LMFDB - Embedded newform 315.4.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(1,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("315.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.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:	$18.5856016518$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.236068 q^{2} -7.94427 q^{4} +5.00000 q^{5} -7.00000 q^{7} +3.76393 q^{8} +O(q^{10})$	\(q-0.236068 q^{2} -7.94427 q^{4} +5.00000 q^{5} -7.00000 q^{7} +3.76393 q^{8} -1.18034 q^{10} +50.4721 q^{11} -80.9706 q^{13} +1.65248 q^{14} +62.6656 q^{16} -76.3870 q^{17} +4.13777 q^{19} -39.7214 q^{20} -11.9149 q^{22} +204.721 q^{23} +25.0000 q^{25} +19.1146 q^{26} +55.6099 q^{28} +91.1672 q^{29} +198.079 q^{31} -44.9048 q^{32} +18.0325 q^{34} -35.0000 q^{35} +155.666 q^{37} -0.976794 q^{38} +18.8197 q^{40} +156.885 q^{41} +354.217 q^{43} -400.964 q^{44} -48.3282 q^{46} +175.659 q^{47} +49.0000 q^{49} -5.90170 q^{50} +643.252 q^{52} -200.302 q^{53} +252.361 q^{55} -26.3475 q^{56} -21.5217 q^{58} +312.498 q^{59} -154.170 q^{61} -46.7601 q^{62} -490.724 q^{64} -404.853 q^{65} +734.715 q^{67} +606.839 q^{68} +8.26238 q^{70} +678.577 q^{71} -60.8003 q^{73} -36.7477 q^{74} -32.8715 q^{76} -353.305 q^{77} -1286.26 q^{79} +313.328 q^{80} -37.0356 q^{82} -116.170 q^{83} -381.935 q^{85} -83.6192 q^{86} +189.974 q^{88} +916.440 q^{89} +566.794 q^{91} -1626.36 q^{92} -41.4676 q^{94} +20.6888 q^{95} -1416.30 q^{97} -11.5673 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 2 q^{4} + 10 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 2 * q^4 + 10 * q^5 - 14 * q^7 + 12 * q^8	$ 2 q + 4 q^{2} + 2 q^{4} + 10 q^{5} - 14 q^{7} + 12 q^{8} + 20 q^{10} + 92 q^{11} + 8 q^{13} - 28 q^{14} + 18 q^{16} + 44 q^{17} - 108 q^{19} + 10 q^{20} + 164 q^{22} + 320 q^{23} + 50 q^{25} + 396 q^{26} - 14 q^{28} + 236 q^{29} - 60 q^{31} - 300 q^{32} + 528 q^{34} - 70 q^{35} + 204 q^{37} - 476 q^{38} + 60 q^{40} - 44 q^{41} + 136 q^{43} + 12 q^{44} + 440 q^{46} - 400 q^{47} + 98 q^{49} + 100 q^{50} + 1528 q^{52} - 16 q^{53} + 460 q^{55} - 84 q^{56} + 592 q^{58} + 464 q^{59} - 684 q^{61} - 1140 q^{62} - 1214 q^{64} + 40 q^{65} + 736 q^{67} + 1804 q^{68} - 140 q^{70} + 740 q^{71} + 424 q^{73} + 168 q^{74} - 1148 q^{76} - 644 q^{77} - 408 q^{79} + 90 q^{80} - 888 q^{82} - 608 q^{83} + 220 q^{85} - 1008 q^{86} + 532 q^{88} + 1332 q^{89} - 56 q^{91} - 480 q^{92} - 2480 q^{94} - 540 q^{95} - 2448 q^{97} + 196 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 2 * q^4 + 10 * q^5 - 14 * q^7 + 12 * q^8 + 20 * q^10 + 92 * q^11 + 8 * q^13 - 28 * q^14 + 18 * q^16 + 44 * q^17 - 108 * q^19 + 10 * q^20 + 164 * q^22 + 320 * q^23 + 50 * q^25 + 396 * q^26 - 14 * q^28 + 236 * q^29 - 60 * q^31 - 300 * q^32 + 528 * q^34 - 70 * q^35 + 204 * q^37 - 476 * q^38 + 60 * q^40 - 44 * q^41 + 136 * q^43 + 12 * q^44 + 440 * q^46 - 400 * q^47 + 98 * q^49 + 100 * q^50 + 1528 * q^52 - 16 * q^53 + 460 * q^55 - 84 * q^56 + 592 * q^58 + 464 * q^59 - 684 * q^61 - 1140 * q^62 - 1214 * q^64 + 40 * q^65 + 736 * q^67 + 1804 * q^68 - 140 * q^70 + 740 * q^71 + 424 * q^73 + 168 * q^74 - 1148 * q^76 - 644 * q^77 - 408 * q^79 + 90 * q^80 - 888 * q^82 - 608 * q^83 + 220 * q^85 - 1008 * q^86 + 532 * q^88 + 1332 * q^89 - 56 * q^91 - 480 * q^92 - 2480 * q^94 - 540 * q^95 - 2448 * 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 $

$2$	−0.236068	−0.0834626	−0.0417313	−	0.999129i	$-0.513287\pi$
$2$	−0.236068	−0.0834626	−0.0417313	+	0.999129i	$0.513287\pi$
$3$	0	0
$3$	0	0
$4$	−7.94427	−0.993034
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	3.76393	0.166344
$9$	0	0
$10$	−1.18034	−0.0373256
$11$	50.4721	1.38345	0.691724	−	0.722162i	$-0.256852\pi$
$11$	50.4721	1.38345	0.691724	+	0.722162i	$0.256852\pi$
$12$	0	0
$13$	−80.9706	−1.72748	−0.863738	−	0.503940i	$-0.831883\pi$
$13$	−80.9706	−1.72748	−0.863738	+	0.503940i	$0.831883\pi$
$14$	1.65248	0.0315459
$15$	0	0
$16$	62.6656	0.979150
$17$	−76.3870	−1.08980	−0.544899	−	0.838502i	$-0.683432\pi$
$17$	−76.3870	−1.08980	−0.544899	+	0.838502i	$0.683432\pi$
$18$	0	0
$19$	4.13777	0.0499615	0.0249808	−	0.999688i	$-0.492048\pi$
$19$	4.13777	0.0499615	0.0249808	+	0.999688i	$0.492048\pi$
$20$	−39.7214	−0.444098
$21$	0	0
$22$	−11.9149	−0.115466
$23$	204.721	1.85597	0.927986	−	0.372615i	$-0.121539\pi$
$23$	204.721	1.85597	0.927986	+	0.372615i	$0.121539\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	19.1146	0.144180
$27$	0	0
$28$	55.6099	0.375332
$29$	91.1672	0.583770	0.291885	−	0.956453i	$-0.405718\pi$
$29$	91.1672	0.583770	0.291885	+	0.956453i	$0.405718\pi$
$30$	0	0
$31$	198.079	1.14761	0.573807	−	0.818991i	$-0.305466\pi$
$31$	198.079	1.14761	0.573807	+	0.818991i	$0.305466\pi$
$32$	−44.9048	−0.248066
$33$	0	0
$34$	18.0325	0.0909574
$35$	−35.0000	−0.169031
$36$	0	0
$37$	155.666	0.691656	0.345828	−	0.938298i	$-0.387598\pi$
$37$	155.666	0.691656	0.345828	+	0.938298i	$0.387598\pi$
$38$	−0.976794	−0.00416992
$39$	0	0
$40$	18.8197	0.0743912
$41$	156.885	0.597595	0.298797	−	0.954317i	$-0.403415\pi$
$41$	156.885	0.597595	0.298797	+	0.954317i	$0.403415\pi$
$42$	0	0
$43$	354.217	1.25622	0.628111	−	0.778124i	$-0.283828\pi$
$43$	354.217	1.25622	0.628111	+	0.778124i	$0.283828\pi$
$44$	−400.964	−1.37381
$45$	0	0
$46$	−48.3282	−0.154904
$47$	175.659	0.545161	0.272580	−	0.962133i	$-0.412123\pi$
$47$	175.659	0.545161	0.272580	+	0.962133i	$0.412123\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−5.90170	−0.0166925
$51$	0	0
$52$	643.252	1.71544
$53$	−200.302	−0.519124	−0.259562	−	0.965726i	$-0.583578\pi$
$53$	−200.302	−0.519124	−0.259562	+	0.965726i	$0.583578\pi$
$54$	0	0
$55$	252.361	0.618696
$56$	−26.3475	−0.0628721
$57$	0	0
$58$	−21.5217	−0.0487230
$59$	312.498	0.689556	0.344778	−	0.938684i	$-0.387954\pi$
$59$	312.498	0.689556	0.344778	+	0.938684i	$0.387954\pi$
$60$	0	0
$61$	−154.170	−0.323598	−0.161799	−	0.986824i	$-0.551730\pi$
$61$	−154.170	−0.323598	−0.161799	+	0.986824i	$0.551730\pi$
$62$	−46.7601	−0.0957829
$63$	0	0
$64$	−490.724	−0.958446
$65$	−404.853	−0.772551
$66$	0	0
$67$	734.715	1.33970	0.669849	−	0.742498i	$-0.266359\pi$
$67$	734.715	1.33970	0.669849	+	0.742498i	$0.266359\pi$
$68$	606.839	1.08221
$69$	0	0
$70$	8.26238	0.0141078
$71$	678.577	1.13426	0.567129	−	0.823629i	$-0.308054\pi$
$71$	678.577	1.13426	0.567129	+	0.823629i	$0.308054\pi$
$72$	0	0
$73$	−60.8003	−0.0974813	−0.0487407	−	0.998811i	$-0.515521\pi$
$73$	−60.8003	−0.0974813	−0.0487407	+	0.998811i	$0.515521\pi$
$74$	−36.7477	−0.0577274
$75$	0	0
$76$	−32.8715	−0.0496135
$77$	−353.305	−0.522894
$78$	0	0
$79$	−1286.26	−1.83184	−0.915919	−	0.401363i	$-0.868537\pi$
$79$	−1286.26	−1.83184	−0.915919	+	0.401363i	$0.868537\pi$
$80$	313.328	0.437889
$81$	0	0
$82$	−37.0356	−0.0498768
$83$	−116.170	−0.153631	−0.0768153	−	0.997045i	$-0.524475\pi$
$83$	−116.170	−0.153631	−0.0768153	+	0.997045i	$0.524475\pi$
$84$	0	0
$85$	−381.935	−0.487372
$86$	−83.6192	−0.104848
$87$	0	0
$88$	189.974	0.230128
$89$	916.440	1.09149	0.545744	−	0.837952i	$-0.316247\pi$
$89$	916.440	1.09149	0.545744	+	0.837952i	$0.316247\pi$
$90$	0	0
$91$	566.794	0.652925
$92$	−1626.36	−1.84304
$93$	0	0
$94$	−41.4676	−0.0455006
$95$	20.6888	0.0223435
$96$	0	0
$97$	−1416.30	−1.48251	−0.741256	−	0.671222i	$-0.765770\pi$
$97$	−1416.30	−1.48251	−0.741256	+	0.671222i	$0.765770\pi$
$98$	−11.5673	−0.0119232
$99$	0	0
$100$	−198.607	−0.198607
$101$	1379.19	1.35876	0.679381	−	0.733785i	$-0.262248\pi$
$101$	1379.19	1.35876	0.679381	+	0.733785i	$0.262248\pi$
$102$	0	0
$103$	−1308.43	−1.25169	−0.625844	−	0.779949i	$-0.715245\pi$
$103$	−1308.43	−1.25169	−0.625844	+	0.779949i	$0.715245\pi$
$104$	−304.768	−0.287355
$105$	0	0
$106$	47.2849	0.0433275
$107$	−1265.65	−1.14351	−0.571754	−	0.820425i	$-0.693737\pi$
$107$	−1265.65	−1.14351	−0.571754	+	0.820425i	$0.693737\pi$
$108$	0	0
$109$	2069.32	1.81839	0.909196	−	0.416368i	$-0.136697\pi$
$109$	2069.32	1.81839	0.909196	+	0.416368i	$0.136697\pi$
$110$	−59.5743	−0.0516380
$111$	0	0
$112$	−438.659	−0.370084
$113$	1953.89	1.62661	0.813303	−	0.581840i	$-0.197667\pi$
$113$	1953.89	1.62661	0.813303	+	0.581840i	$0.197667\pi$
$114$	0	0
$115$	1023.61	0.830016
$116$	−724.257	−0.579703
$117$	0	0
$118$	−73.7709	−0.0575522
$119$	534.709	0.411905
$120$	0	0
$121$	1216.44	0.913927
$122$	36.3947	0.0270083
$123$	0	0
$124$	−1573.59	−1.13962
$125$	125.000	0.0894427
$126$	0	0
$127$	224.251	0.156685	0.0783426	−	0.996926i	$-0.475037\pi$
$127$	224.251	0.156685	0.0783426	+	0.996926i	$0.475037\pi$
$128$	475.083	0.328061
$129$	0	0
$130$	95.5728	0.0644792
$131$	490.898	0.327404	0.163702	−	0.986510i	$-0.447656\pi$
$131$	490.898	0.327404	0.163702	+	0.986510i	$0.447656\pi$
$132$	0	0
$133$	−28.9644	−0.0188837
$134$	−173.443	−0.111815
$135$	0	0
$136$	−287.515	−0.181281
$137$	−1831.12	−1.14192	−0.570961	−	0.820977i	$-0.693429\pi$
$137$	−1831.12	−1.14192	−0.570961	+	0.820977i	$0.693429\pi$
$138$	0	0
$139$	−3050.84	−1.86165	−0.930823	−	0.365469i	$-0.880909\pi$
$139$	−3050.84	−1.86165	−0.930823	+	0.365469i	$0.880909\pi$
$140$	278.050	0.167853
$141$	0	0
$142$	−160.190	−0.0946682
$143$	−4086.76	−2.38987
$144$	0	0
$145$	455.836	0.261070
$146$	14.3530	0.00813605
$147$	0	0
$148$	−1236.65	−0.686838
$149$	−2246.55	−1.23520	−0.617599	−	0.786493i	$-0.711895\pi$
$149$	−2246.55	−1.23520	−0.617599	+	0.786493i	$0.711895\pi$
$150$	0	0
$151$	−1311.53	−0.706826	−0.353413	−	0.935467i	$-0.614979\pi$
$151$	−1311.53	−0.706826	−0.353413	+	0.935467i	$0.614979\pi$
$152$	15.5743	0.00831079
$153$	0	0
$154$	83.4040	0.0436421
$155$	990.395	0.513228
$156$	0	0
$157$	1790.94	0.910398	0.455199	−	0.890390i	$-0.349568\pi$
$157$	1790.94	0.910398	0.455199	+	0.890390i	$0.349568\pi$
$158$	303.644	0.152890
$159$	0	0
$160$	−224.524	−0.110939
$161$	−1433.05	−0.701491
$162$	0	0
$163$	−491.108	−0.235991	−0.117996	−	0.993014i	$-0.537647\pi$
$163$	−491.108	−0.235991	−0.117996	+	0.993014i	$0.537647\pi$
$164$	−1246.34	−0.593432
$165$	0	0
$166$	27.4241	0.0128224
$167$	826.059	0.382769	0.191384	−	0.981515i	$-0.438702\pi$
$167$	826.059	0.382769	0.191384	+	0.981515i	$0.438702\pi$
$168$	0	0
$169$	4359.24	1.98418
$170$	90.1626	0.0406774
$171$	0	0
$172$	−2813.99	−1.24747
$173$	−2918.00	−1.28238	−0.641190	−	0.767382i	$-0.721559\pi$
$173$	−2918.00	−1.28238	−0.641190	+	0.767382i	$0.721559\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	3162.87	1.35460
$177$	0	0
$178$	−216.342	−0.0910984
$179$	−955.745	−0.399082	−0.199541	−	0.979889i	$-0.563945\pi$
$179$	−955.745	−0.399082	−0.199541	+	0.979889i	$0.563945\pi$
$180$	0	0
$181$	206.080	0.0846289	0.0423145	−	0.999104i	$-0.486527\pi$
$181$	206.080	0.0846289	0.0423145	+	0.999104i	$0.486527\pi$
$182$	−133.802	−0.0544948
$183$	0	0
$184$	770.557	0.308730
$185$	778.328	0.309318
$186$	0	0
$187$	−3855.41	−1.50768
$188$	−1395.49	−0.541363
$189$	0	0
$190$	−4.88397	−0.00186485
$191$	2018.63	0.764727	0.382364	−	0.924012i	$-0.375110\pi$
$191$	2018.63	0.764727	0.382364	+	0.924012i	$0.375110\pi$
$192$	0	0
$193$	1031.69	0.384781	0.192390	−	0.981318i	$-0.438376\pi$
$193$	1031.69	0.384781	0.192390	+	0.981318i	$0.438376\pi$
$194$	334.344	0.123734
$195$	0	0
$196$	−389.269	−0.141862
$197$	205.955	0.0744857	0.0372429	−	0.999306i	$-0.488142\pi$
$197$	205.955	0.0744857	0.0372429	+	0.999306i	$0.488142\pi$
$198$	0	0
$199$	−1831.38	−0.652376	−0.326188	−	0.945305i	$-0.605764\pi$
$199$	−1831.38	−0.652376	−0.326188	+	0.945305i	$0.605764\pi$
$200$	94.0983	0.0332688
$201$	0	0
$202$	−325.584	−0.113406
$203$	−638.170	−0.220644
$204$	0	0
$205$	784.427	0.267253
$206$	308.879	0.104469
$207$	0	0
$208$	−5074.07	−1.69146
$209$	208.842	0.0691191
$210$	0	0
$211$	1030.19	0.336119	0.168060	−	0.985777i	$-0.446250\pi$
$211$	1030.19	0.336119	0.168060	+	0.985777i	$0.446250\pi$
$212$	1591.25	0.515508
$213$	0	0
$214$	298.780	0.0954402
$215$	1771.08	0.561800
$216$	0	0
$217$	−1386.55	−0.433757
$218$	−488.500	−0.151768
$219$	0	0
$220$	−2004.82	−0.614387
$221$	6185.10	1.88260
$222$	0	0
$223$	5368.67	1.61217	0.806083	−	0.591803i	$-0.201584\pi$
$223$	5368.67	1.61217	0.806083	+	0.591803i	$0.201584\pi$
$224$	314.334	0.0937603
$225$	0	0
$226$	−461.251	−0.135761
$227$	932.121	0.272542	0.136271	−	0.990672i	$-0.456488\pi$
$227$	932.121	0.272542	0.136271	+	0.990672i	$0.456488\pi$
$228$	0	0
$229$	3163.05	0.912752	0.456376	−	0.889787i	$-0.349147\pi$
$229$	3163.05	0.912752	0.456376	+	0.889787i	$0.349147\pi$
$230$	−241.641	−0.0692753
$231$	0	0
$232$	343.147	0.0971065
$233$	−436.562	−0.122747	−0.0613737	−	0.998115i	$-0.519548\pi$
$233$	−436.562	−0.122747	−0.0613737	+	0.998115i	$0.519548\pi$
$234$	0	0
$235$	878.297	0.243803
$236$	−2482.57	−0.684753
$237$	0	0
$238$	−126.228	−0.0343787
$239$	1980.82	0.536103	0.268051	−	0.963405i	$-0.413620\pi$
$239$	1980.82	0.536103	0.268051	+	0.963405i	$0.413620\pi$
$240$	0	0
$241$	5303.55	1.41756	0.708780	−	0.705430i	$-0.249246\pi$
$241$	5303.55	1.41756	0.708780	+	0.705430i	$0.249246\pi$
$242$	−287.162	−0.0762787
$243$	0	0
$244$	1224.77	0.321344
$245$	245.000	0.0638877
$246$	0	0
$247$	−335.037	−0.0863074
$248$	745.556	0.190899
$249$	0	0
$250$	−29.5085	−0.00746512
$251$	2996.04	0.753420	0.376710	−	0.926331i	$-0.377055\pi$
$251$	2996.04	0.753420	0.376710	+	0.926331i	$0.377055\pi$
$252$	0	0
$253$	10332.7	2.56764
$254$	−52.9384	−0.0130774
$255$	0	0
$256$	3813.64	0.931065
$257$	−968.861	−0.235159	−0.117580	−	0.993063i	$-0.537513\pi$
$257$	−968.861	−0.235159	−0.117580	+	0.993063i	$0.537513\pi$
$258$	0	0
$259$	−1089.66	−0.261421
$260$	3216.26	0.767170
$261$	0	0
$262$	−115.885	−0.0273260
$263$	4830.18	1.13248	0.566239	−	0.824241i	$-0.308398\pi$
$263$	4830.18	1.13248	0.566239	+	0.824241i	$0.308398\pi$
$264$	0	0
$265$	−1001.51	−0.232159
$266$	6.83756	0.00157608
$267$	0	0
$268$	−5836.78	−1.33037
$269$	4774.97	1.08229	0.541143	−	0.840930i	$-0.317992\pi$
$269$	4774.97	1.08229	0.541143	+	0.840930i	$0.317992\pi$
$270$	0	0
$271$	141.909	0.0318094	0.0159047	−	0.999874i	$-0.494937\pi$
$271$	141.909	0.0318094	0.0159047	+	0.999874i	$0.494937\pi$
$272$	−4786.84	−1.06708
$273$	0	0
$274$	432.269	0.0953078
$275$	1261.80	0.276689
$276$	0	0
$277$	−3621.13	−0.785460	−0.392730	−	0.919654i	$-0.628469\pi$
$277$	−3621.13	−0.785460	−0.392730	+	0.919654i	$0.628469\pi$
$278$	720.206	0.155378
$279$	0	0
$280$	−131.738	−0.0281172
$281$	−5790.87	−1.22937	−0.614687	−	0.788771i	$-0.710718\pi$
$281$	−5790.87	−1.22937	−0.614687	+	0.788771i	$0.710718\pi$
$282$	0	0
$283$	526.043	0.110495	0.0552474	−	0.998473i	$-0.482405\pi$
$283$	526.043	0.110495	0.0552474	+	0.998473i	$0.482405\pi$
$284$	−5390.80	−1.12636
$285$	0	0
$286$	964.753	0.199465
$287$	−1098.20	−0.225870
$288$	0	0
$289$	921.972	0.187660
$290$	−107.608	−0.0217896
$291$	0	0
$292$	483.014	0.0968023
$293$	−1914.88	−0.381803	−0.190901	−	0.981609i	$-0.561141\pi$
$293$	−1914.88	−0.381803	−0.190901	+	0.981609i	$0.561141\pi$
$294$	0	0
$295$	1562.49	0.308379
$296$	585.915	0.115053
$297$	0	0
$298$	530.339	0.103093
$299$	−16576.4	−3.20615
$300$	0	0
$301$	−2479.52	−0.474807
$302$	309.610	0.0589936
$303$	0	0
$304$	259.296	0.0489199
$305$	−770.851	−0.144717
$306$	0	0
$307$	−6244.17	−1.16083	−0.580413	−	0.814322i	$-0.697109\pi$
$307$	−6244.17	−1.16083	−0.580413	+	0.814322i	$0.697109\pi$
$308$	2806.75	0.519251
$309$	0	0
$310$	−233.800	−0.0428354
$311$	9658.78	1.76109	0.880545	−	0.473962i	$-0.157177\pi$
$311$	9658.78	1.76109	0.880545	+	0.473962i	$0.157177\pi$
$312$	0	0
$313$	2198.34	0.396988	0.198494	−	0.980102i	$-0.436395\pi$
$313$	2198.34	0.396988	0.198494	+	0.980102i	$0.436395\pi$
$314$	−422.783	−0.0759842
$315$	0	0
$316$	10218.4	1.81908
$317$	−3030.78	−0.536990	−0.268495	−	0.963281i	$-0.586526\pi$
$317$	−3030.78	−0.536990	−0.268495	+	0.963281i	$0.586526\pi$
$318$	0	0
$319$	4601.40	0.807615
$320$	−2453.62	−0.428630
$321$	0	0
$322$	338.297	0.0585483
$323$	−316.072	−0.0544480
$324$	0	0
$325$	−2024.26	−0.345495
$326$	115.935	0.0196965
$327$	0	0
$328$	590.506	0.0994062
$329$	−1229.62	−0.206051
$330$	0	0
$331$	4753.74	0.789393	0.394696	−	0.918812i	$-0.370850\pi$
$331$	4753.74	0.789393	0.394696	+	0.918812i	$0.370850\pi$
$332$	922.888	0.152560
$333$	0	0
$334$	−195.006	−0.0319469
$335$	3673.58	0.599131
$336$	0	0
$337$	8824.40	1.42640	0.713199	−	0.700962i	$-0.247246\pi$
$337$	8824.40	1.42640	0.713199	+	0.700962i	$0.247246\pi$
$338$	−1029.08	−0.165605
$339$	0	0
$340$	3034.20	0.483977
$341$	9997.47	1.58766
$342$	0	0
$343$	−343.000	−0.0539949
$344$	1333.25	0.208965
$345$	0	0
$346$	688.847	0.107031
$347$	3413.97	0.528160	0.264080	−	0.964501i	$-0.414932\pi$
$347$	3413.97	0.528160	0.264080	+	0.964501i	$0.414932\pi$
$348$	0	0
$349$	−5676.32	−0.870621	−0.435310	−	0.900280i	$-0.643361\pi$
$349$	−5676.32	−0.870621	−0.435310	+	0.900280i	$0.643361\pi$
$350$	41.3119	0.00630918
$351$	0	0
$352$	−2266.44	−0.343187
$353$	−6225.80	−0.938713	−0.469357	−	0.883009i	$-0.655514\pi$
$353$	−6225.80	−0.938713	−0.469357	+	0.883009i	$0.655514\pi$
$354$	0	0
$355$	3392.89	0.507256
$356$	−7280.45	−1.08388
$357$	0	0
$358$	225.621	0.0333084
$359$	4907.73	0.721505	0.360752	−	0.932662i	$-0.382520\pi$
$359$	4907.73	0.721505	0.360752	+	0.932662i	$0.382520\pi$
$360$	0	0
$361$	−6841.88	−0.997504
$362$	−48.6490	−0.00706335
$363$	0	0
$364$	−4502.77	−0.648377
$365$	−304.001	−0.0435950
$366$	0	0
$367$	−3906.48	−0.555631	−0.277816	−	0.960634i	$-0.589610\pi$
$367$	−3906.48	−0.555631	−0.277816	+	0.960634i	$0.589610\pi$
$368$	12829.0	1.81728
$369$	0	0
$370$	−183.738	−0.0258165
$371$	1402.11	0.196210
$372$	0	0
$373$	−2102.52	−0.291862	−0.145931	−	0.989295i	$-0.546618\pi$
$373$	−2102.52	−0.291862	−0.145931	+	0.989295i	$0.546618\pi$
$374$	910.140	0.125835
$375$	0	0
$376$	661.170	0.0906842
$377$	−7381.86	−1.00845
$378$	0	0
$379$	−6612.76	−0.896239	−0.448120	−	0.893974i	$-0.647906\pi$
$379$	−6612.76	−0.896239	−0.448120	+	0.893974i	$0.647906\pi$
$380$	−164.358	−0.0221878
$381$	0	0
$382$	−476.534	−0.0638262
$383$	−2.37457	−0.000316801 0	−0.000158401	−	1.00000i	$-0.500050\pi$
$383$	−2.37457	−0.000316801 0	−0.000158401		1.00000i	$0.500050\pi$
$384$	0	0
$385$	−1766.52	−0.233845
$386$	−243.549	−0.0321148
$387$	0	0
$388$	11251.5	1.47218
$389$	−7716.98	−1.00583	−0.502913	−	0.864337i	$-0.667739\pi$
$389$	−7716.98	−1.00583	−0.502913	+	0.864337i	$0.667739\pi$
$390$	0	0
$391$	−15638.0	−2.02263
$392$	184.433	0.0237634
$393$	0	0
$394$	−48.6194	−0.00621678
$395$	−6431.28	−0.819223
$396$	0	0
$397$	6403.95	0.809584	0.404792	−	0.914409i	$-0.367344\pi$
$397$	6403.95	0.809584	0.404792	+	0.914409i	$0.367344\pi$
$398$	432.329	0.0544490
$399$	0	0
$400$	1566.64	0.195830
$401$	10969.9	1.36611	0.683054	−	0.730368i	$-0.260651\pi$
$401$	10969.9	1.36611	0.683054	+	0.730368i	$0.260651\pi$
$402$	0	0
$403$	−16038.6	−1.98248
$404$	−10956.7	−1.34930
$405$	0	0
$406$	150.652	0.0184155
$407$	7856.78	0.956870
$408$	0	0
$409$	−400.353	−0.0484015	−0.0242007	−	0.999707i	$-0.507704\pi$
$409$	−400.353	−0.0484015	−0.0242007	+	0.999707i	$0.507704\pi$
$410$	−185.178	−0.0223056
$411$	0	0
$412$	10394.6	1.24297
$413$	−2187.49	−0.260628
$414$	0	0
$415$	−580.851	−0.0687057
$416$	3635.97	0.428529
$417$	0	0
$418$	−49.3009	−0.00576887
$419$	15815.4	1.84399	0.921995	−	0.387201i	$-0.126558\pi$
$419$	15815.4	1.84399	0.921995	+	0.387201i	$0.126558\pi$
$420$	0	0
$421$	1936.53	0.224182	0.112091	−	0.993698i	$-0.464245\pi$
$421$	1936.53	0.224182	0.112091	+	0.993698i	$0.464245\pi$
$422$	−243.195	−0.0280534
$423$	0	0
$424$	−753.923	−0.0863531
$425$	−1909.67	−0.217960
$426$	0	0
$427$	1079.19	0.122309
$428$	10054.7	1.13554
$429$	0	0
$430$	−418.096	−0.0468893
$431$	−2030.91	−0.226973	−0.113487	−	0.993540i	$-0.536202\pi$
$431$	−2030.91	−0.226973	−0.113487	+	0.993540i	$0.536202\pi$
$432$	0	0
$433$	−10784.1	−1.19689	−0.598443	−	0.801165i	$-0.704214\pi$
$433$	−10784.1	−1.19689	−0.598443	+	0.801165i	$0.704214\pi$
$434$	327.321	0.0362025
$435$	0	0
$436$	−16439.2	−1.80573
$437$	847.089	0.0927272
$438$	0	0
$439$	−6304.19	−0.685382	−0.342691	−	0.939448i	$-0.611338\pi$
$439$	−6304.19	−0.685382	−0.342691	+	0.939448i	$0.611338\pi$
$440$	949.868	0.102916
$441$	0	0
$442$	−1460.10	−0.157127
$443$	−15494.8	−1.66181	−0.830905	−	0.556414i	$-0.812177\pi$
$443$	−15494.8	−1.66181	−0.830905	+	0.556414i	$0.812177\pi$
$444$	0	0
$445$	4582.20	0.488128
$446$	−1267.37	−0.134556
$447$	0	0
$448$	3435.07	0.362259
$449$	242.018	0.0254377	0.0127189	−	0.999919i	$-0.495951\pi$
$449$	242.018	0.0254377	0.0127189	+	0.999919i	$0.495951\pi$
$450$	0	0
$451$	7918.34	0.826741
$452$	−15522.2	−1.61528
$453$	0	0
$454$	−220.044	−0.0227471
$455$	2833.97	0.291997
$456$	0	0
$457$	−11670.4	−1.19457	−0.597283	−	0.802030i	$-0.703753\pi$
$457$	−11670.4	−1.19457	−0.597283	+	0.802030i	$0.703753\pi$
$458$	−746.695	−0.0761807
$459$	0	0
$460$	−8131.81	−0.824234
$461$	−12128.1	−1.22530	−0.612649	−	0.790355i	$-0.709896\pi$
$461$	−12128.1	−1.22530	−0.612649	+	0.790355i	$0.709896\pi$
$462$	0	0
$463$	5161.64	0.518103	0.259051	−	0.965864i	$-0.416590\pi$
$463$	5161.64	0.518103	0.259051	+	0.965864i	$0.416590\pi$
$464$	5713.05	0.571598
$465$	0	0
$466$	103.058	0.0102448
$467$	−11680.2	−1.15738	−0.578691	−	0.815547i	$-0.696436\pi$
$467$	−11680.2	−1.15738	−0.578691	+	0.815547i	$0.696436\pi$
$468$	0	0
$469$	−5143.01	−0.506358
$470$	−207.338	−0.0203485
$471$	0	0
$472$	1176.22	0.114703
$473$	17878.1	1.73792
$474$	0	0
$475$	103.444	0.00999230
$476$	−4247.87	−0.409036
$477$	0	0
$478$	−467.608	−0.0447445
$479$	18458.7	1.76075	0.880373	−	0.474282i	$-0.157292\pi$
$479$	18458.7	1.76075	0.880373	+	0.474282i	$0.157292\pi$
$480$	0	0
$481$	−12604.3	−1.19482
$482$	−1252.00	−0.118313
$483$	0	0
$484$	−9663.70	−0.907560
$485$	−7081.51	−0.663000
$486$	0	0
$487$	8630.11	0.803014	0.401507	−	0.915856i	$-0.368487\pi$
$487$	8630.11	0.803014	0.401507	+	0.915856i	$0.368487\pi$
$488$	−580.286	−0.0538286
$489$	0	0
$490$	−57.8367	−0.00533223
$491$	−17801.6	−1.63621	−0.818103	−	0.575072i	$-0.804974\pi$
$491$	−17801.6	−1.63621	−0.818103	+	0.575072i	$0.804974\pi$
$492$	0	0
$493$	−6963.99	−0.636191
$494$	79.0916	0.00720344
$495$	0	0
$496$	12412.7	1.12369
$497$	−4750.04	−0.428709
$498$	0	0
$499$	200.167	0.0179574	0.00897868	−	0.999960i	$-0.497142\pi$
$499$	200.167	0.0179574	0.00897868	+	0.999960i	$0.497142\pi$
$500$	−993.034	−0.0888197
$501$	0	0
$502$	−707.269	−0.0628824
$503$	−16400.4	−1.45379	−0.726896	−	0.686747i	$-0.759038\pi$
$503$	−16400.4	−1.45379	−0.726896	+	0.686747i	$0.759038\pi$
$504$	0	0
$505$	6895.97	0.607657
$506$	−2439.23	−0.214302
$507$	0	0
$508$	−1781.51	−0.155594
$509$	15006.6	1.30679	0.653394	−	0.757018i	$-0.273344\pi$
$509$	15006.6	1.30679	0.653394	+	0.757018i	$0.273344\pi$
$510$	0	0
$511$	425.602	0.0368445
$512$	−4700.94	−0.405770
$513$	0	0
$514$	228.717	0.0196270
$515$	−6542.17	−0.559772
$516$	0	0
$517$	8865.91	0.754201
$518$	257.234	0.0218189
$519$	0	0
$520$	−1523.84	−0.128509
$521$	−7113.18	−0.598146	−0.299073	−	0.954230i	$-0.596677\pi$
$521$	−7113.18	−0.598146	−0.299073	+	0.954230i	$0.596677\pi$
$522$	0	0
$523$	−8888.46	−0.743146	−0.371573	−	0.928404i	$-0.621181\pi$
$523$	−8888.46	−0.743146	−0.371573	+	0.928404i	$0.621181\pi$
$524$	−3899.83	−0.325123
$525$	0	0
$526$	−1140.25	−0.0945196
$527$	−15130.7	−1.25067
$528$	0	0
$529$	29743.8	2.44463
$530$	236.424	0.0193766
$531$	0	0
$532$	230.101	0.0187521
$533$	−12703.1	−1.03233
$534$	0	0
$535$	−6328.27	−0.511392
$536$	2765.42	0.222850
$537$	0	0
$538$	−1127.22	−0.0903305
$539$	2473.13	0.197635
$540$	0	0
$541$	−653.827	−0.0519597	−0.0259799	−	0.999662i	$-0.508271\pi$
$541$	−653.827	−0.0519597	−0.0259799	+	0.999662i	$0.508271\pi$
$542$	−33.5001	−0.00265489
$543$	0	0
$544$	3430.14	0.270342
$545$	10346.6	0.813210
$546$	0	0
$547$	1138.52	0.0889940	0.0444970	−	0.999010i	$-0.485831\pi$
$547$	1138.52	0.0889940	0.0444970	+	0.999010i	$0.485831\pi$
$548$	14546.9	1.13397
$549$	0	0
$550$	−297.871	−0.0230932
$551$	377.229	0.0291660
$552$	0	0
$553$	9003.80	0.692370
$554$	854.832	0.0655566
$555$	0	0
$556$	24236.7	1.84868
$557$	19804.8	1.50657	0.753283	−	0.657696i	$-0.228469\pi$
$557$	19804.8	1.50657	0.753283	+	0.657696i	$0.228469\pi$
$558$	0	0
$559$	−28681.1	−2.17009
$560$	−2193.30	−0.165507
$561$	0	0
$562$	1367.04	0.102607
$563$	−10276.3	−0.769265	−0.384632	−	0.923070i	$-0.625672\pi$
$563$	−10276.3	−0.769265	−0.384632	+	0.923070i	$0.625672\pi$
$564$	0	0
$565$	9769.45	0.727440
$566$	−124.182	−0.00922219
$567$	0	0
$568$	2554.12	0.188677
$569$	−4139.03	−0.304951	−0.152475	−	0.988307i	$-0.548724\pi$
$569$	−4139.03	−0.304951	−0.152475	+	0.988307i	$0.548724\pi$
$570$	0	0
$571$	−4486.81	−0.328839	−0.164420	−	0.986390i	$-0.552575\pi$
$571$	−4486.81	−0.328839	−0.164420	+	0.986390i	$0.552575\pi$
$572$	32466.3	2.37323
$573$	0	0
$574$	259.249	0.0188517
$575$	5118.03	0.371194
$576$	0	0
$577$	−1104.77	−0.0797093	−0.0398547	−	0.999205i	$-0.512689\pi$
$577$	−1104.77	−0.0797093	−0.0398547	+	0.999205i	$0.512689\pi$
$578$	−217.648	−0.0156626
$579$	0	0
$580$	−3621.28	−0.259251
$581$	813.192	0.0580669
$582$	0	0
$583$	−10109.7	−0.718181
$584$	−228.848	−0.0162154
$585$	0	0
$586$	452.041	0.0318663
$587$	10413.2	0.732199	0.366100	−	0.930576i	$-0.380693\pi$
$587$	10413.2	0.732199	0.366100	+	0.930576i	$0.380693\pi$
$588$	0	0
$589$	819.605	0.0573365
$590$	−368.854	−0.0257381
$591$	0	0
$592$	9754.89	0.677235
$593$	3235.16	0.224034	0.112017	−	0.993706i	$-0.464269\pi$
$593$	3235.16	0.224034	0.112017	+	0.993706i	$0.464269\pi$
$594$	0	0
$595$	2673.54	0.184209
$596$	17847.2	1.22659
$597$	0	0
$598$	3913.16	0.267594
$599$	−569.048	−0.0388158	−0.0194079	−	0.999812i	$-0.506178\pi$
$599$	−569.048	−0.0388158	−0.0194079	+	0.999812i	$0.506178\pi$
$600$	0	0
$601$	3760.89	0.255258	0.127629	−	0.991822i	$-0.459263\pi$
$601$	3760.89	0.255258	0.127629	+	0.991822i	$0.459263\pi$
$602$	585.335	0.0396287
$603$	0	0
$604$	10419.1	0.701902
$605$	6082.18	0.408720
$606$	0	0
$607$	−2224.05	−0.148717	−0.0743585	−	0.997232i	$-0.523691\pi$
$607$	−2224.05	−0.148717	−0.0743585	+	0.997232i	$0.523691\pi$
$608$	−185.806	−0.0123938
$609$	0	0
$610$	181.973	0.0120785
$611$	−14223.2	−0.941753
$612$	0	0
$613$	5914.50	0.389697	0.194849	−	0.980833i	$-0.437578\pi$
$613$	5914.50	0.389697	0.194849	+	0.980833i	$0.437578\pi$
$614$	1474.05	0.0968856
$615$	0	0
$616$	−1329.82	−0.0869802
$617$	18591.2	1.21306	0.606528	−	0.795062i	$-0.292562\pi$
$617$	18591.2	1.21306	0.606528	+	0.795062i	$0.292562\pi$
$618$	0	0
$619$	5125.97	0.332844	0.166422	−	0.986055i	$-0.446779\pi$
$619$	5125.97	0.332844	0.166422	+	0.986055i	$0.446779\pi$
$620$	−7867.96	−0.509653
$621$	0	0
$622$	−2280.13	−0.146985
$623$	−6415.08	−0.412544
$624$	0	0
$625$	625.000	0.0400000
$626$	−518.957	−0.0331337
$627$	0	0
$628$	−14227.7	−0.904056
$629$	−11890.8	−0.753765
$630$	0	0
$631$	−10649.0	−0.671839	−0.335919	−	0.941891i	$-0.609047\pi$
$631$	−10649.0	−0.671839	−0.335919	+	0.941891i	$0.609047\pi$
$632$	−4841.38	−0.304715
$633$	0	0
$634$	715.471	0.0448186
$635$	1121.25	0.0700718
$636$	0	0
$637$	−3967.56	−0.246782
$638$	−1086.24	−0.0674056
$639$	0	0
$640$	2375.41	0.146713
$641$	−24025.7	−1.48044	−0.740218	−	0.672367i	$-0.765278\pi$
$641$	−24025.7	−1.48044	−0.740218	+	0.672367i	$0.765278\pi$
$642$	0	0
$643$	14929.3	0.915638	0.457819	−	0.889045i	$-0.348631\pi$
$643$	14929.3	0.915638	0.457819	+	0.889045i	$0.348631\pi$
$644$	11384.5	0.696605
$645$	0	0
$646$	74.6144	0.00454437
$647$	−14479.1	−0.879801	−0.439901	−	0.898046i	$-0.644986\pi$
$647$	−14479.1	−0.879801	−0.439901	+	0.898046i	$0.644986\pi$
$648$	0	0
$649$	15772.5	0.953965
$650$	477.864	0.0288360
$651$	0	0
$652$	3901.50	0.234347
$653$	−898.168	−0.0538255	−0.0269127	−	0.999638i	$-0.508568\pi$
$653$	−898.168	−0.0538255	−0.0269127	+	0.999638i	$0.508568\pi$
$654$	0	0
$655$	2454.49	0.146420
$656$	9831.33	0.585135
$657$	0	0
$658$	290.273	0.0171976
$659$	30198.0	1.78505	0.892526	−	0.450997i	$-0.148931\pi$
$659$	30198.0	1.78505	0.892526	+	0.450997i	$0.148931\pi$
$660$	0	0
$661$	−19337.8	−1.13790	−0.568952	−	0.822371i	$-0.692651\pi$
$661$	−19337.8	−1.13790	−0.568952	+	0.822371i	$0.692651\pi$
$662$	−1122.20	−0.0658848
$663$	0	0
$664$	−437.257	−0.0255555
$665$	−144.822	−0.00844504
$666$	0	0
$667$	18663.9	1.08346
$668$	−6562.44	−0.380102
$669$	0	0
$670$	−867.214	−0.0500051
$671$	−7781.30	−0.447681
$672$	0	0
$673$	10132.2	0.580336	0.290168	−	0.956976i	$-0.406289\pi$
$673$	10132.2	0.580336	0.290168	+	0.956976i	$0.406289\pi$
$674$	−2083.16	−0.119051
$675$	0	0
$676$	−34631.0	−1.97035
$677$	33177.3	1.88347	0.941733	−	0.336361i	$-0.109196\pi$
$677$	33177.3	1.88347	0.941733	+	0.336361i	$0.109196\pi$
$678$	0	0
$679$	9914.11	0.560337
$680$	−1437.58	−0.0810714
$681$	0	0
$682$	−2360.08	−0.132511
$683$	11423.6	0.639987	0.319994	−	0.947420i	$-0.396319\pi$
$683$	11423.6	0.639987	0.319994	+	0.947420i	$0.396319\pi$
$684$	0	0
$685$	−9155.61	−0.510683
$686$	80.9713	0.00450656
$687$	0	0
$688$	22197.2	1.23003
$689$	16218.6	0.896775
$690$	0	0
$691$	−19737.4	−1.08661	−0.543304	−	0.839536i	$-0.682827\pi$
$691$	−19737.4	−1.08661	−0.543304	+	0.839536i	$0.682827\pi$
$692$	23181.4	1.27345
$693$	0	0
$694$	−805.929	−0.0440816
$695$	−15254.2	−0.832554
$696$	0	0
$697$	−11984.0	−0.651258
$698$	1340.00	0.0726643
$699$	0	0
$700$	1390.25	0.0750663
$701$	−11307.8	−0.609258	−0.304629	−	0.952471i	$-0.598532\pi$
$701$	−11307.8	−0.609258	−0.304629	+	0.952471i	$0.598532\pi$
$702$	0	0
$703$	644.108	0.0345562
$704$	−24767.9	−1.32596
$705$	0	0
$706$	1469.71	0.0783475
$707$	−9654.36	−0.513564
$708$	0	0
$709$	30859.2	1.63461	0.817307	−	0.576202i	$-0.195466\pi$
$709$	30859.2	1.63461	0.817307	+	0.576202i	$0.195466\pi$
$710$	−800.952	−0.0423369
$711$	0	0
$712$	3449.42	0.181562
$713$	40551.0	2.12994
$714$	0	0
$715$	−20433.8	−1.06878
$716$	7592.69	0.396302
$717$	0	0
$718$	−1158.56	−0.0602187
$719$	−33152.4	−1.71958	−0.859789	−	0.510650i	$-0.829405\pi$
$719$	−33152.4	−1.71958	−0.859789	+	0.510650i	$0.829405\pi$
$720$	0	0
$721$	9159.03	0.473093
$722$	1615.15	0.0832543
$723$	0	0
$724$	−1637.16	−0.0840394
$725$	2279.18	0.116754
$726$	0	0
$727$	−16743.0	−0.854146	−0.427073	−	0.904217i	$-0.640455\pi$
$727$	−16743.0	−0.854146	−0.427073	+	0.904217i	$0.640455\pi$
$728$	2133.37	0.108610
$729$	0	0
$730$	71.7650	0.00363855
$731$	−27057.5	−1.36903
$732$	0	0
$733$	−8827.55	−0.444820	−0.222410	−	0.974953i	$-0.571392\pi$
$733$	−8827.55	−0.444820	−0.222410	+	0.974953i	$0.571392\pi$
$734$	922.195	0.0463744
$735$	0	0
$736$	−9192.97	−0.460404
$737$	37082.6	1.85340
$738$	0	0
$739$	36154.0	1.79966	0.899829	−	0.436243i	$-0.143691\pi$
$739$	36154.0	1.79966	0.899829	+	0.436243i	$0.143691\pi$
$740$	−6183.25	−0.307163
$741$	0	0
$742$	−330.994	−0.0163762
$743$	1820.69	0.0898987	0.0449494	−	0.998989i	$-0.485687\pi$
$743$	1820.69	0.0898987	0.0449494	+	0.998989i	$0.485687\pi$
$744$	0	0
$745$	−11232.8	−0.552398
$746$	496.338	0.0243596
$747$	0	0
$748$	30628.5	1.49718
$749$	8859.57	0.432205
$750$	0	0
$751$	27764.4	1.34905	0.674526	−	0.738251i	$-0.264348\pi$
$751$	27764.4	1.34905	0.674526	+	0.738251i	$0.264348\pi$
$752$	11007.8	0.533795
$753$	0	0
$754$	1742.62	0.0841678
$755$	−6557.65	−0.316102
$756$	0	0
$757$	−13518.3	−0.649050	−0.324525	−	0.945877i	$-0.605205\pi$
$757$	−13518.3	−0.649050	−0.324525	+	0.945877i	$0.605205\pi$
$758$	1561.06	0.0748025
$759$	0	0
$760$	77.8714	0.00371670
$761$	−30695.2	−1.46216	−0.731078	−	0.682294i	$-0.760983\pi$
$761$	−30695.2	−1.46216	−0.731078	+	0.682294i	$0.760983\pi$
$762$	0	0
$763$	−14485.2	−0.687288
$764$	−16036.5	−0.759400
$765$	0	0
$766$	0.560560	2.64410e−5 0
$767$	−25303.2	−1.19119
$768$	0	0
$769$	4536.39	0.212726	0.106363	−	0.994327i	$-0.466079\pi$
$769$	4536.39	0.212726	0.106363	+	0.994327i	$0.466079\pi$
$770$	417.020	0.0195173
$771$	0	0
$772$	−8196.03	−0.382100
$773$	−31238.9	−1.45354	−0.726768	−	0.686883i	$-0.758978\pi$
$773$	−31238.9	−1.45354	−0.726768	+	0.686883i	$0.758978\pi$
$774$	0	0
$775$	4951.97	0.229523
$776$	−5330.86	−0.246607
$777$	0	0
$778$	1821.73	0.0839490
$779$	649.155	0.0298567
$780$	0	0
$781$	34249.2	1.56919
$782$	3691.64	0.168814
$783$	0	0
$784$	3070.62	0.139879
$785$	8954.70	0.407143
$786$	0	0
$787$	39597.2	1.79350	0.896752	−	0.442534i	$-0.145921\pi$
$787$	39597.2	1.79350	0.896752	+	0.442534i	$0.145921\pi$
$788$	−1636.16	−0.0739669
$789$	0	0
$790$	1518.22	0.0683745
$791$	−13677.2	−0.614799
$792$	0	0
$793$	12483.3	0.559008
$794$	−1511.77	−0.0675700
$795$	0	0
$796$	14549.0	0.647832
$797$	16567.0	0.736302	0.368151	−	0.929766i	$-0.379991\pi$
$797$	16567.0	0.736302	0.368151	+	0.929766i	$0.379991\pi$
$798$	0	0
$799$	−13418.1	−0.594115
$800$	−1122.62	−0.0496133
$801$	0	0
$802$	−2589.63	−0.114019
$803$	−3068.72	−0.134860
$804$	0	0
$805$	−7165.25	−0.313717
$806$	3786.19	0.165463
$807$	0	0
$808$	5191.20	0.226022
$809$	12141.6	0.527657	0.263828	−	0.964570i	$-0.415015\pi$
$809$	12141.6	0.527657	0.263828	+	0.964570i	$0.415015\pi$
$810$	0	0
$811$	30295.6	1.31174	0.655870	−	0.754873i	$-0.272302\pi$
$811$	30295.6	1.31174	0.655870	+	0.754873i	$0.272302\pi$
$812$	5069.80	0.219107
$813$	0	0
$814$	−1854.73	−0.0798629
$815$	−2455.54	−0.105538
$816$	0	0
$817$	1465.67	0.0627628
$818$	94.5106	0.00403971
$819$	0	0
$820$	−6231.70	−0.265391
$821$	14914.8	0.634018	0.317009	−	0.948423i	$-0.397322\pi$
$821$	14914.8	0.634018	0.317009	+	0.948423i	$0.397322\pi$
$822$	0	0
$823$	−31077.6	−1.31628	−0.658138	−	0.752897i	$-0.728656\pi$
$823$	−31077.6	−1.31628	−0.658138	+	0.752897i	$0.728656\pi$
$824$	−4924.85	−0.208210
$825$	0	0
$826$	516.396	0.0217527
$827$	−15527.8	−0.652908	−0.326454	−	0.945213i	$-0.605854\pi$
$827$	−15527.8	−0.652908	−0.326454	+	0.945213i	$0.605854\pi$
$828$	0	0
$829$	−40221.5	−1.68510	−0.842551	−	0.538617i	$-0.818947\pi$
$829$	−40221.5	−1.68510	−0.842551	+	0.538617i	$0.818947\pi$
$830$	137.120	0.00573436
$831$	0	0
$832$	39734.2	1.65569
$833$	−3742.96	−0.155685
$834$	0	0
$835$	4130.29	0.171179
$836$	−1659.10	−0.0686377
$837$	0	0
$838$	−3733.51	−0.153904
$839$	21153.7	0.870448	0.435224	−	0.900322i	$-0.356669\pi$
$839$	21153.7	0.870448	0.435224	+	0.900322i	$0.356669\pi$
$840$	0	0
$841$	−16077.5	−0.659213
$842$	−457.152	−0.0187108
$843$	0	0
$844$	−8184.10	−0.333778
$845$	21796.2	0.887351
$846$	0	0
$847$	−8515.06	−0.345432
$848$	−12552.0	−0.508301
$849$	0	0
$850$	450.813	0.0181915
$851$	31868.1	1.28369
$852$	0	0
$853$	636.075	0.0255320	0.0127660	−	0.999919i	$-0.495936\pi$
$853$	636.075	0.0255320	0.0127660	+	0.999919i	$0.495936\pi$
$854$	−254.763	−0.0102082
$855$	0	0
$856$	−4763.83	−0.190215
$857$	3941.46	0.157104	0.0785518	−	0.996910i	$-0.474970\pi$
$857$	3941.46	0.157104	0.0785518	+	0.996910i	$0.474970\pi$
$858$	0	0
$859$	−21781.6	−0.865165	−0.432583	−	0.901594i	$-0.642398\pi$
$859$	−21781.6	−0.865165	−0.432583	+	0.901594i	$0.642398\pi$
$860$	−14070.0	−0.557886
$861$	0	0
$862$	479.432	0.0189438
$863$	44697.9	1.76308	0.881538	−	0.472112i	$-0.156508\pi$
$863$	44697.9	1.76308	0.881538	+	0.472112i	$0.156508\pi$
$864$	0	0
$865$	−14590.0	−0.573498
$866$	2545.79	0.0998953
$867$	0	0
$868$	11015.2	0.430736
$869$	−64920.1	−2.53425
$870$	0	0
$871$	−59490.3	−2.31430
$872$	7788.78	0.302478
$873$	0	0
$874$	−199.971	−0.00773926
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−20171.7	−0.776682	−0.388341	−	0.921516i	$-0.626952\pi$
$877$	−20171.7	−0.776682	−0.388341	+	0.921516i	$0.626952\pi$
$878$	1488.22	0.0572038
$879$	0	0
$880$	15814.3	0.605797
$881$	−11577.6	−0.442744	−0.221372	−	0.975189i	$-0.571054\pi$
$881$	−11577.6	−0.442744	−0.221372	+	0.975189i	$0.571054\pi$
$882$	0	0
$883$	−35388.3	−1.34871	−0.674355	−	0.738407i	$-0.735578\pi$
$883$	−35388.3	−1.34871	−0.674355	+	0.738407i	$0.735578\pi$
$884$	−49136.1	−1.86949
$885$	0	0
$886$	3657.83	0.138699
$887$	41705.0	1.57871	0.789356	−	0.613936i	$-0.210415\pi$
$887$	41705.0	1.57871	0.789356	+	0.613936i	$0.210415\pi$
$888$	0	0
$889$	−1569.75	−0.0592215
$890$	−1081.71	−0.0407405
$891$	0	0
$892$	−42650.2	−1.60093
$893$	726.838	0.0272371
$894$	0	0
$895$	−4778.72	−0.178475
$896$	−3325.58	−0.123995
$897$	0	0
$898$	−57.1328	−0.00212310
$899$	18058.3	0.669942
$900$	0	0
$901$	15300.5	0.565740
$902$	−1869.27	−0.0690020
$903$	0	0
$904$	7354.31	0.270576
$905$	1030.40	0.0378472
$906$	0	0
$907$	−28645.1	−1.04867	−0.524336	−	0.851511i	$-0.675687\pi$
$907$	−28645.1	−1.04867	−0.524336	+	0.851511i	$0.675687\pi$
$908$	−7405.02	−0.270643
$909$	0	0
$910$	−669.010	−0.0243708
$911$	−13337.8	−0.485074	−0.242537	−	0.970142i	$-0.577980\pi$
$911$	−13337.8	−0.485074	−0.242537	+	0.970142i	$0.577980\pi$
$912$	0	0
$913$	−5863.36	−0.212540
$914$	2755.00	0.0997016
$915$	0	0
$916$	−25128.1	−0.906394
$917$	−3436.29	−0.123747
$918$	0	0
$919$	−28911.0	−1.03774	−0.518871	−	0.854853i	$-0.673647\pi$
$919$	−28911.0	−1.03774	−0.518871	+	0.854853i	$0.673647\pi$
$920$	3852.79	0.138068
$921$	0	0
$922$	2863.06	0.102267
$923$	−54944.8	−1.95940
$924$	0	0
$925$	3891.64	0.138331
$926$	−1218.50	−0.0432422
$927$	0	0
$928$	−4093.84	−0.144814
$929$	−7093.88	−0.250530	−0.125265	−	0.992123i	$-0.539978\pi$
$929$	−7093.88	−0.250530	−0.125265	+	0.992123i	$0.539978\pi$
$930$	0	0
$931$	202.751	0.00713736
$932$	3468.17	0.121892
$933$	0	0
$934$	2757.33	0.0965981
$935$	−19277.1	−0.674254
$936$	0	0
$937$	−19271.1	−0.671888	−0.335944	−	0.941882i	$-0.609055\pi$
$937$	−19271.1	−0.671888	−0.335944	+	0.941882i	$0.609055\pi$
$938$	1214.10	0.0422620
$939$	0	0
$940$	−6977.43	−0.242105
$941$	18115.2	0.627563	0.313782	−	0.949495i	$-0.398404\pi$
$941$	18115.2	0.627563	0.313782	+	0.949495i	$0.398404\pi$
$942$	0	0
$943$	32117.8	1.10912
$944$	19582.9	0.675180
$945$	0	0
$946$	−4220.44	−0.145051
$947$	2475.55	0.0849468	0.0424734	−	0.999098i	$-0.486476\pi$
$947$	2475.55	0.0849468	0.0424734	+	0.999098i	$0.486476\pi$
$948$	0	0
$949$	4923.04	0.168397
$950$	−24.4199	−0.000833984 0
$951$	0	0
$952$	2012.61	0.0685179
$953$	−12866.6	−0.437344	−0.218672	−	0.975798i	$-0.570173\pi$
$953$	−12866.6	−0.437344	−0.218672	+	0.975798i	$0.570173\pi$
$954$	0	0
$955$	10093.2	0.341997
$956$	−15736.2	−0.532368
$957$	0	0
$958$	−4357.50	−0.146957
$959$	12817.9	0.431606
$960$	0	0
$961$	9444.26	0.317017
$962$	2975.48	0.0997228
$963$	0	0
$964$	−42132.9	−1.40768
$965$	5158.45	0.172079
$966$	0	0
$967$	−2142.86	−0.0712614	−0.0356307	−	0.999365i	$-0.511344\pi$
$967$	−2142.86	−0.0712614	−0.0356307	+	0.999365i	$0.511344\pi$
$968$	4578.58	0.152026
$969$	0	0
$970$	1671.72	0.0553357
$971$	−2879.06	−0.0951529	−0.0475765	−	0.998868i	$-0.515150\pi$
$971$	−2879.06	−0.0951529	−0.0475765	+	0.998868i	$0.515150\pi$
$972$	0	0
$973$	21355.9	0.703636
$974$	−2037.29	−0.0670217
$975$	0	0
$976$	−9661.18	−0.316851
$977$	−48741.1	−1.59607	−0.798037	−	0.602608i	$-0.794128\pi$
$977$	−48741.1	−1.59607	−0.798037	+	0.602608i	$0.794128\pi$
$978$	0	0
$979$	46254.7	1.51002
$980$	−1946.35	−0.0634426
$981$	0	0
$982$	4202.40	0.136562
$983$	45756.8	1.48466	0.742328	−	0.670037i	$-0.233722\pi$
$983$	45756.8	1.48466	0.742328	+	0.670037i	$0.233722\pi$
$984$	0	0
$985$	1029.78	0.0333110
$986$	1643.97	0.0530982
$987$	0	0
$988$	2661.63	0.0857062
$989$	72515.7	2.33151
$990$	0	0
$991$	−51552.1	−1.65248	−0.826240	−	0.563319i	$-0.809524\pi$
$991$	−51552.1	−1.65248	−0.826240	+	0.563319i	$0.809524\pi$
$992$	−8894.70	−0.284684
$993$	0	0
$994$	1121.33	0.0357812
$995$	−9156.88	−0.291751
$996$	0	0
$997$	25565.3	0.812097	0.406048	−	0.913852i	$-0.366906\pi$
$997$	25565.3	0.812097	0.406048	+	0.913852i	$0.366906\pi$
$998$	−47.2531	−0.00149877
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.4.a.l.1.1		2
3.2	odd	2		105.4.a.d.1.2	✓	2
5.4	even	2		1575.4.a.n.1.2		2
7.6	odd	2		2205.4.a.be.1.1		2
12.11	even	2		1680.4.a.bd.1.2		2
15.2	even	4		525.4.d.k.274.3		4
15.8	even	4		525.4.d.k.274.2		4
15.14	odd	2		525.4.a.o.1.1		2
21.20	even	2		735.4.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.d.1.2	✓	2	3.2	odd	2
315.4.a.l.1.1		2	1.1	even	1	trivial
525.4.a.o.1.1		2	15.14	odd	2
525.4.d.k.274.2		4	15.8	even	4
525.4.d.k.274.3		4	15.2	even	4
735.4.a.m.1.2		2	21.20	even	2
1575.4.a.n.1.2		2	5.4	even	2
1680.4.a.bd.1.2		2	12.11	even	2
2205.4.a.be.1.1		2	7.6	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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-0.236068 q^{2} -7.94427 q^{4} +5.00000 q^{5} -7.00000 q^{7} +3.76393 q^{8} +O(q^{10})\)	\(q-0.236068 q^{2} -7.94427 q^{4} +5.00000 q^{5} -7.00000 q^{7} +3.76393 q^{8} -1.18034 q^{10} +50.4721 q^{11} -80.9706 q^{13} +1.65248 q^{14} +62.6656 q^{16} -76.3870 q^{17} +4.13777 q^{19} -39.7214 q^{20} -11.9149 q^{22} +204.721 q^{23} +25.0000 q^{25} +19.1146 q^{26} +55.6099 q^{28} +91.1672 q^{29} +198.079 q^{31} -44.9048 q^{32} +18.0325 q^{34} -35.0000 q^{35} +155.666 q^{37} -0.976794 q^{38} +18.8197 q^{40} +156.885 q^{41} +354.217 q^{43} -400.964 q^{44} -48.3282 q^{46} +175.659 q^{47} +49.0000 q^{49} -5.90170 q^{50} +643.252 q^{52} -200.302 q^{53} +252.361 q^{55} -26.3475 q^{56} -21.5217 q^{58} +312.498 q^{59} -154.170 q^{61} -46.7601 q^{62} -490.724 q^{64} -404.853 q^{65} +734.715 q^{67} +606.839 q^{68} +8.26238 q^{70} +678.577 q^{71} -60.8003 q^{73} -36.7477 q^{74} -32.8715 q^{76} -353.305 q^{77} -1286.26 q^{79} +313.328 q^{80} -37.0356 q^{82} -116.170 q^{83} -381.935 q^{85} -83.6192 q^{86} +189.974 q^{88} +916.440 q^{89} +566.794 q^{91} -1626.36 q^{92} -41.4676 q^{94} +20.6888 q^{95} -1416.30 q^{97} -11.5673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 2 q^{4} + 10 q^{5} - 14 q^{7} + 12 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 2 * q^4 + 10 * q^5 - 14 * q^7 + 12 * q^8	\( 2 q + 4 q^{2} + 2 q^{4} + 10 q^{5} - 14 q^{7} + 12 q^{8} + 20 q^{10} + 92 q^{11} + 8 q^{13} - 28 q^{14} + 18 q^{16} + 44 q^{17} - 108 q^{19} + 10 q^{20} + 164 q^{22} + 320 q^{23} + 50 q^{25} + 396 q^{26} - 14 q^{28} + 236 q^{29} - 60 q^{31} - 300 q^{32} + 528 q^{34} - 70 q^{35} + 204 q^{37} - 476 q^{38} + 60 q^{40} - 44 q^{41} + 136 q^{43} + 12 q^{44} + 440 q^{46} - 400 q^{47} + 98 q^{49} + 100 q^{50} + 1528 q^{52} - 16 q^{53} + 460 q^{55} - 84 q^{56} + 592 q^{58} + 464 q^{59} - 684 q^{61} - 1140 q^{62} - 1214 q^{64} + 40 q^{65} + 736 q^{67} + 1804 q^{68} - 140 q^{70} + 740 q^{71} + 424 q^{73} + 168 q^{74} - 1148 q^{76} - 644 q^{77} - 408 q^{79} + 90 q^{80} - 888 q^{82} - 608 q^{83} + 220 q^{85} - 1008 q^{86} + 532 q^{88} + 1332 q^{89} - 56 q^{91} - 480 q^{92} - 2480 q^{94} - 540 q^{95} - 2448 q^{97} + 196 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 2 * q^4 + 10 * q^5 - 14 * q^7 + 12 * q^8 + 20 * q^10 + 92 * q^11 + 8 * q^13 - 28 * q^14 + 18 * q^16 + 44 * q^17 - 108 * q^19 + 10 * q^20 + 164 * q^22 + 320 * q^23 + 50 * q^25 + 396 * q^26 - 14 * q^28 + 236 * q^29 - 60 * q^31 - 300 * q^32 + 528 * q^34 - 70 * q^35 + 204 * q^37 - 476 * q^38 + 60 * q^40 - 44 * q^41 + 136 * q^43 + 12 * q^44 + 440 * q^46 - 400 * q^47 + 98 * q^49 + 100 * q^50 + 1528 * q^52 - 16 * q^53 + 460 * q^55 - 84 * q^56 + 592 * q^58 + 464 * q^59 - 684 * q^61 - 1140 * q^62 - 1214 * q^64 + 40 * q^65 + 736 * q^67 + 1804 * q^68 - 140 * q^70 + 740 * q^71 + 424 * q^73 + 168 * q^74 - 1148 * q^76 - 644 * q^77 - 408 * q^79 + 90 * q^80 - 888 * q^82 - 608 * q^83 + 220 * q^85 - 1008 * q^86 + 532 * q^88 + 1332 * q^89 - 56 * q^91 - 480 * q^92 - 2480 * q^94 - 540 * q^95 - 2448 * q^97 + 196 * q^98

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