LMFDB - Embedded newform 378.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,4,Mod(1,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, 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("378.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	378.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:	$22.3027219822$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.31662$ of defining polynomial
Character	$\chi$	$=$	378.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -15.9499 q^{5} +7.00000 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} -15.9499 q^{5} +7.00000 q^{7} -8.00000 q^{8} +31.8997 q^{10} -2.05013 q^{11} +79.6992 q^{13} -14.0000 q^{14} +16.0000 q^{16} +67.5990 q^{17} -108.398 q^{19} -63.7995 q^{20} +4.10025 q^{22} +7.74937 q^{23} +129.398 q^{25} -159.398 q^{26} +28.0000 q^{28} -249.198 q^{29} +5.30075 q^{31} -32.0000 q^{32} -135.198 q^{34} -111.649 q^{35} -48.6992 q^{37} +216.797 q^{38} +127.599 q^{40} +45.0476 q^{41} -201.699 q^{43} -8.20050 q^{44} -15.4987 q^{46} +4.49623 q^{47} +49.0000 q^{49} -258.797 q^{50} +318.797 q^{52} -279.198 q^{53} +32.6992 q^{55} -56.0000 q^{56} +498.396 q^{58} -675.293 q^{59} -842.195 q^{61} -10.6015 q^{62} +64.0000 q^{64} -1271.19 q^{65} +329.895 q^{67} +270.396 q^{68} +223.298 q^{70} +811.639 q^{71} +635.895 q^{73} +97.3985 q^{74} -433.594 q^{76} -14.3509 q^{77} -480.105 q^{79} -255.198 q^{80} -90.0952 q^{82} -1005.20 q^{83} -1078.20 q^{85} +403.398 q^{86} +16.4010 q^{88} -1377.54 q^{89} +557.895 q^{91} +30.9975 q^{92} -8.99246 q^{94} +1728.94 q^{95} -1670.20 q^{97} -98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 12 q^{5} + 14 q^{7} - 16 q^{8}+O(q^{10}) $ 2 * q - 4 * q^2 + 8 * q^4 - 12 * q^5 + 14 * q^7 - 16 * q^8	$ 2 q - 4 q^{2} + 8 q^{4} - 12 q^{5} + 14 q^{7} - 16 q^{8} + 24 q^{10} - 24 q^{11} + 40 q^{13} - 28 q^{14} + 32 q^{16} - 24 q^{17} + 22 q^{19} - 48 q^{20} + 48 q^{22} - 84 q^{23} + 20 q^{25} - 80 q^{26} + 56 q^{28} - 180 q^{29} + 130 q^{31} - 64 q^{32} + 48 q^{34} - 84 q^{35} + 22 q^{37} - 44 q^{38} + 96 q^{40} - 288 q^{41} - 284 q^{43} - 96 q^{44} + 168 q^{46} - 588 q^{47} + 98 q^{49} - 40 q^{50} + 160 q^{52} - 240 q^{53} - 54 q^{55} - 112 q^{56} + 360 q^{58} - 276 q^{59} - 968 q^{61} - 260 q^{62} + 128 q^{64} - 1428 q^{65} - 176 q^{67} - 96 q^{68} + 168 q^{70} - 108 q^{71} + 436 q^{73} - 44 q^{74} + 88 q^{76} - 168 q^{77} - 1796 q^{79} - 192 q^{80} + 576 q^{82} - 1692 q^{83} - 1440 q^{85} + 568 q^{86} + 192 q^{88} - 984 q^{89} + 280 q^{91} - 336 q^{92} + 1176 q^{94} + 2244 q^{95} - 2624 q^{97} - 196 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 12 * q^5 + 14 * q^7 - 16 * q^8 + 24 * q^10 - 24 * q^11 + 40 * q^13 - 28 * q^14 + 32 * q^16 - 24 * q^17 + 22 * q^19 - 48 * q^20 + 48 * q^22 - 84 * q^23 + 20 * q^25 - 80 * q^26 + 56 * q^28 - 180 * q^29 + 130 * q^31 - 64 * q^32 + 48 * q^34 - 84 * q^35 + 22 * q^37 - 44 * q^38 + 96 * q^40 - 288 * q^41 - 284 * q^43 - 96 * q^44 + 168 * q^46 - 588 * q^47 + 98 * q^49 - 40 * q^50 + 160 * q^52 - 240 * q^53 - 54 * q^55 - 112 * q^56 + 360 * q^58 - 276 * q^59 - 968 * q^61 - 260 * q^62 + 128 * q^64 - 1428 * q^65 - 176 * q^67 - 96 * q^68 + 168 * q^70 - 108 * q^71 + 436 * q^73 - 44 * q^74 + 88 * q^76 - 168 * q^77 - 1796 * q^79 - 192 * q^80 + 576 * q^82 - 1692 * q^83 - 1440 * q^85 + 568 * q^86 + 192 * q^88 - 984 * q^89 + 280 * q^91 - 336 * q^92 + 1176 * q^94 + 2244 * q^95 - 2624 * q^97 - 196 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−15.9499	−1.42660	−0.713300	−	0.700859i	$-0.752800\pi$
$5$	−15.9499	−1.42660	−0.713300	+	0.700859i	$0.752800\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	31.8997	1.00876
$11$	−2.05013	−0.0561942	−0.0280971	−	0.999605i	$-0.508945\pi$
$11$	−2.05013	−0.0561942	−0.0280971	+	0.999605i	$0.508945\pi$
$12$	0	0
$13$	79.6992	1.70035	0.850177	−	0.526497i	$-0.176495\pi$
$13$	79.6992	1.70035	0.850177	+	0.526497i	$0.176495\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	67.5990	0.964421	0.482211	−	0.876055i	$-0.339834\pi$
$17$	67.5990	0.964421	0.482211	+	0.876055i	$0.339834\pi$
$18$	0	0
$19$	−108.398	−1.30886	−0.654429	−	0.756123i	$-0.727091\pi$
$19$	−108.398	−1.30886	−0.654429	+	0.756123i	$0.727091\pi$
$20$	−63.7995	−0.713300
$21$	0	0
$22$	4.10025	0.0397353
$23$	7.74937	0.0702546	0.0351273	−	0.999383i	$-0.488816\pi$
$23$	7.74937	0.0702546	0.0351273	+	0.999383i	$0.488816\pi$
$24$	0	0
$25$	129.398	1.03519
$26$	−159.398	−1.20233
$27$	0	0
$28$	28.0000	0.188982
$29$	−249.198	−1.59569	−0.797843	−	0.602865i	$-0.794026\pi$
$29$	−249.198	−1.59569	−0.797843	+	0.602865i	$0.794026\pi$
$30$	0	0
$31$	5.30075	0.0307111	0.0153555	−	0.999882i	$-0.495112\pi$
$31$	5.30075	0.0307111	0.0153555	+	0.999882i	$0.495112\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−135.198	−0.681949
$35$	−111.649	−0.539204
$36$	0	0
$37$	−48.6992	−0.216381	−0.108191	−	0.994130i	$-0.534506\pi$
$37$	−48.6992	−0.216381	−0.108191	+	0.994130i	$0.534506\pi$
$38$	216.797	0.925503
$39$	0	0
$40$	127.599	0.504379
$41$	45.0476	0.171592	0.0857958	−	0.996313i	$-0.472657\pi$
$41$	45.0476	0.171592	0.0857958	+	0.996313i	$0.472657\pi$
$42$	0	0
$43$	−201.699	−0.715322	−0.357661	−	0.933851i	$-0.616426\pi$
$43$	−201.699	−0.715322	−0.357661	+	0.933851i	$0.616426\pi$
$44$	−8.20050	−0.0280971
$45$	0	0
$46$	−15.4987	−0.0496775
$47$	4.49623	0.0139541	0.00697705	−	0.999976i	$-0.497779\pi$
$47$	4.49623	0.0139541	0.00697705	+	0.999976i	$0.497779\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−258.797	−0.731988
$51$	0	0
$52$	318.797	0.850177
$53$	−279.198	−0.723600	−0.361800	−	0.932256i	$-0.617838\pi$
$53$	−279.198	−0.723600	−0.361800	+	0.932256i	$0.617838\pi$
$54$	0	0
$55$	32.6992	0.0801666
$56$	−56.0000	−0.133631
$57$	0	0
$58$	498.396	1.12832
$59$	−675.293	−1.49010	−0.745048	−	0.667011i	$-0.767574\pi$
$59$	−675.293	−1.49010	−0.745048	+	0.667011i	$0.767574\pi$
$60$	0	0
$61$	−842.195	−1.76774	−0.883870	−	0.467734i	$-0.845071\pi$
$61$	−842.195	−1.76774	−0.883870	+	0.467734i	$0.845071\pi$
$62$	−10.6015	−0.0217160
$63$	0	0
$64$	64.0000	0.125000
$65$	−1271.19	−2.42572
$66$	0	0
$67$	329.895	0.601538	0.300769	−	0.953697i	$-0.402757\pi$
$67$	329.895	0.601538	0.300769	+	0.953697i	$0.402757\pi$
$68$	270.396	0.482211
$69$	0	0
$70$	223.298	0.381275
$71$	811.639	1.35667	0.678337	−	0.734751i	$-0.262701\pi$
$71$	811.639	1.35667	0.678337	+	0.734751i	$0.262701\pi$
$72$	0	0
$73$	635.895	1.01953	0.509766	−	0.860313i	$-0.329732\pi$
$73$	635.895	1.01953	0.509766	+	0.860313i	$0.329732\pi$
$74$	97.3985	0.153005
$75$	0	0
$76$	−433.594	−0.654429
$77$	−14.3509	−0.0212394
$78$	0	0
$79$	−480.105	−0.683748	−0.341874	−	0.939746i	$-0.611062\pi$
$79$	−480.105	−0.683748	−0.341874	+	0.939746i	$0.611062\pi$
$80$	−255.198	−0.356650
$81$	0	0
$82$	−90.0952	−0.121334
$83$	−1005.20	−1.32934	−0.664668	−	0.747139i	$-0.731427\pi$
$83$	−1005.20	−1.32934	−0.664668	+	0.747139i	$0.731427\pi$
$84$	0	0
$85$	−1078.20	−1.37584
$86$	403.398	0.505809
$87$	0	0
$88$	16.4010	0.0198676
$89$	−1377.54	−1.64066	−0.820330	−	0.571890i	$-0.806210\pi$
$89$	−1377.54	−1.64066	−0.820330	+	0.571890i	$0.806210\pi$
$90$	0	0
$91$	557.895	0.642673
$92$	30.9975	0.0351273
$93$	0	0
$94$	−8.99246	−0.00986704
$95$	1728.94	1.86722
$96$	0	0
$97$	−1670.20	−1.74827	−0.874137	−	0.485679i	$-0.838573\pi$
$97$	−1670.20	−1.74827	−0.874137	+	0.485679i	$0.838573\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	517.594	0.517594
$101$	−1302.40	−1.28310	−0.641551	−	0.767081i	$-0.721709\pi$
$101$	−1302.40	−1.28310	−0.641551	+	0.767081i	$0.721709\pi$
$102$	0	0
$103$	−1802.49	−1.72432	−0.862158	−	0.506640i	$-0.830887\pi$
$103$	−1802.49	−1.72432	−0.862158	+	0.506640i	$0.830887\pi$
$104$	−637.594	−0.601166
$105$	0	0
$106$	558.396	0.511662
$107$	625.093	0.564766	0.282383	−	0.959302i	$-0.408875\pi$
$107$	625.093	0.564766	0.282383	+	0.959302i	$0.408875\pi$
$108$	0	0
$109$	299.902	0.263536	0.131768	−	0.991281i	$-0.457935\pi$
$109$	299.902	0.263536	0.131768	+	0.991281i	$0.457935\pi$
$110$	−65.3985	−0.0566864
$111$	0	0
$112$	112.000	0.0944911
$113$	1492.10	1.24217	0.621084	−	0.783744i	$-0.286693\pi$
$113$	1492.10	1.24217	0.621084	+	0.783744i	$0.286693\pi$
$114$	0	0
$115$	−123.602	−0.100225
$116$	−996.792	−0.797843
$117$	0	0
$118$	1350.59	1.05366
$119$	473.193	0.364517
$120$	0	0
$121$	−1326.80	−0.996842
$122$	1684.39	1.24998
$123$	0	0
$124$	21.2030	0.0153555
$125$	−70.1554	−0.0501991
$126$	0	0
$127$	−1796.48	−1.25521	−0.627606	−	0.778531i	$-0.715965\pi$
$127$	−1796.48	−1.25521	−0.627606	+	0.778531i	$0.715965\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	2542.39	1.71525
$131$	1743.70	1.16296	0.581481	−	0.813560i	$-0.302473\pi$
$131$	1743.70	1.16296	0.581481	+	0.813560i	$0.302473\pi$
$132$	0	0
$133$	−758.789	−0.494702
$134$	−659.789	−0.425352
$135$	0	0
$136$	−540.792	−0.340974
$137$	2238.19	1.39578	0.697889	−	0.716206i	$-0.254123\pi$
$137$	2238.19	1.39578	0.697889	+	0.716206i	$0.254123\pi$
$138$	0	0
$139$	−499.008	−0.304498	−0.152249	−	0.988342i	$-0.548652\pi$
$139$	−499.008	−0.304498	−0.152249	+	0.988342i	$0.548652\pi$
$140$	−446.596	−0.269602
$141$	0	0
$142$	−1623.28	−0.959313
$143$	−163.393	−0.0955500
$144$	0	0
$145$	3974.68	2.27641
$146$	−1271.79	−0.720918
$147$	0	0
$148$	−194.797	−0.108191
$149$	2957.49	1.62609	0.813045	−	0.582201i	$-0.197808\pi$
$149$	2957.49	1.62609	0.813045	+	0.582201i	$0.197808\pi$
$150$	0	0
$151$	1449.79	0.781339	0.390670	−	0.920531i	$-0.372244\pi$
$151$	1449.79	0.781339	0.390670	+	0.920531i	$0.372244\pi$
$152$	867.188	0.462752
$153$	0	0
$154$	28.7018	0.0150185
$155$	−84.5464	−0.0438124
$156$	0	0
$157$	2677.68	1.36116	0.680581	−	0.732673i	$-0.261727\pi$
$157$	2677.68	1.36116	0.680581	+	0.732673i	$0.261727\pi$
$158$	960.211	0.483483
$159$	0	0
$160$	510.396	0.252190
$161$	54.2456	0.0265537
$162$	0	0
$163$	−679.910	−0.326716	−0.163358	−	0.986567i	$-0.552232\pi$
$163$	−679.910	−0.326716	−0.163358	+	0.986567i	$0.552232\pi$
$164$	180.190	0.0857958
$165$	0	0
$166$	2010.40	0.939982
$167$	392.912	0.182063	0.0910313	−	0.995848i	$-0.470984\pi$
$167$	392.912	0.182063	0.0910313	+	0.995848i	$0.470984\pi$
$168$	0	0
$169$	4154.97	1.89120
$170$	2156.39	0.972868
$171$	0	0
$172$	−806.797	−0.357661
$173$	836.842	0.367768	0.183884	−	0.982948i	$-0.441133\pi$
$173$	836.842	0.367768	0.183884	+	0.982948i	$0.441133\pi$
$174$	0	0
$175$	905.789	0.391264
$176$	−32.8020	−0.0140485
$177$	0	0
$178$	2755.08	1.16012
$179$	1417.31	0.591815	0.295908	−	0.955217i	$-0.404378\pi$
$179$	1417.31	0.591815	0.295908	+	0.955217i	$0.404378\pi$
$180$	0	0
$181$	−892.586	−0.366549	−0.183275	−	0.983062i	$-0.558670\pi$
$181$	−892.586	−0.366549	−0.183275	+	0.983062i	$0.558670\pi$
$182$	−1115.79	−0.454439
$183$	0	0
$184$	−61.9950	−0.0248388
$185$	776.747	0.308690
$186$	0	0
$187$	−138.586	−0.0541949
$188$	17.9849	0.00697705
$189$	0	0
$190$	−3457.88	−1.32032
$191$	−1137.74	−0.431017	−0.215509	−	0.976502i	$-0.569141\pi$
$191$	−1137.74	−0.431017	−0.215509	+	0.976502i	$0.569141\pi$
$192$	0	0
$193$	−4019.74	−1.49921	−0.749605	−	0.661885i	$-0.769757\pi$
$193$	−4019.74	−1.49921	−0.749605	+	0.661885i	$0.769757\pi$
$194$	3340.39	1.23622
$195$	0	0
$196$	196.000	0.0714286
$197$	−2814.89	−1.01803	−0.509016	−	0.860757i	$-0.669991\pi$
$197$	−2814.89	−1.01803	−0.509016	+	0.860757i	$0.669991\pi$
$198$	0	0
$199$	−3439.26	−1.22514	−0.612569	−	0.790417i	$-0.709864\pi$
$199$	−3439.26	−1.22514	−0.612569	+	0.790417i	$0.709864\pi$
$200$	−1035.19	−0.365994
$201$	0	0
$202$	2604.79	0.907290
$203$	−1744.39	−0.603113
$204$	0	0
$205$	−718.504	−0.244793
$206$	3604.98	1.21928
$207$	0	0
$208$	1275.19	0.425088
$209$	222.231	0.0735503
$210$	0	0
$211$	1031.56	0.336568	0.168284	−	0.985739i	$-0.446177\pi$
$211$	1031.56	0.336568	0.168284	+	0.985739i	$0.446177\pi$
$212$	−1116.79	−0.361800
$213$	0	0
$214$	−1250.19	−0.399350
$215$	3217.08	1.02048
$216$	0	0
$217$	37.1053	0.0116077
$218$	−599.805	−0.186348
$219$	0	0
$220$	130.797	0.0400833
$221$	5387.59	1.63986
$222$	0	0
$223$	−6293.47	−1.88987	−0.944936	−	0.327254i	$-0.893877\pi$
$223$	−6293.47	−1.88987	−0.944936	+	0.327254i	$0.893877\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	−2984.20	−0.878345
$227$	−6006.10	−1.75612	−0.878058	−	0.478553i	$-0.841161\pi$
$227$	−6006.10	−1.75612	−0.878058	+	0.478553i	$0.841161\pi$
$228$	0	0
$229$	4454.56	1.28544	0.642719	−	0.766102i	$-0.277806\pi$
$229$	4454.56	1.28544	0.642719	+	0.766102i	$0.277806\pi$
$230$	247.203	0.0708699
$231$	0	0
$232$	1993.58	0.564160
$233$	−2627.45	−0.738755	−0.369377	−	0.929279i	$-0.620429\pi$
$233$	−2627.45	−0.738755	−0.369377	+	0.929279i	$0.620429\pi$
$234$	0	0
$235$	−71.7143	−0.0199069
$236$	−2701.17	−0.745048
$237$	0	0
$238$	−946.386	−0.257752
$239$	1598.87	0.432729	0.216364	−	0.976313i	$-0.430580\pi$
$239$	1598.87	0.432729	0.216364	+	0.976313i	$0.430580\pi$
$240$	0	0
$241$	2645.91	0.707212	0.353606	−	0.935395i	$-0.384955\pi$
$241$	2645.91	0.707212	0.353606	+	0.935395i	$0.384955\pi$
$242$	2653.59	0.704874
$243$	0	0
$244$	−3368.78	−0.883870
$245$	−781.544	−0.203800
$246$	0	0
$247$	−8639.28	−2.22552
$248$	−42.4060	−0.0108580
$249$	0	0
$250$	140.311	0.0354961
$251$	−3211.79	−0.807675	−0.403837	−	0.914831i	$-0.632324\pi$
$251$	−3211.79	−0.807675	−0.403837	+	0.914831i	$0.632324\pi$
$252$	0	0
$253$	−15.8872	−0.00394790
$254$	3592.96	0.887569
$255$	0	0
$256$	256.000	0.0625000
$257$	159.584	0.0387338	0.0193669	−	0.999812i	$-0.493835\pi$
$257$	159.584	0.0387338	0.0193669	+	0.999812i	$0.493835\pi$
$258$	0	0
$259$	−340.895	−0.0817844
$260$	−5084.77	−1.21286
$261$	0	0
$262$	−3487.41	−0.822339
$263$	3017.15	0.707398	0.353699	−	0.935359i	$-0.384924\pi$
$263$	3017.15	0.707398	0.353699	+	0.935359i	$0.384924\pi$
$264$	0	0
$265$	4453.17	1.03229
$266$	1517.58	0.349807
$267$	0	0
$268$	1319.58	0.300769
$269$	−3070.84	−0.696031	−0.348015	−	0.937489i	$-0.613144\pi$
$269$	−3070.84	−0.696031	−0.348015	+	0.937489i	$0.613144\pi$
$270$	0	0
$271$	4962.17	1.11229	0.556144	−	0.831086i	$-0.312280\pi$
$271$	4962.17	1.11229	0.556144	+	0.831086i	$0.312280\pi$
$272$	1081.58	0.241105
$273$	0	0
$274$	−4476.38	−0.986964
$275$	−265.283	−0.0581716
$276$	0	0
$277$	2621.29	0.568584	0.284292	−	0.958738i	$-0.408241\pi$
$277$	2621.29	0.568584	0.284292	+	0.958738i	$0.408241\pi$
$278$	998.015	0.215313
$279$	0	0
$280$	893.193	0.190637
$281$	−2508.51	−0.532544	−0.266272	−	0.963898i	$-0.585792\pi$
$281$	−2508.51	−0.532544	−0.266272	+	0.963898i	$0.585792\pi$
$282$	0	0
$283$	−4202.17	−0.882660	−0.441330	−	0.897345i	$-0.645493\pi$
$283$	−4202.17	−0.882660	−0.441330	+	0.897345i	$0.645493\pi$
$284$	3246.56	0.678337
$285$	0	0
$286$	326.787	0.0675640
$287$	315.333	0.0648555
$288$	0	0
$289$	−343.376	−0.0698913
$290$	−7949.35	−1.60966
$291$	0	0
$292$	2543.58	0.509766
$293$	1657.33	0.330452	0.165226	−	0.986256i	$-0.447165\pi$
$293$	1657.33	0.330452	0.165226	+	0.986256i	$0.447165\pi$
$294$	0	0
$295$	10770.8	2.12577
$296$	389.594	0.0765023
$297$	0	0
$298$	−5914.99	−1.14982
$299$	617.619	0.119458
$300$	0	0
$301$	−1411.89	−0.270366
$302$	−2899.58	−0.552490
$303$	0	0
$304$	−1734.38	−0.327215
$305$	13432.9	2.52186
$306$	0	0
$307$	7758.70	1.44239	0.721193	−	0.692734i	$-0.243594\pi$
$307$	7758.70	1.44239	0.721193	+	0.692734i	$0.243594\pi$
$308$	−57.4035	−0.0106197
$309$	0	0
$310$	169.093	0.0309801
$311$	−10895.4	−1.98656	−0.993278	−	0.115755i	$-0.963071\pi$
$311$	−10895.4	−1.98656	−0.993278	+	0.115755i	$0.963071\pi$
$312$	0	0
$313$	1853.29	0.334678	0.167339	−	0.985899i	$-0.446483\pi$
$313$	1853.29	0.334678	0.167339	+	0.985899i	$0.446483\pi$
$314$	−5355.37	−0.962487
$315$	0	0
$316$	−1920.42	−0.341874
$317$	2495.05	0.442070	0.221035	−	0.975266i	$-0.429057\pi$
$317$	2495.05	0.442070	0.221035	+	0.975266i	$0.429057\pi$
$318$	0	0
$319$	510.887	0.0896683
$320$	−1020.79	−0.178325
$321$	0	0
$322$	−108.491	−0.0187763
$323$	−7327.63	−1.26229
$324$	0	0
$325$	10313.0	1.76019
$326$	1359.82	0.231023
$327$	0	0
$328$	−360.381	−0.0606668
$329$	31.4736	0.00527415
$330$	0	0
$331$	−9834.89	−1.63315	−0.816577	−	0.577236i	$-0.804131\pi$
$331$	−9834.89	−1.63315	−0.816577	+	0.577236i	$0.804131\pi$
$332$	−4020.79	−0.664668
$333$	0	0
$334$	−785.825	−0.128738
$335$	−5261.78	−0.858154
$336$	0	0
$337$	−4951.50	−0.800371	−0.400186	−	0.916434i	$-0.631054\pi$
$337$	−4951.50	−0.800371	−0.400186	+	0.916434i	$0.631054\pi$
$338$	−8309.94	−1.33728
$339$	0	0
$340$	−4312.78	−0.687922
$341$	−10.8672	−0.00172578
$342$	0	0
$343$	343.000	0.0539949
$344$	1613.59	0.252905
$345$	0	0
$346$	−1673.68	−0.260051
$347$	−12046.2	−1.86362	−0.931810	−	0.362947i	$-0.881771\pi$
$347$	−12046.2	−1.86362	−0.931810	+	0.362947i	$0.881771\pi$
$348$	0	0
$349$	−3088.59	−0.473720	−0.236860	−	0.971544i	$-0.576118\pi$
$349$	−3088.59	−0.473720	−0.236860	+	0.971544i	$0.576118\pi$
$350$	−1811.58	−0.276666
$351$	0	0
$352$	65.6040	0.00993382
$353$	−4499.67	−0.678452	−0.339226	−	0.940705i	$-0.610165\pi$
$353$	−4499.67	−0.678452	−0.339226	+	0.940705i	$0.610165\pi$
$354$	0	0
$355$	−12945.5	−1.93543
$356$	−5510.16	−0.820330
$357$	0	0
$358$	−2834.63	−0.418477
$359$	5375.29	0.790242	0.395121	−	0.918629i	$-0.370703\pi$
$359$	5375.29	0.790242	0.395121	+	0.918629i	$0.370703\pi$
$360$	0	0
$361$	4891.23	0.713112
$362$	1785.17	0.259190
$363$	0	0
$364$	2231.58	0.321337
$365$	−10142.4	−1.45446
$366$	0	0
$367$	8231.21	1.17075	0.585376	−	0.810762i	$-0.300947\pi$
$367$	8231.21	1.17075	0.585376	+	0.810762i	$0.300947\pi$
$368$	123.990	0.0175636
$369$	0	0
$370$	−1553.49	−0.218276
$371$	−1954.39	−0.273495
$372$	0	0
$373$	7401.41	1.02743	0.513713	−	0.857962i	$-0.328269\pi$
$373$	7401.41	1.02743	0.513713	+	0.857962i	$0.328269\pi$
$374$	277.173	0.0383216
$375$	0	0
$376$	−35.9698	−0.00493352
$377$	−19860.9	−2.71323
$378$	0	0
$379$	−1455.62	−0.197283	−0.0986417	−	0.995123i	$-0.531450\pi$
$379$	−1455.62	−0.197283	−0.0986417	+	0.995123i	$0.531450\pi$
$380$	6915.77	0.933609
$381$	0	0
$382$	2275.49	0.304775
$383$	14034.0	1.87233	0.936167	−	0.351555i	$-0.114347\pi$
$383$	14034.0	1.87233	0.936167	+	0.351555i	$0.114347\pi$
$384$	0	0
$385$	228.895	0.0303001
$386$	8039.49	1.06010
$387$	0	0
$388$	−6680.78	−0.874137
$389$	6141.24	0.800446	0.400223	−	0.916418i	$-0.368933\pi$
$389$	6141.24	0.800446	0.400223	+	0.916418i	$0.368933\pi$
$390$	0	0
$391$	523.850	0.0677550
$392$	−392.000	−0.0505076
$393$	0	0
$394$	5629.77	0.719858
$395$	7657.62	0.975434
$396$	0	0
$397$	−4566.30	−0.577270	−0.288635	−	0.957439i	$-0.593201\pi$
$397$	−4566.30	−0.577270	−0.288635	+	0.957439i	$0.593201\pi$
$398$	6878.51	0.866303
$399$	0	0
$400$	2070.38	0.258797
$401$	10747.5	1.33841	0.669205	−	0.743078i	$-0.266635\pi$
$401$	10747.5	1.33841	0.669205	+	0.743078i	$0.266635\pi$
$402$	0	0
$403$	422.466	0.0522197
$404$	−5209.58	−0.641551
$405$	0	0
$406$	3488.77	0.426465
$407$	99.8396	0.0121594
$408$	0	0
$409$	8753.85	1.05831	0.529156	−	0.848524i	$-0.322508\pi$
$409$	8753.85	1.05831	0.529156	+	0.848524i	$0.322508\pi$
$410$	1437.01	0.173094
$411$	0	0
$412$	−7209.95	−0.862158
$413$	−4727.05	−0.563204
$414$	0	0
$415$	16032.8	1.89643
$416$	−2550.38	−0.300583
$417$	0	0
$418$	−444.461	−0.0520079
$419$	−5449.91	−0.635431	−0.317716	−	0.948186i	$-0.602916\pi$
$419$	−5449.91	−0.635431	−0.317716	+	0.948186i	$0.602916\pi$
$420$	0	0
$421$	3637.17	0.421056	0.210528	−	0.977588i	$-0.432482\pi$
$421$	3637.17	0.421056	0.210528	+	0.977588i	$0.432482\pi$
$422$	−2063.13	−0.237989
$423$	0	0
$424$	2233.58	0.255831
$425$	8747.21	0.998357
$426$	0	0
$427$	−5895.37	−0.668143
$428$	2500.37	0.282383
$429$	0	0
$430$	−6434.16	−0.721587
$431$	6180.92	0.690776	0.345388	−	0.938460i	$-0.387747\pi$
$431$	6180.92	0.690776	0.345388	+	0.938460i	$0.387747\pi$
$432$	0	0
$433$	−30.6766	−0.00340468	−0.00170234	−	0.999999i	$-0.500542\pi$
$433$	−30.6766	−0.00340468	−0.00170234	+	0.999999i	$0.500542\pi$
$434$	−74.2106	−0.00820788
$435$	0	0
$436$	1199.61	0.131768
$437$	−840.020	−0.0919534
$438$	0	0
$439$	−5639.22	−0.613087	−0.306544	−	0.951857i	$-0.599172\pi$
$439$	−5639.22	−0.613087	−0.306544	+	0.951857i	$0.599172\pi$
$440$	−261.594	−0.0283432
$441$	0	0
$442$	−10775.2	−1.15955
$443$	−12967.2	−1.39072	−0.695360	−	0.718662i	$-0.744755\pi$
$443$	−12967.2	−1.39072	−0.695360	+	0.718662i	$0.744755\pi$
$444$	0	0
$445$	21971.6	2.34057
$446$	12586.9	1.33634
$447$	0	0
$448$	448.000	0.0472456
$449$	−13592.0	−1.42861	−0.714305	−	0.699834i	$-0.753257\pi$
$449$	−13592.0	−1.42861	−0.714305	+	0.699834i	$0.753257\pi$
$450$	0	0
$451$	−92.3533	−0.00964245
$452$	5968.40	0.621084
$453$	0	0
$454$	12012.2	1.24176
$455$	−8898.35	−0.916838
$456$	0	0
$457$	16278.1	1.66621	0.833105	−	0.553115i	$-0.186561\pi$
$457$	16278.1	1.66621	0.833105	+	0.553115i	$0.186561\pi$
$458$	−8909.11	−0.908942
$459$	0	0
$460$	−494.406	−0.0501126
$461$	−11133.8	−1.12484	−0.562422	−	0.826850i	$-0.690130\pi$
$461$	−11133.8	−1.12484	−0.562422	+	0.826850i	$0.690130\pi$
$462$	0	0
$463$	460.887	0.0462618	0.0231309	−	0.999732i	$-0.492637\pi$
$463$	460.887	0.0462618	0.0231309	+	0.999732i	$0.492637\pi$
$464$	−3987.17	−0.398922
$465$	0	0
$466$	5254.90	0.522379
$467$	2325.50	0.230431	0.115215	−	0.993341i	$-0.463244\pi$
$467$	2325.50	0.230431	0.115215	+	0.993341i	$0.463244\pi$
$468$	0	0
$469$	2309.26	0.227360
$470$	143.429	0.0140763
$471$	0	0
$472$	5402.35	0.526829
$473$	413.509	0.0401969
$474$	0	0
$475$	−14026.6	−1.35491
$476$	1892.77	0.182259
$477$	0	0
$478$	−3197.73	−0.305985
$479$	16893.1	1.61141	0.805705	−	0.592317i	$-0.201787\pi$
$479$	16893.1	1.61141	0.805705	+	0.592317i	$0.201787\pi$
$480$	0	0
$481$	−3881.29	−0.367925
$482$	−5291.82	−0.500074
$483$	0	0
$484$	−5307.19	−0.498421
$485$	26639.4	2.49409
$486$	0	0
$487$	−7055.38	−0.656489	−0.328244	−	0.944593i	$-0.606457\pi$
$487$	−7055.38	−0.656489	−0.328244	+	0.944593i	$0.606457\pi$
$488$	6737.56	0.624990
$489$	0	0
$490$	1563.09	0.144108
$491$	3366.07	0.309386	0.154693	−	0.987963i	$-0.450561\pi$
$491$	3366.07	0.309386	0.154693	+	0.987963i	$0.450561\pi$
$492$	0	0
$493$	−16845.5	−1.53891
$494$	17278.6	1.57368
$495$	0	0
$496$	84.8121	0.00767777
$497$	5681.47	0.512774
$498$	0	0
$499$	−9503.47	−0.852573	−0.426286	−	0.904588i	$-0.640178\pi$
$499$	−9503.47	−0.852573	−0.426286	+	0.904588i	$0.640178\pi$
$500$	−280.622	−0.0250996
$501$	0	0
$502$	6423.58	0.571112
$503$	14215.6	1.26012	0.630062	−	0.776545i	$-0.283030\pi$
$503$	14215.6	1.26012	0.630062	+	0.776545i	$0.283030\pi$
$504$	0	0
$505$	20773.1	1.83047
$506$	31.7744	0.00279159
$507$	0	0
$508$	−7185.92	−0.627606
$509$	−22265.9	−1.93894	−0.969470	−	0.245211i	$-0.921143\pi$
$509$	−22265.9	−1.93894	−0.969470	+	0.245211i	$0.921143\pi$
$510$	0	0
$511$	4451.26	0.385347
$512$	−512.000	−0.0441942
$513$	0	0
$514$	−319.168	−0.0273889
$515$	28749.5	2.45991
$516$	0	0
$517$	−9.21784	−0.000784139 0
$518$	681.789	0.0578303
$519$	0	0
$520$	10169.5	0.857623
$521$	8414.11	0.707541	0.353771	−	0.935332i	$-0.384899\pi$
$521$	8414.11	0.707541	0.353771	+	0.935332i	$0.384899\pi$
$522$	0	0
$523$	−21008.8	−1.75650	−0.878250	−	0.478202i	$-0.841289\pi$
$523$	−21008.8	−1.75650	−0.878250	+	0.478202i	$0.841289\pi$
$524$	6974.82	0.581481
$525$	0	0
$526$	−6034.31	−0.500206
$527$	358.326	0.0296184
$528$	0	0
$529$	−12106.9	−0.995064
$530$	−8906.35	−0.729938
$531$	0	0
$532$	−3035.16	−0.247351
$533$	3590.26	0.291766
$534$	0	0
$535$	−9970.15	−0.805696
$536$	−2639.16	−0.212676
$537$	0	0
$538$	6141.67	0.492168
$539$	−100.456	−0.00802774
$540$	0	0
$541$	4390.79	0.348937	0.174468	−	0.984663i	$-0.444179\pi$
$541$	4390.79	0.348937	0.174468	+	0.984663i	$0.444179\pi$
$542$	−9924.33	−0.786506
$543$	0	0
$544$	−2163.17	−0.170487
$545$	−4783.40	−0.375961
$546$	0	0
$547$	−5270.11	−0.411944	−0.205972	−	0.978558i	$-0.566036\pi$
$547$	−5270.11	−0.411944	−0.205972	+	0.978558i	$0.566036\pi$
$548$	8952.76	0.697889
$549$	0	0
$550$	530.566	0.0411335
$551$	27012.7	2.08853
$552$	0	0
$553$	−3360.74	−0.258432
$554$	−5242.57	−0.402050
$555$	0	0
$556$	−1996.03	−0.152249
$557$	14664.7	1.11556	0.557778	−	0.829990i	$-0.311654\pi$
$557$	14664.7	1.11556	0.557778	+	0.829990i	$0.311654\pi$
$558$	0	0
$559$	−16075.3	−1.21630
$560$	−1786.39	−0.134801
$561$	0	0
$562$	5017.01	0.376566
$563$	−20513.9	−1.53563	−0.767814	−	0.640673i	$-0.778655\pi$
$563$	−20513.9	−1.53563	−0.767814	+	0.640673i	$0.778655\pi$
$564$	0	0
$565$	−23798.8	−1.77208
$566$	8404.33	0.624135
$567$	0	0
$568$	−6493.11	−0.479657
$569$	−8097.17	−0.596575	−0.298287	−	0.954476i	$-0.596415\pi$
$569$	−8097.17	−0.596575	−0.298287	+	0.954476i	$0.596415\pi$
$570$	0	0
$571$	−6542.87	−0.479528	−0.239764	−	0.970831i	$-0.577070\pi$
$571$	−6542.87	−0.479528	−0.239764	+	0.970831i	$0.577070\pi$
$572$	−653.574	−0.0477750
$573$	0	0
$574$	−630.667	−0.0458598
$575$	1002.76	0.0727267
$576$	0	0
$577$	7609.31	0.549012	0.274506	−	0.961585i	$-0.411486\pi$
$577$	7609.31	0.549012	0.274506	+	0.961585i	$0.411486\pi$
$578$	686.752	0.0494206
$579$	0	0
$580$	15898.7	1.13820
$581$	−7036.39	−0.502441
$582$	0	0
$583$	572.391	0.0406621
$584$	−5087.16	−0.360459
$585$	0	0
$586$	−3314.67	−0.233665
$587$	3654.67	0.256975	0.128487	−	0.991711i	$-0.458988\pi$
$587$	3654.67	0.256975	0.128487	+	0.991711i	$0.458988\pi$
$588$	0	0
$589$	−574.594	−0.0401965
$590$	−21541.7	−1.50315
$591$	0	0
$592$	−779.188	−0.0540953
$593$	−18996.5	−1.31551	−0.657753	−	0.753234i	$-0.728493\pi$
$593$	−18996.5	−1.31551	−0.657753	+	0.753234i	$0.728493\pi$
$594$	0	0
$595$	−7547.37	−0.520020
$596$	11830.0	0.813045
$597$	0	0
$598$	−1235.24	−0.0844693
$599$	9942.89	0.678223	0.339111	−	0.940746i	$-0.389874\pi$
$599$	9942.89	0.678223	0.339111	+	0.940746i	$0.389874\pi$
$600$	0	0
$601$	10272.3	0.697196	0.348598	−	0.937272i	$-0.386658\pi$
$601$	10272.3	0.697196	0.348598	+	0.937272i	$0.386658\pi$
$602$	2823.79	0.191178
$603$	0	0
$604$	5799.16	0.390670
$605$	21162.2	1.42210
$606$	0	0
$607$	−4119.52	−0.275463	−0.137732	−	0.990470i	$-0.543981\pi$
$607$	−4119.52	−0.275463	−0.137732	+	0.990470i	$0.543981\pi$
$608$	3468.75	0.231376
$609$	0	0
$610$	−26865.8	−1.78322
$611$	358.346	0.0237269
$612$	0	0
$613$	19983.8	1.31670	0.658351	−	0.752711i	$-0.271254\pi$
$613$	19983.8	1.31670	0.658351	+	0.752711i	$0.271254\pi$
$614$	−15517.4	−1.01992
$615$	0	0
$616$	114.807	0.00750927
$617$	18882.8	1.23208	0.616040	−	0.787715i	$-0.288736\pi$
$617$	18882.8	1.23208	0.616040	+	0.787715i	$0.288736\pi$
$618$	0	0
$619$	−2651.29	−0.172155	−0.0860777	−	0.996288i	$-0.527433\pi$
$619$	−2651.29	−0.172155	−0.0860777	+	0.996288i	$0.527433\pi$
$620$	−338.185	−0.0219062
$621$	0	0
$622$	21790.7	1.40471
$623$	−9642.77	−0.620112
$624$	0	0
$625$	−15055.8	−0.963574
$626$	−3706.59	−0.236653
$627$	0	0
$628$	10710.7	0.680581
$629$	−3292.02	−0.208683
$630$	0	0
$631$	21458.8	1.35382	0.676911	−	0.736065i	$-0.263318\pi$
$631$	21458.8	1.35382	0.676911	+	0.736065i	$0.263318\pi$
$632$	3840.84	0.241741
$633$	0	0
$634$	−4990.11	−0.312591
$635$	28653.6	1.79069
$636$	0	0
$637$	3905.26	0.242908
$638$	−1021.77	−0.0634051
$639$	0	0
$640$	2041.58	0.126095
$641$	5188.19	0.319690	0.159845	−	0.987142i	$-0.448901\pi$
$641$	5188.19	0.319690	0.159845	+	0.987142i	$0.448901\pi$
$642$	0	0
$643$	3193.92	0.195888	0.0979439	−	0.995192i	$-0.468773\pi$
$643$	3193.92	0.195888	0.0979439	+	0.995192i	$0.468773\pi$
$644$	216.982	0.0132769
$645$	0	0
$646$	14655.3	0.892575
$647$	−14571.4	−0.885410	−0.442705	−	0.896667i	$-0.645981\pi$
$647$	−14571.4	−0.885410	−0.442705	+	0.896667i	$0.645981\pi$
$648$	0	0
$649$	1384.44	0.0837348
$650$	−20625.9	−1.24464
$651$	0	0
$652$	−2719.64	−0.163358
$653$	−15946.0	−0.955612	−0.477806	−	0.878465i	$-0.658568\pi$
$653$	−15946.0	−0.955612	−0.477806	+	0.878465i	$0.658568\pi$
$654$	0	0
$655$	−27811.9	−1.65908
$656$	720.762	0.0428979
$657$	0	0
$658$	−62.9472	−0.00372939
$659$	7232.28	0.427511	0.213755	−	0.976887i	$-0.431430\pi$
$659$	7232.28	0.427511	0.213755	+	0.976887i	$0.431430\pi$
$660$	0	0
$661$	2173.87	0.127918	0.0639588	−	0.997953i	$-0.479627\pi$
$661$	2173.87	0.127918	0.0639588	+	0.997953i	$0.479627\pi$
$662$	19669.8	1.15481
$663$	0	0
$664$	8041.58	0.469991
$665$	12102.6	0.705742
$666$	0	0
$667$	−1931.13	−0.112104
$668$	1571.65	0.0910313
$669$	0	0
$670$	10523.6	0.606807
$671$	1726.61	0.0993367
$672$	0	0
$673$	22215.7	1.27244	0.636221	−	0.771507i	$-0.280497\pi$
$673$	22215.7	1.27244	0.636221	+	0.771507i	$0.280497\pi$
$674$	9902.99	0.565948
$675$	0	0
$676$	16619.9	0.945601
$677$	3258.01	0.184957	0.0924783	−	0.995715i	$-0.470521\pi$
$677$	3258.01	0.184957	0.0924783	+	0.995715i	$0.470521\pi$
$678$	0	0
$679$	−11691.4	−0.660786
$680$	8625.56	0.486434
$681$	0	0
$682$	21.7344	0.00122031
$683$	−13600.9	−0.761968	−0.380984	−	0.924582i	$-0.624415\pi$
$683$	−13600.9	−0.761968	−0.380984	+	0.924582i	$0.624415\pi$
$684$	0	0
$685$	−35698.9	−1.99122
$686$	−686.000	−0.0381802
$687$	0	0
$688$	−3227.19	−0.178831
$689$	−22251.9	−1.23038
$690$	0	0
$691$	29639.3	1.63174	0.815869	−	0.578236i	$-0.196259\pi$
$691$	29639.3	1.63174	0.815869	+	0.578236i	$0.196259\pi$
$692$	3347.37	0.183884
$693$	0	0
$694$	24092.5	1.31778
$695$	7959.11	0.434397
$696$	0	0
$697$	3045.17	0.165487
$698$	6177.17	0.334971
$699$	0	0
$700$	3623.16	0.195632
$701$	−3291.05	−0.177320	−0.0886600	−	0.996062i	$-0.528258\pi$
$701$	−3291.05	−0.177320	−0.0886600	+	0.996062i	$0.528258\pi$
$702$	0	0
$703$	5278.92	0.283213
$704$	−131.208	−0.00702427
$705$	0	0
$706$	8999.35	0.479738
$707$	−9116.77	−0.484967
$708$	0	0
$709$	25692.3	1.36092	0.680461	−	0.732785i	$-0.261780\pi$
$709$	25692.3	1.36092	0.680461	+	0.732785i	$0.261780\pi$
$710$	25891.1	1.36856
$711$	0	0
$712$	11020.3	0.580061
$713$	41.0775	0.00215759
$714$	0	0
$715$	2606.11	0.136312
$716$	5669.25	0.295908
$717$	0	0
$718$	−10750.6	−0.558785
$719$	−28150.4	−1.46013	−0.730063	−	0.683380i	$-0.760509\pi$
$719$	−28150.4	−1.46013	−0.730063	+	0.683380i	$0.760509\pi$
$720$	0	0
$721$	−12617.4	−0.651730
$722$	−9782.47	−0.504246
$723$	0	0
$724$	−3570.35	−0.183275
$725$	−32245.8	−1.65184
$726$	0	0
$727$	7451.10	0.380118	0.190059	−	0.981773i	$-0.439132\pi$
$727$	7451.10	0.380118	0.190059	+	0.981773i	$0.439132\pi$
$728$	−4463.16	−0.227219
$729$	0	0
$730$	20284.9	1.02846
$731$	−13634.7	−0.689872
$732$	0	0
$733$	−10090.3	−0.508450	−0.254225	−	0.967145i	$-0.581820\pi$
$733$	−10090.3	−0.508450	−0.254225	+	0.967145i	$0.581820\pi$
$734$	−16462.4	−0.827846
$735$	0	0
$736$	−247.980	−0.0124194
$737$	−676.326	−0.0338030
$738$	0	0
$739$	−32862.3	−1.63580	−0.817902	−	0.575357i	$-0.804863\pi$
$739$	−32862.3	−1.63580	−0.817902	+	0.575357i	$0.804863\pi$
$740$	3106.99	0.154345
$741$	0	0
$742$	3908.77	0.193390
$743$	−5970.91	−0.294820	−0.147410	−	0.989075i	$-0.547094\pi$
$743$	−5970.91	−0.294820	−0.147410	+	0.989075i	$0.547094\pi$
$744$	0	0
$745$	−47171.7	−2.31978
$746$	−14802.8	−0.726501
$747$	0	0
$748$	−554.346	−0.0270974
$749$	4375.65	0.213462
$750$	0	0
$751$	660.406	0.0320886	0.0160443	−	0.999871i	$-0.494893\pi$
$751$	660.406	0.0320886	0.0160443	+	0.999871i	$0.494893\pi$
$752$	71.9397	0.00348853
$753$	0	0
$754$	39721.8	1.91854
$755$	−23124.0	−1.11466
$756$	0	0
$757$	28659.6	1.37603	0.688013	−	0.725698i	$-0.258483\pi$
$757$	28659.6	1.37603	0.688013	+	0.725698i	$0.258483\pi$
$758$	2911.25	0.139500
$759$	0	0
$760$	−13831.5	−0.660161
$761$	27158.8	1.29370	0.646849	−	0.762618i	$-0.276086\pi$
$761$	27158.8	1.29370	0.646849	+	0.762618i	$0.276086\pi$
$762$	0	0
$763$	2099.32	0.0996073
$764$	−4550.98	−0.215509
$765$	0	0
$766$	−28068.0	−1.32394
$767$	−53820.4	−2.53369
$768$	0	0
$769$	29962.0	1.40502	0.702508	−	0.711676i	$-0.252064\pi$
$769$	29962.0	1.40502	0.702508	+	0.711676i	$0.252064\pi$
$770$	−457.789	−0.0214254
$771$	0	0
$772$	−16079.0	−0.749605
$773$	24540.8	1.14188	0.570939	−	0.820993i	$-0.306579\pi$
$773$	24540.8	1.14188	0.570939	+	0.820993i	$0.306579\pi$
$774$	0	0
$775$	685.910	0.0317917
$776$	13361.6	0.618108
$777$	0	0
$778$	−12282.5	−0.566001
$779$	−4883.09	−0.224589
$780$	0	0
$781$	−1663.96	−0.0762372
$782$	−1047.70	−0.0479100
$783$	0	0
$784$	784.000	0.0357143
$785$	−42708.7	−1.94183
$786$	0	0
$787$	27048.5	1.22513	0.612564	−	0.790421i	$-0.290138\pi$
$787$	27048.5	1.22513	0.612564	+	0.790421i	$0.290138\pi$
$788$	−11259.5	−0.509016
$789$	0	0
$790$	−15315.2	−0.689736
$791$	10444.7	0.469495
$792$	0	0
$793$	−67122.3	−3.00578
$794$	9132.60	0.408191
$795$	0	0
$796$	−13757.0	−0.612569
$797$	−2332.95	−0.103686	−0.0518428	−	0.998655i	$-0.516509\pi$
$797$	−2332.95	−0.103686	−0.0518428	+	0.998655i	$0.516509\pi$
$798$	0	0
$799$	303.941	0.0134576
$800$	−4140.75	−0.182997
$801$	0	0
$802$	−21494.9	−0.946399
$803$	−1303.66	−0.0572918
$804$	0	0
$805$	−865.211	−0.0378816
$806$	−844.932	−0.0369249
$807$	0	0
$808$	10419.2	0.453645
$809$	41284.1	1.79415	0.897077	−	0.441875i	$-0.145686\pi$
$809$	41284.1	1.79415	0.897077	+	0.441875i	$0.145686\pi$
$810$	0	0
$811$	8472.53	0.366844	0.183422	−	0.983034i	$-0.441282\pi$
$811$	8472.53	0.366844	0.183422	+	0.983034i	$0.441282\pi$
$812$	−6977.54	−0.301556
$813$	0	0
$814$	−199.679	−0.00859797
$815$	10844.5	0.466093
$816$	0	0
$817$	21863.9	0.936256
$818$	−17507.7	−0.748340
$819$	0	0
$820$	−2874.02	−0.122396
$821$	24592.8	1.04543	0.522713	−	0.852509i	$-0.324920\pi$
$821$	24592.8	1.04543	0.522713	+	0.852509i	$0.324920\pi$
$822$	0	0
$823$	−3781.52	−0.160165	−0.0800823	−	0.996788i	$-0.525518\pi$
$823$	−3781.52	−0.160165	−0.0800823	+	0.996788i	$0.525518\pi$
$824$	14419.9	0.609638
$825$	0	0
$826$	9454.11	0.398245
$827$	−7970.45	−0.335139	−0.167569	−	0.985860i	$-0.553592\pi$
$827$	−7970.45	−0.335139	−0.167569	+	0.985860i	$0.553592\pi$
$828$	0	0
$829$	11793.9	0.494113	0.247056	−	0.969001i	$-0.420537\pi$
$829$	11793.9	0.494113	0.247056	+	0.969001i	$0.420537\pi$
$830$	−32065.6	−1.34098
$831$	0	0
$832$	5100.75	0.212544
$833$	3312.35	0.137774
$834$	0	0
$835$	−6266.90	−0.259731
$836$	888.922	0.0367751
$837$	0	0
$838$	10899.8	0.449318
$839$	−26401.9	−1.08641	−0.543204	−	0.839601i	$-0.682789\pi$
$839$	−26401.9	−1.08641	−0.543204	+	0.839601i	$0.682789\pi$
$840$	0	0
$841$	37710.6	1.54622
$842$	−7274.33	−0.297732
$843$	0	0
$844$	4126.26	0.168284
$845$	−66271.2	−2.69799
$846$	0	0
$847$	−9287.58	−0.376771
$848$	−4467.17	−0.180900
$849$	0	0
$850$	−17494.4	−0.705945
$851$	−377.389	−0.0152018
$852$	0	0
$853$	608.360	0.0244195	0.0122098	−	0.999925i	$-0.496113\pi$
$853$	608.360	0.0244195	0.0122098	+	0.999925i	$0.496113\pi$
$854$	11790.7	0.472448
$855$	0	0
$856$	−5000.74	−0.199675
$857$	−27737.7	−1.10560	−0.552801	−	0.833313i	$-0.686441\pi$
$857$	−27737.7	−1.10560	−0.552801	+	0.833313i	$0.686441\pi$
$858$	0	0
$859$	−40772.5	−1.61949	−0.809745	−	0.586782i	$-0.800394\pi$
$859$	−40772.5	−1.61949	−0.809745	+	0.586782i	$0.800394\pi$
$860$	12868.3	0.510239
$861$	0	0
$862$	−12361.8	−0.488453
$863$	−37513.6	−1.47969	−0.739847	−	0.672775i	$-0.765102\pi$
$863$	−37513.6	−1.47969	−0.739847	+	0.672775i	$0.765102\pi$
$864$	0	0
$865$	−13347.5	−0.524658
$866$	61.3533	0.00240747
$867$	0	0
$868$	148.421	0.00580385
$869$	984.276	0.0384226
$870$	0	0
$871$	26292.4	1.02283
$872$	−2399.22	−0.0931741
$873$	0	0
$874$	1680.04	0.0650208
$875$	−491.088	−0.0189735
$876$	0	0
$877$	14573.5	0.561133	0.280567	−	0.959835i	$-0.409478\pi$
$877$	14573.5	0.561133	0.280567	+	0.959835i	$0.409478\pi$
$878$	11278.4	0.433518
$879$	0	0
$880$	523.188	0.0200417
$881$	2286.87	0.0874537	0.0437268	−	0.999044i	$-0.486077\pi$
$881$	2286.87	0.0874537	0.0437268	+	0.999044i	$0.486077\pi$
$882$	0	0
$883$	−32853.1	−1.25209	−0.626045	−	0.779787i	$-0.715328\pi$
$883$	−32853.1	−1.25209	−0.626045	+	0.779787i	$0.715328\pi$
$884$	21550.4	0.819929
$885$	0	0
$886$	25934.3	0.983388
$887$	8217.48	0.311066	0.155533	−	0.987831i	$-0.450290\pi$
$887$	8217.48	0.311066	0.155533	+	0.987831i	$0.450290\pi$
$888$	0	0
$889$	−12575.4	−0.474426
$890$	−43943.1	−1.65503
$891$	0	0
$892$	−25173.9	−0.944936
$893$	−487.385	−0.0182639
$894$	0	0
$895$	−22606.0	−0.844284
$896$	−896.000	−0.0334077
$897$	0	0
$898$	27184.0	1.01018
$899$	−1320.94	−0.0490053
$900$	0	0
$901$	−18873.5	−0.697855
$902$	184.707	0.00681824
$903$	0	0
$904$	−11936.8	−0.439173
$905$	14236.6	0.522919
$906$	0	0
$907$	47048.2	1.72239	0.861197	−	0.508272i	$-0.169716\pi$
$907$	47048.2	1.72239	0.861197	+	0.508272i	$0.169716\pi$
$908$	−24024.4	−0.878058
$909$	0	0
$910$	17796.7	0.648302
$911$	46401.3	1.68753	0.843767	−	0.536709i	$-0.180333\pi$
$911$	46401.3	1.68753	0.843767	+	0.536709i	$0.180333\pi$
$912$	0	0
$913$	2060.78	0.0747009
$914$	−32556.2	−1.17819
$915$	0	0
$916$	17818.2	0.642719
$917$	12205.9	0.439559
$918$	0	0
$919$	19749.6	0.708901	0.354450	−	0.935075i	$-0.384668\pi$
$919$	19749.6	0.708901	0.354450	+	0.935075i	$0.384668\pi$
$920$	988.812	0.0354350
$921$	0	0
$922$	22267.6	0.795385
$923$	64687.0	2.30682
$924$	0	0
$925$	−6301.61	−0.223995
$926$	−921.774	−0.0327121
$927$	0	0
$928$	7974.34	0.282080
$929$	−41327.0	−1.45952	−0.729761	−	0.683702i	$-0.760369\pi$
$929$	−41327.0	−1.45952	−0.729761	+	0.683702i	$0.760369\pi$
$930$	0	0
$931$	−5311.53	−0.186980
$932$	−10509.8	−0.369377
$933$	0	0
$934$	−4651.00	−0.162939
$935$	2210.44	0.0773144
$936$	0	0
$937$	−25655.9	−0.894494	−0.447247	−	0.894410i	$-0.647596\pi$
$937$	−25655.9	−0.894494	−0.447247	+	0.894410i	$0.647596\pi$
$938$	−4618.53	−0.160768
$939$	0	0
$940$	−286.857	−0.00995346
$941$	5752.06	0.199269	0.0996343	−	0.995024i	$-0.468233\pi$
$941$	5752.06	0.199269	0.0996343	+	0.995024i	$0.468233\pi$
$942$	0	0
$943$	349.091	0.0120551
$944$	−10804.7	−0.372524
$945$	0	0
$946$	−827.018	−0.0284235
$947$	−43982.1	−1.50922	−0.754608	−	0.656176i	$-0.772173\pi$
$947$	−43982.1	−1.50922	−0.754608	+	0.656176i	$0.772173\pi$
$948$	0	0
$949$	50680.3	1.73356
$950$	28053.2	0.958070
$951$	0	0
$952$	−3785.54	−0.128876
$953$	54822.7	1.86346	0.931731	−	0.363148i	$-0.118298\pi$
$953$	54822.7	1.86346	0.931731	+	0.363148i	$0.118298\pi$
$954$	0	0
$955$	18146.9	0.614889
$956$	6395.47	0.216364
$957$	0	0
$958$	−33786.2	−1.13944
$959$	15667.3	0.527554
$960$	0	0
$961$	−29762.9	−0.999057
$962$	7762.59	0.260162
$963$	0	0
$964$	10583.6	0.353606
$965$	64114.4	2.13877
$966$	0	0
$967$	2879.81	0.0957687	0.0478843	−	0.998853i	$-0.484752\pi$
$967$	2879.81	0.0957687	0.0478843	+	0.998853i	$0.484752\pi$
$968$	10614.4	0.352437
$969$	0	0
$970$	−53278.8	−1.76359
$971$	1582.10	0.0522884	0.0261442	−	0.999658i	$-0.491677\pi$
$971$	1582.10	0.0522884	0.0261442	+	0.999658i	$0.491677\pi$
$972$	0	0
$973$	−3493.05	−0.115090
$974$	14110.8	0.464208
$975$	0	0
$976$	−13475.1	−0.441935
$977$	−13931.5	−0.456199	−0.228100	−	0.973638i	$-0.573251\pi$
$977$	−13931.5	−0.456199	−0.228100	+	0.973638i	$0.573251\pi$
$978$	0	0
$979$	2824.13	0.0921956
$980$	−3126.18	−0.101900
$981$	0	0
$982$	−6732.14	−0.218769
$983$	−27868.4	−0.904237	−0.452118	−	0.891958i	$-0.649332\pi$
$983$	−27868.4	−0.904237	−0.452118	+	0.891958i	$0.649332\pi$
$984$	0	0
$985$	44897.1	1.45233
$986$	33691.1	1.08818
$987$	0	0
$988$	−34557.1	−1.11276
$989$	−1563.04	−0.0502547
$990$	0	0
$991$	19501.0	0.625095	0.312548	−	0.949902i	$-0.398818\pi$
$991$	19501.0	0.625095	0.312548	+	0.949902i	$0.398818\pi$
$992$	−169.624	−0.00542900
$993$	0	0
$994$	−11362.9	−0.362586
$995$	54855.7	1.74778
$996$	0	0
$997$	−6984.47	−0.221866	−0.110933	−	0.993828i	$-0.535384\pi$
$997$	−6984.47	−0.221866	−0.110933	+	0.993828i	$0.535384\pi$
$998$	19006.9	0.602860
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.4.a.m.1.1	✓	2
3.2	odd	2		378.4.a.r.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.4.a.m.1.1	✓	2	1.1	even	1	trivial
378.4.a.r.1.2	yes	2	3.2	odd	2

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	378.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -15.9499 q^{5} +7.00000 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -15.9499 q^{5} +7.00000 q^{7} -8.00000 q^{8} +31.8997 q^{10} -2.05013 q^{11} +79.6992 q^{13} -14.0000 q^{14} +16.0000 q^{16} +67.5990 q^{17} -108.398 q^{19} -63.7995 q^{20} +4.10025 q^{22} +7.74937 q^{23} +129.398 q^{25} -159.398 q^{26} +28.0000 q^{28} -249.198 q^{29} +5.30075 q^{31} -32.0000 q^{32} -135.198 q^{34} -111.649 q^{35} -48.6992 q^{37} +216.797 q^{38} +127.599 q^{40} +45.0476 q^{41} -201.699 q^{43} -8.20050 q^{44} -15.4987 q^{46} +4.49623 q^{47} +49.0000 q^{49} -258.797 q^{50} +318.797 q^{52} -279.198 q^{53} +32.6992 q^{55} -56.0000 q^{56} +498.396 q^{58} -675.293 q^{59} -842.195 q^{61} -10.6015 q^{62} +64.0000 q^{64} -1271.19 q^{65} +329.895 q^{67} +270.396 q^{68} +223.298 q^{70} +811.639 q^{71} +635.895 q^{73} +97.3985 q^{74} -433.594 q^{76} -14.3509 q^{77} -480.105 q^{79} -255.198 q^{80} -90.0952 q^{82} -1005.20 q^{83} -1078.20 q^{85} +403.398 q^{86} +16.4010 q^{88} -1377.54 q^{89} +557.895 q^{91} +30.9975 q^{92} -8.99246 q^{94} +1728.94 q^{95} -1670.20 q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 12 q^{5} + 14 q^{7} - 16 q^{8}+O(q^{10}) \) 2 * q - 4 * q^2 + 8 * q^4 - 12 * q^5 + 14 * q^7 - 16 * q^8	\( 2 q - 4 q^{2} + 8 q^{4} - 12 q^{5} + 14 q^{7} - 16 q^{8} + 24 q^{10} - 24 q^{11} + 40 q^{13} - 28 q^{14} + 32 q^{16} - 24 q^{17} + 22 q^{19} - 48 q^{20} + 48 q^{22} - 84 q^{23} + 20 q^{25} - 80 q^{26} + 56 q^{28} - 180 q^{29} + 130 q^{31} - 64 q^{32} + 48 q^{34} - 84 q^{35} + 22 q^{37} - 44 q^{38} + 96 q^{40} - 288 q^{41} - 284 q^{43} - 96 q^{44} + 168 q^{46} - 588 q^{47} + 98 q^{49} - 40 q^{50} + 160 q^{52} - 240 q^{53} - 54 q^{55} - 112 q^{56} + 360 q^{58} - 276 q^{59} - 968 q^{61} - 260 q^{62} + 128 q^{64} - 1428 q^{65} - 176 q^{67} - 96 q^{68} + 168 q^{70} - 108 q^{71} + 436 q^{73} - 44 q^{74} + 88 q^{76} - 168 q^{77} - 1796 q^{79} - 192 q^{80} + 576 q^{82} - 1692 q^{83} - 1440 q^{85} + 568 q^{86} + 192 q^{88} - 984 q^{89} + 280 q^{91} - 336 q^{92} + 1176 q^{94} + 2244 q^{95} - 2624 q^{97} - 196 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 12 * q^5 + 14 * q^7 - 16 * q^8 + 24 * q^10 - 24 * q^11 + 40 * q^13 - 28 * q^14 + 32 * q^16 - 24 * q^17 + 22 * q^19 - 48 * q^20 + 48 * q^22 - 84 * q^23 + 20 * q^25 - 80 * q^26 + 56 * q^28 - 180 * q^29 + 130 * q^31 - 64 * q^32 + 48 * q^34 - 84 * q^35 + 22 * q^37 - 44 * q^38 + 96 * q^40 - 288 * q^41 - 284 * q^43 - 96 * q^44 + 168 * q^46 - 588 * q^47 + 98 * q^49 - 40 * q^50 + 160 * q^52 - 240 * q^53 - 54 * q^55 - 112 * q^56 + 360 * q^58 - 276 * q^59 - 968 * q^61 - 260 * q^62 + 128 * q^64 - 1428 * q^65 - 176 * q^67 - 96 * q^68 + 168 * q^70 - 108 * q^71 + 436 * q^73 - 44 * q^74 + 88 * q^76 - 168 * q^77 - 1796 * q^79 - 192 * q^80 + 576 * q^82 - 1692 * q^83 - 1440 * q^85 + 568 * q^86 + 192 * q^88 - 984 * q^89 + 280 * q^91 - 336 * q^92 + 1176 * q^94 + 2244 * q^95 - 2624 * q^97 - 196 * q^98

Introduction

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