LMFDB - Embedded newform 3330.2.a.bi.1.2

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:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.470683$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.0586332 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.0586332 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.24914 q^{11} +3.77846 q^{13} -0.0586332 q^{14} +1.00000 q^{16} -7.49828 q^{17} +3.77846 q^{19} -1.00000 q^{20} -1.24914 q^{22} +3.66119 q^{23} +1.00000 q^{25} +3.77846 q^{26} -0.0586332 q^{28} +0.280176 q^{29} +4.41205 q^{31} +1.00000 q^{32} -7.49828 q^{34} +0.0586332 q^{35} -1.00000 q^{37} +3.77846 q^{38} -1.00000 q^{40} +7.14486 q^{41} +11.0276 q^{43} -1.24914 q^{44} +3.66119 q^{46} +6.05520 q^{47} -6.99656 q^{49} +1.00000 q^{50} +3.77846 q^{52} +5.36641 q^{53} +1.24914 q^{55} -0.0586332 q^{56} +0.280176 q^{58} -0.615547 q^{59} +5.02760 q^{61} +4.41205 q^{62} +1.00000 q^{64} -3.77846 q^{65} +1.38445 q^{67} -7.49828 q^{68} +0.0586332 q^{70} +6.61555 q^{71} -14.3319 q^{73} -1.00000 q^{74} +3.77846 q^{76} +0.0732411 q^{77} +5.50172 q^{79} -1.00000 q^{80} +7.14486 q^{82} -14.3940 q^{83} +7.49828 q^{85} +11.0276 q^{86} -1.24914 q^{88} +11.3354 q^{89} -0.221543 q^{91} +3.66119 q^{92} +6.05520 q^{94} -3.77846 q^{95} +9.14486 q^{97} -6.99656 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8} - 3 q^{10} + 5 q^{11} + 3 q^{13} - q^{14} + 3 q^{16} - 5 q^{17} + 3 q^{19} - 3 q^{20} + 5 q^{22} + q^{23} + 3 q^{25} + 3 q^{26} - q^{28} + 10 q^{29} + 12 q^{31} + 3 q^{32} - 5 q^{34} + q^{35} - 3 q^{37} + 3 q^{38} - 3 q^{40} + 6 q^{41} + 16 q^{43} + 5 q^{44} + q^{46} - 16 q^{47} + 14 q^{49} + 3 q^{50} + 3 q^{52} + 9 q^{53} - 5 q^{55} - q^{56} + 10 q^{58} + 14 q^{59} - 2 q^{61} + 12 q^{62} + 3 q^{64} - 3 q^{65} + 20 q^{67} - 5 q^{68} + q^{70} + 4 q^{71} + 17 q^{73} - 3 q^{74} + 3 q^{76} - 11 q^{77} + 34 q^{79} - 3 q^{80} + 6 q^{82} - 19 q^{83} + 5 q^{85} + 16 q^{86} + 5 q^{88} + 9 q^{89} - 9 q^{91} + q^{92} - 16 q^{94} - 3 q^{95} + 12 q^{97} + 14 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - q^7 + 3 * q^8 - 3 * q^10 + 5 * q^11 + 3 * q^13 - q^14 + 3 * q^16 - 5 * q^17 + 3 * q^19 - 3 * q^20 + 5 * q^22 + q^23 + 3 * q^25 + 3 * q^26 - q^28 + 10 * q^29 + 12 * q^31 + 3 * q^32 - 5 * q^34 + q^35 - 3 * q^37 + 3 * q^38 - 3 * q^40 + 6 * q^41 + 16 * q^43 + 5 * q^44 + q^46 - 16 * q^47 + 14 * q^49 + 3 * q^50 + 3 * q^52 + 9 * q^53 - 5 * q^55 - q^56 + 10 * q^58 + 14 * q^59 - 2 * q^61 + 12 * q^62 + 3 * q^64 - 3 * q^65 + 20 * q^67 - 5 * q^68 + q^70 + 4 * q^71 + 17 * q^73 - 3 * q^74 + 3 * q^76 - 11 * q^77 + 34 * q^79 - 3 * q^80 + 6 * q^82 - 19 * q^83 + 5 * q^85 + 16 * q^86 + 5 * q^88 + 9 * q^89 - 9 * q^91 + q^92 - 16 * q^94 - 3 * q^95 + 12 * q^97 + 14 * q^98

Display 100 coefficients 10 coefficients

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
$7$	−0.0586332	−0.0221613	−0.0110806	−	0.999939i	$-0.503527\pi$
$7$	−0.0586332	−0.0221613	−0.0110806	+	0.999939i	$0.503527\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−1.24914	−0.376630	−0.188315	−	0.982109i	$-0.560303\pi$
$11$	−1.24914	−0.376630	−0.188315	+	0.982109i	$0.560303\pi$
$12$	0	0
$13$	3.77846	1.04796	0.523978	−	0.851732i	$-0.324448\pi$
$13$	3.77846	1.04796	0.523978	+	0.851732i	$0.324448\pi$
$14$	−0.0586332	−0.0156704
$15$	0	0
$16$	1.00000	0.250000
$17$	−7.49828	−1.81860	−0.909300	−	0.416141i	$-0.863382\pi$
$17$	−7.49828	−1.81860	−0.909300	+	0.416141i	$0.863382\pi$
$18$	0	0
$19$	3.77846	0.866838	0.433419	−	0.901193i	$-0.357307\pi$
$19$	3.77846	0.866838	0.433419	+	0.901193i	$0.357307\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.24914	−0.266318
$23$	3.66119	0.763411	0.381706	−	0.924284i	$-0.375337\pi$
$23$	3.66119	0.763411	0.381706	+	0.924284i	$0.375337\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.77846	0.741016
$27$	0	0
$28$	−0.0586332	−0.0110806
$29$	0.280176	0.0520274	0.0260137	−	0.999662i	$-0.491719\pi$
$29$	0.280176	0.0520274	0.0260137	+	0.999662i	$0.491719\pi$
$30$	0	0
$31$	4.41205	0.792428	0.396214	−	0.918158i	$-0.370324\pi$
$31$	4.41205	0.792428	0.396214	+	0.918158i	$0.370324\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−7.49828	−1.28594
$35$	0.0586332	0.00991081
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	3.77846	0.612947
$39$	0	0
$40$	−1.00000	−0.158114
$41$	7.14486	1.11584	0.557920	−	0.829895i	$-0.311600\pi$
$41$	7.14486	1.11584	0.557920	+	0.829895i	$0.311600\pi$
$42$	0	0
$43$	11.0276	1.68169	0.840846	−	0.541274i	$-0.182058\pi$
$43$	11.0276	1.68169	0.840846	+	0.541274i	$0.182058\pi$
$44$	−1.24914	−0.188315
$45$	0	0
$46$	3.66119	0.539813
$47$	6.05520	0.883241	0.441621	−	0.897202i	$-0.354404\pi$
$47$	6.05520	0.883241	0.441621	+	0.897202i	$0.354404\pi$
$48$	0	0
$49$	−6.99656	−0.999509
$50$	1.00000	0.141421
$51$	0	0
$52$	3.77846	0.523978
$53$	5.36641	0.737133	0.368566	−	0.929601i	$-0.379849\pi$
$53$	5.36641	0.737133	0.368566	+	0.929601i	$0.379849\pi$
$54$	0	0
$55$	1.24914	0.168434
$56$	−0.0586332	−0.00783519
$57$	0	0
$58$	0.280176	0.0367889
$59$	−0.615547	−0.0801374	−0.0400687	−	0.999197i	$-0.512758\pi$
$59$	−0.615547	−0.0801374	−0.0400687	+	0.999197i	$0.512758\pi$
$60$	0	0
$61$	5.02760	0.643718	0.321859	−	0.946788i	$-0.395692\pi$
$61$	5.02760	0.643718	0.321859	+	0.946788i	$0.395692\pi$
$62$	4.41205	0.560331
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.77846	−0.468660
$66$	0	0
$67$	1.38445	0.169138	0.0845689	−	0.996418i	$-0.473049\pi$
$67$	1.38445	0.169138	0.0845689	+	0.996418i	$0.473049\pi$
$68$	−7.49828	−0.909300
$69$	0	0
$70$	0.0586332	0.00700800
$71$	6.61555	0.785121	0.392561	−	0.919726i	$-0.371589\pi$
$71$	6.61555	0.785121	0.392561	+	0.919726i	$0.371589\pi$
$72$	0	0
$73$	−14.3319	−1.67743	−0.838713	−	0.544574i	$-0.816691\pi$
$73$	−14.3319	−1.67743	−0.838713	+	0.544574i	$0.816691\pi$
$74$	−1.00000	−0.116248
$75$	0	0
$76$	3.77846	0.433419
$77$	0.0732411	0.00834659
$78$	0	0
$79$	5.50172	0.618992	0.309496	−	0.950901i	$-0.399840\pi$
$79$	5.50172	0.618992	0.309496	+	0.950901i	$0.399840\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	7.14486	0.789018
$83$	−14.3940	−1.57995	−0.789974	−	0.613141i	$-0.789906\pi$
$83$	−14.3940	−1.57995	−0.789974	+	0.613141i	$0.789906\pi$
$84$	0	0
$85$	7.49828	0.813303
$86$	11.0276	1.18914
$87$	0	0
$88$	−1.24914	−0.133159
$89$	11.3354	1.20155	0.600773	−	0.799419i	$-0.294859\pi$
$89$	11.3354	1.20155	0.600773	+	0.799419i	$0.294859\pi$
$90$	0	0
$91$	−0.221543	−0.0232240
$92$	3.66119	0.381706
$93$	0	0
$94$	6.05520	0.624546
$95$	−3.77846	−0.387662
$96$	0	0
$97$	9.14486	0.928520	0.464260	−	0.885699i	$-0.346320\pi$
$97$	9.14486	0.928520	0.464260	+	0.885699i	$0.346320\pi$
$98$	−6.99656	−0.706760
$99$	0	0
$100$	1.00000	0.100000
$101$	−5.43965	−0.541265	−0.270633	−	0.962683i	$-0.587233\pi$
$101$	−5.43965	−0.541265	−0.270633	+	0.962683i	$0.587233\pi$
$102$	0	0
$103$	16.2897	1.60507	0.802537	−	0.596602i	$-0.203483\pi$
$103$	16.2897	1.60507	0.802537	+	0.596602i	$0.203483\pi$
$104$	3.77846	0.370508
$105$	0	0
$106$	5.36641	0.521232
$107$	−16.3319	−1.57887	−0.789434	−	0.613836i	$-0.789626\pi$
$107$	−16.3319	−1.57887	−0.789434	+	0.613836i	$0.789626\pi$
$108$	0	0
$109$	12.8061	1.22660	0.613299	−	0.789851i	$-0.289842\pi$
$109$	12.8061	1.22660	0.613299	+	0.789851i	$0.289842\pi$
$110$	1.24914	0.119101
$111$	0	0
$112$	−0.0586332	−0.00554031
$113$	9.89229	0.930588	0.465294	−	0.885156i	$-0.345949\pi$
$113$	9.89229	0.930588	0.465294	+	0.885156i	$0.345949\pi$
$114$	0	0
$115$	−3.66119	−0.341408
$116$	0.280176	0.0260137
$117$	0	0
$118$	−0.615547	−0.0566657
$119$	0.439648	0.0403025
$120$	0	0
$121$	−9.43965	−0.858150
$122$	5.02760	0.455177
$123$	0	0
$124$	4.41205	0.396214
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.394005	−0.0349622	−0.0174811	−	0.999847i	$-0.505565\pi$
$127$	−0.394005	−0.0349622	−0.0174811	+	0.999847i	$0.505565\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.77846	−0.331393
$131$	0.615547	0.0537806	0.0268903	−	0.999638i	$-0.491440\pi$
$131$	0.615547	0.0537806	0.0268903	+	0.999638i	$0.491440\pi$
$132$	0	0
$133$	−0.221543	−0.0192102
$134$	1.38445	0.119598
$135$	0	0
$136$	−7.49828	−0.642972
$137$	9.55691	0.816502	0.408251	−	0.912870i	$-0.366139\pi$
$137$	9.55691	0.816502	0.408251	+	0.912870i	$0.366139\pi$
$138$	0	0
$139$	−1.27674	−0.108292	−0.0541458	−	0.998533i	$-0.517244\pi$
$139$	−1.27674	−0.108292	−0.0541458	+	0.998533i	$0.517244\pi$
$140$	0.0586332	0.00495541
$141$	0	0
$142$	6.61555	0.555164
$143$	−4.71982	−0.394692
$144$	0	0
$145$	−0.280176	−0.0232674
$146$	−14.3319	−1.18612
$147$	0	0
$148$	−1.00000	−0.0821995
$149$	−0.615547	−0.0504276	−0.0252138	−	0.999682i	$-0.508027\pi$
$149$	−0.615547	−0.0504276	−0.0252138	+	0.999682i	$0.508027\pi$
$150$	0	0
$151$	5.39744	0.439238	0.219619	−	0.975586i	$-0.429519\pi$
$151$	5.39744	0.439238	0.219619	+	0.975586i	$0.429519\pi$
$152$	3.77846	0.306473
$153$	0	0
$154$	0.0732411	0.00590193
$155$	−4.41205	−0.354384
$156$	0	0
$157$	19.2147	1.53350	0.766749	−	0.641947i	$-0.221873\pi$
$157$	19.2147	1.53350	0.766749	+	0.641947i	$0.221873\pi$
$158$	5.50172	0.437693
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−0.214667	−0.0169181
$162$	0	0
$163$	−7.63016	−0.597640	−0.298820	−	0.954310i	$-0.596593\pi$
$163$	−7.63016	−0.597640	−0.298820	+	0.954310i	$0.596593\pi$
$164$	7.14486	0.557920
$165$	0	0
$166$	−14.3940	−1.11719
$167$	−19.0096	−1.47100	−0.735502	−	0.677523i	$-0.763053\pi$
$167$	−19.0096	−1.47100	−0.735502	+	0.677523i	$0.763053\pi$
$168$	0	0
$169$	1.27674	0.0982106
$170$	7.49828	0.575092
$171$	0	0
$172$	11.0276	0.840846
$173$	13.7474	1.04520	0.522599	−	0.852579i	$-0.324963\pi$
$173$	13.7474	1.04520	0.522599	+	0.852579i	$0.324963\pi$
$174$	0	0
$175$	−0.0586332	−0.00443225
$176$	−1.24914	−0.0941575
$177$	0	0
$178$	11.3354	0.849622
$179$	10.8241	0.809031	0.404516	−	0.914531i	$-0.367440\pi$
$179$	10.8241	0.809031	0.404516	+	0.914531i	$0.367440\pi$
$180$	0	0
$181$	−23.7294	−1.76379	−0.881895	−	0.471445i	$-0.843732\pi$
$181$	−23.7294	−1.76379	−0.881895	+	0.471445i	$0.843732\pi$
$182$	−0.221543	−0.0164219
$183$	0	0
$184$	3.66119	0.269907
$185$	1.00000	0.0735215
$186$	0	0
$187$	9.36641	0.684940
$188$	6.05520	0.441621
$189$	0	0
$190$	−3.77846	−0.274118
$191$	18.9931	1.37429	0.687147	−	0.726518i	$-0.258863\pi$
$191$	18.9931	1.37429	0.687147	+	0.726518i	$0.258863\pi$
$192$	0	0
$193$	16.9966	1.22344	0.611720	−	0.791075i	$-0.290478\pi$
$193$	16.9966	1.22344	0.611720	+	0.791075i	$0.290478\pi$
$194$	9.14486	0.656563
$195$	0	0
$196$	−6.99656	−0.499754
$197$	−3.39744	−0.242058	−0.121029	−	0.992649i	$-0.538619\pi$
$197$	−3.39744	−0.242058	−0.121029	+	0.992649i	$0.538619\pi$
$198$	0	0
$199$	0.824101	0.0584189	0.0292095	−	0.999573i	$-0.490701\pi$
$199$	0.824101	0.0584189	0.0292095	+	0.999573i	$0.490701\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−5.43965	−0.382732
$203$	−0.0164276	−0.00115299
$204$	0	0
$205$	−7.14486	−0.499019
$206$	16.2897	1.13496
$207$	0	0
$208$	3.77846	0.261989
$209$	−4.71982	−0.326477
$210$	0	0
$211$	−2.39744	−0.165047	−0.0825234	−	0.996589i	$-0.526298\pi$
$211$	−2.39744	−0.165047	−0.0825234	+	0.996589i	$0.526298\pi$
$212$	5.36641	0.368566
$213$	0	0
$214$	−16.3319	−1.11643
$215$	−11.0276	−0.752076
$216$	0	0
$217$	−0.258692	−0.0175612
$218$	12.8061	0.867335
$219$	0	0
$220$	1.24914	0.0842170
$221$	−28.3319	−1.90581
$222$	0	0
$223$	6.66119	0.446066	0.223033	−	0.974811i	$-0.428404\pi$
$223$	6.66119	0.446066	0.223033	+	0.974811i	$0.428404\pi$
$224$	−0.0586332	−0.00391759
$225$	0	0
$226$	9.89229	0.658025
$227$	−11.1043	−0.737017	−0.368508	−	0.929624i	$-0.620131\pi$
$227$	−11.1043	−0.737017	−0.368508	+	0.929624i	$0.620131\pi$
$228$	0	0
$229$	12.7328	0.841408	0.420704	−	0.907198i	$-0.361783\pi$
$229$	12.7328	0.841408	0.420704	+	0.907198i	$0.361783\pi$
$230$	−3.66119	−0.241412
$231$	0	0
$232$	0.280176	0.0183945
$233$	22.2277	1.45618	0.728091	−	0.685480i	$-0.240408\pi$
$233$	22.2277	1.45618	0.728091	+	0.685480i	$0.240408\pi$
$234$	0	0
$235$	−6.05520	−0.394997
$236$	−0.615547	−0.0400687
$237$	0	0
$238$	0.439648	0.0284981
$239$	−20.7164	−1.34003	−0.670016	−	0.742347i	$-0.733713\pi$
$239$	−20.7164	−1.34003	−0.670016	+	0.742347i	$0.733713\pi$
$240$	0	0
$241$	−1.76547	−0.113724	−0.0568619	−	0.998382i	$-0.518109\pi$
$241$	−1.76547	−0.113724	−0.0568619	+	0.998382i	$0.518109\pi$
$242$	−9.43965	−0.606804
$243$	0	0
$244$	5.02760	0.321859
$245$	6.99656	0.446994
$246$	0	0
$247$	14.2767	0.908407
$248$	4.41205	0.280165
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	16.3189	1.03004	0.515021	−	0.857177i	$-0.327784\pi$
$251$	16.3189	1.03004	0.515021	+	0.857177i	$0.327784\pi$
$252$	0	0
$253$	−4.57334	−0.287524
$254$	−0.394005	−0.0247220
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.0456	−0.938521	−0.469261	−	0.883060i	$-0.655480\pi$
$257$	−15.0456	−0.938521	−0.469261	+	0.883060i	$0.655480\pi$
$258$	0	0
$259$	0.0586332	0.00364329
$260$	−3.77846	−0.234330
$261$	0	0
$262$	0.615547	0.0380286
$263$	−30.9035	−1.90559	−0.952794	−	0.303616i	$-0.901806\pi$
$263$	−30.9035	−1.90559	−0.952794	+	0.303616i	$0.901806\pi$
$264$	0	0
$265$	−5.36641	−0.329656
$266$	−0.221543	−0.0135837
$267$	0	0
$268$	1.38445	0.0845689
$269$	10.7198	0.653599	0.326800	−	0.945094i	$-0.394030\pi$
$269$	10.7198	0.653599	0.326800	+	0.945094i	$0.394030\pi$
$270$	0	0
$271$	−2.76891	−0.168199	−0.0840995	−	0.996457i	$-0.526801\pi$
$271$	−2.76891	−0.168199	−0.0840995	+	0.996457i	$0.526801\pi$
$272$	−7.49828	−0.454650
$273$	0	0
$274$	9.55691	0.577354
$275$	−1.24914	−0.0753260
$276$	0	0
$277$	−14.0682	−0.845275	−0.422638	−	0.906299i	$-0.638896\pi$
$277$	−14.0682	−0.845275	−0.422638	+	0.906299i	$0.638896\pi$
$278$	−1.27674	−0.0765737
$279$	0	0
$280$	0.0586332	0.00350400
$281$	−14.3940	−0.858674	−0.429337	−	0.903144i	$-0.641253\pi$
$281$	−14.3940	−0.858674	−0.429337	+	0.903144i	$0.641253\pi$
$282$	0	0
$283$	−32.1234	−1.90954	−0.954768	−	0.297351i	$-0.903897\pi$
$283$	−32.1234	−1.90954	−0.954768	+	0.297351i	$0.903897\pi$
$284$	6.61555	0.392561
$285$	0	0
$286$	−4.71982	−0.279089
$287$	−0.418926	−0.0247284
$288$	0	0
$289$	39.2242	2.30731
$290$	−0.280176	−0.0164525
$291$	0	0
$292$	−14.3319	−0.838713
$293$	−28.0371	−1.63795	−0.818974	−	0.573831i	$-0.805457\pi$
$293$	−28.0371	−1.63795	−0.818974	+	0.573831i	$0.805457\pi$
$294$	0	0
$295$	0.615547	0.0358386
$296$	−1.00000	−0.0581238
$297$	0	0
$298$	−0.615547	−0.0356577
$299$	13.8337	0.800021
$300$	0	0
$301$	−0.646583	−0.0372684
$302$	5.39744	0.310588
$303$	0	0
$304$	3.77846	0.216709
$305$	−5.02760	−0.287879
$306$	0	0
$307$	28.4914	1.62609	0.813045	−	0.582201i	$-0.197808\pi$
$307$	28.4914	1.62609	0.813045	+	0.582201i	$0.197808\pi$
$308$	0.0732411	0.00417330
$309$	0	0
$310$	−4.41205	−0.250588
$311$	−28.9509	−1.64166	−0.820828	−	0.571175i	$-0.806488\pi$
$311$	−28.9509	−1.64166	−0.820828	+	0.571175i	$0.806488\pi$
$312$	0	0
$313$	16.9966	0.960702	0.480351	−	0.877076i	$-0.340509\pi$
$313$	16.9966	0.960702	0.480351	+	0.877076i	$0.340509\pi$
$314$	19.2147	1.08435
$315$	0	0
$316$	5.50172	0.309496
$317$	−15.4707	−0.868920	−0.434460	−	0.900691i	$-0.643061\pi$
$317$	−15.4707	−0.868920	−0.434460	+	0.900691i	$0.643061\pi$
$318$	0	0
$319$	−0.349979	−0.0195951
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−0.214667	−0.0119629
$323$	−28.3319	−1.57643
$324$	0	0
$325$	3.77846	0.209591
$326$	−7.63016	−0.422595
$327$	0	0
$328$	7.14486	0.394509
$329$	−0.355035	−0.0195737
$330$	0	0
$331$	−21.4948	−1.18146	−0.590732	−	0.806868i	$-0.701161\pi$
$331$	−21.4948	−1.18146	−0.590732	+	0.806868i	$0.701161\pi$
$332$	−14.3940	−0.789974
$333$	0	0
$334$	−19.0096	−1.04016
$335$	−1.38445	−0.0756407
$336$	0	0
$337$	−12.5405	−0.683124	−0.341562	−	0.939859i	$-0.610956\pi$
$337$	−12.5405	−0.683124	−0.341562	+	0.939859i	$0.610956\pi$
$338$	1.27674	0.0694454
$339$	0	0
$340$	7.49828	0.406651
$341$	−5.51127	−0.298452
$342$	0	0
$343$	0.820663	0.0443116
$344$	11.0276	0.594568
$345$	0	0
$346$	13.7474	0.739066
$347$	−15.5569	−0.835139	−0.417569	−	0.908645i	$-0.637118\pi$
$347$	−15.5569	−0.835139	−0.417569	+	0.908645i	$0.637118\pi$
$348$	0	0
$349$	25.9379	1.38843	0.694213	−	0.719770i	$-0.255753\pi$
$349$	25.9379	1.38843	0.694213	+	0.719770i	$0.255753\pi$
$350$	−0.0586332	−0.00313407
$351$	0	0
$352$	−1.24914	−0.0665794
$353$	−19.5405	−1.04004	−0.520018	−	0.854156i	$-0.674075\pi$
$353$	−19.5405	−1.04004	−0.520018	+	0.854156i	$0.674075\pi$
$354$	0	0
$355$	−6.61555	−0.351117
$356$	11.3354	0.600773
$357$	0	0
$358$	10.8241	0.572071
$359$	29.3224	1.54758	0.773788	−	0.633445i	$-0.218360\pi$
$359$	29.3224	1.54758	0.773788	+	0.633445i	$0.218360\pi$
$360$	0	0
$361$	−4.72326	−0.248593
$362$	−23.7294	−1.24719
$363$	0	0
$364$	−0.221543	−0.0116120
$365$	14.3319	0.750168
$366$	0	0
$367$	4.26719	0.222745	0.111373	−	0.993779i	$-0.464475\pi$
$367$	4.26719	0.222745	0.111373	+	0.993779i	$0.464475\pi$
$368$	3.66119	0.190853
$369$	0	0
$370$	1.00000	0.0519875
$371$	−0.314649	−0.0163358
$372$	0	0
$373$	−8.32582	−0.431095	−0.215547	−	0.976493i	$-0.569154\pi$
$373$	−8.32582	−0.431095	−0.215547	+	0.976493i	$0.569154\pi$
$374$	9.36641	0.484325
$375$	0	0
$376$	6.05520	0.312273
$377$	1.05863	0.0545224
$378$	0	0
$379$	−2.87930	−0.147899	−0.0739497	−	0.997262i	$-0.523560\pi$
$379$	−2.87930	−0.147899	−0.0739497	+	0.997262i	$0.523560\pi$
$380$	−3.77846	−0.193831
$381$	0	0
$382$	18.9931	0.971773
$383$	8.44920	0.431734	0.215867	−	0.976423i	$-0.430742\pi$
$383$	8.44920	0.431734	0.215867	+	0.976423i	$0.430742\pi$
$384$	0	0
$385$	−0.0732411	−0.00373271
$386$	16.9966	0.865102
$387$	0	0
$388$	9.14486	0.464260
$389$	−3.27674	−0.166137	−0.0830686	−	0.996544i	$-0.526472\pi$
$389$	−3.27674	−0.166137	−0.0830686	+	0.996544i	$0.526472\pi$
$390$	0	0
$391$	−27.4526	−1.38834
$392$	−6.99656	−0.353380
$393$	0	0
$394$	−3.39744	−0.171161
$395$	−5.50172	−0.276822
$396$	0	0
$397$	7.12070	0.357378	0.178689	−	0.983906i	$-0.442814\pi$
$397$	7.12070	0.357378	0.178689	+	0.983906i	$0.442814\pi$
$398$	0.824101	0.0413084
$399$	0	0
$400$	1.00000	0.0500000
$401$	26.7129	1.33398	0.666990	−	0.745066i	$-0.267582\pi$
$401$	26.7129	1.33398	0.666990	+	0.745066i	$0.267582\pi$
$402$	0	0
$403$	16.6707	0.830429
$404$	−5.43965	−0.270633
$405$	0	0
$406$	−0.0164276	−0.000815288 0
$407$	1.24914	0.0619176
$408$	0	0
$409$	−5.53093	−0.273487	−0.136744	−	0.990606i	$-0.543664\pi$
$409$	−5.53093	−0.273487	−0.136744	+	0.990606i	$0.543664\pi$
$410$	−7.14486	−0.352860
$411$	0	0
$412$	16.2897	0.802537
$413$	0.0360915	0.00177595
$414$	0	0
$415$	14.3940	0.706574
$416$	3.77846	0.185254
$417$	0	0
$418$	−4.71982	−0.230854
$419$	−1.10771	−0.0541154	−0.0270577	−	0.999634i	$-0.508614\pi$
$419$	−1.10771	−0.0541154	−0.0270577	+	0.999634i	$0.508614\pi$
$420$	0	0
$421$	14.6707	0.715008	0.357504	−	0.933912i	$-0.383628\pi$
$421$	14.6707	0.715008	0.357504	+	0.933912i	$0.383628\pi$
$422$	−2.39744	−0.116706
$423$	0	0
$424$	5.36641	0.260616
$425$	−7.49828	−0.363720
$426$	0	0
$427$	−0.294784	−0.0142656
$428$	−16.3319	−0.789434
$429$	0	0
$430$	−11.0276	−0.531798
$431$	−8.11383	−0.390829	−0.195415	−	0.980721i	$-0.562605\pi$
$431$	−8.11383	−0.390829	−0.195415	+	0.980721i	$0.562605\pi$
$432$	0	0
$433$	−11.6612	−0.560401	−0.280201	−	0.959941i	$-0.590401\pi$
$433$	−11.6612	−0.560401	−0.280201	+	0.959941i	$0.590401\pi$
$434$	−0.258692	−0.0124176
$435$	0	0
$436$	12.8061	0.613299
$437$	13.8337	0.661753
$438$	0	0
$439$	−15.9690	−0.762157	−0.381079	−	0.924543i	$-0.624447\pi$
$439$	−15.9690	−0.762157	−0.381079	+	0.924543i	$0.624447\pi$
$440$	1.24914	0.0595504
$441$	0	0
$442$	−28.3319	−1.34761
$443$	1.79145	0.0851142	0.0425571	−	0.999094i	$-0.486450\pi$
$443$	1.79145	0.0851142	0.0425571	+	0.999094i	$0.486450\pi$
$444$	0	0
$445$	−11.3354	−0.537348
$446$	6.66119	0.315417
$447$	0	0
$448$	−0.0586332	−0.00277016
$449$	5.70340	0.269160	0.134580	−	0.990903i	$-0.457031\pi$
$449$	5.70340	0.269160	0.134580	+	0.990903i	$0.457031\pi$
$450$	0	0
$451$	−8.92494	−0.420259
$452$	9.89229	0.465294
$453$	0	0
$454$	−11.1043	−0.521150
$455$	0.221543	0.0103861
$456$	0	0
$457$	−26.9035	−1.25849	−0.629245	−	0.777207i	$-0.716636\pi$
$457$	−26.9035	−1.25849	−0.629245	+	0.777207i	$0.716636\pi$
$458$	12.7328	0.594965
$459$	0	0
$460$	−3.66119	−0.170704
$461$	−2.60600	−0.121373	−0.0606867	−	0.998157i	$-0.519329\pi$
$461$	−2.60600	−0.121373	−0.0606867	+	0.998157i	$0.519329\pi$
$462$	0	0
$463$	−1.97402	−0.0917405	−0.0458703	−	0.998947i	$-0.514606\pi$
$463$	−1.97402	−0.0917405	−0.0458703	+	0.998947i	$0.514606\pi$
$464$	0.280176	0.0130068
$465$	0	0
$466$	22.2277	1.02968
$467$	−21.9836	−1.01728	−0.508639	−	0.860980i	$-0.669851\pi$
$467$	−21.9836	−1.01728	−0.508639	+	0.860980i	$0.669851\pi$
$468$	0	0
$469$	−0.0811748	−0.00374831
$470$	−6.05520	−0.279305
$471$	0	0
$472$	−0.615547	−0.0283329
$473$	−13.7750	−0.633376
$474$	0	0
$475$	3.77846	0.173368
$476$	0.439648	0.0201512
$477$	0	0
$478$	−20.7164	−0.947545
$479$	−38.1526	−1.74324	−0.871618	−	0.490185i	$-0.836929\pi$
$479$	−38.1526	−1.74324	−0.871618	+	0.490185i	$0.836929\pi$
$480$	0	0
$481$	−3.77846	−0.172283
$482$	−1.76547	−0.0804148
$483$	0	0
$484$	−9.43965	−0.429075
$485$	−9.14486	−0.415247
$486$	0	0
$487$	−34.0844	−1.54451	−0.772256	−	0.635311i	$-0.780872\pi$
$487$	−34.0844	−1.54451	−0.772256	+	0.635311i	$0.780872\pi$
$488$	5.02760	0.227589
$489$	0	0
$490$	6.99656	0.316072
$491$	25.2181	1.13808	0.569039	−	0.822311i	$-0.307316\pi$
$491$	25.2181	1.13808	0.569039	+	0.822311i	$0.307316\pi$
$492$	0	0
$493$	−2.10084	−0.0946170
$494$	14.2767	0.642341
$495$	0	0
$496$	4.41205	0.198107
$497$	−0.387890	−0.0173993
$498$	0	0
$499$	−28.9183	−1.29456	−0.647280	−	0.762253i	$-0.724093\pi$
$499$	−28.9183	−1.29456	−0.647280	+	0.762253i	$0.724093\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	16.3189	0.728350
$503$	28.5795	1.27429	0.637147	−	0.770742i	$-0.280114\pi$
$503$	28.5795	1.27429	0.637147	+	0.770742i	$0.280114\pi$
$504$	0	0
$505$	5.43965	0.242061
$506$	−4.57334	−0.203310
$507$	0	0
$508$	−0.394005	−0.0174811
$509$	3.39744	0.150589	0.0752945	−	0.997161i	$-0.476010\pi$
$509$	3.39744	0.150589	0.0752945	+	0.997161i	$0.476010\pi$
$510$	0	0
$511$	0.840327	0.0371739
$512$	1.00000	0.0441942
$513$	0	0
$514$	−15.0456	−0.663635
$515$	−16.2897	−0.717811
$516$	0	0
$517$	−7.56379	−0.332655
$518$	0.0586332	0.00257619
$519$	0	0
$520$	−3.77846	−0.165696
$521$	−24.2879	−1.06407	−0.532036	−	0.846722i	$-0.678573\pi$
$521$	−24.2879	−1.06407	−0.532036	+	0.846722i	$0.678573\pi$
$522$	0	0
$523$	−24.7000	−1.08005	−0.540027	−	0.841648i	$-0.681586\pi$
$523$	−24.7000	−1.08005	−0.540027	+	0.841648i	$0.681586\pi$
$524$	0.615547	0.0268903
$525$	0	0
$526$	−30.9035	−1.34745
$527$	−33.0828	−1.44111
$528$	0	0
$529$	−9.59568	−0.417204
$530$	−5.36641	−0.233102
$531$	0	0
$532$	−0.221543	−0.00960510
$533$	26.9966	1.16935
$534$	0	0
$535$	16.3319	0.706091
$536$	1.38445	0.0597992
$537$	0	0
$538$	10.7198	0.462164
$539$	8.73969	0.376445
$540$	0	0
$541$	4.18545	0.179947	0.0899733	−	0.995944i	$-0.471322\pi$
$541$	4.18545	0.179947	0.0899733	+	0.995944i	$0.471322\pi$
$542$	−2.76891	−0.118935
$543$	0	0
$544$	−7.49828	−0.321486
$545$	−12.8061	−0.548551
$546$	0	0
$547$	46.4113	1.98440	0.992202	−	0.124643i	$-0.0397784\pi$
$547$	46.4113	1.98440	0.992202	+	0.124643i	$0.0397784\pi$
$548$	9.55691	0.408251
$549$	0	0
$550$	−1.24914	−0.0532635
$551$	1.05863	0.0450993
$552$	0	0
$553$	−0.322583	−0.0137176
$554$	−14.0682	−0.597700
$555$	0	0
$556$	−1.27674	−0.0541458
$557$	10.9673	0.464701	0.232351	−	0.972632i	$-0.425358\pi$
$557$	10.9673	0.464701	0.232351	+	0.972632i	$0.425358\pi$
$558$	0	0
$559$	41.6673	1.76234
$560$	0.0586332	0.00247770
$561$	0	0
$562$	−14.3940	−0.607174
$563$	−15.6837	−0.660991	−0.330495	−	0.943808i	$-0.607216\pi$
$563$	−15.6837	−0.660991	−0.330495	+	0.943808i	$0.607216\pi$
$564$	0	0
$565$	−9.89229	−0.416172
$566$	−32.1234	−1.35025
$567$	0	0
$568$	6.61555	0.277582
$569$	13.6612	0.572707	0.286353	−	0.958124i	$-0.407557\pi$
$569$	13.6612	0.572707	0.286353	+	0.958124i	$0.407557\pi$
$570$	0	0
$571$	27.2699	1.14121	0.570604	−	0.821225i	$-0.306709\pi$
$571$	27.2699	1.14121	0.570604	+	0.821225i	$0.306709\pi$
$572$	−4.71982	−0.197346
$573$	0	0
$574$	−0.418926	−0.0174856
$575$	3.66119	0.152682
$576$	0	0
$577$	27.8759	1.16049	0.580244	−	0.814443i	$-0.302957\pi$
$577$	27.8759	1.16049	0.580244	+	0.814443i	$0.302957\pi$
$578$	39.2242	1.63151
$579$	0	0
$580$	−0.280176	−0.0116337
$581$	0.843966	0.0350136
$582$	0	0
$583$	−6.70340	−0.277626
$584$	−14.3319	−0.593060
$585$	0	0
$586$	−28.0371	−1.15820
$587$	−36.4198	−1.50321	−0.751603	−	0.659616i	$-0.770719\pi$
$587$	−36.4198	−1.50321	−0.751603	+	0.659616i	$0.770719\pi$
$588$	0	0
$589$	16.6707	0.686906
$590$	0.615547	0.0253417
$591$	0	0
$592$	−1.00000	−0.0410997
$593$	−4.87930	−0.200369	−0.100184	−	0.994969i	$-0.531943\pi$
$593$	−4.87930	−0.200369	−0.100184	+	0.994969i	$0.531943\pi$
$594$	0	0
$595$	−0.439648	−0.0180238
$596$	−0.615547	−0.0252138
$597$	0	0
$598$	13.8337	0.565700
$599$	−12.4983	−0.510666	−0.255333	−	0.966853i	$-0.582185\pi$
$599$	−12.4983	−0.510666	−0.255333	+	0.966853i	$0.582185\pi$
$600$	0	0
$601$	−33.6707	−1.37346	−0.686729	−	0.726913i	$-0.740954\pi$
$601$	−33.6707	−1.37346	−0.686729	+	0.726913i	$0.740954\pi$
$602$	−0.646583	−0.0263528
$603$	0	0
$604$	5.39744	0.219619
$605$	9.43965	0.383776
$606$	0	0
$607$	−16.9053	−0.686164	−0.343082	−	0.939305i	$-0.611471\pi$
$607$	−16.9053	−0.686164	−0.343082	+	0.939305i	$0.611471\pi$
$608$	3.77846	0.153237
$609$	0	0
$610$	−5.02760	−0.203561
$611$	22.8793	0.925597
$612$	0	0
$613$	−45.6578	−1.84410	−0.922050	−	0.387072i	$-0.873487\pi$
$613$	−45.6578	−1.84410	−0.922050	+	0.387072i	$0.873487\pi$
$614$	28.4914	1.14982
$615$	0	0
$616$	0.0732411	0.00295097
$617$	−9.34836	−0.376351	−0.188175	−	0.982135i	$-0.560257\pi$
$617$	−9.34836	−0.376351	−0.188175	+	0.982135i	$0.560257\pi$
$618$	0	0
$619$	33.7129	1.35504	0.677519	−	0.735505i	$-0.263055\pi$
$619$	33.7129	1.35504	0.677519	+	0.735505i	$0.263055\pi$
$620$	−4.41205	−0.177192
$621$	0	0
$622$	−28.9509	−1.16083
$623$	−0.664629	−0.0266278
$624$	0	0
$625$	1.00000	0.0400000
$626$	16.9966	0.679319
$627$	0	0
$628$	19.2147	0.766749
$629$	7.49828	0.298976
$630$	0	0
$631$	−6.14830	−0.244760	−0.122380	−	0.992483i	$-0.539053\pi$
$631$	−6.14830	−0.244760	−0.122380	+	0.992483i	$0.539053\pi$
$632$	5.50172	0.218847
$633$	0	0
$634$	−15.4707	−0.614419
$635$	0.394005	0.0156356
$636$	0	0
$637$	−26.4362	−1.04744
$638$	−0.349979	−0.0138558
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	22.8483	0.902452	0.451226	−	0.892410i	$-0.350987\pi$
$641$	22.8483	0.902452	0.451226	+	0.892410i	$0.350987\pi$
$642$	0	0
$643$	−47.1871	−1.86088	−0.930438	−	0.366449i	$-0.880573\pi$
$643$	−47.1871	−1.86088	−0.930438	+	0.366449i	$0.880573\pi$
$644$	−0.214667	−0.00845907
$645$	0	0
$646$	−28.3319	−1.11471
$647$	33.9992	1.33665	0.668324	−	0.743870i	$-0.267012\pi$
$647$	33.9992	1.33665	0.668324	+	0.743870i	$0.267012\pi$
$648$	0	0
$649$	0.768905	0.0301822
$650$	3.77846	0.148203
$651$	0	0
$652$	−7.63016	−0.298820
$653$	46.9345	1.83669	0.918344	−	0.395782i	$-0.129527\pi$
$653$	46.9345	1.83669	0.918344	+	0.395782i	$0.129527\pi$
$654$	0	0
$655$	−0.615547	−0.0240514
$656$	7.14486	0.278960
$657$	0	0
$658$	−0.355035	−0.0138407
$659$	29.4036	1.14540	0.572700	−	0.819765i	$-0.305896\pi$
$659$	29.4036	1.14540	0.572700	+	0.819765i	$0.305896\pi$
$660$	0	0
$661$	−11.3404	−0.441092	−0.220546	−	0.975377i	$-0.570784\pi$
$661$	−11.3404	−0.441092	−0.220546	+	0.975377i	$0.570784\pi$
$662$	−21.4948	−0.835421
$663$	0	0
$664$	−14.3940	−0.558596
$665$	0.221543	0.00859106
$666$	0	0
$667$	1.02578	0.0397183
$668$	−19.0096	−0.735502
$669$	0	0
$670$	−1.38445	−0.0534861
$671$	−6.28018	−0.242444
$672$	0	0
$673$	−17.5699	−0.677270	−0.338635	−	0.940918i	$-0.609965\pi$
$673$	−17.5699	−0.677270	−0.338635	+	0.940918i	$0.609965\pi$
$674$	−12.5405	−0.483041
$675$	0	0
$676$	1.27674	0.0491053
$677$	−27.3974	−1.05297	−0.526485	−	0.850185i	$-0.676490\pi$
$677$	−27.3974	−1.05297	−0.526485	+	0.850185i	$0.676490\pi$
$678$	0	0
$679$	−0.536192	−0.0205772
$680$	7.49828	0.287546
$681$	0	0
$682$	−5.51127	−0.211037
$683$	30.8888	1.18193	0.590964	−	0.806698i	$-0.298747\pi$
$683$	30.8888	1.18193	0.590964	+	0.806698i	$0.298747\pi$
$684$	0	0
$685$	−9.55691	−0.365151
$686$	0.820663	0.0313330
$687$	0	0
$688$	11.0276	0.420423
$689$	20.2767	0.772482
$690$	0	0
$691$	30.0456	1.14299	0.571495	−	0.820605i	$-0.306364\pi$
$691$	30.0456	1.14299	0.571495	+	0.820605i	$0.306364\pi$
$692$	13.7474	0.522599
$693$	0	0
$694$	−15.5569	−0.590532
$695$	1.27674	0.0484294
$696$	0	0
$697$	−53.5742	−2.02927
$698$	25.9379	0.981765
$699$	0	0
$700$	−0.0586332	−0.00221613
$701$	23.6742	0.894161	0.447081	−	0.894494i	$-0.352464\pi$
$701$	23.6742	0.894161	0.447081	+	0.894494i	$0.352464\pi$
$702$	0	0
$703$	−3.77846	−0.142507
$704$	−1.24914	−0.0470788
$705$	0	0
$706$	−19.5405	−0.735416
$707$	0.318944	0.0119951
$708$	0	0
$709$	−6.79918	−0.255349	−0.127674	−	0.991816i	$-0.540751\pi$
$709$	−6.79918	−0.255349	−0.127674	+	0.991816i	$0.540751\pi$
$710$	−6.61555	−0.248277
$711$	0	0
$712$	11.3354	0.424811
$713$	16.1534	0.604948
$714$	0	0
$715$	4.71982	0.176511
$716$	10.8241	0.404516
$717$	0	0
$718$	29.3224	1.09430
$719$	3.41367	0.127308	0.0636542	−	0.997972i	$-0.479725\pi$
$719$	3.41367	0.127308	0.0636542	+	0.997972i	$0.479725\pi$
$720$	0	0
$721$	−0.955118	−0.0355705
$722$	−4.72326	−0.175782
$723$	0	0
$724$	−23.7294	−0.881895
$725$	0.280176	0.0104055
$726$	0	0
$727$	23.0518	0.854942	0.427471	−	0.904029i	$-0.359405\pi$
$727$	23.0518	0.854942	0.427471	+	0.904029i	$0.359405\pi$
$728$	−0.221543	−0.00821093
$729$	0	0
$730$	14.3319	0.530449
$731$	−82.6880	−3.05833
$732$	0	0
$733$	−17.5405	−0.647873	−0.323936	−	0.946079i	$-0.605006\pi$
$733$	−17.5405	−0.647873	−0.323936	+	0.946079i	$0.605006\pi$
$734$	4.26719	0.157505
$735$	0	0
$736$	3.66119	0.134953
$737$	−1.72938	−0.0637024
$738$	0	0
$739$	−6.06475	−0.223095	−0.111548	−	0.993759i	$-0.535581\pi$
$739$	−6.06475	−0.223095	−0.111548	+	0.993759i	$0.535581\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	−0.314649	−0.0115511
$743$	−3.02760	−0.111072	−0.0555359	−	0.998457i	$-0.517687\pi$
$743$	−3.02760	−0.111072	−0.0555359	+	0.998457i	$0.517687\pi$
$744$	0	0
$745$	0.615547	0.0225519
$746$	−8.32582	−0.304830
$747$	0	0
$748$	9.36641	0.342470
$749$	0.957593	0.0349897
$750$	0	0
$751$	−16.4983	−0.602031	−0.301015	−	0.953619i	$-0.597326\pi$
$751$	−16.4983	−0.602031	−0.301015	+	0.953619i	$0.597326\pi$
$752$	6.05520	0.220810
$753$	0	0
$754$	1.05863	0.0385531
$755$	−5.39744	−0.196433
$756$	0	0
$757$	28.0682	1.02016	0.510078	−	0.860128i	$-0.329617\pi$
$757$	28.0682	1.02016	0.510078	+	0.860128i	$0.329617\pi$
$758$	−2.87930	−0.104581
$759$	0	0
$760$	−3.77846	−0.137059
$761$	33.7276	1.22262	0.611311	−	0.791390i	$-0.290642\pi$
$761$	33.7276	1.22262	0.611311	+	0.791390i	$0.290642\pi$
$762$	0	0
$763$	−0.750859	−0.0271829
$764$	18.9931	0.687147
$765$	0	0
$766$	8.44920	0.305282
$767$	−2.32582	−0.0839805
$768$	0	0
$769$	−32.6707	−1.17814	−0.589069	−	0.808083i	$-0.700505\pi$
$769$	−32.6707	−1.17814	−0.589069	+	0.808083i	$0.700505\pi$
$770$	−0.0732411	−0.00263942
$771$	0	0
$772$	16.9966	0.611720
$773$	34.3370	1.23502	0.617508	−	0.786565i	$-0.288142\pi$
$773$	34.3370	1.23502	0.617508	+	0.786565i	$0.288142\pi$
$774$	0	0
$775$	4.41205	0.158486
$776$	9.14486	0.328281
$777$	0	0
$778$	−3.27674	−0.117477
$779$	26.9966	0.967252
$780$	0	0
$781$	−8.26375	−0.295700
$782$	−27.4526	−0.981704
$783$	0	0
$784$	−6.99656	−0.249877
$785$	−19.2147	−0.685801
$786$	0	0
$787$	−34.7259	−1.23785	−0.618923	−	0.785452i	$-0.712431\pi$
$787$	−34.7259	−1.23785	−0.618923	+	0.785452i	$0.712431\pi$
$788$	−3.39744	−0.121029
$789$	0	0
$790$	−5.50172	−0.195742
$791$	−0.580016	−0.0206230
$792$	0	0
$793$	18.9966	0.674588
$794$	7.12070	0.252704
$795$	0	0
$796$	0.824101	0.0292095
$797$	−35.1398	−1.24472	−0.622358	−	0.782733i	$-0.713825\pi$
$797$	−35.1398	−1.24472	−0.622358	+	0.782733i	$0.713825\pi$
$798$	0	0
$799$	−45.4036	−1.60626
$800$	1.00000	0.0353553
$801$	0	0
$802$	26.7129	0.943267
$803$	17.9026	0.631769
$804$	0	0
$805$	0.214667	0.00756602
$806$	16.6707	0.587202
$807$	0	0
$808$	−5.43965	−0.191366
$809$	27.6612	0.972516	0.486258	−	0.873815i	$-0.338362\pi$
$809$	27.6612	0.972516	0.486258	+	0.873815i	$0.338362\pi$
$810$	0	0
$811$	−2.79488	−0.0981417	−0.0490708	−	0.998795i	$-0.515626\pi$
$811$	−2.79488	−0.0981417	−0.0490708	+	0.998795i	$0.515626\pi$
$812$	−0.0164276	−0.000576496 0
$813$	0	0
$814$	1.24914	0.0437824
$815$	7.63016	0.267273
$816$	0	0
$817$	41.6673	1.45775
$818$	−5.53093	−0.193385
$819$	0	0
$820$	−7.14486	−0.249509
$821$	0.609433	0.0212694	0.0106347	−	0.999943i	$-0.496615\pi$
$821$	0.609433	0.0212694	0.0106347	+	0.999943i	$0.496615\pi$
$822$	0	0
$823$	−38.5113	−1.34242	−0.671209	−	0.741268i	$-0.734225\pi$
$823$	−38.5113	−1.34242	−0.671209	+	0.741268i	$0.734225\pi$
$824$	16.2897	0.567480
$825$	0	0
$826$	0.0360915	0.00125578
$827$	−24.0940	−0.837829	−0.418915	−	0.908026i	$-0.637589\pi$
$827$	−24.0940	−0.837829	−0.418915	+	0.908026i	$0.637589\pi$
$828$	0	0
$829$	54.9716	1.90924	0.954622	−	0.297820i	$-0.0962596\pi$
$829$	54.9716	1.90924	0.954622	+	0.297820i	$0.0962596\pi$
$830$	14.3940	0.499623
$831$	0	0
$832$	3.77846	0.130994
$833$	52.4622	1.81771
$834$	0	0
$835$	19.0096	0.657853
$836$	−4.71982	−0.163239
$837$	0	0
$838$	−1.10771	−0.0382654
$839$	9.08461	0.313636	0.156818	−	0.987628i	$-0.449876\pi$
$839$	9.08461	0.313636	0.156818	+	0.987628i	$0.449876\pi$
$840$	0	0
$841$	−28.9215	−0.997293
$842$	14.6707	0.505587
$843$	0	0
$844$	−2.39744	−0.0825234
$845$	−1.27674	−0.0439211
$846$	0	0
$847$	0.553476	0.0190177
$848$	5.36641	0.184283
$849$	0	0
$850$	−7.49828	−0.257189
$851$	−3.66119	−0.125504
$852$	0	0
$853$	−26.1043	−0.893793	−0.446897	−	0.894586i	$-0.647471\pi$
$853$	−26.1043	−0.893793	−0.446897	+	0.894586i	$0.647471\pi$
$854$	−0.294784	−0.0100873
$855$	0	0
$856$	−16.3319	−0.558214
$857$	−43.8172	−1.49677	−0.748384	−	0.663266i	$-0.769170\pi$
$857$	−43.8172	−1.49677	−0.748384	+	0.663266i	$0.769170\pi$
$858$	0	0
$859$	13.5630	0.462765	0.231382	−	0.972863i	$-0.425675\pi$
$859$	13.5630	0.462765	0.231382	+	0.972863i	$0.425675\pi$
$860$	−11.0276	−0.376038
$861$	0	0
$862$	−8.11383	−0.276358
$863$	−11.4707	−0.390467	−0.195233	−	0.980757i	$-0.562546\pi$
$863$	−11.4707	−0.390467	−0.195233	+	0.980757i	$0.562546\pi$
$864$	0	0
$865$	−13.7474	−0.467426
$866$	−11.6612	−0.396263
$867$	0	0
$868$	−0.258692	−0.00878059
$869$	−6.87242	−0.233131
$870$	0	0
$871$	5.23109	0.177249
$872$	12.8061	0.433668
$873$	0	0
$874$	13.8337	0.467930
$875$	0.0586332	0.00198216
$876$	0	0
$877$	54.2372	1.83146	0.915730	−	0.401794i	$-0.131613\pi$
$877$	54.2372	1.83146	0.915730	+	0.401794i	$0.131613\pi$
$878$	−15.9690	−0.538926
$879$	0	0
$880$	1.24914	0.0421085
$881$	−23.8708	−0.804228	−0.402114	−	0.915590i	$-0.631724\pi$
$881$	−23.8708	−0.804228	−0.402114	+	0.915590i	$0.631724\pi$
$882$	0	0
$883$	−12.6267	−0.424923	−0.212461	−	0.977169i	$-0.568148\pi$
$883$	−12.6267	−0.424923	−0.212461	+	0.977169i	$0.568148\pi$
$884$	−28.3319	−0.952906
$885$	0	0
$886$	1.79145	0.0601848
$887$	−19.0016	−0.638012	−0.319006	−	0.947753i	$-0.603349\pi$
$887$	−19.0016	−0.638012	−0.319006	+	0.947753i	$0.603349\pi$
$888$	0	0
$889$	0.0231017	0.000774807 0
$890$	−11.3354	−0.379963
$891$	0	0
$892$	6.66119	0.223033
$893$	22.8793	0.765626
$894$	0	0
$895$	−10.8241	−0.361810
$896$	−0.0586332	−0.00195880
$897$	0	0
$898$	5.70340	0.190325
$899$	1.23615	0.0412279
$900$	0	0
$901$	−40.2388	−1.34055
$902$	−8.92494	−0.297168
$903$	0	0
$904$	9.89229	0.329013
$905$	23.7294	0.788791
$906$	0	0
$907$	−43.4198	−1.44173	−0.720865	−	0.693075i	$-0.756255\pi$
$907$	−43.4198	−1.44173	−0.720865	+	0.693075i	$0.756255\pi$
$908$	−11.1043	−0.368508
$909$	0	0
$910$	0.221543	0.00734408
$911$	−51.4588	−1.70490	−0.852452	−	0.522805i	$-0.824886\pi$
$911$	−51.4588	−1.70490	−0.852452	+	0.522805i	$0.824886\pi$
$912$	0	0
$913$	17.9801	0.595056
$914$	−26.9035	−0.889887
$915$	0	0
$916$	12.7328	0.420704
$917$	−0.0360915	−0.00119185
$918$	0	0
$919$	31.8398	1.05030	0.525148	−	0.851011i	$-0.324010\pi$
$919$	31.8398	1.05030	0.525148	+	0.851011i	$0.324010\pi$
$920$	−3.66119	−0.120706
$921$	0	0
$922$	−2.60600	−0.0858239
$923$	24.9966	0.822772
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	−1.97402	−0.0648703
$927$	0	0
$928$	0.280176	0.00919723
$929$	16.4741	0.540498	0.270249	−	0.962790i	$-0.412894\pi$
$929$	16.4741	0.540498	0.270249	+	0.962790i	$0.412894\pi$
$930$	0	0
$931$	−26.4362	−0.866412
$932$	22.2277	0.728091
$933$	0	0
$934$	−21.9836	−0.719324
$935$	−9.36641	−0.306314
$936$	0	0
$937$	−37.3155	−1.21904	−0.609522	−	0.792769i	$-0.708639\pi$
$937$	−37.3155	−1.21904	−0.609522	+	0.792769i	$0.708639\pi$
$938$	−0.0811748	−0.00265045
$939$	0	0
$940$	−6.05520	−0.197499
$941$	14.7620	0.481228	0.240614	−	0.970621i	$-0.422651\pi$
$941$	14.7620	0.481228	0.240614	+	0.970621i	$0.422651\pi$
$942$	0	0
$943$	26.1587	0.851845
$944$	−0.615547	−0.0200344
$945$	0	0
$946$	−13.7750	−0.447865
$947$	−13.5665	−0.440851	−0.220425	−	0.975404i	$-0.570745\pi$
$947$	−13.5665	−0.440851	−0.220425	+	0.975404i	$0.570745\pi$
$948$	0	0
$949$	−54.1526	−1.75787
$950$	3.77846	0.122589
$951$	0	0
$952$	0.439648	0.0142491
$953$	23.6381	0.765713	0.382856	−	0.923808i	$-0.374940\pi$
$953$	23.6381	0.765713	0.382856	+	0.923808i	$0.374940\pi$
$954$	0	0
$955$	−18.9931	−0.614603
$956$	−20.7164	−0.670016
$957$	0	0
$958$	−38.1526	−1.23265
$959$	−0.560352	−0.0180947
$960$	0	0
$961$	−11.5338	−0.372058
$962$	−3.77846	−0.121822
$963$	0	0
$964$	−1.76547	−0.0568619
$965$	−16.9966	−0.547139
$966$	0	0
$967$	−13.9119	−0.447378	−0.223689	−	0.974661i	$-0.571810\pi$
$967$	−13.9119	−0.447378	−0.223689	+	0.974661i	$0.571810\pi$
$968$	−9.43965	−0.303402
$969$	0	0
$970$	−9.14486	−0.293624
$971$	−6.52932	−0.209536	−0.104768	−	0.994497i	$-0.533410\pi$
$971$	−6.52932	−0.209536	−0.104768	+	0.994497i	$0.533410\pi$
$972$	0	0
$973$	0.0748592	0.00239988
$974$	−34.0844	−1.09213
$975$	0	0
$976$	5.02760	0.160929
$977$	18.5861	0.594623	0.297311	−	0.954781i	$-0.403910\pi$
$977$	18.5861	0.594623	0.297311	+	0.954781i	$0.403910\pi$
$978$	0	0
$979$	−14.1595	−0.452539
$980$	6.99656	0.223497
$981$	0	0
$982$	25.2181	0.804742
$983$	−8.59139	−0.274023	−0.137011	−	0.990569i	$-0.543750\pi$
$983$	−8.59139	−0.274023	−0.137011	+	0.990569i	$0.543750\pi$
$984$	0	0
$985$	3.39744	0.108252
$986$	−2.10084	−0.0669043
$987$	0	0
$988$	14.2767	0.454204
$989$	40.3741	1.28382
$990$	0	0
$991$	−6.25869	−0.198814	−0.0994070	−	0.995047i	$-0.531695\pi$
$991$	−6.25869	−0.198814	−0.0994070	+	0.995047i	$0.531695\pi$
$992$	4.41205	0.140083
$993$	0	0
$994$	−0.387890	−0.0123031
$995$	−0.824101	−0.0261257
$996$	0	0
$997$	15.7095	0.497525	0.248763	−	0.968564i	$-0.419976\pi$
$997$	15.7095	0.497525	0.248763	+	0.968564i	$0.419976\pi$
$998$	−28.9183	−0.915392
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.bi.1.2	yes	3
3.2	odd	2		3330.2.a.bh.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3330.2.a.bh.1.2	✓	3	3.2	odd	2
3330.2.a.bi.1.2	yes	3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.0586332 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.0586332 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.24914 q^{11} +3.77846 q^{13} -0.0586332 q^{14} +1.00000 q^{16} -7.49828 q^{17} +3.77846 q^{19} -1.00000 q^{20} -1.24914 q^{22} +3.66119 q^{23} +1.00000 q^{25} +3.77846 q^{26} -0.0586332 q^{28} +0.280176 q^{29} +4.41205 q^{31} +1.00000 q^{32} -7.49828 q^{34} +0.0586332 q^{35} -1.00000 q^{37} +3.77846 q^{38} -1.00000 q^{40} +7.14486 q^{41} +11.0276 q^{43} -1.24914 q^{44} +3.66119 q^{46} +6.05520 q^{47} -6.99656 q^{49} +1.00000 q^{50} +3.77846 q^{52} +5.36641 q^{53} +1.24914 q^{55} -0.0586332 q^{56} +0.280176 q^{58} -0.615547 q^{59} +5.02760 q^{61} +4.41205 q^{62} +1.00000 q^{64} -3.77846 q^{65} +1.38445 q^{67} -7.49828 q^{68} +0.0586332 q^{70} +6.61555 q^{71} -14.3319 q^{73} -1.00000 q^{74} +3.77846 q^{76} +0.0732411 q^{77} +5.50172 q^{79} -1.00000 q^{80} +7.14486 q^{82} -14.3940 q^{83} +7.49828 q^{85} +11.0276 q^{86} -1.24914 q^{88} +11.3354 q^{89} -0.221543 q^{91} +3.66119 q^{92} +6.05520 q^{94} -3.77846 q^{95} +9.14486 q^{97} -6.99656 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} - q^{7} + 3 q^{8} - 3 q^{10} + 5 q^{11} + 3 q^{13} - q^{14} + 3 q^{16} - 5 q^{17} + 3 q^{19} - 3 q^{20} + 5 q^{22} + q^{23} + 3 q^{25} + 3 q^{26} - q^{28} + 10 q^{29} + 12 q^{31} + 3 q^{32} - 5 q^{34} + q^{35} - 3 q^{37} + 3 q^{38} - 3 q^{40} + 6 q^{41} + 16 q^{43} + 5 q^{44} + q^{46} - 16 q^{47} + 14 q^{49} + 3 q^{50} + 3 q^{52} + 9 q^{53} - 5 q^{55} - q^{56} + 10 q^{58} + 14 q^{59} - 2 q^{61} + 12 q^{62} + 3 q^{64} - 3 q^{65} + 20 q^{67} - 5 q^{68} + q^{70} + 4 q^{71} + 17 q^{73} - 3 q^{74} + 3 q^{76} - 11 q^{77} + 34 q^{79} - 3 q^{80} + 6 q^{82} - 19 q^{83} + 5 q^{85} + 16 q^{86} + 5 q^{88} + 9 q^{89} - 9 q^{91} + q^{92} - 16 q^{94} - 3 q^{95} + 12 q^{97} + 14 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 - q^7 + 3 * q^8 - 3 * q^10 + 5 * q^11 + 3 * q^13 - q^14 + 3 * q^16 - 5 * q^17 + 3 * q^19 - 3 * q^20 + 5 * q^22 + q^23 + 3 * q^25 + 3 * q^26 - q^28 + 10 * q^29 + 12 * q^31 + 3 * q^32 - 5 * q^34 + q^35 - 3 * q^37 + 3 * q^38 - 3 * q^40 + 6 * q^41 + 16 * q^43 + 5 * q^44 + q^46 - 16 * q^47 + 14 * q^49 + 3 * q^50 + 3 * q^52 + 9 * q^53 - 5 * q^55 - q^56 + 10 * q^58 + 14 * q^59 - 2 * q^61 + 12 * q^62 + 3 * q^64 - 3 * q^65 + 20 * q^67 - 5 * q^68 + q^70 + 4 * q^71 + 17 * q^73 - 3 * q^74 + 3 * q^76 - 11 * q^77 + 34 * q^79 - 3 * q^80 + 6 * q^82 - 19 * q^83 + 5 * q^85 + 16 * q^86 + 5 * q^88 + 9 * q^89 - 9 * q^91 + q^92 - 16 * q^94 - 3 * q^95 + 12 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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