LMFDB - Embedded newform 325.6.a.e.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:	$4$
Coefficient field:	4.4.1878612.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 16x^{2} + 16x + 49 $ x^4 - 2x^3 - 16x^2 + 16*x + 49
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.13795$ 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)

$f(q)$	$=$	\(q+8.75180 q^{2} +26.2145 q^{3} +44.5940 q^{4} +229.424 q^{6} -8.06989 q^{7} +110.220 q^{8} +444.199 q^{9} +591.993 q^{11} +1169.01 q^{12} +169.000 q^{13} -70.6260 q^{14} -462.381 q^{16} +936.856 q^{17} +3887.54 q^{18} -560.050 q^{19} -211.548 q^{21} +5181.00 q^{22} -2970.19 q^{23} +2889.37 q^{24} +1479.05 q^{26} +5274.32 q^{27} -359.869 q^{28} +802.996 q^{29} -4511.01 q^{31} -7573.72 q^{32} +15518.8 q^{33} +8199.18 q^{34} +19808.6 q^{36} -9143.29 q^{37} -4901.45 q^{38} +4430.25 q^{39} +15224.1 q^{41} -1851.42 q^{42} +20927.4 q^{43} +26399.3 q^{44} -25994.5 q^{46} +20002.7 q^{47} -12121.1 q^{48} -16741.9 q^{49} +24559.2 q^{51} +7536.39 q^{52} -19701.8 q^{53} +46159.8 q^{54} -889.467 q^{56} -14681.4 q^{57} +7027.66 q^{58} -41769.8 q^{59} +41527.5 q^{61} -39479.5 q^{62} -3584.63 q^{63} -51487.5 q^{64} +135817. q^{66} +8814.24 q^{67} +41778.2 q^{68} -77862.0 q^{69} -11984.8 q^{71} +48959.8 q^{72} +59144.1 q^{73} -80020.3 q^{74} -24974.9 q^{76} -4777.31 q^{77} +38772.6 q^{78} -69794.0 q^{79} +30323.3 q^{81} +133238. q^{82} +17957.2 q^{83} -9433.77 q^{84} +183152. q^{86} +21050.1 q^{87} +65249.7 q^{88} -110432. q^{89} -1363.81 q^{91} -132453. q^{92} -118254. q^{93} +175060. q^{94} -198541. q^{96} -136379. q^{97} -146522. q^{98} +262962. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 9 q^{2} + 4 q^{3} + 93 q^{4} - 74 q^{6} + 136 q^{7} + 447 q^{8} + 644 q^{9} - 516 q^{11} + 710 q^{12} + 676 q^{13} + 1604 q^{14} - 719 q^{16} - 344 q^{17} + 6905 q^{18} - 5012 q^{19} - 4448 q^{21}+ \cdots + 290668 q^{99}+O(q^{100}) $ 4 * q + 9 * q^2 + 4 * q^3 + 93 * q^4 - 74 * q^6 + 136 * q^7 + 447 * q^8 + 644 * q^9 - 516 * q^11 + 710 * q^12 + 676 * q^13 + 1604 * q^14 - 719 * q^16 - 344 * q^17 + 6905 * q^18 - 5012 * q^19 - 4448 * q^21 + 9318 * q^22 - 708 * q^23 - 2094 * q^24 + 1521 * q^26 + 4264 * q^27 + 7268 * q^28 + 5416 * q^29 - 10124 * q^31 - 9497 * q^32 + 16256 * q^33 + 3022 * q^34 + 36205 * q^36 - 1448 * q^37 - 11922 * q^38 + 676 * q^39 + 21576 * q^41 - 33128 * q^42 + 25740 * q^43 + 24710 * q^44 + 28754 * q^46 + 51096 * q^47 - 43282 * q^48 - 40412 * q^49 + 52984 * q^51 + 15717 * q^52 - 39448 * q^53 + 60340 * q^54 + 30188 * q^56 + 56160 * q^57 - 1374 * q^58 + 37764 * q^59 + 95016 * q^61 - 83102 * q^62 + 43592 * q^63 - 142095 * q^64 + 198500 * q^66 + 50592 * q^67 + 47110 * q^68 - 177648 * q^69 - 89308 * q^71 + 99183 * q^72 + 108312 * q^73 - 16766 * q^74 - 3442 * q^76 - 6208 * q^77 - 12506 * q^78 - 25464 * q^79 - 106348 * q^81 + 206426 * q^82 - 20584 * q^83 - 213448 * q^84 + 363666 * q^86 + 128312 * q^87 - 20558 * q^88 - 148328 * q^89 + 22984 * q^91 + 173474 * q^92 + 10496 * q^93 + 63916 * q^94 + 85154 * q^96 + 77816 * q^97 + 83505 * q^98 + 290668 * 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 $

$2$	8.75180	1.54711	0.773557	−	0.633726i	$-0.218475\pi$
$2$	8.75180	1.54711	0.773557	+	0.633726i	$0.218475\pi$
$3$	26.2145	1.68166	0.840830	−	0.541300i	$-0.182068\pi$
$3$	26.2145	1.68166	0.840830	+	0.541300i	$0.182068\pi$
$4$	44.5940	1.39356
$5$	0	0
$5$	0	0
$6$	229.424	2.60172
$7$	−8.06989	−0.0622476	−0.0311238	−	0.999516i	$-0.509909\pi$
$7$	−8.06989	−0.0622476	−0.0311238	+	0.999516i	$0.509909\pi$
$8$	110.220	0.608888
$9$	444.199	1.82798
$10$	0	0
$11$	591.993	1.47514	0.737572	−	0.675268i	$-0.235972\pi$
$11$	591.993	1.47514	0.737572	+	0.675268i	$0.235972\pi$
$12$	1169.01	2.34350
$13$	169.000	0.277350
$13$	169.000	0.277350
$14$	−70.6260	−0.0963041
$15$	0	0
$16$	−462.381	−0.451544
$17$	936.856	0.786232	0.393116	−	0.919489i	$-0.371397\pi$
$17$	936.856	0.786232	0.393116	+	0.919489i	$0.371397\pi$
$18$	3887.54	2.82809
$19$	−560.050	−0.355912	−0.177956	−	0.984038i	$-0.556949\pi$
$19$	−560.050	−0.355912	−0.177956	+	0.984038i	$0.556949\pi$
$20$	0	0
$21$	−211.548	−0.104679
$22$	5181.00	2.28222
$23$	−2970.19	−1.17075	−0.585376	−	0.810762i	$-0.699053\pi$
$23$	−2970.19	−1.17075	−0.585376	+	0.810762i	$0.699053\pi$
$24$	2889.37	1.02394
$25$	0	0
$26$	1479.05	0.429092
$27$	5274.32	1.39238
$28$	−359.869	−0.0867459
$29$	802.996	0.177304	0.0886520	−	0.996063i	$-0.471744\pi$
$29$	802.996	0.177304	0.0886520	+	0.996063i	$0.471744\pi$
$30$	0	0
$31$	−4511.01	−0.843082	−0.421541	−	0.906809i	$-0.638511\pi$
$31$	−4511.01	−0.843082	−0.421541	+	0.906809i	$0.638511\pi$
$32$	−7573.72	−1.30748
$33$	15518.8	2.48069
$34$	8199.18	1.21639
$35$	0	0
$36$	19808.6	2.54740
$37$	−9143.29	−1.09799	−0.548995	−	0.835826i	$-0.684989\pi$
$37$	−9143.29	−1.09799	−0.548995	+	0.835826i	$0.684989\pi$
$38$	−4901.45	−0.550637
$39$	4430.25	0.466408
$40$	0	0
$41$	15224.1	1.41440	0.707199	−	0.707014i	$-0.249958\pi$
$41$	15224.1	1.41440	0.707199	+	0.707014i	$0.249958\pi$
$42$	−1851.42	−0.161951
$43$	20927.4	1.72601	0.863006	−	0.505193i	$-0.168579\pi$
$43$	20927.4	1.72601	0.863006	+	0.505193i	$0.168579\pi$
$44$	26399.3	2.05571
$45$	0	0
$46$	−25994.5	−1.81129
$47$	20002.7	1.32082	0.660411	−	0.750904i	$-0.270382\pi$
$47$	20002.7	1.32082	0.660411	+	0.750904i	$0.270382\pi$
$48$	−12121.1	−0.759344
$49$	−16741.9	−0.996125
$50$	0	0
$51$	24559.2	1.32217
$52$	7536.39	0.386505
$53$	−19701.8	−0.963423	−0.481711	−	0.876330i	$-0.659985\pi$
$53$	−19701.8	−0.963423	−0.481711	+	0.876330i	$0.659985\pi$
$54$	46159.8	2.15417
$55$	0	0
$56$	−889.467	−0.0379018
$57$	−14681.4	−0.598523
$58$	7027.66	0.274310
$59$	−41769.8	−1.56218	−0.781092	−	0.624416i	$-0.785337\pi$
$59$	−41769.8	−1.56218	−0.781092	+	0.624416i	$0.785337\pi$
$60$	0	0
$61$	41527.5	1.42893	0.714466	−	0.699670i	$-0.246670\pi$
$61$	41527.5	1.42893	0.714466	+	0.699670i	$0.246670\pi$
$62$	−39479.5	−1.30434
$63$	−3584.63	−0.113787
$64$	−51487.5	−1.57127
$65$	0	0
$66$	135817.	3.83791
$67$	8814.24	0.239882	0.119941	−	0.992781i	$-0.461729\pi$
$67$	8814.24	0.239882	0.119941	+	0.992781i	$0.461729\pi$
$68$	41778.2	1.09566
$69$	−77862.0	−1.96881
$70$	0	0
$71$	−11984.8	−0.282152	−0.141076	−	0.989999i	$-0.545056\pi$
$71$	−11984.8	−0.282152	−0.141076	+	0.989999i	$0.545056\pi$
$72$	48959.8	1.11303
$73$	59144.1	1.29899	0.649493	−	0.760368i	$-0.274981\pi$
$73$	59144.1	1.29899	0.649493	+	0.760368i	$0.274981\pi$
$74$	−80020.3	−1.69872
$75$	0	0
$76$	−24974.9	−0.495986
$77$	−4777.31	−0.0918241
$78$	38772.6	0.721587
$79$	−69794.0	−1.25820	−0.629101	−	0.777323i	$-0.716577\pi$
$79$	−69794.0	−1.25820	−0.629101	+	0.777323i	$0.716577\pi$
$80$	0	0
$81$	30323.3	0.513527
$82$	133238.	2.18824
$83$	17957.2	0.286117	0.143058	−	0.989714i	$-0.454306\pi$
$83$	17957.2	0.286117	0.143058	+	0.989714i	$0.454306\pi$
$84$	−9433.77	−0.145877
$85$	0	0
$86$	183152.	2.67034
$87$	21050.1	0.298165
$88$	65249.7	0.898198
$89$	−110432.	−1.47781	−0.738907	−	0.673807i	$-0.764658\pi$
$89$	−110432.	−1.47781	−0.738907	+	0.673807i	$0.764658\pi$
$90$	0	0
$91$	−1363.81	−0.0172644
$92$	−132453.	−1.63152
$93$	−118254.	−1.41778
$94$	175060.	2.04346
$95$	0	0
$96$	−198541.	−2.19873
$97$	−136379.	−1.47170	−0.735850	−	0.677144i	$-0.763217\pi$
$97$	−136379.	−1.47170	−0.735850	+	0.677144i	$0.763217\pi$
$98$	−146522.	−1.54112
$99$	262962.	2.69653
$100$	0	0
$101$	−65451.1	−0.638430	−0.319215	−	0.947682i	$-0.603419\pi$
$101$	−65451.1	−0.638430	−0.319215	+	0.947682i	$0.603419\pi$
$102$	214937.	2.04556
$103$	101771.	0.945215	0.472607	−	0.881273i	$-0.343313\pi$
$103$	101771.	0.945215	0.472607	+	0.881273i	$0.343313\pi$
$104$	18627.3	0.168875
$105$	0	0
$106$	−172427.	−1.49053
$107$	−179471.	−1.51543	−0.757715	−	0.652586i	$-0.773684\pi$
$107$	−179471.	−1.51543	−0.757715	+	0.652586i	$0.773684\pi$
$108$	235203.	1.94037
$109$	−40851.4	−0.329337	−0.164669	−	0.986349i	$-0.552655\pi$
$109$	−40851.4	−0.329337	−0.164669	+	0.986349i	$0.552655\pi$
$110$	0	0
$111$	−239687.	−1.84644
$112$	3731.36	0.0281075
$113$	−227150.	−1.67347	−0.836733	−	0.547612i	$-0.815537\pi$
$113$	−227150.	−1.67347	−0.836733	+	0.547612i	$0.815537\pi$
$114$	−128489.	−0.925984
$115$	0	0
$116$	35808.8	0.247084
$117$	75069.6	0.506990
$118$	−365561.	−2.41688
$119$	−7560.32	−0.0489410
$120$	0	0
$121$	189404.	1.17605
$122$	363441.	2.21072
$123$	399092.	2.37854
$124$	−201164.	−1.17489
$125$	0	0
$126$	−31372.0	−0.176042
$127$	−38400.2	−0.211263	−0.105632	−	0.994405i	$-0.533686\pi$
$127$	−38400.2	−0.211263	−0.105632	+	0.994405i	$0.533686\pi$
$128$	−208249.	−1.12346
$129$	548601.	2.90257
$130$	0	0
$131$	114414.	0.582509	0.291254	−	0.956646i	$-0.405927\pi$
$131$	114414.	0.582509	0.291254	+	0.956646i	$0.405927\pi$
$132$	692045.	3.45700
$133$	4519.54	0.0221547
$134$	77140.5	0.371125
$135$	0	0
$136$	103261.	0.478727
$137$	299097.	1.36148	0.680739	−	0.732526i	$-0.261659\pi$
$137$	299097.	1.36148	0.680739	+	0.732526i	$0.261659\pi$
$138$	−681433.	−3.04597
$139$	−148537.	−0.652074	−0.326037	−	0.945357i	$-0.605713\pi$
$139$	−148537.	−0.652074	−0.326037	+	0.945357i	$0.605713\pi$
$140$	0	0
$141$	524361.	2.22117
$142$	−104888.	−0.436522
$143$	100047.	0.409132
$144$	−205389.	−0.825413
$145$	0	0
$146$	517618.	2.00968
$147$	−438880.	−1.67514
$148$	−407736.	−1.53012
$149$	−295183.	−1.08924	−0.544622	−	0.838682i	$-0.683327\pi$
$149$	−295183.	−1.08924	−0.544622	+	0.838682i	$0.683327\pi$
$150$	0	0
$151$	−406812.	−1.45195	−0.725974	−	0.687722i	$-0.758611\pi$
$151$	−406812.	−1.45195	−0.725974	+	0.687722i	$0.758611\pi$
$152$	−61729.0	−0.216711
$153$	416150.	1.43722
$154$	−41810.1	−0.142062
$155$	0	0
$156$	197563.	0.649970
$157$	103767.	0.335978	0.167989	−	0.985789i	$-0.446273\pi$
$157$	103767.	0.335978	0.167989	+	0.985789i	$0.446273\pi$
$158$	−610824.	−1.94658
$159$	−516473.	−1.62015
$160$	0	0
$161$	23969.1	0.0728765
$162$	265383.	0.794485
$163$	520351.	1.53401	0.767004	−	0.641642i	$-0.221747\pi$
$163$	520351.	1.53401	0.767004	+	0.641642i	$0.221747\pi$
$164$	678904.	1.97105
$165$	0	0
$166$	157158.	0.442655
$167$	225112.	0.624609	0.312305	−	0.949982i	$-0.398899\pi$
$167$	225112.	0.624609	0.312305	+	0.949982i	$0.398899\pi$
$168$	−23316.9	−0.0637379
$169$	28561.0	0.0769231
$170$	0	0
$171$	−248774.	−0.650600
$172$	933237.	2.40531
$173$	−326742.	−0.830023	−0.415011	−	0.909816i	$-0.636222\pi$
$173$	−326742.	−0.830023	−0.415011	+	0.909816i	$0.636222\pi$
$174$	184227.	0.461295
$175$	0	0
$176$	−273726.	−0.666093
$177$	−1.09497e6	−2.62706
$178$	−966479.	−2.28635
$179$	312278.	0.728465	0.364232	−	0.931308i	$-0.381331\pi$
$179$	312278.	0.728465	0.364232	+	0.931308i	$0.381331\pi$
$180$	0	0
$181$	−34110.0	−0.0773901	−0.0386951	−	0.999251i	$-0.512320\pi$
$181$	−34110.0	−0.0773901	−0.0386951	+	0.999251i	$0.512320\pi$
$182$	−11935.8	−0.0267100
$183$	1.08862e6	2.40298
$184$	−327376.	−0.712857
$185$	0	0
$186$	−1.03493e6	−2.19346
$187$	554612.	1.15981
$188$	892002.	1.84065
$189$	−42563.2	−0.0866721
$190$	0	0
$191$	−232313.	−0.460776	−0.230388	−	0.973099i	$-0.574000\pi$
$191$	−232313.	−0.460776	−0.230388	+	0.973099i	$0.574000\pi$
$192$	−1.34972e6	−2.64235
$193$	493813.	0.954265	0.477132	−	0.878831i	$-0.341676\pi$
$193$	493813.	0.954265	0.477132	+	0.878831i	$0.341676\pi$
$194$	−1.19357e6	−2.27689
$195$	0	0
$196$	−746588.	−1.38816
$197$	172838.	0.317302	0.158651	−	0.987335i	$-0.449286\pi$
$197$	172838.	0.317302	0.158651	+	0.987335i	$0.449286\pi$
$198$	2.30139e6	4.17185
$199$	382327.	0.684388	0.342194	−	0.939629i	$-0.388830\pi$
$199$	382327.	0.684388	0.342194	+	0.939629i	$0.388830\pi$
$200$	0	0
$201$	231061.	0.403400
$202$	−572815.	−0.987725
$203$	−6480.09	−0.0110367
$204$	1.09519e6	1.84253
$205$	0	0
$206$	890679.	1.46236
$207$	−1.31936e6	−2.14011
$208$	−78142.5	−0.125236
$209$	−331546.	−0.525022
$210$	0	0
$211$	789495.	1.22080	0.610398	−	0.792095i	$-0.291009\pi$
$211$	789495.	1.22080	0.610398	+	0.792095i	$0.291009\pi$
$212$	−878584.	−1.34259
$213$	−314174.	−0.474484
$214$	−1.57070e6	−2.34454
$215$	0	0
$216$	581338.	0.847802
$217$	36403.4	0.0524798
$218$	−357524.	−0.509523
$219$	1.55043e6	2.18445
$220$	0	0
$221$	158329.	0.218062
$222$	−2.09769e6	−2.85666
$223$	−898082.	−1.20936	−0.604678	−	0.796470i	$-0.706698\pi$
$223$	−898082.	−1.20936	−0.604678	+	0.796470i	$0.706698\pi$
$224$	61119.1	0.0813873
$225$	0	0
$226$	−1.98797e6	−2.58904
$227$	426009.	0.548725	0.274362	−	0.961626i	$-0.411533\pi$
$227$	426009.	0.548725	0.274362	+	0.961626i	$0.411533\pi$
$228$	−654704.	−0.834080
$229$	−221579.	−0.279216	−0.139608	−	0.990207i	$-0.544584\pi$
$229$	−221579.	−0.279216	−0.139608	+	0.990207i	$0.544584\pi$
$230$	0	0
$231$	−125235.	−0.154417
$232$	88506.6	0.107958
$233$	1.48218e6	1.78859	0.894296	−	0.447475i	$-0.147677\pi$
$233$	1.48218e6	1.78859	0.894296	+	0.447475i	$0.147677\pi$
$234$	656994.	0.784372
$235$	0	0
$236$	−1.86268e6	−2.17700
$237$	−1.82961e6	−2.11587
$238$	−66166.4	−0.0757174
$239$	−698853.	−0.791391	−0.395695	−	0.918382i	$-0.629496\pi$
$239$	−698853.	−0.791391	−0.395695	+	0.918382i	$0.629496\pi$
$240$	0	0
$241$	−367154.	−0.407198	−0.203599	−	0.979054i	$-0.565264\pi$
$241$	−367154.	−0.407198	−0.203599	+	0.979054i	$0.565264\pi$
$242$	1.65763e6	1.81949
$243$	−486752.	−0.528800
$244$	1.85188e6	1.99131
$245$	0	0
$246$	3.49277e6	3.67987
$247$	−94648.5	−0.0987123
$248$	−497206.	−0.513342
$249$	470738.	0.481151
$250$	0	0
$251$	469145.	0.470027	0.235013	−	0.971992i	$-0.424487\pi$
$251$	469145.	0.470027	0.235013	+	0.971992i	$0.424487\pi$
$252$	−159853.	−0.158570
$253$	−1.75833e6	−1.72703
$254$	−336071.	−0.326849
$255$	0	0
$256$	−174957.	−0.166852
$257$	1.28756e6	1.21600	0.607999	−	0.793938i	$-0.291972\pi$
$257$	1.28756e6	1.21600	0.607999	+	0.793938i	$0.291972\pi$
$258$	4.80124e6	4.49060
$259$	73785.3	0.0683472
$260$	0	0
$261$	356690.	0.324108
$262$	1.00133e6	0.901208
$263$	470260.	0.419227	0.209613	−	0.977784i	$-0.432779\pi$
$263$	470260.	0.419227	0.209613	+	0.977784i	$0.432779\pi$
$264$	1.71049e6	1.51046
$265$	0	0
$266$	39554.1	0.0342758
$267$	−2.89492e6	−2.48518
$268$	393062.	0.334291
$269$	−1.19043e6	−1.00305	−0.501525	−	0.865143i	$-0.667227\pi$
$269$	−1.19043e6	−1.00305	−0.501525	+	0.865143i	$0.667227\pi$
$270$	0	0
$271$	205159.	0.169694	0.0848472	−	0.996394i	$-0.472960\pi$
$271$	205159.	0.169694	0.0848472	+	0.996394i	$0.472960\pi$
$272$	−433185.	−0.355019
$273$	−35751.6	−0.0290328
$274$	2.61764e6	2.10636
$275$	0	0
$276$	−3.47218e6	−2.74366
$277$	−1.35799e6	−1.06340	−0.531702	−	0.846932i	$-0.678447\pi$
$277$	−1.35799e6	−1.06340	−0.531702	+	0.846932i	$0.678447\pi$
$278$	−1.29996e6	−1.00883
$279$	−2.00379e6	−1.54114
$280$	0	0
$281$	1.13136e6	0.854745	0.427372	−	0.904076i	$-0.359439\pi$
$281$	1.13136e6	0.854745	0.427372	+	0.904076i	$0.359439\pi$
$282$	4.58910e6	3.43641
$283$	896684.	0.665539	0.332769	−	0.943008i	$-0.392017\pi$
$283$	896684.	0.665539	0.332769	+	0.943008i	$0.392017\pi$
$284$	−534449.	−0.393197
$285$	0	0
$286$	875589.	0.632973
$287$	−122857.	−0.0880429
$288$	−3.36424e6	−2.39004
$289$	−542157.	−0.381839
$290$	0	0
$291$	−3.57512e6	−2.47490
$292$	2.63747e6	1.81022
$293$	732944.	0.498772	0.249386	−	0.968404i	$-0.419771\pi$
$293$	732944.	0.498772	0.249386	+	0.968404i	$0.419771\pi$
$294$	−3.84099e6	−2.59164
$295$	0	0
$296$	−1.00778e6	−0.668552
$297$	3.12236e6	2.05396
$298$	−2.58338e6	−1.68519
$299$	−501963.	−0.324708
$300$	0	0
$301$	−168882.	−0.107440
$302$	−3.56033e6	−2.24633
$303$	−1.71577e6	−1.07362
$304$	258957.	0.160710
$305$	0	0
$306$	3.64207e6	2.22354
$307$	1.12669e6	0.682275	0.341138	−	0.940013i	$-0.389188\pi$
$307$	1.12669e6	0.682275	0.341138	+	0.940013i	$0.389188\pi$
$308$	−213040.	−0.127963
$309$	2.66787e6	1.58953
$310$	0	0
$311$	−1.97792e6	−1.15960	−0.579799	−	0.814760i	$-0.696869\pi$
$311$	−1.97792e6	−1.15960	−0.579799	+	0.814760i	$0.696869\pi$
$312$	488304.	0.283990
$313$	−2.48485e6	−1.43364	−0.716818	−	0.697261i	$-0.754402\pi$
$313$	−2.48485e6	−1.43364	−0.716818	+	0.697261i	$0.754402\pi$
$314$	908148.	0.519796
$315$	0	0
$316$	−3.11240e6	−1.75339
$317$	−953844.	−0.533125	−0.266563	−	0.963818i	$-0.585888\pi$
$317$	−953844.	−0.533125	−0.266563	+	0.963818i	$0.585888\pi$
$318$	−4.52007e6	−2.50656
$319$	475368.	0.261549
$320$	0	0
$321$	−4.70475e6	−2.54844
$322$	209773.	0.112748
$323$	−524687.	−0.279830
$324$	1.35224e6	0.715633
$325$	0	0
$326$	4.55401e6	2.37329
$327$	−1.07090e6	−0.553833
$328$	1.67801e6	0.861210
$329$	−161420.	−0.0822179
$330$	0	0
$331$	21507.4	0.0107899	0.00539496	−	0.999985i	$-0.498283\pi$
$331$	21507.4	0.0107899	0.00539496	+	0.999985i	$0.498283\pi$
$332$	800783.	0.398722
$333$	−4.06144e6	−2.00710
$334$	1.97014e6	0.966342
$335$	0	0
$336$	97815.8	0.0472673
$337$	3.16342e6	1.51734	0.758668	−	0.651477i	$-0.225850\pi$
$337$	3.16342e6	1.51734	0.758668	+	0.651477i	$0.225850\pi$
$338$	249960.	0.119009
$339$	−5.95462e6	−2.81420
$340$	0	0
$341$	−2.67049e6	−1.24367
$342$	−2.17722e6	−1.00655
$343$	270736.	0.124254
$344$	2.30663e6	1.05095
$345$	0	0
$346$	−2.85958e6	−1.28414
$347$	2.20243e6	0.981927	0.490963	−	0.871180i	$-0.336645\pi$
$347$	2.20243e6	0.981927	0.490963	+	0.871180i	$0.336645\pi$
$348$	938710.	0.415512
$349$	4.44911e6	1.95528	0.977642	−	0.210276i	$-0.0674361\pi$
$349$	4.44911e6	1.95528	0.977642	+	0.210276i	$0.0674361\pi$
$350$	0	0
$351$	891360.	0.386176
$352$	−4.48359e6	−1.92872
$353$	23142.7	0.00988501	0.00494251	−	0.999988i	$-0.498427\pi$
$353$	23142.7	0.00988501	0.00494251	+	0.999988i	$0.498427\pi$
$354$	−9.58298e6	−4.06436
$355$	0	0
$356$	−4.92461e6	−2.05943
$357$	−198190.	−0.0823021
$358$	2.73299e6	1.12702
$359$	852005.	0.348904	0.174452	−	0.984666i	$-0.444185\pi$
$359$	852005.	0.348904	0.174452	+	0.984666i	$0.444185\pi$
$360$	0	0
$361$	−2.16244e6	−0.873326
$362$	−298524.	−0.119731
$363$	4.96513e6	1.97772
$364$	−60817.8	−0.0240590
$365$	0	0
$366$	9.52741e6	3.71768
$367$	2.60560e6	1.00982	0.504908	−	0.863173i	$-0.331526\pi$
$367$	2.60560e6	1.00982	0.504908	+	0.863173i	$0.331526\pi$
$368$	1.37336e6	0.528647
$369$	6.76253e6	2.58549
$370$	0	0
$371$	158992.	0.0599707
$372$	−5.27342e6	−1.97576
$373$	2.33488e6	0.868947	0.434473	−	0.900685i	$-0.356935\pi$
$373$	2.33488e6	0.868947	0.434473	+	0.900685i	$0.356935\pi$
$374$	4.85385e6	1.79435
$375$	0	0
$376$	2.20471e6	0.804232
$377$	135706.	0.0491753
$378$	−372504.	−0.134092
$379$	−2.67829e6	−0.957767	−0.478883	−	0.877878i	$-0.658958\pi$
$379$	−2.67829e6	−0.957767	−0.478883	+	0.877878i	$0.658958\pi$
$380$	0	0
$381$	−1.00664e6	−0.355273
$382$	−2.03316e6	−0.712874
$383$	2.37423e6	0.827038	0.413519	−	0.910496i	$-0.364300\pi$
$383$	2.37423e6	0.827038	0.413519	+	0.910496i	$0.364300\pi$
$384$	−5.45915e6	−1.88928
$385$	0	0
$386$	4.32175e6	1.47636
$387$	9.29592e6	3.15511
$388$	−6.08171e6	−2.05091
$389$	1.87573e6	0.628487	0.314244	−	0.949342i	$-0.398249\pi$
$389$	1.87573e6	0.628487	0.314244	+	0.949342i	$0.398249\pi$
$390$	0	0
$391$	−2.78264e6	−0.920483
$392$	−1.84530e6	−0.606528
$393$	2.99931e6	0.979582
$394$	1.51264e6	0.490902
$395$	0	0
$396$	1.17266e7	3.75779
$397$	4.12523e6	1.31363	0.656814	−	0.754053i	$-0.271904\pi$
$397$	4.12523e6	1.31363	0.656814	+	0.754053i	$0.271904\pi$
$398$	3.34605e6	1.05883
$399$	118477.	0.0372566
$400$	0	0
$401$	4.32492e6	1.34313	0.671564	−	0.740947i	$-0.265623\pi$
$401$	4.32492e6	1.34313	0.671564	+	0.740947i	$0.265623\pi$
$402$	2.02220e6	0.624106
$403$	−762361.	−0.233829
$404$	−2.91873e6	−0.889693
$405$	0	0
$406$	−56712.5	−0.0170751
$407$	−5.41276e6	−1.61969
$408$	2.70693e6	0.805056
$409$	3.45008e6	1.01981	0.509907	−	0.860229i	$-0.329680\pi$
$409$	3.45008e6	1.01981	0.509907	+	0.860229i	$0.329680\pi$
$410$	0	0
$411$	7.84067e6	2.28954
$412$	4.53837e6	1.31722
$413$	337077.	0.0972421
$414$	−1.15467e7	−3.31100
$415$	0	0
$416$	−1.27996e6	−0.362629
$417$	−3.89381e6	−1.09657
$418$	−2.90162e6	−0.812269
$419$	−1.56769e6	−0.436238	−0.218119	−	0.975922i	$-0.569992\pi$
$419$	−1.56769e6	−0.436238	−0.218119	+	0.975922i	$0.569992\pi$
$420$	0	0
$421$	6.01650e6	1.65439	0.827196	−	0.561913i	$-0.189935\pi$
$421$	6.01650e6	1.65439	0.827196	+	0.561913i	$0.189935\pi$
$422$	6.90950e6	1.88871
$423$	8.88518e6	2.41443
$424$	−2.17155e6	−0.586616
$425$	0	0
$426$	−2.74959e6	−0.734081
$427$	−335122.	−0.0889475
$428$	−8.00335e6	−2.11185
$429$	2.62267e6	0.688020
$430$	0	0
$431$	−4.41465e6	−1.14473	−0.572365	−	0.819999i	$-0.693974\pi$
$431$	−4.41465e6	−1.14473	−0.572365	+	0.819999i	$0.693974\pi$
$432$	−2.43875e6	−0.628720
$433$	−264873.	−0.0678919	−0.0339460	−	0.999424i	$-0.510807\pi$
$433$	−264873.	−0.0678919	−0.0339460	+	0.999424i	$0.510807\pi$
$434$	318595.	0.0811923
$435$	0	0
$436$	−1.82173e6	−0.458953
$437$	1.66346e6	0.416685
$438$	1.35691e7	3.37960
$439$	2.43360e6	0.602682	0.301341	−	0.953516i	$-0.402566\pi$
$439$	2.43360e6	0.602682	0.301341	+	0.953516i	$0.402566\pi$
$440$	0	0
$441$	−7.43672e6	−1.82090
$442$	1.38566e6	0.337366
$443$	−6.67033e6	−1.61487	−0.807436	−	0.589955i	$-0.799145\pi$
$443$	−6.67033e6	−1.61487	−0.807436	+	0.589955i	$0.799145\pi$
$444$	−1.06886e7	−2.57314
$445$	0	0
$446$	−7.85984e6	−1.87101
$447$	−7.73806e6	−1.83174
$448$	415499.	0.0978080
$449$	1.86259e6	0.436014	0.218007	−	0.975947i	$-0.430044\pi$
$449$	1.86259e6	0.436014	0.218007	+	0.975947i	$0.430044\pi$
$450$	0	0
$451$	9.01255e6	2.08644
$452$	−1.01295e7	−2.33208
$453$	−1.06644e7	−2.44168
$454$	3.72835e6	0.848940
$455$	0	0
$456$	−1.61819e6	−0.364433
$457$	4.66486e6	1.04484	0.522418	−	0.852690i	$-0.325030\pi$
$457$	4.66486e6	1.04484	0.522418	+	0.852690i	$0.325030\pi$
$458$	−1.93922e6	−0.431979
$459$	4.94128e6	1.09473
$460$	0	0
$461$	5.59911e6	1.22706	0.613531	−	0.789670i	$-0.289748\pi$
$461$	5.59911e6	1.22706	0.613531	+	0.789670i	$0.289748\pi$
$462$	−1.09603e6	−0.238901
$463$	6.71396e6	1.45555	0.727773	−	0.685818i	$-0.240555\pi$
$463$	6.71396e6	1.45555	0.727773	+	0.685818i	$0.240555\pi$
$464$	−371291.	−0.0800606
$465$	0	0
$466$	1.29718e7	2.76716
$467$	−2.57048e6	−0.545409	−0.272704	−	0.962098i	$-0.587918\pi$
$467$	−2.57048e6	−0.545409	−0.272704	+	0.962098i	$0.587918\pi$
$468$	3.34766e6	0.706523
$469$	−71129.9	−0.0149321
$470$	0	0
$471$	2.72020e6	0.565000
$472$	−4.60388e6	−0.951194
$473$	1.23889e7	2.54612
$474$	−1.60124e7	−3.27349
$475$	0	0
$476$	−337145.	−0.0682024
$477$	−8.75153e6	−1.76112
$478$	−6.11622e6	−1.22437
$479$	1.24698e6	0.248325	0.124163	−	0.992262i	$-0.460376\pi$
$479$	1.24698e6	0.248325	0.124163	+	0.992262i	$0.460376\pi$
$480$	0	0
$481$	−1.54522e6	−0.304528
$482$	−3.21326e6	−0.629981
$483$	628338.	0.122553
$484$	8.44630e6	1.63890
$485$	0	0
$486$	−4.25995e6	−0.818115
$487$	8.90535e6	1.70149	0.850743	−	0.525582i	$-0.176152\pi$
$487$	8.90535e6	1.70149	0.850743	+	0.525582i	$0.176152\pi$
$488$	4.57718e6	0.870059
$489$	1.36407e7	2.57968
$490$	0	0
$491$	−4.97397e6	−0.931107	−0.465553	−	0.885020i	$-0.654145\pi$
$491$	−4.97397e6	−0.931107	−0.465553	+	0.885020i	$0.654145\pi$
$492$	1.77971e7	3.31464
$493$	752292.	0.139402
$494$	−828345.	−0.152719
$495$	0	0
$496$	2.08581e6	0.380689
$497$	96715.7	0.0175633
$498$	4.11981e6	0.744395
$499$	−3.44325e6	−0.619038	−0.309519	−	0.950893i	$-0.600168\pi$
$499$	−3.44325e6	−0.619038	−0.309519	+	0.950893i	$0.600168\pi$
$500$	0	0
$501$	5.90121e6	1.05038
$502$	4.10586e6	0.727185
$503$	3.02775e6	0.533580	0.266790	−	0.963755i	$-0.414037\pi$
$503$	3.02775e6	0.533580	0.266790	+	0.963755i	$0.414037\pi$
$504$	−395100.	−0.0692836
$505$	0	0
$506$	−1.53886e7	−2.67191
$507$	748712.	0.129358
$508$	−1.71242e6	−0.294409
$509$	4.45477e6	0.762134	0.381067	−	0.924547i	$-0.375557\pi$
$509$	4.45477e6	0.762134	0.381067	+	0.924547i	$0.375557\pi$
$510$	0	0
$511$	−477286.	−0.0808587
$512$	5.13279e6	0.865324
$513$	−2.95388e6	−0.495564
$514$	1.12684e7	1.88129
$515$	0	0
$516$	2.44643e7	4.04491
$517$	1.18415e7	1.94840
$518$	645754.	0.105741
$519$	−8.56538e6	−1.39582
$520$	0	0
$521$	−3.88354e6	−0.626806	−0.313403	−	0.949620i	$-0.601469\pi$
$521$	−3.88354e6	−0.626806	−0.313403	+	0.949620i	$0.601469\pi$
$522$	3.12168e6	0.501432
$523$	−1.09959e7	−1.75783	−0.878913	−	0.476983i	$-0.841730\pi$
$523$	−1.09959e7	−1.75783	−0.878913	+	0.476983i	$0.841730\pi$
$524$	5.10220e6	0.811763
$525$	0	0
$526$	4.11563e6	0.648592
$527$	−4.22617e6	−0.662858
$528$	−7.17559e6	−1.12014
$529$	2.38570e6	0.370661
$530$	0	0
$531$	−1.85541e7	−2.85564
$532$	201545.	0.0308739
$533$	2.57287e6	0.392284
$534$	−2.53357e7	−3.84486
$535$	0	0
$536$	971510.	0.146061
$537$	8.18620e6	1.22503
$538$	−1.04184e7	−1.55183
$539$	−9.91107e6	−1.46943
$540$	0	0
$541$	2.61308e6	0.383848	0.191924	−	0.981410i	$-0.438527\pi$
$541$	2.61308e6	0.383848	0.191924	+	0.981410i	$0.438527\pi$
$542$	1.79551e6	0.262537
$543$	−894176.	−0.130144
$544$	−7.09549e6	−1.02798
$545$	0	0
$546$	−312891.	−0.0449170
$547$	−3.49357e6	−0.499230	−0.249615	−	0.968345i	$-0.580304\pi$
$547$	−3.49357e6	−0.499230	−0.249615	+	0.968345i	$0.580304\pi$
$548$	1.33379e7	1.89731
$549$	1.84465e7	2.61206
$550$	0	0
$551$	−449718.	−0.0631047
$552$	−8.58199e6	−1.19878
$553$	563230.	0.0783200
$554$	−1.18849e7	−1.64521
$555$	0	0
$556$	−6.62385e6	−0.908706
$557$	7.52693e6	1.02797	0.513984	−	0.857800i	$-0.328169\pi$
$557$	7.52693e6	1.02797	0.513984	+	0.857800i	$0.328169\pi$
$558$	−1.75367e7	−2.38431
$559$	3.53673e6	0.478710
$560$	0	0
$561$	1.45389e7	1.95040
$562$	9.90147e6	1.32239
$563$	8.11431e6	1.07890	0.539450	−	0.842018i	$-0.318632\pi$
$563$	8.11431e6	1.07890	0.539450	+	0.842018i	$0.318632\pi$
$564$	2.33834e7	3.09534
$565$	0	0
$566$	7.84760e6	1.02966
$567$	−244705.	−0.0319658
$568$	−1.32097e6	−0.171799
$569$	3.42992e6	0.444123	0.222061	−	0.975033i	$-0.428721\pi$
$569$	3.42992e6	0.444123	0.222061	+	0.975033i	$0.428721\pi$
$570$	0	0
$571$	1.17647e7	1.51005	0.755026	−	0.655695i	$-0.227624\pi$
$571$	1.17647e7	1.51005	0.755026	+	0.655695i	$0.227624\pi$
$572$	4.46149e6	0.570151
$573$	−6.08997e6	−0.774869
$574$	−1.07522e6	−0.136212
$575$	0	0
$576$	−2.28707e7	−2.87226
$577$	−8.73061e6	−1.09170	−0.545852	−	0.837882i	$-0.683794\pi$
$577$	−8.73061e6	−1.09170	−0.545852	+	0.837882i	$0.683794\pi$
$578$	−4.74485e6	−0.590749
$579$	1.29450e7	1.60475
$580$	0	0
$581$	−144912.	−0.0178101
$582$	−3.12887e7	−3.82895
$583$	−1.16633e7	−1.42119
$584$	6.51889e6	0.790936
$585$	0	0
$586$	6.41458e6	0.771657
$587$	−6.33682e6	−0.759060	−0.379530	−	0.925179i	$-0.623914\pi$
$587$	−6.33682e6	−0.759060	−0.379530	+	0.925179i	$0.623914\pi$
$588$	−1.95714e7	−2.33442
$589$	2.52639e6	0.300063
$590$	0	0
$591$	4.53085e6	0.533593
$592$	4.22769e6	0.495791
$593$	−1.23132e6	−0.143792	−0.0718960	−	0.997412i	$-0.522905\pi$
$593$	−1.23132e6	−0.143792	−0.0718960	+	0.997412i	$0.522905\pi$
$594$	2.73263e7	3.17771
$595$	0	0
$596$	−1.31634e7	−1.51793
$597$	1.00225e7	1.15091
$598$	−4.39308e6	−0.502361
$599$	−6.75482e6	−0.769213	−0.384607	−	0.923081i	$-0.625663\pi$
$599$	−6.75482e6	−0.769213	−0.384607	+	0.923081i	$0.625663\pi$
$600$	0	0
$601$	−20587.5	−0.00232497	−0.00116248	−	0.999999i	$-0.500370\pi$
$601$	−20587.5	−0.00232497	−0.00116248	+	0.999999i	$0.500370\pi$
$602$	−1.47802e6	−0.166222
$603$	3.91527e6	0.438499
$604$	−1.81414e7	−2.02338
$605$	0	0
$606$	−1.50160e7	−1.66102
$607$	−6.97910e6	−0.768826	−0.384413	−	0.923161i	$-0.625596\pi$
$607$	−6.97910e6	−0.768826	−0.384413	+	0.923161i	$0.625596\pi$
$608$	4.24167e6	0.465348
$609$	−169872.	−0.0185600
$610$	0	0
$611$	3.38046e6	0.366330
$612$	1.85578e7	2.00285
$613$	3.83537e6	0.412246	0.206123	−	0.978526i	$-0.433915\pi$
$613$	3.83537e6	0.412246	0.206123	+	0.978526i	$0.433915\pi$
$614$	9.86059e6	1.05556
$615$	0	0
$616$	−526558.	−0.0559106
$617$	−1.16850e7	−1.23571	−0.617853	−	0.786293i	$-0.711997\pi$
$617$	−1.16850e7	−1.23571	−0.617853	+	0.786293i	$0.711997\pi$
$618$	2.33487e7	2.45918
$619$	−8.33735e6	−0.874583	−0.437292	−	0.899320i	$-0.644062\pi$
$619$	−8.33735e6	−0.874583	−0.437292	+	0.899320i	$0.644062\pi$
$620$	0	0
$621$	−1.56658e7	−1.63013
$622$	−1.73103e7	−1.79403
$623$	891174.	0.0919903
$624$	−2.04846e6	−0.210604
$625$	0	0
$626$	−2.17469e7	−2.21800
$627$	−8.69129e6	−0.882908
$628$	4.62739e6	0.468206
$629$	−8.56595e6	−0.863274
$630$	0	0
$631$	−1.66983e7	−1.66955	−0.834775	−	0.550591i	$-0.814402\pi$
$631$	−1.66983e7	−1.66955	−0.834775	+	0.550591i	$0.814402\pi$
$632$	−7.69273e6	−0.766104
$633$	2.06962e7	2.05296
$634$	−8.34785e6	−0.824805
$635$	0	0
$636$	−2.30316e7	−2.25778
$637$	−2.82938e6	−0.276275
$638$	4.16033e6	0.404646
$639$	−5.32362e6	−0.515768
$640$	0	0
$641$	−1.69010e7	−1.62468	−0.812340	−	0.583184i	$-0.801807\pi$
$641$	−1.69010e7	−1.62468	−0.812340	+	0.583184i	$0.801807\pi$
$642$	−4.11750e7	−3.94272
$643$	1.01899e7	0.971947	0.485974	−	0.873974i	$-0.338465\pi$
$643$	1.01899e7	0.971947	0.485974	+	0.873974i	$0.338465\pi$
$644$	1.06888e6	0.101558
$645$	0	0
$646$	−4.59195e6	−0.432928
$647$	−6.44562e6	−0.605347	−0.302673	−	0.953094i	$-0.597879\pi$
$647$	−6.44562e6	−0.605347	−0.302673	+	0.953094i	$0.597879\pi$
$648$	3.34224e6	0.312680
$649$	−2.47274e7	−2.30445
$650$	0	0
$651$	954295.	0.0882531
$652$	2.32046e7	2.13774
$653$	−1.82109e7	−1.67128	−0.835640	−	0.549277i	$-0.814903\pi$
$653$	−1.82109e7	−1.67128	−0.835640	+	0.549277i	$0.814903\pi$
$654$	−9.37229e6	−0.856844
$655$	0	0
$656$	−7.03934e6	−0.638664
$657$	2.62717e7	2.37452
$658$	−1.41271e6	−0.127201
$659$	4.70706e6	0.422217	0.211109	−	0.977463i	$-0.432293\pi$
$659$	4.70706e6	0.422217	0.211109	+	0.977463i	$0.432293\pi$
$660$	0	0
$661$	−5.15844e6	−0.459213	−0.229607	−	0.973284i	$-0.573744\pi$
$661$	−5.15844e6	−0.459213	−0.229607	+	0.973284i	$0.573744\pi$
$662$	188229.	0.0166932
$663$	4.15050e6	0.366705
$664$	1.97925e6	0.174213
$665$	0	0
$666$	−3.55449e7	−3.10522
$667$	−2.38505e6	−0.207579
$668$	1.00387e7	0.870433
$669$	−2.35428e7	−2.03372
$670$	0	0
$671$	2.45840e7	2.10788
$672$	1.60221e6	0.136866
$673$	−8.41092e6	−0.715823	−0.357911	−	0.933756i	$-0.616511\pi$
$673$	−8.41092e6	−0.715823	−0.357911	+	0.933756i	$0.616511\pi$
$674$	2.76856e7	2.34749
$675$	0	0
$676$	1.27365e6	0.107197
$677$	1.05108e7	0.881378	0.440689	−	0.897660i	$-0.354734\pi$
$677$	1.05108e7	0.881378	0.440689	+	0.897660i	$0.354734\pi$
$678$	−5.21136e7	−4.35389
$679$	1.10057e6	0.0916098
$680$	0	0
$681$	1.11676e7	0.922768
$682$	−2.33716e7	−1.92410
$683$	−1.27854e7	−1.04873	−0.524364	−	0.851494i	$-0.675697\pi$
$683$	−1.27854e7	−1.04873	−0.524364	+	0.851494i	$0.675697\pi$
$684$	−1.10938e7	−0.906652
$685$	0	0
$686$	2.36942e6	0.192235
$687$	−5.80858e6	−0.469546
$688$	−9.67644e6	−0.779371
$689$	−3.32961e6	−0.267205
$690$	0	0
$691$	1.46556e7	1.16764	0.583818	−	0.811884i	$-0.301558\pi$
$691$	1.46556e7	1.16764	0.583818	+	0.811884i	$0.301558\pi$
$692$	−1.45708e7	−1.15669
$693$	−2.12208e6	−0.167853
$694$	1.92753e7	1.51915
$695$	0	0
$696$	2.32015e6	0.181549
$697$	1.42628e7	1.11205
$698$	3.89378e7	3.02505
$699$	3.88546e7	3.00780
$700$	0	0
$701$	1.62766e7	1.25103	0.625517	−	0.780210i	$-0.284888\pi$
$701$	1.62766e7	1.25103	0.625517	+	0.780210i	$0.284888\pi$
$702$	7.80101e6	0.597459
$703$	5.12070e6	0.390788
$704$	−3.04802e7	−2.31786
$705$	0	0
$706$	202540.	0.0152932
$707$	528183.	0.0397407
$708$	−4.88292e7	−3.66098
$709$	−1.32627e7	−0.990873	−0.495436	−	0.868644i	$-0.664992\pi$
$709$	−1.32627e7	−0.990873	−0.495436	+	0.868644i	$0.664992\pi$
$710$	0	0
$711$	−3.10024e7	−2.29997
$712$	−1.21719e7	−0.899823
$713$	1.33986e7	0.987040
$714$	−1.73452e6	−0.127331
$715$	0	0
$716$	1.39257e7	1.01516
$717$	−1.83201e7	−1.33085
$718$	7.45658e6	0.539794
$719$	1.58913e7	1.14640	0.573201	−	0.819415i	$-0.305701\pi$
$719$	1.58913e7	1.14640	0.573201	+	0.819415i	$0.305701\pi$
$720$	0	0
$721$	−821280.	−0.0588373
$722$	−1.89253e7	−1.35114
$723$	−9.62474e6	−0.684768
$724$	−1.52110e6	−0.107848
$725$	0	0
$726$	4.34539e7	3.05976
$727$	−1.21979e7	−0.855950	−0.427975	−	0.903791i	$-0.640773\pi$
$727$	−1.21979e7	−0.855950	−0.427975	+	0.903791i	$0.640773\pi$
$728$	−150320.	−0.0105121
$729$	−2.01285e7	−1.40279
$730$	0	0
$731$	1.96060e7	1.35705
$732$	4.85461e7	3.34870
$733$	3.40342e6	0.233968	0.116984	−	0.993134i	$-0.462677\pi$
$733$	3.40342e6	0.233968	0.116984	+	0.993134i	$0.462677\pi$
$734$	2.28037e7	1.56230
$735$	0	0
$736$	2.24954e7	1.53073
$737$	5.21796e6	0.353861
$738$	5.91843e7	4.00005
$739$	−8.97838e6	−0.604765	−0.302383	−	0.953187i	$-0.597782\pi$
$739$	−8.97838e6	−0.604765	−0.302383	+	0.953187i	$0.597782\pi$
$740$	0	0
$741$	−2.48116e6	−0.166000
$742$	1.39146e6	0.0927816
$743$	−1.67949e7	−1.11611	−0.558054	−	0.829804i	$-0.688452\pi$
$743$	−1.67949e7	−1.11611	−0.558054	+	0.829804i	$0.688452\pi$
$744$	−1.30340e7	−0.863267
$745$	0	0
$746$	2.04344e7	1.34436
$747$	7.97656e6	0.523015
$748$	2.47324e7	1.61626
$749$	1.44831e6	0.0943318
$750$	0	0
$751$	−1.55176e7	−1.00398	−0.501989	−	0.864874i	$-0.667398\pi$
$751$	−1.55176e7	−1.00398	−0.501989	+	0.864874i	$0.667398\pi$
$752$	−9.24888e6	−0.596410
$753$	1.22984e7	0.790425
$754$	1.18768e6	0.0760798
$755$	0	0
$756$	−1.89806e6	−0.120783
$757$	1.23231e7	0.781592	0.390796	−	0.920477i	$-0.372200\pi$
$757$	1.23231e7	0.781592	0.390796	+	0.920477i	$0.372200\pi$
$758$	−2.34399e7	−1.48178
$759$	−4.60938e7	−2.90427
$760$	0	0
$761$	1.01874e7	0.637677	0.318838	−	0.947809i	$-0.396707\pi$
$761$	1.01874e7	0.637677	0.318838	+	0.947809i	$0.396707\pi$
$762$	−8.80992e6	−0.549648
$763$	329666.	0.0205004
$764$	−1.03598e7	−0.642121
$765$	0	0
$766$	2.07788e7	1.27952
$767$	−7.05909e6	−0.433272
$768$	−4.58641e6	−0.280588
$769$	5.61745e6	0.342550	0.171275	−	0.985223i	$-0.445211\pi$
$769$	5.61745e6	0.342550	0.171275	+	0.985223i	$0.445211\pi$
$770$	0	0
$771$	3.37526e7	2.04490
$772$	2.20211e7	1.32983
$773$	−1.86328e7	−1.12158	−0.560790	−	0.827958i	$-0.689502\pi$
$773$	−1.86328e7	−1.12158	−0.560790	+	0.827958i	$0.689502\pi$
$774$	8.13561e7	4.88132
$775$	0	0
$776$	−1.50318e7	−0.896101
$777$	1.93424e6	0.114937
$778$	1.64160e7	0.972342
$779$	−8.52626e6	−0.503402
$780$	0	0
$781$	−7.09489e6	−0.416215
$782$	−2.43531e7	−1.42409
$783$	4.23526e6	0.246874
$784$	7.74113e6	0.449795
$785$	0	0
$786$	2.62494e7	1.51552
$787$	2.13328e7	1.22775	0.613877	−	0.789402i	$-0.289609\pi$
$787$	2.13328e7	1.22775	0.613877	+	0.789402i	$0.289609\pi$
$788$	7.70752e6	0.442180
$789$	1.23276e7	0.704997
$790$	0	0
$791$	1.83307e6	0.104169
$792$	2.89838e7	1.64189
$793$	7.01815e6	0.396314
$794$	3.61032e7	2.03233
$795$	0	0
$796$	1.70495e7	0.953738
$797$	−7.53097e6	−0.419957	−0.209979	−	0.977706i	$-0.567339\pi$
$797$	−7.53097e6	−0.419957	−0.209979	+	0.977706i	$0.567339\pi$
$798$	1.03689e6	0.0576402
$799$	1.87397e7	1.03847
$800$	0	0
$801$	−4.90538e7	−2.70141
$802$	3.78509e7	2.07797
$803$	3.50129e7	1.91619
$804$	1.03039e7	0.562163
$805$	0	0
$806$	−6.67203e6	−0.361760
$807$	−3.12064e7	−1.68679
$808$	−7.21405e6	−0.388732
$809$	2.98814e7	1.60520	0.802601	−	0.596516i	$-0.203449\pi$
$809$	2.98814e7	1.60520	0.802601	+	0.596516i	$0.203449\pi$
$810$	0	0
$811$	−7.90498e6	−0.422035	−0.211018	−	0.977482i	$-0.567678\pi$
$811$	−7.90498e6	−0.422035	−0.211018	+	0.977482i	$0.567678\pi$
$812$	−288973.	−0.0153804
$813$	5.37814e6	0.285368
$814$	−4.73714e7	−2.50585
$815$	0	0
$816$	−1.13557e7	−0.597020
$817$	−1.17204e7	−0.614309
$818$	3.01944e7	1.57777
$819$	−605803.	−0.0315589
$820$	0	0
$821$	−1.45139e7	−0.751494	−0.375747	−	0.926722i	$-0.622614\pi$
$821$	−1.45139e7	−0.751494	−0.375747	+	0.926722i	$0.622614\pi$
$822$	6.86200e7	3.54218
$823$	6.15126e6	0.316566	0.158283	−	0.987394i	$-0.449404\pi$
$823$	6.15126e6	0.316566	0.158283	+	0.987394i	$0.449404\pi$
$824$	1.12172e7	0.575530
$825$	0	0
$826$	2.95003e6	0.150445
$827$	3.97991e6	0.202353	0.101176	−	0.994869i	$-0.467739\pi$
$827$	3.97991e6	0.202353	0.101176	+	0.994869i	$0.467739\pi$
$828$	−5.88354e7	−2.98238
$829$	4.29025e6	0.216819	0.108409	−	0.994106i	$-0.465424\pi$
$829$	4.29025e6	0.216819	0.108409	+	0.994106i	$0.465424\pi$
$830$	0	0
$831$	−3.55991e7	−1.78828
$832$	−8.70139e6	−0.435793
$833$	−1.56847e7	−0.783185
$834$	−3.40779e7	−1.69651
$835$	0	0
$836$	−1.47850e7	−0.731652
$837$	−2.37925e7	−1.17389
$838$	−1.37201e7	−0.674911
$839$	2.83937e7	1.39257	0.696284	−	0.717766i	$-0.254835\pi$
$839$	2.83937e7	1.39257	0.696284	+	0.717766i	$0.254835\pi$
$840$	0	0
$841$	−1.98663e7	−0.968563
$842$	5.26552e7	2.55953
$843$	2.96581e7	1.43739
$844$	3.52068e7	1.70126
$845$	0	0
$846$	7.77613e7	3.73541
$847$	−1.52847e6	−0.0732063
$848$	9.10976e6	0.435028
$849$	2.35061e7	1.11921
$850$	0	0
$851$	2.71573e7	1.28547
$852$	−1.40103e7	−0.661224
$853$	1.61127e7	0.758222	0.379111	−	0.925351i	$-0.376230\pi$
$853$	1.61127e7	0.758222	0.379111	+	0.925351i	$0.376230\pi$
$854$	−2.93292e6	−0.137612
$855$	0	0
$856$	−1.97814e7	−0.922726
$857$	−2.56599e7	−1.19344	−0.596722	−	0.802448i	$-0.703531\pi$
$857$	−2.56599e7	−1.19344	−0.596722	+	0.802448i	$0.703531\pi$
$858$	2.29531e7	1.06445
$859$	−1.56669e7	−0.724439	−0.362219	−	0.932093i	$-0.617981\pi$
$859$	−1.56669e7	−0.724439	−0.362219	+	0.932093i	$0.617981\pi$
$860$	0	0
$861$	−3.22062e6	−0.148058
$862$	−3.86362e7	−1.77103
$863$	2.50939e7	1.14694	0.573471	−	0.819226i	$-0.305597\pi$
$863$	2.50939e7	1.14694	0.573471	+	0.819226i	$0.305597\pi$
$864$	−3.99463e7	−1.82050
$865$	0	0
$866$	−2.31812e6	−0.105037
$867$	−1.42124e7	−0.642124
$868$	1.62337e6	0.0731339
$869$	−4.13176e7	−1.85603
$870$	0	0
$871$	1.48961e6	0.0665313
$872$	−4.50266e6	−0.200530
$873$	−6.05796e7	−2.69024
$874$	1.45582e7	0.644660
$875$	0	0
$876$	6.91400e7	3.04417
$877$	−9.38359e6	−0.411974	−0.205987	−	0.978555i	$-0.566040\pi$
$877$	−9.38359e6	−0.411974	−0.205987	+	0.978555i	$0.566040\pi$
$878$	2.12984e7	0.932419
$879$	1.92137e7	0.838764
$880$	0	0
$881$	2.07853e7	0.902227	0.451114	−	0.892466i	$-0.351027\pi$
$881$	2.07853e7	0.902227	0.451114	+	0.892466i	$0.351027\pi$
$882$	−6.50847e7	−2.81713
$883$	2.64517e6	0.114170	0.0570850	−	0.998369i	$-0.481819\pi$
$883$	2.64517e6	0.114170	0.0570850	+	0.998369i	$0.481819\pi$
$884$	7.06052e6	0.303883
$885$	0	0
$886$	−5.83774e7	−2.49839
$887$	1.19650e7	0.510628	0.255314	−	0.966858i	$-0.417821\pi$
$887$	1.19650e7	0.510628	0.255314	+	0.966858i	$0.417821\pi$
$888$	−2.64184e7	−1.12428
$889$	309885.	0.0131506
$890$	0	0
$891$	1.79512e7	0.757527
$892$	−4.00491e7	−1.68531
$893$	−1.12025e7	−0.470097
$894$	−6.77220e7	−2.83391
$895$	0	0
$896$	1.68055e6	0.0699329
$897$	−1.31587e7	−0.546049
$898$	1.63010e7	0.674563
$899$	−3.62233e6	−0.149482
$900$	0	0
$901$	−1.84578e7	−0.757474
$902$	7.88761e7	3.22797
$903$	−4.42714e6	−0.180678
$904$	−2.50366e7	−1.01895
$905$	0	0
$906$	−9.33323e7	−3.77756
$907$	−9.86913e6	−0.398346	−0.199173	−	0.979964i	$-0.563826\pi$
$907$	−9.86913e6	−0.398346	−0.199173	+	0.979964i	$0.563826\pi$
$908$	1.89975e7	0.764683
$909$	−2.90733e7	−1.16704
$910$	0	0
$911$	1.49818e7	0.598092	0.299046	−	0.954239i	$-0.403332\pi$
$911$	1.49818e7	0.598092	0.299046	+	0.954239i	$0.403332\pi$
$912$	6.78842e6	0.270260
$913$	1.06305e7	0.422063
$914$	4.08259e7	1.61648
$915$	0	0
$916$	−9.88111e6	−0.389105
$917$	−923311.	−0.0362598
$918$	4.32451e7	1.69368
$919$	−3.28227e7	−1.28199	−0.640997	−	0.767543i	$-0.721479\pi$
$919$	−3.28227e7	−1.28199	−0.640997	+	0.767543i	$0.721479\pi$
$920$	0	0
$921$	2.95357e7	1.14735
$922$	4.90023e7	1.89841
$923$	−2.02543e6	−0.0782550
$924$	−5.58472e6	−0.215190
$925$	0	0
$926$	5.87592e7	2.25190
$927$	4.52065e7	1.72783
$928$	−6.08167e6	−0.231821
$929$	2.17184e7	0.825635	0.412818	−	0.910814i	$-0.364545\pi$
$929$	2.17184e7	0.825635	0.412818	+	0.910814i	$0.364545\pi$
$930$	0	0
$931$	9.37629e6	0.354533
$932$	6.60964e7	2.49252
$933$	−5.18501e7	−1.95005
$934$	−2.24963e7	−0.843810
$935$	0	0
$936$	8.27421e6	0.308700
$937$	2.19377e7	0.816286	0.408143	−	0.912918i	$-0.366176\pi$
$937$	2.19377e7	0.816286	0.408143	+	0.912918i	$0.366176\pi$
$938$	−622515.	−0.0231016
$939$	−6.51390e7	−2.41089
$940$	0	0
$941$	−3.70200e7	−1.36289	−0.681447	−	0.731867i	$-0.738649\pi$
$941$	−3.70200e7	−1.36289	−0.681447	+	0.731867i	$0.738649\pi$
$942$	2.38066e7	0.874119
$943$	−4.52185e7	−1.65591
$944$	1.93136e7	0.705395
$945$	0	0
$946$	1.08425e8	3.93914
$947$	2.28075e7	0.826422	0.413211	−	0.910635i	$-0.364407\pi$
$947$	2.28075e7	0.826422	0.413211	+	0.910635i	$0.364407\pi$
$948$	−8.15899e7	−2.94860
$949$	9.99535e6	0.360274
$950$	0	0
$951$	−2.50045e7	−0.896535
$952$	−833302.	−0.0297996
$953$	−1.14558e7	−0.408595	−0.204298	−	0.978909i	$-0.565491\pi$
$953$	−1.14558e7	−0.408595	−0.204298	+	0.978909i	$0.565491\pi$
$954$	−7.65917e7	−2.72465
$955$	0	0
$956$	−3.11647e7	−1.10285
$957$	1.24615e7	0.439836
$958$	1.09133e7	0.384188
$959$	−2.41368e6	−0.0847487
$960$	0	0
$961$	−8.27991e6	−0.289213
$962$	−1.35234e7	−0.471139
$963$	−7.97210e7	−2.77017
$964$	−1.63729e7	−0.567456
$965$	0	0
$966$	5.49909e6	0.189604
$967$	−2.44915e7	−0.842265	−0.421132	−	0.906999i	$-0.638367\pi$
$967$	−2.44915e7	−0.842265	−0.421132	+	0.906999i	$0.638367\pi$
$968$	2.08762e7	0.716084
$969$	−1.37544e7	−0.470578
$970$	0	0
$971$	−1.10404e6	−0.0375783	−0.0187891	−	0.999823i	$-0.505981\pi$
$971$	−1.10404e6	−0.0375783	−0.0187891	+	0.999823i	$0.505981\pi$
$972$	−2.17062e7	−0.736917
$973$	1.19867e6	0.0405900
$974$	7.79378e7	2.63239
$975$	0	0
$976$	−1.92016e7	−0.645226
$977$	2.44544e7	0.819636	0.409818	−	0.912167i	$-0.365592\pi$
$977$	2.44544e7	0.819636	0.409818	+	0.912167i	$0.365592\pi$
$978$	1.19381e8	3.99106
$979$	−6.53749e7	−2.17999
$980$	0	0
$981$	−1.81462e7	−0.602022
$982$	−4.35312e7	−1.44053
$983$	−7.78582e6	−0.256993	−0.128496	−	0.991710i	$-0.541015\pi$
$983$	−7.78582e6	−0.256993	−0.128496	+	0.991710i	$0.541015\pi$
$984$	4.39881e7	1.44826
$985$	0	0
$986$	6.58391e6	0.215671
$987$	−4.23153e6	−0.138263
$988$	−4.22076e6	−0.137562
$989$	−6.21584e7	−2.02073
$990$	0	0
$991$	7.12590e6	0.230492	0.115246	−	0.993337i	$-0.463234\pi$
$991$	7.12590e6	0.230492	0.115246	+	0.993337i	$0.463234\pi$
$992$	3.41652e7	1.10231
$993$	563805.	0.0181450
$994$	846436.	0.0271724
$995$	0	0
$996$	2.09921e7	0.670514
$997$	−4.38741e7	−1.39788	−0.698940	−	0.715180i	$-0.746345\pi$
$997$	−4.38741e7	−1.39788	−0.698940	+	0.715180i	$0.746345\pi$
$998$	−3.01346e7	−0.957722
$999$	−4.82247e7	−1.52882

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.e.1.3		4
5.2	odd	4		325.6.b.e.274.7		8
5.3	odd	4		325.6.b.e.274.2		8
5.4	even	2		65.6.a.c.1.2	✓	4
15.14	odd	2		585.6.a.g.1.3		4
20.19	odd	2		1040.6.a.o.1.4		4
65.64	even	2		845.6.a.f.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.c.1.2	✓	4	5.4	even	2
325.6.a.e.1.3		4	1.1	even	1	trivial
325.6.b.e.274.2		8	5.3	odd	4
325.6.b.e.274.7		8	5.2	odd	4
585.6.a.g.1.3		4	15.14	odd	2
845.6.a.f.1.3		4	65.64	even	2
1040.6.a.o.1.4		4	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)

\(f(q)\)	\(=\)	\(q+8.75180 q^{2} +26.2145 q^{3} +44.5940 q^{4} +229.424 q^{6} -8.06989 q^{7} +110.220 q^{8} +444.199 q^{9} +591.993 q^{11} +1169.01 q^{12} +169.000 q^{13} -70.6260 q^{14} -462.381 q^{16} +936.856 q^{17} +3887.54 q^{18} -560.050 q^{19} -211.548 q^{21} +5181.00 q^{22} -2970.19 q^{23} +2889.37 q^{24} +1479.05 q^{26} +5274.32 q^{27} -359.869 q^{28} +802.996 q^{29} -4511.01 q^{31} -7573.72 q^{32} +15518.8 q^{33} +8199.18 q^{34} +19808.6 q^{36} -9143.29 q^{37} -4901.45 q^{38} +4430.25 q^{39} +15224.1 q^{41} -1851.42 q^{42} +20927.4 q^{43} +26399.3 q^{44} -25994.5 q^{46} +20002.7 q^{47} -12121.1 q^{48} -16741.9 q^{49} +24559.2 q^{51} +7536.39 q^{52} -19701.8 q^{53} +46159.8 q^{54} -889.467 q^{56} -14681.4 q^{57} +7027.66 q^{58} -41769.8 q^{59} +41527.5 q^{61} -39479.5 q^{62} -3584.63 q^{63} -51487.5 q^{64} +135817. q^{66} +8814.24 q^{67} +41778.2 q^{68} -77862.0 q^{69} -11984.8 q^{71} +48959.8 q^{72} +59144.1 q^{73} -80020.3 q^{74} -24974.9 q^{76} -4777.31 q^{77} +38772.6 q^{78} -69794.0 q^{79} +30323.3 q^{81} +133238. q^{82} +17957.2 q^{83} -9433.77 q^{84} +183152. q^{86} +21050.1 q^{87} +65249.7 q^{88} -110432. q^{89} -1363.81 q^{91} -132453. q^{92} -118254. q^{93} +175060. q^{94} -198541. q^{96} -136379. q^{97} -146522. q^{98} +262962. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 9 q^{2} + 4 q^{3} + 93 q^{4} - 74 q^{6} + 136 q^{7} + 447 q^{8} + 644 q^{9} - 516 q^{11} + 710 q^{12} + 676 q^{13} + 1604 q^{14} - 719 q^{16} - 344 q^{17} + 6905 q^{18} - 5012 q^{19} - 4448 q^{21}+ \cdots + 290668 q^{99}+O(q^{100}) \) 4 * q + 9 * q^2 + 4 * q^3 + 93 * q^4 - 74 * q^6 + 136 * q^7 + 447 * q^8 + 644 * q^9 - 516 * q^11 + 710 * q^12 + 676 * q^13 + 1604 * q^14 - 719 * q^16 - 344 * q^17 + 6905 * q^18 - 5012 * q^19 - 4448 * q^21 + 9318 * q^22 - 708 * q^23 - 2094 * q^24 + 1521 * q^26 + 4264 * q^27 + 7268 * q^28 + 5416 * q^29 - 10124 * q^31 - 9497 * q^32 + 16256 * q^33 + 3022 * q^34 + 36205 * q^36 - 1448 * q^37 - 11922 * q^38 + 676 * q^39 + 21576 * q^41 - 33128 * q^42 + 25740 * q^43 + 24710 * q^44 + 28754 * q^46 + 51096 * q^47 - 43282 * q^48 - 40412 * q^49 + 52984 * q^51 + 15717 * q^52 - 39448 * q^53 + 60340 * q^54 + 30188 * q^56 + 56160 * q^57 - 1374 * q^58 + 37764 * q^59 + 95016 * q^61 - 83102 * q^62 + 43592 * q^63 - 142095 * q^64 + 198500 * q^66 + 50592 * q^67 + 47110 * q^68 - 177648 * q^69 - 89308 * q^71 + 99183 * q^72 + 108312 * q^73 - 16766 * q^74 - 3442 * q^76 - 6208 * q^77 - 12506 * q^78 - 25464 * q^79 - 106348 * q^81 + 206426 * q^82 - 20584 * q^83 - 213448 * q^84 + 363666 * q^86 + 128312 * q^87 - 20558 * q^88 - 148328 * q^89 + 22984 * q^91 + 173474 * q^92 + 10496 * q^93 + 63916 * q^94 + 85154 * q^96 + 77816 * q^97 + 83505 * q^98 + 290668 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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