LMFDB - Embedded newform 525.4.d.k.274.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [525,4,Mod(274,525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(525, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("525.274");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 525 = 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	525.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$30.9760027530$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.3
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	525.274
Dual form			525.4.d.k.274.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.236068i q^{2} -3.00000i q^{3} +7.94427 q^{4} +0.708204 q^{6} -7.00000i q^{7} +3.76393i q^{8} -9.00000 q^{9} -50.4721 q^{11} -23.8328i q^{12} +80.9706i q^{13} +1.65248 q^{14} +62.6656 q^{16} +76.3870i q^{17} -2.12461i q^{18} -4.13777 q^{19} -21.0000 q^{21} -11.9149i q^{22} +204.721i q^{23} +11.2918 q^{24} -19.1146 q^{26} +27.0000i q^{27} -55.6099i q^{28} +91.1672 q^{29} +198.079 q^{31} +44.9048i q^{32} +151.416i q^{33} -18.0325 q^{34} -71.4984 q^{36} +155.666i q^{37} -0.976794i q^{38} +242.912 q^{39} -156.885 q^{41} -4.95743i q^{42} -354.217i q^{43} -400.964 q^{44} -48.3282 q^{46} -175.659i q^{47} -187.997i q^{48} -49.0000 q^{49} +229.161 q^{51} +643.252i q^{52} -200.302i q^{53} -6.37384 q^{54} +26.3475 q^{56} +12.4133i q^{57} +21.5217i q^{58} +312.498 q^{59} -154.170 q^{61} +46.7601i q^{62} +63.0000i q^{63} +490.724 q^{64} -35.7446 q^{66} +734.715i q^{67} +606.839i q^{68} +614.164 q^{69} -678.577 q^{71} -33.8754i q^{72} +60.8003i q^{73} -36.7477 q^{74} -32.8715 q^{76} +353.305i q^{77} +57.3437i q^{78} +1286.26 q^{79} +81.0000 q^{81} -37.0356i q^{82} -116.170i q^{83} -166.830 q^{84} +83.6192 q^{86} -273.502i q^{87} -189.974i q^{88} +916.440 q^{89} +566.794 q^{91} +1626.36i q^{92} -594.237i q^{93} +41.4676 q^{94} +134.714 q^{96} -1416.30i q^{97} -11.5673i q^{98} +454.249 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} - 24 q^{6} - 36 q^{9} - 184 q^{11} - 56 q^{14} + 36 q^{16} + 216 q^{19} - 84 q^{21} + 72 q^{24} - 792 q^{26} + 472 q^{29} - 120 q^{31} - 1056 q^{34} + 36 q^{36} - 48 q^{39} + 88 q^{41} + 24 q^{44}+ \cdots + 1656 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 - 24 * q^6 - 36 * q^9 - 184 * q^11 - 56 * q^14 + 36 * q^16 + 216 * q^19 - 84 * q^21 + 72 * q^24 - 792 * q^26 + 472 * q^29 - 120 * q^31 - 1056 * q^34 + 36 * q^36 - 48 * q^39 + 88 * q^41 + 24 * q^44 + 880 * q^46 - 196 * q^49 - 264 * q^51 + 216 * q^54 + 168 * q^56 + 928 * q^59 - 1368 * q^61 + 2428 * q^64 + 984 * q^66 + 1920 * q^69 - 1480 * q^71 + 336 * q^74 - 2296 * q^76 + 816 * q^79 + 324 * q^81 + 84 * q^84 + 2016 * q^86 + 2664 * q^89 - 112 * q^91 + 4960 * q^94 + 1800 * q^96 + 1656 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/525\mathbb{Z}\right)^\times$.

$n$	$127$	$176$	$451$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0.236068i	0.0834626i	0.999129	+	0.0417313i	$0.0132873\pi$
$2$	0.236068i	0.0834626i	−0.999129	+	0.0417313i	$0.986713\pi$
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	7.94427	0.993034
$5$	0	0
$5$	0	0
$6$	0.708204	0.0481872
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	3.76393i	0.166344i
$9$	−9.00000	−0.333333
$10$	0	0
$11$	−50.4721	−1.38345	−0.691724	−	0.722162i	$-0.743148\pi$
$11$	−50.4721	−1.38345	−0.691724	+	0.722162i	$0.743148\pi$
$12$	− 23.8328i	− 0.573328i
$13$	80.9706i	1.72748i	0.503940	+	0.863738i	$0.331883\pi$
$13$	80.9706i	1.72748i	−0.503940	+	0.863738i	$0.668117\pi$
$14$	1.65248	0.0315459
$15$	0	0
$16$	62.6656	0.979150
$17$	76.3870i	1.08980i	0.838502	+	0.544899i	$0.183432\pi$
$17$	76.3870i	1.08980i	−0.838502	+	0.544899i	$0.816568\pi$
$18$	− 2.12461i	− 0.0278209i
$19$	−4.13777	−0.0499615	−0.0249808	−	0.999688i	$-0.507952\pi$
$19$	−4.13777	−0.0499615	−0.0249808	+	0.999688i	$0.507952\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	− 11.9149i	− 0.115466i
$23$	204.721i	1.85597i	0.372615	+	0.927986i	$0.378461\pi$
$23$	204.721i	1.85597i	−0.372615	+	0.927986i	$0.621539\pi$
$24$	11.2918	0.0960387
$25$	0	0
$26$	−19.1146	−0.144180
$27$	27.0000i	0.192450i
$28$	− 55.6099i	− 0.375332i
$29$	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	$0.305466\pi$
$32$	44.9048i	0.248066i
$33$	151.416i	0.798734i
$34$	−18.0325	−0.0909574
$35$	0	0
$36$	−71.4984	−0.331011
$37$	155.666i	0.691656i	0.938298	+	0.345828i	$0.112402\pi$
$37$	155.666i	0.691656i	−0.938298	+	0.345828i	$0.887598\pi$
$38$	− 0.976794i	− 0.00416992i
$39$	242.912	0.997359
$40$	0	0
$41$	−156.885	−0.597595	−0.298797	−	0.954317i	$-0.596585\pi$
$41$	−156.885	−0.597595	−0.298797	+	0.954317i	$0.596585\pi$
$42$	− 4.95743i	− 0.0182130i
$43$	− 354.217i	− 1.25622i	−0.778124	−	0.628111i	$-0.783828\pi$
$43$	− 354.217i	− 1.25622i	0.778124	−	0.628111i	$-0.216172\pi$
$44$	−400.964	−1.37381
$45$	0	0
$46$	−48.3282	−0.154904
$47$	− 175.659i	− 0.545161i	−0.962133	−	0.272580i	$-0.912123\pi$
$47$	− 175.659i	− 0.545161i	0.962133	−	0.272580i	$-0.0878771\pi$
$48$	− 187.997i	− 0.565313i
$49$	−49.0000	−0.142857
$50$	0	0
$51$	229.161	0.629195
$52$	643.252i	1.71544i
$53$	− 200.302i	− 0.519124i	−0.965726	−	0.259562i	$-0.916422\pi$
$53$	− 200.302i	− 0.519124i	0.965726	−	0.259562i	$-0.0835782\pi$
$54$	−6.37384	−0.0160624
$55$	0	0
$56$	26.3475	0.0628721
$57$	12.4133i	0.0288453i
$58$	21.5217i	0.0487230i
$59$	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	$0.551730\pi$
$62$	46.7601i	0.0957829i
$63$	63.0000i	0.125988i
$64$	490.724	0.958446
$65$	0	0
$66$	−35.7446	−0.0666644
$67$	734.715i	1.33970i	0.742498	+	0.669849i	$0.233641\pi$
$67$	734.715i	1.33970i	−0.742498	+	0.669849i	$0.766359\pi$
$68$	606.839i	1.08221i
$69$	614.164	1.07155
$70$	0	0
$71$	−678.577	−1.13426	−0.567129	−	0.823629i	$-0.691946\pi$
$71$	−678.577	−1.13426	−0.567129	+	0.823629i	$0.691946\pi$
$72$	− 33.8754i	− 0.0554480i
$73$	60.8003i	0.0974813i	0.998811	+	0.0487407i	$0.0155208\pi$
$73$	60.8003i	0.0974813i	−0.998811	+	0.0487407i	$0.984479\pi$
$74$	−36.7477	−0.0577274
$75$	0	0
$76$	−32.8715	−0.0496135
$77$	353.305i	0.522894i
$78$	57.3437i	0.0832422i
$79$	1286.26	1.83184	0.915919	−	0.401363i	$-0.131463\pi$
$79$	1286.26	1.83184	0.915919	+	0.401363i	$0.131463\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	− 37.0356i	− 0.0498768i
$83$	− 116.170i	− 0.153631i	−0.997045	−	0.0768153i	$-0.975525\pi$
$83$	− 116.170i	− 0.153631i	0.997045	−	0.0768153i	$-0.0244752\pi$
$84$	−166.830	−0.216698
$85$	0	0
$86$	83.6192	0.104848
$87$	− 273.502i	− 0.337040i
$88$	− 189.974i	− 0.230128i
$89$	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
$92$	1626.36i	1.84304i
$93$	− 594.237i	− 0.662575i
$94$	41.4676	0.0455006
$95$	0	0
$96$	134.714	0.143221
$97$	− 1416.30i	− 1.48251i	−0.671222	−	0.741256i	$-0.734230\pi$
$97$	− 1416.30i	− 1.48251i	0.671222	−	0.741256i	$-0.265770\pi$
$98$	− 11.5673i	− 0.0119232i
$99$	454.249	0.461149
$100$	0	0
$101$	−1379.19	−1.35876	−0.679381	−	0.733785i	$-0.737752\pi$
$101$	−1379.19	−1.35876	−0.679381	+	0.733785i	$0.737752\pi$
$102$	54.0976i	0.0525143i
$103$	1308.43i	1.25169i	0.779949	+	0.625844i	$0.215245\pi$
$103$	1308.43i	1.25169i	−0.779949	+	0.625844i	$0.784755\pi$
$104$	−304.768	−0.287355
$105$	0	0
$106$	47.2849	0.0433275
$107$	1265.65i	1.14351i	0.820425	+	0.571754i	$0.193737\pi$
$107$	1265.65i	1.14351i	−0.820425	+	0.571754i	$0.806263\pi$
$108$	214.495i	0.191109i
$109$	−2069.32	−1.81839	−0.909196	−	0.416368i	$-0.863303\pi$
$109$	−2069.32	−1.81839	−0.909196	+	0.416368i	$0.863303\pi$
$110$	0	0
$111$	466.997	0.399328
$112$	− 438.659i	− 0.370084i
$113$	1953.89i	1.62661i	0.581840	+	0.813303i	$0.302333\pi$
$113$	1953.89i	1.62661i	−0.581840	+	0.813303i	$0.697667\pi$
$114$	−2.93038	−0.00240750
$115$	0	0
$116$	724.257	0.579703
$117$	− 728.735i	− 0.575826i
$118$	73.7709i	0.0575522i
$119$	534.709	0.411905
$120$	0	0
$121$	1216.44	0.913927
$122$	− 36.3947i	− 0.0270083i
$123$	470.656i	0.345022i
$124$	1573.59	1.13962
$125$	0	0
$126$	−14.8723	−0.0105153
$127$	224.251i	0.156685i	0.996926	+	0.0783426i	$0.0249628\pi$
$127$	224.251i	0.156685i	−0.996926	+	0.0783426i	$0.975037\pi$
$128$	475.083i	0.328061i
$129$	−1062.65	−0.725280
$130$	0	0
$131$	−490.898	−0.327404	−0.163702	−	0.986510i	$-0.552344\pi$
$131$	−490.898	−0.327404	−0.163702	+	0.986510i	$0.552344\pi$
$132$	1202.89i	0.793170i
$133$	28.9644i	0.0188837i
$134$	−173.443	−0.111815
$135$	0	0
$136$	−287.515	−0.181281
$137$	1831.12i	1.14192i	0.820977	+	0.570961i	$0.193429\pi$
$137$	1831.12i	1.14192i	−0.820977	+	0.570961i	$0.806571\pi$
$138$	144.984i	0.0894340i
$139$	3050.84	1.86165	0.930823	−	0.365469i	$-0.119091\pi$
$139$	3050.84	1.86165	0.930823	+	0.365469i	$0.119091\pi$
$140$	0	0
$141$	−526.978	−0.314749
$142$	− 160.190i	− 0.0946682i
$143$	− 4086.76i	− 2.38987i
$144$	−563.991	−0.326383
$145$	0	0
$146$	−14.3530	−0.00813605
$147$	147.000i	0.0824786i
$148$	1236.65i	0.686838i
$149$	−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	$0.614979\pi$
$152$	− 15.5743i	− 0.00831079i
$153$	− 687.483i	− 0.363266i
$154$	−83.4040	−0.0436421
$155$	0	0
$156$	1929.76	0.990412
$157$	1790.94i	0.910398i	0.890390	+	0.455199i	$0.150432\pi$
$157$	1790.94i	0.910398i	−0.890390	+	0.455199i	$0.849568\pi$
$158$	303.644i	0.152890i
$159$	−600.906	−0.299716
$160$	0	0
$161$	1433.05	0.701491
$162$	19.1215i	0.00927363i
$163$	491.108i	0.235991i	0.993014	+	0.117996i	$0.0376469\pi$
$163$	491.108i	0.235991i	−0.993014	+	0.117996i	$0.962353\pi$
$164$	−1246.34	−0.593432
$165$	0	0
$166$	27.4241	0.0128224
$167$	− 826.059i	− 0.382769i	−0.981515	−	0.191384i	$-0.938702\pi$
$167$	− 826.059i	− 0.382769i	0.981515	−	0.191384i	$-0.0612977\pi$
$168$	− 79.0426i	− 0.0362992i
$169$	−4359.24	−1.98418
$170$	0	0
$171$	37.2399	0.0166538
$172$	− 2813.99i	− 1.24747i
$173$	− 2918.00i	− 1.28238i	−0.767382	−	0.641190i	$-0.778441\pi$
$173$	− 2918.00i	− 1.28238i	0.767382	−	0.641190i	$-0.221559\pi$
$174$	64.5650	0.0281302
$175$	0	0
$176$	−3162.87	−1.35460
$177$	− 937.495i	− 0.398116i
$178$	216.342i	0.0910984i
$179$	−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	$0.486527\pi$
$182$	133.802i	0.0544948i
$183$	462.511i	0.186829i
$184$	−770.557	−0.308730
$185$	0	0
$186$	140.280	0.0553003
$187$	− 3855.41i	− 1.50768i
$188$	− 1395.49i	− 0.541363i
$189$	189.000	0.0727393
$190$	0	0
$191$	−2018.63	−0.764727	−0.382364	−	0.924012i	$-0.624890\pi$
$191$	−2018.63	−0.764727	−0.382364	+	0.924012i	$0.624890\pi$
$192$	− 1472.17i	− 0.553359i
$193$	− 1031.69i	− 0.384781i	−0.981318	−	0.192390i	$-0.938376\pi$
$193$	− 1031.69i	− 0.384781i	0.981318	−	0.192390i	$-0.0616240\pi$
$194$	334.344	0.123734
$195$	0	0
$196$	−389.269	−0.141862
$197$	− 205.955i	− 0.0744857i	−0.999306	−	0.0372429i	$-0.988142\pi$
$197$	− 205.955i	− 0.0744857i	0.999306	−	0.0372429i	$-0.0118575\pi$
$198$	107.234i	0.0384887i
$199$	1831.38	0.652376	0.326188	−	0.945305i	$-0.394236\pi$
$199$	1831.38	0.652376	0.326188	+	0.945305i	$0.394236\pi$
$200$	0	0
$201$	2204.15	0.773475
$202$	− 325.584i	− 0.113406i
$203$	− 638.170i	− 0.220644i
$204$	1820.52	0.624812
$205$	0	0
$206$	−308.879	−0.104469
$207$	− 1842.49i	− 0.618657i
$208$	5074.07i	1.69146i
$209$	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	$0.446250\pi$
$212$	− 1591.25i	− 0.515508i
$213$	2035.73i	0.654864i
$214$	−298.780	−0.0954402
$215$	0	0
$216$	−101.626	−0.0320129
$217$	− 1386.55i	− 0.433757i
$218$	− 488.500i	− 0.151768i
$219$	182.401	0.0562809
$220$	0	0
$221$	−6185.10	−1.88260
$222$	110.243i	0.0333289i
$223$	− 5368.67i	− 1.61217i	−0.591803	−	0.806083i	$-0.701584\pi$
$223$	− 5368.67i	− 1.61217i	0.591803	−	0.806083i	$-0.298416\pi$
$224$	314.334	0.0937603
$225$	0	0
$226$	−461.251	−0.135761
$227$	− 932.121i	− 0.272542i	−0.990672	−	0.136271i	$-0.956488\pi$
$227$	− 932.121i	− 0.272542i	0.990672	−	0.136271i	$-0.0435118\pi$
$228$	98.6146i	0.0286444i
$229$	−3163.05	−0.912752	−0.456376	−	0.889787i	$-0.650853\pi$
$229$	−3163.05	−0.912752	−0.456376	+	0.889787i	$0.650853\pi$
$230$	0	0
$231$	1059.91	0.301893
$232$	343.147i	0.0971065i
$233$	− 436.562i	− 0.122747i	−0.998115	−	0.0613737i	$-0.980452\pi$
$233$	− 436.562i	− 0.122747i	0.998115	−	0.0613737i	$-0.0195481\pi$
$234$	172.031	0.0480599
$235$	0	0
$236$	2482.57	0.684753
$237$	− 3858.77i	− 1.05761i
$238$	126.228i	0.0343787i
$239$	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	$0.249246\pi$
$242$	287.162i	0.0762787i
$243$	− 243.000i	− 0.0641500i
$244$	−1224.77	−0.321344
$245$	0	0
$246$	−111.107	−0.0287964
$247$	− 335.037i	− 0.0863074i
$248$	745.556i	0.190899i
$249$	−348.511	−0.0886987
$250$	0	0
$251$	−2996.04	−0.753420	−0.376710	−	0.926331i	$-0.622945\pi$
$251$	−2996.04	−0.753420	−0.376710	+	0.926331i	$0.622945\pi$
$252$	500.489i	0.125111i
$253$	− 10332.7i	− 2.56764i
$254$	−52.9384	−0.0130774
$255$	0	0
$256$	3813.64	0.931065
$257$	968.861i	0.235159i	0.993063	+	0.117580i	$0.0375135\pi$
$257$	968.861i	0.235159i	−0.993063	+	0.117580i	$0.962487\pi$
$258$	− 250.858i	− 0.0605338i
$259$	1089.66	0.261421
$260$	0	0
$261$	−820.505	−0.194590
$262$	− 115.885i	− 0.0273260i
$263$	4830.18i	1.13248i	0.824241	+	0.566239i	$0.191602\pi$
$263$	4830.18i	1.13248i	−0.824241	+	0.566239i	$0.808398\pi$
$264$	−569.921	−0.132864
$265$	0	0
$266$	−6.83756	−0.00157608
$267$	− 2749.32i	− 0.630171i
$268$	5836.78i	1.33037i
$269$	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	$0.494937\pi$
$272$	4786.84i	1.06708i
$273$	− 1700.38i	− 0.376966i
$274$	−432.269	−0.0953078
$275$	0	0
$276$	4879.09	1.06408
$277$	− 3621.13i	− 0.785460i	−0.919654	−	0.392730i	$-0.871531\pi$
$277$	− 3621.13i	− 0.785460i	0.919654	−	0.392730i	$-0.128469\pi$
$278$	720.206i	0.155378i
$279$	−1782.71	−0.382538
$280$	0	0
$281$	5790.87	1.22937	0.614687	−	0.788771i	$-0.289282\pi$
$281$	5790.87	1.22937	0.614687	+	0.788771i	$0.289282\pi$
$282$	− 124.403i	− 0.0262698i
$283$	− 526.043i	− 0.110495i	−0.998473	−	0.0552474i	$-0.982405\pi$
$283$	− 526.043i	− 0.110495i	0.998473	−	0.0552474i	$-0.0175948\pi$
$284$	−5390.80	−1.12636
$285$	0	0
$286$	964.753	0.199465
$287$	1098.20i	0.225870i
$288$	− 404.143i	− 0.0826888i
$289$	−921.972	−0.187660
$290$	0	0
$291$	−4248.91	−0.855929
$292$	483.014i	0.0968023i
$293$	− 1914.88i	− 0.381803i	−0.981609	−	0.190901i	$-0.938859\pi$
$293$	− 1914.88i	− 0.381803i	0.981609	−	0.190901i	$-0.0611411\pi$
$294$	−34.7020	−0.00688388
$295$	0	0
$296$	−585.915	−0.115053
$297$	− 1362.75i	− 0.266245i
$298$	− 530.339i	− 0.103093i
$299$	−16576.4	−3.20615
$300$	0	0
$301$	−2479.52	−0.474807
$302$	− 309.610i	− 0.0589936i
$303$	4137.58i	0.784482i
$304$	−259.296	−0.0489199
$305$	0	0
$306$	162.293	0.0303191
$307$	− 6244.17i	− 1.16083i	−0.814322	−	0.580413i	$-0.802891\pi$
$307$	− 6244.17i	− 1.16083i	0.814322	−	0.580413i	$-0.197109\pi$
$308$	2806.75i	0.519251i
$309$	3925.30	0.722662
$310$	0	0
$311$	−9658.78	−1.76109	−0.880545	−	0.473962i	$-0.842823\pi$
$311$	−9658.78	−1.76109	−0.880545	+	0.473962i	$0.842823\pi$
$312$	914.303i	0.165905i
$313$	− 2198.34i	− 0.396988i	−0.980102	−	0.198494i	$-0.936395\pi$
$313$	− 2198.34i	− 0.396988i	0.980102	−	0.198494i	$-0.0636051\pi$
$314$	−422.783	−0.0759842
$315$	0	0
$316$	10218.4	1.81908
$317$	3030.78i	0.536990i	0.963281	+	0.268495i	$0.0865263\pi$
$317$	3030.78i	0.536990i	−0.963281	+	0.268495i	$0.913474\pi$
$318$	− 141.855i	− 0.0250151i
$319$	−4601.40	−0.807615
$320$	0	0
$321$	3796.96	0.660204
$322$	338.297i	0.0585483i
$323$	− 316.072i	− 0.0544480i
$324$	643.486	0.110337
$325$	0	0
$326$	−115.935	−0.0196965
$327$	6207.96i	1.04985i
$328$	− 590.506i	− 0.0994062i
$329$	−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	$0.370850\pi$
$332$	− 922.888i	− 0.152560i
$333$	− 1400.99i	− 0.230552i
$334$	195.006	0.0319469
$335$	0	0
$336$	−1315.98	−0.213668
$337$	8824.40i	1.42640i	0.700962	+	0.713199i	$0.252754\pi$
$337$	8824.40i	1.42640i	−0.700962	+	0.713199i	$0.747246\pi$
$338$	− 1029.08i	− 0.165605i
$339$	5861.67	0.939122
$340$	0	0
$341$	−9997.47	−1.58766
$342$	8.79115i	0.00138997i
$343$	343.000i	0.0539949i
$344$	1333.25	0.208965
$345$	0	0
$346$	688.847	0.107031
$347$	− 3413.97i	− 0.528160i	−0.964501	−	0.264080i	$-0.914932\pi$
$347$	− 3413.97i	− 0.528160i	0.964501	−	0.264080i	$-0.0850683\pi$
$348$	− 2172.77i	− 0.334692i
$349$	5676.32	0.870621	0.435310	−	0.900280i	$-0.356639\pi$
$349$	5676.32	0.870621	0.435310	+	0.900280i	$0.356639\pi$
$350$	0	0
$351$	−2186.21	−0.332453
$352$	− 2266.44i	− 0.343187i
$353$	− 6225.80i	− 0.938713i	−0.883009	−	0.469357i	$-0.844486\pi$
$353$	− 6225.80i	− 0.938713i	0.883009	−	0.469357i	$-0.155514\pi$
$354$	221.313	0.0332278
$355$	0	0
$356$	7280.45	1.08388
$357$	− 1604.13i	− 0.237813i
$358$	− 225.621i	− 0.0333084i
$359$	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
$362$	48.6490i	0.00706335i
$363$	− 3649.31i	− 0.527656i
$364$	4502.77	0.648377
$365$	0	0
$366$	−109.184	−0.0155933
$367$	− 3906.48i	− 0.555631i	−0.960634	−	0.277816i	$-0.910390\pi$
$367$	− 3906.48i	− 0.555631i	0.960634	−	0.277816i	$-0.0896104\pi$
$368$	12829.0i	1.81728i
$369$	1411.97	0.199198
$370$	0	0
$371$	−1402.11	−0.196210
$372$	− 4720.78i	− 0.657960i
$373$	2102.52i	0.291862i	0.989295	+	0.145931i	$0.0466178\pi$
$373$	2102.52i	0.291862i	−0.989295	+	0.145931i	$0.953382\pi$
$374$	910.140	0.125835
$375$	0	0
$376$	661.170	0.0906842
$377$	7381.86i	1.00845i
$378$	44.6168i	0.00607101i
$379$	6612.76	0.896239	0.448120	−	0.893974i	$-0.352094\pi$
$379$	6612.76	0.896239	0.448120	+	0.893974i	$0.352094\pi$
$380$	0	0
$381$	672.752	0.0904623
$382$	− 476.534i	− 0.0638262i
$383$	− 2.37457i	0 0.000316801i	−1.00000		0.000158401i	$-0.999950\pi$
$383$	− 2.37457i	0 0.000316801i	1.00000		0.000158401i	$-5.04204e-5\pi$
$384$	1425.25	0.189406
$385$	0	0
$386$	243.549	0.0321148
$387$	3187.95i	0.418741i
$388$	− 11251.5i	− 1.47218i
$389$	−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
$392$	− 184.433i	− 0.0237634i
$393$	1472.69i	0.189027i
$394$	48.6194	0.00621678
$395$	0	0
$396$	3608.68	0.457937
$397$	6403.95i	0.809584i	0.914409	+	0.404792i	$0.132656\pi$
$397$	6403.95i	0.809584i	−0.914409	+	0.404792i	$0.867344\pi$
$398$	432.329i	0.0544490i
$399$	86.8931	0.0109025
$400$	0	0
$401$	−10969.9	−1.36611	−0.683054	−	0.730368i	$-0.739349\pi$
$401$	−10969.9	−1.36611	−0.683054	+	0.730368i	$0.739349\pi$
$402$	520.328i	0.0645562i
$403$	16038.6i	1.98248i
$404$	−10956.7	−1.34930
$405$	0	0
$406$	150.652	0.0184155
$407$	− 7856.78i	− 0.956870i
$408$	862.546i	0.104663i
$409$	400.353	0.0484015	0.0242007	−	0.999707i	$-0.492296\pi$
$409$	400.353	0.0484015	0.0242007	+	0.999707i	$0.492296\pi$
$410$	0	0
$411$	5493.37	0.659289
$412$	10394.6i	1.24297i
$413$	− 2187.49i	− 0.260628i
$414$	434.953	0.0516348
$415$	0	0
$416$	−3635.97	−0.428529
$417$	− 9152.52i	− 1.07482i
$418$	49.3009i	0.00576887i
$419$	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	$0.464245\pi$
$422$	243.195i	0.0280534i
$423$	1580.93i	0.181720i
$424$	753.923	0.0863531
$425$	0	0
$426$	−480.571	−0.0546567
$427$	1079.19i	0.122309i
$428$	10054.7i	1.13554i
$429$	−12260.3	−1.37979
$430$	0	0
$431$	2030.91	0.226973	0.113487	−	0.993540i	$-0.463798\pi$
$431$	2030.91	0.226973	0.113487	+	0.993540i	$0.463798\pi$
$432$	1691.97i	0.188438i
$433$	10784.1i	1.19689i	0.801165	+	0.598443i	$0.204214\pi$
$433$	10784.1i	1.19689i	−0.801165	+	0.598443i	$0.795786\pi$
$434$	327.321	0.0362025
$435$	0	0
$436$	−16439.2	−1.80573
$437$	− 847.089i	− 0.0927272i
$438$	43.0590i	0.00469735i
$439$	6304.19	0.685382	0.342691	−	0.939448i	$-0.388662\pi$
$439$	6304.19	0.685382	0.342691	+	0.939448i	$0.388662\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	− 1460.10i	− 0.157127i
$443$	− 15494.8i	− 1.66181i	−0.556414	−	0.830905i	$-0.687823\pi$
$443$	− 15494.8i	− 1.66181i	0.556414	−	0.830905i	$-0.312177\pi$
$444$	3709.95	0.396546
$445$	0	0
$446$	1267.37	0.134556
$447$	6739.65i	0.713142i
$448$	− 3435.07i	− 0.362259i
$449$	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
$452$	15522.2i	1.61528i
$453$	3934.59i	0.408086i
$454$	220.044	0.0227471
$455$	0	0
$456$	−46.7228	−0.00479824
$457$	− 11670.4i	− 1.19457i	−0.802030	−	0.597283i	$-0.796247\pi$
$457$	− 11670.4i	− 1.19457i	0.802030	−	0.597283i	$-0.203753\pi$
$458$	− 746.695i	− 0.0761807i
$459$	−2062.45	−0.209732
$460$	0	0
$461$	12128.1	1.22530	0.612649	−	0.790355i	$-0.290104\pi$
$461$	12128.1	1.22530	0.612649	+	0.790355i	$0.290104\pi$
$462$	250.212i	0.0251968i
$463$	− 5161.64i	− 0.518103i	−0.965864	−	0.259051i	$-0.916590\pi$
$463$	− 5161.64i	− 0.518103i	0.965864	−	0.259051i	$-0.0834099\pi$
$464$	5713.05	0.571598
$465$	0	0
$466$	103.058	0.0102448
$467$	11680.2i	1.15738i	0.815547	+	0.578691i	$0.196436\pi$
$467$	11680.2i	1.15738i	−0.815547	+	0.578691i	$0.803564\pi$
$468$	− 5789.27i	− 0.571814i
$469$	5143.01	0.506358
$470$	0	0
$471$	5372.82	0.525619
$472$	1176.22i	0.114703i
$473$	17878.1i	1.73792i
$474$	910.932	0.0882711
$475$	0	0
$476$	4247.87	0.409036
$477$	1802.72i	0.173041i
$478$	467.608i	0.0447445i
$479$	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
$482$	1252.00i	0.118313i
$483$	− 4299.15i	− 0.405006i
$484$	9663.70	0.907560
$485$	0	0
$486$	57.3645	0.00535413
$487$	8630.11i	0.803014i	0.915856	+	0.401507i	$0.131513\pi$
$487$	8630.11i	0.803014i	−0.915856	+	0.401507i	$0.868487\pi$
$488$	− 580.286i	− 0.0538286i
$489$	1473.33	0.136250
$490$	0	0
$491$	17801.6	1.63621	0.818103	−	0.575072i	$-0.195026\pi$
$491$	17801.6	1.63621	0.818103	+	0.575072i	$0.195026\pi$
$492$	3739.02i	0.342618i
$493$	6963.99i	0.636191i
$494$	79.0916	0.00720344
$495$	0	0
$496$	12412.7	1.12369
$497$	4750.04i	0.428709i
$498$	− 82.2723i	− 0.00740303i
$499$	−200.167	−0.0179574	−0.00897868	−	0.999960i	$-0.502858\pi$
$499$	−200.167	−0.0179574	−0.00897868	+	0.999960i	$0.502858\pi$
$500$	0	0
$501$	−2478.18	−0.220992
$502$	− 707.269i	− 0.0628824i
$503$	− 16400.4i	− 1.45379i	−0.686747	−	0.726896i	$-0.740962\pi$
$503$	− 16400.4i	− 1.45379i	0.686747	−	0.726896i	$-0.259038\pi$
$504$	−237.128	−0.0209574
$505$	0	0
$506$	2439.23	0.214302
$507$	13077.7i	1.14556i
$508$	1781.51i	0.155594i
$509$	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
$512$	4700.94i	0.405770i
$513$	− 111.720i	− 0.00961510i
$514$	−228.717	−0.0196270
$515$	0	0
$516$	−8441.98	−0.720228
$517$	8865.91i	0.754201i
$518$	257.234i	0.0218189i
$519$	−8754.01	−0.740382
$520$	0	0
$521$	7113.18	0.598146	0.299073	−	0.954230i	$-0.403323\pi$
$521$	7113.18	0.598146	0.299073	+	0.954230i	$0.403323\pi$
$522$	− 193.695i	− 0.0162410i
$523$	8888.46i	0.743146i	0.928404	+	0.371573i	$0.121181\pi$
$523$	8888.46i	0.743146i	−0.928404	+	0.371573i	$0.878819\pi$
$524$	−3899.83	−0.325123
$525$	0	0
$526$	−1140.25	−0.0945196
$527$	15130.7i	1.25067i
$528$	9488.60i	0.782081i
$529$	−29743.8	−2.44463
$530$	0	0
$531$	−2812.49	−0.229852
$532$	230.101i	0.0187521i
$533$	− 12703.1i	− 1.03233i
$534$	649.026	0.0525957
$535$	0	0
$536$	−2765.42	−0.222850
$537$	2867.23i	0.230410i
$538$	1127.22i	0.0903305i
$539$	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	$0.508271\pi$
$542$	33.5001i	0.00265489i
$543$	− 618.241i	− 0.0488605i
$544$	−3430.14	−0.270342
$545$	0	0
$546$	401.406	0.0314626
$547$	1138.52i	0.0889940i	0.999010	+	0.0444970i	$0.0141685\pi$
$547$	1138.52i	0.0889940i	−0.999010	+	0.0444970i	$0.985831\pi$
$548$	14546.9i	1.13397i
$549$	1387.53	0.107866
$550$	0	0
$551$	−377.229	−0.0291660
$552$	2311.67i	0.178245i
$553$	− 9003.80i	− 0.692370i
$554$	854.832	0.0655566
$555$	0	0
$556$	24236.7	1.84868
$557$	− 19804.8i	− 1.50657i	−0.657696	−	0.753283i	$-0.728469\pi$
$557$	− 19804.8i	− 1.50657i	0.657696	−	0.753283i	$-0.271531\pi$
$558$	− 420.841i	− 0.0319276i
$559$	28681.1	2.17009
$560$	0	0
$561$	−11566.2	−0.870458
$562$	1367.04i	0.102607i
$563$	− 10276.3i	− 0.769265i	−0.923070	−	0.384632i	$-0.874328\pi$
$563$	− 10276.3i	− 0.769265i	0.923070	−	0.384632i	$-0.125672\pi$
$564$	−4186.46	−0.312556
$565$	0	0
$566$	124.182	0.00922219
$567$	− 567.000i	− 0.0419961i
$568$	− 2554.12i	− 0.188677i
$569$	−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	$0.552575\pi$
$572$	− 32466.3i	− 2.37323i
$573$	6055.89i	0.441516i
$574$	−259.249	−0.0188517
$575$	0	0
$576$	−4416.52	−0.319482
$577$	− 1104.77i	− 0.0797093i	−0.999205	−	0.0398547i	$-0.987311\pi$
$577$	− 1104.77i	− 0.0797093i	0.999205	−	0.0398547i	$-0.0126895\pi$
$578$	− 217.648i	− 0.0156626i
$579$	−3095.07	−0.222153
$580$	0	0
$581$	−813.192	−0.0580669
$582$	− 1003.03i	− 0.0714381i
$583$	10109.7i	0.718181i
$584$	−228.848	−0.0162154
$585$	0	0
$586$	452.041	0.0318663
$587$	− 10413.2i	− 0.732199i	−0.930576	−	0.366100i	$-0.880693\pi$
$587$	− 10413.2i	− 0.732199i	0.930576	−	0.366100i	$-0.119307\pi$
$588$	1167.81i	0.0819041i
$589$	−819.605	−0.0573365
$590$	0	0
$591$	−617.865	−0.0430044
$592$	9754.89i	0.677235i
$593$	3235.16i	0.224034i	0.993706	+	0.112017i	$0.0357311\pi$
$593$	3235.16i	0.224034i	−0.993706	+	0.112017i	$0.964269\pi$
$594$	321.701	0.0222215
$595$	0	0
$596$	−17847.2	−1.22659
$597$	− 5494.13i	− 0.376649i
$598$	− 3913.16i	− 0.267594i
$599$	−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	$0.459263\pi$
$602$	− 585.335i	− 0.0396287i
$603$	− 6612.44i	− 0.446566i
$604$	−10419.1	−0.701902
$605$	0	0
$606$	−976.751	−0.0654749
$607$	− 2224.05i	− 0.148717i	−0.997232	−	0.0743585i	$-0.976309\pi$
$607$	− 2224.05i	− 0.148717i	0.997232	−	0.0743585i	$-0.0236909\pi$
$608$	− 185.806i	− 0.0123938i
$609$	−1914.51	−0.127389
$610$	0	0
$611$	14223.2	0.941753
$612$	− 5461.55i	− 0.360735i
$613$	− 5914.50i	− 0.389697i	−0.980833	−	0.194849i	$-0.937578\pi$
$613$	− 5914.50i	− 0.389697i	0.980833	−	0.194849i	$-0.0624216\pi$
$614$	1474.05	0.0968856
$615$	0	0
$616$	−1329.82	−0.0869802
$617$	− 18591.2i	− 1.21306i	−0.795062	−	0.606528i	$-0.792562\pi$
$617$	− 18591.2i	− 1.21306i	0.795062	−	0.606528i	$-0.207438\pi$
$618$	926.638i	0.0603153i
$619$	−5125.97	−0.332844	−0.166422	−	0.986055i	$-0.553221\pi$
$619$	−5125.97	−0.332844	−0.166422	+	0.986055i	$0.553221\pi$
$620$	0	0
$621$	−5527.48	−0.357182
$622$	− 2280.13i	− 0.146985i
$623$	− 6415.08i	− 0.412544i
$624$	15222.2	0.976565
$625$	0	0
$626$	518.957	0.0331337
$627$	− 626.526i	− 0.0399060i
$628$	14227.7i	0.904056i
$629$	−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	$0.609047\pi$
$632$	4841.38i	0.304715i
$633$	− 3090.57i	− 0.194058i
$634$	−715.471	−0.0448186
$635$	0	0
$636$	−4773.76	−0.297629
$637$	− 3967.56i	− 0.246782i
$638$	− 1086.24i	− 0.0674056i
$639$	6107.20	0.378086
$640$	0	0
$641$	24025.7	1.48044	0.740218	−	0.672367i	$-0.234722\pi$
$641$	24025.7	1.48044	0.740218	+	0.672367i	$0.234722\pi$
$642$	896.341i	0.0551024i
$643$	− 14929.3i	− 0.915638i	−0.889045	−	0.457819i	$-0.848631\pi$
$643$	− 14929.3i	− 0.915638i	0.889045	−	0.457819i	$-0.151369\pi$
$644$	11384.5	0.696605
$645$	0	0
$646$	74.6144	0.00454437
$647$	14479.1i	0.879801i	0.898046	+	0.439901i	$0.144986\pi$
$647$	14479.1i	0.879801i	−0.898046	+	0.439901i	$0.855014\pi$
$648$	304.878i	0.0184827i
$649$	−15772.5	−0.953965
$650$	0	0
$651$	−4159.66	−0.250430
$652$	3901.50i	0.234347i
$653$	− 898.168i	− 0.0538255i	−0.999638	−	0.0269127i	$-0.991432\pi$
$653$	− 898.168i	− 0.0538255i	0.999638	−	0.0269127i	$-0.00856762\pi$
$654$	−1465.50	−0.0876232
$655$	0	0
$656$	−9831.33	−0.585135
$657$	− 547.203i	− 0.0324938i
$658$	− 290.273i	− 0.0171976i
$659$	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	$0.692651\pi$
$662$	1122.20i	0.0658848i
$663$	18555.3i	1.08692i
$664$	437.257	0.0255555
$665$	0	0
$666$	330.729	0.0192425
$667$	18663.9i	1.08346i
$668$	− 6562.44i	− 0.380102i
$669$	−16106.0	−0.930784
$670$	0	0
$671$	7781.30	0.447681
$672$	− 943.001i	− 0.0541325i
$673$	− 10132.2i	− 0.580336i	−0.956976	−	0.290168i	$-0.906289\pi$
$673$	− 10132.2i	− 0.580336i	0.956976	−	0.290168i	$-0.0937113\pi$
$674$	−2083.16	−0.119051
$675$	0	0
$676$	−34631.0	−1.97035
$677$	− 33177.3i	− 1.88347i	−0.336361	−	0.941733i	$-0.609196\pi$
$677$	− 33177.3i	− 1.88347i	0.336361	−	0.941733i	$-0.390804\pi$
$678$	1383.75i	0.0783816i
$679$	−9914.11	−0.560337
$680$	0	0
$681$	−2796.36	−0.157352
$682$	− 2360.08i	− 0.132511i
$683$	11423.6i	0.639987i	0.947420	+	0.319994i	$0.103681\pi$
$683$	11423.6i	0.639987i	−0.947420	+	0.319994i	$0.896319\pi$
$684$	295.844	0.0165378
$685$	0	0
$686$	−80.9713	−0.00450656
$687$	9489.15i	0.526978i
$688$	− 22197.2i	− 1.23003i
$689$	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	$0.682827\pi$
$692$	− 23181.4i	− 1.27345i
$693$	− 3179.74i	− 0.174298i
$694$	805.929	0.0440816
$695$	0	0
$696$	1029.44	0.0560645
$697$	− 11984.0i	− 0.651258i
$698$	1340.00i	0.0726643i
$699$	−1309.69	−0.0708682
$700$	0	0
$701$	11307.8	0.609258	0.304629	−	0.952471i	$-0.401468\pi$
$701$	11307.8	0.609258	0.304629	+	0.952471i	$0.401468\pi$
$702$	− 516.093i	− 0.0277474i
$703$	− 644.108i	− 0.0345562i
$704$	−24767.9	−1.32596
$705$	0	0
$706$	1469.71	0.0783475
$707$	9654.36i	0.513564i
$708$	− 7447.72i	− 0.395342i
$709$	−30859.2	−1.63461	−0.817307	−	0.576202i	$-0.804534\pi$
$709$	−30859.2	−1.63461	−0.817307	+	0.576202i	$0.804534\pi$
$710$	0	0
$711$	−11576.3	−0.610613
$712$	3449.42i	0.181562i
$713$	40551.0i	2.12994i
$714$	378.683	0.0198485
$715$	0	0
$716$	−7592.69	−0.396302
$717$	− 5942.46i	− 0.309519i
$718$	1158.56i	0.0602187i
$719$	−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
$722$	− 1615.15i	− 0.0832543i
$723$	− 15910.7i	− 0.818428i
$724$	1637.16	0.0840394
$725$	0	0
$726$	861.485	0.0440395
$727$	− 16743.0i	− 0.854146i	−0.904217	−	0.427073i	$-0.859545\pi$
$727$	− 16743.0i	− 0.854146i	0.904217	−	0.427073i	$-0.140455\pi$
$728$	2133.37i	0.108610i
$729$	−729.000	−0.0370370
$730$	0	0
$731$	27057.5	1.36903
$732$	3674.31i	0.185528i
$733$	8827.55i	0.444820i	0.974953	+	0.222410i	$0.0713924\pi$
$733$	8827.55i	0.444820i	−0.974953	+	0.222410i	$0.928608\pi$
$734$	922.195	0.0463744
$735$	0	0
$736$	−9192.97	−0.460404
$737$	− 37082.6i	− 1.85340i
$738$	333.321i	0.0166256i
$739$	−36154.0	−1.79966	−0.899829	−	0.436243i	$-0.856309\pi$
$739$	−36154.0	−1.79966	−0.899829	+	0.436243i	$0.856309\pi$
$740$	0	0
$741$	−1005.11	−0.0498296
$742$	− 330.994i	− 0.0163762i
$743$	1820.69i	0.0898987i	0.998989	+	0.0449494i	$0.0143126\pi$
$743$	1820.69i	0.0898987i	−0.998989	+	0.0449494i	$0.985687\pi$
$744$	2236.67	0.110215
$745$	0	0
$746$	−496.338	−0.0243596
$747$	1045.53i	0.0512102i
$748$	− 30628.5i	− 1.49718i
$749$	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	$0.264348\pi$
$752$	− 11007.8i	− 0.533795i
$753$	8988.12i	0.434987i
$754$	−1742.62	−0.0841678
$755$	0	0
$756$	1501.47	0.0722326
$757$	− 13518.3i	− 0.649050i	−0.945877	−	0.324525i	$-0.894795\pi$
$757$	− 13518.3i	− 0.649050i	0.945877	−	0.324525i	$-0.105205\pi$
$758$	1561.06i	0.0748025i
$759$	−30998.2	−1.48243
$760$	0	0
$761$	30695.2	1.46216	0.731078	−	0.682294i	$-0.239017\pi$
$761$	30695.2	1.46216	0.731078	+	0.682294i	$0.239017\pi$
$762$	158.815i	0.00755022i
$763$	14485.2i	0.687288i
$764$	−16036.5	−0.759400
$765$	0	0
$766$	0.560560	2.64410e−5 0
$767$	25303.2i	1.19119i
$768$	− 11440.9i	− 0.537551i
$769$	−4536.39	−0.212726	−0.106363	−	0.994327i	$-0.533921\pi$
$769$	−4536.39	−0.212726	−0.106363	+	0.994327i	$0.533921\pi$
$770$	0	0
$771$	2906.58	0.135769
$772$	− 8196.03i	− 0.382100i
$773$	− 31238.9i	− 1.45354i	−0.686883	−	0.726768i	$-0.741022\pi$
$773$	− 31238.9i	− 1.45354i	0.686883	−	0.726768i	$-0.258978\pi$
$774$	−752.573	−0.0349492
$775$	0	0
$776$	5330.86	0.246607
$777$	− 3268.98i	− 0.150932i
$778$	− 1821.73i	− 0.0839490i
$779$	649.155	0.0298567
$780$	0	0
$781$	34249.2	1.56919
$782$	− 3691.64i	− 0.168814i
$783$	2461.51i	0.112347i
$784$	−3070.62	−0.139879
$785$	0	0
$786$	−347.656	−0.0157767
$787$	39597.2i	1.79350i	0.442534	+	0.896752i	$0.354079\pi$
$787$	39597.2i	1.79350i	−0.442534	+	0.896752i	$0.645921\pi$
$788$	− 1636.16i	− 0.0739669i
$789$	14490.5	0.653836
$790$	0	0
$791$	13677.2	0.614799
$792$	1709.76i	0.0767093i
$793$	− 12483.3i	− 0.559008i
$794$	−1511.77	−0.0675700
$795$	0	0
$796$	14549.0	0.647832
$797$	− 16567.0i	− 0.736302i	−0.929766	−	0.368151i	$-0.879991\pi$
$797$	− 16567.0i	− 0.736302i	0.929766	−	0.368151i	$-0.120009\pi$
$798$	20.5127i	0 0.000909951i
$799$	13418.1	0.594115
$800$	0	0
$801$	−8247.96	−0.363829
$802$	− 2589.63i	− 0.114019i
$803$	− 3068.72i	− 0.134860i
$804$	17510.3	0.768087
$805$	0	0
$806$	−3786.19	−0.165463
$807$	− 14324.9i	− 0.624859i
$808$	− 5191.20i	− 0.226022i
$809$	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	$0.272302\pi$
$812$	− 5069.80i	− 0.219107i
$813$	− 425.726i	− 0.0183651i
$814$	1854.73	0.0798629
$815$	0	0
$816$	14360.5	0.616077
$817$	1465.67i	0.0627628i
$818$	94.5106i	0.00403971i
$819$	−5101.15	−0.217642
$820$	0	0
$821$	−14914.8	−0.634018	−0.317009	−	0.948423i	$-0.602678\pi$
$821$	−14914.8	−0.634018	−0.317009	+	0.948423i	$0.602678\pi$
$822$	1296.81i	0.0550260i
$823$	31077.6i	1.31628i	0.752897	+	0.658138i	$0.228656\pi$
$823$	31077.6i	1.31628i	−0.752897	+	0.658138i	$0.771344\pi$
$824$	−4924.85	−0.208210
$825$	0	0
$826$	516.396	0.0217527
$827$	15527.8i	0.652908i	0.945213	+	0.326454i	$0.105854\pi$
$827$	15527.8i	0.652908i	−0.945213	+	0.326454i	$0.894146\pi$
$828$	− 14637.3i	− 0.614348i
$829$	40221.5	1.68510	0.842551	−	0.538617i	$-0.181053\pi$
$829$	40221.5	1.68510	0.842551	+	0.538617i	$0.181053\pi$
$830$	0	0
$831$	−10863.4	−0.453486
$832$	39734.2i	1.65569i
$833$	− 3742.96i	− 0.155685i
$834$	2160.62	0.0897075
$835$	0	0
$836$	1659.10	0.0686377
$837$	5348.13i	0.220858i
$838$	3733.51i	0.153904i
$839$	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
$842$	457.152i	0.0187108i
$843$	− 17372.6i	− 0.709780i
$844$	8184.10	0.333778
$845$	0	0
$846$	−373.208	−0.0151669
$847$	− 8515.06i	− 0.345432i
$848$	− 12552.0i	− 0.508301i
$849$	−1578.13	−0.0637942
$850$	0	0
$851$	−31868.1	−1.28369
$852$	16172.4i	0.650302i
$853$	− 636.075i	− 0.0255320i	−0.999919	−	0.0127660i	$-0.995936\pi$
$853$	− 636.075i	− 0.0255320i	0.999919	−	0.0127660i	$-0.00406366\pi$
$854$	−254.763	−0.0102082
$855$	0	0
$856$	−4763.83	−0.190215
$857$	− 3941.46i	− 0.157104i	−0.996910	−	0.0785518i	$-0.974970\pi$
$857$	− 3941.46i	− 0.157104i	0.996910	−	0.0785518i	$-0.0250296\pi$
$858$	− 2894.26i	− 0.115161i
$859$	21781.6	0.865165	0.432583	−	0.901594i	$-0.357602\pi$
$859$	21781.6	0.865165	0.432583	+	0.901594i	$0.357602\pi$
$860$	0	0
$861$	3294.59	0.130406
$862$	479.432i	0.0189438i
$863$	44697.9i	1.76308i	0.472112	+	0.881538i	$0.343492\pi$
$863$	44697.9i	1.76308i	−0.472112	+	0.881538i	$0.656508\pi$
$864$	−1212.43	−0.0477404
$865$	0	0
$866$	−2545.79	−0.0998953
$867$	2765.92i	0.108345i
$868$	− 11015.2i	− 0.430736i
$869$	−64920.1	−2.53425
$870$	0	0
$871$	−59490.3	−2.31430
$872$	− 7788.78i	− 0.302478i
$873$	12746.7i	0.494171i
$874$	199.971	0.00773926
$875$	0	0
$876$	1449.04	0.0558888
$877$	− 20171.7i	− 0.776682i	−0.921516	−	0.388341i	$-0.873048\pi$
$877$	− 20171.7i	− 0.776682i	0.921516	−	0.388341i	$-0.126952\pi$
$878$	1488.22i	0.0572038i
$879$	−5744.63	−0.220434
$880$	0	0
$881$	11577.6	0.442744	0.221372	−	0.975189i	$-0.428946\pi$
$881$	11577.6	0.442744	0.221372	+	0.975189i	$0.428946\pi$
$882$	104.106i	0.00397441i
$883$	35388.3i	1.34871i	0.738407	+	0.674355i	$0.235578\pi$
$883$	35388.3i	1.34871i	−0.738407	+	0.674355i	$0.764422\pi$
$884$	−49136.1	−1.86949
$885$	0	0
$886$	3657.83	0.138699
$887$	− 41705.0i	− 1.57871i	−0.613936	−	0.789356i	$-0.710415\pi$
$887$	− 41705.0i	− 1.57871i	0.613936	−	0.789356i	$-0.289585\pi$
$888$	1757.74i	0.0664257i
$889$	1569.75	0.0592215
$890$	0	0
$891$	−4088.24	−0.153716
$892$	− 42650.2i	− 1.60093i
$893$	726.838i	0.0272371i
$894$	−1591.02	−0.0595207
$895$	0	0
$896$	3325.58	0.123995
$897$	49729.2i	1.85107i
$898$	57.1328i	0.00212310i
$899$	18058.3	0.669942
$900$	0	0
$901$	15300.5	0.565740
$902$	1869.27i	0.0690020i
$903$	7438.55i	0.274130i
$904$	−7354.31	−0.270576
$905$	0	0
$906$	−928.830	−0.0340600
$907$	− 28645.1i	− 1.04867i	−0.851511	−	0.524336i	$-0.824313\pi$
$907$	− 28645.1i	− 1.04867i	0.851511	−	0.524336i	$-0.175687\pi$
$908$	− 7405.02i	− 0.270643i
$909$	12412.8	0.452921
$910$	0	0
$911$	13337.8	0.485074	0.242537	−	0.970142i	$-0.422020\pi$
$911$	13337.8	0.485074	0.242537	+	0.970142i	$0.422020\pi$
$912$	777.887i	0.0282439i
$913$	5863.36i	0.212540i
$914$	2755.00	0.0997016
$915$	0	0
$916$	−25128.1	−0.906394
$917$	3436.29i	0.123747i
$918$	− 486.878i	− 0.0175048i
$919$	28911.0	1.03774	0.518871	−	0.854853i	$-0.326353\pi$
$919$	28911.0	1.03774	0.518871	+	0.854853i	$0.326353\pi$
$920$	0	0
$921$	−18732.5	−0.670203
$922$	2863.06i	0.102267i
$923$	− 54944.8i	− 1.95940i
$924$	8420.25	0.299790
$925$	0	0
$926$	1218.50	0.0432422
$927$	− 11775.9i	− 0.417229i
$928$	4093.84i	0.144814i
$929$	−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
$932$	− 3468.17i	− 0.121892i
$933$	28976.3i	1.01677i
$934$	−2757.33	−0.0965981
$935$	0	0
$936$	2742.91	0.0957851
$937$	− 19271.1i	− 0.671888i	−0.941882	−	0.335944i	$-0.890945\pi$
$937$	− 19271.1i	− 0.671888i	0.941882	−	0.335944i	$-0.109055\pi$
$938$	1214.10i	0.0422620i
$939$	−6595.01	−0.229201
$940$	0	0
$941$	−18115.2	−0.627563	−0.313782	−	0.949495i	$-0.601596\pi$
$941$	−18115.2	−0.627563	−0.313782	+	0.949495i	$0.601596\pi$
$942$	1268.35i	0.0438695i
$943$	− 32117.8i	− 1.10912i
$944$	19582.9	0.675180
$945$	0	0
$946$	−4220.44	−0.145051
$947$	− 2475.55i	− 0.0849468i	−0.999098	−	0.0424734i	$-0.986476\pi$
$947$	− 2475.55i	− 0.0849468i	0.999098	−	0.0424734i	$-0.0135238\pi$
$948$	− 30655.1i	− 1.05024i
$949$	−4923.04	−0.168397
$950$	0	0
$951$	9092.35	0.310031
$952$	2012.61i	0.0685179i
$953$	− 12866.6i	− 0.437344i	−0.975798	−	0.218672i	$-0.929827\pi$
$953$	− 12866.6i	− 0.437344i	0.975798	−	0.218672i	$-0.0701725\pi$
$954$	−425.564	−0.0144425
$955$	0	0
$956$	15736.2	0.532368
$957$	13804.2i	0.466277i
$958$	4357.50i	0.146957i
$959$	12817.9	0.431606
$960$	0	0
$961$	9444.26	0.317017
$962$	− 2975.48i	− 0.0997228i
$963$	− 11390.9i	− 0.381169i
$964$	42132.9	1.40768
$965$	0	0
$966$	1014.89	0.0338029
$967$	− 2142.86i	− 0.0712614i	−0.999365	−	0.0356307i	$-0.988656\pi$
$967$	− 2142.86i	− 0.0712614i	0.999365	−	0.0356307i	$-0.0113440\pi$
$968$	4578.58i	0.152026i
$969$	−948.215	−0.0314356
$970$	0	0
$971$	2879.06	0.0951529	0.0475765	−	0.998868i	$-0.484850\pi$
$971$	2879.06	0.0951529	0.0475765	+	0.998868i	$0.484850\pi$
$972$	− 1930.46i	− 0.0637032i
$973$	− 21355.9i	− 0.703636i
$974$	−2037.29	−0.0670217
$975$	0	0
$976$	−9661.18	−0.316851
$977$	48741.1i	1.59607i	0.602608	+	0.798037i	$0.294128\pi$
$977$	48741.1i	1.59607i	−0.602608	+	0.798037i	$0.705872\pi$
$978$	347.805i	0.0113718i
$979$	−46254.7	−1.51002
$980$	0	0
$981$	18623.9	0.606131
$982$	4202.40i	0.136562i
$983$	45756.8i	1.48466i	0.670037	+	0.742328i	$0.266278\pi$
$983$	45756.8i	1.48466i	−0.670037	+	0.742328i	$0.733722\pi$
$984$	−1771.52	−0.0573922
$985$	0	0
$986$	−1643.97	−0.0530982
$987$	3688.85i	0.118964i
$988$	− 2661.63i	− 0.0857062i
$989$	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	$0.809524\pi$
$992$	8894.70i	0.284684i
$993$	− 14261.2i	− 0.455756i
$994$	−1121.33	−0.0357812
$995$	0	0
$996$	−2768.67	−0.0880808
$997$	25565.3i	0.812097i	0.913852	+	0.406048i	$0.133094\pi$
$997$	25565.3i	0.812097i	−0.913852	+	0.406048i	$0.866906\pi$
$998$	− 47.2531i	− 0.00149877i
$999$	−4202.97	−0.133109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	525.4.d.k.274.3		4
5.2	odd	4		525.4.a.o.1.1		2
5.3	odd	4		105.4.a.d.1.2	✓	2
5.4	even	2	inner	525.4.d.k.274.2		4
15.2	even	4		1575.4.a.n.1.2		2
15.8	even	4		315.4.a.l.1.1		2
20.3	even	4		1680.4.a.bd.1.2		2
35.13	even	4		735.4.a.m.1.2		2
105.83	odd	4		2205.4.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.d.1.2	✓	2	5.3	odd	4
315.4.a.l.1.1		2	15.8	even	4
525.4.a.o.1.1		2	5.2	odd	4
525.4.d.k.274.2		4	5.4	even	2	inner
525.4.d.k.274.3		4	1.1	even	1	trivial
735.4.a.m.1.2		2	35.13	even	4
1575.4.a.n.1.2		2	15.2	even	4
1680.4.a.bd.1.2		2	20.3	even	4
2205.4.a.be.1.1		2	105.83	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:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	525.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.236068i q^{2} -3.00000i q^{3} +7.94427 q^{4} +0.708204 q^{6} -7.00000i q^{7} +3.76393i q^{8} -9.00000 q^{9} -50.4721 q^{11} -23.8328i q^{12} +80.9706i q^{13} +1.65248 q^{14} +62.6656 q^{16} +76.3870i q^{17} -2.12461i q^{18} -4.13777 q^{19} -21.0000 q^{21} -11.9149i q^{22} +204.721i q^{23} +11.2918 q^{24} -19.1146 q^{26} +27.0000i q^{27} -55.6099i q^{28} +91.1672 q^{29} +198.079 q^{31} +44.9048i q^{32} +151.416i q^{33} -18.0325 q^{34} -71.4984 q^{36} +155.666i q^{37} -0.976794i q^{38} +242.912 q^{39} -156.885 q^{41} -4.95743i q^{42} -354.217i q^{43} -400.964 q^{44} -48.3282 q^{46} -175.659i q^{47} -187.997i q^{48} -49.0000 q^{49} +229.161 q^{51} +643.252i q^{52} -200.302i q^{53} -6.37384 q^{54} +26.3475 q^{56} +12.4133i q^{57} +21.5217i q^{58} +312.498 q^{59} -154.170 q^{61} +46.7601i q^{62} +63.0000i q^{63} +490.724 q^{64} -35.7446 q^{66} +734.715i q^{67} +606.839i q^{68} +614.164 q^{69} -678.577 q^{71} -33.8754i q^{72} +60.8003i q^{73} -36.7477 q^{74} -32.8715 q^{76} +353.305i q^{77} +57.3437i q^{78} +1286.26 q^{79} +81.0000 q^{81} -37.0356i q^{82} -116.170i q^{83} -166.830 q^{84} +83.6192 q^{86} -273.502i q^{87} -189.974i q^{88} +916.440 q^{89} +566.794 q^{91} +1626.36i q^{92} -594.237i q^{93} +41.4676 q^{94} +134.714 q^{96} -1416.30i q^{97} -11.5673i q^{98} +454.249 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} - 24 q^{6} - 36 q^{9} - 184 q^{11} - 56 q^{14} + 36 q^{16} + 216 q^{19} - 84 q^{21} + 72 q^{24} - 792 q^{26} + 472 q^{29} - 120 q^{31} - 1056 q^{34} + 36 q^{36} - 48 q^{39} + 88 q^{41} + 24 q^{44}+ \cdots + 1656 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 - 24 * q^6 - 36 * q^9 - 184 * q^11 - 56 * q^14 + 36 * q^16 + 216 * q^19 - 84 * q^21 + 72 * q^24 - 792 * q^26 + 472 * q^29 - 120 * q^31 - 1056 * q^34 + 36 * q^36 - 48 * q^39 + 88 * q^41 + 24 * q^44 + 880 * q^46 - 196 * q^49 - 264 * q^51 + 216 * q^54 + 168 * q^56 + 928 * q^59 - 1368 * q^61 + 2428 * q^64 + 984 * q^66 + 1920 * q^69 - 1480 * q^71 + 336 * q^74 - 2296 * q^76 + 816 * q^79 + 324 * q^81 + 84 * q^84 + 2016 * q^86 + 2664 * q^89 - 112 * q^91 + 4960 * q^94 + 1800 * q^96 + 1656 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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