LMFDB - Embedded newform 400.4.c.e.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,4,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	400.c (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:	$23.6007640023$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.4.c.e.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.00000i q^{3} +6.00000i q^{7} -22.0000 q^{9} +O(q^{10})$	$q-7.00000i q^{3} +6.00000i q^{7} -22.0000 q^{9} +43.0000 q^{11} -28.0000i q^{13} -91.0000i q^{17} -35.0000 q^{19} +42.0000 q^{21} -162.000i q^{23} -35.0000i q^{27} -160.000 q^{29} -42.0000 q^{31} -301.000i q^{33} +314.000i q^{37} -196.000 q^{39} -203.000 q^{41} -92.0000i q^{43} +196.000i q^{47} +307.000 q^{49} -637.000 q^{51} +82.0000i q^{53} +245.000i q^{57} -280.000 q^{59} -518.000 q^{61} -132.000i q^{63} +141.000i q^{67} -1134.00 q^{69} -412.000 q^{71} -763.000i q^{73} +258.000i q^{77} +510.000 q^{79} -839.000 q^{81} -777.000i q^{83} +1120.00i q^{87} +945.000 q^{89} +168.000 q^{91} +294.000i q^{93} -1246.00i q^{97} -946.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 44 q^{9}+O(q^{10}) $ 2 * q - 44 * q^9	$ 2 q - 44 q^{9} + 86 q^{11} - 70 q^{19} + 84 q^{21} - 320 q^{29} - 84 q^{31} - 392 q^{39} - 406 q^{41} + 614 q^{49} - 1274 q^{51} - 560 q^{59} - 1036 q^{61} - 2268 q^{69} - 824 q^{71} + 1020 q^{79} - 1678 q^{81} + 1890 q^{89} + 336 q^{91} - 1892 q^{99}+O(q^{100}) $ 2 * q - 44 * q^9 + 86 * q^11 - 70 * q^19 + 84 * q^21 - 320 * q^29 - 84 * q^31 - 392 * q^39 - 406 * q^41 + 614 * q^49 - 1274 * q^51 - 560 * q^59 - 1036 * q^61 - 2268 * q^69 - 824 * q^71 + 1020 * q^79 - 1678 * q^81 + 1890 * q^89 + 336 * q^91 - 1892 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 7.00000i	− 1.34715i	−0.739119	−	0.673575i	$-0.764758\pi$
$3$	− 7.00000i	− 1.34715i	0.739119	−	0.673575i	$-0.235242\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	6.00000i	0.323970i	0.986793	+	0.161985i	$0.0517895\pi$
$7$	6.00000i	0.323970i	−0.986793	+	0.161985i	$0.948210\pi$
$8$	0	0
$9$	−22.0000	−0.814815
$10$	0	0
$11$	43.0000	1.17864	0.589318	−	0.807901i	$-0.299397\pi$
$11$	43.0000	1.17864	0.589318	+	0.807901i	$0.299397\pi$
$12$	0	0
$13$	− 28.0000i	− 0.597369i	−0.954352	−	0.298685i	$-0.903452\pi$
$13$	− 28.0000i	− 0.597369i	0.954352	−	0.298685i	$-0.0965479\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 91.0000i	− 1.29828i	−0.760669	−	0.649139i	$-0.775129\pi$
$17$	− 91.0000i	− 1.29828i	0.760669	−	0.649139i	$-0.224871\pi$
$18$	0	0
$19$	−35.0000	−0.422608	−0.211304	−	0.977420i	$-0.567771\pi$
$19$	−35.0000	−0.422608	−0.211304	+	0.977420i	$0.567771\pi$
$20$	0	0
$21$	42.0000	0.436436
$22$	0	0
$23$	− 162.000i	− 1.46867i	−0.678789	−	0.734333i	$-0.737495\pi$
$23$	− 162.000i	− 1.46867i	0.678789	−	0.734333i	$-0.262505\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 35.0000i	− 0.249472i
$28$	0	0
$29$	−160.000	−1.02453	−0.512263	−	0.858829i	$-0.671193\pi$
$29$	−160.000	−1.02453	−0.512263	+	0.858829i	$0.671193\pi$
$30$	0	0
$31$	−42.0000	−0.243336	−0.121668	−	0.992571i	$-0.538824\pi$
$31$	−42.0000	−0.243336	−0.121668	+	0.992571i	$0.538824\pi$
$32$	0	0
$33$	− 301.000i	− 1.58780i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	314.000i	1.39517i	0.716502	+	0.697585i	$0.245742\pi$
$37$	314.000i	1.39517i	−0.716502	+	0.697585i	$0.754258\pi$
$38$	0	0
$39$	−196.000	−0.804747
$40$	0	0
$41$	−203.000	−0.773251	−0.386625	−	0.922237i	$-0.626359\pi$
$41$	−203.000	−0.773251	−0.386625	+	0.922237i	$0.626359\pi$
$42$	0	0
$43$	− 92.0000i	− 0.326276i	−0.986603	−	0.163138i	$-0.947838\pi$
$43$	− 92.0000i	− 0.326276i	0.986603	−	0.163138i	$-0.0521616\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	196.000i	0.608288i	0.952626	+	0.304144i	$0.0983704\pi$
$47$	196.000i	0.608288i	−0.952626	+	0.304144i	$0.901630\pi$
$48$	0	0
$49$	307.000	0.895044
$50$	0	0
$51$	−637.000	−1.74898
$52$	0	0
$53$	82.0000i	0.212520i	0.994338	+	0.106260i	$0.0338876\pi$
$53$	82.0000i	0.212520i	−0.994338	+	0.106260i	$0.966112\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	245.000i	0.569317i
$58$	0	0
$59$	−280.000	−0.617846	−0.308923	−	0.951087i	$-0.599968\pi$
$59$	−280.000	−0.617846	−0.308923	+	0.951087i	$0.599968\pi$
$60$	0	0
$61$	−518.000	−1.08726	−0.543632	−	0.839324i	$-0.682951\pi$
$61$	−518.000	−1.08726	−0.543632	+	0.839324i	$0.682951\pi$
$62$	0	0
$63$	− 132.000i	− 0.263975i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	141.000i	0.257103i	0.991703	+	0.128551i	$0.0410327\pi$
$67$	141.000i	0.257103i	−0.991703	+	0.128551i	$0.958967\pi$
$68$	0	0
$69$	−1134.00	−1.97852
$70$	0	0
$71$	−412.000	−0.688668	−0.344334	−	0.938847i	$-0.611895\pi$
$71$	−412.000	−0.688668	−0.344334	+	0.938847i	$0.611895\pi$
$72$	0	0
$73$	− 763.000i	− 1.22332i	−0.791121	−	0.611660i	$-0.790502\pi$
$73$	− 763.000i	− 1.22332i	0.791121	−	0.611660i	$-0.209498\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	258.000i	0.381842i
$78$	0	0
$79$	510.000	0.726323	0.363161	−	0.931726i	$-0.381697\pi$
$79$	510.000	0.726323	0.363161	+	0.931726i	$0.381697\pi$
$80$	0	0
$81$	−839.000	−1.15089
$82$	0	0
$83$	− 777.000i	− 1.02755i	−0.857924	−	0.513776i	$-0.828246\pi$
$83$	− 777.000i	− 1.02755i	0.857924	−	0.513776i	$-0.171754\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1120.00i	1.38019i
$88$	0	0
$89$	945.000	1.12550	0.562752	−	0.826626i	$-0.309743\pi$
$89$	945.000	1.12550	0.562752	+	0.826626i	$0.309743\pi$
$90$	0	0
$91$	168.000	0.193530
$92$	0	0
$93$	294.000i	0.327811i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1246.00i	− 1.30425i	−0.758112	−	0.652124i	$-0.773878\pi$
$97$	− 1246.00i	− 1.30425i	0.758112	−	0.652124i	$-0.226122\pi$
$98$	0	0
$99$	−946.000	−0.960369
$100$	0	0
$101$	1302.00	1.28271	0.641356	−	0.767244i	$-0.278372\pi$
$101$	1302.00	1.28271	0.641356	+	0.767244i	$0.278372\pi$
$102$	0	0
$103$	− 532.000i	− 0.508927i	−0.967082	−	0.254464i	$-0.918101\pi$
$103$	− 532.000i	− 0.508927i	0.967082	−	0.254464i	$-0.0818989\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1269.00i	− 1.14653i	−0.819370	−	0.573266i	$-0.805676\pi$
$107$	− 1269.00i	− 1.14653i	0.819370	−	0.573266i	$-0.194324\pi$
$108$	0	0
$109$	−1070.00	−0.940251	−0.470126	−	0.882599i	$-0.655791\pi$
$109$	−1070.00	−0.940251	−0.470126	+	0.882599i	$0.655791\pi$
$110$	0	0
$111$	2198.00	1.87950
$112$	0	0
$113$	− 503.000i	− 0.418746i	−0.977836	−	0.209373i	$-0.932858\pi$
$113$	− 503.000i	− 0.418746i	0.977836	−	0.209373i	$-0.0671422\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	616.000i	0.486745i
$118$	0	0
$119$	546.000	0.420603
$120$	0	0
$121$	518.000	0.389181
$122$	0	0
$123$	1421.00i	1.04169i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 874.000i	− 0.610669i	−0.952245	−	0.305334i	$-0.901232\pi$
$127$	− 874.000i	− 0.610669i	0.952245	−	0.305334i	$-0.0987683\pi$
$128$	0	0
$129$	−644.000	−0.439543
$130$	0	0
$131$	−1092.00	−0.728309	−0.364155	−	0.931339i	$-0.618642\pi$
$131$	−1092.00	−0.728309	−0.364155	+	0.931339i	$0.618642\pi$
$132$	0	0
$133$	− 210.000i	− 0.136912i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 411.000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 411.000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	−595.000	−0.363074	−0.181537	−	0.983384i	$-0.558107\pi$
$139$	−595.000	−0.363074	−0.181537	+	0.983384i	$0.558107\pi$
$140$	0	0
$141$	1372.00	0.819456
$142$	0	0
$143$	− 1204.00i	− 0.704081i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2149.00i	− 1.20576i
$148$	0	0
$149$	3200.00	1.75942	0.879712	−	0.475507i	$-0.157735\pi$
$149$	3200.00	1.75942	0.879712	+	0.475507i	$0.157735\pi$
$150$	0	0
$151$	−202.000	−0.108864	−0.0544322	−	0.998517i	$-0.517335\pi$
$151$	−202.000	−0.108864	−0.0544322	+	0.998517i	$0.517335\pi$
$152$	0	0
$153$	2002.00i	1.05786i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 406.000i	− 0.206384i	−0.994661	−	0.103192i	$-0.967094\pi$
$157$	− 406.000i	− 0.206384i	0.994661	−	0.103192i	$-0.0329057\pi$
$158$	0	0
$159$	574.000	0.286297
$160$	0	0
$161$	972.000	0.475803
$162$	0	0
$163$	3803.00i	1.82745i	0.406336	+	0.913724i	$0.366806\pi$
$163$	3803.00i	1.82745i	−0.406336	+	0.913724i	$0.633194\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4116.00i	1.90722i	0.301046	+	0.953610i	$0.402664\pi$
$167$	4116.00i	1.90722i	−0.301046	+	0.953610i	$0.597336\pi$
$168$	0	0
$169$	1413.00	0.643150
$170$	0	0
$171$	770.000	0.344347
$172$	0	0
$173$	1512.00i	0.664481i	0.943195	+	0.332241i	$0.107805\pi$
$173$	1512.00i	0.664481i	−0.943195	+	0.332241i	$0.892195\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1960.00i	0.832331i
$178$	0	0
$179$	2585.00	1.07940	0.539698	−	0.841859i	$-0.318538\pi$
$179$	2585.00	1.07940	0.539698	+	0.841859i	$0.318538\pi$
$180$	0	0
$181$	−2758.00	−1.13260	−0.566300	−	0.824199i	$-0.691626\pi$
$181$	−2758.00	−1.13260	−0.566300	+	0.824199i	$0.691626\pi$
$182$	0	0
$183$	3626.00i	1.46471i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3913.00i	− 1.53020i
$188$	0	0
$189$	210.000	0.0808214
$190$	0	0
$191$	2378.00	0.900869	0.450435	−	0.892809i	$-0.351269\pi$
$191$	2378.00	0.900869	0.450435	+	0.892809i	$0.351269\pi$
$192$	0	0
$193$	3067.00i	1.14387i	0.820298	+	0.571937i	$0.193808\pi$
$193$	3067.00i	1.14387i	−0.820298	+	0.571937i	$0.806192\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2346.00i	− 0.848455i	−0.905556	−	0.424227i	$-0.860546\pi$
$197$	− 2346.00i	− 0.848455i	0.905556	−	0.424227i	$-0.139454\pi$
$198$	0	0
$199$	4900.00	1.74549	0.872743	−	0.488180i	$-0.162339\pi$
$199$	4900.00	1.74549	0.872743	+	0.488180i	$0.162339\pi$
$200$	0	0
$201$	987.000	0.346356
$202$	0	0
$203$	− 960.000i	− 0.331915i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3564.00i	1.19669i
$208$	0	0
$209$	−1505.00	−0.498101
$210$	0	0
$211$	−4307.00	−1.40524	−0.702621	−	0.711564i	$-0.747987\pi$
$211$	−4307.00	−1.40524	−0.702621	+	0.711564i	$0.747987\pi$
$212$	0	0
$213$	2884.00i	0.927739i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 252.000i	− 0.0788335i
$218$	0	0
$219$	−5341.00	−1.64800
$220$	0	0
$221$	−2548.00	−0.775552
$222$	0	0
$223$	− 2212.00i	− 0.664244i	−0.943236	−	0.332122i	$-0.892235\pi$
$223$	− 2212.00i	− 0.664244i	0.943236	−	0.332122i	$-0.107765\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	476.000i	0.139177i	0.997576	+	0.0695886i	$0.0221687\pi$
$227$	476.000i	0.139177i	−0.997576	+	0.0695886i	$0.977831\pi$
$228$	0	0
$229$	2940.00	0.848387	0.424194	−	0.905572i	$-0.360558\pi$
$229$	2940.00	0.848387	0.424194	+	0.905572i	$0.360558\pi$
$230$	0	0
$231$	1806.00	0.514399
$232$	0	0
$233$	1002.00i	0.281730i	0.990029	+	0.140865i	$0.0449884\pi$
$233$	1002.00i	0.281730i	−0.990029	+	0.140865i	$0.955012\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 3570.00i	− 0.978466i
$238$	0	0
$239$	2480.00	0.671204	0.335602	−	0.942004i	$-0.391060\pi$
$239$	2480.00	0.671204	0.335602	+	0.942004i	$0.391060\pi$
$240$	0	0
$241$	1897.00	0.507039	0.253520	−	0.967330i	$-0.418412\pi$
$241$	1897.00	0.507039	0.253520	+	0.967330i	$0.418412\pi$
$242$	0	0
$243$	4928.00i	1.30095i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	980.000i	0.252453i
$248$	0	0
$249$	−5439.00	−1.38427
$250$	0	0
$251$	2373.00	0.596743	0.298371	−	0.954450i	$-0.403557\pi$
$251$	2373.00	0.596743	0.298371	+	0.954450i	$0.403557\pi$
$252$	0	0
$253$	− 6966.00i	− 1.73102i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4494.00i	1.09077i	0.838185	+	0.545385i	$0.183617\pi$
$257$	4494.00i	1.09077i	−0.838185	+	0.545385i	$0.816383\pi$
$258$	0	0
$259$	−1884.00	−0.451993
$260$	0	0
$261$	3520.00	0.834799
$262$	0	0
$263$	− 722.000i	− 0.169279i	−0.996412	−	0.0846396i	$-0.973026\pi$
$263$	− 722.000i	− 0.169279i	0.996412	−	0.0846396i	$-0.0269739\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 6615.00i	− 1.51622i
$268$	0	0
$269$	6160.00	1.39621	0.698107	−	0.715993i	$-0.254026\pi$
$269$	6160.00	1.39621	0.698107	+	0.715993i	$0.254026\pi$
$270$	0	0
$271$	7238.00	1.62243	0.811213	−	0.584751i	$-0.198808\pi$
$271$	7238.00	1.62243	0.811213	+	0.584751i	$0.198808\pi$
$272$	0	0
$273$	− 1176.00i	− 0.260713i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1776.00i	− 0.385233i	−0.981274	−	0.192616i	$-0.938303\pi$
$277$	− 1776.00i	− 0.385233i	0.981274	−	0.192616i	$-0.0616973\pi$
$278$	0	0
$279$	924.000	0.198274
$280$	0	0
$281$	4542.00	0.964246	0.482123	−	0.876104i	$-0.339866\pi$
$281$	4542.00	0.964246	0.482123	+	0.876104i	$0.339866\pi$
$282$	0	0
$283$	− 7077.00i	− 1.48652i	−0.669005	−	0.743258i	$-0.733280\pi$
$283$	− 7077.00i	− 1.48652i	0.669005	−	0.743258i	$-0.266720\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1218.00i	− 0.250510i
$288$	0	0
$289$	−3368.00	−0.685528
$290$	0	0
$291$	−8722.00	−1.75702
$292$	0	0
$293$	− 4158.00i	− 0.829054i	−0.910037	−	0.414527i	$-0.863947\pi$
$293$	− 4158.00i	− 0.829054i	0.910037	−	0.414527i	$-0.136053\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1505.00i	− 0.294037i
$298$	0	0
$299$	−4536.00	−0.877337
$300$	0	0
$301$	552.000	0.105703
$302$	0	0
$303$	− 9114.00i	− 1.72801i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2569.00i	− 0.477591i	−0.971070	−	0.238796i	$-0.923247\pi$
$307$	− 2569.00i	− 0.477591i	0.971070	−	0.238796i	$-0.0767526\pi$
$308$	0	0
$309$	−3724.00	−0.685602
$310$	0	0
$311$	−2982.00	−0.543710	−0.271855	−	0.962338i	$-0.587637\pi$
$311$	−2982.00	−0.543710	−0.271855	+	0.962338i	$0.587637\pi$
$312$	0	0
$313$	2422.00i	0.437379i	0.975795	+	0.218689i	$0.0701781\pi$
$313$	2422.00i	0.437379i	−0.975795	+	0.218689i	$0.929822\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9484.00i	1.68036i	0.542307	+	0.840181i	$0.317551\pi$
$317$	9484.00i	1.68036i	−0.542307	+	0.840181i	$0.682449\pi$
$318$	0	0
$319$	−6880.00	−1.20754
$320$	0	0
$321$	−8883.00	−1.54455
$322$	0	0
$323$	3185.00i	0.548663i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7490.00i	1.26666i
$328$	0	0
$329$	−1176.00	−0.197067
$330$	0	0
$331$	183.000	0.0303885	0.0151942	−	0.999885i	$-0.495163\pi$
$331$	183.000	0.0303885	0.0151942	+	0.999885i	$0.495163\pi$
$332$	0	0
$333$	− 6908.00i	− 1.13681i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 2861.00i	− 0.462459i	−0.972899	−	0.231229i	$-0.925725\pi$
$337$	− 2861.00i	− 0.462459i	0.972899	−	0.231229i	$-0.0742748\pi$
$338$	0	0
$339$	−3521.00	−0.564113
$340$	0	0
$341$	−1806.00	−0.286805
$342$	0	0
$343$	3900.00i	0.613936i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 629.000i	− 0.0973098i	−0.998816	−	0.0486549i	$-0.984507\pi$
$347$	− 629.000i	− 0.0973098i	0.998816	−	0.0486549i	$-0.0154934\pi$
$348$	0	0
$349$	−5950.00	−0.912597	−0.456298	−	0.889827i	$-0.650825\pi$
$349$	−5950.00	−0.912597	−0.456298	+	0.889827i	$0.650825\pi$
$350$	0	0
$351$	−980.000	−0.149027
$352$	0	0
$353$	− 11718.0i	− 1.76682i	−0.468604	−	0.883408i	$-0.655243\pi$
$353$	− 11718.0i	− 1.76682i	0.468604	−	0.883408i	$-0.344757\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 3822.00i	− 0.566615i
$358$	0	0
$359$	8070.00	1.18640	0.593201	−	0.805054i	$-0.297864\pi$
$359$	8070.00	1.18640	0.593201	+	0.805054i	$0.297864\pi$
$360$	0	0
$361$	−5634.00	−0.821403
$362$	0	0
$363$	− 3626.00i	− 0.524286i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8316.00i	1.18281i	0.806374	+	0.591406i	$0.201427\pi$
$367$	8316.00i	1.18281i	−0.806374	+	0.591406i	$0.798573\pi$
$368$	0	0
$369$	4466.00	0.630056
$370$	0	0
$371$	−492.000	−0.0688500
$372$	0	0
$373$	12062.0i	1.67439i	0.546906	+	0.837194i	$0.315805\pi$
$373$	12062.0i	1.67439i	−0.546906	+	0.837194i	$0.684195\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4480.00i	0.612021i
$378$	0	0
$379$	1735.00	0.235148	0.117574	−	0.993064i	$-0.462488\pi$
$379$	1735.00	0.235148	0.117574	+	0.993064i	$0.462488\pi$
$380$	0	0
$381$	−6118.00	−0.822663
$382$	0	0
$383$	− 7602.00i	− 1.01421i	−0.861883	−	0.507107i	$-0.830715\pi$
$383$	− 7602.00i	− 1.01421i	0.861883	−	0.507107i	$-0.169285\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2024.00i	0.265855i
$388$	0	0
$389$	−3030.00	−0.394928	−0.197464	−	0.980310i	$-0.563271\pi$
$389$	−3030.00	−0.394928	−0.197464	+	0.980310i	$0.563271\pi$
$390$	0	0
$391$	−14742.0	−1.90674
$392$	0	0
$393$	7644.00i	0.981142i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1204.00i	0.152209i	0.997100	+	0.0761046i	$0.0242483\pi$
$397$	1204.00i	0.152209i	−0.997100	+	0.0761046i	$0.975752\pi$
$398$	0	0
$399$	−1470.00	−0.184441
$400$	0	0
$401$	1077.00	0.134122	0.0670609	−	0.997749i	$-0.478638\pi$
$401$	1077.00	0.134122	0.0670609	+	0.997749i	$0.478638\pi$
$402$	0	0
$403$	1176.00i	0.145362i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	13502.0i	1.64440i
$408$	0	0
$409$	3955.00	0.478147	0.239074	−	0.971001i	$-0.423156\pi$
$409$	3955.00	0.478147	0.239074	+	0.971001i	$0.423156\pi$
$410$	0	0
$411$	−2877.00	−0.345285
$412$	0	0
$413$	− 1680.00i	− 0.200163i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4165.00i	0.489115i
$418$	0	0
$419$	6265.00	0.730466	0.365233	−	0.930916i	$-0.380989\pi$
$419$	6265.00	0.730466	0.365233	+	0.930916i	$0.380989\pi$
$420$	0	0
$421$	−3788.00	−0.438517	−0.219259	−	0.975667i	$-0.570364\pi$
$421$	−3788.00	−0.438517	−0.219259	+	0.975667i	$0.570364\pi$
$422$	0	0
$423$	− 4312.00i	− 0.495642i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 3108.00i	− 0.352240i
$428$	0	0
$429$	−8428.00	−0.948503
$430$	0	0
$431$	15258.0	1.70523	0.852613	−	0.522544i	$-0.175017\pi$
$431$	15258.0	1.70523	0.852613	+	0.522544i	$0.175017\pi$
$432$	0	0
$433$	− 13573.0i	− 1.50641i	−0.657784	−	0.753206i	$-0.728506\pi$
$433$	− 13573.0i	− 1.50641i	0.657784	−	0.753206i	$-0.271494\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5670.00i	0.620670i
$438$	0	0
$439$	−8120.00	−0.882794	−0.441397	−	0.897312i	$-0.645517\pi$
$439$	−8120.00	−0.882794	−0.441397	+	0.897312i	$0.645517\pi$
$440$	0	0
$441$	−6754.00	−0.729295
$442$	0	0
$443$	6183.00i	0.663122i	0.943434	+	0.331561i	$0.107575\pi$
$443$	6183.00i	0.663122i	−0.943434	+	0.331561i	$0.892425\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 22400.0i	− 2.37021i
$448$	0	0
$449$	1975.00	0.207586	0.103793	−	0.994599i	$-0.466902\pi$
$449$	1975.00	0.207586	0.103793	+	0.994599i	$0.466902\pi$
$450$	0	0
$451$	−8729.00	−0.911380
$452$	0	0
$453$	1414.00i	0.146657i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 11831.0i	− 1.21101i	−0.795842	−	0.605504i	$-0.792971\pi$
$457$	− 11831.0i	− 1.21101i	0.795842	−	0.605504i	$-0.207029\pi$
$458$	0	0
$459$	−3185.00	−0.323885
$460$	0	0
$461$	1932.00	0.195189	0.0975946	−	0.995226i	$-0.468885\pi$
$461$	1932.00	0.195189	0.0975946	+	0.995226i	$0.468885\pi$
$462$	0	0
$463$	9228.00i	0.926267i	0.886289	+	0.463133i	$0.153275\pi$
$463$	9228.00i	0.926267i	−0.886289	+	0.463133i	$0.846725\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	13916.0i	1.37892i	0.724324	+	0.689460i	$0.242152\pi$
$467$	13916.0i	1.37892i	−0.724324	+	0.689460i	$0.757848\pi$
$468$	0	0
$469$	−846.000	−0.0832935
$470$	0	0
$471$	−2842.00	−0.278031
$472$	0	0
$473$	− 3956.00i	− 0.384560i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1804.00i	− 0.173165i
$478$	0	0
$479$	2310.00	0.220348	0.110174	−	0.993912i	$-0.464859\pi$
$479$	2310.00	0.220348	0.110174	+	0.993912i	$0.464859\pi$
$480$	0	0
$481$	8792.00	0.833432
$482$	0	0
$483$	− 6804.00i	− 0.640979i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 17114.0i	− 1.59242i	−0.605019	−	0.796211i	$-0.706835\pi$
$487$	− 17114.0i	− 1.59242i	0.605019	−	0.796211i	$-0.293165\pi$
$488$	0	0
$489$	26621.0	2.46185
$490$	0	0
$491$	17228.0	1.58348	0.791740	−	0.610858i	$-0.209175\pi$
$491$	17228.0	1.58348	0.791740	+	0.610858i	$0.209175\pi$
$492$	0	0
$493$	14560.0i	1.33012i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 2472.00i	− 0.223107i
$498$	0	0
$499$	−12500.0	−1.12140	−0.560698	−	0.828020i	$-0.689467\pi$
$499$	−12500.0	−1.12140	−0.560698	+	0.828020i	$0.689467\pi$
$500$	0	0
$501$	28812.0	2.56931
$502$	0	0
$503$	868.000i	0.0769428i	0.999260	+	0.0384714i	$0.0122488\pi$
$503$	868.000i	0.0769428i	−0.999260	+	0.0384714i	$0.987751\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9891.00i	− 0.866420i
$508$	0	0
$509$	−13370.0	−1.16427	−0.582136	−	0.813091i	$-0.697783\pi$
$509$	−13370.0	−1.16427	−0.582136	+	0.813091i	$0.697783\pi$
$510$	0	0
$511$	4578.00	0.396319
$512$	0	0
$513$	1225.00i	0.105429i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8428.00i	0.716950i
$518$	0	0
$519$	10584.0	0.895156
$520$	0	0
$521$	21637.0	1.81945	0.909726	−	0.415210i	$-0.136292\pi$
$521$	21637.0	1.81945	0.909726	+	0.415210i	$0.136292\pi$
$522$	0	0
$523$	− 287.000i	− 0.0239955i	−0.999928	−	0.0119977i	$-0.996181\pi$
$523$	− 287.000i	− 0.0239955i	0.999928	−	0.0119977i	$-0.00381909\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3822.00i	0.315918i
$528$	0	0
$529$	−14077.0	−1.15698
$530$	0	0
$531$	6160.00	0.503430
$532$	0	0
$533$	5684.00i	0.461916i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 18095.0i	− 1.45411i
$538$	0	0
$539$	13201.0	1.05493
$540$	0	0
$541$	−5328.00	−0.423417	−0.211709	−	0.977333i	$-0.567903\pi$
$541$	−5328.00	−0.423417	−0.211709	+	0.977333i	$0.567903\pi$
$542$	0	0
$543$	19306.0i	1.52578i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	71.0000i	0.00554980i	0.999996	+	0.00277490i	$0.000883279\pi$
$547$	71.0000i	0.00554980i	−0.999996	+	0.00277490i	$0.999117\pi$
$548$	0	0
$549$	11396.0	0.885919
$550$	0	0
$551$	5600.00	0.432973
$552$	0	0
$553$	3060.00i	0.235306i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18444.0i	1.40305i	0.712646	+	0.701524i	$0.247497\pi$
$557$	18444.0i	1.40305i	−0.712646	+	0.701524i	$0.752503\pi$
$558$	0	0
$559$	−2576.00	−0.194907
$560$	0	0
$561$	−27391.0	−2.06141
$562$	0	0
$563$	− 672.000i	− 0.0503045i	−0.999684	−	0.0251522i	$-0.991993\pi$
$563$	− 672.000i	− 0.0503045i	0.999684	−	0.0251522i	$-0.00800705\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 5034.00i	− 0.372854i
$568$	0	0
$569$	10935.0	0.805657	0.402829	−	0.915275i	$-0.368027\pi$
$569$	10935.0	0.805657	0.402829	+	0.915275i	$0.368027\pi$
$570$	0	0
$571$	13588.0	0.995867	0.497934	−	0.867215i	$-0.334092\pi$
$571$	13588.0	0.995867	0.497934	+	0.867215i	$0.334092\pi$
$572$	0	0
$573$	− 16646.0i	− 1.21361i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 8701.00i	− 0.627777i	−0.949460	−	0.313889i	$-0.898368\pi$
$577$	− 8701.00i	− 0.627777i	0.949460	−	0.313889i	$-0.101632\pi$
$578$	0	0
$579$	21469.0	1.54097
$580$	0	0
$581$	4662.00	0.332896
$582$	0	0
$583$	3526.00i	0.250484i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11361.0i	0.798839i	0.916768	+	0.399420i	$0.130788\pi$
$587$	11361.0i	0.798839i	−0.916768	+	0.399420i	$0.869212\pi$
$588$	0	0
$589$	1470.00	0.102836
$590$	0	0
$591$	−16422.0	−1.14300
$592$	0	0
$593$	11417.0i	0.790624i	0.918547	+	0.395312i	$0.129364\pi$
$593$	11417.0i	0.790624i	−0.918547	+	0.395312i	$0.870636\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 34300.0i	− 2.35143i
$598$	0	0
$599$	−21050.0	−1.43586	−0.717930	−	0.696116i	$-0.754910\pi$
$599$	−21050.0	−1.43586	−0.717930	+	0.696116i	$0.754910\pi$
$600$	0	0
$601$	7427.00	0.504083	0.252041	−	0.967716i	$-0.418898\pi$
$601$	7427.00	0.504083	0.252041	+	0.967716i	$0.418898\pi$
$602$	0	0
$603$	− 3102.00i	− 0.209491i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4144.00i	− 0.277100i	−0.990355	−	0.138550i	$-0.955756\pi$
$607$	− 4144.00i	− 0.277100i	0.990355	−	0.138550i	$-0.0442442\pi$
$608$	0	0
$609$	−6720.00	−0.447140
$610$	0	0
$611$	5488.00	0.363373
$612$	0	0
$613$	30122.0i	1.98469i	0.123489	+	0.992346i	$0.460592\pi$
$613$	30122.0i	1.98469i	−0.123489	+	0.992346i	$0.539408\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11934.0i	0.778679i	0.921094	+	0.389339i	$0.127297\pi$
$617$	11934.0i	0.778679i	−0.921094	+	0.389339i	$0.872703\pi$
$618$	0	0
$619$	8540.00	0.554526	0.277263	−	0.960794i	$-0.410573\pi$
$619$	8540.00	0.554526	0.277263	+	0.960794i	$0.410573\pi$
$620$	0	0
$621$	−5670.00	−0.366392
$622$	0	0
$623$	5670.00i	0.364629i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	10535.0i	0.671017i
$628$	0	0
$629$	28574.0	1.81132
$630$	0	0
$631$	3158.00	0.199236	0.0996181	−	0.995026i	$-0.468238\pi$
$631$	3158.00	0.199236	0.0996181	+	0.995026i	$0.468238\pi$
$632$	0	0
$633$	30149.0i	1.89307i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 8596.00i	− 0.534672i
$638$	0	0
$639$	9064.00	0.561137
$640$	0	0
$641$	−4278.00	−0.263605	−0.131803	−	0.991276i	$-0.542076\pi$
$641$	−4278.00	−0.263605	−0.131803	+	0.991276i	$0.542076\pi$
$642$	0	0
$643$	11508.0i	0.705803i	0.935661	+	0.352901i	$0.114805\pi$
$643$	11508.0i	0.705803i	−0.935661	+	0.352901i	$0.885195\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 8204.00i	− 0.498505i	−0.968439	−	0.249252i	$-0.919815\pi$
$647$	− 8204.00i	− 0.498505i	0.968439	−	0.249252i	$-0.0801849\pi$
$648$	0	0
$649$	−12040.0	−0.728215
$650$	0	0
$651$	−1764.00	−0.106201
$652$	0	0
$653$	− 5518.00i	− 0.330683i	−0.986236	−	0.165342i	$-0.947127\pi$
$653$	− 5518.00i	− 0.330683i	0.986236	−	0.165342i	$-0.0528726\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	16786.0i	0.996780i
$658$	0	0
$659$	13295.0	0.785887	0.392944	−	0.919563i	$-0.371457\pi$
$659$	13295.0	0.785887	0.392944	+	0.919563i	$0.371457\pi$
$660$	0	0
$661$	−9968.00	−0.586551	−0.293276	−	0.956028i	$-0.594745\pi$
$661$	−9968.00	−0.586551	−0.293276	+	0.956028i	$0.594745\pi$
$662$	0	0
$663$	17836.0i	1.04479i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	25920.0i	1.50469i
$668$	0	0
$669$	−15484.0	−0.894837
$670$	0	0
$671$	−22274.0	−1.28149
$672$	0	0
$673$	− 15738.0i	− 0.901419i	−0.892671	−	0.450710i	$-0.851171\pi$
$673$	− 15738.0i	− 0.901419i	0.892671	−	0.450710i	$-0.148829\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	19824.0i	1.12540i	0.826660	+	0.562702i	$0.190238\pi$
$677$	19824.0i	1.12540i	−0.826660	+	0.562702i	$0.809762\pi$
$678$	0	0
$679$	7476.00	0.422537
$680$	0	0
$681$	3332.00	0.187493
$682$	0	0
$683$	11073.0i	0.620346i	0.950680	+	0.310173i	$0.100387\pi$
$683$	11073.0i	0.620346i	−0.950680	+	0.310173i	$0.899613\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 20580.0i	− 1.14291i
$688$	0	0
$689$	2296.00	0.126953
$690$	0	0
$691$	6503.00	0.358011	0.179006	−	0.983848i	$-0.442712\pi$
$691$	6503.00	0.358011	0.179006	+	0.983848i	$0.442712\pi$
$692$	0	0
$693$	− 5676.00i	− 0.311130i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18473.0i	1.00389i
$698$	0	0
$699$	7014.00	0.379533
$700$	0	0
$701$	−10148.0	−0.546768	−0.273384	−	0.961905i	$-0.588143\pi$
$701$	−10148.0	−0.546768	−0.273384	+	0.961905i	$0.588143\pi$
$702$	0	0
$703$	− 10990.0i	− 0.589610i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7812.00i	0.415559i
$708$	0	0
$709$	9980.00	0.528641	0.264321	−	0.964435i	$-0.414852\pi$
$709$	9980.00	0.528641	0.264321	+	0.964435i	$0.414852\pi$
$710$	0	0
$711$	−11220.0	−0.591818
$712$	0	0
$713$	6804.00i	0.357380i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 17360.0i	− 0.904214i
$718$	0	0
$719$	−27510.0	−1.42691	−0.713456	−	0.700700i	$-0.752871\pi$
$719$	−27510.0	−1.42691	−0.713456	+	0.700700i	$0.752871\pi$
$720$	0	0
$721$	3192.00	0.164877
$722$	0	0
$723$	− 13279.0i	− 0.683059i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 17024.0i	− 0.868480i	−0.900797	−	0.434240i	$-0.857017\pi$
$727$	− 17024.0i	− 0.868480i	0.900797	−	0.434240i	$-0.142983\pi$
$728$	0	0
$729$	11843.0	0.601687
$730$	0	0
$731$	−8372.00	−0.423597
$732$	0	0
$733$	− 34748.0i	− 1.75095i	−0.483263	−	0.875475i	$-0.660549\pi$
$733$	− 34748.0i	− 1.75095i	0.483263	−	0.875475i	$-0.339451\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6063.00i	0.303030i
$738$	0	0
$739$	−12020.0	−0.598326	−0.299163	−	0.954202i	$-0.596707\pi$
$739$	−12020.0	−0.598326	−0.299163	+	0.954202i	$0.596707\pi$
$740$	0	0
$741$	6860.00	0.340092
$742$	0	0
$743$	− 28642.0i	− 1.41423i	−0.707098	−	0.707115i	$-0.749996\pi$
$743$	− 28642.0i	− 1.41423i	0.707098	−	0.707115i	$-0.250004\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	17094.0i	0.837265i
$748$	0	0
$749$	7614.00	0.371441
$750$	0	0
$751$	−8752.00	−0.425253	−0.212627	−	0.977134i	$-0.568202\pi$
$751$	−8752.00	−0.425253	−0.212627	+	0.977134i	$0.568202\pi$
$752$	0	0
$753$	− 16611.0i	− 0.803902i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10256.0i	− 0.492418i	−0.969217	−	0.246209i	$-0.920815\pi$
$757$	− 10256.0i	− 0.492418i	0.969217	−	0.246209i	$-0.0791850\pi$
$758$	0	0
$759$	−48762.0	−2.33195
$760$	0	0
$761$	33957.0	1.61753	0.808765	−	0.588132i	$-0.200136\pi$
$761$	33957.0	1.61753	0.808765	+	0.588132i	$0.200136\pi$
$762$	0	0
$763$	− 6420.00i	− 0.304613i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7840.00i	0.369082i
$768$	0	0
$769$	−27965.0	−1.31137	−0.655685	−	0.755034i	$-0.727620\pi$
$769$	−27965.0	−1.31137	−0.655685	+	0.755034i	$0.727620\pi$
$770$	0	0
$771$	31458.0	1.46943
$772$	0	0
$773$	9912.00i	0.461203i	0.973048	+	0.230601i	$0.0740694\pi$
$773$	9912.00i	0.461203i	−0.973048	+	0.230601i	$0.925931\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	13188.0i	0.608902i
$778$	0	0
$779$	7105.00	0.326782
$780$	0	0
$781$	−17716.0	−0.811688
$782$	0	0
$783$	5600.00i	0.255591i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 25564.0i	− 1.15789i	−0.815367	−	0.578944i	$-0.803465\pi$
$787$	− 25564.0i	− 1.15789i	0.815367	−	0.578944i	$-0.196535\pi$
$788$	0	0
$789$	−5054.00	−0.228045
$790$	0	0
$791$	3018.00	0.135661
$792$	0	0
$793$	14504.0i	0.649498i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 12446.0i	− 0.553149i	−0.960992	−	0.276575i	$-0.910801\pi$
$797$	− 12446.0i	− 0.553149i	0.960992	−	0.276575i	$-0.0891993\pi$
$798$	0	0
$799$	17836.0	0.789728
$800$	0	0
$801$	−20790.0	−0.917077
$802$	0	0
$803$	− 32809.0i	− 1.44185i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 43120.0i	− 1.88091i
$808$	0	0
$809$	−33970.0	−1.47629	−0.738147	−	0.674640i	$-0.764299\pi$
$809$	−33970.0	−1.47629	−0.738147	+	0.674640i	$0.764299\pi$
$810$	0	0
$811$	−18732.0	−0.811060	−0.405530	−	0.914082i	$-0.632913\pi$
$811$	−18732.0	−0.811060	−0.405530	+	0.914082i	$0.632913\pi$
$812$	0	0
$813$	− 50666.0i	− 2.18565i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3220.00i	0.137887i
$818$	0	0
$819$	−3696.00	−0.157691
$820$	0	0
$821$	6162.00	0.261943	0.130972	−	0.991386i	$-0.458190\pi$
$821$	6162.00	0.261943	0.130972	+	0.991386i	$0.458190\pi$
$822$	0	0
$823$	25388.0i	1.07530i	0.843169	+	0.537649i	$0.180687\pi$
$823$	25388.0i	1.07530i	−0.843169	+	0.537649i	$0.819313\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25201.0i	1.05964i	0.848109	+	0.529821i	$0.177741\pi$
$827$	25201.0i	1.05964i	−0.848109	+	0.529821i	$0.822259\pi$
$828$	0	0
$829$	19740.0	0.827019	0.413509	−	0.910500i	$-0.364303\pi$
$829$	19740.0	0.827019	0.413509	+	0.910500i	$0.364303\pi$
$830$	0	0
$831$	−12432.0	−0.518967
$832$	0	0
$833$	− 27937.0i	− 1.16202i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1470.00i	0.0607057i
$838$	0	0
$839$	29680.0	1.22130	0.610648	−	0.791902i	$-0.290909\pi$
$839$	29680.0	1.22130	0.610648	+	0.791902i	$0.290909\pi$
$840$	0	0
$841$	1211.00	0.0496535
$842$	0	0
$843$	− 31794.0i	− 1.29898i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3108.00i	0.126083i
$848$	0	0
$849$	−49539.0	−2.00256
$850$	0	0
$851$	50868.0	2.04904
$852$	0	0
$853$	− 1218.00i	− 0.0488904i	−0.999701	−	0.0244452i	$-0.992218\pi$
$853$	− 1218.00i	− 0.0488904i	0.999701	−	0.0244452i	$-0.00778193\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 38731.0i	− 1.54379i	−0.635752	−	0.771894i	$-0.719310\pi$
$857$	− 38731.0i	− 1.54379i	0.635752	−	0.771894i	$-0.280690\pi$
$858$	0	0
$859$	−23555.0	−0.935607	−0.467803	−	0.883833i	$-0.654954\pi$
$859$	−23555.0	−0.935607	−0.467803	+	0.883833i	$0.654954\pi$
$860$	0	0
$861$	−8526.00	−0.337474
$862$	0	0
$863$	− 24872.0i	− 0.981058i	−0.871425	−	0.490529i	$-0.836804\pi$
$863$	− 24872.0i	− 0.981058i	0.871425	−	0.490529i	$-0.163196\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	23576.0i	0.923510i
$868$	0	0
$869$	21930.0	0.856069
$870$	0	0
$871$	3948.00	0.153585
$872$	0	0
$873$	27412.0i	1.06272i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	17124.0i	0.659335i	0.944097	+	0.329667i	$0.106937\pi$
$877$	17124.0i	0.659335i	−0.944097	+	0.329667i	$0.893063\pi$
$878$	0	0
$879$	−29106.0	−1.11686
$880$	0	0
$881$	−658.000	−0.0251630	−0.0125815	−	0.999921i	$-0.504005\pi$
$881$	−658.000	−0.0251630	−0.0125815	+	0.999921i	$0.504005\pi$
$882$	0	0
$883$	− 33727.0i	− 1.28540i	−0.766120	−	0.642698i	$-0.777815\pi$
$883$	− 33727.0i	− 1.28540i	0.766120	−	0.642698i	$-0.222185\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36036.0i	1.36412i	0.731298	+	0.682058i	$0.238915\pi$
$887$	36036.0i	1.36412i	−0.731298	+	0.682058i	$0.761085\pi$
$888$	0	0
$889$	5244.00	0.197838
$890$	0	0
$891$	−36077.0	−1.35648
$892$	0	0
$893$	− 6860.00i	− 0.257067i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	31752.0i	1.18190i
$898$	0	0
$899$	6720.00	0.249304
$900$	0	0
$901$	7462.00	0.275910
$902$	0	0
$903$	− 3864.00i	− 0.142399i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	39156.0i	1.43347i	0.697348	+	0.716733i	$0.254363\pi$
$907$	39156.0i	1.43347i	−0.697348	+	0.716733i	$0.745637\pi$
$908$	0	0
$909$	−28644.0	−1.04517
$910$	0	0
$911$	−43532.0	−1.58318	−0.791591	−	0.611051i	$-0.790747\pi$
$911$	−43532.0	−1.58318	−0.791591	+	0.611051i	$0.790747\pi$
$912$	0	0
$913$	− 33411.0i	− 1.21111i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 6552.00i	− 0.235950i
$918$	0	0
$919$	−28610.0	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$919$	−28610.0	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$920$	0	0
$921$	−17983.0	−0.643388
$922$	0	0
$923$	11536.0i	0.411389i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	11704.0i	0.414682i
$928$	0	0
$929$	24290.0	0.857835	0.428918	−	0.903344i	$-0.358895\pi$
$929$	24290.0	0.857835	0.428918	+	0.903344i	$0.358895\pi$
$930$	0	0
$931$	−10745.0	−0.378253
$932$	0	0
$933$	20874.0i	0.732459i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 34461.0i	− 1.20149i	−0.799442	−	0.600743i	$-0.794872\pi$
$937$	− 34461.0i	− 1.20149i	0.799442	−	0.600743i	$-0.205128\pi$
$938$	0	0
$939$	16954.0	0.589215
$940$	0	0
$941$	−40628.0	−1.40748	−0.703738	−	0.710460i	$-0.748487\pi$
$941$	−40628.0	−1.40748	−0.703738	+	0.710460i	$0.748487\pi$
$942$	0	0
$943$	32886.0i	1.13565i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20904.0i	− 0.717306i	−0.933471	−	0.358653i	$-0.883236\pi$
$947$	− 20904.0i	− 0.717306i	0.933471	−	0.358653i	$-0.116764\pi$
$948$	0	0
$949$	−21364.0	−0.730774
$950$	0	0
$951$	66388.0	2.26370
$952$	0	0
$953$	1807.00i	0.0614213i	0.999528	+	0.0307106i	$0.00977704\pi$
$953$	1807.00i	0.0614213i	−0.999528	+	0.0307106i	$0.990223\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	48160.0i	1.62674i
$958$	0	0
$959$	2466.00	0.0830358
$960$	0	0
$961$	−28027.0	−0.940787
$962$	0	0
$963$	27918.0i	0.934211i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 57584.0i	− 1.91497i	−0.288482	−	0.957485i	$-0.593151\pi$
$967$	− 57584.0i	− 1.91497i	0.288482	−	0.957485i	$-0.406849\pi$
$968$	0	0
$969$	22295.0	0.739132
$970$	0	0
$971$	−27237.0	−0.900182	−0.450091	−	0.892983i	$-0.648608\pi$
$971$	−27237.0	−0.900182	−0.450091	+	0.892983i	$0.648608\pi$
$972$	0	0
$973$	− 3570.00i	− 0.117625i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13649.0i	0.446950i	0.974710	+	0.223475i	$0.0717401\pi$
$977$	13649.0i	0.446950i	−0.974710	+	0.223475i	$0.928260\pi$
$978$	0	0
$979$	40635.0	1.32656
$980$	0	0
$981$	23540.0	0.766131
$982$	0	0
$983$	− 16002.0i	− 0.519211i	−0.965715	−	0.259606i	$-0.916407\pi$
$983$	− 16002.0i	− 0.519211i	0.965715	−	0.259606i	$-0.0835925\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8232.00i	0.265479i
$988$	0	0
$989$	−14904.0	−0.479191
$990$	0	0
$991$	−37022.0	−1.18672	−0.593362	−	0.804936i	$-0.702200\pi$
$991$	−37022.0	−1.18672	−0.593362	+	0.804936i	$0.702200\pi$
$992$	0	0
$993$	− 1281.00i	− 0.0409379i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 18396.0i	− 0.584360i	−0.956363	−	0.292180i	$-0.905619\pi$
$997$	− 18396.0i	− 0.584360i	0.956363	−	0.292180i	$-0.0943807\pi$
$998$	0	0
$999$	10990.0	0.348056

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.4.c.e.49.1		2
4.3	odd	2		25.4.b.b.24.1		2
5.2	odd	4		400.4.a.c.1.1		1
5.3	odd	4		400.4.a.s.1.1		1
5.4	even	2	inner	400.4.c.e.49.2		2
12.11	even	2		225.4.b.f.199.2		2
20.3	even	4		25.4.a.a.1.1	✓	1
20.7	even	4		25.4.a.b.1.1	yes	1
20.19	odd	2		25.4.b.b.24.2		2
40.3	even	4		1600.4.a.bt.1.1		1
40.13	odd	4		1600.4.a.h.1.1		1
40.27	even	4		1600.4.a.i.1.1		1
40.37	odd	4		1600.4.a.bs.1.1		1
60.23	odd	4		225.4.a.e.1.1		1
60.47	odd	4		225.4.a.c.1.1		1
60.59	even	2		225.4.b.f.199.1		2
140.27	odd	4		1225.4.a.i.1.1		1
140.83	odd	4		1225.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.4.a.a.1.1	✓	1	20.3	even	4
25.4.a.b.1.1	yes	1	20.7	even	4
25.4.b.b.24.1		2	4.3	odd	2
25.4.b.b.24.2		2	20.19	odd	2
225.4.a.c.1.1		1	60.47	odd	4
225.4.a.e.1.1		1	60.23	odd	4
225.4.b.f.199.1		2	60.59	even	2
225.4.b.f.199.2		2	12.11	even	2
400.4.a.c.1.1		1	5.2	odd	4
400.4.a.s.1.1		1	5.3	odd	4
400.4.c.e.49.1		2	1.1	even	1	trivial
400.4.c.e.49.2		2	5.4	even	2	inner
1225.4.a.h.1.1		1	140.83	odd	4
1225.4.a.i.1.1		1	140.27	odd	4
1600.4.a.h.1.1		1	40.13	odd	4
1600.4.a.i.1.1		1	40.27	even	4
1600.4.a.bs.1.1		1	40.37	odd	4
1600.4.a.bt.1.1		1	40.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	400.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-7.00000i q^{3} +6.00000i q^{7} -22.0000 q^{9} +O(q^{10})\)	\(q-7.00000i q^{3} +6.00000i q^{7} -22.0000 q^{9} +43.0000 q^{11} -28.0000i q^{13} -91.0000i q^{17} -35.0000 q^{19} +42.0000 q^{21} -162.000i q^{23} -35.0000i q^{27} -160.000 q^{29} -42.0000 q^{31} -301.000i q^{33} +314.000i q^{37} -196.000 q^{39} -203.000 q^{41} -92.0000i q^{43} +196.000i q^{47} +307.000 q^{49} -637.000 q^{51} +82.0000i q^{53} +245.000i q^{57} -280.000 q^{59} -518.000 q^{61} -132.000i q^{63} +141.000i q^{67} -1134.00 q^{69} -412.000 q^{71} -763.000i q^{73} +258.000i q^{77} +510.000 q^{79} -839.000 q^{81} -777.000i q^{83} +1120.00i q^{87} +945.000 q^{89} +168.000 q^{91} +294.000i q^{93} -1246.00i q^{97} -946.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 44 q^{9}+O(q^{10}) \) 2 * q - 44 * q^9	\( 2 q - 44 q^{9} + 86 q^{11} - 70 q^{19} + 84 q^{21} - 320 q^{29} - 84 q^{31} - 392 q^{39} - 406 q^{41} + 614 q^{49} - 1274 q^{51} - 560 q^{59} - 1036 q^{61} - 2268 q^{69} - 824 q^{71} + 1020 q^{79} - 1678 q^{81} + 1890 q^{89} + 336 q^{91} - 1892 q^{99}+O(q^{100}) \) 2 * q - 44 * q^9 + 86 * q^11 - 70 * q^19 + 84 * q^21 - 320 * q^29 - 84 * q^31 - 392 * q^39 - 406 * q^41 + 614 * q^49 - 1274 * q^51 - 560 * q^59 - 1036 * q^61 - 2268 * q^69 - 824 * q^71 + 1020 * q^79 - 1678 * q^81 + 1890 * q^89 + 336 * q^91 - 1892 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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