LMFDB - Embedded newform 405.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.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:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288 $ x^7 - 2x^6 - 44x^5 + 74x^4 + 479x^3 - 460x^2 - 1200x + 288
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.31551$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.31551 q^{2} +20.2546 q^{4} -5.00000 q^{5} -13.4337 q^{7} -65.1396 q^{8} +O(q^{10})$	\(q-5.31551 q^{2} +20.2546 q^{4} -5.00000 q^{5} -13.4337 q^{7} -65.1396 q^{8} +26.5775 q^{10} -46.9256 q^{11} -36.1347 q^{13} +71.4071 q^{14} +184.213 q^{16} -54.6071 q^{17} +111.339 q^{19} -101.273 q^{20} +249.434 q^{22} -35.9740 q^{23} +25.0000 q^{25} +192.074 q^{26} -272.095 q^{28} +58.1175 q^{29} -295.667 q^{31} -458.068 q^{32} +290.265 q^{34} +67.1686 q^{35} -53.0417 q^{37} -591.822 q^{38} +325.698 q^{40} +128.359 q^{41} -164.172 q^{43} -950.461 q^{44} +191.220 q^{46} +87.8318 q^{47} -162.535 q^{49} -132.888 q^{50} -731.894 q^{52} -479.247 q^{53} +234.628 q^{55} +875.067 q^{56} -308.924 q^{58} +635.614 q^{59} +48.0257 q^{61} +1571.62 q^{62} +961.163 q^{64} +180.673 q^{65} -28.9183 q^{67} -1106.05 q^{68} -357.035 q^{70} +576.183 q^{71} +835.057 q^{73} +281.944 q^{74} +2255.12 q^{76} +630.386 q^{77} -203.737 q^{79} -921.064 q^{80} -682.294 q^{82} +464.478 q^{83} +273.036 q^{85} +872.656 q^{86} +3056.72 q^{88} +993.782 q^{89} +485.423 q^{91} -728.640 q^{92} -466.871 q^{94} -556.693 q^{95} +881.415 q^{97} +863.956 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8}+O(q^{10}) $ 7 * q - 2 * q^2 + 36 * q^4 - 35 * q^5 + 22 * q^7 - 18 * q^8	$ 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8} + 10 q^{10} - 23 q^{11} + 96 q^{13} + 21 q^{14} + 324 q^{16} - 161 q^{17} + 279 q^{19} - 180 q^{20} + 311 q^{22} - 96 q^{23} + 175 q^{25} + 358 q^{26} + 337 q^{28} + 296 q^{29} + 244 q^{31} + 314 q^{32} + 125 q^{34} - 110 q^{35} + 404 q^{37} - 305 q^{38} + 90 q^{40} + 47 q^{41} + 525 q^{43} - 55 q^{44} + 717 q^{46} - 164 q^{47} + 1225 q^{49} - 50 q^{50} + 1682 q^{52} - 506 q^{53} + 115 q^{55} + 981 q^{56} + 1183 q^{58} + 85 q^{59} + 828 q^{61} + 786 q^{62} + 2236 q^{64} - 480 q^{65} + 1093 q^{67} - 2473 q^{68} - 105 q^{70} - 328 q^{71} + 2085 q^{73} + 1316 q^{74} + 2789 q^{76} - 24 q^{77} + 2110 q^{79} - 1620 q^{80} - 62 q^{82} - 1290 q^{83} + 805 q^{85} + 2569 q^{86} + 2271 q^{88} + 3048 q^{89} + 3338 q^{91} - 2763 q^{92} - 517 q^{94} - 1395 q^{95} + 1787 q^{97} - 1279 q^{98}+O(q^{100}) $ 7 * q - 2 * q^2 + 36 * q^4 - 35 * q^5 + 22 * q^7 - 18 * q^8 + 10 * q^10 - 23 * q^11 + 96 * q^13 + 21 * q^14 + 324 * q^16 - 161 * q^17 + 279 * q^19 - 180 * q^20 + 311 * q^22 - 96 * q^23 + 175 * q^25 + 358 * q^26 + 337 * q^28 + 296 * q^29 + 244 * q^31 + 314 * q^32 + 125 * q^34 - 110 * q^35 + 404 * q^37 - 305 * q^38 + 90 * q^40 + 47 * q^41 + 525 * q^43 - 55 * q^44 + 717 * q^46 - 164 * q^47 + 1225 * q^49 - 50 * q^50 + 1682 * q^52 - 506 * q^53 + 115 * q^55 + 981 * q^56 + 1183 * q^58 + 85 * q^59 + 828 * q^61 + 786 * q^62 + 2236 * q^64 - 480 * q^65 + 1093 * q^67 - 2473 * q^68 - 105 * q^70 - 328 * q^71 + 2085 * q^73 + 1316 * q^74 + 2789 * q^76 - 24 * q^77 + 2110 * q^79 - 1620 * q^80 - 62 * q^82 - 1290 * q^83 + 805 * q^85 + 2569 * q^86 + 2271 * q^88 + 3048 * q^89 + 3338 * q^91 - 2763 * q^92 - 517 * q^94 - 1395 * q^95 + 1787 * q^97 - 1279 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−5.31551	−1.87932	−0.939658	−	0.342115i	$-0.888857\pi$
$2$	−5.31551	−1.87932	−0.939658	+	0.342115i	$0.888857\pi$
$3$	0	0
$3$	0	0
$4$	20.2546	2.53183
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−13.4337	−0.725353	−0.362676	−	0.931915i	$-0.618137\pi$
$7$	−13.4337	−0.725353	−0.362676	+	0.931915i	$0.618137\pi$
$8$	−65.1396	−2.87879
$9$	0	0
$10$	26.5775	0.840456
$11$	−46.9256	−1.28624	−0.643119	−	0.765766i	$-0.722360\pi$
$11$	−46.9256	−1.28624	−0.643119	+	0.765766i	$0.722360\pi$
$12$	0	0
$13$	−36.1347	−0.770920	−0.385460	−	0.922725i	$-0.625957\pi$
$13$	−36.1347	−0.770920	−0.385460	+	0.922725i	$0.625957\pi$
$14$	71.4071	1.36317
$15$	0	0
$16$	184.213	2.87833
$17$	−54.6071	−0.779069	−0.389535	−	0.921012i	$-0.627364\pi$
$17$	−54.6071	−0.779069	−0.389535	+	0.921012i	$0.627364\pi$
$18$	0	0
$19$	111.339	1.34436	0.672180	−	0.740388i	$-0.265358\pi$
$19$	111.339	1.34436	0.672180	+	0.740388i	$0.265358\pi$
$20$	−101.273	−1.13227
$21$	0	0
$22$	249.434	2.41725
$23$	−35.9740	−0.326135	−0.163067	−	0.986615i	$-0.552139\pi$
$23$	−35.9740	−0.326135	−0.163067	+	0.986615i	$0.552139\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	192.074	1.44880
$27$	0	0
$28$	−272.095	−1.83647
$29$	58.1175	0.372143	0.186072	−	0.982536i	$-0.440424\pi$
$29$	58.1175	0.372143	0.186072	+	0.982536i	$0.440424\pi$
$30$	0	0
$31$	−295.667	−1.71301	−0.856506	−	0.516137i	$-0.827369\pi$
$31$	−295.667	−1.71301	−0.856506	+	0.516137i	$0.827369\pi$
$32$	−458.068	−2.53049
$33$	0	0
$34$	290.265	1.46412
$35$	67.1686	0.324388
$36$	0	0
$37$	−53.0417	−0.235676	−0.117838	−	0.993033i	$-0.537596\pi$
$37$	−53.0417	−0.235676	−0.117838	+	0.993033i	$0.537596\pi$
$38$	−591.822	−2.52648
$39$	0	0
$40$	325.698	1.28743
$41$	128.359	0.488935	0.244467	−	0.969658i	$-0.421387\pi$
$41$	128.359	0.488935	0.244467	+	0.969658i	$0.421387\pi$
$42$	0	0
$43$	−164.172	−0.582231	−0.291116	−	0.956688i	$-0.594026\pi$
$43$	−164.172	−0.582231	−0.291116	+	0.956688i	$0.594026\pi$
$44$	−950.461	−3.25653
$45$	0	0
$46$	191.220	0.612910
$47$	87.8318	0.272587	0.136294	−	0.990668i	$-0.456481\pi$
$47$	87.8318	0.272587	0.136294	+	0.990668i	$0.456481\pi$
$48$	0	0
$49$	−162.535	−0.473863
$50$	−132.888	−0.375863
$51$	0	0
$52$	−731.894	−1.95184
$53$	−479.247	−1.24207	−0.621035	−	0.783783i	$-0.713287\pi$
$53$	−479.247	−1.24207	−0.621035	+	0.783783i	$0.713287\pi$
$54$	0	0
$55$	234.628	0.575223
$56$	875.067	2.08814
$57$	0	0
$58$	−308.924	−0.699375
$59$	635.614	1.40254	0.701271	−	0.712895i	$-0.252616\pi$
$59$	635.614	1.40254	0.701271	+	0.712895i	$0.252616\pi$
$60$	0	0
$61$	48.0257	0.100804	0.0504021	−	0.998729i	$-0.483950\pi$
$61$	48.0257	0.100804	0.0504021	+	0.998729i	$0.483950\pi$
$62$	1571.62	3.21929
$63$	0	0
$64$	961.163	1.87727
$65$	180.673	0.344766
$66$	0	0
$67$	−28.9183	−0.0527303	−0.0263652	−	0.999652i	$-0.508393\pi$
$67$	−28.9183	−0.0527303	−0.0263652	+	0.999652i	$0.508393\pi$
$68$	−1106.05	−1.97247
$69$	0	0
$70$	−357.035	−0.609627
$71$	576.183	0.963103	0.481552	−	0.876418i	$-0.340073\pi$
$71$	576.183	0.963103	0.481552	+	0.876418i	$0.340073\pi$
$72$	0	0
$73$	835.057	1.33885	0.669425	−	0.742880i	$-0.266541\pi$
$73$	835.057	1.33885	0.669425	+	0.742880i	$0.266541\pi$
$74$	281.944	0.442909
$75$	0	0
$76$	2255.12	3.40369
$77$	630.386	0.932976
$78$	0	0
$79$	−203.737	−0.290155	−0.145078	−	0.989420i	$-0.546343\pi$
$79$	−203.737	−0.290155	−0.145078	+	0.989420i	$0.546343\pi$
$80$	−921.064	−1.28723
$81$	0	0
$82$	−682.294	−0.918863
$83$	464.478	0.614253	0.307127	−	0.951669i	$-0.400632\pi$
$83$	464.478	0.614253	0.307127	+	0.951669i	$0.400632\pi$
$84$	0	0
$85$	273.036	0.348410
$86$	872.656	1.09420
$87$	0	0
$88$	3056.72	3.70281
$89$	993.782	1.18360	0.591801	−	0.806084i	$-0.298417\pi$
$89$	993.782	1.18360	0.591801	+	0.806084i	$0.298417\pi$
$90$	0	0
$91$	485.423	0.559189
$92$	−728.640	−0.825717
$93$	0	0
$94$	−466.871	−0.512277
$95$	−556.693	−0.601216
$96$	0	0
$97$	881.415	0.922620	0.461310	−	0.887239i	$-0.347380\pi$
$97$	881.415	0.922620	0.461310	+	0.887239i	$0.347380\pi$
$98$	863.956	0.890538
$99$	0	0
$100$	506.366	0.506366
$101$	1209.74	1.19182	0.595909	−	0.803052i	$-0.296792\pi$
$101$	1209.74	1.19182	0.595909	+	0.803052i	$0.296792\pi$
$102$	0	0
$103$	744.843	0.712539	0.356270	−	0.934383i	$-0.384048\pi$
$103$	744.843	0.712539	0.356270	+	0.934383i	$0.384048\pi$
$104$	2353.80	2.21931
$105$	0	0
$106$	2547.44	2.33424
$107$	−1000.14	−0.903622	−0.451811	−	0.892114i	$-0.649222\pi$
$107$	−1000.14	−0.903622	−0.451811	+	0.892114i	$0.649222\pi$
$108$	0	0
$109$	−915.517	−0.804502	−0.402251	−	0.915530i	$-0.631772\pi$
$109$	−915.517	−0.804502	−0.402251	+	0.915530i	$0.631772\pi$
$110$	−1247.17	−1.08103
$111$	0	0
$112$	−2474.66	−2.08780
$113$	−1324.90	−1.10298	−0.551488	−	0.834183i	$-0.685940\pi$
$113$	−1324.90	−1.10298	−0.551488	+	0.834183i	$0.685940\pi$
$114$	0	0
$115$	179.870	0.145852
$116$	1177.15	0.942202
$117$	0	0
$118$	−3378.61	−2.63582
$119$	733.577	0.565100
$120$	0	0
$121$	871.015	0.654407
$122$	−255.281	−0.189443
$123$	0	0
$124$	−5988.62	−4.33705
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−993.635	−0.694259	−0.347129	−	0.937817i	$-0.612843\pi$
$127$	−993.635	−0.694259	−0.347129	+	0.937817i	$0.612843\pi$
$128$	−1444.52	−0.997494
$129$	0	0
$130$	−960.371	−0.647924
$131$	1383.99	0.923051	0.461526	−	0.887127i	$-0.347302\pi$
$131$	1383.99	0.923051	0.461526	+	0.887127i	$0.347302\pi$
$132$	0	0
$133$	−1495.69	−0.975136
$134$	153.715	0.0990970
$135$	0	0
$136$	3557.09	2.24278
$137$	1547.14	0.964823	0.482411	−	0.875945i	$-0.339761\pi$
$137$	1547.14	0.964823	0.482411	+	0.875945i	$0.339761\pi$
$138$	0	0
$139$	−539.505	−0.329210	−0.164605	−	0.986360i	$-0.552635\pi$
$139$	−539.505	−0.329210	−0.164605	+	0.986360i	$0.552635\pi$
$140$	1360.48	0.821294
$141$	0	0
$142$	−3062.70	−1.80998
$143$	1695.64	0.991586
$144$	0	0
$145$	−290.588	−0.166427
$146$	−4438.75	−2.51612
$147$	0	0
$148$	−1074.34	−0.596691
$149$	−1204.67	−0.662351	−0.331176	−	0.943569i	$-0.607445\pi$
$149$	−1204.67	−0.662351	−0.331176	+	0.943569i	$0.607445\pi$
$150$	0	0
$151$	2972.91	1.60220	0.801100	−	0.598531i	$-0.204249\pi$
$151$	2972.91	1.60220	0.801100	+	0.598531i	$0.204249\pi$
$152$	−7252.55	−3.87013
$153$	0	0
$154$	−3350.82	−1.75336
$155$	1478.33	0.766082
$156$	0	0
$157$	−395.761	−0.201179	−0.100590	−	0.994928i	$-0.532073\pi$
$157$	−395.761	−0.201179	−0.100590	+	0.994928i	$0.532073\pi$
$158$	1082.97	0.545293
$159$	0	0
$160$	2290.34	1.13167
$161$	483.265	0.236563
$162$	0	0
$163$	3861.43	1.85553	0.927764	−	0.373169i	$-0.121729\pi$
$163$	3861.43	1.85553	0.927764	+	0.373169i	$0.121729\pi$
$164$	2599.86	1.23790
$165$	0	0
$166$	−2468.93	−1.15438
$167$	2034.97	0.942939	0.471470	−	0.881882i	$-0.343724\pi$
$167$	2034.97	0.942939	0.471470	+	0.881882i	$0.343724\pi$
$168$	0	0
$169$	−891.286	−0.405683
$170$	−1451.32	−0.654773
$171$	0	0
$172$	−3325.24	−1.47411
$173$	1546.92	0.679825	0.339913	−	0.940457i	$-0.389602\pi$
$173$	1546.92	0.679825	0.339913	+	0.940457i	$0.389602\pi$
$174$	0	0
$175$	−335.843	−0.145071
$176$	−8644.31	−3.70221
$177$	0	0
$178$	−5282.45	−2.22436
$179$	−823.973	−0.344059	−0.172030	−	0.985092i	$-0.555033\pi$
$179$	−823.973	−0.344059	−0.172030	+	0.985092i	$0.555033\pi$
$180$	0	0
$181$	−4403.55	−1.80836	−0.904180	−	0.427152i	$-0.859517\pi$
$181$	−4403.55	−1.80836	−0.904180	+	0.427152i	$0.859517\pi$
$182$	−2580.27	−1.05089
$183$	0	0
$184$	2343.33	0.938873
$185$	265.209	0.105397
$186$	0	0
$187$	2562.48	1.00207
$188$	1779.00	0.690144
$189$	0	0
$190$	2959.11	1.12988
$191$	−3046.93	−1.15428	−0.577141	−	0.816645i	$-0.695832\pi$
$191$	−3046.93	−1.15428	−0.577141	+	0.816645i	$0.695832\pi$
$192$	0	0
$193$	1890.90	0.705235	0.352617	−	0.935768i	$-0.385292\pi$
$193$	1890.90	0.705235	0.352617	+	0.935768i	$0.385292\pi$
$194$	−4685.17	−1.73389
$195$	0	0
$196$	−3292.09	−1.19974
$197$	3652.00	1.32078	0.660392	−	0.750921i	$-0.270390\pi$
$197$	3652.00	1.32078	0.660392	+	0.750921i	$0.270390\pi$
$198$	0	0
$199$	3217.26	1.14606	0.573029	−	0.819535i	$-0.305768\pi$
$199$	3217.26	1.14606	0.573029	+	0.819535i	$0.305768\pi$
$200$	−1628.49	−0.575758
$201$	0	0
$202$	−6430.38	−2.23980
$203$	−780.735	−0.269935
$204$	0	0
$205$	−641.795	−0.218658
$206$	−3959.22	−1.33909
$207$	0	0
$208$	−6656.47	−2.21896
$209$	−5224.64	−1.72917
$210$	0	0
$211$	4947.80	1.61432	0.807159	−	0.590335i	$-0.201004\pi$
$211$	4947.80	1.61432	0.807159	+	0.590335i	$0.201004\pi$
$212$	−9706.97	−3.14471
$213$	0	0
$214$	5316.27	1.69819
$215$	820.858	0.260382
$216$	0	0
$217$	3971.91	1.24254
$218$	4866.44	1.51191
$219$	0	0
$220$	4752.31	1.45637
$221$	1973.21	0.600600
$222$	0	0
$223$	1104.11	0.331555	0.165778	−	0.986163i	$-0.446987\pi$
$223$	1104.11	0.331555	0.165778	+	0.986163i	$0.446987\pi$
$224$	6153.56	1.83550
$225$	0	0
$226$	7042.53	2.07284
$227$	−3820.24	−1.11700	−0.558498	−	0.829506i	$-0.688622\pi$
$227$	−3820.24	−1.11700	−0.558498	+	0.829506i	$0.688622\pi$
$228$	0	0
$229$	144.770	0.0417757	0.0208879	−	0.999782i	$-0.493351\pi$
$229$	144.770	0.0417757	0.0208879	+	0.999782i	$0.493351\pi$
$230$	−956.101	−0.274102
$231$	0	0
$232$	−3785.75	−1.07132
$233$	1286.31	0.361670	0.180835	−	0.983513i	$-0.442120\pi$
$233$	1286.31	0.361670	0.180835	+	0.983513i	$0.442120\pi$
$234$	0	0
$235$	−439.159	−0.121905
$236$	12874.1	3.55099
$237$	0	0
$238$	−3899.34	−1.06200
$239$	4219.45	1.14198	0.570991	−	0.820956i	$-0.306559\pi$
$239$	4219.45	1.14198	0.570991	+	0.820956i	$0.306559\pi$
$240$	0	0
$241$	2283.13	0.610247	0.305124	−	0.952313i	$-0.401302\pi$
$241$	2283.13	0.610247	0.305124	+	0.952313i	$0.401302\pi$
$242$	−4629.89	−1.22984
$243$	0	0
$244$	972.742	0.255219
$245$	812.675	0.211918
$246$	0	0
$247$	−4023.19	−1.03639
$248$	19259.6	4.93140
$249$	0	0
$250$	664.439	0.168091
$251$	−7922.23	−1.99222	−0.996109	−	0.0881290i	$-0.971911\pi$
$251$	−7922.23	−1.99222	−0.996109	+	0.0881290i	$0.971911\pi$
$252$	0	0
$253$	1688.10	0.419487
$254$	5281.67	1.30473
$255$	0	0
$256$	−10.9239	−0.00266698
$257$	2262.18	0.549069	0.274535	−	0.961577i	$-0.411476\pi$
$257$	2262.18	0.549069	0.274535	+	0.961577i	$0.411476\pi$
$258$	0	0
$259$	712.548	0.170948
$260$	3659.47	0.872888
$261$	0	0
$262$	−7356.61	−1.73471
$263$	163.470	0.0383269	0.0191635	−	0.999816i	$-0.493900\pi$
$263$	163.470	0.0383269	0.0191635	+	0.999816i	$0.493900\pi$
$264$	0	0
$265$	2396.24	0.555470
$266$	7950.37	1.83259
$267$	0	0
$268$	−585.729	−0.133504
$269$	3304.13	0.748908	0.374454	−	0.927246i	$-0.377830\pi$
$269$	3304.13	0.748908	0.374454	+	0.927246i	$0.377830\pi$
$270$	0	0
$271$	−1954.96	−0.438212	−0.219106	−	0.975701i	$-0.570314\pi$
$271$	−1954.96	−0.438212	−0.219106	+	0.975701i	$0.570314\pi$
$272$	−10059.3	−2.24242
$273$	0	0
$274$	−8223.81	−1.81321
$275$	−1173.14	−0.257247
$276$	0	0
$277$	−4243.35	−0.920426	−0.460213	−	0.887809i	$-0.652227\pi$
$277$	−4243.35	−0.920426	−0.460213	+	0.887809i	$0.652227\pi$
$278$	2867.74	0.618689
$279$	0	0
$280$	−4375.33	−0.933844
$281$	−2622.29	−0.556701	−0.278350	−	0.960480i	$-0.589788\pi$
$281$	−2622.29	−0.556701	−0.278350	+	0.960480i	$0.589788\pi$
$282$	0	0
$283$	−7.07471	−0.00148603	−0.000743017	−	1.00000i	$-0.500237\pi$
$283$	−7.07471	−0.00148603	−0.000743017		1.00000i	$0.500237\pi$
$284$	11670.4	2.43841
$285$	0	0
$286$	−9013.20	−1.86350
$287$	−1724.34	−0.354650
$288$	0	0
$289$	−1931.06	−0.393051
$290$	1544.62	0.312770
$291$	0	0
$292$	16913.8	3.38974
$293$	−4477.49	−0.892757	−0.446378	−	0.894844i	$-0.647286\pi$
$293$	−4477.49	−0.892757	−0.446378	+	0.894844i	$0.647286\pi$
$294$	0	0
$295$	−3178.07	−0.627236
$296$	3455.12	0.678461
$297$	0	0
$298$	6403.43	1.24477
$299$	1299.91	0.251424
$300$	0	0
$301$	2205.44	0.422323
$302$	−15802.5	−3.01104
$303$	0	0
$304$	20510.0	3.86951
$305$	−240.128	−0.0450810
$306$	0	0
$307$	−3889.78	−0.723132	−0.361566	−	0.932347i	$-0.617758\pi$
$307$	−3889.78	−0.723132	−0.361566	+	0.932347i	$0.617758\pi$
$308$	12768.2	2.36214
$309$	0	0
$310$	−7858.10	−1.43971
$311$	−7295.66	−1.33022	−0.665111	−	0.746745i	$-0.731616\pi$
$311$	−7295.66	−1.33022	−0.665111	+	0.746745i	$0.731616\pi$
$312$	0	0
$313$	−636.072	−0.114866	−0.0574328	−	0.998349i	$-0.518291\pi$
$313$	−636.072	−0.114866	−0.0574328	+	0.998349i	$0.518291\pi$
$314$	2103.67	0.378080
$315$	0	0
$316$	−4126.62	−0.734623
$317$	−6146.36	−1.08900	−0.544501	−	0.838760i	$-0.683281\pi$
$317$	−6146.36	−1.08900	−0.544501	+	0.838760i	$0.683281\pi$
$318$	0	0
$319$	−2727.20	−0.478664
$320$	−4805.82	−0.839542
$321$	0	0
$322$	−2568.80	−0.444576
$323$	−6079.89	−1.04735
$324$	0	0
$325$	−903.367	−0.154184
$326$	−20525.5	−3.48712
$327$	0	0
$328$	−8361.25	−1.40754
$329$	−1179.91	−0.197722
$330$	0	0
$331$	11734.4	1.94858	0.974291	−	0.225293i	$-0.0723340\pi$
$331$	11734.4	1.94858	0.974291	+	0.225293i	$0.0723340\pi$
$332$	9407.82	1.55518
$333$	0	0
$334$	−10816.9	−1.77208
$335$	144.592	0.0235817
$336$	0	0
$337$	6283.29	1.01565	0.507823	−	0.861461i	$-0.330450\pi$
$337$	6283.29	1.01565	0.507823	+	0.861461i	$0.330450\pi$
$338$	4737.64	0.762407
$339$	0	0
$340$	5530.24	0.882115
$341$	13874.4	2.20334
$342$	0	0
$343$	6791.22	1.06907
$344$	10694.1	1.67612
$345$	0	0
$346$	−8222.64	−1.27761
$347$	4800.82	0.742713	0.371357	−	0.928490i	$-0.378893\pi$
$347$	4800.82	0.742713	0.371357	+	0.928490i	$0.378893\pi$
$348$	0	0
$349$	−3892.46	−0.597016	−0.298508	−	0.954407i	$-0.596489\pi$
$349$	−3892.46	−0.597016	−0.298508	+	0.954407i	$0.596489\pi$
$350$	1785.18	0.272633
$351$	0	0
$352$	21495.2	3.25482
$353$	−2966.27	−0.447249	−0.223624	−	0.974675i	$-0.571789\pi$
$353$	−2966.27	−0.447249	−0.223624	+	0.974675i	$0.571789\pi$
$354$	0	0
$355$	−2880.91	−0.430713
$356$	20128.7	2.99668
$357$	0	0
$358$	4379.84	0.646596
$359$	−9584.91	−1.40911	−0.704557	−	0.709647i	$-0.748854\pi$
$359$	−9584.91	−1.40911	−0.704557	+	0.709647i	$0.748854\pi$
$360$	0	0
$361$	5537.30	0.807304
$362$	23407.1	3.39848
$363$	0	0
$364$	9832.06	1.41577
$365$	−4175.28	−0.598752
$366$	0	0
$367$	4537.18	0.645338	0.322669	−	0.946512i	$-0.395420\pi$
$367$	4537.18	0.645338	0.322669	+	0.946512i	$0.395420\pi$
$368$	−6626.88	−0.938722
$369$	0	0
$370$	−1409.72	−0.198075
$371$	6438.07	0.900938
$372$	0	0
$373$	13131.3	1.82282	0.911411	−	0.411497i	$-0.134994\pi$
$373$	13131.3	1.82282	0.911411	+	0.411497i	$0.134994\pi$
$374$	−13620.9	−1.88320
$375$	0	0
$376$	−5721.33	−0.784721
$377$	−2100.06	−0.286892
$378$	0	0
$379$	−9451.10	−1.28092	−0.640462	−	0.767990i	$-0.721257\pi$
$379$	−9451.10	−1.28092	−0.640462	+	0.767990i	$0.721257\pi$
$380$	−11275.6	−1.52218
$381$	0	0
$382$	16196.0	2.16926
$383$	−8597.03	−1.14697	−0.573483	−	0.819218i	$-0.694408\pi$
$383$	−8597.03	−1.14697	−0.573483	+	0.819218i	$0.694408\pi$
$384$	0	0
$385$	−3151.93	−0.417240
$386$	−10051.1	−1.32536
$387$	0	0
$388$	17852.7	2.33592
$389$	4442.08	0.578978	0.289489	−	0.957181i	$-0.406515\pi$
$389$	4442.08	0.578978	0.289489	+	0.957181i	$0.406515\pi$
$390$	0	0
$391$	1964.44	0.254082
$392$	10587.5	1.36415
$393$	0	0
$394$	−19412.3	−2.48217
$395$	1018.69	0.129761
$396$	0	0
$397$	10445.6	1.32053	0.660265	−	0.751033i	$-0.270444\pi$
$397$	10445.6	1.32053	0.660265	+	0.751033i	$0.270444\pi$
$398$	−17101.4	−2.15380
$399$	0	0
$400$	4605.32	0.575665
$401$	−12683.4	−1.57950	−0.789751	−	0.613427i	$-0.789790\pi$
$401$	−12683.4	−1.57950	−0.789751	+	0.613427i	$0.789790\pi$
$402$	0	0
$403$	10683.8	1.32059
$404$	24502.8	3.01748
$405$	0	0
$406$	4150.00	0.507293
$407$	2489.02	0.303135
$408$	0	0
$409$	−11112.6	−1.34348	−0.671739	−	0.740788i	$-0.734452\pi$
$409$	−11112.6	−1.34348	−0.671739	+	0.740788i	$0.734452\pi$
$410$	3411.47	0.410928
$411$	0	0
$412$	15086.5	1.80403
$413$	−8538.67	−1.01734
$414$	0	0
$415$	−2322.39	−0.274702
$416$	16552.2	1.95081
$417$	0	0
$418$	27771.6	3.24965
$419$	2018.52	0.235349	0.117675	−	0.993052i	$-0.462456\pi$
$419$	2018.52	0.235349	0.117675	+	0.993052i	$0.462456\pi$
$420$	0	0
$421$	−10957.7	−1.26852	−0.634259	−	0.773121i	$-0.718695\pi$
$421$	−10957.7	−1.26852	−0.634259	+	0.773121i	$0.718695\pi$
$422$	−26300.1	−3.03381
$423$	0	0
$424$	31218.0	3.57565
$425$	−1365.18	−0.155814
$426$	0	0
$427$	−645.164	−0.0731186
$428$	−20257.5	−2.28782
$429$	0	0
$430$	−4363.28	−0.489340
$431$	−2124.75	−0.237460	−0.118730	−	0.992927i	$-0.537882\pi$
$431$	−2124.75	−0.237460	−0.118730	+	0.992927i	$0.537882\pi$
$432$	0	0
$433$	−16169.2	−1.79456	−0.897279	−	0.441463i	$-0.854460\pi$
$433$	−16169.2	−1.79456	−0.897279	+	0.441463i	$0.854460\pi$
$434$	−21112.7	−2.33512
$435$	0	0
$436$	−18543.5	−2.03686
$437$	−4005.30	−0.438443
$438$	0	0
$439$	692.197	0.0752545	0.0376273	−	0.999292i	$-0.488020\pi$
$439$	692.197	0.0752545	0.0376273	+	0.999292i	$0.488020\pi$
$440$	−15283.6	−1.65595
$441$	0	0
$442$	−10488.6	−1.12872
$443$	−6670.51	−0.715408	−0.357704	−	0.933835i	$-0.616440\pi$
$443$	−6670.51	−0.715408	−0.357704	+	0.933835i	$0.616440\pi$
$444$	0	0
$445$	−4968.91	−0.529323
$446$	−5868.92	−0.623097
$447$	0	0
$448$	−12912.0	−1.36169
$449$	−9743.18	−1.02407	−0.512037	−	0.858964i	$-0.671109\pi$
$449$	−9743.18	−1.02407	−0.512037	+	0.858964i	$0.671109\pi$
$450$	0	0
$451$	−6023.33	−0.628886
$452$	−26835.4	−2.79255
$453$	0	0
$454$	20306.5	2.09919
$455$	−2427.12	−0.250077
$456$	0	0
$457$	4442.72	0.454752	0.227376	−	0.973807i	$-0.426985\pi$
$457$	4442.72	0.454752	0.227376	+	0.973807i	$0.426985\pi$
$458$	−769.524	−0.0785098
$459$	0	0
$460$	3643.20	0.369272
$461$	18939.4	1.91344	0.956720	−	0.291011i	$-0.0939915\pi$
$461$	18939.4	1.91344	0.956720	+	0.291011i	$0.0939915\pi$
$462$	0	0
$463$	−16792.2	−1.68553	−0.842764	−	0.538283i	$-0.819073\pi$
$463$	−16792.2	−1.68553	−0.842764	+	0.538283i	$0.819073\pi$
$464$	10706.0	1.07115
$465$	0	0
$466$	−6837.40	−0.679692
$467$	1367.93	0.135547	0.0677735	−	0.997701i	$-0.478410\pi$
$467$	1367.93	0.135547	0.0677735	+	0.997701i	$0.478410\pi$
$468$	0	0
$469$	388.480	0.0382481
$470$	2334.35	0.229097
$471$	0	0
$472$	−41403.6	−4.03762
$473$	7703.86	0.748888
$474$	0	0
$475$	2783.47	0.268872
$476$	14858.3	1.43074
$477$	0	0
$478$	−22428.5	−2.14615
$479$	202.741	0.0193392	0.00966960	−	0.999953i	$-0.496922\pi$
$479$	202.741	0.0193392	0.00966960	+	0.999953i	$0.496922\pi$
$480$	0	0
$481$	1916.65	0.181687
$482$	−12136.0	−1.14685
$483$	0	0
$484$	17642.1	1.65685
$485$	−4407.08	−0.412608
$486$	0	0
$487$	18961.3	1.76431	0.882153	−	0.470964i	$-0.156094\pi$
$487$	18961.3	1.76431	0.882153	+	0.470964i	$0.156094\pi$
$488$	−3128.37	−0.290194
$489$	0	0
$490$	−4319.78	−0.398261
$491$	11712.3	1.07651	0.538257	−	0.842781i	$-0.319083\pi$
$491$	11712.3	1.07651	0.538257	+	0.842781i	$0.319083\pi$
$492$	0	0
$493$	−3173.63	−0.289925
$494$	21385.3	1.94771
$495$	0	0
$496$	−54465.7	−4.93061
$497$	−7740.28	−0.698590
$498$	0	0
$499$	8624.87	0.773752	0.386876	−	0.922132i	$-0.373554\pi$
$499$	8624.87	0.773752	0.386876	+	0.922132i	$0.373554\pi$
$500$	−2531.83	−0.226454
$501$	0	0
$502$	42110.7	3.74401
$503$	4856.12	0.430465	0.215232	−	0.976563i	$-0.430949\pi$
$503$	4856.12	0.430465	0.215232	+	0.976563i	$0.430949\pi$
$504$	0	0
$505$	−6048.69	−0.532997
$506$	−8973.13	−0.788348
$507$	0	0
$508$	−20125.7	−1.75774
$509$	−10850.3	−0.944858	−0.472429	−	0.881369i	$-0.656623\pi$
$509$	−10850.3	−0.944858	−0.472429	+	0.881369i	$0.656623\pi$
$510$	0	0
$511$	−11217.9	−0.971138
$512$	11614.3	1.00251
$513$	0	0
$514$	−12024.6	−1.03187
$515$	−3724.21	−0.318657
$516$	0	0
$517$	−4121.57	−0.350612
$518$	−3787.55	−0.321266
$519$	0	0
$520$	−11769.0	−0.992508
$521$	3559.30	0.299301	0.149651	−	0.988739i	$-0.452185\pi$
$521$	3559.30	0.299301	0.149651	+	0.988739i	$0.452185\pi$
$522$	0	0
$523$	22191.7	1.85540	0.927702	−	0.373321i	$-0.121781\pi$
$523$	22191.7	1.85540	0.927702	+	0.373321i	$0.121781\pi$
$524$	28032.2	2.33701
$525$	0	0
$526$	−868.925	−0.0720284
$527$	16145.5	1.33455
$528$	0	0
$529$	−10872.9	−0.893636
$530$	−12737.2	−1.04390
$531$	0	0
$532$	−30294.7	−2.46888
$533$	−4638.21	−0.376929
$534$	0	0
$535$	5000.72	0.404112
$536$	1883.73	0.151800
$537$	0	0
$538$	−17563.1	−1.40743
$539$	7627.06	0.609500
$540$	0	0
$541$	4257.08	0.338311	0.169155	−	0.985589i	$-0.445896\pi$
$541$	4257.08	0.338311	0.169155	+	0.985589i	$0.445896\pi$
$542$	10391.6	0.823539
$543$	0	0
$544$	25013.8	1.97143
$545$	4577.59	0.359784
$546$	0	0
$547$	−21218.2	−1.65854	−0.829272	−	0.558845i	$-0.811244\pi$
$547$	−21218.2	−1.65854	−0.829272	+	0.558845i	$0.811244\pi$
$548$	31336.7	2.44277
$549$	0	0
$550$	6235.84	0.483449
$551$	6470.73	0.500294
$552$	0	0
$553$	2736.95	0.210465
$554$	22555.5	1.72977
$555$	0	0
$556$	−10927.5	−0.833503
$557$	2506.42	0.190665	0.0953324	−	0.995445i	$-0.469609\pi$
$557$	2506.42	0.190665	0.0953324	+	0.995445i	$0.469609\pi$
$558$	0	0
$559$	5932.29	0.448854
$560$	12373.3	0.933693
$561$	0	0
$562$	13938.8	1.04622
$563$	17279.6	1.29351	0.646757	−	0.762696i	$-0.276125\pi$
$563$	17279.6	1.29351	0.646757	+	0.762696i	$0.276125\pi$
$564$	0	0
$565$	6624.51	0.493266
$566$	37.6057	0.00279273
$567$	0	0
$568$	−37532.3	−2.77257
$569$	−20508.1	−1.51097	−0.755487	−	0.655164i	$-0.772600\pi$
$569$	−20508.1	−1.51097	−0.755487	+	0.655164i	$0.772600\pi$
$570$	0	0
$571$	6825.07	0.500210	0.250105	−	0.968219i	$-0.419535\pi$
$571$	6825.07	0.500210	0.250105	+	0.968219i	$0.419535\pi$
$572$	34344.6	2.51052
$573$	0	0
$574$	9165.74	0.666500
$575$	−899.350	−0.0652270
$576$	0	0
$577$	4249.00	0.306565	0.153283	−	0.988182i	$-0.451016\pi$
$577$	4249.00	0.306565	0.153283	+	0.988182i	$0.451016\pi$
$578$	10264.6	0.738667
$579$	0	0
$580$	−5885.74	−0.421366
$581$	−6239.66	−0.445551
$582$	0	0
$583$	22489.0	1.59760
$584$	−54395.2	−3.85426
$585$	0	0
$586$	23800.1	1.67777
$587$	18688.1	1.31404	0.657018	−	0.753875i	$-0.271817\pi$
$587$	18688.1	1.31404	0.657018	+	0.753875i	$0.271817\pi$
$588$	0	0
$589$	−32919.2	−2.30290
$590$	16893.1	1.17877
$591$	0	0
$592$	−9770.97	−0.678352
$593$	−12434.5	−0.861084	−0.430542	−	0.902570i	$-0.641678\pi$
$593$	−12434.5	−0.861084	−0.430542	+	0.902570i	$0.641678\pi$
$594$	0	0
$595$	−3667.89	−0.252721
$596$	−24400.1	−1.67696
$597$	0	0
$598$	−6909.68	−0.472505
$599$	−6043.04	−0.412207	−0.206103	−	0.978530i	$-0.566078\pi$
$599$	−6043.04	−0.412207	−0.206103	+	0.978530i	$0.566078\pi$
$600$	0	0
$601$	−631.308	−0.0428479	−0.0214239	−	0.999770i	$-0.506820\pi$
$601$	−631.308	−0.0428479	−0.0214239	+	0.999770i	$0.506820\pi$
$602$	−11723.0	−0.793679
$603$	0	0
$604$	60215.2	4.05649
$605$	−4355.08	−0.292660
$606$	0	0
$607$	−9454.87	−0.632226	−0.316113	−	0.948722i	$-0.602378\pi$
$607$	−9454.87	−0.632226	−0.316113	+	0.948722i	$0.602378\pi$
$608$	−51000.7	−3.40190
$609$	0	0
$610$	1276.40	0.0847215
$611$	−3173.77	−0.210143
$612$	0	0
$613$	−2682.66	−0.176757	−0.0883783	−	0.996087i	$-0.528168\pi$
$613$	−2682.66	−0.176757	−0.0883783	+	0.996087i	$0.528168\pi$
$614$	20676.2	1.35899
$615$	0	0
$616$	−41063.1	−2.68584
$617$	−2338.53	−0.152586	−0.0762930	−	0.997085i	$-0.524308\pi$
$617$	−2338.53	−0.152586	−0.0762930	+	0.997085i	$0.524308\pi$
$618$	0	0
$619$	−11930.2	−0.774663	−0.387331	−	0.921941i	$-0.626603\pi$
$619$	−11930.2	−0.774663	−0.387331	+	0.921941i	$0.626603\pi$
$620$	29943.1	1.93959
$621$	0	0
$622$	38780.1	2.49991
$623$	−13350.2	−0.858530
$624$	0	0
$625$	625.000	0.0400000
$626$	3381.05	0.215869
$627$	0	0
$628$	−8015.99	−0.509352
$629$	2896.46	0.183608
$630$	0	0
$631$	4106.81	0.259096	0.129548	−	0.991573i	$-0.458647\pi$
$631$	4106.81	0.259096	0.129548	+	0.991573i	$0.458647\pi$
$632$	13271.4	0.835295
$633$	0	0
$634$	32671.0	2.04658
$635$	4968.17	0.310482
$636$	0	0
$637$	5873.15	0.365310
$638$	14496.5	0.899562
$639$	0	0
$640$	7222.62	0.446093
$641$	14925.3	0.919679	0.459839	−	0.888002i	$-0.347907\pi$
$641$	14925.3	0.919679	0.459839	+	0.888002i	$0.347907\pi$
$642$	0	0
$643$	19005.7	1.16565	0.582823	−	0.812599i	$-0.301948\pi$
$643$	19005.7	1.16565	0.582823	+	0.812599i	$0.301948\pi$
$644$	9788.35	0.598936
$645$	0	0
$646$	32317.7	1.96830
$647$	9252.81	0.562234	0.281117	−	0.959673i	$-0.409295\pi$
$647$	9252.81	0.562234	0.281117	+	0.959673i	$0.409295\pi$
$648$	0	0
$649$	−29826.6	−1.80400
$650$	4801.85	0.289760
$651$	0	0
$652$	78211.9	4.69788
$653$	22950.7	1.37539	0.687695	−	0.726000i	$-0.258623\pi$
$653$	22950.7	1.37539	0.687695	+	0.726000i	$0.258623\pi$
$654$	0	0
$655$	−6919.95	−0.412801
$656$	23645.4	1.40731
$657$	0	0
$658$	6271.81	0.371582
$659$	6517.08	0.385234	0.192617	−	0.981274i	$-0.438302\pi$
$659$	6517.08	0.385234	0.192617	+	0.981274i	$0.438302\pi$
$660$	0	0
$661$	−5196.32	−0.305769	−0.152885	−	0.988244i	$-0.548856\pi$
$661$	−5196.32	−0.305769	−0.152885	+	0.988244i	$0.548856\pi$
$662$	−62374.3	−3.66200
$663$	0	0
$664$	−30255.9	−1.76831
$665$	7478.47	0.436094
$666$	0	0
$667$	−2090.72	−0.121369
$668$	41217.6	2.38736
$669$	0	0
$670$	−768.577	−0.0443175
$671$	−2253.64	−0.129658
$672$	0	0
$673$	28190.0	1.61463	0.807313	−	0.590123i	$-0.200921\pi$
$673$	28190.0	1.61463	0.807313	+	0.590123i	$0.200921\pi$
$674$	−33398.9	−1.90872
$675$	0	0
$676$	−18052.7	−1.02712
$677$	8676.88	0.492584	0.246292	−	0.969196i	$-0.420788\pi$
$677$	8676.88	0.492584	0.246292	+	0.969196i	$0.420788\pi$
$678$	0	0
$679$	−11840.7	−0.669225
$680$	−17785.4	−1.00300
$681$	0	0
$682$	−73749.3	−4.14077
$683$	20624.1	1.15543	0.577714	−	0.816239i	$-0.303945\pi$
$683$	20624.1	1.15543	0.577714	+	0.816239i	$0.303945\pi$
$684$	0	0
$685$	−7735.68	−0.431482
$686$	−36098.8	−2.00912
$687$	0	0
$688$	−30242.5	−1.67585
$689$	17317.4	0.957535
$690$	0	0
$691$	6986.06	0.384605	0.192303	−	0.981336i	$-0.438405\pi$
$691$	6986.06	0.384605	0.192303	+	0.981336i	$0.438405\pi$
$692$	31332.2	1.72120
$693$	0	0
$694$	−25518.8	−1.39579
$695$	2697.52	0.147227
$696$	0	0
$697$	−7009.32	−0.380914
$698$	20690.4	1.12198
$699$	0	0
$700$	−6802.38	−0.367294
$701$	7848.55	0.422875	0.211438	−	0.977391i	$-0.432185\pi$
$701$	7848.55	0.422875	0.211438	+	0.977391i	$0.432185\pi$
$702$	0	0
$703$	−5905.60	−0.316833
$704$	−45103.2	−2.41462
$705$	0	0
$706$	15767.3	0.840522
$707$	−16251.3	−0.864488
$708$	0	0
$709$	−25424.6	−1.34674	−0.673371	−	0.739305i	$-0.735154\pi$
$709$	−25424.6	−1.34674	−0.673371	+	0.739305i	$0.735154\pi$
$710$	15313.5	0.809445
$711$	0	0
$712$	−64734.5	−3.40734
$713$	10636.3	0.558673
$714$	0	0
$715$	−8478.21	−0.443451
$716$	−16689.3	−0.871099
$717$	0	0
$718$	50948.7	2.64817
$719$	16707.2	0.866584	0.433292	−	0.901254i	$-0.357352\pi$
$719$	16707.2	0.866584	0.433292	+	0.901254i	$0.357352\pi$
$720$	0	0
$721$	−10006.0	−0.516843
$722$	−29433.6	−1.51718
$723$	0	0
$724$	−89192.2	−4.57846
$725$	1452.94	0.0744286
$726$	0	0
$727$	1436.48	0.0732821	0.0366410	−	0.999328i	$-0.488334\pi$
$727$	1436.48	0.0732821	0.0366410	+	0.999328i	$0.488334\pi$
$728$	−31620.3	−1.60979
$729$	0	0
$730$	22193.8	1.12524
$731$	8964.95	0.453599
$732$	0	0
$733$	11540.9	0.581545	0.290772	−	0.956792i	$-0.406088\pi$
$733$	11540.9	0.581545	0.290772	+	0.956792i	$0.406088\pi$
$734$	−24117.4	−1.21279
$735$	0	0
$736$	16478.6	0.825282
$737$	1357.01	0.0678237
$738$	0	0
$739$	4127.87	0.205475	0.102737	−	0.994709i	$-0.467240\pi$
$739$	4127.87	0.205475	0.102737	+	0.994709i	$0.467240\pi$
$740$	5371.70	0.266848
$741$	0	0
$742$	−34221.6	−1.69315
$743$	−27497.0	−1.35769	−0.678846	−	0.734281i	$-0.737520\pi$
$743$	−27497.0	−1.35769	−0.678846	+	0.734281i	$0.737520\pi$
$744$	0	0
$745$	6023.35	0.296213
$746$	−69799.5	−3.42566
$747$	0	0
$748$	51902.0	2.53706
$749$	13435.7	0.655445
$750$	0	0
$751$	34394.1	1.67118	0.835592	−	0.549351i	$-0.185125\pi$
$751$	34394.1	1.67118	0.835592	+	0.549351i	$0.185125\pi$
$752$	16179.8	0.784594
$753$	0	0
$754$	11162.9	0.539161
$755$	−14864.6	−0.716526
$756$	0	0
$757$	−30459.8	−1.46246	−0.731229	−	0.682132i	$-0.761053\pi$
$757$	−30459.8	−1.46246	−0.731229	+	0.682132i	$0.761053\pi$
$758$	50237.4	2.40726
$759$	0	0
$760$	36262.8	1.73077
$761$	447.168	0.0213007	0.0106504	−	0.999943i	$-0.496610\pi$
$761$	447.168	0.0213007	0.0106504	+	0.999943i	$0.496610\pi$
$762$	0	0
$763$	12298.8	0.583548
$764$	−61714.3	−2.92244
$765$	0	0
$766$	45697.6	2.15551
$767$	−22967.7	−1.08125
$768$	0	0
$769$	−8057.17	−0.377827	−0.188914	−	0.981994i	$-0.560497\pi$
$769$	−8057.17	−0.377827	−0.188914	+	0.981994i	$0.560497\pi$
$770$	16754.1	0.784125
$771$	0	0
$772$	38299.6	1.78553
$773$	−19600.4	−0.912001	−0.456000	−	0.889980i	$-0.650718\pi$
$773$	−19600.4	−0.912001	−0.456000	+	0.889980i	$0.650718\pi$
$774$	0	0
$775$	−7391.67	−0.342602
$776$	−57415.0	−2.65603
$777$	0	0
$778$	−23611.9	−1.08808
$779$	14291.3	0.657304
$780$	0	0
$781$	−27037.7	−1.23878
$782$	−10442.0	−0.477500
$783$	0	0
$784$	−29941.0	−1.36393
$785$	1978.80	0.0899702
$786$	0	0
$787$	17936.4	0.812404	0.406202	−	0.913783i	$-0.366853\pi$
$787$	17936.4	0.812404	0.406202	+	0.913783i	$0.366853\pi$
$788$	73970.0	3.34400
$789$	0	0
$790$	−5414.84	−0.243862
$791$	17798.4	0.800047
$792$	0	0
$793$	−1735.39	−0.0777119
$794$	−55523.7	−2.48169
$795$	0	0
$796$	65164.4	2.90162
$797$	−17942.6	−0.797440	−0.398720	−	0.917073i	$-0.630546\pi$
$797$	−17942.6	−0.797440	−0.398720	+	0.917073i	$0.630546\pi$
$798$	0	0
$799$	−4796.25	−0.212364
$800$	−11451.7	−0.506099
$801$	0	0
$802$	67418.9	2.96838
$803$	−39185.6	−1.72208
$804$	0	0
$805$	−2416.33	−0.105794
$806$	−56790.0	−2.48181
$807$	0	0
$808$	−78801.9	−3.43099
$809$	29094.2	1.26440	0.632200	−	0.774806i	$-0.282152\pi$
$809$	29094.2	1.26440	0.632200	+	0.774806i	$0.282152\pi$
$810$	0	0
$811$	1438.31	0.0622760	0.0311380	−	0.999515i	$-0.490087\pi$
$811$	1438.31	0.0622760	0.0311380	+	0.999515i	$0.490087\pi$
$812$	−15813.5	−0.683429
$813$	0	0
$814$	−13230.4	−0.569687
$815$	−19307.2	−0.829817
$816$	0	0
$817$	−18278.7	−0.782729
$818$	59069.1	2.52482
$819$	0	0
$820$	−12999.3	−0.553605
$821$	−242.830	−0.0103226	−0.00516128	−	0.999987i	$-0.501643\pi$
$821$	−242.830	−0.0103226	−0.00516128	+	0.999987i	$0.501643\pi$
$822$	0	0
$823$	18171.0	0.769625	0.384812	−	0.922995i	$-0.374266\pi$
$823$	18171.0	0.769625	0.384812	+	0.922995i	$0.374266\pi$
$824$	−48518.7	−2.05125
$825$	0	0
$826$	45387.4	1.91190
$827$	12181.7	0.512213	0.256107	−	0.966649i	$-0.417560\pi$
$827$	12181.7	0.512213	0.256107	+	0.966649i	$0.417560\pi$
$828$	0	0
$829$	−17370.0	−0.727724	−0.363862	−	0.931453i	$-0.618542\pi$
$829$	−17370.0	−0.727724	−0.363862	+	0.931453i	$0.618542\pi$
$830$	12344.7	0.516253
$831$	0	0
$832$	−34731.3	−1.44723
$833$	8875.58	0.369172
$834$	0	0
$835$	−10174.9	−0.421695
$836$	−105823.	−4.37795
$837$	0	0
$838$	−10729.5	−0.442296
$839$	31363.0	1.29055	0.645275	−	0.763950i	$-0.276743\pi$
$839$	31363.0	1.29055	0.645275	+	0.763950i	$0.276743\pi$
$840$	0	0
$841$	−21011.4	−0.861509
$842$	58245.8	2.38395
$843$	0	0
$844$	100216.	4.08717
$845$	4456.43	0.181427
$846$	0	0
$847$	−11701.0	−0.474676
$848$	−88283.5	−3.57508
$849$	0	0
$850$	7256.62	0.292823
$851$	1908.12	0.0768621
$852$	0	0
$853$	−39677.5	−1.59265	−0.796326	−	0.604867i	$-0.793226\pi$
$853$	−39677.5	−1.59265	−0.796326	+	0.604867i	$0.793226\pi$
$854$	3429.37	0.137413
$855$	0	0
$856$	65149.0	2.60134
$857$	33209.3	1.32370	0.661849	−	0.749637i	$-0.269772\pi$
$857$	33209.3	1.32370	0.661849	+	0.749637i	$0.269772\pi$
$858$	0	0
$859$	34404.9	1.36656	0.683282	−	0.730154i	$-0.260552\pi$
$859$	34404.9	1.36656	0.683282	+	0.730154i	$0.260552\pi$
$860$	16626.2	0.659242
$861$	0	0
$862$	11294.1	0.446263
$863$	12999.7	0.512765	0.256382	−	0.966575i	$-0.417469\pi$
$863$	12999.7	0.512765	0.256382	+	0.966575i	$0.417469\pi$
$864$	0	0
$865$	−7734.58	−0.304027
$866$	85947.7	3.37254
$867$	0	0
$868$	80449.5	3.14589
$869$	9560.51	0.373208
$870$	0	0
$871$	1044.95	0.0406509
$872$	59636.4	2.31599
$873$	0	0
$874$	21290.2	0.823972
$875$	1679.22	0.0648775
$876$	0	0
$877$	−22831.5	−0.879093	−0.439546	−	0.898220i	$-0.644861\pi$
$877$	−22831.5	−0.879093	−0.439546	+	0.898220i	$0.644861\pi$
$878$	−3679.38	−0.141427
$879$	0	0
$880$	43221.5	1.65568
$881$	−1889.31	−0.0722502	−0.0361251	−	0.999347i	$-0.511501\pi$
$881$	−1889.31	−0.0722502	−0.0361251	+	0.999347i	$0.511501\pi$
$882$	0	0
$883$	−1778.37	−0.0677768	−0.0338884	−	0.999426i	$-0.510789\pi$
$883$	−1778.37	−0.0677768	−0.0338884	+	0.999426i	$0.510789\pi$
$884$	39966.7	1.52062
$885$	0	0
$886$	35457.2	1.34448
$887$	22984.1	0.870046	0.435023	−	0.900419i	$-0.356740\pi$
$887$	22984.1	0.870046	0.435023	+	0.900419i	$0.356740\pi$
$888$	0	0
$889$	13348.2	0.503582
$890$	26412.3	0.994766
$891$	0	0
$892$	22363.4	0.839441
$893$	9779.08	0.366455
$894$	0	0
$895$	4119.87	0.153868
$896$	19405.3	0.723535
$897$	0	0
$898$	51789.9	1.92456
$899$	−17183.4	−0.637485
$900$	0	0
$901$	26170.3	0.967658
$902$	32017.1	1.18188
$903$	0	0
$904$	86303.6	3.17524
$905$	22017.7	0.808723
$906$	0	0
$907$	45410.1	1.66242	0.831211	−	0.555957i	$-0.187648\pi$
$907$	45410.1	1.66242	0.831211	+	0.555957i	$0.187648\pi$
$908$	−77377.5	−2.82804
$909$	0	0
$910$	12901.4	0.469973
$911$	−30781.8	−1.11948	−0.559740	−	0.828669i	$-0.689099\pi$
$911$	−30781.8	−1.11948	−0.559740	+	0.828669i	$0.689099\pi$
$912$	0	0
$913$	−21795.9	−0.790076
$914$	−23615.3	−0.854623
$915$	0	0
$916$	2932.25	0.105769
$917$	−18592.1	−0.669538
$918$	0	0
$919$	−22457.3	−0.806093	−0.403046	−	0.915180i	$-0.632049\pi$
$919$	−22457.3	−0.806093	−0.403046	+	0.915180i	$0.632049\pi$
$920$	−11716.7	−0.419877
$921$	0	0
$922$	−100673.	−3.59596
$923$	−20820.2	−0.742475
$924$	0	0
$925$	−1326.04	−0.0471352
$926$	89259.1	3.16764
$927$	0	0
$928$	−26621.8	−0.941706
$929$	−43654.2	−1.54171	−0.770855	−	0.637011i	$-0.780171\pi$
$929$	−43654.2	−1.54171	−0.770855	+	0.637011i	$0.780171\pi$
$930$	0	0
$931$	−18096.4	−0.637043
$932$	26053.8	0.915686
$933$	0	0
$934$	−7271.26	−0.254736
$935$	−12812.4	−0.448138
$936$	0	0
$937$	−41123.3	−1.43377	−0.716884	−	0.697193i	$-0.754432\pi$
$937$	−41123.3	−1.43377	−0.716884	+	0.697193i	$0.754432\pi$
$938$	−2064.97	−0.0718803
$939$	0	0
$940$	−8895.01	−0.308642
$941$	22077.7	0.764839	0.382419	−	0.923989i	$-0.375091\pi$
$941$	22077.7	0.764839	0.382419	+	0.923989i	$0.375091\pi$
$942$	0	0
$943$	−4617.59	−0.159459
$944$	117088.	4.03697
$945$	0	0
$946$	−40949.9	−1.40740
$947$	8616.41	0.295666	0.147833	−	0.989012i	$-0.452770\pi$
$947$	8616.41	0.295666	0.147833	+	0.989012i	$0.452770\pi$
$948$	0	0
$949$	−30174.5	−1.03215
$950$	−14795.5	−0.505295
$951$	0	0
$952$	−47784.9	−1.62680
$953$	−29558.0	−1.00470	−0.502350	−	0.864664i	$-0.667531\pi$
$953$	−29558.0	−1.00470	−0.502350	+	0.864664i	$0.667531\pi$
$954$	0	0
$955$	15234.6	0.516211
$956$	85463.4	2.89130
$957$	0	0
$958$	−1077.67	−0.0363445
$959$	−20783.8	−0.699837
$960$	0	0
$961$	57628.0	1.93441
$962$	−10187.9	−0.341448
$963$	0	0
$964$	46244.0	1.54504
$965$	−9454.52	−0.315391
$966$	0	0
$967$	44575.9	1.48238	0.741192	−	0.671293i	$-0.234261\pi$
$967$	44575.9	1.48238	0.741192	+	0.671293i	$0.234261\pi$
$968$	−56737.6	−1.88390
$969$	0	0
$970$	23425.8	0.775421
$971$	43702.8	1.44438	0.722188	−	0.691696i	$-0.243136\pi$
$971$	43702.8	1.44438	0.722188	+	0.691696i	$0.243136\pi$
$972$	0	0
$973$	7247.56	0.238793
$974$	−100789.	−3.31569
$975$	0	0
$976$	8846.95	0.290147
$977$	45341.8	1.48476	0.742381	−	0.669978i	$-0.233696\pi$
$977$	45341.8	1.48476	0.742381	+	0.669978i	$0.233696\pi$
$978$	0	0
$979$	−46633.8	−1.52239
$980$	16460.4	0.536540
$981$	0	0
$982$	−62256.8	−2.02311
$983$	7360.52	0.238824	0.119412	−	0.992845i	$-0.461899\pi$
$983$	7360.52	0.238824	0.119412	+	0.992845i	$0.461899\pi$
$984$	0	0
$985$	−18260.0	−0.590673
$986$	16869.5	0.544861
$987$	0	0
$988$	−81488.1	−2.62397
$989$	5905.91	0.189886
$990$	0	0
$991$	33328.4	1.06833	0.534164	−	0.845381i	$-0.320627\pi$
$991$	33328.4	1.06833	0.534164	+	0.845381i	$0.320627\pi$
$992$	135436.	4.33477
$993$	0	0
$994$	41143.5	1.31287
$995$	−16086.3	−0.512532
$996$	0	0
$997$	−35331.7	−1.12233	−0.561166	−	0.827703i	$-0.689647\pi$
$997$	−35331.7	−1.12233	−0.561166	+	0.827703i	$0.689647\pi$
$998$	−45845.6	−1.45412
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.m.1.1		7
3.2	odd	2		405.4.a.n.1.7		7
5.4	even	2		2025.4.a.bb.1.7		7
9.2	odd	6		135.4.e.c.91.1		14
9.4	even	3		45.4.e.c.16.7	✓	14
9.5	odd	6		135.4.e.c.46.1		14
9.7	even	3		45.4.e.c.31.7	yes	14
15.14	odd	2		2025.4.a.ba.1.1		7
45.4	even	6		225.4.e.d.151.1		14
45.7	odd	12		225.4.k.d.49.2		28
45.13	odd	12		225.4.k.d.124.2		28
45.22	odd	12		225.4.k.d.124.13		28
45.34	even	6		225.4.e.d.76.1		14
45.43	odd	12		225.4.k.d.49.13		28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.c.16.7	✓	14	9.4	even	3
45.4.e.c.31.7	yes	14	9.7	even	3
135.4.e.c.46.1		14	9.5	odd	6
135.4.e.c.91.1		14	9.2	odd	6
225.4.e.d.76.1		14	45.34	even	6
225.4.e.d.151.1		14	45.4	even	6
225.4.k.d.49.2		28	45.7	odd	12
225.4.k.d.49.13		28	45.43	odd	12
225.4.k.d.124.2		28	45.13	odd	12
225.4.k.d.124.13		28	45.22	odd	12
405.4.a.m.1.1		7	1.1	even	1	trivial
405.4.a.n.1.7		7	3.2	odd	2
2025.4.a.ba.1.1		7	15.14	odd	2
2025.4.a.bb.1.7		7	5.4	even	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q-5.31551 q^{2} +20.2546 q^{4} -5.00000 q^{5} -13.4337 q^{7} -65.1396 q^{8} +O(q^{10})\)	\(q-5.31551 q^{2} +20.2546 q^{4} -5.00000 q^{5} -13.4337 q^{7} -65.1396 q^{8} +26.5775 q^{10} -46.9256 q^{11} -36.1347 q^{13} +71.4071 q^{14} +184.213 q^{16} -54.6071 q^{17} +111.339 q^{19} -101.273 q^{20} +249.434 q^{22} -35.9740 q^{23} +25.0000 q^{25} +192.074 q^{26} -272.095 q^{28} +58.1175 q^{29} -295.667 q^{31} -458.068 q^{32} +290.265 q^{34} +67.1686 q^{35} -53.0417 q^{37} -591.822 q^{38} +325.698 q^{40} +128.359 q^{41} -164.172 q^{43} -950.461 q^{44} +191.220 q^{46} +87.8318 q^{47} -162.535 q^{49} -132.888 q^{50} -731.894 q^{52} -479.247 q^{53} +234.628 q^{55} +875.067 q^{56} -308.924 q^{58} +635.614 q^{59} +48.0257 q^{61} +1571.62 q^{62} +961.163 q^{64} +180.673 q^{65} -28.9183 q^{67} -1106.05 q^{68} -357.035 q^{70} +576.183 q^{71} +835.057 q^{73} +281.944 q^{74} +2255.12 q^{76} +630.386 q^{77} -203.737 q^{79} -921.064 q^{80} -682.294 q^{82} +464.478 q^{83} +273.036 q^{85} +872.656 q^{86} +3056.72 q^{88} +993.782 q^{89} +485.423 q^{91} -728.640 q^{92} -466.871 q^{94} -556.693 q^{95} +881.415 q^{97} +863.956 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8}+O(q^{10}) \) 7 * q - 2 * q^2 + 36 * q^4 - 35 * q^5 + 22 * q^7 - 18 * q^8	\( 7 q - 2 q^{2} + 36 q^{4} - 35 q^{5} + 22 q^{7} - 18 q^{8} + 10 q^{10} - 23 q^{11} + 96 q^{13} + 21 q^{14} + 324 q^{16} - 161 q^{17} + 279 q^{19} - 180 q^{20} + 311 q^{22} - 96 q^{23} + 175 q^{25} + 358 q^{26} + 337 q^{28} + 296 q^{29} + 244 q^{31} + 314 q^{32} + 125 q^{34} - 110 q^{35} + 404 q^{37} - 305 q^{38} + 90 q^{40} + 47 q^{41} + 525 q^{43} - 55 q^{44} + 717 q^{46} - 164 q^{47} + 1225 q^{49} - 50 q^{50} + 1682 q^{52} - 506 q^{53} + 115 q^{55} + 981 q^{56} + 1183 q^{58} + 85 q^{59} + 828 q^{61} + 786 q^{62} + 2236 q^{64} - 480 q^{65} + 1093 q^{67} - 2473 q^{68} - 105 q^{70} - 328 q^{71} + 2085 q^{73} + 1316 q^{74} + 2789 q^{76} - 24 q^{77} + 2110 q^{79} - 1620 q^{80} - 62 q^{82} - 1290 q^{83} + 805 q^{85} + 2569 q^{86} + 2271 q^{88} + 3048 q^{89} + 3338 q^{91} - 2763 q^{92} - 517 q^{94} - 1395 q^{95} + 1787 q^{97} - 1279 q^{98}+O(q^{100}) \) 7 * q - 2 * q^2 + 36 * q^4 - 35 * q^5 + 22 * q^7 - 18 * q^8 + 10 * q^10 - 23 * q^11 + 96 * q^13 + 21 * q^14 + 324 * q^16 - 161 * q^17 + 279 * q^19 - 180 * q^20 + 311 * q^22 - 96 * q^23 + 175 * q^25 + 358 * q^26 + 337 * q^28 + 296 * q^29 + 244 * q^31 + 314 * q^32 + 125 * q^34 - 110 * q^35 + 404 * q^37 - 305 * q^38 + 90 * q^40 + 47 * q^41 + 525 * q^43 - 55 * q^44 + 717 * q^46 - 164 * q^47 + 1225 * q^49 - 50 * q^50 + 1682 * q^52 - 506 * q^53 + 115 * q^55 + 981 * q^56 + 1183 * q^58 + 85 * q^59 + 828 * q^61 + 786 * q^62 + 2236 * q^64 - 480 * q^65 + 1093 * q^67 - 2473 * q^68 - 105 * q^70 - 328 * q^71 + 2085 * q^73 + 1316 * q^74 + 2789 * q^76 - 24 * q^77 + 2110 * q^79 - 1620 * q^80 - 62 * q^82 - 1290 * q^83 + 805 * q^85 + 2569 * q^86 + 2271 * q^88 + 3048 * q^89 + 3338 * q^91 - 2763 * q^92 - 517 * q^94 - 1395 * q^95 + 1787 * q^97 - 1279 * q^98

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