LMFDB - Embedded newform 464.8.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,8,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, 8, names="a")

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

chi := DirichletCharacter("464.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 8 $
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:	$144.946651825$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 1101 x^{8} - 1540 x^{7} + 405148 x^{6} + 870160 x^{5} - 54569376 x^{4} - 87078400 x^{3} + \cdots - 9372051456 $ x^10 - 1101x^8 - 1540x^7 + 405148x^6 + 870160x^5 - 54569376x^4 - 87078400x^3 + 2140673280x^2 - 1918315520x - 9372051456
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{26} $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$22.0686$ 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)

$f(q)$	$=$	$q-73.8979 q^{3} +376.792 q^{5} -647.379 q^{7} +3273.90 q^{9} +O(q^{10})$	$q-73.8979 q^{3} +376.792 q^{5} -647.379 q^{7} +3273.90 q^{9} +917.032 q^{11} +5439.90 q^{13} -27844.1 q^{15} -23747.6 q^{17} +29878.0 q^{19} +47840.0 q^{21} -34152.9 q^{23} +63846.9 q^{25} -80319.7 q^{27} -24389.0 q^{29} -215829. q^{31} -67766.7 q^{33} -243927. q^{35} +61163.9 q^{37} -401997. q^{39} -202987. q^{41} -436213. q^{43} +1.23358e6 q^{45} +1.14899e6 q^{47} -404443. q^{49} +1.75490e6 q^{51} +237602. q^{53} +345530. q^{55} -2.20792e6 q^{57} +1.56197e6 q^{59} +2.82667e6 q^{61} -2.11945e6 q^{63} +2.04971e6 q^{65} +1.28367e6 q^{67} +2.52383e6 q^{69} -3.48686e6 q^{71} +68118.2 q^{73} -4.71816e6 q^{75} -593667. q^{77} +770647. q^{79} -1.22456e6 q^{81} +3.66552e6 q^{83} -8.94789e6 q^{85} +1.80230e6 q^{87} -322570. q^{89} -3.52167e6 q^{91} +1.59493e7 q^{93} +1.12578e7 q^{95} -2.17987e6 q^{97} +3.00227e6 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 80 q^{3} + 180 q^{5} - 1040 q^{7} + 10986 q^{9}+O(q^{10}) $ 10 * q - 80 * q^3 + 180 * q^5 - 1040 * q^7 + 10986 * q^9	$ 10 q - 80 q^{3} + 180 q^{5} - 1040 q^{7} + 10986 q^{9} - 7384 q^{11} + 20820 q^{13} - 43516 q^{15} - 11620 q^{17} - 75068 q^{19} + 51480 q^{21} - 62040 q^{23} + 261022 q^{25} + 28060 q^{27} - 243890 q^{29} - 200600 q^{31} - 1068000 q^{33} - 107528 q^{35} - 367740 q^{37} - 392692 q^{39} + 932764 q^{41} - 1443560 q^{43} + 4245684 q^{45} + 286960 q^{47} + 4713194 q^{49} - 1451016 q^{51} + 3953220 q^{53} - 3981316 q^{55} + 2050640 q^{57} - 6712320 q^{59} + 1905196 q^{61} - 3643800 q^{63} + 4667544 q^{65} + 2718200 q^{67} + 1109064 q^{69} - 3447736 q^{71} - 2554460 q^{73} - 1088084 q^{75} - 3967800 q^{77} - 4187744 q^{79} + 5161402 q^{81} - 3498720 q^{83} + 1817072 q^{85} + 1951120 q^{87} - 303268 q^{89} - 27215080 q^{91} + 1097360 q^{93} + 8810536 q^{95} + 4908620 q^{97} + 14408716 q^{99}+O(q^{100}) $ 10 * q - 80 * q^3 + 180 * q^5 - 1040 * q^7 + 10986 * q^9 - 7384 * q^11 + 20820 * q^13 - 43516 * q^15 - 11620 * q^17 - 75068 * q^19 + 51480 * q^21 - 62040 * q^23 + 261022 * q^25 + 28060 * q^27 - 243890 * q^29 - 200600 * q^31 - 1068000 * q^33 - 107528 * q^35 - 367740 * q^37 - 392692 * q^39 + 932764 * q^41 - 1443560 * q^43 + 4245684 * q^45 + 286960 * q^47 + 4713194 * q^49 - 1451016 * q^51 + 3953220 * q^53 - 3981316 * q^55 + 2050640 * q^57 - 6712320 * q^59 + 1905196 * q^61 - 3643800 * q^63 + 4667544 * q^65 + 2718200 * q^67 + 1109064 * q^69 - 3447736 * q^71 - 2554460 * q^73 - 1088084 * q^75 - 3967800 * q^77 - 4187744 * q^79 + 5161402 * q^81 - 3498720 * q^83 + 1817072 * q^85 + 1951120 * q^87 - 303268 * q^89 - 27215080 * q^91 + 1097360 * q^93 + 8810536 * q^95 + 4908620 * q^97 + 14408716 * 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
$3$	−73.8979	−1.58018	−0.790092	−	0.612988i	$-0.789967\pi$
$3$	−73.8979	−1.58018	−0.790092	+	0.612988i	$0.789967\pi$
$4$	0	0
$5$	376.792	1.34805	0.674025	−	0.738708i	$-0.264564\pi$
$5$	376.792	1.34805	0.674025	+	0.738708i	$0.264564\pi$
$6$	0	0
$7$	−647.379	−0.713371	−0.356685	−	0.934225i	$-0.616093\pi$
$7$	−647.379	−0.713371	−0.356685	+	0.934225i	$0.616093\pi$
$8$	0	0
$9$	3273.90	1.49698
$10$	0	0
$11$	917.032	0.207735	0.103868	−	0.994591i	$-0.466878\pi$
$11$	917.032	0.207735	0.103868	+	0.994591i	$0.466878\pi$
$12$	0	0
$13$	5439.90	0.686735	0.343367	−	0.939201i	$-0.388432\pi$
$13$	5439.90	0.686735	0.343367	+	0.939201i	$0.388432\pi$
$14$	0	0
$15$	−27844.1	−2.13017
$16$	0	0
$17$	−23747.6	−1.17233	−0.586163	−	0.810193i	$-0.699362\pi$
$17$	−23747.6	−1.17233	−0.586163	+	0.810193i	$0.699362\pi$
$18$	0	0
$19$	29878.0	0.999340	0.499670	−	0.866216i	$-0.333455\pi$
$19$	29878.0	0.999340	0.499670	+	0.866216i	$0.333455\pi$
$20$	0	0
$21$	47840.0	1.12726
$22$	0	0
$23$	−34152.9	−0.585302	−0.292651	−	0.956219i	$-0.594537\pi$
$23$	−34152.9	−0.585302	−0.292651	+	0.956219i	$0.594537\pi$
$24$	0	0
$25$	63846.9	0.817241
$26$	0	0
$27$	−80319.7	−0.785324
$28$	0	0
$29$	−24389.0	−0.185695
$29$	−24389.0	−0.185695
$30$	0	0
$31$	−215829.	−1.30120	−0.650598	−	0.759422i	$-0.725482\pi$
$31$	−215829.	−1.30120	−0.650598	+	0.759422i	$0.725482\pi$
$32$	0	0
$33$	−67766.7	−0.328260
$34$	0	0
$35$	−243927.	−0.961660
$36$	0	0
$37$	61163.9	0.198513	0.0992565	−	0.995062i	$-0.468354\pi$
$37$	61163.9	0.198513	0.0992565	+	0.995062i	$0.468354\pi$
$38$	0	0
$39$	−401997.	−1.08517
$40$	0	0
$41$	−202987.	−0.459964	−0.229982	−	0.973195i	$-0.573867\pi$
$41$	−202987.	−0.459964	−0.229982	+	0.973195i	$0.573867\pi$
$42$	0	0
$43$	−436213.	−0.836679	−0.418339	−	0.908291i	$-0.637388\pi$
$43$	−436213.	−0.836679	−0.418339	+	0.908291i	$0.637388\pi$
$44$	0	0
$45$	1.23358e6	2.01801
$46$	0	0
$47$	1.14899e6	1.61426	0.807131	−	0.590372i	$-0.201019\pi$
$47$	1.14899e6	1.61426	0.807131	+	0.590372i	$0.201019\pi$
$48$	0	0
$49$	−404443.	−0.491102
$50$	0	0
$51$	1.75490e6	1.85249
$52$	0	0
$53$	237602.	0.219222	0.109611	−	0.993975i	$-0.465040\pi$
$53$	237602.	0.219222	0.109611	+	0.993975i	$0.465040\pi$
$54$	0	0
$55$	345530.	0.280038
$56$	0	0
$57$	−2.20792e6	−1.57914
$58$	0	0
$59$	1.56197e6	0.990126	0.495063	−	0.868857i	$-0.335145\pi$
$59$	1.56197e6	0.990126	0.495063	+	0.868857i	$0.335145\pi$
$60$	0	0
$61$	2.82667e6	1.59449	0.797243	−	0.603659i	$-0.206291\pi$
$61$	2.82667e6	1.59449	0.797243	+	0.603659i	$0.206291\pi$
$62$	0	0
$63$	−2.11945e6	−1.06790
$64$	0	0
$65$	2.04971e6	0.925753
$66$	0	0
$67$	1.28367e6	0.521424	0.260712	−	0.965417i	$-0.416043\pi$
$67$	1.28367e6	0.521424	0.260712	+	0.965417i	$0.416043\pi$
$68$	0	0
$69$	2.52383e6	0.924884
$70$	0	0
$71$	−3.48686e6	−1.15619	−0.578096	−	0.815969i	$-0.696204\pi$
$71$	−3.48686e6	−1.15619	−0.578096	+	0.815969i	$0.696204\pi$
$72$	0	0
$73$	68118.2	0.0204943	0.0102472	−	0.999947i	$-0.496738\pi$
$73$	68118.2	0.0204943	0.0102472	+	0.999947i	$0.496738\pi$
$74$	0	0
$75$	−4.71816e6	−1.29139
$76$	0	0
$77$	−593667.	−0.148192
$78$	0	0
$79$	770647.	0.175857	0.0879287	−	0.996127i	$-0.471975\pi$
$79$	770647.	0.175857	0.0879287	+	0.996127i	$0.471975\pi$
$80$	0	0
$81$	−1.22456e6	−0.256026
$82$	0	0
$83$	3.66552e6	0.703658	0.351829	−	0.936064i	$-0.385560\pi$
$83$	3.66552e6	0.703658	0.351829	+	0.936064i	$0.385560\pi$
$84$	0	0
$85$	−8.94789e6	−1.58035
$86$	0	0
$87$	1.80230e6	0.293433
$88$	0	0
$89$	−322570.	−0.0485019	−0.0242509	−	0.999706i	$-0.507720\pi$
$89$	−322570.	−0.0485019	−0.0242509	+	0.999706i	$0.507720\pi$
$90$	0	0
$91$	−3.52167e6	−0.489897
$92$	0	0
$93$	1.59493e7	2.05613
$94$	0	0
$95$	1.12578e7	1.34716
$96$	0	0
$97$	−2.17987e6	−0.242510	−0.121255	−	0.992621i	$-0.538692\pi$
$97$	−2.17987e6	−0.242510	−0.121255	+	0.992621i	$0.538692\pi$
$98$	0	0
$99$	3.00227e6	0.310976
$100$	0	0
$101$	1.90852e7	1.84320	0.921598	−	0.388147i	$-0.126885\pi$
$101$	1.90852e7	1.84320	0.921598	+	0.388147i	$0.126885\pi$
$102$	0	0
$103$	1.33432e7	1.20317	0.601587	−	0.798808i	$-0.294535\pi$
$103$	1.33432e7	1.20317	0.601587	+	0.798808i	$0.294535\pi$
$104$	0	0
$105$	1.80257e7	1.51960
$106$	0	0
$107$	6.92372e6	0.546382	0.273191	−	0.961960i	$-0.411921\pi$
$107$	6.92372e6	0.546382	0.273191	+	0.961960i	$0.411921\pi$
$108$	0	0
$109$	−2.41593e7	−1.78686	−0.893432	−	0.449199i	$-0.851710\pi$
$109$	−2.41593e7	−1.78686	−0.893432	+	0.449199i	$0.851710\pi$
$110$	0	0
$111$	−4.51988e6	−0.313687
$112$	0	0
$113$	5.03277e6	0.328120	0.164060	−	0.986450i	$-0.447541\pi$
$113$	5.03277e6	0.328120	0.164060	+	0.986450i	$0.447541\pi$
$114$	0	0
$115$	−1.28685e7	−0.789016
$116$	0	0
$117$	1.78097e7	1.02803
$118$	0	0
$119$	1.53737e7	0.836303
$120$	0	0
$121$	−1.86462e7	−0.956846
$122$	0	0
$123$	1.50003e7	0.726828
$124$	0	0
$125$	−5.37985e6	−0.246368
$126$	0	0
$127$	2.20593e7	0.955605	0.477803	−	0.878467i	$-0.341433\pi$
$127$	2.20593e7	0.955605	0.477803	+	0.878467i	$0.341433\pi$
$128$	0	0
$129$	3.22352e7	1.32211
$130$	0	0
$131$	−4.21270e7	−1.63723	−0.818617	−	0.574340i	$-0.805259\pi$
$131$	−4.21270e7	−1.63723	−0.818617	+	0.574340i	$0.805259\pi$
$132$	0	0
$133$	−1.93424e7	−0.712900
$134$	0	0
$135$	−3.02638e7	−1.05866
$136$	0	0
$137$	−1.07382e6	−0.0356789	−0.0178394	−	0.999841i	$-0.505679\pi$
$137$	−1.07382e6	−0.0356789	−0.0178394	+	0.999841i	$0.505679\pi$
$138$	0	0
$139$	−7.04746e6	−0.222577	−0.111289	−	0.993788i	$-0.535498\pi$
$139$	−7.04746e6	−0.222577	−0.111289	+	0.993788i	$0.535498\pi$
$140$	0	0
$141$	−8.49080e7	−2.55083
$142$	0	0
$143$	4.98856e6	0.142659
$144$	0	0
$145$	−9.18957e6	−0.250327
$146$	0	0
$147$	2.98875e7	0.776031
$148$	0	0
$149$	2.10846e6	0.0522173	0.0261086	−	0.999659i	$-0.491688\pi$
$149$	2.10846e6	0.0522173	0.0261086	+	0.999659i	$0.491688\pi$
$150$	0	0
$151$	−3.04554e7	−0.719854	−0.359927	−	0.932980i	$-0.617198\pi$
$151$	−3.04554e7	−0.719854	−0.359927	+	0.932980i	$0.617198\pi$
$152$	0	0
$153$	−7.77472e7	−1.75495
$154$	0	0
$155$	−8.13224e7	−1.75408
$156$	0	0
$157$	−4.33834e7	−0.894694	−0.447347	−	0.894360i	$-0.647631\pi$
$157$	−4.33834e7	−0.894694	−0.447347	+	0.894360i	$0.647631\pi$
$158$	0	0
$159$	−1.75583e7	−0.346411
$160$	0	0
$161$	2.21098e7	0.417537
$162$	0	0
$163$	6.67032e7	1.20640	0.603198	−	0.797591i	$-0.293893\pi$
$163$	6.67032e7	1.20640	0.603198	+	0.797591i	$0.293893\pi$
$164$	0	0
$165$	−2.55339e7	−0.442511
$166$	0	0
$167$	−6.05087e7	−1.00533	−0.502667	−	0.864480i	$-0.667648\pi$
$167$	−6.05087e7	−1.00533	−0.502667	+	0.864480i	$0.667648\pi$
$168$	0	0
$169$	−3.31560e7	−0.528396
$170$	0	0
$171$	9.78175e7	1.49600
$172$	0	0
$173$	−1.01700e8	−1.49335	−0.746675	−	0.665189i	$-0.768351\pi$
$173$	−1.01700e8	−1.49335	−0.746675	+	0.665189i	$0.768351\pi$
$174$	0	0
$175$	−4.13332e7	−0.582996
$176$	0	0
$177$	−1.15426e8	−1.56458
$178$	0	0
$179$	−3.98101e7	−0.518809	−0.259405	−	0.965769i	$-0.583526\pi$
$179$	−3.98101e7	−0.518809	−0.259405	+	0.965769i	$0.583526\pi$
$180$	0	0
$181$	5.63538e7	0.706396	0.353198	−	0.935549i	$-0.385094\pi$
$181$	5.63538e7	0.706396	0.353198	+	0.935549i	$0.385094\pi$
$182$	0	0
$183$	−2.08885e8	−2.51958
$184$	0	0
$185$	2.30460e7	0.267606
$186$	0	0
$187$	−2.17773e7	−0.243533
$188$	0	0
$189$	5.19973e7	0.560227
$190$	0	0
$191$	−1.79471e7	−0.186370	−0.0931851	−	0.995649i	$-0.529705\pi$
$191$	−1.79471e7	−0.186370	−0.0931851	+	0.995649i	$0.529705\pi$
$192$	0	0
$193$	−8.66641e7	−0.867738	−0.433869	−	0.900976i	$-0.642852\pi$
$193$	−8.66641e7	−0.867738	−0.433869	+	0.900976i	$0.642852\pi$
$194$	0	0
$195$	−1.51469e8	−1.46286
$196$	0	0
$197$	−1.48231e8	−1.38136	−0.690681	−	0.723159i	$-0.742689\pi$
$197$	−1.48231e8	−1.38136	−0.690681	+	0.723159i	$0.742689\pi$
$198$	0	0
$199$	−1.80493e8	−1.62358	−0.811791	−	0.583948i	$-0.801507\pi$
$199$	−1.80493e8	−1.62358	−0.811791	+	0.583948i	$0.801507\pi$
$200$	0	0
$201$	−9.48604e7	−0.823946
$202$	0	0
$203$	1.57889e7	0.132470
$204$	0	0
$205$	−7.64837e7	−0.620055
$206$	0	0
$207$	−1.11813e8	−0.876186
$208$	0	0
$209$	2.73991e7	0.207598
$210$	0	0
$211$	−1.01529e8	−0.744049	−0.372024	−	0.928223i	$-0.621336\pi$
$211$	−1.01529e8	−0.744049	−0.372024	+	0.928223i	$0.621336\pi$
$212$	0	0
$213$	2.57672e8	1.82700
$214$	0	0
$215$	−1.64361e8	−1.12789
$216$	0	0
$217$	1.39723e8	0.928236
$218$	0	0
$219$	−5.03379e6	−0.0323848
$220$	0	0
$221$	−1.29184e8	−0.805077
$222$	0	0
$223$	−9.90258e7	−0.597973	−0.298986	−	0.954257i	$-0.596649\pi$
$223$	−9.90258e7	−0.597973	−0.298986	+	0.954257i	$0.596649\pi$
$224$	0	0
$225$	2.09029e8	1.22340
$226$	0	0
$227$	7.06827e7	0.401072	0.200536	−	0.979686i	$-0.435732\pi$
$227$	7.06827e7	0.401072	0.200536	+	0.979686i	$0.435732\pi$
$228$	0	0
$229$	7.97463e7	0.438820	0.219410	−	0.975633i	$-0.429587\pi$
$229$	7.97463e7	0.438820	0.219410	+	0.975633i	$0.429587\pi$
$230$	0	0
$231$	4.38708e7	0.234171
$232$	0	0
$233$	2.78555e8	1.44266	0.721332	−	0.692590i	$-0.243530\pi$
$233$	2.78555e8	1.44266	0.721332	+	0.692590i	$0.243530\pi$
$234$	0	0
$235$	4.32930e8	2.17611
$236$	0	0
$237$	−5.69492e7	−0.277887
$238$	0	0
$239$	−3.03984e8	−1.44032	−0.720159	−	0.693809i	$-0.755931\pi$
$239$	−3.03984e8	−1.44032	−0.720159	+	0.693809i	$0.755931\pi$
$240$	0	0
$241$	1.62667e8	0.748581	0.374291	−	0.927311i	$-0.377886\pi$
$241$	1.62667e8	0.748581	0.374291	+	0.927311i	$0.377886\pi$
$242$	0	0
$243$	2.66152e8	1.18989
$244$	0	0
$245$	−1.52391e8	−0.662030
$246$	0	0
$247$	1.62533e8	0.686282
$248$	0	0
$249$	−2.70874e8	−1.11191
$250$	0	0
$251$	9.86866e7	0.393913	0.196956	−	0.980412i	$-0.436894\pi$
$251$	9.86866e7	0.393913	0.196956	+	0.980412i	$0.436894\pi$
$252$	0	0
$253$	−3.13193e7	−0.121588
$254$	0	0
$255$	6.61230e8	2.49725
$256$	0	0
$257$	1.41643e8	0.520509	0.260254	−	0.965540i	$-0.416194\pi$
$257$	1.41643e8	0.520509	0.260254	+	0.965540i	$0.416194\pi$
$258$	0	0
$259$	−3.95962e7	−0.141613
$260$	0	0
$261$	−7.98472e7	−0.277983
$262$	0	0
$263$	−1.80962e8	−0.613396	−0.306698	−	0.951807i	$-0.599224\pi$
$263$	−1.80962e8	−0.613396	−0.306698	+	0.951807i	$0.599224\pi$
$264$	0	0
$265$	8.95263e7	0.295522
$266$	0	0
$267$	2.38372e7	0.0766419
$268$	0	0
$269$	−2.27845e8	−0.713684	−0.356842	−	0.934165i	$-0.616146\pi$
$269$	−2.27845e8	−0.713684	−0.356842	+	0.934165i	$0.616146\pi$
$270$	0	0
$271$	6.01958e8	1.83727	0.918636	−	0.395104i	$-0.129291\pi$
$271$	6.01958e8	1.83727	0.918636	+	0.395104i	$0.129291\pi$
$272$	0	0
$273$	2.60244e8	0.774127
$274$	0	0
$275$	5.85497e7	0.169770
$276$	0	0
$277$	−1.98459e8	−0.561036	−0.280518	−	0.959849i	$-0.590506\pi$
$277$	−1.98459e8	−0.561036	−0.280518	+	0.959849i	$0.590506\pi$
$278$	0	0
$279$	−7.06601e8	−1.94787
$280$	0	0
$281$	−3.68152e8	−0.989818	−0.494909	−	0.868945i	$-0.664799\pi$
$281$	−3.68152e8	−0.989818	−0.494909	+	0.868945i	$0.664799\pi$
$282$	0	0
$283$	−2.58445e8	−0.677821	−0.338910	−	0.940819i	$-0.610058\pi$
$283$	−2.58445e8	−0.677821	−0.338910	+	0.940819i	$0.610058\pi$
$284$	0	0
$285$	−8.31926e8	−2.12876
$286$	0	0
$287$	1.31409e8	0.328125
$288$	0	0
$289$	1.53609e8	0.374347
$290$	0	0
$291$	1.61088e8	0.383211
$292$	0	0
$293$	−4.91702e8	−1.14200	−0.570999	−	0.820951i	$-0.693444\pi$
$293$	−4.91702e8	−1.14200	−0.570999	+	0.820951i	$0.693444\pi$
$294$	0	0
$295$	5.88537e8	1.33474
$296$	0	0
$297$	−7.36557e7	−0.163139
$298$	0	0
$299$	−1.85788e8	−0.401947
$300$	0	0
$301$	2.82395e8	0.596862
$302$	0	0
$303$	−1.41035e9	−2.91259
$304$	0	0
$305$	1.06507e9	2.14945
$306$	0	0
$307$	4.21676e8	0.831754	0.415877	−	0.909421i	$-0.363475\pi$
$307$	4.21676e8	0.831754	0.415877	+	0.909421i	$0.363475\pi$
$308$	0	0
$309$	−9.86031e8	−1.90124
$310$	0	0
$311$	−8.55447e8	−1.61262	−0.806310	−	0.591494i	$-0.798539\pi$
$311$	−8.55447e8	−1.61262	−0.806310	+	0.591494i	$0.798539\pi$
$312$	0	0
$313$	−417650.	−0.000769853 0	−0.000384926	−	1.00000i	$-0.500123\pi$
$313$	−417650.	−0.000769853 0	−0.000384926		1.00000i	$0.500123\pi$
$314$	0	0
$315$	−7.98593e8	−1.43959
$316$	0	0
$317$	−3.50781e8	−0.618484	−0.309242	−	0.950983i	$-0.600075\pi$
$317$	−3.50781e8	−0.618484	−0.309242	+	0.950983i	$0.600075\pi$
$318$	0	0
$319$	−2.23655e7	−0.0385755
$320$	0	0
$321$	−5.11649e8	−0.863385
$322$	0	0
$323$	−7.09530e8	−1.17155
$324$	0	0
$325$	3.47321e8	0.561228
$326$	0	0
$327$	1.78532e9	2.82357
$328$	0	0
$329$	−7.43832e8	−1.15157
$330$	0	0
$331$	−4.95807e8	−0.751475	−0.375738	−	0.926726i	$-0.622611\pi$
$331$	−4.95807e8	−0.751475	−0.375738	+	0.926726i	$0.622611\pi$
$332$	0	0
$333$	2.00244e8	0.297170
$334$	0	0
$335$	4.83675e8	0.702906
$336$	0	0
$337$	9.54347e8	1.35832	0.679159	−	0.733991i	$-0.262345\pi$
$337$	9.54347e8	1.35832	0.679159	+	0.733991i	$0.262345\pi$
$338$	0	0
$339$	−3.71911e8	−0.518490
$340$	0	0
$341$	−1.97922e8	−0.270304
$342$	0	0
$343$	7.94973e8	1.06371
$344$	0	0
$345$	9.50956e8	1.24679
$346$	0	0
$347$	1.49821e8	0.192495	0.0962474	−	0.995357i	$-0.469316\pi$
$347$	1.49821e8	0.192495	0.0962474	+	0.995357i	$0.469316\pi$
$348$	0	0
$349$	−9.98391e8	−1.25722	−0.628611	−	0.777720i	$-0.716376\pi$
$349$	−9.98391e8	−1.25722	−0.628611	+	0.777720i	$0.716376\pi$
$350$	0	0
$351$	−4.36931e8	−0.539309
$352$	0	0
$353$	1.40557e9	1.70075	0.850373	−	0.526180i	$-0.176376\pi$
$353$	1.40557e9	1.70075	0.850373	+	0.526180i	$0.176376\pi$
$354$	0	0
$355$	−1.31382e9	−1.55861
$356$	0	0
$357$	−1.13608e9	−1.32151
$358$	0	0
$359$	−1.30123e9	−1.48431	−0.742155	−	0.670228i	$-0.766196\pi$
$359$	−1.30123e9	−1.48431	−0.742155	+	0.670228i	$0.766196\pi$
$360$	0	0
$361$	−1.17891e6	−0.00131888
$362$	0	0
$363$	1.37792e9	1.51199
$364$	0	0
$365$	2.56664e7	0.0276274
$366$	0	0
$367$	−8.12443e8	−0.857950	−0.428975	−	0.903316i	$-0.641125\pi$
$367$	−8.12443e8	−0.857950	−0.428975	+	0.903316i	$0.641125\pi$
$368$	0	0
$369$	−6.64558e8	−0.688558
$370$	0	0
$371$	−1.53818e8	−0.156387
$372$	0	0
$373$	1.29059e9	1.28767	0.643837	−	0.765163i	$-0.277342\pi$
$373$	1.29059e9	1.28767	0.643837	+	0.765163i	$0.277342\pi$
$374$	0	0
$375$	3.97560e8	0.389308
$376$	0	0
$377$	−1.32674e8	−0.127523
$378$	0	0
$379$	−4.40462e7	−0.0415596	−0.0207798	−	0.999784i	$-0.506615\pi$
$379$	−4.40462e7	−0.0415596	−0.0207798	+	0.999784i	$0.506615\pi$
$380$	0	0
$381$	−1.63014e9	−1.51003
$382$	0	0
$383$	1.42034e9	1.29180	0.645902	−	0.763421i	$-0.276482\pi$
$383$	1.42034e9	1.29180	0.645902	+	0.763421i	$0.276482\pi$
$384$	0	0
$385$	−2.23689e8	−0.199771
$386$	0	0
$387$	−1.42812e9	−1.25249
$388$	0	0
$389$	8.67673e8	0.747365	0.373682	−	0.927557i	$-0.378095\pi$
$389$	8.67673e8	0.747365	0.373682	+	0.927557i	$0.378095\pi$
$390$	0	0
$391$	8.11048e8	0.686164
$392$	0	0
$393$	3.11309e9	2.58713
$394$	0	0
$395$	2.90373e8	0.237065
$396$	0	0
$397$	1.11899e7	0.00897550	0.00448775	−	0.999990i	$-0.498571\pi$
$397$	1.11899e7	0.00897550	0.00448775	+	0.999990i	$0.498571\pi$
$398$	0	0
$399$	1.42936e9	1.12651
$400$	0	0
$401$	−1.18782e9	−0.919906	−0.459953	−	0.887943i	$-0.652134\pi$
$401$	−1.18782e9	−0.919906	−0.459953	+	0.887943i	$0.652134\pi$
$402$	0	0
$403$	−1.17409e9	−0.893577
$404$	0	0
$405$	−4.61405e8	−0.345136
$406$	0	0
$407$	5.60892e7	0.0412381
$408$	0	0
$409$	−1.10619e9	−0.799462	−0.399731	−	0.916632i	$-0.630896\pi$
$409$	−1.10619e9	−0.799462	−0.399731	+	0.916632i	$0.630896\pi$
$410$	0	0
$411$	7.93533e7	0.0563792
$412$	0	0
$413$	−1.01119e9	−0.706327
$414$	0	0
$415$	1.38114e9	0.948567
$416$	0	0
$417$	5.20792e8	0.351713
$418$	0	0
$419$	−2.88674e9	−1.91716	−0.958581	−	0.284819i	$-0.908066\pi$
$419$	−2.88674e9	−1.91716	−0.958581	+	0.284819i	$0.908066\pi$
$420$	0	0
$421$	9.42877e8	0.615840	0.307920	−	0.951412i	$-0.400367\pi$
$421$	9.42877e8	0.615840	0.307920	+	0.951412i	$0.400367\pi$
$422$	0	0
$423$	3.76168e9	2.41652
$424$	0	0
$425$	−1.51621e9	−0.958072
$426$	0	0
$427$	−1.82993e9	−1.13746
$428$	0	0
$429$	−3.68644e8	−0.225427
$430$	0	0
$431$	−4.00016e8	−0.240662	−0.120331	−	0.992734i	$-0.538396\pi$
$431$	−4.00016e8	−0.240662	−0.120331	+	0.992734i	$0.538396\pi$
$432$	0	0
$433$	−2.34094e9	−1.38575	−0.692873	−	0.721060i	$-0.743655\pi$
$433$	−2.34094e9	−1.38575	−0.692873	+	0.721060i	$0.743655\pi$
$434$	0	0
$435$	6.79090e8	0.395562
$436$	0	0
$437$	−1.02042e9	−0.584916
$438$	0	0
$439$	7.14577e8	0.403110	0.201555	−	0.979477i	$-0.435401\pi$
$439$	7.14577e8	0.403110	0.201555	+	0.979477i	$0.435401\pi$
$440$	0	0
$441$	−1.32411e9	−0.735171
$442$	0	0
$443$	−2.19239e9	−1.19813	−0.599066	−	0.800699i	$-0.704461\pi$
$443$	−2.19239e9	−1.19813	−0.599066	+	0.800699i	$0.704461\pi$
$444$	0	0
$445$	−1.21542e8	−0.0653830
$446$	0	0
$447$	−1.55811e8	−0.0825129
$448$	0	0
$449$	5.35068e8	0.278963	0.139482	−	0.990225i	$-0.455456\pi$
$449$	5.35068e8	0.278963	0.139482	+	0.990225i	$0.455456\pi$
$450$	0	0
$451$	−1.86145e8	−0.0955507
$452$	0	0
$453$	2.25059e9	1.13750
$454$	0	0
$455$	−1.32694e9	−0.660405
$456$	0	0
$457$	3.18626e9	1.56162	0.780808	−	0.624771i	$-0.214808\pi$
$457$	3.18626e9	1.56162	0.780808	+	0.624771i	$0.214808\pi$
$458$	0	0
$459$	1.90740e9	0.920655
$460$	0	0
$461$	3.12207e9	1.48419	0.742094	−	0.670296i	$-0.233833\pi$
$461$	3.12207e9	1.48419	0.742094	+	0.670296i	$0.233833\pi$
$462$	0	0
$463$	−3.15215e9	−1.47595	−0.737977	−	0.674825i	$-0.764219\pi$
$463$	−3.15215e9	−1.47595	−0.737977	+	0.674825i	$0.764219\pi$
$464$	0	0
$465$	6.00956e9	2.77177
$466$	0	0
$467$	−7.68753e8	−0.349283	−0.174642	−	0.984632i	$-0.555877\pi$
$467$	−7.68753e8	−0.349283	−0.174642	+	0.984632i	$0.555877\pi$
$468$	0	0
$469$	−8.31020e8	−0.371969
$470$	0	0
$471$	3.20594e9	1.41378
$472$	0	0
$473$	−4.00021e8	−0.173808
$474$	0	0
$475$	1.90762e9	0.816702
$476$	0	0
$477$	7.77884e8	0.328171
$478$	0	0
$479$	−1.41643e9	−0.588873	−0.294436	−	0.955671i	$-0.595132\pi$
$479$	−1.41643e9	−0.588873	−0.294436	+	0.955671i	$0.595132\pi$
$480$	0	0
$481$	3.32725e8	0.136326
$482$	0	0
$483$	−1.63387e9	−0.659786
$484$	0	0
$485$	−8.21358e8	−0.326916
$486$	0	0
$487$	−3.54792e9	−1.39195	−0.695974	−	0.718067i	$-0.745027\pi$
$487$	−3.54792e9	−1.39195	−0.695974	+	0.718067i	$0.745027\pi$
$488$	0	0
$489$	−4.92923e9	−1.90633
$490$	0	0
$491$	4.84906e8	0.184872	0.0924362	−	0.995719i	$-0.470535\pi$
$491$	4.84906e8	0.184872	0.0924362	+	0.995719i	$0.470535\pi$
$492$	0	0
$493$	5.79180e8	0.217695
$494$	0	0
$495$	1.13123e9	0.419211
$496$	0	0
$497$	2.25732e9	0.824794
$498$	0	0
$499$	4.20411e9	1.51468	0.757342	−	0.653018i	$-0.226497\pi$
$499$	4.20411e9	1.51468	0.757342	+	0.653018i	$0.226497\pi$
$500$	0	0
$501$	4.47147e9	1.58861
$502$	0	0
$503$	6.03464e8	0.211428	0.105714	−	0.994397i	$-0.466287\pi$
$503$	6.03464e8	0.211428	0.105714	+	0.994397i	$0.466287\pi$
$504$	0	0
$505$	7.19114e9	2.48472
$506$	0	0
$507$	2.45016e9	0.834962
$508$	0	0
$509$	−3.03471e9	−1.02001	−0.510006	−	0.860171i	$-0.670357\pi$
$509$	−3.03471e9	−1.02001	−0.510006	+	0.860171i	$0.670357\pi$
$510$	0	0
$511$	−4.40983e7	−0.0146200
$512$	0	0
$513$	−2.39979e9	−0.784806
$514$	0	0
$515$	5.02759e9	1.62194
$516$	0	0
$517$	1.05366e9	0.335339
$518$	0	0
$519$	7.51545e9	2.35977
$520$	0	0
$521$	−4.88230e9	−1.51249	−0.756245	−	0.654289i	$-0.772968\pi$
$521$	−4.88230e9	−1.51249	−0.756245	+	0.654289i	$0.772968\pi$
$522$	0	0
$523$	−4.28368e9	−1.30936	−0.654682	−	0.755904i	$-0.727198\pi$
$523$	−4.28368e9	−1.30936	−0.654682	+	0.755904i	$0.727198\pi$
$524$	0	0
$525$	3.05443e9	0.921241
$526$	0	0
$527$	5.12541e9	1.52543
$528$	0	0
$529$	−2.23841e9	−0.657422
$530$	0	0
$531$	5.11373e9	1.48220
$532$	0	0
$533$	−1.10423e9	−0.315873
$534$	0	0
$535$	2.60880e9	0.736551
$536$	0	0
$537$	2.94188e9	0.819814
$538$	0	0
$539$	−3.70888e8	−0.102019
$540$	0	0
$541$	9.58437e8	0.260239	0.130120	−	0.991498i	$-0.458464\pi$
$541$	9.58437e8	0.260239	0.130120	+	0.991498i	$0.458464\pi$
$542$	0	0
$543$	−4.16443e9	−1.11624
$544$	0	0
$545$	−9.10302e9	−2.40878
$546$	0	0
$547$	1.14779e9	0.299851	0.149926	−	0.988697i	$-0.452097\pi$
$547$	1.14779e9	0.299851	0.149926	+	0.988697i	$0.452097\pi$
$548$	0	0
$549$	9.25424e9	2.38692
$550$	0	0
$551$	−7.28694e8	−0.185573
$552$	0	0
$553$	−4.98900e8	−0.125452
$554$	0	0
$555$	−1.70305e9	−0.422866
$556$	0	0
$557$	−5.96397e9	−1.46232	−0.731159	−	0.682207i	$-0.761020\pi$
$557$	−5.96397e9	−1.46232	−0.731159	+	0.682207i	$0.761020\pi$
$558$	0	0
$559$	−2.37295e9	−0.574576
$560$	0	0
$561$	1.60930e9	0.384827
$562$	0	0
$563$	2.86789e9	0.677302	0.338651	−	0.940912i	$-0.390029\pi$
$563$	2.86789e9	0.677302	0.338651	+	0.940912i	$0.390029\pi$
$564$	0	0
$565$	1.89631e9	0.442322
$566$	0	0
$567$	7.92756e8	0.182641
$568$	0	0
$569$	−5.88781e8	−0.133986	−0.0669932	−	0.997753i	$-0.521341\pi$
$569$	−5.88781e8	−0.133986	−0.0669932	+	0.997753i	$0.521341\pi$
$570$	0	0
$571$	−6.81053e9	−1.53093	−0.765463	−	0.643479i	$-0.777490\pi$
$571$	−6.81053e9	−1.53093	−0.765463	+	0.643479i	$0.777490\pi$
$572$	0	0
$573$	1.32625e9	0.294499
$574$	0	0
$575$	−2.18056e9	−0.478332
$576$	0	0
$577$	6.57528e9	1.42495	0.712474	−	0.701698i	$-0.247575\pi$
$577$	6.57528e9	1.42495	0.712474	+	0.701698i	$0.247575\pi$
$578$	0	0
$579$	6.40430e9	1.37119
$580$	0	0
$581$	−2.37298e9	−0.501970
$582$	0	0
$583$	2.17888e8	0.0455401
$584$	0	0
$585$	6.71054e9	1.38584
$586$	0	0
$587$	−3.22998e9	−0.659123	−0.329561	−	0.944134i	$-0.606901\pi$
$587$	−3.22998e9	−0.659123	−0.329561	+	0.944134i	$0.606901\pi$
$588$	0	0
$589$	−6.44852e9	−1.30034
$590$	0	0
$591$	1.09540e10	2.18281
$592$	0	0
$593$	1.64704e9	0.324350	0.162175	−	0.986762i	$-0.448149\pi$
$593$	1.64704e9	0.324350	0.162175	+	0.986762i	$0.448149\pi$
$594$	0	0
$595$	5.79268e9	1.12738
$596$	0	0
$597$	1.33380e10	2.56556
$598$	0	0
$599$	2.22562e9	0.423114	0.211557	−	0.977366i	$-0.432147\pi$
$599$	2.22562e9	0.423114	0.211557	+	0.977366i	$0.432147\pi$
$600$	0	0
$601$	−3.05479e9	−0.574012	−0.287006	−	0.957929i	$-0.592660\pi$
$601$	−3.05479e9	−0.574012	−0.287006	+	0.957929i	$0.592660\pi$
$602$	0	0
$603$	4.20260e9	0.780562
$604$	0	0
$605$	−7.02574e9	−1.28988
$606$	0	0
$607$	−6.32843e9	−1.14851	−0.574256	−	0.818676i	$-0.694708\pi$
$607$	−6.32843e9	−1.14851	−0.574256	+	0.818676i	$0.694708\pi$
$608$	0	0
$609$	−1.16677e9	−0.209326
$610$	0	0
$611$	6.25039e9	1.10857
$612$	0	0
$613$	−4.08769e9	−0.716747	−0.358373	−	0.933578i	$-0.616669\pi$
$613$	−4.08769e9	−0.716747	−0.358373	+	0.933578i	$0.616669\pi$
$614$	0	0
$615$	5.65198e9	0.979801
$616$	0	0
$617$	5.74557e9	0.984771	0.492385	−	0.870377i	$-0.336125\pi$
$617$	5.74557e9	0.984771	0.492385	+	0.870377i	$0.336125\pi$
$618$	0	0
$619$	−7.82745e9	−1.32649	−0.663244	−	0.748404i	$-0.730821\pi$
$619$	−7.82745e9	−1.32649	−0.663244	+	0.748404i	$0.730821\pi$
$620$	0	0
$621$	2.74315e9	0.459651
$622$	0	0
$623$	2.08825e8	0.0345998
$624$	0	0
$625$	−7.01513e9	−1.14936
$626$	0	0
$627$	−2.02473e9	−0.328043
$628$	0	0
$629$	−1.45249e9	−0.232722
$630$	0	0
$631$	8.67471e9	1.37452	0.687262	−	0.726410i	$-0.258812\pi$
$631$	8.67471e9	1.37452	0.687262	+	0.726410i	$0.258812\pi$
$632$	0	0
$633$	7.50278e9	1.17573
$634$	0	0
$635$	8.31176e9	1.28820
$636$	0	0
$637$	−2.20013e9	−0.337257
$638$	0	0
$639$	−1.14156e10	−1.73080
$640$	0	0
$641$	−2.90390e8	−0.0435490	−0.0217745	−	0.999763i	$-0.506932\pi$
$641$	−2.90390e8	−0.0435490	−0.0217745	+	0.999763i	$0.506932\pi$
$642$	0	0
$643$	−7.73663e9	−1.14766	−0.573830	−	0.818974i	$-0.694543\pi$
$643$	−7.73663e9	−1.14766	−0.573830	+	0.818974i	$0.694543\pi$
$644$	0	0
$645$	1.21460e10	1.78227
$646$	0	0
$647$	−2.11912e9	−0.307604	−0.153802	−	0.988102i	$-0.549152\pi$
$647$	−2.11912e9	−0.307604	−0.153802	+	0.988102i	$0.549152\pi$
$648$	0	0
$649$	1.43238e9	0.205684
$650$	0	0
$651$	−1.03252e10	−1.46678
$652$	0	0
$653$	−6.36948e9	−0.895174	−0.447587	−	0.894240i	$-0.647717\pi$
$653$	−6.36948e9	−0.895174	−0.447587	+	0.894240i	$0.647717\pi$
$654$	0	0
$655$	−1.58731e10	−2.20707
$656$	0	0
$657$	2.23012e8	0.0306796
$658$	0	0
$659$	3.59109e9	0.488796	0.244398	−	0.969675i	$-0.421410\pi$
$659$	3.59109e9	0.488796	0.244398	+	0.969675i	$0.421410\pi$
$660$	0	0
$661$	5.83457e9	0.785786	0.392893	−	0.919584i	$-0.371474\pi$
$661$	5.83457e9	0.785786	0.392893	+	0.919584i	$0.371474\pi$
$662$	0	0
$663$	9.54646e9	1.27217
$664$	0	0
$665$	−7.28804e9	−0.961026
$666$	0	0
$667$	8.32954e8	0.108688
$668$	0	0
$669$	7.31780e9	0.944907
$670$	0	0
$671$	2.59215e9	0.331231
$672$	0	0
$673$	−4.35317e9	−0.550494	−0.275247	−	0.961374i	$-0.588760\pi$
$673$	−4.35317e9	−0.550494	−0.275247	+	0.961374i	$0.588760\pi$
$674$	0	0
$675$	−5.12817e9	−0.641799
$676$	0	0
$677$	1.40182e9	0.173632	0.0868162	−	0.996224i	$-0.472331\pi$
$677$	1.40182e9	0.173632	0.0868162	+	0.996224i	$0.472331\pi$
$678$	0	0
$679$	1.41120e9	0.173000
$680$	0	0
$681$	−5.22331e9	−0.633768
$682$	0	0
$683$	2.09063e9	0.251076	0.125538	−	0.992089i	$-0.459934\pi$
$683$	2.09063e9	0.251076	0.125538	+	0.992089i	$0.459934\pi$
$684$	0	0
$685$	−4.04608e8	−0.0480969
$686$	0	0
$687$	−5.89308e9	−0.693416
$688$	0	0
$689$	1.29253e9	0.150547
$690$	0	0
$691$	−4.31774e9	−0.497833	−0.248916	−	0.968525i	$-0.580074\pi$
$691$	−4.31774e9	−0.497833	−0.248916	+	0.968525i	$0.580074\pi$
$692$	0	0
$693$	−1.94361e9	−0.221841
$694$	0	0
$695$	−2.65542e9	−0.300045
$696$	0	0
$697$	4.82044e9	0.539228
$698$	0	0
$699$	−2.05846e10	−2.27967
$700$	0	0
$701$	1.20285e10	1.31886	0.659428	−	0.751768i	$-0.270799\pi$
$701$	1.20285e10	1.31886	0.659428	+	0.751768i	$0.270799\pi$
$702$	0	0
$703$	1.82745e9	0.198382
$704$	0	0
$705$	−3.19926e10	−3.43865
$706$	0	0
$707$	−1.23553e10	−1.31488
$708$	0	0
$709$	4.69177e9	0.494396	0.247198	−	0.968965i	$-0.420490\pi$
$709$	4.69177e9	0.494396	0.247198	+	0.968965i	$0.420490\pi$
$710$	0	0
$711$	2.52302e9	0.263255
$712$	0	0
$713$	7.37116e9	0.761593
$714$	0	0
$715$	1.87965e9	0.192312
$716$	0	0
$717$	2.24638e10	2.27597
$718$	0	0
$719$	1.45057e9	0.145542	0.0727708	−	0.997349i	$-0.476816\pi$
$719$	1.45057e9	0.145542	0.0727708	+	0.997349i	$0.476816\pi$
$720$	0	0
$721$	−8.63808e9	−0.858309
$722$	0	0
$723$	−1.20207e10	−1.18290
$724$	0	0
$725$	−1.55716e9	−0.151758
$726$	0	0
$727$	1.50239e10	1.45015	0.725073	−	0.688672i	$-0.241806\pi$
$727$	1.50239e10	1.45015	0.725073	+	0.688672i	$0.241806\pi$
$728$	0	0
$729$	−1.69899e10	−1.62422
$730$	0	0
$731$	1.03590e10	0.980860
$732$	0	0
$733$	1.07112e10	1.00456	0.502280	−	0.864705i	$-0.332495\pi$
$733$	1.07112e10	1.00456	0.502280	+	0.864705i	$0.332495\pi$
$734$	0	0
$735$	1.12614e10	1.04613
$736$	0	0
$737$	1.17716e9	0.108318
$738$	0	0
$739$	−1.14070e10	−1.03972	−0.519859	−	0.854252i	$-0.674016\pi$
$739$	−1.14070e10	−1.03972	−0.519859	+	0.854252i	$0.674016\pi$
$740$	0	0
$741$	−1.20109e10	−1.08445
$742$	0	0
$743$	−1.77671e10	−1.58912	−0.794559	−	0.607187i	$-0.792298\pi$
$743$	−1.77671e10	−1.58912	−0.794559	+	0.607187i	$0.792298\pi$
$744$	0	0
$745$	7.94452e8	0.0703916
$746$	0	0
$747$	1.20005e10	1.05336
$748$	0	0
$749$	−4.48227e9	−0.389773
$750$	0	0
$751$	1.00842e9	0.0868765	0.0434382	−	0.999056i	$-0.486169\pi$
$751$	1.00842e9	0.0868765	0.0434382	+	0.999056i	$0.486169\pi$
$752$	0	0
$753$	−7.29273e9	−0.622455
$754$	0	0
$755$	−1.14753e10	−0.970400
$756$	0	0
$757$	−1.41694e10	−1.18718	−0.593591	−	0.804767i	$-0.702290\pi$
$757$	−1.41694e10	−1.18718	−0.593591	+	0.804767i	$0.702290\pi$
$758$	0	0
$759$	2.31443e9	0.192131
$760$	0	0
$761$	−1.23239e10	−1.01368	−0.506842	−	0.862039i	$-0.669187\pi$
$761$	−1.23239e10	−1.01368	−0.506842	+	0.862039i	$0.669187\pi$
$762$	0	0
$763$	1.56402e10	1.27470
$764$	0	0
$765$	−2.92945e10	−2.36576
$766$	0	0
$767$	8.49695e9	0.679954
$768$	0	0
$769$	−7.92692e9	−0.628583	−0.314291	−	0.949327i	$-0.601767\pi$
$769$	−7.92692e9	−0.628583	−0.314291	+	0.949327i	$0.601767\pi$
$770$	0	0
$771$	−1.04671e10	−0.822500
$772$	0	0
$773$	1.76622e10	1.37536	0.687679	−	0.726015i	$-0.258630\pi$
$773$	1.76622e10	1.37536	0.687679	+	0.726015i	$0.258630\pi$
$774$	0	0
$775$	−1.37800e10	−1.06339
$776$	0	0
$777$	2.92608e9	0.223775
$778$	0	0
$779$	−6.06483e9	−0.459661
$780$	0	0
$781$	−3.19756e9	−0.240182
$782$	0	0
$783$	1.95892e9	0.145831
$784$	0	0
$785$	−1.63465e10	−1.20609
$786$	0	0
$787$	−2.48767e9	−0.181920	−0.0909601	−	0.995855i	$-0.528994\pi$
$787$	−2.48767e9	−0.181920	−0.0909601	+	0.995855i	$0.528994\pi$
$788$	0	0
$789$	1.33727e10	0.969279
$790$	0	0
$791$	−3.25811e9	−0.234071
$792$	0	0
$793$	1.53768e10	1.09499
$794$	0	0
$795$	−6.61581e9	−0.466980
$796$	0	0
$797$	−1.94471e10	−1.36066	−0.680332	−	0.732904i	$-0.738165\pi$
$797$	−1.94471e10	−1.36066	−0.680332	+	0.732904i	$0.738165\pi$
$798$	0	0
$799$	−2.72858e10	−1.89244
$800$	0	0
$801$	−1.05606e9	−0.0726065
$802$	0	0
$803$	6.24666e7	0.00425739
$804$	0	0
$805$	8.33081e9	0.562861
$806$	0	0
$807$	1.68372e10	1.12775
$808$	0	0
$809$	1.91310e10	1.27033	0.635167	−	0.772375i	$-0.280931\pi$
$809$	1.91310e10	1.27033	0.635167	+	0.772375i	$0.280931\pi$
$810$	0	0
$811$	−1.21261e9	−0.0798266	−0.0399133	−	0.999203i	$-0.512708\pi$
$811$	−1.21261e9	−0.0798266	−0.0399133	+	0.999203i	$0.512708\pi$
$812$	0	0
$813$	−4.44834e10	−2.90323
$814$	0	0
$815$	2.51332e10	1.62628
$816$	0	0
$817$	−1.30331e10	−0.836127
$818$	0	0
$819$	−1.15296e10	−0.733367
$820$	0	0
$821$	1.93729e9	0.122178	0.0610890	−	0.998132i	$-0.480543\pi$
$821$	1.93729e9	0.122178	0.0610890	+	0.998132i	$0.480543\pi$
$822$	0	0
$823$	−9.15703e9	−0.572605	−0.286302	−	0.958139i	$-0.592426\pi$
$823$	−9.15703e9	−0.572605	−0.286302	+	0.958139i	$0.592426\pi$
$824$	0	0
$825$	−4.32670e9	−0.268267
$826$	0	0
$827$	−2.73124e10	−1.67916	−0.839578	−	0.543239i	$-0.817198\pi$
$827$	−2.73124e10	−1.67916	−0.839578	+	0.543239i	$0.817198\pi$
$828$	0	0
$829$	−9.35892e9	−0.570538	−0.285269	−	0.958447i	$-0.592083\pi$
$829$	−9.35892e9	−0.570538	−0.285269	+	0.958447i	$0.592083\pi$
$830$	0	0
$831$	1.46657e10	0.886540
$832$	0	0
$833$	9.60456e9	0.575731
$834$	0	0
$835$	−2.27992e10	−1.35524
$836$	0	0
$837$	1.73353e10	1.02186
$838$	0	0
$839$	−9.38707e9	−0.548735	−0.274368	−	0.961625i	$-0.588469\pi$
$839$	−9.38707e9	−0.548735	−0.274368	+	0.961625i	$0.588469\pi$
$840$	0	0
$841$	5.94823e8	0.0344828
$842$	0	0
$843$	2.72057e10	1.56410
$844$	0	0
$845$	−1.24929e10	−0.712304
$846$	0	0
$847$	1.20712e10	0.682586
$848$	0	0
$849$	1.90985e10	1.07108
$850$	0	0
$851$	−2.08892e9	−0.116190
$852$	0	0
$853$	−1.00111e10	−0.552280	−0.276140	−	0.961117i	$-0.589055\pi$
$853$	−1.00111e10	−0.552280	−0.276140	+	0.961117i	$0.589055\pi$
$854$	0	0
$855$	3.68568e10	2.01668
$856$	0	0
$857$	1.11853e10	0.607036	0.303518	−	0.952826i	$-0.401839\pi$
$857$	1.11853e10	0.607036	0.303518	+	0.952826i	$0.401839\pi$
$858$	0	0
$859$	1.06684e10	0.574282	0.287141	−	0.957888i	$-0.407295\pi$
$859$	1.06684e10	0.574282	0.287141	+	0.957888i	$0.407295\pi$
$860$	0	0
$861$	−9.71087e9	−0.518498
$862$	0	0
$863$	−1.41277e10	−0.748229	−0.374114	−	0.927383i	$-0.622053\pi$
$863$	−1.41277e10	−0.748229	−0.374114	+	0.927383i	$0.622053\pi$
$864$	0	0
$865$	−3.83199e10	−2.01311
$866$	0	0
$867$	−1.13514e10	−0.591537
$868$	0	0
$869$	7.06708e8	0.0365318
$870$	0	0
$871$	6.98302e9	0.358080
$872$	0	0
$873$	−7.13669e9	−0.363034
$874$	0	0
$875$	3.48280e9	0.175752
$876$	0	0
$877$	4.77476e9	0.239030	0.119515	−	0.992832i	$-0.461866\pi$
$877$	4.77476e9	0.239030	0.119515	+	0.992832i	$0.461866\pi$
$878$	0	0
$879$	3.63357e10	1.80457
$880$	0	0
$881$	−1.54995e10	−0.763664	−0.381832	−	0.924232i	$-0.624707\pi$
$881$	−1.54995e10	−0.763664	−0.381832	+	0.924232i	$0.624707\pi$
$882$	0	0
$883$	1.16646e10	0.570173	0.285086	−	0.958502i	$-0.407978\pi$
$883$	1.16646e10	0.570173	0.285086	+	0.958502i	$0.407978\pi$
$884$	0	0
$885$	−4.34916e10	−2.10913
$886$	0	0
$887$	7.82168e9	0.376329	0.188164	−	0.982138i	$-0.439746\pi$
$887$	7.82168e9	0.376329	0.188164	+	0.982138i	$0.439746\pi$
$888$	0	0
$889$	−1.42807e10	−0.681701
$890$	0	0
$891$	−1.12296e9	−0.0531856
$892$	0	0
$893$	3.43295e10	1.61320
$894$	0	0
$895$	−1.50001e10	−0.699381
$896$	0	0
$897$	1.37294e10	0.635150
$898$	0	0
$899$	5.26384e9	0.241626
$900$	0	0
$901$	−5.64246e9	−0.256999
$902$	0	0
$903$	−2.08684e10	−0.943152
$904$	0	0
$905$	2.12336e10	0.952258
$906$	0	0
$907$	−9.52932e9	−0.424069	−0.212034	−	0.977262i	$-0.568009\pi$
$907$	−9.52932e9	−0.424069	−0.212034	+	0.977262i	$0.568009\pi$
$908$	0	0
$909$	6.24830e10	2.75923
$910$	0	0
$911$	−2.59809e9	−0.113852	−0.0569260	−	0.998378i	$-0.518130\pi$
$911$	−2.59809e9	−0.113852	−0.0569260	+	0.998378i	$0.518130\pi$
$912$	0	0
$913$	3.36140e9	0.146175
$914$	0	0
$915$	−7.87061e10	−3.39652
$916$	0	0
$917$	2.72721e10	1.16796
$918$	0	0
$919$	−8.94350e9	−0.380105	−0.190052	−	0.981774i	$-0.560866\pi$
$919$	−8.94350e9	−0.380105	−0.190052	+	0.981774i	$0.560866\pi$
$920$	0	0
$921$	−3.11610e10	−1.31432
$922$	0	0
$923$	−1.89682e10	−0.793997
$924$	0	0
$925$	3.90513e9	0.162233
$926$	0	0
$927$	4.36842e10	1.80113
$928$	0	0
$929$	2.88780e10	1.18171	0.590856	−	0.806777i	$-0.298790\pi$
$929$	2.88780e10	1.18171	0.590856	+	0.806777i	$0.298790\pi$
$930$	0	0
$931$	−1.20839e10	−0.490778
$932$	0	0
$933$	6.32158e10	2.54824
$934$	0	0
$935$	−8.20550e9	−0.328295
$936$	0	0
$937$	2.86074e10	1.13603	0.568016	−	0.823018i	$-0.307711\pi$
$937$	2.86074e10	1.13603	0.568016	+	0.823018i	$0.307711\pi$
$938$	0	0
$939$	3.08635e7	0.00121651
$940$	0	0
$941$	4.92468e10	1.92670	0.963351	−	0.268246i	$-0.0864438\pi$
$941$	4.92468e10	1.92670	0.963351	+	0.268246i	$0.0864438\pi$
$942$	0	0
$943$	6.93258e9	0.269218
$944$	0	0
$945$	1.95921e10	0.755215
$946$	0	0
$947$	4.35016e10	1.66449	0.832244	−	0.554410i	$-0.187056\pi$
$947$	4.35016e10	1.66449	0.832244	+	0.554410i	$0.187056\pi$
$948$	0	0
$949$	3.70556e8	0.0140742
$950$	0	0
$951$	2.59220e10	0.977318
$952$	0	0
$953$	1.40959e10	0.527556	0.263778	−	0.964583i	$-0.415031\pi$
$953$	1.40959e10	0.527556	0.263778	+	0.964583i	$0.415031\pi$
$954$	0	0
$955$	−6.76230e9	−0.251236
$956$	0	0
$957$	1.65276e9	0.0609563
$958$	0	0
$959$	6.95171e8	0.0254523
$960$	0	0
$961$	1.90694e10	0.693113
$962$	0	0
$963$	2.26676e10	0.817925
$964$	0	0
$965$	−3.26543e10	−1.16975
$966$	0	0
$967$	4.02387e9	0.143104	0.0715519	−	0.997437i	$-0.477205\pi$
$967$	4.02387e9	0.143104	0.0715519	+	0.997437i	$0.477205\pi$
$968$	0	0
$969$	5.24327e10	1.85127
$970$	0	0
$971$	3.56517e10	1.24972	0.624861	−	0.780736i	$-0.285156\pi$
$971$	3.56517e10	1.24972	0.624861	+	0.780736i	$0.285156\pi$
$972$	0	0
$973$	4.56238e9	0.158780
$974$	0	0
$975$	−2.56663e10	−0.886843
$976$	0	0
$977$	−3.79562e10	−1.30212	−0.651060	−	0.759026i	$-0.725676\pi$
$977$	−3.79562e10	−1.30212	−0.651060	+	0.759026i	$0.725676\pi$
$978$	0	0
$979$	−2.95807e8	−0.0100755
$980$	0	0
$981$	−7.90951e10	−2.67490
$982$	0	0
$983$	3.53123e10	1.18574	0.592868	−	0.805299i	$-0.297995\pi$
$983$	3.53123e10	1.18574	0.592868	+	0.805299i	$0.297995\pi$
$984$	0	0
$985$	−5.58523e10	−1.86215
$986$	0	0
$987$	5.49677e10	1.81969
$988$	0	0
$989$	1.48979e10	0.489709
$990$	0	0
$991$	−1.92381e10	−0.627920	−0.313960	−	0.949436i	$-0.601656\pi$
$991$	−1.92381e10	−0.627920	−0.313960	+	0.949436i	$0.601656\pi$
$992$	0	0
$993$	3.66391e10	1.18747
$994$	0	0
$995$	−6.80082e10	−2.18867
$996$	0	0
$997$	−1.85218e10	−0.591901	−0.295951	−	0.955203i	$-0.595636\pi$
$997$	−1.85218e10	−0.591901	−0.295951	+	0.955203i	$0.595636\pi$
$998$	0	0
$999$	−4.91266e9	−0.155897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.8.a.g.1.2		10
4.3	odd	2		29.8.a.b.1.1	✓	10
12.11	even	2		261.8.a.f.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.8.a.b.1.1	✓	10	4.3	odd	2
261.8.a.f.1.10		10	12.11	even	2
464.8.a.g.1.2		10	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	464.a (trivial)

\(f(q)\)	\(=\)	\(q-73.8979 q^{3} +376.792 q^{5} -647.379 q^{7} +3273.90 q^{9} +O(q^{10})\)	\(q-73.8979 q^{3} +376.792 q^{5} -647.379 q^{7} +3273.90 q^{9} +917.032 q^{11} +5439.90 q^{13} -27844.1 q^{15} -23747.6 q^{17} +29878.0 q^{19} +47840.0 q^{21} -34152.9 q^{23} +63846.9 q^{25} -80319.7 q^{27} -24389.0 q^{29} -215829. q^{31} -67766.7 q^{33} -243927. q^{35} +61163.9 q^{37} -401997. q^{39} -202987. q^{41} -436213. q^{43} +1.23358e6 q^{45} +1.14899e6 q^{47} -404443. q^{49} +1.75490e6 q^{51} +237602. q^{53} +345530. q^{55} -2.20792e6 q^{57} +1.56197e6 q^{59} +2.82667e6 q^{61} -2.11945e6 q^{63} +2.04971e6 q^{65} +1.28367e6 q^{67} +2.52383e6 q^{69} -3.48686e6 q^{71} +68118.2 q^{73} -4.71816e6 q^{75} -593667. q^{77} +770647. q^{79} -1.22456e6 q^{81} +3.66552e6 q^{83} -8.94789e6 q^{85} +1.80230e6 q^{87} -322570. q^{89} -3.52167e6 q^{91} +1.59493e7 q^{93} +1.12578e7 q^{95} -2.17987e6 q^{97} +3.00227e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 80 q^{3} + 180 q^{5} - 1040 q^{7} + 10986 q^{9}+O(q^{10}) \) 10 * q - 80 * q^3 + 180 * q^5 - 1040 * q^7 + 10986 * q^9	\( 10 q - 80 q^{3} + 180 q^{5} - 1040 q^{7} + 10986 q^{9} - 7384 q^{11} + 20820 q^{13} - 43516 q^{15} - 11620 q^{17} - 75068 q^{19} + 51480 q^{21} - 62040 q^{23} + 261022 q^{25} + 28060 q^{27} - 243890 q^{29} - 200600 q^{31} - 1068000 q^{33} - 107528 q^{35} - 367740 q^{37} - 392692 q^{39} + 932764 q^{41} - 1443560 q^{43} + 4245684 q^{45} + 286960 q^{47} + 4713194 q^{49} - 1451016 q^{51} + 3953220 q^{53} - 3981316 q^{55} + 2050640 q^{57} - 6712320 q^{59} + 1905196 q^{61} - 3643800 q^{63} + 4667544 q^{65} + 2718200 q^{67} + 1109064 q^{69} - 3447736 q^{71} - 2554460 q^{73} - 1088084 q^{75} - 3967800 q^{77} - 4187744 q^{79} + 5161402 q^{81} - 3498720 q^{83} + 1817072 q^{85} + 1951120 q^{87} - 303268 q^{89} - 27215080 q^{91} + 1097360 q^{93} + 8810536 q^{95} + 4908620 q^{97} + 14408716 q^{99}+O(q^{100}) \) 10 * q - 80 * q^3 + 180 * q^5 - 1040 * q^7 + 10986 * q^9 - 7384 * q^11 + 20820 * q^13 - 43516 * q^15 - 11620 * q^17 - 75068 * q^19 + 51480 * q^21 - 62040 * q^23 + 261022 * q^25 + 28060 * q^27 - 243890 * q^29 - 200600 * q^31 - 1068000 * q^33 - 107528 * q^35 - 367740 * q^37 - 392692 * q^39 + 932764 * q^41 - 1443560 * q^43 + 4245684 * q^45 + 286960 * q^47 + 4713194 * q^49 - 1451016 * q^51 + 3953220 * q^53 - 3981316 * q^55 + 2050640 * q^57 - 6712320 * q^59 + 1905196 * q^61 - 3643800 * q^63 + 4667544 * q^65 + 2718200 * q^67 + 1109064 * q^69 - 3447736 * q^71 - 2554460 * q^73 - 1088084 * q^75 - 3967800 * q^77 - 4187744 * q^79 + 5161402 * q^81 - 3498720 * q^83 + 1817072 * q^85 + 1951120 * q^87 - 303268 * q^89 - 27215080 * q^91 + 1097360 * q^93 + 8810536 * q^95 + 4908620 * q^97 + 14408716 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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