LMFDB - Embedded newform 325.6.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,6,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("325.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	325.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:	$52.1247414392$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 161x^{4} + 328x^{3} + 6584x^{2} - 10688x - 47440 $ x^6 - 2x^5 - 161x^4 + 328x^3 + 6584x^2 - 10688*x - 47440
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.15349$ of defining polynomial
Character	$\chi$	$=$	325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.15349 q^{2} -29.2293 q^{3} -14.7485 q^{4} +121.404 q^{6} -42.5171 q^{7} +194.170 q^{8} +611.349 q^{9} +434.431 q^{11} +431.087 q^{12} +169.000 q^{13} +176.594 q^{14} -334.530 q^{16} +424.475 q^{17} -2539.24 q^{18} -2203.44 q^{19} +1242.74 q^{21} -1804.41 q^{22} +1176.65 q^{23} -5675.43 q^{24} -701.940 q^{26} -10766.6 q^{27} +627.063 q^{28} -3074.40 q^{29} +6242.41 q^{31} -4823.96 q^{32} -12698.1 q^{33} -1763.05 q^{34} -9016.48 q^{36} +10102.1 q^{37} +9151.97 q^{38} -4939.74 q^{39} -7026.52 q^{41} -5161.73 q^{42} +21517.8 q^{43} -6407.20 q^{44} -4887.22 q^{46} -3086.13 q^{47} +9778.08 q^{48} -14999.3 q^{49} -12407.1 q^{51} -2492.49 q^{52} -38178.9 q^{53} +44718.9 q^{54} -8255.53 q^{56} +64404.9 q^{57} +12769.5 q^{58} -17513.0 q^{59} -12844.5 q^{61} -25927.8 q^{62} -25992.8 q^{63} +30741.2 q^{64} +52741.5 q^{66} +46496.3 q^{67} -6260.37 q^{68} -34392.7 q^{69} +55843.2 q^{71} +118705. q^{72} -49566.9 q^{73} -41958.9 q^{74} +32497.4 q^{76} -18470.8 q^{77} +20517.2 q^{78} +8611.37 q^{79} +166141. q^{81} +29184.6 q^{82} +1635.95 q^{83} -18328.6 q^{84} -89374.2 q^{86} +89862.3 q^{87} +84353.3 q^{88} +26554.2 q^{89} -7185.39 q^{91} -17353.9 q^{92} -182461. q^{93} +12818.2 q^{94} +141001. q^{96} -181767. q^{97} +62299.5 q^{98} +265589. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - 20 q^{3} + 134 q^{4} - 52 q^{6} - 172 q^{7} + 138 q^{8} + 1034 q^{9} + 800 q^{11} + 832 q^{12} + 1014 q^{13} + 2108 q^{14} + 322 q^{16} - 4396 q^{17} - 6142 q^{18} + 5304 q^{19} + 1072 q^{21}+ \cdots + 301264 q^{99}+O(q^{100}) $ 6 * q - 2 * q^2 - 20 * q^3 + 134 * q^4 - 52 * q^6 - 172 * q^7 + 138 * q^8 + 1034 * q^9 + 800 * q^11 + 832 * q^12 + 1014 * q^13 + 2108 * q^14 + 322 * q^16 - 4396 * q^17 - 6142 * q^18 + 5304 * q^19 + 1072 * q^21 - 8008 * q^22 + 1140 * q^23 + 4756 * q^24 - 338 * q^26 - 800 * q^27 - 524 * q^28 - 3664 * q^29 + 16664 * q^31 + 9894 * q^32 + 13092 * q^33 + 1380 * q^34 + 1818 * q^36 - 4488 * q^37 + 30112 * q^38 - 3380 * q^39 + 12436 * q^41 + 68336 * q^42 - 1516 * q^43 - 35124 * q^44 - 4556 * q^46 - 212 * q^47 + 16712 * q^48 - 27842 * q^49 - 27664 * q^51 + 22646 * q^52 + 15612 * q^53 - 132912 * q^54 + 44420 * q^56 + 86796 * q^57 + 39664 * q^58 - 11896 * q^59 + 57232 * q^61 + 95488 * q^62 - 207956 * q^63 - 39914 * q^64 + 9908 * q^66 + 25428 * q^67 + 16236 * q^68 + 72404 * q^69 + 214104 * q^71 + 130806 * q^72 - 60808 * q^73 - 182104 * q^74 + 366636 * q^76 + 5192 * q^77 - 8788 * q^78 + 36024 * q^79 + 350514 * q^81 + 1124 * q^82 - 34684 * q^83 + 6024 * q^84 + 29204 * q^86 + 8640 * q^87 + 40520 * q^88 - 52028 * q^89 - 29068 * q^91 + 283232 * q^92 - 26508 * q^93 + 157572 * q^94 + 547908 * q^96 - 129316 * q^97 + 113942 * q^98 + 301264 * 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 $

$2$	−4.15349	−0.734241	−0.367120	−	0.930173i	$-0.619656\pi$
$2$	−4.15349	−0.734241	−0.367120	+	0.930173i	$0.619656\pi$
$3$	−29.2293	−1.87506	−0.937529	−	0.347908i	$-0.886892\pi$
$3$	−29.2293	−1.87506	−0.937529	+	0.347908i	$0.886892\pi$
$4$	−14.7485	−0.460890
$5$	0	0
$5$	0	0
$6$	121.404	1.37674
$7$	−42.5171	−0.327958	−0.163979	−	0.986464i	$-0.552433\pi$
$7$	−42.5171	−0.327958	−0.163979	+	0.986464i	$0.552433\pi$
$8$	194.170	1.07265
$9$	611.349	2.51584
$10$	0	0
$11$	434.431	1.08253	0.541264	−	0.840853i	$-0.317946\pi$
$11$	434.431	1.08253	0.541264	+	0.840853i	$0.317946\pi$
$12$	431.087	0.864196
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	176.594	0.240800
$15$	0	0
$16$	−334.530	−0.326690
$17$	424.475	0.356229	0.178115	−	0.984010i	$-0.443000\pi$
$17$	424.475	0.356229	0.178115	+	0.984010i	$0.443000\pi$
$18$	−2539.24	−1.84723
$19$	−2203.44	−1.40029	−0.700143	−	0.714003i	$-0.746880\pi$
$19$	−2203.44	−1.40029	−0.700143	+	0.714003i	$0.746880\pi$
$20$	0	0
$21$	1242.74	0.614940
$22$	−1804.41	−0.794837
$23$	1176.65	0.463798	0.231899	−	0.972740i	$-0.425506\pi$
$23$	1176.65	0.463798	0.231899	+	0.972740i	$0.425506\pi$
$24$	−5675.43	−2.01127
$25$	0	0
$26$	−701.940	−0.203642
$27$	−10766.6	−2.84229
$28$	627.063	0.151153
$29$	−3074.40	−0.678836	−0.339418	−	0.940636i	$-0.610230\pi$
$29$	−3074.40	−0.678836	−0.339418	+	0.940636i	$0.610230\pi$
$30$	0	0
$31$	6242.41	1.16667	0.583335	−	0.812231i	$-0.301747\pi$
$31$	6242.41	1.16667	0.583335	+	0.812231i	$0.301747\pi$
$32$	−4823.96	−0.832776
$33$	−12698.1	−2.02980
$34$	−1763.05	−0.261558
$35$	0	0
$36$	−9016.48	−1.15953
$37$	10102.1	1.21313	0.606563	−	0.795035i	$-0.292548\pi$
$37$	10102.1	1.21313	0.606563	+	0.795035i	$0.292548\pi$
$38$	9151.97	1.02815
$39$	−4939.74	−0.520047
$40$	0	0
$41$	−7026.52	−0.652801	−0.326400	−	0.945232i	$-0.605836\pi$
$41$	−7026.52	−0.652801	−0.326400	+	0.945232i	$0.605836\pi$
$42$	−5161.73	−0.451514
$43$	21517.8	1.77471	0.887355	−	0.461086i	$-0.152540\pi$
$43$	21517.8	1.77471	0.887355	+	0.461086i	$0.152540\pi$
$44$	−6407.20	−0.498927
$45$	0	0
$46$	−4887.22	−0.340539
$47$	−3086.13	−0.203784	−0.101892	−	0.994795i	$-0.532490\pi$
$47$	−3086.13	−0.203784	−0.101892	+	0.994795i	$0.532490\pi$
$48$	9778.08	0.612562
$49$	−14999.3	−0.892443
$50$	0	0
$51$	−12407.1	−0.667951
$52$	−2492.49	−0.127828
$53$	−38178.9	−1.86695	−0.933477	−	0.358638i	$-0.883241\pi$
$53$	−38178.9	−1.86695	−0.933477	+	0.358638i	$0.883241\pi$
$54$	44718.9	2.08693
$55$	0	0
$56$	−8255.53	−0.351783
$57$	64404.9	2.62562
$58$	12769.5	0.498429
$59$	−17513.0	−0.654984	−0.327492	−	0.944854i	$-0.606203\pi$
$59$	−17513.0	−0.654984	−0.327492	+	0.944854i	$0.606203\pi$
$60$	0	0
$61$	−12844.5	−0.441969	−0.220984	−	0.975277i	$-0.570927\pi$
$61$	−12844.5	−0.441969	−0.220984	+	0.975277i	$0.570927\pi$
$62$	−25927.8	−0.856617
$63$	−25992.8	−0.825091
$64$	30741.2	0.938148
$65$	0	0
$66$	52741.5	1.49036
$67$	46496.3	1.26541	0.632705	−	0.774393i	$-0.281945\pi$
$67$	46496.3	1.26541	0.632705	+	0.774393i	$0.281945\pi$
$68$	−6260.37	−0.164183
$69$	−34392.7	−0.869648
$70$	0	0
$71$	55843.2	1.31469	0.657346	−	0.753589i	$-0.271679\pi$
$71$	55843.2	1.31469	0.657346	+	0.753589i	$0.271679\pi$
$72$	118705.	2.69861
$73$	−49566.9	−1.08864	−0.544320	−	0.838878i	$-0.683212\pi$
$73$	−49566.9	−1.08864	−0.544320	+	0.838878i	$0.683212\pi$
$74$	−41958.9	−0.890727
$75$	0	0
$76$	32497.4	0.645378
$77$	−18470.8	−0.355024
$78$	20517.2	0.381840
$79$	8611.37	0.155240	0.0776202	−	0.996983i	$-0.475268\pi$
$79$	8611.37	0.155240	0.0776202	+	0.996983i	$0.475268\pi$
$80$	0	0
$81$	166141.	2.81362
$82$	29184.6	0.479313
$83$	1635.95	0.0260660	0.0130330	−	0.999915i	$-0.495851\pi$
$83$	1635.95	0.0260660	0.0130330	+	0.999915i	$0.495851\pi$
$84$	−18328.6	−0.283420
$85$	0	0
$86$	−89374.2	−1.30307
$87$	89862.3	1.27286
$88$	84353.3	1.16117
$89$	26554.2	0.355351	0.177676	−	0.984089i	$-0.443142\pi$
$89$	26554.2	0.355351	0.177676	+	0.984089i	$0.443142\pi$
$90$	0	0
$91$	−7185.39	−0.0909592
$92$	−17353.9	−0.213760
$93$	−182461.	−2.18757
$94$	12818.2	0.149627
$95$	0	0
$96$	141001.	1.56150
$97$	−181767.	−1.96149	−0.980745	−	0.195292i	$-0.937435\pi$
$97$	−181767.	−1.96149	−0.980745	+	0.195292i	$0.937435\pi$
$98$	62299.5	0.655268
$99$	265589.	2.72347
$100$	0	0
$101$	30120.5	0.293805	0.146902	−	0.989151i	$-0.453070\pi$
$101$	30120.5	0.293805	0.146902	+	0.989151i	$0.453070\pi$
$102$	51532.8	0.490437
$103$	16097.4	0.149507	0.0747535	−	0.997202i	$-0.476183\pi$
$103$	16097.4	0.149507	0.0747535	+	0.997202i	$0.476183\pi$
$104$	32814.7	0.297498
$105$	0	0
$106$	158576.	1.37079
$107$	215209.	1.81719	0.908597	−	0.417674i	$-0.137155\pi$
$107$	215209.	1.81719	0.908597	+	0.417674i	$0.137155\pi$
$108$	158791.	1.30998
$109$	122039.	0.983856	0.491928	−	0.870636i	$-0.336292\pi$
$109$	122039.	0.983856	0.491928	+	0.870636i	$0.336292\pi$
$110$	0	0
$111$	−295276.	−2.27468
$112$	14223.3	0.107141
$113$	−127320.	−0.937993	−0.468997	−	0.883200i	$-0.655384\pi$
$113$	−127320.	−0.937993	−0.468997	+	0.883200i	$0.655384\pi$
$114$	−267505.	−1.92784
$115$	0	0
$116$	45342.7	0.312869
$117$	103318.	0.697769
$118$	72740.2	0.480916
$119$	−18047.4	−0.116828
$120$	0	0
$121$	27679.5	0.171868
$122$	53349.4	0.324511
$123$	205380.	1.22404
$124$	−92066.2	−0.537707
$125$	0	0
$126$	107961.	0.605815
$127$	−327564.	−1.80213	−0.901067	−	0.433680i	$-0.857215\pi$
$127$	−327564.	−1.80213	−0.901067	+	0.433680i	$0.857215\pi$
$128$	26683.0	0.143949
$129$	−628951.	−3.32769
$130$	0	0
$131$	86617.2	0.440987	0.220494	−	0.975388i	$-0.429233\pi$
$131$	86617.2	0.440987	0.220494	+	0.975388i	$0.429233\pi$
$132$	187278.	0.935517
$133$	93683.8	0.459235
$134$	−193122.	−0.929116
$135$	0	0
$136$	82420.1	0.382108
$137$	−223125.	−1.01565	−0.507827	−	0.861459i	$-0.669551\pi$
$137$	−223125.	−1.01565	−0.507827	+	0.861459i	$0.669551\pi$
$138$	142850.	0.638531
$139$	−268128.	−1.17708	−0.588539	−	0.808469i	$-0.700297\pi$
$139$	−268128.	−1.17708	−0.588539	+	0.808469i	$0.700297\pi$
$140$	0	0
$141$	90205.4	0.382107
$142$	−231944.	−0.965301
$143$	73418.9	0.300239
$144$	−204515.	−0.821900
$145$	0	0
$146$	205876.	0.799324
$147$	438418.	1.67338
$148$	−148990.	−0.559118
$149$	−22314.5	−0.0823420	−0.0411710	−	0.999152i	$-0.513109\pi$
$149$	−22314.5	−0.0823420	−0.0411710	+	0.999152i	$0.513109\pi$
$150$	0	0
$151$	438136.	1.56375	0.781874	−	0.623437i	$-0.214264\pi$
$151$	438136.	1.56375	0.781874	+	0.623437i	$0.214264\pi$
$152$	−427841.	−1.50201
$153$	259503.	0.896217
$154$	76718.2	0.260673
$155$	0	0
$156$	72853.8	0.239685
$157$	452703.	1.46577	0.732883	−	0.680355i	$-0.238174\pi$
$157$	452703.	1.46577	0.732883	+	0.680355i	$0.238174\pi$
$158$	−35767.3	−0.113984
$159$	1.11594e6	3.50065
$160$	0	0
$161$	−50027.9	−0.152106
$162$	−690066.	−2.06587
$163$	−367437.	−1.08321	−0.541606	−	0.840632i	$-0.682183\pi$
$163$	−367437.	−1.08321	−0.541606	+	0.840632i	$0.682183\pi$
$164$	103631.	0.300869
$165$	0	0
$166$	−6794.91	−0.0191388
$167$	−29002.0	−0.0804706	−0.0402353	−	0.999190i	$-0.512811\pi$
$167$	−29002.0	−0.0804706	−0.0402353	+	0.999190i	$0.512811\pi$
$168$	241303.	0.659613
$169$	28561.0	0.0769231
$170$	0	0
$171$	−1.34707e6	−3.52290
$172$	−317356.	−0.817947
$173$	105419.	0.267797	0.133898	−	0.990995i	$-0.457250\pi$
$173$	105419.	0.267797	0.133898	+	0.990995i	$0.457250\pi$
$174$	−373243.	−0.934584
$175$	0	0
$176$	−145330.	−0.353651
$177$	511892.	1.22813
$178$	−110293.	−0.260913
$179$	−299749.	−0.699237	−0.349619	−	0.936892i	$-0.613689\pi$
$179$	−299749.	−0.699237	−0.349619	+	0.936892i	$0.613689\pi$
$180$	0	0
$181$	490770.	1.11348	0.556739	−	0.830688i	$-0.312052\pi$
$181$	490770.	1.11348	0.556739	+	0.830688i	$0.312052\pi$
$182$	29844.5	0.0667860
$183$	375434.	0.828717
$184$	228470.	0.497491
$185$	0	0
$186$	757851.	1.60621
$187$	184405.	0.385629
$188$	45515.8	0.0939221
$189$	457764.	0.932152
$190$	0	0
$191$	−691134.	−1.37082	−0.685408	−	0.728160i	$-0.740376\pi$
$191$	−691134.	−1.37082	−0.685408	+	0.728160i	$0.740376\pi$
$192$	−898544.	−1.75908
$193$	−166093.	−0.320966	−0.160483	−	0.987039i	$-0.551305\pi$
$193$	−166093.	−0.320966	−0.160483	+	0.987039i	$0.551305\pi$
$194$	754969.	1.44021
$195$	0	0
$196$	221217.	0.411318
$197$	656522.	1.20527	0.602634	−	0.798018i	$-0.294118\pi$
$197$	656522.	1.20527	0.602634	+	0.798018i	$0.294118\pi$
$198$	−1.10312e6	−1.99968
$199$	793002.	1.41952	0.709760	−	0.704443i	$-0.248803\pi$
$199$	793002.	1.41952	0.709760	+	0.704443i	$0.248803\pi$
$200$	0	0
$201$	−1.35905e6	−2.37272
$202$	−125105.	−0.215723
$203$	130714.	0.222630
$204$	182986.	0.307852
$205$	0	0
$206$	−66860.3	−0.109774
$207$	719346.	1.16684
$208$	−56535.6	−0.0906075
$209$	−957242.	−1.51585
$210$	0	0
$211$	728395.	1.12632	0.563159	−	0.826349i	$-0.309586\pi$
$211$	728395.	1.12632	0.563159	+	0.826349i	$0.309586\pi$
$212$	563081.	0.860461
$213$	−1.63225e6	−2.46512
$214$	−893870.	−1.33426
$215$	0	0
$216$	−2.09054e6	−3.04877
$217$	−265409.	−0.382619
$218$	−506887.	−0.722387
$219$	1.44880e6	2.04126
$220$	0	0
$221$	71736.3	0.0988003
$222$	1.22643e6	1.67016
$223$	−133554.	−0.179844	−0.0899221	−	0.995949i	$-0.528662\pi$
$223$	−133554.	−0.179844	−0.0899221	+	0.995949i	$0.528662\pi$
$224$	205101.	0.273116
$225$	0	0
$226$	528822.	0.688713
$227$	−1.14153e6	−1.47035	−0.735176	−	0.677876i	$-0.762901\pi$
$227$	−1.14153e6	−1.47035	−0.735176	+	0.677876i	$0.762901\pi$
$228$	−949874.	−1.21012
$229$	−765463.	−0.964573	−0.482287	−	0.876013i	$-0.660194\pi$
$229$	−765463.	−0.964573	−0.482287	+	0.876013i	$0.660194\pi$
$230$	0	0
$231$	539886.	0.665691
$232$	−596954.	−0.728151
$233$	1.07854e6	1.30151	0.650755	−	0.759288i	$-0.274453\pi$
$233$	1.07854e6	1.30151	0.650755	+	0.759288i	$0.274453\pi$
$234$	−429131.	−0.512330
$235$	0	0
$236$	258290.	0.301876
$237$	−251704.	−0.291085
$238$	74960.0	0.0857802
$239$	16098.4	0.0182300	0.00911501	−	0.999958i	$-0.497099\pi$
$239$	16098.4	0.0182300	0.00911501	+	0.999958i	$0.497099\pi$
$240$	0	0
$241$	−76424.5	−0.0847598	−0.0423799	−	0.999102i	$-0.513494\pi$
$241$	−76424.5	−0.0847598	−0.0423799	+	0.999102i	$0.513494\pi$
$242$	−114967.	−0.126193
$243$	−2.23990e6	−2.43340
$244$	189436.	0.203699
$245$	0	0
$246$	−853044.	−0.898739
$247$	−372381.	−0.388370
$248$	1.21209e6	1.25142
$249$	−47817.6	−0.0488753
$250$	0	0
$251$	−816629.	−0.818164	−0.409082	−	0.912498i	$-0.634151\pi$
$251$	−816629.	−0.818164	−0.409082	+	0.912498i	$0.634151\pi$
$252$	383355.	0.380276
$253$	511175.	0.502075
$254$	1.36054e6	1.32320
$255$	0	0
$256$	−1.09455e6	−1.04384
$257$	1.18641e6	1.12047	0.560237	−	0.828332i	$-0.310710\pi$
$257$	1.18641e6	1.12047	0.560237	+	0.828332i	$0.310710\pi$
$258$	2.61234e6	2.44332
$259$	−429510.	−0.397855
$260$	0	0
$261$	−1.87953e6	−1.70784
$262$	−359764.	−0.323791
$263$	1.72216e6	1.53527	0.767635	−	0.640887i	$-0.221433\pi$
$263$	1.72216e6	1.53527	0.767635	+	0.640887i	$0.221433\pi$
$264$	−2.46559e6	−2.17726
$265$	0	0
$266$	−389115.	−0.337189
$267$	−776158.	−0.666304
$268$	−685750.	−0.583215
$269$	932022.	0.785318	0.392659	−	0.919684i	$-0.371555\pi$
$269$	932022.	0.785318	0.392659	+	0.919684i	$0.371555\pi$
$270$	0	0
$271$	−157100.	−0.129943	−0.0649714	−	0.997887i	$-0.520696\pi$
$271$	−157100.	−0.129943	−0.0649714	+	0.997887i	$0.520696\pi$
$272$	−142000.	−0.116377
$273$	210024.	0.170554
$274$	926746.	0.745735
$275$	0	0
$276$	507240.	0.400812
$277$	−432798.	−0.338911	−0.169456	−	0.985538i	$-0.554201\pi$
$277$	−432798.	−0.338911	−0.169456	+	0.985538i	$0.554201\pi$
$278$	1.11367e6	0.864259
$279$	3.81630e6	2.93516
$280$	0	0
$281$	−544542.	−0.411401	−0.205701	−	0.978615i	$-0.565947\pi$
$281$	−544542.	−0.411401	−0.205701	+	0.978615i	$0.565947\pi$
$282$	−374668.	−0.280558
$283$	1.57849e6	1.17159	0.585796	−	0.810458i	$-0.300782\pi$
$283$	1.57849e6	1.17159	0.585796	+	0.810458i	$0.300782\pi$
$284$	−823602.	−0.605929
$285$	0	0
$286$	−304945.	−0.220448
$287$	298747.	0.214091
$288$	−2.94912e6	−2.09513
$289$	−1.23968e6	−0.873101
$290$	0	0
$291$	5.31292e6	3.67791
$292$	731036.	0.501743
$293$	−47169.2	−0.0320989	−0.0160494	−	0.999871i	$-0.505109\pi$
$293$	−47169.2	−0.0320989	−0.0160494	+	0.999871i	$0.505109\pi$
$294$	−1.82097e6	−1.22867
$295$	0	0
$296$	1.96151e6	1.30125
$297$	−4.67734e6	−3.07686
$298$	92683.2	0.0604589
$299$	198854.	0.128634
$300$	0	0
$301$	−914876.	−0.582031
$302$	−1.81979e6	−1.14817
$303$	−880399.	−0.550900
$304$	737117.	0.457459
$305$	0	0
$306$	−1.07784e6	−0.658039
$307$	−1.21647e6	−0.736639	−0.368320	−	0.929699i	$-0.620067\pi$
$307$	−1.21647e6	−0.736639	−0.368320	+	0.929699i	$0.620067\pi$
$308$	272416.	0.163627
$309$	−470514.	−0.280334
$310$	0	0
$311$	745149.	0.436860	0.218430	−	0.975853i	$-0.429906\pi$
$311$	745149.	0.436860	0.218430	+	0.975853i	$0.429906\pi$
$312$	−959148.	−0.557826
$313$	1.98248e6	1.14379	0.571896	−	0.820326i	$-0.306208\pi$
$313$	1.98248e6	1.14379	0.571896	+	0.820326i	$0.306208\pi$
$314$	−1.88030e6	−1.07622
$315$	0	0
$316$	−127005.	−0.0715488
$317$	191631.	0.107107	0.0535535	−	0.998565i	$-0.482945\pi$
$317$	191631.	0.107107	0.0535535	+	0.998565i	$0.482945\pi$
$318$	−4.63505e6	−2.57032
$319$	−1.33561e6	−0.734860
$320$	0	0
$321$	−6.29040e6	−3.40734
$322$	207790.	0.111683
$323$	−935304.	−0.498823
$324$	−2.45033e6	−1.29677
$325$	0	0
$326$	1.52615e6	0.795339
$327$	−3.56710e6	−1.84479
$328$	−1.36434e6	−0.700224
$329$	131213.	0.0668326
$330$	0	0
$331$	1.37008e6	0.687346	0.343673	−	0.939089i	$-0.388329\pi$
$331$	1.37008e6	0.687346	0.343673	+	0.939089i	$0.388329\pi$
$332$	−24127.8	−0.0120136
$333$	6.17589e6	3.05203
$334$	120460.	0.0590848
$335$	0	0
$336$	−415735.	−0.200895
$337$	−2.20098e6	−1.05570	−0.527852	−	0.849337i	$-0.677002\pi$
$337$	−2.20098e6	−1.05570	−0.527852	+	0.849337i	$0.677002\pi$
$338$	−118628.	−0.0564801
$339$	3.72146e6	1.75879
$340$	0	0
$341$	2.71190e6	1.26295
$342$	5.59505e6	2.58666
$343$	1.35231e6	0.620642
$344$	4.17811e6	1.90364
$345$	0	0
$346$	−437859.	−0.196627
$347$	2.30264e6	1.02660	0.513302	−	0.858208i	$-0.328422\pi$
$347$	2.30264e6	1.02660	0.513302	+	0.858208i	$0.328422\pi$
$348$	−1.32533e6	−0.586647
$349$	−642226.	−0.282244	−0.141122	−	0.989992i	$-0.545071\pi$
$349$	−642226.	−0.282244	−0.141122	+	0.989992i	$0.545071\pi$
$350$	0	0
$351$	−1.81955e6	−0.788309
$352$	−2.09568e6	−0.901504
$353$	3.50057e6	1.49521	0.747604	−	0.664145i	$-0.231204\pi$
$353$	3.50057e6	1.49521	0.747604	+	0.664145i	$0.231204\pi$
$354$	−2.12614e6	−0.901746
$355$	0	0
$356$	−391634.	−0.163778
$357$	527513.	0.219060
$358$	1.24500e6	0.513409
$359$	1.77270e6	0.725935	0.362968	−	0.931802i	$-0.381764\pi$
$359$	1.77270e6	0.725935	0.362968	+	0.931802i	$0.381764\pi$
$360$	0	0
$361$	2.37904e6	0.960802
$362$	−2.03841e6	−0.817561
$363$	−809052.	−0.322263
$364$	105974.	0.0419222
$365$	0	0
$366$	−1.55936e6	−0.608478
$367$	509693.	0.197535	0.0987673	−	0.995111i	$-0.468510\pi$
$367$	509693.	0.197535	0.0987673	+	0.995111i	$0.468510\pi$
$368$	−393626.	−0.151518
$369$	−4.29566e6	−1.64234
$370$	0	0
$371$	1.62326e6	0.612283
$372$	2.69103e6	1.00823
$373$	2.55346e6	0.950293	0.475147	−	0.879907i	$-0.342395\pi$
$373$	2.55346e6	0.950293	0.475147	+	0.879907i	$0.342395\pi$
$374$	−765926.	−0.283144
$375$	0	0
$376$	−599233.	−0.218588
$377$	−519573.	−0.188275
$378$	−1.90132e6	−0.684424
$379$	1.42480e6	0.509513	0.254757	−	0.967005i	$-0.418005\pi$
$379$	1.42480e6	0.509513	0.254757	+	0.967005i	$0.418005\pi$
$380$	0	0
$381$	9.57446e6	3.37911
$382$	2.87062e6	1.00651
$383$	−940463.	−0.327601	−0.163800	−	0.986494i	$-0.552375\pi$
$383$	−940463.	−0.327601	−0.163800	+	0.986494i	$0.552375\pi$
$384$	−779924.	−0.269913
$385$	0	0
$386$	689868.	0.235666
$387$	1.31549e7	4.46489
$388$	2.68079e6	0.904032
$389$	5.26222e6	1.76317	0.881586	−	0.472024i	$-0.156476\pi$
$389$	5.26222e6	1.76317	0.881586	+	0.472024i	$0.156476\pi$
$390$	0	0
$391$	499460.	0.165218
$392$	−2.91241e6	−0.957275
$393$	−2.53176e6	−0.826877
$394$	−2.72686e6	−0.884957
$395$	0	0
$396$	−3.91704e6	−1.25522
$397$	66917.7	0.0213091	0.0106545	−	0.999943i	$-0.496608\pi$
$397$	66917.7	0.0213091	0.0106545	+	0.999943i	$0.496608\pi$
$398$	−3.29373e6	−1.04227
$399$	−2.73831e6	−0.861093
$400$	0	0
$401$	−3.26462e6	−1.01384	−0.506922	−	0.861992i	$-0.669217\pi$
$401$	−3.26462e6	−1.01384	−0.506922	+	0.861992i	$0.669217\pi$
$402$	5.64481e6	1.74215
$403$	1.05497e6	0.323576
$404$	−444232.	−0.135412
$405$	0	0
$406$	−542922.	−0.163464
$407$	4.38865e6	1.31324
$408$	−2.40908e6	−0.716474
$409$	5.49644e6	1.62470	0.812350	−	0.583171i	$-0.198188\pi$
$409$	5.49644e6	1.62470	0.812350	+	0.583171i	$0.198188\pi$
$410$	0	0
$411$	6.52176e6	1.90441
$412$	−237412.	−0.0689063
$413$	744602.	0.214807
$414$	−2.98780e6	−0.856743
$415$	0	0
$416$	−815249.	−0.230971
$417$	7.83719e6	2.20709
$418$	3.97590e6	1.11300
$419$	4.68601e6	1.30397	0.651986	−	0.758231i	$-0.273936\pi$
$419$	4.68601e6	1.30397	0.651986	+	0.758231i	$0.273936\pi$
$420$	0	0
$421$	4.86774e6	1.33851	0.669256	−	0.743032i	$-0.266613\pi$
$421$	4.86774e6	1.33851	0.669256	+	0.743032i	$0.266613\pi$
$422$	−3.02538e6	−0.826989
$423$	−1.88671e6	−0.512688
$424$	−7.41318e6	−2.00258
$425$	0	0
$426$	6.77956e6	1.81000
$427$	546109.	0.144947
$428$	−3.17401e6	−0.837527
$429$	−2.14598e6	−0.562966
$430$	0	0
$431$	3.25405e6	0.843783	0.421892	−	0.906646i	$-0.361366\pi$
$431$	3.25405e6	0.843783	0.421892	+	0.906646i	$0.361366\pi$
$432$	3.60175e6	0.928547
$433$	693375.	0.177725	0.0888625	−	0.996044i	$-0.471677\pi$
$433$	693375.	0.177725	0.0888625	+	0.996044i	$0.471677\pi$
$434$	1.10238e6	0.280935
$435$	0	0
$436$	−1.79989e6	−0.453450
$437$	−2.59268e6	−0.649450
$438$	−6.01759e6	−1.49878
$439$	−5.29607e6	−1.31157	−0.655787	−	0.754946i	$-0.727663\pi$
$439$	−5.29607e6	−1.31157	−0.655787	+	0.754946i	$0.727663\pi$
$440$	0	0
$441$	−9.16981e6	−2.24525
$442$	−297956.	−0.0725432
$443$	−1.05676e6	−0.255838	−0.127919	−	0.991785i	$-0.540830\pi$
$443$	−1.05676e6	−0.255838	−0.127919	+	0.991785i	$0.540830\pi$
$444$	4.35487e6	1.04838
$445$	0	0
$446$	554718.	0.132049
$447$	652236.	0.154396
$448$	−1.30703e6	−0.307673
$449$	1.60061e6	0.374687	0.187344	−	0.982294i	$-0.440012\pi$
$449$	1.60061e6	0.374687	0.187344	+	0.982294i	$0.440012\pi$
$450$	0	0
$451$	−3.05254e6	−0.706675
$452$	1.87777e6	0.432312
$453$	−1.28064e7	−2.93212
$454$	4.74132e6	1.07959
$455$	0	0
$456$	1.25055e7	2.81636
$457$	20508.5	0.00459350	0.00229675	−	0.999997i	$-0.499269\pi$
$457$	20508.5	0.00459350	0.00229675	+	0.999997i	$0.499269\pi$
$458$	3.17934e6	0.708229
$459$	−4.57014e6	−1.01251
$460$	0	0
$461$	−5.97691e6	−1.30986	−0.654930	−	0.755690i	$-0.727302\pi$
$461$	−5.97691e6	−1.30986	−0.654930	+	0.755690i	$0.727302\pi$
$462$	−2.24241e6	−0.488777
$463$	758041.	0.164339	0.0821694	−	0.996618i	$-0.473815\pi$
$463$	758041.	0.164339	0.0821694	+	0.996618i	$0.473815\pi$
$464$	1.02848e6	0.221769
$465$	0	0
$466$	−4.47972e6	−0.955621
$467$	5.52758e6	1.17285	0.586426	−	0.810003i	$-0.300535\pi$
$467$	5.52758e6	1.17285	0.586426	+	0.810003i	$0.300535\pi$
$468$	−1.52379e6	−0.321595
$469$	−1.97689e6	−0.415002
$470$	0	0
$471$	−1.32322e7	−2.74839
$472$	−3.40049e6	−0.702566
$473$	9.34802e6	1.92118
$474$	1.04545e6	0.213726
$475$	0	0
$476$	266173.	0.0538451
$477$	−2.33406e7	−4.69696
$478$	−66864.5	−0.0133852
$479$	6.18197e6	1.23109	0.615543	−	0.788103i	$-0.288937\pi$
$479$	6.18197e6	1.23109	0.615543	+	0.788103i	$0.288937\pi$
$480$	0	0
$481$	1.70725e6	0.336461
$482$	317429.	0.0622341
$483$	1.46228e6	0.285208
$484$	−408231.	−0.0792123
$485$	0	0
$486$	9.30343e6	1.78670
$487$	−6.33627e6	−1.21063	−0.605315	−	0.795986i	$-0.706953\pi$
$487$	−6.33627e6	−1.21063	−0.605315	+	0.795986i	$0.706953\pi$
$488$	−2.49400e6	−0.474076
$489$	1.07399e7	2.03108
$490$	0	0
$491$	4.76619e6	0.892211	0.446105	−	0.894980i	$-0.352811\pi$
$491$	4.76619e6	0.892211	0.446105	+	0.894980i	$0.352811\pi$
$492$	−3.02904e6	−0.564148
$493$	−1.30500e6	−0.241821
$494$	1.54668e6	0.285157
$495$	0	0
$496$	−2.08828e6	−0.381140
$497$	−2.37429e6	−0.431164
$498$	198610.	0.0358863
$499$	−6.32136e6	−1.13647	−0.568236	−	0.822865i	$-0.692374\pi$
$499$	−6.32136e6	−1.13647	−0.568236	+	0.822865i	$0.692374\pi$
$500$	0	0
$501$	847708.	0.150887
$502$	3.39186e6	0.600730
$503$	2.42758e6	0.427813	0.213906	−	0.976854i	$-0.431381\pi$
$503$	2.42758e6	0.427813	0.213906	+	0.976854i	$0.431381\pi$
$504$	−5.04701e6	−0.885030
$505$	0	0
$506$	−2.12316e6	−0.368644
$507$	−834817.	−0.144235
$508$	4.83108e6	0.830586
$509$	−5.22784e6	−0.894391	−0.447196	−	0.894436i	$-0.647577\pi$
$509$	−5.22784e6	−0.894391	−0.447196	+	0.894436i	$0.647577\pi$
$510$	0	0
$511$	2.10744e6	0.357028
$512$	3.69234e6	0.622482
$513$	2.37235e7	3.98002
$514$	−4.92775e6	−0.822698
$515$	0	0
$516$	9.27607e6	1.53370
$517$	−1.34071e6	−0.220602
$518$	1.78397e6	0.292121
$519$	−3.08133e6	−0.502134
$520$	0	0
$521$	−1.15960e7	−1.87161	−0.935806	−	0.352516i	$-0.885326\pi$
$521$	−1.15960e7	−1.87161	−0.935806	+	0.352516i	$0.885326\pi$
$522$	7.80662e6	1.25397
$523$	−3.49229e6	−0.558285	−0.279143	−	0.960250i	$-0.590050\pi$
$523$	−3.49229e6	−0.558285	−0.279143	+	0.960250i	$0.590050\pi$
$524$	−1.27747e6	−0.203247
$525$	0	0
$526$	−7.15299e6	−1.12726
$527$	2.64975e6	0.415602
$528$	4.24790e6	0.663116
$529$	−5.05183e6	−0.784891
$530$	0	0
$531$	−1.07066e7	−1.64784
$532$	−1.38169e6	−0.211657
$533$	−1.18748e6	−0.181054
$534$	3.22377e6	0.489227
$535$	0	0
$536$	9.02816e6	1.35734
$537$	8.76143e6	1.31111
$538$	−3.87115e6	−0.576612
$539$	−6.51616e6	−0.966095
$540$	0	0
$541$	216425.	0.0317917	0.0158959	−	0.999874i	$-0.494940\pi$
$541$	216425.	0.0317917	0.0158959	+	0.999874i	$0.494940\pi$
$542$	652513.	0.0954094
$543$	−1.43448e7	−2.08783
$544$	−2.04765e6	−0.296659
$545$	0	0
$546$	−872332.	−0.125228
$547$	658142.	0.0940484	0.0470242	−	0.998894i	$-0.485026\pi$
$547$	658142.	0.0940484	0.0470242	+	0.998894i	$0.485026\pi$
$548$	3.29075e6	0.468105
$549$	−7.85245e6	−1.11192
$550$	0	0
$551$	6.77424e6	0.950565
$552$	−6.67801e6	−0.932824
$553$	−366131.	−0.0509124
$554$	1.79762e6	0.248842
$555$	0	0
$556$	3.95448e6	0.542504
$557$	6.55817e6	0.895663	0.447831	−	0.894118i	$-0.352196\pi$
$557$	6.55817e6	0.895663	0.447831	+	0.894118i	$0.352196\pi$
$558$	−1.58510e7	−2.15511
$559$	3.63652e6	0.492216
$560$	0	0
$561$	−5.39003e6	−0.723076
$562$	2.26175e6	0.302068
$563$	−9.44685e6	−1.25608	−0.628038	−	0.778183i	$-0.716142\pi$
$563$	−9.44685e6	−1.25608	−0.628038	+	0.778183i	$0.716142\pi$
$564$	−1.33039e6	−0.176109
$565$	0	0
$566$	−6.55626e6	−0.860231
$567$	−7.06384e6	−0.922748
$568$	1.08430e7	1.41020
$569$	1.98661e6	0.257236	0.128618	−	0.991694i	$-0.458946\pi$
$569$	1.98661e6	0.257236	0.128618	+	0.991694i	$0.458946\pi$
$570$	0	0
$571$	1.05472e7	1.35378	0.676888	−	0.736086i	$-0.263328\pi$
$571$	1.05472e7	1.35378	0.676888	+	0.736086i	$0.263328\pi$
$572$	−1.08282e6	−0.138377
$573$	2.02013e7	2.57036
$574$	−1.24084e6	−0.157195
$575$	0	0
$576$	1.87936e7	2.36023
$577$	3.57968e6	0.447615	0.223807	−	0.974633i	$-0.428151\pi$
$577$	3.57968e6	0.447615	0.223807	+	0.974633i	$0.428151\pi$
$578$	5.14899e6	0.641066
$579$	4.85478e6	0.601830
$580$	0	0
$581$	−69555.9	−0.00854857
$582$	−2.20672e7	−2.70047
$583$	−1.65861e7	−2.02103
$584$	−9.62437e6	−1.16772
$585$	0	0
$586$	195917.	0.0235683
$587$	8.78869e6	1.05276	0.526379	−	0.850250i	$-0.323549\pi$
$587$	8.78869e6	1.05276	0.526379	+	0.850250i	$0.323549\pi$
$588$	−6.46601e6	−0.771246
$589$	−1.37548e7	−1.63367
$590$	0	0
$591$	−1.91896e7	−2.25995
$592$	−3.37945e6	−0.396316
$593$	−6.61381e6	−0.772352	−0.386176	−	0.922425i	$-0.626204\pi$
$593$	−6.61381e6	−0.772352	−0.386176	+	0.922425i	$0.626204\pi$
$594$	1.94273e7	2.25916
$595$	0	0
$596$	329105.	0.0379506
$597$	−2.31789e7	−2.66168
$598$	−825940.	−0.0944486
$599$	1.20269e7	1.36957	0.684787	−	0.728743i	$-0.259895\pi$
$599$	1.20269e7	1.36957	0.684787	+	0.728743i	$0.259895\pi$
$600$	0	0
$601$	3.83969e6	0.433621	0.216810	−	0.976214i	$-0.430435\pi$
$601$	3.83969e6	0.433621	0.216810	+	0.976214i	$0.430435\pi$
$602$	3.79993e6	0.427351
$603$	2.84255e7	3.18357
$604$	−6.46184e6	−0.720716
$605$	0	0
$606$	3.65673e6	0.404494
$607$	−5.50595e6	−0.606542	−0.303271	−	0.952904i	$-0.598079\pi$
$607$	−5.50595e6	−0.606542	−0.303271	+	0.952904i	$0.598079\pi$
$608$	1.06293e7	1.16613
$609$	−3.82069e6	−0.417444
$610$	0	0
$611$	−521557.	−0.0565195
$612$	−3.82727e6	−0.413058
$613$	−7.77688e6	−0.835899	−0.417950	−	0.908470i	$-0.637251\pi$
$613$	−7.77688e6	−0.835899	−0.417950	+	0.908470i	$0.637251\pi$
$614$	5.05259e6	0.540871
$615$	0	0
$616$	−3.58646e6	−0.380815
$617$	5.24549e6	0.554719	0.277359	−	0.960766i	$-0.410541\pi$
$617$	5.24549e6	0.554719	0.277359	+	0.960766i	$0.410541\pi$
$618$	1.95428e6	0.205833
$619$	1.53925e7	1.61467	0.807333	−	0.590096i	$-0.200910\pi$
$619$	1.53925e7	1.61467	0.807333	+	0.590096i	$0.200910\pi$
$620$	0	0
$621$	−1.26685e7	−1.31825
$622$	−3.09497e6	−0.320760
$623$	−1.12901e6	−0.116540
$624$	1.65249e6	0.169894
$625$	0	0
$626$	−8.23420e6	−0.839819
$627$	2.79795e7	2.84231
$628$	−6.67669e6	−0.675557
$629$	4.28807e6	0.432151
$630$	0	0
$631$	−959665.	−0.0959503	−0.0479752	−	0.998849i	$-0.515277\pi$
$631$	−959665.	−0.0959503	−0.0479752	+	0.998849i	$0.515277\pi$
$632$	1.67207e6	0.166518
$633$	−2.12904e7	−2.11191
$634$	−795939.	−0.0786423
$635$	0	0
$636$	−1.64584e7	−1.61341
$637$	−2.53488e6	−0.247519
$638$	5.54747e6	0.539564
$639$	3.41397e7	3.30756
$640$	0	0
$641$	−4.04738e6	−0.389071	−0.194536	−	0.980895i	$-0.562320\pi$
$641$	−4.04738e6	−0.389071	−0.194536	+	0.980895i	$0.562320\pi$
$642$	2.61271e7	2.50181
$643$	3.34301e6	0.318867	0.159434	−	0.987209i	$-0.449033\pi$
$643$	3.34301e6	0.318867	0.159434	+	0.987209i	$0.449033\pi$
$644$	737835.	0.0701043
$645$	0	0
$646$	3.88478e6	0.366256
$647$	−1.60437e6	−0.150676	−0.0753379	−	0.997158i	$-0.524004\pi$
$647$	−1.60437e6	−0.150676	−0.0753379	+	0.997158i	$0.524004\pi$
$648$	3.22596e7	3.01801
$649$	−7.60820e6	−0.709039
$650$	0	0
$651$	7.75771e6	0.717433
$652$	5.41913e6	0.499242
$653$	−8.15286e6	−0.748216	−0.374108	−	0.927385i	$-0.622051\pi$
$653$	−8.15286e6	−0.748216	−0.374108	+	0.927385i	$0.622051\pi$
$654$	1.48159e7	1.35452
$655$	0	0
$656$	2.35058e6	0.213263
$657$	−3.03027e7	−2.73884
$658$	−544994.	−0.0490713
$659$	−9.64758e6	−0.865376	−0.432688	−	0.901544i	$-0.642435\pi$
$659$	−9.64758e6	−0.865376	−0.432688	+	0.901544i	$0.642435\pi$
$660$	0	0
$661$	−1.08154e7	−0.962802	−0.481401	−	0.876500i	$-0.659872\pi$
$661$	−1.08154e7	−0.962802	−0.481401	+	0.876500i	$0.659872\pi$
$662$	−5.69061e6	−0.504677
$663$	−2.09680e6	−0.185256
$664$	317652.	0.0279596
$665$	0	0
$666$	−2.56515e7	−2.24093
$667$	−3.61750e6	−0.314843
$668$	427736.	0.0370881
$669$	3.90370e6	0.337218
$670$	0	0
$671$	−5.58004e6	−0.478444
$672$	−5.99494e6	−0.512108
$673$	−8.41679e6	−0.716323	−0.358161	−	0.933660i	$-0.616596\pi$
$673$	−8.41679e6	−0.716323	−0.358161	+	0.933660i	$0.616596\pi$
$674$	9.14177e6	0.775141
$675$	0	0
$676$	−421232.	−0.0354531
$677$	−1.30052e7	−1.09055	−0.545276	−	0.838257i	$-0.683575\pi$
$677$	−1.30052e7	−1.09055	−0.545276	+	0.838257i	$0.683575\pi$
$678$	−1.54571e7	−1.29138
$679$	7.72821e6	0.643287
$680$	0	0
$681$	3.33660e7	2.75700
$682$	−1.12639e7	−0.927313
$683$	6.34198e6	0.520204	0.260102	−	0.965581i	$-0.416244\pi$
$683$	6.34198e6	0.520204	0.260102	+	0.965581i	$0.416244\pi$
$684$	1.98673e7	1.62367
$685$	0	0
$686$	−5.61682e6	−0.455701
$687$	2.23739e7	1.80863
$688$	−7.19837e6	−0.579780
$689$	−6.45223e6	−0.517800
$690$	0	0
$691$	−5.39316e6	−0.429683	−0.214841	−	0.976649i	$-0.568923\pi$
$691$	−5.39316e6	−0.429683	−0.214841	+	0.976649i	$0.568923\pi$
$692$	−1.55478e6	−0.123425
$693$	−1.12921e7	−0.893184
$694$	−9.56401e6	−0.753774
$695$	0	0
$696$	1.74485e7	1.36532
$697$	−2.98258e6	−0.232547
$698$	2.66748e6	0.207235
$699$	−3.15250e7	−2.44040
$700$	0	0
$701$	2.51700e6	0.193459	0.0967293	−	0.995311i	$-0.469162\pi$
$701$	2.51700e6	0.193459	0.0967293	+	0.995311i	$0.469162\pi$
$702$	7.55750e6	0.578809
$703$	−2.22593e7	−1.69872
$704$	1.33550e7	1.01557
$705$	0	0
$706$	−1.45396e7	−1.09784
$707$	−1.28064e6	−0.0963556
$708$	−7.54964e6	−0.566035
$709$	−7.70344e6	−0.575532	−0.287766	−	0.957701i	$-0.592913\pi$
$709$	−7.70344e6	−0.575532	−0.287766	+	0.957701i	$0.592913\pi$
$710$	0	0
$711$	5.26456e6	0.390560
$712$	5.15601e6	0.381166
$713$	7.34515e6	0.541099
$714$	−2.19102e6	−0.160843
$715$	0	0
$716$	4.42084e6	0.322272
$717$	−470544.	−0.0341824
$718$	−7.36288e6	−0.533012
$719$	9.67017e6	0.697609	0.348804	−	0.937196i	$-0.386588\pi$
$719$	9.67017e6	0.697609	0.348804	+	0.937196i	$0.386588\pi$
$720$	0	0
$721$	−684413.	−0.0490321
$722$	−9.88133e6	−0.705460
$723$	2.23383e6	0.158929
$724$	−7.23811e6	−0.513191
$725$	0	0
$726$	3.36039e6	0.236618
$727$	−6.56744e6	−0.460850	−0.230425	−	0.973090i	$-0.574012\pi$
$727$	−6.56744e6	−0.460850	−0.230425	+	0.973090i	$0.574012\pi$
$728$	−1.39518e6	−0.0975670
$729$	2.50984e7	1.74915
$730$	0	0
$731$	9.13379e6	0.632204
$732$	−5.53709e6	−0.381947
$733$	−1.34964e7	−0.927808	−0.463904	−	0.885885i	$-0.653552\pi$
$733$	−1.34964e7	−0.927808	−0.463904	+	0.885885i	$0.653552\pi$
$734$	−2.11701e6	−0.145038
$735$	0	0
$736$	−5.67612e6	−0.386240
$737$	2.01994e7	1.36984
$738$	1.78420e7	1.20588
$739$	1.14388e7	0.770492	0.385246	−	0.922814i	$-0.374117\pi$
$739$	1.14388e7	0.770492	0.385246	+	0.922814i	$0.374117\pi$
$740$	0	0
$741$	1.08844e7	0.728215
$742$	−6.74218e6	−0.449563
$743$	1.51089e7	1.00406	0.502031	−	0.864850i	$-0.332586\pi$
$743$	1.51089e7	1.00406	0.502031	+	0.864850i	$0.332586\pi$
$744$	−3.54284e7	−2.34649
$745$	0	0
$746$	−1.06058e7	−0.697744
$747$	1.00014e6	0.0655780
$748$	−2.71970e6	−0.177732
$749$	−9.15007e6	−0.595964
$750$	0	0
$751$	2.10421e6	0.136141	0.0680706	−	0.997681i	$-0.478316\pi$
$751$	2.10421e6	0.136141	0.0680706	+	0.997681i	$0.478316\pi$
$752$	1.03241e6	0.0665742
$753$	2.38695e7	1.53411
$754$	2.15804e6	0.138239
$755$	0	0
$756$	−6.75132e6	−0.429620
$757$	1.09867e7	0.696828	0.348414	−	0.937341i	$-0.386720\pi$
$757$	1.09867e7	0.696828	0.348414	+	0.937341i	$0.386720\pi$
$758$	−5.91789e6	−0.374106
$759$	−1.49413e7	−0.941419
$760$	0	0
$761$	−1.14683e7	−0.717859	−0.358929	−	0.933365i	$-0.616858\pi$
$761$	−1.14683e7	−0.717859	−0.358929	+	0.933365i	$0.616858\pi$
$762$	−3.97675e7	−2.48108
$763$	−5.18873e6	−0.322664
$764$	1.01932e7	0.631795
$765$	0	0
$766$	3.90621e6	0.240538
$767$	−2.95970e6	−0.181660
$768$	3.19928e7	1.95726
$769$	−2.18586e6	−0.133293	−0.0666464	−	0.997777i	$-0.521230\pi$
$769$	−2.18586e6	−0.133293	−0.0666464	+	0.997777i	$0.521230\pi$
$770$	0	0
$771$	−3.46779e7	−2.10095
$772$	2.44963e6	0.147930
$773$	4.06142e6	0.244472	0.122236	−	0.992501i	$-0.460993\pi$
$773$	4.06142e6	0.244472	0.122236	+	0.992501i	$0.460993\pi$
$774$	−5.46389e7	−3.27831
$775$	0	0
$776$	−3.52937e7	−2.10398
$777$	1.25543e7	0.746000
$778$	−2.18566e7	−1.29459
$779$	1.54825e7	0.914108
$780$	0	0
$781$	2.42600e7	1.42319
$782$	−2.07450e6	−0.121310
$783$	3.31007e7	1.92945
$784$	5.01772e6	0.291552
$785$	0	0
$786$	1.05156e7	0.607127
$787$	1.97898e7	1.13895	0.569475	−	0.822008i	$-0.307146\pi$
$787$	1.97898e7	1.13895	0.569475	+	0.822008i	$0.307146\pi$
$788$	−9.68271e6	−0.555496
$789$	−5.03375e7	−2.87872
$790$	0	0
$791$	5.41327e6	0.307623
$792$	5.15694e7	2.92132
$793$	−2.17071e6	−0.122580
$794$	−277942.	−0.0156460
$795$	0	0
$796$	−1.16956e7	−0.654243
$797$	−2.61426e6	−0.145782	−0.0728908	−	0.997340i	$-0.523222\pi$
$797$	−2.61426e6	−0.145782	−0.0728908	+	0.997340i	$0.523222\pi$
$798$	1.13735e7	0.632250
$799$	−1.30999e6	−0.0725939
$800$	0	0
$801$	1.62339e7	0.894007
$802$	1.35596e7	0.744406
$803$	−2.15334e7	−1.17848
$804$	2.00440e7	1.09356
$805$	0	0
$806$	−4.38180e6	−0.237583
$807$	−2.72423e7	−1.47252
$808$	5.84848e6	0.315148
$809$	−134453.	−0.00722267	−0.00361134	−	0.999993i	$-0.501150\pi$
$809$	−134453.	−0.00722267	−0.00361134	+	0.999993i	$0.501150\pi$
$810$	0	0
$811$	−1.38880e7	−0.741459	−0.370729	−	0.928741i	$-0.620892\pi$
$811$	−1.38880e7	−0.741459	−0.370729	+	0.928741i	$0.620892\pi$
$812$	−1.92784e6	−0.102608
$813$	4.59191e6	0.243650
$814$	−1.82282e7	−0.964237
$815$	0	0
$816$	4.15055e6	0.218213
$817$	−4.74132e7	−2.48510
$818$	−2.28294e7	−1.19292
$819$	−4.39278e6	−0.228839
$820$	0	0
$821$	−2.09003e7	−1.08217	−0.541084	−	0.840969i	$-0.681986\pi$
$821$	−2.09003e7	−1.08217	−0.541084	+	0.840969i	$0.681986\pi$
$822$	−2.70881e7	−1.39830
$823$	7.88591e6	0.405837	0.202919	−	0.979196i	$-0.434957\pi$
$823$	7.88591e6	0.405837	0.202919	+	0.979196i	$0.434957\pi$
$824$	3.12562e6	0.160368
$825$	0	0
$826$	−3.09270e6	−0.157720
$827$	3.09009e7	1.57111	0.785556	−	0.618790i	$-0.212377\pi$
$827$	3.09009e7	1.57111	0.785556	+	0.618790i	$0.212377\pi$
$828$	−1.06093e7	−0.537786
$829$	3.72829e7	1.88418	0.942092	−	0.335356i	$-0.108856\pi$
$829$	3.72829e7	1.88418	0.942092	+	0.335356i	$0.108856\pi$
$830$	0	0
$831$	1.26504e7	0.635478
$832$	5.19527e6	0.260196
$833$	−6.36683e6	−0.317915
$834$	−3.25517e7	−1.62054
$835$	0	0
$836$	1.41179e7	0.698640
$837$	−6.72094e7	−3.31602
$838$	−1.94633e7	−0.957429
$839$	3.71338e7	1.82123	0.910615	−	0.413256i	$-0.135608\pi$
$839$	3.71338e7	1.82123	0.910615	+	0.413256i	$0.135608\pi$
$840$	0	0
$841$	−1.10592e7	−0.539181
$842$	−2.02181e7	−0.982790
$843$	1.59166e7	0.771401
$844$	−1.07427e7	−0.519109
$845$	0	0
$846$	7.83642e6	0.376437
$847$	−1.17685e6	−0.0563655
$848$	1.27720e7	0.609915
$849$	−4.61382e7	−2.19680
$850$	0	0
$851$	1.18866e7	0.562645
$852$	2.40733e7	1.13615
$853$	3.69954e7	1.74091	0.870453	−	0.492252i	$-0.163826\pi$
$853$	3.69954e7	1.74091	0.870453	+	0.492252i	$0.163826\pi$
$854$	−2.26826e6	−0.106426
$855$	0	0
$856$	4.17871e7	1.94920
$857$	1.25620e7	0.584260	0.292130	−	0.956379i	$-0.405636\pi$
$857$	1.25620e7	0.584260	0.292130	+	0.956379i	$0.405636\pi$
$858$	8.91331e6	0.413353
$859$	5.95648e6	0.275427	0.137714	−	0.990472i	$-0.456025\pi$
$859$	5.95648e6	0.275427	0.137714	+	0.990472i	$0.456025\pi$
$860$	0	0
$861$	−8.73216e6	−0.401434
$862$	−1.35157e7	−0.619540
$863$	1.36218e7	0.622600	0.311300	−	0.950312i	$-0.399236\pi$
$863$	1.36218e7	0.622600	0.311300	+	0.950312i	$0.399236\pi$
$864$	5.19375e7	2.36699
$865$	0	0
$866$	−2.87993e6	−0.130493
$867$	3.62349e7	1.63711
$868$	3.91439e6	0.176345
$869$	3.74105e6	0.168052
$870$	0	0
$871$	7.85787e6	0.350962
$872$	2.36962e7	1.05533
$873$	−1.11123e8	−4.93480
$874$	1.07687e7	0.476853
$875$	0	0
$876$	−2.13676e7	−0.940798
$877$	2.57256e7	1.12945	0.564725	−	0.825279i	$-0.308982\pi$
$877$	2.57256e7	1.12945	0.564725	+	0.825279i	$0.308982\pi$
$878$	2.19972e7	0.963011
$879$	1.37872e6	0.0601872
$880$	0	0
$881$	4.32713e6	0.187828	0.0939140	−	0.995580i	$-0.470062\pi$
$881$	4.32713e6	0.187828	0.0939140	+	0.995580i	$0.470062\pi$
$882$	3.80868e7	1.64855
$883$	−3.46067e7	−1.49368	−0.746841	−	0.665003i	$-0.768430\pi$
$883$	−3.46067e7	−1.49368	−0.746841	+	0.665003i	$0.768430\pi$
$884$	−1.05800e6	−0.0455361
$885$	0	0
$886$	4.38923e6	0.187847
$887$	−3.18896e7	−1.36094	−0.680472	−	0.732774i	$-0.738225\pi$
$887$	−3.18896e7	−1.36094	−0.680472	+	0.732774i	$0.738225\pi$
$888$	−5.73336e7	−2.43993
$889$	1.39271e7	0.591025
$890$	0	0
$891$	7.21769e7	3.04582
$892$	1.96973e6	0.0828884
$893$	6.80010e6	0.285356
$894$	−2.70906e6	−0.113364
$895$	0	0
$896$	−1.13448e6	−0.0472094
$897$	−5.81236e6	−0.241197
$898$	−6.64811e6	−0.275111
$899$	−1.91917e7	−0.791978
$900$	0	0
$901$	−1.62060e7	−0.665064
$902$	1.26787e7	0.518870
$903$	2.67411e7	1.09134
$904$	−2.47216e7	−1.00613
$905$	0	0
$906$	5.31913e7	2.15288
$907$	2.40329e7	0.970035	0.485018	−	0.874504i	$-0.338813\pi$
$907$	2.40329e7	0.970035	0.485018	+	0.874504i	$0.338813\pi$
$908$	1.68358e7	0.677671
$909$	1.84141e7	0.739166
$910$	0	0
$911$	−1.88212e6	−0.0751366	−0.0375683	−	0.999294i	$-0.511961\pi$
$911$	−1.88212e6	−0.0751366	−0.0375683	+	0.999294i	$0.511961\pi$
$912$	−2.15454e7	−0.857763
$913$	710708.	0.0282172
$914$	−85182.0	−0.00337274
$915$	0	0
$916$	1.12894e7	0.444562
$917$	−3.68271e6	−0.144625
$918$	1.89821e7	0.743424
$919$	−2.08679e6	−0.0815061	−0.0407530	−	0.999169i	$-0.512976\pi$
$919$	−2.08679e6	−0.0815061	−0.0407530	+	0.999169i	$0.512976\pi$
$920$	0	0
$921$	3.55565e7	1.38124
$922$	2.48251e7	0.961752
$923$	9.43750e6	0.364630
$924$	−7.96251e6	−0.306810
$925$	0	0
$926$	−3.14852e6	−0.120664
$927$	9.84111e6	0.376136
$928$	1.48308e7	0.565319
$929$	4.37732e7	1.66406	0.832030	−	0.554730i	$-0.187179\pi$
$929$	4.37732e7	1.66406	0.832030	+	0.554730i	$0.187179\pi$
$930$	0	0
$931$	3.30500e7	1.24968
$932$	−1.59069e7	−0.599853
$933$	−2.17801e7	−0.819137
$934$	−2.29588e7	−0.861156
$935$	0	0
$936$	2.00612e7	0.748458
$937$	1.71324e7	0.637484	0.318742	−	0.947842i	$-0.396740\pi$
$937$	1.71324e7	0.637484	0.318742	+	0.947842i	$0.396740\pi$
$938$	8.21099e6	0.304711
$939$	−5.79463e7	−2.14468
$940$	0	0
$941$	−2.91171e7	−1.07195	−0.535975	−	0.844234i	$-0.680056\pi$
$941$	−2.91171e7	−1.07195	−0.535975	+	0.844234i	$0.680056\pi$
$942$	5.49598e7	2.01798
$943$	−8.26778e6	−0.302768
$944$	5.85864e6	0.213977
$945$	0	0
$946$	−3.88270e7	−1.41061
$947$	3.57121e7	1.29402	0.647009	−	0.762482i	$-0.276019\pi$
$947$	3.57121e7	1.29402	0.647009	+	0.762482i	$0.276019\pi$
$948$	3.71225e6	0.134158
$949$	−8.37680e6	−0.301934
$950$	0	0
$951$	−5.60123e6	−0.200832
$952$	−3.50426e6	−0.125315
$953$	1.21150e7	0.432105	0.216053	−	0.976382i	$-0.430682\pi$
$953$	1.21150e7	0.432105	0.216053	+	0.976382i	$0.430682\pi$
$954$	9.69452e7	3.44870
$955$	0	0
$956$	−237427.	−0.00840204
$957$	3.90390e7	1.37790
$958$	−2.56768e7	−0.903913
$959$	9.48661e6	0.333092
$960$	0	0
$961$	1.03386e7	0.361120
$962$	−7.09105e6	−0.247043
$963$	1.31568e8	4.57177
$964$	1.12715e6	0.0390650
$965$	0	0
$966$	−6.07356e6	−0.209411
$967$	1.23685e6	0.0425353	0.0212676	−	0.999774i	$-0.493230\pi$
$967$	1.23685e6	0.0425353	0.0212676	+	0.999774i	$0.493230\pi$
$968$	5.37452e6	0.184354
$969$	2.73383e7	0.935322
$970$	0	0
$971$	2.65552e7	0.903861	0.451931	−	0.892053i	$-0.350735\pi$
$971$	2.65552e7	0.903861	0.451931	+	0.892053i	$0.350735\pi$
$972$	3.30352e7	1.12153
$973$	1.14000e7	0.386032
$974$	2.63177e7	0.888894
$975$	0	0
$976$	4.29686e6	0.144387
$977$	2.10622e7	0.705939	0.352970	−	0.935635i	$-0.385172\pi$
$977$	2.10622e7	0.705939	0.352970	+	0.935635i	$0.385172\pi$
$978$	−4.46081e7	−1.49131
$979$	1.15360e7	0.384678
$980$	0	0
$981$	7.46083e7	2.47523
$982$	−1.97963e7	−0.655098
$983$	5.95953e7	1.96711	0.983554	−	0.180615i	$-0.0578088\pi$
$983$	5.95953e7	1.96711	0.983554	+	0.180615i	$0.0578088\pi$
$984$	3.98785e7	1.31296
$985$	0	0
$986$	5.42033e6	0.177555
$987$	−3.83527e6	−0.125315
$988$	5.49206e6	0.178996
$989$	2.53190e7	0.823107
$990$	0	0
$991$	7.17026e6	0.231927	0.115963	−	0.993253i	$-0.463004\pi$
$991$	7.17026e6	0.231927	0.115963	+	0.993253i	$0.463004\pi$
$992$	−3.01131e7	−0.971576
$993$	−4.00464e7	−1.28881
$994$	9.86160e6	0.316578
$995$	0	0
$996$	705238.	0.0225262
$997$	5.19624e7	1.65558	0.827791	−	0.561036i	$-0.189597\pi$
$997$	5.19624e7	1.65558	0.827791	+	0.561036i	$0.189597\pi$
$998$	2.62557e7	0.834445
$999$	−1.08765e8	−3.44805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.f.1.3		6
5.2	odd	4		325.6.b.f.274.5		12
5.3	odd	4		325.6.b.f.274.8		12
5.4	even	2		65.6.a.e.1.4	✓	6
15.14	odd	2		585.6.a.k.1.3		6
20.19	odd	2		1040.6.a.r.1.1		6
65.64	even	2		845.6.a.g.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.e.1.4	✓	6	5.4	even	2
325.6.a.f.1.3		6	1.1	even	1	trivial
325.6.b.f.274.5		12	5.2	odd	4
325.6.b.f.274.8		12	5.3	odd	4
585.6.a.k.1.3		6	15.14	odd	2
845.6.a.g.1.3		6	65.64	even	2
1040.6.a.r.1.1		6	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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	325.a (trivial)

\(f(q)\)	\(=\)	\(q-4.15349 q^{2} -29.2293 q^{3} -14.7485 q^{4} +121.404 q^{6} -42.5171 q^{7} +194.170 q^{8} +611.349 q^{9} +434.431 q^{11} +431.087 q^{12} +169.000 q^{13} +176.594 q^{14} -334.530 q^{16} +424.475 q^{17} -2539.24 q^{18} -2203.44 q^{19} +1242.74 q^{21} -1804.41 q^{22} +1176.65 q^{23} -5675.43 q^{24} -701.940 q^{26} -10766.6 q^{27} +627.063 q^{28} -3074.40 q^{29} +6242.41 q^{31} -4823.96 q^{32} -12698.1 q^{33} -1763.05 q^{34} -9016.48 q^{36} +10102.1 q^{37} +9151.97 q^{38} -4939.74 q^{39} -7026.52 q^{41} -5161.73 q^{42} +21517.8 q^{43} -6407.20 q^{44} -4887.22 q^{46} -3086.13 q^{47} +9778.08 q^{48} -14999.3 q^{49} -12407.1 q^{51} -2492.49 q^{52} -38178.9 q^{53} +44718.9 q^{54} -8255.53 q^{56} +64404.9 q^{57} +12769.5 q^{58} -17513.0 q^{59} -12844.5 q^{61} -25927.8 q^{62} -25992.8 q^{63} +30741.2 q^{64} +52741.5 q^{66} +46496.3 q^{67} -6260.37 q^{68} -34392.7 q^{69} +55843.2 q^{71} +118705. q^{72} -49566.9 q^{73} -41958.9 q^{74} +32497.4 q^{76} -18470.8 q^{77} +20517.2 q^{78} +8611.37 q^{79} +166141. q^{81} +29184.6 q^{82} +1635.95 q^{83} -18328.6 q^{84} -89374.2 q^{86} +89862.3 q^{87} +84353.3 q^{88} +26554.2 q^{89} -7185.39 q^{91} -17353.9 q^{92} -182461. q^{93} +12818.2 q^{94} +141001. q^{96} -181767. q^{97} +62299.5 q^{98} +265589. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - 20 q^{3} + 134 q^{4} - 52 q^{6} - 172 q^{7} + 138 q^{8} + 1034 q^{9} + 800 q^{11} + 832 q^{12} + 1014 q^{13} + 2108 q^{14} + 322 q^{16} - 4396 q^{17} - 6142 q^{18} + 5304 q^{19} + 1072 q^{21}+ \cdots + 301264 q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - 20 * q^3 + 134 * q^4 - 52 * q^6 - 172 * q^7 + 138 * q^8 + 1034 * q^9 + 800 * q^11 + 832 * q^12 + 1014 * q^13 + 2108 * q^14 + 322 * q^16 - 4396 * q^17 - 6142 * q^18 + 5304 * q^19 + 1072 * q^21 - 8008 * q^22 + 1140 * q^23 + 4756 * q^24 - 338 * q^26 - 800 * q^27 - 524 * q^28 - 3664 * q^29 + 16664 * q^31 + 9894 * q^32 + 13092 * q^33 + 1380 * q^34 + 1818 * q^36 - 4488 * q^37 + 30112 * q^38 - 3380 * q^39 + 12436 * q^41 + 68336 * q^42 - 1516 * q^43 - 35124 * q^44 - 4556 * q^46 - 212 * q^47 + 16712 * q^48 - 27842 * q^49 - 27664 * q^51 + 22646 * q^52 + 15612 * q^53 - 132912 * q^54 + 44420 * q^56 + 86796 * q^57 + 39664 * q^58 - 11896 * q^59 + 57232 * q^61 + 95488 * q^62 - 207956 * q^63 - 39914 * q^64 + 9908 * q^66 + 25428 * q^67 + 16236 * q^68 + 72404 * q^69 + 214104 * q^71 + 130806 * q^72 - 60808 * q^73 - 182104 * q^74 + 366636 * q^76 + 5192 * q^77 - 8788 * q^78 + 36024 * q^79 + 350514 * q^81 + 1124 * q^82 - 34684 * q^83 + 6024 * q^84 + 29204 * q^86 + 8640 * q^87 + 40520 * q^88 - 52028 * q^89 - 29068 * q^91 + 283232 * q^92 - 26508 * q^93 + 157572 * q^94 + 547908 * q^96 - 129316 * q^97 + 113942 * q^98 + 301264 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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