LMFDB - Embedded newform 980.4.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [980,4,Mod(1,980)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(980, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("980.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	980.a (trivial)

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:	yes
Analytic conductor:	$57.8218718056$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{249}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 62 $ x^2 - x - 62
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.38987$ of defining polynomial
Character	$\chi$	$=$	980.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+7.38987 q^{3} +5.00000 q^{5} +27.6101 q^{9} +O(q^{10})$	$q+7.38987 q^{3} +5.00000 q^{5} +27.6101 q^{9} -62.9493 q^{11} -78.1696 q^{13} +36.9493 q^{15} -21.0507 q^{17} +141.119 q^{19} -202.238 q^{23} +25.0000 q^{25} +4.50880 q^{27} +47.7291 q^{29} -39.4581 q^{31} -465.187 q^{33} -308.374 q^{37} -577.663 q^{39} +233.899 q^{41} -457.220 q^{43} +138.051 q^{45} -73.3899 q^{47} -155.562 q^{51} -364.511 q^{53} -314.747 q^{55} +1042.85 q^{57} +393.322 q^{59} -401.661 q^{61} -390.848 q^{65} +507.255 q^{67} -1494.51 q^{69} +742.441 q^{71} +129.493 q^{73} +184.747 q^{75} -27.7643 q^{79} -712.154 q^{81} +508.445 q^{83} -105.253 q^{85} +352.711 q^{87} +294.511 q^{89} -291.590 q^{93} +705.595 q^{95} -1181.73 q^{97} -1738.04 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 10 q^{5} + 71 q^{9}+O(q^{10}) $ 2 * q - q^3 + 10 * q^5 + 71 * q^9	$ 2 q - q^{3} + 10 q^{5} + 71 q^{9} - 47 q^{11} - 109 q^{13} - 5 q^{15} - 121 q^{17} + 156 q^{19} - 152 q^{23} + 50 q^{25} - 133 q^{27} - 15 q^{29} + 142 q^{31} - 599 q^{33} + 46 q^{37} - 319 q^{39} + 310 q^{41} - 946 q^{43} + 355 q^{45} - 131 q^{47} + 683 q^{51} + 344 q^{53} - 235 q^{55} + 918 q^{57} + 976 q^{59} - 898 q^{61} - 545 q^{65} + 478 q^{67} - 1916 q^{69} + 1548 q^{71} - 530 q^{73} - 25 q^{75} + 623 q^{79} - 730 q^{81} - 782 q^{83} - 605 q^{85} + 879 q^{87} - 484 q^{89} - 1814 q^{93} + 780 q^{95} - 2253 q^{97} - 1046 q^{99}+O(q^{100}) $ 2 * q - q^3 + 10 * q^5 + 71 * q^9 - 47 * q^11 - 109 * q^13 - 5 * q^15 - 121 * q^17 + 156 * q^19 - 152 * q^23 + 50 * q^25 - 133 * q^27 - 15 * q^29 + 142 * q^31 - 599 * q^33 + 46 * q^37 - 319 * q^39 + 310 * q^41 - 946 * q^43 + 355 * q^45 - 131 * q^47 + 683 * q^51 + 344 * q^53 - 235 * q^55 + 918 * q^57 + 976 * q^59 - 898 * q^61 - 545 * q^65 + 478 * q^67 - 1916 * q^69 + 1548 * q^71 - 530 * q^73 - 25 * q^75 + 623 * q^79 - 730 * q^81 - 782 * q^83 - 605 * q^85 + 879 * q^87 - 484 * q^89 - 1814 * q^93 + 780 * q^95 - 2253 * q^97 - 1046 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	7.38987	1.42218	0.711090	−	0.703101i	$-0.248202\pi$
$3$	7.38987	1.42218	0.711090	+	0.703101i	$0.248202\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	27.6101	1.02260
$10$	0	0
$11$	−62.9493	−1.72545	−0.862724	−	0.505674i	$-0.831244\pi$
$11$	−62.9493	−1.72545	−0.862724	+	0.505674i	$0.831244\pi$
$12$	0	0
$13$	−78.1696	−1.66772	−0.833859	−	0.551977i	$-0.813874\pi$
$13$	−78.1696	−1.66772	−0.833859	+	0.551977i	$0.813874\pi$
$14$	0	0
$15$	36.9493	0.636018
$16$	0	0
$17$	−21.0507	−0.300326	−0.150163	−	0.988661i	$-0.547980\pi$
$17$	−21.0507	−0.300326	−0.150163	+	0.988661i	$0.547980\pi$
$18$	0	0
$19$	141.119	1.70394	0.851971	−	0.523589i	$-0.175407\pi$
$19$	141.119	1.70394	0.851971	+	0.523589i	$0.175407\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−202.238	−1.83346	−0.916729	−	0.399511i	$-0.869180\pi$
$23$	−202.238	−1.83346	−0.916729	+	0.399511i	$0.869180\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	4.50880	0.0321378
$28$	0	0
$29$	47.7291	0.305623	0.152811	−	0.988255i	$-0.451167\pi$
$29$	47.7291	0.305623	0.152811	+	0.988255i	$0.451167\pi$
$30$	0	0
$31$	−39.4581	−0.228609	−0.114305	−	0.993446i	$-0.536464\pi$
$31$	−39.4581	−0.228609	−0.114305	+	0.993446i	$0.536464\pi$
$32$	0	0
$33$	−465.187	−2.45390
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−308.374	−1.37017	−0.685087	−	0.728461i	$-0.740236\pi$
$37$	−308.374	−1.37017	−0.685087	+	0.728461i	$0.740236\pi$
$38$	0	0
$39$	−577.663	−2.37180
$40$	0	0
$41$	233.899	0.890947	0.445474	−	0.895295i	$-0.353035\pi$
$41$	233.899	0.890947	0.445474	+	0.895295i	$0.353035\pi$
$42$	0	0
$43$	−457.220	−1.62152	−0.810761	−	0.585377i	$-0.800946\pi$
$43$	−457.220	−1.62152	−0.810761	+	0.585377i	$0.800946\pi$
$44$	0	0
$45$	138.051	0.457320
$46$	0	0
$47$	−73.3899	−0.227766	−0.113883	−	0.993494i	$-0.536329\pi$
$47$	−73.3899	−0.227766	−0.113883	+	0.993494i	$0.536329\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−155.562	−0.427117
$52$	0	0
$53$	−364.511	−0.944706	−0.472353	−	0.881409i	$-0.656595\pi$
$53$	−364.511	−0.944706	−0.472353	+	0.881409i	$0.656595\pi$
$54$	0	0
$55$	−314.747	−0.771644
$56$	0	0
$57$	1042.85	2.42331
$58$	0	0
$59$	393.322	0.867900	0.433950	−	0.900937i	$-0.357119\pi$
$59$	393.322	0.867900	0.433950	+	0.900937i	$0.357119\pi$
$60$	0	0
$61$	−401.661	−0.843072	−0.421536	−	0.906812i	$-0.638509\pi$
$61$	−401.661	−0.843072	−0.421536	+	0.906812i	$0.638509\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−390.848	−0.745827
$66$	0	0
$67$	507.255	0.924942	0.462471	−	0.886634i	$-0.346963\pi$
$67$	507.255	0.924942	0.462471	+	0.886634i	$0.346963\pi$
$68$	0	0
$69$	−1494.51	−2.60751
$70$	0	0
$71$	742.441	1.24101	0.620503	−	0.784204i	$-0.286928\pi$
$71$	742.441	1.24101	0.620503	+	0.784204i	$0.286928\pi$
$72$	0	0
$73$	129.493	0.207617	0.103809	−	0.994597i	$-0.466897\pi$
$73$	129.493	0.207617	0.103809	+	0.994597i	$0.466897\pi$
$74$	0	0
$75$	184.747	0.284436
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−27.7643	−0.0395408	−0.0197704	−	0.999805i	$-0.506294\pi$
$79$	−27.7643	−0.0395408	−0.0197704	+	0.999805i	$0.506294\pi$
$80$	0	0
$81$	−712.154	−0.976892
$82$	0	0
$83$	508.445	0.672398	0.336199	−	0.941791i	$-0.390858\pi$
$83$	508.445	0.672398	0.336199	+	0.941791i	$0.390858\pi$
$84$	0	0
$85$	−105.253	−0.134310
$86$	0	0
$87$	352.711	0.434651
$88$	0	0
$89$	294.511	0.350765	0.175383	−	0.984500i	$-0.443884\pi$
$89$	294.511	0.350765	0.175383	+	0.984500i	$0.443884\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−291.590	−0.325124
$94$	0	0
$95$	705.595	0.762026
$96$	0	0
$97$	−1181.73	−1.23697	−0.618487	−	0.785795i	$-0.712254\pi$
$97$	−1181.73	−1.23697	−0.618487	+	0.785795i	$0.712254\pi$
$98$	0	0
$99$	−1738.04	−1.76444
$100$	0	0
$101$	1064.58	1.04881	0.524405	−	0.851469i	$-0.324288\pi$
$101$	1064.58	1.04881	0.524405	+	0.851469i	$0.324288\pi$
$102$	0	0
$103$	−880.985	−0.842777	−0.421388	−	0.906880i	$-0.638457\pi$
$103$	−880.985	−0.842777	−0.421388	+	0.906880i	$0.638457\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−222.780	−0.201280	−0.100640	−	0.994923i	$-0.532089\pi$
$107$	−222.780	−0.201280	−0.100640	+	0.994923i	$0.532089\pi$
$108$	0	0
$109$	753.425	0.662065	0.331032	−	0.943619i	$-0.392603\pi$
$109$	753.425	0.662065	0.331032	+	0.943619i	$0.392603\pi$
$110$	0	0
$111$	−2278.85	−1.94863
$112$	0	0
$113$	40.6475	0.0338389	0.0169194	−	0.999857i	$-0.494614\pi$
$113$	40.6475	0.0338389	0.0169194	+	0.999857i	$0.494614\pi$
$114$	0	0
$115$	−1011.19	−0.819947
$116$	0	0
$117$	−2158.27	−1.70541
$118$	0	0
$119$	0	0
$120$	0	0
$121$	2631.62	1.97717
$122$	0	0
$123$	1728.48	1.26709
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−678.515	−0.474083	−0.237041	−	0.971500i	$-0.576178\pi$
$127$	−678.515	−0.474083	−0.237041	+	0.971500i	$0.576178\pi$
$128$	0	0
$129$	−3378.80	−2.30610
$130$	0	0
$131$	−2859.63	−1.90723	−0.953616	−	0.301026i	$-0.902671\pi$
$131$	−2859.63	−1.90723	−0.953616	+	0.301026i	$0.902671\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	22.5440	0.0143724
$136$	0	0
$137$	−474.877	−0.296142	−0.148071	−	0.988977i	$-0.547306\pi$
$137$	−474.877	−0.296142	−0.148071	+	0.988977i	$0.547306\pi$
$138$	0	0
$139$	−1624.18	−0.991088	−0.495544	−	0.868583i	$-0.665031\pi$
$139$	−1624.18	−0.991088	−0.495544	+	0.868583i	$0.665031\pi$
$140$	0	0
$141$	−542.341	−0.323925
$142$	0	0
$143$	4920.72	2.87756
$144$	0	0
$145$	238.645	0.136679
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2248.72	1.23639	0.618196	−	0.786024i	$-0.287864\pi$
$149$	2248.72	1.23639	0.618196	+	0.786024i	$0.287864\pi$
$150$	0	0
$151$	2215.43	1.19397	0.596985	−	0.802252i	$-0.296365\pi$
$151$	2215.43	1.19397	0.596985	+	0.802252i	$0.296365\pi$
$152$	0	0
$153$	−581.212	−0.307112
$154$	0	0
$155$	−197.291	−0.102237
$156$	0	0
$157$	2731.22	1.38838	0.694189	−	0.719792i	$-0.255763\pi$
$157$	2731.22	1.38838	0.694189	+	0.719792i	$0.255763\pi$
$158$	0	0
$159$	−2693.69	−1.34354
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2741.22	1.31723	0.658617	−	0.752478i	$-0.271142\pi$
$163$	2741.22	1.31723	0.658617	+	0.752478i	$0.271142\pi$
$164$	0	0
$165$	−2325.94	−1.09742
$166$	0	0
$167$	−1010.07	−0.468033	−0.234016	−	0.972233i	$-0.575187\pi$
$167$	−1010.07	−0.468033	−0.234016	+	0.972233i	$0.575187\pi$
$168$	0	0
$169$	3913.49	1.78129
$170$	0	0
$171$	3896.31	1.74245
$172$	0	0
$173$	−637.698	−0.280250	−0.140125	−	0.990134i	$-0.544750\pi$
$173$	−637.698	−0.280250	−0.140125	+	0.990134i	$0.544750\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2906.59	1.23431
$178$	0	0
$179$	3180.83	1.32819	0.664096	−	0.747648i	$-0.268817\pi$
$179$	3180.83	1.32819	0.664096	+	0.747648i	$0.268817\pi$
$180$	0	0
$181$	891.365	0.366048	0.183024	−	0.983108i	$-0.441411\pi$
$181$	891.365	0.366048	0.183024	+	0.983108i	$0.441411\pi$
$182$	0	0
$183$	−2968.22	−1.19900
$184$	0	0
$185$	−1541.87	−0.612760
$186$	0	0
$187$	1325.13	0.518197
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4517.87	−1.71153	−0.855764	−	0.517366i	$-0.826913\pi$
$191$	−4517.87	−1.71153	−0.855764	+	0.517366i	$0.826913\pi$
$192$	0	0
$193$	−31.6256	−0.0117951	−0.00589756	−	0.999983i	$-0.501877\pi$
$193$	−31.6256	−0.0117951	−0.00589756	+	0.999983i	$0.501877\pi$
$194$	0	0
$195$	−2888.31	−1.06070
$196$	0	0
$197$	−3253.32	−1.17660	−0.588298	−	0.808644i	$-0.700202\pi$
$197$	−3253.32	−1.17660	−0.588298	+	0.808644i	$0.700202\pi$
$198$	0	0
$199$	−5057.78	−1.80169	−0.900846	−	0.434138i	$-0.857053\pi$
$199$	−5057.78	−1.80169	−0.900846	+	0.434138i	$0.857053\pi$
$200$	0	0
$201$	3748.55	1.31543
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1169.49	0.398444
$206$	0	0
$207$	−5583.81	−1.87489
$208$	0	0
$209$	−8883.34	−2.94007
$210$	0	0
$211$	−3717.12	−1.21278	−0.606391	−	0.795166i	$-0.707384\pi$
$211$	−3717.12	−1.21278	−0.606391	+	0.795166i	$0.707384\pi$
$212$	0	0
$213$	5486.54	1.76494
$214$	0	0
$215$	−2286.10	−0.725167
$216$	0	0
$217$	0	0
$218$	0	0
$219$	956.939	0.295269
$220$	0	0
$221$	1645.52	0.500859
$222$	0	0
$223$	1589.75	0.477387	0.238693	−	0.971095i	$-0.423281\pi$
$223$	1589.75	0.477387	0.238693	+	0.971095i	$0.423281\pi$
$224$	0	0
$225$	690.253	0.204520
$226$	0	0
$227$	−2215.05	−0.647657	−0.323828	−	0.946116i	$-0.604970\pi$
$227$	−2215.05	−0.647657	−0.323828	+	0.946116i	$0.604970\pi$
$228$	0	0
$229$	391.154	0.112874	0.0564371	−	0.998406i	$-0.482026\pi$
$229$	391.154	0.112874	0.0564371	+	0.998406i	$0.482026\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2695.51	−0.757892	−0.378946	−	0.925419i	$-0.623713\pi$
$233$	−2695.51	−0.757892	−0.378946	+	0.925419i	$0.623713\pi$
$234$	0	0
$235$	−366.949	−0.101860
$236$	0	0
$237$	−205.174	−0.0562342
$238$	0	0
$239$	−1045.99	−0.283095	−0.141547	−	0.989931i	$-0.545208\pi$
$239$	−1045.99	−0.283095	−0.141547	+	0.989931i	$0.545208\pi$
$240$	0	0
$241$	−1580.94	−0.422562	−0.211281	−	0.977425i	$-0.567764\pi$
$241$	−1580.94	−0.422562	−0.211281	+	0.977425i	$0.567764\pi$
$242$	0	0
$243$	−5384.46	−1.42145
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−11031.2	−2.84170
$248$	0	0
$249$	3757.34	0.956272
$250$	0	0
$251$	6004.18	1.50988	0.754941	−	0.655792i	$-0.227665\pi$
$251$	6004.18	1.50988	0.754941	+	0.655792i	$0.227665\pi$
$252$	0	0
$253$	12730.7	3.16354
$254$	0	0
$255$	−777.808	−0.191013
$256$	0	0
$257$	−3665.51	−0.889682	−0.444841	−	0.895610i	$-0.646740\pi$
$257$	−3665.51	−0.889682	−0.444841	+	0.895610i	$0.646740\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	1317.81	0.312529
$262$	0	0
$263$	1951.94	0.457650	0.228825	−	0.973468i	$-0.426512\pi$
$263$	1951.94	0.457650	0.228825	+	0.973468i	$0.426512\pi$
$264$	0	0
$265$	−1822.55	−0.422485
$266$	0	0
$267$	2176.40	0.498851
$268$	0	0
$269$	−5965.83	−1.35220	−0.676102	−	0.736808i	$-0.736332\pi$
$269$	−5965.83	−1.35220	−0.676102	+	0.736808i	$0.736332\pi$
$270$	0	0
$271$	−7415.88	−1.66230	−0.831148	−	0.556051i	$-0.812316\pi$
$271$	−7415.88	−1.66230	−0.831148	+	0.556051i	$0.812316\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1573.73	−0.345090
$276$	0	0
$277$	−6993.06	−1.51687	−0.758434	−	0.651750i	$-0.774035\pi$
$277$	−6993.06	−1.51687	−0.758434	+	0.651750i	$0.774035\pi$
$278$	0	0
$279$	−1089.44	−0.233775
$280$	0	0
$281$	4144.50	0.879859	0.439929	−	0.898032i	$-0.355003\pi$
$281$	4144.50	0.879859	0.439929	+	0.898032i	$0.355003\pi$
$282$	0	0
$283$	4410.94	0.926512	0.463256	−	0.886224i	$-0.346681\pi$
$283$	4410.94	0.926512	0.463256	+	0.886224i	$0.346681\pi$
$284$	0	0
$285$	5214.25	1.08374
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4469.87	−0.909804
$290$	0	0
$291$	−8732.82	−1.75920
$292$	0	0
$293$	4045.10	0.806544	0.403272	−	0.915080i	$-0.367873\pi$
$293$	4045.10	0.806544	0.403272	+	0.915080i	$0.367873\pi$
$294$	0	0
$295$	1966.61	0.388137
$296$	0	0
$297$	−283.826	−0.0554521
$298$	0	0
$299$	15808.9	3.05769
$300$	0	0
$301$	0	0
$302$	0	0
$303$	7867.11	1.49160
$304$	0	0
$305$	−2008.30	−0.377033
$306$	0	0
$307$	5552.20	1.03219	0.516093	−	0.856533i	$-0.327386\pi$
$307$	5552.20	1.03219	0.516093	+	0.856533i	$0.327386\pi$
$308$	0	0
$309$	−6510.36	−1.19858
$310$	0	0
$311$	6905.16	1.25902	0.629511	−	0.776992i	$-0.283255\pi$
$311$	6905.16	1.25902	0.629511	+	0.776992i	$0.283255\pi$
$312$	0	0
$313$	−6692.18	−1.20851	−0.604256	−	0.796790i	$-0.706530\pi$
$313$	−6692.18	−1.20851	−0.604256	+	0.796790i	$0.706530\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4288.04	−0.759749	−0.379875	−	0.925038i	$-0.624033\pi$
$317$	−4288.04	−0.759749	−0.379875	+	0.925038i	$0.624033\pi$
$318$	0	0
$319$	−3004.51	−0.527337
$320$	0	0
$321$	−1646.31	−0.286256
$322$	0	0
$323$	−2970.65	−0.511738
$324$	0	0
$325$	−1954.24	−0.333544
$326$	0	0
$327$	5567.71	0.941575
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−5510.47	−0.915053	−0.457527	−	0.889196i	$-0.651265\pi$
$331$	−5510.47	−0.915053	−0.457527	+	0.889196i	$0.651265\pi$
$332$	0	0
$333$	−8514.26	−1.40114
$334$	0	0
$335$	2536.28	0.413647
$336$	0	0
$337$	7747.05	1.25225	0.626126	−	0.779722i	$-0.284640\pi$
$337$	7747.05	1.25225	0.626126	+	0.779722i	$0.284640\pi$
$338$	0	0
$339$	300.380	0.0481250
$340$	0	0
$341$	2483.86	0.394454
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7472.55	−1.16611
$346$	0	0
$347$	9404.03	1.45485	0.727427	−	0.686185i	$-0.240716\pi$
$347$	9404.03	1.45485	0.727427	+	0.686185i	$0.240716\pi$
$348$	0	0
$349$	−1067.45	−0.163723	−0.0818617	−	0.996644i	$-0.526087\pi$
$349$	−1067.45	−0.163723	−0.0818617	+	0.996644i	$0.526087\pi$
$350$	0	0
$351$	−352.451	−0.0535967
$352$	0	0
$353$	−7131.35	−1.07525	−0.537625	−	0.843184i	$-0.680679\pi$
$353$	−7131.35	−1.07525	−0.537625	+	0.843184i	$0.680679\pi$
$354$	0	0
$355$	3712.20	0.554995
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6478.24	0.952391	0.476195	−	0.879339i	$-0.342015\pi$
$359$	6478.24	0.952391	0.476195	+	0.879339i	$0.342015\pi$
$360$	0	0
$361$	13055.6	1.90342
$362$	0	0
$363$	19447.3	2.81190
$364$	0	0
$365$	647.467	0.0928492
$366$	0	0
$367$	10123.6	1.43991	0.719955	−	0.694021i	$-0.244162\pi$
$367$	10123.6	1.43991	0.719955	+	0.694021i	$0.244162\pi$
$368$	0	0
$369$	6457.97	0.911080
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1867.57	0.259247	0.129624	−	0.991563i	$-0.458623\pi$
$373$	1867.57	0.259247	0.129624	+	0.991563i	$0.458623\pi$
$374$	0	0
$375$	923.733	0.127204
$376$	0	0
$377$	−3730.96	−0.509693
$378$	0	0
$379$	−7476.27	−1.01327	−0.506636	−	0.862160i	$-0.669111\pi$
$379$	−7476.27	−1.01327	−0.506636	+	0.862160i	$0.669111\pi$
$380$	0	0
$381$	−5014.14	−0.674231
$382$	0	0
$383$	9465.84	1.26288	0.631438	−	0.775426i	$-0.282465\pi$
$383$	9465.84	1.26288	0.631438	+	0.775426i	$0.282465\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12623.9	−1.65816
$388$	0	0
$389$	2961.42	0.385989	0.192995	−	0.981200i	$-0.438180\pi$
$389$	2961.42	0.385989	0.192995	+	0.981200i	$0.438180\pi$
$390$	0	0
$391$	4257.24	0.550634
$392$	0	0
$393$	−21132.3	−2.71243
$394$	0	0
$395$	−138.821	−0.0176832
$396$	0	0
$397$	−1713.78	−0.216655	−0.108327	−	0.994115i	$-0.534550\pi$
$397$	−1713.78	−0.216655	−0.108327	+	0.994115i	$0.534550\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8312.08	1.03513	0.517563	−	0.855645i	$-0.326839\pi$
$401$	8312.08	1.03513	0.517563	+	0.855645i	$0.326839\pi$
$402$	0	0
$403$	3084.43	0.381256
$404$	0	0
$405$	−3560.77	−0.436879
$406$	0	0
$407$	19412.0	2.36417
$408$	0	0
$409$	−7023.72	−0.849146	−0.424573	−	0.905394i	$-0.639576\pi$
$409$	−7023.72	−0.849146	−0.424573	+	0.905394i	$0.639576\pi$
$410$	0	0
$411$	−3509.28	−0.421167
$412$	0	0
$413$	0	0
$414$	0	0
$415$	2542.22	0.300706
$416$	0	0
$417$	−12002.5	−1.40951
$418$	0	0
$419$	−5194.97	−0.605706	−0.302853	−	0.953037i	$-0.597939\pi$
$419$	−5194.97	−0.605706	−0.302853	+	0.953037i	$0.597939\pi$
$420$	0	0
$421$	−8581.61	−0.993449	−0.496725	−	0.867908i	$-0.665464\pi$
$421$	−8581.61	−0.993449	−0.496725	+	0.867908i	$0.665464\pi$
$422$	0	0
$423$	−2026.30	−0.232913
$424$	0	0
$425$	−526.267	−0.0600651
$426$	0	0
$427$	0	0
$428$	0	0
$429$	36363.5	4.09242
$430$	0	0
$431$	−5893.65	−0.658670	−0.329335	−	0.944213i	$-0.606825\pi$
$431$	−5893.65	−0.658670	−0.329335	+	0.944213i	$0.606825\pi$
$432$	0	0
$433$	8696.77	0.965220	0.482610	−	0.875835i	$-0.339689\pi$
$433$	8696.77	0.965220	0.482610	+	0.875835i	$0.339689\pi$
$434$	0	0
$435$	1763.56	0.194382
$436$	0	0
$437$	−28539.6	−3.12410
$438$	0	0
$439$	−8619.55	−0.937104	−0.468552	−	0.883436i	$-0.655224\pi$
$439$	−8619.55	−0.937104	−0.468552	+	0.883436i	$0.655224\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1311.97	−0.140708	−0.0703539	−	0.997522i	$-0.522413\pi$
$443$	−1311.97	−0.140708	−0.0703539	+	0.997522i	$0.522413\pi$
$444$	0	0
$445$	1472.55	0.156867
$446$	0	0
$447$	16617.8	1.75837
$448$	0	0
$449$	−1279.20	−0.134452	−0.0672260	−	0.997738i	$-0.521415\pi$
$449$	−1279.20	−0.134452	−0.0672260	+	0.997738i	$0.521415\pi$
$450$	0	0
$451$	−14723.8	−1.53728
$452$	0	0
$453$	16371.8	1.69804
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6281.47	−0.642965	−0.321482	−	0.946916i	$-0.604181\pi$
$457$	−6281.47	−0.642965	−0.321482	+	0.946916i	$0.604181\pi$
$458$	0	0
$459$	−94.9133	−0.00965179
$460$	0	0
$461$	8283.03	0.836831	0.418415	−	0.908256i	$-0.362586\pi$
$461$	8283.03	0.836831	0.418415	+	0.908256i	$0.362586\pi$
$462$	0	0
$463$	64.4483	0.00646904	0.00323452	−	0.999995i	$-0.498970\pi$
$463$	64.4483	0.00646904	0.00323452	+	0.999995i	$0.498970\pi$
$464$	0	0
$465$	−1457.95	−0.145400
$466$	0	0
$467$	3873.31	0.383801	0.191901	−	0.981414i	$-0.438535\pi$
$467$	3873.31	0.383801	0.191901	+	0.981414i	$0.438535\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	20183.4	1.97453
$472$	0	0
$473$	28781.7	2.79785
$474$	0	0
$475$	3527.97	0.340788
$476$	0	0
$477$	−10064.2	−0.966054
$478$	0	0
$479$	5877.24	0.560622	0.280311	−	0.959909i	$-0.409562\pi$
$479$	5877.24	0.560622	0.280311	+	0.959909i	$0.409562\pi$
$480$	0	0
$481$	24105.5	2.28507
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5908.65	−0.553191
$486$	0	0
$487$	1914.27	0.178119	0.0890595	−	0.996026i	$-0.471614\pi$
$487$	1914.27	0.178119	0.0890595	+	0.996026i	$0.471614\pi$
$488$	0	0
$489$	20257.3	1.87335
$490$	0	0
$491$	8907.18	0.818687	0.409343	−	0.912380i	$-0.365758\pi$
$491$	8907.18	0.818687	0.409343	+	0.912380i	$0.365758\pi$
$492$	0	0
$493$	−1004.73	−0.0917864
$494$	0	0
$495$	−8690.20	−0.789081
$496$	0	0
$497$	0	0
$498$	0	0
$499$	14474.2	1.29850	0.649251	−	0.760574i	$-0.275082\pi$
$499$	14474.2	1.29850	0.649251	+	0.760574i	$0.275082\pi$
$500$	0	0
$501$	−7464.27	−0.665627
$502$	0	0
$503$	−16856.9	−1.49426	−0.747130	−	0.664678i	$-0.768569\pi$
$503$	−16856.9	−1.49426	−0.747130	+	0.664678i	$0.768569\pi$
$504$	0	0
$505$	5322.91	0.469042
$506$	0	0
$507$	28920.1	2.53331
$508$	0	0
$509$	−3867.92	−0.336823	−0.168411	−	0.985717i	$-0.553864\pi$
$509$	−3867.92	−0.336823	−0.168411	+	0.985717i	$0.553864\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	636.277	0.0547609
$514$	0	0
$515$	−4404.92	−0.376901
$516$	0	0
$517$	4619.84	0.392999
$518$	0	0
$519$	−4712.50	−0.398566
$520$	0	0
$521$	22837.3	1.92039	0.960194	−	0.279335i	$-0.0901141\pi$
$521$	22837.3	1.92039	0.960194	+	0.279335i	$0.0901141\pi$
$522$	0	0
$523$	−6247.46	−0.522337	−0.261169	−	0.965293i	$-0.584108\pi$
$523$	−6247.46	−0.522337	−0.261169	+	0.965293i	$0.584108\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	830.620	0.0686573
$528$	0	0
$529$	28733.2	2.36156
$530$	0	0
$531$	10859.7	0.887513
$532$	0	0
$533$	−18283.8	−1.48585
$534$	0	0
$535$	−1113.90	−0.0900150
$536$	0	0
$537$	23505.9	1.88893
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−8094.49	−0.643270	−0.321635	−	0.946864i	$-0.604232\pi$
$541$	−8094.49	−0.643270	−0.321635	+	0.946864i	$0.604232\pi$
$542$	0	0
$543$	6587.07	0.520586
$544$	0	0
$545$	3767.13	0.296084
$546$	0	0
$547$	2748.01	0.214801	0.107401	−	0.994216i	$-0.465747\pi$
$547$	2748.01	0.214801	0.107401	+	0.994216i	$0.465747\pi$
$548$	0	0
$549$	−11089.9	−0.862123
$550$	0	0
$551$	6735.48	0.520764
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−11394.2	−0.871456
$556$	0	0
$557$	−18969.1	−1.44299	−0.721495	−	0.692419i	$-0.756545\pi$
$557$	−18969.1	−1.44299	−0.721495	+	0.692419i	$0.756545\pi$
$558$	0	0
$559$	35740.7	2.70424
$560$	0	0
$561$	9792.50	0.736969
$562$	0	0
$563$	15449.2	1.15650	0.578249	−	0.815860i	$-0.303736\pi$
$563$	15449.2	1.15650	0.578249	+	0.815860i	$0.303736\pi$
$564$	0	0
$565$	203.237	0.0151332
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24995.7	−1.84160	−0.920802	−	0.390030i	$-0.872465\pi$
$569$	−24995.7	−1.84160	−0.920802	+	0.390030i	$0.872465\pi$
$570$	0	0
$571$	20457.5	1.49933	0.749666	−	0.661817i	$-0.230214\pi$
$571$	20457.5	1.49933	0.749666	+	0.661817i	$0.230214\pi$
$572$	0	0
$573$	−33386.5	−2.43410
$574$	0	0
$575$	−5055.95	−0.366691
$576$	0	0
$577$	−24864.5	−1.79397	−0.896985	−	0.442060i	$-0.854248\pi$
$577$	−24864.5	−1.79397	−0.896985	+	0.442060i	$0.854248\pi$
$578$	0	0
$579$	−233.709	−0.0167748
$580$	0	0
$581$	0	0
$582$	0	0
$583$	22945.7	1.63004
$584$	0	0
$585$	−10791.4	−0.762680
$586$	0	0
$587$	−12122.0	−0.852349	−0.426175	−	0.904641i	$-0.640139\pi$
$587$	−12122.0	−0.852349	−0.426175	+	0.904641i	$0.640139\pi$
$588$	0	0
$589$	−5568.29	−0.389537
$590$	0	0
$591$	−24041.6	−1.67333
$592$	0	0
$593$	7330.93	0.507665	0.253832	−	0.967248i	$-0.418309\pi$
$593$	7330.93	0.507665	0.253832	+	0.967248i	$0.418309\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−37376.3	−2.56233
$598$	0	0
$599$	24422.0	1.66587	0.832934	−	0.553373i	$-0.186660\pi$
$599$	24422.0	1.66587	0.832934	+	0.553373i	$0.186660\pi$
$600$	0	0
$601$	−25156.9	−1.70744	−0.853719	−	0.520733i	$-0.825659\pi$
$601$	−25156.9	−1.70744	−0.853719	+	0.520733i	$0.825659\pi$
$602$	0	0
$603$	14005.4	0.945843
$604$	0	0
$605$	13158.1	0.884219
$606$	0	0
$607$	−14437.3	−0.965391	−0.482695	−	0.875788i	$-0.660342\pi$
$607$	−14437.3	−0.965391	−0.482695	+	0.875788i	$0.660342\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5736.86	0.379850
$612$	0	0
$613$	−9373.58	−0.617610	−0.308805	−	0.951125i	$-0.599929\pi$
$613$	−9373.58	−0.617610	−0.308805	+	0.951125i	$0.599929\pi$
$614$	0	0
$615$	8642.40	0.566659
$616$	0	0
$617$	17763.9	1.15907	0.579535	−	0.814947i	$-0.303234\pi$
$617$	17763.9	1.15907	0.579535	+	0.814947i	$0.303234\pi$
$618$	0	0
$619$	14517.6	0.942665	0.471333	−	0.881956i	$-0.343773\pi$
$619$	14517.6	0.942665	0.471333	+	0.881956i	$0.343773\pi$
$620$	0	0
$621$	−911.851	−0.0589232
$622$	0	0
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−65646.7	−4.18130
$628$	0	0
$629$	6491.49	0.411498
$630$	0	0
$631$	−4206.50	−0.265385	−0.132693	−	0.991157i	$-0.542362\pi$
$631$	−4206.50	−0.265385	−0.132693	+	0.991157i	$0.542362\pi$
$632$	0	0
$633$	−27469.0	−1.72480
$634$	0	0
$635$	−3392.58	−0.212016
$636$	0	0
$637$	0	0
$638$	0	0
$639$	20498.9	1.26905
$640$	0	0
$641$	−1896.16	−0.116839	−0.0584195	−	0.998292i	$-0.518606\pi$
$641$	−1896.16	−0.116839	−0.0584195	+	0.998292i	$0.518606\pi$
$642$	0	0
$643$	−1655.83	−0.101555	−0.0507774	−	0.998710i	$-0.516170\pi$
$643$	−1655.83	−0.101555	−0.0507774	+	0.998710i	$0.516170\pi$
$644$	0	0
$645$	−16894.0	−1.03132
$646$	0	0
$647$	6686.24	0.406280	0.203140	−	0.979150i	$-0.434885\pi$
$647$	6686.24	0.406280	0.203140	+	0.979150i	$0.434885\pi$
$648$	0	0
$649$	−24759.3	−1.49752
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4663.09	−0.279450	−0.139725	−	0.990190i	$-0.544622\pi$
$653$	−4663.09	−0.279450	−0.139725	+	0.990190i	$0.544622\pi$
$654$	0	0
$655$	−14298.2	−0.852940
$656$	0	0
$657$	3575.33	0.212309
$658$	0	0
$659$	11637.3	0.687900	0.343950	−	0.938988i	$-0.388235\pi$
$659$	11637.3	0.687900	0.343950	+	0.938988i	$0.388235\pi$
$660$	0	0
$661$	−11343.0	−0.667458	−0.333729	−	0.942669i	$-0.608307\pi$
$661$	−11343.0	−0.667458	−0.333729	+	0.942669i	$0.608307\pi$
$662$	0	0
$663$	12160.2	0.712312
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−9652.63	−0.560347
$668$	0	0
$669$	11748.0	0.678930
$670$	0	0
$671$	25284.3	1.45468
$672$	0	0
$673$	32719.8	1.87408	0.937040	−	0.349223i	$-0.113555\pi$
$673$	32719.8	1.87408	0.937040	+	0.349223i	$0.113555\pi$
$674$	0	0
$675$	112.720	0.00642755
$676$	0	0
$677$	−10244.9	−0.581600	−0.290800	−	0.956784i	$-0.593921\pi$
$677$	−10244.9	−0.581600	−0.290800	+	0.956784i	$0.593921\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−16368.9	−0.921085
$682$	0	0
$683$	−21824.4	−1.22267	−0.611337	−	0.791370i	$-0.709368\pi$
$683$	−21824.4	−1.22267	−0.611337	+	0.791370i	$0.709368\pi$
$684$	0	0
$685$	−2374.38	−0.132439
$686$	0	0
$687$	2890.58	0.160528
$688$	0	0
$689$	28493.7	1.57550
$690$	0	0
$691$	−19889.9	−1.09500	−0.547502	−	0.836804i	$-0.684421\pi$
$691$	−19889.9	−1.09500	−0.547502	+	0.836804i	$0.684421\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8120.90	−0.443228
$696$	0	0
$697$	−4923.72	−0.267574
$698$	0	0
$699$	−19919.5	−1.07786
$700$	0	0
$701$	775.059	0.0417597	0.0208799	−	0.999782i	$-0.493353\pi$
$701$	775.059	0.0417597	0.0208799	+	0.999782i	$0.493353\pi$
$702$	0	0
$703$	−43517.5	−2.33470
$704$	0	0
$705$	−2711.71	−0.144864
$706$	0	0
$707$	0	0
$708$	0	0
$709$	5074.94	0.268820	0.134410	−	0.990926i	$-0.457086\pi$
$709$	5074.94	0.268820	0.134410	+	0.990926i	$0.457086\pi$
$710$	0	0
$711$	−766.575	−0.0404344
$712$	0	0
$713$	7979.93	0.419145
$714$	0	0
$715$	24603.6	1.28689
$716$	0	0
$717$	−7729.75	−0.402612
$718$	0	0
$719$	−14456.5	−0.749843	−0.374921	−	0.927057i	$-0.622330\pi$
$719$	−14456.5	−0.749843	−0.374921	+	0.927057i	$0.622330\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−11683.0	−0.600960
$724$	0	0
$725$	1193.23	0.0611246
$726$	0	0
$727$	−22164.7	−1.13073	−0.565366	−	0.824840i	$-0.691265\pi$
$727$	−22164.7	−1.13073	−0.565366	+	0.824840i	$0.691265\pi$
$728$	0	0
$729$	−20562.3	−1.04467
$730$	0	0
$731$	9624.79	0.486985
$732$	0	0
$733$	5823.52	0.293447	0.146723	−	0.989178i	$-0.453127\pi$
$733$	5823.52	0.293447	0.146723	+	0.989178i	$0.453127\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31931.4	−1.59594
$738$	0	0
$739$	17926.1	0.892315	0.446158	−	0.894954i	$-0.352792\pi$
$739$	17926.1	0.892315	0.446158	+	0.894954i	$0.352792\pi$
$740$	0	0
$741$	−81519.2	−4.04141
$742$	0	0
$743$	11438.2	0.564775	0.282388	−	0.959300i	$-0.408874\pi$
$743$	11438.2	0.564775	0.282388	+	0.959300i	$0.408874\pi$
$744$	0	0
$745$	11243.6	0.552932
$746$	0	0
$747$	14038.2	0.687593
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−31261.9	−1.51899	−0.759496	−	0.650512i	$-0.774554\pi$
$751$	−31261.9	−1.51899	−0.759496	+	0.650512i	$0.774554\pi$
$752$	0	0
$753$	44370.1	2.14733
$754$	0	0
$755$	11077.2	0.533960
$756$	0	0
$757$	−19384.1	−0.930684	−0.465342	−	0.885131i	$-0.654069\pi$
$757$	−19384.1	−0.930684	−0.465342	+	0.885131i	$0.654069\pi$
$758$	0	0
$759$	94078.5	4.49912
$760$	0	0
$761$	10143.0	0.483156	0.241578	−	0.970381i	$-0.422335\pi$
$761$	10143.0	0.483156	0.241578	+	0.970381i	$0.422335\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−2906.06	−0.137345
$766$	0	0
$767$	−30745.8	−1.44741
$768$	0	0
$769$	−12899.6	−0.604906	−0.302453	−	0.953164i	$-0.597805\pi$
$769$	−12899.6	−0.604906	−0.302453	+	0.953164i	$0.597805\pi$
$770$	0	0
$771$	−27087.6	−1.26529
$772$	0	0
$773$	−30948.6	−1.44003	−0.720015	−	0.693959i	$-0.755865\pi$
$773$	−30948.6	−1.44003	−0.720015	+	0.693959i	$0.755865\pi$
$774$	0	0
$775$	−986.453	−0.0457219
$776$	0	0
$777$	0	0
$778$	0	0
$779$	33007.5	1.51812
$780$	0	0
$781$	−46736.1	−2.14129
$782$	0	0
$783$	215.201	0.00982204
$784$	0	0
$785$	13656.1	0.620902
$786$	0	0
$787$	12403.5	0.561801	0.280901	−	0.959737i	$-0.409367\pi$
$787$	12403.5	0.561801	0.280901	+	0.959737i	$0.409367\pi$
$788$	0	0
$789$	14424.6	0.650861
$790$	0	0
$791$	0	0
$792$	0	0
$793$	31397.7	1.40601
$794$	0	0
$795$	−13468.4	−0.600851
$796$	0	0
$797$	−10324.8	−0.458874	−0.229437	−	0.973324i	$-0.573688\pi$
$797$	−10324.8	−0.458874	−0.229437	+	0.973324i	$0.573688\pi$
$798$	0	0
$799$	1544.91	0.0684040
$800$	0	0
$801$	8131.49	0.358692
$802$	0	0
$803$	−8151.52	−0.358233
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−44086.7	−1.92308
$808$	0	0
$809$	−22725.7	−0.987629	−0.493815	−	0.869567i	$-0.664398\pi$
$809$	−22725.7	−0.987629	−0.493815	+	0.869567i	$0.664398\pi$
$810$	0	0
$811$	21908.3	0.948588	0.474294	−	0.880366i	$-0.342703\pi$
$811$	21908.3	0.948588	0.474294	+	0.880366i	$0.342703\pi$
$812$	0	0
$813$	−54802.3	−2.36409
$814$	0	0
$815$	13706.1	0.589085
$816$	0	0
$817$	−64522.4	−2.76298
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21764.6	0.925199	0.462600	−	0.886567i	$-0.346917\pi$
$821$	21764.6	0.925199	0.462600	+	0.886567i	$0.346917\pi$
$822$	0	0
$823$	976.446	0.0413569	0.0206785	−	0.999786i	$-0.493417\pi$
$823$	976.446	0.0413569	0.0206785	+	0.999786i	$0.493417\pi$
$824$	0	0
$825$	−11629.7	−0.490780
$826$	0	0
$827$	−11223.0	−0.471900	−0.235950	−	0.971765i	$-0.575820\pi$
$827$	−11223.0	−0.471900	−0.235950	+	0.971765i	$0.575820\pi$
$828$	0	0
$829$	4969.41	0.208196	0.104098	−	0.994567i	$-0.466804\pi$
$829$	4969.41	0.208196	0.104098	+	0.994567i	$0.466804\pi$
$830$	0	0
$831$	−51677.8	−2.15726
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5050.34	−0.209311
$836$	0	0
$837$	−177.909	−0.00734699
$838$	0	0
$839$	25800.4	1.06166	0.530828	−	0.847480i	$-0.321881\pi$
$839$	25800.4	1.06166	0.530828	+	0.847480i	$0.321881\pi$
$840$	0	0
$841$	−22110.9	−0.906595
$842$	0	0
$843$	30627.3	1.25132
$844$	0	0
$845$	19567.4	0.796616
$846$	0	0
$847$	0	0
$848$	0	0
$849$	32596.2	1.31767
$850$	0	0
$851$	62365.0	2.51216
$852$	0	0
$853$	−43349.3	−1.74004	−0.870019	−	0.493018i	$-0.835894\pi$
$853$	−43349.3	−1.74004	−0.870019	+	0.493018i	$0.835894\pi$
$854$	0	0
$855$	19481.6	0.779246
$856$	0	0
$857$	26656.2	1.06249	0.531247	−	0.847217i	$-0.321724\pi$
$857$	26656.2	1.06249	0.531247	+	0.847217i	$0.321724\pi$
$858$	0	0
$859$	−9081.34	−0.360712	−0.180356	−	0.983601i	$-0.557725\pi$
$859$	−9081.34	−0.360712	−0.180356	+	0.983601i	$0.557725\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24115.6	0.951221	0.475610	−	0.879656i	$-0.342227\pi$
$863$	24115.6	0.951221	0.475610	+	0.879656i	$0.342227\pi$
$864$	0	0
$865$	−3188.49	−0.125332
$866$	0	0
$867$	−33031.7	−1.29391
$868$	0	0
$869$	1747.74	0.0682257
$870$	0	0
$871$	−39652.0	−1.54254
$872$	0	0
$873$	−32627.7	−1.26493
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28052.8	−1.08013	−0.540067	−	0.841622i	$-0.681601\pi$
$877$	−28052.8	−1.08013	−0.540067	+	0.841622i	$0.681601\pi$
$878$	0	0
$879$	29892.8	1.14705
$880$	0	0
$881$	−27264.1	−1.04262	−0.521312	−	0.853366i	$-0.674557\pi$
$881$	−27264.1	−1.04262	−0.521312	+	0.853366i	$0.674557\pi$
$882$	0	0
$883$	11307.9	0.430965	0.215482	−	0.976508i	$-0.430868\pi$
$883$	11307.9	0.430965	0.215482	+	0.976508i	$0.430868\pi$
$884$	0	0
$885$	14533.0	0.552001
$886$	0	0
$887$	41061.0	1.55433	0.777167	−	0.629294i	$-0.216656\pi$
$887$	41061.0	1.55433	0.777167	+	0.629294i	$0.216656\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	44829.6	1.68558
$892$	0	0
$893$	−10356.7	−0.388101
$894$	0	0
$895$	15904.1	0.593985
$896$	0	0
$897$	116825.	4.34859
$898$	0	0
$899$	−1883.30	−0.0698683
$900$	0	0
$901$	7673.20	0.283720
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4456.83	0.163702
$906$	0	0
$907$	51391.6	1.88140	0.940700	−	0.339238i	$-0.110169\pi$
$907$	51391.6	1.88140	0.940700	+	0.339238i	$0.110169\pi$
$908$	0	0
$909$	29393.2	1.07251
$910$	0	0
$911$	10350.4	0.376427	0.188214	−	0.982128i	$-0.439730\pi$
$911$	10350.4	0.376427	0.188214	+	0.982128i	$0.439730\pi$
$912$	0	0
$913$	−32006.3	−1.16019
$914$	0	0
$915$	−14841.1	−0.536209
$916$	0	0
$917$	0	0
$918$	0	0
$919$	22429.4	0.805092	0.402546	−	0.915400i	$-0.368125\pi$
$919$	22429.4	0.805092	0.402546	+	0.915400i	$0.368125\pi$
$920$	0	0
$921$	41030.1	1.46795
$922$	0	0
$923$	−58036.3	−2.06965
$924$	0	0
$925$	−7709.36	−0.274035
$926$	0	0
$927$	−24324.1	−0.861821
$928$	0	0
$929$	−20596.2	−0.727385	−0.363692	−	0.931519i	$-0.618484\pi$
$929$	−20596.2	−0.727385	−0.363692	+	0.931519i	$0.618484\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	51028.2	1.79056
$934$	0	0
$935$	6625.63	0.231745
$936$	0	0
$937$	36416.3	1.26966	0.634828	−	0.772653i	$-0.281071\pi$
$937$	36416.3	1.26966	0.634828	+	0.772653i	$0.281071\pi$
$938$	0	0
$939$	−49454.3	−1.71872
$940$	0	0
$941$	−12263.9	−0.424858	−0.212429	−	0.977176i	$-0.568137\pi$
$941$	−12263.9	−0.424858	−0.212429	+	0.977176i	$0.568137\pi$
$942$	0	0
$943$	−47303.2	−1.63351
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2827.26	−0.0970153	−0.0485076	−	0.998823i	$-0.515447\pi$
$947$	−2827.26	−0.0970153	−0.0485076	+	0.998823i	$0.515447\pi$
$948$	0	0
$949$	−10122.4	−0.346247
$950$	0	0
$951$	−31688.1	−1.08050
$952$	0	0
$953$	−30378.5	−1.03259	−0.516293	−	0.856412i	$-0.672689\pi$
$953$	−30378.5	−1.03259	−0.516293	+	0.856412i	$0.672689\pi$
$954$	0	0
$955$	−22589.4	−0.765419
$956$	0	0
$957$	−22203.0	−0.749968
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−28234.1	−0.947738
$962$	0	0
$963$	−6150.98	−0.205828
$964$	0	0
$965$	−158.128	−0.00527494
$966$	0	0
$967$	−19686.0	−0.654665	−0.327332	−	0.944909i	$-0.606150\pi$
$967$	−19686.0	−0.654665	−0.327332	+	0.944909i	$0.606150\pi$
$968$	0	0
$969$	−21952.7	−0.727783
$970$	0	0
$971$	12712.9	0.420160	0.210080	−	0.977684i	$-0.432628\pi$
$971$	12712.9	0.420160	0.210080	+	0.977684i	$0.432628\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−14441.6	−0.474359
$976$	0	0
$977$	6595.98	0.215992	0.107996	−	0.994151i	$-0.465557\pi$
$977$	6595.98	0.215992	0.107996	+	0.994151i	$0.465557\pi$
$978$	0	0
$979$	−18539.3	−0.605227
$980$	0	0
$981$	20802.2	0.677026
$982$	0	0
$983$	8837.59	0.286750	0.143375	−	0.989668i	$-0.454204\pi$
$983$	8837.59	0.286750	0.143375	+	0.989668i	$0.454204\pi$
$984$	0	0
$985$	−16266.6	−0.526190
$986$	0	0
$987$	0	0
$988$	0	0
$989$	92467.3	2.97299
$990$	0	0
$991$	−14038.6	−0.450002	−0.225001	−	0.974359i	$-0.572238\pi$
$991$	−14038.6	−0.450002	−0.225001	+	0.974359i	$0.572238\pi$
$992$	0	0
$993$	−40721.6	−1.30137
$994$	0	0
$995$	−25288.9	−0.805741
$996$	0	0
$997$	−27214.4	−0.864482	−0.432241	−	0.901758i	$-0.642277\pi$
$997$	−27214.4	−0.864482	−0.432241	+	0.901758i	$0.642277\pi$
$998$	0	0
$999$	−1390.40	−0.0440343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.4.a.p.1.2	✓	2
7.2	even	3		980.4.i.v.361.1		4
7.3	odd	6		980.4.i.t.961.2		4
7.4	even	3		980.4.i.v.961.1		4
7.5	odd	6		980.4.i.t.361.2		4
7.6	odd	2		980.4.a.s.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
980.4.a.p.1.2	✓	2	1.1	even	1	trivial
980.4.a.s.1.1	yes	2	7.6	odd	2
980.4.i.t.361.2		4	7.5	odd	6
980.4.i.t.961.2		4	7.3	odd	6
980.4.i.v.361.1		4	7.2	even	3
980.4.i.v.961.1		4	7.4	even	3

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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	980.a (trivial)

\(f(q)\)	\(=\)	\(q+7.38987 q^{3} +5.00000 q^{5} +27.6101 q^{9} +O(q^{10})\)	\(q+7.38987 q^{3} +5.00000 q^{5} +27.6101 q^{9} -62.9493 q^{11} -78.1696 q^{13} +36.9493 q^{15} -21.0507 q^{17} +141.119 q^{19} -202.238 q^{23} +25.0000 q^{25} +4.50880 q^{27} +47.7291 q^{29} -39.4581 q^{31} -465.187 q^{33} -308.374 q^{37} -577.663 q^{39} +233.899 q^{41} -457.220 q^{43} +138.051 q^{45} -73.3899 q^{47} -155.562 q^{51} -364.511 q^{53} -314.747 q^{55} +1042.85 q^{57} +393.322 q^{59} -401.661 q^{61} -390.848 q^{65} +507.255 q^{67} -1494.51 q^{69} +742.441 q^{71} +129.493 q^{73} +184.747 q^{75} -27.7643 q^{79} -712.154 q^{81} +508.445 q^{83} -105.253 q^{85} +352.711 q^{87} +294.511 q^{89} -291.590 q^{93} +705.595 q^{95} -1181.73 q^{97} -1738.04 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 10 q^{5} + 71 q^{9}+O(q^{10}) \) 2 * q - q^3 + 10 * q^5 + 71 * q^9	\( 2 q - q^{3} + 10 q^{5} + 71 q^{9} - 47 q^{11} - 109 q^{13} - 5 q^{15} - 121 q^{17} + 156 q^{19} - 152 q^{23} + 50 q^{25} - 133 q^{27} - 15 q^{29} + 142 q^{31} - 599 q^{33} + 46 q^{37} - 319 q^{39} + 310 q^{41} - 946 q^{43} + 355 q^{45} - 131 q^{47} + 683 q^{51} + 344 q^{53} - 235 q^{55} + 918 q^{57} + 976 q^{59} - 898 q^{61} - 545 q^{65} + 478 q^{67} - 1916 q^{69} + 1548 q^{71} - 530 q^{73} - 25 q^{75} + 623 q^{79} - 730 q^{81} - 782 q^{83} - 605 q^{85} + 879 q^{87} - 484 q^{89} - 1814 q^{93} + 780 q^{95} - 2253 q^{97} - 1046 q^{99}+O(q^{100}) \) 2 * q - q^3 + 10 * q^5 + 71 * q^9 - 47 * q^11 - 109 * q^13 - 5 * q^15 - 121 * q^17 + 156 * q^19 - 152 * q^23 + 50 * q^25 - 133 * q^27 - 15 * q^29 + 142 * q^31 - 599 * q^33 + 46 * q^37 - 319 * q^39 + 310 * q^41 - 946 * q^43 + 355 * q^45 - 131 * q^47 + 683 * q^51 + 344 * q^53 - 235 * q^55 + 918 * q^57 + 976 * q^59 - 898 * q^61 - 545 * q^65 + 478 * q^67 - 1916 * q^69 + 1548 * q^71 - 530 * q^73 - 25 * q^75 + 623 * q^79 - 730 * q^81 - 782 * q^83 - 605 * q^85 + 879 * q^87 - 484 * q^89 - 1814 * q^93 + 780 * q^95 - 2253 * q^97 - 1046 * q^99

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