LMFDB - Embedded newform 4025.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4025,2,Mod(1,4025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4025, 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("4025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4025 = 5^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4025.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:	$32.1397868136$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 805)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	4025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.801938 q^{2} -1.69202 q^{3} -1.35690 q^{4} +1.35690 q^{6} +1.00000 q^{7} +2.69202 q^{8} -0.137063 q^{9} +O(q^{10})$	\(q-0.801938 q^{2} -1.69202 q^{3} -1.35690 q^{4} +1.35690 q^{6} +1.00000 q^{7} +2.69202 q^{8} -0.137063 q^{9} -3.80194 q^{11} +2.29590 q^{12} +4.54288 q^{13} -0.801938 q^{14} +0.554958 q^{16} -2.91185 q^{17} +0.109916 q^{18} -3.35690 q^{19} -1.69202 q^{21} +3.04892 q^{22} +1.00000 q^{23} -4.55496 q^{24} -3.64310 q^{26} +5.30798 q^{27} -1.35690 q^{28} +2.09783 q^{29} +0.246980 q^{31} -5.82908 q^{32} +6.43296 q^{33} +2.33513 q^{34} +0.185981 q^{36} -5.40581 q^{37} +2.69202 q^{38} -7.68664 q^{39} +2.82371 q^{41} +1.35690 q^{42} +0.801938 q^{43} +5.15883 q^{44} -0.801938 q^{46} -1.97285 q^{47} -0.939001 q^{48} +1.00000 q^{49} +4.92692 q^{51} -6.16421 q^{52} +9.78448 q^{53} -4.25667 q^{54} +2.69202 q^{56} +5.67994 q^{57} -1.68233 q^{58} -6.66487 q^{59} -5.02715 q^{61} -0.198062 q^{62} -0.137063 q^{63} +3.56465 q^{64} -5.15883 q^{66} +5.03684 q^{67} +3.95108 q^{68} -1.69202 q^{69} +5.05861 q^{71} -0.368977 q^{72} +0.972853 q^{73} +4.33513 q^{74} +4.55496 q^{76} -3.80194 q^{77} +6.16421 q^{78} +3.14675 q^{79} -8.57002 q^{81} -2.26444 q^{82} +16.3230 q^{83} +2.29590 q^{84} -0.643104 q^{86} -3.54958 q^{87} -10.2349 q^{88} +7.18598 q^{89} +4.54288 q^{91} -1.35690 q^{92} -0.417895 q^{93} +1.58211 q^{94} +9.86294 q^{96} -12.9758 q^{97} -0.801938 q^{98} +0.521106 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 3 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 + 3 * q^7 + 3 * q^8 + 5 * q^9	$ 3 q + 2 q^{2} + 3 q^{7} + 3 q^{8} + 5 q^{9} - 7 q^{11} - 7 q^{12} - 5 q^{13} + 2 q^{14} + 2 q^{16} - 5 q^{17} + q^{18} - 6 q^{19} + 3 q^{23} - 14 q^{24} - 15 q^{26} + 21 q^{27} - 12 q^{29} - 4 q^{31} - 7 q^{32} + 6 q^{34} - 14 q^{36} - 3 q^{37} + 3 q^{38} - 21 q^{39} + q^{41} - 2 q^{43} + 7 q^{44} + 2 q^{46} - 12 q^{47} + 7 q^{48} + 3 q^{49} - 14 q^{51} - 7 q^{52} + 9 q^{53} + 14 q^{54} + 3 q^{56} - 7 q^{57} - 22 q^{58} - 21 q^{59} - 9 q^{61} - 5 q^{62} + 5 q^{63} - 11 q^{64} - 7 q^{66} - 13 q^{67} + 21 q^{68} - 16 q^{71} - 16 q^{72} + 9 q^{73} + 12 q^{74} + 14 q^{76} - 7 q^{77} + 7 q^{78} - 18 q^{79} - q^{81} - 18 q^{82} + 29 q^{83} - 7 q^{84} - 6 q^{86} - 14 q^{87} - 7 q^{88} + 7 q^{89} - 5 q^{91} - 7 q^{93} - q^{94} + 35 q^{96} - q^{97} + 2 q^{98} - 14 q^{99}+O(q^{100}) $ 3 * q + 2 * q^2 + 3 * q^7 + 3 * q^8 + 5 * q^9 - 7 * q^11 - 7 * q^12 - 5 * q^13 + 2 * q^14 + 2 * q^16 - 5 * q^17 + q^18 - 6 * q^19 + 3 * q^23 - 14 * q^24 - 15 * q^26 + 21 * q^27 - 12 * q^29 - 4 * q^31 - 7 * q^32 + 6 * q^34 - 14 * q^36 - 3 * q^37 + 3 * q^38 - 21 * q^39 + q^41 - 2 * q^43 + 7 * q^44 + 2 * q^46 - 12 * q^47 + 7 * q^48 + 3 * q^49 - 14 * q^51 - 7 * q^52 + 9 * q^53 + 14 * q^54 + 3 * q^56 - 7 * q^57 - 22 * q^58 - 21 * q^59 - 9 * q^61 - 5 * q^62 + 5 * q^63 - 11 * q^64 - 7 * q^66 - 13 * q^67 + 21 * q^68 - 16 * q^71 - 16 * q^72 + 9 * q^73 + 12 * q^74 + 14 * q^76 - 7 * q^77 + 7 * q^78 - 18 * q^79 - q^81 - 18 * q^82 + 29 * q^83 - 7 * q^84 - 6 * q^86 - 14 * q^87 - 7 * q^88 + 7 * q^89 - 5 * q^91 - 7 * q^93 - q^94 + 35 * q^96 - q^97 + 2 * q^98 - 14 * q^99

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 $

$2$	−0.801938	−0.567056	−0.283528	−	0.958964i	$-0.591505\pi$
$2$	−0.801938	−0.567056	−0.283528	+	0.958964i	$0.591505\pi$
$3$	−1.69202	−0.976889	−0.488445	−	0.872595i	$-0.662436\pi$
$3$	−1.69202	−0.976889	−0.488445	+	0.872595i	$0.662436\pi$
$4$	−1.35690	−0.678448
$5$	0	0
$5$	0	0
$6$	1.35690	0.553950
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.69202	0.951773
$9$	−0.137063	−0.0456878
$10$	0	0
$11$	−3.80194	−1.14633	−0.573164	−	0.819441i	$-0.694284\pi$
$11$	−3.80194	−1.14633	−0.573164	+	0.819441i	$0.694284\pi$
$12$	2.29590	0.662768
$13$	4.54288	1.25997	0.629984	−	0.776608i	$-0.283062\pi$
$13$	4.54288	1.25997	0.629984	+	0.776608i	$0.283062\pi$
$14$	−0.801938	−0.214327
$15$	0	0
$16$	0.554958	0.138740
$17$	−2.91185	−0.706228	−0.353114	−	0.935580i	$-0.614877\pi$
$17$	−2.91185	−0.706228	−0.353114	+	0.935580i	$0.614877\pi$
$18$	0.109916	0.0259075
$19$	−3.35690	−0.770125	−0.385062	−	0.922891i	$-0.625820\pi$
$19$	−3.35690	−0.770125	−0.385062	+	0.922891i	$0.625820\pi$
$20$	0	0
$21$	−1.69202	−0.369229
$22$	3.04892	0.650031
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−4.55496	−0.929777
$25$	0	0
$26$	−3.64310	−0.714472
$27$	5.30798	1.02152
$28$	−1.35690	−0.256429
$29$	2.09783	0.389558	0.194779	−	0.980847i	$-0.437601\pi$
$29$	2.09783	0.389558	0.194779	+	0.980847i	$0.437601\pi$
$30$	0	0
$31$	0.246980	0.0443588	0.0221794	−	0.999754i	$-0.492939\pi$
$31$	0.246980	0.0443588	0.0221794	+	0.999754i	$0.492939\pi$
$32$	−5.82908	−1.03045
$33$	6.43296	1.11983
$34$	2.33513	0.400471
$35$	0	0
$36$	0.185981	0.0309968
$37$	−5.40581	−0.888710	−0.444355	−	0.895851i	$-0.646567\pi$
$37$	−5.40581	−0.888710	−0.444355	+	0.895851i	$0.646567\pi$
$38$	2.69202	0.436704
$39$	−7.68664	−1.23085
$40$	0	0
$41$	2.82371	0.440989	0.220495	−	0.975388i	$-0.429233\pi$
$41$	2.82371	0.440989	0.220495	+	0.975388i	$0.429233\pi$
$42$	1.35690	0.209374
$43$	0.801938	0.122294	0.0611472	−	0.998129i	$-0.480524\pi$
$43$	0.801938	0.122294	0.0611472	+	0.998129i	$0.480524\pi$
$44$	5.15883	0.777723
$45$	0	0
$46$	−0.801938	−0.118239
$47$	−1.97285	−0.287770	−0.143885	−	0.989594i	$-0.545960\pi$
$47$	−1.97285	−0.287770	−0.143885	+	0.989594i	$0.545960\pi$
$48$	−0.939001	−0.135533
$49$	1.00000	0.142857
$50$	0	0
$51$	4.92692	0.689907
$52$	−6.16421	−0.854822
$53$	9.78448	1.34400	0.672001	−	0.740550i	$-0.265435\pi$
$53$	9.78448	1.34400	0.672001	+	0.740550i	$0.265435\pi$
$54$	−4.25667	−0.579259
$55$	0	0
$56$	2.69202	0.359737
$57$	5.67994	0.752326
$58$	−1.68233	−0.220901
$59$	−6.66487	−0.867693	−0.433846	−	0.900987i	$-0.642844\pi$
$59$	−6.66487	−0.867693	−0.433846	+	0.900987i	$0.642844\pi$
$60$	0	0
$61$	−5.02715	−0.643660	−0.321830	−	0.946797i	$-0.604298\pi$
$61$	−5.02715	−0.643660	−0.321830	+	0.946797i	$0.604298\pi$
$62$	−0.198062	−0.0251539
$63$	−0.137063	−0.0172684
$64$	3.56465	0.445581
$65$	0	0
$66$	−5.15883	−0.635009
$67$	5.03684	0.615347	0.307674	−	0.951492i	$-0.400449\pi$
$67$	5.03684	0.615347	0.307674	+	0.951492i	$0.400449\pi$
$68$	3.95108	0.479139
$69$	−1.69202	−0.203695
$70$	0	0
$71$	5.05861	0.600346	0.300173	−	0.953885i	$-0.402956\pi$
$71$	5.05861	0.600346	0.300173	+	0.953885i	$0.402956\pi$
$72$	−0.368977	−0.0434844
$73$	0.972853	0.113864	0.0569319	−	0.998378i	$-0.481868\pi$
$73$	0.972853	0.113864	0.0569319	+	0.998378i	$0.481868\pi$
$74$	4.33513	0.503948
$75$	0	0
$76$	4.55496	0.522490
$77$	−3.80194	−0.433271
$78$	6.16421	0.697959
$79$	3.14675	0.354037	0.177019	−	0.984208i	$-0.443355\pi$
$79$	3.14675	0.354037	0.177019	+	0.984208i	$0.443355\pi$
$80$	0	0
$81$	−8.57002	−0.952225
$82$	−2.26444	−0.250065
$83$	16.3230	1.79169	0.895843	−	0.444370i	$-0.146572\pi$
$83$	16.3230	1.79169	0.895843	+	0.444370i	$0.146572\pi$
$84$	2.29590	0.250503
$85$	0	0
$86$	−0.643104	−0.0693477
$87$	−3.54958	−0.380555
$88$	−10.2349	−1.09104
$89$	7.18598	0.761712	0.380856	−	0.924634i	$-0.375629\pi$
$89$	7.18598	0.761712	0.380856	+	0.924634i	$0.375629\pi$
$90$	0	0
$91$	4.54288	0.476223
$92$	−1.35690	−0.141466
$93$	−0.417895	−0.0433337
$94$	1.58211	0.163182
$95$	0	0
$96$	9.86294	1.00663
$97$	−12.9758	−1.31750	−0.658748	−	0.752363i	$-0.728914\pi$
$97$	−12.9758	−1.31750	−0.658748	+	0.752363i	$0.728914\pi$
$98$	−0.801938	−0.0810079
$99$	0.521106	0.0523732
$100$	0	0
$101$	5.57673	0.554905	0.277453	−	0.960739i	$-0.410510\pi$
$101$	5.57673	0.554905	0.277453	+	0.960739i	$0.410510\pi$
$102$	−3.95108	−0.391215
$103$	−2.32975	−0.229557	−0.114778	−	0.993391i	$-0.536616\pi$
$103$	−2.32975	−0.229557	−0.114778	+	0.993391i	$0.536616\pi$
$104$	12.2295	1.19920
$105$	0	0
$106$	−7.84654	−0.762124
$107$	−5.19567	−0.502284	−0.251142	−	0.967950i	$-0.580806\pi$
$107$	−5.19567	−0.502284	−0.251142	+	0.967950i	$0.580806\pi$
$108$	−7.20237	−0.693049
$109$	−3.46681	−0.332060	−0.166030	−	0.986121i	$-0.553095\pi$
$109$	−3.46681	−0.332060	−0.166030	+	0.986121i	$0.553095\pi$
$110$	0	0
$111$	9.14675	0.868171
$112$	0.554958	0.0524386
$113$	20.6310	1.94080	0.970402	−	0.241497i	$-0.0776383\pi$
$113$	20.6310	1.94080	0.970402	+	0.241497i	$0.0776383\pi$
$114$	−4.55496	−0.426611
$115$	0	0
$116$	−2.84654	−0.264295
$117$	−0.622662	−0.0575651
$118$	5.34481	0.492030
$119$	−2.91185	−0.266929
$120$	0	0
$121$	3.45473	0.314066
$122$	4.03146	0.364991
$123$	−4.77777	−0.430797
$124$	−0.335126	−0.0300952
$125$	0	0
$126$	0.109916	0.00979212
$127$	6.66248	0.591200	0.295600	−	0.955312i	$-0.404481\pi$
$127$	6.66248	0.591200	0.295600	+	0.955312i	$0.404481\pi$
$128$	8.79954	0.777777
$129$	−1.35690	−0.119468
$130$	0	0
$131$	8.39612	0.733573	0.366786	−	0.930305i	$-0.380458\pi$
$131$	8.39612	0.733573	0.366786	+	0.930305i	$0.380458\pi$
$132$	−8.72886	−0.759750
$133$	−3.35690	−0.291080
$134$	−4.03923	−0.348936
$135$	0	0
$136$	−7.83877	−0.672169
$137$	15.8659	1.35552	0.677759	−	0.735285i	$-0.262951\pi$
$137$	15.8659	1.35552	0.677759	+	0.735285i	$0.262951\pi$
$138$	1.35690	0.115507
$139$	−13.4426	−1.14019	−0.570095	−	0.821579i	$-0.693094\pi$
$139$	−13.4426	−1.14019	−0.570095	+	0.821579i	$0.693094\pi$
$140$	0	0
$141$	3.33811	0.281120
$142$	−4.05669	−0.340430
$143$	−17.2717	−1.44433
$144$	−0.0760644	−0.00633870
$145$	0	0
$146$	−0.780167	−0.0645671
$147$	−1.69202	−0.139556
$148$	7.33513	0.602944
$149$	−8.41789	−0.689621	−0.344810	−	0.938672i	$-0.612057\pi$
$149$	−8.41789	−0.689621	−0.344810	+	0.938672i	$0.612057\pi$
$150$	0	0
$151$	−0.481878	−0.0392146	−0.0196073	−	0.999808i	$-0.506242\pi$
$151$	−0.481878	−0.0392146	−0.0196073	+	0.999808i	$0.506242\pi$
$152$	−9.03684	−0.732984
$153$	0.399108	0.0322660
$154$	3.04892	0.245689
$155$	0	0
$156$	10.4300	0.835066
$157$	11.6649	0.930958	0.465479	−	0.885059i	$-0.345882\pi$
$157$	11.6649	0.930958	0.465479	+	0.885059i	$0.345882\pi$
$158$	−2.52350	−0.200759
$159$	−16.5555	−1.31294
$160$	0	0
$161$	1.00000	0.0788110
$162$	6.87263	0.539964
$163$	−19.3599	−1.51638	−0.758191	−	0.652032i	$-0.773917\pi$
$163$	−19.3599	−1.51638	−0.758191	+	0.652032i	$0.773917\pi$
$164$	−3.83148	−0.299188
$165$	0	0
$166$	−13.0901	−1.01599
$167$	−20.5362	−1.58914	−0.794568	−	0.607175i	$-0.792303\pi$
$167$	−20.5362	−1.58914	−0.794568	+	0.607175i	$0.792303\pi$
$168$	−4.55496	−0.351423
$169$	7.63773	0.587517
$170$	0	0
$171$	0.460107	0.0351853
$172$	−1.08815	−0.0829704
$173$	−6.57002	−0.499510	−0.249755	−	0.968309i	$-0.580350\pi$
$173$	−6.57002	−0.499510	−0.249755	+	0.968309i	$0.580350\pi$
$174$	2.84654	0.215796
$175$	0	0
$176$	−2.10992	−0.159041
$177$	11.2771	0.847640
$178$	−5.76271	−0.431933
$179$	−15.8998	−1.18840	−0.594202	−	0.804316i	$-0.702532\pi$
$179$	−15.8998	−1.18840	−0.594202	+	0.804316i	$0.702532\pi$
$180$	0	0
$181$	11.0978	0.824896	0.412448	−	0.910981i	$-0.364674\pi$
$181$	11.0978	0.824896	0.412448	+	0.910981i	$0.364674\pi$
$182$	−3.64310	−0.270045
$183$	8.50604	0.628785
$184$	2.69202	0.198458
$185$	0	0
$186$	0.335126	0.0245726
$187$	11.0707	0.809569
$188$	2.67696	0.195237
$189$	5.30798	0.386099
$190$	0	0
$191$	−1.89679	−0.137247	−0.0686234	−	0.997643i	$-0.521861\pi$
$191$	−1.89679	−0.137247	−0.0686234	+	0.997643i	$0.521861\pi$
$192$	−6.03146	−0.435283
$193$	−17.4330	−1.25485	−0.627426	−	0.778676i	$-0.715891\pi$
$193$	−17.4330	−1.25485	−0.627426	+	0.778676i	$0.715891\pi$
$194$	10.4058	0.747094
$195$	0	0
$196$	−1.35690	−0.0969211
$197$	−2.59956	−0.185211	−0.0926056	−	0.995703i	$-0.529520\pi$
$197$	−2.59956	−0.185211	−0.0926056	+	0.995703i	$0.529520\pi$
$198$	−0.417895	−0.0296985
$199$	−21.4263	−1.51887	−0.759433	−	0.650585i	$-0.774524\pi$
$199$	−21.4263	−1.51887	−0.759433	+	0.650585i	$0.774524\pi$
$200$	0	0
$201$	−8.52243	−0.601126
$202$	−4.47219	−0.314662
$203$	2.09783	0.147239
$204$	−6.68532	−0.468066
$205$	0	0
$206$	1.86831	0.130172
$207$	−0.137063	−0.00952656
$208$	2.52111	0.174807
$209$	12.7627	0.882815
$210$	0	0
$211$	−11.9705	−0.824080	−0.412040	−	0.911166i	$-0.635184\pi$
$211$	−11.9705	−0.824080	−0.412040	+	0.911166i	$0.635184\pi$
$212$	−13.2765	−0.911835
$213$	−8.55927	−0.586472
$214$	4.16660	0.284823
$215$	0	0
$216$	14.2892	0.972256
$217$	0.246980	0.0167661
$218$	2.78017	0.188297
$219$	−1.64609	−0.111232
$220$	0	0
$221$	−13.2282	−0.889825
$222$	−7.33513	−0.492301
$223$	−15.1317	−1.01329	−0.506646	−	0.862154i	$-0.669115\pi$
$223$	−15.1317	−1.01329	−0.506646	+	0.862154i	$0.669115\pi$
$224$	−5.82908	−0.389472
$225$	0	0
$226$	−16.5448	−1.10054
$227$	7.42327	0.492700	0.246350	−	0.969181i	$-0.420769\pi$
$227$	7.42327	0.492700	0.246350	+	0.969181i	$0.420769\pi$
$228$	−7.70709	−0.510414
$229$	2.63640	0.174218	0.0871091	−	0.996199i	$-0.472237\pi$
$229$	2.63640	0.174218	0.0871091	+	0.996199i	$0.472237\pi$
$230$	0	0
$231$	6.43296	0.423258
$232$	5.64742	0.370771
$233$	5.36658	0.351577	0.175788	−	0.984428i	$-0.443753\pi$
$233$	5.36658	0.351577	0.175788	+	0.984428i	$0.443753\pi$
$234$	0.499336	0.0326426
$235$	0	0
$236$	9.04354	0.588684
$237$	−5.32437	−0.345855
$238$	2.33513	0.151364
$239$	−10.9825	−0.710402	−0.355201	−	0.934790i	$-0.615588\pi$
$239$	−10.9825	−0.710402	−0.355201	+	0.934790i	$0.615588\pi$
$240$	0	0
$241$	2.99330	0.192815	0.0964075	−	0.995342i	$-0.469265\pi$
$241$	2.99330	0.192815	0.0964075	+	0.995342i	$0.469265\pi$
$242$	−2.77048	−0.178093
$243$	−1.42327	−0.0913029
$244$	6.82132	0.436690
$245$	0	0
$246$	3.83148	0.244286
$247$	−15.2500	−0.970332
$248$	0.664874	0.0422196
$249$	−27.6189	−1.75028
$250$	0	0
$251$	−25.9433	−1.63753	−0.818764	−	0.574131i	$-0.805340\pi$
$251$	−25.9433	−1.63753	−0.818764	+	0.574131i	$0.805340\pi$
$252$	0.185981	0.0117157
$253$	−3.80194	−0.239026
$254$	−5.34290	−0.335243
$255$	0	0
$256$	−14.1860	−0.886624
$257$	−16.3478	−1.01975	−0.509874	−	0.860249i	$-0.670308\pi$
$257$	−16.3478	−1.01975	−0.509874	+	0.860249i	$0.670308\pi$
$258$	1.08815	0.0677450
$259$	−5.40581	−0.335901
$260$	0	0
$261$	−0.287536	−0.0177980
$262$	−6.73317	−0.415977
$263$	−27.6383	−1.70425	−0.852126	−	0.523337i	$-0.824687\pi$
$263$	−27.6383	−1.70425	−0.852126	+	0.523337i	$0.824687\pi$
$264$	17.3177	1.06583
$265$	0	0
$266$	2.69202	0.165058
$267$	−12.1588	−0.744109
$268$	−6.83446	−0.417481
$269$	22.8713	1.39449	0.697244	−	0.716834i	$-0.254409\pi$
$269$	22.8713	1.39449	0.697244	+	0.716834i	$0.254409\pi$
$270$	0	0
$271$	19.7875	1.20200	0.601001	−	0.799248i	$-0.294769\pi$
$271$	19.7875	1.20200	0.601001	+	0.799248i	$0.294769\pi$
$272$	−1.61596	−0.0979818
$273$	−7.68664	−0.465217
$274$	−12.7235	−0.768654
$275$	0	0
$276$	2.29590	0.138197
$277$	14.8780	0.893933	0.446966	−	0.894551i	$-0.352504\pi$
$277$	14.8780	0.893933	0.446966	+	0.894551i	$0.352504\pi$
$278$	10.7802	0.646551
$279$	−0.0338518	−0.00202666
$280$	0	0
$281$	−10.9705	−0.654443	−0.327221	−	0.944948i	$-0.606112\pi$
$281$	−10.9705	−0.654443	−0.327221	+	0.944948i	$0.606112\pi$
$282$	−2.67696	−0.159410
$283$	24.9124	1.48089	0.740446	−	0.672116i	$-0.234615\pi$
$283$	24.9124	1.48089	0.740446	+	0.672116i	$0.234615\pi$
$284$	−6.86400	−0.407304
$285$	0	0
$286$	13.8509	0.819018
$287$	2.82371	0.166678
$288$	0.798954	0.0470788
$289$	−8.52111	−0.501242
$290$	0	0
$291$	21.9554	1.28705
$292$	−1.32006	−0.0772507
$293$	5.30021	0.309642	0.154821	−	0.987943i	$-0.450520\pi$
$293$	5.30021	0.309642	0.154821	+	0.987943i	$0.450520\pi$
$294$	1.35690	0.0791358
$295$	0	0
$296$	−14.5526	−0.845851
$297$	−20.1806	−1.17100
$298$	6.75063	0.391053
$299$	4.54288	0.262721
$300$	0	0
$301$	0.801938	0.0462229
$302$	0.386436	0.0222369
$303$	−9.43594	−0.542081
$304$	−1.86294	−0.106847
$305$	0	0
$306$	−0.320060	−0.0182966
$307$	−34.3032	−1.95779	−0.978893	−	0.204372i	$-0.934485\pi$
$307$	−34.3032	−1.95779	−0.978893	+	0.204372i	$0.934485\pi$
$308$	5.15883	0.293952
$309$	3.94198	0.224252
$310$	0	0
$311$	−9.82371	−0.557051	−0.278526	−	0.960429i	$-0.589846\pi$
$311$	−9.82371	−0.557051	−0.278526	+	0.960429i	$0.589846\pi$
$312$	−20.6926	−1.17149
$313$	−30.5937	−1.72926	−0.864629	−	0.502410i	$-0.832447\pi$
$313$	−30.5937	−1.72926	−0.864629	+	0.502410i	$0.832447\pi$
$314$	−9.35450	−0.527905
$315$	0	0
$316$	−4.26981	−0.240196
$317$	18.2543	1.02526	0.512631	−	0.858609i	$-0.328671\pi$
$317$	18.2543	1.02526	0.512631	+	0.858609i	$0.328671\pi$
$318$	13.2765	0.744510
$319$	−7.97584	−0.446561
$320$	0	0
$321$	8.79118	0.490676
$322$	−0.801938	−0.0446902
$323$	9.77479	0.543884
$324$	11.6286	0.646035
$325$	0	0
$326$	15.5254	0.859873
$327$	5.86592	0.324386
$328$	7.60148	0.419722
$329$	−1.97285	−0.108767
$330$	0	0
$331$	−3.49396	−0.192045	−0.0960227	−	0.995379i	$-0.530612\pi$
$331$	−3.49396	−0.192045	−0.0960227	+	0.995379i	$0.530612\pi$
$332$	−22.1487	−1.21557
$333$	0.740939	0.0406032
$334$	16.4687	0.901129
$335$	0	0
$336$	−0.939001	−0.0512267
$337$	1.09352	0.0595680	0.0297840	−	0.999556i	$-0.490518\pi$
$337$	1.09352	0.0595680	0.0297840	+	0.999556i	$0.490518\pi$
$338$	−6.12498	−0.333155
$339$	−34.9081	−1.89595
$340$	0	0
$341$	−0.939001	−0.0508498
$342$	−0.368977	−0.0199520
$343$	1.00000	0.0539949
$344$	2.15883	0.116397
$345$	0	0
$346$	5.26875	0.283250
$347$	23.6431	1.26923	0.634614	−	0.772829i	$-0.281159\pi$
$347$	23.6431	1.26923	0.634614	+	0.772829i	$0.281159\pi$
$348$	4.81641	0.258187
$349$	−22.8461	−1.22292	−0.611461	−	0.791275i	$-0.709418\pi$
$349$	−22.8461	−1.22292	−0.611461	+	0.791275i	$0.709418\pi$
$350$	0	0
$351$	24.1135	1.28708
$352$	22.1618	1.18123
$353$	15.9584	0.849379	0.424689	−	0.905339i	$-0.360383\pi$
$353$	15.9584	0.849379	0.424689	+	0.905339i	$0.360383\pi$
$354$	−9.04354	−0.480659
$355$	0	0
$356$	−9.75063	−0.516782
$357$	4.92692	0.260760
$358$	12.7506	0.673892
$359$	−23.4480	−1.23754	−0.618770	−	0.785572i	$-0.712369\pi$
$359$	−23.4480	−1.23754	−0.618770	+	0.785572i	$0.712369\pi$
$360$	0	0
$361$	−7.73125	−0.406908
$362$	−8.89977	−0.467762
$363$	−5.84548	−0.306808
$364$	−6.16421	−0.323092
$365$	0	0
$366$	−6.82132	−0.356556
$367$	−17.6256	−0.920051	−0.460026	−	0.887906i	$-0.652160\pi$
$367$	−17.6256	−0.920051	−0.460026	+	0.887906i	$0.652160\pi$
$368$	0.554958	0.0289292
$369$	−0.387027	−0.0201478
$370$	0	0
$371$	9.78448	0.507985
$372$	0.567040	0.0293996
$373$	−27.3153	−1.41433	−0.707166	−	0.707048i	$-0.750026\pi$
$373$	−27.3153	−1.41433	−0.707166	+	0.707048i	$0.750026\pi$
$374$	−8.87800	−0.459071
$375$	0	0
$376$	−5.31096	−0.273892
$377$	9.53020	0.490830
$378$	−4.25667	−0.218939
$379$	−3.77479	−0.193898	−0.0969490	−	0.995289i	$-0.530908\pi$
$379$	−3.77479	−0.193898	−0.0969490	+	0.995289i	$0.530908\pi$
$380$	0	0
$381$	−11.2731	−0.577536
$382$	1.52111	0.0778266
$383$	13.5472	0.692229	0.346114	−	0.938192i	$-0.387501\pi$
$383$	13.5472	0.692229	0.346114	+	0.938192i	$0.387501\pi$
$384$	−14.8890	−0.759802
$385$	0	0
$386$	13.9801	0.711571
$387$	−0.109916	−0.00558736
$388$	17.6069	0.893853
$389$	−31.7211	−1.60832	−0.804161	−	0.594411i	$-0.797385\pi$
$389$	−31.7211	−1.60832	−0.804161	+	0.594411i	$0.797385\pi$
$390$	0	0
$391$	−2.91185	−0.147259
$392$	2.69202	0.135968
$393$	−14.2064	−0.716619
$394$	2.08469	0.105025
$395$	0	0
$396$	−0.707087	−0.0355325
$397$	−30.4523	−1.52836	−0.764180	−	0.645004i	$-0.776856\pi$
$397$	−30.4523	−1.52836	−0.764180	+	0.645004i	$0.776856\pi$
$398$	17.1825	0.861282
$399$	5.67994	0.284353
$400$	0	0
$401$	−30.3967	−1.51794	−0.758970	−	0.651126i	$-0.774297\pi$
$401$	−30.3967	−1.51794	−0.758970	+	0.651126i	$0.774297\pi$
$402$	6.83446	0.340872
$403$	1.12200	0.0558907
$404$	−7.56704	−0.376474
$405$	0	0
$406$	−1.68233	−0.0834928
$407$	20.5526	1.01875
$408$	13.2634	0.656635
$409$	−5.45234	−0.269601	−0.134800	−	0.990873i	$-0.543039\pi$
$409$	−5.45234	−0.269601	−0.134800	+	0.990873i	$0.543039\pi$
$410$	0	0
$411$	−26.8455	−1.32419
$412$	3.16123	0.155742
$413$	−6.66487	−0.327957
$414$	0.109916	0.00540209
$415$	0	0
$416$	−26.4808	−1.29833
$417$	22.7453	1.11384
$418$	−10.2349	−0.500605
$419$	−25.1008	−1.22626	−0.613128	−	0.789984i	$-0.710089\pi$
$419$	−25.1008	−1.22626	−0.613128	+	0.789984i	$0.710089\pi$
$420$	0	0
$421$	4.95539	0.241511	0.120756	−	0.992682i	$-0.461468\pi$
$421$	4.95539	0.241511	0.120756	+	0.992682i	$0.461468\pi$
$422$	9.59956	0.467299
$423$	0.270406	0.0131476
$424$	26.3400	1.27918
$425$	0	0
$426$	6.86400	0.332562
$427$	−5.02715	−0.243281
$428$	7.04998	0.340774
$429$	29.2241	1.41095
$430$	0	0
$431$	−23.9487	−1.15357	−0.576784	−	0.816897i	$-0.695693\pi$
$431$	−23.9487	−1.15357	−0.576784	+	0.816897i	$0.695693\pi$
$432$	2.94571	0.141725
$433$	14.8804	0.715106	0.357553	−	0.933893i	$-0.383611\pi$
$433$	14.8804	0.715106	0.357553	+	0.933893i	$0.383611\pi$
$434$	−0.198062	−0.00950729
$435$	0	0
$436$	4.70410	0.225286
$437$	−3.35690	−0.160582
$438$	1.32006	0.0630749
$439$	39.3303	1.87713	0.938567	−	0.345096i	$-0.112154\pi$
$439$	39.3303	1.87713	0.938567	+	0.345096i	$0.112154\pi$
$440$	0	0
$441$	−0.137063	−0.00652683
$442$	10.6082	0.504580
$443$	11.9172	0.566205	0.283102	−	0.959090i	$-0.408636\pi$
$443$	11.9172	0.566205	0.283102	+	0.959090i	$0.408636\pi$
$444$	−12.4112	−0.589009
$445$	0	0
$446$	12.1347	0.574593
$447$	14.2433	0.673683
$448$	3.56465	0.168414
$449$	14.9323	0.704699	0.352349	−	0.935869i	$-0.385383\pi$
$449$	14.9323	0.704699	0.352349	+	0.935869i	$0.385383\pi$
$450$	0	0
$451$	−10.7356	−0.505518
$452$	−27.9941	−1.31673
$453$	0.815347	0.0383084
$454$	−5.95300	−0.279388
$455$	0	0
$456$	15.2905	0.716044
$457$	−14.5265	−0.679520	−0.339760	−	0.940512i	$-0.610346\pi$
$457$	−14.5265	−0.679520	−0.339760	+	0.940512i	$0.610346\pi$
$458$	−2.11423	−0.0987914
$459$	−15.4561	−0.721427
$460$	0	0
$461$	−21.6698	−1.00926	−0.504631	−	0.863335i	$-0.668371\pi$
$461$	−21.6698	−1.00926	−0.504631	+	0.863335i	$0.668371\pi$
$462$	−5.15883	−0.240011
$463$	−1.96615	−0.0913747	−0.0456873	−	0.998956i	$-0.514548\pi$
$463$	−1.96615	−0.0913747	−0.0456873	+	0.998956i	$0.514548\pi$
$464$	1.16421	0.0540471
$465$	0	0
$466$	−4.30367	−0.199364
$467$	−14.4644	−0.669333	−0.334667	−	0.942337i	$-0.608624\pi$
$467$	−14.4644	−0.669333	−0.334667	+	0.942337i	$0.608624\pi$
$468$	0.844887	0.0390549
$469$	5.03684	0.232579
$470$	0	0
$471$	−19.7372	−0.909443
$472$	−17.9420	−0.825847
$473$	−3.04892	−0.140189
$474$	4.26981	0.196119
$475$	0	0
$476$	3.95108	0.181098
$477$	−1.34109	−0.0614044
$478$	8.80731	0.402837
$479$	16.4306	0.750732	0.375366	−	0.926877i	$-0.377517\pi$
$479$	16.4306	0.750732	0.375366	+	0.926877i	$0.377517\pi$
$480$	0	0
$481$	−24.5579	−1.11975
$482$	−2.40044	−0.109337
$483$	−1.69202	−0.0769896
$484$	−4.68771	−0.213078
$485$	0	0
$486$	1.14138	0.0517738
$487$	31.0452	1.40679	0.703396	−	0.710798i	$-0.251666\pi$
$487$	31.0452	1.40679	0.703396	+	0.710798i	$0.251666\pi$
$488$	−13.5332	−0.612619
$489$	32.7573	1.48134
$490$	0	0
$491$	−20.0411	−0.904444	−0.452222	−	0.891905i	$-0.649369\pi$
$491$	−20.0411	−0.904444	−0.452222	+	0.891905i	$0.649369\pi$
$492$	6.48294	0.292274
$493$	−6.10859	−0.275117
$494$	12.2295	0.550232
$495$	0	0
$496$	0.137063	0.00615433
$497$	5.05861	0.226909
$498$	22.1487	0.992506
$499$	6.67456	0.298794	0.149397	−	0.988777i	$-0.452267\pi$
$499$	6.67456	0.298794	0.149397	+	0.988777i	$0.452267\pi$
$500$	0	0
$501$	34.7476	1.55241
$502$	20.8049	0.928569
$503$	15.4808	0.690255	0.345128	−	0.938556i	$-0.387836\pi$
$503$	15.4808	0.690255	0.345128	+	0.938556i	$0.387836\pi$
$504$	−0.368977	−0.0164356
$505$	0	0
$506$	3.04892	0.135541
$507$	−12.9232	−0.573939
$508$	−9.04029	−0.401098
$509$	29.2717	1.29745	0.648723	−	0.761024i	$-0.275303\pi$
$509$	29.2717	1.29745	0.648723	+	0.761024i	$0.275303\pi$
$510$	0	0
$511$	0.972853	0.0430365
$512$	−6.22282	−0.275012
$513$	−17.8183	−0.786699
$514$	13.1099	0.578254
$515$	0	0
$516$	1.84117	0.0810528
$517$	7.50066	0.329879
$518$	4.33513	0.190474
$519$	11.1166	0.487966
$520$	0	0
$521$	7.29829	0.319744	0.159872	−	0.987138i	$-0.448892\pi$
$521$	7.29829	0.319744	0.159872	+	0.987138i	$0.448892\pi$
$522$	0.230586	0.0100925
$523$	24.9457	1.09080	0.545400	−	0.838176i	$-0.316378\pi$
$523$	24.9457	1.09080	0.545400	+	0.838176i	$0.316378\pi$
$524$	−11.3927	−0.497691
$525$	0	0
$526$	22.1642	0.966405
$527$	−0.719169	−0.0313275
$528$	3.57002	0.155365
$529$	1.00000	0.0434783
$530$	0	0
$531$	0.913510	0.0396430
$532$	4.55496	0.197482
$533$	12.8278	0.555632
$534$	9.75063	0.421951
$535$	0	0
$536$	13.5593	0.585671
$537$	26.9028	1.16094
$538$	−18.3414	−0.790752
$539$	−3.80194	−0.163761
$540$	0	0
$541$	35.7385	1.53652	0.768260	−	0.640138i	$-0.221123\pi$
$541$	35.7385	1.53652	0.768260	+	0.640138i	$0.221123\pi$
$542$	−15.8683	−0.681602
$543$	−18.7778	−0.805831
$544$	16.9734	0.727730
$545$	0	0
$546$	6.16421	0.263804
$547$	2.98925	0.127811	0.0639055	−	0.997956i	$-0.479644\pi$
$547$	2.98925	0.127811	0.0639055	+	0.997956i	$0.479644\pi$
$548$	−21.5284	−0.919648
$549$	0.689038	0.0294074
$550$	0	0
$551$	−7.04221	−0.300008
$552$	−4.55496	−0.193872
$553$	3.14675	0.133814
$554$	−11.9312	−0.506909
$555$	0	0
$556$	18.2403	0.773560
$557$	−43.5273	−1.84431	−0.922156	−	0.386818i	$-0.873574\pi$
$557$	−43.5273	−1.84431	−0.922156	+	0.386818i	$0.873574\pi$
$558$	0.0271471	0.00114923
$559$	3.64310	0.154087
$560$	0	0
$561$	−18.7318	−0.790859
$562$	8.79763	0.371105
$563$	29.1879	1.23012	0.615062	−	0.788479i	$-0.289131\pi$
$563$	29.1879	1.23012	0.615062	+	0.788479i	$0.289131\pi$
$564$	−4.52947	−0.190725
$565$	0	0
$566$	−19.9782	−0.839748
$567$	−8.57002	−0.359907
$568$	13.6179	0.571393
$569$	7.09544	0.297456	0.148728	−	0.988878i	$-0.452482\pi$
$569$	7.09544	0.297456	0.148728	+	0.988878i	$0.452482\pi$
$570$	0	0
$571$	−7.33752	−0.307066	−0.153533	−	0.988144i	$-0.549065\pi$
$571$	−7.33752	−0.307066	−0.153533	+	0.988144i	$0.549065\pi$
$572$	23.4359	0.979906
$573$	3.20941	0.134075
$574$	−2.26444	−0.0945158
$575$	0	0
$576$	−0.488582	−0.0203576
$577$	23.3967	0.974018	0.487009	−	0.873397i	$-0.338088\pi$
$577$	23.3967	0.974018	0.487009	+	0.873397i	$0.338088\pi$
$578$	6.83340	0.284232
$579$	29.4969	1.22585
$580$	0	0
$581$	16.3230	0.677194
$582$	−17.6069	−0.729828
$583$	−37.2000	−1.54067
$584$	2.61894	0.108373
$585$	0	0
$586$	−4.25044	−0.175584
$587$	−10.4058	−0.429494	−0.214747	−	0.976670i	$-0.568893\pi$
$587$	−10.4058	−0.429494	−0.214747	+	0.976670i	$0.568893\pi$
$588$	2.29590	0.0946812
$589$	−0.829085	−0.0341618
$590$	0	0
$591$	4.39852	0.180931
$592$	−3.00000	−0.123299
$593$	−2.21254	−0.0908580	−0.0454290	−	0.998968i	$-0.514465\pi$
$593$	−2.21254	−0.0908580	−0.0454290	+	0.998968i	$0.514465\pi$
$594$	16.1836	0.664021
$595$	0	0
$596$	11.4222	0.467872
$597$	36.2537	1.48376
$598$	−3.64310	−0.148978
$599$	−18.5851	−0.759366	−0.379683	−	0.925117i	$-0.623967\pi$
$599$	−18.5851	−0.759366	−0.379683	+	0.925117i	$0.623967\pi$
$600$	0	0
$601$	23.2610	0.948835	0.474418	−	0.880300i	$-0.342659\pi$
$601$	23.2610	0.948835	0.474418	+	0.880300i	$0.342659\pi$
$602$	−0.643104	−0.0262110
$603$	−0.690366	−0.0281139
$604$	0.653858	0.0266051
$605$	0	0
$606$	7.56704	0.307390
$607$	16.8672	0.684620	0.342310	−	0.939587i	$-0.388791\pi$
$607$	16.8672	0.684620	0.342310	+	0.939587i	$0.388791\pi$
$608$	19.5676	0.793572
$609$	−3.54958	−0.143836
$610$	0	0
$611$	−8.96243	−0.362581
$612$	−0.541549	−0.0218908
$613$	−3.51573	−0.141999	−0.0709995	−	0.997476i	$-0.522619\pi$
$613$	−3.51573	−0.141999	−0.0709995	+	0.997476i	$0.522619\pi$
$614$	27.5090	1.11017
$615$	0	0
$616$	−10.2349	−0.412376
$617$	37.4523	1.50777	0.753887	−	0.657004i	$-0.228177\pi$
$617$	37.4523	1.50777	0.753887	+	0.657004i	$0.228177\pi$
$618$	−3.16123	−0.127163
$619$	−9.90084	−0.397948	−0.198974	−	0.980005i	$-0.563761\pi$
$619$	−9.90084	−0.397948	−0.198974	+	0.980005i	$0.563761\pi$
$620$	0	0
$621$	5.30798	0.213002
$622$	7.87800	0.315879
$623$	7.18598	0.287900
$624$	−4.26577	−0.170767
$625$	0	0
$626$	24.5343	0.980586
$627$	−21.5948	−0.862412
$628$	−15.8280	−0.631607
$629$	15.7409	0.627632
$630$	0	0
$631$	−48.0411	−1.91249	−0.956244	−	0.292571i	$-0.905489\pi$
$631$	−48.0411	−1.91249	−0.956244	+	0.292571i	$0.905489\pi$
$632$	8.47112	0.336963
$633$	20.2543	0.805035
$634$	−14.6388	−0.581381
$635$	0	0
$636$	22.4642	0.890762
$637$	4.54288	0.179995
$638$	6.39612	0.253225
$639$	−0.693349	−0.0274285
$640$	0	0
$641$	−39.4131	−1.55672	−0.778362	−	0.627816i	$-0.783949\pi$
$641$	−39.4131	−1.55672	−0.778362	+	0.627816i	$0.783949\pi$
$642$	−7.04998	−0.278241
$643$	−32.6819	−1.28885	−0.644423	−	0.764669i	$-0.722903\pi$
$643$	−32.6819	−1.28885	−0.644423	+	0.764669i	$0.722903\pi$
$644$	−1.35690	−0.0534692
$645$	0	0
$646$	−7.83877	−0.308412
$647$	39.2131	1.54163	0.770814	−	0.637061i	$-0.219850\pi$
$647$	39.2131	1.54163	0.770814	+	0.637061i	$0.219850\pi$
$648$	−23.0707	−0.906302
$649$	25.3394	0.994660
$650$	0	0
$651$	−0.417895	−0.0163786
$652$	26.2693	1.02879
$653$	13.0024	0.508823	0.254411	−	0.967096i	$-0.418118\pi$
$653$	13.0024	0.508823	0.254411	+	0.967096i	$0.418118\pi$
$654$	−4.70410	−0.183945
$655$	0	0
$656$	1.56704	0.0611826
$657$	−0.133342	−0.00520219
$658$	1.58211	0.0616769
$659$	42.5763	1.65854	0.829268	−	0.558852i	$-0.188758\pi$
$659$	42.5763	1.65854	0.829268	+	0.558852i	$0.188758\pi$
$660$	0	0
$661$	34.0863	1.32581	0.662903	−	0.748706i	$-0.269324\pi$
$661$	34.0863	1.32581	0.662903	+	0.748706i	$0.269324\pi$
$662$	2.80194	0.108900
$663$	22.3824	0.869260
$664$	43.9420	1.70528
$665$	0	0
$666$	−0.594187	−0.0230243
$667$	2.09783	0.0812285
$668$	27.8654	1.07815
$669$	25.6031	0.989875
$670$	0	0
$671$	19.1129	0.737845
$672$	9.86294	0.380471
$673$	−47.3099	−1.82366	−0.911831	−	0.410565i	$-0.865331\pi$
$673$	−47.3099	−1.82366	−0.911831	+	0.410565i	$0.865331\pi$
$674$	−0.876937	−0.0337784
$675$	0	0
$676$	−10.3636	−0.398600
$677$	18.1909	0.699132	0.349566	−	0.936912i	$-0.386329\pi$
$677$	18.1909	0.699132	0.349566	+	0.936912i	$0.386329\pi$
$678$	27.9941	1.07511
$679$	−12.9758	−0.497967
$680$	0	0
$681$	−12.5603	−0.481313
$682$	0.753020	0.0288346
$683$	−45.5437	−1.74268	−0.871341	−	0.490678i	$-0.836749\pi$
$683$	−45.5437	−1.74268	−0.871341	+	0.490678i	$0.836749\pi$
$684$	−0.624318	−0.0238714
$685$	0	0
$686$	−0.801938	−0.0306181
$687$	−4.46084	−0.170192
$688$	0.445042	0.0169671
$689$	44.4497	1.69340
$690$	0	0
$691$	−29.3569	−1.11679	−0.558394	−	0.829576i	$-0.688582\pi$
$691$	−29.3569	−1.11679	−0.558394	+	0.829576i	$0.688582\pi$
$692$	8.91484	0.338891
$693$	0.521106	0.0197952
$694$	−18.9603	−0.719723
$695$	0	0
$696$	−9.55555	−0.362202
$697$	−8.22223	−0.311439
$698$	18.3211	0.693465
$699$	−9.08038	−0.343451
$700$	0	0
$701$	1.41896	0.0535934	0.0267967	−	0.999641i	$-0.491469\pi$
$701$	1.41896	0.0535934	0.0267967	+	0.999641i	$0.491469\pi$
$702$	−19.3375	−0.729848
$703$	18.1468	0.684418
$704$	−13.5526	−0.510782
$705$	0	0
$706$	−12.7976	−0.481645
$707$	5.57673	0.209734
$708$	−15.3019	−0.575079
$709$	35.4566	1.33160	0.665801	−	0.746129i	$-0.268090\pi$
$709$	35.4566	1.33160	0.665801	+	0.746129i	$0.268090\pi$
$710$	0	0
$711$	−0.431304	−0.0161752
$712$	19.3448	0.724978
$713$	0.246980	0.00924946
$714$	−3.95108	−0.147866
$715$	0	0
$716$	21.5743	0.806271
$717$	18.5827	0.693984
$718$	18.8039	0.701754
$719$	−5.12067	−0.190969	−0.0954844	−	0.995431i	$-0.530440\pi$
$719$	−5.12067	−0.190969	−0.0954844	+	0.995431i	$0.530440\pi$
$720$	0	0
$721$	−2.32975	−0.0867644
$722$	6.19998	0.230739
$723$	−5.06472	−0.188359
$724$	−15.0586	−0.559649
$725$	0	0
$726$	4.68771	0.173977
$727$	−3.26337	−0.121032	−0.0605159	−	0.998167i	$-0.519275\pi$
$727$	−3.26337	−0.121032	−0.0605159	+	0.998167i	$0.519275\pi$
$728$	12.2295	0.453256
$729$	28.1183	1.04142
$730$	0	0
$731$	−2.33513	−0.0863677
$732$	−11.5418	−0.426598
$733$	−7.48832	−0.276587	−0.138294	−	0.990391i	$-0.544162\pi$
$733$	−7.48832	−0.276587	−0.138294	+	0.990391i	$0.544162\pi$
$734$	14.1347	0.521720
$735$	0	0
$736$	−5.82908	−0.214863
$737$	−19.1497	−0.705390
$738$	0.310371	0.0114249
$739$	−34.9681	−1.28632	−0.643161	−	0.765731i	$-0.722377\pi$
$739$	−34.9681	−1.28632	−0.643161	+	0.765731i	$0.722377\pi$
$740$	0	0
$741$	25.8033	0.947907
$742$	−7.84654	−0.288056
$743$	−21.6310	−0.793565	−0.396783	−	0.917913i	$-0.629873\pi$
$743$	−21.6310	−0.793565	−0.396783	+	0.917913i	$0.629873\pi$
$744$	−1.12498	−0.0412438
$745$	0	0
$746$	21.9051	0.802004
$747$	−2.23729	−0.0818582
$748$	−15.0218	−0.549250
$749$	−5.19567	−0.189846
$750$	0	0
$751$	42.4663	1.54962	0.774809	−	0.632195i	$-0.217846\pi$
$751$	42.4663	1.54962	0.774809	+	0.632195i	$0.217846\pi$
$752$	−1.09485	−0.0399251
$753$	43.8966	1.59968
$754$	−7.64263	−0.278328
$755$	0	0
$756$	−7.20237	−0.261948
$757$	−39.5273	−1.43665	−0.718323	−	0.695710i	$-0.755090\pi$
$757$	−39.5273	−1.43665	−0.718323	+	0.695710i	$0.755090\pi$
$758$	3.02715	0.109951
$759$	6.43296	0.233502
$760$	0	0
$761$	−15.2118	−0.551427	−0.275714	−	0.961240i	$-0.588914\pi$
$761$	−15.2118	−0.551427	−0.275714	+	0.961240i	$0.588914\pi$
$762$	9.04029	0.327495
$763$	−3.46681	−0.125507
$764$	2.57374	0.0931148
$765$	0	0
$766$	−10.8640	−0.392532
$767$	−30.2777	−1.09326
$768$	24.0030	0.866133
$769$	−23.4228	−0.844648	−0.422324	−	0.906445i	$-0.638785\pi$
$769$	−23.4228	−0.844648	−0.422324	+	0.906445i	$0.638785\pi$
$770$	0	0
$771$	27.6608	0.996180
$772$	23.6547	0.851352
$773$	−41.2185	−1.48253	−0.741263	−	0.671214i	$-0.765773\pi$
$773$	−41.2185	−1.48253	−0.741263	+	0.671214i	$0.765773\pi$
$774$	0.0881460	0.00316834
$775$	0	0
$776$	−34.9312	−1.25396
$777$	9.14675	0.328138
$778$	25.4383	0.912009
$779$	−9.47889	−0.339617
$780$	0	0
$781$	−19.2325	−0.688193
$782$	2.33513	0.0835039
$783$	11.1353	0.397942
$784$	0.554958	0.0198199
$785$	0	0
$786$	11.3927	0.406363
$787$	−32.3430	−1.15290	−0.576452	−	0.817131i	$-0.695563\pi$
$787$	−32.3430	−1.15290	−0.576452	+	0.817131i	$0.695563\pi$
$788$	3.52734	0.125656
$789$	46.7646	1.66486
$790$	0	0
$791$	20.6310	0.733555
$792$	1.40283	0.0498474
$793$	−22.8377	−0.810991
$794$	24.4209	0.866665
$795$	0	0
$796$	29.0732	1.03047
$797$	9.50365	0.336636	0.168318	−	0.985733i	$-0.446166\pi$
$797$	9.50365	0.336636	0.168318	+	0.985733i	$0.446166\pi$
$798$	−4.55496	−0.161244
$799$	5.74466	0.203231
$800$	0	0
$801$	−0.984935	−0.0348009
$802$	24.3763	0.860756
$803$	−3.69873	−0.130525
$804$	11.5641	0.407833
$805$	0	0
$806$	−0.899772	−0.0316931
$807$	−38.6987	−1.36226
$808$	15.0127	0.528144
$809$	20.9597	0.736904	0.368452	−	0.929647i	$-0.379888\pi$
$809$	20.9597	0.736904	0.368452	+	0.929647i	$0.379888\pi$
$810$	0	0
$811$	16.5690	0.581815	0.290907	−	0.956751i	$-0.406043\pi$
$811$	16.5690	0.581815	0.290907	+	0.956751i	$0.406043\pi$
$812$	−2.84654	−0.0998941
$813$	−33.4808	−1.17422
$814$	−16.4819	−0.577690
$815$	0	0
$816$	2.73423	0.0957173
$817$	−2.69202	−0.0941819
$818$	4.37244	0.152879
$819$	−0.622662	−0.0217576
$820$	0	0
$821$	49.5182	1.72820	0.864099	−	0.503321i	$-0.167889\pi$
$821$	49.5182	1.72820	0.864099	+	0.503321i	$0.167889\pi$
$822$	21.5284	0.750889
$823$	−47.4543	−1.65415	−0.827076	−	0.562091i	$-0.809997\pi$
$823$	−47.4543	−1.65415	−0.827076	+	0.562091i	$0.809997\pi$
$824$	−6.27173	−0.218486
$825$	0	0
$826$	5.34481	0.185970
$827$	20.4316	0.710477	0.355239	−	0.934776i	$-0.384400\pi$
$827$	20.4316	0.710477	0.355239	+	0.934776i	$0.384400\pi$
$828$	0.185981	0.00646328
$829$	33.7517	1.17224	0.586122	−	0.810223i	$-0.300654\pi$
$829$	33.7517	1.17224	0.586122	+	0.810223i	$0.300654\pi$
$830$	0	0
$831$	−25.1739	−0.873273
$832$	16.1938	0.561417
$833$	−2.91185	−0.100890
$834$	−18.2403	−0.631609
$835$	0	0
$836$	−17.3177	−0.598944
$837$	1.31096	0.0453135
$838$	20.1293	0.695355
$839$	−35.0814	−1.21115	−0.605573	−	0.795790i	$-0.707056\pi$
$839$	−35.0814	−1.21115	−0.605573	+	0.795790i	$0.707056\pi$
$840$	0	0
$841$	−24.5991	−0.848244
$842$	−3.97392	−0.136950
$843$	18.5623	0.639318
$844$	16.2427	0.559096
$845$	0	0
$846$	−0.216849	−0.00745541
$847$	3.45473	0.118706
$848$	5.42998	0.186466
$849$	−42.1524	−1.44667
$850$	0	0
$851$	−5.40581	−0.185309
$852$	11.6140	0.397890
$853$	−31.1675	−1.06715	−0.533577	−	0.845752i	$-0.679152\pi$
$853$	−31.1675	−1.06715	−0.533577	+	0.845752i	$0.679152\pi$
$854$	4.03146	0.137954
$855$	0	0
$856$	−13.9869	−0.478061
$857$	−18.2543	−0.623554	−0.311777	−	0.950155i	$-0.600924\pi$
$857$	−18.2543	−0.623554	−0.311777	+	0.950155i	$0.600924\pi$
$858$	−23.4359	−0.800090
$859$	−7.95215	−0.271324	−0.135662	−	0.990755i	$-0.543316\pi$
$859$	−7.95215	−0.271324	−0.135662	+	0.990755i	$0.543316\pi$
$860$	0	0
$861$	−4.77777	−0.162826
$862$	19.2054	0.654137
$863$	−17.8941	−0.609123	−0.304562	−	0.952493i	$-0.598510\pi$
$863$	−17.8941	−0.609123	−0.304562	+	0.952493i	$0.598510\pi$
$864$	−30.9407	−1.05262
$865$	0	0
$866$	−11.9332	−0.405505
$867$	14.4179	0.489657
$868$	−0.335126	−0.0113749
$869$	−11.9638	−0.405843
$870$	0	0
$871$	22.8817	0.775318
$872$	−9.33273	−0.316046
$873$	1.77851	0.0601935
$874$	2.69202	0.0910590
$875$	0	0
$876$	2.23357	0.0754653
$877$	−22.8079	−0.770168	−0.385084	−	0.922881i	$-0.625828\pi$
$877$	−22.8079	−0.770168	−0.385084	+	0.922881i	$0.625828\pi$
$878$	−31.5405	−1.06444
$879$	−8.96807	−0.302485
$880$	0	0
$881$	7.64502	0.257567	0.128784	−	0.991673i	$-0.458893\pi$
$881$	7.64502	0.257567	0.128784	+	0.991673i	$0.458893\pi$
$882$	0.109916	0.00370107
$883$	−35.0331	−1.17896	−0.589479	−	0.807784i	$-0.700667\pi$
$883$	−35.0331	−1.17896	−0.589479	+	0.807784i	$0.700667\pi$
$884$	17.9493	0.603700
$885$	0	0
$886$	−9.55688	−0.321070
$887$	49.1594	1.65061	0.825306	−	0.564686i	$-0.191003\pi$
$887$	49.1594	1.65061	0.825306	+	0.564686i	$0.191003\pi$
$888$	24.6233	0.826302
$889$	6.66248	0.223452
$890$	0	0
$891$	32.5827	1.09156
$892$	20.5321	0.687466
$893$	6.62266	0.221619
$894$	−11.4222	−0.382016
$895$	0	0
$896$	8.79954	0.293972
$897$	−7.68664	−0.256650
$898$	−11.9748	−0.399603
$899$	0.518122	0.0172803
$900$	0	0
$901$	−28.4910	−0.949172
$902$	8.60925	0.286657
$903$	−1.35690	−0.0451547
$904$	55.5392	1.84720
$905$	0	0
$906$	−0.653858	−0.0217230
$907$	20.6558	0.685864	0.342932	−	0.939360i	$-0.388580\pi$
$907$	20.6558	0.685864	0.342932	+	0.939360i	$0.388580\pi$
$908$	−10.0726	−0.334271
$909$	−0.764365	−0.0253524
$910$	0	0
$911$	−58.9342	−1.95258	−0.976289	−	0.216472i	$-0.930545\pi$
$911$	−58.9342	−1.95258	−0.976289	+	0.216472i	$0.930545\pi$
$912$	3.15213	0.104377
$913$	−62.0592	−2.05386
$914$	11.6493	0.385326
$915$	0	0
$916$	−3.57732	−0.118198
$917$	8.39612	0.277264
$918$	12.3948	0.409089
$919$	−50.4644	−1.66467	−0.832334	−	0.554275i	$-0.812996\pi$
$919$	−50.4644	−1.66467	−0.832334	+	0.554275i	$0.812996\pi$
$920$	0	0
$921$	58.0417	1.91254
$922$	17.3778	0.572308
$923$	22.9806	0.756416
$924$	−8.72886	−0.287158
$925$	0	0
$926$	1.57673	0.0518145
$927$	0.319323	0.0104879
$928$	−12.2285	−0.401419
$929$	29.8834	0.980442	0.490221	−	0.871598i	$-0.336916\pi$
$929$	29.8834	0.980442	0.490221	+	0.871598i	$0.336916\pi$
$930$	0	0
$931$	−3.35690	−0.110018
$932$	−7.28190	−0.238526
$933$	16.6219	0.544177
$934$	11.5996	0.379549
$935$	0	0
$936$	−1.67622	−0.0547889
$937$	−31.4728	−1.02817	−0.514085	−	0.857739i	$-0.671869\pi$
$937$	−31.4728	−1.02817	−0.514085	+	0.857739i	$0.671869\pi$
$938$	−4.03923	−0.131885
$939$	51.7652	1.68929
$940$	0	0
$941$	30.6805	1.00016	0.500078	−	0.865980i	$-0.333305\pi$
$941$	30.6805	1.00016	0.500078	+	0.865980i	$0.333305\pi$
$942$	15.8280	0.515705
$943$	2.82371	0.0919526
$944$	−3.69873	−0.120383
$945$	0	0
$946$	2.44504	0.0794952
$947$	1.12008	0.0363977	0.0181988	−	0.999834i	$-0.494207\pi$
$947$	1.12008	0.0363977	0.0181988	+	0.999834i	$0.494207\pi$
$948$	7.22462	0.234645
$949$	4.41955	0.143465
$950$	0	0
$951$	−30.8866	−1.00157
$952$	−7.83877	−0.254056
$953$	−18.2000	−0.589555	−0.294778	−	0.955566i	$-0.595246\pi$
$953$	−18.2000	−0.589555	−0.294778	+	0.955566i	$0.595246\pi$
$954$	1.07547	0.0348197
$955$	0	0
$956$	14.9022	0.481970
$957$	13.4953	0.436241
$958$	−13.1763	−0.425707
$959$	15.8659	0.512337
$960$	0	0
$961$	−30.9390	−0.998032
$962$	19.6939	0.634958
$963$	0.712136	0.0229483
$964$	−4.06159	−0.130815
$965$	0	0
$966$	1.35690	0.0436574
$967$	−26.2524	−0.844219	−0.422109	−	0.906545i	$-0.638710\pi$
$967$	−26.2524	−0.844219	−0.422109	+	0.906545i	$0.638710\pi$
$968$	9.30021	0.298920
$969$	−16.5392	−0.531314
$970$	0	0
$971$	−21.0358	−0.675070	−0.337535	−	0.941313i	$-0.609593\pi$
$971$	−21.0358	−0.675070	−0.337535	+	0.941313i	$0.609593\pi$
$972$	1.93123	0.0619443
$973$	−13.4426	−0.430952
$974$	−24.8963	−0.797729
$975$	0	0
$976$	−2.78986	−0.0893011
$977$	−58.3021	−1.86525	−0.932625	−	0.360847i	$-0.882488\pi$
$977$	−58.3021	−1.86525	−0.932625	+	0.360847i	$0.882488\pi$
$978$	−26.2693	−0.840001
$979$	−27.3207	−0.873172
$980$	0	0
$981$	0.475173	0.0151711
$982$	16.0718	0.512870
$983$	1.15106	0.0367132	0.0183566	−	0.999832i	$-0.494157\pi$
$983$	1.15106	0.0367132	0.0183566	+	0.999832i	$0.494157\pi$
$984$	−12.8619	−0.410021
$985$	0	0
$986$	4.89871	0.156007
$987$	3.33811	0.106253
$988$	20.6926	0.658320
$989$	0.801938	0.0255001
$990$	0	0
$991$	−14.2174	−0.451632	−0.225816	−	0.974170i	$-0.572505\pi$
$991$	−14.2174	−0.451632	−0.225816	+	0.974170i	$0.572505\pi$
$992$	−1.43967	−0.0457094
$993$	5.91185	0.187607
$994$	−4.05669	−0.128670
$995$	0	0
$996$	37.4760	1.18747
$997$	8.06292	0.255355	0.127678	−	0.991816i	$-0.459248\pi$
$997$	8.06292	0.255355	0.127678	+	0.991816i	$0.459248\pi$
$998$	−5.35258	−0.169433
$999$	−28.6939	−0.907836

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.k.1.1		3
5.4	even	2		805.2.a.f.1.3	✓	3
15.14	odd	2		7245.2.a.ba.1.1		3
35.34	odd	2		5635.2.a.r.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
805.2.a.f.1.3	✓	3	5.4	even	2
4025.2.a.k.1.1		3	1.1	even	1	trivial
5635.2.a.r.1.3		3	35.34	odd	2
7245.2.a.ba.1.1		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.801938 q^{2} -1.69202 q^{3} -1.35690 q^{4} +1.35690 q^{6} +1.00000 q^{7} +2.69202 q^{8} -0.137063 q^{9} +O(q^{10})\)	\(q-0.801938 q^{2} -1.69202 q^{3} -1.35690 q^{4} +1.35690 q^{6} +1.00000 q^{7} +2.69202 q^{8} -0.137063 q^{9} -3.80194 q^{11} +2.29590 q^{12} +4.54288 q^{13} -0.801938 q^{14} +0.554958 q^{16} -2.91185 q^{17} +0.109916 q^{18} -3.35690 q^{19} -1.69202 q^{21} +3.04892 q^{22} +1.00000 q^{23} -4.55496 q^{24} -3.64310 q^{26} +5.30798 q^{27} -1.35690 q^{28} +2.09783 q^{29} +0.246980 q^{31} -5.82908 q^{32} +6.43296 q^{33} +2.33513 q^{34} +0.185981 q^{36} -5.40581 q^{37} +2.69202 q^{38} -7.68664 q^{39} +2.82371 q^{41} +1.35690 q^{42} +0.801938 q^{43} +5.15883 q^{44} -0.801938 q^{46} -1.97285 q^{47} -0.939001 q^{48} +1.00000 q^{49} +4.92692 q^{51} -6.16421 q^{52} +9.78448 q^{53} -4.25667 q^{54} +2.69202 q^{56} +5.67994 q^{57} -1.68233 q^{58} -6.66487 q^{59} -5.02715 q^{61} -0.198062 q^{62} -0.137063 q^{63} +3.56465 q^{64} -5.15883 q^{66} +5.03684 q^{67} +3.95108 q^{68} -1.69202 q^{69} +5.05861 q^{71} -0.368977 q^{72} +0.972853 q^{73} +4.33513 q^{74} +4.55496 q^{76} -3.80194 q^{77} +6.16421 q^{78} +3.14675 q^{79} -8.57002 q^{81} -2.26444 q^{82} +16.3230 q^{83} +2.29590 q^{84} -0.643104 q^{86} -3.54958 q^{87} -10.2349 q^{88} +7.18598 q^{89} +4.54288 q^{91} -1.35690 q^{92} -0.417895 q^{93} +1.58211 q^{94} +9.86294 q^{96} -12.9758 q^{97} -0.801938 q^{98} +0.521106 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 3 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 + 3 * q^7 + 3 * q^8 + 5 * q^9	\( 3 q + 2 q^{2} + 3 q^{7} + 3 q^{8} + 5 q^{9} - 7 q^{11} - 7 q^{12} - 5 q^{13} + 2 q^{14} + 2 q^{16} - 5 q^{17} + q^{18} - 6 q^{19} + 3 q^{23} - 14 q^{24} - 15 q^{26} + 21 q^{27} - 12 q^{29} - 4 q^{31} - 7 q^{32} + 6 q^{34} - 14 q^{36} - 3 q^{37} + 3 q^{38} - 21 q^{39} + q^{41} - 2 q^{43} + 7 q^{44} + 2 q^{46} - 12 q^{47} + 7 q^{48} + 3 q^{49} - 14 q^{51} - 7 q^{52} + 9 q^{53} + 14 q^{54} + 3 q^{56} - 7 q^{57} - 22 q^{58} - 21 q^{59} - 9 q^{61} - 5 q^{62} + 5 q^{63} - 11 q^{64} - 7 q^{66} - 13 q^{67} + 21 q^{68} - 16 q^{71} - 16 q^{72} + 9 q^{73} + 12 q^{74} + 14 q^{76} - 7 q^{77} + 7 q^{78} - 18 q^{79} - q^{81} - 18 q^{82} + 29 q^{83} - 7 q^{84} - 6 q^{86} - 14 q^{87} - 7 q^{88} + 7 q^{89} - 5 q^{91} - 7 q^{93} - q^{94} + 35 q^{96} - q^{97} + 2 q^{98} - 14 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 + 3 * q^7 + 3 * q^8 + 5 * q^9 - 7 * q^11 - 7 * q^12 - 5 * q^13 + 2 * q^14 + 2 * q^16 - 5 * q^17 + q^18 - 6 * q^19 + 3 * q^23 - 14 * q^24 - 15 * q^26 + 21 * q^27 - 12 * q^29 - 4 * q^31 - 7 * q^32 + 6 * q^34 - 14 * q^36 - 3 * q^37 + 3 * q^38 - 21 * q^39 + q^41 - 2 * q^43 + 7 * q^44 + 2 * q^46 - 12 * q^47 + 7 * q^48 + 3 * q^49 - 14 * q^51 - 7 * q^52 + 9 * q^53 + 14 * q^54 + 3 * q^56 - 7 * q^57 - 22 * q^58 - 21 * q^59 - 9 * q^61 - 5 * q^62 + 5 * q^63 - 11 * q^64 - 7 * q^66 - 13 * q^67 + 21 * q^68 - 16 * q^71 - 16 * q^72 + 9 * q^73 + 12 * q^74 + 14 * q^76 - 7 * q^77 + 7 * q^78 - 18 * q^79 - q^81 - 18 * q^82 + 29 * q^83 - 7 * q^84 - 6 * q^86 - 14 * q^87 - 7 * q^88 + 7 * q^89 - 5 * q^91 - 7 * q^93 - q^94 + 35 * q^96 - q^97 + 2 * q^98 - 14 * q^99

Introduction

L-functions

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