LMFDB - Embedded newform 1840.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$5.72015$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.72015 q^{3} +5.00000 q^{5} -26.6008 q^{7} +18.1605 q^{9} +O(q^{10})$	$q+6.72015 q^{3} +5.00000 q^{5} -26.6008 q^{7} +18.1605 q^{9} +39.6008 q^{11} -23.1605 q^{13} +33.6008 q^{15} -2.95893 q^{17} -32.3620 q^{19} -178.761 q^{21} +23.0000 q^{23} +25.0000 q^{25} -59.4031 q^{27} -162.798 q^{29} +241.243 q^{31} +266.123 q^{33} -133.004 q^{35} -180.164 q^{37} -155.642 q^{39} -353.922 q^{41} -365.761 q^{43} +90.8023 q^{45} +195.291 q^{47} +364.601 q^{49} -19.8844 q^{51} -461.687 q^{53} +198.004 q^{55} -217.478 q^{57} +290.888 q^{59} -301.049 q^{61} -483.082 q^{63} -115.802 q^{65} +366.732 q^{67} +154.564 q^{69} +8.14513 q^{71} +360.650 q^{73} +168.004 q^{75} -1053.41 q^{77} -1243.87 q^{79} -889.530 q^{81} +1481.70 q^{83} -14.7946 q^{85} -1094.03 q^{87} +829.628 q^{89} +616.086 q^{91} +1621.19 q^{93} -161.810 q^{95} -390.191 q^{97} +719.168 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 10 q^{5} - q^{7} + 5 q^{9}+O(q^{10}) $ 2 * q + 3 * q^3 + 10 * q^5 - q^7 + 5 * q^9	$ 2 q + 3 q^{3} + 10 q^{5} - q^{7} + 5 q^{9} + 27 q^{11} - 15 q^{13} + 15 q^{15} - 79 q^{17} + 71 q^{19} - 274 q^{21} + 46 q^{23} + 50 q^{25} + 90 q^{27} - 430 q^{29} + 305 q^{31} + 313 q^{33} - 5 q^{35} - 68 q^{37} - 186 q^{39} - 593 q^{41} - 648 q^{43} + 25 q^{45} - 382 q^{47} + 677 q^{49} + 263 q^{51} - 464 q^{53} + 135 q^{55} - 602 q^{57} + 18 q^{59} - 7 q^{61} - 820 q^{63} - 75 q^{65} - 60 q^{67} + 69 q^{69} + 1029 q^{71} + 74 q^{73} + 75 q^{75} - 1376 q^{77} - 692 q^{79} - 1090 q^{81} + 1460 q^{83} - 395 q^{85} - 100 q^{87} - 220 q^{89} + 825 q^{91} + 1384 q^{93} + 355 q^{95} + 1339 q^{97} + 885 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 10 * q^5 - q^7 + 5 * q^9 + 27 * q^11 - 15 * q^13 + 15 * q^15 - 79 * q^17 + 71 * q^19 - 274 * q^21 + 46 * q^23 + 50 * q^25 + 90 * q^27 - 430 * q^29 + 305 * q^31 + 313 * q^33 - 5 * q^35 - 68 * q^37 - 186 * q^39 - 593 * q^41 - 648 * q^43 + 25 * q^45 - 382 * q^47 + 677 * q^49 + 263 * q^51 - 464 * q^53 + 135 * q^55 - 602 * q^57 + 18 * q^59 - 7 * q^61 - 820 * q^63 - 75 * q^65 - 60 * q^67 + 69 * q^69 + 1029 * q^71 + 74 * q^73 + 75 * q^75 - 1376 * q^77 - 692 * q^79 - 1090 * q^81 + 1460 * q^83 - 395 * q^85 - 100 * q^87 - 220 * q^89 + 825 * q^91 + 1384 * q^93 + 355 * q^95 + 1339 * q^97 + 885 * 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 $

$2$	0	0
$2$	0	0
$3$	6.72015	1.29329	0.646647	−	0.762789i	$-0.276171\pi$
$3$	6.72015	1.29329	0.646647	+	0.762789i	$0.276171\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−26.6008	−1.43631	−0.718153	−	0.695885i	$-0.755012\pi$
$7$	−26.6008	−1.43631	−0.718153	+	0.695885i	$0.755012\pi$
$8$	0	0
$9$	18.1605	0.672610
$10$	0	0
$11$	39.6008	1.08546	0.542731	−	0.839907i	$-0.317390\pi$
$11$	39.6008	1.08546	0.542731	+	0.839907i	$0.317390\pi$
$12$	0	0
$13$	−23.1605	−0.494120	−0.247060	−	0.969000i	$-0.579464\pi$
$13$	−23.1605	−0.494120	−0.247060	+	0.969000i	$0.579464\pi$
$14$	0	0
$15$	33.6008	0.578379
$16$	0	0
$17$	−2.95893	−0.0422144	−0.0211072	−	0.999777i	$-0.506719\pi$
$17$	−2.95893	−0.0422144	−0.0211072	+	0.999777i	$0.506719\pi$
$18$	0	0
$19$	−32.3620	−0.390755	−0.195378	−	0.980728i	$-0.562593\pi$
$19$	−32.3620	−0.390755	−0.195378	+	0.980728i	$0.562593\pi$
$20$	0	0
$21$	−178.761	−1.85757
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−59.4031	−0.423412
$28$	0	0
$29$	−162.798	−1.04245	−0.521223	−	0.853421i	$-0.674524\pi$
$29$	−162.798	−1.04245	−0.521223	+	0.853421i	$0.674524\pi$
$30$	0	0
$31$	241.243	1.39769	0.698846	−	0.715272i	$-0.253697\pi$
$31$	241.243	1.39769	0.698846	+	0.715272i	$0.253697\pi$
$32$	0	0
$33$	266.123	1.40382
$34$	0	0
$35$	−133.004	−0.642336
$36$	0	0
$37$	−180.164	−0.800509	−0.400254	−	0.916404i	$-0.631078\pi$
$37$	−180.164	−0.800509	−0.400254	+	0.916404i	$0.631078\pi$
$38$	0	0
$39$	−155.642	−0.639042
$40$	0	0
$41$	−353.922	−1.34813	−0.674064	−	0.738673i	$-0.735453\pi$
$41$	−353.922	−1.34813	−0.674064	+	0.738673i	$0.735453\pi$
$42$	0	0
$43$	−365.761	−1.29716	−0.648582	−	0.761145i	$-0.724638\pi$
$43$	−365.761	−1.29716	−0.648582	+	0.761145i	$0.724638\pi$
$44$	0	0
$45$	90.8023	0.300800
$46$	0	0
$47$	195.291	0.606089	0.303044	−	0.952976i	$-0.401997\pi$
$47$	195.291	0.606089	0.303044	+	0.952976i	$0.401997\pi$
$48$	0	0
$49$	364.601	1.06298
$50$	0	0
$51$	−19.8844	−0.0545957
$52$	0	0
$53$	−461.687	−1.19656	−0.598279	−	0.801288i	$-0.704148\pi$
$53$	−461.687	−1.19656	−0.598279	+	0.801288i	$0.704148\pi$
$54$	0	0
$55$	198.004	0.485433
$56$	0	0
$57$	−217.478	−0.505361
$58$	0	0
$59$	290.888	0.641872	0.320936	−	0.947101i	$-0.396003\pi$
$59$	290.888	0.641872	0.320936	+	0.947101i	$0.396003\pi$
$60$	0	0
$61$	−301.049	−0.631891	−0.315945	−	0.948777i	$-0.602322\pi$
$61$	−301.049	−0.631891	−0.315945	+	0.948777i	$0.602322\pi$
$62$	0	0
$63$	−483.082	−0.966073
$64$	0	0
$65$	−115.802	−0.220977
$66$	0	0
$67$	366.732	0.668707	0.334354	−	0.942448i	$-0.391482\pi$
$67$	366.732	0.668707	0.334354	+	0.942448i	$0.391482\pi$
$68$	0	0
$69$	154.564	0.269670
$70$	0	0
$71$	8.14513	0.0136148	0.00680739	−	0.999977i	$-0.497833\pi$
$71$	8.14513	0.0136148	0.00680739	+	0.999977i	$0.497833\pi$
$72$	0	0
$73$	360.650	0.578231	0.289115	−	0.957294i	$-0.406639\pi$
$73$	360.650	0.578231	0.289115	+	0.957294i	$0.406639\pi$
$74$	0	0
$75$	168.004	0.258659
$76$	0	0
$77$	−1053.41	−1.55906
$78$	0	0
$79$	−1243.87	−1.77147	−0.885734	−	0.464194i	$-0.846344\pi$
$79$	−1243.87	−1.77147	−0.885734	+	0.464194i	$0.846344\pi$
$80$	0	0
$81$	−889.530	−1.22021
$82$	0	0
$83$	1481.70	1.95949	0.979747	−	0.200241i	$-0.0641727\pi$
$83$	1481.70	1.95949	0.979747	+	0.200241i	$0.0641727\pi$
$84$	0	0
$85$	−14.7946	−0.0188789
$86$	0	0
$87$	−1094.03	−1.34819
$88$	0	0
$89$	829.628	0.988094	0.494047	−	0.869435i	$-0.335517\pi$
$89$	829.628	0.988094	0.494047	+	0.869435i	$0.335517\pi$
$90$	0	0
$91$	616.086	0.709707
$92$	0	0
$93$	1621.19	1.80763
$94$	0	0
$95$	−161.810	−0.174751
$96$	0	0
$97$	−390.191	−0.408432	−0.204216	−	0.978926i	$-0.565464\pi$
$97$	−390.191	−0.408432	−0.204216	+	0.978926i	$0.565464\pi$
$98$	0	0
$99$	719.168	0.730092
$100$	0	0
$101$	−1461.87	−1.44022	−0.720108	−	0.693862i	$-0.755908\pi$
$101$	−1461.87	−1.44022	−0.720108	+	0.693862i	$0.755908\pi$
$102$	0	0
$103$	−317.534	−0.303763	−0.151881	−	0.988399i	$-0.548533\pi$
$103$	−317.534	−0.303763	−0.151881	+	0.988399i	$0.548533\pi$
$104$	0	0
$105$	−893.806	−0.830729
$106$	0	0
$107$	47.6199	0.0430242	0.0215121	−	0.999769i	$-0.493152\pi$
$107$	47.6199	0.0430242	0.0215121	+	0.999769i	$0.493152\pi$
$108$	0	0
$109$	−1437.21	−1.26293	−0.631465	−	0.775404i	$-0.717546\pi$
$109$	−1437.21	−1.26293	−0.631465	+	0.775404i	$0.717546\pi$
$110$	0	0
$111$	−1210.73	−1.03529
$112$	0	0
$113$	−1575.76	−1.31182	−0.655908	−	0.754841i	$-0.727714\pi$
$113$	−1575.76	−1.31182	−0.655908	+	0.754841i	$0.727714\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−420.605	−0.332350
$118$	0	0
$119$	78.7097	0.0606329
$120$	0	0
$121$	237.221	0.178227
$122$	0	0
$123$	−2378.41	−1.74353
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−2202.29	−1.53875	−0.769377	−	0.638795i	$-0.779433\pi$
$127$	−2202.29	−1.53875	−0.769377	+	0.638795i	$0.779433\pi$
$128$	0	0
$129$	−2457.97	−1.67761
$130$	0	0
$131$	−2067.76	−1.37909	−0.689546	−	0.724242i	$-0.742190\pi$
$131$	−2067.76	−1.37909	−0.689546	+	0.724242i	$0.742190\pi$
$132$	0	0
$133$	860.854	0.561244
$134$	0	0
$135$	−297.015	−0.189356
$136$	0	0
$137$	2225.67	1.38797	0.693984	−	0.719990i	$-0.255854\pi$
$137$	2225.67	1.38797	0.693984	+	0.719990i	$0.255854\pi$
$138$	0	0
$139$	−2803.89	−1.71096	−0.855478	−	0.517840i	$-0.826736\pi$
$139$	−2803.89	−1.71096	−0.855478	+	0.517840i	$0.826736\pi$
$140$	0	0
$141$	1312.39	0.783851
$142$	0	0
$143$	−917.172	−0.536348
$144$	0	0
$145$	−813.992	−0.466196
$146$	0	0
$147$	2450.17	1.37474
$148$	0	0
$149$	−1322.84	−0.727324	−0.363662	−	0.931531i	$-0.618474\pi$
$149$	−1322.84	−0.727324	−0.363662	+	0.931531i	$0.618474\pi$
$150$	0	0
$151$	−2052.35	−1.10608	−0.553039	−	0.833156i	$-0.686532\pi$
$151$	−2052.35	−1.10608	−0.553039	+	0.833156i	$0.686532\pi$
$152$	0	0
$153$	−53.7355	−0.0283938
$154$	0	0
$155$	1206.21	0.625067
$156$	0	0
$157$	−381.171	−0.193763	−0.0968814	−	0.995296i	$-0.530887\pi$
$157$	−381.171	−0.193763	−0.0968814	+	0.995296i	$0.530887\pi$
$158$	0	0
$159$	−3102.61	−1.54750
$160$	0	0
$161$	−611.818	−0.299491
$162$	0	0
$163$	2358.16	1.13316	0.566581	−	0.824006i	$-0.308266\pi$
$163$	2358.16	1.13316	0.566581	+	0.824006i	$0.308266\pi$
$164$	0	0
$165$	1330.62	0.627808
$166$	0	0
$167$	144.267	0.0668487	0.0334244	−	0.999441i	$-0.489359\pi$
$167$	144.267	0.0668487	0.0334244	+	0.999441i	$0.489359\pi$
$168$	0	0
$169$	−1660.59	−0.755846
$170$	0	0
$171$	−587.709	−0.262826
$172$	0	0
$173$	3210.33	1.41085	0.705424	−	0.708786i	$-0.250757\pi$
$173$	3210.33	1.41085	0.705424	+	0.708786i	$0.250757\pi$
$174$	0	0
$175$	−665.019	−0.287261
$176$	0	0
$177$	1954.81	0.830129
$178$	0	0
$179$	4070.43	1.69965	0.849827	−	0.527062i	$-0.176706\pi$
$179$	4070.43	1.69965	0.849827	+	0.527062i	$0.176706\pi$
$180$	0	0
$181$	−4123.13	−1.69320	−0.846602	−	0.532226i	$-0.821356\pi$
$181$	−4123.13	−1.69320	−0.846602	+	0.532226i	$0.821356\pi$
$182$	0	0
$183$	−2023.09	−0.817221
$184$	0	0
$185$	−900.821	−0.357998
$186$	0	0
$187$	−117.176	−0.0458221
$188$	0	0
$189$	1580.17	0.608149
$190$	0	0
$191$	1827.02	0.692140	0.346070	−	0.938209i	$-0.387516\pi$
$191$	1827.02	0.692140	0.346070	+	0.938209i	$0.387516\pi$
$192$	0	0
$193$	2358.93	0.879792	0.439896	−	0.898049i	$-0.355015\pi$
$193$	2358.93	0.879792	0.439896	+	0.898049i	$0.355015\pi$
$194$	0	0
$195$	−778.209	−0.285788
$196$	0	0
$197$	1236.21	0.447089	0.223545	−	0.974694i	$-0.428237\pi$
$197$	1236.21	0.447089	0.223545	+	0.974694i	$0.428237\pi$
$198$	0	0
$199$	30.6572	0.0109207	0.00546037	−	0.999985i	$-0.498262\pi$
$199$	30.6572	0.0109207	0.00546037	+	0.999985i	$0.498262\pi$
$200$	0	0
$201$	2464.49	0.864835
$202$	0	0
$203$	4330.56	1.49727
$204$	0	0
$205$	−1769.61	−0.602902
$206$	0	0
$207$	417.691	0.140249
$208$	0	0
$209$	−1281.56	−0.424150
$210$	0	0
$211$	416.808	0.135992	0.0679959	−	0.997686i	$-0.478340\pi$
$211$	416.808	0.135992	0.0679959	+	0.997686i	$0.478340\pi$
$212$	0	0
$213$	54.7366	0.0176079
$214$	0	0
$215$	−1828.81	−0.580110
$216$	0	0
$217$	−6417.24	−2.00751
$218$	0	0
$219$	2423.62	0.747822
$220$	0	0
$221$	68.5301	0.0208590
$222$	0	0
$223$	2604.60	0.782139	0.391069	−	0.920361i	$-0.372105\pi$
$223$	2604.60	0.782139	0.391069	+	0.920361i	$0.372105\pi$
$224$	0	0
$225$	454.011	0.134522
$226$	0	0
$227$	−5092.32	−1.48894	−0.744470	−	0.667656i	$-0.767298\pi$
$227$	−5092.32	−1.48894	−0.744470	+	0.667656i	$0.767298\pi$
$228$	0	0
$229$	809.702	0.233653	0.116827	−	0.993152i	$-0.462728\pi$
$229$	809.702	0.233653	0.116827	+	0.993152i	$0.462728\pi$
$230$	0	0
$231$	−7079.08	−2.01632
$232$	0	0
$233$	6122.68	1.72150	0.860752	−	0.509025i	$-0.169994\pi$
$233$	6122.68	1.72150	0.860752	+	0.509025i	$0.169994\pi$
$234$	0	0
$235$	976.457	0.271051
$236$	0	0
$237$	−8358.97	−2.29103
$238$	0	0
$239$	4503.21	1.21878	0.609390	−	0.792871i	$-0.291414\pi$
$239$	4503.21	1.21878	0.609390	+	0.792871i	$0.291414\pi$
$240$	0	0
$241$	−7119.43	−1.90292	−0.951459	−	0.307776i	$-0.900415\pi$
$241$	−7119.43	−1.90292	−0.951459	+	0.307776i	$0.900415\pi$
$242$	0	0
$243$	−4373.90	−1.15467
$244$	0	0
$245$	1823.00	0.475377
$246$	0	0
$247$	749.519	0.193080
$248$	0	0
$249$	9957.26	2.53420
$250$	0	0
$251$	451.556	0.113554	0.0567768	−	0.998387i	$-0.481918\pi$
$251$	451.556	0.113554	0.0567768	+	0.998387i	$0.481918\pi$
$252$	0	0
$253$	910.818	0.226334
$254$	0	0
$255$	−99.4222	−0.0244159
$256$	0	0
$257$	−2413.83	−0.585878	−0.292939	−	0.956131i	$-0.594633\pi$
$257$	−2413.83	−0.585878	−0.292939	+	0.956131i	$0.594633\pi$
$258$	0	0
$259$	4792.51	1.14978
$260$	0	0
$261$	−2956.50	−0.701159
$262$	0	0
$263$	−1214.24	−0.284689	−0.142345	−	0.989817i	$-0.545464\pi$
$263$	−1214.24	−0.284689	−0.142345	+	0.989817i	$0.545464\pi$
$264$	0	0
$265$	−2308.43	−0.535117
$266$	0	0
$267$	5575.22	1.27790
$268$	0	0
$269$	5286.97	1.19833	0.599167	−	0.800624i	$-0.295498\pi$
$269$	5286.97	1.19833	0.599167	+	0.800624i	$0.295498\pi$
$270$	0	0
$271$	−4648.30	−1.04193	−0.520967	−	0.853577i	$-0.674428\pi$
$271$	−4648.30	−1.04193	−0.520967	+	0.853577i	$0.674428\pi$
$272$	0	0
$273$	4140.19	0.917860
$274$	0	0
$275$	990.019	0.217092
$276$	0	0
$277$	−46.5805	−0.0101038	−0.00505190	−	0.999987i	$-0.501608\pi$
$277$	−46.5805	−0.0101038	−0.00505190	+	0.999987i	$0.501608\pi$
$278$	0	0
$279$	4381.08	0.940101
$280$	0	0
$281$	7012.40	1.48870	0.744350	−	0.667790i	$-0.232759\pi$
$281$	7012.40	1.48870	0.744350	+	0.667790i	$0.232759\pi$
$282$	0	0
$283$	7210.00	1.51445	0.757227	−	0.653152i	$-0.226554\pi$
$283$	7210.00	1.51445	0.757227	+	0.653152i	$0.226554\pi$
$284$	0	0
$285$	−1087.39	−0.226005
$286$	0	0
$287$	9414.59	1.93633
$288$	0	0
$289$	−4904.24	−0.998218
$290$	0	0
$291$	−2622.14	−0.528223
$292$	0	0
$293$	−392.296	−0.0782190	−0.0391095	−	0.999235i	$-0.512452\pi$
$293$	−392.296	−0.0782190	−0.0391095	+	0.999235i	$0.512452\pi$
$294$	0	0
$295$	1454.44	0.287054
$296$	0	0
$297$	−2352.41	−0.459598
$298$	0	0
$299$	−532.691	−0.103031
$300$	0	0
$301$	9729.53	1.86313
$302$	0	0
$303$	−9824.02	−1.86262
$304$	0	0
$305$	−1505.24	−0.282590
$306$	0	0
$307$	−5640.01	−1.04851	−0.524255	−	0.851562i	$-0.675656\pi$
$307$	−5640.01	−1.04851	−0.524255	+	0.851562i	$0.675656\pi$
$308$	0	0
$309$	−2133.88	−0.392854
$310$	0	0
$311$	−5270.57	−0.960985	−0.480493	−	0.876999i	$-0.659542\pi$
$311$	−5270.57	−0.960985	−0.480493	+	0.876999i	$0.659542\pi$
$312$	0	0
$313$	2219.57	0.400823	0.200411	−	0.979712i	$-0.435772\pi$
$313$	2219.57	0.400823	0.200411	+	0.979712i	$0.435772\pi$
$314$	0	0
$315$	−2415.41	−0.432041
$316$	0	0
$317$	−7556.36	−1.33882	−0.669412	−	0.742891i	$-0.733454\pi$
$317$	−7556.36	−1.33882	−0.669412	+	0.742891i	$0.733454\pi$
$318$	0	0
$319$	−6446.94	−1.13153
$320$	0	0
$321$	320.013	0.0556430
$322$	0	0
$323$	95.7568	0.0164955
$324$	0	0
$325$	−579.011	−0.0988239
$326$	0	0
$327$	−9658.25	−1.63334
$328$	0	0
$329$	−5194.90	−0.870529
$330$	0	0
$331$	−5161.93	−0.857177	−0.428588	−	0.903500i	$-0.640989\pi$
$331$	−5161.93	−0.857177	−0.428588	+	0.903500i	$0.640989\pi$
$332$	0	0
$333$	−3271.87	−0.538430
$334$	0	0
$335$	1833.66	0.299055
$336$	0	0
$337$	5589.58	0.903513	0.451756	−	0.892141i	$-0.350798\pi$
$337$	5589.58	0.903513	0.451756	+	0.892141i	$0.350798\pi$
$338$	0	0
$339$	−10589.4	−1.69656
$340$	0	0
$341$	9553.39	1.51714
$342$	0	0
$343$	−574.597	−0.0904528
$344$	0	0
$345$	772.818	0.120600
$346$	0	0
$347$	−9794.09	−1.51520	−0.757600	−	0.652719i	$-0.773628\pi$
$347$	−9794.09	−1.51520	−0.757600	+	0.652719i	$0.773628\pi$
$348$	0	0
$349$	8022.81	1.23052	0.615260	−	0.788325i	$-0.289051\pi$
$349$	8022.81	1.23052	0.615260	+	0.788325i	$0.289051\pi$
$350$	0	0
$351$	1375.80	0.209216
$352$	0	0
$353$	4388.90	0.661749	0.330875	−	0.943675i	$-0.392656\pi$
$353$	4388.90	0.661749	0.330875	+	0.943675i	$0.392656\pi$
$354$	0	0
$355$	40.7257	0.00608872
$356$	0	0
$357$	528.941	0.0784161
$358$	0	0
$359$	4538.82	0.667270	0.333635	−	0.942702i	$-0.391725\pi$
$359$	4538.82	0.667270	0.333635	+	0.942702i	$0.391725\pi$
$360$	0	0
$361$	−5811.70	−0.847310
$362$	0	0
$363$	1594.16	0.230500
$364$	0	0
$365$	1803.25	0.258593
$366$	0	0
$367$	−9270.56	−1.31858	−0.659290	−	0.751888i	$-0.729143\pi$
$367$	−9270.56	−1.31858	−0.659290	+	0.751888i	$0.729143\pi$
$368$	0	0
$369$	−6427.38	−0.906764
$370$	0	0
$371$	12281.2	1.71862
$372$	0	0
$373$	−8132.28	−1.12888	−0.564442	−	0.825473i	$-0.690909\pi$
$373$	−8132.28	−1.12888	−0.564442	+	0.825473i	$0.690909\pi$
$374$	0	0
$375$	840.019	0.115676
$376$	0	0
$377$	3770.49	0.515093
$378$	0	0
$379$	7678.93	1.04074	0.520370	−	0.853941i	$-0.325794\pi$
$379$	7678.93	1.04074	0.520370	+	0.853941i	$0.325794\pi$
$380$	0	0
$381$	−14799.7	−1.99006
$382$	0	0
$383$	8597.37	1.14701	0.573505	−	0.819202i	$-0.305583\pi$
$383$	8597.37	1.14701	0.573505	+	0.819202i	$0.305583\pi$
$384$	0	0
$385$	−5267.05	−0.697231
$386$	0	0
$387$	−6642.39	−0.872485
$388$	0	0
$389$	−833.374	−0.108621	−0.0543107	−	0.998524i	$-0.517296\pi$
$389$	−833.374	−0.108621	−0.0543107	+	0.998524i	$0.517296\pi$
$390$	0	0
$391$	−68.0553	−0.00880232
$392$	0	0
$393$	−13895.7	−1.78357
$394$	0	0
$395$	−6219.33	−0.792224
$396$	0	0
$397$	−295.817	−0.0373970	−0.0186985	−	0.999825i	$-0.505952\pi$
$397$	−295.817	−0.0373970	−0.0186985	+	0.999825i	$0.505952\pi$
$398$	0	0
$399$	5785.07	0.725854
$400$	0	0
$401$	5080.23	0.632655	0.316327	−	0.948650i	$-0.397550\pi$
$401$	5080.23	0.632655	0.316327	+	0.948650i	$0.397550\pi$
$402$	0	0
$403$	−5587.29	−0.690627
$404$	0	0
$405$	−4447.65	−0.545693
$406$	0	0
$407$	−7134.64	−0.868922
$408$	0	0
$409$	−6043.43	−0.730632	−0.365316	−	0.930884i	$-0.619039\pi$
$409$	−6043.43	−0.730632	−0.365316	+	0.930884i	$0.619039\pi$
$410$	0	0
$411$	14956.8	1.79505
$412$	0	0
$413$	−7737.85	−0.921924
$414$	0	0
$415$	7408.51	0.876312
$416$	0	0
$417$	−18842.6	−2.21277
$418$	0	0
$419$	9872.50	1.15108	0.575541	−	0.817773i	$-0.304792\pi$
$419$	9872.50	1.15108	0.575541	+	0.817773i	$0.304792\pi$
$420$	0	0
$421$	4088.28	0.473279	0.236639	−	0.971598i	$-0.423954\pi$
$421$	4088.28	0.473279	0.236639	+	0.971598i	$0.423954\pi$
$422$	0	0
$423$	3546.58	0.407661
$424$	0	0
$425$	−73.9732	−0.00844289
$426$	0	0
$427$	8008.13	0.907589
$428$	0	0
$429$	−6163.54	−0.693656
$430$	0	0
$431$	−5325.61	−0.595187	−0.297593	−	0.954693i	$-0.596184\pi$
$431$	−5325.61	−0.595187	−0.297593	+	0.954693i	$0.596184\pi$
$432$	0	0
$433$	2448.60	0.271760	0.135880	−	0.990725i	$-0.456614\pi$
$433$	2448.60	0.271760	0.135880	+	0.990725i	$0.456614\pi$
$434$	0	0
$435$	−5470.15	−0.602928
$436$	0	0
$437$	−744.326	−0.0814781
$438$	0	0
$439$	−3960.18	−0.430544	−0.215272	−	0.976554i	$-0.569064\pi$
$439$	−3960.18	−0.430544	−0.215272	+	0.976554i	$0.569064\pi$
$440$	0	0
$441$	6621.32	0.714968
$442$	0	0
$443$	6174.42	0.662202	0.331101	−	0.943595i	$-0.392580\pi$
$443$	6174.42	0.662202	0.331101	+	0.943595i	$0.392580\pi$
$444$	0	0
$445$	4148.14	0.441889
$446$	0	0
$447$	−8889.68	−0.940644
$448$	0	0
$449$	−6003.87	−0.631048	−0.315524	−	0.948918i	$-0.602180\pi$
$449$	−6003.87	−0.631048	−0.315524	+	0.948918i	$0.602180\pi$
$450$	0	0
$451$	−14015.6	−1.46334
$452$	0	0
$453$	−13792.1	−1.43048
$454$	0	0
$455$	3080.43	0.317391
$456$	0	0
$457$	6044.87	0.618746	0.309373	−	0.950941i	$-0.399881\pi$
$457$	6044.87	0.618746	0.309373	+	0.950941i	$0.399881\pi$
$458$	0	0
$459$	175.769	0.0178741
$460$	0	0
$461$	−4620.62	−0.466819	−0.233410	−	0.972378i	$-0.574988\pi$
$461$	−4620.62	−0.466819	−0.233410	+	0.972378i	$0.574988\pi$
$462$	0	0
$463$	10322.9	1.03616	0.518082	−	0.855331i	$-0.326646\pi$
$463$	10322.9	1.03616	0.518082	+	0.855331i	$0.326646\pi$
$464$	0	0
$465$	8105.94	0.808395
$466$	0	0
$467$	−16496.0	−1.63456	−0.817282	−	0.576238i	$-0.804520\pi$
$467$	−16496.0	−1.63456	−0.817282	+	0.576238i	$0.804520\pi$
$468$	0	0
$469$	−9755.34	−0.960469
$470$	0	0
$471$	−2561.53	−0.250592
$472$	0	0
$473$	−14484.4	−1.40802
$474$	0	0
$475$	−809.050	−0.0781511
$476$	0	0
$477$	−8384.44	−0.804816
$478$	0	0
$479$	−3767.40	−0.359367	−0.179684	−	0.983724i	$-0.557507\pi$
$479$	−3767.40	−0.359367	−0.179684	+	0.983724i	$0.557507\pi$
$480$	0	0
$481$	4172.69	0.395547
$482$	0	0
$483$	−4111.51	−0.387329
$484$	0	0
$485$	−1950.96	−0.182656
$486$	0	0
$487$	2901.21	0.269951	0.134976	−	0.990849i	$-0.456904\pi$
$487$	2901.21	0.269951	0.134976	+	0.990849i	$0.456904\pi$
$488$	0	0
$489$	15847.2	1.46551
$490$	0	0
$491$	13508.0	1.24157	0.620783	−	0.783982i	$-0.286815\pi$
$491$	13508.0	1.24157	0.620783	+	0.783982i	$0.286815\pi$
$492$	0	0
$493$	481.709	0.0440062
$494$	0	0
$495$	3595.84	0.326507
$496$	0	0
$497$	−216.667	−0.0195550
$498$	0	0
$499$	7470.55	0.670196	0.335098	−	0.942183i	$-0.391231\pi$
$499$	7470.55	0.670196	0.335098	+	0.942183i	$0.391231\pi$
$500$	0	0
$501$	969.498	0.0864551
$502$	0	0
$503$	−6962.88	−0.617215	−0.308608	−	0.951189i	$-0.599863\pi$
$503$	−6962.88	−0.617215	−0.308608	+	0.951189i	$0.599863\pi$
$504$	0	0
$505$	−7309.37	−0.644085
$506$	0	0
$507$	−11159.4	−0.977531
$508$	0	0
$509$	12963.8	1.12890	0.564451	−	0.825467i	$-0.309088\pi$
$509$	12963.8	1.12890	0.564451	+	0.825467i	$0.309088\pi$
$510$	0	0
$511$	−9593.55	−0.830516
$512$	0	0
$513$	1922.40	0.165450
$514$	0	0
$515$	−1587.67	−0.135847
$516$	0	0
$517$	7733.69	0.657886
$518$	0	0
$519$	21573.9	1.82464
$520$	0	0
$521$	−2256.37	−0.189738	−0.0948688	−	0.995490i	$-0.530243\pi$
$521$	−2256.37	−0.189738	−0.0948688	+	0.995490i	$0.530243\pi$
$522$	0	0
$523$	−20058.2	−1.67703	−0.838514	−	0.544880i	$-0.816575\pi$
$523$	−20058.2	−1.67703	−0.838514	+	0.544880i	$0.816575\pi$
$524$	0	0
$525$	−4469.03	−0.371513
$526$	0	0
$527$	−713.819	−0.0590028
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	5282.66	0.431729
$532$	0	0
$533$	8196.99	0.666137
$534$	0	0
$535$	238.100	0.0192410
$536$	0	0
$537$	27353.9	2.19815
$538$	0	0
$539$	14438.5	1.15382
$540$	0	0
$541$	3300.93	0.262325	0.131163	−	0.991361i	$-0.458129\pi$
$541$	3300.93	0.262325	0.131163	+	0.991361i	$0.458129\pi$
$542$	0	0
$543$	−27708.1	−2.18981
$544$	0	0
$545$	−7186.03	−0.564800
$546$	0	0
$547$	18198.7	1.42252	0.711260	−	0.702929i	$-0.248125\pi$
$547$	18198.7	1.42252	0.711260	+	0.702929i	$0.248125\pi$
$548$	0	0
$549$	−5467.18	−0.425016
$550$	0	0
$551$	5268.48	0.407341
$552$	0	0
$553$	33087.8	2.54437
$554$	0	0
$555$	−6053.66	−0.462997
$556$	0	0
$557$	5899.11	0.448749	0.224375	−	0.974503i	$-0.427966\pi$
$557$	5899.11	0.448749	0.224375	+	0.974503i	$0.427966\pi$
$558$	0	0
$559$	8471.20	0.640954
$560$	0	0
$561$	−787.439	−0.0592615
$562$	0	0
$563$	18106.3	1.35540	0.677698	−	0.735340i	$-0.262977\pi$
$563$	18106.3	1.35540	0.677698	+	0.735340i	$0.262977\pi$
$564$	0	0
$565$	−7878.81	−0.586662
$566$	0	0
$567$	23662.2	1.75259
$568$	0	0
$569$	5985.76	0.441012	0.220506	−	0.975386i	$-0.429229\pi$
$569$	5985.76	0.441012	0.220506	+	0.975386i	$0.429229\pi$
$570$	0	0
$571$	24457.6	1.79250	0.896252	−	0.443546i	$-0.146280\pi$
$571$	24457.6	1.79250	0.896252	+	0.443546i	$0.146280\pi$
$572$	0	0
$573$	12277.9	0.895140
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	8812.01	0.635787	0.317893	−	0.948126i	$-0.397025\pi$
$577$	8812.01	0.635787	0.317893	+	0.948126i	$0.397025\pi$
$578$	0	0
$579$	15852.4	1.13783
$580$	0	0
$581$	−39414.4	−2.81443
$582$	0	0
$583$	−18283.1	−1.29882
$584$	0	0
$585$	−2103.02	−0.148631
$586$	0	0
$587$	−21952.3	−1.54356	−0.771779	−	0.635891i	$-0.780633\pi$
$587$	−21952.3	−1.54356	−0.771779	+	0.635891i	$0.780633\pi$
$588$	0	0
$589$	−7807.09	−0.546156
$590$	0	0
$591$	8307.55	0.578218
$592$	0	0
$593$	916.597	0.0634741	0.0317370	−	0.999496i	$-0.489896\pi$
$593$	916.597	0.0634741	0.0317370	+	0.999496i	$0.489896\pi$
$594$	0	0
$595$	393.549	0.0271158
$596$	0	0
$597$	206.021	0.0141237
$598$	0	0
$599$	7327.24	0.499804	0.249902	−	0.968271i	$-0.419602\pi$
$599$	7327.24	0.499804	0.249902	+	0.968271i	$0.419602\pi$
$600$	0	0
$601$	4378.57	0.297181	0.148590	−	0.988899i	$-0.452526\pi$
$601$	4378.57	0.297181	0.148590	+	0.988899i	$0.452526\pi$
$602$	0	0
$603$	6660.02	0.449779
$604$	0	0
$605$	1186.10	0.0797057
$606$	0	0
$607$	13150.8	0.879367	0.439683	−	0.898153i	$-0.355091\pi$
$607$	13150.8	0.879367	0.439683	+	0.898153i	$0.355091\pi$
$608$	0	0
$609$	29102.1	1.93641
$610$	0	0
$611$	−4523.04	−0.299480
$612$	0	0
$613$	23672.8	1.55976	0.779881	−	0.625928i	$-0.215279\pi$
$613$	23672.8	1.55976	0.779881	+	0.625928i	$0.215279\pi$
$614$	0	0
$615$	−11892.0	−0.779729
$616$	0	0
$617$	5581.73	0.364201	0.182101	−	0.983280i	$-0.441710\pi$
$617$	5581.73	0.364201	0.182101	+	0.983280i	$0.441710\pi$
$618$	0	0
$619$	−24249.7	−1.57460	−0.787301	−	0.616568i	$-0.788522\pi$
$619$	−24249.7	−1.57460	−0.787301	+	0.616568i	$0.788522\pi$
$620$	0	0
$621$	−1366.27	−0.0882875
$622$	0	0
$623$	−22068.7	−1.41921
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−8612.28	−0.548551
$628$	0	0
$629$	533.093	0.0337930
$630$	0	0
$631$	−15717.9	−0.991633	−0.495816	−	0.868427i	$-0.665131\pi$
$631$	−15717.9	−0.991633	−0.495816	+	0.868427i	$0.665131\pi$
$632$	0	0
$633$	2801.02	0.175877
$634$	0	0
$635$	−11011.5	−0.688152
$636$	0	0
$637$	−8444.32	−0.525237
$638$	0	0
$639$	147.919	0.00915743
$640$	0	0
$641$	−17043.9	−1.05022	−0.525111	−	0.851034i	$-0.675976\pi$
$641$	−17043.9	−1.05022	−0.525111	+	0.851034i	$0.675976\pi$
$642$	0	0
$643$	6648.00	0.407732	0.203866	−	0.978999i	$-0.434649\pi$
$643$	6648.00	0.407732	0.203866	+	0.978999i	$0.434649\pi$
$644$	0	0
$645$	−12289.9	−0.750252
$646$	0	0
$647$	1899.44	0.115417	0.0577085	−	0.998333i	$-0.481621\pi$
$647$	1899.44	0.115417	0.0577085	+	0.998333i	$0.481621\pi$
$648$	0	0
$649$	11519.4	0.696727
$650$	0	0
$651$	−43124.8	−2.59631
$652$	0	0
$653$	14942.8	0.895492	0.447746	−	0.894161i	$-0.352227\pi$
$653$	14942.8	0.895492	0.447746	+	0.894161i	$0.352227\pi$
$654$	0	0
$655$	−10338.8	−0.616749
$656$	0	0
$657$	6549.56	0.388923
$658$	0	0
$659$	−13828.4	−0.817419	−0.408710	−	0.912664i	$-0.634021\pi$
$659$	−13828.4	−0.817419	−0.408710	+	0.912664i	$0.634021\pi$
$660$	0	0
$661$	−17897.0	−1.05312	−0.526559	−	0.850138i	$-0.676518\pi$
$661$	−17897.0	−1.05312	−0.526559	+	0.850138i	$0.676518\pi$
$662$	0	0
$663$	460.533	0.0269768
$664$	0	0
$665$	4304.27	0.250996
$666$	0	0
$667$	−3744.36	−0.217365
$668$	0	0
$669$	17503.3	1.01154
$670$	0	0
$671$	−11921.8	−0.685893
$672$	0	0
$673$	−13486.7	−0.772475	−0.386238	−	0.922399i	$-0.626226\pi$
$673$	−13486.7	−0.772475	−0.386238	+	0.922399i	$0.626226\pi$
$674$	0	0
$675$	−1485.08	−0.0846824
$676$	0	0
$677$	−11202.2	−0.635949	−0.317974	−	0.948099i	$-0.603003\pi$
$677$	−11202.2	−0.635949	−0.317974	+	0.948099i	$0.603003\pi$
$678$	0	0
$679$	10379.4	0.586633
$680$	0	0
$681$	−34221.2	−1.92564
$682$	0	0
$683$	12993.0	0.727909	0.363955	−	0.931417i	$-0.381426\pi$
$683$	12993.0	0.727909	0.363955	+	0.931417i	$0.381426\pi$
$684$	0	0
$685$	11128.3	0.620718
$686$	0	0
$687$	5441.32	0.302183
$688$	0	0
$689$	10692.9	0.591243
$690$	0	0
$691$	19132.4	1.05330	0.526650	−	0.850082i	$-0.323448\pi$
$691$	19132.4	1.05330	0.526650	+	0.850082i	$0.323448\pi$
$692$	0	0
$693$	−19130.4	−1.04864
$694$	0	0
$695$	−14019.4	−0.765162
$696$	0	0
$697$	1047.23	0.0569105
$698$	0	0
$699$	41145.4	2.22641
$700$	0	0
$701$	21743.7	1.17154	0.585768	−	0.810479i	$-0.300793\pi$
$701$	21743.7	1.17154	0.585768	+	0.810479i	$0.300793\pi$
$702$	0	0
$703$	5830.48	0.312803
$704$	0	0
$705$	6561.94	0.350549
$706$	0	0
$707$	38887.0	2.06859
$708$	0	0
$709$	−4085.34	−0.216401	−0.108200	−	0.994129i	$-0.534509\pi$
$709$	−4085.34	−0.216401	−0.108200	+	0.994129i	$0.534509\pi$
$710$	0	0
$711$	−22589.2	−1.19151
$712$	0	0
$713$	5548.58	0.291439
$714$	0	0
$715$	−4585.86	−0.239862
$716$	0	0
$717$	30262.3	1.57624
$718$	0	0
$719$	−34076.1	−1.76749	−0.883744	−	0.467971i	$-0.844985\pi$
$719$	−34076.1	−1.76749	−0.883744	+	0.467971i	$0.844985\pi$
$720$	0	0
$721$	8446.65	0.436296
$722$	0	0
$723$	−47843.7	−2.46103
$724$	0	0
$725$	−4069.96	−0.208489
$726$	0	0
$727$	880.475	0.0449175	0.0224587	−	0.999748i	$-0.492851\pi$
$727$	880.475	0.0449175	0.0224587	+	0.999748i	$0.492851\pi$
$728$	0	0
$729$	−5375.94	−0.273126
$730$	0	0
$731$	1082.26	0.0547590
$732$	0	0
$733$	−14731.6	−0.742324	−0.371162	−	0.928568i	$-0.621041\pi$
$733$	−14731.6	−0.742324	−0.371162	+	0.928568i	$0.621041\pi$
$734$	0	0
$735$	12250.9	0.614803
$736$	0	0
$737$	14522.9	0.725856
$738$	0	0
$739$	−6380.78	−0.317619	−0.158810	−	0.987309i	$-0.550766\pi$
$739$	−6380.78	−0.317619	−0.158810	+	0.987309i	$0.550766\pi$
$740$	0	0
$741$	5036.88	0.249709
$742$	0	0
$743$	11649.5	0.575208	0.287604	−	0.957749i	$-0.407141\pi$
$743$	11649.5	0.575208	0.287604	+	0.957749i	$0.407141\pi$
$744$	0	0
$745$	−6614.20	−0.325269
$746$	0	0
$747$	26908.4	1.31797
$748$	0	0
$749$	−1266.73	−0.0617960
$750$	0	0
$751$	−14143.0	−0.687199	−0.343600	−	0.939116i	$-0.611646\pi$
$751$	−14143.0	−0.687199	−0.343600	+	0.939116i	$0.611646\pi$
$752$	0	0
$753$	3034.52	0.146858
$754$	0	0
$755$	−10261.7	−0.494653
$756$	0	0
$757$	−30054.3	−1.44299	−0.721495	−	0.692420i	$-0.756545\pi$
$757$	−30054.3	−1.44299	−0.721495	+	0.692420i	$0.756545\pi$
$758$	0	0
$759$	6120.83	0.292717
$760$	0	0
$761$	29327.5	1.39701	0.698503	−	0.715607i	$-0.253850\pi$
$761$	29327.5	1.39701	0.698503	+	0.715607i	$0.253850\pi$
$762$	0	0
$763$	38230.8	1.81395
$764$	0	0
$765$	−268.677	−0.0126981
$766$	0	0
$767$	−6737.11	−0.317161
$768$	0	0
$769$	−16343.8	−0.766413	−0.383207	−	0.923663i	$-0.625180\pi$
$769$	−16343.8	−0.766413	−0.383207	+	0.923663i	$0.625180\pi$
$770$	0	0
$771$	−16221.3	−0.757712
$772$	0	0
$773$	−32875.3	−1.52968	−0.764839	−	0.644221i	$-0.777182\pi$
$773$	−32875.3	−1.52968	−0.764839	+	0.644221i	$0.777182\pi$
$774$	0	0
$775$	6031.07	0.279538
$776$	0	0
$777$	32206.4	1.48700
$778$	0	0
$779$	11453.6	0.526788
$780$	0	0
$781$	322.554	0.0147783
$782$	0	0
$783$	9670.73	0.441384
$784$	0	0
$785$	−1905.85	−0.0866533
$786$	0	0
$787$	15927.1	0.721397	0.360699	−	0.932682i	$-0.382538\pi$
$787$	15927.1	0.721397	0.360699	+	0.932682i	$0.382538\pi$
$788$	0	0
$789$	−8159.88	−0.368187
$790$	0	0
$791$	41916.5	1.88417
$792$	0	0
$793$	6972.43	0.312230
$794$	0	0
$795$	−15513.0	−0.692063
$796$	0	0
$797$	−21390.5	−0.950676	−0.475338	−	0.879803i	$-0.657674\pi$
$797$	−21390.5	−0.950676	−0.475338	+	0.879803i	$0.657674\pi$
$798$	0	0
$799$	−577.853	−0.0255857
$800$	0	0
$801$	15066.4	0.664601
$802$	0	0
$803$	14282.0	0.627647
$804$	0	0
$805$	−3059.09	−0.133936
$806$	0	0
$807$	35529.2	1.54980
$808$	0	0
$809$	23294.4	1.01234	0.506172	−	0.862432i	$-0.331060\pi$
$809$	23294.4	1.01234	0.506172	+	0.862432i	$0.331060\pi$
$810$	0	0
$811$	33524.8	1.45156	0.725779	−	0.687928i	$-0.241479\pi$
$811$	33524.8	1.45156	0.725779	+	0.687928i	$0.241479\pi$
$812$	0	0
$813$	−31237.3	−1.34753
$814$	0	0
$815$	11790.8	0.506766
$816$	0	0
$817$	11836.8	0.506874
$818$	0	0
$819$	11188.4	0.477356
$820$	0	0
$821$	−12493.6	−0.531095	−0.265548	−	0.964098i	$-0.585553\pi$
$821$	−12493.6	−0.531095	−0.265548	+	0.964098i	$0.585553\pi$
$822$	0	0
$823$	−6236.39	−0.264140	−0.132070	−	0.991240i	$-0.542162\pi$
$823$	−6236.39	−0.264140	−0.132070	+	0.991240i	$0.542162\pi$
$824$	0	0
$825$	6653.08	0.280764
$826$	0	0
$827$	31346.5	1.31805	0.659023	−	0.752122i	$-0.270970\pi$
$827$	31346.5	1.31805	0.659023	+	0.752122i	$0.270970\pi$
$828$	0	0
$829$	1452.79	0.0608657	0.0304328	−	0.999537i	$-0.490311\pi$
$829$	1452.79	0.0608657	0.0304328	+	0.999537i	$0.490311\pi$
$830$	0	0
$831$	−313.028	−0.0130672
$832$	0	0
$833$	−1078.83	−0.0448729
$834$	0	0
$835$	721.336	0.0298957
$836$	0	0
$837$	−14330.6	−0.591800
$838$	0	0
$839$	−18327.7	−0.754161	−0.377081	−	0.926180i	$-0.623072\pi$
$839$	−18327.7	−0.754161	−0.377081	+	0.926180i	$0.623072\pi$
$840$	0	0
$841$	2114.34	0.0866924
$842$	0	0
$843$	47124.4	1.92533
$844$	0	0
$845$	−8302.97	−0.338024
$846$	0	0
$847$	−6310.25	−0.255989
$848$	0	0
$849$	48452.3	1.95863
$850$	0	0
$851$	−4143.78	−0.166918
$852$	0	0
$853$	48657.4	1.95311	0.976553	−	0.215279i	$-0.0690660\pi$
$853$	48657.4	1.95311	0.976553	+	0.215279i	$0.0690660\pi$
$854$	0	0
$855$	−2938.54	−0.117539
$856$	0	0
$857$	−2116.61	−0.0843666	−0.0421833	−	0.999110i	$-0.513431\pi$
$857$	−2116.61	−0.0843666	−0.0421833	+	0.999110i	$0.513431\pi$
$858$	0	0
$859$	−16391.3	−0.651063	−0.325531	−	0.945531i	$-0.605543\pi$
$859$	−16391.3	−0.651063	−0.325531	+	0.945531i	$0.605543\pi$
$860$	0	0
$861$	63267.5	2.50424
$862$	0	0
$863$	33241.8	1.31120	0.655600	−	0.755109i	$-0.272416\pi$
$863$	33241.8	1.31120	0.655600	+	0.755109i	$0.272416\pi$
$864$	0	0
$865$	16051.6	0.630950
$866$	0	0
$867$	−32957.3	−1.29099
$868$	0	0
$869$	−49258.1	−1.92286
$870$	0	0
$871$	−8493.67	−0.330422
$872$	0	0
$873$	−7086.05	−0.274715
$874$	0	0
$875$	−3325.10	−0.128467
$876$	0	0
$877$	14732.4	0.567249	0.283625	−	0.958935i	$-0.408463\pi$
$877$	14732.4	0.567249	0.283625	+	0.958935i	$0.408463\pi$
$878$	0	0
$879$	−2636.29	−0.101160
$880$	0	0
$881$	−604.107	−0.0231020	−0.0115510	−	0.999933i	$-0.503677\pi$
$881$	−604.107	−0.0231020	−0.0115510	+	0.999933i	$0.503677\pi$
$882$	0	0
$883$	−40030.5	−1.52563	−0.762816	−	0.646616i	$-0.776184\pi$
$883$	−40030.5	−1.52563	−0.762816	+	0.646616i	$0.776184\pi$
$884$	0	0
$885$	9774.07	0.371245
$886$	0	0
$887$	35364.3	1.33869	0.669344	−	0.742953i	$-0.266575\pi$
$887$	35364.3	1.33869	0.669344	+	0.742953i	$0.266575\pi$
$888$	0	0
$889$	58582.7	2.21012
$890$	0	0
$891$	−35226.1	−1.32449
$892$	0	0
$893$	−6320.02	−0.236832
$894$	0	0
$895$	20352.1	0.760108
$896$	0	0
$897$	−3579.76	−0.133249
$898$	0	0
$899$	−39273.9	−1.45702
$900$	0	0
$901$	1366.10	0.0505120
$902$	0	0
$903$	65383.9	2.40957
$904$	0	0
$905$	−20615.7	−0.757224
$906$	0	0
$907$	−46960.3	−1.71918	−0.859588	−	0.510988i	$-0.829280\pi$
$907$	−46960.3	−1.71918	−0.859588	+	0.510988i	$0.829280\pi$
$908$	0	0
$909$	−26548.3	−0.968704
$910$	0	0
$911$	23027.3	0.837461	0.418731	−	0.908110i	$-0.362475\pi$
$911$	23027.3	0.837461	0.418731	+	0.908110i	$0.362475\pi$
$912$	0	0
$913$	58676.5	2.12696
$914$	0	0
$915$	−10115.5	−0.365472
$916$	0	0
$917$	55004.0	1.98080
$918$	0	0
$919$	20203.7	0.725198	0.362599	−	0.931945i	$-0.381889\pi$
$919$	20203.7	0.725198	0.362599	+	0.931945i	$0.381889\pi$
$920$	0	0
$921$	−37901.7	−1.35603
$922$	0	0
$923$	−188.645	−0.00672733
$924$	0	0
$925$	−4504.11	−0.160102
$926$	0	0
$927$	−5766.56	−0.204314
$928$	0	0
$929$	28382.1	1.00235	0.501177	−	0.865345i	$-0.332901\pi$
$929$	28382.1	1.00235	0.501177	+	0.865345i	$0.332901\pi$
$930$	0	0
$931$	−11799.2	−0.415363
$932$	0	0
$933$	−35419.0	−1.24284
$934$	0	0
$935$	−585.879	−0.0204923
$936$	0	0
$937$	855.936	0.0298423	0.0149211	−	0.999889i	$-0.495250\pi$
$937$	855.936	0.0298423	0.0149211	+	0.999889i	$0.495250\pi$
$938$	0	0
$939$	14915.8	0.518381
$940$	0	0
$941$	−15957.0	−0.552797	−0.276399	−	0.961043i	$-0.589141\pi$
$941$	−15957.0	−0.552797	−0.276399	+	0.961043i	$0.589141\pi$
$942$	0	0
$943$	−8140.20	−0.281104
$944$	0	0
$945$	7900.84	0.271973
$946$	0	0
$947$	51719.6	1.77472	0.887361	−	0.461074i	$-0.152536\pi$
$947$	51719.6	1.77472	0.887361	+	0.461074i	$0.152536\pi$
$948$	0	0
$949$	−8352.81	−0.285715
$950$	0	0
$951$	−50779.9	−1.73149
$952$	0	0
$953$	−25040.6	−0.851147	−0.425574	−	0.904924i	$-0.639928\pi$
$953$	−25040.6	−0.851147	−0.425574	+	0.904924i	$0.639928\pi$
$954$	0	0
$955$	9135.11	0.309534
$956$	0	0
$957$	−43324.5	−1.46341
$958$	0	0
$959$	−59204.5	−1.99355
$960$	0	0
$961$	28407.0	0.953543
$962$	0	0
$963$	864.800	0.0289385
$964$	0	0
$965$	11794.7	0.393455
$966$	0	0
$967$	35852.9	1.19230	0.596148	−	0.802874i	$-0.296697\pi$
$967$	35852.9	1.19230	0.596148	+	0.802874i	$0.296697\pi$
$968$	0	0
$969$	643.500	0.0213335
$970$	0	0
$971$	19740.9	0.652437	0.326218	−	0.945294i	$-0.394225\pi$
$971$	19740.9	0.652437	0.326218	+	0.945294i	$0.394225\pi$
$972$	0	0
$973$	74585.6	2.45746
$974$	0	0
$975$	−3891.05	−0.127808
$976$	0	0
$977$	−28263.7	−0.925524	−0.462762	−	0.886483i	$-0.653141\pi$
$977$	−28263.7	−0.925524	−0.462762	+	0.886483i	$0.653141\pi$
$978$	0	0
$979$	32853.9	1.07254
$980$	0	0
$981$	−26100.3	−0.849459
$982$	0	0
$983$	24050.0	0.780340	0.390170	−	0.920743i	$-0.372416\pi$
$983$	24050.0	0.780340	0.390170	+	0.920743i	$0.372416\pi$
$984$	0	0
$985$	6181.07	0.199944
$986$	0	0
$987$	−34910.5	−1.12585
$988$	0	0
$989$	−8412.51	−0.270477
$990$	0	0
$991$	52738.0	1.69049	0.845246	−	0.534378i	$-0.179454\pi$
$991$	52738.0	1.69049	0.845246	+	0.534378i	$0.179454\pi$
$992$	0	0
$993$	−34689.0	−1.10858
$994$	0	0
$995$	153.286	0.00488391
$996$	0	0
$997$	6903.86	0.219305	0.109653	−	0.993970i	$-0.465026\pi$
$997$	6903.86	0.219305	0.109653	+	0.993970i	$0.465026\pi$
$998$	0	0
$999$	10702.3	0.338945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.h.1.2		2
4.3	odd	2		115.4.a.c.1.1	✓	2
12.11	even	2		1035.4.a.g.1.2		2
20.3	even	4		575.4.b.f.24.3		4
20.7	even	4		575.4.b.f.24.2		4
20.19	odd	2		575.4.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.c.1.1	✓	2	4.3	odd	2
575.4.a.h.1.2		2	20.19	odd	2
575.4.b.f.24.2		4	20.7	even	4
575.4.b.f.24.3		4	20.3	even	4
1035.4.a.g.1.2		2	12.11	even	2
1840.4.a.h.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q+6.72015 q^{3} +5.00000 q^{5} -26.6008 q^{7} +18.1605 q^{9} +O(q^{10})\)	\(q+6.72015 q^{3} +5.00000 q^{5} -26.6008 q^{7} +18.1605 q^{9} +39.6008 q^{11} -23.1605 q^{13} +33.6008 q^{15} -2.95893 q^{17} -32.3620 q^{19} -178.761 q^{21} +23.0000 q^{23} +25.0000 q^{25} -59.4031 q^{27} -162.798 q^{29} +241.243 q^{31} +266.123 q^{33} -133.004 q^{35} -180.164 q^{37} -155.642 q^{39} -353.922 q^{41} -365.761 q^{43} +90.8023 q^{45} +195.291 q^{47} +364.601 q^{49} -19.8844 q^{51} -461.687 q^{53} +198.004 q^{55} -217.478 q^{57} +290.888 q^{59} -301.049 q^{61} -483.082 q^{63} -115.802 q^{65} +366.732 q^{67} +154.564 q^{69} +8.14513 q^{71} +360.650 q^{73} +168.004 q^{75} -1053.41 q^{77} -1243.87 q^{79} -889.530 q^{81} +1481.70 q^{83} -14.7946 q^{85} -1094.03 q^{87} +829.628 q^{89} +616.086 q^{91} +1621.19 q^{93} -161.810 q^{95} -390.191 q^{97} +719.168 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 10 q^{5} - q^{7} + 5 q^{9}+O(q^{10}) \) 2 * q + 3 * q^3 + 10 * q^5 - q^7 + 5 * q^9	\( 2 q + 3 q^{3} + 10 q^{5} - q^{7} + 5 q^{9} + 27 q^{11} - 15 q^{13} + 15 q^{15} - 79 q^{17} + 71 q^{19} - 274 q^{21} + 46 q^{23} + 50 q^{25} + 90 q^{27} - 430 q^{29} + 305 q^{31} + 313 q^{33} - 5 q^{35} - 68 q^{37} - 186 q^{39} - 593 q^{41} - 648 q^{43} + 25 q^{45} - 382 q^{47} + 677 q^{49} + 263 q^{51} - 464 q^{53} + 135 q^{55} - 602 q^{57} + 18 q^{59} - 7 q^{61} - 820 q^{63} - 75 q^{65} - 60 q^{67} + 69 q^{69} + 1029 q^{71} + 74 q^{73} + 75 q^{75} - 1376 q^{77} - 692 q^{79} - 1090 q^{81} + 1460 q^{83} - 395 q^{85} - 100 q^{87} - 220 q^{89} + 825 q^{91} + 1384 q^{93} + 355 q^{95} + 1339 q^{97} + 885 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 10 * q^5 - q^7 + 5 * q^9 + 27 * q^11 - 15 * q^13 + 15 * q^15 - 79 * q^17 + 71 * q^19 - 274 * q^21 + 46 * q^23 + 50 * q^25 + 90 * q^27 - 430 * q^29 + 305 * q^31 + 313 * q^33 - 5 * q^35 - 68 * q^37 - 186 * q^39 - 593 * q^41 - 648 * q^43 + 25 * q^45 - 382 * q^47 + 677 * q^49 + 263 * q^51 - 464 * q^53 + 135 * q^55 - 602 * q^57 + 18 * q^59 - 7 * q^61 - 820 * q^63 - 75 * q^65 - 60 * q^67 + 69 * q^69 + 1029 * q^71 + 74 * q^73 + 75 * q^75 - 1376 * q^77 - 692 * q^79 - 1090 * q^81 + 1460 * q^83 - 395 * q^85 - 100 * q^87 - 220 * q^89 + 825 * q^91 + 1384 * q^93 + 355 * q^95 + 1339 * q^97 + 885 * q^99

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