LMFDB - Embedded newform 325.6.a.f.1.4

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			1.4
Root			$-2.16876$ 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)

$f(q)$	$=$	\(q+2.16876 q^{2} +2.56930 q^{3} -27.2965 q^{4} +5.57220 q^{6} +75.5966 q^{7} -128.600 q^{8} -236.399 q^{9} +624.896 q^{11} -70.1329 q^{12} +169.000 q^{13} +163.951 q^{14} +594.585 q^{16} -2344.47 q^{17} -512.692 q^{18} -283.916 q^{19} +194.231 q^{21} +1355.25 q^{22} -2043.60 q^{23} -330.412 q^{24} +366.520 q^{26} -1231.72 q^{27} -2063.52 q^{28} +6172.51 q^{29} +687.967 q^{31} +5404.71 q^{32} +1605.55 q^{33} -5084.60 q^{34} +6452.85 q^{36} +2797.24 q^{37} -615.745 q^{38} +434.212 q^{39} +8236.40 q^{41} +421.239 q^{42} +13268.3 q^{43} -17057.5 q^{44} -4432.07 q^{46} +15489.4 q^{47} +1527.67 q^{48} -11092.1 q^{49} -6023.66 q^{51} -4613.11 q^{52} +9603.36 q^{53} -2671.30 q^{54} -9721.71 q^{56} -729.465 q^{57} +13386.7 q^{58} +40189.6 q^{59} +32587.1 q^{61} +1492.03 q^{62} -17870.9 q^{63} -7305.22 q^{64} +3482.05 q^{66} -15914.0 q^{67} +63995.9 q^{68} -5250.62 q^{69} +60206.9 q^{71} +30400.8 q^{72} +35469.1 q^{73} +6066.54 q^{74} +7749.90 q^{76} +47240.1 q^{77} +941.701 q^{78} +65026.6 q^{79} +54280.2 q^{81} +17862.8 q^{82} -86013.0 q^{83} -5301.81 q^{84} +28775.7 q^{86} +15859.1 q^{87} -80361.5 q^{88} -139114. q^{89} +12775.8 q^{91} +55783.0 q^{92} +1767.59 q^{93} +33592.8 q^{94} +13886.3 q^{96} +8013.02 q^{97} -24056.2 q^{98} -147725. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.16876	0.383386	0.191693	−	0.981455i	$-0.438602\pi$
$2$	2.16876	0.383386	0.191693	+	0.981455i	$0.438602\pi$
$3$	2.56930	0.164821	0.0824104	−	0.996598i	$-0.473738\pi$
$3$	2.56930	0.164821	0.0824104	+	0.996598i	$0.473738\pi$
$4$	−27.2965	−0.853015
$5$	0	0
$5$	0	0
$6$	5.57220	0.0631900
$7$	75.5966	0.583119	0.291560	−	0.956553i	$-0.405826\pi$
$7$	75.5966	0.583119	0.291560	+	0.956553i	$0.405826\pi$
$8$	−128.600	−0.710420
$9$	−236.399	−0.972834
$10$	0	0
$11$	624.896	1.55713	0.778567	−	0.627561i	$-0.215947\pi$
$11$	624.896	1.55713	0.778567	+	0.627561i	$0.215947\pi$
$12$	−70.1329	−0.140595
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	163.951	0.223560
$15$	0	0
$16$	594.585	0.580650
$17$	−2344.47	−1.96754	−0.983769	−	0.179439i	$-0.942572\pi$
$17$	−2344.47	−1.96754	−0.983769	+	0.179439i	$0.942572\pi$
$18$	−512.692	−0.372971
$19$	−283.916	−0.180429	−0.0902143	−	0.995922i	$-0.528755\pi$
$19$	−283.916	−0.180429	−0.0902143	+	0.995922i	$0.528755\pi$
$20$	0	0
$21$	194.231	0.0961102
$22$	1355.25	0.596984
$23$	−2043.60	−0.805519	−0.402760	−	0.915306i	$-0.631949\pi$
$23$	−2043.60	−0.805519	−0.402760	+	0.915306i	$0.631949\pi$
$24$	−330.412	−0.117092
$25$	0	0
$26$	366.520	0.106332
$27$	−1231.72	−0.325164
$28$	−2063.52	−0.497410
$29$	6172.51	1.36291	0.681455	−	0.731860i	$-0.261348\pi$
$29$	6172.51	1.36291	0.681455	+	0.731860i	$0.261348\pi$
$30$	0	0
$31$	687.967	0.128577	0.0642885	−	0.997931i	$-0.479522\pi$
$31$	687.967	0.128577	0.0642885	+	0.997931i	$0.479522\pi$
$32$	5404.71	0.933033
$33$	1605.55	0.256648
$34$	−5084.60	−0.754327
$35$	0	0
$36$	6452.85	0.829842
$37$	2797.24	0.335912	0.167956	−	0.985795i	$-0.446283\pi$
$37$	2797.24	0.335912	0.167956	+	0.985795i	$0.446283\pi$
$38$	−615.745	−0.0691738
$39$	434.212	0.0457131
$40$	0	0
$41$	8236.40	0.765205	0.382602	−	0.923913i	$-0.375028\pi$
$41$	8236.40	0.765205	0.382602	+	0.923913i	$0.375028\pi$
$42$	421.239	0.0368473
$43$	13268.3	1.09432	0.547158	−	0.837029i	$-0.315710\pi$
$43$	13268.3	1.09432	0.547158	+	0.837029i	$0.315710\pi$
$44$	−17057.5	−1.32826
$45$	0	0
$46$	−4432.07	−0.308825
$47$	15489.4	1.02280	0.511400	−	0.859343i	$-0.329127\pi$
$47$	15489.4	1.02280	0.511400	+	0.859343i	$0.329127\pi$
$48$	1527.67	0.0957032
$49$	−11092.1	−0.659972
$50$	0	0
$51$	−6023.66	−0.324291
$52$	−4613.11	−0.236584
$53$	9603.36	0.469606	0.234803	−	0.972043i	$-0.424556\pi$
$53$	9603.36	0.469606	0.234803	+	0.972043i	$0.424556\pi$
$54$	−2671.30	−0.124663
$55$	0	0
$56$	−9721.71	−0.414260
$57$	−729.465	−0.0297384
$58$	13386.7	0.522521
$59$	40189.6	1.50308	0.751542	−	0.659685i	$-0.229310\pi$
$59$	40189.6	1.50308	0.751542	+	0.659685i	$0.229310\pi$
$60$	0	0
$61$	32587.1	1.12130	0.560649	−	0.828053i	$-0.310552\pi$
$61$	32587.1	1.12130	0.560649	+	0.828053i	$0.310552\pi$
$62$	1492.03	0.0492946
$63$	−17870.9	−0.567278
$64$	−7305.22	−0.222938
$65$	0	0
$66$	3482.05	0.0983953
$67$	−15914.0	−0.433105	−0.216553	−	0.976271i	$-0.569481\pi$
$67$	−15914.0	−0.433105	−0.216553	+	0.976271i	$0.569481\pi$
$68$	63995.9	1.67834
$69$	−5250.62	−0.132766
$70$	0	0
$71$	60206.9	1.41743	0.708713	−	0.705496i	$-0.249276\pi$
$71$	60206.9	1.41743	0.708713	+	0.705496i	$0.249276\pi$
$72$	30400.8	0.691121
$73$	35469.1	0.779010	0.389505	−	0.921024i	$-0.372646\pi$
$73$	35469.1	0.779010	0.389505	+	0.921024i	$0.372646\pi$
$74$	6066.54	0.128784
$75$	0	0
$76$	7749.90	0.153908
$77$	47240.1	0.907995
$78$	941.701	0.0175258
$79$	65026.6	1.17226	0.586129	−	0.810218i	$-0.300651\pi$
$79$	65026.6	1.17226	0.586129	+	0.810218i	$0.300651\pi$
$80$	0	0
$81$	54280.2	0.919240
$82$	17862.8	0.293369
$83$	−86013.0	−1.37047	−0.685234	−	0.728323i	$-0.740300\pi$
$83$	−86013.0	−1.37047	−0.685234	+	0.728323i	$0.740300\pi$
$84$	−5301.81	−0.0819834
$85$	0	0
$86$	28775.7	0.419545
$87$	15859.1	0.224636
$88$	−80361.5	−1.10622
$89$	−139114.	−1.86163	−0.930817	−	0.365485i	$-0.880903\pi$
$89$	−139114.	−1.86163	−0.930817	+	0.365485i	$0.880903\pi$
$90$	0	0
$91$	12775.8	0.161728
$92$	55783.0	0.687120
$93$	1767.59	0.0211922
$94$	33592.8	0.392127
$95$	0	0
$96$	13886.3	0.153783
$97$	8013.02	0.0864703	0.0432351	−	0.999065i	$-0.486234\pi$
$97$	8013.02	0.0864703	0.0432351	+	0.999065i	$0.486234\pi$
$98$	−24056.2	−0.253024
$99$	−147725.	−1.51483
$100$	0	0
$101$	127679.	1.24542	0.622709	−	0.782454i	$-0.286032\pi$
$101$	127679.	1.24542	0.622709	+	0.782454i	$0.286032\pi$
$102$	−13063.9	−0.124329
$103$	−136390.	−1.26675	−0.633373	−	0.773847i	$-0.718330\pi$
$103$	−136390.	−1.26675	−0.633373	+	0.773847i	$0.718330\pi$
$104$	−21733.4	−0.197035
$105$	0	0
$106$	20827.4	0.180040
$107$	−33127.6	−0.279725	−0.139862	−	0.990171i	$-0.544666\pi$
$107$	−33127.6	−0.279725	−0.139862	+	0.990171i	$0.544666\pi$
$108$	33621.6	0.277370
$109$	53441.1	0.430834	0.215417	−	0.976522i	$-0.430889\pi$
$109$	53441.1	0.430834	0.215417	+	0.976522i	$0.430889\pi$
$110$	0	0
$111$	7186.95	0.0553652
$112$	44948.7	0.338588
$113$	181691.	1.33855	0.669277	−	0.743013i	$-0.266604\pi$
$113$	181691.	1.33855	0.669277	+	0.743013i	$0.266604\pi$
$114$	−1582.03	−0.0114013
$115$	0	0
$116$	−168488.	−1.16258
$117$	−39951.4	−0.269816
$118$	87161.5	0.576262
$119$	−177234.	−1.14731
$120$	0	0
$121$	229444.	1.42467
$122$	70673.6	0.429890
$123$	21161.8	0.126122
$124$	−18779.1	−0.109678
$125$	0	0
$126$	−38757.8	−0.217487
$127$	290716.	1.59941	0.799704	−	0.600395i	$-0.204990\pi$
$127$	290716.	1.59941	0.799704	+	0.600395i	$0.204990\pi$
$128$	−188794.	−1.01850
$129$	34090.2	0.180366
$130$	0	0
$131$	−216572.	−1.10262	−0.551308	−	0.834302i	$-0.685871\pi$
$131$	−216572.	−1.10262	−0.551308	+	0.834302i	$0.685871\pi$
$132$	−43825.8	−0.218925
$133$	−21463.1	−0.105211
$134$	−34513.7	−0.166047
$135$	0	0
$136$	301499.	1.39778
$137$	200864.	0.914326	0.457163	−	0.889383i	$-0.348866\pi$
$137$	200864.	0.914326	0.457163	+	0.889383i	$0.348866\pi$
$138$	−11387.3	−0.0509008
$139$	−11515.0	−0.0505508	−0.0252754	−	0.999681i	$-0.508046\pi$
$139$	−11515.0	−0.0505508	−0.0252754	+	0.999681i	$0.508046\pi$
$140$	0	0
$141$	39797.0	0.168579
$142$	130574.	0.543422
$143$	105607.	0.431871
$144$	−140559.	−0.564876
$145$	0	0
$146$	76924.0	0.298662
$147$	−28499.1	−0.108777
$148$	−76354.7	−0.286538
$149$	386747.	1.42712	0.713561	−	0.700593i	$-0.247081\pi$
$149$	386747.	1.42712	0.713561	+	0.700593i	$0.247081\pi$
$150$	0	0
$151$	−22668.6	−0.0809062	−0.0404531	−	0.999181i	$-0.512880\pi$
$151$	−22668.6	−0.0809062	−0.0404531	+	0.999181i	$0.512880\pi$
$152$	36511.5	0.128180
$153$	554231.	1.91409
$154$	102452.	0.348113
$155$	0	0
$156$	−11852.5	−0.0389939
$157$	237735.	0.769741	0.384870	−	0.922971i	$-0.374246\pi$
$157$	237735.	0.769741	0.384870	+	0.922971i	$0.374246\pi$
$158$	141027.	0.449427
$159$	24673.9	0.0774008
$160$	0	0
$161$	−154489.	−0.469714
$162$	117721.	0.352424
$163$	−508182.	−1.49813	−0.749066	−	0.662495i	$-0.769498\pi$
$163$	−508182.	−1.49813	−0.749066	+	0.662495i	$0.769498\pi$
$164$	−224825.	−0.652731
$165$	0	0
$166$	−186541.	−0.525418
$167$	−307248.	−0.852507	−0.426253	−	0.904604i	$-0.640167\pi$
$167$	−307248.	−0.852507	−0.426253	+	0.904604i	$0.640167\pi$
$168$	−24978.0	−0.0682786
$169$	28561.0	0.0769231
$170$	0	0
$171$	67117.3	0.175527
$172$	−362177.	−0.933468
$173$	4598.14	0.0116807	0.00584033	−	0.999983i	$-0.498141\pi$
$173$	4598.14	0.0116807	0.00584033	+	0.999983i	$0.498141\pi$
$174$	34394.5	0.0861223
$175$	0	0
$176$	371554.	0.904150
$177$	103259.	0.247740
$178$	−301704.	−0.713725
$179$	557239.	1.29990	0.649948	−	0.759978i	$-0.274791\pi$
$179$	557239.	1.29990	0.649948	+	0.759978i	$0.274791\pi$
$180$	0	0
$181$	−410955.	−0.932389	−0.466195	−	0.884682i	$-0.654375\pi$
$181$	−410955.	−0.932389	−0.466195	+	0.884682i	$0.654375\pi$
$182$	27707.7	0.0620043
$183$	83726.1	0.184813
$184$	262806.	0.572257
$185$	0	0
$186$	3833.49	0.00812478
$187$	−1.46505e6	−3.06372
$188$	−422807.	−0.872464
$189$	−93113.9	−0.189609
$190$	0	0
$191$	−361321.	−0.716655	−0.358327	−	0.933596i	$-0.616653\pi$
$191$	−361321.	−0.716655	−0.358327	+	0.933596i	$0.616653\pi$
$192$	−18769.3	−0.0367448
$193$	149297.	0.288509	0.144254	−	0.989541i	$-0.453922\pi$
$193$	149297.	0.288509	0.144254	+	0.989541i	$0.453922\pi$
$194$	17378.3	0.0331515
$195$	0	0
$196$	302777.	0.562966
$197$	−139691.	−0.256449	−0.128225	−	0.991745i	$-0.540928\pi$
$197$	−139691.	−0.256449	−0.128225	+	0.991745i	$0.540928\pi$
$198$	−320379.	−0.580766
$199$	−433871.	−0.776655	−0.388327	−	0.921521i	$-0.626947\pi$
$199$	−433871.	−0.776655	−0.388327	+	0.921521i	$0.626947\pi$
$200$	0	0
$201$	−40888.0	−0.0713848
$202$	276904.	0.477476
$203$	466621.	0.794739
$204$	164425.	0.276625
$205$	0	0
$206$	−295797.	−0.485653
$207$	483104.	0.783636
$208$	100485.	0.161043
$209$	−177418.	−0.280952
$210$	0	0
$211$	392100.	0.606305	0.303153	−	0.952942i	$-0.401961\pi$
$211$	392100.	0.606305	0.303153	+	0.952942i	$0.401961\pi$
$212$	−262138.	−0.400581
$213$	154690.	0.233621
$214$	−71845.8	−0.107243
$215$	0	0
$216$	158399.	0.231003
$217$	52008.0	0.0749757
$218$	115901.	0.165176
$219$	91130.9	0.128397
$220$	0	0
$221$	−396216.	−0.545697
$222$	15586.8	0.0212263
$223$	594720.	0.800848	0.400424	−	0.916330i	$-0.368863\pi$
$223$	594720.	0.800848	0.400424	+	0.916330i	$0.368863\pi$
$224$	408578.	0.544070
$225$	0	0
$226$	394043.	0.513183
$227$	−473048.	−0.609313	−0.304657	−	0.952462i	$-0.598542\pi$
$227$	−473048.	−0.609313	−0.304657	+	0.952462i	$0.598542\pi$
$228$	19911.8	0.0253673
$229$	−905716.	−1.14131	−0.570655	−	0.821190i	$-0.693311\pi$
$229$	−905716.	−1.14131	−0.570655	+	0.821190i	$0.693311\pi$
$230$	0	0
$231$	121374.	0.149656
$232$	−793784.	−0.968239
$233$	746250.	0.900522	0.450261	−	0.892897i	$-0.351331\pi$
$233$	746250.	0.900522	0.450261	+	0.892897i	$0.351331\pi$
$234$	−86644.9	−0.103444
$235$	0	0
$236$	−1.09703e6	−1.28215
$237$	167073.	0.193212
$238$	−384379.	−0.439863
$239$	909186.	1.02958	0.514788	−	0.857318i	$-0.327871\pi$
$239$	909186.	1.02958	0.514788	+	0.857318i	$0.327871\pi$
$240$	0	0
$241$	−575680.	−0.638467	−0.319233	−	0.947676i	$-0.603425\pi$
$241$	−575680.	−0.638467	−0.319233	+	0.947676i	$0.603425\pi$
$242$	497609.	0.546198
$243$	438770.	0.476674
$244$	−889513.	−0.956484
$245$	0	0
$246$	45894.8	0.0483533
$247$	−47981.8	−0.0500419
$248$	−88472.4	−0.0913437
$249$	−220993.	−0.225882
$250$	0	0
$251$	−1.77777e6	−1.78111	−0.890555	−	0.454876i	$-0.849683\pi$
$251$	−1.77777e6	−1.78111	−0.890555	+	0.454876i	$0.849683\pi$
$252$	487814.	0.483897
$253$	−1.27704e6	−1.25430
$254$	630492.	0.613191
$255$	0	0
$256$	−175681.	−0.167543
$257$	−1.60384e6	−1.51470	−0.757352	−	0.653006i	$-0.773507\pi$
$257$	−1.60384e6	−1.51470	−0.757352	+	0.653006i	$0.773507\pi$
$258$	73933.3	0.0691498
$259$	211462.	0.195877
$260$	0	0
$261$	−1.45917e6	−1.32589
$262$	−469693.	−0.422728
$263$	−1.67245e6	−1.49095	−0.745477	−	0.666532i	$-0.767778\pi$
$263$	−1.67245e6	−1.49095	−0.745477	+	0.666532i	$0.767778\pi$
$264$	−206473.	−0.182328
$265$	0	0
$266$	−46548.2	−0.0403366
$267$	−357425.	−0.306836
$268$	434397.	0.369445
$269$	−64944.5	−0.0547220	−0.0273610	−	0.999626i	$-0.508710\pi$
$269$	−64944.5	−0.0547220	−0.0273610	+	0.999626i	$0.508710\pi$
$270$	0	0
$271$	1.37951e6	1.14104	0.570519	−	0.821284i	$-0.306742\pi$
$271$	1.37951e6	1.14104	0.570519	+	0.821284i	$0.306742\pi$
$272$	−1.39399e6	−1.14245
$273$	32825.0	0.0266562
$274$	435626.	0.350540
$275$	0	0
$276$	143323.	0.113252
$277$	2.40558e6	1.88374	0.941870	−	0.335978i	$-0.109067\pi$
$277$	2.40558e6	1.88374	0.941870	+	0.335978i	$0.109067\pi$
$278$	−24973.3	−0.0193805
$279$	−162634.	−0.125084
$280$	0	0
$281$	−1.70763e6	−1.29011	−0.645055	−	0.764136i	$-0.723166\pi$
$281$	−1.70763e6	−1.29011	−0.645055	+	0.764136i	$0.723166\pi$
$282$	86310.2	0.0646307
$283$	−513628.	−0.381226	−0.190613	−	0.981665i	$-0.561048\pi$
$283$	−513628.	−0.381226	−0.190613	+	0.981665i	$0.561048\pi$
$284$	−1.64344e6	−1.20909
$285$	0	0
$286$	229037.	0.165574
$287$	622644.	0.446206
$288$	−1.27767e6	−0.907687
$289$	4.07670e6	2.87121
$290$	0	0
$291$	20587.9	0.0142521
$292$	−968182.	−0.664508
$293$	2.29105e6	1.55907	0.779535	−	0.626359i	$-0.215456\pi$
$293$	2.29105e6	1.55907	0.779535	+	0.626359i	$0.215456\pi$
$294$	−61807.6	−0.0417036
$295$	0	0
$296$	−359724.	−0.238638
$297$	−769697.	−0.506324
$298$	838761.	0.547139
$299$	−345368.	−0.223411
$300$	0	0
$301$	1.00304e6	0.638117
$302$	−49162.7	−0.0310183
$303$	328045.	0.205271
$304$	−168812.	−0.104766
$305$	0	0
$306$	1.20199e6	0.733835
$307$	−244483.	−0.148048	−0.0740240	−	0.997256i	$-0.523584\pi$
$307$	−244483.	−0.148048	−0.0740240	+	0.997256i	$0.523584\pi$
$308$	−1.28949e6	−0.774534
$309$	−350427.	−0.208786
$310$	0	0
$311$	1.21897e6	0.714648	0.357324	−	0.933981i	$-0.383689\pi$
$311$	1.21897e6	0.714648	0.357324	+	0.933981i	$0.383689\pi$
$312$	−55839.6	−0.0324755
$313$	1.21980e6	0.703766	0.351883	−	0.936044i	$-0.385542\pi$
$313$	1.21980e6	0.703766	0.351883	+	0.936044i	$0.385542\pi$
$314$	515591.	0.295108
$315$	0	0
$316$	−1.77500e6	−0.999954
$317$	−2.20055e6	−1.22994	−0.614970	−	0.788551i	$-0.710832\pi$
$317$	−2.20055e6	−1.22994	−0.614970	+	0.788551i	$0.710832\pi$
$318$	53511.8	0.0296744
$319$	3.85718e6	2.12223
$320$	0	0
$321$	−85114.9	−0.0461044
$322$	−335050.	−0.180082
$323$	665633.	0.355000
$324$	−1.48166e6	−0.784126
$325$	0	0
$326$	−1.10212e6	−0.574363
$327$	137306.	0.0710103
$328$	−1.05920e6	−0.543617
$329$	1.17095e6	0.596414
$330$	0	0
$331$	−755217.	−0.378880	−0.189440	−	0.981892i	$-0.560667\pi$
$331$	−755217.	−0.378880	−0.189440	+	0.981892i	$0.560667\pi$
$332$	2.34785e6	1.16903
$333$	−661263.	−0.326786
$334$	−666347.	−0.326839
$335$	0	0
$336$	115487.	0.0558064
$337$	1.10179e6	0.528475	0.264238	−	0.964458i	$-0.414880\pi$
$337$	1.10179e6	0.528475	0.264238	+	0.964458i	$0.414880\pi$
$338$	61941.9	0.0294912
$339$	466818.	0.220622
$340$	0	0
$341$	429908.	0.200212
$342$	145561.	0.0672947
$343$	−2.10908e6	−0.967962
$344$	−1.70629e6	−0.777424
$345$	0	0
$346$	9972.27	0.00447820
$347$	−1.25898e6	−0.561301	−0.280650	−	0.959810i	$-0.590550\pi$
$347$	−1.25898e6	−0.561301	−0.280650	+	0.959810i	$0.590550\pi$
$348$	−432896.	−0.191618
$349$	−674977.	−0.296637	−0.148318	−	0.988940i	$-0.547386\pi$
$349$	−674977.	−0.296637	−0.148318	+	0.988940i	$0.547386\pi$
$350$	0	0
$351$	−208161.	−0.0901843
$352$	3.37738e6	1.45286
$353$	−1.05470e6	−0.450496	−0.225248	−	0.974301i	$-0.572319\pi$
$353$	−1.05470e6	−0.450496	−0.225248	+	0.974301i	$0.572319\pi$
$354$	223944.	0.0949799
$355$	0	0
$356$	3.79731e6	1.58800
$357$	−455369.	−0.189100
$358$	1.20852e6	0.498362
$359$	1.94312e6	0.795728	0.397864	−	0.917444i	$-0.369752\pi$
$359$	1.94312e6	0.795728	0.397864	+	0.917444i	$0.369752\pi$
$360$	0	0
$361$	−2.39549e6	−0.967446
$362$	−891262.	−0.357465
$363$	589511.	0.234815
$364$	−348735.	−0.137957
$365$	0	0
$366$	181582.	0.0708548
$367$	3.34298e6	1.29559	0.647797	−	0.761813i	$-0.275690\pi$
$367$	3.34298e6	1.29559	0.647797	+	0.761813i	$0.275690\pi$
$368$	−1.21509e6	−0.467725
$369$	−1.94707e6	−0.744417
$370$	0	0
$371$	725981.	0.273836
$372$	−48249.1	−0.0180772
$373$	1.66367e6	0.619149	0.309575	−	0.950875i	$-0.399813\pi$
$373$	1.66367e6	0.619149	0.309575	+	0.950875i	$0.399813\pi$
$374$	−3.17735e6	−1.17459
$375$	0	0
$376$	−1.99194e6	−0.726618
$377$	1.04315e6	0.378003
$378$	−201942.	−0.0726936
$379$	−5.46932e6	−1.95585	−0.977924	−	0.208962i	$-0.932991\pi$
$379$	−5.46932e6	−1.95585	−0.977924	+	0.208962i	$0.932991\pi$
$380$	0	0
$381$	746936.	0.263616
$382$	−783619.	−0.274756
$383$	1.39999e6	0.487674	0.243837	−	0.969816i	$-0.421594\pi$
$383$	1.39999e6	0.487674	0.243837	+	0.969816i	$0.421594\pi$
$384$	−485068.	−0.167871
$385$	0	0
$386$	323790.	0.110610
$387$	−3.13660e6	−1.06459
$388$	−218727.	−0.0737605
$389$	−296544.	−0.0993607	−0.0496803	−	0.998765i	$-0.515820\pi$
$389$	−296544.	−0.0993607	−0.0496803	+	0.998765i	$0.515820\pi$
$390$	0	0
$391$	4.79116e6	1.58489
$392$	1.42645e6	0.468857
$393$	−556439.	−0.181734
$394$	−302955.	−0.0983191
$395$	0	0
$396$	4.03236e6	1.29218
$397$	1.62560e6	0.517653	0.258826	−	0.965924i	$-0.416664\pi$
$397$	1.62560e6	0.517653	0.258826	+	0.965924i	$0.416664\pi$
$398$	−940962.	−0.297759
$399$	−55145.1	−0.0173410
$400$	0	0
$401$	870992.	0.270491	0.135246	−	0.990812i	$-0.456818\pi$
$401$	870992.	0.270491	0.135246	+	0.990812i	$0.456818\pi$
$402$	−88676.2	−0.0273679
$403$	116266.	0.0356608
$404$	−3.48518e6	−1.06236
$405$	0	0
$406$	1.01199e6	0.304692
$407$	1.74798e6	0.523060
$408$	774642.	0.230383
$409$	478324.	0.141388	0.0706942	−	0.997498i	$-0.477479\pi$
$409$	478324.	0.141388	0.0706942	+	0.997498i	$0.477479\pi$
$410$	0	0
$411$	516081.	0.150700
$412$	3.72297e6	1.08055
$413$	3.03820e6	0.876477
$414$	1.04774e6	0.300435
$415$	0	0
$416$	913395.	0.258777
$417$	−29585.6	−0.00833182
$418$	−384777.	−0.107713
$419$	1.58332e6	0.440589	0.220295	−	0.975433i	$-0.429298\pi$
$419$	1.58332e6	0.440589	0.220295	+	0.975433i	$0.429298\pi$
$420$	0	0
$421$	−2.63883e6	−0.725614	−0.362807	−	0.931864i	$-0.618182\pi$
$421$	−2.63883e6	−0.725614	−0.362807	+	0.931864i	$0.618182\pi$
$422$	850371.	0.232449
$423$	−3.66168e6	−0.995015
$424$	−1.23499e6	−0.333617
$425$	0	0
$426$	335485.	0.0895672
$427$	2.46348e6	0.653851
$428$	904268.	0.238609
$429$	271337.	0.0711814
$430$	0	0
$431$	−2.05662e6	−0.533287	−0.266644	−	0.963795i	$-0.585915\pi$
$431$	−2.05662e6	−0.533287	−0.266644	+	0.963795i	$0.585915\pi$
$432$	−732363.	−0.188806
$433$	3.35571e6	0.860131	0.430066	−	0.902798i	$-0.358490\pi$
$433$	3.35571e6	0.860131	0.430066	+	0.902798i	$0.358490\pi$
$434$	112793.	0.0287447
$435$	0	0
$436$	−1.45876e6	−0.367507
$437$	580210.	0.145339
$438$	197641.	0.0492257
$439$	594295.	0.147177	0.0735886	−	0.997289i	$-0.476555\pi$
$439$	594295.	0.147177	0.0735886	+	0.997289i	$0.476555\pi$
$440$	0	0
$441$	2.62217e6	0.642043
$442$	−859298.	−0.209213
$443$	32710.8	0.00791921	0.00395960	−	0.999992i	$-0.498740\pi$
$443$	32710.8	0.00791921	0.00395960	+	0.999992i	$0.498740\pi$
$444$	−196178.	−0.0472274
$445$	0	0
$446$	1.28980e6	0.307034
$447$	993669.	0.235219
$448$	−552250.	−0.129999
$449$	1.14362e6	0.267711	0.133856	−	0.991001i	$-0.457264\pi$
$449$	1.14362e6	0.267711	0.133856	+	0.991001i	$0.457264\pi$
$450$	0	0
$451$	5.14689e6	1.19153
$452$	−4.95951e6	−1.14181
$453$	−58242.4	−0.0133350
$454$	−1.02593e6	−0.233602
$455$	0	0
$456$	93809.1	0.0211268
$457$	1.26412e6	0.283139	0.141569	−	0.989928i	$-0.454785\pi$
$457$	1.26412e6	0.283139	0.141569	+	0.989928i	$0.454785\pi$
$458$	−1.96428e6	−0.437562
$459$	2.88774e6	0.639773
$460$	0	0
$461$	2.38521e6	0.522727	0.261364	−	0.965240i	$-0.415828\pi$
$461$	2.38521e6	0.522727	0.261364	+	0.965240i	$0.415828\pi$
$462$	263231.	0.0573762
$463$	5.27680e6	1.14398	0.571990	−	0.820261i	$-0.306172\pi$
$463$	5.27680e6	1.14398	0.571990	+	0.820261i	$0.306172\pi$
$464$	3.67009e6	0.791373
$465$	0	0
$466$	1.61844e6	0.345248
$467$	5.46999e6	1.16063	0.580316	−	0.814391i	$-0.302929\pi$
$467$	5.46999e6	1.16063	0.580316	+	0.814391i	$0.302929\pi$
$468$	1.09053e6	0.230157
$469$	−1.20305e6	−0.252552
$470$	0	0
$471$	610814.	0.126869
$472$	−5.16837e6	−1.06782
$473$	8.29128e6	1.70400
$474$	362341.	0.0740750
$475$	0	0
$476$	4.83788e6	0.978672
$477$	−2.27022e6	−0.456848
$478$	1.97181e6	0.394725
$479$	8.31612e6	1.65608	0.828042	−	0.560666i	$-0.189455\pi$
$479$	8.31612e6	1.65608	0.828042	+	0.560666i	$0.189455\pi$
$480$	0	0
$481$	472733.	0.0931651
$482$	−1.24851e6	−0.244779
$483$	−396929.	−0.0774186
$484$	−6.26302e6	−1.21526
$485$	0	0
$486$	951587.	0.182750
$487$	−1.23823e6	−0.236581	−0.118291	−	0.992979i	$-0.537741\pi$
$487$	−1.23823e6	−0.236581	−0.118291	+	0.992979i	$0.537741\pi$
$488$	−4.19069e6	−0.796593
$489$	−1.30567e6	−0.246923
$490$	0	0
$491$	472497.	0.0884494	0.0442247	−	0.999022i	$-0.485918\pi$
$491$	472497.	0.0884494	0.0442247	+	0.999022i	$0.485918\pi$
$492$	−577643.	−0.107584
$493$	−1.44713e7	−2.68158
$494$	−104061.	−0.0191854
$495$	0	0
$496$	409055.	0.0746582
$497$	4.55144e6	0.826529
$498$	−479281.	−0.0865998
$499$	3.43591e6	0.617718	0.308859	−	0.951108i	$-0.400053\pi$
$499$	3.43591e6	0.617718	0.308859	+	0.951108i	$0.400053\pi$
$500$	0	0
$501$	−789413.	−0.140511
$502$	−3.85555e6	−0.682853
$503$	9.25786e6	1.63151	0.815756	−	0.578396i	$-0.196321\pi$
$503$	9.25786e6	1.63151	0.815756	+	0.578396i	$0.196321\pi$
$504$	2.29820e6	0.403006
$505$	0	0
$506$	−2.76959e6	−0.480882
$507$	73381.8	0.0126785
$508$	−7.93552e6	−1.36432
$509$	3.39691e6	0.581152	0.290576	−	0.956852i	$-0.406153\pi$
$509$	3.39691e6	0.581152	0.290576	+	0.956852i	$0.406153\pi$
$510$	0	0
$511$	2.68135e6	0.454256
$512$	5.66039e6	0.954271
$513$	349705.	0.0586689
$514$	−3.47834e6	−0.580717
$515$	0	0
$516$	−930541.	−0.153855
$517$	9.67928e6	1.59264
$518$	458610.	0.0750964
$519$	11814.0	0.00192522
$520$	0	0
$521$	7.22852e6	1.16669	0.583344	−	0.812225i	$-0.301744\pi$
$521$	7.22852e6	1.16669	0.583344	+	0.812225i	$0.301744\pi$
$522$	−3.16460e6	−0.508326
$523$	5.69535e6	0.910471	0.455236	−	0.890371i	$-0.349555\pi$
$523$	5.69535e6	0.910471	0.455236	+	0.890371i	$0.349555\pi$
$524$	5.91166e6	0.940549
$525$	0	0
$526$	−3.62714e6	−0.571611
$527$	−1.61292e6	−0.252980
$528$	954635.	0.149023
$529$	−2.26005e6	−0.351139
$530$	0	0
$531$	−9.50076e6	−1.46225
$532$	585866.	0.0897469
$533$	1.39195e6	0.212230
$534$	−775168.	−0.117637
$535$	0	0
$536$	2.04654e6	0.307687
$537$	1.43171e6	0.214250
$538$	−140849.	−0.0209796
$539$	−6.93144e6	−1.02767
$540$	0	0
$541$	−3.69650e6	−0.542998	−0.271499	−	0.962439i	$-0.587519\pi$
$541$	−3.69650e6	−0.542998	−0.271499	+	0.962439i	$0.587519\pi$
$542$	2.99181e6	0.437458
$543$	−1.05587e6	−0.153677
$544$	−1.26712e7	−1.83578
$545$	0	0
$546$	71189.5	0.0102196
$547$	−1.94896e6	−0.278507	−0.139253	−	0.990257i	$-0.544470\pi$
$547$	−1.94896e6	−0.278507	−0.139253	+	0.990257i	$0.544470\pi$
$548$	−5.48288e6	−0.779934
$549$	−7.70355e6	−1.09084
$550$	0	0
$551$	−1.75247e6	−0.245908
$552$	675229.	0.0943199
$553$	4.91579e6	0.683566
$554$	5.21713e6	0.722200
$555$	0	0
$556$	314320.	0.0431206
$557$	508289.	0.0694180	0.0347090	−	0.999397i	$-0.488950\pi$
$557$	508289.	0.0694180	0.0347090	+	0.999397i	$0.488950\pi$
$558$	−352715.	−0.0479555
$559$	2.24234e6	0.303509
$560$	0	0
$561$	−3.76416e6	−0.504965
$562$	−3.70343e6	−0.494611
$563$	1.76345e6	0.234473	0.117237	−	0.993104i	$-0.462596\pi$
$563$	1.76345e6	0.234473	0.117237	+	0.993104i	$0.462596\pi$
$564$	−1.08632e6	−0.143800
$565$	0	0
$566$	−1.11394e6	−0.146157
$567$	4.10340e6	0.536027
$568$	−7.74260e6	−1.00697
$569$	1.08114e7	1.39992	0.699960	−	0.714182i	$-0.253201\pi$
$569$	1.08114e7	1.39992	0.699960	+	0.714182i	$0.253201\pi$
$570$	0	0
$571$	−8.48510e6	−1.08910	−0.544549	−	0.838729i	$-0.683299\pi$
$571$	−8.48510e6	−1.08910	−0.544549	+	0.838729i	$0.683299\pi$
$572$	−2.88271e6	−0.368393
$573$	−928343.	−0.118120
$574$	1.35037e6	0.171069
$575$	0	0
$576$	1.72695e6	0.216881
$577$	−8.97573e6	−1.12235	−0.561177	−	0.827695i	$-0.689651\pi$
$577$	−8.97573e6	−1.12235	−0.561177	+	0.827695i	$0.689651\pi$
$578$	8.84139e6	1.10078
$579$	383590.	0.0475523
$580$	0	0
$581$	−6.50229e6	−0.799146
$582$	44650.1	0.00546406
$583$	6.00110e6	0.731239
$584$	−4.56132e6	−0.553425
$585$	0	0
$586$	4.96874e6	0.597726
$587$	−9.29097e6	−1.11293	−0.556463	−	0.830873i	$-0.687842\pi$
$587$	−9.29097e6	−1.11293	−0.556463	+	0.830873i	$0.687842\pi$
$588$	777925.	0.0927885
$589$	−195325.	−0.0231990
$590$	0	0
$591$	−358907.	−0.0422682
$592$	1.66320e6	0.195047
$593$	−6.88126e6	−0.803584	−0.401792	−	0.915731i	$-0.631613\pi$
$593$	−6.88126e6	−0.803584	−0.401792	+	0.915731i	$0.631613\pi$
$594$	−1.66929e6	−0.194118
$595$	0	0
$596$	−1.05568e7	−1.21736
$597$	−1.11475e6	−0.128009
$598$	−749020.	−0.0856526
$599$	2.36690e6	0.269533	0.134767	−	0.990877i	$-0.456972\pi$
$599$	2.36690e6	0.269533	0.134767	+	0.990877i	$0.456972\pi$
$600$	0	0
$601$	−1.66780e7	−1.88346	−0.941731	−	0.336368i	$-0.890801\pi$
$601$	−1.66780e7	−1.88346	−0.941731	+	0.336368i	$0.890801\pi$
$602$	2.17534e6	0.244645
$603$	3.76206e6	0.421340
$604$	618772.	0.0690142
$605$	0	0
$606$	711451.	0.0786979
$607$	−148356.	−0.0163430	−0.00817152	−	0.999967i	$-0.502601\pi$
$607$	−148356.	−0.0163430	−0.00817152	+	0.999967i	$0.502601\pi$
$608$	−1.53448e6	−0.168346
$609$	1.19889e6	0.130989
$610$	0	0
$611$	2.61771e6	0.283674
$612$	−1.51285e7	−1.63275
$613$	−1.61218e7	−1.73286	−0.866429	−	0.499301i	$-0.833590\pi$
$613$	−1.61218e7	−1.73286	−0.866429	+	0.499301i	$0.833590\pi$
$614$	−530225.	−0.0567595
$615$	0	0
$616$	−6.07506e6	−0.645058
$617$	9.27027e6	0.980346	0.490173	−	0.871625i	$-0.336934\pi$
$617$	9.27027e6	0.980346	0.490173	+	0.871625i	$0.336934\pi$
$618$	−759992.	−0.0800457
$619$	−2.39889e6	−0.251643	−0.125821	−	0.992053i	$-0.540157\pi$
$619$	−2.39889e6	−0.251643	−0.125821	+	0.992053i	$0.540157\pi$
$620$	0	0
$621$	2.51714e6	0.261926
$622$	2.64365e6	0.273986
$623$	−1.05165e7	−1.08555
$624$	258176.	0.0265433
$625$	0	0
$626$	2.64546e6	0.269814
$627$	−455840.	−0.0463067
$628$	−6.48934e6	−0.656600
$629$	−6.55805e6	−0.660919
$630$	0	0
$631$	5.93145e6	0.593045	0.296523	−	0.955026i	$-0.404173\pi$
$631$	5.93145e6	0.593045	0.296523	+	0.955026i	$0.404173\pi$
$632$	−8.36240e6	−0.832796
$633$	1.00742e6	0.0999317
$634$	−4.77247e6	−0.471542
$635$	0	0
$636$	−673511.	−0.0660240
$637$	−1.87457e6	−0.183043
$638$	8.36530e6	0.813635
$639$	−1.42328e7	−1.37892
$640$	0	0
$641$	−1.85865e7	−1.78670	−0.893351	−	0.449360i	$-0.851652\pi$
$641$	−1.85865e7	−1.78670	−0.893351	+	0.449360i	$0.851652\pi$
$642$	−184594.	−0.0176758
$643$	−1.36174e7	−1.29887	−0.649437	−	0.760416i	$-0.724995\pi$
$643$	−1.36174e7	−1.29887	−0.649437	+	0.760416i	$0.724995\pi$
$644$	4.21701e6	0.400673
$645$	0	0
$646$	1.44360e6	0.136102
$647$	−1.49003e6	−0.139937	−0.0699686	−	0.997549i	$-0.522290\pi$
$647$	−1.49003e6	−0.139937	−0.0699686	+	0.997549i	$0.522290\pi$
$648$	−6.98043e6	−0.653047
$649$	2.51143e7	2.34050
$650$	0	0
$651$	133624.	0.0123576
$652$	1.38716e7	1.27793
$653$	1.02829e7	0.943698	0.471849	−	0.881679i	$-0.343587\pi$
$653$	1.02829e7	0.943698	0.471849	+	0.881679i	$0.343587\pi$
$654$	297785.	0.0272244
$655$	0	0
$656$	4.89724e6	0.444316
$657$	−8.38485e6	−0.757848
$658$	2.53951e6	0.228657
$659$	1.29392e7	1.16063	0.580313	−	0.814393i	$-0.302930\pi$
$659$	1.29392e7	1.16063	0.580313	+	0.814393i	$0.302930\pi$
$660$	0	0
$661$	1.78038e7	1.58492	0.792461	−	0.609922i	$-0.208799\pi$
$661$	1.78038e7	1.58492	0.792461	+	0.609922i	$0.208799\pi$
$662$	−1.63788e6	−0.145257
$663$	−1.01800e6	−0.0899422
$664$	1.10613e7	0.973608
$665$	0	0
$666$	−1.43412e6	−0.125285
$667$	−1.26141e7	−1.09785
$668$	8.38679e6	0.727201
$669$	1.52801e6	0.131996
$670$	0	0
$671$	2.03636e7	1.74601
$672$	1.04976e6	0.0896740
$673$	−5.55797e6	−0.473019	−0.236509	−	0.971629i	$-0.576003\pi$
$673$	−5.55797e6	−0.473019	−0.236509	+	0.971629i	$0.576003\pi$
$674$	2.38952e6	0.202610
$675$	0	0
$676$	−779615.	−0.0656165
$677$	−1.41700e7	−1.18822	−0.594110	−	0.804384i	$-0.702496\pi$
$677$	−1.41700e7	−1.18822	−0.594110	+	0.804384i	$0.702496\pi$
$678$	1.01242e6	0.0845833
$679$	605757.	0.0504225
$680$	0	0
$681$	−1.21540e6	−0.100427
$682$	932367.	0.0767584
$683$	2.01906e7	1.65614	0.828072	−	0.560621i	$-0.189438\pi$
$683$	2.01906e7	1.65614	0.828072	+	0.560621i	$0.189438\pi$
$684$	−1.83207e6	−0.149727
$685$	0	0
$686$	−4.57409e6	−0.371103
$687$	−2.32706e6	−0.188112
$688$	7.88911e6	0.635414
$689$	1.62297e6	0.130245
$690$	0	0
$691$	−8.18523e6	−0.652132	−0.326066	−	0.945347i	$-0.605723\pi$
$691$	−8.18523e6	−0.652132	−0.326066	+	0.945347i	$0.605723\pi$
$692$	−125513.	−0.00996378
$693$	−1.11675e7	−0.883329
$694$	−2.73043e6	−0.215195
$695$	0	0
$696$	−2.03947e6	−0.159586
$697$	−1.93100e7	−1.50557
$698$	−1.46386e6	−0.113726
$699$	1.91734e6	0.148425
$700$	0	0
$701$	−6.38587e6	−0.490823	−0.245411	−	0.969419i	$-0.578923\pi$
$701$	−6.38587e6	−0.490823	−0.245411	+	0.969419i	$0.578923\pi$
$702$	−451450.	−0.0345754
$703$	−794180.	−0.0606081
$704$	−4.56501e6	−0.347144
$705$	0	0
$706$	−2.28739e6	−0.172714
$707$	9.65208e6	0.726227
$708$	−2.81861e6	−0.211326
$709$	1.36770e7	1.02182	0.510911	−	0.859633i	$-0.329308\pi$
$709$	1.36770e7	1.02182	0.510911	+	0.859633i	$0.329308\pi$
$710$	0	0
$711$	−1.53722e7	−1.14041
$712$	1.78900e7	1.32254
$713$	−1.40593e6	−0.103571
$714$	−987585.	−0.0724985
$715$	0	0
$716$	−1.52107e7	−1.10883
$717$	2.33597e6	0.169695
$718$	4.21417e6	0.305071
$719$	1.17397e6	0.0846907	0.0423453	−	0.999103i	$-0.486517\pi$
$719$	1.17397e6	0.0846907	0.0423453	+	0.999103i	$0.486517\pi$
$720$	0	0
$721$	−1.03106e7	−0.738664
$722$	−5.19524e6	−0.370905
$723$	−1.47910e6	−0.105233
$724$	1.12176e7	0.795342
$725$	0	0
$726$	1.27851e6	0.0900248
$727$	−2.09397e7	−1.46938	−0.734691	−	0.678402i	$-0.762673\pi$
$727$	−2.09397e7	−1.46938	−0.734691	+	0.678402i	$0.762673\pi$
$728$	−1.64297e6	−0.114895
$729$	−1.20628e7	−0.840675
$730$	0	0
$731$	−3.11071e7	−2.15311
$732$	−2.28543e6	−0.157648
$733$	8.07907e6	0.555394	0.277697	−	0.960669i	$-0.410429\pi$
$733$	8.07907e6	0.555394	0.277697	+	0.960669i	$0.410429\pi$
$734$	7.25013e6	0.496713
$735$	0	0
$736$	−1.10450e7	−0.751576
$737$	−9.94462e6	−0.674403
$738$	−4.22273e6	−0.285399
$739$	−1.66515e7	−1.12161	−0.560806	−	0.827947i	$-0.689509\pi$
$739$	−1.66515e7	−1.12161	−0.560806	+	0.827947i	$0.689509\pi$
$740$	0	0
$741$	−123280.	−0.00824794
$742$	1.57448e6	0.104985
$743$	−6.99037e6	−0.464545	−0.232273	−	0.972651i	$-0.574616\pi$
$743$	−6.99037e6	−0.464545	−0.232273	+	0.972651i	$0.574616\pi$
$744$	−227312.	−0.0150553
$745$	0	0
$746$	3.60810e6	0.237373
$747$	2.03334e7	1.33324
$748$	3.99908e7	2.61340
$749$	−2.50434e6	−0.163113
$750$	0	0
$751$	1.16465e7	0.753520	0.376760	−	0.926311i	$-0.377038\pi$
$751$	1.16465e7	0.753520	0.376760	+	0.926311i	$0.377038\pi$
$752$	9.20979e6	0.593889
$753$	−4.56762e6	−0.293564
$754$	2.26235e6	0.144921
$755$	0	0
$756$	2.54168e6	0.161740
$757$	905942.	0.0574593	0.0287297	−	0.999587i	$-0.490854\pi$
$757$	905942.	0.0574593	0.0287297	+	0.999587i	$0.490854\pi$
$758$	−1.18616e7	−0.749845
$759$	−3.28109e6	−0.206735
$760$	0	0
$761$	1.01329e7	0.634267	0.317134	−	0.948381i	$-0.397280\pi$
$761$	1.01329e7	0.634267	0.317134	+	0.948381i	$0.397280\pi$
$762$	1.61993e6	0.101067
$763$	4.03997e6	0.251227
$764$	9.86280e6	0.611317
$765$	0	0
$766$	3.03625e6	0.186967
$767$	6.79204e6	0.416881
$768$	−451378.	−0.0276145
$769$	1.25661e7	0.766274	0.383137	−	0.923692i	$-0.374844\pi$
$769$	1.25661e7	0.766274	0.383137	+	0.923692i	$0.374844\pi$
$770$	0	0
$771$	−4.12075e6	−0.249655
$772$	−4.07529e6	−0.246102
$773$	3.34145e6	0.201135	0.100567	−	0.994930i	$-0.467934\pi$
$773$	3.34145e6	0.201135	0.100567	+	0.994930i	$0.467934\pi$
$774$	−6.80253e6	−0.408148
$775$	0	0
$776$	−1.03047e6	−0.0614302
$777$	543309.	0.0322845
$778$	−643132.	−0.0380935
$779$	−2.33844e6	−0.138065
$780$	0	0
$781$	3.76231e7	2.20712
$782$	1.03909e7	0.607625
$783$	−7.60281e6	−0.443169
$784$	−6.59523e6	−0.383213
$785$	0	0
$786$	−1.20678e6	−0.0696743
$787$	−1.19270e7	−0.686424	−0.343212	−	0.939258i	$-0.611515\pi$
$787$	−1.19270e7	−0.686424	−0.343212	+	0.939258i	$0.611515\pi$
$788$	3.81306e6	0.218755
$789$	−4.29703e6	−0.245740
$790$	0	0
$791$	1.37352e7	0.780537
$792$	1.89974e7	1.07617
$793$	5.50722e6	0.310992
$794$	3.52554e6	0.198461
$795$	0	0
$796$	1.18432e7	0.662498
$797$	1.42516e7	0.794729	0.397365	−	0.917661i	$-0.369925\pi$
$797$	1.42516e7	0.794729	0.397365	+	0.917661i	$0.369925\pi$
$798$	−119596.	−0.00664831
$799$	−3.63146e7	−2.01240
$800$	0	0
$801$	3.28863e7	1.81106
$802$	1.88897e6	0.103703
$803$	2.21645e7	1.21302
$804$	1.11610e6	0.0608923
$805$	0	0
$806$	252154.	0.0136719
$807$	−166862.	−0.00901932
$808$	−1.64194e7	−0.884770
$809$	−2.13109e7	−1.14480	−0.572400	−	0.819974i	$-0.693988\pi$
$809$	−2.13109e7	−1.14480	−0.572400	+	0.819974i	$0.693988\pi$
$810$	0	0
$811$	1.86425e7	0.995294	0.497647	−	0.867380i	$-0.334198\pi$
$811$	1.86425e7	0.995294	0.497647	+	0.867380i	$0.334198\pi$
$812$	−1.27371e7	−0.677924
$813$	3.54437e6	0.188067
$814$	3.79095e6	0.200534
$815$	0	0
$816$	−3.58158e6	−0.188300
$817$	−3.76707e6	−0.197446
$818$	1.03737e6	0.0542064
$819$	−3.02019e6	−0.157335
$820$	0	0
$821$	−3.17151e7	−1.64213	−0.821066	−	0.570833i	$-0.806620\pi$
$821$	−3.17151e7	−1.64213	−0.821066	+	0.570833i	$0.806620\pi$
$822$	1.11925e6	0.0577762
$823$	−3.21047e7	−1.65222	−0.826112	−	0.563506i	$-0.809452\pi$
$823$	−3.21047e7	−1.65222	−0.826112	+	0.563506i	$0.809452\pi$
$824$	1.75397e7	0.899922
$825$	0	0
$826$	6.58912e6	0.336029
$827$	3.37394e7	1.71543	0.857717	−	0.514123i	$-0.171882\pi$
$827$	3.37394e7	1.71543	0.857717	+	0.514123i	$0.171882\pi$
$828$	−1.31870e7	−0.668454
$829$	7.35563e6	0.371735	0.185868	−	0.982575i	$-0.440490\pi$
$829$	7.35563e6	0.371735	0.185868	+	0.982575i	$0.440490\pi$
$830$	0	0
$831$	6.18067e6	0.310480
$832$	−1.23458e6	−0.0618318
$833$	2.60053e7	1.29852
$834$	−64164.0	−0.00319430
$835$	0	0
$836$	4.84288e6	0.239656
$837$	−847383.	−0.0418086
$838$	3.43384e6	0.168916
$839$	−1.82258e7	−0.893885	−0.446942	−	0.894563i	$-0.647487\pi$
$839$	−1.82258e7	−0.893885	−0.446942	+	0.894563i	$0.647487\pi$
$840$	0	0
$841$	1.75888e7	0.857523
$842$	−5.72298e6	−0.278190
$843$	−4.38741e6	−0.212637
$844$	−1.07030e7	−0.517187
$845$	0	0
$846$	−7.94130e6	−0.381475
$847$	1.73452e7	0.830751
$848$	5.71001e6	0.272676
$849$	−1.31967e6	−0.0628340
$850$	0	0
$851$	−5.71643e6	−0.270583
$852$	−4.22249e6	−0.199283
$853$	−1.85516e7	−0.872987	−0.436494	−	0.899707i	$-0.643780\pi$
$853$	−1.85516e7	−0.872987	−0.436494	+	0.899707i	$0.643780\pi$
$854$	5.34269e6	0.250677
$855$	0	0
$856$	4.26021e6	0.198722
$857$	710710.	0.0330552	0.0165276	−	0.999863i	$-0.494739\pi$
$857$	710710.	0.0330552	0.0165276	+	0.999863i	$0.494739\pi$
$858$	588466.	0.0272900
$859$	−2.15616e7	−0.997006	−0.498503	−	0.866888i	$-0.666117\pi$
$859$	−2.15616e7	−0.997006	−0.498503	+	0.866888i	$0.666117\pi$
$860$	0	0
$861$	1.59976e6	0.0735440
$862$	−4.46032e6	−0.204455
$863$	1.43049e7	0.653817	0.326909	−	0.945056i	$-0.393993\pi$
$863$	1.43049e7	0.653817	0.326909	+	0.945056i	$0.393993\pi$
$864$	−6.65708e6	−0.303389
$865$	0	0
$866$	7.27773e6	0.329762
$867$	1.04743e7	0.473235
$868$	−1.41964e6	−0.0639554
$869$	4.06349e7	1.82536
$870$	0	0
$871$	−2.68947e6	−0.120122
$872$	−6.87252e6	−0.306073
$873$	−1.89427e6	−0.0841212
$874$	1.25833e6	0.0557208
$875$	0	0
$876$	−2.48755e6	−0.109525
$877$	−3.64324e7	−1.59952	−0.799759	−	0.600322i	$-0.795039\pi$
$877$	−3.64324e7	−1.59952	−0.799759	+	0.600322i	$0.795039\pi$
$878$	1.28888e6	0.0564257
$879$	5.88640e6	0.256967
$880$	0	0
$881$	−1.29195e7	−0.560797	−0.280398	−	0.959884i	$-0.590467\pi$
$881$	−1.29195e7	−0.560797	−0.280398	+	0.959884i	$0.590467\pi$
$882$	5.68685e6	0.246150
$883$	−1.74399e7	−0.752735	−0.376367	−	0.926471i	$-0.622827\pi$
$883$	−1.74399e7	−0.752735	−0.376367	+	0.926471i	$0.622827\pi$
$884$	1.08153e7	0.465488
$885$	0	0
$886$	70941.9	0.00303612
$887$	−5.45819e6	−0.232938	−0.116469	−	0.993194i	$-0.537157\pi$
$887$	−5.45819e6	−0.232938	−0.116469	+	0.993194i	$0.537157\pi$
$888$	−924240.	−0.0393326
$889$	2.19771e7	0.932645
$890$	0	0
$891$	3.39195e7	1.43138
$892$	−1.62338e7	−0.683136
$893$	−4.39769e6	−0.184542
$894$	2.15503e6	0.0901798
$895$	0	0
$896$	−1.42722e7	−0.593910
$897$	−887355.	−0.0368227
$898$	2.48024e6	0.102637
$899$	4.24649e6	0.175239
$900$	0	0
$901$	−2.25148e7	−0.923967
$902$	1.11624e7	0.456815
$903$	2.57710e6	0.105175
$904$	−2.33654e7	−0.950937
$905$	0	0
$906$	−126314.	−0.00511246
$907$	3.84724e7	1.55285	0.776427	−	0.630207i	$-0.217030\pi$
$907$	3.84724e7	1.55285	0.776427	+	0.630207i	$0.217030\pi$
$908$	1.29125e7	0.519753
$909$	−3.01831e7	−1.21158
$910$	0	0
$911$	2.86336e7	1.14309	0.571543	−	0.820572i	$-0.306345\pi$
$911$	2.86336e7	1.14309	0.571543	+	0.820572i	$0.306345\pi$
$912$	−433729.	−0.0172676
$913$	−5.37492e7	−2.13400
$914$	2.74158e6	0.108551
$915$	0	0
$916$	2.47229e7	0.973555
$917$	−1.63721e7	−0.642957
$918$	6.26280e6	0.245280
$919$	−2.14158e7	−0.836460	−0.418230	−	0.908341i	$-0.637349\pi$
$919$	−2.14158e7	−0.836460	−0.418230	+	0.908341i	$0.637349\pi$
$920$	0	0
$921$	−628150.	−0.0244014
$922$	5.17296e6	0.200406
$923$	1.01750e7	0.393123
$924$	−3.31308e6	−0.127659
$925$	0	0
$926$	1.14441e7	0.438586
$927$	3.22424e7	1.23233
$928$	3.33606e7	1.27164
$929$	1.93670e6	0.0736245	0.0368123	−	0.999322i	$-0.488280\pi$
$929$	1.93670e6	0.0736245	0.0368123	+	0.999322i	$0.488280\pi$
$930$	0	0
$931$	3.14923e6	0.119078
$932$	−2.03700e7	−0.768159
$933$	3.13190e6	0.117789
$934$	1.18631e7	0.444970
$935$	0	0
$936$	5.13774e6	0.191683
$937$	−5.93574e6	−0.220865	−0.110432	−	0.993884i	$-0.535224\pi$
$937$	−5.93574e6	−0.220865	−0.110432	+	0.993884i	$0.535224\pi$
$938$	−2.60912e6	−0.0968249
$939$	3.13404e6	0.115995
$940$	0	0
$941$	4.84523e7	1.78378	0.891888	−	0.452256i	$-0.149381\pi$
$941$	4.84523e7	1.78378	0.891888	+	0.452256i	$0.149381\pi$
$942$	1.32471e6	0.0486399
$943$	−1.68319e7	−0.616387
$944$	2.38961e7	0.872766
$945$	0	0
$946$	1.79818e7	0.653289
$947$	1.16568e7	0.422382	0.211191	−	0.977445i	$-0.432266\pi$
$947$	1.16568e7	0.422382	0.211191	+	0.977445i	$0.432266\pi$
$948$	−4.56050e6	−0.164813
$949$	5.99428e6	0.216059
$950$	0	0
$951$	−5.65389e6	−0.202720
$952$	2.27923e7	0.815072
$953$	2.70508e7	0.964825	0.482413	−	0.875944i	$-0.339761\pi$
$953$	2.70508e7	0.964825	0.482413	+	0.875944i	$0.339761\pi$
$954$	−4.92356e6	−0.175149
$955$	0	0
$956$	−2.48176e7	−0.878243
$957$	9.91026e6	0.349788
$958$	1.80357e7	0.634919
$959$	1.51847e7	0.533161
$960$	0	0
$961$	−2.81559e7	−0.983468
$962$	1.02524e6	0.0357182
$963$	7.83133e6	0.272126
$964$	1.57140e7	0.544622
$965$	0	0
$966$	−860844.	−0.0296812
$967$	6.67822e6	0.229665	0.114832	−	0.993385i	$-0.463367\pi$
$967$	6.67822e6	0.229665	0.114832	+	0.993385i	$0.463367\pi$
$968$	−2.95065e7	−1.01211
$969$	1.71021e6	0.0585114
$970$	0	0
$971$	−1.10643e7	−0.376597	−0.188298	−	0.982112i	$-0.560297\pi$
$971$	−1.10643e7	−0.376597	−0.188298	+	0.982112i	$0.560297\pi$
$972$	−1.19769e7	−0.406610
$973$	−870497.	−0.0294771
$974$	−2.68543e6	−0.0907020
$975$	0	0
$976$	1.93758e7	0.651082
$977$	4.37043e7	1.46483	0.732416	−	0.680857i	$-0.238393\pi$
$977$	4.37043e7	1.46483	0.732416	+	0.680857i	$0.238393\pi$
$978$	−2.83169e6	−0.0946670
$979$	−8.69315e7	−2.89882
$980$	0	0
$981$	−1.26334e7	−0.419130
$982$	1.02473e6	0.0339103
$983$	5.65161e7	1.86547	0.932735	−	0.360563i	$-0.117415\pi$
$983$	5.65161e7	1.86547	0.932735	+	0.360563i	$0.117415\pi$
$984$	−2.72140e6	−0.0895994
$985$	0	0
$986$	−3.13848e7	−1.02808
$987$	3.00852e6	0.0983015
$988$	1.30973e6	0.0426865
$989$	−2.71150e7	−0.881492
$990$	0	0
$991$	2.40368e7	0.777487	0.388744	−	0.921346i	$-0.372909\pi$
$991$	2.40368e7	0.777487	0.388744	+	0.921346i	$0.372909\pi$
$992$	3.71826e6	0.119967
$993$	−1.94038e6	−0.0624473
$994$	9.87098e6	0.316880
$995$	0	0
$996$	6.03234e6	0.192680
$997$	4.18542e7	1.33353	0.666763	−	0.745270i	$-0.267679\pi$
$997$	4.18542e7	1.33353	0.666763	+	0.745270i	$0.267679\pi$
$998$	7.45166e6	0.236825
$999$	−3.44541e6	−0.109226

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.f.1.4		6
5.2	odd	4		325.6.b.f.274.7		12
5.3	odd	4		325.6.b.f.274.6		12
5.4	even	2		65.6.a.e.1.3	✓	6
15.14	odd	2		585.6.a.k.1.4		6
20.19	odd	2		1040.6.a.r.1.4		6
65.64	even	2		845.6.a.g.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.e.1.3	✓	6	5.4	even	2
325.6.a.f.1.4		6	1.1	even	1	trivial
325.6.b.f.274.6		12	5.3	odd	4
325.6.b.f.274.7		12	5.2	odd	4
585.6.a.k.1.4		6	15.14	odd	2
845.6.a.g.1.4		6	65.64	even	2
1040.6.a.r.1.4		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)

\(f(q)\)	\(=\)	\(q+2.16876 q^{2} +2.56930 q^{3} -27.2965 q^{4} +5.57220 q^{6} +75.5966 q^{7} -128.600 q^{8} -236.399 q^{9} +624.896 q^{11} -70.1329 q^{12} +169.000 q^{13} +163.951 q^{14} +594.585 q^{16} -2344.47 q^{17} -512.692 q^{18} -283.916 q^{19} +194.231 q^{21} +1355.25 q^{22} -2043.60 q^{23} -330.412 q^{24} +366.520 q^{26} -1231.72 q^{27} -2063.52 q^{28} +6172.51 q^{29} +687.967 q^{31} +5404.71 q^{32} +1605.55 q^{33} -5084.60 q^{34} +6452.85 q^{36} +2797.24 q^{37} -615.745 q^{38} +434.212 q^{39} +8236.40 q^{41} +421.239 q^{42} +13268.3 q^{43} -17057.5 q^{44} -4432.07 q^{46} +15489.4 q^{47} +1527.67 q^{48} -11092.1 q^{49} -6023.66 q^{51} -4613.11 q^{52} +9603.36 q^{53} -2671.30 q^{54} -9721.71 q^{56} -729.465 q^{57} +13386.7 q^{58} +40189.6 q^{59} +32587.1 q^{61} +1492.03 q^{62} -17870.9 q^{63} -7305.22 q^{64} +3482.05 q^{66} -15914.0 q^{67} +63995.9 q^{68} -5250.62 q^{69} +60206.9 q^{71} +30400.8 q^{72} +35469.1 q^{73} +6066.54 q^{74} +7749.90 q^{76} +47240.1 q^{77} +941.701 q^{78} +65026.6 q^{79} +54280.2 q^{81} +17862.8 q^{82} -86013.0 q^{83} -5301.81 q^{84} +28775.7 q^{86} +15859.1 q^{87} -80361.5 q^{88} -139114. q^{89} +12775.8 q^{91} +55783.0 q^{92} +1767.59 q^{93} +33592.8 q^{94} +13886.3 q^{96} +8013.02 q^{97} -24056.2 q^{98} -147725. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - 20 q^{3} + 134 q^{4} - 52 q^{6} - 172 q^{7} + 138 q^{8} + 1034 q^{9} + 800 q^{11} + 832 q^{12} + 1014 q^{13} + 2108 q^{14} + 322 q^{16} - 4396 q^{17} - 6142 q^{18} + 5304 q^{19} + 1072 q^{21}+ \cdots + 301264 q^{99}+O(q^{100}) \) 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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