LMFDB - Embedded newform 4425.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4425,2,Mod(1,4425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4425, 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("4425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4425 = 3 \cdot 5^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4425.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:	$35.3338028944$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1957.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} - x + 1 $ x^4 - 4*x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.396339$ of defining polynomial
Character	$\chi$	$=$	4425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.12676 q^{2} -1.00000 q^{3} +2.52310 q^{4} +2.12676 q^{6} +1.07652 q^{7} -1.11250 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.12676 q^{2} -1.00000 q^{3} +2.52310 q^{4} +2.12676 q^{6} +1.07652 q^{7} -1.11250 q^{8} +1.00000 q^{9} -5.96967 q^{11} -2.52310 q^{12} -1.80694 q^{13} -2.28949 q^{14} -2.68018 q^{16} +6.81259 q^{17} -2.12676 q^{18} -2.36036 q^{19} -1.07652 q^{21} +12.6960 q^{22} +2.46084 q^{23} +1.11250 q^{24} +3.84292 q^{26} -1.00000 q^{27} +2.71616 q^{28} +3.47851 q^{29} -5.62133 q^{31} +7.92509 q^{32} +5.96967 q^{33} -14.4887 q^{34} +2.52310 q^{36} -4.84452 q^{37} +5.01991 q^{38} +1.80694 q^{39} +4.52310 q^{41} +2.28949 q^{42} +10.1786 q^{43} -15.0621 q^{44} -5.23360 q^{46} -8.58355 q^{47} +2.68018 q^{48} -5.84111 q^{49} -6.81259 q^{51} -4.55907 q^{52} +13.7623 q^{53} +2.12676 q^{54} -1.19762 q^{56} +2.36036 q^{57} -7.39794 q^{58} -1.00000 q^{59} +13.1888 q^{61} +11.9552 q^{62} +1.07652 q^{63} -11.4944 q^{64} -12.6960 q^{66} -6.23765 q^{67} +17.1888 q^{68} -2.46084 q^{69} +11.6331 q^{71} -1.11250 q^{72} -15.0249 q^{73} +10.3031 q^{74} -5.95541 q^{76} -6.42646 q^{77} -3.84292 q^{78} -15.9369 q^{79} +1.00000 q^{81} -9.61953 q^{82} +5.55566 q^{83} -2.71616 q^{84} -21.6474 q^{86} -3.47851 q^{87} +6.64124 q^{88} -17.8907 q^{89} -1.94520 q^{91} +6.20893 q^{92} +5.62133 q^{93} +18.2551 q^{94} -7.92509 q^{96} -3.09939 q^{97} +12.4226 q^{98} -5.96967 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - 4 q^{3} + q^{4} + q^{6} + q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - q^2 - 4 * q^3 + q^4 + q^6 + q^7 - 3 * q^8 + 4 * q^9	$ 4 q - q^{2} - 4 q^{3} + q^{4} + q^{6} + q^{7} - 3 q^{8} + 4 q^{9} - 9 q^{11} - q^{12} + 2 q^{13} + 4 q^{14} - 9 q^{16} + 5 q^{17} - q^{18} - 6 q^{19} - q^{21} + 16 q^{22} - 2 q^{23} + 3 q^{24} + 8 q^{26} - 4 q^{27} + 11 q^{28} - 4 q^{29} - 18 q^{31} + 8 q^{32} + 9 q^{33} - 20 q^{34} + q^{36} + 15 q^{37} + q^{38} - 2 q^{39} + 9 q^{41} - 4 q^{42} + 2 q^{43} - 9 q^{44} - 25 q^{46} - q^{47} + 9 q^{48} + 11 q^{49} - 5 q^{51} - 11 q^{52} + 37 q^{53} + q^{54} - 7 q^{56} + 6 q^{57} - q^{58} - 4 q^{59} - 6 q^{61} - 15 q^{62} + q^{63} + 5 q^{64} - 16 q^{66} - 43 q^{67} + 10 q^{68} + 2 q^{69} - 5 q^{71} - 3 q^{72} - 14 q^{73} + 8 q^{74} - 15 q^{76} + 15 q^{77} - 8 q^{78} + 7 q^{79} + 4 q^{81} - 7 q^{82} + 15 q^{83} - 11 q^{84} - 29 q^{86} + 4 q^{87} + 3 q^{88} - 19 q^{89} - 19 q^{91} + q^{92} + 18 q^{93} + 31 q^{94} - 8 q^{96} - 10 q^{97} + 49 q^{98} - 9 q^{99}+O(q^{100}) $ 4 * q - q^2 - 4 * q^3 + q^4 + q^6 + q^7 - 3 * q^8 + 4 * q^9 - 9 * q^11 - q^12 + 2 * q^13 + 4 * q^14 - 9 * q^16 + 5 * q^17 - q^18 - 6 * q^19 - q^21 + 16 * q^22 - 2 * q^23 + 3 * q^24 + 8 * q^26 - 4 * q^27 + 11 * q^28 - 4 * q^29 - 18 * q^31 + 8 * q^32 + 9 * q^33 - 20 * q^34 + q^36 + 15 * q^37 + q^38 - 2 * q^39 + 9 * q^41 - 4 * q^42 + 2 * q^43 - 9 * q^44 - 25 * q^46 - q^47 + 9 * q^48 + 11 * q^49 - 5 * q^51 - 11 * q^52 + 37 * q^53 + q^54 - 7 * q^56 + 6 * q^57 - q^58 - 4 * q^59 - 6 * q^61 - 15 * q^62 + q^63 + 5 * q^64 - 16 * q^66 - 43 * q^67 + 10 * q^68 + 2 * q^69 - 5 * q^71 - 3 * q^72 - 14 * q^73 + 8 * q^74 - 15 * q^76 + 15 * q^77 - 8 * q^78 + 7 * q^79 + 4 * q^81 - 7 * q^82 + 15 * q^83 - 11 * q^84 - 29 * q^86 + 4 * q^87 + 3 * q^88 - 19 * q^89 - 19 * q^91 + q^92 + 18 * q^93 + 31 * q^94 - 8 * q^96 - 10 * q^97 + 49 * 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 $

$2$	−2.12676	−1.50384	−0.751922	−	0.659252i	$-0.770873\pi$
$2$	−2.12676	−1.50384	−0.751922	+	0.659252i	$0.770873\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.52310	1.26155
$5$	0	0
$5$	0	0
$6$	2.12676	0.868245
$7$	1.07652	0.406886	0.203443	−	0.979087i	$-0.434787\pi$
$7$	1.07652	0.406886	0.203443	+	0.979087i	$0.434787\pi$
$8$	−1.11250	−0.393327
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.96967	−1.79992	−0.899962	−	0.435968i	$-0.856406\pi$
$11$	−5.96967	−1.79992	−0.899962	+	0.435968i	$0.856406\pi$
$12$	−2.52310	−0.728355
$13$	−1.80694	−0.501154	−0.250577	−	0.968097i	$-0.580620\pi$
$13$	−1.80694	−0.501154	−0.250577	+	0.968097i	$0.580620\pi$
$14$	−2.28949	−0.611893
$15$	0	0
$16$	−2.68018	−0.670045
$17$	6.81259	1.65230	0.826148	−	0.563454i	$-0.190528\pi$
$17$	6.81259	1.65230	0.826148	+	0.563454i	$0.190528\pi$
$18$	−2.12676	−0.501281
$19$	−2.36036	−0.541504	−0.270752	−	0.962649i	$-0.587272\pi$
$19$	−2.36036	−0.541504	−0.270752	+	0.962649i	$0.587272\pi$
$20$	0	0
$21$	−1.07652	−0.234916
$22$	12.6960	2.70681
$23$	2.46084	0.513120	0.256560	−	0.966528i	$-0.417411\pi$
$23$	2.46084	0.513120	0.256560	+	0.966528i	$0.417411\pi$
$24$	1.11250	0.227088
$25$	0	0
$26$	3.84292	0.753658
$27$	−1.00000	−0.192450
$28$	2.71616	0.513306
$29$	3.47851	0.645943	0.322971	−	0.946409i	$-0.395318\pi$
$29$	3.47851	0.645943	0.322971	+	0.946409i	$0.395318\pi$
$30$	0	0
$31$	−5.62133	−1.00962	−0.504811	−	0.863230i	$-0.668438\pi$
$31$	−5.62133	−1.00962	−0.504811	+	0.863230i	$0.668438\pi$
$32$	7.92509	1.40097
$33$	5.96967	1.03919
$34$	−14.4887	−2.48480
$35$	0	0
$36$	2.52310	0.420516
$37$	−4.84452	−0.796434	−0.398217	−	0.917291i	$-0.630371\pi$
$37$	−4.84452	−0.796434	−0.398217	+	0.917291i	$0.630371\pi$
$38$	5.01991	0.814337
$39$	1.80694	0.289341
$40$	0	0
$41$	4.52310	0.706389	0.353194	−	0.935550i	$-0.385095\pi$
$41$	4.52310	0.706389	0.353194	+	0.935550i	$0.385095\pi$
$42$	2.28949	0.353276
$43$	10.1786	1.55222	0.776111	−	0.630596i	$-0.217190\pi$
$43$	10.1786	1.55222	0.776111	+	0.630596i	$0.217190\pi$
$44$	−15.0621	−2.27069
$45$	0	0
$46$	−5.23360	−0.771653
$47$	−8.58355	−1.25204	−0.626019	−	0.779807i	$-0.715317\pi$
$47$	−8.58355	−1.25204	−0.626019	+	0.779807i	$0.715317\pi$
$48$	2.68018	0.386851
$49$	−5.84111	−0.834444
$50$	0	0
$51$	−6.81259	−0.953953
$52$	−4.55907	−0.632230
$53$	13.7623	1.89040	0.945202	−	0.326486i	$-0.105865\pi$
$53$	13.7623	1.89040	0.945202	+	0.326486i	$0.105865\pi$
$54$	2.12676	0.289415
$55$	0	0
$56$	−1.19762	−0.160039
$57$	2.36036	0.312637
$58$	−7.39794	−0.971397
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	13.1888	1.68865	0.844327	−	0.535828i	$-0.180000\pi$
$61$	13.1888	1.68865	0.844327	+	0.535828i	$0.180000\pi$
$62$	11.9552	1.51831
$63$	1.07652	0.135629
$64$	−11.4944	−1.43680
$65$	0	0
$66$	−12.6960	−1.56277
$67$	−6.23765	−0.762050	−0.381025	−	0.924565i	$-0.624429\pi$
$67$	−6.23765	−0.762050	−0.381025	+	0.924565i	$0.624429\pi$
$68$	17.1888	2.08445
$69$	−2.46084	−0.296250
$70$	0	0
$71$	11.6331	1.38060	0.690300	−	0.723523i	$-0.257478\pi$
$71$	11.6331	1.38060	0.690300	+	0.723523i	$0.257478\pi$
$72$	−1.11250	−0.131109
$73$	−15.0249	−1.75853	−0.879267	−	0.476329i	$-0.841967\pi$
$73$	−15.0249	−1.75853	−0.879267	+	0.476329i	$0.841967\pi$
$74$	10.3031	1.19771
$75$	0	0
$76$	−5.95541	−0.683133
$77$	−6.42646	−0.732363
$78$	−3.84292	−0.435125
$79$	−15.9369	−1.79304	−0.896521	−	0.443002i	$-0.853914\pi$
$79$	−15.9369	−1.79304	−0.896521	+	0.443002i	$0.853914\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−9.61953	−1.06230
$83$	5.55566	0.609813	0.304907	−	0.952382i	$-0.401375\pi$
$83$	5.55566	0.609813	0.304907	+	0.952382i	$0.401375\pi$
$84$	−2.71616	−0.296357
$85$	0	0
$86$	−21.6474	−2.33430
$87$	−3.47851	−0.372935
$88$	6.64124	0.707959
$89$	−17.8907	−1.89641	−0.948206	−	0.317657i	$-0.897104\pi$
$89$	−17.8907	−1.89641	−0.948206	+	0.317657i	$0.897104\pi$
$90$	0	0
$91$	−1.94520	−0.203912
$92$	6.20893	0.647325
$93$	5.62133	0.582905
$94$	18.2551	1.88287
$95$	0	0
$96$	−7.92509	−0.808851
$97$	−3.09939	−0.314695	−0.157347	−	0.987543i	$-0.550294\pi$
$97$	−3.09939	−0.314695	−0.157347	+	0.987543i	$0.550294\pi$
$98$	12.4226	1.25487
$99$	−5.96967	−0.599975
$100$	0	0
$101$	−3.18446	−0.316865	−0.158433	−	0.987370i	$-0.550644\pi$
$101$	−3.18446	−0.316865	−0.158433	+	0.987370i	$0.550644\pi$
$102$	14.4887	1.43460
$103$	−7.62974	−0.751780	−0.375890	−	0.926664i	$-0.622663\pi$
$103$	−7.62974	−0.751780	−0.375890	+	0.926664i	$0.622663\pi$
$104$	2.01021	0.197118
$105$	0	0
$106$	−29.2692	−2.84287
$107$	6.13132	0.592737	0.296368	−	0.955074i	$-0.404224\pi$
$107$	6.13132	0.592737	0.296368	+	0.955074i	$0.404224\pi$
$108$	−2.52310	−0.242785
$109$	13.5446	1.29734	0.648669	−	0.761070i	$-0.275326\pi$
$109$	13.5446	1.29734	0.648669	+	0.761070i	$0.275326\pi$
$110$	0	0
$111$	4.84452	0.459821
$112$	−2.88526	−0.272632
$113$	−6.46020	−0.607725	−0.303862	−	0.952716i	$-0.598276\pi$
$113$	−6.46020	−0.607725	−0.303862	+	0.952716i	$0.598276\pi$
$114$	−5.01991	−0.470158
$115$	0	0
$116$	8.77661	0.814888
$117$	−1.80694	−0.167051
$118$	2.12676	0.195784
$119$	7.33388	0.672295
$120$	0	0
$121$	24.6370	2.23973
$122$	−28.0494	−2.53947
$123$	−4.52310	−0.407834
$124$	−14.1832	−1.27369
$125$	0	0
$126$	−2.28949	−0.203964
$127$	9.19742	0.816139	0.408070	−	0.912951i	$-0.366202\pi$
$127$	9.19742	0.816139	0.408070	+	0.912951i	$0.366202\pi$
$128$	8.59557	0.759748
$129$	−10.1786	−0.896176
$130$	0	0
$131$	−18.8015	−1.64270	−0.821349	−	0.570426i	$-0.806778\pi$
$131$	−18.8015	−1.64270	−0.821349	+	0.570426i	$0.806778\pi$
$132$	15.0621	1.31098
$133$	−2.54097	−0.220330
$134$	13.2660	1.14600
$135$	0	0
$136$	−7.57899	−0.649893
$137$	12.3864	1.05824	0.529122	−	0.848546i	$-0.322521\pi$
$137$	12.3864	1.05824	0.529122	+	0.848546i	$0.322521\pi$
$138$	5.23360	0.445514
$139$	−2.29586	−0.194732	−0.0973662	−	0.995249i	$-0.531042\pi$
$139$	−2.29586	−0.194732	−0.0973662	+	0.995249i	$0.531042\pi$
$140$	0	0
$141$	8.58355	0.722865
$142$	−24.7409	−2.07621
$143$	10.7868	0.902039
$144$	−2.68018	−0.223348
$145$	0	0
$146$	31.9544	2.64456
$147$	5.84111	0.481766
$148$	−12.2232	−1.00474
$149$	12.1444	0.994910	0.497455	−	0.867490i	$-0.334268\pi$
$149$	12.1444	0.994910	0.497455	+	0.867490i	$0.334268\pi$
$150$	0	0
$151$	20.2527	1.64814	0.824070	−	0.566488i	$-0.191698\pi$
$151$	20.2527	1.64814	0.824070	+	0.566488i	$0.191698\pi$
$152$	2.62589	0.212988
$153$	6.81259	0.550765
$154$	13.6675	1.10136
$155$	0	0
$156$	4.55907	0.365018
$157$	−1.86515	−0.148855	−0.0744275	−	0.997226i	$-0.523713\pi$
$157$	−1.86515	−0.148855	−0.0744275	+	0.997226i	$0.523713\pi$
$158$	33.8939	2.69646
$159$	−13.7623	−1.09143
$160$	0	0
$161$	2.64914	0.208781
$162$	−2.12676	−0.167094
$163$	2.75265	0.215604	0.107802	−	0.994172i	$-0.465619\pi$
$163$	2.75265	0.215604	0.107802	+	0.994172i	$0.465619\pi$
$164$	11.4122	0.891143
$165$	0	0
$166$	−11.8155	−0.917064
$167$	−13.0084	−1.00662	−0.503310	−	0.864106i	$-0.667885\pi$
$167$	−13.0084	−1.00662	−0.503310	+	0.864106i	$0.667885\pi$
$168$	1.19762	0.0923987
$169$	−9.73498	−0.748845
$170$	0	0
$171$	−2.36036	−0.180501
$172$	25.6816	1.95820
$173$	15.7264	1.19565	0.597827	−	0.801625i	$-0.296031\pi$
$173$	15.7264	1.19565	0.597827	+	0.801625i	$0.296031\pi$
$174$	7.39794	0.560837
$175$	0	0
$176$	15.9998	1.20603
$177$	1.00000	0.0751646
$178$	38.0492	2.85191
$179$	11.3733	0.850078	0.425039	−	0.905175i	$-0.360260\pi$
$179$	11.3733	0.850078	0.425039	+	0.905175i	$0.360260\pi$
$180$	0	0
$181$	18.6583	1.38686	0.693429	−	0.720525i	$-0.256099\pi$
$181$	18.6583	1.38686	0.693429	+	0.720525i	$0.256099\pi$
$182$	4.13697	0.306653
$183$	−13.1888	−0.974945
$184$	−2.73767	−0.201824
$185$	0	0
$186$	−11.9552	−0.876599
$187$	−40.6689	−2.97401
$188$	−21.6571	−1.57951
$189$	−1.07652	−0.0783052
$190$	0	0
$191$	−0.386126	−0.0279391	−0.0139696	−	0.999902i	$-0.504447\pi$
$191$	−0.386126	−0.0279391	−0.0139696	+	0.999902i	$0.504447\pi$
$192$	11.4944	0.829535
$193$	18.2226	1.31169	0.655844	−	0.754896i	$-0.272313\pi$
$193$	18.2226	1.31169	0.655844	+	0.754896i	$0.272313\pi$
$194$	6.59164	0.473252
$195$	0	0
$196$	−14.7377	−1.05269
$197$	−2.03578	−0.145043	−0.0725215	−	0.997367i	$-0.523105\pi$
$197$	−2.03578	−0.145043	−0.0725215	+	0.997367i	$0.523105\pi$
$198$	12.6960	0.902269
$199$	−22.8708	−1.62127	−0.810634	−	0.585553i	$-0.800877\pi$
$199$	−22.8708	−1.62127	−0.810634	+	0.585553i	$0.800877\pi$
$200$	0	0
$201$	6.23765	0.439970
$202$	6.77256	0.476516
$203$	3.74468	0.262825
$204$	−17.1888	−1.20346
$205$	0	0
$206$	16.2266	1.13056
$207$	2.46084	0.171040
$208$	4.84292	0.335796
$209$	14.0906	0.974665
$210$	0	0
$211$	−25.1968	−1.73462	−0.867309	−	0.497770i	$-0.834152\pi$
$211$	−25.1968	−1.73462	−0.867309	+	0.497770i	$0.834152\pi$
$212$	34.7237	2.38484
$213$	−11.6331	−0.797090
$214$	−13.0398	−0.891384
$215$	0	0
$216$	1.11250	0.0756958
$217$	−6.05147	−0.410800
$218$	−28.8061	−1.95100
$219$	15.0249	1.01529
$220$	0	0
$221$	−12.3099	−0.828055
$222$	−10.3031	−0.691500
$223$	−12.7931	−0.856690	−0.428345	−	0.903615i	$-0.640903\pi$
$223$	−12.7931	−0.856690	−0.428345	+	0.903615i	$0.640903\pi$
$224$	8.53150	0.570035
$225$	0	0
$226$	13.7393	0.913923
$227$	−19.8486	−1.31740	−0.658698	−	0.752408i	$-0.728892\pi$
$227$	−19.8486	−1.31740	−0.658698	+	0.752408i	$0.728892\pi$
$228$	5.95541	0.394407
$229$	−11.6179	−0.767734	−0.383867	−	0.923388i	$-0.625408\pi$
$229$	−11.6179	−0.767734	−0.383867	+	0.923388i	$0.625408\pi$
$230$	0	0
$231$	6.42646	0.422830
$232$	−3.86983	−0.254067
$233$	13.5172	0.885544	0.442772	−	0.896634i	$-0.353995\pi$
$233$	13.5172	0.885544	0.442772	+	0.896634i	$0.353995\pi$
$234$	3.84292	0.251219
$235$	0	0
$236$	−2.52310	−0.164240
$237$	15.9369	1.03521
$238$	−15.5974	−1.01103
$239$	−11.1754	−0.722876	−0.361438	−	0.932396i	$-0.617714\pi$
$239$	−11.1754	−0.722876	−0.361438	+	0.932396i	$0.617714\pi$
$240$	0	0
$241$	1.80803	0.116465	0.0582327	−	0.998303i	$-0.481453\pi$
$241$	1.80803	0.116465	0.0582327	+	0.998303i	$0.481453\pi$
$242$	−52.3969	−3.36820
$243$	−1.00000	−0.0641500
$244$	33.2766	2.13032
$245$	0	0
$246$	9.61953	0.613319
$247$	4.26502	0.271377
$248$	6.25372	0.397111
$249$	−5.55566	−0.352076
$250$	0	0
$251$	3.94006	0.248695	0.124347	−	0.992239i	$-0.460316\pi$
$251$	3.94006	0.248695	0.124347	+	0.992239i	$0.460316\pi$
$252$	2.71616	0.171102
$253$	−14.6904	−0.923577
$254$	−19.5607	−1.22735
$255$	0	0
$256$	4.70806	0.294254
$257$	−5.78413	−0.360804	−0.180402	−	0.983593i	$-0.557740\pi$
$257$	−5.78413	−0.360804	−0.180402	+	0.983593i	$0.557740\pi$
$258$	21.6474	1.34771
$259$	−5.21521	−0.324058
$260$	0	0
$261$	3.47851	0.215314
$262$	39.9863	2.47036
$263$	23.0444	1.42098	0.710489	−	0.703709i	$-0.248474\pi$
$263$	23.0444	1.42098	0.710489	+	0.703709i	$0.248474\pi$
$264$	−6.64124	−0.408740
$265$	0	0
$266$	5.40403	0.331342
$267$	17.8907	1.09489
$268$	−15.7382	−0.961363
$269$	−5.20192	−0.317167	−0.158583	−	0.987346i	$-0.550693\pi$
$269$	−5.20192	−0.317167	−0.158583	+	0.987346i	$0.550693\pi$
$270$	0	0
$271$	−1.51744	−0.0921782	−0.0460891	−	0.998937i	$-0.514676\pi$
$271$	−1.51744	−0.0921782	−0.0460891	+	0.998937i	$0.514676\pi$
$272$	−18.2590	−1.10711
$273$	1.94520	0.117729
$274$	−26.3429	−1.59144
$275$	0	0
$276$	−6.20893	−0.373733
$277$	0.0598152	0.00359395	0.00179697	−	0.999998i	$-0.499428\pi$
$277$	0.0598152	0.00359395	0.00179697	+	0.999998i	$0.499428\pi$
$278$	4.88274	0.292847
$279$	−5.62133	−0.336540
$280$	0	0
$281$	−30.5027	−1.81964	−0.909820	−	0.415003i	$-0.863780\pi$
$281$	−30.5027	−1.81964	−0.909820	+	0.415003i	$0.863780\pi$
$282$	−18.2551	−1.08708
$283$	22.6416	1.34590	0.672950	−	0.739688i	$-0.265027\pi$
$283$	22.6416	1.34590	0.672950	+	0.739688i	$0.265027\pi$
$284$	29.3515	1.74169
$285$	0	0
$286$	−22.9409	−1.35653
$287$	4.86920	0.287420
$288$	7.92509	0.466990
$289$	29.4114	1.73008
$290$	0	0
$291$	3.09939	0.181689
$292$	−37.9093	−2.21848
$293$	−13.4493	−0.785718	−0.392859	−	0.919599i	$-0.628514\pi$
$293$	−13.4493	−0.785718	−0.392859	+	0.919599i	$0.628514\pi$
$294$	−12.4226	−0.724502
$295$	0	0
$296$	5.38952	0.313259
$297$	5.96967	0.346396
$298$	−25.8282	−1.49619
$299$	−4.44658	−0.257152
$300$	0	0
$301$	10.9574	0.631577
$302$	−43.0725	−2.47854
$303$	3.18446	0.182942
$304$	6.32619	0.362832
$305$	0	0
$306$	−14.4887	−0.828265
$307$	25.4116	1.45031	0.725157	−	0.688583i	$-0.241767\pi$
$307$	25.4116	1.45031	0.725157	+	0.688583i	$0.241767\pi$
$308$	−16.2146	−0.923911
$309$	7.62974	0.434041
$310$	0	0
$311$	−26.2825	−1.49035	−0.745173	−	0.666871i	$-0.767633\pi$
$311$	−26.2825	−1.49035	−0.745173	+	0.666871i	$0.767633\pi$
$312$	−2.01021	−0.113806
$313$	−20.0369	−1.13255	−0.566277	−	0.824215i	$-0.691617\pi$
$313$	−20.0369	−1.13255	−0.566277	+	0.824215i	$0.691617\pi$
$314$	3.96672	0.223855
$315$	0	0
$316$	−40.2103	−2.26201
$317$	−25.5503	−1.43505	−0.717524	−	0.696533i	$-0.754725\pi$
$317$	−25.5503	−1.43505	−0.717524	+	0.696533i	$0.754725\pi$
$318$	29.2692	1.64133
$319$	−20.7656	−1.16265
$320$	0	0
$321$	−6.13132	−0.342217
$322$	−5.63407	−0.313974
$323$	−16.0802	−0.894724
$324$	2.52310	0.140172
$325$	0	0
$326$	−5.85422	−0.324235
$327$	−13.5446	−0.749019
$328$	−5.03193	−0.277842
$329$	−9.24035	−0.509437
$330$	0	0
$331$	−22.7085	−1.24817	−0.624086	−	0.781356i	$-0.714528\pi$
$331$	−22.7085	−1.24817	−0.624086	+	0.781356i	$0.714528\pi$
$332$	14.0175	0.769309
$333$	−4.84452	−0.265478
$334$	27.6657	1.51380
$335$	0	0
$336$	2.88526	0.157404
$337$	5.93479	0.323288	0.161644	−	0.986849i	$-0.448320\pi$
$337$	5.93479	0.323288	0.161644	+	0.986849i	$0.448320\pi$
$338$	20.7039	1.12615
$339$	6.46020	0.350870
$340$	0	0
$341$	33.5575	1.81724
$342$	5.01991	0.271446
$343$	−13.8237	−0.746409
$344$	−11.3237	−0.610531
$345$	0	0
$346$	−33.4462	−1.79808
$347$	−5.93349	−0.318526	−0.159263	−	0.987236i	$-0.550912\pi$
$347$	−5.93349	−0.318526	−0.159263	+	0.987236i	$0.550912\pi$
$348$	−8.77661	−0.470476
$349$	10.9234	0.584714	0.292357	−	0.956309i	$-0.405560\pi$
$349$	10.9234	0.584714	0.292357	+	0.956309i	$0.405560\pi$
$350$	0	0
$351$	1.80694	0.0964472
$352$	−47.3102	−2.52164
$353$	−12.9953	−0.691670	−0.345835	−	0.938295i	$-0.612404\pi$
$353$	−12.9953	−0.691670	−0.345835	+	0.938295i	$0.612404\pi$
$354$	−2.12676	−0.113036
$355$	0	0
$356$	−45.1400	−2.39241
$357$	−7.33388	−0.388150
$358$	−24.1882	−1.27838
$359$	21.8271	1.15199	0.575995	−	0.817453i	$-0.304615\pi$
$359$	21.8271	1.15199	0.575995	+	0.817453i	$0.304615\pi$
$360$	0	0
$361$	−13.4287	−0.706774
$362$	−39.6816	−2.08562
$363$	−24.6370	−1.29311
$364$	−4.90793	−0.257245
$365$	0	0
$366$	28.0494	1.46617
$367$	−28.8798	−1.50751	−0.753756	−	0.657154i	$-0.771760\pi$
$367$	−28.8798	−1.50751	−0.753756	+	0.657154i	$0.771760\pi$
$368$	−6.59549	−0.343813
$369$	4.52310	0.235463
$370$	0	0
$371$	14.8154	0.769178
$372$	14.1832	0.735363
$373$	−16.3404	−0.846076	−0.423038	−	0.906112i	$-0.639036\pi$
$373$	−16.3404	−0.846076	−0.423038	+	0.906112i	$0.639036\pi$
$374$	86.4929	4.47244
$375$	0	0
$376$	9.54917	0.492461
$377$	−6.28545	−0.323717
$378$	2.28949	0.117759
$379$	−27.4840	−1.41176	−0.705878	−	0.708333i	$-0.749447\pi$
$379$	−27.4840	−1.41176	−0.705878	+	0.708333i	$0.749447\pi$
$380$	0	0
$381$	−9.19742	−0.471198
$382$	0.821197	0.0420161
$383$	15.9532	0.815169	0.407585	−	0.913167i	$-0.366371\pi$
$383$	15.9532	0.815169	0.407585	+	0.913167i	$0.366371\pi$
$384$	−8.59557	−0.438641
$385$	0	0
$386$	−38.7549	−1.97257
$387$	10.1786	0.517407
$388$	−7.82005	−0.397003
$389$	−3.77431	−0.191365	−0.0956826	−	0.995412i	$-0.530503\pi$
$389$	−3.77431	−0.191365	−0.0956826	+	0.995412i	$0.530503\pi$
$390$	0	0
$391$	16.7647	0.847826
$392$	6.49822	0.328210
$393$	18.8015	0.948412
$394$	4.32960	0.218122
$395$	0	0
$396$	−15.0621	−0.756897
$397$	−17.2196	−0.864227	−0.432113	−	0.901819i	$-0.642232\pi$
$397$	−17.2196	−0.864227	−0.432113	+	0.901819i	$0.642232\pi$
$398$	48.6406	2.43813
$399$	2.54097	0.127208
$400$	0	0
$401$	−22.9640	−1.14677	−0.573384	−	0.819287i	$-0.694370\pi$
$401$	−22.9640	−1.14677	−0.573384	+	0.819287i	$0.694370\pi$
$402$	−13.2660	−0.661646
$403$	10.1574	0.505976
$404$	−8.03468	−0.399740
$405$	0	0
$406$	−7.96402	−0.395248
$407$	28.9202	1.43352
$408$	7.57899	0.375216
$409$	−18.2150	−0.900675	−0.450338	−	0.892858i	$-0.648696\pi$
$409$	−18.2150	−0.900675	−0.450338	+	0.892858i	$0.648696\pi$
$410$	0	0
$411$	−12.3864	−0.610978
$412$	−19.2506	−0.948407
$413$	−1.07652	−0.0529720
$414$	−5.23360	−0.257218
$415$	0	0
$416$	−14.3201	−0.702102
$417$	2.29586	0.112429
$418$	−29.9672	−1.46575
$419$	−9.31918	−0.455272	−0.227636	−	0.973746i	$-0.573100\pi$
$419$	−9.31918	−0.455272	−0.227636	+	0.973746i	$0.573100\pi$
$420$	0	0
$421$	8.23200	0.401203	0.200602	−	0.979673i	$-0.435710\pi$
$421$	8.23200	0.401203	0.200602	+	0.979673i	$0.435710\pi$
$422$	53.5874	2.60860
$423$	−8.58355	−0.417346
$424$	−15.3106	−0.743547
$425$	0	0
$426$	24.7409	1.19870
$427$	14.1980	0.687089
$428$	15.4699	0.747766
$429$	−10.7868	−0.520793
$430$	0	0
$431$	−0.961781	−0.0463274	−0.0231637	−	0.999732i	$-0.507374\pi$
$431$	−0.961781	−0.0463274	−0.0231637	+	0.999732i	$0.507374\pi$
$432$	2.68018	0.128950
$433$	2.08558	0.100227	0.0501133	−	0.998744i	$-0.484042\pi$
$433$	2.08558	0.100227	0.0501133	+	0.998744i	$0.484042\pi$
$434$	12.8700	0.617780
$435$	0	0
$436$	34.1743	1.63665
$437$	−5.80846	−0.277856
$438$	−31.9544	−1.52684
$439$	8.21959	0.392300	0.196150	−	0.980574i	$-0.437156\pi$
$439$	8.21959	0.392300	0.196150	+	0.980574i	$0.437156\pi$
$440$	0	0
$441$	−5.84111	−0.278148
$442$	26.1802	1.24527
$443$	−8.11186	−0.385406	−0.192703	−	0.981257i	$-0.561725\pi$
$443$	−8.11186	−0.385406	−0.192703	+	0.981257i	$0.561725\pi$
$444$	12.2232	0.580087
$445$	0	0
$446$	27.2078	1.28833
$447$	−12.1444	−0.574412
$448$	−12.3739	−0.584612
$449$	−10.9326	−0.515941	−0.257971	−	0.966153i	$-0.583054\pi$
$449$	−10.9326	−0.515941	−0.257971	+	0.966153i	$0.583054\pi$
$450$	0	0
$451$	−27.0014	−1.27145
$452$	−16.2997	−0.766674
$453$	−20.2527	−0.951554
$454$	42.2131	1.98116
$455$	0	0
$456$	−2.62589	−0.122969
$457$	−12.0346	−0.562956	−0.281478	−	0.959568i	$-0.590825\pi$
$457$	−12.0346	−0.562956	−0.281478	+	0.959568i	$0.590825\pi$
$458$	24.7085	1.15455
$459$	−6.81259	−0.317984
$460$	0	0
$461$	−11.1440	−0.519029	−0.259514	−	0.965739i	$-0.583563\pi$
$461$	−11.1440	−0.519029	−0.259514	+	0.965739i	$0.583563\pi$
$462$	−13.6675	−0.635871
$463$	−30.3128	−1.40876	−0.704378	−	0.709825i	$-0.748774\pi$
$463$	−30.3128	−1.40876	−0.704378	+	0.709825i	$0.748774\pi$
$464$	−9.32303	−0.432811
$465$	0	0
$466$	−28.7479	−1.33172
$467$	31.0098	1.43496	0.717481	−	0.696578i	$-0.245295\pi$
$467$	31.0098	1.43496	0.717481	+	0.696578i	$0.245295\pi$
$468$	−4.55907	−0.210743
$469$	−6.71494	−0.310067
$470$	0	0
$471$	1.86515	0.0859415
$472$	1.11250	0.0512068
$473$	−60.7629	−2.79388
$474$	−33.8939	−1.55680
$475$	0	0
$476$	18.5041	0.848133
$477$	13.7623	0.630135
$478$	23.7673	1.08709
$479$	−13.3696	−0.610874	−0.305437	−	0.952212i	$-0.598803\pi$
$479$	−13.3696	−0.610874	−0.305437	+	0.952212i	$0.598803\pi$
$480$	0	0
$481$	8.75374	0.399136
$482$	−3.84524	−0.175146
$483$	−2.64914	−0.120540
$484$	62.1615	2.82552
$485$	0	0
$486$	2.12676	0.0964717
$487$	−38.9389	−1.76449	−0.882245	−	0.470791i	$-0.843968\pi$
$487$	−38.9389	−1.76449	−0.882245	+	0.470791i	$0.843968\pi$
$488$	−14.6725	−0.664194
$489$	−2.75265	−0.124479
$490$	0	0
$491$	30.6998	1.38546	0.692731	−	0.721197i	$-0.256408\pi$
$491$	30.6998	1.38546	0.692731	+	0.721197i	$0.256408\pi$
$492$	−11.4122	−0.514502
$493$	23.6976	1.06729
$494$	−9.07066	−0.408108
$495$	0	0
$496$	15.0662	0.676492
$497$	12.5233	0.561747
$498$	11.8155	0.529467
$499$	−23.3400	−1.04484	−0.522422	−	0.852687i	$-0.674971\pi$
$499$	−23.3400	−1.04484	−0.522422	+	0.852687i	$0.674971\pi$
$500$	0	0
$501$	13.0084	0.581173
$502$	−8.37955	−0.373998
$503$	24.4290	1.08923	0.544617	−	0.838685i	$-0.316675\pi$
$503$	24.4290	1.08923	0.544617	+	0.838685i	$0.316675\pi$
$504$	−1.19762	−0.0533464
$505$	0	0
$506$	31.2429	1.38892
$507$	9.73498	0.432346
$508$	23.2060	1.02960
$509$	−30.9821	−1.37326	−0.686630	−	0.727007i	$-0.740911\pi$
$509$	−30.9821	−1.37326	−0.686630	+	0.727007i	$0.740911\pi$
$510$	0	0
$511$	−16.1746	−0.715523
$512$	−27.2040	−1.20226
$513$	2.36036	0.104212
$514$	12.3014	0.542593
$515$	0	0
$516$	−25.6816	−1.13057
$517$	51.2410	2.25358
$518$	11.0915	0.487332
$519$	−15.7264	−0.690311
$520$	0	0
$521$	3.55515	0.155754	0.0778770	−	0.996963i	$-0.475186\pi$
$521$	3.55515	0.155754	0.0778770	+	0.996963i	$0.475186\pi$
$522$	−7.39794	−0.323799
$523$	−18.0330	−0.788528	−0.394264	−	0.918997i	$-0.629000\pi$
$523$	−18.0330	−0.788528	−0.394264	+	0.918997i	$0.629000\pi$
$524$	−47.4381	−2.07234
$525$	0	0
$526$	−49.0098	−2.13693
$527$	−38.2958	−1.66819
$528$	−15.9998	−0.696302
$529$	−16.9443	−0.736708
$530$	0	0
$531$	−1.00000	−0.0433963
$532$	−6.41111	−0.277957
$533$	−8.17295	−0.354010
$534$	−38.0492	−1.64655
$535$	0	0
$536$	6.93937	0.299735
$537$	−11.3733	−0.490793
$538$	11.0632	0.476970
$539$	34.8695	1.50194
$540$	0	0
$541$	−37.8425	−1.62698	−0.813489	−	0.581581i	$-0.802434\pi$
$541$	−37.8425	−1.62698	−0.813489	+	0.581581i	$0.802434\pi$
$542$	3.22723	0.138622
$543$	−18.6583	−0.800702
$544$	53.9903	2.31482
$545$	0	0
$546$	−4.13697	−0.177046
$547$	14.9032	0.637217	0.318608	−	0.947886i	$-0.396785\pi$
$547$	14.9032	0.637217	0.318608	+	0.947886i	$0.396785\pi$
$548$	31.2522	1.33503
$549$	13.1888	0.562885
$550$	0	0
$551$	−8.21053	−0.349780
$552$	2.73767	0.116523
$553$	−17.1564	−0.729563
$554$	−0.127212	−0.00540474
$555$	0	0
$556$	−5.79268	−0.245664
$557$	35.4139	1.50053	0.750267	−	0.661135i	$-0.229925\pi$
$557$	35.4139	1.50053	0.750267	+	0.661135i	$0.229925\pi$
$558$	11.9552	0.506104
$559$	−18.3921	−0.777902
$560$	0	0
$561$	40.6689	1.71704
$562$	64.8719	2.73645
$563$	9.07626	0.382519	0.191259	−	0.981540i	$-0.438743\pi$
$563$	9.07626	0.382519	0.191259	+	0.981540i	$0.438743\pi$
$564$	21.6571	0.911929
$565$	0	0
$566$	−48.1531	−2.02403
$567$	1.07652	0.0452095
$568$	−12.9418	−0.543028
$569$	21.3617	0.895529	0.447765	−	0.894151i	$-0.352220\pi$
$569$	21.3617	0.895529	0.447765	+	0.894151i	$0.352220\pi$
$570$	0	0
$571$	25.9783	1.08716	0.543580	−	0.839358i	$-0.317069\pi$
$571$	25.9783	1.08716	0.543580	+	0.839358i	$0.317069\pi$
$572$	27.2162	1.13797
$573$	0.386126	0.0161307
$574$	−10.3556	−0.432234
$575$	0	0
$576$	−11.4944	−0.478932
$577$	−21.8099	−0.907958	−0.453979	−	0.891012i	$-0.649996\pi$
$577$	−21.8099	−0.907958	−0.453979	+	0.891012i	$0.649996\pi$
$578$	−62.5508	−2.60177
$579$	−18.2226	−0.757303
$580$	0	0
$581$	5.98077	0.248124
$582$	−6.59164	−0.273232
$583$	−82.1567	−3.40258
$584$	16.7152	0.691679
$585$	0	0
$586$	28.6035	1.18160
$587$	−14.9555	−0.617278	−0.308639	−	0.951179i	$-0.599874\pi$
$587$	−14.9555	−0.617278	−0.308639	+	0.951179i	$0.599874\pi$
$588$	14.7377	0.607771
$589$	13.2684	0.546714
$590$	0	0
$591$	2.03578	0.0837406
$592$	12.9842	0.533647
$593$	−15.2066	−0.624460	−0.312230	−	0.950006i	$-0.601076\pi$
$593$	−15.2066	−0.624460	−0.312230	+	0.950006i	$0.601076\pi$
$594$	−12.6960	−0.520925
$595$	0	0
$596$	30.6416	1.25513
$597$	22.8708	0.936040
$598$	9.45679	0.386717
$599$	−19.3122	−0.789073	−0.394537	−	0.918880i	$-0.629095\pi$
$599$	−19.3122	−0.789073	−0.394537	+	0.918880i	$0.629095\pi$
$600$	0	0
$601$	−18.4015	−0.750614	−0.375307	−	0.926901i	$-0.622463\pi$
$601$	−18.4015	−0.750614	−0.375307	+	0.926901i	$0.622463\pi$
$602$	−23.3038	−0.949793
$603$	−6.23765	−0.254017
$604$	51.0994	2.07921
$605$	0	0
$606$	−6.77256	−0.275117
$607$	−19.5994	−0.795516	−0.397758	−	0.917490i	$-0.630212\pi$
$607$	−19.5994	−0.795516	−0.397758	+	0.917490i	$0.630212\pi$
$608$	−18.7061	−0.758631
$609$	−3.74468	−0.151742
$610$	0	0
$611$	15.5099	0.627464
$612$	17.1888	0.694817
$613$	−16.4002	−0.662397	−0.331199	−	0.943561i	$-0.607453\pi$
$613$	−16.4002	−0.662397	−0.331199	+	0.943561i	$0.607453\pi$
$614$	−54.0442	−2.18105
$615$	0	0
$616$	7.14942	0.288058
$617$	16.0812	0.647405	0.323702	−	0.946159i	$-0.395072\pi$
$617$	16.0812	0.647405	0.323702	+	0.946159i	$0.395072\pi$
$618$	−16.2266	−0.652730
$619$	17.0842	0.686672	0.343336	−	0.939213i	$-0.388443\pi$
$619$	17.0842	0.686672	0.343336	+	0.939213i	$0.388443\pi$
$620$	0	0
$621$	−2.46084	−0.0987500
$622$	55.8966	2.24125
$623$	−19.2597	−0.771623
$624$	−4.84292	−0.193872
$625$	0	0
$626$	42.6137	1.70318
$627$	−14.0906	−0.562723
$628$	−4.70595	−0.187788
$629$	−33.0037	−1.31594
$630$	0	0
$631$	6.67998	0.265926	0.132963	−	0.991121i	$-0.457551\pi$
$631$	6.67998	0.265926	0.132963	+	0.991121i	$0.457551\pi$
$632$	17.7298	0.705252
$633$	25.1968	1.00148
$634$	54.3393	2.15809
$635$	0	0
$636$	−34.7237	−1.37689
$637$	10.5545	0.418185
$638$	44.1633	1.74844
$639$	11.6331	0.460200
$640$	0	0
$641$	36.3244	1.43473	0.717363	−	0.696699i	$-0.245349\pi$
$641$	36.3244	1.43473	0.717363	+	0.696699i	$0.245349\pi$
$642$	13.0398	0.514641
$643$	5.43347	0.214275	0.107137	−	0.994244i	$-0.465831\pi$
$643$	5.43347	0.214275	0.107137	+	0.994244i	$0.465831\pi$
$644$	6.68402	0.263387
$645$	0	0
$646$	34.1986	1.34553
$647$	−16.1037	−0.633100	−0.316550	−	0.948576i	$-0.602525\pi$
$647$	−16.1037	−0.633100	−0.316550	+	0.948576i	$0.602525\pi$
$648$	−1.11250	−0.0437030
$649$	5.96967	0.234330
$650$	0	0
$651$	6.05147	0.237176
$652$	6.94520	0.271995
$653$	−29.3315	−1.14783	−0.573914	−	0.818915i	$-0.694576\pi$
$653$	−29.3315	−1.14783	−0.573914	+	0.818915i	$0.694576\pi$
$654$	28.8061	1.12641
$655$	0	0
$656$	−12.1227	−0.473312
$657$	−15.0249	−0.586178
$658$	19.6520	0.766114
$659$	−49.7516	−1.93805	−0.969024	−	0.246968i	$-0.920566\pi$
$659$	−49.7516	−1.93805	−0.969024	+	0.246968i	$0.920566\pi$
$660$	0	0
$661$	22.7832	0.886164	0.443082	−	0.896481i	$-0.353885\pi$
$661$	22.7832	0.886164	0.443082	+	0.896481i	$0.353885\pi$
$662$	48.2955	1.87706
$663$	12.3099	0.478078
$664$	−6.18066	−0.239856
$665$	0	0
$666$	10.3031	0.399238
$667$	8.56004	0.331446
$668$	−32.8214	−1.26990
$669$	12.7931	0.494610
$670$	0	0
$671$	−78.7329	−3.03945
$672$	−8.53150	−0.329110
$673$	−14.5226	−0.559807	−0.279903	−	0.960028i	$-0.590302\pi$
$673$	−14.5226	−0.559807	−0.279903	+	0.960028i	$0.590302\pi$
$674$	−12.6218	−0.486175
$675$	0	0
$676$	−24.5623	−0.944703
$677$	−42.1035	−1.61817	−0.809084	−	0.587693i	$-0.800036\pi$
$677$	−42.1035	−1.61817	−0.809084	+	0.587693i	$0.800036\pi$
$678$	−13.7393	−0.527654
$679$	−3.33655	−0.128045
$680$	0	0
$681$	19.8486	0.760599
$682$	−71.3687	−2.73285
$683$	28.7808	1.10127	0.550634	−	0.834747i	$-0.314386\pi$
$683$	28.7808	1.10127	0.550634	+	0.834747i	$0.314386\pi$
$684$	−5.95541	−0.227711
$685$	0	0
$686$	29.3996	1.12248
$687$	11.6179	0.443251
$688$	−27.2805	−1.04006
$689$	−24.8677	−0.947384
$690$	0	0
$691$	11.3837	0.433056	0.216528	−	0.976276i	$-0.430527\pi$
$691$	11.3837	0.433056	0.216528	+	0.976276i	$0.430527\pi$
$692$	39.6791	1.50837
$693$	−6.42646	−0.244121
$694$	12.6191	0.479014
$695$	0	0
$696$	3.86983	0.146686
$697$	30.8140	1.16716
$698$	−23.2313	−0.879319
$699$	−13.5172	−0.511269
$700$	0	0
$701$	1.08969	0.0411569	0.0205785	−	0.999788i	$-0.493449\pi$
$701$	1.08969	0.0411569	0.0205785	+	0.999788i	$0.493449\pi$
$702$	−3.84292	−0.145042
$703$	11.4348	0.431272
$704$	68.6176	2.58612
$705$	0	0
$706$	27.6378	1.04016
$707$	−3.42812	−0.128928
$708$	2.52310	0.0948237
$709$	7.52215	0.282500	0.141250	−	0.989974i	$-0.454888\pi$
$709$	7.52215	0.282500	0.141250	+	0.989974i	$0.454888\pi$
$710$	0	0
$711$	−15.9369	−0.597681
$712$	19.9034	0.745910
$713$	−13.8332	−0.518057
$714$	15.5974	0.583717
$715$	0	0
$716$	28.6958	1.07241
$717$	11.1754	0.417353
$718$	−46.4209	−1.73241
$719$	−1.60083	−0.0597008	−0.0298504	−	0.999554i	$-0.509503\pi$
$719$	−1.60083	−0.0597008	−0.0298504	+	0.999554i	$0.509503\pi$
$720$	0	0
$721$	−8.21355	−0.305889
$722$	28.5596	1.06288
$723$	−1.80803	−0.0672413
$724$	47.0766	1.74959
$725$	0	0
$726$	52.3969	1.94463
$727$	1.43860	0.0533549	0.0266775	−	0.999644i	$-0.491507\pi$
$727$	1.43860	0.0533549	0.0266775	+	0.999644i	$0.491507\pi$
$728$	2.16403	0.0802043
$729$	1.00000	0.0370370
$730$	0	0
$731$	69.3426	2.56473
$732$	−33.2766	−1.22994
$733$	1.33382	0.0492657	0.0246329	−	0.999697i	$-0.492158\pi$
$733$	1.33382	0.0492657	0.0246329	+	0.999697i	$0.492158\pi$
$734$	61.4203	2.26706
$735$	0	0
$736$	19.5023	0.718866
$737$	37.2367	1.37163
$738$	−9.61953	−0.354100
$739$	19.0843	0.702027	0.351013	−	0.936370i	$-0.385837\pi$
$739$	19.0843	0.702027	0.351013	+	0.936370i	$0.385837\pi$
$740$	0	0
$741$	−4.26502	−0.156679
$742$	−31.5088	−1.15672
$743$	23.2378	0.852511	0.426255	−	0.904603i	$-0.359833\pi$
$743$	23.2378	0.852511	0.426255	+	0.904603i	$0.359833\pi$
$744$	−6.25372	−0.229272
$745$	0	0
$746$	34.7522	1.27237
$747$	5.55566	0.203271
$748$	−102.612	−3.75185
$749$	6.60048	0.241176
$750$	0	0
$751$	−43.3368	−1.58138	−0.790690	−	0.612216i	$-0.790278\pi$
$751$	−43.3368	−1.58138	−0.790690	+	0.612216i	$0.790278\pi$
$752$	23.0054	0.838922
$753$	−3.94006	−0.143584
$754$	13.3676	0.486820
$755$	0	0
$756$	−2.71616	−0.0987857
$757$	2.74063	0.0996099	0.0498050	−	0.998759i	$-0.484140\pi$
$757$	2.74063	0.0996099	0.0498050	+	0.998759i	$0.484140\pi$
$758$	58.4517	2.12306
$759$	14.6904	0.533227
$760$	0	0
$761$	27.8826	1.01074	0.505371	−	0.862902i	$-0.331356\pi$
$761$	27.8826	1.01074	0.505371	+	0.862902i	$0.331356\pi$
$762$	19.5607	0.708609
$763$	14.5810	0.527868
$764$	−0.974233	−0.0352465
$765$	0	0
$766$	−33.9285	−1.22589
$767$	1.80694	0.0652447
$768$	−4.70806	−0.169888
$769$	−23.6674	−0.853468	−0.426734	−	0.904377i	$-0.640336\pi$
$769$	−23.6674	−0.853468	−0.426734	+	0.904377i	$0.640336\pi$
$770$	0	0
$771$	5.78413	0.208310
$772$	45.9772	1.65476
$773$	1.69622	0.0610089	0.0305045	−	0.999535i	$-0.490289\pi$
$773$	1.69622	0.0610089	0.0305045	+	0.999535i	$0.490289\pi$
$774$	−21.6474	−0.778100
$775$	0	0
$776$	3.44806	0.123778
$777$	5.21521	0.187095
$778$	8.02704	0.287783
$779$	−10.6761	−0.382512
$780$	0	0
$781$	−69.4461	−2.48498
$782$	−35.6544	−1.27500
$783$	−3.47851	−0.124312
$784$	15.6552	0.559115
$785$	0	0
$786$	−39.9863	−1.42626
$787$	8.05910	0.287276	0.143638	−	0.989630i	$-0.454120\pi$
$787$	8.05910	0.287276	0.143638	+	0.989630i	$0.454120\pi$
$788$	−5.13646	−0.182979
$789$	−23.0444	−0.820402
$790$	0	0
$791$	−6.95453	−0.247274
$792$	6.64124	0.235986
$793$	−23.8314	−0.846276
$794$	36.6219	1.29966
$795$	0	0
$796$	−57.7052	−2.04531
$797$	−4.53279	−0.160560	−0.0802799	−	0.996772i	$-0.525581\pi$
$797$	−4.53279	−0.160560	−0.0802799	+	0.996772i	$0.525581\pi$
$798$	−5.40403	−0.191300
$799$	−58.4762	−2.06874
$800$	0	0
$801$	−17.8907	−0.632137
$802$	48.8389	1.72456
$803$	89.6939	3.16523
$804$	15.7382	0.555043
$805$	0	0
$806$	−21.6023	−0.760909
$807$	5.20192	0.183116
$808$	3.54270	0.124632
$809$	1.53481	0.0539609	0.0269804	−	0.999636i	$-0.491411\pi$
$809$	1.53481	0.0539609	0.0269804	+	0.999636i	$0.491411\pi$
$810$	0	0
$811$	−28.4481	−0.998946	−0.499473	−	0.866329i	$-0.666473\pi$
$811$	−28.4481	−0.998946	−0.499473	+	0.866329i	$0.666473\pi$
$812$	9.44818	0.331566
$813$	1.51744	0.0532191
$814$	−61.5062	−2.15579
$815$	0	0
$816$	18.2590	0.639192
$817$	−24.0252	−0.840534
$818$	38.7390	1.35448
$819$	−1.94520	−0.0679708
$820$	0	0
$821$	−0.565159	−0.0197242	−0.00986209	−	0.999951i	$-0.503139\pi$
$821$	−0.565159	−0.0197242	−0.00986209	+	0.999951i	$0.503139\pi$
$822$	26.3429	0.918816
$823$	−9.37487	−0.326787	−0.163394	−	0.986561i	$-0.552244\pi$
$823$	−9.37487	−0.326787	−0.163394	+	0.986561i	$0.552244\pi$
$824$	8.48806	0.295696
$825$	0	0
$826$	2.28949	0.0796617
$827$	−34.7921	−1.20984	−0.604919	−	0.796287i	$-0.706795\pi$
$827$	−34.7921	−1.20984	−0.604919	+	0.796287i	$0.706795\pi$
$828$	6.20893	0.215775
$829$	−18.2182	−0.632745	−0.316372	−	0.948635i	$-0.602465\pi$
$829$	−18.2182	−0.632745	−0.316372	+	0.948635i	$0.602465\pi$
$830$	0	0
$831$	−0.0598152	−0.00207497
$832$	20.7696	0.720057
$833$	−39.7931	−1.37875
$834$	−4.88274	−0.169075
$835$	0	0
$836$	35.5519	1.22959
$837$	5.62133	0.194302
$838$	19.8196	0.684658
$839$	−30.1691	−1.04155	−0.520776	−	0.853693i	$-0.674357\pi$
$839$	−30.1691	−1.04155	−0.520776	+	0.853693i	$0.674357\pi$
$840$	0	0
$841$	−16.9000	−0.582758
$842$	−17.5075	−0.603347
$843$	30.5027	1.05057
$844$	−63.5739	−2.18830
$845$	0	0
$846$	18.2551	0.627624
$847$	26.5222	0.911313
$848$	−36.8856	−1.26666
$849$	−22.6416	−0.777056
$850$	0	0
$851$	−11.9216	−0.408666
$852$	−29.3515	−1.00557
$853$	54.1964	1.85565	0.927824	−	0.373017i	$-0.121677\pi$
$853$	54.1964	1.85565	0.927824	+	0.373017i	$0.121677\pi$
$854$	−30.1957	−1.03328
$855$	0	0
$856$	−6.82107	−0.233140
$857$	−4.80102	−0.164000	−0.0819999	−	0.996632i	$-0.526131\pi$
$857$	−4.80102	−0.164000	−0.0819999	+	0.996632i	$0.526131\pi$
$858$	22.9409	0.783191
$859$	−8.87947	−0.302963	−0.151482	−	0.988460i	$-0.548404\pi$
$859$	−8.87947	−0.302963	−0.151482	+	0.988460i	$0.548404\pi$
$860$	0	0
$861$	−4.86920	−0.165942
$862$	2.04547	0.0696691
$863$	13.7913	0.469462	0.234731	−	0.972060i	$-0.424579\pi$
$863$	13.7913	0.469462	0.234731	+	0.972060i	$0.424579\pi$
$864$	−7.92509	−0.269617
$865$	0	0
$866$	−4.43553	−0.150725
$867$	−29.4114	−0.998862
$868$	−15.2684	−0.518244
$869$	95.1381	3.22734
$870$	0	0
$871$	11.2710	0.381905
$872$	−15.0683	−0.510278
$873$	−3.09939	−0.104898
$874$	12.3532	0.417853
$875$	0	0
$876$	37.9093	1.28084
$877$	27.9166	0.942676	0.471338	−	0.881953i	$-0.343771\pi$
$877$	27.9166	0.942676	0.471338	+	0.881953i	$0.343771\pi$
$878$	−17.4811	−0.589958
$879$	13.4493	0.453635
$880$	0	0
$881$	−58.9152	−1.98490	−0.992452	−	0.122634i	$-0.960866\pi$
$881$	−58.9152	−1.98490	−0.992452	+	0.122634i	$0.960866\pi$
$882$	12.4226	0.418291
$883$	6.49773	0.218666	0.109333	−	0.994005i	$-0.465129\pi$
$883$	6.49773	0.218666	0.109333	+	0.994005i	$0.465129\pi$
$884$	−31.0591	−1.04463
$885$	0	0
$886$	17.2520	0.579591
$887$	−7.62029	−0.255864	−0.127932	−	0.991783i	$-0.540834\pi$
$887$	−7.62029	−0.255864	−0.127932	+	0.991783i	$0.540834\pi$
$888$	−5.38952	−0.180860
$889$	9.90119	0.332075
$890$	0	0
$891$	−5.96967	−0.199992
$892$	−32.2782	−1.08076
$893$	20.2603	0.677984
$894$	25.8282	0.863826
$895$	0	0
$896$	9.25329	0.309131
$897$	4.44658	0.148467
$898$	23.2510	0.775896
$899$	−19.5539	−0.652158
$900$	0	0
$901$	93.7572	3.12351
$902$	57.4254	1.91206
$903$	−10.9574	−0.364641
$904$	7.18696	0.239035
$905$	0	0
$906$	43.0725	1.43099
$907$	9.02743	0.299751	0.149875	−	0.988705i	$-0.452113\pi$
$907$	9.02743	0.299751	0.149875	+	0.988705i	$0.452113\pi$
$908$	−50.0798	−1.66196
$909$	−3.18446	−0.105622
$910$	0	0
$911$	42.5127	1.40851	0.704255	−	0.709947i	$-0.251281\pi$
$911$	42.5127	1.40851	0.704255	+	0.709947i	$0.251281\pi$
$912$	−6.32619	−0.209481
$913$	−33.1655	−1.09762
$914$	25.5947	0.846598
$915$	0	0
$916$	−29.3131	−0.968533
$917$	−20.2402	−0.668390
$918$	14.4887	0.478199
$919$	32.1017	1.05894	0.529469	−	0.848330i	$-0.322391\pi$
$919$	32.1017	1.05894	0.529469	+	0.848330i	$0.322391\pi$
$920$	0	0
$921$	−25.4116	−0.837340
$922$	23.7006	0.780539
$923$	−21.0204	−0.691894
$924$	16.2146	0.533420
$925$	0	0
$926$	64.4680	2.11855
$927$	−7.62974	−0.250593
$928$	27.5675	0.904947
$929$	23.6128	0.774712	0.387356	−	0.921930i	$-0.373388\pi$
$929$	23.6128	0.774712	0.387356	+	0.921930i	$0.373388\pi$
$930$	0	0
$931$	13.7871	0.451854
$932$	34.1053	1.11716
$933$	26.2825	0.860452
$934$	−65.9503	−2.15796
$935$	0	0
$936$	2.01021	0.0657058
$937$	23.8658	0.779661	0.389831	−	0.920887i	$-0.372534\pi$
$937$	23.8658	0.779661	0.389831	+	0.920887i	$0.372534\pi$
$938$	14.2811	0.466293
$939$	20.0369	0.653880
$940$	0	0
$941$	−36.2577	−1.18197	−0.590984	−	0.806683i	$-0.701261\pi$
$941$	−36.2577	−1.18197	−0.590984	+	0.806683i	$0.701261\pi$
$942$	−3.96672	−0.129243
$943$	11.1306	0.362462
$944$	2.68018	0.0872324
$945$	0	0
$946$	129.228	4.20156
$947$	−60.0497	−1.95135	−0.975676	−	0.219219i	$-0.929649\pi$
$947$	−60.0497	−1.95135	−0.975676	+	0.219219i	$0.929649\pi$
$948$	40.2103	1.30597
$949$	27.1491	0.881297
$950$	0	0
$951$	25.5503	0.828526
$952$	−8.15892	−0.264432
$953$	−19.8885	−0.644251	−0.322125	−	0.946697i	$-0.604397\pi$
$953$	−19.8885	−0.644251	−0.322125	+	0.946697i	$0.604397\pi$
$954$	−29.2692	−0.947625
$955$	0	0
$956$	−28.1966	−0.911943
$957$	20.7656	0.671255
$958$	28.4339	0.918659
$959$	13.3342	0.430585
$960$	0	0
$961$	0.599385	0.0193350
$962$	−18.6171	−0.600239
$963$	6.13132	0.197579
$964$	4.56183	0.146927
$965$	0	0
$966$	5.63407	0.181273
$967$	−33.9400	−1.09144	−0.545718	−	0.837969i	$-0.683743\pi$
$967$	−33.9400	−1.09144	−0.545718	+	0.837969i	$0.683743\pi$
$968$	−27.4086	−0.880945
$969$	16.0802	0.516569
$970$	0	0
$971$	19.2121	0.616547	0.308273	−	0.951298i	$-0.400249\pi$
$971$	19.2121	0.616547	0.308273	+	0.951298i	$0.400249\pi$
$972$	−2.52310	−0.0809283
$973$	−2.47154	−0.0792338
$974$	82.8135	2.65352
$975$	0	0
$976$	−35.3484	−1.13147
$977$	19.5916	0.626790	0.313395	−	0.949623i	$-0.398534\pi$
$977$	19.5916	0.626790	0.313395	+	0.949623i	$0.398534\pi$
$978$	5.85422	0.187197
$979$	106.802	3.41340
$980$	0	0
$981$	13.5446	0.432446
$982$	−65.2909	−2.08352
$983$	42.5569	1.35735	0.678677	−	0.734437i	$-0.262554\pi$
$983$	42.5569	1.35735	0.678677	+	0.734437i	$0.262554\pi$
$984$	5.03193	0.160412
$985$	0	0
$986$	−50.3991	−1.60504
$987$	9.24035	0.294123
$988$	10.7611	0.342355
$989$	25.0479	0.796476
$990$	0	0
$991$	−13.2476	−0.420824	−0.210412	−	0.977613i	$-0.567481\pi$
$991$	−13.2476	−0.420824	−0.210412	+	0.977613i	$0.567481\pi$
$992$	−44.5495	−1.41445
$993$	22.7085	0.720632
$994$	−26.6340	−0.844780
$995$	0	0
$996$	−14.0175	−0.444160
$997$	−28.2375	−0.894290	−0.447145	−	0.894461i	$-0.647559\pi$
$997$	−28.2375	−0.894290	−0.447145	+	0.894461i	$0.647559\pi$
$998$	49.6386	1.57128
$999$	4.84452	0.153274

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4425.2.a.ba.1.1	✓	4
5.4	even	2		4425.2.a.bb.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4425.2.a.ba.1.1	✓	4	1.1	even	1	trivial
4425.2.a.bb.1.4	yes	4	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4425 = 3 \cdot 5^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4425.a (trivial)

\(f(q)\)	\(=\)	\(q-2.12676 q^{2} -1.00000 q^{3} +2.52310 q^{4} +2.12676 q^{6} +1.07652 q^{7} -1.11250 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.12676 q^{2} -1.00000 q^{3} +2.52310 q^{4} +2.12676 q^{6} +1.07652 q^{7} -1.11250 q^{8} +1.00000 q^{9} -5.96967 q^{11} -2.52310 q^{12} -1.80694 q^{13} -2.28949 q^{14} -2.68018 q^{16} +6.81259 q^{17} -2.12676 q^{18} -2.36036 q^{19} -1.07652 q^{21} +12.6960 q^{22} +2.46084 q^{23} +1.11250 q^{24} +3.84292 q^{26} -1.00000 q^{27} +2.71616 q^{28} +3.47851 q^{29} -5.62133 q^{31} +7.92509 q^{32} +5.96967 q^{33} -14.4887 q^{34} +2.52310 q^{36} -4.84452 q^{37} +5.01991 q^{38} +1.80694 q^{39} +4.52310 q^{41} +2.28949 q^{42} +10.1786 q^{43} -15.0621 q^{44} -5.23360 q^{46} -8.58355 q^{47} +2.68018 q^{48} -5.84111 q^{49} -6.81259 q^{51} -4.55907 q^{52} +13.7623 q^{53} +2.12676 q^{54} -1.19762 q^{56} +2.36036 q^{57} -7.39794 q^{58} -1.00000 q^{59} +13.1888 q^{61} +11.9552 q^{62} +1.07652 q^{63} -11.4944 q^{64} -12.6960 q^{66} -6.23765 q^{67} +17.1888 q^{68} -2.46084 q^{69} +11.6331 q^{71} -1.11250 q^{72} -15.0249 q^{73} +10.3031 q^{74} -5.95541 q^{76} -6.42646 q^{77} -3.84292 q^{78} -15.9369 q^{79} +1.00000 q^{81} -9.61953 q^{82} +5.55566 q^{83} -2.71616 q^{84} -21.6474 q^{86} -3.47851 q^{87} +6.64124 q^{88} -17.8907 q^{89} -1.94520 q^{91} +6.20893 q^{92} +5.62133 q^{93} +18.2551 q^{94} -7.92509 q^{96} -3.09939 q^{97} +12.4226 q^{98} -5.96967 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 4 q^{3} + q^{4} + q^{6} + q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - q^2 - 4 * q^3 + q^4 + q^6 + q^7 - 3 * q^8 + 4 * q^9	\( 4 q - q^{2} - 4 q^{3} + q^{4} + q^{6} + q^{7} - 3 q^{8} + 4 q^{9} - 9 q^{11} - q^{12} + 2 q^{13} + 4 q^{14} - 9 q^{16} + 5 q^{17} - q^{18} - 6 q^{19} - q^{21} + 16 q^{22} - 2 q^{23} + 3 q^{24} + 8 q^{26} - 4 q^{27} + 11 q^{28} - 4 q^{29} - 18 q^{31} + 8 q^{32} + 9 q^{33} - 20 q^{34} + q^{36} + 15 q^{37} + q^{38} - 2 q^{39} + 9 q^{41} - 4 q^{42} + 2 q^{43} - 9 q^{44} - 25 q^{46} - q^{47} + 9 q^{48} + 11 q^{49} - 5 q^{51} - 11 q^{52} + 37 q^{53} + q^{54} - 7 q^{56} + 6 q^{57} - q^{58} - 4 q^{59} - 6 q^{61} - 15 q^{62} + q^{63} + 5 q^{64} - 16 q^{66} - 43 q^{67} + 10 q^{68} + 2 q^{69} - 5 q^{71} - 3 q^{72} - 14 q^{73} + 8 q^{74} - 15 q^{76} + 15 q^{77} - 8 q^{78} + 7 q^{79} + 4 q^{81} - 7 q^{82} + 15 q^{83} - 11 q^{84} - 29 q^{86} + 4 q^{87} + 3 q^{88} - 19 q^{89} - 19 q^{91} + q^{92} + 18 q^{93} + 31 q^{94} - 8 q^{96} - 10 q^{97} + 49 q^{98} - 9 q^{99}+O(q^{100}) \) 4 * q - q^2 - 4 * q^3 + q^4 + q^6 + q^7 - 3 * q^8 + 4 * q^9 - 9 * q^11 - q^12 + 2 * q^13 + 4 * q^14 - 9 * q^16 + 5 * q^17 - q^18 - 6 * q^19 - q^21 + 16 * q^22 - 2 * q^23 + 3 * q^24 + 8 * q^26 - 4 * q^27 + 11 * q^28 - 4 * q^29 - 18 * q^31 + 8 * q^32 + 9 * q^33 - 20 * q^34 + q^36 + 15 * q^37 + q^38 - 2 * q^39 + 9 * q^41 - 4 * q^42 + 2 * q^43 - 9 * q^44 - 25 * q^46 - q^47 + 9 * q^48 + 11 * q^49 - 5 * q^51 - 11 * q^52 + 37 * q^53 + q^54 - 7 * q^56 + 6 * q^57 - q^58 - 4 * q^59 - 6 * q^61 - 15 * q^62 + q^63 + 5 * q^64 - 16 * q^66 - 43 * q^67 + 10 * q^68 + 2 * q^69 - 5 * q^71 - 3 * q^72 - 14 * q^73 + 8 * q^74 - 15 * q^76 + 15 * q^77 - 8 * q^78 + 7 * q^79 + 4 * q^81 - 7 * q^82 + 15 * q^83 - 11 * q^84 - 29 * q^86 + 4 * q^87 + 3 * q^88 - 19 * q^89 - 19 * q^91 + q^92 + 18 * q^93 + 31 * q^94 - 8 * q^96 - 10 * q^97 + 49 * q^98 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more