LMFDB - Embedded newform 6600.2.d.bb.1849.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6600,2,Mod(1849,6600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6600.1849");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6600.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$52.7012653340$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1320)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	6600.1849
Dual form			6600.2.d.bb.1849.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -2.82843i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -2.82843i q^{7} -1.00000 q^{9} +1.00000 q^{11} -4.82843i q^{13} +4.82843i q^{17} -2.82843 q^{21} +5.65685i q^{23} +1.00000i q^{27} +3.65685 q^{29} +5.65685 q^{31} -1.00000i q^{33} -2.00000i q^{37} -4.82843 q^{39} +11.6569 q^{41} -12.4853i q^{43} +5.65685i q^{47} -1.00000 q^{49} +4.82843 q^{51} -11.6569i q^{53} +9.65685 q^{59} -7.65685 q^{61} +2.82843i q^{63} -9.65685i q^{67} +5.65685 q^{69} +5.65685 q^{71} -3.17157i q^{73} -2.82843i q^{77} +4.00000 q^{79} +1.00000 q^{81} -5.17157i q^{83} -3.65685i q^{87} -2.00000 q^{89} -13.6569 q^{91} -5.65685i q^{93} -11.6569i q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 4 q^{11} - 8 q^{29} - 8 q^{39} + 24 q^{41} - 4 q^{49} + 8 q^{51} + 16 q^{59} - 8 q^{61} + 16 q^{79} + 4 q^{81} - 8 q^{89} - 32 q^{91} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^9 + 4 * q^11 - 8 * q^29 - 8 * q^39 + 24 * q^41 - 4 * q^49 + 8 * q^51 + 16 * q^59 - 8 * q^61 + 16 * q^79 + 4 * q^81 - 8 * q^89 - 32 * q^91 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6600\mathbb{Z}\right)^\times$.

$n$	$1201$	$2201$	$2377$	$3301$	$4951$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 2.82843i	− 1.06904i	−0.845154	−	0.534522i	$-0.820491\pi$
$7$	− 2.82843i	− 1.06904i	0.845154	−	0.534522i	$-0.179509\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	− 4.82843i	− 1.33916i	−0.742738	−	0.669582i	$-0.766473\pi$
$13$	− 4.82843i	− 1.33916i	0.742738	−	0.669582i	$-0.233527\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.82843i	1.17107i	0.810649	+	0.585533i	$0.199115\pi$
$17$	4.82843i	1.17107i	−0.810649	+	0.585533i	$0.800885\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−2.82843	−0.617213
$22$	0	0
$23$	5.65685i	1.17954i	0.807573	+	0.589768i	$0.200781\pi$
$23$	5.65685i	1.17954i	−0.807573	+	0.589768i	$0.799219\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	3.65685	0.679061	0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685	0.679061	0.339530	+	0.940595i	$0.389732\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	− 1.00000i	− 0.174078i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	−4.82843	−0.773167
$40$	0	0
$41$	11.6569	1.82049	0.910247	−	0.414065i	$-0.135891\pi$
$41$	11.6569	1.82049	0.910247	+	0.414065i	$0.135891\pi$
$42$	0	0
$43$	− 12.4853i	− 1.90399i	−0.306117	−	0.951994i	$-0.599030\pi$
$43$	− 12.4853i	− 1.90399i	0.306117	−	0.951994i	$-0.400970\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.65685i	0.825137i	0.910927	+	0.412568i	$0.135368\pi$
$47$	5.65685i	0.825137i	−0.910927	+	0.412568i	$0.864632\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	4.82843	0.676115
$52$	0	0
$53$	− 11.6569i	− 1.60119i	−0.599204	−	0.800596i	$-0.704516\pi$
$53$	− 11.6569i	− 1.60119i	0.599204	−	0.800596i	$-0.295484\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.65685	1.25722	0.628608	−	0.777723i	$-0.283625\pi$
$59$	9.65685	1.25722	0.628608	+	0.777723i	$0.283625\pi$
$60$	0	0
$61$	−7.65685	−0.980360	−0.490180	−	0.871621i	$-0.663069\pi$
$61$	−7.65685	−0.980360	−0.490180	+	0.871621i	$0.663069\pi$
$62$	0	0
$63$	2.82843i	0.356348i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 9.65685i	− 1.17977i	−0.807486	−	0.589886i	$-0.799173\pi$
$67$	− 9.65685i	− 1.17977i	0.807486	−	0.589886i	$-0.200827\pi$
$68$	0	0
$69$	5.65685	0.681005
$70$	0	0
$71$	5.65685	0.671345	0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685	0.671345	0.335673	+	0.941979i	$0.391036\pi$
$72$	0	0
$73$	− 3.17157i	− 0.371205i	−0.982625	−	0.185602i	$-0.940576\pi$
$73$	− 3.17157i	− 0.371205i	0.982625	−	0.185602i	$-0.0594236\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 2.82843i	− 0.322329i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 5.17157i	− 0.567654i	−0.958876	−	0.283827i	$-0.908396\pi$
$83$	− 5.17157i	− 0.567654i	0.958876	−	0.283827i	$-0.0916041\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 3.65685i	− 0.392056i
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−13.6569	−1.43163
$92$	0	0
$93$	− 5.65685i	− 0.586588i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 11.6569i	− 1.18357i	−0.806094	−	0.591787i	$-0.798423\pi$
$97$	− 11.6569i	− 1.18357i	0.806094	−	0.591787i	$-0.201577\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−9.31371	−0.926749	−0.463374	−	0.886163i	$-0.653361\pi$
$101$	−9.31371	−0.926749	−0.463374	+	0.886163i	$0.653361\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.48528i	0.820303i	0.912017	+	0.410152i	$0.134524\pi$
$107$	8.48528i	0.820303i	−0.912017	+	0.410152i	$0.865476\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	− 10.0000i	− 0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$	− 10.0000i	− 0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.82843i	0.446388i
$118$	0	0
$119$	13.6569	1.25192
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	− 11.6569i	− 1.05106i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.1421i	1.25491i	0.778652	+	0.627456i	$0.215904\pi$
$127$	14.1421i	1.25491i	−0.778652	+	0.627456i	$0.784096\pi$
$128$	0	0
$129$	−12.4853	−1.09927
$130$	0	0
$131$	−9.65685	−0.843723	−0.421862	−	0.906660i	$-0.638623\pi$
$131$	−9.65685	−0.843723	−0.421862	+	0.906660i	$0.638623\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.00000i	0.170872i	0.996344	+	0.0854358i	$0.0272282\pi$
$137$	2.00000i	0.170872i	−0.996344	+	0.0854358i	$0.972772\pi$
$138$	0	0
$139$	−5.65685	−0.479808	−0.239904	−	0.970797i	$-0.577116\pi$
$139$	−5.65685	−0.479808	−0.239904	+	0.970797i	$0.577116\pi$
$140$	0	0
$141$	5.65685	0.476393
$142$	0	0
$143$	− 4.82843i	− 0.403773i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.00000i	0.0824786i
$148$	0	0
$149$	−21.3137	−1.74609	−0.873044	−	0.487642i	$-0.837857\pi$
$149$	−21.3137	−1.74609	−0.873044	+	0.487642i	$0.837857\pi$
$150$	0	0
$151$	−23.3137	−1.89724	−0.948621	−	0.316414i	$-0.897521\pi$
$151$	−23.3137	−1.89724	−0.948621	+	0.316414i	$0.897521\pi$
$152$	0	0
$153$	− 4.82843i	− 0.390355i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 23.6569i	− 1.88802i	−0.329913	−	0.944011i	$-0.607019\pi$
$157$	− 23.6569i	− 1.88802i	0.329913	−	0.944011i	$-0.392981\pi$
$158$	0	0
$159$	−11.6569	−0.924449
$160$	0	0
$161$	16.0000	1.26098
$162$	0	0
$163$	15.3137i	1.19946i	0.800202	+	0.599731i	$0.204726\pi$
$163$	15.3137i	1.19946i	−0.800202	+	0.599731i	$0.795274\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1.17157i	− 0.0906590i	−0.998972	−	0.0453295i	$-0.985566\pi$
$167$	− 1.17157i	− 0.0906590i	0.998972	−	0.0453295i	$-0.0144338\pi$
$168$	0	0
$169$	−10.3137	−0.793362
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.7990i	1.04912i	0.851374	+	0.524559i	$0.175770\pi$
$173$	13.7990i	1.04912i	−0.851374	+	0.524559i	$0.824230\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 9.65685i	− 0.725854i
$178$	0	0
$179$	−7.31371	−0.546652	−0.273326	−	0.961921i	$-0.588124\pi$
$179$	−7.31371	−0.546652	−0.273326	+	0.961921i	$0.588124\pi$
$180$	0	0
$181$	9.31371	0.692283	0.346141	−	0.938182i	$-0.387492\pi$
$181$	9.31371	0.692283	0.346141	+	0.938182i	$0.387492\pi$
$182$	0	0
$183$	7.65685i	0.566011i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.82843i	0.353090i
$188$	0	0
$189$	2.82843	0.205738
$190$	0	0
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	0	0
$193$	− 3.17157i	− 0.228295i	−0.993464	−	0.114147i	$-0.963586\pi$
$193$	− 3.17157i	− 0.228295i	0.993464	−	0.114147i	$-0.0364136\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.1421i	0.865091i	0.901612	+	0.432546i	$0.142385\pi$
$197$	12.1421i	0.865091i	−0.901612	+	0.432546i	$0.857615\pi$
$198$	0	0
$199$	19.3137	1.36911	0.684556	−	0.728960i	$-0.259996\pi$
$199$	19.3137	1.36911	0.684556	+	0.728960i	$0.259996\pi$
$200$	0	0
$201$	−9.65685	−0.681142
$202$	0	0
$203$	− 10.3431i	− 0.725947i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 5.65685i	− 0.393179i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−4.68629	−0.322618	−0.161309	−	0.986904i	$-0.551572\pi$
$211$	−4.68629	−0.322618	−0.161309	+	0.986904i	$0.551572\pi$
$212$	0	0
$213$	− 5.65685i	− 0.387601i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 16.0000i	− 1.08615i
$218$	0	0
$219$	−3.17157	−0.214315
$220$	0	0
$221$	23.3137	1.56825
$222$	0	0
$223$	24.9706i	1.67215i	0.548613	+	0.836076i	$0.315156\pi$
$223$	24.9706i	1.67215i	−0.548613	+	0.836076i	$0.684844\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.48528i	− 0.563188i	−0.959534	−	0.281594i	$-0.909137\pi$
$227$	− 8.48528i	− 0.563188i	0.959534	−	0.281594i	$-0.0908631\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−2.82843	−0.186097
$232$	0	0
$233$	− 18.4853i	− 1.21101i	−0.795841	−	0.605506i	$-0.792971\pi$
$233$	− 18.4853i	− 1.21101i	0.795841	−	0.605506i	$-0.207029\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.00000i	− 0.259828i
$238$	0	0
$239$	10.3431	0.669042	0.334521	−	0.942388i	$-0.391425\pi$
$239$	10.3431	0.669042	0.334521	+	0.942388i	$0.391425\pi$
$240$	0	0
$241$	7.65685	0.493221	0.246611	−	0.969115i	$-0.420683\pi$
$241$	7.65685	0.493221	0.246611	+	0.969115i	$0.420683\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−5.17157	−0.327735
$250$	0	0
$251$	−17.6569	−1.11449	−0.557245	−	0.830348i	$-0.688142\pi$
$251$	−17.6569	−1.11449	−0.557245	+	0.830348i	$0.688142\pi$
$252$	0	0
$253$	5.65685i	0.355643i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 22.9706i	− 1.43286i	−0.697657	−	0.716432i	$-0.745774\pi$
$257$	− 22.9706i	− 1.43286i	0.697657	−	0.716432i	$-0.254226\pi$
$258$	0	0
$259$	−5.65685	−0.351500
$260$	0	0
$261$	−3.65685	−0.226354
$262$	0	0
$263$	− 12.4853i	− 0.769875i	−0.922943	−	0.384938i	$-0.874223\pi$
$263$	− 12.4853i	− 0.769875i	0.922943	−	0.384938i	$-0.125777\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.00000i	0.122398i
$268$	0	0
$269$	24.6274	1.50156	0.750780	−	0.660552i	$-0.229678\pi$
$269$	24.6274	1.50156	0.750780	+	0.660552i	$0.229678\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	13.6569i	0.826550i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 19.1716i	− 1.15191i	−0.817482	−	0.575954i	$-0.804631\pi$
$277$	− 19.1716i	− 1.15191i	0.817482	−	0.575954i	$-0.195369\pi$
$278$	0	0
$279$	−5.65685	−0.338667
$280$	0	0
$281$	−26.9706	−1.60893	−0.804464	−	0.594001i	$-0.797548\pi$
$281$	−26.9706	−1.60893	−0.804464	+	0.594001i	$0.797548\pi$
$282$	0	0
$283$	− 23.7990i	− 1.41470i	−0.706862	−	0.707352i	$-0.749890\pi$
$283$	− 23.7990i	− 1.41470i	0.706862	−	0.707352i	$-0.250110\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 32.9706i	− 1.94619i
$288$	0	0
$289$	−6.31371	−0.371395
$290$	0	0
$291$	−11.6569	−0.683337
$292$	0	0
$293$	− 27.1716i	− 1.58738i	−0.608322	−	0.793690i	$-0.708157\pi$
$293$	− 27.1716i	− 1.58738i	0.608322	−	0.793690i	$-0.291843\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.00000i	0.0580259i
$298$	0	0
$299$	27.3137	1.57959
$300$	0	0
$301$	−35.3137	−2.03545
$302$	0	0
$303$	9.31371i	0.535059i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5.85786i	0.334326i	0.985929	+	0.167163i	$0.0534606\pi$
$307$	5.85786i	0.334326i	−0.985929	+	0.167163i	$0.946539\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	8.97056	0.508674	0.254337	−	0.967116i	$-0.418143\pi$
$311$	8.97056	0.508674	0.254337	+	0.967116i	$0.418143\pi$
$312$	0	0
$313$	− 5.31371i	− 0.300349i	−0.988660	−	0.150174i	$-0.952017\pi$
$313$	− 5.31371i	− 0.300349i	0.988660	−	0.150174i	$-0.0479835\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.6274i	1.15855i	0.815132	+	0.579276i	$0.196664\pi$
$317$	20.6274i	1.15855i	−0.815132	+	0.579276i	$0.803336\pi$
$318$	0	0
$319$	3.65685	0.204745
$320$	0	0
$321$	8.48528	0.473602
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.00000i	− 0.110600i
$328$	0	0
$329$	16.0000	0.882109
$330$	0	0
$331$	20.9706	1.15265	0.576323	−	0.817222i	$-0.304487\pi$
$331$	20.9706	1.15265	0.576323	+	0.817222i	$0.304487\pi$
$332$	0	0
$333$	2.00000i	0.109599i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11.1716i	0.608554i	0.952584	+	0.304277i	$0.0984149\pi$
$337$	11.1716i	0.608554i	−0.952584	+	0.304277i	$0.901585\pi$
$338$	0	0
$339$	−10.0000	−0.543125
$340$	0	0
$341$	5.65685	0.306336
$342$	0	0
$343$	− 16.9706i	− 0.916324i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.85786i	0.0997354i	0.998756	+	0.0498677i	$0.0158800\pi$
$347$	1.85786i	0.0997354i	−0.998756	+	0.0498677i	$0.984120\pi$
$348$	0	0
$349$	−3.65685	−0.195747	−0.0978735	−	0.995199i	$-0.531204\pi$
$349$	−3.65685	−0.195747	−0.0978735	+	0.995199i	$0.531204\pi$
$350$	0	0
$351$	4.82843	0.257722
$352$	0	0
$353$	0.343146i	0.0182638i	0.999958	+	0.00913190i	$0.00290682\pi$
$353$	0.343146i	0.0182638i	−0.999958	+	0.00913190i	$0.997093\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 13.6569i	− 0.722797i
$358$	0	0
$359$	−5.65685	−0.298557	−0.149279	−	0.988795i	$-0.547695\pi$
$359$	−5.65685	−0.298557	−0.149279	+	0.988795i	$0.547695\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	− 1.00000i	− 0.0524864i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.97056i	0.468260i	0.972205	+	0.234130i	$0.0752241\pi$
$367$	8.97056i	0.468260i	−0.972205	+	0.234130i	$0.924776\pi$
$368$	0	0
$369$	−11.6569	−0.606832
$370$	0	0
$371$	−32.9706	−1.71175
$372$	0	0
$373$	31.4558i	1.62872i	0.580359	+	0.814361i	$0.302912\pi$
$373$	31.4558i	1.62872i	−0.580359	+	0.814361i	$0.697088\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 17.6569i	− 0.909374i
$378$	0	0
$379$	−8.68629	−0.446185	−0.223092	−	0.974797i	$-0.571615\pi$
$379$	−8.68629	−0.446185	−0.223092	+	0.974797i	$0.571615\pi$
$380$	0	0
$381$	14.1421	0.724524
$382$	0	0
$383$	− 22.6274i	− 1.15621i	−0.815963	−	0.578103i	$-0.803793\pi$
$383$	− 22.6274i	− 1.15621i	0.815963	−	0.578103i	$-0.196207\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	12.4853i	0.634663i
$388$	0	0
$389$	18.0000	0.912636	0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000	0.912636	0.456318	+	0.889817i	$0.349168\pi$
$390$	0	0
$391$	−27.3137	−1.38131
$392$	0	0
$393$	9.65685i	0.487124i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.02944i	− 0.0516660i	−0.999666	−	0.0258330i	$-0.991776\pi$
$397$	− 1.02944i	− 0.0516660i	0.999666	−	0.0258330i	$-0.00822381\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.68629	0.333897	0.166949	−	0.985966i	$-0.446609\pi$
$401$	6.68629	0.333897	0.166949	+	0.985966i	$0.446609\pi$
$402$	0	0
$403$	− 27.3137i	− 1.36059i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.00000i	− 0.0991363i
$408$	0	0
$409$	36.6274	1.81111	0.905555	−	0.424230i	$-0.139455\pi$
$409$	36.6274	1.81111	0.905555	+	0.424230i	$0.139455\pi$
$410$	0	0
$411$	2.00000	0.0986527
$412$	0	0
$413$	− 27.3137i	− 1.34402i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.65685i	0.277017i
$418$	0	0
$419$	−18.6274	−0.910009	−0.455004	−	0.890489i	$-0.650362\pi$
$419$	−18.6274	−0.910009	−0.455004	+	0.890489i	$0.650362\pi$
$420$	0	0
$421$	−18.0000	−0.877266	−0.438633	−	0.898666i	$-0.644537\pi$
$421$	−18.0000	−0.877266	−0.438633	+	0.898666i	$0.644537\pi$
$422$	0	0
$423$	− 5.65685i	− 0.275046i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	21.6569i	1.04805i
$428$	0	0
$429$	−4.82843	−0.233119
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	− 4.34315i	− 0.208718i	−0.994540	−	0.104359i	$-0.966721\pi$
$433$	− 4.34315i	− 0.208718i	0.994540	−	0.104359i	$-0.0332791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−14.3431	−0.684561	−0.342280	−	0.939598i	$-0.611199\pi$
$439$	−14.3431	−0.684561	−0.342280	+	0.939598i	$0.611199\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	− 36.0000i	− 1.71041i	−0.518289	−	0.855206i	$-0.673431\pi$
$443$	− 36.0000i	− 1.71041i	0.518289	−	0.855206i	$-0.326569\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	21.3137i	1.00810i
$448$	0	0
$449$	33.3137	1.57217	0.786086	−	0.618118i	$-0.212104\pi$
$449$	33.3137	1.57217	0.786086	+	0.618118i	$0.212104\pi$
$450$	0	0
$451$	11.6569	0.548900
$452$	0	0
$453$	23.3137i	1.09537i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 37.7990i	− 1.76816i	−0.467334	−	0.884081i	$-0.654785\pi$
$457$	− 37.7990i	− 1.76816i	0.467334	−	0.884081i	$-0.345215\pi$
$458$	0	0
$459$	−4.82843	−0.225372
$460$	0	0
$461$	13.3137	0.620081	0.310041	−	0.950723i	$-0.399657\pi$
$461$	13.3137	0.620081	0.310041	+	0.950723i	$0.399657\pi$
$462$	0	0
$463$	18.3431i	0.852478i	0.904611	+	0.426239i	$0.140162\pi$
$463$	18.3431i	0.852478i	−0.904611	+	0.426239i	$0.859838\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	36.9706i	1.71079i	0.517973	+	0.855397i	$0.326687\pi$
$467$	36.9706i	1.71079i	−0.517973	+	0.855397i	$0.673313\pi$
$468$	0	0
$469$	−27.3137	−1.26123
$470$	0	0
$471$	−23.6569	−1.09005
$472$	0	0
$473$	− 12.4853i	− 0.574074i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	11.6569i	0.533731i
$478$	0	0
$479$	5.65685	0.258468	0.129234	−	0.991614i	$-0.458748\pi$
$479$	5.65685	0.258468	0.129234	+	0.991614i	$0.458748\pi$
$480$	0	0
$481$	−9.65685	−0.440315
$482$	0	0
$483$	− 16.0000i	− 0.728025i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	29.6569i	1.34388i	0.740605	+	0.671940i	$0.234539\pi$
$487$	29.6569i	1.34388i	−0.740605	+	0.671940i	$0.765461\pi$
$488$	0	0
$489$	15.3137	0.692510
$490$	0	0
$491$	−23.3137	−1.05213	−0.526066	−	0.850443i	$-0.676334\pi$
$491$	−23.3137	−1.05213	−0.526066	+	0.850443i	$0.676334\pi$
$492$	0	0
$493$	17.6569i	0.795225i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 16.0000i	− 0.717698i
$498$	0	0
$499$	29.9411	1.34035	0.670174	−	0.742204i	$-0.266219\pi$
$499$	29.9411	1.34035	0.670174	+	0.742204i	$0.266219\pi$
$500$	0	0
$501$	−1.17157	−0.0523420
$502$	0	0
$503$	23.7990i	1.06114i	0.847640	+	0.530572i	$0.178023\pi$
$503$	23.7990i	1.06114i	−0.847640	+	0.530572i	$0.821977\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	10.3137i	0.458048i
$508$	0	0
$509$	−9.31371	−0.412823	−0.206411	−	0.978465i	$-0.566179\pi$
$509$	−9.31371	−0.412823	−0.206411	+	0.978465i	$0.566179\pi$
$510$	0	0
$511$	−8.97056	−0.396834
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5.65685i	0.248788i
$518$	0	0
$519$	13.7990	0.605708
$520$	0	0
$521$	27.9411	1.22412	0.612061	−	0.790810i	$-0.290340\pi$
$521$	27.9411	1.22412	0.612061	+	0.790810i	$0.290340\pi$
$522$	0	0
$523$	7.79899i	0.341026i	0.985355	+	0.170513i	$0.0545425\pi$
$523$	7.79899i	0.341026i	−0.985355	+	0.170513i	$0.945458\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	27.3137i	1.18980i
$528$	0	0
$529$	−9.00000	−0.391304
$530$	0	0
$531$	−9.65685	−0.419072
$532$	0	0
$533$	− 56.2843i	− 2.43794i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.31371i	0.315610i
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−6.68629	−0.287466	−0.143733	−	0.989616i	$-0.545911\pi$
$541$	−6.68629	−0.287466	−0.143733	+	0.989616i	$0.545911\pi$
$542$	0	0
$543$	− 9.31371i	− 0.399689i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 15.7990i	− 0.675516i	−0.941233	−	0.337758i	$-0.890331\pi$
$547$	− 15.7990i	− 0.675516i	0.941233	−	0.337758i	$-0.109669\pi$
$548$	0	0
$549$	7.65685	0.326787
$550$	0	0
$551$	0	0
$552$	0	0
$553$	− 11.3137i	− 0.481108i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.51472i	0.233666i	0.993152	+	0.116833i	$0.0372742\pi$
$557$	5.51472i	0.233666i	−0.993152	+	0.116833i	$0.962726\pi$
$558$	0	0
$559$	−60.2843	−2.54975
$560$	0	0
$561$	4.82843	0.203856
$562$	0	0
$563$	− 15.5147i	− 0.653867i	−0.945047	−	0.326934i	$-0.893985\pi$
$563$	− 15.5147i	− 0.653867i	0.945047	−	0.326934i	$-0.106015\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.82843i	− 0.118783i
$568$	0	0
$569$	14.6863	0.615681	0.307841	−	0.951438i	$-0.400394\pi$
$569$	14.6863	0.615681	0.307841	+	0.951438i	$0.400394\pi$
$570$	0	0
$571$	18.3431	0.767637	0.383818	−	0.923409i	$-0.374609\pi$
$571$	18.3431	0.767637	0.383818	+	0.923409i	$0.374609\pi$
$572$	0	0
$573$	24.0000i	1.00261i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.9706i	1.12280i	0.827545	+	0.561400i	$0.189737\pi$
$577$	26.9706i	1.12280i	−0.827545	+	0.561400i	$0.810263\pi$
$578$	0	0
$579$	−3.17157	−0.131806
$580$	0	0
$581$	−14.6274	−0.606848
$582$	0	0
$583$	− 11.6569i	− 0.482778i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.65685i	− 0.398581i	−0.979940	−	0.199291i	$-0.936136\pi$
$587$	− 9.65685i	− 0.398581i	0.979940	−	0.199291i	$-0.0638637\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	12.1421	0.499461
$592$	0	0
$593$	41.7990i	1.71648i	0.513250	+	0.858239i	$0.328442\pi$
$593$	41.7990i	1.71648i	−0.513250	+	0.858239i	$0.671558\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 19.3137i	− 0.790457i
$598$	0	0
$599$	−12.6863	−0.518348	−0.259174	−	0.965831i	$-0.583450\pi$
$599$	−12.6863	−0.518348	−0.259174	+	0.965831i	$0.583450\pi$
$600$	0	0
$601$	28.3431	1.15614	0.578071	−	0.815987i	$-0.303806\pi$
$601$	28.3431	1.15614	0.578071	+	0.815987i	$0.303806\pi$
$602$	0	0
$603$	9.65685i	0.393258i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 47.1127i	− 1.91225i	−0.292966	−	0.956123i	$-0.594642\pi$
$607$	− 47.1127i	− 1.91225i	0.292966	−	0.956123i	$-0.405358\pi$
$608$	0	0
$609$	−10.3431	−0.419125
$610$	0	0
$611$	27.3137	1.10499
$612$	0	0
$613$	− 7.17157i	− 0.289657i	−0.989457	−	0.144829i	$-0.953737\pi$
$613$	− 7.17157i	− 0.289657i	0.989457	−	0.144829i	$-0.0462631\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.31371i	− 0.374956i	−0.982269	−	0.187478i	$-0.939969\pi$
$617$	− 9.31371i	− 0.374956i	0.982269	−	0.187478i	$-0.0600313\pi$
$618$	0	0
$619$	−14.3431	−0.576500	−0.288250	−	0.957555i	$-0.593073\pi$
$619$	−14.3431	−0.576500	−0.288250	+	0.957555i	$0.593073\pi$
$620$	0	0
$621$	−5.65685	−0.227002
$622$	0	0
$623$	5.65685i	0.226637i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.65685	0.385044
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	4.68629i	0.186263i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.82843i	0.191309i
$638$	0	0
$639$	−5.65685	−0.223782
$640$	0	0
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	0	0
$643$	− 23.3137i	− 0.919403i	−0.888074	−	0.459701i	$-0.847956\pi$
$643$	− 23.3137i	− 0.919403i	0.888074	−	0.459701i	$-0.152044\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 30.6274i	− 1.20409i	−0.798463	−	0.602044i	$-0.794353\pi$
$647$	− 30.6274i	− 1.20409i	0.798463	−	0.602044i	$-0.205647\pi$
$648$	0	0
$649$	9.65685	0.379065
$650$	0	0
$651$	−16.0000	−0.627089
$652$	0	0
$653$	18.9706i	0.742375i	0.928558	+	0.371188i	$0.121049\pi$
$653$	18.9706i	0.742375i	−0.928558	+	0.371188i	$0.878951\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.17157i	0.123735i
$658$	0	0
$659$	0.686292	0.0267341	0.0133671	−	0.999911i	$-0.495745\pi$
$659$	0.686292	0.0267341	0.0133671	+	0.999911i	$0.495745\pi$
$660$	0	0
$661$	−27.9411	−1.08678	−0.543392	−	0.839479i	$-0.682860\pi$
$661$	−27.9411	−1.08678	−0.543392	+	0.839479i	$0.682860\pi$
$662$	0	0
$663$	− 23.3137i	− 0.905429i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.6863i	0.800976i
$668$	0	0
$669$	24.9706	0.965418
$670$	0	0
$671$	−7.65685	−0.295590
$672$	0	0
$673$	− 41.7990i	− 1.61123i	−0.592438	−	0.805616i	$-0.701834\pi$
$673$	− 41.7990i	− 1.61123i	0.592438	−	0.805616i	$-0.298166\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.14214i	− 0.312928i	−0.987684	−	0.156464i	$-0.949991\pi$
$677$	− 8.14214i	− 0.312928i	0.987684	−	0.156464i	$-0.0500095\pi$
$678$	0	0
$679$	−32.9706	−1.26529
$680$	0	0
$681$	−8.48528	−0.325157
$682$	0	0
$683$	− 25.6569i	− 0.981732i	−0.871235	−	0.490866i	$-0.836680\pi$
$683$	− 25.6569i	− 0.981732i	0.871235	−	0.490866i	$-0.163320\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 10.0000i	− 0.381524i
$688$	0	0
$689$	−56.2843	−2.14426
$690$	0	0
$691$	23.3137	0.886895	0.443448	−	0.896300i	$-0.353755\pi$
$691$	23.3137	0.886895	0.443448	+	0.896300i	$0.353755\pi$
$692$	0	0
$693$	2.82843i	0.107443i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	56.2843i	2.13192i
$698$	0	0
$699$	−18.4853	−0.699178
$700$	0	0
$701$	42.9706	1.62298	0.811488	−	0.584369i	$-0.198658\pi$
$701$	42.9706	1.62298	0.811488	+	0.584369i	$0.198658\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	26.3431i	0.990736i
$708$	0	0
$709$	−2.68629	−0.100886	−0.0504429	−	0.998727i	$-0.516063\pi$
$709$	−2.68629	−0.100886	−0.0504429	+	0.998727i	$0.516063\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	32.0000i	1.19841i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 10.3431i	− 0.386272i
$718$	0	0
$719$	12.6863	0.473119	0.236559	−	0.971617i	$-0.423980\pi$
$719$	12.6863	0.473119	0.236559	+	0.971617i	$0.423980\pi$
$720$	0	0
$721$	22.6274	0.842689
$722$	0	0
$723$	− 7.65685i	− 0.284761i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	11.3137i	0.419602i	0.977744	+	0.209801i	$0.0672817\pi$
$727$	11.3137i	0.419602i	−0.977744	+	0.209801i	$0.932718\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	60.2843	2.22969
$732$	0	0
$733$	− 34.4853i	− 1.27374i	−0.770970	−	0.636871i	$-0.780228\pi$
$733$	− 34.4853i	− 1.27374i	0.770970	−	0.636871i	$-0.219772\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 9.65685i	− 0.355715i
$738$	0	0
$739$	−5.65685	−0.208091	−0.104045	−	0.994573i	$-0.533179\pi$
$739$	−5.65685	−0.208091	−0.104045	+	0.994573i	$0.533179\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 9.17157i	− 0.336472i	−0.985747	−	0.168236i	$-0.946193\pi$
$743$	− 9.17157i	− 0.336472i	0.985747	−	0.168236i	$-0.0538071\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	5.17157i	0.189218i
$748$	0	0
$749$	24.0000	0.876941
$750$	0	0
$751$	22.6274	0.825686	0.412843	−	0.910802i	$-0.364536\pi$
$751$	22.6274	0.825686	0.412843	+	0.910802i	$0.364536\pi$
$752$	0	0
$753$	17.6569i	0.643452i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 42.9706i	− 1.56179i	−0.624661	−	0.780896i	$-0.714763\pi$
$757$	− 42.9706i	− 1.56179i	0.624661	−	0.780896i	$-0.285237\pi$
$758$	0	0
$759$	5.65685	0.205331
$760$	0	0
$761$	44.6274	1.61774	0.808871	−	0.587986i	$-0.200079\pi$
$761$	44.6274	1.61774	0.808871	+	0.587986i	$0.200079\pi$
$762$	0	0
$763$	− 5.65685i	− 0.204792i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 46.6274i	− 1.68362i
$768$	0	0
$769$	−42.9706	−1.54956	−0.774779	−	0.632232i	$-0.782139\pi$
$769$	−42.9706	−1.54956	−0.774779	+	0.632232i	$0.782139\pi$
$770$	0	0
$771$	−22.9706	−0.827265
$772$	0	0
$773$	− 16.3431i	− 0.587822i	−0.955833	−	0.293911i	$-0.905043\pi$
$773$	− 16.3431i	− 0.587822i	0.955833	−	0.293911i	$-0.0949569\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5.65685i	0.202939i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	5.65685	0.202418
$782$	0	0
$783$	3.65685i	0.130685i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	54.4264i	1.94009i	0.242922	+	0.970046i	$0.421894\pi$
$787$	54.4264i	1.94009i	−0.242922	+	0.970046i	$0.578106\pi$
$788$	0	0
$789$	−12.4853	−0.444488
$790$	0	0
$791$	−28.2843	−1.00567
$792$	0	0
$793$	36.9706i	1.31286i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	46.0000i	1.62940i	0.579880	+	0.814702i	$0.303099\pi$
$797$	46.0000i	1.62940i	−0.579880	+	0.814702i	$0.696901\pi$
$798$	0	0
$799$	−27.3137	−0.966290
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	− 3.17157i	− 0.111922i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 24.6274i	− 0.866926i
$808$	0	0
$809$	23.6569	0.831731	0.415865	−	0.909426i	$-0.363479\pi$
$809$	23.6569	0.831731	0.415865	+	0.909426i	$0.363479\pi$
$810$	0	0
$811$	37.6569	1.32231	0.661155	−	0.750249i	$-0.270066\pi$
$811$	37.6569	1.32231	0.661155	+	0.750249i	$0.270066\pi$
$812$	0	0
$813$	12.0000i	0.420858i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	13.6569	0.477209
$820$	0	0
$821$	20.3431	0.709981	0.354990	−	0.934870i	$-0.384484\pi$
$821$	20.3431	0.709981	0.354990	+	0.934870i	$0.384484\pi$
$822$	0	0
$823$	− 14.6274i	− 0.509880i	−0.966957	−	0.254940i	$-0.917944\pi$
$823$	− 14.6274i	− 0.509880i	0.966957	−	0.254940i	$-0.0820557\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.1127i	0.525520i	0.964861	+	0.262760i	$0.0846327\pi$
$827$	15.1127i	0.525520i	−0.964861	+	0.262760i	$0.915367\pi$
$828$	0	0
$829$	11.9411	0.414732	0.207366	−	0.978263i	$-0.433511\pi$
$829$	11.9411	0.414732	0.207366	+	0.978263i	$0.433511\pi$
$830$	0	0
$831$	−19.1716	−0.665054
$832$	0	0
$833$	− 4.82843i	− 0.167295i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.65685i	0.195529i
$838$	0	0
$839$	8.97056	0.309698	0.154849	−	0.987938i	$-0.450511\pi$
$839$	8.97056	0.309698	0.154849	+	0.987938i	$0.450511\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	26.9706i	0.928916i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2.82843i	− 0.0971859i
$848$	0	0
$849$	−23.7990	−0.816779
$850$	0	0
$851$	11.3137	0.387829
$852$	0	0
$853$	− 11.8579i	− 0.406006i	−0.979178	−	0.203003i	$-0.934930\pi$
$853$	− 11.8579i	− 0.406006i	0.979178	−	0.203003i	$-0.0650700\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 46.4853i	− 1.58791i	−0.607979	−	0.793953i	$-0.708019\pi$
$857$	− 46.4853i	− 1.58791i	0.607979	−	0.793953i	$-0.291981\pi$
$858$	0	0
$859$	17.6569	0.602444	0.301222	−	0.953554i	$-0.402605\pi$
$859$	17.6569	0.602444	0.301222	+	0.953554i	$0.402605\pi$
$860$	0	0
$861$	−32.9706	−1.12363
$862$	0	0
$863$	11.3137i	0.385123i	0.981285	+	0.192562i	$0.0616795\pi$
$863$	11.3137i	0.385123i	−0.981285	+	0.192562i	$0.938320\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.31371i	0.214425i
$868$	0	0
$869$	4.00000	0.135691
$870$	0	0
$871$	−46.6274	−1.57991
$872$	0	0
$873$	11.6569i	0.394525i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.79899i	0.195818i	0.995195	+	0.0979090i	$0.0312154\pi$
$877$	5.79899i	0.195818i	−0.995195	+	0.0979090i	$0.968785\pi$
$878$	0	0
$879$	−27.1716	−0.916474
$880$	0	0
$881$	3.37258	0.113625	0.0568126	−	0.998385i	$-0.481906\pi$
$881$	3.37258	0.113625	0.0568126	+	0.998385i	$0.481906\pi$
$882$	0	0
$883$	41.2548i	1.38834i	0.719813	+	0.694168i	$0.244227\pi$
$883$	41.2548i	1.38834i	−0.719813	+	0.694168i	$0.755773\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.4558i	1.52626i	0.646246	+	0.763129i	$0.276338\pi$
$887$	45.4558i	1.52626i	−0.646246	+	0.763129i	$0.723662\pi$
$888$	0	0
$889$	40.0000	1.34156
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 27.3137i	− 0.911978i
$898$	0	0
$899$	20.6863	0.689926
$900$	0	0
$901$	56.2843	1.87510
$902$	0	0
$903$	35.3137i	1.17517i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 9.65685i	− 0.320651i	−0.987064	−	0.160325i	$-0.948746\pi$
$907$	− 9.65685i	− 0.320651i	0.987064	−	0.160325i	$-0.0512543\pi$
$908$	0	0
$909$	9.31371	0.308916
$910$	0	0
$911$	−12.6863	−0.420316	−0.210158	−	0.977667i	$-0.567398\pi$
$911$	−12.6863	−0.420316	−0.210158	+	0.977667i	$0.567398\pi$
$912$	0	0
$913$	− 5.17157i	− 0.171154i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	27.3137i	0.901978i
$918$	0	0
$919$	6.34315	0.209241	0.104621	−	0.994512i	$-0.466637\pi$
$919$	6.34315	0.209241	0.104621	+	0.994512i	$0.466637\pi$
$920$	0	0
$921$	5.85786	0.193023
$922$	0	0
$923$	− 27.3137i	− 0.899042i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	−16.6274	−0.545528	−0.272764	−	0.962081i	$-0.587938\pi$
$929$	−16.6274	−0.545528	−0.272764	+	0.962081i	$0.587938\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 8.97056i	− 0.293683i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 25.1127i	− 0.820396i	−0.911996	−	0.410198i	$-0.865460\pi$
$937$	− 25.1127i	− 0.820396i	0.911996	−	0.410198i	$-0.134540\pi$
$938$	0	0
$939$	−5.31371	−0.173406
$940$	0	0
$941$	−17.3137	−0.564411	−0.282205	−	0.959354i	$-0.591066\pi$
$941$	−17.3137	−0.564411	−0.282205	+	0.959354i	$0.591066\pi$
$942$	0	0
$943$	65.9411i	2.14734i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.9706i	0.421487i	0.977541	+	0.210743i	$0.0675884\pi$
$947$	12.9706i	0.421487i	−0.977541	+	0.210743i	$0.932412\pi$
$948$	0	0
$949$	−15.3137	−0.497104
$950$	0	0
$951$	20.6274	0.668890
$952$	0	0
$953$	39.4558i	1.27810i	0.769165	+	0.639050i	$0.220672\pi$
$953$	39.4558i	1.27810i	−0.769165	+	0.639050i	$0.779328\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 3.65685i	− 0.118209i
$958$	0	0
$959$	5.65685	0.182669
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	− 8.48528i	− 0.273434i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	38.1421i	1.22657i	0.789862	+	0.613284i	$0.210152\pi$
$967$	38.1421i	1.22657i	−0.789862	+	0.613284i	$0.789848\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37.9411	−1.21759	−0.608794	−	0.793328i	$-0.708347\pi$
$971$	−37.9411	−1.21759	−0.608794	+	0.793328i	$0.708347\pi$
$972$	0	0
$973$	16.0000i	0.512936i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 17.3137i	− 0.553915i	−0.960882	−	0.276957i	$-0.910674\pi$
$977$	− 17.3137i	− 0.553915i	0.960882	−	0.276957i	$-0.0893261\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	− 9.94113i	− 0.317073i	−0.987353	−	0.158536i	$-0.949323\pi$
$983$	− 9.94113i	− 0.317073i	0.987353	−	0.158536i	$-0.0506775\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 16.0000i	− 0.509286i
$988$	0	0
$989$	70.6274	2.24582
$990$	0	0
$991$	21.6569	0.687953	0.343976	−	0.938978i	$-0.388226\pi$
$991$	21.6569	0.687953	0.343976	+	0.938978i	$0.388226\pi$
$992$	0	0
$993$	− 20.9706i	− 0.665481i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 31.4558i	− 0.996217i	−0.867115	−	0.498108i	$-0.834028\pi$
$997$	− 31.4558i	− 0.996217i	0.867115	−	0.498108i	$-0.165972\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6600.2.d.bb.1849.1		4
5.2	odd	4		6600.2.a.bh.1.2		2
5.3	odd	4		1320.2.a.q.1.1	✓	2
5.4	even	2	inner	6600.2.d.bb.1849.4		4
15.8	even	4		3960.2.a.x.1.1		2
20.3	even	4		2640.2.a.z.1.2		2
60.23	odd	4		7920.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.a.q.1.1	✓	2	5.3	odd	4
2640.2.a.z.1.2		2	20.3	even	4
3960.2.a.x.1.1		2	15.8	even	4
6600.2.a.bh.1.2		2	5.2	odd	4
6600.2.d.bb.1849.1		4	1.1	even	1	trivial
6600.2.d.bb.1849.4		4	5.4	even	2	inner
7920.2.a.bt.1.2		2	60.23	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6600.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -2.82843i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -2.82843i q^{7} -1.00000 q^{9} +1.00000 q^{11} -4.82843i q^{13} +4.82843i q^{17} -2.82843 q^{21} +5.65685i q^{23} +1.00000i q^{27} +3.65685 q^{29} +5.65685 q^{31} -1.00000i q^{33} -2.00000i q^{37} -4.82843 q^{39} +11.6569 q^{41} -12.4853i q^{43} +5.65685i q^{47} -1.00000 q^{49} +4.82843 q^{51} -11.6569i q^{53} +9.65685 q^{59} -7.65685 q^{61} +2.82843i q^{63} -9.65685i q^{67} +5.65685 q^{69} +5.65685 q^{71} -3.17157i q^{73} -2.82843i q^{77} +4.00000 q^{79} +1.00000 q^{81} -5.17157i q^{83} -3.65685i q^{87} -2.00000 q^{89} -13.6569 q^{91} -5.65685i q^{93} -11.6569i q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 4 q^{11} - 8 q^{29} - 8 q^{39} + 24 q^{41} - 4 q^{49} + 8 q^{51} + 16 q^{59} - 8 q^{61} + 16 q^{79} + 4 q^{81} - 8 q^{89} - 32 q^{91} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^9 + 4 * q^11 - 8 * q^29 - 8 * q^39 + 24 * q^41 - 4 * q^49 + 8 * q^51 + 16 * q^59 - 8 * q^61 + 16 * q^79 + 4 * q^81 - 8 * q^89 - 32 * q^91 - 4 * q^99

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