LMFDB - Embedded newform 7245.2.a.bv.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7245,2,Mod(1,7245)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7245, 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("7245.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7245.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:	$57.8516162644$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 16x^{8} + 30x^{7} + 87x^{6} - 143x^{5} - 196x^{4} + 244x^{3} + 160x^{2} - 89x - 12 $ x^10 - 2x^9 - 16x^8 + 30x^7 + 87x^6 - 143x^5 - 196x^4 + 244x^3 + 160x^2 - 89*x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2415)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.88703$ of defining polynomial
Character	$\chi$	$=$	7245.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.88703 q^{2} +1.56089 q^{4} -1.00000 q^{5} +1.00000 q^{7} +0.828610 q^{8} +O(q^{10})$	\(q-1.88703 q^{2} +1.56089 q^{4} -1.00000 q^{5} +1.00000 q^{7} +0.828610 q^{8} +1.88703 q^{10} +2.76344 q^{11} +5.73169 q^{13} -1.88703 q^{14} -4.68540 q^{16} -4.82653 q^{17} +4.18325 q^{19} -1.56089 q^{20} -5.21470 q^{22} +1.00000 q^{23} +1.00000 q^{25} -10.8159 q^{26} +1.56089 q^{28} +8.32979 q^{29} +5.10382 q^{31} +7.18428 q^{32} +9.10782 q^{34} -1.00000 q^{35} +3.98441 q^{37} -7.89393 q^{38} -0.828610 q^{40} +3.49426 q^{41} +6.13541 q^{43} +4.31343 q^{44} -1.88703 q^{46} -10.7297 q^{47} +1.00000 q^{49} -1.88703 q^{50} +8.94656 q^{52} +7.60272 q^{53} -2.76344 q^{55} +0.828610 q^{56} -15.7186 q^{58} +0.943685 q^{59} -14.7835 q^{61} -9.63107 q^{62} -4.18618 q^{64} -5.73169 q^{65} -3.94831 q^{67} -7.53369 q^{68} +1.88703 q^{70} -0.189223 q^{71} +14.6936 q^{73} -7.51872 q^{74} +6.52961 q^{76} +2.76344 q^{77} +8.77235 q^{79} +4.68540 q^{80} -6.59378 q^{82} +9.38891 q^{83} +4.82653 q^{85} -11.5777 q^{86} +2.28981 q^{88} +8.69381 q^{89} +5.73169 q^{91} +1.56089 q^{92} +20.2472 q^{94} -4.18325 q^{95} +8.78399 q^{97} -1.88703 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8}+O(q^{10}) $ 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8	$ 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8} + 2 q^{10} - 9 q^{11} + 14 q^{13} - 2 q^{14} + 20 q^{16} - 8 q^{17} + 13 q^{19} - 16 q^{20} + 10 q^{23} + 10 q^{25} + 11 q^{26} + 16 q^{28} - 10 q^{29} + 8 q^{31} + 11 q^{32} - 5 q^{34} - 10 q^{35} + 8 q^{37} + 10 q^{38} + 6 q^{40} + 5 q^{41} + 4 q^{43} - 3 q^{44} - 2 q^{46} - q^{47} + 10 q^{49} - 2 q^{50} + 14 q^{52} - 9 q^{53} + 9 q^{55} - 6 q^{56} - 28 q^{58} + 17 q^{59} + 19 q^{61} + 28 q^{62} + 24 q^{64} - 14 q^{65} - 8 q^{68} + 2 q^{70} + 6 q^{73} - 3 q^{74} + 15 q^{76} - 9 q^{77} + 32 q^{79} - 20 q^{80} + 14 q^{82} + 2 q^{83} + 8 q^{85} - 2 q^{86} - 3 q^{88} - 10 q^{89} + 14 q^{91} + 16 q^{92} - 10 q^{94} - 13 q^{95} + 18 q^{97} - 2 q^{98}+O(q^{100}) $ 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8 + 2 * q^10 - 9 * q^11 + 14 * q^13 - 2 * q^14 + 20 * q^16 - 8 * q^17 + 13 * q^19 - 16 * q^20 + 10 * q^23 + 10 * q^25 + 11 * q^26 + 16 * q^28 - 10 * q^29 + 8 * q^31 + 11 * q^32 - 5 * q^34 - 10 * q^35 + 8 * q^37 + 10 * q^38 + 6 * q^40 + 5 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 - q^47 + 10 * q^49 - 2 * q^50 + 14 * q^52 - 9 * q^53 + 9 * q^55 - 6 * q^56 - 28 * q^58 + 17 * q^59 + 19 * q^61 + 28 * q^62 + 24 * q^64 - 14 * q^65 - 8 * q^68 + 2 * q^70 + 6 * q^73 - 3 * q^74 + 15 * q^76 - 9 * q^77 + 32 * q^79 - 20 * q^80 + 14 * q^82 + 2 * q^83 + 8 * q^85 - 2 * q^86 - 3 * q^88 - 10 * q^89 + 14 * q^91 + 16 * q^92 - 10 * q^94 - 13 * q^95 + 18 * q^97 - 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.88703	−1.33433	−0.667167	−	0.744908i	$-0.732493\pi$
$2$	−1.88703	−1.33433	−0.667167	+	0.744908i	$0.732493\pi$
$3$	0	0
$3$	0	0
$4$	1.56089	0.780446
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.828610	0.292958
$9$	0	0
$10$	1.88703	0.596732
$11$	2.76344	0.833209	0.416604	−	0.909088i	$-0.363220\pi$
$11$	2.76344	0.833209	0.416604	+	0.909088i	$0.363220\pi$
$12$	0	0
$13$	5.73169	1.58969	0.794843	−	0.606816i	$-0.207553\pi$
$13$	5.73169	1.58969	0.794843	+	0.606816i	$0.207553\pi$
$14$	−1.88703	−0.504331
$15$	0	0
$16$	−4.68540	−1.17135
$17$	−4.82653	−1.17061	−0.585303	−	0.810815i	$-0.699024\pi$
$17$	−4.82653	−1.17061	−0.585303	+	0.810815i	$0.699024\pi$
$18$	0	0
$19$	4.18325	0.959704	0.479852	−	0.877350i	$-0.340690\pi$
$19$	4.18325	0.959704	0.479852	+	0.877350i	$0.340690\pi$
$20$	−1.56089	−0.349026
$21$	0	0
$22$	−5.21470	−1.11178
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	−10.8159	−2.12117
$27$	0	0
$28$	1.56089	0.294981
$29$	8.32979	1.54680	0.773402	−	0.633916i	$-0.218554\pi$
$29$	8.32979	1.54680	0.773402	+	0.633916i	$0.218554\pi$
$30$	0	0
$31$	5.10382	0.916673	0.458336	−	0.888779i	$-0.348446\pi$
$31$	5.10382	0.916673	0.458336	+	0.888779i	$0.348446\pi$
$32$	7.18428	1.27001
$33$	0	0
$34$	9.10782	1.56198
$35$	−1.00000	−0.169031
$36$	0	0
$37$	3.98441	0.655033	0.327517	−	0.944845i	$-0.393788\pi$
$37$	3.98441	0.655033	0.327517	+	0.944845i	$0.393788\pi$
$38$	−7.89393	−1.28056
$39$	0	0
$40$	−0.828610	−0.131015
$41$	3.49426	0.545712	0.272856	−	0.962055i	$-0.412032\pi$
$41$	3.49426	0.545712	0.272856	+	0.962055i	$0.412032\pi$
$42$	0	0
$43$	6.13541	0.935641	0.467820	−	0.883824i	$-0.345039\pi$
$43$	6.13541	0.935641	0.467820	+	0.883824i	$0.345039\pi$
$44$	4.31343	0.650275
$45$	0	0
$46$	−1.88703	−0.278228
$47$	−10.7297	−1.56508	−0.782540	−	0.622600i	$-0.786076\pi$
$47$	−10.7297	−1.56508	−0.782540	+	0.622600i	$0.786076\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−1.88703	−0.266867
$51$	0	0
$52$	8.94656	1.24066
$53$	7.60272	1.04431	0.522157	−	0.852850i	$-0.325128\pi$
$53$	7.60272	1.04431	0.522157	+	0.852850i	$0.325128\pi$
$54$	0	0
$55$	−2.76344	−0.372622
$56$	0.828610	0.110728
$57$	0	0
$58$	−15.7186	−2.06395
$59$	0.943685	0.122857	0.0614286	−	0.998111i	$-0.480434\pi$
$59$	0.943685	0.122857	0.0614286	+	0.998111i	$0.480434\pi$
$60$	0	0
$61$	−14.7835	−1.89283	−0.946417	−	0.322948i	$-0.895326\pi$
$61$	−14.7835	−1.89283	−0.946417	+	0.322948i	$0.895326\pi$
$62$	−9.63107	−1.22315
$63$	0	0
$64$	−4.18618	−0.523272
$65$	−5.73169	−0.710929
$66$	0	0
$67$	−3.94831	−0.482363	−0.241182	−	0.970480i	$-0.577535\pi$
$67$	−3.94831	−0.482363	−0.241182	+	0.970480i	$0.577535\pi$
$68$	−7.53369	−0.913595
$69$	0	0
$70$	1.88703	0.225544
$71$	−0.189223	−0.0224567	−0.0112283	−	0.999937i	$-0.503574\pi$
$71$	−0.189223	−0.0224567	−0.0112283	+	0.999937i	$0.503574\pi$
$72$	0	0
$73$	14.6936	1.71976	0.859879	−	0.510497i	$-0.170539\pi$
$73$	14.6936	1.71976	0.859879	+	0.510497i	$0.170539\pi$
$74$	−7.51872	−0.874033
$75$	0	0
$76$	6.52961	0.748997
$77$	2.76344	0.314923
$78$	0	0
$79$	8.77235	0.986966	0.493483	−	0.869755i	$-0.335723\pi$
$79$	8.77235	0.986966	0.493483	+	0.869755i	$0.335723\pi$
$80$	4.68540	0.523844
$81$	0	0
$82$	−6.59378	−0.728161
$83$	9.38891	1.03057	0.515283	−	0.857020i	$-0.327687\pi$
$83$	9.38891	1.03057	0.515283	+	0.857020i	$0.327687\pi$
$84$	0	0
$85$	4.82653	0.523511
$86$	−11.5777	−1.24846
$87$	0	0
$88$	2.28981	0.244095
$89$	8.69381	0.921542	0.460771	−	0.887519i	$-0.347573\pi$
$89$	8.69381	0.921542	0.460771	+	0.887519i	$0.347573\pi$
$90$	0	0
$91$	5.73169	0.600845
$92$	1.56089	0.162734
$93$	0	0
$94$	20.2472	2.08834
$95$	−4.18325	−0.429192
$96$	0	0
$97$	8.78399	0.891880	0.445940	−	0.895063i	$-0.352870\pi$
$97$	8.78399	0.891880	0.445940	+	0.895063i	$0.352870\pi$
$98$	−1.88703	−0.190619
$99$	0	0
$100$	1.56089	0.156089
$101$	8.41512	0.837335	0.418668	−	0.908139i	$-0.362497\pi$
$101$	8.41512	0.837335	0.418668	+	0.908139i	$0.362497\pi$
$102$	0	0
$103$	8.44094	0.831710	0.415855	−	0.909431i	$-0.363482\pi$
$103$	8.44094	0.831710	0.415855	+	0.909431i	$0.363482\pi$
$104$	4.74934	0.465711
$105$	0	0
$106$	−14.3466	−1.39346
$107$	10.1994	0.986009	0.493004	−	0.870027i	$-0.335899\pi$
$107$	10.1994	0.986009	0.493004	+	0.870027i	$0.335899\pi$
$108$	0	0
$109$	−13.6876	−1.31104	−0.655519	−	0.755179i	$-0.727550\pi$
$109$	−13.6876	−1.31104	−0.655519	+	0.755179i	$0.727550\pi$
$110$	5.21470	0.497202
$111$	0	0
$112$	−4.68540	−0.442729
$113$	−17.7930	−1.67383	−0.836913	−	0.547336i	$-0.815642\pi$
$113$	−17.7930	−1.67383	−0.836913	+	0.547336i	$0.815642\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	13.0019	1.20720
$117$	0	0
$118$	−1.78076	−0.163933
$119$	−4.82653	−0.442447
$120$	0	0
$121$	−3.36340	−0.305763
$122$	27.8970	2.52567
$123$	0	0
$124$	7.96651	0.715414
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.4250	−1.54622	−0.773108	−	0.634274i	$-0.781299\pi$
$127$	−17.4250	−1.54622	−0.773108	+	0.634274i	$0.781299\pi$
$128$	−6.46911	−0.571794
$129$	0	0
$130$	10.8159	0.948616
$131$	21.2884	1.85997	0.929987	−	0.367591i	$-0.119817\pi$
$131$	21.2884	1.85997	0.929987	+	0.367591i	$0.119817\pi$
$132$	0	0
$133$	4.18325	0.362734
$134$	7.45060	0.643634
$135$	0	0
$136$	−3.99931	−0.342938
$137$	−16.8918	−1.44316	−0.721580	−	0.692331i	$-0.756584\pi$
$137$	−16.8918	−1.44316	−0.721580	+	0.692331i	$0.756584\pi$
$138$	0	0
$139$	−13.9494	−1.18317	−0.591584	−	0.806243i	$-0.701497\pi$
$139$	−13.9494	−1.18317	−0.591584	+	0.806243i	$0.701497\pi$
$140$	−1.56089	−0.131920
$141$	0	0
$142$	0.357070	0.0299647
$143$	15.8392	1.32454
$144$	0	0
$145$	−8.32979	−0.691751
$146$	−27.7274	−2.29473
$147$	0	0
$148$	6.21924	0.511218
$149$	2.66203	0.218082	0.109041	−	0.994037i	$-0.465222\pi$
$149$	2.66203	0.218082	0.109041	+	0.994037i	$0.465222\pi$
$150$	0	0
$151$	−2.18060	−0.177454	−0.0887272	−	0.996056i	$-0.528280\pi$
$151$	−2.18060	−0.177454	−0.0887272	+	0.996056i	$0.528280\pi$
$152$	3.46628	0.281153
$153$	0	0
$154$	−5.21470	−0.420213
$155$	−5.10382	−0.409948
$156$	0	0
$157$	−12.5707	−1.00325	−0.501624	−	0.865086i	$-0.667264\pi$
$157$	−12.5707	−1.00325	−0.501624	+	0.865086i	$0.667264\pi$
$158$	−16.5537	−1.31694
$159$	0	0
$160$	−7.18428	−0.567967
$161$	1.00000	0.0788110
$162$	0	0
$163$	−20.3288	−1.59227	−0.796137	−	0.605117i	$-0.793126\pi$
$163$	−20.3288	−1.59227	−0.796137	+	0.605117i	$0.793126\pi$
$164$	5.45416	0.425899
$165$	0	0
$166$	−17.7172	−1.37512
$167$	12.0470	0.932227	0.466114	−	0.884725i	$-0.345654\pi$
$167$	12.0470	0.932227	0.466114	+	0.884725i	$0.345654\pi$
$168$	0	0
$169$	19.8523	1.52710
$170$	−9.10782	−0.698538
$171$	0	0
$172$	9.57671	0.730217
$173$	−20.7346	−1.57642	−0.788209	−	0.615407i	$-0.788992\pi$
$173$	−20.7346	−1.57642	−0.788209	+	0.615407i	$0.788992\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−12.9478	−0.975979
$177$	0	0
$178$	−16.4055	−1.22964
$179$	−1.40660	−0.105135	−0.0525673	−	0.998617i	$-0.516740\pi$
$179$	−1.40660	−0.105135	−0.0525673	+	0.998617i	$0.516740\pi$
$180$	0	0
$181$	7.52723	0.559494	0.279747	−	0.960074i	$-0.409749\pi$
$181$	7.52723	0.559494	0.279747	+	0.960074i	$0.409749\pi$
$182$	−10.8159	−0.801727
$183$	0	0
$184$	0.828610	0.0610859
$185$	−3.98441	−0.292940
$186$	0	0
$187$	−13.3378	−0.975359
$188$	−16.7478	−1.22146
$189$	0	0
$190$	7.89393	0.572686
$191$	−5.72468	−0.414223	−0.207112	−	0.978317i	$-0.566406\pi$
$191$	−5.72468	−0.414223	−0.207112	+	0.978317i	$0.566406\pi$
$192$	0	0
$193$	−7.49392	−0.539424	−0.269712	−	0.962941i	$-0.586928\pi$
$193$	−7.49392	−0.539424	−0.269712	+	0.962941i	$0.586928\pi$
$194$	−16.5757	−1.19006
$195$	0	0
$196$	1.56089	0.111492
$197$	22.5090	1.60370	0.801848	−	0.597528i	$-0.203850\pi$
$197$	22.5090	1.60370	0.801848	+	0.597528i	$0.203850\pi$
$198$	0	0
$199$	−11.3634	−0.805532	−0.402766	−	0.915303i	$-0.631951\pi$
$199$	−11.3634	−0.805532	−0.402766	+	0.915303i	$0.631951\pi$
$200$	0.828610	0.0585916
$201$	0	0
$202$	−15.8796	−1.11728
$203$	8.32979	0.584637
$204$	0	0
$205$	−3.49426	−0.244050
$206$	−15.9283	−1.10978
$207$	0	0
$208$	−26.8553	−1.86208
$209$	11.5602	0.799633
$210$	0	0
$211$	−10.2089	−0.702807	−0.351403	−	0.936224i	$-0.614295\pi$
$211$	−10.2089	−0.702807	−0.351403	+	0.936224i	$0.614295\pi$
$212$	11.8670	0.815031
$213$	0	0
$214$	−19.2465	−1.31566
$215$	−6.13541	−0.418431
$216$	0	0
$217$	5.10382	0.346470
$218$	25.8290	1.74936
$219$	0	0
$220$	−4.31343	−0.290812
$221$	−27.6642	−1.86089
$222$	0	0
$223$	4.68318	0.313609	0.156805	−	0.987630i	$-0.449881\pi$
$223$	4.68318	0.313609	0.156805	+	0.987630i	$0.449881\pi$
$224$	7.18428	0.480020
$225$	0	0
$226$	33.5760	2.23344
$227$	1.11647	0.0741027	0.0370513	−	0.999313i	$-0.488203\pi$
$227$	1.11647	0.0741027	0.0370513	+	0.999313i	$0.488203\pi$
$228$	0	0
$229$	−9.16628	−0.605725	−0.302862	−	0.953034i	$-0.597942\pi$
$229$	−9.16628	−0.605725	−0.302862	+	0.953034i	$0.597942\pi$
$230$	1.88703	0.124427
$231$	0	0
$232$	6.90215	0.453148
$233$	−10.6638	−0.698607	−0.349303	−	0.937010i	$-0.613582\pi$
$233$	−10.6638	−0.698607	−0.349303	+	0.937010i	$0.613582\pi$
$234$	0	0
$235$	10.7297	0.699925
$236$	1.47299	0.0958835
$237$	0	0
$238$	9.10782	0.590372
$239$	26.2437	1.69756	0.848780	−	0.528746i	$-0.177337\pi$
$239$	26.2437	1.69756	0.848780	+	0.528746i	$0.177337\pi$
$240$	0	0
$241$	24.8799	1.60265	0.801327	−	0.598227i	$-0.204128\pi$
$241$	24.8799	1.60265	0.801327	+	0.598227i	$0.204128\pi$
$242$	6.34684	0.407990
$243$	0	0
$244$	−23.0755	−1.47725
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	23.9771	1.52563
$248$	4.22907	0.268546
$249$	0	0
$250$	1.88703	0.119346
$251$	−15.4628	−0.976003	−0.488001	−	0.872843i	$-0.662274\pi$
$251$	−15.4628	−0.976003	−0.488001	+	0.872843i	$0.662274\pi$
$252$	0	0
$253$	2.76344	0.173736
$254$	32.8815	2.06317
$255$	0	0
$256$	20.5798	1.28624
$257$	25.5019	1.59076	0.795382	−	0.606109i	$-0.207270\pi$
$257$	25.5019	1.59076	0.795382	+	0.606109i	$0.207270\pi$
$258$	0	0
$259$	3.98441	0.247579
$260$	−8.94656	−0.554842
$261$	0	0
$262$	−40.1719	−2.48183
$263$	−19.8175	−1.22200	−0.610999	−	0.791631i	$-0.709232\pi$
$263$	−19.8175	−1.22200	−0.610999	+	0.791631i	$0.709232\pi$
$264$	0	0
$265$	−7.60272	−0.467031
$266$	−7.89393	−0.484008
$267$	0	0
$268$	−6.16290	−0.376459
$269$	−26.7170	−1.62897	−0.814483	−	0.580188i	$-0.802979\pi$
$269$	−26.7170	−1.62897	−0.814483	+	0.580188i	$0.802979\pi$
$270$	0	0
$271$	−9.96933	−0.605593	−0.302797	−	0.953055i	$-0.597920\pi$
$271$	−9.96933	−0.605593	−0.302797	+	0.953055i	$0.597920\pi$
$272$	22.6142	1.37119
$273$	0	0
$274$	31.8753	1.92566
$275$	2.76344	0.166642
$276$	0	0
$277$	2.62532	0.157740	0.0788700	−	0.996885i	$-0.474869\pi$
$277$	2.62532	0.157740	0.0788700	+	0.996885i	$0.474869\pi$
$278$	26.3229	1.57874
$279$	0	0
$280$	−0.828610	−0.0495189
$281$	21.0523	1.25588	0.627939	−	0.778263i	$-0.283899\pi$
$281$	21.0523	1.25588	0.627939	+	0.778263i	$0.283899\pi$
$282$	0	0
$283$	14.9895	0.891033	0.445516	−	0.895274i	$-0.353020\pi$
$283$	14.9895	0.891033	0.445516	+	0.895274i	$0.353020\pi$
$284$	−0.295357	−0.0175262
$285$	0	0
$286$	−29.8891	−1.76738
$287$	3.49426	0.206260
$288$	0	0
$289$	6.29538	0.370317
$290$	15.7186	0.923027
$291$	0	0
$292$	22.9352	1.34218
$293$	−15.9314	−0.930723	−0.465361	−	0.885121i	$-0.654076\pi$
$293$	−15.9314	−0.930723	−0.465361	+	0.885121i	$0.654076\pi$
$294$	0	0
$295$	−0.943685	−0.0549434
$296$	3.30152	0.191897
$297$	0	0
$298$	−5.02334	−0.290994
$299$	5.73169	0.331472
$300$	0	0
$301$	6.13541	0.353639
$302$	4.11486	0.236783
$303$	0	0
$304$	−19.6002	−1.12415
$305$	14.7835	0.846501
$306$	0	0
$307$	−11.2105	−0.639818	−0.319909	−	0.947448i	$-0.603652\pi$
$307$	−11.2105	−0.639818	−0.319909	+	0.947448i	$0.603652\pi$
$308$	4.31343	0.245781
$309$	0	0
$310$	9.63107	0.547008
$311$	23.6492	1.34103	0.670513	−	0.741898i	$-0.266074\pi$
$311$	23.6492	1.34103	0.670513	+	0.741898i	$0.266074\pi$
$312$	0	0
$313$	5.05393	0.285665	0.142832	−	0.989747i	$-0.454379\pi$
$313$	5.05393	0.285665	0.142832	+	0.989747i	$0.454379\pi$
$314$	23.7212	1.33867
$315$	0	0
$316$	13.6927	0.770274
$317$	−31.6774	−1.77918	−0.889590	−	0.456761i	$-0.849010\pi$
$317$	−31.6774	−1.77918	−0.889590	+	0.456761i	$0.849010\pi$
$318$	0	0
$319$	23.0189	1.28881
$320$	4.18618	0.234014
$321$	0	0
$322$	−1.88703	−0.105160
$323$	−20.1906	−1.12343
$324$	0	0
$325$	5.73169	0.317937
$326$	38.3611	2.12462
$327$	0	0
$328$	2.89538	0.159870
$329$	−10.7297	−0.591545
$330$	0	0
$331$	−33.5974	−1.84668	−0.923341	−	0.383982i	$-0.874552\pi$
$331$	−33.5974	−1.84668	−0.923341	+	0.383982i	$0.874552\pi$
$332$	14.6551	0.804302
$333$	0	0
$334$	−22.7331	−1.24390
$335$	3.94831	0.215719
$336$	0	0
$337$	−18.5561	−1.01081	−0.505407	−	0.862881i	$-0.668658\pi$
$337$	−18.5561	−1.01081	−0.505407	+	0.862881i	$0.668658\pi$
$338$	−37.4619	−2.03766
$339$	0	0
$340$	7.53369	0.408572
$341$	14.1041	0.763780
$342$	0	0
$343$	1.00000	0.0539949
$344$	5.08386	0.274103
$345$	0	0
$346$	39.1268	2.10347
$347$	−9.38261	−0.503685	−0.251842	−	0.967768i	$-0.581036\pi$
$347$	−9.38261	−0.503685	−0.251842	+	0.967768i	$0.581036\pi$
$348$	0	0
$349$	18.4351	0.986808	0.493404	−	0.869800i	$-0.335753\pi$
$349$	18.4351	0.986808	0.493404	+	0.869800i	$0.335753\pi$
$350$	−1.88703	−0.100866
$351$	0	0
$352$	19.8533	1.05819
$353$	−29.5543	−1.57302	−0.786509	−	0.617579i	$-0.788114\pi$
$353$	−29.5543	−1.57302	−0.786509	+	0.617579i	$0.788114\pi$
$354$	0	0
$355$	0.189223	0.0100429
$356$	13.5701	0.719214
$357$	0	0
$358$	2.65431	0.140285
$359$	1.02873	0.0542944	0.0271472	−	0.999631i	$-0.491358\pi$
$359$	1.02873	0.0542944	0.0271472	+	0.999631i	$0.491358\pi$
$360$	0	0
$361$	−1.50041	−0.0789691
$362$	−14.2041	−0.746552
$363$	0	0
$364$	8.94656	0.468927
$365$	−14.6936	−0.769100
$366$	0	0
$367$	−20.8876	−1.09032	−0.545161	−	0.838331i	$-0.683532\pi$
$367$	−20.8876	−1.09032	−0.545161	+	0.838331i	$0.683532\pi$
$368$	−4.68540	−0.244243
$369$	0	0
$370$	7.51872	0.390880
$371$	7.60272	0.394713
$372$	0	0
$373$	5.93540	0.307323	0.153662	−	0.988124i	$-0.450893\pi$
$373$	5.93540	0.307323	0.153662	+	0.988124i	$0.450893\pi$
$374$	25.1689	1.30145
$375$	0	0
$376$	−8.89070	−0.458503
$377$	47.7438	2.45893
$378$	0	0
$379$	−24.6244	−1.26487	−0.632435	−	0.774613i	$-0.717945\pi$
$379$	−24.6244	−1.26487	−0.632435	+	0.774613i	$0.717945\pi$
$380$	−6.52961	−0.334962
$381$	0	0
$382$	10.8027	0.552712
$383$	−15.1784	−0.775579	−0.387789	−	0.921748i	$-0.626761\pi$
$383$	−15.1784	−0.775579	−0.387789	+	0.921748i	$0.626761\pi$
$384$	0	0
$385$	−2.76344	−0.140838
$386$	14.1413	0.719771
$387$	0	0
$388$	13.7109	0.696064
$389$	35.8307	1.81669	0.908344	−	0.418225i	$-0.137347\pi$
$389$	35.8307	1.81669	0.908344	+	0.418225i	$0.137347\pi$
$390$	0	0
$391$	−4.82653	−0.244088
$392$	0.828610	0.0418511
$393$	0	0
$394$	−42.4751	−2.13987
$395$	−8.77235	−0.441385
$396$	0	0
$397$	−11.9122	−0.597857	−0.298928	−	0.954276i	$-0.596629\pi$
$397$	−11.9122	−0.597857	−0.298928	+	0.954276i	$0.596629\pi$
$398$	21.4432	1.07485
$399$	0	0
$400$	−4.68540	−0.234270
$401$	−13.8828	−0.693276	−0.346638	−	0.937999i	$-0.612677\pi$
$401$	−13.8828	−0.693276	−0.346638	+	0.937999i	$0.612677\pi$
$402$	0	0
$403$	29.2535	1.45722
$404$	13.1351	0.653495
$405$	0	0
$406$	−15.7186	−0.780100
$407$	11.0107	0.545780
$408$	0	0
$409$	33.6644	1.66460	0.832298	−	0.554328i	$-0.187025\pi$
$409$	33.6644	1.66460	0.832298	+	0.554328i	$0.187025\pi$
$410$	6.59378	0.325644
$411$	0	0
$412$	13.1754	0.649105
$413$	0.943685	0.0464357
$414$	0	0
$415$	−9.38891	−0.460883
$416$	41.1781	2.01892
$417$	0	0
$418$	−21.8144	−1.06698
$419$	−8.67518	−0.423810	−0.211905	−	0.977290i	$-0.567967\pi$
$419$	−8.67518	−0.423810	−0.211905	+	0.977290i	$0.567967\pi$
$420$	0	0
$421$	35.4978	1.73006	0.865029	−	0.501721i	$-0.167300\pi$
$421$	35.4978	1.73006	0.865029	+	0.501721i	$0.167300\pi$
$422$	19.2644	0.937778
$423$	0	0
$424$	6.29969	0.305940
$425$	−4.82653	−0.234121
$426$	0	0
$427$	−14.7835	−0.715424
$428$	15.9201	0.769527
$429$	0	0
$430$	11.5777	0.558327
$431$	10.2558	0.494002	0.247001	−	0.969015i	$-0.420555\pi$
$431$	10.2558	0.494002	0.247001	+	0.969015i	$0.420555\pi$
$432$	0	0
$433$	−1.48709	−0.0714650	−0.0357325	−	0.999361i	$-0.511376\pi$
$433$	−1.48709	−0.0714650	−0.0357325	+	0.999361i	$0.511376\pi$
$434$	−9.63107	−0.462306
$435$	0	0
$436$	−21.3649	−1.02319
$437$	4.18325	0.200112
$438$	0	0
$439$	13.1518	0.627702	0.313851	−	0.949472i	$-0.398381\pi$
$439$	13.1518	0.627702	0.313851	+	0.949472i	$0.398381\pi$
$440$	−2.28981	−0.109163
$441$	0	0
$442$	52.2032	2.48305
$443$	30.6473	1.45610	0.728048	−	0.685526i	$-0.240428\pi$
$443$	30.6473	1.45610	0.728048	+	0.685526i	$0.240428\pi$
$444$	0	0
$445$	−8.69381	−0.412126
$446$	−8.83732	−0.418459
$447$	0	0
$448$	−4.18618	−0.197778
$449$	2.87735	0.135790	0.0678952	−	0.997692i	$-0.478372\pi$
$449$	2.87735	0.135790	0.0678952	+	0.997692i	$0.478372\pi$
$450$	0	0
$451$	9.65618	0.454692
$452$	−27.7730	−1.30633
$453$	0	0
$454$	−2.10681	−0.0988777
$455$	−5.73169	−0.268706
$456$	0	0
$457$	11.4730	0.536684	0.268342	−	0.963324i	$-0.413524\pi$
$457$	11.4730	0.536684	0.268342	+	0.963324i	$0.413524\pi$
$458$	17.2971	0.808239
$459$	0	0
$460$	−1.56089	−0.0727770
$461$	4.97184	0.231561	0.115781	−	0.993275i	$-0.463063\pi$
$461$	4.97184	0.231561	0.115781	+	0.993275i	$0.463063\pi$
$462$	0	0
$463$	7.48612	0.347910	0.173955	−	0.984754i	$-0.444345\pi$
$463$	7.48612	0.347910	0.173955	+	0.984754i	$0.444345\pi$
$464$	−39.0284	−1.81185
$465$	0	0
$466$	20.1229	0.932174
$467$	−30.6919	−1.42025	−0.710125	−	0.704075i	$-0.751362\pi$
$467$	−30.6919	−1.42025	−0.710125	+	0.704075i	$0.751362\pi$
$468$	0	0
$469$	−3.94831	−0.182316
$470$	−20.2472	−0.933934
$471$	0	0
$472$	0.781946	0.0359920
$473$	16.9548	0.779584
$474$	0	0
$475$	4.18325	0.191941
$476$	−7.53369	−0.345306
$477$	0	0
$478$	−49.5226	−2.26511
$479$	38.3822	1.75373	0.876864	−	0.480738i	$-0.159631\pi$
$479$	38.3822	1.75373	0.876864	+	0.480738i	$0.159631\pi$
$480$	0	0
$481$	22.8374	1.04130
$482$	−46.9491	−2.13847
$483$	0	0
$484$	−5.24990	−0.238632
$485$	−8.78399	−0.398861
$486$	0	0
$487$	10.9845	0.497754	0.248877	−	0.968535i	$-0.419939\pi$
$487$	10.9845	0.497754	0.248877	+	0.968535i	$0.419939\pi$
$488$	−12.2498	−0.554520
$489$	0	0
$490$	1.88703	0.0852475
$491$	28.8020	1.29982	0.649909	−	0.760012i	$-0.274807\pi$
$491$	28.8020	1.29982	0.649909	+	0.760012i	$0.274807\pi$
$492$	0	0
$493$	−40.2040	−1.81070
$494$	−45.2456	−2.03569
$495$	0	0
$496$	−23.9134	−1.07374
$497$	−0.189223	−0.00848782
$498$	0	0
$499$	13.4792	0.603414	0.301707	−	0.953401i	$-0.402444\pi$
$499$	13.4792	0.603414	0.301707	+	0.953401i	$0.402444\pi$
$500$	−1.56089	−0.0698052
$501$	0	0
$502$	29.1788	1.30231
$503$	−25.5683	−1.14003	−0.570017	−	0.821633i	$-0.693063\pi$
$503$	−25.5683	−1.14003	−0.570017	+	0.821633i	$0.693063\pi$
$504$	0	0
$505$	−8.41512	−0.374468
$506$	−5.21470	−0.231822
$507$	0	0
$508$	−27.1985	−1.20674
$509$	40.1726	1.78062	0.890310	−	0.455355i	$-0.150487\pi$
$509$	40.1726	1.78062	0.890310	+	0.455355i	$0.150487\pi$
$510$	0	0
$511$	14.6936	0.650008
$512$	−25.8965	−1.14447
$513$	0	0
$514$	−48.1229	−2.12261
$515$	−8.44094	−0.371952
$516$	0	0
$517$	−29.6508	−1.30404
$518$	−7.51872	−0.330354
$519$	0	0
$520$	−4.74934	−0.208272
$521$	32.8518	1.43926	0.719632	−	0.694356i	$-0.244311\pi$
$521$	32.8518	1.43926	0.719632	+	0.694356i	$0.244311\pi$
$522$	0	0
$523$	−14.5456	−0.636036	−0.318018	−	0.948085i	$-0.603017\pi$
$523$	−14.5456	−0.636036	−0.318018	+	0.948085i	$0.603017\pi$
$524$	33.2289	1.45161
$525$	0	0
$526$	37.3962	1.63055
$527$	−24.6337	−1.07306
$528$	0	0
$529$	1.00000	0.0434783
$530$	14.3466	0.623175
$531$	0	0
$532$	6.52961	0.283094
$533$	20.0280	0.867510
$534$	0	0
$535$	−10.1994	−0.440956
$536$	−3.27161	−0.141312
$537$	0	0
$538$	50.4159	2.17358
$539$	2.76344	0.119030
$540$	0	0
$541$	7.06454	0.303728	0.151864	−	0.988401i	$-0.451472\pi$
$541$	7.06454	0.303728	0.151864	+	0.988401i	$0.451472\pi$
$542$	18.8124	0.808064
$543$	0	0
$544$	−34.6751	−1.48668
$545$	13.6876	0.586314
$546$	0	0
$547$	−13.1927	−0.564078	−0.282039	−	0.959403i	$-0.591011\pi$
$547$	−13.1927	−0.564078	−0.282039	+	0.959403i	$0.591011\pi$
$548$	−26.3662	−1.12631
$549$	0	0
$550$	−5.21470	−0.222356
$551$	34.8456	1.48447
$552$	0	0
$553$	8.77235	0.373038
$554$	−4.95406	−0.210478
$555$	0	0
$556$	−21.7734	−0.923400
$557$	−35.6655	−1.51120	−0.755598	−	0.655035i	$-0.772654\pi$
$557$	−35.6655	−1.51120	−0.755598	+	0.655035i	$0.772654\pi$
$558$	0	0
$559$	35.1663	1.48737
$560$	4.68540	0.197994
$561$	0	0
$562$	−39.7265	−1.67576
$563$	−12.1370	−0.511512	−0.255756	−	0.966741i	$-0.582324\pi$
$563$	−12.1370	−0.511512	−0.255756	+	0.966741i	$0.582324\pi$
$564$	0	0
$565$	17.7930	0.748558
$566$	−28.2857	−1.18893
$567$	0	0
$568$	−0.156792	−0.00657885
$569$	6.55611	0.274846	0.137423	−	0.990512i	$-0.456118\pi$
$569$	6.55611	0.274846	0.137423	+	0.990512i	$0.456118\pi$
$570$	0	0
$571$	−11.9562	−0.500353	−0.250177	−	0.968200i	$-0.580489\pi$
$571$	−11.9562	−0.500353	−0.250177	+	0.968200i	$0.580489\pi$
$572$	24.7233	1.03373
$573$	0	0
$574$	−6.59378	−0.275219
$575$	1.00000	0.0417029
$576$	0	0
$577$	−20.8650	−0.868621	−0.434310	−	0.900763i	$-0.643008\pi$
$577$	−20.8650	−0.868621	−0.434310	+	0.900763i	$0.643008\pi$
$578$	−11.8796	−0.494126
$579$	0	0
$580$	−13.0019	−0.539875
$581$	9.38891	0.389517
$582$	0	0
$583$	21.0097	0.870131
$584$	12.1753	0.503817
$585$	0	0
$586$	30.0631	1.24189
$587$	10.3965	0.429109	0.214554	−	0.976712i	$-0.431170\pi$
$587$	10.3965	0.429109	0.214554	+	0.976712i	$0.431170\pi$
$588$	0	0
$589$	21.3505	0.879734
$590$	1.78076	0.0733129
$591$	0	0
$592$	−18.6686	−0.767273
$593$	−5.97284	−0.245275	−0.122638	−	0.992452i	$-0.539135\pi$
$593$	−5.97284	−0.245275	−0.122638	+	0.992452i	$0.539135\pi$
$594$	0	0
$595$	4.82653	0.197868
$596$	4.15515	0.170201
$597$	0	0
$598$	−10.8159	−0.442295
$599$	−18.7374	−0.765591	−0.382796	−	0.923833i	$-0.625039\pi$
$599$	−18.7374	−0.765591	−0.382796	+	0.923833i	$0.625039\pi$
$600$	0	0
$601$	16.7255	0.682247	0.341123	−	0.940019i	$-0.389193\pi$
$601$	16.7255	0.682247	0.341123	+	0.940019i	$0.389193\pi$
$602$	−11.5777	−0.471872
$603$	0	0
$604$	−3.40368	−0.138494
$605$	3.36340	0.136741
$606$	0	0
$607$	−6.97185	−0.282979	−0.141489	−	0.989940i	$-0.545189\pi$
$607$	−6.97185	−0.282979	−0.141489	+	0.989940i	$0.545189\pi$
$608$	30.0537	1.21884
$609$	0	0
$610$	−27.8970	−1.12951
$611$	−61.4991	−2.48799
$612$	0	0
$613$	25.0901	1.01338	0.506690	−	0.862128i	$-0.330869\pi$
$613$	25.0901	1.01338	0.506690	+	0.862128i	$0.330869\pi$
$614$	21.1546	0.853730
$615$	0	0
$616$	2.28981	0.0922592
$617$	14.0523	0.565724	0.282862	−	0.959161i	$-0.408716\pi$
$617$	14.0523	0.565724	0.282862	+	0.959161i	$0.408716\pi$
$618$	0	0
$619$	11.7007	0.470289	0.235145	−	0.971960i	$-0.424444\pi$
$619$	11.7007	0.470289	0.235145	+	0.971960i	$0.424444\pi$
$620$	−7.96651	−0.319943
$621$	0	0
$622$	−44.6269	−1.78938
$623$	8.69381	0.348310
$624$	0	0
$625$	1.00000	0.0400000
$626$	−9.53693	−0.381172
$627$	0	0
$628$	−19.6214	−0.782981
$629$	−19.2309	−0.766786
$630$	0	0
$631$	21.6403	0.861486	0.430743	−	0.902475i	$-0.358251\pi$
$631$	21.6403	0.861486	0.430743	+	0.902475i	$0.358251\pi$
$632$	7.26885	0.289139
$633$	0	0
$634$	59.7763	2.37402
$635$	17.4250	0.691489
$636$	0	0
$637$	5.73169	0.227098
$638$	−43.4374	−1.71970
$639$	0	0
$640$	6.46911	0.255714
$641$	−25.7386	−1.01661	−0.508307	−	0.861176i	$-0.669728\pi$
$641$	−25.7386	−1.01661	−0.508307	+	0.861176i	$0.669728\pi$
$642$	0	0
$643$	20.7802	0.819490	0.409745	−	0.912200i	$-0.365618\pi$
$643$	20.7802	0.819490	0.409745	+	0.912200i	$0.365618\pi$
$644$	1.56089	0.0615078
$645$	0	0
$646$	38.1003	1.49904
$647$	−10.4913	−0.412456	−0.206228	−	0.978504i	$-0.566119\pi$
$647$	−10.4913	−0.412456	−0.206228	+	0.978504i	$0.566119\pi$
$648$	0	0
$649$	2.60782	0.102366
$650$	−10.8159	−0.424234
$651$	0	0
$652$	−31.7311	−1.24268
$653$	32.9451	1.28924	0.644621	−	0.764502i	$-0.277015\pi$
$653$	32.9451	1.28924	0.644621	+	0.764502i	$0.277015\pi$
$654$	0	0
$655$	−21.2884	−0.831806
$656$	−16.3720	−0.639219
$657$	0	0
$658$	20.2472	0.789318
$659$	−26.8160	−1.04460	−0.522301	−	0.852761i	$-0.674926\pi$
$659$	−26.8160	−1.04460	−0.522301	+	0.852761i	$0.674926\pi$
$660$	0	0
$661$	33.9080	1.31887	0.659434	−	0.751762i	$-0.270796\pi$
$661$	33.9080	1.31887	0.659434	+	0.751762i	$0.270796\pi$
$662$	63.3994	2.46409
$663$	0	0
$664$	7.77974	0.301912
$665$	−4.18325	−0.162220
$666$	0	0
$667$	8.32979	0.322531
$668$	18.8041	0.727553
$669$	0	0
$670$	−7.45060	−0.287842
$671$	−40.8533	−1.57713
$672$	0	0
$673$	38.9754	1.50239	0.751196	−	0.660080i	$-0.229477\pi$
$673$	38.9754	1.50239	0.751196	+	0.660080i	$0.229477\pi$
$674$	35.0159	1.34876
$675$	0	0
$676$	30.9873	1.19182
$677$	13.6260	0.523691	0.261845	−	0.965110i	$-0.415669\pi$
$677$	13.6260	0.523691	0.261845	+	0.965110i	$0.415669\pi$
$678$	0	0
$679$	8.78399	0.337099
$680$	3.99931	0.153367
$681$	0	0
$682$	−26.6149	−1.01914
$683$	−0.835032	−0.0319516	−0.0159758	−	0.999872i	$-0.505085\pi$
$683$	−0.835032	−0.0319516	−0.0159758	+	0.999872i	$0.505085\pi$
$684$	0	0
$685$	16.8918	0.645401
$686$	−1.88703	−0.0720472
$687$	0	0
$688$	−28.7468	−1.09596
$689$	43.5764	1.66013
$690$	0	0
$691$	−46.4748	−1.76799	−0.883993	−	0.467500i	$-0.845155\pi$
$691$	−46.4748	−1.76799	−0.883993	+	0.467500i	$0.845155\pi$
$692$	−32.3644	−1.23031
$693$	0	0
$694$	17.7053	0.672084
$695$	13.9494	0.529129
$696$	0	0
$697$	−16.8651	−0.638813
$698$	−34.7876	−1.31673
$699$	0	0
$700$	1.56089	0.0589962
$701$	−23.1813	−0.875545	−0.437772	−	0.899086i	$-0.644232\pi$
$701$	−23.1813	−0.875545	−0.437772	+	0.899086i	$0.644232\pi$
$702$	0	0
$703$	16.6678	0.628638
$704$	−11.5683	−0.435995
$705$	0	0
$706$	55.7700	2.09893
$707$	8.41512	0.316483
$708$	0	0
$709$	19.2114	0.721499	0.360749	−	0.932663i	$-0.382521\pi$
$709$	19.2114	0.721499	0.360749	+	0.932663i	$0.382521\pi$
$710$	−0.357070	−0.0134006
$711$	0	0
$712$	7.20378	0.269973
$713$	5.10382	0.191139
$714$	0	0
$715$	−15.8392	−0.592352
$716$	−2.19556	−0.0820519
$717$	0	0
$718$	−1.94125	−0.0724469
$719$	36.1631	1.34866	0.674328	−	0.738432i	$-0.264433\pi$
$719$	36.1631	1.34866	0.674328	+	0.738432i	$0.264433\pi$
$720$	0	0
$721$	8.44094	0.314357
$722$	2.83133	0.105371
$723$	0	0
$724$	11.7492	0.436655
$725$	8.32979	0.309361
$726$	0	0
$727$	5.62523	0.208628	0.104314	−	0.994544i	$-0.466735\pi$
$727$	5.62523	0.208628	0.104314	+	0.994544i	$0.466735\pi$
$728$	4.74934	0.176022
$729$	0	0
$730$	27.7274	1.02624
$731$	−29.6127	−1.09527
$732$	0	0
$733$	−4.23048	−0.156256	−0.0781281	−	0.996943i	$-0.524894\pi$
$733$	−4.23048	−0.156256	−0.0781281	+	0.996943i	$0.524894\pi$
$734$	39.4155	1.45485
$735$	0	0
$736$	7.18428	0.264816
$737$	−10.9109	−0.401909
$738$	0	0
$739$	41.0958	1.51173	0.755866	−	0.654726i	$-0.227216\pi$
$739$	41.0958	1.51173	0.755866	+	0.654726i	$0.227216\pi$
$740$	−6.21924	−0.228624
$741$	0	0
$742$	−14.3466	−0.526679
$743$	37.6532	1.38136	0.690682	−	0.723159i	$-0.257311\pi$
$743$	37.6532	1.38136	0.690682	+	0.723159i	$0.257311\pi$
$744$	0	0
$745$	−2.66203	−0.0975293
$746$	−11.2003	−0.410072
$747$	0	0
$748$	−20.8189	−0.761215
$749$	10.1994	0.372676
$750$	0	0
$751$	−18.4973	−0.674977	−0.337489	−	0.941330i	$-0.609577\pi$
$751$	−18.4973	−0.674977	−0.337489	+	0.941330i	$0.609577\pi$
$752$	50.2727	1.83326
$753$	0	0
$754$	−90.0941	−3.28103
$755$	2.18060	0.0793600
$756$	0	0
$757$	−16.8077	−0.610887	−0.305444	−	0.952210i	$-0.598805\pi$
$757$	−16.8077	−0.610887	−0.305444	+	0.952210i	$0.598805\pi$
$758$	46.4671	1.68776
$759$	0	0
$760$	−3.46628	−0.125735
$761$	41.2345	1.49475	0.747374	−	0.664404i	$-0.231314\pi$
$761$	41.2345	1.49475	0.747374	+	0.664404i	$0.231314\pi$
$762$	0	0
$763$	−13.6876	−0.495526
$764$	−8.93561	−0.323279
$765$	0	0
$766$	28.6421	1.03488
$767$	5.40891	0.195304
$768$	0	0
$769$	25.8135	0.930859	0.465429	−	0.885085i	$-0.345900\pi$
$769$	25.8135	0.930859	0.465429	+	0.885085i	$0.345900\pi$
$770$	5.21470	0.187925
$771$	0	0
$772$	−11.6972	−0.420991
$773$	−47.9254	−1.72376	−0.861879	−	0.507114i	$-0.830712\pi$
$773$	−47.9254	−1.72376	−0.861879	+	0.507114i	$0.830712\pi$
$774$	0	0
$775$	5.10382	0.183335
$776$	7.27850	0.261283
$777$	0	0
$778$	−67.6137	−2.42407
$779$	14.6174	0.523721
$780$	0	0
$781$	−0.522907	−0.0187111
$782$	9.10782	0.325695
$783$	0	0
$784$	−4.68540	−0.167336
$785$	12.5707	0.448666
$786$	0	0
$787$	26.6300	0.949255	0.474628	−	0.880187i	$-0.342583\pi$
$787$	26.6300	0.949255	0.474628	+	0.880187i	$0.342583\pi$
$788$	35.1341	1.25160
$789$	0	0
$790$	16.5537	0.588954
$791$	−17.7930	−0.632647
$792$	0	0
$793$	−84.7345	−3.00901
$794$	22.4787	0.797740
$795$	0	0
$796$	−17.7371	−0.628675
$797$	5.39533	0.191112	0.0955562	−	0.995424i	$-0.469537\pi$
$797$	5.39533	0.191112	0.0955562	+	0.995424i	$0.469537\pi$
$798$	0	0
$799$	51.7870	1.83209
$800$	7.18428	0.254003
$801$	0	0
$802$	26.1974	0.925061
$803$	40.6050	1.43292
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	−55.2023	−1.94442
$807$	0	0
$808$	6.97285	0.245304
$809$	12.9256	0.454441	0.227221	−	0.973843i	$-0.427036\pi$
$809$	12.9256	0.454441	0.227221	+	0.973843i	$0.427036\pi$
$810$	0	0
$811$	40.5729	1.42471	0.712354	−	0.701821i	$-0.247629\pi$
$811$	40.5729	1.42471	0.712354	+	0.701821i	$0.247629\pi$
$812$	13.0019	0.456278
$813$	0	0
$814$	−20.7775	−0.728252
$815$	20.3288	0.712086
$816$	0	0
$817$	25.6659	0.897938
$818$	−63.5258	−2.22113
$819$	0	0
$820$	−5.45416	−0.190468
$821$	−28.0878	−0.980270	−0.490135	−	0.871647i	$-0.663052\pi$
$821$	−28.0878	−0.980270	−0.490135	+	0.871647i	$0.663052\pi$
$822$	0	0
$823$	−26.8209	−0.934916	−0.467458	−	0.884015i	$-0.654830\pi$
$823$	−26.8209	−0.934916	−0.467458	+	0.884015i	$0.654830\pi$
$824$	6.99425	0.243656
$825$	0	0
$826$	−1.78076	−0.0619607
$827$	42.9862	1.49478	0.747389	−	0.664386i	$-0.231307\pi$
$827$	42.9862	1.49478	0.747389	+	0.664386i	$0.231307\pi$
$828$	0	0
$829$	−5.46109	−0.189672	−0.0948358	−	0.995493i	$-0.530233\pi$
$829$	−5.46109	−0.189672	−0.0948358	+	0.995493i	$0.530233\pi$
$830$	17.7172	0.614972
$831$	0	0
$832$	−23.9939	−0.831838
$833$	−4.82653	−0.167229
$834$	0	0
$835$	−12.0470	−0.416905
$836$	18.0442	0.624071
$837$	0	0
$838$	16.3703	0.565504
$839$	−18.8628	−0.651218	−0.325609	−	0.945505i	$-0.605569\pi$
$839$	−18.8628	−0.651218	−0.325609	+	0.945505i	$0.605569\pi$
$840$	0	0
$841$	40.3854	1.39260
$842$	−66.9856	−2.30848
$843$	0	0
$844$	−15.9349	−0.548503
$845$	−19.8523	−0.682939
$846$	0	0
$847$	−3.36340	−0.115568
$848$	−35.6218	−1.22326
$849$	0	0
$850$	9.10782	0.312396
$851$	3.98441	0.136584
$852$	0	0
$853$	0.949259	0.0325020	0.0162510	−	0.999868i	$-0.494827\pi$
$853$	0.949259	0.0325020	0.0162510	+	0.999868i	$0.494827\pi$
$854$	27.8970	0.954614
$855$	0	0
$856$	8.45128	0.288859
$857$	−9.44081	−0.322492	−0.161246	−	0.986914i	$-0.551551\pi$
$857$	−9.44081	−0.322492	−0.161246	+	0.986914i	$0.551551\pi$
$858$	0	0
$859$	−24.7189	−0.843397	−0.421698	−	0.906736i	$-0.638566\pi$
$859$	−24.7189	−0.843397	−0.421698	+	0.906736i	$0.638566\pi$
$860$	−9.57671	−0.326563
$861$	0	0
$862$	−19.3529	−0.659164
$863$	−21.1933	−0.721429	−0.360715	−	0.932676i	$-0.617467\pi$
$863$	−21.1933	−0.721429	−0.360715	+	0.932676i	$0.617467\pi$
$864$	0	0
$865$	20.7346	0.704996
$866$	2.80619	0.0953582
$867$	0	0
$868$	7.96651	0.270401
$869$	24.2419	0.822349
$870$	0	0
$871$	−22.6305	−0.766806
$872$	−11.3417	−0.384079
$873$	0	0
$874$	−7.89393	−0.267016
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	25.0644	0.846364	0.423182	−	0.906045i	$-0.360913\pi$
$877$	25.0644	0.846364	0.423182	+	0.906045i	$0.360913\pi$
$878$	−24.8179	−0.837564
$879$	0	0
$880$	12.9478	0.436471
$881$	−49.9375	−1.68244	−0.841219	−	0.540695i	$-0.818161\pi$
$881$	−49.9375	−1.68244	−0.841219	+	0.540695i	$0.818161\pi$
$882$	0	0
$883$	5.34720	0.179948	0.0899739	−	0.995944i	$-0.471322\pi$
$883$	5.34720	0.179948	0.0899739	+	0.995944i	$0.471322\pi$
$884$	−43.1808	−1.45233
$885$	0	0
$886$	−57.8324	−1.94292
$887$	30.5018	1.02415	0.512076	−	0.858940i	$-0.328877\pi$
$887$	30.5018	1.02415	0.512076	+	0.858940i	$0.328877\pi$
$888$	0	0
$889$	−17.4250	−0.584415
$890$	16.4055	0.549914
$891$	0	0
$892$	7.30994	0.244755
$893$	−44.8848	−1.50201
$894$	0	0
$895$	1.40660	0.0470176
$896$	−6.46911	−0.216118
$897$	0	0
$898$	−5.42965	−0.181190
$899$	42.5137	1.41791
$900$	0	0
$901$	−36.6947	−1.22248
$902$	−18.2215	−0.606710
$903$	0	0
$904$	−14.7435	−0.490360
$905$	−7.52723	−0.250213
$906$	0	0
$907$	3.92377	0.130287	0.0651433	−	0.997876i	$-0.479250\pi$
$907$	3.92377	0.130287	0.0651433	+	0.997876i	$0.479250\pi$
$908$	1.74269	0.0578332
$909$	0	0
$910$	10.8159	0.358543
$911$	−8.44911	−0.279932	−0.139966	−	0.990156i	$-0.544699\pi$
$911$	−8.44911	−0.279932	−0.139966	+	0.990156i	$0.544699\pi$
$912$	0	0
$913$	25.9457	0.858677
$914$	−21.6499	−0.716115
$915$	0	0
$916$	−14.3076	−0.472736
$917$	21.2884	0.703004
$918$	0	0
$919$	5.55929	0.183384	0.0916920	−	0.995787i	$-0.470772\pi$
$919$	5.55929	0.183384	0.0916920	+	0.995787i	$0.470772\pi$
$920$	−0.828610	−0.0273185
$921$	0	0
$922$	−9.38202	−0.308980
$923$	−1.08457	−0.0356990
$924$	0	0
$925$	3.98441	0.131007
$926$	−14.1266	−0.464228
$927$	0	0
$928$	59.8436	1.96446
$929$	−51.1223	−1.67727	−0.838635	−	0.544694i	$-0.816646\pi$
$929$	−51.1223	−1.67727	−0.838635	+	0.544694i	$0.816646\pi$
$930$	0	0
$931$	4.18325	0.137101
$932$	−16.6450	−0.545225
$933$	0	0
$934$	57.9166	1.89509
$935$	13.3378	0.436194
$936$	0	0
$937$	22.7884	0.744464	0.372232	−	0.928140i	$-0.378593\pi$
$937$	22.7884	0.744464	0.372232	+	0.928140i	$0.378593\pi$
$938$	7.45060	0.243271
$939$	0	0
$940$	16.7478	0.546254
$941$	−1.02341	−0.0333621	−0.0166810	−	0.999861i	$-0.505310\pi$
$941$	−1.02341	−0.0333621	−0.0166810	+	0.999861i	$0.505310\pi$
$942$	0	0
$943$	3.49426	0.113789
$944$	−4.42154	−0.143909
$945$	0	0
$946$	−31.9943	−1.04023
$947$	28.3311	0.920638	0.460319	−	0.887754i	$-0.347735\pi$
$947$	28.3311	0.920638	0.460319	+	0.887754i	$0.347735\pi$
$948$	0	0
$949$	84.2193	2.73388
$950$	−7.89393	−0.256113
$951$	0	0
$952$	−3.99931	−0.129618
$953$	−28.9150	−0.936648	−0.468324	−	0.883557i	$-0.655142\pi$
$953$	−28.9150	−0.936648	−0.468324	+	0.883557i	$0.655142\pi$
$954$	0	0
$955$	5.72468	0.185246
$956$	40.9635	1.32486
$957$	0	0
$958$	−72.4285	−2.34006
$959$	−16.8918	−0.545463
$960$	0	0
$961$	−4.95105	−0.159711
$962$	−43.0950	−1.38944
$963$	0	0
$964$	38.8348	1.25079
$965$	7.49392	0.241238
$966$	0	0
$967$	35.6299	1.14578	0.572890	−	0.819632i	$-0.305822\pi$
$967$	35.6299	1.14578	0.572890	+	0.819632i	$0.305822\pi$
$968$	−2.78694	−0.0895757
$969$	0	0
$970$	16.5757	0.532213
$971$	−8.82435	−0.283187	−0.141593	−	0.989925i	$-0.545223\pi$
$971$	−8.82435	−0.283187	−0.141593	+	0.989925i	$0.545223\pi$
$972$	0	0
$973$	−13.9494	−0.447196
$974$	−20.7281	−0.664170
$975$	0	0
$976$	69.2666	2.21717
$977$	−32.1161	−1.02749	−0.513743	−	0.857944i	$-0.671741\pi$
$977$	−32.1161	−1.02749	−0.513743	+	0.857944i	$0.671741\pi$
$978$	0	0
$979$	24.0248	0.767837
$980$	−1.56089	−0.0498609
$981$	0	0
$982$	−54.3504	−1.73439
$983$	37.5582	1.19792	0.598960	−	0.800779i	$-0.295581\pi$
$983$	37.5582	1.19792	0.598960	+	0.800779i	$0.295581\pi$
$984$	0	0
$985$	−22.5090	−0.717195
$986$	75.8662	2.41607
$987$	0	0
$988$	37.4257	1.19067
$989$	6.13541	0.195095
$990$	0	0
$991$	38.3530	1.21832	0.609161	−	0.793046i	$-0.291506\pi$
$991$	38.3530	1.21832	0.609161	+	0.793046i	$0.291506\pi$
$992$	36.6673	1.16419
$993$	0	0
$994$	0.357070	0.0113256
$995$	11.3634	0.360245
$996$	0	0
$997$	−4.68847	−0.148485	−0.0742426	−	0.997240i	$-0.523654\pi$
$997$	−4.68847	−0.148485	−0.0742426	+	0.997240i	$0.523654\pi$
$998$	−25.4358	−0.805155
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7245.2.a.bv.1.4		10
3.2	odd	2		2415.2.a.w.1.7	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2415.2.a.w.1.7	✓	10	3.2	odd	2
7245.2.a.bv.1.4		10	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:	\( N \)	\(=\)	\( 7245 = 3^{2} \cdot 5 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7245.a (trivial)

\(f(q)\)	\(=\)	\(q-1.88703 q^{2} +1.56089 q^{4} -1.00000 q^{5} +1.00000 q^{7} +0.828610 q^{8} +O(q^{10})\)	\(q-1.88703 q^{2} +1.56089 q^{4} -1.00000 q^{5} +1.00000 q^{7} +0.828610 q^{8} +1.88703 q^{10} +2.76344 q^{11} +5.73169 q^{13} -1.88703 q^{14} -4.68540 q^{16} -4.82653 q^{17} +4.18325 q^{19} -1.56089 q^{20} -5.21470 q^{22} +1.00000 q^{23} +1.00000 q^{25} -10.8159 q^{26} +1.56089 q^{28} +8.32979 q^{29} +5.10382 q^{31} +7.18428 q^{32} +9.10782 q^{34} -1.00000 q^{35} +3.98441 q^{37} -7.89393 q^{38} -0.828610 q^{40} +3.49426 q^{41} +6.13541 q^{43} +4.31343 q^{44} -1.88703 q^{46} -10.7297 q^{47} +1.00000 q^{49} -1.88703 q^{50} +8.94656 q^{52} +7.60272 q^{53} -2.76344 q^{55} +0.828610 q^{56} -15.7186 q^{58} +0.943685 q^{59} -14.7835 q^{61} -9.63107 q^{62} -4.18618 q^{64} -5.73169 q^{65} -3.94831 q^{67} -7.53369 q^{68} +1.88703 q^{70} -0.189223 q^{71} +14.6936 q^{73} -7.51872 q^{74} +6.52961 q^{76} +2.76344 q^{77} +8.77235 q^{79} +4.68540 q^{80} -6.59378 q^{82} +9.38891 q^{83} +4.82653 q^{85} -11.5777 q^{86} +2.28981 q^{88} +8.69381 q^{89} +5.73169 q^{91} +1.56089 q^{92} +20.2472 q^{94} -4.18325 q^{95} +8.78399 q^{97} -1.88703 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8}+O(q^{10}) \) 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8	\( 10 q - 2 q^{2} + 16 q^{4} - 10 q^{5} + 10 q^{7} - 6 q^{8} + 2 q^{10} - 9 q^{11} + 14 q^{13} - 2 q^{14} + 20 q^{16} - 8 q^{17} + 13 q^{19} - 16 q^{20} + 10 q^{23} + 10 q^{25} + 11 q^{26} + 16 q^{28} - 10 q^{29} + 8 q^{31} + 11 q^{32} - 5 q^{34} - 10 q^{35} + 8 q^{37} + 10 q^{38} + 6 q^{40} + 5 q^{41} + 4 q^{43} - 3 q^{44} - 2 q^{46} - q^{47} + 10 q^{49} - 2 q^{50} + 14 q^{52} - 9 q^{53} + 9 q^{55} - 6 q^{56} - 28 q^{58} + 17 q^{59} + 19 q^{61} + 28 q^{62} + 24 q^{64} - 14 q^{65} - 8 q^{68} + 2 q^{70} + 6 q^{73} - 3 q^{74} + 15 q^{76} - 9 q^{77} + 32 q^{79} - 20 q^{80} + 14 q^{82} + 2 q^{83} + 8 q^{85} - 2 q^{86} - 3 q^{88} - 10 q^{89} + 14 q^{91} + 16 q^{92} - 10 q^{94} - 13 q^{95} + 18 q^{97} - 2 q^{98}+O(q^{100}) \) 10 * q - 2 * q^2 + 16 * q^4 - 10 * q^5 + 10 * q^7 - 6 * q^8 + 2 * q^10 - 9 * q^11 + 14 * q^13 - 2 * q^14 + 20 * q^16 - 8 * q^17 + 13 * q^19 - 16 * q^20 + 10 * q^23 + 10 * q^25 + 11 * q^26 + 16 * q^28 - 10 * q^29 + 8 * q^31 + 11 * q^32 - 5 * q^34 - 10 * q^35 + 8 * q^37 + 10 * q^38 + 6 * q^40 + 5 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 - q^47 + 10 * q^49 - 2 * q^50 + 14 * q^52 - 9 * q^53 + 9 * q^55 - 6 * q^56 - 28 * q^58 + 17 * q^59 + 19 * q^61 + 28 * q^62 + 24 * q^64 - 14 * q^65 - 8 * q^68 + 2 * q^70 + 6 * q^73 - 3 * q^74 + 15 * q^76 - 9 * q^77 + 32 * q^79 - 20 * q^80 + 14 * q^82 + 2 * q^83 + 8 * q^85 - 2 * q^86 - 3 * q^88 - 10 * q^89 + 14 * q^91 + 16 * q^92 - 10 * q^94 - 13 * q^95 + 18 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

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Database

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