LMFDB - Embedded newform 4025.2.a.v.1.5

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 9x^{5} + 28x^{4} - 22x^{3} - 16x^{2} + 7x + 3 $ x^8 - x^7 - 10x^6 + 9x^5 + 28x^4 - 22x^3 - 16x^2 + 7x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.651939$ 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)

$f(q)$	$=$	$q+0.651939 q^{2} +2.38619 q^{3} -1.57498 q^{4} +1.55565 q^{6} -1.00000 q^{7} -2.33067 q^{8} +2.69390 q^{9} +O(q^{10})$	\(q+0.651939 q^{2} +2.38619 q^{3} -1.57498 q^{4} +1.55565 q^{6} -1.00000 q^{7} -2.33067 q^{8} +2.69390 q^{9} +2.86336 q^{11} -3.75819 q^{12} -0.353935 q^{13} -0.651939 q^{14} +1.63050 q^{16} +4.74269 q^{17} +1.75626 q^{18} -4.16177 q^{19} -2.38619 q^{21} +1.86674 q^{22} +1.00000 q^{23} -5.56141 q^{24} -0.230744 q^{26} -0.730420 q^{27} +1.57498 q^{28} -1.30729 q^{29} -0.275454 q^{31} +5.72432 q^{32} +6.83251 q^{33} +3.09195 q^{34} -4.24282 q^{36} +7.76097 q^{37} -2.71322 q^{38} -0.844555 q^{39} +11.8109 q^{41} -1.55565 q^{42} -2.76959 q^{43} -4.50972 q^{44} +0.651939 q^{46} +3.46539 q^{47} +3.89067 q^{48} +1.00000 q^{49} +11.3170 q^{51} +0.557439 q^{52} +8.57275 q^{53} -0.476189 q^{54} +2.33067 q^{56} -9.93077 q^{57} -0.852276 q^{58} +11.7387 q^{59} +8.35523 q^{61} -0.179579 q^{62} -2.69390 q^{63} +0.470914 q^{64} +4.45438 q^{66} -1.62871 q^{67} -7.46962 q^{68} +2.38619 q^{69} +0.755603 q^{71} -6.27858 q^{72} -6.60981 q^{73} +5.05968 q^{74} +6.55469 q^{76} -2.86336 q^{77} -0.550599 q^{78} +3.73268 q^{79} -9.82461 q^{81} +7.69998 q^{82} +3.36397 q^{83} +3.75819 q^{84} -1.80561 q^{86} -3.11945 q^{87} -6.67353 q^{88} -13.3680 q^{89} +0.353935 q^{91} -1.57498 q^{92} -0.657285 q^{93} +2.25923 q^{94} +13.6593 q^{96} -15.4998 q^{97} +0.651939 q^{98} +7.71359 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7}+O(q^{10}) $ 8 * q + q^2 + 4 * q^3 + 5 * q^4 - q^6 - 8 * q^7	$ 8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7} + 3 q^{11} + 9 q^{12} + 5 q^{13} - q^{14} - q^{16} + 5 q^{17} + 2 q^{18} - 2 q^{19} - 4 q^{21} + 21 q^{22} + 8 q^{23} - 6 q^{24} + 18 q^{26} + 7 q^{27} - 5 q^{28} - 9 q^{29} - 3 q^{31} + 6 q^{32} + 4 q^{33} - 10 q^{34} + 16 q^{36} + 6 q^{37} + 4 q^{38} - 2 q^{39} - 7 q^{41} + q^{42} + 8 q^{43} + 4 q^{44} + q^{46} + 22 q^{47} + 9 q^{48} + 8 q^{49} - 12 q^{51} + 11 q^{52} + 21 q^{53} - 15 q^{54} + 8 q^{57} + 16 q^{58} + 14 q^{59} + 8 q^{61} + 12 q^{62} - 40 q^{64} + 55 q^{66} + 21 q^{67} + 3 q^{68} + 4 q^{69} + 11 q^{71} - q^{72} + 26 q^{73} - 41 q^{74} + 21 q^{76} - 3 q^{77} + 17 q^{78} - 16 q^{79} - 20 q^{81} - q^{82} + 20 q^{83} - 9 q^{84} + 14 q^{86} - 29 q^{87} + 32 q^{88} + 15 q^{89} - 5 q^{91} + 5 q^{92} + 19 q^{93} + 21 q^{94} + 52 q^{96} + q^{97} + q^{98} + 15 q^{99}+O(q^{100}) $ 8 * q + q^2 + 4 * q^3 + 5 * q^4 - q^6 - 8 * q^7 + 3 * q^11 + 9 * q^12 + 5 * q^13 - q^14 - q^16 + 5 * q^17 + 2 * q^18 - 2 * q^19 - 4 * q^21 + 21 * q^22 + 8 * q^23 - 6 * q^24 + 18 * q^26 + 7 * q^27 - 5 * q^28 - 9 * q^29 - 3 * q^31 + 6 * q^32 + 4 * q^33 - 10 * q^34 + 16 * q^36 + 6 * q^37 + 4 * q^38 - 2 * q^39 - 7 * q^41 + q^42 + 8 * q^43 + 4 * q^44 + q^46 + 22 * q^47 + 9 * q^48 + 8 * q^49 - 12 * q^51 + 11 * q^52 + 21 * q^53 - 15 * q^54 + 8 * q^57 + 16 * q^58 + 14 * q^59 + 8 * q^61 + 12 * q^62 - 40 * q^64 + 55 * q^66 + 21 * q^67 + 3 * q^68 + 4 * q^69 + 11 * q^71 - q^72 + 26 * q^73 - 41 * q^74 + 21 * q^76 - 3 * q^77 + 17 * q^78 - 16 * q^79 - 20 * q^81 - q^82 + 20 * q^83 - 9 * q^84 + 14 * q^86 - 29 * q^87 + 32 * q^88 + 15 * q^89 - 5 * q^91 + 5 * q^92 + 19 * q^93 + 21 * q^94 + 52 * q^96 + q^97 + q^98 + 15 * 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 $

$2$	0.651939	0.460991	0.230495	−	0.973073i	$-0.425965\pi$
$2$	0.651939	0.460991	0.230495	+	0.973073i	$0.425965\pi$
$3$	2.38619	1.37767	0.688833	−	0.724920i	$-0.258123\pi$
$3$	2.38619	1.37767	0.688833	+	0.724920i	$0.258123\pi$
$4$	−1.57498	−0.787488
$5$	0	0
$5$	0	0
$6$	1.55565	0.635092
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−2.33067	−0.824015
$9$	2.69390	0.897966
$10$	0	0
$11$	2.86336	0.863335	0.431667	−	0.902033i	$-0.357925\pi$
$11$	2.86336	0.863335	0.431667	+	0.902033i	$0.357925\pi$
$12$	−3.75819	−1.08490
$13$	−0.353935	−0.0981638	−0.0490819	−	0.998795i	$-0.515630\pi$
$13$	−0.353935	−0.0981638	−0.0490819	+	0.998795i	$0.515630\pi$
$14$	−0.651939	−0.174238
$15$	0	0
$16$	1.63050	0.407624
$17$	4.74269	1.15027	0.575136	−	0.818058i	$-0.304949\pi$
$17$	4.74269	1.15027	0.575136	+	0.818058i	$0.304949\pi$
$18$	1.75626	0.413954
$19$	−4.16177	−0.954776	−0.477388	−	0.878693i	$-0.658416\pi$
$19$	−4.16177	−0.954776	−0.477388	+	0.878693i	$0.658416\pi$
$20$	0	0
$21$	−2.38619	−0.520709
$22$	1.86674	0.397989
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−5.56141	−1.13522
$25$	0	0
$26$	−0.230744	−0.0452526
$27$	−0.730420	−0.140569
$28$	1.57498	0.297642
$29$	−1.30729	−0.242758	−0.121379	−	0.992606i	$-0.538732\pi$
$29$	−1.30729	−0.242758	−0.121379	+	0.992606i	$0.538732\pi$
$30$	0	0
$31$	−0.275454	−0.0494730	−0.0247365	−	0.999694i	$-0.507875\pi$
$31$	−0.275454	−0.0494730	−0.0247365	+	0.999694i	$0.507875\pi$
$32$	5.72432	1.01193
$33$	6.83251	1.18939
$34$	3.09195	0.530265
$35$	0	0
$36$	−4.24282	−0.707137
$37$	7.76097	1.27590	0.637948	−	0.770080i	$-0.279784\pi$
$37$	7.76097	1.27590	0.637948	+	0.770080i	$0.279784\pi$
$38$	−2.71322	−0.440143
$39$	−0.844555	−0.135237
$40$	0	0
$41$	11.8109	1.84455	0.922275	−	0.386534i	$-0.126328\pi$
$41$	11.8109	1.84455	0.922275	+	0.386534i	$0.126328\pi$
$42$	−1.55565	−0.240042
$43$	−2.76959	−0.422359	−0.211180	−	0.977447i	$-0.567730\pi$
$43$	−2.76959	−0.422359	−0.211180	+	0.977447i	$0.567730\pi$
$44$	−4.50972	−0.679866
$45$	0	0
$46$	0.651939	0.0961232
$47$	3.46539	0.505480	0.252740	−	0.967534i	$-0.418668\pi$
$47$	3.46539	0.505480	0.252740	+	0.967534i	$0.418668\pi$
$48$	3.89067	0.561570
$49$	1.00000	0.142857
$50$	0	0
$51$	11.3170	1.58469
$52$	0.557439	0.0773028
$53$	8.57275	1.17756	0.588779	−	0.808294i	$-0.299609\pi$
$53$	8.57275	1.17756	0.588779	+	0.808294i	$0.299609\pi$
$54$	−0.476189	−0.0648012
$55$	0	0
$56$	2.33067	0.311448
$57$	−9.93077	−1.31536
$58$	−0.852276	−0.111909
$59$	11.7387	1.52825	0.764126	−	0.645067i	$-0.223171\pi$
$59$	11.7387	1.52825	0.764126	+	0.645067i	$0.223171\pi$
$60$	0	0
$61$	8.35523	1.06978	0.534889	−	0.844922i	$-0.320353\pi$
$61$	8.35523	1.06978	0.534889	+	0.844922i	$0.320353\pi$
$62$	−0.179579	−0.0228066
$63$	−2.69390	−0.339399
$64$	0.470914	0.0588642
$65$	0	0
$66$	4.45438	0.548297
$67$	−1.62871	−0.198978	−0.0994892	−	0.995039i	$-0.531721\pi$
$67$	−1.62871	−0.198978	−0.0994892	+	0.995039i	$0.531721\pi$
$68$	−7.46962	−0.905825
$69$	2.38619	0.287263
$70$	0	0
$71$	0.755603	0.0896736	0.0448368	−	0.998994i	$-0.485723\pi$
$71$	0.755603	0.0896736	0.0448368	+	0.998994i	$0.485723\pi$
$72$	−6.27858	−0.739937
$73$	−6.60981	−0.773619	−0.386810	−	0.922160i	$-0.626423\pi$
$73$	−6.60981	−0.773619	−0.386810	+	0.922160i	$0.626423\pi$
$74$	5.05968	0.588176
$75$	0	0
$76$	6.55469	0.751874
$77$	−2.86336	−0.326310
$78$	−0.550599	−0.0623430
$79$	3.73268	0.419959	0.209980	−	0.977706i	$-0.432660\pi$
$79$	3.73268	0.419959	0.209980	+	0.977706i	$0.432660\pi$
$80$	0	0
$81$	−9.82461	−1.09162
$82$	7.69998	0.850321
$83$	3.36397	0.369244	0.184622	−	0.982810i	$-0.440894\pi$
$83$	3.36397	0.369244	0.184622	+	0.982810i	$0.440894\pi$
$84$	3.75819	0.410052
$85$	0	0
$86$	−1.80561	−0.194704
$87$	−3.11945	−0.334440
$88$	−6.67353	−0.711401
$89$	−13.3680	−1.41701	−0.708503	−	0.705708i	$-0.750629\pi$
$89$	−13.3680	−1.41701	−0.708503	+	0.705708i	$0.750629\pi$
$90$	0	0
$91$	0.353935	0.0371024
$92$	−1.57498	−0.164203
$93$	−0.657285	−0.0681573
$94$	2.25923	0.233021
$95$	0	0
$96$	13.6593	1.39410
$97$	−15.4998	−1.57377	−0.786885	−	0.617099i	$-0.788308\pi$
$97$	−15.4998	−1.57377	−0.786885	+	0.617099i	$0.788308\pi$
$98$	0.651939	0.0658558
$99$	7.71359	0.775245
$100$	0	0
$101$	14.2876	1.42167	0.710834	−	0.703360i	$-0.248318\pi$
$101$	14.2876	1.42167	0.710834	+	0.703360i	$0.248318\pi$
$102$	7.37797	0.730528
$103$	7.26200	0.715546	0.357773	−	0.933809i	$-0.383536\pi$
$103$	7.26200	0.715546	0.357773	+	0.933809i	$0.383536\pi$
$104$	0.824904	0.0808885
$105$	0	0
$106$	5.58891	0.542843
$107$	12.7892	1.23637	0.618187	−	0.786031i	$-0.287867\pi$
$107$	12.7892	1.23637	0.618187	+	0.786031i	$0.287867\pi$
$108$	1.15039	0.110697
$109$	−1.39334	−0.133458	−0.0667288	−	0.997771i	$-0.521256\pi$
$109$	−1.39334	−0.133458	−0.0667288	+	0.997771i	$0.521256\pi$
$110$	0	0
$111$	18.5191	1.75776
$112$	−1.63050	−0.154068
$113$	−9.96960	−0.937861	−0.468930	−	0.883235i	$-0.655361\pi$
$113$	−9.96960	−0.937861	−0.468930	+	0.883235i	$0.655361\pi$
$114$	−6.47426	−0.606370
$115$	0	0
$116$	2.05896	0.191169
$117$	−0.953464	−0.0881478
$118$	7.65294	0.704510
$119$	−4.74269	−0.434762
$120$	0	0
$121$	−2.80118	−0.254653
$122$	5.44710	0.493158
$123$	28.1830	2.54118
$124$	0.433833	0.0389594
$125$	0	0
$126$	−1.75626	−0.156460
$127$	−4.28578	−0.380301	−0.190151	−	0.981755i	$-0.560898\pi$
$127$	−4.28578	−0.380301	−0.190151	+	0.981755i	$0.560898\pi$
$128$	−11.1416	−0.984790
$129$	−6.60877	−0.581870
$130$	0	0
$131$	9.89892	0.864872	0.432436	−	0.901665i	$-0.357654\pi$
$131$	9.89892	0.864872	0.432436	+	0.901665i	$0.357654\pi$
$132$	−10.7610	−0.936628
$133$	4.16177	0.360871
$134$	−1.06182	−0.0917272
$135$	0	0
$136$	−11.0536	−0.947841
$137$	7.58744	0.648239	0.324119	−	0.946016i	$-0.394932\pi$
$137$	7.58744	0.648239	0.324119	+	0.946016i	$0.394932\pi$
$138$	1.55565	0.132426
$139$	8.54023	0.724373	0.362186	−	0.932106i	$-0.382030\pi$
$139$	8.54023	0.724373	0.362186	+	0.932106i	$0.382030\pi$
$140$	0	0
$141$	8.26908	0.696382
$142$	0.492607	0.0413387
$143$	−1.01344	−0.0847483
$144$	4.39239	0.366033
$145$	0	0
$146$	−4.30919	−0.356631
$147$	2.38619	0.196810
$148$	−12.2233	−1.00475
$149$	−0.273337	−0.0223927	−0.0111963	−	0.999937i	$-0.503564\pi$
$149$	−0.273337	−0.0223927	−0.0111963	+	0.999937i	$0.503564\pi$
$150$	0	0
$151$	−11.9419	−0.971815	−0.485908	−	0.874010i	$-0.661511\pi$
$151$	−11.9419	−0.971815	−0.485908	+	0.874010i	$0.661511\pi$
$152$	9.69970	0.786750
$153$	12.7763	1.03290
$154$	−1.86674	−0.150426
$155$	0	0
$156$	1.33015	0.106498
$157$	−13.2610	−1.05834	−0.529172	−	0.848514i	$-0.677497\pi$
$157$	−13.2610	−1.05834	−0.529172	+	0.848514i	$0.677497\pi$
$158$	2.43348	0.193597
$159$	20.4562	1.62228
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	−6.40505	−0.503228
$163$	19.3954	1.51916	0.759581	−	0.650413i	$-0.225404\pi$
$163$	19.3954	1.51916	0.759581	+	0.650413i	$0.225404\pi$
$164$	−18.6019	−1.45256
$165$	0	0
$166$	2.19310	0.170218
$167$	−0.682953	−0.0528485	−0.0264243	−	0.999651i	$-0.508412\pi$
$167$	−0.682953	−0.0528485	−0.0264243	+	0.999651i	$0.508412\pi$
$168$	5.56141	0.429072
$169$	−12.8747	−0.990364
$170$	0	0
$171$	−11.2114	−0.857356
$172$	4.36204	0.332603
$173$	12.2636	0.932385	0.466192	−	0.884683i	$-0.345625\pi$
$173$	12.2636	0.932385	0.466192	+	0.884683i	$0.345625\pi$
$174$	−2.03369	−0.154174
$175$	0	0
$176$	4.66870	0.351916
$177$	28.0108	2.10542
$178$	−8.71513	−0.653227
$179$	20.1013	1.50244	0.751219	−	0.660053i	$-0.229466\pi$
$179$	20.1013	1.50244	0.751219	+	0.660053i	$0.229466\pi$
$180$	0	0
$181$	−9.54773	−0.709677	−0.354839	−	0.934928i	$-0.615464\pi$
$181$	−9.54773	−0.709677	−0.354839	+	0.934928i	$0.615464\pi$
$182$	0.230744	0.0171039
$183$	19.9372	1.47380
$184$	−2.33067	−0.171819
$185$	0	0
$186$	−0.428510	−0.0314199
$187$	13.5800	0.993070
$188$	−5.45791	−0.398059
$189$	0.730420	0.0531302
$190$	0	0
$191$	−13.6207	−0.985558	−0.492779	−	0.870154i	$-0.664019\pi$
$191$	−13.6207	−0.985558	−0.492779	+	0.870154i	$0.664019\pi$
$192$	1.12369	0.0810952
$193$	9.08238	0.653764	0.326882	−	0.945065i	$-0.394002\pi$
$193$	9.08238	0.653764	0.326882	+	0.945065i	$0.394002\pi$
$194$	−10.1050	−0.725494
$195$	0	0
$196$	−1.57498	−0.112498
$197$	17.9479	1.27874	0.639368	−	0.768901i	$-0.279196\pi$
$197$	17.9479	1.27874	0.639368	+	0.768901i	$0.279196\pi$
$198$	5.02879	0.357381
$199$	−0.102119	−0.00723901	−0.00361950	−	0.999993i	$-0.501152\pi$
$199$	−0.102119	−0.00723901	−0.00361950	+	0.999993i	$0.501152\pi$
$200$	0	0
$201$	−3.88641	−0.274126
$202$	9.31463	0.655375
$203$	1.30729	0.0917540
$204$	−17.8239	−1.24792
$205$	0	0
$206$	4.73438	0.329860
$207$	2.69390	0.187239
$208$	−0.577090	−0.0400140
$209$	−11.9166	−0.824291
$210$	0	0
$211$	−12.9081	−0.888627	−0.444314	−	0.895871i	$-0.646552\pi$
$211$	−12.9081	−0.888627	−0.444314	+	0.895871i	$0.646552\pi$
$212$	−13.5019	−0.927312
$213$	1.80301	0.123540
$214$	8.33775	0.569957
$215$	0	0
$216$	1.70237	0.115831
$217$	0.275454	0.0186990
$218$	−0.908372	−0.0615227
$219$	−15.7722	−1.06579
$220$	0	0
$221$	−1.67860	−0.112915
$222$	12.0734	0.810310
$223$	7.22195	0.483618	0.241809	−	0.970324i	$-0.422259\pi$
$223$	7.22195	0.483618	0.241809	+	0.970324i	$0.422259\pi$
$224$	−5.72432	−0.382472
$225$	0	0
$226$	−6.49957	−0.432345
$227$	−5.30284	−0.351962	−0.175981	−	0.984394i	$-0.556310\pi$
$227$	−5.30284	−0.351962	−0.175981	+	0.984394i	$0.556310\pi$
$228$	15.6407	1.03583
$229$	18.4259	1.21762	0.608809	−	0.793317i	$-0.291648\pi$
$229$	18.4259	1.21762	0.608809	+	0.793317i	$0.291648\pi$
$230$	0	0
$231$	−6.83251	−0.449546
$232$	3.04687	0.200037
$233$	−6.04063	−0.395735	−0.197867	−	0.980229i	$-0.563402\pi$
$233$	−6.04063	−0.395735	−0.197867	+	0.980229i	$0.563402\pi$
$234$	−0.621600	−0.0406353
$235$	0	0
$236$	−18.4882	−1.20348
$237$	8.90688	0.578564
$238$	−3.09195	−0.200421
$239$	0.955399	0.0617996	0.0308998	−	0.999522i	$-0.490163\pi$
$239$	0.955399	0.0617996	0.0308998	+	0.999522i	$0.490163\pi$
$240$	0	0
$241$	0.976650	0.0629116	0.0314558	−	0.999505i	$-0.489986\pi$
$241$	0.976650	0.0629116	0.0314558	+	0.999505i	$0.489986\pi$
$242$	−1.82620	−0.117393
$243$	−21.2521	−1.36332
$244$	−13.1593	−0.842437
$245$	0	0
$246$	18.3736	1.17146
$247$	1.47300	0.0937245
$248$	0.641992	0.0407665
$249$	8.02707	0.508695
$250$	0	0
$251$	8.62574	0.544452	0.272226	−	0.962233i	$-0.412240\pi$
$251$	8.62574	0.544452	0.272226	+	0.962233i	$0.412240\pi$
$252$	4.24282	0.267273
$253$	2.86336	0.180018
$254$	−2.79407	−0.175315
$255$	0	0
$256$	−8.20549	−0.512843
$257$	−22.9498	−1.43157	−0.715785	−	0.698320i	$-0.753931\pi$
$257$	−22.9498	−1.43157	−0.715785	+	0.698320i	$0.753931\pi$
$258$	−4.30852	−0.268237
$259$	−7.76097	−0.482243
$260$	0	0
$261$	−3.52171	−0.217989
$262$	6.45349	0.398698
$263$	29.6718	1.82964	0.914822	−	0.403858i	$-0.132331\pi$
$263$	29.6718	1.82964	0.914822	+	0.403858i	$0.132331\pi$
$264$	−15.9243	−0.980074
$265$	0	0
$266$	2.71322	0.166358
$267$	−31.8986	−1.95216
$268$	2.56518	0.156693
$269$	−25.9055	−1.57949	−0.789743	−	0.613438i	$-0.789786\pi$
$269$	−25.9055	−1.57949	−0.789743	+	0.613438i	$0.789786\pi$
$270$	0	0
$271$	−21.4377	−1.30225	−0.651124	−	0.758971i	$-0.725702\pi$
$271$	−21.4377	−1.30225	−0.651124	+	0.758971i	$0.725702\pi$
$272$	7.73295	0.468879
$273$	0.844555	0.0511148
$274$	4.94655	0.298832
$275$	0	0
$276$	−3.75819	−0.226216
$277$	−5.47056	−0.328694	−0.164347	−	0.986403i	$-0.552552\pi$
$277$	−5.47056	−0.328694	−0.164347	+	0.986403i	$0.552552\pi$
$278$	5.56771	0.333929
$279$	−0.742045	−0.0444251
$280$	0	0
$281$	15.7416	0.939065	0.469532	−	0.882915i	$-0.344423\pi$
$281$	15.7416	0.939065	0.469532	+	0.882915i	$0.344423\pi$
$282$	5.39094	0.321026
$283$	17.4821	1.03920	0.519602	−	0.854408i	$-0.326080\pi$
$283$	17.4821	1.03920	0.519602	+	0.854408i	$0.326080\pi$
$284$	−1.19006	−0.0706169
$285$	0	0
$286$	−0.660703	−0.0390682
$287$	−11.8109	−0.697175
$288$	15.4207	0.908675
$289$	5.49313	0.323125
$290$	0	0
$291$	−36.9856	−2.16813
$292$	10.4103	0.609216
$293$	22.5324	1.31636	0.658178	−	0.752862i	$-0.271327\pi$
$293$	22.5324	1.31636	0.658178	+	0.752862i	$0.271327\pi$
$294$	1.55565	0.0907274
$295$	0	0
$296$	−18.0882	−1.05136
$297$	−2.09145	−0.121358
$298$	−0.178199	−0.0103228
$299$	−0.353935	−0.0204686
$300$	0	0
$301$	2.76959	0.159637
$302$	−7.78537	−0.447998
$303$	34.0929	1.95858
$304$	−6.78576	−0.389190
$305$	0	0
$306$	8.32939	0.476159
$307$	−14.8992	−0.850344	−0.425172	−	0.905112i	$-0.639786\pi$
$307$	−14.8992	−0.850344	−0.425172	+	0.905112i	$0.639786\pi$
$308$	4.50972	0.256965
$309$	17.3285	0.985785
$310$	0	0
$311$	−18.4177	−1.04437	−0.522187	−	0.852831i	$-0.674884\pi$
$311$	−18.4177	−1.04437	−0.522187	+	0.852831i	$0.674884\pi$
$312$	1.96838	0.111437
$313$	−25.7564	−1.45584	−0.727919	−	0.685664i	$-0.759512\pi$
$313$	−25.7564	−1.45584	−0.727919	+	0.685664i	$0.759512\pi$
$314$	−8.64538	−0.487887
$315$	0	0
$316$	−5.87888	−0.330713
$317$	16.0330	0.900505	0.450252	−	0.892901i	$-0.351334\pi$
$317$	16.0330	0.900505	0.450252	+	0.892901i	$0.351334\pi$
$318$	13.3362	0.747857
$319$	−3.74325	−0.209582
$320$	0	0
$321$	30.5173	1.70331
$322$	−0.651939	−0.0363312
$323$	−19.7380	−1.09825
$324$	15.4735	0.859640
$325$	0	0
$326$	12.6446	0.700319
$327$	−3.32477	−0.183860
$328$	−27.5272	−1.51994
$329$	−3.46539	−0.191053
$330$	0	0
$331$	−18.5035	−1.01705	−0.508523	−	0.861048i	$-0.669808\pi$
$331$	−18.5035	−1.01705	−0.508523	+	0.861048i	$0.669808\pi$
$332$	−5.29817	−0.290775
$333$	20.9072	1.14571
$334$	−0.445244	−0.0243627
$335$	0	0
$336$	−3.89067	−0.212254
$337$	−13.0653	−0.711712	−0.355856	−	0.934541i	$-0.615811\pi$
$337$	−13.0653	−0.711712	−0.355856	+	0.934541i	$0.615811\pi$
$338$	−8.39354	−0.456549
$339$	−23.7893	−1.29206
$340$	0	0
$341$	−0.788724	−0.0427118
$342$	−7.30914	−0.395233
$343$	−1.00000	−0.0539949
$344$	6.45500	0.348030
$345$	0	0
$346$	7.99512	0.429821
$347$	7.28070	0.390849	0.195424	−	0.980719i	$-0.437392\pi$
$347$	7.28070	0.390849	0.195424	+	0.980719i	$0.437392\pi$
$348$	4.91306	0.263367
$349$	−15.7564	−0.843420	−0.421710	−	0.906731i	$-0.638570\pi$
$349$	−15.7564	−0.843420	−0.421710	+	0.906731i	$0.638570\pi$
$350$	0	0
$351$	0.258521	0.0137988
$352$	16.3908	0.873631
$353$	−13.4427	−0.715482	−0.357741	−	0.933821i	$-0.616453\pi$
$353$	−13.4427	−0.715482	−0.357741	+	0.933821i	$0.616453\pi$
$354$	18.2613	0.970580
$355$	0	0
$356$	21.0543	1.11587
$357$	−11.3170	−0.598957
$358$	13.1048	0.692610
$359$	2.96947	0.156722	0.0783612	−	0.996925i	$-0.475031\pi$
$359$	2.96947	0.156722	0.0783612	+	0.996925i	$0.475031\pi$
$360$	0	0
$361$	−1.67966	−0.0884033
$362$	−6.22454	−0.327155
$363$	−6.68414	−0.350827
$364$	−0.557439	−0.0292177
$365$	0	0
$366$	12.9978	0.679407
$367$	8.64514	0.451273	0.225636	−	0.974212i	$-0.427554\pi$
$367$	8.64514	0.451273	0.225636	+	0.974212i	$0.427554\pi$
$368$	1.63050	0.0849955
$369$	31.8173	1.65634
$370$	0	0
$371$	−8.57275	−0.445075
$372$	1.03521	0.0536730
$373$	−31.8219	−1.64767	−0.823837	−	0.566827i	$-0.808171\pi$
$373$	−31.8219	−1.64767	−0.823837	+	0.566827i	$0.808171\pi$
$374$	8.85335	0.457796
$375$	0	0
$376$	−8.07668	−0.416523
$377$	0.462697	0.0238301
$378$	0.476189	0.0244925
$379$	26.2130	1.34647	0.673234	−	0.739429i	$-0.264904\pi$
$379$	26.2130	1.34647	0.673234	+	0.739429i	$0.264904\pi$
$380$	0	0
$381$	−10.2267	−0.523928
$382$	−8.87986	−0.454333
$383$	16.8240	0.859664	0.429832	−	0.902909i	$-0.358573\pi$
$383$	16.8240	0.859664	0.429832	+	0.902909i	$0.358573\pi$
$384$	−26.5860	−1.35671
$385$	0	0
$386$	5.92116	0.301379
$387$	−7.46100	−0.379264
$388$	24.4119	1.23933
$389$	9.88794	0.501338	0.250669	−	0.968073i	$-0.419349\pi$
$389$	9.88794	0.501338	0.250669	+	0.968073i	$0.419349\pi$
$390$	0	0
$391$	4.74269	0.239848
$392$	−2.33067	−0.117716
$393$	23.6207	1.19151
$394$	11.7010	0.589486
$395$	0	0
$396$	−12.1487	−0.610496
$397$	12.6857	0.636675	0.318338	−	0.947977i	$-0.396875\pi$
$397$	12.6857	0.636675	0.318338	+	0.947977i	$0.396875\pi$
$398$	−0.0665752	−0.00333712
$399$	9.93077	0.497160
$400$	0	0
$401$	30.3580	1.51601	0.758003	−	0.652251i	$-0.226175\pi$
$401$	30.3580	1.51601	0.758003	+	0.652251i	$0.226175\pi$
$402$	−2.53370	−0.126370
$403$	0.0974928	0.00485646
$404$	−22.5026	−1.11955
$405$	0	0
$406$	0.852276	0.0422978
$407$	22.2224	1.10152
$408$	−26.3761	−1.30581
$409$	−34.5070	−1.70626	−0.853130	−	0.521699i	$-0.825298\pi$
$409$	−34.5070	−1.70626	−0.853130	+	0.521699i	$0.825298\pi$
$410$	0	0
$411$	18.1051	0.893057
$412$	−11.4375	−0.563484
$413$	−11.7387	−0.577625
$414$	1.75626	0.0863153
$415$	0	0
$416$	−2.02604	−0.0993346
$417$	20.3786	0.997944
$418$	−7.76893	−0.379991
$419$	−17.9612	−0.877463	−0.438731	−	0.898618i	$-0.644572\pi$
$419$	−17.9612	−0.877463	−0.438731	+	0.898618i	$0.644572\pi$
$420$	0	0
$421$	5.18195	0.252553	0.126277	−	0.991995i	$-0.459697\pi$
$421$	5.18195	0.252553	0.126277	+	0.991995i	$0.459697\pi$
$422$	−8.41527	−0.409649
$423$	9.33541	0.453903
$424$	−19.9802	−0.970325
$425$	0	0
$426$	1.17545	0.0569509
$427$	−8.35523	−0.404338
$428$	−20.1426	−0.973629
$429$	−2.41826	−0.116755
$430$	0	0
$431$	−30.8620	−1.48657	−0.743286	−	0.668974i	$-0.766734\pi$
$431$	−30.8620	−1.48657	−0.743286	+	0.668974i	$0.766734\pi$
$432$	−1.19095	−0.0572995
$433$	36.7775	1.76742	0.883708	−	0.468040i	$-0.155040\pi$
$433$	36.7775	1.76742	0.883708	+	0.468040i	$0.155040\pi$
$434$	0.179579	0.00862008
$435$	0	0
$436$	2.19447	0.105096
$437$	−4.16177	−0.199085
$438$	−10.2825	−0.491319
$439$	−17.3533	−0.828228	−0.414114	−	0.910225i	$-0.635909\pi$
$439$	−17.3533	−0.828228	−0.414114	+	0.910225i	$0.635909\pi$
$440$	0	0
$441$	2.69390	0.128281
$442$	−1.09435	−0.0520528
$443$	−20.4177	−0.970074	−0.485037	−	0.874494i	$-0.661194\pi$
$443$	−20.4177	−0.970074	−0.485037	+	0.874494i	$0.661194\pi$
$444$	−29.1672	−1.38421
$445$	0	0
$446$	4.70827	0.222943
$447$	−0.652234	−0.0308496
$448$	−0.470914	−0.0222486
$449$	−25.7809	−1.21667	−0.608337	−	0.793679i	$-0.708163\pi$
$449$	−25.7809	−1.21667	−0.608337	+	0.793679i	$0.708163\pi$
$450$	0	0
$451$	33.8188	1.59246
$452$	15.7019	0.738554
$453$	−28.4955	−1.33884
$454$	−3.45713	−0.162251
$455$	0	0
$456$	23.1453	1.08388
$457$	−24.5564	−1.14870	−0.574350	−	0.818610i	$-0.694745\pi$
$457$	−24.5564	−1.14870	−0.574350	+	0.818610i	$0.694745\pi$
$458$	12.0126	0.561310
$459$	−3.46416	−0.161693
$460$	0	0
$461$	−12.9461	−0.602961	−0.301480	−	0.953472i	$-0.597481\pi$
$461$	−12.9461	−0.602961	−0.301480	+	0.953472i	$0.597481\pi$
$462$	−4.45438	−0.207237
$463$	−15.7702	−0.732902	−0.366451	−	0.930437i	$-0.619427\pi$
$463$	−15.7702	−0.732902	−0.366451	+	0.930437i	$0.619427\pi$
$464$	−2.13154	−0.0989542
$465$	0	0
$466$	−3.93812	−0.182430
$467$	−0.0437045	−0.00202240	−0.00101120	−	0.999999i	$-0.500322\pi$
$467$	−0.0437045	−0.00202240	−0.00101120	+	0.999999i	$0.500322\pi$
$468$	1.50168	0.0694153
$469$	1.62871	0.0752068
$470$	0	0
$471$	−31.6433	−1.45805
$472$	−27.3591	−1.25930
$473$	−7.93034	−0.364637
$474$	5.80674	0.266713
$475$	0	0
$476$	7.46962	0.342370
$477$	23.0941	1.05741
$478$	0.622862	0.0284890
$479$	−31.0038	−1.41660	−0.708299	−	0.705913i	$-0.750537\pi$
$479$	−31.0038	−1.41660	−0.708299	+	0.705913i	$0.750537\pi$
$480$	0	0
$481$	−2.74688	−0.125247
$482$	0.636717	0.0290017
$483$	−2.38619	−0.108575
$484$	4.41179	0.200536
$485$	0	0
$486$	−13.8551	−0.628480
$487$	30.1026	1.36408	0.682038	−	0.731316i	$-0.261094\pi$
$487$	30.1026	1.36408	0.682038	+	0.731316i	$0.261094\pi$
$488$	−19.4733	−0.881513
$489$	46.2810	2.09290
$490$	0	0
$491$	−16.4179	−0.740930	−0.370465	−	0.928847i	$-0.620802\pi$
$491$	−16.4179	−0.740930	−0.370465	+	0.928847i	$0.620802\pi$
$492$	−44.3875	−2.00114
$493$	−6.20009	−0.279238
$494$	0.960304	0.0432061
$495$	0	0
$496$	−0.449127	−0.0201664
$497$	−0.755603	−0.0338934
$498$	5.23316	0.234504
$499$	−30.1437	−1.34942	−0.674709	−	0.738084i	$-0.735731\pi$
$499$	−30.1437	−1.34942	−0.674709	+	0.738084i	$0.735731\pi$
$500$	0	0
$501$	−1.62966	−0.0728077
$502$	5.62346	0.250987
$503$	5.01211	0.223479	0.111740	−	0.993738i	$-0.464358\pi$
$503$	5.01211	0.223479	0.111740	+	0.993738i	$0.464358\pi$
$504$	6.27858	0.279670
$505$	0	0
$506$	1.86674	0.0829865
$507$	−30.7215	−1.36439
$508$	6.74999	0.299482
$509$	−1.78352	−0.0790530	−0.0395265	−	0.999219i	$-0.512585\pi$
$509$	−1.78352	−0.0790530	−0.0395265	+	0.999219i	$0.512585\pi$
$510$	0	0
$511$	6.60981	0.292401
$512$	16.9338	0.748374
$513$	3.03984	0.134212
$514$	−14.9619	−0.659941
$515$	0	0
$516$	10.4087	0.458216
$517$	9.92266	0.436398
$518$	−5.05968	−0.222310
$519$	29.2633	1.28452
$520$	0	0
$521$	−5.95951	−0.261091	−0.130545	−	0.991442i	$-0.541673\pi$
$521$	−5.95951	−0.261091	−0.130545	+	0.991442i	$0.541673\pi$
$522$	−2.29594	−0.100491
$523$	−18.1146	−0.792095	−0.396047	−	0.918230i	$-0.629618\pi$
$523$	−18.1146	−0.792095	−0.396047	+	0.918230i	$0.629618\pi$
$524$	−15.5905	−0.681076
$525$	0	0
$526$	19.3442	0.843449
$527$	−1.30639	−0.0569074
$528$	11.1404	0.484823
$529$	1.00000	0.0434783
$530$	0	0
$531$	31.6229	1.37232
$532$	−6.55469	−0.284182
$533$	−4.18028	−0.181068
$534$	−20.7959	−0.899929
$535$	0	0
$536$	3.79598	0.163961
$537$	47.9654	2.06986
$538$	−16.8888	−0.728128
$539$	2.86336	0.123334
$540$	0	0
$541$	−28.5317	−1.22667	−0.613337	−	0.789821i	$-0.710173\pi$
$541$	−28.5317	−1.22667	−0.613337	+	0.789821i	$0.710173\pi$
$542$	−13.9761	−0.600325
$543$	−22.7827	−0.977698
$544$	27.1487	1.16399
$545$	0	0
$546$	0.550599	0.0235634
$547$	22.7052	0.970806	0.485403	−	0.874291i	$-0.338673\pi$
$547$	22.7052	0.970806	0.485403	+	0.874291i	$0.338673\pi$
$548$	−11.9500	−0.510480
$549$	22.5081	0.960623
$550$	0	0
$551$	5.44066	0.231780
$552$	−5.56141	−0.236709
$553$	−3.73268	−0.158730
$554$	−3.56647	−0.151525
$555$	0	0
$556$	−13.4506	−0.570434
$557$	−7.53910	−0.319442	−0.159721	−	0.987162i	$-0.551059\pi$
$557$	−7.53910	−0.319442	−0.159721	+	0.987162i	$0.551059\pi$
$558$	−0.483768	−0.0204795
$559$	0.980256	0.0414604
$560$	0	0
$561$	32.4045	1.36812
$562$	10.2626	0.432900
$563$	−20.9082	−0.881175	−0.440587	−	0.897710i	$-0.645230\pi$
$563$	−20.9082	−0.881175	−0.440587	+	0.897710i	$0.645230\pi$
$564$	−13.0236	−0.548392
$565$	0	0
$566$	11.3973	0.479064
$567$	9.82461	0.412595
$568$	−1.76106	−0.0738924
$569$	23.3003	0.976799	0.488399	−	0.872620i	$-0.337581\pi$
$569$	23.3003	0.976799	0.488399	+	0.872620i	$0.337581\pi$
$570$	0	0
$571$	−43.3434	−1.81387	−0.906933	−	0.421275i	$-0.861583\pi$
$571$	−43.3434	−1.81387	−0.906933	+	0.421275i	$0.861583\pi$
$572$	1.59615	0.0667382
$573$	−32.5015	−1.35777
$574$	−7.69998	−0.321391
$575$	0	0
$576$	1.26859	0.0528580
$577$	21.0757	0.877393	0.438697	−	0.898635i	$-0.355440\pi$
$577$	21.0757	0.877393	0.438697	+	0.898635i	$0.355440\pi$
$578$	3.58119	0.148958
$579$	21.6723	0.900669
$580$	0	0
$581$	−3.36397	−0.139561
$582$	−24.1123	−0.999489
$583$	24.5469	1.01663
$584$	15.4053	0.637474
$585$	0	0
$586$	14.6898	0.606828
$587$	4.31958	0.178288	0.0891442	−	0.996019i	$-0.471587\pi$
$587$	4.31958	0.178288	0.0891442	+	0.996019i	$0.471587\pi$
$588$	−3.75819	−0.154985
$589$	1.14638	0.0472356
$590$	0	0
$591$	42.8271	1.76167
$592$	12.6542	0.520086
$593$	−36.5048	−1.49907	−0.749536	−	0.661963i	$-0.769723\pi$
$593$	−36.5048	−1.49907	−0.749536	+	0.661963i	$0.769723\pi$
$594$	−1.36350	−0.0559451
$595$	0	0
$596$	0.430499	0.0176339
$597$	−0.243675	−0.00997294
$598$	−0.230744	−0.00943582
$599$	5.43768	0.222178	0.111089	−	0.993810i	$-0.464566\pi$
$599$	5.43768	0.222178	0.111089	+	0.993810i	$0.464566\pi$
$600$	0	0
$601$	−22.1151	−0.902093	−0.451046	−	0.892500i	$-0.648949\pi$
$601$	−22.1151	−0.902093	−0.451046	+	0.892500i	$0.648949\pi$
$602$	1.80561	0.0735911
$603$	−4.38757	−0.178676
$604$	18.8081	0.765292
$605$	0	0
$606$	22.2265	0.902889
$607$	27.0386	1.09746	0.548731	−	0.835999i	$-0.315111\pi$
$607$	27.0386	1.09746	0.548731	+	0.835999i	$0.315111\pi$
$608$	−23.8233	−0.966163
$609$	3.11945	0.126406
$610$	0	0
$611$	−1.22652	−0.0496198
$612$	−20.1224	−0.813400
$613$	−39.7891	−1.60707	−0.803534	−	0.595259i	$-0.797050\pi$
$613$	−39.7891	−1.60707	−0.803534	+	0.595259i	$0.797050\pi$
$614$	−9.71340	−0.392001
$615$	0	0
$616$	6.67353	0.268884
$617$	−24.0782	−0.969350	−0.484675	−	0.874694i	$-0.661062\pi$
$617$	−24.0782	−0.969350	−0.484675	+	0.874694i	$0.661062\pi$
$618$	11.2971	0.454437
$619$	14.5254	0.583824	0.291912	−	0.956445i	$-0.405709\pi$
$619$	14.5254	0.583824	0.291912	+	0.956445i	$0.405709\pi$
$620$	0	0
$621$	−0.730420	−0.0293107
$622$	−12.0072	−0.481447
$623$	13.3680	0.535578
$624$	−1.37704	−0.0551259
$625$	0	0
$626$	−16.7916	−0.671127
$627$	−28.4354	−1.13560
$628$	20.8858	0.833433
$629$	36.8079	1.46763
$630$	0	0
$631$	25.6182	1.01984	0.509922	−	0.860221i	$-0.329674\pi$
$631$	25.6182	1.01984	0.509922	+	0.860221i	$0.329674\pi$
$632$	−8.69963	−0.346053
$633$	−30.8011	−1.22423
$634$	10.4526	0.415124
$635$	0	0
$636$	−32.2180	−1.27753
$637$	−0.353935	−0.0140234
$638$	−2.44037	−0.0966152
$639$	2.03552	0.0805238
$640$	0	0
$641$	0.316131	0.0124864	0.00624322	−	0.999981i	$-0.498013\pi$
$641$	0.316131	0.0124864	0.00624322	+	0.999981i	$0.498013\pi$
$642$	19.8954	0.785211
$643$	4.13925	0.163236	0.0816180	−	0.996664i	$-0.473991\pi$
$643$	4.13925	0.163236	0.0816180	+	0.996664i	$0.473991\pi$
$644$	1.57498	0.0620627
$645$	0	0
$646$	−12.8680	−0.506284
$647$	9.73220	0.382612	0.191306	−	0.981530i	$-0.438728\pi$
$647$	9.73220	0.382612	0.191306	+	0.981530i	$0.438728\pi$
$648$	22.8979	0.899514
$649$	33.6122	1.31939
$650$	0	0
$651$	0.657285	0.0257610
$652$	−30.5472	−1.19632
$653$	24.6537	0.964775	0.482388	−	0.875958i	$-0.339770\pi$
$653$	24.6537	0.964775	0.482388	+	0.875958i	$0.339770\pi$
$654$	−2.16755	−0.0847578
$655$	0	0
$656$	19.2576	0.751884
$657$	−17.8061	−0.694684
$658$	−2.25923	−0.0880738
$659$	−6.51299	−0.253710	−0.126855	−	0.991921i	$-0.540488\pi$
$659$	−6.51299	−0.253710	−0.126855	+	0.991921i	$0.540488\pi$
$660$	0	0
$661$	−27.1145	−1.05463	−0.527317	−	0.849669i	$-0.676802\pi$
$661$	−27.1145	−1.05463	−0.527317	+	0.849669i	$0.676802\pi$
$662$	−12.0632	−0.468849
$663$	−4.00547	−0.155559
$664$	−7.84029	−0.304262
$665$	0	0
$666$	13.6303	0.528162
$667$	−1.30729	−0.0506186
$668$	1.07563	0.0416176
$669$	17.2329	0.666264
$670$	0	0
$671$	23.9240	0.923576
$672$	−13.6593	−0.526919
$673$	−18.6626	−0.719392	−0.359696	−	0.933070i	$-0.617120\pi$
$673$	−18.6626	−0.719392	−0.359696	+	0.933070i	$0.617120\pi$
$674$	−8.51778	−0.328093
$675$	0	0
$676$	20.2774	0.779899
$677$	8.11307	0.311811	0.155905	−	0.987772i	$-0.450171\pi$
$677$	8.11307	0.311811	0.155905	+	0.987772i	$0.450171\pi$
$678$	−15.5092	−0.595627
$679$	15.4998	0.594830
$680$	0	0
$681$	−12.6536	−0.484886
$682$	−0.514200	−0.0196897
$683$	25.2742	0.967090	0.483545	−	0.875320i	$-0.339349\pi$
$683$	25.2742	0.967090	0.483545	+	0.875320i	$0.339349\pi$
$684$	17.6576	0.675157
$685$	0	0
$686$	−0.651939	−0.0248912
$687$	43.9677	1.67747
$688$	−4.51582	−0.172164
$689$	−3.03419	−0.115594
$690$	0	0
$691$	−8.02172	−0.305161	−0.152580	−	0.988291i	$-0.548758\pi$
$691$	−8.02172	−0.305161	−0.152580	+	0.988291i	$0.548758\pi$
$692$	−19.3149	−0.734241
$693$	−7.71359	−0.293015
$694$	4.74658	0.180178
$695$	0	0
$696$	7.27040	0.275584
$697$	56.0154	2.12173
$698$	−10.2722	−0.388809
$699$	−14.4141	−0.545191
$700$	0	0
$701$	12.6848	0.479098	0.239549	−	0.970884i	$-0.423000\pi$
$701$	12.6848	0.479098	0.239549	+	0.970884i	$0.423000\pi$
$702$	0.168540	0.00636113
$703$	−32.2994	−1.21819
$704$	1.34839	0.0508195
$705$	0	0
$706$	−8.76382	−0.329831
$707$	−14.2876	−0.537340
$708$	−44.1163	−1.65799
$709$	37.3305	1.40198	0.700989	−	0.713172i	$-0.252742\pi$
$709$	37.3305	1.40198	0.700989	+	0.713172i	$0.252742\pi$
$710$	0	0
$711$	10.0555	0.377109
$712$	31.1564	1.16763
$713$	−0.275454	−0.0103158
$714$	−7.37797	−0.276114
$715$	0	0
$716$	−31.6590	−1.18315
$717$	2.27976	0.0851392
$718$	1.93591	0.0722476
$719$	47.5659	1.77391	0.886954	−	0.461859i	$-0.152817\pi$
$719$	47.5659	1.77391	0.886954	+	0.461859i	$0.152817\pi$
$720$	0	0
$721$	−7.26200	−0.270451
$722$	−1.09504	−0.0407531
$723$	2.33047	0.0866712
$724$	15.0374	0.558862
$725$	0	0
$726$	−4.35766	−0.161728
$727$	42.7962	1.58722	0.793611	−	0.608425i	$-0.208198\pi$
$727$	42.7962	1.58722	0.793611	+	0.608425i	$0.208198\pi$
$728$	−0.824904	−0.0305730
$729$	−21.2377	−0.786582
$730$	0	0
$731$	−13.1353	−0.485828
$732$	−31.4005	−1.16060
$733$	39.5165	1.45957	0.729787	−	0.683674i	$-0.239619\pi$
$733$	39.5165	1.45957	0.729787	+	0.683674i	$0.239619\pi$
$734$	5.63611	0.208033
$735$	0	0
$736$	5.72432	0.211001
$737$	−4.66358	−0.171785
$738$	20.7430	0.763559
$739$	−36.9529	−1.35933	−0.679667	−	0.733520i	$-0.737876\pi$
$739$	−36.9529	−1.35933	−0.679667	+	0.733520i	$0.737876\pi$
$740$	0	0
$741$	3.51485	0.129121
$742$	−5.58891	−0.205175
$743$	18.7266	0.687011	0.343506	−	0.939151i	$-0.388386\pi$
$743$	18.7266	0.687011	0.343506	+	0.939151i	$0.388386\pi$
$744$	1.53191	0.0561627
$745$	0	0
$746$	−20.7459	−0.759562
$747$	9.06219	0.331568
$748$	−21.3882	−0.782030
$749$	−12.7892	−0.467305
$750$	0	0
$751$	−41.0833	−1.49915	−0.749575	−	0.661919i	$-0.769742\pi$
$751$	−41.0833	−1.49915	−0.749575	+	0.661919i	$0.769742\pi$
$752$	5.65031	0.206046
$753$	20.5827	0.750074
$754$	0.301650	0.0109855
$755$	0	0
$756$	−1.15039	−0.0418394
$757$	−6.31603	−0.229560	−0.114780	−	0.993391i	$-0.536616\pi$
$757$	−6.31603	−0.229560	−0.114780	+	0.993391i	$0.536616\pi$
$758$	17.0893	0.620710
$759$	6.83251	0.248005
$760$	0	0
$761$	34.9208	1.26588	0.632939	−	0.774202i	$-0.281849\pi$
$761$	34.9208	1.26588	0.632939	+	0.774202i	$0.281849\pi$
$762$	−6.66717	−0.241526
$763$	1.39334	0.0504423
$764$	21.4522	0.776115
$765$	0	0
$766$	10.9682	0.396297
$767$	−4.15474	−0.150019
$768$	−19.5799	−0.706527
$769$	−14.5251	−0.523789	−0.261894	−	0.965097i	$-0.584347\pi$
$769$	−14.5251	−0.523789	−0.261894	+	0.965097i	$0.584347\pi$
$770$	0	0
$771$	−54.7626	−1.97223
$772$	−14.3045	−0.514831
$773$	−26.1385	−0.940137	−0.470069	−	0.882630i	$-0.655771\pi$
$773$	−26.1385	−0.940137	−0.470069	+	0.882630i	$0.655771\pi$
$774$	−4.86412	−0.174837
$775$	0	0
$776$	36.1250	1.29681
$777$	−18.5191	−0.664370
$778$	6.44633	0.231112
$779$	−49.1542	−1.76113
$780$	0	0
$781$	2.16356	0.0774184
$782$	3.09195	0.110568
$783$	0.954873	0.0341244
$784$	1.63050	0.0582320
$785$	0	0
$786$	15.3992	0.549273
$787$	4.06084	0.144753	0.0723767	−	0.997377i	$-0.476942\pi$
$787$	4.06084	0.144753	0.0723767	+	0.997377i	$0.476942\pi$
$788$	−28.2675	−1.00699
$789$	70.8026	2.52064
$790$	0	0
$791$	9.96960	0.354478
$792$	−17.9778	−0.638814
$793$	−2.95721	−0.105013
$794$	8.27028	0.293501
$795$	0	0
$796$	0.160835	0.00570063
$797$	11.5894	0.410518	0.205259	−	0.978708i	$-0.434196\pi$
$797$	11.5894	0.410518	0.205259	+	0.978708i	$0.434196\pi$
$798$	6.47426	0.229186
$799$	16.4353	0.581439
$800$	0	0
$801$	−36.0120	−1.27242
$802$	19.7916	0.698864
$803$	−18.9262	−0.667893
$804$	6.12099	0.215871
$805$	0	0
$806$	0.0635594	0.00223878
$807$	−61.8154	−2.17601
$808$	−33.2996	−1.17148
$809$	−43.5663	−1.53171	−0.765855	−	0.643014i	$-0.777684\pi$
$809$	−43.5663	−1.53171	−0.765855	+	0.643014i	$0.777684\pi$
$810$	0	0
$811$	−17.2940	−0.607274	−0.303637	−	0.952788i	$-0.598201\pi$
$811$	−17.2940	−0.607274	−0.303637	+	0.952788i	$0.598201\pi$
$812$	−2.05896	−0.0722552
$813$	−51.1544	−1.79406
$814$	14.4877	0.507793
$815$	0	0
$816$	18.4523	0.645959
$817$	11.5264	0.403258
$818$	−22.4964	−0.786570
$819$	0.953464	0.0333167
$820$	0	0
$821$	6.83003	0.238370	0.119185	−	0.992872i	$-0.461972\pi$
$821$	6.83003	0.238370	0.119185	+	0.992872i	$0.461972\pi$
$822$	11.8034	0.411691
$823$	36.1463	1.25998	0.629990	−	0.776604i	$-0.283059\pi$
$823$	36.1463	1.25998	0.629990	+	0.776604i	$0.283059\pi$
$824$	−16.9253	−0.589621
$825$	0	0
$826$	−7.65294	−0.266280
$827$	−2.07579	−0.0721824	−0.0360912	−	0.999348i	$-0.511491\pi$
$827$	−2.07579	−0.0721824	−0.0360912	+	0.999348i	$0.511491\pi$
$828$	−4.24282	−0.147448
$829$	14.9828	0.520375	0.260188	−	0.965558i	$-0.416216\pi$
$829$	14.9828	0.520375	0.260188	+	0.965558i	$0.416216\pi$
$830$	0	0
$831$	−13.0538	−0.452831
$832$	−0.166673	−0.00577834
$833$	4.74269	0.164325
$834$	13.2856	0.460043
$835$	0	0
$836$	18.7684	0.649119
$837$	0.201197	0.00695439
$838$	−11.7096	−0.404502
$839$	−11.1509	−0.384970	−0.192485	−	0.981300i	$-0.561655\pi$
$839$	−11.1509	−0.384970	−0.192485	+	0.981300i	$0.561655\pi$
$840$	0	0
$841$	−27.2910	−0.941068
$842$	3.37832	0.116425
$843$	37.5624	1.29372
$844$	20.3299	0.699783
$845$	0	0
$846$	6.08612	0.209245
$847$	2.80118	0.0962497
$848$	13.9778	0.480001
$849$	41.7157	1.43168
$850$	0	0
$851$	7.76097	0.266043
$852$	−2.83970	−0.0972865
$853$	49.4279	1.69238	0.846189	−	0.532883i	$-0.178891\pi$
$853$	49.4279	1.69238	0.846189	+	0.532883i	$0.178891\pi$
$854$	−5.44710	−0.186396
$855$	0	0
$856$	−29.8072	−1.01879
$857$	−40.5065	−1.38368	−0.691838	−	0.722053i	$-0.743199\pi$
$857$	−40.5065	−1.38368	−0.691838	+	0.722053i	$0.743199\pi$
$858$	−1.57656	−0.0538229
$859$	39.8116	1.35836	0.679178	−	0.733974i	$-0.262336\pi$
$859$	39.8116	1.35836	0.679178	+	0.733974i	$0.262336\pi$
$860$	0	0
$861$	−28.1830	−0.960474
$862$	−20.1202	−0.685296
$863$	16.9478	0.576911	0.288455	−	0.957493i	$-0.406858\pi$
$863$	16.9478	0.576911	0.288455	+	0.957493i	$0.406858\pi$
$864$	−4.18116	−0.142246
$865$	0	0
$866$	23.9767	0.814762
$867$	13.1077	0.445159
$868$	−0.433833	−0.0147253
$869$	10.6880	0.362566
$870$	0	0
$871$	0.576457	0.0195325
$872$	3.24741	0.109971
$873$	−41.7550	−1.41319
$874$	−2.71322	−0.0917761
$875$	0	0
$876$	24.8409	0.839296
$877$	28.8892	0.975519	0.487759	−	0.872978i	$-0.337814\pi$
$877$	28.8892	0.975519	0.487759	+	0.872978i	$0.337814\pi$
$878$	−11.3133	−0.381805
$879$	53.7666	1.81350
$880$	0	0
$881$	8.86183	0.298563	0.149281	−	0.988795i	$-0.452304\pi$
$881$	8.86183	0.298563	0.149281	+	0.988795i	$0.452304\pi$
$882$	1.75626	0.0591363
$883$	−51.1548	−1.72149	−0.860747	−	0.509032i	$-0.830003\pi$
$883$	−51.1548	−1.72149	−0.860747	+	0.509032i	$0.830003\pi$
$884$	2.64376	0.0889193
$885$	0	0
$886$	−13.3111	−0.447195
$887$	−0.577892	−0.0194037	−0.00970186	−	0.999953i	$-0.503088\pi$
$887$	−0.577892	−0.0194037	−0.00970186	+	0.999953i	$0.503088\pi$
$888$	−43.1619	−1.44842
$889$	4.28578	0.143740
$890$	0	0
$891$	−28.1314	−0.942437
$892$	−11.3744	−0.380843
$893$	−14.4222	−0.482620
$894$	−0.425217	−0.0142214
$895$	0	0
$896$	11.1416	0.372216
$897$	−0.844555	−0.0281989
$898$	−16.8076	−0.560876
$899$	0.360099	0.0120100
$900$	0	0
$901$	40.6579	1.35451
$902$	22.0478	0.734111
$903$	6.60877	0.219926
$904$	23.2358	0.772812
$905$	0	0
$906$	−18.5774	−0.617192
$907$	19.1910	0.637228	0.318614	−	0.947885i	$-0.396783\pi$
$907$	19.1910	0.637228	0.318614	+	0.947885i	$0.396783\pi$
$908$	8.35185	0.277166
$909$	38.4893	1.27661
$910$	0	0
$911$	18.0336	0.597481	0.298740	−	0.954334i	$-0.403434\pi$
$911$	18.0336	0.597481	0.298740	+	0.954334i	$0.403434\pi$
$912$	−16.1921	−0.536174
$913$	9.63225	0.318781
$914$	−16.0093	−0.529540
$915$	0	0
$916$	−29.0203	−0.958859
$917$	−9.89892	−0.326891
$918$	−2.25842	−0.0745390
$919$	−32.7118	−1.07906	−0.539531	−	0.841966i	$-0.681399\pi$
$919$	−32.7118	−1.07906	−0.539531	+	0.841966i	$0.681399\pi$
$920$	0	0
$921$	−35.5524	−1.17149
$922$	−8.44008	−0.277959
$923$	−0.267434	−0.00880271
$924$	10.7610	0.354012
$925$	0	0
$926$	−10.2812	−0.337861
$927$	19.5631	0.642536
$928$	−7.48337	−0.245654
$929$	−21.5139	−0.705850	−0.352925	−	0.935652i	$-0.614813\pi$
$929$	−21.5139	−0.705850	−0.352925	+	0.935652i	$0.614813\pi$
$930$	0	0
$931$	−4.16177	−0.136397
$932$	9.51384	0.311636
$933$	−43.9482	−1.43880
$934$	−0.0284927	−0.000932309 0
$935$	0	0
$936$	2.22221	0.0726351
$937$	15.2891	0.499473	0.249736	−	0.968314i	$-0.419656\pi$
$937$	15.2891	0.499473	0.249736	+	0.968314i	$0.419656\pi$
$938$	1.06182	0.0346696
$939$	−61.4596	−2.00566
$940$	0	0
$941$	17.7826	0.579695	0.289847	−	0.957073i	$-0.406395\pi$
$941$	17.7826	0.579695	0.289847	+	0.957073i	$0.406395\pi$
$942$	−20.6295	−0.672146
$943$	11.8109	0.384615
$944$	19.1400	0.622953
$945$	0	0
$946$	−5.17010	−0.168094
$947$	15.5324	0.504737	0.252368	−	0.967631i	$-0.418791\pi$
$947$	15.5324	0.504737	0.252368	+	0.967631i	$0.418791\pi$
$948$	−14.0281	−0.455612
$949$	2.33944	0.0759415
$950$	0	0
$951$	38.2578	1.24060
$952$	11.0536	0.358250
$953$	46.1468	1.49484	0.747420	−	0.664352i	$-0.231292\pi$
$953$	46.1468	1.49484	0.747420	+	0.664352i	$0.231292\pi$
$954$	15.0560	0.487455
$955$	0	0
$956$	−1.50473	−0.0486664
$957$	−8.93210	−0.288734
$958$	−20.2126	−0.653038
$959$	−7.58744	−0.245011
$960$	0	0
$961$	−30.9241	−0.997552
$962$	−1.79080	−0.0577376
$963$	34.4527	1.11022
$964$	−1.53820	−0.0495421
$965$	0	0
$966$	−1.55565	−0.0500522
$967$	42.1181	1.35443	0.677213	−	0.735787i	$-0.263187\pi$
$967$	42.1181	1.35443	0.677213	+	0.735787i	$0.263187\pi$
$968$	6.52862	0.209838
$969$	−47.0986	−1.51302
$970$	0	0
$971$	−30.5447	−0.980227	−0.490113	−	0.871659i	$-0.663045\pi$
$971$	−30.5447	−0.980227	−0.490113	+	0.871659i	$0.663045\pi$
$972$	33.4716	1.07360
$973$	−8.54023	−0.273787
$974$	19.6250	0.628827
$975$	0	0
$976$	13.6232	0.436067
$977$	12.7366	0.407479	0.203740	−	0.979025i	$-0.434690\pi$
$977$	12.7366	0.407479	0.203740	+	0.979025i	$0.434690\pi$
$978$	30.1724	0.964807
$979$	−38.2774	−1.22335
$980$	0	0
$981$	−3.75351	−0.119840
$982$	−10.7035	−0.341562
$983$	−37.3247	−1.19047	−0.595237	−	0.803551i	$-0.702942\pi$
$983$	−37.3247	−1.19047	−0.595237	+	0.803551i	$0.702942\pi$
$984$	−65.6852	−2.09397
$985$	0	0
$986$	−4.04208	−0.128726
$987$	−8.26908	−0.263208
$988$	−2.31993	−0.0738069
$989$	−2.76959	−0.0880680
$990$	0	0
$991$	−21.6356	−0.687277	−0.343638	−	0.939102i	$-0.611659\pi$
$991$	−21.6356	−0.687277	−0.343638	+	0.939102i	$0.611659\pi$
$992$	−1.57679	−0.0500630
$993$	−44.1529	−1.40115
$994$	−0.492607	−0.0156246
$995$	0	0
$996$	−12.6424	−0.400591
$997$	−30.2771	−0.958885	−0.479443	−	0.877573i	$-0.659161\pi$
$997$	−30.2771	−0.958885	−0.479443	+	0.877573i	$0.659161\pi$
$998$	−19.6519	−0.622069
$999$	−5.66877	−0.179352

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.v.1.5	yes	8
5.4	even	2		4025.2.a.u.1.4	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4025.2.a.u.1.4	✓	8	5.4	even	2
4025.2.a.v.1.5	yes	8	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:	\( N \)	\(=\)	\( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.651939 q^{2} +2.38619 q^{3} -1.57498 q^{4} +1.55565 q^{6} -1.00000 q^{7} -2.33067 q^{8} +2.69390 q^{9} +O(q^{10})\)	\(q+0.651939 q^{2} +2.38619 q^{3} -1.57498 q^{4} +1.55565 q^{6} -1.00000 q^{7} -2.33067 q^{8} +2.69390 q^{9} +2.86336 q^{11} -3.75819 q^{12} -0.353935 q^{13} -0.651939 q^{14} +1.63050 q^{16} +4.74269 q^{17} +1.75626 q^{18} -4.16177 q^{19} -2.38619 q^{21} +1.86674 q^{22} +1.00000 q^{23} -5.56141 q^{24} -0.230744 q^{26} -0.730420 q^{27} +1.57498 q^{28} -1.30729 q^{29} -0.275454 q^{31} +5.72432 q^{32} +6.83251 q^{33} +3.09195 q^{34} -4.24282 q^{36} +7.76097 q^{37} -2.71322 q^{38} -0.844555 q^{39} +11.8109 q^{41} -1.55565 q^{42} -2.76959 q^{43} -4.50972 q^{44} +0.651939 q^{46} +3.46539 q^{47} +3.89067 q^{48} +1.00000 q^{49} +11.3170 q^{51} +0.557439 q^{52} +8.57275 q^{53} -0.476189 q^{54} +2.33067 q^{56} -9.93077 q^{57} -0.852276 q^{58} +11.7387 q^{59} +8.35523 q^{61} -0.179579 q^{62} -2.69390 q^{63} +0.470914 q^{64} +4.45438 q^{66} -1.62871 q^{67} -7.46962 q^{68} +2.38619 q^{69} +0.755603 q^{71} -6.27858 q^{72} -6.60981 q^{73} +5.05968 q^{74} +6.55469 q^{76} -2.86336 q^{77} -0.550599 q^{78} +3.73268 q^{79} -9.82461 q^{81} +7.69998 q^{82} +3.36397 q^{83} +3.75819 q^{84} -1.80561 q^{86} -3.11945 q^{87} -6.67353 q^{88} -13.3680 q^{89} +0.353935 q^{91} -1.57498 q^{92} -0.657285 q^{93} +2.25923 q^{94} +13.6593 q^{96} -15.4998 q^{97} +0.651939 q^{98} +7.71359 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7}+O(q^{10}) \) 8 * q + q^2 + 4 * q^3 + 5 * q^4 - q^6 - 8 * q^7	\( 8 q + q^{2} + 4 q^{3} + 5 q^{4} - q^{6} - 8 q^{7} + 3 q^{11} + 9 q^{12} + 5 q^{13} - q^{14} - q^{16} + 5 q^{17} + 2 q^{18} - 2 q^{19} - 4 q^{21} + 21 q^{22} + 8 q^{23} - 6 q^{24} + 18 q^{26} + 7 q^{27} - 5 q^{28} - 9 q^{29} - 3 q^{31} + 6 q^{32} + 4 q^{33} - 10 q^{34} + 16 q^{36} + 6 q^{37} + 4 q^{38} - 2 q^{39} - 7 q^{41} + q^{42} + 8 q^{43} + 4 q^{44} + q^{46} + 22 q^{47} + 9 q^{48} + 8 q^{49} - 12 q^{51} + 11 q^{52} + 21 q^{53} - 15 q^{54} + 8 q^{57} + 16 q^{58} + 14 q^{59} + 8 q^{61} + 12 q^{62} - 40 q^{64} + 55 q^{66} + 21 q^{67} + 3 q^{68} + 4 q^{69} + 11 q^{71} - q^{72} + 26 q^{73} - 41 q^{74} + 21 q^{76} - 3 q^{77} + 17 q^{78} - 16 q^{79} - 20 q^{81} - q^{82} + 20 q^{83} - 9 q^{84} + 14 q^{86} - 29 q^{87} + 32 q^{88} + 15 q^{89} - 5 q^{91} + 5 q^{92} + 19 q^{93} + 21 q^{94} + 52 q^{96} + q^{97} + q^{98} + 15 q^{99}+O(q^{100}) \) 8 * q + q^2 + 4 * q^3 + 5 * q^4 - q^6 - 8 * q^7 + 3 * q^11 + 9 * q^12 + 5 * q^13 - q^14 - q^16 + 5 * q^17 + 2 * q^18 - 2 * q^19 - 4 * q^21 + 21 * q^22 + 8 * q^23 - 6 * q^24 + 18 * q^26 + 7 * q^27 - 5 * q^28 - 9 * q^29 - 3 * q^31 + 6 * q^32 + 4 * q^33 - 10 * q^34 + 16 * q^36 + 6 * q^37 + 4 * q^38 - 2 * q^39 - 7 * q^41 + q^42 + 8 * q^43 + 4 * q^44 + q^46 + 22 * q^47 + 9 * q^48 + 8 * q^49 - 12 * q^51 + 11 * q^52 + 21 * q^53 - 15 * q^54 + 8 * q^57 + 16 * q^58 + 14 * q^59 + 8 * q^61 + 12 * q^62 - 40 * q^64 + 55 * q^66 + 21 * q^67 + 3 * q^68 + 4 * q^69 + 11 * q^71 - q^72 + 26 * q^73 - 41 * q^74 + 21 * q^76 - 3 * q^77 + 17 * q^78 - 16 * q^79 - 20 * q^81 - q^82 + 20 * q^83 - 9 * q^84 + 14 * q^86 - 29 * q^87 + 32 * q^88 + 15 * q^89 - 5 * q^91 + 5 * q^92 + 19 * q^93 + 21 * q^94 + 52 * q^96 + q^97 + q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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