LMFDB - Embedded newform 340.2.c.a.101.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [340,2,Mod(101,340)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(340, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("340.101"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 340 = 2^{2} \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	340.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.71491366872$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.37161216.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 15x^{4} + 51x^{2} + 1 $ x^6 + 15x^4 + 51x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			101.6
Root			$2.27307i$ of defining polynomial
Character	$\chi$	$=$	340.101
Dual form			340.2.c.a.101.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.27307i q^{3} -1.00000i q^{5} +1.27307i q^{7} -7.71301 q^{9} +5.43993i q^{11} -0.166860 q^{13} +3.27307 q^{15} +(2.27307 - 3.43993i) q^{17} -2.54615 q^{19} -4.16686 q^{21} -3.27307i q^{23} -1.00000 q^{25} -15.4260i q^{27} +6.54615i q^{29} +5.98608i q^{31} -17.8053 q^{33} +1.27307 q^{35} +4.00000i q^{37} -0.546146i q^{39} -8.54615i q^{41} +2.71301 q^{43} +7.71301i q^{45} +10.7130 q^{47} +5.37929 q^{49} +(11.2592 + 7.43993i) q^{51} +12.8799 q^{53} +5.43993 q^{55} -8.33372i q^{57} -6.21243 q^{59} +7.42601i q^{61} -9.81922i q^{63} +0.166860i q^{65} -6.37929 q^{67} +10.7130 q^{69} +2.89379i q^{71} +12.5461i q^{73} -3.27307i q^{75} -6.92543 q^{77} -3.77365i q^{79} +27.3514 q^{81} +13.5929 q^{83} +(-3.43993 - 2.27307i) q^{85} -21.4260 q^{87} -5.83314 q^{89} -0.212425i q^{91} -19.5929 q^{93} +2.54615i q^{95} +1.45385i q^{97} -41.9582i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 14 q^{9} + 4 q^{15} - 2 q^{17} + 16 q^{19} - 24 q^{21} - 6 q^{25} - 12 q^{33} - 8 q^{35} - 16 q^{43} + 32 q^{47} + 2 q^{49} + 4 q^{51} + 44 q^{53} + 16 q^{55} - 8 q^{59} - 8 q^{67} + 32 q^{69} + 20 q^{77}+ \cdots - 52 q^{93}+O(q^{100}) $ 6 * q - 14 * q^9 + 4 * q^15 - 2 * q^17 + 16 * q^19 - 24 * q^21 - 6 * q^25 - 12 * q^33 - 8 * q^35 - 16 * q^43 + 32 * q^47 + 2 * q^49 + 4 * q^51 + 44 * q^53 + 16 * q^55 - 8 * q^59 - 8 * q^67 + 32 * q^69 + 20 * q^77 + 38 * q^81 + 16 * q^83 - 4 * q^85 - 64 * q^87 - 36 * q^89 - 52 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/340\mathbb{Z}\right)^\times$.

$n$	$137$	$171$	$241$
$\chi(n)$	$1$	$1$	$-1$

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
$3$			3.27307i			1.88971i	0.327490	+	0.944855i	$0.393797\pi$
$3$			3.27307i			1.88971i	−0.327490	+	0.944855i	$0.606203\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$			1.27307i			0.481176i	0.970627	+	0.240588i	$0.0773403\pi$
$7$			1.27307i			0.481176i	−0.970627	+	0.240588i	$0.922660\pi$
$8$		0			0
$9$	−7.71301			−2.57100
$10$		0			0
$11$			5.43993i			1.64020i	0.572219	+	0.820101i	$0.306083\pi$
$11$			5.43993i			1.64020i	−0.572219	+	0.820101i	$0.693917\pi$
$12$		0			0
$13$	−0.166860			−0.0462787			−0.0231394	−	0.999732i	$-0.507366\pi$
$13$	−0.166860			−0.0462787			−0.0231394	+	0.999732i	$0.507366\pi$
$14$		0			0
$15$	3.27307			0.845104
$16$		0			0
$17$	2.27307	−	3.43993i	0.551301	−	0.834306i
$17$	2.27307	−	3.43993i	0.551301	−	0.834306i
$18$		0			0
$19$	−2.54615			−0.584126			−0.292063	−	0.956399i	$-0.594342\pi$
$19$	−2.54615			−0.584126			−0.292063	+	0.956399i	$0.594342\pi$
$20$		0			0
$21$	−4.16686			−0.909283
$22$		0			0
$23$		−	3.27307i		−	0.682483i	−0.939976	−	0.341241i	$-0.889153\pi$
$23$		−	3.27307i		−	0.682483i	0.939976	−	0.341241i	$-0.110847\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	15.4260i		−	2.96874i
$28$		0			0
$29$			6.54615i			1.21559i	0.794094	+	0.607794i	$0.207946\pi$
$29$			6.54615i			1.21559i	−0.794094	+	0.607794i	$0.792054\pi$
$30$		0			0
$31$			5.98608i			1.07513i	0.843222	+	0.537566i	$0.180656\pi$
$31$			5.98608i			1.07513i	−0.843222	+	0.537566i	$0.819344\pi$
$32$		0			0
$33$	−17.8053			−3.09950
$34$		0			0
$35$	1.27307			0.215189
$36$		0			0
$37$			4.00000i			0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$			4.00000i			0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$		0			0
$39$		−	0.546146i		−	0.0874533i
$40$		0			0
$41$		−	8.54615i		−	1.33468i	−0.744752	−	0.667342i	$-0.767432\pi$
$41$		−	8.54615i		−	1.33468i	0.744752	−	0.667342i	$-0.232568\pi$
$42$		0			0
$43$	2.71301			0.413730			0.206865	−	0.978370i	$-0.433674\pi$
$43$	2.71301			0.413730			0.206865	+	0.978370i	$0.433674\pi$
$44$		0			0
$45$			7.71301i			1.14979i
$46$		0			0
$47$	10.7130			1.56265			0.781326	−	0.624123i	$-0.214544\pi$
$47$	10.7130			1.56265			0.781326	+	0.624123i	$0.214544\pi$
$48$		0			0
$49$	5.37929			0.768469
$50$		0			0
$51$	11.2592	+	7.43993i	1.57660	+	1.04180i
$52$		0			0
$53$	12.8799			1.76919			0.884593	−	0.466364i	$-0.154436\pi$
$53$	12.8799			1.76919			0.884593	+	0.466364i	$0.154436\pi$
$54$		0			0
$55$	5.43993			0.733520
$56$		0			0
$57$		−	8.33372i		−	1.10383i
$58$		0			0
$59$	−6.21243			−0.808789			−0.404394	−	0.914585i	$-0.632518\pi$
$59$	−6.21243			−0.808789			−0.404394	+	0.914585i	$0.632518\pi$
$60$		0			0
$61$			7.42601i			0.950803i	0.879769	+	0.475402i	$0.157697\pi$
$61$			7.42601i			0.950803i	−0.879769	+	0.475402i	$0.842303\pi$
$62$		0			0
$63$		−	9.81922i		−	1.23711i
$64$		0			0
$65$			0.166860i			0.0206965i
$66$		0			0
$67$	−6.37929			−0.779354			−0.389677	−	0.920952i	$-0.627413\pi$
$67$	−6.37929			−0.779354			−0.389677	+	0.920952i	$0.627413\pi$
$68$		0			0
$69$	10.7130			1.28969
$70$		0			0
$71$			2.89379i			0.343429i	0.985147	+	0.171715i	$0.0549307\pi$
$71$			2.89379i			0.343429i	−0.985147	+	0.171715i	$0.945069\pi$
$72$		0			0
$73$			12.5461i			1.46842i	0.678925	+	0.734208i	$0.262446\pi$
$73$			12.5461i			1.46842i	−0.678925	+	0.734208i	$0.737554\pi$
$74$		0			0
$75$		−	3.27307i		−	0.377942i
$76$		0			0
$77$	−6.92543			−0.789226
$78$		0			0
$79$		−	3.77365i		−	0.424569i	−0.977208	−	0.212285i	$-0.931910\pi$
$79$		−	3.77365i		−	0.424569i	0.977208	−	0.212285i	$-0.0680904\pi$
$80$		0			0
$81$	27.3514			3.03905
$82$		0			0
$83$	13.5929			1.49201			0.746006	−	0.665939i	$-0.231969\pi$
$83$	13.5929			1.49201			0.746006	+	0.665939i	$0.231969\pi$
$84$		0			0
$85$	−3.43993	−	2.27307i	−0.373113	−	0.246549i
$86$		0			0
$87$	−21.4260			−2.29711
$88$		0			0
$89$	−5.83314			−0.618312			−0.309156	−	0.951011i	$-0.600046\pi$
$89$	−5.83314			−0.618312			−0.309156	+	0.951011i	$0.600046\pi$
$90$		0			0
$91$		−	0.212425i		−	0.0222682i
$92$		0			0
$93$	−19.5929			−2.03169
$94$		0			0
$95$			2.54615i			0.261229i
$96$		0			0
$97$			1.45385i			0.147617i	0.997272	+	0.0738083i	$0.0235153\pi$
$97$			1.45385i			0.147617i	−0.997272	+	0.0738083i	$0.976485\pi$
$98$		0			0
$99$		−	41.9582i		−	4.21696i
$100$		0			0
$101$	11.5929			1.15353			0.576767	−	0.816909i	$-0.304314\pi$
$101$	11.5929			1.15353			0.576767	+	0.816909i	$0.304314\pi$
$102$		0			0
$103$	8.92543			0.879449			0.439724	−	0.898133i	$-0.355076\pi$
$103$	8.92543			0.879449			0.439724	+	0.898133i	$0.355076\pi$
$104$		0			0
$105$			4.16686i			0.406644i
$106$		0			0
$107$			1.84706i			0.178562i	0.996006	+	0.0892811i	$0.0284569\pi$
$107$			1.84706i			0.178562i	−0.996006	+	0.0892811i	$0.971543\pi$
$108$		0			0
$109$		−	10.0000i		−	0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$		−	10.0000i		−	0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$		0			0
$111$	−13.0923			−1.24267
$112$		0			0
$113$		−	17.9722i		−	1.69068i	−0.534230	−	0.845339i	$-0.679398\pi$
$113$		−	17.9722i		−	1.69068i	0.534230	−	0.845339i	$-0.320602\pi$
$114$		0			0
$115$	−3.27307			−0.305216
$116$		0			0
$117$	1.28699			0.118983
$118$		0			0
$119$	4.37929	+	2.89379i	0.401448	+	0.265273i
$120$		0			0
$121$	−18.5929			−1.69026
$122$		0			0
$123$	27.9722			2.52216
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	−11.8053			−1.04755			−0.523775	−	0.851856i	$-0.675477\pi$
$127$	−11.8053			−1.04755			−0.523775	+	0.851856i	$0.675477\pi$
$128$		0			0
$129$			8.87987i			0.781829i
$130$		0			0
$131$		−	4.56007i		−	0.398415i	−0.979957	−	0.199207i	$-0.936163\pi$
$131$		−	4.56007i		−	0.398415i	0.979957	−	0.199207i	$-0.0638368\pi$
$132$		0			0
$133$		−	3.24143i		−	0.281068i
$134$		0			0
$135$	−15.4260			−1.32766
$136$		0			0
$137$	−8.16686			−0.697742			−0.348871	−	0.937171i	$-0.613435\pi$
$137$	−8.16686			−0.697742			−0.348871	+	0.937171i	$0.613435\pi$
$138$		0			0
$139$		−	8.31980i		−	0.705676i	−0.935684	−	0.352838i	$-0.885217\pi$
$139$		−	8.31980i		−	0.705676i	0.935684	−	0.352838i	$-0.114783\pi$
$140$		0			0
$141$			35.0644i			2.95296i
$142$		0			0
$143$		−	0.907709i		−	0.0759064i
$144$		0			0
$145$	6.54615			0.543628
$146$		0			0
$147$			17.6068i			1.45218i
$148$		0			0
$149$	21.9722			1.80003			0.900015	−	0.435860i	$-0.143556\pi$
$149$	21.9722			1.80003			0.900015	+	0.435860i	$0.143556\pi$
$150$		0			0
$151$	−10.8799			−0.885391			−0.442695	−	0.896672i	$-0.645978\pi$
$151$	−10.8799			−0.885391			−0.442695	+	0.896672i	$0.645978\pi$
$152$		0			0
$153$	−17.5322	+	26.5322i	−1.41740	+	2.14500i
$154$		0			0
$155$	5.98608			0.480813
$156$		0			0
$157$	−13.9722			−1.11510			−0.557550	−	0.830144i	$-0.688258\pi$
$157$	−13.9722			−1.11510			−0.557550	+	0.830144i	$0.688258\pi$
$158$		0			0
$159$			42.1567i			3.34325i
$160$		0			0
$161$	4.16686			0.328395
$162$		0			0
$163$		−	8.93935i		−	0.700184i	−0.936715	−	0.350092i	$-0.886150\pi$
$163$		−	8.93935i		−	0.700184i	0.936715	−	0.350092i	$-0.113850\pi$
$164$		0			0
$165$			17.8053i			1.38614i
$166$		0			0
$167$		−	2.39321i		−	0.185192i	−0.995704	−	0.0925959i	$-0.970484\pi$
$167$		−	2.39321i		−	0.185192i	0.995704	−	0.0925959i	$-0.0295165\pi$
$168$		0			0
$169$	−12.9722			−0.997858
$170$		0			0
$171$	19.6384			1.50179
$172$		0			0
$173$		−	1.42601i		−	0.108418i	−0.998530	−	0.0542088i	$-0.982736\pi$
$173$		−	1.42601i		−	0.108418i	0.998530	−	0.0542088i	$-0.0172637\pi$
$174$		0			0
$175$		−	1.27307i		−	0.0962353i
$176$		0			0
$177$		−	20.3337i		−	1.52838i
$178$		0			0
$179$	13.0923			0.978564			0.489282	−	0.872126i	$-0.337259\pi$
$179$	13.0923			0.978564			0.489282	+	0.872126i	$0.337259\pi$
$180$		0			0
$181$		−	19.9722i		−	1.48452i	−0.670113	−	0.742259i	$-0.733754\pi$
$181$		−	19.9722i		−	1.48452i	0.670113	−	0.742259i	$-0.266246\pi$
$182$		0			0
$183$	−24.3059			−1.79674
$184$		0			0
$185$	4.00000			0.294086
$186$		0			0
$187$	18.7130	+	12.3654i	1.36843	+	0.904245i
$188$		0			0
$189$	19.6384			1.42849
$190$		0			0
$191$	−20.3059			−1.46928			−0.734641	−	0.678456i	$-0.762649\pi$
$191$	−20.3059			−1.46928			−0.734641	+	0.678456i	$0.762649\pi$
$192$		0			0
$193$		−	12.2124i		−	0.879070i	−0.898226	−	0.439535i	$-0.855143\pi$
$193$		−	12.2124i		−	0.879070i	0.898226	−	0.439535i	$-0.144857\pi$
$194$		0			0
$195$	−0.546146			−0.0391103
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$			13.3186i			0.944133i	0.881563	+	0.472067i	$0.156492\pi$
$199$			13.3186i			0.944133i	−0.881563	+	0.472067i	$0.843508\pi$
$200$		0			0
$201$		−	20.8799i		−	1.47275i
$202$		0			0
$203$	−8.33372			−0.584913
$204$		0			0
$205$	−8.54615			−0.596889
$206$		0			0
$207$			25.2452i			1.75466i
$208$		0			0
$209$		−	13.8509i		−	0.958084i
$210$		0			0
$211$			20.8659i			1.43647i	0.695800	+	0.718235i	$0.255050\pi$
$211$			20.8659i			1.43647i	−0.695800	+	0.718235i	$0.744950\pi$
$212$		0			0
$213$	−9.47158			−0.648982
$214$		0			0
$215$		−	2.71301i		−	0.185025i
$216$		0			0
$217$	−7.62071			−0.517328
$218$		0			0
$219$	−41.0644			−2.77488
$220$		0			0
$221$	−0.379285	+	0.573988i	−0.0255135	+	0.0386106i
$222$		0			0
$223$	13.2592			0.887898			0.443949	−	0.896052i	$-0.353577\pi$
$223$	13.2592			0.887898			0.443949	+	0.896052i	$0.353577\pi$
$224$		0			0
$225$	7.71301			0.514200
$226$		0			0
$227$		−	9.27307i		−	0.615475i	−0.951471	−	0.307738i	$-0.900428\pi$
$227$		−	9.27307i		−	0.615475i	0.951471	−	0.307738i	$-0.0995719\pi$
$228$		0			0
$229$	−14.1669			−0.936172			−0.468086	−	0.883683i	$-0.655056\pi$
$229$	−14.1669			−0.936172			−0.468086	+	0.883683i	$0.655056\pi$
$230$		0			0
$231$		−	22.6674i		−	1.49141i
$232$		0			0
$233$		−	29.0644i		−	1.90408i	−0.305981	−	0.952038i	$-0.598984\pi$
$233$		−	29.0644i		−	1.90408i	0.305981	−	0.952038i	$-0.401016\pi$
$234$		0			0
$235$		−	10.7130i		−	0.698839i
$236$		0			0
$237$	12.3514			0.802312
$238$		0			0
$239$	−5.42601			−0.350980			−0.175490	−	0.984481i	$-0.556151\pi$
$239$	−5.42601			−0.350980			−0.175490	+	0.984481i	$0.556151\pi$
$240$		0			0
$241$			20.2124i			1.30200i	0.759079	+	0.650998i	$0.225650\pi$
$241$			20.2124i			1.30200i	−0.759079	+	0.650998i	$0.774350\pi$
$242$		0			0
$243$			43.2452i			2.77418i
$244$		0			0
$245$		−	5.37929i		−	0.343670i
$246$		0			0
$247$	0.424850			0.0270326
$248$		0			0
$249$			44.4905i			2.81947i
$250$		0			0
$251$	−19.3047			−1.21850			−0.609251	−	0.792977i	$-0.708530\pi$
$251$	−19.3047			−1.21850			−0.609251	+	0.792977i	$0.708530\pi$
$252$		0			0
$253$	17.8053			1.11941
$254$		0			0
$255$	7.43993	−	11.2592i	0.465907	−	0.705075i
$256$		0			0
$257$	1.59287			0.0993606			0.0496803	−	0.998765i	$-0.484180\pi$
$257$	1.59287			0.0993606			0.0496803	+	0.998765i	$0.484180\pi$
$258$		0			0
$259$	−5.09229			−0.316420
$260$		0			0
$261$		−	50.4905i		−	3.12528i
$262$		0			0
$263$	−2.37929			−0.146713			−0.0733565	−	0.997306i	$-0.523371\pi$
$263$	−2.37929			−0.146713			−0.0733565	+	0.997306i	$0.523371\pi$
$264$		0			0
$265$		−	12.8799i		−	0.791204i
$266$		0			0
$267$		−	19.0923i		−	1.16843i
$268$		0			0
$269$		−	18.4249i		−	1.12338i	−0.827347	−	0.561691i	$-0.810151\pi$
$269$		−	18.4249i		−	1.12338i	0.827347	−	0.561691i	$-0.189849\pi$
$270$		0			0
$271$	25.0923			1.52425			0.762124	−	0.647431i	$-0.224157\pi$
$271$	25.0923			1.52425			0.762124	+	0.647431i	$0.224157\pi$
$272$		0			0
$273$	0.695283			0.0420805
$274$		0			0
$275$		−	5.43993i		−	0.328040i
$276$		0			0
$277$		−	12.1846i		−	0.732101i	−0.930595	−	0.366050i	$-0.880710\pi$
$277$		−	12.1846i		−	0.732101i	0.930595	−	0.366050i	$-0.119290\pi$
$278$		0			0
$279$		−	46.1707i		−	2.76417i
$280$		0			0
$281$	4.21243			0.251292			0.125646	−	0.992075i	$-0.459900\pi$
$281$	4.21243			0.251292			0.125646	+	0.992075i	$0.459900\pi$
$282$		0			0
$283$			16.4867i			0.980030i	0.871714	+	0.490015i	$0.163009\pi$
$283$			16.4867i			0.980030i	−0.871714	+	0.490015i	$0.836991\pi$
$284$		0			0
$285$	−8.33372			−0.493647
$286$		0			0
$287$	10.8799			0.642218
$288$		0			0
$289$	−6.66628	−	15.6384i	−0.392134	−	0.919908i
$290$		0			0
$291$	−4.75857			−0.278952
$292$		0			0
$293$	−4.57399			−0.267215			−0.133608	−	0.991034i	$-0.542656\pi$
$293$	−4.57399			−0.267215			−0.133608	+	0.991034i	$0.542656\pi$
$294$		0			0
$295$			6.21243i			0.361701i
$296$		0			0
$297$	83.9165			4.86933
$298$		0			0
$299$			0.546146i			0.0315844i
$300$		0			0
$301$			3.45385i			0.199077i
$302$		0			0
$303$			37.9443i			2.17984i
$304$		0			0
$305$	7.42601			0.425212
$306$		0			0
$307$	15.0189			0.857173			0.428586	−	0.903501i	$-0.359012\pi$
$307$	15.0189			0.857173			0.428586	+	0.903501i	$0.359012\pi$
$308$		0			0
$309$			29.2136i			1.66190i
$310$		0			0
$311$			6.22635i			0.353064i	0.984295	+	0.176532i	$0.0564879\pi$
$311$			6.22635i			0.353064i	−0.984295	+	0.176532i	$0.943512\pi$
$312$		0			0
$313$			4.54615i			0.256963i	0.991712	+	0.128482i	$0.0410103\pi$
$313$			4.54615i			0.256963i	−0.991712	+	0.128482i	$0.958990\pi$
$314$		0			0
$315$	−9.81922			−0.553250
$316$		0			0
$317$			11.4260i			0.641749i	0.947122	+	0.320874i	$0.103977\pi$
$317$			11.4260i			0.641749i	−0.947122	+	0.320874i	$0.896023\pi$
$318$		0			0
$319$	−35.6106			−1.99381
$320$		0			0
$321$	−6.04557			−0.337431
$322$		0			0
$323$	−5.78757	+	8.75857i	−0.322029	+	0.487340i
$324$		0			0
$325$	0.166860			0.00925574
$326$		0			0
$327$	32.7307			1.81001
$328$		0			0
$329$			13.6384i			0.751911i
$330$		0			0
$331$	3.63844			0.199987			0.0999933	−	0.994988i	$-0.468118\pi$
$331$	3.63844			0.199987			0.0999933	+	0.994988i	$0.468118\pi$
$332$		0			0
$333$		−	30.8520i		−	1.69068i
$334$		0			0
$335$			6.37929i			0.348538i
$336$		0			0
$337$			7.97216i			0.434271i	0.976141	+	0.217136i	$0.0696714\pi$
$337$			7.97216i			0.434271i	−0.976141	+	0.217136i	$0.930329\pi$
$338$		0			0
$339$	58.8242			3.19489
$340$		0			0
$341$	−32.5639			−1.76343
$342$		0			0
$343$			15.7597i			0.850946i
$344$		0			0
$345$		−	10.7130i		−	0.576769i
$346$		0			0
$347$			0.152939i			0.00821020i	0.999992	+	0.00410510i	$0.00130670\pi$
$347$			0.152939i			0.00821020i	−0.999992	+	0.00410510i	$0.998693\pi$
$348$		0			0
$349$	−4.21243			−0.225486			−0.112743	−	0.993624i	$-0.535964\pi$
$349$	−4.21243			−0.225486			−0.112743	+	0.993624i	$0.535964\pi$
$350$		0			0
$351$			2.57399i			0.137389i
$352$		0			0
$353$	6.66744			0.354872			0.177436	−	0.984132i	$-0.443220\pi$
$353$	6.66744			0.354872			0.177436	+	0.984132i	$0.443220\pi$
$354$		0			0
$355$	2.89379			0.153586
$356$		0			0
$357$	−9.47158	+	14.3337i	−0.501289	+	0.758621i
$358$		0			0
$359$	−2.51830			−0.132911			−0.0664555	−	0.997789i	$-0.521169\pi$
$359$	−2.51830			−0.132911			−0.0664555	+	0.997789i	$0.521169\pi$
$360$		0			0
$361$	−12.5171			−0.658797
$362$		0			0
$363$		−	60.8558i		−	3.19410i
$364$		0			0
$365$	12.5461			0.656695
$366$		0			0
$367$		−	19.1518i		−	0.999715i	−0.866108	−	0.499857i	$-0.833386\pi$
$367$		−	19.1518i		−	0.999715i	0.866108	−	0.499857i	$-0.166614\pi$
$368$		0			0
$369$			65.9165i			3.43147i
$370$		0			0
$371$			16.3970i			0.851290i
$372$		0			0
$373$	24.9254			1.29059			0.645295	−	0.763934i	$-0.276734\pi$
$373$	24.9254			1.29059			0.645295	+	0.763934i	$0.276734\pi$
$374$		0			0
$375$	−3.27307			−0.169021
$376$		0			0
$377$		−	1.09229i		−	0.0562559i
$378$		0			0
$379$		−	7.41209i		−	0.380734i	−0.981713	−	0.190367i	$-0.939032\pi$
$379$		−	7.41209i		−	0.380734i	0.981713	−	0.190367i	$-0.0609677\pi$
$380$		0			0
$381$		−	38.6396i		−	1.97957i
$382$		0			0
$383$	30.0177			1.53383			0.766917	−	0.641746i	$-0.221790\pi$
$383$	30.0177			1.53383			0.766917	+	0.641746i	$0.221790\pi$
$384$		0			0
$385$			6.92543i			0.352953i
$386$		0			0
$387$	−20.9254			−1.06370
$388$		0			0
$389$	24.6573			1.25018			0.625088	−	0.780554i	$-0.285063\pi$
$389$	24.6573			1.25018			0.625088	+	0.780554i	$0.285063\pi$
$390$		0			0
$391$	−11.2592	−	7.43993i	−0.569400	−	0.376254i
$392$		0			0
$393$	14.9254			0.752888
$394$		0			0
$395$	−3.77365			−0.189873
$396$		0			0
$397$			16.8520i			0.845779i	0.906181	+	0.422889i	$0.138984\pi$
$397$			16.8520i			0.845779i	−0.906181	+	0.422889i	$0.861016\pi$
$398$		0			0
$399$	10.6094			0.531136
$400$		0			0
$401$		−	22.5183i		−	1.12451i	−0.826964	−	0.562255i	$-0.809934\pi$
$401$		−	22.5183i		−	1.12451i	0.826964	−	0.562255i	$-0.190066\pi$
$402$		0			0
$403$		−	0.998839i		−	0.0497557i
$404$		0			0
$405$		−	27.3514i		−	1.35910i
$406$		0			0
$407$	−21.7597			−1.07859
$408$		0			0
$409$	14.0000			0.692255			0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000			0.692255			0.346128	+	0.938187i	$0.387496\pi$
$410$		0			0
$411$		−	26.7307i		−	1.31853i
$412$		0			0
$413$		−	7.90887i		−	0.389170i
$414$		0			0
$415$		−	13.5929i		−	0.667248i
$416$		0			0
$417$	27.2313			1.33352
$418$		0			0
$419$			35.4399i			1.73135i	0.500603	+	0.865677i	$0.333112\pi$
$419$			35.4399i			1.73135i	−0.500603	+	0.865677i	$0.666888\pi$
$420$		0			0
$421$	−27.2313			−1.32717			−0.663586	−	0.748100i	$-0.730966\pi$
$421$	−27.2313			−1.32717			−0.663586	+	0.748100i	$0.730966\pi$
$422$		0			0
$423$	−82.6295			−4.01758
$424$		0			0
$425$	−2.27307	+	3.43993i	−0.110260	+	0.166861i
$426$		0			0
$427$	−9.45385			−0.457504
$428$		0			0
$429$	2.97100			0.143441
$430$		0			0
$431$			8.56007i			0.412324i	0.978518	+	0.206162i	$0.0660974\pi$
$431$			8.56007i			0.412324i	−0.978518	+	0.206162i	$0.933903\pi$
$432$		0			0
$433$	6.68516			0.321268			0.160634	−	0.987014i	$-0.448646\pi$
$433$	6.68516			0.321268			0.160634	+	0.987014i	$0.448646\pi$
$434$		0			0
$435$			21.4260i			1.02730i
$436$		0			0
$437$			8.33372i			0.398656i
$438$		0			0
$439$		−	35.2908i		−	1.68434i	−0.539214	−	0.842169i	$-0.681279\pi$
$439$		−	35.2908i		−	1.68434i	0.539214	−	0.842169i	$-0.318721\pi$
$440$		0			0
$441$	−41.4905			−1.97574
$442$		0			0
$443$	−27.7775			−1.31975			−0.659873	−	0.751377i	$-0.729390\pi$
$443$	−27.7775			−1.31975			−0.659873	+	0.751377i	$0.729390\pi$
$444$		0			0
$445$			5.83314i			0.276517i
$446$		0			0
$447$			71.9165i			3.40153i
$448$		0			0
$449$			6.30588i			0.297593i	0.988868	+	0.148796i	$0.0475399\pi$
$449$			6.30588i			0.297593i	−0.988868	+	0.148796i	$0.952460\pi$
$450$		0			0
$451$	46.4905			2.18915
$452$		0			0
$453$		−	35.6106i		−	1.67313i
$454$		0			0
$455$	−0.212425			−0.00995865
$456$		0			0
$457$	15.1657			0.709421			0.354711	−	0.934976i	$-0.384579\pi$
$457$	15.1657			0.709421			0.354711	+	0.934976i	$0.384579\pi$
$458$		0			0
$459$	−53.0644	−	35.0644i	−2.47684	−	1.63667i
$460$		0			0
$461$	29.3047			1.36486			0.682428	−	0.730952i	$-0.260924\pi$
$461$	29.3047			1.36486			0.682428	+	0.730952i	$0.260924\pi$
$462$		0			0
$463$	−17.2592			−0.802101			−0.401050	−	0.916056i	$-0.631355\pi$
$463$	−17.2592			−0.802101			−0.401050	+	0.916056i	$0.631355\pi$
$464$		0			0
$465$			19.5929i			0.908598i
$466$		0			0
$467$	−16.8065			−0.777710			−0.388855	−	0.921299i	$-0.627129\pi$
$467$	−16.8065			−0.777710			−0.388855	+	0.921299i	$0.627129\pi$
$468$		0			0
$469$		−	8.12130i		−	0.375007i
$470$		0			0
$471$		−	45.7319i		−	2.10721i
$472$		0			0
$473$			14.7586i			0.678600i
$474$		0			0
$475$	2.54615			0.116825
$476$		0			0
$477$	−99.3425			−4.54858
$478$		0			0
$479$			12.5044i			0.571340i	0.958328	+	0.285670i	$0.0922161\pi$
$479$			12.5044i			0.571340i	−0.958328	+	0.285670i	$0.907784\pi$
$480$		0			0
$481$		−	0.667441i		−	0.0304327i
$482$		0			0
$483$			13.6384i			0.620570i
$484$		0			0
$485$	1.45385			0.0660161
$486$		0			0
$487$		−	24.4867i		−	1.10960i	−0.831985	−	0.554798i	$-0.812795\pi$
$487$		−	24.4867i		−	1.10960i	0.831985	−	0.554798i	$-0.187205\pi$
$488$		0			0
$489$	29.2592			1.32314
$490$		0			0
$491$	−14.0935			−0.636029			−0.318014	−	0.948086i	$-0.603016\pi$
$491$	−14.0935			−0.636029			−0.318014	+	0.948086i	$0.603016\pi$
$492$		0			0
$493$	22.5183	+	14.8799i	1.01417	+	0.670155i
$494$		0			0
$495$	−41.9582			−1.88588
$496$		0			0
$497$	−3.68400			−0.165250
$498$		0			0
$499$		−	5.89495i		−	0.263894i	−0.991257	−	0.131947i	$-0.957877\pi$
$499$		−	5.89495i		−	0.263894i	0.991257	−	0.131947i	$-0.0421229\pi$
$500$		0			0
$501$	7.83314			0.349959
$502$		0			0
$503$			15.0328i			0.670280i	0.942168	+	0.335140i	$0.108784\pi$
$503$			15.0328i			0.670280i	−0.942168	+	0.335140i	$0.891216\pi$
$504$		0			0
$505$		−	11.5929i		−	0.515876i
$506$		0			0
$507$		−	42.4588i		−	1.88566i
$508$		0			0
$509$	14.4249			0.639370			0.319685	−	0.947524i	$-0.396423\pi$
$509$	14.4249			0.639370			0.319685	+	0.947524i	$0.396423\pi$
$510$		0			0
$511$	−15.9722			−0.706567
$512$		0			0
$513$			39.2769i			1.73412i
$514$		0			0
$515$		−	8.92543i		−	0.393301i
$516$		0			0
$517$			58.2780i			2.56307i
$518$		0			0
$519$	4.66744			0.204878
$520$		0			0
$521$			9.42601i			0.412961i	0.978451	+	0.206481i	$0.0662010\pi$
$521$			9.42601i			0.412961i	−0.978451	+	0.206481i	$0.933799\pi$
$522$		0			0
$523$	19.7142			0.862040			0.431020	−	0.902342i	$-0.358154\pi$
$523$	19.7142			0.862040			0.431020	+	0.902342i	$0.358154\pi$
$524$		0			0
$525$	4.16686			0.181857
$526$		0			0
$527$	20.5917	+	13.6068i	0.896989	+	0.592721i
$528$		0			0
$529$	12.2870			0.534217
$530$		0			0
$531$	47.9165			2.07940
$532$		0			0
$533$			1.42601i			0.0617674i
$534$		0			0
$535$	1.84706			0.0798554
$536$		0			0
$537$			42.8520i			1.84920i
$538$		0			0
$539$			29.2630i			1.26044i
$540$		0			0
$541$			15.2136i			0.654083i	0.945010	+	0.327042i	$0.106052\pi$
$541$			15.2136i			0.654083i	−0.945010	+	0.327042i	$0.893948\pi$
$542$		0			0
$543$	65.3703			2.80531
$544$		0			0
$545$	−10.0000			−0.428353
$546$		0			0
$547$		−	27.5790i		−	1.17919i	−0.807699	−	0.589595i	$-0.799287\pi$
$547$		−	27.5790i		−	1.17919i	0.807699	−	0.589595i	$-0.200713\pi$
$548$		0			0
$549$		−	57.2769i		−	2.44452i
$550$		0			0
$551$		−	16.6674i		−	0.710057i
$552$		0			0
$553$	4.80414			0.204293
$554$		0			0
$555$			13.0923i			0.555737i
$556$		0			0
$557$	−36.1112			−1.53008			−0.765040	−	0.643983i	$-0.777281\pi$
$557$	−36.1112			−1.53008			−0.765040	+	0.643983i	$0.777281\pi$
$558$		0			0
$559$	−0.452693			−0.0191469
$560$		0			0
$561$	−40.4727	+	61.2490i	−1.70876	+	2.58594i
$562$		0			0
$563$	−24.1112			−1.01616			−0.508082	−	0.861308i	$-0.669645\pi$
$563$	−24.1112			−1.01616			−0.508082	+	0.861308i	$0.669645\pi$
$564$		0			0
$565$	−17.9722			−0.756094
$566$		0			0
$567$			34.8204i			1.46232i
$568$		0			0
$569$	−23.7597			−0.996060			−0.498030	−	0.867160i	$-0.665943\pi$
$569$	−23.7597			−0.996060			−0.498030	+	0.867160i	$0.665943\pi$
$570$		0			0
$571$			30.2920i			1.26768i	0.773465	+	0.633839i	$0.218522\pi$
$571$			30.2920i			1.26768i	−0.773465	+	0.633839i	$0.781478\pi$
$572$		0			0
$573$		−	66.4626i		−	2.77652i
$574$		0			0
$575$			3.27307i			0.136497i
$576$		0			0
$577$	10.6852			0.444829			0.222415	−	0.974952i	$-0.428606\pi$
$577$	10.6852			0.444829			0.222415	+	0.974952i	$0.428606\pi$
$578$		0			0
$579$	39.9722			1.66119
$580$		0			0
$581$			17.3047i			0.717921i
$582$		0			0
$583$			70.0656i			2.90182i
$584$		0			0
$585$		−	1.28699i		−	0.0532107i
$586$		0			0
$587$	1.64856			0.0680432			0.0340216	−	0.999421i	$-0.489168\pi$
$587$	1.64856			0.0680432			0.0340216	+	0.999421i	$0.489168\pi$
$588$		0			0
$589$		−	15.2414i		−	0.628012i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.21359			−0.0498360			−0.0249180	−	0.999689i	$-0.507932\pi$
$593$	−1.21359			−0.0498360			−0.0249180	+	0.999689i	$0.507932\pi$
$594$		0			0
$595$	2.89379	−	4.37929i	0.118634	−	0.179533i
$596$		0			0
$597$	−43.5929			−1.78414
$598$		0			0
$599$	−8.36156			−0.341644			−0.170822	−	0.985302i	$-0.554642\pi$
$599$	−8.36156			−0.341644			−0.170822	+	0.985302i	$0.554642\pi$
$600$		0			0
$601$			7.06445i			0.288165i	0.989566	+	0.144082i	$0.0460231\pi$
$601$			7.06445i			0.288165i	−0.989566	+	0.144082i	$0.953977\pi$
$602$		0			0
$603$	49.2035			2.00372
$604$		0			0
$605$			18.5929i			0.755908i
$606$		0			0
$607$			22.9115i			0.929950i	0.885324	+	0.464975i	$0.153937\pi$
$607$			22.9115i			0.929950i	−0.885324	+	0.464975i	$0.846063\pi$
$608$		0			0
$609$		−	27.2769i		−	1.10531i
$610$		0			0
$611$	−1.78757			−0.0723175
$612$		0			0
$613$	−0.573988			−0.0231832			−0.0115916	−	0.999933i	$-0.503690\pi$
$613$	−0.573988			−0.0231832			−0.0115916	+	0.999933i	$0.503690\pi$
$614$		0			0
$615$		−	27.9722i		−	1.12795i
$616$		0			0
$617$			11.7319i			0.472308i	0.971716	+	0.236154i	$0.0758870\pi$
$617$			11.7319i			0.472308i	−0.971716	+	0.236154i	$0.924113\pi$
$618$		0			0
$619$		−	2.50438i		−	0.100660i	−0.998733	−	0.0503298i	$-0.983973\pi$
$619$		−	2.50438i		−	0.100660i	0.998733	−	0.0503298i	$-0.0160272\pi$
$620$		0			0
$621$	−50.4905			−2.02611
$622$		0			0
$623$		−	7.42601i		−	0.297517i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	45.3349			1.81050
$628$		0			0
$629$	13.7597	+	9.09229i	0.548636	+	0.362533i
$630$		0			0
$631$	38.0656			1.51537			0.757684	−	0.652622i	$-0.226331\pi$
$631$	38.0656			1.51537			0.757684	+	0.652622i	$0.226331\pi$
$632$		0			0
$633$	−68.2958			−2.71451
$634$		0			0
$635$			11.8053i			0.468479i
$636$		0			0
$637$	−0.897589			−0.0355638
$638$		0			0
$639$		−	22.3198i		−	0.882958i
$640$		0			0
$641$		−	18.7307i		−	0.739819i	−0.929068	−	0.369910i	$-0.879389\pi$
$641$		−	18.7307i		−	0.739819i	0.929068	−	0.369910i	$-0.120611\pi$
$642$		0			0
$643$		−	8.72693i		−	0.344156i	−0.985083	−	0.172078i	$-0.944952\pi$
$643$		−	8.72693i		−	0.344156i	0.985083	−	0.172078i	$-0.0550482\pi$
$644$		0			0
$645$	8.87987			0.349644
$646$		0			0
$647$	14.6573			0.576239			0.288119	−	0.957594i	$-0.406970\pi$
$647$	14.6573			0.576239			0.288119	+	0.957594i	$0.406970\pi$
$648$		0			0
$649$		−	33.7952i		−	1.32658i
$650$		0			0
$651$		−	24.9432i		−	0.977599i
$652$		0			0
$653$			17.4260i			0.681933i	0.940076	+	0.340966i	$0.110754\pi$
$653$			17.4260i			0.681933i	−0.940076	+	0.340966i	$0.889246\pi$
$654$		0			0
$655$	−4.56007			−0.178177
$656$		0			0
$657$		−	96.7685i		−	3.77530i
$658$		0			0
$659$	−39.8532			−1.55246			−0.776230	−	0.630450i	$-0.782870\pi$
$659$	−39.8532			−1.55246			−0.776230	+	0.630450i	$0.782870\pi$
$660$		0			0
$661$	40.8242			1.58788			0.793938	−	0.607998i	$-0.208027\pi$
$661$	40.8242			1.58788			0.793938	+	0.607998i	$0.208027\pi$
$662$		0			0
$663$	−1.87870	−	1.24143i	−0.0729628	−	0.0482131i
$664$		0			0
$665$	−3.24143			−0.125697
$666$		0			0
$667$	21.4260			0.829618
$668$		0			0
$669$			43.3982i			1.67787i
$670$		0			0
$671$	−40.3970			−1.55951
$672$		0			0
$673$		−	25.5473i		−	0.984776i	−0.870376	−	0.492388i	$-0.836124\pi$
$673$		−	25.5473i		−	0.984776i	0.870376	−	0.492388i	$-0.163876\pi$
$674$		0			0
$675$			15.4260i			0.593747i
$676$		0			0
$677$		−	3.33488i		−	0.128170i	−0.997944	−	0.0640850i	$-0.979587\pi$
$677$		−	3.33488i		−	0.128170i	0.997944	−	0.0640850i	$-0.0204129\pi$
$678$		0			0
$679$	−1.85086			−0.0710296
$680$		0			0
$681$	30.3514			1.16307
$682$		0			0
$683$		−	46.0973i		−	1.76386i	−0.471378	−	0.881931i	$-0.656243\pi$
$683$		−	46.0973i		−	1.76386i	0.471378	−	0.881931i	$-0.343757\pi$
$684$		0			0
$685$			8.16686i			0.312040i
$686$		0			0
$687$		−	46.3692i		−	1.76909i
$688$		0			0
$689$	−2.14914			−0.0818756
$690$		0			0
$691$		−	18.9849i		−	0.722220i	−0.932523	−	0.361110i	$-0.882398\pi$
$691$		−	18.9849i		−	0.722220i	0.932523	−	0.361110i	$-0.117602\pi$
$692$		0			0
$693$	53.4159			2.02910
$694$		0			0
$695$	−8.31980			−0.315588
$696$		0			0
$697$	−29.3982	−	19.4260i	−1.11354	−	0.735813i
$698$		0			0
$699$	95.1301			3.59815
$700$		0			0
$701$	42.8976			1.62022			0.810110	−	0.586278i	$-0.199407\pi$
$701$	42.8976			1.62022			0.810110	+	0.586278i	$0.199407\pi$
$702$		0			0
$703$		−	10.1846i		−	0.384119i
$704$		0			0
$705$	35.0644			1.32060
$706$		0			0
$707$			14.7586i			0.555053i
$708$		0			0
$709$			25.5450i			0.959362i	0.877443	+	0.479681i	$0.159248\pi$
$709$			25.5450i			0.959362i	−0.877443	+	0.479681i	$0.840752\pi$
$710$		0			0
$711$			29.1062i			1.09157i
$712$		0			0
$713$	19.5929			0.733759
$714$		0			0
$715$	−0.907709			−0.0339464
$716$		0			0
$717$		−	17.7597i		−	0.663249i
$718$		0			0
$719$		−	5.10621i		−	0.190430i	−0.995457	−	0.0952148i	$-0.969646\pi$
$719$		−	5.10621i		−	0.190430i	0.995457	−	0.0952148i	$-0.0303538\pi$
$720$		0			0
$721$			11.3627i			0.423170i
$722$		0			0
$723$	−66.1567			−2.46040
$724$		0			0
$725$		−	6.54615i		−	0.243118i
$726$		0			0
$727$	23.3526			0.866100			0.433050	−	0.901370i	$-0.357437\pi$
$727$	23.3526			0.866100			0.433050	+	0.901370i	$0.357437\pi$
$728$		0			0
$729$	−59.4905			−2.20335
$730$		0			0
$731$	6.16686	−	9.33256i	0.228090	−	0.345177i
$732$		0			0
$733$	−35.3703			−1.30643			−0.653216	−	0.757171i	$-0.726581\pi$
$733$	−35.3703			−1.30643			−0.653216	+	0.757171i	$0.726581\pi$
$734$		0			0
$735$	17.6068			0.649436
$736$		0			0
$737$		−	34.7029i		−	1.27830i
$738$		0			0
$739$	−22.3036			−0.820450			−0.410225	−	0.911984i	$-0.634550\pi$
$739$	−22.3036			−0.820450			−0.410225	+	0.911984i	$0.634550\pi$
$740$		0			0
$741$			1.39057i			0.0510837i
$742$		0			0
$743$		−	10.9394i		−	0.401326i	−0.979660	−	0.200663i	$-0.935690\pi$
$743$		−	10.9394i		−	0.401326i	0.979660	−	0.200663i	$-0.0643096\pi$
$744$		0			0
$745$		−	21.9722i		−	0.804998i
$746$		0			0
$747$	−104.842			−3.83596
$748$		0			0
$749$	−2.35144			−0.0859199
$750$		0			0
$751$		−	14.9216i		−	0.544498i	−0.962227	−	0.272249i	$-0.912233\pi$
$751$		−	14.9216i		−	0.544498i	0.962227	−	0.272249i	$-0.0877674\pi$
$752$		0			0
$753$		−	63.1857i		−	2.30262i
$754$		0			0
$755$			10.8799i			0.395959i
$756$		0			0
$757$	22.2603			0.809065			0.404532	−	0.914524i	$-0.367434\pi$
$757$	22.2603			0.809065			0.404532	+	0.914524i	$0.367434\pi$
$758$		0			0
$759$			58.2780i			2.11536i
$760$		0			0
$761$	−19.5929			−0.710241			−0.355121	−	0.934821i	$-0.615560\pi$
$761$	−19.5929			−0.710241			−0.355121	+	0.934821i	$0.615560\pi$
$762$		0			0
$763$	12.7307			0.460883
$764$		0			0
$765$	26.5322	+	17.5322i	0.959275	+	0.633879i
$766$		0			0
$767$	1.03661			0.0374297
$768$		0			0
$769$	−37.1100			−1.33822			−0.669111	−	0.743163i	$-0.733325\pi$
$769$	−37.1100			−1.33822			−0.669111	+	0.743163i	$0.733325\pi$
$770$		0			0
$771$			5.21359i			0.187763i
$772$		0			0
$773$	−38.7763			−1.39469			−0.697343	−	0.716737i	$-0.745635\pi$
$773$	−38.7763			−1.39469			−0.697343	+	0.716737i	$0.745635\pi$
$774$		0			0
$775$		−	5.98608i		−	0.215026i
$776$		0			0
$777$		−	16.6674i		−	0.597941i
$778$		0			0
$779$			21.7597i			0.779623i
$780$		0			0
$781$	−15.7420			−0.563293
$782$		0			0
$783$	100.981			3.60876
$784$		0			0
$785$			13.9722i			0.498688i
$786$		0			0
$787$		−	10.6080i		−	0.378133i	−0.981964	−	0.189066i	$-0.939454\pi$
$787$		−	10.6080i		−	0.378133i	0.981964	−	0.189066i	$-0.0605461\pi$
$788$		0			0
$789$		−	7.78757i		−	0.277245i
$790$		0			0
$791$	22.8799			0.813514
$792$		0			0
$793$		−	1.23911i		−	0.0440019i
$794$		0			0
$795$	42.1567			1.49515
$796$		0			0
$797$	9.66628			0.342397			0.171199	−	0.985237i	$-0.445236\pi$
$797$	9.66628			0.342397			0.171199	+	0.985237i	$0.445236\pi$
$798$		0			0
$799$	24.3514	−	36.8520i	0.861492	−	1.30373i
$800$		0			0
$801$	44.9910			1.58968
$802$		0			0
$803$	−68.2502			−2.40850
$804$		0			0
$805$		−	4.16686i		−	0.146863i
$806$		0			0
$807$	60.3059			2.12287
$808$		0			0
$809$			42.5183i			1.49486i	0.664338	+	0.747432i	$0.268713\pi$
$809$			42.5183i			1.49486i	−0.664338	+	0.747432i	$0.731287\pi$
$810$		0			0
$811$		−	6.65120i		−	0.233555i	−0.993158	−	0.116778i	$-0.962744\pi$
$811$		−	6.65120i		−	0.233555i	0.993158	−	0.116778i	$-0.0372565\pi$
$812$		0			0
$813$			82.1289i			2.88039i
$814$		0			0
$815$	−8.93935			−0.313132
$816$		0			0
$817$	−6.90771			−0.241670
$818$		0			0
$819$			1.63844i			0.0572516i
$820$		0			0
$821$			44.7963i			1.56340i	0.623653	+	0.781702i	$0.285648\pi$
$821$			44.7963i			1.56340i	−0.623653	+	0.781702i	$0.714352\pi$
$822$		0			0
$823$		−	9.36653i		−	0.326497i	−0.986585	−	0.163248i	$-0.947803\pi$
$823$		−	9.36653i		−	0.326497i	0.986585	−	0.163248i	$-0.0521972\pi$
$824$		0			0
$825$	17.8053			0.619901
$826$		0			0
$827$			44.7902i			1.55751i	0.627329	+	0.778754i	$0.284148\pi$
$827$			44.7902i			1.55751i	−0.627329	+	0.778754i	$0.715852\pi$
$828$		0			0
$829$	36.8520			1.27992			0.639962	−	0.768407i	$-0.278950\pi$
$829$	36.8520			1.27992			0.639962	+	0.768407i	$0.278950\pi$
$830$		0			0
$831$	39.8810			1.38346
$832$		0			0
$833$	12.2275	−	18.5044i	0.423658	−	0.641139i
$834$		0			0
$835$	−2.39321			−0.0828203
$836$		0			0
$837$	92.3413			3.19178
$838$		0			0
$839$		−	11.6245i		−	0.401323i	−0.979661	−	0.200661i	$-0.935691\pi$
$839$		−	11.6245i		−	0.401323i	0.979661	−	0.200661i	$-0.0643091\pi$
$840$		0			0
$841$	−13.8520			−0.477656
$842$		0			0
$843$			13.7876i			0.474869i
$844$		0			0
$845$			12.9722i			0.446256i
$846$		0			0
$847$		−	23.6701i		−	0.813314i
$848$		0			0
$849$	−53.9620			−1.85197
$850$		0			0
$851$	13.0923			0.448798
$852$		0			0
$853$			49.2792i			1.68729i	0.536903	+	0.843644i	$0.319594\pi$
$853$			49.2792i			1.68729i	−0.536903	+	0.843644i	$0.680406\pi$
$854$		0			0
$855$		−	19.6384i		−	0.671620i
$856$		0			0
$857$		−	23.6408i		−	0.807553i	−0.914858	−	0.403776i	$-0.867697\pi$
$857$		−	23.6408i		−	0.807553i	0.914858	−	0.403776i	$-0.132303\pi$
$858$		0			0
$859$	−14.0935			−0.480862			−0.240431	−	0.970666i	$-0.577289\pi$
$859$	−14.0935			−0.480862			−0.240431	+	0.970666i	$0.577289\pi$
$860$		0			0
$861$			35.6106i			1.21361i
$862$		0			0
$863$	4.28583			0.145891			0.0729457	−	0.997336i	$-0.476760\pi$
$863$	4.28583			0.145891			0.0729457	+	0.997336i	$0.476760\pi$
$864$		0			0
$865$	−1.42601			−0.0484859
$866$		0			0
$867$	51.1857	−	21.8192i	1.73836	−	0.741020i
$868$		0			0
$869$	20.5284			0.696379
$870$		0			0
$871$	1.06445			0.0360675
$872$		0			0
$873$		−	11.2136i		−	0.379522i
$874$		0			0
$875$	−1.27307			−0.0430377
$876$		0			0
$877$			7.85086i			0.265105i	0.991176	+	0.132552i	$0.0423173\pi$
$877$			7.85086i			0.265105i	−0.991176	+	0.132552i	$0.957683\pi$
$878$		0			0
$879$		−	14.9710i		−	0.504959i
$880$		0			0
$881$		−	5.30472i		−	0.178720i	−0.995999	−	0.0893602i	$-0.971518\pi$
$881$		−	5.30472i		−	0.178720i	0.995999	−	0.0893602i	$-0.0284822\pi$
$882$		0			0
$883$	3.74201			0.125929			0.0629643	−	0.998016i	$-0.479945\pi$
$883$	3.74201			0.125929			0.0629643	+	0.998016i	$0.479945\pi$
$884$		0			0
$885$	−20.3337			−0.683511
$886$		0			0
$887$		−	4.75477i		−	0.159650i	−0.996809	−	0.0798248i	$-0.974564\pi$
$887$		−	4.75477i		−	0.159650i	0.996809	−	0.0798248i	$-0.0254361\pi$
$888$		0			0
$889$		−	15.0290i		−	0.504057i
$890$		0			0
$891$			148.790i			4.98465i
$892$		0			0
$893$	−27.2769			−0.912786
$894$		0			0
$895$		−	13.0923i		−	0.437627i
$896$		0			0
$897$	−1.78757			−0.0596854
$898$		0			0
$899$	−39.1857			−1.30692
$900$		0			0
$901$	29.2769	−	44.3059i	0.975354	−	1.47604i
$902$		0			0
$903$	−11.3047			−0.376197
$904$		0			0
$905$	−19.9722			−0.663897
$906$		0			0
$907$		−	34.9771i		−	1.16140i	−0.814119	−	0.580698i	$-0.802780\pi$
$907$		−	34.9771i		−	1.16140i	0.814119	−	0.580698i	$-0.197220\pi$
$908$		0			0
$909$	−89.4159			−2.96574
$910$		0			0
$911$		−	20.2920i		−	0.672303i	−0.941808	−	0.336151i	$-0.890875\pi$
$911$		−	20.2920i		−	0.672303i	0.941808	−	0.336151i	$-0.109125\pi$
$912$		0			0
$913$			73.9443i			2.44720i
$914$		0			0
$915$			24.3059i			0.803528i
$916$		0			0
$917$	5.80530			0.191708
$918$		0			0
$919$	−26.8242			−0.884848			−0.442424	−	0.896806i	$-0.645881\pi$
$919$	−26.8242			−0.884848			−0.442424	+	0.896806i	$0.645881\pi$
$920$		0			0
$921$			49.1579i			1.61981i
$922$		0			0
$923$		−	0.482858i		−	0.0158935i
$924$		0			0
$925$		−	4.00000i		−	0.131519i
$926$		0			0
$927$	−68.8419			−2.26106
$928$		0			0
$929$			0.546146i			0.0179185i	0.999960	+	0.00895923i	$0.00285185\pi$
$929$			0.546146i			0.0179185i	−0.999960	+	0.00895923i	$0.997148\pi$
$930$		0			0
$931$	−13.6964			−0.448883
$932$		0			0
$933$	−20.3793			−0.667188
$934$		0			0
$935$	12.3654	−	18.7130i	0.404391	−	0.611981i
$936$		0			0
$937$	−22.7864			−0.744400			−0.372200	−	0.928153i	$-0.621396\pi$
$937$	−22.7864			−0.744400			−0.372200	+	0.928153i	$0.621396\pi$
$938$		0			0
$939$	−14.8799			−0.485586
$940$		0			0
$941$		−	8.69528i		−	0.283458i	−0.989905	−	0.141729i	$-0.954734\pi$
$941$		−	8.69528i		−	0.283458i	0.989905	−	0.141729i	$-0.0452662\pi$
$942$		0			0
$943$	−27.9722			−0.910899
$944$		0			0
$945$		−	19.6384i		−	0.638838i
$946$		0			0
$947$		−	55.3665i		−	1.79917i	−0.436745	−	0.899585i	$-0.643869\pi$
$947$		−	55.3665i		−	1.79917i	0.436745	−	0.899585i	$-0.356131\pi$
$948$		0			0
$949$		−	2.09345i		−	0.0679564i
$950$		0			0
$951$	−37.3982			−1.21272
$952$		0			0
$953$	50.4426			1.63400			0.816998	−	0.576641i	$-0.195637\pi$
$953$	50.4426			1.63400			0.816998	+	0.576641i	$0.195637\pi$
$954$		0			0
$955$			20.3059i			0.657083i
$956$		0			0
$957$		−	116.556i		−	3.76772i
$958$		0			0
$959$		−	10.3970i		−	0.335737i
$960$		0			0
$961$	−4.83314			−0.155908
$962$		0			0
$963$		−	14.2464i		−	0.459084i
$964$		0			0
$965$	−12.2124			−0.393132
$966$		0			0
$967$	51.1379			1.64448			0.822241	−	0.569139i	$-0.192723\pi$
$967$	51.1379			1.64448			0.822241	+	0.569139i	$0.192723\pi$
$968$		0			0
$969$	−28.6674	−	18.9432i	−0.920931	−	0.608542i
$970$		0			0
$971$	−36.1869			−1.16129			−0.580647	−	0.814156i	$-0.697200\pi$
$971$	−36.1869			−1.16129			−0.580647	+	0.814156i	$0.697200\pi$
$972$		0			0
$973$	10.5917			0.339555
$974$		0			0
$975$			0.546146i			0.0174907i
$976$		0			0
$977$	−25.5828			−0.818465			−0.409232	−	0.912430i	$-0.634203\pi$
$977$	−25.5828			−0.818465			−0.409232	+	0.912430i	$0.634203\pi$
$978$		0			0
$979$		−	31.7319i		−	1.01416i
$980$		0			0
$981$			77.1301i			2.46257i
$982$		0			0
$983$			18.7548i			0.598184i	0.954224	+	0.299092i	$0.0966838\pi$
$983$			18.7548i			0.598184i	−0.954224	+	0.299092i	$0.903316\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−44.6396			−1.42089
$988$		0			0
$989$		−	8.87987i		−	0.282363i
$990$		0			0
$991$		−	20.7168i		−	0.658091i	−0.944314	−	0.329046i	$-0.893273\pi$
$991$		−	20.7168i		−	0.658091i	0.944314	−	0.329046i	$-0.106727\pi$
$992$		0			0
$993$			11.9089i			0.377917i
$994$		0			0
$995$	13.3186			0.422229
$996$		0			0
$997$			20.9455i			0.663350i	0.943394	+	0.331675i	$0.107614\pi$
$997$			20.9455i			0.663350i	−0.943394	+	0.331675i	$0.892386\pi$
$998$		0			0
$999$	61.7040			1.95223

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	340.2.c.a.101.6	yes	6
3.2	odd	2		3060.2.e.j.1801.6		6
4.3	odd	2		1360.2.c.e.1121.1		6
5.2	odd	4		1700.2.g.c.849.6		6
5.3	odd	4		1700.2.g.b.849.2		6
5.4	even	2		1700.2.c.b.101.1		6
17.4	even	4		5780.2.a.k.1.3		3
17.13	even	4		5780.2.a.i.1.1		3
17.16	even	2	inner	340.2.c.a.101.1	✓	6
51.50	odd	2		3060.2.e.j.1801.1		6
68.67	odd	2		1360.2.c.e.1121.6		6
85.33	odd	4		1700.2.g.c.849.5		6
85.67	odd	4		1700.2.g.b.849.1		6
85.84	even	2		1700.2.c.b.101.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.c.a.101.1	✓	6	17.16	even	2	inner
340.2.c.a.101.6	yes	6	1.1	even	1	trivial
1360.2.c.e.1121.1		6	4.3	odd	2
1360.2.c.e.1121.6		6	68.67	odd	2
1700.2.c.b.101.1		6	5.4	even	2
1700.2.c.b.101.6		6	85.84	even	2
1700.2.g.b.849.1		6	85.67	odd	4
1700.2.g.b.849.2		6	5.3	odd	4
1700.2.g.c.849.5		6	85.33	odd	4
1700.2.g.c.849.6		6	5.2	odd	4
3060.2.e.j.1801.1		6	51.50	odd	2
3060.2.e.j.1801.6		6	3.2	odd	2
5780.2.a.i.1.1		3	17.13	even	4
5780.2.a.k.1.3		3	17.4	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 340 = 2^{2} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	340.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.27307i q^{3} -1.00000i q^{5} +1.27307i q^{7} -7.71301 q^{9} +5.43993i q^{11} -0.166860 q^{13} +3.27307 q^{15} +(2.27307 - 3.43993i) q^{17} -2.54615 q^{19} -4.16686 q^{21} -3.27307i q^{23} -1.00000 q^{25} -15.4260i q^{27} +6.54615i q^{29} +5.98608i q^{31} -17.8053 q^{33} +1.27307 q^{35} +4.00000i q^{37} -0.546146i q^{39} -8.54615i q^{41} +2.71301 q^{43} +7.71301i q^{45} +10.7130 q^{47} +5.37929 q^{49} +(11.2592 + 7.43993i) q^{51} +12.8799 q^{53} +5.43993 q^{55} -8.33372i q^{57} -6.21243 q^{59} +7.42601i q^{61} -9.81922i q^{63} +0.166860i q^{65} -6.37929 q^{67} +10.7130 q^{69} +2.89379i q^{71} +12.5461i q^{73} -3.27307i q^{75} -6.92543 q^{77} -3.77365i q^{79} +27.3514 q^{81} +13.5929 q^{83} +(-3.43993 - 2.27307i) q^{85} -21.4260 q^{87} -5.83314 q^{89} -0.212425i q^{91} -19.5929 q^{93} +2.54615i q^{95} +1.45385i q^{97} -41.9582i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 14 q^{9} + 4 q^{15} - 2 q^{17} + 16 q^{19} - 24 q^{21} - 6 q^{25} - 12 q^{33} - 8 q^{35} - 16 q^{43} + 32 q^{47} + 2 q^{49} + 4 q^{51} + 44 q^{53} + 16 q^{55} - 8 q^{59} - 8 q^{67} + 32 q^{69} + 20 q^{77}+ \cdots - 52 q^{93}+O(q^{100}) \) 6 * q - 14 * q^9 + 4 * q^15 - 2 * q^17 + 16 * q^19 - 24 * q^21 - 6 * q^25 - 12 * q^33 - 8 * q^35 - 16 * q^43 + 32 * q^47 + 2 * q^49 + 4 * q^51 + 44 * q^53 + 16 * q^55 - 8 * q^59 - 8 * q^67 + 32 * q^69 + 20 * q^77 + 38 * q^81 + 16 * q^83 - 4 * q^85 - 64 * q^87 - 36 * q^89 - 52 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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