LMFDB - Embedded newform 7448.2.a.bm.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7448,2,Mod(1,7448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7448 = 2^{3} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7448.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:	$59.4725794254$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.98211824.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 6x^{4} + 15x^{3} + 8x^{2} - 9x + 1 $ x^6 - 3x^5 - 6x^4 + 15x^3 + 8x^2 - 9*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$3.28578$ of defining polynomial
Character	$\chi$	$=$	7448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.28578 q^{3} -1.50233 q^{5} +2.22478 q^{9} +O(q^{10})$	$q+2.28578 q^{3} -1.50233 q^{5} +2.22478 q^{9} +2.10336 q^{11} -3.79902 q^{13} -3.43400 q^{15} +2.43400 q^{17} +1.00000 q^{19} +3.12609 q^{23} -2.74300 q^{25} -1.77197 q^{27} -8.00114 q^{29} -8.70171 q^{31} +4.80782 q^{33} +3.59835 q^{37} -8.68371 q^{39} -0.873155 q^{41} +5.89147 q^{43} -3.34236 q^{45} -4.76756 q^{47} +5.56358 q^{51} -11.4157 q^{53} -3.15994 q^{55} +2.28578 q^{57} -1.76374 q^{59} -12.6632 q^{61} +5.70738 q^{65} +1.44840 q^{67} +7.14554 q^{69} -2.47693 q^{71} +1.04070 q^{73} -6.26989 q^{75} +12.9444 q^{79} -10.7247 q^{81} +6.67051 q^{83} -3.65667 q^{85} -18.2888 q^{87} -11.5172 q^{89} -19.8902 q^{93} -1.50233 q^{95} +2.84904 q^{97} +4.67952 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - q^{5} + 3 q^{9}+O(q^{10}) $ 6 * q - 3 * q^3 - q^5 + 3 * q^9	$ 6 q - 3 q^{3} - q^{5} + 3 q^{9} + 3 q^{11} - 6 q^{13} - 8 q^{15} + 2 q^{17} + 6 q^{19} + 2 q^{23} + 5 q^{25} - 6 q^{27} - q^{29} - 14 q^{31} - 19 q^{33} - 7 q^{37} + 22 q^{39} + 21 q^{41} + q^{43} + 4 q^{45} - 23 q^{47} + 2 q^{51} - 15 q^{53} - 2 q^{55} - 3 q^{57} - 25 q^{59} - 21 q^{61} + 6 q^{65} + 2 q^{67} + 20 q^{69} + 13 q^{71} + 2 q^{73} - 24 q^{75} - 17 q^{79} - 2 q^{81} - 4 q^{83} + 40 q^{85} - 30 q^{87} + q^{89} + 6 q^{93} - q^{95} + 13 q^{97} + 41 q^{99}+O(q^{100}) $ 6 * q - 3 * q^3 - q^5 + 3 * q^9 + 3 * q^11 - 6 * q^13 - 8 * q^15 + 2 * q^17 + 6 * q^19 + 2 * q^23 + 5 * q^25 - 6 * q^27 - q^29 - 14 * q^31 - 19 * q^33 - 7 * q^37 + 22 * q^39 + 21 * q^41 + q^43 + 4 * q^45 - 23 * q^47 + 2 * q^51 - 15 * q^53 - 2 * q^55 - 3 * q^57 - 25 * q^59 - 21 * q^61 + 6 * q^65 + 2 * q^67 + 20 * q^69 + 13 * q^71 + 2 * q^73 - 24 * q^75 - 17 * q^79 - 2 * q^81 - 4 * q^83 + 40 * q^85 - 30 * q^87 + q^89 + 6 * q^93 - q^95 + 13 * q^97 + 41 * 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
$3$	2.28578	1.31969	0.659847	−	0.751400i	$-0.270621\pi$
$3$	2.28578	1.31969	0.659847	+	0.751400i	$0.270621\pi$
$4$	0	0
$5$	−1.50233	−0.671863	−0.335931	−	0.941886i	$-0.609051\pi$
$5$	−1.50233	−0.671863	−0.335931	+	0.941886i	$0.609051\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.22478	0.741595
$10$	0	0
$11$	2.10336	0.634187	0.317093	−	0.948394i	$-0.397293\pi$
$11$	2.10336	0.634187	0.317093	+	0.948394i	$0.397293\pi$
$12$	0	0
$13$	−3.79902	−1.05366	−0.526829	−	0.849971i	$-0.676619\pi$
$13$	−3.79902	−1.05366	−0.526829	+	0.849971i	$0.676619\pi$
$14$	0	0
$15$	−3.43400	−0.886654
$16$	0	0
$17$	2.43400	0.590331	0.295165	−	0.955446i	$-0.404625\pi$
$17$	2.43400	0.590331	0.295165	+	0.955446i	$0.404625\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.12609	0.651834	0.325917	−	0.945398i	$-0.394327\pi$
$23$	3.12609	0.651834	0.325917	+	0.945398i	$0.394327\pi$
$24$	0	0
$25$	−2.74300	−0.548600
$26$	0	0
$27$	−1.77197	−0.341016
$28$	0	0
$29$	−8.00114	−1.48577	−0.742887	−	0.669416i	$-0.766544\pi$
$29$	−8.00114	−1.48577	−0.742887	+	0.669416i	$0.766544\pi$
$30$	0	0
$31$	−8.70171	−1.56287	−0.781437	−	0.623984i	$-0.785513\pi$
$31$	−8.70171	−1.56287	−0.781437	+	0.623984i	$0.785513\pi$
$32$	0	0
$33$	4.80782	0.836933
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.59835	0.591565	0.295783	−	0.955255i	$-0.404420\pi$
$37$	3.59835	0.591565	0.295783	+	0.955255i	$0.404420\pi$
$38$	0	0
$39$	−8.68371	−1.39051
$40$	0	0
$41$	−0.873155	−0.136364	−0.0681820	−	0.997673i	$-0.521720\pi$
$41$	−0.873155	−0.136364	−0.0681820	+	0.997673i	$0.521720\pi$
$42$	0	0
$43$	5.89147	0.898441	0.449220	−	0.893421i	$-0.351702\pi$
$43$	5.89147	0.898441	0.449220	+	0.893421i	$0.351702\pi$
$44$	0	0
$45$	−3.34236	−0.498250
$46$	0	0
$47$	−4.76756	−0.695420	−0.347710	−	0.937602i	$-0.613041\pi$
$47$	−4.76756	−0.695420	−0.347710	+	0.937602i	$0.613041\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	5.56358	0.779057
$52$	0	0
$53$	−11.4157	−1.56806	−0.784032	−	0.620720i	$-0.786840\pi$
$53$	−11.4157	−1.56806	−0.784032	+	0.620720i	$0.786840\pi$
$54$	0	0
$55$	−3.15994	−0.426087
$56$	0	0
$57$	2.28578	0.302759
$58$	0	0
$59$	−1.76374	−0.229620	−0.114810	−	0.993387i	$-0.536626\pi$
$59$	−1.76374	−0.229620	−0.114810	+	0.993387i	$0.536626\pi$
$60$	0	0
$61$	−12.6632	−1.62135	−0.810676	−	0.585495i	$-0.800900\pi$
$61$	−12.6632	−1.62135	−0.810676	+	0.585495i	$0.800900\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.70738	0.707914
$66$	0	0
$67$	1.44840	0.176950	0.0884751	−	0.996078i	$-0.471801\pi$
$67$	1.44840	0.176950	0.0884751	+	0.996078i	$0.471801\pi$
$68$	0	0
$69$	7.14554	0.860222
$70$	0	0
$71$	−2.47693	−0.293957	−0.146979	−	0.989140i	$-0.546955\pi$
$71$	−2.47693	−0.293957	−0.146979	+	0.989140i	$0.546955\pi$
$72$	0	0
$73$	1.04070	0.121804	0.0609022	−	0.998144i	$-0.480602\pi$
$73$	1.04070	0.121804	0.0609022	+	0.998144i	$0.480602\pi$
$74$	0	0
$75$	−6.26989	−0.723985
$76$	0	0
$77$	0	0
$78$	0	0
$79$	12.9444	1.45635	0.728177	−	0.685389i	$-0.240368\pi$
$79$	12.9444	1.45635	0.728177	+	0.685389i	$0.240368\pi$
$80$	0	0
$81$	−10.7247	−1.19163
$82$	0	0
$83$	6.67051	0.732183	0.366092	−	0.930579i	$-0.380696\pi$
$83$	6.67051	0.732183	0.366092	+	0.930579i	$0.380696\pi$
$84$	0	0
$85$	−3.65667	−0.396621
$86$	0	0
$87$	−18.2888	−1.96077
$88$	0	0
$89$	−11.5172	−1.22082	−0.610409	−	0.792086i	$-0.708995\pi$
$89$	−11.5172	−1.22082	−0.610409	+	0.792086i	$0.708995\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−19.8902	−2.06252
$94$	0	0
$95$	−1.50233	−0.154136
$96$	0	0
$97$	2.84904	0.289276	0.144638	−	0.989485i	$-0.453798\pi$
$97$	2.84904	0.289276	0.144638	+	0.989485i	$0.453798\pi$
$98$	0	0
$99$	4.67952	0.470310
$100$	0	0
$101$	7.86632	0.782728	0.391364	−	0.920236i	$-0.372003\pi$
$101$	7.86632	0.782728	0.391364	+	0.920236i	$0.372003\pi$
$102$	0	0
$103$	6.21795	0.612672	0.306336	−	0.951923i	$-0.400897\pi$
$103$	6.21795	0.612672	0.306336	+	0.951923i	$0.400897\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.57240	0.345357	0.172678	−	0.984978i	$-0.444758\pi$
$107$	3.57240	0.345357	0.172678	+	0.984978i	$0.444758\pi$
$108$	0	0
$109$	−3.09929	−0.296858	−0.148429	−	0.988923i	$-0.547422\pi$
$109$	−3.09929	−0.296858	−0.148429	+	0.988923i	$0.547422\pi$
$110$	0	0
$111$	8.22503	0.780686
$112$	0	0
$113$	−11.5136	−1.08311	−0.541556	−	0.840665i	$-0.682164\pi$
$113$	−11.5136	−1.08311	−0.541556	+	0.840665i	$0.682164\pi$
$114$	0	0
$115$	−4.69642	−0.437943
$116$	0	0
$117$	−8.45199	−0.781387
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−6.57588	−0.597807
$122$	0	0
$123$	−1.99584	−0.179959
$124$	0	0
$125$	11.6326	1.04045
$126$	0	0
$127$	8.69629	0.771671	0.385835	−	0.922568i	$-0.373913\pi$
$127$	8.69629	0.771671	0.385835	+	0.922568i	$0.373913\pi$
$128$	0	0
$129$	13.4666	1.18567
$130$	0	0
$131$	−1.15640	−0.101035	−0.0505176	−	0.998723i	$-0.516087\pi$
$131$	−1.15640	−0.101035	−0.0505176	+	0.998723i	$0.516087\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.66209	0.229116
$136$	0	0
$137$	14.9033	1.27327	0.636636	−	0.771164i	$-0.280325\pi$
$137$	14.9033	1.27327	0.636636	+	0.771164i	$0.280325\pi$
$138$	0	0
$139$	−5.69572	−0.483105	−0.241553	−	0.970388i	$-0.577657\pi$
$139$	−5.69572	−0.483105	−0.241553	+	0.970388i	$0.577657\pi$
$140$	0	0
$141$	−10.8976	−0.917743
$142$	0	0
$143$	−7.99070	−0.668216
$144$	0	0
$145$	12.0204	0.998237
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.89970	−0.729092	−0.364546	−	0.931185i	$-0.618776\pi$
$149$	−8.89970	−0.729092	−0.364546	+	0.931185i	$0.618776\pi$
$150$	0	0
$151$	−21.1484	−1.72103	−0.860516	−	0.509424i	$-0.829859\pi$
$151$	−21.1484	−1.72103	−0.860516	+	0.509424i	$0.829859\pi$
$152$	0	0
$153$	5.41512	0.437786
$154$	0	0
$155$	13.0729	1.05004
$156$	0	0
$157$	−13.6136	−1.08649	−0.543244	−	0.839575i	$-0.682804\pi$
$157$	−13.6136	−1.08649	−0.543244	+	0.839575i	$0.682804\pi$
$158$	0	0
$159$	−26.0937	−2.06937
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.2106	0.799756	0.399878	−	0.916568i	$-0.369052\pi$
$163$	10.2106	0.799756	0.399878	+	0.916568i	$0.369052\pi$
$164$	0	0
$165$	−7.22293	−0.562304
$166$	0	0
$167$	−8.15068	−0.630718	−0.315359	−	0.948972i	$-0.602125\pi$
$167$	−8.15068	−0.630718	−0.315359	+	0.948972i	$0.602125\pi$
$168$	0	0
$169$	1.43254	0.110195
$170$	0	0
$171$	2.22478	0.170133
$172$	0	0
$173$	−0.659491	−0.0501401	−0.0250701	−	0.999686i	$-0.507981\pi$
$173$	−0.659491	−0.0501401	−0.0250701	+	0.999686i	$0.507981\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.03152	−0.303028
$178$	0	0
$179$	−17.9091	−1.33859	−0.669294	−	0.742998i	$-0.733403\pi$
$179$	−17.9091	−1.33859	−0.669294	+	0.742998i	$0.733403\pi$
$180$	0	0
$181$	−2.28178	−0.169604	−0.0848018	−	0.996398i	$-0.527026\pi$
$181$	−2.28178	−0.169604	−0.0848018	+	0.996398i	$0.527026\pi$
$182$	0	0
$183$	−28.9452	−2.13969
$184$	0	0
$185$	−5.40592	−0.397451
$186$	0	0
$187$	5.11957	0.374380
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.5754	0.909924	0.454962	−	0.890511i	$-0.349653\pi$
$191$	12.5754	0.909924	0.454962	+	0.890511i	$0.349653\pi$
$192$	0	0
$193$	−4.34103	−0.312474	−0.156237	−	0.987720i	$-0.549936\pi$
$193$	−4.34103	−0.312474	−0.156237	+	0.987720i	$0.549936\pi$
$194$	0	0
$195$	13.0458	0.934230
$196$	0	0
$197$	−18.9665	−1.35131	−0.675655	−	0.737218i	$-0.736139\pi$
$197$	−18.9665	−1.35131	−0.675655	+	0.737218i	$0.736139\pi$
$198$	0	0
$199$	−2.07987	−0.147438	−0.0737192	−	0.997279i	$-0.523487\pi$
$199$	−2.07987	−0.147438	−0.0737192	+	0.997279i	$0.523487\pi$
$200$	0	0
$201$	3.31072	0.233520
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.31177	0.0916179
$206$	0	0
$207$	6.95487	0.483397
$208$	0	0
$209$	2.10336	0.145492
$210$	0	0
$211$	14.2072	0.978062	0.489031	−	0.872266i	$-0.337350\pi$
$211$	14.2072	0.978062	0.489031	+	0.872266i	$0.337350\pi$
$212$	0	0
$213$	−5.66171	−0.387934
$214$	0	0
$215$	−8.85094	−0.603629
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2.37880	0.160745
$220$	0	0
$221$	−9.24680	−0.622007
$222$	0	0
$223$	18.8650	1.26330	0.631648	−	0.775255i	$-0.282379\pi$
$223$	18.8650	1.26330	0.631648	+	0.775255i	$0.282379\pi$
$224$	0	0
$225$	−6.10258	−0.406839
$226$	0	0
$227$	−1.24034	−0.0823246	−0.0411623	−	0.999152i	$-0.513106\pi$
$227$	−1.24034	−0.0823246	−0.0411623	+	0.999152i	$0.513106\pi$
$228$	0	0
$229$	−18.0353	−1.19180	−0.595902	−	0.803057i	$-0.703205\pi$
$229$	−18.0353	−1.19180	−0.595902	+	0.803057i	$0.703205\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	20.8402	1.36528	0.682642	−	0.730753i	$-0.260831\pi$
$233$	20.8402	1.36528	0.682642	+	0.730753i	$0.260831\pi$
$234$	0	0
$235$	7.16246	0.467227
$236$	0	0
$237$	29.5880	1.92194
$238$	0	0
$239$	5.24639	0.339361	0.169680	−	0.985499i	$-0.445726\pi$
$239$	5.24639	0.339361	0.169680	+	0.985499i	$0.445726\pi$
$240$	0	0
$241$	−25.6952	−1.65518	−0.827588	−	0.561337i	$-0.810287\pi$
$241$	−25.6952	−1.65518	−0.827588	+	0.561337i	$0.810287\pi$
$242$	0	0
$243$	−19.1983	−1.23157
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.79902	−0.241726
$248$	0	0
$249$	15.2473	0.966258
$250$	0	0
$251$	−16.9859	−1.07214	−0.536070	−	0.844174i	$-0.680092\pi$
$251$	−16.9859	−1.07214	−0.536070	+	0.844174i	$0.680092\pi$
$252$	0	0
$253$	6.57528	0.413385
$254$	0	0
$255$	−8.35834	−0.523419
$256$	0	0
$257$	−17.0370	−1.06274	−0.531370	−	0.847140i	$-0.678323\pi$
$257$	−17.0370	−1.06274	−0.531370	+	0.847140i	$0.678323\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−17.8008	−1.10184
$262$	0	0
$263$	−12.6143	−0.777832	−0.388916	−	0.921273i	$-0.627150\pi$
$263$	−12.6143	−0.777832	−0.388916	+	0.921273i	$0.627150\pi$
$264$	0	0
$265$	17.1501	1.05352
$266$	0	0
$267$	−26.3257	−1.61111
$268$	0	0
$269$	0.686018	0.0418273	0.0209136	−	0.999781i	$-0.493342\pi$
$269$	0.686018	0.0418273	0.0209136	+	0.999781i	$0.493342\pi$
$270$	0	0
$271$	27.1831	1.65126	0.825628	−	0.564214i	$-0.190821\pi$
$271$	27.1831	1.65126	0.825628	+	0.564214i	$0.190821\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.76952	−0.347915
$276$	0	0
$277$	14.8938	0.894881	0.447440	−	0.894314i	$-0.352336\pi$
$277$	14.8938	0.894881	0.447440	+	0.894314i	$0.352336\pi$
$278$	0	0
$279$	−19.3594	−1.15902
$280$	0	0
$281$	−16.2910	−0.971842	−0.485921	−	0.874003i	$-0.661516\pi$
$281$	−16.2910	−0.971842	−0.485921	+	0.874003i	$0.661516\pi$
$282$	0	0
$283$	−5.60997	−0.333478	−0.166739	−	0.986001i	$-0.553324\pi$
$283$	−5.60997	−0.333478	−0.166739	+	0.986001i	$0.553324\pi$
$284$	0	0
$285$	−3.43400	−0.203412
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−11.0757	−0.651509
$290$	0	0
$291$	6.51227	0.381756
$292$	0	0
$293$	−19.6149	−1.14591	−0.572957	−	0.819586i	$-0.694204\pi$
$293$	−19.6149	−1.14591	−0.572957	+	0.819586i	$0.694204\pi$
$294$	0	0
$295$	2.64972	0.154273
$296$	0	0
$297$	−3.72710	−0.216268
$298$	0	0
$299$	−11.8761	−0.686810
$300$	0	0
$301$	0	0
$302$	0	0
$303$	17.9807	1.03296
$304$	0	0
$305$	19.0243	1.08933
$306$	0	0
$307$	6.48600	0.370176	0.185088	−	0.982722i	$-0.440743\pi$
$307$	6.48600	0.370176	0.185088	+	0.982722i	$0.440743\pi$
$308$	0	0
$309$	14.2128	0.808541
$310$	0	0
$311$	−0.166456	−0.00943886	−0.00471943	−	0.999989i	$-0.501502\pi$
$311$	−0.166456	−0.00943886	−0.00471943	+	0.999989i	$0.501502\pi$
$312$	0	0
$313$	1.65711	0.0936655	0.0468328	−	0.998903i	$-0.485087\pi$
$313$	1.65711	0.0936655	0.0468328	+	0.998903i	$0.485087\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.65335	0.0928613	0.0464307	−	0.998922i	$-0.485215\pi$
$317$	1.65335	0.0928613	0.0464307	+	0.998922i	$0.485215\pi$
$318$	0	0
$319$	−16.8293	−0.942259
$320$	0	0
$321$	8.16572	0.455766
$322$	0	0
$323$	2.43400	0.135431
$324$	0	0
$325$	10.4207	0.578037
$326$	0	0
$327$	−7.08429	−0.391762
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−28.0907	−1.54400	−0.772001	−	0.635621i	$-0.780744\pi$
$331$	−28.0907	−1.54400	−0.772001	+	0.635621i	$0.780744\pi$
$332$	0	0
$333$	8.00555	0.438702
$334$	0	0
$335$	−2.17598	−0.118886
$336$	0	0
$337$	1.05092	0.0572472	0.0286236	−	0.999590i	$-0.490888\pi$
$337$	1.05092	0.0572472	0.0286236	+	0.999590i	$0.490888\pi$
$338$	0	0
$339$	−26.3176	−1.42938
$340$	0	0
$341$	−18.3028	−0.991154
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−10.7350	−0.577951
$346$	0	0
$347$	25.1323	1.34917	0.674586	−	0.738196i	$-0.264322\pi$
$347$	25.1323	1.34917	0.674586	+	0.738196i	$0.264322\pi$
$348$	0	0
$349$	−21.4346	−1.14737	−0.573684	−	0.819077i	$-0.694486\pi$
$349$	−21.4346	−1.14737	−0.573684	+	0.819077i	$0.694486\pi$
$350$	0	0
$351$	6.73176	0.359315
$352$	0	0
$353$	−12.1316	−0.645699	−0.322849	−	0.946450i	$-0.604641\pi$
$353$	−12.1316	−0.645699	−0.322849	+	0.946450i	$0.604641\pi$
$354$	0	0
$355$	3.72117	0.197499
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.4220	1.02505	0.512526	−	0.858672i	$-0.328710\pi$
$359$	19.4220	1.02505	0.512526	+	0.858672i	$0.328710\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−15.0310	−0.788923
$364$	0	0
$365$	−1.56347	−0.0818359
$366$	0	0
$367$	−11.6956	−0.610507	−0.305253	−	0.952271i	$-0.598741\pi$
$367$	−11.6956	−0.610507	−0.305253	+	0.952271i	$0.598741\pi$
$368$	0	0
$369$	−1.94258	−0.101127
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−8.66393	−0.448602	−0.224301	−	0.974520i	$-0.572010\pi$
$373$	−8.66393	−0.448602	−0.224301	+	0.974520i	$0.572010\pi$
$374$	0	0
$375$	26.5894	1.37307
$376$	0	0
$377$	30.3965	1.56550
$378$	0	0
$379$	2.71551	0.139487	0.0697433	−	0.997565i	$-0.477782\pi$
$379$	2.71551	0.139487	0.0697433	+	0.997565i	$0.477782\pi$
$380$	0	0
$381$	19.8778	1.01837
$382$	0	0
$383$	22.1482	1.13172	0.565859	−	0.824502i	$-0.308545\pi$
$383$	22.1482	1.13172	0.565859	+	0.824502i	$0.308545\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	13.1072	0.666279
$388$	0	0
$389$	29.0037	1.47055	0.735274	−	0.677770i	$-0.237054\pi$
$389$	29.0037	1.47055	0.735274	+	0.677770i	$0.237054\pi$
$390$	0	0
$391$	7.60888	0.384798
$392$	0	0
$393$	−2.64328	−0.133336
$394$	0	0
$395$	−19.4467	−0.978471
$396$	0	0
$397$	−31.5938	−1.58565	−0.792823	−	0.609452i	$-0.791390\pi$
$397$	−31.5938	−1.58565	−0.792823	+	0.609452i	$0.791390\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−27.7045	−1.38350	−0.691748	−	0.722139i	$-0.743159\pi$
$401$	−27.7045	−1.38350	−0.691748	+	0.722139i	$0.743159\pi$
$402$	0	0
$403$	33.0580	1.64673
$404$	0	0
$405$	16.1120	0.800613
$406$	0	0
$407$	7.56863	0.375163
$408$	0	0
$409$	0.474535	0.0234643	0.0117321	−	0.999931i	$-0.496265\pi$
$409$	0.474535	0.0234643	0.0117321	+	0.999931i	$0.496265\pi$
$410$	0	0
$411$	34.0656	1.68033
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−10.0213	−0.491927
$416$	0	0
$417$	−13.0192	−0.637551
$418$	0	0
$419$	−21.0544	−1.02857	−0.514287	−	0.857618i	$-0.671943\pi$
$419$	−21.0544	−1.02857	−0.514287	+	0.857618i	$0.671943\pi$
$420$	0	0
$421$	8.89324	0.433430	0.216715	−	0.976235i	$-0.430466\pi$
$421$	8.89324	0.433430	0.216715	+	0.976235i	$0.430466\pi$
$422$	0	0
$423$	−10.6068	−0.515720
$424$	0	0
$425$	−6.67645	−0.323856
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−18.2650	−0.881841
$430$	0	0
$431$	−20.5442	−0.989578	−0.494789	−	0.869013i	$-0.664755\pi$
$431$	−20.5442	−0.989578	−0.494789	+	0.869013i	$0.664755\pi$
$432$	0	0
$433$	−11.6228	−0.558555	−0.279277	−	0.960210i	$-0.590095\pi$
$433$	−11.6228	−0.558555	−0.279277	+	0.960210i	$0.590095\pi$
$434$	0	0
$435$	27.4759	1.31737
$436$	0	0
$437$	3.12609	0.149541
$438$	0	0
$439$	−39.7296	−1.89619	−0.948096	−	0.317986i	$-0.896994\pi$
$439$	−39.7296	−1.89619	−0.948096	+	0.317986i	$0.896994\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18.2858	0.868782	0.434391	−	0.900724i	$-0.356964\pi$
$443$	18.2858	0.868782	0.434391	+	0.900724i	$0.356964\pi$
$444$	0	0
$445$	17.3026	0.820223
$446$	0	0
$447$	−20.3427	−0.962179
$448$	0	0
$449$	26.7431	1.26209	0.631043	−	0.775748i	$-0.282627\pi$
$449$	26.7431	1.26209	0.631043	+	0.775748i	$0.282627\pi$
$450$	0	0
$451$	−1.83656	−0.0864802
$452$	0	0
$453$	−48.3405	−2.27124
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.3288	1.55905	0.779527	−	0.626369i	$-0.215460\pi$
$457$	33.3288	1.55905	0.779527	+	0.626369i	$0.215460\pi$
$458$	0	0
$459$	−4.31298	−0.201312
$460$	0	0
$461$	24.7784	1.15405	0.577023	−	0.816728i	$-0.304214\pi$
$461$	24.7784	1.15405	0.577023	+	0.816728i	$0.304214\pi$
$462$	0	0
$463$	13.3230	0.619172	0.309586	−	0.950872i	$-0.399810\pi$
$463$	13.3230	0.619172	0.309586	+	0.950872i	$0.399810\pi$
$464$	0	0
$465$	29.8816	1.38573
$466$	0	0
$467$	−14.9325	−0.690994	−0.345497	−	0.938420i	$-0.612290\pi$
$467$	−14.9325	−0.690994	−0.345497	+	0.938420i	$0.612290\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−31.1178	−1.43383
$472$	0	0
$473$	12.3919	0.569779
$474$	0	0
$475$	−2.74300	−0.125858
$476$	0	0
$477$	−25.3974	−1.16287
$478$	0	0
$479$	−16.5824	−0.757671	−0.378836	−	0.925464i	$-0.623675\pi$
$479$	−16.5824	−0.757671	−0.378836	+	0.925464i	$0.623675\pi$
$480$	0	0
$481$	−13.6702	−0.623308
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.28020	−0.194354
$486$	0	0
$487$	11.9003	0.539252	0.269626	−	0.962965i	$-0.413100\pi$
$487$	11.9003	0.539252	0.269626	+	0.962965i	$0.413100\pi$
$488$	0	0
$489$	23.3392	1.05543
$490$	0	0
$491$	34.1036	1.53907	0.769537	−	0.638602i	$-0.220487\pi$
$491$	34.1036	1.53907	0.769537	+	0.638602i	$0.220487\pi$
$492$	0	0
$493$	−19.4748	−0.877099
$494$	0	0
$495$	−7.03019	−0.315984
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−25.6631	−1.14884	−0.574419	−	0.818562i	$-0.694772\pi$
$499$	−25.6631	−1.14884	−0.574419	+	0.818562i	$0.694772\pi$
$500$	0	0
$501$	−18.6306	−0.832356
$502$	0	0
$503$	0.480489	0.0214239	0.0107120	−	0.999943i	$-0.496590\pi$
$503$	0.480489	0.0214239	0.0107120	+	0.999943i	$0.496590\pi$
$504$	0	0
$505$	−11.8178	−0.525886
$506$	0	0
$507$	3.27447	0.145424
$508$	0	0
$509$	1.82404	0.0808490	0.0404245	−	0.999183i	$-0.487129\pi$
$509$	1.82404	0.0808490	0.0404245	+	0.999183i	$0.487129\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1.77197	−0.0782345
$514$	0	0
$515$	−9.34141	−0.411632
$516$	0	0
$517$	−10.0279	−0.441026
$518$	0	0
$519$	−1.50745	−0.0661697
$520$	0	0
$521$	−2.21626	−0.0970961	−0.0485481	−	0.998821i	$-0.515459\pi$
$521$	−2.21626	−0.0970961	−0.0485481	+	0.998821i	$0.515459\pi$
$522$	0	0
$523$	−8.84272	−0.386665	−0.193333	−	0.981133i	$-0.561930\pi$
$523$	−8.84272	−0.386665	−0.193333	+	0.981133i	$0.561930\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−21.1799	−0.922612
$528$	0	0
$529$	−13.2276	−0.575112
$530$	0	0
$531$	−3.92394	−0.170285
$532$	0	0
$533$	3.31713	0.143681
$534$	0	0
$535$	−5.36693	−0.232033
$536$	0	0
$537$	−40.9362	−1.76653
$538$	0	0
$539$	0	0
$540$	0	0
$541$	8.05024	0.346107	0.173053	−	0.984912i	$-0.444637\pi$
$541$	8.05024	0.346107	0.173053	+	0.984912i	$0.444637\pi$
$542$	0	0
$543$	−5.21565	−0.223825
$544$	0	0
$545$	4.65616	0.199448
$546$	0	0
$547$	−6.58372	−0.281500	−0.140750	−	0.990045i	$-0.544951\pi$
$547$	−6.58372	−0.281500	−0.140750	+	0.990045i	$0.544951\pi$
$548$	0	0
$549$	−28.1728	−1.20239
$550$	0	0
$551$	−8.00114	−0.340860
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−12.3567	−0.524514
$556$	0	0
$557$	−5.99440	−0.253991	−0.126995	−	0.991903i	$-0.540533\pi$
$557$	−5.99440	−0.253991	−0.126995	+	0.991903i	$0.540533\pi$
$558$	0	0
$559$	−22.3818	−0.946649
$560$	0	0
$561$	11.7022	0.494068
$562$	0	0
$563$	−18.0238	−0.759613	−0.379807	−	0.925066i	$-0.624009\pi$
$563$	−18.0238	−0.759613	−0.379807	+	0.925066i	$0.624009\pi$
$564$	0	0
$565$	17.2973	0.727702
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.3279	1.06180	0.530901	−	0.847434i	$-0.321854\pi$
$569$	25.3279	1.06180	0.530901	+	0.847434i	$0.321854\pi$
$570$	0	0
$571$	27.7300	1.16046	0.580232	−	0.814451i	$-0.302962\pi$
$571$	27.7300	1.16046	0.580232	+	0.814451i	$0.302962\pi$
$572$	0	0
$573$	28.7446	1.20082
$574$	0	0
$575$	−8.57486	−0.357596
$576$	0	0
$577$	35.1453	1.46312	0.731559	−	0.681778i	$-0.238793\pi$
$577$	35.1453	1.46312	0.731559	+	0.681778i	$0.238793\pi$
$578$	0	0
$579$	−9.92263	−0.412370
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−24.0113	−0.994446
$584$	0	0
$585$	12.6977	0.524985
$586$	0	0
$587$	2.91495	0.120313	0.0601564	−	0.998189i	$-0.480840\pi$
$587$	2.91495	0.120313	0.0601564	+	0.998189i	$0.480840\pi$
$588$	0	0
$589$	−8.70171	−0.358548
$590$	0	0
$591$	−43.3533	−1.78332
$592$	0	0
$593$	29.9287	1.22903	0.614513	−	0.788907i	$-0.289353\pi$
$593$	29.9287	1.22903	0.614513	+	0.788907i	$0.289353\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.75413	−0.194574
$598$	0	0
$599$	−20.9090	−0.854317	−0.427158	−	0.904177i	$-0.640485\pi$
$599$	−20.9090	−0.854317	−0.427158	+	0.904177i	$0.640485\pi$
$600$	0	0
$601$	37.4286	1.52674	0.763372	−	0.645960i	$-0.223543\pi$
$601$	37.4286	1.52674	0.763372	+	0.645960i	$0.223543\pi$
$602$	0	0
$603$	3.22238	0.131225
$604$	0	0
$605$	9.87914	0.401644
$606$	0	0
$607$	4.12339	0.167363	0.0836816	−	0.996493i	$-0.473332\pi$
$607$	4.12339	0.167363	0.0836816	+	0.996493i	$0.473332\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	18.1121	0.732735
$612$	0	0
$613$	−6.01803	−0.243066	−0.121533	−	0.992587i	$-0.538781\pi$
$613$	−6.01803	−0.243066	−0.121533	+	0.992587i	$0.538781\pi$
$614$	0	0
$615$	2.99841	0.120908
$616$	0	0
$617$	42.3840	1.70631	0.853157	−	0.521654i	$-0.174685\pi$
$617$	42.3840	1.70631	0.853157	+	0.521654i	$0.174685\pi$
$618$	0	0
$619$	−37.8235	−1.52025	−0.760127	−	0.649774i	$-0.774863\pi$
$619$	−37.8235	−1.52025	−0.760127	+	0.649774i	$0.774863\pi$
$620$	0	0
$621$	−5.53934	−0.222286
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−3.76094	−0.150438
$626$	0	0
$627$	4.80782	0.192006
$628$	0	0
$629$	8.75838	0.349219
$630$	0	0
$631$	34.2901	1.36507	0.682534	−	0.730854i	$-0.260878\pi$
$631$	34.2901	1.36507	0.682534	+	0.730854i	$0.260878\pi$
$632$	0	0
$633$	32.4745	1.29074
$634$	0	0
$635$	−13.0647	−0.518457
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−5.51063	−0.217997
$640$	0	0
$641$	−4.82023	−0.190388	−0.0951939	−	0.995459i	$-0.530347\pi$
$641$	−4.82023	−0.190388	−0.0951939	+	0.995459i	$0.530347\pi$
$642$	0	0
$643$	27.2145	1.07324	0.536618	−	0.843825i	$-0.319702\pi$
$643$	27.2145	1.07324	0.536618	+	0.843825i	$0.319702\pi$
$644$	0	0
$645$	−20.2313	−0.796606
$646$	0	0
$647$	21.3241	0.838336	0.419168	−	0.907909i	$-0.362322\pi$
$647$	21.3241	0.838336	0.419168	+	0.907909i	$0.362322\pi$
$648$	0	0
$649$	−3.70978	−0.145622
$650$	0	0
$651$	0	0
$652$	0	0
$653$	14.4961	0.567276	0.283638	−	0.958931i	$-0.408459\pi$
$653$	14.4961	0.567276	0.283638	+	0.958931i	$0.408459\pi$
$654$	0	0
$655$	1.73730	0.0678818
$656$	0	0
$657$	2.31533	0.0903295
$658$	0	0
$659$	38.5594	1.50206	0.751030	−	0.660268i	$-0.229557\pi$
$659$	38.5594	1.50206	0.751030	+	0.660268i	$0.229557\pi$
$660$	0	0
$661$	6.74065	0.262181	0.131090	−	0.991370i	$-0.458152\pi$
$661$	6.74065	0.262181	0.131090	+	0.991370i	$0.458152\pi$
$662$	0	0
$663$	−21.1361	−0.820859
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−25.0123	−0.968479
$668$	0	0
$669$	43.1213	1.66717
$670$	0	0
$671$	−26.6352	−1.02824
$672$	0	0
$673$	33.8643	1.30537	0.652686	−	0.757629i	$-0.273642\pi$
$673$	33.8643	1.30537	0.652686	+	0.757629i	$0.273642\pi$
$674$	0	0
$675$	4.86052	0.187082
$676$	0	0
$677$	26.8461	1.03178	0.515890	−	0.856655i	$-0.327461\pi$
$677$	26.8461	1.03178	0.515890	+	0.856655i	$0.327461\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−2.83515	−0.108643
$682$	0	0
$683$	28.2987	1.08282	0.541411	−	0.840758i	$-0.317890\pi$
$683$	28.2987	1.08282	0.541411	+	0.840758i	$0.317890\pi$
$684$	0	0
$685$	−22.3896	−0.855464
$686$	0	0
$687$	−41.2246	−1.57282
$688$	0	0
$689$	43.3684	1.65220
$690$	0	0
$691$	14.6108	0.555822	0.277911	−	0.960607i	$-0.410358\pi$
$691$	14.6108	0.555822	0.277911	+	0.960607i	$0.410358\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.55686	0.324580
$696$	0	0
$697$	−2.12526	−0.0804998
$698$	0	0
$699$	47.6360	1.80176
$700$	0	0
$701$	16.2496	0.613739	0.306869	−	0.951752i	$-0.400719\pi$
$701$	16.2496	0.613739	0.306869	+	0.951752i	$0.400719\pi$
$702$	0	0
$703$	3.59835	0.135714
$704$	0	0
$705$	16.3718	0.616597
$706$	0	0
$707$	0	0
$708$	0	0
$709$	18.5432	0.696404	0.348202	−	0.937420i	$-0.386792\pi$
$709$	18.5432	0.696404	0.348202	+	0.937420i	$0.386792\pi$
$710$	0	0
$711$	28.7984	1.08002
$712$	0	0
$713$	−27.2023	−1.01873
$714$	0	0
$715$	12.0047	0.448950
$716$	0	0
$717$	11.9921	0.447853
$718$	0	0
$719$	36.4717	1.36017	0.680083	−	0.733135i	$-0.261944\pi$
$719$	36.4717	1.36017	0.680083	+	0.733135i	$0.261944\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−58.7336	−2.18433
$724$	0	0
$725$	21.9471	0.815096
$726$	0	0
$727$	−49.4445	−1.83380	−0.916898	−	0.399122i	$-0.869315\pi$
$727$	−49.4445	−1.83380	−0.916898	+	0.399122i	$0.869315\pi$
$728$	0	0
$729$	−11.7091	−0.433670
$730$	0	0
$731$	14.3398	0.530377
$732$	0	0
$733$	20.9039	0.772102	0.386051	−	0.922477i	$-0.373839\pi$
$733$	20.9039	0.772102	0.386051	+	0.922477i	$0.373839\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.04651	0.112219
$738$	0	0
$739$	−31.5546	−1.16076	−0.580378	−	0.814347i	$-0.697095\pi$
$739$	−31.5546	−1.16076	−0.580378	+	0.814347i	$0.697095\pi$
$740$	0	0
$741$	−8.68371	−0.319004
$742$	0	0
$743$	−25.9123	−0.950628	−0.475314	−	0.879816i	$-0.657666\pi$
$743$	−25.9123	−0.950628	−0.475314	+	0.879816i	$0.657666\pi$
$744$	0	0
$745$	13.3703	0.489850
$746$	0	0
$747$	14.8404	0.542983
$748$	0	0
$749$	0	0
$750$	0	0
$751$	47.0277	1.71607	0.858033	−	0.513594i	$-0.171686\pi$
$751$	47.0277	1.71607	0.858033	+	0.513594i	$0.171686\pi$
$752$	0	0
$753$	−38.8260	−1.41490
$754$	0	0
$755$	31.7719	1.15630
$756$	0	0
$757$	−21.8924	−0.795693	−0.397846	−	0.917452i	$-0.630242\pi$
$757$	−21.8924	−0.795693	−0.397846	+	0.917452i	$0.630242\pi$
$758$	0	0
$759$	15.0296	0.545542
$760$	0	0
$761$	11.1869	0.405527	0.202763	−	0.979228i	$-0.435008\pi$
$761$	11.1869	0.405527	0.202763	+	0.979228i	$0.435008\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−8.13530	−0.294132
$766$	0	0
$767$	6.70049	0.241941
$768$	0	0
$769$	−6.03222	−0.217527	−0.108764	−	0.994068i	$-0.534689\pi$
$769$	−6.03222	−0.217527	−0.108764	+	0.994068i	$0.534689\pi$
$770$	0	0
$771$	−38.9428	−1.40249
$772$	0	0
$773$	−28.1013	−1.01073	−0.505367	−	0.862905i	$-0.668643\pi$
$773$	−28.1013	−1.01073	−0.505367	+	0.862905i	$0.668643\pi$
$774$	0	0
$775$	23.8688	0.857393
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.873155	−0.0312840
$780$	0	0
$781$	−5.20987	−0.186424
$782$	0	0
$783$	14.1778	0.506673
$784$	0	0
$785$	20.4522	0.729970
$786$	0	0
$787$	−39.0182	−1.39085	−0.695424	−	0.718599i	$-0.744784\pi$
$787$	−39.0182	−1.39085	−0.695424	+	0.718599i	$0.744784\pi$
$788$	0	0
$789$	−28.8335	−1.02650
$790$	0	0
$791$	0	0
$792$	0	0
$793$	48.1076	1.70835
$794$	0	0
$795$	39.2014	1.39033
$796$	0	0
$797$	33.0551	1.17087	0.585435	−	0.810719i	$-0.300924\pi$
$797$	33.0551	1.17087	0.585435	+	0.810719i	$0.300924\pi$
$798$	0	0
$799$	−11.6042	−0.410528
$800$	0	0
$801$	−25.6232	−0.905353
$802$	0	0
$803$	2.18896	0.0772468
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.56809	0.0551992
$808$	0	0
$809$	−11.1926	−0.393512	−0.196756	−	0.980452i	$-0.563041\pi$
$809$	−11.1926	−0.393512	−0.196756	+	0.980452i	$0.563041\pi$
$810$	0	0
$811$	−41.3203	−1.45095	−0.725475	−	0.688248i	$-0.758380\pi$
$811$	−41.3203	−1.45095	−0.725475	+	0.688248i	$0.758380\pi$
$812$	0	0
$813$	62.1346	2.17916
$814$	0	0
$815$	−15.3397	−0.537327
$816$	0	0
$817$	5.89147	0.206116
$818$	0	0
$819$	0	0
$820$	0	0
$821$	37.6569	1.31423	0.657117	−	0.753789i	$-0.271776\pi$
$821$	37.6569	1.31423	0.657117	+	0.753789i	$0.271776\pi$
$822$	0	0
$823$	−27.6497	−0.963809	−0.481905	−	0.876224i	$-0.660055\pi$
$823$	−27.6497	−0.963809	−0.481905	+	0.876224i	$0.660055\pi$
$824$	0	0
$825$	−13.1878	−0.459142
$826$	0	0
$827$	16.2955	0.566651	0.283326	−	0.959024i	$-0.408562\pi$
$827$	16.2955	0.566651	0.283326	+	0.959024i	$0.408562\pi$
$828$	0	0
$829$	−5.19887	−0.180564	−0.0902821	−	0.995916i	$-0.528777\pi$
$829$	−5.19887	−0.180564	−0.0902821	+	0.995916i	$0.528777\pi$
$830$	0	0
$831$	34.0439	1.18097
$832$	0	0
$833$	0	0
$834$	0	0
$835$	12.2450	0.423756
$836$	0	0
$837$	15.4192	0.532965
$838$	0	0
$839$	56.8378	1.96226	0.981129	−	0.193356i	$-0.0619373\pi$
$839$	56.8378	1.96226	0.981129	+	0.193356i	$0.0619373\pi$
$840$	0	0
$841$	35.0183	1.20753
$842$	0	0
$843$	−37.2377	−1.28254
$844$	0	0
$845$	−2.15215	−0.0740362
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−12.8231	−0.440089
$850$	0	0
$851$	11.2488	0.385602
$852$	0	0
$853$	6.37920	0.218420	0.109210	−	0.994019i	$-0.465168\pi$
$853$	6.37920	0.218420	0.109210	+	0.994019i	$0.465168\pi$
$854$	0	0
$855$	−3.34236	−0.114306
$856$	0	0
$857$	29.0289	0.991609	0.495805	−	0.868434i	$-0.334873\pi$
$857$	29.0289	0.991609	0.495805	+	0.868434i	$0.334873\pi$
$858$	0	0
$859$	32.9712	1.12496	0.562482	−	0.826810i	$-0.309847\pi$
$859$	32.9712	1.12496	0.562482	+	0.826810i	$0.309847\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	28.6959	0.976818	0.488409	−	0.872615i	$-0.337577\pi$
$863$	28.6959	0.976818	0.488409	+	0.872615i	$0.337577\pi$
$864$	0	0
$865$	0.990773	0.0336873
$866$	0	0
$867$	−25.3165	−0.859794
$868$	0	0
$869$	27.2267	0.923601
$870$	0	0
$871$	−5.50250	−0.186445
$872$	0	0
$873$	6.33849	0.214526
$874$	0	0
$875$	0	0
$876$	0	0
$877$	15.8596	0.535540	0.267770	−	0.963483i	$-0.413713\pi$
$877$	15.8596	0.535540	0.267770	+	0.963483i	$0.413713\pi$
$878$	0	0
$879$	−44.8353	−1.51226
$880$	0	0
$881$	−11.0864	−0.373511	−0.186755	−	0.982406i	$-0.559797\pi$
$881$	−11.0864	−0.373511	−0.186755	+	0.982406i	$0.559797\pi$
$882$	0	0
$883$	38.3939	1.29206	0.646029	−	0.763313i	$-0.276429\pi$
$883$	38.3939	1.29206	0.646029	+	0.763313i	$0.276429\pi$
$884$	0	0
$885$	6.05668	0.203593
$886$	0	0
$887$	−11.8483	−0.397827	−0.198914	−	0.980017i	$-0.563741\pi$
$887$	−11.8483	−0.397827	−0.198914	+	0.980017i	$0.563741\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−22.5579	−0.755717
$892$	0	0
$893$	−4.76756	−0.159540
$894$	0	0
$895$	26.9054	0.899347
$896$	0	0
$897$	−27.1460	−0.906380
$898$	0	0
$899$	69.6236	2.32208
$900$	0	0
$901$	−27.7857	−0.925677
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.42800	0.113950
$906$	0	0
$907$	29.6864	0.985720	0.492860	−	0.870109i	$-0.335951\pi$
$907$	29.6864	0.985720	0.492860	+	0.870109i	$0.335951\pi$
$908$	0	0
$909$	17.5009	0.580467
$910$	0	0
$911$	−20.5983	−0.682452	−0.341226	−	0.939981i	$-0.610842\pi$
$911$	−20.5983	−0.682452	−0.341226	+	0.939981i	$0.610842\pi$
$912$	0	0
$913$	14.0305	0.464341
$914$	0	0
$915$	43.4853	1.43758
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−27.4117	−0.904228	−0.452114	−	0.891960i	$-0.649330\pi$
$919$	−27.4117	−0.904228	−0.452114	+	0.891960i	$0.649330\pi$
$920$	0	0
$921$	14.8256	0.488519
$922$	0	0
$923$	9.40989	0.309730
$924$	0	0
$925$	−9.87028	−0.324533
$926$	0	0
$927$	13.8336	0.454355
$928$	0	0
$929$	−17.3485	−0.569187	−0.284594	−	0.958648i	$-0.591859\pi$
$929$	−17.3485	−0.569187	−0.284594	+	0.958648i	$0.591859\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−0.380482	−0.0124564
$934$	0	0
$935$	−7.69129	−0.251532
$936$	0	0
$937$	−41.6883	−1.36190	−0.680948	−	0.732332i	$-0.738432\pi$
$937$	−41.6883	−1.36190	−0.680948	+	0.732332i	$0.738432\pi$
$938$	0	0
$939$	3.78779	0.123610
$940$	0	0
$941$	17.2816	0.563363	0.281681	−	0.959508i	$-0.409108\pi$
$941$	17.2816	0.563363	0.281681	+	0.959508i	$0.409108\pi$
$942$	0	0
$943$	−2.72956	−0.0888867
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.3014	1.21213	0.606066	−	0.795414i	$-0.292747\pi$
$947$	37.3014	1.21213	0.606066	+	0.795414i	$0.292747\pi$
$948$	0	0
$949$	−3.95363	−0.128340
$950$	0	0
$951$	3.77919	0.122549
$952$	0	0
$953$	−42.4534	−1.37520	−0.687600	−	0.726089i	$-0.741336\pi$
$953$	−42.4534	−1.37520	−0.687600	+	0.726089i	$0.741336\pi$
$954$	0	0
$955$	−18.8924	−0.611344
$956$	0	0
$957$	−38.4680	−1.24349
$958$	0	0
$959$	0	0
$960$	0	0
$961$	44.7198	1.44257
$962$	0	0
$963$	7.94782	0.256115
$964$	0	0
$965$	6.52166	0.209940
$966$	0	0
$967$	−29.1861	−0.938561	−0.469280	−	0.883049i	$-0.655487\pi$
$967$	−29.1861	−0.938561	−0.469280	+	0.883049i	$0.655487\pi$
$968$	0	0
$969$	5.56358	0.178728
$970$	0	0
$971$	−47.6945	−1.53059	−0.765294	−	0.643680i	$-0.777407\pi$
$971$	−47.6945	−1.53059	−0.765294	+	0.643680i	$0.777407\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	23.8194	0.762832
$976$	0	0
$977$	9.91351	0.317161	0.158581	−	0.987346i	$-0.449308\pi$
$977$	9.91351	0.317161	0.158581	+	0.987346i	$0.449308\pi$
$978$	0	0
$979$	−24.2248	−0.774227
$980$	0	0
$981$	−6.89525	−0.220149
$982$	0	0
$983$	−16.9219	−0.539725	−0.269863	−	0.962899i	$-0.586978\pi$
$983$	−16.9219	−0.539725	−0.269863	+	0.962899i	$0.586978\pi$
$984$	0	0
$985$	28.4940	0.907894
$986$	0	0
$987$	0	0
$988$	0	0
$989$	18.4172	0.585634
$990$	0	0
$991$	−61.6939	−1.95977	−0.979886	−	0.199558i	$-0.936049\pi$
$991$	−61.6939	−1.95977	−0.979886	+	0.199558i	$0.936049\pi$
$992$	0	0
$993$	−64.2090	−2.03761
$994$	0	0
$995$	3.12466	0.0990583
$996$	0	0
$997$	−43.4308	−1.37547	−0.687733	−	0.725964i	$-0.741394\pi$
$997$	−43.4308	−1.37547	−0.687733	+	0.725964i	$0.741394\pi$
$998$	0	0
$999$	−6.37618	−0.201733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7448.2.a.bm.1.6	✓	6
7.6	odd	2		7448.2.a.bn.1.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7448.2.a.bm.1.6	✓	6	1.1	even	1	trivial
7448.2.a.bn.1.1	yes	6	7.6	odd	2

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:	\( N \)	\(=\)	\( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7448.a (trivial)

\(f(q)\)	\(=\)	\(q+2.28578 q^{3} -1.50233 q^{5} +2.22478 q^{9} +O(q^{10})\)	\(q+2.28578 q^{3} -1.50233 q^{5} +2.22478 q^{9} +2.10336 q^{11} -3.79902 q^{13} -3.43400 q^{15} +2.43400 q^{17} +1.00000 q^{19} +3.12609 q^{23} -2.74300 q^{25} -1.77197 q^{27} -8.00114 q^{29} -8.70171 q^{31} +4.80782 q^{33} +3.59835 q^{37} -8.68371 q^{39} -0.873155 q^{41} +5.89147 q^{43} -3.34236 q^{45} -4.76756 q^{47} +5.56358 q^{51} -11.4157 q^{53} -3.15994 q^{55} +2.28578 q^{57} -1.76374 q^{59} -12.6632 q^{61} +5.70738 q^{65} +1.44840 q^{67} +7.14554 q^{69} -2.47693 q^{71} +1.04070 q^{73} -6.26989 q^{75} +12.9444 q^{79} -10.7247 q^{81} +6.67051 q^{83} -3.65667 q^{85} -18.2888 q^{87} -11.5172 q^{89} -19.8902 q^{93} -1.50233 q^{95} +2.84904 q^{97} +4.67952 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - q^{5} + 3 q^{9}+O(q^{10}) \) 6 * q - 3 * q^3 - q^5 + 3 * q^9	\( 6 q - 3 q^{3} - q^{5} + 3 q^{9} + 3 q^{11} - 6 q^{13} - 8 q^{15} + 2 q^{17} + 6 q^{19} + 2 q^{23} + 5 q^{25} - 6 q^{27} - q^{29} - 14 q^{31} - 19 q^{33} - 7 q^{37} + 22 q^{39} + 21 q^{41} + q^{43} + 4 q^{45} - 23 q^{47} + 2 q^{51} - 15 q^{53} - 2 q^{55} - 3 q^{57} - 25 q^{59} - 21 q^{61} + 6 q^{65} + 2 q^{67} + 20 q^{69} + 13 q^{71} + 2 q^{73} - 24 q^{75} - 17 q^{79} - 2 q^{81} - 4 q^{83} + 40 q^{85} - 30 q^{87} + q^{89} + 6 q^{93} - q^{95} + 13 q^{97} + 41 q^{99}+O(q^{100}) \) 6 * q - 3 * q^3 - q^5 + 3 * q^9 + 3 * q^11 - 6 * q^13 - 8 * q^15 + 2 * q^17 + 6 * q^19 + 2 * q^23 + 5 * q^25 - 6 * q^27 - q^29 - 14 * q^31 - 19 * q^33 - 7 * q^37 + 22 * q^39 + 21 * q^41 + q^43 + 4 * q^45 - 23 * q^47 + 2 * q^51 - 15 * q^53 - 2 * q^55 - 3 * q^57 - 25 * q^59 - 21 * q^61 + 6 * q^65 + 2 * q^67 + 20 * q^69 + 13 * q^71 + 2 * q^73 - 24 * q^75 - 17 * q^79 - 2 * q^81 - 4 * q^83 + 40 * q^85 - 30 * q^87 + q^89 + 6 * q^93 - q^95 + 13 * q^97 + 41 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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