LMFDB - Embedded newform 3822.2.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3822,2,Mod(1,3822)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3822, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3822.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3822.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$30.5188236525$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3822.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} +3.00000 q^{11} -1.00000 q^{12} -1.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} -3.00000 q^{19} -2.00000 q^{20} -3.00000 q^{22} +1.00000 q^{24} -1.00000 q^{25} +1.00000 q^{26} -1.00000 q^{27} +3.00000 q^{29} -2.00000 q^{30} -4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{33} -1.00000 q^{34} +1.00000 q^{36} -2.00000 q^{37} +3.00000 q^{38} +1.00000 q^{39} +2.00000 q^{40} +2.00000 q^{41} +6.00000 q^{43} +3.00000 q^{44} -2.00000 q^{45} -9.00000 q^{47} -1.00000 q^{48} +1.00000 q^{50} -1.00000 q^{51} -1.00000 q^{52} +1.00000 q^{53} +1.00000 q^{54} -6.00000 q^{55} +3.00000 q^{57} -3.00000 q^{58} +11.0000 q^{59} +2.00000 q^{60} +11.0000 q^{61} +4.00000 q^{62} +1.00000 q^{64} +2.00000 q^{65} +3.00000 q^{66} -7.00000 q^{67} +1.00000 q^{68} +15.0000 q^{71} -1.00000 q^{72} -12.0000 q^{73} +2.00000 q^{74} +1.00000 q^{75} -3.00000 q^{76} -1.00000 q^{78} +2.00000 q^{79} -2.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -2.00000 q^{85} -6.00000 q^{86} -3.00000 q^{87} -3.00000 q^{88} +10.0000 q^{89} +2.00000 q^{90} +4.00000 q^{93} +9.00000 q^{94} +6.00000 q^{95} +1.00000 q^{96} -12.0000 q^{97} +3.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	1.00000	0.408248
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−1.00000	−0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	−1.00000	−0.235702
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	−3.00000	−0.639602
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	−1.00000	−0.200000
$26$	1.00000	0.196116
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	−2.00000	−0.365148
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	−1.00000	−0.176777
$33$	−3.00000	−0.522233
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	3.00000	0.486664
$39$	1.00000	0.160128
$40$	2.00000	0.316228
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	3.00000	0.452267
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	1.00000	0.141421
$51$	−1.00000	−0.140028
$52$	−1.00000	−0.138675
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	1.00000	0.136083
$55$	−6.00000	−0.809040
$56$	0	0
$57$	3.00000	0.397360
$58$	−3.00000	−0.393919
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	2.00000	0.258199
$61$	11.0000	1.40841	0.704203	−	0.709999i	$-0.251305\pi$
$61$	11.0000	1.40841	0.704203	+	0.709999i	$0.251305\pi$
$62$	4.00000	0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	2.00000	0.248069
$66$	3.00000	0.369274
$67$	−7.00000	−0.855186	−0.427593	−	0.903971i	$-0.640638\pi$
$67$	−7.00000	−0.855186	−0.427593	+	0.903971i	$0.640638\pi$
$68$	1.00000	0.121268
$69$	0	0
$70$	0	0
$71$	15.0000	1.78017	0.890086	−	0.455792i	$-0.150644\pi$
$71$	15.0000	1.78017	0.890086	+	0.455792i	$0.150644\pi$
$72$	−1.00000	−0.117851
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	−3.00000	−0.344124
$77$	0	0
$78$	−1.00000	−0.113228
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	−6.00000	−0.646997
$87$	−3.00000	−0.321634
$88$	−3.00000	−0.319801
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	0	0
$93$	4.00000	0.414781
$94$	9.00000	0.928279
$95$	6.00000	0.615587
$96$	1.00000	0.102062
$97$	−12.0000	−1.21842	−0.609208	−	0.793011i	$-0.708512\pi$
$97$	−12.0000	−1.21842	−0.609208	+	0.793011i	$0.708512\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	−1.00000	−0.100000
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	1.00000	0.0990148
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	−1.00000	−0.0962250
$109$	−8.00000	−0.766261	−0.383131	−	0.923694i	$-0.625154\pi$
$109$	−8.00000	−0.766261	−0.383131	+	0.923694i	$0.625154\pi$
$110$	6.00000	0.572078
$111$	2.00000	0.189832
$112$	0	0
$113$	3.00000	0.282216	0.141108	−	0.989994i	$-0.454933\pi$
$113$	3.00000	0.282216	0.141108	+	0.989994i	$0.454933\pi$
$114$	−3.00000	−0.280976
$115$	0	0
$116$	3.00000	0.278543
$117$	−1.00000	−0.0924500
$118$	−11.0000	−1.01263
$119$	0	0
$120$	−2.00000	−0.182574
$121$	−2.00000	−0.181818
$122$	−11.0000	−0.995893
$123$	−2.00000	−0.180334
$124$	−4.00000	−0.359211
$125$	12.0000	1.07331
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	−6.00000	−0.528271
$130$	−2.00000	−0.175412
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	−3.00000	−0.261116
$133$	0	0
$134$	7.00000	0.604708
$135$	2.00000	0.172133
$136$	−1.00000	−0.0857493
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	−15.0000	−1.25877
$143$	−3.00000	−0.250873
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	12.0000	0.993127
$147$	0	0
$148$	−2.00000	−0.164399
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	−1.00000	−0.0816497
$151$	−17.0000	−1.38344	−0.691720	−	0.722166i	$-0.743147\pi$
$151$	−17.0000	−1.38344	−0.691720	+	0.722166i	$0.743147\pi$
$152$	3.00000	0.243332
$153$	1.00000	0.0808452
$154$	0	0
$155$	8.00000	0.642575
$156$	1.00000	0.0800641
$157$	−11.0000	−0.877896	−0.438948	−	0.898513i	$-0.644649\pi$
$157$	−11.0000	−0.877896	−0.438948	+	0.898513i	$0.644649\pi$
$158$	−2.00000	−0.159111
$159$	−1.00000	−0.0793052
$160$	2.00000	0.158114
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−25.0000	−1.95815	−0.979076	−	0.203497i	$-0.934769\pi$
$163$	−25.0000	−1.95815	−0.979076	+	0.203497i	$0.934769\pi$
$164$	2.00000	0.156174
$165$	6.00000	0.467099
$166$	0	0
$167$	−19.0000	−1.47026	−0.735132	−	0.677924i	$-0.762880\pi$
$167$	−19.0000	−1.47026	−0.735132	+	0.677924i	$0.762880\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.00000	0.153393
$171$	−3.00000	−0.229416
$172$	6.00000	0.457496
$173$	7.00000	0.532200	0.266100	−	0.963945i	$-0.414265\pi$
$173$	7.00000	0.532200	0.266100	+	0.963945i	$0.414265\pi$
$174$	3.00000	0.227429
$175$	0	0
$176$	3.00000	0.226134
$177$	−11.0000	−0.826811
$178$	−10.0000	−0.749532
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	−2.00000	−0.149071
$181$	−5.00000	−0.371647	−0.185824	−	0.982583i	$-0.559495\pi$
$181$	−5.00000	−0.371647	−0.185824	+	0.982583i	$0.559495\pi$
$182$	0	0
$183$	−11.0000	−0.813143
$184$	0	0
$185$	4.00000	0.294086
$186$	−4.00000	−0.293294
$187$	3.00000	0.219382
$188$	−9.00000	−0.656392
$189$	0	0
$190$	−6.00000	−0.435286
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	12.0000	0.861550
$195$	−2.00000	−0.143223
$196$	0	0
$197$	8.00000	0.569976	0.284988	−	0.958531i	$-0.408010\pi$
$197$	8.00000	0.569976	0.284988	+	0.958531i	$0.408010\pi$
$198$	−3.00000	−0.213201
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	1.00000	0.0707107
$201$	7.00000	0.493742
$202$	14.0000	0.985037
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	−4.00000	−0.279372
$206$	−14.0000	−0.975426
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−9.00000	−0.622543
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	1.00000	0.0686803
$213$	−15.0000	−1.02778
$214$	4.00000	0.273434
$215$	−12.0000	−0.818393
$216$	1.00000	0.0680414
$217$	0	0
$218$	8.00000	0.541828
$219$	12.0000	0.810885
$220$	−6.00000	−0.404520
$221$	−1.00000	−0.0672673
$222$	−2.00000	−0.134231
$223$	−1.00000	−0.0669650	−0.0334825	−	0.999439i	$-0.510660\pi$
$223$	−1.00000	−0.0669650	−0.0334825	+	0.999439i	$0.510660\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−3.00000	−0.199557
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	3.00000	0.198680
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	0	0
$231$	0	0
$232$	−3.00000	−0.196960
$233$	7.00000	0.458585	0.229293	−	0.973358i	$-0.426359\pi$
$233$	7.00000	0.458585	0.229293	+	0.973358i	$0.426359\pi$
$234$	1.00000	0.0653720
$235$	18.0000	1.17419
$236$	11.0000	0.716039
$237$	−2.00000	−0.129914
$238$	0	0
$239$	−11.0000	−0.711531	−0.355765	−	0.934575i	$-0.615780\pi$
$239$	−11.0000	−0.711531	−0.355765	+	0.934575i	$0.615780\pi$
$240$	2.00000	0.129099
$241$	12.0000	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$241$	12.0000	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$242$	2.00000	0.128565
$243$	−1.00000	−0.0641500
$244$	11.0000	0.704203
$245$	0	0
$246$	2.00000	0.127515
$247$	3.00000	0.190885
$248$	4.00000	0.254000
$249$	0	0
$250$	−12.0000	−0.758947
$251$	−10.0000	−0.631194	−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000	−0.631194	−0.315597	+	0.948893i	$0.602205\pi$
$252$	0	0
$253$	0	0
$254$	−8.00000	−0.501965
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	6.00000	0.373544
$259$	0	0
$260$	2.00000	0.124035
$261$	3.00000	0.185695
$262$	12.0000	0.741362
$263$	10.0000	0.616626	0.308313	−	0.951285i	$-0.400236\pi$
$263$	10.0000	0.616626	0.308313	+	0.951285i	$0.400236\pi$
$264$	3.00000	0.184637
$265$	−2.00000	−0.122859
$266$	0	0
$267$	−10.0000	−0.611990
$268$	−7.00000	−0.427593
$269$	−21.0000	−1.28039	−0.640196	−	0.768211i	$-0.721147\pi$
$269$	−21.0000	−1.28039	−0.640196	+	0.768211i	$0.721147\pi$
$270$	−2.00000	−0.121716
$271$	15.0000	0.911185	0.455593	−	0.890188i	$-0.349427\pi$
$271$	15.0000	0.911185	0.455593	+	0.890188i	$0.349427\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	−18.0000	−1.08742
$275$	−3.00000	−0.180907
$276$	0	0
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	14.0000	0.839664
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	−9.00000	−0.535942
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	15.0000	0.890086
$285$	−6.00000	−0.355409
$286$	3.00000	0.177394
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−16.0000	−0.941176
$290$	6.00000	0.352332
$291$	12.0000	0.703452
$292$	−12.0000	−0.702247
$293$	−12.0000	−0.701047	−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000	−0.701047	−0.350524	+	0.936554i	$0.613996\pi$
$294$	0	0
$295$	−22.0000	−1.28089
$296$	2.00000	0.116248
$297$	−3.00000	−0.174078
$298$	0	0
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	17.0000	0.978240
$303$	14.0000	0.804279
$304$	−3.00000	−0.172062
$305$	−22.0000	−1.25972
$306$	−1.00000	−0.0571662
$307$	−15.0000	−0.856095	−0.428048	−	0.903756i	$-0.640798\pi$
$307$	−15.0000	−0.856095	−0.428048	+	0.903756i	$0.640798\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	−8.00000	−0.454369
$311$	14.0000	0.793867	0.396934	−	0.917847i	$-0.370074\pi$
$311$	14.0000	0.793867	0.396934	+	0.917847i	$0.370074\pi$
$312$	−1.00000	−0.0566139
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	11.0000	0.620766
$315$	0	0
$316$	2.00000	0.112509
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	1.00000	0.0560772
$319$	9.00000	0.503903
$320$	−2.00000	−0.111803
$321$	4.00000	0.223258
$322$	0	0
$323$	−3.00000	−0.166924
$324$	1.00000	0.0555556
$325$	1.00000	0.0554700
$326$	25.0000	1.38462
$327$	8.00000	0.442401
$328$	−2.00000	−0.110432
$329$	0	0
$330$	−6.00000	−0.330289
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	−2.00000	−0.109599
$334$	19.0000	1.03963
$335$	14.0000	0.764902
$336$	0	0
$337$	7.00000	0.381314	0.190657	−	0.981657i	$-0.438938\pi$
$337$	7.00000	0.381314	0.190657	+	0.981657i	$0.438938\pi$
$338$	−1.00000	−0.0543928
$339$	−3.00000	−0.162938
$340$	−2.00000	−0.108465
$341$	−12.0000	−0.649836
$342$	3.00000	0.162221
$343$	0	0
$344$	−6.00000	−0.323498
$345$	0	0
$346$	−7.00000	−0.376322
$347$	2.00000	0.107366	0.0536828	−	0.998558i	$-0.482904\pi$
$347$	2.00000	0.107366	0.0536828	+	0.998558i	$0.482904\pi$
$348$	−3.00000	−0.160817
$349$	20.0000	1.07058	0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000	1.07058	0.535288	+	0.844670i	$0.320203\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−3.00000	−0.159901
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	11.0000	0.584643
$355$	−30.0000	−1.59223
$356$	10.0000	0.529999
$357$	0	0
$358$	6.00000	0.317110
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	2.00000	0.105409
$361$	−10.0000	−0.526316
$362$	5.00000	0.262794
$363$	2.00000	0.104973
$364$	0	0
$365$	24.0000	1.25622
$366$	11.0000	0.574979
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	−4.00000	−0.207950
$371$	0	0
$372$	4.00000	0.207390
$373$	−11.0000	−0.569558	−0.284779	−	0.958593i	$-0.591920\pi$
$373$	−11.0000	−0.569558	−0.284779	+	0.958593i	$0.591920\pi$
$374$	−3.00000	−0.155126
$375$	−12.0000	−0.619677
$376$	9.00000	0.464140
$377$	−3.00000	−0.154508
$378$	0	0
$379$	36.0000	1.84920	0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000	1.84920	0.924598	+	0.380945i	$0.124401\pi$
$380$	6.00000	0.307794
$381$	−8.00000	−0.409852
$382$	12.0000	0.613973
$383$	−32.0000	−1.63512	−0.817562	−	0.575841i	$-0.804675\pi$
$383$	−32.0000	−1.63512	−0.817562	+	0.575841i	$0.804675\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	6.00000	0.304997
$388$	−12.0000	−0.609208
$389$	13.0000	0.659126	0.329563	−	0.944134i	$-0.393099\pi$
$389$	13.0000	0.659126	0.329563	+	0.944134i	$0.393099\pi$
$390$	2.00000	0.101274
$391$	0	0
$392$	0	0
$393$	12.0000	0.605320
$394$	−8.00000	−0.403034
$395$	−4.00000	−0.201262
$396$	3.00000	0.150756
$397$	−20.0000	−1.00377	−0.501886	−	0.864934i	$-0.667360\pi$
$397$	−20.0000	−1.00377	−0.501886	+	0.864934i	$0.667360\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	−7.00000	−0.349128
$403$	4.00000	0.199254
$404$	−14.0000	−0.696526
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	−6.00000	−0.297409
$408$	1.00000	0.0495074
$409$	−38.0000	−1.87898	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−38.0000	−1.87898	−0.939490	+	0.342578i	$0.888700\pi$
$410$	4.00000	0.197546
$411$	−18.0000	−0.887875
$412$	14.0000	0.689730
$413$	0	0
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	14.0000	0.685583
$418$	9.00000	0.440204
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	16.0000	0.778868
$423$	−9.00000	−0.437595
$424$	−1.00000	−0.0485643
$425$	−1.00000	−0.0485071
$426$	15.0000	0.726752
$427$	0	0
$428$	−4.00000	−0.193347
$429$	3.00000	0.144841
$430$	12.0000	0.578691
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	−1.00000	−0.0481125
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	−8.00000	−0.383131
$437$	0	0
$438$	−12.0000	−0.573382
$439$	16.0000	0.763638	0.381819	−	0.924237i	$-0.375298\pi$
$439$	16.0000	0.763638	0.381819	+	0.924237i	$0.375298\pi$
$440$	6.00000	0.286039
$441$	0	0
$442$	1.00000	0.0475651
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	2.00000	0.0949158
$445$	−20.0000	−0.948091
$446$	1.00000	0.0473514
$447$	0	0
$448$	0	0
$449$	−40.0000	−1.88772	−0.943858	−	0.330350i	$-0.892833\pi$
$449$	−40.0000	−1.88772	−0.943858	+	0.330350i	$0.892833\pi$
$450$	1.00000	0.0471405
$451$	6.00000	0.282529
$452$	3.00000	0.141108
$453$	17.0000	0.798730
$454$	−8.00000	−0.375459
$455$	0	0
$456$	−3.00000	−0.140488
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	2.00000	0.0934539
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−20.0000	−0.931493	−0.465746	−	0.884918i	$-0.654214\pi$
$461$	−20.0000	−0.931493	−0.465746	+	0.884918i	$0.654214\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	3.00000	0.139272
$465$	−8.00000	−0.370991
$466$	−7.00000	−0.324269
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	−1.00000	−0.0462250
$469$	0	0
$470$	−18.0000	−0.830278
$471$	11.0000	0.506853
$472$	−11.0000	−0.506316
$473$	18.0000	0.827641
$474$	2.00000	0.0918630
$475$	3.00000	0.137649
$476$	0	0
$477$	1.00000	0.0457869
$478$	11.0000	0.503128
$479$	15.0000	0.685367	0.342684	−	0.939451i	$-0.388664\pi$
$479$	15.0000	0.685367	0.342684	+	0.939451i	$0.388664\pi$
$480$	−2.00000	−0.0912871
$481$	2.00000	0.0911922
$482$	−12.0000	−0.546585
$483$	0	0
$484$	−2.00000	−0.0909091
$485$	24.0000	1.08978
$486$	1.00000	0.0453609
$487$	−33.0000	−1.49537	−0.747686	−	0.664052i	$-0.768835\pi$
$487$	−33.0000	−1.49537	−0.747686	+	0.664052i	$0.768835\pi$
$488$	−11.0000	−0.497947
$489$	25.0000	1.13054
$490$	0	0
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	−2.00000	−0.0901670
$493$	3.00000	0.135113
$494$	−3.00000	−0.134976
$495$	−6.00000	−0.269680
$496$	−4.00000	−0.179605
$497$	0	0
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	12.0000	0.536656
$501$	19.0000	0.848857
$502$	10.0000	0.446322
$503$	−20.0000	−0.891756	−0.445878	−	0.895094i	$-0.647108\pi$
$503$	−20.0000	−0.891756	−0.445878	+	0.895094i	$0.647108\pi$
$504$	0	0
$505$	28.0000	1.24598
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	8.00000	0.354943
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	−2.00000	−0.0885615
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	3.00000	0.132453
$514$	26.0000	1.14681
$515$	−28.0000	−1.23383
$516$	−6.00000	−0.264135
$517$	−27.0000	−1.18746
$518$	0	0
$519$	−7.00000	−0.307266
$520$	−2.00000	−0.0877058
$521$	−38.0000	−1.66481	−0.832405	−	0.554168i	$-0.813037\pi$
$521$	−38.0000	−1.66481	−0.832405	+	0.554168i	$0.813037\pi$
$522$	−3.00000	−0.131306
$523$	6.00000	0.262362	0.131181	−	0.991358i	$-0.458123\pi$
$523$	6.00000	0.262362	0.131181	+	0.991358i	$0.458123\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−10.0000	−0.436021
$527$	−4.00000	−0.174243
$528$	−3.00000	−0.130558
$529$	−23.0000	−1.00000
$530$	2.00000	0.0868744
$531$	11.0000	0.477359
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	10.0000	0.432742
$535$	8.00000	0.345870
$536$	7.00000	0.302354
$537$	6.00000	0.258919
$538$	21.0000	0.905374
$539$	0	0
$540$	2.00000	0.0860663
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	−15.0000	−0.644305
$543$	5.00000	0.214571
$544$	−1.00000	−0.0428746
$545$	16.0000	0.685365
$546$	0	0
$547$	44.0000	1.88130	0.940652	−	0.339372i	$-0.110215\pi$
$547$	44.0000	1.88130	0.940652	+	0.339372i	$0.110215\pi$
$548$	18.0000	0.768922
$549$	11.0000	0.469469
$550$	3.00000	0.127920
$551$	−9.00000	−0.383413
$552$	0	0
$553$	0	0
$554$	−5.00000	−0.212430
$555$	−4.00000	−0.169791
$556$	−14.0000	−0.593732
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	4.00000	0.169334
$559$	−6.00000	−0.253773
$560$	0	0
$561$	−3.00000	−0.126660
$562$	24.0000	1.01238
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	9.00000	0.378968
$565$	−6.00000	−0.252422
$566$	4.00000	0.168133
$567$	0	0
$568$	−15.0000	−0.629386
$569$	39.0000	1.63497	0.817483	−	0.575953i	$-0.195369\pi$
$569$	39.0000	1.63497	0.817483	+	0.575953i	$0.195369\pi$
$570$	6.00000	0.251312
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	−3.00000	−0.125436
$573$	12.0000	0.501307
$574$	0	0
$575$	0	0
$576$	1.00000	0.0416667
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	16.0000	0.665512
$579$	14.0000	0.581820
$580$	−6.00000	−0.249136
$581$	0	0
$582$	−12.0000	−0.497416
$583$	3.00000	0.124247
$584$	12.0000	0.496564
$585$	2.00000	0.0826898
$586$	12.0000	0.495715
$587$	39.0000	1.60970	0.804851	−	0.593477i	$-0.202245\pi$
$587$	39.0000	1.60970	0.804851	+	0.593477i	$0.202245\pi$
$588$	0	0
$589$	12.0000	0.494451
$590$	22.0000	0.905726
$591$	−8.00000	−0.329076
$592$	−2.00000	−0.0821995
$593$	−34.0000	−1.39621	−0.698106	−	0.715994i	$-0.745974\pi$
$593$	−34.0000	−1.39621	−0.698106	+	0.715994i	$0.745974\pi$
$594$	3.00000	0.123091
$595$	0	0
$596$	0	0
$597$	−12.0000	−0.491127
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	−1.00000	−0.0408248
$601$	−35.0000	−1.42768	−0.713840	−	0.700309i	$-0.753046\pi$
$601$	−35.0000	−1.42768	−0.713840	+	0.700309i	$0.753046\pi$
$602$	0	0
$603$	−7.00000	−0.285062
$604$	−17.0000	−0.691720
$605$	4.00000	0.162623
$606$	−14.0000	−0.568711
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	3.00000	0.121666
$609$	0	0
$610$	22.0000	0.890754
$611$	9.00000	0.364101
$612$	1.00000	0.0404226
$613$	−16.0000	−0.646234	−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000	−0.646234	−0.323117	+	0.946359i	$0.604731\pi$
$614$	15.0000	0.605351
$615$	4.00000	0.161296
$616$	0	0
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	14.0000	0.563163
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	8.00000	0.321288
$621$	0	0
$622$	−14.0000	−0.561349
$623$	0	0
$624$	1.00000	0.0400320
$625$	−19.0000	−0.760000
$626$	14.0000	0.559553
$627$	9.00000	0.359425
$628$	−11.0000	−0.438948
$629$	−2.00000	−0.0797452
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	−2.00000	−0.0795557
$633$	16.0000	0.635943
$634$	−18.0000	−0.714871
$635$	−16.0000	−0.634941
$636$	−1.00000	−0.0396526
$637$	0	0
$638$	−9.00000	−0.356313
$639$	15.0000	0.593391
$640$	2.00000	0.0790569
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	−4.00000	−0.157867
$643$	1.00000	0.0394362	0.0197181	−	0.999806i	$-0.493723\pi$
$643$	1.00000	0.0394362	0.0197181	+	0.999806i	$0.493723\pi$
$644$	0	0
$645$	12.0000	0.472500
$646$	3.00000	0.118033
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	−1.00000	−0.0392837
$649$	33.0000	1.29536
$650$	−1.00000	−0.0392232
$651$	0	0
$652$	−25.0000	−0.979076
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	−8.00000	−0.312825
$655$	24.0000	0.937758
$656$	2.00000	0.0780869
$657$	−12.0000	−0.468165
$658$	0	0
$659$	30.0000	1.16863	0.584317	−	0.811525i	$-0.301362\pi$
$659$	30.0000	1.16863	0.584317	+	0.811525i	$0.301362\pi$
$660$	6.00000	0.233550
$661$	38.0000	1.47803	0.739014	−	0.673690i	$-0.235292\pi$
$661$	38.0000	1.47803	0.739014	+	0.673690i	$0.235292\pi$
$662$	28.0000	1.08825
$663$	1.00000	0.0388368
$664$	0	0
$665$	0	0
$666$	2.00000	0.0774984
$667$	0	0
$668$	−19.0000	−0.735132
$669$	1.00000	0.0386622
$670$	−14.0000	−0.540867
$671$	33.0000	1.27395
$672$	0	0
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	−7.00000	−0.269630
$675$	1.00000	0.0384900
$676$	1.00000	0.0384615
$677$	1.00000	0.0384331	0.0192166	−	0.999815i	$-0.493883\pi$
$677$	1.00000	0.0384331	0.0192166	+	0.999815i	$0.493883\pi$
$678$	3.00000	0.115214
$679$	0	0
$680$	2.00000	0.0766965
$681$	−8.00000	−0.306561
$682$	12.0000	0.459504
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	−3.00000	−0.114708
$685$	−36.0000	−1.37549
$686$	0	0
$687$	2.00000	0.0763048
$688$	6.00000	0.228748
$689$	−1.00000	−0.0380970
$690$	0	0
$691$	35.0000	1.33146	0.665731	−	0.746191i	$-0.268120\pi$
$691$	35.0000	1.33146	0.665731	+	0.746191i	$0.268120\pi$
$692$	7.00000	0.266100
$693$	0	0
$694$	−2.00000	−0.0759190
$695$	28.0000	1.06210
$696$	3.00000	0.113715
$697$	2.00000	0.0757554
$698$	−20.0000	−0.757011
$699$	−7.00000	−0.264764
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−1.00000	−0.0377426
$703$	6.00000	0.226294
$704$	3.00000	0.113067
$705$	−18.0000	−0.677919
$706$	18.0000	0.677439
$707$	0	0
$708$	−11.0000	−0.413405
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	30.0000	1.12588
$711$	2.00000	0.0750059
$712$	−10.0000	−0.374766
$713$	0	0
$714$	0	0
$715$	6.00000	0.224387
$716$	−6.00000	−0.224231
$717$	11.0000	0.410803
$718$	−16.0000	−0.597115
$719$	38.0000	1.41716	0.708580	−	0.705630i	$-0.249336\pi$
$719$	38.0000	1.41716	0.708580	+	0.705630i	$0.249336\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	10.0000	0.372161
$723$	−12.0000	−0.446285
$724$	−5.00000	−0.185824
$725$	−3.00000	−0.111417
$726$	−2.00000	−0.0742270
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−24.0000	−0.888280
$731$	6.00000	0.221918
$732$	−11.0000	−0.406572
$733$	−8.00000	−0.295487	−0.147743	−	0.989026i	$-0.547201\pi$
$733$	−8.00000	−0.295487	−0.147743	+	0.989026i	$0.547201\pi$
$734$	−18.0000	−0.664392
$735$	0	0
$736$	0	0
$737$	−21.0000	−0.773545
$738$	−2.00000	−0.0736210
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	4.00000	0.147043
$741$	−3.00000	−0.110208
$742$	0	0
$743$	−35.0000	−1.28403	−0.642013	−	0.766694i	$-0.721900\pi$
$743$	−35.0000	−1.28403	−0.642013	+	0.766694i	$0.721900\pi$
$744$	−4.00000	−0.146647
$745$	0	0
$746$	11.0000	0.402739
$747$	0	0
$748$	3.00000	0.109691
$749$	0	0
$750$	12.0000	0.438178
$751$	28.0000	1.02173	0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000	1.02173	0.510867	+	0.859660i	$0.329324\pi$
$752$	−9.00000	−0.328196
$753$	10.0000	0.364420
$754$	3.00000	0.109254
$755$	34.0000	1.23739
$756$	0	0
$757$	19.0000	0.690567	0.345283	−	0.938498i	$-0.387783\pi$
$757$	19.0000	0.690567	0.345283	+	0.938498i	$0.387783\pi$
$758$	−36.0000	−1.30758
$759$	0	0
$760$	−6.00000	−0.217643
$761$	28.0000	1.01500	0.507500	−	0.861652i	$-0.330570\pi$
$761$	28.0000	1.01500	0.507500	+	0.861652i	$0.330570\pi$
$762$	8.00000	0.289809
$763$	0	0
$764$	−12.0000	−0.434145
$765$	−2.00000	−0.0723102
$766$	32.0000	1.15621
$767$	−11.0000	−0.397187
$768$	−1.00000	−0.0360844
$769$	−4.00000	−0.144244	−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000	−0.144244	−0.0721218	+	0.997396i	$0.522977\pi$
$770$	0	0
$771$	26.0000	0.936367
$772$	−14.0000	−0.503871
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	−6.00000	−0.215666
$775$	4.00000	0.143684
$776$	12.0000	0.430775
$777$	0	0
$778$	−13.0000	−0.466073
$779$	−6.00000	−0.214972
$780$	−2.00000	−0.0716115
$781$	45.0000	1.61023
$782$	0	0
$783$	−3.00000	−0.107211
$784$	0	0
$785$	22.0000	0.785214
$786$	−12.0000	−0.428026
$787$	7.00000	0.249523	0.124762	−	0.992187i	$-0.460183\pi$
$787$	7.00000	0.249523	0.124762	+	0.992187i	$0.460183\pi$
$788$	8.00000	0.284988
$789$	−10.0000	−0.356009
$790$	4.00000	0.142314
$791$	0	0
$792$	−3.00000	−0.106600
$793$	−11.0000	−0.390621
$794$	20.0000	0.709773
$795$	2.00000	0.0709327
$796$	12.0000	0.425329
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	−9.00000	−0.318397
$800$	1.00000	0.0353553
$801$	10.0000	0.353333
$802$	−12.0000	−0.423735
$803$	−36.0000	−1.27041
$804$	7.00000	0.246871
$805$	0	0
$806$	−4.00000	−0.140894
$807$	21.0000	0.739235
$808$	14.0000	0.492518
$809$	7.00000	0.246107	0.123053	−	0.992400i	$-0.460731\pi$
$809$	7.00000	0.246107	0.123053	+	0.992400i	$0.460731\pi$
$810$	2.00000	0.0702728
$811$	12.0000	0.421377	0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000	0.421377	0.210688	+	0.977553i	$0.432429\pi$
$812$	0	0
$813$	−15.0000	−0.526073
$814$	6.00000	0.210300
$815$	50.0000	1.75142
$816$	−1.00000	−0.0350070
$817$	−18.0000	−0.629740
$818$	38.0000	1.32864
$819$	0	0
$820$	−4.00000	−0.139686
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	18.0000	0.627822
$823$	26.0000	0.906303	0.453152	−	0.891434i	$-0.350300\pi$
$823$	26.0000	0.906303	0.453152	+	0.891434i	$0.350300\pi$
$824$	−14.0000	−0.487713
$825$	3.00000	0.104447
$826$	0	0
$827$	−3.00000	−0.104320	−0.0521601	−	0.998639i	$-0.516611\pi$
$827$	−3.00000	−0.104320	−0.0521601	+	0.998639i	$0.516611\pi$
$828$	0	0
$829$	53.0000	1.84077	0.920383	−	0.391018i	$-0.127877\pi$
$829$	53.0000	1.84077	0.920383	+	0.391018i	$0.127877\pi$
$830$	0	0
$831$	−5.00000	−0.173448
$832$	−1.00000	−0.0346688
$833$	0	0
$834$	−14.0000	−0.484780
$835$	38.0000	1.31504
$836$	−9.00000	−0.311272
$837$	4.00000	0.138260
$838$	6.00000	0.207267
$839$	23.0000	0.794048	0.397024	−	0.917808i	$-0.370043\pi$
$839$	23.0000	0.794048	0.397024	+	0.917808i	$0.370043\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	24.0000	0.826604
$844$	−16.0000	−0.550743
$845$	−2.00000	−0.0688021
$846$	9.00000	0.309426
$847$	0	0
$848$	1.00000	0.0343401
$849$	4.00000	0.137280
$850$	1.00000	0.0342997
$851$	0	0
$852$	−15.0000	−0.513892
$853$	−32.0000	−1.09566	−0.547830	−	0.836590i	$-0.684546\pi$
$853$	−32.0000	−1.09566	−0.547830	+	0.836590i	$0.684546\pi$
$854$	0	0
$855$	6.00000	0.205196
$856$	4.00000	0.136717
$857$	7.00000	0.239115	0.119558	−	0.992827i	$-0.461852\pi$
$857$	7.00000	0.239115	0.119558	+	0.992827i	$0.461852\pi$
$858$	−3.00000	−0.102418
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	−12.0000	−0.409197
$861$	0	0
$862$	40.0000	1.36241
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	1.00000	0.0340207
$865$	−14.0000	−0.476014
$866$	11.0000	0.373795
$867$	16.0000	0.543388
$868$	0	0
$869$	6.00000	0.203536
$870$	−6.00000	−0.203419
$871$	7.00000	0.237186
$872$	8.00000	0.270914
$873$	−12.0000	−0.406138
$874$	0	0
$875$	0	0
$876$	12.0000	0.405442
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	−16.0000	−0.539974
$879$	12.0000	0.404750
$880$	−6.00000	−0.202260
$881$	−10.0000	−0.336909	−0.168454	−	0.985709i	$-0.553878\pi$
$881$	−10.0000	−0.336909	−0.168454	+	0.985709i	$0.553878\pi$
$882$	0	0
$883$	−14.0000	−0.471138	−0.235569	−	0.971858i	$-0.575695\pi$
$883$	−14.0000	−0.471138	−0.235569	+	0.971858i	$0.575695\pi$
$884$	−1.00000	−0.0336336
$885$	22.0000	0.739522
$886$	−6.00000	−0.201574
$887$	−6.00000	−0.201460	−0.100730	−	0.994914i	$-0.532118\pi$
$887$	−6.00000	−0.201460	−0.100730	+	0.994914i	$0.532118\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	20.0000	0.670402
$891$	3.00000	0.100504
$892$	−1.00000	−0.0334825
$893$	27.0000	0.903521
$894$	0	0
$895$	12.0000	0.401116
$896$	0	0
$897$	0	0
$898$	40.0000	1.33482
$899$	−12.0000	−0.400222
$900$	−1.00000	−0.0333333
$901$	1.00000	0.0333148
$902$	−6.00000	−0.199778
$903$	0	0
$904$	−3.00000	−0.0997785
$905$	10.0000	0.332411
$906$	−17.0000	−0.564787
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	8.00000	0.265489
$909$	−14.0000	−0.464351
$910$	0	0
$911$	−14.0000	−0.463841	−0.231920	−	0.972735i	$-0.574501\pi$
$911$	−14.0000	−0.463841	−0.231920	+	0.972735i	$0.574501\pi$
$912$	3.00000	0.0993399
$913$	0	0
$914$	32.0000	1.05847
$915$	22.0000	0.727298
$916$	−2.00000	−0.0660819
$917$	0	0
$918$	1.00000	0.0330049
$919$	46.0000	1.51740	0.758700	−	0.651440i	$-0.225835\pi$
$919$	46.0000	1.51740	0.758700	+	0.651440i	$0.225835\pi$
$920$	0	0
$921$	15.0000	0.494267
$922$	20.0000	0.658665
$923$	−15.0000	−0.493731
$924$	0	0
$925$	2.00000	0.0657596
$926$	32.0000	1.05159
$927$	14.0000	0.459820
$928$	−3.00000	−0.0984798
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	8.00000	0.262330
$931$	0	0
$932$	7.00000	0.229293
$933$	−14.0000	−0.458339
$934$	0	0
$935$	−6.00000	−0.196221
$936$	1.00000	0.0326860
$937$	−13.0000	−0.424691	−0.212346	−	0.977195i	$-0.568110\pi$
$937$	−13.0000	−0.424691	−0.212346	+	0.977195i	$0.568110\pi$
$938$	0	0
$939$	14.0000	0.456873
$940$	18.0000	0.587095
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	−11.0000	−0.358399
$943$	0	0
$944$	11.0000	0.358020
$945$	0	0
$946$	−18.0000	−0.585230
$947$	−33.0000	−1.07236	−0.536178	−	0.844105i	$-0.680132\pi$
$947$	−33.0000	−1.07236	−0.536178	+	0.844105i	$0.680132\pi$
$948$	−2.00000	−0.0649570
$949$	12.0000	0.389536
$950$	−3.00000	−0.0973329
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−33.0000	−1.06897	−0.534487	−	0.845176i	$-0.679495\pi$
$953$	−33.0000	−1.06897	−0.534487	+	0.845176i	$0.679495\pi$
$954$	−1.00000	−0.0323762
$955$	24.0000	0.776622
$956$	−11.0000	−0.355765
$957$	−9.00000	−0.290929
$958$	−15.0000	−0.484628
$959$	0	0
$960$	2.00000	0.0645497
$961$	−15.0000	−0.483871
$962$	−2.00000	−0.0644826
$963$	−4.00000	−0.128898
$964$	12.0000	0.386494
$965$	28.0000	0.901352
$966$	0	0
$967$	21.0000	0.675314	0.337657	−	0.941269i	$-0.390366\pi$
$967$	21.0000	0.675314	0.337657	+	0.941269i	$0.390366\pi$
$968$	2.00000	0.0642824
$969$	3.00000	0.0963739
$970$	−24.0000	−0.770594
$971$	10.0000	0.320915	0.160458	−	0.987043i	$-0.448703\pi$
$971$	10.0000	0.320915	0.160458	+	0.987043i	$0.448703\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	33.0000	1.05739
$975$	−1.00000	−0.0320256
$976$	11.0000	0.352101
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	−25.0000	−0.799412
$979$	30.0000	0.958804
$980$	0	0
$981$	−8.00000	−0.255420
$982$	−22.0000	−0.702048
$983$	21.0000	0.669796	0.334898	−	0.942254i	$-0.391298\pi$
$983$	21.0000	0.669796	0.334898	+	0.942254i	$0.391298\pi$
$984$	2.00000	0.0637577
$985$	−16.0000	−0.509802
$986$	−3.00000	−0.0955395
$987$	0	0
$988$	3.00000	0.0954427
$989$	0	0
$990$	6.00000	0.190693
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	4.00000	0.127000
$993$	28.0000	0.888553
$994$	0	0
$995$	−24.0000	−0.760851
$996$	0	0
$997$	11.0000	0.348373	0.174187	−	0.984713i	$-0.444270\pi$
$997$	11.0000	0.348373	0.174187	+	0.984713i	$0.444270\pi$
$998$	−28.0000	−0.886325
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3822.2.a.c.1.1		1
7.2	even	3		546.2.i.g.235.1	yes	2
7.4	even	3		546.2.i.g.79.1	✓	2
7.6	odd	2		3822.2.a.q.1.1		1
21.2	odd	6		1638.2.j.b.235.1		2
21.11	odd	6		1638.2.j.b.1171.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.i.g.79.1	✓	2	7.4	even	3
546.2.i.g.235.1	yes	2	7.2	even	3
1638.2.j.b.235.1		2	21.2	odd	6
1638.2.j.b.1171.1		2	21.11	odd	6
3822.2.a.c.1.1		1	1.1	even	1	trivial
3822.2.a.q.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3822.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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