LMFDB - Embedded newform 225.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,Mod(1,225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, 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("225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 225 = 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	225.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:	$13.2754297513$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.35890$ of defining polynomial
Character	$\chi$	$=$	225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.35890 q^{2} +20.7178 q^{4} +4.43560 q^{7} -68.1534 q^{8} +O(q^{10})$	$q-5.35890 q^{2} +20.7178 q^{4} +4.43560 q^{7} -68.1534 q^{8} +3.43560 q^{11} -78.7424 q^{13} -23.7699 q^{14} +199.485 q^{16} +53.1780 q^{17} +20.4356 q^{19} -18.4110 q^{22} +118.307 q^{23} +421.972 q^{26} +91.8958 q^{28} -168.049 q^{29} -61.0492 q^{31} -523.792 q^{32} -284.975 q^{34} +246.614 q^{37} -109.512 q^{38} -422.663 q^{41} -362.436 q^{43} +71.1780 q^{44} -633.994 q^{46} -170.515 q^{47} -323.325 q^{49} -1631.37 q^{52} -546.049 q^{53} -302.301 q^{56} +900.559 q^{58} +216.970 q^{59} +130.902 q^{61} +327.156 q^{62} +1211.07 q^{64} +614.890 q^{67} +1101.73 q^{68} -324.822 q^{71} +88.8712 q^{73} -1321.58 q^{74} +423.381 q^{76} +15.2389 q^{77} -1137.42 q^{79} +2265.01 q^{82} -758.909 q^{83} +1942.26 q^{86} -234.148 q^{88} -195.681 q^{89} -349.269 q^{91} +2451.06 q^{92} +913.774 q^{94} -521.000 q^{97} +1732.67 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 24 q^{4} - 26 q^{7} - 84 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 24 * q^4 - 26 * q^7 - 84 * q^8	$ 2 q - 2 q^{2} + 24 q^{4} - 26 q^{7} - 84 q^{8} - 28 q^{11} - 18 q^{13} - 126 q^{14} + 120 q^{16} - 68 q^{17} + 6 q^{19} - 124 q^{22} + 132 q^{23} + 626 q^{26} - 8 q^{28} - 92 q^{29} + 122 q^{31} - 664 q^{32} - 692 q^{34} + 284 q^{37} - 158 q^{38} - 392 q^{41} - 690 q^{43} - 32 q^{44} - 588 q^{46} - 620 q^{47} + 260 q^{49} - 1432 q^{52} - 848 q^{53} + 180 q^{56} + 1156 q^{58} - 124 q^{59} + 750 q^{61} + 942 q^{62} + 1376 q^{64} + 358 q^{67} + 704 q^{68} - 824 q^{71} + 108 q^{73} - 1196 q^{74} + 376 q^{76} + 972 q^{77} - 880 q^{79} + 2368 q^{82} + 156 q^{83} + 842 q^{86} + 264 q^{88} + 864 q^{89} - 2198 q^{91} + 2496 q^{92} - 596 q^{94} - 1042 q^{97} + 3692 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 24 * q^4 - 26 * q^7 - 84 * q^8 - 28 * q^11 - 18 * q^13 - 126 * q^14 + 120 * q^16 - 68 * q^17 + 6 * q^19 - 124 * q^22 + 132 * q^23 + 626 * q^26 - 8 * q^28 - 92 * q^29 + 122 * q^31 - 664 * q^32 - 692 * q^34 + 284 * q^37 - 158 * q^38 - 392 * q^41 - 690 * q^43 - 32 * q^44 - 588 * q^46 - 620 * q^47 + 260 * q^49 - 1432 * q^52 - 848 * q^53 + 180 * q^56 + 1156 * q^58 - 124 * q^59 + 750 * q^61 + 942 * q^62 + 1376 * q^64 + 358 * q^67 + 704 * q^68 - 824 * q^71 + 108 * q^73 - 1196 * q^74 + 376 * q^76 + 972 * q^77 - 880 * q^79 + 2368 * q^82 + 156 * q^83 + 842 * q^86 + 264 * q^88 + 864 * q^89 - 2198 * q^91 + 2496 * q^92 - 596 * q^94 - 1042 * q^97 + 3692 * 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 $

$2$	−5.35890	−1.89466	−0.947328	−	0.320264i	$-0.896228\pi$
$2$	−5.35890	−1.89466	−0.947328	+	0.320264i	$0.896228\pi$
$3$	0	0
$3$	0	0
$4$	20.7178	2.58972
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.43560	0.239500	0.119750	−	0.992804i	$-0.461791\pi$
$7$	4.43560	0.239500	0.119750	+	0.992804i	$0.461791\pi$
$8$	−68.1534	−3.01198
$9$	0	0
$10$	0	0
$11$	3.43560	0.0941701	0.0470851	−	0.998891i	$-0.485007\pi$
$11$	3.43560	0.0941701	0.0470851	+	0.998891i	$0.485007\pi$
$12$	0	0
$13$	−78.7424	−1.67994	−0.839970	−	0.542634i	$-0.817427\pi$
$13$	−78.7424	−1.67994	−0.839970	+	0.542634i	$0.817427\pi$
$14$	−23.7699	−0.453770
$15$	0	0
$16$	199.485	3.11695
$17$	53.1780	0.758680	0.379340	−	0.925257i	$-0.376151\pi$
$17$	53.1780	0.758680	0.379340	+	0.925257i	$0.376151\pi$
$18$	0	0
$19$	20.4356	0.246750	0.123375	−	0.992360i	$-0.460628\pi$
$19$	20.4356	0.246750	0.123375	+	0.992360i	$0.460628\pi$
$20$	0	0
$21$	0	0
$22$	−18.4110	−0.178420
$23$	118.307	1.07255	0.536275	−	0.844043i	$-0.319831\pi$
$23$	118.307	1.07255	0.536275	+	0.844043i	$0.319831\pi$
$24$	0	0
$25$	0	0
$26$	421.972	3.18291
$27$	0	0
$28$	91.8958	0.620238
$29$	−168.049	−1.07607	−0.538034	−	0.842923i	$-0.680833\pi$
$29$	−168.049	−1.07607	−0.538034	+	0.842923i	$0.680833\pi$
$30$	0	0
$31$	−61.0492	−0.353702	−0.176851	−	0.984238i	$-0.556591\pi$
$31$	−61.0492	−0.353702	−0.176851	+	0.984238i	$0.556591\pi$
$32$	−523.792	−2.89357
$33$	0	0
$34$	−284.975	−1.43744
$35$	0	0
$36$	0	0
$37$	246.614	1.09576	0.547879	−	0.836558i	$-0.315436\pi$
$37$	246.614	1.09576	0.547879	+	0.836558i	$0.315436\pi$
$38$	−109.512	−0.467506
$39$	0	0
$40$	0	0
$41$	−422.663	−1.60997	−0.804986	−	0.593294i	$-0.797827\pi$
$41$	−422.663	−1.60997	−0.804986	+	0.593294i	$0.797827\pi$
$42$	0	0
$43$	−362.436	−1.28537	−0.642685	−	0.766131i	$-0.722180\pi$
$43$	−362.436	−1.28537	−0.642685	+	0.766131i	$0.722180\pi$
$44$	71.1780	0.243875
$45$	0	0
$46$	−633.994	−2.03212
$47$	−170.515	−0.529196	−0.264598	−	0.964359i	$-0.585239\pi$
$47$	−170.515	−0.529196	−0.264598	+	0.964359i	$0.585239\pi$
$48$	0	0
$49$	−323.325	−0.942640
$50$	0	0
$51$	0	0
$52$	−1631.37	−4.35058
$53$	−546.049	−1.41520	−0.707600	−	0.706613i	$-0.750222\pi$
$53$	−546.049	−1.41520	−0.707600	+	0.706613i	$0.750222\pi$
$54$	0	0
$55$	0	0
$56$	−302.301	−0.721369
$57$	0	0
$58$	900.559	2.03878
$59$	216.970	0.478763	0.239382	−	0.970926i	$-0.423055\pi$
$59$	216.970	0.478763	0.239382	+	0.970926i	$0.423055\pi$
$60$	0	0
$61$	130.902	0.274758	0.137379	−	0.990519i	$-0.456132\pi$
$61$	130.902	0.274758	0.137379	+	0.990519i	$0.456132\pi$
$62$	327.156	0.670143
$63$	0	0
$64$	1211.07	2.36537
$65$	0	0
$66$	0	0
$67$	614.890	1.12121	0.560603	−	0.828085i	$-0.310570\pi$
$67$	614.890	1.12121	0.560603	+	0.828085i	$0.310570\pi$
$68$	1101.73	1.96477
$69$	0	0
$70$	0	0
$71$	−324.822	−0.542948	−0.271474	−	0.962446i	$-0.587511\pi$
$71$	−324.822	−0.542948	−0.271474	+	0.962446i	$0.587511\pi$
$72$	0	0
$73$	88.8712	0.142487	0.0712437	−	0.997459i	$-0.477303\pi$
$73$	88.8712	0.142487	0.0712437	+	0.997459i	$0.477303\pi$
$74$	−1321.58	−2.07608
$75$	0	0
$76$	423.381	0.639014
$77$	15.2389	0.0225537
$78$	0	0
$79$	−1137.42	−1.61988	−0.809938	−	0.586516i	$-0.800499\pi$
$79$	−1137.42	−1.61988	−0.809938	+	0.586516i	$0.800499\pi$
$80$	0	0
$81$	0	0
$82$	2265.01	3.05034
$83$	−758.909	−1.00363	−0.501813	−	0.864976i	$-0.667334\pi$
$83$	−758.909	−1.00363	−0.501813	+	0.864976i	$0.667334\pi$
$84$	0	0
$85$	0	0
$86$	1942.26	2.43534
$87$	0	0
$88$	−234.148	−0.283639
$89$	−195.681	−0.233058	−0.116529	−	0.993187i	$-0.537177\pi$
$89$	−195.681	−0.233058	−0.116529	+	0.993187i	$0.537177\pi$
$90$	0	0
$91$	−349.269	−0.402345
$92$	2451.06	2.77761
$93$	0	0
$94$	913.774	1.00264
$95$	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	$0.587909\pi$
$98$	1732.67	1.78598
$99$	0	0
$100$	0	0
$101$	−660.920	−0.651129	−0.325565	−	0.945520i	$-0.605554\pi$
$101$	−660.920	−0.651129	−0.325565	+	0.945520i	$0.605554\pi$
$102$	0	0
$103$	−1530.75	−1.46436	−0.732181	−	0.681110i	$-0.761497\pi$
$103$	−1530.75	−1.46436	−0.732181	+	0.681110i	$0.761497\pi$
$104$	5366.56	5.05995
$105$	0	0
$106$	2926.22	2.68132
$107$	−264.625	−0.239087	−0.119543	−	0.992829i	$-0.538143\pi$
$107$	−264.625	−0.239087	−0.119543	+	0.992829i	$0.538143\pi$
$108$	0	0
$109$	1117.61	0.982091	0.491046	−	0.871134i	$-0.336615\pi$
$109$	1117.61	0.982091	0.491046	+	0.871134i	$0.336615\pi$
$110$	0	0
$111$	0	0
$112$	884.834	0.746508
$113$	−934.061	−0.777602	−0.388801	−	0.921322i	$-0.627111\pi$
$113$	−934.061	−0.777602	−0.388801	+	0.921322i	$0.627111\pi$
$114$	0	0
$115$	0	0
$116$	−3481.61	−2.78672
$117$	0	0
$118$	−1162.72	−0.907092
$119$	235.876	0.181704
$120$	0	0
$121$	−1319.20	−0.991132
$122$	−701.489	−0.520572
$123$	0	0
$124$	−1264.80	−0.915990
$125$	0	0
$126$	0	0
$127$	−630.356	−0.440433	−0.220217	−	0.975451i	$-0.570676\pi$
$127$	−630.356	−0.440433	−0.220217	+	0.975451i	$0.570676\pi$
$128$	−2299.66	−1.58799
$129$	0	0
$130$	0	0
$131$	2163.06	1.44265	0.721325	−	0.692597i	$-0.243534\pi$
$131$	2163.06	1.44265	0.721325	+	0.692597i	$0.243534\pi$
$132$	0	0
$133$	90.6440	0.0590965
$134$	−3295.13	−2.12430
$135$	0	0
$136$	−3624.26	−2.28513
$137$	1118.61	0.697588	0.348794	−	0.937199i	$-0.386591\pi$
$137$	1118.61	0.697588	0.348794	+	0.937199i	$0.386591\pi$
$138$	0	0
$139$	−166.478	−0.101586	−0.0507930	−	0.998709i	$-0.516175\pi$
$139$	−166.478	−0.101586	−0.0507930	+	0.998709i	$0.516175\pi$
$140$	0	0
$141$	0	0
$142$	1740.69	1.02870
$143$	−270.527	−0.158200
$144$	0	0
$145$	0	0
$146$	−476.252	−0.269965
$147$	0	0
$148$	5109.29	2.83771
$149$	653.143	0.359111	0.179555	−	0.983748i	$-0.442534\pi$
$149$	653.143	0.359111	0.179555	+	0.983748i	$0.442534\pi$
$150$	0	0
$151$	−1929.38	−1.03981	−0.519903	−	0.854225i	$-0.674032\pi$
$151$	−1929.38	−1.03981	−0.519903	+	0.854225i	$0.674032\pi$
$152$	−1392.76	−0.743206
$153$	0	0
$154$	−81.6638	−0.0427315
$155$	0	0
$156$	0	0
$157$	−2169.75	−1.10296	−0.551480	−	0.834188i	$-0.685937\pi$
$157$	−2169.75	−1.10296	−0.551480	+	0.834188i	$0.685937\pi$
$158$	6095.34	3.06911
$159$	0	0
$160$	0	0
$161$	524.761	0.256876
$162$	0	0
$163$	763.738	0.366997	0.183499	−	0.983020i	$-0.441258\pi$
$163$	763.738	0.366997	0.183499	+	0.983020i	$0.441258\pi$
$164$	−8756.64	−4.16938
$165$	0	0
$166$	4066.91	1.90153
$167$	−2564.28	−1.18820	−0.594102	−	0.804389i	$-0.702493\pi$
$167$	−2564.28	−1.18820	−0.594102	+	0.804389i	$0.702493\pi$
$168$	0	0
$169$	4003.36	1.82220
$170$	0	0
$171$	0	0
$172$	−7508.87	−3.32875
$173$	51.8290	0.0227774	0.0113887	−	0.999935i	$-0.496375\pi$
$173$	51.8290	0.0227774	0.0113887	+	0.999935i	$0.496375\pi$
$174$	0	0
$175$	0	0
$176$	685.349	0.293523
$177$	0	0
$178$	1048.64	0.441566
$179$	3956.63	1.65214	0.826068	−	0.563571i	$-0.190573\pi$
$179$	3956.63	1.65214	0.826068	+	0.563571i	$0.190573\pi$
$180$	0	0
$181$	1804.04	0.740848	0.370424	−	0.928863i	$-0.379212\pi$
$181$	1804.04	0.740848	0.370424	+	0.928863i	$0.379212\pi$
$182$	1871.70	0.762305
$183$	0	0
$184$	−8063.01	−3.23050
$185$	0	0
$186$	0	0
$187$	182.698	0.0714449
$188$	−3532.70	−1.37047
$189$	0	0
$190$	0	0
$191$	−3666.75	−1.38909	−0.694547	−	0.719448i	$-0.744395\pi$
$191$	−3666.75	−1.38909	−0.694547	+	0.719448i	$0.744395\pi$
$192$	0	0
$193$	2716.98	1.01333	0.506664	−	0.862144i	$-0.330879\pi$
$193$	2716.98	1.01333	0.506664	+	0.862144i	$0.330879\pi$
$194$	2791.99	1.03326
$195$	0	0
$196$	−6698.59	−2.44118
$197$	2034.30	0.735723	0.367862	−	0.929881i	$-0.380090\pi$
$197$	2034.30	0.735723	0.367862	+	0.929881i	$0.380090\pi$
$198$	0	0
$199$	−1551.27	−0.552596	−0.276298	−	0.961072i	$-0.589108\pi$
$199$	−1551.27	−0.552596	−0.276298	+	0.961072i	$0.589108\pi$
$200$	0	0
$201$	0	0
$202$	3541.81	1.23367
$203$	−745.398	−0.257718
$204$	0	0
$205$	0	0
$206$	8203.13	2.77446
$207$	0	0
$208$	−15707.9	−5.23629
$209$	70.2084	0.0232365
$210$	0	0
$211$	3192.51	1.04162	0.520809	−	0.853673i	$-0.325630\pi$
$211$	3192.51	1.04162	0.520809	+	0.853673i	$0.325630\pi$
$212$	−11312.9	−3.66498
$213$	0	0
$214$	1418.10	0.452988
$215$	0	0
$216$	0	0
$217$	−270.789	−0.0847115
$218$	−5989.18	−1.86073
$219$	0	0
$220$	0	0
$221$	−4187.36	−1.27454
$222$	0	0
$223$	1555.55	0.467120	0.233560	−	0.972342i	$-0.424963\pi$
$223$	1555.55	0.467120	0.233560	+	0.972342i	$0.424963\pi$
$224$	−2323.33	−0.693008
$225$	0	0
$226$	5005.54	1.47329
$227$	−6206.86	−1.81482	−0.907409	−	0.420248i	$-0.861943\pi$
$227$	−6206.86	−1.81482	−0.907409	+	0.420248i	$0.861943\pi$
$228$	0	0
$229$	4679.51	1.35035	0.675176	−	0.737657i	$-0.264068\pi$
$229$	4679.51	1.35035	0.675176	+	0.737657i	$0.264068\pi$
$230$	0	0
$231$	0	0
$232$	11453.1	3.24110
$233$	−3244.53	−0.912259	−0.456129	−	0.889913i	$-0.650765\pi$
$233$	−3244.53	−0.912259	−0.456129	+	0.889913i	$0.650765\pi$
$234$	0	0
$235$	0	0
$236$	4495.13	1.23986
$237$	0	0
$238$	−1264.04	−0.344266
$239$	3658.62	0.990193	0.495097	−	0.868838i	$-0.335133\pi$
$239$	3658.62	0.990193	0.495097	+	0.868838i	$0.335133\pi$
$240$	0	0
$241$	1931.01	0.516131	0.258065	−	0.966127i	$-0.416915\pi$
$241$	1931.01	0.516131	0.258065	+	0.966127i	$0.416915\pi$
$242$	7069.44	1.87786
$243$	0	0
$244$	2711.99	0.711548
$245$	0	0
$246$	0	0
$247$	−1609.15	−0.414525
$248$	4160.71	1.06534
$249$	0	0
$250$	0	0
$251$	5843.34	1.46944	0.734718	−	0.678373i	$-0.237315\pi$
$251$	5843.34	1.46944	0.734718	+	0.678373i	$0.237315\pi$
$252$	0	0
$253$	406.454	0.101002
$254$	3378.01	0.834470
$255$	0	0
$256$	2635.09	0.643333
$257$	4506.11	1.09371	0.546855	−	0.837227i	$-0.315825\pi$
$257$	4506.11	1.09371	0.546855	+	0.837227i	$0.315825\pi$
$258$	0	0
$259$	1093.88	0.262434
$260$	0	0
$261$	0	0
$262$	−11591.6	−2.73333
$263$	5340.16	1.25205	0.626024	−	0.779804i	$-0.284681\pi$
$263$	5340.16	1.25205	0.626024	+	0.779804i	$0.284681\pi$
$264$	0	0
$265$	0	0
$266$	−485.752	−0.111968
$267$	0	0
$268$	12739.2	2.90361
$269$	−2809.79	−0.636863	−0.318431	−	0.947946i	$-0.603156\pi$
$269$	−2809.79	−0.636863	−0.318431	+	0.947946i	$0.603156\pi$
$270$	0	0
$271$	3102.95	0.695537	0.347769	−	0.937580i	$-0.386940\pi$
$271$	3102.95	0.695537	0.347769	+	0.937580i	$0.386940\pi$
$272$	10608.2	2.36477
$273$	0	0
$274$	−5994.54	−1.32169
$275$	0	0
$276$	0	0
$277$	4598.93	0.997555	0.498777	−	0.866730i	$-0.333783\pi$
$277$	4598.93	0.997555	0.498777	+	0.866730i	$0.333783\pi$
$278$	892.138	0.192471
$279$	0	0
$280$	0	0
$281$	−2571.83	−0.545987	−0.272994	−	0.962016i	$-0.588014\pi$
$281$	−2571.83	−0.545987	−0.272994	+	0.962016i	$0.588014\pi$
$282$	0	0
$283$	−5575.31	−1.17109	−0.585544	−	0.810641i	$-0.699119\pi$
$283$	−5575.31	−1.17109	−0.585544	+	0.810641i	$0.699119\pi$
$284$	−6729.60	−1.40608
$285$	0	0
$286$	1449.73	0.299735
$287$	−1874.76	−0.385588
$288$	0	0
$289$	−2085.10	−0.424405
$290$	0	0
$291$	0	0
$292$	1841.22	0.369003
$293$	5794.27	1.15531	0.577654	−	0.816282i	$-0.303968\pi$
$293$	5794.27	1.15531	0.577654	+	0.816282i	$0.303968\pi$
$294$	0	0
$295$	0	0
$296$	−16807.6	−3.30040
$297$	0	0
$298$	−3500.13	−0.680392
$299$	−9315.76	−1.80182
$300$	0	0
$301$	−1607.62	−0.307846
$302$	10339.4	1.97008
$303$	0	0
$304$	4076.59	0.769107
$305$	0	0
$306$	0	0
$307$	1404.47	0.261099	0.130550	−	0.991442i	$-0.458326\pi$
$307$	1404.47	0.261099	0.130550	+	0.991442i	$0.458326\pi$
$308$	315.717	0.0584079
$309$	0	0
$310$	0	0
$311$	4096.75	0.746963	0.373481	−	0.927638i	$-0.378164\pi$
$311$	4096.75	0.746963	0.373481	+	0.927638i	$0.378164\pi$
$312$	0	0
$313$	−974.611	−0.176001	−0.0880004	−	0.996120i	$-0.528048\pi$
$313$	−974.611	−0.176001	−0.0880004	+	0.996120i	$0.528048\pi$
$314$	11627.5	2.08973
$315$	0	0
$316$	−23564.9	−4.19503
$317$	−2071.69	−0.367058	−0.183529	−	0.983014i	$-0.558752\pi$
$317$	−2071.69	−0.367058	−0.183529	+	0.983014i	$0.558752\pi$
$318$	0	0
$319$	−577.349	−0.101333
$320$	0	0
$321$	0	0
$322$	−2812.14	−0.486691
$323$	1086.72	0.187204
$324$	0	0
$325$	0	0
$326$	−4092.79	−0.695334
$327$	0	0
$328$	28805.9	4.84921
$329$	−756.337	−0.126742
$330$	0	0
$331$	−6159.17	−1.02278	−0.511388	−	0.859350i	$-0.670868\pi$
$331$	−6159.17	−1.02278	−0.511388	+	0.859350i	$0.670868\pi$
$332$	−15722.9	−2.59912
$333$	0	0
$334$	13741.7	2.25124
$335$	0	0
$336$	0	0
$337$	−2791.26	−0.451186	−0.225593	−	0.974222i	$-0.572432\pi$
$337$	−2791.26	−0.451186	−0.225593	+	0.974222i	$0.572432\pi$
$338$	−21453.6	−3.45243
$339$	0	0
$340$	0	0
$341$	−209.740	−0.0333081
$342$	0	0
$343$	−2955.55	−0.465262
$344$	24701.2	3.87151
$345$	0	0
$346$	−277.746	−0.0431553
$347$	940.848	0.145554	0.0727772	−	0.997348i	$-0.476814\pi$
$347$	940.848	0.145554	0.0727772	+	0.997348i	$0.476814\pi$
$348$	0	0
$349$	−3519.62	−0.539831	−0.269915	−	0.962884i	$-0.586996\pi$
$349$	−3519.62	−0.539831	−0.269915	+	0.962884i	$0.586996\pi$
$350$	0	0
$351$	0	0
$352$	−1799.54	−0.272487
$353$	5021.60	0.757147	0.378573	−	0.925571i	$-0.376415\pi$
$353$	5021.60	0.757147	0.378573	+	0.925571i	$0.376415\pi$
$354$	0	0
$355$	0	0
$356$	−4054.09	−0.603557
$357$	0	0
$358$	−21203.2	−3.13023
$359$	−6811.99	−1.00146	−0.500728	−	0.865604i	$-0.666934\pi$
$359$	−6811.99	−1.00146	−0.500728	+	0.865604i	$0.666934\pi$
$360$	0	0
$361$	−6441.39	−0.939115
$362$	−9667.69	−1.40365
$363$	0	0
$364$	−7236.09	−1.04196
$365$	0	0
$366$	0	0
$367$	−3748.07	−0.533099	−0.266550	−	0.963821i	$-0.585884\pi$
$367$	−3748.07	−0.533099	−0.266550	+	0.963821i	$0.585884\pi$
$368$	23600.4	3.34309
$369$	0	0
$370$	0	0
$371$	−2422.05	−0.338940
$372$	0	0
$373$	898.302	0.124698	0.0623489	−	0.998054i	$-0.480141\pi$
$373$	898.302	0.124698	0.0623489	+	0.998054i	$0.480141\pi$
$374$	−979.060	−0.135364
$375$	0	0
$376$	11621.2	1.59393
$377$	13232.6	1.80773
$378$	0	0
$379$	−9378.99	−1.27115	−0.635576	−	0.772038i	$-0.719237\pi$
$379$	−9378.99	−1.27115	−0.635576	+	0.772038i	$0.719237\pi$
$380$	0	0
$381$	0	0
$382$	19649.8	2.63186
$383$	−9446.29	−1.26027	−0.630134	−	0.776486i	$-0.717000\pi$
$383$	−9446.29	−1.26027	−0.630134	+	0.776486i	$0.717000\pi$
$384$	0	0
$385$	0	0
$386$	−14560.0	−1.91991
$387$	0	0
$388$	−10794.0	−1.41232
$389$	7643.23	0.996214	0.498107	−	0.867116i	$-0.334029\pi$
$389$	7643.23	0.996214	0.498107	+	0.867116i	$0.334029\pi$
$390$	0	0
$391$	6291.32	0.813723
$392$	22035.7	2.83922
$393$	0	0
$394$	−10901.6	−1.39394
$395$	0	0
$396$	0	0
$397$	12013.6	1.51876	0.759378	−	0.650650i	$-0.225503\pi$
$397$	12013.6	1.51876	0.759378	+	0.650650i	$0.225503\pi$
$398$	8313.10	1.04698
$399$	0	0
$400$	0	0
$401$	8538.51	1.06332	0.531662	−	0.846957i	$-0.321568\pi$
$401$	8538.51	1.06332	0.531662	+	0.846957i	$0.321568\pi$
$402$	0	0
$403$	4807.16	0.594197
$404$	−13692.8	−1.68625
$405$	0	0
$406$	3994.51	0.488287
$407$	847.265	0.103188
$408$	0	0
$409$	−12267.6	−1.48312	−0.741558	−	0.670889i	$-0.765913\pi$
$409$	−12267.6	−1.48312	−0.741558	+	0.670889i	$0.765913\pi$
$410$	0	0
$411$	0	0
$412$	−31713.8	−3.79229
$413$	962.389	0.114664
$414$	0	0
$415$	0	0
$416$	41244.6	4.86102
$417$	0	0
$418$	−376.240	−0.0440251
$419$	−15493.0	−1.80641	−0.903204	−	0.429212i	$-0.858791\pi$
$419$	−15493.0	−1.80641	−0.903204	+	0.429212i	$0.858791\pi$
$420$	0	0
$421$	7510.67	0.869472	0.434736	−	0.900558i	$-0.356842\pi$
$421$	7510.67	0.869472	0.434736	+	0.900558i	$0.356842\pi$
$422$	−17108.3	−1.97351
$423$	0	0
$424$	37215.1	4.26256
$425$	0	0
$426$	0	0
$427$	580.627	0.0658045
$428$	−5482.45	−0.619169
$429$	0	0
$430$	0	0
$431$	15675.3	1.75186	0.875932	−	0.482434i	$-0.160247\pi$
$431$	15675.3	1.75186	0.875932	+	0.482434i	$0.160247\pi$
$432$	0	0
$433$	9604.78	1.06600	0.532998	−	0.846117i	$-0.321065\pi$
$433$	9604.78	1.06600	0.532998	+	0.846117i	$0.321065\pi$
$434$	1451.13	0.160499
$435$	0	0
$436$	23154.5	2.54335
$437$	2417.67	0.264652
$438$	0	0
$439$	−6362.06	−0.691673	−0.345837	−	0.938295i	$-0.612405\pi$
$439$	−6362.06	−0.691673	−0.345837	+	0.938295i	$0.612405\pi$
$440$	0	0
$441$	0	0
$442$	22439.6	2.41481
$443$	931.658	0.0999196	0.0499598	−	0.998751i	$-0.484091\pi$
$443$	931.658	0.0999196	0.0499598	+	0.998751i	$0.484091\pi$
$444$	0	0
$445$	0	0
$446$	−8336.06	−0.885031
$447$	0	0
$448$	5371.81	0.566505
$449$	18684.1	1.96383	0.981914	−	0.189329i	$-0.0606314\pi$
$449$	18684.1	1.96383	0.981914	+	0.189329i	$0.0606314\pi$
$450$	0	0
$451$	−1452.10	−0.151611
$452$	−19351.7	−2.01378
$453$	0	0
$454$	33261.9	3.43846
$455$	0	0
$456$	0	0
$457$	11565.9	1.18387	0.591936	−	0.805985i	$-0.298364\pi$
$457$	11565.9	1.18387	0.591936	+	0.805985i	$0.298364\pi$
$458$	−25077.0	−2.55845
$459$	0	0
$460$	0	0
$461$	−19401.0	−1.96008	−0.980039	−	0.198806i	$-0.936294\pi$
$461$	−19401.0	−1.96008	−0.980039	+	0.198806i	$0.936294\pi$
$462$	0	0
$463$	−1576.28	−0.158220	−0.0791099	−	0.996866i	$-0.525208\pi$
$463$	−1576.28	−0.158220	−0.0791099	+	0.996866i	$0.525208\pi$
$464$	−33523.2	−3.35405
$465$	0	0
$466$	17387.1	1.72842
$467$	3256.55	0.322687	0.161344	−	0.986898i	$-0.448417\pi$
$467$	3256.55	0.322687	0.161344	+	0.986898i	$0.448417\pi$
$468$	0	0
$469$	2727.40	0.268528
$470$	0	0
$471$	0	0
$472$	−14787.2	−1.44203
$473$	−1245.18	−0.121043
$474$	0	0
$475$	0	0
$476$	4886.83	0.470562
$477$	0	0
$478$	−19606.2	−1.87608
$479$	−8291.59	−0.790924	−0.395462	−	0.918482i	$-0.629415\pi$
$479$	−8291.59	−0.790924	−0.395462	+	0.918482i	$0.629415\pi$
$480$	0	0
$481$	−19418.9	−1.84081
$482$	−10348.1	−0.977891
$483$	0	0
$484$	−27330.9	−2.56676
$485$	0	0
$486$	0	0
$487$	−4758.55	−0.442773	−0.221387	−	0.975186i	$-0.571058\pi$
$487$	−4758.55	−0.442773	−0.221387	+	0.975186i	$0.571058\pi$
$488$	−8921.39	−0.827567
$489$	0	0
$490$	0	0
$491$	−3906.46	−0.359055	−0.179528	−	0.983753i	$-0.557457\pi$
$491$	−3906.46	−0.359055	−0.179528	+	0.983753i	$0.557457\pi$
$492$	0	0
$493$	−8936.52	−0.816390
$494$	8623.26	0.785382
$495$	0	0
$496$	−12178.4	−1.10247
$497$	−1440.78	−0.130036
$498$	0	0
$499$	−3093.31	−0.277506	−0.138753	−	0.990327i	$-0.544309\pi$
$499$	−3093.31	−0.277506	−0.138753	+	0.990327i	$0.544309\pi$
$500$	0	0
$501$	0	0
$502$	−31313.9	−2.78408
$503$	−18153.9	−1.60923	−0.804616	−	0.593796i	$-0.797629\pi$
$503$	−18153.9	−1.60923	−0.804616	+	0.593796i	$0.797629\pi$
$504$	0	0
$505$	0	0
$506$	−2178.15	−0.191365
$507$	0	0
$508$	−13059.6	−1.14060
$509$	−2281.32	−0.198660	−0.0993298	−	0.995055i	$-0.531670\pi$
$509$	−2281.32	−0.198660	−0.0993298	+	0.995055i	$0.531670\pi$
$510$	0	0
$511$	394.197	0.0341257
$512$	4276.07	0.369097
$513$	0	0
$514$	−24147.8	−2.07221
$515$	0	0
$516$	0	0
$517$	−585.821	−0.0498344
$518$	−5861.98	−0.497221
$519$	0	0
$520$	0	0
$521$	16691.9	1.40362	0.701809	−	0.712366i	$-0.252376\pi$
$521$	16691.9	1.40362	0.701809	+	0.712366i	$0.252376\pi$
$522$	0	0
$523$	17090.4	1.42889	0.714446	−	0.699690i	$-0.246679\pi$
$523$	17090.4	1.42889	0.714446	+	0.699690i	$0.246679\pi$
$524$	44813.8	3.73607
$525$	0	0
$526$	−28617.4	−2.37220
$527$	−3246.47	−0.268346
$528$	0	0
$529$	1829.50	0.150365
$530$	0	0
$531$	0	0
$532$	1877.94	0.153044
$533$	33281.5	2.70465
$534$	0	0
$535$	0	0
$536$	−41906.8	−3.37705
$537$	0	0
$538$	15057.4	1.20664
$539$	−1110.82	−0.0887685
$540$	0	0
$541$	5271.40	0.418919	0.209459	−	0.977817i	$-0.432830\pi$
$541$	5271.40	0.418919	0.209459	+	0.977817i	$0.432830\pi$
$542$	−16628.4	−1.31780
$543$	0	0
$544$	−27854.2	−2.19529
$545$	0	0
$546$	0	0
$547$	−15182.2	−1.18673	−0.593366	−	0.804933i	$-0.702201\pi$
$547$	−15182.2	−1.18673	−0.593366	+	0.804933i	$0.702201\pi$
$548$	23175.2	1.80656
$549$	0	0
$550$	0	0
$551$	−3434.18	−0.265519
$552$	0	0
$553$	−5045.15	−0.387960
$554$	−24645.2	−1.89002
$555$	0	0
$556$	−3449.05	−0.263080
$557$	12241.2	0.931198	0.465599	−	0.884996i	$-0.345839\pi$
$557$	12241.2	0.931198	0.465599	+	0.884996i	$0.345839\pi$
$558$	0	0
$559$	28539.0	2.15934
$560$	0	0
$561$	0	0
$562$	13782.2	1.03446
$563$	14196.4	1.06271	0.531355	−	0.847149i	$-0.321683\pi$
$563$	14196.4	1.06271	0.531355	+	0.847149i	$0.321683\pi$
$564$	0	0
$565$	0	0
$566$	29877.5	2.21881
$567$	0	0
$568$	22137.7	1.63535
$569$	−9150.05	−0.674148	−0.337074	−	0.941478i	$-0.609437\pi$
$569$	−9150.05	−0.674148	−0.337074	+	0.941478i	$0.609437\pi$
$570$	0	0
$571$	23582.1	1.72833	0.864167	−	0.503206i	$-0.167846\pi$
$571$	23582.1	1.72833	0.864167	+	0.503206i	$0.167846\pi$
$572$	−5604.72	−0.409695
$573$	0	0
$574$	10046.7	0.730556
$575$	0	0
$576$	0	0
$577$	−3906.22	−0.281834	−0.140917	−	0.990021i	$-0.545005\pi$
$577$	−3906.22	−0.281834	−0.140917	+	0.990021i	$0.545005\pi$
$578$	11173.9	0.804102
$579$	0	0
$580$	0	0
$581$	−3366.21	−0.240368
$582$	0	0
$583$	−1876.00	−0.133270
$584$	−6056.87	−0.429170
$585$	0	0
$586$	−31050.9	−2.18891
$587$	−25938.0	−1.82381	−0.911905	−	0.410401i	$-0.865389\pi$
$587$	−25938.0	−1.82381	−0.911905	+	0.410401i	$0.865389\pi$
$588$	0	0
$589$	−1247.58	−0.0872759
$590$	0	0
$591$	0	0
$592$	49195.7	3.41542
$593$	1908.23	0.132145	0.0660723	−	0.997815i	$-0.478953\pi$
$593$	1908.23	0.132145	0.0660723	+	0.997815i	$0.478953\pi$
$594$	0	0
$595$	0	0
$596$	13531.7	0.929999
$597$	0	0
$598$	49922.2	3.41383
$599$	3495.41	0.238429	0.119214	−	0.992869i	$-0.461962\pi$
$599$	3495.41	0.238429	0.119214	+	0.992869i	$0.461962\pi$
$600$	0	0
$601$	−18267.2	−1.23983	−0.619913	−	0.784671i	$-0.712832\pi$
$601$	−18267.2	−1.23983	−0.619913	+	0.784671i	$0.712832\pi$
$602$	8615.06	0.583262
$603$	0	0
$604$	−39972.5	−2.69281
$605$	0	0
$606$	0	0
$607$	−11538.2	−0.771534	−0.385767	−	0.922596i	$-0.626063\pi$
$607$	−11538.2	−0.771534	−0.385767	+	0.922596i	$0.626063\pi$
$608$	−10704.0	−0.713987
$609$	0	0
$610$	0	0
$611$	13426.8	0.889017
$612$	0	0
$613$	−21136.9	−1.39268	−0.696340	−	0.717713i	$-0.745189\pi$
$613$	−21136.9	−1.39268	−0.696340	+	0.717713i	$0.745189\pi$
$614$	−7526.43	−0.494694
$615$	0	0
$616$	−1038.58	−0.0679314
$617$	15673.2	1.02266	0.511329	−	0.859385i	$-0.329153\pi$
$617$	15673.2	1.02266	0.511329	+	0.859385i	$0.329153\pi$
$618$	0	0
$619$	22923.7	1.48850	0.744249	−	0.667902i	$-0.232807\pi$
$619$	22923.7	1.48850	0.744249	+	0.667902i	$0.232807\pi$
$620$	0	0
$621$	0	0
$622$	−21954.1	−1.41524
$623$	−867.964	−0.0558174
$624$	0	0
$625$	0	0
$626$	5222.84	0.333461
$627$	0	0
$628$	−44952.4	−2.85636
$629$	13114.4	0.831329
$630$	0	0
$631$	9108.23	0.574632	0.287316	−	0.957836i	$-0.407237\pi$
$631$	9108.23	0.574632	0.287316	+	0.957836i	$0.407237\pi$
$632$	77519.3	4.87904
$633$	0	0
$634$	11102.0	0.695449
$635$	0	0
$636$	0	0
$637$	25459.4	1.58358
$638$	3093.96	0.191992
$639$	0	0
$640$	0	0
$641$	−20103.5	−1.23875	−0.619375	−	0.785095i	$-0.712614\pi$
$641$	−20103.5	−1.23875	−0.619375	+	0.785095i	$0.712614\pi$
$642$	0	0
$643$	−5934.92	−0.363997	−0.181999	−	0.983299i	$-0.558257\pi$
$643$	−5934.92	−0.363997	−0.181999	+	0.983299i	$0.558257\pi$
$644$	10871.9	0.665237
$645$	0	0
$646$	−5823.64	−0.354688
$647$	−14193.7	−0.862460	−0.431230	−	0.902242i	$-0.641920\pi$
$647$	−14193.7	−0.862460	−0.431230	+	0.902242i	$0.641920\pi$
$648$	0	0
$649$	745.420	0.0450852
$650$	0	0
$651$	0	0
$652$	15823.0	0.950422
$653$	−4795.80	−0.287403	−0.143701	−	0.989621i	$-0.545900\pi$
$653$	−4795.80	−0.287403	−0.143701	+	0.989621i	$0.545900\pi$
$654$	0	0
$655$	0	0
$656$	−84314.8	−5.01820
$657$	0	0
$658$	4053.13	0.240133
$659$	−4399.57	−0.260065	−0.130032	−	0.991510i	$-0.541508\pi$
$659$	−4399.57	−0.260065	−0.130032	+	0.991510i	$0.541508\pi$
$660$	0	0
$661$	24096.0	1.41789	0.708945	−	0.705263i	$-0.249171\pi$
$661$	24096.0	1.41789	0.708945	+	0.705263i	$0.249171\pi$
$662$	33006.4	1.93781
$663$	0	0
$664$	51722.2	3.02291
$665$	0	0
$666$	0	0
$667$	−19881.4	−1.15414
$668$	−53126.3	−3.07712
$669$	0	0
$670$	0	0
$671$	449.725	0.0258740
$672$	0	0
$673$	27648.3	1.58360	0.791800	−	0.610781i	$-0.209144\pi$
$673$	27648.3	1.58360	0.791800	+	0.610781i	$0.209144\pi$
$674$	14958.1	0.854843
$675$	0	0
$676$	82940.9	4.71898
$677$	−27605.5	−1.56716	−0.783580	−	0.621292i	$-0.786608\pi$
$677$	−27605.5	−1.56716	−0.783580	+	0.621292i	$0.786608\pi$
$678$	0	0
$679$	−2310.95	−0.130613
$680$	0	0
$681$	0	0
$682$	1123.98	0.0631075
$683$	−14949.4	−0.837513	−0.418756	−	0.908099i	$-0.637534\pi$
$683$	−14949.4	−0.837513	−0.418756	+	0.908099i	$0.637534\pi$
$684$	0	0
$685$	0	0
$686$	15838.5	0.881511
$687$	0	0
$688$	−72300.4	−4.00643
$689$	42997.2	2.37745
$690$	0	0
$691$	8884.30	0.489110	0.244555	−	0.969635i	$-0.421358\pi$
$691$	8884.30	0.489110	0.244555	+	0.969635i	$0.421358\pi$
$692$	1073.78	0.0589871
$693$	0	0
$694$	−5041.91	−0.275775
$695$	0	0
$696$	0	0
$697$	−22476.4	−1.22145
$698$	18861.3	1.02279
$699$	0	0
$700$	0	0
$701$	−10556.9	−0.568798	−0.284399	−	0.958706i	$-0.591794\pi$
$701$	−10556.9	−0.568798	−0.284399	+	0.958706i	$0.591794\pi$
$702$	0	0
$703$	5039.70	0.270378
$704$	4160.74	0.222747
$705$	0	0
$706$	−26910.2	−1.43453
$707$	−2931.58	−0.155945
$708$	0	0
$709$	25351.9	1.34289	0.671445	−	0.741055i	$-0.265674\pi$
$709$	25351.9	1.34289	0.671445	+	0.741055i	$0.265674\pi$
$710$	0	0
$711$	0	0
$712$	13336.4	0.701968
$713$	−7222.53	−0.379363
$714$	0	0
$715$	0	0
$716$	81972.6	4.27858
$717$	0	0
$718$	36504.8	1.89742
$719$	−9719.94	−0.504162	−0.252081	−	0.967706i	$-0.581115\pi$
$719$	−9719.94	−0.504162	−0.252081	+	0.967706i	$0.581115\pi$
$720$	0	0
$721$	−6789.79	−0.350714
$722$	34518.7	1.77930
$723$	0	0
$724$	37375.8	1.91859
$725$	0	0
$726$	0	0
$727$	27509.3	1.40339	0.701694	−	0.712479i	$-0.252428\pi$
$727$	27509.3	1.40339	0.701694	+	0.712479i	$0.252428\pi$
$728$	23803.9	1.21186
$729$	0	0
$730$	0	0
$731$	−19273.6	−0.975184
$732$	0	0
$733$	7240.49	0.364848	0.182424	−	0.983220i	$-0.441606\pi$
$733$	7240.49	0.364848	0.182424	+	0.983220i	$0.441606\pi$
$734$	20085.5	1.01004
$735$	0	0
$736$	−61968.1	−3.10350
$737$	2112.51	0.105584
$738$	0	0
$739$	−15875.3	−0.790234	−0.395117	−	0.918631i	$-0.629296\pi$
$739$	−15875.3	−0.790234	−0.395117	+	0.918631i	$0.629296\pi$
$740$	0	0
$741$	0	0
$742$	12979.5	0.642175
$743$	−25714.3	−1.26967	−0.634836	−	0.772647i	$-0.718932\pi$
$743$	−25714.3	−1.26967	−0.634836	+	0.772647i	$0.718932\pi$
$744$	0	0
$745$	0	0
$746$	−4813.91	−0.236260
$747$	0	0
$748$	3785.10	0.185023
$749$	−1173.77	−0.0572612
$750$	0	0
$751$	−9709.09	−0.471757	−0.235879	−	0.971783i	$-0.575797\pi$
$751$	−9709.09	−0.471757	−0.235879	+	0.971783i	$0.575797\pi$
$752$	−34015.2	−1.64948
$753$	0	0
$754$	−70912.1	−3.42502
$755$	0	0
$756$	0	0
$757$	9567.13	0.459344	0.229672	−	0.973268i	$-0.426235\pi$
$757$	9567.13	0.459344	0.229672	+	0.973268i	$0.426235\pi$
$758$	50261.1	2.40840
$759$	0	0
$760$	0	0
$761$	12322.5	0.586980	0.293490	−	0.955962i	$-0.405183\pi$
$761$	12322.5	0.586980	0.293490	+	0.955962i	$0.405183\pi$
$762$	0	0
$763$	4957.28	0.235211
$764$	−75967.0	−3.59737
$765$	0	0
$766$	50621.7	2.38778
$767$	−17084.7	−0.804293
$768$	0	0
$769$	2575.56	0.120776	0.0603881	−	0.998175i	$-0.480766\pi$
$769$	2575.56	0.120776	0.0603881	+	0.998175i	$0.480766\pi$
$770$	0	0
$771$	0	0
$772$	56289.8	2.62424
$773$	−6606.23	−0.307386	−0.153693	−	0.988119i	$-0.549117\pi$
$773$	−6606.23	−0.307386	−0.153693	+	0.988119i	$0.549117\pi$
$774$	0	0
$775$	0	0
$776$	35507.9	1.64260
$777$	0	0
$778$	−40959.3	−1.88748
$779$	−8637.37	−0.397260
$780$	0	0
$781$	−1115.96	−0.0511294
$782$	−33714.5	−1.54173
$783$	0	0
$784$	−64498.5	−2.93816
$785$	0	0
$786$	0	0
$787$	−16417.0	−0.743587	−0.371793	−	0.928315i	$-0.621257\pi$
$787$	−16417.0	−0.743587	−0.371793	+	0.928315i	$0.621257\pi$
$788$	42146.1	1.90532
$789$	0	0
$790$	0	0
$791$	−4143.12	−0.186235
$792$	0	0
$793$	−10307.5	−0.461577
$794$	−64379.8	−2.87752
$795$	0	0
$796$	−32138.9	−1.43107
$797$	−3944.19	−0.175295	−0.0876477	−	0.996152i	$-0.527935\pi$
$797$	−3944.19	−0.175295	−0.0876477	+	0.996152i	$0.527935\pi$
$798$	0	0
$799$	−9067.66	−0.401490
$800$	0	0
$801$	0	0
$802$	−45757.0	−2.01463
$803$	305.325	0.0134181
$804$	0	0
$805$	0	0
$806$	−25761.1	−1.12580
$807$	0	0
$808$	45044.0	1.96119
$809$	17960.7	0.780549	0.390275	−	0.920699i	$-0.372380\pi$
$809$	17960.7	0.780549	0.390275	+	0.920699i	$0.372380\pi$
$810$	0	0
$811$	−13162.5	−0.569912	−0.284956	−	0.958541i	$-0.591979\pi$
$811$	−13162.5	−0.569912	−0.284956	+	0.958541i	$0.591979\pi$
$812$	−15443.0	−0.667418
$813$	0	0
$814$	−4540.41	−0.195505
$815$	0	0
$816$	0	0
$817$	−7406.59	−0.317165
$818$	65740.9	2.81000
$819$	0	0
$820$	0	0
$821$	−26502.4	−1.12660	−0.563302	−	0.826251i	$-0.690469\pi$
$821$	−26502.4	−1.12660	−0.563302	+	0.826251i	$0.690469\pi$
$822$	0	0
$823$	−6937.86	−0.293850	−0.146925	−	0.989148i	$-0.546938\pi$
$823$	−6937.86	−0.293850	−0.146925	+	0.989148i	$0.546938\pi$
$824$	104326.	4.41063
$825$	0	0
$826$	−5157.35	−0.217248
$827$	41197.9	1.73228	0.866138	−	0.499805i	$-0.166595\pi$
$827$	41197.9	1.73228	0.866138	+	0.499805i	$0.166595\pi$
$828$	0	0
$829$	−693.324	−0.0290472	−0.0145236	−	0.999895i	$-0.504623\pi$
$829$	−693.324	−0.0290472	−0.0145236	+	0.999895i	$0.504623\pi$
$830$	0	0
$831$	0	0
$832$	−95362.4	−3.97367
$833$	−17193.8	−0.715162
$834$	0	0
$835$	0	0
$836$	1454.56	0.0601760
$837$	0	0
$838$	83025.7	3.42252
$839$	6491.28	0.267108	0.133554	−	0.991042i	$-0.457361\pi$
$839$	6491.28	0.267108	0.133554	+	0.991042i	$0.457361\pi$
$840$	0	0
$841$	3851.52	0.157921
$842$	−40248.9	−1.64735
$843$	0	0
$844$	66141.8	2.69750
$845$	0	0
$846$	0	0
$847$	−5851.42	−0.237376
$848$	−108928.	−4.41111
$849$	0	0
$850$	0	0
$851$	29176.1	1.17526
$852$	0	0
$853$	1116.68	0.0448233	0.0224117	−	0.999749i	$-0.492866\pi$
$853$	1116.68	0.0448233	0.0224117	+	0.999749i	$0.492866\pi$
$854$	−3111.52	−0.124677
$855$	0	0
$856$	18035.1	0.720126
$857$	44383.9	1.76911	0.884554	−	0.466438i	$-0.154463\pi$
$857$	44383.9	1.76911	0.884554	+	0.466438i	$0.154463\pi$
$858$	0	0
$859$	−25579.3	−1.01601	−0.508006	−	0.861354i	$-0.669617\pi$
$859$	−25579.3	−1.01601	−0.508006	+	0.861354i	$0.669617\pi$
$860$	0	0
$861$	0	0
$862$	−84002.5	−3.31918
$863$	−11194.8	−0.441570	−0.220785	−	0.975323i	$-0.570862\pi$
$863$	−11194.8	−0.441570	−0.220785	+	0.975323i	$0.570862\pi$
$864$	0	0
$865$	0	0
$866$	−51471.0	−2.01970
$867$	0	0
$868$	−5610.16	−0.219379
$869$	−3907.73	−0.152544
$870$	0	0
$871$	−48417.9	−1.88356
$872$	−76169.2	−2.95804
$873$	0	0
$874$	−12956.0	−0.501424
$875$	0	0
$876$	0	0
$877$	5721.75	0.220308	0.110154	−	0.993915i	$-0.464866\pi$
$877$	5721.75	0.220308	0.110154	+	0.993915i	$0.464866\pi$
$878$	34093.6	1.31048
$879$	0	0
$880$	0	0
$881$	34682.8	1.32633	0.663163	−	0.748475i	$-0.269214\pi$
$881$	34682.8	1.32633	0.663163	+	0.748475i	$0.269214\pi$
$882$	0	0
$883$	−37990.4	−1.44788	−0.723941	−	0.689862i	$-0.757671\pi$
$883$	−37990.4	−1.44788	−0.723941	+	0.689862i	$0.757671\pi$
$884$	−86752.9	−3.30070
$885$	0	0
$886$	−4992.66	−0.189313
$887$	−28299.0	−1.07124	−0.535618	−	0.844460i	$-0.679921\pi$
$887$	−28299.0	−1.07124	−0.535618	+	0.844460i	$0.679921\pi$
$888$	0	0
$889$	−2796.00	−0.105484
$890$	0	0
$891$	0	0
$892$	32227.7	1.20971
$893$	−3484.58	−0.130579
$894$	0	0
$895$	0	0
$896$	−10200.4	−0.380324
$897$	0	0
$898$	−100126.	−3.72078
$899$	10259.3	0.380607
$900$	0	0
$901$	−29037.8	−1.07368
$902$	7781.65	0.287251
$903$	0	0
$904$	63659.4	2.34212
$905$	0	0
$906$	0	0
$907$	17388.0	0.636559	0.318280	−	0.947997i	$-0.396895\pi$
$907$	17388.0	0.636559	0.318280	+	0.947997i	$0.396895\pi$
$908$	−128592.	−4.69988
$909$	0	0
$910$	0	0
$911$	−23555.3	−0.856663	−0.428332	−	0.903622i	$-0.640899\pi$
$911$	−23555.3	−0.856663	−0.428332	+	0.903622i	$0.640899\pi$
$912$	0	0
$913$	−2607.30	−0.0945117
$914$	−61980.4	−2.24303
$915$	0	0
$916$	96949.1	3.49704
$917$	9594.44	0.345514
$918$	0	0
$919$	−5983.09	−0.214760	−0.107380	−	0.994218i	$-0.534246\pi$
$919$	−5983.09	−0.214760	−0.107380	+	0.994218i	$0.534246\pi$
$920$	0	0
$921$	0	0
$922$	103968.	3.71368
$923$	25577.3	0.912119
$924$	0	0
$925$	0	0
$926$	8447.10	0.299772
$927$	0	0
$928$	88022.7	3.11367
$929$	−20576.7	−0.726694	−0.363347	−	0.931654i	$-0.618366\pi$
$929$	−20576.7	−0.726694	−0.363347	+	0.931654i	$0.618366\pi$
$930$	0	0
$931$	−6607.35	−0.232596
$932$	−67219.5	−2.36250
$933$	0	0
$934$	−17451.5	−0.611382
$935$	0	0
$936$	0	0
$937$	11228.6	0.391485	0.195743	−	0.980655i	$-0.437288\pi$
$937$	11228.6	0.391485	0.195743	+	0.980655i	$0.437288\pi$
$938$	−14615.9	−0.508769
$939$	0	0
$940$	0	0
$941$	−38567.6	−1.33610	−0.668049	−	0.744118i	$-0.732870\pi$
$941$	−38567.6	−1.33610	−0.668049	+	0.744118i	$0.732870\pi$
$942$	0	0
$943$	−50003.9	−1.72678
$944$	43282.1	1.49228
$945$	0	0
$946$	6672.81	0.229336
$947$	4606.17	0.158057	0.0790287	−	0.996872i	$-0.474818\pi$
$947$	4606.17	0.158057	0.0790287	+	0.996872i	$0.474818\pi$
$948$	0	0
$949$	−6997.93	−0.239370
$950$	0	0
$951$	0	0
$952$	−16075.8	−0.547288
$953$	−25559.7	−0.868795	−0.434397	−	0.900721i	$-0.643039\pi$
$953$	−25559.7	−0.868795	−0.434397	+	0.900721i	$0.643039\pi$
$954$	0	0
$955$	0	0
$956$	75798.5	2.56433
$957$	0	0
$958$	44433.8	1.49853
$959$	4961.72	0.167072
$960$	0	0
$961$	−26064.0	−0.874895
$962$	104064.	3.48769
$963$	0	0
$964$	40006.4	1.33664
$965$	0	0
$966$	0	0
$967$	−37895.8	−1.26023	−0.630117	−	0.776500i	$-0.716993\pi$
$967$	−37895.8	−1.26023	−0.630117	+	0.776500i	$0.716993\pi$
$968$	89907.7	2.98527
$969$	0	0
$970$	0	0
$971$	46761.0	1.54545	0.772726	−	0.634740i	$-0.218893\pi$
$971$	46761.0	1.54545	0.772726	+	0.634740i	$0.218893\pi$
$972$	0	0
$973$	−738.428	−0.0243298
$974$	25500.6	0.838903
$975$	0	0
$976$	26112.9	0.856407
$977$	3070.29	0.100540	0.0502698	−	0.998736i	$-0.483992\pi$
$977$	3070.29	0.100540	0.0502698	+	0.998736i	$0.483992\pi$
$978$	0	0
$979$	−672.282	−0.0219471
$980$	0	0
$981$	0	0
$982$	20934.3	0.680287
$983$	16319.0	0.529498	0.264749	−	0.964317i	$-0.414711\pi$
$983$	16319.0	0.529498	0.264749	+	0.964317i	$0.414711\pi$
$984$	0	0
$985$	0	0
$986$	47889.9	1.54678
$987$	0	0
$988$	−33338.0	−1.07350
$989$	−42878.6	−1.37862
$990$	0	0
$991$	5105.79	0.163664	0.0818319	−	0.996646i	$-0.473923\pi$
$991$	5105.79	0.163664	0.0818319	+	0.996646i	$0.473923\pi$
$992$	31977.0	1.02346
$993$	0	0
$994$	7720.99	0.246373
$995$	0	0
$996$	0	0
$997$	−7206.97	−0.228934	−0.114467	−	0.993427i	$-0.536516\pi$
$997$	−7206.97	−0.228934	−0.114467	+	0.993427i	$0.536516\pi$
$998$	16576.7	0.525779
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	225.4.a.j.1.1		2
3.2	odd	2		75.4.a.e.1.2	yes	2
5.2	odd	4		225.4.b.h.199.1		4
5.3	odd	4		225.4.b.h.199.4		4
5.4	even	2		225.4.a.n.1.2		2
12.11	even	2		1200.4.a.bu.1.1		2
15.2	even	4		75.4.b.c.49.4		4
15.8	even	4		75.4.b.c.49.1		4
15.14	odd	2		75.4.a.d.1.1	✓	2
60.23	odd	4		1200.4.f.v.49.4		4
60.47	odd	4		1200.4.f.v.49.1		4
60.59	even	2		1200.4.a.bl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.4.a.d.1.1	✓	2	15.14	odd	2
75.4.a.e.1.2	yes	2	3.2	odd	2
75.4.b.c.49.1		4	15.8	even	4
75.4.b.c.49.4		4	15.2	even	4
225.4.a.j.1.1		2	1.1	even	1	trivial
225.4.a.n.1.2		2	5.4	even	2
225.4.b.h.199.1		4	5.2	odd	4
225.4.b.h.199.4		4	5.3	odd	4
1200.4.a.bl.1.2		2	60.59	even	2
1200.4.a.bu.1.1		2	12.11	even	2
1200.4.f.v.49.1		4	60.47	odd	4
1200.4.f.v.49.4		4	60.23	odd	4

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:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	225.a (trivial)

\(f(q)\)	\(=\)	\(q-5.35890 q^{2} +20.7178 q^{4} +4.43560 q^{7} -68.1534 q^{8} +O(q^{10})\)	\(q-5.35890 q^{2} +20.7178 q^{4} +4.43560 q^{7} -68.1534 q^{8} +3.43560 q^{11} -78.7424 q^{13} -23.7699 q^{14} +199.485 q^{16} +53.1780 q^{17} +20.4356 q^{19} -18.4110 q^{22} +118.307 q^{23} +421.972 q^{26} +91.8958 q^{28} -168.049 q^{29} -61.0492 q^{31} -523.792 q^{32} -284.975 q^{34} +246.614 q^{37} -109.512 q^{38} -422.663 q^{41} -362.436 q^{43} +71.1780 q^{44} -633.994 q^{46} -170.515 q^{47} -323.325 q^{49} -1631.37 q^{52} -546.049 q^{53} -302.301 q^{56} +900.559 q^{58} +216.970 q^{59} +130.902 q^{61} +327.156 q^{62} +1211.07 q^{64} +614.890 q^{67} +1101.73 q^{68} -324.822 q^{71} +88.8712 q^{73} -1321.58 q^{74} +423.381 q^{76} +15.2389 q^{77} -1137.42 q^{79} +2265.01 q^{82} -758.909 q^{83} +1942.26 q^{86} -234.148 q^{88} -195.681 q^{89} -349.269 q^{91} +2451.06 q^{92} +913.774 q^{94} -521.000 q^{97} +1732.67 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 24 q^{4} - 26 q^{7} - 84 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 24 * q^4 - 26 * q^7 - 84 * q^8	\( 2 q - 2 q^{2} + 24 q^{4} - 26 q^{7} - 84 q^{8} - 28 q^{11} - 18 q^{13} - 126 q^{14} + 120 q^{16} - 68 q^{17} + 6 q^{19} - 124 q^{22} + 132 q^{23} + 626 q^{26} - 8 q^{28} - 92 q^{29} + 122 q^{31} - 664 q^{32} - 692 q^{34} + 284 q^{37} - 158 q^{38} - 392 q^{41} - 690 q^{43} - 32 q^{44} - 588 q^{46} - 620 q^{47} + 260 q^{49} - 1432 q^{52} - 848 q^{53} + 180 q^{56} + 1156 q^{58} - 124 q^{59} + 750 q^{61} + 942 q^{62} + 1376 q^{64} + 358 q^{67} + 704 q^{68} - 824 q^{71} + 108 q^{73} - 1196 q^{74} + 376 q^{76} + 972 q^{77} - 880 q^{79} + 2368 q^{82} + 156 q^{83} + 842 q^{86} + 264 q^{88} + 864 q^{89} - 2198 q^{91} + 2496 q^{92} - 596 q^{94} - 1042 q^{97} + 3692 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 24 * q^4 - 26 * q^7 - 84 * q^8 - 28 * q^11 - 18 * q^13 - 126 * q^14 + 120 * q^16 - 68 * q^17 + 6 * q^19 - 124 * q^22 + 132 * q^23 + 626 * q^26 - 8 * q^28 - 92 * q^29 + 122 * q^31 - 664 * q^32 - 692 * q^34 + 284 * q^37 - 158 * q^38 - 392 * q^41 - 690 * q^43 - 32 * q^44 - 588 * q^46 - 620 * q^47 + 260 * q^49 - 1432 * q^52 - 848 * q^53 + 180 * q^56 + 1156 * q^58 - 124 * q^59 + 750 * q^61 + 942 * q^62 + 1376 * q^64 + 358 * q^67 + 704 * q^68 - 824 * q^71 + 108 * q^73 - 1196 * q^74 + 376 * q^76 + 972 * q^77 - 880 * q^79 + 2368 * q^82 + 156 * q^83 + 842 * q^86 + 264 * q^88 + 864 * q^89 - 2198 * q^91 + 2496 * q^92 - 596 * q^94 - 1042 * q^97 + 3692 * q^98

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