LMFDB - Embedded newform 464.6.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,6,Mod(1,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("464.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	464.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:	$74.4180923932$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.3257317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 34x^{2} - 27x + 10 $ x^4 - 34x^2 - 27x + 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$6.17343$ of defining polynomial
Character	$\chi$	$=$	464.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.07413 q^{3} -44.4758 q^{5} +36.7447 q^{7} -192.957 q^{9} +O(q^{10})$	$q-7.07413 q^{3} -44.4758 q^{5} +36.7447 q^{7} -192.957 q^{9} +302.283 q^{11} -373.472 q^{13} +314.627 q^{15} +280.365 q^{17} -1371.41 q^{19} -259.937 q^{21} -1861.10 q^{23} -1146.91 q^{25} +3084.01 q^{27} -841.000 q^{29} -1472.03 q^{31} -2138.39 q^{33} -1634.25 q^{35} -11730.4 q^{37} +2641.99 q^{39} -2177.39 q^{41} +9679.03 q^{43} +8581.90 q^{45} +15909.1 q^{47} -15456.8 q^{49} -1983.34 q^{51} +24359.3 q^{53} -13444.3 q^{55} +9701.53 q^{57} -36304.7 q^{59} -22316.1 q^{61} -7090.14 q^{63} +16610.5 q^{65} +54808.6 q^{67} +13165.7 q^{69} -27790.4 q^{71} +31685.5 q^{73} +8113.36 q^{75} +11107.3 q^{77} +55328.4 q^{79} +25071.8 q^{81} +46888.8 q^{83} -12469.5 q^{85} +5949.34 q^{87} +2564.30 q^{89} -13723.1 q^{91} +10413.3 q^{93} +60994.5 q^{95} +34940.3 q^{97} -58327.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9}+O(q^{10}) $ 4 * q + 28 * q^3 - 68 * q^5 + 208 * q^7 - 280 * q^9	$ 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9} + 124 q^{11} - 460 q^{13} - 932 q^{15} + 184 q^{17} + 2392 q^{19} + 992 q^{21} + 1192 q^{23} + 1824 q^{25} - 2468 q^{27} - 3364 q^{29} + 19212 q^{31} - 10580 q^{33} + 22944 q^{35} - 10928 q^{37} + 8732 q^{39} - 1120 q^{41} + 21420 q^{43} - 8344 q^{45} - 23772 q^{47} + 10452 q^{49} - 12744 q^{51} + 8860 q^{53} + 52652 q^{55} + 48944 q^{57} + 10840 q^{59} + 49448 q^{61} - 27488 q^{63} + 97836 q^{65} + 7840 q^{67} + 58792 q^{69} + 48744 q^{71} - 74992 q^{73} + 90448 q^{75} + 128656 q^{77} + 106076 q^{79} - 59692 q^{81} - 62888 q^{83} + 23848 q^{85} - 23548 q^{87} + 107568 q^{89} + 268896 q^{91} + 221460 q^{93} - 147352 q^{95} - 49520 q^{97} - 166720 q^{99}+O(q^{100}) $ 4 * q + 28 * q^3 - 68 * q^5 + 208 * q^7 - 280 * q^9 + 124 * q^11 - 460 * q^13 - 932 * q^15 + 184 * q^17 + 2392 * q^19 + 992 * q^21 + 1192 * q^23 + 1824 * q^25 - 2468 * q^27 - 3364 * q^29 + 19212 * q^31 - 10580 * q^33 + 22944 * q^35 - 10928 * q^37 + 8732 * q^39 - 1120 * q^41 + 21420 * q^43 - 8344 * q^45 - 23772 * q^47 + 10452 * q^49 - 12744 * q^51 + 8860 * q^53 + 52652 * q^55 + 48944 * q^57 + 10840 * q^59 + 49448 * q^61 - 27488 * q^63 + 97836 * q^65 + 7840 * q^67 + 58792 * q^69 + 48744 * q^71 - 74992 * q^73 + 90448 * q^75 + 128656 * q^77 + 106076 * q^79 - 59692 * q^81 - 62888 * q^83 + 23848 * q^85 - 23548 * q^87 + 107568 * q^89 + 268896 * q^91 + 221460 * q^93 - 147352 * q^95 - 49520 * q^97 - 166720 * q^99

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	0
$2$	0	0
$3$	−7.07413	−0.453806	−0.226903	−	0.973917i	$-0.572860\pi$
$3$	−7.07413	−0.453806	−0.226903	+	0.973917i	$0.572860\pi$
$4$	0	0
$5$	−44.4758	−0.795607	−0.397803	−	0.917471i	$-0.630227\pi$
$5$	−44.4758	−0.795607	−0.397803	+	0.917471i	$0.630227\pi$
$6$	0	0
$7$	36.7447	0.283433	0.141716	−	0.989907i	$-0.454738\pi$
$7$	36.7447	0.283433	0.141716	+	0.989907i	$0.454738\pi$
$8$	0	0
$9$	−192.957	−0.794060
$10$	0	0
$11$	302.283	0.753237	0.376619	−	0.926368i	$-0.377087\pi$
$11$	302.283	0.753237	0.376619	+	0.926368i	$0.377087\pi$
$12$	0	0
$13$	−373.472	−0.612915	−0.306457	−	0.951884i	$-0.599144\pi$
$13$	−373.472	−0.612915	−0.306457	+	0.951884i	$0.599144\pi$
$14$	0	0
$15$	314.627	0.361051
$16$	0	0
$17$	280.365	0.235289	0.117645	−	0.993056i	$-0.462466\pi$
$17$	280.365	0.235289	0.117645	+	0.993056i	$0.462466\pi$
$18$	0	0
$19$	−1371.41	−0.871532	−0.435766	−	0.900060i	$-0.643522\pi$
$19$	−1371.41	−0.871532	−0.435766	+	0.900060i	$0.643522\pi$
$20$	0	0
$21$	−259.937	−0.128623
$22$	0	0
$23$	−1861.10	−0.733586	−0.366793	−	0.930303i	$-0.619544\pi$
$23$	−1861.10	−0.733586	−0.366793	+	0.930303i	$0.619544\pi$
$24$	0	0
$25$	−1146.91	−0.367010
$26$	0	0
$27$	3084.01	0.814155
$28$	0	0
$29$	−841.000	−0.185695
$29$	−841.000	−0.185695
$30$	0	0
$31$	−1472.03	−0.275114	−0.137557	−	0.990494i	$-0.543925\pi$
$31$	−1472.03	−0.275114	−0.137557	+	0.990494i	$0.543925\pi$
$32$	0	0
$33$	−2138.39	−0.341823
$34$	0	0
$35$	−1634.25	−0.225501
$36$	0	0
$37$	−11730.4	−1.40867	−0.704335	−	0.709868i	$-0.748755\pi$
$37$	−11730.4	−1.40867	−0.704335	+	0.709868i	$0.748755\pi$
$38$	0	0
$39$	2641.99	0.278144
$40$	0	0
$41$	−2177.39	−0.202291	−0.101145	−	0.994872i	$-0.532251\pi$
$41$	−2177.39	−0.202291	−0.101145	+	0.994872i	$0.532251\pi$
$42$	0	0
$43$	9679.03	0.798290	0.399145	−	0.916888i	$-0.369307\pi$
$43$	9679.03	0.798290	0.399145	+	0.916888i	$0.369307\pi$
$44$	0	0
$45$	8581.90	0.631760
$46$	0	0
$47$	15909.1	1.05051	0.525255	−	0.850945i	$-0.323970\pi$
$47$	15909.1	1.05051	0.525255	+	0.850945i	$0.323970\pi$
$48$	0	0
$49$	−15456.8	−0.919666
$50$	0	0
$51$	−1983.34	−0.106776
$52$	0	0
$53$	24359.3	1.19117	0.595586	−	0.803291i	$-0.296920\pi$
$53$	24359.3	1.19117	0.595586	+	0.803291i	$0.296920\pi$
$54$	0	0
$55$	−13444.3	−0.599280
$56$	0	0
$57$	9701.53	0.395506
$58$	0	0
$59$	−36304.7	−1.35779	−0.678894	−	0.734236i	$-0.737541\pi$
$59$	−36304.7	−1.35779	−0.678894	+	0.734236i	$0.737541\pi$
$60$	0	0
$61$	−22316.1	−0.767880	−0.383940	−	0.923358i	$-0.625433\pi$
$61$	−22316.1	−0.767880	−0.383940	+	0.923358i	$0.625433\pi$
$62$	0	0
$63$	−7090.14	−0.225063
$64$	0	0
$65$	16610.5	0.487639
$66$	0	0
$67$	54808.6	1.49163	0.745817	−	0.666151i	$-0.232060\pi$
$67$	54808.6	1.49163	0.745817	+	0.666151i	$0.232060\pi$
$68$	0	0
$69$	13165.7	0.332905
$70$	0	0
$71$	−27790.4	−0.654258	−0.327129	−	0.944980i	$-0.606081\pi$
$71$	−27790.4	−0.654258	−0.327129	+	0.944980i	$0.606081\pi$
$72$	0	0
$73$	31685.5	0.695910	0.347955	−	0.937511i	$-0.386876\pi$
$73$	31685.5	0.695910	0.347955	+	0.937511i	$0.386876\pi$
$74$	0	0
$75$	8113.36	0.166551
$76$	0	0
$77$	11107.3	0.213492
$78$	0	0
$79$	55328.4	0.997426	0.498713	−	0.866767i	$-0.333806\pi$
$79$	55328.4	0.997426	0.498713	+	0.866767i	$0.333806\pi$
$80$	0	0
$81$	25071.8	0.424592
$82$	0	0
$83$	46888.8	0.747092	0.373546	−	0.927612i	$-0.378142\pi$
$83$	46888.8	0.747092	0.373546	+	0.927612i	$0.378142\pi$
$84$	0	0
$85$	−12469.5	−0.187198
$86$	0	0
$87$	5949.34	0.0842696
$88$	0	0
$89$	2564.30	0.0343157	0.0171579	−	0.999853i	$-0.494538\pi$
$89$	2564.30	0.0343157	0.0171579	+	0.999853i	$0.494538\pi$
$90$	0	0
$91$	−13723.1	−0.173720
$92$	0	0
$93$	10413.3	0.124848
$94$	0	0
$95$	60994.5	0.693396
$96$	0	0
$97$	34940.3	0.377048	0.188524	−	0.982069i	$-0.439630\pi$
$97$	34940.3	0.377048	0.188524	+	0.982069i	$0.439630\pi$
$98$	0	0
$99$	−58327.5	−0.598116
$100$	0	0
$101$	170224.	1.66042	0.830210	−	0.557451i	$-0.188221\pi$
$101$	170224.	1.66042	0.830210	+	0.557451i	$0.188221\pi$
$102$	0	0
$103$	−83962.9	−0.779820	−0.389910	−	0.920853i	$-0.627494\pi$
$103$	−83962.9	−0.779820	−0.389910	+	0.920853i	$0.627494\pi$
$104$	0	0
$105$	11560.9	0.102334
$106$	0	0
$107$	−16842.4	−0.142215	−0.0711074	−	0.997469i	$-0.522653\pi$
$107$	−16842.4	−0.142215	−0.0711074	+	0.997469i	$0.522653\pi$
$108$	0	0
$109$	−3855.63	−0.0310834	−0.0155417	−	0.999879i	$-0.504947\pi$
$109$	−3855.63	−0.0310834	−0.0155417	+	0.999879i	$0.504947\pi$
$110$	0	0
$111$	82982.5	0.639262
$112$	0	0
$113$	107319.	0.790644	0.395322	−	0.918543i	$-0.370633\pi$
$113$	107319.	0.790644	0.395322	+	0.918543i	$0.370633\pi$
$114$	0	0
$115$	82774.0	0.583646
$116$	0	0
$117$	72064.0	0.486691
$118$	0	0
$119$	10301.9	0.0666886
$120$	0	0
$121$	−69676.1	−0.432634
$122$	0	0
$123$	15403.1	0.0918006
$124$	0	0
$125$	189996.	1.08760
$126$	0	0
$127$	14823.9	0.0815554	0.0407777	−	0.999168i	$-0.487016\pi$
$127$	14823.9	0.0815554	0.0407777	+	0.999168i	$0.487016\pi$
$128$	0	0
$129$	−68470.7	−0.362269
$130$	0	0
$131$	313171.	1.59442	0.797211	−	0.603701i	$-0.206308\pi$
$131$	313171.	1.59442	0.797211	+	0.603701i	$0.206308\pi$
$132$	0	0
$133$	−50392.1	−0.247020
$134$	0	0
$135$	−137164.	−0.647747
$136$	0	0
$137$	−254312.	−1.15762	−0.578810	−	0.815463i	$-0.696483\pi$
$137$	−254312.	−1.15762	−0.578810	+	0.815463i	$0.696483\pi$
$138$	0	0
$139$	390931.	1.71618	0.858091	−	0.513498i	$-0.171651\pi$
$139$	390931.	1.71618	0.858091	+	0.513498i	$0.171651\pi$
$140$	0	0
$141$	−112543.	−0.476727
$142$	0	0
$143$	−112894.	−0.461670
$144$	0	0
$145$	37404.1	0.147740
$146$	0	0
$147$	109344.	0.417350
$148$	0	0
$149$	72345.7	0.266961	0.133480	−	0.991051i	$-0.457385\pi$
$149$	72345.7	0.266961	0.133480	+	0.991051i	$0.457385\pi$
$150$	0	0
$151$	227419.	0.811679	0.405840	−	0.913944i	$-0.366979\pi$
$151$	227419.	0.811679	0.405840	+	0.913944i	$0.366979\pi$
$152$	0	0
$153$	−54098.4	−0.186834
$154$	0	0
$155$	65469.6	0.218882
$156$	0	0
$157$	539281.	1.74609	0.873044	−	0.487641i	$-0.162143\pi$
$157$	539281.	1.74609	0.873044	+	0.487641i	$0.162143\pi$
$158$	0	0
$159$	−172321.	−0.540561
$160$	0	0
$161$	−68385.7	−0.207922
$162$	0	0
$163$	−87996.5	−0.259416	−0.129708	−	0.991552i	$-0.541404\pi$
$163$	−87996.5	−0.259416	−0.129708	+	0.991552i	$0.541404\pi$
$164$	0	0
$165$	95106.4	0.271957
$166$	0	0
$167$	−199897.	−0.554644	−0.277322	−	0.960777i	$-0.589447\pi$
$167$	−199897.	−0.554644	−0.277322	+	0.960777i	$0.589447\pi$
$168$	0	0
$169$	−231811.	−0.624335
$170$	0	0
$171$	264623.	0.692049
$172$	0	0
$173$	−313797.	−0.797139	−0.398569	−	0.917138i	$-0.630493\pi$
$173$	−313797.	−0.797139	−0.398569	+	0.917138i	$0.630493\pi$
$174$	0	0
$175$	−42142.7	−0.104023
$176$	0	0
$177$	256824.	0.616172
$178$	0	0
$179$	59814.7	0.139533	0.0697663	−	0.997563i	$-0.477775\pi$
$179$	59814.7	0.139533	0.0697663	+	0.997563i	$0.477775\pi$
$180$	0	0
$181$	402231.	0.912597	0.456298	−	0.889827i	$-0.349175\pi$
$181$	402231.	0.912597	0.456298	+	0.889827i	$0.349175\pi$
$182$	0	0
$183$	157867.	0.348468
$184$	0	0
$185$	521719.	1.12075
$186$	0	0
$187$	84749.6	0.177229
$188$	0	0
$189$	113321.	0.230758
$190$	0	0
$191$	903529.	1.79209	0.896043	−	0.443967i	$-0.146429\pi$
$191$	903529.	1.79209	0.896043	+	0.443967i	$0.146429\pi$
$192$	0	0
$193$	1.02870e6	1.98791	0.993956	−	0.109780i	$-0.0350148\pi$
$193$	1.02870e6	1.98791	0.993956	+	0.109780i	$0.0350148\pi$
$194$	0	0
$195$	−117505.	−0.221293
$196$	0	0
$197$	−866099.	−1.59002	−0.795009	−	0.606598i	$-0.792534\pi$
$197$	−866099.	−1.59002	−0.795009	+	0.606598i	$0.792534\pi$
$198$	0	0
$199$	−285836.	−0.511663	−0.255832	−	0.966721i	$-0.582349\pi$
$199$	−285836.	−0.511663	−0.255832	+	0.966721i	$0.582349\pi$
$200$	0	0
$201$	−387723.	−0.676911
$202$	0	0
$203$	−30902.3	−0.0526321
$204$	0	0
$205$	96840.9	0.160944
$206$	0	0
$207$	359112.	0.582512
$208$	0	0
$209$	−414554.	−0.656470
$210$	0	0
$211$	918930.	1.42094	0.710471	−	0.703727i	$-0.248482\pi$
$211$	918930.	1.42094	0.710471	+	0.703727i	$0.248482\pi$
$212$	0	0
$213$	196593.	0.296906
$214$	0	0
$215$	−430482.	−0.635125
$216$	0	0
$217$	−54089.3	−0.0779762
$218$	0	0
$219$	−224147.	−0.315808
$220$	0	0
$221$	−104709.	−0.144212
$222$	0	0
$223$	−486134.	−0.654626	−0.327313	−	0.944916i	$-0.606143\pi$
$223$	−486134.	−0.654626	−0.327313	+	0.944916i	$0.606143\pi$
$224$	0	0
$225$	221303.	0.291428
$226$	0	0
$227$	557639.	0.718271	0.359136	−	0.933285i	$-0.383072\pi$
$227$	557639.	0.718271	0.359136	+	0.933285i	$0.383072\pi$
$228$	0	0
$229$	440289.	0.554816	0.277408	−	0.960752i	$-0.410525\pi$
$229$	440289.	0.554816	0.277408	+	0.960752i	$0.410525\pi$
$230$	0	0
$231$	−78574.4	−0.0968838
$232$	0	0
$233$	1.46183e6	1.76403	0.882015	−	0.471221i	$-0.156187\pi$
$233$	1.46183e6	1.76403	0.882015	+	0.471221i	$0.156187\pi$
$234$	0	0
$235$	−707568.	−0.835792
$236$	0	0
$237$	−391400.	−0.452637
$238$	0	0
$239$	−1.51292e6	−1.71325	−0.856627	−	0.515936i	$-0.827444\pi$
$239$	−1.51292e6	−1.71325	−0.856627	+	0.515936i	$0.827444\pi$
$240$	0	0
$241$	−800679.	−0.888006	−0.444003	−	0.896025i	$-0.646442\pi$
$241$	−800679.	−0.888006	−0.444003	+	0.896025i	$0.646442\pi$
$242$	0	0
$243$	−926776.	−1.00684
$244$	0	0
$245$	687454.	0.731692
$246$	0	0
$247$	512184.	0.534175
$248$	0	0
$249$	−331698.	−0.339035
$250$	0	0
$251$	−1.66023e6	−1.66335	−0.831676	−	0.555261i	$-0.812618\pi$
$251$	−1.66023e6	−1.66335	−0.831676	+	0.555261i	$0.812618\pi$
$252$	0	0
$253$	−562580.	−0.552564
$254$	0	0
$255$	88210.6	0.0849514
$256$	0	0
$257$	−1.00723e6	−0.951250	−0.475625	−	0.879648i	$-0.657778\pi$
$257$	−1.00723e6	−0.951250	−0.475625	+	0.879648i	$0.657778\pi$
$258$	0	0
$259$	−431031.	−0.399263
$260$	0	0
$261$	162277.	0.147453
$262$	0	0
$263$	−1.01326e6	−0.903298	−0.451649	−	0.892196i	$-0.649164\pi$
$263$	−1.01326e6	−0.903298	−0.451649	+	0.892196i	$0.649164\pi$
$264$	0	0
$265$	−1.08340e6	−0.947705
$266$	0	0
$267$	−18140.2	−0.0155727
$268$	0	0
$269$	−1.45518e6	−1.22613	−0.613064	−	0.790034i	$-0.710063\pi$
$269$	−1.45518e6	−1.22613	−0.613064	+	0.790034i	$0.710063\pi$
$270$	0	0
$271$	1.61146e6	1.33290	0.666450	−	0.745550i	$-0.267813\pi$
$271$	1.61146e6	1.33290	0.666450	+	0.745550i	$0.267813\pi$
$272$	0	0
$273$	97079.2	0.0788351
$274$	0	0
$275$	−346690.	−0.276446
$276$	0	0
$277$	2.06358e6	1.61593	0.807966	−	0.589230i	$-0.200569\pi$
$277$	2.06358e6	1.61593	0.807966	+	0.589230i	$0.200569\pi$
$278$	0	0
$279$	284038.	0.218457
$280$	0	0
$281$	21664.7	0.0163676	0.00818382	−	0.999967i	$-0.497395\pi$
$281$	21664.7	0.0163676	0.00818382	+	0.999967i	$0.497395\pi$
$282$	0	0
$283$	552726.	0.410245	0.205123	−	0.978736i	$-0.434241\pi$
$283$	552726.	0.410245	0.205123	+	0.978736i	$0.434241\pi$
$284$	0	0
$285$	−431483.	−0.314667
$286$	0	0
$287$	−80007.4	−0.0573358
$288$	0	0
$289$	−1.34125e6	−0.944639
$290$	0	0
$291$	−247172.	−0.171107
$292$	0	0
$293$	−1.53503e6	−1.04459	−0.522297	−	0.852764i	$-0.674925\pi$
$293$	−1.53503e6	−1.04459	−0.522297	+	0.852764i	$0.674925\pi$
$294$	0	0
$295$	1.61468e6	1.08027
$296$	0	0
$297$	932244.	0.613251
$298$	0	0
$299$	695071.	0.449626
$300$	0	0
$301$	355653.	0.226261
$302$	0	0
$303$	−1.20419e6	−0.753508
$304$	0	0
$305$	992525.	0.610930
$306$	0	0
$307$	2.69140e6	1.62979	0.814895	−	0.579609i	$-0.196795\pi$
$307$	2.69140e6	1.62979	0.814895	+	0.579609i	$0.196795\pi$
$308$	0	0
$309$	593964.	0.353887
$310$	0	0
$311$	1.88553e6	1.10543	0.552717	−	0.833369i	$-0.313591\pi$
$311$	1.88553e6	1.10543	0.552717	+	0.833369i	$0.313591\pi$
$312$	0	0
$313$	−2.48142e6	−1.43166	−0.715828	−	0.698276i	$-0.753951\pi$
$313$	−2.48142e6	−1.43166	−0.715828	+	0.698276i	$0.753951\pi$
$314$	0	0
$315$	315339.	0.179061
$316$	0	0
$317$	204476.	0.114286	0.0571432	−	0.998366i	$-0.481801\pi$
$317$	204476.	0.114286	0.0571432	+	0.998366i	$0.481801\pi$
$318$	0	0
$319$	−254220.	−0.139873
$320$	0	0
$321$	119145.	0.0645379
$322$	0	0
$323$	−384496.	−0.205062
$324$	0	0
$325$	428338.	0.224946
$326$	0	0
$327$	27275.2	0.0141058
$328$	0	0
$329$	584574.	0.297749
$330$	0	0
$331$	−1.58800e6	−0.796672	−0.398336	−	0.917240i	$-0.630412\pi$
$331$	−1.58800e6	−0.796672	−0.398336	+	0.917240i	$0.630412\pi$
$332$	0	0
$333$	2.26346e6	1.11857
$334$	0	0
$335$	−2.43766e6	−1.18675
$336$	0	0
$337$	−813616.	−0.390252	−0.195126	−	0.980778i	$-0.562512\pi$
$337$	−813616.	−0.390252	−0.195126	+	0.980778i	$0.562512\pi$
$338$	0	0
$339$	−759189.	−0.358799
$340$	0	0
$341$	−444969.	−0.207226
$342$	0	0
$343$	−1.18553e6	−0.544096
$344$	0	0
$345$	−585554.	−0.264862
$346$	0	0
$347$	−516038.	−0.230069	−0.115034	−	0.993362i	$-0.536698\pi$
$347$	−516038.	−0.230069	−0.115034	+	0.993362i	$0.536698\pi$
$348$	0	0
$349$	−2.65164e6	−1.16533	−0.582667	−	0.812711i	$-0.697991\pi$
$349$	−2.65164e6	−1.16533	−0.582667	+	0.812711i	$0.697991\pi$
$350$	0	0
$351$	−1.15179e6	−0.499007
$352$	0	0
$353$	391323.	0.167147	0.0835734	−	0.996502i	$-0.473367\pi$
$353$	391323.	0.167147	0.0835734	+	0.996502i	$0.473367\pi$
$354$	0	0
$355$	1.23600e6	0.520532
$356$	0	0
$357$	−72877.3	−0.0302637
$358$	0	0
$359$	−3.03988e6	−1.24486	−0.622430	−	0.782675i	$-0.713854\pi$
$359$	−3.03988e6	−1.24486	−0.622430	+	0.782675i	$0.713854\pi$
$360$	0	0
$361$	−595334.	−0.240432
$362$	0	0
$363$	492898.	0.196332
$364$	0	0
$365$	−1.40924e6	−0.553670
$366$	0	0
$367$	−1.02897e6	−0.398785	−0.199392	−	0.979920i	$-0.563897\pi$
$367$	−1.02897e6	−0.398785	−0.199392	+	0.979920i	$0.563897\pi$
$368$	0	0
$369$	420141.	0.160631
$370$	0	0
$371$	895074.	0.337617
$372$	0	0
$373$	−4.42358e6	−1.64627	−0.823136	−	0.567844i	$-0.807778\pi$
$373$	−4.42358e6	−1.64627	−0.823136	+	0.567844i	$0.807778\pi$
$374$	0	0
$375$	−1.34406e6	−0.493560
$376$	0	0
$377$	314090.	0.113815
$378$	0	0
$379$	4.23499e6	1.51445	0.757223	−	0.653156i	$-0.226555\pi$
$379$	4.23499e6	1.51445	0.757223	+	0.653156i	$0.226555\pi$
$380$	0	0
$381$	−104866.	−0.0370103
$382$	0	0
$383$	894484.	0.311584	0.155792	−	0.987790i	$-0.450207\pi$
$383$	894484.	0.311584	0.155792	+	0.987790i	$0.450207\pi$
$384$	0	0
$385$	−494005.	−0.169856
$386$	0	0
$387$	−1.86763e6	−0.633891
$388$	0	0
$389$	3.60641e6	1.20837	0.604186	−	0.796843i	$-0.293498\pi$
$389$	3.60641e6	1.20837	0.604186	+	0.796843i	$0.293498\pi$
$390$	0	0
$391$	−521789.	−0.172605
$392$	0	0
$393$	−2.21541e6	−0.723557
$394$	0	0
$395$	−2.46077e6	−0.793559
$396$	0	0
$397$	−1.96195e6	−0.624757	−0.312379	−	0.949958i	$-0.601126\pi$
$397$	−1.96195e6	−0.624757	−0.312379	+	0.949958i	$0.601126\pi$
$398$	0	0
$399$	356480.	0.112099
$400$	0	0
$401$	−721116.	−0.223947	−0.111973	−	0.993711i	$-0.535717\pi$
$401$	−721116.	−0.223947	−0.111973	+	0.993711i	$0.535717\pi$
$402$	0	0
$403$	549762.	0.168621
$404$	0	0
$405$	−1.11509e6	−0.337809
$406$	0	0
$407$	−3.54590e6	−1.06106
$408$	0	0
$409$	1.33383e6	0.394269	0.197135	−	0.980376i	$-0.436836\pi$
$409$	1.33383e6	0.394269	0.197135	+	0.980376i	$0.436836\pi$
$410$	0	0
$411$	1.79904e6	0.525334
$412$	0	0
$413$	−1.33400e6	−0.384842
$414$	0	0
$415$	−2.08542e6	−0.594391
$416$	0	0
$417$	−2.76550e6	−0.778813
$418$	0	0
$419$	4.51294e6	1.25581	0.627905	−	0.778290i	$-0.283913\pi$
$419$	4.51294e6	1.25581	0.627905	+	0.778290i	$0.283913\pi$
$420$	0	0
$421$	982181.	0.270076	0.135038	−	0.990840i	$-0.456884\pi$
$421$	982181.	0.270076	0.135038	+	0.990840i	$0.456884\pi$
$422$	0	0
$423$	−3.06976e6	−0.834168
$424$	0	0
$425$	−321553.	−0.0863535
$426$	0	0
$427$	−819998.	−0.217642
$428$	0	0
$429$	798629.	0.209508
$430$	0	0
$431$	−5.31784e6	−1.37893	−0.689465	−	0.724319i	$-0.742154\pi$
$431$	−5.31784e6	−1.37893	−0.689465	+	0.724319i	$0.742154\pi$
$432$	0	0
$433$	679278.	0.174112	0.0870558	−	0.996203i	$-0.472254\pi$
$433$	679278.	0.174112	0.0870558	+	0.996203i	$0.472254\pi$
$434$	0	0
$435$	−264602.	−0.0670454
$436$	0	0
$437$	2.55234e6	0.639344
$438$	0	0
$439$	−3.53619e6	−0.875739	−0.437870	−	0.899039i	$-0.644267\pi$
$439$	−3.53619e6	−0.875739	−0.437870	+	0.899039i	$0.644267\pi$
$440$	0	0
$441$	2.98250e6	0.730270
$442$	0	0
$443$	1.35298e6	0.327553	0.163776	−	0.986497i	$-0.447632\pi$
$443$	1.35298e6	0.327553	0.163776	+	0.986497i	$0.447632\pi$
$444$	0	0
$445$	−114049.	−0.0273018
$446$	0	0
$447$	−511783.	−0.121148
$448$	0	0
$449$	−2.04554e6	−0.478843	−0.239421	−	0.970916i	$-0.576958\pi$
$449$	−2.04554e6	−0.478843	−0.239421	+	0.970916i	$0.576958\pi$
$450$	0	0
$451$	−658186.	−0.152373
$452$	0	0
$453$	−1.60879e6	−0.368345
$454$	0	0
$455$	610347.	0.138213
$456$	0	0
$457$	8.48379e6	1.90020	0.950100	−	0.311944i	$-0.100980\pi$
$457$	8.48379e6	1.90020	0.950100	+	0.311944i	$0.100980\pi$
$458$	0	0
$459$	864651.	0.191562
$460$	0	0
$461$	−2.06795e6	−0.453198	−0.226599	−	0.973988i	$-0.572761\pi$
$461$	−2.06795e6	−0.453198	−0.226599	+	0.973988i	$0.572761\pi$
$462$	0	0
$463$	368290.	0.0798430	0.0399215	−	0.999203i	$-0.487289\pi$
$463$	368290.	0.0798430	0.0399215	+	0.999203i	$0.487289\pi$
$464$	0	0
$465$	−463141.	−0.0993300
$466$	0	0
$467$	2.78365e6	0.590640	0.295320	−	0.955398i	$-0.404574\pi$
$467$	2.78365e6	0.590640	0.295320	+	0.955398i	$0.404574\pi$
$468$	0	0
$469$	2.01393e6	0.422777
$470$	0	0
$471$	−3.81494e6	−0.792385
$472$	0	0
$473$	2.92580e6	0.601302
$474$	0	0
$475$	1.57288e6	0.319861
$476$	0	0
$477$	−4.70028e6	−0.945863
$478$	0	0
$479$	−1.49820e6	−0.298352	−0.149176	−	0.988811i	$-0.547662\pi$
$479$	−1.49820e6	−0.298352	−0.149176	+	0.988811i	$0.547662\pi$
$480$	0	0
$481$	4.38099e6	0.863394
$482$	0	0
$483$	483770.	0.0943563
$484$	0	0
$485$	−1.55399e6	−0.299982
$486$	0	0
$487$	−7.95675e6	−1.52024	−0.760122	−	0.649781i	$-0.774861\pi$
$487$	−7.95675e6	−1.52024	−0.760122	+	0.649781i	$0.774861\pi$
$488$	0	0
$489$	622498.	0.117724
$490$	0	0
$491$	1.87342e6	0.350697	0.175348	−	0.984506i	$-0.443895\pi$
$491$	1.87342e6	0.350697	0.175348	+	0.984506i	$0.443895\pi$
$492$	0	0
$493$	−235787.	−0.0436921
$494$	0	0
$495$	2.59416e6	0.475865
$496$	0	0
$497$	−1.02115e6	−0.185438
$498$	0	0
$499$	352286.	0.0633350	0.0316675	−	0.999498i	$-0.489918\pi$
$499$	352286.	0.0633350	0.0316675	+	0.999498i	$0.489918\pi$
$500$	0	0
$501$	1.41410e6	0.251701
$502$	0	0
$503$	3.57266e6	0.629611	0.314805	−	0.949156i	$-0.398061\pi$
$503$	3.57266e6	0.629611	0.314805	+	0.949156i	$0.398061\pi$
$504$	0	0
$505$	−7.57085e6	−1.32104
$506$	0	0
$507$	1.63986e6	0.283327
$508$	0	0
$509$	−1.05719e7	−1.80867	−0.904337	−	0.426819i	$-0.859634\pi$
$509$	−1.05719e7	−1.80867	−0.904337	+	0.426819i	$0.859634\pi$
$510$	0	0
$511$	1.16427e6	0.197243
$512$	0	0
$513$	−4.22945e6	−0.709562
$514$	0	0
$515$	3.73431e6	0.620430
$516$	0	0
$517$	4.80903e6	0.791282
$518$	0	0
$519$	2.21984e6	0.361746
$520$	0	0
$521$	7.42701e6	1.19872	0.599362	−	0.800478i	$-0.295421\pi$
$521$	7.42701e6	1.19872	0.599362	+	0.800478i	$0.295421\pi$
$522$	0	0
$523$	1.03292e7	1.65125	0.825623	−	0.564222i	$-0.190824\pi$
$523$	1.03292e7	1.65125	0.825623	+	0.564222i	$0.190824\pi$
$524$	0	0
$525$	298123.	0.0472060
$526$	0	0
$527$	−412706.	−0.0647313
$528$	0	0
$529$	−2.97263e6	−0.461851
$530$	0	0
$531$	7.00523e6	1.07817
$532$	0	0
$533$	813193.	0.123987
$534$	0	0
$535$	749079.	0.113147
$536$	0	0
$537$	−423137.	−0.0633207
$538$	0	0
$539$	−4.67233e6	−0.692726
$540$	0	0
$541$	4.60833e6	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$541$	4.60833e6	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$542$	0	0
$543$	−2.84543e6	−0.414142
$544$	0	0
$545$	171482.	0.0247302
$546$	0	0
$547$	1.79146e6	0.255999	0.127999	−	0.991774i	$-0.459144\pi$
$547$	1.79146e6	0.255999	0.127999	+	0.991774i	$0.459144\pi$
$548$	0	0
$549$	4.30604e6	0.609743
$550$	0	0
$551$	1.15336e6	0.161839
$552$	0	0
$553$	2.03303e6	0.282703
$554$	0	0
$555$	−3.69071e6	−0.508601
$556$	0	0
$557$	1.07840e7	1.47280	0.736399	−	0.676548i	$-0.236525\pi$
$557$	1.07840e7	1.47280	0.736399	+	0.676548i	$0.236525\pi$
$558$	0	0
$559$	−3.61485e6	−0.489284
$560$	0	0
$561$	−599530.	−0.0804273
$562$	0	0
$563$	1.40598e7	1.86942	0.934710	−	0.355411i	$-0.115659\pi$
$563$	1.40598e7	1.86942	0.934710	+	0.355411i	$0.115659\pi$
$564$	0	0
$565$	−4.77310e6	−0.629041
$566$	0	0
$567$	921255.	0.120343
$568$	0	0
$569$	4.09890e6	0.530746	0.265373	−	0.964146i	$-0.414505\pi$
$569$	4.09890e6	0.530746	0.265373	+	0.964146i	$0.414505\pi$
$570$	0	0
$571$	2.46148e6	0.315941	0.157971	−	0.987444i	$-0.449505\pi$
$571$	2.46148e6	0.315941	0.157971	+	0.987444i	$0.449505\pi$
$572$	0	0
$573$	−6.39168e6	−0.813259
$574$	0	0
$575$	2.13451e6	0.269233
$576$	0	0
$577$	7.09785e6	0.887539	0.443770	−	0.896141i	$-0.353641\pi$
$577$	7.09785e6	0.887539	0.443770	+	0.896141i	$0.353641\pi$
$578$	0	0
$579$	−7.27718e6	−0.902125
$580$	0	0
$581$	1.72292e6	0.211750
$582$	0	0
$583$	7.36339e6	0.897235
$584$	0	0
$585$	−3.20510e6	−0.387215
$586$	0	0
$587$	1.43549e7	1.71951	0.859753	−	0.510710i	$-0.170617\pi$
$587$	1.43549e7	1.71951	0.859753	+	0.510710i	$0.170617\pi$
$588$	0	0
$589$	2.01876e6	0.239770
$590$	0	0
$591$	6.12690e6	0.721559
$592$	0	0
$593$	1.01492e7	1.18521	0.592606	−	0.805493i	$-0.298099\pi$
$593$	1.01492e7	1.18521	0.592606	+	0.805493i	$0.298099\pi$
$594$	0	0
$595$	−458187.	−0.0530579
$596$	0	0
$597$	2.02204e6	0.232196
$598$	0	0
$599$	8.76777e6	0.998440	0.499220	−	0.866475i	$-0.333620\pi$
$599$	8.76777e6	0.998440	0.499220	+	0.866475i	$0.333620\pi$
$600$	0	0
$601$	1.49619e7	1.68966	0.844831	−	0.535034i	$-0.179701\pi$
$601$	1.49619e7	1.68966	0.844831	+	0.535034i	$0.179701\pi$
$602$	0	0
$603$	−1.05757e7	−1.18445
$604$	0	0
$605$	3.09890e6	0.344206
$606$	0	0
$607$	−7.03290e6	−0.774752	−0.387376	−	0.921922i	$-0.626618\pi$
$607$	−7.03290e6	−0.774752	−0.387376	+	0.921922i	$0.626618\pi$
$608$	0	0
$609$	218607.	0.0238847
$610$	0	0
$611$	−5.94159e6	−0.643873
$612$	0	0
$613$	−806845.	−0.0867239	−0.0433620	−	0.999059i	$-0.513807\pi$
$613$	−806845.	−0.0867239	−0.0433620	+	0.999059i	$0.513807\pi$
$614$	0	0
$615$	−685065.	−0.0730372
$616$	0	0
$617$	7.36483e6	0.778843	0.389421	−	0.921060i	$-0.372675\pi$
$617$	7.36483e6	0.778843	0.389421	+	0.921060i	$0.372675\pi$
$618$	0	0
$619$	1.30528e7	1.36923	0.684617	−	0.728903i	$-0.259969\pi$
$619$	1.30528e7	1.36923	0.684617	+	0.728903i	$0.259969\pi$
$620$	0	0
$621$	−5.73967e6	−0.597253
$622$	0	0
$623$	94224.4	0.00972620
$624$	0	0
$625$	−4.86615e6	−0.498294
$626$	0	0
$627$	2.93261e6	0.297910
$628$	0	0
$629$	−3.28880e6	−0.331445
$630$	0	0
$631$	−2.89105e6	−0.289056	−0.144528	−	0.989501i	$-0.546166\pi$
$631$	−2.89105e6	−0.289056	−0.144528	+	0.989501i	$0.546166\pi$
$632$	0	0
$633$	−6.50063e6	−0.644831
$634$	0	0
$635$	−659304.	−0.0648860
$636$	0	0
$637$	5.77270e6	0.563677
$638$	0	0
$639$	5.36234e6	0.519520
$640$	0	0
$641$	−1.31397e7	−1.26311	−0.631555	−	0.775331i	$-0.717583\pi$
$641$	−1.31397e7	−1.26311	−0.631555	+	0.775331i	$0.717583\pi$
$642$	0	0
$643$	1.17266e7	1.11852	0.559259	−	0.828993i	$-0.311086\pi$
$643$	1.17266e7	1.11852	0.559259	+	0.828993i	$0.311086\pi$
$644$	0	0
$645$	3.04529e6	0.288223
$646$	0	0
$647$	−1.10662e7	−1.03929	−0.519646	−	0.854382i	$-0.673936\pi$
$647$	−1.10662e7	−1.03929	−0.519646	+	0.854382i	$0.673936\pi$
$648$	0	0
$649$	−1.09743e7	−1.02274
$650$	0	0
$651$	382635.	0.0353860
$652$	0	0
$653$	−3.44597e6	−0.316249	−0.158124	−	0.987419i	$-0.550545\pi$
$653$	−3.44597e6	−0.316249	−0.158124	+	0.987419i	$0.550545\pi$
$654$	0	0
$655$	−1.39285e7	−1.26853
$656$	0	0
$657$	−6.11392e6	−0.552594
$658$	0	0
$659$	9.12181e6	0.818215	0.409108	−	0.912486i	$-0.365840\pi$
$659$	9.12181e6	0.818215	0.409108	+	0.912486i	$0.365840\pi$
$660$	0	0
$661$	1.68113e7	1.49657	0.748287	−	0.663375i	$-0.230877\pi$
$661$	1.68113e7	1.49657	0.748287	+	0.663375i	$0.230877\pi$
$662$	0	0
$663$	740723.	0.0654443
$664$	0	0
$665$	2.24123e6	0.196531
$666$	0	0
$667$	1.56519e6	0.136224
$668$	0	0
$669$	3.43897e6	0.297073
$670$	0	0
$671$	−6.74577e6	−0.578396
$672$	0	0
$673$	8.53511e6	0.726393	0.363196	−	0.931713i	$-0.381685\pi$
$673$	8.53511e6	0.726393	0.363196	+	0.931713i	$0.381685\pi$
$674$	0	0
$675$	−3.53708e6	−0.298803
$676$	0	0
$677$	−912690.	−0.0765335	−0.0382667	−	0.999268i	$-0.512184\pi$
$677$	−912690.	−0.0765335	−0.0382667	+	0.999268i	$0.512184\pi$
$678$	0	0
$679$	1.28387e6	0.106868
$680$	0	0
$681$	−3.94481e6	−0.325955
$682$	0	0
$683$	4.20567e6	0.344972	0.172486	−	0.985012i	$-0.444820\pi$
$683$	4.20567e6	0.344972	0.172486	+	0.985012i	$0.444820\pi$
$684$	0	0
$685$	1.13107e7	0.921010
$686$	0	0
$687$	−3.11466e6	−0.251779
$688$	0	0
$689$	−9.09751e6	−0.730087
$690$	0	0
$691$	−6.23030e6	−0.496379	−0.248190	−	0.968711i	$-0.579836\pi$
$691$	−6.23030e6	−0.496379	−0.248190	+	0.968711i	$0.579836\pi$
$692$	0	0
$693$	−2.14323e6	−0.169525
$694$	0	0
$695$	−1.73870e7	−1.36541
$696$	0	0
$697$	−610464.	−0.0475968
$698$	0	0
$699$	−1.03412e7	−0.800527
$700$	0	0
$701$	1.69453e7	1.30243	0.651215	−	0.758893i	$-0.274259\pi$
$701$	1.69453e7	1.30243	0.651215	+	0.758893i	$0.274259\pi$
$702$	0	0
$703$	1.60872e7	1.22770
$704$	0	0
$705$	5.00542e6	0.379287
$706$	0	0
$707$	6.25484e6	0.470617
$708$	0	0
$709$	−1.87326e7	−1.39953	−0.699766	−	0.714372i	$-0.746712\pi$
$709$	−1.87326e7	−1.39953	−0.699766	+	0.714372i	$0.746712\pi$
$710$	0	0
$711$	−1.06760e7	−0.792016
$712$	0	0
$713$	2.73960e6	0.201820
$714$	0	0
$715$	5.02106e6	0.367308
$716$	0	0
$717$	1.07026e7	0.777485
$718$	0	0
$719$	−2.15611e6	−0.155542	−0.0777711	−	0.996971i	$-0.524780\pi$
$719$	−2.15611e6	−0.155542	−0.0777711	+	0.996971i	$0.524780\pi$
$720$	0	0
$721$	−3.08519e6	−0.221026
$722$	0	0
$723$	5.66410e6	0.402982
$724$	0	0
$725$	964548.	0.0681520
$726$	0	0
$727$	−4.87466e6	−0.342065	−0.171032	−	0.985265i	$-0.554710\pi$
$727$	−4.87466e6	−0.342065	−0.171032	+	0.985265i	$0.554710\pi$
$728$	0	0
$729$	463697.	0.0323159
$730$	0	0
$731$	2.71367e6	0.187829
$732$	0	0
$733$	5.15920e6	0.354669	0.177334	−	0.984151i	$-0.443253\pi$
$733$	5.15920e6	0.354669	0.177334	+	0.984151i	$0.443253\pi$
$734$	0	0
$735$	−4.86314e6	−0.332046
$736$	0	0
$737$	1.65677e7	1.12355
$738$	0	0
$739$	4.60821e6	0.310400	0.155200	−	0.987883i	$-0.450398\pi$
$739$	4.60821e6	0.310400	0.155200	+	0.987883i	$0.450398\pi$
$740$	0	0
$741$	−3.62325e6	−0.242411
$742$	0	0
$743$	−2.50473e7	−1.66452	−0.832261	−	0.554384i	$-0.812954\pi$
$743$	−2.50473e7	−1.66452	−0.832261	+	0.554384i	$0.812954\pi$
$744$	0	0
$745$	−3.21763e6	−0.212396
$746$	0	0
$747$	−9.04751e6	−0.593236
$748$	0	0
$749$	−618870.	−0.0403083
$750$	0	0
$751$	1.87279e7	1.21168	0.605841	−	0.795586i	$-0.292837\pi$
$751$	1.87279e7	1.21168	0.605841	+	0.795586i	$0.292837\pi$
$752$	0	0
$753$	1.17447e7	0.754839
$754$	0	0
$755$	−1.01146e7	−0.645778
$756$	0	0
$757$	−5.50814e6	−0.349354	−0.174677	−	0.984626i	$-0.555888\pi$
$757$	−5.50814e6	−0.349354	−0.174677	+	0.984626i	$0.555888\pi$
$758$	0	0
$759$	3.97976e6	0.250757
$760$	0	0
$761$	−2.67826e7	−1.67645	−0.838226	−	0.545323i	$-0.816407\pi$
$761$	−2.67826e7	−1.67645	−0.838226	+	0.545323i	$0.816407\pi$
$762$	0	0
$763$	−141674.	−0.00881005
$764$	0	0
$765$	2.40607e6	0.148646
$766$	0	0
$767$	1.35588e7	0.832209
$768$	0	0
$769$	1.76374e7	1.07552	0.537761	−	0.843097i	$-0.319270\pi$
$769$	1.76374e7	1.07552	0.537761	+	0.843097i	$0.319270\pi$
$770$	0	0
$771$	7.12526e6	0.431683
$772$	0	0
$773$	−6.59079e6	−0.396724	−0.198362	−	0.980129i	$-0.563562\pi$
$773$	−6.59079e6	−0.396724	−0.198362	+	0.980129i	$0.563562\pi$
$774$	0	0
$775$	1.68828e6	0.100970
$776$	0	0
$777$	3.04917e6	0.181188
$778$	0	0
$779$	2.98609e6	0.176303
$780$	0	0
$781$	−8.40056e6	−0.492811
$782$	0	0
$783$	−2.59366e6	−0.151185
$784$	0	0
$785$	−2.39849e7	−1.38920
$786$	0	0
$787$	−1.83730e7	−1.05741	−0.528704	−	0.848806i	$-0.677322\pi$
$787$	−1.83730e7	−1.05741	−0.528704	+	0.848806i	$0.677322\pi$
$788$	0	0
$789$	7.16792e6	0.409922
$790$	0	0
$791$	3.94341e6	0.224094
$792$	0	0
$793$	8.33444e6	0.470645
$794$	0	0
$795$	7.66409e6	0.430074
$796$	0	0
$797$	−3.06769e7	−1.71067	−0.855334	−	0.518078i	$-0.826648\pi$
$797$	−3.06769e7	−1.71067	−0.855334	+	0.518078i	$0.826648\pi$
$798$	0	0
$799$	4.46035e6	0.247174
$800$	0	0
$801$	−494798.	−0.0272488
$802$	0	0
$803$	9.57797e6	0.524185
$804$	0	0
$805$	3.04151e6	0.165424
$806$	0	0
$807$	1.02941e7	0.556423
$808$	0	0
$809$	−2.81720e6	−0.151338	−0.0756688	−	0.997133i	$-0.524109\pi$
$809$	−2.81720e6	−0.151338	−0.0756688	+	0.997133i	$0.524109\pi$
$810$	0	0
$811$	1.57819e7	0.842574	0.421287	−	0.906927i	$-0.361579\pi$
$811$	1.57819e7	0.842574	0.421287	+	0.906927i	$0.361579\pi$
$812$	0	0
$813$	−1.13997e7	−0.604877
$814$	0	0
$815$	3.91371e6	0.206393
$816$	0	0
$817$	−1.32739e7	−0.695735
$818$	0	0
$819$	2.64797e6	0.137944
$820$	0	0
$821$	−2.40563e6	−0.124558	−0.0622789	−	0.998059i	$-0.519837\pi$
$821$	−2.40563e6	−0.124558	−0.0622789	+	0.998059i	$0.519837\pi$
$822$	0	0
$823$	−4.47397e6	−0.230247	−0.115123	−	0.993351i	$-0.536726\pi$
$823$	−4.47397e6	−0.230247	−0.115123	+	0.993351i	$0.536726\pi$
$824$	0	0
$825$	2.45253e6	0.125453
$826$	0	0
$827$	−2.36054e7	−1.20018	−0.600091	−	0.799932i	$-0.704869\pi$
$827$	−2.36054e7	−1.20018	−0.600091	+	0.799932i	$0.704869\pi$
$828$	0	0
$829$	−7.52668e6	−0.380380	−0.190190	−	0.981747i	$-0.560910\pi$
$829$	−7.52668e6	−0.380380	−0.190190	+	0.981747i	$0.560910\pi$
$830$	0	0
$831$	−1.45981e7	−0.733319
$832$	0	0
$833$	−4.33356e6	−0.216388
$834$	0	0
$835$	8.89056e6	0.441279
$836$	0	0
$837$	−4.53976e6	−0.223985
$838$	0	0
$839$	7.28878e6	0.357478	0.178739	−	0.983896i	$-0.442798\pi$
$839$	7.28878e6	0.357478	0.178739	+	0.983896i	$0.442798\pi$
$840$	0	0
$841$	707281.	0.0344828
$842$	0	0
$843$	−153259.	−0.00742773
$844$	0	0
$845$	1.03100e7	0.496725
$846$	0	0
$847$	−2.56023e6	−0.122623
$848$	0	0
$849$	−3.91005e6	−0.186172
$850$	0	0
$851$	2.18315e7	1.03338
$852$	0	0
$853$	2.18767e7	1.02946	0.514729	−	0.857353i	$-0.327893\pi$
$853$	2.18767e7	1.02946	0.514729	+	0.857353i	$0.327893\pi$
$854$	0	0
$855$	−1.17693e7	−0.550599
$856$	0	0
$857$	−1.62738e7	−0.756897	−0.378448	−	0.925622i	$-0.623542\pi$
$857$	−1.62738e7	−0.756897	−0.378448	+	0.925622i	$0.623542\pi$
$858$	0	0
$859$	−1.50220e7	−0.694615	−0.347308	−	0.937751i	$-0.612904\pi$
$859$	−1.50220e7	−0.694615	−0.347308	+	0.937751i	$0.612904\pi$
$860$	0	0
$861$	565983.	0.0260193
$862$	0	0
$863$	−2.24278e7	−1.02508	−0.512542	−	0.858662i	$-0.671296\pi$
$863$	−2.24278e7	−1.02508	−0.512542	+	0.858662i	$0.671296\pi$
$864$	0	0
$865$	1.39564e7	0.634209
$866$	0	0
$867$	9.48819e6	0.428682
$868$	0	0
$869$	1.67248e7	0.751298
$870$	0	0
$871$	−2.04695e7	−0.914244
$872$	0	0
$873$	−6.74196e6	−0.299399
$874$	0	0
$875$	6.98136e6	0.308262
$876$	0	0
$877$	1.25450e7	0.550771	0.275385	−	0.961334i	$-0.411195\pi$
$877$	1.25450e7	0.550771	0.275385	+	0.961334i	$0.411195\pi$
$878$	0	0
$879$	1.08590e7	0.474043
$880$	0	0
$881$	7.60341e6	0.330042	0.165021	−	0.986290i	$-0.447231\pi$
$881$	7.60341e6	0.330042	0.165021	+	0.986290i	$0.447231\pi$
$882$	0	0
$883$	−2.15220e7	−0.928923	−0.464462	−	0.885593i	$-0.653752\pi$
$883$	−2.15220e7	−0.928923	−0.464462	+	0.885593i	$0.653752\pi$
$884$	0	0
$885$	−1.14224e7	−0.490231
$886$	0	0
$887$	333279.	0.0142232	0.00711162	−	0.999975i	$-0.497736\pi$
$887$	333279.	0.0142232	0.00711162	+	0.999975i	$0.497736\pi$
$888$	0	0
$889$	544700.	0.0231155
$890$	0	0
$891$	7.57876e6	0.319819
$892$	0	0
$893$	−2.18178e7	−0.915552
$894$	0	0
$895$	−2.66031e6	−0.111013
$896$	0	0
$897$	−4.91702e6	−0.204043
$898$	0	0
$899$	1.23798e6	0.0510873
$900$	0	0
$901$	6.82950e6	0.280270
$902$	0	0
$903$	−2.51594e6	−0.102679
$904$	0	0
$905$	−1.78895e7	−0.726068
$906$	0	0
$907$	−1.65687e7	−0.668759	−0.334380	−	0.942439i	$-0.608527\pi$
$907$	−1.65687e7	−0.668759	−0.334380	+	0.942439i	$0.608527\pi$
$908$	0	0
$909$	−3.28459e7	−1.31847
$910$	0	0
$911$	−1.30648e7	−0.521561	−0.260781	−	0.965398i	$-0.583980\pi$
$911$	−1.30648e7	−0.521561	−0.260781	+	0.965398i	$0.583980\pi$
$912$	0	0
$913$	1.41737e7	0.562737
$914$	0	0
$915$	−7.02125e6	−0.277244
$916$	0	0
$917$	1.15074e7	0.451911
$918$	0	0
$919$	7.30805e6	0.285439	0.142719	−	0.989763i	$-0.454415\pi$
$919$	7.30805e6	0.285439	0.142719	+	0.989763i	$0.454415\pi$
$920$	0	0
$921$	−1.90393e7	−0.739608
$922$	0	0
$923$	1.03789e7	0.401004
$924$	0	0
$925$	1.34537e7	0.516996
$926$	0	0
$927$	1.62012e7	0.619224
$928$	0	0
$929$	2.69866e7	1.02591	0.512955	−	0.858416i	$-0.328551\pi$
$929$	2.69866e7	1.02591	0.512955	+	0.858416i	$0.328551\pi$
$930$	0	0
$931$	2.11976e7	0.801518
$932$	0	0
$933$	−1.33385e7	−0.501652
$934$	0	0
$935$	−3.76930e6	−0.141004
$936$	0	0
$937$	−2.15536e7	−0.801993	−0.400997	−	0.916080i	$-0.631336\pi$
$937$	−2.15536e7	−0.801993	−0.400997	+	0.916080i	$0.631336\pi$
$938$	0	0
$939$	1.75539e7	0.649694
$940$	0	0
$941$	−1.22668e7	−0.451602	−0.225801	−	0.974173i	$-0.572500\pi$
$941$	−1.22668e7	−0.451602	−0.225801	+	0.974173i	$0.572500\pi$
$942$	0	0
$943$	4.05234e6	0.148398
$944$	0	0
$945$	−5.04005e6	−0.183593
$946$	0	0
$947$	−5.13280e7	−1.85986	−0.929928	−	0.367741i	$-0.880131\pi$
$947$	−5.13280e7	−1.85986	−0.929928	+	0.367741i	$0.880131\pi$
$948$	0	0
$949$	−1.18336e7	−0.426533
$950$	0	0
$951$	−1.44649e6	−0.0518638
$952$	0	0
$953$	−3.30301e7	−1.17809	−0.589044	−	0.808101i	$-0.700496\pi$
$953$	−3.30301e7	−1.17809	−0.589044	+	0.808101i	$0.700496\pi$
$954$	0	0
$955$	−4.01852e7	−1.42580
$956$	0	0
$957$	1.79838e6	0.0634750
$958$	0	0
$959$	−9.34463e6	−0.328107
$960$	0	0
$961$	−2.64623e7	−0.924312
$962$	0	0
$963$	3.24986e6	0.112927
$964$	0	0
$965$	−4.57524e7	−1.58160
$966$	0	0
$967$	7.77281e6	0.267308	0.133654	−	0.991028i	$-0.457329\pi$
$967$	7.77281e6	0.267308	0.133654	+	0.991028i	$0.457329\pi$
$968$	0	0
$969$	2.71997e6	0.0930583
$970$	0	0
$971$	−2.31111e7	−0.786635	−0.393317	−	0.919403i	$-0.628673\pi$
$971$	−2.31111e7	−0.786635	−0.393317	+	0.919403i	$0.628673\pi$
$972$	0	0
$973$	1.43647e7	0.486422
$974$	0	0
$975$	−3.03012e6	−0.102082
$976$	0	0
$977$	1.04792e7	0.351231	0.175616	−	0.984459i	$-0.443808\pi$
$977$	1.04792e7	0.351231	0.175616	+	0.984459i	$0.443808\pi$
$978$	0	0
$979$	775143.	0.0258479
$980$	0	0
$981$	743969.	0.0246821
$982$	0	0
$983$	−3.94189e7	−1.30113	−0.650565	−	0.759450i	$-0.725468\pi$
$983$	−3.94189e7	−1.30113	−0.650565	+	0.759450i	$0.725468\pi$
$984$	0	0
$985$	3.85204e7	1.26503
$986$	0	0
$987$	−4.13535e6	−0.135120
$988$	0	0
$989$	−1.80137e7	−0.585615
$990$	0	0
$991$	4.53294e7	1.46621	0.733104	−	0.680117i	$-0.238071\pi$
$991$	4.53294e7	1.46621	0.733104	+	0.680117i	$0.238071\pi$
$992$	0	0
$993$	1.12337e7	0.361534
$994$	0	0
$995$	1.27128e7	0.407083
$996$	0	0
$997$	4.74723e7	1.51253	0.756263	−	0.654268i	$-0.227023\pi$
$997$	4.74723e7	1.51253	0.756263	+	0.654268i	$0.227023\pi$
$998$	0	0
$999$	−3.61768e7	−1.14687

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.6.a.i.1.1		4
4.3	odd	2		29.6.a.a.1.2	✓	4
12.11	even	2		261.6.a.a.1.3		4
20.19	odd	2		725.6.a.a.1.3		4
116.115	odd	2		841.6.a.a.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.6.a.a.1.2	✓	4	4.3	odd	2
261.6.a.a.1.3		4	12.11	even	2
464.6.a.i.1.1		4	1.1	even	1	trivial
725.6.a.a.1.3		4	20.19	odd	2
841.6.a.a.1.3		4	116.115	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)

\(f(q)\)	\(=\)	\(q-7.07413 q^{3} -44.4758 q^{5} +36.7447 q^{7} -192.957 q^{9} +O(q^{10})\)	\(q-7.07413 q^{3} -44.4758 q^{5} +36.7447 q^{7} -192.957 q^{9} +302.283 q^{11} -373.472 q^{13} +314.627 q^{15} +280.365 q^{17} -1371.41 q^{19} -259.937 q^{21} -1861.10 q^{23} -1146.91 q^{25} +3084.01 q^{27} -841.000 q^{29} -1472.03 q^{31} -2138.39 q^{33} -1634.25 q^{35} -11730.4 q^{37} +2641.99 q^{39} -2177.39 q^{41} +9679.03 q^{43} +8581.90 q^{45} +15909.1 q^{47} -15456.8 q^{49} -1983.34 q^{51} +24359.3 q^{53} -13444.3 q^{55} +9701.53 q^{57} -36304.7 q^{59} -22316.1 q^{61} -7090.14 q^{63} +16610.5 q^{65} +54808.6 q^{67} +13165.7 q^{69} -27790.4 q^{71} +31685.5 q^{73} +8113.36 q^{75} +11107.3 q^{77} +55328.4 q^{79} +25071.8 q^{81} +46888.8 q^{83} -12469.5 q^{85} +5949.34 q^{87} +2564.30 q^{89} -13723.1 q^{91} +10413.3 q^{93} +60994.5 q^{95} +34940.3 q^{97} -58327.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9}+O(q^{10}) \) 4 * q + 28 * q^3 - 68 * q^5 + 208 * q^7 - 280 * q^9	\( 4 q + 28 q^{3} - 68 q^{5} + 208 q^{7} - 280 q^{9} + 124 q^{11} - 460 q^{13} - 932 q^{15} + 184 q^{17} + 2392 q^{19} + 992 q^{21} + 1192 q^{23} + 1824 q^{25} - 2468 q^{27} - 3364 q^{29} + 19212 q^{31} - 10580 q^{33} + 22944 q^{35} - 10928 q^{37} + 8732 q^{39} - 1120 q^{41} + 21420 q^{43} - 8344 q^{45} - 23772 q^{47} + 10452 q^{49} - 12744 q^{51} + 8860 q^{53} + 52652 q^{55} + 48944 q^{57} + 10840 q^{59} + 49448 q^{61} - 27488 q^{63} + 97836 q^{65} + 7840 q^{67} + 58792 q^{69} + 48744 q^{71} - 74992 q^{73} + 90448 q^{75} + 128656 q^{77} + 106076 q^{79} - 59692 q^{81} - 62888 q^{83} + 23848 q^{85} - 23548 q^{87} + 107568 q^{89} + 268896 q^{91} + 221460 q^{93} - 147352 q^{95} - 49520 q^{97} - 166720 q^{99}+O(q^{100}) \) 4 * q + 28 * q^3 - 68 * q^5 + 208 * q^7 - 280 * q^9 + 124 * q^11 - 460 * q^13 - 932 * q^15 + 184 * q^17 + 2392 * q^19 + 992 * q^21 + 1192 * q^23 + 1824 * q^25 - 2468 * q^27 - 3364 * q^29 + 19212 * q^31 - 10580 * q^33 + 22944 * q^35 - 10928 * q^37 + 8732 * q^39 - 1120 * q^41 + 21420 * q^43 - 8344 * q^45 - 23772 * q^47 + 10452 * q^49 - 12744 * q^51 + 8860 * q^53 + 52652 * q^55 + 48944 * q^57 + 10840 * q^59 + 49448 * q^61 - 27488 * q^63 + 97836 * q^65 + 7840 * q^67 + 58792 * q^69 + 48744 * q^71 - 74992 * q^73 + 90448 * q^75 + 128656 * q^77 + 106076 * q^79 - 59692 * q^81 - 62888 * q^83 + 23848 * q^85 - 23548 * q^87 + 107568 * q^89 + 268896 * q^91 + 221460 * q^93 - 147352 * q^95 - 49520 * q^97 - 166720 * q^99

Introduction

L-functions

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