LMFDB - Embedded newform 1925.2.a.bd.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1925,2,Mod(1,1925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1925, 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("1925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1925 = 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1925.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:	$15.3712023891$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 13x^{5} + 12x^{4} + 47x^{3} - 37x^{2} - 35x + 6 $ x^7 - x^6 - 13x^5 + 12x^4 + 47x^3 - 37x^2 - 35*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.60926$ of defining polynomial
Character	$\chi$	$=$	1925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.60926 q^{2} -2.47161 q^{3} +4.80822 q^{4} +6.44906 q^{6} +1.00000 q^{7} -7.32737 q^{8} +3.10886 q^{9} +O(q^{10})$	\(q-2.60926 q^{2} -2.47161 q^{3} +4.80822 q^{4} +6.44906 q^{6} +1.00000 q^{7} -7.32737 q^{8} +3.10886 q^{9} +1.00000 q^{11} -11.8840 q^{12} +5.11245 q^{13} -2.60926 q^{14} +9.50254 q^{16} -0.171628 q^{17} -8.11180 q^{18} +5.80822 q^{19} -2.47161 q^{21} -2.60926 q^{22} +4.19392 q^{23} +18.1104 q^{24} -13.3397 q^{26} -0.269047 q^{27} +4.80822 q^{28} -7.18733 q^{29} +4.25514 q^{31} -10.1398 q^{32} -2.47161 q^{33} +0.447822 q^{34} +14.9481 q^{36} +8.41748 q^{37} -15.1551 q^{38} -12.6360 q^{39} +11.5185 q^{41} +6.44906 q^{42} +2.38415 q^{43} +4.80822 q^{44} -10.9430 q^{46} +2.98716 q^{47} -23.4866 q^{48} +1.00000 q^{49} +0.424198 q^{51} +24.5818 q^{52} +2.49456 q^{53} +0.702013 q^{54} -7.32737 q^{56} -14.3557 q^{57} +18.7536 q^{58} -8.65163 q^{59} +1.20773 q^{61} -11.1028 q^{62} +3.10886 q^{63} +7.45236 q^{64} +6.44906 q^{66} -9.48784 q^{67} -0.825226 q^{68} -10.3657 q^{69} +7.15610 q^{71} -22.7797 q^{72} -16.3866 q^{73} -21.9634 q^{74} +27.9272 q^{76} +1.00000 q^{77} +32.9705 q^{78} -11.9576 q^{79} -8.66159 q^{81} -30.0547 q^{82} -6.44282 q^{83} -11.8840 q^{84} -6.22086 q^{86} +17.7643 q^{87} -7.32737 q^{88} -11.8342 q^{89} +5.11245 q^{91} +20.1653 q^{92} -10.5171 q^{93} -7.79426 q^{94} +25.0617 q^{96} -7.86534 q^{97} -2.60926 q^{98} +3.10886 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 13 q^{4} + 3 q^{6} + 7 q^{7} + 9 q^{9}+O(q^{10}) $ 7 * q + q^2 + 13 * q^4 + 3 * q^6 + 7 * q^7 + 9 * q^9	$ 7 q + q^{2} + 13 q^{4} + 3 q^{6} + 7 q^{7} + 9 q^{9} + 7 q^{11} - 9 q^{12} - 3 q^{13} + q^{14} + 21 q^{16} - 2 q^{17} + 8 q^{18} + 20 q^{19} + q^{22} + 11 q^{23} + 18 q^{24} - 13 q^{26} - 12 q^{27} + 13 q^{28} + 4 q^{29} + 6 q^{31} + q^{32} - 7 q^{34} + 12 q^{36} + 19 q^{37} + 3 q^{38} - 10 q^{39} + 24 q^{41} + 3 q^{42} + 13 q^{44} + 33 q^{46} - q^{47} - 15 q^{48} + 7 q^{49} + 19 q^{51} - 29 q^{52} + 7 q^{53} + 9 q^{54} - 9 q^{57} + 37 q^{58} + 9 q^{59} + 18 q^{61} - 40 q^{62} + 9 q^{63} + 8 q^{64} + 3 q^{66} + 18 q^{68} - 15 q^{69} + 18 q^{71} - 64 q^{72} + 5 q^{73} - 24 q^{74} + 88 q^{76} + 7 q^{77} + 79 q^{78} + 25 q^{79} - q^{81} - 60 q^{82} - 17 q^{83} - 9 q^{84} - 41 q^{86} - 24 q^{87} - 16 q^{89} - 3 q^{91} + 28 q^{92} + 26 q^{93} - 31 q^{94} + 17 q^{96} - 4 q^{97} + q^{98} + 9 q^{99}+O(q^{100}) $ 7 * q + q^2 + 13 * q^4 + 3 * q^6 + 7 * q^7 + 9 * q^9 + 7 * q^11 - 9 * q^12 - 3 * q^13 + q^14 + 21 * q^16 - 2 * q^17 + 8 * q^18 + 20 * q^19 + q^22 + 11 * q^23 + 18 * q^24 - 13 * q^26 - 12 * q^27 + 13 * q^28 + 4 * q^29 + 6 * q^31 + q^32 - 7 * q^34 + 12 * q^36 + 19 * q^37 + 3 * q^38 - 10 * q^39 + 24 * q^41 + 3 * q^42 + 13 * q^44 + 33 * q^46 - q^47 - 15 * q^48 + 7 * q^49 + 19 * q^51 - 29 * q^52 + 7 * q^53 + 9 * q^54 - 9 * q^57 + 37 * q^58 + 9 * q^59 + 18 * q^61 - 40 * q^62 + 9 * q^63 + 8 * q^64 + 3 * q^66 + 18 * q^68 - 15 * q^69 + 18 * q^71 - 64 * q^72 + 5 * q^73 - 24 * q^74 + 88 * q^76 + 7 * q^77 + 79 * q^78 + 25 * q^79 - q^81 - 60 * q^82 - 17 * q^83 - 9 * q^84 - 41 * q^86 - 24 * q^87 - 16 * q^89 - 3 * q^91 + 28 * q^92 + 26 * q^93 - 31 * q^94 + 17 * q^96 - 4 * q^97 + q^98 + 9 * 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 $

$2$	−2.60926	−1.84502	−0.922512	−	0.385969i	$-0.873867\pi$
$2$	−2.60926	−1.84502	−0.922512	+	0.385969i	$0.873867\pi$
$3$	−2.47161	−1.42698	−0.713492	−	0.700663i	$-0.752888\pi$
$3$	−2.47161	−1.42698	−0.713492	+	0.700663i	$0.752888\pi$
$4$	4.80822	2.40411
$5$	0	0
$5$	0	0
$6$	6.44906	2.63282
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−7.32737	−2.59062
$9$	3.10886	1.03629
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−11.8840	−3.43063
$13$	5.11245	1.41794	0.708970	−	0.705239i	$-0.249160\pi$
$13$	5.11245	1.41794	0.708970	+	0.705239i	$0.249160\pi$
$14$	−2.60926	−0.697353
$15$	0	0
$16$	9.50254	2.37564
$17$	−0.171628	−0.0416259	−0.0208130	−	0.999783i	$-0.506625\pi$
$17$	−0.171628	−0.0416259	−0.0208130	+	0.999783i	$0.506625\pi$
$18$	−8.11180	−1.91197
$19$	5.80822	1.33250	0.666249	−	0.745730i	$-0.267899\pi$
$19$	5.80822	1.33250	0.666249	+	0.745730i	$0.267899\pi$
$20$	0	0
$21$	−2.47161	−0.539349
$22$	−2.60926	−0.556295
$23$	4.19392	0.874493	0.437247	−	0.899342i	$-0.355954\pi$
$23$	4.19392	0.874493	0.437247	+	0.899342i	$0.355954\pi$
$24$	18.1104	3.69677
$25$	0	0
$26$	−13.3397	−2.61613
$27$	−0.269047	−0.0517782
$28$	4.80822	0.908668
$29$	−7.18733	−1.33465	−0.667327	−	0.744765i	$-0.732561\pi$
$29$	−7.18733	−1.33465	−0.667327	+	0.744765i	$0.732561\pi$
$30$	0	0
$31$	4.25514	0.764246	0.382123	−	0.924111i	$-0.375193\pi$
$31$	4.25514	0.764246	0.382123	+	0.924111i	$0.375193\pi$
$32$	−10.1398	−1.79249
$33$	−2.47161	−0.430252
$34$	0.447822	0.0768008
$35$	0	0
$36$	14.9481	2.49134
$37$	8.41748	1.38382	0.691912	−	0.721981i	$-0.256768\pi$
$37$	8.41748	1.38382	0.691912	+	0.721981i	$0.256768\pi$
$38$	−15.1551	−2.45849
$39$	−12.6360	−2.02338
$40$	0	0
$41$	11.5185	1.79889	0.899443	−	0.437037i	$-0.143972\pi$
$41$	11.5185	1.79889	0.899443	+	0.437037i	$0.143972\pi$
$42$	6.44906	0.995112
$43$	2.38415	0.363579	0.181790	−	0.983337i	$-0.441811\pi$
$43$	2.38415	0.363579	0.181790	+	0.983337i	$0.441811\pi$
$44$	4.80822	0.724867
$45$	0	0
$46$	−10.9430	−1.61346
$47$	2.98716	0.435722	0.217861	−	0.975980i	$-0.430092\pi$
$47$	2.98716	0.435722	0.217861	+	0.975980i	$0.430092\pi$
$48$	−23.4866	−3.39000
$49$	1.00000	0.142857
$50$	0	0
$51$	0.424198	0.0593995
$52$	24.5818	3.40888
$53$	2.49456	0.342654	0.171327	−	0.985214i	$-0.445195\pi$
$53$	2.49456	0.342654	0.171327	+	0.985214i	$0.445195\pi$
$54$	0.702013	0.0955319
$55$	0	0
$56$	−7.32737	−0.979161
$57$	−14.3557	−1.90145
$58$	18.7536	2.46247
$59$	−8.65163	−1.12635	−0.563173	−	0.826339i	$-0.690420\pi$
$59$	−8.65163	−1.12635	−0.563173	+	0.826339i	$0.690420\pi$
$60$	0	0
$61$	1.20773	0.154634	0.0773170	−	0.997007i	$-0.475365\pi$
$61$	1.20773	0.154634	0.0773170	+	0.997007i	$0.475365\pi$
$62$	−11.1028	−1.41005
$63$	3.10886	0.391679
$64$	7.45236	0.931545
$65$	0	0
$66$	6.44906	0.793825
$67$	−9.48784	−1.15912	−0.579562	−	0.814928i	$-0.696776\pi$
$67$	−9.48784	−1.15912	−0.579562	+	0.814928i	$0.696776\pi$
$68$	−0.825226	−0.100073
$69$	−10.3657	−1.24789
$70$	0	0
$71$	7.15610	0.849273	0.424636	−	0.905364i	$-0.360402\pi$
$71$	7.15610	0.849273	0.424636	+	0.905364i	$0.360402\pi$
$72$	−22.7797	−2.68462
$73$	−16.3866	−1.91791	−0.958953	−	0.283566i	$-0.908482\pi$
$73$	−16.3866	−1.91791	−0.958953	+	0.283566i	$0.908482\pi$
$74$	−21.9634	−2.55319
$75$	0	0
$76$	27.9272	3.20347
$77$	1.00000	0.113961
$78$	32.9705	3.73318
$79$	−11.9576	−1.34534	−0.672668	−	0.739945i	$-0.734852\pi$
$79$	−11.9576	−1.34534	−0.672668	+	0.739945i	$0.734852\pi$
$80$	0	0
$81$	−8.66159	−0.962398
$82$	−30.0547	−3.31899
$83$	−6.44282	−0.707191	−0.353596	−	0.935398i	$-0.615041\pi$
$83$	−6.44282	−0.707191	−0.353596	+	0.935398i	$0.615041\pi$
$84$	−11.8840	−1.29666
$85$	0	0
$86$	−6.22086	−0.670812
$87$	17.7643	1.90453
$88$	−7.32737	−0.781100
$89$	−11.8342	−1.25442	−0.627209	−	0.778851i	$-0.715803\pi$
$89$	−11.8342	−1.25442	−0.627209	+	0.778851i	$0.715803\pi$
$90$	0	0
$91$	5.11245	0.535931
$92$	20.1653	2.10238
$93$	−10.5171	−1.09057
$94$	−7.79426	−0.803917
$95$	0	0
$96$	25.0617	2.55785
$97$	−7.86534	−0.798604	−0.399302	−	0.916819i	$-0.630748\pi$
$97$	−7.86534	−0.798604	−0.399302	+	0.916819i	$0.630748\pi$
$98$	−2.60926	−0.263575
$99$	3.10886	0.312452
$100$	0	0
$101$	−3.83701	−0.381797	−0.190899	−	0.981610i	$-0.561140\pi$
$101$	−3.83701	−0.381797	−0.190899	+	0.981610i	$0.561140\pi$
$102$	−1.10684	−0.109594
$103$	7.92387	0.780762	0.390381	−	0.920653i	$-0.372343\pi$
$103$	7.92387	0.780762	0.390381	+	0.920653i	$0.372343\pi$
$104$	−37.4608	−3.67334
$105$	0	0
$106$	−6.50894	−0.632204
$107$	8.27563	0.800035	0.400018	−	0.916507i	$-0.369004\pi$
$107$	8.27563	0.800035	0.400018	+	0.916507i	$0.369004\pi$
$108$	−1.29364	−0.124480
$109$	12.4289	1.19047	0.595237	−	0.803550i	$-0.297058\pi$
$109$	12.4289	1.19047	0.595237	+	0.803550i	$0.297058\pi$
$110$	0	0
$111$	−20.8047	−1.97470
$112$	9.50254	0.897906
$113$	4.76099	0.447876	0.223938	−	0.974603i	$-0.428109\pi$
$113$	4.76099	0.447876	0.223938	+	0.974603i	$0.428109\pi$
$114$	37.4576	3.50822
$115$	0	0
$116$	−34.5583	−3.20865
$117$	15.8939	1.46939
$118$	22.5743	2.07813
$119$	−0.171628	−0.0157331
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.15128	−0.285303
$123$	−28.4692	−2.56698
$124$	20.4597	1.83733
$125$	0	0
$126$	−8.11180	−0.722657
$127$	19.0780	1.69290	0.846450	−	0.532468i	$-0.178735\pi$
$127$	19.0780	1.69290	0.846450	+	0.532468i	$0.178735\pi$
$128$	0.834565	0.0737658
$129$	−5.89269	−0.518822
$130$	0	0
$131$	17.9357	1.56705	0.783525	−	0.621360i	$-0.213420\pi$
$131$	17.9357	1.56705	0.783525	+	0.621360i	$0.213420\pi$
$132$	−11.8840	−1.03437
$133$	5.80822	0.503637
$134$	24.7562	2.13861
$135$	0	0
$136$	1.25758	0.107837
$137$	−18.1829	−1.55347	−0.776734	−	0.629829i	$-0.783125\pi$
$137$	−18.1829	−1.55347	−0.776734	+	0.629829i	$0.783125\pi$
$138$	27.0469	2.30238
$139$	−16.6742	−1.41428	−0.707142	−	0.707072i	$-0.750016\pi$
$139$	−16.6742	−1.41428	−0.707142	+	0.707072i	$0.750016\pi$
$140$	0	0
$141$	−7.38309	−0.621768
$142$	−18.6721	−1.56693
$143$	5.11245	0.427525
$144$	29.5420	2.46184
$145$	0	0
$146$	42.7568	3.53858
$147$	−2.47161	−0.203855
$148$	40.4731	3.32687
$149$	20.3180	1.66451	0.832256	−	0.554391i	$-0.187049\pi$
$149$	20.3180	1.66451	0.832256	+	0.554391i	$0.187049\pi$
$150$	0	0
$151$	22.1214	1.80022	0.900108	−	0.435666i	$-0.143487\pi$
$151$	22.1214	1.80022	0.900108	+	0.435666i	$0.143487\pi$
$152$	−42.5590	−3.45199
$153$	−0.533567	−0.0431363
$154$	−2.60926	−0.210260
$155$	0	0
$156$	−60.7566	−4.86442
$157$	−8.93032	−0.712717	−0.356358	−	0.934349i	$-0.615982\pi$
$157$	−8.93032	−0.712717	−0.356358	+	0.934349i	$0.615982\pi$
$158$	31.2005	2.48217
$159$	−6.16557	−0.488962
$160$	0	0
$161$	4.19392	0.330527
$162$	22.6003	1.77565
$163$	10.7957	0.845581	0.422790	−	0.906227i	$-0.361051\pi$
$163$	10.7957	0.845581	0.422790	+	0.906227i	$0.361051\pi$
$164$	55.3835	4.32472
$165$	0	0
$166$	16.8110	1.30478
$167$	2.11041	0.163308	0.0816541	−	0.996661i	$-0.473980\pi$
$167$	2.11041	0.163308	0.0816541	+	0.996661i	$0.473980\pi$
$168$	18.1104	1.39725
$169$	13.1372	1.01055
$170$	0	0
$171$	18.0569	1.38085
$172$	11.4635	0.874085
$173$	−14.6817	−1.11623	−0.558113	−	0.829765i	$-0.688474\pi$
$173$	−14.6817	−1.11623	−0.558113	+	0.829765i	$0.688474\pi$
$174$	−46.3515	−3.51390
$175$	0	0
$176$	9.50254	0.716281
$177$	21.3834	1.60728
$178$	30.8783	2.31443
$179$	−5.56483	−0.415935	−0.207967	−	0.978136i	$-0.566685\pi$
$179$	−5.56483	−0.415935	−0.207967	+	0.978136i	$0.566685\pi$
$180$	0	0
$181$	−11.0565	−0.821822	−0.410911	−	0.911675i	$-0.634789\pi$
$181$	−11.0565	−0.821822	−0.410911	+	0.911675i	$0.634789\pi$
$182$	−13.3397	−0.988805
$183$	−2.98504	−0.220660
$184$	−30.7304	−2.26548
$185$	0	0
$186$	27.4417	2.01212
$187$	−0.171628	−0.0125507
$188$	14.3629	1.04752
$189$	−0.269047	−0.0195703
$190$	0	0
$191$	−24.6307	−1.78222	−0.891109	−	0.453789i	$-0.850072\pi$
$191$	−24.6307	−1.78222	−0.891109	+	0.453789i	$0.850072\pi$
$192$	−18.4193	−1.32930
$193$	17.2410	1.24103	0.620517	−	0.784193i	$-0.286923\pi$
$193$	17.2410	1.24103	0.620517	+	0.784193i	$0.286923\pi$
$194$	20.5227	1.47344
$195$	0	0
$196$	4.80822	0.343444
$197$	−0.639350	−0.0455518	−0.0227759	−	0.999741i	$-0.507250\pi$
$197$	−0.639350	−0.0455518	−0.0227759	+	0.999741i	$0.507250\pi$
$198$	−8.11180	−0.576481
$199$	−3.28183	−0.232642	−0.116321	−	0.993212i	$-0.537110\pi$
$199$	−3.28183	−0.232642	−0.116321	+	0.993212i	$0.537110\pi$
$200$	0	0
$201$	23.4502	1.65405
$202$	10.0118	0.704424
$203$	−7.18733	−0.504452
$204$	2.03964	0.142803
$205$	0	0
$206$	−20.6754	−1.44052
$207$	13.0383	0.906224
$208$	48.5813	3.36851
$209$	5.80822	0.401763
$210$	0	0
$211$	22.8893	1.57576	0.787882	−	0.615826i	$-0.211178\pi$
$211$	22.8893	1.57576	0.787882	+	0.615826i	$0.211178\pi$
$212$	11.9944	0.823777
$213$	−17.6871	−1.21190
$214$	−21.5932	−1.47608
$215$	0	0
$216$	1.97141	0.134137
$217$	4.25514	0.288858
$218$	−32.4302	−2.19645
$219$	40.5013	2.73682
$220$	0	0
$221$	−0.877441	−0.0590230
$222$	54.2849	3.64336
$223$	0.0470236	0.00314893	0.00157447	−	0.999999i	$-0.499499\pi$
$223$	0.0470236	0.00314893	0.00157447	+	0.999999i	$0.499499\pi$
$224$	−10.1398	−0.677497
$225$	0	0
$226$	−12.4226	−0.826342
$227$	−5.63671	−0.374121	−0.187061	−	0.982348i	$-0.559896\pi$
$227$	−5.63671	−0.374121	−0.187061	+	0.982348i	$0.559896\pi$
$228$	−69.0252	−4.57130
$229$	−2.44614	−0.161646	−0.0808229	−	0.996728i	$-0.525755\pi$
$229$	−2.44614	−0.161646	−0.0808229	+	0.996728i	$0.525755\pi$
$230$	0	0
$231$	−2.47161	−0.162620
$232$	52.6642	3.45757
$233$	17.5181	1.14765	0.573823	−	0.818979i	$-0.305460\pi$
$233$	17.5181	1.14765	0.573823	+	0.818979i	$0.305460\pi$
$234$	−41.4712	−2.71106
$235$	0	0
$236$	−41.5989	−2.70786
$237$	29.5545	1.91977
$238$	0.447822	0.0290280
$239$	7.71643	0.499135	0.249567	−	0.968357i	$-0.419712\pi$
$239$	7.71643	0.499135	0.249567	+	0.968357i	$0.419712\pi$
$240$	0	0
$241$	0.954055	0.0614561	0.0307280	−	0.999528i	$-0.490217\pi$
$241$	0.954055	0.0614561	0.0307280	+	0.999528i	$0.490217\pi$
$242$	−2.60926	−0.167729
$243$	22.2152	1.42511
$244$	5.80703	0.371757
$245$	0	0
$246$	74.2835	4.73614
$247$	29.6943	1.88940
$248$	−31.1790	−1.97987
$249$	15.9241	1.00915
$250$	0	0
$251$	22.7157	1.43380	0.716900	−	0.697176i	$-0.245561\pi$
$251$	22.7157	1.43380	0.716900	+	0.697176i	$0.245561\pi$
$252$	14.9481	0.941639
$253$	4.19392	0.263670
$254$	−49.7794	−3.12344
$255$	0	0
$256$	−17.0823	−1.06764
$257$	−5.46472	−0.340880	−0.170440	−	0.985368i	$-0.554519\pi$
$257$	−5.46472	−0.340880	−0.170440	+	0.985368i	$0.554519\pi$
$258$	15.3755	0.957239
$259$	8.41748	0.523037
$260$	0	0
$261$	−22.3444	−1.38308
$262$	−46.7989	−2.89124
$263$	−17.5287	−1.08087	−0.540433	−	0.841387i	$-0.681740\pi$
$263$	−17.5287	−1.08087	−0.540433	+	0.841387i	$0.681740\pi$
$264$	18.1104	1.11462
$265$	0	0
$266$	−15.1551	−0.929221
$267$	29.2494	1.79003
$268$	−45.6196	−2.78666
$269$	5.68504	0.346623	0.173312	−	0.984867i	$-0.444553\pi$
$269$	5.68504	0.346623	0.173312	+	0.984867i	$0.444553\pi$
$270$	0	0
$271$	17.8547	1.08460	0.542298	−	0.840186i	$-0.317554\pi$
$271$	17.8547	1.08460	0.542298	+	0.840186i	$0.317554\pi$
$272$	−1.63090	−0.0988880
$273$	−12.6360	−0.764765
$274$	47.4438	2.86618
$275$	0	0
$276$	−49.8408	−3.00006
$277$	−17.6392	−1.05984	−0.529919	−	0.848048i	$-0.677778\pi$
$277$	−17.6392	−1.05984	−0.529919	+	0.848048i	$0.677778\pi$
$278$	43.5072	2.60939
$279$	13.2286	0.791977
$280$	0	0
$281$	−16.9556	−1.01149	−0.505744	−	0.862684i	$-0.668782\pi$
$281$	−16.9556	−1.01149	−0.505744	+	0.862684i	$0.668782\pi$
$282$	19.2644	1.14718
$283$	−17.2019	−1.02255	−0.511274	−	0.859418i	$-0.670826\pi$
$283$	−17.2019	−1.02255	−0.511274	+	0.859418i	$0.670826\pi$
$284$	34.4081	2.04175
$285$	0	0
$286$	−13.3397	−0.788793
$287$	11.5185	0.679915
$288$	−31.5233	−1.85753
$289$	−16.9705	−0.998267
$290$	0	0
$291$	19.4400	1.13960
$292$	−78.7903	−4.61086
$293$	−8.34872	−0.487738	−0.243869	−	0.969808i	$-0.578417\pi$
$293$	−8.34872	−0.487738	−0.243869	+	0.969808i	$0.578417\pi$
$294$	6.44906	0.376117
$295$	0	0
$296$	−61.6780	−3.58496
$297$	−0.269047	−0.0156117
$298$	−53.0148	−3.07106
$299$	21.4412	1.23998
$300$	0	0
$301$	2.38415	0.137420
$302$	−57.7205	−3.32144
$303$	9.48360	0.544818
$304$	55.1929	3.16553
$305$	0	0
$306$	1.39221	0.0795875
$307$	29.8638	1.70442	0.852208	−	0.523203i	$-0.175263\pi$
$307$	29.8638	1.70442	0.852208	+	0.523203i	$0.175263\pi$
$308$	4.80822	0.273974
$309$	−19.5847	−1.11414
$310$	0	0
$311$	−7.41946	−0.420719	−0.210360	−	0.977624i	$-0.567463\pi$
$311$	−7.41946	−0.420719	−0.210360	+	0.977624i	$0.567463\pi$
$312$	92.5886	5.24180
$313$	−7.28097	−0.411545	−0.205772	−	0.978600i	$-0.565971\pi$
$313$	−7.28097	−0.411545	−0.205772	+	0.978600i	$0.565971\pi$
$314$	23.3015	1.31498
$315$	0	0
$316$	−57.4948	−3.23433
$317$	13.8576	0.778321	0.389160	−	0.921170i	$-0.372765\pi$
$317$	13.8576	0.778321	0.389160	+	0.921170i	$0.372765\pi$
$318$	16.0876	0.902145
$319$	−7.18733	−0.402413
$320$	0	0
$321$	−20.4541	−1.14164
$322$	−10.9430	−0.609831
$323$	−0.996854	−0.0554664
$324$	−41.6468	−2.31371
$325$	0	0
$326$	−28.1686	−1.56012
$327$	−30.7194	−1.69879
$328$	−84.4003	−4.66022
$329$	2.98716	0.164687
$330$	0	0
$331$	11.5802	0.636505	0.318253	−	0.948006i	$-0.396904\pi$
$331$	11.5802	0.636505	0.318253	+	0.948006i	$0.396904\pi$
$332$	−30.9785	−1.70017
$333$	26.1687	1.43404
$334$	−5.50660	−0.301307
$335$	0	0
$336$	−23.4866	−1.28130
$337$	2.35237	0.128142	0.0640710	−	0.997945i	$-0.479592\pi$
$337$	2.35237	0.128142	0.0640710	+	0.997945i	$0.479592\pi$
$338$	−34.2783	−1.86449
$339$	−11.7673	−0.639112
$340$	0	0
$341$	4.25514	0.230429
$342$	−47.1151	−2.54769
$343$	1.00000	0.0539949
$344$	−17.4695	−0.941894
$345$	0	0
$346$	38.3082	2.05946
$347$	5.45653	0.292922	0.146461	−	0.989216i	$-0.453212\pi$
$347$	5.45653	0.292922	0.146461	+	0.989216i	$0.453212\pi$
$348$	85.4145	4.57870
$349$	20.6567	1.10573	0.552865	−	0.833271i	$-0.313535\pi$
$349$	20.6567	1.10573	0.552865	+	0.833271i	$0.313535\pi$
$350$	0	0
$351$	−1.37549	−0.0734183
$352$	−10.1398	−0.540455
$353$	6.61704	0.352190	0.176095	−	0.984373i	$-0.443653\pi$
$353$	6.61704	0.352190	0.176095	+	0.984373i	$0.443653\pi$
$354$	−55.7949	−2.96547
$355$	0	0
$356$	−56.9012	−3.01576
$357$	0.424198	0.0224509
$358$	14.5201	0.767409
$359$	28.1367	1.48500	0.742500	−	0.669846i	$-0.233640\pi$
$359$	28.1367	1.48500	0.742500	+	0.669846i	$0.233640\pi$
$360$	0	0
$361$	14.7354	0.775549
$362$	28.8492	1.51628
$363$	−2.47161	−0.129726
$364$	24.5818	1.28844
$365$	0	0
$366$	7.78873	0.407123
$367$	1.99523	0.104150	0.0520751	−	0.998643i	$-0.483416\pi$
$367$	1.99523	0.104150	0.0520751	+	0.998643i	$0.483416\pi$
$368$	39.8529	2.07748
$369$	35.8093	1.86416
$370$	0	0
$371$	2.49456	0.129511
$372$	−50.5683	−2.62184
$373$	16.6806	0.863690	0.431845	−	0.901948i	$-0.357863\pi$
$373$	16.6806	0.863690	0.431845	+	0.901948i	$0.357863\pi$
$374$	0.447822	0.0231563
$375$	0	0
$376$	−21.8880	−1.12879
$377$	−36.7449	−1.89246
$378$	0.702013	0.0361077
$379$	37.3758	1.91986	0.959932	−	0.280234i	$-0.0904120\pi$
$379$	37.3758	1.91986	0.959932	+	0.280234i	$0.0904120\pi$
$380$	0	0
$381$	−47.1534	−2.41574
$382$	64.2679	3.28823
$383$	−13.7153	−0.700818	−0.350409	−	0.936597i	$-0.613957\pi$
$383$	−13.7153	−0.700818	−0.350409	+	0.936597i	$0.613957\pi$
$384$	−2.06272	−0.105263
$385$	0	0
$386$	−44.9862	−2.28974
$387$	7.41197	0.376772
$388$	−37.8183	−1.91993
$389$	9.21380	0.467158	0.233579	−	0.972338i	$-0.424956\pi$
$389$	9.21380	0.467158	0.233579	+	0.972338i	$0.424956\pi$
$390$	0	0
$391$	−0.719795	−0.0364016
$392$	−7.32737	−0.370088
$393$	−44.3301	−2.23616
$394$	1.66823	0.0840441
$395$	0	0
$396$	14.9481	0.751168
$397$	−34.4764	−1.73032	−0.865160	−	0.501497i	$-0.832783\pi$
$397$	−34.4764	−1.73032	−0.865160	+	0.501497i	$0.832783\pi$
$398$	8.56313	0.429231
$399$	−14.3557	−0.718682
$400$	0	0
$401$	−25.6483	−1.28082	−0.640408	−	0.768035i	$-0.721235\pi$
$401$	−25.6483	−1.28082	−0.640408	+	0.768035i	$0.721235\pi$
$402$	−61.1877	−3.05176
$403$	21.7542	1.08365
$404$	−18.4492	−0.917882
$405$	0	0
$406$	18.7536	0.930725
$407$	8.41748	0.417239
$408$	−3.10825	−0.153881
$409$	−29.7880	−1.47292	−0.736461	−	0.676480i	$-0.763504\pi$
$409$	−29.7880	−1.47292	−0.736461	+	0.676480i	$0.763504\pi$
$410$	0	0
$411$	44.9410	2.21677
$412$	38.0997	1.87704
$413$	−8.65163	−0.425719
$414$	−34.0203	−1.67200
$415$	0	0
$416$	−51.8395	−2.54164
$417$	41.2120	2.01816
$418$	−15.1551	−0.741262
$419$	−0.115754	−0.00565497	−0.00282748	−	0.999996i	$-0.500900\pi$
$419$	−0.115754	−0.00565497	−0.00282748	+	0.999996i	$0.500900\pi$
$420$	0	0
$421$	2.75864	0.134448	0.0672241	−	0.997738i	$-0.478586\pi$
$421$	2.75864	0.134448	0.0672241	+	0.997738i	$0.478586\pi$
$422$	−59.7241	−2.90732
$423$	9.28664	0.451532
$424$	−18.2785	−0.887684
$425$	0	0
$426$	46.1502	2.23598
$427$	1.20773	0.0584462
$428$	39.7911	1.92337
$429$	−12.6360	−0.610071
$430$	0	0
$431$	33.8904	1.63244	0.816222	−	0.577739i	$-0.196065\pi$
$431$	33.8904	1.63244	0.816222	+	0.577739i	$0.196065\pi$
$432$	−2.55663	−0.123006
$433$	−31.0106	−1.49028	−0.745138	−	0.666911i	$-0.767616\pi$
$433$	−31.0106	−1.49028	−0.745138	+	0.666911i	$0.767616\pi$
$434$	−11.1028	−0.532949
$435$	0	0
$436$	59.7610	2.86203
$437$	24.3592	1.16526
$438$	−105.678	−5.04950
$439$	−20.1736	−0.962831	−0.481416	−	0.876492i	$-0.659877\pi$
$439$	−20.1736	−0.962831	−0.481416	+	0.876492i	$0.659877\pi$
$440$	0	0
$441$	3.10886	0.148041
$442$	2.28947	0.108899
$443$	19.8930	0.945144	0.472572	−	0.881292i	$-0.343326\pi$
$443$	19.8930	0.945144	0.472572	+	0.881292i	$0.343326\pi$
$444$	−100.034	−4.74739
$445$	0	0
$446$	−0.122697	−0.00580985
$447$	−50.2181	−2.37523
$448$	7.45236	0.352091
$449$	−4.07472	−0.192298	−0.0961491	−	0.995367i	$-0.530653\pi$
$449$	−4.07472	−0.192298	−0.0961491	+	0.995367i	$0.530653\pi$
$450$	0	0
$451$	11.5185	0.542385
$452$	22.8919	1.07674
$453$	−54.6755	−2.56888
$454$	14.7076	0.690262
$455$	0	0
$456$	105.189	4.92593
$457$	−8.50031	−0.397628	−0.198814	−	0.980037i	$-0.563709\pi$
$457$	−8.50031	−0.397628	−0.198814	+	0.980037i	$0.563709\pi$
$458$	6.38262	0.298240
$459$	0.0461761	0.00215531
$460$	0	0
$461$	−16.7248	−0.778951	−0.389475	−	0.921037i	$-0.627344\pi$
$461$	−16.7248	−0.778951	−0.389475	+	0.921037i	$0.627344\pi$
$462$	6.44906	0.300038
$463$	20.1656	0.937177	0.468589	−	0.883417i	$-0.344763\pi$
$463$	20.1656	0.937177	0.468589	+	0.883417i	$0.344763\pi$
$464$	−68.2979	−3.17065
$465$	0	0
$466$	−45.7091	−2.11743
$467$	10.7077	0.495496	0.247748	−	0.968825i	$-0.420310\pi$
$467$	10.7077	0.495496	0.247748	+	0.968825i	$0.420310\pi$
$468$	76.4213	3.53257
$469$	−9.48784	−0.438108
$470$	0	0
$471$	22.0723	1.01704
$472$	63.3937	2.91793
$473$	2.38415	0.109623
$474$	−77.1153	−3.54203
$475$	0	0
$476$	−0.825226	−0.0378242
$477$	7.75521	0.355087
$478$	−20.1342	−0.920915
$479$	7.74225	0.353753	0.176876	−	0.984233i	$-0.443401\pi$
$479$	7.74225	0.353753	0.176876	+	0.984233i	$0.443401\pi$
$480$	0	0
$481$	43.0340	1.96218
$482$	−2.48937	−0.113388
$483$	−10.3657	−0.471657
$484$	4.80822	0.218555
$485$	0	0
$486$	−57.9652	−2.62935
$487$	3.51272	0.159176	0.0795882	−	0.996828i	$-0.474639\pi$
$487$	3.51272	0.159176	0.0795882	+	0.996828i	$0.474639\pi$
$488$	−8.84948	−0.400597
$489$	−26.6826	−1.20663
$490$	0	0
$491$	−21.4062	−0.966049	−0.483025	−	0.875607i	$-0.660462\pi$
$491$	−21.4062	−0.966049	−0.483025	+	0.875607i	$0.660462\pi$
$492$	−136.886	−6.17131
$493$	1.23355	0.0555562
$494$	−77.4799	−3.48599
$495$	0	0
$496$	40.4347	1.81557
$497$	7.15610	0.320995
$498$	−41.5501	−1.86191
$499$	−24.6770	−1.10469	−0.552346	−	0.833615i	$-0.686267\pi$
$499$	−24.6770	−1.10469	−0.552346	+	0.833615i	$0.686267\pi$
$500$	0	0
$501$	−5.21610	−0.233038
$502$	−59.2710	−2.64539
$503$	22.9562	1.02356	0.511782	−	0.859115i	$-0.328985\pi$
$503$	22.9562	1.02356	0.511782	+	0.859115i	$0.328985\pi$
$504$	−22.7797	−1.01469
$505$	0	0
$506$	−10.9430	−0.486477
$507$	−32.4700	−1.44204
$508$	91.7313	4.06992
$509$	20.1345	0.892448	0.446224	−	0.894921i	$-0.352768\pi$
$509$	20.1345	0.892448	0.446224	+	0.894921i	$0.352768\pi$
$510$	0	0
$511$	−16.3866	−0.724900
$512$	42.9030	1.89606
$513$	−1.56269	−0.0689943
$514$	14.2589	0.628932
$515$	0	0
$516$	−28.3333	−1.24731
$517$	2.98716	0.131375
$518$	−21.9634	−0.965015
$519$	36.2873	1.59284
$520$	0	0
$521$	21.5461	0.943953	0.471977	−	0.881611i	$-0.343541\pi$
$521$	21.5461	0.943953	0.471977	+	0.881611i	$0.343541\pi$
$522$	58.3022	2.55182
$523$	−31.9757	−1.39820	−0.699100	−	0.715024i	$-0.746416\pi$
$523$	−31.9757	−1.39820	−0.699100	+	0.715024i	$0.746416\pi$
$524$	86.2388	3.76736
$525$	0	0
$526$	45.7369	1.99422
$527$	−0.730302	−0.0318124
$528$	−23.4866	−1.02212
$529$	−5.41102	−0.235262
$530$	0	0
$531$	−26.8967	−1.16722
$532$	27.9272	1.21080
$533$	58.8878	2.55071
$534$	−76.3192	−3.30266
$535$	0	0
$536$	69.5209	3.00284
$537$	13.7541	0.593533
$538$	−14.8337	−0.639528
$539$	1.00000	0.0430730
$540$	0	0
$541$	24.4339	1.05050	0.525248	−	0.850949i	$-0.323973\pi$
$541$	24.4339	1.05050	0.525248	+	0.850949i	$0.323973\pi$
$542$	−46.5875	−2.00111
$543$	27.3273	1.17273
$544$	1.74028	0.0746139
$545$	0	0
$546$	32.9705	1.41101
$547$	−16.2879	−0.696420	−0.348210	−	0.937417i	$-0.613210\pi$
$547$	−16.2879	−0.696420	−0.348210	+	0.937417i	$0.613210\pi$
$548$	−87.4273	−3.73471
$549$	3.75466	0.160245
$550$	0	0
$551$	−41.7456	−1.77842
$552$	75.9536	3.23280
$553$	−11.9576	−0.508489
$554$	46.0253	1.95543
$555$	0	0
$556$	−80.1730	−3.40009
$557$	−28.5051	−1.20780	−0.603900	−	0.797060i	$-0.706387\pi$
$557$	−28.5051	−1.20780	−0.603900	+	0.797060i	$0.706387\pi$
$558$	−34.5169	−1.46122
$559$	12.1889	0.515533
$560$	0	0
$561$	0.424198	0.0179096
$562$	44.2416	1.86622
$563$	33.7433	1.42211	0.711056	−	0.703136i	$-0.248217\pi$
$563$	33.7433	1.42211	0.711056	+	0.703136i	$0.248217\pi$
$564$	−35.4995	−1.49480
$565$	0	0
$566$	44.8842	1.88662
$567$	−8.66159	−0.363752
$568$	−52.4354	−2.20014
$569$	25.3772	1.06387	0.531934	−	0.846786i	$-0.321465\pi$
$569$	25.3772	1.06387	0.531934	+	0.846786i	$0.321465\pi$
$570$	0	0
$571$	−25.2662	−1.05736	−0.528678	−	0.848822i	$-0.677312\pi$
$571$	−25.2662	−1.05736	−0.528678	+	0.848822i	$0.677312\pi$
$572$	24.5818	1.02782
$573$	60.8776	2.54320
$574$	−30.0547	−1.25446
$575$	0	0
$576$	23.1683	0.965346
$577$	−14.8071	−0.616426	−0.308213	−	0.951317i	$-0.599731\pi$
$577$	−14.8071	−0.616426	−0.308213	+	0.951317i	$0.599731\pi$
$578$	44.2805	1.84183
$579$	−42.6130	−1.77094
$580$	0	0
$581$	−6.44282	−0.267293
$582$	−50.7241	−2.10258
$583$	2.49456	0.103314
$584$	120.071	4.96856
$585$	0	0
$586$	21.7840	0.899887
$587$	−7.47206	−0.308405	−0.154203	−	0.988039i	$-0.549281\pi$
$587$	−7.47206	−0.308405	−0.154203	+	0.988039i	$0.549281\pi$
$588$	−11.8840	−0.490090
$589$	24.7148	1.01836
$590$	0	0
$591$	1.58022	0.0650017
$592$	79.9875	3.28746
$593$	14.1328	0.580364	0.290182	−	0.956971i	$-0.406284\pi$
$593$	14.1328	0.580364	0.290182	+	0.956971i	$0.406284\pi$
$594$	0.702013	0.0288040
$595$	0	0
$596$	97.6932	4.00167
$597$	8.11139	0.331977
$598$	−55.9457	−2.28779
$599$	−2.69669	−0.110184	−0.0550919	−	0.998481i	$-0.517545\pi$
$599$	−2.69669	−0.110184	−0.0550919	+	0.998481i	$0.517545\pi$
$600$	0	0
$601$	−7.55560	−0.308199	−0.154100	−	0.988055i	$-0.549248\pi$
$601$	−7.55560	−0.308199	−0.154100	+	0.988055i	$0.549248\pi$
$602$	−6.22086	−0.253543
$603$	−29.4963	−1.20118
$604$	106.365	4.32792
$605$	0	0
$606$	−24.7451	−1.00520
$607$	6.85769	0.278345	0.139173	−	0.990268i	$-0.455556\pi$
$607$	6.85769	0.278345	0.139173	+	0.990268i	$0.455556\pi$
$608$	−58.8944	−2.38848
$609$	17.7643	0.719845
$610$	0	0
$611$	15.2717	0.617827
$612$	−2.56551	−0.103704
$613$	18.4152	0.743783	0.371891	−	0.928276i	$-0.378709\pi$
$613$	18.4152	0.743783	0.371891	+	0.928276i	$0.378709\pi$
$614$	−77.9223	−3.14469
$615$	0	0
$616$	−7.32737	−0.295228
$617$	30.7870	1.23944	0.619720	−	0.784823i	$-0.287246\pi$
$617$	30.7870	1.23944	0.619720	+	0.784823i	$0.287246\pi$
$618$	51.1016	2.05561
$619$	−7.01207	−0.281839	−0.140919	−	0.990021i	$-0.545006\pi$
$619$	−7.01207	−0.281839	−0.140919	+	0.990021i	$0.545006\pi$
$620$	0	0
$621$	−1.12836	−0.0452797
$622$	19.3593	0.776236
$623$	−11.8342	−0.474125
$624$	−120.074	−4.80681
$625$	0	0
$626$	18.9979	0.759310
$627$	−14.3557	−0.573310
$628$	−42.9389	−1.71345
$629$	−1.44468	−0.0576030
$630$	0	0
$631$	−12.6870	−0.505061	−0.252530	−	0.967589i	$-0.581263\pi$
$631$	−12.6870	−0.505061	−0.252530	+	0.967589i	$0.581263\pi$
$632$	87.6178	3.48525
$633$	−56.5734	−2.24859
$634$	−36.1580	−1.43602
$635$	0	0
$636$	−29.6454	−1.17552
$637$	5.11245	0.202563
$638$	18.7536	0.742461
$639$	22.2473	0.880089
$640$	0	0
$641$	−47.6105	−1.88050	−0.940251	−	0.340483i	$-0.889410\pi$
$641$	−47.6105	−1.88050	−0.940251	+	0.340483i	$0.889410\pi$
$642$	53.3701	2.10635
$643$	34.7051	1.36864	0.684318	−	0.729184i	$-0.260100\pi$
$643$	34.7051	1.36864	0.684318	+	0.729184i	$0.260100\pi$
$644$	20.1653	0.794624
$645$	0	0
$646$	2.60105	0.102337
$647$	22.5710	0.887356	0.443678	−	0.896186i	$-0.353673\pi$
$647$	22.5710	0.887356	0.443678	+	0.896186i	$0.353673\pi$
$648$	63.4666	2.49320
$649$	−8.65163	−0.339606
$650$	0	0
$651$	−10.5171	−0.412196
$652$	51.9079	2.03287
$653$	−2.65223	−0.103790	−0.0518949	−	0.998653i	$-0.516526\pi$
$653$	−2.65223	−0.103790	−0.0518949	+	0.998653i	$0.516526\pi$
$654$	80.1549	3.13430
$655$	0	0
$656$	109.455	4.27350
$657$	−50.9435	−1.98750
$658$	−7.79426	−0.303852
$659$	−1.71822	−0.0669325	−0.0334663	−	0.999440i	$-0.510655\pi$
$659$	−1.71822	−0.0669325	−0.0334663	+	0.999440i	$0.510655\pi$
$660$	0	0
$661$	24.2995	0.945139	0.472570	−	0.881293i	$-0.343327\pi$
$661$	24.2995	0.945139	0.472570	+	0.881293i	$0.343327\pi$
$662$	−30.2157	−1.17437
$663$	2.16869	0.0842250
$664$	47.2089	1.83206
$665$	0	0
$666$	−68.2809	−2.64583
$667$	−30.1431	−1.16715
$668$	10.1473	0.392611
$669$	−0.116224	−0.00449348
$670$	0	0
$671$	1.20773	0.0466239
$672$	25.0617	0.966777
$673$	−33.8264	−1.30391	−0.651956	−	0.758257i	$-0.726051\pi$
$673$	−33.8264	−1.30391	−0.651956	+	0.758257i	$0.726051\pi$
$674$	−6.13794	−0.236425
$675$	0	0
$676$	63.1665	2.42948
$677$	22.7861	0.875742	0.437871	−	0.899038i	$-0.355733\pi$
$677$	22.7861	0.875742	0.437871	+	0.899038i	$0.355733\pi$
$678$	30.7039	1.17918
$679$	−7.86534	−0.301844
$680$	0	0
$681$	13.9317	0.533865
$682$	−11.1028	−0.425147
$683$	25.0646	0.959069	0.479534	−	0.877523i	$-0.340806\pi$
$683$	25.0646	0.959069	0.479534	+	0.877523i	$0.340806\pi$
$684$	86.8216	3.31971
$685$	0	0
$686$	−2.60926	−0.0996219
$687$	6.04591	0.230666
$688$	22.6555	0.863732
$689$	12.7533	0.485862
$690$	0	0
$691$	−45.9733	−1.74891	−0.874453	−	0.485111i	$-0.838779\pi$
$691$	−45.9733	−1.74891	−0.874453	+	0.485111i	$0.838779\pi$
$692$	−70.5927	−2.68353
$693$	3.10886	0.118096
$694$	−14.2375	−0.540447
$695$	0	0
$696$	−130.165	−4.93390
$697$	−1.97690	−0.0748803
$698$	−53.8987	−2.04010
$699$	−43.2978	−1.63767
$700$	0	0
$701$	−22.9309	−0.866089	−0.433045	−	0.901372i	$-0.642561\pi$
$701$	−22.9309	−0.866089	−0.433045	+	0.901372i	$0.642561\pi$
$702$	3.58901	0.135458
$703$	48.8906	1.84394
$704$	7.45236	0.280871
$705$	0	0
$706$	−17.2656	−0.649798
$707$	−3.83701	−0.144306
$708$	102.816	3.86407
$709$	7.89657	0.296562	0.148281	−	0.988945i	$-0.452626\pi$
$709$	7.89657	0.296562	0.148281	+	0.988945i	$0.452626\pi$
$710$	0	0
$711$	−37.1744	−1.39415
$712$	86.7132	3.24971
$713$	17.8457	0.668328
$714$	−1.10684	−0.0414225
$715$	0	0
$716$	−26.7569	−0.999953
$717$	−19.0720	−0.712257
$718$	−73.4160	−2.73986
$719$	−29.1677	−1.08777	−0.543885	−	0.839160i	$-0.683047\pi$
$719$	−29.1677	−1.08777	−0.543885	+	0.839160i	$0.683047\pi$
$720$	0	0
$721$	7.92387	0.295100
$722$	−38.4485	−1.43091
$723$	−2.35805	−0.0876969
$724$	−53.1620	−1.97575
$725$	0	0
$726$	6.44906	0.239347
$727$	−31.5979	−1.17190	−0.585950	−	0.810347i	$-0.699279\pi$
$727$	−31.5979	−1.17190	−0.585950	+	0.810347i	$0.699279\pi$
$728$	−37.4608	−1.38839
$729$	−28.9226	−1.07121
$730$	0	0
$731$	−0.409187	−0.0151343
$732$	−14.3527	−0.530492
$733$	−25.6983	−0.949188	−0.474594	−	0.880205i	$-0.657405\pi$
$733$	−25.6983	−0.949188	−0.474594	+	0.880205i	$0.657405\pi$
$734$	−5.20607	−0.192160
$735$	0	0
$736$	−42.5257	−1.56752
$737$	−9.48784	−0.349489
$738$	−93.4357	−3.43942
$739$	−27.3100	−1.00461	−0.502307	−	0.864689i	$-0.667515\pi$
$739$	−27.3100	−1.00461	−0.502307	+	0.864689i	$0.667515\pi$
$740$	0	0
$741$	−73.3926	−2.69615
$742$	−6.50894	−0.238951
$743$	−29.9040	−1.09707	−0.548536	−	0.836127i	$-0.684815\pi$
$743$	−29.9040	−1.09707	−0.548536	+	0.836127i	$0.684815\pi$
$744$	77.0623	2.82524
$745$	0	0
$746$	−43.5240	−1.59353
$747$	−20.0298	−0.732852
$748$	−0.825226	−0.0301732
$749$	8.27563	0.302385
$750$	0	0
$751$	28.6885	1.04686	0.523429	−	0.852070i	$-0.324653\pi$
$751$	28.6885	1.04686	0.523429	+	0.852070i	$0.324653\pi$
$752$	28.3856	1.03512
$753$	−56.1442	−2.04601
$754$	95.8768	3.49163
$755$	0	0
$756$	−1.29364	−0.0470492
$757$	−6.87567	−0.249900	−0.124950	−	0.992163i	$-0.539877\pi$
$757$	−6.87567	−0.249900	−0.124950	+	0.992163i	$0.539877\pi$
$758$	−97.5229	−3.54219
$759$	−10.3657	−0.376252
$760$	0	0
$761$	−5.73360	−0.207843	−0.103922	−	0.994586i	$-0.533139\pi$
$761$	−5.73360	−0.207843	−0.103922	+	0.994586i	$0.533139\pi$
$762$	123.035	4.45710
$763$	12.4289	0.449957
$764$	−118.430	−4.28465
$765$	0	0
$766$	35.7867	1.29303
$767$	−44.2310	−1.59709
$768$	42.2208	1.52351
$769$	17.6190	0.635356	0.317678	−	0.948199i	$-0.397097\pi$
$769$	17.6190	0.635356	0.317678	+	0.948199i	$0.397097\pi$
$770$	0	0
$771$	13.5067	0.486431
$772$	82.8985	2.98358
$773$	−18.1980	−0.654537	−0.327268	−	0.944931i	$-0.606128\pi$
$773$	−18.1980	−0.654537	−0.327268	+	0.944931i	$0.606128\pi$
$774$	−19.3397	−0.695153
$775$	0	0
$776$	57.6322	2.06888
$777$	−20.8047	−0.746365
$778$	−24.0412	−0.861917
$779$	66.9020	2.39701
$780$	0	0
$781$	7.15610	0.256065
$782$	1.87813	0.0671618
$783$	1.93373	0.0691059
$784$	9.50254	0.339377
$785$	0	0
$786$	115.669	4.12576
$787$	24.1959	0.862492	0.431246	−	0.902234i	$-0.358074\pi$
$787$	24.1959	0.862492	0.431246	+	0.902234i	$0.358074\pi$
$788$	−3.07413	−0.109512
$789$	43.3241	1.54238
$790$	0	0
$791$	4.76099	0.169281
$792$	−22.7797	−0.809442
$793$	6.17447	0.219262
$794$	89.9577	3.19248
$795$	0	0
$796$	−15.7797	−0.559298
$797$	37.7734	1.33800	0.669002	−	0.743261i	$-0.266722\pi$
$797$	37.7734	1.33800	0.669002	+	0.743261i	$0.266722\pi$
$798$	37.4576	1.32598
$799$	−0.512680	−0.0181373
$800$	0	0
$801$	−36.7907	−1.29993
$802$	66.9231	2.36314
$803$	−16.3866	−0.578270
$804$	112.754	3.97652
$805$	0	0
$806$	−56.7623	−1.99937
$807$	−14.0512	−0.494626
$808$	28.1152	0.989089
$809$	51.5617	1.81281	0.906407	−	0.422406i	$-0.138814\pi$
$809$	51.5617	1.81281	0.906407	+	0.422406i	$0.138814\pi$
$810$	0	0
$811$	25.2624	0.887083	0.443542	−	0.896254i	$-0.353722\pi$
$811$	25.2624	0.887083	0.443542	+	0.896254i	$0.353722\pi$
$812$	−34.5583	−1.21276
$813$	−44.1299	−1.54770
$814$	−21.9634	−0.769815
$815$	0	0
$816$	4.03096	0.141112
$817$	13.8477	0.484468
$818$	77.7245	2.71757
$819$	15.8939	0.555377
$820$	0	0
$821$	−14.8433	−0.518036	−0.259018	−	0.965872i	$-0.583399\pi$
$821$	−14.8433	−0.518036	−0.259018	+	0.965872i	$0.583399\pi$
$822$	−117.263	−4.09000
$823$	2.20027	0.0766965	0.0383482	−	0.999264i	$-0.487790\pi$
$823$	2.20027	0.0766965	0.0383482	+	0.999264i	$0.487790\pi$
$824$	−58.0611	−2.02266
$825$	0	0
$826$	22.5743	0.785461
$827$	−16.0867	−0.559391	−0.279696	−	0.960089i	$-0.590234\pi$
$827$	−16.0867	−0.559391	−0.279696	+	0.960089i	$0.590234\pi$
$828$	62.6910	2.17866
$829$	7.64549	0.265539	0.132769	−	0.991147i	$-0.457613\pi$
$829$	7.64549	0.265539	0.132769	+	0.991147i	$0.457613\pi$
$830$	0	0
$831$	43.5973	1.51237
$832$	38.0998	1.32087
$833$	−0.171628	−0.00594656
$834$	−107.533	−3.72355
$835$	0	0
$836$	27.9272	0.965883
$837$	−1.14483	−0.0395713
$838$	0.302033	0.0104335
$839$	−31.7523	−1.09621	−0.548105	−	0.836409i	$-0.684651\pi$
$839$	−31.7523	−1.09621	−0.548105	+	0.836409i	$0.684651\pi$
$840$	0	0
$841$	22.6577	0.781299
$842$	−7.19801	−0.248060
$843$	41.9077	1.44338
$844$	110.057	3.78831
$845$	0	0
$846$	−24.2312	−0.833087
$847$	1.00000	0.0343604
$848$	23.7046	0.814020
$849$	42.5164	1.45916
$850$	0	0
$851$	35.3022	1.21015
$852$	−85.0434	−2.91354
$853$	18.8338	0.644858	0.322429	−	0.946594i	$-0.395501\pi$
$853$	18.8338	0.644858	0.322429	+	0.946594i	$0.395501\pi$
$854$	−3.15128	−0.107835
$855$	0	0
$856$	−60.6386	−2.07258
$857$	37.6888	1.28742	0.643712	−	0.765268i	$-0.277394\pi$
$857$	37.6888	1.28742	0.643712	+	0.765268i	$0.277394\pi$
$858$	32.9705	1.12560
$859$	27.5987	0.941656	0.470828	−	0.882225i	$-0.343955\pi$
$859$	27.5987	0.941656	0.470828	+	0.882225i	$0.343955\pi$
$860$	0	0
$861$	−28.4692	−0.970229
$862$	−88.4287	−3.01190
$863$	−40.7038	−1.38557	−0.692787	−	0.721142i	$-0.743617\pi$
$863$	−40.7038	−1.38557	−0.692787	+	0.721142i	$0.743617\pi$
$864$	2.72810	0.0928117
$865$	0	0
$866$	80.9147	2.74959
$867$	41.9446	1.42451
$868$	20.4597	0.694446
$869$	−11.9576	−0.405634
$870$	0	0
$871$	−48.5061	−1.64357
$872$	−91.0712	−3.08406
$873$	−24.4522	−0.827581
$874$	−63.5595	−2.14993
$875$	0	0
$876$	194.739	6.57962
$877$	3.37845	0.114082	0.0570411	−	0.998372i	$-0.481833\pi$
$877$	3.37845	0.114082	0.0570411	+	0.998372i	$0.481833\pi$
$878$	52.6380	1.77645
$879$	20.6348	0.695994
$880$	0	0
$881$	−30.9852	−1.04392	−0.521959	−	0.852971i	$-0.674799\pi$
$881$	−30.9852	−1.04392	−0.521959	+	0.852971i	$0.674799\pi$
$882$	−8.11180	−0.273139
$883$	−46.4259	−1.56236	−0.781179	−	0.624308i	$-0.785381\pi$
$883$	−46.4259	−1.56236	−0.781179	+	0.624308i	$0.785381\pi$
$884$	−4.21893	−0.141898
$885$	0	0
$886$	−51.9059	−1.74381
$887$	−20.4698	−0.687308	−0.343654	−	0.939096i	$-0.611665\pi$
$887$	−20.4698	−0.687308	−0.343654	+	0.939096i	$0.611665\pi$
$888$	152.444	5.11568
$889$	19.0780	0.639856
$890$	0	0
$891$	−8.66159	−0.290174
$892$	0.226100	0.00757038
$893$	17.3501	0.580598
$894$	131.032	4.38236
$895$	0	0
$896$	0.834565	0.0278809
$897$	−52.9944	−1.76943
$898$	10.6320	0.354794
$899$	−30.5831	−1.02000
$900$	0	0
$901$	−0.428136	−0.0142633
$902$	−30.0547	−1.00071
$903$	−5.89269	−0.196096
$904$	−34.8855	−1.16027
$905$	0	0
$906$	142.663	4.73965
$907$	−22.3480	−0.742054	−0.371027	−	0.928622i	$-0.620994\pi$
$907$	−22.3480	−0.742054	−0.371027	+	0.928622i	$0.620994\pi$
$908$	−27.1025	−0.899429
$909$	−11.9287	−0.395651
$910$	0	0
$911$	27.1924	0.900925	0.450463	−	0.892795i	$-0.351259\pi$
$911$	27.1924	0.900925	0.450463	+	0.892795i	$0.351259\pi$
$912$	−136.415	−4.51716
$913$	−6.44282	−0.213226
$914$	22.1795	0.733632
$915$	0	0
$916$	−11.7616	−0.388614
$917$	17.9357	0.592289
$918$	−0.120485	−0.00397660
$919$	31.4961	1.03896	0.519480	−	0.854483i	$-0.326126\pi$
$919$	31.4961	1.03896	0.519480	+	0.854483i	$0.326126\pi$
$920$	0	0
$921$	−73.8116	−2.43218
$922$	43.6392	1.43718
$923$	36.5852	1.20422
$924$	−11.8840	−0.390956
$925$	0	0
$926$	−52.6173	−1.72911
$927$	24.6342	0.809092
$928$	72.8784	2.39235
$929$	−37.6543	−1.23540	−0.617700	−	0.786414i	$-0.711935\pi$
$929$	−37.6543	−1.23540	−0.617700	+	0.786414i	$0.711935\pi$
$930$	0	0
$931$	5.80822	0.190357
$932$	84.2307	2.75907
$933$	18.3380	0.600360
$934$	−27.9393	−0.914201
$935$	0	0
$936$	−116.460	−3.80662
$937$	10.8443	0.354268	0.177134	−	0.984187i	$-0.443317\pi$
$937$	10.8443	0.354268	0.177134	+	0.984187i	$0.443317\pi$
$938$	24.7562	0.808319
$939$	17.9957	0.587268
$940$	0	0
$941$	43.1240	1.40580	0.702902	−	0.711287i	$-0.251887\pi$
$941$	43.1240	1.40580	0.702902	+	0.711287i	$0.251887\pi$
$942$	−57.5922	−1.87646
$943$	48.3077	1.57311
$944$	−82.2125	−2.67579
$945$	0	0
$946$	−6.22086	−0.202257
$947$	18.2395	0.592703	0.296351	−	0.955079i	$-0.404230\pi$
$947$	18.2395	0.592703	0.296351	+	0.955079i	$0.404230\pi$
$948$	142.105	4.61535
$949$	−83.7757	−2.71947
$950$	0	0
$951$	−34.2506	−1.11065
$952$	1.25758	0.0407585
$953$	54.9268	1.77925	0.889627	−	0.456688i	$-0.150965\pi$
$953$	54.9268	1.77925	0.889627	+	0.456688i	$0.150965\pi$
$954$	−20.2353	−0.655144
$955$	0	0
$956$	37.1023	1.19997
$957$	17.7643	0.574237
$958$	−20.2015	−0.652682
$959$	−18.1829	−0.587156
$960$	0	0
$961$	−12.8938	−0.415928
$962$	−112.287	−3.62027
$963$	25.7277	0.829065
$964$	4.58731	0.147747
$965$	0	0
$966$	27.0469	0.870219
$967$	35.0089	1.12581	0.562905	−	0.826521i	$-0.309684\pi$
$967$	35.0089	1.12581	0.562905	+	0.826521i	$0.309684\pi$
$968$	−7.32737	−0.235511
$969$	2.46383	0.0791497
$970$	0	0
$971$	53.2034	1.70738	0.853690	−	0.520782i	$-0.174359\pi$
$971$	53.2034	1.70738	0.853690	+	0.520782i	$0.174359\pi$
$972$	106.816	3.42611
$973$	−16.6742	−0.534549
$974$	−9.16558	−0.293684
$975$	0	0
$976$	11.4765	0.367354
$977$	−7.44446	−0.238170	−0.119085	−	0.992884i	$-0.537996\pi$
$977$	−7.44446	−0.238170	−0.119085	+	0.992884i	$0.537996\pi$
$978$	69.6219	2.22626
$979$	−11.8342	−0.378221
$980$	0	0
$981$	38.6397	1.23367
$982$	55.8543	1.78238
$983$	22.0152	0.702177	0.351088	−	0.936342i	$-0.385812\pi$
$983$	22.0152	0.702177	0.351088	+	0.936342i	$0.385812\pi$
$984$	208.605	6.65007
$985$	0	0
$986$	−3.21864	−0.102502
$987$	−7.38309	−0.235006
$988$	142.777	4.54233
$989$	9.99893	0.317948
$990$	0	0
$991$	−0.0768274	−0.00244050	−0.00122025	−	0.999999i	$-0.500388\pi$
$991$	−0.0768274	−0.00244050	−0.00122025	+	0.999999i	$0.500388\pi$
$992$	−43.1465	−1.36990
$993$	−28.6217	−0.908283
$994$	−18.6721	−0.592243
$995$	0	0
$996$	76.5667	2.42611
$997$	11.5613	0.366149	0.183074	−	0.983099i	$-0.441395\pi$
$997$	11.5613	0.366149	0.183074	+	0.983099i	$0.441395\pi$
$998$	64.3885	2.03818
$999$	−2.26470	−0.0716519

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1925.2.a.bd.1.1	yes	7
5.2	odd	4		1925.2.b.r.1849.2		14
5.3	odd	4		1925.2.b.r.1849.13		14
5.4	even	2		1925.2.a.bb.1.7	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1925.2.a.bb.1.7	✓	7	5.4	even	2
1925.2.a.bd.1.1	yes	7	1.1	even	1	trivial
1925.2.b.r.1849.2		14	5.2	odd	4
1925.2.b.r.1849.13		14	5.3	odd	4

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:	\( N \)	\(=\)	\( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1925.a (trivial)

\(f(q)\)	\(=\)	\(q-2.60926 q^{2} -2.47161 q^{3} +4.80822 q^{4} +6.44906 q^{6} +1.00000 q^{7} -7.32737 q^{8} +3.10886 q^{9} +O(q^{10})\)	\(q-2.60926 q^{2} -2.47161 q^{3} +4.80822 q^{4} +6.44906 q^{6} +1.00000 q^{7} -7.32737 q^{8} +3.10886 q^{9} +1.00000 q^{11} -11.8840 q^{12} +5.11245 q^{13} -2.60926 q^{14} +9.50254 q^{16} -0.171628 q^{17} -8.11180 q^{18} +5.80822 q^{19} -2.47161 q^{21} -2.60926 q^{22} +4.19392 q^{23} +18.1104 q^{24} -13.3397 q^{26} -0.269047 q^{27} +4.80822 q^{28} -7.18733 q^{29} +4.25514 q^{31} -10.1398 q^{32} -2.47161 q^{33} +0.447822 q^{34} +14.9481 q^{36} +8.41748 q^{37} -15.1551 q^{38} -12.6360 q^{39} +11.5185 q^{41} +6.44906 q^{42} +2.38415 q^{43} +4.80822 q^{44} -10.9430 q^{46} +2.98716 q^{47} -23.4866 q^{48} +1.00000 q^{49} +0.424198 q^{51} +24.5818 q^{52} +2.49456 q^{53} +0.702013 q^{54} -7.32737 q^{56} -14.3557 q^{57} +18.7536 q^{58} -8.65163 q^{59} +1.20773 q^{61} -11.1028 q^{62} +3.10886 q^{63} +7.45236 q^{64} +6.44906 q^{66} -9.48784 q^{67} -0.825226 q^{68} -10.3657 q^{69} +7.15610 q^{71} -22.7797 q^{72} -16.3866 q^{73} -21.9634 q^{74} +27.9272 q^{76} +1.00000 q^{77} +32.9705 q^{78} -11.9576 q^{79} -8.66159 q^{81} -30.0547 q^{82} -6.44282 q^{83} -11.8840 q^{84} -6.22086 q^{86} +17.7643 q^{87} -7.32737 q^{88} -11.8342 q^{89} +5.11245 q^{91} +20.1653 q^{92} -10.5171 q^{93} -7.79426 q^{94} +25.0617 q^{96} -7.86534 q^{97} -2.60926 q^{98} +3.10886 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 13 q^{4} + 3 q^{6} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) 7 * q + q^2 + 13 * q^4 + 3 * q^6 + 7 * q^7 + 9 * q^9	\( 7 q + q^{2} + 13 q^{4} + 3 q^{6} + 7 q^{7} + 9 q^{9} + 7 q^{11} - 9 q^{12} - 3 q^{13} + q^{14} + 21 q^{16} - 2 q^{17} + 8 q^{18} + 20 q^{19} + q^{22} + 11 q^{23} + 18 q^{24} - 13 q^{26} - 12 q^{27} + 13 q^{28} + 4 q^{29} + 6 q^{31} + q^{32} - 7 q^{34} + 12 q^{36} + 19 q^{37} + 3 q^{38} - 10 q^{39} + 24 q^{41} + 3 q^{42} + 13 q^{44} + 33 q^{46} - q^{47} - 15 q^{48} + 7 q^{49} + 19 q^{51} - 29 q^{52} + 7 q^{53} + 9 q^{54} - 9 q^{57} + 37 q^{58} + 9 q^{59} + 18 q^{61} - 40 q^{62} + 9 q^{63} + 8 q^{64} + 3 q^{66} + 18 q^{68} - 15 q^{69} + 18 q^{71} - 64 q^{72} + 5 q^{73} - 24 q^{74} + 88 q^{76} + 7 q^{77} + 79 q^{78} + 25 q^{79} - q^{81} - 60 q^{82} - 17 q^{83} - 9 q^{84} - 41 q^{86} - 24 q^{87} - 16 q^{89} - 3 q^{91} + 28 q^{92} + 26 q^{93} - 31 q^{94} + 17 q^{96} - 4 q^{97} + q^{98} + 9 q^{99}+O(q^{100}) \) 7 * q + q^2 + 13 * q^4 + 3 * q^6 + 7 * q^7 + 9 * q^9 + 7 * q^11 - 9 * q^12 - 3 * q^13 + q^14 + 21 * q^16 - 2 * q^17 + 8 * q^18 + 20 * q^19 + q^22 + 11 * q^23 + 18 * q^24 - 13 * q^26 - 12 * q^27 + 13 * q^28 + 4 * q^29 + 6 * q^31 + q^32 - 7 * q^34 + 12 * q^36 + 19 * q^37 + 3 * q^38 - 10 * q^39 + 24 * q^41 + 3 * q^42 + 13 * q^44 + 33 * q^46 - q^47 - 15 * q^48 + 7 * q^49 + 19 * q^51 - 29 * q^52 + 7 * q^53 + 9 * q^54 - 9 * q^57 + 37 * q^58 + 9 * q^59 + 18 * q^61 - 40 * q^62 + 9 * q^63 + 8 * q^64 + 3 * q^66 + 18 * q^68 - 15 * q^69 + 18 * q^71 - 64 * q^72 + 5 * q^73 - 24 * q^74 + 88 * q^76 + 7 * q^77 + 79 * q^78 + 25 * q^79 - q^81 - 60 * q^82 - 17 * q^83 - 9 * q^84 - 41 * q^86 - 24 * q^87 - 16 * q^89 - 3 * q^91 + 28 * q^92 + 26 * q^93 - 31 * q^94 + 17 * q^96 - 4 * q^97 + q^98 + 9 * q^99

Introduction

L-functions

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