LMFDB - Embedded newform 4025.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.1397868136$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 805)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ 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)

$f(q)$	$=$	$q+2.24698 q^{2} -1.35690 q^{3} +3.04892 q^{4} -3.04892 q^{6} +1.00000 q^{7} +2.35690 q^{8} -1.15883 q^{9} +O(q^{10})$	\(q+2.24698 q^{2} -1.35690 q^{3} +3.04892 q^{4} -3.04892 q^{6} +1.00000 q^{7} +2.35690 q^{8} -1.15883 q^{9} -0.753020 q^{11} -4.13706 q^{12} -3.58211 q^{13} +2.24698 q^{14} -0.801938 q^{16} +2.85086 q^{17} -2.60388 q^{18} +1.04892 q^{19} -1.35690 q^{21} -1.69202 q^{22} +1.00000 q^{23} -3.19806 q^{24} -8.04892 q^{26} +5.64310 q^{27} +3.04892 q^{28} -7.38404 q^{29} -1.44504 q^{31} -6.51573 q^{32} +1.02177 q^{33} +6.40581 q^{34} -3.53319 q^{36} +3.74094 q^{37} +2.35690 q^{38} +4.86054 q^{39} -8.70171 q^{41} -3.04892 q^{42} -2.24698 q^{43} -2.29590 q^{44} +2.24698 q^{46} +1.76271 q^{47} +1.08815 q^{48} +1.00000 q^{49} -3.86831 q^{51} -10.9215 q^{52} -12.2446 q^{53} +12.6799 q^{54} +2.35690 q^{56} -1.42327 q^{57} -16.5918 q^{58} -2.59419 q^{59} -8.76271 q^{61} -3.24698 q^{62} -1.15883 q^{63} -13.0368 q^{64} +2.29590 q^{66} -6.47219 q^{67} +8.69202 q^{68} -1.35690 q^{69} -14.9269 q^{71} -2.73125 q^{72} -2.76271 q^{73} +8.40581 q^{74} +3.19806 q^{76} -0.753020 q^{77} +10.9215 q^{78} -11.0761 q^{79} -4.18060 q^{81} -19.5526 q^{82} +13.6256 q^{83} -4.13706 q^{84} -5.04892 q^{86} +10.0194 q^{87} -1.77479 q^{88} +3.46681 q^{89} -3.58211 q^{91} +3.04892 q^{92} +1.96077 q^{93} +3.96077 q^{94} +8.84117 q^{96} +0.560335 q^{97} +2.24698 q^{98} +0.872625 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 3 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) $ 3 * q + 2 * q^2 + 3 * q^7 + 3 * q^8 + 5 * q^9	$ 3 q + 2 q^{2} + 3 q^{7} + 3 q^{8} + 5 q^{9} - 7 q^{11} - 7 q^{12} - 5 q^{13} + 2 q^{14} + 2 q^{16} - 5 q^{17} + q^{18} - 6 q^{19} + 3 q^{23} - 14 q^{24} - 15 q^{26} + 21 q^{27} - 12 q^{29} - 4 q^{31} - 7 q^{32} + 6 q^{34} - 14 q^{36} - 3 q^{37} + 3 q^{38} - 21 q^{39} + q^{41} - 2 q^{43} + 7 q^{44} + 2 q^{46} - 12 q^{47} + 7 q^{48} + 3 q^{49} - 14 q^{51} - 7 q^{52} + 9 q^{53} + 14 q^{54} + 3 q^{56} - 7 q^{57} - 22 q^{58} - 21 q^{59} - 9 q^{61} - 5 q^{62} + 5 q^{63} - 11 q^{64} - 7 q^{66} - 13 q^{67} + 21 q^{68} - 16 q^{71} - 16 q^{72} + 9 q^{73} + 12 q^{74} + 14 q^{76} - 7 q^{77} + 7 q^{78} - 18 q^{79} - q^{81} - 18 q^{82} + 29 q^{83} - 7 q^{84} - 6 q^{86} - 14 q^{87} - 7 q^{88} + 7 q^{89} - 5 q^{91} - 7 q^{93} - q^{94} + 35 q^{96} - q^{97} + 2 q^{98} - 14 q^{99}+O(q^{100}) $ 3 * q + 2 * q^2 + 3 * q^7 + 3 * q^8 + 5 * q^9 - 7 * q^11 - 7 * q^12 - 5 * q^13 + 2 * q^14 + 2 * q^16 - 5 * q^17 + q^18 - 6 * q^19 + 3 * q^23 - 14 * q^24 - 15 * q^26 + 21 * q^27 - 12 * q^29 - 4 * q^31 - 7 * q^32 + 6 * q^34 - 14 * q^36 - 3 * q^37 + 3 * q^38 - 21 * q^39 + q^41 - 2 * q^43 + 7 * q^44 + 2 * q^46 - 12 * q^47 + 7 * q^48 + 3 * q^49 - 14 * q^51 - 7 * q^52 + 9 * q^53 + 14 * q^54 + 3 * q^56 - 7 * q^57 - 22 * q^58 - 21 * q^59 - 9 * q^61 - 5 * q^62 + 5 * q^63 - 11 * q^64 - 7 * q^66 - 13 * q^67 + 21 * q^68 - 16 * q^71 - 16 * q^72 + 9 * q^73 + 12 * q^74 + 14 * q^76 - 7 * q^77 + 7 * q^78 - 18 * q^79 - q^81 - 18 * q^82 + 29 * q^83 - 7 * q^84 - 6 * q^86 - 14 * q^87 - 7 * q^88 + 7 * q^89 - 5 * q^91 - 7 * q^93 - q^94 + 35 * q^96 - q^97 + 2 * q^98 - 14 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.24698	1.58885	0.794427	−	0.607359i	$-0.207771\pi$
$2$	2.24698	1.58885	0.794427	+	0.607359i	$0.207771\pi$
$3$	−1.35690	−0.783404	−0.391702	−	0.920092i	$-0.628114\pi$
$3$	−1.35690	−0.783404	−0.391702	+	0.920092i	$0.628114\pi$
$4$	3.04892	1.52446
$5$	0	0
$5$	0	0
$6$	−3.04892	−1.24472
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.35690	0.833289
$9$	−1.15883	−0.386278
$10$	0	0
$11$	−0.753020	−0.227044	−0.113522	−	0.993535i	$-0.536213\pi$
$11$	−0.753020	−0.227044	−0.113522	+	0.993535i	$0.536213\pi$
$12$	−4.13706	−1.19427
$13$	−3.58211	−0.993497	−0.496749	−	0.867894i	$-0.665473\pi$
$13$	−3.58211	−0.993497	−0.496749	+	0.867894i	$0.665473\pi$
$14$	2.24698	0.600531
$15$	0	0
$16$	−0.801938	−0.200484
$17$	2.85086	0.691434	0.345717	−	0.938339i	$-0.387636\pi$
$17$	2.85086	0.691434	0.345717	+	0.938339i	$0.387636\pi$
$18$	−2.60388	−0.613739
$19$	1.04892	0.240638	0.120319	−	0.992735i	$-0.461608\pi$
$19$	1.04892	0.240638	0.120319	+	0.992735i	$0.461608\pi$
$20$	0	0
$21$	−1.35690	−0.296099
$22$	−1.69202	−0.360740
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−3.19806	−0.652802
$25$	0	0
$26$	−8.04892	−1.57852
$27$	5.64310	1.08602
$28$	3.04892	0.576191
$29$	−7.38404	−1.37118	−0.685591	−	0.727987i	$-0.740456\pi$
$29$	−7.38404	−1.37118	−0.685591	+	0.727987i	$0.740456\pi$
$30$	0	0
$31$	−1.44504	−0.259537	−0.129769	−	0.991544i	$-0.541423\pi$
$31$	−1.44504	−0.259537	−0.129769	+	0.991544i	$0.541423\pi$
$32$	−6.51573	−1.15183
$33$	1.02177	0.177867
$34$	6.40581	1.09859
$35$	0	0
$36$	−3.53319	−0.588865
$37$	3.74094	0.615007	0.307503	−	0.951547i	$-0.400506\pi$
$37$	3.74094	0.615007	0.307503	+	0.951547i	$0.400506\pi$
$38$	2.35690	0.382339
$39$	4.86054	0.778310
$40$	0	0
$41$	−8.70171	−1.35898	−0.679489	−	0.733685i	$-0.737799\pi$
$41$	−8.70171	−1.35898	−0.679489	+	0.733685i	$0.737799\pi$
$42$	−3.04892	−0.470458
$43$	−2.24698	−0.342661	−0.171331	−	0.985214i	$-0.554807\pi$
$43$	−2.24698	−0.342661	−0.171331	+	0.985214i	$0.554807\pi$
$44$	−2.29590	−0.346119
$45$	0	0
$46$	2.24698	0.331299
$47$	1.76271	0.257118	0.128559	−	0.991702i	$-0.458965\pi$
$47$	1.76271	0.257118	0.128559	+	0.991702i	$0.458965\pi$
$48$	1.08815	0.157060
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.86831	−0.541672
$52$	−10.9215	−1.51455
$53$	−12.2446	−1.68192	−0.840962	−	0.541095i	$-0.818010\pi$
$53$	−12.2446	−1.68192	−0.840962	+	0.541095i	$0.818010\pi$
$54$	12.6799	1.72552
$55$	0	0
$56$	2.35690	0.314953
$57$	−1.42327	−0.188517
$58$	−16.5918	−2.17861
$59$	−2.59419	−0.337734	−0.168867	−	0.985639i	$-0.554011\pi$
$59$	−2.59419	−0.337734	−0.168867	+	0.985639i	$0.554011\pi$
$60$	0	0
$61$	−8.76271	−1.12195	−0.560975	−	0.827833i	$-0.689574\pi$
$61$	−8.76271	−1.12195	−0.560975	+	0.827833i	$0.689574\pi$
$62$	−3.24698	−0.412367
$63$	−1.15883	−0.145999
$64$	−13.0368	−1.62960
$65$	0	0
$66$	2.29590	0.282605
$67$	−6.47219	−0.790704	−0.395352	−	0.918530i	$-0.629377\pi$
$67$	−6.47219	−0.790704	−0.395352	+	0.918530i	$0.629377\pi$
$68$	8.69202	1.05406
$69$	−1.35690	−0.163351
$70$	0	0
$71$	−14.9269	−1.77150	−0.885750	−	0.464163i	$-0.846355\pi$
$71$	−14.9269	−1.77150	−0.885750	+	0.464163i	$0.846355\pi$
$72$	−2.73125	−0.321881
$73$	−2.76271	−0.323351	−0.161675	−	0.986844i	$-0.551690\pi$
$73$	−2.76271	−0.323351	−0.161675	+	0.986844i	$0.551690\pi$
$74$	8.40581	0.977156
$75$	0	0
$76$	3.19806	0.366843
$77$	−0.753020	−0.0858146
$78$	10.9215	1.23662
$79$	−11.0761	−1.24615	−0.623077	−	0.782160i	$-0.714118\pi$
$79$	−11.0761	−1.24615	−0.623077	+	0.782160i	$0.714118\pi$
$80$	0	0
$81$	−4.18060	−0.464512
$82$	−19.5526	−2.15922
$83$	13.6256	1.49561	0.747804	−	0.663919i	$-0.231108\pi$
$83$	13.6256	1.49561	0.747804	+	0.663919i	$0.231108\pi$
$84$	−4.13706	−0.451391
$85$	0	0
$86$	−5.04892	−0.544439
$87$	10.0194	1.07419
$88$	−1.77479	−0.189193
$89$	3.46681	0.367481	0.183741	−	0.982975i	$-0.441179\pi$
$89$	3.46681	0.367481	0.183741	+	0.982975i	$0.441179\pi$
$90$	0	0
$91$	−3.58211	−0.375507
$92$	3.04892	0.317872
$93$	1.96077	0.203323
$94$	3.96077	0.408522
$95$	0	0
$96$	8.84117	0.902348
$97$	0.560335	0.0568934	0.0284467	−	0.999595i	$-0.490944\pi$
$97$	0.560335	0.0568934	0.0284467	+	0.999595i	$0.490944\pi$
$98$	2.24698	0.226979
$99$	0.872625	0.0877021
$100$	0	0
$101$	−4.25667	−0.423554	−0.211777	−	0.977318i	$-0.567925\pi$
$101$	−4.25667	−0.423554	−0.211777	+	0.977318i	$0.567925\pi$
$102$	−8.69202	−0.860638
$103$	5.81163	0.572637	0.286318	−	0.958135i	$-0.407569\pi$
$103$	5.81163	0.572637	0.286318	+	0.958135i	$0.407569\pi$
$104$	−8.44265	−0.827870
$105$	0	0
$106$	−27.5133	−2.67233
$107$	13.7681	1.33101	0.665506	−	0.746393i	$-0.268216\pi$
$107$	13.7681	1.33101	0.665506	+	0.746393i	$0.268216\pi$
$108$	17.2054	1.65559
$109$	3.65279	0.349874	0.174937	−	0.984580i	$-0.444028\pi$
$109$	3.65279	0.349874	0.174937	+	0.984580i	$0.444028\pi$
$110$	0	0
$111$	−5.07606	−0.481799
$112$	−0.801938	−0.0757760
$113$	18.2687	1.71858	0.859290	−	0.511489i	$-0.170906\pi$
$113$	18.2687	1.71858	0.859290	+	0.511489i	$0.170906\pi$
$114$	−3.19806	−0.299526
$115$	0	0
$116$	−22.5133	−2.09031
$117$	4.15106	0.383766
$118$	−5.82908	−0.536611
$119$	2.85086	0.261337
$120$	0	0
$121$	−10.4330	−0.948451
$122$	−19.6896	−1.78262
$123$	11.8073	1.06463
$124$	−4.40581	−0.395654
$125$	0	0
$126$	−2.60388	−0.231972
$127$	−19.4209	−1.72332	−0.861662	−	0.507482i	$-0.830576\pi$
$127$	−19.4209	−1.72332	−0.861662	+	0.507482i	$0.830576\pi$
$128$	−16.2620	−1.43738
$129$	3.04892	0.268442
$130$	0	0
$131$	14.4940	1.26634	0.633172	−	0.774011i	$-0.281753\pi$
$131$	14.4940	1.26634	0.633172	+	0.774011i	$0.281753\pi$
$132$	3.11529	0.271151
$133$	1.04892	0.0909527
$134$	−14.5429	−1.25631
$135$	0	0
$136$	6.71917	0.576164
$137$	5.04354	0.430899	0.215449	−	0.976515i	$-0.430878\pi$
$137$	5.04354	0.430899	0.215449	+	0.976515i	$0.430878\pi$
$138$	−3.04892	−0.259541
$139$	7.21313	0.611810	0.305905	−	0.952062i	$-0.401041\pi$
$139$	7.21313	0.611810	0.305905	+	0.952062i	$0.401041\pi$
$140$	0	0
$141$	−2.39181	−0.201427
$142$	−33.5405	−2.81465
$143$	2.69740	0.225568
$144$	0.929312	0.0774427
$145$	0	0
$146$	−6.20775	−0.513757
$147$	−1.35690	−0.111915
$148$	11.4058	0.937552
$149$	−6.03923	−0.494753	−0.247376	−	0.968919i	$-0.579568\pi$
$149$	−6.03923	−0.494753	−0.247376	+	0.968919i	$0.579568\pi$
$150$	0	0
$151$	9.67025	0.786954	0.393477	−	0.919334i	$-0.371272\pi$
$151$	9.67025	0.786954	0.393477	+	0.919334i	$0.371272\pi$
$152$	2.47219	0.200521
$153$	−3.30367	−0.267086
$154$	−1.69202	−0.136347
$155$	0	0
$156$	14.8194	1.18650
$157$	7.59419	0.606082	0.303041	−	0.952978i	$-0.401998\pi$
$157$	7.59419	0.606082	0.303041	+	0.952978i	$0.401998\pi$
$158$	−24.8877	−1.97996
$159$	16.6146	1.31763
$160$	0	0
$161$	1.00000	0.0788110
$162$	−9.39373	−0.738041
$163$	−5.15346	−0.403650	−0.201825	−	0.979422i	$-0.564687\pi$
$163$	−5.15346	−0.403650	−0.201825	+	0.979422i	$0.564687\pi$
$164$	−26.5308	−2.07171
$165$	0	0
$166$	30.6165	2.37630
$167$	−17.8552	−1.38167	−0.690837	−	0.723010i	$-0.742758\pi$
$167$	−17.8552	−1.38167	−0.690837	+	0.723010i	$0.742758\pi$
$168$	−3.19806	−0.246736
$169$	−0.168522	−0.0129633
$170$	0	0
$171$	−1.21552	−0.0929532
$172$	−6.85086	−0.522373
$173$	−2.18060	−0.165788	−0.0828941	−	0.996558i	$-0.526416\pi$
$173$	−2.18060	−0.165788	−0.0828941	+	0.996558i	$0.526416\pi$
$174$	22.5133	1.70673
$175$	0	0
$176$	0.603875	0.0455188
$177$	3.52004	0.264583
$178$	7.78986	0.583874
$179$	−3.36898	−0.251809	−0.125905	−	0.992042i	$-0.540183\pi$
$179$	−3.36898	−0.251809	−0.125905	+	0.992042i	$0.540183\pi$
$180$	0	0
$181$	1.61596	0.120113	0.0600566	−	0.998195i	$-0.480872\pi$
$181$	1.61596	0.120113	0.0600566	+	0.998195i	$0.480872\pi$
$182$	−8.04892	−0.596625
$183$	11.8901	0.878940
$184$	2.35690	0.173753
$185$	0	0
$186$	4.40581	0.323050
$187$	−2.14675	−0.156986
$188$	5.37435	0.391965
$189$	5.64310	0.410475
$190$	0	0
$191$	0.833397	0.0603025	0.0301512	−	0.999545i	$-0.490401\pi$
$191$	0.833397	0.0603025	0.0301512	+	0.999545i	$0.490401\pi$
$192$	17.6896	1.27664
$193$	−12.0218	−0.865346	−0.432673	−	0.901551i	$-0.642430\pi$
$193$	−12.0218	−0.865346	−0.432673	+	0.901551i	$0.642430\pi$
$194$	1.25906	0.0903953
$195$	0	0
$196$	3.04892	0.217780
$197$	−23.9584	−1.70696	−0.853482	−	0.521122i	$-0.825513\pi$
$197$	−23.9584	−1.70696	−0.853482	+	0.521122i	$0.825513\pi$
$198$	1.96077	0.139346
$199$	−21.4590	−1.52119	−0.760596	−	0.649226i	$-0.775093\pi$
$199$	−21.4590	−1.52119	−0.760596	+	0.649226i	$0.775093\pi$
$200$	0	0
$201$	8.78209	0.619441
$202$	−9.56465	−0.672966
$203$	−7.38404	−0.518258
$204$	−11.7942	−0.825757
$205$	0	0
$206$	13.0586	0.909836
$207$	−1.15883	−0.0805445
$208$	2.87263	0.199181
$209$	−0.789856	−0.0546355
$210$	0	0
$211$	13.7778	0.948501	0.474251	−	0.880390i	$-0.342719\pi$
$211$	13.7778	0.948501	0.474251	+	0.880390i	$0.342719\pi$
$212$	−37.3327	−2.56402
$213$	20.2543	1.38780
$214$	30.9366	2.11478
$215$	0	0
$216$	13.3002	0.904965
$217$	−1.44504	−0.0980958
$218$	8.20775	0.555899
$219$	3.74871	0.253314
$220$	0	0
$221$	−10.2121	−0.686938
$222$	−11.4058	−0.765508
$223$	−3.94139	−0.263935	−0.131968	−	0.991254i	$-0.542129\pi$
$223$	−3.94139	−0.263935	−0.131968	+	0.991254i	$0.542129\pi$
$224$	−6.51573	−0.435350
$225$	0	0
$226$	41.0495	2.73057
$227$	17.2567	1.14537	0.572683	−	0.819777i	$-0.305903\pi$
$227$	17.2567	1.14537	0.572683	+	0.819777i	$0.305903\pi$
$228$	−4.33944	−0.287386
$229$	12.4862	0.825111	0.412555	−	0.910933i	$-0.364636\pi$
$229$	12.4862	0.825111	0.412555	+	0.910933i	$0.364636\pi$
$230$	0	0
$231$	1.02177	0.0672275
$232$	−17.4034	−1.14259
$233$	−14.2838	−0.935764	−0.467882	−	0.883791i	$-0.654983\pi$
$233$	−14.2838	−0.935764	−0.467882	+	0.883791i	$0.654983\pi$
$234$	9.32736	0.609748
$235$	0	0
$236$	−7.90946	−0.514862
$237$	15.0291	0.976243
$238$	6.40581	0.415227
$239$	7.99761	0.517322	0.258661	−	0.965968i	$-0.416719\pi$
$239$	7.99761	0.517322	0.258661	+	0.965968i	$0.416719\pi$
$240$	0	0
$241$	8.43727	0.543492	0.271746	−	0.962369i	$-0.412399\pi$
$241$	8.43727	0.543492	0.271746	+	0.962369i	$0.412399\pi$
$242$	−23.4426	−1.50695
$243$	−11.2567	−0.722116
$244$	−26.7168	−1.71037
$245$	0	0
$246$	26.5308	1.69154
$247$	−3.75733	−0.239073
$248$	−3.40581	−0.216269
$249$	−18.4886	−1.17167
$250$	0	0
$251$	3.54048	0.223473	0.111737	−	0.993738i	$-0.464359\pi$
$251$	3.54048	0.223473	0.111737	+	0.993738i	$0.464359\pi$
$252$	−3.53319	−0.222570
$253$	−0.753020	−0.0473420
$254$	−43.6383	−2.73811
$255$	0	0
$256$	−10.4668	−0.654176
$257$	4.62671	0.288606	0.144303	−	0.989534i	$-0.453906\pi$
$257$	4.62671	0.288606	0.144303	+	0.989534i	$0.453906\pi$
$258$	6.85086	0.426516
$259$	3.74094	0.232451
$260$	0	0
$261$	8.55688	0.529657
$262$	32.5676	2.01203
$263$	11.9812	0.738793	0.369397	−	0.929272i	$-0.379564\pi$
$263$	11.9812	0.738793	0.369397	+	0.929272i	$0.379564\pi$
$264$	2.40821	0.148215
$265$	0	0
$266$	2.35690	0.144511
$267$	−4.70410	−0.287886
$268$	−19.7332	−1.20540
$269$	24.2610	1.47922	0.739609	−	0.673037i	$-0.235010\pi$
$269$	24.2610	1.47922	0.739609	+	0.673037i	$0.235010\pi$
$270$	0	0
$271$	−12.0422	−0.731512	−0.365756	−	0.930711i	$-0.619190\pi$
$271$	−12.0422	−0.731512	−0.365756	+	0.930711i	$0.619190\pi$
$272$	−2.28621	−0.138622
$273$	4.86054	0.294173
$274$	11.3327	0.684635
$275$	0	0
$276$	−4.13706	−0.249022
$277$	10.8237	0.650334	0.325167	−	0.945657i	$-0.394580\pi$
$277$	10.8237	0.650334	0.325167	+	0.945657i	$0.394580\pi$
$278$	16.2078	0.972076
$279$	1.67456	0.100253
$280$	0	0
$281$	14.7778	0.881568	0.440784	−	0.897613i	$-0.354700\pi$
$281$	14.7778	0.881568	0.440784	+	0.897613i	$0.354700\pi$
$282$	−5.37435	−0.320038
$283$	−12.6635	−0.752770	−0.376385	−	0.926463i	$-0.622833\pi$
$283$	−12.6635	−0.752770	−0.376385	+	0.926463i	$0.622833\pi$
$284$	−45.5109	−2.70058
$285$	0	0
$286$	6.06100	0.358394
$287$	−8.70171	−0.513646
$288$	7.55065	0.444926
$289$	−8.87263	−0.521919
$290$	0	0
$291$	−0.760316	−0.0445705
$292$	−8.42327	−0.492935
$293$	−28.5894	−1.67021	−0.835105	−	0.550090i	$-0.814593\pi$
$293$	−28.5894	−1.67021	−0.835105	+	0.550090i	$0.814593\pi$
$294$	−3.04892	−0.177816
$295$	0	0
$296$	8.81700	0.512478
$297$	−4.24937	−0.246574
$298$	−13.5700	−0.786090
$299$	−3.58211	−0.207158
$300$	0	0
$301$	−2.24698	−0.129514
$302$	21.7289	1.25036
$303$	5.77586	0.331814
$304$	−0.841166	−0.0482442
$305$	0	0
$306$	−7.42327	−0.424360
$307$	9.38703	0.535746	0.267873	−	0.963454i	$-0.413679\pi$
$307$	9.38703	0.535746	0.267873	+	0.963454i	$0.413679\pi$
$308$	−2.29590	−0.130821
$309$	−7.88577	−0.448606
$310$	0	0
$311$	1.70171	0.0964951	0.0482476	−	0.998835i	$-0.484636\pi$
$311$	1.70171	0.0964951	0.0482476	+	0.998835i	$0.484636\pi$
$312$	11.4558	0.648557
$313$	31.7415	1.79414	0.897069	−	0.441891i	$-0.145692\pi$
$313$	31.7415	1.79414	0.897069	+	0.441891i	$0.145692\pi$
$314$	17.0640	0.962976
$315$	0	0
$316$	−33.7700	−1.89971
$317$	−20.6950	−1.16235	−0.581174	−	0.813780i	$-0.697406\pi$
$317$	−20.6950	−1.16235	−0.581174	+	0.813780i	$0.697406\pi$
$318$	37.3327	2.09352
$319$	5.56033	0.311319
$320$	0	0
$321$	−18.6819	−1.04272
$322$	2.24698	0.125219
$323$	2.99031	0.166385
$324$	−12.7463	−0.708129
$325$	0	0
$326$	−11.5797	−0.641341
$327$	−4.95646	−0.274093
$328$	−20.5090	−1.13242
$329$	1.76271	0.0971813
$330$	0	0
$331$	−0.109916	−0.00604154	−0.00302077	−	0.999995i	$-0.500962\pi$
$331$	−0.109916	−0.00604154	−0.00302077	+	0.999995i	$0.500962\pi$
$332$	41.5435	2.27999
$333$	−4.33513	−0.237563
$334$	−40.1202	−2.19528
$335$	0	0
$336$	1.08815	0.0593632
$337$	19.0683	1.03872	0.519358	−	0.854557i	$-0.326171\pi$
$337$	19.0683	1.03872	0.519358	+	0.854557i	$0.326171\pi$
$338$	−0.378666	−0.0205967
$339$	−24.7888	−1.34634
$340$	0	0
$341$	1.08815	0.0589264
$342$	−2.73125	−0.147689
$343$	1.00000	0.0539949
$344$	−5.29590	−0.285536
$345$	0	0
$346$	−4.89977	−0.263413
$347$	28.0489	1.50574	0.752872	−	0.658167i	$-0.228668\pi$
$347$	28.0489	1.50574	0.752872	+	0.658167i	$0.228668\pi$
$348$	30.5483	1.63756
$349$	28.9691	1.55068	0.775341	−	0.631543i	$-0.217578\pi$
$349$	28.9691	1.55068	0.775341	+	0.631543i	$0.217578\pi$
$350$	0	0
$351$	−20.2142	−1.07895
$352$	4.90648	0.261516
$353$	−16.5579	−0.881290	−0.440645	−	0.897681i	$-0.645250\pi$
$353$	−16.5579	−0.881290	−0.440645	+	0.897681i	$0.645250\pi$
$354$	7.90946	0.420383
$355$	0	0
$356$	10.5700	0.560210
$357$	−3.86831	−0.204733
$358$	−7.57002	−0.400088
$359$	−15.0043	−0.791897	−0.395949	−	0.918273i	$-0.629584\pi$
$359$	−15.0043	−0.791897	−0.395949	+	0.918273i	$0.629584\pi$
$360$	0	0
$361$	−17.8998	−0.942093
$362$	3.63102	0.190842
$363$	14.1564	0.743020
$364$	−10.9215	−0.572444
$365$	0	0
$366$	26.7168	1.39651
$367$	−3.05131	−0.159277	−0.0796385	−	0.996824i	$-0.525377\pi$
$367$	−3.05131	−0.159277	−0.0796385	+	0.996824i	$0.525377\pi$
$368$	−0.801938	−0.0418039
$369$	10.0838	0.524943
$370$	0	0
$371$	−12.2446	−0.635707
$372$	5.97823	0.309957
$373$	9.60686	0.497424	0.248712	−	0.968577i	$-0.419993\pi$
$373$	9.60686	0.497424	0.248712	+	0.968577i	$0.419993\pi$
$374$	−4.82371	−0.249428
$375$	0	0
$376$	4.15452	0.214253
$377$	26.4504	1.36227
$378$	12.6799	0.652186
$379$	3.00969	0.154597	0.0772987	−	0.997008i	$-0.475371\pi$
$379$	3.00969	0.154597	0.0772987	+	0.997008i	$0.475371\pi$
$380$	0	0
$381$	26.3521	1.35006
$382$	1.87263	0.0958118
$383$	−22.0344	−1.12591	−0.562954	−	0.826488i	$-0.690335\pi$
$383$	−22.0344	−1.12591	−0.562954	+	0.826488i	$0.690335\pi$
$384$	22.0659	1.12605
$385$	0	0
$386$	−27.0127	−1.37491
$387$	2.60388	0.132362
$388$	1.70841	0.0867316
$389$	14.3478	0.727462	0.363731	−	0.931504i	$-0.381503\pi$
$389$	14.3478	0.727462	0.363731	+	0.931504i	$0.381503\pi$
$390$	0	0
$391$	2.85086	0.144174
$392$	2.35690	0.119041
$393$	−19.6668	−0.992058
$394$	−53.8340	−2.71212
$395$	0	0
$396$	2.66056	0.133698
$397$	5.44803	0.273429	0.136714	−	0.990611i	$-0.456346\pi$
$397$	5.44803	0.273429	0.136714	+	0.990611i	$0.456346\pi$
$398$	−48.2180	−2.41695
$399$	−1.42327	−0.0712527
$400$	0	0
$401$	−4.68127	−0.233771	−0.116886	−	0.993145i	$-0.537291\pi$
$401$	−4.68127	−0.233771	−0.116886	+	0.993145i	$0.537291\pi$
$402$	19.7332	0.984201
$403$	5.17629	0.257849
$404$	−12.9782	−0.645691
$405$	0	0
$406$	−16.5918	−0.823437
$407$	−2.81700	−0.139634
$408$	−9.11721	−0.451369
$409$	30.4480	1.50556	0.752779	−	0.658273i	$-0.228713\pi$
$409$	30.4480	1.50556	0.752779	+	0.658273i	$0.228713\pi$
$410$	0	0
$411$	−6.84356	−0.337568
$412$	17.7192	0.872961
$413$	−2.59419	−0.127652
$414$	−2.60388	−0.127973
$415$	0	0
$416$	23.3400	1.14434
$417$	−9.78746	−0.479294
$418$	−1.77479	−0.0868078
$419$	−5.81833	−0.284244	−0.142122	−	0.989849i	$-0.545393\pi$
$419$	−5.81833	−0.284244	−0.142122	+	0.989849i	$0.545393\pi$
$420$	0	0
$421$	−17.7603	−0.865585	−0.432792	−	0.901494i	$-0.642472\pi$
$421$	−17.7603	−0.865585	−0.432792	+	0.901494i	$0.642472\pi$
$422$	30.9584	1.50703
$423$	−2.04269	−0.0993188
$424$	−28.8592	−1.40153
$425$	0	0
$426$	45.5109	2.20501
$427$	−8.76271	−0.424057
$428$	41.9778	2.02907
$429$	−3.66009	−0.176711
$430$	0	0
$431$	−6.67696	−0.321618	−0.160809	−	0.986986i	$-0.551410\pi$
$431$	−6.67696	−0.321618	−0.160809	+	0.986986i	$0.551410\pi$
$432$	−4.52542	−0.217729
$433$	32.8388	1.57813	0.789065	−	0.614309i	$-0.210565\pi$
$433$	32.8388	1.57813	0.789065	+	0.614309i	$0.210565\pi$
$434$	−3.24698	−0.155860
$435$	0	0
$436$	11.1371	0.533369
$437$	1.04892	0.0501765
$438$	8.42327	0.402479
$439$	−0.624318	−0.0297971	−0.0148985	−	0.999889i	$-0.504743\pi$
$439$	−0.624318	−0.0297971	−0.0148985	+	0.999889i	$0.504743\pi$
$440$	0	0
$441$	−1.15883	−0.0551826
$442$	−22.9463	−1.09144
$443$	18.3666	0.872623	0.436311	−	0.899796i	$-0.356285\pi$
$443$	18.3666	0.872623	0.436311	+	0.899796i	$0.356285\pi$
$444$	−15.4765	−0.734482
$445$	0	0
$446$	−8.85623	−0.419355
$447$	8.19460	0.387591
$448$	−13.0368	−0.615933
$449$	18.3491	0.865949	0.432974	−	0.901406i	$-0.357464\pi$
$449$	18.3491	0.865949	0.432974	+	0.901406i	$0.357464\pi$
$450$	0	0
$451$	6.55257	0.308548
$452$	55.6999	2.61990
$453$	−13.1215	−0.616503
$454$	38.7754	1.81982
$455$	0	0
$456$	−3.35450	−0.157089
$457$	−27.0901	−1.26722	−0.633610	−	0.773653i	$-0.718428\pi$
$457$	−27.0901	−1.26722	−0.633610	+	0.773653i	$0.718428\pi$
$458$	28.0562	1.31098
$459$	16.0877	0.750908
$460$	0	0
$461$	41.6708	1.94080	0.970402	−	0.241494i	$-0.0776374\pi$
$461$	41.6708	1.94080	0.970402	+	0.241494i	$0.0776374\pi$
$462$	2.29590	0.106815
$463$	−3.67456	−0.170771	−0.0853857	−	0.996348i	$-0.527212\pi$
$463$	−3.67456	−0.170771	−0.0853857	+	0.996348i	$0.527212\pi$
$464$	5.92154	0.274901
$465$	0	0
$466$	−32.0954	−1.48679
$467$	14.6679	0.678748	0.339374	−	0.940652i	$-0.389785\pi$
$467$	14.6679	0.678748	0.339374	+	0.940652i	$0.389785\pi$
$468$	12.6563	0.585035
$469$	−6.47219	−0.298858
$470$	0	0
$471$	−10.3045	−0.474807
$472$	−6.11423	−0.281430
$473$	1.69202	0.0777992
$474$	33.7700	1.55111
$475$	0	0
$476$	8.69202	0.398398
$477$	14.1894	0.649690
$478$	17.9705	0.821950
$479$	−10.9933	−0.502296	−0.251148	−	0.967949i	$-0.580808\pi$
$479$	−10.9933	−0.502296	−0.251148	+	0.967949i	$0.580808\pi$
$480$	0	0
$481$	−13.4004	−0.611007
$482$	18.9584	0.863530
$483$	−1.35690	−0.0617409
$484$	−31.8092	−1.44587
$485$	0	0
$486$	−25.2935	−1.14734
$487$	21.9476	0.994542	0.497271	−	0.867595i	$-0.334336\pi$
$487$	21.9476	0.994542	0.497271	+	0.867595i	$0.334336\pi$
$488$	−20.6528	−0.934908
$489$	6.99270	0.316221
$490$	0	0
$491$	18.9245	0.854052	0.427026	−	0.904239i	$-0.359561\pi$
$491$	18.9245	0.854052	0.427026	+	0.904239i	$0.359561\pi$
$492$	35.9995	1.62298
$493$	−21.0508	−0.948082
$494$	−8.44265	−0.379853
$495$	0	0
$496$	1.15883	0.0520332
$497$	−14.9269	−0.669564
$498$	−41.5435	−1.86161
$499$	−12.6407	−0.565876	−0.282938	−	0.959138i	$-0.591309\pi$
$499$	−12.6407	−0.565876	−0.282938	+	0.959138i	$0.591309\pi$
$500$	0	0
$501$	24.2276	1.08241
$502$	7.95539	0.355067
$503$	−34.3400	−1.53115	−0.765573	−	0.643349i	$-0.777544\pi$
$503$	−34.3400	−1.53115	−0.765573	+	0.643349i	$0.777544\pi$
$504$	−2.73125	−0.121660
$505$	0	0
$506$	−1.69202	−0.0752195
$507$	0.228667	0.0101555
$508$	−59.2127	−2.62714
$509$	9.30260	0.412331	0.206165	−	0.978517i	$-0.433902\pi$
$509$	9.30260	0.412331	0.206165	+	0.978517i	$0.433902\pi$
$510$	0	0
$511$	−2.76271	−0.122215
$512$	9.00538	0.397985
$513$	5.91915	0.261337
$514$	10.3961	0.458553
$515$	0	0
$516$	9.29590	0.409229
$517$	−1.32736	−0.0583770
$518$	8.40581	0.369330
$519$	2.95885	0.129879
$520$	0	0
$521$	22.8780	1.00230	0.501152	−	0.865359i	$-0.332910\pi$
$521$	22.8780	1.00230	0.501152	+	0.865359i	$0.332910\pi$
$522$	19.2271	0.841549
$523$	17.4746	0.764110	0.382055	−	0.924140i	$-0.375216\pi$
$523$	17.4746	0.764110	0.382055	+	0.924140i	$0.375216\pi$
$524$	44.1909	1.93049
$525$	0	0
$526$	26.9215	1.17384
$527$	−4.11960	−0.179453
$528$	−0.819396	−0.0356596
$529$	1.00000	0.0434783
$530$	0	0
$531$	3.00623	0.130459
$532$	3.19806	0.138654
$533$	31.1704	1.35014
$534$	−10.5700	−0.457410
$535$	0	0
$536$	−15.2543	−0.658884
$537$	4.57135	0.197268
$538$	54.5139	2.35026
$539$	−0.753020	−0.0324349
$540$	0	0
$541$	8.64981	0.371884	0.185942	−	0.982561i	$-0.440466\pi$
$541$	8.64981	0.371884	0.185942	+	0.982561i	$0.440466\pi$
$542$	−27.0586	−1.16227
$543$	−2.19269	−0.0940971
$544$	−18.5754	−0.796414
$545$	0	0
$546$	10.9215	0.467399
$547$	−21.4349	−0.916489	−0.458245	−	0.888826i	$-0.651522\pi$
$547$	−21.4349	−0.916489	−0.458245	+	0.888826i	$0.651522\pi$
$548$	15.3773	0.656887
$549$	10.1545	0.433384
$550$	0	0
$551$	−7.74525	−0.329959
$552$	−3.19806	−0.136119
$553$	−11.0761	−0.471002
$554$	24.3207	1.03329
$555$	0	0
$556$	21.9922	0.932678
$557$	33.0471	1.40025	0.700126	−	0.714020i	$-0.253127\pi$
$557$	33.0471	1.40025	0.700126	+	0.714020i	$0.253127\pi$
$558$	3.76271	0.159288
$559$	8.04892	0.340433
$560$	0	0
$561$	2.91292	0.122984
$562$	33.2054	1.40068
$563$	−24.0006	−1.01150	−0.505752	−	0.862679i	$-0.668785\pi$
$563$	−24.0006	−1.01150	−0.505752	+	0.862679i	$0.668785\pi$
$564$	−7.29244	−0.307067
$565$	0	0
$566$	−28.4547	−1.19604
$567$	−4.18060	−0.175569
$568$	−35.1812	−1.47617
$569$	−24.3991	−1.02286	−0.511432	−	0.859324i	$-0.670885\pi$
$569$	−24.3991	−1.02286	−0.511432	+	0.859324i	$0.670885\pi$
$570$	0	0
$571$	−33.4209	−1.39862	−0.699310	−	0.714818i	$-0.746509\pi$
$571$	−33.4209	−1.39862	−0.699310	+	0.714818i	$0.746509\pi$
$572$	8.22414	0.343869
$573$	−1.13083	−0.0472412
$574$	−19.5526	−0.816108
$575$	0	0
$576$	15.1075	0.629480
$577$	−2.31873	−0.0965301	−0.0482650	−	0.998835i	$-0.515369\pi$
$577$	−2.31873	−0.0965301	−0.0482650	+	0.998835i	$0.515369\pi$
$578$	−19.9366	−0.829254
$579$	16.3123	0.677916
$580$	0	0
$581$	13.6256	0.565287
$582$	−1.70841	−0.0708161
$583$	9.22042	0.381871
$584$	−6.51142	−0.269444
$585$	0	0
$586$	−64.2398	−2.65372
$587$	−1.25906	−0.0519670	−0.0259835	−	0.999662i	$-0.508272\pi$
$587$	−1.25906	−0.0519670	−0.0259835	+	0.999662i	$0.508272\pi$
$588$	−4.13706	−0.170610
$589$	−1.51573	−0.0624545
$590$	0	0
$591$	32.5090	1.33724
$592$	−3.00000	−0.123299
$593$	−34.0422	−1.39795	−0.698973	−	0.715148i	$-0.746359\pi$
$593$	−34.0422	−1.39795	−0.698973	+	0.715148i	$0.746359\pi$
$594$	−9.54825	−0.391770
$595$	0	0
$596$	−18.4131	−0.754230
$597$	29.1177	1.19171
$598$	−8.04892	−0.329145
$599$	−11.1631	−0.456114	−0.228057	−	0.973648i	$-0.573237\pi$
$599$	−11.1631	−0.456114	−0.228057	+	0.973648i	$0.573237\pi$
$600$	0	0
$601$	−21.1323	−0.862004	−0.431002	−	0.902351i	$-0.641840\pi$
$601$	−21.1323	−0.862004	−0.431002	+	0.902351i	$0.641840\pi$
$602$	−5.04892	−0.205779
$603$	7.50019	0.305431
$604$	29.4838	1.19968
$605$	0	0
$606$	12.9782	0.527205
$607$	−11.6112	−0.471283	−0.235641	−	0.971840i	$-0.575719\pi$
$607$	−11.6112	−0.471283	−0.235641	+	0.971840i	$0.575719\pi$
$608$	−6.83446	−0.277174
$609$	10.0194	0.406006
$610$	0	0
$611$	−6.31421	−0.255446
$612$	−10.0726	−0.407161
$613$	8.34481	0.337044	0.168522	−	0.985698i	$-0.446101\pi$
$613$	8.34481	0.337044	0.168522	+	0.985698i	$0.446101\pi$
$614$	21.0925	0.851222
$615$	0	0
$616$	−1.77479	−0.0715084
$617$	1.55197	0.0624801	0.0312401	−	0.999512i	$-0.490054\pi$
$617$	1.55197	0.0624801	0.0312401	+	0.999512i	$0.490054\pi$
$618$	−17.7192	−0.712769
$619$	−37.0388	−1.48871	−0.744357	−	0.667782i	$-0.767244\pi$
$619$	−37.0388	−1.48871	−0.744357	+	0.667782i	$0.767244\pi$
$620$	0	0
$621$	5.64310	0.226450
$622$	3.82371	0.153317
$623$	3.46681	0.138895
$624$	−3.89785	−0.156039
$625$	0	0
$626$	71.3226	2.85062
$627$	1.07175	0.0428017
$628$	23.1540	0.923947
$629$	10.6649	0.425236
$630$	0	0
$631$	−9.07547	−0.361289	−0.180644	−	0.983548i	$-0.557818\pi$
$631$	−9.07547	−0.361289	−0.180644	+	0.983548i	$0.557818\pi$
$632$	−26.1051	−1.03841
$633$	−18.6950	−0.743060
$634$	−46.5013	−1.84680
$635$	0	0
$636$	50.6566	2.00867
$637$	−3.58211	−0.141928
$638$	12.4940	0.494641
$639$	17.2978	0.684291
$640$	0	0
$641$	6.99090	0.276124	0.138062	−	0.990424i	$-0.455913\pi$
$641$	6.99090	0.276124	0.138062	+	0.990424i	$0.455913\pi$
$642$	−41.9778	−1.65673
$643$	23.8907	0.942156	0.471078	−	0.882091i	$-0.343865\pi$
$643$	23.8907	0.942156	0.471078	+	0.882091i	$0.343865\pi$
$644$	3.04892	0.120144
$645$	0	0
$646$	6.71917	0.264362
$647$	39.2295	1.54227	0.771136	−	0.636671i	$-0.219689\pi$
$647$	39.2295	1.54227	0.771136	+	0.636671i	$0.219689\pi$
$648$	−9.85325	−0.387072
$649$	1.95348	0.0766806
$650$	0	0
$651$	1.96077	0.0768487
$652$	−15.7125	−0.615348
$653$	35.0151	1.37025	0.685123	−	0.728428i	$-0.259749\pi$
$653$	35.0151	1.37025	0.685123	+	0.728428i	$0.259749\pi$
$654$	−11.1371	−0.435494
$655$	0	0
$656$	6.97823	0.272454
$657$	3.20152	0.124903
$658$	3.96077	0.154407
$659$	−38.7391	−1.50906	−0.754531	−	0.656264i	$-0.772136\pi$
$659$	−38.7391	−1.50906	−0.754531	+	0.656264i	$0.772136\pi$
$660$	0	0
$661$	−13.9769	−0.543638	−0.271819	−	0.962348i	$-0.587625\pi$
$661$	−13.9769	−0.543638	−0.271819	+	0.962348i	$0.587625\pi$
$662$	−0.246980	−0.00959913
$663$	13.8567	0.538150
$664$	32.1142	1.24627
$665$	0	0
$666$	−9.74094	−0.377454
$667$	−7.38404	−0.285911
$668$	−54.4389	−2.10631
$669$	5.34806	0.206768
$670$	0	0
$671$	6.59850	0.254732
$672$	8.84117	0.341055
$673$	1.82430	0.0703216	0.0351608	−	0.999382i	$-0.488806\pi$
$673$	1.82430	0.0703216	0.0351608	+	0.999382i	$0.488806\pi$
$674$	42.8461	1.65037
$675$	0	0
$676$	−0.513811	−0.0197619
$677$	−44.7982	−1.72174	−0.860868	−	0.508829i	$-0.830079\pi$
$677$	−44.7982	−1.72174	−0.860868	+	0.508829i	$0.830079\pi$
$678$	−55.6999	−2.13914
$679$	0.560335	0.0215037
$680$	0	0
$681$	−23.4155	−0.897284
$682$	2.44504	0.0936255
$683$	51.7193	1.97898	0.989492	−	0.144589i	$-0.0461861\pi$
$683$	51.7193	1.97898	0.989492	+	0.144589i	$0.0461861\pi$
$684$	−3.70602	−0.141703
$685$	0	0
$686$	2.24698	0.0857901
$687$	−16.9425	−0.646395
$688$	1.80194	0.0686982
$689$	43.8614	1.67099
$690$	0	0
$691$	−24.9511	−0.949184	−0.474592	−	0.880206i	$-0.657404\pi$
$691$	−24.9511	−0.949184	−0.474592	+	0.880206i	$0.657404\pi$
$692$	−6.64848	−0.252737
$693$	0.872625	0.0331483
$694$	63.0253	2.39241
$695$	0	0
$696$	23.6146	0.895110
$697$	−24.8073	−0.939644
$698$	65.0930	2.46381
$699$	19.3817	0.733081
$700$	0	0
$701$	38.7090	1.46202	0.731009	−	0.682367i	$-0.239050\pi$
$701$	38.7090	1.46202	0.731009	+	0.682367i	$0.239050\pi$
$702$	−45.4209	−1.71430
$703$	3.92394	0.147994
$704$	9.81700	0.369992
$705$	0	0
$706$	−37.2054	−1.40024
$707$	−4.25667	−0.160088
$708$	10.7323	0.403345
$709$	−27.9004	−1.04782	−0.523910	−	0.851774i	$-0.675527\pi$
$709$	−27.9004	−1.04782	−0.523910	+	0.851774i	$0.675527\pi$
$710$	0	0
$711$	12.8353	0.481362
$712$	8.17092	0.306218
$713$	−1.44504	−0.0541172
$714$	−8.69202	−0.325291
$715$	0	0
$716$	−10.2717	−0.383873
$717$	−10.8519	−0.405272
$718$	−33.7144	−1.25821
$719$	−26.8310	−1.00063	−0.500314	−	0.865844i	$-0.666782\pi$
$719$	−26.8310	−1.00063	−0.500314	+	0.865844i	$0.666782\pi$
$720$	0	0
$721$	5.81163	0.216436
$722$	−40.2204	−1.49685
$723$	−11.4485	−0.425774
$724$	4.92692	0.183108
$725$	0	0
$726$	31.8092	1.18055
$727$	19.1172	0.709018	0.354509	−	0.935053i	$-0.384648\pi$
$727$	19.1172	0.709018	0.354509	+	0.935053i	$0.384648\pi$
$728$	−8.44265	−0.312905
$729$	27.8159	1.03022
$730$	0	0
$731$	−6.40581	−0.236928
$732$	36.2519	1.33991
$733$	−49.2170	−1.81787	−0.908935	−	0.416938i	$-0.863103\pi$
$733$	−49.2170	−1.81787	−0.908935	+	0.416938i	$0.863103\pi$
$734$	−6.85623	−0.253068
$735$	0	0
$736$	−6.51573	−0.240173
$737$	4.87369	0.179525
$738$	22.6582	0.834059
$739$	12.7928	0.470592	0.235296	−	0.971924i	$-0.424394\pi$
$739$	12.7928	0.470592	0.235296	+	0.971924i	$0.424394\pi$
$740$	0	0
$741$	5.09831	0.187291
$742$	−27.5133	−1.01005
$743$	−19.2687	−0.706902	−0.353451	−	0.935453i	$-0.614992\pi$
$743$	−19.2687	−0.706902	−0.353451	+	0.935453i	$0.614992\pi$
$744$	4.62133	0.169426
$745$	0	0
$746$	21.5864	0.790335
$747$	−15.7899	−0.577721
$748$	−6.54527	−0.239319
$749$	13.7681	0.503075
$750$	0	0
$751$	−36.1353	−1.31859	−0.659297	−	0.751882i	$-0.729146\pi$
$751$	−36.1353	−1.31859	−0.659297	+	0.751882i	$0.729146\pi$
$752$	−1.41358	−0.0515481
$753$	−4.80407	−0.175070
$754$	59.4336	2.16444
$755$	0	0
$756$	17.2054	0.625753
$757$	37.0471	1.34650	0.673250	−	0.739415i	$-0.264898\pi$
$757$	37.0471	1.34650	0.673250	+	0.739415i	$0.264898\pi$
$758$	6.76271	0.245633
$759$	1.02177	0.0370879
$760$	0	0
$761$	−32.8842	−1.19205	−0.596026	−	0.802965i	$-0.703255\pi$
$761$	−32.8842	−1.19205	−0.596026	+	0.802965i	$0.703255\pi$
$762$	59.2127	2.14505
$763$	3.65279	0.132240
$764$	2.54096	0.0919286
$765$	0	0
$766$	−49.5109	−1.78890
$767$	9.29265	0.335538
$768$	14.2024	0.512484
$769$	38.2258	1.37846	0.689229	−	0.724544i	$-0.257949\pi$
$769$	38.2258	1.37846	0.689229	+	0.724544i	$0.257949\pi$
$770$	0	0
$771$	−6.27796	−0.226095
$772$	−36.6534	−1.31918
$773$	−53.4470	−1.92235	−0.961177	−	0.275933i	$-0.911013\pi$
$773$	−53.4470	−1.92235	−0.961177	+	0.275933i	$0.911013\pi$
$774$	5.85086	0.210305
$775$	0	0
$776$	1.32065	0.0474086
$777$	−5.07606	−0.182103
$778$	32.2392	1.15583
$779$	−9.12737	−0.327022
$780$	0	0
$781$	11.2403	0.402209
$782$	6.40581	0.229071
$783$	−41.6689	−1.48913
$784$	−0.801938	−0.0286406
$785$	0	0
$786$	−44.1909	−1.57624
$787$	32.6568	1.16409	0.582045	−	0.813156i	$-0.302253\pi$
$787$	32.6568	1.16409	0.582045	+	0.813156i	$0.302253\pi$
$788$	−73.0471	−2.60220
$789$	−16.2573	−0.578774
$790$	0	0
$791$	18.2687	0.649562
$792$	2.05669	0.0730812
$793$	31.3889	1.11465
$794$	12.2416	0.434438
$795$	0	0
$796$	−65.4268	−2.31899
$797$	−9.12498	−0.323223	−0.161612	−	0.986854i	$-0.551669\pi$
$797$	−9.12498	−0.323223	−0.161612	+	0.986854i	$0.551669\pi$
$798$	−3.19806	−0.113210
$799$	5.02523	0.177780
$800$	0	0
$801$	−4.01746	−0.141950
$802$	−10.5187	−0.371429
$803$	2.08038	0.0734149
$804$	26.7759	0.944312
$805$	0	0
$806$	11.6310	0.409685
$807$	−32.9196	−1.15883
$808$	−10.0325	−0.352943
$809$	−29.2127	−1.02706	−0.513531	−	0.858071i	$-0.671663\pi$
$809$	−29.2127	−1.02706	−0.513531	+	0.858071i	$0.671663\pi$
$810$	0	0
$811$	−27.4892	−0.965275	−0.482638	−	0.875820i	$-0.660321\pi$
$811$	−27.4892	−0.965275	−0.482638	+	0.875820i	$0.660321\pi$
$812$	−22.5133	−0.790063
$813$	16.3400	0.573070
$814$	−6.32975	−0.221858
$815$	0	0
$816$	3.10215	0.108597
$817$	−2.35690	−0.0824573
$818$	68.4161	2.39211
$819$	4.15106	0.145050
$820$	0	0
$821$	−43.6249	−1.52252	−0.761260	−	0.648447i	$-0.775419\pi$
$821$	−43.6249	−1.52252	−0.761260	+	0.648447i	$0.775419\pi$
$822$	−15.3773	−0.536346
$823$	37.9154	1.32165	0.660824	−	0.750541i	$-0.270207\pi$
$823$	37.9154	1.32165	0.660824	+	0.750541i	$0.270207\pi$
$824$	13.6974	0.477171
$825$	0	0
$826$	−5.82908	−0.202820
$827$	32.6765	1.13627	0.568136	−	0.822934i	$-0.307665\pi$
$827$	32.6765	1.13627	0.568136	+	0.822934i	$0.307665\pi$
$828$	−3.53319	−0.122787
$829$	53.0998	1.84423	0.922115	−	0.386915i	$-0.126459\pi$
$829$	53.0998	1.84423	0.922115	+	0.386915i	$0.126459\pi$
$830$	0	0
$831$	−14.6866	−0.509474
$832$	46.6993	1.61901
$833$	2.85086	0.0987763
$834$	−21.9922	−0.761529
$835$	0	0
$836$	−2.40821	−0.0832896
$837$	−8.15452	−0.281862
$838$	−13.0737	−0.451622
$839$	−46.2881	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$839$	−46.2881	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$840$	0	0
$841$	25.5241	0.880141
$842$	−39.9071	−1.37529
$843$	−20.0519	−0.690624
$844$	42.0073	1.44595
$845$	0	0
$846$	−4.58987	−0.157803
$847$	−10.4330	−0.358481
$848$	9.81940	0.337199
$849$	17.1831	0.589723
$850$	0	0
$851$	3.74094	0.128238
$852$	61.7536	2.11564
$853$	31.2006	1.06829	0.534144	−	0.845394i	$-0.320634\pi$
$853$	31.2006	1.06829	0.534144	+	0.845394i	$0.320634\pi$
$854$	−19.6896	−0.673765
$855$	0	0
$856$	32.4499	1.10912
$857$	20.6950	0.706928	0.353464	−	0.935448i	$-0.385004\pi$
$857$	20.6950	0.706928	0.353464	+	0.935448i	$0.385004\pi$
$858$	−8.22414	−0.280768
$859$	−52.3618	−1.78656	−0.893281	−	0.449499i	$-0.851602\pi$
$859$	−52.3618	−1.78656	−0.893281	+	0.449499i	$0.851602\pi$
$860$	0	0
$861$	11.8073	0.402392
$862$	−15.0030	−0.511004
$863$	−50.4760	−1.71822	−0.859112	−	0.511788i	$-0.828983\pi$
$863$	−50.4760	−1.71822	−0.859112	+	0.511788i	$0.828983\pi$
$864$	−36.7689	−1.25090
$865$	0	0
$866$	73.7881	2.50742
$867$	12.0392	0.408874
$868$	−4.40581	−0.149543
$869$	8.34050	0.282932
$870$	0	0
$871$	23.1841	0.785562
$872$	8.60925	0.291546
$873$	−0.649335	−0.0219767
$874$	2.35690	0.0797232
$875$	0	0
$876$	11.4295	0.386167
$877$	−0.157769	−0.00532747	−0.00266373	−	0.999996i	$-0.500848\pi$
$877$	−0.157769	−0.00532747	−0.00266373	+	0.999996i	$0.500848\pi$
$878$	−1.40283	−0.0473432
$879$	38.7928	1.30845
$880$	0	0
$881$	−37.4185	−1.26066	−0.630330	−	0.776327i	$-0.717081\pi$
$881$	−37.4185	−1.26066	−0.630330	+	0.776327i	$0.717081\pi$
$882$	−2.60388	−0.0876770
$883$	−19.1675	−0.645036	−0.322518	−	0.946563i	$-0.604529\pi$
$883$	−19.1675	−0.645036	−0.322518	+	0.946563i	$0.604529\pi$
$884$	−31.1357	−1.04721
$885$	0	0
$886$	41.2693	1.38647
$887$	9.89141	0.332121	0.166061	−	0.986116i	$-0.446895\pi$
$887$	9.89141	0.332121	0.166061	+	0.986116i	$0.446895\pi$
$888$	−11.9638	−0.401477
$889$	−19.4209	−0.651355
$890$	0	0
$891$	3.14808	0.105465
$892$	−12.0170	−0.402358
$893$	1.84894	0.0618723
$894$	18.4131	0.615826
$895$	0	0
$896$	−16.2620	−0.543277
$897$	4.86054	0.162289
$898$	41.2301	1.37587
$899$	10.6703	0.355873
$900$	0	0
$901$	−34.9075	−1.16294
$902$	14.7235	0.490238
$903$	3.04892	0.101462
$904$	43.0575	1.43207
$905$	0	0
$906$	−29.4838	−0.979534
$907$	0.0163935	0.000544336 0	0.000272168	−	1.00000i	$-0.499913\pi$
$907$	0.0163935	0.000544336 0	0.000272168		1.00000i	$0.499913\pi$
$908$	52.6142	1.74606
$909$	4.93277	0.163610
$910$	0	0
$911$	−12.8817	−0.426791	−0.213395	−	0.976966i	$-0.568452\pi$
$911$	−12.8817	−0.426791	−0.213395	+	0.976966i	$0.568452\pi$
$912$	1.14138	0.0377947
$913$	−10.2604	−0.339569
$914$	−60.8708	−2.01343
$915$	0	0
$916$	38.0694	1.25785
$917$	14.4940	0.478633
$918$	36.1487	1.19308
$919$	−21.3321	−0.703682	−0.351841	−	0.936060i	$-0.614444\pi$
$919$	−21.3321	−0.703682	−0.351841	+	0.936060i	$0.614444\pi$
$920$	0	0
$921$	−12.7372	−0.419706
$922$	93.6335	3.08366
$923$	53.4698	1.75998
$924$	3.11529	0.102486
$925$	0	0
$926$	−8.25667	−0.271331
$927$	−6.73471	−0.221197
$928$	48.1124	1.57937
$929$	38.0411	1.24809	0.624045	−	0.781389i	$-0.285488\pi$
$929$	38.0411	1.24809	0.624045	+	0.781389i	$0.285488\pi$
$930$	0	0
$931$	1.04892	0.0343769
$932$	−43.5502	−1.42653
$933$	−2.30904	−0.0755947
$934$	32.9584	1.07843
$935$	0	0
$936$	9.78363	0.319788
$937$	−4.75196	−0.155240	−0.0776198	−	0.996983i	$-0.524732\pi$
$937$	−4.75196	−0.155240	−0.0776198	+	0.996983i	$0.524732\pi$
$938$	−14.5429	−0.474842
$939$	−43.0700	−1.40553
$940$	0	0
$941$	−8.23596	−0.268485	−0.134242	−	0.990949i	$-0.542860\pi$
$941$	−8.23596	−0.268485	−0.134242	+	0.990949i	$0.542860\pi$
$942$	−23.1540	−0.754400
$943$	−8.70171	−0.283367
$944$	2.08038	0.0677105
$945$	0	0
$946$	3.80194	0.123612
$947$	54.6437	1.77568	0.887841	−	0.460151i	$-0.152205\pi$
$947$	54.6437	1.77568	0.887841	+	0.460151i	$0.152205\pi$
$948$	45.8224	1.48824
$949$	9.89631	0.321248
$950$	0	0
$951$	28.0810	0.910588
$952$	6.71917	0.217770
$953$	28.2204	0.914149	0.457075	−	0.889428i	$-0.348897\pi$
$953$	28.2204	0.914149	0.457075	+	0.889428i	$0.348897\pi$
$954$	31.8834	1.03226
$955$	0	0
$956$	24.3840	0.788636
$957$	−7.54480	−0.243889
$958$	−24.7017	−0.798076
$959$	5.04354	0.162864
$960$	0	0
$961$	−28.9119	−0.932640
$962$	−30.1105	−0.970802
$963$	−15.9549	−0.514140
$964$	25.7245	0.828532
$965$	0	0
$966$	−3.04892	−0.0980973
$967$	−36.7724	−1.18252	−0.591260	−	0.806481i	$-0.701369\pi$
$967$	−36.7724	−1.18252	−0.591260	+	0.806481i	$0.701369\pi$
$968$	−24.5894	−0.790333
$969$	−4.05754	−0.130347
$970$	0	0
$971$	30.1420	0.967302	0.483651	−	0.875261i	$-0.339310\pi$
$971$	30.1420	0.967302	0.483651	+	0.875261i	$0.339310\pi$
$972$	−34.3207	−1.10084
$973$	7.21313	0.231242
$974$	49.3159	1.58018
$975$	0	0
$976$	7.02715	0.224933
$977$	25.0568	0.801638	0.400819	−	0.916157i	$-0.368726\pi$
$977$	25.0568	0.801638	0.400819	+	0.916157i	$0.368726\pi$
$978$	15.7125	0.502429
$979$	−2.61058	−0.0834345
$980$	0	0
$981$	−4.23298	−0.135149
$982$	42.5230	1.35696
$983$	−40.5284	−1.29266	−0.646328	−	0.763060i	$-0.723696\pi$
$983$	−40.5284	−1.29266	−0.646328	+	0.763060i	$0.723696\pi$
$984$	27.8286	0.887144
$985$	0	0
$986$	−47.3008	−1.50636
$987$	−2.39181	−0.0761322
$988$	−11.4558	−0.364457
$989$	−2.24698	−0.0714498
$990$	0	0
$991$	13.2228	0.420037	0.210018	−	0.977697i	$-0.432648\pi$
$991$	13.2228	0.420037	0.210018	+	0.977697i	$0.432648\pi$
$992$	9.41550	0.298942
$993$	0.149145	0.00473297
$994$	−33.5405	−1.06384
$995$	0	0
$996$	−56.3702	−1.78616
$997$	−39.3793	−1.24715	−0.623577	−	0.781762i	$-0.714321\pi$
$997$	−39.3793	−1.24715	−0.623577	+	0.781762i	$0.714321\pi$
$998$	−28.4034	−0.899095
$999$	21.1105	0.667907

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.24698 q^{2} -1.35690 q^{3} +3.04892 q^{4} -3.04892 q^{6} +1.00000 q^{7} +2.35690 q^{8} -1.15883 q^{9} +O(q^{10})\)	\(q+2.24698 q^{2} -1.35690 q^{3} +3.04892 q^{4} -3.04892 q^{6} +1.00000 q^{7} +2.35690 q^{8} -1.15883 q^{9} -0.753020 q^{11} -4.13706 q^{12} -3.58211 q^{13} +2.24698 q^{14} -0.801938 q^{16} +2.85086 q^{17} -2.60388 q^{18} +1.04892 q^{19} -1.35690 q^{21} -1.69202 q^{22} +1.00000 q^{23} -3.19806 q^{24} -8.04892 q^{26} +5.64310 q^{27} +3.04892 q^{28} -7.38404 q^{29} -1.44504 q^{31} -6.51573 q^{32} +1.02177 q^{33} +6.40581 q^{34} -3.53319 q^{36} +3.74094 q^{37} +2.35690 q^{38} +4.86054 q^{39} -8.70171 q^{41} -3.04892 q^{42} -2.24698 q^{43} -2.29590 q^{44} +2.24698 q^{46} +1.76271 q^{47} +1.08815 q^{48} +1.00000 q^{49} -3.86831 q^{51} -10.9215 q^{52} -12.2446 q^{53} +12.6799 q^{54} +2.35690 q^{56} -1.42327 q^{57} -16.5918 q^{58} -2.59419 q^{59} -8.76271 q^{61} -3.24698 q^{62} -1.15883 q^{63} -13.0368 q^{64} +2.29590 q^{66} -6.47219 q^{67} +8.69202 q^{68} -1.35690 q^{69} -14.9269 q^{71} -2.73125 q^{72} -2.76271 q^{73} +8.40581 q^{74} +3.19806 q^{76} -0.753020 q^{77} +10.9215 q^{78} -11.0761 q^{79} -4.18060 q^{81} -19.5526 q^{82} +13.6256 q^{83} -4.13706 q^{84} -5.04892 q^{86} +10.0194 q^{87} -1.77479 q^{88} +3.46681 q^{89} -3.58211 q^{91} +3.04892 q^{92} +1.96077 q^{93} +3.96077 q^{94} +8.84117 q^{96} +0.560335 q^{97} +2.24698 q^{98} +0.872625 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 3 q^{7} + 3 q^{8} + 5 q^{9}+O(q^{10}) \) 3 * q + 2 * q^2 + 3 * q^7 + 3 * q^8 + 5 * q^9	\( 3 q + 2 q^{2} + 3 q^{7} + 3 q^{8} + 5 q^{9} - 7 q^{11} - 7 q^{12} - 5 q^{13} + 2 q^{14} + 2 q^{16} - 5 q^{17} + q^{18} - 6 q^{19} + 3 q^{23} - 14 q^{24} - 15 q^{26} + 21 q^{27} - 12 q^{29} - 4 q^{31} - 7 q^{32} + 6 q^{34} - 14 q^{36} - 3 q^{37} + 3 q^{38} - 21 q^{39} + q^{41} - 2 q^{43} + 7 q^{44} + 2 q^{46} - 12 q^{47} + 7 q^{48} + 3 q^{49} - 14 q^{51} - 7 q^{52} + 9 q^{53} + 14 q^{54} + 3 q^{56} - 7 q^{57} - 22 q^{58} - 21 q^{59} - 9 q^{61} - 5 q^{62} + 5 q^{63} - 11 q^{64} - 7 q^{66} - 13 q^{67} + 21 q^{68} - 16 q^{71} - 16 q^{72} + 9 q^{73} + 12 q^{74} + 14 q^{76} - 7 q^{77} + 7 q^{78} - 18 q^{79} - q^{81} - 18 q^{82} + 29 q^{83} - 7 q^{84} - 6 q^{86} - 14 q^{87} - 7 q^{88} + 7 q^{89} - 5 q^{91} - 7 q^{93} - q^{94} + 35 q^{96} - q^{97} + 2 q^{98} - 14 q^{99}+O(q^{100}) \) 3 * q + 2 * q^2 + 3 * q^7 + 3 * q^8 + 5 * q^9 - 7 * q^11 - 7 * q^12 - 5 * q^13 + 2 * q^14 + 2 * q^16 - 5 * q^17 + q^18 - 6 * q^19 + 3 * q^23 - 14 * q^24 - 15 * q^26 + 21 * q^27 - 12 * q^29 - 4 * q^31 - 7 * q^32 + 6 * q^34 - 14 * q^36 - 3 * q^37 + 3 * q^38 - 21 * q^39 + q^41 - 2 * q^43 + 7 * q^44 + 2 * q^46 - 12 * q^47 + 7 * q^48 + 3 * q^49 - 14 * q^51 - 7 * q^52 + 9 * q^53 + 14 * q^54 + 3 * q^56 - 7 * q^57 - 22 * q^58 - 21 * q^59 - 9 * q^61 - 5 * q^62 + 5 * q^63 - 11 * q^64 - 7 * q^66 - 13 * q^67 + 21 * q^68 - 16 * q^71 - 16 * q^72 + 9 * q^73 + 12 * q^74 + 14 * q^76 - 7 * q^77 + 7 * q^78 - 18 * q^79 - q^81 - 18 * q^82 + 29 * q^83 - 7 * q^84 - 6 * q^86 - 14 * q^87 - 7 * q^88 + 7 * q^89 - 5 * q^91 - 7 * q^93 - q^94 + 35 * q^96 - q^97 + 2 * q^98 - 14 * q^99

Introduction

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Database

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