LMFDB - Embedded newform 4025.2.a.u.1.7

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:	$8$
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			1.7
Root			$-2.07739$ 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)

$f(q)$	$=$	$q+2.07739 q^{2} +0.298210 q^{3} +2.31556 q^{4} +0.619500 q^{6} +1.00000 q^{7} +0.655538 q^{8} -2.91107 q^{9} +O(q^{10})$	\(q+2.07739 q^{2} +0.298210 q^{3} +2.31556 q^{4} +0.619500 q^{6} +1.00000 q^{7} +0.655538 q^{8} -2.91107 q^{9} -0.993361 q^{11} +0.690523 q^{12} +1.85378 q^{13} +2.07739 q^{14} -3.26931 q^{16} -7.16751 q^{17} -6.04744 q^{18} -2.81417 q^{19} +0.298210 q^{21} -2.06360 q^{22} -1.00000 q^{23} +0.195488 q^{24} +3.85103 q^{26} -1.76274 q^{27} +2.31556 q^{28} -4.82054 q^{29} -1.53913 q^{31} -8.10271 q^{32} -0.296230 q^{33} -14.8897 q^{34} -6.74075 q^{36} -10.5690 q^{37} -5.84613 q^{38} +0.552816 q^{39} -9.37957 q^{41} +0.619500 q^{42} +12.5692 q^{43} -2.30018 q^{44} -2.07739 q^{46} +3.48711 q^{47} -0.974941 q^{48} +1.00000 q^{49} -2.13742 q^{51} +4.29253 q^{52} +6.38141 q^{53} -3.66191 q^{54} +0.655538 q^{56} -0.839214 q^{57} -10.0141 q^{58} +7.66700 q^{59} -0.599994 q^{61} -3.19737 q^{62} -2.91107 q^{63} -10.2939 q^{64} -0.615387 q^{66} +0.642727 q^{67} -16.5968 q^{68} -0.298210 q^{69} +10.5715 q^{71} -1.90832 q^{72} -1.17644 q^{73} -21.9559 q^{74} -6.51637 q^{76} -0.993361 q^{77} +1.14842 q^{78} +2.91935 q^{79} +8.20754 q^{81} -19.4850 q^{82} +5.83299 q^{83} +0.690523 q^{84} +26.1112 q^{86} -1.43753 q^{87} -0.651186 q^{88} +4.38245 q^{89} +1.85378 q^{91} -2.31556 q^{92} -0.458984 q^{93} +7.24410 q^{94} -2.41631 q^{96} +1.70164 q^{97} +2.07739 q^{98} +2.89174 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.07739	1.46894	0.734469	−	0.678642i	$-0.237431\pi$
$2$	2.07739	1.46894	0.734469	+	0.678642i	$0.237431\pi$
$3$	0.298210	0.172172	0.0860859	−	0.996288i	$-0.472564\pi$
$3$	0.298210	0.172172	0.0860859	+	0.996288i	$0.472564\pi$
$4$	2.31556	1.15778
$5$	0	0
$5$	0	0
$6$	0.619500	0.252910
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.655538	0.231768
$9$	−2.91107	−0.970357
$10$	0	0
$11$	−0.993361	−0.299509	−0.149755	−	0.988723i	$-0.547848\pi$
$11$	−0.993361	−0.299509	−0.149755	+	0.988723i	$0.547848\pi$
$12$	0.690523	0.199337
$13$	1.85378	0.514146	0.257073	−	0.966392i	$-0.417242\pi$
$13$	1.85378	0.514146	0.257073	+	0.966392i	$0.417242\pi$
$14$	2.07739	0.555206
$15$	0	0
$16$	−3.26931	−0.817327
$17$	−7.16751	−1.73838	−0.869188	−	0.494482i	$-0.835358\pi$
$17$	−7.16751	−1.73838	−0.869188	+	0.494482i	$0.835358\pi$
$18$	−6.04744	−1.42539
$19$	−2.81417	−0.645614	−0.322807	−	0.946465i	$-0.604627\pi$
$19$	−2.81417	−0.645614	−0.322807	+	0.946465i	$0.604627\pi$
$20$	0	0
$21$	0.298210	0.0650748
$22$	−2.06360	−0.439961
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0.195488	0.0399039
$25$	0	0
$26$	3.85103	0.755248
$27$	−1.76274	−0.339240
$28$	2.31556	0.437599
$29$	−4.82054	−0.895151	−0.447575	−	0.894246i	$-0.647712\pi$
$29$	−4.82054	−0.895151	−0.447575	+	0.894246i	$0.647712\pi$
$30$	0	0
$31$	−1.53913	−0.276436	−0.138218	−	0.990402i	$-0.544137\pi$
$31$	−1.53913	−0.276436	−0.138218	+	0.990402i	$0.544137\pi$
$32$	−8.10271	−1.43237
$33$	−0.296230	−0.0515671
$34$	−14.8897	−2.55357
$35$	0	0
$36$	−6.74075	−1.12346
$37$	−10.5690	−1.73753	−0.868765	−	0.495225i	$-0.835086\pi$
$37$	−10.5690	−1.73753	−0.868765	+	0.495225i	$0.835086\pi$
$38$	−5.84613	−0.948367
$39$	0.552816	0.0885214
$40$	0	0
$41$	−9.37957	−1.46484	−0.732421	−	0.680852i	$-0.761610\pi$
$41$	−9.37957	−1.46484	−0.732421	+	0.680852i	$0.761610\pi$
$42$	0.619500	0.0955909
$43$	12.5692	1.91679	0.958393	−	0.285453i	$-0.0921440\pi$
$43$	12.5692	1.91679	0.958393	+	0.285453i	$0.0921440\pi$
$44$	−2.30018	−0.346766
$45$	0	0
$46$	−2.07739	−0.306295
$47$	3.48711	0.508648	0.254324	−	0.967119i	$-0.418147\pi$
$47$	3.48711	0.508648	0.254324	+	0.967119i	$0.418147\pi$
$48$	−0.974941	−0.140721
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.13742	−0.299299
$52$	4.29253	0.595267
$53$	6.38141	0.876555	0.438277	−	0.898840i	$-0.355589\pi$
$53$	6.38141	0.876555	0.438277	+	0.898840i	$0.355589\pi$
$54$	−3.66191	−0.498322
$55$	0	0
$56$	0.655538	0.0876000
$57$	−0.839214	−0.111157
$58$	−10.0141	−1.31492
$59$	7.66700	0.998159	0.499079	−	0.866556i	$-0.333672\pi$
$59$	7.66700	0.998159	0.499079	+	0.866556i	$0.333672\pi$
$60$	0	0
$61$	−0.599994	−0.0768214	−0.0384107	−	0.999262i	$-0.512230\pi$
$61$	−0.599994	−0.0768214	−0.0384107	+	0.999262i	$0.512230\pi$
$62$	−3.19737	−0.406067
$63$	−2.91107	−0.366760
$64$	−10.2939	−1.28674
$65$	0	0
$66$	−0.615387	−0.0757489
$67$	0.642727	0.0785216	0.0392608	−	0.999229i	$-0.487500\pi$
$67$	0.642727	0.0785216	0.0392608	+	0.999229i	$0.487500\pi$
$68$	−16.5968	−2.01265
$69$	−0.298210	−0.0359003
$70$	0	0
$71$	10.5715	1.25461	0.627304	−	0.778775i	$-0.284158\pi$
$71$	10.5715	1.25461	0.627304	+	0.778775i	$0.284158\pi$
$72$	−1.90832	−0.224897
$73$	−1.17644	−0.137692	−0.0688460	−	0.997627i	$-0.521932\pi$
$73$	−1.17644	−0.137692	−0.0688460	+	0.997627i	$0.521932\pi$
$74$	−21.9559	−2.55232
$75$	0	0
$76$	−6.51637	−0.747479
$77$	−0.993361	−0.113204
$78$	1.14842	0.130032
$79$	2.91935	0.328453	0.164226	−	0.986423i	$-0.447487\pi$
$79$	2.91935	0.328453	0.164226	+	0.986423i	$0.447487\pi$
$80$	0	0
$81$	8.20754	0.911949
$82$	−19.4850	−2.15176
$83$	5.83299	0.640253	0.320127	−	0.947375i	$-0.396275\pi$
$83$	5.83299	0.640253	0.320127	+	0.947375i	$0.396275\pi$
$84$	0.690523	0.0753423
$85$	0	0
$86$	26.1112	2.81564
$87$	−1.43753	−0.154120
$88$	−0.651186	−0.0694166
$89$	4.38245	0.464539	0.232269	−	0.972651i	$-0.425385\pi$
$89$	4.38245	0.464539	0.232269	+	0.972651i	$0.425385\pi$
$90$	0	0
$91$	1.85378	0.194329
$92$	−2.31556	−0.241414
$93$	−0.458984	−0.0475944
$94$	7.24410	0.747172
$95$	0	0
$96$	−2.41631	−0.246614
$97$	1.70164	0.172775	0.0863876	−	0.996262i	$-0.472468\pi$
$97$	1.70164	0.172775	0.0863876	+	0.996262i	$0.472468\pi$
$98$	2.07739	0.209848
$99$	2.89174	0.290631
$100$	0	0
$101$	−17.7515	−1.76634	−0.883169	−	0.469055i	$-0.844594\pi$
$101$	−17.7515	−1.76634	−0.883169	+	0.469055i	$0.844594\pi$
$102$	−4.44027	−0.439652
$103$	−10.2506	−1.01002	−0.505010	−	0.863114i	$-0.668511\pi$
$103$	−10.2506	−1.01002	−0.505010	+	0.863114i	$0.668511\pi$
$104$	1.21522	0.119162
$105$	0	0
$106$	13.2567	1.28760
$107$	−8.58885	−0.830316	−0.415158	−	0.909749i	$-0.636274\pi$
$107$	−8.58885	−0.830316	−0.415158	+	0.909749i	$0.636274\pi$
$108$	−4.08173	−0.392765
$109$	−19.4159	−1.85970	−0.929852	−	0.367935i	$-0.880065\pi$
$109$	−19.4159	−1.85970	−0.929852	+	0.367935i	$0.880065\pi$
$110$	0	0
$111$	−3.15178	−0.299154
$112$	−3.26931	−0.308920
$113$	4.64453	0.436921	0.218460	−	0.975846i	$-0.429897\pi$
$113$	4.64453	0.436921	0.218460	+	0.975846i	$0.429897\pi$
$114$	−1.74338	−0.163282
$115$	0	0
$116$	−11.1622	−1.03639
$117$	−5.39648	−0.498905
$118$	15.9274	1.46623
$119$	−7.16751	−0.657044
$120$	0	0
$121$	−10.0132	−0.910294
$122$	−1.24642	−0.112846
$123$	−2.79708	−0.252205
$124$	−3.56394	−0.320052
$125$	0	0
$126$	−6.04744	−0.538748
$127$	−17.5660	−1.55873	−0.779366	−	0.626569i	$-0.784459\pi$
$127$	−17.5660	−1.55873	−0.779366	+	0.626569i	$0.784459\pi$
$128$	−5.17903	−0.457766
$129$	3.74827	0.330016
$130$	0	0
$131$	13.3723	1.16834	0.584172	−	0.811630i	$-0.301419\pi$
$131$	13.3723	1.16834	0.584172	+	0.811630i	$0.301419\pi$
$132$	−0.685939	−0.0597033
$133$	−2.81417	−0.244019
$134$	1.33520	0.115343
$135$	0	0
$136$	−4.69857	−0.402899
$137$	−5.20003	−0.444268	−0.222134	−	0.975016i	$-0.571302\pi$
$137$	−5.20003	−0.444268	−0.222134	+	0.975016i	$0.571302\pi$
$138$	−0.619500	−0.0527353
$139$	8.63108	0.732079	0.366039	−	0.930599i	$-0.380714\pi$
$139$	8.63108	0.732079	0.366039	+	0.930599i	$0.380714\pi$
$140$	0	0
$141$	1.03989	0.0875748
$142$	21.9612	1.84294
$143$	−1.84147	−0.153992
$144$	9.51718	0.793099
$145$	0	0
$146$	−2.44393	−0.202261
$147$	0.298210	0.0245960
$148$	−24.4731	−2.01168
$149$	3.58784	0.293928	0.146964	−	0.989142i	$-0.453050\pi$
$149$	3.58784	0.293928	0.146964	+	0.989142i	$0.453050\pi$
$150$	0	0
$151$	−4.23764	−0.344854	−0.172427	−	0.985022i	$-0.555161\pi$
$151$	−4.23764	−0.344854	−0.172427	+	0.985022i	$0.555161\pi$
$152$	−1.84479	−0.149633
$153$	20.8651	1.68684
$154$	−2.06360	−0.166290
$155$	0	0
$156$	1.28008	0.102488
$157$	−7.05240	−0.562843	−0.281421	−	0.959584i	$-0.590806\pi$
$157$	−7.05240	−0.562843	−0.281421	+	0.959584i	$0.590806\pi$
$158$	6.06464	0.482477
$159$	1.90300	0.150918
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	17.0503	1.33960
$163$	−19.7321	−1.54554	−0.772769	−	0.634688i	$-0.781129\pi$
$163$	−19.7321	−1.54554	−0.772769	+	0.634688i	$0.781129\pi$
$164$	−21.7189	−1.69596
$165$	0	0
$166$	12.1174	0.940493
$167$	19.0253	1.47222	0.736109	−	0.676863i	$-0.236661\pi$
$167$	19.0253	1.47222	0.736109	+	0.676863i	$0.236661\pi$
$168$	0.195488	0.0150822
$169$	−9.56350	−0.735654
$170$	0	0
$171$	8.19224	0.626476
$172$	29.1047	2.21921
$173$	20.6370	1.56900	0.784500	−	0.620129i	$-0.212920\pi$
$173$	20.6370	1.56900	0.784500	+	0.620129i	$0.212920\pi$
$174$	−2.98632	−0.226392
$175$	0	0
$176$	3.24760	0.244797
$177$	2.28638	0.171855
$178$	9.10407	0.682379
$179$	−14.0344	−1.04898	−0.524490	−	0.851417i	$-0.675744\pi$
$179$	−14.0344	−1.04898	−0.524490	+	0.851417i	$0.675744\pi$
$180$	0	0
$181$	23.3513	1.73569	0.867846	−	0.496833i	$-0.165504\pi$
$181$	23.3513	1.73569	0.867846	+	0.496833i	$0.165504\pi$
$182$	3.85103	0.285457
$183$	−0.178924	−0.0132265
$184$	−0.655538	−0.0483269
$185$	0	0
$186$	−0.953490	−0.0699133
$187$	7.11992	0.520660
$188$	8.07461	0.588902
$189$	−1.76274	−0.128221
$190$	0	0
$191$	13.8019	0.998674	0.499337	−	0.866408i	$-0.333577\pi$
$191$	13.8019	0.998674	0.499337	+	0.866408i	$0.333577\pi$
$192$	−3.06974	−0.221540
$193$	−0.492508	−0.0354515	−0.0177257	−	0.999843i	$-0.505643\pi$
$193$	−0.492508	−0.0354515	−0.0177257	+	0.999843i	$0.505643\pi$
$194$	3.53497	0.253796
$195$	0	0
$196$	2.31556	0.165397
$197$	−19.1684	−1.36569	−0.682846	−	0.730562i	$-0.739258\pi$
$197$	−19.1684	−1.36569	−0.682846	+	0.730562i	$0.739258\pi$
$198$	6.00728	0.426919
$199$	3.57382	0.253342	0.126671	−	0.991945i	$-0.459571\pi$
$199$	3.57382	0.253342	0.126671	+	0.991945i	$0.459571\pi$
$200$	0	0
$201$	0.191668	0.0135192
$202$	−36.8768	−2.59464
$203$	−4.82054	−0.338335
$204$	−4.94933	−0.346522
$205$	0	0
$206$	−21.2945	−1.48366
$207$	2.91107	0.202333
$208$	−6.06057	−0.420225
$209$	2.79548	0.193368
$210$	0	0
$211$	2.46333	0.169583	0.0847914	−	0.996399i	$-0.472978\pi$
$211$	2.46333	0.169583	0.0847914	+	0.996399i	$0.472978\pi$
$212$	14.7765	1.01486
$213$	3.15253	0.216008
$214$	−17.8424	−1.21968
$215$	0	0
$216$	−1.15554	−0.0786249
$217$	−1.53913	−0.104483
$218$	−40.3344	−2.73179
$219$	−0.350827	−0.0237067
$220$	0	0
$221$	−13.2870	−0.893778
$222$	−6.54748	−0.439438
$223$	−4.73704	−0.317215	−0.158608	−	0.987342i	$-0.550700\pi$
$223$	−4.73704	−0.317215	−0.158608	+	0.987342i	$0.550700\pi$
$224$	−8.10271	−0.541385
$225$	0	0
$226$	9.64851	0.641809
$227$	27.2245	1.80696	0.903478	−	0.428635i	$-0.141005\pi$
$227$	27.2245	1.80696	0.903478	+	0.428635i	$0.141005\pi$
$228$	−1.94325	−0.128695
$229$	−24.7985	−1.63873	−0.819366	−	0.573270i	$-0.805675\pi$
$229$	−24.7985	−1.63873	−0.819366	+	0.573270i	$0.805675\pi$
$230$	0	0
$231$	−0.296230	−0.0194905
$232$	−3.16004	−0.207467
$233$	−23.9620	−1.56980	−0.784901	−	0.619622i	$-0.787286\pi$
$233$	−23.9620	−1.56980	−0.784901	+	0.619622i	$0.787286\pi$
$234$	−11.2106	−0.732860
$235$	0	0
$236$	17.7534	1.15565
$237$	0.870581	0.0565503
$238$	−14.8897	−0.965157
$239$	−8.04766	−0.520560	−0.260280	−	0.965533i	$-0.583815\pi$
$239$	−8.04766	−0.520560	−0.260280	+	0.965533i	$0.583815\pi$
$240$	0	0
$241$	9.63222	0.620466	0.310233	−	0.950661i	$-0.399593\pi$
$241$	9.63222	0.620466	0.310233	+	0.950661i	$0.399593\pi$
$242$	−20.8014	−1.33717
$243$	7.73580	0.496252
$244$	−1.38932	−0.0889422
$245$	0	0
$246$	−5.81064	−0.370473
$247$	−5.21684	−0.331940
$248$	−1.00896	−0.0640689
$249$	1.73946	0.110234
$250$	0	0
$251$	−25.1553	−1.58779	−0.793894	−	0.608056i	$-0.791949\pi$
$251$	−25.1553	−1.58779	−0.793894	+	0.608056i	$0.791949\pi$
$252$	−6.74075	−0.424628
$253$	0.993361	0.0624520
$254$	−36.4915	−2.28968
$255$	0	0
$256$	9.82891	0.614307
$257$	15.3143	0.955278	0.477639	−	0.878556i	$-0.341493\pi$
$257$	15.3143	0.955278	0.477639	+	0.878556i	$0.341493\pi$
$258$	7.78662	0.484774
$259$	−10.5690	−0.656725
$260$	0	0
$261$	14.0329	0.868616
$262$	27.7795	1.71622
$263$	8.09808	0.499349	0.249675	−	0.968330i	$-0.419676\pi$
$263$	8.09808	0.499349	0.249675	+	0.968330i	$0.419676\pi$
$264$	−0.194190	−0.0119516
$265$	0	0
$266$	−5.84613	−0.358449
$267$	1.30689	0.0799805
$268$	1.48827	0.0909106
$269$	19.2842	1.17578	0.587888	−	0.808942i	$-0.299959\pi$
$269$	19.2842	1.17578	0.587888	+	0.808942i	$0.299959\pi$
$270$	0	0
$271$	3.10068	0.188353	0.0941763	−	0.995556i	$-0.469978\pi$
$271$	3.10068	0.188353	0.0941763	+	0.995556i	$0.469978\pi$
$272$	23.4328	1.42082
$273$	0.552816	0.0334579
$274$	−10.8025	−0.652603
$275$	0	0
$276$	−0.690523	−0.0415646
$277$	−4.48060	−0.269213	−0.134607	−	0.990899i	$-0.542977\pi$
$277$	−4.48060	−0.269213	−0.134607	+	0.990899i	$0.542977\pi$
$278$	17.9301	1.07538
$279$	4.48051	0.268241
$280$	0	0
$281$	24.4264	1.45716	0.728579	−	0.684962i	$-0.240181\pi$
$281$	24.4264	1.45716	0.728579	+	0.684962i	$0.240181\pi$
$282$	2.16027	0.128642
$283$	−33.3345	−1.98153	−0.990765	−	0.135592i	$-0.956706\pi$
$283$	−33.3345	−1.98153	−0.990765	+	0.135592i	$0.956706\pi$
$284$	24.4789	1.45256
$285$	0	0
$286$	−3.82546	−0.226204
$287$	−9.37957	−0.553658
$288$	23.5876	1.38991
$289$	34.3731	2.02195
$290$	0	0
$291$	0.507446	0.0297470
$292$	−2.72412	−0.159417
$293$	3.55759	0.207837	0.103918	−	0.994586i	$-0.466862\pi$
$293$	3.55759	0.207837	0.103918	+	0.994586i	$0.466862\pi$
$294$	0.619500	0.0361300
$295$	0	0
$296$	−6.92837	−0.402703
$297$	1.75104	0.101606
$298$	7.45336	0.431761
$299$	−1.85378	−0.107207
$300$	0	0
$301$	12.5692	0.724477
$302$	−8.80323	−0.506569
$303$	−5.29367	−0.304114
$304$	9.20037	0.527678
$305$	0	0
$306$	43.3450	2.47787
$307$	−27.6550	−1.57836	−0.789178	−	0.614165i	$-0.789493\pi$
$307$	−27.6550	−1.57836	−0.789178	+	0.614165i	$0.789493\pi$
$308$	−2.30018	−0.131065
$309$	−3.05683	−0.173897
$310$	0	0
$311$	27.8032	1.57657	0.788287	−	0.615308i	$-0.210968\pi$
$311$	27.8032	1.57657	0.788287	+	0.615308i	$0.210968\pi$
$312$	0.362392	0.0205164
$313$	−27.8407	−1.57365	−0.786824	−	0.617178i	$-0.788276\pi$
$313$	−27.8407	−1.57365	−0.786824	+	0.617178i	$0.788276\pi$
$314$	−14.6506	−0.826781
$315$	0	0
$316$	6.75993	0.380276
$317$	14.6354	0.822007	0.411003	−	0.911634i	$-0.365178\pi$
$317$	14.6354	0.822007	0.411003	+	0.911634i	$0.365178\pi$
$318$	3.95328	0.221689
$319$	4.78853	0.268106
$320$	0	0
$321$	−2.56128	−0.142957
$322$	−2.07739	−0.115769
$323$	20.1706	1.12232
$324$	19.0050	1.05584
$325$	0	0
$326$	−40.9913	−2.27030
$327$	−5.79001	−0.320188
$328$	−6.14866	−0.339503
$329$	3.48711	0.192251
$330$	0	0
$331$	11.4857	0.631309	0.315655	−	0.948874i	$-0.397776\pi$
$331$	11.4857	0.631309	0.315655	+	0.948874i	$0.397776\pi$
$332$	13.5066	0.741272
$333$	30.7671	1.68602
$334$	39.5229	2.16260
$335$	0	0
$336$	−0.974941	−0.0531874
$337$	−1.68540	−0.0918098	−0.0459049	−	0.998946i	$-0.514617\pi$
$337$	−1.68540	−0.0918098	−0.0459049	+	0.998946i	$0.514617\pi$
$338$	−19.8671	−1.08063
$339$	1.38505	0.0752254
$340$	0	0
$341$	1.52891	0.0827951
$342$	17.0185	0.920255
$343$	1.00000	0.0539949
$344$	8.23959	0.444249
$345$	0	0
$346$	42.8711	2.30476
$347$	−10.1978	−0.547446	−0.273723	−	0.961809i	$-0.588255\pi$
$347$	−10.1978	−0.547446	−0.273723	+	0.961809i	$0.588255\pi$
$348$	−3.32869	−0.178437
$349$	8.14441	0.435960	0.217980	−	0.975953i	$-0.430053\pi$
$349$	8.14441	0.435960	0.217980	+	0.975953i	$0.430053\pi$
$350$	0	0
$351$	−3.26773	−0.174419
$352$	8.04891	0.429008
$353$	−26.2377	−1.39649	−0.698245	−	0.715859i	$-0.746035\pi$
$353$	−26.2377	−1.39649	−0.698245	+	0.715859i	$0.746035\pi$
$354$	4.74971	0.252444
$355$	0	0
$356$	10.1478	0.537833
$357$	−2.13742	−0.113124
$358$	−29.1549	−1.54089
$359$	0.301901	0.0159337	0.00796687	−	0.999968i	$-0.497464\pi$
$359$	0.301901	0.0159337	0.00796687	+	0.999968i	$0.497464\pi$
$360$	0	0
$361$	−11.0805	−0.583182
$362$	48.5099	2.54962
$363$	−2.98605	−0.156727
$364$	4.29253	0.224990
$365$	0	0
$366$	−0.371696	−0.0194289
$367$	13.2434	0.691299	0.345649	−	0.938364i	$-0.387659\pi$
$367$	13.2434	0.691299	0.345649	+	0.938364i	$0.387659\pi$
$368$	3.26931	0.170424
$369$	27.3046	1.42142
$370$	0	0
$371$	6.38141	0.331306
$372$	−1.06280	−0.0551038
$373$	−18.1485	−0.939694	−0.469847	−	0.882748i	$-0.655691\pi$
$373$	−18.1485	−0.939694	−0.469847	+	0.882748i	$0.655691\pi$
$374$	14.7909	0.764817
$375$	0	0
$376$	2.28594	0.117888
$377$	−8.93621	−0.460238
$378$	−3.66191	−0.188348
$379$	10.4836	0.538508	0.269254	−	0.963069i	$-0.413223\pi$
$379$	10.4836	0.538508	0.269254	+	0.963069i	$0.413223\pi$
$380$	0	0
$381$	−5.23836	−0.268370
$382$	28.6720	1.46699
$383$	−12.0027	−0.613309	−0.306655	−	0.951821i	$-0.599210\pi$
$383$	−12.0027	−0.613309	−0.306655	+	0.951821i	$0.599210\pi$
$384$	−1.54444	−0.0788144
$385$	0	0
$386$	−1.02313	−0.0520760
$387$	−36.5898	−1.85997
$388$	3.94024	0.200036
$389$	33.3345	1.69013	0.845063	−	0.534667i	$-0.179563\pi$
$389$	33.3345	1.69013	0.845063	+	0.534667i	$0.179563\pi$
$390$	0	0
$391$	7.16751	0.362476
$392$	0.655538	0.0331097
$393$	3.98776	0.201156
$394$	−39.8203	−2.00612
$395$	0	0
$396$	6.69600	0.336487
$397$	−26.5014	−1.33007	−0.665035	−	0.746813i	$-0.731583\pi$
$397$	−26.5014	−1.33007	−0.665035	+	0.746813i	$0.731583\pi$
$398$	7.42424	0.372143
$399$	−0.839214	−0.0420132
$400$	0	0
$401$	−24.0298	−1.19999	−0.599994	−	0.800004i	$-0.704830\pi$
$401$	−24.0298	−1.19999	−0.599994	+	0.800004i	$0.704830\pi$
$402$	0.398169	0.0198589
$403$	−2.85321	−0.142128
$404$	−41.1046	−2.04503
$405$	0	0
$406$	−10.0141	−0.496994
$407$	10.4988	0.520407
$408$	−1.40116	−0.0693679
$409$	14.4436	0.714192	0.357096	−	0.934068i	$-0.383767\pi$
$409$	14.4436	0.714192	0.357096	+	0.934068i	$0.383767\pi$
$410$	0	0
$411$	−1.55070	−0.0764905
$412$	−23.7358	−1.16938
$413$	7.66700	0.377269
$414$	6.04744	0.297215
$415$	0	0
$416$	−15.0206	−0.736447
$417$	2.57388	0.126043
$418$	5.80731	0.284045
$419$	−8.42230	−0.411456	−0.205728	−	0.978609i	$-0.565956\pi$
$419$	−8.42230	−0.411456	−0.205728	+	0.978609i	$0.565956\pi$
$420$	0	0
$421$	−0.0309373	−0.00150779	−0.000753895	−	1.00000i	$-0.500240\pi$
$421$	−0.0309373	−0.00150779	−0.000753895		1.00000i	$0.500240\pi$
$422$	5.11731	0.249107
$423$	−10.1512	−0.493570
$424$	4.18326	0.203157
$425$	0	0
$426$	6.54905	0.317302
$427$	−0.599994	−0.0290358
$428$	−19.8880	−0.961322
$429$	−0.549146	−0.0265130
$430$	0	0
$431$	12.2045	0.587869	0.293935	−	0.955826i	$-0.405035\pi$
$431$	12.2045	0.587869	0.293935	+	0.955826i	$0.405035\pi$
$432$	5.76294	0.277270
$433$	31.3645	1.50728	0.753641	−	0.657287i	$-0.228296\pi$
$433$	31.3645	1.50728	0.753641	+	0.657287i	$0.228296\pi$
$434$	−3.19737	−0.153479
$435$	0	0
$436$	−44.9586	−2.15313
$437$	2.81417	0.134620
$438$	−0.728805	−0.0348236
$439$	23.2752	1.11086	0.555432	−	0.831562i	$-0.312553\pi$
$439$	23.2752	1.11086	0.555432	+	0.831562i	$0.312553\pi$
$440$	0	0
$441$	−2.91107	−0.138622
$442$	−27.6023	−1.31291
$443$	2.43539	0.115709	0.0578544	−	0.998325i	$-0.481574\pi$
$443$	2.43539	0.115709	0.0578544	+	0.998325i	$0.481574\pi$
$444$	−7.29813	−0.346354
$445$	0	0
$446$	−9.84068	−0.465970
$447$	1.06993	0.0506060
$448$	−10.2939	−0.486341
$449$	−4.60139	−0.217153	−0.108576	−	0.994088i	$-0.534629\pi$
$449$	−4.60139	−0.217153	−0.108576	+	0.994088i	$0.534629\pi$
$450$	0	0
$451$	9.31729	0.438734
$452$	10.7547	0.505857
$453$	−1.26371	−0.0593741
$454$	56.5560	2.65431
$455$	0	0
$456$	−0.550137	−0.0257625
$457$	−38.0985	−1.78217	−0.891086	−	0.453834i	$-0.850056\pi$
$457$	−38.0985	−1.78217	−0.891086	+	0.453834i	$0.850056\pi$
$458$	−51.5163	−2.40720
$459$	12.6345	0.589726
$460$	0	0
$461$	0.378240	0.0176164	0.00880820	−	0.999961i	$-0.497196\pi$
$461$	0.378240	0.0176164	0.00880820	+	0.999961i	$0.497196\pi$
$462$	−0.615387	−0.0286304
$463$	−13.6530	−0.634508	−0.317254	−	0.948341i	$-0.602761\pi$
$463$	−13.6530	−0.634508	−0.317254	+	0.948341i	$0.602761\pi$
$464$	15.7598	0.731631
$465$	0	0
$466$	−49.7784	−2.30594
$467$	9.71913	0.449748	0.224874	−	0.974388i	$-0.427803\pi$
$467$	9.71913	0.449748	0.224874	+	0.974388i	$0.427803\pi$
$468$	−12.4959	−0.577622
$469$	0.642727	0.0296784
$470$	0	0
$471$	−2.10310	−0.0969057
$472$	5.02601	0.231341
$473$	−12.4858	−0.574095
$474$	1.80854	0.0830689
$475$	0	0
$476$	−16.5968	−0.760712
$477$	−18.5767	−0.850571
$478$	−16.7182	−0.764671
$479$	34.4600	1.57452	0.787259	−	0.616622i	$-0.211499\pi$
$479$	34.4600	1.57452	0.787259	+	0.616622i	$0.211499\pi$
$480$	0	0
$481$	−19.5926	−0.893344
$482$	20.0099	0.911426
$483$	−0.298210	−0.0135690
$484$	−23.1862	−1.05392
$485$	0	0
$486$	16.0703	0.728963
$487$	−35.2140	−1.59570	−0.797848	−	0.602858i	$-0.794028\pi$
$487$	−35.2140	−1.59570	−0.797848	+	0.602858i	$0.794028\pi$
$488$	−0.393319	−0.0178047
$489$	−5.88431	−0.266098
$490$	0	0
$491$	−27.4004	−1.23656	−0.618281	−	0.785957i	$-0.712170\pi$
$491$	−27.4004	−1.23656	−0.618281	+	0.785957i	$0.712170\pi$
$492$	−6.47681	−0.291997
$493$	34.5512	1.55611
$494$	−10.8374	−0.487599
$495$	0	0
$496$	5.03189	0.225938
$497$	10.5715	0.474197
$498$	3.61353	0.161926
$499$	−25.9319	−1.16087	−0.580436	−	0.814306i	$-0.697118\pi$
$499$	−25.9319	−1.16087	−0.580436	+	0.814306i	$0.697118\pi$
$500$	0	0
$501$	5.67353	0.253475
$502$	−52.2574	−2.33236
$503$	22.6403	1.00948	0.504740	−	0.863271i	$-0.331588\pi$
$503$	22.6403	1.00948	0.504740	+	0.863271i	$0.331588\pi$
$504$	−1.90832	−0.0850032
$505$	0	0
$506$	2.06360	0.0917382
$507$	−2.85194	−0.126659
$508$	−40.6751	−1.80467
$509$	−12.2856	−0.544551	−0.272275	−	0.962219i	$-0.587776\pi$
$509$	−12.2856	−0.544551	−0.272275	+	0.962219i	$0.587776\pi$
$510$	0	0
$511$	−1.17644	−0.0520427
$512$	30.7765	1.36014
$513$	4.96065	0.219018
$514$	31.8137	1.40324
$515$	0	0
$516$	8.67933	0.382086
$517$	−3.46396	−0.152345
$518$	−21.9559	−0.964688
$519$	6.15416	0.270137
$520$	0	0
$521$	13.9977	0.613252	0.306626	−	0.951830i	$-0.400800\pi$
$521$	13.9977	0.613252	0.306626	+	0.951830i	$0.400800\pi$
$522$	29.1519	1.27594
$523$	9.48038	0.414548	0.207274	−	0.978283i	$-0.433541\pi$
$523$	9.48038	0.414548	0.207274	+	0.978283i	$0.433541\pi$
$524$	30.9643	1.35268
$525$	0	0
$526$	16.8229	0.733513
$527$	11.0317	0.480549
$528$	0.968468	0.0421472
$529$	1.00000	0.0434783
$530$	0	0
$531$	−22.3192	−0.968570
$532$	−6.51637	−0.282520
$533$	−17.3876	−0.753143
$534$	2.71493	0.117486
$535$	0	0
$536$	0.421332	0.0181988
$537$	−4.18520	−0.180605
$538$	40.0608	1.72714
$539$	−0.993361	−0.0427871
$540$	0	0
$541$	−13.8876	−0.597073	−0.298537	−	0.954398i	$-0.596498\pi$
$541$	−13.8876	−0.597073	−0.298537	+	0.954398i	$0.596498\pi$
$542$	6.44132	0.276678
$543$	6.96361	0.298837
$544$	58.0762	2.49000
$545$	0	0
$546$	1.14842	0.0491477
$547$	−23.4473	−1.00253	−0.501267	−	0.865293i	$-0.667132\pi$
$547$	−23.4473	−1.00253	−0.501267	+	0.865293i	$0.667132\pi$
$548$	−12.0410	−0.514365
$549$	1.74663	0.0745442
$550$	0	0
$551$	13.5658	0.577922
$552$	−0.195488	−0.00832053
$553$	2.91935	0.124143
$554$	−9.30797	−0.395458
$555$	0	0
$556$	19.9858	0.847585
$557$	30.0479	1.27317	0.636586	−	0.771206i	$-0.280346\pi$
$557$	30.0479	1.27317	0.636586	+	0.771206i	$0.280346\pi$
$558$	9.30778	0.394030
$559$	23.3005	0.985507
$560$	0	0
$561$	2.12323	0.0896430
$562$	50.7433	2.14048
$563$	−10.0913	−0.425299	−0.212649	−	0.977129i	$-0.568209\pi$
$563$	−10.0913	−0.425299	−0.212649	+	0.977129i	$0.568209\pi$
$564$	2.40793	0.101392
$565$	0	0
$566$	−69.2488	−2.91074
$567$	8.20754	0.344684
$568$	6.93003	0.290778
$569$	5.94085	0.249053	0.124527	−	0.992216i	$-0.460259\pi$
$569$	5.94085	0.249053	0.124527	+	0.992216i	$0.460259\pi$
$570$	0	0
$571$	38.5350	1.61264	0.806320	−	0.591480i	$-0.201456\pi$
$571$	38.5350	1.61264	0.806320	+	0.591480i	$0.201456\pi$
$572$	−4.26403	−0.178288
$573$	4.11588	0.171943
$574$	−19.4850	−0.813290
$575$	0	0
$576$	29.9662	1.24859
$577$	−42.5350	−1.77076	−0.885378	−	0.464871i	$-0.846101\pi$
$577$	−42.5350	−1.77076	−0.885378	+	0.464871i	$0.846101\pi$
$578$	71.4065	2.97012
$579$	−0.146871	−0.00610375
$580$	0	0
$581$	5.83299	0.241993
$582$	1.05416	0.0436965
$583$	−6.33904	−0.262536
$584$	−0.771202	−0.0319126
$585$	0	0
$586$	7.39051	0.305299
$587$	−6.45934	−0.266606	−0.133303	−	0.991075i	$-0.542558\pi$
$587$	−6.45934	−0.266606	−0.133303	+	0.991075i	$0.542558\pi$
$588$	0.690523	0.0284767
$589$	4.33137	0.178471
$590$	0	0
$591$	−5.71622	−0.235134
$592$	34.5532	1.42013
$593$	10.8641	0.446134	0.223067	−	0.974803i	$-0.428393\pi$
$593$	10.8641	0.446134	0.223067	+	0.974803i	$0.428393\pi$
$594$	3.63759	0.149252
$595$	0	0
$596$	8.30786	0.340303
$597$	1.06575	0.0436183
$598$	−3.85103	−0.157480
$599$	−2.31346	−0.0945254	−0.0472627	−	0.998882i	$-0.515050\pi$
$599$	−2.31346	−0.0945254	−0.0472627	+	0.998882i	$0.515050\pi$
$600$	0	0
$601$	29.3276	1.19630	0.598150	−	0.801384i	$-0.295903\pi$
$601$	29.3276	1.19630	0.598150	+	0.801384i	$0.295903\pi$
$602$	26.1112	1.06421
$603$	−1.87102	−0.0761939
$604$	−9.81249	−0.399265
$605$	0	0
$606$	−10.9970	−0.446724
$607$	−27.0080	−1.09622	−0.548111	−	0.836406i	$-0.684653\pi$
$607$	−27.0080	−1.09622	−0.548111	+	0.836406i	$0.684653\pi$
$608$	22.8024	0.924758
$609$	−1.43753	−0.0582518
$610$	0	0
$611$	6.46434	0.261519
$612$	48.3144	1.95299
$613$	13.1561	0.531372	0.265686	−	0.964060i	$-0.414402\pi$
$613$	13.1561	0.531372	0.265686	+	0.964060i	$0.414402\pi$
$614$	−57.4503	−2.31851
$615$	0	0
$616$	−0.651186	−0.0262370
$617$	−39.9124	−1.60681	−0.803406	−	0.595431i	$-0.796981\pi$
$617$	−39.9124	−1.60681	−0.803406	+	0.595431i	$0.796981\pi$
$618$	−6.35023	−0.255444
$619$	−9.49735	−0.381731	−0.190865	−	0.981616i	$-0.561129\pi$
$619$	−9.49735	−0.381731	−0.190865	+	0.981616i	$0.561129\pi$
$620$	0	0
$621$	1.76274	0.0707364
$622$	57.7581	2.31589
$623$	4.38245	0.175579
$624$	−1.80733	−0.0723509
$625$	0	0
$626$	−57.8360	−2.31159
$627$	0.833642	0.0332924
$628$	−16.3302	−0.651648
$629$	75.7532	3.02048
$630$	0	0
$631$	0.734808	0.0292523	0.0146261	−	0.999893i	$-0.495344\pi$
$631$	0.734808	0.0292523	0.0146261	+	0.999893i	$0.495344\pi$
$632$	1.91375	0.0761247
$633$	0.734591	0.0291974
$634$	30.4035	1.20748
$635$	0	0
$636$	4.40651	0.174730
$637$	1.85378	0.0734494
$638$	9.94765	0.393831
$639$	−30.7744	−1.21742
$640$	0	0
$641$	−29.0820	−1.14867	−0.574334	−	0.818621i	$-0.694739\pi$
$641$	−29.0820	−1.14867	−0.574334	+	0.818621i	$0.694739\pi$
$642$	−5.32079	−0.209995
$643$	−7.73214	−0.304926	−0.152463	−	0.988309i	$-0.548720\pi$
$643$	−7.73214	−0.304926	−0.152463	+	0.988309i	$0.548720\pi$
$644$	−2.31556	−0.0912458
$645$	0	0
$646$	41.9022	1.64862
$647$	9.05936	0.356160	0.178080	−	0.984016i	$-0.443011\pi$
$647$	9.05936	0.356160	0.178080	+	0.984016i	$0.443011\pi$
$648$	5.38036	0.211360
$649$	−7.61610	−0.298958
$650$	0	0
$651$	−0.458984	−0.0179890
$652$	−45.6908	−1.78939
$653$	−15.9458	−0.624007	−0.312003	−	0.950081i	$-0.601000\pi$
$653$	−15.9458	−0.624007	−0.312003	+	0.950081i	$0.601000\pi$
$654$	−12.0281	−0.470337
$655$	0	0
$656$	30.6647	1.19725
$657$	3.42470	0.133610
$658$	7.24410	0.282405
$659$	6.88170	0.268073	0.134037	−	0.990976i	$-0.457206\pi$
$659$	6.88170	0.268073	0.134037	+	0.990976i	$0.457206\pi$
$660$	0	0
$661$	−30.2502	−1.17660	−0.588298	−	0.808644i	$-0.700202\pi$
$661$	−30.2502	−1.17660	−0.588298	+	0.808644i	$0.700202\pi$
$662$	23.8602	0.927354
$663$	−3.96231	−0.153883
$664$	3.82374	0.148390
$665$	0	0
$666$	63.9152	2.47666
$667$	4.82054	0.186652
$668$	44.0541	1.70450
$669$	−1.41263	−0.0546155
$670$	0	0
$671$	0.596011	0.0230087
$672$	−2.41631	−0.0932112
$673$	−6.95862	−0.268235	−0.134117	−	0.990965i	$-0.542820\pi$
$673$	−6.95862	−0.268235	−0.134117	+	0.990965i	$0.542820\pi$
$674$	−3.50124	−0.134863
$675$	0	0
$676$	−22.1448	−0.851725
$677$	−18.0549	−0.693908	−0.346954	−	0.937882i	$-0.612784\pi$
$677$	−18.0549	−0.693908	−0.346954	+	0.937882i	$0.612784\pi$
$678$	2.87728	0.110501
$679$	1.70164	0.0653029
$680$	0	0
$681$	8.11863	0.311107
$682$	3.17615	0.121621
$683$	−36.4491	−1.39469	−0.697343	−	0.716738i	$-0.745634\pi$
$683$	−36.4491	−1.39469	−0.697343	+	0.716738i	$0.745634\pi$
$684$	18.9696	0.725321
$685$	0	0
$686$	2.07739	0.0793152
$687$	−7.39518	−0.282144
$688$	−41.0926	−1.56664
$689$	11.8297	0.450677
$690$	0	0
$691$	46.7539	1.77860	0.889302	−	0.457321i	$-0.151191\pi$
$691$	46.7539	1.77860	0.889302	+	0.457321i	$0.151191\pi$
$692$	47.7861	1.81655
$693$	2.89174	0.109848
$694$	−21.1848	−0.804164
$695$	0	0
$696$	−0.942358	−0.0357200
$697$	67.2281	2.54645
$698$	16.9191	0.640399
$699$	−7.14571	−0.270276
$700$	0	0
$701$	5.95624	0.224964	0.112482	−	0.993654i	$-0.464120\pi$
$701$	5.95624	0.224964	0.112482	+	0.993654i	$0.464120\pi$
$702$	−6.78837	−0.256210
$703$	29.7429	1.12177
$704$	10.2255	0.385390
$705$	0	0
$706$	−54.5059	−2.05136
$707$	−17.7515	−0.667613
$708$	5.29424	0.198970
$709$	27.2995	1.02525	0.512627	−	0.858611i	$-0.328672\pi$
$709$	27.2995	1.02525	0.512627	+	0.858611i	$0.328672\pi$
$710$	0	0
$711$	−8.49844	−0.318716
$712$	2.87286	0.107665
$713$	1.53913	0.0576408
$714$	−4.44027	−0.166173
$715$	0	0
$716$	−32.4975	−1.21449
$717$	−2.39990	−0.0896258
$718$	0.627167	0.0234057
$719$	18.4674	0.688719	0.344359	−	0.938838i	$-0.388096\pi$
$719$	18.4674	0.688719	0.344359	+	0.938838i	$0.388096\pi$
$720$	0	0
$721$	−10.2506	−0.381751
$722$	−23.0185	−0.856659
$723$	2.87243	0.106827
$724$	54.0714	2.00955
$725$	0	0
$726$	−6.20320	−0.230222
$727$	−3.46127	−0.128371	−0.0641856	−	0.997938i	$-0.520445\pi$
$727$	−3.46127	−0.128371	−0.0641856	+	0.997938i	$0.520445\pi$
$728$	1.21522	0.0450392
$729$	−22.3157	−0.826509
$730$	0	0
$731$	−90.0898	−3.33209
$732$	−0.414310	−0.0153133
$733$	25.2147	0.931324	0.465662	−	0.884963i	$-0.345816\pi$
$733$	25.2147	0.931324	0.465662	+	0.884963i	$0.345816\pi$
$734$	27.5117	1.01548
$735$	0	0
$736$	8.10271	0.298670
$737$	−0.638459	−0.0235180
$738$	56.7223	2.08798
$739$	−0.243657	−0.00896308	−0.00448154	−	0.999990i	$-0.501427\pi$
$739$	−0.243657	−0.00896308	−0.00448154	+	0.999990i	$0.501427\pi$
$740$	0	0
$741$	−1.55572	−0.0571507
$742$	13.2567	0.486669
$743$	18.9236	0.694240	0.347120	−	0.937821i	$-0.387160\pi$
$743$	18.9236	0.694240	0.347120	+	0.937821i	$0.387160\pi$
$744$	−0.300882	−0.0110309
$745$	0	0
$746$	−37.7016	−1.38035
$747$	−16.9802	−0.621274
$748$	16.4866	0.602809
$749$	−8.58885	−0.313830
$750$	0	0
$751$	21.4786	0.783764	0.391882	−	0.920015i	$-0.371824\pi$
$751$	21.4786	0.783764	0.391882	+	0.920015i	$0.371824\pi$
$752$	−11.4004	−0.415731
$753$	−7.50157	−0.273372
$754$	−18.5640	−0.676061
$755$	0	0
$756$	−4.08173	−0.148451
$757$	29.6310	1.07696	0.538479	−	0.842639i	$-0.318999\pi$
$757$	29.6310	1.07696	0.538479	+	0.842639i	$0.318999\pi$
$758$	21.7786	0.791035
$759$	0.296230	0.0107525
$760$	0	0
$761$	33.3358	1.20842	0.604210	−	0.796825i	$-0.293489\pi$
$761$	33.3358	1.20842	0.604210	+	0.796825i	$0.293489\pi$
$762$	−10.8821	−0.394218
$763$	−19.4159	−0.702902
$764$	31.9592	1.15624
$765$	0	0
$766$	−24.9343	−0.900914
$767$	14.2129	0.513199
$768$	2.93108	0.105766
$769$	−10.8543	−0.391415	−0.195708	−	0.980662i	$-0.562700\pi$
$769$	−10.8543	−0.391415	−0.195708	+	0.980662i	$0.562700\pi$
$770$	0	0
$771$	4.56687	0.164472
$772$	−1.14043	−0.0410450
$773$	−34.3678	−1.23612	−0.618062	−	0.786129i	$-0.712082\pi$
$773$	−34.3678	−1.23612	−0.618062	+	0.786129i	$0.712082\pi$
$774$	−76.0114	−2.73217
$775$	0	0
$776$	1.11549	0.0400437
$777$	−3.15178	−0.113069
$778$	69.2488	2.48269
$779$	26.3957	0.945723
$780$	0	0
$781$	−10.5013	−0.375767
$782$	14.8897	0.532455
$783$	8.49736	0.303671
$784$	−3.26931	−0.116761
$785$	0	0
$786$	8.28414	0.295485
$787$	−4.59388	−0.163754	−0.0818771	−	0.996642i	$-0.526091\pi$
$787$	−4.59388	−0.163754	−0.0818771	+	0.996642i	$0.526091\pi$
$788$	−44.3856	−1.58117
$789$	2.41493	0.0859738
$790$	0	0
$791$	4.64453	0.165140
$792$	1.89565	0.0673589
$793$	−1.11226	−0.0394974
$794$	−55.0539	−1.95379
$795$	0	0
$796$	8.27540	0.293314
$797$	−10.3975	−0.368300	−0.184150	−	0.982898i	$-0.558953\pi$
$797$	−10.3975	−0.368300	−0.184150	+	0.982898i	$0.558953\pi$
$798$	−1.74338	−0.0617148
$799$	−24.9939	−0.884221
$800$	0	0
$801$	−12.7576	−0.450768
$802$	−49.9192	−1.76271
$803$	1.16863	0.0412401
$804$	0.443818	0.0156522
$805$	0	0
$806$	−5.92723	−0.208778
$807$	5.75074	0.202436
$808$	−11.6368	−0.409380
$809$	−5.46014	−0.191968	−0.0959842	−	0.995383i	$-0.530600\pi$
$809$	−5.46014	−0.191968	−0.0959842	+	0.995383i	$0.530600\pi$
$810$	0	0
$811$	32.7801	1.15107	0.575533	−	0.817779i	$-0.304795\pi$
$811$	32.7801	1.15107	0.575533	+	0.817779i	$0.304795\pi$
$812$	−11.1622	−0.391718
$813$	0.924654	0.0324290
$814$	21.8101	0.764445
$815$	0	0
$816$	6.98789	0.244625
$817$	−35.3718	−1.23750
$818$	30.0051	1.04910
$819$	−5.39648	−0.188568
$820$	0	0
$821$	−36.4259	−1.27127	−0.635637	−	0.771988i	$-0.719262\pi$
$821$	−36.4259	−1.27127	−0.635637	+	0.771988i	$0.719262\pi$
$822$	−3.22142	−0.112360
$823$	44.7894	1.56126	0.780630	−	0.624993i	$-0.214898\pi$
$823$	44.7894	1.56126	0.780630	+	0.624993i	$0.214898\pi$
$824$	−6.71964	−0.234090
$825$	0	0
$826$	15.9274	0.554184
$827$	48.8330	1.69809	0.849045	−	0.528320i	$-0.177178\pi$
$827$	48.8330	1.69809	0.849045	+	0.528320i	$0.177178\pi$
$828$	6.74075	0.234257
$829$	−32.9943	−1.14594	−0.572970	−	0.819577i	$-0.694209\pi$
$829$	−32.9943	−1.14594	−0.572970	+	0.819577i	$0.694209\pi$
$830$	0	0
$831$	−1.33616	−0.0463509
$832$	−19.0826	−0.661570
$833$	−7.16751	−0.248339
$834$	5.34695	0.185150
$835$	0	0
$836$	6.47310	0.223877
$837$	2.71309	0.0937780
$838$	−17.4964	−0.604404
$839$	34.9466	1.20649	0.603246	−	0.797555i	$-0.293874\pi$
$839$	34.9466	1.20649	0.603246	+	0.797555i	$0.293874\pi$
$840$	0	0
$841$	−5.76244	−0.198705
$842$	−0.0642689	−0.00221485
$843$	7.28421	0.250882
$844$	5.70399	0.196339
$845$	0	0
$846$	−21.0881	−0.725024
$847$	−10.0132	−0.344059
$848$	−20.8628	−0.716431
$849$	−9.94069	−0.341164
$850$	0	0
$851$	10.5690	0.362300
$852$	7.29987	0.250090
$853$	−2.13264	−0.0730203	−0.0365102	−	0.999333i	$-0.511624\pi$
$853$	−2.13264	−0.0730203	−0.0365102	+	0.999333i	$0.511624\pi$
$854$	−1.24642	−0.0426517
$855$	0	0
$856$	−5.63032	−0.192440
$857$	28.0616	0.958566	0.479283	−	0.877661i	$-0.340897\pi$
$857$	28.0616	0.958566	0.479283	+	0.877661i	$0.340897\pi$
$858$	−1.14079	−0.0389460
$859$	−13.2339	−0.451534	−0.225767	−	0.974181i	$-0.572489\pi$
$859$	−13.2339	−0.451534	−0.225767	+	0.974181i	$0.572489\pi$
$860$	0	0
$861$	−2.79708	−0.0953244
$862$	25.3535	0.863543
$863$	−13.3156	−0.453268	−0.226634	−	0.973980i	$-0.572772\pi$
$863$	−13.3156	−0.453268	−0.226634	+	0.973980i	$0.572772\pi$
$864$	14.2830	0.485917
$865$	0	0
$866$	65.1563	2.21410
$867$	10.2504	0.348123
$868$	−3.56394	−0.120968
$869$	−2.89997	−0.0983747
$870$	0	0
$871$	1.19147	0.0403715
$872$	−12.7278	−0.431019
$873$	−4.95359	−0.167654
$874$	5.84613	0.197748
$875$	0	0
$876$	−0.812360	−0.0274471
$877$	19.9266	0.672872	0.336436	−	0.941706i	$-0.390778\pi$
$877$	19.9266	0.672872	0.336436	+	0.941706i	$0.390778\pi$
$878$	48.3517	1.63179
$879$	1.06091	0.0357836
$880$	0	0
$881$	−44.8358	−1.51055	−0.755277	−	0.655405i	$-0.772498\pi$
$881$	−44.8358	−1.51055	−0.755277	+	0.655405i	$0.772498\pi$
$882$	−6.04744	−0.203628
$883$	−0.0116754	−0.000392909 0	−0.000196455	−	1.00000i	$-0.500063\pi$
$883$	−0.0116754	−0.000392909 0	−0.000196455		1.00000i	$0.500063\pi$
$884$	−30.7668	−1.03480
$885$	0	0
$886$	5.05926	0.169969
$887$	−43.5198	−1.46125	−0.730625	−	0.682779i	$-0.760771\pi$
$887$	−43.5198	−1.46125	−0.730625	+	0.682779i	$0.760771\pi$
$888$	−2.06611	−0.0693342
$889$	−17.5660	−0.589145
$890$	0	0
$891$	−8.15305	−0.273137
$892$	−10.9689	−0.367265
$893$	−9.81332	−0.328390
$894$	2.22267	0.0743371
$895$	0	0
$896$	−5.17903	−0.173019
$897$	−0.552816	−0.0184580
$898$	−9.55889	−0.318984
$899$	7.41943	0.247452
$900$	0	0
$901$	−45.7388	−1.52378
$902$	19.3557	0.644473
$903$	3.74827	0.124734
$904$	3.04467	0.101264
$905$	0	0
$906$	−2.62521	−0.0872169
$907$	−25.9229	−0.860757	−0.430379	−	0.902649i	$-0.641620\pi$
$907$	−25.9229	−0.860757	−0.430379	+	0.902649i	$0.641620\pi$
$908$	63.0400	2.09206
$909$	51.6758	1.71398
$910$	0	0
$911$	4.06454	0.134664	0.0673321	−	0.997731i	$-0.478551\pi$
$911$	4.06454	0.134664	0.0673321	+	0.997731i	$0.478551\pi$
$912$	2.74365	0.0908512
$913$	−5.79426	−0.191762
$914$	−79.1455	−2.61790
$915$	0	0
$916$	−57.4224	−1.89729
$917$	13.3723	0.441592
$918$	26.2467	0.866271
$919$	−58.6773	−1.93558	−0.967792	−	0.251750i	$-0.918994\pi$
$919$	−58.6773	−1.93558	−0.967792	+	0.251750i	$0.918994\pi$
$920$	0	0
$921$	−8.24701	−0.271748
$922$	0.785753	0.0258774
$923$	19.5972	0.645051
$924$	−0.685939	−0.0225657
$925$	0	0
$926$	−28.3626	−0.932053
$927$	29.8401	0.980079
$928$	39.0594	1.28219
$929$	8.04634	0.263992	0.131996	−	0.991250i	$-0.457861\pi$
$929$	8.04634	0.263992	0.131996	+	0.991250i	$0.457861\pi$
$930$	0	0
$931$	−2.81417	−0.0922306
$932$	−55.4854	−1.81748
$933$	8.29119	0.271442
$934$	20.1904	0.660651
$935$	0	0
$936$	−3.53760	−0.115630
$937$	49.6078	1.62061	0.810307	−	0.586005i	$-0.199300\pi$
$937$	49.6078	1.62061	0.810307	+	0.586005i	$0.199300\pi$
$938$	1.33520	0.0435957
$939$	−8.30237	−0.270938
$940$	0	0
$941$	−37.1651	−1.21155	−0.605774	−	0.795637i	$-0.707137\pi$
$941$	−37.1651	−1.21155	−0.605774	+	0.795637i	$0.707137\pi$
$942$	−4.36896	−0.142348
$943$	9.37957	0.305441
$944$	−25.0658	−0.815822
$945$	0	0
$946$	−25.9378	−0.843311
$947$	23.8951	0.776487	0.388243	−	0.921557i	$-0.373082\pi$
$947$	23.8951	0.776487	0.388243	+	0.921557i	$0.373082\pi$
$948$	2.01588	0.0654727
$949$	−2.18086	−0.0707938
$950$	0	0
$951$	4.36443	0.141526
$952$	−4.69857	−0.152282
$953$	−29.3745	−0.951533	−0.475766	−	0.879572i	$-0.657829\pi$
$953$	−29.3745	−0.951533	−0.475766	+	0.879572i	$0.657829\pi$
$954$	−38.5912	−1.24944
$955$	0	0
$956$	−18.6348	−0.602694
$957$	1.42799	0.0461603
$958$	71.5870	2.31287
$959$	−5.20003	−0.167918
$960$	0	0
$961$	−28.6311	−0.923583
$962$	−40.7014	−1.31227
$963$	25.0028	0.805702
$964$	22.3040	0.718363
$965$	0	0
$966$	−0.619500	−0.0199321
$967$	13.9363	0.448161	0.224080	−	0.974571i	$-0.428062\pi$
$967$	13.9363	0.448161	0.224080	+	0.974571i	$0.428062\pi$
$968$	−6.56406	−0.210977
$969$	6.01507	0.193232
$970$	0	0
$971$	−17.1508	−0.550395	−0.275198	−	0.961388i	$-0.588743\pi$
$971$	−17.1508	−0.550395	−0.275198	+	0.961388i	$0.588743\pi$
$972$	17.9127	0.574550
$973$	8.63108	0.276700
$974$	−73.1532	−2.34398
$975$	0	0
$976$	1.96157	0.0627882
$977$	42.0538	1.34542	0.672709	−	0.739907i	$-0.265130\pi$
$977$	42.0538	1.34542	0.672709	+	0.739907i	$0.265130\pi$
$978$	−12.2240	−0.390881
$979$	−4.35335	−0.139134
$980$	0	0
$981$	56.5210	1.80458
$982$	−56.9213	−1.81643
$983$	48.0416	1.53229	0.766144	−	0.642669i	$-0.222173\pi$
$983$	48.0416	1.53229	0.766144	+	0.642669i	$0.222173\pi$
$984$	−1.83360	−0.0584529
$985$	0	0
$986$	71.7764	2.28583
$987$	1.03989	0.0331002
$988$	−12.0799	−0.384313
$989$	−12.5692	−0.399677
$990$	0	0
$991$	−32.0960	−1.01956	−0.509781	−	0.860304i	$-0.670274\pi$
$991$	−32.0960	−1.01956	−0.509781	+	0.860304i	$0.670274\pi$
$992$	12.4711	0.395958
$993$	3.42514	0.108694
$994$	21.9612	0.696566
$995$	0	0
$996$	4.02781	0.127626
$997$	−51.4259	−1.62867	−0.814337	−	0.580392i	$-0.802899\pi$
$997$	−51.4259	−1.62867	−0.814337	+	0.580392i	$0.802899\pi$
$998$	−53.8708	−1.70525
$999$	18.6304	0.589439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.u.1.7	✓	8
5.4	even	2		4025.2.a.v.1.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4025.2.a.u.1.7	✓	8	1.1	even	1	trivial
4025.2.a.v.1.2	yes	8	5.4	even	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)

\(f(q)\)	\(=\)	\(q+2.07739 q^{2} +0.298210 q^{3} +2.31556 q^{4} +0.619500 q^{6} +1.00000 q^{7} +0.655538 q^{8} -2.91107 q^{9} +O(q^{10})\)	\(q+2.07739 q^{2} +0.298210 q^{3} +2.31556 q^{4} +0.619500 q^{6} +1.00000 q^{7} +0.655538 q^{8} -2.91107 q^{9} -0.993361 q^{11} +0.690523 q^{12} +1.85378 q^{13} +2.07739 q^{14} -3.26931 q^{16} -7.16751 q^{17} -6.04744 q^{18} -2.81417 q^{19} +0.298210 q^{21} -2.06360 q^{22} -1.00000 q^{23} +0.195488 q^{24} +3.85103 q^{26} -1.76274 q^{27} +2.31556 q^{28} -4.82054 q^{29} -1.53913 q^{31} -8.10271 q^{32} -0.296230 q^{33} -14.8897 q^{34} -6.74075 q^{36} -10.5690 q^{37} -5.84613 q^{38} +0.552816 q^{39} -9.37957 q^{41} +0.619500 q^{42} +12.5692 q^{43} -2.30018 q^{44} -2.07739 q^{46} +3.48711 q^{47} -0.974941 q^{48} +1.00000 q^{49} -2.13742 q^{51} +4.29253 q^{52} +6.38141 q^{53} -3.66191 q^{54} +0.655538 q^{56} -0.839214 q^{57} -10.0141 q^{58} +7.66700 q^{59} -0.599994 q^{61} -3.19737 q^{62} -2.91107 q^{63} -10.2939 q^{64} -0.615387 q^{66} +0.642727 q^{67} -16.5968 q^{68} -0.298210 q^{69} +10.5715 q^{71} -1.90832 q^{72} -1.17644 q^{73} -21.9559 q^{74} -6.51637 q^{76} -0.993361 q^{77} +1.14842 q^{78} +2.91935 q^{79} +8.20754 q^{81} -19.4850 q^{82} +5.83299 q^{83} +0.690523 q^{84} +26.1112 q^{86} -1.43753 q^{87} -0.651186 q^{88} +4.38245 q^{89} +1.85378 q^{91} -2.31556 q^{92} -0.458984 q^{93} +7.24410 q^{94} -2.41631 q^{96} +1.70164 q^{97} +2.07739 q^{98} +2.89174 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} - 4 q^{3} + 5 q^{4} - q^{6} + 8 q^{7}+O(q^{10}) \) 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

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Database

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