LMFDB - Embedded newform 1850.2.a.bc.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1850,2,Mod(1,1850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1850, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1850.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1850 = 2 \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1850.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:	$14.7723243739$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 1 $ x^3 - x^2 - 7*x + 1
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			1.2
Root			$0.140435$ of defining polynomial
Character	$\chi$	$=$	1850.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +0.140435 q^{3} +1.00000 q^{4} +0.140435 q^{6} +1.14044 q^{7} +1.00000 q^{8} -2.98028 q^{9} +3.12071 q^{11} +0.140435 q^{12} +2.85956 q^{13} +1.14044 q^{14} +1.00000 q^{16} -2.14044 q^{17} -2.98028 q^{18} +4.26115 q^{19} +0.160157 q^{21} +3.12071 q^{22} +1.71913 q^{23} +0.140435 q^{24} +2.85956 q^{26} -0.839843 q^{27} +1.14044 q^{28} -4.28087 q^{29} +6.40158 q^{31} +1.00000 q^{32} +0.438259 q^{33} -2.14044 q^{34} -2.98028 q^{36} +1.00000 q^{37} +4.26115 q^{38} +0.401584 q^{39} -3.24143 q^{41} +0.160157 q^{42} +10.6994 q^{43} +3.12071 q^{44} +1.71913 q^{46} +0.140435 q^{48} -5.69941 q^{49} -0.300593 q^{51} +2.85956 q^{52} +7.96056 q^{53} -0.839843 q^{54} +1.14044 q^{56} +0.598416 q^{57} -4.28087 q^{58} +2.69941 q^{59} +4.57869 q^{61} +6.40158 q^{62} -3.39881 q^{63} +1.00000 q^{64} +0.438259 q^{66} +10.3819 q^{67} -2.14044 q^{68} +0.241427 q^{69} -7.10099 q^{71} -2.98028 q^{72} -0.261149 q^{73} +1.00000 q^{74} +4.26115 q^{76} +3.55897 q^{77} +0.401584 q^{78} -8.52230 q^{79} +8.82289 q^{81} -3.24143 q^{82} -1.16016 q^{83} +0.160157 q^{84} +10.6994 q^{86} -0.601186 q^{87} +3.12071 q^{88} -13.6430 q^{89} +3.26115 q^{91} +1.71913 q^{92} +0.899009 q^{93} +0.140435 q^{96} +14.2414 q^{97} -5.69941 q^{98} -9.30059 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9} - 5 q^{11} + q^{12} + 8 q^{13} + 4 q^{14} + 3 q^{16} - 7 q^{17} + 6 q^{18} - q^{19} + 16 q^{21} - 5 q^{22} + 4 q^{23} + q^{24}+ \cdots - 44 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 4 * q^7 + 3 * q^8 + 6 * q^9 - 5 * q^11 + q^12 + 8 * q^13 + 4 * q^14 + 3 * q^16 - 7 * q^17 + 6 * q^18 - q^19 + 16 * q^21 - 5 * q^22 + 4 * q^23 + q^24 + 8 * q^26 + 13 * q^27 + 4 * q^28 - 14 * q^29 + 6 * q^31 + 3 * q^32 - q^33 - 7 * q^34 + 6 * q^36 + 3 * q^37 - q^38 - 12 * q^39 + 19 * q^41 + 16 * q^42 + 16 * q^43 - 5 * q^44 + 4 * q^46 + q^48 - q^49 - 17 * q^51 + 8 * q^52 - 6 * q^53 + 13 * q^54 + 4 * q^56 + 15 * q^57 - 14 * q^58 - 8 * q^59 + 12 * q^61 + 6 * q^62 + 22 * q^63 + 3 * q^64 - q^66 + 3 * q^67 - 7 * q^68 - 28 * q^69 + 8 * q^71 + 6 * q^72 + 13 * q^73 + 3 * q^74 - q^76 - 6 * q^77 - 12 * q^78 + 2 * q^79 + 15 * q^81 + 19 * q^82 - 19 * q^83 + 16 * q^84 + 16 * q^86 - 34 * q^87 - 5 * q^88 + q^89 - 4 * q^91 + 4 * q^92 + 32 * q^93 + q^96 + 14 * q^97 - q^98 - 44 * 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
$3$	0.140435	0.0810804	0.0405402	−	0.999178i	$-0.487092\pi$
$3$	0.140435	0.0810804	0.0405402	+	0.999178i	$0.487092\pi$
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.140435	0.0573325
$7$	1.14044	0.431044	0.215522	−	0.976499i	$-0.430855\pi$
$7$	1.14044	0.431044	0.215522	+	0.976499i	$0.430855\pi$
$8$	1.00000	0.353553
$9$	−2.98028	−0.993426
$10$	0	0
$11$	3.12071	0.940930	0.470465	−	0.882419i	$-0.344086\pi$
$11$	3.12071	0.940930	0.470465	+	0.882419i	$0.344086\pi$
$12$	0.140435	0.0405402
$13$	2.85956	0.793101	0.396550	−	0.918013i	$-0.370207\pi$
$13$	2.85956	0.793101	0.396550	+	0.918013i	$0.370207\pi$
$14$	1.14044	0.304794
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.14044	−0.519132	−0.259566	−	0.965725i	$-0.583579\pi$
$17$	−2.14044	−0.519132	−0.259566	+	0.965725i	$0.583579\pi$
$18$	−2.98028	−0.702458
$19$	4.26115	0.977575	0.488787	−	0.872403i	$-0.337439\pi$
$19$	4.26115	0.977575	0.488787	+	0.872403i	$0.337439\pi$
$20$	0	0
$21$	0.160157	0.0349492
$22$	3.12071	0.665338
$23$	1.71913	0.358463	0.179232	−	0.983807i	$-0.442639\pi$
$23$	1.71913	0.358463	0.179232	+	0.983807i	$0.442639\pi$
$24$	0.140435	0.0286662
$25$	0	0
$26$	2.85956	0.560807
$27$	−0.839843	−0.161628
$28$	1.14044	0.215522
$29$	−4.28087	−0.794938	−0.397469	−	0.917616i	$-0.630111\pi$
$29$	−4.28087	−0.794938	−0.397469	+	0.917616i	$0.630111\pi$
$30$	0	0
$31$	6.40158	1.14976	0.574879	−	0.818238i	$-0.305049\pi$
$31$	6.40158	1.14976	0.574879	+	0.818238i	$0.305049\pi$
$32$	1.00000	0.176777
$33$	0.438259	0.0762910
$34$	−2.14044	−0.367082
$35$	0	0
$36$	−2.98028	−0.496713
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	4.26115	0.691250
$39$	0.401584	0.0643049
$40$	0	0
$41$	−3.24143	−0.506226	−0.253113	−	0.967437i	$-0.581454\pi$
$41$	−3.24143	−0.506226	−0.253113	+	0.967437i	$0.581454\pi$
$42$	0.160157	0.0247128
$43$	10.6994	1.63164	0.815822	−	0.578303i	$-0.196285\pi$
$43$	10.6994	1.63164	0.815822	+	0.578303i	$0.196285\pi$
$44$	3.12071	0.470465
$45$	0	0
$46$	1.71913	0.253472
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0.140435	0.0202701
$49$	−5.69941	−0.814201
$50$	0	0
$51$	−0.300593	−0.0420914
$52$	2.85956	0.396550
$53$	7.96056	1.09347	0.546733	−	0.837307i	$-0.315871\pi$
$53$	7.96056	1.09347	0.546733	+	0.837307i	$0.315871\pi$
$54$	−0.839843	−0.114288
$55$	0	0
$56$	1.14044	0.152397
$57$	0.598416	0.0792621
$58$	−4.28087	−0.562106
$59$	2.69941	0.351433	0.175716	−	0.984441i	$-0.443776\pi$
$59$	2.69941	0.351433	0.175716	+	0.984441i	$0.443776\pi$
$60$	0	0
$61$	4.57869	0.586242	0.293121	−	0.956075i	$-0.405306\pi$
$61$	4.57869	0.586242	0.293121	+	0.956075i	$0.405306\pi$
$62$	6.40158	0.813002
$63$	−3.39881	−0.428210
$64$	1.00000	0.125000
$65$	0	0
$66$	0.438259	0.0539459
$67$	10.3819	1.26835	0.634173	−	0.773191i	$-0.281341\pi$
$67$	10.3819	1.26835	0.634173	+	0.773191i	$0.281341\pi$
$68$	−2.14044	−0.259566
$69$	0.241427	0.0290643
$70$	0	0
$71$	−7.10099	−0.842733	−0.421366	−	0.906891i	$-0.638449\pi$
$71$	−7.10099	−0.842733	−0.421366	+	0.906891i	$0.638449\pi$
$72$	−2.98028	−0.351229
$73$	−0.261149	−0.0305651	−0.0152826	−	0.999883i	$-0.504865\pi$
$73$	−0.261149	−0.0305651	−0.0152826	+	0.999883i	$0.504865\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	4.26115	0.488787
$77$	3.55897	0.405582
$78$	0.401584	0.0454704
$79$	−8.52230	−0.958833	−0.479417	−	0.877587i	$-0.659152\pi$
$79$	−8.52230	−0.958833	−0.479417	+	0.877587i	$0.659152\pi$
$80$	0	0
$81$	8.82289	0.980321
$82$	−3.24143	−0.357956
$83$	−1.16016	−0.127344	−0.0636719	−	0.997971i	$-0.520281\pi$
$83$	−1.16016	−0.127344	−0.0636719	+	0.997971i	$0.520281\pi$
$84$	0.160157	0.0174746
$85$	0	0
$86$	10.6994	1.15375
$87$	−0.601186	−0.0644539
$88$	3.12071	0.332669
$89$	−13.6430	−1.44616	−0.723078	−	0.690766i	$-0.757273\pi$
$89$	−13.6430	−1.44616	−0.723078	+	0.690766i	$0.757273\pi$
$90$	0	0
$91$	3.26115	0.341861
$92$	1.71913	0.179232
$93$	0.899009	0.0932229
$94$	0	0
$95$	0	0
$96$	0.140435	0.0143331
$97$	14.2414	1.44600	0.722999	−	0.690849i	$-0.242763\pi$
$97$	14.2414	1.44600	0.722999	+	0.690849i	$0.242763\pi$
$98$	−5.69941	−0.575727
$99$	−9.30059	−0.934745
$100$	0	0
$101$	13.9606	1.38913	0.694564	−	0.719431i	$-0.255597\pi$
$101$	13.9606	1.38913	0.694564	+	0.719431i	$0.255597\pi$
$102$	−0.300593	−0.0297631
$103$	−10.3621	−1.02101	−0.510506	−	0.859874i	$-0.670542\pi$
$103$	−10.3621	−1.02101	−0.510506	+	0.859874i	$0.670542\pi$
$104$	2.85956	0.280403
$105$	0	0
$106$	7.96056	0.773198
$107$	4.70218	0.454577	0.227288	−	0.973828i	$-0.427014\pi$
$107$	4.70218	0.454577	0.227288	+	0.973828i	$0.427014\pi$
$108$	−0.839843	−0.0808139
$109$	−5.96056	−0.570918	−0.285459	−	0.958391i	$-0.592146\pi$
$109$	−5.96056	−0.570918	−0.285459	+	0.958391i	$0.592146\pi$
$110$	0	0
$111$	0.140435	0.0133295
$112$	1.14044	0.107761
$113$	−4.83984	−0.455294	−0.227647	−	0.973744i	$-0.573103\pi$
$113$	−4.83984	−0.455294	−0.227647	+	0.973744i	$0.573103\pi$
$114$	0.598416	0.0560468
$115$	0	0
$116$	−4.28087	−0.397469
$117$	−8.52230	−0.787887
$118$	2.69941	0.248501
$119$	−2.44103	−0.223769
$120$	0	0
$121$	−1.26115	−0.114650
$122$	4.57869	0.414535
$123$	−0.455211	−0.0410450
$124$	6.40158	0.574879
$125$	0	0
$126$	−3.39881	−0.302790
$127$	0.899009	0.0797741	0.0398871	−	0.999204i	$-0.487300\pi$
$127$	0.899009	0.0797741	0.0398871	+	0.999204i	$0.487300\pi$
$128$	1.00000	0.0883883
$129$	1.50258	0.132294
$130$	0	0
$131$	−0.980278	−0.0856473	−0.0428236	−	0.999083i	$-0.513635\pi$
$131$	−0.980278	−0.0856473	−0.0428236	+	0.999083i	$0.513635\pi$
$132$	0.438259	0.0381455
$133$	4.85956	0.421378
$134$	10.3819	0.896856
$135$	0	0
$136$	−2.14044	−0.183541
$137$	−6.67969	−0.570684	−0.285342	−	0.958426i	$-0.592107\pi$
$137$	−6.67969	−0.570684	−0.285342	+	0.958426i	$0.592107\pi$
$138$	0.241427	0.0205516
$139$	−9.68245	−0.821255	−0.410628	−	0.911803i	$-0.634690\pi$
$139$	−9.68245	−0.821255	−0.410628	+	0.911803i	$0.634690\pi$
$140$	0	0
$141$	0	0
$142$	−7.10099	−0.595902
$143$	8.92388	0.746252
$144$	−2.98028	−0.248356
$145$	0	0
$146$	−0.261149	−0.0216128
$147$	−0.800398	−0.0660157
$148$	1.00000	0.0821995
$149$	−1.59842	−0.130947	−0.0654737	−	0.997854i	$-0.520856\pi$
$149$	−1.59842	−0.130947	−0.0654737	+	0.997854i	$0.520856\pi$
$150$	0	0
$151$	−16.4829	−1.34136	−0.670678	−	0.741749i	$-0.733997\pi$
$151$	−16.4829	−1.34136	−0.670678	+	0.741749i	$0.733997\pi$
$152$	4.26115	0.345625
$153$	6.37909	0.515719
$154$	3.55897	0.286790
$155$	0	0
$156$	0.401584	0.0321525
$157$	−10.1207	−0.807721	−0.403860	−	0.914821i	$-0.632332\pi$
$157$	−10.1207	−0.807721	−0.403860	+	0.914821i	$0.632332\pi$
$158$	−8.52230	−0.677998
$159$	1.11794	0.0886587
$160$	0	0
$161$	1.96056	0.154513
$162$	8.82289	0.693192
$163$	11.7389	0.919458	0.459729	−	0.888059i	$-0.347947\pi$
$163$	11.7389	0.919458	0.459729	+	0.888059i	$0.347947\pi$
$164$	−3.24143	−0.253113
$165$	0	0
$166$	−1.16016	−0.0900457
$167$	10.6430	0.823581	0.411790	−	0.911279i	$-0.364904\pi$
$167$	10.6430	0.823581	0.411790	+	0.911279i	$0.364904\pi$
$168$	0.160157	0.0123564
$169$	−4.82289	−0.370992
$170$	0	0
$171$	−12.6994	−0.971148
$172$	10.6994	0.815822
$173$	−11.3988	−0.866636	−0.433318	−	0.901241i	$-0.642657\pi$
$173$	−11.3988	−0.866636	−0.433318	+	0.901241i	$0.642657\pi$
$174$	−0.601186	−0.0455758
$175$	0	0
$176$	3.12071	0.235233
$177$	0.379092	0.0284943
$178$	−13.6430	−1.02259
$179$	−18.9211	−1.41423	−0.707115	−	0.707098i	$-0.750004\pi$
$179$	−18.9211	−1.41423	−0.707115	+	0.707098i	$0.750004\pi$
$180$	0	0
$181$	−18.8032	−1.39763	−0.698814	−	0.715303i	$-0.746289\pi$
$181$	−18.8032	−1.39763	−0.698814	+	0.715303i	$0.746289\pi$
$182$	3.26115	0.241732
$183$	0.643011	0.0475327
$184$	1.71913	0.126736
$185$	0	0
$186$	0.899009	0.0659185
$187$	−6.67969	−0.488467
$188$	0	0
$189$	−0.957786	−0.0696687
$190$	0	0
$191$	15.0446	1.08859	0.544294	−	0.838894i	$-0.316797\pi$
$191$	15.0446	1.08859	0.544294	+	0.838894i	$0.316797\pi$
$192$	0.140435	0.0101350
$193$	25.3227	1.82277	0.911384	−	0.411558i	$-0.135015\pi$
$193$	25.3227	1.82277	0.911384	+	0.411558i	$0.135015\pi$
$194$	14.2414	1.02247
$195$	0	0
$196$	−5.69941	−0.407101
$197$	−2.68245	−0.191117	−0.0955585	−	0.995424i	$-0.530464\pi$
$197$	−2.68245	−0.191117	−0.0955585	+	0.995424i	$0.530464\pi$
$198$	−9.30059	−0.660964
$199$	2.24143	0.158891	0.0794453	−	0.996839i	$-0.474685\pi$
$199$	2.24143	0.158891	0.0794453	+	0.996839i	$0.474685\pi$
$200$	0	0
$201$	1.45798	0.102838
$202$	13.9606	0.982261
$203$	−4.88206	−0.342653
$204$	−0.300593	−0.0210457
$205$	0	0
$206$	−10.3621	−0.721964
$207$	−5.12348	−0.356107
$208$	2.85956	0.198275
$209$	13.2978	0.919830
$210$	0	0
$211$	−1.61814	−0.111397	−0.0556986	−	0.998448i	$-0.517739\pi$
$211$	−1.61814	−0.111397	−0.0556986	+	0.998448i	$0.517739\pi$
$212$	7.96056	0.546733
$213$	−0.997230	−0.0683291
$214$	4.70218	0.321434
$215$	0	0
$216$	−0.839843	−0.0571440
$217$	7.30059	0.495597
$218$	−5.96056	−0.403700
$219$	−0.0366745	−0.00247823
$220$	0	0
$221$	−6.12071	−0.411724
$222$	0.140435	0.00942540
$223$	−3.39881	−0.227601	−0.113801	−	0.993504i	$-0.536303\pi$
$223$	−3.39881	−0.227601	−0.113801	+	0.993504i	$0.536303\pi$
$224$	1.14044	0.0761985
$225$	0	0
$226$	−4.83984	−0.321942
$227$	2.45798	0.163142	0.0815710	−	0.996668i	$-0.474006\pi$
$227$	2.45798	0.163142	0.0815710	+	0.996668i	$0.474006\pi$
$228$	0.598416	0.0396311
$229$	12.7637	0.843451	0.421725	−	0.906724i	$-0.361425\pi$
$229$	12.7637	0.843451	0.421725	+	0.906724i	$0.361425\pi$
$230$	0	0
$231$	0.499806	0.0328848
$232$	−4.28087	−0.281053
$233$	−0.103761	−0.00679760	−0.00339880	−	0.999994i	$-0.501082\pi$
$233$	−0.103761	−0.00679760	−0.00339880	+	0.999994i	$0.501082\pi$
$234$	−8.52230	−0.557120
$235$	0	0
$236$	2.69941	0.175716
$237$	−1.19683	−0.0777426
$238$	−2.44103	−0.158228
$239$	−3.67969	−0.238019	−0.119010	−	0.992893i	$-0.537972\pi$
$239$	−3.67969	−0.238019	−0.119010	+	0.992893i	$0.537972\pi$
$240$	0	0
$241$	−6.14044	−0.395540	−0.197770	−	0.980248i	$-0.563370\pi$
$241$	−6.14044	−0.395540	−0.197770	+	0.980248i	$0.563370\pi$
$242$	−1.26115	−0.0810697
$243$	3.75857	0.241113
$244$	4.57869	0.293121
$245$	0	0
$246$	−0.455211	−0.0290232
$247$	12.1850	0.775315
$248$	6.40158	0.406501
$249$	−0.162927	−0.0103251
$250$	0	0
$251$	−9.00000	−0.568075	−0.284037	−	0.958813i	$-0.591674\pi$
$251$	−9.00000	−0.568075	−0.284037	+	0.958813i	$0.591674\pi$
$252$	−3.39881	−0.214105
$253$	5.36491	0.337289
$254$	0.899009	0.0564088
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.5223	−1.03063	−0.515316	−	0.857000i	$-0.672326\pi$
$257$	−16.5223	−1.03063	−0.515316	+	0.857000i	$0.672326\pi$
$258$	1.50258	0.0935462
$259$	1.14044	0.0708632
$260$	0	0
$261$	12.7582	0.789712
$262$	−0.980278	−0.0605618
$263$	−22.5223	−1.38878	−0.694392	−	0.719597i	$-0.744327\pi$
$263$	−22.5223	−1.38878	−0.694392	+	0.719597i	$0.744327\pi$
$264$	0.438259	0.0269729
$265$	0	0
$266$	4.85956	0.297959
$267$	−1.91596	−0.117255
$268$	10.3819	0.634173
$269$	27.2860	1.66366	0.831829	−	0.555032i	$-0.187294\pi$
$269$	27.2860	1.66366	0.831829	+	0.555032i	$0.187294\pi$
$270$	0	0
$271$	−26.1456	−1.58823	−0.794116	−	0.607767i	$-0.792066\pi$
$271$	−26.1456	−1.58823	−0.794116	+	0.607767i	$0.792066\pi$
$272$	−2.14044	−0.129783
$273$	0.457981	0.0277182
$274$	−6.67969	−0.403535
$275$	0	0
$276$	0.241427	0.0145322
$277$	−20.5223	−1.23307	−0.616533	−	0.787329i	$-0.711463\pi$
$277$	−20.5223	−1.23307	−0.616533	+	0.787329i	$0.711463\pi$
$278$	−9.68245	−0.580715
$279$	−19.0785	−1.14220
$280$	0	0
$281$	−14.4580	−0.862491	−0.431245	−	0.902235i	$-0.641926\pi$
$281$	−14.4580	−0.862491	−0.431245	+	0.902235i	$0.641926\pi$
$282$	0	0
$283$	6.85679	0.407594	0.203797	−	0.979013i	$-0.434672\pi$
$283$	6.85679	0.407594	0.203797	+	0.979013i	$0.434672\pi$
$284$	−7.10099	−0.421366
$285$	0	0
$286$	8.92388	0.527680
$287$	−3.69664	−0.218206
$288$	−2.98028	−0.175615
$289$	−12.4185	−0.730502
$290$	0	0
$291$	2.00000	0.117242
$292$	−0.261149	−0.0152826
$293$	−19.2048	−1.12195	−0.560977	−	0.827832i	$-0.689574\pi$
$293$	−19.2048	−1.12195	−0.560977	+	0.827832i	$0.689574\pi$
$294$	−0.800398	−0.0466802
$295$	0	0
$296$	1.00000	0.0581238
$297$	−2.62091	−0.152080
$298$	−1.59842	−0.0925938
$299$	4.91596	0.284297
$300$	0	0
$301$	12.2020	0.703311
$302$	−16.4829	−0.948482
$303$	1.96056	0.112631
$304$	4.26115	0.244394
$305$	0	0
$306$	6.37909	0.364668
$307$	16.2781	0.929040	0.464520	−	0.885563i	$-0.346227\pi$
$307$	16.2781	0.929040	0.464520	+	0.885563i	$0.346227\pi$
$308$	3.55897	0.202791
$309$	−1.45521	−0.0827841
$310$	0	0
$311$	−25.4801	−1.44484	−0.722421	−	0.691453i	$-0.756971\pi$
$311$	−25.4801	−1.44484	−0.722421	+	0.691453i	$0.756971\pi$
$312$	0.401584	0.0227352
$313$	19.0197	1.07506	0.537529	−	0.843245i	$-0.319358\pi$
$313$	19.0197	1.07506	0.537529	+	0.843245i	$0.319358\pi$
$314$	−10.1207	−0.571145
$315$	0	0
$316$	−8.52230	−0.479417
$317$	15.1653	0.851769	0.425884	−	0.904778i	$-0.359963\pi$
$317$	15.1653	0.851769	0.425884	+	0.904778i	$0.359963\pi$
$318$	1.11794	0.0626912
$319$	−13.3594	−0.747981
$320$	0	0
$321$	0.660352	0.0368573
$322$	1.96056	0.109258
$323$	−9.12071	−0.507490
$324$	8.82289	0.490161
$325$	0	0
$326$	11.7389	0.650155
$327$	−0.837073	−0.0462902
$328$	−3.24143	−0.178978
$329$	0	0
$330$	0	0
$331$	−23.4185	−1.28720	−0.643600	−	0.765362i	$-0.722560\pi$
$331$	−23.4185	−1.28720	−0.643600	+	0.765362i	$0.722560\pi$
$332$	−1.16016	−0.0636719
$333$	−2.98028	−0.163318
$334$	10.6430	0.582360
$335$	0	0
$336$	0.160157	0.00873731
$337$	−28.9211	−1.57543	−0.787717	−	0.616038i	$-0.788737\pi$
$337$	−28.9211	−1.57543	−0.787717	+	0.616038i	$0.788737\pi$
$338$	−4.82289	−0.262331
$339$	−0.679685	−0.0369154
$340$	0	0
$341$	19.9775	1.08184
$342$	−12.6994	−0.686705
$343$	−14.4829	−0.782001
$344$	10.6994	0.576873
$345$	0	0
$346$	−11.3988	−0.612804
$347$	−32.8817	−1.76518	−0.882590	−	0.470143i	$-0.844202\pi$
$347$	−32.8817	−1.76518	−0.882590	+	0.470143i	$0.844202\pi$
$348$	−0.601186	−0.0322269
$349$	15.2048	0.813892	0.406946	−	0.913452i	$-0.366594\pi$
$349$	15.2048	0.813892	0.406946	+	0.913452i	$0.366594\pi$
$350$	0	0
$351$	−2.40158	−0.128187
$352$	3.12071	0.166335
$353$	−18.4829	−0.983743	−0.491872	−	0.870668i	$-0.663687\pi$
$353$	−18.4829	−0.983743	−0.491872	+	0.870668i	$0.663687\pi$
$354$	0.379092	0.0201485
$355$	0	0
$356$	−13.6430	−0.723078
$357$	−0.342807	−0.0181433
$358$	−18.9211	−1.00001
$359$	22.5223	1.18868	0.594341	−	0.804213i	$-0.297413\pi$
$359$	22.5223	1.18868	0.594341	+	0.804213i	$0.297413\pi$
$360$	0	0
$361$	−0.842612	−0.0443480
$362$	−18.8032	−0.988273
$363$	−0.177110	−0.00929586
$364$	3.26115	0.170931
$365$	0	0
$366$	0.643011	0.0336107
$367$	15.3424	0.800868	0.400434	−	0.916326i	$-0.368859\pi$
$367$	15.3424	0.800868	0.400434	+	0.916326i	$0.368859\pi$
$368$	1.71913	0.0896158
$369$	9.66035	0.502898
$370$	0	0
$371$	9.07850	0.471332
$372$	0.899009	0.0466114
$373$	12.6430	0.654630	0.327315	−	0.944915i	$-0.393856\pi$
$373$	12.6430	0.654630	0.327315	+	0.944915i	$0.393856\pi$
$374$	−6.67969	−0.345398
$375$	0	0
$376$	0	0
$377$	−12.2414	−0.630466
$378$	−0.957786	−0.0492632
$379$	15.7807	0.810599	0.405299	−	0.914184i	$-0.367167\pi$
$379$	15.7807	0.810599	0.405299	+	0.914184i	$0.367167\pi$
$380$	0	0
$381$	0.126253	0.00646812
$382$	15.0446	0.769748
$383$	−27.8004	−1.42053	−0.710267	−	0.703932i	$-0.751426\pi$
$383$	−27.8004	−1.42053	−0.710267	+	0.703932i	$0.751426\pi$
$384$	0.140435	0.00716656
$385$	0	0
$386$	25.3227	1.28889
$387$	−31.8872	−1.62092
$388$	14.2414	0.722999
$389$	9.66273	0.489920	0.244960	−	0.969533i	$-0.421225\pi$
$389$	9.66273	0.489920	0.244960	+	0.969533i	$0.421225\pi$
$390$	0	0
$391$	−3.67969	−0.186090
$392$	−5.69941	−0.287864
$393$	−0.137666	−0.00694432
$394$	−2.68245	−0.135140
$395$	0	0
$396$	−9.30059	−0.467372
$397$	−23.4801	−1.17843	−0.589216	−	0.807976i	$-0.700563\pi$
$397$	−23.4801	−1.17843	−0.589216	+	0.807976i	$0.700563\pi$
$398$	2.24143	0.112353
$399$	0.682455	0.0341655
$400$	0	0
$401$	20.2781	1.01264	0.506320	−	0.862346i	$-0.331005\pi$
$401$	20.2781	1.01264	0.506320	+	0.862346i	$0.331005\pi$
$402$	1.45798	0.0727175
$403$	18.3057	0.911874
$404$	13.9606	0.694564
$405$	0	0
$406$	−4.88206	−0.242292
$407$	3.12071	0.154688
$408$	−0.300593	−0.0148816
$409$	4.17434	0.206408	0.103204	−	0.994660i	$-0.467091\pi$
$409$	4.17434	0.206408	0.103204	+	0.994660i	$0.467091\pi$
$410$	0	0
$411$	−0.938064	−0.0462713
$412$	−10.3621	−0.510506
$413$	3.07850	0.151483
$414$	−5.12348	−0.251805
$415$	0	0
$416$	2.85956	0.140202
$417$	−1.35976	−0.0665877
$418$	13.2978	0.650418
$419$	−12.3175	−0.601751	−0.300876	−	0.953663i	$-0.597279\pi$
$419$	−12.3175	−0.601751	−0.300876	+	0.953663i	$0.597279\pi$
$420$	0	0
$421$	23.8872	1.16419	0.582096	−	0.813120i	$-0.302233\pi$
$421$	23.8872	1.16419	0.582096	+	0.813120i	$0.302233\pi$
$422$	−1.61814	−0.0787697
$423$	0	0
$424$	7.96056	0.386599
$425$	0	0
$426$	−0.997230	−0.0483160
$427$	5.22170	0.252696
$428$	4.70218	0.227288
$429$	1.25323	0.0605064
$430$	0	0
$431$	−2.80317	−0.135024	−0.0675119	−	0.997718i	$-0.521506\pi$
$431$	−2.80317	−0.135024	−0.0675119	+	0.997718i	$0.521506\pi$
$432$	−0.839843	−0.0404069
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	7.30059	0.350440
$435$	0	0
$436$	−5.96056	−0.285459
$437$	7.32547	0.350425
$438$	−0.0366745	−0.00175238
$439$	30.7976	1.46989	0.734945	−	0.678126i	$-0.237208\pi$
$439$	30.7976	1.46989	0.734945	+	0.678126i	$0.237208\pi$
$440$	0	0
$441$	16.9858	0.808848
$442$	−6.12071	−0.291133
$443$	−16.5984	−0.788615	−0.394307	−	0.918979i	$-0.629015\pi$
$443$	−16.5984	−0.788615	−0.394307	+	0.918979i	$0.629015\pi$
$444$	0.140435	0.00666477
$445$	0	0
$446$	−3.39881	−0.160939
$447$	−0.224474	−0.0106173
$448$	1.14044	0.0538805
$449$	9.75580	0.460405	0.230202	−	0.973143i	$-0.426061\pi$
$449$	9.75580	0.460405	0.230202	+	0.973143i	$0.426061\pi$
$450$	0	0
$451$	−10.1156	−0.476323
$452$	−4.83984	−0.227647
$453$	−2.31478	−0.108758
$454$	2.45798	0.115359
$455$	0	0
$456$	0.598416	0.0280234
$457$	−7.99723	−0.374095	−0.187047	−	0.982351i	$-0.559892\pi$
$457$	−7.99723	−0.374095	−0.187047	+	0.982351i	$0.559892\pi$
$458$	12.7637	0.596410
$459$	1.79763	0.0839061
$460$	0	0
$461$	−20.6848	−0.963389	−0.481694	−	0.876339i	$-0.659978\pi$
$461$	−20.6848	−0.963389	−0.481694	+	0.876339i	$0.659978\pi$
$462$	0.499806	0.0232531
$463$	−7.52507	−0.349720	−0.174860	−	0.984593i	$-0.555947\pi$
$463$	−7.52507	−0.349720	−0.174860	+	0.984593i	$0.555947\pi$
$464$	−4.28087	−0.198734
$465$	0	0
$466$	−0.103761	−0.00480663
$467$	6.84261	0.316638	0.158319	−	0.987388i	$-0.449393\pi$
$467$	6.84261	0.316638	0.158319	+	0.987388i	$0.449393\pi$
$468$	−8.52230	−0.393943
$469$	11.8398	0.546713
$470$	0	0
$471$	−1.42131	−0.0654903
$472$	2.69941	0.124250
$473$	33.3898	1.53526
$474$	−1.19683	−0.0549723
$475$	0	0
$476$	−2.44103	−0.111884
$477$	−23.7247	−1.08628
$478$	−3.67969	−0.168305
$479$	−16.4016	−0.749408	−0.374704	−	0.927145i	$-0.622256\pi$
$479$	−16.4016	−0.749408	−0.374704	+	0.927145i	$0.622256\pi$
$480$	0	0
$481$	2.85956	0.130385
$482$	−6.14044	−0.279689
$483$	0.275331	0.0125280
$484$	−1.26115	−0.0573249
$485$	0	0
$486$	3.75857	0.170492
$487$	13.7270	0.622032	0.311016	−	0.950405i	$-0.399331\pi$
$487$	13.7270	0.622032	0.311016	+	0.950405i	$0.399331\pi$
$488$	4.57869	0.207268
$489$	1.64855	0.0745500
$490$	0	0
$491$	3.40435	0.153636	0.0768182	−	0.997045i	$-0.475524\pi$
$491$	3.40435	0.153636	0.0768182	+	0.997045i	$0.475524\pi$
$492$	−0.455211	−0.0205225
$493$	9.16293	0.412677
$494$	12.1850	0.548230
$495$	0	0
$496$	6.40158	0.287440
$497$	−8.09822	−0.363255
$498$	−0.162927	−0.00730094
$499$	−29.4631	−1.31895	−0.659475	−	0.751726i	$-0.729222\pi$
$499$	−29.4631	−1.31895	−0.659475	+	0.751726i	$0.729222\pi$
$500$	0	0
$501$	1.49466	0.0667763
$502$	−9.00000	−0.401690
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	−3.39881	−0.151395
$505$	0	0
$506$	5.36491	0.238499
$507$	−0.677304	−0.0300801
$508$	0.899009	0.0398871
$509$	12.9633	0.574589	0.287295	−	0.957842i	$-0.407244\pi$
$509$	12.9633	0.574589	0.287295	+	0.957842i	$0.407244\pi$
$510$	0	0
$511$	−0.297823	−0.0131749
$512$	1.00000	0.0441942
$513$	−3.57869	−0.158003
$514$	−16.5223	−0.728767
$515$	0	0
$516$	1.50258	0.0661472
$517$	0	0
$518$	1.14044	0.0501079
$519$	−1.60080	−0.0702672
$520$	0	0
$521$	−2.01972	−0.0884856	−0.0442428	−	0.999021i	$-0.514088\pi$
$521$	−2.01972	−0.0884856	−0.0442428	+	0.999021i	$0.514088\pi$
$522$	12.7582	0.558411
$523$	15.5562	0.680225	0.340113	−	0.940385i	$-0.389535\pi$
$523$	15.5562	0.680225	0.340113	+	0.940385i	$0.389535\pi$
$524$	−0.980278	−0.0428236
$525$	0	0
$526$	−22.5223	−0.982019
$527$	−13.7022	−0.596876
$528$	0.438259	0.0190728
$529$	−20.0446	−0.871504
$530$	0	0
$531$	−8.04498	−0.349123
$532$	4.85956	0.210689
$533$	−9.26907	−0.401488
$534$	−1.91596	−0.0829118
$535$	0	0
$536$	10.3819	0.448428
$537$	−2.65719	−0.114666
$538$	27.2860	1.17638
$539$	−17.7862	−0.766107
$540$	0	0
$541$	−0.544789	−0.0234223	−0.0117112	−	0.999931i	$-0.503728\pi$
$541$	−0.544789	−0.0234223	−0.0117112	+	0.999931i	$0.503728\pi$
$542$	−26.1456	−1.12305
$543$	−2.64063	−0.113320
$544$	−2.14044	−0.0917704
$545$	0	0
$546$	0.457981	0.0195998
$547$	−35.6261	−1.52326	−0.761630	−	0.648012i	$-0.775601\pi$
$547$	−35.6261	−1.52326	−0.761630	+	0.648012i	$0.775601\pi$
$548$	−6.67969	−0.285342
$549$	−13.6458	−0.582388
$550$	0	0
$551$	−18.2414	−0.777111
$552$	0.241427	0.0102758
$553$	−9.71913	−0.413299
$554$	−20.5223	−0.871909
$555$	0	0
$556$	−9.68245	−0.410628
$557$	31.9606	1.35421	0.677106	−	0.735885i	$-0.263234\pi$
$557$	31.9606	1.35421	0.677106	+	0.735885i	$0.263234\pi$
$558$	−19.0785	−0.807657
$559$	30.5956	1.29406
$560$	0	0
$561$	−0.938064	−0.0396051
$562$	−14.4580	−0.609873
$563$	14.6655	0.618077	0.309039	−	0.951049i	$-0.399993\pi$
$563$	14.6655	0.618077	0.309039	+	0.951049i	$0.399993\pi$
$564$	0	0
$565$	0	0
$566$	6.85679	0.288213
$567$	10.0619	0.422562
$568$	−7.10099	−0.297951
$569$	9.99723	0.419106	0.209553	−	0.977797i	$-0.432799\pi$
$569$	9.99723	0.419106	0.209553	+	0.977797i	$0.432799\pi$
$570$	0	0
$571$	27.6008	1.15506	0.577529	−	0.816370i	$-0.304017\pi$
$571$	27.6008	1.15506	0.577529	+	0.816370i	$0.304017\pi$
$572$	8.92388	0.373126
$573$	2.11279	0.0882632
$574$	−3.69664	−0.154295
$575$	0	0
$576$	−2.98028	−0.124178
$577$	−3.78622	−0.157622	−0.0788111	−	0.996890i	$-0.525112\pi$
$577$	−3.78622	−0.157622	−0.0788111	+	0.996890i	$0.525112\pi$
$578$	−12.4185	−0.516543
$579$	3.55620	0.147791
$580$	0	0
$581$	−1.32308	−0.0548908
$582$	2.00000	0.0829027
$583$	24.8426	1.02888
$584$	−0.261149	−0.0108064
$585$	0	0
$586$	−19.2048	−0.793341
$587$	19.1771	0.791524	0.395762	−	0.918353i	$-0.370481\pi$
$587$	19.1771	0.791524	0.395762	+	0.918353i	$0.370481\pi$
$588$	−0.800398	−0.0330079
$589$	27.2781	1.12397
$590$	0	0
$591$	−0.376712	−0.0154958
$592$	1.00000	0.0410997
$593$	−12.4383	−0.510778	−0.255389	−	0.966838i	$-0.582204\pi$
$593$	−12.4383	−0.510778	−0.255389	+	0.966838i	$0.582204\pi$
$594$	−2.62091	−0.107537
$595$	0	0
$596$	−1.59842	−0.0654737
$597$	0.314776	0.0128829
$598$	4.91596	0.201029
$599$	−35.3819	−1.44566	−0.722832	−	0.691024i	$-0.757160\pi$
$599$	−35.3819	−1.44566	−0.722832	+	0.691024i	$0.757160\pi$
$600$	0	0
$601$	22.2951	0.909434	0.454717	−	0.890636i	$-0.349740\pi$
$601$	22.2951	0.909434	0.454717	+	0.890636i	$0.349740\pi$
$602$	12.2020	0.497316
$603$	−30.9408	−1.26001
$604$	−16.4829	−0.670678
$605$	0	0
$606$	1.96056	0.0796421
$607$	−9.72705	−0.394809	−0.197404	−	0.980322i	$-0.563251\pi$
$607$	−9.72705	−0.394809	−0.197404	+	0.980322i	$0.563251\pi$
$608$	4.26115	0.172812
$609$	−0.685613	−0.0277825
$610$	0	0
$611$	0	0
$612$	6.37909	0.257860
$613$	10.3203	0.416834	0.208417	−	0.978040i	$-0.433169\pi$
$613$	10.3203	0.416834	0.208417	+	0.978040i	$0.433169\pi$
$614$	16.2781	0.656931
$615$	0	0
$616$	3.55897	0.143395
$617$	−3.78345	−0.152316	−0.0761579	−	0.997096i	$-0.524265\pi$
$617$	−3.78345	−0.152316	−0.0761579	+	0.997096i	$0.524265\pi$
$618$	−1.45521	−0.0585372
$619$	27.1179	1.08996	0.544981	−	0.838448i	$-0.316537\pi$
$619$	27.1179	1.08996	0.544981	+	0.838448i	$0.316537\pi$
$620$	0	0
$621$	−1.44380	−0.0579376
$622$	−25.4801	−1.02166
$623$	−15.5590	−0.623357
$624$	0.401584	0.0160762
$625$	0	0
$626$	19.0197	0.760181
$627$	1.86748	0.0745802
$628$	−10.1207	−0.403860
$629$	−2.14044	−0.0853447
$630$	0	0
$631$	−11.1179	−0.442598	−0.221299	−	0.975206i	$-0.571030\pi$
$631$	−11.1179	−0.442598	−0.221299	+	0.975206i	$0.571030\pi$
$632$	−8.52230	−0.338999
$633$	−0.227244	−0.00903213
$634$	15.1653	0.602291
$635$	0	0
$636$	1.11794	0.0443293
$637$	−16.2978	−0.645743
$638$	−13.3594	−0.528903
$639$	21.1629	0.837192
$640$	0	0
$641$	27.0446	1.06820	0.534099	−	0.845422i	$-0.320651\pi$
$641$	27.0446	1.06820	0.534099	+	0.845422i	$0.320651\pi$
$642$	0.660352	0.0260620
$643$	29.7834	1.17454	0.587272	−	0.809389i	$-0.300202\pi$
$643$	29.7834	1.17454	0.587272	+	0.809389i	$0.300202\pi$
$644$	1.96056	0.0772567
$645$	0	0
$646$	−9.12071	−0.358850
$647$	19.3570	0.761002	0.380501	−	0.924781i	$-0.375752\pi$
$647$	19.3570	0.761002	0.380501	+	0.924781i	$0.375752\pi$
$648$	8.82289	0.346596
$649$	8.42408	0.330674
$650$	0	0
$651$	1.02526	0.0401832
$652$	11.7389	0.459729
$653$	−10.5392	−0.412433	−0.206216	−	0.978506i	$-0.566115\pi$
$653$	−10.5392	−0.412433	−0.206216	+	0.978506i	$0.566115\pi$
$654$	−0.837073	−0.0327321
$655$	0	0
$656$	−3.24143	−0.126556
$657$	0.778296	0.0303642
$658$	0	0
$659$	9.22447	0.359334	0.179667	−	0.983727i	$-0.442498\pi$
$659$	9.22447	0.359334	0.179667	+	0.983727i	$0.442498\pi$
$660$	0	0
$661$	27.3594	1.06416	0.532078	−	0.846695i	$-0.321411\pi$
$661$	27.3594	1.06416	0.532078	+	0.846695i	$0.321411\pi$
$662$	−23.4185	−0.910187
$663$	−0.859565	−0.0333827
$664$	−1.16016	−0.0450228
$665$	0	0
$666$	−2.98028	−0.115483
$667$	−7.35937	−0.284956
$668$	10.6430	0.411790
$669$	−0.477314	−0.0184540
$670$	0	0
$671$	14.2888	0.551613
$672$	0.160157	0.00617821
$673$	−13.1484	−0.506832	−0.253416	−	0.967357i	$-0.581554\pi$
$673$	−13.1484	−0.506832	−0.253416	+	0.967357i	$0.581554\pi$
$674$	−28.9211	−1.11400
$675$	0	0
$676$	−4.82289	−0.185496
$677$	29.8817	1.14845	0.574223	−	0.818699i	$-0.305304\pi$
$677$	29.8817	1.14845	0.574223	+	0.818699i	$0.305304\pi$
$678$	−0.679685	−0.0261031
$679$	16.2414	0.623289
$680$	0	0
$681$	0.345187	0.0132276
$682$	19.9775	0.764978
$683$	−26.9854	−1.03257	−0.516284	−	0.856417i	$-0.672685\pi$
$683$	−26.9854	−1.03257	−0.516284	+	0.856417i	$0.672685\pi$
$684$	−12.6994	−0.485574
$685$	0	0
$686$	−14.4829	−0.552958
$687$	1.79248	0.0683873
$688$	10.6994	0.407911
$689$	22.7637	0.867229
$690$	0	0
$691$	−30.7270	−1.16891	−0.584456	−	0.811425i	$-0.698692\pi$
$691$	−30.7270	−1.16891	−0.584456	+	0.811425i	$0.698692\pi$
$692$	−11.3988	−0.433318
$693$	−10.6067	−0.402916
$694$	−32.8817	−1.24817
$695$	0	0
$696$	−0.601186	−0.0227879
$697$	6.93806	0.262798
$698$	15.2048	0.575508
$699$	−0.0145717	−0.000551152 0
$700$	0	0
$701$	−27.7858	−1.04946	−0.524728	−	0.851270i	$-0.675833\pi$
$701$	−27.7858	−1.04946	−0.524728	+	0.851270i	$0.675833\pi$
$702$	−2.40158	−0.0906419
$703$	4.26115	0.160712
$704$	3.12071	0.117616
$705$	0	0
$706$	−18.4829	−0.695611
$707$	15.9211	0.598775
$708$	0.379092	0.0142472
$709$	9.71913	0.365010	0.182505	−	0.983205i	$-0.441579\pi$
$709$	9.71913	0.365010	0.182505	+	0.983205i	$0.441579\pi$
$710$	0	0
$711$	25.3988	0.952530
$712$	−13.6430	−0.511293
$713$	11.0052	0.412146
$714$	−0.342807	−0.0128292
$715$	0	0
$716$	−18.9211	−0.707115
$717$	−0.516758	−0.0192987
$718$	22.5223	0.840525
$719$	5.41577	0.201974	0.100987	−	0.994888i	$-0.467800\pi$
$719$	5.41577	0.201974	0.100987	+	0.994888i	$0.467800\pi$
$720$	0	0
$721$	−11.8174	−0.440101
$722$	−0.842612	−0.0313588
$723$	−0.862334	−0.0320706
$724$	−18.8032	−0.698814
$725$	0	0
$726$	−0.177110	−0.00657316
$727$	13.7586	0.510277	0.255139	−	0.966904i	$-0.417879\pi$
$727$	13.7586	0.510277	0.255139	+	0.966904i	$0.417879\pi$
$728$	3.26115	0.120866
$729$	−25.9408	−0.960772
$730$	0	0
$731$	−22.9014	−0.847038
$732$	0.643011	0.0237664
$733$	52.4095	1.93579	0.967895	−	0.251356i	$-0.0808766\pi$
$733$	52.4095	1.93579	0.967895	+	0.251356i	$0.0808766\pi$
$734$	15.3424	0.566299
$735$	0	0
$736$	1.71913	0.0633679
$737$	32.3988	1.19343
$738$	9.66035	0.355602
$739$	50.0892	1.84256	0.921280	−	0.388899i	$-0.127145\pi$
$739$	50.0892	1.84256	0.921280	+	0.388899i	$0.127145\pi$
$740$	0	0
$741$	1.71121	0.0628628
$742$	9.07850	0.333282
$743$	−40.7976	−1.49672	−0.748360	−	0.663293i	$-0.769158\pi$
$743$	−40.7976	−1.49672	−0.748360	+	0.663293i	$0.769158\pi$
$744$	0.899009	0.0329593
$745$	0	0
$746$	12.6430	0.462894
$747$	3.45759	0.126507
$748$	−6.67969	−0.244233
$749$	5.36253	0.195943
$750$	0	0
$751$	2.84261	0.103728	0.0518642	−	0.998654i	$-0.483484\pi$
$751$	2.84261	0.103728	0.0518642	+	0.998654i	$0.483484\pi$
$752$	0	0
$753$	−1.26392	−0.0460597
$754$	−12.2414	−0.445806
$755$	0	0
$756$	−0.957786	−0.0348343
$757$	11.4383	0.415731	0.207865	−	0.978157i	$-0.433348\pi$
$757$	11.4383	0.415731	0.207865	+	0.978157i	$0.433348\pi$
$758$	15.7807	0.573180
$759$	0.753423	0.0273475
$760$	0	0
$761$	43.2663	1.56840	0.784201	−	0.620507i	$-0.213073\pi$
$761$	43.2663	1.56840	0.784201	+	0.620507i	$0.213073\pi$
$762$	0.126253	0.00457365
$763$	−6.79763	−0.246091
$764$	15.0446	0.544294
$765$	0	0
$766$	−27.8004	−1.00447
$767$	7.71913	0.278722
$768$	0.140435	0.00506752
$769$	26.5590	0.957741	0.478871	−	0.877886i	$-0.341046\pi$
$769$	26.5590	0.957741	0.478871	+	0.877886i	$0.341046\pi$
$770$	0	0
$771$	−2.32031	−0.0835641
$772$	25.3227	0.911384
$773$	12.3621	0.444635	0.222318	−	0.974974i	$-0.428638\pi$
$773$	12.3621	0.444635	0.222318	+	0.974974i	$0.428638\pi$
$774$	−31.8872	−1.14616
$775$	0	0
$776$	14.2414	0.511237
$777$	0.160157	0.00574562
$778$	9.66273	0.346426
$779$	−13.8122	−0.494873
$780$	0	0
$781$	−22.1602	−0.792953
$782$	−3.67969	−0.131585
$783$	3.59526	0.128484
$784$	−5.69941	−0.203550
$785$	0	0
$786$	−0.137666	−0.00491037
$787$	6.03944	0.215283	0.107641	−	0.994190i	$-0.465670\pi$
$787$	6.03944	0.215283	0.107641	+	0.994190i	$0.465670\pi$
$788$	−2.68245	−0.0955585
$789$	−3.16293	−0.112603
$790$	0	0
$791$	−5.51953	−0.196252
$792$	−9.30059	−0.330482
$793$	13.0931	0.464949
$794$	−23.4801	−0.833277
$795$	0	0
$796$	2.24143	0.0794453
$797$	35.6233	1.26184	0.630921	−	0.775847i	$-0.282677\pi$
$797$	35.6233	1.26184	0.630921	+	0.775847i	$0.282677\pi$
$798$	0.682455	0.0241586
$799$	0	0
$800$	0	0
$801$	40.6600	1.43665
$802$	20.2781	0.716045
$803$	−0.814970	−0.0287597
$804$	1.45798	0.0514190
$805$	0	0
$806$	18.3057	0.644792
$807$	3.83192	0.134890
$808$	13.9606	0.491131
$809$	−23.6402	−0.831147	−0.415573	−	0.909560i	$-0.636419\pi$
$809$	−23.6402	−0.831147	−0.415573	+	0.909560i	$0.636419\pi$
$810$	0	0
$811$	−3.27572	−0.115026	−0.0575130	−	0.998345i	$-0.518317\pi$
$811$	−3.27572	−0.115026	−0.0575130	+	0.998345i	$0.518317\pi$
$812$	−4.88206	−0.171327
$813$	−3.67176	−0.128774
$814$	3.12071	0.109381
$815$	0	0
$816$	−0.300593	−0.0105229
$817$	45.5918	1.59505
$818$	4.17434	0.145952
$819$	−9.71913	−0.339614
$820$	0	0
$821$	−30.3621	−1.05965	−0.529823	−	0.848108i	$-0.677742\pi$
$821$	−30.3621	−1.05965	−0.529823	+	0.848108i	$0.677742\pi$
$822$	−0.938064	−0.0327187
$823$	−14.2978	−0.498391	−0.249195	−	0.968453i	$-0.580166\pi$
$823$	−14.2978	−0.498391	−0.249195	+	0.968453i	$0.580166\pi$
$824$	−10.3621	−0.360982
$825$	0	0
$826$	3.07850	0.107115
$827$	7.90139	0.274758	0.137379	−	0.990519i	$-0.456132\pi$
$827$	7.90139	0.274758	0.137379	+	0.990519i	$0.456132\pi$
$828$	−5.12348	−0.178053
$829$	−43.3030	−1.50397	−0.751987	−	0.659178i	$-0.770905\pi$
$829$	−43.3030	−1.50397	−0.751987	+	0.659178i	$0.770905\pi$
$830$	0	0
$831$	−2.88206	−0.0999774
$832$	2.85956	0.0991376
$833$	12.1992	0.422678
$834$	−1.35976	−0.0470846
$835$	0	0
$836$	13.2978	0.459915
$837$	−5.37632	−0.185833
$838$	−12.3175	−0.425503
$839$	11.3819	0.392946	0.196473	−	0.980509i	$-0.437051\pi$
$839$	11.3819	0.392946	0.196473	+	0.980509i	$0.437051\pi$
$840$	0	0
$841$	−10.6741	−0.368074
$842$	23.8872	0.823208
$843$	−2.03041	−0.0699311
$844$	−1.61814	−0.0556986
$845$	0	0
$846$	0	0
$847$	−1.43826	−0.0494191
$848$	7.96056	0.273367
$849$	0.962937	0.0330479
$850$	0	0
$851$	1.71913	0.0589310
$852$	−0.997230	−0.0341645
$853$	10.7862	0.369313	0.184656	−	0.982803i	$-0.440883\pi$
$853$	10.7862	0.369313	0.184656	+	0.982803i	$0.440883\pi$
$854$	5.22170	0.178683
$855$	0	0
$856$	4.70218	0.160717
$857$	−2.38186	−0.0813629	−0.0406814	−	0.999172i	$-0.512953\pi$
$857$	−2.38186	−0.0813629	−0.0406814	+	0.999172i	$0.512953\pi$
$858$	1.25323	0.0427845
$859$	−48.1428	−1.64261	−0.821306	−	0.570488i	$-0.806754\pi$
$859$	−48.1428	−1.64261	−0.821306	+	0.570488i	$0.806754\pi$
$860$	0	0
$861$	−0.519139	−0.0176922
$862$	−2.80317	−0.0954763
$863$	29.8308	1.01545	0.507726	−	0.861518i	$-0.330486\pi$
$863$	29.8308	1.01545	0.507726	+	0.861518i	$0.330486\pi$
$864$	−0.839843	−0.0285720
$865$	0	0
$866$	5.00000	0.169907
$867$	−1.74400	−0.0592294
$868$	7.30059	0.247798
$869$	−26.5956	−0.902196
$870$	0	0
$871$	29.6876	1.00593
$872$	−5.96056	−0.201850
$873$	−42.4434	−1.43649
$874$	7.32547	0.247788
$875$	0	0
$876$	−0.0366745	−0.00123912
$877$	−48.9578	−1.65319	−0.826593	−	0.562799i	$-0.809724\pi$
$877$	−48.9578	−1.65319	−0.826593	+	0.562799i	$0.809724\pi$
$878$	30.7976	1.03937
$879$	−2.69703	−0.0909684
$880$	0	0
$881$	3.50811	0.118191	0.0590957	−	0.998252i	$-0.481178\pi$
$881$	3.50811	0.118191	0.0590957	+	0.998252i	$0.481178\pi$
$882$	16.9858	0.571942
$883$	−48.8817	−1.64500	−0.822500	−	0.568766i	$-0.807421\pi$
$883$	−48.8817	−1.64500	−0.822500	+	0.568766i	$0.807421\pi$
$884$	−6.12071	−0.205862
$885$	0	0
$886$	−16.5984	−0.557635
$887$	46.3701	1.55695	0.778477	−	0.627673i	$-0.215992\pi$
$887$	46.3701	1.55695	0.778477	+	0.627673i	$0.215992\pi$
$888$	0.140435	0.00471270
$889$	1.02526	0.0343862
$890$	0	0
$891$	27.5337	0.922414
$892$	−3.39881	−0.113801
$893$	0	0
$894$	−0.224474	−0.00750754
$895$	0	0
$896$	1.14044	0.0380993
$897$	0.690375	0.0230509
$898$	9.75580	0.325555
$899$	−27.4044	−0.913986
$900$	0	0
$901$	−17.0391	−0.567653
$902$	−10.1156	−0.336811
$903$	1.71359	0.0570247
$904$	−4.83984	−0.160971
$905$	0	0
$906$	−2.31478	−0.0769033
$907$	45.4631	1.50958	0.754789	−	0.655967i	$-0.227739\pi$
$907$	45.4631	1.50958	0.754789	+	0.655967i	$0.227739\pi$
$908$	2.45798	0.0815710
$909$	−41.6063	−1.38000
$910$	0	0
$911$	17.7610	0.588447	0.294223	−	0.955737i	$-0.404939\pi$
$911$	17.7610	0.588447	0.294223	+	0.955737i	$0.404939\pi$
$912$	0.598416	0.0198155
$913$	−3.62052	−0.119822
$914$	−7.99723	−0.264525
$915$	0	0
$916$	12.7637	0.421725
$917$	−1.11794	−0.0369178
$918$	1.79763	0.0593306
$919$	26.4829	0.873589	0.436794	−	0.899561i	$-0.356114\pi$
$919$	26.4829	0.873589	0.436794	+	0.899561i	$0.356114\pi$
$920$	0	0
$921$	2.28602	0.0753270
$922$	−20.6848	−0.681219
$923$	−20.3057	−0.668372
$924$	0.499806	0.0164424
$925$	0	0
$926$	−7.52507	−0.247289
$927$	30.8821	1.01430
$928$	−4.28087	−0.140526
$929$	59.3842	1.94833	0.974167	−	0.225829i	$-0.0725091\pi$
$929$	59.3842	1.94833	0.974167	+	0.225829i	$0.0725091\pi$
$930$	0	0
$931$	−24.2860	−0.795942
$932$	−0.103761	−0.00339880
$933$	−3.57830	−0.117148
$934$	6.84261	0.223897
$935$	0	0
$936$	−8.52230	−0.278560
$937$	15.1771	0.495815	0.247907	−	0.968784i	$-0.420257\pi$
$937$	15.1771	0.495815	0.247907	+	0.968784i	$0.420257\pi$
$938$	11.8398	0.386585
$939$	2.67104	0.0871662
$940$	0	0
$941$	12.8426	0.418657	0.209329	−	0.977845i	$-0.432872\pi$
$941$	12.8426	0.418657	0.209329	+	0.977845i	$0.432872\pi$
$942$	−1.42131	−0.0463087
$943$	−5.57243	−0.181463
$944$	2.69941	0.0878582
$945$	0	0
$946$	33.3898	1.08560
$947$	−9.04459	−0.293910	−0.146955	−	0.989143i	$-0.546947\pi$
$947$	−9.04459	−0.293910	−0.146955	+	0.989143i	$0.546947\pi$
$948$	−1.19683	−0.0388713
$949$	−0.746771	−0.0242412
$950$	0	0
$951$	2.12975	0.0690617
$952$	−2.44103	−0.0791142
$953$	−13.7783	−0.446323	−0.223161	−	0.974782i	$-0.571638\pi$
$953$	−13.7783	−0.446323	−0.223161	+	0.974782i	$0.571638\pi$
$954$	−23.7247	−0.768115
$955$	0	0
$956$	−3.67969	−0.119010
$957$	−1.87613	−0.0606466
$958$	−16.4016	−0.529911
$959$	−7.61775	−0.245990
$960$	0	0
$961$	9.98028	0.321944
$962$	2.85956	0.0921961
$963$	−14.0138	−0.451588
$964$	−6.14044	−0.197770
$965$	0	0
$966$	0.275331	0.00885864
$967$	60.1964	1.93579	0.967894	−	0.251360i	$-0.0808779\pi$
$967$	60.1964	1.93579	0.967894	+	0.251360i	$0.0808779\pi$
$968$	−1.26115	−0.0405349
$969$	−1.28087	−0.0411475
$970$	0	0
$971$	−28.4123	−0.911793	−0.455897	−	0.890033i	$-0.650681\pi$
$971$	−28.4123	−0.911793	−0.455897	+	0.890033i	$0.650681\pi$
$972$	3.75857	0.120556
$973$	−11.0422	−0.353997
$974$	13.7270	0.439843
$975$	0	0
$976$	4.57869	0.146560
$977$	−55.7468	−1.78350	−0.891749	−	0.452531i	$-0.850521\pi$
$977$	−55.7468	−1.78350	−0.891749	+	0.452531i	$0.850521\pi$
$978$	1.64855	0.0527148
$979$	−42.5759	−1.36073
$980$	0	0
$981$	17.7641	0.567164
$982$	3.40435	0.108637
$983$	37.1179	1.18388	0.591939	−	0.805983i	$-0.298363\pi$
$983$	37.1179	1.18388	0.591939	+	0.805983i	$0.298363\pi$
$984$	−0.455211	−0.0145116
$985$	0	0
$986$	9.16293	0.291807
$987$	0	0
$988$	12.1850	0.387657
$989$	18.3937	0.584884
$990$	0	0
$991$	−38.0418	−1.20844	−0.604219	−	0.796818i	$-0.706515\pi$
$991$	−38.0418	−1.20844	−0.604219	+	0.796818i	$0.706515\pi$
$992$	6.40158	0.203250
$993$	−3.28879	−0.104367
$994$	−8.09822	−0.256860
$995$	0	0
$996$	−0.162927	−0.00516254
$997$	11.6288	0.368289	0.184144	−	0.982899i	$-0.441049\pi$
$997$	11.6288	0.368289	0.184144	+	0.982899i	$0.441049\pi$
$998$	−29.4631	−0.932639
$999$	−0.839843	−0.0265714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1850.2.a.bc.1.2	yes	3
5.2	odd	4		1850.2.b.p.149.5		6
5.3	odd	4		1850.2.b.p.149.2		6
5.4	even	2		1850.2.a.y.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1850.2.a.y.1.2	✓	3	5.4	even	2
1850.2.a.bc.1.2	yes	3	1.1	even	1	trivial
1850.2.b.p.149.2		6	5.3	odd	4
1850.2.b.p.149.5		6	5.2	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:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.140435 q^{3} +1.00000 q^{4} +0.140435 q^{6} +1.14044 q^{7} +1.00000 q^{8} -2.98028 q^{9} +3.12071 q^{11} +0.140435 q^{12} +2.85956 q^{13} +1.14044 q^{14} +1.00000 q^{16} -2.14044 q^{17} -2.98028 q^{18} +4.26115 q^{19} +0.160157 q^{21} +3.12071 q^{22} +1.71913 q^{23} +0.140435 q^{24} +2.85956 q^{26} -0.839843 q^{27} +1.14044 q^{28} -4.28087 q^{29} +6.40158 q^{31} +1.00000 q^{32} +0.438259 q^{33} -2.14044 q^{34} -2.98028 q^{36} +1.00000 q^{37} +4.26115 q^{38} +0.401584 q^{39} -3.24143 q^{41} +0.160157 q^{42} +10.6994 q^{43} +3.12071 q^{44} +1.71913 q^{46} +0.140435 q^{48} -5.69941 q^{49} -0.300593 q^{51} +2.85956 q^{52} +7.96056 q^{53} -0.839843 q^{54} +1.14044 q^{56} +0.598416 q^{57} -4.28087 q^{58} +2.69941 q^{59} +4.57869 q^{61} +6.40158 q^{62} -3.39881 q^{63} +1.00000 q^{64} +0.438259 q^{66} +10.3819 q^{67} -2.14044 q^{68} +0.241427 q^{69} -7.10099 q^{71} -2.98028 q^{72} -0.261149 q^{73} +1.00000 q^{74} +4.26115 q^{76} +3.55897 q^{77} +0.401584 q^{78} -8.52230 q^{79} +8.82289 q^{81} -3.24143 q^{82} -1.16016 q^{83} +0.160157 q^{84} +10.6994 q^{86} -0.601186 q^{87} +3.12071 q^{88} -13.6430 q^{89} +3.26115 q^{91} +1.71913 q^{92} +0.899009 q^{93} +0.140435 q^{96} +14.2414 q^{97} -5.69941 q^{98} -9.30059 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + q^{3} + 3 q^{4} + q^{6} + 4 q^{7} + 3 q^{8} + 6 q^{9} - 5 q^{11} + q^{12} + 8 q^{13} + 4 q^{14} + 3 q^{16} - 7 q^{17} + 6 q^{18} - q^{19} + 16 q^{21} - 5 q^{22} + 4 q^{23} + q^{24}+ \cdots - 44 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + q^3 + 3 * q^4 + q^6 + 4 * q^7 + 3 * q^8 + 6 * q^9 - 5 * q^11 + q^12 + 8 * q^13 + 4 * q^14 + 3 * q^16 - 7 * q^17 + 6 * q^18 - q^19 + 16 * q^21 - 5 * q^22 + 4 * q^23 + q^24 + 8 * q^26 + 13 * q^27 + 4 * q^28 - 14 * q^29 + 6 * q^31 + 3 * q^32 - q^33 - 7 * q^34 + 6 * q^36 + 3 * q^37 - q^38 - 12 * q^39 + 19 * q^41 + 16 * q^42 + 16 * q^43 - 5 * q^44 + 4 * q^46 + q^48 - q^49 - 17 * q^51 + 8 * q^52 - 6 * q^53 + 13 * q^54 + 4 * q^56 + 15 * q^57 - 14 * q^58 - 8 * q^59 + 12 * q^61 + 6 * q^62 + 22 * q^63 + 3 * q^64 - q^66 + 3 * q^67 - 7 * q^68 - 28 * q^69 + 8 * q^71 + 6 * q^72 + 13 * q^73 + 3 * q^74 - q^76 - 6 * q^77 - 12 * q^78 + 2 * q^79 + 15 * q^81 + 19 * q^82 - 19 * q^83 + 16 * q^84 + 16 * q^86 - 34 * q^87 - 5 * q^88 + q^89 - 4 * q^91 + 4 * q^92 + 32 * q^93 + q^96 + 14 * q^97 - q^98 - 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more