LMFDB - Embedded newform 7605.2.a.co.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.3352656.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 10x^{2} + 6x - 6 $ x^5 - 2x^4 - 6x^3 + 10x^2 + 6x - 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.626791$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.626791 q^{2} -1.60713 q^{4} +1.00000 q^{5} +4.43127 q^{7} -2.26092 q^{8} +O(q^{10})$	$q+0.626791 q^{2} -1.60713 q^{4} +1.00000 q^{5} +4.43127 q^{7} -2.26092 q^{8} +0.626791 q^{10} +6.05806 q^{11} +2.77748 q^{14} +1.79714 q^{16} -5.84106 q^{17} -3.60713 q^{19} -1.60713 q^{20} +3.79714 q^{22} +2.26092 q^{23} +1.00000 q^{25} -7.12164 q^{28} +8.08506 q^{29} +6.45826 q^{31} +5.64827 q^{32} -3.66112 q^{34} +4.43127 q^{35} -1.79714 q^{37} -2.26092 q^{38} -2.26092 q^{40} -7.99175 q^{41} +6.48751 q^{43} -9.73610 q^{44} +1.41713 q^{46} -3.22841 q^{47} +12.6361 q^{49} +0.626791 q^{50} +10.0768 q^{53} +6.05806 q^{55} -10.0187 q^{56} +5.06764 q^{58} +3.12921 q^{59} -2.45826 q^{61} +4.04798 q^{62} -0.0539916 q^{64} -4.33864 q^{67} +9.38735 q^{68} +2.77748 q^{70} +8.49484 q^{71} -0.819388 q^{73} -1.12643 q^{74} +5.79714 q^{76} +26.8449 q^{77} -8.42187 q^{79} +1.79714 q^{80} -5.00916 q^{82} +2.35423 q^{83} -5.84106 q^{85} +4.06631 q^{86} -13.6968 q^{88} +0.773413 q^{89} -3.63360 q^{92} -2.02354 q^{94} -3.60713 q^{95} -6.32104 q^{97} +7.92022 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + q^{7} + 6 q^{8}+O(q^{10}) $ 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 + q^7 + 6 * q^8	$ 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{11} - 4 q^{14} + 4 q^{16} - 4 q^{19} + 6 q^{20} + 14 q^{22} - 6 q^{23} + 5 q^{25} - 2 q^{28} + 16 q^{29} + 9 q^{31} + 14 q^{32} + q^{35} - 4 q^{37} + 6 q^{38} + 6 q^{40} + 6 q^{41} + 15 q^{43} + 14 q^{44} - 16 q^{46} + 10 q^{47} + 10 q^{49} + 2 q^{50} - 20 q^{53} + 8 q^{55} - 2 q^{56} - 4 q^{58} + 12 q^{59} + 11 q^{61} - 22 q^{62} + 4 q^{64} + 5 q^{67} + 50 q^{68} - 4 q^{70} + 10 q^{71} + q^{73} - 26 q^{74} + 24 q^{76} + 42 q^{77} - 17 q^{79} + 4 q^{80} + 16 q^{82} + 16 q^{83} + 44 q^{86} + 20 q^{88} + 4 q^{89} - 34 q^{92} - 16 q^{94} - 4 q^{95} - 11 q^{97} + 30 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 + q^7 + 6 * q^8 + 2 * q^10 + 8 * q^11 - 4 * q^14 + 4 * q^16 - 4 * q^19 + 6 * q^20 + 14 * q^22 - 6 * q^23 + 5 * q^25 - 2 * q^28 + 16 * q^29 + 9 * q^31 + 14 * q^32 + q^35 - 4 * q^37 + 6 * q^38 + 6 * q^40 + 6 * q^41 + 15 * q^43 + 14 * q^44 - 16 * q^46 + 10 * q^47 + 10 * q^49 + 2 * q^50 - 20 * q^53 + 8 * q^55 - 2 * q^56 - 4 * q^58 + 12 * q^59 + 11 * q^61 - 22 * q^62 + 4 * q^64 + 5 * q^67 + 50 * q^68 - 4 * q^70 + 10 * q^71 + q^73 - 26 * q^74 + 24 * q^76 + 42 * q^77 - 17 * q^79 + 4 * q^80 + 16 * q^82 + 16 * q^83 + 44 * q^86 + 20 * q^88 + 4 * q^89 - 34 * q^92 - 16 * q^94 - 4 * q^95 - 11 * q^97 + 30 * 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 $

$2$	0.626791	0.443208	0.221604	−	0.975137i	$-0.428871\pi$
$2$	0.626791	0.443208	0.221604	+	0.975137i	$0.428871\pi$
$3$	0	0
$3$	0	0
$4$	−1.60713	−0.803566
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.43127	1.67486	0.837431	−	0.546543i	$-0.184057\pi$
$7$	4.43127	1.67486	0.837431	+	0.546543i	$0.184057\pi$
$8$	−2.26092	−0.799356
$9$	0	0
$10$	0.626791	0.198209
$11$	6.05806	1.82657	0.913287	−	0.407317i	$-0.133536\pi$
$11$	6.05806	1.82657	0.913287	+	0.407317i	$0.133536\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	2.77748	0.742313
$15$	0	0
$16$	1.79714	0.449285
$17$	−5.84106	−1.41666	−0.708332	−	0.705879i	$-0.750552\pi$
$17$	−5.84106	−1.41666	−0.708332	+	0.705879i	$0.750552\pi$
$18$	0	0
$19$	−3.60713	−0.827533	−0.413766	−	0.910383i	$-0.635787\pi$
$19$	−3.60713	−0.827533	−0.413766	+	0.910383i	$0.635787\pi$
$20$	−1.60713	−0.359366
$21$	0	0
$22$	3.79714	0.809553
$23$	2.26092	0.471434	0.235717	−	0.971822i	$-0.424256\pi$
$23$	2.26092	0.471434	0.235717	+	0.971822i	$0.424256\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	−7.12164	−1.34586
$29$	8.08506	1.50136	0.750679	−	0.660668i	$-0.229727\pi$
$29$	8.08506	1.50136	0.750679	+	0.660668i	$0.229727\pi$
$30$	0	0
$31$	6.45826	1.15994	0.579969	−	0.814638i	$-0.303065\pi$
$31$	6.45826	1.15994	0.579969	+	0.814638i	$0.303065\pi$
$32$	5.64827	0.998483
$33$	0	0
$34$	−3.66112	−0.627878
$35$	4.43127	0.749021
$36$	0	0
$37$	−1.79714	−0.295448	−0.147724	−	0.989029i	$-0.547195\pi$
$37$	−1.79714	−0.295448	−0.147724	+	0.989029i	$0.547195\pi$
$38$	−2.26092	−0.366770
$39$	0	0
$40$	−2.26092	−0.357483
$41$	−7.99175	−1.24810	−0.624051	−	0.781384i	$-0.714514\pi$
$41$	−7.99175	−1.24810	−0.624051	+	0.781384i	$0.714514\pi$
$42$	0	0
$43$	6.48751	0.989336	0.494668	−	0.869082i	$-0.335290\pi$
$43$	6.48751	0.989336	0.494668	+	0.869082i	$0.335290\pi$
$44$	−9.73610	−1.46777
$45$	0	0
$46$	1.41713	0.208944
$47$	−3.22841	−0.470912	−0.235456	−	0.971885i	$-0.575658\pi$
$47$	−3.22841	−0.470912	−0.235456	+	0.971885i	$0.575658\pi$
$48$	0	0
$49$	12.6361	1.80516
$50$	0.626791	0.0886417
$51$	0	0
$52$	0	0
$53$	10.0768	1.38416	0.692078	−	0.721823i	$-0.256696\pi$
$53$	10.0768	1.38416	0.692078	+	0.721823i	$0.256696\pi$
$54$	0	0
$55$	6.05806	0.816869
$56$	−10.0187	−1.33881
$57$	0	0
$58$	5.06764	0.665414
$59$	3.12921	0.407388	0.203694	−	0.979035i	$-0.434705\pi$
$59$	3.12921	0.407388	0.203694	+	0.979035i	$0.434705\pi$
$60$	0	0
$61$	−2.45826	−0.314748	−0.157374	−	0.987539i	$-0.550303\pi$
$61$	−2.45826	−0.314748	−0.157374	+	0.987539i	$0.550303\pi$
$62$	4.04798	0.514095
$63$	0	0
$64$	−0.0539916	−0.00674895
$65$	0	0
$66$	0	0
$67$	−4.33864	−0.530049	−0.265025	−	0.964242i	$-0.585380\pi$
$67$	−4.33864	−0.530049	−0.265025	+	0.964242i	$0.585380\pi$
$68$	9.38735	1.13838
$69$	0	0
$70$	2.77748	0.331972
$71$	8.49484	1.00815	0.504076	−	0.863659i	$-0.331833\pi$
$71$	8.49484	1.00815	0.504076	+	0.863659i	$0.331833\pi$
$72$	0	0
$73$	−0.819388	−0.0959021	−0.0479511	−	0.998850i	$-0.515269\pi$
$73$	−0.819388	−0.0959021	−0.0479511	+	0.998850i	$0.515269\pi$
$74$	−1.12643	−0.130945
$75$	0	0
$76$	5.79714	0.664978
$77$	26.8449	3.05926
$78$	0	0
$79$	−8.42187	−0.947535	−0.473767	−	0.880650i	$-0.657106\pi$
$79$	−8.42187	−0.947535	−0.473767	+	0.880650i	$0.657106\pi$
$80$	1.79714	0.200926
$81$	0	0
$82$	−5.00916	−0.553169
$83$	2.35423	0.258410	0.129205	−	0.991618i	$-0.458757\pi$
$83$	2.35423	0.258410	0.129205	+	0.991618i	$0.458757\pi$
$84$	0	0
$85$	−5.84106	−0.633552
$86$	4.06631	0.438482
$87$	0	0
$88$	−13.6968	−1.46008
$89$	0.773413	0.0819816	0.0409908	−	0.999160i	$-0.486949\pi$
$89$	0.773413	0.0819816	0.0409908	+	0.999160i	$0.486949\pi$
$90$	0	0
$91$	0	0
$92$	−3.63360	−0.378829
$93$	0	0
$94$	−2.02354	−0.208712
$95$	−3.60713	−0.370084
$96$	0	0
$97$	−6.32104	−0.641804	−0.320902	−	0.947112i	$-0.603986\pi$
$97$	−6.32104	−0.641804	−0.320902	+	0.947112i	$0.603986\pi$
$98$	7.92022	0.800063
$99$	0	0
$100$	−1.60713	−0.160713
$101$	−8.91695	−0.887270	−0.443635	−	0.896208i	$-0.646311\pi$
$101$	−8.91695	−0.887270	−0.443635	+	0.896208i	$0.646311\pi$
$102$	0	0
$103$	0.112968	0.0111311	0.00556556	−	0.999985i	$-0.498228\pi$
$103$	0.112968	0.0111311	0.00556556	+	0.999985i	$0.498228\pi$
$104$	0	0
$105$	0	0
$106$	6.31605	0.613469
$107$	−7.75025	−0.749245	−0.374622	−	0.927177i	$-0.622228\pi$
$107$	−7.75025	−0.749245	−0.374622	+	0.927177i	$0.622228\pi$
$108$	0	0
$109$	−11.4744	−1.09905	−0.549525	−	0.835477i	$-0.685191\pi$
$109$	−11.4744	−1.09905	−0.549525	+	0.835477i	$0.685191\pi$
$110$	3.79714	0.362043
$111$	0	0
$112$	7.96361	0.752490
$113$	6.24739	0.587705	0.293852	−	0.955851i	$-0.405063\pi$
$113$	6.24739	0.587705	0.293852	+	0.955851i	$0.405063\pi$
$114$	0	0
$115$	2.26092	0.210832
$116$	−12.9938	−1.20644
$117$	0	0
$118$	1.96136	0.180558
$119$	−25.8833	−2.37272
$120$	0	0
$121$	25.7001	2.33637
$122$	−1.54082	−0.139499
$123$	0	0
$124$	−10.3793	−0.932087
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.2034	0.994143	0.497071	−	0.867710i	$-0.334409\pi$
$127$	11.2034	0.994143	0.497071	+	0.867710i	$0.334409\pi$
$128$	−11.3304	−1.00147
$129$	0	0
$130$	0	0
$131$	−0.494893	−0.0432390	−0.0216195	−	0.999766i	$-0.506882\pi$
$131$	−0.494893	−0.0432390	−0.0216195	+	0.999766i	$0.506882\pi$
$132$	0	0
$133$	−15.9842	−1.38600
$134$	−2.71942	−0.234922
$135$	0	0
$136$	13.2062	1.13242
$137$	23.1498	1.97782	0.988909	−	0.148521i	$-0.0474512\pi$
$137$	23.1498	1.97782	0.988909	+	0.148521i	$0.0474512\pi$
$138$	0	0
$139$	−11.9574	−1.01421	−0.507107	−	0.861883i	$-0.669285\pi$
$139$	−11.9574	−1.01421	−0.507107	+	0.861883i	$0.669285\pi$
$140$	−7.12164	−0.601888
$141$	0	0
$142$	5.32450	0.446822
$143$	0	0
$144$	0	0
$145$	8.08506	0.671427
$146$	−0.513585	−0.0425046
$147$	0	0
$148$	2.88824	0.237412
$149$	−21.2179	−1.73823	−0.869117	−	0.494606i	$-0.835312\pi$
$149$	−21.2179	−1.73823	−0.869117	+	0.494606i	$0.835312\pi$
$150$	0	0
$151$	19.4623	1.58382	0.791909	−	0.610639i	$-0.209087\pi$
$151$	19.4623	1.58382	0.791909	+	0.610639i	$0.209087\pi$
$152$	8.15544	0.661493
$153$	0	0
$154$	16.8261	1.35589
$155$	6.45826	0.518740
$156$	0	0
$157$	20.6547	1.64842	0.824212	−	0.566282i	$-0.191618\pi$
$157$	20.6547	1.64842	0.824212	+	0.566282i	$0.191618\pi$
$158$	−5.27876	−0.419955
$159$	0	0
$160$	5.64827	0.446535
$161$	10.0187	0.789587
$162$	0	0
$163$	−4.85980	−0.380649	−0.190324	−	0.981721i	$-0.560954\pi$
$163$	−4.85980	−0.380649	−0.190324	+	0.981721i	$0.560954\pi$
$164$	12.8438	1.00293
$165$	0	0
$166$	1.47561	0.114530
$167$	2.57995	0.199642	0.0998212	−	0.995005i	$-0.468173\pi$
$167$	2.57995	0.199642	0.0998212	+	0.995005i	$0.468173\pi$
$168$	0	0
$169$	0	0
$170$	−3.66112	−0.280795
$171$	0	0
$172$	−10.4263	−0.794997
$173$	−10.2468	−0.779048	−0.389524	−	0.921016i	$-0.627360\pi$
$173$	−10.2468	−0.779048	−0.389524	+	0.921016i	$0.627360\pi$
$174$	0	0
$175$	4.43127	0.334972
$176$	10.8872	0.820652
$177$	0	0
$178$	0.484769	0.0363349
$179$	−10.6028	−0.792492	−0.396246	−	0.918144i	$-0.629687\pi$
$179$	−10.6028	−0.792492	−0.396246	+	0.918144i	$0.629687\pi$
$180$	0	0
$181$	16.8618	1.25333	0.626664	−	0.779289i	$-0.284420\pi$
$181$	16.8618	1.25333	0.626664	+	0.779289i	$0.284420\pi$
$182$	0	0
$183$	0	0
$184$	−5.11176	−0.376844
$185$	−1.79714	−0.132128
$186$	0	0
$187$	−35.3855	−2.58764
$188$	5.18848	0.378409
$189$	0	0
$190$	−2.26092	−0.164024
$191$	−10.9910	−0.795279	−0.397640	−	0.917542i	$-0.630171\pi$
$191$	−10.9910	−0.795279	−0.397640	+	0.917542i	$0.630171\pi$
$192$	0	0
$193$	−6.65173	−0.478802	−0.239401	−	0.970921i	$-0.576951\pi$
$193$	−6.65173	−0.478802	−0.239401	+	0.970921i	$0.576951\pi$
$194$	−3.96197	−0.284453
$195$	0	0
$196$	−20.3079	−1.45057
$197$	23.9860	1.70893	0.854466	−	0.519508i	$-0.173885\pi$
$197$	23.9860	1.70893	0.854466	+	0.519508i	$0.173885\pi$
$198$	0	0
$199$	−20.9103	−1.48229	−0.741147	−	0.671343i	$-0.765718\pi$
$199$	−20.9103	−1.48229	−0.741147	+	0.671343i	$0.765718\pi$
$200$	−2.26092	−0.159871
$201$	0	0
$202$	−5.58907	−0.393246
$203$	35.8270	2.51457
$204$	0	0
$205$	−7.99175	−0.558168
$206$	0.0708076	0.00493340
$207$	0	0
$208$	0	0
$209$	−21.8522	−1.51155
$210$	0	0
$211$	13.1308	0.903961	0.451981	−	0.892028i	$-0.350718\pi$
$211$	13.1308	0.903961	0.451981	+	0.892028i	$0.350718\pi$
$212$	−16.1948	−1.11226
$213$	0	0
$214$	−4.85779	−0.332072
$215$	6.48751	0.442444
$216$	0	0
$217$	28.6183	1.94274
$218$	−7.19207	−0.487108
$219$	0	0
$220$	−9.73610	−0.656408
$221$	0	0
$222$	0	0
$223$	−21.2007	−1.41970	−0.709851	−	0.704351i	$-0.751238\pi$
$223$	−21.2007	−1.41970	−0.709851	+	0.704351i	$0.751238\pi$
$224$	25.0290	1.67232
$225$	0	0
$226$	3.91581	0.260476
$227$	9.23944	0.613243	0.306622	−	0.951832i	$-0.400801\pi$
$227$	9.23944	0.613243	0.306622	+	0.951832i	$0.400801\pi$
$228$	0	0
$229$	3.16502	0.209150	0.104575	−	0.994517i	$-0.466652\pi$
$229$	3.16502	0.209150	0.104575	+	0.994517i	$0.466652\pi$
$230$	1.41713	0.0934425
$231$	0	0
$232$	−18.2797	−1.20012
$233$	2.83723	0.185873	0.0929364	−	0.995672i	$-0.470375\pi$
$233$	2.83723	0.185873	0.0929364	+	0.995672i	$0.470375\pi$
$234$	0	0
$235$	−3.22841	−0.210598
$236$	−5.02905	−0.327364
$237$	0	0
$238$	−16.2234	−1.05161
$239$	0.634366	0.0410337	0.0205169	−	0.999790i	$-0.493469\pi$
$239$	0.634366	0.0410337	0.0205169	+	0.999790i	$0.493469\pi$
$240$	0	0
$241$	−0.898466	−0.0578753	−0.0289376	−	0.999581i	$-0.509212\pi$
$241$	−0.898466	−0.0578753	−0.0289376	+	0.999581i	$0.509212\pi$
$242$	16.1086	1.03550
$243$	0	0
$244$	3.95076	0.252921
$245$	12.6361	0.807293
$246$	0	0
$247$	0	0
$248$	−14.6016	−0.927204
$249$	0	0
$250$	0.626791	0.0396418
$251$	14.4568	0.912503	0.456252	−	0.889851i	$-0.349192\pi$
$251$	14.4568	0.912503	0.456252	+	0.889851i	$0.349192\pi$
$252$	0	0
$253$	13.6968	0.861110
$254$	7.02221	0.440613
$255$	0	0
$256$	−6.99380	−0.437113
$257$	−0.418504	−0.0261056	−0.0130528	−	0.999915i	$-0.504155\pi$
$257$	−0.418504	−0.0261056	−0.0130528	+	0.999915i	$0.504155\pi$
$258$	0	0
$259$	−7.96361	−0.494835
$260$	0	0
$261$	0	0
$262$	−0.310195	−0.0191639
$263$	2.49052	0.153572	0.0767861	−	0.997048i	$-0.475534\pi$
$263$	2.49052	0.153572	0.0767861	+	0.997048i	$0.475534\pi$
$264$	0	0
$265$	10.0768	0.619013
$266$	−10.0187	−0.614288
$267$	0	0
$268$	6.97277	0.425930
$269$	1.76858	0.107832	0.0539160	−	0.998545i	$-0.482830\pi$
$269$	1.76858	0.107832	0.0539160	+	0.998545i	$0.482830\pi$
$270$	0	0
$271$	−20.4035	−1.23943	−0.619714	−	0.784828i	$-0.712751\pi$
$271$	−20.4035	−1.23943	−0.619714	+	0.784828i	$0.712751\pi$
$272$	−10.4972	−0.636486
$273$	0	0
$274$	14.5101	0.876586
$275$	6.05806	0.365315
$276$	0	0
$277$	−4.84494	−0.291104	−0.145552	−	0.989351i	$-0.546496\pi$
$277$	−4.84494	−0.291104	−0.145552	+	0.989351i	$0.546496\pi$
$278$	−7.49480	−0.449508
$279$	0	0
$280$	−10.0187	−0.598734
$281$	5.20237	0.310347	0.155174	−	0.987887i	$-0.450406\pi$
$281$	5.20237	0.310347	0.155174	+	0.987887i	$0.450406\pi$
$282$	0	0
$283$	10.0203	0.595647	0.297824	−	0.954621i	$-0.403739\pi$
$283$	10.0203	0.595647	0.297824	+	0.954621i	$0.403739\pi$
$284$	−13.6523	−0.810117
$285$	0	0
$286$	0	0
$287$	−35.4136	−2.09040
$288$	0	0
$289$	17.1179	1.00694
$290$	5.06764	0.297582
$291$	0	0
$292$	1.31687	0.0770637
$293$	7.50698	0.438562	0.219281	−	0.975662i	$-0.429629\pi$
$293$	7.50698	0.438562	0.219281	+	0.975662i	$0.429629\pi$
$294$	0	0
$295$	3.12921	0.182190
$296$	4.06319	0.236168
$297$	0	0
$298$	−13.2992	−0.770400
$299$	0	0
$300$	0	0
$301$	28.7479	1.65700
$302$	12.1988	0.701962
$303$	0	0
$304$	−6.48252	−0.371798
$305$	−2.45826	−0.140760
$306$	0	0
$307$	−11.4404	−0.652940	−0.326470	−	0.945208i	$-0.605859\pi$
$307$	−11.4404	−0.652940	−0.326470	+	0.945208i	$0.605859\pi$
$308$	−43.1433	−2.45832
$309$	0	0
$310$	4.04798	0.229910
$311$	24.2351	1.37424	0.687122	−	0.726542i	$-0.258874\pi$
$311$	24.2351	1.37424	0.687122	+	0.726542i	$0.258874\pi$
$312$	0	0
$313$	−18.8796	−1.06714	−0.533570	−	0.845756i	$-0.679150\pi$
$313$	−18.8796	−1.06714	−0.533570	+	0.845756i	$0.679150\pi$
$314$	12.9462	0.730595
$315$	0	0
$316$	13.5351	0.761407
$317$	22.3459	1.25507	0.627534	−	0.778589i	$-0.284064\pi$
$317$	22.3459	1.25507	0.627534	+	0.778589i	$0.284064\pi$
$318$	0	0
$319$	48.9797	2.74234
$320$	−0.0539916	−0.00301822
$321$	0	0
$322$	6.27966	0.349952
$323$	21.0695	1.17234
$324$	0	0
$325$	0	0
$326$	−3.04608	−0.168707
$327$	0	0
$328$	18.0687	0.997677
$329$	−14.3059	−0.788712
$330$	0	0
$331$	−22.6887	−1.24708	−0.623541	−	0.781790i	$-0.714307\pi$
$331$	−22.6887	−1.24708	−0.623541	+	0.781790i	$0.714307\pi$
$332$	−3.78356	−0.207650
$333$	0	0
$334$	1.61709	0.0884832
$335$	−4.33864	−0.237045
$336$	0	0
$337$	−0.166960	−0.00909489	−0.00454745	−	0.999990i	$-0.501448\pi$
$337$	−0.166960	−0.00909489	−0.00454745	+	0.999990i	$0.501448\pi$
$338$	0	0
$339$	0	0
$340$	9.38735	0.509101
$341$	39.1245	2.11871
$342$	0	0
$343$	24.9752	1.34854
$344$	−14.6677	−0.790831
$345$	0	0
$346$	−6.42259	−0.345281
$347$	14.6343	0.785611	0.392806	−	0.919622i	$-0.371505\pi$
$347$	14.6343	0.785611	0.392806	+	0.919622i	$0.371505\pi$
$348$	0	0
$349$	2.22712	0.119215	0.0596074	−	0.998222i	$-0.481015\pi$
$349$	2.22712	0.119215	0.0596074	+	0.998222i	$0.481015\pi$
$350$	2.77748	0.148463
$351$	0	0
$352$	34.2176	1.82380
$353$	5.93939	0.316122	0.158061	−	0.987429i	$-0.449476\pi$
$353$	5.93939	0.316122	0.158061	+	0.987429i	$0.449476\pi$
$354$	0	0
$355$	8.49484	0.450859
$356$	−1.24298	−0.0658777
$357$	0	0
$358$	−6.64576	−0.351239
$359$	10.9234	0.576516	0.288258	−	0.957553i	$-0.406924\pi$
$359$	10.9234	0.576516	0.288258	+	0.957553i	$0.406924\pi$
$360$	0	0
$361$	−5.98860	−0.315189
$362$	10.5688	0.555486
$363$	0	0
$364$	0	0
$365$	−0.819388	−0.0428887
$366$	0	0
$367$	8.64448	0.451238	0.225619	−	0.974216i	$-0.427560\pi$
$367$	8.64448	0.451238	0.225619	+	0.974216i	$0.427560\pi$
$368$	4.06319	0.211808
$369$	0	0
$370$	−1.12643	−0.0585604
$371$	44.6530	2.31827
$372$	0	0
$373$	18.3052	0.947808	0.473904	−	0.880577i	$-0.342844\pi$
$373$	18.3052	0.947808	0.473904	+	0.880577i	$0.342844\pi$
$374$	−22.1793	−1.14686
$375$	0	0
$376$	7.29917	0.376426
$377$	0	0
$378$	0	0
$379$	−30.0062	−1.54131	−0.770657	−	0.637250i	$-0.780072\pi$
$379$	−30.0062	−1.54131	−0.770657	+	0.637250i	$0.780072\pi$
$380$	5.79714	0.297387
$381$	0	0
$382$	−6.88905	−0.352474
$383$	14.5360	0.742754	0.371377	−	0.928482i	$-0.378886\pi$
$383$	14.5360	0.742754	0.371377	+	0.928482i	$0.378886\pi$
$384$	0	0
$385$	26.8449	1.36814
$386$	−4.16925	−0.212209
$387$	0	0
$388$	10.1587	0.515732
$389$	24.5259	1.24351	0.621756	−	0.783211i	$-0.286419\pi$
$389$	24.5259	1.24351	0.621756	+	0.783211i	$0.286419\pi$
$390$	0	0
$391$	−13.2062	−0.667864
$392$	−28.5693	−1.44297
$393$	0	0
$394$	15.0342	0.757413
$395$	−8.42187	−0.423750
$396$	0	0
$397$	−9.44090	−0.473825	−0.236913	−	0.971531i	$-0.576136\pi$
$397$	−9.44090	−0.473825	−0.236913	+	0.971531i	$0.576136\pi$
$398$	−13.1064	−0.656965
$399$	0	0
$400$	1.79714	0.0898570
$401$	0.488375	0.0243883	0.0121941	−	0.999926i	$-0.496118\pi$
$401$	0.488375	0.0243883	0.0121941	+	0.999926i	$0.496118\pi$
$402$	0	0
$403$	0	0
$404$	14.3307	0.712980
$405$	0	0
$406$	22.4561	1.11448
$407$	−10.8872	−0.539658
$408$	0	0
$409$	20.7203	1.02455	0.512277	−	0.858820i	$-0.328802\pi$
$409$	20.7203	1.02455	0.512277	+	0.858820i	$0.328802\pi$
$410$	−5.00916	−0.247385
$411$	0	0
$412$	−0.181555	−0.00894458
$413$	13.8664	0.682319
$414$	0	0
$415$	2.35423	0.115565
$416$	0	0
$417$	0	0
$418$	−13.6968	−0.669932
$419$	4.54748	0.222159	0.111079	−	0.993812i	$-0.464569\pi$
$419$	4.54748	0.222159	0.111079	+	0.993812i	$0.464569\pi$
$420$	0	0
$421$	31.0991	1.51568	0.757840	−	0.652441i	$-0.226255\pi$
$421$	31.0991	1.51568	0.757840	+	0.652441i	$0.226255\pi$
$422$	8.23027	0.400643
$423$	0	0
$424$	−22.7828	−1.10643
$425$	−5.84106	−0.283333
$426$	0	0
$427$	−10.8932	−0.527160
$428$	12.4557	0.602068
$429$	0	0
$430$	4.06631	0.196095
$431$	34.2162	1.64813	0.824067	−	0.566492i	$-0.191700\pi$
$431$	34.2162	1.64813	0.824067	+	0.566492i	$0.191700\pi$
$432$	0	0
$433$	−5.56326	−0.267353	−0.133676	−	0.991025i	$-0.542678\pi$
$433$	−5.56326	−0.267353	−0.133676	+	0.991025i	$0.542678\pi$
$434$	17.9377	0.861037
$435$	0	0
$436$	18.4409	0.883159
$437$	−8.15544	−0.390127
$438$	0	0
$439$	−2.81949	−0.134567	−0.0672834	−	0.997734i	$-0.521433\pi$
$439$	−2.81949	−0.134567	−0.0672834	+	0.997734i	$0.521433\pi$
$440$	−13.6968	−0.652969
$441$	0	0
$442$	0	0
$443$	−15.0635	−0.715686	−0.357843	−	0.933782i	$-0.616488\pi$
$443$	−15.0635	−0.715686	−0.357843	+	0.933782i	$0.616488\pi$
$444$	0	0
$445$	0.773413	0.0366633
$446$	−13.2884	−0.629224
$447$	0	0
$448$	−0.239251	−0.0113036
$449$	3.84456	0.181436	0.0907181	−	0.995877i	$-0.471084\pi$
$449$	3.84456	0.181436	0.0907181	+	0.995877i	$0.471084\pi$
$450$	0	0
$451$	−48.4145	−2.27975
$452$	−10.0404	−0.472260
$453$	0	0
$454$	5.79120	0.271795
$455$	0	0
$456$	0	0
$457$	30.4280	1.42336	0.711682	−	0.702502i	$-0.247934\pi$
$457$	30.4280	1.42336	0.711682	+	0.702502i	$0.247934\pi$
$458$	1.98381	0.0926973
$459$	0	0
$460$	−3.63360	−0.169417
$461$	30.7921	1.43413	0.717066	−	0.697006i	$-0.245485\pi$
$461$	30.7921	1.43413	0.717066	+	0.697006i	$0.245485\pi$
$462$	0	0
$463$	−12.9034	−0.599674	−0.299837	−	0.953991i	$-0.596932\pi$
$463$	−12.9034	−0.599674	−0.299837	+	0.953991i	$0.596932\pi$
$464$	14.5300	0.674537
$465$	0	0
$466$	1.77835	0.0823804
$467$	10.2346	0.473601	0.236800	−	0.971558i	$-0.423901\pi$
$467$	10.2346	0.473601	0.236800	+	0.971558i	$0.423901\pi$
$468$	0	0
$469$	−19.2257	−0.887759
$470$	−2.02354	−0.0933389
$471$	0	0
$472$	−7.07489	−0.325648
$473$	39.3017	1.80709
$474$	0	0
$475$	−3.60713	−0.165507
$476$	41.5979	1.90664
$477$	0	0
$478$	0.397615	0.0181865
$479$	35.0043	1.59939	0.799693	−	0.600408i	$-0.204995\pi$
$479$	35.0043	1.59939	0.799693	+	0.600408i	$0.204995\pi$
$480$	0	0
$481$	0	0
$482$	−0.563151	−0.0256508
$483$	0	0
$484$	−41.3034	−1.87743
$485$	−6.32104	−0.287024
$486$	0	0
$487$	−1.53185	−0.0694147	−0.0347074	−	0.999398i	$-0.511050\pi$
$487$	−1.53185	−0.0694147	−0.0347074	+	0.999398i	$0.511050\pi$
$488$	5.55794	0.251596
$489$	0	0
$490$	7.92022	0.357799
$491$	−25.0248	−1.12935	−0.564676	−	0.825313i	$-0.690999\pi$
$491$	−25.0248	−1.12935	−0.564676	+	0.825313i	$0.690999\pi$
$492$	0	0
$493$	−47.2253	−2.12692
$494$	0	0
$495$	0	0
$496$	11.6064	0.521143
$497$	37.6429	1.68852
$498$	0	0
$499$	27.7838	1.24377	0.621887	−	0.783107i	$-0.286366\pi$
$499$	27.7838	1.24377	0.621887	+	0.783107i	$0.286366\pi$
$500$	−1.60713	−0.0718732
$501$	0	0
$502$	9.06138	0.404429
$503$	43.0139	1.91789	0.958947	−	0.283587i	$-0.0915245\pi$
$503$	43.0139	1.91789	0.958947	+	0.283587i	$0.0915245\pi$
$504$	0	0
$505$	−8.91695	−0.396799
$506$	8.58503	0.381651
$507$	0	0
$508$	−18.0054	−0.798860
$509$	1.13704	0.0503983	0.0251992	−	0.999682i	$-0.491978\pi$
$509$	1.13704	0.0503983	0.0251992	+	0.999682i	$0.491978\pi$
$510$	0	0
$511$	−3.63093	−0.160623
$512$	18.2771	0.807742
$513$	0	0
$514$	−0.262315	−0.0115702
$515$	0.112968	0.00497798
$516$	0	0
$517$	−19.5579	−0.860155
$518$	−4.99152	−0.219315
$519$	0	0
$520$	0	0
$521$	30.7311	1.34635	0.673177	−	0.739481i	$-0.264929\pi$
$521$	30.7311	1.34635	0.673177	+	0.739481i	$0.264929\pi$
$522$	0	0
$523$	−17.2654	−0.754962	−0.377481	−	0.926017i	$-0.623210\pi$
$523$	−17.2654	−0.754962	−0.377481	+	0.926017i	$0.623210\pi$
$524$	0.795358	0.0347454
$525$	0	0
$526$	1.56104	0.0680645
$527$	−37.7231	−1.64324
$528$	0	0
$529$	−17.8882	−0.777750
$530$	6.31605	0.274352
$531$	0	0
$532$	25.6887	1.11375
$533$	0	0
$534$	0	0
$535$	−7.75025	−0.335072
$536$	9.80931	0.423698
$537$	0	0
$538$	1.10853	0.0477921
$539$	76.5505	3.29726
$540$	0	0
$541$	−17.8625	−0.767970	−0.383985	−	0.923339i	$-0.625449\pi$
$541$	−17.8625	−0.767970	−0.383985	+	0.923339i	$0.625449\pi$
$542$	−12.7888	−0.549325
$543$	0	0
$544$	−32.9919	−1.41451
$545$	−11.4744	−0.491510
$546$	0	0
$547$	−4.97952	−0.212909	−0.106455	−	0.994318i	$-0.533950\pi$
$547$	−4.97952	−0.212909	−0.106455	+	0.994318i	$0.533950\pi$
$548$	−37.2048	−1.58931
$549$	0	0
$550$	3.79714	0.161911
$551$	−29.1639	−1.24242
$552$	0	0
$553$	−37.3196	−1.58699
$554$	−3.03676	−0.129020
$555$	0	0
$556$	19.2171	0.814989
$557$	−36.9312	−1.56482	−0.782412	−	0.622761i	$-0.786011\pi$
$557$	−36.9312	−1.56482	−0.782412	+	0.622761i	$0.786011\pi$
$558$	0	0
$559$	0	0
$560$	7.96361	0.336524
$561$	0	0
$562$	3.26080	0.137549
$563$	−9.36568	−0.394716	−0.197358	−	0.980331i	$-0.563236\pi$
$563$	−9.36568	−0.394716	−0.197358	+	0.980331i	$0.563236\pi$
$564$	0	0
$565$	6.24739	0.262830
$566$	6.28066	0.263996
$567$	0	0
$568$	−19.2062	−0.805873
$569$	−0.436833	−0.0183130	−0.00915650	−	0.999958i	$-0.502915\pi$
$569$	−0.436833	−0.0183130	−0.00915650	+	0.999958i	$0.502915\pi$
$570$	0	0
$571$	−4.69561	−0.196505	−0.0982525	−	0.995162i	$-0.531325\pi$
$571$	−4.69561	−0.196505	−0.0982525	+	0.995162i	$0.531325\pi$
$572$	0	0
$573$	0	0
$574$	−22.1969	−0.926482
$575$	2.26092	0.0942869
$576$	0	0
$577$	−30.2121	−1.25775	−0.628873	−	0.777508i	$-0.716483\pi$
$577$	−30.2121	−1.25775	−0.628873	+	0.777508i	$0.716483\pi$
$578$	10.7294	0.446283
$579$	0	0
$580$	−12.9938	−0.539536
$581$	10.4322	0.432801
$582$	0	0
$583$	61.0459	2.52826
$584$	1.85257	0.0766599
$585$	0	0
$586$	4.70531	0.194375
$587$	−43.7538	−1.80591	−0.902956	−	0.429733i	$-0.858608\pi$
$587$	−43.7538	−1.80591	−0.902956	+	0.429733i	$0.858608\pi$
$588$	0	0
$589$	−23.2958	−0.959887
$590$	1.96136	0.0807480
$591$	0	0
$592$	−3.22971	−0.132740
$593$	−5.94151	−0.243989	−0.121994	−	0.992531i	$-0.538929\pi$
$593$	−5.94151	−0.243989	−0.121994	+	0.992531i	$0.538929\pi$
$594$	0	0
$595$	−25.8833	−1.06111
$596$	34.0999	1.39679
$597$	0	0
$598$	0	0
$599$	−37.0505	−1.51384	−0.756922	−	0.653506i	$-0.773298\pi$
$599$	−37.0505	−1.51384	−0.756922	+	0.653506i	$0.773298\pi$
$600$	0	0
$601$	−37.2822	−1.52077	−0.760387	−	0.649470i	$-0.774991\pi$
$601$	−37.2822	−1.52077	−0.760387	+	0.649470i	$0.774991\pi$
$602$	18.0189	0.734397
$603$	0	0
$604$	−31.2785	−1.27270
$605$	25.7001	1.04486
$606$	0	0
$607$	24.4929	0.994135	0.497068	−	0.867712i	$-0.334410\pi$
$607$	24.4929	0.994135	0.497068	+	0.867712i	$0.334410\pi$
$608$	−20.3741	−0.826277
$609$	0	0
$610$	−1.54082	−0.0623859
$611$	0	0
$612$	0	0
$613$	9.47202	0.382571	0.191286	−	0.981534i	$-0.438734\pi$
$613$	9.47202	0.382571	0.191286	+	0.981534i	$0.438734\pi$
$614$	−7.17076	−0.289388
$615$	0	0
$616$	−60.6941	−2.44544
$617$	1.19807	0.0482326	0.0241163	−	0.999709i	$-0.492323\pi$
$617$	1.19807	0.0482326	0.0241163	+	0.999709i	$0.492323\pi$
$618$	0	0
$619$	−19.2559	−0.773959	−0.386979	−	0.922088i	$-0.626482\pi$
$619$	−19.2559	−0.773959	−0.386979	+	0.922088i	$0.626482\pi$
$620$	−10.3793	−0.416842
$621$	0	0
$622$	15.1903	0.609077
$623$	3.42720	0.137308
$624$	0	0
$625$	1.00000	0.0400000
$626$	−11.8336	−0.472966
$627$	0	0
$628$	−33.1948	−1.32462
$629$	10.4972	0.418551
$630$	0	0
$631$	6.25990	0.249203	0.124601	−	0.992207i	$-0.460235\pi$
$631$	6.25990	0.249203	0.124601	+	0.992207i	$0.460235\pi$
$632$	19.0412	0.757417
$633$	0	0
$634$	14.0062	0.556257
$635$	11.2034	0.444594
$636$	0	0
$637$	0	0
$638$	30.7001	1.21543
$639$	0	0
$640$	−11.3304	−0.447873
$641$	8.84417	0.349324	0.174662	−	0.984628i	$-0.444117\pi$
$641$	8.84417	0.349324	0.174662	+	0.984628i	$0.444117\pi$
$642$	0	0
$643$	−32.1012	−1.26595	−0.632973	−	0.774174i	$-0.718166\pi$
$643$	−32.1012	−1.26595	−0.632973	+	0.774174i	$0.718166\pi$
$644$	−16.1014	−0.634486
$645$	0	0
$646$	13.2062	0.519589
$647$	−14.7567	−0.580145	−0.290072	−	0.957005i	$-0.593679\pi$
$647$	−14.7567	−0.580145	−0.290072	+	0.957005i	$0.593679\pi$
$648$	0	0
$649$	18.9569	0.744125
$650$	0	0
$651$	0	0
$652$	7.81034	0.305877
$653$	26.6491	1.04286	0.521430	−	0.853294i	$-0.325399\pi$
$653$	26.6491	1.04286	0.521430	+	0.853294i	$0.325399\pi$
$654$	0	0
$655$	−0.494893	−0.0193371
$656$	−14.3623	−0.560753
$657$	0	0
$658$	−8.96684	−0.349564
$659$	2.44953	0.0954202	0.0477101	−	0.998861i	$-0.484808\pi$
$659$	2.44953	0.0954202	0.0477101	+	0.998861i	$0.484808\pi$
$660$	0	0
$661$	16.4711	0.640652	0.320326	−	0.947307i	$-0.396208\pi$
$661$	16.4711	0.640652	0.320326	+	0.947307i	$0.396208\pi$
$662$	−14.2211	−0.552718
$663$	0	0
$664$	−5.32272	−0.206562
$665$	−15.9842	−0.619840
$666$	0	0
$667$	18.2797	0.707791
$668$	−4.14632	−0.160426
$669$	0	0
$670$	−2.71942	−0.105060
$671$	−14.8923	−0.574911
$672$	0	0
$673$	−1.38749	−0.0534838	−0.0267419	−	0.999642i	$-0.508513\pi$
$673$	−1.38749	−0.0534838	−0.0267419	+	0.999642i	$0.508513\pi$
$674$	−0.104649	−0.00403093
$675$	0	0
$676$	0	0
$677$	23.0636	0.886407	0.443203	−	0.896421i	$-0.353842\pi$
$677$	23.0636	0.886407	0.443203	+	0.896421i	$0.353842\pi$
$678$	0	0
$679$	−28.0102	−1.07493
$680$	13.2062	0.506433
$681$	0	0
$682$	24.5229	0.939032
$683$	21.0160	0.804155	0.402078	−	0.915606i	$-0.368288\pi$
$683$	21.0160	0.804155	0.402078	+	0.915606i	$0.368288\pi$
$684$	0	0
$685$	23.1498	0.884507
$686$	15.6543	0.597683
$687$	0	0
$688$	11.6590	0.444494
$689$	0	0
$690$	0	0
$691$	−24.0578	−0.915200	−0.457600	−	0.889158i	$-0.651291\pi$
$691$	−24.0578	−0.915200	−0.457600	+	0.889158i	$0.651291\pi$
$692$	16.4679	0.626017
$693$	0	0
$694$	9.17266	0.348190
$695$	−11.9574	−0.453571
$696$	0	0
$697$	46.6802	1.76814
$698$	1.39594	0.0528370
$699$	0	0
$700$	−7.12164	−0.269173
$701$	20.0188	0.756099	0.378050	−	0.925785i	$-0.376595\pi$
$701$	20.0188	0.756099	0.378050	+	0.925785i	$0.376595\pi$
$702$	0	0
$703$	6.48252	0.244493
$704$	−0.327085	−0.0123275
$705$	0	0
$706$	3.72276	0.140108
$707$	−39.5134	−1.48605
$708$	0	0
$709$	17.0592	0.640672	0.320336	−	0.947304i	$-0.396204\pi$
$709$	17.0592	0.640672	0.320336	+	0.947304i	$0.396204\pi$
$710$	5.32450	0.199825
$711$	0	0
$712$	−1.74862	−0.0655325
$713$	14.6016	0.546835
$714$	0	0
$715$	0	0
$716$	17.0401	0.636820
$717$	0	0
$718$	6.84671	0.255517
$719$	−42.1119	−1.57051	−0.785254	−	0.619174i	$-0.787468\pi$
$719$	−42.1119	−1.57051	−0.785254	+	0.619174i	$0.787468\pi$
$720$	0	0
$721$	0.500593	0.0186431
$722$	−3.75360	−0.139695
$723$	0	0
$724$	−27.0992	−1.00713
$725$	8.08506	0.300271
$726$	0	0
$727$	3.62167	0.134320	0.0671601	−	0.997742i	$-0.478606\pi$
$727$	3.62167	0.134320	0.0671601	+	0.997742i	$0.478606\pi$
$728$	0	0
$729$	0	0
$730$	−0.513585	−0.0190086
$731$	−37.8939	−1.40156
$732$	0	0
$733$	−10.2438	−0.378362	−0.189181	−	0.981942i	$-0.560583\pi$
$733$	−10.2438	−0.378362	−0.189181	+	0.981942i	$0.560583\pi$
$734$	5.41829	0.199993
$735$	0	0
$736$	12.7703	0.470719
$737$	−26.2837	−0.968174
$738$	0	0
$739$	9.26945	0.340982	0.170491	−	0.985359i	$-0.445465\pi$
$739$	9.26945	0.340982	0.170491	+	0.985359i	$0.445465\pi$
$740$	2.88824	0.106174
$741$	0	0
$742$	27.9881	1.02748
$743$	44.7671	1.64234	0.821172	−	0.570680i	$-0.193320\pi$
$743$	44.7671	1.64234	0.821172	+	0.570680i	$0.193320\pi$
$744$	0	0
$745$	−21.2179	−0.777362
$746$	11.4735	0.420076
$747$	0	0
$748$	56.8691	2.07934
$749$	−34.3434	−1.25488
$750$	0	0
$751$	15.3403	0.559776	0.279888	−	0.960033i	$-0.409703\pi$
$751$	15.3403	0.559776	0.279888	+	0.960033i	$0.409703\pi$
$752$	−5.80190	−0.211574
$753$	0	0
$754$	0	0
$755$	19.4623	0.708305
$756$	0	0
$757$	6.00296	0.218181	0.109091	−	0.994032i	$-0.465206\pi$
$757$	6.00296	0.218181	0.109091	+	0.994032i	$0.465206\pi$
$758$	−18.8076	−0.683123
$759$	0	0
$760$	8.15544	0.295829
$761$	−20.9651	−0.759984	−0.379992	−	0.924990i	$-0.624073\pi$
$761$	−20.9651	−0.759984	−0.379992	+	0.924990i	$0.624073\pi$
$762$	0	0
$763$	−50.8462	−1.84076
$764$	17.6640	0.639060
$765$	0	0
$766$	9.11103	0.329195
$767$	0	0
$768$	0	0
$769$	−43.8152	−1.58002	−0.790008	−	0.613096i	$-0.789924\pi$
$769$	−43.8152	−1.58002	−0.790008	+	0.613096i	$0.789924\pi$
$770$	16.8261	0.606372
$771$	0	0
$772$	10.6902	0.384749
$773$	−3.37220	−0.121290	−0.0606449	−	0.998159i	$-0.519316\pi$
$773$	−3.37220	−0.121290	−0.0606449	+	0.998159i	$0.519316\pi$
$774$	0	0
$775$	6.45826	0.231988
$776$	14.2914	0.513030
$777$	0	0
$778$	15.3726	0.551135
$779$	28.8273	1.03284
$780$	0	0
$781$	51.4623	1.84146
$782$	−8.27751	−0.296003
$783$	0	0
$784$	22.7089	0.811032
$785$	20.6547	0.737198
$786$	0	0
$787$	−35.7844	−1.27558	−0.637789	−	0.770211i	$-0.720151\pi$
$787$	−35.7844	−1.27558	−0.637789	+	0.770211i	$0.720151\pi$
$788$	−38.5487	−1.37324
$789$	0	0
$790$	−5.27876	−0.187810
$791$	27.6838	0.984324
$792$	0	0
$793$	0	0
$794$	−5.91748	−0.210003
$795$	0	0
$796$	33.6057	1.19112
$797$	−28.3251	−1.00333	−0.501663	−	0.865063i	$-0.667278\pi$
$797$	−28.3251	−1.00333	−0.501663	+	0.865063i	$0.667278\pi$
$798$	0	0
$799$	18.8573	0.667124
$800$	5.64827	0.199697
$801$	0	0
$802$	0.306109	0.0108091
$803$	−4.96390	−0.175172
$804$	0	0
$805$	10.0187	0.353114
$806$	0	0
$807$	0	0
$808$	20.1605	0.709245
$809$	−29.1396	−1.02450	−0.512248	−	0.858838i	$-0.671187\pi$
$809$	−29.1396	−1.02450	−0.512248	+	0.858838i	$0.671187\pi$
$810$	0	0
$811$	−16.3733	−0.574946	−0.287473	−	0.957789i	$-0.592815\pi$
$811$	−16.3733	−0.574946	−0.287473	+	0.957789i	$0.592815\pi$
$812$	−57.5788	−2.02062
$813$	0	0
$814$	−6.82399	−0.239181
$815$	−4.85980	−0.170231
$816$	0	0
$817$	−23.4013	−0.818708
$818$	12.9873	0.454091
$819$	0	0
$820$	12.8438	0.448525
$821$	35.1612	1.22713	0.613566	−	0.789643i	$-0.289734\pi$
$821$	35.1612	1.22713	0.613566	+	0.789643i	$0.289734\pi$
$822$	0	0
$823$	19.6847	0.686164	0.343082	−	0.939305i	$-0.388529\pi$
$823$	19.6847	0.686164	0.343082	+	0.939305i	$0.388529\pi$
$824$	−0.255413	−0.00889772
$825$	0	0
$826$	8.69132	0.302410
$827$	11.6881	0.406436	0.203218	−	0.979134i	$-0.434860\pi$
$827$	11.6881	0.406436	0.203218	+	0.979134i	$0.434860\pi$
$828$	0	0
$829$	−45.5203	−1.58099	−0.790493	−	0.612472i	$-0.790175\pi$
$829$	−45.5203	−1.58099	−0.790493	+	0.612472i	$0.790175\pi$
$830$	1.47561	0.0512192
$831$	0	0
$832$	0	0
$833$	−73.8084	−2.55731
$834$	0	0
$835$	2.57995	0.0892828
$836$	35.1194	1.21463
$837$	0	0
$838$	2.85032	0.0984626
$839$	29.0477	1.00284	0.501420	−	0.865204i	$-0.332811\pi$
$839$	29.0477	1.00284	0.501420	+	0.865204i	$0.332811\pi$
$840$	0	0
$841$	36.3681	1.25407
$842$	19.4927	0.671762
$843$	0	0
$844$	−21.1029	−0.726393
$845$	0	0
$846$	0	0
$847$	113.884	3.91310
$848$	18.1094	0.621880
$849$	0	0
$850$	−3.66112	−0.125576
$851$	−4.06319	−0.139284
$852$	0	0
$853$	−48.3320	−1.65486	−0.827429	−	0.561571i	$-0.810197\pi$
$853$	−48.3320	−1.65486	−0.827429	+	0.561571i	$0.810197\pi$
$854$	−6.82778	−0.233642
$855$	0	0
$856$	17.5227	0.598913
$857$	−41.2151	−1.40788	−0.703940	−	0.710260i	$-0.748578\pi$
$857$	−41.2151	−1.40788	−0.703940	+	0.710260i	$0.748578\pi$
$858$	0	0
$859$	−52.5448	−1.79280	−0.896402	−	0.443242i	$-0.853828\pi$
$859$	−52.5448	−1.79280	−0.896402	+	0.443242i	$0.853828\pi$
$860$	−10.4263	−0.355533
$861$	0	0
$862$	21.4464	0.730467
$863$	−22.9888	−0.782549	−0.391275	−	0.920274i	$-0.627966\pi$
$863$	−22.9888	−0.782549	−0.391275	+	0.920274i	$0.627966\pi$
$864$	0	0
$865$	−10.2468	−0.348401
$866$	−3.48700	−0.118493
$867$	0	0
$868$	−45.9934	−1.56112
$869$	−51.0202	−1.73074
$870$	0	0
$871$	0	0
$872$	25.9427	0.878532
$873$	0	0
$874$	−5.11176	−0.172908
$875$	4.43127	0.149804
$876$	0	0
$877$	−52.2984	−1.76599	−0.882995	−	0.469381i	$-0.844477\pi$
$877$	−52.2984	−1.76599	−0.882995	+	0.469381i	$0.844477\pi$
$878$	−1.76723	−0.0596412
$879$	0	0
$880$	10.8872	0.367007
$881$	−33.6966	−1.13527	−0.567634	−	0.823281i	$-0.692141\pi$
$881$	−33.6966	−1.13527	−0.567634	+	0.823281i	$0.692141\pi$
$882$	0	0
$883$	−20.4392	−0.687835	−0.343917	−	0.939000i	$-0.611754\pi$
$883$	−20.4392	−0.687835	−0.343917	+	0.939000i	$0.611754\pi$
$884$	0	0
$885$	0	0
$886$	−9.44165	−0.317198
$887$	−20.2005	−0.678268	−0.339134	−	0.940738i	$-0.610134\pi$
$887$	−20.2005	−0.678268	−0.339134	+	0.940738i	$0.610134\pi$
$888$	0	0
$889$	49.6454	1.66505
$890$	0.484769	0.0162495
$891$	0	0
$892$	34.0723	1.14083
$893$	11.6453	0.389695
$894$	0	0
$895$	−10.6028	−0.354413
$896$	−50.2080	−1.67733
$897$	0	0
$898$	2.40974	0.0804140
$899$	52.2154	1.74148
$900$	0	0
$901$	−58.8592	−1.96088
$902$	−30.3458	−1.01040
$903$	0	0
$904$	−14.1248	−0.469785
$905$	16.8618	0.560505
$906$	0	0
$907$	−41.8029	−1.38804	−0.694022	−	0.719954i	$-0.744163\pi$
$907$	−41.8029	−1.38804	−0.694022	+	0.719954i	$0.744163\pi$
$908$	−14.8490	−0.492782
$909$	0	0
$910$	0	0
$911$	−3.42332	−0.113420	−0.0567098	−	0.998391i	$-0.518061\pi$
$911$	−3.42332	−0.113420	−0.0567098	+	0.998391i	$0.518061\pi$
$912$	0	0
$913$	14.2621	0.472005
$914$	19.0720	0.630847
$915$	0	0
$916$	−5.08661	−0.168066
$917$	−2.19300	−0.0724193
$918$	0	0
$919$	7.05207	0.232626	0.116313	−	0.993213i	$-0.462892\pi$
$919$	7.05207	0.232626	0.116313	+	0.993213i	$0.462892\pi$
$920$	−5.11176	−0.168530
$921$	0	0
$922$	19.3002	0.635619
$923$	0	0
$924$	0	0
$925$	−1.79714	−0.0590896
$926$	−8.08776	−0.265780
$927$	0	0
$928$	45.6666	1.49908
$929$	−10.0420	−0.329467	−0.164733	−	0.986338i	$-0.552676\pi$
$929$	−10.0420	−0.329467	−0.164733	+	0.986338i	$0.552676\pi$
$930$	0	0
$931$	−45.5802	−1.49383
$932$	−4.55980	−0.149361
$933$	0	0
$934$	6.41496	0.209904
$935$	−35.3855	−1.15723
$936$	0	0
$937$	5.84613	0.190985	0.0954924	−	0.995430i	$-0.469557\pi$
$937$	5.84613	0.190985	0.0954924	+	0.995430i	$0.469557\pi$
$938$	−12.0505	−0.393462
$939$	0	0
$940$	5.18848	0.169230
$941$	−46.1640	−1.50490	−0.752452	−	0.658647i	$-0.771129\pi$
$941$	−46.1640	−1.50490	−0.752452	+	0.658647i	$0.771129\pi$
$942$	0	0
$943$	−18.0687	−0.588398
$944$	5.62363	0.183033
$945$	0	0
$946$	24.6340	0.800919
$947$	−14.4830	−0.470635	−0.235318	−	0.971919i	$-0.575613\pi$
$947$	−14.4830	−0.470635	−0.235318	+	0.971919i	$0.575613\pi$
$948$	0	0
$949$	0	0
$950$	−2.26092	−0.0733539
$951$	0	0
$952$	58.5200	1.89665
$953$	−38.4423	−1.24527	−0.622635	−	0.782513i	$-0.713938\pi$
$953$	−38.4423	−1.24527	−0.622635	+	0.782513i	$0.713938\pi$
$954$	0	0
$955$	−10.9910	−0.355660
$956$	−1.01951	−0.0329733
$957$	0	0
$958$	21.9404	0.708862
$959$	102.583	3.31257
$960$	0	0
$961$	10.7092	0.345457
$962$	0	0
$963$	0	0
$964$	1.44395	0.0465066
$965$	−6.65173	−0.214127
$966$	0	0
$967$	44.2627	1.42339	0.711696	−	0.702487i	$-0.247927\pi$
$967$	44.2627	1.42339	0.711696	+	0.702487i	$0.247927\pi$
$968$	−58.1058	−1.86759
$969$	0	0
$970$	−3.96197	−0.127211
$971$	−44.4059	−1.42505	−0.712527	−	0.701644i	$-0.752450\pi$
$971$	−44.4059	−1.42505	−0.712527	+	0.701644i	$0.752450\pi$
$972$	0	0
$973$	−52.9865	−1.69867
$974$	−0.960150	−0.0307652
$975$	0	0
$976$	−4.41784	−0.141412
$977$	−3.38438	−0.108276	−0.0541379	−	0.998533i	$-0.517241\pi$
$977$	−3.38438	−0.108276	−0.0541379	+	0.998533i	$0.517241\pi$
$978$	0	0
$979$	4.68538	0.149745
$980$	−20.3079	−0.648714
$981$	0	0
$982$	−15.6853	−0.500538
$983$	39.6424	1.26440	0.632198	−	0.774807i	$-0.282153\pi$
$983$	39.6424	1.26440	0.632198	+	0.774807i	$0.282153\pi$
$984$	0	0
$985$	23.9860	0.764257
$986$	−29.6004	−0.942669
$987$	0	0
$988$	0	0
$989$	14.6677	0.466407
$990$	0	0
$991$	−45.3443	−1.44041	−0.720205	−	0.693761i	$-0.755952\pi$
$991$	−45.3443	−1.44041	−0.720205	+	0.693761i	$0.755952\pi$
$992$	36.4780	1.15818
$993$	0	0
$994$	23.5943	0.748365
$995$	−20.9103	−0.662902
$996$	0	0
$997$	53.9968	1.71010	0.855048	−	0.518549i	$-0.173528\pi$
$997$	53.9968	1.71010	0.855048	+	0.518549i	$0.173528\pi$
$998$	17.4147	0.551251
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.co.1.3		5
3.2	odd	2		7605.2.a.cm.1.3		5
13.3	even	3		585.2.j.h.451.3	yes	10
13.9	even	3		585.2.j.h.406.3	✓	10
13.12	even	2		7605.2.a.cl.1.3		5
39.29	odd	6		585.2.j.i.451.3	yes	10
39.35	odd	6		585.2.j.i.406.3	yes	10
39.38	odd	2		7605.2.a.cn.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.j.h.406.3	✓	10	13.9	even	3
585.2.j.h.451.3	yes	10	13.3	even	3
585.2.j.i.406.3	yes	10	39.35	odd	6
585.2.j.i.451.3	yes	10	39.29	odd	6
7605.2.a.cl.1.3		5	13.12	even	2
7605.2.a.cm.1.3		5	3.2	odd	2
7605.2.a.cn.1.3		5	39.38	odd	2
7605.2.a.co.1.3		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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q+0.626791 q^{2} -1.60713 q^{4} +1.00000 q^{5} +4.43127 q^{7} -2.26092 q^{8} +O(q^{10})\)	\(q+0.626791 q^{2} -1.60713 q^{4} +1.00000 q^{5} +4.43127 q^{7} -2.26092 q^{8} +0.626791 q^{10} +6.05806 q^{11} +2.77748 q^{14} +1.79714 q^{16} -5.84106 q^{17} -3.60713 q^{19} -1.60713 q^{20} +3.79714 q^{22} +2.26092 q^{23} +1.00000 q^{25} -7.12164 q^{28} +8.08506 q^{29} +6.45826 q^{31} +5.64827 q^{32} -3.66112 q^{34} +4.43127 q^{35} -1.79714 q^{37} -2.26092 q^{38} -2.26092 q^{40} -7.99175 q^{41} +6.48751 q^{43} -9.73610 q^{44} +1.41713 q^{46} -3.22841 q^{47} +12.6361 q^{49} +0.626791 q^{50} +10.0768 q^{53} +6.05806 q^{55} -10.0187 q^{56} +5.06764 q^{58} +3.12921 q^{59} -2.45826 q^{61} +4.04798 q^{62} -0.0539916 q^{64} -4.33864 q^{67} +9.38735 q^{68} +2.77748 q^{70} +8.49484 q^{71} -0.819388 q^{73} -1.12643 q^{74} +5.79714 q^{76} +26.8449 q^{77} -8.42187 q^{79} +1.79714 q^{80} -5.00916 q^{82} +2.35423 q^{83} -5.84106 q^{85} +4.06631 q^{86} -13.6968 q^{88} +0.773413 q^{89} -3.63360 q^{92} -2.02354 q^{94} -3.60713 q^{95} -6.32104 q^{97} +7.92022 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + q^{7} + 6 q^{8}+O(q^{10}) \) 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 + q^7 + 6 * q^8	\( 5 q + 2 q^{2} + 6 q^{4} + 5 q^{5} + q^{7} + 6 q^{8} + 2 q^{10} + 8 q^{11} - 4 q^{14} + 4 q^{16} - 4 q^{19} + 6 q^{20} + 14 q^{22} - 6 q^{23} + 5 q^{25} - 2 q^{28} + 16 q^{29} + 9 q^{31} + 14 q^{32} + q^{35} - 4 q^{37} + 6 q^{38} + 6 q^{40} + 6 q^{41} + 15 q^{43} + 14 q^{44} - 16 q^{46} + 10 q^{47} + 10 q^{49} + 2 q^{50} - 20 q^{53} + 8 q^{55} - 2 q^{56} - 4 q^{58} + 12 q^{59} + 11 q^{61} - 22 q^{62} + 4 q^{64} + 5 q^{67} + 50 q^{68} - 4 q^{70} + 10 q^{71} + q^{73} - 26 q^{74} + 24 q^{76} + 42 q^{77} - 17 q^{79} + 4 q^{80} + 16 q^{82} + 16 q^{83} + 44 q^{86} + 20 q^{88} + 4 q^{89} - 34 q^{92} - 16 q^{94} - 4 q^{95} - 11 q^{97} + 30 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 6 * q^4 + 5 * q^5 + q^7 + 6 * q^8 + 2 * q^10 + 8 * q^11 - 4 * q^14 + 4 * q^16 - 4 * q^19 + 6 * q^20 + 14 * q^22 - 6 * q^23 + 5 * q^25 - 2 * q^28 + 16 * q^29 + 9 * q^31 + 14 * q^32 + q^35 - 4 * q^37 + 6 * q^38 + 6 * q^40 + 6 * q^41 + 15 * q^43 + 14 * q^44 - 16 * q^46 + 10 * q^47 + 10 * q^49 + 2 * q^50 - 20 * q^53 + 8 * q^55 - 2 * q^56 - 4 * q^58 + 12 * q^59 + 11 * q^61 - 22 * q^62 + 4 * q^64 + 5 * q^67 + 50 * q^68 - 4 * q^70 + 10 * q^71 + q^73 - 26 * q^74 + 24 * q^76 + 42 * q^77 - 17 * q^79 + 4 * q^80 + 16 * q^82 + 16 * q^83 + 44 * q^86 + 20 * q^88 + 4 * q^89 - 34 * q^92 - 16 * q^94 - 4 * q^95 - 11 * q^97 + 30 * q^98

Introduction

L-functions

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