LMFDB - Embedded newform 1250.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1250,2,Mod(1,1250)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1250, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1250.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1250 = 2 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1250.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:	$9.98130025266$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.7625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 4x + 16 $ x^4 - x^3 - 9x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 50)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.71472$ of defining polynomial
Character	$\chi$	$=$	1250.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.71472 q^{3} +1.00000 q^{4} +1.71472 q^{6} +2.77447 q^{7} -1.00000 q^{8} -0.0597522 q^{9} +2.77447 q^{11} -1.71472 q^{12} -5.67779 q^{13} -2.77447 q^{14} +1.00000 q^{16} -5.15643 q^{17} +0.0597522 q^{18} +1.41238 q^{19} -4.75742 q^{21} -2.77447 q^{22} +0.654963 q^{23} +1.71472 q^{24} +5.67779 q^{26} +5.24660 q^{27} +2.77447 q^{28} +4.09668 q^{29} -7.12710 q^{31} -1.00000 q^{32} -4.75742 q^{33} +5.15643 q^{34} -0.0597522 q^{36} +1.04746 q^{37} -1.41238 q^{38} +9.73579 q^{39} +9.10722 q^{41} +4.75742 q^{42} -9.24660 q^{43} +2.77447 q^{44} -0.654963 q^{46} +2.77447 q^{47} -1.71472 q^{48} +0.697669 q^{49} +8.84181 q^{51} -5.67779 q^{52} +0.526111 q^{53} -5.24660 q^{54} -2.77447 q^{56} -2.42184 q^{57} -4.09668 q^{58} -3.78206 q^{59} -10.8325 q^{61} +7.12710 q^{62} -0.165781 q^{63} +1.00000 q^{64} +4.75742 q^{66} +4.32340 q^{67} -5.15643 q^{68} -1.12307 q^{69} -13.2466 q^{71} +0.0597522 q^{72} -4.21324 q^{73} -1.04746 q^{74} +1.41238 q^{76} +7.69767 q^{77} -9.73579 q^{78} -9.90157 q^{79} -8.81717 q^{81} -9.10722 q^{82} -4.67603 q^{83} -4.75742 q^{84} +9.24660 q^{86} -7.02464 q^{87} -2.77447 q^{88} -9.18401 q^{89} -15.7528 q^{91} +0.654963 q^{92} +12.2209 q^{93} -2.77447 q^{94} +1.71472 q^{96} +0.0901699 q^{97} -0.697669 q^{98} -0.165781 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - q^{3} + 4 q^{4} + q^{6} - 2 q^{7} - 4 q^{8} + 7 q^{9} - 2 q^{11} - q^{12} - 11 q^{13} + 2 q^{14} + 4 q^{16} - 12 q^{17} - 7 q^{18} - 5 q^{19} - 7 q^{21} + 2 q^{22} + 4 q^{23} + q^{24}+ \cdots - 21 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 - q^3 + 4 * q^4 + q^6 - 2 * q^7 - 4 * q^8 + 7 * q^9 - 2 * q^11 - q^12 - 11 * q^13 + 2 * q^14 + 4 * q^16 - 12 * q^17 - 7 * q^18 - 5 * q^19 - 7 * q^21 + 2 * q^22 + 4 * q^23 + q^24 + 11 * q^26 - 10 * q^27 - 2 * q^28 + 15 * q^29 - 12 * q^31 - 4 * q^32 - 7 * q^33 + 12 * q^34 + 7 * q^36 - 12 * q^37 + 5 * q^38 - 11 * q^39 + 13 * q^41 + 7 * q^42 - 6 * q^43 - 2 * q^44 - 4 * q^46 - 2 * q^47 - q^48 - 2 * q^49 + 13 * q^51 - 11 * q^52 - 11 * q^53 + 10 * q^54 + 2 * q^56 - 15 * q^58 + 8 * q^61 + 12 * q^62 - 21 * q^63 + 4 * q^64 + 7 * q^66 - 22 * q^67 - 12 * q^68 - 31 * q^69 - 22 * q^71 - 7 * q^72 - 21 * q^73 + 12 * q^74 - 5 * q^76 + 26 * q^77 + 11 * q^78 - 10 * q^79 - 16 * q^81 - 13 * q^82 + 24 * q^83 - 7 * q^84 + 6 * q^86 - 25 * q^87 + 2 * q^88 - 5 * q^89 - 12 * q^91 + 4 * q^92 + 23 * q^93 + 2 * q^94 + q^96 - 22 * q^97 + 2 * q^98 - 21 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.71472	−0.989991	−0.494996	−	0.868895i	$-0.664830\pi$
$3$	−1.71472	−0.989991	−0.494996	+	0.868895i	$0.664830\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.71472	0.700029
$7$	2.77447	1.04865	0.524325	−	0.851518i	$-0.324318\pi$
$7$	2.77447	1.04865	0.524325	+	0.851518i	$0.324318\pi$
$8$	−1.00000	−0.353553
$9$	−0.0597522	−0.0199174
$10$	0	0
$11$	2.77447	0.836533	0.418267	−	0.908324i	$-0.362638\pi$
$11$	2.77447	0.836533	0.418267	+	0.908324i	$0.362638\pi$
$12$	−1.71472	−0.494996
$13$	−5.67779	−1.57473	−0.787367	−	0.616484i	$-0.788556\pi$
$13$	−5.67779	−1.57473	−0.787367	+	0.616484i	$0.788556\pi$
$14$	−2.77447	−0.741508
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.15643	−1.25062	−0.625309	−	0.780377i	$-0.715027\pi$
$17$	−5.15643	−1.25062	−0.625309	+	0.780377i	$0.715027\pi$
$18$	0.0597522	0.0140837
$19$	1.41238	0.324023	0.162012	−	0.986789i	$-0.448202\pi$
$19$	1.41238	0.324023	0.162012	+	0.986789i	$0.448202\pi$
$20$	0	0
$21$	−4.75742	−1.03815
$22$	−2.77447	−0.591518
$23$	0.654963	0.136569	0.0682846	−	0.997666i	$-0.478247\pi$
$23$	0.654963	0.136569	0.0682846	+	0.997666i	$0.478247\pi$
$24$	1.71472	0.350015
$25$	0	0
$26$	5.67779	1.11351
$27$	5.24660	1.00971
$28$	2.77447	0.524325
$29$	4.09668	0.760735	0.380367	−	0.924836i	$-0.375798\pi$
$29$	4.09668	0.760735	0.380367	+	0.924836i	$0.375798\pi$
$30$	0	0
$31$	−7.12710	−1.28006	−0.640032	−	0.768348i	$-0.721079\pi$
$31$	−7.12710	−1.28006	−0.640032	+	0.768348i	$0.721079\pi$
$32$	−1.00000	−0.176777
$33$	−4.75742	−0.828161
$34$	5.15643	0.884321
$35$	0	0
$36$	−0.0597522	−0.00995870
$37$	1.04746	0.172202	0.0861010	−	0.996286i	$-0.472559\pi$
$37$	1.04746	0.172202	0.0861010	+	0.996286i	$0.472559\pi$
$38$	−1.41238	−0.229119
$39$	9.73579	1.55897
$40$	0	0
$41$	9.10722	1.42231	0.711154	−	0.703036i	$-0.248173\pi$
$41$	9.10722	1.42231	0.711154	+	0.703036i	$0.248173\pi$
$42$	4.75742	0.734086
$43$	−9.24660	−1.41009	−0.705047	−	0.709161i	$-0.749074\pi$
$43$	−9.24660	−1.41009	−0.705047	+	0.709161i	$0.749074\pi$
$44$	2.77447	0.418267
$45$	0	0
$46$	−0.654963	−0.0965690
$47$	2.77447	0.404698	0.202349	−	0.979314i	$-0.435143\pi$
$47$	2.77447	0.404698	0.202349	+	0.979314i	$0.435143\pi$
$48$	−1.71472	−0.247498
$49$	0.697669	0.0996670
$50$	0	0
$51$	8.84181	1.23810
$52$	−5.67779	−0.787367
$53$	0.526111	0.0722669	0.0361335	−	0.999347i	$-0.488496\pi$
$53$	0.526111	0.0722669	0.0361335	+	0.999347i	$0.488496\pi$
$54$	−5.24660	−0.713972
$55$	0	0
$56$	−2.77447	−0.370754
$57$	−2.42184	−0.320780
$58$	−4.09668	−0.537921
$59$	−3.78206	−0.492382	−0.246191	−	0.969221i	$-0.579179\pi$
$59$	−3.78206	−0.492382	−0.246191	+	0.969221i	$0.579179\pi$
$60$	0	0
$61$	−10.8325	−1.38696	−0.693478	−	0.720478i	$-0.743922\pi$
$61$	−10.8325	−1.38696	−0.693478	+	0.720478i	$0.743922\pi$
$62$	7.12710	0.905142
$63$	−0.165781	−0.0208864
$64$	1.00000	0.125000
$65$	0	0
$66$	4.75742	0.585598
$67$	4.32340	0.528188	0.264094	−	0.964497i	$-0.414927\pi$
$67$	4.32340	0.528188	0.264094	+	0.964497i	$0.414927\pi$
$68$	−5.15643	−0.625309
$69$	−1.12307	−0.135202
$70$	0	0
$71$	−13.2466	−1.57208	−0.786041	−	0.618174i	$-0.787873\pi$
$71$	−13.2466	−1.57208	−0.786041	+	0.618174i	$0.787873\pi$
$72$	0.0597522	0.00704186
$73$	−4.21324	−0.493123	−0.246561	−	0.969127i	$-0.579301\pi$
$73$	−4.21324	−0.493123	−0.246561	+	0.969127i	$0.579301\pi$
$74$	−1.04746	−0.121765
$75$	0	0
$76$	1.41238	0.162012
$77$	7.69767	0.877231
$78$	−9.73579	−1.10236
$79$	−9.90157	−1.11401	−0.557007	−	0.830508i	$-0.688050\pi$
$79$	−9.90157	−1.11401	−0.557007	+	0.830508i	$0.688050\pi$
$80$	0	0
$81$	−8.81717	−0.979686
$82$	−9.10722	−1.00572
$83$	−4.67603	−0.513261	−0.256631	−	0.966510i	$-0.582612\pi$
$83$	−4.67603	−0.513261	−0.256631	+	0.966510i	$0.582612\pi$
$84$	−4.75742	−0.519077
$85$	0	0
$86$	9.24660	0.997087
$87$	−7.02464	−0.753121
$88$	−2.77447	−0.295759
$89$	−9.18401	−0.973504	−0.486752	−	0.873540i	$-0.661818\pi$
$89$	−9.18401	−0.973504	−0.486752	+	0.873540i	$0.661818\pi$
$90$	0	0
$91$	−15.7528	−1.65135
$92$	0.654963	0.0682846
$93$	12.2209	1.26725
$94$	−2.77447	−0.286164
$95$	0	0
$96$	1.71472	0.175007
$97$	0.0901699	0.00915537	0.00457769	−	0.999990i	$-0.498543\pi$
$97$	0.0901699	0.00915537	0.00457769	+	0.999990i	$0.498543\pi$
$98$	−0.697669	−0.0704752
$99$	−0.165781	−0.0166616
$100$	0	0
$101$	−7.90632	−0.786709	−0.393354	−	0.919387i	$-0.628685\pi$
$101$	−7.90632	−0.786709	−0.393354	+	0.919387i	$0.628685\pi$
$102$	−8.84181	−0.875470
$103$	−7.83422	−0.771929	−0.385964	−	0.922514i	$-0.626131\pi$
$103$	−7.83422	−0.771929	−0.385964	+	0.922514i	$0.626131\pi$
$104$	5.67779	0.556753
$105$	0	0
$106$	−0.526111	−0.0511004
$107$	10.2220	0.988194	0.494097	−	0.869407i	$-0.335499\pi$
$107$	10.2220	0.988194	0.494097	+	0.869407i	$0.335499\pi$
$108$	5.24660	0.504855
$109$	3.91091	0.374598	0.187299	−	0.982303i	$-0.440027\pi$
$109$	3.91091	0.374598	0.187299	+	0.982303i	$0.440027\pi$
$110$	0	0
$111$	−1.79610	−0.170479
$112$	2.77447	0.262163
$113$	6.37545	0.599752	0.299876	−	0.953978i	$-0.403055\pi$
$113$	6.37545	0.599752	0.299876	+	0.953978i	$0.403055\pi$
$114$	2.42184	0.226826
$115$	0	0
$116$	4.09668	0.380367
$117$	0.339260	0.0313646
$118$	3.78206	0.348167
$119$	−14.3064	−1.31146
$120$	0	0
$121$	−3.30233	−0.300212
$122$	10.8325	0.980725
$123$	−15.6163	−1.40807
$124$	−7.12710	−0.640032
$125$	0	0
$126$	0.165781	0.0147689
$127$	−9.54893	−0.847331	−0.423665	−	0.905819i	$-0.639257\pi$
$127$	−9.54893	−0.847331	−0.423665	+	0.905819i	$0.639257\pi$
$128$	−1.00000	−0.0883883
$129$	15.8553	1.39598
$130$	0	0
$131$	5.59923	0.489207	0.244604	−	0.969623i	$-0.421342\pi$
$131$	5.59923	0.489207	0.244604	+	0.969623i	$0.421342\pi$
$132$	−4.75742	−0.414080
$133$	3.91861	0.339787
$134$	−4.32340	−0.373485
$135$	0	0
$136$	5.15643	0.442161
$137$	−7.98414	−0.682131	−0.341066	−	0.940039i	$-0.610788\pi$
$137$	−7.98414	−0.682131	−0.341066	+	0.940039i	$0.610788\pi$
$138$	1.12307	0.0956025
$139$	−22.0884	−1.87352	−0.936758	−	0.349979i	$-0.886189\pi$
$139$	−22.0884	−1.87352	−0.936758	+	0.349979i	$0.886189\pi$
$140$	0	0
$141$	−4.75742	−0.400647
$142$	13.2466	1.11163
$143$	−15.7528	−1.31732
$144$	−0.0597522	−0.00497935
$145$	0	0
$146$	4.21324	0.348691
$147$	−1.19630	−0.0986694
$148$	1.04746	0.0861010
$149$	4.18401	0.342768	0.171384	−	0.985204i	$-0.445176\pi$
$149$	4.18401	0.342768	0.171384	+	0.985204i	$0.445176\pi$
$150$	0	0
$151$	−0.331561	−0.0269821	−0.0134910	−	0.999909i	$-0.504294\pi$
$151$	−0.331561	−0.0269821	−0.0134910	+	0.999909i	$0.504294\pi$
$152$	−1.41238	−0.114559
$153$	0.308108	0.0249091
$154$	−7.69767	−0.620296
$155$	0	0
$156$	9.73579	0.779487
$157$	−3.60750	−0.287910	−0.143955	−	0.989584i	$-0.545982\pi$
$157$	−3.60750	−0.287910	−0.143955	+	0.989584i	$0.545982\pi$
$158$	9.90157	0.787726
$159$	−0.902131	−0.0715436
$160$	0	0
$161$	1.81717	0.143213
$162$	8.81717	0.692743
$163$	−5.09385	−0.398981	−0.199490	−	0.979900i	$-0.563929\pi$
$163$	−5.09385	−0.398981	−0.199490	+	0.979900i	$0.563929\pi$
$164$	9.10722	0.711154
$165$	0	0
$166$	4.67603	0.362931
$167$	−20.8629	−1.61442	−0.807209	−	0.590265i	$-0.799023\pi$
$167$	−20.8629	−1.61442	−0.807209	+	0.590265i	$0.799023\pi$
$168$	4.75742	0.367043
$169$	19.2373	1.47979
$170$	0	0
$171$	−0.0843930	−0.00645370
$172$	−9.24660	−0.705047
$173$	11.9877	0.911409	0.455704	−	0.890131i	$-0.349387\pi$
$173$	11.9877	0.911409	0.455704	+	0.890131i	$0.349387\pi$
$174$	7.02464	0.532537
$175$	0	0
$176$	2.77447	0.209133
$177$	6.48516	0.487454
$178$	9.18401	0.688371
$179$	−8.54134	−0.638410	−0.319205	−	0.947686i	$-0.603416\pi$
$179$	−8.54134	−0.638410	−0.319205	+	0.947686i	$0.603416\pi$
$180$	0	0
$181$	7.82844	0.581884	0.290942	−	0.956741i	$-0.406031\pi$
$181$	7.82844	0.581884	0.290942	+	0.956741i	$0.406031\pi$
$182$	15.7528	1.16768
$183$	18.5746	1.37307
$184$	−0.654963	−0.0482845
$185$	0	0
$186$	−12.2209	−0.896083
$187$	−14.3064	−1.04618
$188$	2.77447	0.202349
$189$	14.5565	1.05883
$190$	0	0
$191$	3.36611	0.243563	0.121782	−	0.992557i	$-0.461139\pi$
$191$	3.36611	0.243563	0.121782	+	0.992557i	$0.461139\pi$
$192$	−1.71472	−0.123749
$193$	15.4211	1.11004	0.555018	−	0.831839i	$-0.312712\pi$
$193$	15.4211	1.11004	0.555018	+	0.831839i	$0.312712\pi$
$194$	−0.0901699	−0.00647382
$195$	0	0
$196$	0.697669	0.0498335
$197$	−8.80489	−0.627322	−0.313661	−	0.949535i	$-0.601556\pi$
$197$	−8.80489	−0.627322	−0.313661	+	0.949535i	$0.601556\pi$
$198$	0.165781	0.0117815
$199$	−17.6222	−1.24920	−0.624601	−	0.780944i	$-0.714738\pi$
$199$	−17.6222	−1.24920	−0.624601	+	0.780944i	$0.714738\pi$
$200$	0	0
$201$	−7.41340	−0.522901
$202$	7.90632	0.556287
$203$	11.3661	0.797744
$204$	8.84181	0.619051
$205$	0	0
$206$	7.83422	0.545836
$207$	−0.0391355	−0.00272010
$208$	−5.67779	−0.393684
$209$	3.91861	0.271056
$210$	0	0
$211$	6.10144	0.420040	0.210020	−	0.977697i	$-0.432647\pi$
$211$	6.10144	0.420040	0.210020	+	0.977697i	$0.432647\pi$
$212$	0.526111	0.0361335
$213$	22.7142	1.55635
$214$	−10.2220	−0.698759
$215$	0	0
$216$	−5.24660	−0.356986
$217$	−19.7739	−1.34234
$218$	−3.91091	−0.264880
$219$	7.22451	0.488187
$220$	0	0
$221$	29.2771	1.96939
$222$	1.79610	0.120547
$223$	5.02866	0.336744	0.168372	−	0.985723i	$-0.446149\pi$
$223$	5.02866	0.336744	0.168372	+	0.985723i	$0.446149\pi$
$224$	−2.77447	−0.185377
$225$	0	0
$226$	−6.37545	−0.424089
$227$	17.8382	1.18397	0.591983	−	0.805951i	$-0.298345\pi$
$227$	17.8382	1.18397	0.591983	+	0.805951i	$0.298345\pi$
$228$	−2.42184	−0.160390
$229$	−21.5566	−1.42450	−0.712251	−	0.701925i	$-0.752324\pi$
$229$	−21.5566	−1.42450	−0.712251	+	0.701925i	$0.752324\pi$
$230$	0	0
$231$	−13.1993	−0.868451
$232$	−4.09668	−0.268960
$233$	−14.7065	−0.963452	−0.481726	−	0.876322i	$-0.659990\pi$
$233$	−14.7065	−0.963452	−0.481726	+	0.876322i	$0.659990\pi$
$234$	−0.339260	−0.0221781
$235$	0	0
$236$	−3.78206	−0.246191
$237$	16.9784	1.10286
$238$	14.3064	0.927343
$239$	6.94025	0.448927	0.224464	−	0.974482i	$-0.427937\pi$
$239$	6.94025	0.448927	0.224464	+	0.974482i	$0.427937\pi$
$240$	0	0
$241$	3.31003	0.213218	0.106609	−	0.994301i	$-0.466001\pi$
$241$	3.31003	0.213218	0.106609	+	0.994301i	$0.466001\pi$
$242$	3.30233	0.212282
$243$	−0.620870	−0.0398288
$244$	−10.8325	−0.693478
$245$	0	0
$246$	15.6163	0.995658
$247$	−8.01921	−0.510250
$248$	7.12710	0.452571
$249$	8.01806	0.508124
$250$	0	0
$251$	−9.46454	−0.597397	−0.298698	−	0.954348i	$-0.596552\pi$
$251$	−9.46454	−0.597397	−0.298698	+	0.954348i	$0.596552\pi$
$252$	−0.165781	−0.0104432
$253$	1.81717	0.114245
$254$	9.54893	0.599153
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.07963	0.441615	0.220808	−	0.975317i	$-0.429131\pi$
$257$	7.07963	0.441615	0.220808	+	0.975317i	$0.429131\pi$
$258$	−15.8553	−0.987107
$259$	2.90615	0.180580
$260$	0	0
$261$	−0.244786	−0.0151519
$262$	−5.59923	−0.345922
$263$	−1.43345	−0.0883906	−0.0441953	−	0.999023i	$-0.514072\pi$
$263$	−1.43345	−0.0883906	−0.0441953	+	0.999023i	$0.514072\pi$
$264$	4.75742	0.292799
$265$	0	0
$266$	−3.91861	−0.240266
$267$	15.7480	0.963760
$268$	4.32340	0.264094
$269$	−14.2830	−0.870848	−0.435424	−	0.900226i	$-0.643402\pi$
$269$	−14.2830	−0.870848	−0.435424	+	0.900226i	$0.643402\pi$
$270$	0	0
$271$	30.7855	1.87009	0.935044	−	0.354533i	$-0.115360\pi$
$271$	30.7855	1.87009	0.935044	+	0.354533i	$0.115360\pi$
$272$	−5.15643	−0.312655
$273$	27.0116	1.63482
$274$	7.98414	0.482340
$275$	0	0
$276$	−1.12307	−0.0676012
$277$	−3.18969	−0.191650	−0.0958249	−	0.995398i	$-0.530549\pi$
$277$	−3.18969	−0.191650	−0.0958249	+	0.995398i	$0.530549\pi$
$278$	22.0884	1.32478
$279$	0.425860	0.0254956
$280$	0	0
$281$	20.3889	1.21630	0.608151	−	0.793822i	$-0.291912\pi$
$281$	20.3889	1.21630	0.608151	+	0.793822i	$0.291912\pi$
$282$	4.75742	0.283300
$283$	13.2969	0.790419	0.395209	−	0.918591i	$-0.370672\pi$
$283$	13.2969	0.790419	0.395209	+	0.918591i	$0.370672\pi$
$284$	−13.2466	−0.786041
$285$	0	0
$286$	15.7528	0.931484
$287$	25.2677	1.49150
$288$	0.0597522	0.00352093
$289$	9.58880	0.564047
$290$	0	0
$291$	−0.154616	−0.00906374
$292$	−4.21324	−0.246561
$293$	26.0420	1.52139	0.760696	−	0.649108i	$-0.224858\pi$
$293$	26.0420	1.52139	0.760696	+	0.649108i	$0.224858\pi$
$294$	1.19630	0.0697698
$295$	0	0
$296$	−1.04746	−0.0608826
$297$	14.5565	0.844655
$298$	−4.18401	−0.242373
$299$	−3.71874	−0.215060
$300$	0	0
$301$	−25.6544	−1.47869
$302$	0.331561	0.0190792
$303$	13.5571	0.778835
$304$	1.41238	0.0810058
$305$	0	0
$306$	−0.308108	−0.0176134
$307$	−25.1000	−1.43253	−0.716267	−	0.697826i	$-0.754151\pi$
$307$	−25.1000	−1.43253	−0.716267	+	0.697826i	$0.754151\pi$
$308$	7.69767	0.438615
$309$	13.4335	0.764203
$310$	0	0
$311$	0.723328	0.0410162	0.0205081	−	0.999790i	$-0.493472\pi$
$311$	0.723328	0.0410162	0.0205081	+	0.999790i	$0.493472\pi$
$312$	−9.73579	−0.551180
$313$	−21.6163	−1.22182	−0.610912	−	0.791698i	$-0.709197\pi$
$313$	−21.6163	−1.22182	−0.610912	+	0.791698i	$0.709197\pi$
$314$	3.60750	0.203583
$315$	0	0
$316$	−9.90157	−0.557007
$317$	18.8337	1.05780	0.528902	−	0.848683i	$-0.322604\pi$
$317$	18.8337	1.05780	0.528902	+	0.848683i	$0.322604\pi$
$318$	0.902131	0.0505890
$319$	11.3661	0.636380
$320$	0	0
$321$	−17.5278	−0.978304
$322$	−1.81717	−0.101267
$323$	−7.28286	−0.405229
$324$	−8.81717	−0.489843
$325$	0	0
$326$	5.09385	0.282122
$327$	−6.70610	−0.370848
$328$	−9.10722	−0.502862
$329$	7.69767	0.424386
$330$	0	0
$331$	16.8759	0.927584	0.463792	−	0.885944i	$-0.346489\pi$
$331$	16.8759	0.927584	0.463792	+	0.885944i	$0.346489\pi$
$332$	−4.67603	−0.256631
$333$	−0.0625883	−0.00342982
$334$	20.8629	1.14157
$335$	0	0
$336$	−4.75742	−0.259539
$337$	−19.3521	−1.05417	−0.527087	−	0.849811i	$-0.676716\pi$
$337$	−19.3521	−1.05417	−0.527087	+	0.849811i	$0.676716\pi$
$338$	−19.2373	−1.04637
$339$	−10.9321	−0.593750
$340$	0	0
$341$	−19.7739	−1.07082
$342$	0.0843930	0.00456345
$343$	−17.4856	−0.944134
$344$	9.24660	0.498543
$345$	0	0
$346$	−11.9877	−0.644463
$347$	9.03868	0.485222	0.242611	−	0.970124i	$-0.421996\pi$
$347$	9.03868	0.485222	0.242611	+	0.970124i	$0.421996\pi$
$348$	−7.02464	−0.376560
$349$	36.7305	1.96614	0.983068	−	0.183240i	$-0.0586584\pi$
$349$	36.7305	1.96614	0.983068	+	0.183240i	$0.0586584\pi$
$350$	0	0
$351$	−29.7891	−1.59002
$352$	−2.77447	−0.147880
$353$	−18.3737	−0.977933	−0.488967	−	0.872302i	$-0.662626\pi$
$353$	−18.3737	−0.977933	−0.488967	+	0.872302i	$0.662626\pi$
$354$	−6.48516	−0.344682
$355$	0	0
$356$	−9.18401	−0.486752
$357$	24.5313	1.29834
$358$	8.54134	0.451424
$359$	−24.4259	−1.28915	−0.644574	−	0.764542i	$-0.722965\pi$
$359$	−24.4259	−1.28915	−0.644574	+	0.764542i	$0.722965\pi$
$360$	0	0
$361$	−17.0052	−0.895009
$362$	−7.82844	−0.411454
$363$	5.66256	0.297207
$364$	−15.7528	−0.825673
$365$	0	0
$366$	−18.5746	−0.970910
$367$	−14.5224	−0.758065	−0.379032	−	0.925383i	$-0.623743\pi$
$367$	−14.5224	−0.758065	−0.379032	+	0.925383i	$0.623743\pi$
$368$	0.654963	0.0341423
$369$	−0.544176	−0.0283287
$370$	0	0
$371$	1.45968	0.0757827
$372$	12.2209	0.633626
$373$	23.2887	1.20585	0.602923	−	0.797800i	$-0.294003\pi$
$373$	23.2887	1.20585	0.602923	+	0.797800i	$0.294003\pi$
$374$	14.3064	0.739764
$375$	0	0
$376$	−2.77447	−0.143082
$377$	−23.2601	−1.19796
$378$	−14.5565	−0.748707
$379$	28.6176	1.46999	0.734993	−	0.678075i	$-0.237185\pi$
$379$	28.6176	1.46999	0.734993	+	0.678075i	$0.237185\pi$
$380$	0	0
$381$	16.3737	0.838850
$382$	−3.36611	−0.172225
$383$	16.6157	0.849023	0.424512	−	0.905422i	$-0.360446\pi$
$383$	16.6157	0.849023	0.424512	+	0.905422i	$0.360446\pi$
$384$	1.71472	0.0875037
$385$	0	0
$386$	−15.4211	−0.784913
$387$	0.552505	0.0280854
$388$	0.0901699	0.00457769
$389$	26.3767	1.33735	0.668677	−	0.743553i	$-0.266861\pi$
$389$	26.3767	1.33735	0.668677	+	0.743553i	$0.266861\pi$
$390$	0	0
$391$	−3.37727	−0.170796
$392$	−0.697669	−0.0352376
$393$	−9.60109	−0.484311
$394$	8.80489	0.443584
$395$	0	0
$396$	−0.165781	−0.00833078
$397$	17.9086	0.898807	0.449403	−	0.893329i	$-0.351637\pi$
$397$	17.9086	0.898807	0.449403	+	0.893329i	$0.351637\pi$
$398$	17.6222	0.883319
$399$	−6.71930	−0.336386
$400$	0	0
$401$	32.0164	1.59882	0.799411	−	0.600785i	$-0.205145\pi$
$401$	32.0164	1.59882	0.799411	+	0.600785i	$0.205145\pi$
$402$	7.41340	0.369747
$403$	40.4661	2.01576
$404$	−7.90632	−0.393354
$405$	0	0
$406$	−11.3661	−0.564090
$407$	2.90615	0.144053
$408$	−8.84181	−0.437735
$409$	18.0344	0.891743	0.445871	−	0.895097i	$-0.352894\pi$
$409$	18.0344	0.891743	0.445871	+	0.895097i	$0.352894\pi$
$410$	0	0
$411$	13.6905	0.675304
$412$	−7.83422	−0.385964
$413$	−10.4932	−0.516337
$414$	0.0391355	0.00192340
$415$	0	0
$416$	5.67779	0.278376
$417$	37.8753	1.85476
$418$	−3.91861	−0.191666
$419$	−19.5300	−0.954104	−0.477052	−	0.878875i	$-0.658295\pi$
$419$	−19.5300	−0.954104	−0.477052	+	0.878875i	$0.658295\pi$
$420$	0	0
$421$	−19.5537	−0.952989	−0.476494	−	0.879178i	$-0.658093\pi$
$421$	−19.5537	−0.952989	−0.476494	+	0.879178i	$0.658093\pi$
$422$	−6.10144	−0.297013
$423$	−0.165781	−0.00806052
$424$	−0.526111	−0.0255502
$425$	0	0
$426$	−22.7142	−1.10050
$427$	−30.0543	−1.45443
$428$	10.2220	0.494097
$429$	27.0116	1.30413
$430$	0	0
$431$	−21.1963	−1.02099	−0.510495	−	0.859881i	$-0.670538\pi$
$431$	−21.1963	−1.02099	−0.510495	+	0.859881i	$0.670538\pi$
$432$	5.24660	0.252427
$433$	−8.54967	−0.410871	−0.205435	−	0.978671i	$-0.565861\pi$
$433$	−8.54967	−0.410871	−0.205435	+	0.978671i	$0.565861\pi$
$434$	19.7739	0.949178
$435$	0	0
$436$	3.91091	0.187299
$437$	0.925059	0.0442516
$438$	−7.22451	−0.345201
$439$	−28.0714	−1.33977	−0.669887	−	0.742463i	$-0.733657\pi$
$439$	−28.0714	−1.33977	−0.669887	+	0.742463i	$0.733657\pi$
$440$	0	0
$441$	−0.0416872	−0.00198511
$442$	−29.2771	−1.39257
$443$	33.0546	1.57047	0.785236	−	0.619197i	$-0.212542\pi$
$443$	33.0546	1.57047	0.785236	+	0.619197i	$0.212542\pi$
$444$	−1.79610	−0.0852393
$445$	0	0
$446$	−5.02866	−0.238114
$447$	−7.17439	−0.339337
$448$	2.77447	0.131081
$449$	−37.1628	−1.75382	−0.876911	−	0.480652i	$-0.840400\pi$
$449$	−37.1628	−1.75382	−0.876911	+	0.480652i	$0.840400\pi$
$450$	0	0
$451$	25.2677	1.18981
$452$	6.37545	0.299876
$453$	0.568533	0.0267120
$454$	−17.8382	−0.837190
$455$	0	0
$456$	2.42184	0.113413
$457$	−29.3139	−1.37125	−0.685624	−	0.727956i	$-0.740471\pi$
$457$	−29.3139	−1.37125	−0.685624	+	0.727956i	$0.740471\pi$
$458$	21.5566	1.00728
$459$	−27.0538	−1.26276
$460$	0	0
$461$	8.90741	0.414859	0.207430	−	0.978250i	$-0.433490\pi$
$461$	8.90741	0.414859	0.207430	+	0.978250i	$0.433490\pi$
$462$	13.1993	0.614087
$463$	18.2771	0.849410	0.424705	−	0.905332i	$-0.360378\pi$
$463$	18.2771	0.849410	0.424705	+	0.905332i	$0.360378\pi$
$464$	4.09668	0.190184
$465$	0	0
$466$	14.7065	0.681263
$467$	36.6841	1.69754	0.848768	−	0.528765i	$-0.177345\pi$
$467$	36.6841	1.69754	0.848768	+	0.528765i	$0.177345\pi$
$468$	0.339260	0.0156823
$469$	11.9951	0.553884
$470$	0	0
$471$	6.18583	0.285028
$472$	3.78206	0.174084
$473$	−25.6544	−1.17959
$474$	−16.9784	−0.779842
$475$	0	0
$476$	−14.3064	−0.655731
$477$	−0.0314363	−0.00143937
$478$	−6.94025	−0.317440
$479$	−12.5898	−0.575242	−0.287621	−	0.957744i	$-0.592864\pi$
$479$	−12.5898	−0.575242	−0.287621	+	0.957744i	$0.592864\pi$
$480$	0	0
$481$	−5.94728	−0.271173
$482$	−3.31003	−0.150768
$483$	−3.11593	−0.141780
$484$	−3.30233	−0.150106
$485$	0	0
$486$	0.620870	0.0281632
$487$	18.3256	0.830411	0.415205	−	0.909728i	$-0.363710\pi$
$487$	18.3256	0.830411	0.415205	+	0.909728i	$0.363710\pi$
$488$	10.8325	0.490363
$489$	8.73449	0.394987
$490$	0	0
$491$	21.7047	0.979519	0.489760	−	0.871857i	$-0.337085\pi$
$491$	21.7047	0.979519	0.489760	+	0.871857i	$0.337085\pi$
$492$	−15.6163	−0.704036
$493$	−21.1243	−0.951389
$494$	8.01921	0.360801
$495$	0	0
$496$	−7.12710	−0.320016
$497$	−36.7523	−1.64856
$498$	−8.01806	−0.359298
$499$	15.8391	0.709055	0.354527	−	0.935046i	$-0.384642\pi$
$499$	15.8391	0.709055	0.354527	+	0.935046i	$0.384642\pi$
$500$	0	0
$501$	35.7739	1.59826
$502$	9.46454	0.422423
$503$	−27.5530	−1.22853	−0.614263	−	0.789101i	$-0.710547\pi$
$503$	−27.5530	−1.22853	−0.614263	+	0.789101i	$0.710547\pi$
$504$	0.165781	0.00738445
$505$	0	0
$506$	−1.81717	−0.0807832
$507$	−32.9864	−1.46498
$508$	−9.54893	−0.423665
$509$	−27.6120	−1.22388	−0.611941	−	0.790903i	$-0.709611\pi$
$509$	−27.6120	−1.22388	−0.611941	+	0.790903i	$0.709611\pi$
$510$	0	0
$511$	−11.6895	−0.517113
$512$	−1.00000	−0.0441942
$513$	7.41022	0.327169
$514$	−7.07963	−0.312269
$515$	0	0
$516$	15.8553	0.697990
$517$	7.69767	0.338543
$518$	−2.90615	−0.127689
$519$	−20.5555	−0.902287
$520$	0	0
$521$	17.0892	0.748689	0.374345	−	0.927290i	$-0.377868\pi$
$521$	17.0892	0.748689	0.374345	+	0.927290i	$0.377868\pi$
$522$	0.244786	0.0107140
$523$	−36.3152	−1.58795	−0.793977	−	0.607947i	$-0.791993\pi$
$523$	−36.3152	−1.58795	−0.793977	+	0.607947i	$0.791993\pi$
$524$	5.59923	0.244604
$525$	0	0
$526$	1.43345	0.0625016
$527$	36.7504	1.60087
$528$	−4.75742	−0.207040
$529$	−22.5710	−0.981349
$530$	0	0
$531$	0.225987	0.00980698
$532$	3.91861	0.169893
$533$	−51.7088	−2.23976
$534$	−15.7480	−0.681481
$535$	0	0
$536$	−4.32340	−0.186743
$537$	14.6460	0.632020
$538$	14.2830	0.615782
$539$	1.93566	0.0833747
$540$	0	0
$541$	24.9366	1.07211	0.536055	−	0.844183i	$-0.319914\pi$
$541$	24.9366	1.07211	0.536055	+	0.844183i	$0.319914\pi$
$542$	−30.7855	−1.32235
$543$	−13.4235	−0.576060
$544$	5.15643	0.221080
$545$	0	0
$546$	−27.0116	−1.15599
$547$	−6.67201	−0.285275	−0.142637	−	0.989775i	$-0.545558\pi$
$547$	−6.67201	−0.285275	−0.142637	+	0.989775i	$0.545558\pi$
$548$	−7.98414	−0.341066
$549$	0.647264	0.0276245
$550$	0	0
$551$	5.78609	0.246496
$552$	1.12307	0.0478012
$553$	−27.4716	−1.16821
$554$	3.18969	0.135517
$555$	0	0
$556$	−22.0884	−0.936758
$557$	20.4328	0.865765	0.432882	−	0.901450i	$-0.357497\pi$
$557$	20.4328	0.865765	0.432882	+	0.901450i	$0.357497\pi$
$558$	−0.425860	−0.0180281
$559$	52.5002	2.22052
$560$	0	0
$561$	24.5313	1.03571
$562$	−20.3889	−0.860055
$563$	0.602805	0.0254052	0.0127026	−	0.999919i	$-0.495957\pi$
$563$	0.602805	0.0254052	0.0127026	+	0.999919i	$0.495957\pi$
$564$	−4.75742	−0.200324
$565$	0	0
$566$	−13.2969	−0.558911
$567$	−24.4630	−1.02735
$568$	13.2466	0.555815
$569$	33.8776	1.42022	0.710112	−	0.704089i	$-0.248644\pi$
$569$	33.8776	1.42022	0.710112	+	0.704089i	$0.248644\pi$
$570$	0	0
$571$	−6.23012	−0.260722	−0.130361	−	0.991467i	$-0.541614\pi$
$571$	−6.23012	−0.260722	−0.130361	+	0.991467i	$0.541614\pi$
$572$	−15.7528	−0.658659
$573$	−5.77192	−0.241125
$574$	−25.2677	−1.05465
$575$	0	0
$576$	−0.0597522	−0.00248967
$577$	−3.78104	−0.157407	−0.0787034	−	0.996898i	$-0.525078\pi$
$577$	−3.78104	−0.157407	−0.0787034	+	0.996898i	$0.525078\pi$
$578$	−9.58880	−0.398842
$579$	−26.4428	−1.09893
$580$	0	0
$581$	−12.9735	−0.538232
$582$	0.154616	0.00640903
$583$	1.45968	0.0604537
$584$	4.21324	0.174345
$585$	0	0
$586$	−26.0420	−1.07579
$587$	−16.3526	−0.674945	−0.337473	−	0.941335i	$-0.609572\pi$
$587$	−16.3526	−0.674945	−0.337473	+	0.941335i	$0.609572\pi$
$588$	−1.19630	−0.0493347
$589$	−10.0662	−0.414770
$590$	0	0
$591$	15.0979	0.621043
$592$	1.04746	0.0430505
$593$	−9.53314	−0.391479	−0.195740	−	0.980656i	$-0.562711\pi$
$593$	−9.53314	−0.391479	−0.195740	+	0.980656i	$0.562711\pi$
$594$	−14.5565	−0.597262
$595$	0	0
$596$	4.18401	0.171384
$597$	30.2170	1.23670
$598$	3.71874	0.152071
$599$	−16.0682	−0.656528	−0.328264	−	0.944586i	$-0.606464\pi$
$599$	−16.0682	−0.656528	−0.328264	+	0.944586i	$0.606464\pi$
$600$	0	0
$601$	15.0380	0.613413	0.306707	−	0.951804i	$-0.400773\pi$
$601$	15.0380	0.613413	0.306707	+	0.951804i	$0.400773\pi$
$602$	25.6544	1.04560
$603$	−0.258333	−0.0105201
$604$	−0.331561	−0.0134910
$605$	0	0
$606$	−13.5571	−0.550719
$607$	44.8348	1.81979	0.909894	−	0.414840i	$-0.136163\pi$
$607$	44.8348	1.81979	0.909894	+	0.414840i	$0.136163\pi$
$608$	−1.41238	−0.0572797
$609$	−19.4896	−0.789760
$610$	0	0
$611$	−15.7528	−0.637291
$612$	0.308108	0.0124545
$613$	15.5488	0.628011	0.314006	−	0.949421i	$-0.398329\pi$
$613$	15.5488	0.628011	0.314006	+	0.949421i	$0.398329\pi$
$614$	25.1000	1.01296
$615$	0	0
$616$	−7.69767	−0.310148
$617$	−17.5994	−0.708525	−0.354263	−	0.935146i	$-0.615268\pi$
$617$	−17.5994	−0.708525	−0.354263	+	0.935146i	$0.615268\pi$
$618$	−13.4335	−0.540373
$619$	−37.2887	−1.49876	−0.749381	−	0.662140i	$-0.769649\pi$
$619$	−37.2887	−1.49876	−0.749381	+	0.662140i	$0.769649\pi$
$620$	0	0
$621$	3.43633	0.137895
$622$	−0.723328	−0.0290028
$623$	−25.4807	−1.02086
$624$	9.73579	0.389743
$625$	0	0
$626$	21.6163	0.863960
$627$	−6.71930	−0.268343
$628$	−3.60750	−0.143955
$629$	−5.40118	−0.215359
$630$	0	0
$631$	8.55551	0.340589	0.170295	−	0.985393i	$-0.445528\pi$
$631$	8.55551	0.340589	0.170295	+	0.985393i	$0.445528\pi$
$632$	9.90157	0.393863
$633$	−10.4622	−0.415836
$634$	−18.8337	−0.747980
$635$	0	0
$636$	−0.902131	−0.0357718
$637$	−3.96121	−0.156949
$638$	−11.3661	−0.449989
$639$	0.791514	0.0313118
$640$	0	0
$641$	−22.8156	−0.901162	−0.450581	−	0.892736i	$-0.648783\pi$
$641$	−22.8156	−0.901162	−0.450581	+	0.892736i	$0.648783\pi$
$642$	17.5278	0.691765
$643$	−34.7745	−1.37137	−0.685686	−	0.727898i	$-0.740497\pi$
$643$	−34.7745	−1.37137	−0.685686	+	0.727898i	$0.740497\pi$
$644$	1.81717	0.0716067
$645$	0	0
$646$	7.28286	0.286540
$647$	16.4581	0.647034	0.323517	−	0.946222i	$-0.395135\pi$
$647$	16.4581	0.647034	0.323517	+	0.946222i	$0.395135\pi$
$648$	8.81717	0.346371
$649$	−10.4932	−0.411894
$650$	0	0
$651$	33.9066	1.32890
$652$	−5.09385	−0.199490
$653$	−30.2092	−1.18218	−0.591088	−	0.806607i	$-0.701301\pi$
$653$	−30.2092	−1.18218	−0.591088	+	0.806607i	$0.701301\pi$
$654$	6.70610	0.262229
$655$	0	0
$656$	9.10722	0.355577
$657$	0.251751	0.00982173
$658$	−7.69767	−0.300086
$659$	−28.4942	−1.10998	−0.554989	−	0.831858i	$-0.687277\pi$
$659$	−28.4942	−1.10998	−0.554989	+	0.831858i	$0.687277\pi$
$660$	0	0
$661$	50.7342	1.97333	0.986666	−	0.162759i	$-0.0520393\pi$
$661$	50.7342	1.97333	0.986666	+	0.162759i	$0.0520393\pi$
$662$	−16.8759	−0.655901
$663$	−50.2019	−1.94968
$664$	4.67603	0.181465
$665$	0	0
$666$	0.0625883	0.00242525
$667$	2.68317	0.103893
$668$	−20.8629	−0.807209
$669$	−8.62273	−0.333374
$670$	0	0
$671$	−30.0543	−1.16023
$672$	4.75742	0.183521
$673$	20.4447	0.788084	0.394042	−	0.919093i	$-0.371077\pi$
$673$	20.4447	0.788084	0.394042	+	0.919093i	$0.371077\pi$
$674$	19.3521	0.745414
$675$	0	0
$676$	19.2373	0.739894
$677$	−8.53360	−0.327973	−0.163986	−	0.986463i	$-0.552435\pi$
$677$	−8.53360	−0.327973	−0.163986	+	0.986463i	$0.552435\pi$
$678$	10.9321	0.419844
$679$	0.250174	0.00960078
$680$	0	0
$681$	−30.5875	−1.17212
$682$	19.7739	0.757182
$683$	48.5684	1.85842	0.929210	−	0.369553i	$-0.120489\pi$
$683$	48.5684	1.85842	0.929210	+	0.369553i	$0.120489\pi$
$684$	−0.0843930	−0.00322685
$685$	0	0
$686$	17.4856	0.667604
$687$	36.9635	1.41024
$688$	−9.24660	−0.352523
$689$	−2.98715	−0.113801
$690$	0	0
$691$	44.8076	1.70456	0.852281	−	0.523084i	$-0.175219\pi$
$691$	44.8076	1.70456	0.852281	+	0.523084i	$0.175219\pi$
$692$	11.9877	0.455704
$693$	−0.459953	−0.0174722
$694$	−9.03868	−0.343104
$695$	0	0
$696$	7.02464	0.266268
$697$	−46.9608	−1.77877
$698$	−36.7305	−1.39027
$699$	25.2174	0.953809
$700$	0	0
$701$	2.21913	0.0838152	0.0419076	−	0.999121i	$-0.486656\pi$
$701$	2.21913	0.0838152	0.0419076	+	0.999121i	$0.486656\pi$
$702$	29.7891	1.12432
$703$	1.47942	0.0557974
$704$	2.77447	0.104567
$705$	0	0
$706$	18.3737	0.691503
$707$	−21.9358	−0.824982
$708$	6.48516	0.243727
$709$	34.3587	1.29037	0.645184	−	0.764027i	$-0.276781\pi$
$709$	34.3587	1.29037	0.645184	+	0.764027i	$0.276781\pi$
$710$	0	0
$711$	0.591640	0.0221882
$712$	9.18401	0.344186
$713$	−4.66799	−0.174817
$714$	−24.5313	−0.918062
$715$	0	0
$716$	−8.54134	−0.319205
$717$	−11.9005	−0.444434
$718$	24.4259	0.911565
$719$	11.9900	0.447151	0.223575	−	0.974687i	$-0.428227\pi$
$719$	11.9900	0.447151	0.223575	+	0.974687i	$0.428227\pi$
$720$	0	0
$721$	−21.7358	−0.809483
$722$	17.0052	0.632867
$723$	−5.67576	−0.211084
$724$	7.82844	0.290942
$725$	0	0
$726$	−5.66256	−0.210157
$727$	−42.1827	−1.56447	−0.782235	−	0.622983i	$-0.785920\pi$
$727$	−42.1827	−1.56447	−0.782235	+	0.622983i	$0.785920\pi$
$728$	15.7528	0.583839
$729$	27.5161	1.01912
$730$	0	0
$731$	47.6795	1.76349
$732$	18.5746	0.686537
$733$	0.322383	0.0119075	0.00595375	−	0.999982i	$-0.498105\pi$
$733$	0.322383	0.0119075	0.00595375	+	0.999982i	$0.498105\pi$
$734$	14.5224	0.536033
$735$	0	0
$736$	−0.654963	−0.0241423
$737$	11.9951	0.441847
$738$	0.544176	0.0200314
$739$	−5.49978	−0.202313	−0.101156	−	0.994871i	$-0.532254\pi$
$739$	−5.49978	−0.202313	−0.101156	+	0.994871i	$0.532254\pi$
$740$	0	0
$741$	13.7507	0.505143
$742$	−1.45968	−0.0535865
$743$	6.03812	0.221517	0.110759	−	0.993847i	$-0.464672\pi$
$743$	6.03812	0.221517	0.110759	+	0.993847i	$0.464672\pi$
$744$	−12.2209	−0.448042
$745$	0	0
$746$	−23.2887	−0.852662
$747$	0.279403	0.0102228
$748$	−14.3064	−0.523092
$749$	28.3605	1.03627
$750$	0	0
$751$	−13.5719	−0.495244	−0.247622	−	0.968857i	$-0.579649\pi$
$751$	−13.5719	−0.495244	−0.247622	+	0.968857i	$0.579649\pi$
$752$	2.77447	0.101174
$753$	16.2290	0.591417
$754$	23.2601	0.847082
$755$	0	0
$756$	14.5565	0.529416
$757$	−33.0661	−1.20181	−0.600904	−	0.799321i	$-0.705193\pi$
$757$	−33.0661	−1.20181	−0.600904	+	0.799321i	$0.705193\pi$
$758$	−28.6176	−1.03944
$759$	−3.11593	−0.113101
$760$	0	0
$761$	5.18860	0.188087	0.0940434	−	0.995568i	$-0.470021\pi$
$761$	5.18860	0.188087	0.0940434	+	0.995568i	$0.470021\pi$
$762$	−16.3737	−0.593157
$763$	10.8507	0.392822
$764$	3.36611	0.121782
$765$	0	0
$766$	−16.6157	−0.600350
$767$	21.4737	0.775372
$768$	−1.71472	−0.0618745
$769$	33.0698	1.19253	0.596264	−	0.802789i	$-0.296651\pi$
$769$	33.0698	1.19253	0.596264	+	0.802789i	$0.296651\pi$
$770$	0	0
$771$	−12.1396	−0.437195
$772$	15.4211	0.555018
$773$	1.13095	0.0406776	0.0203388	−	0.999793i	$-0.493526\pi$
$773$	1.13095	0.0406776	0.0203388	+	0.999793i	$0.493526\pi$
$774$	−0.552505	−0.0198594
$775$	0	0
$776$	−0.0901699	−0.00323691
$777$	−4.98323	−0.178772
$778$	−26.3767	−0.945652
$779$	12.8629	0.460861
$780$	0	0
$781$	−36.7523	−1.31510
$782$	3.37727	0.120771
$783$	21.4937	0.768121
$784$	0.697669	0.0249167
$785$	0	0
$786$	9.60109	0.342460
$787$	−15.7610	−0.561819	−0.280909	−	0.959734i	$-0.590636\pi$
$787$	−15.7610	−0.561819	−0.280909	+	0.959734i	$0.590636\pi$
$788$	−8.80489	−0.313661
$789$	2.45797	0.0875059
$790$	0	0
$791$	17.6885	0.628930
$792$	0.165781	0.00589075
$793$	61.5044	2.18409
$794$	−17.9086	−0.635552
$795$	0	0
$796$	−17.6222	−0.624601
$797$	23.1397	0.819649	0.409824	−	0.912164i	$-0.365590\pi$
$797$	23.1397	0.819649	0.409824	+	0.912164i	$0.365590\pi$
$798$	6.71930	0.237861
$799$	−14.3064	−0.506122
$800$	0	0
$801$	0.548765	0.0193897
$802$	−32.0164	−1.13054
$803$	−11.6895	−0.412514
$804$	−7.41340	−0.261451
$805$	0	0
$806$	−40.4661	−1.42536
$807$	24.4912	0.862132
$808$	7.90632	0.278144
$809$	33.5400	1.17921	0.589603	−	0.807694i	$-0.299284\pi$
$809$	33.5400	1.17921	0.589603	+	0.807694i	$0.299284\pi$
$810$	0	0
$811$	−11.9626	−0.420064	−0.210032	−	0.977695i	$-0.567357\pi$
$811$	−11.9626	−0.420064	−0.210032	+	0.977695i	$0.567357\pi$
$812$	11.3661	0.398872
$813$	−52.7884	−1.85137
$814$	−2.90615	−0.101861
$815$	0	0
$816$	8.84181	0.309525
$817$	−13.0598	−0.456903
$818$	−18.0344	−0.630557
$819$	0.941266	0.0328905
$820$	0	0
$821$	−10.6941	−0.373227	−0.186613	−	0.982433i	$-0.559751\pi$
$821$	−10.6941	−0.373227	−0.186613	+	0.982433i	$0.559751\pi$
$822$	−13.6905	−0.477512
$823$	12.0211	0.419028	0.209514	−	0.977806i	$-0.432812\pi$
$823$	12.0211	0.419028	0.209514	+	0.977806i	$0.432812\pi$
$824$	7.83422	0.272918
$825$	0	0
$826$	10.4932	0.365105
$827$	4.13974	0.143953	0.0719764	−	0.997406i	$-0.477069\pi$
$827$	4.13974	0.143953	0.0719764	+	0.997406i	$0.477069\pi$
$828$	−0.0391355	−0.00136005
$829$	−20.3503	−0.706796	−0.353398	−	0.935473i	$-0.614974\pi$
$829$	−20.3503	−0.706796	−0.353398	+	0.935473i	$0.614974\pi$
$830$	0	0
$831$	5.46940	0.189732
$832$	−5.67779	−0.196842
$833$	−3.59748	−0.124645
$834$	−37.8753	−1.31152
$835$	0	0
$836$	3.91861	0.135528
$837$	−37.3931	−1.29249
$838$	19.5300	0.674654
$839$	3.13515	0.108237	0.0541186	−	0.998535i	$-0.482765\pi$
$839$	3.13515	0.108237	0.0541186	+	0.998535i	$0.482765\pi$
$840$	0	0
$841$	−12.2172	−0.421283
$842$	19.5537	0.673865
$843$	−34.9612	−1.20413
$844$	6.10144	0.210020
$845$	0	0
$846$	0.165781	0.00569965
$847$	−9.16221	−0.314817
$848$	0.526111	0.0180667
$849$	−22.8004	−0.782508
$850$	0	0
$851$	0.686050	0.0235175
$852$	22.7142	0.778174
$853$	−23.3471	−0.799388	−0.399694	−	0.916649i	$-0.630884\pi$
$853$	−23.3471	−0.799388	−0.399694	+	0.916649i	$0.630884\pi$
$854$	30.0543	1.02844
$855$	0	0
$856$	−10.2220	−0.349379
$857$	−15.3036	−0.522762	−0.261381	−	0.965236i	$-0.584178\pi$
$857$	−15.3036	−0.522762	−0.261381	+	0.965236i	$0.584178\pi$
$858$	−27.0116	−0.922161
$859$	−2.96132	−0.101039	−0.0505194	−	0.998723i	$-0.516088\pi$
$859$	−2.96132	−0.101039	−0.0505194	+	0.998723i	$0.516088\pi$
$860$	0	0
$861$	−43.3269	−1.47658
$862$	21.1963	0.721949
$863$	37.8220	1.28748	0.643739	−	0.765246i	$-0.277382\pi$
$863$	37.8220	1.28748	0.643739	+	0.765246i	$0.277382\pi$
$864$	−5.24660	−0.178493
$865$	0	0
$866$	8.54967	0.290530
$867$	−16.4421	−0.558402
$868$	−19.7739	−0.671170
$869$	−27.4716	−0.931909
$870$	0	0
$871$	−24.5474	−0.831755
$872$	−3.91091	−0.132440
$873$	−0.00538785	−0.000182351 0
$874$	−0.925059	−0.0312906
$875$	0	0
$876$	7.22451	0.244094
$877$	15.6011	0.526810	0.263405	−	0.964685i	$-0.415154\pi$
$877$	15.6011	0.526810	0.263405	+	0.964685i	$0.415154\pi$
$878$	28.0714	0.947363
$879$	−44.6547	−1.50616
$880$	0	0
$881$	1.16438	0.0392289	0.0196144	−	0.999808i	$-0.493756\pi$
$881$	1.16438	0.0392289	0.0196144	+	0.999808i	$0.493756\pi$
$882$	0.0416872	0.00140368
$883$	−48.8851	−1.64511	−0.822557	−	0.568682i	$-0.807453\pi$
$883$	−48.8851	−1.64511	−0.822557	+	0.568682i	$0.807453\pi$
$884$	29.2771	0.984696
$885$	0	0
$886$	−33.0546	−1.11049
$887$	−8.62892	−0.289731	−0.144865	−	0.989451i	$-0.546275\pi$
$887$	−8.62892	−0.289731	−0.144865	+	0.989451i	$0.546275\pi$
$888$	1.79610	0.0602733
$889$	−26.4932	−0.888554
$890$	0	0
$891$	−24.4630	−0.819540
$892$	5.02866	0.168372
$893$	3.91861	0.131131
$894$	7.17439	0.239948
$895$	0	0
$896$	−2.77447	−0.0926884
$897$	6.37658	0.212908
$898$	37.1628	1.24014
$899$	−29.1975	−0.973790
$900$	0	0
$901$	−2.71286	−0.0903784
$902$	−25.2677	−0.841322
$903$	43.9900	1.46389
$904$	−6.37545	−0.212044
$905$	0	0
$906$	−0.568533	−0.0188882
$907$	−29.5085	−0.979815	−0.489907	−	0.871774i	$-0.662969\pi$
$907$	−29.5085	−0.979815	−0.489907	+	0.871774i	$0.662969\pi$
$908$	17.8382	0.591983
$909$	0.472420	0.0156692
$910$	0	0
$911$	−38.4088	−1.27254	−0.636270	−	0.771466i	$-0.719524\pi$
$911$	−38.4088	−1.27254	−0.636270	+	0.771466i	$0.719524\pi$
$912$	−2.42184	−0.0801950
$913$	−12.9735	−0.429360
$914$	29.3139	0.969619
$915$	0	0
$916$	−21.5566	−0.712251
$917$	15.5349	0.513007
$918$	27.0538	0.892907
$919$	41.6905	1.37524	0.687622	−	0.726069i	$-0.258655\pi$
$919$	41.6905	1.37524	0.687622	+	0.726069i	$0.258655\pi$
$920$	0	0
$921$	43.0394	1.41820
$922$	−8.90741	−0.293350
$923$	75.2114	2.47561
$924$	−13.1993	−0.434225
$925$	0	0
$926$	−18.2771	−0.600624
$927$	0.468112	0.0153748
$928$	−4.09668	−0.134480
$929$	28.1047	0.922086	0.461043	−	0.887378i	$-0.347475\pi$
$929$	28.1047	0.922086	0.461043	+	0.887378i	$0.347475\pi$
$930$	0	0
$931$	0.985376	0.0322944
$932$	−14.7065	−0.481726
$933$	−1.24030	−0.0406056
$934$	−36.6841	−1.20034
$935$	0	0
$936$	−0.339260	−0.0110891
$937$	12.8754	0.420620	0.210310	−	0.977635i	$-0.432553\pi$
$937$	12.8754	0.420620	0.210310	+	0.977635i	$0.432553\pi$
$938$	−11.9951	−0.391655
$939$	37.0658	1.20960
$940$	0	0
$941$	10.1510	0.330913	0.165457	−	0.986217i	$-0.447090\pi$
$941$	10.1510	0.330913	0.165457	+	0.986217i	$0.447090\pi$
$942$	−6.18583	−0.201545
$943$	5.96489	0.194243
$944$	−3.78206	−0.123096
$945$	0	0
$946$	25.6544	0.834096
$947$	57.3750	1.86444	0.932218	−	0.361896i	$-0.117871\pi$
$947$	57.3750	1.86444	0.932218	+	0.361896i	$0.117871\pi$
$948$	16.9784	0.551432
$949$	23.9219	0.776538
$950$	0	0
$951$	−32.2944	−1.04722
$952$	14.3064	0.463672
$953$	−5.17264	−0.167558	−0.0837791	−	0.996484i	$-0.526699\pi$
$953$	−5.17264	−0.167558	−0.0837791	+	0.996484i	$0.526699\pi$
$954$	0.0314363	0.00101779
$955$	0	0
$956$	6.94025	0.224464
$957$	−19.4896	−0.630010
$958$	12.5898	0.406757
$959$	−22.1517	−0.715317
$960$	0	0
$961$	19.7955	0.638566
$962$	5.94728	0.191748
$963$	−0.610785	−0.0196823
$964$	3.31003	0.106609
$965$	0	0
$966$	3.11593	0.100254
$967$	41.3979	1.33127	0.665634	−	0.746279i	$-0.268161\pi$
$967$	41.3979	1.33127	0.665634	+	0.746279i	$0.268161\pi$
$968$	3.30233	0.106141
$969$	12.4880	0.401173
$970$	0	0
$971$	43.0848	1.38266	0.691329	−	0.722540i	$-0.257025\pi$
$971$	43.0848	1.38266	0.691329	+	0.722540i	$0.257025\pi$
$972$	−0.620870	−0.0199144
$973$	−61.2836	−1.96466
$974$	−18.3256	−0.587189
$975$	0	0
$976$	−10.8325	−0.346739
$977$	−40.4614	−1.29448	−0.647238	−	0.762288i	$-0.724076\pi$
$977$	−40.4614	−1.29448	−0.647238	+	0.762288i	$0.724076\pi$
$978$	−8.73449	−0.279298
$979$	−25.4807	−0.814368
$980$	0	0
$981$	−0.233686	−0.00746101
$982$	−21.7047	−0.692625
$983$	−40.3786	−1.28788	−0.643938	−	0.765078i	$-0.722701\pi$
$983$	−40.3786	−1.28788	−0.643938	+	0.765078i	$0.722701\pi$
$984$	15.6163	0.497829
$985$	0	0
$986$	21.1243	0.672734
$987$	−13.1993	−0.420139
$988$	−8.01921	−0.255125
$989$	−6.05618	−0.192575
$990$	0	0
$991$	−58.1435	−1.84699	−0.923494	−	0.383613i	$-0.874680\pi$
$991$	−58.1435	−1.84699	−0.923494	+	0.383613i	$0.874680\pi$
$992$	7.12710	0.226286
$993$	−28.9374	−0.918300
$994$	36.7523	1.16571
$995$	0	0
$996$	8.01806	0.254062
$997$	−39.2968	−1.24454	−0.622271	−	0.782802i	$-0.713790\pi$
$997$	−39.2968	−1.24454	−0.622271	+	0.782802i	$0.713790\pi$
$998$	−15.8391	−0.501377
$999$	5.49563	0.173874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1250.2.a.f.1.2		4
4.3	odd	2		10000.2.a.x.1.3		4
5.2	odd	4		1250.2.b.e.1249.3		8
5.3	odd	4		1250.2.b.e.1249.6		8
5.4	even	2		1250.2.a.l.1.3		4
20.19	odd	2		10000.2.a.t.1.2		4
25.3	odd	20		250.2.e.c.49.4		16
25.4	even	10		50.2.d.b.41.2	yes	8
25.6	even	5		250.2.d.d.51.1		8
25.8	odd	20		250.2.e.c.199.1		16
25.17	odd	20		250.2.e.c.199.4		16
25.19	even	10		50.2.d.b.11.2	✓	8
25.21	even	5		250.2.d.d.201.1		8
25.22	odd	20		250.2.e.c.49.1		16
75.29	odd	10		450.2.h.e.91.1		8
75.44	odd	10		450.2.h.e.361.1		8
100.19	odd	10		400.2.u.d.161.1		8
100.79	odd	10		400.2.u.d.241.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
50.2.d.b.11.2	✓	8	25.19	even	10
50.2.d.b.41.2	yes	8	25.4	even	10
250.2.d.d.51.1		8	25.6	even	5
250.2.d.d.201.1		8	25.21	even	5
250.2.e.c.49.1		16	25.22	odd	20
250.2.e.c.49.4		16	25.3	odd	20
250.2.e.c.199.1		16	25.8	odd	20
250.2.e.c.199.4		16	25.17	odd	20
400.2.u.d.161.1		8	100.19	odd	10
400.2.u.d.241.1		8	100.79	odd	10
450.2.h.e.91.1		8	75.29	odd	10
450.2.h.e.361.1		8	75.44	odd	10
1250.2.a.f.1.2		4	1.1	even	1	trivial
1250.2.a.l.1.3		4	5.4	even	2
1250.2.b.e.1249.3		8	5.2	odd	4
1250.2.b.e.1249.6		8	5.3	odd	4
10000.2.a.t.1.2		4	20.19	odd	2
10000.2.a.x.1.3		4	4.3	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:	\( N \)	\(=\)	\( 1250 = 2 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1250.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.71472 q^{3} +1.00000 q^{4} +1.71472 q^{6} +2.77447 q^{7} -1.00000 q^{8} -0.0597522 q^{9} +2.77447 q^{11} -1.71472 q^{12} -5.67779 q^{13} -2.77447 q^{14} +1.00000 q^{16} -5.15643 q^{17} +0.0597522 q^{18} +1.41238 q^{19} -4.75742 q^{21} -2.77447 q^{22} +0.654963 q^{23} +1.71472 q^{24} +5.67779 q^{26} +5.24660 q^{27} +2.77447 q^{28} +4.09668 q^{29} -7.12710 q^{31} -1.00000 q^{32} -4.75742 q^{33} +5.15643 q^{34} -0.0597522 q^{36} +1.04746 q^{37} -1.41238 q^{38} +9.73579 q^{39} +9.10722 q^{41} +4.75742 q^{42} -9.24660 q^{43} +2.77447 q^{44} -0.654963 q^{46} +2.77447 q^{47} -1.71472 q^{48} +0.697669 q^{49} +8.84181 q^{51} -5.67779 q^{52} +0.526111 q^{53} -5.24660 q^{54} -2.77447 q^{56} -2.42184 q^{57} -4.09668 q^{58} -3.78206 q^{59} -10.8325 q^{61} +7.12710 q^{62} -0.165781 q^{63} +1.00000 q^{64} +4.75742 q^{66} +4.32340 q^{67} -5.15643 q^{68} -1.12307 q^{69} -13.2466 q^{71} +0.0597522 q^{72} -4.21324 q^{73} -1.04746 q^{74} +1.41238 q^{76} +7.69767 q^{77} -9.73579 q^{78} -9.90157 q^{79} -8.81717 q^{81} -9.10722 q^{82} -4.67603 q^{83} -4.75742 q^{84} +9.24660 q^{86} -7.02464 q^{87} -2.77447 q^{88} -9.18401 q^{89} -15.7528 q^{91} +0.654963 q^{92} +12.2209 q^{93} -2.77447 q^{94} +1.71472 q^{96} +0.0901699 q^{97} -0.697669 q^{98} -0.165781 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - q^{3} + 4 q^{4} + q^{6} - 2 q^{7} - 4 q^{8} + 7 q^{9} - 2 q^{11} - q^{12} - 11 q^{13} + 2 q^{14} + 4 q^{16} - 12 q^{17} - 7 q^{18} - 5 q^{19} - 7 q^{21} + 2 q^{22} + 4 q^{23} + q^{24}+ \cdots - 21 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 - q^3 + 4 * q^4 + q^6 - 2 * q^7 - 4 * q^8 + 7 * q^9 - 2 * q^11 - q^12 - 11 * q^13 + 2 * q^14 + 4 * q^16 - 12 * q^17 - 7 * q^18 - 5 * q^19 - 7 * q^21 + 2 * q^22 + 4 * q^23 + q^24 + 11 * q^26 - 10 * q^27 - 2 * q^28 + 15 * q^29 - 12 * q^31 - 4 * q^32 - 7 * q^33 + 12 * q^34 + 7 * q^36 - 12 * q^37 + 5 * q^38 - 11 * q^39 + 13 * q^41 + 7 * q^42 - 6 * q^43 - 2 * q^44 - 4 * q^46 - 2 * q^47 - q^48 - 2 * q^49 + 13 * q^51 - 11 * q^52 - 11 * q^53 + 10 * q^54 + 2 * q^56 - 15 * q^58 + 8 * q^61 + 12 * q^62 - 21 * q^63 + 4 * q^64 + 7 * q^66 - 22 * q^67 - 12 * q^68 - 31 * q^69 - 22 * q^71 - 7 * q^72 - 21 * q^73 + 12 * q^74 - 5 * q^76 + 26 * q^77 + 11 * q^78 - 10 * q^79 - 16 * q^81 - 13 * q^82 + 24 * q^83 - 7 * q^84 + 6 * q^86 - 25 * q^87 + 2 * q^88 - 5 * q^89 - 12 * q^91 + 4 * q^92 + 23 * q^93 + 2 * q^94 + q^96 - 22 * q^97 + 2 * q^98 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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