LMFDB - Embedded newform 1900.3.e.e.1101.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,3,Mod(1101,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1900.1101");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1900.e (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:	$51.7712502285$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 20x^{10} + 44x^{8} + 270x^{6} + 36676x^{4} + 71664x^{2} + 687241 $ x^12 - 20x^10 + 44x^8 + 270x^6 + 36676x^4 + 71664*x^2 + 687241
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22} $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1101.3
Root			$-1.34185 + 1.63363i$ of defining polynomial
Character	$\chi$	$=$	1900.1101
Dual form			1900.3.e.e.1101.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.26725i q^{3} -6.30368 q^{7} -1.67495 q^{9} +O(q^{10})$	$q-3.26725i q^{3} -6.30368 q^{7} -1.67495 q^{9} +4.53070 q^{11} -11.5357i q^{13} +1.08572 q^{17} +(-16.8558 - 8.76832i) q^{19} +20.5957i q^{21} +31.5184 q^{23} -23.9328i q^{27} +38.1324i q^{29} -58.2398i q^{31} -14.8030i q^{33} -40.2691i q^{37} -37.6901 q^{39} +18.0249i q^{41} +2.74764 q^{43} -76.1195 q^{47} -9.26364 q^{49} -3.54733i q^{51} -8.06781i q^{53} +(-28.6483 + 55.0720i) q^{57} +95.3957i q^{59} -4.28507 q^{61} +10.5583 q^{63} -70.7364i q^{67} -102.979i q^{69} +93.3131i q^{71} +14.2908 q^{73} -28.5601 q^{77} -6.11815i q^{79} -93.2691 q^{81} -89.0257 q^{83} +124.588 q^{87} -92.3366i q^{89} +72.7174i q^{91} -190.284 q^{93} -64.6843i q^{97} -7.58869 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 76 q^{9}+O(q^{10}) $ 12 * q - 76 * q^9	$ 12 q - 76 q^{9} - 24 q^{11} - 68 q^{19} + 264 q^{39} - 212 q^{49} - 600 q^{61} + 492 q^{81} - 136 q^{99}+O(q^{100}) $ 12 * q - 76 * q^9 - 24 * q^11 - 68 * q^19 + 264 * q^39 - 212 * q^49 - 600 * q^61 + 492 * q^81 - 136 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$.

$n$	$77$	$401$	$951$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	3.26725i		−	1.08908i	−0.838733	−	0.544542i	$-0.816703\pi$
$3$		−	3.26725i		−	1.08908i	0.838733	−	0.544542i	$-0.183297\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−6.30368			−0.900525			−0.450263	−	0.892896i	$-0.648670\pi$
$7$	−6.30368			−0.900525			−0.450263	+	0.892896i	$0.648670\pi$
$8$		0			0
$9$	−1.67495			−0.186105
$10$		0			0
$11$	4.53070			0.411882			0.205941	−	0.978564i	$-0.433974\pi$
$11$	4.53070			0.411882			0.205941	+	0.978564i	$0.433974\pi$
$12$		0			0
$13$		−	11.5357i		−	0.887362i	−0.896185	−	0.443681i	$-0.853672\pi$
$13$		−	11.5357i		−	0.887362i	0.896185	−	0.443681i	$-0.146328\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.08572			0.0638659			0.0319330	−	0.999490i	$-0.489834\pi$
$17$	1.08572			0.0638659			0.0319330	+	0.999490i	$0.489834\pi$
$18$		0			0
$19$	−16.8558	−	8.76832i	−0.887145	−	0.461491i
$19$	−16.8558	−	8.76832i	−0.887145	−	0.461491i
$20$		0			0
$21$			20.5957i			0.980748i
$22$		0			0
$23$	31.5184			1.37036			0.685182	−	0.728372i	$-0.259722\pi$
$23$	31.5184			1.37036			0.685182	+	0.728372i	$0.259722\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	23.9328i		−	0.886400i
$28$		0			0
$29$			38.1324i			1.31491i	0.753494	+	0.657454i	$0.228367\pi$
$29$			38.1324i			1.31491i	−0.753494	+	0.657454i	$0.771633\pi$
$30$		0			0
$31$		−	58.2398i		−	1.87870i	−0.342955	−	0.939352i	$-0.611428\pi$
$31$		−	58.2398i		−	1.87870i	0.342955	−	0.939352i	$-0.388572\pi$
$32$		0			0
$33$		−	14.8030i		−	0.448575i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	40.2691i		−	1.08835i	−0.838971	−	0.544177i	$-0.816842\pi$
$37$		−	40.2691i		−	1.08835i	0.838971	−	0.544177i	$-0.183158\pi$
$38$		0			0
$39$	−37.6901			−0.966413
$40$		0			0
$41$			18.0249i			0.439632i	0.975541	+	0.219816i	$0.0705456\pi$
$41$			18.0249i			0.439632i	−0.975541	+	0.219816i	$0.929454\pi$
$42$		0			0
$43$	2.74764			0.0638986			0.0319493	−	0.999489i	$-0.489828\pi$
$43$	2.74764			0.0638986			0.0319493	+	0.999489i	$0.489828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−76.1195			−1.61956			−0.809782	−	0.586731i	$-0.800414\pi$
$47$	−76.1195			−1.61956			−0.809782	+	0.586731i	$0.800414\pi$
$48$		0			0
$49$	−9.26364			−0.189054
$50$		0			0
$51$		−	3.54733i		−	0.0695554i
$52$		0			0
$53$		−	8.06781i		−	0.152223i	−0.997099	−	0.0761115i	$-0.975750\pi$
$53$		−	8.06781i		−	0.152223i	0.997099	−	0.0761115i	$-0.0242505\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−28.6483	+	55.0720i	−0.502602	+	0.966176i
$58$		0			0
$59$			95.3957i			1.61688i	0.588581	+	0.808438i	$0.299687\pi$
$59$			95.3957i			1.61688i	−0.588581	+	0.808438i	$0.700313\pi$
$60$		0			0
$61$	−4.28507			−0.0702470			−0.0351235	−	0.999383i	$-0.511182\pi$
$61$	−4.28507			−0.0702470			−0.0351235	+	0.999383i	$0.511182\pi$
$62$		0			0
$63$	10.5583			0.167593
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	70.7364i		−	1.05577i	−0.849317	−	0.527883i	$-0.822986\pi$
$67$		−	70.7364i		−	1.05577i	0.849317	−	0.527883i	$-0.177014\pi$
$68$		0			0
$69$		−	102.979i		−	1.49244i
$70$		0			0
$71$			93.3131i			1.31427i	0.753773	+	0.657135i	$0.228232\pi$
$71$			93.3131i			1.31427i	−0.753773	+	0.657135i	$0.771768\pi$
$72$		0			0
$73$	14.2908			0.195765			0.0978824	−	0.995198i	$-0.468793\pi$
$73$	14.2908			0.195765			0.0978824	+	0.995198i	$0.468793\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−28.5601			−0.370910
$78$		0			0
$79$		−	6.11815i		−	0.0774449i	−0.999250	−	0.0387224i	$-0.987671\pi$
$79$		−	6.11815i		−	0.0774449i	0.999250	−	0.0387224i	$-0.0123288\pi$
$80$		0			0
$81$	−93.2691			−1.15147
$82$		0			0
$83$	−89.0257			−1.07260			−0.536299	−	0.844028i	$-0.680178\pi$
$83$	−89.0257			−1.07260			−0.536299	+	0.844028i	$0.680178\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	124.588			1.43205
$88$		0			0
$89$		−	92.3366i		−	1.03749i	−0.854929	−	0.518745i	$-0.826399\pi$
$89$		−	92.3366i		−	1.03749i	0.854929	−	0.518745i	$-0.173601\pi$
$90$		0			0
$91$			72.7174i			0.799092i
$92$		0			0
$93$	−190.284			−2.04607
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	64.6843i		−	0.666849i	−0.942777	−	0.333424i	$-0.891796\pi$
$97$		−	64.6843i		−	0.666849i	0.942777	−	0.333424i	$-0.108204\pi$
$98$		0			0
$99$	−7.58869			−0.0766535
$100$		0			0
$101$	−155.672			−1.54131			−0.770654	−	0.637254i	$-0.780070\pi$
$101$	−155.672			−1.54131			−0.770654	+	0.637254i	$0.780070\pi$
$102$		0			0
$103$		−	130.821i		−	1.27010i	−0.772469	−	0.635052i	$-0.780978\pi$
$103$		−	130.821i		−	1.27010i	0.772469	−	0.635052i	$-0.219022\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	95.1516i		−	0.889268i	−0.895712	−	0.444634i	$-0.853334\pi$
$107$		−	95.1516i		−	0.889268i	0.895712	−	0.444634i	$-0.146666\pi$
$108$		0			0
$109$			155.589i			1.42742i	0.700443	+	0.713709i	$0.252986\pi$
$109$			155.589i			1.42742i	−0.700443	+	0.713709i	$0.747014\pi$
$110$		0			0
$111$	−131.569			−1.18531
$112$		0			0
$113$			174.840i			1.54725i	0.633642	+	0.773626i	$0.281559\pi$
$113$			174.840i			1.54725i	−0.633642	+	0.773626i	$0.718441\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			19.3217i			0.165143i
$118$		0			0
$119$	−6.84404			−0.0575129
$120$		0			0
$121$	−100.473			−0.830353
$122$		0			0
$123$	58.8919			0.478796
$124$		0			0
$125$		0			0
$126$		0			0
$127$			80.8200i			0.636378i	0.948027	+	0.318189i	$0.103075\pi$
$127$			80.8200i			0.636378i	−0.948027	+	0.318189i	$0.896925\pi$
$128$		0			0
$129$		−	8.97723i		−	0.0695910i
$130$		0			0
$131$	−129.957			−0.992039			−0.496020	−	0.868311i	$-0.665206\pi$
$131$	−129.957			−0.992039			−0.496020	+	0.868311i	$0.665206\pi$
$132$		0			0
$133$	106.253	+	55.2727i	0.798897	+	0.415584i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−58.8037			−0.429224			−0.214612	−	0.976699i	$-0.568849\pi$
$137$	−58.8037			−0.429224			−0.214612	+	0.976699i	$0.568849\pi$
$138$		0			0
$139$	−2.48444			−0.0178736			−0.00893682	−	0.999960i	$-0.502845\pi$
$139$	−2.48444			−0.0178736			−0.00893682	+	0.999960i	$0.502845\pi$
$140$		0			0
$141$			248.702i			1.76384i
$142$		0			0
$143$		−	52.2649i		−	0.365489i
$144$		0			0
$145$		0			0
$146$		0			0
$147$			30.2667i			0.205896i
$148$		0			0
$149$	90.8294			0.609594			0.304797	−	0.952417i	$-0.401411\pi$
$149$	90.8294			0.609594			0.304797	+	0.952417i	$0.401411\pi$
$150$		0			0
$151$			44.1209i			0.292192i	0.989270	+	0.146096i	$0.0466708\pi$
$151$			44.1209i			0.292192i	−0.989270	+	0.146096i	$0.953329\pi$
$152$		0			0
$153$	−1.81853			−0.0118858
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−9.36283			−0.0596358			−0.0298179	−	0.999555i	$-0.509493\pi$
$157$	−9.36283			−0.0596358			−0.0298179	+	0.999555i	$0.509493\pi$
$158$		0			0
$159$	−26.3596			−0.165784
$160$		0			0
$161$	−198.682			−1.23405
$162$		0			0
$163$	35.8739			0.220085			0.110043	−	0.993927i	$-0.464901\pi$
$163$	35.8739			0.220085			0.110043	+	0.993927i	$0.464901\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			195.223i			1.16900i	0.811393	+	0.584501i	$0.198710\pi$
$167$			195.223i			1.16900i	−0.811393	+	0.584501i	$0.801290\pi$
$168$		0			0
$169$	35.9274			0.212588
$170$		0			0
$171$	28.2325	+	14.6865i	0.165102	+	0.0858859i
$172$		0			0
$173$			236.304i			1.36592i	0.730455	+	0.682960i	$0.239308\pi$
$173$			236.304i			1.36592i	−0.730455	+	0.682960i	$0.760692\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	311.682			1.76091
$178$		0			0
$179$		−	125.945i		−	0.703604i	−0.936074	−	0.351802i	$-0.885569\pi$
$179$		−	125.945i		−	0.703604i	0.936074	−	0.351802i	$-0.114431\pi$
$180$		0			0
$181$			11.4185i			0.0630856i	0.999502	+	0.0315428i	$0.0100421\pi$
$181$			11.4185i			0.0630856i	−0.999502	+	0.0315428i	$0.989958\pi$
$182$		0			0
$183$			14.0004i			0.0765049i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	4.91908			0.0263052
$188$		0			0
$189$			150.865i			0.798226i
$190$		0			0
$191$	36.3733			0.190436			0.0952180	−	0.995456i	$-0.469645\pi$
$191$	36.3733			0.190436			0.0952180	+	0.995456i	$0.469645\pi$
$192$		0			0
$193$			150.814i			0.781422i	0.920513	+	0.390711i	$0.127771\pi$
$193$			150.814i			0.781422i	−0.920513	+	0.390711i	$0.872229\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	257.585			1.30754			0.653768	−	0.756695i	$-0.273187\pi$
$197$	257.585			1.30754			0.653768	+	0.756695i	$0.273187\pi$
$198$		0			0
$199$	−128.758			−0.647027			−0.323513	−	0.946224i	$-0.604864\pi$
$199$	−128.758			−0.647027			−0.323513	+	0.946224i	$0.604864\pi$
$200$		0			0
$201$	−231.114			−1.14982
$202$		0			0
$203$		−	240.374i		−	1.18411i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−52.7917			−0.255032
$208$		0			0
$209$	−76.3685	−	39.7267i	−0.365399	−	0.190080i
$210$		0			0
$211$		−	161.636i		−	0.766048i	−0.923738	−	0.383024i	$-0.874883\pi$
$211$		−	161.636i		−	0.766048i	0.923738	−	0.383024i	$-0.125117\pi$
$212$		0			0
$213$	304.878			1.43135
$214$		0			0
$215$		0			0
$216$		0			0
$217$			367.125i			1.69182i
$218$		0			0
$219$		−	46.6918i		−	0.213204i
$220$		0			0
$221$		−	12.5246i		−	0.0566722i
$222$		0			0
$223$		−	386.480i		−	1.73310i	−0.499094	−	0.866548i	$-0.666334\pi$
$223$		−	386.480i		−	1.73310i	0.499094	−	0.866548i	$-0.333666\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	52.4767i		−	0.231175i	−0.993297	−	0.115587i	$-0.963125\pi$
$227$		−	52.4767i		−	0.231175i	0.993297	−	0.115587i	$-0.0368751\pi$
$228$		0			0
$229$	−394.025			−1.72063			−0.860316	−	0.509762i	$-0.829734\pi$
$229$	−394.025			−1.72063			−0.860316	+	0.509762i	$0.829734\pi$
$230$		0			0
$231$			93.3131i			0.403953i
$232$		0			0
$233$	22.2691			0.0955754			0.0477877	−	0.998858i	$-0.484783\pi$
$233$	22.2691			0.0955754			0.0477877	+	0.998858i	$0.484783\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−19.9895			−0.0843440
$238$		0			0
$239$	−83.5976			−0.349781			−0.174890	−	0.984588i	$-0.555957\pi$
$239$	−83.5976			−0.349781			−0.174890	+	0.984588i	$0.555957\pi$
$240$		0			0
$241$			214.605i			0.890477i	0.895412	+	0.445238i	$0.146881\pi$
$241$			214.605i			0.890477i	−0.895412	+	0.445238i	$0.853119\pi$
$242$		0			0
$243$			89.3385i			0.367648i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−101.149	+	194.443i	−0.409509	+	0.787219i
$248$		0			0
$249$			290.870i			1.16815i
$250$		0			0
$251$	−24.6770			−0.0983146			−0.0491573	−	0.998791i	$-0.515654\pi$
$251$	−24.6770			−0.0983146			−0.0491573	+	0.998791i	$0.515654\pi$
$252$		0			0
$253$	142.801			0.564429
$254$		0			0
$255$		0			0
$256$		0			0
$257$			152.548i			0.593573i	0.954944	+	0.296787i	$0.0959150\pi$
$257$			152.548i			0.593573i	−0.954944	+	0.296787i	$0.904085\pi$
$258$		0			0
$259$			253.843i			0.980090i
$260$		0			0
$261$		−	63.8697i		−	0.244712i
$262$		0			0
$263$	−345.570			−1.31395			−0.656977	−	0.753911i	$-0.728165\pi$
$263$	−345.570			−1.31395			−0.656977	+	0.753911i	$0.728165\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−301.687			−1.12991
$268$		0			0
$269$			1.39433i			0.00518338i	0.999997	+	0.00259169i	$0.000824962\pi$
$269$			1.39433i			0.00518338i	−0.999997	+	0.00259169i	$0.999175\pi$
$270$		0			0
$271$	407.864			1.50503			0.752516	−	0.658574i	$-0.228840\pi$
$271$	407.864			1.50503			0.752516	+	0.658574i	$0.228840\pi$
$272$		0			0
$273$	237.586			0.870279
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−330.614			−1.19355			−0.596777	−	0.802407i	$-0.703552\pi$
$277$	−330.614			−1.19355			−0.596777	+	0.802407i	$0.703552\pi$
$278$		0			0
$279$			97.5487i			0.349637i
$280$		0			0
$281$			79.1238i			0.281579i	0.990040	+	0.140790i	$0.0449641\pi$
$281$			79.1238i			0.281579i	−0.990040	+	0.140790i	$0.955036\pi$
$282$		0			0
$283$	15.0236			0.0530870			0.0265435	−	0.999648i	$-0.491550\pi$
$283$	15.0236			0.0530870			0.0265435	+	0.999648i	$0.491550\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	113.623i		−	0.395899i
$288$		0			0
$289$	−287.821			−0.995921
$290$		0			0
$291$	−211.340			−0.726255
$292$		0			0
$293$		−	77.7057i		−	0.265207i	−0.991169	−	0.132604i	$-0.957666\pi$
$293$		−	77.7057i		−	0.265207i	0.991169	−	0.132604i	$-0.0423337\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		−	108.432i		−	0.365092i
$298$		0			0
$299$		−	363.587i		−	1.21601i
$300$		0			0
$301$	−17.3202			−0.0575423
$302$		0			0
$303$			508.620i			1.67861i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	212.669i		−	0.692734i	−0.938099	−	0.346367i	$-0.887415\pi$
$307$		−	212.669i		−	0.692734i	0.938099	−	0.346367i	$-0.112585\pi$
$308$		0			0
$309$	−427.425			−1.38325
$310$		0			0
$311$	104.314			0.335415			0.167707	−	0.985837i	$-0.446364\pi$
$311$	104.314			0.335415			0.167707	+	0.985837i	$0.446364\pi$
$312$		0			0
$313$	−489.532			−1.56400			−0.781999	−	0.623279i	$-0.785800\pi$
$313$	−489.532			−1.56400			−0.781999	+	0.623279i	$0.785800\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			146.496i			0.462133i	0.972938	+	0.231067i	$0.0742216\pi$
$317$			146.496i			0.462133i	−0.972938	+	0.231067i	$0.925778\pi$
$318$		0			0
$319$			172.766i			0.541588i
$320$		0			0
$321$	−310.885			−0.968488
$322$		0			0
$323$	−18.3006	−	9.51995i	−0.0566584	−	0.0294735i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	508.347			1.55458
$328$		0			0
$329$	479.833			1.45846
$330$		0			0
$331$		−	319.048i		−	0.963892i	−0.876201	−	0.481946i	$-0.839930\pi$
$331$		−	319.048i		−	0.963892i	0.876201	−	0.481946i	$-0.160070\pi$
$332$		0			0
$333$			67.4486i			0.202548i
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	386.447i		−	1.14673i	−0.819301	−	0.573363i	$-0.805638\pi$
$337$		−	386.447i		−	1.14673i	0.819301	−	0.573363i	$-0.194362\pi$
$338$		0			0
$339$	571.245			1.68509
$340$		0			0
$341$		−	263.867i		−	0.773805i
$342$		0			0
$343$	367.275			1.07077
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−624.138			−1.79867			−0.899334	−	0.437262i	$-0.855948\pi$
$347$	−624.138			−1.79867			−0.899334	+	0.437262i	$0.855948\pi$
$348$		0			0
$349$	−509.597			−1.46016			−0.730081	−	0.683360i	$-0.760518\pi$
$349$	−509.597			−1.46016			−0.730081	+	0.683360i	$0.760518\pi$
$350$		0			0
$351$	−276.082			−0.786558
$352$		0			0
$353$	−65.8979			−0.186680			−0.0933398	−	0.995634i	$-0.529754\pi$
$353$	−65.8979			−0.186680			−0.0933398	+	0.995634i	$0.529754\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			22.3612i			0.0626364i
$358$		0			0
$359$	548.958			1.52913			0.764566	−	0.644546i	$-0.222953\pi$
$359$	548.958			1.52913			0.764566	+	0.644546i	$0.222953\pi$
$360$		0			0
$361$	207.233	+	295.593i	0.574053	+	0.818818i
$362$		0			0
$363$			328.270i			0.904325i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	234.257			0.638303			0.319151	−	0.947704i	$-0.396602\pi$
$367$	234.257			0.638303			0.319151	+	0.947704i	$0.396602\pi$
$368$		0			0
$369$		−	30.1908i		−	0.0818178i
$370$		0			0
$371$			50.8569i			0.137081i
$372$		0			0
$373$		−	167.444i		−	0.448911i	−0.974484	−	0.224455i	$-0.927940\pi$
$373$		−	167.444i		−	0.448911i	0.974484	−	0.224455i	$-0.0720603\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	439.884			1.16680
$378$		0			0
$379$			69.3701i			0.183034i	0.995803	+	0.0915172i	$0.0291717\pi$
$379$			69.3701i			0.183034i	−0.995803	+	0.0915172i	$0.970828\pi$
$380$		0			0
$381$	264.059			0.693069
$382$		0			0
$383$		−	547.099i		−	1.42846i	−0.699912	−	0.714229i	$-0.746777\pi$
$383$		−	547.099i		−	1.42846i	0.699912	−	0.714229i	$-0.253223\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.60215			−0.0118919
$388$		0			0
$389$	−49.2968			−0.126727			−0.0633635	−	0.997991i	$-0.520183\pi$
$389$	−49.2968			−0.126727			−0.0633635	+	0.997991i	$0.520183\pi$
$390$		0			0
$391$	34.2202			0.0875196
$392$		0			0
$393$			424.603i			1.08041i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−333.006			−0.838805			−0.419402	−	0.907800i	$-0.637760\pi$
$397$	−333.006			−0.838805			−0.419402	+	0.907800i	$0.637760\pi$
$398$		0			0
$399$	180.590	−	347.156i	0.452606	−	0.870066i
$400$		0			0
$401$			338.509i			0.844161i	0.906558	+	0.422081i	$0.138700\pi$
$401$			338.509i			0.844161i	−0.906558	+	0.422081i	$0.861300\pi$
$402$		0			0
$403$	−671.838			−1.66709
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	182.447i		−	0.448273i
$408$		0			0
$409$		−	162.742i		−	0.397903i	−0.980009	−	0.198951i	$-0.936246\pi$
$409$		−	162.742i		−	0.397903i	0.980009	−	0.198951i	$-0.0637536\pi$
$410$		0			0
$411$			192.127i			0.467461i
$412$		0			0
$413$		−	601.344i		−	1.45604i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			8.11728i			0.0194659i
$418$		0			0
$419$	240.614			0.574257			0.287128	−	0.957892i	$-0.407299\pi$
$419$	240.614			0.574257			0.287128	+	0.957892i	$0.407299\pi$
$420$		0			0
$421$		−	753.729i		−	1.79033i	−0.445735	−	0.895165i	$-0.647057\pi$
$421$		−	753.729i		−	1.79033i	0.445735	−	0.895165i	$-0.352943\pi$
$422$		0			0
$423$	127.496			0.301409
$424$		0			0
$425$		0			0
$426$		0			0
$427$	27.0117			0.0632592
$428$		0			0
$429$	−170.763			−0.398048
$430$		0			0
$431$			198.792i			0.461235i	0.973045	+	0.230617i	$0.0740745\pi$
$431$			198.792i			0.461235i	−0.973045	+	0.230617i	$0.925925\pi$
$432$		0			0
$433$		−	592.322i		−	1.36795i	−0.729506	−	0.683975i	$-0.760250\pi$
$433$		−	592.322i		−	1.36795i	0.729506	−	0.683975i	$-0.239750\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−531.266	−	276.363i	−1.21571	−	0.632410i
$438$		0			0
$439$		−	837.288i		−	1.90726i	−0.300977	−	0.953631i	$-0.597313\pi$
$439$		−	837.288i		−	1.90726i	0.300977	−	0.953631i	$-0.402687\pi$
$440$		0			0
$441$	15.5161			0.0351839
$442$		0			0
$443$	−736.917			−1.66347			−0.831734	−	0.555174i	$-0.812652\pi$
$443$	−736.917			−1.66347			−0.831734	+	0.555174i	$0.812652\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		−	296.763i		−	0.663899i
$448$		0			0
$449$			369.746i			0.823489i	0.911299	+	0.411744i	$0.135080\pi$
$449$			369.746i			0.823489i	−0.911299	+	0.411744i	$0.864920\pi$
$450$		0			0
$451$			81.6655i			0.181076i
$452$		0			0
$453$	144.154			0.318221
$454$		0			0
$455$		0			0
$456$		0			0
$457$	799.508			1.74947			0.874735	−	0.484601i	$-0.161035\pi$
$457$	799.508			1.74947			0.874735	+	0.484601i	$0.161035\pi$
$458$		0			0
$459$		−	25.9843i		−	0.0566108i
$460$		0			0
$461$	721.581			1.56525			0.782625	−	0.622493i	$-0.213880\pi$
$461$	721.581			1.56525			0.782625	+	0.622493i	$0.213880\pi$
$462$		0			0
$463$	850.242			1.83638			0.918188	−	0.396145i	$-0.129652\pi$
$463$	850.242			1.83638			0.918188	+	0.396145i	$0.129652\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−125.893			−0.269579			−0.134789	−	0.990874i	$-0.543036\pi$
$467$	−125.893			−0.269579			−0.134789	+	0.990874i	$0.543036\pi$
$468$		0			0
$469$			445.899i			0.950745i
$470$		0			0
$471$			30.5907i			0.0649485i
$472$		0			0
$473$	12.4487			0.0263187
$474$		0			0
$475$		0			0
$476$		0			0
$477$			13.5132i			0.0283295i
$478$		0			0
$479$	549.803			1.14782			0.573908	−	0.818920i	$-0.305427\pi$
$479$	549.803			1.14782			0.573908	+	0.818920i	$0.305427\pi$
$480$		0			0
$481$	−464.532			−0.965764
$482$		0			0
$483$			649.144i			1.34398i
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	165.606i		−	0.340054i	−0.985439	−	0.170027i	$-0.945615\pi$
$487$		−	165.606i		−	0.340054i	0.985439	−	0.170027i	$-0.0543855\pi$
$488$		0			0
$489$		−	117.209i		−	0.239692i
$490$		0			0
$491$	−344.284			−0.701190			−0.350595	−	0.936527i	$-0.614021\pi$
$491$	−344.284			−0.701190			−0.350595	+	0.936527i	$0.614021\pi$
$492$		0			0
$493$			41.4011i			0.0839779i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	588.216i		−	1.18353i
$498$		0			0
$499$	−136.376			−0.273298			−0.136649	−	0.990620i	$-0.543633\pi$
$499$	−136.376			−0.273298			−0.136649	+	0.990620i	$0.543633\pi$
$500$		0			0
$501$	637.844			1.27314
$502$		0			0
$503$	−175.677			−0.349258			−0.174629	−	0.984634i	$-0.555873\pi$
$503$	−175.677			−0.349258			−0.174629	+	0.984634i	$0.555873\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	117.384i		−	0.231527i
$508$		0			0
$509$		−	374.070i		−	0.734912i	−0.930041	−	0.367456i	$-0.880229\pi$
$509$		−	374.070i		−	0.734912i	0.930041	−	0.367456i	$-0.119771\pi$
$510$		0			0
$511$	−90.0848			−0.176291
$512$		0			0
$513$	−209.851	+	403.406i	−0.409065	+	0.786366i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−344.875			−0.667069
$518$		0			0
$519$	772.066			1.48760
$520$		0			0
$521$		−	160.601i		−	0.308255i	−0.988051	−	0.154127i	$-0.950743\pi$
$521$		−	160.601i		−	0.308255i	0.988051	−	0.154127i	$-0.0492566\pi$
$522$		0			0
$523$		−	671.370i		−	1.28369i	−0.766834	−	0.641845i	$-0.778169\pi$
$523$		−	671.370i		−	1.28369i	0.766834	−	0.641845i	$-0.221831\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	63.2322i		−	0.119985i
$528$		0			0
$529$	464.409			0.877900
$530$		0			0
$531$		−	159.783i		−	0.300909i
$532$		0			0
$533$	207.930			0.390112
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−411.495			−0.766284
$538$		0			0
$539$	−41.9708			−0.0778680
$540$		0			0
$541$	59.9401			0.110795			0.0553975	−	0.998464i	$-0.482357\pi$
$541$	59.9401			0.110795			0.0553975	+	0.998464i	$0.482357\pi$
$542$		0			0
$543$	37.3071			0.0687056
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	754.492i		−	1.37933i	−0.724130	−	0.689664i	$-0.757758\pi$
$547$		−	754.492i		−	1.37933i	0.724130	−	0.689664i	$-0.242242\pi$
$548$		0			0
$549$	7.17726			0.0130733
$550$		0			0
$551$	334.357	−	642.750i	0.606818	−	1.16652i
$552$		0			0
$553$			38.5668i			0.0697411i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	758.951			1.36257			0.681285	−	0.732018i	$-0.261421\pi$
$557$	758.951			1.36257			0.681285	+	0.732018i	$0.261421\pi$
$558$		0			0
$559$		−	31.6960i		−	0.0567012i
$560$		0			0
$561$		−	16.0719i		−	0.0286486i
$562$		0			0
$563$		−	200.922i		−	0.356877i	−0.983951	−	0.178438i	$-0.942895\pi$
$563$		−	200.922i		−	0.356877i	0.983951	−	0.178438i	$-0.0571046\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	587.938			1.03693
$568$		0			0
$569$		−	353.921i		−	0.622006i	−0.950409	−	0.311003i	$-0.899335\pi$
$569$		−	353.921i		−	0.622006i	0.950409	−	0.311003i	$-0.100665\pi$
$570$		0			0
$571$	820.990			1.43781			0.718905	−	0.695108i	$-0.244643\pi$
$571$	820.990			1.43781			0.718905	+	0.695108i	$0.244643\pi$
$572$		0			0
$573$		−	118.841i		−	0.207401i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−549.666			−0.952627			−0.476314	−	0.879276i	$-0.658027\pi$
$577$	−549.666			−0.952627			−0.476314	+	0.879276i	$0.658027\pi$
$578$		0			0
$579$	492.749			0.851035
$580$		0			0
$581$	561.189			0.965903
$582$		0			0
$583$		−	36.5529i		−	0.0626979i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	206.107			0.351120			0.175560	−	0.984469i	$-0.443826\pi$
$587$	206.107			0.351120			0.175560	+	0.984469i	$0.443826\pi$
$588$		0			0
$589$	−510.666	+	981.676i	−0.867004	+	1.66668i
$590$		0			0
$591$		−	841.594i		−	1.42402i
$592$		0			0
$593$	332.865			0.561324			0.280662	−	0.959807i	$-0.409446\pi$
$593$	332.865			0.561324			0.280662	+	0.959807i	$0.409446\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			420.686i			0.704667i
$598$		0			0
$599$		−	77.8882i		−	0.130030i	−0.997884	−	0.0650152i	$-0.979290\pi$
$599$		−	77.8882i		−	0.130030i	0.997884	−	0.0650152i	$-0.0207096\pi$
$600$		0			0
$601$			454.271i			0.755859i	0.925834	+	0.377929i	$0.123364\pi$
$601$			454.271i			0.755859i	−0.925834	+	0.377929i	$0.876636\pi$
$602$		0			0
$603$			118.480i			0.196484i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			874.377i			1.44049i	0.693720	+	0.720245i	$0.255971\pi$
$607$			874.377i			1.44049i	−0.693720	+	0.720245i	$0.744029\pi$
$608$		0			0
$609$	−785.363			−1.28959
$610$		0			0
$611$			878.092i			1.43714i
$612$		0			0
$613$	980.965			1.60027			0.800134	−	0.599821i	$-0.204761\pi$
$613$	980.965			1.60027			0.800134	+	0.599821i	$0.204761\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−806.424			−1.30701			−0.653504	−	0.756923i	$-0.726702\pi$
$617$	−806.424			−1.30701			−0.653504	+	0.756923i	$0.726702\pi$
$618$		0			0
$619$	−221.549			−0.357915			−0.178957	−	0.983857i	$-0.557272\pi$
$619$	−221.549			−0.357915			−0.178957	+	0.983857i	$0.557272\pi$
$620$		0			0
$621$		−	754.323i		−	1.21469i
$622$		0			0
$623$			582.060i			0.934286i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−129.797	+	249.515i	−0.207013	+	0.397951i
$628$		0			0
$629$		−	43.7210i		−	0.0695087i
$630$		0			0
$631$	590.815			0.936316			0.468158	−	0.883645i	$-0.344918\pi$
$631$	590.815			0.936316			0.468158	+	0.883645i	$0.344918\pi$
$632$		0			0
$633$	−528.107			−0.834292
$634$		0			0
$635$		0			0
$636$		0			0
$637$			106.863i			0.167759i
$638$		0			0
$639$		−	156.295i		−	0.244592i
$640$		0			0
$641$		−	1110.32i		−	1.73217i	−0.499896	−	0.866086i	$-0.666628\pi$
$641$		−	1110.32i		−	1.73217i	0.499896	−	0.866086i	$-0.333372\pi$
$642$		0			0
$643$	−430.490			−0.669503			−0.334752	−	0.942306i	$-0.608652\pi$
$643$	−430.490			−0.669503			−0.334752	+	0.942306i	$0.608652\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−102.290			−0.158099			−0.0790497	−	0.996871i	$-0.525189\pi$
$647$	−102.290			−0.158099			−0.0790497	+	0.996871i	$0.525189\pi$
$648$		0			0
$649$			432.210i			0.665962i
$650$		0			0
$651$	1199.49			1.84254
$652$		0			0
$653$	71.2294			0.109080			0.0545401	−	0.998512i	$-0.482631\pi$
$653$	71.2294			0.109080			0.0545401	+	0.998512i	$0.482631\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−23.9364			−0.0364329
$658$		0			0
$659$		−	1003.48i		−	1.52273i	−0.648325	−	0.761364i	$-0.724530\pi$
$659$		−	1003.48i		−	1.52273i	0.648325	−	0.761364i	$-0.275470\pi$
$660$		0			0
$661$		−	620.531i		−	0.938775i	−0.882992	−	0.469388i	$-0.844475\pi$
$661$		−	620.531i		−	0.938775i	0.882992	−	0.469388i	$-0.155525\pi$
$662$		0			0
$663$	−40.9209			−0.0617208
$664$		0			0
$665$		0			0
$666$		0			0
$667$			1201.87i			1.80190i
$668$		0			0
$669$	−1262.73			−1.88749
$670$		0			0
$671$	−19.4144			−0.0289335
$672$		0			0
$673$			973.102i			1.44592i	0.690891	+	0.722959i	$0.257218\pi$
$673$			973.102i			1.44592i	−0.690891	+	0.722959i	$0.742782\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			518.327i			0.765623i	0.923827	+	0.382811i	$0.125044\pi$
$677$			518.327i			0.765623i	−0.923827	+	0.382811i	$0.874956\pi$
$678$		0			0
$679$			407.749i			0.600514i
$680$		0			0
$681$	−171.455			−0.251769
$682$		0			0
$683$		−	423.707i		−	0.620362i	−0.950678	−	0.310181i	$-0.899610\pi$
$683$		−	423.707i		−	0.620362i	0.950678	−	0.310181i	$-0.100390\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$			1287.38i			1.87391i
$688$		0			0
$689$	−93.0680			−0.135077
$690$		0			0
$691$	789.329			1.14230			0.571150	−	0.820846i	$-0.306498\pi$
$691$	789.329			1.14230			0.571150	+	0.820846i	$0.306498\pi$
$692$		0			0
$693$	47.8367			0.0690284
$694$		0			0
$695$		0			0
$696$		0			0
$697$			19.5700i			0.0280775i
$698$		0			0
$699$		−	72.7587i		−	0.104090i
$700$		0			0
$701$	307.669			0.438900			0.219450	−	0.975624i	$-0.429574\pi$
$701$	307.669			0.438900			0.219450	+	0.975624i	$0.429574\pi$
$702$		0			0
$703$	−353.092	+	678.766i	−0.502265	+	0.965527i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	981.307			1.38799
$708$		0			0
$709$	767.955			1.08315			0.541576	−	0.840651i	$-0.317828\pi$
$709$	767.955			1.08315			0.541576	+	0.840651i	$0.317828\pi$
$710$		0			0
$711$			10.2476i			0.0144129i
$712$		0			0
$713$		−	1835.63i		−	2.57451i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			273.134i			0.380941i
$718$		0			0
$719$	1329.40			1.84895			0.924476	−	0.381239i	$-0.124503\pi$
$719$	1329.40			1.84895			0.924476	+	0.381239i	$0.124503\pi$
$720$		0			0
$721$			824.652i			1.14376i
$722$		0			0
$723$	701.169			0.969804
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1117.94			−1.53775			−0.768875	−	0.639399i	$-0.779183\pi$
$727$	−1117.94			−1.53775			−0.768875	+	0.639399i	$0.779183\pi$
$728$		0			0
$729$	−547.530			−0.751070
$730$		0			0
$731$	2.98317			0.00408094
$732$		0			0
$733$	−1173.92			−1.60153			−0.800767	−	0.598976i	$-0.795575\pi$
$733$	−1173.92			−1.60153			−0.800767	+	0.598976i	$0.795575\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	320.486i		−	0.434852i
$738$		0			0
$739$	415.270			0.561935			0.280968	−	0.959717i	$-0.409345\pi$
$739$	415.270			0.561935			0.280968	+	0.959717i	$0.409345\pi$
$740$		0			0
$741$	635.295	+	330.479i	0.857348	+	0.445990i
$742$		0			0
$743$			1113.69i			1.49892i	0.662052	+	0.749458i	$0.269686\pi$
$743$			1113.69i			1.49892i	−0.662052	+	0.749458i	$0.730314\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	149.113			0.199616
$748$		0			0
$749$			599.805i			0.800808i
$750$		0			0
$751$		−	434.422i		−	0.578458i	−0.957260	−	0.289229i	$-0.906601\pi$
$751$		−	434.422i		−	0.578458i	0.957260	−	0.289229i	$-0.0933989\pi$
$752$		0			0
$753$			80.6259i			0.107073i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−90.2103			−0.119168			−0.0595841	−	0.998223i	$-0.518977\pi$
$757$	−90.2103			−0.119168			−0.0595841	+	0.998223i	$0.518977\pi$
$758$		0			0
$759$		−	466.566i		−	0.614711i
$760$		0			0
$761$	846.768			1.11270			0.556352	−	0.830947i	$-0.312201\pi$
$761$	846.768			1.11270			0.556352	+	0.830947i	$0.312201\pi$
$762$		0			0
$763$		−	980.780i		−	1.28543i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1100.46			1.43475
$768$		0			0
$769$	−154.558			−0.200986			−0.100493	−	0.994938i	$-0.532042\pi$
$769$	−154.558			−0.200986			−0.100493	+	0.994938i	$0.532042\pi$
$770$		0			0
$771$	498.414			0.646452
$772$		0			0
$773$			1094.68i			1.41615i	0.706137	+	0.708076i	$0.250436\pi$
$773$			1094.68i			1.41615i	−0.706137	+	0.708076i	$0.749564\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	829.370			1.06740
$778$		0			0
$779$	158.048	−	303.823i	0.202886	−	0.390017i
$780$		0			0
$781$			422.774i			0.541324i
$782$		0			0
$783$	912.614			1.16554
$784$		0			0
$785$		0			0
$786$		0			0
$787$		−	1085.31i		−	1.37905i	−0.724262	−	0.689524i	$-0.757820\pi$
$787$		−	1085.31i		−	1.37905i	0.724262	−	0.689524i	$-0.242180\pi$
$788$		0			0
$789$			1129.06i			1.43101i
$790$		0			0
$791$		−	1102.13i		−	1.39334i
$792$		0			0
$793$			49.4313i			0.0623345i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			788.528i			0.989370i	0.869072	+	0.494685i	$0.164717\pi$
$797$			788.528i			0.989370i	−0.869072	+	0.494685i	$0.835283\pi$
$798$		0			0
$799$	−82.6445			−0.103435
$800$		0			0
$801$			154.659i			0.193082i
$802$		0			0
$803$	64.7475			0.0806320
$804$		0			0
$805$		0			0
$806$		0			0
$807$	4.55563			0.00564514
$808$		0			0
$809$	−159.009			−0.196550			−0.0982752	−	0.995159i	$-0.531333\pi$
$809$	−159.009			−0.196550			−0.0982752	+	0.995159i	$0.531333\pi$
$810$		0			0
$811$		−	948.179i		−	1.16915i	−0.811340	−	0.584574i	$-0.801262\pi$
$811$		−	948.179i		−	1.16915i	0.811340	−	0.584574i	$-0.198738\pi$
$812$		0			0
$813$		−	1332.59i		−	1.63911i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−46.3135	−	24.0922i	−0.0566873	−	0.0294886i
$818$		0			0
$819$		−	121.798i		−	0.148715i
$820$		0			0
$821$	352.733			0.429638			0.214819	−	0.976654i	$-0.431084\pi$
$821$	352.733			0.429638			0.214819	+	0.976654i	$0.431084\pi$
$822$		0			0
$823$	993.008			1.20657			0.603286	−	0.797525i	$-0.293858\pi$
$823$	993.008			1.20657			0.603286	+	0.797525i	$0.293858\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1388.92i		−	1.67947i	−0.542994	−	0.839737i	$-0.682709\pi$
$827$		−	1388.92i		−	1.67947i	0.542994	−	0.839737i	$-0.317291\pi$
$828$		0			0
$829$		−	1346.99i		−	1.62483i	−0.583077	−	0.812417i	$-0.698151\pi$
$829$		−	1346.99i		−	1.62483i	0.583077	−	0.812417i	$-0.301849\pi$
$830$		0			0
$831$			1080.20i			1.29988i
$832$		0			0
$833$	−10.0577			−0.0120741
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1393.84			−1.66528
$838$		0			0
$839$		−	23.1544i		−	0.0275976i	−0.999905	−	0.0137988i	$-0.995608\pi$
$839$		−	23.1544i		−	0.0275976i	0.999905	−	0.0137988i	$-0.00439244\pi$
$840$		0			0
$841$	−613.077			−0.728986
$842$		0			0
$843$	258.518			0.306664
$844$		0			0
$845$		0			0
$846$		0			0
$847$	633.348			0.747754
$848$		0			0
$849$		−	49.0860i		−	0.0578163i
$850$		0			0
$851$		−	1269.22i		−	1.49144i
$852$		0			0
$853$	−1344.49			−1.57619			−0.788097	−	0.615551i	$-0.788933\pi$
$853$	−1344.49			−1.57619			−0.788097	+	0.615551i	$0.788933\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			612.032i			0.714156i	0.934074	+	0.357078i	$0.116227\pi$
$857$			612.032i			0.714156i	−0.934074	+	0.357078i	$0.883773\pi$
$858$		0			0
$859$	−405.145			−0.471647			−0.235824	−	0.971796i	$-0.575779\pi$
$859$	−405.145			−0.471647			−0.235824	+	0.971796i	$0.575779\pi$
$860$		0			0
$861$	−371.236			−0.431168
$862$		0			0
$863$			1715.67i			1.98802i	0.109267	+	0.994012i	$0.465150\pi$
$863$			1715.67i			1.98802i	−0.109267	+	0.994012i	$0.534850\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			940.385i			1.08464i
$868$		0			0
$869$		−	27.7195i		−	0.0318982i
$870$		0			0
$871$	−815.994			−0.936848
$872$		0			0
$873$			108.343i			0.124104i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	806.571i		−	0.919693i	−0.887998	−	0.459847i	$-0.847904\pi$
$877$		−	806.571i		−	0.919693i	0.887998	−	0.459847i	$-0.152096\pi$
$878$		0			0
$879$	−253.884			−0.288833
$880$		0			0
$881$	−762.909			−0.865958			−0.432979	−	0.901404i	$-0.642538\pi$
$881$	−762.909			−0.865958			−0.432979	+	0.901404i	$0.642538\pi$
$882$		0			0
$883$	−292.155			−0.330866			−0.165433	−	0.986221i	$-0.552902\pi$
$883$	−292.155			−0.330866			−0.165433	+	0.986221i	$0.552902\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	6.76046i		−	0.00762171i	−0.999993	−	0.00381086i	$-0.998787\pi$
$887$		−	6.76046i		−	0.00762171i	0.999993	−	0.00381086i	$-0.00121304\pi$
$888$		0			0
$889$		−	509.463i		−	0.573074i
$890$		0			0
$891$	−422.575			−0.474270
$892$		0			0
$893$	1283.05	+	667.440i	1.43679	+	0.747413i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−1187.93			−1.32434
$898$		0			0
$899$	2220.82			2.47032
$900$		0			0
$901$		−	8.75940i		−	0.00972186i
$902$		0			0
$903$			56.5896i			0.0626684i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	123.248i		−	0.135885i	−0.997689	−	0.0679425i	$-0.978357\pi$
$907$		−	123.248i		−	0.135885i	0.997689	−	0.0679425i	$-0.0216435\pi$
$908$		0			0
$909$	260.743			0.286846
$910$		0			0
$911$			102.149i			0.112129i	0.998427	+	0.0560645i	$0.0178552\pi$
$911$			102.149i			0.112129i	−0.998427	+	0.0560645i	$0.982145\pi$
$912$		0			0
$913$	−403.349			−0.441784
$914$		0			0
$915$		0			0
$916$		0			0
$917$	819.208			0.893357
$918$		0			0
$919$	−793.186			−0.863096			−0.431548	−	0.902090i	$-0.642033\pi$
$919$	−793.186			−0.863096			−0.431548	+	0.902090i	$0.642033\pi$
$920$		0			0
$921$	−694.844			−0.754446
$922$		0			0
$923$	1076.43			1.16623
$924$		0			0
$925$		0			0
$926$		0			0
$927$			219.118i			0.236373i
$928$		0			0
$929$	527.324			0.567625			0.283813	−	0.958880i	$-0.408401\pi$
$929$	527.324			0.567625			0.283813	+	0.958880i	$0.408401\pi$
$930$		0			0
$931$	156.146	+	81.2266i	0.167718	+	0.0872466i
$932$		0			0
$933$		−	340.820i		−	0.365295i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1358.67			1.45002			0.725012	−	0.688736i	$-0.241834\pi$
$937$	1358.67			1.45002			0.725012	+	0.688736i	$0.241834\pi$
$938$		0			0
$939$			1599.42i			1.70333i
$940$		0			0
$941$			767.460i			0.815579i	0.913076	+	0.407790i	$0.133700\pi$
$941$			767.460i			0.815579i	−0.913076	+	0.407790i	$0.866300\pi$
$942$		0			0
$943$			568.116i			0.602456i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−588.458			−0.621392			−0.310696	−	0.950509i	$-0.600562\pi$
$947$	−588.458			−0.621392			−0.310696	+	0.950509i	$0.600562\pi$
$948$		0			0
$949$		−	164.855i		−	0.173714i
$950$		0			0
$951$	478.641			0.503302
$952$		0			0
$953$		−	1523.39i		−	1.59852i	−0.600987	−	0.799259i	$-0.705226\pi$
$953$		−	1523.39i		−	1.59852i	0.600987	−	0.799259i	$-0.294774\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	564.472			0.589835
$958$		0			0
$959$	370.679			0.386527
$960$		0			0
$961$	−2430.88			−2.52953
$962$		0			0
$963$			159.374i			0.165497i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	18.3510			0.0189772			0.00948861	−	0.999955i	$-0.496980\pi$
$967$	18.3510			0.0189772			0.00948861	+	0.999955i	$0.496980\pi$
$968$		0			0
$969$	−31.1041	+	59.7929i	−0.0320992	+	0.0617057i
$970$		0			0
$971$			1560.15i			1.60675i	0.595476	+	0.803373i	$0.296964\pi$
$971$			1560.15i			1.60675i	−0.595476	+	0.803373i	$0.703036\pi$
$972$		0			0
$973$	15.6611			0.0160957
$974$		0			0
$975$		0			0
$976$		0			0
$977$			577.097i			0.590683i	0.955392	+	0.295341i	$0.0954334\pi$
$977$			577.097i			0.590683i	−0.955392	+	0.295341i	$0.904567\pi$
$978$		0			0
$979$		−	418.350i		−	0.427324i
$980$		0			0
$981$		−	260.603i		−	0.265650i
$982$		0			0
$983$		−	1162.84i		−	1.18295i	−0.806324	−	0.591474i	$-0.798546\pi$
$983$		−	1162.84i		−	1.18295i	0.806324	−	0.591474i	$-0.201454\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		−	1567.74i		−	1.58838i
$988$		0			0
$989$	86.6012			0.0875644
$990$		0			0
$991$			469.236i			0.473497i	0.971571	+	0.236749i	$0.0760818\pi$
$991$			469.236i			0.473497i	−0.971571	+	0.236749i	$0.923918\pi$
$992$		0			0
$993$	−1042.41			−1.04976
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1618.73			1.62360			0.811802	−	0.583933i	$-0.198487\pi$
$997$	1618.73			1.62360			0.811802	+	0.583933i	$0.198487\pi$
$998$		0			0
$999$	−963.752			−0.964717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.3.e.e.1101.3		12
5.2	odd	4		380.3.g.c.189.4	yes	12
5.3	odd	4		380.3.g.c.189.9	yes	12
5.4	even	2	inner	1900.3.e.e.1101.10		12
15.2	even	4		3420.3.h.e.2089.1		12
15.8	even	4		3420.3.h.e.2089.4		12
19.18	odd	2	inner	1900.3.e.e.1101.9		12
95.18	even	4		380.3.g.c.189.3	✓	12
95.37	even	4		380.3.g.c.189.10	yes	12
95.94	odd	2	inner	1900.3.e.e.1101.4		12
285.113	odd	4		3420.3.h.e.2089.3		12
285.227	odd	4		3420.3.h.e.2089.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.g.c.189.3	✓	12	95.18	even	4
380.3.g.c.189.4	yes	12	5.2	odd	4
380.3.g.c.189.9	yes	12	5.3	odd	4
380.3.g.c.189.10	yes	12	95.37	even	4
1900.3.e.e.1101.3		12	1.1	even	1	trivial
1900.3.e.e.1101.4		12	95.94	odd	2	inner
1900.3.e.e.1101.9		12	19.18	odd	2	inner
1900.3.e.e.1101.10		12	5.4	even	2	inner
3420.3.h.e.2089.1		12	15.2	even	4
3420.3.h.e.2089.2		12	285.227	odd	4
3420.3.h.e.2089.3		12	285.113	odd	4
3420.3.h.e.2089.4		12	15.8	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:	\( N \)	\(=\)	\( 1900 = 2^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1900.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.26725i q^{3} -6.30368 q^{7} -1.67495 q^{9} +O(q^{10})\)	\(q-3.26725i q^{3} -6.30368 q^{7} -1.67495 q^{9} +4.53070 q^{11} -11.5357i q^{13} +1.08572 q^{17} +(-16.8558 - 8.76832i) q^{19} +20.5957i q^{21} +31.5184 q^{23} -23.9328i q^{27} +38.1324i q^{29} -58.2398i q^{31} -14.8030i q^{33} -40.2691i q^{37} -37.6901 q^{39} +18.0249i q^{41} +2.74764 q^{43} -76.1195 q^{47} -9.26364 q^{49} -3.54733i q^{51} -8.06781i q^{53} +(-28.6483 + 55.0720i) q^{57} +95.3957i q^{59} -4.28507 q^{61} +10.5583 q^{63} -70.7364i q^{67} -102.979i q^{69} +93.3131i q^{71} +14.2908 q^{73} -28.5601 q^{77} -6.11815i q^{79} -93.2691 q^{81} -89.0257 q^{83} +124.588 q^{87} -92.3366i q^{89} +72.7174i q^{91} -190.284 q^{93} -64.6843i q^{97} -7.58869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 76 q^{9}+O(q^{10}) \) 12 * q - 76 * q^9	\( 12 q - 76 q^{9} - 24 q^{11} - 68 q^{19} + 264 q^{39} - 212 q^{49} - 600 q^{61} + 492 q^{81} - 136 q^{99}+O(q^{100}) \) 12 * q - 76 * q^9 - 24 * q^11 - 68 * q^19 + 264 * q^39 - 212 * q^49 - 600 * q^61 + 492 * q^81 - 136 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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