LMFDB - Embedded newform 1700.2.g.c.849.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1700,2,Mod(849,1700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1700, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 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("1700.849"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1700 = 2^{2} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1700.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$13.5745683436$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 340)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			849.6
Root			$-2.27307i$ of defining polynomial
Character	$\chi$	$=$	1700.849
Dual form			1700.2.g.c.849.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.27307 q^{3} -1.27307 q^{7} +7.71301 q^{9} +5.43993i q^{11} +0.166860i q^{13} +(3.43993 + 2.27307i) q^{17} +2.54615 q^{19} -4.16686 q^{21} -3.27307 q^{23} +15.4260 q^{27} -6.54615i q^{29} +5.98608i q^{31} +17.8053i q^{33} -4.00000 q^{37} +0.546146i q^{39} -8.54615i q^{41} -2.71301i q^{43} +10.7130i q^{47} -5.37929 q^{49} +(11.2592 + 7.43993i) q^{51} -12.8799i q^{53} +8.33372 q^{57} +6.21243 q^{59} +7.42601i q^{61} -9.81922 q^{63} -6.37929i q^{67} -10.7130 q^{69} +2.89379i q^{71} +12.5461 q^{73} -6.92543i q^{77} +3.77365i q^{79} +27.3514 q^{81} -13.5929i q^{83} -21.4260i q^{87} +5.83314 q^{89} -0.212425i q^{91} +19.5929i q^{93} -1.45385 q^{97} +41.9582i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} + 8 q^{7} + 14 q^{9} + 4 q^{17} - 16 q^{19} - 24 q^{21} - 4 q^{23} + 28 q^{27} - 24 q^{37} - 2 q^{49} + 4 q^{51} + 48 q^{57} + 8 q^{59} - 12 q^{63} - 32 q^{69} + 44 q^{73} + 38 q^{81} + 36 q^{89}+ \cdots - 40 q^{97}+O(q^{100}) $ 6 * q + 4 * q^3 + 8 * q^7 + 14 * q^9 + 4 * q^17 - 16 * q^19 - 24 * q^21 - 4 * q^23 + 28 * q^27 - 24 * q^37 - 2 * q^49 + 4 * q^51 + 48 * q^57 + 8 * q^59 - 12 * q^63 - 32 * q^69 + 44 * q^73 + 38 * q^81 + 36 * q^89 - 40 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1700\mathbb{Z}\right)^\times$.

$n$	$477$	$851$	$1601$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	3.27307			1.88971			0.944855	−	0.327490i	$-0.106203\pi$
$3$	3.27307			1.88971			0.944855	+	0.327490i	$0.106203\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.27307			−0.481176			−0.240588	−	0.970627i	$-0.577340\pi$
$7$	−1.27307			−0.481176			−0.240588	+	0.970627i	$0.577340\pi$
$8$		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
$13$			0.166860i			0.0462787i	0.999732	+	0.0231394i	$0.00736614\pi$
$13$			0.166860i			0.0462787i	−0.999732	+	0.0231394i	$0.992634\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.43993	+	2.27307i	0.834306	+	0.551301i
$17$	3.43993	+	2.27307i	0.834306	+	0.551301i
$18$		0			0
$19$	2.54615			0.584126			0.292063	−	0.956399i	$-0.405658\pi$
$19$	2.54615			0.584126			0.292063	+	0.956399i	$0.405658\pi$
$20$		0			0
$21$	−4.16686			−0.909283
$22$		0			0
$23$	−3.27307			−0.682483			−0.341241	−	0.939976i	$-0.610847\pi$
$23$	−3.27307			−0.682483			−0.341241	+	0.939976i	$0.610847\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	15.4260			2.96874
$28$		0			0
$29$		−	6.54615i		−	1.21559i	−0.794094	−	0.607794i	$-0.792054\pi$
$29$		−	6.54615i		−	1.21559i	0.794094	−	0.607794i	$-0.207946\pi$
$30$		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
$33$			17.8053i			3.09950i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		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
$43$		−	2.71301i		−	0.413730i	−0.978370	−	0.206865i	$-0.933674\pi$
$43$		−	2.71301i		−	0.413730i	0.978370	−	0.206865i	$-0.0663260\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.7130i			1.56265i	0.624123	+	0.781326i	$0.285456\pi$
$47$			10.7130i			1.56265i	−0.624123	+	0.781326i	$0.714544\pi$
$48$		0			0
$49$	−5.37929			−0.768469
$50$		0			0
$51$	11.2592	+	7.43993i	1.57660	+	1.04180i
$52$		0			0
$53$		−	12.8799i		−	1.76919i	−0.466364	−	0.884593i	$-0.654436\pi$
$53$		−	12.8799i		−	1.76919i	0.466364	−	0.884593i	$-0.345564\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	8.33372			1.10383
$58$		0			0
$59$	6.21243			0.808789			0.404394	−	0.914585i	$-0.367482\pi$
$59$	6.21243			0.808789			0.404394	+	0.914585i	$0.367482\pi$
$60$		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
$63$	−9.81922			−1.23711
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	6.37929i		−	0.779354i	−0.920952	−	0.389677i	$-0.872587\pi$
$67$		−	6.37929i		−	0.779354i	0.920952	−	0.389677i	$-0.127413\pi$
$68$		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
$73$	12.5461			1.46842			0.734208	−	0.678925i	$-0.237554\pi$
$73$	12.5461			1.46842			0.734208	+	0.678925i	$0.237554\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	6.92543i		−	0.789226i
$78$		0			0
$79$			3.77365i			0.424569i	0.977208	+	0.212285i	$0.0680904\pi$
$79$			3.77365i			0.424569i	−0.977208	+	0.212285i	$0.931910\pi$
$80$		0			0
$81$	27.3514			3.03905
$82$		0			0
$83$		−	13.5929i		−	1.49201i	−0.665939	−	0.746006i	$-0.731969\pi$
$83$		−	13.5929i		−	1.49201i	0.665939	−	0.746006i	$-0.268031\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	21.4260i		−	2.29711i
$88$		0			0
$89$	5.83314			0.618312			0.309156	−	0.951011i	$-0.399954\pi$
$89$	5.83314			0.618312			0.309156	+	0.951011i	$0.399954\pi$
$90$		0			0
$91$		−	0.212425i		−	0.0222682i
$92$		0			0
$93$			19.5929i			2.03169i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.45385			−0.147617			−0.0738083	−	0.997272i	$-0.523515\pi$
$97$	−1.45385			−0.147617			−0.0738083	+	0.997272i	$0.523515\pi$
$98$		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
$103$		−	8.92543i		−	0.879449i	−0.898133	−	0.439724i	$-0.855076\pi$
$103$		−	8.92543i		−	0.879449i	0.898133	−	0.439724i	$-0.144924\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.84706			−0.178562			−0.0892811	−	0.996006i	$-0.528457\pi$
$107$	−1.84706			−0.178562			−0.0892811	+	0.996006i	$0.528457\pi$
$108$		0			0
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$		0			0
$111$	−13.0923			−1.24267
$112$		0			0
$113$	−17.9722			−1.69068			−0.845339	−	0.534230i	$-0.820602\pi$
$113$	−17.9722			−1.69068			−0.845339	+	0.534230i	$0.820602\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			1.28699i			0.118983i
$118$		0			0
$119$	−4.37929	−	2.89379i	−0.401448	−	0.265273i
$120$		0			0
$121$	−18.5929			−1.69026
$122$		0			0
$123$		−	27.9722i		−	2.52216i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		−	11.8053i		−	1.04755i	−0.851856	−	0.523775i	$-0.824523\pi$
$127$		−	11.8053i		−	1.04755i	0.851856	−	0.523775i	$-0.175477\pi$
$128$		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
$133$	−3.24143			−0.281068
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	8.16686i		−	0.697742i	−0.937171	−	0.348871i	$-0.886565\pi$
$137$		−	8.16686i		−	0.697742i	0.937171	−	0.348871i	$-0.113435\pi$
$138$		0			0
$139$			8.31980i			0.705676i	0.935684	+	0.352838i	$0.114783\pi$
$139$			8.31980i			0.705676i	−0.935684	+	0.352838i	$0.885217\pi$
$140$		0			0
$141$			35.0644i			2.95296i
$142$		0			0
$143$	−0.907709			−0.0759064
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−17.6068			−1.45218
$148$		0			0
$149$	−21.9722			−1.80003			−0.900015	−	0.435860i	$-0.856444\pi$
$149$	−21.9722			−1.80003			−0.900015	+	0.435860i	$0.856444\pi$
$150$		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
$153$	26.5322	+	17.5322i	2.14500	+	1.41740i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		−	13.9722i		−	1.11510i	−0.830144	−	0.557550i	$-0.811742\pi$
$157$		−	13.9722i		−	1.11510i	0.830144	−	0.557550i	$-0.188258\pi$
$158$		0			0
$159$		−	42.1567i		−	3.34325i
$160$		0			0
$161$	4.16686			0.328395
$162$		0			0
$163$	−8.93935			−0.700184			−0.350092	−	0.936715i	$-0.613850\pi$
$163$	−8.93935			−0.700184			−0.350092	+	0.936715i	$0.613850\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2.39321			0.185192			0.0925959	−	0.995704i	$-0.470484\pi$
$167$	2.39321			0.185192			0.0925959	+	0.995704i	$0.470484\pi$
$168$		0			0
$169$	12.9722			0.997858
$170$		0			0
$171$	19.6384			1.50179
$172$		0			0
$173$	−1.42601			−0.108418			−0.0542088	−	0.998530i	$-0.517264\pi$
$173$	−1.42601			−0.108418			−0.0542088	+	0.998530i	$0.517264\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	20.3337			1.52838
$178$		0			0
$179$	−13.0923			−0.978564			−0.489282	−	0.872126i	$-0.662741\pi$
$179$	−13.0923			−0.978564			−0.489282	+	0.872126i	$0.662741\pi$
$180$		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
$183$			24.3059i			1.79674i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−12.3654	+	18.7130i	−0.904245	+	1.36843i
$188$		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
$193$	−12.2124			−0.879070			−0.439535	−	0.898226i	$-0.644857\pi$
$193$	−12.2124			−0.879070			−0.439535	+	0.898226i	$0.644857\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		−	13.3186i		−	0.944133i	−0.881563	−	0.472067i	$-0.843508\pi$
$199$		−	13.3186i		−	0.944133i	0.881563	−	0.472067i	$-0.156492\pi$
$200$		0			0
$201$		−	20.8799i		−	1.47275i
$202$		0			0
$203$			8.33372i			0.584913i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−25.2452			−1.75466
$208$		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
$213$			9.47158i			0.648982i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	7.62071i		−	0.517328i
$218$		0			0
$219$	41.0644			2.77488
$220$		0			0
$221$	−0.379285	+	0.573988i	−0.0255135	+	0.0386106i
$222$		0			0
$223$		−	13.2592i		−	0.887898i	−0.896052	−	0.443949i	$-0.853577\pi$
$223$		−	13.2592i		−	0.887898i	0.896052	−	0.443949i	$-0.146423\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.27307			0.615475			0.307738	−	0.951471i	$-0.400428\pi$
$227$	9.27307			0.615475			0.307738	+	0.951471i	$0.400428\pi$
$228$		0			0
$229$	14.1669			0.936172			0.468086	−	0.883683i	$-0.344944\pi$
$229$	14.1669			0.936172			0.468086	+	0.883683i	$0.344944\pi$
$230$		0			0
$231$		−	22.6674i		−	1.49141i
$232$		0			0
$233$	−29.0644			−1.90408			−0.952038	−	0.305981i	$-0.901016\pi$
$233$	−29.0644			−1.90408			−0.952038	+	0.305981i	$0.901016\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			12.3514i			0.802312i
$238$		0			0
$239$	5.42601			0.350980			0.175490	−	0.984481i	$-0.443849\pi$
$239$	5.42601			0.350980			0.175490	+	0.984481i	$0.443849\pi$
$240$		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
$243$	43.2452			2.77418
$244$		0			0
$245$		0			0
$246$		0			0
$247$			0.424850i			0.0270326i
$248$		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
$253$		−	17.8053i		−	1.11941i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			1.59287i			0.0993606i	0.998765	+	0.0496803i	$0.0158202\pi$
$257$			1.59287i			0.0993606i	−0.998765	+	0.0496803i	$0.984180\pi$
$258$		0			0
$259$	5.09229			0.316420
$260$		0			0
$261$		−	50.4905i		−	3.12528i
$262$		0			0
$263$			2.37929i			0.146713i	0.997306	+	0.0733565i	$0.0233711\pi$
$263$			2.37929i			0.146713i	−0.997306	+	0.0733565i	$0.976629\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	19.0923			1.16843
$268$		0			0
$269$			18.4249i			1.12338i	0.827347	+	0.561691i	$0.189849\pi$
$269$			18.4249i			1.12338i	−0.827347	+	0.561691i	$0.810151\pi$
$270$		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
$273$		−	0.695283i		−	0.0420805i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	12.1846			0.732101			0.366050	−	0.930595i	$-0.380710\pi$
$277$	12.1846			0.732101			0.366050	+	0.930595i	$0.380710\pi$
$278$		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
$283$	16.4867			0.980030			0.490015	−	0.871714i	$-0.336991\pi$
$283$	16.4867			0.980030			0.490015	+	0.871714i	$0.336991\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			10.8799i			0.642218i
$288$		0			0
$289$	6.66628	+	15.6384i	0.392134	+	0.919908i
$290$		0			0
$291$	−4.75857			−0.278952
$292$		0			0
$293$			4.57399i			0.267215i	0.991034	+	0.133608i	$0.0426562\pi$
$293$			4.57399i			0.267215i	−0.991034	+	0.133608i	$0.957344\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			83.9165i			4.86933i
$298$		0			0
$299$		−	0.546146i		−	0.0315844i
$300$		0			0
$301$			3.45385i			0.199077i
$302$		0			0
$303$	37.9443			2.17984
$304$		0			0
$305$		0			0
$306$		0			0
$307$			15.0189i			0.857173i	0.903501	+	0.428586i	$0.140988\pi$
$307$			15.0189i			0.857173i	−0.903501	+	0.428586i	$0.859012\pi$
$308$		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
$313$	4.54615			0.256963			0.128482	−	0.991712i	$-0.458990\pi$
$313$	4.54615			0.256963			0.128482	+	0.991712i	$0.458990\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.4260			−0.641749			−0.320874	−	0.947122i	$-0.603977\pi$
$317$	−11.4260			−0.641749			−0.320874	+	0.947122i	$0.603977\pi$
$318$		0			0
$319$	35.6106			1.99381
$320$		0			0
$321$	−6.04557			−0.337431
$322$		0			0
$323$	8.75857	+	5.78757i	0.487340	+	0.322029i
$324$		0			0
$325$		0			0
$326$		0			0
$327$			32.7307i			1.81001i
$328$		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
$333$	−30.8520			−1.69068
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−7.97216			−0.434271			−0.217136	−	0.976141i	$-0.569671\pi$
$337$	−7.97216			−0.434271			−0.217136	+	0.976141i	$0.569671\pi$
$338$		0			0
$339$	−58.8242			−3.19489
$340$		0			0
$341$	−32.5639			−1.76343
$342$		0			0
$343$	15.7597			0.850946
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.152939			−0.00821020			−0.00410510	−	0.999992i	$-0.501307\pi$
$347$	−0.152939			−0.00821020			−0.00410510	+	0.999992i	$0.501307\pi$
$348$		0			0
$349$	4.21243			0.225486			0.112743	−	0.993624i	$-0.464036\pi$
$349$	4.21243			0.225486			0.112743	+	0.993624i	$0.464036\pi$
$350$		0			0
$351$			2.57399i			0.137389i
$352$		0			0
$353$		−	6.66744i		−	0.354872i	−0.984132	−	0.177436i	$-0.943220\pi$
$353$		−	6.66744i		−	0.354872i	0.984132	−	0.177436i	$-0.0567803\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−14.3337	−	9.47158i	−0.758621	−	0.501289i
$358$		0			0
$359$	2.51830			0.132911			0.0664555	−	0.997789i	$-0.478831\pi$
$359$	2.51830			0.132911			0.0664555	+	0.997789i	$0.478831\pi$
$360$		0			0
$361$	−12.5171			−0.658797
$362$		0			0
$363$	−60.8558			−3.19410
$364$		0			0
$365$		0			0
$366$		0			0
$367$	19.1518			0.999715			0.499857	−	0.866108i	$-0.333386\pi$
$367$	19.1518			0.999715			0.499857	+	0.866108i	$0.333386\pi$
$368$		0			0
$369$		−	65.9165i		−	3.43147i
$370$		0			0
$371$			16.3970i			0.851290i
$372$		0			0
$373$		−	24.9254i		−	1.29059i	−0.763934	−	0.645295i	$-0.776734\pi$
$373$		−	24.9254i		−	1.29059i	0.763934	−	0.645295i	$-0.223266\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.09229			0.0562559
$378$		0			0
$379$			7.41209i			0.380734i	0.981713	+	0.190367i	$0.0609677\pi$
$379$			7.41209i			0.380734i	−0.981713	+	0.190367i	$0.939032\pi$
$380$		0			0
$381$		−	38.6396i		−	1.97957i
$382$		0			0
$383$		−	30.0177i		−	1.53383i	−0.641746	−	0.766917i	$-0.721790\pi$
$383$		−	30.0177i		−	1.53383i	0.641746	−	0.766917i	$-0.278210\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	20.9254i		−	1.06370i
$388$		0			0
$389$	−24.6573			−1.25018			−0.625088	−	0.780554i	$-0.714937\pi$
$389$	−24.6573			−1.25018			−0.625088	+	0.780554i	$0.714937\pi$
$390$		0			0
$391$	−11.2592	−	7.43993i	−0.569400	−	0.376254i
$392$		0			0
$393$		−	14.9254i		−	0.752888i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−16.8520			−0.845779			−0.422889	−	0.906181i	$-0.638984\pi$
$397$	−16.8520			−0.845779			−0.422889	+	0.906181i	$0.638984\pi$
$398$		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
$403$	−0.998839			−0.0497557
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	21.7597i		−	1.07859i
$408$		0			0
$409$	−14.0000			−0.692255			−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000			−0.692255			−0.346128	+	0.938187i	$0.612504\pi$
$410$		0			0
$411$		−	26.7307i		−	1.31853i
$412$		0			0
$413$	−7.90887			−0.389170
$414$		0			0
$415$		0			0
$416$		0			0
$417$			27.2313i			1.33352i
$418$		0			0
$419$		−	35.4399i		−	1.73135i	−0.500603	−	0.865677i	$-0.666888\pi$
$419$		−	35.4399i		−	1.73135i	0.500603	−	0.865677i	$-0.333112\pi$
$420$		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
$423$			82.6295i			4.01758i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	9.45385i		−	0.457504i
$428$		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
$433$		−	6.68516i		−	0.321268i	−0.987014	−	0.160634i	$-0.948646\pi$
$433$		−	6.68516i		−	0.321268i	0.987014	−	0.160634i	$-0.0513540\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−8.33372			−0.398656
$438$		0			0
$439$			35.2908i			1.68434i	0.539214	+	0.842169i	$0.318721\pi$
$439$			35.2908i			1.68434i	−0.539214	+	0.842169i	$0.681279\pi$
$440$		0			0
$441$	−41.4905			−1.97574
$442$		0			0
$443$			27.7775i			1.31975i	0.751377	+	0.659873i	$0.229390\pi$
$443$			27.7775i			1.31975i	−0.751377	+	0.659873i	$0.770610\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−71.9165			−3.40153
$448$		0			0
$449$		−	6.30588i		−	0.297593i	−0.988868	−	0.148796i	$-0.952460\pi$
$449$		−	6.30588i		−	0.297593i	0.988868	−	0.148796i	$-0.0475399\pi$
$450$		0			0
$451$	46.4905			2.18915
$452$		0			0
$453$	−35.6106			−1.67313
$454$		0			0
$455$		0			0
$456$		0			0
$457$			15.1657i			0.709421i	0.934976	+	0.354711i	$0.115421\pi$
$457$			15.1657i			0.709421i	−0.934976	+	0.354711i	$0.884579\pi$
$458$		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
$463$			17.2592i			0.802101i	0.916056	+	0.401050i	$0.131355\pi$
$463$			17.2592i			0.802101i	−0.916056	+	0.401050i	$0.868645\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	16.8065i		−	0.777710i	−0.921299	−	0.388855i	$-0.872871\pi$
$467$		−	16.8065i		−	0.777710i	0.921299	−	0.388855i	$-0.127129\pi$
$468$		0			0
$469$			8.12130i			0.375007i
$470$		0			0
$471$		−	45.7319i		−	2.10721i
$472$		0			0
$473$	14.7586			0.678600
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	99.3425i		−	4.54858i
$478$		0			0
$479$		−	12.5044i		−	0.571340i	−0.958328	−	0.285670i	$-0.907784\pi$
$479$		−	12.5044i		−	0.571340i	0.958328	−	0.285670i	$-0.0922161\pi$
$480$		0			0
$481$		−	0.667441i		−	0.0304327i
$482$		0			0
$483$	13.6384			0.620570
$484$		0			0
$485$		0			0
$486$		0			0
$487$	24.4867			1.10960			0.554798	−	0.831985i	$-0.312795\pi$
$487$	24.4867			1.10960			0.554798	+	0.831985i	$0.312795\pi$
$488$		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
$493$	14.8799	−	22.5183i	0.670155	−	1.01417i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	3.68400i		−	0.165250i
$498$		0			0
$499$			5.89495i			0.263894i	0.991257	+	0.131947i	$0.0421229\pi$
$499$			5.89495i			0.263894i	−0.991257	+	0.131947i	$0.957877\pi$
$500$		0			0
$501$	7.83314			0.349959
$502$		0			0
$503$	15.0328			0.670280			0.335140	−	0.942168i	$-0.391216\pi$
$503$	15.0328			0.670280			0.335140	+	0.942168i	$0.391216\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	42.4588			1.88566
$508$		0			0
$509$	−14.4249			−0.639370			−0.319685	−	0.947524i	$-0.603577\pi$
$509$	−14.4249			−0.639370			−0.319685	+	0.947524i	$0.603577\pi$
$510$		0			0
$511$	−15.9722			−0.706567
$512$		0			0
$513$	39.2769			1.73412
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−58.2780			−2.56307
$518$		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
$523$		−	19.7142i		−	0.862040i	−0.902342	−	0.431020i	$-0.858154\pi$
$523$		−	19.7142i		−	0.862040i	0.902342	−	0.431020i	$-0.141846\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−13.6068	+	20.5917i	−0.592721	+	0.896989i
$528$		0			0
$529$	−12.2870			−0.534217
$530$		0			0
$531$	47.9165			2.07940
$532$		0			0
$533$	1.42601			0.0617674
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−42.8520			−1.84920
$538$		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
$543$		−	65.3703i		−	2.80531i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	27.5790			1.17919			0.589595	−	0.807699i	$-0.299287\pi$
$547$	27.5790			1.17919			0.589595	+	0.807699i	$0.299287\pi$
$548$		0			0
$549$			57.2769i			2.44452i
$550$		0			0
$551$		−	16.6674i		−	0.710057i
$552$		0			0
$553$		−	4.80414i		−	0.204293i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	36.1112i		−	1.53008i	−0.643983	−	0.765040i	$-0.722719\pi$
$557$		−	36.1112i		−	1.53008i	0.643983	−	0.765040i	$-0.277281\pi$
$558$		0			0
$559$	0.452693			0.0191469
$560$		0			0
$561$	−40.4727	+	61.2490i	−1.70876	+	2.58594i
$562$		0			0
$563$			24.1112i			1.01616i	0.861308	+	0.508082i	$0.169645\pi$
$563$			24.1112i			1.01616i	−0.861308	+	0.508082i	$0.830355\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−34.8204			−1.46232
$568$		0			0
$569$	23.7597			0.996060			0.498030	−	0.867160i	$-0.334057\pi$
$569$	23.7597			0.996060			0.498030	+	0.867160i	$0.334057\pi$
$570$		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
$573$	−66.4626			−2.77652
$574$		0			0
$575$		0			0
$576$		0			0
$577$			10.6852i			0.444829i	0.974952	+	0.222415i	$0.0713939\pi$
$577$			10.6852i			0.444829i	−0.974952	+	0.222415i	$0.928606\pi$
$578$		0			0
$579$	−39.9722			−1.66119
$580$		0			0
$581$			17.3047i			0.717921i
$582$		0			0
$583$	70.0656			2.90182
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1.64856i			0.0680432i	0.999421	+	0.0340216i	$0.0108315\pi$
$587$			1.64856i			0.0680432i	−0.999421	+	0.0340216i	$0.989168\pi$
$588$		0			0
$589$			15.2414i			0.628012i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			1.21359i			0.0498360i	0.999689	+	0.0249180i	$0.00793247\pi$
$593$			1.21359i			0.0498360i	−0.999689	+	0.0249180i	$0.992068\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	43.5929i		−	1.78414i
$598$		0			0
$599$	8.36156			0.341644			0.170822	−	0.985302i	$-0.445358\pi$
$599$	8.36156			0.341644			0.170822	+	0.985302i	$0.445358\pi$
$600$		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
$603$		−	49.2035i		−	2.00372i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−22.9115			−0.929950			−0.464975	−	0.885324i	$-0.653937\pi$
$607$	−22.9115			−0.929950			−0.464975	+	0.885324i	$0.653937\pi$
$608$		0			0
$609$			27.2769i			1.10531i
$610$		0			0
$611$	−1.78757			−0.0723175
$612$		0			0
$613$			0.573988i			0.0231832i	0.999933	+	0.0115916i	$0.00368980\pi$
$613$			0.573988i			0.0231832i	−0.999933	+	0.0115916i	$0.996310\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−11.7319			−0.472308			−0.236154	−	0.971716i	$-0.575887\pi$
$617$	−11.7319			−0.472308			−0.236154	+	0.971716i	$0.575887\pi$
$618$		0			0
$619$			2.50438i			0.100660i	0.998733	+	0.0503298i	$0.0160272\pi$
$619$			2.50438i			0.100660i	−0.998733	+	0.0503298i	$0.983973\pi$
$620$		0			0
$621$	−50.4905			−2.02611
$622$		0			0
$623$	−7.42601			−0.297517
$624$		0			0
$625$		0			0
$626$		0			0
$627$			45.3349i			1.81050i
$628$		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
$633$			68.2958i			2.71451i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	0.897589i		−	0.0355638i
$638$		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
$643$	−8.72693			−0.344156			−0.172078	−	0.985083i	$-0.555048\pi$
$643$	−8.72693			−0.344156			−0.172078	+	0.985083i	$0.555048\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			14.6573i			0.576239i	0.957594	+	0.288119i	$0.0930300\pi$
$647$			14.6573i			0.576239i	−0.957594	+	0.288119i	$0.906970\pi$
$648$		0			0
$649$			33.7952i			1.32658i
$650$		0			0
$651$		−	24.9432i		−	0.977599i
$652$		0			0
$653$	17.4260			0.681933			0.340966	−	0.940076i	$-0.389246\pi$
$653$	17.4260			0.681933			0.340966	+	0.940076i	$0.389246\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	96.7685			3.77530
$658$		0			0
$659$	39.8532			1.55246			0.776230	−	0.630450i	$-0.217130\pi$
$659$	39.8532			1.55246			0.776230	+	0.630450i	$0.217130\pi$
$660$		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
$663$	−1.24143	+	1.87870i	−0.0482131	+	0.0729628i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			21.4260i			0.829618i
$668$		0			0
$669$		−	43.3982i		−	1.67787i
$670$		0			0
$671$	−40.3970			−1.55951
$672$		0			0
$673$	−25.5473			−0.984776			−0.492388	−	0.870376i	$-0.663876\pi$
$673$	−25.5473			−0.984776			−0.492388	+	0.870376i	$0.663876\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.33488			0.128170			0.0640850	−	0.997944i	$-0.479587\pi$
$677$	3.33488			0.128170			0.0640850	+	0.997944i	$0.479587\pi$
$678$		0			0
$679$	1.85086			0.0710296
$680$		0			0
$681$	30.3514			1.16307
$682$		0			0
$683$	−46.0973			−1.76386			−0.881931	−	0.471378i	$-0.843757\pi$
$683$	−46.0973			−1.76386			−0.881931	+	0.471378i	$0.843757\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	46.3692			1.76909
$688$		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
$693$		−	53.4159i		−	2.02910i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	19.4260	−	29.3982i	0.735813	−	1.11354i
$698$		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
$703$	−10.1846			−0.384119
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14.7586			−0.555053
$708$		0			0
$709$		−	25.5450i		−	0.959362i	−0.877443	−	0.479681i	$-0.840752\pi$
$709$		−	25.5450i		−	0.959362i	0.877443	−	0.479681i	$-0.159248\pi$
$710$		0			0
$711$			29.1062i			1.09157i
$712$		0			0
$713$		−	19.5929i		−	0.733759i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	17.7597			0.663249
$718$		0			0
$719$			5.10621i			0.190430i	0.995457	+	0.0952148i	$0.0303538\pi$
$719$			5.10621i			0.190430i	−0.995457	+	0.0952148i	$0.969646\pi$
$720$		0			0
$721$			11.3627i			0.423170i
$722$		0			0
$723$			66.1567i			2.46040i
$724$		0			0
$725$		0			0
$726$		0			0
$727$			23.3526i			0.866100i	0.901370	+	0.433050i	$0.142563\pi$
$727$			23.3526i			0.866100i	−0.901370	+	0.433050i	$0.857437\pi$
$728$		0			0
$729$	59.4905			2.20335
$730$		0			0
$731$	6.16686	−	9.33256i	0.228090	−	0.345177i
$732$		0			0
$733$			35.3703i			1.30643i	0.757171	+	0.653216i	$0.226581\pi$
$733$			35.3703i			1.30643i	−0.757171	+	0.653216i	$0.773419\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	34.7029			1.27830
$738$		0			0
$739$	22.3036			0.820450			0.410225	−	0.911984i	$-0.365450\pi$
$739$	22.3036			0.820450			0.410225	+	0.911984i	$0.365450\pi$
$740$		0			0
$741$			1.39057i			0.0510837i
$742$		0			0
$743$	−10.9394			−0.401326			−0.200663	−	0.979660i	$-0.564310\pi$
$743$	−10.9394			−0.401326			−0.200663	+	0.979660i	$0.564310\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		−	104.842i		−	3.83596i
$748$		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
$753$	−63.1857			−2.30262
$754$		0			0
$755$		0			0
$756$		0			0
$757$			22.2603i			0.809065i	0.914524	+	0.404532i	$0.132566\pi$
$757$			22.2603i			0.809065i	−0.914524	+	0.404532i	$0.867434\pi$
$758$		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
$763$		−	12.7307i		−	0.460883i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1.03661i			0.0374297i
$768$		0			0
$769$	37.1100			1.33822			0.669111	−	0.743163i	$-0.266675\pi$
$769$	37.1100			1.33822			0.669111	+	0.743163i	$0.266675\pi$
$770$		0			0
$771$			5.21359i			0.187763i
$772$		0			0
$773$			38.7763i			1.39469i	0.716737	+	0.697343i	$0.245635\pi$
$773$			38.7763i			1.39469i	−0.716737	+	0.697343i	$0.754365\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	16.6674			0.597941
$778$		0			0
$779$		−	21.7597i		−	0.779623i
$780$		0			0
$781$	−15.7420			−0.563293
$782$		0			0
$783$		−	100.981i		−	3.60876i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	10.6080			0.378133			0.189066	−	0.981964i	$-0.439454\pi$
$787$	10.6080			0.378133			0.189066	+	0.981964i	$0.439454\pi$
$788$		0			0
$789$			7.78757i			0.277245i
$790$		0			0
$791$	22.8799			0.813514
$792$		0			0
$793$	−1.23911			−0.0440019
$794$		0			0
$795$		0			0
$796$		0			0
$797$			9.66628i			0.342397i	0.985237	+	0.171199i	$0.0547640\pi$
$797$			9.66628i			0.342397i	−0.985237	+	0.171199i	$0.945236\pi$
$798$		0			0
$799$	−24.3514	+	36.8520i	−0.861492	+	1.30373i
$800$		0			0
$801$	44.9910			1.58968
$802$		0			0
$803$			68.2502i			2.40850i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			60.3059i			2.12287i
$808$		0			0
$809$		−	42.5183i		−	1.49486i	−0.664338	−	0.747432i	$-0.731287\pi$
$809$		−	42.5183i		−	1.49486i	0.664338	−	0.747432i	$-0.268713\pi$
$810$		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
$813$	82.1289			2.88039
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	6.90771i		−	0.241670i
$818$		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
$823$	−9.36653			−0.326497			−0.163248	−	0.986585i	$-0.552197\pi$
$823$	−9.36653			−0.326497			−0.163248	+	0.986585i	$0.552197\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−44.7902			−1.55751			−0.778754	−	0.627329i	$-0.784148\pi$
$827$	−44.7902			−1.55751			−0.778754	+	0.627329i	$0.784148\pi$
$828$		0			0
$829$	−36.8520			−1.27992			−0.639962	−	0.768407i	$-0.721050\pi$
$829$	−36.8520			−1.27992			−0.639962	+	0.768407i	$0.721050\pi$
$830$		0			0
$831$	39.8810			1.38346
$832$		0			0
$833$	−18.5044	−	12.2275i	−0.641139	−	0.423658i
$834$		0			0
$835$		0			0
$836$		0			0
$837$			92.3413i			3.19178i
$838$		0			0
$839$			11.6245i			0.401323i	0.979661	+	0.200661i	$0.0643091\pi$
$839$			11.6245i			0.401323i	−0.979661	+	0.200661i	$0.935691\pi$
$840$		0			0
$841$	−13.8520			−0.477656
$842$		0			0
$843$	13.7876			0.474869
$844$		0			0
$845$		0			0
$846$		0			0
$847$	23.6701			0.813314
$848$		0			0
$849$	53.9620			1.85197
$850$		0			0
$851$	13.0923			0.448798
$852$		0			0
$853$	49.2792			1.68729			0.843644	−	0.536903i	$-0.180406\pi$
$853$	49.2792			1.68729			0.843644	+	0.536903i	$0.180406\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	23.6408			0.807553			0.403776	−	0.914858i	$-0.367697\pi$
$857$	23.6408			0.807553			0.403776	+	0.914858i	$0.367697\pi$
$858$		0			0
$859$	14.0935			0.480862			0.240431	−	0.970666i	$-0.422711\pi$
$859$	14.0935			0.480862			0.240431	+	0.970666i	$0.422711\pi$
$860$		0			0
$861$			35.6106i			1.21361i
$862$		0			0
$863$		−	4.28583i		−	0.145891i	−0.997336	−	0.0729457i	$-0.976760\pi$
$863$		−	4.28583i		−	0.145891i	0.997336	−	0.0729457i	$-0.0232400\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	21.8192	+	51.1857i	0.741020	+	1.73836i
$868$		0			0
$869$	−20.5284			−0.696379
$870$		0			0
$871$	1.06445			0.0360675
$872$		0			0
$873$	−11.2136			−0.379522
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−7.85086			−0.265105			−0.132552	−	0.991176i	$-0.542317\pi$
$877$	−7.85086			−0.265105			−0.132552	+	0.991176i	$0.542317\pi$
$878$		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
$883$		−	3.74201i		−	0.125929i	−0.998016	−	0.0629643i	$-0.979945\pi$
$883$		−	3.74201i		−	0.125929i	0.998016	−	0.0629643i	$-0.0200554\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	4.75477			0.159650			0.0798248	−	0.996809i	$-0.474564\pi$
$887$	4.75477			0.159650			0.0798248	+	0.996809i	$0.474564\pi$
$888$		0			0
$889$			15.0290i			0.504057i
$890$		0			0
$891$			148.790i			4.98465i
$892$		0			0
$893$			27.2769i			0.912786i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		−	1.78757i		−	0.0596854i
$898$		0			0
$899$	39.1857			1.30692
$900$		0			0
$901$	29.2769	−	44.3059i	0.975354	−	1.47604i
$902$		0			0
$903$			11.3047i			0.376197i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	34.9771			1.16140			0.580698	−	0.814119i	$-0.302780\pi$
$907$	34.9771			1.16140			0.580698	+	0.814119i	$0.302780\pi$
$908$		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
$913$	73.9443			2.44720
$914$		0			0
$915$		0			0
$916$		0			0
$917$			5.80530i			0.191708i
$918$		0			0
$919$	26.8242			0.884848			0.442424	−	0.896806i	$-0.354119\pi$
$919$	26.8242			0.884848			0.442424	+	0.896806i	$0.354119\pi$
$920$		0			0
$921$			49.1579i			1.61981i
$922$		0			0
$923$	−0.482858			−0.0158935
$924$		0			0
$925$		0			0
$926$		0			0
$927$		−	68.8419i		−	2.26106i
$928$		0			0
$929$		−	0.546146i		−	0.0179185i	−0.999960	−	0.00895923i	$-0.997148\pi$
$929$		−	0.546146i		−	0.0179185i	0.999960	−	0.00895923i	$-0.00285185\pi$
$930$		0			0
$931$	−13.6964			−0.448883
$932$		0			0
$933$			20.3793i			0.667188i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	22.7864i		−	0.744400i	−0.928153	−	0.372200i	$-0.878604\pi$
$937$		−	22.7864i		−	0.744400i	0.928153	−	0.372200i	$-0.121396\pi$
$938$		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
$943$			27.9722i			0.910899i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	55.3665			1.79917			0.899585	−	0.436745i	$-0.143869\pi$
$947$	55.3665			1.79917			0.899585	+	0.436745i	$0.143869\pi$
$948$		0			0
$949$			2.09345i			0.0679564i
$950$		0			0
$951$	−37.3982			−1.21272
$952$		0			0
$953$		−	50.4426i		−	1.63400i	−0.576641	−	0.816998i	$-0.695637\pi$
$953$		−	50.4426i		−	1.63400i	0.576641	−	0.816998i	$-0.304363\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	116.556			3.76772
$958$		0			0
$959$			10.3970i			0.335737i
$960$		0			0
$961$	−4.83314			−0.155908
$962$		0			0
$963$	−14.2464			−0.459084
$964$		0			0
$965$		0			0
$966$		0			0
$967$			51.1379i			1.64448i	0.569139	+	0.822241i	$0.307277\pi$
$967$			51.1379i			1.64448i	−0.569139	+	0.822241i	$0.692723\pi$
$968$		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
$973$		−	10.5917i		−	0.339555i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	25.5828i		−	0.818465i	−0.912430	−	0.409232i	$-0.865797\pi$
$977$		−	25.5828i		−	0.818465i	0.912430	−	0.409232i	$-0.134203\pi$
$978$		0			0
$979$			31.7319i			1.01416i
$980$		0			0
$981$			77.1301i			2.46257i
$982$		0			0
$983$	18.7548			0.598184			0.299092	−	0.954224i	$-0.403316\pi$
$983$	18.7548			0.598184			0.299092	+	0.954224i	$0.403316\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	44.6396i		−	1.42089i
$988$		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
$993$	11.9089			0.377917
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−20.9455			−0.663350			−0.331675	−	0.943394i	$-0.607614\pi$
$997$	−20.9455			−0.663350			−0.331675	+	0.943394i	$0.607614\pi$
$998$		0			0
$999$	−61.7040			−1.95223

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
340.2.c.a.101.1	✓	6	85.33	odd	4
340.2.c.a.101.6	yes	6	5.3	odd	4
1360.2.c.e.1121.1		6	20.3	even	4
1360.2.c.e.1121.6		6	340.203	even	4
1700.2.c.b.101.1		6	5.2	odd	4
1700.2.c.b.101.6		6	85.67	odd	4
1700.2.g.b.849.1		6	17.16	even	2
1700.2.g.b.849.2		6	5.4	even	2
1700.2.g.c.849.5		6	85.84	even	2	inner
1700.2.g.c.849.6		6	1.1	even	1	trivial
3060.2.e.j.1801.1		6	255.203	even	4
3060.2.e.j.1801.6		6	15.8	even	4
5780.2.a.i.1.1		3	85.13	odd	4
5780.2.a.k.1.3		3	85.38	odd	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:	\( N \)	\(=\)	\( 1700 = 2^{2} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1700.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.27307 q^{3} -1.27307 q^{7} +7.71301 q^{9} +5.43993i q^{11} +0.166860i q^{13} +(3.43993 + 2.27307i) q^{17} +2.54615 q^{19} -4.16686 q^{21} -3.27307 q^{23} +15.4260 q^{27} -6.54615i q^{29} +5.98608i q^{31} +17.8053i q^{33} -4.00000 q^{37} +0.546146i q^{39} -8.54615i q^{41} -2.71301i q^{43} +10.7130i q^{47} -5.37929 q^{49} +(11.2592 + 7.43993i) q^{51} -12.8799i q^{53} +8.33372 q^{57} +6.21243 q^{59} +7.42601i q^{61} -9.81922 q^{63} -6.37929i q^{67} -10.7130 q^{69} +2.89379i q^{71} +12.5461 q^{73} -6.92543i q^{77} +3.77365i q^{79} +27.3514 q^{81} -13.5929i q^{83} -21.4260i q^{87} +5.83314 q^{89} -0.212425i q^{91} +19.5929i q^{93} -1.45385 q^{97} +41.9582i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} + 8 q^{7} + 14 q^{9} + 4 q^{17} - 16 q^{19} - 24 q^{21} - 4 q^{23} + 28 q^{27} - 24 q^{37} - 2 q^{49} + 4 q^{51} + 48 q^{57} + 8 q^{59} - 12 q^{63} - 32 q^{69} + 44 q^{73} + 38 q^{81} + 36 q^{89}+ \cdots - 40 q^{97}+O(q^{100}) \) 6 * q + 4 * q^3 + 8 * q^7 + 14 * q^9 + 4 * q^17 - 16 * q^19 - 24 * q^21 - 4 * q^23 + 28 * q^27 - 24 * q^37 - 2 * q^49 + 4 * q^51 + 48 * q^57 + 8 * q^59 - 12 * q^63 - 32 * q^69 + 44 * q^73 + 38 * q^81 + 36 * q^89 - 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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