LMFDB - Embedded newform 2100.4.k.g.1849.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2100,4,Mod(1849,2100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2100, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2100.1849");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2100.k (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:	$123.904011012$
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 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1849.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	2100.1849
Dual form			2100.4.k.g.1849.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} +O(q^{10})$	$q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} +4.00000 q^{11} -54.0000i q^{13} -14.0000i q^{17} -92.0000 q^{19} -21.0000 q^{21} +152.000i q^{23} +27.0000i q^{27} +106.000 q^{29} -144.000 q^{31} -12.0000i q^{33} +158.000i q^{37} -162.000 q^{39} -390.000 q^{41} +508.000i q^{43} -528.000i q^{47} -49.0000 q^{49} -42.0000 q^{51} -606.000i q^{53} +276.000i q^{57} +364.000 q^{59} +678.000 q^{61} +63.0000i q^{63} +844.000i q^{67} +456.000 q^{69} -8.00000 q^{71} +422.000i q^{73} -28.0000i q^{77} -384.000 q^{79} +81.0000 q^{81} +548.000i q^{83} -318.000i q^{87} -1194.00 q^{89} -378.000 q^{91} +432.000i q^{93} -1502.00i q^{97} -36.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{9}+O(q^{10}) $ 2 * q - 18 * q^9	$ 2 q - 18 q^{9} + 8 q^{11} - 184 q^{19} - 42 q^{21} + 212 q^{29} - 288 q^{31} - 324 q^{39} - 780 q^{41} - 98 q^{49} - 84 q^{51} + 728 q^{59} + 1356 q^{61} + 912 q^{69} - 16 q^{71} - 768 q^{79} + 162 q^{81} - 2388 q^{89} - 756 q^{91} - 72 q^{99}+O(q^{100}) $ 2 * q - 18 * q^9 + 8 * q^11 - 184 * q^19 - 42 * q^21 + 212 * q^29 - 288 * q^31 - 324 * q^39 - 780 * q^41 - 98 * q^49 - 84 * q^51 + 728 * q^59 + 1356 * q^61 + 912 * q^69 - 16 * q^71 - 768 * q^79 + 162 * q^81 - 2388 * q^89 - 756 * q^91 - 72 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times$.

$n$	$701$	$1051$	$1177$	$1501$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	− 3.00000i	− 0.577350i
$3$	− 3.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 7.00000i	− 0.377964i
$7$	− 7.00000i	− 0.377964i
$8$	0	0
$9$	−9.00000	−0.333333
$10$	0	0
$11$	4.00000	0.109640	0.0548202	−	0.998496i	$-0.482541\pi$
$11$	4.00000	0.109640	0.0548202	+	0.998496i	$0.482541\pi$
$12$	0	0
$13$	− 54.0000i	− 1.15207i	−0.817425	−	0.576035i	$-0.804599\pi$
$13$	− 54.0000i	− 1.15207i	0.817425	−	0.576035i	$-0.195401\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 14.0000i	− 0.199735i	−0.995001	−	0.0998676i	$-0.968158\pi$
$17$	− 14.0000i	− 0.199735i	0.995001	−	0.0998676i	$-0.0318419\pi$
$18$	0	0
$19$	−92.0000	−1.11086	−0.555428	−	0.831565i	$-0.687445\pi$
$19$	−92.0000	−1.11086	−0.555428	+	0.831565i	$0.687445\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	152.000i	1.37801i	0.724757	+	0.689004i	$0.241952\pi$
$23$	152.000i	1.37801i	−0.724757	+	0.689004i	$0.758048\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	27.0000i	0.192450i
$28$	0	0
$29$	106.000	0.678748	0.339374	−	0.940651i	$-0.389785\pi$
$29$	106.000	0.678748	0.339374	+	0.940651i	$0.389785\pi$
$30$	0	0
$31$	−144.000	−0.834296	−0.417148	−	0.908839i	$-0.636970\pi$
$31$	−144.000	−0.834296	−0.417148	+	0.908839i	$0.636970\pi$
$32$	0	0
$33$	− 12.0000i	− 0.0633010i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	158.000i	0.702028i	0.936370	+	0.351014i	$0.114163\pi$
$37$	158.000i	0.702028i	−0.936370	+	0.351014i	$0.885837\pi$
$38$	0	0
$39$	−162.000	−0.665148
$40$	0	0
$41$	−390.000	−1.48556	−0.742778	−	0.669538i	$-0.766492\pi$
$41$	−390.000	−1.48556	−0.742778	+	0.669538i	$0.766492\pi$
$42$	0	0
$43$	508.000i	1.80161i	0.434223	+	0.900806i	$0.357023\pi$
$43$	508.000i	1.80161i	−0.434223	+	0.900806i	$0.642977\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 528.000i	− 1.63865i	−0.573327	−	0.819327i	$-0.694347\pi$
$47$	− 528.000i	− 1.63865i	0.573327	−	0.819327i	$-0.305653\pi$
$48$	0	0
$49$	−49.0000	−0.142857
$50$	0	0
$51$	−42.0000	−0.115317
$52$	0	0
$53$	− 606.000i	− 1.57058i	−0.619131	−	0.785288i	$-0.712515\pi$
$53$	− 606.000i	− 1.57058i	0.619131	−	0.785288i	$-0.287485\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	276.000i	0.641353i
$58$	0	0
$59$	364.000	0.803199	0.401600	−	0.915815i	$-0.368454\pi$
$59$	364.000	0.803199	0.401600	+	0.915815i	$0.368454\pi$
$60$	0	0
$61$	678.000	1.42310	0.711549	−	0.702636i	$-0.247994\pi$
$61$	678.000	1.42310	0.711549	+	0.702636i	$0.247994\pi$
$62$	0	0
$63$	63.0000i	0.125988i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	844.000i	1.53897i	0.638665	+	0.769485i	$0.279487\pi$
$67$	844.000i	1.53897i	−0.638665	+	0.769485i	$0.720513\pi$
$68$	0	0
$69$	456.000	0.795593
$70$	0	0
$71$	−8.00000	−0.0133722	−0.00668609	−	0.999978i	$-0.502128\pi$
$71$	−8.00000	−0.0133722	−0.00668609	+	0.999978i	$0.502128\pi$
$72$	0	0
$73$	422.000i	0.676594i	0.941039	+	0.338297i	$0.109851\pi$
$73$	422.000i	0.676594i	−0.941039	+	0.338297i	$0.890149\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 28.0000i	− 0.0414402i
$78$	0	0
$79$	−384.000	−0.546878	−0.273439	−	0.961889i	$-0.588161\pi$
$79$	−384.000	−0.546878	−0.273439	+	0.961889i	$0.588161\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	548.000i	0.724709i	0.932040	+	0.362354i	$0.118027\pi$
$83$	548.000i	0.724709i	−0.932040	+	0.362354i	$0.881973\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 318.000i	− 0.391876i
$88$	0	0
$89$	−1194.00	−1.42206	−0.711032	−	0.703159i	$-0.751772\pi$
$89$	−1194.00	−1.42206	−0.711032	+	0.703159i	$0.751772\pi$
$90$	0	0
$91$	−378.000	−0.435441
$92$	0	0
$93$	432.000i	0.481681i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1502.00i	− 1.57222i	−0.618089	−	0.786108i	$-0.712093\pi$
$97$	− 1502.00i	− 1.57222i	0.618089	−	0.786108i	$-0.287907\pi$
$98$	0	0
$99$	−36.0000	−0.0365468
$100$	0	0
$101$	398.000	0.392104	0.196052	−	0.980594i	$-0.437188\pi$
$101$	398.000	0.392104	0.196052	+	0.980594i	$0.437188\pi$
$102$	0	0
$103$	− 1160.00i	− 1.10969i	−0.831953	−	0.554846i	$-0.812777\pi$
$103$	− 1160.00i	− 1.10969i	0.831953	−	0.554846i	$-0.187223\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	324.000i	0.292731i	0.989231	+	0.146366i	$0.0467576\pi$
$107$	324.000i	0.292731i	−0.989231	+	0.146366i	$0.953242\pi$
$108$	0	0
$109$	938.000	0.824258	0.412129	−	0.911126i	$-0.364785\pi$
$109$	938.000	0.824258	0.412129	+	0.911126i	$0.364785\pi$
$110$	0	0
$111$	474.000	0.405316
$112$	0	0
$113$	622.000i	0.517813i	0.965902	+	0.258906i	$0.0833621\pi$
$113$	622.000i	0.517813i	−0.965902	+	0.258906i	$0.916638\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	486.000i	0.384023i
$118$	0	0
$119$	−98.0000	−0.0754928
$120$	0	0
$121$	−1315.00	−0.987979
$122$	0	0
$123$	1170.00i	0.857686i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1200.00i	0.838447i	0.907883	+	0.419224i	$0.137698\pi$
$127$	1200.00i	0.838447i	−0.907883	+	0.419224i	$0.862302\pi$
$128$	0	0
$129$	1524.00	1.04016
$130$	0	0
$131$	−1396.00	−0.931062	−0.465531	−	0.885032i	$-0.654137\pi$
$131$	−1396.00	−0.931062	−0.465531	+	0.885032i	$0.654137\pi$
$132$	0	0
$133$	644.000i	0.419864i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2810.00i	1.75237i	0.481976	+	0.876184i	$0.339919\pi$
$137$	2810.00i	1.75237i	−0.481976	+	0.876184i	$0.660081\pi$
$138$	0	0
$139$	−4.00000	−0.00244083	−0.00122042	−	0.999999i	$-0.500388\pi$
$139$	−4.00000	−0.00244083	−0.00122042	+	0.999999i	$0.500388\pi$
$140$	0	0
$141$	−1584.00	−0.946077
$142$	0	0
$143$	− 216.000i	− 0.126313i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	147.000i	0.0824786i
$148$	0	0
$149$	−1374.00	−0.755453	−0.377726	−	0.925917i	$-0.623294\pi$
$149$	−1374.00	−0.755453	−0.377726	+	0.925917i	$0.623294\pi$
$150$	0	0
$151$	2104.00	1.13391	0.566957	−	0.823747i	$-0.308120\pi$
$151$	2104.00	1.13391	0.566957	+	0.823747i	$0.308120\pi$
$152$	0	0
$153$	126.000i	0.0665784i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3206.00i	1.62972i	0.579655	+	0.814862i	$0.303187\pi$
$157$	3206.00i	1.62972i	−0.579655	+	0.814862i	$0.696813\pi$
$158$	0	0
$159$	−1818.00	−0.906772
$160$	0	0
$161$	1064.00	0.520838
$162$	0	0
$163$	− 332.000i	− 0.159535i	−0.996813	−	0.0797676i	$-0.974582\pi$
$163$	− 332.000i	− 0.159535i	0.996813	−	0.0797676i	$-0.0254178\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1496.00i	0.693197i	0.938014	+	0.346599i	$0.112663\pi$
$167$	1496.00i	0.693197i	−0.938014	+	0.346599i	$0.887337\pi$
$168$	0	0
$169$	−719.000	−0.327264
$170$	0	0
$171$	828.000	0.370285
$172$	0	0
$173$	3322.00i	1.45992i	0.683487	+	0.729962i	$0.260462\pi$
$173$	3322.00i	1.45992i	−0.683487	+	0.729962i	$0.739538\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1092.00i	− 0.463727i
$178$	0	0
$179$	900.000	0.375805	0.187903	−	0.982188i	$-0.439831\pi$
$179$	900.000	0.375805	0.187903	+	0.982188i	$0.439831\pi$
$180$	0	0
$181$	1902.00	0.781075	0.390537	−	0.920587i	$-0.372289\pi$
$181$	1902.00	0.781075	0.390537	+	0.920587i	$0.372289\pi$
$182$	0	0
$183$	− 2034.00i	− 0.821626i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 56.0000i	− 0.0218991i
$188$	0	0
$189$	189.000	0.0727393
$190$	0	0
$191$	4128.00	1.56383	0.781915	−	0.623385i	$-0.214243\pi$
$191$	4128.00	1.56383	0.781915	+	0.623385i	$0.214243\pi$
$192$	0	0
$193$	1342.00i	0.500514i	0.968179	+	0.250257i	$0.0805152\pi$
$193$	1342.00i	0.500514i	−0.968179	+	0.250257i	$0.919485\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 3506.00i	− 1.26798i	−0.773341	−	0.633990i	$-0.781416\pi$
$197$	− 3506.00i	− 1.26798i	0.773341	−	0.633990i	$-0.218584\pi$
$198$	0	0
$199$	−680.000	−0.242231	−0.121115	−	0.992638i	$-0.538647\pi$
$199$	−680.000	−0.242231	−0.121115	+	0.992638i	$0.538647\pi$
$200$	0	0
$201$	2532.00	0.888525
$202$	0	0
$203$	− 742.000i	− 0.256543i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1368.00i	− 0.459336i
$208$	0	0
$209$	−368.000	−0.121795
$210$	0	0
$211$	5372.00	1.75272	0.876360	−	0.481657i	$-0.159965\pi$
$211$	5372.00	1.75272	0.876360	+	0.481657i	$0.159965\pi$
$212$	0	0
$213$	24.0000i	0.00772044i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1008.00i	0.315334i
$218$	0	0
$219$	1266.00	0.390632
$220$	0	0
$221$	−756.000	−0.230109
$222$	0	0
$223$	1072.00i	0.321912i	0.986962	+	0.160956i	$0.0514578\pi$
$223$	1072.00i	0.321912i	−0.986962	+	0.160956i	$0.948542\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2868.00i	− 0.838572i	−0.907854	−	0.419286i	$-0.862280\pi$
$227$	− 2868.00i	− 0.838572i	0.907854	−	0.419286i	$-0.137720\pi$
$228$	0	0
$229$	−4798.00	−1.38454	−0.692272	−	0.721636i	$-0.743390\pi$
$229$	−4798.00	−1.38454	−0.692272	+	0.721636i	$0.743390\pi$
$230$	0	0
$231$	−84.0000	−0.0239255
$232$	0	0
$233$	5126.00i	1.44127i	0.693316	+	0.720634i	$0.256149\pi$
$233$	5126.00i	1.44127i	−0.693316	+	0.720634i	$0.743851\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1152.00i	0.315740i
$238$	0	0
$239$	528.000	0.142902	0.0714508	−	0.997444i	$-0.477237\pi$
$239$	528.000	0.142902	0.0714508	+	0.997444i	$0.477237\pi$
$240$	0	0
$241$	−814.000	−0.217570	−0.108785	−	0.994065i	$-0.534696\pi$
$241$	−814.000	−0.217570	−0.108785	+	0.994065i	$0.534696\pi$
$242$	0	0
$243$	− 243.000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4968.00i	1.27978i
$248$	0	0
$249$	1644.00	0.418411
$250$	0	0
$251$	−1932.00	−0.485844	−0.242922	−	0.970046i	$-0.578106\pi$
$251$	−1932.00	−0.485844	−0.242922	+	0.970046i	$0.578106\pi$
$252$	0	0
$253$	608.000i	0.151086i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3294.00i	− 0.799510i	−0.916622	−	0.399755i	$-0.869095\pi$
$257$	− 3294.00i	− 0.799510i	0.916622	−	0.399755i	$-0.130905\pi$
$258$	0	0
$259$	1106.00	0.265342
$260$	0	0
$261$	−954.000	−0.226249
$262$	0	0
$263$	7080.00i	1.65997i	0.557787	+	0.829984i	$0.311650\pi$
$263$	7080.00i	1.65997i	−0.557787	+	0.829984i	$0.688350\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3582.00i	0.821029i
$268$	0	0
$269$	−7814.00	−1.77111	−0.885554	−	0.464537i	$-0.846221\pi$
$269$	−7814.00	−1.77111	−0.885554	+	0.464537i	$0.846221\pi$
$270$	0	0
$271$	3168.00	0.710119	0.355060	−	0.934844i	$-0.384461\pi$
$271$	3168.00	0.710119	0.355060	+	0.934844i	$0.384461\pi$
$272$	0	0
$273$	1134.00i	0.251402i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 7858.00i	− 1.70448i	−0.523150	−	0.852241i	$-0.675243\pi$
$277$	− 7858.00i	− 1.70448i	0.523150	−	0.852241i	$-0.324757\pi$
$278$	0	0
$279$	1296.00	0.278099
$280$	0	0
$281$	6730.00	1.42875	0.714374	−	0.699764i	$-0.246712\pi$
$281$	6730.00	1.42875	0.714374	+	0.699764i	$0.246712\pi$
$282$	0	0
$283$	3020.00i	0.634348i	0.948367	+	0.317174i	$0.102734\pi$
$283$	3020.00i	0.634348i	−0.948367	+	0.317174i	$0.897266\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2730.00i	0.561487i
$288$	0	0
$289$	4717.00	0.960106
$290$	0	0
$291$	−4506.00	−0.907720
$292$	0	0
$293$	6834.00i	1.36262i	0.731997	+	0.681308i	$0.238589\pi$
$293$	6834.00i	1.36262i	−0.731997	+	0.681308i	$0.761411\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	108.000i	0.0211003i
$298$	0	0
$299$	8208.00	1.58756
$300$	0	0
$301$	3556.00	0.680945
$302$	0	0
$303$	− 1194.00i	− 0.226381i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2332.00i	0.433532i	0.976224	+	0.216766i	$0.0695508\pi$
$307$	2332.00i	0.433532i	−0.976224	+	0.216766i	$0.930449\pi$
$308$	0	0
$309$	−3480.00	−0.640681
$310$	0	0
$311$	8840.00	1.61180	0.805901	−	0.592050i	$-0.201681\pi$
$311$	8840.00	1.61180	0.805901	+	0.592050i	$0.201681\pi$
$312$	0	0
$313$	1046.00i	0.188893i	0.995530	+	0.0944464i	$0.0301081\pi$
$313$	1046.00i	0.188893i	−0.995530	+	0.0944464i	$0.969892\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7542.00i	1.33628i	0.744035	+	0.668140i	$0.232909\pi$
$317$	7542.00i	1.33628i	−0.744035	+	0.668140i	$0.767091\pi$
$318$	0	0
$319$	424.000	0.0744183
$320$	0	0
$321$	972.000	0.169009
$322$	0	0
$323$	1288.00i	0.221877i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2814.00i	− 0.475885i
$328$	0	0
$329$	−3696.00	−0.619353
$330$	0	0
$331$	2756.00	0.457654	0.228827	−	0.973467i	$-0.426511\pi$
$331$	2756.00	0.457654	0.228827	+	0.973467i	$0.426511\pi$
$332$	0	0
$333$	− 1422.00i	− 0.234009i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3954.00i	0.639134i	0.947564	+	0.319567i	$0.103537\pi$
$337$	3954.00i	0.639134i	−0.947564	+	0.319567i	$0.896463\pi$
$338$	0	0
$339$	1866.00	0.298959
$340$	0	0
$341$	−576.000	−0.0914726
$342$	0	0
$343$	343.000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6900.00i	1.06747i	0.845652	+	0.533734i	$0.179212\pi$
$347$	6900.00i	1.06747i	−0.845652	+	0.533734i	$0.820788\pi$
$348$	0	0
$349$	2426.00	0.372094	0.186047	−	0.982541i	$-0.440432\pi$
$349$	2426.00	0.372094	0.186047	+	0.982541i	$0.440432\pi$
$350$	0	0
$351$	1458.00	0.221716
$352$	0	0
$353$	1470.00i	0.221644i	0.993840	+	0.110822i	$0.0353483\pi$
$353$	1470.00i	0.221644i	−0.993840	+	0.110822i	$0.964652\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	294.000i	0.0435858i
$358$	0	0
$359$	−6872.00	−1.01028	−0.505140	−	0.863038i	$-0.668559\pi$
$359$	−6872.00	−1.01028	−0.505140	+	0.863038i	$0.668559\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	3945.00i	0.570410i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 7072.00i	− 1.00587i	−0.864323	−	0.502937i	$-0.832253\pi$
$367$	− 7072.00i	− 1.00587i	0.864323	−	0.502937i	$-0.167747\pi$
$368$	0	0
$369$	3510.00	0.495185
$370$	0	0
$371$	−4242.00	−0.593622
$372$	0	0
$373$	818.000i	0.113551i	0.998387	+	0.0567754i	$0.0180819\pi$
$373$	818.000i	0.113551i	−0.998387	+	0.0567754i	$0.981918\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5724.00i	− 0.781966i
$378$	0	0
$379$	5132.00	0.695549	0.347775	−	0.937578i	$-0.386937\pi$
$379$	5132.00	0.695549	0.347775	+	0.937578i	$0.386937\pi$
$380$	0	0
$381$	3600.00	0.484078
$382$	0	0
$383$	8576.00i	1.14416i	0.820198	+	0.572080i	$0.193863\pi$
$383$	8576.00i	1.14416i	−0.820198	+	0.572080i	$0.806137\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 4572.00i	− 0.600537i
$388$	0	0
$389$	3730.00	0.486166	0.243083	−	0.970006i	$-0.421841\pi$
$389$	3730.00	0.486166	0.243083	+	0.970006i	$0.421841\pi$
$390$	0	0
$391$	2128.00	0.275237
$392$	0	0
$393$	4188.00i	0.537549i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	6678.00i	0.844230i	0.906542	+	0.422115i	$0.138712\pi$
$397$	6678.00i	0.844230i	−0.906542	+	0.422115i	$0.861288\pi$
$398$	0	0
$399$	1932.00	0.242408
$400$	0	0
$401$	−3054.00	−0.380323	−0.190161	−	0.981753i	$-0.560901\pi$
$401$	−3054.00	−0.380323	−0.190161	+	0.981753i	$0.560901\pi$
$402$	0	0
$403$	7776.00i	0.961167i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	632.000i	0.0769707i
$408$	0	0
$409$	−266.000	−0.0321586	−0.0160793	−	0.999871i	$-0.505118\pi$
$409$	−266.000	−0.0321586	−0.0160793	+	0.999871i	$0.505118\pi$
$410$	0	0
$411$	8430.00	1.01173
$412$	0	0
$413$	− 2548.00i	− 0.303581i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.0000i	0.00140921i
$418$	0	0
$419$	−8844.00	−1.03116	−0.515582	−	0.856840i	$-0.672424\pi$
$419$	−8844.00	−1.03116	−0.515582	+	0.856840i	$0.672424\pi$
$420$	0	0
$421$	−4482.00	−0.518858	−0.259429	−	0.965762i	$-0.583534\pi$
$421$	−4482.00	−0.518858	−0.259429	+	0.965762i	$0.583534\pi$
$422$	0	0
$423$	4752.00i	0.546218i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4746.00i	− 0.537881i
$428$	0	0
$429$	−648.000	−0.0729271
$430$	0	0
$431$	9936.00	1.11044	0.555221	−	0.831703i	$-0.312634\pi$
$431$	9936.00	1.11044	0.555221	+	0.831703i	$0.312634\pi$
$432$	0	0
$433$	11758.0i	1.30497i	0.757800	+	0.652487i	$0.226274\pi$
$433$	11758.0i	1.30497i	−0.757800	+	0.652487i	$0.773726\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 13984.0i	− 1.53077i
$438$	0	0
$439$	4104.00	0.446180	0.223090	−	0.974798i	$-0.428386\pi$
$439$	4104.00	0.446180	0.223090	+	0.974798i	$0.428386\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	− 9748.00i	− 1.04547i	−0.852496	−	0.522733i	$-0.824912\pi$
$443$	− 9748.00i	− 1.04547i	0.852496	−	0.522733i	$-0.175088\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4122.00i	0.436161i
$448$	0	0
$449$	478.000	0.0502410	0.0251205	−	0.999684i	$-0.492003\pi$
$449$	478.000	0.0502410	0.0251205	+	0.999684i	$0.492003\pi$
$450$	0	0
$451$	−1560.00	−0.162877
$452$	0	0
$453$	− 6312.00i	− 0.654666i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 11174.0i	− 1.14376i	−0.820338	−	0.571879i	$-0.806215\pi$
$457$	− 11174.0i	− 1.14376i	0.820338	−	0.571879i	$-0.193785\pi$
$458$	0	0
$459$	378.000	0.0384391
$460$	0	0
$461$	−11674.0	−1.17942	−0.589710	−	0.807615i	$-0.700758\pi$
$461$	−11674.0	−1.17942	−0.589710	+	0.807615i	$0.700758\pi$
$462$	0	0
$463$	− 10528.0i	− 1.05676i	−0.849009	−	0.528378i	$-0.822801\pi$
$463$	− 10528.0i	− 1.05676i	0.849009	−	0.528378i	$-0.177199\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	16604.0i	1.64527i	0.568569	+	0.822635i	$0.307497\pi$
$467$	16604.0i	1.64527i	−0.568569	+	0.822635i	$0.692503\pi$
$468$	0	0
$469$	5908.00	0.581676
$470$	0	0
$471$	9618.00	0.940922
$472$	0	0
$473$	2032.00i	0.197530i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5454.00i	0.523525i
$478$	0	0
$479$	8576.00	0.818053	0.409027	−	0.912522i	$-0.365868\pi$
$479$	8576.00	0.818053	0.409027	+	0.912522i	$0.365868\pi$
$480$	0	0
$481$	8532.00	0.808785
$482$	0	0
$483$	− 3192.00i	− 0.300706i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9704.00i	0.902937i	0.892287	+	0.451468i	$0.149100\pi$
$487$	9704.00i	0.902937i	−0.892287	+	0.451468i	$0.850900\pi$
$488$	0	0
$489$	−996.000	−0.0921077
$490$	0	0
$491$	−4092.00	−0.376109	−0.188054	−	0.982159i	$-0.560218\pi$
$491$	−4092.00	−0.376109	−0.188054	+	0.982159i	$0.560218\pi$
$492$	0	0
$493$	− 1484.00i	− 0.135570i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	56.0000i	0.00505421i
$498$	0	0
$499$	−17884.0	−1.60440	−0.802202	−	0.597052i	$-0.796338\pi$
$499$	−17884.0	−1.60440	−0.802202	+	0.597052i	$0.796338\pi$
$500$	0	0
$501$	4488.00	0.400218
$502$	0	0
$503$	7704.00i	0.682911i	0.939898	+	0.341456i	$0.110920\pi$
$503$	7704.00i	0.682911i	−0.939898	+	0.341456i	$0.889080\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2157.00i	0.188946i
$508$	0	0
$509$	−14358.0	−1.25031	−0.625154	−	0.780501i	$-0.714964\pi$
$509$	−14358.0	−1.25031	−0.625154	+	0.780501i	$0.714964\pi$
$510$	0	0
$511$	2954.00	0.255729
$512$	0	0
$513$	− 2484.00i	− 0.213784i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2112.00i	− 0.179663i
$518$	0	0
$519$	9966.00	0.842888
$520$	0	0
$521$	5082.00	0.427344	0.213672	−	0.976905i	$-0.431458\pi$
$521$	5082.00	0.427344	0.213672	+	0.976905i	$0.431458\pi$
$522$	0	0
$523$	1756.00i	0.146816i	0.997302	+	0.0734078i	$0.0233875\pi$
$523$	1756.00i	0.146816i	−0.997302	+	0.0734078i	$0.976613\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2016.00i	0.166638i
$528$	0	0
$529$	−10937.0	−0.898907
$530$	0	0
$531$	−3276.00	−0.267733
$532$	0	0
$533$	21060.0i	1.71146i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2700.00i	− 0.216971i
$538$	0	0
$539$	−196.000	−0.0156629
$540$	0	0
$541$	16230.0	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	16230.0	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	− 5706.00i	− 0.450954i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	17676.0i	1.38167i	0.723014	+	0.690833i	$0.242756\pi$
$547$	17676.0i	1.38167i	−0.723014	+	0.690833i	$0.757244\pi$
$548$	0	0
$549$	−6102.00	−0.474366
$550$	0	0
$551$	−9752.00	−0.753991
$552$	0	0
$553$	2688.00i	0.206701i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 12250.0i	− 0.931866i	−0.884820	−	0.465933i	$-0.845719\pi$
$557$	− 12250.0i	− 0.931866i	0.884820	−	0.465933i	$-0.154281\pi$
$558$	0	0
$559$	27432.0	2.07558
$560$	0	0
$561$	−168.000	−0.0126434
$562$	0	0
$563$	10052.0i	0.752471i	0.926524	+	0.376236i	$0.122782\pi$
$563$	10052.0i	0.752471i	−0.926524	+	0.376236i	$0.877218\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 567.000i	− 0.0419961i
$568$	0	0
$569$	−25674.0	−1.89158	−0.945791	−	0.324776i	$-0.894711\pi$
$569$	−25674.0	−1.89158	−0.945791	+	0.324776i	$0.894711\pi$
$570$	0	0
$571$	3732.00	0.273519	0.136759	−	0.990604i	$-0.456331\pi$
$571$	3732.00	0.273519	0.136759	+	0.990604i	$0.456331\pi$
$572$	0	0
$573$	− 12384.0i	− 0.902878i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1214.00i	− 0.0875901i	−0.999041	−	0.0437950i	$-0.986055\pi$
$577$	− 1214.00i	− 0.0875901i	0.999041	−	0.0437950i	$-0.0139449\pi$
$578$	0	0
$579$	4026.00	0.288972
$580$	0	0
$581$	3836.00	0.273914
$582$	0	0
$583$	− 2424.00i	− 0.172199i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7108.00i	0.499793i	0.968273	+	0.249897i	$0.0803966\pi$
$587$	7108.00i	0.499793i	−0.968273	+	0.249897i	$0.919603\pi$
$588$	0	0
$589$	13248.0	0.926782
$590$	0	0
$591$	−10518.0	−0.732069
$592$	0	0
$593$	− 6162.00i	− 0.426717i	−0.976974	−	0.213358i	$-0.931560\pi$
$593$	− 6162.00i	− 0.426717i	0.976974	−	0.213358i	$-0.0684402\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2040.00i	0.139852i
$598$	0	0
$599$	−2472.00	−0.168620	−0.0843098	−	0.996440i	$-0.526869\pi$
$599$	−2472.00	−0.168620	−0.0843098	+	0.996440i	$0.526869\pi$
$600$	0	0
$601$	−13750.0	−0.933235	−0.466617	−	0.884459i	$-0.654528\pi$
$601$	−13750.0	−0.933235	−0.466617	+	0.884459i	$0.654528\pi$
$602$	0	0
$603$	− 7596.00i	− 0.512990i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 11376.0i	− 0.760688i	−0.924845	−	0.380344i	$-0.875806\pi$
$607$	− 11376.0i	− 0.760688i	0.924845	−	0.380344i	$-0.124194\pi$
$608$	0	0
$609$	−2226.00	−0.148115
$610$	0	0
$611$	−28512.0	−1.88784
$612$	0	0
$613$	− 20382.0i	− 1.34294i	−0.741032	−	0.671469i	$-0.765664\pi$
$613$	− 20382.0i	− 1.34294i	0.741032	−	0.671469i	$-0.234336\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21178.0i	1.38184i	0.722932	+	0.690919i	$0.242794\pi$
$617$	21178.0i	1.38184i	−0.722932	+	0.690919i	$0.757206\pi$
$618$	0	0
$619$	4700.00	0.305184	0.152592	−	0.988289i	$-0.451238\pi$
$619$	4700.00	0.305184	0.152592	+	0.988289i	$0.451238\pi$
$620$	0	0
$621$	−4104.00	−0.265198
$622$	0	0
$623$	8358.00i	0.537490i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1104.00i	0.0703182i
$628$	0	0
$629$	2212.00	0.140220
$630$	0	0
$631$	−21736.0	−1.37131	−0.685655	−	0.727927i	$-0.740484\pi$
$631$	−21736.0	−1.37131	−0.685655	+	0.727927i	$0.740484\pi$
$632$	0	0
$633$	− 16116.0i	− 1.01193i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2646.00i	0.164581i
$638$	0	0
$639$	72.0000	0.00445740
$640$	0	0
$641$	−13022.0	−0.802399	−0.401200	−	0.915991i	$-0.631407\pi$
$641$	−13022.0	−0.802399	−0.401200	+	0.915991i	$0.631407\pi$
$642$	0	0
$643$	− 3308.00i	− 0.202885i	−0.994841	−	0.101442i	$-0.967654\pi$
$643$	− 3308.00i	− 0.202885i	0.994841	−	0.101442i	$-0.0323457\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 13800.0i	− 0.838538i	−0.907862	−	0.419269i	$-0.862286\pi$
$647$	− 13800.0i	− 0.838538i	0.907862	−	0.419269i	$-0.137714\pi$
$648$	0	0
$649$	1456.00	0.0880632
$650$	0	0
$651$	3024.00	0.182058
$652$	0	0
$653$	2682.00i	0.160727i	0.996766	+	0.0803635i	$0.0256081\pi$
$653$	2682.00i	0.160727i	−0.996766	+	0.0803635i	$0.974392\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 3798.00i	− 0.225531i
$658$	0	0
$659$	−23836.0	−1.40898	−0.704491	−	0.709713i	$-0.748825\pi$
$659$	−23836.0	−1.40898	−0.704491	+	0.709713i	$0.748825\pi$
$660$	0	0
$661$	−11282.0	−0.663871	−0.331936	−	0.943302i	$-0.607702\pi$
$661$	−11282.0	−0.663871	−0.331936	+	0.943302i	$0.607702\pi$
$662$	0	0
$663$	2268.00i	0.132853i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16112.0i	0.935321i
$668$	0	0
$669$	3216.00	0.185856
$670$	0	0
$671$	2712.00	0.156029
$672$	0	0
$673$	13726.0i	0.786179i	0.919500	+	0.393089i	$0.128594\pi$
$673$	13726.0i	0.786179i	−0.919500	+	0.393089i	$0.871406\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4974.00i	0.282373i	0.989983	+	0.141186i	$0.0450917\pi$
$677$	4974.00i	0.282373i	−0.989983	+	0.141186i	$0.954908\pi$
$678$	0	0
$679$	−10514.0	−0.594242
$680$	0	0
$681$	−8604.00	−0.484150
$682$	0	0
$683$	8988.00i	0.503538i	0.967787	+	0.251769i	$0.0810123\pi$
$683$	8988.00i	0.503538i	−0.967787	+	0.251769i	$0.918988\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	14394.0i	0.799367i
$688$	0	0
$689$	−32724.0	−1.80941
$690$	0	0
$691$	10172.0	0.560002	0.280001	−	0.960000i	$-0.409665\pi$
$691$	10172.0	0.560002	0.280001	+	0.960000i	$0.409665\pi$
$692$	0	0
$693$	252.000i	0.0138134i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5460.00i	0.296718i
$698$	0	0
$699$	15378.0	0.832116
$700$	0	0
$701$	27446.0	1.47877	0.739387	−	0.673280i	$-0.235115\pi$
$701$	27446.0	1.47877	0.739387	+	0.673280i	$0.235115\pi$
$702$	0	0
$703$	− 14536.0i	− 0.779852i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2786.00i	− 0.148201i
$708$	0	0
$709$	−3998.00	−0.211774	−0.105887	−	0.994378i	$-0.533768\pi$
$709$	−3998.00	−0.211774	−0.105887	+	0.994378i	$0.533768\pi$
$710$	0	0
$711$	3456.00	0.182293
$712$	0	0
$713$	− 21888.0i	− 1.14967i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1584.00i	− 0.0825043i
$718$	0	0
$719$	−25872.0	−1.34195	−0.670976	−	0.741480i	$-0.734124\pi$
$719$	−25872.0	−1.34195	−0.670976	+	0.741480i	$0.734124\pi$
$720$	0	0
$721$	−8120.00	−0.419424
$722$	0	0
$723$	2442.00i	0.125614i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	12088.0i	0.616670i	0.951278	+	0.308335i	$0.0997718\pi$
$727$	12088.0i	0.616670i	−0.951278	+	0.308335i	$0.900228\pi$
$728$	0	0
$729$	−729.000	−0.0370370
$730$	0	0
$731$	7112.00	0.359845
$732$	0	0
$733$	− 7974.00i	− 0.401810i	−0.979611	−	0.200905i	$-0.935612\pi$
$733$	− 7974.00i	− 0.401810i	0.979611	−	0.200905i	$-0.0643882\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3376.00i	0.168733i
$738$	0	0
$739$	31764.0	1.58113	0.790567	−	0.612376i	$-0.209786\pi$
$739$	31764.0	1.58113	0.790567	+	0.612376i	$0.209786\pi$
$740$	0	0
$741$	14904.0	0.738883
$742$	0	0
$743$	− 888.000i	− 0.0438460i	−0.999760	−	0.0219230i	$-0.993021\pi$
$743$	− 888.000i	− 0.0438460i	0.999760	−	0.0219230i	$-0.00697886\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 4932.00i	− 0.241570i
$748$	0	0
$749$	2268.00	0.110642
$750$	0	0
$751$	−34656.0	−1.68391	−0.841954	−	0.539549i	$-0.818595\pi$
$751$	−34656.0	−1.68391	−0.841954	+	0.539549i	$0.818595\pi$
$752$	0	0
$753$	5796.00i	0.280502i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 22866.0i	− 1.09786i	−0.835869	−	0.548929i	$-0.815036\pi$
$757$	− 22866.0i	− 1.09786i	0.835869	−	0.548929i	$-0.184964\pi$
$758$	0	0
$759$	1824.00	0.0872293
$760$	0	0
$761$	22570.0	1.07511	0.537557	−	0.843227i	$-0.319347\pi$
$761$	22570.0	1.07511	0.537557	+	0.843227i	$0.319347\pi$
$762$	0	0
$763$	− 6566.00i	− 0.311540i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 19656.0i	− 0.925342i
$768$	0	0
$769$	1790.00	0.0839389	0.0419695	−	0.999119i	$-0.486637\pi$
$769$	1790.00	0.0839389	0.0419695	+	0.999119i	$0.486637\pi$
$770$	0	0
$771$	−9882.00	−0.461597
$772$	0	0
$773$	− 2990.00i	− 0.139124i	−0.997578	−	0.0695620i	$-0.977840\pi$
$773$	− 2990.00i	− 0.139124i	0.997578	−	0.0695620i	$-0.0221602\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3318.00i	− 0.153195i
$778$	0	0
$779$	35880.0	1.65024
$780$	0	0
$781$	−32.0000	−0.00146613
$782$	0	0
$783$	2862.00i	0.130625i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 30756.0i	− 1.39305i	−0.717531	−	0.696527i	$-0.754728\pi$
$787$	− 30756.0i	− 1.39305i	0.717531	−	0.696527i	$-0.245272\pi$
$788$	0	0
$789$	21240.0	0.958383
$790$	0	0
$791$	4354.00	0.195715
$792$	0	0
$793$	− 36612.0i	− 1.63951i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15126.0i	0.672259i	0.941816	+	0.336129i	$0.109118\pi$
$797$	15126.0i	0.672259i	−0.941816	+	0.336129i	$0.890882\pi$
$798$	0	0
$799$	−7392.00	−0.327297
$800$	0	0
$801$	10746.0	0.474022
$802$	0	0
$803$	1688.00i	0.0741821i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	23442.0i	1.02255i
$808$	0	0
$809$	6502.00	0.282569	0.141284	−	0.989969i	$-0.454877\pi$
$809$	6502.00	0.282569	0.141284	+	0.989969i	$0.454877\pi$
$810$	0	0
$811$	−8252.00	−0.357296	−0.178648	−	0.983913i	$-0.557172\pi$
$811$	−8252.00	−0.357296	−0.178648	+	0.983913i	$0.557172\pi$
$812$	0	0
$813$	− 9504.00i	− 0.409987i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 46736.0i	− 2.00133i
$818$	0	0
$819$	3402.00	0.145147
$820$	0	0
$821$	−21986.0	−0.934612	−0.467306	−	0.884096i	$-0.654775\pi$
$821$	−21986.0	−0.934612	−0.467306	+	0.884096i	$0.654775\pi$
$822$	0	0
$823$	− 3736.00i	− 0.158237i	−0.996865	−	0.0791183i	$-0.974790\pi$
$823$	− 3736.00i	− 0.158237i	0.996865	−	0.0791183i	$-0.0252105\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 23820.0i	− 1.00158i	−0.865570	−	0.500788i	$-0.833044\pi$
$827$	− 23820.0i	− 1.00158i	0.865570	−	0.500788i	$-0.166956\pi$
$828$	0	0
$829$	−7942.00	−0.332735	−0.166367	−	0.986064i	$-0.553204\pi$
$829$	−7942.00	−0.332735	−0.166367	+	0.986064i	$0.553204\pi$
$830$	0	0
$831$	−23574.0	−0.984083
$832$	0	0
$833$	686.000i	0.0285336i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 3888.00i	− 0.160560i
$838$	0	0
$839$	−21016.0	−0.864783	−0.432391	−	0.901686i	$-0.642330\pi$
$839$	−21016.0	−0.864783	−0.432391	+	0.901686i	$0.642330\pi$
$840$	0	0
$841$	−13153.0	−0.539301
$842$	0	0
$843$	− 20190.0i	− 0.824888i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9205.00i	0.373421i
$848$	0	0
$849$	9060.00	0.366241
$850$	0	0
$851$	−24016.0	−0.967401
$852$	0	0
$853$	− 24878.0i	− 0.998601i	−0.866429	−	0.499300i	$-0.833590\pi$
$853$	− 24878.0i	− 0.998601i	0.866429	−	0.499300i	$-0.166410\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 6390.00i	− 0.254700i	−0.991858	−	0.127350i	$-0.959353\pi$
$857$	− 6390.00i	− 0.254700i	0.991858	−	0.127350i	$-0.0406472\pi$
$858$	0	0
$859$	46444.0	1.84476	0.922380	−	0.386284i	$-0.126241\pi$
$859$	46444.0	1.84476	0.922380	+	0.386284i	$0.126241\pi$
$860$	0	0
$861$	8190.00	0.324175
$862$	0	0
$863$	25408.0i	1.00220i	0.865389	+	0.501100i	$0.167071\pi$
$863$	25408.0i	1.00220i	−0.865389	+	0.501100i	$0.832929\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 14151.0i	− 0.554317i
$868$	0	0
$869$	−1536.00	−0.0599600
$870$	0	0
$871$	45576.0	1.77300
$872$	0	0
$873$	13518.0i	0.524072i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1078.00i	0.0415068i	0.999785	+	0.0207534i	$0.00660649\pi$
$877$	1078.00i	0.0415068i	−0.999785	+	0.0207534i	$0.993394\pi$
$878$	0	0
$879$	20502.0	0.786707
$880$	0	0
$881$	−45006.0	−1.72110	−0.860551	−	0.509364i	$-0.829881\pi$
$881$	−45006.0	−1.72110	−0.860551	+	0.509364i	$0.829881\pi$
$882$	0	0
$883$	− 4028.00i	− 0.153514i	−0.997050	−	0.0767571i	$-0.975543\pi$
$883$	− 4028.00i	− 0.153514i	0.997050	−	0.0767571i	$-0.0244566\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 29304.0i	− 1.10928i	−0.832090	−	0.554640i	$-0.812856\pi$
$887$	− 29304.0i	− 1.10928i	0.832090	−	0.554640i	$-0.187144\pi$
$888$	0	0
$889$	8400.00	0.316903
$890$	0	0
$891$	324.000	0.0121823
$892$	0	0
$893$	48576.0i	1.82031i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 24624.0i	− 0.916579i
$898$	0	0
$899$	−15264.0	−0.566277
$900$	0	0
$901$	−8484.00	−0.313699
$902$	0	0
$903$	− 10668.0i	− 0.393144i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	50916.0i	1.86399i	0.362472	+	0.931995i	$0.381933\pi$
$907$	50916.0i	1.86399i	−0.362472	+	0.931995i	$0.618067\pi$
$908$	0	0
$909$	−3582.00	−0.130701
$910$	0	0
$911$	−24432.0	−0.888549	−0.444275	−	0.895891i	$-0.646539\pi$
$911$	−24432.0	−0.888549	−0.444275	+	0.895891i	$0.646539\pi$
$912$	0	0
$913$	2192.00i	0.0794574i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9772.00i	0.351908i
$918$	0	0
$919$	20360.0	0.730810	0.365405	−	0.930849i	$-0.380930\pi$
$919$	20360.0	0.730810	0.365405	+	0.930849i	$0.380930\pi$
$920$	0	0
$921$	6996.00	0.250300
$922$	0	0
$923$	432.000i	0.0154057i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	10440.0i	0.369897i
$928$	0	0
$929$	−23202.0	−0.819411	−0.409706	−	0.912218i	$-0.634369\pi$
$929$	−23202.0	−0.819411	−0.409706	+	0.912218i	$0.634369\pi$
$930$	0	0
$931$	4508.00	0.158694
$932$	0	0
$933$	− 26520.0i	− 0.930574i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1990.00i	− 0.0693815i	−0.999398	−	0.0346908i	$-0.988955\pi$
$937$	− 1990.00i	− 0.0693815i	0.999398	−	0.0346908i	$-0.0110446\pi$
$938$	0	0
$939$	3138.00	0.109057
$940$	0	0
$941$	−51130.0	−1.77130	−0.885648	−	0.464356i	$-0.846286\pi$
$941$	−51130.0	−1.77130	−0.885648	+	0.464356i	$0.846286\pi$
$942$	0	0
$943$	− 59280.0i	− 2.04711i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 47044.0i	− 1.61428i	−0.590359	−	0.807141i	$-0.701014\pi$
$947$	− 47044.0i	− 1.61428i	0.590359	−	0.807141i	$-0.298986\pi$
$948$	0	0
$949$	22788.0	0.779483
$950$	0	0
$951$	22626.0	0.771502
$952$	0	0
$953$	− 46858.0i	− 1.59274i	−0.604811	−	0.796369i	$-0.706751\pi$
$953$	− 46858.0i	− 1.59274i	0.604811	−	0.796369i	$-0.293249\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1272.00i	− 0.0429654i
$958$	0	0
$959$	19670.0	0.662333
$960$	0	0
$961$	−9055.00	−0.303951
$962$	0	0
$963$	− 2916.00i	− 0.0975771i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	30632.0i	1.01867i	0.860567	+	0.509337i	$0.170110\pi$
$967$	30632.0i	1.01867i	−0.860567	+	0.509337i	$0.829890\pi$
$968$	0	0
$969$	3864.00	0.128101
$970$	0	0
$971$	−3804.00	−0.125722	−0.0628611	−	0.998022i	$-0.520022\pi$
$971$	−3804.00	−0.125722	−0.0628611	+	0.998022i	$0.520022\pi$
$972$	0	0
$973$	28.0000i	0 0.000922548i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 49326.0i	− 1.61523i	−0.589711	−	0.807614i	$-0.700758\pi$
$977$	− 49326.0i	− 1.61523i	0.589711	−	0.807614i	$-0.299242\pi$
$978$	0	0
$979$	−4776.00	−0.155916
$980$	0	0
$981$	−8442.00	−0.274753
$982$	0	0
$983$	− 11112.0i	− 0.360547i	−0.983617	−	0.180274i	$-0.942302\pi$
$983$	− 11112.0i	− 0.360547i	0.983617	−	0.180274i	$-0.0576983\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	11088.0i	0.357584i
$988$	0	0
$989$	−77216.0	−2.48263
$990$	0	0
$991$	13616.0	0.436455	0.218227	−	0.975898i	$-0.429973\pi$
$991$	13616.0	0.436455	0.218227	+	0.975898i	$0.429973\pi$
$992$	0	0
$993$	− 8268.00i	− 0.264227i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 56674.0i	− 1.80028i	−0.435596	−	0.900142i	$-0.643462\pi$
$997$	− 56674.0i	− 1.80028i	0.435596	−	0.900142i	$-0.356538\pi$
$998$	0	0
$999$	−4266.00	−0.135105

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2100.4.k.g.1849.1		2
5.2	odd	4		2100.4.a.g.1.1		1
5.3	odd	4		84.4.a.b.1.1	✓	1
5.4	even	2	inner	2100.4.k.g.1849.2		2
15.8	even	4		252.4.a.a.1.1		1
20.3	even	4		336.4.a.e.1.1		1
35.3	even	12		588.4.i.h.373.1		2
35.13	even	4		588.4.a.a.1.1		1
35.18	odd	12		588.4.i.a.373.1		2
35.23	odd	12		588.4.i.a.361.1		2
35.33	even	12		588.4.i.h.361.1		2
40.3	even	4		1344.4.a.p.1.1		1
40.13	odd	4		1344.4.a.b.1.1		1
60.23	odd	4		1008.4.a.d.1.1		1
105.23	even	12		1764.4.k.n.361.1		2
105.38	odd	12		1764.4.k.c.1549.1		2
105.53	even	12		1764.4.k.n.1549.1		2
105.68	odd	12		1764.4.k.c.361.1		2
105.83	odd	4		1764.4.a.l.1.1		1
140.83	odd	4		2352.4.a.v.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.a.b.1.1	✓	1	5.3	odd	4
252.4.a.a.1.1		1	15.8	even	4
336.4.a.e.1.1		1	20.3	even	4
588.4.a.a.1.1		1	35.13	even	4
588.4.i.a.361.1		2	35.23	odd	12
588.4.i.a.373.1		2	35.18	odd	12
588.4.i.h.361.1		2	35.33	even	12
588.4.i.h.373.1		2	35.3	even	12
1008.4.a.d.1.1		1	60.23	odd	4
1344.4.a.b.1.1		1	40.13	odd	4
1344.4.a.p.1.1		1	40.3	even	4
1764.4.a.l.1.1		1	105.83	odd	4
1764.4.k.c.361.1		2	105.68	odd	12
1764.4.k.c.1549.1		2	105.38	odd	12
1764.4.k.n.361.1		2	105.23	even	12
1764.4.k.n.1549.1		2	105.53	even	12
2100.4.a.g.1.1		1	5.2	odd	4
2100.4.k.g.1849.1		2	1.1	even	1	trivial
2100.4.k.g.1849.2		2	5.4	even	2	inner
2352.4.a.v.1.1		1	140.83	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2100.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{3} -7.00000i q^{7} -9.00000 q^{9} +4.00000 q^{11} -54.0000i q^{13} -14.0000i q^{17} -92.0000 q^{19} -21.0000 q^{21} +152.000i q^{23} +27.0000i q^{27} +106.000 q^{29} -144.000 q^{31} -12.0000i q^{33} +158.000i q^{37} -162.000 q^{39} -390.000 q^{41} +508.000i q^{43} -528.000i q^{47} -49.0000 q^{49} -42.0000 q^{51} -606.000i q^{53} +276.000i q^{57} +364.000 q^{59} +678.000 q^{61} +63.0000i q^{63} +844.000i q^{67} +456.000 q^{69} -8.00000 q^{71} +422.000i q^{73} -28.0000i q^{77} -384.000 q^{79} +81.0000 q^{81} +548.000i q^{83} -318.000i q^{87} -1194.00 q^{89} -378.000 q^{91} +432.000i q^{93} -1502.00i q^{97} -36.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{9}+O(q^{10}) \) 2 * q - 18 * q^9	\( 2 q - 18 q^{9} + 8 q^{11} - 184 q^{19} - 42 q^{21} + 212 q^{29} - 288 q^{31} - 324 q^{39} - 780 q^{41} - 98 q^{49} - 84 q^{51} + 728 q^{59} + 1356 q^{61} + 912 q^{69} - 16 q^{71} - 768 q^{79} + 162 q^{81} - 2388 q^{89} - 756 q^{91} - 72 q^{99}+O(q^{100}) \) 2 * q - 18 * q^9 + 8 * q^11 - 184 * q^19 - 42 * q^21 + 212 * q^29 - 288 * q^31 - 324 * q^39 - 780 * q^41 - 98 * q^49 - 84 * q^51 + 728 * q^59 + 1356 * q^61 + 912 * q^69 - 16 * q^71 - 768 * q^79 + 162 * q^81 - 2388 * q^89 - 756 * q^91 - 72 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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