LMFDB - Embedded newform 2400.4.a.bv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2400,4,Mod(1,2400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2400, 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("2400.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2400.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:	$141.604584014$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.304244.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 164x - 780 $ x^3 - 164*x - 780
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8.49763$ of defining polynomial
Character	$\chi$	$=$	2400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} -13.9953 q^{7} +9.00000 q^{9} -43.3861 q^{11} -25.3909 q^{13} -85.3766 q^{17} -92.3766 q^{19} -41.9858 q^{21} -31.3861 q^{23} +27.0000 q^{27} -220.130 q^{29} +111.976 q^{31} -130.158 q^{33} -123.972 q^{37} -76.1726 q^{39} +431.693 q^{41} +278.348 q^{43} +340.326 q^{47} -147.133 q^{49} -256.130 q^{51} +367.728 q^{53} -277.130 q^{57} +638.596 q^{59} -195.651 q^{61} -125.957 q^{63} -11.5701 q^{67} -94.1584 q^{69} +110.542 q^{71} +734.226 q^{73} +607.200 q^{77} +195.256 q^{79} +81.0000 q^{81} -392.994 q^{83} -660.390 q^{87} +317.620 q^{89} +355.352 q^{91} +335.929 q^{93} -993.323 q^{97} -390.475 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 9 q^{3} + 9 q^{7} + 27 q^{9} + 4 q^{11} + 7 q^{13} - 20 q^{17} - 41 q^{19} + 27 q^{21} + 40 q^{23} + 81 q^{27} + 48 q^{29} + 81 q^{31} + 12 q^{33} - 66 q^{37} + 21 q^{39} + 254 q^{41} + 293 q^{43}+ \cdots + 36 q^{99}+O(q^{100}) $ 3 * q + 9 * q^3 + 9 * q^7 + 27 * q^9 + 4 * q^11 + 7 * q^13 - 20 * q^17 - 41 * q^19 + 27 * q^21 + 40 * q^23 + 81 * q^27 + 48 * q^29 + 81 * q^31 + 12 * q^33 - 66 * q^37 + 21 * q^39 + 254 * q^41 + 293 * q^43 + 318 * q^47 + 310 * q^49 - 60 * q^51 + 282 * q^53 - 123 * q^57 - 102 * q^59 + 913 * q^61 + 81 * q^63 + 153 * q^67 + 120 * q^69 - 162 * q^71 + 566 * q^73 - 164 * q^77 + 1160 * q^79 + 243 * q^81 + 914 * q^83 + 144 * q^87 + 1232 * q^89 - 1467 * q^91 + 243 * q^93 - 2089 * q^97 + 36 * 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
$7$	−13.9953	−0.755673	−0.377836	−	0.925872i	$-0.623332\pi$
$7$	−13.9953	−0.755673	−0.377836	+	0.925872i	$0.623332\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−43.3861	−1.18922	−0.594610	−	0.804015i	$-0.702693\pi$
$11$	−43.3861	−1.18922	−0.594610	+	0.804015i	$0.702693\pi$
$12$	0	0
$13$	−25.3909	−0.541705	−0.270852	−	0.962621i	$-0.587306\pi$
$13$	−25.3909	−0.541705	−0.270852	+	0.962621i	$0.587306\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−85.3766	−1.21805	−0.609026	−	0.793150i	$-0.708439\pi$
$17$	−85.3766	−1.21805	−0.609026	+	0.793150i	$0.708439\pi$
$18$	0	0
$19$	−92.3766	−1.11540	−0.557701	−	0.830042i	$-0.688317\pi$
$19$	−92.3766	−1.11540	−0.557701	+	0.830042i	$0.688317\pi$
$20$	0	0
$21$	−41.9858	−0.436288
$22$	0	0
$23$	−31.3861	−0.284542	−0.142271	−	0.989828i	$-0.545440\pi$
$23$	−31.3861	−0.284542	−0.142271	+	0.989828i	$0.545440\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−220.130	−1.40956	−0.704778	−	0.709428i	$-0.748953\pi$
$29$	−220.130	−1.40956	−0.704778	+	0.709428i	$0.748953\pi$
$30$	0	0
$31$	111.976	0.648759	0.324380	−	0.945927i	$-0.394844\pi$
$31$	111.976	0.648759	0.324380	+	0.945927i	$0.394844\pi$
$32$	0	0
$33$	−130.158	−0.686596
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−123.972	−0.550832	−0.275416	−	0.961325i	$-0.588816\pi$
$37$	−123.972	−0.550832	−0.275416	+	0.961325i	$0.588816\pi$
$38$	0	0
$39$	−76.1726	−0.312753
$40$	0	0
$41$	431.693	1.64437	0.822185	−	0.569220i	$-0.192755\pi$
$41$	431.693	1.64437	0.822185	+	0.569220i	$0.192755\pi$
$42$	0	0
$43$	278.348	0.987156	0.493578	−	0.869702i	$-0.335689\pi$
$43$	278.348	0.987156	0.493578	+	0.869702i	$0.335689\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	340.326	1.05621	0.528103	−	0.849180i	$-0.322903\pi$
$47$	340.326	1.05621	0.528103	+	0.849180i	$0.322903\pi$
$48$	0	0
$49$	−147.133	−0.428959
$50$	0	0
$51$	−256.130	−0.703242
$52$	0	0
$53$	367.728	0.953043	0.476522	−	0.879163i	$-0.341897\pi$
$53$	367.728	0.953043	0.476522	+	0.879163i	$0.341897\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−277.130	−0.643978
$58$	0	0
$59$	638.596	1.40912	0.704560	−	0.709644i	$-0.251144\pi$
$59$	638.596	1.40912	0.704560	+	0.709644i	$0.251144\pi$
$60$	0	0
$61$	−195.651	−0.410664	−0.205332	−	0.978692i	$-0.565827\pi$
$61$	−195.651	−0.410664	−0.205332	+	0.978692i	$0.565827\pi$
$62$	0	0
$63$	−125.957	−0.251891
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−11.5701	−0.0210972	−0.0105486	−	0.999944i	$-0.503358\pi$
$67$	−11.5701	−0.0210972	−0.0105486	+	0.999944i	$0.503358\pi$
$68$	0	0
$69$	−94.1584	−0.164280
$70$	0	0
$71$	110.542	0.184773	0.0923865	−	0.995723i	$-0.470550\pi$
$71$	110.542	0.184773	0.0923865	+	0.995723i	$0.470550\pi$
$72$	0	0
$73$	734.226	1.17719	0.588593	−	0.808429i	$-0.299682\pi$
$73$	734.226	1.17719	0.588593	+	0.808429i	$0.299682\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	607.200	0.898661
$78$	0	0
$79$	195.256	0.278076	0.139038	−	0.990287i	$-0.455599\pi$
$79$	195.256	0.278076	0.139038	+	0.990287i	$0.455599\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−392.994	−0.519719	−0.259860	−	0.965646i	$-0.583676\pi$
$83$	−392.994	−0.519719	−0.259860	+	0.965646i	$0.583676\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−660.390	−0.813807
$88$	0	0
$89$	317.620	0.378289	0.189144	−	0.981949i	$-0.439429\pi$
$89$	317.620	0.378289	0.189144	+	0.981949i	$0.439429\pi$
$90$	0	0
$91$	355.352	0.409352
$92$	0	0
$93$	335.929	0.374561
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−993.323	−1.03976	−0.519880	−	0.854240i	$-0.674023\pi$
$97$	−993.323	−1.03976	−0.519880	+	0.854240i	$0.674023\pi$
$98$	0	0
$99$	−390.475	−0.396406
$100$	0	0
$101$	11.2649	0.0110980	0.00554900	−	0.999985i	$-0.498234\pi$
$101$	11.2649	0.0110980	0.00554900	+	0.999985i	$0.498234\pi$
$102$	0	0
$103$	1497.69	1.43274	0.716370	−	0.697721i	$-0.245802\pi$
$103$	1497.69	1.43274	0.716370	+	0.697721i	$0.245802\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.6081	0.0150053	0.00750264	−	0.999972i	$-0.497612\pi$
$107$	16.6081	0.0150053	0.00750264	+	0.999972i	$0.497612\pi$
$108$	0	0
$109$	1670.46	1.46790	0.733949	−	0.679204i	$-0.237675\pi$
$109$	1670.46	1.46790	0.733949	+	0.679204i	$0.237675\pi$
$110$	0	0
$111$	−371.915	−0.318023
$112$	0	0
$113$	−322.209	−0.268238	−0.134119	−	0.990965i	$-0.542820\pi$
$113$	−322.209	−0.268238	−0.134119	+	0.990965i	$0.542820\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−228.518	−0.180568
$118$	0	0
$119$	1194.87	0.920448
$120$	0	0
$121$	551.357	0.414243
$122$	0	0
$123$	1295.08	0.949378
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1322.37	0.923950	0.461975	−	0.886893i	$-0.347141\pi$
$127$	1322.37	0.923950	0.461975	+	0.886893i	$0.347141\pi$
$128$	0	0
$129$	835.045	0.569935
$130$	0	0
$131$	−1545.31	−1.03064	−0.515322	−	0.856997i	$-0.672328\pi$
$131$	−1545.31	−1.03064	−0.515322	+	0.856997i	$0.672328\pi$
$132$	0	0
$133$	1292.83	0.842880
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1125.56	−0.701919	−0.350959	−	0.936391i	$-0.614145\pi$
$137$	−1125.56	−0.701919	−0.350959	+	0.936391i	$0.614145\pi$
$138$	0	0
$139$	−1933.99	−1.18013	−0.590067	−	0.807354i	$-0.700899\pi$
$139$	−1933.99	−1.18013	−0.590067	+	0.807354i	$0.700899\pi$
$140$	0	0
$141$	1020.98	0.609801
$142$	0	0
$143$	1101.61	0.644206
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−441.398	−0.247659
$148$	0	0
$149$	303.869	0.167073	0.0835367	−	0.996505i	$-0.473378\pi$
$149$	303.869	0.167073	0.0835367	+	0.996505i	$0.473378\pi$
$150$	0	0
$151$	−2290.31	−1.23432	−0.617162	−	0.786836i	$-0.711718\pi$
$151$	−2290.31	−1.23432	−0.617162	+	0.786836i	$0.711718\pi$
$152$	0	0
$153$	−768.390	−0.406017
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1976.47	1.00471	0.502355	−	0.864661i	$-0.332467\pi$
$157$	1976.47	1.00471	0.502355	+	0.864661i	$0.332467\pi$
$158$	0	0
$159$	1103.18	0.550240
$160$	0	0
$161$	439.257	0.215021
$162$	0	0
$163$	−82.0214	−0.0394136	−0.0197068	−	0.999806i	$-0.506273\pi$
$163$	−82.0214	−0.0394136	−0.0197068	+	0.999806i	$0.506273\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−929.769	−0.430825	−0.215412	−	0.976523i	$-0.569110\pi$
$167$	−929.769	−0.430825	−0.215412	+	0.976523i	$0.569110\pi$
$168$	0	0
$169$	−1552.30	−0.706556
$170$	0	0
$171$	−831.390	−0.371801
$172$	0	0
$173$	−465.080	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$173$	−465.080	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1915.79	0.813556
$178$	0	0
$179$	3917.60	1.63584	0.817919	−	0.575334i	$-0.195128\pi$
$179$	3917.60	1.63584	0.817919	+	0.575334i	$0.195128\pi$
$180$	0	0
$181$	3846.50	1.57960	0.789802	−	0.613362i	$-0.210183\pi$
$181$	3846.50	1.57960	0.789802	+	0.613362i	$0.210183\pi$
$182$	0	0
$183$	−586.952	−0.237097
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3704.16	1.44853
$188$	0	0
$189$	−377.872	−0.145429
$190$	0	0
$191$	757.545	0.286985	0.143492	−	0.989651i	$-0.454167\pi$
$191$	757.545	0.286985	0.143492	+	0.989651i	$0.454167\pi$
$192$	0	0
$193$	−3505.79	−1.30752	−0.653762	−	0.756700i	$-0.726810\pi$
$193$	−3505.79	−1.30752	−0.653762	+	0.756700i	$0.726810\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2015.09	−0.728779	−0.364390	−	0.931247i	$-0.618722\pi$
$197$	−2015.09	−0.728779	−0.364390	+	0.931247i	$0.618722\pi$
$198$	0	0
$199$	3225.89	1.14913	0.574566	−	0.818458i	$-0.305171\pi$
$199$	3225.89	1.14913	0.574566	+	0.818458i	$0.305171\pi$
$200$	0	0
$201$	−34.7104	−0.0121805
$202$	0	0
$203$	3080.77	1.06516
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−282.475	−0.0948473
$208$	0	0
$209$	4007.87	1.32646
$210$	0	0
$211$	−2091.98	−0.682550	−0.341275	−	0.939964i	$-0.610859\pi$
$211$	−2091.98	−0.682550	−0.341275	+	0.939964i	$0.610859\pi$
$212$	0	0
$213$	331.625	0.106679
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1567.14	−0.490250
$218$	0	0
$219$	2202.68	0.679649
$220$	0	0
$221$	2167.79	0.659825
$222$	0	0
$223$	6171.25	1.85317	0.926587	−	0.376081i	$-0.122729\pi$
$223$	6171.25	1.85317	0.926587	+	0.376081i	$0.122729\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4440.71	1.29841	0.649207	−	0.760611i	$-0.275101\pi$
$227$	4440.71	1.29841	0.649207	+	0.760611i	$0.275101\pi$
$228$	0	0
$229$	−5771.96	−1.66560	−0.832798	−	0.553576i	$-0.813263\pi$
$229$	−5771.96	−1.66560	−0.832798	+	0.553576i	$0.813263\pi$
$230$	0	0
$231$	1821.60	0.518842
$232$	0	0
$233$	−131.028	−0.0368408	−0.0184204	−	0.999830i	$-0.505864\pi$
$233$	−131.028	−0.0368408	−0.0184204	+	0.999830i	$0.505864\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	585.769	0.160548
$238$	0	0
$239$	−3674.77	−0.994566	−0.497283	−	0.867588i	$-0.665669\pi$
$239$	−3674.77	−0.994566	−0.497283	+	0.867588i	$0.665669\pi$
$240$	0	0
$241$	−5713.18	−1.52705	−0.763524	−	0.645780i	$-0.776532\pi$
$241$	−5713.18	−1.52705	−0.763524	+	0.645780i	$0.776532\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2345.52	0.604219
$248$	0	0
$249$	−1178.98	−0.300060
$250$	0	0
$251$	−4897.68	−1.23163	−0.615814	−	0.787891i	$-0.711173\pi$
$251$	−4897.68	−1.23163	−0.615814	+	0.787891i	$0.711173\pi$
$252$	0	0
$253$	1361.72	0.338383
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−211.524	−0.0513405	−0.0256702	−	0.999670i	$-0.508172\pi$
$257$	−211.524	−0.0513405	−0.0256702	+	0.999670i	$0.508172\pi$
$258$	0	0
$259$	1735.01	0.416249
$260$	0	0
$261$	−1981.17	−0.469852
$262$	0	0
$263$	7121.69	1.66974	0.834871	−	0.550446i	$-0.185542\pi$
$263$	7121.69	1.66974	0.834871	+	0.550446i	$0.185542\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	952.861	0.218405
$268$	0	0
$269$	−285.049	−0.0646087	−0.0323044	−	0.999478i	$-0.510285\pi$
$269$	−285.049	−0.0646087	−0.0323044	+	0.999478i	$0.510285\pi$
$270$	0	0
$271$	3616.79	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$271$	3616.79	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$272$	0	0
$273$	1066.06	0.236339
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3355.98	−0.727947	−0.363973	−	0.931409i	$-0.618580\pi$
$277$	−3355.98	−0.727947	−0.363973	+	0.931409i	$0.618580\pi$
$278$	0	0
$279$	1007.79	0.216253
$280$	0	0
$281$	−4125.61	−0.875848	−0.437924	−	0.899012i	$-0.644286\pi$
$281$	−4125.61	−0.875848	−0.437924	+	0.899012i	$0.644286\pi$
$282$	0	0
$283$	3678.07	0.772575	0.386287	−	0.922378i	$-0.373757\pi$
$283$	3678.07	0.772575	0.386287	+	0.922378i	$0.373757\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6041.66	−1.24261
$288$	0	0
$289$	2376.17	0.483650
$290$	0	0
$291$	−2979.97	−0.600305
$292$	0	0
$293$	−847.397	−0.168961	−0.0844803	−	0.996425i	$-0.526923\pi$
$293$	−847.397	−0.168961	−0.0844803	+	0.996425i	$0.526923\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1171.43	−0.228865
$298$	0	0
$299$	796.922	0.154138
$300$	0	0
$301$	−3895.55	−0.745967
$302$	0	0
$303$	33.7947	0.00640743
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2049.62	−0.381035	−0.190518	−	0.981684i	$-0.561017\pi$
$307$	−2049.62	−0.381035	−0.190518	+	0.981684i	$0.561017\pi$
$308$	0	0
$309$	4493.08	0.827193
$310$	0	0
$311$	−8754.65	−1.59624	−0.798120	−	0.602498i	$-0.794172\pi$
$311$	−8754.65	−1.59624	−0.798120	+	0.602498i	$0.794172\pi$
$312$	0	0
$313$	8221.23	1.48464	0.742318	−	0.670047i	$-0.233726\pi$
$313$	8221.23	1.48464	0.742318	+	0.670047i	$0.233726\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9177.22	1.62601	0.813003	−	0.582260i	$-0.197831\pi$
$317$	9177.22	1.62601	0.813003	+	0.582260i	$0.197831\pi$
$318$	0	0
$319$	9550.59	1.67627
$320$	0	0
$321$	49.8242	0.00866330
$322$	0	0
$323$	7886.81	1.35862
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5011.38	0.847492
$328$	0	0
$329$	−4762.95	−0.798146
$330$	0	0
$331$	−8137.57	−1.35130	−0.675651	−	0.737222i	$-0.736137\pi$
$331$	−8137.57	−1.35130	−0.675651	+	0.737222i	$0.736137\pi$
$332$	0	0
$333$	−1115.74	−0.183611
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2490.26	−0.402532	−0.201266	−	0.979537i	$-0.564506\pi$
$337$	−2490.26	−0.402532	−0.201266	+	0.979537i	$0.564506\pi$
$338$	0	0
$339$	−966.626	−0.154867
$340$	0	0
$341$	−4858.22	−0.771517
$342$	0	0
$343$	6859.53	1.07983
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4551.26	0.704104	0.352052	−	0.935980i	$-0.385484\pi$
$347$	4551.26	0.704104	0.352052	+	0.935980i	$0.385484\pi$
$348$	0	0
$349$	−7826.48	−1.20041	−0.600203	−	0.799847i	$-0.704914\pi$
$349$	−7826.48	−1.20041	−0.600203	+	0.799847i	$0.704914\pi$
$350$	0	0
$351$	−685.554	−0.104251
$352$	0	0
$353$	4816.88	0.726280	0.363140	−	0.931735i	$-0.381705\pi$
$353$	4816.88	0.726280	0.363140	+	0.931735i	$0.381705\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3584.60	0.531421
$358$	0	0
$359$	5261.91	0.773574	0.386787	−	0.922169i	$-0.373585\pi$
$359$	5261.91	0.773574	0.386787	+	0.922169i	$0.373585\pi$
$360$	0	0
$361$	1674.44	0.244124
$362$	0	0
$363$	1654.07	0.239163
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1964.39	0.279401	0.139700	−	0.990194i	$-0.455386\pi$
$367$	1964.39	0.279401	0.139700	+	0.990194i	$0.455386\pi$
$368$	0	0
$369$	3885.24	0.548124
$370$	0	0
$371$	−5146.44	−0.720189
$372$	0	0
$373$	−4896.84	−0.679756	−0.339878	−	0.940470i	$-0.610386\pi$
$373$	−4896.84	−0.679756	−0.339878	+	0.940470i	$0.610386\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5589.29	0.763563
$378$	0	0
$379$	7414.06	1.00484	0.502420	−	0.864623i	$-0.332443\pi$
$379$	7414.06	1.00484	0.502420	+	0.864623i	$0.332443\pi$
$380$	0	0
$381$	3967.12	0.533443
$382$	0	0
$383$	12359.7	1.64896	0.824478	−	0.565894i	$-0.191469\pi$
$383$	12359.7	1.64896	0.824478	+	0.565894i	$0.191469\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2505.13	0.329052
$388$	0	0
$389$	−6344.48	−0.826935	−0.413468	−	0.910519i	$-0.635682\pi$
$389$	−6344.48	−0.826935	−0.413468	+	0.910519i	$0.635682\pi$
$390$	0	0
$391$	2679.64	0.346587
$392$	0	0
$393$	−4635.93	−0.595043
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1236.73	0.156347	0.0781734	−	0.996940i	$-0.475091\pi$
$397$	1236.73	0.156347	0.0781734	+	0.996940i	$0.475091\pi$
$398$	0	0
$399$	3878.50	0.486637
$400$	0	0
$401$	−1868.99	−0.232750	−0.116375	−	0.993205i	$-0.537127\pi$
$401$	−1868.99	−0.232750	−0.116375	+	0.993205i	$0.537127\pi$
$402$	0	0
$403$	−2843.18	−0.351436
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5378.65	0.655060
$408$	0	0
$409$	−5205.01	−0.629269	−0.314635	−	0.949213i	$-0.601882\pi$
$409$	−5205.01	−0.629269	−0.314635	+	0.949213i	$0.601882\pi$
$410$	0	0
$411$	−3376.67	−0.405253
$412$	0	0
$413$	−8937.31	−1.06483
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−5801.96	−0.681350
$418$	0	0
$419$	−3557.56	−0.414793	−0.207397	−	0.978257i	$-0.566499\pi$
$419$	−3557.56	−0.414793	−0.207397	+	0.978257i	$0.566499\pi$
$420$	0	0
$421$	−10753.8	−1.24492	−0.622459	−	0.782653i	$-0.713866\pi$
$421$	−10753.8	−1.24492	−0.622459	+	0.782653i	$0.713866\pi$
$422$	0	0
$423$	3062.94	0.352069
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2738.18	0.310328
$428$	0	0
$429$	3304.84	0.371932
$430$	0	0
$431$	6087.82	0.680371	0.340185	−	0.940358i	$-0.389510\pi$
$431$	6087.82	0.680371	0.340185	+	0.940358i	$0.389510\pi$
$432$	0	0
$433$	−5043.70	−0.559780	−0.279890	−	0.960032i	$-0.590298\pi$
$433$	−5043.70	−0.559780	−0.279890	+	0.960032i	$0.590298\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2899.35	0.317379
$438$	0	0
$439$	6465.16	0.702882	0.351441	−	0.936210i	$-0.385692\pi$
$439$	6465.16	0.702882	0.351441	+	0.936210i	$0.385692\pi$
$440$	0	0
$441$	−1324.20	−0.142986
$442$	0	0
$443$	−13626.5	−1.46143	−0.730715	−	0.682682i	$-0.760813\pi$
$443$	−13626.5	−1.46143	−0.730715	+	0.682682i	$0.760813\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	911.608	0.0964599
$448$	0	0
$449$	9030.40	0.949155	0.474578	−	0.880214i	$-0.342601\pi$
$449$	9030.40	0.949155	0.474578	+	0.880214i	$0.342601\pi$
$450$	0	0
$451$	−18729.5	−1.95552
$452$	0	0
$453$	−6870.93	−0.712637
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−520.068	−0.0532336	−0.0266168	−	0.999646i	$-0.508473\pi$
$457$	−520.068	−0.0532336	−0.0266168	+	0.999646i	$0.508473\pi$
$458$	0	0
$459$	−2305.17	−0.234414
$460$	0	0
$461$	15812.8	1.59756	0.798782	−	0.601621i	$-0.205478\pi$
$461$	15812.8	1.59756	0.798782	+	0.601621i	$0.205478\pi$
$462$	0	0
$463$	9743.53	0.978013	0.489007	−	0.872280i	$-0.337359\pi$
$463$	9743.53	0.978013	0.489007	+	0.872280i	$0.337359\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17272.1	1.71147	0.855734	−	0.517415i	$-0.173106\pi$
$467$	17272.1	1.71147	0.855734	+	0.517415i	$0.173106\pi$
$468$	0	0
$469$	161.927	0.0159426
$470$	0	0
$471$	5929.41	0.580070
$472$	0	0
$473$	−12076.5	−1.17394
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3309.55	0.317681
$478$	0	0
$479$	14442.2	1.37762	0.688812	−	0.724940i	$-0.258132\pi$
$479$	14442.2	1.37762	0.688812	+	0.724940i	$0.258132\pi$
$480$	0	0
$481$	3147.75	0.298389
$482$	0	0
$483$	1317.77	0.124142
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−18605.5	−1.73120	−0.865599	−	0.500737i	$-0.833062\pi$
$487$	−18605.5	−1.73120	−0.865599	+	0.500737i	$0.833062\pi$
$488$	0	0
$489$	−246.064	−0.0227554
$490$	0	0
$491$	−8083.62	−0.742991	−0.371496	−	0.928435i	$-0.621155\pi$
$491$	−8083.62	−0.742991	−0.371496	+	0.928435i	$0.621155\pi$
$492$	0	0
$493$	18794.0	1.71691
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1547.06	−0.139628
$498$	0	0
$499$	−175.643	−0.0157572	−0.00787861	−	0.999969i	$-0.502508\pi$
$499$	−175.643	−0.0157572	−0.00787861	+	0.999969i	$0.502508\pi$
$500$	0	0
$501$	−2789.31	−0.248737
$502$	0	0
$503$	17108.0	1.51652	0.758258	−	0.651954i	$-0.226051\pi$
$503$	17108.0	1.51652	0.758258	+	0.651954i	$0.226051\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−4656.91	−0.407930
$508$	0	0
$509$	17505.3	1.52438	0.762189	−	0.647355i	$-0.224125\pi$
$509$	17505.3	1.52438	0.762189	+	0.647355i	$0.224125\pi$
$510$	0	0
$511$	−10275.7	−0.889568
$512$	0	0
$513$	−2494.17	−0.214659
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−14765.4	−1.25606
$518$	0	0
$519$	−1395.24	−0.118004
$520$	0	0
$521$	6321.21	0.531549	0.265774	−	0.964035i	$-0.414372\pi$
$521$	6321.21	0.531549	0.265774	+	0.964035i	$0.414372\pi$
$522$	0	0
$523$	22467.1	1.87843	0.939214	−	0.343333i	$-0.111556\pi$
$523$	22467.1	1.87843	0.939214	+	0.343333i	$0.111556\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9560.16	−0.790222
$528$	0	0
$529$	−11181.9	−0.919036
$530$	0	0
$531$	5747.36	0.469707
$532$	0	0
$533$	−10961.1	−0.890764
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11752.8	0.944451
$538$	0	0
$539$	6383.53	0.510126
$540$	0	0
$541$	−14522.2	−1.15408	−0.577039	−	0.816716i	$-0.695792\pi$
$541$	−14522.2	−1.15408	−0.577039	+	0.816716i	$0.695792\pi$
$542$	0	0
$543$	11539.5	0.911985
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14832.1	1.15937	0.579686	−	0.814840i	$-0.303175\pi$
$547$	14832.1	1.15937	0.579686	+	0.814840i	$0.303175\pi$
$548$	0	0
$549$	−1760.86	−0.136888
$550$	0	0
$551$	20334.9	1.57222
$552$	0	0
$553$	−2732.66	−0.210135
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12339.9	−0.938704	−0.469352	−	0.883011i	$-0.655512\pi$
$557$	−12339.9	−0.938704	−0.469352	+	0.883011i	$0.655512\pi$
$558$	0	0
$559$	−7067.51	−0.534747
$560$	0	0
$561$	11112.5	0.836310
$562$	0	0
$563$	7909.79	0.592110	0.296055	−	0.955171i	$-0.404329\pi$
$563$	7909.79	0.592110	0.296055	+	0.955171i	$0.404329\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1133.62	−0.0839636
$568$	0	0
$569$	11381.7	0.838568	0.419284	−	0.907855i	$-0.362281\pi$
$569$	11381.7	0.838568	0.419284	+	0.907855i	$0.362281\pi$
$570$	0	0
$571$	−14294.5	−1.04765	−0.523823	−	0.851827i	$-0.675495\pi$
$571$	−14294.5	−1.04765	−0.523823	+	0.851827i	$0.675495\pi$
$572$	0	0
$573$	2272.64	0.165691
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4197.35	0.302839	0.151420	−	0.988470i	$-0.451616\pi$
$577$	4197.35	0.302839	0.151420	+	0.988470i	$0.451616\pi$
$578$	0	0
$579$	−10517.4	−0.754899
$580$	0	0
$581$	5500.05	0.392738
$582$	0	0
$583$	−15954.3	−1.13338
$584$	0	0
$585$	0	0
$586$	0	0
$587$	203.919	0.0143384	0.00716918	−	0.999974i	$-0.497718\pi$
$587$	203.919	0.0143384	0.00716918	+	0.999974i	$0.497718\pi$
$588$	0	0
$589$	−10344.0	−0.723628
$590$	0	0
$591$	−6045.28	−0.420761
$592$	0	0
$593$	8172.28	0.565928	0.282964	−	0.959131i	$-0.408682\pi$
$593$	8172.28	0.565928	0.282964	+	0.959131i	$0.408682\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	9677.67	0.663452
$598$	0	0
$599$	5634.82	0.384362	0.192181	−	0.981360i	$-0.438444\pi$
$599$	5634.82	0.384362	0.192181	+	0.981360i	$0.438444\pi$
$600$	0	0
$601$	1962.41	0.133192	0.0665959	−	0.997780i	$-0.478786\pi$
$601$	1962.41	0.133192	0.0665959	+	0.997780i	$0.478786\pi$
$602$	0	0
$603$	−104.131	−0.00703242
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−2032.86	−0.135933	−0.0679665	−	0.997688i	$-0.521651\pi$
$607$	−2032.86	−0.135933	−0.0679665	+	0.997688i	$0.521651\pi$
$608$	0	0
$609$	9242.32	0.614972
$610$	0	0
$611$	−8641.18	−0.572152
$612$	0	0
$613$	21125.9	1.39196	0.695978	−	0.718064i	$-0.254971\pi$
$613$	21125.9	1.39196	0.695978	+	0.718064i	$0.254971\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	16122.1	1.05195	0.525973	−	0.850501i	$-0.323701\pi$
$617$	16122.1	1.05195	0.525973	+	0.850501i	$0.323701\pi$
$618$	0	0
$619$	−16312.2	−1.05920	−0.529598	−	0.848249i	$-0.677657\pi$
$619$	−16312.2	−1.05920	−0.529598	+	0.848249i	$0.677657\pi$
$620$	0	0
$621$	−847.426	−0.0547601
$622$	0	0
$623$	−4445.18	−0.285863
$624$	0	0
$625$	0	0
$626$	0	0
$627$	12023.6	0.765831
$628$	0	0
$629$	10584.3	0.670942
$630$	0	0
$631$	28427.6	1.79348	0.896739	−	0.442560i	$-0.145930\pi$
$631$	28427.6	1.79348	0.896739	+	0.442560i	$0.145930\pi$
$632$	0	0
$633$	−6275.95	−0.394070
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3735.83	0.232369
$638$	0	0
$639$	994.875	0.0615910
$640$	0	0
$641$	−14792.9	−0.911521	−0.455760	−	0.890103i	$-0.650633\pi$
$641$	−14792.9	−0.911521	−0.455760	+	0.890103i	$0.650633\pi$
$642$	0	0
$643$	−14400.1	−0.883177	−0.441588	−	0.897218i	$-0.645585\pi$
$643$	−14400.1	−0.883177	−0.441588	+	0.897218i	$0.645585\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19302.0	1.17286	0.586430	−	0.810000i	$-0.300533\pi$
$647$	19302.0	1.17286	0.586430	+	0.810000i	$0.300533\pi$
$648$	0	0
$649$	−27706.2	−1.67575
$650$	0	0
$651$	−4701.41	−0.283046
$652$	0	0
$653$	25140.8	1.50664	0.753321	−	0.657653i	$-0.228451\pi$
$653$	25140.8	1.50664	0.753321	+	0.657653i	$0.228451\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	6608.03	0.392395
$658$	0	0
$659$	11509.5	0.680342	0.340171	−	0.940364i	$-0.389515\pi$
$659$	11509.5	0.680342	0.340171	+	0.940364i	$0.389515\pi$
$660$	0	0
$661$	22601.6	1.32996	0.664979	−	0.746862i	$-0.268441\pi$
$661$	22601.6	1.32996	0.664979	+	0.746862i	$0.268441\pi$
$662$	0	0
$663$	6503.36	0.380950
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6909.03	0.401077
$668$	0	0
$669$	18513.8	1.06993
$670$	0	0
$671$	8488.53	0.488370
$672$	0	0
$673$	32353.7	1.85311	0.926555	−	0.376160i	$-0.122756\pi$
$673$	32353.7	1.85311	0.926555	+	0.376160i	$0.122756\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15287.2	−0.867853	−0.433927	−	0.900948i	$-0.642872\pi$
$677$	−15287.2	−0.867853	−0.433927	+	0.900948i	$0.642872\pi$
$678$	0	0
$679$	13901.8	0.785718
$680$	0	0
$681$	13322.1	0.749640
$682$	0	0
$683$	−19333.5	−1.08313	−0.541563	−	0.840660i	$-0.682167\pi$
$683$	−19333.5	−1.08313	−0.541563	+	0.840660i	$0.682167\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−17315.9	−0.961633
$688$	0	0
$689$	−9336.93	−0.516268
$690$	0	0
$691$	−20845.7	−1.14762	−0.573812	−	0.818987i	$-0.694536\pi$
$691$	−20845.7	−1.14762	−0.573812	+	0.818987i	$0.694536\pi$
$692$	0	0
$693$	5464.80	0.299554
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−36856.5	−2.00293
$698$	0	0
$699$	−393.083	−0.0212701
$700$	0	0
$701$	−17650.0	−0.950974	−0.475487	−	0.879723i	$-0.657728\pi$
$701$	−17650.0	−0.950974	−0.475487	+	0.879723i	$0.657728\pi$
$702$	0	0
$703$	11452.1	0.614400
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−157.655	−0.00838646
$708$	0	0
$709$	8592.23	0.455131	0.227566	−	0.973763i	$-0.426923\pi$
$709$	8592.23	0.455131	0.227566	+	0.973763i	$0.426923\pi$
$710$	0	0
$711$	1757.31	0.0926921
$712$	0	0
$713$	−3514.50	−0.184599
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11024.3	−0.574213
$718$	0	0
$719$	2820.32	0.146287	0.0731434	−	0.997321i	$-0.476697\pi$
$719$	2820.32	0.146287	0.0731434	+	0.997321i	$0.476697\pi$
$720$	0	0
$721$	−20960.6	−1.08268
$722$	0	0
$723$	−17139.5	−0.881641
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−16483.9	−0.840929	−0.420465	−	0.907309i	$-0.638133\pi$
$727$	−16483.9	−0.840929	−0.420465	+	0.907309i	$0.638133\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−23764.4	−1.20241
$732$	0	0
$733$	−747.009	−0.0376418	−0.0188209	−	0.999823i	$-0.505991\pi$
$733$	−747.009	−0.0376418	−0.0188209	+	0.999823i	$0.505991\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	501.983	0.0250893
$738$	0	0
$739$	22384.1	1.11423	0.557114	−	0.830436i	$-0.311909\pi$
$739$	22384.1	1.11423	0.557114	+	0.830436i	$0.311909\pi$
$740$	0	0
$741$	7036.57	0.348846
$742$	0	0
$743$	−1177.47	−0.0581389	−0.0290694	−	0.999577i	$-0.509254\pi$
$743$	−1177.47	−0.0581389	−0.0290694	+	0.999577i	$0.509254\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−3536.95	−0.173240
$748$	0	0
$749$	−232.434	−0.0113391
$750$	0	0
$751$	30821.1	1.49757	0.748786	−	0.662812i	$-0.230637\pi$
$751$	30821.1	1.49757	0.748786	+	0.662812i	$0.230637\pi$
$752$	0	0
$753$	−14693.0	−0.711081
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−19590.0	−0.940568	−0.470284	−	0.882515i	$-0.655849\pi$
$757$	−19590.0	−0.940568	−0.470284	+	0.882515i	$0.655849\pi$
$758$	0	0
$759$	4085.17	0.195365
$760$	0	0
$761$	21456.5	1.02207	0.511036	−	0.859560i	$-0.329262\pi$
$761$	21456.5	1.02207	0.511036	+	0.859560i	$0.329262\pi$
$762$	0	0
$763$	−23378.5	−1.10925
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16214.5	−0.763327
$768$	0	0
$769$	−39916.3	−1.87180	−0.935902	−	0.352260i	$-0.885413\pi$
$769$	−39916.3	−1.87180	−0.935902	+	0.352260i	$0.885413\pi$
$770$	0	0
$771$	−634.572	−0.0296414
$772$	0	0
$773$	−40881.3	−1.90220	−0.951098	−	0.308890i	$-0.900043\pi$
$773$	−40881.3	−1.90220	−0.951098	+	0.308890i	$0.900043\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5205.04	0.240321
$778$	0	0
$779$	−39878.4	−1.83414
$780$	0	0
$781$	−4795.98	−0.219736
$782$	0	0
$783$	−5943.51	−0.271269
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1246.46	0.0564570	0.0282285	−	0.999601i	$-0.491013\pi$
$787$	1246.46	0.0564570	0.0282285	+	0.999601i	$0.491013\pi$
$788$	0	0
$789$	21365.1	0.964026
$790$	0	0
$791$	4509.39	0.202700
$792$	0	0
$793$	4967.75	0.222459
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3737.97	0.166130	0.0830650	−	0.996544i	$-0.473529\pi$
$797$	3737.97	0.166130	0.0830650	+	0.996544i	$0.473529\pi$
$798$	0	0
$799$	−29055.9	−1.28651
$800$	0	0
$801$	2858.58	0.126096
$802$	0	0
$803$	−31855.2	−1.39993
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−855.147	−0.0373019
$808$	0	0
$809$	32369.8	1.40675	0.703374	−	0.710820i	$-0.251676\pi$
$809$	32369.8	1.40675	0.703374	+	0.710820i	$0.251676\pi$
$810$	0	0
$811$	27132.6	1.17479	0.587394	−	0.809301i	$-0.300154\pi$
$811$	27132.6	1.17479	0.587394	+	0.809301i	$0.300154\pi$
$812$	0	0
$813$	10850.4	0.468067
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−25712.9	−1.10108
$818$	0	0
$819$	3198.17	0.136451
$820$	0	0
$821$	24168.2	1.02738	0.513688	−	0.857977i	$-0.328279\pi$
$821$	24168.2	1.02738	0.513688	+	0.857977i	$0.328279\pi$
$822$	0	0
$823$	28730.8	1.21688	0.608440	−	0.793600i	$-0.291796\pi$
$823$	28730.8	1.21688	0.608440	+	0.793600i	$0.291796\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13095.1	−0.550617	−0.275309	−	0.961356i	$-0.588780\pi$
$827$	−13095.1	−0.550617	−0.275309	+	0.961356i	$0.588780\pi$
$828$	0	0
$829$	−43637.6	−1.82822	−0.914112	−	0.405462i	$-0.867111\pi$
$829$	−43637.6	−1.82822	−0.914112	+	0.405462i	$0.867111\pi$
$830$	0	0
$831$	−10067.9	−0.420280
$832$	0	0
$833$	12561.7	0.522494
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3023.36	0.124854
$838$	0	0
$839$	−13467.7	−0.554181	−0.277091	−	0.960844i	$-0.589370\pi$
$839$	−13467.7	−0.554181	−0.277091	+	0.960844i	$0.589370\pi$
$840$	0	0
$841$	24068.2	0.986846
$842$	0	0
$843$	−12376.8	−0.505671
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7716.38	−0.313032
$848$	0	0
$849$	11034.2	0.446046
$850$	0	0
$851$	3890.99	0.156735
$852$	0	0
$853$	−6921.82	−0.277841	−0.138921	−	0.990304i	$-0.544363\pi$
$853$	−6921.82	−0.277841	−0.138921	+	0.990304i	$0.544363\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34133.9	1.36055	0.680275	−	0.732957i	$-0.261860\pi$
$857$	34133.9	1.36055	0.680275	+	0.732957i	$0.261860\pi$
$858$	0	0
$859$	−33686.4	−1.33803	−0.669014	−	0.743250i	$-0.733284\pi$
$859$	−33686.4	−1.33803	−0.669014	+	0.743250i	$0.733284\pi$
$860$	0	0
$861$	−18125.0	−0.717419
$862$	0	0
$863$	17696.4	0.698022	0.349011	−	0.937119i	$-0.386518\pi$
$863$	17696.4	0.698022	0.349011	+	0.937119i	$0.386518\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7128.52	0.279235
$868$	0	0
$869$	−8471.41	−0.330694
$870$	0	0
$871$	293.776	0.0114285
$872$	0	0
$873$	−8939.90	−0.346586
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−27975.0	−1.07714	−0.538568	−	0.842582i	$-0.681034\pi$
$877$	−27975.0	−1.07714	−0.538568	+	0.842582i	$0.681034\pi$
$878$	0	0
$879$	−2542.19	−0.0975495
$880$	0	0
$881$	28810.0	1.10174	0.550870	−	0.834591i	$-0.314296\pi$
$881$	28810.0	1.10174	0.550870	+	0.834591i	$0.314296\pi$
$882$	0	0
$883$	−17068.6	−0.650513	−0.325256	−	0.945626i	$-0.605451\pi$
$883$	−17068.6	−0.650513	−0.325256	+	0.945626i	$0.605451\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31393.8	−1.18839	−0.594195	−	0.804321i	$-0.702529\pi$
$887$	−31393.8	−1.18839	−0.594195	+	0.804321i	$0.702529\pi$
$888$	0	0
$889$	−18507.0	−0.698204
$890$	0	0
$891$	−3514.28	−0.132135
$892$	0	0
$893$	−31438.2	−1.17810
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2390.77	0.0889914
$898$	0	0
$899$	−24649.3	−0.914462
$900$	0	0
$901$	−31395.4	−1.16086
$902$	0	0
$903$	−11686.7	−0.430684
$904$	0	0
$905$	0	0
$906$	0	0
$907$	38309.8	1.40249	0.701243	−	0.712922i	$-0.252629\pi$
$907$	38309.8	1.40249	0.701243	+	0.712922i	$0.252629\pi$
$908$	0	0
$909$	101.384	0.00369933
$910$	0	0
$911$	40656.9	1.47862	0.739311	−	0.673364i	$-0.235151\pi$
$911$	40656.9	1.47862	0.739311	+	0.673364i	$0.235151\pi$
$912$	0	0
$913$	17050.5	0.618060
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21627.0	0.778830
$918$	0	0
$919$	10516.8	0.377494	0.188747	−	0.982026i	$-0.439557\pi$
$919$	10516.8	0.377494	0.188747	+	0.982026i	$0.439557\pi$
$920$	0	0
$921$	−6148.85	−0.219991
$922$	0	0
$923$	−2806.75	−0.100092
$924$	0	0
$925$	0	0
$926$	0	0
$927$	13479.2	0.477580
$928$	0	0
$929$	33457.0	1.18158	0.590790	−	0.806826i	$-0.298816\pi$
$929$	33457.0	1.18158	0.590790	+	0.806826i	$0.298816\pi$
$930$	0	0
$931$	13591.6	0.478462
$932$	0	0
$933$	−26264.0	−0.921590
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−7276.37	−0.253691	−0.126846	−	0.991922i	$-0.540485\pi$
$937$	−7276.37	−0.253691	−0.126846	+	0.991922i	$0.540485\pi$
$938$	0	0
$939$	24663.7	0.857155
$940$	0	0
$941$	23369.0	0.809573	0.404786	−	0.914411i	$-0.367346\pi$
$941$	23369.0	0.809573	0.404786	+	0.914411i	$0.367346\pi$
$942$	0	0
$943$	−13549.2	−0.467892
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32741.6	−1.12350	−0.561752	−	0.827306i	$-0.689872\pi$
$947$	−32741.6	−1.12350	−0.561752	+	0.827306i	$0.689872\pi$
$948$	0	0
$949$	−18642.6	−0.637688
$950$	0	0
$951$	27531.6	0.938775
$952$	0	0
$953$	37093.1	1.26082	0.630410	−	0.776262i	$-0.282887\pi$
$953$	37093.1	1.26082	0.630410	+	0.776262i	$0.282887\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	28651.8	0.967795
$958$	0	0
$959$	15752.5	0.530421
$960$	0	0
$961$	−17252.3	−0.579112
$962$	0	0
$963$	149.473	0.00500176
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17450.9	−0.580334	−0.290167	−	0.956976i	$-0.593711\pi$
$967$	−17450.9	−0.580334	−0.290167	+	0.956976i	$0.593711\pi$
$968$	0	0
$969$	23660.4	0.784399
$970$	0	0
$971$	−37250.4	−1.23112	−0.615562	−	0.788088i	$-0.711071\pi$
$971$	−37250.4	−1.23112	−0.615562	+	0.788088i	$0.711071\pi$
$972$	0	0
$973$	27066.6	0.891795
$974$	0	0
$975$	0	0
$976$	0	0
$977$	39716.1	1.30054	0.650271	−	0.759702i	$-0.274655\pi$
$977$	39716.1	1.30054	0.650271	+	0.759702i	$0.274655\pi$
$978$	0	0
$979$	−13780.3	−0.449868
$980$	0	0
$981$	15034.1	0.489300
$982$	0	0
$983$	−60348.7	−1.95811	−0.979057	−	0.203587i	$-0.934740\pi$
$983$	−60348.7	−1.95811	−0.979057	+	0.203587i	$0.934740\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−14288.9	−0.460810
$988$	0	0
$989$	−8736.27	−0.280887
$990$	0	0
$991$	60820.4	1.94957	0.974784	−	0.223149i	$-0.0716337\pi$
$991$	60820.4	1.94957	0.974784	+	0.223149i	$0.0716337\pi$
$992$	0	0
$993$	−24412.7	−0.780175
$994$	0	0
$995$	0	0
$996$	0	0
$997$	5973.09	0.189739	0.0948694	−	0.995490i	$-0.469757\pi$
$997$	5973.09	0.189739	0.0948694	+	0.995490i	$0.469757\pi$
$998$	0	0
$999$	−3347.23	−0.106008

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.4.a.bv.1.1	yes	3
4.3	odd	2		2400.4.a.bg.1.3	✓	3
5.4	even	2		2400.4.a.bh.1.3	yes	3
20.19	odd	2		2400.4.a.bu.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2400.4.a.bg.1.3	✓	3	4.3	odd	2
2400.4.a.bh.1.3	yes	3	5.4	even	2
2400.4.a.bu.1.1	yes	3	20.19	odd	2
2400.4.a.bv.1.1	yes	3	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2400.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} -13.9953 q^{7} +9.00000 q^{9} -43.3861 q^{11} -25.3909 q^{13} -85.3766 q^{17} -92.3766 q^{19} -41.9858 q^{21} -31.3861 q^{23} +27.0000 q^{27} -220.130 q^{29} +111.976 q^{31} -130.158 q^{33} -123.972 q^{37} -76.1726 q^{39} +431.693 q^{41} +278.348 q^{43} +340.326 q^{47} -147.133 q^{49} -256.130 q^{51} +367.728 q^{53} -277.130 q^{57} +638.596 q^{59} -195.651 q^{61} -125.957 q^{63} -11.5701 q^{67} -94.1584 q^{69} +110.542 q^{71} +734.226 q^{73} +607.200 q^{77} +195.256 q^{79} +81.0000 q^{81} -392.994 q^{83} -660.390 q^{87} +317.620 q^{89} +355.352 q^{91} +335.929 q^{93} -993.323 q^{97} -390.475 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 9 q^{3} + 9 q^{7} + 27 q^{9} + 4 q^{11} + 7 q^{13} - 20 q^{17} - 41 q^{19} + 27 q^{21} + 40 q^{23} + 81 q^{27} + 48 q^{29} + 81 q^{31} + 12 q^{33} - 66 q^{37} + 21 q^{39} + 254 q^{41} + 293 q^{43}+ \cdots + 36 q^{99}+O(q^{100}) \) 3 * q + 9 * q^3 + 9 * q^7 + 27 * q^9 + 4 * q^11 + 7 * q^13 - 20 * q^17 - 41 * q^19 + 27 * q^21 + 40 * q^23 + 81 * q^27 + 48 * q^29 + 81 * q^31 + 12 * q^33 - 66 * q^37 + 21 * q^39 + 254 * q^41 + 293 * q^43 + 318 * q^47 + 310 * q^49 - 60 * q^51 + 282 * q^53 - 123 * q^57 - 102 * q^59 + 913 * q^61 + 81 * q^63 + 153 * q^67 + 120 * q^69 - 162 * q^71 + 566 * q^73 - 164 * q^77 + 1160 * q^79 + 243 * q^81 + 914 * q^83 + 144 * q^87 + 1232 * q^89 - 1467 * q^91 + 243 * q^93 - 2089 * q^97 + 36 * q^99

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