LMFDB - Embedded newform 1900.2.c.g.1749.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,2,Mod(1749,1900)]

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, 1, 0]))

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

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

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1900 = 2^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1900.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$15.1715763840$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.4227136.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 22x^{2} + 9 $ x^6 + 9x^4 + 22x^2 + 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1749.3
Root			$0.713538i$ of defining polynomial
Character	$\chi$	$=$	1900.1749
Dual form			1900.2.c.g.1749.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.286462i q^{3} +0.286462i q^{7} +2.91794 q^{9} +O(q^{10})$	$q-0.286462i q^{3} +0.286462i q^{7} +2.91794 q^{9} +4.26819 q^{11} -3.20440i q^{13} +0.286462i q^{17} +1.00000 q^{19} +0.0820605 q^{21} -0.936212i q^{23} -1.69527i q^{27} -2.26819 q^{29} -4.18613 q^{31} -1.22267i q^{33} +8.67699i q^{37} -0.917939 q^{39} +1.08206 q^{41} -4.42708i q^{43} -0.759053i q^{47} +6.91794 q^{49} +0.0820605 q^{51} -4.42708i q^{53} -0.286462i q^{57} +4.70050 q^{59} +10.1861 q^{61} +0.835879i q^{63} +9.82284i q^{67} -0.268189 q^{69} +4.83588 q^{71} -10.4726i q^{73} +1.22267i q^{77} -5.10407 q^{79} +8.26819 q^{81} -15.7355i q^{83} +0.649750i q^{87} +9.75382 q^{89} +0.917939 q^{91} +1.19917i q^{93} -9.61320i q^{97} +12.4543 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{9}+O(q^{10}) $ 6 * q - 2 * q^9	$ 6 q - 2 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 14 q^{29} + 22 q^{31} + 14 q^{39} + 26 q^{41} + 22 q^{49} + 20 q^{51} + 12 q^{59} + 14 q^{61} + 26 q^{69} - 10 q^{71} + 36 q^{79} + 22 q^{81} - 14 q^{91}+O(q^{100}) $ 6 * q - 2 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 14 * q^29 + 22 * q^31 + 14 * q^39 + 26 * q^41 + 22 * q^49 + 20 * q^51 + 12 * q^59 + 14 * q^61 + 26 * q^69 - 10 * q^71 + 36 * q^79 + 22 * q^81 - 14 * q^91

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
$3$	− 0.286462i	− 0.165389i	−0.996575	−	0.0826945i	$-0.973647\pi$
$3$	− 0.286462i	− 0.165389i	0.996575	−	0.0826945i	$-0.0263526\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.286462i	0.108272i	0.998534	+	0.0541362i	$0.0172405\pi$
$7$	0.286462i	0.108272i	−0.998534	+	0.0541362i	$0.982759\pi$
$8$	0	0
$9$	2.91794	0.972646
$10$	0	0
$11$	4.26819	1.28691	0.643454	−	0.765485i	$-0.277501\pi$
$11$	4.26819	1.28691	0.643454	+	0.765485i	$0.277501\pi$
$12$	0	0
$13$	− 3.20440i	− 0.888741i	−0.895843	−	0.444371i	$-0.853427\pi$
$13$	− 3.20440i	− 0.888741i	0.895843	−	0.444371i	$-0.146573\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.286462i	0.0694773i	0.999396	+	0.0347386i	$0.0110599\pi$
$17$	0.286462i	0.0694773i	−0.999396	+	0.0347386i	$0.988940\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.0820605	0.0179071
$22$	0	0
$23$	− 0.936212i	− 0.195214i	−0.995225	−	0.0976069i	$-0.968881\pi$
$23$	− 0.936212i	− 0.195214i	0.995225	−	0.0976069i	$-0.0311188\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.69527i	− 0.326254i
$28$	0	0
$29$	−2.26819	−0.421192	−0.210596	−	0.977573i	$-0.567540\pi$
$29$	−2.26819	−0.421192	−0.210596	+	0.977573i	$0.567540\pi$
$30$	0	0
$31$	−4.18613	−0.751851	−0.375925	−	0.926650i	$-0.622675\pi$
$31$	−4.18613	−0.751851	−0.375925	+	0.926650i	$0.622675\pi$
$32$	0	0
$33$	− 1.22267i	− 0.212840i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.67699i	1.42649i	0.700915	+	0.713244i	$0.252775\pi$
$37$	8.67699i	1.42649i	−0.700915	+	0.713244i	$0.747225\pi$
$38$	0	0
$39$	−0.917939	−0.146988
$40$	0	0
$41$	1.08206	0.168989	0.0844947	−	0.996424i	$-0.473072\pi$
$41$	1.08206	0.168989	0.0844947	+	0.996424i	$0.473072\pi$
$42$	0	0
$43$	− 4.42708i	− 0.675123i	−0.941304	−	0.337561i	$-0.890398\pi$
$43$	− 4.42708i	− 0.675123i	0.941304	−	0.337561i	$-0.109602\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.759053i	− 0.110719i	−0.998466	−	0.0553596i	$-0.982369\pi$
$47$	− 0.759053i	− 0.110719i	0.998466	−	0.0553596i	$-0.0176305\pi$
$48$	0	0
$49$	6.91794	0.988277
$50$	0	0
$51$	0.0820605	0.0114908
$52$	0	0
$53$	− 4.42708i	− 0.608106i	−0.952655	−	0.304053i	$-0.901660\pi$
$53$	− 4.42708i	− 0.608106i	0.952655	−	0.304053i	$-0.0983399\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 0.286462i	− 0.0379428i
$58$	0	0
$59$	4.70050	0.611953	0.305976	−	0.952039i	$-0.401017\pi$
$59$	4.70050	0.611953	0.305976	+	0.952039i	$0.401017\pi$
$60$	0	0
$61$	10.1861	1.30420	0.652100	−	0.758133i	$-0.273888\pi$
$61$	10.1861	1.30420	0.652100	+	0.758133i	$0.273888\pi$
$62$	0	0
$63$	0.835879i	0.105311i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.82284i	1.20005i	0.799981	+	0.600025i	$0.204843\pi$
$67$	9.82284i	1.20005i	−0.799981	+	0.600025i	$0.795157\pi$
$68$	0	0
$69$	−0.268189	−0.0322862
$70$	0	0
$71$	4.83588	0.573913	0.286957	−	0.957944i	$-0.407356\pi$
$71$	4.83588	0.573913	0.286957	+	0.957944i	$0.407356\pi$
$72$	0	0
$73$	− 10.4726i	− 1.22572i	−0.790190	−	0.612862i	$-0.790018\pi$
$73$	− 10.4726i	− 1.22572i	0.790190	−	0.612862i	$-0.209982\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.22267i	0.139337i
$78$	0	0
$79$	−5.10407	−0.574253	−0.287126	−	0.957893i	$-0.592700\pi$
$79$	−5.10407	−0.574253	−0.287126	+	0.957893i	$0.592700\pi$
$80$	0	0
$81$	8.26819	0.918688
$82$	0	0
$83$	− 15.7355i	− 1.72720i	−0.504177	−	0.863600i	$-0.668204\pi$
$83$	− 15.7355i	− 1.72720i	0.504177	−	0.863600i	$-0.331796\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.649750i	0.0696605i
$88$	0	0
$89$	9.75382	1.03390	0.516951	−	0.856015i	$-0.327067\pi$
$89$	9.75382	1.03390	0.516951	+	0.856015i	$0.327067\pi$
$90$	0	0
$91$	0.917939	0.0962262
$92$	0	0
$93$	1.19917i	0.124348i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 9.61320i	− 0.976073i	−0.872823	−	0.488037i	$-0.837713\pi$
$97$	− 9.61320i	− 0.976073i	0.872823	−	0.488037i	$-0.162287\pi$
$98$	0	0
$99$	12.4543	1.25171
$100$	0	0
$101$	0.731811	0.0728179	0.0364089	−	0.999337i	$-0.488408\pi$
$101$	0.731811	0.0728179	0.0364089	+	0.999337i	$0.488408\pi$
$102$	0	0
$103$	3.10407i	0.305853i	0.988238	+	0.152926i	$0.0488698\pi$
$103$	3.10407i	0.305853i	−0.988238	+	0.152926i	$0.951130\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.1731i	1.37016i	0.728466	+	0.685082i	$0.240234\pi$
$107$	14.1731i	1.37016i	−0.728466	+	0.685082i	$0.759766\pi$
$108$	0	0
$109$	9.61844	0.921279	0.460640	−	0.887587i	$-0.347620\pi$
$109$	9.61844	0.921279	0.460640	+	0.887587i	$0.347620\pi$
$110$	0	0
$111$	2.48563	0.235925
$112$	0	0
$113$	− 8.77733i	− 0.825701i	−0.910799	−	0.412851i	$-0.864533\pi$
$113$	− 8.77733i	− 0.825701i	0.910799	−	0.412851i	$-0.135467\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 9.35025i	− 0.864431i
$118$	0	0
$119$	−0.0820605	−0.00752248
$120$	0	0
$121$	7.21744	0.656131
$122$	0	0
$123$	− 0.309969i	− 0.0279490i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.77209i	0.600926i	0.953793	+	0.300463i	$0.0971412\pi$
$127$	6.77209i	0.600926i	−0.953793	+	0.300463i	$0.902859\pi$
$128$	0	0
$129$	−1.26819	−0.111658
$130$	0	0
$131$	−2.91794	−0.254942	−0.127471	−	0.991842i	$-0.540686\pi$
$131$	−2.91794	−0.254942	−0.127471	+	0.991842i	$0.540686\pi$
$132$	0	0
$133$	0.286462i	0.0248394i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.4360i	− 1.23335i	−0.787216	−	0.616677i	$-0.788478\pi$
$137$	− 14.4360i	− 1.23335i	0.787216	−	0.616677i	$-0.211522\pi$
$138$	0	0
$139$	2.29950	0.195041	0.0975205	−	0.995234i	$-0.468909\pi$
$139$	2.29950	0.195041	0.0975205	+	0.995234i	$0.468909\pi$
$140$	0	0
$141$	−0.217440	−0.0183117
$142$	0	0
$143$	− 13.6770i	− 1.14373i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.98173i	− 0.163450i
$148$	0	0
$149$	−6.75382	−0.553294	−0.276647	−	0.960972i	$-0.589223\pi$
$149$	−6.75382	−0.553294	−0.276647	+	0.960972i	$0.589223\pi$
$150$	0	0
$151$	−11.8866	−0.967320	−0.483660	−	0.875256i	$-0.660693\pi$
$151$	−11.8866	−0.967320	−0.483660	+	0.875256i	$0.660693\pi$
$152$	0	0
$153$	0.835879i	0.0675768i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 14.1731i	− 1.13114i	−0.824702	−	0.565568i	$-0.808657\pi$
$157$	− 14.1731i	− 1.13114i	0.824702	−	0.565568i	$-0.191343\pi$
$158$	0	0
$159$	−1.26819	−0.100574
$160$	0	0
$161$	0.268189	0.0211363
$162$	0	0
$163$	− 7.89443i	− 0.618340i	−0.951007	−	0.309170i	$-0.899949\pi$
$163$	− 7.89443i	− 0.618340i	0.951007	−	0.309170i	$-0.100051\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	20.4726i	1.58422i	0.610381	+	0.792108i	$0.291017\pi$
$167$	20.4726i	1.58422i	−0.610381	+	0.792108i	$0.708983\pi$
$168$	0	0
$169$	2.73181	0.210139
$170$	0	0
$171$	2.91794	0.223140
$172$	0	0
$173$	24.0492i	1.82843i	0.405229	+	0.914215i	$0.367192\pi$
$173$	24.0492i	1.82843i	−0.405229	+	0.914215i	$0.632808\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 1.34651i	− 0.101210i
$178$	0	0
$179$	−18.6718	−1.39559	−0.697796	−	0.716296i	$-0.745836\pi$
$179$	−18.6718	−1.39559	−0.697796	+	0.716296i	$0.745836\pi$
$180$	0	0
$181$	−3.53638	−0.262857	−0.131428	−	0.991326i	$-0.541956\pi$
$181$	−3.53638	−0.262857	−0.131428	+	0.991326i	$0.541956\pi$
$182$	0	0
$183$	− 2.91794i	− 0.215700i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.22267i	0.0894108i
$188$	0	0
$189$	0.485629	0.0353243
$190$	0	0
$191$	−14.2682	−1.03241	−0.516205	−	0.856465i	$-0.672656\pi$
$191$	−14.2682	−1.03241	−0.516205	+	0.856465i	$0.672656\pi$
$192$	0	0
$193$	8.18089i	0.588874i	0.955671	+	0.294437i	$0.0951321\pi$
$193$	8.18089i	0.588874i	−0.955671	+	0.294437i	$0.904868\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	12.6404i	0.900595i	0.892879	+	0.450297i	$0.148682\pi$
$197$	12.6404i	0.900595i	−0.892879	+	0.450297i	$0.851318\pi$
$198$	0	0
$199$	4.10407	0.290930	0.145465	−	0.989363i	$-0.453532\pi$
$199$	4.10407	0.290930	0.145465	+	0.989363i	$0.453532\pi$
$200$	0	0
$201$	2.81387	0.198475
$202$	0	0
$203$	− 0.649750i	− 0.0456035i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 2.73181i	− 0.189874i
$208$	0	0
$209$	4.26819	0.295237
$210$	0	0
$211$	1.53638	0.105769	0.0528843	−	0.998601i	$-0.483159\pi$
$211$	1.53638	0.105769	0.0528843	+	0.998601i	$0.483159\pi$
$212$	0	0
$213$	− 1.38530i	− 0.0949189i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.19917i	− 0.0814048i
$218$	0	0
$219$	−3.00000	−0.202721
$220$	0	0
$221$	0.917939	0.0617473
$222$	0	0
$223$	− 1.40880i	− 0.0943404i	−0.998887	−	0.0471702i	$-0.984980\pi$
$223$	− 1.40880i	− 0.0943404i	0.998887	−	0.0471702i	$-0.0150203\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.9179i	1.52112i	0.649269	+	0.760559i	$0.275075\pi$
$227$	22.9179i	1.52112i	−0.649269	+	0.760559i	$0.724925\pi$
$228$	0	0
$229$	−3.78513	−0.250128	−0.125064	−	0.992149i	$-0.539914\pi$
$229$	−3.78513	−0.250128	−0.125064	+	0.992149i	$0.539914\pi$
$230$	0	0
$231$	0.350250	0.0230447
$232$	0	0
$233$	− 2.26819i	− 0.148594i	−0.997236	−	0.0742970i	$-0.976329\pi$
$233$	− 2.26819i	− 0.148594i	0.997236	−	0.0742970i	$-0.0236713\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.46212i	0.0949750i
$238$	0	0
$239$	2.18613	0.141409	0.0707045	−	0.997497i	$-0.477475\pi$
$239$	2.18613	0.141409	0.0707045	+	0.997497i	$0.477475\pi$
$240$	0	0
$241$	19.9907	1.28771	0.643857	−	0.765146i	$-0.277333\pi$
$241$	19.9907	1.28771	0.643857	+	0.765146i	$0.277333\pi$
$242$	0	0
$243$	− 7.45432i	− 0.478195i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 3.20440i	− 0.203891i
$248$	0	0
$249$	−4.50764	−0.285660
$250$	0	0
$251$	1.35025	0.0852270	0.0426135	−	0.999092i	$-0.486432\pi$
$251$	1.35025	0.0852270	0.0426135	+	0.999092i	$0.486432\pi$
$252$	0	0
$253$	− 3.99593i	− 0.251222i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.9582i	− 0.745933i	−0.927845	−	0.372967i	$-0.878341\pi$
$257$	− 11.9582i	− 0.745933i	0.927845	−	0.372967i	$-0.121659\pi$
$258$	0	0
$259$	−2.48563	−0.154449
$260$	0	0
$261$	−6.61844	−0.409671
$262$	0	0
$263$	8.31370i	0.512645i	0.966591	+	0.256322i	$0.0825109\pi$
$263$	8.31370i	0.512645i	−0.966591	+	0.256322i	$0.917489\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.79410i	− 0.170996i
$268$	0	0
$269$	5.02201	0.306197	0.153099	−	0.988211i	$-0.451075\pi$
$269$	5.02201	0.306197	0.153099	+	0.988211i	$0.451075\pi$
$270$	0	0
$271$	6.96869	0.423318	0.211659	−	0.977344i	$-0.432113\pi$
$271$	6.96869	0.423318	0.211659	+	0.977344i	$0.432113\pi$
$272$	0	0
$273$	− 0.262955i	− 0.0159148i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 12.7083i	− 0.763568i	−0.924252	−	0.381784i	$-0.875310\pi$
$277$	− 12.7083i	− 0.763568i	0.924252	−	0.381784i	$-0.124690\pi$
$278$	0	0
$279$	−12.2149	−0.731285
$280$	0	0
$281$	−16.5364	−0.986478	−0.493239	−	0.869894i	$-0.664187\pi$
$281$	−16.5364	−0.986478	−0.493239	+	0.869894i	$0.664187\pi$
$282$	0	0
$283$	26.1548i	1.55474i	0.629042	+	0.777371i	$0.283447\pi$
$283$	26.1548i	1.55474i	−0.629042	+	0.777371i	$0.716553\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.309969i	0.0182969i
$288$	0	0
$289$	16.9179	0.995173
$290$	0	0
$291$	−2.75382	−0.161432
$292$	0	0
$293$	3.95449i	0.231023i	0.993306	+	0.115512i	$0.0368508\pi$
$293$	3.95449i	0.231023i	−0.993306	+	0.115512i	$0.963149\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 7.23571i	− 0.419859i
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	1.26819	0.0730972
$302$	0	0
$303$	− 0.209636i	− 0.0120433i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 20.7355i	− 1.18344i	−0.806144	−	0.591720i	$-0.798449\pi$
$307$	− 20.7355i	− 1.18344i	0.806144	−	0.591720i	$-0.201551\pi$
$308$	0	0
$309$	0.889198	0.0505847
$310$	0	0
$311$	−1.97799	−0.112162	−0.0560808	−	0.998426i	$-0.517860\pi$
$311$	−1.97799	−0.112162	−0.0560808	+	0.998426i	$0.517860\pi$
$312$	0	0
$313$	8.67699i	0.490453i	0.969466	+	0.245226i	$0.0788623\pi$
$313$	8.67699i	0.490453i	−0.969466	+	0.245226i	$0.921138\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.34502i	− 0.131709i	−0.997829	−	0.0658546i	$-0.979023\pi$
$317$	− 2.34502i	− 0.131709i	0.997829	−	0.0658546i	$-0.0209773\pi$
$318$	0	0
$319$	−9.68106	−0.542035
$320$	0	0
$321$	4.06005	0.226610
$322$	0	0
$323$	0.286462i	0.0159392i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 2.75532i	− 0.152369i
$328$	0	0
$329$	0.217440	0.0119878
$330$	0	0
$331$	−22.8579	−1.25638	−0.628192	−	0.778059i	$-0.716205\pi$
$331$	−22.8579	−1.25638	−0.628192	+	0.778059i	$0.716205\pi$
$332$	0	0
$333$	25.3189i	1.38747i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 0.386795i	− 0.0210701i	−0.999945	−	0.0105350i	$-0.996647\pi$
$337$	− 0.386795i	− 0.0210701i	0.999945	−	0.0105350i	$-0.00335347\pi$
$338$	0	0
$339$	−2.51437	−0.136562
$340$	0	0
$341$	−17.8672	−0.967563
$342$	0	0
$343$	3.98696i	0.215276i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	14.1406i	0.759108i	0.925170	+	0.379554i	$0.123923\pi$
$347$	14.1406i	0.759108i	−0.925170	+	0.379554i	$0.876077\pi$
$348$	0	0
$349$	2.81387	0.150623	0.0753115	−	0.997160i	$-0.476005\pi$
$349$	2.81387	0.150623	0.0753115	+	0.997160i	$0.476005\pi$
$350$	0	0
$351$	−5.43231	−0.289955
$352$	0	0
$353$	13.3137i	0.708617i	0.935129	+	0.354308i	$0.115284\pi$
$353$	13.3137i	0.708617i	−0.935129	+	0.354308i	$0.884716\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.0235072i	0.00124413i
$358$	0	0
$359$	−14.1861	−0.748715	−0.374358	−	0.927284i	$-0.622137\pi$
$359$	−14.1861	−0.748715	−0.374358	+	0.927284i	$0.622137\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	− 2.06752i	− 0.108517i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 28.0272i	− 1.46301i	−0.681836	−	0.731505i	$-0.738818\pi$
$367$	− 28.0272i	− 1.46301i	0.681836	−	0.731505i	$-0.261182\pi$
$368$	0	0
$369$	3.15739	0.164367
$370$	0	0
$371$	1.26819	0.0658411
$372$	0	0
$373$	24.3502i	1.26081i	0.776267	+	0.630404i	$0.217111\pi$
$373$	24.3502i	1.26081i	−0.776267	+	0.630404i	$0.782889\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	7.26819i	0.374331i
$378$	0	0
$379$	−28.8866	−1.48381	−0.741903	−	0.670507i	$-0.766077\pi$
$379$	−28.8866	−1.48381	−0.741903	+	0.670507i	$0.766077\pi$
$380$	0	0
$381$	1.93995	0.0993865
$382$	0	0
$383$	10.3398i	0.528338i	0.964476	+	0.264169i	$0.0850977\pi$
$383$	10.3398i	0.528338i	−0.964476	+	0.264169i	$0.914902\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 12.9179i	− 0.656656i
$388$	0	0
$389$	−33.9086	−1.71924	−0.859618	−	0.510937i	$-0.829298\pi$
$389$	−33.9086	−1.71924	−0.859618	+	0.510937i	$0.829298\pi$
$390$	0	0
$391$	0.268189	0.0135629
$392$	0	0
$393$	0.835879i	0.0421645i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 33.8866i	− 1.70072i	−0.526200	−	0.850361i	$-0.676384\pi$
$397$	− 33.8866i	− 1.70072i	0.526200	−	0.850361i	$-0.323616\pi$
$398$	0	0
$399$	0.0820605	0.00410816
$400$	0	0
$401$	−10.5051	−0.524598	−0.262299	−	0.964987i	$-0.584481\pi$
$401$	−10.5051	−0.524598	−0.262299	+	0.964987i	$0.584481\pi$
$402$	0	0
$403$	13.4140i	0.668201i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.0350i	1.83576i
$408$	0	0
$409$	6.78256	0.335376	0.167688	−	0.985840i	$-0.446370\pi$
$409$	6.78256	0.335376	0.167688	+	0.985840i	$0.446370\pi$
$410$	0	0
$411$	−4.13538	−0.203983
$412$	0	0
$413$	1.34651i	0.0662577i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 0.658720i	− 0.0322576i
$418$	0	0
$419$	−18.6498	−0.911100	−0.455550	−	0.890210i	$-0.650557\pi$
$419$	−18.6498	−0.911100	−0.455550	+	0.890210i	$0.650557\pi$
$420$	0	0
$421$	9.64975	0.470300	0.235150	−	0.971959i	$-0.424442\pi$
$421$	9.64975	0.470300	0.235150	+	0.971959i	$0.424442\pi$
$422$	0	0
$423$	− 2.21487i	− 0.107691i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.91794i	0.141209i
$428$	0	0
$429$	−3.91794	−0.189160
$430$	0	0
$431$	−33.7758	−1.62692	−0.813462	−	0.581618i	$-0.802420\pi$
$431$	−33.7758	−1.62692	−0.813462	+	0.581618i	$0.802420\pi$
$432$	0	0
$433$	− 1.08986i	− 0.0523755i	−0.999657	−	0.0261878i	$-0.991663\pi$
$433$	− 1.08986i	− 0.0523755i	0.999657	−	0.0261878i	$-0.00833678\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 0.936212i	− 0.0447851i
$438$	0	0
$439$	12.9687	0.618962	0.309481	−	0.950906i	$-0.399845\pi$
$439$	12.9687	0.618962	0.309481	+	0.950906i	$0.399845\pi$
$440$	0	0
$441$	20.1861	0.961244
$442$	0	0
$443$	31.9437i	1.51769i	0.651271	+	0.758845i	$0.274236\pi$
$443$	31.9437i	1.51769i	−0.651271	+	0.758845i	$0.725764\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.93471i	0.0915088i
$448$	0	0
$449$	5.05075	0.238360	0.119180	−	0.992873i	$-0.461974\pi$
$449$	5.05075	0.238360	0.119180	+	0.992873i	$0.461974\pi$
$450$	0	0
$451$	4.61844	0.217474
$452$	0	0
$453$	3.40507i	0.159984i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	35.0948i	1.64166i	0.571170	+	0.820832i	$0.306490\pi$
$457$	35.0948i	1.64166i	−0.571170	+	0.820832i	$0.693510\pi$
$458$	0	0
$459$	0.485629	0.0226672
$460$	0	0
$461$	−27.8866	−1.29881	−0.649405	−	0.760443i	$-0.724982\pi$
$461$	−27.8866	−1.29881	−0.649405	+	0.760443i	$0.724982\pi$
$462$	0	0
$463$	28.0037i	1.30144i	0.759316	+	0.650722i	$0.225534\pi$
$463$	28.0037i	1.30144i	−0.759316	+	0.650722i	$0.774466\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	17.4543i	0.807690i	0.914828	+	0.403845i	$0.132326\pi$
$467$	17.4543i	0.807690i	−0.914828	+	0.403845i	$0.867674\pi$
$468$	0	0
$469$	−2.81387	−0.129933
$470$	0	0
$471$	−4.06005	−0.187077
$472$	0	0
$473$	− 18.8956i	− 0.868821i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 12.9179i	− 0.591472i
$478$	0	0
$479$	−26.0440	−1.18998	−0.594991	−	0.803733i	$-0.702844\pi$
$479$	−26.0440	−1.18998	−0.594991	+	0.803733i	$0.702844\pi$
$480$	0	0
$481$	27.8046	1.26778
$482$	0	0
$483$	− 0.0768261i	− 0.00349571i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 21.9907i	− 0.996494i	−0.867035	−	0.498247i	$-0.833977\pi$
$487$	− 21.9907i	− 0.996494i	0.867035	−	0.498247i	$-0.166023\pi$
$488$	0	0
$489$	−2.26146	−0.102267
$490$	0	0
$491$	−33.9907	−1.53398	−0.766989	−	0.641660i	$-0.778246\pi$
$491$	−33.9907	−1.53398	−0.766989	+	0.641660i	$0.778246\pi$
$492$	0	0
$493$	− 0.649750i	− 0.0292633i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.38530i	0.0621390i
$498$	0	0
$499$	−6.83331	−0.305901	−0.152950	−	0.988234i	$-0.548877\pi$
$499$	−6.83331	−0.305901	−0.152950	+	0.988234i	$0.548877\pi$
$500$	0	0
$501$	5.86462	0.262012
$502$	0	0
$503$	− 34.4685i	− 1.53688i	−0.639925	−	0.768438i	$-0.721034\pi$
$503$	− 34.4685i	− 1.53688i	0.639925	−	0.768438i	$-0.278966\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 0.782560i	− 0.0347547i
$508$	0	0
$509$	−18.9179	−0.838523	−0.419261	−	0.907866i	$-0.637711\pi$
$509$	−18.9179	−0.838523	−0.419261	+	0.907866i	$0.637711\pi$
$510$	0	0
$511$	3.00000	0.132712
$512$	0	0
$513$	− 1.69527i	− 0.0748478i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 3.23978i	− 0.142485i
$518$	0	0
$519$	6.88920	0.302402
$520$	0	0
$521$	−1.62774	−0.0713127	−0.0356563	−	0.999364i	$-0.511352\pi$
$521$	−1.62774	−0.0713127	−0.0356563	+	0.999364i	$0.511352\pi$
$522$	0	0
$523$	1.16669i	0.0510158i	0.999675	+	0.0255079i	$0.00812030\pi$
$523$	1.16669i	0.0510158i	−0.999675	+	0.0255079i	$0.991880\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.19917i	− 0.0522365i
$528$	0	0
$529$	22.1235	0.961892
$530$	0	0
$531$	13.7158	0.595214
$532$	0	0
$533$	− 3.46736i	− 0.150188i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	5.34875i	0.230816i
$538$	0	0
$539$	29.5271	1.27182
$540$	0	0
$541$	−23.9399	−1.02926	−0.514629	−	0.857413i	$-0.672070\pi$
$541$	−23.9399	−1.02926	−0.514629	+	0.857413i	$0.672070\pi$
$542$	0	0
$543$	1.01304i	0.0434736i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.8139i	0.633395i	0.948527	+	0.316698i	$0.102574\pi$
$547$	14.8139i	0.633395i	−0.948527	+	0.316698i	$0.897426\pi$
$548$	0	0
$549$	29.7225	1.26853
$550$	0	0
$551$	−2.26819	−0.0966281
$552$	0	0
$553$	− 1.46212i	− 0.0621757i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 39.3592i	− 1.66770i	−0.551988	−	0.833852i	$-0.686131\pi$
$557$	− 39.3592i	− 1.66770i	0.551988	−	0.833852i	$-0.313869\pi$
$558$	0	0
$559$	−14.1861	−0.600009
$560$	0	0
$561$	0.350250	0.0147876
$562$	0	0
$563$	− 9.55722i	− 0.402789i	−0.979510	−	0.201394i	$-0.935453\pi$
$563$	− 9.55722i	− 0.402789i	0.979510	−	0.201394i	$-0.0645473\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.36852i	0.0994686i
$568$	0	0
$569$	−21.2995	−0.892922	−0.446461	−	0.894803i	$-0.647316\pi$
$569$	−21.2995	−0.892922	−0.446461	+	0.894803i	$0.647316\pi$
$570$	0	0
$571$	−17.4543	−0.730440	−0.365220	−	0.930921i	$-0.619006\pi$
$571$	−17.4543	−0.730440	−0.365220	+	0.930921i	$0.619006\pi$
$572$	0	0
$573$	4.08729i	0.170749i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 37.2861i	− 1.55224i	−0.630584	−	0.776121i	$-0.717185\pi$
$577$	− 37.2861i	− 1.55224i	0.630584	−	0.776121i	$-0.282815\pi$
$578$	0	0
$579$	2.34352	0.0973932
$580$	0	0
$581$	4.50764	0.187008
$582$	0	0
$583$	− 18.8956i	− 0.782576i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 7.73181i	− 0.319126i	−0.987188	−	0.159563i	$-0.948991\pi$
$587$	− 7.73181i	− 0.319126i	0.987188	−	0.159563i	$-0.0510085\pi$
$588$	0	0
$589$	−4.18613	−0.172486
$590$	0	0
$591$	3.62101	0.148948
$592$	0	0
$593$	6.74858i	0.277131i	0.990353	+	0.138566i	$0.0442492\pi$
$593$	6.74858i	0.277131i	−0.990353	+	0.138566i	$0.955751\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 1.17566i	− 0.0481166i
$598$	0	0
$599$	16.9687	0.693322	0.346661	−	0.937991i	$-0.387315\pi$
$599$	16.9687	0.693322	0.346661	+	0.937991i	$0.387315\pi$
$600$	0	0
$601$	31.7251	1.29409	0.647046	−	0.762451i	$-0.276004\pi$
$601$	31.7251	1.29409	0.647046	+	0.762451i	$0.276004\pi$
$602$	0	0
$603$	28.6625i	1.16723i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.84518i	0.156071i	0.996951	+	0.0780356i	$0.0248648\pi$
$607$	3.84518i	0.156071i	−0.996951	+	0.0780356i	$0.975135\pi$
$608$	0	0
$609$	−0.186129	−0.00754232
$610$	0	0
$611$	−2.43231	−0.0984007
$612$	0	0
$613$	32.1208i	1.29735i	0.761066	+	0.648674i	$0.224676\pi$
$613$	32.1208i	1.29735i	−0.761066	+	0.648674i	$0.775324\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	35.2630i	1.41963i	0.704387	+	0.709817i	$0.251222\pi$
$617$	35.2630i	1.41963i	−0.704387	+	0.709817i	$0.748778\pi$
$618$	0	0
$619$	−9.29020	−0.373405	−0.186702	−	0.982417i	$-0.559780\pi$
$619$	−9.29020	−0.373405	−0.186702	+	0.982417i	$0.559780\pi$
$620$	0	0
$621$	−1.58713	−0.0636893
$622$	0	0
$623$	2.79410i	0.111943i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.22267i	− 0.0488289i
$628$	0	0
$629$	−2.48563	−0.0991085
$630$	0	0
$631$	−26.5271	−1.05603	−0.528013	−	0.849236i	$-0.677063\pi$
$631$	−26.5271	−1.05603	−0.528013	+	0.849236i	$0.677063\pi$
$632$	0	0
$633$	− 0.440114i	− 0.0174930i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 22.1679i	− 0.878322i
$638$	0	0
$639$	14.1108	0.558215
$640$	0	0
$641$	−7.86462	−0.310634	−0.155317	−	0.987865i	$-0.549640\pi$
$641$	−7.86462	−0.310634	−0.155317	+	0.987865i	$0.549640\pi$
$642$	0	0
$643$	1.67176i	0.0659277i	0.999457	+	0.0329638i	$0.0104946\pi$
$643$	1.67176i	0.0659277i	−0.999457	+	0.0329638i	$0.989505\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.4909i	1.51323i	0.653859	+	0.756616i	$0.273149\pi$
$647$	38.4909i	1.51323i	−0.653859	+	0.756616i	$0.726851\pi$
$648$	0	0
$649$	20.0626	0.787527
$650$	0	0
$651$	−0.343516	−0.0134634
$652$	0	0
$653$	25.9985i	1.01740i	0.860944	+	0.508700i	$0.169874\pi$
$653$	25.9985i	1.01740i	−0.860944	+	0.508700i	$0.830126\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 30.5584i	− 1.19220i
$658$	0	0
$659$	−46.9594	−1.82928	−0.914639	−	0.404272i	$-0.867525\pi$
$659$	−46.9594	−1.82928	−0.914639	+	0.404272i	$0.867525\pi$
$660$	0	0
$661$	−35.7225	−1.38944	−0.694722	−	0.719278i	$-0.744473\pi$
$661$	−35.7225	−1.38944	−0.694722	+	0.719278i	$0.744473\pi$
$662$	0	0
$663$	− 0.262955i	− 0.0102123i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.12351i	0.0822225i
$668$	0	0
$669$	−0.403569	−0.0156029
$670$	0	0
$671$	43.4763	1.67838
$672$	0	0
$673$	− 27.3450i	− 1.05407i	−0.849843	−	0.527036i	$-0.823303\pi$
$673$	− 27.3450i	− 1.05407i	0.849843	−	0.527036i	$-0.176697\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 39.9944i	− 1.53711i	−0.639783	−	0.768555i	$-0.720976\pi$
$677$	− 39.9944i	− 1.53711i	0.639783	−	0.768555i	$-0.279024\pi$
$678$	0	0
$679$	2.75382	0.105682
$680$	0	0
$681$	6.56512	0.251576
$682$	0	0
$683$	− 21.0130i	− 0.804042i	−0.915631	−	0.402021i	$-0.868308\pi$
$683$	− 21.0130i	− 0.804042i	0.915631	−	0.402021i	$-0.131692\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.08430i	0.0413685i
$688$	0	0
$689$	−14.1861	−0.540448
$690$	0	0
$691$	−10.7445	−0.408741	−0.204370	−	0.978894i	$-0.565515\pi$
$691$	−10.7445	−0.408741	−0.204370	+	0.978894i	$0.565515\pi$
$692$	0	0
$693$	3.56769i	0.135525i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0.309969i	0.0117409i
$698$	0	0
$699$	−0.649750	−0.0245758
$700$	0	0
$701$	42.3029	1.59776	0.798879	−	0.601491i	$-0.205427\pi$
$701$	42.3029	1.59776	0.798879	+	0.601491i	$0.205427\pi$
$702$	0	0
$703$	8.67699i	0.327259i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.209636i	0.00788417i
$708$	0	0
$709$	40.5584	1.52320	0.761601	−	0.648046i	$-0.224414\pi$
$709$	40.5584	1.52320	0.761601	+	0.648046i	$0.224414\pi$
$710$	0	0
$711$	−14.8934	−0.558545
$712$	0	0
$713$	3.91911i	0.146772i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 0.626243i	− 0.0233875i
$718$	0	0
$719$	−16.8866	−0.629765	−0.314882	−	0.949131i	$-0.601965\pi$
$719$	−16.8866	−0.629765	−0.314882	+	0.949131i	$0.601965\pi$
$720$	0	0
$721$	−0.889198	−0.0331155
$722$	0	0
$723$	− 5.72658i	− 0.212974i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	27.4398i	1.01769i	0.860860	+	0.508843i	$0.169926\pi$
$727$	27.4398i	1.01769i	−0.860860	+	0.508843i	$0.830074\pi$
$728$	0	0
$729$	22.6692	0.839600
$730$	0	0
$731$	1.26819	0.0469057
$732$	0	0
$733$	14.0078i	0.517390i	0.965959	+	0.258695i	$0.0832925\pi$
$733$	14.0078i	0.517390i	−0.965959	+	0.258695i	$0.916708\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	41.9257i	1.54435i
$738$	0	0
$739$	19.7851	0.727808	0.363904	−	0.931437i	$-0.381444\pi$
$739$	19.7851	0.727808	0.363904	+	0.931437i	$0.381444\pi$
$740$	0	0
$741$	−0.917939	−0.0337213
$742$	0	0
$743$	− 41.8344i	− 1.53475i	−0.641196	−	0.767377i	$-0.721561\pi$
$743$	− 41.8344i	− 1.53475i	0.641196	−	0.767377i	$-0.278439\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 45.9154i	− 1.67996i
$748$	0	0
$749$	−4.06005	−0.148351
$750$	0	0
$751$	48.4983	1.76973	0.884865	−	0.465848i	$-0.154251\pi$
$751$	48.4983	1.76973	0.884865	+	0.465848i	$0.154251\pi$
$752$	0	0
$753$	− 0.386795i	− 0.0140956i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	32.2537i	1.17228i	0.810210	+	0.586139i	$0.199353\pi$
$757$	32.2537i	1.17228i	−0.810210	+	0.586139i	$0.800647\pi$
$758$	0	0
$759$	−1.14468	−0.0415493
$760$	0	0
$761$	−25.6184	−0.928668	−0.464334	−	0.885660i	$-0.653706\pi$
$761$	−25.6184	−0.928668	−0.464334	+	0.885660i	$0.653706\pi$
$762$	0	0
$763$	2.75532i	0.0997492i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 15.0623i	− 0.543868i
$768$	0	0
$769$	−42.3096	−1.52572	−0.762862	−	0.646561i	$-0.776207\pi$
$769$	−42.3096	−1.52572	−0.762862	+	0.646561i	$0.776207\pi$
$770$	0	0
$771$	−3.42558	−0.123369
$772$	0	0
$773$	0.889198i	0.0319822i	0.999872	+	0.0159911i	$0.00509035\pi$
$773$	0.889198i	0.0319822i	−0.999872	+	0.0159911i	$0.994910\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.712038i	0.0255442i
$778$	0	0
$779$	1.08206	0.0387688
$780$	0	0
$781$	20.6404	0.738573
$782$	0	0
$783$	3.84518i	0.137416i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	50.6650i	1.80601i	0.429627	+	0.903007i	$0.358645\pi$
$787$	50.6650i	1.80601i	−0.429627	+	0.903007i	$0.641355\pi$
$788$	0	0
$789$	2.38156	0.0847858
$790$	0	0
$791$	2.51437	0.0894007
$792$	0	0
$793$	− 32.6404i	− 1.15910i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.4909i	1.18631i	0.805089	+	0.593154i	$0.202117\pi$
$797$	33.4909i	1.18631i	−0.805089	+	0.593154i	$0.797883\pi$
$798$	0	0
$799$	0.217440	0.00769247
$800$	0	0
$801$	28.4611	1.00562
$802$	0	0
$803$	− 44.6990i	− 1.57739i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.43861i	− 0.0506416i
$808$	0	0
$809$	6.35025	0.223263	0.111631	−	0.993750i	$-0.464392\pi$
$809$	6.35025	0.223263	0.111631	+	0.993750i	$0.464392\pi$
$810$	0	0
$811$	−1.08463	−0.0380865	−0.0190433	−	0.999819i	$-0.506062\pi$
$811$	−1.08463	−0.0380865	−0.0190433	+	0.999819i	$0.506062\pi$
$812$	0	0
$813$	− 1.99627i	− 0.0700121i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 4.42708i	− 0.154884i
$818$	0	0
$819$	2.67849	0.0935941
$820$	0	0
$821$	−12.3630	−0.431470	−0.215735	−	0.976452i	$-0.569215\pi$
$821$	−12.3630	−0.431470	−0.215735	+	0.976452i	$0.569215\pi$
$822$	0	0
$823$	− 27.0765i	− 0.943827i	−0.881645	−	0.471914i	$-0.843563\pi$
$823$	− 27.0765i	− 0.943827i	0.881645	−	0.471914i	$-0.156437\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 17.0765i	− 0.593808i	−0.954907	−	0.296904i	$-0.904046\pi$
$827$	− 17.0765i	− 0.593808i	0.954907	−	0.296904i	$-0.0959541\pi$
$828$	0	0
$829$	28.5804	0.992638	0.496319	−	0.868140i	$-0.334685\pi$
$829$	28.5804	0.992638	0.496319	+	0.868140i	$0.334685\pi$
$830$	0	0
$831$	−3.64045	−0.126286
$832$	0	0
$833$	1.98173i	0.0686628i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.09660i	0.245294i
$838$	0	0
$839$	55.9019	1.92995	0.964974	−	0.262346i	$-0.0844960\pi$
$839$	55.9019	1.92995	0.964974	+	0.262346i	$0.0844960\pi$
$840$	0	0
$841$	−23.8553	−0.822597
$842$	0	0
$843$	4.73705i	0.163153i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.06752i	0.0710409i
$848$	0	0
$849$	7.49236	0.257137
$850$	0	0
$851$	8.12351	0.278470
$852$	0	0
$853$	4.93214i	0.168873i	0.996429	+	0.0844367i	$0.0269091\pi$
$853$	4.93214i	0.168873i	−0.996429	+	0.0844367i	$0.973091\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.9411i	1.36436i	0.731183	+	0.682181i	$0.238968\pi$
$857$	39.9411i	1.36436i	−0.731183	+	0.682181i	$0.761032\pi$
$858$	0	0
$859$	−1.99070	−0.0679217	−0.0339608	−	0.999423i	$-0.510812\pi$
$859$	−1.99070	−0.0679217	−0.0339608	+	0.999423i	$0.510812\pi$
$860$	0	0
$861$	0.0887944	0.00302611
$862$	0	0
$863$	− 31.9817i	− 1.08867i	−0.838868	−	0.544335i	$-0.816782\pi$
$863$	− 31.9817i	− 1.08867i	0.838868	−	0.544335i	$-0.183218\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 4.84635i	− 0.164591i
$868$	0	0
$869$	−21.7851	−0.739010
$870$	0	0
$871$	31.4763	1.06653
$872$	0	0
$873$	− 28.0507i	− 0.949374i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 22.2917i	− 0.752737i	−0.926470	−	0.376369i	$-0.877173\pi$
$877$	− 22.2917i	− 0.752737i	0.926470	−	0.376369i	$-0.122827\pi$
$878$	0	0
$879$	1.13281	0.0382087
$880$	0	0
$881$	35.6912	1.20247	0.601233	−	0.799073i	$-0.294676\pi$
$881$	35.6912	1.20247	0.601233	+	0.799073i	$0.294676\pi$
$882$	0	0
$883$	− 17.9948i	− 0.605572i	−0.953059	−	0.302786i	$-0.902083\pi$
$883$	− 17.9948i	− 0.605572i	0.953059	−	0.302786i	$-0.0979168\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	24.3566i	0.817813i	0.912576	+	0.408907i	$0.134090\pi$
$887$	24.3566i	0.817813i	−0.912576	+	0.408907i	$0.865910\pi$
$888$	0	0
$889$	−1.93995	−0.0650637
$890$	0	0
$891$	35.2902	1.18227
$892$	0	0
$893$	− 0.759053i	− 0.0254007i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0.859386i	0.0286941i
$898$	0	0
$899$	9.49493	0.316674
$900$	0	0
$901$	1.26819	0.0422495
$902$	0	0
$903$	− 0.363288i	− 0.0120895i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	5.73821i	0.190534i	0.995452	+	0.0952671i	$0.0303705\pi$
$907$	5.73821i	0.190534i	−0.995452	+	0.0952671i	$0.969629\pi$
$908$	0	0
$909$	2.13538	0.0708261
$910$	0	0
$911$	−46.8291	−1.55152	−0.775759	−	0.631029i	$-0.782633\pi$
$911$	−46.8291	−1.55152	−0.775759	+	0.631029i	$0.782633\pi$
$912$	0	0
$913$	− 67.1623i	− 2.22275i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 0.835879i	− 0.0276032i
$918$	0	0
$919$	−2.32151	−0.0765795	−0.0382897	−	0.999267i	$-0.512191\pi$
$919$	−2.32151	−0.0765795	−0.0382897	+	0.999267i	$0.512191\pi$
$920$	0	0
$921$	−5.93995	−0.195728
$922$	0	0
$923$	− 15.4961i	− 0.510060i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.05748i	0.297487i
$928$	0	0
$929$	25.5610	0.838628	0.419314	−	0.907841i	$-0.362271\pi$
$929$	25.5610	0.838628	0.419314	+	0.907841i	$0.362271\pi$
$930$	0	0
$931$	6.91794	0.226726
$932$	0	0
$933$	0.566620i	0.0185503i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 25.7501i	− 0.841219i	−0.907242	−	0.420609i	$-0.861816\pi$
$937$	− 25.7501i	− 0.841219i	0.907242	−	0.420609i	$-0.138184\pi$
$938$	0	0
$939$	2.48563	0.0811154
$940$	0	0
$941$	−29.1261	−0.949483	−0.474741	−	0.880125i	$-0.657458\pi$
$941$	−29.1261	−0.949483	−0.474741	+	0.880125i	$0.657458\pi$
$942$	0	0
$943$	− 1.01304i	− 0.0329891i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 6.21711i	− 0.202029i	−0.994885	−	0.101014i	$-0.967791\pi$
$947$	− 6.21711i	− 0.202029i	0.994885	−	0.101014i	$-0.0322088\pi$
$948$	0	0
$949$	−33.5584	−1.08935
$950$	0	0
$951$	−0.671758	−0.0217832
$952$	0	0
$953$	− 47.8266i	− 1.54925i	−0.632418	−	0.774627i	$-0.717937\pi$
$953$	− 47.8266i	− 1.54925i	0.632418	−	0.774627i	$-0.282063\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.77326i	0.0896467i
$958$	0	0
$959$	4.13538	0.133538
$960$	0	0
$961$	−13.4763	−0.434720
$962$	0	0
$963$	41.3562i	1.33269i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 37.2301i	− 1.19724i	−0.801033	−	0.598620i	$-0.795716\pi$
$967$	− 37.2301i	− 1.19724i	0.801033	−	0.598620i	$-0.204284\pi$
$968$	0	0
$969$	0.0820605	0.00263616
$970$	0	0
$971$	37.8359	1.21421	0.607106	−	0.794621i	$-0.292331\pi$
$971$	37.8359	1.21421	0.607106	+	0.794621i	$0.292331\pi$
$972$	0	0
$973$	0.658720i	0.0211176i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	55.8684i	1.78739i	0.448678	+	0.893694i	$0.351895\pi$
$977$	55.8684i	1.78739i	−0.448678	+	0.893694i	$0.648105\pi$
$978$	0	0
$979$	41.6311	1.33054
$980$	0	0
$981$	28.0660	0.896079
$982$	0	0
$983$	6.93738i	0.221268i	0.993861	+	0.110634i	$0.0352881\pi$
$983$	6.93738i	0.221268i	−0.993861	+	0.110634i	$0.964712\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 0.0622883i	− 0.00198266i
$988$	0	0
$989$	−4.14468	−0.131793
$990$	0	0
$991$	36.5897	1.16231	0.581155	−	0.813793i	$-0.302601\pi$
$991$	36.5897	1.16231	0.581155	+	0.813793i	$0.302601\pi$
$992$	0	0
$993$	6.54792i	0.207792i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33.7538i	1.06899i	0.845170	+	0.534497i	$0.179499\pi$
$997$	33.7538i	1.06899i	−0.845170	+	0.534497i	$0.820501\pi$
$998$	0	0
$999$	14.7098	0.465398

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1900.2.c.g.1749.3		6
5.2	odd	4		1900.2.a.f.1.2	✓	3
5.3	odd	4		1900.2.a.h.1.2	yes	3
5.4	even	2	inner	1900.2.c.g.1749.4		6
20.3	even	4		7600.2.a.bj.1.2		3
20.7	even	4		7600.2.a.by.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1900.2.a.f.1.2	✓	3	5.2	odd	4
1900.2.a.h.1.2	yes	3	5.3	odd	4
1900.2.c.g.1749.3		6	1.1	even	1	trivial
1900.2.c.g.1749.4		6	5.4	even	2	inner
7600.2.a.bj.1.2		3	20.3	even	4
7600.2.a.by.1.2		3	20.7	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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1900.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.286462i q^{3} +0.286462i q^{7} +2.91794 q^{9} +O(q^{10})\)	\(q-0.286462i q^{3} +0.286462i q^{7} +2.91794 q^{9} +4.26819 q^{11} -3.20440i q^{13} +0.286462i q^{17} +1.00000 q^{19} +0.0820605 q^{21} -0.936212i q^{23} -1.69527i q^{27} -2.26819 q^{29} -4.18613 q^{31} -1.22267i q^{33} +8.67699i q^{37} -0.917939 q^{39} +1.08206 q^{41} -4.42708i q^{43} -0.759053i q^{47} +6.91794 q^{49} +0.0820605 q^{51} -4.42708i q^{53} -0.286462i q^{57} +4.70050 q^{59} +10.1861 q^{61} +0.835879i q^{63} +9.82284i q^{67} -0.268189 q^{69} +4.83588 q^{71} -10.4726i q^{73} +1.22267i q^{77} -5.10407 q^{79} +8.26819 q^{81} -15.7355i q^{83} +0.649750i q^{87} +9.75382 q^{89} +0.917939 q^{91} +1.19917i q^{93} -9.61320i q^{97} +12.4543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9}+O(q^{10}) \) 6 * q - 2 * q^9	\( 6 q - 2 q^{9} - 2 q^{11} + 6 q^{19} + 20 q^{21} + 14 q^{29} + 22 q^{31} + 14 q^{39} + 26 q^{41} + 22 q^{49} + 20 q^{51} + 12 q^{59} + 14 q^{61} + 26 q^{69} - 10 q^{71} + 36 q^{79} + 22 q^{81} - 14 q^{91}+O(q^{100}) \) 6 * q - 2 * q^9 - 2 * q^11 + 6 * q^19 + 20 * q^21 + 14 * q^29 + 22 * q^31 + 14 * q^39 + 26 * q^41 + 22 * q^49 + 20 * q^51 + 12 * q^59 + 14 * q^61 + 26 * q^69 - 10 * q^71 + 36 * q^79 + 22 * q^81 - 14 * q^91

Introduction

L-functions

Modular forms

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Representations

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Database

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