LMFDB - Embedded newform 1840.4.a.h.1.1

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			1.1
Root			$-4.72015$ 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)

$f(q)$	$=$	$q-3.72015 q^{3} +5.00000 q^{5} +25.6008 q^{7} -13.1605 q^{9} +O(q^{10})$	$q-3.72015 q^{3} +5.00000 q^{5} +25.6008 q^{7} -13.1605 q^{9} -12.6008 q^{11} +8.16046 q^{13} -18.6008 q^{15} -76.0411 q^{17} +103.362 q^{19} -95.2388 q^{21} +23.0000 q^{23} +25.0000 q^{25} +149.403 q^{27} -267.202 q^{29} +63.7574 q^{31} +46.8768 q^{33} +128.004 q^{35} +112.164 q^{37} -30.3582 q^{39} -239.078 q^{41} -282.239 q^{43} -65.8023 q^{45} -577.291 q^{47} +312.399 q^{49} +282.884 q^{51} -2.31326 q^{53} -63.0038 q^{55} -384.522 q^{57} -272.888 q^{59} +294.049 q^{61} -336.918 q^{63} +40.8023 q^{65} -426.732 q^{67} -85.5635 q^{69} +1020.85 q^{71} -286.650 q^{73} -93.0038 q^{75} -322.589 q^{77} +551.866 q^{79} -200.470 q^{81} -21.7021 q^{83} -380.205 q^{85} +994.031 q^{87} -1049.63 q^{89} +208.914 q^{91} -237.187 q^{93} +516.810 q^{95} +1729.19 q^{97} +165.832 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.72015	−0.715944	−0.357972	−	0.933732i	$-0.616532\pi$
$3$	−3.72015	−0.715944	−0.357972	+	0.933732i	$0.616532\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	25.6008	1.38231	0.691156	−	0.722706i	$-0.257102\pi$
$7$	25.6008	1.38231	0.691156	+	0.722706i	$0.257102\pi$
$8$	0	0
$9$	−13.1605	−0.487424
$10$	0	0
$11$	−12.6008	−0.345389	−0.172694	−	0.984975i	$-0.555247\pi$
$11$	−12.6008	−0.345389	−0.172694	+	0.984975i	$0.555247\pi$
$12$	0	0
$13$	8.16046	0.174100	0.0870502	−	0.996204i	$-0.472256\pi$
$13$	8.16046	0.174100	0.0870502	+	0.996204i	$0.472256\pi$
$14$	0	0
$15$	−18.6008	−0.320180
$16$	0	0
$17$	−76.0411	−1.08486	−0.542431	−	0.840100i	$-0.682496\pi$
$17$	−76.0411	−1.08486	−0.542431	+	0.840100i	$0.682496\pi$
$18$	0	0
$19$	103.362	1.24805	0.624023	−	0.781406i	$-0.285497\pi$
$19$	103.362	1.24805	0.624023	+	0.781406i	$0.285497\pi$
$20$	0	0
$21$	−95.2388	−0.989657
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	149.403	1.06491
$28$	0	0
$29$	−267.202	−1.71097	−0.855484	−	0.517829i	$-0.826740\pi$
$29$	−267.202	−1.71097	−0.855484	+	0.517829i	$0.826740\pi$
$30$	0	0
$31$	63.7574	0.369392	0.184696	−	0.982796i	$-0.440870\pi$
$31$	63.7574	0.369392	0.184696	+	0.982796i	$0.440870\pi$
$32$	0	0
$33$	46.8768	0.247279
$34$	0	0
$35$	128.004	0.618188
$36$	0	0
$37$	112.164	0.498370	0.249185	−	0.968456i	$-0.419837\pi$
$37$	112.164	0.498370	0.249185	+	0.968456i	$0.419837\pi$
$38$	0	0
$39$	−30.3582	−0.124646
$40$	0	0
$41$	−239.078	−0.910677	−0.455339	−	0.890318i	$-0.650482\pi$
$41$	−239.078	−0.910677	−0.455339	+	0.890318i	$0.650482\pi$
$42$	0	0
$43$	−282.239	−1.00095	−0.500477	−	0.865750i	$-0.666842\pi$
$43$	−282.239	−1.00095	−0.500477	+	0.865750i	$0.666842\pi$
$44$	0	0
$45$	−65.8023	−0.217983
$46$	0	0
$47$	−577.291	−1.79163	−0.895815	−	0.444427i	$-0.853407\pi$
$47$	−577.291	−1.79163	−0.895815	+	0.444427i	$0.853407\pi$
$48$	0	0
$49$	312.399	0.910785
$50$	0	0
$51$	282.884	0.776701
$52$	0	0
$53$	−2.31326	−0.00599529	−0.00299764	−	0.999996i	$-0.500954\pi$
$53$	−2.31326	−0.00599529	−0.00299764	+	0.999996i	$0.500954\pi$
$54$	0	0
$55$	−63.0038	−0.154462
$56$	0	0
$57$	−384.522	−0.893531
$58$	0	0
$59$	−272.888	−0.602153	−0.301077	−	0.953600i	$-0.597346\pi$
$59$	−272.888	−0.602153	−0.301077	+	0.953600i	$0.597346\pi$
$60$	0	0
$61$	294.049	0.617198	0.308599	−	0.951192i	$-0.400140\pi$
$61$	294.049	0.617198	0.308599	+	0.951192i	$0.400140\pi$
$62$	0	0
$63$	−336.918	−0.673772
$64$	0	0
$65$	40.8023	0.0778600
$66$	0	0
$67$	−426.732	−0.778113	−0.389056	−	0.921214i	$-0.627199\pi$
$67$	−426.732	−0.778113	−0.389056	+	0.921214i	$0.627199\pi$
$68$	0	0
$69$	−85.5635	−0.149285
$70$	0	0
$71$	1020.85	1.70638	0.853191	−	0.521598i	$-0.174664\pi$
$71$	1020.85	1.70638	0.853191	+	0.521598i	$0.174664\pi$
$72$	0	0
$73$	−286.650	−0.459586	−0.229793	−	0.973240i	$-0.573805\pi$
$73$	−286.650	−0.459586	−0.229793	+	0.973240i	$0.573805\pi$
$74$	0	0
$75$	−93.0038	−0.143189
$76$	0	0
$77$	−322.589	−0.477435
$78$	0	0
$79$	551.866	0.785947	0.392974	−	0.919550i	$-0.371446\pi$
$79$	551.866	0.785947	0.392974	+	0.919550i	$0.371446\pi$
$80$	0	0
$81$	−200.470	−0.274993
$82$	0	0
$83$	−21.7021	−0.0287001	−0.0143501	−	0.999897i	$-0.504568\pi$
$83$	−21.7021	−0.0287001	−0.0143501	+	0.999897i	$0.504568\pi$
$84$	0	0
$85$	−380.205	−0.485165
$86$	0	0
$87$	994.031	1.22496
$88$	0	0
$89$	−1049.63	−1.25012	−0.625058	−	0.780578i	$-0.714925\pi$
$89$	−1049.63	−1.25012	−0.625058	+	0.780578i	$0.714925\pi$
$90$	0	0
$91$	208.914	0.240661
$92$	0	0
$93$	−237.187	−0.264464
$94$	0	0
$95$	516.810	0.558143
$96$	0	0
$97$	1729.19	1.81003	0.905014	−	0.425381i	$-0.139860\pi$
$97$	1729.19	1.81003	0.905014	+	0.425381i	$0.139860\pi$
$98$	0	0
$99$	165.832	0.168351
$100$	0	0
$101$	855.874	0.843195	0.421597	−	0.906783i	$-0.361470\pi$
$101$	855.874	0.843195	0.421597	+	0.906783i	$0.361470\pi$
$102$	0	0
$103$	632.534	0.605101	0.302551	−	0.953133i	$-0.402162\pi$
$103$	632.534	0.605101	0.302551	+	0.953133i	$0.402162\pi$
$104$	0	0
$105$	−476.194	−0.442588
$106$	0	0
$107$	−1309.62	−1.18323	−0.591616	−	0.806220i	$-0.701510\pi$
$107$	−1309.62	−1.18323	−0.591616	+	0.806220i	$0.701510\pi$
$108$	0	0
$109$	1726.21	1.51689	0.758443	−	0.651739i	$-0.225960\pi$
$109$	1726.21	1.51689	0.758443	+	0.651739i	$0.225960\pi$
$110$	0	0
$111$	−417.268	−0.356805
$112$	0	0
$113$	−1492.24	−1.24228	−0.621142	−	0.783698i	$-0.713331\pi$
$113$	−1492.24	−1.24228	−0.621142	+	0.783698i	$0.713331\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−107.395	−0.0848608
$118$	0	0
$119$	−1946.71	−1.49962
$120$	0	0
$121$	−1172.22	−0.880707
$122$	0	0
$123$	889.408	0.651994
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1368.29	0.956034	0.478017	−	0.878351i	$-0.341356\pi$
$127$	1368.29	0.956034	0.478017	+	0.878351i	$0.341356\pi$
$128$	0	0
$129$	1049.97	0.716627
$130$	0	0
$131$	−1984.24	−1.32339	−0.661694	−	0.749774i	$-0.730162\pi$
$131$	−1984.24	−1.32339	−0.661694	+	0.749774i	$0.730162\pi$
$132$	0	0
$133$	2646.15	1.72519
$134$	0	0
$135$	747.015	0.476243
$136$	0	0
$137$	273.331	0.170455	0.0852273	−	0.996362i	$-0.472838\pi$
$137$	273.331	0.170455	0.0852273	+	0.996362i	$0.472838\pi$
$138$	0	0
$139$	557.889	0.340428	0.170214	−	0.985407i	$-0.445554\pi$
$139$	557.889	0.340428	0.170214	+	0.985407i	$0.445554\pi$
$140$	0	0
$141$	2147.61	1.28271
$142$	0	0
$143$	−102.828	−0.0601323
$144$	0	0
$145$	−1336.01	−0.765168
$146$	0	0
$147$	−1162.17	−0.652071
$148$	0	0
$149$	−1354.16	−0.744545	−0.372272	−	0.928124i	$-0.621421\pi$
$149$	−1354.16	−0.744545	−0.372272	+	0.928124i	$0.621421\pi$
$150$	0	0
$151$	−162.652	−0.0876586	−0.0438293	−	0.999039i	$-0.513956\pi$
$151$	−162.652	−0.0876586	−0.0438293	+	0.999039i	$0.513956\pi$
$152$	0	0
$153$	1000.74	0.528789
$154$	0	0
$155$	318.787	0.165197
$156$	0	0
$157$	−2364.83	−1.20213	−0.601063	−	0.799201i	$-0.705256\pi$
$157$	−2364.83	−1.20213	−0.601063	+	0.799201i	$0.705256\pi$
$158$	0	0
$159$	8.60567	0.00429229
$160$	0	0
$161$	588.818	0.288232
$162$	0	0
$163$	−471.162	−0.226406	−0.113203	−	0.993572i	$-0.536111\pi$
$163$	−471.162	−0.226406	−0.113203	+	0.993572i	$0.536111\pi$
$164$	0	0
$165$	234.384	0.110586
$166$	0	0
$167$	3735.73	1.73102	0.865508	−	0.500895i	$-0.166996\pi$
$167$	3735.73	1.73102	0.865508	+	0.500895i	$0.166996\pi$
$168$	0	0
$169$	−2130.41	−0.969689
$170$	0	0
$171$	−1360.29	−0.608328
$172$	0	0
$173$	−2709.33	−1.19067	−0.595336	−	0.803477i	$-0.702981\pi$
$173$	−2709.33	−1.19067	−0.595336	+	0.803477i	$0.702981\pi$
$174$	0	0
$175$	640.019	0.276462
$176$	0	0
$177$	1015.19	0.431108
$178$	0	0
$179$	−3300.43	−1.37813	−0.689066	−	0.724699i	$-0.741979\pi$
$179$	−3300.43	−1.37813	−0.689066	+	0.724699i	$0.741979\pi$
$180$	0	0
$181$	−583.868	−0.239771	−0.119886	−	0.992788i	$-0.538253\pi$
$181$	−583.868	−0.239771	−0.119886	+	0.992788i	$0.538253\pi$
$182$	0	0
$183$	−1093.91	−0.441879
$184$	0	0
$185$	560.821	0.222878
$186$	0	0
$187$	958.176	0.374699
$188$	0	0
$189$	3824.83	1.47204
$190$	0	0
$191$	3058.98	1.15885	0.579424	−	0.815026i	$-0.303278\pi$
$191$	3058.98	1.15885	0.579424	+	0.815026i	$0.303278\pi$
$192$	0	0
$193$	−1336.93	−0.498625	−0.249313	−	0.968423i	$-0.580205\pi$
$193$	−1336.93	−0.498625	−0.249313	+	0.968423i	$0.580205\pi$
$194$	0	0
$195$	−151.791	−0.0557434
$196$	0	0
$197$	−2449.21	−0.885783	−0.442892	−	0.896575i	$-0.646047\pi$
$197$	−2449.21	−0.885783	−0.442892	+	0.896575i	$0.646047\pi$
$198$	0	0
$199$	−1138.66	−0.405614	−0.202807	−	0.979219i	$-0.565006\pi$
$199$	−1138.66	−0.405614	−0.202807	+	0.979219i	$0.565006\pi$
$200$	0	0
$201$	1587.51	0.557085
$202$	0	0
$203$	−6840.56	−2.36509
$204$	0	0
$205$	−1195.39	−0.407267
$206$	0	0
$207$	−302.691	−0.101635
$208$	0	0
$209$	−1302.44	−0.431061
$210$	0	0
$211$	−5596.81	−1.82607	−0.913034	−	0.407884i	$-0.866267\pi$
$211$	−5596.81	−1.82607	−0.913034	+	0.407884i	$0.866267\pi$
$212$	0	0
$213$	−3797.74	−1.22167
$214$	0	0
$215$	−1411.19	−0.447640
$216$	0	0
$217$	1632.24	0.510615
$218$	0	0
$219$	1066.38	0.329038
$220$	0	0
$221$	−620.530	−0.188875
$222$	0	0
$223$	−5580.60	−1.67581	−0.837903	−	0.545820i	$-0.816218\pi$
$223$	−5580.60	−1.67581	−0.837903	+	0.545820i	$0.816218\pi$
$224$	0	0
$225$	−329.011	−0.0974849
$226$	0	0
$227$	566.323	0.165587	0.0827934	−	0.996567i	$-0.473616\pi$
$227$	566.323	0.165587	0.0827934	+	0.996567i	$0.473616\pi$
$228$	0	0
$229$	−693.702	−0.200180	−0.100090	−	0.994978i	$-0.531913\pi$
$229$	−693.702	−0.200180	−0.100090	+	0.994978i	$0.531913\pi$
$230$	0	0
$231$	1200.08	0.341816
$232$	0	0
$233$	−2208.68	−0.621011	−0.310506	−	0.950572i	$-0.600498\pi$
$233$	−2208.68	−0.621011	−0.310506	+	0.950572i	$0.600498\pi$
$234$	0	0
$235$	−2886.46	−0.801241
$236$	0	0
$237$	−2053.03	−0.562694
$238$	0	0
$239$	3876.79	1.04924	0.524621	−	0.851336i	$-0.324207\pi$
$239$	3876.79	1.04924	0.524621	+	0.851336i	$0.324207\pi$
$240$	0	0
$241$	−2024.57	−0.541136	−0.270568	−	0.962701i	$-0.587211\pi$
$241$	−2024.57	−0.541136	−0.270568	+	0.962701i	$0.587211\pi$
$242$	0	0
$243$	−3288.10	−0.868033
$244$	0	0
$245$	1562.00	0.407315
$246$	0	0
$247$	843.481	0.217285
$248$	0	0
$249$	80.7350	0.0205477
$250$	0	0
$251$	733.444	0.184441	0.0922203	−	0.995739i	$-0.470604\pi$
$251$	733.444	0.184441	0.0922203	+	0.995739i	$0.470604\pi$
$252$	0	0
$253$	−289.818	−0.0720185
$254$	0	0
$255$	1414.42	0.347351
$256$	0	0
$257$	2367.83	0.574713	0.287356	−	0.957824i	$-0.407224\pi$
$257$	2367.83	0.574713	0.287356	+	0.957824i	$0.407224\pi$
$258$	0	0
$259$	2871.49	0.688903
$260$	0	0
$261$	3516.50	0.833968
$262$	0	0
$263$	−6632.76	−1.55511	−0.777554	−	0.628816i	$-0.783540\pi$
$263$	−6632.76	−1.55511	−0.777554	+	0.628816i	$0.783540\pi$
$264$	0	0
$265$	−11.5663	−0.00268117
$266$	0	0
$267$	3904.78	0.895013
$268$	0	0
$269$	−6092.97	−1.38102	−0.690511	−	0.723322i	$-0.742614\pi$
$269$	−6092.97	−1.38102	−0.690511	+	0.723322i	$0.742614\pi$
$270$	0	0
$271$	1981.30	0.444115	0.222058	−	0.975034i	$-0.428723\pi$
$271$	1981.30	0.444115	0.222058	+	0.975034i	$0.428723\pi$
$272$	0	0
$273$	−777.192	−0.172300
$274$	0	0
$275$	−315.019	−0.0690777
$276$	0	0
$277$	−4097.42	−0.888773	−0.444386	−	0.895835i	$-0.646578\pi$
$277$	−4097.42	−0.888773	−0.444386	+	0.895835i	$0.646578\pi$
$278$	0	0
$279$	−839.077	−0.180051
$280$	0	0
$281$	−3866.40	−0.820819	−0.410409	−	0.911901i	$-0.634614\pi$
$281$	−3866.40	−0.820819	−0.410409	+	0.911901i	$0.634614\pi$
$282$	0	0
$283$	−3982.00	−0.836415	−0.418208	−	0.908351i	$-0.637342\pi$
$283$	−3982.00	−0.836415	−0.418208	+	0.908351i	$0.637342\pi$
$284$	0	0
$285$	−1922.61	−0.399599
$286$	0	0
$287$	−6120.59	−1.25884
$288$	0	0
$289$	869.245	0.176927
$290$	0	0
$291$	−6432.86	−1.29588
$292$	0	0
$293$	−7491.70	−1.49375	−0.746877	−	0.664962i	$-0.768448\pi$
$293$	−7491.70	−1.49375	−0.746877	+	0.664962i	$0.768448\pi$
$294$	0	0
$295$	−1364.44	−0.269291
$296$	0	0
$297$	−1882.59	−0.367809
$298$	0	0
$299$	187.691	0.0363024
$300$	0	0
$301$	−7225.53	−1.38363
$302$	0	0
$303$	−3183.98	−0.603680
$304$	0	0
$305$	1470.24	0.276019
$306$	0	0
$307$	5813.01	1.08067	0.540335	−	0.841450i	$-0.318297\pi$
$307$	5813.01	1.08067	0.540335	+	0.841450i	$0.318297\pi$
$308$	0	0
$309$	−2353.12	−0.433218
$310$	0	0
$311$	−4769.43	−0.869613	−0.434807	−	0.900524i	$-0.643183\pi$
$311$	−4769.43	−0.869613	−0.434807	+	0.900524i	$0.643183\pi$
$312$	0	0
$313$	7053.43	1.27375	0.636875	−	0.770967i	$-0.280227\pi$
$313$	7053.43	1.27375	0.636875	+	0.770967i	$0.280227\pi$
$314$	0	0
$315$	−1684.59	−0.301320
$316$	0	0
$317$	8845.36	1.56721	0.783604	−	0.621261i	$-0.213379\pi$
$317$	8845.36	1.56721	0.783604	+	0.621261i	$0.213379\pi$
$318$	0	0
$319$	3366.94	0.590949
$320$	0	0
$321$	4871.99	0.847127
$322$	0	0
$323$	−7859.76	−1.35396
$324$	0	0
$325$	204.011	0.0348201
$326$	0	0
$327$	−6421.75	−1.08601
$328$	0	0
$329$	−14779.1	−2.47659
$330$	0	0
$331$	−4264.07	−0.708079	−0.354040	−	0.935230i	$-0.615192\pi$
$331$	−4264.07	−0.708079	−0.354040	+	0.935230i	$0.615192\pi$
$332$	0	0
$333$	−1476.13	−0.242918
$334$	0	0
$335$	−2133.66	−0.347983
$336$	0	0
$337$	−1290.58	−0.208613	−0.104306	−	0.994545i	$-0.533262\pi$
$337$	−1290.58	−0.208613	−0.104306	+	0.994545i	$0.533262\pi$
$338$	0	0
$339$	5551.36	0.889405
$340$	0	0
$341$	−803.392	−0.127584
$342$	0	0
$343$	−783.403	−0.123323
$344$	0	0
$345$	−427.818	−0.0667621
$346$	0	0
$347$	1805.09	0.279257	0.139629	−	0.990204i	$-0.455409\pi$
$347$	1805.09	0.279257	0.139629	+	0.990204i	$0.455409\pi$
$348$	0	0
$349$	−3586.81	−0.550136	−0.275068	−	0.961425i	$-0.588700\pi$
$349$	−3586.81	−0.550136	−0.275068	+	0.961425i	$0.588700\pi$
$350$	0	0
$351$	1219.20	0.185402
$352$	0	0
$353$	−2292.90	−0.345719	−0.172859	−	0.984947i	$-0.555301\pi$
$353$	−2292.90	−0.345719	−0.172859	+	0.984947i	$0.555301\pi$
$354$	0	0
$355$	5104.27	0.763118
$356$	0	0
$357$	7242.06	1.07364
$358$	0	0
$359$	3077.18	0.452388	0.226194	−	0.974082i	$-0.427372\pi$
$359$	3077.18	0.452388	0.226194	+	0.974082i	$0.427372\pi$
$360$	0	0
$361$	3824.70	0.557618
$362$	0	0
$363$	4360.84	0.630537
$364$	0	0
$365$	−1433.25	−0.205533
$366$	0	0
$367$	1378.56	0.196076	0.0980382	−	0.995183i	$-0.468743\pi$
$367$	1378.56	0.196076	0.0980382	+	0.995183i	$0.468743\pi$
$368$	0	0
$369$	3146.38	0.443886
$370$	0	0
$371$	−59.2211	−0.00828736
$372$	0	0
$373$	11182.3	1.55227	0.776135	−	0.630567i	$-0.217178\pi$
$373$	11182.3	1.55227	0.776135	+	0.630567i	$0.217178\pi$
$374$	0	0
$375$	−465.019	−0.0640360
$376$	0	0
$377$	−2180.49	−0.297880
$378$	0	0
$379$	−12021.9	−1.62935	−0.814677	−	0.579915i	$-0.803086\pi$
$379$	−12021.9	−1.62935	−0.814677	+	0.579915i	$0.803086\pi$
$380$	0	0
$381$	−5090.26	−0.684467
$382$	0	0
$383$	13274.6	1.77102	0.885512	−	0.464617i	$-0.153808\pi$
$383$	13274.6	1.77102	0.885512	+	0.464617i	$0.153808\pi$
$384$	0	0
$385$	−1612.95	−0.213515
$386$	0	0
$387$	3714.39	0.487889
$388$	0	0
$389$	11016.4	1.43587	0.717934	−	0.696112i	$-0.245088\pi$
$389$	11016.4	1.43587	0.717934	+	0.696112i	$0.245088\pi$
$390$	0	0
$391$	−1748.94	−0.226210
$392$	0	0
$393$	7381.67	0.947471
$394$	0	0
$395$	2759.33	0.351486
$396$	0	0
$397$	−1893.18	−0.239335	−0.119668	−	0.992814i	$-0.538183\pi$
$397$	−1893.18	−0.239335	−0.119668	+	0.992814i	$0.538183\pi$
$398$	0	0
$399$	−9844.07	−1.23514
$400$	0	0
$401$	11281.8	1.40495	0.702475	−	0.711709i	$-0.252078\pi$
$401$	11281.8	1.40495	0.702475	+	0.711709i	$0.252078\pi$
$402$	0	0
$403$	520.290	0.0643113
$404$	0	0
$405$	−1002.35	−0.122981
$406$	0	0
$407$	−1413.36	−0.172131
$408$	0	0
$409$	−14677.6	−1.77447	−0.887236	−	0.461316i	$-0.847377\pi$
$409$	−14677.6	−1.77447	−0.887236	+	0.461316i	$0.847377\pi$
$410$	0	0
$411$	−1016.83	−0.122036
$412$	0	0
$413$	−6986.15	−0.832363
$414$	0	0
$415$	−108.510	−0.0128351
$416$	0	0
$417$	−2075.43	−0.243728
$418$	0	0
$419$	−2196.50	−0.256100	−0.128050	−	0.991768i	$-0.540872\pi$
$419$	−2196.50	−0.256100	−0.128050	+	0.991768i	$0.540872\pi$
$420$	0	0
$421$	−6571.28	−0.760723	−0.380362	−	0.924838i	$-0.624200\pi$
$421$	−6571.28	−0.760723	−0.380362	+	0.924838i	$0.624200\pi$
$422$	0	0
$423$	7597.42	0.873284
$424$	0	0
$425$	−1901.03	−0.216973
$426$	0	0
$427$	7527.87	0.853160
$428$	0	0
$429$	382.536	0.0430513
$430$	0	0
$431$	583.607	0.0652235	0.0326118	−	0.999468i	$-0.489618\pi$
$431$	583.607	0.0652235	0.0326118	+	0.999468i	$0.489618\pi$
$432$	0	0
$433$	−2677.60	−0.297176	−0.148588	−	0.988899i	$-0.547473\pi$
$433$	−2677.60	−0.297176	−0.148588	+	0.988899i	$0.547473\pi$
$434$	0	0
$435$	4970.15	0.547817
$436$	0	0
$437$	2377.33	0.260236
$438$	0	0
$439$	5509.18	0.598949	0.299475	−	0.954104i	$-0.403189\pi$
$439$	5509.18	0.598949	0.299475	+	0.954104i	$0.403189\pi$
$440$	0	0
$441$	−4111.32	−0.443939
$442$	0	0
$443$	13054.6	1.40009	0.700047	−	0.714097i	$-0.253162\pi$
$443$	13054.6	1.40009	0.700047	+	0.714097i	$0.253162\pi$
$444$	0	0
$445$	−5248.14	−0.559069
$446$	0	0
$447$	5037.68	0.533052
$448$	0	0
$449$	−11819.1	−1.24227	−0.621135	−	0.783704i	$-0.713328\pi$
$449$	−11819.1	−1.24227	−0.621135	+	0.783704i	$0.713328\pi$
$450$	0	0
$451$	3012.57	0.314537
$452$	0	0
$453$	605.091	0.0627587
$454$	0	0
$455$	1044.57	0.107627
$456$	0	0
$457$	−4144.87	−0.424264	−0.212132	−	0.977241i	$-0.568041\pi$
$457$	−4144.87	−0.424264	−0.212132	+	0.977241i	$0.568041\pi$
$458$	0	0
$459$	−11360.8	−1.15528
$460$	0	0
$461$	−6061.38	−0.612379	−0.306189	−	0.951971i	$-0.599054\pi$
$461$	−6061.38	−0.612379	−0.306189	+	0.951971i	$0.599054\pi$
$462$	0	0
$463$	5729.13	0.575066	0.287533	−	0.957771i	$-0.407165\pi$
$463$	5729.13	0.575066	0.287533	+	0.957771i	$0.407165\pi$
$464$	0	0
$465$	−1185.94	−0.118272
$466$	0	0
$467$	−8958.05	−0.887643	−0.443821	−	0.896115i	$-0.646378\pi$
$467$	−8958.05	−0.887643	−0.443821	+	0.896115i	$0.646378\pi$
$468$	0	0
$469$	−10924.7	−1.07559
$470$	0	0
$471$	8797.53	0.860655
$472$	0	0
$473$	3556.42	0.345718
$474$	0	0
$475$	2584.05	0.249609
$476$	0	0
$477$	30.4435	0.00292225
$478$	0	0
$479$	−6356.60	−0.606347	−0.303174	−	0.952935i	$-0.598046\pi$
$479$	−6356.60	−0.606347	−0.303174	+	0.952935i	$0.598046\pi$
$480$	0	0
$481$	915.312	0.0867664
$482$	0	0
$483$	−2190.49	−0.206358
$484$	0	0
$485$	8645.96	0.809469
$486$	0	0
$487$	−13991.2	−1.30185	−0.650926	−	0.759141i	$-0.725619\pi$
$487$	−13991.2	−1.30185	−0.650926	+	0.759141i	$0.725619\pi$
$488$	0	0
$489$	1752.79	0.162094
$490$	0	0
$491$	5301.96	0.487320	0.243660	−	0.969861i	$-0.421652\pi$
$491$	5301.96	0.487320	0.243660	+	0.969861i	$0.421652\pi$
$492$	0	0
$493$	20318.3	1.85617
$494$	0	0
$495$	829.159	0.0752888
$496$	0	0
$497$	26134.7	2.35875
$498$	0	0
$499$	8013.45	0.718900	0.359450	−	0.933164i	$-0.382964\pi$
$499$	8013.45	0.718900	0.359450	+	0.933164i	$0.382964\pi$
$500$	0	0
$501$	−13897.5	−1.23931
$502$	0	0
$503$	11881.9	1.05325	0.526627	−	0.850097i	$-0.323456\pi$
$503$	11881.9	1.05325	0.526627	+	0.850097i	$0.323456\pi$
$504$	0	0
$505$	4279.37	0.377088
$506$	0	0
$507$	7925.44	0.694243
$508$	0	0
$509$	−12113.8	−1.05488	−0.527441	−	0.849592i	$-0.676848\pi$
$509$	−12113.8	−1.05488	−0.527441	+	0.849592i	$0.676848\pi$
$510$	0	0
$511$	−7338.45	−0.635291
$512$	0	0
$513$	15442.6	1.32906
$514$	0	0
$515$	3162.67	0.270609
$516$	0	0
$517$	7274.31	0.618808
$518$	0	0
$519$	10079.1	0.852454
$520$	0	0
$521$	−15327.6	−1.28890	−0.644449	−	0.764647i	$-0.722913\pi$
$521$	−15327.6	−1.28890	−0.644449	+	0.764647i	$0.722913\pi$
$522$	0	0
$523$	−14545.8	−1.21614	−0.608070	−	0.793883i	$-0.708056\pi$
$523$	−14545.8	−1.21614	−0.608070	+	0.793883i	$0.708056\pi$
$524$	0	0
$525$	−2380.97	−0.197931
$526$	0	0
$527$	−4848.18	−0.400740
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	3591.34	0.293504
$532$	0	0
$533$	−1950.99	−0.158549
$534$	0	0
$535$	−6548.10	−0.529157
$536$	0	0
$537$	12278.1	0.986665
$538$	0	0
$539$	−3936.47	−0.314575
$540$	0	0
$541$	−5468.93	−0.434617	−0.217308	−	0.976103i	$-0.569728\pi$
$541$	−5468.93	−0.434617	−0.217308	+	0.976103i	$0.569728\pi$
$542$	0	0
$543$	2172.08	0.171663
$544$	0	0
$545$	8631.03	0.678372
$546$	0	0
$547$	19566.3	1.52943	0.764713	−	0.644371i	$-0.222881\pi$
$547$	19566.3	1.52943	0.764713	+	0.644371i	$0.222881\pi$
$548$	0	0
$549$	−3869.82	−0.300837
$550$	0	0
$551$	−27618.5	−2.13537
$552$	0	0
$553$	14128.2	1.08642
$554$	0	0
$555$	−2086.34	−0.159568
$556$	0	0
$557$	−9803.11	−0.745729	−0.372865	−	0.927886i	$-0.621624\pi$
$557$	−9803.11	−0.745729	−0.372865	+	0.927886i	$0.621624\pi$
$558$	0	0
$559$	−2303.20	−0.174266
$560$	0	0
$561$	−3564.56	−0.268264
$562$	0	0
$563$	4909.73	0.367532	0.183766	−	0.982970i	$-0.441171\pi$
$563$	4909.73	0.367532	0.183766	+	0.982970i	$0.441171\pi$
$564$	0	0
$565$	−7461.19	−0.555566
$566$	0	0
$567$	−5132.18	−0.380126
$568$	0	0
$569$	11498.2	0.847155	0.423578	−	0.905860i	$-0.360774\pi$
$569$	11498.2	0.847155	0.423578	+	0.905860i	$0.360774\pi$
$570$	0	0
$571$	−14766.6	−1.08225	−0.541124	−	0.840943i	$-0.682001\pi$
$571$	−14766.6	−1.08225	−0.541124	+	0.840943i	$0.682001\pi$
$572$	0	0
$573$	−11379.9	−0.829670
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	13364.0	0.964212	0.482106	−	0.876113i	$-0.339872\pi$
$577$	13364.0	0.964212	0.482106	+	0.876113i	$0.339872\pi$
$578$	0	0
$579$	4973.60	0.356988
$580$	0	0
$581$	−555.590	−0.0396725
$582$	0	0
$583$	29.1488	0.00207070
$584$	0	0
$585$	−536.977	−0.0379509
$586$	0	0
$587$	−12524.7	−0.880664	−0.440332	−	0.897835i	$-0.645139\pi$
$587$	−12524.7	−0.880664	−0.440332	+	0.897835i	$0.645139\pi$
$588$	0	0
$589$	6590.09	0.461019
$590$	0	0
$591$	9111.45	0.634171
$592$	0	0
$593$	−17938.6	−1.24224	−0.621121	−	0.783714i	$-0.713323\pi$
$593$	−17938.6	−1.24224	−0.621121	+	0.783714i	$0.713323\pi$
$594$	0	0
$595$	−9733.55	−0.670650
$596$	0	0
$597$	4235.98	0.290397
$598$	0	0
$599$	26735.8	1.82370	0.911848	−	0.410528i	$-0.134656\pi$
$599$	26735.8	1.82370	0.911848	+	0.410528i	$0.134656\pi$
$600$	0	0
$601$	−21043.6	−1.42826	−0.714131	−	0.700012i	$-0.753178\pi$
$601$	−21043.6	−1.42826	−0.714131	+	0.700012i	$0.753178\pi$
$602$	0	0
$603$	5615.98	0.379271
$604$	0	0
$605$	−5861.10	−0.393864
$606$	0	0
$607$	−5098.83	−0.340947	−0.170474	−	0.985362i	$-0.554530\pi$
$607$	−5098.83	−0.340947	−0.170474	+	0.985362i	$0.554530\pi$
$608$	0	0
$609$	25447.9	1.69327
$610$	0	0
$611$	−4710.96	−0.311923
$612$	0	0
$613$	−12784.8	−0.842369	−0.421185	−	0.906975i	$-0.638386\pi$
$613$	−12784.8	−0.842369	−0.421185	+	0.906975i	$0.638386\pi$
$614$	0	0
$615$	4447.04	0.291580
$616$	0	0
$617$	−8940.73	−0.583372	−0.291686	−	0.956514i	$-0.594216\pi$
$617$	−8940.73	−0.583372	−0.291686	+	0.956514i	$0.594216\pi$
$618$	0	0
$619$	−1333.26	−0.0865725	−0.0432863	−	0.999063i	$-0.513783\pi$
$619$	−1333.26	−0.0865725	−0.0432863	+	0.999063i	$0.513783\pi$
$620$	0	0
$621$	3436.27	0.222050
$622$	0	0
$623$	−26871.3	−1.72805
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4845.28	0.308615
$628$	0	0
$629$	−8529.09	−0.540663
$630$	0	0
$631$	16667.9	1.05157	0.525784	−	0.850618i	$-0.323772\pi$
$631$	16667.9	1.05157	0.525784	+	0.850618i	$0.323772\pi$
$632$	0	0
$633$	20821.0	1.30736
$634$	0	0
$635$	6841.46	0.427551
$636$	0	0
$637$	2549.32	0.158568
$638$	0	0
$639$	−13434.9	−0.831733
$640$	0	0
$641$	27431.9	1.69032	0.845158	−	0.534516i	$-0.179506\pi$
$641$	27431.9	1.69032	0.845158	+	0.534516i	$0.179506\pi$
$642$	0	0
$643$	−9618.00	−0.589886	−0.294943	−	0.955515i	$-0.595301\pi$
$643$	−9618.00	−0.589886	−0.294943	+	0.955515i	$0.595301\pi$
$644$	0	0
$645$	5249.86	0.320485
$646$	0	0
$647$	12548.6	0.762496	0.381248	−	0.924473i	$-0.375494\pi$
$647$	12548.6	0.762496	0.381248	+	0.924473i	$0.375494\pi$
$648$	0	0
$649$	3438.60	0.207977
$650$	0	0
$651$	−6072.18	−0.365572
$652$	0	0
$653$	−8829.79	−0.529152	−0.264576	−	0.964365i	$-0.585232\pi$
$653$	−8829.79	−0.529152	−0.264576	+	0.964365i	$0.585232\pi$
$654$	0	0
$655$	−9921.19	−0.591837
$656$	0	0
$657$	3772.44	0.224014
$658$	0	0
$659$	−5935.56	−0.350860	−0.175430	−	0.984492i	$-0.556132\pi$
$659$	−5935.56	−0.350860	−0.175430	+	0.984492i	$0.556132\pi$
$660$	0	0
$661$	−1182.03	−0.0695549	−0.0347775	−	0.999395i	$-0.511072\pi$
$661$	−1182.03	−0.0695549	−0.0347775	+	0.999395i	$0.511072\pi$
$662$	0	0
$663$	2308.47	0.135224
$664$	0	0
$665$	13230.7	0.771527
$666$	0	0
$667$	−6145.64	−0.356762
$668$	0	0
$669$	20760.7	1.19978
$670$	0	0
$671$	−3705.24	−0.213173
$672$	0	0
$673$	−6053.25	−0.346710	−0.173355	−	0.984859i	$-0.555461\pi$
$673$	−6053.25	−0.346710	−0.173355	+	0.984859i	$0.555461\pi$
$674$	0	0
$675$	3735.08	0.212982
$676$	0	0
$677$	16694.2	0.947728	0.473864	−	0.880598i	$-0.342859\pi$
$677$	16694.2	0.947728	0.473864	+	0.880598i	$0.342859\pi$
$678$	0	0
$679$	44268.6	2.50202
$680$	0	0
$681$	−2106.81	−0.118551
$682$	0	0
$683$	4150.03	0.232499	0.116249	−	0.993220i	$-0.462913\pi$
$683$	4150.03	0.232499	0.116249	+	0.993220i	$0.462913\pi$
$684$	0	0
$685$	1366.66	0.0762296
$686$	0	0
$687$	2580.68	0.143317
$688$	0	0
$689$	−18.8772	−0.00104378
$690$	0	0
$691$	−33382.4	−1.83781	−0.918904	−	0.394481i	$-0.870924\pi$
$691$	−33382.4	−1.83781	−0.918904	+	0.394481i	$0.870924\pi$
$692$	0	0
$693$	4245.42	0.232713
$694$	0	0
$695$	2789.45	0.152244
$696$	0	0
$697$	18179.8	0.987960
$698$	0	0
$699$	8216.64	0.444609
$700$	0	0
$701$	−29528.7	−1.59099	−0.795494	−	0.605961i	$-0.792789\pi$
$701$	−29528.7	−1.59099	−0.795494	+	0.605961i	$0.792789\pi$
$702$	0	0
$703$	11593.5	0.621989
$704$	0	0
$705$	10738.1	0.573644
$706$	0	0
$707$	21911.0	1.16556
$708$	0	0
$709$	27016.3	1.43106	0.715529	−	0.698583i	$-0.246186\pi$
$709$	27016.3	1.43106	0.715529	+	0.698583i	$0.246186\pi$
$710$	0	0
$711$	−7262.82	−0.383090
$712$	0	0
$713$	1466.42	0.0770237
$714$	0	0
$715$	−514.140	−0.0268920
$716$	0	0
$717$	−14422.3	−0.751198
$718$	0	0
$719$	−24752.9	−1.28390	−0.641952	−	0.766745i	$-0.721875\pi$
$719$	−24752.9	−1.28390	−0.641952	+	0.766745i	$0.721875\pi$
$720$	0	0
$721$	16193.4	0.836438
$722$	0	0
$723$	7531.69	0.387423
$724$	0	0
$725$	−6680.04	−0.342194
$726$	0	0
$727$	17574.5	0.896565	0.448283	−	0.893892i	$-0.352036\pi$
$727$	17574.5	0.896565	0.448283	+	0.893892i	$0.352036\pi$
$728$	0	0
$729$	17644.9	0.896456
$730$	0	0
$731$	21461.7	1.08590
$732$	0	0
$733$	27739.6	1.39780	0.698898	−	0.715221i	$-0.253674\pi$
$733$	27739.6	1.39780	0.698898	+	0.715221i	$0.253674\pi$
$734$	0	0
$735$	−5810.87	−0.291615
$736$	0	0
$737$	5377.15	0.268751
$738$	0	0
$739$	3140.78	0.156340	0.0781701	−	0.996940i	$-0.475092\pi$
$739$	3140.78	0.156340	0.0781701	+	0.996940i	$0.475092\pi$
$740$	0	0
$741$	−3137.88	−0.155564
$742$	0	0
$743$	35881.5	1.77169	0.885844	−	0.463984i	$-0.153580\pi$
$743$	35881.5	1.77169	0.885844	+	0.463984i	$0.153580\pi$
$744$	0	0
$745$	−6770.80	−0.332970
$746$	0	0
$747$	285.609	0.0139891
$748$	0	0
$749$	−33527.3	−1.63559
$750$	0	0
$751$	7009.03	0.340563	0.170282	−	0.985395i	$-0.445532\pi$
$751$	7009.03	0.340563	0.170282	+	0.985395i	$0.445532\pi$
$752$	0	0
$753$	−2728.52	−0.132049
$754$	0	0
$755$	−813.261	−0.0392021
$756$	0	0
$757$	29998.3	1.44030	0.720150	−	0.693818i	$-0.244073\pi$
$757$	29998.3	1.44030	0.720150	+	0.693818i	$0.244073\pi$
$758$	0	0
$759$	1078.17	0.0515612
$760$	0	0
$761$	5283.49	0.251677	0.125839	−	0.992051i	$-0.459838\pi$
$761$	5283.49	0.251677	0.125839	+	0.992051i	$0.459838\pi$
$762$	0	0
$763$	44192.2	2.09681
$764$	0	0
$765$	5003.68	0.236481
$766$	0	0
$767$	−2226.89	−0.104835
$768$	0	0
$769$	−15216.2	−0.713538	−0.356769	−	0.934193i	$-0.616122\pi$
$769$	−15216.2	−0.713538	−0.356769	+	0.934193i	$0.616122\pi$
$770$	0	0
$771$	−8808.69	−0.411462
$772$	0	0
$773$	15651.3	0.728250	0.364125	−	0.931350i	$-0.381368\pi$
$773$	15651.3	0.728250	0.364125	+	0.931350i	$0.381368\pi$
$774$	0	0
$775$	1593.93	0.0738785
$776$	0	0
$777$	−10682.4	−0.493216
$778$	0	0
$779$	−24711.6	−1.13657
$780$	0	0
$781$	−12863.6	−0.589365
$782$	0	0
$783$	−39920.7	−1.82203
$784$	0	0
$785$	−11824.1	−0.537607
$786$	0	0
$787$	−9401.09	−0.425810	−0.212905	−	0.977073i	$-0.568293\pi$
$787$	−9401.09	−0.425810	−0.212905	+	0.977073i	$0.568293\pi$
$788$	0	0
$789$	24674.9	1.11337
$790$	0	0
$791$	−38202.5	−1.71722
$792$	0	0
$793$	2399.57	0.107454
$794$	0	0
$795$	43.0283	0.00191957
$796$	0	0
$797$	37388.5	1.66169	0.830845	−	0.556504i	$-0.187857\pi$
$797$	37388.5	1.66169	0.830845	+	0.556504i	$0.187857\pi$
$798$	0	0
$799$	43897.9	1.94367
$800$	0	0
$801$	13813.6	0.609337
$802$	0	0
$803$	3612.00	0.158736
$804$	0	0
$805$	2944.09	0.128901
$806$	0	0
$807$	22666.8	0.988734
$808$	0	0
$809$	24390.6	1.05999	0.529993	−	0.848002i	$-0.322195\pi$
$809$	24390.6	1.05999	0.529993	+	0.848002i	$0.322195\pi$
$810$	0	0
$811$	−23750.8	−1.02836	−0.514181	−	0.857682i	$-0.671904\pi$
$811$	−23750.8	−1.02836	−0.514181	+	0.857682i	$0.671904\pi$
$812$	0	0
$813$	−7370.73	−0.317962
$814$	0	0
$815$	−2355.81	−0.101252
$816$	0	0
$817$	−29172.8	−1.24924
$818$	0	0
$819$	−2749.40	−0.117304
$820$	0	0
$821$	3041.59	0.129296	0.0646481	−	0.997908i	$-0.479408\pi$
$821$	3041.59	0.129296	0.0646481	+	0.997908i	$0.479408\pi$
$822$	0	0
$823$	−28411.6	−1.20336	−0.601681	−	0.798737i	$-0.705502\pi$
$823$	−28411.6	−1.20336	−0.601681	+	0.798737i	$0.705502\pi$
$824$	0	0
$825$	1171.92	0.0494558
$826$	0	0
$827$	24351.5	1.02392	0.511962	−	0.859008i	$-0.328919\pi$
$827$	24351.5	1.02392	0.511962	+	0.859008i	$0.328919\pi$
$828$	0	0
$829$	−47240.8	−1.97918	−0.989590	−	0.143914i	$-0.954031\pi$
$829$	−47240.8	−1.97918	−0.989590	+	0.143914i	$0.954031\pi$
$830$	0	0
$831$	15243.0	0.636311
$832$	0	0
$833$	−23755.2	−0.988077
$834$	0	0
$835$	18678.7	0.774134
$836$	0	0
$837$	9525.55	0.393371
$838$	0	0
$839$	−2646.33	−0.108893	−0.0544466	−	0.998517i	$-0.517339\pi$
$839$	−2646.33	−0.108893	−0.0544466	+	0.998517i	$0.517339\pi$
$840$	0	0
$841$	47007.7	1.92741
$842$	0	0
$843$	14383.6	0.587660
$844$	0	0
$845$	−10652.0	−0.433658
$846$	0	0
$847$	−30009.7	−1.21741
$848$	0	0
$849$	14813.7	0.598827
$850$	0	0
$851$	2579.78	0.103917
$852$	0	0
$853$	−40826.4	−1.63877	−0.819385	−	0.573244i	$-0.805685\pi$
$853$	−40826.4	−1.63877	−0.819385	+	0.573244i	$0.805685\pi$
$854$	0	0
$855$	−6801.46	−0.272053
$856$	0	0
$857$	1516.61	0.0604510	0.0302255	−	0.999543i	$-0.490377\pi$
$857$	1516.61	0.0604510	0.0302255	+	0.999543i	$0.490377\pi$
$858$	0	0
$859$	−6514.74	−0.258766	−0.129383	−	0.991595i	$-0.541300\pi$
$859$	−6514.74	−0.258766	−0.129383	+	0.991595i	$0.541300\pi$
$860$	0	0
$861$	22769.5	0.901258
$862$	0	0
$863$	7120.18	0.280850	0.140425	−	0.990091i	$-0.455153\pi$
$863$	7120.18	0.280850	0.140425	+	0.990091i	$0.455153\pi$
$864$	0	0
$865$	−13546.6	−0.532485
$866$	0	0
$867$	−3233.72	−0.126670
$868$	0	0
$869$	−6953.94	−0.271457
$870$	0	0
$871$	−3482.33	−0.135470
$872$	0	0
$873$	−22757.0	−0.882252
$874$	0	0
$875$	3200.10	0.123638
$876$	0	0
$877$	−34013.4	−1.30964	−0.654818	−	0.755787i	$-0.727255\pi$
$877$	−34013.4	−1.30964	−0.654818	+	0.755787i	$0.727255\pi$
$878$	0	0
$879$	27870.3	1.06944
$880$	0	0
$881$	6704.11	0.256376	0.128188	−	0.991750i	$-0.459084\pi$
$881$	6704.11	0.256376	0.128188	+	0.991750i	$0.459084\pi$
$882$	0	0
$883$	−47808.5	−1.82207	−0.911033	−	0.412333i	$-0.864714\pi$
$883$	−47808.5	−1.82207	−0.911033	+	0.412333i	$0.864714\pi$
$884$	0	0
$885$	5075.93	0.192797
$886$	0	0
$887$	38955.7	1.47464	0.737320	−	0.675544i	$-0.236091\pi$
$887$	38955.7	1.47464	0.737320	+	0.675544i	$0.236091\pi$
$888$	0	0
$889$	35029.3	1.32154
$890$	0	0
$891$	2526.07	0.0949794
$892$	0	0
$893$	−59670.0	−2.23604
$894$	0	0
$895$	−16502.1	−0.616319
$896$	0	0
$897$	−698.238	−0.0259905
$898$	0	0
$899$	−17036.1	−0.632019
$900$	0	0
$901$	175.903	0.00650407
$902$	0	0
$903$	26880.1	0.990601
$904$	0	0
$905$	−2919.34	−0.107229
$906$	0	0
$907$	−7725.66	−0.282830	−0.141415	−	0.989950i	$-0.545165\pi$
$907$	−7725.66	−0.282830	−0.141415	+	0.989950i	$0.545165\pi$
$908$	0	0
$909$	−11263.7	−0.410994
$910$	0	0
$911$	13150.7	0.478269	0.239135	−	0.970986i	$-0.423136\pi$
$911$	13150.7	0.478269	0.239135	+	0.970986i	$0.423136\pi$
$912$	0	0
$913$	273.463	0.00991270
$914$	0	0
$915$	−5469.53	−0.197614
$916$	0	0
$917$	−50798.0	−1.82933
$918$	0	0
$919$	−33083.7	−1.18752	−0.593759	−	0.804643i	$-0.702357\pi$
$919$	−33083.7	−1.18752	−0.593759	+	0.804643i	$0.702357\pi$
$920$	0	0
$921$	−21625.3	−0.773699
$922$	0	0
$923$	8330.65	0.297082
$924$	0	0
$925$	2804.11	0.0996740
$926$	0	0
$927$	−8324.44	−0.294941
$928$	0	0
$929$	−1310.12	−0.0462686	−0.0231343	−	0.999732i	$-0.507365\pi$
$929$	−1310.12	−0.0462686	−0.0231343	+	0.999732i	$0.507365\pi$
$930$	0	0
$931$	32290.2	1.13670
$932$	0	0
$933$	17743.0	0.622594
$934$	0	0
$935$	4790.88	0.167571
$936$	0	0
$937$	40623.1	1.41633	0.708163	−	0.706049i	$-0.249524\pi$
$937$	40623.1	1.41633	0.708163	+	0.706049i	$0.249524\pi$
$938$	0	0
$939$	−26239.8	−0.911933
$940$	0	0
$941$	−10434.0	−0.361466	−0.180733	−	0.983532i	$-0.557847\pi$
$941$	−10434.0	−0.361466	−0.180733	+	0.983532i	$0.557847\pi$
$942$	0	0
$943$	−5498.80	−0.189889
$944$	0	0
$945$	19124.2	0.658317
$946$	0	0
$947$	51333.4	1.76147	0.880734	−	0.473612i	$-0.157050\pi$
$947$	51333.4	1.76147	0.880734	+	0.473612i	$0.157050\pi$
$948$	0	0
$949$	−2339.19	−0.0800141
$950$	0	0
$951$	−32906.1	−1.12203
$952$	0	0
$953$	40263.6	1.36859	0.684294	−	0.729206i	$-0.260110\pi$
$953$	40263.6	1.36859	0.684294	+	0.729206i	$0.260110\pi$
$954$	0	0
$955$	15294.9	0.518252
$956$	0	0
$957$	−12525.5	−0.423086
$958$	0	0
$959$	6997.49	0.235621
$960$	0	0
$961$	−25726.0	−0.863549
$962$	0	0
$963$	17235.2	0.576736
$964$	0	0
$965$	−6684.67	−0.222992
$966$	0	0
$967$	8353.12	0.277785	0.138893	−	0.990307i	$-0.455646\pi$
$967$	8353.12	0.277785	0.138893	+	0.990307i	$0.455646\pi$
$968$	0	0
$969$	29239.5	0.969358
$970$	0	0
$971$	28020.1	0.926063	0.463032	−	0.886342i	$-0.346762\pi$
$971$	28020.1	0.926063	0.463032	+	0.886342i	$0.346762\pi$
$972$	0	0
$973$	14282.4	0.470578
$974$	0	0
$975$	−758.954	−0.0249292
$976$	0	0
$977$	−50115.3	−1.64107	−0.820537	−	0.571593i	$-0.806326\pi$
$977$	−50115.3	−1.64107	−0.820537	+	0.571593i	$0.806326\pi$
$978$	0	0
$979$	13226.1	0.431776
$980$	0	0
$981$	−22717.7	−0.739367
$982$	0	0
$983$	38113.0	1.23664	0.618320	−	0.785926i	$-0.287814\pi$
$983$	38113.0	1.23664	0.618320	+	0.785926i	$0.287814\pi$
$984$	0	0
$985$	−12246.1	−0.396134
$986$	0	0
$987$	54980.5	1.77310
$988$	0	0
$989$	−6491.49	−0.208713
$990$	0	0
$991$	5245.02	0.168127	0.0840634	−	0.996460i	$-0.473210\pi$
$991$	5245.02	0.168127	0.0840634	+	0.996460i	$0.473210\pi$
$992$	0	0
$993$	15863.0	0.506945
$994$	0	0
$995$	−5693.29	−0.181396
$996$	0	0
$997$	21896.1	0.695544	0.347772	−	0.937579i	$-0.386938\pi$
$997$	21896.1	0.695544	0.347772	+	0.937579i	$0.386938\pi$
$998$	0	0
$999$	16757.7	0.530721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.h.1.1		2
4.3	odd	2		115.4.a.c.1.2	✓	2
12.11	even	2		1035.4.a.g.1.1		2
20.3	even	4		575.4.b.f.24.4		4
20.7	even	4		575.4.b.f.24.1		4
20.19	odd	2		575.4.a.h.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.4.a.c.1.2	✓	2	4.3	odd	2
575.4.a.h.1.1		2	20.19	odd	2
575.4.b.f.24.1		4	20.7	even	4
575.4.b.f.24.4		4	20.3	even	4
1035.4.a.g.1.1		2	12.11	even	2
1840.4.a.h.1.1		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)

\(f(q)\)	\(=\)	\(q-3.72015 q^{3} +5.00000 q^{5} +25.6008 q^{7} -13.1605 q^{9} +O(q^{10})\)	\(q-3.72015 q^{3} +5.00000 q^{5} +25.6008 q^{7} -13.1605 q^{9} -12.6008 q^{11} +8.16046 q^{13} -18.6008 q^{15} -76.0411 q^{17} +103.362 q^{19} -95.2388 q^{21} +23.0000 q^{23} +25.0000 q^{25} +149.403 q^{27} -267.202 q^{29} +63.7574 q^{31} +46.8768 q^{33} +128.004 q^{35} +112.164 q^{37} -30.3582 q^{39} -239.078 q^{41} -282.239 q^{43} -65.8023 q^{45} -577.291 q^{47} +312.399 q^{49} +282.884 q^{51} -2.31326 q^{53} -63.0038 q^{55} -384.522 q^{57} -272.888 q^{59} +294.049 q^{61} -336.918 q^{63} +40.8023 q^{65} -426.732 q^{67} -85.5635 q^{69} +1020.85 q^{71} -286.650 q^{73} -93.0038 q^{75} -322.589 q^{77} +551.866 q^{79} -200.470 q^{81} -21.7021 q^{83} -380.205 q^{85} +994.031 q^{87} -1049.63 q^{89} +208.914 q^{91} -237.187 q^{93} +516.810 q^{95} +1729.19 q^{97} +165.832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 10 q^{5} - q^{7} + 5 q^{9}+O(q^{10}) \) 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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