LMFDB - Embedded newform 1200.4.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,4,Mod(1,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, 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("1200.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1200.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:	$70.8022920069$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{19}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 19 $ x^2 - 19
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.35890$ of defining polynomial
Character	$\chi$	$=$	1200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +30.4356 q^{7} +9.00000 q^{9} -31.4356 q^{11} +60.7424 q^{13} +121.178 q^{17} +14.4356 q^{19} +91.3068 q^{21} +13.6932 q^{23} +27.0000 q^{27} -76.0492 q^{29} -183.049 q^{31} -94.3068 q^{33} +37.3864 q^{37} +182.227 q^{39} -30.6627 q^{41} +327.564 q^{43} -449.485 q^{47} +583.325 q^{49} +363.534 q^{51} +301.951 q^{53} +43.3068 q^{57} -340.970 q^{59} +619.098 q^{61} +273.920 q^{63} +256.890 q^{67} +41.0796 q^{69} -499.178 q^{71} +19.1288 q^{73} -956.761 q^{77} -257.424 q^{79} +81.0000 q^{81} +914.909 q^{83} -228.148 q^{87} -1059.68 q^{89} +1848.73 q^{91} -549.148 q^{93} -521.000 q^{97} -282.920 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} + 26 q^{7} + 18 q^{9} - 28 q^{11} - 18 q^{13} + 68 q^{17} - 6 q^{19} + 78 q^{21} + 132 q^{23} + 54 q^{27} + 92 q^{29} - 122 q^{31} - 84 q^{33} + 284 q^{37} - 54 q^{39} + 392 q^{41} + 690 q^{43}+ \cdots - 252 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 + 26 * q^7 + 18 * q^9 - 28 * q^11 - 18 * q^13 + 68 * q^17 - 6 * q^19 + 78 * q^21 + 132 * q^23 + 54 * q^27 + 92 * q^29 - 122 * q^31 - 84 * q^33 + 284 * q^37 - 54 * q^39 + 392 * q^41 + 690 * q^43 - 620 * q^47 + 260 * q^49 + 204 * q^51 + 848 * q^53 - 18 * q^57 - 124 * q^59 + 750 * q^61 + 234 * q^63 - 358 * q^67 + 396 * q^69 - 824 * q^71 + 108 * q^73 - 972 * q^77 + 880 * q^79 + 162 * q^81 + 156 * q^83 + 276 * q^87 - 864 * q^89 + 2198 * q^91 - 366 * q^93 - 1042 * q^97 - 252 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	30.4356	1.64337	0.821684	−	0.569944i	$-0.193035\pi$
$7$	30.4356	1.64337	0.821684	+	0.569944i	$0.193035\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−31.4356	−0.861654	−0.430827	−	0.902435i	$-0.641778\pi$
$11$	−31.4356	−0.861654	−0.430827	+	0.902435i	$0.641778\pi$
$12$	0	0
$13$	60.7424	1.29592	0.647958	−	0.761676i	$-0.275623\pi$
$13$	60.7424	1.29592	0.647958	+	0.761676i	$0.275623\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	121.178	1.72882	0.864411	−	0.502786i	$-0.167692\pi$
$17$	121.178	1.72882	0.864411	+	0.502786i	$0.167692\pi$
$18$	0	0
$19$	14.4356	0.174303	0.0871514	−	0.996195i	$-0.472224\pi$
$19$	14.4356	0.174303	0.0871514	+	0.996195i	$0.472224\pi$
$20$	0	0
$21$	91.3068	0.948799
$22$	0	0
$23$	13.6932	0.124141	0.0620703	−	0.998072i	$-0.480230\pi$
$23$	13.6932	0.124141	0.0620703	+	0.998072i	$0.480230\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−76.0492	−0.486965	−0.243482	−	0.969905i	$-0.578290\pi$
$29$	−76.0492	−0.486965	−0.243482	+	0.969905i	$0.578290\pi$
$30$	0	0
$31$	−183.049	−1.06054	−0.530268	−	0.847830i	$-0.677909\pi$
$31$	−183.049	−1.06054	−0.530268	+	0.847830i	$0.677909\pi$
$32$	0	0
$33$	−94.3068	−0.497476
$34$	0	0
$35$	0	0
$36$	0	0
$37$	37.3864	0.166116	0.0830580	−	0.996545i	$-0.473531\pi$
$37$	37.3864	0.166116	0.0830580	+	0.996545i	$0.473531\pi$
$38$	0	0
$39$	182.227	0.748197
$40$	0	0
$41$	−30.6627	−0.116798	−0.0583990	−	0.998293i	$-0.518600\pi$
$41$	−30.6627	−0.116798	−0.0583990	+	0.998293i	$0.518600\pi$
$42$	0	0
$43$	327.564	1.16170	0.580850	−	0.814011i	$-0.302720\pi$
$43$	327.564	1.16170	0.580850	+	0.814011i	$0.302720\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−449.485	−1.39498	−0.697490	−	0.716594i	$-0.745700\pi$
$47$	−449.485	−1.39498	−0.697490	+	0.716594i	$0.745700\pi$
$48$	0	0
$49$	583.325	1.70066
$50$	0	0
$51$	363.534	0.998136
$52$	0	0
$53$	301.951	0.782569	0.391284	−	0.920270i	$-0.372031\pi$
$53$	301.951	0.782569	0.391284	+	0.920270i	$0.372031\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	43.3068	0.100634
$58$	0	0
$59$	−340.970	−0.752381	−0.376190	−	0.926542i	$-0.622766\pi$
$59$	−340.970	−0.752381	−0.376190	+	0.926542i	$0.622766\pi$
$60$	0	0
$61$	619.098	1.29947	0.649733	−	0.760163i	$-0.274881\pi$
$61$	619.098	1.29947	0.649733	+	0.760163i	$0.274881\pi$
$62$	0	0
$63$	273.920	0.547789
$64$	0	0
$65$	0	0
$66$	0	0
$67$	256.890	0.468419	0.234210	−	0.972186i	$-0.424750\pi$
$67$	256.890	0.468419	0.234210	+	0.972186i	$0.424750\pi$
$68$	0	0
$69$	41.0796	0.0716726
$70$	0	0
$71$	−499.178	−0.834388	−0.417194	−	0.908818i	$-0.636986\pi$
$71$	−499.178	−0.834388	−0.417194	+	0.908818i	$0.636986\pi$
$72$	0	0
$73$	19.1288	0.0306693	0.0153346	−	0.999882i	$-0.495119\pi$
$73$	19.1288	0.0306693	0.0153346	+	0.999882i	$0.495119\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−956.761	−1.41601
$78$	0	0
$79$	−257.424	−0.366613	−0.183307	−	0.983056i	$-0.558680\pi$
$79$	−257.424	−0.366613	−0.183307	+	0.983056i	$0.558680\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	914.909	1.20993	0.604965	−	0.796252i	$-0.293187\pi$
$83$	914.909	1.20993	0.604965	+	0.796252i	$0.293187\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−228.148	−0.281149
$88$	0	0
$89$	−1059.68	−1.26209	−0.631045	−	0.775746i	$-0.717374\pi$
$89$	−1059.68	−1.26209	−0.631045	+	0.775746i	$0.717374\pi$
$90$	0	0
$91$	1848.73	2.12967
$92$	0	0
$93$	−549.148	−0.612300
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−521.000	−0.545356	−0.272678	−	0.962105i	$-0.587909\pi$
$97$	−521.000	−0.545356	−0.272678	+	0.962105i	$0.587909\pi$
$98$	0	0
$99$	−282.920	−0.287218
$100$	0	0
$101$	347.080	0.341938	0.170969	−	0.985276i	$-0.445310\pi$
$101$	347.080	0.341938	0.170969	+	0.985276i	$0.445310\pi$
$102$	0	0
$103$	−770.749	−0.737322	−0.368661	−	0.929564i	$-0.620184\pi$
$103$	−770.749	−0.737322	−0.368661	+	0.929564i	$0.620184\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1415.37	−1.27878	−0.639390	−	0.768883i	$-0.720813\pi$
$107$	−1415.37	−1.27878	−0.639390	+	0.768883i	$0.720813\pi$
$108$	0	0
$109$	908.386	0.798235	0.399118	−	0.916900i	$-0.369317\pi$
$109$	908.386	0.798235	0.399118	+	0.916900i	$0.369317\pi$
$110$	0	0
$111$	112.159	0.0959071
$112$	0	0
$113$	2049.94	1.70657	0.853283	−	0.521447i	$-0.174608\pi$
$113$	2049.94	1.70657	0.853283	+	0.521447i	$0.174608\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	546.681	0.431972
$118$	0	0
$119$	3688.12	2.84109
$120$	0	0
$121$	−342.803	−0.257553
$122$	0	0
$123$	−91.9882	−0.0674333
$124$	0	0
$125$	0	0
$126$	0	0
$127$	281.644	0.196786	0.0983932	−	0.995148i	$-0.468630\pi$
$127$	281.644	0.196786	0.0983932	+	0.995148i	$0.468630\pi$
$128$	0	0
$129$	982.693	0.670708
$130$	0	0
$131$	−243.056	−0.162106	−0.0810531	−	0.996710i	$-0.525828\pi$
$131$	−243.056	−0.162106	−0.0810531	+	0.996710i	$0.525828\pi$
$132$	0	0
$133$	439.356	0.286444
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−909.386	−0.567110	−0.283555	−	0.958956i	$-0.591514\pi$
$137$	−909.386	−0.567110	−0.283555	+	0.958956i	$0.591514\pi$
$138$	0	0
$139$	2049.52	1.25063	0.625317	−	0.780371i	$-0.284970\pi$
$139$	2049.52	1.25063	0.625317	+	0.780371i	$0.284970\pi$
$140$	0	0
$141$	−1348.45	−0.805392
$142$	0	0
$143$	−1909.47	−1.11663
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1749.98	0.981875
$148$	0	0
$149$	3601.14	1.97998	0.989990	−	0.141136i	$-0.0450753\pi$
$149$	3601.14	1.97998	0.989990	+	0.141136i	$0.0450753\pi$
$150$	0	0
$151$	−1383.38	−0.745550	−0.372775	−	0.927922i	$-0.621594\pi$
$151$	−1383.38	−0.745550	−0.372775	+	0.927922i	$0.621594\pi$
$152$	0	0
$153$	1090.60	0.576274
$154$	0	0
$155$	0	0
$156$	0	0
$157$	131.749	0.0669729	0.0334864	−	0.999439i	$-0.489339\pi$
$157$	131.749	0.0669729	0.0334864	+	0.999439i	$0.489339\pi$
$158$	0	0
$159$	905.852	0.451816
$160$	0	0
$161$	416.761	0.204009
$162$	0	0
$163$	2897.74	1.39244	0.696222	−	0.717827i	$-0.254863\pi$
$163$	2897.74	1.39244	0.696222	+	0.717827i	$0.254863\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	260.283	0.120607	0.0603034	−	0.998180i	$-0.480793\pi$
$167$	260.283	0.120607	0.0603034	+	0.998180i	$0.480793\pi$
$168$	0	0
$169$	1492.64	0.679398
$170$	0	0
$171$	129.920	0.0581009
$172$	0	0
$173$	1935.83	0.850742	0.425371	−	0.905019i	$-0.360144\pi$
$173$	1935.83	0.850742	0.425371	+	0.905019i	$0.360144\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1022.91	−0.434387
$178$	0	0
$179$	−576.627	−0.240777	−0.120389	−	0.992727i	$-0.538414\pi$
$179$	−576.627	−0.240777	−0.120389	+	0.992727i	$0.538414\pi$
$180$	0	0
$181$	−1962.04	−0.805733	−0.402866	−	0.915259i	$-0.631986\pi$
$181$	−1962.04	−0.805733	−0.402866	+	0.915259i	$0.631986\pi$
$182$	0	0
$183$	1857.30	0.750247
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3809.30	−1.48965
$188$	0	0
$189$	821.761	0.316266
$190$	0	0
$191$	4318.75	1.63609	0.818047	−	0.575152i	$-0.195057\pi$
$191$	4318.75	1.63609	0.818047	+	0.575152i	$0.195057\pi$
$192$	0	0
$193$	−2.97647	−0.00111011	−0.000555054	−	1.00000i	$-0.500177\pi$
$193$	−2.97647	−0.00111011	−0.000555054		1.00000i	$0.500177\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−569.705	−0.206040	−0.103020	−	0.994679i	$-0.532851\pi$
$197$	−569.705	−0.206040	−0.103020	+	0.994679i	$0.532851\pi$
$198$	0	0
$199$	3050.73	1.08674	0.543368	−	0.839494i	$-0.317149\pi$
$199$	3050.73	1.08674	0.543368	+	0.839494i	$0.317149\pi$
$200$	0	0
$201$	770.670	0.270442
$202$	0	0
$203$	−2314.60	−0.800262
$204$	0	0
$205$	0	0
$206$	0	0
$207$	123.239	0.0413802
$208$	0	0
$209$	−453.792	−0.150189
$210$	0	0
$211$	50.5104	0.0164800	0.00824000	−	0.999966i	$-0.497377\pi$
$211$	50.5104	0.0164800	0.00824000	+	0.999966i	$0.497377\pi$
$212$	0	0
$213$	−1497.53	−0.481734
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−5571.21	−1.74285
$218$	0	0
$219$	57.3864	0.0177069
$220$	0	0
$221$	7360.64	2.24041
$222$	0	0
$223$	5453.55	1.63765	0.818827	−	0.574040i	$-0.194625\pi$
$223$	5453.55	1.63765	0.818827	+	0.574040i	$0.194625\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−4777.14	−1.39678	−0.698392	−	0.715715i	$-0.746101\pi$
$227$	−4777.14	−1.39678	−0.698392	+	0.715715i	$0.746101\pi$
$228$	0	0
$229$	−2085.51	−0.601808	−0.300904	−	0.953654i	$-0.597288\pi$
$229$	−2085.51	−0.601808	−0.300904	+	0.953654i	$0.597288\pi$
$230$	0	0
$231$	−2870.28	−0.817536
$232$	0	0
$233$	−6484.53	−1.82324	−0.911622	−	0.411030i	$-0.865169\pi$
$233$	−6484.53	−1.82324	−0.911622	+	0.411030i	$0.865169\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−772.271	−0.211664
$238$	0	0
$239$	−2234.62	−0.604792	−0.302396	−	0.953182i	$-0.597786\pi$
$239$	−2234.62	−0.604792	−0.302396	+	0.953182i	$0.597786\pi$
$240$	0	0
$241$	−2393.01	−0.639616	−0.319808	−	0.947482i	$-0.603618\pi$
$241$	−2393.01	−0.639616	−0.319808	+	0.947482i	$0.603618\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	876.852	0.225882
$248$	0	0
$249$	2744.73	0.698554
$250$	0	0
$251$	612.661	0.154067	0.0770335	−	0.997029i	$-0.475455\pi$
$251$	612.661	0.154067	0.0770335	+	0.997029i	$0.475455\pi$
$252$	0	0
$253$	−430.454	−0.106966
$254$	0	0
$255$	0	0
$256$	0	0
$257$	306.112	0.0742987	0.0371493	−	0.999310i	$-0.488172\pi$
$257$	306.112	0.0742987	0.0371493	+	0.999310i	$0.488172\pi$
$258$	0	0
$259$	1137.88	0.272990
$260$	0	0
$261$	−684.443	−0.162322
$262$	0	0
$263$	283.839	0.0665484	0.0332742	−	0.999446i	$-0.489407\pi$
$263$	283.839	0.0665484	0.0332742	+	0.999446i	$0.489407\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3179.04	−0.728668
$268$	0	0
$269$	2426.21	0.549920	0.274960	−	0.961456i	$-0.411335\pi$
$269$	2426.21	0.549920	0.274960	+	0.961456i	$0.411335\pi$
$270$	0	0
$271$	174.946	0.0392148	0.0196074	−	0.999808i	$-0.493758\pi$
$271$	174.946	0.0392148	0.0196074	+	0.999808i	$0.493758\pi$
$272$	0	0
$273$	5546.19	1.22956
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7807.07	1.69344	0.846718	−	0.532042i	$-0.178575\pi$
$277$	7807.07	1.69344	0.846718	+	0.532042i	$0.178575\pi$
$278$	0	0
$279$	−1647.44	−0.353512
$280$	0	0
$281$	584.171	0.124017	0.0620084	−	0.998076i	$-0.480249\pi$
$281$	584.171	0.124017	0.0620084	+	0.998076i	$0.480249\pi$
$282$	0	0
$283$	−5897.31	−1.23872	−0.619362	−	0.785106i	$-0.712609\pi$
$283$	−5897.31	−1.23872	−0.619362	+	0.785106i	$0.712609\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−933.239	−0.191942
$288$	0	0
$289$	9771.10	1.98883
$290$	0	0
$291$	−1563.00	−0.314861
$292$	0	0
$293$	−1609.73	−0.320960	−0.160480	−	0.987039i	$-0.551304\pi$
$293$	−1609.73	−0.320960	−0.160480	+	0.987039i	$0.551304\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−848.761	−0.165825
$298$	0	0
$299$	831.758	0.160876
$300$	0	0
$301$	9969.62	1.90910
$302$	0	0
$303$	1041.24	0.197418
$304$	0	0
$305$	0	0
$306$	0	0
$307$	234.473	0.0435898	0.0217949	−	0.999762i	$-0.493062\pi$
$307$	234.473	0.0435898	0.0217949	+	0.999762i	$0.493062\pi$
$308$	0	0
$309$	−2312.25	−0.425693
$310$	0	0
$311$	1795.25	0.327329	0.163665	−	0.986516i	$-0.447669\pi$
$311$	1795.25	0.327329	0.163665	+	0.986516i	$0.447669\pi$
$312$	0	0
$313$	8440.61	1.52425	0.762127	−	0.647427i	$-0.224155\pi$
$313$	8440.61	1.52425	0.762127	+	0.647427i	$0.224155\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10551.7	−1.86953	−0.934766	−	0.355264i	$-0.884391\pi$
$317$	−10551.7	−1.86953	−0.934766	+	0.355264i	$0.884391\pi$
$318$	0	0
$319$	2390.65	0.419595
$320$	0	0
$321$	−4246.12	−0.738304
$322$	0	0
$323$	1749.28	0.301339
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2725.16	0.460861
$328$	0	0
$329$	−13680.3	−2.29247
$330$	0	0
$331$	−6743.17	−1.11975	−0.559876	−	0.828576i	$-0.689151\pi$
$331$	−6743.17	−1.11975	−0.559876	+	0.828576i	$0.689151\pi$
$332$	0	0
$333$	336.478	0.0553720
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8437.26	1.36382	0.681909	−	0.731437i	$-0.261150\pi$
$337$	8437.26	1.36382	0.681909	+	0.731437i	$0.261150\pi$
$338$	0	0
$339$	6149.82	0.985287
$340$	0	0
$341$	5754.26	0.913814
$342$	0	0
$343$	7314.45	1.15144
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1848.85	−0.286027	−0.143013	−	0.989721i	$-0.545679\pi$
$347$	−1848.85	−0.286027	−0.143013	+	0.989721i	$0.545679\pi$
$348$	0	0
$349$	−1148.38	−0.176136	−0.0880678	−	0.996114i	$-0.528069\pi$
$349$	−1148.38	−0.176136	−0.0880678	+	0.996114i	$0.528069\pi$
$350$	0	0
$351$	1640.04	0.249399
$352$	0	0
$353$	5753.60	0.867516	0.433758	−	0.901029i	$-0.357187\pi$
$353$	5753.60	0.867516	0.433758	+	0.901029i	$0.357187\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	11064.4	1.64030
$358$	0	0
$359$	−5452.01	−0.801521	−0.400761	−	0.916183i	$-0.631254\pi$
$359$	−5452.01	−0.801521	−0.400761	+	0.916183i	$0.631254\pi$
$360$	0	0
$361$	−6650.61	−0.969619
$362$	0	0
$363$	−1028.41	−0.148698
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8385.93	1.19276	0.596379	−	0.802703i	$-0.296606\pi$
$367$	8385.93	1.19276	0.596379	+	0.802703i	$0.296606\pi$
$368$	0	0
$369$	−275.965	−0.0389327
$370$	0	0
$371$	9190.05	1.28605
$372$	0	0
$373$	−2728.30	−0.378730	−0.189365	−	0.981907i	$-0.560643\pi$
$373$	−2728.30	−0.378730	−0.189365	+	0.981907i	$0.560643\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4619.41	−0.631065
$378$	0	0
$379$	−3348.99	−0.453895	−0.226947	−	0.973907i	$-0.572875\pi$
$379$	−3348.99	−0.453895	−0.226947	+	0.973907i	$0.572875\pi$
$380$	0	0
$381$	844.932	0.113615
$382$	0	0
$383$	10430.3	1.39155	0.695774	−	0.718261i	$-0.255062\pi$
$383$	10430.3	1.39155	0.695774	+	0.718261i	$0.255062\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2948.08	0.387233
$388$	0	0
$389$	9827.23	1.28088	0.640438	−	0.768010i	$-0.278753\pi$
$389$	9827.23	1.28088	0.640438	+	0.768010i	$0.278753\pi$
$390$	0	0
$391$	1659.32	0.214617
$392$	0	0
$393$	−729.168	−0.0935921
$394$	0	0
$395$	0	0
$396$	0	0
$397$	436.382	0.0551672	0.0275836	−	0.999619i	$-0.491219\pi$
$397$	436.382	0.0551672	0.0275836	+	0.999619i	$0.491219\pi$
$398$	0	0
$399$	1318.07	0.165378
$400$	0	0
$401$	−14501.5	−1.80591	−0.902955	−	0.429736i	$-0.858607\pi$
$401$	−14501.5	−1.80591	−0.902955	+	0.429736i	$0.858607\pi$
$402$	0	0
$403$	−11118.8	−1.37436
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1175.26	−0.143134
$408$	0	0
$409$	−12058.4	−1.45782	−0.728911	−	0.684609i	$-0.759973\pi$
$409$	−12058.4	−1.45782	−0.728911	+	0.684609i	$0.759973\pi$
$410$	0	0
$411$	−2728.16	−0.327421
$412$	0	0
$413$	−10377.6	−1.23644
$414$	0	0
$415$	0	0
$416$	0	0
$417$	6148.57	0.722054
$418$	0	0
$419$	−6042.95	−0.704577	−0.352288	−	0.935892i	$-0.614596\pi$
$419$	−6042.95	−0.704577	−0.352288	+	0.935892i	$0.614596\pi$
$420$	0	0
$421$	−9994.67	−1.15703	−0.578516	−	0.815671i	$-0.696368\pi$
$421$	−9994.67	−1.15703	−0.578516	+	0.815671i	$0.696368\pi$
$422$	0	0
$423$	−4045.36	−0.464994
$424$	0	0
$425$	0	0
$426$	0	0
$427$	18842.6	2.13550
$428$	0	0
$429$	−5728.42	−0.644687
$430$	0	0
$431$	−9327.32	−1.04242	−0.521208	−	0.853430i	$-0.674518\pi$
$431$	−9327.32	−1.04242	−0.521208	+	0.853430i	$0.674518\pi$
$432$	0	0
$433$	7861.22	0.872485	0.436243	−	0.899829i	$-0.356309\pi$
$433$	7861.22	0.872485	0.436243	+	0.899829i	$0.356309\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	197.670	0.0216380
$438$	0	0
$439$	−7412.06	−0.805828	−0.402914	−	0.915238i	$-0.632003\pi$
$439$	−7412.06	−0.805828	−0.402914	+	0.915238i	$0.632003\pi$
$440$	0	0
$441$	5249.93	0.566886
$442$	0	0
$443$	−3043.66	−0.326430	−0.163215	−	0.986591i	$-0.552186\pi$
$443$	−3043.66	−0.326430	−0.163215	+	0.986591i	$0.552186\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10803.4	1.14314
$448$	0	0
$449$	−9547.87	−1.00355	−0.501773	−	0.864999i	$-0.667319\pi$
$449$	−9547.87	−1.00355	−0.501773	+	0.864999i	$0.667319\pi$
$450$	0	0
$451$	963.902	0.100639
$452$	0	0
$453$	−4150.14	−0.430443
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13401.9	−1.37180	−0.685901	−	0.727695i	$-0.740592\pi$
$457$	−13401.9	−1.37180	−0.685901	+	0.727695i	$0.740592\pi$
$458$	0	0
$459$	3271.81	0.332712
$460$	0	0
$461$	−4137.03	−0.417962	−0.208981	−	0.977920i	$-0.567015\pi$
$461$	−4137.03	−0.417962	−0.208981	+	0.977920i	$0.567015\pi$
$462$	0	0
$463$	−13976.3	−1.40288	−0.701439	−	0.712729i	$-0.747459\pi$
$463$	−13976.3	−1.40288	−0.701439	+	0.712729i	$0.747459\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10796.5	−1.06982	−0.534908	−	0.844910i	$-0.679654\pi$
$467$	−10796.5	−1.06982	−0.534908	+	0.844910i	$0.679654\pi$
$468$	0	0
$469$	7818.60	0.769785
$470$	0	0
$471$	395.248	0.0386668
$472$	0	0
$473$	−10297.2	−1.00098
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2717.56	0.260856
$478$	0	0
$479$	−14568.4	−1.38966	−0.694830	−	0.719174i	$-0.744521\pi$
$479$	−14568.4	−1.38966	−0.694830	+	0.719174i	$0.744521\pi$
$480$	0	0
$481$	2270.94	0.215272
$482$	0	0
$483$	1250.28	0.117784
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11456.6	−1.06601	−0.533004	−	0.846113i	$-0.678937\pi$
$487$	−11456.6	−1.06601	−0.533004	+	0.846113i	$0.678937\pi$
$488$	0	0
$489$	8693.21	0.803928
$490$	0	0
$491$	19666.5	1.80761	0.903804	−	0.427948i	$-0.140763\pi$
$491$	19666.5	1.80761	0.903804	+	0.427948i	$0.140763\pi$
$492$	0	0
$493$	−9215.48	−0.841875
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15192.8	−1.37121
$498$	0	0
$499$	−8379.31	−0.751722	−0.375861	−	0.926676i	$-0.622653\pi$
$499$	−8379.31	−0.751722	−0.375861	+	0.926676i	$0.622653\pi$
$500$	0	0
$501$	780.850	0.0696323
$502$	0	0
$503$	−15678.1	−1.38976	−0.694881	−	0.719124i	$-0.744543\pi$
$503$	−15678.1	−1.38976	−0.694881	+	0.719124i	$0.744543\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4477.91	0.392251
$508$	0	0
$509$	−17037.3	−1.48363	−0.741813	−	0.670606i	$-0.766034\pi$
$509$	−17037.3	−1.48363	−0.741813	+	0.670606i	$0.766034\pi$
$510$	0	0
$511$	582.197	0.0504009
$512$	0	0
$513$	389.761	0.0335446
$514$	0	0
$515$	0	0
$516$	0	0
$517$	14129.8	1.20199
$518$	0	0
$519$	5807.49	0.491176
$520$	0	0
$521$	−8776.12	−0.737982	−0.368991	−	0.929433i	$-0.620297\pi$
$521$	−8776.12	−0.737982	−0.368991	+	0.929433i	$0.620297\pi$
$522$	0	0
$523$	11120.4	0.929753	0.464877	−	0.885375i	$-0.346099\pi$
$523$	11120.4	0.929753	0.464877	+	0.885375i	$0.346099\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−22181.5	−1.83348
$528$	0	0
$529$	−11979.5	−0.984589
$530$	0	0
$531$	−3068.73	−0.250794
$532$	0	0
$533$	−1862.53	−0.151360
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1729.88	−0.139013
$538$	0	0
$539$	−18337.2	−1.46538
$540$	0	0
$541$	21730.6	1.72693	0.863467	−	0.504405i	$-0.168288\pi$
$541$	21730.6	1.72693	0.863467	+	0.504405i	$0.168288\pi$
$542$	0	0
$543$	−5886.13	−0.465190
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−6926.17	−0.541392	−0.270696	−	0.962665i	$-0.587254\pi$
$547$	−6926.17	−0.541392	−0.270696	+	0.962665i	$0.587254\pi$
$548$	0	0
$549$	5571.89	0.433155
$550$	0	0
$551$	−1097.82	−0.0848793
$552$	0	0
$553$	−7834.85	−0.602480
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6589.22	0.501246	0.250623	−	0.968085i	$-0.419364\pi$
$557$	6589.22	0.501246	0.250623	+	0.968085i	$0.419364\pi$
$558$	0	0
$559$	19897.0	1.50547
$560$	0	0
$561$	−11427.9	−0.860047
$562$	0	0
$563$	3839.63	0.287426	0.143713	−	0.989619i	$-0.454096\pi$
$563$	3839.63	0.287426	0.143713	+	0.989619i	$0.454096\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2465.28	0.182596
$568$	0	0
$569$	−20874.0	−1.53794	−0.768968	−	0.639287i	$-0.779229\pi$
$569$	−20874.0	−1.53794	−0.768968	+	0.639287i	$0.779229\pi$
$570$	0	0
$571$	−21175.9	−1.55199	−0.775994	−	0.630740i	$-0.782752\pi$
$571$	−21175.9	−1.55199	−0.775994	+	0.630740i	$0.782752\pi$
$572$	0	0
$573$	12956.3	0.944599
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14924.2	1.07678	0.538391	−	0.842695i	$-0.319032\pi$
$577$	14924.2	1.07678	0.538391	+	0.842695i	$0.319032\pi$
$578$	0	0
$579$	−8.92941	−0.000640922 0
$580$	0	0
$581$	27845.8	1.98836
$582$	0	0
$583$	−9492.00	−0.674303
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25218.0	1.77318	0.886592	−	0.462552i	$-0.153066\pi$
$587$	25218.0	1.77318	0.886592	+	0.462552i	$0.153066\pi$
$588$	0	0
$589$	−2642.42	−0.184854
$590$	0	0
$591$	−1709.11	−0.118957
$592$	0	0
$593$	−5011.77	−0.347063	−0.173532	−	0.984828i	$-0.555518\pi$
$593$	−5011.77	−0.347063	−0.173532	+	0.984828i	$0.555518\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9152.19	0.627428
$598$	0	0
$599$	−4943.41	−0.337199	−0.168600	−	0.985685i	$-0.553925\pi$
$599$	−4943.41	−0.337199	−0.168600	+	0.985685i	$0.553925\pi$
$600$	0	0
$601$	−24334.8	−1.65164	−0.825821	−	0.563932i	$-0.809288\pi$
$601$	−24334.8	−1.65164	−0.825821	+	0.563932i	$0.809288\pi$
$602$	0	0
$603$	2312.01	0.156140
$604$	0	0
$605$	0	0
$606$	0	0
$607$	28973.8	1.93741	0.968707	−	0.248207i	$-0.0798415\pi$
$607$	28973.8	1.93741	0.968707	+	0.248207i	$0.0798415\pi$
$608$	0	0
$609$	−6943.81	−0.462032
$610$	0	0
$611$	−27302.8	−1.80778
$612$	0	0
$613$	−15139.1	−0.997490	−0.498745	−	0.866749i	$-0.666206\pi$
$613$	−15139.1	−0.997490	−0.498745	+	0.866749i	$0.666206\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13894.8	−0.906617	−0.453309	−	0.891354i	$-0.649756\pi$
$617$	−13894.8	−0.906617	−0.453309	+	0.891354i	$0.649756\pi$
$618$	0	0
$619$	4589.69	0.298021	0.149011	−	0.988836i	$-0.452391\pi$
$619$	4589.69	0.298021	0.149011	+	0.988836i	$0.452391\pi$
$620$	0	0
$621$	369.717	0.0238909
$622$	0	0
$623$	−32252.0	−2.07408
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1361.37	−0.0867114
$628$	0	0
$629$	4530.41	0.287185
$630$	0	0
$631$	−3005.77	−0.189632	−0.0948160	−	0.995495i	$-0.530226\pi$
$631$	−3005.77	−0.189632	−0.0948160	+	0.995495i	$0.530226\pi$
$632$	0	0
$633$	151.531	0.00951473
$634$	0	0
$635$	0	0
$636$	0	0
$637$	35432.6	2.20391
$638$	0	0
$639$	−4492.60	−0.278129
$640$	0	0
$641$	−5631.47	−0.347004	−0.173502	−	0.984834i	$-0.555508\pi$
$641$	−5631.47	−0.347004	−0.173502	+	0.984834i	$0.555508\pi$
$642$	0	0
$643$	11305.1	0.693358	0.346679	−	0.937984i	$-0.387309\pi$
$643$	11305.1	0.693358	0.346679	+	0.937984i	$0.387309\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−8614.30	−0.523436	−0.261718	−	0.965144i	$-0.584289\pi$
$647$	−8614.30	−0.523436	−0.261718	+	0.965144i	$0.584289\pi$
$648$	0	0
$649$	10718.6	0.648291
$650$	0	0
$651$	−16713.6	−1.00623
$652$	0	0
$653$	−12639.8	−0.757479	−0.378739	−	0.925503i	$-0.623642\pi$
$653$	−12639.8	−0.757479	−0.378739	+	0.925503i	$0.623642\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	172.159	0.0102231
$658$	0	0
$659$	−13640.4	−0.806306	−0.403153	−	0.915133i	$-0.632086\pi$
$659$	−13640.4	−0.806306	−0.403153	+	0.915133i	$0.632086\pi$
$660$	0	0
$661$	−17052.0	−1.00340	−0.501699	−	0.865042i	$-0.667291\pi$
$661$	−17052.0	−1.00340	−0.501699	+	0.865042i	$0.667291\pi$
$662$	0	0
$663$	22081.9	1.29350
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1041.36	−0.0604521
$668$	0	0
$669$	16360.7	0.945500
$670$	0	0
$671$	−19461.7	−1.11969
$672$	0	0
$673$	16419.7	0.940467	0.470234	−	0.882542i	$-0.344170\pi$
$673$	16419.7	0.940467	0.470234	+	0.882542i	$0.344170\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8670.47	0.492221	0.246110	−	0.969242i	$-0.420847\pi$
$677$	8670.47	0.492221	0.246110	+	0.969242i	$0.420847\pi$
$678$	0	0
$679$	−15856.9	−0.896220
$680$	0	0
$681$	−14331.4	−0.806434
$682$	0	0
$683$	5973.36	0.334647	0.167324	−	0.985902i	$-0.446488\pi$
$683$	5973.36	0.334647	0.167324	+	0.985902i	$0.446488\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−6256.52	−0.347454
$688$	0	0
$689$	18341.2	1.01414
$690$	0	0
$691$	15316.3	0.843212	0.421606	−	0.906779i	$-0.361467\pi$
$691$	15316.3	0.843212	0.421606	+	0.906779i	$0.361467\pi$
$692$	0	0
$693$	−8610.85	−0.472005
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−3715.65	−0.201923
$698$	0	0
$699$	−19453.6	−1.05265
$700$	0	0
$701$	34583.1	1.86332	0.931660	−	0.363333i	$-0.118361\pi$
$701$	34583.1	1.86332	0.931660	+	0.363333i	$0.118361\pi$
$702$	0	0
$703$	539.695	0.0289545
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10563.6	0.561929
$708$	0	0
$709$	11194.1	0.592955	0.296477	−	0.955040i	$-0.404188\pi$
$709$	11194.1	0.592955	0.296477	+	0.955040i	$0.404188\pi$
$710$	0	0
$711$	−2316.81	−0.122204
$712$	0	0
$713$	−2506.53	−0.131655
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6703.85	−0.349177
$718$	0	0
$719$	15491.9	0.803549	0.401774	−	0.915739i	$-0.368394\pi$
$719$	15491.9	0.803549	0.401774	+	0.915739i	$0.368394\pi$
$720$	0	0
$721$	−23458.2	−1.21169
$722$	0	0
$723$	−7179.04	−0.369283
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−6272.72	−0.320003	−0.160002	−	0.987117i	$-0.551150\pi$
$727$	−6272.72	−0.320003	−0.160002	+	0.987117i	$0.551150\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	39693.6	2.00837
$732$	0	0
$733$	−24980.5	−1.25877	−0.629383	−	0.777095i	$-0.716692\pi$
$733$	−24980.5	−1.25877	−0.629383	+	0.777095i	$0.716692\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8075.49	−0.403615
$738$	0	0
$739$	30660.7	1.52621	0.763107	−	0.646272i	$-0.223673\pi$
$739$	30660.7	1.52621	0.763107	+	0.646272i	$0.223673\pi$
$740$	0	0
$741$	2630.56	0.130413
$742$	0	0
$743$	−17205.7	−0.849551	−0.424776	−	0.905299i	$-0.639647\pi$
$743$	−17205.7	−0.849551	−0.424776	+	0.905299i	$0.639647\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	8234.18	0.403310
$748$	0	0
$749$	−43077.8	−2.10151
$750$	0	0
$751$	−18397.1	−0.893901	−0.446950	−	0.894559i	$-0.647490\pi$
$751$	−18397.1	−0.893901	−0.446950	+	0.894559i	$0.647490\pi$
$752$	0	0
$753$	1837.98	0.0889506
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−22305.1	−1.07093	−0.535465	−	0.844557i	$-0.679864\pi$
$757$	−22305.1	−1.07093	−0.535465	+	0.844557i	$0.679864\pi$
$758$	0	0
$759$	−1291.36	−0.0617569
$760$	0	0
$761$	14458.5	0.688727	0.344364	−	0.938836i	$-0.388095\pi$
$761$	14458.5	0.688727	0.344364	+	0.938836i	$0.388095\pi$
$762$	0	0
$763$	27647.3	1.31179
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−20711.3	−0.975022
$768$	0	0
$769$	−39897.6	−1.87093	−0.935463	−	0.353424i	$-0.885017\pi$
$769$	−39897.6	−1.87093	−0.935463	+	0.353424i	$0.885017\pi$
$770$	0	0
$771$	918.337	0.0428964
$772$	0	0
$773$	−20070.2	−0.933863	−0.466931	−	0.884294i	$-0.654641\pi$
$773$	−20070.2	−0.933863	−0.466931	+	0.884294i	$0.654641\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3413.63	0.157611
$778$	0	0
$779$	−442.635	−0.0203582
$780$	0	0
$781$	15692.0	0.718953
$782$	0	0
$783$	−2053.33	−0.0937164
$784$	0	0
$785$	0	0
$786$	0	0
$787$	10733.0	0.486137	0.243069	−	0.970009i	$-0.421846\pi$
$787$	10733.0	0.486137	0.243069	+	0.970009i	$0.421846\pi$
$788$	0	0
$789$	851.516	0.0384218
$790$	0	0
$791$	62391.1	2.80452
$792$	0	0
$793$	37605.5	1.68400
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14335.8	0.637140	0.318570	−	0.947899i	$-0.396797\pi$
$797$	14335.8	0.637140	0.318570	+	0.947899i	$0.396797\pi$
$798$	0	0
$799$	−54467.7	−2.41167
$800$	0	0
$801$	−9537.13	−0.420697
$802$	0	0
$803$	−601.325	−0.0264263
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7278.63	0.317497
$808$	0	0
$809$	20920.7	0.909187	0.454593	−	0.890699i	$-0.349785\pi$
$809$	20920.7	0.909187	0.454593	+	0.890699i	$0.349785\pi$
$810$	0	0
$811$	−12816.5	−0.554931	−0.277465	−	0.960736i	$-0.589494\pi$
$811$	−12816.5	−0.554931	−0.277465	+	0.960736i	$0.589494\pi$
$812$	0	0
$813$	524.838	0.0226407
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4728.59	0.202488
$818$	0	0
$819$	16638.6	0.709889
$820$	0	0
$821$	7253.55	0.308344	0.154172	−	0.988044i	$-0.450729\pi$
$821$	7253.55	0.308344	0.154172	+	0.988044i	$0.450729\pi$
$822$	0	0
$823$	35288.1	1.49461	0.747307	−	0.664479i	$-0.231346\pi$
$823$	35288.1	1.49461	0.747307	+	0.664479i	$0.231346\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32205.9	−1.35418	−0.677092	−	0.735899i	$-0.736760\pi$
$827$	−32205.9	−1.35418	−0.677092	+	0.735899i	$0.736760\pi$
$828$	0	0
$829$	29993.3	1.25659	0.628294	−	0.777976i	$-0.283754\pi$
$829$	29993.3	1.25659	0.628294	+	0.777976i	$0.283754\pi$
$830$	0	0
$831$	23421.2	0.977706
$832$	0	0
$833$	70686.2	2.94013
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−4942.33	−0.204100
$838$	0	0
$839$	35608.7	1.46526	0.732628	−	0.680629i	$-0.238294\pi$
$839$	35608.7	1.46526	0.732628	+	0.680629i	$0.238294\pi$
$840$	0	0
$841$	−18605.5	−0.762865
$842$	0	0
$843$	1752.51	0.0716011
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10433.4	−0.423255
$848$	0	0
$849$	−17691.9	−0.715177
$850$	0	0
$851$	511.940	0.0206217
$852$	0	0
$853$	11229.3	0.450744	0.225372	−	0.974273i	$-0.427640\pi$
$853$	11229.3	0.450744	0.225372	+	0.974273i	$0.427640\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22136.1	−0.882327	−0.441164	−	0.897427i	$-0.645434\pi$
$857$	−22136.1	−0.882327	−0.441164	+	0.897427i	$0.645434\pi$
$858$	0	0
$859$	820.727	0.0325994	0.0162997	−	0.999867i	$-0.494811\pi$
$859$	820.727	0.0325994	0.0162997	+	0.999867i	$0.494811\pi$
$860$	0	0
$861$	−2799.72	−0.110818
$862$	0	0
$863$	−245.223	−0.00967264	−0.00483632	−	0.999988i	$-0.501539\pi$
$863$	−245.223	−0.00967264	−0.00483632	+	0.999988i	$0.501539\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	29313.3	1.14825
$868$	0	0
$869$	8092.27	0.315894
$870$	0	0
$871$	15604.1	0.607032
$872$	0	0
$873$	−4689.00	−0.181785
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37727.8	−1.45265	−0.726326	−	0.687350i	$-0.758774\pi$
$877$	−37727.8	−1.45265	−0.726326	+	0.687350i	$0.758774\pi$
$878$	0	0
$879$	−4829.19	−0.185306
$880$	0	0
$881$	21738.8	0.831326	0.415663	−	0.909519i	$-0.363550\pi$
$881$	21738.8	0.831326	0.415663	+	0.909519i	$0.363550\pi$
$882$	0	0
$883$	−44340.4	−1.68989	−0.844946	−	0.534852i	$-0.820368\pi$
$883$	−44340.4	−1.68989	−0.844946	+	0.534852i	$0.820368\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−681.008	−0.0257790	−0.0128895	−	0.999917i	$-0.504103\pi$
$887$	−681.008	−0.0257790	−0.0128895	+	0.999917i	$0.504103\pi$
$888$	0	0
$889$	8572.00	0.323392
$890$	0	0
$891$	−2546.28	−0.0957393
$892$	0	0
$893$	−6488.58	−0.243149
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2495.28	0.0928816
$898$	0	0
$899$	13920.7	0.516443
$900$	0	0
$901$	36589.8	1.35292
$902$	0	0
$903$	29908.9	1.10222
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5348.01	0.195786	0.0978929	−	0.995197i	$-0.468790\pi$
$907$	5348.01	0.195786	0.0978929	+	0.995197i	$0.468790\pi$
$908$	0	0
$909$	3123.72	0.113979
$910$	0	0
$911$	−14488.7	−0.526930	−0.263465	−	0.964669i	$-0.584865\pi$
$911$	−14488.7	−0.526930	−0.263465	+	0.964669i	$0.584865\pi$
$912$	0	0
$913$	−28760.7	−1.04254
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−7397.56	−0.266400
$918$	0	0
$919$	22546.9	0.809308	0.404654	−	0.914470i	$-0.367392\pi$
$919$	22546.9	0.809308	0.404654	+	0.914470i	$0.367392\pi$
$920$	0	0
$921$	703.419	0.0251666
$922$	0	0
$923$	−30321.3	−1.08130
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−6936.74	−0.245774
$928$	0	0
$929$	−13980.7	−0.493747	−0.246874	−	0.969048i	$-0.579403\pi$
$929$	−13980.7	−0.493747	−0.246874	+	0.969048i	$0.579403\pi$
$930$	0	0
$931$	8420.65	0.296429
$932$	0	0
$933$	5385.75	0.188984
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−26362.6	−0.919133	−0.459567	−	0.888143i	$-0.651995\pi$
$937$	−26362.6	−0.919133	−0.459567	+	0.888143i	$0.651995\pi$
$938$	0	0
$939$	25321.8	0.880029
$940$	0	0
$941$	−14715.6	−0.509792	−0.254896	−	0.966968i	$-0.582041\pi$
$941$	−14715.6	−0.509792	−0.254896	+	0.966968i	$0.582041\pi$
$942$	0	0
$943$	−419.871	−0.0144994
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11818.2	−0.405532	−0.202766	−	0.979227i	$-0.564993\pi$
$947$	−11818.2	−0.405532	−0.202766	+	0.979227i	$0.564993\pi$
$948$	0	0
$949$	1161.93	0.0397448
$950$	0	0
$951$	−31655.1	−1.07937
$952$	0	0
$953$	43832.3	1.48989	0.744945	−	0.667125i	$-0.232476\pi$
$953$	43832.3	1.48989	0.744945	+	0.667125i	$0.232476\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	7171.95	0.242253
$958$	0	0
$959$	−27677.7	−0.931971
$960$	0	0
$961$	3716.00	0.124736
$962$	0	0
$963$	−12738.4	−0.426260
$964$	0	0
$965$	0	0
$966$	0	0
$967$	10696.2	0.355706	0.177853	−	0.984057i	$-0.443085\pi$
$967$	10696.2	0.355706	0.177853	+	0.984057i	$0.443085\pi$
$968$	0	0
$969$	5247.83	0.173978
$970$	0	0
$971$	−27933.0	−0.923187	−0.461593	−	0.887092i	$-0.652722\pi$
$971$	−27933.0	−0.923187	−0.461593	+	0.887092i	$0.652722\pi$
$972$	0	0
$973$	62378.4	2.05525
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24341.7	−0.797094	−0.398547	−	0.917148i	$-0.630485\pi$
$977$	−24341.7	−0.797094	−0.398547	+	0.917148i	$0.630485\pi$
$978$	0	0
$979$	33311.7	1.08748
$980$	0	0
$981$	8175.48	0.266078
$982$	0	0
$983$	12553.0	0.407301	0.203651	−	0.979044i	$-0.434719\pi$
$983$	12553.0	0.407301	0.203651	+	0.979044i	$0.434719\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−41041.0	−1.32356
$988$	0	0
$989$	4485.41	0.144214
$990$	0	0
$991$	45631.8	1.46271	0.731353	−	0.681999i	$-0.238889\pi$
$991$	45631.8	1.46271	0.731353	+	0.681999i	$0.238889\pi$
$992$	0	0
$993$	−20229.5	−0.646490
$994$	0	0
$995$	0	0
$996$	0	0
$997$	34499.0	1.09588	0.547941	−	0.836517i	$-0.315412\pi$
$997$	34499.0	1.09588	0.547941	+	0.836517i	$0.315412\pi$
$998$	0	0
$999$	1009.43	0.0319690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.4.a.bu.1.2		2
4.3	odd	2		75.4.a.e.1.1	yes	2
5.2	odd	4		1200.4.f.v.49.2		4
5.3	odd	4		1200.4.f.v.49.3		4
5.4	even	2		1200.4.a.bl.1.1		2
12.11	even	2		225.4.a.j.1.2		2
20.3	even	4		75.4.b.c.49.3		4
20.7	even	4		75.4.b.c.49.2		4
20.19	odd	2		75.4.a.d.1.2	✓	2
60.23	odd	4		225.4.b.h.199.2		4
60.47	odd	4		225.4.b.h.199.3		4
60.59	even	2		225.4.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.a.d.1.2	✓	2	20.19	odd	2
75.4.a.e.1.1	yes	2	4.3	odd	2
75.4.b.c.49.2		4	20.7	even	4
75.4.b.c.49.3		4	20.3	even	4
225.4.a.j.1.2		2	12.11	even	2
225.4.a.n.1.1		2	60.59	even	2
225.4.b.h.199.2		4	60.23	odd	4
225.4.b.h.199.3		4	60.47	odd	4
1200.4.a.bl.1.1		2	5.4	even	2
1200.4.a.bu.1.2		2	1.1	even	1	trivial
1200.4.f.v.49.2		4	5.2	odd	4
1200.4.f.v.49.3		4	5.3	odd	4

Overview	Random
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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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1200.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +30.4356 q^{7} +9.00000 q^{9} -31.4356 q^{11} +60.7424 q^{13} +121.178 q^{17} +14.4356 q^{19} +91.3068 q^{21} +13.6932 q^{23} +27.0000 q^{27} -76.0492 q^{29} -183.049 q^{31} -94.3068 q^{33} +37.3864 q^{37} +182.227 q^{39} -30.6627 q^{41} +327.564 q^{43} -449.485 q^{47} +583.325 q^{49} +363.534 q^{51} +301.951 q^{53} +43.3068 q^{57} -340.970 q^{59} +619.098 q^{61} +273.920 q^{63} +256.890 q^{67} +41.0796 q^{69} -499.178 q^{71} +19.1288 q^{73} -956.761 q^{77} -257.424 q^{79} +81.0000 q^{81} +914.909 q^{83} -228.148 q^{87} -1059.68 q^{89} +1848.73 q^{91} -549.148 q^{93} -521.000 q^{97} -282.920 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} + 26 q^{7} + 18 q^{9} - 28 q^{11} - 18 q^{13} + 68 q^{17} - 6 q^{19} + 78 q^{21} + 132 q^{23} + 54 q^{27} + 92 q^{29} - 122 q^{31} - 84 q^{33} + 284 q^{37} - 54 q^{39} + 392 q^{41} + 690 q^{43}+ \cdots - 252 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 + 26 * q^7 + 18 * q^9 - 28 * q^11 - 18 * q^13 + 68 * q^17 - 6 * q^19 + 78 * q^21 + 132 * q^23 + 54 * q^27 + 92 * q^29 - 122 * q^31 - 84 * q^33 + 284 * q^37 - 54 * q^39 + 392 * q^41 + 690 * q^43 - 620 * q^47 + 260 * q^49 + 204 * q^51 + 848 * q^53 - 18 * q^57 - 124 * q^59 + 750 * q^61 + 234 * q^63 - 358 * q^67 + 396 * q^69 - 824 * q^71 + 108 * q^73 - 972 * q^77 + 880 * q^79 + 162 * q^81 + 156 * q^83 + 276 * q^87 - 864 * q^89 + 2198 * q^91 - 366 * q^93 - 1042 * q^97 - 252 * q^99

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