LMFDB - Embedded newform 304.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,6,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, 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("304.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 304 = 2^{4} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	304.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:	$48.7566812231$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 454x + 3760 $ x^3 - x^2 - 454*x + 3760
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$10.7990$ of defining polynomial
Character	$\chi$	$=$	304.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-14.7990 q^{3} -19.0956 q^{5} -212.287 q^{7} -23.9901 q^{9} +O(q^{10})$	$q-14.7990 q^{3} -19.0956 q^{5} -212.287 q^{7} -23.9901 q^{9} +662.625 q^{11} +1112.31 q^{13} +282.595 q^{15} -522.290 q^{17} +361.000 q^{19} +3141.63 q^{21} +3213.52 q^{23} -2760.36 q^{25} +3951.18 q^{27} -3725.71 q^{29} -4832.50 q^{31} -9806.17 q^{33} +4053.74 q^{35} -7152.96 q^{37} -16461.1 q^{39} +10812.9 q^{41} +10418.4 q^{43} +458.106 q^{45} -11998.2 q^{47} +28258.8 q^{49} +7729.35 q^{51} +24975.0 q^{53} -12653.2 q^{55} -5342.43 q^{57} -23279.2 q^{59} -15102.7 q^{61} +5092.80 q^{63} -21240.2 q^{65} -59216.5 q^{67} -47556.9 q^{69} -14643.8 q^{71} -18953.3 q^{73} +40850.5 q^{75} -140667. q^{77} +3628.29 q^{79} -52643.9 q^{81} -22655.2 q^{83} +9973.42 q^{85} +55136.7 q^{87} +65622.3 q^{89} -236129. q^{91} +71516.1 q^{93} -6893.50 q^{95} -59429.2 q^{97} -15896.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 13 q^{3} + 81 q^{5} - 228 q^{7} + 236 q^{9}+O(q^{10}) $ 3 * q - 13 * q^3 + 81 * q^5 - 228 * q^7 + 236 * q^9	$ 3 q - 13 q^{3} + 81 q^{5} - 228 q^{7} + 236 q^{9} - 363 q^{11} + 501 q^{13} + 670 q^{15} - 1206 q^{17} + 1083 q^{19} - 2085 q^{21} + 1077 q^{23} - 3882 q^{25} + 5087 q^{27} - 8349 q^{29} + 7332 q^{31} - 15784 q^{33} + 1185 q^{35} - 1650 q^{37} - 773 q^{39} + 10140 q^{41} - 3777 q^{43} + 14005 q^{45} - 33231 q^{47} + 31269 q^{49} - 46935 q^{51} + 31029 q^{53} - 66003 q^{55} - 4693 q^{57} - 20409 q^{59} + 17115 q^{61} - 6327 q^{63} - 45348 q^{65} + 789 q^{67} - 151147 q^{69} - 19164 q^{71} - 76260 q^{73} + 69607 q^{75} - 97209 q^{77} - 68358 q^{79} - 197713 q^{81} - 6762 q^{83} - 45837 q^{85} - 66805 q^{87} - 85506 q^{89} - 345033 q^{91} + 15688 q^{93} + 29241 q^{95} + 105024 q^{97} - 158317 q^{99}+O(q^{100}) $ 3 * q - 13 * q^3 + 81 * q^5 - 228 * q^7 + 236 * q^9 - 363 * q^11 + 501 * q^13 + 670 * q^15 - 1206 * q^17 + 1083 * q^19 - 2085 * q^21 + 1077 * q^23 - 3882 * q^25 + 5087 * q^27 - 8349 * q^29 + 7332 * q^31 - 15784 * q^33 + 1185 * q^35 - 1650 * q^37 - 773 * q^39 + 10140 * q^41 - 3777 * q^43 + 14005 * q^45 - 33231 * q^47 + 31269 * q^49 - 46935 * q^51 + 31029 * q^53 - 66003 * q^55 - 4693 * q^57 - 20409 * q^59 + 17115 * q^61 - 6327 * q^63 - 45348 * q^65 + 789 * q^67 - 151147 * q^69 - 19164 * q^71 - 76260 * q^73 + 69607 * q^75 - 97209 * q^77 - 68358 * q^79 - 197713 * q^81 - 6762 * q^83 - 45837 * q^85 - 66805 * q^87 - 85506 * q^89 - 345033 * q^91 + 15688 * q^93 + 29241 * q^95 + 105024 * q^97 - 158317 * 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
$3$	−14.7990	−0.949355	−0.474678	−	0.880160i	$-0.657435\pi$
$3$	−14.7990	−0.949355	−0.474678	+	0.880160i	$0.657435\pi$
$4$	0	0
$5$	−19.0956	−0.341592	−0.170796	−	0.985306i	$-0.554634\pi$
$5$	−19.0956	−0.341592	−0.170796	+	0.985306i	$0.554634\pi$
$6$	0	0
$7$	−212.287	−1.63749	−0.818745	−	0.574158i	$-0.805330\pi$
$7$	−212.287	−1.63749	−0.818745	+	0.574158i	$0.805330\pi$
$8$	0	0
$9$	−23.9901	−0.0987248
$10$	0	0
$11$	662.625	1.65115	0.825574	−	0.564294i	$-0.190852\pi$
$11$	662.625	1.65115	0.825574	+	0.564294i	$0.190852\pi$
$12$	0	0
$13$	1112.31	1.82544	0.912720	−	0.408586i	$-0.133978\pi$
$13$	1112.31	1.82544	0.912720	+	0.408586i	$0.133978\pi$
$14$	0	0
$15$	282.595	0.324292
$16$	0	0
$17$	−522.290	−0.438318	−0.219159	−	0.975689i	$-0.570331\pi$
$17$	−522.290	−0.438318	−0.219159	+	0.975689i	$0.570331\pi$
$18$	0	0
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	3141.63	1.55456
$22$	0	0
$23$	3213.52	1.26667	0.633333	−	0.773880i	$-0.281686\pi$
$23$	3213.52	1.26667	0.633333	+	0.773880i	$0.281686\pi$
$24$	0	0
$25$	−2760.36	−0.883315
$26$	0	0
$27$	3951.18	1.04308
$28$	0	0
$29$	−3725.71	−0.822648	−0.411324	−	0.911489i	$-0.634934\pi$
$29$	−3725.71	−0.822648	−0.411324	+	0.911489i	$0.634934\pi$
$30$	0	0
$31$	−4832.50	−0.903167	−0.451583	−	0.892229i	$-0.649141\pi$
$31$	−4832.50	−0.903167	−0.451583	+	0.892229i	$0.649141\pi$
$32$	0	0
$33$	−9806.17	−1.56753
$34$	0	0
$35$	4053.74	0.559353
$36$	0	0
$37$	−7152.96	−0.858977	−0.429489	−	0.903072i	$-0.641306\pi$
$37$	−7152.96	−0.858977	−0.429489	+	0.903072i	$0.641306\pi$
$38$	0	0
$39$	−16461.1	−1.73299
$40$	0	0
$41$	10812.9	1.00457	0.502287	−	0.864701i	$-0.332492\pi$
$41$	10812.9	1.00457	0.502287	+	0.864701i	$0.332492\pi$
$42$	0	0
$43$	10418.4	0.859270	0.429635	−	0.903003i	$-0.358642\pi$
$43$	10418.4	0.859270	0.429635	+	0.903003i	$0.358642\pi$
$44$	0	0
$45$	458.106	0.0337236
$46$	0	0
$47$	−11998.2	−0.792267	−0.396134	−	0.918193i	$-0.629648\pi$
$47$	−11998.2	−0.792267	−0.396134	+	0.918193i	$0.629648\pi$
$48$	0	0
$49$	28258.8	1.68137
$50$	0	0
$51$	7729.35	0.416119
$52$	0	0
$53$	24975.0	1.22128	0.610640	−	0.791908i	$-0.290912\pi$
$53$	24975.0	1.22128	0.610640	+	0.791908i	$0.290912\pi$
$54$	0	0
$55$	−12653.2	−0.564019
$56$	0	0
$57$	−5342.43	−0.217797
$58$	0	0
$59$	−23279.2	−0.870638	−0.435319	−	0.900276i	$-0.643364\pi$
$59$	−23279.2	−0.870638	−0.435319	+	0.900276i	$0.643364\pi$
$60$	0	0
$61$	−15102.7	−0.519673	−0.259836	−	0.965653i	$-0.583669\pi$
$61$	−15102.7	−0.519673	−0.259836	+	0.965653i	$0.583669\pi$
$62$	0	0
$63$	5092.80	0.161661
$64$	0	0
$65$	−21240.2	−0.623556
$66$	0	0
$67$	−59216.5	−1.61159	−0.805797	−	0.592192i	$-0.798263\pi$
$67$	−59216.5	−1.61159	−0.805797	+	0.592192i	$0.798263\pi$
$68$	0	0
$69$	−47556.9	−1.20252
$70$	0	0
$71$	−14643.8	−0.344754	−0.172377	−	0.985031i	$-0.555145\pi$
$71$	−14643.8	−0.344754	−0.172377	+	0.985031i	$0.555145\pi$
$72$	0	0
$73$	−18953.3	−0.416272	−0.208136	−	0.978100i	$-0.566740\pi$
$73$	−18953.3	−0.416272	−0.208136	+	0.978100i	$0.566740\pi$
$74$	0	0
$75$	40850.5	0.838579
$76$	0	0
$77$	−140667.	−2.70374
$78$	0	0
$79$	3628.29	0.0654085	0.0327043	−	0.999465i	$-0.489588\pi$
$79$	3628.29	0.0654085	0.0327043	+	0.999465i	$0.489588\pi$
$80$	0	0
$81$	−52643.9	−0.891529
$82$	0	0
$83$	−22655.2	−0.360971	−0.180485	−	0.983578i	$-0.557767\pi$
$83$	−22655.2	−0.360971	−0.180485	+	0.983578i	$0.557767\pi$
$84$	0	0
$85$	9973.42	0.149726
$86$	0	0
$87$	55136.7	0.780986
$88$	0	0
$89$	65622.3	0.878166	0.439083	−	0.898447i	$-0.355303\pi$
$89$	65622.3	0.878166	0.439083	+	0.898447i	$0.355303\pi$
$90$	0	0
$91$	−236129.	−2.98914
$92$	0	0
$93$	71516.1	0.857426
$94$	0	0
$95$	−6893.50	−0.0783666
$96$	0	0
$97$	−59429.2	−0.641314	−0.320657	−	0.947195i	$-0.603904\pi$
$97$	−59429.2	−0.641314	−0.320657	+	0.947195i	$0.603904\pi$
$98$	0	0
$99$	−15896.5	−0.163009
$100$	0	0
$101$	−24934.1	−0.243215	−0.121608	−	0.992578i	$-0.538805\pi$
$101$	−24934.1	−0.243215	−0.121608	+	0.992578i	$0.538805\pi$
$102$	0	0
$103$	−25045.1	−0.232611	−0.116306	−	0.993213i	$-0.537105\pi$
$103$	−25045.1	−0.232611	−0.116306	+	0.993213i	$0.537105\pi$
$104$	0	0
$105$	−59991.3	−0.531025
$106$	0	0
$107$	27876.0	0.235381	0.117690	−	0.993050i	$-0.462451\pi$
$107$	27876.0	0.235381	0.117690	+	0.993050i	$0.462451\pi$
$108$	0	0
$109$	−109560.	−0.883257	−0.441629	−	0.897198i	$-0.645599\pi$
$109$	−109560.	−0.883257	−0.441629	+	0.897198i	$0.645599\pi$
$110$	0	0
$111$	105857.	0.815475
$112$	0	0
$113$	−93201.6	−0.686637	−0.343319	−	0.939219i	$-0.611551\pi$
$113$	−93201.6	−0.686637	−0.343319	+	0.939219i	$0.611551\pi$
$114$	0	0
$115$	−61364.1	−0.432683
$116$	0	0
$117$	−26684.5	−0.180216
$118$	0	0
$119$	110875.	0.717741
$120$	0	0
$121$	278021.	1.72629
$122$	0	0
$123$	−160020.	−0.953698
$124$	0	0
$125$	112384.	0.643325
$126$	0	0
$127$	−60981.0	−0.335495	−0.167747	−	0.985830i	$-0.553649\pi$
$127$	−60981.0	−0.335495	−0.167747	+	0.985830i	$0.553649\pi$
$128$	0	0
$129$	−154182.	−0.815752
$130$	0	0
$131$	356070.	1.81283	0.906416	−	0.422387i	$-0.138808\pi$
$131$	356070.	1.81283	0.906416	+	0.422387i	$0.138808\pi$
$132$	0	0
$133$	−76635.6	−0.375666
$134$	0	0
$135$	−75450.1	−0.356308
$136$	0	0
$137$	47760.0	0.217402	0.108701	−	0.994075i	$-0.465331\pi$
$137$	47760.0	0.217402	0.108701	+	0.994075i	$0.465331\pi$
$138$	0	0
$139$	−413927.	−1.81713	−0.908566	−	0.417741i	$-0.862822\pi$
$139$	−413927.	−1.81713	−0.908566	+	0.417741i	$0.862822\pi$
$140$	0	0
$141$	177561.	0.752143
$142$	0	0
$143$	737044.	3.01407
$144$	0	0
$145$	71144.6	0.281010
$146$	0	0
$147$	−418201.	−1.59622
$148$	0	0
$149$	−168897.	−0.623242	−0.311621	−	0.950206i	$-0.600872\pi$
$149$	−168897.	−0.623242	−0.311621	+	0.950206i	$0.600872\pi$
$150$	0	0
$151$	−132860.	−0.474191	−0.237096	−	0.971486i	$-0.576195\pi$
$151$	−132860.	−0.474191	−0.237096	+	0.971486i	$0.576195\pi$
$152$	0	0
$153$	12529.8	0.0432729
$154$	0	0
$155$	92279.5	0.308515
$156$	0	0
$157$	−129994.	−0.420894	−0.210447	−	0.977605i	$-0.567492\pi$
$157$	−129994.	−0.420894	−0.210447	+	0.977605i	$0.567492\pi$
$158$	0	0
$159$	−369604.	−1.15943
$160$	0	0
$161$	−682190.	−2.07415
$162$	0	0
$163$	542110.	1.59815	0.799076	−	0.601230i	$-0.205322\pi$
$163$	542110.	1.59815	0.799076	+	0.601230i	$0.205322\pi$
$164$	0	0
$165$	187255.	0.535454
$166$	0	0
$167$	−525577.	−1.45829	−0.729147	−	0.684357i	$-0.760083\pi$
$167$	−525577.	−1.45829	−0.729147	+	0.684357i	$0.760083\pi$
$168$	0	0
$169$	865940.	2.33223
$170$	0	0
$171$	−8660.44	−0.0226490
$172$	0	0
$173$	650536.	1.65256	0.826278	−	0.563263i	$-0.190454\pi$
$173$	650536.	1.65256	0.826278	+	0.563263i	$0.190454\pi$
$174$	0	0
$175$	585988.	1.44642
$176$	0	0
$177$	344508.	0.826545
$178$	0	0
$179$	−647733.	−1.51100	−0.755498	−	0.655151i	$-0.772605\pi$
$179$	−647733.	−1.51100	−0.755498	+	0.655151i	$0.772605\pi$
$180$	0	0
$181$	−442268.	−1.00343	−0.501717	−	0.865032i	$-0.667298\pi$
$181$	−442268.	−1.00343	−0.501717	+	0.865032i	$0.667298\pi$
$182$	0	0
$183$	223505.	0.493354
$184$	0	0
$185$	136590.	0.293420
$186$	0	0
$187$	−346082.	−0.723727
$188$	0	0
$189$	−838785.	−1.70803
$190$	0	0
$191$	186520.	0.369949	0.184974	−	0.982743i	$-0.440780\pi$
$191$	186520.	0.369949	0.184974	+	0.982743i	$0.440780\pi$
$192$	0	0
$193$	−623123.	−1.20415	−0.602075	−	0.798440i	$-0.705659\pi$
$193$	−623123.	−1.20415	−0.602075	+	0.798440i	$0.705659\pi$
$194$	0	0
$195$	314333.	0.591976
$196$	0	0
$197$	63660.0	0.116869	0.0584347	−	0.998291i	$-0.481389\pi$
$197$	63660.0	0.116869	0.0584347	+	0.998291i	$0.481389\pi$
$198$	0	0
$199$	−531810.	−0.951971	−0.475986	−	0.879453i	$-0.657909\pi$
$199$	−531810.	−0.951971	−0.475986	+	0.879453i	$0.657909\pi$
$200$	0	0
$201$	876344.	1.52997
$202$	0	0
$203$	790920.	1.34708
$204$	0	0
$205$	−206478.	−0.343155
$206$	0	0
$207$	−77092.9	−0.125051
$208$	0	0
$209$	239208.	0.378799
$210$	0	0
$211$	950256.	1.46938	0.734691	−	0.678402i	$-0.237327\pi$
$211$	950256.	1.46938	0.734691	+	0.678402i	$0.237327\pi$
$212$	0	0
$213$	216714.	0.327294
$214$	0	0
$215$	−198945.	−0.293520
$216$	0	0
$217$	1.02588e6	1.47893
$218$	0	0
$219$	280489.	0.395190
$220$	0	0
$221$	−580948.	−0.800122
$222$	0	0
$223$	412023.	0.554829	0.277415	−	0.960750i	$-0.410522\pi$
$223$	412023.	0.554829	0.277415	+	0.960750i	$0.410522\pi$
$224$	0	0
$225$	66221.4	0.0872051
$226$	0	0
$227$	−216720.	−0.279148	−0.139574	−	0.990212i	$-0.544573\pi$
$227$	−216720.	−0.279148	−0.139574	+	0.990212i	$0.544573\pi$
$228$	0	0
$229$	−467558.	−0.589178	−0.294589	−	0.955624i	$-0.595183\pi$
$229$	−467558.	−0.589178	−0.294589	+	0.955624i	$0.595183\pi$
$230$	0	0
$231$	2.08172e6	2.56681
$232$	0	0
$233$	−964515.	−1.16391	−0.581955	−	0.813221i	$-0.697712\pi$
$233$	−964515.	−1.16391	−0.581955	+	0.813221i	$0.697712\pi$
$234$	0	0
$235$	229113.	0.270632
$236$	0	0
$237$	−53695.0	−0.0620959
$238$	0	0
$239$	−1.66583e6	−1.88641	−0.943206	−	0.332207i	$-0.892207\pi$
$239$	−1.66583e6	−1.88641	−0.943206	+	0.332207i	$0.892207\pi$
$240$	0	0
$241$	−395797.	−0.438965	−0.219482	−	0.975616i	$-0.570437\pi$
$241$	−395797.	−0.438965	−0.219482	+	0.975616i	$0.570437\pi$
$242$	0	0
$243$	−181062.	−0.196703
$244$	0	0
$245$	−539618.	−0.574343
$246$	0	0
$247$	401544.	0.418784
$248$	0	0
$249$	335273.	0.342690
$250$	0	0
$251$	52655.6	0.0527546	0.0263773	−	0.999652i	$-0.491603\pi$
$251$	52655.6	0.0527546	0.0263773	+	0.999652i	$0.491603\pi$
$252$	0	0
$253$	2.12936e6	2.09145
$254$	0	0
$255$	−147596.	−0.142143
$256$	0	0
$257$	369622.	0.349080	0.174540	−	0.984650i	$-0.444156\pi$
$257$	369622.	0.349080	0.174540	+	0.984650i	$0.444156\pi$
$258$	0	0
$259$	1.51848e6	1.40657
$260$	0	0
$261$	89380.3	0.0812158
$262$	0	0
$263$	266855.	0.237895	0.118948	−	0.992901i	$-0.462048\pi$
$263$	266855.	0.237895	0.118948	+	0.992901i	$0.462048\pi$
$264$	0	0
$265$	−476912.	−0.417180
$266$	0	0
$267$	−971144.	−0.833691
$268$	0	0
$269$	−283145.	−0.238577	−0.119289	−	0.992860i	$-0.538061\pi$
$269$	−283145.	−0.238577	−0.119289	+	0.992860i	$0.538061\pi$
$270$	0	0
$271$	−1.26094e6	−1.04296	−0.521482	−	0.853262i	$-0.674621\pi$
$271$	−1.26094e6	−1.04296	−0.521482	+	0.853262i	$0.674621\pi$
$272$	0	0
$273$	3.49447e6	2.83775
$274$	0	0
$275$	−1.82908e6	−1.45848
$276$	0	0
$277$	1.86541e6	1.46074	0.730372	−	0.683050i	$-0.239347\pi$
$277$	1.86541e6	1.46074	0.730372	+	0.683050i	$0.239347\pi$
$278$	0	0
$279$	115932.	0.0891650
$280$	0	0
$281$	−432083.	−0.326439	−0.163219	−	0.986590i	$-0.552188\pi$
$281$	−432083.	−0.326439	−0.163219	+	0.986590i	$0.552188\pi$
$282$	0	0
$283$	1.32987e6	0.987056	0.493528	−	0.869730i	$-0.335707\pi$
$283$	1.32987e6	0.987056	0.493528	+	0.869730i	$0.335707\pi$
$284$	0	0
$285$	102017.	0.0743977
$286$	0	0
$287$	−2.29544e6	−1.64498
$288$	0	0
$289$	−1.14707e6	−0.807878
$290$	0	0
$291$	879492.	0.608835
$292$	0	0
$293$	−701589.	−0.477435	−0.238717	−	0.971089i	$-0.576727\pi$
$293$	−701589.	−0.477435	−0.238717	+	0.971089i	$0.576727\pi$
$294$	0	0
$295$	444529.	0.297403
$296$	0	0
$297$	2.61815e6	1.72228
$298$	0	0
$299$	3.57443e6	2.31222
$300$	0	0
$301$	−2.21169e6	−1.40705
$302$	0	0
$303$	368999.	0.230897
$304$	0	0
$305$	288395.	0.177516
$306$	0	0
$307$	667303.	0.404089	0.202044	−	0.979376i	$-0.435241\pi$
$307$	667303.	0.404089	0.202044	+	0.979376i	$0.435241\pi$
$308$	0	0
$309$	370643.	0.220831
$310$	0	0
$311$	−242281.	−0.142042	−0.0710211	−	0.997475i	$-0.522626\pi$
$311$	−242281.	−0.142042	−0.0710211	+	0.997475i	$0.522626\pi$
$312$	0	0
$313$	2.02771e6	1.16989	0.584945	−	0.811073i	$-0.301116\pi$
$313$	2.02771e6	1.16989	0.584945	+	0.811073i	$0.301116\pi$
$314$	0	0
$315$	−97249.9	−0.0552221
$316$	0	0
$317$	−983497.	−0.549699	−0.274849	−	0.961487i	$-0.588628\pi$
$317$	−983497.	−0.549699	−0.274849	+	0.961487i	$0.588628\pi$
$318$	0	0
$319$	−2.46875e6	−1.35831
$320$	0	0
$321$	−412536.	−0.223460
$322$	0	0
$323$	−188547.	−0.100557
$324$	0	0
$325$	−3.07037e6	−1.61244
$326$	0	0
$327$	1.62138e6	0.838525
$328$	0	0
$329$	2.54706e6	1.29733
$330$	0	0
$331$	−69223.1	−0.0347281	−0.0173640	−	0.999849i	$-0.505527\pi$
$331$	−69223.1	−0.0347281	−0.0173640	+	0.999849i	$0.505527\pi$
$332$	0	0
$333$	171601.	0.0848024
$334$	0	0
$335$	1.13077e6	0.550508
$336$	0	0
$337$	567994.	0.272439	0.136219	−	0.990679i	$-0.456505\pi$
$337$	567994.	0.272439	0.136219	+	0.990679i	$0.456505\pi$
$338$	0	0
$339$	1.37929e6	0.651862
$340$	0	0
$341$	−3.20214e6	−1.49126
$342$	0	0
$343$	−2.43107e6	−1.11574
$344$	0	0
$345$	908126.	0.410770
$346$	0	0
$347$	46773.6	0.0208534	0.0104267	−	0.999946i	$-0.496681\pi$
$347$	46773.6	0.0208534	0.0104267	+	0.999946i	$0.496681\pi$
$348$	0	0
$349$	−3.10376e6	−1.36403	−0.682016	−	0.731337i	$-0.738897\pi$
$349$	−3.10376e6	−1.36403	−0.682016	+	0.731337i	$0.738897\pi$
$350$	0	0
$351$	4.39494e6	1.90408
$352$	0	0
$353$	−1.92152e6	−0.820743	−0.410372	−	0.911918i	$-0.634601\pi$
$353$	−1.92152e6	−0.820743	−0.410372	+	0.911918i	$0.634601\pi$
$354$	0	0
$355$	279633.	0.117765
$356$	0	0
$357$	−1.64084e6	−0.681391
$358$	0	0
$359$	−1.88950e6	−0.773766	−0.386883	−	0.922129i	$-0.626448\pi$
$359$	−1.88950e6	−0.773766	−0.386883	+	0.922129i	$0.626448\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	0	0
$363$	−4.11442e6	−1.63886
$364$	0	0
$365$	361924.	0.142195
$366$	0	0
$367$	−1.15810e6	−0.448831	−0.224415	−	0.974494i	$-0.572047\pi$
$367$	−1.15810e6	−0.448831	−0.224415	+	0.974494i	$0.572047\pi$
$368$	0	0
$369$	−259403.	−0.0991765
$370$	0	0
$371$	−5.30187e6	−1.99983
$372$	0	0
$373$	−5.01675e6	−1.86703	−0.933513	−	0.358543i	$-0.883274\pi$
$373$	−5.01675e6	−1.86703	−0.933513	+	0.358543i	$0.883274\pi$
$374$	0	0
$375$	−1.66317e6	−0.610744
$376$	0	0
$377$	−4.14415e6	−1.50169
$378$	0	0
$379$	−3.83992e6	−1.37317	−0.686584	−	0.727050i	$-0.740891\pi$
$379$	−3.83992e6	−1.37317	−0.686584	+	0.727050i	$0.740891\pi$
$380$	0	0
$381$	902457.	0.318503
$382$	0	0
$383$	4.41598e6	1.53826	0.769131	−	0.639091i	$-0.220689\pi$
$383$	4.41598e6	1.53826	0.769131	+	0.639091i	$0.220689\pi$
$384$	0	0
$385$	2.68611e6	0.923575
$386$	0	0
$387$	−249939.	−0.0848313
$388$	0	0
$389$	−1.32552e6	−0.444133	−0.222067	−	0.975032i	$-0.571280\pi$
$389$	−1.32552e6	−0.444133	−0.222067	+	0.975032i	$0.571280\pi$
$390$	0	0
$391$	−1.67839e6	−0.555202
$392$	0	0
$393$	−5.26948e6	−1.72102
$394$	0	0
$395$	−69284.3	−0.0223430
$396$	0	0
$397$	244751.	0.0779379	0.0389689	−	0.999240i	$-0.487593\pi$
$397$	244751.	0.0779379	0.0389689	+	0.999240i	$0.487593\pi$
$398$	0	0
$399$	1.13413e6	0.356640
$400$	0	0
$401$	−686533.	−0.213206	−0.106603	−	0.994302i	$-0.533997\pi$
$401$	−686533.	−0.213206	−0.106603	+	0.994302i	$0.533997\pi$
$402$	0	0
$403$	−5.37524e6	−1.64868
$404$	0	0
$405$	1.00527e6	0.304539
$406$	0	0
$407$	−4.73973e6	−1.41830
$408$	0	0
$409$	2.48415e6	0.734293	0.367147	−	0.930163i	$-0.380335\pi$
$409$	2.48415e6	0.734293	0.367147	+	0.930163i	$0.380335\pi$
$410$	0	0
$411$	−706799.	−0.206391
$412$	0	0
$413$	4.94187e6	1.42566
$414$	0	0
$415$	432614.	0.123305
$416$	0	0
$417$	6.12570e6	1.72510
$418$	0	0
$419$	−116076.	−0.0323004	−0.0161502	−	0.999870i	$-0.505141\pi$
$419$	−116076.	−0.0323004	−0.0161502	+	0.999870i	$0.505141\pi$
$420$	0	0
$421$	192169.	0.0528419	0.0264209	−	0.999651i	$-0.491589\pi$
$421$	192169.	0.0528419	0.0264209	+	0.999651i	$0.491589\pi$
$422$	0	0
$423$	287839.	0.0782165
$424$	0	0
$425$	1.44171e6	0.387173
$426$	0	0
$427$	3.20611e6	0.850959
$428$	0	0
$429$	−1.09075e7	−2.86142
$430$	0	0
$431$	−3.25385e6	−0.843731	−0.421865	−	0.906659i	$-0.638624\pi$
$431$	−3.25385e6	−0.843731	−0.421865	+	0.906659i	$0.638624\pi$
$432$	0	0
$433$	−247577.	−0.0634585	−0.0317292	−	0.999497i	$-0.510101\pi$
$433$	−247577.	−0.0634585	−0.0317292	+	0.999497i	$0.510101\pi$
$434$	0	0
$435$	−1.05287e6	−0.266778
$436$	0	0
$437$	1.16008e6	0.290593
$438$	0	0
$439$	3.55674e6	0.880828	0.440414	−	0.897795i	$-0.354832\pi$
$439$	3.55674e6	0.880828	0.440414	+	0.897795i	$0.354832\pi$
$440$	0	0
$441$	−677932.	−0.165993
$442$	0	0
$443$	2.71438e6	0.657145	0.328573	−	0.944479i	$-0.393432\pi$
$443$	2.71438e6	0.657145	0.328573	+	0.944479i	$0.393432\pi$
$444$	0	0
$445$	−1.25310e6	−0.299975
$446$	0	0
$447$	2.49951e6	0.591678
$448$	0	0
$449$	−2.35366e6	−0.550969	−0.275484	−	0.961306i	$-0.588838\pi$
$449$	−2.35366e6	−0.550969	−0.275484	+	0.961306i	$0.588838\pi$
$450$	0	0
$451$	7.16489e6	1.65870
$452$	0	0
$453$	1.96620e6	0.450176
$454$	0	0
$455$	4.50902e6	1.02107
$456$	0	0
$457$	7.61497e6	1.70560	0.852802	−	0.522235i	$-0.174902\pi$
$457$	7.61497e6	1.70560	0.852802	+	0.522235i	$0.174902\pi$
$458$	0	0
$459$	−2.06366e6	−0.457201
$460$	0	0
$461$	−7.99940e6	−1.75309	−0.876546	−	0.481318i	$-0.840158\pi$
$461$	−7.99940e6	−1.75309	−0.876546	+	0.481318i	$0.840158\pi$
$462$	0	0
$463$	−2.55307e6	−0.553491	−0.276745	−	0.960943i	$-0.589256\pi$
$463$	−2.55307e6	−0.553491	−0.276745	+	0.960943i	$0.589256\pi$
$464$	0	0
$465$	−1.36564e6	−0.292890
$466$	0	0
$467$	6.01000e6	1.27521	0.637605	−	0.770363i	$-0.279925\pi$
$467$	6.01000e6	1.27521	0.637605	+	0.770363i	$0.279925\pi$
$468$	0	0
$469$	1.25709e7	2.63897
$470$	0	0
$471$	1.92377e6	0.399578
$472$	0	0
$473$	6.90349e6	1.41878
$474$	0	0
$475$	−996490.	−0.202646
$476$	0	0
$477$	−599153.	−0.120571
$478$	0	0
$479$	−9.37118e6	−1.86619	−0.933094	−	0.359632i	$-0.882902\pi$
$479$	−9.37118e6	−1.86619	−0.933094	+	0.359632i	$0.882902\pi$
$480$	0	0
$481$	−7.95631e6	−1.56801
$482$	0	0
$483$	1.00957e7	1.96911
$484$	0	0
$485$	1.13484e6	0.219068
$486$	0	0
$487$	−909566.	−0.173785	−0.0868924	−	0.996218i	$-0.527694\pi$
$487$	−909566.	−0.173785	−0.0868924	+	0.996218i	$0.527694\pi$
$488$	0	0
$489$	−8.02267e6	−1.51721
$490$	0	0
$491$	−5.72830e6	−1.07231	−0.536157	−	0.844118i	$-0.680124\pi$
$491$	−5.72830e6	−1.07231	−0.536157	+	0.844118i	$0.680124\pi$
$492$	0	0
$493$	1.94590e6	0.360581
$494$	0	0
$495$	303552.	0.0556827
$496$	0	0
$497$	3.10870e6	0.564531
$498$	0	0
$499$	−2.66518e6	−0.479154	−0.239577	−	0.970877i	$-0.577009\pi$
$499$	−2.66518e6	−0.479154	−0.239577	+	0.970877i	$0.577009\pi$
$500$	0	0
$501$	7.77801e6	1.38444
$502$	0	0
$503$	−4.39821e6	−0.775098	−0.387549	−	0.921849i	$-0.626678\pi$
$503$	−4.39821e6	−0.775098	−0.387549	+	0.921849i	$0.626678\pi$
$504$	0	0
$505$	476131.	0.0830803
$506$	0	0
$507$	−1.28150e7	−2.21411
$508$	0	0
$509$	−4.66199e6	−0.797585	−0.398792	−	0.917041i	$-0.630571\pi$
$509$	−4.66199e6	−0.797585	−0.398792	+	0.917041i	$0.630571\pi$
$510$	0	0
$511$	4.02354e6	0.681641
$512$	0	0
$513$	1.42638e6	0.239299
$514$	0	0
$515$	478251.	0.0794581
$516$	0	0
$517$	−7.95031e6	−1.30815
$518$	0	0
$519$	−9.62727e6	−1.56886
$520$	0	0
$521$	1.26923e6	0.204855	0.102427	−	0.994740i	$-0.467339\pi$
$521$	1.26923e6	0.204855	0.102427	+	0.994740i	$0.467339\pi$
$522$	0	0
$523$	−5.54928e6	−0.887119	−0.443560	−	0.896245i	$-0.646285\pi$
$523$	−5.54928e6	−0.887119	−0.443560	+	0.896245i	$0.646285\pi$
$524$	0	0
$525$	−8.67203e6	−1.37316
$526$	0	0
$527$	2.52397e6	0.395874
$528$	0	0
$529$	3.89040e6	0.604442
$530$	0	0
$531$	558471.	0.0859536
$532$	0	0
$533$	1.20273e7	1.83379
$534$	0	0
$535$	−532308.	−0.0804041
$536$	0	0
$537$	9.58579e6	1.43447
$538$	0	0
$539$	1.87250e7	2.77619
$540$	0	0
$541$	−9.66859e6	−1.42027	−0.710134	−	0.704067i	$-0.751366\pi$
$541$	−9.66859e6	−1.42027	−0.710134	+	0.704067i	$0.751366\pi$
$542$	0	0
$543$	6.54511e6	0.952615
$544$	0	0
$545$	2.09212e6	0.301714
$546$	0	0
$547$	3.56127e6	0.508905	0.254453	−	0.967085i	$-0.418105\pi$
$547$	3.56127e6	0.508905	0.254453	+	0.967085i	$0.418105\pi$
$548$	0	0
$549$	362316.	0.0513046
$550$	0	0
$551$	−1.34498e6	−0.188729
$552$	0	0
$553$	−770239.	−0.107106
$554$	0	0
$555$	−2.02139e6	−0.278560
$556$	0	0
$557$	−1.12732e7	−1.53960	−0.769801	−	0.638284i	$-0.779644\pi$
$557$	−1.12732e7	−1.53960	−0.769801	+	0.638284i	$0.779644\pi$
$558$	0	0
$559$	1.15885e7	1.56855
$560$	0	0
$561$	5.12166e6	0.687074
$562$	0	0
$563$	2.63453e6	0.350293	0.175147	−	0.984542i	$-0.443960\pi$
$563$	2.63453e6	0.350293	0.175147	+	0.984542i	$0.443960\pi$
$564$	0	0
$565$	1.77974e6	0.234550
$566$	0	0
$567$	1.11756e7	1.45987
$568$	0	0
$569$	2.77738e6	0.359629	0.179814	−	0.983701i	$-0.442450\pi$
$569$	2.77738e6	0.359629	0.179814	+	0.983701i	$0.442450\pi$
$570$	0	0
$571$	−3.91874e6	−0.502986	−0.251493	−	0.967859i	$-0.580922\pi$
$571$	−3.91874e6	−0.502986	−0.251493	+	0.967859i	$0.580922\pi$
$572$	0	0
$573$	−2.76030e6	−0.351213
$574$	0	0
$575$	−8.87048e6	−1.11886
$576$	0	0
$577$	938331.	0.117332	0.0586660	−	0.998278i	$-0.481315\pi$
$577$	938331.	0.117332	0.0586660	+	0.998278i	$0.481315\pi$
$578$	0	0
$579$	9.22159e6	1.14317
$580$	0	0
$581$	4.80940e6	0.591086
$582$	0	0
$583$	1.65490e7	2.01652
$584$	0	0
$585$	509555.	0.0615604
$586$	0	0
$587$	−1.21860e7	−1.45971	−0.729856	−	0.683601i	$-0.760413\pi$
$587$	−1.21860e7	−1.45971	−0.729856	+	0.683601i	$0.760413\pi$
$588$	0	0
$589$	−1.74453e6	−0.207201
$590$	0	0
$591$	−942103.	−0.110951
$592$	0	0
$593$	−8.86284e6	−1.03499	−0.517495	−	0.855686i	$-0.673135\pi$
$593$	−8.86284e6	−1.03499	−0.517495	+	0.855686i	$0.673135\pi$
$594$	0	0
$595$	−2.11723e6	−0.245174
$596$	0	0
$597$	7.87025e6	0.903759
$598$	0	0
$599$	6.26482e6	0.713414	0.356707	−	0.934216i	$-0.383899\pi$
$599$	6.26482e6	0.713414	0.356707	+	0.934216i	$0.383899\pi$
$600$	0	0
$601$	9.59570e6	1.08365	0.541827	−	0.840490i	$-0.317733\pi$
$601$	9.59570e6	1.08365	0.541827	+	0.840490i	$0.317733\pi$
$602$	0	0
$603$	1.42061e6	0.159104
$604$	0	0
$605$	−5.30897e6	−0.589687
$606$	0	0
$607$	1.06282e7	1.17082	0.585408	−	0.810739i	$-0.300934\pi$
$607$	1.06282e7	1.17082	0.585408	+	0.810739i	$0.300934\pi$
$608$	0	0
$609$	−1.17048e7	−1.27886
$610$	0	0
$611$	−1.33457e7	−1.44624
$612$	0	0
$613$	−1.43183e7	−1.53900	−0.769502	−	0.638645i	$-0.779495\pi$
$613$	−1.43183e7	−1.53900	−0.769502	+	0.638645i	$0.779495\pi$
$614$	0	0
$615$	3.05567e6	0.325776
$616$	0	0
$617$	1.71005e7	1.80841	0.904204	−	0.427100i	$-0.140465\pi$
$617$	1.71005e7	1.80841	0.904204	+	0.427100i	$0.140465\pi$
$618$	0	0
$619$	1.54652e7	1.62229	0.811146	−	0.584844i	$-0.198844\pi$
$619$	1.54652e7	1.62229	0.811146	+	0.584844i	$0.198844\pi$
$620$	0	0
$621$	1.26972e7	1.32123
$622$	0	0
$623$	−1.39308e7	−1.43799
$624$	0	0
$625$	6.48008e6	0.663560
$626$	0	0
$627$	−3.54003e6	−0.359615
$628$	0	0
$629$	3.73592e6	0.376505
$630$	0	0
$631$	−1.73001e6	−0.172972	−0.0864860	−	0.996253i	$-0.527564\pi$
$631$	−1.73001e6	−0.172972	−0.0864860	+	0.996253i	$0.527564\pi$
$632$	0	0
$633$	−1.40628e7	−1.39496
$634$	0	0
$635$	1.16447e6	0.114602
$636$	0	0
$637$	3.14325e7	3.06924
$638$	0	0
$639$	351308.	0.0340358
$640$	0	0
$641$	−8.59864e6	−0.826579	−0.413290	−	0.910600i	$-0.635620\pi$
$641$	−8.59864e6	−0.826579	−0.413290	+	0.910600i	$0.635620\pi$
$642$	0	0
$643$	−1.02223e7	−0.975039	−0.487519	−	0.873112i	$-0.662098\pi$
$643$	−1.02223e7	−0.975039	−0.487519	+	0.873112i	$0.662098\pi$
$644$	0	0
$645$	2.94419e6	0.278655
$646$	0	0
$647$	2.86656e6	0.269215	0.134608	−	0.990899i	$-0.457023\pi$
$647$	2.86656e6	0.269215	0.134608	+	0.990899i	$0.457023\pi$
$648$	0	0
$649$	−1.54254e7	−1.43755
$650$	0	0
$651$	−1.51820e7	−1.40403
$652$	0	0
$653$	−1.59807e7	−1.46660	−0.733300	−	0.679905i	$-0.762021\pi$
$653$	−1.59807e7	−1.46660	−0.733300	+	0.679905i	$0.762021\pi$
$654$	0	0
$655$	−6.79937e6	−0.619249
$656$	0	0
$657$	454692.	0.0410964
$658$	0	0
$659$	−3.24758e6	−0.291304	−0.145652	−	0.989336i	$-0.546528\pi$
$659$	−3.24758e6	−0.291304	−0.145652	+	0.989336i	$0.546528\pi$
$660$	0	0
$661$	1.91717e7	1.70670	0.853350	−	0.521339i	$-0.174567\pi$
$661$	1.91717e7	1.70670	0.853350	+	0.521339i	$0.174567\pi$
$662$	0	0
$663$	8.59744e6	0.759600
$664$	0	0
$665$	1.46340e6	0.128324
$666$	0	0
$667$	−1.19727e7	−1.04202
$668$	0	0
$669$	−6.09752e6	−0.526730
$670$	0	0
$671$	−1.00074e7	−0.858057
$672$	0	0
$673$	8.11070e6	0.690273	0.345136	−	0.938553i	$-0.387833\pi$
$673$	8.11070e6	0.690273	0.345136	+	0.938553i	$0.387833\pi$
$674$	0	0
$675$	−1.09067e7	−0.921368
$676$	0	0
$677$	4.53293e6	0.380109	0.190054	−	0.981774i	$-0.439134\pi$
$677$	4.53293e6	0.380109	0.190054	+	0.981774i	$0.439134\pi$
$678$	0	0
$679$	1.26161e7	1.05014
$680$	0	0
$681$	3.20724e6	0.265011
$682$	0	0
$683$	2.12901e7	1.74633	0.873163	−	0.487427i	$-0.162065\pi$
$683$	2.12901e7	1.74633	0.873163	+	0.487427i	$0.162065\pi$
$684$	0	0
$685$	−912004.	−0.0742627
$686$	0	0
$687$	6.91938e6	0.559339
$688$	0	0
$689$	2.77799e7	2.22937
$690$	0	0
$691$	−4.77800e6	−0.380672	−0.190336	−	0.981719i	$-0.560958\pi$
$691$	−4.77800e6	−0.380672	−0.190336	+	0.981719i	$0.560958\pi$
$692$	0	0
$693$	3.37461e6	0.266926
$694$	0	0
$695$	7.90417e6	0.620718
$696$	0	0
$697$	−5.64746e6	−0.440323
$698$	0	0
$699$	1.42738e7	1.10496
$700$	0	0
$701$	−1.17811e7	−0.905509	−0.452754	−	0.891635i	$-0.649559\pi$
$701$	−1.17811e7	−0.905509	−0.452754	+	0.891635i	$0.649559\pi$
$702$	0	0
$703$	−2.58222e6	−0.197063
$704$	0	0
$705$	−3.39064e6	−0.256926
$706$	0	0
$707$	5.29319e6	0.398262
$708$	0	0
$709$	−4.51131e6	−0.337044	−0.168522	−	0.985698i	$-0.553899\pi$
$709$	−4.51131e6	−0.337044	−0.168522	+	0.985698i	$0.553899\pi$
$710$	0	0
$711$	−87043.2	−0.00645745
$712$	0	0
$713$	−1.55294e7	−1.14401
$714$	0	0
$715$	−1.40743e7	−1.02958
$716$	0	0
$717$	2.46526e7	1.79088
$718$	0	0
$719$	−8.00238e6	−0.577294	−0.288647	−	0.957436i	$-0.593205\pi$
$719$	−8.00238e6	−0.577294	−0.288647	+	0.957436i	$0.593205\pi$
$720$	0	0
$721$	5.31676e6	0.380898
$722$	0	0
$723$	5.85739e6	0.416733
$724$	0	0
$725$	1.02843e7	0.726658
$726$	0	0
$727$	2.21846e7	1.55674	0.778370	−	0.627806i	$-0.216047\pi$
$727$	2.21846e7	1.55674	0.778370	+	0.627806i	$0.216047\pi$
$728$	0	0
$729$	1.54720e7	1.07827
$730$	0	0
$731$	−5.44142e6	−0.376633
$732$	0	0
$733$	−3.50846e6	−0.241188	−0.120594	−	0.992702i	$-0.538480\pi$
$733$	−3.50846e6	−0.241188	−0.120594	+	0.992702i	$0.538480\pi$
$734$	0	0
$735$	7.98580e6	0.545255
$736$	0	0
$737$	−3.92383e7	−2.66098
$738$	0	0
$739$	1.68883e7	1.13756	0.568781	−	0.822489i	$-0.307415\pi$
$739$	1.68883e7	1.13756	0.568781	+	0.822489i	$0.307415\pi$
$740$	0	0
$741$	−5.94244e6	−0.397575
$742$	0	0
$743$	−2.72139e7	−1.80850	−0.904250	−	0.427003i	$-0.859569\pi$
$743$	−2.72139e7	−1.80850	−0.904250	+	0.427003i	$0.859569\pi$
$744$	0	0
$745$	3.22519e6	0.212895
$746$	0	0
$747$	543501.	0.0356368
$748$	0	0
$749$	−5.91771e6	−0.385433
$750$	0	0
$751$	5.12279e6	0.331442	0.165721	−	0.986173i	$-0.447005\pi$
$751$	5.12279e6	0.331442	0.165721	+	0.986173i	$0.447005\pi$
$752$	0	0
$753$	−779250.	−0.0500829
$754$	0	0
$755$	2.53705e6	0.161980
$756$	0	0
$757$	−1.84692e7	−1.17141	−0.585706	−	0.810524i	$-0.699182\pi$
$757$	−1.84692e7	−1.17141	−0.585706	+	0.810524i	$0.699182\pi$
$758$	0	0
$759$	−3.15124e7	−1.98553
$760$	0	0
$761$	1.50050e6	0.0939237	0.0469619	−	0.998897i	$-0.485046\pi$
$761$	1.50050e6	0.0939237	0.0469619	+	0.998897i	$0.485046\pi$
$762$	0	0
$763$	2.32582e7	1.44632
$764$	0	0
$765$	−239264.	−0.0147817
$766$	0	0
$767$	−2.58937e7	−1.58930
$768$	0	0
$769$	−7.45292e6	−0.454475	−0.227238	−	0.973839i	$-0.572969\pi$
$769$	−7.45292e6	−0.454475	−0.227238	+	0.973839i	$0.572969\pi$
$770$	0	0
$771$	−5.47003e6	−0.331401
$772$	0	0
$773$	8.78766e6	0.528962	0.264481	−	0.964391i	$-0.414799\pi$
$773$	8.78766e6	0.528962	0.264481	+	0.964391i	$0.414799\pi$
$774$	0	0
$775$	1.33394e7	0.797781
$776$	0	0
$777$	−2.24720e7	−1.33533
$778$	0	0
$779$	3.90345e6	0.230465
$780$	0	0
$781$	−9.70338e6	−0.569240
$782$	0	0
$783$	−1.47210e7	−0.858088
$784$	0	0
$785$	2.48230e6	0.143774
$786$	0	0
$787$	2.19416e7	1.26279	0.631394	−	0.775462i	$-0.282483\pi$
$787$	2.19416e7	1.26279	0.631394	+	0.775462i	$0.282483\pi$
$788$	0	0
$789$	−3.94918e6	−0.225847
$790$	0	0
$791$	1.97855e7	1.12436
$792$	0	0
$793$	−1.67989e7	−0.948631
$794$	0	0
$795$	7.05781e6	0.396052
$796$	0	0
$797$	7.18246e6	0.400523	0.200261	−	0.979743i	$-0.435821\pi$
$797$	7.18246e6	0.400523	0.200261	+	0.979743i	$0.435821\pi$
$798$	0	0
$799$	6.26654e6	0.347265
$800$	0	0
$801$	−1.57429e6	−0.0866968
$802$	0	0
$803$	−1.25589e7	−0.687327
$804$	0	0
$805$	1.30268e7	0.708514
$806$	0	0
$807$	4.19026e6	0.226494
$808$	0	0
$809$	1.72882e7	0.928705	0.464353	−	0.885650i	$-0.346287\pi$
$809$	1.72882e7	0.928705	0.464353	+	0.885650i	$0.346287\pi$
$810$	0	0
$811$	2.44923e7	1.30761	0.653804	−	0.756664i	$-0.273172\pi$
$811$	2.44923e7	1.30761	0.653804	+	0.756664i	$0.273172\pi$
$812$	0	0
$813$	1.86606e7	0.990144
$814$	0	0
$815$	−1.03519e7	−0.545916
$816$	0	0
$817$	3.76104e6	0.197130
$818$	0	0
$819$	5.66477e6	0.295102
$820$	0	0
$821$	−2.92010e7	−1.51196	−0.755980	−	0.654595i	$-0.772839\pi$
$821$	−2.92010e7	−1.51196	−0.755980	+	0.654595i	$0.772839\pi$
$822$	0	0
$823$	−2.10175e7	−1.08164	−0.540818	−	0.841140i	$-0.681885\pi$
$823$	−2.10175e7	−1.08164	−0.540818	+	0.841140i	$0.681885\pi$
$824$	0	0
$825$	2.70686e7	1.38462
$826$	0	0
$827$	3.22352e7	1.63895	0.819477	−	0.573112i	$-0.194264\pi$
$827$	3.22352e7	1.63895	0.819477	+	0.573112i	$0.194264\pi$
$828$	0	0
$829$	2.14967e7	1.08639	0.543193	−	0.839608i	$-0.317215\pi$
$829$	2.14967e7	1.08639	0.543193	+	0.839608i	$0.317215\pi$
$830$	0	0
$831$	−2.76061e7	−1.38676
$832$	0	0
$833$	−1.47593e7	−0.736975
$834$	0	0
$835$	1.00362e7	0.498142
$836$	0	0
$837$	−1.90941e7	−0.942075
$838$	0	0
$839$	2.97457e6	0.145888	0.0729439	−	0.997336i	$-0.476761\pi$
$839$	2.97457e6	0.145888	0.0729439	+	0.997336i	$0.476761\pi$
$840$	0	0
$841$	−6.63022e6	−0.323249
$842$	0	0
$843$	6.39439e6	0.309906
$844$	0	0
$845$	−1.65356e7	−0.796671
$846$	0	0
$847$	−5.90202e7	−2.82678
$848$	0	0
$849$	−1.96807e7	−0.937067
$850$	0	0
$851$	−2.29862e7	−1.08804
$852$	0	0
$853$	2.16578e7	1.01916	0.509579	−	0.860424i	$-0.329801\pi$
$853$	2.16578e7	1.01916	0.509579	+	0.860424i	$0.329801\pi$
$854$	0	0
$855$	165376.	0.00773673
$856$	0	0
$857$	−3.76821e7	−1.75260	−0.876301	−	0.481763i	$-0.839997\pi$
$857$	−3.76821e7	−1.75260	−0.876301	+	0.481763i	$0.839997\pi$
$858$	0	0
$859$	−3.33168e7	−1.54057	−0.770283	−	0.637702i	$-0.779885\pi$
$859$	−3.33168e7	−1.54057	−0.770283	+	0.637702i	$0.779885\pi$
$860$	0	0
$861$	3.39701e7	1.56167
$862$	0	0
$863$	−2.83574e7	−1.29610	−0.648051	−	0.761597i	$-0.724415\pi$
$863$	−2.83574e7	−1.29610	−0.648051	+	0.761597i	$0.724415\pi$
$864$	0	0
$865$	−1.24224e7	−0.564500
$866$	0	0
$867$	1.69755e7	0.766963
$868$	0	0
$869$	2.40420e6	0.107999
$870$	0	0
$871$	−6.58671e7	−2.94187
$872$	0	0
$873$	1.42572e6	0.0633136
$874$	0	0
$875$	−2.38577e7	−1.05344
$876$	0	0
$877$	1.87261e7	0.822146	0.411073	−	0.911602i	$-0.365154\pi$
$877$	1.87261e7	0.822146	0.411073	+	0.911602i	$0.365154\pi$
$878$	0	0
$879$	1.03828e7	0.453255
$880$	0	0
$881$	−1.81827e7	−0.789256	−0.394628	−	0.918841i	$-0.629127\pi$
$881$	−1.81827e7	−0.789256	−0.394628	+	0.918841i	$0.629127\pi$
$882$	0	0
$883$	−1.36080e6	−0.0587343	−0.0293672	−	0.999569i	$-0.509349\pi$
$883$	−1.36080e6	−0.0587343	−0.0293672	+	0.999569i	$0.509349\pi$
$884$	0	0
$885$	−6.57858e6	−0.282341
$886$	0	0
$887$	−2.00032e7	−0.853669	−0.426834	−	0.904330i	$-0.640371\pi$
$887$	−2.00032e7	−0.853669	−0.426834	+	0.904330i	$0.640371\pi$
$888$	0	0
$889$	1.29455e7	0.549369
$890$	0	0
$891$	−3.48831e7	−1.47205
$892$	0	0
$893$	−4.33135e6	−0.181759
$894$	0	0
$895$	1.23688e7	0.516144
$896$	0	0
$897$	−5.28980e7	−2.19512
$898$	0	0
$899$	1.80045e7	0.742989
$900$	0	0
$901$	−1.30442e7	−0.535309
$902$	0	0
$903$	3.27308e7	1.33579
$904$	0	0
$905$	8.44536e6	0.342765
$906$	0	0
$907$	−3.24238e7	−1.30872	−0.654358	−	0.756185i	$-0.727061\pi$
$907$	−3.24238e7	−1.30872	−0.654358	+	0.756185i	$0.727061\pi$
$908$	0	0
$909$	598173.	0.0240114
$910$	0	0
$911$	9.99133e6	0.398866	0.199433	−	0.979911i	$-0.436090\pi$
$911$	9.99133e6	0.398866	0.199433	+	0.979911i	$0.436090\pi$
$912$	0	0
$913$	−1.50119e7	−0.596016
$914$	0	0
$915$	−4.26795e6	−0.168526
$916$	0	0
$917$	−7.55891e7	−2.96849
$918$	0	0
$919$	4.37067e7	1.70710	0.853550	−	0.521012i	$-0.174445\pi$
$919$	4.37067e7	1.70710	0.853550	+	0.521012i	$0.174445\pi$
$920$	0	0
$921$	−9.87540e6	−0.383624
$922$	0	0
$923$	−1.62885e7	−0.629328
$924$	0	0
$925$	1.97448e7	0.758748
$926$	0	0
$927$	600836.	0.0229645
$928$	0	0
$929$	1.59094e7	0.604805	0.302402	−	0.953180i	$-0.402211\pi$
$929$	1.59094e7	0.604805	0.302402	+	0.953180i	$0.402211\pi$
$930$	0	0
$931$	1.02014e7	0.385733
$932$	0	0
$933$	3.58551e6	0.134849
$934$	0	0
$935$	6.60864e6	0.247220
$936$	0	0
$937$	1.88286e7	0.700597	0.350299	−	0.936638i	$-0.386080\pi$
$937$	1.88286e7	0.700597	0.350299	+	0.936638i	$0.386080\pi$
$938$	0	0
$939$	−3.00080e7	−1.11064
$940$	0	0
$941$	5.94474e6	0.218856	0.109428	−	0.993995i	$-0.465098\pi$
$941$	5.94474e6	0.218856	0.109428	+	0.993995i	$0.465098\pi$
$942$	0	0
$943$	3.47475e7	1.27246
$944$	0	0
$945$	1.60171e7	0.583450
$946$	0	0
$947$	1.44485e7	0.523539	0.261770	−	0.965130i	$-0.415694\pi$
$947$	1.44485e7	0.523539	0.261770	+	0.965130i	$0.415694\pi$
$948$	0	0
$949$	−2.10819e7	−0.759879
$950$	0	0
$951$	1.45548e7	0.521860
$952$	0	0
$953$	−1.88659e7	−0.672891	−0.336445	−	0.941703i	$-0.609225\pi$
$953$	−1.88659e7	−0.672891	−0.336445	+	0.941703i	$0.609225\pi$
$954$	0	0
$955$	−3.56170e6	−0.126372
$956$	0	0
$957$	3.65350e7	1.28952
$958$	0	0
$959$	−1.01388e7	−0.355993
$960$	0	0
$961$	−5.27605e6	−0.184290
$962$	0	0
$963$	−668748.	−0.0232379
$964$	0	0
$965$	1.18989e7	0.411328
$966$	0	0
$967$	9.39594e6	0.323128	0.161564	−	0.986862i	$-0.448346\pi$
$967$	9.39594e6	0.323128	0.161564	+	0.986862i	$0.448346\pi$
$968$	0	0
$969$	2.79030e6	0.0954643
$970$	0	0
$971$	3.60091e6	0.122564	0.0612822	−	0.998120i	$-0.480481\pi$
$971$	3.60091e6	0.122564	0.0612822	+	0.998120i	$0.480481\pi$
$972$	0	0
$973$	8.78713e7	2.97553
$974$	0	0
$975$	4.54384e7	1.53078
$976$	0	0
$977$	1.05610e7	0.353972	0.176986	−	0.984213i	$-0.443365\pi$
$977$	1.05610e7	0.353972	0.176986	+	0.984213i	$0.443365\pi$
$978$	0	0
$979$	4.34830e7	1.44998
$980$	0	0
$981$	2.62837e6	0.0871994
$982$	0	0
$983$	2.16758e7	0.715469	0.357735	−	0.933823i	$-0.383549\pi$
$983$	2.16758e7	0.715469	0.357735	+	0.933823i	$0.383549\pi$
$984$	0	0
$985$	−1.21562e6	−0.0399217
$986$	0	0
$987$	−3.76940e7	−1.23163
$988$	0	0
$989$	3.34798e7	1.08841
$990$	0	0
$991$	−9.04602e6	−0.292599	−0.146300	−	0.989240i	$-0.546736\pi$
$991$	−9.04602e6	−0.292599	−0.146300	+	0.989240i	$0.546736\pi$
$992$	0	0
$993$	1.02443e6	0.0329693
$994$	0	0
$995$	1.01552e7	0.325186
$996$	0	0
$997$	4.49579e7	1.43241	0.716207	−	0.697888i	$-0.245877\pi$
$997$	4.49579e7	1.43241	0.716207	+	0.697888i	$0.245877\pi$
$998$	0	0
$999$	−2.82627e7	−0.895982

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	304.6.a.h.1.2		3
4.3	odd	2		38.6.a.d.1.2	✓	3
12.11	even	2		342.6.a.l.1.3		3
20.19	odd	2		950.6.a.f.1.2		3
76.75	even	2		722.6.a.d.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.a.d.1.2	✓	3	4.3	odd	2
304.6.a.h.1.2		3	1.1	even	1	trivial
342.6.a.l.1.3		3	12.11	even	2
722.6.a.d.1.2		3	76.75	even	2
950.6.a.f.1.2		3	20.19	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:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	304.a (trivial)

\(f(q)\)	\(=\)	\(q-14.7990 q^{3} -19.0956 q^{5} -212.287 q^{7} -23.9901 q^{9} +O(q^{10})\)	\(q-14.7990 q^{3} -19.0956 q^{5} -212.287 q^{7} -23.9901 q^{9} +662.625 q^{11} +1112.31 q^{13} +282.595 q^{15} -522.290 q^{17} +361.000 q^{19} +3141.63 q^{21} +3213.52 q^{23} -2760.36 q^{25} +3951.18 q^{27} -3725.71 q^{29} -4832.50 q^{31} -9806.17 q^{33} +4053.74 q^{35} -7152.96 q^{37} -16461.1 q^{39} +10812.9 q^{41} +10418.4 q^{43} +458.106 q^{45} -11998.2 q^{47} +28258.8 q^{49} +7729.35 q^{51} +24975.0 q^{53} -12653.2 q^{55} -5342.43 q^{57} -23279.2 q^{59} -15102.7 q^{61} +5092.80 q^{63} -21240.2 q^{65} -59216.5 q^{67} -47556.9 q^{69} -14643.8 q^{71} -18953.3 q^{73} +40850.5 q^{75} -140667. q^{77} +3628.29 q^{79} -52643.9 q^{81} -22655.2 q^{83} +9973.42 q^{85} +55136.7 q^{87} +65622.3 q^{89} -236129. q^{91} +71516.1 q^{93} -6893.50 q^{95} -59429.2 q^{97} -15896.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 13 q^{3} + 81 q^{5} - 228 q^{7} + 236 q^{9}+O(q^{10}) \) 3 * q - 13 * q^3 + 81 * q^5 - 228 * q^7 + 236 * q^9	\( 3 q - 13 q^{3} + 81 q^{5} - 228 q^{7} + 236 q^{9} - 363 q^{11} + 501 q^{13} + 670 q^{15} - 1206 q^{17} + 1083 q^{19} - 2085 q^{21} + 1077 q^{23} - 3882 q^{25} + 5087 q^{27} - 8349 q^{29} + 7332 q^{31} - 15784 q^{33} + 1185 q^{35} - 1650 q^{37} - 773 q^{39} + 10140 q^{41} - 3777 q^{43} + 14005 q^{45} - 33231 q^{47} + 31269 q^{49} - 46935 q^{51} + 31029 q^{53} - 66003 q^{55} - 4693 q^{57} - 20409 q^{59} + 17115 q^{61} - 6327 q^{63} - 45348 q^{65} + 789 q^{67} - 151147 q^{69} - 19164 q^{71} - 76260 q^{73} + 69607 q^{75} - 97209 q^{77} - 68358 q^{79} - 197713 q^{81} - 6762 q^{83} - 45837 q^{85} - 66805 q^{87} - 85506 q^{89} - 345033 q^{91} + 15688 q^{93} + 29241 q^{95} + 105024 q^{97} - 158317 q^{99}+O(q^{100}) \) 3 * q - 13 * q^3 + 81 * q^5 - 228 * q^7 + 236 * q^9 - 363 * q^11 + 501 * q^13 + 670 * q^15 - 1206 * q^17 + 1083 * q^19 - 2085 * q^21 + 1077 * q^23 - 3882 * q^25 + 5087 * q^27 - 8349 * q^29 + 7332 * q^31 - 15784 * q^33 + 1185 * q^35 - 1650 * q^37 - 773 * q^39 + 10140 * q^41 - 3777 * q^43 + 14005 * q^45 - 33231 * q^47 + 31269 * q^49 - 46935 * q^51 + 31029 * q^53 - 66003 * q^55 - 4693 * q^57 - 20409 * q^59 + 17115 * q^61 - 6327 * q^63 - 45348 * q^65 + 789 * q^67 - 151147 * q^69 - 19164 * q^71 - 76260 * q^73 + 69607 * q^75 - 97209 * q^77 - 68358 * q^79 - 197713 * q^81 - 6762 * q^83 - 45837 * q^85 - 66805 * q^87 - 85506 * q^89 - 345033 * q^91 + 15688 * q^93 + 29241 * q^95 + 105024 * q^97 - 158317 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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