LMFDB - Embedded newform 405.4.a.k.1.4

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:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 38x^{4} + 42x^{3} + 393x^{2} - 72x - 432 $ x^6 - 2x^5 - 38x^4 + 42x^3 + 393x^2 - 72*x - 432
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.14915$ 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)

$f(q)$	$=$	$q+0.149150 q^{2} -7.97775 q^{4} -5.00000 q^{5} +20.1424 q^{7} -2.38308 q^{8} +O(q^{10})$	\(q+0.149150 q^{2} -7.97775 q^{4} -5.00000 q^{5} +20.1424 q^{7} -2.38308 q^{8} -0.745751 q^{10} +9.89377 q^{11} -11.8279 q^{13} +3.00425 q^{14} +63.4666 q^{16} +6.09177 q^{17} -62.6618 q^{19} +39.8888 q^{20} +1.47566 q^{22} -12.0761 q^{23} +25.0000 q^{25} -1.76413 q^{26} -160.691 q^{28} -140.585 q^{29} +178.103 q^{31} +28.5307 q^{32} +0.908588 q^{34} -100.712 q^{35} -216.112 q^{37} -9.34602 q^{38} +11.9154 q^{40} -411.770 q^{41} -48.9176 q^{43} -78.9300 q^{44} -1.80115 q^{46} -615.419 q^{47} +62.7179 q^{49} +3.72875 q^{50} +94.3600 q^{52} -705.120 q^{53} -49.4688 q^{55} -48.0011 q^{56} -20.9683 q^{58} -494.540 q^{59} +666.202 q^{61} +26.5640 q^{62} -503.477 q^{64} +59.1394 q^{65} +277.818 q^{67} -48.5986 q^{68} -15.0212 q^{70} +239.308 q^{71} -919.015 q^{73} -32.2331 q^{74} +499.900 q^{76} +199.285 q^{77} +516.223 q^{79} -317.333 q^{80} -61.4155 q^{82} +652.161 q^{83} -30.4588 q^{85} -7.29606 q^{86} -23.5777 q^{88} +543.003 q^{89} -238.242 q^{91} +96.3400 q^{92} -91.7899 q^{94} +313.309 q^{95} +1133.39 q^{97} +9.35439 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8}+O(q^{10}) $ 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8	$ 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8} + 20 q^{10} - 88 q^{11} + 20 q^{13} - 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} - 170 q^{20} - 74 q^{22} - 210 q^{23} + 150 q^{25} - 4 q^{26} + 352 q^{28} - 296 q^{29} - 104 q^{31} - 722 q^{32} - 428 q^{34} - 200 q^{35} - 204 q^{37} + 20 q^{38} + 330 q^{40} - 344 q^{41} + 512 q^{43} - 716 q^{44} - 186 q^{46} - 238 q^{47} + 68 q^{49} - 100 q^{50} - 468 q^{52} - 850 q^{53} + 440 q^{55} - 2316 q^{56} + 890 q^{58} - 1840 q^{59} - 364 q^{61} - 1038 q^{62} - 990 q^{64} - 100 q^{65} + 88 q^{67} - 236 q^{68} + 900 q^{70} - 1364 q^{71} + 836 q^{73} - 1316 q^{74} - 2106 q^{76} - 840 q^{77} - 680 q^{79} - 290 q^{80} + 1742 q^{82} - 2148 q^{83} + 620 q^{85} - 2872 q^{86} + 1296 q^{88} - 3000 q^{89} - 3058 q^{91} - 1002 q^{92} - 3662 q^{94} + 230 q^{95} - 612 q^{97} - 1982 q^{98}+O(q^{100}) $ 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8 + 20 * q^10 - 88 * q^11 + 20 * q^13 - 180 * q^14 + 58 * q^16 - 124 * q^17 - 46 * q^19 - 170 * q^20 - 74 * q^22 - 210 * q^23 + 150 * q^25 - 4 * q^26 + 352 * q^28 - 296 * q^29 - 104 * q^31 - 722 * q^32 - 428 * q^34 - 200 * q^35 - 204 * q^37 + 20 * q^38 + 330 * q^40 - 344 * q^41 + 512 * q^43 - 716 * q^44 - 186 * q^46 - 238 * q^47 + 68 * q^49 - 100 * q^50 - 468 * q^52 - 850 * q^53 + 440 * q^55 - 2316 * q^56 + 890 * q^58 - 1840 * q^59 - 364 * q^61 - 1038 * q^62 - 990 * q^64 - 100 * q^65 + 88 * q^67 - 236 * q^68 + 900 * q^70 - 1364 * q^71 + 836 * q^73 - 1316 * q^74 - 2106 * q^76 - 840 * q^77 - 680 * q^79 - 290 * q^80 + 1742 * q^82 - 2148 * q^83 + 620 * q^85 - 2872 * q^86 + 1296 * q^88 - 3000 * q^89 - 3058 * q^91 - 1002 * q^92 - 3662 * q^94 + 230 * q^95 - 612 * q^97 - 1982 * 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 $

$2$	0.149150	0.0527325	0.0263663	−	0.999652i	$-0.491606\pi$
$2$	0.149150	0.0527325	0.0263663	+	0.999652i	$0.491606\pi$
$3$	0	0
$3$	0	0
$4$	−7.97775	−0.997219
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	20.1424	1.08759	0.543795	−	0.839218i	$-0.316987\pi$
$7$	20.1424	1.08759	0.543795	+	0.839218i	$0.316987\pi$
$8$	−2.38308	−0.105318
$9$	0	0
$10$	−0.745751	−0.0235827
$11$	9.89377	0.271189	0.135595	−	0.990764i	$-0.456706\pi$
$11$	9.89377	0.271189	0.135595	+	0.990764i	$0.456706\pi$
$12$	0	0
$13$	−11.8279	−0.252343	−0.126172	−	0.992008i	$-0.540269\pi$
$13$	−11.8279	−0.252343	−0.126172	+	0.992008i	$0.540269\pi$
$14$	3.00425	0.0573514
$15$	0	0
$16$	63.4666	0.991666
$17$	6.09177	0.0869101	0.0434550	−	0.999055i	$-0.486163\pi$
$17$	6.09177	0.0869101	0.0434550	+	0.999055i	$0.486163\pi$
$18$	0	0
$19$	−62.6618	−0.756610	−0.378305	−	0.925681i	$-0.623493\pi$
$19$	−62.6618	−0.756610	−0.378305	+	0.925681i	$0.623493\pi$
$20$	39.8888	0.445970
$21$	0	0
$22$	1.47566	0.0143005
$23$	−12.0761	−0.109480	−0.0547399	−	0.998501i	$-0.517433\pi$
$23$	−12.0761	−0.109480	−0.0547399	+	0.998501i	$0.517433\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−1.76413	−0.0133067
$27$	0	0
$28$	−160.691	−1.08457
$29$	−140.585	−0.900206	−0.450103	−	0.892977i	$-0.648613\pi$
$29$	−140.585	−0.900206	−0.450103	+	0.892977i	$0.648613\pi$
$30$	0	0
$31$	178.103	1.03188	0.515938	−	0.856626i	$-0.327443\pi$
$31$	178.103	1.03188	0.515938	+	0.856626i	$0.327443\pi$
$32$	28.5307	0.157612
$33$	0	0
$34$	0.908588	0.00458299
$35$	−100.712	−0.486385
$36$	0	0
$37$	−216.112	−0.960231	−0.480116	−	0.877205i	$-0.659405\pi$
$37$	−216.112	−0.960231	−0.480116	+	0.877205i	$0.659405\pi$
$38$	−9.34602	−0.0398980
$39$	0	0
$40$	11.9154	0.0470998
$41$	−411.770	−1.56848	−0.784239	−	0.620459i	$-0.786946\pi$
$41$	−411.770	−1.56848	−0.784239	+	0.620459i	$0.786946\pi$
$42$	0	0
$43$	−48.9176	−0.173485	−0.0867425	−	0.996231i	$-0.527646\pi$
$43$	−48.9176	−0.173485	−0.0867425	+	0.996231i	$0.527646\pi$
$44$	−78.9300	−0.270435
$45$	0	0
$46$	−1.80115	−0.00577315
$47$	−615.419	−1.90996	−0.954980	−	0.296670i	$-0.904124\pi$
$47$	−615.419	−1.90996	−0.954980	+	0.296670i	$0.904124\pi$
$48$	0	0
$49$	62.7179	0.182851
$50$	3.72875	0.0105465
$51$	0	0
$52$	94.3600	0.251642
$53$	−705.120	−1.82746	−0.913732	−	0.406316i	$-0.866813\pi$
$53$	−705.120	−1.82746	−0.913732	+	0.406316i	$0.866813\pi$
$54$	0	0
$55$	−49.4688	−0.121280
$56$	−48.0011	−0.114543
$57$	0	0
$58$	−20.9683	−0.0474701
$59$	−494.540	−1.09125	−0.545624	−	0.838030i	$-0.683707\pi$
$59$	−494.540	−1.09125	−0.545624	+	0.838030i	$0.683707\pi$
$60$	0	0
$61$	666.202	1.39833	0.699167	−	0.714958i	$-0.253554\pi$
$61$	666.202	1.39833	0.699167	+	0.714958i	$0.253554\pi$
$62$	26.5640	0.0544135
$63$	0	0
$64$	−503.477	−0.983354
$65$	59.1394	0.112851
$66$	0	0
$67$	277.818	0.506581	0.253290	−	0.967390i	$-0.418487\pi$
$67$	277.818	0.506581	0.253290	+	0.967390i	$0.418487\pi$
$68$	−48.5986	−0.0866684
$69$	0	0
$70$	−15.0212	−0.0256483
$71$	239.308	0.400009	0.200005	−	0.979795i	$-0.435904\pi$
$71$	239.308	0.400009	0.200005	+	0.979795i	$0.435904\pi$
$72$	0	0
$73$	−919.015	−1.47346	−0.736730	−	0.676187i	$-0.763631\pi$
$73$	−919.015	−1.47346	−0.736730	+	0.676187i	$0.763631\pi$
$74$	−32.2331	−0.0506354
$75$	0	0
$76$	499.900	0.754507
$77$	199.285	0.294943
$78$	0	0
$79$	516.223	0.735185	0.367592	−	0.929987i	$-0.380182\pi$
$79$	516.223	0.735185	0.367592	+	0.929987i	$0.380182\pi$
$80$	−317.333	−0.443486
$81$	0	0
$82$	−61.4155	−0.0827099
$83$	652.161	0.862458	0.431229	−	0.902243i	$-0.358080\pi$
$83$	652.161	0.862458	0.431229	+	0.902243i	$0.358080\pi$
$84$	0	0
$85$	−30.4588	−0.0388674
$86$	−7.29606	−0.00914831
$87$	0	0
$88$	−23.5777	−0.0285612
$89$	543.003	0.646721	0.323361	−	0.946276i	$-0.395187\pi$
$89$	543.003	0.646721	0.323361	+	0.946276i	$0.395187\pi$
$90$	0	0
$91$	−238.242	−0.274446
$92$	96.3400	0.109175
$93$	0	0
$94$	−91.7899	−0.100717
$95$	313.309	0.338367
$96$	0	0
$97$	1133.39	1.18638	0.593188	−	0.805064i	$-0.297869\pi$
$97$	1133.39	1.18638	0.593188	+	0.805064i	$0.297869\pi$
$98$	9.35439	0.00964220
$99$	0	0
$100$	−199.444	−0.199444
$101$	−1199.24	−1.18147	−0.590735	−	0.806866i	$-0.701162\pi$
$101$	−1199.24	−1.18147	−0.590735	+	0.806866i	$0.701162\pi$
$102$	0	0
$103$	−1245.93	−1.19190	−0.595949	−	0.803022i	$-0.703224\pi$
$103$	−1245.93	−1.19190	−0.595949	+	0.803022i	$0.703224\pi$
$104$	28.1869	0.0265764
$105$	0	0
$106$	−105.169	−0.0963669
$107$	−1159.14	−1.04728	−0.523638	−	0.851941i	$-0.675426\pi$
$107$	−1159.14	−1.04728	−0.523638	+	0.851941i	$0.675426\pi$
$108$	0	0
$109$	−118.618	−0.104234	−0.0521171	−	0.998641i	$-0.516597\pi$
$109$	−118.618	−0.104234	−0.0521171	+	0.998641i	$0.516597\pi$
$110$	−7.37829	−0.00639538
$111$	0	0
$112$	1278.37	1.07853
$113$	−383.283	−0.319081	−0.159541	−	0.987191i	$-0.551001\pi$
$113$	−383.283	−0.319081	−0.159541	+	0.987191i	$0.551001\pi$
$114$	0	0
$115$	60.3804	0.0489609
$116$	1121.55	0.897702
$117$	0	0
$118$	−73.7608	−0.0575443
$119$	122.703	0.0945225
$120$	0	0
$121$	−1233.11	−0.926456
$122$	99.3641	0.0737378
$123$	0	0
$124$	−1420.86	−1.02901
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2570.69	1.79616	0.898079	−	0.439834i	$-0.144963\pi$
$127$	2570.69	1.79616	0.898079	+	0.439834i	$0.144963\pi$
$128$	−303.340	−0.209466
$129$	0	0
$130$	8.82065	0.00595094
$131$	658.057	0.438891	0.219446	−	0.975625i	$-0.429575\pi$
$131$	658.057	0.438891	0.219446	+	0.975625i	$0.429575\pi$
$132$	0	0
$133$	−1262.16	−0.822882
$134$	41.4367	0.0267133
$135$	0	0
$136$	−14.5172	−0.00915323
$137$	626.734	0.390843	0.195421	−	0.980719i	$-0.437393\pi$
$137$	626.734	0.390843	0.195421	+	0.980719i	$0.437393\pi$
$138$	0	0
$139$	−973.745	−0.594187	−0.297093	−	0.954848i	$-0.596017\pi$
$139$	−973.745	−0.594187	−0.297093	+	0.954848i	$0.596017\pi$
$140$	803.457	0.485032
$141$	0	0
$142$	35.6929	0.0210935
$143$	−117.022	−0.0684329
$144$	0	0
$145$	702.925	0.402584
$146$	−137.071	−0.0776993
$147$	0	0
$148$	1724.09	0.957561
$149$	−2975.84	−1.63618	−0.818089	−	0.575092i	$-0.804966\pi$
$149$	−2975.84	−1.63618	−0.818089	+	0.575092i	$0.804966\pi$
$150$	0	0
$151$	−2076.43	−1.11906	−0.559529	−	0.828811i	$-0.689018\pi$
$151$	−2076.43	−1.11906	−0.559529	+	0.828811i	$0.689018\pi$
$152$	149.328	0.0796850
$153$	0	0
$154$	29.7233	0.0155531
$155$	−890.513	−0.461469
$156$	0	0
$157$	352.350	0.179112	0.0895559	−	0.995982i	$-0.471455\pi$
$157$	352.350	0.179112	0.0895559	+	0.995982i	$0.471455\pi$
$158$	76.9947	0.0387682
$159$	0	0
$160$	−142.654	−0.0704860
$161$	−243.242	−0.119069
$162$	0	0
$163$	2772.05	1.33205	0.666023	−	0.745931i	$-0.267995\pi$
$163$	2772.05	1.33205	0.666023	+	0.745931i	$0.267995\pi$
$164$	3285.00	1.56412
$165$	0	0
$166$	97.2699	0.0454796
$167$	3835.44	1.77722	0.888608	−	0.458667i	$-0.151673\pi$
$167$	3835.44	1.77722	0.888608	+	0.458667i	$0.151673\pi$
$168$	0	0
$169$	−2057.10	−0.936323
$170$	−4.54294	−0.00204958
$171$	0	0
$172$	390.252	0.173003
$173$	1135.45	0.499000	0.249500	−	0.968375i	$-0.419734\pi$
$173$	1135.45	0.499000	0.249500	+	0.968375i	$0.419734\pi$
$174$	0	0
$175$	503.561	0.217518
$176$	627.924	0.268929
$177$	0	0
$178$	80.9890	0.0341033
$179$	3192.57	1.33309	0.666546	−	0.745464i	$-0.267772\pi$
$179$	3192.57	1.33309	0.666546	+	0.745464i	$0.267772\pi$
$180$	0	0
$181$	990.641	0.406816	0.203408	−	0.979094i	$-0.434798\pi$
$181$	990.641	0.406816	0.203408	+	0.979094i	$0.434798\pi$
$182$	−35.5339	−0.0144722
$183$	0	0
$184$	28.7783	0.0115303
$185$	1080.56	0.429428
$186$	0	0
$187$	60.2706	0.0235691
$188$	4909.66	1.90465
$189$	0	0
$190$	46.7301	0.0178429
$191$	3283.72	1.24399	0.621994	−	0.783022i	$-0.286323\pi$
$191$	3283.72	1.24399	0.621994	+	0.783022i	$0.286323\pi$
$192$	0	0
$193$	−2368.01	−0.883176	−0.441588	−	0.897218i	$-0.645585\pi$
$193$	−2368.01	−0.883176	−0.441588	+	0.897218i	$0.645585\pi$
$194$	169.045	0.0625606
$195$	0	0
$196$	−500.348	−0.182343
$197$	−3754.00	−1.35767	−0.678836	−	0.734290i	$-0.737515\pi$
$197$	−3754.00	−1.35767	−0.678836	+	0.734290i	$0.737515\pi$
$198$	0	0
$199$	1905.82	0.678894	0.339447	−	0.940625i	$-0.389760\pi$
$199$	1905.82	0.678894	0.339447	+	0.940625i	$0.389760\pi$
$200$	−59.5771	−0.0210637
$201$	0	0
$202$	−178.866	−0.0623019
$203$	−2831.72	−0.979054
$204$	0	0
$205$	2058.85	0.701445
$206$	−185.831	−0.0628518
$207$	0	0
$208$	−750.676	−0.250240
$209$	−619.961	−0.205185
$210$	0	0
$211$	−3689.19	−1.20367	−0.601834	−	0.798621i	$-0.705563\pi$
$211$	−3689.19	−1.20367	−0.601834	+	0.798621i	$0.705563\pi$
$212$	5625.27	1.82238
$213$	0	0
$214$	−172.886	−0.0552255
$215$	244.588	0.0775849
$216$	0	0
$217$	3587.42	1.12226
$218$	−17.6919	−0.00549653
$219$	0	0
$220$	394.650	0.120942
$221$	−72.0528	−0.0219312
$222$	0	0
$223$	169.717	0.0509644	0.0254822	−	0.999675i	$-0.491888\pi$
$223$	169.717	0.0509644	0.0254822	+	0.999675i	$0.491888\pi$
$224$	574.679	0.171417
$225$	0	0
$226$	−57.1667	−0.0168260
$227$	−3607.46	−1.05478	−0.527391	−	0.849623i	$-0.676830\pi$
$227$	−3607.46	−1.05478	−0.527391	+	0.849623i	$0.676830\pi$
$228$	0	0
$229$	−6093.90	−1.75850	−0.879250	−	0.476361i	$-0.841955\pi$
$229$	−6093.90	−1.75850	−0.879250	+	0.476361i	$0.841955\pi$
$230$	9.00575	0.00258183
$231$	0	0
$232$	335.026	0.0948083
$233$	3930.28	1.10507	0.552534	−	0.833490i	$-0.313661\pi$
$233$	3930.28	1.10507	0.552534	+	0.833490i	$0.313661\pi$
$234$	0	0
$235$	3077.10	0.854160
$236$	3945.32	1.08821
$237$	0	0
$238$	18.3012	0.00498441
$239$	−3662.24	−0.991174	−0.495587	−	0.868558i	$-0.665047\pi$
$239$	−3662.24	−0.991174	−0.495587	+	0.868558i	$0.665047\pi$
$240$	0	0
$241$	222.829	0.0595588	0.0297794	−	0.999556i	$-0.490520\pi$
$241$	222.829	0.0595588	0.0297794	+	0.999556i	$0.490520\pi$
$242$	−183.919	−0.0488544
$243$	0	0
$244$	−5314.80	−1.39445
$245$	−313.590	−0.0817735
$246$	0	0
$247$	741.156	0.190926
$248$	−424.434	−0.108676
$249$	0	0
$250$	−18.6438	−0.00471654
$251$	−4748.92	−1.19422	−0.597110	−	0.802159i	$-0.703684\pi$
$251$	−4748.92	−1.19422	−0.597110	+	0.802159i	$0.703684\pi$
$252$	0	0
$253$	−119.478	−0.0296898
$254$	383.419	0.0947160
$255$	0	0
$256$	3982.58	0.972309
$257$	5652.15	1.37187	0.685936	−	0.727661i	$-0.259393\pi$
$257$	5652.15	1.37187	0.685936	+	0.727661i	$0.259393\pi$
$258$	0	0
$259$	−4353.02	−1.04434
$260$	−471.800	−0.112538
$261$	0	0
$262$	98.1494	0.0231438
$263$	−3289.86	−0.771337	−0.385669	−	0.922637i	$-0.626029\pi$
$263$	−3289.86	−0.771337	−0.385669	+	0.922637i	$0.626029\pi$
$264$	0	0
$265$	3525.60	0.817267
$266$	−188.252	−0.0433926
$267$	0	0
$268$	−2216.37	−0.505172
$269$	5452.66	1.23589	0.617945	−	0.786221i	$-0.287965\pi$
$269$	5452.66	1.23589	0.617945	+	0.786221i	$0.287965\pi$
$270$	0	0
$271$	−238.923	−0.0535555	−0.0267778	−	0.999641i	$-0.508525\pi$
$271$	−238.923	−0.0535555	−0.0267778	+	0.999641i	$0.508525\pi$
$272$	386.624	0.0861857
$273$	0	0
$274$	93.4775	0.0206101
$275$	247.344	0.0542379
$276$	0	0
$277$	3757.75	0.815094	0.407547	−	0.913184i	$-0.366384\pi$
$277$	3757.75	0.815094	0.407547	+	0.913184i	$0.366384\pi$
$278$	−145.234	−0.0313330
$279$	0	0
$280$	240.006	0.0512253
$281$	472.982	0.100412	0.0502060	−	0.998739i	$-0.484012\pi$
$281$	472.982	0.100412	0.0502060	+	0.998739i	$0.484012\pi$
$282$	0	0
$283$	2353.89	0.494431	0.247215	−	0.968961i	$-0.420484\pi$
$283$	2353.89	0.494431	0.247215	+	0.968961i	$0.420484\pi$
$284$	−1909.14	−0.398897
$285$	0	0
$286$	−17.4539	−0.00360864
$287$	−8294.05	−1.70586
$288$	0	0
$289$	−4875.89	−0.992447
$290$	104.841	0.0212293
$291$	0	0
$292$	7331.68	1.46936
$293$	−4143.06	−0.826075	−0.413038	−	0.910714i	$-0.635532\pi$
$293$	−4143.06	−0.826075	−0.413038	+	0.910714i	$0.635532\pi$
$294$	0	0
$295$	2472.70	0.488021
$296$	515.013	0.101130
$297$	0	0
$298$	−443.847	−0.0862798
$299$	142.834	0.0276265
$300$	0	0
$301$	−985.319	−0.188681
$302$	−309.700	−0.0590108
$303$	0	0
$304$	−3976.93	−0.750305
$305$	−3331.01	−0.625354
$306$	0	0
$307$	224.413	0.0417196	0.0208598	−	0.999782i	$-0.493360\pi$
$307$	224.413	0.0417196	0.0208598	+	0.999782i	$0.493360\pi$
$308$	−1589.84	−0.294123
$309$	0	0
$310$	−132.820	−0.0243344
$311$	−6076.12	−1.10786	−0.553931	−	0.832562i	$-0.686873\pi$
$311$	−6076.12	−1.10786	−0.553931	+	0.832562i	$0.686873\pi$
$312$	0	0
$313$	3456.33	0.624164	0.312082	−	0.950055i	$-0.398974\pi$
$313$	3456.33	0.624164	0.312082	+	0.950055i	$0.398974\pi$
$314$	52.5530	0.00944502
$315$	0	0
$316$	−4118.30	−0.733140
$317$	−7632.04	−1.35223	−0.676117	−	0.736795i	$-0.736338\pi$
$317$	−7632.04	−1.35223	−0.676117	+	0.736795i	$0.736338\pi$
$318$	0	0
$319$	−1390.91	−0.244126
$320$	2517.39	0.439769
$321$	0	0
$322$	−36.2795	−0.00627882
$323$	−381.721	−0.0657571
$324$	0	0
$325$	−295.697	−0.0504687
$326$	413.451	0.0702422
$327$	0	0
$328$	981.282	0.165190
$329$	−12396.0	−2.07725
$330$	0	0
$331$	1190.38	0.197671	0.0988357	−	0.995104i	$-0.468488\pi$
$331$	1190.38	0.197671	0.0988357	+	0.995104i	$0.468488\pi$
$332$	−5202.78	−0.860059
$333$	0	0
$334$	572.056	0.0937172
$335$	−1389.09	−0.226550
$336$	0	0
$337$	695.217	0.112377	0.0561883	−	0.998420i	$-0.482105\pi$
$337$	695.217	0.112377	0.0561883	+	0.998420i	$0.482105\pi$
$338$	−306.817	−0.0493747
$339$	0	0
$340$	242.993	0.0387593
$341$	1762.11	0.279834
$342$	0	0
$343$	−5645.57	−0.888723
$344$	116.575	0.0182712
$345$	0	0
$346$	169.353	0.0263135
$347$	10232.4	1.58301	0.791506	−	0.611161i	$-0.209297\pi$
$347$	10232.4	1.58301	0.791506	+	0.611161i	$0.209297\pi$
$348$	0	0
$349$	7997.60	1.22665	0.613326	−	0.789830i	$-0.289831\pi$
$349$	7997.60	1.22665	0.613326	+	0.789830i	$0.289831\pi$
$350$	75.1062	0.0114703
$351$	0	0
$352$	282.276	0.0427426
$353$	5436.70	0.819735	0.409867	−	0.912145i	$-0.365575\pi$
$353$	5436.70	0.819735	0.409867	+	0.912145i	$0.365575\pi$
$354$	0	0
$355$	−1196.54	−0.178890
$356$	−4331.94	−0.644923
$357$	0	0
$358$	476.172	0.0702974
$359$	−4767.51	−0.700890	−0.350445	−	0.936583i	$-0.613970\pi$
$359$	−4767.51	−0.700890	−0.350445	+	0.936583i	$0.613970\pi$
$360$	0	0
$361$	−2932.50	−0.427541
$362$	147.754	0.0214525
$363$	0	0
$364$	1900.64	0.273683
$365$	4595.08	0.658952
$366$	0	0
$367$	12721.4	1.80940	0.904701	−	0.426046i	$-0.140094\pi$
$367$	12721.4	1.80940	0.904701	+	0.426046i	$0.140094\pi$
$368$	−766.428	−0.108567
$369$	0	0
$370$	161.166	0.0226449
$371$	−14202.8	−1.98753
$372$	0	0
$373$	−9306.12	−1.29183	−0.645915	−	0.763409i	$-0.723524\pi$
$373$	−9306.12	−1.29183	−0.645915	+	0.763409i	$0.723524\pi$
$374$	8.98936	0.00124286
$375$	0	0
$376$	1466.60	0.201154
$377$	1662.82	0.227161
$378$	0	0
$379$	−11623.8	−1.57539	−0.787694	−	0.616067i	$-0.788725\pi$
$379$	−11623.8	−1.57539	−0.787694	+	0.616067i	$0.788725\pi$
$380$	−2499.50	−0.337426
$381$	0	0
$382$	489.768	0.0655987
$383$	−11642.9	−1.55333	−0.776664	−	0.629915i	$-0.783090\pi$
$383$	−11642.9	−1.55333	−0.776664	+	0.629915i	$0.783090\pi$
$384$	0	0
$385$	−996.423	−0.131902
$386$	−353.189	−0.0465721
$387$	0	0
$388$	−9041.91	−1.18308
$389$	6979.31	0.909679	0.454839	−	0.890573i	$-0.349697\pi$
$389$	6979.31	0.909679	0.454839	+	0.890573i	$0.349697\pi$
$390$	0	0
$391$	−73.5647	−0.00951490
$392$	−149.462	−0.0192576
$393$	0	0
$394$	−559.910	−0.0715935
$395$	−2581.11	−0.328785
$396$	0	0
$397$	2884.53	0.364661	0.182330	−	0.983237i	$-0.441636\pi$
$397$	2884.53	0.364661	0.182330	+	0.983237i	$0.441636\pi$
$398$	284.253	0.0357998
$399$	0	0
$400$	1586.66	0.198333
$401$	10994.4	1.36916	0.684579	−	0.728939i	$-0.259986\pi$
$401$	10994.4	1.36916	0.684579	+	0.728939i	$0.259986\pi$
$402$	0	0
$403$	−2106.58	−0.260387
$404$	9567.21	1.17818
$405$	0	0
$406$	−422.352	−0.0516280
$407$	−2138.16	−0.260404
$408$	0	0
$409$	−2846.53	−0.344137	−0.172069	−	0.985085i	$-0.555045\pi$
$409$	−2846.53	−0.344137	−0.172069	+	0.985085i	$0.555045\pi$
$410$	307.078	0.0369890
$411$	0	0
$412$	9939.76	1.18858
$413$	−9961.25	−1.18683
$414$	0	0
$415$	−3260.81	−0.385703
$416$	−337.458	−0.0397722
$417$	0	0
$418$	−92.4673	−0.0108199
$419$	−8305.12	−0.968333	−0.484167	−	0.874976i	$-0.660877\pi$
$419$	−8305.12	−0.968333	−0.484167	+	0.874976i	$0.660877\pi$
$420$	0	0
$421$	3571.48	0.413452	0.206726	−	0.978399i	$-0.433719\pi$
$421$	3571.48	0.413452	0.206726	+	0.978399i	$0.433719\pi$
$422$	−550.243	−0.0634725
$423$	0	0
$424$	1680.36	0.192466
$425$	152.294	0.0173820
$426$	0	0
$427$	13418.9	1.52081
$428$	9247.35	1.04436
$429$	0	0
$430$	36.4803	0.00409125
$431$	13408.7	1.49854	0.749272	−	0.662262i	$-0.230403\pi$
$431$	13408.7	1.49854	0.749272	+	0.662262i	$0.230403\pi$
$432$	0	0
$433$	−1698.17	−0.188473	−0.0942364	−	0.995550i	$-0.530041\pi$
$433$	−1698.17	−0.188473	−0.0942364	+	0.995550i	$0.530041\pi$
$434$	535.064	0.0591795
$435$	0	0
$436$	946.303	0.103944
$437$	756.709	0.0828336
$438$	0	0
$439$	−10730.2	−1.16657	−0.583283	−	0.812269i	$-0.698232\pi$
$439$	−10730.2	−1.16657	−0.583283	+	0.812269i	$0.698232\pi$
$440$	117.888	0.0127730
$441$	0	0
$442$	−10.7467	−0.00115649
$443$	8439.52	0.905133	0.452566	−	0.891731i	$-0.350509\pi$
$443$	8439.52	0.905133	0.452566	+	0.891731i	$0.350509\pi$
$444$	0	0
$445$	−2715.01	−0.289223
$446$	25.3133	0.00268748
$447$	0	0
$448$	−10141.3	−1.06949
$449$	3860.69	0.405784	0.202892	−	0.979201i	$-0.434966\pi$
$449$	3860.69	0.405784	0.202892	+	0.979201i	$0.434966\pi$
$450$	0	0
$451$	−4073.95	−0.425355
$452$	3057.73	0.318194
$453$	0	0
$454$	−538.053	−0.0556214
$455$	1191.21	0.122736
$456$	0	0
$457$	−3472.87	−0.355479	−0.177739	−	0.984078i	$-0.556878\pi$
$457$	−3472.87	−0.355479	−0.177739	+	0.984078i	$0.556878\pi$
$458$	−908.906	−0.0927301
$459$	0	0
$460$	−481.700	−0.0488247
$461$	−6841.91	−0.691235	−0.345618	−	0.938375i	$-0.612331\pi$
$461$	−6841.91	−0.691235	−0.345618	+	0.938375i	$0.612331\pi$
$462$	0	0
$463$	6180.53	0.620375	0.310188	−	0.950675i	$-0.399608\pi$
$463$	6180.53	0.620375	0.310188	+	0.950675i	$0.399608\pi$
$464$	−8922.45	−0.892703
$465$	0	0
$466$	586.202	0.0582731
$467$	−10977.3	−1.08772	−0.543861	−	0.839175i	$-0.683038\pi$
$467$	−10977.3	−1.08772	−0.543861	+	0.839175i	$0.683038\pi$
$468$	0	0
$469$	5595.94	0.550952
$470$	458.949	0.0450420
$471$	0	0
$472$	1178.53	0.114929
$473$	−483.979	−0.0470473
$474$	0	0
$475$	−1566.54	−0.151322
$476$	−978.895	−0.0942596
$477$	0	0
$478$	−546.223	−0.0522671
$479$	−5285.65	−0.504191	−0.252096	−	0.967702i	$-0.581120\pi$
$479$	−5285.65	−0.504191	−0.252096	+	0.967702i	$0.581120\pi$
$480$	0	0
$481$	2556.14	0.242308
$482$	33.2350	0.00314069
$483$	0	0
$484$	9837.48	0.923880
$485$	−5666.95	−0.530563
$486$	0	0
$487$	15418.1	1.43462	0.717312	−	0.696752i	$-0.245372\pi$
$487$	15418.1	1.43462	0.717312	+	0.696752i	$0.245372\pi$
$488$	−1587.62	−0.147270
$489$	0	0
$490$	−46.7719	−0.00431212
$491$	−13782.4	−1.26678	−0.633392	−	0.773831i	$-0.718338\pi$
$491$	−13782.4	−1.26678	−0.633392	+	0.773831i	$0.718338\pi$
$492$	0	0
$493$	−856.411	−0.0782369
$494$	110.544	0.0100680
$495$	0	0
$496$	11303.6	1.02328
$497$	4820.25	0.435046
$498$	0	0
$499$	−2302.63	−0.206573	−0.103287	−	0.994652i	$-0.532936\pi$
$499$	−2302.63	−0.206573	−0.103287	+	0.994652i	$0.532936\pi$
$500$	997.219	0.0891940
$501$	0	0
$502$	−708.302	−0.0629742
$503$	−830.769	−0.0736425	−0.0368213	−	0.999322i	$-0.511723\pi$
$503$	−830.769	−0.0736425	−0.0368213	+	0.999322i	$0.511723\pi$
$504$	0	0
$505$	5996.18	0.528369
$506$	−17.8202	−0.00156562
$507$	0	0
$508$	−20508.4	−1.79116
$509$	13864.3	1.20732	0.603658	−	0.797243i	$-0.293709\pi$
$509$	13864.3	1.20732	0.603658	+	0.797243i	$0.293709\pi$
$510$	0	0
$511$	−18511.2	−1.60252
$512$	3020.72	0.260739
$513$	0	0
$514$	843.019	0.0723423
$515$	6229.67	0.533033
$516$	0	0
$517$	−6088.81	−0.517961
$518$	−649.253	−0.0550706
$519$	0	0
$520$	−140.934	−0.0118853
$521$	2262.83	0.190281	0.0951403	−	0.995464i	$-0.469670\pi$
$521$	2262.83	0.190281	0.0951403	+	0.995464i	$0.469670\pi$
$522$	0	0
$523$	8379.47	0.700590	0.350295	−	0.936639i	$-0.386081\pi$
$523$	8379.47	0.700590	0.350295	+	0.936639i	$0.386081\pi$
$524$	−5249.82	−0.437671
$525$	0	0
$526$	−490.684	−0.0406746
$527$	1084.96	0.0896805
$528$	0	0
$529$	−12021.2	−0.988014
$530$	525.844	0.0430966
$531$	0	0
$532$	10069.2	0.820593
$533$	4870.36	0.395795
$534$	0	0
$535$	5795.71	0.468356
$536$	−662.065	−0.0533523
$537$	0	0
$538$	813.265	0.0651717
$539$	620.517	0.0495873
$540$	0	0
$541$	11880.6	0.944155	0.472078	−	0.881557i	$-0.343504\pi$
$541$	11880.6	0.944155	0.472078	+	0.881557i	$0.343504\pi$
$542$	−35.6354	−0.00282412
$543$	0	0
$544$	173.803	0.0136980
$545$	593.089	0.0466149
$546$	0	0
$547$	−8668.47	−0.677581	−0.338791	−	0.940862i	$-0.610018\pi$
$547$	−8668.47	−0.677581	−0.338791	+	0.940862i	$0.610018\pi$
$548$	−4999.93	−0.389756
$549$	0	0
$550$	36.8914	0.00286010
$551$	8809.30	0.681105
$552$	0	0
$553$	10398.0	0.799579
$554$	560.468	0.0429820
$555$	0	0
$556$	7768.30	0.592534
$557$	19918.2	1.51519	0.757594	−	0.652726i	$-0.226375\pi$
$557$	19918.2	1.51519	0.757594	+	0.652726i	$0.226375\pi$
$558$	0	0
$559$	578.591	0.0437778
$560$	−6391.86	−0.482331
$561$	0	0
$562$	70.5454	0.00529498
$563$	14482.2	1.08411	0.542055	−	0.840343i	$-0.317647\pi$
$563$	14482.2	1.08411	0.542055	+	0.840343i	$0.317647\pi$
$564$	0	0
$565$	1916.41	0.142698
$566$	351.082	0.0260726
$567$	0	0
$568$	−570.292	−0.0421284
$569$	16633.2	1.22548	0.612741	−	0.790284i	$-0.290067\pi$
$569$	16633.2	1.22548	0.612741	+	0.790284i	$0.290067\pi$
$570$	0	0
$571$	830.361	0.0608573	0.0304287	−	0.999537i	$-0.490313\pi$
$571$	830.361	0.0608573	0.0304287	+	0.999537i	$0.490313\pi$
$572$	933.576	0.0682426
$573$	0	0
$574$	−1237.06	−0.0899544
$575$	−301.902	−0.0218960
$576$	0	0
$577$	−21152.6	−1.52616	−0.763080	−	0.646305i	$-0.776314\pi$
$577$	−21152.6	−1.52616	−0.763080	+	0.646305i	$0.776314\pi$
$578$	−727.240	−0.0523342
$579$	0	0
$580$	−5607.76	−0.401465
$581$	13136.1	0.938000
$582$	0	0
$583$	−6976.29	−0.495589
$584$	2190.09	0.155183
$585$	0	0
$586$	−617.938	−0.0435610
$587$	−17054.5	−1.19917	−0.599587	−	0.800309i	$-0.704669\pi$
$587$	−17054.5	−1.19917	−0.599587	+	0.800309i	$0.704669\pi$
$588$	0	0
$589$	−11160.2	−0.780729
$590$	368.804	0.0257346
$591$	0	0
$592$	−13715.9	−0.952228
$593$	24693.4	1.71001	0.855006	−	0.518619i	$-0.173554\pi$
$593$	24693.4	1.71001	0.855006	+	0.518619i	$0.173554\pi$
$594$	0	0
$595$	−613.516	−0.0422717
$596$	23740.5	1.63163
$597$	0	0
$598$	21.3038	0.00145682
$599$	14641.2	0.998702	0.499351	−	0.866400i	$-0.333572\pi$
$599$	14641.2	0.998702	0.499351	+	0.866400i	$0.333572\pi$
$600$	0	0
$601$	13123.7	0.890730	0.445365	−	0.895349i	$-0.353074\pi$
$601$	13123.7	0.890730	0.445365	+	0.895349i	$0.353074\pi$
$602$	−146.961	−0.00994961
$603$	0	0
$604$	16565.3	1.11595
$605$	6165.57	0.414324
$606$	0	0
$607$	−10585.5	−0.707831	−0.353915	−	0.935277i	$-0.615150\pi$
$607$	−10585.5	−0.707831	−0.353915	+	0.935277i	$0.615150\pi$
$608$	−1787.79	−0.119251
$609$	0	0
$610$	−496.821	−0.0329765
$611$	7279.11	0.481966
$612$	0	0
$613$	17835.2	1.17513	0.587566	−	0.809176i	$-0.300086\pi$
$613$	17835.2	1.17513	0.587566	+	0.809176i	$0.300086\pi$
$614$	33.4712	0.00219998
$615$	0	0
$616$	−474.912	−0.0310629
$617$	18779.6	1.22535	0.612674	−	0.790336i	$-0.290094\pi$
$617$	18779.6	1.22535	0.612674	+	0.790336i	$0.290094\pi$
$618$	0	0
$619$	3290.29	0.213648	0.106824	−	0.994278i	$-0.465932\pi$
$619$	3290.29	0.213648	0.106824	+	0.994278i	$0.465932\pi$
$620$	7104.29	0.460186
$621$	0	0
$622$	−906.255	−0.0584204
$623$	10937.4	0.703367
$624$	0	0
$625$	625.000	0.0400000
$626$	515.512	0.0329138
$627$	0	0
$628$	−2810.96	−0.178614
$629$	−1316.50	−0.0834538
$630$	0	0
$631$	2094.43	0.132136	0.0660682	−	0.997815i	$-0.478955\pi$
$631$	2094.43	0.132136	0.0660682	+	0.997815i	$0.478955\pi$
$632$	−1230.20	−0.0774285
$633$	0	0
$634$	−1138.32	−0.0713067
$635$	−12853.5	−0.803266
$636$	0	0
$637$	−741.820	−0.0461413
$638$	−207.455	−0.0128734
$639$	0	0
$640$	1516.70	0.0936762
$641$	4205.73	0.259152	0.129576	−	0.991569i	$-0.458638\pi$
$641$	4205.73	0.259152	0.129576	+	0.991569i	$0.458638\pi$
$642$	0	0
$643$	7034.19	0.431417	0.215709	−	0.976458i	$-0.430794\pi$
$643$	7034.19	0.431417	0.215709	+	0.976458i	$0.430794\pi$
$644$	1940.52	0.118738
$645$	0	0
$646$	−56.9338	−0.00346754
$647$	11051.7	0.671543	0.335772	−	0.941943i	$-0.391003\pi$
$647$	11051.7	0.671543	0.335772	+	0.941943i	$0.391003\pi$
$648$	0	0
$649$	−4892.87	−0.295935
$650$	−44.1033	−0.00266134
$651$	0	0
$652$	−22114.7	−1.32834
$653$	−2555.51	−0.153147	−0.0765734	−	0.997064i	$-0.524398\pi$
$653$	−2555.51	−0.153147	−0.0765734	+	0.997064i	$0.524398\pi$
$654$	0	0
$655$	−3290.29	−0.196278
$656$	−26133.6	−1.55541
$657$	0	0
$658$	−1848.87	−0.109539
$659$	−4667.18	−0.275884	−0.137942	−	0.990440i	$-0.544049\pi$
$659$	−4667.18	−0.275884	−0.137942	+	0.990440i	$0.544049\pi$
$660$	0	0
$661$	−4534.83	−0.266845	−0.133422	−	0.991059i	$-0.542597\pi$
$661$	−4534.83	−0.266845	−0.133422	+	0.991059i	$0.542597\pi$
$662$	177.545	0.0104237
$663$	0	0
$664$	−1554.15	−0.0908327
$665$	6310.81	0.368004
$666$	0	0
$667$	1697.71	0.0985544
$668$	−30598.2	−1.77227
$669$	0	0
$670$	−207.183	−0.0119466
$671$	6591.25	0.379214
$672$	0	0
$673$	27420.0	1.57052	0.785262	−	0.619164i	$-0.212528\pi$
$673$	27420.0	1.57052	0.785262	+	0.619164i	$0.212528\pi$
$674$	103.692	0.00592590
$675$	0	0
$676$	16411.0	0.933719
$677$	26519.9	1.50553	0.752764	−	0.658291i	$-0.228720\pi$
$677$	26519.9	1.50553	0.752764	+	0.658291i	$0.228720\pi$
$678$	0	0
$679$	22829.3	1.29029
$680$	72.5860	0.00409345
$681$	0	0
$682$	262.818	0.0147564
$683$	4297.61	0.240767	0.120383	−	0.992727i	$-0.461588\pi$
$683$	4297.61	0.240767	0.120383	+	0.992727i	$0.461588\pi$
$684$	0	0
$685$	−3133.67	−0.174790
$686$	−842.037	−0.0468646
$687$	0	0
$688$	−3104.63	−0.172039
$689$	8340.07	0.461149
$690$	0	0
$691$	−21901.2	−1.20573	−0.602866	−	0.797842i	$-0.705975\pi$
$691$	−21901.2	−1.20573	−0.602866	+	0.797842i	$0.705975\pi$
$692$	−9058.37	−0.497612
$693$	0	0
$694$	1526.17	0.0834763
$695$	4868.72	0.265728
$696$	0	0
$697$	−2508.41	−0.136317
$698$	1192.84	0.0646845
$699$	0	0
$700$	−4017.29	−0.216913
$701$	−11225.3	−0.604814	−0.302407	−	0.953179i	$-0.597790\pi$
$701$	−11225.3	−0.604814	−0.302407	+	0.953179i	$0.597790\pi$
$702$	0	0
$703$	13541.9	0.726521
$704$	−4981.29	−0.266675
$705$	0	0
$706$	810.885	0.0432267
$707$	−24155.5	−1.28495
$708$	0	0
$709$	11558.8	0.612269	0.306134	−	0.951988i	$-0.400964\pi$
$709$	11558.8	0.612269	0.306134	+	0.951988i	$0.400964\pi$
$710$	−178.464	−0.00943331
$711$	0	0
$712$	−1294.02	−0.0681117
$713$	−2150.78	−0.112970
$714$	0	0
$715$	585.112	0.0306041
$716$	−25469.5	−1.32939
$717$	0	0
$718$	−711.075	−0.0369597
$719$	−17315.4	−0.898128	−0.449064	−	0.893500i	$-0.648243\pi$
$719$	−17315.4	−0.898128	−0.449064	+	0.893500i	$0.648243\pi$
$720$	0	0
$721$	−25096.2	−1.29630
$722$	−437.383	−0.0225453
$723$	0	0
$724$	−7903.09	−0.405685
$725$	−3514.62	−0.180041
$726$	0	0
$727$	−31457.0	−1.60478	−0.802390	−	0.596800i	$-0.796439\pi$
$727$	−31457.0	−1.60478	−0.802390	+	0.596800i	$0.796439\pi$
$728$	567.752	0.0289042
$729$	0	0
$730$	685.357	0.0347482
$731$	−297.995	−0.0150776
$732$	0	0
$733$	16092.4	0.810893	0.405447	−	0.914119i	$-0.367116\pi$
$733$	16092.4	0.810893	0.405447	+	0.914119i	$0.367116\pi$
$734$	1897.40	0.0954144
$735$	0	0
$736$	−344.539	−0.0172553
$737$	2748.67	0.137379
$738$	0	0
$739$	13798.5	0.686857	0.343429	−	0.939179i	$-0.388412\pi$
$739$	13798.5	0.686857	0.343429	+	0.939179i	$0.388412\pi$
$740$	−8620.43	−0.428234
$741$	0	0
$742$	−2118.35	−0.104808
$743$	−19368.8	−0.956356	−0.478178	−	0.878263i	$-0.658703\pi$
$743$	−19368.8	−0.956356	−0.478178	+	0.878263i	$0.658703\pi$
$744$	0	0
$745$	14879.2	0.731721
$746$	−1388.01	−0.0681215
$747$	0	0
$748$	−480.824	−0.0235035
$749$	−23348.0	−1.13901
$750$	0	0
$751$	2350.90	0.114228	0.0571141	−	0.998368i	$-0.481810\pi$
$751$	2350.90	0.114228	0.0571141	+	0.998368i	$0.481810\pi$
$752$	−39058.6	−1.89404
$753$	0	0
$754$	248.010	0.0119788
$755$	10382.2	0.500458
$756$	0	0
$757$	−32608.1	−1.56560	−0.782801	−	0.622272i	$-0.786210\pi$
$757$	−32608.1	−1.56560	−0.782801	+	0.622272i	$0.786210\pi$
$758$	−1733.68	−0.0830742
$759$	0	0
$760$	−746.642	−0.0356362
$761$	32977.3	1.57086	0.785431	−	0.618949i	$-0.212441\pi$
$761$	32977.3	1.57086	0.785431	+	0.618949i	$0.212441\pi$
$762$	0	0
$763$	−2389.25	−0.113364
$764$	−26196.7	−1.24053
$765$	0	0
$766$	−1736.54	−0.0819110
$767$	5849.37	0.275369
$768$	0	0
$769$	26787.7	1.25616	0.628082	−	0.778147i	$-0.283840\pi$
$769$	26787.7	1.25616	0.628082	+	0.778147i	$0.283840\pi$
$770$	−148.617	−0.00695555
$771$	0	0
$772$	18891.4	0.880720
$773$	1474.56	0.0686108	0.0343054	−	0.999411i	$-0.489078\pi$
$773$	1474.56	0.0686108	0.0343054	+	0.999411i	$0.489078\pi$
$774$	0	0
$775$	4452.56	0.206375
$776$	−2700.97	−0.124947
$777$	0	0
$778$	1040.97	0.0479697
$779$	25802.2	1.18673
$780$	0	0
$781$	2367.66	0.108478
$782$	−10.9722	−0.000501745 0
$783$	0	0
$784$	3980.49	0.181327
$785$	−1761.75	−0.0801013
$786$	0	0
$787$	17264.3	0.781966	0.390983	−	0.920398i	$-0.372135\pi$
$787$	17264.3	0.781966	0.390983	+	0.920398i	$0.372135\pi$
$788$	29948.5	1.35390
$789$	0	0
$790$	−384.973	−0.0173376
$791$	−7720.25	−0.347030
$792$	0	0
$793$	−7879.76	−0.352861
$794$	430.228	0.0192295
$795$	0	0
$796$	−15204.1	−0.677006
$797$	−43561.3	−1.93603	−0.968017	−	0.250885i	$-0.919278\pi$
$797$	−43561.3	−1.93603	−0.968017	+	0.250885i	$0.919278\pi$
$798$	0	0
$799$	−3748.99	−0.165995
$800$	713.268	0.0315223
$801$	0	0
$802$	1639.81	0.0721992
$803$	−9092.53	−0.399587
$804$	0	0
$805$	1216.21	0.0532493
$806$	−314.196	−0.0137309
$807$	0	0
$808$	2857.88	0.124431
$809$	−1854.39	−0.0805896	−0.0402948	−	0.999188i	$-0.512830\pi$
$809$	−1854.39	−0.0805896	−0.0402948	+	0.999188i	$0.512830\pi$
$810$	0	0
$811$	−42771.0	−1.85190	−0.925952	−	0.377642i	$-0.876735\pi$
$811$	−42771.0	−1.85190	−0.925952	+	0.377642i	$0.876735\pi$
$812$	22590.8	0.976332
$813$	0	0
$814$	−318.907	−0.0137318
$815$	−13860.2	−0.595709
$816$	0	0
$817$	3065.26	0.131261
$818$	−424.561	−0.0181472
$819$	0	0
$820$	−16425.0	−0.699494
$821$	−25816.7	−1.09745	−0.548727	−	0.836001i	$-0.684888\pi$
$821$	−25816.7	−1.09745	−0.548727	+	0.836001i	$0.684888\pi$
$822$	0	0
$823$	41350.6	1.75139	0.875694	−	0.482867i	$-0.160404\pi$
$823$	41350.6	1.75139	0.875694	+	0.482867i	$0.160404\pi$
$824$	2969.17	0.125529
$825$	0	0
$826$	−1485.72	−0.0625846
$827$	14681.0	0.617302	0.308651	−	0.951175i	$-0.400122\pi$
$827$	14681.0	0.617302	0.308651	+	0.951175i	$0.400122\pi$
$828$	0	0
$829$	−31867.1	−1.33509	−0.667546	−	0.744568i	$-0.732655\pi$
$829$	−31867.1	−1.33509	−0.667546	+	0.744568i	$0.732655\pi$
$830$	−486.350	−0.0203391
$831$	0	0
$832$	5955.07	0.248143
$833$	382.063	0.0158916
$834$	0	0
$835$	−19177.2	−0.794796
$836$	4945.90	0.204614
$837$	0	0
$838$	−1238.71	−0.0510627
$839$	1524.16	0.0627173	0.0313587	−	0.999508i	$-0.490017\pi$
$839$	1524.16	0.0627173	0.0313587	+	0.999508i	$0.490017\pi$
$840$	0	0
$841$	−4624.88	−0.189630
$842$	532.686	0.0218024
$843$	0	0
$844$	29431.4	1.20032
$845$	10285.5	0.418736
$846$	0	0
$847$	−24837.9	−1.00760
$848$	−44751.5	−1.81223
$849$	0	0
$850$	22.7147	0.000916598 0
$851$	2609.78	0.105126
$852$	0	0
$853$	43298.5	1.73800	0.868999	−	0.494814i	$-0.164764\pi$
$853$	43298.5	1.73800	0.868999	+	0.494814i	$0.164764\pi$
$854$	2001.44	0.0801964
$855$	0	0
$856$	2762.33	0.110297
$857$	13372.0	0.532999	0.266500	−	0.963835i	$-0.414133\pi$
$857$	13372.0	0.532999	0.266500	+	0.963835i	$0.414133\pi$
$858$	0	0
$859$	16341.6	0.649091	0.324545	−	0.945870i	$-0.394789\pi$
$859$	16341.6	0.649091	0.324545	+	0.945870i	$0.394789\pi$
$860$	−1951.26	−0.0773691
$861$	0	0
$862$	1999.90	0.0790221
$863$	−48077.3	−1.89637	−0.948186	−	0.317715i	$-0.897084\pi$
$863$	−48077.3	−1.89637	−0.948186	+	0.317715i	$0.897084\pi$
$864$	0	0
$865$	−5677.27	−0.223159
$866$	−253.282	−0.00993865
$867$	0	0
$868$	−28619.6	−1.11914
$869$	5107.39	0.199374
$870$	0	0
$871$	−3286.00	−0.127832
$872$	282.676	0.0109778
$873$	0	0
$874$	112.863	0.00436803
$875$	−2517.81	−0.0972770
$876$	0	0
$877$	−50885.9	−1.95929	−0.979643	−	0.200749i	$-0.935663\pi$
$877$	−50885.9	−1.95929	−0.979643	+	0.200749i	$0.935663\pi$
$878$	−1600.40	−0.0615160
$879$	0	0
$880$	−3139.62	−0.120269
$881$	−2826.26	−0.108081	−0.0540404	−	0.998539i	$-0.517210\pi$
$881$	−2826.26	−0.108081	−0.0540404	+	0.998539i	$0.517210\pi$
$882$	0	0
$883$	−2235.30	−0.0851913	−0.0425956	−	0.999092i	$-0.513563\pi$
$883$	−2235.30	−0.0851913	−0.0425956	+	0.999092i	$0.513563\pi$
$884$	574.819	0.0218702
$885$	0	0
$886$	1258.76	0.0477300
$887$	−8053.31	−0.304852	−0.152426	−	0.988315i	$-0.548709\pi$
$887$	−8053.31	−0.304852	−0.152426	+	0.988315i	$0.548709\pi$
$888$	0	0
$889$	51780.0	1.95348
$890$	−404.945	−0.0152514
$891$	0	0
$892$	−1353.96	−0.0508227
$893$	38563.3	1.44510
$894$	0	0
$895$	−15962.8	−0.596177
$896$	−6110.00	−0.227813
$897$	0	0
$898$	575.822	0.0213980
$899$	−25038.5	−0.928901
$900$	0	0
$901$	−4295.43	−0.158825
$902$	−607.631	−0.0224300
$903$	0	0
$904$	913.395	0.0336052
$905$	−4953.20	−0.181934
$906$	0	0
$907$	−20577.5	−0.753324	−0.376662	−	0.926351i	$-0.622928\pi$
$907$	−20577.5	−0.753324	−0.376662	+	0.926351i	$0.622928\pi$
$908$	28779.4	1.05185
$909$	0	0
$910$	177.670	0.00647218
$911$	−43267.9	−1.57358	−0.786790	−	0.617221i	$-0.788258\pi$
$911$	−43267.9	−1.57358	−0.786790	+	0.617221i	$0.788258\pi$
$912$	0	0
$913$	6452.33	0.233889
$914$	−517.979	−0.0187453
$915$	0	0
$916$	48615.6	1.75361
$917$	13254.9	0.477333
$918$	0	0
$919$	39601.9	1.42149	0.710743	−	0.703452i	$-0.248359\pi$
$919$	39601.9	1.42149	0.710743	+	0.703452i	$0.248359\pi$
$920$	−143.892	−0.00515648
$921$	0	0
$922$	−1020.47	−0.0364506
$923$	−2830.51	−0.100940
$924$	0	0
$925$	−5402.79	−0.192046
$926$	921.827	0.0327140
$927$	0	0
$928$	−4010.99	−0.141883
$929$	−8484.80	−0.299653	−0.149826	−	0.988712i	$-0.547871\pi$
$929$	−8484.80	−0.299653	−0.149826	+	0.988712i	$0.547871\pi$
$930$	0	0
$931$	−3930.02	−0.138347
$932$	−31354.8	−1.10200
$933$	0	0
$934$	−1637.26	−0.0573584
$935$	−301.353	−0.0105404
$936$	0	0
$937$	−31244.9	−1.08935	−0.544677	−	0.838646i	$-0.683348\pi$
$937$	−31244.9	−1.08935	−0.544677	+	0.838646i	$0.683348\pi$
$938$	834.635	0.0290531
$939$	0	0
$940$	−24548.3	−0.851785
$941$	19514.9	0.676055	0.338027	−	0.941136i	$-0.390240\pi$
$941$	19514.9	0.676055	0.338027	+	0.941136i	$0.390240\pi$
$942$	0	0
$943$	4972.56	0.171717
$944$	−31386.8	−1.08215
$945$	0	0
$946$	−72.1855	−0.00248092
$947$	−29363.7	−1.00759	−0.503797	−	0.863822i	$-0.668064\pi$
$947$	−29363.7	−1.00759	−0.503797	+	0.863822i	$0.668064\pi$
$948$	0	0
$949$	10870.0	0.371818
$950$	−233.650	−0.00797960
$951$	0	0
$952$	−292.412	−0.00995496
$953$	−21469.5	−0.729763	−0.364881	−	0.931054i	$-0.618890\pi$
$953$	−21469.5	−0.729763	−0.364881	+	0.931054i	$0.618890\pi$
$954$	0	0
$955$	−16418.6	−0.556328
$956$	29216.4	0.988417
$957$	0	0
$958$	−788.356	−0.0265873
$959$	12623.9	0.425077
$960$	0	0
$961$	1929.53	0.0647690
$962$	381.249	0.0127775
$963$	0	0
$964$	−1777.67	−0.0593932
$965$	11840.0	0.394968
$966$	0	0
$967$	22389.4	0.744566	0.372283	−	0.928119i	$-0.378575\pi$
$967$	22389.4	0.744566	0.372283	+	0.928119i	$0.378575\pi$
$968$	2938.61	0.0975729
$969$	0	0
$970$	−845.227	−0.0279779
$971$	9483.42	0.313427	0.156713	−	0.987644i	$-0.449910\pi$
$971$	9483.42	0.313427	0.156713	+	0.987644i	$0.449910\pi$
$972$	0	0
$973$	−19613.6	−0.646231
$974$	2299.62	0.0756514
$975$	0	0
$976$	42281.6	1.38668
$977$	−46343.6	−1.51757	−0.758784	−	0.651342i	$-0.774206\pi$
$977$	−46343.6	−1.51757	−0.758784	+	0.651342i	$0.774206\pi$
$978$	0	0
$979$	5372.34	0.175384
$980$	2501.74	0.0815461
$981$	0	0
$982$	−2055.65	−0.0668007
$983$	41130.3	1.33454	0.667270	−	0.744816i	$-0.267463\pi$
$983$	41130.3	1.33454	0.667270	+	0.744816i	$0.267463\pi$
$984$	0	0
$985$	18770.0	0.607169
$986$	−127.734	−0.00412563
$987$	0	0
$988$	−5912.76	−0.190395
$989$	590.732	0.0189931
$990$	0	0
$991$	25380.8	0.813568	0.406784	−	0.913524i	$-0.366650\pi$
$991$	25380.8	0.813568	0.406784	+	0.913524i	$0.366650\pi$
$992$	5081.40	0.162636
$993$	0	0
$994$	718.941	0.0229411
$995$	−9529.09	−0.303611
$996$	0	0
$997$	−28351.0	−0.900588	−0.450294	−	0.892880i	$-0.648681\pi$
$997$	−28351.0	−0.900588	−0.450294	+	0.892880i	$0.648681\pi$
$998$	−343.438	−0.0108931
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.k.1.4	✓	6
3.2	odd	2		405.4.a.l.1.3	yes	6
5.4	even	2		2025.4.a.z.1.3		6
9.2	odd	6		405.4.e.w.271.4		12
9.4	even	3		405.4.e.x.136.3		12
9.5	odd	6		405.4.e.w.136.4		12
9.7	even	3		405.4.e.x.271.3		12
15.14	odd	2		2025.4.a.y.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.4	✓	6	1.1	even	1	trivial
405.4.a.l.1.3	yes	6	3.2	odd	2
405.4.e.w.136.4		12	9.5	odd	6
405.4.e.w.271.4		12	9.2	odd	6
405.4.e.x.136.3		12	9.4	even	3
405.4.e.x.271.3		12	9.7	even	3
2025.4.a.y.1.4		6	15.14	odd	2
2025.4.a.z.1.3		6	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)

\(f(q)\)	\(=\)	\(q+0.149150 q^{2} -7.97775 q^{4} -5.00000 q^{5} +20.1424 q^{7} -2.38308 q^{8} +O(q^{10})\)	\(q+0.149150 q^{2} -7.97775 q^{4} -5.00000 q^{5} +20.1424 q^{7} -2.38308 q^{8} -0.745751 q^{10} +9.89377 q^{11} -11.8279 q^{13} +3.00425 q^{14} +63.4666 q^{16} +6.09177 q^{17} -62.6618 q^{19} +39.8888 q^{20} +1.47566 q^{22} -12.0761 q^{23} +25.0000 q^{25} -1.76413 q^{26} -160.691 q^{28} -140.585 q^{29} +178.103 q^{31} +28.5307 q^{32} +0.908588 q^{34} -100.712 q^{35} -216.112 q^{37} -9.34602 q^{38} +11.9154 q^{40} -411.770 q^{41} -48.9176 q^{43} -78.9300 q^{44} -1.80115 q^{46} -615.419 q^{47} +62.7179 q^{49} +3.72875 q^{50} +94.3600 q^{52} -705.120 q^{53} -49.4688 q^{55} -48.0011 q^{56} -20.9683 q^{58} -494.540 q^{59} +666.202 q^{61} +26.5640 q^{62} -503.477 q^{64} +59.1394 q^{65} +277.818 q^{67} -48.5986 q^{68} -15.0212 q^{70} +239.308 q^{71} -919.015 q^{73} -32.2331 q^{74} +499.900 q^{76} +199.285 q^{77} +516.223 q^{79} -317.333 q^{80} -61.4155 q^{82} +652.161 q^{83} -30.4588 q^{85} -7.29606 q^{86} -23.5777 q^{88} +543.003 q^{89} -238.242 q^{91} +96.3400 q^{92} -91.7899 q^{94} +313.309 q^{95} +1133.39 q^{97} +9.35439 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8}+O(q^{10}) \) 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8	\( 6 q - 4 q^{2} + 34 q^{4} - 30 q^{5} + 40 q^{7} - 66 q^{8} + 20 q^{10} - 88 q^{11} + 20 q^{13} - 180 q^{14} + 58 q^{16} - 124 q^{17} - 46 q^{19} - 170 q^{20} - 74 q^{22} - 210 q^{23} + 150 q^{25} - 4 q^{26} + 352 q^{28} - 296 q^{29} - 104 q^{31} - 722 q^{32} - 428 q^{34} - 200 q^{35} - 204 q^{37} + 20 q^{38} + 330 q^{40} - 344 q^{41} + 512 q^{43} - 716 q^{44} - 186 q^{46} - 238 q^{47} + 68 q^{49} - 100 q^{50} - 468 q^{52} - 850 q^{53} + 440 q^{55} - 2316 q^{56} + 890 q^{58} - 1840 q^{59} - 364 q^{61} - 1038 q^{62} - 990 q^{64} - 100 q^{65} + 88 q^{67} - 236 q^{68} + 900 q^{70} - 1364 q^{71} + 836 q^{73} - 1316 q^{74} - 2106 q^{76} - 840 q^{77} - 680 q^{79} - 290 q^{80} + 1742 q^{82} - 2148 q^{83} + 620 q^{85} - 2872 q^{86} + 1296 q^{88} - 3000 q^{89} - 3058 q^{91} - 1002 q^{92} - 3662 q^{94} + 230 q^{95} - 612 q^{97} - 1982 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 + 34 * q^4 - 30 * q^5 + 40 * q^7 - 66 * q^8 + 20 * q^10 - 88 * q^11 + 20 * q^13 - 180 * q^14 + 58 * q^16 - 124 * q^17 - 46 * q^19 - 170 * q^20 - 74 * q^22 - 210 * q^23 + 150 * q^25 - 4 * q^26 + 352 * q^28 - 296 * q^29 - 104 * q^31 - 722 * q^32 - 428 * q^34 - 200 * q^35 - 204 * q^37 + 20 * q^38 + 330 * q^40 - 344 * q^41 + 512 * q^43 - 716 * q^44 - 186 * q^46 - 238 * q^47 + 68 * q^49 - 100 * q^50 - 468 * q^52 - 850 * q^53 + 440 * q^55 - 2316 * q^56 + 890 * q^58 - 1840 * q^59 - 364 * q^61 - 1038 * q^62 - 990 * q^64 - 100 * q^65 + 88 * q^67 - 236 * q^68 + 900 * q^70 - 1364 * q^71 + 836 * q^73 - 1316 * q^74 - 2106 * q^76 - 840 * q^77 - 680 * q^79 - 290 * q^80 + 1742 * q^82 - 2148 * q^83 + 620 * q^85 - 2872 * q^86 + 1296 * q^88 - 3000 * q^89 - 3058 * q^91 - 1002 * q^92 - 3662 * q^94 + 230 * q^95 - 612 * q^97 - 1982 * q^98

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