Newform orbit 507.2.p.b

• Label: 507.2.p.b
• Level: $507$
• Weight: $2$
• Character orbit: 507.p
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $192$
• Inner twists: $2$

Newspace parameters

Level:	$N$	$=$	$507 = 3 \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	507.p (of order $26$, degree $12$, minimal)

Newform invariants

Self dual:	no
Analytic conductor:	$4.04841538248$
Analytic rank:	$0$
Dimension:	$192$
Relative dimension:	$16$ over $\Q(\zeta_{26})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{26}]$

$q$-expansion

The algebraic $q$-expansion of this newform has not been computed, but we have computed the trace expansion.

$\operatorname{Tr}(f)(q) =$ 192 * q + 16 * q^3 + 18 * q^4 - 16 * q^9 - 18 * q^12 - 63 * q^13 - 10 * q^14 - 6 * q^16 + 12 * q^17 + 16 * q^22 - 52 * q^23 + 58 * q^25 + 51 * q^26 + 16 * q^27 - 49 * q^29 - 26 * q^31 - 13 * q^33 - 65 * q^34 + 39 * q^35 + 18 * q^36 + 77 * q^38 - 2 * q^39 - 55 * q^42 - 76 * q^43 + 39 * q^44 + 6 * q^48 - 58 * q^49 + 52 * q^50 - 12 * q^51 + 63 * q^52 - 73 * q^53 + 37 * q^55 - 10 * q^56 + 13 * q^57 - 26 * q^58 - 104 * q^59 - 13 * q^60 + 8 * q^61 + 53 * q^62 + 42 * q^64 + 52 * q^65 - 42 * q^66 + 26 * q^67 - 34 * q^68 - 39 * q^71 + 52 * q^73 + 59 * q^74 - 6 * q^75 - 130 * q^76 - 52 * q^77 + 53 * q^78 + 14 * q^79 - 16 * q^81 + 41 * q^82 - 78 * q^83 + 91 * q^85 + 169 * q^86 - 42 * q^87 - 270 * q^88 + 80 * q^91 - 54 * q^92 - 91 * q^93 + 25 * q^94 - 58 * q^95 - 65 * q^96 + 130 * q^97 + 104 * q^98

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

Label	$a_{2}$			$a_{3}$			$a_{4}$			$a_{5}$			$a_{6}$			$a_{7}$			$a_{8}$			$a_{9}$			$a_{10}$
25.1	−2.52295	−	0.956827i	−0.568065	−	0.822984i	3.95272	+	3.50180i	4.16354	+	0.505545i	0.645743	+	2.61988i	0.140914	+	0.268489i	−4.11396	−	7.83849i	−0.354605	+	0.935016i	−10.0207	−	5.25925i
25.2	−2.42142	−	0.918325i	−0.568065	−	0.822984i	3.52295	+	3.12106i	−2.78849	−	0.338584i	0.619759	+	2.51446i	0.357250	+	0.680683i	−3.25740	−	6.20646i	−0.354605	+	0.935016i	6.44118	+	3.38059i
25.3	−2.13008	−	0.807834i	−0.568065	−	0.822984i	2.38764	+	2.11526i	−1.00892	−	0.122506i	0.545191	+	2.21193i	1.86515	+	3.55375i	−1.25969	−	2.40014i	−0.354605	+	0.935016i	2.05013	+	1.07599i
25.4	−1.50260	−	0.569861i	−0.568065	−	0.822984i	0.436045	+	0.386302i	2.17889	+	0.264565i	0.384588	+	1.56033i	0.728343	+	1.38774i	1.05859	+	2.01697i	−0.354605	+	0.935016i	−3.12324	−	1.63920i
25.5	−1.31279	−	0.497877i	−0.568065	−	0.822984i	−0.0214779	−	0.0190278i	1.11474	+	0.135354i	0.336007	+	1.36323i	−0.369513	−	0.704048i	1.32369	+	2.52209i	−0.354605	+	0.935016i	−1.39603	−	0.732692i
25.6	−1.13089	−	0.428890i	−0.568065	−	0.822984i	−0.402056	−	0.356191i	−4.00621	−	0.486442i	0.289449	+	1.17434i	−2.12665	−	4.05199i	1.42607	+	2.71715i	−0.354605	+	0.935016i	4.32195	+	2.26833i
25.7	−0.468954	−	0.177851i	−0.568065	−	0.822984i	−1.30873	−	1.15944i	−1.41049	−	0.171264i	0.120028	+	0.486973i	2.16100	+	4.11744i	0.873690	+	1.66468i	−0.354605	+	0.935016i	0.630994	+	0.331171i
25.8	−0.103579	−	0.0392824i	−0.568065	−	0.822984i	−1.48784	−	1.31811i	−1.67597	−	0.203500i	0.0265109	+	0.107559i	0.366680	+	0.698651i	0.205293	+	0.391152i	−0.354605	+	0.935016i	0.165602	+	0.0869145i
25.9	0.283929	+	0.107680i	−0.568065	−	0.822984i	−1.42800	−	1.26510i	−0.879635	−	0.106807i	−0.0726712	−	0.294839i	−0.935242	−	1.78196i	−0.551463	−	1.05073i	−0.354605	+	0.935016i	−0.238253	−	0.125045i
25.10	0.494430	+	0.187512i	−0.568065	−	0.822984i	−1.28772	−	1.14082i	3.75871	+	0.456390i	−0.126548	−	0.513427i	1.26721	+	2.41446i	−0.914254	−	1.74197i	−0.354605	+	0.935016i	1.77284	+	0.930459i
25.11	0.917259	+	0.347870i	−0.568065	−	0.822984i	−0.776671	−	0.688071i	0.986534	+	0.119787i	−0.234771	−	0.952502i	−0.506792	−	0.965611i	−1.38484	−	2.63860i	−0.354605	+	0.935016i	0.863237	+	0.453062i
25.12	1.43480	+	0.544146i	−0.568065	−	0.822984i	0.265521	+	0.235231i	−3.88465	−	0.471682i	−0.367233	−	1.48992i	2.17406	+	4.14233i	−1.17328	−	2.23550i	−0.354605	+	0.935016i	−5.31702	−	2.79059i
25.13	1.78796	+	0.678085i	−0.568065	−	0.822984i	1.23999	+	1.09854i	−2.39960	−	0.291364i	−0.457626	−	1.85666i	−1.52079	−	2.89763i	−0.305152	−	0.581418i	−0.354605	+	0.935016i	−4.09283	−	2.14808i
25.14	1.98795	+	0.753931i	−0.568065	−	0.822984i	1.88652	+	1.67131i	3.81359	+	0.463053i	−0.508813	−	2.06433i	−2.20899	−	4.20888i	0.514149	+	0.979628i	−0.354605	+	0.935016i	7.23212	+	3.79571i
25.15	2.12140	+	0.804542i	−0.568065	−	0.822984i	2.35604	+	2.08727i	2.11775	+	0.257141i	−0.542969	−	2.20291i	1.12326	+	2.14019i	1.21005	+	2.30557i	−0.354605	+	0.935016i	4.28571	+	2.24932i
25.16	2.56554	+	0.972980i	−0.568065	−	0.822984i	4.13827	+	3.66618i	−1.46104	−	0.177402i	−0.656644	−	2.66411i	0.543789	+	1.03610i	4.49950	+	8.57307i	−0.354605	+	0.935016i	−3.57574	−	1.87669i
64.1	−2.62119	−	0.318270i	0.748511	−	0.663123i	4.82745	+	1.18986i	−0.880115	−	1.67692i	−2.17304	+	1.49994i	−2.02452	−	0.767800i	−7.33724	−	2.78265i	0.120537	−	0.992709i	1.77323	+	4.67563i
64.2	−2.60940	−	0.316839i	0.748511	−	0.663123i	4.76672	+	1.17489i	1.60429	+	3.05673i	−2.16327	+	1.49320i	2.76033	+	1.04686i	−7.15054	−	2.71184i	0.120537	−	0.992709i	−3.21776	−	8.48454i
64.3	−2.10986	−	0.256183i	0.748511	−	0.663123i	2.44400	+	0.602391i	1.00545	+	1.91572i	−1.74913	+	1.20734i	−2.58799	−	0.981496i	−1.02769	−	0.389750i	0.120537	−	0.992709i	−1.63057	−	4.29947i
64.4	−1.61341	−	0.195904i	0.748511	−	0.663123i	0.622835	+	0.153515i	−1.54944	−	2.95221i	−1.33756	+	0.923254i	−1.00058	−	0.379471i	2.06448	+	0.782953i	0.120537	−	0.992709i	1.92153	+	5.06666i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
169.h	even	26	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	507.2.p.b	✓	192
169.h	even	26	1	inner	507.2.p.b	✓	192

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
507.2.p.b	✓	192	1.a	even	1	1	trivial
507.2.p.b	✓	192	169.h	even	26	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^192 - 25*T2^190 + 373*T2^188 - 4435*T2^186 + 130*T2^185 + 46719*T2^184 - 3237*T2^183 - 452089*T2^182 + 30342*T2^181 + 4097993*T2^180 - 126126*T2^179 - 35334345*T2^178 + 102921*T2^177 + 293001477*T2^176 + 2370056*T2^175 - 2367435386*T2^174 - 67053701*T2^173 + 18828052116*T2^172 + 1391977275*T2^171 - 147941691182*T2^170 - 20959993301*T2^169 + 1148310338544*T2^168 + 256460285133*T2^167 - 8609238841176*T2^166 - 2753881247884*T2^165 + 61810224768699*T2^164 + 27020153387279*T2^163 - 426713436955249*T2^162 - 243475898575702*T2^161 + 2842029500387214*T2^160 + 2018681268997081*T2^159 - 18271535417426490*T2^158 - 15598752538263492*T2^157 + 113689015051134102*T2^156 + 111770971407940949*T2^155 - 688258295693614920*T2^154 - 741095711034525834*T2^153 + 4078741655299038701*T2^152 + 4630252818250379242*T2^151 - 23839022453406479836*T2^150 - 27710256489420567634*T2^149 + 138006138455427064791*T2^148 + 160186290820815548705*T2^147 - 787338442172778651625*T2^146 - 905488215491493956153*T2^145 + 4396564198485652838809*T2^144 + 5013020794599540258718*T2^143 - 23803533883199671615767*T2^142 - 27191782203337579090302*T2^141 + 125400769395592970318746*T2^140 + 143110578162681891651890*T2^139 - 646945500503222114490826*T2^138 - 727257383718780154956131*T2^137 + 3288655457781500311308917*T2^136 + 3531043733321655612187607*T2^135 - 16447721334534269760210246*T2^134 - 16191984052466419077055708*T2^133 + 80753476252076471092716990*T2^132 + 68618296720709549558596267*T2^131 - 388299929803609905346555821*T2^130 - 261904471481783222185593214*T2^129 + 1821991702123495979631725894*T2^128 + 869578771140672966538780940*T2^127 - 8285510684960913173593571436*T2^126 - 2262163742968979679405783129*T2^125 + 36094150182909969544868687387*T2^124 + 2613636444323895703515354305*T2^123 - 149123488629780449453196901045*T2^122 + 18077924509181198980676774338*T2^121 + 579840429446733774614483661721*T2^120 - 180261855475313540419038741219*T2^119 - 2110113335823466511044227341638*T2^118 + 1048647493691646918525655318041*T2^117 + 7244274136496666938676141662859*T2^116 - 4914006429925968992739464368117*T2^115 - 23732007221233162518352507779516*T2^114 + 19724160702895461757985343941391*T2^113 + 75402685893727875408661864753033*T2^112 - 70017714903552709419574381631425*T2^111 - 235140461811771563752434634523406*T2^110 + 223789498932878388802207618312480*T2^109 + 723838680581087141100868293176022*T2^108 - 646665639112120327990653972874798*T2^107 - 2197373054471484816808296704006051*T2^106 + 1687555472736007331343506133750561*T2^105 + 6507535932673564081826610592750342*T2^104 - 3902764623303605387616062913900138*T2^103 - 18581570985467057208501807191477671*T2^102 + 7527252828283421172927890651238022*T2^101 + 50701763929924742022742557529408656*T2^100 - 10141898315458125919639081849380469*T2^99 - 130513019152440157644418688518874924*T2^98 - 121443093700301651686229116063028*T2^97 + 312379399536533847127426313289205798*T2^96 + 61630059039577419935547252278771677*T2^95 - 686363715263037200123180163603323847*T2^94 - 267762557739696275513416408729409878*T2^93 + 1376896202030230925179959066715156086*T2^92 + 801116506476988430303469484506808164*T2^91 - 2511348860435101381590680864002476966*T2^90 - 2015122914714204944730771727742286868*T2^89 + 4151305604133594826119977197084258267*T2^88 + 4485057304878044410534305841989520241*T2^87 - 6373125605879051121210202912532480216*T2^86 - 8938635800826921532679489334253140146*T2^85 + 9322456247934671424024544760062763379*T2^84 + 16096697387265392802518981035341593193*T2^83 - 12574370066061322668363393067795414793*T2^82 - 26091691902074985505378729356399982984*T2^81 + 15777254282743135501374031450394974064*T2^80 + 40622473760562946802816068172225689808*T2^79 - 17035545294055648536713136785517264727*T2^78 - 56704759463711299423505321362452046749*T2^77 + 26558572969409383479615617292466518752*T2^76 + 82912827774753024727270565196453666956*T2^75 - 32077444549107110451449546705646387120*T2^74 - 89636174115212130366435757160701452773*T2^73 + 72391200705296748580869101126141455206*T2^72 + 147877583107582254482802561278313372789*T2^71 - 31037605986227697713320909493220913569*T2^70 - 133914655269259341193048143845666821786*T2^69 + 109779220781989026509767818976828146432*T2^68 + 289697623353840768700361635019763856660*T2^67 - 222688692273943374195789508962887599*T2^66 - 209909849084450018734953825310702140348*T2^65 + 147674441874375795428279333777650981400*T2^64 + 363750666193603047795155086434220001498*T2^63 + 55663503107187785292394126955573261690*T2^62 - 273056129724523479836450826754730119539*T2^61 - 41451410219597517997604163696623897329*T2^60 + 406231509877224251101034961860203852519*T2^59 + 95657268863624434644019905819978767094*T2^58 - 443017271566143862593790823349485984797*T2^57 + 9687386114841673163704104163918087539*T2^56 + 313850200951052360306354385331628174681*T2^55 - 9430782332272369812756457251227911531*T2^54 - 91222514012552133389787943190903446036*T2^53 - 16341357644296054647984645749634440655*T2^52 + 126049174364019440577858468358109138760*T2^51 + 46787074587924411038673510686607267493*T2^50 - 214122232387579196851368715363035175818*T2^49 - 49542826443192056419826208238173072582*T2^48 + 208322867720896804001342823007191894350*T2^47 - 27221956619895313123741063758231031882*T2^46 - 237348281874562118498801696597446805823*T2^45 + 19503454790531978686542736817501520560*T2^44 + 174777247486597508382966180876410037737*T2^43 + 11842733169401461659882626201642835214*T2^42 - 44585456622968093643780407091120178179*T2^41 + 30949646456960273660282246807975440184*T2^40 + 16445241268550988698977520769032347606*T2^39 - 22401419357714880015625893304246622399*T2^38 + 7016137575816321408045229520655235695*T2^37 + 32592021581029049010847992951444176082*T2^36 + 13208133386463897424547807244355128709*T2^35 - 8266000092855418359829583955431999478*T2^34 - 8367079319056883268566373522920857848*T2^33 - 1338398484091528807690091085931930931*T2^32 + 1356417618069894011325788134792265553*T2^31 + 864776559817024160283075546515435108*T2^30 + 194426717877139094166482542227871406*T2^29 - 21753077060444844924887524995398773*T2^28 - 44301844009256086529115794462734252*T2^27 - 26342631326933377101170790865139540*T2^26 - 8155176246529773046020773443588158*T2^25 + 23837107231934012090602302797334*T2^24 + 1132973164335507988400929675952454*T2^23 + 487222248753333199793628886767901*T2^22 + 112706990438093299128274564177176*T2^21 + 20068277420286779077639373329981*T2^20 + 2283120081977324919744900184973*T2^19 - 1375817883653089140611178803244*T2^18 - 950436477445731177770636184326*T2^17 - 173926828037189852886047647562*T2^16 + 47596165979770334662040426212*T2^15 + 34838387004837238004265757921*T2^14 + 9604959060700136609818412564*T2^13 + 1616515054318541746943539030*T2^12 + 181348327939258432997290155*T2^11 + 12502940035204700904367467*T2^10 - 38821590902236545836082*T2^9 - 150856670991420598820031*T2^8 - 23473142053041500162928*T2^7 - 1837358176247373180960*T2^6 - 33189895631118677724*T2^5 + 8430215296512853116*T2^4 + 927899126851857498*T2^3 + 44169562390260411*T2^2 + 454034063713113*T2 + 2565164201769