LMFDB - Embedded newform 425.6.a.k.1.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,6,Mod(1,425)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("425.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

sage:traces = [15,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$68.1631234205$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 378 x^{13} + 106 x^{12} + 55677 x^{11} + 23739 x^{10} - 4018640 x^{9} + \cdots - 45034730496 $ x^15 - x^14 - 378x^13 + 106x^12 + 55677x^11 + 23739x^10 - 4018640x^9 - 4229556x^8 + 146332768x^7 + 216845216x^6 - 2522980352x^5 - 3824088960x^4 + 19716500736x^3 + 24929041152x^2 - 56205914112*x - 45034730496
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{9}\cdot 5^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$3.10924$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.10924 q^{2} +28.9596 q^{3} -22.3326 q^{4} +90.0423 q^{6} +107.101 q^{7} -168.933 q^{8} +595.658 q^{9} +264.087 q^{11} -646.744 q^{12} +169.699 q^{13} +333.004 q^{14} +189.390 q^{16} +289.000 q^{17} +1852.04 q^{18} +398.563 q^{19} +3101.61 q^{21} +821.110 q^{22} -1500.82 q^{23} -4892.23 q^{24} +527.636 q^{26} +10212.8 q^{27} -2391.86 q^{28} +2515.15 q^{29} -6203.63 q^{31} +5994.72 q^{32} +7647.85 q^{33} +898.570 q^{34} -13302.6 q^{36} +12783.5 q^{37} +1239.23 q^{38} +4914.42 q^{39} -6233.26 q^{41} +9643.66 q^{42} -11387.6 q^{43} -5897.76 q^{44} -4666.41 q^{46} +13788.6 q^{47} +5484.67 q^{48} -5336.28 q^{49} +8369.32 q^{51} -3789.83 q^{52} -3302.37 q^{53} +31754.1 q^{54} -18093.0 q^{56} +11542.2 q^{57} +7820.21 q^{58} +42681.3 q^{59} +16538.6 q^{61} -19288.6 q^{62} +63795.8 q^{63} +12578.5 q^{64} +23779.0 q^{66} +68740.7 q^{67} -6454.13 q^{68} -43463.1 q^{69} +71268.3 q^{71} -100626. q^{72} +82764.5 q^{73} +39747.1 q^{74} -8900.95 q^{76} +28284.1 q^{77} +15280.1 q^{78} -46521.2 q^{79} +151014. q^{81} -19380.7 q^{82} -88857.9 q^{83} -69267.2 q^{84} -35406.8 q^{86} +72837.7 q^{87} -44613.1 q^{88} -87636.6 q^{89} +18175.0 q^{91} +33517.2 q^{92} -179654. q^{93} +42872.0 q^{94} +173605. q^{96} -10566.6 q^{97} -16591.8 q^{98} +157305. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{2} + 9 q^{3} + 277 q^{4} + 169 q^{6} + 181 q^{7} + 753 q^{8} + 1826 q^{9} + 172 q^{11} - 2109 q^{12} - 389 q^{13} + 3635 q^{14} + 6837 q^{16} + 4335 q^{17} + 6742 q^{18} + 5150 q^{19} - 6891 q^{21}+ \cdots - 183214 q^{99}+O(q^{100}) $ 15 * q + q^2 + 9 * q^3 + 277 * q^4 + 169 * q^6 + 181 * q^7 + 753 * q^8 + 1826 * q^9 + 172 * q^11 - 2109 * q^12 - 389 * q^13 + 3635 * q^14 + 6837 * q^16 + 4335 * q^17 + 6742 * q^18 + 5150 * q^19 - 6891 * q^21 - 10522 * q^22 + 3140 * q^23 + 9263 * q^24 + 3181 * q^26 + 1611 * q^27 + 16371 * q^28 + 15214 * q^29 + 18873 * q^31 + 16857 * q^32 - 14700 * q^33 + 289 * q^34 + 83010 * q^36 + 12122 * q^37 - 5302 * q^38 + 17995 * q^39 - 2336 * q^41 - 65201 * q^42 - 660 * q^43 + 40740 * q^44 - 18610 * q^46 + 11494 * q^47 - 48879 * q^48 + 89846 * q^49 + 2601 * q^51 - 71027 * q^52 - 76605 * q^53 - 33675 * q^54 + 71385 * q^56 + 112044 * q^57 - 11270 * q^58 + 7268 * q^59 + 137458 * q^61 + 212145 * q^62 - 25488 * q^63 + 116341 * q^64 + 87900 * q^66 - 5036 * q^67 + 80053 * q^68 - 57176 * q^69 + 180961 * q^71 + 588818 * q^72 - 176062 * q^73 + 11010 * q^74 + 390158 * q^76 + 319456 * q^77 - 339079 * q^78 + 127813 * q^79 + 419847 * q^81 + 200996 * q^82 - 318176 * q^83 - 373745 * q^84 + 214564 * q^86 + 138204 * q^87 - 722070 * q^88 + 97682 * q^89 + 229269 * q^91 + 202080 * q^92 - 514385 * q^93 + 104594 * q^94 + 706465 * q^96 + 381812 * q^97 - 399302 * q^98 - 183214 * q^99

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 $

$2$	3.10924	0.549641	0.274821	−	0.961496i	$-0.411382\pi$
$2$	3.10924	0.549641	0.274821	+	0.961496i	$0.411382\pi$
$3$	28.9596	1.85776	0.928879	−	0.370383i	$-0.120774\pi$
$3$	28.9596	1.85776	0.928879	+	0.370383i	$0.120774\pi$
$4$	−22.3326	−0.697895
$5$	0	0
$5$	0	0
$6$	90.0423	1.02110
$7$	107.101	0.826134	0.413067	−	0.910701i	$-0.364458\pi$
$7$	107.101	0.826134	0.413067	+	0.910701i	$0.364458\pi$
$8$	−168.933	−0.933233
$9$	595.658	2.45127
$10$	0	0
$11$	264.087	0.658060	0.329030	−	0.944320i	$-0.393278\pi$
$11$	264.087	0.658060	0.329030	+	0.944320i	$0.393278\pi$
$12$	−646.744	−1.29652
$13$	169.699	0.278498	0.139249	−	0.990257i	$-0.455531\pi$
$13$	169.699	0.278498	0.139249	+	0.990257i	$0.455531\pi$
$14$	333.004	0.454077
$15$	0	0
$16$	189.390	0.184952
$17$	289.000	0.242536
$17$	289.000	0.242536
$18$	1852.04	1.34732
$19$	398.563	0.253287	0.126643	−	0.991948i	$-0.459580\pi$
$19$	398.563	0.253287	0.126643	+	0.991948i	$0.459580\pi$
$20$	0	0
$21$	3101.61	1.53476
$22$	821.110	0.361697
$23$	−1500.82	−0.591573	−0.295787	−	0.955254i	$-0.595582\pi$
$23$	−1500.82	−0.591573	−0.295787	+	0.955254i	$0.595582\pi$
$24$	−4892.23	−1.73372
$25$	0	0
$26$	527.636	0.153074
$27$	10212.8	2.69610
$28$	−2391.86	−0.576554
$29$	2515.15	0.555353	0.277676	−	0.960675i	$-0.410436\pi$
$29$	2515.15	0.555353	0.277676	+	0.960675i	$0.410436\pi$
$30$	0	0
$31$	−6203.63	−1.15942	−0.579711	−	0.814822i	$-0.696834\pi$
$31$	−6203.63	−1.15942	−0.579711	+	0.814822i	$0.696834\pi$
$32$	5994.72	1.03489
$33$	7647.85	1.22252
$34$	898.570	0.133308
$35$	0	0
$36$	−13302.6	−1.71073
$37$	12783.5	1.53514	0.767568	−	0.640967i	$-0.221466\pi$
$37$	12783.5	1.53514	0.767568	+	0.640967i	$0.221466\pi$
$38$	1239.23	0.139217
$39$	4914.42	0.517381
$40$	0	0
$41$	−6233.26	−0.579103	−0.289551	−	0.957162i	$-0.593506\pi$
$41$	−6233.26	−0.579103	−0.289551	+	0.957162i	$0.593506\pi$
$42$	9643.66	0.843565
$43$	−11387.6	−0.939207	−0.469603	−	0.882877i	$-0.655603\pi$
$43$	−11387.6	−0.939207	−0.469603	+	0.882877i	$0.655603\pi$
$44$	−5897.76	−0.459256
$45$	0	0
$46$	−4666.41	−0.325153
$47$	13788.6	0.910489	0.455245	−	0.890366i	$-0.349552\pi$
$47$	13788.6	0.910489	0.455245	+	0.890366i	$0.349552\pi$
$48$	5484.67	0.343595
$49$	−5336.28	−0.317503
$50$	0	0
$51$	8369.32	0.450573
$52$	−3789.83	−0.194362
$53$	−3302.37	−0.161486	−0.0807431	−	0.996735i	$-0.525729\pi$
$53$	−3302.37	−0.161486	−0.0807431	+	0.996735i	$0.525729\pi$
$54$	31754.1	1.48189
$55$	0	0
$56$	−18093.0	−0.770975
$57$	11542.2	0.470546
$58$	7820.21	0.305245
$59$	42681.3	1.59627	0.798137	−	0.602475i	$-0.205819\pi$
$59$	42681.3	1.59627	0.798137	+	0.602475i	$0.205819\pi$
$60$	0	0
$61$	16538.6	0.569082	0.284541	−	0.958664i	$-0.408159\pi$
$61$	16538.6	0.569082	0.284541	+	0.958664i	$0.408159\pi$
$62$	−19288.6	−0.637266
$63$	63795.8	2.02507
$64$	12578.5	0.383866
$65$	0	0
$66$	23779.0	0.671945
$67$	68740.7	1.87080	0.935400	−	0.353593i	$-0.115040\pi$
$67$	68740.7	1.87080	0.935400	+	0.353593i	$0.115040\pi$
$68$	−6454.13	−0.169264
$69$	−43463.1	−1.09900
$70$	0	0
$71$	71268.3	1.67784	0.838920	−	0.544256i	$-0.183188\pi$
$71$	71268.3	1.67784	0.838920	+	0.544256i	$0.183188\pi$
$72$	−100626.	−2.28760
$73$	82764.5	1.81776	0.908881	−	0.417056i	$-0.136938\pi$
$73$	82764.5	1.81776	0.908881	+	0.417056i	$0.136938\pi$
$74$	39747.1	0.843774
$75$	0	0
$76$	−8900.95	−0.176767
$77$	28284.1	0.543645
$78$	15280.1	0.284374
$79$	−46521.2	−0.838655	−0.419327	−	0.907835i	$-0.637734\pi$
$79$	−46521.2	−0.838655	−0.419327	+	0.907835i	$0.637734\pi$
$80$	0	0
$81$	151014.	2.55744
$82$	−19380.7	−0.318299
$83$	−88857.9	−1.41580	−0.707898	−	0.706314i	$-0.750357\pi$
$83$	−88857.9	−1.41580	−0.707898	+	0.706314i	$0.750357\pi$
$84$	−69267.2	−1.07110
$85$	0	0
$86$	−35406.8	−0.516227
$87$	72837.7	1.03171
$88$	−44613.1	−0.614123
$89$	−87636.6	−1.17276	−0.586382	−	0.810035i	$-0.699448\pi$
$89$	−87636.6	−1.17276	−0.586382	+	0.810035i	$0.699448\pi$
$90$	0	0
$91$	18175.0	0.230076
$92$	33517.2	0.412856
$93$	−179654.	−2.15392
$94$	42872.0	0.500442
$95$	0	0
$96$	173605.	1.92257
$97$	−10566.6	−0.114026	−0.0570131	−	0.998373i	$-0.518158\pi$
$97$	−10566.6	−0.114026	−0.0570131	+	0.998373i	$0.518158\pi$
$98$	−16591.8	−0.174513
$99$	157305.	1.61308
$100$	0	0
$101$	−17364.1	−0.169374	−0.0846872	−	0.996408i	$-0.526989\pi$
$101$	−17364.1	−0.169374	−0.0846872	+	0.996408i	$0.526989\pi$
$102$	26022.2	0.247653
$103$	−49057.2	−0.455627	−0.227813	−	0.973705i	$-0.573158\pi$
$103$	−49057.2	−0.455627	−0.227813	+	0.973705i	$0.573158\pi$
$104$	−28667.8	−0.259903
$105$	0	0
$106$	−10267.9	−0.0887595
$107$	−112460.	−0.949597	−0.474798	−	0.880095i	$-0.657479\pi$
$107$	−112460.	−0.949597	−0.474798	+	0.880095i	$0.657479\pi$
$108$	−228079.	−1.88159
$109$	−90096.5	−0.726343	−0.363171	−	0.931722i	$-0.618306\pi$
$109$	−90096.5	−0.726343	−0.363171	+	0.931722i	$0.618306\pi$
$110$	0	0
$111$	370206.	2.85191
$112$	20284.0	0.152795
$113$	109851.	0.809298	0.404649	−	0.914472i	$-0.367394\pi$
$113$	109851.	0.809298	0.404649	+	0.914472i	$0.367394\pi$
$114$	35887.5	0.258631
$115$	0	0
$116$	−56169.9	−0.387578
$117$	101083.	0.682672
$118$	132706.	0.877378
$119$	30952.3	0.200367
$120$	0	0
$121$	−91309.0	−0.566957
$122$	51422.5	0.312791
$123$	−180513.	−1.07583
$124$	138543.	0.809154
$125$	0	0
$126$	198356.	1.11306
$127$	−163715.	−0.900697	−0.450348	−	0.892853i	$-0.648700\pi$
$127$	−163715.	−0.900697	−0.450348	+	0.892853i	$0.648700\pi$
$128$	−152721.	−0.823901
$129$	−329780.	−1.74482
$130$	0	0
$131$	−128543.	−0.654441	−0.327220	−	0.944948i	$-0.606112\pi$
$131$	−128543.	−0.654441	−0.327220	+	0.944948i	$0.606112\pi$
$132$	−170797.	−0.853187
$133$	42686.6	0.209249
$134$	213731.	1.02827
$135$	0	0
$136$	−48821.7	−0.226342
$137$	−250442.	−1.14000	−0.570001	−	0.821644i	$-0.693057\pi$
$137$	−250442.	−1.14000	−0.570001	+	0.821644i	$0.693057\pi$
$138$	−135137.	−0.604056
$139$	−67891.7	−0.298043	−0.149022	−	0.988834i	$-0.547612\pi$
$139$	−67891.7	−0.298043	−0.149022	+	0.988834i	$0.547612\pi$
$140$	0	0
$141$	399311.	1.69147
$142$	221590.	0.922209
$143$	44815.4	0.183268
$144$	112812.	0.453365
$145$	0	0
$146$	257335.	0.999117
$147$	−154536.	−0.589844
$148$	−285490.	−1.07136
$149$	244123.	0.900832	0.450416	−	0.892819i	$-0.351276\pi$
$149$	244123.	0.900832	0.450416	+	0.892819i	$0.351276\pi$
$150$	0	0
$151$	−167293.	−0.597084	−0.298542	−	0.954397i	$-0.596500\pi$
$151$	−167293.	−0.597084	−0.298542	+	0.954397i	$0.596500\pi$
$152$	−67330.4	−0.236376
$153$	172145.	0.594519
$154$	87942.1	0.298810
$155$	0	0
$156$	−109752.	−0.361078
$157$	338789.	1.09693	0.548467	−	0.836172i	$-0.315212\pi$
$157$	338789.	1.09693	0.548467	+	0.836172i	$0.315212\pi$
$158$	−144646.	−0.460959
$159$	−95635.2	−0.300002
$160$	0	0
$161$	−160740.	−0.488719
$162$	469539.	1.40567
$163$	584768.	1.72391	0.861955	−	0.506986i	$-0.169240\pi$
$163$	584768.	1.72391	0.861955	+	0.506986i	$0.169240\pi$
$164$	139205.	0.404153
$165$	0	0
$166$	−276281.	−0.778180
$167$	173760.	0.482125	0.241063	−	0.970510i	$-0.422504\pi$
$167$	173760.	0.482125	0.241063	+	0.970510i	$0.422504\pi$
$168$	−523965.	−1.43229
$169$	−342495.	−0.922439
$170$	0	0
$171$	237407.	0.620873
$172$	254315.	0.655467
$173$	−619045.	−1.57256	−0.786280	−	0.617870i	$-0.787996\pi$
$173$	−619045.	−1.57256	−0.786280	+	0.617870i	$0.787996\pi$
$174$	226470.	0.567071
$175$	0	0
$176$	50015.5	0.121709
$177$	1.23603e6	2.96549
$178$	−272483.	−0.644599
$179$	−381186.	−0.889211	−0.444605	−	0.895727i	$-0.646656\pi$
$179$	−381186.	−0.889211	−0.444605	+	0.895727i	$0.646656\pi$
$180$	0	0
$181$	−592374.	−1.34400	−0.672000	−	0.740551i	$-0.734565\pi$
$181$	−592374.	−1.34400	−0.672000	+	0.740551i	$0.734565\pi$
$182$	56510.6	0.126459
$183$	478951.	1.05722
$184$	253538.	0.552076
$185$	0	0
$186$	−558589.	−1.18389
$187$	76321.1	0.159603
$188$	−307935.	−0.635426
$189$	1.09381e6	2.22734
$190$	0	0
$191$	−9643.58	−0.0191274	−0.00956368	−	0.999954i	$-0.503044\pi$
$191$	−9643.58	−0.0191274	−0.00956368	+	0.999954i	$0.503044\pi$
$192$	364269.	0.713131
$193$	−795730.	−1.53770	−0.768851	−	0.639428i	$-0.779171\pi$
$193$	−795730.	−1.53770	−0.768851	+	0.639428i	$0.779171\pi$
$194$	−32854.0	−0.0626735
$195$	0	0
$196$	119173.	0.221584
$197$	628294.	1.15345	0.576723	−	0.816940i	$-0.304331\pi$
$197$	628294.	1.15345	0.576723	+	0.816940i	$0.304331\pi$
$198$	489100.	0.886615
$199$	414085.	0.741237	0.370618	−	0.928785i	$-0.379146\pi$
$199$	414085.	0.741237	0.370618	+	0.928785i	$0.379146\pi$
$200$	0	0
$201$	1.99070e6	3.47549
$202$	−53989.0	−0.0930952
$203$	269376.	0.458796
$204$	−186909.	−0.314452
$205$	0	0
$206$	−152530.	−0.250431
$207$	−893974.	−1.45010
$208$	32139.4	0.0515086
$209$	105255.	0.166678
$210$	0	0
$211$	1.00593e6	1.55547	0.777735	−	0.628593i	$-0.216369\pi$
$211$	1.00593e6	1.55547	0.777735	+	0.628593i	$0.216369\pi$
$212$	73750.5	0.112700
$213$	2.06390e6	3.11702
$214$	−349665.	−0.521937
$215$	0	0
$216$	−1.72528e6	−2.51609
$217$	−664417.	−0.957837
$218$	−280132.	−0.399228
$219$	2.39683e6	3.37696
$220$	0	0
$221$	49043.1	0.0675456
$222$	1.15106e6	1.56753
$223$	618909.	0.833421	0.416710	−	0.909039i	$-0.363183\pi$
$223$	618909.	0.833421	0.416710	+	0.909039i	$0.363183\pi$
$224$	642043.	0.854957
$225$	0	0
$226$	341554.	0.444824
$227$	−400031.	−0.515263	−0.257631	−	0.966243i	$-0.582942\pi$
$227$	−400031.	−0.515263	−0.257631	+	0.966243i	$0.582942\pi$
$228$	−257768.	−0.328391
$229$	−329434.	−0.415126	−0.207563	−	0.978222i	$-0.566553\pi$
$229$	−329434.	−0.415126	−0.207563	+	0.978222i	$0.566553\pi$
$230$	0	0
$231$	819096.	1.00996
$232$	−424892.	−0.518273
$233$	−1.62517e6	−1.96114	−0.980572	−	0.196160i	$-0.937153\pi$
$233$	−1.62517e6	−1.96114	−0.980572	+	0.196160i	$0.937153\pi$
$234$	314290.	0.375224
$235$	0	0
$236$	−953186.	−1.11403
$237$	−1.34723e6	−1.55802
$238$	96238.2	0.110130
$239$	−433026.	−0.490364	−0.245182	−	0.969477i	$-0.578848\pi$
$239$	−433026.	−0.490364	−0.245182	+	0.969477i	$0.578848\pi$
$240$	0	0
$241$	−1.55825e6	−1.72820	−0.864102	−	0.503318i	$-0.832113\pi$
$241$	−1.55825e6	−1.72820	−0.864102	+	0.503318i	$0.832113\pi$
$242$	−283902.	−0.311623
$243$	1.89159e6	2.05500
$244$	−369351.	−0.397159
$245$	0	0
$246$	−561257.	−0.591322
$247$	67635.8	0.0705398
$248$	1.04800e6	1.08201
$249$	−2.57329e6	−2.63021
$250$	0	0
$251$	−100993.	−0.101183	−0.0505915	−	0.998719i	$-0.516111\pi$
$251$	−100993.	−0.101183	−0.0505915	+	0.998719i	$0.516111\pi$
$252$	−1.42473e6	−1.41329
$253$	−396347.	−0.389291
$254$	−509029.	−0.495060
$255$	0	0
$256$	−877361.	−0.836716
$257$	−443479.	−0.418832	−0.209416	−	0.977827i	$-0.567156\pi$
$257$	−443479.	−0.418832	−0.209416	+	0.977827i	$0.567156\pi$
$258$	−1.02537e6	−0.959024
$259$	1.36914e6	1.26823
$260$	0	0
$261$	1.49817e6	1.36132
$262$	−399671.	−0.359708
$263$	−968462.	−0.863363	−0.431681	−	0.902026i	$-0.642080\pi$
$263$	−968462.	−0.863363	−0.431681	+	0.902026i	$0.642080\pi$
$264$	−1.29198e6	−1.14089
$265$	0	0
$266$	132723.	0.115012
$267$	−2.53792e6	−2.17871
$268$	−1.53516e6	−1.30562
$269$	−1.60323e6	−1.35087	−0.675436	−	0.737419i	$-0.736045\pi$
$269$	−1.60323e6	−1.35087	−0.675436	+	0.737419i	$0.736045\pi$
$270$	0	0
$271$	−170946.	−0.141395	−0.0706976	−	0.997498i	$-0.522523\pi$
$271$	−170946.	−0.141395	−0.0706976	+	0.997498i	$0.522523\pi$
$272$	54733.8	0.0448573
$273$	526342.	0.427426
$274$	−778684.	−0.626592
$275$	0	0
$276$	970645.	0.766986
$277$	1.57270e6	1.23154	0.615769	−	0.787927i	$-0.288846\pi$
$277$	1.57270e6	1.23154	0.615769	+	0.787927i	$0.288846\pi$
$278$	−211092.	−0.163817
$279$	−3.69524e6	−2.84205
$280$	0	0
$281$	2.17716e6	1.64484	0.822421	−	0.568879i	$-0.192623\pi$
$281$	2.17716e6	1.64484	0.822421	+	0.568879i	$0.192623\pi$
$282$	1.24156e6	0.929701
$283$	303527.	0.225285	0.112642	−	0.993636i	$-0.464069\pi$
$283$	303527.	0.225285	0.112642	+	0.993636i	$0.464069\pi$
$284$	−1.59161e6	−1.17095
$285$	0	0
$286$	139342.	0.100732
$287$	−667591.	−0.478416
$288$	3.57080e6	2.53679
$289$	83521.0	0.0588235
$290$	0	0
$291$	−306004.	−0.211833
$292$	−1.84835e6	−1.26861
$293$	−300633.	−0.204582	−0.102291	−	0.994755i	$-0.532617\pi$
$293$	−300633.	−0.204582	−0.102291	+	0.994755i	$0.532617\pi$
$294$	−480491.	−0.324203
$295$	0	0
$296$	−2.15957e6	−1.43264
$297$	2.69707e6	1.77420
$298$	759038.	0.495134
$299$	−254688.	−0.164752
$300$	0	0
$301$	−1.21963e6	−0.775910
$302$	−520154.	−0.328182
$303$	−502856.	−0.314657
$304$	75483.9	0.0468458
$305$	0	0
$306$	535240.	0.326772
$307$	−2.34927e6	−1.42261	−0.711307	−	0.702881i	$-0.751897\pi$
$307$	−2.34927e6	−1.42261	−0.711307	+	0.702881i	$0.751897\pi$
$308$	−631658.	−0.379407
$309$	−1.42068e6	−0.846445
$310$	0	0
$311$	−2.06822e6	−1.21254	−0.606269	−	0.795260i	$-0.707335\pi$
$311$	−2.06822e6	−1.21254	−0.606269	+	0.795260i	$0.707335\pi$
$312$	−830209.	−0.482837
$313$	−52661.4	−0.0303830	−0.0151915	−	0.999885i	$-0.504836\pi$
$313$	−52661.4	−0.0303830	−0.0151915	+	0.999885i	$0.504836\pi$
$314$	1.05338e6	0.602920
$315$	0	0
$316$	1.03894e6	0.585293
$317$	1.87117e6	1.04584	0.522920	−	0.852382i	$-0.324843\pi$
$317$	1.87117e6	1.04584	0.522920	+	0.852382i	$0.324843\pi$
$318$	−297353.	−0.164894
$319$	664219.	0.365455
$320$	0	0
$321$	−3.25680e6	−1.76412
$322$	−499779.	−0.268620
$323$	115185.	0.0614311
$324$	−3.37254e6	−1.78482
$325$	0	0
$326$	1.81818e6	0.947531
$327$	−2.60916e6	−1.34937
$328$	1.05300e6	0.540438
$329$	1.47678e6	0.752186
$330$	0	0
$331$	179875.	0.0902402	0.0451201	−	0.998982i	$-0.485633\pi$
$331$	179875.	0.0902402	0.0451201	+	0.998982i	$0.485633\pi$
$332$	1.98443e6	0.988077
$333$	7.61462e6	3.76303
$334$	540263.	0.264996
$335$	0	0
$336$	587416.	0.283856
$337$	−3.93749e6	−1.88862	−0.944310	−	0.329057i	$-0.893269\pi$
$337$	−3.93749e6	−1.88862	−0.944310	+	0.329057i	$0.893269\pi$
$338$	−1.06490e6	−0.507010
$339$	3.18125e6	1.50348
$340$	0	0
$341$	−1.63830e6	−0.762969
$342$	738155.	0.341257
$343$	−2.37158e6	−1.08843
$344$	1.92374e6	0.876499
$345$	0	0
$346$	−1.92476e6	−0.864344
$347$	2.64474e6	1.17912	0.589562	−	0.807723i	$-0.299301\pi$
$347$	2.64474e6	1.17912	0.589562	+	0.807723i	$0.299301\pi$
$348$	−1.62666e6	−0.720026
$349$	503558.	0.221302	0.110651	−	0.993859i	$-0.464706\pi$
$349$	503558.	0.221302	0.110651	+	0.993859i	$0.464706\pi$
$350$	0	0
$351$	1.73311e6	0.750858
$352$	1.58313e6	0.681019
$353$	3.75677e6	1.60464	0.802321	−	0.596893i	$-0.203598\pi$
$353$	3.75677e6	1.60464	0.802321	+	0.596893i	$0.203598\pi$
$354$	3.84312e6	1.62996
$355$	0	0
$356$	1.95716e6	0.818465
$357$	896366.	0.372233
$358$	−1.18520e6	−0.488747
$359$	4.69428e6	1.92235	0.961176	−	0.275935i	$-0.0889876\pi$
$359$	4.69428e6	1.92235	0.961176	+	0.275935i	$0.0889876\pi$
$360$	0	0
$361$	−2.31725e6	−0.935846
$362$	−1.84183e6	−0.738718
$363$	−2.64427e6	−1.05327
$364$	−405896.	−0.160569
$365$	0	0
$366$	1.48917e6	0.581089
$367$	−4.77737e6	−1.85150	−0.925751	−	0.378134i	$-0.876566\pi$
$367$	−4.77737e6	−1.85150	−0.925751	+	0.378134i	$0.876566\pi$
$368$	−284241.	−0.109412
$369$	−3.71289e6	−1.41953
$370$	0	0
$371$	−353688.	−0.133409
$372$	4.01216e6	1.50321
$373$	−802904.	−0.298808	−0.149404	−	0.988776i	$-0.547735\pi$
$373$	−802904.	−0.298808	−0.149404	+	0.988776i	$0.547735\pi$
$374$	237301.	0.0877243
$375$	0	0
$376$	−2.32935e6	−0.849698
$377$	426819.	0.154664
$378$	3.40091e6	1.22424
$379$	−1.93264e6	−0.691118	−0.345559	−	0.938397i	$-0.612311\pi$
$379$	−1.93264e6	−0.691118	−0.345559	+	0.938397i	$0.612311\pi$
$380$	0	0
$381$	−4.74111e6	−1.67328
$382$	−29984.2	−0.0105132
$383$	−3.51775e6	−1.22537	−0.612686	−	0.790326i	$-0.709911\pi$
$383$	−3.51775e6	−1.22537	−0.612686	+	0.790326i	$0.709911\pi$
$384$	−4.42275e6	−1.53061
$385$	0	0
$386$	−2.47411e6	−0.845185
$387$	−6.78311e6	−2.30225
$388$	235979.	0.0795783
$389$	4.85594e6	1.62704	0.813522	−	0.581534i	$-0.197547\pi$
$389$	4.85594e6	1.62704	0.813522	+	0.581534i	$0.197547\pi$
$390$	0	0
$391$	−433737.	−0.143478
$392$	901474.	0.296304
$393$	−3.72255e6	−1.21579
$394$	1.95352e6	0.633982
$395$	0	0
$396$	−3.51304e6	−1.12576
$397$	440244.	0.140190	0.0700950	−	0.997540i	$-0.477670\pi$
$397$	440244.	0.140190	0.0700950	+	0.997540i	$0.477670\pi$
$398$	1.28749e6	0.407414
$399$	1.23619e6	0.388734
$400$	0	0
$401$	−1.21487e6	−0.377284	−0.188642	−	0.982046i	$-0.560409\pi$
$401$	−1.21487e6	−0.377284	−0.188642	+	0.982046i	$0.560409\pi$
$402$	6.18957e6	1.91027
$403$	−1.05275e6	−0.322896
$404$	387785.	0.118206
$405$	0	0
$406$	837556.	0.252173
$407$	3.37597e6	1.01021
$408$	−1.41386e6	−0.420489
$409$	439240.	0.129836	0.0649178	−	0.997891i	$-0.479321\pi$
$409$	439240.	0.129836	0.0649178	+	0.997891i	$0.479321\pi$
$410$	0	0
$411$	−7.25270e6	−2.11785
$412$	1.09558e6	0.317980
$413$	4.57123e6	1.31874
$414$	−2.77958e6	−0.797037
$415$	0	0
$416$	1.01730e6	0.288214
$417$	−1.96611e6	−0.553693
$418$	327264.	0.0916130
$419$	−2.90977e6	−0.809700	−0.404850	−	0.914383i	$-0.632676\pi$
$419$	−2.90977e6	−0.809700	−0.404850	+	0.914383i	$0.632676\pi$
$420$	0	0
$421$	2.67032e6	0.734274	0.367137	−	0.930167i	$-0.380338\pi$
$421$	2.67032e6	0.734274	0.367137	+	0.930167i	$0.380338\pi$
$422$	3.12768e6	0.854950
$423$	8.21327e6	2.23185
$424$	557879.	0.150704
$425$	0	0
$426$	6.41716e6	1.71324
$427$	1.77131e6	0.470137
$428$	2.51153e6	0.662718
$429$	1.29783e6	0.340468
$430$	0	0
$431$	−1.57114e6	−0.407402	−0.203701	−	0.979033i	$-0.565297\pi$
$431$	−1.57114e6	−0.407402	−0.203701	+	0.979033i	$0.565297\pi$
$432$	1.93421e6	0.498648
$433$	−1.15199e6	−0.295277	−0.147638	−	0.989041i	$-0.547167\pi$
$433$	−1.15199e6	−0.295277	−0.147638	+	0.989041i	$0.547167\pi$
$434$	−2.06583e6	−0.526467
$435$	0	0
$436$	2.01209e6	0.506911
$437$	−598170.	−0.149838
$438$	7.45231e6	1.85612
$439$	6.80704e6	1.68576	0.842882	−	0.538098i	$-0.180857\pi$
$439$	6.80704e6	1.68576	0.842882	+	0.538098i	$0.180857\pi$
$440$	0	0
$441$	−3.17859e6	−0.778285
$442$	152487.	0.0371258
$443$	−4.57955e6	−1.10870	−0.554349	−	0.832284i	$-0.687033\pi$
$443$	−4.57955e6	−1.10870	−0.554349	+	0.832284i	$0.687033\pi$
$444$	−8.26768e6	−1.99033
$445$	0	0
$446$	1.92434e6	0.458082
$447$	7.06971e6	1.67353
$448$	1.34718e6	0.317125
$449$	4.84555e6	1.13430	0.567149	−	0.823615i	$-0.308046\pi$
$449$	4.84555e6	1.13430	0.567149	+	0.823615i	$0.308046\pi$
$450$	0	0
$451$	−1.64612e6	−0.381084
$452$	−2.45327e6	−0.564805
$453$	−4.84473e6	−1.10924
$454$	−1.24379e6	−0.283209
$455$	0	0
$456$	−1.94986e6	−0.439129
$457$	−2.63730e6	−0.590702	−0.295351	−	0.955389i	$-0.595437\pi$
$457$	−2.63730e6	−0.590702	−0.295351	+	0.955389i	$0.595437\pi$
$458$	−1.02429e6	−0.228170
$459$	2.95150e6	0.653900
$460$	0	0
$461$	−3.30857e6	−0.725083	−0.362541	−	0.931968i	$-0.618091\pi$
$461$	−3.30857e6	−0.725083	−0.362541	+	0.931968i	$0.618091\pi$
$462$	2.54677e6	0.555116
$463$	−1.92055e6	−0.416365	−0.208182	−	0.978090i	$-0.566755\pi$
$463$	−1.92055e6	−0.416365	−0.208182	+	0.978090i	$0.566755\pi$
$464$	476345.	0.102713
$465$	0	0
$466$	−5.05305e6	−1.07793
$467$	−4.41186e6	−0.936116	−0.468058	−	0.883698i	$-0.655046\pi$
$467$	−4.41186e6	−0.936116	−0.468058	+	0.883698i	$0.655046\pi$
$468$	−2.25744e6	−0.476433
$469$	7.36223e6	1.54553
$470$	0	0
$471$	9.81120e6	2.03784
$472$	−7.21029e6	−1.48970
$473$	−3.00732e6	−0.618054
$474$	−4.18888e6	−0.856351
$475$	0	0
$476$	−691247.	−0.139835
$477$	−1.96708e6	−0.395846
$478$	−1.34638e6	−0.269524
$479$	−4.96082e6	−0.987905	−0.493952	−	0.869489i	$-0.664448\pi$
$479$	−4.96082e6	−0.987905	−0.493952	+	0.869489i	$0.664448\pi$
$480$	0	0
$481$	2.16936e6	0.427532
$482$	−4.84498e6	−0.949891
$483$	−4.65496e6	−0.907921
$484$	2.03917e6	0.395676
$485$	0	0
$486$	5.88141e6	1.12951
$487$	−2.11174e6	−0.403475	−0.201738	−	0.979440i	$-0.564659\pi$
$487$	−2.11174e6	−0.403475	−0.201738	+	0.979440i	$0.564659\pi$
$488$	−2.79392e6	−0.531085
$489$	1.69346e7	3.20261
$490$	0	0
$491$	4.83234e6	0.904595	0.452297	−	0.891867i	$-0.350605\pi$
$491$	4.83234e6	0.904595	0.452297	+	0.891867i	$0.350605\pi$
$492$	4.03132e6	0.750818
$493$	726879.	0.134693
$494$	210296.	0.0387716
$495$	0	0
$496$	−1.17491e6	−0.214437
$497$	7.63294e6	1.38612
$498$	−8.00097e6	−1.44567
$499$	−3.12941e6	−0.562614	−0.281307	−	0.959618i	$-0.590768\pi$
$499$	−3.12941e6	−0.562614	−0.281307	+	0.959618i	$0.590768\pi$
$500$	0	0
$501$	5.03203e6	0.895672
$502$	−314012.	−0.0556143
$503$	−1.10480e7	−1.94699	−0.973493	−	0.228716i	$-0.926547\pi$
$503$	−1.10480e7	−1.94699	−0.973493	+	0.228716i	$0.926547\pi$
$504$	−1.07772e7	−1.88986
$505$	0	0
$506$	−1.23234e6	−0.213970
$507$	−9.91852e6	−1.71367
$508$	3.65618e6	0.628591
$509$	9.47337e6	1.62073	0.810364	−	0.585927i	$-0.199269\pi$
$509$	9.47337e6	1.62073	0.810364	+	0.585927i	$0.199269\pi$
$510$	0	0
$511$	8.86420e6	1.50171
$512$	2.15916e6	0.364007
$513$	4.07045e6	0.682887
$514$	−1.37888e6	−0.230207
$515$	0	0
$516$	7.36486e6	1.21770
$517$	3.64138e6	0.599156
$518$	4.25697e6	0.697070
$519$	−1.79273e7	−2.92144
$520$	0	0
$521$	−2.20028e6	−0.355127	−0.177564	−	0.984109i	$-0.556822\pi$
$521$	−2.20028e6	−0.355127	−0.177564	+	0.984109i	$0.556822\pi$
$522$	4.65817e6	0.748236
$523$	4.41712e6	0.706131	0.353066	−	0.935599i	$-0.385139\pi$
$523$	4.41712e6	0.706131	0.353066	+	0.935599i	$0.385139\pi$
$524$	2.87070e6	0.456731
$525$	0	0
$526$	−3.01118e6	−0.474540
$527$	−1.79285e6	−0.281201
$528$	1.44843e6	0.226106
$529$	−4.18389e6	−0.650041
$530$	0	0
$531$	2.54234e7	3.91289
$532$	−953305.	−0.146034
$533$	−1.05778e6	−0.161279
$534$	−7.89100e6	−1.19751
$535$	0	0
$536$	−1.16126e7	−1.74589
$537$	−1.10390e7	−1.65194
$538$	−4.98482e6	−0.742495
$539$	−1.40924e6	−0.208936
$540$	0	0
$541$	1.16392e7	1.70974	0.854871	−	0.518841i	$-0.173636\pi$
$541$	1.16392e7	1.70974	0.854871	+	0.518841i	$0.173636\pi$
$542$	−531511.	−0.0777166
$543$	−1.71549e7	−2.49683
$544$	1.73247e6	0.250998
$545$	0	0
$546$	1.63652e6	0.234931
$547$	−552465.	−0.0789472	−0.0394736	−	0.999221i	$-0.512568\pi$
$547$	−552465.	−0.0789472	−0.0394736	+	0.999221i	$0.512568\pi$
$548$	5.59303e6	0.795601
$549$	9.85135e6	1.39497
$550$	0	0
$551$	1.00245e6	0.140664
$552$	7.34236e6	1.02562
$553$	−4.98249e6	−0.692841
$554$	4.88991e6	0.676904
$555$	0	0
$556$	1.51620e6	0.208003
$557$	886782.	0.121110	0.0605548	−	0.998165i	$-0.480713\pi$
$557$	886782.	0.121110	0.0605548	+	0.998165i	$0.480713\pi$
$558$	−1.14894e7	−1.56211
$559$	−1.93247e6	−0.261567
$560$	0	0
$561$	2.21023e6	0.296504
$562$	6.76931e6	0.904073
$563$	−2.83094e6	−0.376409	−0.188205	−	0.982130i	$-0.560267\pi$
$563$	−2.83094e6	−0.376409	−0.188205	+	0.982130i	$0.560267\pi$
$564$	−8.91768e6	−1.18047
$565$	0	0
$566$	943739.	0.123826
$567$	1.61738e7	2.11279
$568$	−1.20396e7	−1.56581
$569$	4.65481e6	0.602727	0.301364	−	0.953509i	$-0.402558\pi$
$569$	4.65481e6	0.602727	0.301364	+	0.953509i	$0.402558\pi$
$570$	0	0
$571$	−2.12068e6	−0.272197	−0.136099	−	0.990695i	$-0.543456\pi$
$571$	−2.12068e6	−0.272197	−0.136099	+	0.990695i	$0.543456\pi$
$572$	−1.00084e6	−0.127902
$573$	−279274.	−0.0355340
$574$	−2.07570e6	−0.262957
$575$	0	0
$576$	7.49250e6	0.940959
$577$	−1.25576e7	−1.57024	−0.785120	−	0.619343i	$-0.787399\pi$
$577$	−1.25576e7	−1.57024	−0.785120	+	0.619343i	$0.787399\pi$
$578$	259687.	0.0323318
$579$	−2.30440e7	−2.85668
$580$	0	0
$581$	−9.51681e6	−1.16964
$582$	−951439.	−0.116432
$583$	−872112.	−0.106268
$584$	−1.39817e7	−1.69639
$585$	0	0
$586$	−934739.	−0.112447
$587$	−4.14828e6	−0.496905	−0.248452	−	0.968644i	$-0.579922\pi$
$587$	−4.14828e6	−0.496905	−0.248452	+	0.968644i	$0.579922\pi$
$588$	3.45120e6	0.411649
$589$	−2.47253e6	−0.293666
$590$	0	0
$591$	1.81951e7	2.14282
$592$	2.42108e6	0.283926
$593$	1.00342e7	1.17178	0.585888	−	0.810392i	$-0.300746\pi$
$593$	1.00342e7	1.17178	0.585888	+	0.810392i	$0.300746\pi$
$594$	8.38584e6	0.975171
$595$	0	0
$596$	−5.45191e6	−0.628685
$597$	1.19917e7	1.37704
$598$	−791885.	−0.0905544
$599$	3.04010e6	0.346194	0.173097	−	0.984905i	$-0.444623\pi$
$599$	3.04010e6	0.346194	0.173097	+	0.984905i	$0.444623\pi$
$600$	0	0
$601$	4.79829e6	0.541877	0.270938	−	0.962597i	$-0.412666\pi$
$601$	4.79829e6	0.541877	0.270938	+	0.962597i	$0.412666\pi$
$602$	−3.79212e6	−0.426472
$603$	4.09459e7	4.58583
$604$	3.73609e6	0.416701
$605$	0	0
$606$	−1.56350e6	−0.172948
$607$	−1.41879e6	−0.156295	−0.0781475	−	0.996942i	$-0.524901\pi$
$607$	−1.41879e6	−0.156295	−0.0781475	+	0.996942i	$0.524901\pi$
$608$	2.38927e6	0.262124
$609$	7.80103e6	0.852332
$610$	0	0
$611$	2.33991e6	0.253569
$612$	−3.84445e6	−0.414912
$613$	1.00420e7	1.07937	0.539684	−	0.841868i	$-0.318544\pi$
$613$	1.00420e7	1.07937	0.539684	+	0.841868i	$0.318544\pi$
$614$	−7.30445e6	−0.781927
$615$	0	0
$616$	−4.77812e6	−0.507348
$617$	6.10343e6	0.645447	0.322724	−	0.946493i	$-0.395402\pi$
$617$	6.10343e6	0.645447	0.322724	+	0.946493i	$0.395402\pi$
$618$	−4.41722e6	−0.465241
$619$	9.79469e6	1.02746	0.513729	−	0.857952i	$-0.328264\pi$
$619$	9.79469e6	1.02746	0.513729	+	0.857952i	$0.328264\pi$
$620$	0	0
$621$	−1.53276e7	−1.59494
$622$	−6.43059e6	−0.666461
$623$	−9.38601e6	−0.968859
$624$	930744.	0.0956905
$625$	0	0
$626$	−163737.	−0.0166998
$627$	3.04815e6	0.309647
$628$	−7.56605e6	−0.765544
$629$	3.69445e6	0.372325
$630$	0	0
$631$	−5.10429e6	−0.510343	−0.255171	−	0.966896i	$-0.582132\pi$
$631$	−5.10429e6	−0.510343	−0.255171	+	0.966896i	$0.582132\pi$
$632$	7.85898e6	0.782660
$633$	2.91313e7	2.88969
$634$	5.81792e6	0.574837
$635$	0	0
$636$	2.13578e6	0.209370
$637$	−905562.	−0.0884239
$638$	2.06522e6	0.200869
$639$	4.24515e7	4.11283
$640$	0	0
$641$	−2.55088e6	−0.245214	−0.122607	−	0.992455i	$-0.539125\pi$
$641$	−2.55088e6	−0.245214	−0.122607	+	0.992455i	$0.539125\pi$
$642$	−1.01262e7	−0.969633
$643$	1.21014e7	1.15427	0.577136	−	0.816648i	$-0.304170\pi$
$643$	1.21014e7	1.15427	0.577136	+	0.816648i	$0.304170\pi$
$644$	3.58974e6	0.341074
$645$	0	0
$646$	358137.	0.0337650
$647$	5.82166e6	0.546747	0.273373	−	0.961908i	$-0.411861\pi$
$647$	5.82166e6	0.546747	0.273373	+	0.961908i	$0.411861\pi$
$648$	−2.55113e7	−2.38668
$649$	1.12716e7	1.05044
$650$	0	0
$651$	−1.92413e7	−1.77943
$652$	−1.30594e7	−1.20311
$653$	−9.97911e6	−0.915818	−0.457909	−	0.888999i	$-0.651401\pi$
$653$	−9.97911e6	−0.915818	−0.457909	+	0.888999i	$0.651401\pi$
$654$	−8.11249e6	−0.741669
$655$	0	0
$656$	−1.18052e6	−0.107106
$657$	4.92993e7	4.45582
$658$	4.59165e6	0.413432
$659$	1.28125e7	1.14927	0.574633	−	0.818411i	$-0.305145\pi$
$659$	1.28125e7	1.14927	0.574633	+	0.818411i	$0.305145\pi$
$660$	0	0
$661$	2.09491e7	1.86493	0.932463	−	0.361265i	$-0.117655\pi$
$661$	2.09491e7	1.86493	0.932463	+	0.361265i	$0.117655\pi$
$662$	559274.	0.0495997
$663$	1.42027e6	0.125483
$664$	1.50110e7	1.32127
$665$	0	0
$666$	2.36757e7	2.06832
$667$	−3.77479e6	−0.328532
$668$	−3.88053e6	−0.336473
$669$	1.79233e7	1.54829
$670$	0	0
$671$	4.36763e6	0.374490
$672$	1.85933e7	1.58830
$673$	3.87942e6	0.330164	0.165082	−	0.986280i	$-0.447211\pi$
$673$	3.87942e6	0.330164	0.165082	+	0.986280i	$0.447211\pi$
$674$	−1.22426e7	−1.03806
$675$	0	0
$676$	7.64882e6	0.643765
$677$	−6.42054e6	−0.538393	−0.269197	−	0.963085i	$-0.586758\pi$
$677$	−6.42054e6	−0.538393	−0.269197	+	0.963085i	$0.586758\pi$
$678$	9.89125e6	0.826375
$679$	−1.13170e6	−0.0942010
$680$	0	0
$681$	−1.15847e7	−0.957233
$682$	−5.09386e6	−0.419359
$683$	5.44303e6	0.446467	0.223233	−	0.974765i	$-0.428339\pi$
$683$	5.44303e6	0.446467	0.223233	+	0.974765i	$0.428339\pi$
$684$	−5.30192e6	−0.433304
$685$	0	0
$686$	−7.37380e6	−0.598248
$687$	−9.54028e6	−0.771204
$688$	−2.15670e6	−0.173708
$689$	−560409.	−0.0449736
$690$	0	0
$691$	−5.89266e6	−0.469479	−0.234740	−	0.972058i	$-0.575424\pi$
$691$	−5.89266e6	−0.469479	−0.234740	+	0.972058i	$0.575424\pi$
$692$	1.38249e7	1.09748
$693$	1.68476e7	1.33262
$694$	8.22313e6	0.648095
$695$	0	0
$696$	−1.23047e7	−0.962827
$697$	−1.80141e6	−0.140453
$698$	1.56568e6	0.121637
$699$	−4.70643e7	−3.64333
$700$	0	0
$701$	8.12271e6	0.624318	0.312159	−	0.950030i	$-0.398948\pi$
$701$	8.12271e6	0.624318	0.312159	+	0.950030i	$0.398948\pi$
$702$	5.38865e6	0.412702
$703$	5.09504e6	0.388830
$704$	3.32183e6	0.252607
$705$	0	0
$706$	1.16807e7	0.881977
$707$	−1.85972e6	−0.139926
$708$	−2.76039e7	−2.06960
$709$	−2.46598e7	−1.84236	−0.921179	−	0.389139i	$-0.872773\pi$
$709$	−2.46598e7	−1.84236	−0.921179	+	0.389139i	$0.872773\pi$
$710$	0	0
$711$	−2.77107e7	−2.05577
$712$	1.48047e7	1.09446
$713$	9.31052e6	0.685883
$714$	2.78702e6	0.204595
$715$	0	0
$716$	8.51289e6	0.620576
$717$	−1.25402e7	−0.910978
$718$	1.45956e7	1.05660
$719$	−1.78878e7	−1.29043	−0.645215	−	0.764001i	$-0.723232\pi$
$719$	−1.78878e7	−1.29043	−0.645215	+	0.764001i	$0.723232\pi$
$720$	0	0
$721$	−5.25409e6	−0.376409
$722$	−7.20488e6	−0.514379
$723$	−4.51263e7	−3.21058
$724$	1.32293e7	0.937971
$725$	0	0
$726$	−8.22168e6	−0.578920
$727$	1.67266e7	1.17374	0.586868	−	0.809683i	$-0.300361\pi$
$727$	1.67266e7	1.17374	0.586868	+	0.809683i	$0.300361\pi$
$728$	−3.07037e6	−0.214715
$729$	1.80833e7	1.26026
$730$	0	0
$731$	−3.29102e6	−0.227791
$732$	−1.06962e7	−0.737825
$733$	1.61327e7	1.10904	0.554519	−	0.832171i	$-0.312902\pi$
$733$	1.61327e7	1.10904	0.554519	+	0.832171i	$0.312902\pi$
$734$	−1.48540e7	−1.01766
$735$	0	0
$736$	−8.99699e6	−0.612213
$737$	1.81535e7	1.23110
$738$	−1.15443e7	−0.780235
$739$	1.67085e6	0.112545	0.0562725	−	0.998415i	$-0.482078\pi$
$739$	1.67085e6	0.112545	0.0562725	+	0.998415i	$0.482078\pi$
$740$	0	0
$741$	1.95870e6	0.131046
$742$	−1.09970e6	−0.0733272
$743$	−1.96655e7	−1.30687	−0.653437	−	0.756981i	$-0.726674\pi$
$743$	−1.96655e7	−1.30687	−0.653437	+	0.756981i	$0.726674\pi$
$744$	3.03496e7	2.01011
$745$	0	0
$746$	−2.49642e6	−0.164237
$747$	−5.29289e7	−3.47049
$748$	−1.70445e6	−0.111386
$749$	−1.20446e7	−0.784494
$750$	0	0
$751$	2.27263e7	1.47037	0.735187	−	0.677864i	$-0.237094\pi$
$751$	2.27263e7	1.47037	0.735187	+	0.677864i	$0.237094\pi$
$752$	2.61142e6	0.168396
$753$	−2.92472e6	−0.187973
$754$	1.32708e6	0.0850100
$755$	0	0
$756$	−2.44276e7	−1.55445
$757$	2.55716e7	1.62188	0.810940	−	0.585129i	$-0.198956\pi$
$757$	2.55716e7	1.62188	0.810940	+	0.585129i	$0.198956\pi$
$758$	−6.00903e6	−0.379867
$759$	−1.14780e7	−0.723208
$760$	0	0
$761$	−2.09253e7	−1.30982	−0.654909	−	0.755708i	$-0.727293\pi$
$761$	−2.09253e7	−1.30982	−0.654909	+	0.755708i	$0.727293\pi$
$762$	−1.47413e7	−0.919702
$763$	−9.64946e6	−0.600056
$764$	215367.	0.0133489
$765$	0	0
$766$	−1.09375e7	−0.673515
$767$	7.24299e6	0.444559
$768$	−2.54080e7	−1.55442
$769$	1.76503e7	1.07631	0.538154	−	0.842847i	$-0.319122\pi$
$769$	1.76503e7	1.07631	0.538154	+	0.842847i	$0.319122\pi$
$770$	0	0
$771$	−1.28430e7	−0.778089
$772$	1.77707e7	1.07315
$773$	−5.09512e6	−0.306694	−0.153347	−	0.988172i	$-0.549005\pi$
$773$	−5.09512e6	−0.306694	−0.153347	+	0.988172i	$0.549005\pi$
$774$	−2.10903e7	−1.26541
$775$	0	0
$776$	1.78505e6	0.106413
$777$	3.96496e7	2.35606
$778$	1.50983e7	0.894291
$779$	−2.48434e6	−0.146679
$780$	0	0
$781$	1.88210e7	1.10412
$782$	−1.34859e6	−0.0788612
$783$	2.56868e7	1.49729
$784$	−1.01064e6	−0.0587227
$785$	0	0
$786$	−1.15743e7	−0.668250
$787$	−1.94455e7	−1.11914	−0.559568	−	0.828784i	$-0.689033\pi$
$787$	−1.94455e7	−1.11914	−0.559568	+	0.828784i	$0.689033\pi$
$788$	−1.40315e7	−0.804984
$789$	−2.80463e7	−1.60392
$790$	0	0
$791$	1.17652e7	0.668589
$792$	−2.65741e7	−1.50538
$793$	2.80659e6	0.158488
$794$	1.36882e6	0.0770542
$795$	0	0
$796$	−9.24761e6	−0.517305
$797$	−3.01184e7	−1.67952	−0.839762	−	0.542954i	$-0.817306\pi$
$797$	−3.01184e7	−1.67952	−0.839762	+	0.542954i	$0.817306\pi$
$798$	3.84360e6	0.213664
$799$	3.98490e6	0.220826
$800$	0	0
$801$	−5.22014e7	−2.87475
$802$	−3.77732e6	−0.207371
$803$	2.18570e7	1.19620
$804$	−4.44576e7	−2.42553
$805$	0	0
$806$	−3.27325e6	−0.177477
$807$	−4.64288e7	−2.50959
$808$	2.93337e6	0.158066
$809$	−3.30956e6	−0.177787	−0.0888934	−	0.996041i	$-0.528333\pi$
$809$	−3.30956e6	−0.177787	−0.0888934	+	0.996041i	$0.528333\pi$
$810$	0	0
$811$	4.44276e6	0.237192	0.118596	−	0.992943i	$-0.462161\pi$
$811$	4.44276e6	0.237192	0.118596	+	0.992943i	$0.462161\pi$
$812$	−6.01588e6	−0.320191
$813$	−4.95051e6	−0.262678
$814$	1.04967e7	0.555254
$815$	0	0
$816$	1.58507e6	0.0833341
$817$	−4.53867e6	−0.237889
$818$	1.36570e6	0.0713629
$819$	1.08261e7	0.563978
$820$	0	0
$821$	−3.43695e6	−0.177957	−0.0889786	−	0.996034i	$-0.528360\pi$
$821$	−3.43695e6	−0.177957	−0.0889786	+	0.996034i	$0.528360\pi$
$822$	−2.25504e7	−1.16406
$823$	−1.47868e7	−0.760980	−0.380490	−	0.924785i	$-0.624245\pi$
$823$	−1.47868e7	−0.760980	−0.380490	+	0.924785i	$0.624245\pi$
$824$	8.28738e6	0.425206
$825$	0	0
$826$	1.42131e7	0.724832
$827$	1.50566e7	0.765534	0.382767	−	0.923845i	$-0.374971\pi$
$827$	1.50566e7	0.765534	0.382767	+	0.923845i	$0.374971\pi$
$828$	1.99648e7	1.01202
$829$	2.64799e7	1.33823	0.669115	−	0.743159i	$-0.266673\pi$
$829$	2.64799e7	1.33823	0.669115	+	0.743159i	$0.266673\pi$
$830$	0	0
$831$	4.55449e7	2.28790
$832$	2.13457e6	0.106906
$833$	−1.54218e6	−0.0770058
$834$	−6.11312e6	−0.304332
$835$	0	0
$836$	−2.35063e6	−0.116324
$837$	−6.33565e7	−3.12592
$838$	−9.04718e6	−0.445044
$839$	−1.16506e7	−0.571405	−0.285703	−	0.958318i	$-0.592227\pi$
$839$	−1.16506e7	−0.571405	−0.285703	+	0.958318i	$0.592227\pi$
$840$	0	0
$841$	−1.41852e7	−0.691583
$842$	8.30267e6	0.403587
$843$	6.30496e7	3.05572
$844$	−2.24650e7	−1.08555
$845$	0	0
$846$	2.55370e7	1.22672
$847$	−9.77933e6	−0.468383
$848$	−625437.	−0.0298671
$849$	8.79002e6	0.418524
$850$	0	0
$851$	−1.91858e7	−0.908146
$852$	−4.60923e7	−2.17535
$853$	1.30715e7	0.615111	0.307556	−	0.951530i	$-0.400489\pi$
$853$	1.30715e7	0.615111	0.307556	+	0.951530i	$0.400489\pi$
$854$	5.50743e6	0.258407
$855$	0	0
$856$	1.89982e7	0.886195
$857$	1.58656e7	0.737911	0.368956	−	0.929447i	$-0.379715\pi$
$857$	1.58656e7	0.737911	0.368956	+	0.929447i	$0.379715\pi$
$858$	4.03528e6	0.187135
$859$	−3.10164e7	−1.43420	−0.717099	−	0.696972i	$-0.754530\pi$
$859$	−3.10164e7	−1.43420	−0.717099	+	0.696972i	$0.754530\pi$
$860$	0	0
$861$	−1.93332e7	−0.888782
$862$	−4.88506e6	−0.223925
$863$	1.30372e7	0.595879	0.297939	−	0.954585i	$-0.403701\pi$
$863$	1.30372e7	0.595879	0.297939	+	0.954585i	$0.403701\pi$
$864$	6.12230e7	2.79017
$865$	0	0
$866$	−3.58182e6	−0.162296
$867$	2.41873e6	0.109280
$868$	1.48382e7	0.668469
$869$	−1.22856e7	−0.551885
$870$	0	0
$871$	1.16653e7	0.521013
$872$	1.52203e7	0.677847
$873$	−6.29406e6	−0.279509
$874$	−1.85985e6	−0.0823570
$875$	0	0
$876$	−5.35274e7	−2.35676
$877$	−1.66227e7	−0.729797	−0.364899	−	0.931047i	$-0.618896\pi$
$877$	−1.66227e7	−0.729797	−0.364899	+	0.931047i	$0.618896\pi$
$878$	2.11647e7	0.926565
$879$	−8.70619e6	−0.380063
$880$	0	0
$881$	1.83375e7	0.795976	0.397988	−	0.917391i	$-0.369709\pi$
$881$	1.83375e7	0.795976	0.397988	+	0.917391i	$0.369709\pi$
$882$	−9.88301e6	−0.427777
$883$	1.34948e7	0.582457	0.291229	−	0.956653i	$-0.405936\pi$
$883$	1.34948e7	0.582457	0.291229	+	0.956653i	$0.405936\pi$
$884$	−1.09526e6	−0.0471397
$885$	0	0
$886$	−1.42389e7	−0.609386
$887$	2.98004e7	1.27178	0.635891	−	0.771779i	$-0.280633\pi$
$887$	2.98004e7	1.27178	0.635891	+	0.771779i	$0.280633\pi$
$888$	−6.25401e7	−2.66150
$889$	−1.75341e7	−0.744096
$890$	0	0
$891$	3.98809e7	1.68295
$892$	−1.38219e7	−0.581640
$893$	5.49561e6	0.230615
$894$	2.19814e7	0.919839
$895$	0	0
$896$	−1.63567e7	−0.680652
$897$	−7.37565e6	−0.306069
$898$	1.50660e7	0.623457
$899$	−1.56031e7	−0.643888
$900$	0	0
$901$	−954384.	−0.0391662
$902$	−5.11819e6	−0.209460
$903$	−3.53200e7	−1.44145
$904$	−1.85575e7	−0.755264
$905$	0	0
$906$	−1.50634e7	−0.609682
$907$	3.17531e6	0.128165	0.0640823	−	0.997945i	$-0.479588\pi$
$907$	3.17531e6	0.128165	0.0640823	+	0.997945i	$0.479588\pi$
$908$	8.93373e6	0.359599
$909$	−1.03430e7	−0.415182
$910$	0	0
$911$	3.06040e7	1.22175	0.610876	−	0.791726i	$-0.290817\pi$
$911$	3.06040e7	1.22175	0.610876	+	0.791726i	$0.290817\pi$
$912$	2.18598e6	0.0870281
$913$	−2.34662e7	−0.931679
$914$	−8.19999e6	−0.324674
$915$	0	0
$916$	7.35713e6	0.289714
$917$	−1.37671e7	−0.540656
$918$	9.17693e6	0.359411
$919$	1.93193e7	0.754575	0.377288	−	0.926096i	$-0.376857\pi$
$919$	1.93193e7	0.754575	0.377288	+	0.926096i	$0.376857\pi$
$920$	0	0
$921$	−6.80339e7	−2.64287
$922$	−1.02871e7	−0.398535
$923$	1.20942e7	0.467274
$924$	−1.82926e7	−0.704847
$925$	0	0
$926$	−5.97146e6	−0.228851
$927$	−2.92213e7	−1.11686
$928$	1.50776e7	0.574729
$929$	−4.05375e7	−1.54105	−0.770527	−	0.637407i	$-0.780007\pi$
$929$	−4.05375e7	−1.54105	−0.770527	+	0.637407i	$0.780007\pi$
$930$	0	0
$931$	−2.12684e6	−0.0804194
$932$	3.62944e7	1.36867
$933$	−5.98947e7	−2.25260
$934$	−1.37175e7	−0.514528
$935$	0	0
$936$	−1.70762e7	−0.637092
$937$	4.17191e7	1.55234	0.776168	−	0.630527i	$-0.217161\pi$
$937$	4.17191e7	1.55234	0.776168	+	0.630527i	$0.217161\pi$
$938$	2.28909e7	0.849487
$939$	−1.52505e6	−0.0564443
$940$	0	0
$941$	1.08124e6	0.0398058	0.0199029	−	0.999802i	$-0.493664\pi$
$941$	1.08124e6	0.0398058	0.0199029	+	0.999802i	$0.493664\pi$
$942$	3.05054e7	1.12008
$943$	9.35500e6	0.342582
$944$	8.08343e6	0.295233
$945$	0	0
$946$	−9.35047e6	−0.339708
$947$	1.42216e7	0.515315	0.257658	−	0.966236i	$-0.417049\pi$
$947$	1.42216e7	0.515315	0.257658	+	0.966236i	$0.417049\pi$
$948$	3.00873e7	1.08733
$949$	1.40451e7	0.506242
$950$	0	0
$951$	5.41883e7	1.94292
$952$	−5.22887e6	−0.186989
$953$	−3.49208e7	−1.24552	−0.622762	−	0.782412i	$-0.713989\pi$
$953$	−3.49208e7	−1.24552	−0.622762	+	0.782412i	$0.713989\pi$
$954$	−6.11612e6	−0.217573
$955$	0	0
$956$	9.67060e6	0.342223
$957$	1.92355e7	0.678928
$958$	−1.54244e7	−0.542993
$959$	−2.68227e7	−0.941794
$960$	0	0
$961$	9.85582e6	0.344258
$962$	6.74506e6	0.234989
$963$	−6.69877e7	−2.32771
$964$	3.47998e7	1.20610
$965$	0	0
$966$	−1.44734e7	−0.499031
$967$	−4.78677e7	−1.64618	−0.823089	−	0.567913i	$-0.807751\pi$
$967$	−4.78677e7	−1.64618	−0.823089	+	0.567913i	$0.807751\pi$
$968$	1.54251e7	0.529103
$969$	3.33570e6	0.114124
$970$	0	0
$971$	−2.74029e7	−0.932712	−0.466356	−	0.884597i	$-0.654433\pi$
$971$	−2.74029e7	−0.932712	−0.466356	+	0.884597i	$0.654433\pi$
$972$	−4.22442e7	−1.43417
$973$	−7.27130e6	−0.246224
$974$	−6.56589e6	−0.221767
$975$	0	0
$976$	3.13225e6	0.105252
$977$	−1.11984e7	−0.375334	−0.187667	−	0.982233i	$-0.560093\pi$
$977$	−1.11984e7	−0.375334	−0.187667	+	0.982233i	$0.560093\pi$
$978$	5.26538e7	1.76028
$979$	−2.31437e7	−0.771749
$980$	0	0
$981$	−5.36666e7	−1.78046
$982$	1.50249e7	0.497203
$983$	4.29207e7	1.41672	0.708358	−	0.705853i	$-0.249436\pi$
$983$	4.29207e7	1.41672	0.708358	+	0.705853i	$0.249436\pi$
$984$	3.04946e7	1.00400
$985$	0	0
$986$	2.26004e6	0.0740327
$987$	4.27668e7	1.39738
$988$	−1.51048e6	−0.0492293
$989$	1.70907e7	0.555610
$990$	0	0
$991$	5.42427e7	1.75451	0.877257	−	0.480021i	$-0.159371\pi$
$991$	5.42427e7	1.75451	0.877257	+	0.480021i	$0.159371\pi$
$992$	−3.71890e7	−1.19987
$993$	5.20910e6	0.167644
$994$	2.37326e7	0.761868
$995$	0	0
$996$	5.74683e7	1.83561
$997$	7.27608e6	0.231825	0.115912	−	0.993259i	$-0.463021\pi$
$997$	7.27608e6	0.231825	0.115912	+	0.993259i	$0.463021\pi$
$998$	−9.73007e6	−0.309236
$999$	1.30556e8	4.13888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.6.a.k.1.10	yes	15
5.4	even	2		425.6.a.j.1.6	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.6.a.j.1.6	✓	15	5.4	even	2
425.6.a.k.1.10	yes	15	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q+3.10924 q^{2} +28.9596 q^{3} -22.3326 q^{4} +90.0423 q^{6} +107.101 q^{7} -168.933 q^{8} +595.658 q^{9} +264.087 q^{11} -646.744 q^{12} +169.699 q^{13} +333.004 q^{14} +189.390 q^{16} +289.000 q^{17} +1852.04 q^{18} +398.563 q^{19} +3101.61 q^{21} +821.110 q^{22} -1500.82 q^{23} -4892.23 q^{24} +527.636 q^{26} +10212.8 q^{27} -2391.86 q^{28} +2515.15 q^{29} -6203.63 q^{31} +5994.72 q^{32} +7647.85 q^{33} +898.570 q^{34} -13302.6 q^{36} +12783.5 q^{37} +1239.23 q^{38} +4914.42 q^{39} -6233.26 q^{41} +9643.66 q^{42} -11387.6 q^{43} -5897.76 q^{44} -4666.41 q^{46} +13788.6 q^{47} +5484.67 q^{48} -5336.28 q^{49} +8369.32 q^{51} -3789.83 q^{52} -3302.37 q^{53} +31754.1 q^{54} -18093.0 q^{56} +11542.2 q^{57} +7820.21 q^{58} +42681.3 q^{59} +16538.6 q^{61} -19288.6 q^{62} +63795.8 q^{63} +12578.5 q^{64} +23779.0 q^{66} +68740.7 q^{67} -6454.13 q^{68} -43463.1 q^{69} +71268.3 q^{71} -100626. q^{72} +82764.5 q^{73} +39747.1 q^{74} -8900.95 q^{76} +28284.1 q^{77} +15280.1 q^{78} -46521.2 q^{79} +151014. q^{81} -19380.7 q^{82} -88857.9 q^{83} -69267.2 q^{84} -35406.8 q^{86} +72837.7 q^{87} -44613.1 q^{88} -87636.6 q^{89} +18175.0 q^{91} +33517.2 q^{92} -179654. q^{93} +42872.0 q^{94} +173605. q^{96} -10566.6 q^{97} -16591.8 q^{98} +157305. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{2} + 9 q^{3} + 277 q^{4} + 169 q^{6} + 181 q^{7} + 753 q^{8} + 1826 q^{9} + 172 q^{11} - 2109 q^{12} - 389 q^{13} + 3635 q^{14} + 6837 q^{16} + 4335 q^{17} + 6742 q^{18} + 5150 q^{19} - 6891 q^{21}+ \cdots - 183214 q^{99}+O(q^{100}) \) 15 * q + q^2 + 9 * q^3 + 277 * q^4 + 169 * q^6 + 181 * q^7 + 753 * q^8 + 1826 * q^9 + 172 * q^11 - 2109 * q^12 - 389 * q^13 + 3635 * q^14 + 6837 * q^16 + 4335 * q^17 + 6742 * q^18 + 5150 * q^19 - 6891 * q^21 - 10522 * q^22 + 3140 * q^23 + 9263 * q^24 + 3181 * q^26 + 1611 * q^27 + 16371 * q^28 + 15214 * q^29 + 18873 * q^31 + 16857 * q^32 - 14700 * q^33 + 289 * q^34 + 83010 * q^36 + 12122 * q^37 - 5302 * q^38 + 17995 * q^39 - 2336 * q^41 - 65201 * q^42 - 660 * q^43 + 40740 * q^44 - 18610 * q^46 + 11494 * q^47 - 48879 * q^48 + 89846 * q^49 + 2601 * q^51 - 71027 * q^52 - 76605 * q^53 - 33675 * q^54 + 71385 * q^56 + 112044 * q^57 - 11270 * q^58 + 7268 * q^59 + 137458 * q^61 + 212145 * q^62 - 25488 * q^63 + 116341 * q^64 + 87900 * q^66 - 5036 * q^67 + 80053 * q^68 - 57176 * q^69 + 180961 * q^71 + 588818 * q^72 - 176062 * q^73 + 11010 * q^74 + 390158 * q^76 + 319456 * q^77 - 339079 * q^78 + 127813 * q^79 + 419847 * q^81 + 200996 * q^82 - 318176 * q^83 - 373745 * q^84 + 214564 * q^86 + 138204 * q^87 - 722070 * q^88 + 97682 * q^89 + 229269 * q^91 + 202080 * q^92 - 514385 * q^93 + 104594 * q^94 + 706465 * q^96 + 381812 * q^97 - 399302 * q^98 - 183214 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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