LMFDB - Embedded newform 2178.4.a.bz.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,4,Mod(1,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2178.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2178.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:	$128.506159993$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.12421225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 355x^{2} + 356x + 30964 $ x^4 - 2x^3 - 355x^2 + 356*x + 30964
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 11 $
Twist minimal:	no (minimal twist has level 66)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$12.8052$ of defining polynomial
Character	$\chi$	$=$	2178.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +6.18714 q^{5} -25.7192 q^{7} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} +6.18714 q^{5} -25.7192 q^{7} +8.00000 q^{8} +12.3743 q^{10} -30.7630 q^{13} -51.4384 q^{14} +16.0000 q^{16} +47.9850 q^{17} +46.7480 q^{19} +24.7485 q^{20} -2.04459 q^{23} -86.7194 q^{25} -61.5261 q^{26} -102.877 q^{28} +59.0080 q^{29} +304.896 q^{31} +32.0000 q^{32} +95.9699 q^{34} -159.128 q^{35} -303.823 q^{37} +93.4960 q^{38} +49.4971 q^{40} +145.553 q^{41} +284.084 q^{43} -4.08917 q^{46} -577.251 q^{47} +318.477 q^{49} -173.439 q^{50} -123.052 q^{52} -655.216 q^{53} -205.754 q^{56} +118.016 q^{58} +23.1802 q^{59} +631.874 q^{61} +609.792 q^{62} +64.0000 q^{64} -190.335 q^{65} -655.918 q^{67} +191.940 q^{68} -318.256 q^{70} -528.602 q^{71} -447.652 q^{73} -607.646 q^{74} +186.992 q^{76} -985.464 q^{79} +98.9942 q^{80} +291.106 q^{82} +128.459 q^{83} +296.889 q^{85} +568.167 q^{86} -1425.15 q^{89} +791.201 q^{91} -8.17835 q^{92} -1154.50 q^{94} +289.236 q^{95} -142.553 q^{97} +636.954 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} - 21 q^{7} + 32 q^{8}+O(q^{10}) $ 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 - 21 * q^7 + 32 * q^8	$ 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} - 21 q^{7} + 32 q^{8} - 40 q^{10} - 38 q^{13} - 42 q^{14} + 64 q^{16} + 85 q^{17} - 5 q^{19} - 80 q^{20} - 316 q^{23} + 318 q^{25} - 76 q^{26} - 84 q^{28} + 123 q^{29} + 436 q^{31} + 128 q^{32} + 170 q^{34} - 129 q^{35} + 336 q^{37} - 10 q^{38} - 160 q^{40} - 394 q^{41} + 680 q^{43} - 632 q^{46} - 1085 q^{47} - 311 q^{49} + 636 q^{50} - 152 q^{52} - 880 q^{53} - 168 q^{56} + 246 q^{58} - 724 q^{59} - 355 q^{61} + 872 q^{62} + 256 q^{64} - 2019 q^{65} - 869 q^{67} + 340 q^{68} - 258 q^{70} + 331 q^{71} - 1135 q^{73} + 672 q^{74} - 20 q^{76} - 1083 q^{79} - 320 q^{80} - 788 q^{82} - 169 q^{83} - 1104 q^{85} + 1360 q^{86} + 258 q^{89} + 874 q^{91} - 1264 q^{92} - 2170 q^{94} + 1555 q^{95} - 248 q^{97} - 622 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 - 21 * q^7 + 32 * q^8 - 40 * q^10 - 38 * q^13 - 42 * q^14 + 64 * q^16 + 85 * q^17 - 5 * q^19 - 80 * q^20 - 316 * q^23 + 318 * q^25 - 76 * q^26 - 84 * q^28 + 123 * q^29 + 436 * q^31 + 128 * q^32 + 170 * q^34 - 129 * q^35 + 336 * q^37 - 10 * q^38 - 160 * q^40 - 394 * q^41 + 680 * q^43 - 632 * q^46 - 1085 * q^47 - 311 * q^49 + 636 * q^50 - 152 * q^52 - 880 * q^53 - 168 * q^56 + 246 * q^58 - 724 * q^59 - 355 * q^61 + 872 * q^62 + 256 * q^64 - 2019 * q^65 - 869 * q^67 + 340 * q^68 - 258 * q^70 + 331 * q^71 - 1135 * q^73 + 672 * q^74 - 20 * q^76 - 1083 * q^79 - 320 * q^80 - 788 * q^82 - 169 * q^83 - 1104 * q^85 + 1360 * q^86 + 258 * q^89 + 874 * q^91 - 1264 * q^92 - 2170 * q^94 + 1555 * q^95 - 248 * q^97 - 622 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	6.18714	0.553394	0.276697	−	0.960957i	$-0.410760\pi$
$5$	6.18714	0.553394	0.276697	+	0.960957i	$0.410760\pi$
$6$	0	0
$7$	−25.7192	−1.38871	−0.694353	−	0.719634i	$-0.744309\pi$
$7$	−25.7192	−1.38871	−0.694353	+	0.719634i	$0.744309\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	12.3743	0.391309
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−30.7630	−0.656318	−0.328159	−	0.944623i	$-0.606428\pi$
$13$	−30.7630	−0.656318	−0.328159	+	0.944623i	$0.606428\pi$
$14$	−51.4384	−0.981964
$15$	0	0
$16$	16.0000	0.250000
$17$	47.9850	0.684592	0.342296	−	0.939592i	$-0.388795\pi$
$17$	47.9850	0.684592	0.342296	+	0.939592i	$0.388795\pi$
$18$	0	0
$19$	46.7480	0.564459	0.282230	−	0.959347i	$-0.408926\pi$
$19$	46.7480	0.564459	0.282230	+	0.959347i	$0.408926\pi$
$20$	24.7485	0.276697
$21$	0	0
$22$	0	0
$23$	−2.04459	−0.0185359	−0.00926795	−	0.999957i	$-0.502950\pi$
$23$	−2.04459	−0.0185359	−0.00926795	+	0.999957i	$0.502950\pi$
$24$	0	0
$25$	−86.7194	−0.693755
$26$	−61.5261	−0.464087
$27$	0	0
$28$	−102.877	−0.694353
$29$	59.0080	0.377845	0.188923	−	0.981992i	$-0.439500\pi$
$29$	59.0080	0.377845	0.188923	+	0.981992i	$0.439500\pi$
$30$	0	0
$31$	304.896	1.76648	0.883241	−	0.468920i	$-0.155357\pi$
$31$	304.896	1.76648	0.883241	+	0.468920i	$0.155357\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	95.9699	0.484079
$35$	−159.128	−0.768502
$36$	0	0
$37$	−303.823	−1.34995	−0.674976	−	0.737840i	$-0.735846\pi$
$37$	−303.823	−1.34995	−0.674976	+	0.737840i	$0.735846\pi$
$38$	93.4960	0.399133
$39$	0	0
$40$	49.4971	0.195654
$41$	145.553	0.554428	0.277214	−	0.960808i	$-0.410589\pi$
$41$	145.553	0.554428	0.277214	+	0.960808i	$0.410589\pi$
$42$	0	0
$43$	284.084	1.00750	0.503748	−	0.863850i	$-0.331954\pi$
$43$	284.084	1.00750	0.503748	+	0.863850i	$0.331954\pi$
$44$	0	0
$45$	0	0
$46$	−4.08917	−0.0131069
$47$	−577.251	−1.79151	−0.895753	−	0.444552i	$-0.853363\pi$
$47$	−577.251	−1.79151	−0.895753	+	0.444552i	$0.853363\pi$
$48$	0	0
$49$	318.477	0.928505
$50$	−173.439	−0.490559
$51$	0	0
$52$	−123.052	−0.328159
$53$	−655.216	−1.69813	−0.849065	−	0.528289i	$-0.822834\pi$
$53$	−655.216	−1.69813	−0.849065	+	0.528289i	$0.822834\pi$
$54$	0	0
$55$	0	0
$56$	−205.754	−0.490982
$57$	0	0
$58$	118.016	0.267177
$59$	23.1802	0.0511492	0.0255746	−	0.999673i	$-0.491858\pi$
$59$	23.1802	0.0511492	0.0255746	+	0.999673i	$0.491858\pi$
$60$	0	0
$61$	631.874	1.32628	0.663140	−	0.748495i	$-0.269223\pi$
$61$	631.874	1.32628	0.663140	+	0.748495i	$0.269223\pi$
$62$	609.792	1.24909
$63$	0	0
$64$	64.0000	0.125000
$65$	−190.335	−0.363202
$66$	0	0
$67$	−655.918	−1.19602	−0.598009	−	0.801490i	$-0.704041\pi$
$67$	−655.918	−1.19602	−0.598009	+	0.801490i	$0.704041\pi$
$68$	191.940	0.342296
$69$	0	0
$70$	−318.256	−0.543413
$71$	−528.602	−0.883571	−0.441786	−	0.897121i	$-0.645655\pi$
$71$	−528.602	−0.883571	−0.441786	+	0.897121i	$0.645655\pi$
$72$	0	0
$73$	−447.652	−0.717723	−0.358861	−	0.933391i	$-0.616835\pi$
$73$	−447.652	−0.717723	−0.358861	+	0.933391i	$0.616835\pi$
$74$	−607.646	−0.954560
$75$	0	0
$76$	186.992	0.282230
$77$	0	0
$78$	0	0
$79$	−985.464	−1.40346	−0.701730	−	0.712443i	$-0.747589\pi$
$79$	−985.464	−1.40346	−0.701730	+	0.712443i	$0.747589\pi$
$80$	98.9942	0.138349
$81$	0	0
$82$	291.106	0.392040
$83$	128.459	0.169882	0.0849411	−	0.996386i	$-0.472930\pi$
$83$	128.459	0.169882	0.0849411	+	0.996386i	$0.472930\pi$
$84$	0	0
$85$	296.889	0.378849
$86$	568.167	0.712408
$87$	0	0
$88$	0	0
$89$	−1425.15	−1.69736	−0.848682	−	0.528904i	$-0.822603\pi$
$89$	−1425.15	−1.69736	−0.848682	+	0.528904i	$0.822603\pi$
$90$	0	0
$91$	791.201	0.911432
$92$	−8.17835	−0.00926795
$93$	0	0
$94$	−1154.50	−1.26679
$95$	289.236	0.312368
$96$	0	0
$97$	−142.553	−0.149217	−0.0746084	−	0.997213i	$-0.523771\pi$
$97$	−142.553	−0.149217	−0.0746084	+	0.997213i	$0.523771\pi$
$98$	636.954	0.656552
$99$	0	0
$100$	−346.877	−0.346877
$101$	293.830	0.289477	0.144738	−	0.989470i	$-0.453766\pi$
$101$	293.830	0.289477	0.144738	+	0.989470i	$0.453766\pi$
$102$	0	0
$103$	−1001.55	−0.958112	−0.479056	−	0.877784i	$-0.659021\pi$
$103$	−1001.55	−0.958112	−0.479056	+	0.877784i	$0.659021\pi$
$104$	−246.104	−0.232043
$105$	0	0
$106$	−1310.43	−1.20076
$107$	−1604.00	−1.44920	−0.724602	−	0.689168i	$-0.757976\pi$
$107$	−1604.00	−1.44920	−0.724602	+	0.689168i	$0.757976\pi$
$108$	0	0
$109$	323.134	0.283951	0.141975	−	0.989870i	$-0.454655\pi$
$109$	323.134	0.283951	0.141975	+	0.989870i	$0.454655\pi$
$110$	0	0
$111$	0	0
$112$	−411.507	−0.347177
$113$	464.460	0.386661	0.193331	−	0.981134i	$-0.438071\pi$
$113$	464.460	0.386661	0.193331	+	0.981134i	$0.438071\pi$
$114$	0	0
$115$	−12.6501	−0.0102577
$116$	236.032	0.188923
$117$	0	0
$118$	46.3604	0.0361680
$119$	−1234.13	−0.950697
$120$	0	0
$121$	0	0
$122$	1263.75	0.937822
$123$	0	0
$124$	1219.58	0.883241
$125$	−1309.94	−0.937314
$126$	0	0
$127$	327.549	0.228861	0.114430	−	0.993431i	$-0.463496\pi$
$127$	327.549	0.228861	0.114430	+	0.993431i	$0.463496\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−380.670	−0.256823
$131$	−2871.12	−1.91489	−0.957447	−	0.288610i	$-0.906807\pi$
$131$	−2871.12	−1.91489	−0.957447	+	0.288610i	$0.906807\pi$
$132$	0	0
$133$	−1202.32	−0.783868
$134$	−1311.84	−0.845712
$135$	0	0
$136$	383.880	0.242040
$137$	−1143.36	−0.713018	−0.356509	−	0.934292i	$-0.616033\pi$
$137$	−1143.36	−0.713018	−0.356509	+	0.934292i	$0.616033\pi$
$138$	0	0
$139$	336.573	0.205379	0.102690	−	0.994713i	$-0.467255\pi$
$139$	336.573	0.205379	0.102690	+	0.994713i	$0.467255\pi$
$140$	−636.513	−0.384251
$141$	0	0
$142$	−1057.20	−0.624779
$143$	0	0
$144$	0	0
$145$	365.091	0.209097
$146$	−895.305	−0.507507
$147$	0	0
$148$	−1215.29	−0.674976
$149$	−858.353	−0.471940	−0.235970	−	0.971760i	$-0.575827\pi$
$149$	−858.353	−0.471940	−0.235970	+	0.971760i	$0.575827\pi$
$150$	0	0
$151$	3086.90	1.66363	0.831814	−	0.555054i	$-0.187302\pi$
$151$	3086.90	1.66363	0.831814	+	0.555054i	$0.187302\pi$
$152$	373.984	0.199566
$153$	0	0
$154$	0	0
$155$	1886.43	0.977560
$156$	0	0
$157$	1737.12	0.883039	0.441519	−	0.897252i	$-0.354440\pi$
$157$	1737.12	0.883039	0.441519	+	0.897252i	$0.354440\pi$
$158$	−1970.93	−0.992396
$159$	0	0
$160$	197.988	0.0978272
$161$	52.5851	0.0257409
$162$	0	0
$163$	−2382.91	−1.14506	−0.572528	−	0.819885i	$-0.694037\pi$
$163$	−2382.91	−1.14506	−0.572528	+	0.819885i	$0.694037\pi$
$164$	582.212	0.277214
$165$	0	0
$166$	256.918	0.120125
$167$	−2961.84	−1.37242	−0.686209	−	0.727404i	$-0.740726\pi$
$167$	−2961.84	−1.37242	−0.686209	+	0.727404i	$0.740726\pi$
$168$	0	0
$169$	−1250.64	−0.569247
$170$	593.779	0.267887
$171$	0	0
$172$	1136.33	0.503748
$173$	314.894	0.138387	0.0691936	−	0.997603i	$-0.477957\pi$
$173$	314.894	0.138387	0.0691936	+	0.997603i	$0.477957\pi$
$174$	0	0
$175$	2230.35	0.963422
$176$	0	0
$177$	0	0
$178$	−2850.30	−1.20022
$179$	2412.21	1.00725	0.503624	−	0.863923i	$-0.332000\pi$
$179$	2412.21	1.00725	0.503624	+	0.863923i	$0.332000\pi$
$180$	0	0
$181$	3908.18	1.60493	0.802466	−	0.596698i	$-0.203521\pi$
$181$	3908.18	1.60493	0.802466	+	0.596698i	$0.203521\pi$
$182$	1582.40	0.644480
$183$	0	0
$184$	−16.3567	−0.00655343
$185$	−1879.80	−0.747056
$186$	0	0
$187$	0	0
$188$	−2309.01	−0.895753
$189$	0	0
$190$	578.472	0.220878
$191$	912.609	0.345728	0.172864	−	0.984946i	$-0.444698\pi$
$191$	912.609	0.345728	0.172864	+	0.984946i	$0.444698\pi$
$192$	0	0
$193$	−533.160	−0.198848	−0.0994241	−	0.995045i	$-0.531700\pi$
$193$	−533.160	−0.198848	−0.0994241	+	0.995045i	$0.531700\pi$
$194$	−285.105	−0.105512
$195$	0	0
$196$	1273.91	0.464252
$197$	−2700.49	−0.976659	−0.488329	−	0.872659i	$-0.662394\pi$
$197$	−2700.49	−0.976659	−0.488329	+	0.872659i	$0.662394\pi$
$198$	0	0
$199$	4409.62	1.57080	0.785401	−	0.618987i	$-0.212457\pi$
$199$	4409.62	1.57080	0.785401	+	0.618987i	$0.212457\pi$
$200$	−693.755	−0.245279
$201$	0	0
$202$	587.659	0.204691
$203$	−1517.64	−0.524716
$204$	0	0
$205$	900.555	0.306817
$206$	−2003.10	−0.677488
$207$	0	0
$208$	−492.208	−0.164079
$209$	0	0
$210$	0	0
$211$	−5804.51	−1.89383	−0.946917	−	0.321479i	$-0.895820\pi$
$211$	−5804.51	−1.89383	−0.946917	+	0.321479i	$0.895820\pi$
$212$	−2620.86	−0.849065
$213$	0	0
$214$	−3208.01	−1.02474
$215$	1757.66	0.557543
$216$	0	0
$217$	−7841.68	−2.45312
$218$	646.268	0.200784
$219$	0	0
$220$	0	0
$221$	−1476.16	−0.449310
$222$	0	0
$223$	−3466.84	−1.04106	−0.520530	−	0.853843i	$-0.674266\pi$
$223$	−3466.84	−1.04106	−0.520530	+	0.853843i	$0.674266\pi$
$224$	−823.014	−0.245491
$225$	0	0
$226$	928.920	0.273411
$227$	4036.58	1.18025	0.590127	−	0.807311i	$-0.299078\pi$
$227$	4036.58	1.18025	0.590127	+	0.807311i	$0.299078\pi$
$228$	0	0
$229$	−3078.22	−0.888272	−0.444136	−	0.895959i	$-0.646489\pi$
$229$	−3078.22	−0.888272	−0.444136	+	0.895959i	$0.646489\pi$
$230$	−25.3003	−0.00725326
$231$	0	0
$232$	472.064	0.133589
$233$	−1687.61	−0.474501	−0.237251	−	0.971449i	$-0.576246\pi$
$233$	−1687.61	−0.474501	−0.237251	+	0.971449i	$0.576246\pi$
$234$	0	0
$235$	−3571.53	−0.991409
$236$	92.7208	0.0255746
$237$	0	0
$238$	−2468.27	−0.672244
$239$	−2421.76	−0.655442	−0.327721	−	0.944775i	$-0.606281\pi$
$239$	−2421.76	−0.655442	−0.327721	+	0.944775i	$0.606281\pi$
$240$	0	0
$241$	6935.97	1.85388	0.926940	−	0.375209i	$-0.122429\pi$
$241$	6935.97	1.85388	0.926940	+	0.375209i	$0.122429\pi$
$242$	0	0
$243$	0	0
$244$	2527.49	0.663140
$245$	1970.46	0.513829
$246$	0	0
$247$	−1438.11	−0.370464
$248$	2439.17	0.624545
$249$	0	0
$250$	−2619.87	−0.662781
$251$	764.367	0.192217	0.0961085	−	0.995371i	$-0.469360\pi$
$251$	764.367	0.192217	0.0961085	+	0.995371i	$0.469360\pi$
$252$	0	0
$253$	0	0
$254$	655.099	0.161829
$255$	0	0
$256$	256.000	0.0625000
$257$	−1828.90	−0.443906	−0.221953	−	0.975057i	$-0.571243\pi$
$257$	−1828.90	−0.443906	−0.221953	+	0.975057i	$0.571243\pi$
$258$	0	0
$259$	7814.09	1.87469
$260$	−761.340	−0.181601
$261$	0	0
$262$	−5742.24	−1.35403
$263$	7853.83	1.84140	0.920700	−	0.390271i	$-0.127619\pi$
$263$	7853.83	1.84140	0.920700	+	0.390271i	$0.127619\pi$
$264$	0	0
$265$	−4053.91	−0.939735
$266$	−2404.64	−0.554278
$267$	0	0
$268$	−2623.67	−0.598009
$269$	7656.82	1.73548	0.867741	−	0.497017i	$-0.165571\pi$
$269$	7656.82	1.73548	0.867741	+	0.497017i	$0.165571\pi$
$270$	0	0
$271$	7351.92	1.64796	0.823981	−	0.566618i	$-0.191749\pi$
$271$	7351.92	1.64796	0.823981	+	0.566618i	$0.191749\pi$
$272$	767.759	0.171148
$273$	0	0
$274$	−2286.71	−0.504180
$275$	0	0
$276$	0	0
$277$	−4238.60	−0.919396	−0.459698	−	0.888075i	$-0.652042\pi$
$277$	−4238.60	−0.919396	−0.459698	+	0.888075i	$0.652042\pi$
$278$	673.146	0.145225
$279$	0	0
$280$	−1273.03	−0.271706
$281$	−4243.25	−0.900822	−0.450411	−	0.892821i	$-0.648723\pi$
$281$	−4243.25	−0.900822	−0.450411	+	0.892821i	$0.648723\pi$
$282$	0	0
$283$	−4069.93	−0.854883	−0.427442	−	0.904043i	$-0.640585\pi$
$283$	−4069.93	−0.854883	−0.427442	+	0.904043i	$0.640585\pi$
$284$	−2114.41	−0.441786
$285$	0	0
$286$	0	0
$287$	−3743.50	−0.769937
$288$	0	0
$289$	−2610.44	−0.531334
$290$	730.181	0.147854
$291$	0	0
$292$	−1790.61	−0.358861
$293$	−7903.75	−1.57591	−0.787955	−	0.615733i	$-0.788860\pi$
$293$	−7903.75	−1.57591	−0.787955	+	0.615733i	$0.788860\pi$
$294$	0	0
$295$	143.419	0.0283057
$296$	−2430.59	−0.477280
$297$	0	0
$298$	−1716.71	−0.333712
$299$	62.8977	0.0121654
$300$	0	0
$301$	−7306.40	−1.39912
$302$	6173.79	1.17636
$303$	0	0
$304$	747.968	0.141115
$305$	3909.49	0.733956
$306$	0	0
$307$	4887.27	0.908570	0.454285	−	0.890856i	$-0.349895\pi$
$307$	4887.27	0.908570	0.454285	+	0.890856i	$0.349895\pi$
$308$	0	0
$309$	0	0
$310$	3772.86	0.691240
$311$	6555.16	1.19521	0.597603	−	0.801792i	$-0.296120\pi$
$311$	6555.16	1.19521	0.597603	+	0.801792i	$0.296120\pi$
$312$	0	0
$313$	−6473.07	−1.16894	−0.584472	−	0.811414i	$-0.698698\pi$
$313$	−6473.07	−1.16894	−0.584472	+	0.811414i	$0.698698\pi$
$314$	3474.23	0.624403
$315$	0	0
$316$	−3941.86	−0.701730
$317$	−3732.52	−0.661322	−0.330661	−	0.943750i	$-0.607272\pi$
$317$	−3732.52	−0.661322	−0.330661	+	0.943750i	$0.607272\pi$
$318$	0	0
$319$	0	0
$320$	395.977	0.0691743
$321$	0	0
$322$	105.170	0.0182016
$323$	2243.20	0.386424
$324$	0	0
$325$	2667.75	0.455324
$326$	−4765.82	−0.809676
$327$	0	0
$328$	1164.42	0.196020
$329$	14846.4	2.48788
$330$	0	0
$331$	6093.87	1.01193	0.505966	−	0.862554i	$-0.331136\pi$
$331$	6093.87	1.01193	0.505966	+	0.862554i	$0.331136\pi$
$332$	513.837	0.0849411
$333$	0	0
$334$	−5923.67	−0.970446
$335$	−4058.26	−0.661869
$336$	0	0
$337$	3220.27	0.520531	0.260266	−	0.965537i	$-0.416190\pi$
$337$	3220.27	0.520531	0.260266	+	0.965537i	$0.416190\pi$
$338$	−2501.27	−0.402518
$339$	0	0
$340$	1187.56	0.189425
$341$	0	0
$342$	0	0
$343$	630.707	0.0992857
$344$	2272.67	0.356204
$345$	0	0
$346$	629.789	0.0978545
$347$	−2922.59	−0.452141	−0.226071	−	0.974111i	$-0.572588\pi$
$347$	−2922.59	−0.452141	−0.226071	+	0.974111i	$0.572588\pi$
$348$	0	0
$349$	−2398.61	−0.367893	−0.183946	−	0.982936i	$-0.558887\pi$
$349$	−2398.61	−0.367893	−0.183946	+	0.982936i	$0.558887\pi$
$350$	4460.70	0.681242
$351$	0	0
$352$	0	0
$353$	8037.61	1.21189	0.605947	−	0.795505i	$-0.292794\pi$
$353$	8037.61	1.21189	0.605947	+	0.795505i	$0.292794\pi$
$354$	0	0
$355$	−3270.53	−0.488963
$356$	−5700.59	−0.848682
$357$	0	0
$358$	4824.43	0.712231
$359$	6245.09	0.918115	0.459058	−	0.888406i	$-0.348187\pi$
$359$	6245.09	0.918115	0.459058	+	0.888406i	$0.348187\pi$
$360$	0	0
$361$	−4673.63	−0.681386
$362$	7816.36	1.13486
$363$	0	0
$364$	3164.80	0.455716
$365$	−2769.69	−0.397184
$366$	0	0
$367$	3605.89	0.512878	0.256439	−	0.966560i	$-0.417451\pi$
$367$	3605.89	0.512878	0.256439	+	0.966560i	$0.417451\pi$
$368$	−32.7134	−0.00463398
$369$	0	0
$370$	−3759.59	−0.528248
$371$	16851.6	2.35820
$372$	0	0
$373$	1535.30	0.213123	0.106561	−	0.994306i	$-0.466016\pi$
$373$	1535.30	0.213123	0.106561	+	0.994306i	$0.466016\pi$
$374$	0	0
$375$	0	0
$376$	−4618.01	−0.633393
$377$	−1815.27	−0.247987
$378$	0	0
$379$	−12947.9	−1.75485	−0.877426	−	0.479713i	$-0.840741\pi$
$379$	−12947.9	−1.75485	−0.877426	+	0.479713i	$0.840741\pi$
$380$	1156.94	0.156184
$381$	0	0
$382$	1825.22	0.244467
$383$	3063.26	0.408683	0.204341	−	0.978900i	$-0.434495\pi$
$383$	3063.26	0.408683	0.204341	+	0.978900i	$0.434495\pi$
$384$	0	0
$385$	0	0
$386$	−1066.32	−0.140607
$387$	0	0
$388$	−570.211	−0.0746084
$389$	−3662.10	−0.477316	−0.238658	−	0.971104i	$-0.576707\pi$
$389$	−3662.10	−0.477316	−0.238658	+	0.971104i	$0.576707\pi$
$390$	0	0
$391$	−98.1094	−0.0126895
$392$	2547.82	0.328276
$393$	0	0
$394$	−5400.97	−0.690602
$395$	−6097.20	−0.776667
$396$	0	0
$397$	−5438.40	−0.687520	−0.343760	−	0.939057i	$-0.611701\pi$
$397$	−5438.40	−0.687520	−0.343760	+	0.939057i	$0.611701\pi$
$398$	8819.24	1.11072
$399$	0	0
$400$	−1387.51	−0.173439
$401$	−3275.54	−0.407912	−0.203956	−	0.978980i	$-0.565380\pi$
$401$	−3275.54	−0.407912	−0.203956	+	0.978980i	$0.565380\pi$
$402$	0	0
$403$	−9379.52	−1.15937
$404$	1175.32	0.144738
$405$	0	0
$406$	−3035.28	−0.371030
$407$	0	0
$408$	0	0
$409$	28.2549	0.00341593	0.00170797	−	0.999999i	$-0.499456\pi$
$409$	28.2549	0.00341593	0.00170797	+	0.999999i	$0.499456\pi$
$410$	1801.11	0.216952
$411$	0	0
$412$	−4006.20	−0.479056
$413$	−596.176	−0.0710312
$414$	0	0
$415$	794.795	0.0940119
$416$	−984.417	−0.116022
$417$	0	0
$418$	0	0
$419$	−2788.66	−0.325143	−0.162572	−	0.986697i	$-0.551979\pi$
$419$	−2788.66	−0.325143	−0.162572	+	0.986697i	$0.551979\pi$
$420$	0	0
$421$	−8006.82	−0.926908	−0.463454	−	0.886121i	$-0.653390\pi$
$421$	−8006.82	−0.926908	−0.463454	+	0.886121i	$0.653390\pi$
$422$	−11609.0	−1.33914
$423$	0	0
$424$	−5241.73	−0.600380
$425$	−4161.22	−0.474939
$426$	0	0
$427$	−16251.3	−1.84181
$428$	−6416.01	−0.724602
$429$	0	0
$430$	3515.33	0.394242
$431$	−11372.5	−1.27099	−0.635495	−	0.772105i	$-0.719204\pi$
$431$	−11372.5	−1.27099	−0.635495	+	0.772105i	$0.719204\pi$
$432$	0	0
$433$	−2336.59	−0.259329	−0.129664	−	0.991558i	$-0.541390\pi$
$433$	−2336.59	−0.259329	−0.129664	+	0.991558i	$0.541390\pi$
$434$	−15683.4	−1.73462
$435$	0	0
$436$	1292.54	0.141975
$437$	−95.5803	−0.0104628
$438$	0	0
$439$	−1689.97	−0.183731	−0.0918655	−	0.995771i	$-0.529283\pi$
$439$	−1689.97	−0.183731	−0.0918655	+	0.995771i	$0.529283\pi$
$440$	0	0
$441$	0	0
$442$	−2952.33	−0.317710
$443$	−6847.10	−0.734347	−0.367173	−	0.930153i	$-0.619674\pi$
$443$	−6847.10	−0.734347	−0.367173	+	0.930153i	$0.619674\pi$
$444$	0	0
$445$	−8817.58	−0.939311
$446$	−6933.67	−0.736141
$447$	0	0
$448$	−1646.03	−0.173588
$449$	−16416.6	−1.72550	−0.862749	−	0.505633i	$-0.831259\pi$
$449$	−16416.6	−1.72550	−0.862749	+	0.505633i	$0.831259\pi$
$450$	0	0
$451$	0	0
$452$	1857.84	0.193331
$453$	0	0
$454$	8073.17	0.834565
$455$	4895.26	0.504381
$456$	0	0
$457$	−14619.7	−1.49646	−0.748228	−	0.663442i	$-0.769095\pi$
$457$	−14619.7	−1.49646	−0.748228	+	0.663442i	$0.769095\pi$
$458$	−6156.43	−0.628103
$459$	0	0
$460$	−50.6005	−0.00512883
$461$	−932.481	−0.0942082	−0.0471041	−	0.998890i	$-0.514999\pi$
$461$	−932.481	−0.0942082	−0.0471041	+	0.998890i	$0.514999\pi$
$462$	0	0
$463$	5806.81	0.582862	0.291431	−	0.956592i	$-0.405869\pi$
$463$	5806.81	0.582862	0.291431	+	0.956592i	$0.405869\pi$
$464$	944.128	0.0944613
$465$	0	0
$466$	−3375.21	−0.335523
$467$	2143.97	0.212444	0.106222	−	0.994342i	$-0.466125\pi$
$467$	2143.97	0.212444	0.106222	+	0.994342i	$0.466125\pi$
$468$	0	0
$469$	16869.7	1.66092
$470$	−7143.07	−0.701032
$471$	0	0
$472$	185.442	0.0180840
$473$	0	0
$474$	0	0
$475$	−4053.96	−0.391596
$476$	−4936.54	−0.475348
$477$	0	0
$478$	−4843.52	−0.463467
$479$	3401.22	0.324437	0.162219	−	0.986755i	$-0.448135\pi$
$479$	3401.22	0.324437	0.162219	+	0.986755i	$0.448135\pi$
$480$	0	0
$481$	9346.52	0.885997
$482$	13871.9	1.31089
$483$	0	0
$484$	0	0
$485$	−881.993	−0.0825758
$486$	0	0
$487$	9632.81	0.896313	0.448156	−	0.893955i	$-0.352081\pi$
$487$	9632.81	0.896313	0.448156	+	0.893955i	$0.352081\pi$
$488$	5054.99	0.468911
$489$	0	0
$490$	3940.92	0.363332
$491$	5594.16	0.514177	0.257088	−	0.966388i	$-0.417237\pi$
$491$	5594.16	0.514177	0.257088	+	0.966388i	$0.417237\pi$
$492$	0	0
$493$	2831.50	0.258670
$494$	−2876.22	−0.261958
$495$	0	0
$496$	4878.33	0.441620
$497$	13595.2	1.22702
$498$	0	0
$499$	−2295.31	−0.205917	−0.102958	−	0.994686i	$-0.532831\pi$
$499$	−2295.31	−0.205917	−0.102958	+	0.994686i	$0.532831\pi$
$500$	−5239.75	−0.468657
$501$	0	0
$502$	1528.73	0.135918
$503$	−6271.41	−0.555921	−0.277961	−	0.960592i	$-0.589658\pi$
$503$	−6271.41	−0.555921	−0.277961	+	0.960592i	$0.589658\pi$
$504$	0	0
$505$	1817.96	0.160195
$506$	0	0
$507$	0	0
$508$	1310.20	0.114430
$509$	−8882.04	−0.773457	−0.386729	−	0.922194i	$-0.626395\pi$
$509$	−8882.04	−0.773457	−0.386729	+	0.922194i	$0.626395\pi$
$510$	0	0
$511$	11513.3	0.996706
$512$	512.000	0.0441942
$513$	0	0
$514$	−3657.81	−0.313889
$515$	−6196.72	−0.530214
$516$	0	0
$517$	0	0
$518$	15628.2	1.32560
$519$	0	0
$520$	−1522.68	−0.128411
$521$	−506.901	−0.0426252	−0.0213126	−	0.999773i	$-0.506785\pi$
$521$	−506.901	−0.0426252	−0.0213126	+	0.999773i	$0.506785\pi$
$522$	0	0
$523$	−8415.97	−0.703642	−0.351821	−	0.936067i	$-0.614437\pi$
$523$	−8415.97	−0.703642	−0.351821	+	0.936067i	$0.614437\pi$
$524$	−11484.5	−0.957447
$525$	0	0
$526$	15707.7	1.30207
$527$	14630.4	1.20932
$528$	0	0
$529$	−12162.8	−0.999656
$530$	−8107.82	−0.664493
$531$	0	0
$532$	−4809.28	−0.391934
$533$	−4477.65	−0.363881
$534$	0	0
$535$	−9924.18	−0.801981
$536$	−5247.35	−0.422856
$537$	0	0
$538$	15313.6	1.22717
$539$	0	0
$540$	0	0
$541$	2292.00	0.182145	0.0910727	−	0.995844i	$-0.470970\pi$
$541$	2292.00	0.182145	0.0910727	+	0.995844i	$0.470970\pi$
$542$	14703.8	1.16528
$543$	0	0
$544$	1535.52	0.121020
$545$	1999.28	0.157137
$546$	0	0
$547$	9334.79	0.729665	0.364833	−	0.931073i	$-0.381126\pi$
$547$	9334.79	0.729665	0.364833	+	0.931073i	$0.381126\pi$
$548$	−4573.43	−0.356509
$549$	0	0
$550$	0	0
$551$	2758.51	0.213278
$552$	0	0
$553$	25345.3	1.94899
$554$	−8477.20	−0.650111
$555$	0	0
$556$	1346.29	0.102690
$557$	−3681.18	−0.280030	−0.140015	−	0.990149i	$-0.544715\pi$
$557$	−3681.18	−0.280030	−0.140015	+	0.990149i	$0.544715\pi$
$558$	0	0
$559$	−8739.27	−0.661238
$560$	−2546.05	−0.192125
$561$	0	0
$562$	−8486.50	−0.636978
$563$	23961.3	1.79369	0.896846	−	0.442342i	$-0.145852\pi$
$563$	23961.3	1.79369	0.896846	+	0.442342i	$0.145852\pi$
$564$	0	0
$565$	2873.68	0.213976
$566$	−8139.85	−0.604494
$567$	0	0
$568$	−4228.82	−0.312390
$569$	3727.10	0.274601	0.137301	−	0.990529i	$-0.456157\pi$
$569$	3727.10	0.274601	0.137301	+	0.990529i	$0.456157\pi$
$570$	0	0
$571$	23463.5	1.71965	0.859823	−	0.510593i	$-0.170574\pi$
$571$	23463.5	1.71965	0.859823	+	0.510593i	$0.170574\pi$
$572$	0	0
$573$	0	0
$574$	−7487.01	−0.544428
$575$	177.305	0.0128594
$576$	0	0
$577$	−1118.13	−0.0806729	−0.0403365	−	0.999186i	$-0.512843\pi$
$577$	−1118.13	−0.0806729	−0.0403365	+	0.999186i	$0.512843\pi$
$578$	−5220.89	−0.375710
$579$	0	0
$580$	1460.36	0.104549
$581$	−3303.87	−0.235917
$582$	0	0
$583$	0	0
$584$	−3581.22	−0.253753
$585$	0	0
$586$	−15807.5	−1.11434
$587$	20399.2	1.43436	0.717178	−	0.696890i	$-0.245434\pi$
$587$	20399.2	1.43436	0.717178	+	0.696890i	$0.245434\pi$
$588$	0	0
$589$	14253.3	0.997106
$590$	286.838	0.0200151
$591$	0	0
$592$	−4861.17	−0.337488
$593$	−2722.97	−0.188565	−0.0942826	−	0.995545i	$-0.530056\pi$
$593$	−2722.97	−0.188565	−0.0942826	+	0.995545i	$0.530056\pi$
$594$	0	0
$595$	−7635.76	−0.526110
$596$	−3433.41	−0.235970
$597$	0	0
$598$	125.795	0.00860227
$599$	444.486	0.0303192	0.0151596	−	0.999885i	$-0.495174\pi$
$599$	444.486	0.0303192	0.0151596	+	0.999885i	$0.495174\pi$
$600$	0	0
$601$	−3564.00	−0.241895	−0.120947	−	0.992659i	$-0.538593\pi$
$601$	−3564.00	−0.241895	−0.120947	+	0.992659i	$0.538593\pi$
$602$	−14612.8	−0.989325
$603$	0	0
$604$	12347.6	0.831814
$605$	0	0
$606$	0	0
$607$	−9182.13	−0.613989	−0.306994	−	0.951711i	$-0.599323\pi$
$607$	−9182.13	−0.613989	−0.306994	+	0.951711i	$0.599323\pi$
$608$	1495.94	0.0997832
$609$	0	0
$610$	7818.97	0.518985
$611$	17758.0	1.17580
$612$	0	0
$613$	15822.6	1.04253	0.521263	−	0.853396i	$-0.325461\pi$
$613$	15822.6	1.04253	0.521263	+	0.853396i	$0.325461\pi$
$614$	9774.53	0.642456
$615$	0	0
$616$	0	0
$617$	−15442.0	−1.00757	−0.503784	−	0.863830i	$-0.668059\pi$
$617$	−15442.0	−1.00757	−0.503784	+	0.863830i	$0.668059\pi$
$618$	0	0
$619$	4205.61	0.273082	0.136541	−	0.990634i	$-0.456401\pi$
$619$	4205.61	0.273082	0.136541	+	0.990634i	$0.456401\pi$
$620$	7545.73	0.488780
$621$	0	0
$622$	13110.3	0.845138
$623$	36653.7	2.35714
$624$	0	0
$625$	2735.17	0.175051
$626$	−12946.1	−0.826568
$627$	0	0
$628$	6948.47	0.441519
$629$	−14578.9	−0.924166
$630$	0	0
$631$	18506.4	1.16755	0.583777	−	0.811914i	$-0.301574\pi$
$631$	18506.4	1.16755	0.583777	+	0.811914i	$0.301574\pi$
$632$	−7883.71	−0.496198
$633$	0	0
$634$	−7465.03	−0.467625
$635$	2026.59	0.126650
$636$	0	0
$637$	−9797.32	−0.609394
$638$	0	0
$639$	0	0
$640$	791.953	0.0489136
$641$	−20000.4	−1.23240	−0.616201	−	0.787589i	$-0.711329\pi$
$641$	−20000.4	−1.23240	−0.616201	+	0.787589i	$0.711329\pi$
$642$	0	0
$643$	22453.5	1.37710	0.688552	−	0.725187i	$-0.258247\pi$
$643$	22453.5	1.37710	0.688552	+	0.725187i	$0.258247\pi$
$644$	210.341	0.0128705
$645$	0	0
$646$	4486.40	0.273243
$647$	−12963.8	−0.787727	−0.393863	−	0.919169i	$-0.628862\pi$
$647$	−12963.8	−0.787727	−0.393863	+	0.919169i	$0.628862\pi$
$648$	0	0
$649$	0	0
$650$	5335.50	0.321962
$651$	0	0
$652$	−9531.65	−0.572528
$653$	5075.91	0.304189	0.152095	−	0.988366i	$-0.451398\pi$
$653$	5075.91	0.304189	0.152095	+	0.988366i	$0.451398\pi$
$654$	0	0
$655$	−17764.0	−1.05969
$656$	2328.85	0.138607
$657$	0	0
$658$	29692.9	1.75919
$659$	29578.3	1.74842	0.874210	−	0.485548i	$-0.161380\pi$
$659$	29578.3	1.74842	0.874210	+	0.485548i	$0.161380\pi$
$660$	0	0
$661$	13835.2	0.814113	0.407056	−	0.913403i	$-0.366555\pi$
$661$	13835.2	0.814113	0.407056	+	0.913403i	$0.366555\pi$
$662$	12187.7	0.715543
$663$	0	0
$664$	1027.67	0.0600625
$665$	−7438.92	−0.433788
$666$	0	0
$667$	−120.647	−0.00700371
$668$	−11847.3	−0.686209
$669$	0	0
$670$	−8116.51	−0.468012
$671$	0	0
$672$	0	0
$673$	14592.6	0.835812	0.417906	−	0.908490i	$-0.362764\pi$
$673$	14592.6	0.835812	0.417906	+	0.908490i	$0.362764\pi$
$674$	6440.53	0.368071
$675$	0	0
$676$	−5002.54	−0.284624
$677$	−6957.20	−0.394959	−0.197479	−	0.980307i	$-0.563276\pi$
$677$	−6957.20	−0.394959	−0.197479	+	0.980307i	$0.563276\pi$
$678$	0	0
$679$	3666.34	0.207218
$680$	2375.12	0.133943
$681$	0	0
$682$	0	0
$683$	−4828.48	−0.270508	−0.135254	−	0.990811i	$-0.543185\pi$
$683$	−4828.48	−0.270508	−0.135254	+	0.990811i	$0.543185\pi$
$684$	0	0
$685$	−7074.10	−0.394580
$686$	1261.41	0.0702056
$687$	0	0
$688$	4545.34	0.251874
$689$	20156.4	1.11451
$690$	0	0
$691$	−13876.3	−0.763934	−0.381967	−	0.924176i	$-0.624753\pi$
$691$	−13876.3	−0.763934	−0.381967	+	0.924176i	$0.624753\pi$
$692$	1259.58	0.0691936
$693$	0	0
$694$	−5845.19	−0.319712
$695$	2082.42	0.113656
$696$	0	0
$697$	6984.35	0.379557
$698$	−4797.22	−0.260140
$699$	0	0
$700$	8921.41	0.481711
$701$	10253.4	0.552445	0.276222	−	0.961094i	$-0.410917\pi$
$701$	10253.4	0.552445	0.276222	+	0.961094i	$0.410917\pi$
$702$	0	0
$703$	−14203.1	−0.761993
$704$	0	0
$705$	0	0
$706$	16075.2	0.856939
$707$	−7557.06	−0.401998
$708$	0	0
$709$	28775.1	1.52422	0.762111	−	0.647447i	$-0.224163\pi$
$709$	28775.1	1.52422	0.762111	+	0.647447i	$0.224163\pi$
$710$	−6541.07	−0.345749
$711$	0	0
$712$	−11401.2	−0.600109
$713$	−623.386	−0.0327433
$714$	0	0
$715$	0	0
$716$	9648.85	0.503624
$717$	0	0
$718$	12490.2	0.649206
$719$	−10314.7	−0.535011	−0.267506	−	0.963556i	$-0.586199\pi$
$719$	−10314.7	−0.535011	−0.267506	+	0.963556i	$0.586199\pi$
$720$	0	0
$721$	25759.0	1.33054
$722$	−9347.25	−0.481813
$723$	0	0
$724$	15632.7	0.802466
$725$	−5117.14	−0.262132
$726$	0	0
$727$	16485.6	0.841013	0.420506	−	0.907290i	$-0.361852\pi$
$727$	16485.6	0.841013	0.420506	+	0.907290i	$0.361852\pi$
$728$	6329.60	0.322240
$729$	0	0
$730$	−5539.37	−0.280851
$731$	13631.7	0.689724
$732$	0	0
$733$	−10119.9	−0.509943	−0.254971	−	0.966949i	$-0.582066\pi$
$733$	−10119.9	−0.509943	−0.254971	+	0.966949i	$0.582066\pi$
$734$	7211.79	0.362659
$735$	0	0
$736$	−65.4268	−0.00327672
$737$	0	0
$738$	0	0
$739$	2713.38	0.135065	0.0675326	−	0.997717i	$-0.478487\pi$
$739$	2713.38	0.135065	0.0675326	+	0.997717i	$0.478487\pi$
$740$	−7519.18	−0.373528
$741$	0	0
$742$	33703.3	1.66750
$743$	13700.9	0.676497	0.338249	−	0.941057i	$-0.390166\pi$
$743$	13700.9	0.676497	0.338249	+	0.941057i	$0.390166\pi$
$744$	0	0
$745$	−5310.75	−0.261169
$746$	3070.60	0.150700
$747$	0	0
$748$	0	0
$749$	41253.7	2.01252
$750$	0	0
$751$	−7831.68	−0.380535	−0.190268	−	0.981732i	$-0.560936\pi$
$751$	−7831.68	−0.380535	−0.190268	+	0.981732i	$0.560936\pi$
$752$	−9236.02	−0.447877
$753$	0	0
$754$	−3630.53	−0.175353
$755$	19099.0	0.920643
$756$	0	0
$757$	36879.2	1.77067	0.885335	−	0.464953i	$-0.153929\pi$
$757$	36879.2	1.77067	0.885335	+	0.464953i	$0.153929\pi$
$758$	−25895.8	−1.24087
$759$	0	0
$760$	2313.89	0.110439
$761$	−30071.5	−1.43245	−0.716224	−	0.697871i	$-0.754131\pi$
$761$	−30071.5	−1.43245	−0.716224	+	0.697871i	$0.754131\pi$
$762$	0	0
$763$	−8310.75	−0.394324
$764$	3650.44	0.172864
$765$	0	0
$766$	6126.53	0.288982
$767$	−713.093	−0.0335701
$768$	0	0
$769$	−23609.8	−1.10714	−0.553571	−	0.832802i	$-0.686735\pi$
$769$	−23609.8	−1.10714	−0.553571	+	0.832802i	$0.686735\pi$
$770$	0	0
$771$	0	0
$772$	−2132.64	−0.0994241
$773$	−16132.1	−0.750622	−0.375311	−	0.926899i	$-0.622464\pi$
$773$	−16132.1	−0.750622	−0.375311	+	0.926899i	$0.622464\pi$
$774$	0	0
$775$	−26440.4	−1.22550
$776$	−1140.42	−0.0527561
$777$	0	0
$778$	−7324.20	−0.337513
$779$	6804.30	0.312952
$780$	0	0
$781$	0	0
$782$	−196.219	−0.00897285
$783$	0	0
$784$	5095.63	0.232126
$785$	10747.8	0.488668
$786$	0	0
$787$	−12473.2	−0.564959	−0.282480	−	0.959273i	$-0.591157\pi$
$787$	−12473.2	−0.564959	−0.282480	+	0.959273i	$0.591157\pi$
$788$	−10801.9	−0.488329
$789$	0	0
$790$	−12194.4	−0.549186
$791$	−11945.5	−0.536959
$792$	0	0
$793$	−19438.3	−0.870461
$794$	−10876.8	−0.486150
$795$	0	0
$796$	17638.5	0.785401
$797$	10468.1	0.465246	0.232623	−	0.972567i	$-0.425269\pi$
$797$	10468.1	0.465246	0.232623	+	0.972567i	$0.425269\pi$
$798$	0	0
$799$	−27699.4	−1.22645
$800$	−2775.02	−0.122640
$801$	0	0
$802$	−6551.08	−0.288437
$803$	0	0
$804$	0	0
$805$	325.351	0.0142449
$806$	−18759.0	−0.819800
$807$	0	0
$808$	2350.64	0.102345
$809$	−17641.7	−0.766686	−0.383343	−	0.923606i	$-0.625227\pi$
$809$	−17641.7	−0.766686	−0.383343	+	0.923606i	$0.625227\pi$
$810$	0	0
$811$	17854.5	0.773066	0.386533	−	0.922276i	$-0.373672\pi$
$811$	17854.5	0.773066	0.386533	+	0.922276i	$0.373672\pi$
$812$	−6070.56	−0.262358
$813$	0	0
$814$	0	0
$815$	−14743.4	−0.633667
$816$	0	0
$817$	13280.3	0.568691
$818$	56.5099	0.00241543
$819$	0	0
$820$	3602.22	0.153409
$821$	14573.5	0.619511	0.309755	−	0.950816i	$-0.399753\pi$
$821$	14573.5	0.619511	0.309755	+	0.950816i	$0.399753\pi$
$822$	0	0
$823$	18565.3	0.786326	0.393163	−	0.919469i	$-0.371381\pi$
$823$	18565.3	0.786326	0.393163	+	0.919469i	$0.371381\pi$
$824$	−8012.39	−0.338744
$825$	0	0
$826$	−1192.35	−0.0502267
$827$	38779.4	1.63058	0.815291	−	0.579052i	$-0.196577\pi$
$827$	38779.4	1.63058	0.815291	+	0.579052i	$0.196577\pi$
$828$	0	0
$829$	12060.0	0.505263	0.252631	−	0.967563i	$-0.418704\pi$
$829$	12060.0	0.505263	0.252631	+	0.967563i	$0.418704\pi$
$830$	1589.59	0.0664764
$831$	0	0
$832$	−1968.83	−0.0820397
$833$	15282.1	0.635647
$834$	0	0
$835$	−18325.3	−0.759488
$836$	0	0
$837$	0	0
$838$	−5577.32	−0.229911
$839$	28708.8	1.18133	0.590666	−	0.806916i	$-0.298865\pi$
$839$	28708.8	1.18133	0.590666	+	0.806916i	$0.298865\pi$
$840$	0	0
$841$	−20907.1	−0.857233
$842$	−16013.6	−0.655423
$843$	0	0
$844$	−23218.0	−0.946917
$845$	−7737.85	−0.315018
$846$	0	0
$847$	0	0
$848$	−10483.5	−0.424532
$849$	0	0
$850$	−8322.45	−0.335833
$851$	621.193	0.0250226
$852$	0	0
$853$	−8281.48	−0.332418	−0.166209	−	0.986091i	$-0.553153\pi$
$853$	−8281.48	−0.332418	−0.166209	+	0.986091i	$0.553153\pi$
$854$	−32502.6	−1.30236
$855$	0	0
$856$	−12832.0	−0.512371
$857$	−21003.8	−0.837195	−0.418597	−	0.908172i	$-0.637478\pi$
$857$	−21003.8	−0.837195	−0.418597	+	0.908172i	$0.637478\pi$
$858$	0	0
$859$	120.604	0.00479040	0.00239520	−	0.999997i	$-0.499238\pi$
$859$	120.604	0.00479040	0.00239520	+	0.999997i	$0.499238\pi$
$860$	7030.66	0.278771
$861$	0	0
$862$	−22745.1	−0.898725
$863$	24086.3	0.950065	0.475033	−	0.879968i	$-0.342436\pi$
$863$	24086.3	0.950065	0.475033	+	0.879968i	$0.342436\pi$
$864$	0	0
$865$	1948.29	0.0765827
$866$	−4673.18	−0.183373
$867$	0	0
$868$	−31366.7	−1.22656
$869$	0	0
$870$	0	0
$871$	20178.0	0.784967
$872$	2585.07	0.100392
$873$	0	0
$874$	−191.161	−0.00739829
$875$	33690.5	1.30165
$876$	0	0
$877$	3898.55	0.150108	0.0750539	−	0.997179i	$-0.476087\pi$
$877$	3898.55	0.150108	0.0750539	+	0.997179i	$0.476087\pi$
$878$	−3379.94	−0.129918
$879$	0	0
$880$	0	0
$881$	−27751.9	−1.06128	−0.530639	−	0.847598i	$-0.678048\pi$
$881$	−27751.9	−1.06128	−0.530639	+	0.847598i	$0.678048\pi$
$882$	0	0
$883$	−9931.42	−0.378504	−0.189252	−	0.981929i	$-0.560606\pi$
$883$	−9931.42	−0.378504	−0.189252	+	0.981929i	$0.560606\pi$
$884$	−5904.65	−0.224655
$885$	0	0
$886$	−13694.2	−0.519261
$887$	−16262.0	−0.615586	−0.307793	−	0.951453i	$-0.599590\pi$
$887$	−16262.0	−0.615586	−0.307793	+	0.951453i	$0.599590\pi$
$888$	0	0
$889$	−8424.31	−0.317820
$890$	−17635.2	−0.664193
$891$	0	0
$892$	−13867.3	−0.520530
$893$	−26985.3	−1.01123
$894$	0	0
$895$	14924.7	0.557405
$896$	−3292.06	−0.122745
$897$	0	0
$898$	−32833.2	−1.22011
$899$	17991.3	0.667457
$900$	0	0
$901$	−31440.5	−1.16253
$902$	0	0
$903$	0	0
$904$	3715.68	0.136705
$905$	24180.4	0.888160
$906$	0	0
$907$	31520.9	1.15395	0.576976	−	0.816761i	$-0.304233\pi$
$907$	31520.9	1.15395	0.576976	+	0.816761i	$0.304233\pi$
$908$	16146.3	0.590127
$909$	0	0
$910$	9790.53	0.356652
$911$	48867.3	1.77722	0.888609	−	0.458665i	$-0.151672\pi$
$911$	48867.3	1.77722	0.888609	+	0.458665i	$0.151672\pi$
$912$	0	0
$913$	0	0
$914$	−29239.4	−1.05815
$915$	0	0
$916$	−12312.9	−0.444136
$917$	73843.0	2.65922
$918$	0	0
$919$	−18472.9	−0.663073	−0.331536	−	0.943442i	$-0.607567\pi$
$919$	−18472.9	−0.663073	−0.331536	+	0.943442i	$0.607567\pi$
$920$	−101.201	−0.00362663
$921$	0	0
$922$	−1864.96	−0.0666153
$923$	16261.4	0.579904
$924$	0	0
$925$	26347.4	0.936536
$926$	11613.6	0.412146
$927$	0	0
$928$	1888.26	0.0667943
$929$	40016.6	1.41324	0.706622	−	0.707591i	$-0.250218\pi$
$929$	40016.6	1.41324	0.706622	+	0.707591i	$0.250218\pi$
$930$	0	0
$931$	14888.2	0.524103
$932$	−6750.42	−0.237251
$933$	0	0
$934$	4287.95	0.150220
$935$	0	0
$936$	0	0
$937$	20667.1	0.720562	0.360281	−	0.932844i	$-0.382681\pi$
$937$	20667.1	0.720562	0.360281	+	0.932844i	$0.382681\pi$
$938$	33739.4	1.17445
$939$	0	0
$940$	−14286.1	−0.495705
$941$	−47773.6	−1.65502	−0.827510	−	0.561451i	$-0.810243\pi$
$941$	−47773.6	−1.65502	−0.827510	+	0.561451i	$0.810243\pi$
$942$	0	0
$943$	−297.595	−0.0102768
$944$	370.883	0.0127873
$945$	0	0
$946$	0	0
$947$	−46098.8	−1.58185	−0.790924	−	0.611914i	$-0.790400\pi$
$947$	−46098.8	−1.58185	−0.790924	+	0.611914i	$0.790400\pi$
$948$	0	0
$949$	13771.1	0.471054
$950$	−8107.91	−0.276900
$951$	0	0
$952$	−9873.08	−0.336122
$953$	−57230.1	−1.94529	−0.972646	−	0.232290i	$-0.925378\pi$
$953$	−57230.1	−1.94529	−0.972646	+	0.232290i	$0.925378\pi$
$954$	0	0
$955$	5646.44	0.191324
$956$	−9687.03	−0.327721
$957$	0	0
$958$	6802.43	0.229412
$959$	29406.2	0.990173
$960$	0	0
$961$	63170.5	2.12046
$962$	18693.0	0.626495
$963$	0	0
$964$	27743.9	0.926940
$965$	−3298.73	−0.110041
$966$	0	0
$967$	43977.3	1.46247	0.731237	−	0.682123i	$-0.238943\pi$
$967$	43977.3	1.46247	0.731237	+	0.682123i	$0.238943\pi$
$968$	0	0
$969$	0	0
$970$	−1763.99	−0.0583899
$971$	−3968.54	−0.131160	−0.0655800	−	0.997847i	$-0.520890\pi$
$971$	−3968.54	−0.131160	−0.0655800	+	0.997847i	$0.520890\pi$
$972$	0	0
$973$	−8656.38	−0.285212
$974$	19265.6	0.633789
$975$	0	0
$976$	10110.0	0.331570
$977$	−2691.26	−0.0881278	−0.0440639	−	0.999029i	$-0.514031\pi$
$977$	−2691.26	−0.0881278	−0.0440639	+	0.999029i	$0.514031\pi$
$978$	0	0
$979$	0	0
$980$	7881.85	0.256915
$981$	0	0
$982$	11188.3	0.363578
$983$	−45874.5	−1.48847	−0.744237	−	0.667916i	$-0.767187\pi$
$983$	−45874.5	−1.48847	−0.744237	+	0.667916i	$0.767187\pi$
$984$	0	0
$985$	−16708.3	−0.540477
$986$	5663.00	0.182907
$987$	0	0
$988$	−5752.44	−0.185232
$989$	−580.834	−0.0186749
$990$	0	0
$991$	−1102.69	−0.0353462	−0.0176731	−	0.999844i	$-0.505626\pi$
$991$	−1102.69	−0.0353462	−0.0176731	+	0.999844i	$0.505626\pi$
$992$	9756.67	0.312273
$993$	0	0
$994$	27190.5	0.867635
$995$	27282.9	0.869273
$996$	0	0
$997$	46230.7	1.46855	0.734274	−	0.678853i	$-0.237523\pi$
$997$	46230.7	1.46855	0.734274	+	0.678853i	$0.237523\pi$
$998$	−4590.63	−0.145605
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.bz.1.3		4
3.2	odd	2		726.4.a.w.1.2		4
11.7	odd	10		198.4.f.f.181.2		8
11.8	odd	10		198.4.f.f.163.2		8
11.10	odd	2		2178.4.a.bu.1.3		4
33.8	even	10		66.4.e.c.31.1	✓	8
33.29	even	10		66.4.e.c.49.1	yes	8
33.32	even	2		726.4.a.z.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
66.4.e.c.31.1	✓	8	33.8	even	10
66.4.e.c.49.1	yes	8	33.29	even	10
198.4.f.f.163.2		8	11.8	odd	10
198.4.f.f.181.2		8	11.7	odd	10
726.4.a.w.1.2		4	3.2	odd	2
726.4.a.z.1.2		4	33.32	even	2
2178.4.a.bu.1.3		4	11.10	odd	2
2178.4.a.bz.1.3		4	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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +6.18714 q^{5} -25.7192 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +6.18714 q^{5} -25.7192 q^{7} +8.00000 q^{8} +12.3743 q^{10} -30.7630 q^{13} -51.4384 q^{14} +16.0000 q^{16} +47.9850 q^{17} +46.7480 q^{19} +24.7485 q^{20} -2.04459 q^{23} -86.7194 q^{25} -61.5261 q^{26} -102.877 q^{28} +59.0080 q^{29} +304.896 q^{31} +32.0000 q^{32} +95.9699 q^{34} -159.128 q^{35} -303.823 q^{37} +93.4960 q^{38} +49.4971 q^{40} +145.553 q^{41} +284.084 q^{43} -4.08917 q^{46} -577.251 q^{47} +318.477 q^{49} -173.439 q^{50} -123.052 q^{52} -655.216 q^{53} -205.754 q^{56} +118.016 q^{58} +23.1802 q^{59} +631.874 q^{61} +609.792 q^{62} +64.0000 q^{64} -190.335 q^{65} -655.918 q^{67} +191.940 q^{68} -318.256 q^{70} -528.602 q^{71} -447.652 q^{73} -607.646 q^{74} +186.992 q^{76} -985.464 q^{79} +98.9942 q^{80} +291.106 q^{82} +128.459 q^{83} +296.889 q^{85} +568.167 q^{86} -1425.15 q^{89} +791.201 q^{91} -8.17835 q^{92} -1154.50 q^{94} +289.236 q^{95} -142.553 q^{97} +636.954 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} - 21 q^{7} + 32 q^{8}+O(q^{10}) \) 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 - 21 * q^7 + 32 * q^8	\( 4 q + 8 q^{2} + 16 q^{4} - 20 q^{5} - 21 q^{7} + 32 q^{8} - 40 q^{10} - 38 q^{13} - 42 q^{14} + 64 q^{16} + 85 q^{17} - 5 q^{19} - 80 q^{20} - 316 q^{23} + 318 q^{25} - 76 q^{26} - 84 q^{28} + 123 q^{29} + 436 q^{31} + 128 q^{32} + 170 q^{34} - 129 q^{35} + 336 q^{37} - 10 q^{38} - 160 q^{40} - 394 q^{41} + 680 q^{43} - 632 q^{46} - 1085 q^{47} - 311 q^{49} + 636 q^{50} - 152 q^{52} - 880 q^{53} - 168 q^{56} + 246 q^{58} - 724 q^{59} - 355 q^{61} + 872 q^{62} + 256 q^{64} - 2019 q^{65} - 869 q^{67} + 340 q^{68} - 258 q^{70} + 331 q^{71} - 1135 q^{73} + 672 q^{74} - 20 q^{76} - 1083 q^{79} - 320 q^{80} - 788 q^{82} - 169 q^{83} - 1104 q^{85} + 1360 q^{86} + 258 q^{89} + 874 q^{91} - 1264 q^{92} - 2170 q^{94} + 1555 q^{95} - 248 q^{97} - 622 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 - 20 * q^5 - 21 * q^7 + 32 * q^8 - 40 * q^10 - 38 * q^13 - 42 * q^14 + 64 * q^16 + 85 * q^17 - 5 * q^19 - 80 * q^20 - 316 * q^23 + 318 * q^25 - 76 * q^26 - 84 * q^28 + 123 * q^29 + 436 * q^31 + 128 * q^32 + 170 * q^34 - 129 * q^35 + 336 * q^37 - 10 * q^38 - 160 * q^40 - 394 * q^41 + 680 * q^43 - 632 * q^46 - 1085 * q^47 - 311 * q^49 + 636 * q^50 - 152 * q^52 - 880 * q^53 - 168 * q^56 + 246 * q^58 - 724 * q^59 - 355 * q^61 + 872 * q^62 + 256 * q^64 - 2019 * q^65 - 869 * q^67 + 340 * q^68 - 258 * q^70 + 331 * q^71 - 1135 * q^73 + 672 * q^74 - 20 * q^76 - 1083 * q^79 - 320 * q^80 - 788 * q^82 - 169 * q^83 - 1104 * q^85 + 1360 * q^86 + 258 * q^89 + 874 * q^91 - 1264 * q^92 - 2170 * q^94 + 1555 * q^95 - 248 * q^97 - 622 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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