LMFDB - Embedded newform 3440.2.a.w.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3440,2,Mod(1,3440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3440, 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("3440.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3440 = 2^{4} \cdot 5 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3440.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:	$27.4685382953$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1933097.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 7x^{3} + 13x^{2} + 5x - 4 $ x^5 - 2x^4 - 7x^3 + 13x^2 + 5x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 215)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.50989$ of defining polynomial
Character	$\chi$	$=$	3440.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.26173 q^{3} +1.00000 q^{5} -3.13519 q^{7} +7.63887 q^{9} +O(q^{10})$	$q+3.26173 q^{3} +1.00000 q^{5} -3.13519 q^{7} +7.63887 q^{9} +0.248163 q^{11} -2.67669 q^{13} +3.26173 q^{15} -0.902631 q^{17} +6.19474 q^{19} -10.2261 q^{21} +7.69647 q^{23} +1.00000 q^{25} +15.1307 q^{27} +2.32872 q^{29} -7.73623 q^{31} +0.809440 q^{33} -3.13519 q^{35} +3.67425 q^{37} -8.73063 q^{39} +9.45452 q^{41} +1.00000 q^{43} +7.63887 q^{45} +9.61888 q^{47} +2.82943 q^{49} -2.94414 q^{51} -7.09737 q^{53} +0.248163 q^{55} +20.2055 q^{57} +3.26173 q^{59} -3.35338 q^{61} -23.9493 q^{63} -2.67669 q^{65} +0.613476 q^{67} +25.1038 q^{69} +8.46512 q^{71} -1.33737 q^{73} +3.26173 q^{75} -0.778039 q^{77} -6.65447 q^{79} +26.4357 q^{81} -10.7162 q^{83} -0.902631 q^{85} +7.59564 q^{87} -1.34850 q^{89} +8.39193 q^{91} -25.2335 q^{93} +6.19474 q^{95} +4.49827 q^{97} +1.89568 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} + 5 q^{5} - 5 q^{7} + 18 q^{9}+O(q^{10}) $ 5 * q + q^3 + 5 * q^5 - 5 * q^7 + 18 * q^9	$ 5 q + q^{3} + 5 q^{5} - 5 q^{7} + 18 q^{9} + 6 q^{11} + 5 q^{13} + q^{15} - 17 q^{17} + 6 q^{19} + 20 q^{21} - q^{23} + 5 q^{25} + 22 q^{27} + 6 q^{29} - 6 q^{31} - 20 q^{33} - 5 q^{35} + 5 q^{37} + 14 q^{39} + 2 q^{41} + 5 q^{43} + 18 q^{45} + 18 q^{49} + 10 q^{51} - 23 q^{53} + 6 q^{55} + 28 q^{57} + q^{59} + 20 q^{61} - 26 q^{63} + 5 q^{65} - 21 q^{67} + 10 q^{69} - 4 q^{71} + 5 q^{73} + q^{75} - 26 q^{77} - 41 q^{79} + 41 q^{81} + 7 q^{83} - 17 q^{85} + 40 q^{87} + 20 q^{89} + 42 q^{91} - 36 q^{93} + 6 q^{95} + 37 q^{97} - 12 q^{99}+O(q^{100}) $ 5 * q + q^3 + 5 * q^5 - 5 * q^7 + 18 * q^9 + 6 * q^11 + 5 * q^13 + q^15 - 17 * q^17 + 6 * q^19 + 20 * q^21 - q^23 + 5 * q^25 + 22 * q^27 + 6 * q^29 - 6 * q^31 - 20 * q^33 - 5 * q^35 + 5 * q^37 + 14 * q^39 + 2 * q^41 + 5 * q^43 + 18 * q^45 + 18 * q^49 + 10 * q^51 - 23 * q^53 + 6 * q^55 + 28 * q^57 + q^59 + 20 * q^61 - 26 * q^63 + 5 * q^65 - 21 * q^67 + 10 * q^69 - 4 * q^71 + 5 * q^73 + q^75 - 26 * q^77 - 41 * q^79 + 41 * q^81 + 7 * q^83 - 17 * q^85 + 40 * q^87 + 20 * q^89 + 42 * q^91 - 36 * q^93 + 6 * q^95 + 37 * q^97 - 12 * 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
$3$	3.26173	1.88316	0.941580	−	0.336791i	$-0.109341\pi$
$3$	3.26173	1.88316	0.941580	+	0.336791i	$0.109341\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.13519	−1.18499	−0.592495	−	0.805574i	$-0.701857\pi$
$7$	−3.13519	−1.18499	−0.592495	+	0.805574i	$0.701857\pi$
$8$	0	0
$9$	7.63887	2.54629
$10$	0	0
$11$	0.248163	0.0748240	0.0374120	−	0.999300i	$-0.488089\pi$
$11$	0.248163	0.0748240	0.0374120	+	0.999300i	$0.488089\pi$
$12$	0	0
$13$	−2.67669	−0.742380	−0.371190	−	0.928557i	$-0.621050\pi$
$13$	−2.67669	−0.742380	−0.371190	+	0.928557i	$0.621050\pi$
$14$	0	0
$15$	3.26173	0.842174
$16$	0	0
$17$	−0.902631	−0.218920	−0.109460	−	0.993991i	$-0.534912\pi$
$17$	−0.902631	−0.218920	−0.109460	+	0.993991i	$0.534912\pi$
$18$	0	0
$19$	6.19474	1.42117	0.710585	−	0.703611i	$-0.248430\pi$
$19$	6.19474	1.42117	0.710585	+	0.703611i	$0.248430\pi$
$20$	0	0
$21$	−10.2261	−2.23153
$22$	0	0
$23$	7.69647	1.60482	0.802412	−	0.596770i	$-0.203550\pi$
$23$	7.69647	1.60482	0.802412	+	0.596770i	$0.203550\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	15.1307	2.91191
$28$	0	0
$29$	2.32872	0.432432	0.216216	−	0.976346i	$-0.430628\pi$
$29$	2.32872	0.432432	0.216216	+	0.976346i	$0.430628\pi$
$30$	0	0
$31$	−7.73623	−1.38947	−0.694734	−	0.719266i	$-0.744478\pi$
$31$	−7.73623	−1.38947	−0.694734	+	0.719266i	$0.744478\pi$
$32$	0	0
$33$	0.809440	0.140905
$34$	0	0
$35$	−3.13519	−0.529944
$36$	0	0
$37$	3.67425	0.604043	0.302021	−	0.953301i	$-0.402339\pi$
$37$	3.67425	0.604043	0.302021	+	0.953301i	$0.402339\pi$
$38$	0	0
$39$	−8.73063	−1.39802
$40$	0	0
$41$	9.45452	1.47655	0.738274	−	0.674501i	$-0.235641\pi$
$41$	9.45452	1.47655	0.738274	+	0.674501i	$0.235641\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	0	0
$45$	7.63887	1.13873
$46$	0	0
$47$	9.61888	1.40306	0.701529	−	0.712641i	$-0.252501\pi$
$47$	9.61888	1.40306	0.701529	+	0.712641i	$0.252501\pi$
$48$	0	0
$49$	2.82943	0.404204
$50$	0	0
$51$	−2.94414	−0.412262
$52$	0	0
$53$	−7.09737	−0.974899	−0.487449	−	0.873151i	$-0.662073\pi$
$53$	−7.09737	−0.974899	−0.487449	+	0.873151i	$0.662073\pi$
$54$	0	0
$55$	0.248163	0.0334623
$56$	0	0
$57$	20.2055	2.67629
$58$	0	0
$59$	3.26173	0.424641	0.212320	−	0.977200i	$-0.431898\pi$
$59$	3.26173	0.424641	0.212320	+	0.977200i	$0.431898\pi$
$60$	0	0
$61$	−3.35338	−0.429356	−0.214678	−	0.976685i	$-0.568870\pi$
$61$	−3.35338	−0.429356	−0.214678	+	0.976685i	$0.568870\pi$
$62$	0	0
$63$	−23.9493	−3.01733
$64$	0	0
$65$	−2.67669	−0.332002
$66$	0	0
$67$	0.613476	0.0749480	0.0374740	−	0.999298i	$-0.488069\pi$
$67$	0.613476	0.0749480	0.0374740	+	0.999298i	$0.488069\pi$
$68$	0	0
$69$	25.1038	3.02214
$70$	0	0
$71$	8.46512	1.00463	0.502313	−	0.864686i	$-0.332483\pi$
$71$	8.46512	1.00463	0.502313	+	0.864686i	$0.332483\pi$
$72$	0	0
$73$	−1.33737	−0.156528	−0.0782638	−	0.996933i	$-0.524938\pi$
$73$	−1.33737	−0.156528	−0.0782638	+	0.996933i	$0.524938\pi$
$74$	0	0
$75$	3.26173	0.376632
$76$	0	0
$77$	−0.778039	−0.0886657
$78$	0	0
$79$	−6.65447	−0.748686	−0.374343	−	0.927290i	$-0.622132\pi$
$79$	−6.65447	−0.748686	−0.374343	+	0.927290i	$0.622132\pi$
$80$	0	0
$81$	26.4357	2.93730
$82$	0	0
$83$	−10.7162	−1.17626	−0.588131	−	0.808766i	$-0.700136\pi$
$83$	−10.7162	−1.17626	−0.588131	+	0.808766i	$0.700136\pi$
$84$	0	0
$85$	−0.902631	−0.0979041
$86$	0	0
$87$	7.59564	0.814338
$88$	0	0
$89$	−1.34850	−0.142940	−0.0714702	−	0.997443i	$-0.522769\pi$
$89$	−1.34850	−0.142940	−0.0714702	+	0.997443i	$0.522769\pi$
$90$	0	0
$91$	8.39193	0.879713
$92$	0	0
$93$	−25.2335	−2.61659
$94$	0	0
$95$	6.19474	0.635567
$96$	0	0
$97$	4.49827	0.456730	0.228365	−	0.973576i	$-0.426662\pi$
$97$	4.49827	0.456730	0.228365	+	0.973576i	$0.426662\pi$
$98$	0	0
$99$	1.89568	0.190523
$100$	0	0
$101$	6.23256	0.620163	0.310081	−	0.950710i	$-0.399644\pi$
$101$	6.23256	0.620163	0.310081	+	0.950710i	$0.399644\pi$
$102$	0	0
$103$	−7.29211	−0.718513	−0.359256	−	0.933239i	$-0.616970\pi$
$103$	−7.29211	−0.718513	−0.359256	+	0.933239i	$0.616970\pi$
$104$	0	0
$105$	−10.2261	−0.997969
$106$	0	0
$107$	−5.69841	−0.550886	−0.275443	−	0.961317i	$-0.588825\pi$
$107$	−5.69841	−0.550886	−0.275443	+	0.961317i	$0.588825\pi$
$108$	0	0
$109$	13.2648	1.27053	0.635267	−	0.772292i	$-0.280890\pi$
$109$	13.2648	1.27053	0.635267	+	0.772292i	$0.280890\pi$
$110$	0	0
$111$	11.9844	1.13751
$112$	0	0
$113$	8.71451	0.819792	0.409896	−	0.912132i	$-0.365565\pi$
$113$	8.71451	0.819792	0.409896	+	0.912132i	$0.365565\pi$
$114$	0	0
$115$	7.69647	0.717699
$116$	0	0
$117$	−20.4469	−1.89031
$118$	0	0
$119$	2.82992	0.259418
$120$	0	0
$121$	−10.9384	−0.994401
$122$	0	0
$123$	30.8381	2.78058
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.3875	−1.63163	−0.815815	−	0.578313i	$-0.803711\pi$
$127$	−18.3875	−1.63163	−0.815815	+	0.578313i	$0.803711\pi$
$128$	0	0
$129$	3.26173	0.287179
$130$	0	0
$131$	−6.10277	−0.533202	−0.266601	−	0.963807i	$-0.585901\pi$
$131$	−6.10277	−0.533202	−0.266601	+	0.963807i	$0.585901\pi$
$132$	0	0
$133$	−19.4217	−1.68407
$134$	0	0
$135$	15.1307	1.30224
$136$	0	0
$137$	−9.57167	−0.817763	−0.408882	−	0.912587i	$-0.634081\pi$
$137$	−9.57167	−0.817763	−0.408882	+	0.912587i	$0.634081\pi$
$138$	0	0
$139$	7.86277	0.666911	0.333456	−	0.942766i	$-0.391785\pi$
$139$	7.86277	0.666911	0.333456	+	0.942766i	$0.391785\pi$
$140$	0	0
$141$	31.3742	2.64218
$142$	0	0
$143$	−0.664255	−0.0555478
$144$	0	0
$145$	2.32872	0.193389
$146$	0	0
$147$	9.22881	0.761180
$148$	0	0
$149$	−1.09931	−0.0900592	−0.0450296	−	0.998986i	$-0.514338\pi$
$149$	−1.09931	−0.0900592	−0.0450296	+	0.998986i	$0.514338\pi$
$150$	0	0
$151$	−7.87683	−0.641007	−0.320504	−	0.947247i	$-0.603852\pi$
$151$	−7.87683	−0.641007	−0.320504	+	0.947247i	$0.603852\pi$
$152$	0	0
$153$	−6.89508	−0.557434
$154$	0	0
$155$	−7.73623	−0.621389
$156$	0	0
$157$	3.04822	0.243274	0.121637	−	0.992575i	$-0.461186\pi$
$157$	3.04822	0.243274	0.121637	+	0.992575i	$0.461186\pi$
$158$	0	0
$159$	−23.1497	−1.83589
$160$	0	0
$161$	−24.1299	−1.90170
$162$	0	0
$163$	−7.04394	−0.551724	−0.275862	−	0.961197i	$-0.588963\pi$
$163$	−7.04394	−0.551724	−0.275862	+	0.961197i	$0.588963\pi$
$164$	0	0
$165$	0.809440	0.0630148
$166$	0	0
$167$	−1.89121	−0.146346	−0.0731730	−	0.997319i	$-0.523313\pi$
$167$	−1.89121	−0.146346	−0.0731730	+	0.997319i	$0.523313\pi$
$168$	0	0
$169$	−5.83534	−0.448873
$170$	0	0
$171$	47.3208	3.61871
$172$	0	0
$173$	7.44534	0.566059	0.283029	−	0.959111i	$-0.408661\pi$
$173$	7.44534	0.566059	0.283029	+	0.959111i	$0.408661\pi$
$174$	0	0
$175$	−3.13519	−0.236998
$176$	0	0
$177$	10.6389	0.799666
$178$	0	0
$179$	10.1784	0.760771	0.380385	−	0.924828i	$-0.375791\pi$
$179$	10.1784	0.760771	0.380385	+	0.924828i	$0.375791\pi$
$180$	0	0
$181$	13.0309	0.968580	0.484290	−	0.874908i	$-0.339078\pi$
$181$	13.0309	0.968580	0.484290	+	0.874908i	$0.339078\pi$
$182$	0	0
$183$	−10.9378	−0.808545
$184$	0	0
$185$	3.67425	0.270136
$186$	0	0
$187$	−0.224000	−0.0163805
$188$	0	0
$189$	−47.4377	−3.45058
$190$	0	0
$191$	13.0247	0.942431	0.471216	−	0.882018i	$-0.343815\pi$
$191$	13.0247	0.942431	0.471216	+	0.882018i	$0.343815\pi$
$192$	0	0
$193$	11.6569	0.839083	0.419541	−	0.907736i	$-0.362191\pi$
$193$	11.6569	0.839083	0.419541	+	0.907736i	$0.362191\pi$
$194$	0	0
$195$	−8.73063	−0.625213
$196$	0	0
$197$	−15.3026	−1.09026	−0.545132	−	0.838350i	$-0.683521\pi$
$197$	−15.3026	−1.09026	−0.545132	+	0.838350i	$0.683521\pi$
$198$	0	0
$199$	−0.718193	−0.0509113	−0.0254557	−	0.999676i	$-0.508104\pi$
$199$	−0.718193	−0.0509113	−0.0254557	+	0.999676i	$0.508104\pi$
$200$	0	0
$201$	2.00099	0.141139
$202$	0	0
$203$	−7.30097	−0.512428
$204$	0	0
$205$	9.45452	0.660332
$206$	0	0
$207$	58.7923	4.08635
$208$	0	0
$209$	1.53730	0.106338
$210$	0	0
$211$	−20.2812	−1.39621	−0.698107	−	0.715993i	$-0.745974\pi$
$211$	−20.2812	−1.39621	−0.698107	+	0.715993i	$0.745974\pi$
$212$	0	0
$213$	27.6109	1.89187
$214$	0	0
$215$	1.00000	0.0681994
$216$	0	0
$217$	24.2546	1.64651
$218$	0	0
$219$	−4.36214	−0.294766
$220$	0	0
$221$	2.41606	0.162522
$222$	0	0
$223$	−22.4003	−1.50003	−0.750017	−	0.661419i	$-0.769955\pi$
$223$	−22.4003	−1.50003	−0.750017	+	0.661419i	$0.769955\pi$
$224$	0	0
$225$	7.63887	0.509258
$226$	0	0
$227$	23.0343	1.52884	0.764419	−	0.644720i	$-0.223026\pi$
$227$	23.0343	1.52884	0.764419	+	0.644720i	$0.223026\pi$
$228$	0	0
$229$	12.6648	0.836911	0.418456	−	0.908237i	$-0.362572\pi$
$229$	12.6648	0.836911	0.418456	+	0.908237i	$0.362572\pi$
$230$	0	0
$231$	−2.53775	−0.166972
$232$	0	0
$233$	−30.2611	−1.98247	−0.991236	−	0.132105i	$-0.957826\pi$
$233$	−30.2611	−1.98247	−0.991236	+	0.132105i	$0.957826\pi$
$234$	0	0
$235$	9.61888	0.627467
$236$	0	0
$237$	−21.7051	−1.40990
$238$	0	0
$239$	−18.5033	−1.19688	−0.598438	−	0.801169i	$-0.704212\pi$
$239$	−18.5033	−1.19688	−0.598438	+	0.801169i	$0.704212\pi$
$240$	0	0
$241$	−16.0840	−1.03606	−0.518031	−	0.855362i	$-0.673335\pi$
$241$	−16.0840	−1.03606	−0.518031	+	0.855362i	$0.673335\pi$
$242$	0	0
$243$	40.8338	2.61949
$244$	0	0
$245$	2.82943	0.180765
$246$	0	0
$247$	−16.5814	−1.05505
$248$	0	0
$249$	−34.9535	−2.21509
$250$	0	0
$251$	12.4459	0.785576	0.392788	−	0.919629i	$-0.371511\pi$
$251$	12.4459	0.785576	0.392788	+	0.919629i	$0.371511\pi$
$252$	0	0
$253$	1.90998	0.120079
$254$	0	0
$255$	−2.94414	−0.184369
$256$	0	0
$257$	4.68647	0.292334	0.146167	−	0.989260i	$-0.453306\pi$
$257$	4.68647	0.292334	0.146167	+	0.989260i	$0.453306\pi$
$258$	0	0
$259$	−11.5195	−0.715785
$260$	0	0
$261$	17.7888	1.10110
$262$	0	0
$263$	−17.3959	−1.07268	−0.536339	−	0.844003i	$-0.680193\pi$
$263$	−17.3959	−1.07268	−0.536339	+	0.844003i	$0.680193\pi$
$264$	0	0
$265$	−7.09737	−0.435988
$266$	0	0
$267$	−4.39843	−0.269180
$268$	0	0
$269$	−20.0133	−1.22023	−0.610115	−	0.792313i	$-0.708877\pi$
$269$	−20.0133	−1.22023	−0.610115	+	0.792313i	$0.708877\pi$
$270$	0	0
$271$	−14.0575	−0.853933	−0.426966	−	0.904267i	$-0.640418\pi$
$271$	−14.0575	−0.853933	−0.426966	+	0.904267i	$0.640418\pi$
$272$	0	0
$273$	27.3722	1.65664
$274$	0	0
$275$	0.248163	0.0149648
$276$	0	0
$277$	−10.8297	−0.650695	−0.325347	−	0.945595i	$-0.605481\pi$
$277$	−10.8297	−0.650695	−0.325347	+	0.945595i	$0.605481\pi$
$278$	0	0
$279$	−59.0961	−3.53799
$280$	0	0
$281$	−7.76915	−0.463469	−0.231734	−	0.972779i	$-0.574440\pi$
$281$	−7.76915	−0.463469	−0.231734	+	0.972779i	$0.574440\pi$
$282$	0	0
$283$	−7.32924	−0.435678	−0.217839	−	0.975985i	$-0.569901\pi$
$283$	−7.32924	−0.435678	−0.217839	+	0.975985i	$0.569901\pi$
$284$	0	0
$285$	20.2055	1.19687
$286$	0	0
$287$	−29.6417	−1.74970
$288$	0	0
$289$	−16.1853	−0.952074
$290$	0	0
$291$	14.6721	0.860095
$292$	0	0
$293$	−2.25307	−0.131626	−0.0658129	−	0.997832i	$-0.520964\pi$
$293$	−2.25307	−0.131626	−0.0658129	+	0.997832i	$0.520964\pi$
$294$	0	0
$295$	3.26173	0.189905
$296$	0	0
$297$	3.75488	0.217880
$298$	0	0
$299$	−20.6010	−1.19139
$300$	0	0
$301$	−3.13519	−0.180709
$302$	0	0
$303$	20.3289	1.16787
$304$	0	0
$305$	−3.35338	−0.192014
$306$	0	0
$307$	−17.5142	−0.999586	−0.499793	−	0.866145i	$-0.666591\pi$
$307$	−17.5142	−0.999586	−0.499793	+	0.866145i	$0.666591\pi$
$308$	0	0
$309$	−23.7849	−1.35307
$310$	0	0
$311$	11.8628	0.672676	0.336338	−	0.941741i	$-0.390812\pi$
$311$	11.8628	0.672676	0.336338	+	0.941741i	$0.390812\pi$
$312$	0	0
$313$	−28.1905	−1.59342	−0.796709	−	0.604363i	$-0.793428\pi$
$313$	−28.1905	−1.59342	−0.796709	+	0.604363i	$0.793428\pi$
$314$	0	0
$315$	−23.9493	−1.34939
$316$	0	0
$317$	−6.60104	−0.370752	−0.185376	−	0.982668i	$-0.559350\pi$
$317$	−6.60104	−0.370752	−0.185376	+	0.982668i	$0.559350\pi$
$318$	0	0
$319$	0.577901	0.0323563
$320$	0	0
$321$	−18.5867	−1.03741
$322$	0	0
$323$	−5.59156	−0.311123
$324$	0	0
$325$	−2.67669	−0.148476
$326$	0	0
$327$	43.2661	2.39262
$328$	0	0
$329$	−30.1570	−1.66261
$330$	0	0
$331$	9.68209	0.532176	0.266088	−	0.963949i	$-0.414269\pi$
$331$	9.68209	0.532176	0.266088	+	0.963949i	$0.414269\pi$
$332$	0	0
$333$	28.0671	1.53807
$334$	0	0
$335$	0.613476	0.0335178
$336$	0	0
$337$	29.2544	1.59359	0.796794	−	0.604251i	$-0.206527\pi$
$337$	29.2544	1.59359	0.796794	+	0.604251i	$0.206527\pi$
$338$	0	0
$339$	28.4244	1.54380
$340$	0	0
$341$	−1.91985	−0.103966
$342$	0	0
$343$	13.0756	0.706013
$344$	0	0
$345$	25.1038	1.35154
$346$	0	0
$347$	−8.34899	−0.448197	−0.224099	−	0.974566i	$-0.571944\pi$
$347$	−8.34899	−0.448197	−0.224099	+	0.974566i	$0.571944\pi$
$348$	0	0
$349$	−22.2663	−1.19189	−0.595944	−	0.803026i	$-0.703222\pi$
$349$	−22.2663	−1.19189	−0.595944	+	0.803026i	$0.703222\pi$
$350$	0	0
$351$	−40.5002	−2.16174
$352$	0	0
$353$	18.3351	0.975880	0.487940	−	0.872877i	$-0.337748\pi$
$353$	18.3351	0.975880	0.487940	+	0.872877i	$0.337748\pi$
$354$	0	0
$355$	8.46512	0.449282
$356$	0	0
$357$	9.23043	0.488526
$358$	0	0
$359$	28.8857	1.52453	0.762265	−	0.647265i	$-0.224087\pi$
$359$	28.8857	1.52453	0.762265	+	0.647265i	$0.224087\pi$
$360$	0	0
$361$	19.3748	1.01973
$362$	0	0
$363$	−35.6781	−1.87262
$364$	0	0
$365$	−1.33737	−0.0700013
$366$	0	0
$367$	−22.7231	−1.18613	−0.593067	−	0.805153i	$-0.702083\pi$
$367$	−22.7231	−1.18613	−0.593067	+	0.805153i	$0.702083\pi$
$368$	0	0
$369$	72.2218	3.75972
$370$	0	0
$371$	22.2516	1.15525
$372$	0	0
$373$	−30.1880	−1.56308	−0.781539	−	0.623857i	$-0.785565\pi$
$373$	−30.1880	−1.56308	−0.781539	+	0.623857i	$0.785565\pi$
$374$	0	0
$375$	3.26173	0.168435
$376$	0	0
$377$	−6.23325	−0.321029
$378$	0	0
$379$	−14.7592	−0.758128	−0.379064	−	0.925370i	$-0.623754\pi$
$379$	−14.7592	−0.758128	−0.379064	+	0.925370i	$0.623754\pi$
$380$	0	0
$381$	−59.9751	−3.07262
$382$	0	0
$383$	14.0955	0.720248	0.360124	−	0.932904i	$-0.382734\pi$
$383$	14.0955	0.720248	0.360124	+	0.932904i	$0.382734\pi$
$384$	0	0
$385$	−0.778039	−0.0396525
$386$	0	0
$387$	7.63887	0.388305
$388$	0	0
$389$	30.9837	1.57094	0.785468	−	0.618902i	$-0.212422\pi$
$389$	30.9837	1.57094	0.785468	+	0.618902i	$0.212422\pi$
$390$	0	0
$391$	−6.94707	−0.351328
$392$	0	0
$393$	−19.9056	−1.00410
$394$	0	0
$395$	−6.65447	−0.334823
$396$	0	0
$397$	31.0726	1.55949	0.779744	−	0.626099i	$-0.215349\pi$
$397$	31.0726	1.55949	0.779744	+	0.626099i	$0.215349\pi$
$398$	0	0
$399$	−63.3483	−3.17138
$400$	0	0
$401$	−4.92956	−0.246170	−0.123085	−	0.992396i	$-0.539279\pi$
$401$	−4.92956	−0.246170	−0.123085	+	0.992396i	$0.539279\pi$
$402$	0	0
$403$	20.7075	1.03151
$404$	0	0
$405$	26.4357	1.31360
$406$	0	0
$407$	0.911813	0.0451969
$408$	0	0
$409$	−7.61888	−0.376729	−0.188365	−	0.982099i	$-0.560319\pi$
$409$	−7.61888	−0.376729	−0.188365	+	0.982099i	$0.560319\pi$
$410$	0	0
$411$	−31.2202	−1.53998
$412$	0	0
$413$	−10.2261	−0.503195
$414$	0	0
$415$	−10.7162	−0.526040
$416$	0	0
$417$	25.6462	1.25590
$418$	0	0
$419$	−21.8373	−1.06682	−0.533410	−	0.845857i	$-0.679090\pi$
$419$	−21.8373	−1.06682	−0.533410	+	0.845857i	$0.679090\pi$
$420$	0	0
$421$	−13.5012	−0.658009	−0.329005	−	0.944328i	$-0.606713\pi$
$421$	−13.5012	−0.658009	−0.329005	+	0.944328i	$0.606713\pi$
$422$	0	0
$423$	73.4773	3.57259
$424$	0	0
$425$	−0.902631	−0.0437840
$426$	0	0
$427$	10.5135	0.508783
$428$	0	0
$429$	−2.16662	−0.104605
$430$	0	0
$431$	7.29975	0.351617	0.175808	−	0.984424i	$-0.443746\pi$
$431$	7.29975	0.351617	0.175808	+	0.984424i	$0.443746\pi$
$432$	0	0
$433$	10.1310	0.486865	0.243433	−	0.969918i	$-0.421726\pi$
$433$	10.1310	0.486865	0.243433	+	0.969918i	$0.421726\pi$
$434$	0	0
$435$	7.59564	0.364183
$436$	0	0
$437$	47.6776	2.28073
$438$	0	0
$439$	17.6650	0.843103	0.421552	−	0.906804i	$-0.361486\pi$
$439$	17.6650	0.843103	0.421552	+	0.906804i	$0.361486\pi$
$440$	0	0
$441$	21.6136	1.02922
$442$	0	0
$443$	−36.7934	−1.74811	−0.874054	−	0.485828i	$-0.838518\pi$
$443$	−36.7934	−1.74811	−0.874054	+	0.485828i	$0.838518\pi$
$444$	0	0
$445$	−1.34850	−0.0639249
$446$	0	0
$447$	−3.58566	−0.169596
$448$	0	0
$449$	0.788959	0.0372333	0.0186166	−	0.999827i	$-0.494074\pi$
$449$	0.788959	0.0372333	0.0186166	+	0.999827i	$0.494074\pi$
$450$	0	0
$451$	2.34626	0.110481
$452$	0	0
$453$	−25.6921	−1.20712
$454$	0	0
$455$	8.39193	0.393420
$456$	0	0
$457$	23.7442	1.11070	0.555352	−	0.831615i	$-0.312583\pi$
$457$	23.7442	1.11070	0.555352	+	0.831615i	$0.312583\pi$
$458$	0	0
$459$	−13.6575	−0.637475
$460$	0	0
$461$	10.5205	0.489988	0.244994	−	0.969525i	$-0.421214\pi$
$461$	10.5205	0.489988	0.244994	+	0.969525i	$0.421214\pi$
$462$	0	0
$463$	33.7411	1.56808	0.784042	−	0.620708i	$-0.213155\pi$
$463$	33.7411	1.56808	0.784042	+	0.620708i	$0.213155\pi$
$464$	0	0
$465$	−25.2335	−1.17017
$466$	0	0
$467$	−19.2665	−0.891548	−0.445774	−	0.895146i	$-0.647071\pi$
$467$	−19.2665	−0.891548	−0.445774	+	0.895146i	$0.647071\pi$
$468$	0	0
$469$	−1.92336	−0.0888127
$470$	0	0
$471$	9.94245	0.458124
$472$	0	0
$473$	0.248163	0.0114105
$474$	0	0
$475$	6.19474	0.284234
$476$	0	0
$477$	−54.2158	−2.48237
$478$	0	0
$479$	9.69667	0.443052	0.221526	−	0.975154i	$-0.428896\pi$
$479$	9.69667	0.443052	0.221526	+	0.975154i	$0.428896\pi$
$480$	0	0
$481$	−9.83482	−0.448429
$482$	0	0
$483$	−78.7052	−3.58121
$484$	0	0
$485$	4.49827	0.204256
$486$	0	0
$487$	0.683509	0.0309728	0.0154864	−	0.999880i	$-0.495070\pi$
$487$	0.683509	0.0309728	0.0154864	+	0.999880i	$0.495070\pi$
$488$	0	0
$489$	−22.9754	−1.03898
$490$	0	0
$491$	29.1038	1.31344	0.656718	−	0.754137i	$-0.271944\pi$
$491$	29.1038	1.31344	0.656718	+	0.754137i	$0.271944\pi$
$492$	0	0
$493$	−2.10197	−0.0946680
$494$	0	0
$495$	1.89568	0.0852047
$496$	0	0
$497$	−26.5398	−1.19047
$498$	0	0
$499$	−30.7637	−1.37717	−0.688586	−	0.725155i	$-0.741768\pi$
$499$	−30.7637	−1.37717	−0.688586	+	0.725155i	$0.741768\pi$
$500$	0	0
$501$	−6.16860	−0.275593
$502$	0	0
$503$	−4.02345	−0.179397	−0.0896983	−	0.995969i	$-0.528590\pi$
$503$	−4.02345	−0.179397	−0.0896983	+	0.995969i	$0.528590\pi$
$504$	0	0
$505$	6.23256	0.277345
$506$	0	0
$507$	−19.0333	−0.845298
$508$	0	0
$509$	12.8168	0.568094	0.284047	−	0.958810i	$-0.408323\pi$
$509$	12.8168	0.568094	0.284047	+	0.958810i	$0.408323\pi$
$510$	0	0
$511$	4.19292	0.185484
$512$	0	0
$513$	93.7308	4.13832
$514$	0	0
$515$	−7.29211	−0.321329
$516$	0	0
$517$	2.38705	0.104982
$518$	0	0
$519$	24.2847	1.06598
$520$	0	0
$521$	−10.8754	−0.476460	−0.238230	−	0.971209i	$-0.576567\pi$
$521$	−10.8754	−0.476460	−0.238230	+	0.971209i	$0.576567\pi$
$522$	0	0
$523$	35.9454	1.57178	0.785892	−	0.618364i	$-0.212204\pi$
$523$	35.9454	1.57178	0.785892	+	0.618364i	$0.212204\pi$
$524$	0	0
$525$	−10.2261	−0.446305
$526$	0	0
$527$	6.98296	0.304183
$528$	0	0
$529$	36.2356	1.57546
$530$	0	0
$531$	24.9159	1.08126
$532$	0	0
$533$	−25.3068	−1.09616
$534$	0	0
$535$	−5.69841	−0.246364
$536$	0	0
$537$	33.1992	1.43265
$538$	0	0
$539$	0.702159	0.0302441
$540$	0	0
$541$	26.1499	1.12427	0.562135	−	0.827045i	$-0.309980\pi$
$541$	26.1499	1.12427	0.562135	+	0.827045i	$0.309980\pi$
$542$	0	0
$543$	42.5033	1.82399
$544$	0	0
$545$	13.2648	0.568200
$546$	0	0
$547$	−0.231141	−0.00988287	−0.00494143	−	0.999988i	$-0.501573\pi$
$547$	−0.231141	−0.00988287	−0.00494143	+	0.999988i	$0.501573\pi$
$548$	0	0
$549$	−25.6160	−1.09326
$550$	0	0
$551$	14.4258	0.614559
$552$	0	0
$553$	20.8630	0.887186
$554$	0	0
$555$	11.9844	0.508709
$556$	0	0
$557$	−3.96726	−0.168098	−0.0840491	−	0.996462i	$-0.526785\pi$
$557$	−3.96726	−0.168098	−0.0840491	+	0.996462i	$0.526785\pi$
$558$	0	0
$559$	−2.67669	−0.113212
$560$	0	0
$561$	−0.730626	−0.0308470
$562$	0	0
$563$	−12.1492	−0.512026	−0.256013	−	0.966673i	$-0.582409\pi$
$563$	−12.1492	−0.512026	−0.256013	+	0.966673i	$0.582409\pi$
$564$	0	0
$565$	8.71451	0.366622
$566$	0	0
$567$	−82.8809	−3.48067
$568$	0	0
$569$	−42.4665	−1.78029	−0.890143	−	0.455681i	$-0.849396\pi$
$569$	−42.4665	−1.78029	−0.890143	+	0.455681i	$0.849396\pi$
$570$	0	0
$571$	23.4374	0.980824	0.490412	−	0.871491i	$-0.336846\pi$
$571$	23.4374	0.980824	0.490412	+	0.871491i	$0.336846\pi$
$572$	0	0
$573$	42.4829	1.77475
$574$	0	0
$575$	7.69647	0.320965
$576$	0	0
$577$	−41.0241	−1.70785	−0.853927	−	0.520393i	$-0.825786\pi$
$577$	−41.0241	−1.70785	−0.853927	+	0.520393i	$0.825786\pi$
$578$	0	0
$579$	38.0217	1.58013
$580$	0	0
$581$	33.5975	1.39386
$582$	0	0
$583$	−1.76130	−0.0729458
$584$	0	0
$585$	−20.4469	−0.845374
$586$	0	0
$587$	29.4970	1.21747	0.608735	−	0.793374i	$-0.291677\pi$
$587$	29.4970	1.21747	0.608735	+	0.793374i	$0.291677\pi$
$588$	0	0
$589$	−47.9239	−1.97467
$590$	0	0
$591$	−49.9129	−2.05314
$592$	0	0
$593$	17.7334	0.728223	0.364111	−	0.931355i	$-0.381373\pi$
$593$	17.7334	0.728223	0.364111	+	0.931355i	$0.381373\pi$
$594$	0	0
$595$	2.82992	0.116015
$596$	0	0
$597$	−2.34255	−0.0958742
$598$	0	0
$599$	16.9852	0.693997	0.346998	−	0.937866i	$-0.387201\pi$
$599$	16.9852	0.693997	0.346998	+	0.937866i	$0.387201\pi$
$600$	0	0
$601$	5.70922	0.232884	0.116442	−	0.993197i	$-0.462851\pi$
$601$	5.70922	0.232884	0.116442	+	0.993197i	$0.462851\pi$
$602$	0	0
$603$	4.68626	0.190839
$604$	0	0
$605$	−10.9384	−0.444710
$606$	0	0
$607$	−4.37105	−0.177415	−0.0887077	−	0.996058i	$-0.528274\pi$
$607$	−4.37105	−0.177415	−0.0887077	+	0.996058i	$0.528274\pi$
$608$	0	0
$609$	−23.8138	−0.964983
$610$	0	0
$611$	−25.7467	−1.04160
$612$	0	0
$613$	35.0848	1.41706	0.708532	−	0.705679i	$-0.249358\pi$
$613$	35.0848	1.41706	0.708532	+	0.705679i	$0.249358\pi$
$614$	0	0
$615$	30.8381	1.24351
$616$	0	0
$617$	23.0537	0.928108	0.464054	−	0.885807i	$-0.346394\pi$
$617$	23.0537	0.928108	0.464054	+	0.885807i	$0.346394\pi$
$618$	0	0
$619$	29.3047	1.17785	0.588927	−	0.808186i	$-0.299551\pi$
$619$	29.3047	1.17785	0.588927	+	0.808186i	$0.299551\pi$
$620$	0	0
$621$	116.453	4.67310
$622$	0	0
$623$	4.22780	0.169383
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.01427	0.200251
$628$	0	0
$629$	−3.31649	−0.132237
$630$	0	0
$631$	31.7883	1.26547	0.632737	−	0.774367i	$-0.281931\pi$
$631$	31.7883	1.26547	0.632737	+	0.774367i	$0.281931\pi$
$632$	0	0
$633$	−66.1517	−2.62929
$634$	0	0
$635$	−18.3875	−0.729687
$636$	0	0
$637$	−7.57349	−0.300073
$638$	0	0
$639$	64.6639	2.55807
$640$	0	0
$641$	42.7217	1.68740	0.843702	−	0.536811i	$-0.180371\pi$
$641$	42.7217	1.68740	0.843702	+	0.536811i	$0.180371\pi$
$642$	0	0
$643$	5.88234	0.231977	0.115988	−	0.993251i	$-0.462996\pi$
$643$	5.88234	0.231977	0.115988	+	0.993251i	$0.462996\pi$
$644$	0	0
$645$	3.26173	0.128430
$646$	0	0
$647$	−39.6468	−1.55868	−0.779338	−	0.626604i	$-0.784445\pi$
$647$	−39.6468	−1.55868	−0.779338	+	0.626604i	$0.784445\pi$
$648$	0	0
$649$	0.809440	0.0317733
$650$	0	0
$651$	79.1118	3.10064
$652$	0	0
$653$	−39.0899	−1.52971	−0.764854	−	0.644204i	$-0.777189\pi$
$653$	−39.0899	−1.52971	−0.764854	+	0.644204i	$0.777189\pi$
$654$	0	0
$655$	−6.10277	−0.238455
$656$	0	0
$657$	−10.2160	−0.398564
$658$	0	0
$659$	49.9625	1.94626	0.973131	−	0.230254i	$-0.0739558\pi$
$659$	49.9625	1.94626	0.973131	+	0.230254i	$0.0739558\pi$
$660$	0	0
$661$	−12.3002	−0.478421	−0.239210	−	0.970968i	$-0.576889\pi$
$661$	−12.3002	−0.478421	−0.239210	+	0.970968i	$0.576889\pi$
$662$	0	0
$663$	7.88053	0.306055
$664$	0	0
$665$	−19.4217	−0.753141
$666$	0	0
$667$	17.9229	0.693977
$668$	0	0
$669$	−73.0636	−2.82480
$670$	0	0
$671$	−0.832184	−0.0321261
$672$	0	0
$673$	14.3751	0.554118	0.277059	−	0.960853i	$-0.410640\pi$
$673$	14.3751	0.554118	0.277059	+	0.960853i	$0.410640\pi$
$674$	0	0
$675$	15.1307	0.582381
$676$	0	0
$677$	−19.8543	−0.763062	−0.381531	−	0.924356i	$-0.624603\pi$
$677$	−19.8543	−0.763062	−0.381531	+	0.924356i	$0.624603\pi$
$678$	0	0
$679$	−14.1029	−0.541221
$680$	0	0
$681$	75.1315	2.87905
$682$	0	0
$683$	−33.8908	−1.29680	−0.648398	−	0.761302i	$-0.724561\pi$
$683$	−33.8908	−1.29680	−0.648398	+	0.761302i	$0.724561\pi$
$684$	0	0
$685$	−9.57167	−0.365715
$686$	0	0
$687$	41.3090	1.57604
$688$	0	0
$689$	18.9974	0.723745
$690$	0	0
$691$	−14.3619	−0.546354	−0.273177	−	0.961964i	$-0.588074\pi$
$691$	−14.3619	−0.546354	−0.273177	+	0.961964i	$0.588074\pi$
$692$	0	0
$693$	−5.94333	−0.225769
$694$	0	0
$695$	7.86277	0.298252
$696$	0	0
$697$	−8.53394	−0.323246
$698$	0	0
$699$	−98.7035	−3.73331
$700$	0	0
$701$	11.8354	0.447015	0.223508	−	0.974702i	$-0.428249\pi$
$701$	11.8354	0.447015	0.223508	+	0.974702i	$0.428249\pi$
$702$	0	0
$703$	22.7610	0.858448
$704$	0	0
$705$	31.3742	1.18162
$706$	0	0
$707$	−19.5403	−0.734887
$708$	0	0
$709$	−22.9382	−0.861462	−0.430731	−	0.902480i	$-0.641744\pi$
$709$	−22.9382	−0.861462	−0.430731	+	0.902480i	$0.641744\pi$
$710$	0	0
$711$	−50.8326	−1.90637
$712$	0	0
$713$	−59.5417	−2.22985
$714$	0	0
$715$	−0.664255	−0.0248417
$716$	0	0
$717$	−60.3526	−2.25391
$718$	0	0
$719$	−40.1157	−1.49606	−0.748032	−	0.663663i	$-0.769001\pi$
$719$	−40.1157	−1.49606	−0.748032	+	0.663663i	$0.769001\pi$
$720$	0	0
$721$	22.8622	0.851431
$722$	0	0
$723$	−52.4616	−1.95107
$724$	0	0
$725$	2.32872	0.0864864
$726$	0	0
$727$	−10.3290	−0.383082	−0.191541	−	0.981485i	$-0.561348\pi$
$727$	−10.3290	−0.383082	−0.191541	+	0.981485i	$0.561348\pi$
$728$	0	0
$729$	53.8817	1.99562
$730$	0	0
$731$	−0.902631	−0.0333850
$732$	0	0
$733$	−27.0182	−0.997938	−0.498969	−	0.866620i	$-0.666288\pi$
$733$	−27.0182	−0.997938	−0.498969	+	0.866620i	$0.666288\pi$
$734$	0	0
$735$	9.22881	0.340410
$736$	0	0
$737$	0.152242	0.00560791
$738$	0	0
$739$	−20.7027	−0.761560	−0.380780	−	0.924666i	$-0.624345\pi$
$739$	−20.7027	−0.761560	−0.380780	+	0.924666i	$0.624345\pi$
$740$	0	0
$741$	−54.0839	−1.98682
$742$	0	0
$743$	−40.3655	−1.48087	−0.740433	−	0.672130i	$-0.765380\pi$
$743$	−40.3655	−1.48087	−0.740433	+	0.672130i	$0.765380\pi$
$744$	0	0
$745$	−1.09931	−0.0402757
$746$	0	0
$747$	−81.8600	−2.99510
$748$	0	0
$749$	17.8656	0.652795
$750$	0	0
$751$	−9.23655	−0.337046	−0.168523	−	0.985698i	$-0.553900\pi$
$751$	−9.23655	−0.337046	−0.168523	+	0.985698i	$0.553900\pi$
$752$	0	0
$753$	40.5950	1.47936
$754$	0	0
$755$	−7.87683	−0.286667
$756$	0	0
$757$	10.6745	0.387973	0.193986	−	0.981004i	$-0.437858\pi$
$757$	10.6745	0.387973	0.193986	+	0.981004i	$0.437858\pi$
$758$	0	0
$759$	6.22983	0.226129
$760$	0	0
$761$	−21.3301	−0.773217	−0.386608	−	0.922244i	$-0.626353\pi$
$761$	−21.3301	−0.773217	−0.386608	+	0.922244i	$0.626353\pi$
$762$	0	0
$763$	−41.5876	−1.50557
$764$	0	0
$765$	−6.89508	−0.249292
$766$	0	0
$767$	−8.73063	−0.315245
$768$	0	0
$769$	25.6062	0.923385	0.461692	−	0.887040i	$-0.347242\pi$
$769$	25.6062	0.923385	0.461692	+	0.887040i	$0.347242\pi$
$770$	0	0
$771$	15.2860	0.550512
$772$	0	0
$773$	−21.4476	−0.771416	−0.385708	−	0.922621i	$-0.626043\pi$
$773$	−21.4476	−0.771416	−0.385708	+	0.922621i	$0.626043\pi$
$774$	0	0
$775$	−7.73623	−0.277894
$776$	0	0
$777$	−37.5734	−1.34794
$778$	0	0
$779$	58.5683	2.09843
$780$	0	0
$781$	2.10073	0.0751700
$782$	0	0
$783$	35.2351	1.25920
$784$	0	0
$785$	3.04822	0.108796
$786$	0	0
$787$	43.6900	1.55738	0.778691	−	0.627408i	$-0.215884\pi$
$787$	43.6900	1.55738	0.778691	+	0.627408i	$0.215884\pi$
$788$	0	0
$789$	−56.7407	−2.02002
$790$	0	0
$791$	−27.3217	−0.971446
$792$	0	0
$793$	8.97594	0.318745
$794$	0	0
$795$	−23.1497	−0.821035
$796$	0	0
$797$	−46.8097	−1.65809	−0.829043	−	0.559185i	$-0.811114\pi$
$797$	−46.8097	−1.65809	−0.829043	+	0.559185i	$0.811114\pi$
$798$	0	0
$799$	−8.68230	−0.307158
$800$	0	0
$801$	−10.3010	−0.363968
$802$	0	0
$803$	−0.331886	−0.0117120
$804$	0	0
$805$	−24.1299	−0.850467
$806$	0	0
$807$	−65.2778	−2.29789
$808$	0	0
$809$	2.62610	0.0923288	0.0461644	−	0.998934i	$-0.485300\pi$
$809$	2.62610	0.0923288	0.0461644	+	0.998934i	$0.485300\pi$
$810$	0	0
$811$	−36.1962	−1.27102	−0.635510	−	0.772093i	$-0.719210\pi$
$811$	−36.1962	−1.27102	−0.635510	+	0.772093i	$0.719210\pi$
$812$	0	0
$813$	−45.8518	−1.60809
$814$	0	0
$815$	−7.04394	−0.246739
$816$	0	0
$817$	6.19474	0.216726
$818$	0	0
$819$	64.1048	2.24000
$820$	0	0
$821$	−31.4660	−1.09817	−0.549085	−	0.835767i	$-0.685024\pi$
$821$	−31.4660	−1.09817	−0.549085	+	0.835767i	$0.685024\pi$
$822$	0	0
$823$	43.9257	1.53115	0.765577	−	0.643344i	$-0.222454\pi$
$823$	43.9257	1.53115	0.765577	+	0.643344i	$0.222454\pi$
$824$	0	0
$825$	0.809440	0.0281811
$826$	0	0
$827$	−28.7775	−1.00069	−0.500346	−	0.865825i	$-0.666794\pi$
$827$	−28.7775	−1.00069	−0.500346	+	0.865825i	$0.666794\pi$
$828$	0	0
$829$	31.2180	1.08425	0.542123	−	0.840299i	$-0.317621\pi$
$829$	31.2180	1.08425	0.542123	+	0.840299i	$0.317621\pi$
$830$	0	0
$831$	−35.3236	−1.22536
$832$	0	0
$833$	−2.55393	−0.0884883
$834$	0	0
$835$	−1.89121	−0.0654479
$836$	0	0
$837$	−117.055	−4.04600
$838$	0	0
$839$	16.8160	0.580554	0.290277	−	0.956943i	$-0.406253\pi$
$839$	16.8160	0.580554	0.290277	+	0.956943i	$0.406253\pi$
$840$	0	0
$841$	−23.5771	−0.813003
$842$	0	0
$843$	−25.3408	−0.872785
$844$	0	0
$845$	−5.83534	−0.200742
$846$	0	0
$847$	34.2940	1.17836
$848$	0	0
$849$	−23.9060	−0.820452
$850$	0	0
$851$	28.2787	0.969383
$852$	0	0
$853$	−20.6282	−0.706295	−0.353148	−	0.935568i	$-0.614889\pi$
$853$	−20.6282	−0.706295	−0.353148	+	0.935568i	$0.614889\pi$
$854$	0	0
$855$	47.3208	1.61834
$856$	0	0
$857$	−32.8803	−1.12317	−0.561585	−	0.827419i	$-0.689808\pi$
$857$	−32.8803	−1.12317	−0.561585	+	0.827419i	$0.689808\pi$
$858$	0	0
$859$	43.1433	1.47203	0.736016	−	0.676964i	$-0.236705\pi$
$859$	43.1433	1.47203	0.736016	+	0.676964i	$0.236705\pi$
$860$	0	0
$861$	−96.6833	−3.29496
$862$	0	0
$863$	45.6650	1.55445	0.777227	−	0.629221i	$-0.216626\pi$
$863$	45.6650	1.55445	0.777227	+	0.629221i	$0.216626\pi$
$864$	0	0
$865$	7.44534	0.253149
$866$	0	0
$867$	−52.7919	−1.79291
$868$	0	0
$869$	−1.65139	−0.0560197
$870$	0	0
$871$	−1.64208	−0.0556399
$872$	0	0
$873$	34.3617	1.16297
$874$	0	0
$875$	−3.13519	−0.105989
$876$	0	0
$877$	15.9707	0.539294	0.269647	−	0.962959i	$-0.413093\pi$
$877$	15.9707	0.539294	0.269647	+	0.962959i	$0.413093\pi$
$878$	0	0
$879$	−7.34891	−0.247872
$880$	0	0
$881$	13.7670	0.463823	0.231912	−	0.972737i	$-0.425502\pi$
$881$	13.7670	0.463823	0.231912	+	0.972737i	$0.425502\pi$
$882$	0	0
$883$	−12.0482	−0.405455	−0.202728	−	0.979235i	$-0.564981\pi$
$883$	−12.0482	−0.405455	−0.202728	+	0.979235i	$0.564981\pi$
$884$	0	0
$885$	10.6389	0.357622
$886$	0	0
$887$	14.1405	0.474791	0.237395	−	0.971413i	$-0.423706\pi$
$887$	14.1405	0.474791	0.237395	+	0.971413i	$0.423706\pi$
$888$	0	0
$889$	57.6484	1.93347
$890$	0	0
$891$	6.56036	0.219780
$892$	0	0
$893$	59.5864	1.99398
$894$	0	0
$895$	10.1784	0.340227
$896$	0	0
$897$	−67.1950	−2.24358
$898$	0	0
$899$	−18.0155	−0.600850
$900$	0	0
$901$	6.40630	0.213425
$902$	0	0
$903$	−10.2261	−0.340305
$904$	0	0
$905$	13.0309	0.433162
$906$	0	0
$907$	−2.93772	−0.0975455	−0.0487727	−	0.998810i	$-0.515531\pi$
$907$	−2.93772	−0.0975455	−0.0487727	+	0.998810i	$0.515531\pi$
$908$	0	0
$909$	47.6097	1.57911
$910$	0	0
$911$	−1.04732	−0.0346992	−0.0173496	−	0.999849i	$-0.505523\pi$
$911$	−1.04732	−0.0346992	−0.0173496	+	0.999849i	$0.505523\pi$
$912$	0	0
$913$	−2.65938	−0.0880125
$914$	0	0
$915$	−10.9378	−0.361592
$916$	0	0
$917$	19.1334	0.631839
$918$	0	0
$919$	−15.7325	−0.518966	−0.259483	−	0.965748i	$-0.583552\pi$
$919$	−15.7325	−0.518966	−0.259483	+	0.965748i	$0.583552\pi$
$920$	0	0
$921$	−57.1264	−1.88238
$922$	0	0
$923$	−22.6585	−0.745813
$924$	0	0
$925$	3.67425	0.120809
$926$	0	0
$927$	−55.7034	−1.82954
$928$	0	0
$929$	−26.5084	−0.869711	−0.434855	−	0.900500i	$-0.643201\pi$
$929$	−26.5084	−0.869711	−0.434855	+	0.900500i	$0.643201\pi$
$930$	0	0
$931$	17.5275	0.574442
$932$	0	0
$933$	38.6931	1.26676
$934$	0	0
$935$	−0.224000	−0.00732557
$936$	0	0
$937$	25.7747	0.842021	0.421011	−	0.907056i	$-0.361676\pi$
$937$	25.7747	0.842021	0.421011	+	0.907056i	$0.361676\pi$
$938$	0	0
$939$	−91.9496	−3.00066
$940$	0	0
$941$	−33.5036	−1.09219	−0.546093	−	0.837725i	$-0.683885\pi$
$941$	−33.5036	−1.09219	−0.546093	+	0.837725i	$0.683885\pi$
$942$	0	0
$943$	72.7664	2.36960
$944$	0	0
$945$	−47.4377	−1.54315
$946$	0	0
$947$	12.5066	0.406411	0.203205	−	0.979136i	$-0.434864\pi$
$947$	12.5066	0.406411	0.203205	+	0.979136i	$0.434864\pi$
$948$	0	0
$949$	3.57973	0.116203
$950$	0	0
$951$	−21.5308	−0.698184
$952$	0	0
$953$	13.3246	0.431628	0.215814	−	0.976435i	$-0.430760\pi$
$953$	13.3246	0.431628	0.215814	+	0.976435i	$0.430760\pi$
$954$	0	0
$955$	13.0247	0.421468
$956$	0	0
$957$	1.88496	0.0609320
$958$	0	0
$959$	30.0090	0.969042
$960$	0	0
$961$	28.8493	0.930623
$962$	0	0
$963$	−43.5294	−1.40272
$964$	0	0
$965$	11.6569	0.375249
$966$	0	0
$967$	35.1437	1.13014	0.565072	−	0.825042i	$-0.308848\pi$
$967$	35.1437	1.13014	0.565072	+	0.825042i	$0.308848\pi$
$968$	0	0
$969$	−18.2382	−0.585894
$970$	0	0
$971$	−54.0062	−1.73314	−0.866571	−	0.499054i	$-0.833681\pi$
$971$	−54.0062	−1.73314	−0.866571	+	0.499054i	$0.833681\pi$
$972$	0	0
$973$	−24.6513	−0.790284
$974$	0	0
$975$	−8.73063	−0.279604
$976$	0	0
$977$	0.447973	0.0143319	0.00716596	−	0.999974i	$-0.497719\pi$
$977$	0.447973	0.0143319	0.00716596	+	0.999974i	$0.497719\pi$
$978$	0	0
$979$	−0.334647	−0.0106954
$980$	0	0
$981$	101.328	3.23515
$982$	0	0
$983$	−43.5886	−1.39026	−0.695130	−	0.718884i	$-0.744653\pi$
$983$	−43.5886	−1.39026	−0.695130	+	0.718884i	$0.744653\pi$
$984$	0	0
$985$	−15.3026	−0.487581
$986$	0	0
$987$	−98.3640	−3.13096
$988$	0	0
$989$	7.69647	0.244733
$990$	0	0
$991$	47.3742	1.50489	0.752445	−	0.658655i	$-0.228874\pi$
$991$	47.3742	1.50489	0.752445	+	0.658655i	$0.228874\pi$
$992$	0	0
$993$	31.5803	1.00217
$994$	0	0
$995$	−0.718193	−0.0227682
$996$	0	0
$997$	−9.73338	−0.308259	−0.154130	−	0.988051i	$-0.549257\pi$
$997$	−9.73338	−0.308259	−0.154130	+	0.988051i	$0.549257\pi$
$998$	0	0
$999$	55.5940	1.75892

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3440.2.a.w.1.5		5
4.3	odd	2		215.2.a.c.1.5	✓	5
12.11	even	2		1935.2.a.u.1.1		5
20.3	even	4		1075.2.b.h.474.1		10
20.7	even	4		1075.2.b.h.474.10		10
20.19	odd	2		1075.2.a.m.1.1		5
60.59	even	2		9675.2.a.ch.1.5		5
172.171	even	2		9245.2.a.l.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.2.a.c.1.5	✓	5	4.3	odd	2
1075.2.a.m.1.1		5	20.19	odd	2
1075.2.b.h.474.1		10	20.3	even	4
1075.2.b.h.474.10		10	20.7	even	4
1935.2.a.u.1.1		5	12.11	even	2
3440.2.a.w.1.5		5	1.1	even	1	trivial
9245.2.a.l.1.1		5	172.171	even	2
9675.2.a.ch.1.5		5	60.59	even	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:	\( N \)	\(=\)	\( 3440 = 2^{4} \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3440.a (trivial)

\(f(q)\)	\(=\)	\(q+3.26173 q^{3} +1.00000 q^{5} -3.13519 q^{7} +7.63887 q^{9} +O(q^{10})\)	\(q+3.26173 q^{3} +1.00000 q^{5} -3.13519 q^{7} +7.63887 q^{9} +0.248163 q^{11} -2.67669 q^{13} +3.26173 q^{15} -0.902631 q^{17} +6.19474 q^{19} -10.2261 q^{21} +7.69647 q^{23} +1.00000 q^{25} +15.1307 q^{27} +2.32872 q^{29} -7.73623 q^{31} +0.809440 q^{33} -3.13519 q^{35} +3.67425 q^{37} -8.73063 q^{39} +9.45452 q^{41} +1.00000 q^{43} +7.63887 q^{45} +9.61888 q^{47} +2.82943 q^{49} -2.94414 q^{51} -7.09737 q^{53} +0.248163 q^{55} +20.2055 q^{57} +3.26173 q^{59} -3.35338 q^{61} -23.9493 q^{63} -2.67669 q^{65} +0.613476 q^{67} +25.1038 q^{69} +8.46512 q^{71} -1.33737 q^{73} +3.26173 q^{75} -0.778039 q^{77} -6.65447 q^{79} +26.4357 q^{81} -10.7162 q^{83} -0.902631 q^{85} +7.59564 q^{87} -1.34850 q^{89} +8.39193 q^{91} -25.2335 q^{93} +6.19474 q^{95} +4.49827 q^{97} +1.89568 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} + 5 q^{5} - 5 q^{7} + 18 q^{9}+O(q^{10}) \) 5 * q + q^3 + 5 * q^5 - 5 * q^7 + 18 * q^9	\( 5 q + q^{3} + 5 q^{5} - 5 q^{7} + 18 q^{9} + 6 q^{11} + 5 q^{13} + q^{15} - 17 q^{17} + 6 q^{19} + 20 q^{21} - q^{23} + 5 q^{25} + 22 q^{27} + 6 q^{29} - 6 q^{31} - 20 q^{33} - 5 q^{35} + 5 q^{37} + 14 q^{39} + 2 q^{41} + 5 q^{43} + 18 q^{45} + 18 q^{49} + 10 q^{51} - 23 q^{53} + 6 q^{55} + 28 q^{57} + q^{59} + 20 q^{61} - 26 q^{63} + 5 q^{65} - 21 q^{67} + 10 q^{69} - 4 q^{71} + 5 q^{73} + q^{75} - 26 q^{77} - 41 q^{79} + 41 q^{81} + 7 q^{83} - 17 q^{85} + 40 q^{87} + 20 q^{89} + 42 q^{91} - 36 q^{93} + 6 q^{95} + 37 q^{97} - 12 q^{99}+O(q^{100}) \) 5 * q + q^3 + 5 * q^5 - 5 * q^7 + 18 * q^9 + 6 * q^11 + 5 * q^13 + q^15 - 17 * q^17 + 6 * q^19 + 20 * q^21 - q^23 + 5 * q^25 + 22 * q^27 + 6 * q^29 - 6 * q^31 - 20 * q^33 - 5 * q^35 + 5 * q^37 + 14 * q^39 + 2 * q^41 + 5 * q^43 + 18 * q^45 + 18 * q^49 + 10 * q^51 - 23 * q^53 + 6 * q^55 + 28 * q^57 + q^59 + 20 * q^61 - 26 * q^63 + 5 * q^65 - 21 * q^67 + 10 * q^69 - 4 * q^71 + 5 * q^73 + q^75 - 26 * q^77 - 41 * q^79 + 41 * q^81 + 7 * q^83 - 17 * q^85 + 40 * q^87 + 20 * q^89 + 42 * q^91 - 36 * q^93 + 6 * q^95 + 37 * q^97 - 12 * q^99

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