LMFDB - Embedded newform 3330.2.a.bk.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.23544108.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 20x^{3} + 39x^{2} + 9x - 36 $ x^5 - x^4 - 20x^3 + 39x^2 + 9*x - 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.43959$ of defining polynomial
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.23800 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.01784 q^{11} -3.86299 q^{13} +4.23800 q^{14} +1.00000 q^{16} -6.23800 q^{17} -7.49220 q^{19} -1.00000 q^{20} +2.01784 q^{22} -5.86299 q^{23} +1.00000 q^{25} +3.86299 q^{26} -4.23800 q^{28} -0.375011 q^{29} +9.47436 q^{31} -1.00000 q^{32} +6.23800 q^{34} +4.23800 q^{35} +1.00000 q^{37} +7.49220 q^{38} +1.00000 q^{40} -6.72433 q^{41} +8.72433 q^{43} -2.01784 q^{44} +5.86299 q^{46} -11.3195 q^{47} +10.9607 q^{49} -1.00000 q^{50} -3.86299 q^{52} +0.861339 q^{53} +2.01784 q^{55} +4.23800 q^{56} +0.375011 q^{58} +4.03569 q^{59} +4.12247 q^{61} -9.47436 q^{62} +1.00000 q^{64} +3.86299 q^{65} +12.5117 q^{67} -6.23800 q^{68} -4.23800 q^{70} +14.1987 q^{71} -10.3714 q^{73} -1.00000 q^{74} -7.49220 q^{76} +8.55164 q^{77} -12.1987 q^{79} -1.00000 q^{80} +6.72433 q^{82} -1.82730 q^{83} +6.23800 q^{85} -8.72433 q^{86} +2.01784 q^{88} -13.0890 q^{89} +16.3714 q^{91} -5.86299 q^{92} +11.3195 q^{94} +7.49220 q^{95} +6.59517 q^{97} -10.9607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8} + 5 q^{10} - 5 q^{11} + 5 q^{13} - 3 q^{14} + 5 q^{16} - 7 q^{17} - q^{19} - 5 q^{20} + 5 q^{22} - 5 q^{23} + 5 q^{25} - 5 q^{26} + 3 q^{28} - 2 q^{29}+ \cdots - 16 q^{98}+O(q^{100}) $ 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8 + 5 * q^10 - 5 * q^11 + 5 * q^13 - 3 * q^14 + 5 * q^16 - 7 * q^17 - q^19 - 5 * q^20 + 5 * q^22 - 5 * q^23 + 5 * q^25 - 5 * q^26 + 3 * q^28 - 2 * q^29 + 16 * q^31 - 5 * q^32 + 7 * q^34 - 3 * q^35 + 5 * q^37 + q^38 + 5 * q^40 - 2 * q^41 + 12 * q^43 - 5 * q^44 + 5 * q^46 - 6 * q^47 + 16 * q^49 - 5 * q^50 + 5 * q^52 - 3 * q^53 + 5 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^59 + 16 * q^61 - 16 * q^62 + 5 * q^64 - 5 * q^65 + 4 * q^67 - 7 * q^68 + 3 * q^70 + 8 * q^71 - 3 * q^73 - 5 * q^74 - q^76 - 3 * q^77 + 2 * q^79 - 5 * q^80 + 2 * q^82 + 5 * q^83 + 7 * q^85 - 12 * q^86 + 5 * q^88 + 7 * q^89 + 33 * q^91 - 5 * q^92 + 6 * q^94 + q^95 + 14 * q^97 - 16 * q^98

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.23800	−1.60182	−0.800908	−	0.598788i	$-0.795649\pi$
$7$	−4.23800	−1.60182	−0.800908	+	0.598788i	$0.795649\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−2.01784	−0.608403	−0.304202	−	0.952608i	$-0.598390\pi$
$11$	−2.01784	−0.608403	−0.304202	+	0.952608i	$0.598390\pi$
$12$	0	0
$13$	−3.86299	−1.07140	−0.535701	−	0.844408i	$-0.679953\pi$
$13$	−3.86299	−1.07140	−0.535701	+	0.844408i	$0.679953\pi$
$14$	4.23800	1.13265
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.23800	−1.51294	−0.756469	−	0.654029i	$-0.773077\pi$
$17$	−6.23800	−1.51294	−0.756469	+	0.654029i	$0.773077\pi$
$18$	0	0
$19$	−7.49220	−1.71883	−0.859414	−	0.511280i	$-0.829172\pi$
$19$	−7.49220	−1.71883	−0.859414	+	0.511280i	$0.829172\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.01784	0.430206
$23$	−5.86299	−1.22252	−0.611259	−	0.791430i	$-0.709337\pi$
$23$	−5.86299	−1.22252	−0.611259	+	0.791430i	$0.709337\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.86299	0.757595
$27$	0	0
$28$	−4.23800	−0.800908
$29$	−0.375011	−0.0696378	−0.0348189	−	0.999394i	$-0.511085\pi$
$29$	−0.375011	−0.0696378	−0.0348189	+	0.999394i	$0.511085\pi$
$30$	0	0
$31$	9.47436	1.70164	0.850822	−	0.525454i	$-0.176104\pi$
$31$	9.47436	1.70164	0.850822	+	0.525454i	$0.176104\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.23800	1.06981
$35$	4.23800	0.716354
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	7.49220	1.21540
$39$	0	0
$40$	1.00000	0.158114
$41$	−6.72433	−1.05016	−0.525082	−	0.851052i	$-0.675965\pi$
$41$	−6.72433	−1.05016	−0.525082	+	0.851052i	$0.675965\pi$
$42$	0	0
$43$	8.72433	1.33045	0.665224	−	0.746644i	$-0.268336\pi$
$43$	8.72433	1.33045	0.665224	+	0.746644i	$0.268336\pi$
$44$	−2.01784	−0.304202
$45$	0	0
$46$	5.86299	0.864451
$47$	−11.3195	−1.65112	−0.825560	−	0.564315i	$-0.809140\pi$
$47$	−11.3195	−1.65112	−0.825560	+	0.564315i	$0.809140\pi$
$48$	0	0
$49$	10.9607	1.56581
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.86299	−0.535701
$53$	0.861339	0.118314	0.0591570	−	0.998249i	$-0.481159\pi$
$53$	0.861339	0.118314	0.0591570	+	0.998249i	$0.481159\pi$
$54$	0	0
$55$	2.01784	0.272086
$56$	4.23800	0.566327
$57$	0	0
$58$	0.375011	0.0492414
$59$	4.03569	0.525402	0.262701	−	0.964877i	$-0.415387\pi$
$59$	4.03569	0.525402	0.262701	+	0.964877i	$0.415387\pi$
$60$	0	0
$61$	4.12247	0.527828	0.263914	−	0.964546i	$-0.414986\pi$
$61$	4.12247	0.527828	0.263914	+	0.964546i	$0.414986\pi$
$62$	−9.47436	−1.20324
$63$	0	0
$64$	1.00000	0.125000
$65$	3.86299	0.479145
$66$	0	0
$67$	12.5117	1.52855	0.764274	−	0.644892i	$-0.223098\pi$
$67$	12.5117	1.52855	0.764274	+	0.644892i	$0.223098\pi$
$68$	−6.23800	−0.756469
$69$	0	0
$70$	−4.23800	−0.506538
$71$	14.1987	1.68507	0.842537	−	0.538638i	$-0.181061\pi$
$71$	14.1987	1.68507	0.842537	+	0.538638i	$0.181061\pi$
$72$	0	0
$73$	−10.3714	−1.21388	−0.606939	−	0.794748i	$-0.707603\pi$
$73$	−10.3714	−1.21388	−0.606939	+	0.794748i	$0.707603\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	−7.49220	−0.859414
$77$	8.55164	0.974549
$78$	0	0
$79$	−12.1987	−1.37246	−0.686230	−	0.727385i	$-0.740736\pi$
$79$	−12.1987	−1.37246	−0.686230	+	0.727385i	$0.740736\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	6.72433	0.742578
$83$	−1.82730	−0.200573	−0.100286	−	0.994959i	$-0.531976\pi$
$83$	−1.82730	−0.200573	−0.100286	+	0.994959i	$0.531976\pi$
$84$	0	0
$85$	6.23800	0.676607
$86$	−8.72433	−0.940769
$87$	0	0
$88$	2.01784	0.215103
$89$	−13.0890	−1.38743	−0.693717	−	0.720248i	$-0.744028\pi$
$89$	−13.0890	−1.38743	−0.693717	+	0.720248i	$0.744028\pi$
$90$	0	0
$91$	16.3714	1.71619
$92$	−5.86299	−0.611259
$93$	0	0
$94$	11.3195	1.16752
$95$	7.49220	0.768684
$96$	0	0
$97$	6.59517	0.669638	0.334819	−	0.942282i	$-0.391325\pi$
$97$	6.59517	0.669638	0.334819	+	0.942282i	$0.391325\pi$
$98$	−10.9607	−1.10720
$99$	0	0
$100$	1.00000	0.100000
$101$	−13.0422	−1.29775	−0.648873	−	0.760897i	$-0.724759\pi$
$101$	−13.0422	−1.29775	−0.648873	+	0.760897i	$0.724759\pi$
$102$	0	0
$103$	−9.31950	−0.918278	−0.459139	−	0.888364i	$-0.651842\pi$
$103$	−9.31950	−0.918278	−0.459139	+	0.888364i	$0.651842\pi$
$104$	3.86299	0.378798
$105$	0	0
$106$	−0.861339	−0.0836607
$107$	−5.11297	−0.494290	−0.247145	−	0.968979i	$-0.579492\pi$
$107$	−5.11297	−0.494290	−0.247145	+	0.968979i	$0.579492\pi$
$108$	0	0
$109$	14.1808	1.35828	0.679139	−	0.734009i	$-0.262353\pi$
$109$	14.1808	1.35828	0.679139	+	0.734009i	$0.262353\pi$
$110$	−2.01784	−0.192394
$111$	0	0
$112$	−4.23800	−0.400454
$113$	−3.66068	−0.344368	−0.172184	−	0.985065i	$-0.555082\pi$
$113$	−3.66068	−0.344368	−0.172184	+	0.985065i	$0.555082\pi$
$114$	0	0
$115$	5.86299	0.546727
$116$	−0.375011	−0.0348189
$117$	0	0
$118$	−4.03569	−0.371515
$119$	26.4367	2.42345
$120$	0	0
$121$	−6.92830	−0.629846
$122$	−4.12247	−0.373231
$123$	0	0
$124$	9.47436	0.850822
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.609707	−0.0541028	−0.0270514	−	0.999634i	$-0.508612\pi$
$127$	−0.609707	−0.0541028	−0.0270514	+	0.999634i	$0.508612\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.86299	−0.338807
$131$	20.9163	1.82747	0.913734	−	0.406313i	$-0.133186\pi$
$131$	20.9163	1.82747	0.913734	+	0.406313i	$0.133186\pi$
$132$	0	0
$133$	31.7520	2.75325
$134$	−12.5117	−1.08085
$135$	0	0
$136$	6.23800	0.534904
$137$	2.87918	0.245985	0.122993	−	0.992408i	$-0.460751\pi$
$137$	2.87918	0.245985	0.122993	+	0.992408i	$0.460751\pi$
$138$	0	0
$139$	−15.8594	−1.34517	−0.672587	−	0.740018i	$-0.734817\pi$
$139$	−15.8594	−1.34517	−0.672587	+	0.740018i	$0.734817\pi$
$140$	4.23800	0.358177
$141$	0	0
$142$	−14.1987	−1.19153
$143$	7.79492	0.651844
$144$	0	0
$145$	0.375011	0.0311430
$146$	10.3714	0.858342
$147$	0	0
$148$	1.00000	0.0821995
$149$	−5.35189	−0.438444	−0.219222	−	0.975675i	$-0.570352\pi$
$149$	−5.35189	−0.438444	−0.219222	+	0.975675i	$0.570352\pi$
$150$	0	0
$151$	17.0857	1.39042	0.695208	−	0.718809i	$-0.255312\pi$
$151$	17.0857	1.39042	0.695208	+	0.718809i	$0.255312\pi$
$152$	7.49220	0.607698
$153$	0	0
$154$	−8.55164	−0.689110
$155$	−9.47436	−0.760999
$156$	0	0
$157$	1.16072	0.0926358	0.0463179	−	0.998927i	$-0.485251\pi$
$157$	1.16072	0.0926358	0.0463179	+	0.998927i	$0.485251\pi$
$158$	12.1987	0.970475
$159$	0	0
$160$	1.00000	0.0790569
$161$	24.8474	1.95825
$162$	0	0
$163$	9.36973	0.733894	0.366947	−	0.930242i	$-0.380403\pi$
$163$	9.36973	0.733894	0.366947	+	0.930242i	$0.380403\pi$
$164$	−6.72433	−0.525082
$165$	0	0
$166$	1.82730	0.141826
$167$	5.86299	0.453692	0.226846	−	0.973931i	$-0.427159\pi$
$167$	5.86299	0.453692	0.226846	+	0.973931i	$0.427159\pi$
$168$	0	0
$169$	1.92272	0.147901
$170$	−6.23800	−0.478433
$171$	0	0
$172$	8.72433	0.665224
$173$	−2.36138	−0.179533	−0.0897663	−	0.995963i	$-0.528612\pi$
$173$	−2.36138	−0.179533	−0.0897663	+	0.995963i	$0.528612\pi$
$174$	0	0
$175$	−4.23800	−0.320363
$176$	−2.01784	−0.152101
$177$	0	0
$178$	13.0890	0.981064
$179$	−13.9247	−1.04078	−0.520390	−	0.853929i	$-0.674213\pi$
$179$	−13.9247	−1.04078	−0.520390	+	0.853929i	$0.674213\pi$
$180$	0	0
$181$	−2.51170	−0.186693	−0.0933466	−	0.995634i	$-0.529756\pi$
$181$	−2.51170	−0.186693	−0.0933466	+	0.995634i	$0.529756\pi$
$182$	−16.3714	−1.21353
$183$	0	0
$184$	5.86299	0.432226
$185$	−1.00000	−0.0735215
$186$	0	0
$187$	12.5873	0.920476
$188$	−11.3195	−0.825560
$189$	0	0
$190$	−7.49220	−0.543541
$191$	25.6867	1.85862	0.929311	−	0.369298i	$-0.120402\pi$
$191$	25.6867	1.85862	0.929311	+	0.369298i	$0.120402\pi$
$192$	0	0
$193$	−6.03384	−0.434325	−0.217163	−	0.976135i	$-0.569680\pi$
$193$	−6.03384	−0.434325	−0.217163	+	0.976135i	$0.569680\pi$
$194$	−6.59517	−0.473506
$195$	0	0
$196$	10.9607	0.782906
$197$	−22.3747	−1.59413	−0.797065	−	0.603893i	$-0.793615\pi$
$197$	−22.3747	−1.59413	−0.797065	+	0.603893i	$0.793615\pi$
$198$	0	0
$199$	−18.1663	−1.28778	−0.643888	−	0.765120i	$-0.722680\pi$
$199$	−18.1663	−1.28778	−0.643888	+	0.765120i	$0.722680\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	13.0422	0.917645
$203$	1.58930	0.111547
$204$	0	0
$205$	6.72433	0.469648
$206$	9.31950	0.649321
$207$	0	0
$208$	−3.86299	−0.267850
$209$	15.1181	1.04574
$210$	0	0
$211$	−17.3594	−1.19507	−0.597536	−	0.801842i	$-0.703853\pi$
$211$	−17.3594	−1.19507	−0.597536	+	0.801842i	$0.703853\pi$
$212$	0.861339	0.0591570
$213$	0	0
$214$	5.11297	0.349516
$215$	−8.72433	−0.594995
$216$	0	0
$217$	−40.1524	−2.72572
$218$	−14.1808	−0.960448
$219$	0	0
$220$	2.01784	0.136043
$221$	24.0974	1.62096
$222$	0	0
$223$	0.652333	0.0436834	0.0218417	−	0.999761i	$-0.493047\pi$
$223$	0.652333	0.0436834	0.0218417	+	0.999761i	$0.493047\pi$
$224$	4.23800	0.283164
$225$	0	0
$226$	3.66068	0.243505
$227$	−10.1367	−0.672796	−0.336398	−	0.941720i	$-0.609209\pi$
$227$	−10.1367	−0.672796	−0.336398	+	0.941720i	$0.609209\pi$
$228$	0	0
$229$	−26.1747	−1.72968	−0.864838	−	0.502051i	$-0.832579\pi$
$229$	−26.1747	−1.72968	−0.864838	+	0.502051i	$0.832579\pi$
$230$	−5.86299	−0.386594
$231$	0	0
$232$	0.375011	0.0246207
$233$	−27.0812	−1.77415	−0.887074	−	0.461628i	$-0.847265\pi$
$233$	−27.0812	−1.77415	−0.887074	+	0.461628i	$0.847265\pi$
$234$	0	0
$235$	11.3195	0.738403
$236$	4.03569	0.262701
$237$	0	0
$238$	−26.4367	−1.71364
$239$	12.3426	0.798378	0.399189	−	0.916869i	$-0.369292\pi$
$239$	12.3426	0.798378	0.399189	+	0.916869i	$0.369292\pi$
$240$	0	0
$241$	12.5000	0.805193	0.402596	−	0.915378i	$-0.368108\pi$
$241$	12.5000	0.805193	0.402596	+	0.915378i	$0.368108\pi$
$242$	6.92830	0.445368
$243$	0	0
$244$	4.12247	0.263914
$245$	−10.9607	−0.700252
$246$	0	0
$247$	28.9423	1.84156
$248$	−9.47436	−0.601622
$249$	0	0
$250$	1.00000	0.0632456
$251$	−9.41298	−0.594142	−0.297071	−	0.954855i	$-0.596010\pi$
$251$	−9.41298	−0.594142	−0.297071	+	0.954855i	$0.596010\pi$
$252$	0	0
$253$	11.8306	0.743784
$254$	0.609707	0.0382564
$255$	0	0
$256$	1.00000	0.0625000
$257$	−6.26431	−0.390757	−0.195379	−	0.980728i	$-0.562594\pi$
$257$	−6.26431	−0.390757	−0.195379	+	0.980728i	$0.562594\pi$
$258$	0	0
$259$	−4.23800	−0.263337
$260$	3.86299	0.239573
$261$	0	0
$262$	−20.9163	−1.29222
$263$	25.6392	1.58098	0.790491	−	0.612474i	$-0.209826\pi$
$263$	25.6392	1.58098	0.790491	+	0.612474i	$0.209826\pi$
$264$	0	0
$265$	−0.861339	−0.0529117
$266$	−31.7520	−1.94684
$267$	0	0
$268$	12.5117	0.764274
$269$	−6.74549	−0.411280	−0.205640	−	0.978628i	$-0.565928\pi$
$269$	−6.74549	−0.411280	−0.205640	+	0.978628i	$0.565928\pi$
$270$	0	0
$271$	−14.6390	−0.889256	−0.444628	−	0.895715i	$-0.646664\pi$
$271$	−14.6390	−0.889256	−0.444628	+	0.895715i	$0.646664\pi$
$272$	−6.23800	−0.378235
$273$	0	0
$274$	−2.87918	−0.173938
$275$	−2.01784	−0.121681
$276$	0	0
$277$	17.0533	1.02464	0.512318	−	0.858796i	$-0.328787\pi$
$277$	17.0533	1.02464	0.512318	+	0.858796i	$0.328787\pi$
$278$	15.8594	0.951182
$279$	0	0
$280$	−4.23800	−0.253269
$281$	26.0617	1.55471	0.777355	−	0.629062i	$-0.216561\pi$
$281$	26.0617	1.55471	0.777355	+	0.629062i	$0.216561\pi$
$282$	0	0
$283$	15.1214	0.898874	0.449437	−	0.893312i	$-0.351625\pi$
$283$	15.1214	0.898874	0.449437	+	0.893312i	$0.351625\pi$
$284$	14.1987	0.842537
$285$	0	0
$286$	−7.79492	−0.460923
$287$	28.4978	1.68217
$288$	0	0
$289$	21.9127	1.28898
$290$	−0.375011	−0.0220214
$291$	0	0
$292$	−10.3714	−0.606939
$293$	−6.82896	−0.398952	−0.199476	−	0.979903i	$-0.563924\pi$
$293$	−6.82896	−0.398952	−0.199476	+	0.979903i	$0.563924\pi$
$294$	0	0
$295$	−4.03569	−0.234967
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	5.35189	0.310026
$299$	22.6487	1.30981
$300$	0	0
$301$	−36.9738	−2.13113
$302$	−17.0857	−0.983173
$303$	0	0
$304$	−7.49220	−0.429707
$305$	−4.12247	−0.236052
$306$	0	0
$307$	−8.03569	−0.458621	−0.229311	−	0.973353i	$-0.573647\pi$
$307$	−8.03569	−0.458621	−0.229311	+	0.973353i	$0.573647\pi$
$308$	8.55164	0.487275
$309$	0	0
$310$	9.47436	0.538107
$311$	−25.5614	−1.44945	−0.724727	−	0.689036i	$-0.758034\pi$
$311$	−25.5614	−1.44945	−0.724727	+	0.689036i	$0.758034\pi$
$312$	0	0
$313$	20.4182	1.15411	0.577053	−	0.816707i	$-0.304203\pi$
$313$	20.4182	1.15411	0.577053	+	0.816707i	$0.304203\pi$
$314$	−1.16072	−0.0655034
$315$	0	0
$316$	−12.1987	−0.686230
$317$	−0.280705	−0.0157660	−0.00788298	−	0.999969i	$-0.502509\pi$
$317$	−0.280705	−0.0157660	−0.00788298	+	0.999969i	$0.502509\pi$
$318$	0	0
$319$	0.756714	0.0423679
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−24.8474	−1.38469
$323$	46.7364	2.60048
$324$	0	0
$325$	−3.86299	−0.214280
$326$	−9.36973	−0.518941
$327$	0	0
$328$	6.72433	0.371289
$329$	47.9721	2.64479
$330$	0	0
$331$	−1.85184	−0.101786	−0.0508932	−	0.998704i	$-0.516207\pi$
$331$	−1.85184	−0.101786	−0.0508932	+	0.998704i	$0.516207\pi$
$332$	−1.82730	−0.100286
$333$	0	0
$334$	−5.86299	−0.320809
$335$	−12.5117	−0.683587
$336$	0	0
$337$	−13.4344	−0.731819	−0.365910	−	0.930650i	$-0.619242\pi$
$337$	−13.4344	−0.731819	−0.365910	+	0.930650i	$0.619242\pi$
$338$	−1.92272	−0.104582
$339$	0	0
$340$	6.23800	0.338303
$341$	−19.1178	−1.03529
$342$	0	0
$343$	−16.7854	−0.906326
$344$	−8.72433	−0.470384
$345$	0	0
$346$	2.36138	0.126949
$347$	−13.2260	−0.710011	−0.355005	−	0.934864i	$-0.615521\pi$
$347$	−13.2260	−0.710011	−0.355005	+	0.934864i	$0.615521\pi$
$348$	0	0
$349$	34.1591	1.82849	0.914246	−	0.405159i	$-0.132784\pi$
$349$	34.1591	1.82849	0.914246	+	0.405159i	$0.132784\pi$
$350$	4.23800	0.226531
$351$	0	0
$352$	2.01784	0.107551
$353$	9.08934	0.483777	0.241888	−	0.970304i	$-0.422233\pi$
$353$	9.08934	0.483777	0.241888	+	0.970304i	$0.422233\pi$
$354$	0	0
$355$	−14.1987	−0.753588
$356$	−13.0890	−0.693717
$357$	0	0
$358$	13.9247	0.735942
$359$	19.3214	1.01974	0.509871	−	0.860251i	$-0.329693\pi$
$359$	19.3214	1.01974	0.509871	+	0.860251i	$0.329693\pi$
$360$	0	0
$361$	37.1331	1.95437
$362$	2.51170	0.132012
$363$	0	0
$364$	16.3714	0.858094
$365$	10.3714	0.542863
$366$	0	0
$367$	3.77043	0.196815	0.0984074	−	0.995146i	$-0.468625\pi$
$367$	3.77043	0.196815	0.0984074	+	0.995146i	$0.468625\pi$
$368$	−5.86299	−0.305630
$369$	0	0
$370$	1.00000	0.0519875
$371$	−3.65036	−0.189517
$372$	0	0
$373$	−12.0390	−0.623356	−0.311678	−	0.950188i	$-0.600891\pi$
$373$	−12.0390	−0.623356	−0.311678	+	0.950188i	$0.600891\pi$
$374$	−12.5873	−0.650875
$375$	0	0
$376$	11.3195	0.583759
$377$	1.44867	0.0746101
$378$	0	0
$379$	38.1624	1.96027	0.980134	−	0.198334i	$-0.0635532\pi$
$379$	38.1624	1.96027	0.980134	+	0.198334i	$0.0635532\pi$
$380$	7.49220	0.384342
$381$	0	0
$382$	−25.6867	−1.31424
$383$	33.1280	1.69276	0.846381	−	0.532577i	$-0.178776\pi$
$383$	33.1280	1.69276	0.846381	+	0.532577i	$0.178776\pi$
$384$	0	0
$385$	−8.55164	−0.435832
$386$	6.03384	0.307114
$387$	0	0
$388$	6.59517	0.334819
$389$	−13.1178	−0.665098	−0.332549	−	0.943086i	$-0.607909\pi$
$389$	−13.1178	−0.665098	−0.332549	+	0.943086i	$0.607909\pi$
$390$	0	0
$391$	36.5734	1.84960
$392$	−10.9607	−0.553598
$393$	0	0
$394$	22.3747	1.12722
$395$	12.1987	0.613783
$396$	0	0
$397$	−2.74159	−0.137596	−0.0687982	−	0.997631i	$-0.521916\pi$
$397$	−2.74159	−0.137596	−0.0687982	+	0.997631i	$0.521916\pi$
$398$	18.1663	0.910595
$399$	0	0
$400$	1.00000	0.0500000
$401$	−10.3423	−0.516470	−0.258235	−	0.966082i	$-0.583141\pi$
$401$	−10.3423	−0.516470	−0.258235	+	0.966082i	$0.583141\pi$
$402$	0	0
$403$	−36.5994	−1.82314
$404$	−13.0422	−0.648873
$405$	0	0
$406$	−1.58930	−0.0788756
$407$	−2.01784	−0.100021
$408$	0	0
$409$	−7.68699	−0.380097	−0.190049	−	0.981775i	$-0.560865\pi$
$409$	−7.68699	−0.380097	−0.190049	+	0.981775i	$0.560865\pi$
$410$	−6.72433	−0.332091
$411$	0	0
$412$	−9.31950	−0.459139
$413$	−17.1033	−0.841597
$414$	0	0
$415$	1.82730	0.0896988
$416$	3.86299	0.189399
$417$	0	0
$418$	−15.1181	−0.739450
$419$	6.70318	0.327472	0.163736	−	0.986504i	$-0.447646\pi$
$419$	6.70318	0.327472	0.163736	+	0.986504i	$0.447646\pi$
$420$	0	0
$421$	11.1208	0.541995	0.270998	−	0.962580i	$-0.412646\pi$
$421$	11.1208	0.541995	0.270998	+	0.962580i	$0.412646\pi$
$422$	17.3594	0.845043
$423$	0	0
$424$	−0.861339	−0.0418303
$425$	−6.23800	−0.302588
$426$	0	0
$427$	−17.4710	−0.845483
$428$	−5.11297	−0.247145
$429$	0	0
$430$	8.72433	0.420725
$431$	−26.7035	−1.28626	−0.643130	−	0.765757i	$-0.722365\pi$
$431$	−26.7035	−1.28626	−0.643130	+	0.765757i	$0.722365\pi$
$432$	0	0
$433$	−3.38698	−0.162768	−0.0813840	−	0.996683i	$-0.525934\pi$
$433$	−3.38698	−0.162768	−0.0813840	+	0.996683i	$0.525934\pi$
$434$	40.1524	1.92737
$435$	0	0
$436$	14.1808	0.679139
$437$	43.9267	2.10130
$438$	0	0
$439$	−27.5847	−1.31655	−0.658274	−	0.752779i	$-0.728713\pi$
$439$	−27.5847	−1.31655	−0.658274	+	0.752779i	$0.728713\pi$
$440$	−2.01784	−0.0961970
$441$	0	0
$442$	−24.0974	−1.14620
$443$	−15.6474	−0.743428	−0.371714	−	0.928347i	$-0.621230\pi$
$443$	−15.6474	−0.743428	−0.371714	+	0.928347i	$0.621230\pi$
$444$	0	0
$445$	13.0890	0.620479
$446$	−0.652333	−0.0308888
$447$	0	0
$448$	−4.23800	−0.200227
$449$	13.6903	0.646085	0.323043	−	0.946384i	$-0.395294\pi$
$449$	13.6903	0.646085	0.323043	+	0.946384i	$0.395294\pi$
$450$	0	0
$451$	13.5687	0.638923
$452$	−3.66068	−0.172184
$453$	0	0
$454$	10.1367	0.475738
$455$	−16.3714	−0.767502
$456$	0	0
$457$	4.66324	0.218137	0.109069	−	0.994034i	$-0.465213\pi$
$457$	4.66324	0.218137	0.109069	+	0.994034i	$0.465213\pi$
$458$	26.1747	1.22307
$459$	0	0
$460$	5.86299	0.273363
$461$	22.8048	1.06212	0.531062	−	0.847333i	$-0.321793\pi$
$461$	22.8048	1.06212	0.531062	+	0.847333i	$0.321793\pi$
$462$	0	0
$463$	15.8551	0.736851	0.368426	−	0.929657i	$-0.379897\pi$
$463$	15.8551	0.736851	0.368426	+	0.929657i	$0.379897\pi$
$464$	−0.375011	−0.0174095
$465$	0	0
$466$	27.0812	1.25451
$467$	27.2841	1.26256	0.631278	−	0.775556i	$-0.282531\pi$
$467$	27.2841	1.26256	0.631278	+	0.775556i	$0.282531\pi$
$468$	0	0
$469$	−53.0246	−2.44845
$470$	−11.3195	−0.522130
$471$	0	0
$472$	−4.03569	−0.185758
$473$	−17.6044	−0.809449
$474$	0	0
$475$	−7.49220	−0.343766
$476$	26.4367	1.21172
$477$	0	0
$478$	−12.3426	−0.564539
$479$	38.3033	1.75012	0.875062	−	0.484012i	$-0.160821\pi$
$479$	38.3033	1.75012	0.875062	+	0.484012i	$0.160821\pi$
$480$	0	0
$481$	−3.86299	−0.176137
$482$	−12.5000	−0.569357
$483$	0	0
$484$	−6.92830	−0.314923
$485$	−6.59517	−0.299471
$486$	0	0
$487$	7.66820	0.347480	0.173740	−	0.984792i	$-0.444415\pi$
$487$	7.66820	0.347480	0.173740	+	0.984792i	$0.444415\pi$
$488$	−4.12247	−0.186615
$489$	0	0
$490$	10.9607	0.495153
$491$	−42.1786	−1.90350	−0.951748	−	0.306882i	$-0.900714\pi$
$491$	−42.1786	−1.90350	−0.951748	+	0.306882i	$0.900714\pi$
$492$	0	0
$493$	2.33932	0.105358
$494$	−28.9423	−1.30218
$495$	0	0
$496$	9.47436	0.425411
$497$	−60.1741	−2.69918
$498$	0	0
$499$	−4.70318	−0.210543	−0.105272	−	0.994444i	$-0.533571\pi$
$499$	−4.70318	−0.210543	−0.105272	+	0.994444i	$0.533571\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	9.41298	0.420122
$503$	25.7650	1.14880	0.574402	−	0.818573i	$-0.305235\pi$
$503$	25.7650	1.14880	0.574402	+	0.818573i	$0.305235\pi$
$504$	0	0
$505$	13.0422	0.580369
$506$	−11.8306	−0.525935
$507$	0	0
$508$	−0.609707	−0.0270514
$509$	9.86114	0.437087	0.218544	−	0.975827i	$-0.429869\pi$
$509$	9.86114	0.437087	0.218544	+	0.975827i	$0.429869\pi$
$510$	0	0
$511$	43.9540	1.94441
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	6.26431	0.276307
$515$	9.31950	0.410666
$516$	0	0
$517$	22.8410	1.00455
$518$	4.23800	0.186207
$519$	0	0
$520$	−3.86299	−0.169403
$521$	−19.1374	−0.838425	−0.419212	−	0.907888i	$-0.637694\pi$
$521$	−19.1374	−0.838425	−0.419212	+	0.907888i	$0.637694\pi$
$522$	0	0
$523$	−31.9313	−1.39626	−0.698129	−	0.715972i	$-0.745984\pi$
$523$	−31.9313	−1.39626	−0.698129	+	0.715972i	$0.745984\pi$
$524$	20.9163	0.913734
$525$	0	0
$526$	−25.6392	−1.11792
$527$	−59.1011	−2.57448
$528$	0	0
$529$	11.3747	0.494552
$530$	0.861339	0.0374142
$531$	0	0
$532$	31.7520	1.37662
$533$	25.9761	1.12515
$534$	0	0
$535$	5.11297	0.221053
$536$	−12.5117	−0.540423
$537$	0	0
$538$	6.74549	0.290819
$539$	−22.1170	−0.952645
$540$	0	0
$541$	−16.6825	−0.717238	−0.358619	−	0.933484i	$-0.616752\pi$
$541$	−16.6825	−0.717238	−0.358619	+	0.933484i	$0.616752\pi$
$542$	14.6390	0.628799
$543$	0	0
$544$	6.23800	0.267452
$545$	−14.1808	−0.607441
$546$	0	0
$547$	15.5600	0.665297	0.332648	−	0.943051i	$-0.392058\pi$
$547$	15.5600	0.665297	0.332648	+	0.943051i	$0.392058\pi$
$548$	2.87918	0.122993
$549$	0	0
$550$	2.01784	0.0860412
$551$	2.80966	0.119695
$552$	0	0
$553$	51.6981	2.19843
$554$	−17.0533	−0.724527
$555$	0	0
$556$	−15.8594	−0.672587
$557$	−30.6986	−1.30074	−0.650372	−	0.759616i	$-0.725387\pi$
$557$	−30.6986	−1.30074	−0.650372	+	0.759616i	$0.725387\pi$
$558$	0	0
$559$	−33.7020	−1.42544
$560$	4.23800	0.179088
$561$	0	0
$562$	−26.0617	−1.09935
$563$	−15.0173	−0.632905	−0.316452	−	0.948608i	$-0.602492\pi$
$563$	−15.0173	−0.632905	−0.316452	+	0.948608i	$0.602492\pi$
$564$	0	0
$565$	3.66068	0.154006
$566$	−15.1214	−0.635600
$567$	0	0
$568$	−14.1987	−0.595764
$569$	−16.8084	−0.704645	−0.352322	−	0.935879i	$-0.614608\pi$
$569$	−16.8084	−0.704645	−0.352322	+	0.935879i	$0.614608\pi$
$570$	0	0
$571$	40.0050	1.67416	0.837079	−	0.547082i	$-0.184261\pi$
$571$	40.0050	1.67416	0.837079	+	0.547082i	$0.184261\pi$
$572$	7.79492	0.325922
$573$	0	0
$574$	−28.4978	−1.18947
$575$	−5.86299	−0.244504
$576$	0	0
$577$	−36.7442	−1.52968	−0.764841	−	0.644219i	$-0.777182\pi$
$577$	−36.7442	−1.52968	−0.764841	+	0.644219i	$0.777182\pi$
$578$	−21.9127	−0.911448
$579$	0	0
$580$	0.375011	0.0155715
$581$	7.74412	0.321280
$582$	0	0
$583$	−1.73805	−0.0719826
$584$	10.3714	0.429171
$585$	0	0
$586$	6.82896	0.282102
$587$	−7.28472	−0.300673	−0.150336	−	0.988635i	$-0.548036\pi$
$587$	−7.28472	−0.300673	−0.150336	+	0.988635i	$0.548036\pi$
$588$	0	0
$589$	−70.9838	−2.92484
$590$	4.03569	0.166147
$591$	0	0
$592$	1.00000	0.0410997
$593$	31.0025	1.27312	0.636561	−	0.771226i	$-0.280356\pi$
$593$	31.0025	1.27312	0.636561	+	0.771226i	$0.280356\pi$
$594$	0	0
$595$	−26.4367	−1.08380
$596$	−5.35189	−0.219222
$597$	0	0
$598$	−22.6487	−0.926174
$599$	16.5117	0.674650	0.337325	−	0.941388i	$-0.390478\pi$
$599$	16.5117	0.674650	0.337325	+	0.941388i	$0.390478\pi$
$600$	0	0
$601$	−13.9997	−0.571059	−0.285529	−	0.958370i	$-0.592169\pi$
$601$	−13.9997	−0.571059	−0.285529	+	0.958370i	$0.592169\pi$
$602$	36.9738	1.50694
$603$	0	0
$604$	17.0857	0.695208
$605$	6.92830	0.281676
$606$	0	0
$607$	−36.5422	−1.48320	−0.741602	−	0.670841i	$-0.765933\pi$
$607$	−36.5422	−1.48320	−0.741602	+	0.670841i	$0.765933\pi$
$608$	7.49220	0.303849
$609$	0	0
$610$	4.12247	0.166914
$611$	43.7272	1.76901
$612$	0	0
$613$	5.90231	0.238392	0.119196	−	0.992871i	$-0.461968\pi$
$613$	5.90231	0.238392	0.119196	+	0.992871i	$0.461968\pi$
$614$	8.03569	0.324294
$615$	0	0
$616$	−8.55164	−0.344555
$617$	−19.4115	−0.781478	−0.390739	−	0.920501i	$-0.627781\pi$
$617$	−19.4115	−0.781478	−0.390739	+	0.920501i	$0.627781\pi$
$618$	0	0
$619$	−28.6723	−1.15244	−0.576219	−	0.817295i	$-0.695473\pi$
$619$	−28.6723	−1.15244	−0.576219	+	0.817295i	$0.695473\pi$
$620$	−9.47436	−0.380499
$621$	0	0
$622$	25.5614	1.02492
$623$	55.4713	2.22241
$624$	0	0
$625$	1.00000	0.0400000
$626$	−20.4182	−0.816076
$627$	0	0
$628$	1.16072	0.0463179
$629$	−6.23800	−0.248726
$630$	0	0
$631$	−17.5167	−0.697327	−0.348664	−	0.937248i	$-0.613364\pi$
$631$	−17.5167	−0.697327	−0.348664	+	0.937248i	$0.613364\pi$
$632$	12.1987	0.485238
$633$	0	0
$634$	0.280705	0.0111482
$635$	0.609707	0.0241955
$636$	0	0
$637$	−42.3410	−1.67761
$638$	−0.756714	−0.0299586
$639$	0	0
$640$	1.00000	0.0395285
$641$	−29.7444	−1.17483	−0.587417	−	0.809284i	$-0.699855\pi$
$641$	−29.7444	−1.17483	−0.587417	+	0.809284i	$0.699855\pi$
$642$	0	0
$643$	16.3541	0.644944	0.322472	−	0.946579i	$-0.395486\pi$
$643$	16.3541	0.644944	0.322472	+	0.946579i	$0.395486\pi$
$644$	24.8474	0.979124
$645$	0	0
$646$	−46.7364	−1.83882
$647$	−9.67605	−0.380405	−0.190202	−	0.981745i	$-0.560914\pi$
$647$	−9.67605	−0.380405	−0.190202	+	0.981745i	$0.560914\pi$
$648$	0	0
$649$	−8.14340	−0.319656
$650$	3.86299	0.151519
$651$	0	0
$652$	9.36973	0.366947
$653$	30.9006	1.20924	0.604618	−	0.796516i	$-0.293326\pi$
$653$	30.9006	1.20924	0.604618	+	0.796516i	$0.293326\pi$
$654$	0	0
$655$	−20.9163	−0.817269
$656$	−6.72433	−0.262541
$657$	0	0
$658$	−47.9721	−1.87015
$659$	30.5779	1.19115	0.595573	−	0.803301i	$-0.296925\pi$
$659$	30.5779	1.19115	0.595573	+	0.803301i	$0.296925\pi$
$660$	0	0
$661$	13.4749	0.524115	0.262057	−	0.965052i	$-0.415599\pi$
$661$	13.4749	0.524115	0.262057	+	0.965052i	$0.415599\pi$
$662$	1.85184	0.0719738
$663$	0	0
$664$	1.82730	0.0709131
$665$	−31.7520	−1.23129
$666$	0	0
$667$	2.19869	0.0851335
$668$	5.86299	0.226846
$669$	0	0
$670$	12.5117	0.483369
$671$	−8.31851	−0.321132
$672$	0	0
$673$	−21.4937	−0.828519	−0.414260	−	0.910159i	$-0.635959\pi$
$673$	−21.4937	−0.828519	−0.414260	+	0.910159i	$0.635959\pi$
$674$	13.4344	0.517474
$675$	0	0
$676$	1.92272	0.0739507
$677$	27.9981	1.07605	0.538027	−	0.842927i	$-0.319170\pi$
$677$	27.9981	1.07605	0.538027	+	0.842927i	$0.319170\pi$
$678$	0	0
$679$	−27.9504	−1.07264
$680$	−6.23800	−0.239217
$681$	0	0
$682$	19.1178	0.732058
$683$	−20.6366	−0.789639	−0.394820	−	0.918759i	$-0.629193\pi$
$683$	−20.6366	−0.789639	−0.394820	+	0.918759i	$0.629193\pi$
$684$	0	0
$685$	−2.87918	−0.110008
$686$	16.7854	0.640869
$687$	0	0
$688$	8.72433	0.332612
$689$	−3.32735	−0.126762
$690$	0	0
$691$	33.2874	1.26631	0.633156	−	0.774024i	$-0.281759\pi$
$691$	33.2874	1.26631	0.633156	+	0.774024i	$0.281759\pi$
$692$	−2.36138	−0.0897663
$693$	0	0
$694$	13.2260	0.502053
$695$	15.8594	0.601580
$696$	0	0
$697$	41.9464	1.58883
$698$	−34.1591	−1.29294
$699$	0	0
$700$	−4.23800	−0.160182
$701$	3.29741	0.124542	0.0622708	−	0.998059i	$-0.480166\pi$
$701$	3.29741	0.124542	0.0622708	+	0.998059i	$0.480166\pi$
$702$	0	0
$703$	−7.49220	−0.282574
$704$	−2.01784	−0.0760504
$705$	0	0
$706$	−9.08934	−0.342082
$707$	55.2728	2.07875
$708$	0	0
$709$	−27.8744	−1.04685	−0.523423	−	0.852073i	$-0.675345\pi$
$709$	−27.8744	−1.04685	−0.523423	+	0.852073i	$0.675345\pi$
$710$	14.1987	0.532867
$711$	0	0
$712$	13.0890	0.490532
$713$	−55.5481	−2.08029
$714$	0	0
$715$	−7.79492	−0.291514
$716$	−13.9247	−0.520390
$717$	0	0
$718$	−19.3214	−0.721067
$719$	34.0982	1.27165	0.635825	−	0.771833i	$-0.280660\pi$
$719$	34.0982	1.27165	0.635825	+	0.771833i	$0.280660\pi$
$720$	0	0
$721$	39.4961	1.47091
$722$	−37.1331	−1.38195
$723$	0	0
$724$	−2.51170	−0.0933466
$725$	−0.375011	−0.0139276
$726$	0	0
$727$	−33.3122	−1.23548	−0.617741	−	0.786381i	$-0.711952\pi$
$727$	−33.3122	−1.23548	−0.617741	+	0.786381i	$0.711952\pi$
$728$	−16.3714	−0.606764
$729$	0	0
$730$	−10.3714	−0.383862
$731$	−54.4224	−2.01289
$732$	0	0
$733$	−17.5569	−0.648480	−0.324240	−	0.945975i	$-0.605108\pi$
$733$	−17.5569	−0.648480	−0.324240	+	0.945975i	$0.605108\pi$
$734$	−3.77043	−0.139169
$735$	0	0
$736$	5.86299	0.216113
$737$	−25.2467	−0.929973
$738$	0	0
$739$	−18.4464	−0.678561	−0.339281	−	0.940685i	$-0.610184\pi$
$739$	−18.4464	−0.678561	−0.339281	+	0.940685i	$0.610184\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	3.65036	0.134009
$743$	19.0679	0.699532	0.349766	−	0.936837i	$-0.386261\pi$
$743$	19.0679	0.699532	0.349766	+	0.936837i	$0.386261\pi$
$744$	0	0
$745$	5.35189	0.196078
$746$	12.0390	0.440779
$747$	0	0
$748$	12.5873	0.460238
$749$	21.6688	0.791761
$750$	0	0
$751$	−35.9571	−1.31209	−0.656046	−	0.754721i	$-0.727772\pi$
$751$	−35.9571	−1.31209	−0.656046	+	0.754721i	$0.727772\pi$
$752$	−11.3195	−0.412780
$753$	0	0
$754$	−1.44867	−0.0527573
$755$	−17.0857	−0.621813
$756$	0	0
$757$	−22.5940	−0.821194	−0.410597	−	0.911817i	$-0.634680\pi$
$757$	−22.5940	−0.821194	−0.410597	+	0.911817i	$0.634680\pi$
$758$	−38.1624	−1.38612
$759$	0	0
$760$	−7.49220	−0.271771
$761$	25.4347	0.922008	0.461004	−	0.887398i	$-0.347489\pi$
$761$	25.4347	0.922008	0.461004	+	0.887398i	$0.347489\pi$
$762$	0	0
$763$	−60.0985	−2.17571
$764$	25.6867	0.929311
$765$	0	0
$766$	−33.1280	−1.19696
$767$	−15.5898	−0.562917
$768$	0	0
$769$	−2.20263	−0.0794290	−0.0397145	−	0.999211i	$-0.512645\pi$
$769$	−2.20263	−0.0794290	−0.0397145	+	0.999211i	$0.512645\pi$
$770$	8.55164	0.308180
$771$	0	0
$772$	−6.03384	−0.217163
$773$	20.5939	0.740713	0.370356	−	0.928890i	$-0.379236\pi$
$773$	20.5939	0.740713	0.370356	+	0.928890i	$0.379236\pi$
$774$	0	0
$775$	9.47436	0.340329
$776$	−6.59517	−0.236753
$777$	0	0
$778$	13.1178	0.470295
$779$	50.3800	1.80505
$780$	0	0
$781$	−28.6508	−1.02520
$782$	−36.5734	−1.30786
$783$	0	0
$784$	10.9607	0.391453
$785$	−1.16072	−0.0414280
$786$	0	0
$787$	5.94036	0.211751	0.105876	−	0.994379i	$-0.466235\pi$
$787$	5.94036	0.211751	0.105876	+	0.994379i	$0.466235\pi$
$788$	−22.3747	−0.797065
$789$	0	0
$790$	−12.1987	−0.434010
$791$	15.5140	0.551613
$792$	0	0
$793$	−15.9251	−0.565516
$794$	2.74159	0.0972953
$795$	0	0
$796$	−18.1663	−0.643888
$797$	−13.3214	−0.471867	−0.235933	−	0.971769i	$-0.575815\pi$
$797$	−13.3214	−0.471867	−0.235933	+	0.971769i	$0.575815\pi$
$798$	0	0
$799$	70.6111	2.49804
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	10.3423	0.365200
$803$	20.9278	0.738528
$804$	0	0
$805$	−24.8474	−0.875756
$806$	36.5994	1.28916
$807$	0	0
$808$	13.0422	0.458822
$809$	−3.52974	−0.124099	−0.0620496	−	0.998073i	$-0.519764\pi$
$809$	−3.52974	−0.124099	−0.0620496	+	0.998073i	$0.519764\pi$
$810$	0	0
$811$	−3.46105	−0.121534	−0.0607669	−	0.998152i	$-0.519355\pi$
$811$	−3.46105	−0.121534	−0.0607669	+	0.998152i	$0.519355\pi$
$812$	1.58930	0.0557735
$813$	0	0
$814$	2.01784	0.0707254
$815$	−9.36973	−0.328207
$816$	0	0
$817$	−65.3644	−2.28681
$818$	7.68699	0.268769
$819$	0	0
$820$	6.72433	0.234824
$821$	6.13452	0.214096	0.107048	−	0.994254i	$-0.465860\pi$
$821$	6.13452	0.214096	0.107048	+	0.994254i	$0.465860\pi$
$822$	0	0
$823$	−7.18104	−0.250315	−0.125158	−	0.992137i	$-0.539944\pi$
$823$	−7.18104	−0.250315	−0.125158	+	0.992137i	$0.539944\pi$
$824$	9.31950	0.324660
$825$	0	0
$826$	17.1033	0.595099
$827$	−34.6127	−1.20360	−0.601801	−	0.798646i	$-0.705550\pi$
$827$	−34.6127	−1.20360	−0.601801	+	0.798646i	$0.705550\pi$
$828$	0	0
$829$	−21.3225	−0.740561	−0.370281	−	0.928920i	$-0.620738\pi$
$829$	−21.3225	−0.740561	−0.370281	+	0.928920i	$0.620738\pi$
$830$	−1.82730	−0.0634266
$831$	0	0
$832$	−3.86299	−0.133925
$833$	−68.3728	−2.36898
$834$	0	0
$835$	−5.86299	−0.202897
$836$	15.1181	0.522870
$837$	0	0
$838$	−6.70318	−0.231557
$839$	−38.1267	−1.31628	−0.658139	−	0.752896i	$-0.728656\pi$
$839$	−38.1267	−1.31628	−0.658139	+	0.752896i	$0.728656\pi$
$840$	0	0
$841$	−28.8594	−0.995151
$842$	−11.1208	−0.383249
$843$	0	0
$844$	−17.3594	−0.597536
$845$	−1.92272	−0.0661435
$846$	0	0
$847$	29.3622	1.00890
$848$	0.861339	0.0295785
$849$	0	0
$850$	6.23800	0.213962
$851$	−5.86299	−0.200981
$852$	0	0
$853$	25.8306	0.884423	0.442212	−	0.896911i	$-0.354194\pi$
$853$	25.8306	0.884423	0.442212	+	0.896911i	$0.354194\pi$
$854$	17.4710	0.597847
$855$	0	0
$856$	5.11297	0.174758
$857$	−39.5354	−1.35050	−0.675252	−	0.737587i	$-0.735965\pi$
$857$	−39.5354	−1.35050	−0.675252	+	0.737587i	$0.735965\pi$
$858$	0	0
$859$	−48.9118	−1.66885	−0.834424	−	0.551122i	$-0.814200\pi$
$859$	−48.9118	−1.66885	−0.834424	+	0.551122i	$0.814200\pi$
$860$	−8.72433	−0.297497
$861$	0	0
$862$	26.7035	0.909523
$863$	−33.7426	−1.14861	−0.574305	−	0.818641i	$-0.694728\pi$
$863$	−33.7426	−1.14861	−0.574305	+	0.818641i	$0.694728\pi$
$864$	0	0
$865$	2.36138	0.0802895
$866$	3.38698	0.115094
$867$	0	0
$868$	−40.1524	−1.36286
$869$	24.6151	0.835009
$870$	0	0
$871$	−48.3326	−1.63769
$872$	−14.1808	−0.480224
$873$	0	0
$874$	−43.9267	−1.48584
$875$	4.23800	0.143271
$876$	0	0
$877$	−41.2829	−1.39402	−0.697012	−	0.717059i	$-0.745488\pi$
$877$	−41.2829	−1.39402	−0.697012	+	0.717059i	$0.745488\pi$
$878$	27.5847	0.930940
$879$	0	0
$880$	2.01784	0.0680215
$881$	−53.0893	−1.78862	−0.894312	−	0.447443i	$-0.852335\pi$
$881$	−53.0893	−1.78862	−0.894312	+	0.447443i	$0.852335\pi$
$882$	0	0
$883$	11.6180	0.390976	0.195488	−	0.980706i	$-0.437371\pi$
$883$	11.6180	0.390976	0.195488	+	0.980706i	$0.437371\pi$
$884$	24.0974	0.810482
$885$	0	0
$886$	15.6474	0.525683
$887$	28.7458	0.965189	0.482595	−	0.875844i	$-0.339694\pi$
$887$	28.7458	0.965189	0.482595	+	0.875844i	$0.339694\pi$
$888$	0	0
$889$	2.58394	0.0866627
$890$	−13.0890	−0.438745
$891$	0	0
$892$	0.652333	0.0218417
$893$	84.8080	2.83799
$894$	0	0
$895$	13.9247	0.465451
$896$	4.23800	0.141582
$897$	0	0
$898$	−13.6903	−0.456851
$899$	−3.55299	−0.118499
$900$	0	0
$901$	−5.37304	−0.179002
$902$	−13.5687	−0.451787
$903$	0	0
$904$	3.66068	0.121752
$905$	2.51170	0.0834917
$906$	0	0
$907$	24.6046	0.816982	0.408491	−	0.912762i	$-0.366055\pi$
$907$	24.6046	0.816982	0.408491	+	0.912762i	$0.366055\pi$
$908$	−10.1367	−0.336398
$909$	0	0
$910$	16.3714	0.542706
$911$	41.3494	1.36997	0.684983	−	0.728559i	$-0.259809\pi$
$911$	41.3494	1.36997	0.684983	+	0.728559i	$0.259809\pi$
$912$	0	0
$913$	3.68722	0.122029
$914$	−4.66324	−0.154246
$915$	0	0
$916$	−26.1747	−0.864838
$917$	−88.6435	−2.92727
$918$	0	0
$919$	49.8444	1.64422	0.822108	−	0.569332i	$-0.192798\pi$
$919$	49.8444	1.64422	0.822108	+	0.569332i	$0.192798\pi$
$920$	−5.86299	−0.193297
$921$	0	0
$922$	−22.8048	−0.751035
$923$	−54.8494	−1.80539
$924$	0	0
$925$	1.00000	0.0328798
$926$	−15.8551	−0.521033
$927$	0	0
$928$	0.375011	0.0123103
$929$	−15.1440	−0.496859	−0.248429	−	0.968650i	$-0.579914\pi$
$929$	−15.1440	−0.496859	−0.248429	+	0.968650i	$0.579914\pi$
$930$	0	0
$931$	−82.1196	−2.69136
$932$	−27.0812	−0.887074
$933$	0	0
$934$	−27.2841	−0.892763
$935$	−12.5873	−0.411650
$936$	0	0
$937$	−33.2735	−1.08700	−0.543498	−	0.839410i	$-0.682901\pi$
$937$	−33.2735	−1.08700	−0.543498	+	0.839410i	$0.682901\pi$
$938$	53.0246	1.73132
$939$	0	0
$940$	11.3195	0.369201
$941$	47.6286	1.55265	0.776323	−	0.630335i	$-0.217082\pi$
$941$	47.6286	1.55265	0.776323	+	0.630335i	$0.217082\pi$
$942$	0	0
$943$	39.4247	1.28385
$944$	4.03569	0.131351
$945$	0	0
$946$	17.6044	0.572367
$947$	−20.6366	−0.670601	−0.335300	−	0.942111i	$-0.608838\pi$
$947$	−20.6366	−0.670601	−0.335300	+	0.942111i	$0.608838\pi$
$948$	0	0
$949$	40.0646	1.30055
$950$	7.49220	0.243079
$951$	0	0
$952$	−26.4367	−0.856818
$953$	−22.8435	−0.739973	−0.369987	−	0.929037i	$-0.620638\pi$
$953$	−22.8435	−0.739973	−0.369987	+	0.929037i	$0.620638\pi$
$954$	0	0
$955$	−25.6867	−0.831201
$956$	12.3426	0.399189
$957$	0	0
$958$	−38.3033	−1.23752
$959$	−12.2020	−0.394023
$960$	0	0
$961$	58.7634	1.89559
$962$	3.86299	0.124548
$963$	0	0
$964$	12.5000	0.402596
$965$	6.03384	0.194236
$966$	0	0
$967$	0.0934721	0.00300586	0.00150293	−	0.999999i	$-0.499522\pi$
$967$	0.0934721	0.00300586	0.00150293	+	0.999999i	$0.499522\pi$
$968$	6.92830	0.222684
$969$	0	0
$970$	6.59517	0.211758
$971$	−34.5628	−1.10917	−0.554586	−	0.832126i	$-0.687123\pi$
$971$	−34.5628	−1.10917	−0.554586	+	0.832126i	$0.687123\pi$
$972$	0	0
$973$	67.2121	2.15472
$974$	−7.66820	−0.245705
$975$	0	0
$976$	4.12247	0.131957
$977$	61.6264	1.97160	0.985802	−	0.167915i	$-0.0537033\pi$
$977$	61.6264	1.97160	0.985802	+	0.167915i	$0.0537033\pi$
$978$	0	0
$979$	26.4116	0.844119
$980$	−10.9607	−0.350126
$981$	0	0
$982$	42.1786	1.34597
$983$	6.38987	0.203805	0.101903	−	0.994794i	$-0.467507\pi$
$983$	6.38987	0.203805	0.101903	+	0.994794i	$0.467507\pi$
$984$	0	0
$985$	22.3747	0.712917
$986$	−2.33932	−0.0744992
$987$	0	0
$988$	28.9423	0.920778
$989$	−51.1507	−1.62650
$990$	0	0
$991$	−24.0777	−0.764852	−0.382426	−	0.923986i	$-0.624911\pi$
$991$	−24.0777	−0.764852	−0.382426	+	0.923986i	$0.624911\pi$
$992$	−9.47436	−0.300811
$993$	0	0
$994$	60.1741	1.90861
$995$	18.1663	0.575911
$996$	0	0
$997$	50.1397	1.58794	0.793970	−	0.607957i	$-0.208011\pi$
$997$	50.1397	1.58794	0.793970	+	0.607957i	$0.208011\pi$
$998$	4.70318	0.148877
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bk.1.1	✓	5
3.2	odd	2		3330.2.a.bl.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3330.2.a.bk.1.1	✓	5	1.1	even	1	trivial
3330.2.a.bl.1.1	yes	5	3.2	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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.23800 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.01784 q^{11} -3.86299 q^{13} +4.23800 q^{14} +1.00000 q^{16} -6.23800 q^{17} -7.49220 q^{19} -1.00000 q^{20} +2.01784 q^{22} -5.86299 q^{23} +1.00000 q^{25} +3.86299 q^{26} -4.23800 q^{28} -0.375011 q^{29} +9.47436 q^{31} -1.00000 q^{32} +6.23800 q^{34} +4.23800 q^{35} +1.00000 q^{37} +7.49220 q^{38} +1.00000 q^{40} -6.72433 q^{41} +8.72433 q^{43} -2.01784 q^{44} +5.86299 q^{46} -11.3195 q^{47} +10.9607 q^{49} -1.00000 q^{50} -3.86299 q^{52} +0.861339 q^{53} +2.01784 q^{55} +4.23800 q^{56} +0.375011 q^{58} +4.03569 q^{59} +4.12247 q^{61} -9.47436 q^{62} +1.00000 q^{64} +3.86299 q^{65} +12.5117 q^{67} -6.23800 q^{68} -4.23800 q^{70} +14.1987 q^{71} -10.3714 q^{73} -1.00000 q^{74} -7.49220 q^{76} +8.55164 q^{77} -12.1987 q^{79} -1.00000 q^{80} +6.72433 q^{82} -1.82730 q^{83} +6.23800 q^{85} -8.72433 q^{86} +2.01784 q^{88} -13.0890 q^{89} +16.3714 q^{91} -5.86299 q^{92} +11.3195 q^{94} +7.49220 q^{95} +6.59517 q^{97} -10.9607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{2} + 5 q^{4} - 5 q^{5} + 3 q^{7} - 5 q^{8} + 5 q^{10} - 5 q^{11} + 5 q^{13} - 3 q^{14} + 5 q^{16} - 7 q^{17} - q^{19} - 5 q^{20} + 5 q^{22} - 5 q^{23} + 5 q^{25} - 5 q^{26} + 3 q^{28} - 2 q^{29}+ \cdots - 16 q^{98}+O(q^{100}) \) 5 * q - 5 * q^2 + 5 * q^4 - 5 * q^5 + 3 * q^7 - 5 * q^8 + 5 * q^10 - 5 * q^11 + 5 * q^13 - 3 * q^14 + 5 * q^16 - 7 * q^17 - q^19 - 5 * q^20 + 5 * q^22 - 5 * q^23 + 5 * q^25 - 5 * q^26 + 3 * q^28 - 2 * q^29 + 16 * q^31 - 5 * q^32 + 7 * q^34 - 3 * q^35 + 5 * q^37 + q^38 + 5 * q^40 - 2 * q^41 + 12 * q^43 - 5 * q^44 + 5 * q^46 - 6 * q^47 + 16 * q^49 - 5 * q^50 + 5 * q^52 - 3 * q^53 + 5 * q^55 - 3 * q^56 + 2 * q^58 + 10 * q^59 + 16 * q^61 - 16 * q^62 + 5 * q^64 - 5 * q^65 + 4 * q^67 - 7 * q^68 + 3 * q^70 + 8 * q^71 - 3 * q^73 - 5 * q^74 - q^76 - 3 * q^77 + 2 * q^79 - 5 * q^80 + 2 * q^82 + 5 * q^83 + 7 * q^85 - 12 * q^86 + 5 * q^88 + 7 * q^89 + 33 * q^91 - 5 * q^92 + 6 * q^94 + q^95 + 14 * q^97 - 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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