LMFDB - Embedded newform 1840.3.c.b.1151.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(1151,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1840.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.c (of order $2$, degree $1$, 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:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.16
Character	$\chi$	$=$	1840.1151
Dual form			1840.3.c.b.1151.41

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.96566i q^{3} +2.23607 q^{5} +5.20785i q^{7} +0.204832 q^{9} +O(q^{10})$	$q-2.96566i q^{3} +2.23607 q^{5} +5.20785i q^{7} +0.204832 q^{9} +8.56871i q^{11} -24.2685 q^{13} -6.63143i q^{15} -0.0733004 q^{17} -12.2064i q^{19} +15.4447 q^{21} -4.79583i q^{23} +5.00000 q^{25} -27.2984i q^{27} +38.7346 q^{29} +0.309308i q^{31} +25.4119 q^{33} +11.6451i q^{35} -52.8381 q^{37} +71.9721i q^{39} +42.5042 q^{41} +44.2355i q^{43} +0.458018 q^{45} +60.5520i q^{47} +21.8783 q^{49} +0.217384i q^{51} +64.4729 q^{53} +19.1602i q^{55} -36.2000 q^{57} +72.7250i q^{59} +30.7027 q^{61} +1.06673i q^{63} -54.2659 q^{65} +94.2551i q^{67} -14.2228 q^{69} +77.7405i q^{71} +47.0837 q^{73} -14.8283i q^{75} -44.6246 q^{77} +7.99809i q^{79} -79.1146 q^{81} -23.9977i q^{83} -0.163905 q^{85} -114.874i q^{87} +128.503 q^{89} -126.387i q^{91} +0.917303 q^{93} -27.2943i q^{95} -156.389 q^{97} +1.75514i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 120 q^{9}+O(q^{10}) $ 56 * q - 120 * q^9	$ 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) $ 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 2.96566i	− 0.988555i	−0.869304	−	0.494277i	$-0.835433\pi$
$3$	− 2.96566i	− 0.988555i	0.869304	−	0.494277i	$-0.164567\pi$
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	5.20785i	0.743979i	0.928237	+	0.371989i	$0.121324\pi$
$7$	5.20785i	0.743979i	−0.928237	+	0.371989i	$0.878676\pi$
$8$	0	0
$9$	0.204832	0.0227591
$10$	0	0
$11$	8.56871i	0.778974i	0.921032	+	0.389487i	$0.127348\pi$
$11$	8.56871i	0.778974i	−0.921032	+	0.389487i	$0.872652\pi$
$12$	0	0
$13$	−24.2685	−1.86680	−0.933402	−	0.358832i	$-0.883175\pi$
$13$	−24.2685	−1.86680	−0.933402	+	0.358832i	$0.883175\pi$
$14$	0	0
$15$	− 6.63143i	− 0.442095i
$16$	0	0
$17$	−0.0733004	−0.00431179	−0.00215589	−	0.999998i	$-0.500686\pi$
$17$	−0.0733004	−0.00431179	−0.00215589	+	0.999998i	$0.500686\pi$
$18$	0	0
$19$	− 12.2064i	− 0.642441i	−0.947004	−	0.321220i	$-0.895907\pi$
$19$	− 12.2064i	− 0.642441i	0.947004	−	0.321220i	$-0.104093\pi$
$20$	0	0
$21$	15.4447	0.735464
$22$	0	0
$23$	− 4.79583i	− 0.208514i
$23$	− 4.79583i	− 0.208514i
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 27.2984i	− 1.01105i
$28$	0	0
$29$	38.7346	1.33568	0.667839	−	0.744306i	$-0.267220\pi$
$29$	38.7346	1.33568	0.667839	+	0.744306i	$0.267220\pi$
$30$	0	0
$31$	0.309308i	0.00997767i	0.999988	+	0.00498883i	$0.00158800\pi$
$31$	0.309308i	0.00997767i	−0.999988	+	0.00498883i	$0.998412\pi$
$32$	0	0
$33$	25.4119	0.770058
$34$	0	0
$35$	11.6451i	0.332717i
$36$	0	0
$37$	−52.8381	−1.42806	−0.714028	−	0.700117i	$-0.753131\pi$
$37$	−52.8381	−1.42806	−0.714028	+	0.700117i	$0.753131\pi$
$38$	0	0
$39$	71.9721i	1.84544i
$40$	0	0
$41$	42.5042	1.03669	0.518343	−	0.855173i	$-0.326549\pi$
$41$	42.5042	1.03669	0.518343	+	0.855173i	$0.326549\pi$
$42$	0	0
$43$	44.2355i	1.02873i	0.857571	+	0.514366i	$0.171973\pi$
$43$	44.2355i	1.02873i	−0.857571	+	0.514366i	$0.828027\pi$
$44$	0	0
$45$	0.458018	0.0101782
$46$	0	0
$47$	60.5520i	1.28834i	0.764882	+	0.644170i	$0.222797\pi$
$47$	60.5520i	1.28834i	−0.764882	+	0.644170i	$0.777203\pi$
$48$	0	0
$49$	21.8783	0.446496
$50$	0	0
$51$	0.217384i	0.00426244i
$52$	0	0
$53$	64.4729	1.21647	0.608235	−	0.793757i	$-0.291878\pi$
$53$	64.4729	1.21647	0.608235	+	0.793757i	$0.291878\pi$
$54$	0	0
$55$	19.1602i	0.348368i
$56$	0	0
$57$	−36.2000	−0.635088
$58$	0	0
$59$	72.7250i	1.23263i	0.787501	+	0.616313i	$0.211374\pi$
$59$	72.7250i	1.23263i	−0.787501	+	0.616313i	$0.788626\pi$
$60$	0	0
$61$	30.7027	0.503323	0.251662	−	0.967815i	$-0.419023\pi$
$61$	30.7027	0.503323	0.251662	+	0.967815i	$0.419023\pi$
$62$	0	0
$63$	1.06673i	0.0169323i
$64$	0	0
$65$	−54.2659	−0.834860
$66$	0	0
$67$	94.2551i	1.40679i	0.710798	+	0.703397i	$0.248334\pi$
$67$	94.2551i	1.40679i	−0.710798	+	0.703397i	$0.751666\pi$
$68$	0	0
$69$	−14.2228	−0.206128
$70$	0	0
$71$	77.7405i	1.09494i	0.836826	+	0.547468i	$0.184408\pi$
$71$	77.7405i	1.09494i	−0.836826	+	0.547468i	$0.815592\pi$
$72$	0	0
$73$	47.0837	0.644983	0.322491	−	0.946572i	$-0.395480\pi$
$73$	47.0837	0.644983	0.322491	+	0.946572i	$0.395480\pi$
$74$	0	0
$75$	− 14.8283i	− 0.197711i
$76$	0	0
$77$	−44.6246	−0.579540
$78$	0	0
$79$	7.99809i	0.101242i	0.998718	+	0.0506208i	$0.0161200\pi$
$79$	7.99809i	0.101242i	−0.998718	+	0.0506208i	$0.983880\pi$
$80$	0	0
$81$	−79.1146	−0.976723
$82$	0	0
$83$	− 23.9977i	− 0.289129i	−0.989495	−	0.144564i	$-0.953822\pi$
$83$	− 23.9977i	− 0.289129i	0.989495	−	0.144564i	$-0.0461781\pi$
$84$	0	0
$85$	−0.163905	−0.00192829
$86$	0	0
$87$	− 114.874i	− 1.32039i
$88$	0	0
$89$	128.503	1.44385	0.721926	−	0.691970i	$-0.243257\pi$
$89$	128.503	1.44385	0.721926	+	0.691970i	$0.243257\pi$
$90$	0	0
$91$	− 126.387i	− 1.38886i
$92$	0	0
$93$	0.917303	0.00986347
$94$	0	0
$95$	− 27.2943i	− 0.287308i
$96$	0	0
$97$	−156.389	−1.61226	−0.806130	−	0.591738i	$-0.798442\pi$
$97$	−156.389	−1.61226	−0.806130	+	0.591738i	$0.798442\pi$
$98$	0	0
$99$	1.75514i	0.0177287i
$100$	0	0
$101$	13.7097	0.135739	0.0678697	−	0.997694i	$-0.478380\pi$
$101$	13.7097	0.135739	0.0678697	+	0.997694i	$0.478380\pi$
$102$	0	0
$103$	37.3232i	0.362361i	0.983450	+	0.181181i	$0.0579918\pi$
$103$	37.3232i	0.362361i	−0.983450	+	0.181181i	$0.942008\pi$
$104$	0	0
$105$	34.5355	0.328909
$106$	0	0
$107$	166.970i	1.56047i	0.625489	+	0.780233i	$0.284900\pi$
$107$	166.970i	1.56047i	−0.625489	+	0.780233i	$0.715100\pi$
$108$	0	0
$109$	−158.278	−1.45209	−0.726047	−	0.687645i	$-0.758644\pi$
$109$	−158.278	−1.45209	−0.726047	+	0.687645i	$0.758644\pi$
$110$	0	0
$111$	156.700i	1.41171i
$112$	0	0
$113$	212.657	1.88192	0.940960	−	0.338517i	$-0.109925\pi$
$113$	212.657	1.88192	0.940960	+	0.338517i	$0.109925\pi$
$114$	0	0
$115$	− 10.7238i	− 0.0932505i
$116$	0	0
$117$	−4.97095	−0.0424868
$118$	0	0
$119$	− 0.381737i	− 0.00320788i
$120$	0	0
$121$	47.5772	0.393200
$122$	0	0
$123$	− 126.053i	− 1.02482i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	188.238i	1.48219i	0.671401	+	0.741094i	$0.265693\pi$
$127$	188.238i	1.48219i	−0.671401	+	0.741094i	$0.734307\pi$
$128$	0	0
$129$	131.188	1.01696
$130$	0	0
$131$	82.8011i	0.632070i	0.948748	+	0.316035i	$0.102352\pi$
$131$	82.8011i	0.632070i	−0.948748	+	0.316035i	$0.897648\pi$
$132$	0	0
$133$	63.5690	0.477962
$134$	0	0
$135$	− 61.0412i	− 0.452157i
$136$	0	0
$137$	102.414	0.747550	0.373775	−	0.927519i	$-0.378063\pi$
$137$	102.414	0.747550	0.373775	+	0.927519i	$0.378063\pi$
$138$	0	0
$139$	− 229.379i	− 1.65021i	−0.564983	−	0.825103i	$-0.691117\pi$
$139$	− 229.379i	− 1.65021i	0.564983	−	0.825103i	$-0.308883\pi$
$140$	0	0
$141$	179.577	1.27359
$142$	0	0
$143$	− 207.949i	− 1.45419i
$144$	0	0
$145$	86.6133	0.597333
$146$	0	0
$147$	− 64.8837i	− 0.441385i
$148$	0	0
$149$	−121.594	−0.816069	−0.408034	−	0.912967i	$-0.633786\pi$
$149$	−121.594	−0.816069	−0.408034	+	0.912967i	$0.633786\pi$
$150$	0	0
$151$	184.596i	1.22249i	0.791442	+	0.611244i	$0.209331\pi$
$151$	184.596i	1.22249i	−0.791442	+	0.611244i	$0.790669\pi$
$152$	0	0
$153$	−0.0150142	−9.81323e−5 0
$154$	0	0
$155$	0.691633i	0.00446215i
$156$	0	0
$157$	167.174	1.06480	0.532402	−	0.846491i	$-0.321289\pi$
$157$	167.174	1.06480	0.532402	+	0.846491i	$0.321289\pi$
$158$	0	0
$159$	− 191.205i	− 1.20255i
$160$	0	0
$161$	24.9760	0.155130
$162$	0	0
$163$	221.671i	1.35994i	0.733239	+	0.679971i	$0.238008\pi$
$163$	221.671i	1.35994i	−0.733239	+	0.679971i	$0.761992\pi$
$164$	0	0
$165$	56.8228	0.344381
$166$	0	0
$167$	− 14.7687i	− 0.0884354i	−0.999022	−	0.0442177i	$-0.985920\pi$
$167$	− 14.7687i	− 0.0884354i	0.999022	−	0.0442177i	$-0.0140795\pi$
$168$	0	0
$169$	419.958	2.48496
$170$	0	0
$171$	− 2.50025i	− 0.0146214i
$172$	0	0
$173$	−234.152	−1.35348	−0.676740	−	0.736223i	$-0.736608\pi$
$173$	−234.152	−1.35348	−0.676740	+	0.736223i	$0.736608\pi$
$174$	0	0
$175$	26.0393i	0.148796i
$176$	0	0
$177$	215.678	1.21852
$178$	0	0
$179$	62.3947i	0.348574i	0.984695	+	0.174287i	$0.0557620\pi$
$179$	62.3947i	0.348574i	−0.984695	+	0.174287i	$0.944238\pi$
$180$	0	0
$181$	−4.43616	−0.0245092	−0.0122546	−	0.999925i	$-0.503901\pi$
$181$	−4.43616	−0.0245092	−0.0122546	+	0.999925i	$0.503901\pi$
$182$	0	0
$183$	− 91.0539i	− 0.497563i
$184$	0	0
$185$	−118.150	−0.638646
$186$	0	0
$187$	− 0.628090i	− 0.00335877i
$188$	0	0
$189$	142.166	0.752202
$190$	0	0
$191$	− 76.7376i	− 0.401768i	−0.979615	−	0.200884i	$-0.935619\pi$
$191$	− 76.7376i	− 0.401768i	0.979615	−	0.200884i	$-0.0643814\pi$
$192$	0	0
$193$	−51.9757	−0.269304	−0.134652	−	0.990893i	$-0.542992\pi$
$193$	−51.9757	−0.269304	−0.134652	+	0.990893i	$0.542992\pi$
$194$	0	0
$195$	160.935i	0.825305i
$196$	0	0
$197$	−13.9953	−0.0710422	−0.0355211	−	0.999369i	$-0.511309\pi$
$197$	−13.9953	−0.0710422	−0.0355211	+	0.999369i	$0.511309\pi$
$198$	0	0
$199$	76.7411i	0.385634i	0.981235	+	0.192817i	$0.0617623\pi$
$199$	76.7411i	0.385634i	−0.981235	+	0.192817i	$0.938238\pi$
$200$	0	0
$201$	279.529	1.39069
$202$	0	0
$203$	201.724i	0.993716i
$204$	0	0
$205$	95.0422	0.463620
$206$	0	0
$207$	− 0.982339i	− 0.00474560i
$208$	0	0
$209$	104.593	0.500445
$210$	0	0
$211$	− 152.609i	− 0.723263i	−0.932321	−	0.361632i	$-0.882220\pi$
$211$	− 152.609i	− 0.723263i	0.932321	−	0.361632i	$-0.117780\pi$
$212$	0	0
$213$	230.552	1.08240
$214$	0	0
$215$	98.9135i	0.460063i
$216$	0	0
$217$	−1.61083	−0.00742317
$218$	0	0
$219$	− 139.635i	− 0.637601i
$220$	0	0
$221$	1.77889	0.00804926
$222$	0	0
$223$	− 371.851i	− 1.66749i	−0.552148	−	0.833746i	$-0.686192\pi$
$223$	− 371.851i	− 1.66749i	0.552148	−	0.833746i	$-0.313808\pi$
$224$	0	0
$225$	1.02416	0.00455182
$226$	0	0
$227$	413.927i	1.82347i	0.410781	+	0.911734i	$0.365256\pi$
$227$	413.927i	1.82347i	−0.410781	+	0.911734i	$0.634744\pi$
$228$	0	0
$229$	−347.138	−1.51589	−0.757943	−	0.652320i	$-0.773796\pi$
$229$	−347.138	−1.51589	−0.757943	+	0.652320i	$0.773796\pi$
$230$	0	0
$231$	132.342i	0.572907i
$232$	0	0
$233$	42.9647	0.184398	0.0921989	−	0.995741i	$-0.470610\pi$
$233$	42.9647	0.184398	0.0921989	+	0.995741i	$0.470610\pi$
$234$	0	0
$235$	135.398i	0.576163i
$236$	0	0
$237$	23.7197	0.100083
$238$	0	0
$239$	− 394.741i	− 1.65164i	−0.563936	−	0.825819i	$-0.690713\pi$
$239$	− 394.741i	− 1.65164i	0.563936	−	0.825819i	$-0.309287\pi$
$240$	0	0
$241$	49.5604	0.205645	0.102822	−	0.994700i	$-0.467213\pi$
$241$	49.5604	0.205645	0.102822	+	0.994700i	$0.467213\pi$
$242$	0	0
$243$	− 11.0587i	− 0.0455093i
$244$	0	0
$245$	48.9213	0.199679
$246$	0	0
$247$	296.230i	1.19931i
$248$	0	0
$249$	−71.1691	−0.285820
$250$	0	0
$251$	− 158.751i	− 0.632475i	−0.948680	−	0.316237i	$-0.897580\pi$
$251$	− 158.751i	− 0.632475i	0.948680	−	0.316237i	$-0.102420\pi$
$252$	0	0
$253$	41.0941	0.162427
$254$	0	0
$255$	0.486086i	0.00190622i
$256$	0	0
$257$	17.0485	0.0663365	0.0331682	−	0.999450i	$-0.489440\pi$
$257$	17.0485	0.0663365	0.0331682	+	0.999450i	$0.489440\pi$
$258$	0	0
$259$	− 275.173i	− 1.06244i
$260$	0	0
$261$	7.93408	0.0303988
$262$	0	0
$263$	− 116.966i	− 0.444738i	−0.974963	−	0.222369i	$-0.928621\pi$
$263$	− 116.966i	− 0.444738i	0.974963	−	0.222369i	$-0.0713790\pi$
$264$	0	0
$265$	144.166	0.544022
$266$	0	0
$267$	− 381.096i	− 1.42733i
$268$	0	0
$269$	167.641	0.623202	0.311601	−	0.950213i	$-0.399135\pi$
$269$	167.641	0.623202	0.311601	+	0.950213i	$0.399135\pi$
$270$	0	0
$271$	− 165.336i	− 0.610095i	−0.952337	−	0.305047i	$-0.901328\pi$
$271$	− 165.336i	− 0.610095i	0.952337	−	0.305047i	$-0.0986723\pi$
$272$	0	0
$273$	−374.820	−1.37297
$274$	0	0
$275$	42.8436i	0.155795i
$276$	0	0
$277$	−407.401	−1.47076	−0.735381	−	0.677654i	$-0.762997\pi$
$277$	−407.401	−1.47076	−0.735381	+	0.677654i	$0.762997\pi$
$278$	0	0
$279$	0.0633560i	0 0.000227083i
$280$	0	0
$281$	264.934	0.942827	0.471413	−	0.881912i	$-0.343744\pi$
$281$	264.934	0.942827	0.471413	+	0.881912i	$0.343744\pi$
$282$	0	0
$283$	− 37.9887i	− 0.134236i	−0.997745	−	0.0671179i	$-0.978620\pi$
$283$	− 37.9887i	− 0.134236i	0.997745	−	0.0671179i	$-0.0213803\pi$
$284$	0	0
$285$	−80.9457	−0.284020
$286$	0	0
$287$	221.355i	0.771273i
$288$	0	0
$289$	−288.995	−0.999981
$290$	0	0
$291$	463.798i	1.59381i
$292$	0	0
$293$	380.362	1.29816	0.649081	−	0.760719i	$-0.275154\pi$
$293$	380.362	1.29816	0.649081	+	0.760719i	$0.275154\pi$
$294$	0	0
$295$	162.618i	0.551247i
$296$	0	0
$297$	233.912	0.787584
$298$	0	0
$299$	116.387i	0.389256i
$300$	0	0
$301$	−230.372	−0.765355
$302$	0	0
$303$	− 40.6583i	− 0.134186i
$304$	0	0
$305$	68.6533	0.225093
$306$	0	0
$307$	− 369.321i	− 1.20300i	−0.798873	−	0.601500i	$-0.794570\pi$
$307$	− 369.321i	− 1.20300i	0.798873	−	0.601500i	$-0.205430\pi$
$308$	0	0
$309$	110.688	0.358214
$310$	0	0
$311$	342.687i	1.10189i	0.834542	+	0.550944i	$0.185732\pi$
$311$	342.687i	1.10189i	−0.834542	+	0.550944i	$0.814268\pi$
$312$	0	0
$313$	494.191	1.57889	0.789443	−	0.613824i	$-0.210370\pi$
$313$	494.191	1.57889	0.789443	+	0.613824i	$0.210370\pi$
$314$	0	0
$315$	2.38529i	0.00757234i
$316$	0	0
$317$	−154.861	−0.488521	−0.244260	−	0.969710i	$-0.578545\pi$
$317$	−154.861	−0.488521	−0.244260	+	0.969710i	$0.578545\pi$
$318$	0	0
$319$	331.906i	1.04046i
$320$	0	0
$321$	495.177	1.54261
$322$	0	0
$323$	0.894732i	0.00277007i
$324$	0	0
$325$	−121.342	−0.373361
$326$	0	0
$327$	469.400i	1.43547i
$328$	0	0
$329$	−315.346	−0.958497
$330$	0	0
$331$	− 524.889i	− 1.58577i	−0.609372	−	0.792884i	$-0.708579\pi$
$331$	− 524.889i	− 1.58577i	0.609372	−	0.792884i	$-0.291421\pi$
$332$	0	0
$333$	−10.8229	−0.0325012
$334$	0	0
$335$	210.761i	0.629137i
$336$	0	0
$337$	39.2846	0.116572	0.0582858	−	0.998300i	$-0.481437\pi$
$337$	39.2846	0.116572	0.0582858	+	0.998300i	$0.481437\pi$
$338$	0	0
$339$	− 630.669i	− 1.86038i
$340$	0	0
$341$	−2.65037	−0.00777234
$342$	0	0
$343$	369.124i	1.07616i
$344$	0	0
$345$	−31.8032	−0.0921832
$346$	0	0
$347$	− 532.895i	− 1.53572i	−0.640617	−	0.767860i	$-0.721322\pi$
$347$	− 532.895i	− 1.53572i	0.640617	−	0.767860i	$-0.278678\pi$
$348$	0	0
$349$	310.849	0.890685	0.445343	−	0.895360i	$-0.353082\pi$
$349$	310.849	0.890685	0.445343	+	0.895360i	$0.353082\pi$
$350$	0	0
$351$	662.491i	1.88744i
$352$	0	0
$353$	−665.455	−1.88514	−0.942572	−	0.334004i	$-0.891600\pi$
$353$	−665.455	−1.88514	−0.942572	+	0.334004i	$0.891600\pi$
$354$	0	0
$355$	173.833i	0.489670i
$356$	0	0
$357$	−1.13211	−0.00317116
$358$	0	0
$359$	397.130i	1.10621i	0.833111	+	0.553105i	$0.186557\pi$
$359$	397.130i	1.10621i	−0.833111	+	0.553105i	$0.813443\pi$
$360$	0	0
$361$	212.004	0.587270
$362$	0	0
$363$	− 141.098i	− 0.388700i
$364$	0	0
$365$	105.282	0.288445
$366$	0	0
$367$	183.711i	0.500575i	0.968172	+	0.250287i	$0.0805251\pi$
$367$	183.711i	0.500575i	−0.968172	+	0.250287i	$0.919475\pi$
$368$	0	0
$369$	8.70620	0.0235940
$370$	0	0
$371$	335.765i	0.905028i
$372$	0	0
$373$	192.970	0.517347	0.258673	−	0.965965i	$-0.416715\pi$
$373$	192.970	0.517347	0.258673	+	0.965965i	$0.416715\pi$
$374$	0	0
$375$	− 33.1571i	− 0.0884190i
$376$	0	0
$377$	−940.030	−2.49345
$378$	0	0
$379$	141.314i	0.372860i	0.982468	+	0.186430i	$0.0596917\pi$
$379$	141.314i	0.372860i	−0.982468	+	0.186430i	$0.940308\pi$
$380$	0	0
$381$	558.251	1.46523
$382$	0	0
$383$	300.331i	0.784153i	0.919933	+	0.392077i	$0.128243\pi$
$383$	300.331i	0.784153i	−0.919933	+	0.392077i	$0.871757\pi$
$384$	0	0
$385$	−99.7836	−0.259178
$386$	0	0
$387$	9.06083i	0.0234130i
$388$	0	0
$389$	−161.556	−0.415312	−0.207656	−	0.978202i	$-0.566583\pi$
$389$	−161.556	−0.415312	−0.207656	+	0.978202i	$0.566583\pi$
$390$	0	0
$391$	0.351536i	0 0.000899070i
$392$	0	0
$393$	245.560	0.624835
$394$	0	0
$395$	17.8843i	0.0452767i
$396$	0	0
$397$	−206.635	−0.520492	−0.260246	−	0.965542i	$-0.583804\pi$
$397$	−206.635	−0.520492	−0.260246	+	0.965542i	$0.583804\pi$
$398$	0	0
$399$	− 188.524i	− 0.472492i
$400$	0	0
$401$	−430.856	−1.07445	−0.537227	−	0.843438i	$-0.680528\pi$
$401$	−430.856	−1.07445	−0.537227	+	0.843438i	$0.680528\pi$
$402$	0	0
$403$	− 7.50642i	− 0.0186263i
$404$	0	0
$405$	−176.906	−0.436804
$406$	0	0
$407$	− 452.754i	− 1.11242i
$408$	0	0
$409$	550.299	1.34547	0.672737	−	0.739882i	$-0.265119\pi$
$409$	550.299	1.34547	0.672737	+	0.739882i	$0.265119\pi$
$410$	0	0
$411$	− 303.727i	− 0.738994i
$412$	0	0
$413$	−378.741	−0.917048
$414$	0	0
$415$	− 53.6605i	− 0.129302i
$416$	0	0
$417$	−680.260	−1.63132
$418$	0	0
$419$	− 70.1294i	− 0.167373i	−0.996492	−	0.0836867i	$-0.973331\pi$
$419$	− 70.1294i	− 0.167373i	0.996492	−	0.0836867i	$-0.0266695\pi$
$420$	0	0
$421$	−419.561	−0.996583	−0.498292	−	0.867010i	$-0.666039\pi$
$421$	−419.561	−0.996583	−0.498292	+	0.867010i	$0.666039\pi$
$422$	0	0
$423$	12.4030i	0.0293214i
$424$	0	0
$425$	−0.366502	−0.000862357 0
$426$	0	0
$427$	159.895i	0.374462i
$428$	0	0
$429$	−616.708	−1.43755
$430$	0	0
$431$	124.498i	0.288860i	0.989515	+	0.144430i	$0.0461348\pi$
$431$	124.498i	0.288860i	−0.989515	+	0.144430i	$0.953865\pi$
$432$	0	0
$433$	−522.407	−1.20648	−0.603241	−	0.797559i	$-0.706124\pi$
$433$	−522.407	−1.20648	−0.603241	+	0.797559i	$0.706124\pi$
$434$	0	0
$435$	− 256.866i	− 0.590497i
$436$	0	0
$437$	−58.5397	−0.133958
$438$	0	0
$439$	− 198.715i	− 0.452653i	−0.974052	−	0.226326i	$-0.927328\pi$
$439$	− 198.715i	− 0.452653i	0.974052	−	0.226326i	$-0.0726716\pi$
$440$	0	0
$441$	4.48137	0.0101618
$442$	0	0
$443$	− 503.231i	− 1.13596i	−0.823041	−	0.567981i	$-0.807725\pi$
$443$	− 503.231i	− 1.13596i	0.823041	−	0.567981i	$-0.192275\pi$
$444$	0	0
$445$	287.341	0.645710
$446$	0	0
$447$	360.608i	0.806729i
$448$	0	0
$449$	−744.455	−1.65803	−0.829014	−	0.559227i	$-0.811098\pi$
$449$	−744.455	−1.65803	−0.829014	+	0.559227i	$0.811098\pi$
$450$	0	0
$451$	364.206i	0.807552i
$452$	0	0
$453$	547.449	1.20850
$454$	0	0
$455$	− 282.609i	− 0.621118i
$456$	0	0
$457$	−786.608	−1.72124	−0.860622	−	0.509245i	$-0.829925\pi$
$457$	−786.608	−1.72124	−0.860622	+	0.509245i	$0.829925\pi$
$458$	0	0
$459$	2.00099i	0.00435945i
$460$	0	0
$461$	406.369	0.881494	0.440747	−	0.897631i	$-0.354714\pi$
$461$	406.369	0.881494	0.440747	+	0.897631i	$0.354714\pi$
$462$	0	0
$463$	619.039i	1.33702i	0.743705	+	0.668508i	$0.233067\pi$
$463$	619.039i	1.33702i	−0.743705	+	0.668508i	$0.766933\pi$
$464$	0	0
$465$	2.05115	0.00441108
$466$	0	0
$467$	187.393i	0.401270i	0.979666	+	0.200635i	$0.0643006\pi$
$467$	187.393i	0.401270i	−0.979666	+	0.200635i	$0.935699\pi$
$468$	0	0
$469$	−490.867	−1.04662
$470$	0	0
$471$	− 495.783i	− 1.05262i
$472$	0	0
$473$	−379.041	−0.801355
$474$	0	0
$475$	− 61.0319i	− 0.128488i
$476$	0	0
$477$	13.2061	0.0276857
$478$	0	0
$479$	− 286.158i	− 0.597407i	−0.954346	−	0.298704i	$-0.903446\pi$
$479$	− 286.158i	− 0.597407i	0.954346	−	0.298704i	$-0.0965542\pi$
$480$	0	0
$481$	1282.30	2.66590
$482$	0	0
$483$	− 74.0704i	− 0.153355i
$484$	0	0
$485$	−349.697	−0.721025
$486$	0	0
$487$	− 76.9119i	− 0.157930i	−0.996877	−	0.0789650i	$-0.974838\pi$
$487$	− 76.9119i	− 0.157930i	0.996877	−	0.0789650i	$-0.0251615\pi$
$488$	0	0
$489$	657.401	1.34438
$490$	0	0
$491$	− 102.828i	− 0.209425i	−0.994503	−	0.104713i	$-0.966608\pi$
$491$	− 102.828i	− 0.209425i	0.994503	−	0.104713i	$-0.0333923\pi$
$492$	0	0
$493$	−2.83926	−0.00575916
$494$	0	0
$495$	3.92462i	0.00792853i
$496$	0	0
$497$	−404.861	−0.814610
$498$	0	0
$499$	− 567.766i	− 1.13781i	−0.822404	−	0.568904i	$-0.807368\pi$
$499$	− 567.766i	− 1.13781i	0.822404	−	0.568904i	$-0.192632\pi$
$500$	0	0
$501$	−43.7990	−0.0874232
$502$	0	0
$503$	81.2990i	0.161628i	0.996729	+	0.0808141i	$0.0257520\pi$
$503$	81.2990i	0.161628i	−0.996729	+	0.0808141i	$0.974248\pi$
$504$	0	0
$505$	30.6558	0.0607045
$506$	0	0
$507$	− 1245.45i	− 2.45652i
$508$	0	0
$509$	313.260	0.615442	0.307721	−	0.951477i	$-0.400434\pi$
$509$	313.260	0.615442	0.307721	+	0.951477i	$0.400434\pi$
$510$	0	0
$511$	245.205i	0.479853i
$512$	0	0
$513$	−333.215	−0.649542
$514$	0	0
$515$	83.4572i	0.162053i
$516$	0	0
$517$	−518.852	−1.00358
$518$	0	0
$519$	694.416i	1.33799i
$520$	0	0
$521$	−475.368	−0.912414	−0.456207	−	0.889874i	$-0.650792\pi$
$521$	−475.368	−0.912414	−0.456207	+	0.889874i	$0.650792\pi$
$522$	0	0
$523$	291.425i	0.557218i	0.960405	+	0.278609i	$0.0898733\pi$
$523$	291.425i	0.557218i	−0.960405	+	0.278609i	$0.910127\pi$
$524$	0	0
$525$	77.2237	0.147093
$526$	0	0
$527$	− 0.0226724i	0 4.30216e-5i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	14.8964i	0.0280534i
$532$	0	0
$533$	−1031.51	−1.93529
$534$	0	0
$535$	373.356i	0.697862i
$536$	0	0
$537$	185.042	0.344584
$538$	0	0
$539$	187.469i	0.347808i
$540$	0	0
$541$	72.3527	0.133739	0.0668694	−	0.997762i	$-0.478699\pi$
$541$	72.3527	0.133739	0.0668694	+	0.997762i	$0.478699\pi$
$542$	0	0
$543$	13.1562i	0.0242287i
$544$	0	0
$545$	−353.921	−0.649396
$546$	0	0
$547$	− 335.458i	− 0.613269i	−0.951827	−	0.306635i	$-0.900797\pi$
$547$	− 335.458i	− 0.613269i	0.951827	−	0.306635i	$-0.0992030\pi$
$548$	0	0
$549$	6.28889	0.0114552
$550$	0	0
$551$	− 472.810i	− 0.858094i
$552$	0	0
$553$	−41.6529	−0.0753217
$554$	0	0
$555$	350.392i	0.631337i
$556$	0	0
$557$	−533.748	−0.958255	−0.479127	−	0.877745i	$-0.659047\pi$
$557$	−533.748	−0.958255	−0.479127	+	0.877745i	$0.659047\pi$
$558$	0	0
$559$	− 1073.53i	− 1.92044i
$560$	0	0
$561$	−1.86270	−0.00332033
$562$	0	0
$563$	− 1052.35i	− 1.86918i	−0.355728	−	0.934590i	$-0.615767\pi$
$563$	− 1052.35i	− 1.86918i	0.355728	−	0.934590i	$-0.384233\pi$
$564$	0	0
$565$	475.516	0.841620
$566$	0	0
$567$	− 412.017i	− 0.726661i
$568$	0	0
$569$	545.406	0.958535	0.479267	−	0.877669i	$-0.340902\pi$
$569$	545.406	0.958535	0.479267	+	0.877669i	$0.340902\pi$
$570$	0	0
$571$	− 247.420i	− 0.433311i	−0.976248	−	0.216655i	$-0.930485\pi$
$571$	− 247.420i	− 0.433311i	0.976248	−	0.216655i	$-0.0695148\pi$
$572$	0	0
$573$	−227.578	−0.397170
$574$	0	0
$575$	− 23.9792i	− 0.0417029i
$576$	0	0
$577$	−612.716	−1.06190	−0.530950	−	0.847403i	$-0.678165\pi$
$577$	−612.716	−1.06190	−0.530950	+	0.847403i	$0.678165\pi$
$578$	0	0
$579$	154.143i	0.266222i
$580$	0	0
$581$	124.976	0.215106
$582$	0	0
$583$	552.449i	0.947598i
$584$	0	0
$585$	−11.1154	−0.0190007
$586$	0	0
$587$	405.263i	0.690396i	0.938530	+	0.345198i	$0.112188\pi$
$587$	405.263i	0.690396i	−0.938530	+	0.345198i	$0.887812\pi$
$588$	0	0
$589$	3.77553	0.00641006
$590$	0	0
$591$	41.5054i	0.0702292i
$592$	0	0
$593$	−214.980	−0.362530	−0.181265	−	0.983434i	$-0.558019\pi$
$593$	−214.980	−0.362530	−0.181265	+	0.983434i	$0.558019\pi$
$594$	0	0
$595$	− 0.853591i	− 0.00143461i
$596$	0	0
$597$	227.588	0.381220
$598$	0	0
$599$	211.915i	0.353781i	0.984231	+	0.176890i	$0.0566038\pi$
$599$	211.915i	0.353781i	−0.984231	+	0.176890i	$0.943396\pi$
$600$	0	0
$601$	−674.364	−1.12207	−0.561035	−	0.827792i	$-0.689597\pi$
$601$	−674.364	−1.12207	−0.561035	+	0.827792i	$0.689597\pi$
$602$	0	0
$603$	19.3064i	0.0320173i
$604$	0	0
$605$	106.386	0.175844
$606$	0	0
$607$	387.791i	0.638866i	0.947609	+	0.319433i	$0.103492\pi$
$607$	387.791i	0.638866i	−0.947609	+	0.319433i	$0.896508\pi$
$608$	0	0
$609$	598.247	0.982342
$610$	0	0
$611$	− 1469.50i	− 2.40508i
$612$	0	0
$613$	642.540	1.04819	0.524094	−	0.851660i	$-0.324404\pi$
$613$	642.540	1.04819	0.524094	+	0.851660i	$0.324404\pi$
$614$	0	0
$615$	− 281.863i	− 0.458314i
$616$	0	0
$617$	393.878	0.638375	0.319188	−	0.947692i	$-0.396590\pi$
$617$	393.878	0.638375	0.319188	+	0.947692i	$0.396590\pi$
$618$	0	0
$619$	949.368i	1.53371i	0.641819	+	0.766857i	$0.278180\pi$
$619$	949.368i	1.53371i	−0.641819	+	0.766857i	$0.721820\pi$
$620$	0	0
$621$	−130.919	−0.210819
$622$	0	0
$623$	669.224i	1.07420i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 310.188i	− 0.494717i
$628$	0	0
$629$	3.87305	0.00615747
$630$	0	0
$631$	358.618i	0.568333i	0.958775	+	0.284166i	$0.0917168\pi$
$631$	358.618i	0.568333i	−0.958775	+	0.284166i	$0.908283\pi$
$632$	0	0
$633$	−452.586	−0.714986
$634$	0	0
$635$	420.913i	0.662855i
$636$	0	0
$637$	−530.952	−0.833520
$638$	0	0
$639$	15.9237i	0.0249198i
$640$	0	0
$641$	664.620	1.03685	0.518424	−	0.855124i	$-0.326519\pi$
$641$	664.620	1.03685	0.518424	+	0.855124i	$0.326519\pi$
$642$	0	0
$643$	− 927.368i	− 1.44225i	−0.692804	−	0.721126i	$-0.743625\pi$
$643$	− 927.368i	− 1.44225i	0.692804	−	0.721126i	$-0.256375\pi$
$644$	0	0
$645$	293.344	0.454798
$646$	0	0
$647$	− 1288.69i	− 1.99180i	−0.0904837	−	0.995898i	$-0.528841\pi$
$647$	− 1288.69i	− 1.99180i	0.0904837	−	0.995898i	$-0.471159\pi$
$648$	0	0
$649$	−623.159	−0.960183
$650$	0	0
$651$	4.77718i	0.00733821i
$652$	0	0
$653$	243.959	0.373597	0.186799	−	0.982398i	$-0.440189\pi$
$653$	243.959	0.373597	0.186799	+	0.982398i	$0.440189\pi$
$654$	0	0
$655$	185.149i	0.282670i
$656$	0	0
$657$	9.64424	0.0146792
$658$	0	0
$659$	254.349i	0.385963i	0.981202	+	0.192981i	$0.0618157\pi$
$659$	254.349i	0.385963i	−0.981202	+	0.192981i	$0.938184\pi$
$660$	0	0
$661$	434.785	0.657768	0.328884	−	0.944370i	$-0.393327\pi$
$661$	434.785	0.657768	0.328884	+	0.944370i	$0.393327\pi$
$662$	0	0
$663$	− 5.27558i	− 0.00795714i
$664$	0	0
$665$	142.145	0.213751
$666$	0	0
$667$	− 185.765i	− 0.278508i
$668$	0	0
$669$	−1102.78	−1.64841
$670$	0	0
$671$	263.083i	0.392075i
$672$	0	0
$673$	753.315	1.11934	0.559670	−	0.828716i	$-0.310928\pi$
$673$	753.315	1.11934	0.559670	+	0.828716i	$0.310928\pi$
$674$	0	0
$675$	− 136.492i	− 0.202211i
$676$	0	0
$677$	681.787	1.00707	0.503535	−	0.863975i	$-0.332032\pi$
$677$	681.787	1.00707	0.503535	+	0.863975i	$0.332032\pi$
$678$	0	0
$679$	− 814.452i	− 1.19949i
$680$	0	0
$681$	1227.57	1.80260
$682$	0	0
$683$	52.7233i	0.0771937i	0.999255	+	0.0385968i	$0.0122888\pi$
$683$	52.7233i	0.0771937i	−0.999255	+	0.0385968i	$0.987711\pi$
$684$	0	0
$685$	229.005	0.334315
$686$	0	0
$687$	1029.50i	1.49854i
$688$	0	0
$689$	−1564.66	−2.27091
$690$	0	0
$691$	908.695i	1.31504i	0.753436	+	0.657521i	$0.228395\pi$
$691$	908.695i	1.31504i	−0.753436	+	0.657521i	$0.771605\pi$
$692$	0	0
$693$	−9.14053	−0.0131898
$694$	0	0
$695$	− 512.906i	− 0.737994i
$696$	0	0
$697$	−3.11557	−0.00446997
$698$	0	0
$699$	− 127.419i	− 0.182287i
$700$	0	0
$701$	3.78257	0.00539596	0.00269798	−	0.999996i	$-0.499141\pi$
$701$	3.78257	0.00539596	0.00269798	+	0.999996i	$0.499141\pi$
$702$	0	0
$703$	644.961i	0.917441i
$704$	0	0
$705$	401.546	0.569569
$706$	0	0
$707$	71.3980i	0.100987i
$708$	0	0
$709$	−169.124	−0.238539	−0.119270	−	0.992862i	$-0.538055\pi$
$709$	−169.124	−0.238539	−0.119270	+	0.992862i	$0.538055\pi$
$710$	0	0
$711$	1.63826i	0.00230417i
$712$	0	0
$713$	1.48339	0.00208049
$714$	0	0
$715$	− 464.989i	− 0.650334i
$716$	0	0
$717$	−1170.67	−1.63273
$718$	0	0
$719$	703.821i	0.978889i	0.872035	+	0.489444i	$0.162800\pi$
$719$	703.821i	0.978889i	−0.872035	+	0.489444i	$0.837200\pi$
$720$	0	0
$721$	−194.374	−0.269589
$722$	0	0
$723$	− 146.979i	− 0.203291i
$724$	0	0
$725$	193.673	0.267135
$726$	0	0
$727$	− 193.747i	− 0.266502i	−0.991082	−	0.133251i	$-0.957458\pi$
$727$	− 193.747i	− 0.266502i	0.991082	−	0.133251i	$-0.0425416\pi$
$728$	0	0
$729$	−744.828	−1.02171
$730$	0	0
$731$	− 3.24248i	− 0.00443567i
$732$	0	0
$733$	1193.35	1.62804	0.814020	−	0.580836i	$-0.197274\pi$
$733$	1193.35	1.62804	0.814020	+	0.580836i	$0.197274\pi$
$734$	0	0
$735$	− 145.084i	− 0.197394i
$736$	0	0
$737$	−807.645	−1.09585
$738$	0	0
$739$	− 794.890i	− 1.07563i	−0.843063	−	0.537815i	$-0.819250\pi$
$739$	− 794.890i	− 1.07563i	0.843063	−	0.537815i	$-0.180750\pi$
$740$	0	0
$741$	878.519	1.18559
$742$	0	0
$743$	− 956.847i	− 1.28782i	−0.765103	−	0.643908i	$-0.777312\pi$
$743$	− 956.847i	− 1.28782i	0.765103	−	0.643908i	$-0.222688\pi$
$744$	0	0
$745$	−271.893	−0.364957
$746$	0	0
$747$	− 4.91549i	− 0.00658031i
$748$	0	0
$749$	−869.555	−1.16095
$750$	0	0
$751$	643.234i	0.856503i	0.903660	+	0.428251i	$0.140870\pi$
$751$	643.234i	0.856503i	−0.903660	+	0.428251i	$0.859130\pi$
$752$	0	0
$753$	−470.803	−0.625236
$754$	0	0
$755$	412.768i	0.546713i
$756$	0	0
$757$	−942.304	−1.24479	−0.622394	−	0.782704i	$-0.713840\pi$
$757$	−942.304	−1.24479	−0.622394	+	0.782704i	$0.713840\pi$
$758$	0	0
$759$	− 121.871i	− 0.160568i
$760$	0	0
$761$	218.000	0.286466	0.143233	−	0.989689i	$-0.454250\pi$
$761$	218.000	0.286466	0.143233	+	0.989689i	$0.454250\pi$
$762$	0	0
$763$	− 824.289i	− 1.08033i
$764$	0	0
$765$	−0.0335729	−4.38861e−5 0
$766$	0	0
$767$	− 1764.92i	− 2.30107i
$768$	0	0
$769$	798.158	1.03792	0.518958	−	0.854800i	$-0.326320\pi$
$769$	798.158	1.03792	0.518958	+	0.854800i	$0.326320\pi$
$770$	0	0
$771$	− 50.5601i	− 0.0655772i
$772$	0	0
$773$	−1318.56	−1.70577	−0.852884	−	0.522100i	$-0.825149\pi$
$773$	−1318.56	−1.70577	−0.852884	+	0.522100i	$0.825149\pi$
$774$	0	0
$775$	1.54654i	0.00199553i
$776$	0	0
$777$	−816.070	−1.05028
$778$	0	0
$779$	− 518.822i	− 0.666010i
$780$	0	0
$781$	−666.136	−0.852927
$782$	0	0
$783$	− 1057.40i	− 1.35044i
$784$	0	0
$785$	373.813	0.476195
$786$	0	0
$787$	− 879.554i	− 1.11760i	−0.829301	−	0.558802i	$-0.811261\pi$
$787$	− 879.554i	− 1.11760i	0.829301	−	0.558802i	$-0.188739\pi$
$788$	0	0
$789$	−346.883	−0.439648
$790$	0	0
$791$	1107.49i	1.40011i
$792$	0	0
$793$	−745.107	−0.939606
$794$	0	0
$795$	− 427.547i	− 0.537795i
$796$	0	0
$797$	671.307	0.842293	0.421146	−	0.906993i	$-0.361628\pi$
$797$	671.307	0.842293	0.421146	+	0.906993i	$0.361628\pi$
$798$	0	0
$799$	− 4.43848i	− 0.00555505i
$800$	0	0
$801$	26.3215	0.0328607
$802$	0	0
$803$	403.447i	0.502424i
$804$	0	0
$805$	55.8480	0.0693764
$806$	0	0
$807$	− 497.168i	− 0.616069i
$808$	0	0
$809$	−635.552	−0.785603	−0.392801	−	0.919623i	$-0.628494\pi$
$809$	−635.552	−0.785603	−0.392801	+	0.919623i	$0.628494\pi$
$810$	0	0
$811$	1553.57i	1.91563i	0.287392	+	0.957813i	$0.407212\pi$
$811$	1553.57i	1.91563i	−0.287392	+	0.957813i	$0.592788\pi$
$812$	0	0
$813$	−490.330	−0.603112
$814$	0	0
$815$	495.671i	0.608185i
$816$	0	0
$817$	539.955	0.660900
$818$	0	0
$819$	− 25.8880i	− 0.0316092i
$820$	0	0
$821$	14.0415	0.0171029	0.00855146	−	0.999963i	$-0.497278\pi$
$821$	14.0415	0.0171029	0.00855146	+	0.999963i	$0.497278\pi$
$822$	0	0
$823$	1260.66i	1.53178i	0.642970	+	0.765892i	$0.277702\pi$
$823$	1260.66i	1.53178i	−0.642970	+	0.765892i	$0.722298\pi$
$824$	0	0
$825$	127.060	0.154012
$826$	0	0
$827$	− 816.660i	− 0.987497i	−0.869605	−	0.493749i	$-0.835626\pi$
$827$	− 816.660i	− 0.987497i	0.869605	−	0.493749i	$-0.164374\pi$
$828$	0	0
$829$	559.001	0.674308	0.337154	−	0.941450i	$-0.390536\pi$
$829$	559.001	0.674308	0.337154	+	0.941450i	$0.390536\pi$
$830$	0	0
$831$	1208.22i	1.45393i
$832$	0	0
$833$	−1.60369	−0.00192519
$834$	0	0
$835$	− 33.0238i	− 0.0395495i
$836$	0	0
$837$	8.44362	0.0100880
$838$	0	0
$839$	− 563.881i	− 0.672087i	−0.941846	−	0.336044i	$-0.890911\pi$
$839$	− 563.881i	− 0.672087i	0.941846	−	0.336044i	$-0.109089\pi$
$840$	0	0
$841$	659.372	0.784034
$842$	0	0
$843$	− 785.706i	− 0.932036i
$844$	0	0
$845$	939.054	1.11131
$846$	0	0
$847$	247.775i	0.292532i
$848$	0	0
$849$	−112.662	−0.132699
$850$	0	0
$851$	253.402i	0.297770i
$852$	0	0
$853$	−1060.97	−1.24381	−0.621904	−	0.783093i	$-0.713641\pi$
$853$	−1060.97	−1.24381	−0.621904	+	0.783093i	$0.713641\pi$
$854$	0	0
$855$	− 5.59074i	− 0.00653887i
$856$	0	0
$857$	1316.58	1.53626	0.768132	−	0.640291i	$-0.221186\pi$
$857$	1316.58	1.53626	0.768132	+	0.640291i	$0.221186\pi$
$858$	0	0
$859$	− 247.400i	− 0.288009i	−0.989577	−	0.144004i	$-0.954002\pi$
$859$	− 247.400i	− 0.288009i	0.989577	−	0.144004i	$-0.0459980\pi$
$860$	0	0
$861$	656.466	0.762446
$862$	0	0
$863$	− 199.534i	− 0.231210i	−0.993295	−	0.115605i	$-0.963119\pi$
$863$	− 199.534i	− 0.231210i	0.993295	−	0.115605i	$-0.0368807\pi$
$864$	0	0
$865$	−523.580	−0.605294
$866$	0	0
$867$	857.061i	0.988537i
$868$	0	0
$869$	−68.5333	−0.0788646
$870$	0	0
$871$	− 2287.43i	− 2.62621i
$872$	0	0
$873$	−32.0335	−0.0366936
$874$	0	0
$875$	58.2255i	0.0665435i
$876$	0	0
$877$	946.057	1.07874	0.539371	−	0.842068i	$-0.318662\pi$
$877$	946.057	1.07874	0.539371	+	0.842068i	$0.318662\pi$
$878$	0	0
$879$	− 1128.02i	− 1.28330i
$880$	0	0
$881$	1333.23	1.51331	0.756657	−	0.653812i	$-0.226831\pi$
$881$	1333.23	1.51331	0.756657	+	0.653812i	$0.226831\pi$
$882$	0	0
$883$	218.566i	0.247527i	0.992312	+	0.123764i	$0.0394964\pi$
$883$	218.566i	0.247527i	−0.992312	+	0.123764i	$0.960504\pi$
$884$	0	0
$885$	482.270	0.544938
$886$	0	0
$887$	− 1398.72i	− 1.57691i	−0.615090	−	0.788457i	$-0.710880\pi$
$887$	− 1398.72i	− 1.57691i	0.615090	−	0.788457i	$-0.289120\pi$
$888$	0	0
$889$	−980.316	−1.10272
$890$	0	0
$891$	− 677.910i	− 0.760841i
$892$	0	0
$893$	739.120	0.827682
$894$	0	0
$895$	139.519i	0.155887i
$896$	0	0
$897$	345.166	0.384801
$898$	0	0
$899$	11.9809i	0.0133269i
$900$	0	0
$901$	−4.72589	−0.00524516
$902$	0	0
$903$	683.206i	0.756595i
$904$	0	0
$905$	−9.91956	−0.0109608
$906$	0	0
$907$	− 343.241i	− 0.378435i	−0.981935	−	0.189218i	$-0.939405\pi$
$907$	− 343.241i	− 0.378435i	0.981935	−	0.189218i	$-0.0605952\pi$
$908$	0	0
$909$	2.80818	0.00308930
$910$	0	0
$911$	1318.38i	1.44718i	0.690231	+	0.723589i	$0.257509\pi$
$911$	1318.38i	1.44718i	−0.690231	+	0.723589i	$0.742491\pi$
$912$	0	0
$913$	205.629	0.225224
$914$	0	0
$915$	− 203.603i	− 0.222517i
$916$	0	0
$917$	−431.216	−0.470246
$918$	0	0
$919$	319.808i	0.347996i	0.984746	+	0.173998i	$0.0556686\pi$
$919$	319.808i	0.347996i	−0.984746	+	0.173998i	$0.944331\pi$
$920$	0	0
$921$	−1095.28	−1.18923
$922$	0	0
$923$	− 1886.64i	− 2.04403i
$924$	0	0
$925$	−264.190	−0.285611
$926$	0	0
$927$	7.64498i	0.00824701i
$928$	0	0
$929$	−1644.70	−1.77040	−0.885200	−	0.465211i	$-0.845979\pi$
$929$	−1644.70	−1.77040	−0.885200	+	0.465211i	$0.845979\pi$
$930$	0	0
$931$	− 267.055i	− 0.286847i
$932$	0	0
$933$	1016.30	1.08928
$934$	0	0
$935$	− 1.40445i	− 0.00150209i
$936$	0	0
$937$	588.388	0.627949	0.313975	−	0.949431i	$-0.398339\pi$
$937$	588.388	0.627949	0.313975	+	0.949431i	$0.398339\pi$
$938$	0	0
$939$	− 1465.61i	− 1.56081i
$940$	0	0
$941$	373.648	0.397075	0.198538	−	0.980093i	$-0.436381\pi$
$941$	373.648	0.397075	0.198538	+	0.980093i	$0.436381\pi$
$942$	0	0
$943$	− 203.843i	− 0.216164i
$944$	0	0
$945$	317.893	0.336395
$946$	0	0
$947$	568.386i	0.600197i	0.953908	+	0.300098i	$0.0970195\pi$
$947$	568.386i	0.600197i	−0.953908	+	0.300098i	$0.902980\pi$
$948$	0	0
$949$	−1142.65	−1.20406
$950$	0	0
$951$	459.266i	0.482929i
$952$	0	0
$953$	231.170	0.242570	0.121285	−	0.992618i	$-0.461298\pi$
$953$	231.170	0.242570	0.121285	+	0.992618i	$0.461298\pi$
$954$	0	0
$955$	− 171.591i	− 0.179676i
$956$	0	0
$957$	984.322	1.02855
$958$	0	0
$959$	533.359i	0.556161i
$960$	0	0
$961$	960.904	0.999900
$962$	0	0
$963$	34.2007i	0.0355148i
$964$	0	0
$965$	−116.221	−0.120437
$966$	0	0
$967$	1838.78i	1.90153i	0.309908	+	0.950767i	$0.399702\pi$
$967$	1838.78i	1.90153i	−0.309908	+	0.950767i	$0.600298\pi$
$968$	0	0
$969$	2.65347	0.00273836
$970$	0	0
$971$	475.034i	0.489222i	0.969621	+	0.244611i	$0.0786602\pi$
$971$	475.034i	0.489222i	−0.969621	+	0.244611i	$0.921340\pi$
$972$	0	0
$973$	1194.57	1.22772
$974$	0	0
$975$	359.861i	0.369088i
$976$	0	0
$977$	−142.591	−0.145948	−0.0729738	−	0.997334i	$-0.523249\pi$
$977$	−142.591	−0.145948	−0.0729738	+	0.997334i	$0.523249\pi$
$978$	0	0
$979$	1101.10i	1.12472i
$980$	0	0
$981$	−32.4204	−0.0330483
$982$	0	0
$983$	299.203i	0.304377i	0.988351	+	0.152188i	$0.0486321\pi$
$983$	299.203i	0.304377i	−0.988351	+	0.152188i	$0.951368\pi$
$984$	0	0
$985$	−31.2945	−0.0317711
$986$	0	0
$987$	935.210i	0.947527i
$988$	0	0
$989$	212.146	0.214505
$990$	0	0
$991$	1103.58i	1.11360i	0.830646	+	0.556802i	$0.187972\pi$
$991$	1103.58i	1.11360i	−0.830646	+	0.556802i	$0.812028\pi$
$992$	0	0
$993$	−1556.65	−1.56762
$994$	0	0
$995$	171.598i	0.172461i
$996$	0	0
$997$	138.703	0.139121	0.0695604	−	0.997578i	$-0.477840\pi$
$997$	138.703	0.139121	0.0695604	+	0.997578i	$0.477840\pi$
$998$	0	0
$999$	1442.40i	1.44384i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.c.b.1151.16	✓	56
4.3	odd	2	inner	1840.3.c.b.1151.41	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.3.c.b.1151.16	✓	56	1.1	even	1	trivial
1840.3.c.b.1151.41	yes	56	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.96566i q^{3} +2.23607 q^{5} +5.20785i q^{7} +0.204832 q^{9} +O(q^{10})\)	\(q-2.96566i q^{3} +2.23607 q^{5} +5.20785i q^{7} +0.204832 q^{9} +8.56871i q^{11} -24.2685 q^{13} -6.63143i q^{15} -0.0733004 q^{17} -12.2064i q^{19} +15.4447 q^{21} -4.79583i q^{23} +5.00000 q^{25} -27.2984i q^{27} +38.7346 q^{29} +0.309308i q^{31} +25.4119 q^{33} +11.6451i q^{35} -52.8381 q^{37} +71.9721i q^{39} +42.5042 q^{41} +44.2355i q^{43} +0.458018 q^{45} +60.5520i q^{47} +21.8783 q^{49} +0.217384i q^{51} +64.4729 q^{53} +19.1602i q^{55} -36.2000 q^{57} +72.7250i q^{59} +30.7027 q^{61} +1.06673i q^{63} -54.2659 q^{65} +94.2551i q^{67} -14.2228 q^{69} +77.7405i q^{71} +47.0837 q^{73} -14.8283i q^{75} -44.6246 q^{77} +7.99809i q^{79} -79.1146 q^{81} -23.9977i q^{83} -0.163905 q^{85} -114.874i q^{87} +128.503 q^{89} -126.387i q^{91} +0.917303 q^{93} -27.2943i q^{95} -156.389 q^{97} +1.75514i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 120 q^{9}+O(q^{10}) \) 56 * q - 120 * q^9	\( 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) \) 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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