LMFDB - Embedded newform 3344.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.11491 q^{3} +2.11491 q^{5} -3.11491 q^{7} +1.47283 q^{9} +O(q^{10})$	$q-2.11491 q^{3} +2.11491 q^{5} -3.11491 q^{7} +1.47283 q^{9} +1.00000 q^{11} -1.83076 q^{13} -4.47283 q^{15} +4.22982 q^{17} +1.00000 q^{19} +6.58774 q^{21} -2.64207 q^{23} -0.527166 q^{25} +3.22982 q^{27} +1.24302 q^{29} -0.811313 q^{31} -2.11491 q^{33} -6.58774 q^{35} +7.58774 q^{37} +3.87189 q^{39} -2.54661 q^{41} -8.41850 q^{43} +3.11491 q^{45} +9.17548 q^{47} +2.70265 q^{49} -8.94567 q^{51} -4.94567 q^{53} +2.11491 q^{55} -2.11491 q^{57} -1.69641 q^{59} -0.715853 q^{61} -4.58774 q^{63} -3.87189 q^{65} +8.21661 q^{67} +5.58774 q^{69} -8.11491 q^{71} -2.22982 q^{73} +1.11491 q^{75} -3.11491 q^{77} +3.51396 q^{79} -11.2493 q^{81} -3.95887 q^{83} +8.94567 q^{85} -2.62887 q^{87} -17.7089 q^{89} +5.70265 q^{91} +1.71585 q^{93} +2.11491 q^{95} -1.69641 q^{97} +1.47283 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{7} - q^{9}+O(q^{10}) $ 3 * q - 3 * q^7 - q^9	$ 3 q - 3 q^{7} - q^{9} + 3 q^{11} - q^{13} - 8 q^{15} + 3 q^{19} + 8 q^{21} - 7 q^{23} - 7 q^{25} - 3 q^{27} + 11 q^{29} - 6 q^{31} - 8 q^{35} + 11 q^{37} - 2 q^{39} - 5 q^{41} - 9 q^{43} + 3 q^{45} + 4 q^{47} - 10 q^{49} - 16 q^{51} - 4 q^{53} - 15 q^{59} - 4 q^{61} - 2 q^{63} + 2 q^{65} - 8 q^{67} + 5 q^{69} - 18 q^{71} + 6 q^{73} - 3 q^{75} - 3 q^{77} - 4 q^{79} - 13 q^{81} - 21 q^{83} + 16 q^{85} + 13 q^{87} - 7 q^{89} - q^{91} + 7 q^{93} - 15 q^{97} - q^{99}+O(q^{100}) $ 3 * q - 3 * q^7 - q^9 + 3 * q^11 - q^13 - 8 * q^15 + 3 * q^19 + 8 * q^21 - 7 * q^23 - 7 * q^25 - 3 * q^27 + 11 * q^29 - 6 * q^31 - 8 * q^35 + 11 * q^37 - 2 * q^39 - 5 * q^41 - 9 * q^43 + 3 * q^45 + 4 * q^47 - 10 * q^49 - 16 * q^51 - 4 * q^53 - 15 * q^59 - 4 * q^61 - 2 * q^63 + 2 * q^65 - 8 * q^67 + 5 * q^69 - 18 * q^71 + 6 * q^73 - 3 * q^75 - 3 * q^77 - 4 * q^79 - 13 * q^81 - 21 * q^83 + 16 * q^85 + 13 * q^87 - 7 * q^89 - q^91 + 7 * q^93 - 15 * q^97 - q^99

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 $

$2$	0	0
$2$	0	0
$3$	−2.11491	−1.22104	−0.610521	−	0.792000i	$-0.709040\pi$
$3$	−2.11491	−1.22104	−0.610521	+	0.792000i	$0.709040\pi$
$4$	0	0
$5$	2.11491	0.945815	0.472908	−	0.881112i	$-0.343204\pi$
$5$	2.11491	0.945815	0.472908	+	0.881112i	$0.343204\pi$
$6$	0	0
$7$	−3.11491	−1.17732	−0.588662	−	0.808379i	$-0.700345\pi$
$7$	−3.11491	−1.17732	−0.588662	+	0.808379i	$0.700345\pi$
$8$	0	0
$9$	1.47283	0.490945
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.83076	−0.507762	−0.253881	−	0.967236i	$-0.581707\pi$
$13$	−1.83076	−0.507762	−0.253881	+	0.967236i	$0.581707\pi$
$14$	0	0
$15$	−4.47283	−1.15488
$16$	0	0
$17$	4.22982	1.02588	0.512940	−	0.858424i	$-0.328556\pi$
$17$	4.22982	1.02588	0.512940	+	0.858424i	$0.328556\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	6.58774	1.43756
$22$	0	0
$23$	−2.64207	−0.550910	−0.275455	−	0.961314i	$-0.588829\pi$
$23$	−2.64207	−0.550910	−0.275455	+	0.961314i	$0.588829\pi$
$24$	0	0
$25$	−0.527166	−0.105433
$26$	0	0
$27$	3.22982	0.621578
$28$	0	0
$29$	1.24302	0.230823	0.115411	−	0.993318i	$-0.463181\pi$
$29$	1.24302	0.230823	0.115411	+	0.993318i	$0.463181\pi$
$30$	0	0
$31$	−0.811313	−0.145716	−0.0728581	−	0.997342i	$-0.523212\pi$
$31$	−0.811313	−0.145716	−0.0728581	+	0.997342i	$0.523212\pi$
$32$	0	0
$33$	−2.11491	−0.368158
$34$	0	0
$35$	−6.58774	−1.11353
$36$	0	0
$37$	7.58774	1.24742	0.623709	−	0.781657i	$-0.285625\pi$
$37$	7.58774	1.24742	0.623709	+	0.781657i	$0.285625\pi$
$38$	0	0
$39$	3.87189	0.619998
$40$	0	0
$41$	−2.54661	−0.397714	−0.198857	−	0.980029i	$-0.563723\pi$
$41$	−2.54661	−0.397714	−0.198857	+	0.980029i	$0.563723\pi$
$42$	0	0
$43$	−8.41850	−1.28381	−0.641905	−	0.766784i	$-0.721856\pi$
$43$	−8.41850	−1.28381	−0.641905	+	0.766784i	$0.721856\pi$
$44$	0	0
$45$	3.11491	0.464343
$46$	0	0
$47$	9.17548	1.33838	0.669191	−	0.743091i	$-0.266641\pi$
$47$	9.17548	1.33838	0.669191	+	0.743091i	$0.266641\pi$
$48$	0	0
$49$	2.70265	0.386093
$50$	0	0
$51$	−8.94567	−1.25264
$52$	0	0
$53$	−4.94567	−0.679340	−0.339670	−	0.940545i	$-0.610315\pi$
$53$	−4.94567	−0.679340	−0.339670	+	0.940545i	$0.610315\pi$
$54$	0	0
$55$	2.11491	0.285174
$56$	0	0
$57$	−2.11491	−0.280126
$58$	0	0
$59$	−1.69641	−0.220853	−0.110427	−	0.993884i	$-0.535222\pi$
$59$	−1.69641	−0.220853	−0.110427	+	0.993884i	$0.535222\pi$
$60$	0	0
$61$	−0.715853	−0.0916556	−0.0458278	−	0.998949i	$-0.514593\pi$
$61$	−0.715853	−0.0916556	−0.0458278	+	0.998949i	$0.514593\pi$
$62$	0	0
$63$	−4.58774	−0.578001
$64$	0	0
$65$	−3.87189	−0.480249
$66$	0	0
$67$	8.21661	1.00382	0.501909	−	0.864920i	$-0.332631\pi$
$67$	8.21661	1.00382	0.501909	+	0.864920i	$0.332631\pi$
$68$	0	0
$69$	5.58774	0.672685
$70$	0	0
$71$	−8.11491	−0.963062	−0.481531	−	0.876429i	$-0.659919\pi$
$71$	−8.11491	−0.963062	−0.481531	+	0.876429i	$0.659919\pi$
$72$	0	0
$73$	−2.22982	−0.260980	−0.130490	−	0.991450i	$-0.541655\pi$
$73$	−2.22982	−0.260980	−0.130490	+	0.991450i	$0.541655\pi$
$74$	0	0
$75$	1.11491	0.128738
$76$	0	0
$77$	−3.11491	−0.354977
$78$	0	0
$79$	3.51396	0.395352	0.197676	−	0.980267i	$-0.436661\pi$
$79$	3.51396	0.395352	0.197676	+	0.980267i	$0.436661\pi$
$80$	0	0
$81$	−11.2493	−1.24992
$82$	0	0
$83$	−3.95887	−0.434543	−0.217271	−	0.976111i	$-0.569716\pi$
$83$	−3.95887	−0.434543	−0.217271	+	0.976111i	$0.569716\pi$
$84$	0	0
$85$	8.94567	0.970294
$86$	0	0
$87$	−2.62887	−0.281844
$88$	0	0
$89$	−17.7089	−1.87714	−0.938569	−	0.345091i	$-0.887848\pi$
$89$	−17.7089	−1.87714	−0.938569	+	0.345091i	$0.887848\pi$
$90$	0	0
$91$	5.70265	0.597800
$92$	0	0
$93$	1.71585	0.177926
$94$	0	0
$95$	2.11491	0.216985
$96$	0	0
$97$	−1.69641	−0.172244	−0.0861220	−	0.996285i	$-0.527447\pi$
$97$	−1.69641	−0.172244	−0.0861220	+	0.996285i	$0.527447\pi$
$98$	0	0
$99$	1.47283	0.148025
$100$	0	0
$101$	−11.9736	−1.19142	−0.595708	−	0.803201i	$-0.703129\pi$
$101$	−11.9736	−1.19142	−0.595708	+	0.803201i	$0.703129\pi$
$102$	0	0
$103$	9.36417	0.922679	0.461340	−	0.887224i	$-0.347369\pi$
$103$	9.36417	0.922679	0.461340	+	0.887224i	$0.347369\pi$
$104$	0	0
$105$	13.9325	1.35967
$106$	0	0
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	−0.108664	−0.0104082	−0.00520408	−	0.999986i	$-0.501657\pi$
$109$	−0.108664	−0.0104082	−0.00520408	+	0.999986i	$0.501657\pi$
$110$	0	0
$111$	−16.0474	−1.52315
$112$	0	0
$113$	12.1949	1.14720	0.573601	−	0.819135i	$-0.305546\pi$
$113$	12.1949	1.14720	0.573601	+	0.819135i	$0.305546\pi$
$114$	0	0
$115$	−5.58774	−0.521060
$116$	0	0
$117$	−2.69641	−0.249283
$118$	0	0
$119$	−13.1755	−1.20779
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.38585	0.485626
$124$	0	0
$125$	−11.6894	−1.04554
$126$	0	0
$127$	13.5529	1.20262	0.601311	−	0.799015i	$-0.294645\pi$
$127$	13.5529	1.20262	0.601311	+	0.799015i	$0.294645\pi$
$128$	0	0
$129$	17.8044	1.56759
$130$	0	0
$131$	−15.1972	−1.32778	−0.663891	−	0.747829i	$-0.731096\pi$
$131$	−15.1972	−1.32778	−0.663891	+	0.747829i	$0.731096\pi$
$132$	0	0
$133$	−3.11491	−0.270097
$134$	0	0
$135$	6.83076	0.587898
$136$	0	0
$137$	9.18869	0.785042	0.392521	−	0.919743i	$-0.371603\pi$
$137$	9.18869	0.785042	0.392521	+	0.919743i	$0.371603\pi$
$138$	0	0
$139$	−1.01320	−0.0859388	−0.0429694	−	0.999076i	$-0.513682\pi$
$139$	−1.01320	−0.0859388	−0.0429694	+	0.999076i	$0.513682\pi$
$140$	0	0
$141$	−19.4053	−1.63422
$142$	0	0
$143$	−1.83076	−0.153096
$144$	0	0
$145$	2.62887	0.218316
$146$	0	0
$147$	−5.71585	−0.471436
$148$	0	0
$149$	22.2034	1.81897	0.909487	−	0.415732i	$-0.136475\pi$
$149$	22.2034	1.81897	0.909487	+	0.415732i	$0.136475\pi$
$150$	0	0
$151$	−16.3510	−1.33062	−0.665311	−	0.746566i	$-0.731701\pi$
$151$	−16.3510	−1.33062	−0.665311	+	0.746566i	$0.731701\pi$
$152$	0	0
$153$	6.22982	0.503651
$154$	0	0
$155$	−1.71585	−0.137821
$156$	0	0
$157$	−17.2904	−1.37992	−0.689962	−	0.723846i	$-0.742373\pi$
$157$	−17.2904	−1.37992	−0.689962	+	0.723846i	$0.742373\pi$
$158$	0	0
$159$	10.4596	0.829503
$160$	0	0
$161$	8.22982	0.648600
$162$	0	0
$163$	−9.51396	−0.745191	−0.372596	−	0.927994i	$-0.621532\pi$
$163$	−9.51396	−0.745191	−0.372596	+	0.927994i	$0.621532\pi$
$164$	0	0
$165$	−4.47283	−0.348210
$166$	0	0
$167$	7.74378	0.599231	0.299616	−	0.954060i	$-0.403142\pi$
$167$	7.74378	0.599231	0.299616	+	0.954060i	$0.403142\pi$
$168$	0	0
$169$	−9.64832	−0.742178
$170$	0	0
$171$	1.47283	0.112630
$172$	0	0
$173$	−8.41850	−0.640047	−0.320023	−	0.947410i	$-0.603691\pi$
$173$	−8.41850	−0.640047	−0.320023	+	0.947410i	$0.603691\pi$
$174$	0	0
$175$	1.64207	0.124129
$176$	0	0
$177$	3.58774	0.269671
$178$	0	0
$179$	7.41850	0.554485	0.277242	−	0.960800i	$-0.410579\pi$
$179$	7.41850	0.554485	0.277242	+	0.960800i	$0.410579\pi$
$180$	0	0
$181$	−15.1406	−1.12539	−0.562696	−	0.826664i	$-0.690236\pi$
$181$	−15.1406	−1.12539	−0.562696	+	0.826664i	$0.690236\pi$
$182$	0	0
$183$	1.51396	0.111915
$184$	0	0
$185$	16.0474	1.17983
$186$	0	0
$187$	4.22982	0.309315
$188$	0	0
$189$	−10.0606	−0.731799
$190$	0	0
$191$	−5.73530	−0.414992	−0.207496	−	0.978236i	$-0.566531\pi$
$191$	−5.73530	−0.414992	−0.207496	+	0.978236i	$0.566531\pi$
$192$	0	0
$193$	4.30984	0.310229	0.155114	−	0.987897i	$-0.450425\pi$
$193$	4.30984	0.310229	0.155114	+	0.987897i	$0.450425\pi$
$194$	0	0
$195$	8.18869	0.586404
$196$	0	0
$197$	−20.3510	−1.44995	−0.724973	−	0.688777i	$-0.758148\pi$
$197$	−20.3510	−1.44995	−0.724973	+	0.688777i	$0.758148\pi$
$198$	0	0
$199$	−9.97359	−0.707009	−0.353505	−	0.935433i	$-0.615010\pi$
$199$	−9.97359	−0.707009	−0.353505	+	0.935433i	$0.615010\pi$
$200$	0	0
$201$	−17.3774	−1.22571
$202$	0	0
$203$	−3.87189	−0.271753
$204$	0	0
$205$	−5.38585	−0.376164
$206$	0	0
$207$	−3.89134	−0.270467
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−5.05433	−0.347955	−0.173977	−	0.984750i	$-0.555662\pi$
$211$	−5.05433	−0.347955	−0.173977	+	0.984750i	$0.555662\pi$
$212$	0	0
$213$	17.1623	1.17594
$214$	0	0
$215$	−17.8044	−1.21425
$216$	0	0
$217$	2.52717	0.171555
$218$	0	0
$219$	4.71585	0.318668
$220$	0	0
$221$	−7.74378	−0.520903
$222$	0	0
$223$	10.1560	0.680098	0.340049	−	0.940408i	$-0.389556\pi$
$223$	10.1560	0.680098	0.340049	+	0.940408i	$0.389556\pi$
$224$	0	0
$225$	−0.776428	−0.0517619
$226$	0	0
$227$	−7.25774	−0.481713	−0.240857	−	0.970561i	$-0.577428\pi$
$227$	−7.25774	−0.481713	−0.240857	+	0.970561i	$0.577428\pi$
$228$	0	0
$229$	−16.4728	−1.08856	−0.544278	−	0.838905i	$-0.683196\pi$
$229$	−16.4728	−1.08856	−0.544278	+	0.838905i	$0.683196\pi$
$230$	0	0
$231$	6.58774	0.433442
$232$	0	0
$233$	−16.9193	−1.10842	−0.554209	−	0.832378i	$-0.686979\pi$
$233$	−16.9193	−1.10842	−0.554209	+	0.832378i	$0.686979\pi$
$234$	0	0
$235$	19.4053	1.26586
$236$	0	0
$237$	−7.43171	−0.482741
$238$	0	0
$239$	−28.9993	−1.87581	−0.937904	−	0.346895i	$-0.887236\pi$
$239$	−28.9993	−1.87581	−0.937904	+	0.346895i	$0.887236\pi$
$240$	0	0
$241$	−11.0955	−0.714721	−0.357361	−	0.933966i	$-0.616323\pi$
$241$	−11.0955	−0.714721	−0.357361	+	0.933966i	$0.616323\pi$
$242$	0	0
$243$	14.1017	0.904625
$244$	0	0
$245$	5.71585	0.365172
$246$	0	0
$247$	−1.83076	−0.116488
$248$	0	0
$249$	8.37265	0.530595
$250$	0	0
$251$	−25.4791	−1.60823	−0.804113	−	0.594477i	$-0.797359\pi$
$251$	−25.4791	−1.60823	−0.804113	+	0.594477i	$0.797359\pi$
$252$	0	0
$253$	−2.64207	−0.166106
$254$	0	0
$255$	−18.9193	−1.18477
$256$	0	0
$257$	−1.36640	−0.0852339	−0.0426170	−	0.999091i	$-0.513570\pi$
$257$	−1.36640	−0.0852339	−0.0426170	+	0.999091i	$0.513570\pi$
$258$	0	0
$259$	−23.6351	−1.46861
$260$	0	0
$261$	1.83076	0.113321
$262$	0	0
$263$	−7.10795	−0.438295	−0.219147	−	0.975692i	$-0.570328\pi$
$263$	−7.10795	−0.438295	−0.219147	+	0.975692i	$0.570328\pi$
$264$	0	0
$265$	−10.4596	−0.642530
$266$	0	0
$267$	37.4527	2.29207
$268$	0	0
$269$	−8.48604	−0.517403	−0.258701	−	0.965957i	$-0.583295\pi$
$269$	−8.48604	−0.517403	−0.258701	+	0.965957i	$0.583295\pi$
$270$	0	0
$271$	7.72210	0.469084	0.234542	−	0.972106i	$-0.424641\pi$
$271$	7.72210	0.469084	0.234542	+	0.972106i	$0.424641\pi$
$272$	0	0
$273$	−12.0606	−0.729939
$274$	0	0
$275$	−0.527166	−0.0317893
$276$	0	0
$277$	20.8106	1.25039	0.625194	−	0.780470i	$-0.285020\pi$
$277$	20.8106	1.25039	0.625194	+	0.780470i	$0.285020\pi$
$278$	0	0
$279$	−1.19493	−0.0715386
$280$	0	0
$281$	−4.67472	−0.278871	−0.139435	−	0.990231i	$-0.544529\pi$
$281$	−4.67472	−0.278871	−0.139435	+	0.990231i	$0.544529\pi$
$282$	0	0
$283$	−6.88509	−0.409276	−0.204638	−	0.978838i	$-0.565602\pi$
$283$	−6.88509	−0.409276	−0.204638	+	0.978838i	$0.565602\pi$
$284$	0	0
$285$	−4.47283	−0.264948
$286$	0	0
$287$	7.93246	0.468239
$288$	0	0
$289$	0.891336	0.0524315
$290$	0	0
$291$	3.58774	0.210317
$292$	0	0
$293$	−8.97984	−0.524608	−0.262304	−	0.964985i	$-0.584482\pi$
$293$	−8.97984	−0.524608	−0.262304	+	0.964985i	$0.584482\pi$
$294$	0	0
$295$	−3.58774	−0.208886
$296$	0	0
$297$	3.22982	0.187413
$298$	0	0
$299$	4.83700	0.279731
$300$	0	0
$301$	26.2229	1.51146
$302$	0	0
$303$	25.3230	1.45477
$304$	0	0
$305$	−1.51396	−0.0866892
$306$	0	0
$307$	−11.8649	−0.677167	−0.338584	−	0.940936i	$-0.609948\pi$
$307$	−11.8649	−0.677167	−0.338584	+	0.940936i	$0.609948\pi$
$308$	0	0
$309$	−19.8044	−1.12663
$310$	0	0
$311$	11.6226	0.659059	0.329529	−	0.944145i	$-0.393110\pi$
$311$	11.6226	0.659059	0.329529	+	0.944145i	$0.393110\pi$
$312$	0	0
$313$	14.9783	0.846625	0.423312	−	0.905984i	$-0.360867\pi$
$313$	14.9783	0.846625	0.423312	+	0.905984i	$0.360867\pi$
$314$	0	0
$315$	−9.70265	−0.546682
$316$	0	0
$317$	−7.21037	−0.404975	−0.202487	−	0.979285i	$-0.564902\pi$
$317$	−7.21037	−0.404975	−0.202487	+	0.979285i	$0.564902\pi$
$318$	0	0
$319$	1.24302	0.0695957
$320$	0	0
$321$	21.1491	1.18043
$322$	0	0
$323$	4.22982	0.235353
$324$	0	0
$325$	0.965115	0.0535349
$326$	0	0
$327$	0.229815	0.0127088
$328$	0	0
$329$	−28.5808	−1.57571
$330$	0	0
$331$	−9.43795	−0.518757	−0.259378	−	0.965776i	$-0.583518\pi$
$331$	−9.43795	−0.518757	−0.259378	+	0.965776i	$0.583518\pi$
$332$	0	0
$333$	11.1755	0.612413
$334$	0	0
$335$	17.3774	0.949427
$336$	0	0
$337$	−17.0451	−0.928508	−0.464254	−	0.885702i	$-0.653677\pi$
$337$	−17.0451	−0.928508	−0.464254	+	0.885702i	$0.653677\pi$
$338$	0	0
$339$	−25.7911	−1.40078
$340$	0	0
$341$	−0.811313	−0.0439351
$342$	0	0
$343$	13.3859	0.722768
$344$	0	0
$345$	11.8176	0.636236
$346$	0	0
$347$	−32.2423	−1.73086	−0.865429	−	0.501032i	$-0.832954\pi$
$347$	−32.2423	−1.73086	−0.865429	+	0.501032i	$0.832954\pi$
$348$	0	0
$349$	8.56829	0.458650	0.229325	−	0.973350i	$-0.426348\pi$
$349$	8.56829	0.458650	0.229325	+	0.973350i	$0.426348\pi$
$350$	0	0
$351$	−5.91302	−0.315614
$352$	0	0
$353$	−4.15604	−0.221203	−0.110602	−	0.993865i	$-0.535278\pi$
$353$	−4.15604	−0.221203	−0.110602	+	0.993865i	$0.535278\pi$
$354$	0	0
$355$	−17.1623	−0.910879
$356$	0	0
$357$	27.8649	1.47477
$358$	0	0
$359$	4.97984	0.262826	0.131413	−	0.991328i	$-0.458049\pi$
$359$	4.97984	0.262826	0.131413	+	0.991328i	$0.458049\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.11491	−0.111004
$364$	0	0
$365$	−4.71585	−0.246839
$366$	0	0
$367$	18.1949	0.949768	0.474884	−	0.880049i	$-0.342490\pi$
$367$	18.1949	0.949768	0.474884	+	0.880049i	$0.342490\pi$
$368$	0	0
$369$	−3.75074	−0.195256
$370$	0	0
$371$	15.4053	0.799803
$372$	0	0
$373$	−28.2012	−1.46020	−0.730101	−	0.683340i	$-0.760527\pi$
$373$	−28.2012	−1.46020	−0.730101	+	0.683340i	$0.760527\pi$
$374$	0	0
$375$	24.7221	1.27664
$376$	0	0
$377$	−2.27567	−0.117203
$378$	0	0
$379$	8.24302	0.423415	0.211708	−	0.977333i	$-0.432098\pi$
$379$	8.24302	0.423415	0.211708	+	0.977333i	$0.432098\pi$
$380$	0	0
$381$	−28.6630	−1.46845
$382$	0	0
$383$	16.6204	0.849262	0.424631	−	0.905366i	$-0.360404\pi$
$383$	16.6204	0.849262	0.424631	+	0.905366i	$0.360404\pi$
$384$	0	0
$385$	−6.58774	−0.335742
$386$	0	0
$387$	−12.3991	−0.630279
$388$	0	0
$389$	7.08698	0.359324	0.179662	−	0.983728i	$-0.442500\pi$
$389$	7.08698	0.359324	0.179662	+	0.983728i	$0.442500\pi$
$390$	0	0
$391$	−11.1755	−0.565168
$392$	0	0
$393$	32.1406	1.62128
$394$	0	0
$395$	7.43171	0.373930
$396$	0	0
$397$	−12.2515	−0.614885	−0.307442	−	0.951567i	$-0.599473\pi$
$397$	−12.2515	−0.614885	−0.307442	+	0.951567i	$0.599473\pi$
$398$	0	0
$399$	6.58774	0.329800
$400$	0	0
$401$	30.3246	1.51434	0.757168	−	0.653220i	$-0.226582\pi$
$401$	30.3246	1.51434	0.757168	+	0.653220i	$0.226582\pi$
$402$	0	0
$403$	1.48532	0.0739891
$404$	0	0
$405$	−23.7911	−1.18219
$406$	0	0
$407$	7.58774	0.376110
$408$	0	0
$409$	−11.7026	−0.578659	−0.289329	−	0.957230i	$-0.593432\pi$
$409$	−11.7026	−0.578659	−0.289329	+	0.957230i	$0.593432\pi$
$410$	0	0
$411$	−19.4332	−0.958570
$412$	0	0
$413$	5.28415	0.260016
$414$	0	0
$415$	−8.37265	−0.410997
$416$	0	0
$417$	2.14283	0.104935
$418$	0	0
$419$	24.9582	1.21929	0.609643	−	0.792676i	$-0.291313\pi$
$419$	24.9582	1.21929	0.609643	+	0.792676i	$0.291313\pi$
$420$	0	0
$421$	−18.3774	−0.895658	−0.447829	−	0.894119i	$-0.647803\pi$
$421$	−18.3774	−0.895658	−0.447829	+	0.894119i	$0.647803\pi$
$422$	0	0
$423$	13.5140	0.657071
$424$	0	0
$425$	−2.22982	−0.108162
$426$	0	0
$427$	2.22982	0.107908
$428$	0	0
$429$	3.87189	0.186937
$430$	0	0
$431$	17.0668	0.822080	0.411040	−	0.911617i	$-0.365166\pi$
$431$	17.0668	0.822080	0.411040	+	0.911617i	$0.365166\pi$
$432$	0	0
$433$	−0.424745	−0.0204119	−0.0102060	−	0.999948i	$-0.503249\pi$
$433$	−0.424745	−0.0204119	−0.0102060	+	0.999948i	$0.503249\pi$
$434$	0	0
$435$	−5.55982	−0.266573
$436$	0	0
$437$	−2.64207	−0.126388
$438$	0	0
$439$	21.1366	1.00879	0.504397	−	0.863472i	$-0.331715\pi$
$439$	21.1366	1.00879	0.504397	+	0.863472i	$0.331715\pi$
$440$	0	0
$441$	3.98055	0.189550
$442$	0	0
$443$	39.2229	1.86353	0.931767	−	0.363057i	$-0.118267\pi$
$443$	39.2229	1.86353	0.931767	+	0.363057i	$0.118267\pi$
$444$	0	0
$445$	−37.4527	−1.77543
$446$	0	0
$447$	−46.9582	−2.22104
$448$	0	0
$449$	−10.3470	−0.488303	−0.244152	−	0.969737i	$-0.578509\pi$
$449$	−10.3470	−0.488303	−0.244152	+	0.969737i	$0.578509\pi$
$450$	0	0
$451$	−2.54661	−0.119915
$452$	0	0
$453$	34.5808	1.62475
$454$	0	0
$455$	12.0606	0.565408
$456$	0	0
$457$	−0.594702	−0.0278190	−0.0139095	−	0.999903i	$-0.504428\pi$
$457$	−0.594702	−0.0278190	−0.0139095	+	0.999903i	$0.504428\pi$
$458$	0	0
$459$	13.6615	0.637665
$460$	0	0
$461$	−13.5962	−0.633239	−0.316620	−	0.948553i	$-0.602548\pi$
$461$	−13.5962	−0.633239	−0.316620	+	0.948553i	$0.602548\pi$
$462$	0	0
$463$	25.0364	1.16354	0.581770	−	0.813353i	$-0.302360\pi$
$463$	25.0364	1.16354	0.581770	+	0.813353i	$0.302360\pi$
$464$	0	0
$465$	3.62887	0.168285
$466$	0	0
$467$	−24.6282	−1.13965	−0.569827	−	0.821764i	$-0.692990\pi$
$467$	−24.6282	−1.13965	−0.569827	+	0.821764i	$0.692990\pi$
$468$	0	0
$469$	−25.5940	−1.18182
$470$	0	0
$471$	36.5676	1.68495
$472$	0	0
$473$	−8.41850	−0.387083
$474$	0	0
$475$	−0.527166	−0.0241880
$476$	0	0
$477$	−7.28415	−0.333518
$478$	0	0
$479$	−10.5855	−0.483664	−0.241832	−	0.970318i	$-0.577748\pi$
$479$	−10.5855	−0.483664	−0.241832	+	0.970318i	$0.577748\pi$
$480$	0	0
$481$	−13.8913	−0.633390
$482$	0	0
$483$	−17.4053	−0.791968
$484$	0	0
$485$	−3.58774	−0.162911
$486$	0	0
$487$	25.6289	1.16135	0.580677	−	0.814134i	$-0.302788\pi$
$487$	25.6289	1.16135	0.580677	+	0.814134i	$0.302788\pi$
$488$	0	0
$489$	20.1212	0.909910
$490$	0	0
$491$	−18.2081	−0.821722	−0.410861	−	0.911698i	$-0.634772\pi$
$491$	−18.2081	−0.821722	−0.410861	+	0.911698i	$0.634772\pi$
$492$	0	0
$493$	5.25774	0.236797
$494$	0	0
$495$	3.11491	0.140005
$496$	0	0
$497$	25.2772	1.13384
$498$	0	0
$499$	−35.9861	−1.61096	−0.805479	−	0.592624i	$-0.798092\pi$
$499$	−35.9861	−1.61096	−0.805479	+	0.592624i	$0.798092\pi$
$500$	0	0
$501$	−16.3774	−0.731687
$502$	0	0
$503$	−36.7042	−1.63656	−0.818279	−	0.574821i	$-0.805072\pi$
$503$	−36.7042	−1.63656	−0.818279	+	0.574821i	$0.805072\pi$
$504$	0	0
$505$	−25.3230	−1.12686
$506$	0	0
$507$	20.4053	0.906231
$508$	0	0
$509$	16.0474	0.711287	0.355644	−	0.934622i	$-0.384262\pi$
$509$	16.0474	0.711287	0.355644	+	0.934622i	$0.384262\pi$
$510$	0	0
$511$	6.94567	0.307258
$512$	0	0
$513$	3.22982	0.142600
$514$	0	0
$515$	19.8044	0.872684
$516$	0	0
$517$	9.17548	0.403537
$518$	0	0
$519$	17.8044	0.781524
$520$	0	0
$521$	41.3704	1.81247	0.906235	−	0.422774i	$-0.138943\pi$
$521$	41.3704	1.81247	0.906235	+	0.422774i	$0.138943\pi$
$522$	0	0
$523$	28.1072	1.22904	0.614522	−	0.788900i	$-0.289349\pi$
$523$	28.1072	1.22904	0.614522	+	0.788900i	$0.289349\pi$
$524$	0	0
$525$	−3.47283	−0.151567
$526$	0	0
$527$	−3.43171	−0.149487
$528$	0	0
$529$	−16.0194	−0.696498
$530$	0	0
$531$	−2.49852	−0.108427
$532$	0	0
$533$	4.66224	0.201944
$534$	0	0
$535$	−21.1491	−0.914354
$536$	0	0
$537$	−15.6894	−0.677050
$538$	0	0
$539$	2.70265	0.116411
$540$	0	0
$541$	19.8788	0.854658	0.427329	−	0.904096i	$-0.359455\pi$
$541$	19.8788	0.854658	0.427329	+	0.904096i	$0.359455\pi$
$542$	0	0
$543$	32.0210	1.37415
$544$	0	0
$545$	−0.229815	−0.00984420
$546$	0	0
$547$	−8.75475	−0.374326	−0.187163	−	0.982329i	$-0.559929\pi$
$547$	−8.75475	−0.374326	−0.187163	+	0.982329i	$0.559929\pi$
$548$	0	0
$549$	−1.05433	−0.0449978
$550$	0	0
$551$	1.24302	0.0529544
$552$	0	0
$553$	−10.9457	−0.465457
$554$	0	0
$555$	−33.9387	−1.44062
$556$	0	0
$557$	12.6072	0.534184	0.267092	−	0.963671i	$-0.413937\pi$
$557$	12.6072	0.534184	0.267092	+	0.963671i	$0.413937\pi$
$558$	0	0
$559$	15.4123	0.651869
$560$	0	0
$561$	−8.94567	−0.377686
$562$	0	0
$563$	−4.25622	−0.179378	−0.0896892	−	0.995970i	$-0.528587\pi$
$563$	−4.25622	−0.179378	−0.0896892	+	0.995970i	$0.528587\pi$
$564$	0	0
$565$	25.7911	1.08504
$566$	0	0
$567$	35.0404	1.47156
$568$	0	0
$569$	−0.897579	−0.0376285	−0.0188142	−	0.999823i	$-0.505989\pi$
$569$	−0.897579	−0.0376285	−0.0188142	+	0.999823i	$0.505989\pi$
$570$	0	0
$571$	7.68320	0.321532	0.160766	−	0.986993i	$-0.448604\pi$
$571$	7.68320	0.321532	0.160766	+	0.986993i	$0.448604\pi$
$572$	0	0
$573$	12.1296	0.506722
$574$	0	0
$575$	1.39281	0.0580843
$576$	0	0
$577$	−0.121868	−0.00507344	−0.00253672	−	0.999997i	$-0.500807\pi$
$577$	−0.121868	−0.00507344	−0.00253672	+	0.999997i	$0.500807\pi$
$578$	0	0
$579$	−9.11491	−0.378803
$580$	0	0
$581$	12.3315	0.511598
$582$	0	0
$583$	−4.94567	−0.204829
$584$	0	0
$585$	−5.70265	−0.235776
$586$	0	0
$587$	−20.6770	−0.853429	−0.426715	−	0.904386i	$-0.640329\pi$
$587$	−20.6770	−0.853429	−0.426715	+	0.904386i	$0.640329\pi$
$588$	0	0
$589$	−0.811313	−0.0334296
$590$	0	0
$591$	43.0404	1.77045
$592$	0	0
$593$	−3.06682	−0.125939	−0.0629696	−	0.998015i	$-0.520057\pi$
$593$	−3.06682	−0.125939	−0.0629696	+	0.998015i	$0.520057\pi$
$594$	0	0
$595$	−27.8649	−1.14235
$596$	0	0
$597$	21.0932	0.863288
$598$	0	0
$599$	−8.60942	−0.351771	−0.175886	−	0.984411i	$-0.556279\pi$
$599$	−8.60942	−0.351771	−0.175886	+	0.984411i	$0.556279\pi$
$600$	0	0
$601$	22.4768	0.916850	0.458425	−	0.888733i	$-0.348414\pi$
$601$	22.4768	0.916850	0.458425	+	0.888733i	$0.348414\pi$
$602$	0	0
$603$	12.1017	0.492819
$604$	0	0
$605$	2.11491	0.0859832
$606$	0	0
$607$	35.2966	1.43265	0.716323	−	0.697769i	$-0.245824\pi$
$607$	35.2966	1.43265	0.716323	+	0.697769i	$0.245824\pi$
$608$	0	0
$609$	8.18869	0.331822
$610$	0	0
$611$	−16.7981	−0.679579
$612$	0	0
$613$	−2.09474	−0.0846059	−0.0423029	−	0.999105i	$-0.513469\pi$
$613$	−2.09474	−0.0846059	−0.0423029	+	0.999105i	$0.513469\pi$
$614$	0	0
$615$	11.3906	0.459312
$616$	0	0
$617$	−11.1010	−0.446909	−0.223454	−	0.974714i	$-0.571733\pi$
$617$	−11.1010	−0.446909	−0.223454	+	0.974714i	$0.571733\pi$
$618$	0	0
$619$	21.4402	0.861754	0.430877	−	0.902411i	$-0.358204\pi$
$619$	21.4402	0.861754	0.430877	+	0.902411i	$0.358204\pi$
$620$	0	0
$621$	−8.53341	−0.342434
$622$	0	0
$623$	55.1616	2.21000
$624$	0	0
$625$	−22.0863	−0.883451
$626$	0	0
$627$	−2.11491	−0.0844613
$628$	0	0
$629$	32.0947	1.27970
$630$	0	0
$631$	−26.1296	−1.04020	−0.520102	−	0.854104i	$-0.674106\pi$
$631$	−26.1296	−1.04020	−0.520102	+	0.854104i	$0.674106\pi$
$632$	0	0
$633$	10.6894	0.424867
$634$	0	0
$635$	28.6630	1.13746
$636$	0	0
$637$	−4.94790	−0.196043
$638$	0	0
$639$	−11.9519	−0.472810
$640$	0	0
$641$	44.5893	1.76117	0.880585	−	0.473888i	$-0.157150\pi$
$641$	44.5893	1.76117	0.880585	+	0.473888i	$0.157150\pi$
$642$	0	0
$643$	27.6825	1.09169	0.545845	−	0.837886i	$-0.316209\pi$
$643$	27.6825	1.09169	0.545845	+	0.837886i	$0.316209\pi$
$644$	0	0
$645$	37.6546	1.48265
$646$	0	0
$647$	49.9512	1.96378	0.981892	−	0.189441i	$-0.0606677\pi$
$647$	49.9512	1.96378	0.981892	+	0.189441i	$0.0606677\pi$
$648$	0	0
$649$	−1.69641	−0.0665898
$650$	0	0
$651$	−5.34472	−0.209476
$652$	0	0
$653$	12.6638	0.495571	0.247786	−	0.968815i	$-0.420297\pi$
$653$	12.6638	0.495571	0.247786	+	0.968815i	$0.420297\pi$
$654$	0	0
$655$	−32.1406	−1.25584
$656$	0	0
$657$	−3.28415	−0.128127
$658$	0	0
$659$	16.1909	0.630709	0.315354	−	0.948974i	$-0.397877\pi$
$659$	16.1909	0.630709	0.315354	+	0.948974i	$0.397877\pi$
$660$	0	0
$661$	28.4372	1.10608	0.553040	−	0.833155i	$-0.313468\pi$
$661$	28.4372	1.10608	0.553040	+	0.833155i	$0.313468\pi$
$662$	0	0
$663$	16.3774	0.636044
$664$	0	0
$665$	−6.58774	−0.255462
$666$	0	0
$667$	−3.28415	−0.127163
$668$	0	0
$669$	−21.4791	−0.830429
$670$	0	0
$671$	−0.715853	−0.0276352
$672$	0	0
$673$	20.2904	0.782137	0.391069	−	0.920362i	$-0.372106\pi$
$673$	20.2904	0.782137	0.391069	+	0.920362i	$0.372106\pi$
$674$	0	0
$675$	−1.70265	−0.0655350
$676$	0	0
$677$	27.1513	1.04351	0.521755	−	0.853095i	$-0.325278\pi$
$677$	27.1513	1.04351	0.521755	+	0.853095i	$0.325278\pi$
$678$	0	0
$679$	5.28415	0.202787
$680$	0	0
$681$	15.3494	0.588192
$682$	0	0
$683$	5.13659	0.196546	0.0982731	−	0.995159i	$-0.468668\pi$
$683$	5.13659	0.196546	0.0982731	+	0.995159i	$0.468668\pi$
$684$	0	0
$685$	19.4332	0.742505
$686$	0	0
$687$	34.8385	1.32917
$688$	0	0
$689$	9.05433	0.344943
$690$	0	0
$691$	−48.7104	−1.85303	−0.926516	−	0.376256i	$-0.877211\pi$
$691$	−48.7104	−1.85303	−0.926516	+	0.376256i	$0.877211\pi$
$692$	0	0
$693$	−4.58774	−0.174274
$694$	0	0
$695$	−2.14283	−0.0812823
$696$	0	0
$697$	−10.7717	−0.408007
$698$	0	0
$699$	35.7827	1.35342
$700$	0	0
$701$	37.2313	1.40621	0.703104	−	0.711087i	$-0.251797\pi$
$701$	37.2313	1.40621	0.703104	+	0.711087i	$0.251797\pi$
$702$	0	0
$703$	7.58774	0.286177
$704$	0	0
$705$	−41.0404	−1.54567
$706$	0	0
$707$	37.2966	1.40268
$708$	0	0
$709$	−45.1748	−1.69657	−0.848287	−	0.529537i	$-0.822366\pi$
$709$	−45.1748	−1.69657	−0.848287	+	0.529537i	$0.822366\pi$
$710$	0	0
$711$	5.17548	0.194096
$712$	0	0
$713$	2.14355	0.0802766
$714$	0	0
$715$	−3.87189	−0.144800
$716$	0	0
$717$	61.3308	2.29044
$718$	0	0
$719$	−0.789632	−0.0294483	−0.0147241	−	0.999892i	$-0.504687\pi$
$719$	−0.789632	−0.0294483	−0.0147241	+	0.999892i	$0.504687\pi$
$720$	0	0
$721$	−29.1685	−1.08629
$722$	0	0
$723$	23.4659	0.872705
$724$	0	0
$725$	−0.655277	−0.0243364
$726$	0	0
$727$	26.2338	0.972959	0.486479	−	0.873692i	$-0.338281\pi$
$727$	26.2338	0.972959	0.486479	+	0.873692i	$0.338281\pi$
$728$	0	0
$729$	3.92399	0.145333
$730$	0	0
$731$	−35.6087	−1.31704
$732$	0	0
$733$	−11.7438	−0.433766	−0.216883	−	0.976198i	$-0.569589\pi$
$733$	−11.7438	−0.433766	−0.216883	+	0.976198i	$0.569589\pi$
$734$	0	0
$735$	−12.0885	−0.445891
$736$	0	0
$737$	8.21661	0.302663
$738$	0	0
$739$	−32.7959	−1.20642	−0.603208	−	0.797584i	$-0.706111\pi$
$739$	−32.7959	−1.20642	−0.603208	+	0.797584i	$0.706111\pi$
$740$	0	0
$741$	3.87189	0.142237
$742$	0	0
$743$	−11.7438	−0.430837	−0.215419	−	0.976522i	$-0.569112\pi$
$743$	−11.7438	−0.430837	−0.215419	+	0.976522i	$0.569112\pi$
$744$	0	0
$745$	46.9582	1.72041
$746$	0	0
$747$	−5.83076	−0.213336
$748$	0	0
$749$	31.1491	1.13816
$750$	0	0
$751$	33.5738	1.22513	0.612563	−	0.790422i	$-0.290139\pi$
$751$	33.5738	1.22513	0.612563	+	0.790422i	$0.290139\pi$
$752$	0	0
$753$	53.8859	1.96371
$754$	0	0
$755$	−34.5808	−1.25852
$756$	0	0
$757$	11.9061	0.432733	0.216366	−	0.976312i	$-0.430579\pi$
$757$	11.9061	0.432733	0.216366	+	0.976312i	$0.430579\pi$
$758$	0	0
$759$	5.58774	0.202822
$760$	0	0
$761$	46.4332	1.68320	0.841602	−	0.540099i	$-0.181613\pi$
$761$	46.4332	1.68320	0.841602	+	0.540099i	$0.181613\pi$
$762$	0	0
$763$	0.338479	0.0122538
$764$	0	0
$765$	13.1755	0.476361
$766$	0	0
$767$	3.10571	0.112141
$768$	0	0
$769$	3.96111	0.142841	0.0714206	−	0.997446i	$-0.477247\pi$
$769$	3.96111	0.142841	0.0714206	+	0.997446i	$0.477247\pi$
$770$	0	0
$771$	2.88982	0.104074
$772$	0	0
$773$	−1.78267	−0.0641182	−0.0320591	−	0.999486i	$-0.510206\pi$
$773$	−1.78267	−0.0641182	−0.0320591	+	0.999486i	$0.510206\pi$
$774$	0	0
$775$	0.427697	0.0153633
$776$	0	0
$777$	49.9861	1.79324
$778$	0	0
$779$	−2.54661	−0.0912419
$780$	0	0
$781$	−8.11491	−0.290374
$782$	0	0
$783$	4.01472	0.143474
$784$	0	0
$785$	−36.5676	−1.30515
$786$	0	0
$787$	−26.4860	−0.944125	−0.472063	−	0.881565i	$-0.656490\pi$
$787$	−26.4860	−0.944125	−0.472063	+	0.881565i	$0.656490\pi$
$788$	0	0
$789$	15.0327	0.535177
$790$	0	0
$791$	−37.9861	−1.35063
$792$	0	0
$793$	1.31055	0.0465392
$794$	0	0
$795$	22.1212	0.784556
$796$	0	0
$797$	40.2772	1.42669	0.713346	−	0.700812i	$-0.247179\pi$
$797$	40.2772	1.42669	0.713346	+	0.700812i	$0.247179\pi$
$798$	0	0
$799$	38.8106	1.37302
$800$	0	0
$801$	−26.0823	−0.921571
$802$	0	0
$803$	−2.22982	−0.0786885
$804$	0	0
$805$	17.4053	0.613456
$806$	0	0
$807$	17.9472	0.631771
$808$	0	0
$809$	−12.8415	−0.451482	−0.225741	−	0.974187i	$-0.572480\pi$
$809$	−12.8415	−0.451482	−0.225741	+	0.974187i	$0.572480\pi$
$810$	0	0
$811$	−16.8634	−0.592154	−0.296077	−	0.955164i	$-0.595679\pi$
$811$	−16.8634	−0.592154	−0.296077	+	0.955164i	$0.595679\pi$
$812$	0	0
$813$	−16.3315	−0.572771
$814$	0	0
$815$	−20.1212	−0.704813
$816$	0	0
$817$	−8.41850	−0.294526
$818$	0	0
$819$	8.39905	0.293487
$820$	0	0
$821$	15.0543	0.525400	0.262700	−	0.964878i	$-0.415387\pi$
$821$	15.0543	0.525400	0.262700	+	0.964878i	$0.415387\pi$
$822$	0	0
$823$	−36.1949	−1.26168	−0.630838	−	0.775915i	$-0.717289\pi$
$823$	−36.1949	−1.26168	−0.630838	+	0.775915i	$0.717289\pi$
$824$	0	0
$825$	1.11491	0.0388161
$826$	0	0
$827$	−36.4596	−1.26783	−0.633913	−	0.773405i	$-0.718552\pi$
$827$	−36.4596	−1.26783	−0.633913	+	0.773405i	$0.718552\pi$
$828$	0	0
$829$	14.5070	0.503849	0.251924	−	0.967747i	$-0.418937\pi$
$829$	14.5070	0.503849	0.251924	+	0.967747i	$0.418937\pi$
$830$	0	0
$831$	−44.0125	−1.52678
$832$	0	0
$833$	11.4317	0.396085
$834$	0	0
$835$	16.3774	0.566762
$836$	0	0
$837$	−2.62039	−0.0905740
$838$	0	0
$839$	0.249979	0.00863025	0.00431512	−	0.999991i	$-0.498626\pi$
$839$	0.249979	0.00863025	0.00431512	+	0.999991i	$0.498626\pi$
$840$	0	0
$841$	−27.4549	−0.946721
$842$	0	0
$843$	9.88661	0.340513
$844$	0	0
$845$	−20.4053	−0.701964
$846$	0	0
$847$	−3.11491	−0.107029
$848$	0	0
$849$	14.5613	0.499744
$850$	0	0
$851$	−20.0474	−0.687215
$852$	0	0
$853$	21.5793	0.738860	0.369430	−	0.929259i	$-0.379553\pi$
$853$	21.5793	0.738860	0.369430	+	0.929259i	$0.379553\pi$
$854$	0	0
$855$	3.11491	0.106528
$856$	0	0
$857$	46.2765	1.58077	0.790387	−	0.612608i	$-0.209879\pi$
$857$	46.2765	1.58077	0.790387	+	0.612608i	$0.209879\pi$
$858$	0	0
$859$	−9.91373	−0.338252	−0.169126	−	0.985594i	$-0.554095\pi$
$859$	−9.91373	−0.338252	−0.169126	+	0.985594i	$0.554095\pi$
$860$	0	0
$861$	−16.7764	−0.571739
$862$	0	0
$863$	−42.9923	−1.46348	−0.731738	−	0.681586i	$-0.761290\pi$
$863$	−42.9923	−1.46348	−0.731738	+	0.681586i	$0.761290\pi$
$864$	0	0
$865$	−17.8044	−0.605366
$866$	0	0
$867$	−1.88509	−0.0640211
$868$	0	0
$869$	3.51396	0.119203
$870$	0	0
$871$	−15.0426	−0.509701
$872$	0	0
$873$	−2.49852	−0.0845622
$874$	0	0
$875$	36.4115	1.23093
$876$	0	0
$877$	8.43099	0.284694	0.142347	−	0.989817i	$-0.454535\pi$
$877$	8.43099	0.284694	0.142347	+	0.989817i	$0.454535\pi$
$878$	0	0
$879$	18.9915	0.640568
$880$	0	0
$881$	16.9714	0.571780	0.285890	−	0.958263i	$-0.407711\pi$
$881$	16.9714	0.571780	0.285890	+	0.958263i	$0.407711\pi$
$882$	0	0
$883$	−30.8929	−1.03963	−0.519814	−	0.854280i	$-0.673999\pi$
$883$	−30.8929	−1.03963	−0.519814	+	0.854280i	$0.673999\pi$
$884$	0	0
$885$	7.58774	0.255059
$886$	0	0
$887$	−51.8774	−1.74187	−0.870937	−	0.491395i	$-0.836487\pi$
$887$	−51.8774	−1.74187	−0.870937	+	0.491395i	$0.836487\pi$
$888$	0	0
$889$	−42.2159	−1.41588
$890$	0	0
$891$	−11.2493	−0.376864
$892$	0	0
$893$	9.17548	0.307046
$894$	0	0
$895$	15.6894	0.524440
$896$	0	0
$897$	−10.2298	−0.341564
$898$	0	0
$899$	−1.00848	−0.0336346
$900$	0	0
$901$	−20.9193	−0.696922
$902$	0	0
$903$	−55.4589	−1.84556
$904$	0	0
$905$	−32.0210	−1.06441
$906$	0	0
$907$	−34.0558	−1.13081	−0.565403	−	0.824815i	$-0.691279\pi$
$907$	−34.0558	−1.13081	−0.565403	+	0.824815i	$0.691279\pi$
$908$	0	0
$909$	−17.6351	−0.584920
$910$	0	0
$911$	−13.6740	−0.453040	−0.226520	−	0.974007i	$-0.572735\pi$
$911$	−13.6740	−0.453040	−0.226520	+	0.974007i	$0.572735\pi$
$912$	0	0
$913$	−3.95887	−0.131020
$914$	0	0
$915$	3.20189	0.105851
$916$	0	0
$917$	47.3378	1.56323
$918$	0	0
$919$	38.5132	1.27043	0.635217	−	0.772333i	$-0.280911\pi$
$919$	38.5132	1.27043	0.635217	+	0.772333i	$0.280911\pi$
$920$	0	0
$921$	25.0932	0.826850
$922$	0	0
$923$	14.8565	0.489006
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	13.7919	0.452984
$928$	0	0
$929$	−8.11642	−0.266291	−0.133146	−	0.991096i	$-0.542508\pi$
$929$	−8.11642	−0.266291	−0.133146	+	0.991096i	$0.542508\pi$
$930$	0	0
$931$	2.70265	0.0885757
$932$	0	0
$933$	−24.5808	−0.804739
$934$	0	0
$935$	8.94567	0.292555
$936$	0	0
$937$	−35.7438	−1.16770	−0.583849	−	0.811862i	$-0.698454\pi$
$937$	−35.7438	−1.16770	−0.583849	+	0.811862i	$0.698454\pi$
$938$	0	0
$939$	−31.6778	−1.03376
$940$	0	0
$941$	−37.5653	−1.22460	−0.612298	−	0.790627i	$-0.709755\pi$
$941$	−37.5653	−1.22460	−0.612298	+	0.790627i	$0.709755\pi$
$942$	0	0
$943$	6.72834	0.219105
$944$	0	0
$945$	−21.2772	−0.692147
$946$	0	0
$947$	−48.8844	−1.58853	−0.794264	−	0.607572i	$-0.792143\pi$
$947$	−48.8844	−1.58853	−0.794264	+	0.607572i	$0.792143\pi$
$948$	0	0
$949$	4.08226	0.132516
$950$	0	0
$951$	15.2493	0.494491
$952$	0	0
$953$	−22.7019	−0.735388	−0.367694	−	0.929947i	$-0.619853\pi$
$953$	−22.7019	−0.735388	−0.367694	+	0.929947i	$0.619853\pi$
$954$	0	0
$955$	−12.1296	−0.392506
$956$	0	0
$957$	−2.62887	−0.0849793
$958$	0	0
$959$	−28.6219	−0.924250
$960$	0	0
$961$	−30.3418	−0.978767
$962$	0	0
$963$	−14.7283	−0.474614
$964$	0	0
$965$	9.11491	0.293419
$966$	0	0
$967$	4.45963	0.143412	0.0717060	−	0.997426i	$-0.477156\pi$
$967$	4.45963	0.143412	0.0717060	+	0.997426i	$0.477156\pi$
$968$	0	0
$969$	−8.94567	−0.287376
$970$	0	0
$971$	28.0760	0.901002	0.450501	−	0.892776i	$-0.351245\pi$
$971$	28.0760	0.901002	0.450501	+	0.892776i	$0.351245\pi$
$972$	0	0
$973$	3.15604	0.101178
$974$	0	0
$975$	−2.04113	−0.0653684
$976$	0	0
$977$	−9.31456	−0.297999	−0.149000	−	0.988837i	$-0.547605\pi$
$977$	−9.31456	−0.297999	−0.149000	+	0.988837i	$0.547605\pi$
$978$	0	0
$979$	−17.7089	−0.565979
$980$	0	0
$981$	−0.160045	−0.00510983
$982$	0	0
$983$	−8.17772	−0.260829	−0.130414	−	0.991460i	$-0.541631\pi$
$983$	−8.17772	−0.260829	−0.130414	+	0.991460i	$0.541631\pi$
$984$	0	0
$985$	−43.0404	−1.37138
$986$	0	0
$987$	60.4457	1.92401
$988$	0	0
$989$	22.2423	0.707264
$990$	0	0
$991$	−25.4395	−0.808111	−0.404056	−	0.914734i	$-0.632400\pi$
$991$	−25.4395	−0.808111	−0.404056	+	0.914734i	$0.632400\pi$
$992$	0	0
$993$	19.9604	0.633424
$994$	0	0
$995$	−21.0932	−0.668700
$996$	0	0
$997$	44.1381	1.39787	0.698934	−	0.715186i	$-0.253658\pi$
$997$	44.1381	1.39787	0.698934	+	0.715186i	$0.253658\pi$
$998$	0	0
$999$	24.5070	0.775367

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.r.1.1		3
4.3	odd	2		1672.2.a.e.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.e.1.3	✓	3	4.3	odd	2
3344.2.a.r.1.1		3	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-2.11491 q^{3} +2.11491 q^{5} -3.11491 q^{7} +1.47283 q^{9} +O(q^{10})\)	\(q-2.11491 q^{3} +2.11491 q^{5} -3.11491 q^{7} +1.47283 q^{9} +1.00000 q^{11} -1.83076 q^{13} -4.47283 q^{15} +4.22982 q^{17} +1.00000 q^{19} +6.58774 q^{21} -2.64207 q^{23} -0.527166 q^{25} +3.22982 q^{27} +1.24302 q^{29} -0.811313 q^{31} -2.11491 q^{33} -6.58774 q^{35} +7.58774 q^{37} +3.87189 q^{39} -2.54661 q^{41} -8.41850 q^{43} +3.11491 q^{45} +9.17548 q^{47} +2.70265 q^{49} -8.94567 q^{51} -4.94567 q^{53} +2.11491 q^{55} -2.11491 q^{57} -1.69641 q^{59} -0.715853 q^{61} -4.58774 q^{63} -3.87189 q^{65} +8.21661 q^{67} +5.58774 q^{69} -8.11491 q^{71} -2.22982 q^{73} +1.11491 q^{75} -3.11491 q^{77} +3.51396 q^{79} -11.2493 q^{81} -3.95887 q^{83} +8.94567 q^{85} -2.62887 q^{87} -17.7089 q^{89} +5.70265 q^{91} +1.71585 q^{93} +2.11491 q^{95} -1.69641 q^{97} +1.47283 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{7} - q^{9}+O(q^{10}) \) 3 * q - 3 * q^7 - q^9	\( 3 q - 3 q^{7} - q^{9} + 3 q^{11} - q^{13} - 8 q^{15} + 3 q^{19} + 8 q^{21} - 7 q^{23} - 7 q^{25} - 3 q^{27} + 11 q^{29} - 6 q^{31} - 8 q^{35} + 11 q^{37} - 2 q^{39} - 5 q^{41} - 9 q^{43} + 3 q^{45} + 4 q^{47} - 10 q^{49} - 16 q^{51} - 4 q^{53} - 15 q^{59} - 4 q^{61} - 2 q^{63} + 2 q^{65} - 8 q^{67} + 5 q^{69} - 18 q^{71} + 6 q^{73} - 3 q^{75} - 3 q^{77} - 4 q^{79} - 13 q^{81} - 21 q^{83} + 16 q^{85} + 13 q^{87} - 7 q^{89} - q^{91} + 7 q^{93} - 15 q^{97} - q^{99}+O(q^{100}) \) 3 * q - 3 * q^7 - q^9 + 3 * q^11 - q^13 - 8 * q^15 + 3 * q^19 + 8 * q^21 - 7 * q^23 - 7 * q^25 - 3 * q^27 + 11 * q^29 - 6 * q^31 - 8 * q^35 + 11 * q^37 - 2 * q^39 - 5 * q^41 - 9 * q^43 + 3 * q^45 + 4 * q^47 - 10 * q^49 - 16 * q^51 - 4 * q^53 - 15 * q^59 - 4 * q^61 - 2 * q^63 + 2 * q^65 - 8 * q^67 + 5 * q^69 - 18 * q^71 + 6 * q^73 - 3 * q^75 - 3 * q^77 - 4 * q^79 - 13 * q^81 - 21 * q^83 + 16 * q^85 + 13 * q^87 - 7 * q^89 - q^91 + 7 * q^93 - 15 * q^97 - q^99

Introduction

L-functions

Modular forms

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Database

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