LMFDB - Embedded newform 4225.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.67513 q^{2} +0.481194 q^{3} +5.15633 q^{4} +1.28726 q^{6} +0.806063 q^{7} +8.44358 q^{8} -2.76845 q^{9} +3.67513 q^{11} +2.48119 q^{12} +2.15633 q^{14} +12.2750 q^{16} +1.35026 q^{17} -7.40597 q^{18} +1.67513 q^{19} +0.387873 q^{21} +9.83146 q^{22} -6.48119 q^{23} +4.06300 q^{24} -2.77575 q^{27} +4.15633 q^{28} +2.41819 q^{29} +5.28726 q^{31} +15.9502 q^{32} +1.76845 q^{33} +3.61213 q^{34} -14.2750 q^{36} +3.76845 q^{37} +4.48119 q^{38} +8.31265 q^{41} +1.03761 q^{42} +6.79384 q^{43} +18.9502 q^{44} -17.3380 q^{46} +3.19394 q^{47} +5.90668 q^{48} -6.35026 q^{49} +0.649738 q^{51} -5.73813 q^{53} -7.42548 q^{54} +6.80606 q^{56} +0.806063 q^{57} +6.46898 q^{58} -5.98778 q^{59} -1.76845 q^{61} +14.1441 q^{62} -2.23155 q^{63} +18.1187 q^{64} +4.73084 q^{66} +9.89446 q^{67} +6.96239 q^{68} -3.11871 q^{69} -8.56230 q^{71} -23.3757 q^{72} -11.7685 q^{73} +10.0811 q^{74} +8.63752 q^{76} +2.96239 q^{77} -2.26187 q^{79} +6.96968 q^{81} +22.2374 q^{82} +3.84367 q^{83} +2.00000 q^{84} +18.1744 q^{86} +1.16362 q^{87} +31.0313 q^{88} -2.77575 q^{89} -33.4191 q^{92} +2.54420 q^{93} +8.54420 q^{94} +7.67513 q^{96} -1.87399 q^{97} -16.9878 q^{98} -10.1744 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 4 q^{3} + 5 q^{4} - 2 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} + 6 q^{11} + 2 q^{12} - 4 q^{14} + 5 q^{16} - 6 q^{17} + 5 q^{18} + 2 q^{21} + 14 q^{22} - 14 q^{23} + 8 q^{24} - 10 q^{27} + 2 q^{28}+ \cdots + 8 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 4 * q^3 + 5 * q^4 - 2 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9 + 6 * q^11 + 2 * q^12 - 4 * q^14 + 5 * q^16 - 6 * q^17 + 5 * q^18 + 2 * q^21 + 14 * q^22 - 14 * q^23 + 8 * q^24 - 10 * q^27 + 2 * q^28 + 6 * q^29 + 10 * q^31 + 11 * q^32 - 6 * q^33 + 10 * q^34 - 11 * q^36 + 8 * q^38 + 4 * q^41 + 14 * q^42 - 6 * q^43 + 20 * q^44 - 16 * q^46 + 10 * q^47 + 24 * q^48 - 9 * q^49 + 12 * q^51 - 8 * q^53 - 34 * q^54 + 20 * q^56 + 2 * q^57 - 12 * q^58 + 8 * q^59 + 6 * q^61 + 6 * q^62 - 18 * q^63 + 33 * q^64 - 8 * q^66 + 10 * q^67 + 10 * q^68 + 12 * q^69 + 12 * q^71 - 45 * q^72 - 24 * q^73 - 2 * q^74 + 10 * q^76 - 2 * q^77 - 16 * q^79 + 23 * q^81 + 24 * q^82 + 22 * q^83 + 6 * q^84 + 16 * q^86 + 6 * q^87 + 24 * q^88 - 10 * q^89 - 32 * q^92 - 2 * q^93 + 16 * q^94 + 18 * q^96 - 14 * q^97 - 25 * q^98 + 8 * q^99

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 $

$2$	2.67513	1.89160	0.945802	−	0.324745i	$-0.105279\pi$
$2$	2.67513	1.89160	0.945802	+	0.324745i	$0.105279\pi$
$3$	0.481194	0.277818	0.138909	−	0.990305i	$-0.455641\pi$
$3$	0.481194	0.277818	0.138909	+	0.990305i	$0.455641\pi$
$4$	5.15633	2.57816
$5$	0	0
$5$	0	0
$6$	1.28726	0.525521
$7$	0.806063	0.304663	0.152332	−	0.988329i	$-0.451322\pi$
$7$	0.806063	0.304663	0.152332	+	0.988329i	$0.451322\pi$
$8$	8.44358	2.98526
$9$	−2.76845	−0.922817
$10$	0	0
$11$	3.67513	1.10809	0.554047	−	0.832486i	$-0.313083\pi$
$11$	3.67513	1.10809	0.554047	+	0.832486i	$0.313083\pi$
$12$	2.48119	0.716259
$13$	0	0
$13$	0	0
$14$	2.15633	0.576302
$15$	0	0
$16$	12.2750	3.06876
$17$	1.35026	0.327487	0.163743	−	0.986503i	$-0.447643\pi$
$17$	1.35026	0.327487	0.163743	+	0.986503i	$0.447643\pi$
$18$	−7.40597	−1.74560
$19$	1.67513	0.384301	0.192151	−	0.981365i	$-0.438454\pi$
$19$	1.67513	0.384301	0.192151	+	0.981365i	$0.438454\pi$
$20$	0	0
$21$	0.387873	0.0846409
$22$	9.83146	2.09607
$23$	−6.48119	−1.35142	−0.675711	−	0.737166i	$-0.736163\pi$
$23$	−6.48119	−1.35142	−0.675711	+	0.737166i	$0.736163\pi$
$24$	4.06300	0.829357
$25$	0	0
$26$	0	0
$27$	−2.77575	−0.534193
$28$	4.15633	0.785472
$29$	2.41819	0.449047	0.224523	−	0.974469i	$-0.427917\pi$
$29$	2.41819	0.449047	0.224523	+	0.974469i	$0.427917\pi$
$30$	0	0
$31$	5.28726	0.949620	0.474810	−	0.880088i	$-0.342517\pi$
$31$	5.28726	0.949620	0.474810	+	0.880088i	$0.342517\pi$
$32$	15.9502	2.81962
$33$	1.76845	0.307848
$34$	3.61213	0.619475
$35$	0	0
$36$	−14.2750	−2.37917
$37$	3.76845	0.619530	0.309765	−	0.950813i	$-0.399750\pi$
$37$	3.76845	0.619530	0.309765	+	0.950813i	$0.399750\pi$
$38$	4.48119	0.726946
$39$	0	0
$40$	0	0
$41$	8.31265	1.29822	0.649109	−	0.760695i	$-0.275142\pi$
$41$	8.31265	1.29822	0.649109	+	0.760695i	$0.275142\pi$
$42$	1.03761	0.160107
$43$	6.79384	1.03605	0.518026	−	0.855365i	$-0.326667\pi$
$43$	6.79384	1.03605	0.518026	+	0.855365i	$0.326667\pi$
$44$	18.9502	2.85685
$45$	0	0
$46$	−17.3380	−2.55635
$47$	3.19394	0.465884	0.232942	−	0.972491i	$-0.425165\pi$
$47$	3.19394	0.465884	0.232942	+	0.972491i	$0.425165\pi$
$48$	5.90668	0.852556
$49$	−6.35026	−0.907180
$50$	0	0
$51$	0.649738	0.0909816
$52$	0	0
$53$	−5.73813	−0.788193	−0.394097	−	0.919069i	$-0.628943\pi$
$53$	−5.73813	−0.788193	−0.394097	+	0.919069i	$0.628943\pi$
$54$	−7.42548	−1.01048
$55$	0	0
$56$	6.80606	0.909498
$57$	0.806063	0.106766
$58$	6.46898	0.849418
$59$	−5.98778	−0.779543	−0.389771	−	0.920912i	$-0.627446\pi$
$59$	−5.98778	−0.779543	−0.389771	+	0.920912i	$0.627446\pi$
$60$	0	0
$61$	−1.76845	−0.226427	−0.113214	−	0.993571i	$-0.536114\pi$
$61$	−1.76845	−0.226427	−0.113214	+	0.993571i	$0.536114\pi$
$62$	14.1441	1.79630
$63$	−2.23155	−0.281149
$64$	18.1187	2.26484
$65$	0	0
$66$	4.73084	0.582326
$67$	9.89446	1.20880	0.604400	−	0.796681i	$-0.293413\pi$
$67$	9.89446	1.20880	0.604400	+	0.796681i	$0.293413\pi$
$68$	6.96239	0.844314
$69$	−3.11871	−0.375449
$70$	0	0
$71$	−8.56230	−1.01616	−0.508079	−	0.861311i	$-0.669644\pi$
$71$	−8.56230	−1.01616	−0.508079	+	0.861311i	$0.669644\pi$
$72$	−23.3757	−2.75485
$73$	−11.7685	−1.37739	−0.688697	−	0.725050i	$-0.741817\pi$
$73$	−11.7685	−1.37739	−0.688697	+	0.725050i	$0.741817\pi$
$74$	10.0811	1.17190
$75$	0	0
$76$	8.63752	0.990791
$77$	2.96239	0.337596
$78$	0	0
$79$	−2.26187	−0.254480	−0.127240	−	0.991872i	$-0.540612\pi$
$79$	−2.26187	−0.254480	−0.127240	+	0.991872i	$0.540612\pi$
$80$	0	0
$81$	6.96968	0.774409
$82$	22.2374	2.45571
$83$	3.84367	0.421898	0.210949	−	0.977497i	$-0.432345\pi$
$83$	3.84367	0.421898	0.210949	+	0.977497i	$0.432345\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	18.1744	1.95980
$87$	1.16362	0.124753
$88$	31.0313	3.30794
$89$	−2.77575	−0.294229	−0.147114	−	0.989120i	$-0.546999\pi$
$89$	−2.77575	−0.294229	−0.147114	+	0.989120i	$0.546999\pi$
$90$	0	0
$91$	0	0
$92$	−33.4191	−3.48419
$93$	2.54420	0.263821
$94$	8.54420	0.881267
$95$	0	0
$96$	7.67513	0.783340
$97$	−1.87399	−0.190275	−0.0951375	−	0.995464i	$-0.530329\pi$
$97$	−1.87399	−0.190275	−0.0951375	+	0.995464i	$0.530329\pi$
$98$	−16.9878	−1.71603
$99$	−10.1744	−1.02257
$100$	0	0
$101$	10.4993	1.04472	0.522359	−	0.852725i	$-0.325052\pi$
$101$	10.4993	1.04472	0.522359	+	0.852725i	$0.325052\pi$
$102$	1.73813	0.172101
$103$	−15.3684	−1.51429	−0.757145	−	0.653247i	$-0.773406\pi$
$103$	−15.3684	−1.51429	−0.757145	+	0.653247i	$0.773406\pi$
$104$	0	0
$105$	0	0
$106$	−15.3503	−1.49095
$107$	−11.1309	−1.07607	−0.538034	−	0.842923i	$-0.680833\pi$
$107$	−11.1309	−1.07607	−0.538034	+	0.842923i	$0.680833\pi$
$108$	−14.3127	−1.37724
$109$	−9.58769	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$109$	−9.58769	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$110$	0	0
$111$	1.81336	0.172116
$112$	9.89446	0.934939
$113$	−0.574515	−0.0540459	−0.0270229	−	0.999635i	$-0.508603\pi$
$113$	−0.574515	−0.0540459	−0.0270229	+	0.999635i	$0.508603\pi$
$114$	2.15633	0.201958
$115$	0	0
$116$	12.4690	1.15772
$117$	0	0
$118$	−16.0181	−1.47459
$119$	1.08840	0.0997732
$120$	0	0
$121$	2.50659	0.227872
$122$	−4.73084	−0.428310
$123$	4.00000	0.360668
$124$	27.2628	2.44827
$125$	0	0
$126$	−5.96968	−0.531822
$127$	4.29455	0.381080	0.190540	−	0.981679i	$-0.438976\pi$
$127$	4.29455	0.381080	0.190540	+	0.981679i	$0.438976\pi$
$128$	16.5696	1.46456
$129$	3.26916	0.287833
$130$	0	0
$131$	−0.836381	−0.0730749	−0.0365375	−	0.999332i	$-0.511633\pi$
$131$	−0.836381	−0.0730749	−0.0365375	+	0.999332i	$0.511633\pi$
$132$	9.11871	0.793682
$133$	1.35026	0.117083
$134$	26.4690	2.28657
$135$	0	0
$136$	11.4010	0.977632
$137$	−14.9380	−1.27624	−0.638118	−	0.769939i	$-0.720287\pi$
$137$	−14.9380	−1.27624	−0.638118	+	0.769939i	$0.720287\pi$
$138$	−8.34297	−0.710201
$139$	−8.43866	−0.715758	−0.357879	−	0.933768i	$-0.616500\pi$
$139$	−8.43866	−0.715758	−0.357879	+	0.933768i	$0.616500\pi$
$140$	0	0
$141$	1.53690	0.129431
$142$	−22.9053	−1.92217
$143$	0	0
$144$	−33.9829	−2.83190
$145$	0	0
$146$	−31.4821	−2.60548
$147$	−3.05571	−0.252031
$148$	19.4314	1.59725
$149$	11.3503	0.929850	0.464925	−	0.885350i	$-0.346081\pi$
$149$	11.3503	0.929850	0.464925	+	0.885350i	$0.346081\pi$
$150$	0	0
$151$	−13.9878	−1.13831	−0.569155	−	0.822230i	$-0.692729\pi$
$151$	−13.9878	−1.13831	−0.569155	+	0.822230i	$0.692729\pi$
$152$	14.1441	1.14724
$153$	−3.73813	−0.302210
$154$	7.92478	0.638597
$155$	0	0
$156$	0	0
$157$	−2.77575	−0.221529	−0.110764	−	0.993847i	$-0.535330\pi$
$157$	−2.77575	−0.221529	−0.110764	+	0.993847i	$0.535330\pi$
$158$	−6.05079	−0.481375
$159$	−2.76116	−0.218974
$160$	0	0
$161$	−5.22425	−0.411729
$162$	18.6448	1.46487
$163$	2.23155	0.174788	0.0873942	−	0.996174i	$-0.472146\pi$
$163$	2.23155	0.174788	0.0873942	+	0.996174i	$0.472146\pi$
$164$	42.8627	3.34702
$165$	0	0
$166$	10.2823	0.798064
$167$	−15.6932	−1.21438	−0.607189	−	0.794557i	$-0.707703\pi$
$167$	−15.6932	−1.21438	−0.607189	+	0.794557i	$0.707703\pi$
$168$	3.27504	0.252675
$169$	0	0
$170$	0	0
$171$	−4.63752	−0.354640
$172$	35.0313	2.67111
$173$	25.5877	1.94540	0.972698	−	0.232075i	$-0.0745513\pi$
$173$	25.5877	1.94540	0.972698	+	0.232075i	$0.0745513\pi$
$174$	3.11283	0.235983
$175$	0	0
$176$	45.1124	3.40047
$177$	−2.88129	−0.216571
$178$	−7.42548	−0.556564
$179$	12.1260	0.906340	0.453170	−	0.891424i	$-0.350293\pi$
$179$	12.1260	0.906340	0.453170	+	0.891424i	$0.350293\pi$
$180$	0	0
$181$	−2.73084	−0.202982	−0.101491	−	0.994836i	$-0.532361\pi$
$181$	−2.73084	−0.202982	−0.101491	+	0.994836i	$0.532361\pi$
$182$	0	0
$183$	−0.850969	−0.0629054
$184$	−54.7245	−4.03434
$185$	0	0
$186$	6.80606	0.499045
$187$	4.96239	0.362886
$188$	16.4690	1.20112
$189$	−2.23743	−0.162749
$190$	0	0
$191$	20.6253	1.49239	0.746197	−	0.665725i	$-0.231878\pi$
$191$	20.6253	1.49239	0.746197	+	0.665725i	$0.231878\pi$
$192$	8.71862	0.629212
$193$	−21.7889	−1.56840	−0.784200	−	0.620508i	$-0.786927\pi$
$193$	−21.7889	−1.56840	−0.784200	+	0.620508i	$0.786927\pi$
$194$	−5.01317	−0.359925
$195$	0	0
$196$	−32.7440	−2.33886
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	−27.2179	−1.93429
$199$	−16.7513	−1.18747	−0.593734	−	0.804661i	$-0.702347\pi$
$199$	−16.7513	−1.18747	−0.593734	+	0.804661i	$0.702347\pi$
$200$	0	0
$201$	4.76116	0.335826
$202$	28.0870	1.97619
$203$	1.94921	0.136808
$204$	3.35026	0.234565
$205$	0	0
$206$	−41.1124	−2.86443
$207$	17.9429	1.24712
$208$	0	0
$209$	6.15633	0.425842
$210$	0	0
$211$	−4.90175	−0.337451	−0.168725	−	0.985663i	$-0.553965\pi$
$211$	−4.90175	−0.337451	−0.168725	+	0.985663i	$0.553965\pi$
$212$	−29.5877	−2.03209
$213$	−4.12013	−0.282307
$214$	−29.7767	−2.03549
$215$	0	0
$216$	−23.4372	−1.59470
$217$	4.26187	0.289314
$218$	−25.6483	−1.73712
$219$	−5.66291	−0.382664
$220$	0	0
$221$	0	0
$222$	4.85097	0.325576
$223$	24.9076	1.66794	0.833969	−	0.551811i	$-0.186063\pi$
$223$	24.9076	1.66794	0.833969	+	0.551811i	$0.186063\pi$
$224$	12.8568	0.859034
$225$	0	0
$226$	−1.53690	−0.102233
$227$	9.95509	0.660743	0.330371	−	0.943851i	$-0.392826\pi$
$227$	9.95509	0.660743	0.330371	+	0.943851i	$0.392826\pi$
$228$	4.15633	0.275259
$229$	−5.35026	−0.353555	−0.176778	−	0.984251i	$-0.556567\pi$
$229$	−5.35026	−0.353555	−0.176778	+	0.984251i	$0.556567\pi$
$230$	0	0
$231$	1.42548	0.0937900
$232$	20.4182	1.34052
$233$	10.7612	0.704987	0.352493	−	0.935814i	$-0.385334\pi$
$233$	10.7612	0.704987	0.352493	+	0.935814i	$0.385334\pi$
$234$	0	0
$235$	0	0
$236$	−30.8749	−2.00979
$237$	−1.08840	−0.0706990
$238$	2.91160	0.188731
$239$	11.8618	0.767274	0.383637	−	0.923484i	$-0.374671\pi$
$239$	11.8618	0.767274	0.383637	+	0.923484i	$0.374671\pi$
$240$	0	0
$241$	28.6253	1.84392	0.921959	−	0.387288i	$-0.126588\pi$
$241$	28.6253	1.84392	0.921959	+	0.387288i	$0.126588\pi$
$242$	6.70545	0.431043
$243$	11.6810	0.749337
$244$	−9.11871	−0.583766
$245$	0	0
$246$	10.7005	0.682240
$247$	0	0
$248$	44.6434	2.83486
$249$	1.84955	0.117211
$250$	0	0
$251$	−19.3865	−1.22366	−0.611831	−	0.790988i	$-0.709567\pi$
$251$	−19.3865	−1.22366	−0.611831	+	0.790988i	$0.709567\pi$
$252$	−11.5066	−0.724847
$253$	−23.8192	−1.49750
$254$	11.4885	0.720852
$255$	0	0
$256$	8.08840	0.505525
$257$	−22.8627	−1.42614	−0.713069	−	0.701094i	$-0.752695\pi$
$257$	−22.8627	−1.42614	−0.713069	+	0.701094i	$0.752695\pi$
$258$	8.74543	0.544467
$259$	3.03761	0.188748
$260$	0	0
$261$	−6.69464	−0.414388
$262$	−2.23743	−0.138229
$263$	−21.8822	−1.34932	−0.674658	−	0.738130i	$-0.735709\pi$
$263$	−21.8822	−1.34932	−0.674658	+	0.738130i	$0.735709\pi$
$264$	14.9321	0.919005
$265$	0	0
$266$	3.61213	0.221474
$267$	−1.33567	−0.0817419
$268$	51.0191	3.11648
$269$	22.7513	1.38717	0.693586	−	0.720374i	$-0.256030\pi$
$269$	22.7513	1.38717	0.693586	+	0.720374i	$0.256030\pi$
$270$	0	0
$271$	−0.123638	−0.00751049	−0.00375525	−	0.999993i	$-0.501195\pi$
$271$	−0.123638	−0.00751049	−0.00375525	+	0.999993i	$0.501195\pi$
$272$	16.5745	1.00498
$273$	0	0
$274$	−39.9610	−2.41413
$275$	0	0
$276$	−16.0811	−0.967969
$277$	15.3503	0.922308	0.461154	−	0.887320i	$-0.347436\pi$
$277$	15.3503	0.922308	0.461154	+	0.887320i	$0.347436\pi$
$278$	−22.5745	−1.35393
$279$	−14.6375	−0.876325
$280$	0	0
$281$	−13.9248	−0.830683	−0.415341	−	0.909666i	$-0.636338\pi$
$281$	−13.9248	−0.830683	−0.415341	+	0.909666i	$0.636338\pi$
$282$	4.11142	0.244831
$283$	−20.3815	−1.21156	−0.605778	−	0.795634i	$-0.707138\pi$
$283$	−20.3815	−1.21156	−0.605778	+	0.795634i	$0.707138\pi$
$284$	−44.1500	−2.61982
$285$	0	0
$286$	0	0
$287$	6.70052	0.395519
$288$	−44.1573	−2.60199
$289$	−15.1768	−0.892753
$290$	0	0
$291$	−0.901754	−0.0528618
$292$	−60.6820	−3.55114
$293$	5.38058	0.314337	0.157168	−	0.987572i	$-0.449763\pi$
$293$	5.38058	0.314337	0.157168	+	0.987572i	$0.449763\pi$
$294$	−8.17442	−0.476742
$295$	0	0
$296$	31.8192	1.84946
$297$	−10.2012	−0.591935
$298$	30.3634	1.75891
$299$	0	0
$300$	0	0
$301$	5.47627	0.315647
$302$	−37.4191	−2.15323
$303$	5.05220	0.290241
$304$	20.5623	1.17933
$305$	0	0
$306$	−10.0000	−0.571662
$307$	−19.1695	−1.09406	−0.547031	−	0.837113i	$-0.684242\pi$
$307$	−19.1695	−1.09406	−0.547031	+	0.837113i	$0.684242\pi$
$308$	15.2750	0.870376
$309$	−7.39517	−0.420696
$310$	0	0
$311$	−25.2506	−1.43183	−0.715915	−	0.698187i	$-0.753990\pi$
$311$	−25.2506	−1.43183	−0.715915	+	0.698187i	$0.753990\pi$
$312$	0	0
$313$	−2.81194	−0.158940	−0.0794702	−	0.996837i	$-0.525323\pi$
$313$	−2.81194	−0.158940	−0.0794702	+	0.996837i	$0.525323\pi$
$314$	−7.42548	−0.419044
$315$	0	0
$316$	−11.6629	−0.656090
$317$	−23.7685	−1.33497	−0.667485	−	0.744624i	$-0.732629\pi$
$317$	−23.7685	−1.33497	−0.667485	+	0.744624i	$0.732629\pi$
$318$	−7.38646	−0.414212
$319$	8.88717	0.497586
$320$	0	0
$321$	−5.35614	−0.298951
$322$	−13.9756	−0.778828
$323$	2.26187	0.125854
$324$	35.9380	1.99655
$325$	0	0
$326$	5.96968	0.330630
$327$	−4.61354	−0.255129
$328$	70.1886	3.87551
$329$	2.57452	0.141938
$330$	0	0
$331$	11.8011	0.648649	0.324325	−	0.945946i	$-0.394863\pi$
$331$	11.8011	0.648649	0.324325	+	0.945946i	$0.394863\pi$
$332$	19.8192	1.08772
$333$	−10.4328	−0.571713
$334$	−41.9814	−2.29712
$335$	0	0
$336$	4.76116	0.259742
$337$	−16.1114	−0.877645	−0.438822	−	0.898574i	$-0.644604\pi$
$337$	−16.1114	−0.877645	−0.438822	+	0.898574i	$0.644604\pi$
$338$	0	0
$339$	−0.276454	−0.0150149
$340$	0	0
$341$	19.4314	1.05227
$342$	−12.4060	−0.670838
$343$	−10.7612	−0.581048
$344$	57.3644	3.09288
$345$	0	0
$346$	68.4504	3.67992
$347$	−27.4944	−1.47598	−0.737988	−	0.674814i	$-0.764224\pi$
$347$	−27.4944	−1.47598	−0.737988	+	0.674814i	$0.764224\pi$
$348$	6.00000	0.321634
$349$	17.6023	0.942228	0.471114	−	0.882072i	$-0.343852\pi$
$349$	17.6023	0.942228	0.471114	+	0.882072i	$0.343852\pi$
$350$	0	0
$351$	0	0
$352$	58.6190	3.12440
$353$	15.7685	0.839270	0.419635	−	0.907693i	$-0.362158\pi$
$353$	15.7685	0.839270	0.419635	+	0.907693i	$0.362158\pi$
$354$	−7.70782	−0.409666
$355$	0	0
$356$	−14.3127	−0.758569
$357$	0.523730	0.0277187
$358$	32.4387	1.71444
$359$	14.8242	0.782389	0.391195	−	0.920308i	$-0.372062\pi$
$359$	14.8242	0.782389	0.391195	+	0.920308i	$0.372062\pi$
$360$	0	0
$361$	−16.1939	−0.852312
$362$	−7.30536	−0.383961
$363$	1.20616	0.0633067
$364$	0	0
$365$	0	0
$366$	−2.27645	−0.118992
$367$	27.0313	1.41102	0.705510	−	0.708700i	$-0.250718\pi$
$367$	27.0313	1.41102	0.705510	+	0.708700i	$0.250718\pi$
$368$	−79.5569	−4.14719
$369$	−23.0132	−1.19802
$370$	0	0
$371$	−4.62530	−0.240134
$372$	13.1187	0.680174
$373$	−12.9525	−0.670657	−0.335329	−	0.942101i	$-0.608847\pi$
$373$	−12.9525	−0.670657	−0.335329	+	0.942101i	$0.608847\pi$
$374$	13.2750	0.686436
$375$	0	0
$376$	26.9683	1.39078
$377$	0	0
$378$	−5.98541	−0.307856
$379$	30.2858	1.55568	0.777840	−	0.628463i	$-0.216316\pi$
$379$	30.2858	1.55568	0.777840	+	0.628463i	$0.216316\pi$
$380$	0	0
$381$	2.06651	0.105871
$382$	55.1754	2.82302
$383$	−21.0943	−1.07787	−0.538934	−	0.842348i	$-0.681173\pi$
$383$	−21.0943	−1.07787	−0.538934	+	0.842348i	$0.681173\pi$
$384$	7.97319	0.406880
$385$	0	0
$386$	−58.2882	−2.96679
$387$	−18.8084	−0.956086
$388$	−9.66291	−0.490560
$389$	−6.77575	−0.343544	−0.171772	−	0.985137i	$-0.554949\pi$
$389$	−6.77575	−0.343544	−0.171772	+	0.985137i	$0.554949\pi$
$390$	0	0
$391$	−8.75131	−0.442573
$392$	−53.6190	−2.70817
$393$	−0.402462	−0.0203015
$394$	5.35026	0.269542
$395$	0	0
$396$	−52.4626	−2.63635
$397$	10.4690	0.525423	0.262711	−	0.964874i	$-0.415383\pi$
$397$	10.4690	0.525423	0.262711	+	0.964874i	$0.415383\pi$
$398$	−44.8119	−2.24622
$399$	0.649738	0.0325276
$400$	0	0
$401$	−5.01317	−0.250346	−0.125173	−	0.992135i	$-0.539949\pi$
$401$	−5.01317	−0.250346	−0.125173	+	0.992135i	$0.539949\pi$
$402$	12.7367	0.635250
$403$	0	0
$404$	54.1378	2.69345
$405$	0	0
$406$	5.21440	0.258787
$407$	13.8496	0.686497
$408$	5.48612	0.271603
$409$	14.3879	0.711435	0.355717	−	0.934594i	$-0.384237\pi$
$409$	14.3879	0.711435	0.355717	+	0.934594i	$0.384237\pi$
$410$	0	0
$411$	−7.18806	−0.354561
$412$	−79.2443	−3.90408
$413$	−4.82653	−0.237498
$414$	47.9995	2.35905
$415$	0	0
$416$	0	0
$417$	−4.06063	−0.198850
$418$	16.4690	0.805524
$419$	17.4617	0.853059	0.426529	−	0.904474i	$-0.359736\pi$
$419$	17.4617	0.853059	0.426529	+	0.904474i	$0.359736\pi$
$420$	0	0
$421$	−2.88717	−0.140712	−0.0703559	−	0.997522i	$-0.522414\pi$
$421$	−2.88717	−0.140712	−0.0703559	+	0.997522i	$0.522414\pi$
$422$	−13.1128	−0.638323
$423$	−8.84226	−0.429925
$424$	−48.4504	−2.35296
$425$	0	0
$426$	−11.0219	−0.534012
$427$	−1.42548	−0.0689840
$428$	−57.3947	−2.77428
$429$	0	0
$430$	0	0
$431$	−0.889535	−0.0428474	−0.0214237	−	0.999770i	$-0.506820\pi$
$431$	−0.889535	−0.0428474	−0.0214237	+	0.999770i	$0.506820\pi$
$432$	−34.0724	−1.63931
$433$	25.2506	1.21347	0.606733	−	0.794906i	$-0.292480\pi$
$433$	25.2506	1.21347	0.606733	+	0.794906i	$0.292480\pi$
$434$	11.4010	0.547268
$435$	0	0
$436$	−49.4372	−2.36761
$437$	−10.8568	−0.519354
$438$	−15.1490	−0.723849
$439$	28.8119	1.37512	0.687560	−	0.726128i	$-0.258682\pi$
$439$	28.8119	1.37512	0.687560	+	0.726128i	$0.258682\pi$
$440$	0	0
$441$	17.5804	0.837162
$442$	0	0
$443$	−36.9805	−1.75700	−0.878498	−	0.477746i	$-0.841454\pi$
$443$	−36.9805	−1.75700	−0.878498	+	0.477746i	$0.841454\pi$
$444$	9.35026	0.443744
$445$	0	0
$446$	66.6312	3.15508
$447$	5.46168	0.258329
$448$	14.6048	0.690013
$449$	12.6859	0.598686	0.299343	−	0.954146i	$-0.403232\pi$
$449$	12.6859	0.598686	0.299343	+	0.954146i	$0.403232\pi$
$450$	0	0
$451$	30.5501	1.43855
$452$	−2.96239	−0.139339
$453$	−6.73084	−0.316242
$454$	26.6312	1.24986
$455$	0	0
$456$	6.80606	0.318723
$457$	−25.0494	−1.17176	−0.585880	−	0.810398i	$-0.699251\pi$
$457$	−25.0494	−1.17176	−0.585880	+	0.810398i	$0.699251\pi$
$458$	−14.3127	−0.668786
$459$	−3.74798	−0.174941
$460$	0	0
$461$	36.8872	1.71801	0.859003	−	0.511970i	$-0.171084\pi$
$461$	36.8872	1.71801	0.859003	+	0.511970i	$0.171084\pi$
$462$	3.81336	0.177413
$463$	−39.0191	−1.81337	−0.906685	−	0.421809i	$-0.861395\pi$
$463$	−39.0191	−1.81337	−0.906685	+	0.421809i	$0.861395\pi$
$464$	29.6834	1.37802
$465$	0	0
$466$	28.7875	1.33356
$467$	−32.7694	−1.51639	−0.758194	−	0.652029i	$-0.773918\pi$
$467$	−32.7694	−1.51639	−0.758194	+	0.652029i	$0.773918\pi$
$468$	0	0
$469$	7.97556	0.368277
$470$	0	0
$471$	−1.33567	−0.0615446
$472$	−50.5583	−2.32714
$473$	24.9683	1.14804
$474$	−2.91160	−0.133734
$475$	0	0
$476$	5.61213	0.257231
$477$	15.8858	0.727359
$478$	31.7318	1.45138
$479$	−16.8749	−0.771036	−0.385518	−	0.922700i	$-0.625977\pi$
$479$	−16.8749	−0.771036	−0.385518	+	0.922700i	$0.625977\pi$
$480$	0	0
$481$	0	0
$482$	76.5764	3.48796
$483$	−2.51388	−0.114386
$484$	12.9248	0.587490
$485$	0	0
$486$	31.2482	1.41745
$487$	−9.24472	−0.418918	−0.209459	−	0.977817i	$-0.567170\pi$
$487$	−9.24472	−0.418918	−0.209459	+	0.977817i	$0.567170\pi$
$488$	−14.9321	−0.675943
$489$	1.07381	0.0485593
$490$	0	0
$491$	−25.7499	−1.16208	−0.581038	−	0.813876i	$-0.697353\pi$
$491$	−25.7499	−1.16208	−0.581038	+	0.813876i	$0.697353\pi$
$492$	20.6253	0.929860
$493$	3.26519	0.147057
$494$	0	0
$495$	0	0
$496$	64.9013	2.91415
$497$	−6.90175	−0.309586
$498$	4.94780	0.221716
$499$	27.7015	1.24009	0.620044	−	0.784567i	$-0.287115\pi$
$499$	27.7015	1.24009	0.620044	+	0.784567i	$0.287115\pi$
$500$	0	0
$501$	−7.55149	−0.337376
$502$	−51.8613	−2.31468
$503$	−2.35519	−0.105013	−0.0525063	−	0.998621i	$-0.516721\pi$
$503$	−2.35519	−0.105013	−0.0525063	+	0.998621i	$0.516721\pi$
$504$	−18.8423	−0.839301
$505$	0	0
$506$	−63.7196	−2.83268
$507$	0	0
$508$	22.1441	0.982486
$509$	−21.5125	−0.953523	−0.476762	−	0.879033i	$-0.658190\pi$
$509$	−21.5125	−0.953523	−0.476762	+	0.879033i	$0.658190\pi$
$510$	0	0
$511$	−9.48612	−0.419641
$512$	−11.5017	−0.508306
$513$	−4.64974	−0.205291
$514$	−61.1608	−2.69769
$515$	0	0
$516$	16.8568	0.742081
$517$	11.7381	0.516243
$518$	8.12601	0.357036
$519$	12.3127	0.540465
$520$	0	0
$521$	37.7440	1.65360	0.826798	−	0.562499i	$-0.190160\pi$
$521$	37.7440	1.65360	0.826798	+	0.562499i	$0.190160\pi$
$522$	−17.9090	−0.783858
$523$	−23.7416	−1.03815	−0.519075	−	0.854729i	$-0.673723\pi$
$523$	−23.7416	−1.03815	−0.519075	+	0.854729i	$0.673723\pi$
$524$	−4.31265	−0.188399
$525$	0	0
$526$	−58.5379	−2.55237
$527$	7.13918	0.310988
$528$	21.7078	0.944712
$529$	19.0059	0.826343
$530$	0	0
$531$	16.5769	0.719376
$532$	6.96239	0.301858
$533$	0	0
$534$	−3.57310	−0.154623
$535$	0	0
$536$	83.5447	3.60858
$537$	5.83497	0.251797
$538$	60.8627	2.62398
$539$	−23.3380	−1.00524
$540$	0	0
$541$	−13.0376	−0.560531	−0.280265	−	0.959923i	$-0.590422\pi$
$541$	−13.0376	−0.560531	−0.280265	+	0.959923i	$0.590422\pi$
$542$	−0.330749	−0.0142069
$543$	−1.31406	−0.0563919
$544$	21.5369	0.923387
$545$	0	0
$546$	0	0
$547$	−8.43041	−0.360458	−0.180229	−	0.983625i	$-0.557684\pi$
$547$	−8.43041	−0.360458	−0.180229	+	0.983625i	$0.557684\pi$
$548$	−77.0249	−3.29034
$549$	4.89587	0.208951
$550$	0	0
$551$	4.05079	0.172569
$552$	−26.3331	−1.12081
$553$	−1.82321	−0.0775306
$554$	41.0640	1.74464
$555$	0	0
$556$	−43.5125	−1.84534
$557$	13.6932	0.580201	0.290100	−	0.956996i	$-0.406311\pi$
$557$	13.6932	0.580201	0.290100	+	0.956996i	$0.406311\pi$
$558$	−39.1573	−1.65766
$559$	0	0
$560$	0	0
$561$	2.38787	0.100816
$562$	−37.2506	−1.57132
$563$	8.86907	0.373787	0.186893	−	0.982380i	$-0.440158\pi$
$563$	8.86907	0.373787	0.186893	+	0.982380i	$0.440158\pi$
$564$	7.92478	0.333693
$565$	0	0
$566$	−54.5233	−2.29178
$567$	5.61801	0.235934
$568$	−72.2965	−3.03349
$569$	−32.7816	−1.37428	−0.687139	−	0.726526i	$-0.741134\pi$
$569$	−32.7816	−1.37428	−0.687139	+	0.726526i	$0.741134\pi$
$570$	0	0
$571$	40.2882	1.68601	0.843005	−	0.537906i	$-0.180785\pi$
$571$	40.2882	1.68601	0.843005	+	0.537906i	$0.180785\pi$
$572$	0	0
$573$	9.92478	0.414614
$574$	17.9248	0.748166
$575$	0	0
$576$	−50.1608	−2.09003
$577$	28.8568	1.20133	0.600663	−	0.799502i	$-0.294903\pi$
$577$	28.8568	1.20133	0.600663	+	0.799502i	$0.294903\pi$
$578$	−40.5999	−1.68873
$579$	−10.4847	−0.435729
$580$	0	0
$581$	3.09825	0.128537
$582$	−2.41231	−0.0999935
$583$	−21.0884	−0.873392
$584$	−99.3679	−4.11187
$585$	0	0
$586$	14.3938	0.594600
$587$	41.6786	1.72026	0.860131	−	0.510074i	$-0.170382\pi$
$587$	41.6786	1.72026	0.860131	+	0.510074i	$0.170382\pi$
$588$	−15.7562	−0.649776
$589$	8.85685	0.364940
$590$	0	0
$591$	0.962389	0.0395874
$592$	46.2579	1.90119
$593$	22.4993	0.923935	0.461968	−	0.886897i	$-0.347144\pi$
$593$	22.4993	0.923935	0.461968	+	0.886897i	$0.347144\pi$
$594$	−27.2896	−1.11971
$595$	0	0
$596$	58.5256	2.39730
$597$	−8.06063	−0.329900
$598$	0	0
$599$	4.15045	0.169583	0.0847913	−	0.996399i	$-0.472978\pi$
$599$	4.15045	0.169583	0.0847913	+	0.996399i	$0.472978\pi$
$600$	0	0
$601$	27.9248	1.13908	0.569538	−	0.821965i	$-0.307122\pi$
$601$	27.9248	1.13908	0.569538	+	0.821965i	$0.307122\pi$
$602$	14.6497	0.597079
$603$	−27.3923	−1.11550
$604$	−72.1255	−2.93475
$605$	0	0
$606$	13.5153	0.549021
$607$	−8.19489	−0.332620	−0.166310	−	0.986073i	$-0.553185\pi$
$607$	−8.19489	−0.332620	−0.166310	+	0.986073i	$0.553185\pi$
$608$	26.7186	1.08358
$609$	0.937951	0.0380077
$610$	0	0
$611$	0	0
$612$	−19.2750	−0.779147
$613$	33.1392	1.33848	0.669239	−	0.743047i	$-0.266620\pi$
$613$	33.1392	1.33848	0.669239	+	0.743047i	$0.266620\pi$
$614$	−51.2809	−2.06953
$615$	0	0
$616$	25.0132	1.00781
$617$	−29.0132	−1.16803	−0.584013	−	0.811744i	$-0.698518\pi$
$617$	−29.0132	−1.16803	−0.584013	+	0.811744i	$0.698518\pi$
$618$	−19.7830	−0.795791
$619$	−12.2134	−0.490900	−0.245450	−	0.969409i	$-0.578936\pi$
$619$	−12.2134	−0.490900	−0.245450	+	0.969409i	$0.578936\pi$
$620$	0	0
$621$	17.9902	0.721920
$622$	−67.5487	−2.70845
$623$	−2.23743	−0.0896406
$624$	0	0
$625$	0	0
$626$	−7.52232	−0.300652
$627$	2.96239	0.118306
$628$	−14.3127	−0.571137
$629$	5.08840	0.202888
$630$	0	0
$631$	1.22188	0.0486424	0.0243212	−	0.999704i	$-0.492258\pi$
$631$	1.22188	0.0486424	0.0243212	+	0.999704i	$0.492258\pi$
$632$	−19.0982	−0.759687
$633$	−2.35870	−0.0937498
$634$	−63.5837	−2.52523
$635$	0	0
$636$	−14.2374	−0.564551
$637$	0	0
$638$	23.7743	0.941235
$639$	23.7043	0.937728
$640$	0	0
$641$	−22.1016	−0.872960	−0.436480	−	0.899714i	$-0.643775\pi$
$641$	−22.1016	−0.872960	−0.436480	+	0.899714i	$0.643775\pi$
$642$	−14.3284	−0.565496
$643$	11.6688	0.460172	0.230086	−	0.973170i	$-0.426099\pi$
$643$	11.6688	0.460172	0.230086	+	0.973170i	$0.426099\pi$
$644$	−26.9380	−1.06150
$645$	0	0
$646$	6.05079	0.238065
$647$	−11.9575	−0.470096	−0.235048	−	0.971984i	$-0.575525\pi$
$647$	−11.9575	−0.470096	−0.235048	+	0.971984i	$0.575525\pi$
$648$	58.8491	2.31181
$649$	−22.0059	−0.863806
$650$	0	0
$651$	2.05079	0.0803766
$652$	11.5066	0.450633
$653$	10.9986	0.430408	0.215204	−	0.976569i	$-0.430958\pi$
$653$	10.9986	0.430408	0.215204	+	0.976569i	$0.430958\pi$
$654$	−12.3418	−0.482604
$655$	0	0
$656$	102.038	3.98392
$657$	32.5804	1.27108
$658$	6.88717	0.268490
$659$	−2.63989	−0.102835	−0.0514177	−	0.998677i	$-0.516374\pi$
$659$	−2.63989	−0.102835	−0.0514177	+	0.998677i	$0.516374\pi$
$660$	0	0
$661$	−18.3028	−0.711896	−0.355948	−	0.934506i	$-0.615842\pi$
$661$	−18.3028	−0.711896	−0.355948	+	0.934506i	$0.615842\pi$
$662$	31.5696	1.22699
$663$	0	0
$664$	32.4544	1.25947
$665$	0	0
$666$	−27.9090	−1.08145
$667$	−15.6728	−0.606852
$668$	−80.9194	−3.13087
$669$	11.9854	0.463383
$670$	0	0
$671$	−6.49929	−0.250902
$672$	6.18664	0.238655
$673$	−6.71037	−0.258666	−0.129333	−	0.991601i	$-0.541284\pi$
$673$	−6.71037	−0.258666	−0.129333	+	0.991601i	$0.541284\pi$
$674$	−43.1002	−1.66016
$675$	0	0
$676$	0	0
$677$	1.57593	0.0605679	0.0302840	−	0.999541i	$-0.490359\pi$
$677$	1.57593	0.0605679	0.0302840	+	0.999541i	$0.490359\pi$
$678$	−0.739549	−0.0284022
$679$	−1.51056	−0.0579698
$680$	0	0
$681$	4.79033	0.183566
$682$	51.9814	1.99047
$683$	15.1939	0.581380	0.290690	−	0.956817i	$-0.406115\pi$
$683$	15.1939	0.581380	0.290690	+	0.956817i	$0.406115\pi$
$684$	−23.9126	−0.914320
$685$	0	0
$686$	−28.7875	−1.09911
$687$	−2.57452	−0.0982239
$688$	83.3947	3.17939
$689$	0	0
$690$	0	0
$691$	18.7127	0.711866	0.355933	−	0.934511i	$-0.384163\pi$
$691$	18.7127	0.711866	0.355933	+	0.934511i	$0.384163\pi$
$692$	131.938	5.01555
$693$	−8.20123	−0.311539
$694$	−73.5510	−2.79196
$695$	0	0
$696$	9.82512	0.372420
$697$	11.2243	0.425149
$698$	47.0884	1.78232
$699$	5.17821	0.195858
$700$	0	0
$701$	24.3028	0.917904	0.458952	−	0.888461i	$-0.348225\pi$
$701$	24.3028	0.917904	0.458952	+	0.888461i	$0.348225\pi$
$702$	0	0
$703$	6.31265	0.238086
$704$	66.5886	2.50965
$705$	0	0
$706$	42.1827	1.58757
$707$	8.46310	0.318287
$708$	−14.8568	−0.558355
$709$	9.66291	0.362898	0.181449	−	0.983400i	$-0.441921\pi$
$709$	9.66291	0.362898	0.181449	+	0.983400i	$0.441921\pi$
$710$	0	0
$711$	6.26187	0.234838
$712$	−23.4372	−0.878348
$713$	−34.2677	−1.28334
$714$	1.40105	0.0524329
$715$	0	0
$716$	62.5256	2.33669
$717$	5.70782	0.213162
$718$	39.6566	1.47997
$719$	28.4142	1.05967	0.529836	−	0.848100i	$-0.322254\pi$
$719$	28.4142	1.05967	0.529836	+	0.848100i	$0.322254\pi$
$720$	0	0
$721$	−12.3879	−0.461349
$722$	−43.3209	−1.61224
$723$	13.7743	0.512273
$724$	−14.0811	−0.523320
$725$	0	0
$726$	3.22662	0.119751
$727$	34.8545	1.29268	0.646341	−	0.763049i	$-0.276299\pi$
$727$	34.8545	1.29268	0.646341	+	0.763049i	$0.276299\pi$
$728$	0	0
$729$	−15.2882	−0.566230
$730$	0	0
$731$	9.17347	0.339293
$732$	−4.38787	−0.162180
$733$	6.25202	0.230923	0.115462	−	0.993312i	$-0.463165\pi$
$733$	6.25202	0.230923	0.115462	+	0.993312i	$0.463165\pi$
$734$	72.3122	2.66909
$735$	0	0
$736$	−103.376	−3.81050
$737$	36.3634	1.33946
$738$	−61.5633	−2.26617
$739$	32.0846	1.18025	0.590126	−	0.807311i	$-0.299078\pi$
$739$	32.0846	1.18025	0.590126	+	0.807311i	$0.299078\pi$
$740$	0	0
$741$	0	0
$742$	−12.3733	−0.454238
$743$	−30.5442	−1.12056	−0.560279	−	0.828304i	$-0.689306\pi$
$743$	−30.5442	−1.12056	−0.560279	+	0.828304i	$0.689306\pi$
$744$	21.4821	0.787574
$745$	0	0
$746$	−34.6497	−1.26862
$747$	−10.6410	−0.389335
$748$	25.5877	0.935579
$749$	−8.97224	−0.327838
$750$	0	0
$751$	28.1622	1.02765	0.513827	−	0.857894i	$-0.328227\pi$
$751$	28.1622	1.02765	0.513827	+	0.857894i	$0.328227\pi$
$752$	39.2057	1.42968
$753$	−9.32865	−0.339955
$754$	0	0
$755$	0	0
$756$	−11.5369	−0.419593
$757$	−35.4109	−1.28703	−0.643515	−	0.765433i	$-0.722525\pi$
$757$	−35.4109	−1.28703	−0.643515	+	0.765433i	$0.722525\pi$
$758$	81.0186	2.94273
$759$	−11.4617	−0.416033
$760$	0	0
$761$	19.2388	0.697407	0.348704	−	0.937233i	$-0.386622\pi$
$761$	19.2388	0.697407	0.348704	+	0.937233i	$0.386622\pi$
$762$	5.52820	0.200265
$763$	−7.72829	−0.279783
$764$	106.351	3.84764
$765$	0	0
$766$	−56.4299	−2.03890
$767$	0	0
$768$	3.89209	0.140444
$769$	−48.9643	−1.76570	−0.882849	−	0.469657i	$-0.844378\pi$
$769$	−48.9643	−1.76570	−0.882849	+	0.469657i	$0.844378\pi$
$770$	0	0
$771$	−11.0014	−0.396206
$772$	−112.351	−4.04359
$773$	46.1681	1.66055	0.830275	−	0.557354i	$-0.188183\pi$
$773$	46.1681	1.66055	0.830275	+	0.557354i	$0.188183\pi$
$774$	−50.3150	−1.80854
$775$	0	0
$776$	−15.8232	−0.568020
$777$	1.46168	0.0524375
$778$	−18.1260	−0.649849
$779$	13.9248	0.498907
$780$	0	0
$781$	−31.4676	−1.12600
$782$	−23.4109	−0.837172
$783$	−6.71228	−0.239877
$784$	−77.9497	−2.78392
$785$	0	0
$786$	−1.07664	−0.0384024
$787$	−22.6458	−0.807234	−0.403617	−	0.914928i	$-0.632247\pi$
$787$	−22.6458	−0.807234	−0.403617	+	0.914928i	$0.632247\pi$
$788$	10.3127	0.367373
$789$	−10.5296	−0.374864
$790$	0	0
$791$	−0.463096	−0.0164658
$792$	−85.9086	−3.05263
$793$	0	0
$794$	28.0059	0.993891
$795$	0	0
$796$	−86.3752	−3.06149
$797$	−8.23743	−0.291785	−0.145892	−	0.989300i	$-0.546605\pi$
$797$	−8.23743	−0.291785	−0.145892	+	0.989300i	$0.546605\pi$
$798$	1.73813	0.0615293
$799$	4.31265	0.152571
$800$	0	0
$801$	7.68452	0.271519
$802$	−13.4109	−0.473555
$803$	−43.2506	−1.52628
$804$	24.5501	0.865814
$805$	0	0
$806$	0	0
$807$	10.9478	0.385381
$808$	88.6516	3.11875
$809$	44.1319	1.55159	0.775797	−	0.630982i	$-0.217348\pi$
$809$	44.1319	1.55159	0.775797	+	0.630982i	$0.217348\pi$
$810$	0	0
$811$	−22.6883	−0.796694	−0.398347	−	0.917235i	$-0.630416\pi$
$811$	−22.6883	−0.796694	−0.398347	+	0.917235i	$0.630416\pi$
$812$	10.0508	0.352713
$813$	−0.0594941	−0.00208655
$814$	37.0494	1.29858
$815$	0	0
$816$	7.97556	0.279201
$817$	11.3806	0.398156
$818$	38.4894	1.34575
$819$	0	0
$820$	0	0
$821$	50.2736	1.75456	0.877281	−	0.479978i	$-0.159355\pi$
$821$	50.2736	1.75456	0.877281	+	0.479978i	$0.159355\pi$
$822$	−19.2290	−0.670688
$823$	−5.13093	−0.178853	−0.0894265	−	0.995993i	$-0.528503\pi$
$823$	−5.13093	−0.178853	−0.0894265	+	0.995993i	$0.528503\pi$
$824$	−129.764	−4.52054
$825$	0	0
$826$	−12.9116	−0.449252
$827$	18.6946	0.650076	0.325038	−	0.945701i	$-0.394623\pi$
$827$	18.6946	0.650076	0.325038	+	0.945701i	$0.394623\pi$
$828$	92.5193	3.21527
$829$	−3.44121	−0.119518	−0.0597591	−	0.998213i	$-0.519033\pi$
$829$	−3.44121	−0.119518	−0.0597591	+	0.998213i	$0.519033\pi$
$830$	0	0
$831$	7.38646	0.256233
$832$	0	0
$833$	−8.57452	−0.297089
$834$	−10.8627	−0.376146
$835$	0	0
$836$	31.7440	1.09789
$837$	−14.6761	−0.507280
$838$	46.7123	1.61365
$839$	52.6248	1.81681	0.908406	−	0.418090i	$-0.137300\pi$
$839$	52.6248	1.81681	0.908406	+	0.418090i	$0.137300\pi$
$840$	0	0
$841$	−23.1524	−0.798357
$842$	−7.72355	−0.266171
$843$	−6.70052	−0.230778
$844$	−25.2750	−0.870003
$845$	0	0
$846$	−23.6542	−0.813248
$847$	2.02047	0.0694241
$848$	−70.4358	−2.41878
$849$	−9.80748	−0.336592
$850$	0	0
$851$	−24.4241	−0.837246
$852$	−21.2447	−0.727832
$853$	−6.31853	−0.216342	−0.108171	−	0.994132i	$-0.534499\pi$
$853$	−6.31853	−0.216342	−0.108171	+	0.994132i	$0.534499\pi$
$854$	−3.81336	−0.130490
$855$	0	0
$856$	−93.9850	−3.21234
$857$	0.775746	0.0264990	0.0132495	−	0.999912i	$-0.495782\pi$
$857$	0.775746	0.0264990	0.0132495	+	0.999912i	$0.495782\pi$
$858$	0	0
$859$	3.24869	0.110844	0.0554220	−	0.998463i	$-0.482350\pi$
$859$	3.24869	0.110844	0.0554220	+	0.998463i	$0.482350\pi$
$860$	0	0
$861$	3.22425	0.109882
$862$	−2.37962	−0.0810503
$863$	−19.9208	−0.678112	−0.339056	−	0.940766i	$-0.610108\pi$
$863$	−19.9208	−0.678112	−0.339056	+	0.940766i	$0.610108\pi$
$864$	−44.2736	−1.50622
$865$	0	0
$866$	67.5487	2.29540
$867$	−7.30299	−0.248022
$868$	21.9756	0.745899
$869$	−8.31265	−0.281987
$870$	0	0
$871$	0	0
$872$	−80.9544	−2.74146
$873$	5.18806	0.175589
$874$	−29.0435	−0.982411
$875$	0	0
$876$	−29.1998	−0.986570
$877$	22.1378	0.747539	0.373770	−	0.927522i	$-0.378065\pi$
$877$	22.1378	0.747539	0.373770	+	0.927522i	$0.378065\pi$
$878$	77.0757	2.60118
$879$	2.58910	0.0873283
$880$	0	0
$881$	2.23155	0.0751828	0.0375914	−	0.999293i	$-0.488031\pi$
$881$	2.23155	0.0751828	0.0375914	+	0.999293i	$0.488031\pi$
$882$	47.0299	1.58358
$883$	4.30440	0.144855	0.0724273	−	0.997374i	$-0.476925\pi$
$883$	4.30440	0.144855	0.0724273	+	0.997374i	$0.476925\pi$
$884$	0	0
$885$	0	0
$886$	−98.9276	−3.32354
$887$	−15.9330	−0.534979	−0.267489	−	0.963561i	$-0.586194\pi$
$887$	−15.9330	−0.534979	−0.267489	+	0.963561i	$0.586194\pi$
$888$	15.3112	0.513811
$889$	3.46168	0.116101
$890$	0	0
$891$	25.6145	0.858118
$892$	128.432	4.30022
$893$	5.35026	0.179040
$894$	14.6107	0.488655
$895$	0	0
$896$	13.3561	0.446197
$897$	0	0
$898$	33.9365	1.13248
$899$	12.7856	0.426423
$900$	0	0
$901$	−7.74798	−0.258123
$902$	81.7255	2.72116
$903$	2.63515	0.0876923
$904$	−4.85097	−0.161341
$905$	0	0
$906$	−18.0059	−0.598205
$907$	51.9086	1.72360	0.861798	−	0.507251i	$-0.169338\pi$
$907$	51.9086	1.72360	0.861798	+	0.507251i	$0.169338\pi$
$908$	51.3317	1.70350
$909$	−29.0668	−0.964085
$910$	0	0
$911$	−9.67750	−0.320630	−0.160315	−	0.987066i	$-0.551251\pi$
$911$	−9.67750	−0.320630	−0.160315	+	0.987066i	$0.551251\pi$
$912$	9.89446	0.327638
$913$	14.1260	0.467503
$914$	−67.0103	−2.21651
$915$	0	0
$916$	−27.5877	−0.911523
$917$	−0.674176	−0.0222632
$918$	−10.0263	−0.330919
$919$	13.5515	0.447022	0.223511	−	0.974701i	$-0.428248\pi$
$919$	13.5515	0.447022	0.223511	+	0.974701i	$0.428248\pi$
$920$	0	0
$921$	−9.22425	−0.303949
$922$	98.6780	3.24979
$923$	0	0
$924$	7.35026	0.241806
$925$	0	0
$926$	−104.381	−3.43017
$927$	42.5466	1.39741
$928$	38.5705	1.26614
$929$	9.44992	0.310042	0.155021	−	0.987911i	$-0.450455\pi$
$929$	9.44992	0.310042	0.155021	+	0.987911i	$0.450455\pi$
$930$	0	0
$931$	−10.6375	−0.348631
$932$	55.4880	1.81757
$933$	−12.1504	−0.397788
$934$	−87.6625	−2.86840
$935$	0	0
$936$	0	0
$937$	−16.0409	−0.524035	−0.262017	−	0.965063i	$-0.584388\pi$
$937$	−16.0409	−0.524035	−0.262017	+	0.965063i	$0.584388\pi$
$938$	21.3357	0.696634
$939$	−1.35309	−0.0441565
$940$	0	0
$941$	21.6747	0.706574	0.353287	−	0.935515i	$-0.385064\pi$
$941$	21.6747	0.706574	0.353287	+	0.935515i	$0.385064\pi$
$942$	−3.57310	−0.116418
$943$	−53.8759	−1.75444
$944$	−73.5002	−2.39223
$945$	0	0
$946$	66.7934	2.17164
$947$	−4.63118	−0.150493	−0.0752466	−	0.997165i	$-0.523974\pi$
$947$	−4.63118	−0.150493	−0.0752466	+	0.997165i	$0.523974\pi$
$948$	−5.61213	−0.182273
$949$	0	0
$950$	0	0
$951$	−11.4372	−0.370878
$952$	9.18997	0.297849
$953$	−26.2981	−0.851878	−0.425939	−	0.904752i	$-0.640056\pi$
$953$	−26.2981	−0.851878	−0.425939	+	0.904752i	$0.640056\pi$
$954$	42.4965	1.37587
$955$	0	0
$956$	61.1632	1.97816
$957$	4.27645	0.138238
$958$	−45.1427	−1.45849
$959$	−12.0409	−0.388822
$960$	0	0
$961$	−3.04491	−0.0982228
$962$	0	0
$963$	30.8155	0.993014
$964$	147.601	4.75392
$965$	0	0
$966$	−6.72496	−0.216372
$967$	−11.9405	−0.383981	−0.191990	−	0.981397i	$-0.561494\pi$
$967$	−11.9405	−0.383981	−0.191990	+	0.981397i	$0.561494\pi$
$968$	21.1646	0.680255
$969$	1.08840	0.0349643
$970$	0	0
$971$	30.1524	0.967635	0.483818	−	0.875169i	$-0.339250\pi$
$971$	30.1524	0.967635	0.483818	+	0.875169i	$0.339250\pi$
$972$	60.2311	1.93191
$973$	−6.80209	−0.218065
$974$	−24.7308	−0.792427
$975$	0	0
$976$	−21.7078	−0.694850
$977$	26.9321	0.861633	0.430817	−	0.902439i	$-0.358226\pi$
$977$	26.9321	0.861633	0.430817	+	0.902439i	$0.358226\pi$
$978$	2.87258	0.0918549
$979$	−10.2012	−0.326033
$980$	0	0
$981$	26.5431	0.847455
$982$	−68.8843	−2.19819
$983$	−20.5902	−0.656727	−0.328363	−	0.944551i	$-0.606497\pi$
$983$	−20.5902	−0.656727	−0.328363	+	0.944551i	$0.606497\pi$
$984$	33.7743	1.07669
$985$	0	0
$986$	8.73481	0.278173
$987$	1.23884	0.0394328
$988$	0	0
$989$	−44.0322	−1.40014
$990$	0	0
$991$	−48.1378	−1.52915	−0.764573	−	0.644537i	$-0.777050\pi$
$991$	−48.1378	−1.52915	−0.764573	+	0.644537i	$0.777050\pi$
$992$	84.3327	2.67756
$993$	5.67864	0.180206
$994$	−18.4631	−0.585614
$995$	0	0
$996$	9.53690	0.302188
$997$	−33.4255	−1.05860	−0.529298	−	0.848436i	$-0.677545\pi$
$997$	−33.4255	−1.05860	−0.529298	+	0.848436i	$0.677545\pi$
$998$	74.1051	2.34576
$999$	−10.4603	−0.330948

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bh.1.3		3
5.2	odd	4		845.2.b.c.339.6		6
5.3	odd	4		845.2.b.c.339.1		6
5.4	even	2		4225.2.a.ba.1.1		3
13.12	even	2		325.2.a.j.1.1		3
39.38	odd	2		2925.2.a.bj.1.3		3
52.51	odd	2		5200.2.a.cj.1.1		3
65.2	even	12		845.2.l.d.654.2		12
65.3	odd	12		845.2.n.g.529.6		12
65.7	even	12		845.2.l.e.699.5		12
65.8	even	4		845.2.d.b.844.6		6
65.12	odd	4		65.2.b.a.14.1	✓	6
65.17	odd	12		845.2.n.f.484.1		12
65.18	even	4		845.2.d.a.844.2		6
65.22	odd	12		845.2.n.g.484.6		12
65.23	odd	12		845.2.n.f.529.1		12
65.28	even	12		845.2.l.e.654.5		12
65.32	even	12		845.2.l.d.699.1		12
65.33	even	12		845.2.l.d.699.2		12
65.37	even	12		845.2.l.e.654.6		12
65.38	odd	4		65.2.b.a.14.6	yes	6
65.42	odd	12		845.2.n.g.529.1		12
65.43	odd	12		845.2.n.f.484.6		12
65.47	even	4		845.2.d.a.844.1		6
65.48	odd	12		845.2.n.g.484.1		12
65.57	even	4		845.2.d.b.844.5		6
65.58	even	12		845.2.l.e.699.6		12
65.62	odd	12		845.2.n.f.529.6		12
65.63	even	12		845.2.l.d.654.1		12
65.64	even	2		325.2.a.k.1.3		3
195.38	even	4		585.2.c.b.469.1		6
195.77	even	4		585.2.c.b.469.6		6
195.194	odd	2		2925.2.a.bf.1.1		3
260.103	even	4		1040.2.d.c.209.3		6
260.207	even	4		1040.2.d.c.209.4		6
260.259	odd	2		5200.2.a.cb.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.b.a.14.1	✓	6	65.12	odd	4
65.2.b.a.14.6	yes	6	65.38	odd	4
325.2.a.j.1.1		3	13.12	even	2
325.2.a.k.1.3		3	65.64	even	2
585.2.c.b.469.1		6	195.38	even	4
585.2.c.b.469.6		6	195.77	even	4
845.2.b.c.339.1		6	5.3	odd	4
845.2.b.c.339.6		6	5.2	odd	4
845.2.d.a.844.1		6	65.47	even	4
845.2.d.a.844.2		6	65.18	even	4
845.2.d.b.844.5		6	65.57	even	4
845.2.d.b.844.6		6	65.8	even	4
845.2.l.d.654.1		12	65.63	even	12
845.2.l.d.654.2		12	65.2	even	12
845.2.l.d.699.1		12	65.32	even	12
845.2.l.d.699.2		12	65.33	even	12
845.2.l.e.654.5		12	65.28	even	12
845.2.l.e.654.6		12	65.37	even	12
845.2.l.e.699.5		12	65.7	even	12
845.2.l.e.699.6		12	65.58	even	12
845.2.n.f.484.1		12	65.17	odd	12
845.2.n.f.484.6		12	65.43	odd	12
845.2.n.f.529.1		12	65.23	odd	12
845.2.n.f.529.6		12	65.62	odd	12
845.2.n.g.484.1		12	65.48	odd	12
845.2.n.g.484.6		12	65.22	odd	12
845.2.n.g.529.1		12	65.42	odd	12
845.2.n.g.529.6		12	65.3	odd	12
1040.2.d.c.209.3		6	260.103	even	4
1040.2.d.c.209.4		6	260.207	even	4
2925.2.a.bf.1.1		3	195.194	odd	2
2925.2.a.bj.1.3		3	39.38	odd	2
4225.2.a.ba.1.1		3	5.4	even	2
4225.2.a.bh.1.3		3	1.1	even	1	trivial
5200.2.a.cb.1.3		3	260.259	odd	2
5200.2.a.cj.1.1		3	52.51	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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.67513 q^{2} +0.481194 q^{3} +5.15633 q^{4} +1.28726 q^{6} +0.806063 q^{7} +8.44358 q^{8} -2.76845 q^{9} +3.67513 q^{11} +2.48119 q^{12} +2.15633 q^{14} +12.2750 q^{16} +1.35026 q^{17} -7.40597 q^{18} +1.67513 q^{19} +0.387873 q^{21} +9.83146 q^{22} -6.48119 q^{23} +4.06300 q^{24} -2.77575 q^{27} +4.15633 q^{28} +2.41819 q^{29} +5.28726 q^{31} +15.9502 q^{32} +1.76845 q^{33} +3.61213 q^{34} -14.2750 q^{36} +3.76845 q^{37} +4.48119 q^{38} +8.31265 q^{41} +1.03761 q^{42} +6.79384 q^{43} +18.9502 q^{44} -17.3380 q^{46} +3.19394 q^{47} +5.90668 q^{48} -6.35026 q^{49} +0.649738 q^{51} -5.73813 q^{53} -7.42548 q^{54} +6.80606 q^{56} +0.806063 q^{57} +6.46898 q^{58} -5.98778 q^{59} -1.76845 q^{61} +14.1441 q^{62} -2.23155 q^{63} +18.1187 q^{64} +4.73084 q^{66} +9.89446 q^{67} +6.96239 q^{68} -3.11871 q^{69} -8.56230 q^{71} -23.3757 q^{72} -11.7685 q^{73} +10.0811 q^{74} +8.63752 q^{76} +2.96239 q^{77} -2.26187 q^{79} +6.96968 q^{81} +22.2374 q^{82} +3.84367 q^{83} +2.00000 q^{84} +18.1744 q^{86} +1.16362 q^{87} +31.0313 q^{88} -2.77575 q^{89} -33.4191 q^{92} +2.54420 q^{93} +8.54420 q^{94} +7.67513 q^{96} -1.87399 q^{97} -16.9878 q^{98} -10.1744 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 4 q^{3} + 5 q^{4} - 2 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} + 6 q^{11} + 2 q^{12} - 4 q^{14} + 5 q^{16} - 6 q^{17} + 5 q^{18} + 2 q^{21} + 14 q^{22} - 14 q^{23} + 8 q^{24} - 10 q^{27} + 2 q^{28}+ \cdots + 8 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 4 * q^3 + 5 * q^4 - 2 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9 + 6 * q^11 + 2 * q^12 - 4 * q^14 + 5 * q^16 - 6 * q^17 + 5 * q^18 + 2 * q^21 + 14 * q^22 - 14 * q^23 + 8 * q^24 - 10 * q^27 + 2 * q^28 + 6 * q^29 + 10 * q^31 + 11 * q^32 - 6 * q^33 + 10 * q^34 - 11 * q^36 + 8 * q^38 + 4 * q^41 + 14 * q^42 - 6 * q^43 + 20 * q^44 - 16 * q^46 + 10 * q^47 + 24 * q^48 - 9 * q^49 + 12 * q^51 - 8 * q^53 - 34 * q^54 + 20 * q^56 + 2 * q^57 - 12 * q^58 + 8 * q^59 + 6 * q^61 + 6 * q^62 - 18 * q^63 + 33 * q^64 - 8 * q^66 + 10 * q^67 + 10 * q^68 + 12 * q^69 + 12 * q^71 - 45 * q^72 - 24 * q^73 - 2 * q^74 + 10 * q^76 - 2 * q^77 - 16 * q^79 + 23 * q^81 + 24 * q^82 + 22 * q^83 + 6 * q^84 + 16 * q^86 + 6 * q^87 + 24 * q^88 - 10 * q^89 - 32 * q^92 - 2 * q^93 + 16 * q^94 + 18 * q^96 - 14 * q^97 - 25 * q^98 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more