LMFDB - Embedded newform 2178.4.a.ca.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,4,Mod(1,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2178.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2178.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:	$128.506159993$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{67})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 35x^{2} + 256 $ x^4 - 35*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.95870$ of defining polynomial
Character	$\chi$	$=$	2178.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} +16.6082 q^{5} -7.94790 q^{7} +8.00000 q^{8} +O(q^{10})$	$q+2.00000 q^{2} +4.00000 q^{4} +16.6082 q^{5} -7.94790 q^{7} +8.00000 q^{8} +33.2163 q^{10} -30.4646 q^{13} -15.8958 q^{14} +16.0000 q^{16} -122.299 q^{17} -152.630 q^{19} +66.4326 q^{20} +132.865 q^{23} +150.831 q^{25} -60.9291 q^{26} -31.7916 q^{28} +93.3632 q^{29} +88.3632 q^{31} +32.0000 q^{32} -244.597 q^{34} -132.000 q^{35} -219.597 q^{37} -305.260 q^{38} +132.865 q^{40} -95.8309 q^{41} -371.693 q^{43} +265.730 q^{46} -95.3748 q^{47} -279.831 q^{49} +301.662 q^{50} -121.858 q^{52} -62.6468 q^{53} -63.5832 q^{56} +186.726 q^{58} -99.8443 q^{59} -615.229 q^{61} +176.726 q^{62} +64.0000 q^{64} -505.960 q^{65} -1045.35 q^{67} -489.194 q^{68} -264.000 q^{70} +94.9842 q^{71} -459.120 q^{73} -439.194 q^{74} -610.521 q^{76} +646.153 q^{79} +265.730 q^{80} -191.662 q^{82} -434.129 q^{83} -2031.15 q^{85} -743.386 q^{86} +1628.67 q^{89} +242.129 q^{91} +531.461 q^{92} -190.750 q^{94} -2534.91 q^{95} -48.9502 q^{97} -559.662 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8}+O(q^{10}) $ 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8	$ 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} + 64 q^{16} - 234 q^{17} + 178 q^{25} - 222 q^{29} - 242 q^{31} + 128 q^{32} - 468 q^{34} - 528 q^{35} - 368 q^{37} + 42 q^{41} - 694 q^{49} + 356 q^{50} - 444 q^{58} - 484 q^{62} + 256 q^{64} - 918 q^{65} - 1034 q^{67} - 936 q^{68} - 1056 q^{70} - 736 q^{74} + 84 q^{82} - 1056 q^{83} + 288 q^{91} - 5376 q^{95} - 3088 q^{97} - 1388 q^{98}+O(q^{100}) $ 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 + 64 * q^16 - 234 * q^17 + 178 * q^25 - 222 * q^29 - 242 * q^31 + 128 * q^32 - 468 * q^34 - 528 * q^35 - 368 * q^37 + 42 * q^41 - 694 * q^49 + 356 * q^50 - 444 * q^58 - 484 * q^62 + 256 * q^64 - 918 * q^65 - 1034 * q^67 - 936 * q^68 - 1056 * q^70 - 736 * q^74 + 84 * q^82 - 1056 * q^83 + 288 * q^91 - 5376 * q^95 - 3088 * q^97 - 1388 * q^98

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	16.6082	1.48548	0.742739	−	0.669581i	$-0.233526\pi$
$5$	16.6082	1.48548	0.742739	+	0.669581i	$0.233526\pi$
$6$	0	0
$7$	−7.94790	−0.429146	−0.214573	−	0.976708i	$-0.568836\pi$
$7$	−7.94790	−0.429146	−0.214573	+	0.976708i	$0.568836\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	33.2163	1.05039
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−30.4646	−0.649950	−0.324975	−	0.945723i	$-0.605356\pi$
$13$	−30.4646	−0.649950	−0.324975	+	0.945723i	$0.605356\pi$
$14$	−15.8958	−0.303452
$15$	0	0
$16$	16.0000	0.250000
$17$	−122.299	−1.74481	−0.872404	−	0.488785i	$-0.837440\pi$
$17$	−122.299	−1.74481	−0.872404	+	0.488785i	$0.837440\pi$
$18$	0	0
$19$	−152.630	−1.84293	−0.921467	−	0.388456i	$-0.873009\pi$
$19$	−152.630	−1.84293	−0.921467	+	0.388456i	$0.873009\pi$
$20$	66.4326	0.742739
$21$	0	0
$22$	0	0
$23$	132.865	1.20454	0.602268	−	0.798294i	$-0.294264\pi$
$23$	132.865	1.20454	0.602268	+	0.798294i	$0.294264\pi$
$24$	0	0
$25$	150.831	1.20665
$26$	−60.9291	−0.459584
$27$	0	0
$28$	−31.7916	−0.214573
$29$	93.3632	0.597831	0.298916	−	0.954280i	$-0.403375\pi$
$29$	93.3632	0.597831	0.298916	+	0.954280i	$0.403375\pi$
$30$	0	0
$31$	88.3632	0.511952	0.255976	−	0.966683i	$-0.417603\pi$
$31$	88.3632	0.511952	0.255976	+	0.966683i	$0.417603\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−244.597	−1.23377
$35$	−132.000	−0.637488
$36$	0	0
$37$	−219.597	−0.975717	−0.487858	−	0.872923i	$-0.662222\pi$
$37$	−219.597	−0.975717	−0.487858	+	0.872923i	$0.662222\pi$
$38$	−305.260	−1.30315
$39$	0	0
$40$	132.865	0.525196
$41$	−95.8309	−0.365031	−0.182515	−	0.983203i	$-0.558424\pi$
$41$	−95.8309	−0.365031	−0.182515	+	0.983203i	$0.558424\pi$
$42$	0	0
$43$	−371.693	−1.31820	−0.659100	−	0.752055i	$-0.729063\pi$
$43$	−371.693	−1.31820	−0.659100	+	0.752055i	$0.729063\pi$
$44$	0	0
$45$	0	0
$46$	265.730	0.851735
$47$	−95.3748	−0.295997	−0.147998	−	0.988988i	$-0.547283\pi$
$47$	−95.3748	−0.295997	−0.147998	+	0.988988i	$0.547283\pi$
$48$	0	0
$49$	−279.831	−0.815833
$50$	301.662	0.853228
$51$	0	0
$52$	−121.858	−0.324975
$53$	−62.6468	−0.162362	−0.0811811	−	0.996699i	$-0.525869\pi$
$53$	−62.6468	−0.162362	−0.0811811	+	0.996699i	$0.525869\pi$
$54$	0	0
$55$	0	0
$56$	−63.5832	−0.151726
$57$	0	0
$58$	186.726	0.422731
$59$	−99.8443	−0.220316	−0.110158	−	0.993914i	$-0.535136\pi$
$59$	−99.8443	−0.220316	−0.110158	+	0.993914i	$0.535136\pi$
$60$	0	0
$61$	−615.229	−1.29134	−0.645672	−	0.763615i	$-0.723423\pi$
$61$	−615.229	−1.29134	−0.645672	+	0.763615i	$0.723423\pi$
$62$	176.726	0.362004
$63$	0	0
$64$	64.0000	0.125000
$65$	−505.960	−0.965487
$66$	0	0
$67$	−1045.35	−1.90611	−0.953057	−	0.302791i	$-0.902081\pi$
$67$	−1045.35	−1.90611	−0.953057	+	0.302791i	$0.902081\pi$
$68$	−489.194	−0.872404
$69$	0	0
$70$	−264.000	−0.450772
$71$	94.9842	0.158768	0.0793842	−	0.996844i	$-0.474705\pi$
$71$	94.9842	0.158768	0.0793842	+	0.996844i	$0.474705\pi$
$72$	0	0
$73$	−459.120	−0.736108	−0.368054	−	0.929804i	$-0.619976\pi$
$73$	−459.120	−0.736108	−0.368054	+	0.929804i	$0.619976\pi$
$74$	−439.194	−0.689936
$75$	0	0
$76$	−610.521	−0.921467
$77$	0	0
$78$	0	0
$79$	646.153	0.920226	0.460113	−	0.887860i	$-0.347809\pi$
$79$	646.153	0.920226	0.460113	+	0.887860i	$0.347809\pi$
$80$	265.730	0.371370
$81$	0	0
$82$	−191.662	−0.258116
$83$	−434.129	−0.574119	−0.287060	−	0.957913i	$-0.592678\pi$
$83$	−434.129	−0.574119	−0.287060	+	0.957913i	$0.592678\pi$
$84$	0	0
$85$	−2031.15	−2.59188
$86$	−743.386	−0.932109
$87$	0	0
$88$	0	0
$89$	1628.67	1.93976	0.969882	−	0.243574i	$-0.0783199\pi$
$89$	1628.67	1.93976	0.969882	+	0.243574i	$0.0783199\pi$
$90$	0	0
$91$	242.129	0.278924
$92$	531.461	0.602268
$93$	0	0
$94$	−190.750	−0.209301
$95$	−2534.91	−2.73764
$96$	0	0
$97$	−48.9502	−0.0512386	−0.0256193	−	0.999672i	$-0.508156\pi$
$97$	−48.9502	−0.0512386	−0.0256193	+	0.999672i	$0.508156\pi$
$98$	−559.662	−0.576881
$99$	0	0
$100$	603.323	0.603323
$101$	1622.57	1.59853	0.799265	−	0.600979i	$-0.205223\pi$
$101$	1622.57	1.59853	0.799265	+	0.600979i	$0.205223\pi$
$102$	0	0
$103$	−953.766	−0.912402	−0.456201	−	0.889877i	$-0.650790\pi$
$103$	−953.766	−0.912402	−0.456201	+	0.889877i	$0.650790\pi$
$104$	−243.717	−0.229792
$105$	0	0
$106$	−125.294	−0.114807
$107$	173.711	0.156947	0.0784735	−	0.996916i	$-0.474995\pi$
$107$	173.711	0.156947	0.0784735	+	0.996916i	$0.474995\pi$
$108$	0	0
$109$	676.646	0.594596	0.297298	−	0.954785i	$-0.403915\pi$
$109$	676.646	0.594596	0.297298	+	0.954785i	$0.403915\pi$
$110$	0	0
$111$	0	0
$112$	−127.166	−0.107287
$113$	1216.67	1.01287	0.506436	−	0.862277i	$-0.330962\pi$
$113$	1216.67	1.01287	0.506436	+	0.862277i	$0.330962\pi$
$114$	0	0
$115$	2206.65	1.78931
$116$	373.453	0.298916
$117$	0	0
$118$	−199.689	−0.155787
$119$	972.017	0.748778
$120$	0	0
$121$	0	0
$122$	−1230.46	−0.913118
$123$	0	0
$124$	353.453	0.255976
$125$	429.003	0.306969
$126$	0	0
$127$	1964.64	1.37270	0.686352	−	0.727270i	$-0.259211\pi$
$127$	1964.64	1.37270	0.686352	+	0.727270i	$0.259211\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−1011.92	−0.682702
$131$	1927.16	1.28532	0.642661	−	0.766151i	$-0.277830\pi$
$131$	1927.16	1.28532	0.642661	+	0.766151i	$0.277830\pi$
$132$	0	0
$133$	1213.09	0.790889
$134$	−2090.70	−1.34783
$135$	0	0
$136$	−978.388	−0.616883
$137$	−1505.43	−0.938815	−0.469407	−	0.882982i	$-0.655532\pi$
$137$	−1505.43	−0.938815	−0.469407	+	0.882982i	$0.655532\pi$
$138$	0	0
$139$	859.126	0.524245	0.262123	−	0.965035i	$-0.415577\pi$
$139$	859.126	0.524245	0.262123	+	0.965035i	$0.415577\pi$
$140$	−528.000	−0.318744
$141$	0	0
$142$	189.968	0.112266
$143$	0	0
$144$	0	0
$145$	1550.59	0.888066
$146$	−918.240	−0.520507
$147$	0	0
$148$	−878.388	−0.487858
$149$	−1988.01	−1.09305	−0.546524	−	0.837443i	$-0.684049\pi$
$149$	−1988.01	−1.09305	−0.546524	+	0.837443i	$0.684049\pi$
$150$	0	0
$151$	1089.10	0.586952	0.293476	−	0.955966i	$-0.405188\pi$
$151$	1089.10	0.586952	0.293476	+	0.955966i	$0.405188\pi$
$152$	−1221.04	−0.651576
$153$	0	0
$154$	0	0
$155$	1467.55	0.760493
$156$	0	0
$157$	−2432.80	−1.23668	−0.618340	−	0.785911i	$-0.712194\pi$
$157$	−2432.80	−1.23668	−0.618340	+	0.785911i	$0.712194\pi$
$158$	1292.31	0.650698
$159$	0	0
$160$	531.461	0.262598
$161$	−1056.00	−0.516922
$162$	0	0
$163$	−3018.54	−1.45049	−0.725247	−	0.688489i	$-0.758274\pi$
$163$	−3018.54	−1.45049	−0.725247	+	0.688489i	$0.758274\pi$
$164$	−383.323	−0.182515
$165$	0	0
$166$	−868.259	−0.405964
$167$	−1857.97	−0.860922	−0.430461	−	0.902609i	$-0.641649\pi$
$167$	−1857.97	−0.860922	−0.430461	+	0.902609i	$0.641649\pi$
$168$	0	0
$169$	−1268.91	−0.577565
$170$	−4062.31	−1.83273
$171$	0	0
$172$	−1486.77	−0.659100
$173$	−3205.79	−1.40885	−0.704427	−	0.709776i	$-0.748796\pi$
$173$	−3205.79	−1.40885	−0.704427	+	0.709776i	$0.748796\pi$
$174$	0	0
$175$	−1198.79	−0.517828
$176$	0	0
$177$	0	0
$178$	3257.35	1.37162
$179$	−2663.34	−1.11211	−0.556053	−	0.831147i	$-0.687685\pi$
$179$	−2663.34	−1.11211	−0.556053	+	0.831147i	$0.687685\pi$
$180$	0	0
$181$	−1219.10	−0.500635	−0.250318	−	0.968164i	$-0.580535\pi$
$181$	−1219.10	−0.500635	−0.250318	+	0.968164i	$0.580535\pi$
$182$	484.259	0.197229
$183$	0	0
$184$	1062.92	0.425868
$185$	−3647.10	−1.44941
$186$	0	0
$187$	0	0
$188$	−381.499	−0.147998
$189$	0	0
$190$	−5069.81	−1.93580
$191$	3845.93	1.45697	0.728487	−	0.685060i	$-0.240224\pi$
$191$	3845.93	1.45697	0.728487	+	0.685060i	$0.240224\pi$
$192$	0	0
$193$	4114.71	1.53463	0.767315	−	0.641271i	$-0.221592\pi$
$193$	4114.71	1.53463	0.767315	+	0.641271i	$0.221592\pi$
$194$	−97.9004	−0.0362311
$195$	0	0
$196$	−1119.32	−0.407917
$197$	1759.23	0.636245	0.318122	−	0.948050i	$-0.396948\pi$
$197$	1759.23	0.636245	0.318122	+	0.948050i	$0.396948\pi$
$198$	0	0
$199$	4733.34	1.68612	0.843059	−	0.537821i	$-0.180752\pi$
$199$	4733.34	1.68612	0.843059	+	0.537821i	$0.180752\pi$
$200$	1206.65	0.426614
$201$	0	0
$202$	3245.13	1.13033
$203$	−742.042	−0.256557
$204$	0	0
$205$	−1591.57	−0.542246
$206$	−1907.53	−0.645165
$207$	0	0
$208$	−487.433	−0.162487
$209$	0	0
$210$	0	0
$211$	−2690.09	−0.877693	−0.438847	−	0.898562i	$-0.644613\pi$
$211$	−2690.09	−0.877693	−0.438847	+	0.898562i	$0.644613\pi$
$212$	−250.587	−0.0811811
$213$	0	0
$214$	347.423	0.110978
$215$	−6173.13	−1.95816
$216$	0	0
$217$	−702.302	−0.219702
$218$	1353.29	0.420443
$219$	0	0
$220$	0	0
$221$	3725.77	1.13404
$222$	0	0
$223$	−2151.11	−0.645959	−0.322980	−	0.946406i	$-0.604685\pi$
$223$	−2151.11	−0.645959	−0.322980	+	0.946406i	$0.604685\pi$
$224$	−254.333	−0.0758631
$225$	0	0
$226$	2433.34	0.716209
$227$	−3651.48	−1.06765	−0.533827	−	0.845594i	$-0.679247\pi$
$227$	−3651.48	−1.06765	−0.533827	+	0.845594i	$0.679247\pi$
$228$	0	0
$229$	−5270.37	−1.52085	−0.760427	−	0.649423i	$-0.775010\pi$
$229$	−5270.37	−1.52085	−0.760427	+	0.649423i	$0.775010\pi$
$230$	4413.29	1.26523
$231$	0	0
$232$	746.906	0.211365
$233$	5977.04	1.68055	0.840277	−	0.542157i	$-0.182392\pi$
$233$	5977.04	1.68055	0.840277	+	0.542157i	$0.182392\pi$
$234$	0	0
$235$	−1584.00	−0.439697
$236$	−399.377	−0.110158
$237$	0	0
$238$	1944.03	0.529466
$239$	−4999.06	−1.35298	−0.676491	−	0.736451i	$-0.736500\pi$
$239$	−4999.06	−1.35298	−0.676491	+	0.736451i	$0.736500\pi$
$240$	0	0
$241$	3380.18	0.903472	0.451736	−	0.892152i	$-0.350805\pi$
$241$	3380.18	0.903472	0.451736	+	0.892152i	$0.350805\pi$
$242$	0	0
$243$	0	0
$244$	−2460.91	−0.645672
$245$	−4647.47	−1.21190
$246$	0	0
$247$	4649.81	1.19782
$248$	706.906	0.181002
$249$	0	0
$250$	858.006	0.217060
$251$	1967.12	0.494674	0.247337	−	0.968929i	$-0.420444\pi$
$251$	1967.12	0.494674	0.247337	+	0.968929i	$0.420444\pi$
$252$	0	0
$253$	0	0
$254$	3929.27	0.970648
$255$	0	0
$256$	256.000	0.0625000
$257$	5392.06	1.30874	0.654372	−	0.756173i	$-0.272933\pi$
$257$	5392.06	1.30874	0.654372	+	0.756173i	$0.272933\pi$
$258$	0	0
$259$	1745.34	0.418725
$260$	−2023.84	−0.482743
$261$	0	0
$262$	3854.33	0.908860
$263$	−1654.17	−0.387834	−0.193917	−	0.981018i	$-0.562119\pi$
$263$	−1654.17	−0.387834	−0.193917	+	0.981018i	$0.562119\pi$
$264$	0	0
$265$	−1040.45	−0.241186
$266$	2426.18	0.559243
$267$	0	0
$268$	−4181.39	−0.953057
$269$	−6184.97	−1.40188	−0.700938	−	0.713223i	$-0.747235\pi$
$269$	−6184.97	−1.40188	−0.700938	+	0.713223i	$0.747235\pi$
$270$	0	0
$271$	3004.47	0.673463	0.336731	−	0.941601i	$-0.390679\pi$
$271$	3004.47	0.673463	0.336731	+	0.941601i	$0.390679\pi$
$272$	−1956.78	−0.436202
$273$	0	0
$274$	−3010.86	−0.663842
$275$	0	0
$276$	0	0
$277$	−152.584	−0.0330971	−0.0165485	−	0.999863i	$-0.505268\pi$
$277$	−152.584	−0.0330971	−0.0165485	+	0.999863i	$0.505268\pi$
$278$	1718.25	0.370697
$279$	0	0
$280$	−1056.00	−0.225386
$281$	3885.25	0.824821	0.412411	−	0.910998i	$-0.364687\pi$
$281$	3885.25	0.824821	0.412411	+	0.910998i	$0.364687\pi$
$282$	0	0
$283$	−5598.17	−1.17589	−0.587945	−	0.808901i	$-0.700063\pi$
$283$	−5598.17	−1.17589	−0.587945	+	0.808901i	$0.700063\pi$
$284$	379.937	0.0793842
$285$	0	0
$286$	0	0
$287$	761.654	0.156652
$288$	0	0
$289$	10043.9	2.04436
$290$	3101.18	0.627957
$291$	0	0
$292$	−1836.48	−0.368054
$293$	−6879.65	−1.37172	−0.685859	−	0.727734i	$-0.740573\pi$
$293$	−6879.65	−1.37172	−0.685859	+	0.727734i	$0.740573\pi$
$294$	0	0
$295$	−1658.23	−0.327274
$296$	−1756.78	−0.344968
$297$	0	0
$298$	−3976.02	−0.772902
$299$	−4047.68	−0.782888
$300$	0	0
$301$	2954.18	0.565701
$302$	2178.20	0.415038
$303$	0	0
$304$	−2442.08	−0.460734
$305$	−10217.8	−1.91826
$306$	0	0
$307$	7549.75	1.40354	0.701770	−	0.712403i	$-0.252393\pi$
$307$	7549.75	1.40354	0.701770	+	0.712403i	$0.252393\pi$
$308$	0	0
$309$	0	0
$310$	2935.10	0.537750
$311$	5249.13	0.957077	0.478538	−	0.878067i	$-0.341167\pi$
$311$	5249.13	0.957077	0.478538	+	0.878067i	$0.341167\pi$
$312$	0	0
$313$	1439.42	0.259939	0.129970	−	0.991518i	$-0.458512\pi$
$313$	1439.42	0.259939	0.129970	+	0.991518i	$0.458512\pi$
$314$	−4865.60	−0.874464
$315$	0	0
$316$	2584.61	0.460113
$317$	−8329.33	−1.47578	−0.737889	−	0.674922i	$-0.764177\pi$
$317$	−8329.33	−1.47578	−0.737889	+	0.674922i	$0.764177\pi$
$318$	0	0
$319$	0	0
$320$	1062.92	0.185685
$321$	0	0
$322$	−2112.00	−0.365519
$323$	18666.4	3.21557
$324$	0	0
$325$	−4595.00	−0.784260
$326$	−6037.08	−1.02565
$327$	0	0
$328$	−766.647	−0.129058
$329$	758.030	0.127026
$330$	0	0
$331$	−1669.61	−0.277250	−0.138625	−	0.990345i	$-0.544268\pi$
$331$	−1669.61	−0.277250	−0.138625	+	0.990345i	$0.544268\pi$
$332$	−1736.52	−0.287060
$333$	0	0
$334$	−3715.94	−0.608764
$335$	−17361.3	−2.83149
$336$	0	0
$337$	3150.74	0.509294	0.254647	−	0.967034i	$-0.418041\pi$
$337$	3150.74	0.509294	0.254647	+	0.967034i	$0.418041\pi$
$338$	−2537.82	−0.408400
$339$	0	0
$340$	−8124.61	−1.29594
$341$	0	0
$342$	0	0
$343$	4950.20	0.779258
$344$	−2973.54	−0.466054
$345$	0	0
$346$	−6411.58	−0.996210
$347$	3422.33	0.529453	0.264727	−	0.964324i	$-0.414718\pi$
$347$	3422.33	0.529453	0.264727	+	0.964324i	$0.414718\pi$
$348$	0	0
$349$	−4752.29	−0.728894	−0.364447	−	0.931224i	$-0.618742\pi$
$349$	−4752.29	−0.728894	−0.364447	+	0.931224i	$0.618742\pi$
$350$	−2397.58	−0.366160
$351$	0	0
$352$	0	0
$353$	−4337.22	−0.653958	−0.326979	−	0.945032i	$-0.606031\pi$
$353$	−4337.22	−0.653958	−0.326979	+	0.945032i	$0.606031\pi$
$354$	0	0
$355$	1577.51	0.235847
$356$	6514.69	0.969882
$357$	0	0
$358$	−5326.67	−0.786378
$359$	2794.95	0.410896	0.205448	−	0.978668i	$-0.434135\pi$
$359$	2794.95	0.410896	0.205448	+	0.978668i	$0.434135\pi$
$360$	0	0
$361$	16437.0	2.39641
$362$	−2438.20	−0.354002
$363$	0	0
$364$	968.517	0.139462
$365$	−7625.13	−1.09347
$366$	0	0
$367$	8348.88	1.18749	0.593744	−	0.804654i	$-0.297649\pi$
$367$	8348.88	1.18749	0.593744	+	0.804654i	$0.297649\pi$
$368$	2125.84	0.301134
$369$	0	0
$370$	−7294.20	−1.02489
$371$	497.911	0.0696772
$372$	0	0
$373$	−6856.52	−0.951789	−0.475895	−	0.879502i	$-0.657876\pi$
$373$	−6856.52	−0.951789	−0.475895	+	0.879502i	$0.657876\pi$
$374$	0	0
$375$	0	0
$376$	−762.999	−0.104651
$377$	−2844.27	−0.388560
$378$	0	0
$379$	−9740.86	−1.32020	−0.660098	−	0.751180i	$-0.729485\pi$
$379$	−9740.86	−1.32020	−0.660098	+	0.751180i	$0.729485\pi$
$380$	−10139.6	−1.36882
$381$	0	0
$382$	7691.87	1.03024
$383$	4595.70	0.613131	0.306565	−	0.951850i	$-0.400820\pi$
$383$	4595.70	0.613131	0.306565	+	0.951850i	$0.400820\pi$
$384$	0	0
$385$	0	0
$386$	8229.42	1.08515
$387$	0	0
$388$	−195.801	−0.0256193
$389$	−10148.5	−1.32275	−0.661375	−	0.750055i	$-0.730027\pi$
$389$	−10148.5	−1.32275	−0.661375	+	0.750055i	$0.730027\pi$
$390$	0	0
$391$	−16249.2	−2.10168
$392$	−2238.65	−0.288441
$393$	0	0
$394$	3518.47	0.449893
$395$	10731.4	1.36698
$396$	0	0
$397$	−9345.78	−1.18149	−0.590744	−	0.806859i	$-0.701166\pi$
$397$	−9345.78	−1.18149	−0.590744	+	0.806859i	$0.701166\pi$
$398$	9466.68	1.19227
$399$	0	0
$400$	2413.29	0.301662
$401$	−4977.53	−0.619866	−0.309933	−	0.950758i	$-0.600307\pi$
$401$	−4977.53	−0.619866	−0.309933	+	0.950758i	$0.600307\pi$
$402$	0	0
$403$	−2691.95	−0.332743
$404$	6490.27	0.799265
$405$	0	0
$406$	−1484.08	−0.181413
$407$	0	0
$408$	0	0
$409$	−8775.41	−1.06092	−0.530460	−	0.847710i	$-0.677981\pi$
$409$	−8775.41	−1.06092	−0.530460	+	0.847710i	$0.677981\pi$
$410$	−3183.15	−0.383425
$411$	0	0
$412$	−3815.06	−0.456201
$413$	793.552	0.0945476
$414$	0	0
$415$	−7210.09	−0.852842
$416$	−974.866	−0.114896
$417$	0	0
$418$	0	0
$419$	15336.7	1.78818	0.894089	−	0.447889i	$-0.147824\pi$
$419$	15336.7	1.78818	0.894089	+	0.447889i	$0.147824\pi$
$420$	0	0
$421$	16485.8	1.90848	0.954238	−	0.299049i	$-0.0966693\pi$
$421$	16485.8	1.90848	0.954238	+	0.299049i	$0.0966693\pi$
$422$	−5380.18	−0.620623
$423$	0	0
$424$	−501.174	−0.0574037
$425$	−18446.4	−2.10537
$426$	0	0
$427$	4889.78	0.554175
$428$	694.846	0.0784735
$429$	0	0
$430$	−12346.3	−1.38463
$431$	−5732.34	−0.640643	−0.320321	−	0.947309i	$-0.603791\pi$
$431$	−5732.34	−0.640643	−0.320321	+	0.947309i	$0.603791\pi$
$432$	0	0
$433$	−15574.0	−1.72849	−0.864247	−	0.503068i	$-0.832204\pi$
$433$	−15574.0	−1.72849	−0.864247	+	0.503068i	$0.832204\pi$
$434$	−1404.60	−0.155353
$435$	0	0
$436$	2706.58	0.297298
$437$	−20279.2	−2.21988
$438$	0	0
$439$	4132.18	0.449244	0.224622	−	0.974446i	$-0.427885\pi$
$439$	4132.18	0.449244	0.224622	+	0.974446i	$0.427885\pi$
$440$	0	0
$441$	0	0
$442$	7451.54	0.801886
$443$	3737.93	0.400891	0.200445	−	0.979705i	$-0.435761\pi$
$443$	3737.93	0.400891	0.200445	+	0.979705i	$0.435761\pi$
$444$	0	0
$445$	27049.3	2.88148
$446$	−4302.22	−0.456762
$447$	0	0
$448$	−508.666	−0.0536433
$449$	−11083.9	−1.16499	−0.582496	−	0.812834i	$-0.697924\pi$
$449$	−11083.9	−1.16499	−0.582496	+	0.812834i	$0.697924\pi$
$450$	0	0
$451$	0	0
$452$	4866.68	0.506436
$453$	0	0
$454$	−7302.97	−0.754945
$455$	4021.32	0.414335
$456$	0	0
$457$	−1078.34	−0.110377	−0.0551886	−	0.998476i	$-0.517576\pi$
$457$	−1078.34	−0.110377	−0.0551886	+	0.998476i	$0.517576\pi$
$458$	−10540.7	−1.07541
$459$	0	0
$460$	8826.59	0.894656
$461$	−2439.55	−0.246467	−0.123233	−	0.992378i	$-0.539326\pi$
$461$	−2439.55	−0.246467	−0.123233	+	0.992378i	$0.539326\pi$
$462$	0	0
$463$	11819.7	1.18641	0.593204	−	0.805052i	$-0.297863\pi$
$463$	11819.7	1.18641	0.593204	+	0.805052i	$0.297863\pi$
$464$	1493.81	0.149458
$465$	0	0
$466$	11954.1	1.18833
$467$	−14738.6	−1.46043	−0.730215	−	0.683218i	$-0.760580\pi$
$467$	−14738.6	−1.46043	−0.730215	+	0.683218i	$0.760580\pi$
$468$	0	0
$469$	8308.33	0.818002
$470$	−3168.00	−0.310913
$471$	0	0
$472$	−798.754	−0.0778933
$473$	0	0
$474$	0	0
$475$	−23021.3	−2.22377
$476$	3888.07	0.374389
$477$	0	0
$478$	−9998.13	−0.956702
$479$	−19922.1	−1.90034	−0.950169	−	0.311736i	$-0.899090\pi$
$479$	−19922.1	−1.90034	−0.950169	+	0.311736i	$0.899090\pi$
$480$	0	0
$481$	6689.93	0.634167
$482$	6760.36	0.638851
$483$	0	0
$484$	0	0
$485$	−812.973	−0.0761138
$486$	0	0
$487$	16527.4	1.53784	0.768921	−	0.639344i	$-0.220794\pi$
$487$	16527.4	1.53784	0.768921	+	0.639344i	$0.220794\pi$
$488$	−4921.83	−0.456559
$489$	0	0
$490$	−9294.95	−0.856945
$491$	6031.66	0.554389	0.277195	−	0.960814i	$-0.410595\pi$
$491$	6031.66	0.554389	0.277195	+	0.960814i	$0.410595\pi$
$492$	0	0
$493$	−11418.2	−1.04310
$494$	9299.62	0.846983
$495$	0	0
$496$	1413.81	0.127988
$497$	−754.925	−0.0681348
$498$	0	0
$499$	11130.0	0.998493	0.499246	−	0.866460i	$-0.333610\pi$
$499$	11130.0	0.998493	0.499246	+	0.866460i	$0.333610\pi$
$500$	1716.01	0.153485
$501$	0	0
$502$	3934.23	0.349788
$503$	9164.64	0.812388	0.406194	−	0.913787i	$-0.366856\pi$
$503$	9164.64	0.812388	0.406194	+	0.913787i	$0.366856\pi$
$504$	0	0
$505$	26947.9	2.37458
$506$	0	0
$507$	0	0
$508$	7858.55	0.686352
$509$	−20396.8	−1.77617	−0.888087	−	0.459675i	$-0.847966\pi$
$509$	−20396.8	−1.77617	−0.888087	+	0.459675i	$0.847966\pi$
$510$	0	0
$511$	3649.04	0.315898
$512$	512.000	0.0441942
$513$	0	0
$514$	10784.1	0.925422
$515$	−15840.3	−1.35535
$516$	0	0
$517$	0	0
$518$	3490.67	0.296084
$519$	0	0
$520$	−4047.68	−0.341351
$521$	−976.486	−0.0821125	−0.0410563	−	0.999157i	$-0.513072\pi$
$521$	−976.486	−0.0821125	−0.0410563	+	0.999157i	$0.513072\pi$
$522$	0	0
$523$	21798.2	1.82250	0.911250	−	0.411854i	$-0.135119\pi$
$523$	21798.2	1.82250	0.911250	+	0.411854i	$0.135119\pi$
$524$	7708.66	0.642661
$525$	0	0
$526$	−3308.34	−0.274240
$527$	−10806.7	−0.893257
$528$	0	0
$529$	5486.17	0.450906
$530$	−2080.90	−0.170544
$531$	0	0
$532$	4852.36	0.395444
$533$	2919.44	0.237252
$534$	0	0
$535$	2885.03	0.233141
$536$	−8362.79	−0.673913
$537$	0	0
$538$	−12369.9	−0.991275
$539$	0	0
$540$	0	0
$541$	−14412.4	−1.14536	−0.572680	−	0.819779i	$-0.694096\pi$
$541$	−14412.4	−1.14536	−0.572680	+	0.819779i	$0.694096\pi$
$542$	6008.94	0.476210
$543$	0	0
$544$	−3913.55	−0.308441
$545$	11237.8	0.883259
$546$	0	0
$547$	5193.23	0.405935	0.202967	−	0.979185i	$-0.434941\pi$
$547$	5193.23	0.405935	0.202967	+	0.979185i	$0.434941\pi$
$548$	−6021.72	−0.469407
$549$	0	0
$550$	0	0
$551$	−14250.0	−1.10176
$552$	0	0
$553$	−5135.56	−0.394912
$554$	−305.168	−0.0234032
$555$	0	0
$556$	3436.50	0.262123
$557$	5635.70	0.428712	0.214356	−	0.976756i	$-0.431235\pi$
$557$	5635.70	0.428712	0.214356	+	0.976756i	$0.431235\pi$
$558$	0	0
$559$	11323.5	0.856765
$560$	−2112.00	−0.159372
$561$	0	0
$562$	7770.51	0.583237
$563$	−16130.5	−1.20750	−0.603748	−	0.797175i	$-0.706327\pi$
$563$	−16130.5	−1.20750	−0.603748	+	0.797175i	$0.706327\pi$
$564$	0	0
$565$	20206.6	1.50460
$566$	−11196.3	−0.831479
$567$	0	0
$568$	759.874	0.0561331
$569$	−12305.4	−0.906624	−0.453312	−	0.891352i	$-0.649758\pi$
$569$	−12305.4	−0.906624	−0.453312	+	0.891352i	$0.649758\pi$
$570$	0	0
$571$	2413.84	0.176911	0.0884555	−	0.996080i	$-0.471807\pi$
$571$	2413.84	0.176911	0.0884555	+	0.996080i	$0.471807\pi$
$572$	0	0
$573$	0	0
$574$	1523.31	0.110769
$575$	20040.2	1.45345
$576$	0	0
$577$	−16175.9	−1.16709	−0.583545	−	0.812081i	$-0.698335\pi$
$577$	−16175.9	−1.16709	−0.583545	+	0.812081i	$0.698335\pi$
$578$	20087.9	1.44558
$579$	0	0
$580$	6202.36	0.444033
$581$	3450.42	0.246381
$582$	0	0
$583$	0	0
$584$	−3672.96	−0.260254
$585$	0	0
$586$	−13759.3	−0.969952
$587$	14369.1	1.01035	0.505174	−	0.863017i	$-0.331428\pi$
$587$	14369.1	1.01035	0.505174	+	0.863017i	$0.331428\pi$
$588$	0	0
$589$	−13486.9	−0.943493
$590$	−3316.46	−0.231418
$591$	0	0
$592$	−3513.55	−0.243929
$593$	−23258.6	−1.61065	−0.805325	−	0.592834i	$-0.798009\pi$
$593$	−23258.6	−1.61065	−0.805325	+	0.592834i	$0.798009\pi$
$594$	0	0
$595$	16143.4	1.11229
$596$	−7952.04	−0.546524
$597$	0	0
$598$	−8095.36	−0.553585
$599$	2760.49	0.188298	0.0941491	−	0.995558i	$-0.469987\pi$
$599$	2760.49	0.188298	0.0941491	+	0.995558i	$0.469987\pi$
$600$	0	0
$601$	−4582.71	−0.311036	−0.155518	−	0.987833i	$-0.549705\pi$
$601$	−4582.71	−0.311036	−0.155518	+	0.987833i	$0.549705\pi$
$602$	5908.36	0.400011
$603$	0	0
$604$	4356.40	0.293476
$605$	0	0
$606$	0	0
$607$	−6255.92	−0.418320	−0.209160	−	0.977881i	$-0.567073\pi$
$607$	−6255.92	−0.418320	−0.209160	+	0.977881i	$0.567073\pi$
$608$	−4884.17	−0.325788
$609$	0	0
$610$	−20435.6	−1.35642
$611$	2905.55	0.192383
$612$	0	0
$613$	5323.44	0.350753	0.175377	−	0.984501i	$-0.443886\pi$
$613$	5323.44	0.350753	0.175377	+	0.984501i	$0.443886\pi$
$614$	15099.5	0.992453
$615$	0	0
$616$	0	0
$617$	11579.9	0.755577	0.377788	−	0.925892i	$-0.376685\pi$
$617$	11579.9	0.755577	0.377788	+	0.925892i	$0.376685\pi$
$618$	0	0
$619$	−10031.5	−0.651372	−0.325686	−	0.945478i	$-0.605595\pi$
$619$	−10031.5	−0.651372	−0.325686	+	0.945478i	$0.605595\pi$
$620$	5870.20	0.380247
$621$	0	0
$622$	10498.3	0.676756
$623$	−12944.5	−0.832443
$624$	0	0
$625$	−11728.9	−0.750650
$626$	2878.85	0.183805
$627$	0	0
$628$	−9731.20	−0.618340
$629$	26856.4	1.70244
$630$	0	0
$631$	172.677	0.0108941	0.00544703	−	0.999985i	$-0.498266\pi$
$631$	172.677	0.0108941	0.00544703	+	0.999985i	$0.498266\pi$
$632$	5169.22	0.325349
$633$	0	0
$634$	−16658.7	−1.04353
$635$	32629.0	2.03912
$636$	0	0
$637$	8524.92	0.530251
$638$	0	0
$639$	0	0
$640$	2125.84	0.131299
$641$	9366.34	0.577142	0.288571	−	0.957458i	$-0.406820\pi$
$641$	9366.34	0.577142	0.288571	+	0.957458i	$0.406820\pi$
$642$	0	0
$643$	26713.7	1.63839	0.819195	−	0.573516i	$-0.194421\pi$
$643$	26713.7	1.63839	0.819195	+	0.573516i	$0.194421\pi$
$644$	−4224.00	−0.258461
$645$	0	0
$646$	37332.9	2.27375
$647$	31017.8	1.88475	0.942376	−	0.334555i	$-0.108586\pi$
$647$	31017.8	1.88475	0.942376	+	0.334555i	$0.108586\pi$
$648$	0	0
$649$	0	0
$650$	−9189.99	−0.554556
$651$	0	0
$652$	−12074.2	−0.725247
$653$	−7167.57	−0.429538	−0.214769	−	0.976665i	$-0.568900\pi$
$653$	−7167.57	−0.429538	−0.214769	+	0.976665i	$0.568900\pi$
$654$	0	0
$655$	32006.6	1.90932
$656$	−1533.29	−0.0912577
$657$	0	0
$658$	1516.06	0.0898209
$659$	29107.5	1.72059	0.860293	−	0.509799i	$-0.170280\pi$
$659$	29107.5	1.72059	0.860293	+	0.509799i	$0.170280\pi$
$660$	0	0
$661$	4536.32	0.266933	0.133466	−	0.991053i	$-0.457389\pi$
$661$	4536.32	0.266933	0.133466	+	0.991053i	$0.457389\pi$
$662$	−3339.21	−0.196046
$663$	0	0
$664$	−3473.03	−0.202982
$665$	20147.2	1.17485
$666$	0	0
$667$	12404.7	0.720109
$668$	−7431.88	−0.430461
$669$	0	0
$670$	−34722.6	−2.00217
$671$	0	0
$672$	0	0
$673$	249.843	0.0143102	0.00715509	−	0.999974i	$-0.497722\pi$
$673$	249.843	0.0143102	0.00715509	+	0.999974i	$0.497722\pi$
$674$	6301.49	0.360125
$675$	0	0
$676$	−5075.64	−0.288783
$677$	1385.30	0.0786433	0.0393217	−	0.999227i	$-0.487480\pi$
$677$	1385.30	0.0786433	0.0393217	+	0.999227i	$0.487480\pi$
$678$	0	0
$679$	389.051	0.0219888
$680$	−16249.2	−0.916367
$681$	0	0
$682$	0	0
$683$	19640.5	1.10033	0.550163	−	0.835057i	$-0.314566\pi$
$683$	19640.5	1.10033	0.550163	+	0.835057i	$0.314566\pi$
$684$	0	0
$685$	−25002.4	−1.39459
$686$	9900.40	0.551019
$687$	0	0
$688$	−5947.09	−0.329550
$689$	1908.51	0.105527
$690$	0	0
$691$	−10.2738	−0.000565609 0	−0.000282804	−	1.00000i	$-0.500090\pi$
$691$	−10.2738	−0.000565609 0	−0.000282804		1.00000i	$0.500090\pi$
$692$	−12823.2	−0.704427
$693$	0	0
$694$	6844.66	0.374380
$695$	14268.5	0.778755
$696$	0	0
$697$	11720.0	0.636909
$698$	−9504.57	−0.515406
$699$	0	0
$700$	−4795.16	−0.258914
$701$	−3738.10	−0.201407	−0.100703	−	0.994916i	$-0.532109\pi$
$701$	−3738.10	−0.201407	−0.100703	+	0.994916i	$0.532109\pi$
$702$	0	0
$703$	33517.1	1.79818
$704$	0	0
$705$	0	0
$706$	−8674.44	−0.462418
$707$	−12896.0	−0.686003
$708$	0	0
$709$	20147.5	1.06721	0.533606	−	0.845733i	$-0.320836\pi$
$709$	20147.5	1.06721	0.533606	+	0.845733i	$0.320836\pi$
$710$	3155.02	0.166769
$711$	0	0
$712$	13029.4	0.685810
$713$	11740.4	0.616664
$714$	0	0
$715$	0	0
$716$	−10653.3	−0.556053
$717$	0	0
$718$	5589.89	0.290547
$719$	−3246.70	−0.168402	−0.0842012	−	0.996449i	$-0.526834\pi$
$719$	−3246.70	−0.168402	−0.0842012	+	0.996449i	$0.526834\pi$
$720$	0	0
$721$	7580.44	0.391554
$722$	32873.9	1.69452
$723$	0	0
$724$	−4876.40	−0.250318
$725$	14082.0	0.721371
$726$	0	0
$727$	11919.4	0.608070	0.304035	−	0.952661i	$-0.401666\pi$
$727$	11919.4	0.608070	0.304035	+	0.952661i	$0.401666\pi$
$728$	1937.03	0.0986144
$729$	0	0
$730$	−15250.3	−0.773202
$731$	45457.5	2.30001
$732$	0	0
$733$	7064.58	0.355984	0.177992	−	0.984032i	$-0.443040\pi$
$733$	7064.58	0.355984	0.177992	+	0.984032i	$0.443040\pi$
$734$	16697.8	0.839680
$735$	0	0
$736$	4251.69	0.212934
$737$	0	0
$738$	0	0
$739$	7117.84	0.354309	0.177154	−	0.984183i	$-0.443311\pi$
$739$	7117.84	0.354309	0.177154	+	0.984183i	$0.443311\pi$
$740$	−14588.4	−0.724703
$741$	0	0
$742$	995.821	0.0492692
$743$	−11747.2	−0.580029	−0.290015	−	0.957022i	$-0.593660\pi$
$743$	−11747.2	−0.580029	−0.290015	+	0.957022i	$0.593660\pi$
$744$	0	0
$745$	−33017.2	−1.62370
$746$	−13713.0	−0.673016
$747$	0	0
$748$	0	0
$749$	−1380.64	−0.0673532
$750$	0	0
$751$	32868.1	1.59704	0.798519	−	0.601969i	$-0.205617\pi$
$751$	32868.1	1.59704	0.798519	+	0.601969i	$0.205617\pi$
$752$	−1526.00	−0.0739992
$753$	0	0
$754$	−5688.54	−0.274754
$755$	18088.0	0.871905
$756$	0	0
$757$	−16268.7	−0.781103	−0.390552	−	0.920581i	$-0.627716\pi$
$757$	−16268.7	−0.781103	−0.390552	+	0.920581i	$0.627716\pi$
$758$	−19481.7	−0.933519
$759$	0	0
$760$	−20279.2	−0.967902
$761$	28600.4	1.36237	0.681185	−	0.732111i	$-0.261465\pi$
$761$	28600.4	1.36237	0.681185	+	0.732111i	$0.261465\pi$
$762$	0	0
$763$	−5377.92	−0.255169
$764$	15383.7	0.728487
$765$	0	0
$766$	9191.39	0.433549
$767$	3041.71	0.143194
$768$	0	0
$769$	10635.0	0.498708	0.249354	−	0.968412i	$-0.419782\pi$
$769$	10635.0	0.498708	0.249354	+	0.968412i	$0.419782\pi$
$770$	0	0
$771$	0	0
$772$	16458.8	0.767315
$773$	8752.15	0.407235	0.203618	−	0.979050i	$-0.434730\pi$
$773$	8752.15	0.407235	0.203618	+	0.979050i	$0.434730\pi$
$774$	0	0
$775$	13327.9	0.617745
$776$	−391.602	−0.0181156
$777$	0	0
$778$	−20297.0	−0.935326
$779$	14626.7	0.672728
$780$	0	0
$781$	0	0
$782$	−32498.4	−1.48612
$783$	0	0
$784$	−4477.29	−0.203958
$785$	−40404.3	−1.83706
$786$	0	0
$787$	−21369.9	−0.967922	−0.483961	−	0.875090i	$-0.660802\pi$
$787$	−21369.9	−0.967922	−0.483961	+	0.875090i	$0.660802\pi$
$788$	7036.94	0.318122
$789$	0	0
$790$	21462.8	0.966598
$791$	−9669.97	−0.434671
$792$	0	0
$793$	18742.7	0.839308
$794$	−18691.6	−0.835439
$795$	0	0
$796$	18933.4	0.843059
$797$	5755.08	0.255779	0.127889	−	0.991788i	$-0.459180\pi$
$797$	5755.08	0.255779	0.127889	+	0.991788i	$0.459180\pi$
$798$	0	0
$799$	11664.2	0.516458
$800$	4826.59	0.213307
$801$	0	0
$802$	−9955.07	−0.438311
$803$	0	0
$804$	0	0
$805$	−17538.2	−0.767877
$806$	−5383.89	−0.235285
$807$	0	0
$808$	12980.5	0.565166
$809$	13104.4	0.569502	0.284751	−	0.958601i	$-0.408089\pi$
$809$	13104.4	0.569502	0.284751	+	0.958601i	$0.408089\pi$
$810$	0	0
$811$	−40095.2	−1.73605	−0.868023	−	0.496525i	$-0.834609\pi$
$811$	−40095.2	−1.73605	−0.868023	+	0.496525i	$0.834609\pi$
$812$	−2968.17	−0.128279
$813$	0	0
$814$	0	0
$815$	−50132.4	−2.15468
$816$	0	0
$817$	56731.6	2.42936
$818$	−17550.8	−0.750184
$819$	0	0
$820$	−6366.29	−0.271123
$821$	34670.7	1.47383	0.736915	−	0.675985i	$-0.236282\pi$
$821$	34670.7	1.47383	0.736915	+	0.675985i	$0.236282\pi$
$822$	0	0
$823$	−18530.1	−0.784835	−0.392417	−	0.919787i	$-0.628361\pi$
$823$	−18530.1	−0.784835	−0.392417	+	0.919787i	$0.628361\pi$
$824$	−7630.13	−0.322583
$825$	0	0
$826$	1587.10	0.0668552
$827$	−18256.2	−0.767630	−0.383815	−	0.923410i	$-0.625390\pi$
$827$	−18256.2	−0.767630	−0.383815	+	0.923410i	$0.625390\pi$
$828$	0	0
$829$	−25387.4	−1.06362	−0.531810	−	0.846864i	$-0.678488\pi$
$829$	−25387.4	−1.06362	−0.531810	+	0.846864i	$0.678488\pi$
$830$	−14420.2	−0.603050
$831$	0	0
$832$	−1949.73	−0.0812437
$833$	34222.9	1.42347
$834$	0	0
$835$	−30857.5	−1.27888
$836$	0	0
$837$	0	0
$838$	30673.4	1.26443
$839$	−11165.9	−0.459465	−0.229732	−	0.973254i	$-0.573785\pi$
$839$	−11165.9	−0.459465	−0.229732	+	0.973254i	$0.573785\pi$
$840$	0	0
$841$	−15672.3	−0.642598
$842$	32971.6	1.34950
$843$	0	0
$844$	−10760.4	−0.438847
$845$	−21074.3	−0.857961
$846$	0	0
$847$	0	0
$848$	−1002.35	−0.0405906
$849$	0	0
$850$	−36892.8	−1.48872
$851$	−29176.8	−1.17529
$852$	0	0
$853$	−14987.8	−0.601608	−0.300804	−	0.953686i	$-0.597255\pi$
$853$	−14987.8	−0.601608	−0.300804	+	0.953686i	$0.597255\pi$
$854$	9779.55	0.391861
$855$	0	0
$856$	1389.69	0.0554891
$857$	18355.3	0.731627	0.365813	−	0.930688i	$-0.380791\pi$
$857$	18355.3	0.731627	0.365813	+	0.930688i	$0.380791\pi$
$858$	0	0
$859$	−24259.8	−0.963602	−0.481801	−	0.876281i	$-0.660017\pi$
$859$	−24259.8	−0.963602	−0.481801	+	0.876281i	$0.660017\pi$
$860$	−24692.5	−0.979080
$861$	0	0
$862$	−11464.7	−0.453003
$863$	−1655.60	−0.0653039	−0.0326520	−	0.999467i	$-0.510395\pi$
$863$	−1655.60	−0.0653039	−0.0326520	+	0.999467i	$0.510395\pi$
$864$	0	0
$865$	−53242.3	−2.09282
$866$	−31148.0	−1.22223
$867$	0	0
$868$	−2809.21	−0.109851
$869$	0	0
$870$	0	0
$871$	31846.1	1.23888
$872$	5413.17	0.210221
$873$	0	0
$874$	−40558.5	−1.56969
$875$	−3409.67	−0.131735
$876$	0	0
$877$	−17107.3	−0.658690	−0.329345	−	0.944210i	$-0.606828\pi$
$877$	−17107.3	−0.658690	−0.329345	+	0.944210i	$0.606828\pi$
$878$	8264.36	0.317664
$879$	0	0
$880$	0	0
$881$	−3961.89	−0.151509	−0.0757546	−	0.997126i	$-0.524137\pi$
$881$	−3961.89	−0.151509	−0.0757546	+	0.997126i	$0.524137\pi$
$882$	0	0
$883$	3511.56	0.133832	0.0669158	−	0.997759i	$-0.478684\pi$
$883$	3511.56	0.133832	0.0669158	+	0.997759i	$0.478684\pi$
$884$	14903.1	0.567019
$885$	0	0
$886$	7475.86	0.283472
$887$	−1269.41	−0.0480527	−0.0240263	−	0.999711i	$-0.507649\pi$
$887$	−1269.41	−0.0480527	−0.0240263	+	0.999711i	$0.507649\pi$
$888$	0	0
$889$	−15614.7	−0.589091
$890$	54098.5	2.03751
$891$	0	0
$892$	−8604.44	−0.322980
$893$	14557.1	0.545503
$894$	0	0
$895$	−44233.1	−1.65201
$896$	−1017.33	−0.0379315
$897$	0	0
$898$	−22167.8	−0.823773
$899$	8249.87	0.306061
$900$	0	0
$901$	7661.61	0.283291
$902$	0	0
$903$	0	0
$904$	9733.36	0.358105
$905$	−20247.0	−0.743683
$906$	0	0
$907$	18719.3	0.685299	0.342649	−	0.939463i	$-0.388676\pi$
$907$	18719.3	0.685299	0.342649	+	0.939463i	$0.388676\pi$
$908$	−14605.9	−0.533827
$909$	0	0
$910$	8042.64	0.292979
$911$	30563.1	1.11153	0.555763	−	0.831341i	$-0.312426\pi$
$911$	30563.1	1.11153	0.555763	+	0.831341i	$0.312426\pi$
$912$	0	0
$913$	0	0
$914$	−2156.67	−0.0780485
$915$	0	0
$916$	−21081.5	−0.760427
$917$	−15316.9	−0.551591
$918$	0	0
$919$	−12572.0	−0.451265	−0.225633	−	0.974212i	$-0.572445\pi$
$919$	−12572.0	−0.451265	−0.225633	+	0.974212i	$0.572445\pi$
$920$	17653.2	0.632617
$921$	0	0
$922$	−4879.10	−0.174278
$923$	−2893.65	−0.103191
$924$	0	0
$925$	−33122.0	−1.17735
$926$	23639.4	0.838918
$927$	0	0
$928$	2987.62	0.105683
$929$	33731.5	1.19128	0.595638	−	0.803253i	$-0.296899\pi$
$929$	33731.5	1.19128	0.595638	+	0.803253i	$0.296899\pi$
$930$	0	0
$931$	42710.6	1.50353
$932$	23908.2	0.840277
$933$	0	0
$934$	−29477.2	−1.03268
$935$	0	0
$936$	0	0
$937$	2839.98	0.0990161	0.0495081	−	0.998774i	$-0.484235\pi$
$937$	2839.98	0.0990161	0.0495081	+	0.998774i	$0.484235\pi$
$938$	16616.7	0.578415
$939$	0	0
$940$	−6336.00	−0.219848
$941$	569.661	0.0197348	0.00986738	−	0.999951i	$-0.496859\pi$
$941$	569.661	0.0197348	0.00986738	+	0.999951i	$0.496859\pi$
$942$	0	0
$943$	−12732.6	−0.439693
$944$	−1597.51	−0.0550789
$945$	0	0
$946$	0	0
$947$	−12116.3	−0.415762	−0.207881	−	0.978154i	$-0.566657\pi$
$947$	−12116.3	−0.415762	−0.207881	+	0.978154i	$0.566657\pi$
$948$	0	0
$949$	13986.9	0.478434
$950$	−46042.7	−1.57244
$951$	0	0
$952$	7776.13	0.264733
$953$	45104.6	1.53314	0.766570	−	0.642160i	$-0.221962\pi$
$953$	45104.6	1.53314	0.766570	+	0.642160i	$0.221962\pi$
$954$	0	0
$955$	63873.9	2.16430
$956$	−19996.3	−0.676491
$957$	0	0
$958$	−39844.1	−1.34374
$959$	11965.0	0.402889
$960$	0	0
$961$	−21982.9	−0.737906
$962$	13379.9	0.448424
$963$	0	0
$964$	13520.7	0.451736
$965$	68337.8	2.27966
$966$	0	0
$967$	−45608.1	−1.51671	−0.758354	−	0.651843i	$-0.773996\pi$
$967$	−45608.1	−1.51671	−0.758354	+	0.651843i	$0.773996\pi$
$968$	0	0
$969$	0	0
$970$	−1625.95	−0.0538206
$971$	20090.5	0.663990	0.331995	−	0.943281i	$-0.392278\pi$
$971$	20090.5	0.663990	0.331995	+	0.943281i	$0.392278\pi$
$972$	0	0
$973$	−6828.25	−0.224978
$974$	33054.8	1.08742
$975$	0	0
$976$	−9843.66	−0.322836
$977$	−18382.1	−0.601940	−0.300970	−	0.953634i	$-0.597310\pi$
$977$	−18382.1	−0.601940	−0.300970	+	0.953634i	$0.597310\pi$
$978$	0	0
$979$	0	0
$980$	−18589.9	−0.605952
$981$	0	0
$982$	12063.3	0.392012
$983$	−23085.2	−0.749037	−0.374519	−	0.927219i	$-0.622192\pi$
$983$	−23085.2	−0.749037	−0.374519	+	0.927219i	$0.622192\pi$
$984$	0	0
$985$	29217.6	0.945128
$986$	−22836.4	−0.737584
$987$	0	0
$988$	18599.2	0.598908
$989$	−49385.1	−1.58782
$990$	0	0
$991$	−18135.6	−0.581328	−0.290664	−	0.956825i	$-0.593876\pi$
$991$	−18135.6	−0.581328	−0.290664	+	0.956825i	$0.593876\pi$
$992$	2827.62	0.0905011
$993$	0	0
$994$	−1509.85	−0.0481786
$995$	78612.0	2.50469
$996$	0	0
$997$	−20164.0	−0.640523	−0.320262	−	0.947329i	$-0.603771\pi$
$997$	−20164.0	−0.640523	−0.320262	+	0.947329i	$0.603771\pi$
$998$	22260.0	0.706041
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.4.a.ca.1.4	yes	4
3.2	odd	2		2178.4.a.bv.1.1	✓	4
11.10	odd	2		2178.4.a.bv.1.4	yes	4
33.32	even	2	inner	2178.4.a.ca.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2178.4.a.bv.1.1	✓	4	3.2	odd	2
2178.4.a.bv.1.4	yes	4	11.10	odd	2
2178.4.a.ca.1.1	yes	4	33.32	even	2	inner
2178.4.a.ca.1.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2178.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} +16.6082 q^{5} -7.94790 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} +16.6082 q^{5} -7.94790 q^{7} +8.00000 q^{8} +33.2163 q^{10} -30.4646 q^{13} -15.8958 q^{14} +16.0000 q^{16} -122.299 q^{17} -152.630 q^{19} +66.4326 q^{20} +132.865 q^{23} +150.831 q^{25} -60.9291 q^{26} -31.7916 q^{28} +93.3632 q^{29} +88.3632 q^{31} +32.0000 q^{32} -244.597 q^{34} -132.000 q^{35} -219.597 q^{37} -305.260 q^{38} +132.865 q^{40} -95.8309 q^{41} -371.693 q^{43} +265.730 q^{46} -95.3748 q^{47} -279.831 q^{49} +301.662 q^{50} -121.858 q^{52} -62.6468 q^{53} -63.5832 q^{56} +186.726 q^{58} -99.8443 q^{59} -615.229 q^{61} +176.726 q^{62} +64.0000 q^{64} -505.960 q^{65} -1045.35 q^{67} -489.194 q^{68} -264.000 q^{70} +94.9842 q^{71} -459.120 q^{73} -439.194 q^{74} -610.521 q^{76} +646.153 q^{79} +265.730 q^{80} -191.662 q^{82} -434.129 q^{83} -2031.15 q^{85} -743.386 q^{86} +1628.67 q^{89} +242.129 q^{91} +531.461 q^{92} -190.750 q^{94} -2534.91 q^{95} -48.9502 q^{97} -559.662 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8}+O(q^{10}) \) 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8	\( 4 q + 8 q^{2} + 16 q^{4} + 32 q^{8} + 64 q^{16} - 234 q^{17} + 178 q^{25} - 222 q^{29} - 242 q^{31} + 128 q^{32} - 468 q^{34} - 528 q^{35} - 368 q^{37} + 42 q^{41} - 694 q^{49} + 356 q^{50} - 444 q^{58} - 484 q^{62} + 256 q^{64} - 918 q^{65} - 1034 q^{67} - 936 q^{68} - 1056 q^{70} - 736 q^{74} + 84 q^{82} - 1056 q^{83} + 288 q^{91} - 5376 q^{95} - 3088 q^{97} - 1388 q^{98}+O(q^{100}) \) 4 * q + 8 * q^2 + 16 * q^4 + 32 * q^8 + 64 * q^16 - 234 * q^17 + 178 * q^25 - 222 * q^29 - 242 * q^31 + 128 * q^32 - 468 * q^34 - 528 * q^35 - 368 * q^37 + 42 * q^41 - 694 * q^49 + 356 * q^50 - 444 * q^58 - 484 * q^62 + 256 * q^64 - 918 * q^65 - 1034 * q^67 - 936 * q^68 - 1056 * q^70 - 736 * q^74 + 84 * q^82 - 1056 * q^83 + 288 * q^91 - 5376 * q^95 - 3088 * q^97 - 1388 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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