0.169957
$794$	−60.8078	−2.15799
$795$	0	0
$796$	−19.4413	−0.689078
$797$	39.1687	1.38743	0.693713	−	0.720252i	$-0.255974\pi$
$797$	39.1687	1.38743	0.693713	+	0.720252i	$0.255974\pi$
$798$	0	0
$799$	1.67037	0.0590935
$800$	−2.24955	−0.0795338
$801$	0	0
$802$	−47.1060	−1.66337
$803$	−0.202033	−0.00712960
$804$	0	0
$805$	11.5962	0.408714
$806$	−75.0111	−2.64215
$807$	0	0
$808$	−7.39394	−0.260118
$809$	−18.2003	−0.639887	−0.319944	−	0.947437i	$-0.603664\pi$
$809$	−18.2003	−0.639887	−0.319944	+	0.947437i	$0.603664\pi$
$810$	0	0
$811$	−54.6434	−1.91879	−0.959395	−	0.282065i	$-0.908981\pi$
$811$	−54.6434	−1.91879	−0.959395	+	0.282065i	$0.908981\pi$
$812$	101.226	3.55234
$813$	0	0
$814$	14.5716	0.510733
$815$	−6.09722	−0.213576
$816$	0	0
$817$	10.9515	0.383144
$818$	−68.7056	−2.40224
$819$	0	0
$820$	20.5240	0.716729
$821$	−18.3540	−0.640560	−0.320280	−	0.947323i	$-0.603777\pi$
$821$	−18.3540	−0.640560	−0.320280	+	0.947323i	$0.603777\pi$
$822$	0	0
$823$	9.63416	0.335826	0.167913	−	0.985802i	$-0.446297\pi$
$823$	9.63416	0.335826	0.167913	+	0.985802i	$0.446297\pi$
$824$	−23.2270	−0.809150
$825$	0	0
$826$	−38.5063	−1.33981
$827$	−26.2829	−0.913947	−0.456973	−	0.889480i	$-0.651067\pi$
$827$	−26.2829	−0.913947	−0.456973	+	0.889480i	$0.651067\pi$
$828$	0	0
$829$	−45.2690	−1.57226	−0.786128	−	0.618064i	$-0.787917\pi$
$829$	−45.2690	−1.57226	−0.786128	+	0.618064i	$0.787917\pi$
$830$	−32.8607	−1.14061
$831$	0	0
$832$	−49.8281	−1.72748
$833$	7.77197	0.269283
$834$	0	0
$835$	1.06720	0.0369321
$836$	3.74982	0.129690
$837$	0	0
$838$	6.72648	0.232362
$839$	−23.0656	−0.796315	−0.398157	−	0.917317i	$-0.630350\pi$
$839$	−23.0656	−0.796315	−0.398157	+	0.917317i	$0.630350\pi$
$840$	0	0
$841$	58.0162	2.00056
$842$	7.82104	0.269531
$843$	0	0
$844$	−80.7810	−2.78060
$845$	−9.44675	−0.324978
$846$	0	0
$847$	−2.89389	−0.0994354
$848$	9.76309	0.335266
$849$	0	0
$850$	13.5573	0.465013
$851$	24.3508	0.834736
$852$	0	0
$853$	−10.1175	−0.346418	−0.173209	−	0.984885i	$-0.555414\pi$
$853$	−10.1175	−0.346418	−0.173209	+	0.984885i	$0.555414\pi$
$854$	7.00983	0.239872
$855$	0	0
$856$	18.1291	0.619641
$857$	−17.5975	−0.601119	−0.300559	−	0.953763i	$-0.597173\pi$
$857$	−17.5975	−0.601119	−0.300559	+	0.953763i	$0.597173\pi$
$858$	0	0
$859$	51.9873	1.77378	0.886891	−	0.461978i	$-0.152860\pi$
$859$	51.9873	1.77378	0.886891	+	0.461978i	$0.152860\pi$
$860$	−41.0661	−1.40034
$861$	0	0
$862$	−78.5560	−2.67563
$863$	42.2968	1.43980	0.719900	−	0.694078i	$-0.244188\pi$
$863$	42.2968	1.43980	0.719900	+	0.694078i	$0.244188\pi$
$864$	0	0
$865$	−13.0289	−0.442997
$866$	−52.4310	−1.78168
$867$	0	0
$868$	71.6497	2.43195
$869$	7.28690	0.247191
$870$	0	0
$871$	33.1081	1.12182
$872$	−54.5625	−1.84772
$873$	0	0
$874$	9.60863	0.325017
$875$	2.89389	0.0978315
$876$	0	0
$877$	−4.60606	−0.155535	−0.0777677	−	0.996972i	$-0.524779\pi$
$877$	−4.60606	−0.155535	−0.0777677	+	0.996972i	$0.524779\pi$
$878$	73.4423	2.47856
$879$	0	0
$880$	−2.56149	−0.0863477
$881$	−45.1124	−1.51987	−0.759937	−	0.649997i	$-0.774770\pi$
$881$	−45.1124	−1.51987	−0.759937	+	0.649997i	$0.774770\pi$
$882$	0	0
$883$	−40.3849	−1.35906	−0.679530	−	0.733648i	$-0.737816\pi$
$883$	−40.3849	−1.35906	−0.679530	+	0.733648i	$0.737816\pi$
$884$	100.446	3.37837
$885$	0	0
$886$	−10.9083	−0.366471
$887$	−50.2462	−1.68710	−0.843551	−	0.537050i	$-0.819539\pi$
$887$	−50.2462	−1.68710	−0.843551	+	0.537050i	$0.819539\pi$
$888$	0	0
$889$	28.3672	0.951407
$890$	−37.9229	−1.27118
$891$	0	0
$892$	−101.095	−3.38491
$893$	0.295438	0.00988645
$894$	0	0
$895$	20.6273	0.689494
$896$	59.9606	2.00314
$897$	0	0
$898$	−27.6495	−0.922674
$899$	61.5916	2.05420
$900$	0	0
$901$	21.5498	0.717926
$902$	−13.1244	−0.436994
$903$	0	0
$904$	71.3490	2.37303
$905$	20.5908	0.684462
$906$	0	0
$907$	58.4194	1.93979	0.969893	−	0.243532i	$-0.0783062\pi$
$907$	58.4194	1.93979	0.969893	+	0.243532i	$0.0783062\pi$
$908$	−33.8447	−1.12318
$909$	0	0
$910$	32.8766	1.08985
$911$	39.2409	1.30011	0.650054	−	0.759888i	$-0.274746\pi$
$911$	39.2409	1.30011	0.650054	+	0.759888i	$0.274746\pi$
$912$	0	0
$913$	13.7041	0.453539
$914$	27.2826	0.902427
$915$	0	0
$916$	−51.6337	−1.70603
$917$	−38.8889	−1.28422
$918$	0	0
$919$	18.4113	0.607334	0.303667	−	0.952778i	$-0.401789\pi$
$919$	18.4113	0.607334	0.303667	+	0.952778i	$0.401789\pi$
$920$	−16.8133	−0.554319
$921$	0	0
$922$	−10.4851	−0.345308
$923$	4.83657	0.159198
$924$	0	0
$925$	6.07686	0.199806
$926$	−22.3478	−0.734393
$927$	0	0
$928$	20.9844	0.688847
$929$	32.8679	1.07836	0.539180	−	0.842191i	$-0.318734\pi$
$929$	32.8679	1.07836	0.539180	+	0.842191i	$0.318734\pi$
$930$	0	0
$931$	1.37462	0.0450515
$932$	9.20907	0.301653
$933$	0	0
$934$	67.3110	2.20248
$935$	−5.65389	−0.184902
$936$	0	0
$937$	4.34616	0.141983	0.0709914	−	0.997477i	$-0.477384\pi$
$937$	4.34616	0.141983	0.0709914	+	0.997477i	$0.477384\pi$
$938$	−48.4917	−1.58331
$939$	0	0
$940$	−1.10784	−0.0361337
$941$	55.1238	1.79699	0.898493	−	0.438989i	$-0.144663\pi$
$941$	55.1238	1.79699	0.898493	+	0.438989i	$0.144663\pi$
$942$	0	0
$943$	−21.9324	−0.714218
$944$	14.2139	0.462625
$945$	0	0
$946$	26.2603	0.853797
$947$	−16.6794	−0.542007	−0.271004	−	0.962578i	$-0.587356\pi$
$947$	−16.6794	−0.542007	−0.271004	+	0.962578i	$0.587356\pi$
$948$	0	0
$949$	−0.957193	−0.0310718
$950$	2.39788	0.0777974
$951$	0	0
$952$	−68.6514	−2.22500
$953$	−51.8274	−1.67885	−0.839427	−	0.543472i	$-0.817109\pi$
$953$	−51.8274	−1.67885	−0.839427	+	0.543472i	$0.817109\pi$
$954$	0	0
$955$	8.40047	0.271833
$956$	85.6605	2.77046
$957$	0	0
$958$	13.5524	0.437859
$959$	−46.9163	−1.51501
$960$	0	0
$961$	12.5957	0.406312
$962$	69.0371	2.22585
$963$	0	0
$964$	60.4399	1.94664
$965$	−11.4982	−0.370141
$966$	0	0
$967$	−28.7146	−0.923398	−0.461699	−	0.887037i	$-0.652760\pi$
$967$	−28.7146	−0.923398	−0.461699	+	0.887037i	$0.652760\pi$
$968$	4.19584	0.134860
$969$	0	0
$970$	−11.5412	−0.370564
$971$	−43.9744	−1.41121	−0.705603	−	0.708607i	$-0.749324\pi$
$971$	−43.9744	−1.41121	−0.705603	+	0.708607i	$0.749324\pi$
$972$	0	0
$973$	−55.0918	−1.76616
$974$	−14.6114	−0.468180
$975$	0	0
$976$	−2.58756	−0.0828258
$977$	−19.2496	−0.615848	−0.307924	−	0.951411i	$-0.599634\pi$
$977$	−19.2496	−0.615848	−0.307924	+	0.951411i	$0.599634\pi$
$978$	0	0
$979$	15.8152	0.505455
$980$	−5.15458	−0.164657
$981$	0	0
$982$	−20.5327	−0.655224
$983$	12.7410	0.406373	0.203187	−	0.979140i	$-0.434870\pi$
$983$	12.7410	0.406373	0.203187	+	0.979140i	$0.434870\pi$
$984$	0	0
$985$	27.0376	0.861488
$986$	−126.466	−4.02750
$987$	0	0
$988$	17.7659	0.565208
$989$	43.8842	1.39544
$990$	0	0
$991$	27.9827	0.888900	0.444450	−	0.895804i	$-0.353399\pi$
$991$	27.9827	0.888900	0.444450	+	0.895804i	$0.353399\pi$
$992$	14.8531	0.471588
$993$	0	0
$994$	−7.08387	−0.224687
$995$	5.18460	0.164363
$996$	0	0
$997$	−47.7121	−1.51106	−0.755528	−	0.655116i	$-0.772620\pi$
$997$	−47.7121	−1.51106	−0.755528	+	0.655116i	$0.772620\pi$
$998$	−3.25002	−0.102878
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.v.1.5		5
3.2	odd	2		1045.2.a.d.1.1	✓	5
15.14	odd	2		5225.2.a.j.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.d.1.1	✓	5	3.2	odd	2
5225.2.a.j.1.5		5	15.14	odd	2
9405.2.a.v.1.5		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q+2.39788 q^{2} +3.74982 q^{4} -1.00000 q^{5} -2.89389 q^{7} +4.19584 q^{8} +O(q^{10})\)	\(q+2.39788 q^{2} +3.74982 q^{4} -1.00000 q^{5} -2.89389 q^{7} +4.19584 q^{8} -2.39788 q^{10} +1.00000 q^{11} +4.73780 q^{13} -6.93920 q^{14} +2.56149 q^{16} +5.65389 q^{17} +1.00000 q^{19} -3.74982 q^{20} +2.39788 q^{22} +4.00714 q^{23} +1.00000 q^{25} +11.3607 q^{26} -10.8516 q^{28} -9.32825 q^{29} -6.60270 q^{31} -2.24955 q^{32} +13.5573 q^{34} +2.89389 q^{35} +6.07686 q^{37} +2.39788 q^{38} -4.19584 q^{40} -5.47333 q^{41} +10.9515 q^{43} +3.74982 q^{44} +9.60863 q^{46} +0.295438 q^{47} +1.37462 q^{49} +2.39788 q^{50} +17.7659 q^{52} +3.81149 q^{53} -1.00000 q^{55} -12.1423 q^{56} -22.3680 q^{58} +5.54910 q^{59} -1.01018 q^{61} -15.8325 q^{62} -10.5171 q^{64} -4.73780 q^{65} +6.98807 q^{67} +21.2010 q^{68} +6.93920 q^{70} +1.02085 q^{71} -0.202033 q^{73} +14.5716 q^{74} +3.74982 q^{76} -2.89389 q^{77} +7.28690 q^{79} -2.56149 q^{80} -13.1244 q^{82} +13.7041 q^{83} -5.65389 q^{85} +26.2603 q^{86} +4.19584 q^{88} +15.8152 q^{89} -13.7107 q^{91} +15.0260 q^{92} +0.708423 q^{94} -1.00000 q^{95} +4.81308 q^{97} +3.29618 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8}+O(q^{10}) \) 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8	\( 5 q + 3 q^{2} + 5 q^{4} - 5 q^{5} - 11 q^{7} - 3 q^{8} - 3 q^{10} + 5 q^{11} + q^{13} - 3 q^{16} + 3 q^{17} + 5 q^{19} - 5 q^{20} + 3 q^{22} + 8 q^{23} + 5 q^{25} + 16 q^{26} - 22 q^{28} - 11 q^{29} - 5 q^{31} + 2 q^{32} + 4 q^{34} + 11 q^{35} - 9 q^{37} + 3 q^{38} + 3 q^{40} - 15 q^{41} - 13 q^{43} + 5 q^{44} + 18 q^{46} + 20 q^{47} + 20 q^{49} + 3 q^{50} + q^{52} + 5 q^{53} - 5 q^{55} - 33 q^{58} + 17 q^{59} + 3 q^{61} - 14 q^{62} - 17 q^{64} - q^{65} - 28 q^{67} + 25 q^{68} + 6 q^{71} - 16 q^{73} + 21 q^{74} + 5 q^{76} - 11 q^{77} + 3 q^{79} + 3 q^{80} + 2 q^{82} + 33 q^{83} - 3 q^{85} - 10 q^{86} - 3 q^{88} + 16 q^{89} - 22 q^{91} + 19 q^{92} - 10 q^{94} - 5 q^{95} - 14 q^{97} - 10 q^{98}+O(q^{100}) \) 5 * q + 3 * q^2 + 5 * q^4 - 5 * q^5 - 11 * q^7 - 3 * q^8 - 3 * q^10 + 5 * q^11 + q^13 - 3 * q^16 + 3 * q^17 + 5 * q^19 - 5 * q^20 + 3 * q^22 + 8 * q^23 + 5 * q^25 + 16 * q^26 - 22 * q^28 - 11 * q^29 - 5 * q^31 + 2 * q^32 + 4 * q^34 + 11 * q^35 - 9 * q^37 + 3 * q^38 + 3 * q^40 - 15 * q^41 - 13 * q^43 + 5 * q^44 + 18 * q^46 + 20 * q^47 + 20 * q^49 + 3 * q^50 + q^52 + 5 * q^53 - 5 * q^55 - 33 * q^58 + 17 * q^59 + 3 * q^61 - 14 * q^62 - 17 * q^64 - q^65 - 28 * q^67 + 25 * q^68 + 6 * q^71 - 16 * q^73 + 21 * q^74 + 5 * q^76 - 11 * q^77 + 3 * q^79 + 3 * q^80 + 2 * q^82 + 33 * q^83 - 3 * q^85 - 10 * q^86 - 3 * q^88 + 16 * q^89 - 22 * q^91 + 19 * q^92 - 10 * q^94 - 5 * q^95 - 14 * q^97 - 10 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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