LMFDB - Embedded newform 325.6.a.i.1.5

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4 x^{8} - 181 x^{7} + 688 x^{6} + 10455 x^{5} - 37904 x^{4} - 197375 x^{3} + 702868 x^{2} + \cdots - 366960 $ x^9 - 4x^8 - 181x^7 + 688x^6 + 10455x^5 - 37904x^4 - 197375x^3 + 702868x^2 + 262812x - 366960
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.603392$ 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)

$f(q)$	$=$	\(q+0.396608 q^{2} -11.4974 q^{3} -31.8427 q^{4} -4.55994 q^{6} +8.04695 q^{7} -25.3205 q^{8} -110.811 q^{9} +335.903 q^{11} +366.107 q^{12} -169.000 q^{13} +3.19148 q^{14} +1008.92 q^{16} +38.1592 q^{17} -43.9484 q^{18} +1178.86 q^{19} -92.5186 q^{21} +133.222 q^{22} +1679.35 q^{23} +291.119 q^{24} -67.0267 q^{26} +4067.89 q^{27} -256.237 q^{28} -6705.42 q^{29} +7858.95 q^{31} +1210.40 q^{32} -3861.99 q^{33} +15.1342 q^{34} +3528.51 q^{36} -1715.81 q^{37} +467.545 q^{38} +1943.05 q^{39} -10076.9 q^{41} -36.6936 q^{42} +10822.3 q^{43} -10696.0 q^{44} +666.042 q^{46} +12999.2 q^{47} -11600.0 q^{48} -16742.2 q^{49} -438.730 q^{51} +5381.42 q^{52} +2639.19 q^{53} +1613.36 q^{54} -203.753 q^{56} -13553.8 q^{57} -2659.42 q^{58} +4956.81 q^{59} -43787.4 q^{61} +3116.92 q^{62} -891.687 q^{63} -31805.5 q^{64} -1531.70 q^{66} -9170.90 q^{67} -1215.09 q^{68} -19308.0 q^{69} -13030.5 q^{71} +2805.78 q^{72} -41726.8 q^{73} -680.503 q^{74} -37538.1 q^{76} +2702.99 q^{77} +770.630 q^{78} -49809.7 q^{79} -19843.0 q^{81} -3996.58 q^{82} -86196.9 q^{83} +2946.04 q^{84} +4292.19 q^{86} +77094.6 q^{87} -8505.22 q^{88} -67287.2 q^{89} -1359.93 q^{91} -53474.9 q^{92} -90357.2 q^{93} +5155.58 q^{94} -13916.4 q^{96} +175537. q^{97} -6640.10 q^{98} -37221.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 11 q^{3} + 91 q^{4} - 83 q^{6} - 12 q^{7} + 639 q^{8} + 562 q^{9} - 1422 q^{11} - 1567 q^{12} - 1521 q^{13} - 342 q^{14} - 1061 q^{16} - 648 q^{17} - 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) $ 9 * q + 5 * q^2 + 11 * q^3 + 91 * q^4 - 83 * q^6 - 12 * q^7 + 639 * q^8 + 562 * q^9 - 1422 * q^11 - 1567 * q^12 - 1521 * q^13 - 342 * q^14 - 1061 * q^16 - 648 * q^17 - 418 * q^18 - 408 * q^19 - 3912 * q^21 - 4345 * q^22 - 1839 * q^23 - 8469 * q^24 - 845 * q^26 + 7649 * q^27 + 2836 * q^28 - 8737 * q^29 + 748 * q^31 + 423 * q^32 + 356 * q^33 - 17789 * q^34 + 512 * q^36 + 15486 * q^37 + 3425 * q^38 - 1859 * q^39 - 28676 * q^41 - 6876 * q^42 - 28665 * q^43 - 30599 * q^44 - 12056 * q^46 + 29452 * q^47 - 64759 * q^48 - 40907 * q^49 - 31006 * q^51 - 15379 * q^52 - 75977 * q^53 - 102761 * q^54 - 23002 * q^56 + 38038 * q^57 - 142384 * q^58 - 88142 * q^59 + 28165 * q^61 + 137308 * q^62 - 41492 * q^63 - 100845 * q^64 + 42577 * q^66 + 94754 * q^67 - 89267 * q^68 - 181747 * q^69 - 70562 * q^71 + 263778 * q^72 - 60602 * q^73 - 135676 * q^74 + 46373 * q^76 + 140292 * q^77 + 14027 * q^78 - 164073 * q^79 - 69935 * q^81 + 72887 * q^82 + 22458 * q^83 - 345656 * q^84 - 294920 * q^86 + 87031 * q^87 - 430607 * q^88 - 252698 * q^89 + 2028 * q^91 + 237824 * q^92 - 56556 * q^93 - 501606 * q^94 - 319181 * q^96 - 137986 * q^97 - 378699 * q^98 - 757776 * 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 $

$2$	0.396608	0.0701110	0.0350555	−	0.999385i	$-0.488839\pi$
$2$	0.396608	0.0701110	0.0350555	+	0.999385i	$0.488839\pi$
$3$	−11.4974	−0.737556	−0.368778	−	0.929517i	$-0.620224\pi$
$3$	−11.4974	−0.737556	−0.368778	+	0.929517i	$0.620224\pi$
$4$	−31.8427	−0.995084
$5$	0	0
$5$	0	0
$6$	−4.55994	−0.0517108
$7$	8.04695	0.0620706	0.0310353	−	0.999518i	$-0.490120\pi$
$7$	8.04695	0.0620706	0.0310353	+	0.999518i	$0.490120\pi$
$8$	−25.3205	−0.139877
$9$	−110.811	−0.456011
$10$	0	0
$11$	335.903	0.837012	0.418506	−	0.908214i	$-0.362554\pi$
$11$	335.903	0.837012	0.418506	+	0.908214i	$0.362554\pi$
$12$	366.107	0.733931
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	3.19148	0.00435183
$15$	0	0
$16$	1008.92	0.985278
$17$	38.1592	0.0320241	0.0160121	−	0.999872i	$-0.494903\pi$
$17$	38.1592	0.0320241	0.0160121	+	0.999872i	$0.494903\pi$
$18$	−43.9484	−0.0319714
$19$	1178.86	0.749167	0.374584	−	0.927193i	$-0.377786\pi$
$19$	1178.86	0.749167	0.374584	+	0.927193i	$0.377786\pi$
$20$	0	0
$21$	−92.5186	−0.0457806
$22$	133.222	0.0586837
$23$	1679.35	0.661943	0.330972	−	0.943641i	$-0.392624\pi$
$23$	1679.35	0.661943	0.330972	+	0.943641i	$0.392624\pi$
$24$	291.119	0.103167
$25$	0	0
$26$	−67.0267	−0.0194453
$27$	4067.89	1.07389
$28$	−256.237	−0.0617655
$29$	−6705.42	−1.48058	−0.740288	−	0.672290i	$-0.765311\pi$
$29$	−6705.42	−1.48058	−0.740288	+	0.672290i	$0.765311\pi$
$30$	0	0
$31$	7858.95	1.46879	0.734396	−	0.678721i	$-0.237465\pi$
$31$	7858.95	1.46879	0.734396	+	0.678721i	$0.237465\pi$
$32$	1210.40	0.208956
$33$	−3861.99	−0.617343
$34$	15.1342	0.00224524
$35$	0	0
$36$	3528.51	0.453769
$37$	−1715.81	−0.206046	−0.103023	−	0.994679i	$-0.532852\pi$
$37$	−1715.81	−0.206046	−0.103023	+	0.994679i	$0.532852\pi$
$38$	467.545	0.0525249
$39$	1943.05	0.204561
$40$	0	0
$41$	−10076.9	−0.936198	−0.468099	−	0.883676i	$-0.655061\pi$
$41$	−10076.9	−0.936198	−0.468099	+	0.883676i	$0.655061\pi$
$42$	−36.6936	−0.00320972
$43$	10822.3	0.892579	0.446289	−	0.894889i	$-0.352745\pi$
$43$	10822.3	0.892579	0.446289	+	0.894889i	$0.352745\pi$
$44$	−10696.0	−0.832897
$45$	0	0
$46$	666.042	0.0464095
$47$	12999.2	0.858364	0.429182	−	0.903218i	$-0.358802\pi$
$47$	12999.2	0.858364	0.429182	+	0.903218i	$0.358802\pi$
$48$	−11600.0	−0.726697
$49$	−16742.2	−0.996147
$50$	0	0
$51$	−438.730	−0.0236196
$52$	5381.42	0.275987
$53$	2639.19	0.129057	0.0645285	−	0.997916i	$-0.479446\pi$
$53$	2639.19	0.129057	0.0645285	+	0.997916i	$0.479446\pi$
$54$	1613.36	0.0752915
$55$	0	0
$56$	−203.753	−0.00868227
$57$	−13553.8	−0.552553
$58$	−2659.42	−0.103805
$59$	4956.81	0.185384	0.0926919	−	0.995695i	$-0.470453\pi$
$59$	4956.81	0.185384	0.0926919	+	0.995695i	$0.470453\pi$
$60$	0	0
$61$	−43787.4	−1.50669	−0.753346	−	0.657625i	$-0.771561\pi$
$61$	−43787.4	−1.50669	−0.753346	+	0.657625i	$0.771561\pi$
$62$	3116.92	0.102979
$63$	−891.687	−0.0283049
$64$	−31805.5	−0.970627
$65$	0	0
$66$	−1531.70	−0.0432825
$67$	−9170.90	−0.249589	−0.124794	−	0.992183i	$-0.539827\pi$
$67$	−9170.90	−0.249589	−0.124794	+	0.992183i	$0.539827\pi$
$68$	−1215.09	−0.0318667
$69$	−19308.0	−0.488220
$70$	0	0
$71$	−13030.5	−0.306772	−0.153386	−	0.988166i	$-0.549018\pi$
$71$	−13030.5	−0.306772	−0.153386	+	0.988166i	$0.549018\pi$
$72$	2805.78	0.0637856
$73$	−41726.8	−0.916447	−0.458224	−	0.888837i	$-0.651514\pi$
$73$	−41726.8	−0.916447	−0.458224	+	0.888837i	$0.651514\pi$
$74$	−680.503	−0.0144461
$75$	0	0
$76$	−37538.1	−0.745485
$77$	2702.99	0.0519538
$78$	770.630	0.0143420
$79$	−49809.7	−0.897938	−0.448969	−	0.893547i	$-0.648209\pi$
$79$	−49809.7	−0.897938	−0.448969	+	0.893547i	$0.648209\pi$
$80$	0	0
$81$	−19843.0	−0.336043
$82$	−3996.58	−0.0656378
$83$	−86196.9	−1.37340	−0.686699	−	0.726942i	$-0.740941\pi$
$83$	−86196.9	−1.37340	−0.686699	+	0.726942i	$0.740941\pi$
$84$	2946.04	0.0455555
$85$	0	0
$86$	4292.19	0.0625796
$87$	77094.6	1.09201
$88$	−8505.22	−0.117079
$89$	−67287.2	−0.900446	−0.450223	−	0.892916i	$-0.648656\pi$
$89$	−67287.2	−0.900446	−0.450223	+	0.892916i	$0.648656\pi$
$90$	0	0
$91$	−1359.93	−0.0172153
$92$	−53474.9	−0.658689
$93$	−90357.2	−1.08332
$94$	5155.58	0.0601808
$95$	0	0
$96$	−13916.4	−0.154117
$97$	175537.	1.89426	0.947130	−	0.320850i	$-0.103969\pi$
$97$	175537.	1.89426	0.947130	+	0.320850i	$0.103969\pi$
$98$	−6640.10	−0.0698409
$99$	−37221.6	−0.381687
$100$	0	0
$101$	−20550.4	−0.200455	−0.100227	−	0.994965i	$-0.531957\pi$
$101$	−20550.4	−0.200455	−0.100227	+	0.994965i	$0.531957\pi$
$102$	−174.004	−0.00165599
$103$	−73661.4	−0.684143	−0.342071	−	0.939674i	$-0.611128\pi$
$103$	−73661.4	−0.684143	−0.342071	+	0.939674i	$0.611128\pi$
$104$	4279.17	0.0387950
$105$	0	0
$106$	1046.72	0.00904831
$107$	141079.	1.19125	0.595626	−	0.803262i	$-0.296904\pi$
$107$	141079.	1.19125	0.595626	+	0.803262i	$0.296904\pi$
$108$	−129533.	−1.06861
$109$	−3329.82	−0.0268444	−0.0134222	−	0.999910i	$-0.504273\pi$
$109$	−3329.82	−0.0268444	−0.0134222	+	0.999910i	$0.504273\pi$
$110$	0	0
$111$	19727.3	0.151971
$112$	8118.76	0.0611568
$113$	80229.6	0.591070	0.295535	−	0.955332i	$-0.404502\pi$
$113$	80229.6	0.591070	0.295535	+	0.955332i	$0.404502\pi$
$114$	−5375.54	−0.0387400
$115$	0	0
$116$	213519.	1.47330
$117$	18727.0	0.126475
$118$	1965.91	0.0129974
$119$	307.065	0.00198776
$120$	0	0
$121$	−48220.5	−0.299411
$122$	−17366.4	−0.105636
$123$	115858.	0.690499
$124$	−250250.	−1.46157
$125$	0	0
$126$	−353.650	−0.00198448
$127$	−91691.7	−0.504453	−0.252227	−	0.967668i	$-0.581163\pi$
$127$	−91691.7	−0.504453	−0.252227	+	0.967668i	$0.581163\pi$
$128$	−51347.2	−0.277008
$129$	−124427.	−0.658327
$130$	0	0
$131$	−150911.	−0.768319	−0.384160	−	0.923267i	$-0.625509\pi$
$131$	−150911.	−0.768319	−0.384160	+	0.923267i	$0.625509\pi$
$132$	122976.	0.614309
$133$	9486.23	0.0465013
$134$	−3637.25	−0.0174989
$135$	0	0
$136$	−966.211	−0.00447945
$137$	20446.3	0.0930710	0.0465355	−	0.998917i	$-0.485182\pi$
$137$	20446.3	0.0930710	0.0465355	+	0.998917i	$0.485182\pi$
$138$	−7657.72	−0.0342296
$139$	−142881.	−0.627245	−0.313623	−	0.949548i	$-0.601543\pi$
$139$	−142881.	−0.627245	−0.313623	+	0.949548i	$0.601543\pi$
$140$	0	0
$141$	−149456.	−0.633092
$142$	−5168.00	−0.0215081
$143$	−56767.5	−0.232145
$144$	−111800.	−0.449297
$145$	0	0
$146$	−16549.2	−0.0642530
$147$	192492.	0.734714
$148$	54636.0	0.205033
$149$	−175152.	−0.646321	−0.323161	−	0.946344i	$-0.604745\pi$
$149$	−175152.	−0.646321	−0.323161	+	0.946344i	$0.604745\pi$
$150$	0	0
$151$	−3588.76	−0.0128086	−0.00640431	−	0.999979i	$-0.502039\pi$
$151$	−3588.76	−0.0128086	−0.00640431	+	0.999979i	$0.502039\pi$
$152$	−29849.4	−0.104792
$153$	−4228.45	−0.0146034
$154$	1072.03	0.00364253
$155$	0	0
$156$	−61872.1	−0.203556
$157$	232402.	0.752473	0.376236	−	0.926524i	$-0.377218\pi$
$157$	232402.	0.752473	0.376236	+	0.926524i	$0.377218\pi$
$158$	−19754.9	−0.0629554
$159$	−30343.7	−0.0951867
$160$	0	0
$161$	13513.6	0.0410872
$162$	−7869.89	−0.0235603
$163$	144816.	0.426921	0.213460	−	0.976952i	$-0.431527\pi$
$163$	144816.	0.426921	0.213460	+	0.976952i	$0.431527\pi$
$164$	320876.	0.931596
$165$	0	0
$166$	−34186.3	−0.0962903
$167$	−413472.	−1.14724	−0.573621	−	0.819121i	$-0.694462\pi$
$167$	−413472.	−1.14724	−0.573621	+	0.819121i	$0.694462\pi$
$168$	2342.62	0.00640366
$169$	28561.0	0.0769231
$170$	0	0
$171$	−130630.	−0.341628
$172$	−344610.	−0.888191
$173$	686615.	1.74421	0.872103	−	0.489322i	$-0.162756\pi$
$173$	686615.	1.74421	0.872103	+	0.489322i	$0.162756\pi$
$174$	30576.3	0.0765618
$175$	0	0
$176$	338900.	0.824689
$177$	−56990.2	−0.136731
$178$	−26686.6	−0.0631312
$179$	217412.	0.507168	0.253584	−	0.967313i	$-0.418391\pi$
$179$	217412.	0.507168	0.253584	+	0.967313i	$0.418391\pi$
$180$	0	0
$181$	−282439.	−0.640807	−0.320404	−	0.947281i	$-0.603819\pi$
$181$	−282439.	−0.640807	−0.320404	+	0.947281i	$0.603819\pi$
$182$	−539.360	−0.00120698
$183$	503439.	1.11127
$184$	−42521.9	−0.0925908
$185$	0	0
$186$	−35836.4	−0.0759524
$187$	12817.8	0.0268046
$188$	−413929.	−0.854145
$189$	32734.1	0.0666570
$190$	0	0
$191$	−141669.	−0.280990	−0.140495	−	0.990081i	$-0.544869\pi$
$191$	−141669.	−0.280990	−0.140495	+	0.990081i	$0.544869\pi$
$192$	365680.	0.715892
$193$	−817556.	−1.57988	−0.789941	−	0.613183i	$-0.789889\pi$
$193$	−817556.	−1.57988	−0.789941	+	0.613183i	$0.789889\pi$
$194$	69619.4	0.132808
$195$	0	0
$196$	533118.	0.991251
$197$	−566426.	−1.03987	−0.519933	−	0.854207i	$-0.674043\pi$
$197$	−566426.	−1.03987	−0.519933	+	0.854207i	$0.674043\pi$
$198$	−14762.4	−0.0267604
$199$	400781.	0.717422	0.358711	−	0.933449i	$-0.383216\pi$
$199$	400781.	0.717422	0.358711	+	0.933449i	$0.383216\pi$
$200$	0	0
$201$	105441.	0.184086
$202$	−8150.43	−0.0140541
$203$	−53958.1	−0.0919002
$204$	13970.4	0.0235035
$205$	0	0
$206$	−29214.7	−0.0479659
$207$	−186089.	−0.301853
$208$	−170508.	−0.273267
$209$	395982.	0.627062
$210$	0	0
$211$	389440.	0.602191	0.301095	−	0.953594i	$-0.402648\pi$
$211$	389440.	0.602191	0.301095	+	0.953594i	$0.402648\pi$
$212$	−84039.0	−0.128423
$213$	149816.	0.226261
$214$	55953.1	0.0835198
$215$	0	0
$216$	−103001.	−0.150213
$217$	63240.6	0.0911688
$218$	−1320.63	−0.00188209
$219$	479748.	0.675931
$220$	0	0
$221$	−6448.91	−0.00888190
$222$	7823.99	0.0106548
$223$	−312613.	−0.420964	−0.210482	−	0.977598i	$-0.567503\pi$
$223$	−312613.	−0.420964	−0.210482	+	0.977598i	$0.567503\pi$
$224$	9740.05	0.0129700
$225$	0	0
$226$	31819.7	0.0414405
$227$	−863698.	−1.11249	−0.556246	−	0.831017i	$-0.687759\pi$
$227$	−863698.	−1.11249	−0.556246	+	0.831017i	$0.687759\pi$
$228$	431589.	0.549837
$229$	−255333.	−0.321749	−0.160875	−	0.986975i	$-0.551431\pi$
$229$	−255333.	−0.321749	−0.160875	+	0.986975i	$0.551431\pi$
$230$	0	0
$231$	−31077.2	−0.0383189
$232$	169785.	0.207099
$233$	566258.	0.683321	0.341660	−	0.939823i	$-0.389011\pi$
$233$	566258.	0.683321	0.341660	+	0.939823i	$0.389011\pi$
$234$	7427.27	0.00886727
$235$	0	0
$236$	−157838.	−0.184473
$237$	572681.	0.662280
$238$	121.784	0.000139364 0
$239$	1.70328e6	1.92882	0.964410	−	0.264412i	$-0.0851777\pi$
$239$	1.70328e6	1.92882	0.964410	+	0.264412i	$0.0851777\pi$
$240$	0	0
$241$	189220.	0.209857	0.104929	−	0.994480i	$-0.466539\pi$
$241$	189220.	0.209857	0.104929	+	0.994480i	$0.466539\pi$
$242$	−19124.6	−0.0209920
$243$	−760355.	−0.826039
$244$	1.39431e6	1.49928
$245$	0	0
$246$	45950.1	0.0484116
$247$	−199228.	−0.207782
$248$	−198993.	−0.205451
$249$	991037.	1.01296
$250$	0	0
$251$	807394.	0.808911	0.404456	−	0.914558i	$-0.367461\pi$
$251$	807394.	0.808911	0.404456	+	0.914558i	$0.367461\pi$
$252$	28393.7	0.0281657
$253$	564097.	0.554054
$254$	−36365.6	−0.0353677
$255$	0	0
$256$	997412.	0.951206
$257$	−1.71583e6	−1.62047	−0.810234	−	0.586107i	$-0.800660\pi$
$257$	−1.71583e6	−1.62047	−0.810234	+	0.586107i	$0.800660\pi$
$258$	−49348.8	−0.0461559
$259$	−13807.0	−0.0127894
$260$	0	0
$261$	743032.	0.675159
$262$	−59852.3	−0.0538676
$263$	2.09428e6	1.86701	0.933504	−	0.358568i	$-0.116735\pi$
$263$	2.09428e6	1.86701	0.933504	+	0.358568i	$0.116735\pi$
$264$	97787.6	0.0863523
$265$	0	0
$266$	3762.31	0.00326025
$267$	773626.	0.664129
$268$	292026.	0.248362
$269$	−431530.	−0.363605	−0.181803	−	0.983335i	$-0.558193\pi$
$269$	−431530.	−0.363605	−0.181803	+	0.983335i	$0.558193\pi$
$270$	0	0
$271$	−290063.	−0.239921	−0.119961	−	0.992779i	$-0.538277\pi$
$271$	−290063.	−0.239921	−0.119961	+	0.992779i	$0.538277\pi$
$272$	38499.8	0.0315527
$273$	15635.7	0.0126972
$274$	8109.18	0.00652530
$275$	0	0
$276$	614820.	0.485820
$277$	−1.89373e6	−1.48292	−0.741462	−	0.670995i	$-0.765867\pi$
$277$	−1.89373e6	−1.48292	−0.741462	+	0.670995i	$0.765867\pi$
$278$	−56667.7	−0.0439768
$279$	−870856.	−0.669786
$280$	0	0
$281$	−655309.	−0.495086	−0.247543	−	0.968877i	$-0.579623\pi$
$281$	−655309.	−0.495086	−0.247543	+	0.968877i	$0.579623\pi$
$282$	−59275.6	−0.0443867
$283$	−695619.	−0.516304	−0.258152	−	0.966104i	$-0.583113\pi$
$283$	−695619.	−0.516304	−0.258152	+	0.966104i	$0.583113\pi$
$284$	414926.	0.305264
$285$	0	0
$286$	−22514.4	−0.0162759
$287$	−81088.4	−0.0581104
$288$	−134126.	−0.0952863
$289$	−1.41840e6	−0.998974
$290$	0	0
$291$	−2.01821e6	−1.39712
$292$	1.32869e6	0.911943
$293$	2.44727e6	1.66538	0.832689	−	0.553742i	$-0.186800\pi$
$293$	2.44727e6	1.66538	0.832689	+	0.553742i	$0.186800\pi$
$294$	76343.7	0.0515116
$295$	0	0
$296$	43445.2	0.0288212
$297$	1.36641e6	0.898858
$298$	−69466.5	−0.0453142
$299$	−283809.	−0.183590
$300$	0	0
$301$	87086.1	0.0554029
$302$	−1423.33	−0.000898025 0
$303$	236275.	0.147847
$304$	1.18938e6	0.738138
$305$	0	0
$306$	−1677.04	−0.00102386
$307$	−1.41799e6	−0.858674	−0.429337	−	0.903144i	$-0.641253\pi$
$307$	−1.41799e6	−0.858674	−0.429337	+	0.903144i	$0.641253\pi$
$308$	−86070.5	−0.0516984
$309$	846911.	0.504594
$310$	0	0
$311$	−146871.	−0.0861062	−0.0430531	−	0.999073i	$-0.513708\pi$
$311$	−146871.	−0.0861062	−0.0430531	+	0.999073i	$0.513708\pi$
$312$	−49199.1	−0.0286135
$313$	2.60983e6	1.50575	0.752873	−	0.658165i	$-0.228667\pi$
$313$	2.60983e6	1.50575	0.752873	+	0.658165i	$0.228667\pi$
$314$	92172.4	0.0527566
$315$	0	0
$316$	1.58608e6	0.893525
$317$	−2.85498e6	−1.59571	−0.797857	−	0.602847i	$-0.794033\pi$
$317$	−2.85498e6	−1.59571	−0.797857	+	0.602847i	$0.794033\pi$
$318$	−12034.6	−0.00667364
$319$	−2.25237e6	−1.23926
$320$	0	0
$321$	−1.62204e6	−0.878615
$322$	5359.60	0.00288066
$323$	44984.4	0.0239914
$324$	631855.	0.334391
$325$	0	0
$326$	57435.1	0.0299319
$327$	38284.1	0.0197993
$328$	255153.	0.130953
$329$	104604.	0.0532792
$330$	0	0
$331$	−2.25339e6	−1.13049	−0.565245	−	0.824923i	$-0.691218\pi$
$331$	−2.25339e6	−1.13049	−0.565245	+	0.824923i	$0.691218\pi$
$332$	2.74474e6	1.36665
$333$	190130.	0.0939594
$334$	−163986.	−0.0804342
$335$	0	0
$336$	−93344.3	−0.0451066
$337$	795896.	0.381752	0.190876	−	0.981614i	$-0.438867\pi$
$337$	795896.	0.381752	0.190876	+	0.981614i	$0.438867\pi$
$338$	11327.5	0.00539315
$339$	−922429.	−0.435947
$340$	0	0
$341$	2.63984e6	1.22940
$342$	−51809.0	−0.0239519
$343$	−269969.	−0.123902
$344$	−274025.	−0.124852
$345$	0	0
$346$	272317.	0.122288
$347$	869632.	0.387714	0.193857	−	0.981030i	$-0.437900\pi$
$347$	869632.	0.387714	0.193857	+	0.981030i	$0.437900\pi$
$348$	−2.45490e6	−1.08664
$349$	−268145.	−0.117844	−0.0589219	−	0.998263i	$-0.518766\pi$
$349$	−268145.	−0.117844	−0.0589219	+	0.998263i	$0.518766\pi$
$350$	0	0
$351$	−687473.	−0.297843
$352$	406578.	0.174899
$353$	−2.22293e6	−0.949485	−0.474743	−	0.880125i	$-0.657459\pi$
$353$	−2.22293e6	−0.949485	−0.474743	+	0.880125i	$0.657459\pi$
$354$	−22602.7	−0.00958634
$355$	0	0
$356$	2.14261e6	0.896020
$357$	−3530.44	−0.00146608
$358$	86227.4	0.0355581
$359$	356227.	0.145878	0.0729391	−	0.997336i	$-0.476762\pi$
$359$	356227.	0.145878	0.0729391	+	0.997336i	$0.476762\pi$
$360$	0	0
$361$	−1.08639e6	−0.438749
$362$	−112017.	−0.0449276
$363$	554409.	0.220833
$364$	43304.0	0.0171307
$365$	0	0
$366$	199668.	0.0779122
$367$	−1.68973e6	−0.654867	−0.327433	−	0.944874i	$-0.606184\pi$
$367$	−1.68973e6	−0.654867	−0.327433	+	0.944874i	$0.606184\pi$
$368$	1.69433e6	0.652198
$369$	1.11663e6	0.426917
$370$	0	0
$371$	21237.4	0.00801064
$372$	2.87722e6	1.07799
$373$	−2.01242e6	−0.748938	−0.374469	−	0.927239i	$-0.622175\pi$
$373$	−2.01242e6	−0.748938	−0.374469	+	0.927239i	$0.622175\pi$
$374$	5083.63	0.00187930
$375$	0	0
$376$	−329146.	−0.120066
$377$	1.13322e6	0.410638
$378$	12982.6	0.00467339
$379$	−3.07308e6	−1.09894	−0.549472	−	0.835512i	$-0.685171\pi$
$379$	−3.07308e6	−1.09894	−0.549472	+	0.835512i	$0.685171\pi$
$380$	0	0
$381$	1.05421e6	0.372063
$382$	−56186.9	−0.0197005
$383$	−5.65657e6	−1.97041	−0.985204	−	0.171387i	$-0.945175\pi$
$383$	−5.65657e6	−1.97041	−0.985204	+	0.171387i	$0.945175\pi$
$384$	590358.	0.204309
$385$	0	0
$386$	−324249.	−0.110767
$387$	−1.19922e6	−0.407026
$388$	−5.58958e6	−1.88495
$389$	−1.20141e6	−0.402547	−0.201274	−	0.979535i	$-0.564508\pi$
$389$	−1.20141e6	−0.402547	−0.201274	+	0.979535i	$0.564508\pi$
$390$	0	0
$391$	64082.6	0.0211981
$392$	423922.	0.139338
$393$	1.73507e6	0.566678
$394$	−224649.	−0.0729060
$395$	0	0
$396$	1.18524e6	0.379810
$397$	1.91438e6	0.609610	0.304805	−	0.952415i	$-0.401409\pi$
$397$	1.91438e6	0.609610	0.304805	+	0.952415i	$0.401409\pi$
$398$	158953.	0.0502992
$399$	−109067.	−0.0342973
$400$	0	0
$401$	−5.02610e6	−1.56088	−0.780441	−	0.625229i	$-0.785006\pi$
$401$	−5.02610e6	−1.56088	−0.780441	+	0.625229i	$0.785006\pi$
$402$	41818.8	0.0129064
$403$	−1.32816e6	−0.407370
$404$	654379.	0.199469
$405$	0	0
$406$	−21400.2	−0.00644322
$407$	−576345.	−0.172463
$408$	11108.9	0.00330385
$409$	−2.26919e6	−0.670754	−0.335377	−	0.942084i	$-0.608864\pi$
$409$	−2.26919e6	−0.670754	−0.335377	+	0.942084i	$0.608864\pi$
$410$	0	0
$411$	−235079.	−0.0686450
$412$	2.34558e6	0.680780
$413$	39887.1	0.0115069
$414$	−73804.5	−0.0211632
$415$	0	0
$416$	−204558.	−0.0579540
$417$	1.64275e6	0.462629
$418$	157050.	0.0439639
$419$	−3.92074e6	−1.09102	−0.545510	−	0.838104i	$-0.683664\pi$
$419$	−3.92074e6	−1.09102	−0.545510	+	0.838104i	$0.683664\pi$
$420$	0	0
$421$	1.77811e6	0.488939	0.244469	−	0.969657i	$-0.421386\pi$
$421$	1.77811e6	0.488939	0.244469	+	0.969657i	$0.421386\pi$
$422$	154455.	0.0422202
$423$	−1.44045e6	−0.391424
$424$	−66825.7	−0.0180521
$425$	0	0
$426$	59418.3	0.0158634
$427$	−352355.	−0.0935212
$428$	−4.49234e6	−1.18540
$429$	652677.	0.171220
$430$	0	0
$431$	−4.45663e6	−1.15561	−0.577807	−	0.816173i	$-0.696091\pi$
$431$	−4.45663e6	−1.15561	−0.577807	+	0.816173i	$0.696091\pi$
$432$	4.10419e6	1.05808
$433$	1.25271e6	0.321093	0.160547	−	0.987028i	$-0.448674\pi$
$433$	1.25271e6	0.321093	0.160547	+	0.987028i	$0.448674\pi$
$434$	25081.7	0.00639194
$435$	0	0
$436$	106030.	0.0267125
$437$	1.97972e6	0.495906
$438$	190272.	0.0473902
$439$	3.75453e6	0.929810	0.464905	−	0.885360i	$-0.346088\pi$
$439$	3.75453e6	0.929810	0.464905	+	0.885360i	$0.346088\pi$
$440$	0	0
$441$	1.85522e6	0.454254
$442$	−2557.69	−0.000622719 0
$443$	2.31259e6	0.559874	0.279937	−	0.960018i	$-0.409686\pi$
$443$	2.31259e6	0.559874	0.279937	+	0.960018i	$0.409686\pi$
$444$	−628170.	−0.151224
$445$	0	0
$446$	−123985.	−0.0295142
$447$	2.01378e6	0.476698
$448$	−255937.	−0.0602474
$449$	−7.80175e6	−1.82632	−0.913159	−	0.407603i	$-0.866365\pi$
$449$	−7.80175e6	−1.82632	−0.913159	+	0.407603i	$0.866365\pi$
$450$	0	0
$451$	−3.38486e6	−0.783609
$452$	−2.55473e6	−0.588164
$453$	41261.3	0.00944708
$454$	−342549.	−0.0779980
$455$	0	0
$456$	343189.	0.0772896
$457$	−4.88928e6	−1.09510	−0.547551	−	0.836772i	$-0.684440\pi$
$457$	−4.88928e6	−1.09510	−0.547551	+	0.836772i	$0.684440\pi$
$458$	−101267.	−0.0225582
$459$	155228.	0.0343904
$460$	0	0
$461$	−4.08764e6	−0.895818	−0.447909	−	0.894079i	$-0.647831\pi$
$461$	−4.08764e6	−0.895818	−0.447909	+	0.894079i	$0.647831\pi$
$462$	−12325.5	−0.00268657
$463$	−3.13129e6	−0.678845	−0.339423	−	0.940634i	$-0.610232\pi$
$463$	−3.13129e6	−0.678845	−0.339423	+	0.940634i	$0.610232\pi$
$464$	−6.76526e6	−1.45878
$465$	0	0
$466$	224582.	0.0479083
$467$	−5.78075e6	−1.22657	−0.613284	−	0.789862i	$-0.710152\pi$
$467$	−5.78075e6	−1.22657	−0.613284	+	0.789862i	$0.710152\pi$
$468$	−596318.	−0.125853
$469$	−73797.7	−0.0154921
$470$	0	0
$471$	−2.67201e6	−0.554991
$472$	−125509.	−0.0259310
$473$	3.63522e6	0.747099
$474$	227130.	0.0464331
$475$	0	0
$476$	−9777.79	−0.00197799
$477$	−292451.	−0.0588514
$478$	675535.	0.135231
$479$	−5.98034e6	−1.19093	−0.595467	−	0.803380i	$-0.703033\pi$
$479$	−5.98034e6	−1.19093	−0.595467	+	0.803380i	$0.703033\pi$
$480$	0	0
$481$	289972.	0.0571469
$482$	75045.9	0.0147133
$483$	−155371.	−0.0303041
$484$	1.53547e6	0.297940
$485$	0	0
$486$	−301563.	−0.0579144
$487$	8.62843e6	1.64858	0.824289	−	0.566169i	$-0.191575\pi$
$487$	8.62843e6	1.64858	0.824289	+	0.566169i	$0.191575\pi$
$488$	1.10872e6	0.210752
$489$	−1.66500e6	−0.314878
$490$	0	0
$491$	−4.67336e6	−0.874834	−0.437417	−	0.899259i	$-0.644107\pi$
$491$	−4.67336e6	−0.874834	−0.437417	+	0.899259i	$0.644107\pi$
$492$	−3.68923e6	−0.687105
$493$	−255873.	−0.0474142
$494$	−79015.2	−0.0145678
$495$	0	0
$496$	7.92909e6	1.44717
$497$	−104856.	−0.0190415
$498$	393053.	0.0710195
$499$	1.02014e7	1.83404	0.917018	−	0.398847i	$-0.130589\pi$
$499$	1.02014e7	1.83404	0.917018	+	0.398847i	$0.130589\pi$
$500$	0	0
$501$	4.75383e6	0.846155
$502$	320218.	0.0567136
$503$	−1.14659e6	−0.202064	−0.101032	−	0.994883i	$-0.532214\pi$
$503$	−1.14659e6	−0.202064	−0.101032	+	0.994883i	$0.532214\pi$
$504$	22578.0	0.00395921
$505$	0	0
$506$	223725.	0.0388453
$507$	−328376.	−0.0567351
$508$	2.91971e6	0.501974
$509$	−2.19167e6	−0.374957	−0.187478	−	0.982269i	$-0.560031\pi$
$509$	−2.19167e6	−0.374957	−0.187478	+	0.982269i	$0.560031\pi$
$510$	0	0
$511$	−335773.	−0.0568844
$512$	2.03869e6	0.343698
$513$	4.79548e6	0.804523
$514$	−680509.	−0.113613
$515$	0	0
$516$	3.96210e6	0.655091
$517$	4.36646e6	0.718461
$518$	−5475.97	−0.000896679 0
$519$	−7.89426e6	−1.28645
$520$	0	0
$521$	−6.90308e6	−1.11416	−0.557082	−	0.830458i	$-0.688079\pi$
$521$	−6.90308e6	−1.11416	−0.557082	+	0.830458i	$0.688079\pi$
$522$	294692.	0.0473361
$523$	1.59809e6	0.255473	0.127737	−	0.991808i	$-0.459229\pi$
$523$	1.59809e6	0.255473	0.127737	+	0.991808i	$0.459229\pi$
$524$	4.80540e6	0.764542
$525$	0	0
$526$	830609.	0.130898
$527$	299892.	0.0470368
$528$	−3.89646e6	−0.608254
$529$	−3.61614e6	−0.561831
$530$	0	0
$531$	−549267.	−0.0845371
$532$	−302067.	−0.0462727
$533$	1.70300e6	0.259655
$534$	306826.	0.0465628
$535$	0	0
$536$	232212.	0.0349118
$537$	−2.49967e6	−0.374065
$538$	−171148.	−0.0254927
$539$	−5.62376e6	−0.833787
$540$	0	0
$541$	−279387.	−0.0410405	−0.0205202	−	0.999789i	$-0.506532\pi$
$541$	−279387.	−0.0410405	−0.0205202	+	0.999789i	$0.506532\pi$
$542$	−115041.	−0.0168211
$543$	3.24730e6	0.472631
$544$	46188.1	0.00669164
$545$	0	0
$546$	6201.22	0.000890216 0
$547$	−606222.	−0.0866290	−0.0433145	−	0.999061i	$-0.513792\pi$
$547$	−606222.	−0.0866290	−0.0433145	+	0.999061i	$0.513792\pi$
$548$	−651067.	−0.0926135
$549$	4.85211e6	0.687068
$550$	0	0
$551$	−7.90475e6	−1.10920
$552$	488890.	0.0682909
$553$	−400816.	−0.0557356
$554$	−751068.	−0.103969
$555$	0	0
$556$	4.54972e6	0.624162
$557$	3.37279e6	0.460629	0.230315	−	0.973116i	$-0.426024\pi$
$557$	3.37279e6	0.460629	0.230315	+	0.973116i	$0.426024\pi$
$558$	−345388.	−0.0469593
$559$	−1.82896e6	−0.247557
$560$	0	0
$561$	−147371.	−0.0197699
$562$	−259901.	−0.0347110
$563$	3.75518e6	0.499298	0.249649	−	0.968336i	$-0.419685\pi$
$563$	3.75518e6	0.499298	0.249649	+	0.968336i	$0.419685\pi$
$564$	4.75910e6	0.629980
$565$	0	0
$566$	−275888.	−0.0361986
$567$	−159676.	−0.0208584
$568$	329939.	0.0429104
$569$	5.11904e6	0.662839	0.331420	−	0.943483i	$-0.392472\pi$
$569$	5.11904e6	0.662839	0.331420	+	0.943483i	$0.392472\pi$
$570$	0	0
$571$	1.23612e7	1.58662	0.793308	−	0.608821i	$-0.208357\pi$
$571$	1.23612e7	1.58662	0.793308	+	0.608821i	$0.208357\pi$
$572$	1.80763e6	0.231004
$573$	1.62882e6	0.207246
$574$	−32160.3	−0.00407418
$575$	0	0
$576$	3.52439e6	0.442617
$577$	−1.46773e6	−0.183530	−0.0917651	−	0.995781i	$-0.529251\pi$
$577$	−1.46773e6	−0.183530	−0.0917651	+	0.995781i	$0.529251\pi$
$578$	−562549.	−0.0700391
$579$	9.39974e6	1.16525
$580$	0	0
$581$	−693622.	−0.0852476
$582$	−800439.	−0.0979537
$583$	886511.	0.108022
$584$	1.05654e6	0.128190
$585$	0	0
$586$	970606.	0.116761
$587$	−6.01770e6	−0.720834	−0.360417	−	0.932791i	$-0.617366\pi$
$587$	−6.01770e6	−0.720834	−0.360417	+	0.932791i	$0.617366\pi$
$588$	−6.12945e6	−0.731103
$589$	9.26461e6	1.10037
$590$	0	0
$591$	6.51240e6	0.766959
$592$	−1.73112e6	−0.203013
$593$	1.28954e7	1.50590	0.752951	−	0.658076i	$-0.228629\pi$
$593$	1.28954e7	1.50590	0.752951	+	0.658076i	$0.228629\pi$
$594$	541930.	0.0630199
$595$	0	0
$596$	5.57730e6	0.643144
$597$	−4.60793e6	−0.529139
$598$	−112561.	−0.0128717
$599$	1.07891e7	1.22863	0.614313	−	0.789062i	$-0.289433\pi$
$599$	1.07891e7	1.22863	0.614313	+	0.789062i	$0.289433\pi$
$600$	0	0
$601$	3.51995e6	0.397512	0.198756	−	0.980049i	$-0.436310\pi$
$601$	3.51995e6	0.397512	0.198756	+	0.980049i	$0.436310\pi$
$602$	34539.0	0.00388435
$603$	1.01623e6	0.113815
$604$	114276.	0.0127457
$605$	0	0
$606$	93708.5	0.0103657
$607$	1.43143e6	0.157688	0.0788439	−	0.996887i	$-0.474877\pi$
$607$	1.43143e6	0.157688	0.0788439	+	0.996887i	$0.474877\pi$
$608$	1.42690e6	0.156543
$609$	620376.	0.0677816
$610$	0	0
$611$	−2.19686e6	−0.238067
$612$	134645.	0.0145316
$613$	−1.06458e7	−1.14426	−0.572131	−	0.820162i	$-0.693883\pi$
$613$	−1.06458e7	−1.14426	−0.572131	+	0.820162i	$0.693883\pi$
$614$	−562387.	−0.0602025
$615$	0	0
$616$	−68441.1	−0.00726716
$617$	−9.96269e6	−1.05357	−0.526785	−	0.849998i	$-0.676603\pi$
$617$	−9.96269e6	−1.05357	−0.526785	+	0.849998i	$0.676603\pi$
$618$	335892.	0.0353776
$619$	1.14066e7	1.19655	0.598273	−	0.801292i	$-0.295854\pi$
$619$	1.14066e7	1.19655	0.598273	+	0.801292i	$0.295854\pi$
$620$	0	0
$621$	6.83139e6	0.710854
$622$	−58250.1	−0.00603699
$623$	−541457.	−0.0558912
$624$	1.96039e6	0.201550
$625$	0	0
$626$	1.03508e6	0.105569
$627$	−4.55275e6	−0.462493
$628$	−7.40031e6	−0.748774
$629$	−65474.0	−0.00659845
$630$	0	0
$631$	−1.56307e7	−1.56280	−0.781402	−	0.624028i	$-0.785495\pi$
$631$	−1.56307e7	−1.56280	−0.781402	+	0.624028i	$0.785495\pi$
$632$	1.26121e6	0.125601
$633$	−4.47753e6	−0.444150
$634$	−1.13231e6	−0.111877
$635$	0	0
$636$	966227.	0.0947188
$637$	2.82944e6	0.276282
$638$	−893306.	−0.0868857
$639$	1.44392e6	0.139891
$640$	0	0
$641$	4.75966e6	0.457542	0.228771	−	0.973480i	$-0.426529\pi$
$641$	4.75966e6	0.457542	0.228771	+	0.973480i	$0.426529\pi$
$642$	−643313.	−0.0616006
$643$	946165.	0.0902484	0.0451242	−	0.998981i	$-0.485632\pi$
$643$	946165.	0.0902484	0.0451242	+	0.998981i	$0.485632\pi$
$644$	−430310.	−0.0408852
$645$	0	0
$646$	17841.2	0.00168206
$647$	−870173.	−0.0817231	−0.0408616	−	0.999165i	$-0.513010\pi$
$647$	−870173.	−0.0817231	−0.0408616	+	0.999165i	$0.513010\pi$
$648$	502435.	0.0470048
$649$	1.66500e6	0.155168
$650$	0	0
$651$	−727100.	−0.0672421
$652$	−4.61133e6	−0.424822
$653$	−2.57122e6	−0.235970	−0.117985	−	0.993015i	$-0.537643\pi$
$653$	−2.57122e6	−0.235970	−0.117985	+	0.993015i	$0.537643\pi$
$654$	15183.8	0.00138815
$655$	0	0
$656$	−1.01668e7	−0.922415
$657$	4.62377e6	0.417910
$658$	41486.7	0.00373546
$659$	1.09142e7	0.978988	0.489494	−	0.872007i	$-0.337182\pi$
$659$	1.09142e7	0.978988	0.489494	+	0.872007i	$0.337182\pi$
$660$	0	0
$661$	4.31733e6	0.384337	0.192168	−	0.981362i	$-0.438448\pi$
$661$	4.31733e6	0.384337	0.192168	+	0.981362i	$0.438448\pi$
$662$	−893712.	−0.0792597
$663$	74145.5	0.00655090
$664$	2.18255e6	0.192107
$665$	0	0
$666$	75407.0	0.00658758
$667$	−1.12607e7	−0.980057
$668$	1.31661e7	1.14160
$669$	3.59422e6	0.310484
$670$	0	0
$671$	−1.47083e7	−1.26112
$672$	−111985.	−0.00956613
$673$	2.02793e7	1.72590	0.862949	−	0.505292i	$-0.168615\pi$
$673$	2.02793e7	1.72590	0.862949	+	0.505292i	$0.168615\pi$
$674$	315658.	0.0267650
$675$	0	0
$676$	−909459.	−0.0765450
$677$	1.37469e7	1.15275	0.576373	−	0.817186i	$-0.304467\pi$
$677$	1.37469e7	1.15275	0.576373	+	0.817186i	$0.304467\pi$
$678$	−365842.	−0.0305647
$679$	1.41254e6	0.117578
$680$	0	0
$681$	9.93025e6	0.820526
$682$	1.04698e6	0.0861942
$683$	−1.37681e7	−1.12933	−0.564665	−	0.825320i	$-0.690995\pi$
$683$	−1.37681e7	−1.12933	−0.564665	+	0.825320i	$0.690995\pi$
$684$	4.15963e6	0.339949
$685$	0	0
$686$	−107072.	−0.00868690
$687$	2.93565e6	0.237308
$688$	1.09188e7	0.879438
$689$	−446023.	−0.0357939
$690$	0	0
$691$	6.60598e6	0.526310	0.263155	−	0.964754i	$-0.415237\pi$
$691$	6.60598e6	0.526310	0.263155	+	0.964754i	$0.415237\pi$
$692$	−2.18637e7	−1.73563
$693$	−299520.	−0.0236915
$694$	344903.	0.0271830
$695$	0	0
$696$	−1.95207e6	−0.152747
$697$	−384527.	−0.0299809
$698$	−106348.	−0.00826214
$699$	−6.51047e6	−0.503987
$700$	0	0
$701$	−1.37908e7	−1.05997	−0.529985	−	0.848007i	$-0.677803\pi$
$701$	−1.37908e7	−1.05997	−0.529985	+	0.848007i	$0.677803\pi$
$702$	−272657.	−0.0208821
$703$	−2.02270e6	−0.154363
$704$	−1.06836e7	−0.812426
$705$	0	0
$706$	−881630.	−0.0665694
$707$	−165368.	−0.0124423
$708$	1.81472e6	0.136059
$709$	1.96433e7	1.46757	0.733785	−	0.679382i	$-0.237752\pi$
$709$	1.96433e7	1.46757	0.733785	+	0.679382i	$0.237752\pi$
$710$	0	0
$711$	5.51945e6	0.409470
$712$	1.70375e6	0.125952
$713$	1.31979e7	0.972257
$714$	−1400.20	−0.000102788 0
$715$	0	0
$716$	−6.92300e6	−0.504675
$717$	−1.95832e7	−1.42261
$718$	141282.	0.0102277
$719$	−2.26567e7	−1.63446	−0.817229	−	0.576312i	$-0.804491\pi$
$719$	−2.26567e7	−1.63446	−0.817229	+	0.576312i	$0.804491\pi$
$720$	0	0
$721$	−592749.	−0.0424652
$722$	−430869.	−0.0307611
$723$	−2.17553e6	−0.154781
$724$	8.99361e6	0.637657
$725$	0	0
$726$	219883.	0.0154828
$727$	−1.36449e7	−0.957491	−0.478746	−	0.877954i	$-0.658908\pi$
$727$	−1.36449e7	−0.957491	−0.478746	+	0.877954i	$0.658908\pi$
$728$	34434.2	0.00240803
$729$	1.35639e7	0.945293
$730$	0	0
$731$	412969.	0.0285841
$732$	−1.60309e7	−1.10581
$733$	1.07727e7	0.740569	0.370284	−	0.928918i	$-0.379260\pi$
$733$	1.07727e7	0.740569	0.370284	+	0.928918i	$0.379260\pi$
$734$	−670161.	−0.0459134
$735$	0	0
$736$	2.03269e6	0.138317
$737$	−3.08053e6	−0.208909
$738$	442864.	0.0299316
$739$	8.30186e6	0.559196	0.279598	−	0.960117i	$-0.409799\pi$
$739$	8.30186e6	0.559196	0.279598	+	0.960117i	$0.409799\pi$
$740$	0	0
$741$	2.29059e6	0.153251
$742$	8422.93	0.000561634 0
$743$	2.21659e7	1.47304	0.736519	−	0.676417i	$-0.236468\pi$
$743$	2.21659e7	1.47304	0.736519	+	0.676417i	$0.236468\pi$
$744$	2.28789e6	0.151531
$745$	0	0
$746$	−798140.	−0.0525088
$747$	9.55153e6	0.626284
$748$	−408153.	−0.0266728
$749$	1.13526e6	0.0739417
$750$	0	0
$751$	2.47886e7	1.60380	0.801902	−	0.597455i	$-0.203821\pi$
$751$	2.47886e7	1.60380	0.801902	+	0.597455i	$0.203821\pi$
$752$	1.31152e7	0.845727
$753$	−9.28290e6	−0.596618
$754$	449442.	0.0287902
$755$	0	0
$756$	−1.04234e6	−0.0663293
$757$	1.63378e6	0.103623	0.0518113	−	0.998657i	$-0.483501\pi$
$757$	1.63378e6	0.103623	0.0518113	+	0.998657i	$0.483501\pi$
$758$	−1.21881e6	−0.0770481
$759$	−6.48562e6	−0.408646
$760$	0	0
$761$	−1.29552e7	−0.810931	−0.405465	−	0.914110i	$-0.632891\pi$
$761$	−1.29552e7	−0.810931	−0.405465	+	0.914110i	$0.632891\pi$
$762$	418109.	0.0260857
$763$	−26794.9	−0.00166625
$764$	4.51111e6	0.279608
$765$	0	0
$766$	−2.24344e6	−0.138147
$767$	−837700.	−0.0514162
$768$	−1.14676e7	−0.701568
$769$	−1.14055e7	−0.695502	−0.347751	−	0.937587i	$-0.613055\pi$
$769$	−1.14055e7	−0.695502	−0.347751	+	0.937587i	$0.613055\pi$
$770$	0	0
$771$	1.97275e7	1.19519
$772$	2.60332e7	1.57212
$773$	6.83570e6	0.411467	0.205733	−	0.978608i	$-0.434042\pi$
$773$	6.83570e6	0.411467	0.205733	+	0.978608i	$0.434042\pi$
$774$	−475620.	−0.0285370
$775$	0	0
$776$	−4.44469e6	−0.264964
$777$	158744.	0.00943291
$778$	−476488.	−0.0282230
$779$	−1.18793e7	−0.701369
$780$	0	0
$781$	−4.37698e6	−0.256772
$782$	25415.6	0.00148622
$783$	−2.72769e7	−1.58998
$784$	−1.68917e7	−0.981481
$785$	0	0
$786$	688144.	0.0397304
$787$	2.71866e7	1.56465	0.782327	−	0.622868i	$-0.214033\pi$
$787$	2.71866e7	1.56465	0.782327	+	0.622868i	$0.214033\pi$
$788$	1.80365e7	1.03475
$789$	−2.40787e7	−1.37702
$790$	0	0
$791$	645603.	0.0366880
$792$	942469.	0.0533893
$793$	7.40006e6	0.417881
$794$	759258.	0.0427403
$795$	0	0
$796$	−1.27620e7	−0.713896
$797$	−1.81329e7	−1.01117	−0.505583	−	0.862778i	$-0.668722\pi$
$797$	−1.81329e7	−1.01117	−0.505583	+	0.862778i	$0.668722\pi$
$798$	−43256.7	−0.00240462
$799$	496039.	0.0274884
$800$	0	0
$801$	7.45614e6	0.410613
$802$	−1.99339e6	−0.109435
$803$	−1.40161e7	−0.767077
$804$	−3.35753e6	−0.183181
$805$	0	0
$806$	−526760.	−0.0285611
$807$	4.96146e6	0.268179
$808$	520346.	0.0280391
$809$	3.89183e6	0.209065	0.104533	−	0.994521i	$-0.466665\pi$
$809$	3.89183e6	0.209065	0.104533	+	0.994521i	$0.466665\pi$
$810$	0	0
$811$	−1.40881e7	−0.752142	−0.376071	−	0.926591i	$-0.622725\pi$
$811$	−1.40881e7	−0.752142	−0.376071	+	0.926591i	$0.622725\pi$
$812$	1.71817e6	0.0914485
$813$	3.33496e6	0.176955
$814$	−228583.	−0.0120916
$815$	0	0
$816$	−442646.	−0.0232719
$817$	1.27579e7	0.668691
$818$	−899980.	−0.0470272
$819$	150695.	0.00785036
$820$	0	0
$821$	3.32912e7	1.72374	0.861871	−	0.507128i	$-0.169293\pi$
$821$	3.32912e7	1.72374	0.861871	+	0.507128i	$0.169293\pi$
$822$	−93234.1	−0.00481277
$823$	−3.41282e7	−1.75636	−0.878180	−	0.478330i	$-0.841242\pi$
$823$	−3.41282e7	−1.75636	−0.878180	+	0.478330i	$0.841242\pi$
$824$	1.86514e6	0.0956961
$825$	0	0
$826$	15819.5	0.000806759 0
$827$	1.68225e7	0.855317	0.427658	−	0.903940i	$-0.359339\pi$
$827$	1.68225e7	0.855317	0.427658	+	0.903940i	$0.359339\pi$
$828$	5.92559e6	0.300370
$829$	−2.49836e7	−1.26261	−0.631305	−	0.775534i	$-0.717480\pi$
$829$	−2.49836e7	−1.26261	−0.631305	+	0.775534i	$0.717480\pi$
$830$	0	0
$831$	2.17729e7	1.09374
$832$	5.37513e6	0.269204
$833$	−638871.	−0.0319007
$834$	651529.	0.0324354
$835$	0	0
$836$	−1.26092e7	−0.623979
$837$	3.19693e7	1.57732
$838$	−1.55500e6	−0.0764925
$839$	3.58243e7	1.75700	0.878501	−	0.477740i	$-0.158544\pi$
$839$	3.58243e7	1.75700	0.878501	+	0.477740i	$0.158544\pi$
$840$	0	0
$841$	2.44514e7	1.19211
$842$	705214.	0.0342800
$843$	7.53433e6	0.365153
$844$	−1.24008e7	−0.599231
$845$	0	0
$846$	−571293.	−0.0274431
$847$	−388028.	−0.0185846
$848$	2.66274e6	0.127157
$849$	7.99778e6	0.380803
$850$	0	0
$851$	−2.88144e6	−0.136391
$852$	−4.77056e6	−0.225149
$853$	−1.13433e7	−0.533786	−0.266893	−	0.963726i	$-0.585997\pi$
$853$	−1.13433e7	−0.533786	−0.266893	+	0.963726i	$0.585997\pi$
$854$	−139747.	−0.00655687
$855$	0	0
$856$	−3.57220e6	−0.166629
$857$	−2.46384e7	−1.14594	−0.572968	−	0.819578i	$-0.694208\pi$
$857$	−2.46384e7	−1.14594	−0.572968	+	0.819578i	$0.694208\pi$
$858$	258857.	0.0120044
$859$	1.82989e6	0.0846140	0.0423070	−	0.999105i	$-0.486529\pi$
$859$	1.82989e6	0.0846140	0.0423070	+	0.999105i	$0.486529\pi$
$860$	0	0
$861$	932302.	0.0428597
$862$	−1.76753e6	−0.0810213
$863$	−2.26407e6	−0.103482	−0.0517408	−	0.998661i	$-0.516477\pi$
$863$	−2.26407e6	−0.103482	−0.0517408	+	0.998661i	$0.516477\pi$
$864$	4.92379e6	0.224396
$865$	0	0
$866$	496835.	0.0225122
$867$	1.63079e7	0.736800
$868$	−2.01375e6	−0.0907207
$869$	−1.67312e7	−0.751585
$870$	0	0
$871$	1.54988e6	0.0692234
$872$	84312.7	0.00375493
$873$	−1.94514e7	−0.863803
$874$	785171.	0.0347685
$875$	0	0
$876$	−1.52765e7	−0.672609
$877$	−1.77367e7	−0.778708	−0.389354	−	0.921088i	$-0.627302\pi$
$877$	−1.77367e7	−0.778708	−0.389354	+	0.921088i	$0.627302\pi$
$878$	1.48908e6	0.0651899
$879$	−2.81371e7	−1.22831
$880$	0	0
$881$	−3.79053e7	−1.64536	−0.822679	−	0.568506i	$-0.807522\pi$
$881$	−3.79053e7	−1.64536	−0.822679	+	0.568506i	$0.807522\pi$
$882$	735794.	0.0318482
$883$	−3.29131e7	−1.42058	−0.710291	−	0.703908i	$-0.751437\pi$
$883$	−3.29131e7	−1.42058	−0.710291	+	0.703908i	$0.751437\pi$
$884$	205351.	0.00883824
$885$	0	0
$886$	917192.	0.0392533
$887$	−9.54706e6	−0.407437	−0.203719	−	0.979029i	$-0.565303\pi$
$887$	−9.54706e6	−0.407437	−0.203719	+	0.979029i	$0.565303\pi$
$888$	−499505.	−0.0212573
$889$	−737838.	−0.0313117
$890$	0	0
$891$	−6.66531e6	−0.281272
$892$	9.95444e6	0.418894
$893$	1.53242e7	0.643058
$894$	798681.	0.0334218
$895$	0	0
$896$	−413188.	−0.0171940
$897$	3.26306e6	0.135408
$898$	−3.09424e6	−0.128045
$899$	−5.26975e7	−2.17466
$900$	0	0
$901$	100710.	0.00413293
$902$	−1.34246e6	−0.0549396
$903$	−1.00126e6	−0.0408627
$904$	−2.03145e6	−0.0826773
$905$	0	0
$906$	16364.6	0.000662344 0
$907$	2.11768e7	0.854758	0.427379	−	0.904073i	$-0.359437\pi$
$907$	2.11768e7	0.854758	0.427379	+	0.904073i	$0.359437\pi$
$908$	2.75025e7	1.10702
$909$	2.27720e6	0.0914095
$910$	0	0
$911$	−4.30962e7	−1.72045	−0.860227	−	0.509912i	$-0.829678\pi$
$911$	−4.30962e7	−1.72045	−0.860227	+	0.509912i	$0.829678\pi$
$912$	−1.36748e7	−0.544418
$913$	−2.89537e7	−1.14955
$914$	−1.93913e6	−0.0767787
$915$	0	0
$916$	8.13048e6	0.320168
$917$	−1.21437e6	−0.0476900
$918$	61564.4	0.00241114
$919$	1.40263e6	0.0547841	0.0273920	−	0.999625i	$-0.491280\pi$
$919$	1.40263e6	0.0547841	0.0273920	+	0.999625i	$0.491280\pi$
$920$	0	0
$921$	1.63032e7	0.633320
$922$	−1.62119e6	−0.0628067
$923$	2.20216e6	0.0850832
$924$	989583.	0.0381305
$925$	0	0
$926$	−1.24189e6	−0.0475945
$927$	8.16247e6	0.311977
$928$	−8.11626e6	−0.309375
$929$	1.27299e7	0.483934	0.241967	−	0.970285i	$-0.422207\pi$
$929$	1.27299e7	0.483934	0.241967	+	0.970285i	$0.422207\pi$
$930$	0	0
$931$	−1.97368e7	−0.746281
$932$	−1.80312e7	−0.679962
$933$	1.68863e6	0.0635081
$934$	−2.29269e6	−0.0859960
$935$	0	0
$936$	−474177.	−0.0176909
$937$	4.02938e7	1.49930	0.749651	−	0.661833i	$-0.230221\pi$
$937$	4.02938e7	1.49930	0.749651	+	0.661833i	$0.230221\pi$
$938$	−29268.7	−0.00108617
$939$	−3.00062e7	−1.11057
$940$	0	0
$941$	−2.47470e7	−0.911064	−0.455532	−	0.890219i	$-0.650551\pi$
$941$	−2.47470e7	−0.911064	−0.455532	+	0.890219i	$0.650551\pi$
$942$	−1.05974e6	−0.0389110
$943$	−1.69226e7	−0.619710
$944$	5.00104e6	0.182655
$945$	0	0
$946$	1.44176e6	0.0523798
$947$	−3.28379e7	−1.18987	−0.594937	−	0.803772i	$-0.702823\pi$
$947$	−3.28379e7	−1.18987	−0.594937	+	0.803772i	$0.702823\pi$
$948$	−1.82357e7	−0.659025
$949$	7.05182e6	0.254177
$950$	0	0
$951$	3.28248e7	1.17693
$952$	−7775.05	−0.000278042 0
$953$	−2.12237e7	−0.756989	−0.378494	−	0.925604i	$-0.623558\pi$
$953$	−2.12237e7	−0.756989	−0.378494	+	0.925604i	$0.623558\pi$
$954$	−115988.	−0.00412613
$955$	0	0
$956$	−5.42371e7	−1.91934
$957$	2.58963e7	0.914023
$958$	−2.37185e6	−0.0834975
$959$	164531.	0.00577697
$960$	0	0
$961$	3.31340e7	1.15735
$962$	115005.	0.00400663
$963$	−1.56331e7	−0.543224
$964$	−6.02526e6	−0.208825
$965$	0	0
$966$	−61621.3	−0.00212465
$967$	5.66776e7	1.94915	0.974575	−	0.224060i	$-0.0719313\pi$
$967$	5.66776e7	1.94915	0.974575	+	0.224060i	$0.0719313\pi$
$968$	1.22097e6	0.0418809
$969$	−517202.	−0.0176950
$970$	0	0
$971$	−3.20888e7	−1.09221	−0.546104	−	0.837718i	$-0.683889\pi$
$971$	−3.20888e7	−1.09221	−0.546104	+	0.837718i	$0.683889\pi$
$972$	2.42118e7	0.821979
$973$	−1.14976e6	−0.0389335
$974$	3.42210e6	0.115583
$975$	0	0
$976$	−4.41781e7	−1.48451
$977$	−5.01796e7	−1.68186	−0.840931	−	0.541142i	$-0.817992\pi$
$977$	−5.01796e7	−1.68186	−0.840931	+	0.541142i	$0.817992\pi$
$978$	−660352.	−0.0220764
$979$	−2.26020e7	−0.753684
$980$	0	0
$981$	368979.	0.0122414
$982$	−1.85349e6	−0.0613355
$983$	1.44682e7	0.477563	0.238781	−	0.971073i	$-0.423252\pi$
$983$	1.44682e7	0.477563	0.238781	+	0.971073i	$0.423252\pi$
$984$	−2.93358e6	−0.0965852
$985$	0	0
$986$	−101481.	−0.00332425
$987$	−1.20267e6	−0.0392964
$988$	6.34394e6	0.206760
$989$	1.81743e7	0.590836
$990$	0	0
$991$	2.92271e7	0.945369	0.472685	−	0.881232i	$-0.343285\pi$
$991$	2.92271e7	0.945369	0.472685	+	0.881232i	$0.343285\pi$
$992$	9.51250e6	0.306913
$993$	2.59080e7	0.833799
$994$	−41586.6	−0.00133502
$995$	0	0
$996$	−3.15573e7	−1.00798
$997$	−3.35666e7	−1.06947	−0.534736	−	0.845019i	$-0.679589\pi$
$997$	−3.35666e7	−1.06947	−0.534736	+	0.845019i	$0.679589\pi$
$998$	4.04595e6	0.128586
$999$	−6.97972e6	−0.221271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.6.a.i.1.5	yes	9
5.2	odd	4		325.6.b.h.274.10		18
5.3	odd	4		325.6.b.h.274.9		18
5.4	even	2		325.6.a.h.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.6.a.h.1.5	✓	9	5.4	even	2
325.6.a.i.1.5	yes	9	1.1	even	1	trivial
325.6.b.h.274.9		18	5.3	odd	4
325.6.b.h.274.10		18	5.2	odd	4

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)

\(f(q)\)	\(=\)	\(q+0.396608 q^{2} -11.4974 q^{3} -31.8427 q^{4} -4.55994 q^{6} +8.04695 q^{7} -25.3205 q^{8} -110.811 q^{9} +335.903 q^{11} +366.107 q^{12} -169.000 q^{13} +3.19148 q^{14} +1008.92 q^{16} +38.1592 q^{17} -43.9484 q^{18} +1178.86 q^{19} -92.5186 q^{21} +133.222 q^{22} +1679.35 q^{23} +291.119 q^{24} -67.0267 q^{26} +4067.89 q^{27} -256.237 q^{28} -6705.42 q^{29} +7858.95 q^{31} +1210.40 q^{32} -3861.99 q^{33} +15.1342 q^{34} +3528.51 q^{36} -1715.81 q^{37} +467.545 q^{38} +1943.05 q^{39} -10076.9 q^{41} -36.6936 q^{42} +10822.3 q^{43} -10696.0 q^{44} +666.042 q^{46} +12999.2 q^{47} -11600.0 q^{48} -16742.2 q^{49} -438.730 q^{51} +5381.42 q^{52} +2639.19 q^{53} +1613.36 q^{54} -203.753 q^{56} -13553.8 q^{57} -2659.42 q^{58} +4956.81 q^{59} -43787.4 q^{61} +3116.92 q^{62} -891.687 q^{63} -31805.5 q^{64} -1531.70 q^{66} -9170.90 q^{67} -1215.09 q^{68} -19308.0 q^{69} -13030.5 q^{71} +2805.78 q^{72} -41726.8 q^{73} -680.503 q^{74} -37538.1 q^{76} +2702.99 q^{77} +770.630 q^{78} -49809.7 q^{79} -19843.0 q^{81} -3996.58 q^{82} -86196.9 q^{83} +2946.04 q^{84} +4292.19 q^{86} +77094.6 q^{87} -8505.22 q^{88} -67287.2 q^{89} -1359.93 q^{91} -53474.9 q^{92} -90357.2 q^{93} +5155.58 q^{94} -13916.4 q^{96} +175537. q^{97} -6640.10 q^{98} -37221.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{3} + 91 q^{4} - 83 q^{6} - 12 q^{7} + 639 q^{8} + 562 q^{9} - 1422 q^{11} - 1567 q^{12} - 1521 q^{13} - 342 q^{14} - 1061 q^{16} - 648 q^{17} - 418 q^{18} - 408 q^{19} - 3912 q^{21}+ \cdots - 757776 q^{99}+O(q^{100}) \) 9 * q + 5 * q^2 + 11 * q^3 + 91 * q^4 - 83 * q^6 - 12 * q^7 + 639 * q^8 + 562 * q^9 - 1422 * q^11 - 1567 * q^12 - 1521 * q^13 - 342 * q^14 - 1061 * q^16 - 648 * q^17 - 418 * q^18 - 408 * q^19 - 3912 * q^21 - 4345 * q^22 - 1839 * q^23 - 8469 * q^24 - 845 * q^26 + 7649 * q^27 + 2836 * q^28 - 8737 * q^29 + 748 * q^31 + 423 * q^32 + 356 * q^33 - 17789 * q^34 + 512 * q^36 + 15486 * q^37 + 3425 * q^38 - 1859 * q^39 - 28676 * q^41 - 6876 * q^42 - 28665 * q^43 - 30599 * q^44 - 12056 * q^46 + 29452 * q^47 - 64759 * q^48 - 40907 * q^49 - 31006 * q^51 - 15379 * q^52 - 75977 * q^53 - 102761 * q^54 - 23002 * q^56 + 38038 * q^57 - 142384 * q^58 - 88142 * q^59 + 28165 * q^61 + 137308 * q^62 - 41492 * q^63 - 100845 * q^64 + 42577 * q^66 + 94754 * q^67 - 89267 * q^68 - 181747 * q^69 - 70562 * q^71 + 263778 * q^72 - 60602 * q^73 - 135676 * q^74 + 46373 * q^76 + 140292 * q^77 + 14027 * q^78 - 164073 * q^79 - 69935 * q^81 + 72887 * q^82 + 22458 * q^83 - 345656 * q^84 - 294920 * q^86 + 87031 * q^87 - 430607 * q^88 - 252698 * q^89 + 2028 * q^91 + 237824 * q^92 - 56556 * q^93 - 501606 * q^94 - 319181 * q^96 - 137986 * q^97 - 378699 * q^98 - 757776 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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